1001.0014.txt

Preprint typeset in JHEP style - HYPER VERSION UCB-PTH-10/01
arXiv:1001.0014
Jet Shapes and Jet Algorithms in SCET
Stephen D. Ellis, Christopher K. Vermilion, and Jonathan R. Walsh
University of Washington, Seattle, WA 98195-1560, USA
E-mail: sdellis@u.washington.edu ,verm@uw.edu ,jrwalsh@u.washington.edu
Andrew Hornig and Christopher Lee
Theoretical Physics Group, Lawrence Berkeley National Laboratory,
and Center for Theoretical Physics, University of California, Berkeley, CA 94720, USA
E-mail: ahornig@uw.edu ,clee137@mit.edu
Abstract: Jet shapes are weighted sums over the four-momenta of the constituents of a
jet and reveal details of its internal structure, potentially allowing discrimination of its par-
tonic origin. In this work we make predictions for quark and gluon jet shape distributions
inN-jet nal states in e+ecollisions, dened with a cone or recombination algorithm,
where we measure some jet shape observable on a subset of these jets. Using the framework
of Soft-Collinear Eective Theory, we prove a factorization theorem for jet shape distri-
butions and demonstrate the consistent renormalization-group running of the functions in
the factorization theorem for any number of measured and unmeasured jets, any number
of quark and gluon jets, and any angular size Rof the jets, as long as Ris much smaller
than the angular separation between jets. We calculate the jet and soft functions for angu-
larity jet shapes ato one-loop order ( O(s)) and resum a subset of the large logarithms of
aneeded for next-to-leading logarithmic (NLL) accuracy for both cone and k T-type jets.
We compare our predictions for the resummed adistribution of a quark or a gluon jet
produced in a 3-jet nal state in e+eannihilation to the output of a Monte Carlo event
generator and nd that the dependence on aandRis very similar.
Keywords: Jets, Factorization, Resummation, Eective Field Theory .arXiv:1001.0014v3  [hep-ph]  15 Nov 2010Contents
1. Introduction 2
1.1 Motivation and Objectives 2
1.2 Soft-Collinear Eective Theory and Factorization 4
1.3 Power Corrections to Factorized Jet Shape Distributions 6
1.4 Resummation and Logarithmic Accuracy 7
1.5 Detailed Outline of This Work 11
2. Jet Shapes and Jet Algorithms 14
2.1 Jet Shapes 14
2.2 Jet Algorithms 14
2.3 Do Jet Algorithms Respect Factorization? 16
3. Factorization of Jet Shape Distributions in e+etoNJets 17
3.1 Overview of SCET 17
3.2 Jet Shape Distribution in e+e!3 Jets 20
3.3 Jet Shapes in e+e!Njets 26
3.4 Do Jet Algorithms Induce Large Power Corrections to Factorization? 27
4. Jet Functions at O(s)for Jet Shapes 30
4.1 Phase Space Cuts 30
4.2 Quark Jet Function 32
4.2.1 Measured Quark Jet 33
4.2.2 Gluon Outside Measured Quark Jet 34
4.2.3 Unmeasured Quark Jet 35
4.3 Gluon Jet Function 35
4.3.1 Measured Gluon Jet 36
4.3.2 Unmeasured Gluon Jet 37
5. Soft Functions at O(s)for Jet Shapes 37
5.1 Phase Space Cuts 37
5.2 Calculation of contributions to the N-Jet Soft Function 38
5.2.1 Inclusive Contribution: Sincl
ij 40
5.2.2 Soft gluon inside jet kwithEg>:Sk
ij 40
5.2.3 Soft gluon inside measured jet k:Smeas
ij(k
a) 41
5.3 TotalN-Jet Soft Function in the large- tLimit 42
{ 1 {6. Resummation and Consistency Relations at NLL 43
6.1 General Form of Renormalization Group Equations and Solutions 43
6.2 RG Evolution of Hard, Jet, and Soft Functions 46
6.2.1 Hard Function 46
6.2.2 Jet Functions 47
6.2.3 Soft Function 48
6.3 Consistency Relation among Anomalous Dimensions 49
6.4 Refactorization of the Soft Function 50
6.5 Total Resummed Distribution 52
7. Plots of Distributions and Comparisons to Monte Carlo 54
8. Conclusions 61
A. Jet Function Calculations 62
A.1 Finite Pieces of the Quark Jet Function 62
A.2 Finite Pieces of the Gluon Jet Function 65
B. Soft function calculations 67
B.1Sincl
ij 67
B.2Si
ijandSmeas
ij(i
a) 69
B.2.1 Common Integrals 69
B.2.2Smeas
ij(i
a) 70
B.2.3Si
ij 70
B.3Smeas
ij(k
a) andSk
ijfork6=i;j 70
B.3.1 Common Integrals 71
B.3.2Sk
ij 72
B.3.3Smeas
ij(k
a) 73
B.3.4Sk
ij+Smeas
ij(k
a) 73
C. Convolutions and Finite Terms in the Resummed Distribution 74
D. Color Algebra for n= 2;3Jets 77
1. Introduction
1.1 Motivation and Objectives
Jets provide troves of information about physics within and beyond the Standard Model of
particle physics. On the one hand, jets display the behavior of Quantum Chromodynamics
(QCD) over a wide range of energy scales, from the energy of the hard scattering, through
intermediate scales of branching and showering, to the lowest scale of hadronization. On
{ 2 {the other hand, jets contain signatures of exotic physics when produced by the decays of
heavy, strongly-interacting particles such as top quarks or particles beyond the Standard
Model.
Recently, several groups have explored strategies to probe jet substructure to distin-
guish jets produced by light partons in QCD from those produced by heavier particles
[1,2,3,4,5,6,7,8], and methods to \clean" jets of soft radiation to more easily iden-
tify their origin, such as \ltering" or \pruning" for jets from heavy particles [ 5,9,10] or
\trimming" for jets from light partons [ 11]. Another type of strategy, explored in [ 12], to
probe jet substructure is the use of jet shapes , which are modications of event shapes [ 13]
such as thrust. Jet shapes are continuous variables constructed by taking a weighted sum
over the four-momenta of all particles constituting a jet. Dierent choices of weighting
functions produce dierent jet shapes, and can be designed to probe regions closer to or
further from the jet axis with greater sensitivity.1While such jet shapes may integrate over
some of the detailed substructure for which some other methods search, they are better
suited to analytical calculation and understanding from the underlying theory of QCD.
In this paper, we consider measuring the shape of one or more jets in an e+ecollision
at center-of-mass energy QproducingNjets with an angular size Raccording to a cone or
recombination jet algorithm, with an energy cut  on the radiation allowed outside of jets.
We use this exclusive characterization of an N-jet nal state looking forward to extension
of our results to a hadron collider environment, where such a nal state denition is more
typical. For the jet shape observable we choose the angularity aof a jet, dened by (cf.
[12,17]),
a1
2EJX
i2Jpi
Tei(1a); (1.1)
whereais a parameter taking values 1<a< 2 (for IR safety, although factorizability
will require a < 1), the sum is over all particles in the jet, EJis the jet energy, pT
is the transverse momentum relative to the jet direction, and =ln tan(=2) is the
(pseudo)rapidity measured from the jet direction. The jet is dened by a jet algorithm, such
as a cone algorithm, the details of which we will discuss below. We complete the calculation
for the jet shape afor jets dened by cone or recombination algorithms, but our logic and
methods could be applied to a wider spectrum of jet shapes and jet algorithms. We have
organized our results in such a way that the pieces independent of the choice of jet shape
and dependent only on the jet algorithm are easily identiable, requiring recalculation only
of the observable-dependent pieces to extend our results to other choices of jet shapes.
Reliable theoretical prediction of jet observables in the presence of jet algorithms is
made challenging by the presence of many scales. Logarithms of ratios of these scales can
become large and spoil the behavior of perturbative expansions predicting these quantities.
These scales are determined by the jet energy !, the cut on the angular size of a jet R,
the measured value of the jet shape such as a, and any other cut or selection parameters
introduced by the jet algorithm.
1The original \jet shape," to which the name properly belongs, is the quantity 	( r=R), the fraction of
the total energy of a jet of radius Rthat is contained in a subjet of radius r[14,15,16] . This observable
falls into the larger class of jet shapes we have described here and for which we have hijacked the name.
{ 3 {Precisely this separation of scales, however, allows us to take advantage of the powerful
tools of factorization and eective eld theory. Factorization separates the calculation of
a hard scattering cross section into hard, jet and soft functions each depending only on
physics at a single scale [ 18,19]. Renormalization group (RG) evolution of these functions
between scales resums logarithms of these scales to all orders in s, with the logarithmic
accuracy determined by the order to which the anomalous dimensions in the running are
calculated [ 20]. Eective eld theory organizes these concepts and tools into a conceptually
simple framework unifying many ingredients going into traditional methods, such as power
counting, gauge invariance, and resummation through RG evolution. The rules of eective
theory facilitate proofs of factorization and achievement of logarithmic resummation at
leading order in the power counting and make straightforward the improvement of results
order-by-order in power counting and logarithmic accuracy of resummation.
1.2 Soft-Collinear Eective Theory and Factorization
Soft-Collinear Eective Theory (SCET) [ 21,22,23,24] has been successfully applied to the
analysis of many hard scattering cross sections [ 25] including the production of jets. SCET
is constructed by integrating out of QCD all degrees of freedom except those collinear to
a lightlike direction nand those which are soft, that is, have much lower energy than the
energy of the hard scattering or of the jets. Using this formalism, the factorization and
calculation of two-jet cross sections and event shape distributions in SCET were developed
in [26,27,28,29]. Later, these techniques were extended to the factorization of jet cross
sections and observables using jet algorithms in [ 30]. Calculations in SCET of two-jet rates
using jet algorithms have been performed in [ 27,31], and more recently in [ 32]. Calculations
of cross sections with more than two jet directions have been given in [ 33,34,35].
Building on many of the ideas in these previous studies, in this paper, we will demon-
strate a factorization theorem for jet shape distributions in e+e!Njet events,
d(P1;:::;PN)
d1


dM=(0)(P1;:::;PN)H(n1;!1;


nN;!N;) (1.2)
h
Jn1;!1(1;)


JnM;!M(M;)i

Sn1


nN(1;:::;M;R;;)
JnM+1;!M+1(R;)


JnN;!N(R;);
where theNjets have three-momenta Pi, andMNof the jets' shapes 1;:::;Mare
measured.(0)is the Born cross-section, His a hard function dependent on the directions
niand energies !iof theNjets,Jn;!() is the jet function for a jet whose shape is measured
to be,Jn;!(R) is the jet function for a jet with size Rwhose shape is not measured, and
Sis the soft function connecting all Njets, dependent on all jets' shapes i, sizesR, and
total energy  that is left outside of all jets. The symbol \ 
" stands for a set of convolution
integrals in the variables ibetween the measured jet functions and the soft function. All
terms in the factorization theorem depend on the factorization scale .
SCET is typically constructed as a power expansion in a small parameter formed by
the ratio of soft to collinear or collinear to hard scales, determined by the kinematics of the
process under study. is roughly the typical transverse momentum pTof the constituent
{ 4 {of a jet (relative to the jet direction) divided by the jet energy EJ. This is set either by
the measured value of the jet shape afor a measured jet or the algorithm measure R
for an unmeasured jet. Thus we encounter in this work the new twist that the size of
may be dierent for dierent jets. We will comment on further implications of this in
subsequent sections. Still, in each separate collinear sector, the momentum pnof collinear
modes in the light-cone direction nin SCET is separated into a large \label" momentum ~ pn
containingO(EJ) andO(EJ) components and a \residual" component of O(2EJ), the
same size as soft momenta. Eective theory elds have dynamical momenta only of this
soft or residual scale. This fact, along with the fact that soft quarks and soft gluons can be
shown to decouple from collinear modes at the level of the Lagrangian [ 24], makes possible
the factorization of a jet shape distribution into hard, jet, and soft functions depending
only on the dynamics at those respective scales.
In using SCET for jets in multiple directions and using jet algorithms to dene the
jets, we will encounter the need for several additional criteria to ensure the validity of the
N-jet factorization theorem.
First, to ensure that the algorithm does not group nal-state particles into fewer than
Njets, the jets must be \well separated." This allows us to use as the eective theory
Lagrangian a sum of Ncopies of the collinear part of the SCET Lagrangian for a
single direction nand a soft part, and to construct a basis of N-jet operators built
from elds from each of these sectors to produce the nal state. Our calculations will
reveal the precise quantitative condition that jets must satisfy to be \well separated".
Second, to ensure that the jet algorithm does not nd more than Njets, we place an
energy cut  on the total energy outside of the observed jets. We will take this energy
 to scale as a soft momentum so that we will be able to identify the total energy of
each jet with the \label" momentum on the SCET collinear jet eld producing the
jet. Corrections to this identication are subleading in the SCET power counting.
Third (and related to the above two), we will assume that the N-jet restriction on the
nal state can itself be factorized into a product of N1-jet restrictions, one in each
collinear sector, and a 0-jet restriction in the soft sector. We represent the energetic
particles in the ith jet by collinear elds in the SCET Lagrangian in the nicollinear
sector and soft particles everywhere with elds in the soft part of the Lagrangian.
We then stipulate that the jet algorithm acting on states in the nicollinear sector
nd exactly one jet in that sector, and when acting on the soft nal state nd no
additional jet in that sector.
Fourth, the way in which a jet algorithm combines particles in the process of nding
a jet must respect the order of steps envisioned by factorization. In particular, fac-
torization requires that the jet directions and energies be determined by the collinear
particles alone, so that the soft function knows only about the directions and colors of
the jets, not the details of any collinear recombinations. Ideally, all energetic collinear
particles should be recombined rst, with soft particles within a radius Rof the jet
{ 5 {axis being recombined into the jet only afterwards. Jet algorithms in use at experi-
ments do not have this precise behavior, but we will discuss in Sec. 3.4the extent to
which common algorithms meet this requirement and estimate the size of the power
corrections due to their failure to do so. In general, we will nd that for suciently
largeR, infrared-safe cone algorithms and k T-type recombination algorithms satisfy
the requirements of factorizability, with anti-k Tallowing smaller values of Rthan k T.
After enforcing the above requirements, a key test of the consistency of Eq. ( 1.2) will
be the independence of the physical cross section on the factorization scale . This requires
the anomalous dimensions of the hard, jet, and soft functions to sum to zero,
0 = [
H() +
JM+1(R;) +


+
JN(R;)](1)


(M)
+
J1(1;)(2)


(N) +


+(1)


(M1)
JM(M;)
+
S(1;:::;M;R;):(1.3)
It seems highly nontrivial that this condition would be satised for any number, size, and

avors of jets (and that the soft anomalous dimension be independent of ), but we will
demonstrate that it does hold at O(s), up to corrections of O(1=t2) which violate Eq. ( 1.3),
wheretis a measure of the separation between jets. In particular, for a pair of jets, i;j,
with 3-vector directions separated by a polar angle  ij, the separation tijis given by
tk;l=tan( k;l=2)
tan(R=2): (1.4)
Now dene t(no indices) as the minimum of tijover all jet pairs. This quanties the
qualitative condition of jets being well-separated, t1, that is required to justify the
factorization theorem Eq. ( 1.2). The factorization theorem is valid up to corrections of
O() in the SCET power expansion parameter and corrections of O(1=t2) in the separa-
tion parameter. As an example of the magnitude of t, for three jets in a Mercedes-Benz
conguration (  = 2=3 for all pairs of jets), 1 =t2= 0:04 forR= 0:7 and 1=t2= 0:1 for
R= 1, so these corrections are indeed small. More generally, for non-overlapping jets,
 >2R, we have 1 =t2<1=4.
Notice that for back-to-back jets (  =),t!1 . Thus, for all cases previously con-
sidered in the literature, the jets are innitely separated according to this measure, and no
additional criterion regarding jet separation is required for consistency of the factorization
and running. A key insight of our work is that for an N-jet cross-section described by
Eq. ( 1.2), the factorization theorem receives corrections not only in the usual SCET power
counting parameter , but also corrections due to jet separation beginning at O(1=t2).
1.3 Power Corrections to Factorized Jet Shape Distributions
As always, there are power corrections to the factorization theorem which we must ensure
are small. One class of power corrections arises from approximating the jet axis of the
measured jet with the collinear direction ni, which labels that jet in the SCET Lagrangian.
This direction niis the direction of the parent parton initiating the jet. The jet observable
must be such that the dierence between the parent parton direction and the jet axis
{ 6 {identied by the algorithm makes a subleading correction to the calculated value of the jet
observable. In the context of angularity event shapes, such corrections were estimated in
[17,29] and found to be negligible for a<1, and we nd the same condition for jet shapes.
In the presence of algorithms, however, there are additional power corrections due to
the dierence in the soft particles that are included or excluded in a jet by the actual
algorithm and in its approximated form in the factorization theorem. We study the eect
of this dierence on the measurement of jet shapes, and nd that for suciently large Rthe
power corrections due to the action of the algorithm on soft particles remain small enough
not to spoil the factorization for infrared-safe cone and k T-type recombination algorithms.
Algorithm-related power corrections to jet momenta were studied more quantitatively in
[36], and their estimated Rdependence is consistent with our observations.
We do not address in this work the issue of power corrections to jet shapes due to
hadronization. Event shape distributions are known to receive power corrections of the
order 1=(aQ), enhanced in the endpoint region but suppressed by large energy. The
endpoints of our jet shape distribution near a!0, therefore, will have to be corrected by
a nonperturbative shape function. Such functions have been constructed for event shapes
in [37,38]. The shift in the rst moment of event shape distributions induced by these
shape functions was postulated to take a universal form in [ 39,40] based on the behavior
of single soft gluon emission, and the universality was proven to all orders in soft gluon
emission at leading order in the SCET power counting in [ 41,42]. This universality relied
on the boost invariance of the soft function describing soft gluon radiation from two back-
to-back collinear jets. The extent to which such universality may survive for jet shapes
with multiple jets in arbitrary directions is an open question that must be addressed in
order to construct appropriate soft shape function models to deal adequately with the
power corrections to jet shapes from hadronization. Nonperturbative power corrections to
jet observables from hadronization and the underlying event in hadron collisions were also
studied in [ 36], and hadronization corrections were found to scale like 1 =R. In this work,
we focus only on the perturbative calculation and resummation of large logarithms of jet
shapes, and leave inclusion of nonperturbative power corrections for future work.
1.4 Resummation and Logarithmic Accuracy
Knowing the anomalous dimensions of the hard, jet, and soft functions in the factorization
theorem allows us to resum logarithms of ratios of the hard, jet, and soft scales. We
take this opportunity to explain the order of accuracy to which we are able to resum
these logarithms. For an event shape distribution d=d (i.e. Eq. ( 1.2) with two jets and
integrated against (12)), the accuracy of logarithmic resummation [ 43] is typically
characterized by counting logs in the exponent ln R() of the \radiator,"
R() =1
0Z
0d0d
d0; (1.5)
where they appear in the form n
slnmwithmn+ 1. At leading-logarithmic (LL)
accuracy all the terms with m=n+ 1 are summed; next-to-leading-logarithmic (NLL)
accuracy sums also the m=nterms, and so on. In more traditional methods in QCD, event
{ 7 {shapes that have been resummed include NLL resummation of thrust in [ 43,44], jet masses
in [43,45,46,47], jet broadening in [ 48,49], theC-parameter in [ 50], and angularities in
[17]. Resummation of an event shape distribution using the modern SCET method was
rst illustrated with the thrust distribution to LL accuracy in [ 51]. Heavy quark jet mass
distributions were resummed in SCET to NLL accuracy as part of a proposed method to
extract the top quark mass in [ 52]. The N3LL resummed thrust distribution in SCET was
compared to LEP data to extract a value for the strong coupling sto high precision in
[53]. Angularities were resummed to NLL accuracy in SCET in [ 54] directly in a-space
instead of in moment space as in [ 17].
Summation of logarithms in eective eld theory is achieved by RG evolution. In the
factorized radiator of the thrust distribution Eq. ( 1.5), one nds that the hard function
contains logarithms of =Q, the jet functions contain logarithms of =(Qp), and the soft
function contains logarithms of =(Q). Thus, evaluating these functions respectively at
the hard scale H=Q, jet scaleJ=Qpand soft scale S=Qeliminates large
logarithms in each function. They can then be RG-evolved to the common factorization
scaleafter calculating their anomalous dimensions. The solutions of the RG evolution
equations are of the form that logarithms of are resummed to all orders in sto a
logarithmic accuracy determined by the order in sto which the anomalous dimensions
and hard/jet/soft functions are known. This underlying hierarchy of scales is illustrated
Fig. 1[in this case, with only one (measured) jet scale and soft scale and !=Q] along
with a table that lists the order in sto which various quantities must be known in order
to achieve a given NkLL accuracy in the exponent of the radiator Eq. ( 1.5). The power of
the EFT framework is to organize of the logs of arising in Eq. ( 1.5) into those that arise
from ratios of the jet to the hard scale and those that arise from ratios of the soft to the
hard scale, which then allows RG evolution to resum them.
For the multijet shape distribution in Eq. ( 1.2), the strategy to sum logs is the same,
but is complicated by the presence of additional scales. This also makes trickier the clas-
sication of logarithmic accuracy into the standard NkLL scheme. Our aim will be to
sum as many logs of the jet shapes aas possible, while not worrying about any others.
For instance, phase space cuts induce logs of Rand =!(where!is a typical hard jet
energy), and the presence of multiple jets induces logs of jet separations ni
njor ratios of
jet energies !i=!j. We will not aim to sum these types logs systematically in this paper,
only those of a(though we sum subsets of the others incidentally). In particular, resum-
ming the phase space logs of Ror =!is complicated by how the phase space cuts act
order-by-order in perturbation theory2, and the fact that a simple angular cut Ris less
restrictive than a small jet mass or angularity on how collimated a jet must be. That is,
an angular cut allows particles in a jet to be anywhere within an angle Rof the jet axis
regardless of their energy, while a small jet mass or angularity forces harder particles to be
closer to the jet axis. The former allows hard particles to lie along the edges of a jet, and
2The JADE algorithm is one well-known example in which resummability even of leading logarithms of
the jet mass cut yis spoiled by the dierences in the jet phase space at dierent orders in perturbation
theory [ 55]. Another example that will not work is using a k T-type algorithm with Rrandomly chosen for
each recombination. This is clearly such that resummation of logarithms of Rcannot be achieved.
{ 8 {hard scale
“unmeasured”
jet scale
soft scalesµµH=ω
µmeas
J=ωτ1
2−aaµunmeas
J =ωtanR
2
“measured”
jet scale
µSγmeas
Jγunmeas
J
γS
EFT
countingmatching/
matrix element
LL tree 1-loop tree 1-loop
NLL tree 2-loop 1-loop 2-loop
NNLL 1-loop 3-loop 2-loop 3-loopΓcusp γH,J,S β[αs]
µmeas
S =ωτa/tan1−a(R/2)µΛR= 2Λtan(R/2)µΛ=2ΛFigure 1: An illustration of generic scales along with a table of log-accuracy versus perturbative
order. A cross section with jets of energy !, radiusR, and energy  outside the jets, with
some jets' shapes abeing measured and others' shapes left unmeasured, induces measured and
unmeasured jet scales at meas
J andunmeas
J . Dynamics at these scales are described by separate
collinear modes in SCET. Soft dynamics occur at several soft scales, andRinduced by the
soft out-of-jet energy cut  and jet radius R, andmeas
S induced by the measured jet shape a. RG
evolution in SCET resums logs of ratios of jet scales to the hard scale Hindividually, and logs
of the ratio of a \common" soft scale Sto the hard scale. Remaining logs of ratios of soft scales
to one another are not resummed in current formulations of SCET. The accuracy of logarithmic
resummation of these ratios of scales is determined by the order to which anomalous dimensions and
matching coecients or matrix elements (i.e. hard/jet/soft functions) are calculated in perturbation
theory. In this paper we perform the RG evolution indicated by the arrows to NLL accuracy.
soft radiation from such congurations that escapes the jets can lead to logs of  =!that
are not captured in our treatment. These are not issues we solve in this paper, in which
we focus on resumming logs of jet shapes a. (Some exploration of phase space logarithms
in SCET was carried out in [ 31,32].)
A way to understand how we sum logs and which ones we capture is presented in Fig. 1.
The factorization theorem Eq. ( 1.2) organizes logs in the multijet cross section into those
in the hard function, those in measured jet functions, those in unmeasured jet functions,
and those in the soft function, much like for the simple thrust distribution. The dierence
is the presence now of multiple jet and soft scales. Logarithms in jet functions can still be
minimized by choices of individual jet scales, meas
J!1=(2a)
a for a jet whose shape a
is measured, and unmeas
J!tan(R=2) for a jet whose shape is not measured but has a
radiusR. Thus logs arising from ratios of these scales to the hard scale can be summed
{ 9 {completely to a desired NkLL order. The complication is in the soft function. The soft
function is sensitive to soft radiation into measured and unmeasured jets and outside of all
jets. As we will see by explicit calculation, this induces logs of tan1a(R=2)=(!a) from
radiation into measured jets, and logs of =(2) and=(2 tanR
2) from radiation from
unmeasured jets cut o by the energy . In addition, though not illustrated in Fig. 1,
there can be logs of multiple jet shapes to one another, i
a=j
a. No single choice of a soft
scaleSwill minimize all of these logs.
In the present work, we will start with the simple strategy of choosing a single soft
scaleS!a=tan1a(R=2) for a jet whose shape awe are measuring and logs of which
we aim to resum. We will calculate hard/jet/soft functions and anomalous dimensions
corresponding to \NLL" accuracy listed in Fig. 1. By this we do not mean all potentially
large logs in Eq. ( 1.2) are resummed to NLL, but only those logs of ratios of a jet scale to
the hard scale or of the (common) soft scale to the hard scale. We do not attempt to sum
logs of ratios of soft scales to one another completely to NLL accuracy (which can contain
a). In the case that all jets' shapes are measured and are similar to one another, i
aj
a,
our resummation of large logs of i
awould be complete to NLL accuracy.
We will nevertheless venture to propose a framework to \refactorize" the soft function
into further pieces dependent on only a single soft scale at a time and perform additional
RG running between these scales to resum the additional logarithms, and will implement
it at the level of the O(s) soft functions we calculate. However, one cannot really address
mixed logarithms such as log( i
a=j
a) that arise for multiple jets until O(2
s), the rst order
at which two soft gluons can probe two dierent physical regions. This lies beyond the
scope of the present work. (We note, however, that our implementation of refactorization
using the one-loop soft function does already seem to tame logarithmic dependence on 
in our numerical studies of jet shape distributions.)
These issues are related to some types of \non-global" logarithms described by [ 46,56,
57,58] that spoil the simple characterization of NLL accuracy. In [ 59] these were identied
as next-to-leading logs of  R2=(!ia) and =Q(whenR1) that appear at O(2
s) in jet
shape distributions. These authors organized the radiator for a single jet shape distribution
into a \global" and \non-global" part [ 58,59],
R(i
a;R;;!i;Q) =Rgl
i
a;R;
Q
RngR2
!iia;
Q
: (1.6)
In this language, the calculations we undertake in this paper resum logs in the global part
to NLL accuracy but not in the non-global part. The rst argument in Rngis related
to ratios of soft scales illustrated in Fig. 1, and the second argument arises when there
are unmeasured jets. In the case that all jets are measured, R1, and !ii
a, the
non-global logs vanish.
While summing all global and non-global logs to at least NLL accuracy will be impor-
tant for precision jet phenomenology, what we contribute in this paper are key developments
and calculations necessary to resum even global logs of jet shapes for jets dened with al-
gorithms. We also believe the eective theory approach and the idea of refactorizing the
soft function will help us understand and resum many types of non-global logarithms.
{ 10 {1.5 Detailed Outline of This Work
In this paper, we will formulate and prove a factorization theorem for distributions in the
jet shape variables we introduced above, calculate the jet and soft functions appearing in
the factorization theorem to O(s) in SCET, and use the renormalization group evolution
of these functions to sum global logs of ato NLL accuracy. We consider Njets (dened
with a cone or k Talgorithm) produced in an e+ecollision, with Mof the jets' shapes
(angularities) being measured. The key formal result is our demonstration of Eq. ( 1.3),
the consistency of the anomalous dimensions of hard, jet and soft functions to O(s) for
any number of total jets, any numbers of quark and gluon jets, any number of these jets
whose shapes are measured, and any value of the distance measure Rin cone or k T-
type algorithms (as long as t1). These results lead to accurate predictions for the
shape of the adistribution near the peak, but not necessarily the endpoints for very
smalla(where hadronization corrections dominate) and very large a(where xed-order
NLO QCD corrections take over, which are not yet calculated and not captured by NLL
resummation).3
In Sec. 2we describe in detail the jet shapes and jet algorithms that we use. We describe
features of an \ideal" jet algorithm that would respect exactly the order of operations
envisioned in the factorization theorem derived in SCET, and the extent to which cone
and recombination algorithms actually in use resemble this idealization.
In Sec. 3, using the tools of SCET, we will derive in detail a factorization theorem
for exclusive 3-jet production where we measure the angularity jet shape of one of the
jets, and then perform the straightforward extension to N-jet production with MN
measured jets. We will give a review of the necessary technical details of SCET in Sec. 3.1.
In the process of justifying the factorization theorem, we identify the new requirements
listed above on N-jet nal states and jet algorithms that must be satised for factorization
to hold. In Sec. 3.4we explore in some detail the power corrections to the factorization
theorem due to soft radiation and the action of jet algorithms that cause tension with these
requirements, and argue that for suciently large Rin infrared-safe cone and recombination
algorithms, these corrections are suciently small.
Next we calculate to O(s) the jet and soft functions corresponding to Ncone or
kT-type jets, with Mjets' shapes measured.
In Sec. 4we calculate the jet functions for measured quark jets, Jq
!(a), unmeasured
quark jets,Jq
!, measured gluon jets, Jg
!(a), and unmeasured gluon jets, Jg
!, where!= 2EJ
is the label momentum of the collinear jet eld in each jet function. We nd that in collinear
sectors for measured jets, the collinear scale (and thus the SCET power counting parameter
in that sector i) is given by !i1=(2a)
a , and in unmeasured jet sectors, itan(R=2).
In studying power corrections, however, as mentioned above, we nd that Rmust be
parametrically larger than a. So, in collinear sectors for measured jets, iis set by the
3Jet shapes were also studied in the QCD factorization approach in [ 60]. In that work QCD jet functions
for quark and gluon jets dened with an algorithm and whose jet masses m2
Jare measured were calculated
toO(s). The jet mass2corresponds to afora= 0,0=m2=!2(01). A xed-order QCD jet function
as dened in [ 60] is given by the convolution of our xed-order SCET jet function and soft function for a
measured jet away from a= 0.
{ 11 {shapeawithR0
i, while in unmeasured jet sectors, itan(R=2). Thus one should
understand tan( R=2) to be signicantly less than 1 but much larger than any jet shape a.
In Sec. 5we calculate the soft function. To do this, we split the soft function into several
contributions from dierent parts of phase space in order to facilitate the calculation and
elucidate its intuitive structure. We nd it most convenient to split the soft function into
an observable-independent part that arises from soft emission out of the jets, Sunmeas, and
a part that depends on our choice of angularities as the observable that arises from soft
emission into measured jet i,Smeas(i
a).Sunmeasis hence sensitive to the scale  while
Smeas(i
a) is sensitive to the scale !ii
a.
In Sec. 6, having calculated all the jet and soft function contributions to O(s), we
extract the anomalous dimensions and perform renormalization-group (RG) evolution. We
nd the hard anomalous dimension from existing results in the literature. The hard, jet,
and soft anomalous dimensions have to satisfy the consistency condition Eq. ( 1.3) in order
for the physical cross section to be independent of the arbitrary factorization scale . Our
calculations reveal that, as long as the jet separation parameter tEq. ( 1.4) between all
pairs of jets is much larger than 1, the condition is satised.
Even after requiring t1, the satisfaction of the consistency condition is non-trivial.
The hard function knows only about the direction of each jet and the jet function knows
only the jet size R; the soft function knows about both. Furthermore, it is not sucient
only to include regions of phase space where radiation enters the measured jets. We learn
from our results in this Section that it is crucial to include soft radiation outside of all jets
with an upper energy cuto of . Only after including all of these contributions from the
various parts of phase space do the jet, hard, and soft anomalous dimensions cancel and
we arrive at a consistent factorization theorem.
We conclude Sec. 6by proposing in Sec. 6.4a strategy to sum logs due to a hierarchy
of scales in the soft function, by \refactorizing" it into multiple pieces, each sensitive to a
single scale, as suggested by the discussion surrounding Fig. 1. Our current implementation
of this procedure does tame the logarithmic dependence of jet shape distributions on the
ratio =!in our numerical studies, but we leave for further development the resummation
of all \non-global" logs of ratios of multiple soft scales that begin at NLL and O(2
s).
To help the reader nd the results of the calculations in Sec. 4through Sec. 6just
outlined, Table 1provides a summary with equation numbers.
In Sec. 7we compare our resummed perturbative predictions for jet shape distributions
to the output of a Monte Carlo event generator. We test both the accuracy of these
predictions and assess the extent to which hadronization corrections aect jet shapes. We
will illustrate our results in the case of e+e!3 jets, with the jets constrained to be
in a conguration where each has equal energy and are maximally separated. In both
the eective theory and Monte Carlo, we can take the jets to have been produced by an
underlying hard process e+e!qqg. After placing cuts on jets to ensure each parton
corresponds to a nearby jet, we measure the angularity jet shape of one of the jets. We
compare our resummed theoretical predictions with the Monte Carlo output for quark and
gluon jet shapes with various values of aandR. We nd that the dependencies on aandR
of the shapes of the distribution and the peak value of aagree well between the theory and
{ 12 {Category Contribution Symbol Location
measured quark jet function Jq
!(a) Eq. ( 4.11)
unmeas. quark jet function Jq
! Eq. ( 4.17)
measured gluon jet function Jg
!(a) Eq. ( 4.25)
unmeas. gluon jet function Jg
! Eq. ( 4.26)
NLO contributions summary of divergent| Table 2before resummation: parts of soft func. (any t)
total universalSunmeasEq. ( 5.20)soft func. (large t)
total measuredSmeas(i
a) Eq. ( 5.22)soft func. (large t)
anomalous dimensions: | | Table 3
measured jet function fi
J(i
a;i
J)Eq. ( 6.42a )
NLO contributions measured soft function fS(i
a;i
J)Eq. ( 6.42b )
after resummation: unmeas. jet function Ji
!(J) Eq. ( 6.43a )
universal soft function Sunmeas(
S)Eq. ( 6.43b )
Total NLL Distribution: | | Eq. ( 6.40)
Table 1: Directory of main results: the xed-order NLO quark and gluon jet functions for jets
whose shapes aare measured or not; the xed-order NLO contributions to the soft functions from
parts of phase space where a soft gluon enters a measured jet, Smeas(a), or not,Sunmeas; their
anomalous dimensions; the contributions the jet and soft functions make to the nite part of the
NLL resummed distributions; and the full NLL resummed jet shape distribution itself.
Monte Carlo, with small but noticeable corrections due to hadronization. We can estimate
these corrections by comparing output with hadronization turned on or o in Monte Carlo.
In Sec. 8, we give our conclusions and outlook. We also collect a number of technical
details and results for O(s) nite pieces of jet and soft functions in the Appendices.
Our work is, to our knowledge, the rst achieving factorization and resummation of a
jet observable distribution in an exclusive N-jet nal state dened by a non-hemisphere jet
algorithm.4Having demonstrated the consistency of this factorization for any number of
quark and gluon jets, measured and unmeasured jets, and phase space cuts in cone and k T-
type algorithms, and having constructed a framework to resum logarithms of jet shapes in
the presence of these phase space cuts, we hope to have provided a starting point for future
precision calculations of many jet observables both in e+eand hadron-hadron collisions.
The case of ppcollisions will require a number of modications, including turning two of
our outgoing jet functions into incoming \beam functions" introduced in [ 62]. We leave
this generalization for future work.
The reader wishing to follow the general structure of our ideas and logic and understand
the basis of the nal results of the paper without working through all the technical details
may read Secs. 1and2, and then skip to Sec. 7. Some short less technical discussion also
appears in Sec. 3.4.
4Dijet cross sections for cone jets were factorized and resummed in [ 61].
{ 13 {2. Jet Shapes and Jet Algorithms
2.1 Jet Shapes
Event shapes, such as thrust, characterize events based on the distribution of energy in
the nal state by assigning diering weights to events with diering energy distributions.
Events that are two-jet like, with two very collimated back-to-back jets, produce values of
the observable at one end of the distribution, while spherical events with a broad energy
distribution produce values of the observable at the other end of the distribution. While
event shapes can quantify the global geometry of events, they are not sensitive to the
detailed structure of jets in the event. Two classes of events may have similar values of
an event shape but characteristically dierent structure in terms of number of jets and the
energy distribution within those jets.
Jet shapes, which are event shape-like observables applied to single jets, are an eective
tool to measure the structure of individual jets. These observables can be used to not only
quantify QCD-like events, but study more complex, non-QCD topologies, as illustrated
for light quark vs. top quark and Zjets in [ 12,60]. Broad jets, with wide-angle energy
depositions, and very collimated jets, with a narrow energy prole, take on distinct values
for jet shape observables. In this work, we consider the example of the class of jet shapes
called angularities, dened in Eq. ( 1.1) and denoted a. Every value of acorresponds to
a dierent jet shape. As adecreases, the angularity weights particles at the periphery
of the jet more, and is therefore more sensitive to wide-angle radiation. Simultaneous
measurements of the angularity of a jet for dierent values of acan be an additional probe
of the structure of the jet.
2.2 Jet Algorithms
A key component of the distribution of jet shapes is the jet algorithm, which builds jets
from the nal state particles in an event. (We are using the term \particle" generically here
to refer to actual individual tracks, to cells/towers in a calorimeter, to partons in a pertur-
bative calculation, and to combinations of these objects within a jet.) Since the underlying
jet is not intrinsically well dened, there is no unique jet algorithm and a wide variety of jet
algorithms have been proposed and implemented in experiments. The details of each algo-
rithm are motivated by particular properties desired of jets, and dierent algorithms have
dierent strengths and weaknesses. In this work we will calculate angularity distributions
for jets coming from a variety of algorithms. Because we calculate (only) at next-to-leading
order, there are at most 2 particles in a jet, and jet algorithms that implement the same
phase space cuts at NLO simplify to the same algorithm. At this order the two standard
classes of algorithms, cone algorithms and recombination algorithms, each simplify to a
generic jet algorithm at NLO. At NLO the cone algorithms place a constraint on the sep-
aration between each particle and the jet axis, while the recombination algorithms place a
constraint on the separation between the two particles.
Cone algorithms build jets by grouping particles within a xed geometric shape, the
cone, and nding \stable" cones. A cone contains all of the particles within an angle Rof
the cone axis, and the angular parameter Rsets the size of the jet. In found jets (stable
{ 14 {cones), the direction of the total three-momentum of particles in the cone equals the cone
(jet) axis. Dierent cone algorithms employ dierent methods to nd stable cones and
deal with dierently the \split/merge" problem of overlapping stable cones. The SISCone
algorithm [ 63] is a modern implementation of the cone algorithm that nds all stable cones
and is free of infrared unsafety issues. In the next-to-leading order calculation we perform,
there are at most two particles in a jet, and we only consider congurations where all jets
are well-separated. Therefore, it is straightforward to nd all stable cones, there are no
issues with overlapping stable cones, and the phase space cuts of all cone algorithms are
equivalent. This simplies all standard cone algorithms to a generic cone-type algorithm, in
which each particle is constrained to be within an angle Rof the jet axis. For a two-particle
jet, if we label the particles 1 and 2 and the jet axis n, then the cone-like constraints for
the two particles to be in a jet are
cone jet:1n<R and2n<R: (2.1)
This denes our cone-type algorithm.
Recombination algorithms build jets by recursively merging pairs of particles. Two
distance metrics, dened by the algorithm, determine when particles are merged and when
jets are formed. A pairwise metric pair(the recombination metric) denes a distance
between pairs of particles, and a single particle metric jet(the beam, or promotion, metric)
denes a distance for each single particle. Using these metrics, a recombination algorithm
builds jets with the following procedure:5
0. Begin with a list Lof particles.
1. Find the smallest distance for all pairs of particles (using pair) and all single particles
(usingjet).
2a. If the smallest distance is from a pair, merge those particles together by adding their
four momenta. Replace the pair in Lwith the new particle.
2b. If the smallest distance is from a single particle, promote that particle to a jet and
remove it from L.
3. Loop back to step 1 until all particles have been merged into jets.
The k T, Cambridge-Aachen, and anti-k Talgorithms are common recombination algo-
rithms, and their distance metrics are part of a general class of recombination algorithms.
Fore+ecolliders, a class of recombination algorithms can be dened by the parameter :
pair(i;j) = min
E
i;E
j
ij
R
jet(i) =E
i; (2.2)
5This denes an inclusive recombination algorithm more typically applied to hadron-hadron colliders.
We are applying it here to the simpler case of e+ecollisions in order to facilitate the eventual transition
to LHC studies. Exclusive recombination algorithms, more typical of e+ecollisions, are described along
with other jet algorithms in [ 64] and their description in SCET is given in [ 32].
{ 15 {where= 1 for k T, 0 for Cambridge-Aachen, and 1 for the anti-k Talgorithm. The
parameterRsets the maximum angle between two particles for a single recombination.6
In the multijet congurations we consider the jets are separated by an angle larger than
R, so that only the pairwise metric is relevant for describing the phase space constraints
for particles in each jet. For a two-particle (NLO) jet, the only phase space constraint
imposed by this class of recombination algorithms is that the two particles be separated
by an angle less than R:
kTjet:12<R: (2.3)
This denes a generic recombination algorithm suitable for our calculation. We will denote
this as the k T-type algorithm.
The congurations with two widely separated energetic particles best distinguish cone-
type jets from k T-type jets at NLO. For instance, the case where the two energetic particles
are at opposite edges of a cone jet (at an angle 2 Rapart) is not a single k Tjet. However, it
is important to note that these congurations will not be accurately described in this SCET
calculation for R, as such congurations are power suppressed in our description of
jets. Our concern is in accurately describing the congurations with narrow jets (small a),
and not the wide angle congurations above.
Because jets are reliable degrees of freedom and provide a useful description of an
event when they have large energy, in the description of an event we impose a cut  on
the minimum energy of jets. An N-jet event, therefore, is one where Njets have energy
larger than the cuto , with any number of jets having energy less than the cuto. In
our calculation, we impose the same constraint: any jet with energy less than  is not
considered when we count the number of jets in the nal state. This imposes phase space
cuts: for a gluon radiated outside of all jets in the event, that gluon is required to have
energyEg< to maintain the same number of jets in the event. The proper division of
phase space in calculating the jet and soft functions is a key part of the discussion below,
and careful treatment of the phase space cuts is needed.
2.3 Do Jet Algorithms Respect Factorization?
The factorization theorem places specic requirements on the structure of jet algorithms
used to describe events. As in Eq. ( 1.2), the factorization theorem divides the cross section
into separately calculable hard, jet, and soft functions. The hard function depends only on
the conguration of jets, while the jet and soft functions describe the degrees of freedom
in each jet in terms of the observable . While the soft function is global, the jet function
depends only on the collinear degrees of freedom in a single jet. The limited dependence
of the hard and jet functions implies constraints on the jet algorithm.
Because jets are built from the long distance degrees of freedom arising from evolution
of energetic partons to lower energies, the conguration of jets in an event depends on
dynamics across all energy scales. This naively breaks factorization in SCET, since the
6We useRfor both cone and k Talgorithms for ease of notation. For k T, this parameter is sometimes
referred to as D. We emphasize that having the same size Rfor dierent algorithms does not in general
guarantee the same sized jets.
{ 16 {conguration of jets in the hard function would depend on dynamics at low energy in the
soft function. However, we can describe a jet algorithm that respects factorization, and in
Sec.3.4we will parameterize the power corrections that arise from various algorithms.
The primary constraint on the jet algorithm in order to satisfy the factorization the-
orem is that the phase space cuts on the collinear particles in the jet are determined only
by the collinear degrees of freedom. This essentially ensures that the jet functions are
independent of dynamics in the soft function. Correspondingly the soft function can only
know about the jet directions and their color representations. The direction of the jet is
naturally set by the collinear particles, as soft particles have energy parametrically lower
than the collinear ones and change the jet direction by a power suppressed amount. The
further restriction that the phase space cuts on the collinear degrees of freedom are in-
dependent of the soft degrees of freedom places a strong constraint on the action of the
jet algorithm. Cone algorithms already implement this constraint: the jet boundary (the
cone) is determined by the location of the jet axis, which is the direction of the sum of
all collinear particles up to a power correction. Recombination algorithms, however, are
constrained by the factorization theorem to operate in a specic way: all collinear particles
must be recombined before soft particles. As discussed in Sec. 3.4, commonly used algo-
rithms obey this constraint up to power corrections in the observable for measured jets.
Of particular note is the anti-kT algorithm, which exhibits behavior very close to what is
required by the factorization theorem (similarly to the way cone algorithms behave).
3. Factorization of Jet Shape Distributions in e+etoNJets
In this Section we formulate a factorization theorem for jet shape distributions in e+e
annihilation to Njets. All the formal aspects we need to describe an N-jet cross section
appear already in the 3-jet cross section, so we will give explicit details only for that
case. We will use the framework of Soft-Collinear Eective Theory (SCET), developed in
[21,22,23,24], to formulate the factorization theorem. We begin with a basic review of
the relevant aspects of the eective theory.
3.1 Overview of SCET
SCET is the eective eld theory for QCD with all degrees of freedom integrated out, other
than those traveling with large energy but small virtuality along a light-like trajectory
n, and those with small, or soft, momenta in all components. A particularly useful set
of coordinates is light-cone coordinates, which uses light-like directions nand n, with
n2= n2= 0 andn
n= 2. In Minkowski coordinates, we take n= (1;0;0;1) and
n= (1;0;0;1), corresponding to collinear particles moving in the + zdirection. A generic
four-vector pcan be decomposed into components
p= n
pn
2+n
pn
2+p
?: (3.1)
In terms of these components, p= (n
p;n
p;p?), collinear and soft momenta scale with
some small parameter as
pn=E(1;2;); psE(2;2;2); (3.2)
{ 17 {whereEis a large energy scale, for example, the center-of-mass energy in an e+ecollision.
is then the ratio of the typical transverse momentum of the constituents of the jet to the
total jet energy. Quark and gluon elds in QCD are divided into collinear and soft eective
theory elds with these respective momentum scalings:
q(x) =qn(x) +qs(x); A(x) =A
n(x) +A
s(x): (3.3)
We factor out a phase containing the largest components of the collinear momentum from
the eldsqn;An. Dening the \label" momentum ~ p
n= n
~pnn
2+ ~p
?, where n
~pncontains
theO(1) part of the large light-cone component of the collinear momentum pn, and ~p?the
O() transverse component, we can partition the collinear elds qn;Aninto their labeled
components,
qn(x) =X
~p6=0ei~p
xqn;p(x); A
n(x) =X
~p6=0ei~p
xA
n;p(x): (3.4)
The sums are over a discrete set of O(1;) label momenta into which momentum space is
partitioned. The bin ~ p= 0 is omitted to avoid double-counting the soft mode in Eq. ( 3.3)
[65]. The labeled elds qn;p;An;pnow have spacetime 
uctuations in xwhich are conjugate
to \residual" momenta kof the order E2, describing remaining 
uctuations within each
labeled momentum partition [ 23,65]. It will be convenient to dene label operators P=
n
Pn=2 +P
?which pick out just the label components of momentum of a collinear eld:
Pn;p(x) = ~pn;p(x): (3.5)
Ordinary derivatives @acting on eective theory elds n;p(x) are of order E2.
The nal step to construct the eective theory elds is to isolate the two large compo-
nents of the Dirac spinor qn;pfor a fermion with lightlike momentum along n. The large
components n;pand the small  n;pcan be separated by the projections
n;p=n =n =
4qn;p;n;p=n =n =
4qn;p; (3.6)
and we have qn;p=n;p+ n;p. One can show, substituting these denitions into the QCD
Lagrangian, that the elds  n;phave an eective mass of order Eand can be integrated
out of the theory. The eective theory Lagrangian at leading order in is [22,23,24]
LSCET =L+LAn+Ls; (3.7)
where the collinear quark Lagrangian Lis
L=n(x)
in
D+iD =c
?Wn(x)1
in
PWy
n(x)iD =c
?n =
2n(x); (3.8)
whereWnis the Wilson line of collinear gluons,
Wn(x) =X
permsexp
g1
n
Pn
An(x)
; (3.9)
{ 18 {the collinear gluon Lagrangian LAnis
LAn=1
2g2Trh
iD+gA
n;iD+gA
ni2
+ 2 Tr
cnh
iD;h
iD+gA
n;cnii
+1
Trh
iD;A
ni
;(3.10)
wherecnis the collinear ghost eld and the gauge-xing parameter; and the soft La-
grangianLsis
Ls= qsiD =sqs(x)1
2TrG
sGs(x); (3.11)
which is identical to the form of the full QCD Lagrangian (the usual gauge-xing terms
are implicit). In the collinear Lagrangians, we have dened several covariant derivative
operators,
D=@igA
nigA
s; iD
c=P+gA
n; iD=P+in
Dn
2: (3.12)
In addition, there is an implicit sum over the label momenta of each collinear eld and the
requirement that the total label momentum of each term in the Lagrangian be zero.
Note the soft quarks do not couple to collinear particles at leading order in . Mean-
while, the coupling of the soft gluon eld to a collinear eld is in the component n
Asonly,
according to Eqs. ( 3.8) and ( 3.10), which makes possible the decoupling of such interac-
tions through a eld redenition of the soft gluon eld given in [ 24]. We will utilize this
soft-collinear decoupling to simplify the proof of factorization below.
The SCET Lagrangian Eq. ( 3.7) may be extended to include collinear particles in
more than one direction [ 25]. One adds multiple copies of the collinear quark and gluon
Lagrangians Eqs. ( 3.8) and ( 3.10) together. The collinear elds in each direction nicon-
stitute their own independent set of quark and gluon elds, and are governed in principle
by dierent expansion parameters associated with the transverse momentum of each jet,
set either by the angular cut Rin the jet algorithm or by the measured value of the jet
shapea. Each collinear sector may be paired with its own associated soft eld Aswith
momentum of order E2with the appropriate . For the purposes of keeping the notation
tractable while proving the factorization theorem in this section, we will for simplicity take
all's to be the same, with a single soft gluon eld Ascoupling to collinear modes in
all sectors. In Sec. 6.4, we will discuss how to \refactorize" the soft function further into
separate soft functions each depending only on one of the various possible soft scales.
The eective theory containing Ncollinear sectors and the soft sector is appropriate to
describe QCD processes with strongly-interacting particles collimated in Nwell-separated
directions. Thus, in addition to the power counting in the small parameter within each
sector, guaranteeing that the particles in each direction are well collimated, we will nd in
calculating an N-jet cross section the need for another parameter that guarantees that the
dierent directions niare well separated. This latter condition requires t1, wheretis
dened in Eq. ( 1.4).7
7This condition is a consequence of our insistence on using operators with exactly Ndirections to create
{ 19 {3.2 Jet Shape Distribution in e+e!3 Jets
Consider a 3-jet cross section dierential in the jet 3-momenta P1;2;3, where we measure
the shape1
aof one of the jets, which we will call jet 1. The full theory cross section for
e+e!
!3 jets at center-of-mass energy Qis
d
d1ad3P1;2;3=1
2Q2X
XjhXjj(0)j0iLj2(2)44(QpX)N(J(X));3

1
aa(jet 1)
3Y
j=13
PjP(jetj)

;(3.13)
where theJ(X) is the jet algorithm acting on nal state X, andN(J(X)) is the number
of jets identied by the algorithm [ 30].P(jetj) is the 3-momentum of jet j, and is also a
function of the output of the jet algorithm J(X).Lis the leptonic part of the amplitude
fore+e!
!qqg. The current jis
j=X
a;fqa
qa; (3.14)
summing over colors aand 
avors f.
When the three jet directions are well separated, we can match the QCD current j(x)
onto a basis of three-jet operators in SCET [ 34,66]. We build these operators from quark
jet eldsn, related to collinear quark elds nbyn=Wy
nn, whereWnis given by
Eq. ( 3.9), and a gluon jet eld B?
nrelated to gluons Anby
B?
n=1
gWy
n(P?+A?
n)Wn: (3.15)
The matching relation is
j(x) =X
n1n2n3X
~p1~p2~p3ei(~p1~p2+~p3)
xC
(n1;~p1;n2;~p2;n3;~p3)
n1;p1(gB?
n3;p3)
n2;p2(x);
(3.16)
with sums over Dirac spinor indices ;and Lorentz index , and over label directions n1;2;3
and label momenta ~ p1;2;3. Sums over colors and 
avors are implied. We have chosen to
produce a quark in direction n1, antiquark in n2, and gluon in n3. The matching coecients
C
are found by equating QCD matrix elements of jto SCET matrix elements of the
right-hand side of Eq. ( 3.16). These coecients have been found at tree level in [ 66]. The
number of independent Dirac and Lorentz structures that can actually appear with nonzero
coecients is considerably smaller than suggested by Eq. ( 3.16) due to symmetries. We
will keep the form of these coecients general to make extension to Njets transparent,
which would require the introduction of a basis of Njet elds in Eq. ( 3.16), with specied
the nal state. We could move away from the large- tlimit and account for corrections to it by using a basis
of operators with arbitrary numbers of jets and properly accounting for the regions of overlap between an
Njet operator and ( N1)-jet operators. This is outside the scope of the present work, where we limit
ourselves to kinematics well described by an N-jet operator, and thus, limit ourselves to the large- tlimit.
{ 20 {numbers of quark, antiquark, and gluon elds. We will not write the details for an N-jet
cross section here, but the procedures are obvious extensions of all the steps in factorizing
the 3-jet cross section below.
As a nal step before factorization, we redene the collinear elds to decouple collinear-
soft interactions in the Lagrangian [ 24]:
n(x) =Yy
n(x)(0)
n(x) (3.17a)
n(x) = (0)
n(x)Yn(x) (3.17b)
An(x) =Yn(x)A(0)
n(x); (3.17c)
whereYnis a Wilson line of soft gluons along the light-cone direction n,
Yn(x) =Pexp
igZ1
0dsn
As(ns+x)
; (3.18)
withAsin the fundamental representation.8Ynis similar but in the adjoint representation.
The new elds (0)
n;A(0)
ndo not have interactions with soft elds in the SCET Lagrangian
at leading order in . Henceforth, we use only these redened elds, but for simplicity
drop the (0) superscripts.
The cross section in SCET can now be written,
d
d1ad3P1;2;3=NFL2
6Q2X
XN(J(X));3(1
aa(jet 1))3Y
j=13(PjP(jetj))
X
n1;2;3X
~p1;2;3Z
d4xei(Q~p1+~p2~p3)
xC
(n1;2;3; ~p1;2;3)C

(n1;2;3; ~p1;2;3)
h0jTn
a
n2;p2Yab
n2YAB
n3(gB?B
n3;p3)TA
bcYycd
n1d
n1;p1(x)o
jXi
hXjTn
e
n1;p1Yef
n1YCD
n3(gB?D
n3;p3)TC
fgYygh
n2h
n2;p2(0)o
j0i: (3.19)
To proceed to factorize this cross section, it is convenient to rewrite the remaining delta
functions that depend on the nal state Xin terms of operators acting on X. Those
quantities depending on the jet algorithm Jcan be rewritten in terms of an operator
containing the momentum 
ow operator,
E(n) = lim
R!1Z1
0dtniTi(t;Rn); (3.20)
whereTis the energy-momentum tensor, evaluated at time tand position Rn. The
operatorE(n) measures the 
ow of four-momentum Pin the direction n(cf. [ 29,68,69]),
and the jet algorithm Jcan be written as an operator ^Jacting on the momentum 
ow in
all directions [ 30]. Correspondingly we can dene an operator for the 3-momentum of the
8The path choice (0 to 1) in Eq. ( 3.18) is convenient for outgoing particles. The physical cross section
is independent of whether the path goes to 1 if the transformation of the external states Xis also taken
into account [ 67].
{ 21 {jet,^P(Jj(^J)). In addition, the event shape a(jet 1) can also be expressed as an operator
^a(J1(^J)), built from the momentum 
ow operator, acting on the state jXi(cf. [ 29]):
^a(J1(^J)) =Z
de(1a)ET()(min(J1(^J))): (3.21)
The operator is constructed to count only particles actually entering the jet in direction n1
determined by the action of the jet algorithm (for simplicity we will suppress the argument
J1(^J) of ^ain the following, but add a superscript for the jet number). Using these
operators, we can eliminate the Xdependence in the delta functions in Eq. ( 3.19) and
perform the sum over states X, obtaining
d
d1ad3P1;2;3=L2
6Q2X
n1;2;3X
~p1;2;3Z
d4xei(Q~p1+~p2~p3)
xC
(n1;2;3; ~p1;2;3)C

(n1;2;3; ~p1;2;3)
h0jTn
a
n2;p2Yab
n2YAB
n3(gB?B
n3;p3)TA
bcYycd
n1d
n1;p1(x)o
N(^J);3(1
a^1
a)3Y
j=13(Pj^P(Jj(^J)))
Tn
e
n1;p1Yef
n1YCD
n3(gB?D
n3;p3)TC
fgYygh
n2h
n2;p2(0)o
j0i: (3.22)
The matrix element can be calculated as the sum over cuts of time-ordered Feynman
graphs, with the delta function operators inside the matrix element enforcing phase space
restrictions from the jet algorithm and jet shape measurement on the nal state created
by the cut.
The operators ^ aand ^Jdepend linearly on the energy-momentum tensor, which itself
splits linearly in SCET into separate collinear and soft pieces,
T=X
iTni+Ts
; (3.23)
which will aid us to factorize the full matrix element in Eq. ( 3.22) into separate collinear
and soft matrix elements. To achieve this factorization, however, we must make some more
approximations:
1. The contribution of soft particles and residual collinear momenta to the momentum
P(jetj) of each jet can be neglected, and the jet momentum is just given by the
label momentum ~ pjof the collinear state jXji. Thus the energy and jet axis of each
jet is approximated to be that of the parent collinear parton initiating the jet. In
particular, the squared mass of the jet is order 2compared to its energy. So in
this approximation we take the jet energy to be equal to the magnitude of its 3-
momentum. On the other hand, we keep the leading non-zero contribution to the
angularity even though it is also of order 2. These approximations also require that
we treat the energy of any particles outside all of the jets, and thus the cuto , as
a soft or residual energy.
{ 22 {2. The Kronecker delta restricting the total number of jets to 3 can be factored into three
separate Kronecker deltas restricting the number of jets in each collinear direction ni
to 1, and one Kronecker delta restricting the soft particles not to create an additional
jet. This approximation requires the separation between jets to be much larger than
the size of any individual jet so that dierent jets do not overlap. Factoring the
restriction on the number of jets in this way is one reason that the parameter tijin
Eq. ( 1.4) is required to be large.
We describe to what extent the algorithms we consider actually satisfy these two approxi-
mations in Sec. 3.4. For now we assume these approximations and facts hold, which allows
us to factor the cross section Eq. ( 3.19),
d
d1adE1;2;3d2
1;2;3=L2
6Q2X
n1;2;3X
!1;2;3C
(n1;2;3;!1;2;3)C

(n1;2;3;!1;2;3)
Z
d4xei(Q!1n1=2+!2n2=2!3n3=2)
xZ
dJdS(1
aJS)
h0jf
n1;!1(x)N(^J);1(J^n1a)e
n1;!1(0)j0i
E1!1
2
2(
1n1)
h0ja
n2;!2(x)N(^J);1h
n2;!2(0)j0i
E2!2
2
2(
2n2)
h0j(gB?A
n3;!3)(x)N(^J);1(gB?B
n3;!3(0)j0i
E3!3
2
2(
3n3)
h0jYyab
n2Yybc
n3TA
cdYyde
n3Yyef
n1(x)N(^J);0(S^s
a)Ygh
n1Yhi
n3TB
ijYjk
n3Ykl
n2(0)j0i(3.24)
We have rewritten the cross section to be dierential in Ei(the magnitude of Pi) and 
i
(the direction of Pi). In the sum over label directions, nican be chosen to align with Pi
such that ~p?
i= 0. In Eq. ( 3.24) we have written the label momentum as !ini
~pi. In
Eq. ( 1.1) we approximate the jet axis by this niand the jet energy by  ni
~pi=2, so that
they do not depend on soft momenta at all. The operators ^ n1aand ^s
aare dened to count
only particles inside the measured jet identied by the algorithm.
In the soft matrix element in Eq. ( 3.24), we have introduced the soft Wilson line Ynin
the antifundamental representation to remove the time- and anti-time-ordering operators
T;Tin Eq. ( 3.19) [27], and related Wilson lines Ynin the adjoint representation to those
in the fundamental representation by [ 24]
YAB
nTB=Yy
nTAYn: (3.25)
Dening the jet functions by the relations
Zd4k1
(2)4eik1
xJn1;!1(J;n1
k1)n =1
2

ef=h0jf
n1;!1(x)N(^J);1(J^n1a)e
n1;!1(0)j0i
(3.26a)Zd4k2
(2)4eik2
xJn2;!2(n2
k2)n =2
2
ah=h0ja
n2;!2(x)N(^J);1h
n2;!2(0)j0i (3.26b)
Zd4k3
(2)4eik3
xJn3;!3(n3
k3)g
?AB=!3h0j(gB?A
n3;!3)N(^J);1(gB?B
n3;!3)j0i;(3.26c)
{ 23 {and the soft function by
Zd4r
(2)4eir
xS(s;r) =1
NCCFTrh0jYy
n2Yy
n3TAYy
n3Yy
n1(x)N(^J);0(S^s
a)
Yn1Yn3TBYn3Yn2(0)j0i(3.27)
we can express the cross section Eq. ( 3.24) as
d
d1adE1;2;3d2
1;2;3=L2NFNCCF
6Q2X
n1;2;3X
!1;2;3Z
d4xei[Q(!1n1!2n2+!3n3)=2]
x(3.28)
C
(n1;2;3;!1;2;3)C

(n1;2;3;!1;2;3)n =1
2

n =2
2
g
?
Z
dJdS(1
aJS)3Y
i=1
Ei!i
2
2(
ini)
Zd4k1
(2)4eik1
xZd4k2
(2)4eik2
xZd4k3
(2)4eik3
xZd4r
(2)4eir
x
Jn1;!1(J;n1
k1)Jn2;!2(n2
k2)Jn3;!3(n3
k3)S(S;r);
where now Pi=Ei(1;ni). The quark and antiquark jet functions are now for a single

avor, and we have summed over NF
avors to obtain the factor in front. The jet functions
depend only on the smallest component of momentum ni
kiin each collinear direction.
The residual and soft momenta appearing in the exponentials can be reabsorbed into the
label momenta by a series of reparameterizations of the label momenta and directions,
under which the SCET Lagrangian is invariant [ 70]. The three-jet operator Eq. ( 3.16)
will receive corrections of order 2(which we can drop) under the reparameterizations we
perform below, or can be constructed from the outset to be explicitly reparameterization
invariant (RPI) [ 66].
First, collect the residual and soft momenta together by dening K=k1+k2+k3+r.
We can decompose xinn1light-cone coordinates, so
eiK
x=ei(n1
Kn1
x=2+n1
Kn1
x=2+K?1
x?1): (3.29)
Performing a type-A transformation (in the language of [ 70]) on the label momentum
~p1=!1n1=2,
!1!!1+ n1
K; (3.30)
and a type-IB transformation on the vector n1itself,
n1!n1+ 
?;
?=2
!1K?; (3.31)
shifts the label momentum on the jet function 1 by !1n1=2!(!1+ n1
K)n1=2 +K?1.
The summation variables n1;!1can then be shifted to eliminate  n1
KandK?1from the
exponentials entirely. We drop all corrections suppressed by 2due to these shifts.
It remains to absorb the n1
Kcomponent of residual and soft momentum, appearing
in the exponential factor ein1
Kn1
x=2. This cannot be achieved by RPI transformations
{ 24 {in then1sector since this momentum is purely residual|there is no label momentum
in this direction. However, in a multijet cross section,  n1can be written as a linear
combination of n2;n3, and, say, n?2(a unit vector transverse to n2;n2), so that RPI
transformations on !2;!3andn2similar to those above can absorb n1
Kinto the label
momenta also. Then, the x-dependent residual and soft exponentials all disappear, and we
can combine the nine super
uous  ni
kiandk?iintegrals with the nine discrete sums over
label directions and momenta to give continuous integrals over total momenta. Performing
these with the remaining energy and direction delta functions in Eq. ( 3.28) and thex
integral with the remaining exponential gives the momentum conservation delta function
4(QE1n1E2n2E3n3).
The resulting cross section Eq. ( 3.28) takes the form
d
d1adE1;2;3d2n1;2;3=d(0)
dE1;2;3d2n1;2;3H(n1;2;3;E1;2;3)Z
dJdS(1
aJS)
Zdn1
k1
2Zdn2
k2
2Zdn3
k3
2Zd4r
(2)4
Jn1;2E1(J;n1
k1)Jn2;2E2(n2
k2)Jn3;2E3(n3
k3)S(S;r);(3.32)
where we used that the matching coecients C
(ni; ~pi) are such that, by construction,
the right-hand side at tree-level is simply the Born cross section (denoted by (0)) for
e+e!qqgtimes(1
a). The hard function H= 1 +O(s) is determined by calculating
the matching coecients Corder-by-order in perturbation theory.
The above may be easily modied to describe the antiquark or gluon jet angularities,
by moving the appropriate delta function (i
aa(jeti)) into the J2orJ3jet functions.
In addition, we may choose from among various jet algorithms. The choice determines
what -function restrictions must be inserted into the nal state phase space integrations
created by cutting the jet and soft diagrams to determine which particles make it into the
jet. As long as the algorithm is such that the approximations enumerated above hold, it
will not violate factorization of the jet shape cross section. We will discuss factorization
and its potential breakdown in the context of particular jet algorithms in more detail in
Sec.3.4.
Another check of the validity of the factorization theorem is that the factorized jet and
soft functions be separately IR safe, which is a stronger condition than the full cross section
being IR safe. If the observable [ 54,71] or algorithm [ 32] too strongly weights nal states
with narrow jets whose invariant masses are the same as the virtuality of soft particles,
then the jet and soft functions for the observable in standard SCET with dimensional
regularization become IR divergent. When this occurs it does not necessarily mean that
factorization is not possible; but at least not in the standard form derived from the version
of SCET we utilized above. It could be, for example, that a scheme to further separate
modes by dening the theory with an explicit cuto [ 32] or by factorizing modes by rapidity
instead of virtuality [ 65] can restore a version of the factorization theorem. We leave an
explicit study of which algorithms and observables give IR safe jet and soft functions in
SCET in dimensional regularization for a separate publication. However, we note here that
{ 25 {the algorithms and observables ( afora <1) that we consider in this paper, at least at
NLO, do give rise to IR safe jet and soft functions.
3.3 Jet Shapes in e+e!Njets
To generalize the result Eq. ( 3.32) to an arbitary number Nof jets, we simply add more
quark and gluon jet elds to the operator matching in Eq. ( 3.16), resulting in the corre-
sponding number of additional quark and gluon jet functions in Eq. ( 3.32), along with a
hard coecient and a soft function for Njet directions. Consider an event with 2 Nqquark
and antiquark jets and Nggluon jets, where 2 Nq+Ng=N. Furthermore, we can choose
to measure the angularity shape for any number of these jets. Achieving a factorization
theorem that remains consistent for any of these combinations is a nontrivial task and thus
a powerful test of the validity of the eective theory.
For anN= 2Nq+Ngjet event, we generalize the matching of the QCD current in
Eq. ( 3.16) to:
j(x) =X
n1


nNX
~p1


~pNei(~p1+


+~pN)
xCa1


aNqb1


bNqA1


ANg
1


Nq1


Nq1


Ng(n1;~p1;


;nN;~pN)
NqY
i=1iaini;pi(x)NqY
j=1jbj
nj;~pj(x)NgY
k=1(gB?kAk
nk;~pk)(x);(3.33)
with implicit sums over Dirac spinor indices i;j, Lorentz indices k, (anti-)fundamental
color indices ai;bj, and adjoint color indices Ak. TheNjet cross section dierential in M
jet shapes, with M <N , factors in SCET into the form
d(E1;n1;


;EN;nN)
d1a1


dMaM=(0)(E1;n1;


;EN;nN)HaibjAk(n1;E1;


nN;EN)
MY
l=1Z
dl
Jdl
S(l
al
Jl
S)Zdn1
k1
2


ZdnN
kN
2
Jf1
n1;2E1(1
J;n1
k1)


JfM
nM;2EM(M
J;nM
kM)
JfM+1
nM+1;2EM+1(nM+1
kM+1)


JfN
nN;2EN(nN
kN)
Zd4r
(2)4SaibjAk(1
S;:::;M
S;r); (3.34)
where(0)is the Born cross section for e+etoNqquarks,Nqantiquarks, and Nggluons;
the color indices on the hard and soft functions HandSallow for color mixing; and fiis the

avor of each jet function (quark, antiquark, or gluon). Hitself is determined by calculating
the matching coecients Cin Eq. ( 3.33). The jet functions have the same denitions given
in Eq. ( 3.26), and the soft function is given by the appropriate generalization of Eq. ( 3.27)
with Wilson lines in the directions and color representations corresponding to the choice
of elds in Eq. ( 3.33). We rearrange the order of 
avor and color indices in the hard and
soft functions to agree with the choices of 
avor indices on the jet functions.
{ 26 {3.4 Do Jet Algorithms Induce Large Power Corrections to Factorization?
In this section we explore when power corrections to the factorization theorem above be-
come large, in particular those that are due to the action of jet algorithms. We will argue
that power corrections to jet angularities induced by the commonly-used cone and recom-
bination algorithms remain suppressed as long as Ris suciently large. In particular, we
need in general that RsatisesR&and, for the case of the k Talgorithm, we require
R.9These power corrections are associated with assumptions we made about the
action of the jet algorithm on nal states in deriving Eq. ( 3.34). In general, the size of
these power corrections depends both on the algorithm and the observable.
Power corrections to the pTof a jet arising from perturbative emissions (as well as
from hadronization and the underlying event in ppcollisions) for various jet algorithms
were explored in [ 36]. These power corrections arise for similar reasons as those we discuss
below, namely, perturbative emissions changing which partons get combined into the jet.
Ref. [ 36] nds that such power corrections scale like ln Rfor smallR. This result is
consistent with our qualitative discussion below, where we argue that power corrections to
jet angularities arising from the jet algorithms we use are minimized when Ris suciently
large. For us, Rshould be at least O() (and for the case of the k Talgorithm,R).
One set of power corrections that is independent of the choice of algorithm arises
from the approximation of setting the jet axis equal to the label direction n. Since this
neglects the eects of soft particles, it is valid up to O(2) corrections. It was argued in
Refs. [ 17,29,41] for the case of hemisphere jet algorithms that these corrections in turn
induce corrections to the angularity aof order2(2a), which, for a2, are subleading
fora < 1. Essentially the same arguments can be applied to all of the algorithms we
consider.
Jet algorithm-dependent power corrections arise from the dierence in soft particles
included in a jet by a given algorithm and those included in the soft function in the
factorization theorem. The algorithms also dier amongst themselves in which soft particles
they include in a jet. For observables that scale as O(1), such as the jet energies and 3-
momenta, the contribution of soft momenta can be neglected since they scale as O(2).
Clearly then, these observables are not dependent on our choice of jet algorithm and so
the assumptions we made about factorization of the algorithm in deriving Eq. ( 3.34) are
trivially satised.
However, for observables that scale as O(2) such as angularities, soft contributions
become important and so the details of the algorithms we consider become relevant. We
now estimate how closely the phase space region included in the soft function in the SCET
factorization theorem approximates the region of soft particles actually included by the
jet algorithm. We will argue that unless R&O() for the anti-k Tand infrared-safe cone
algorithms, and Rfor the k Talgorithm, the mismatch in areas is suciently large
9We noted earlier that there may be dierent 's for the SCET modes describing dierent jets. For
measured jets, 2a, while for unmeasured jets, tan(R=2), and for soft gluons outside jets, 2=Q.
In this section, we mean by the expansion parameter associated with measured jets, and ensuring Ris
much larger than this . But ifRistoolarge, the separation parameter t/1=tan(R=2) becomes too small.
We will consider R0:4 to 1 to be safe.
{ 27 {(B)Rθij}θij
anti-k Talg .Parent Location (before collinear splitting)Additional Soft Region of AlgorithmSoft Region Common to Both
Algorithm and SCET Factorization Thm.
Daughter Location (after collinear splitting)
(A)θij}
RR
kTalg .Figure 2: Dierence between regions of soft radiation included in the SCET factorization theorem
and the actual (A) k Tand (B) anti-k Talgorithms. We illustrate how the regions of soft radiation
included by the algorithms change when a single, energetic parent particle splits into two collinear
daughters. Both the algorithm and the soft function merge soft particles contained in the large
white circle. The algorithm also merges the hatched area and hence contains a region of phase
space which is dierent than that included in the SCET soft function (since, as explained in the
text, the shape and size of the region used in SCET cannot depend on the details of collinear
splittings).
so as to cause a leading-order power correction to the measured jet angularities. For R
larger than these bounds, the corrections are negligible. This miscounting arises due to the
fact that factorization requires that collinear particles be combined rst, and that the soft
function only knows about the parent collinear direction. When algorithms do not obey
this ordering, factorization may be violated.
To determine the size of the soft particle phase space region for each jet algorithm that
is not included by the factorization theorem, we consider the situation depicted in Fig. 2.
A parent collinear particle splits into two daughter collinear particles. In the factorization
theorem, since collinear particles are combined rst, the region of phase space where soft
particles are combined into the jet is a circle of radius Rabout the parent particle direction.
However, in a jet algorithm, soft particles in an additional region outside of this may also
be combined into the jet (the hatched regions in Fig. 2). If the area of this region is of the
same order as the area included by the factorization theorem, then the power corrections
to jet angularities induced by the jet algorithm will be leading-order.
Because soft particles have momenta that are parametrically smaller than collinear
particle momenta, we determine the omitted region of soft particle phase space by con-
sidering the dominant action of the jet algorithm. The k Talgorithm serves as a useful
example. The k Tmetric between a pair of soft particles is O(2)ss, the metric between a
soft particle and collinear particle is O(2)cs, and the metric between a pair of collinear
particles isO(0)cc. Therefore, collinear-collinear recombinations only occur if the angle
ccbetween the collinear particles is smaller than the separation between any soft parti-
cle and its nearest neighbor by a factor of O(2). Given that the typical angle between
collinear particles is O(), the dominant action of the jet algorithm is to rst merge all
soft particles with their nearest neighbors, while collinear-collinear recombinations occur
late in the operation of the algorithm. This description will suce to accurately determine
the area of the omitted soft phase space for the k Talgorithm. Since collinear particles
are combined last, on average soft particles within circles of radius Rabout the collinear
{ 28 {daughter particles are included in the k Talgorithm jet,10as shown in Fig. 2A. The area
that the k Talgorithm includes that the soft function does not is represented by the hatched
region, which is an area of O(ijR). This area must be parametrically smaller than the
area included by the soft function (of O(R2)) for the associated power corrections to be
small. We thus require that Rijin the SCET power counting.
The anti-k Talgorithm combines particles in a manner that is closer to respecting
factorization. It nds the hardest particle rst and merges particles at successively larger
distances from this particle. For the example of two collinear daughters, it will merge all
soft particles with the hardest daughter that are closer than the distance to the softer
daughter before merging the two daughters and then merging all soft particles a distance
Rfrom the merged daughters (i.e., the parent particle), as shown in Fig. 2B. As the Figure
illustrates, the hatched area of the anti-k Tjet tends to be smaller than that of the k T
jet. In fact, for R > 3ij=2, this region vanishes completely (and this case of having only
two collinear daughters is a worst-case scenario). This leads us to expect that, for any
number of collinear splittings, for R&(i.e., not necessarily parametrically larger), power
corrections due the action of the anti-k Talgorithm vanish.
Cone algorithms such as the SISCone algorithm can also include regions that dier
from the lowest-order region at higher orders in perturbation theory. We now argue that
an arithmetic bound R&is sucient to minimize the power correction from these
dierences, as for the anti-k Talgorithm. This situation arises due to the fact that stable
solutions may exist with overlapping cones when collinear splittings are larger than the cone
radius, i.e., R<ij. In these cases, the amount of radiation that falls into the overlapping
region is used to decide whether the cones are split or merged. In either case, the boundary
of the resulting jet(s) has roughly the appearance of Fig. 2A and the dierence between
the region of soft radiation assumed in SCET and that by the actual algorithm is O(1).
However, for R > ijfor the case of a single collinear splitting, all of the collinear
radiation lies within a region of size Rand there will always be a stable cone that includes
this radiation and thus the algorithm and the SCET soft function will be sensitive to soft
particles in the same region of phase space.
In summary, we have argued that for all the algorithms we consider (k T, anti-k T,
infrared-safe cone), power corrections are negligible for suciently large R. While anti-k T
and cone allow simply R&instead ofRas for k T, we will in fact always consider
R(for thein a measured jet sector) in the remainder of this paper, guaranteeing
small power corrections for all these algorithms. ( Rstill determines the scale in an
unmeasured jet sector.) Our focus will remain on resumming logs of jet shapes such as
angularities ain the presence of jet algorithms, without worrying about resumming logs
ofRthemselves.11
10Soft particles in this region can also be removed from this region by merging with other soft particles
outside of the region and vice-versa, but this average area suces for our discussion.
11Because small R(.0.3) jets cannot be well resolved in current experiments, resummation of logarithms
ofRis not of primary practical importance in the near future.
{ 29 {4. Jet Functions at O(s)for Jet Shapes
In this section, we calculate the quark and gluon jet functions for jet shapes at next-to-
leading order in perturbation theory. The jet functions can be divided into two categories:
those for measured jets, which are xed to have a specic angularity a, and those for
unmeasured jets, which are not. We will denote the quark jet function by Jq
!, the gluon
jet function by Jg
!, where!is the label momentum, and a jet function Jq;g(a) with an
argument of adenotes a measured jet. We will calculate the jet functions for the two
classes of jet algorithms, k T-type and cone-type algorithms.
In the course of these calculations, we will demonstrate the crucial role of zero-bin
subtractions [ 65] from collinear jet functions in obtaining the consistent anomalous dimen-
sions and the correct nite parts. In this case zero-bin subtractions are not merely scaleless
integrals converting IR to UV divergences, but in fact contribute part (sometimes the most
important part) of the correct nonzero result, as was already pointed out by [ 32,72]. The
relation of zero-bin subtractions in SCET to eikonal jet subtractions from soft functions in
traditional methods of QCD factorization was explored in [ 41,73,74]. In addition, we nd
that the zero-bin subtraction removes the contribution of collinear emissions that escape a
jet, leaving only power-suppressed pieces in  =!i.
4.1 Phase Space Cuts
To calculate the jet functions for a particular algorithm,
Figure 3: A representa-
tive diagram for the NLO
quark and gluon jet func-
tions. The incoming mo-
mentum isl=n
2!+n
2l+and
particles in the loop carry
momentum q(\particle 1")
andlq(\particle 2").we must impose phase space restrictions in the matrix ele-
ment. From the jet function denitions, Eq. ( 3.26), these cuts
take two forms. One kind, imposed by the operator N(^J);1
in Eq. ( 3.26), is common to every jet function. It is the set
of phase space restrictions related to the jet algorithm, and
requires exactly one jet to arise from each collinear sector of
SCET. The other, imposed by the operator (a^a), is im-
plemented only on measured jets and restricts the kinematics
of the cut nal states to produce a xed value of the jet shape.
In this section we describe these phase space cuts in detail.
The typical form of the NLO diagrams in the jet functions
is shown in Fig. 3. As shown in the gure, the momentum

owing through the graph has label momentum ln
l=!and residual momentum
l+n
l, and the loop momentum is q. We will label \particle 1" as the particle in the
loop with momentum qand \particle 2" as the particle in the loop with momentum lq.
For the quark jet, we take particle 1 as the emitted gluon and particle 2 as the quark.
As usual, the total forward scattering matrix element can be written as a sum over
all cuts. Cutting through the loops corresponds to the interference of two real emission
diagrams, each with two nal state particles, whereas cutting through a lone propagator
that is connected to a current corresponds to the interference between a tree-level diagram
and a virtual diagram, each with a single nal state particle. Thus, the phase space
restrictions and measurements we impose act dierently depending on where the diagrams
{ 30 {are cut. In addition, since we will be working in dimensional regularization (with d= 42),
which sets scaleless integrals to zero, the only diagrams that contribute are the cuts through
the loops. This means that we only need to focus on the form of phase-space restrictions
and angularities in the case of nal states with two particles.
The regions of phase space for two particles created by cutting through a loop in the
jet function diagrams can be divided into three contributions:
1. Both particles are inside the jet.
2. Particle 1 exits the jet with energy E1<.
3. Particle 2 exits the jet with energy E2<.
In contributions (2) and (3), the jet has only one particle, which is the remaining particle
withE > .
It is well known that collinear integrations of jet functions can be allowed to extend
over all values of loop momenta so long as a \zero-bin subtraction" is taken from the result
to avoid double counting the soft region already accounted for in the soft function [ 65]. We
will demonstrate that contributions (2) and (3) are power suppressed by O(=!), which
scales as2, after the zero-bin subtraction.
The phase space cuts that enforce both particles to be in the jet depend on the jet
algorithm. There are two classes of jet algorithms that we consider, cone-type algorithms
and (inclusive) k T-type algorithms, and all the algorithms in each class yield the same
phase space cuts. We label the phase space restrictions as  coneand  kT, generically  alg.
For the cone-type algorithms,
conecone(q;l+) = 
tan2R
2>q+
q

tan2R
2>l+q+
!q
: (4.1)
These  functions demand that both particles are within Rof the label direction. For the
kT-type algorithms, the only restriction is that the relative angle of the particles be less
thanR:
kTkT(q;l+) = 0
@cosR<~ q
~lq2
qq
l2+q22~ q
~l1
A
= 
tan2R
2>q+!2
q(!q)2
: (4.2)
In the second line we took the collinear scaling of q(q+q). While this is not strictly
needed, it makes the calculations signicantly simpler.
For the phase space restrictions of zero-bin subtractions, we take the soft limit of
the above restrictions. The zero-bin subtractions are the same for all the algorithms we
consider. For the case of particle 1, which has momentum q, the zero-bin phase space cuts
are given by
(0)
alg= (0)
cone= (0)
kT= 
tan2R
2>q+
q
: (4.3)
{ 31 {(B) (A) (D) (C) (A) (A)Figure 4: Diagrams contributing to the quark jet function. (A) and (B) Wilson line emission
diagrams; (C) and (D) QCD-like diagrams. The momentum assignments are the same as in Fig. 3.
The zero bin of particle 2 is given by the replacement q!lq.
For all the jet algorithms we consider, the zero-bin subtractions of the unmeasured jet
functions are scaleless integrals.12However, for the measured jet functions, the zero-bin
subtractions give nonzero contributions that are needed for the consistency of the eective
theory.
In the case of a measured jet, in addition to the phase space restrictions we also demand
that the jet contributes to the angularity by an amount awith the use of the delta function
R=(a^a), which is given in terms of qandlby
RR(q;l+) =
a1
!(!q)a=2(l+q+)1a=21
!(q)a=2(q+)1a=2
:(4.4)
In the zero-bin subtraction of particle 1, the on-shell conditions can be used to write the
corresponding zero-bin -function as
(0)
R=
a1
!(q)a=2(q+)1a=2
; (4.5)
(and for particle 2 with q!lq).
4.2 Quark Jet Function
The diagrams corresponding to the quark jet function are shown in Fig. 4. The fully
inclusive quark jet function is dened as
Z
d4xeil
xh0ja
n;!(x)b
n;!(0)j0iabn =
2
Jq
!(l+); (4.6)
and has been computed to NLO (see, e.g., [ 75,76]) and to NNLO [ 77]. Below we compute
the quark jet function at NLO with phase space cuts for the jet algorithm for both the
measured jet, Jq
!(a), and the unmeasured jet, Jq
!. As discussed above, we will nd that
the only nonzero contributions come from cuts through the loop when both cut particles
are inside the jet.
12Note that algorithms do exist that give nonzero zero-bin contributions to unmeasured jet functions [ 32].
{ 32 {4.2.1 Measured Quark Jet
The measured quark jet function includes contributions from naive Wilson line graphs (A)
and (B) and QCD-like graphs (C) and (D) in Fig. 4. The sum of these contributions is
~Jq
!(a) =g22CFZdl+
21
(l+)2Zddq
(2)d
4l+
q+ (d2)l+q+
!q
2(qq+q2
?)
(q)(q+)2
l+q+q2
?
!q
(!q)(l+q+) algR:(4.7)
The contribution proportional to d2 comes from the QCD-like graphs (C) and (D) in
Fig.4. Only the Wilson line graphs have a nonzero zero-bin limit, which comes from taking
the scaling limit q2of the naive contribution:
Jq(0)
!(a) = 4g22CFZdl+
21
l+Zddq
(2)d1
q2(qq+q2
?)(q)(q+)
2
l+q+

(l+q+) (0)
alg(0)
R:(4.8)
All jet algorithms that we use yield the same zero-bin contribution, since the phase space
cuts are the same.
To evaluate these integrals, we can analytically extract the coecient of (a) by
integrating over aand using the fact that the remainder is a plus distribution, as dened
in Eq. ( A.2). We nd the naive contribution is
~Jq
!(a) =sCF
21
(1) 
42
!2tan2R
2!1
2+3
2
(a) +s
2~Jq
alg(a): (4.9)
The only dierence between the jet algorithms that we consider resides in the nite distri-
bution ~Jq
alg(a), which is a complicated function of athat we give in Appendix A. Note
that the divergent part of the naive contribution is proportional to (a). This is due to
the fact that the jet algorithm regulates the distribution for a>0. The divergent plus
distributions come entirely from the zero-bin subtraction, which is given by
Jq(0)
!(a) =sCF
1
(1) 
42tan2(1a)R
2
!2!1
(1a)1
1+2a: (4.10)
Adding the leading-order contribution to all of the NLO graphs and expanding in
powers of, adopting the MS scheme, we nd the total quark jet function,
Jq
!(a) =(a) +~Jq
!(a)Jq(0)
!(a) =(
1 +sCF
"
1a
2
1a1
2+1a
2
1a1
ln2
!2+3
4#)
(a)
sCF
"
1
1
1a(a)
a#
++s
2Jq
alg(a):(4.11)
This agrees with the standard jet function J(k+) given in [ 75,76] by setting a= 0 and
k+=!a. We have shown the divergent terms explicitly, and collect the nite pieces in
Jq
alg(a), which we give in Eq. ( A.14). Note that there is no jet algorithm dependence in
the divergent parts of the jet function at this order in perturbation theory.
{ 33 {4.2.2 Gluon Outside Measured Quark Jet
In this section we calculate the contribution to the quark jet function from the region of
phase space in which the gluon exits the jet carrying an energy Eg<. This cut causes
the contribution to be power suppressed by  =!, which scales as 2. However, we elect
to evaluate this case explicitly as it provides a clear example of the zero-bin subtraction
giving the proper scaling to the total contribution. We only evaluate this contribution
for the cone algorithm; the details of the k Talgorithm calculation are similar. Note that
the contribution when the quark is out of the jet is power suppressed at the level of the
Lagrangian given in Sec. 3.1, in which soft quarks do not couple to collinear partons at
leading order in .
For the cone algorithm, the gluon exits the jet when the angle between the jet axis,
n1, and the gluon is greater than R. When the gluon is not in the jet, the cone axis is the
quark direction, and so it makes no contribution to the angularity. Therefore, this region
of phase space contributes only to the (a) part of the angularity distribution.
For the naive contributions, requiring the gluon to be outside the jet and have energy
less than , we have the integral
~Jq;out
!(a) =g22CFZdl+
21
(l+)2Zddq
(2)d
4l+
q+ (d2)l+q+
!q
2(qq+q2
?)
(q)(q+)2
l+q+q2
?
!q
(!q)(l+q+)
q+
qtan2R
2

2q

(a): (4.12)
Note that the theta function requiring q<2 is more restrictive than q<!. Evaluating
Eq. ( 4.12) yields a contribution that scales with  only below the leading term in 1 =:
~Jq;out
!(a) =sCF
21
(1) 
42
(2 tanR
2)2!
(a)1
2+1
4
!22
!2
+8
!
(4.13)
The zero-bin subtraction of Eq. 4.12 is
~Jq;out(0)
! (a) =g22CFZdl+
21
(l+)2Zddq
(2)d
4l+
q+ (d2)l+q+
!q
2(qq+q2
?)
(q)(q+)2
l+q+

q+
qtan2R
2

2q

(a):(4.14)
Evaluating Eq. ( 4.14), we nd the zero bin will exactly remove the leading term in 1 =:
~Jq;out(0)
! (a) =sCF
21
(1) 
42
(2 tanR
2)2!
(a)1
2(4.15)
Therefore, the dierence is power suppressed only after the zero bin is included. Because
other contributions when one particle is outside of the jet are similarly power suppressed,
we will drop them in our remaining discussion of the jet functions.
{ 34 {4.2.3 Unmeasured Quark Jet
When the angularity of a jet is not measured, the jet function has no adependence. The
naive and zero-bin contributions are the same as Eqs. ( 4.7) and ( 4.8) except for the factor
ofR. The zero-bin contribution is
Jq(0)
!= 2g22CFn
nZdl+
21
l+Zddq
(2)d1
q2(qq+q2
?)(q)(q+)
2
l+q+

(l+q+) (0)
alg:(4.16)
This integral is scaleless and therefore equal to 0 in dimensional regularization. This implies
that the NLO part of the quark jet function for an unmeasured jet is just the naive result.
We nd, making the divergent part explicit, in the MS scheme,
Jq
!= 1 + ~Jq
!= 1 +sCF
2(
1
2+3
2+1
ln 
2
!2tan2R
2!)
+s
2Jq
alg; (4.17)
where the nite part Jq
algis given in Eq. ( A.18).13
4.3 Gluon Jet Function
The diagrams needed for the gluon jet function at NLO are shown in Fig. 5. The fully
inclusive jet function, dened as
Z
d4xeil
xh0jBA
?;!(x)BB
?;!(0)j0i1
!g
?ABJg
!(l+); (4.18)
(withJg
!(l+) normalized to 2 (l+) at tree-level) has been calculated to NLO in Feynman
gauge in [ 34,78,79] and was reported to give the same result in both Rand light-cone
gauges in [ 35]. Since our phase space restrictions and the observables act dierently on cuts
through loops and on cuts through external propagators, it is useful to reproduce these
results by explicitly cutting the diagrams.
After some algebra, we nd that the sum of all cuts through the loops of the na ve
collinear graphs gives
Zdl+
2~Jg
!(l+) =22g2
(2)d1Zdl+
l+Z
ddq(q2)((lq)2) (!q)

TRNf
12
1q+q
!l+
CA
2!
q!
!qq+q
!l+
:(4.19)
We also record the zero bin that needs to be subtracted from Eq. ( 4.19). To leading-power
it is given by
Zdl+
2Jg(0)
!(l+) =2CA2g2
(2)d1Zdl+
l+Z
ddq
(q2)(l+q+)(q)1
q
+((lq)2)(q+)(!q)1
!q
: (4.20)
13The unmeasured jet function Eq. ( 4.17) is not simply obtained by integrating the measured jet function
Eq. ( 4.11) overa. This is due to the dierent relative scaling of Rwith the SCET expansion parameter
iin a measured and unmeasured jet sector, as noted earlier. Namely, R0
iin a measured jet sector
(wherepa) whilektan(R=2) in an unmeasured jet sector.
{ 35 {Figure 5: Diagrams contributing to the gluon jet function. (A) sunset and (B) tadpole gluon
loops; (C) ghost loop; (D) sunset and (E) tadpole collinear quark loops; (F) and (G) Wilson line
emission loops. Diagrams (F) and (G) each have mirror diagrams (not shown). The momentum
assignments are the same as in Fig. 3.
Without inserting any additional constraints, this integral is scaleless and zero in dimen-
sional regularization. Therefore, in the absence of phase-space restrictions, the na ve inte-
gral Eq. ( 4.19) gives the standard (inclusive) gluon jet function
Jg
!(l+)
2!=s
42(!l+)1
TRNf4
3+20
9
CA4
+11
3+67
92

;(4.21)
in the MS scheme. The measured and unmeasured jet functions are obtained by inserting
algRand  alg, respectively, into Eqs. ( 4.19) and ( 4.20).
4.3.1 Measured Gluon Jet
The naive contribution to the measured gluon jet can be written as
~Jg
!(a) =s
21
(1)42
!21
1a
21
a1+2
2aZ1
0dx(xa1+ (1x)a1)2
2a(4.22)

TRNf
12
1x(1x)
CA
21
x(1x)x(1x)
alg(x);
wherexq=!. This gives
~Jg
!(a) =s
21
(1) 
42
!2tan2R
2!
(a)"
CA1
2+11
61

2
3TRNf#
+s
2~Jg
alg(a);
(4.23)
where, as for the quark jet function, the nite distributions ~Jg
alg(a) dier among the
algorithms we consider. They are given in Appendix A.
The zero-bin result is
Jg(0)
!(a) =sCA
1
(1) 
42tan2(1a)R
2
!2!1
a1+21
(1a): (4.24)
{ 36 {Subtracting the zero-bin from the naive integral and adding the leading-order contribution,
we obtain in MS
Jg
!(a) =(a) +~Jg
!(a)Jg(0)
!(a)
=(
1 +sCA
"
1a=2
1a1
2+1
ln2
!2
+11
121
#
s
3TRNf1
)
(a)
sCA
1
1a1
(a)
a
++s
2Jg
alg(a): (4.25)
The nite distribution Jg
alg(a) is given in Eq. ( A.14).
4.3.2 Unmeasured Gluon Jet
As for the quark jet function, for unmeasured jets the naive and zero-bin contributions are
given by the measured jet contributions without the Rconstraint. Also, as it was for the
quark jet function, the zero-bin contribution to the unmeasured jet function is a scaleless
integral. Thus, the nal answer is just the naive result, which is given by
Jg
!= 1 +s
2"
CA 
1
2+11
61
+1
ln2
!2tan2R
2!
2
3TRNf#
+s
2Jg
alg; (4.26)
with the nite part Jg
alggiven in Eq. ( A.29) in the Appendix.
5. Soft Functions at O(s)for Jet Shapes
In this section we compute the soft function for a generic Njet event. In Sec. 5.1, we
describe the phase space cuts that we impose on soft particles emitted into the nal state.
We then give an outline of how we disentangle the various contributions to the N-jet
soft function in Sec. 5.2and proceed to calculate these contributions in the remaining
subsections.
5.1 Phase Space Cuts
Soft particles in the nal state must satisfy the phase space cuts required by the operator
N(^J);0in Eq. ( 3.27), which constrains the soft particles to not create an extra jet. A soft
particle is allowed in the nal state if it is emitted into one of the jets as dened by the
jet algorithm, or if it is not in a jet but has energy less than a cuto . At NLO, only a
single soft gluon can be emitted. Therefore, for both cone-type and k T-type algorithms,
the constraint that the soft gluon be in a jet is simply that the angle of the gluon with
respect to the jet axis satises gJ<R. Thus, our requirement on soft gluons is that they
obey one of the two following conditions:
in jeti:gJi<R
out of all jets: Eg< andgJi>R for alli: (5.1)
{ 37 {Figure 6: Soft function real-emission diagrams. Diagrams (A) and (C) are interference diagrams
of Wilson line emission from lines iandjand (B) and (D) are from lines iandk. The shaded
region in the center represents the region of phase space corresponding to jet kdened by the jet
algorithm, and so the gluons in diagrams (A) and (B) are inside jet kand those in (C) and (D) are
not. Each diagram has a corresponding mirror diagram (not shown).
These conditions can be written in terms of theta functions on the gluon momentum k.
We denote the energy restriction for out-of-jet gluons as
(k0<); (5.2)
and we denote the requirement that a gluon be in jet iin terms of the light-cone components
kabout the direction of jet i,ni, as
i
Rk+
k<tan2R
2
: (5.3)
For the case when the soft gluon is in a measured jet, we demand that it contributes
an amount ato the angularity of a jet with label momentum !with the use of the delta
function
R
a1
!(k)a=2(k+)1a=2
: (5.4)
5.2 Calculation of contributions to the N-Jet Soft Function
The topology of the various graphs that we need to compute the soft function is shown
in Fig. 6. The next-to-leading order part S(1)of the soft function Sis the sum of the
interference of soft gluon emissions from Wilson lines in directions iandj,Sij, over all
linesiandjwithi6=j(since fori=j, the diagram is proportional to n2
i= 0),
S(1)=X
i6=jSij: (5.5)
It is conceptually straightforward to see that Sijis just the sum of the following three
classications of the nal state cut gluon:
The gluon is in a measured jet and thus contributes to the jet angularity.
The gluon is outside of all the jets and has energy Eg<.
The gluon is in an unmeasured jet and has any energy.
{ 38 {However, the second contribution is technically dicult to compute due to the complicated
shape of the space with all jets cut out of it, like Swiss cheese. A division of phase space
leading to a simpler calculation is the following:
Smeas
ij(k
a): The gluon is in measured jet kand contributes to the jet's angularity k
a.
Sk
ij: The gluon is in jet kwith energy Eg> (and does not contribute to k
a).
Sk
ij: The gluon is in jet kwith energy Eg< (and does not contribute to k
a).
Sincl
ij: The gluon is anywhere with Eg< (and does not contribute to any angularity).
In terms of these pieces, the NLO soft function with Mmeasured jets and NMunmea-
sured jets is given by
S(1)(1
a;2
a;:::;M
a) =X
i6=j2
664X
k2measSmeas
ij(k
a)MY
l=1
l6=k(l
a)3
775
+X
i6=j" 
Sincl
ijX
k2measSk
ij+X
k=2measSk
ij!MY
l=1(l
a)3
5:(5.6)
From the denitions above, it is easy to see that the term in large parentheses on the
second line is equivalent to the sum of the last two contributions on the original list above,
i.e., the contributions from a gluon not in any jet with Eg< and from a gluon in an
unmeasured jet with any energy.
We can simplify this expression by noting that the contribution from a gluon in jet
kwith no energy restriction involves a scaleless integral over the energy that vanishes in
dimensional regularization and thus
Sk
ij+Sk
ij= 0: (5.7)
Using this relation, the soft function simplies to
S(1)(1
a;:::;M
a) =Sunmeas
(1)MY
l=1(l
a) +X
k2measSmeas
(1)(k
a)MY
l=1
l6=k(l
a); (5.8)
where the rst term in Eq. ( 5.8) is a universal contribution that is needed for every N-jet
observable, dened as
Sunmeas
(1)X
i6=j
Sincl
ij+NX
k=1Sk
ij
: (5.9)
The second term, dened as,
Smeas
(1)(k
a)X
i6=jSmeas
ij(k
a); (5.10)
{ 39 {depends on our choice of angularities as the observable.
We now proceed to set up the one-loop expressions for the contributions in Eq. ( 5.8).
The integrals involved are highly nontrivial and so in this section we simply quote the result
of each integral, referring the reader to Appendix Bfor details. Most of these integrals are
most easily written in terms of the variable tij, dened in Eq. ( 1.4) astijtan ij
2=tanR
2,
where ijis the angle between jet directions iandj. (That is, ni
nj= 1cos ij.) In
Table 2, we summarize the divergent parts of the soft function.
The Feynman rules for the emission of a soft gluon from fundamental- and adjoint-
representation Wilson lines (corresponding to quark and gluon jets, respectively) have
the same kinematic structure. The dierence is entirely encoded in the color charge of the
Wilson lines which, using the color space formalism of [ 80,81], we denote as Tifor emission
from Wilson line i. Thus, we can consider the N-jet soft function without specifying the
color representation of the nal-state jets.
5.2.1 Inclusive Contribution: Sincl
ij
The contribution to the soft function from a gluon going in any direction with energy
Eg< is given by the integral
Sincl
ij=g22Ti
TjZddk
(2)dni
nj
(ni
k)(nj
k)2(k2)(k0) : (5.11)
We evaulate this integral in Sec. B.1of the Appendix and nd
Sincl
ij=s
2Ti
Tj
(1)42
421
21
lnni
nj
22
6Li2
12
ni
nj
:(5.12)
Note that this calculation is related to the inclusive, timelike soft function that has
applications elsewhere (see, e.g., [ 82,83,84]), generalized for non back-to-back jets:
dSincl
ij
d=g22Ti
TjZddk
(2)dni
nj
(ni
k)(nj
k)2(k2)(k0)(k0): (5.13)
5.2.2 Soft gluon inside jet kwithEg>:Sk
ij
Using Eq. ( 5.7), the contribution Sk
ijfrom a gluon emitted into jet kfrom linesiandjis
given by the integral
Sk
ij=g22Ti
TjZddk
(2)dni
nj
(ni
k)(nj
k)2(k2)(k0) k
R: (5.14)
Much like for the Smeas
ij contribution, if k=i;j, there is an additional divergence (arising
fromnk
k!0) relative to the case k6=i;j, and so we evaluate these two cases separately
below.
{ 40 {Case 1:k=iorjThe integrals for this case are performed in Sec. B.2of the Appendix,
with the result that Sj
ijis
Sj
ij=Si
ij=sTi
Tj
4"
1
21
(1)42
42 
t2
ij
t2
ij1tan2R
2!
+ Li 21
t2
ij1
+ 2 Li 21
cos2 ij
2(t2
ij1)#
: (5.15)
Case 2:k6=i;j These contributions are at most 1 =divergent since the matrix element
does not have the nk
k!0 singularity. We show in Appendix B.3.2 that the result takes
the form
Sk
ij
k6=i;j=s
4Ti
Tj1
lnt2
ikt2
jk2tiktjkcosij+ 1
(t2
ik1)(t2
jk1)
+F(tik;tjk;ij)
;(5.16)
whereijis the angle between the i-kandj-kplanes and the nite function Fis given in
Eq. ( B.33) and isO(1=t2).
5.2.3 Soft gluon inside measured jet k:Smeas
ij(k
a)
The contribution of a gluon emitted into jet 1 (the measured jet) from lines iandjis given
by the integral
Smeas
ij(k
a) =g22Ti
TjZddk
(2)dni
nj
(ni
k)(nj
k)2(k2)(k0) k
RR: (5.17)
The singularity structure of this integral depends on whether or not k=iorj. Thus, we
evaluate the case k=iorjand the case k6=i;jseparately below.
Case 1:k=iorjWe consider rst Smeas
ij(i
a). Using the results of Sec. B.2of the
Appendix, we obtain the result in terms of tij,
Smeas
ij(i
a) =Smeas
ji(i
a)
=s
2Ti
Tj1
1
1a1
ia1+21
(1)42
!2t2
ij
t2
ij1tan2R
2(1a)
+1 +a
2(i
a) Li21
t2
ij1
: (5.18)
Case 2:k6=i;j The remaining contributions to the observed jet angularity are Smeas
ij
fork6=i;j. Using the results from Sec. B.3.3 in the Appendix, this contribution is
Smeas
ij(k
a)
i;j6=k=s
2Ti
Tj1
ka1+2
lnt2
ikt2
jk2tiktjkcosij+ 1
(t2
ik1)(t2
jk1)
+(k
a)G(tik;tjk;ij)
; (5.19)
whereGisO(1=t2) and is given in Eq. ( B.36) and, again, ijis the angle between the i-k
andj-kplanes.
{ 41 {contribution divergent terms
Sincl
ij 1
s
2Ti
Tj
1
lnni
nj
2+ ln2
42
Si
ij1
s
4Ti
Tj
1
lnt2
ijtan2(R=2)
t2
ij1+ ln2
42
Sk
ij 1
s
4Ti
Tjlnt2
ikt2
jk2tiktjkcosij+1
(t2
ik1)(t2
jk1)
Sunmeas
(1)1
s
2hPN
i=1T2
iln tan2(R=2) +P
i6=jTi
Tjln(ni
nj=2)i
+O(1=t2)
Smeas
ij(i
a)1
s
4Ti
Tjh
1
1a1
+ ln2
!2
i

+ lnt2
ijtan2(R=2)
t2
ij1
(i
a)2
1a
1
ia
+i
Smeas
ij(k
a)1
s
4Ti
Tjlnt2
ikt2
jk2tiktjkcosij+1
(t2
ik1)(t2
jk1)(k
a)
Smeas
(1)(i
a)1
s
2T2
ih
1
1a1
+ ln2
!2
i

+ ln tan2(R=2)
(i
a)2
1a
1
ia
+i
+O(1=t2)
Table 2: Summary of the divergent parts of the soft function at NLO. The rst block contains the
the observable-independent contributions: soft gluons emitted by jets iandjin any direction with
energyEg< in row 1; soft gluons entering jet kwithEg> (withk=iorjin the second row
andk6=i;jin the third). In the last row of the rst block, we summed over iandjand took the
large-tlimit to get the total Sunmeas
(1). Similarly, in the second block we give the contributions to
the angularities k
a(withk=iorjin the rst row and k6=i;jin the second) and summed over i
andjand took the large- tlimit to get Smeas
(1)in the third row.
5.3 Total N-Jet Soft Function in the large- tLimit
In this section, we add together the necessary ingredients calculated above to obtain the
N-jet soft function from Eq. ( 5.8). The results for the divergent pieces are summarized in
Table 2. In Sec. 6we use Table 2to show that the consistency of factorization is explicitly
violated by terms of order 1 =t2, and so in this section we give the full soft function (including
the nite terms) to O(1=t2).
Using color-conservation (P
iTi= 0), we nd that the observable-independent part,
Sunmeas
(1), dened in Eq. ( 5.9), is given for large tby
Sunmeas
(1) =s
2X
iT2
i"
1
ln2
42
1
ln 
2
42tan2R
2!
(5.20)
+1
2ln22
42
1
2ln2 
2
42tan2R
2!
2
6#
+s
2X
i6=jTi
Tj"
1
lnni
nj
2+ ln2
42
lnni
nj
2
+ Li 2
12
ni
nj#
+O(1=t2):
{ 42 {We see that the nite parts of this contribution are sensitive to two scales, 2 and 2 tanR
2.
For simplicity, in this paper, since we take tan( R=2)O(1), we will choose only a single
scale
Sto attempt to minimize logs in Eq. ( 5.20), where

S2 tan1=2R
2; (5.21)
chosen as the geometric mean of the two.
The remaining part of the soft function that is dependent on angularities as our choice
of jet observable is the sum over Smeas
(1)(i
a) (dened in Eq. ( 5.10)) for each jet i, where
Smeas
(1)(i
a) is given by
Smeas
(1)(i
a) =s
2T2
i1
1a(1
2+1
ln2
!2
itan2(1a)R
2
2
12
+1
2ln22
!2
itan2(1a)R
2
(i
a) (5.22)
2" 
1
+ ln2tan2(1a)R
2
(!iia)2!
(i
a)
ia#
+)
+O(1=t2):
The nite part of this contribution is sensitive to the scale i
S, where
i
S!ii
a
tan1aR
2; (5.23)
which, in principle, diers for each jet iand from the scale 
Sin the unmeasured part of
the soft function Eq. ( 5.20). After discussing resummation of large logarithms through RG
evolution, we will describe in Sec. 6.4a framework to \refactorize" the soft function into
pieces depending on multiple separated soft scales. This framework will enable us to resum
logarithms of all of these potentially disparate scales.
6. Resummation and Consistency Relations at NLL
The factorized cross section Eq. ( 3.34) is written in terms of hard, jet, and soft functions
evaluated at a factorization scale . Since the physical cross section is independent of ,
the anomalous dimensions of these functions are closely related. We derive and verify this
relation in Sec. 6.3. In Sec. 6.1and Sec. 6.2, we work out the form of the renormalization-
group equations (RGEs) satised by the hard, jet, and soft functions, and obtain their
solutions so that we can express each function at the scale in terms of its value at a scale
0where logarithms in it are minimized. In Sec. 6.4, we present a framework to refactorize
the soft function and give the total resummed distribution in Sec. 6.5.
6.1 General Form of Renormalization Group Equations and Solutions
We will build solutions for the hard, jet, and soft functions from two forms of RGEs. The
rst form is for a function which does not depend on the observable aand is multiplicatively
renormalized,
Fbare=ZF()F(); (6.1)
{ 43 {and satises the RGE,
d
dF() =
F()F(); (6.2)
where the anomalous dimension 
Fis found from the Z-factor by the relation

F() =1
ZF()d
dZF(); (6.3)
and takes the general form,

F() = F[] ln2
!2+
F[]: (6.4)
We call  F[] the \cusp part" of the anomalous dimension and 
F[] the \non-cusp part".
They have the perturbative expansions
F[s] =s
4
0
F+s
42
1
F+


 (6.5)
and

F[s] =s
4

0
F+s
42

1
F+


: (6.6)
The RGE Eq. ( 6.2) has the solution
F() =UF(; 0)F(0); (6.7)
where the evolution kernel UFis given by
UF(; 0) =eKF(;0)0
!!F(;0)
; (6.8)
whereKFand!Fwill be dened below in Eq. ( 6.15).
The second form of RGE is for a function dependent on the jet shape aand is renor-
malized through a convolution,
Fbare(a) =Z
d0
aZF(a0
a;)F(0
a;); (6.9)
and satisfying the RGE
d
dF(a;) =Z
d0
a
F(a0
a;)F(0
a;); (6.10)
with an anomalous dimension calculated from

F(a;) =Z
d0Z1
F(a0
a;)d
dZF(0
a;); (6.11)
and taking the general form

F(a;) =F[s]2
jF(a)
a
+ln2
!2(a)
+
F[s](a): (6.12)
{ 44 {The solution of an RGE of the form Eq. ( 6.10) has the solution [ 82,85,86,87,52]
F(a;) =Z
d0UF(a0
a;; 0)F(0
a;0); (6.13)
where the evolution kernel UFis given to all orders in sby the expression
UF(a;; 0) =eKF+
E!F
(!F)0
!jF!F(a)
(a)1+!F
+; (6.14)
where
Eis the Euler constant.
In Eqs. ( 6.8) and ( 6.14), the exponents !FandKFare given in terms of the cusp and
non-cusp parts of the anomalous dimensions by the expressions
!F(; 0)2
jFZs()
s(0)d
[]F[]; (6.15a)
KF(; 0)Zs()
s(0)d
[]
F[] + 2Zs()
s(0)d
[]F[]Z
s(0)d0
[0]: (6.15b)
In the case of Eq. ( 6.8) or if F[] happens to be zero, we dene jFto be 1. To achieve NLL
accuracy in the evolution kernels UF, we need the cusp part of the anomalous dimension
to two loops and the non-cusp part to one loop, in which case the parameters !F;KFin
Eq. ( 6.15) are given explicitly by
!F(; 0) =0
F
jF0"
lnr+ 
1
cusp
0cusp1
0!
s(0)
4(r1)#
; (6.16a)
KF(;0) =
0
F
20lnr20
F
(0)2r1rlnr
s()
+ 
1
cusp
0cusp1
0!
1r+ lnr
4+1
80ln2r
: (6.16b)
Herer=s()
s(0), and0;1are the one-loop and two-loop coecients of the beta function,
[s] =ds
d=2s
0s
4
+1s
42
+



; (6.17)
where (with TR= 1=2)
0=11CA
32Nf
3and1=34C2
A
310CANf
32CFNf: (6.18)
The two-loop running coupling s() at any scale in terms of its value at another scale
Qis given by
1
s()=1
s(Q)+0
2ln
Q
+1
40ln
1 +0
2s(Q) ln
Q
: (6.19)
{ 45 {In Eq. ( 6.16), we have used the fact that, for the hard, jet, and soft functions for which
we will solve, the cusp part of the anomalous dimension  F[s] is proportional to thecusp
anomalous dimension  cusp[s], where
cusp[s] =s
4
0
cusp+s
42
1
cusp+


: (6.20)
The ratio of the one-loop and two-loop coecients of  cuspis [88]
1
cusp
0cusp=67
92
3
CA10Nf
9: (6.21)
1
cuspand1are needed in the expressions of !FandKFfor complete NLL resummation
since we formally take 2
slnaO(s).
6.2 RG Evolution of Hard, Jet, and Soft Functions
6.2.1 Hard Function
The hard function is related to the matching coecient CNof theN-jet operator in
Eq. ( 3.33). If there are multiple operators with dierent color structures, CNis a vec-
tor of coecients. The hard function is then a matrix Hab=Cy
bCa. The hard function is
contracted in the cross section Eq. ( 3.34) with a matrix of soft functions.
The anomalous dimensions of the matching coecients Cahave been calculated in the
literature (for example, Table III of Ref. [ 89]). For an operator with Nlegs with color
charges T2
i, the anomalous dimension is

CN(s) =NX
i=1
T2
i(s) ln
!i+1
2
i(s)
1
2(s)X
i6=jTi
Tjlnni
nji0+
2
;
(6.22)
where Tiis a matrix of color charges in the space of operators, and 
iis given for quarks
and gluons by

q=3sCF
2; 
g=s
11CA2Nf
6: (6.23)
The coecient ( s) is the cusp anomalous dimension and is given to one-loop order by
(s) =s=. The anomalous dimension of the hard function itself is given by 
H=

y
CN+
CN, and takes the form of Eq. ( 6.4), with cusp and non-cusp parts  H[s] and

H[s] given to one loop in Table 3, with the result

H(s) =(s)T2ln2
!2
HNX
i=1
i(s)(s)X
i6=jTi
Tjlnni
nj
2; (6.24)
where T2=PN
i=1T2
iis the sum of all the Casimirs, and the eective hard scale  !H
appearing as the scale !in the logarithm in Eq. ( 6.4) is given by the color-weighted average
of the jet energies,
!H=NY
i=1!T2
i=T2
i (6.25)
{ 46 {F[s] 
F[s] jF!

HP
iT2
iP
i
iP
i6=jTi
Tjlnni
nj
21 !H

Ji(i
a) T2
i2a
1a
i 2a!i

meas
S(i
a)T2
i1
1a0 1!itan1+aR
2

Ji T2
i 
i 1!itanR
2

unmeas
S 0 P
iT2
iln tan2R
2+ P
i6=jTi
Tjlnni
nj
21 |
O(1=t2) 0 P
i6=jTi
Tjh
i=2meas2 lnt2
ij
t2
ij11 |
+P
k6=i;j
k=2measlnt2
ikt2
jk2tiktjkcosij+1
(t2
ik1)(t2
jk1)i
Table 3: Anomalous dimensions for the jet and soft functions. We give the cusp and non-cusp
parts of the anomalous dimensions,  F[s] and
F[s].  is the cusp anomalous dimension itself,
equal tos=at one loop. 
iis given for quark and gluon jets in Eq. ( 6.23). The third column
gives the value of jFappearing in Eq. ( 6.12) or Eq. ( 6.15). The last column gives the values of
!appearing in the logarithm ln 2=!2multiplying the cusp part of the anomalous dimension in
Eqs. ( 6.4) and ( 6.12). The scale  !His the color-weighted averages of all jet energies dened in
Eq. ( 6.25). All rows except for the last are given in the large- tlimit and in the last row we give
the remaining terms that are present for arbitrary t. This last row explictly violates consistency at
O(1=t2). The rst group of rows are needed for measured jets and the second group for unmeasured
jets. In the large- tlimit, for any number of measured and unmeasured jets, the consistency relation
Eq. ( 6.33) is satised.
6.2.2 Jet Functions
There are two forms of jet functions, those for measured and those for unmeasured jets.
Unmeasured jet functions Jq;g
!() satisfy multiplicative RGEs, with anomalous dimensions
given by the innite parts of Eqs. ( 4.17) and ( 4.26),

Ji= (s)T2
iln2
!2
itan2R
2+
i; (6.26)
which falls into the general form Eq. ( 6.4), with cusp and non-cusp parts of the anomalous
dimension given in Table 3, and the scale !in Eq. ( 6.4) being!itanR
2. The part 
iis
given by Eq. ( 6.23).
Measured jet functions satisfy RGEs of the form Eq. ( 6.10), with anomalous dimensions
given by the innite parts of Eqs. ( 4.11) and ( 4.25),

Ji(i
a) =
T2
i(s)2a
1aln2
!2
i+
i
(i
a)2(s)T2
i1
1a(i
a)
ia
+; (6.27)
{ 47 {which takes the form Eq. ( 6.12) with cusp and non-cusp parts of the anomalous dimension
split up as in Table 3, and the scale !in Eq. ( 6.12) being!i.
6.2.3 Soft Function
The totalN-jet soft function is given by Eq. ( 5.20) for unmeasured jets added to the
sum over measured jets of Eq. ( 5.22). This soft function depends on the Mjet shapes
1
a;:::;M
a, and satises the RGE
d
dS(1;:::;M;) =Z
d0
1


d0
M
S(10
1;:::;M0
M;)S(0
1;:::;0
M;):(6.28)
From the innite parts of the soft function given in Table 2, we nd the anomalous di-
mension
S(1;:::;M;) decomposes, as required by the consistency condition Eq. ( 6.33)
given below, into a sum of terms,

S(1;:::;M;) =
unmeas
S ()(1)


(M) +MX
k=1
meas
S(k;)Y
j6=k(j); (6.29)
where the pieces 
unmeas
S () and
meas
S(k;) are given in terms of their cusp and non-cusp
parts in Table 3, with the result

unmeas
S () =NX
i(s)T2
iln tan2R
2+ (s)X
i6=jTi
Tjlnni
nj
2; (6.30)
which takes the form of Eq. ( 6.4) with no cusp part, and

meas
S(k;) =(s)T2
k1
1a
ln 
2tan2(1a)R
2
!2
k!
(k)2(k)
k
+
; (6.31)
which takes the form of Eq. ( 6.12) with no non-cusp part, and the scale !in Eq. ( 6.12)
being!k=tan1aR
2.
The solution of the RGE Eq. ( 6.28) is given by
S(1;:::;M;) =Z
d0
1


d0
MS(0
1;:::;0
M;0)Uunmeas
S (; 0)MY
k=1Uk
S(k0
k;; 0);
(6.32)
whereUunmeas
S is given by the form of Eq. ( 6.8) andUk
S(k
a) by the form of Eq. ( 6.14).
The solution Eq. ( 6.32) is appropriate if all the scales appearing in the soft function
are similar, and thus all large logarithms in the nite part can be minimized at a single
scale0. As we noted in Sec. 5.3, however, the potentially disparate scales !ii
a, set by the
jet shapes of the measured jets, and , set by the cuto on particles outside jets, appear
together in the soft function, and logarithms of ratios of these scales may be large. In this
case, the soft function should be \refactorized" into pieces depending only on one of these
scales at a time. We describe a framework for doing so below in Sec. 6.4.
But rst, we verify the consistency of the anomalous dimensions for the hard, jet, and
soft functions to the order we have calculated them.
{ 48 {6.3 Consistency Relation among Anomalous Dimensions
We summarize the anomalous dimensions of the hard, jet, and soft functions in Table 3. We
separate contributions to the jet and soft anomalous dimensions that arise from measured
jets, from unmeasured jets, and those that are universally present. In all rows except the
last row, we take the large- tlimit and give the additional terms that arise (from the soft
function) for arbitrary t.
Consistency of the eective theory requires that the anomalous dimensions satisfy
0 =

H() +
unmeas
S () +X
i=2meas
Ji()
(i
a) +X
i2meas

Ji(i
a;)) +
meas
S(i
a;)
:
(6.33)
From the results tabulated in Table 3, up to corrections of O(1=t2), we see that, remark-
ably, this relation is indeed satised! This is highly nontrivial, as jet and soft anomalous
dimensions depend on the jet radius Rand the jet shape a, while the hard function does
not; in addition, the hard and soft functions know the directions niof allNjets, while the
jet functions do not. These dependencies cancel precisely between the R-dependent pieces
of unmeasured jet contributions to the jet and soft functions, between a-dependent pieces
of the measured jet contributions, and between ni
nj-dependent pieces of the hard and
soft functions. The sum of all jet and soft anomalous dimensions then precisely matches
the hard anomalous dimensions, satisfying Eq. ( 6.33).
Making the satisfaction of consistency even more nontrivial, individual contributions
to the innite part of the soft function, and therefore its anomalous dimension, given by
Table 2depend on the energy cut parameter  as well. However, these terms cancel in the
sum over the contributions Sincl
ijandSi
ijin the rst two rows of Table 2. The double poles
of the form1
ln  arise from regions of phase space where a soft gluon can become both
collinear to a jet direction (giving a 1 =) and soft (giving a ln ). These regions exist in
the integral over all directions giving Sincl
ijbut are subtracted back out in the contributions
Si
ij. In the surviving \Swiss cheese" region the regions giving these double poles are cut
out.
TheO(1=t2) terms that violate consistency arise whenever there are unmeasured jets.
While this limit is not required for the contribution of measured jets to the anomalous
dimension to satisfy the consistency condition Eq. ( 6.33), the nite parts of measured
jet contributions to the soft function contain large logarithms of != that can not be
minimized with a scale choice but are suppressed by O(1=t2) (cf. Eq. ( B.37) of Appendix
B). This is the manifestation of the fact that jets need to be well-separated, as explained
in Sec. 3. For the remainder of the paper, we work strictly in the large- tlimit.
It may seem mysterious that the calculations of the hard, jet, and soft functions them-
selves and requiring their consistency lead to the condition of a large separation parameter
t. Although we already specied qualitatively in the proof of factorization the requirement
of well-separated jets, it may not be immediately apparent where it is implemented in the
actual calculations. It enters in the denition of the collinear jet functions. In the large- t
limit, theNjets are innitely separated from one another according to the measure given
{ 49 {by Eq. ( 1.4). And indeed, when N-jet operators are constructed in SCET, each collinear
jet eld contains a Wilson line Wn, dened below in Eq. ( 3.9), of collinear gluons in the
directionnthat were emitted from the back-to-back direction  n, for which the separation
measuret!1 . (This is similar to QCD factorization proofs of hard scattering cross
sections, e.g. in [ 17], in which this direction  nis chosen to be along some arbitrary path
that is separated by an O(1) amount from the jet direction n.) Furthermore, the ni-
collinear jet function knows only its own color representation, and not those of the other
jets. Meanwhile, the hard and soft functions we calculate \know" about all Njets and
their precise directions and color representations. Therefore it is no surprise that, when
we actually calculate the anomalous dimensions of these functions, we nd that they are
consistent with one another only in the limit that the separation parameter t!1 .
6.4 Refactorization of the Soft Function
Our results for the soft function in Sec. 5.3make clear that in general there are multiple
scales that appear in the soft function: the 1
S;:::;M
Sassociated with the Mmeasured
jets and the scale 
Sassociated with the out-of-jet cuto  (see Eq. ( 5.21)). When these
scales are all comparable, we can RG evolve the soft function from the single scale S.
However, when any of them dier considerably from the others, we need to \refactorize"
the soft function into multiple contributions, each of which is sensitive to a single energy
scale. For illustration, take the scales i
Sto be such that 1
S2
S


M
Sas in
Fig.7. We also take l1
S
Sl
Sfor our discussion, but the result is independent of
whether
Slies in the range 1
S<
S<M
Sor not.
We can express the soft function appearing in Eq. ( 3.34) as
Figure 7: Soft
scales.S(1
a;2
a;:::;M
a;) =h0jOy
S(^)MY
i=1(i
a^i
a)OSj0i; (6.34)
where the operator i
areturns the contribution to aof nal-state soft
particles entering jet i, and ^ returns the energy of nal-state soft particles
outside of all Njets. We have kept the dependence on the scales i
Sand
on  implicit on the left-hand side.
There areNWilson lines appearing in the operator OS,
OS=Y1:::YMYM+1:::YN; (6.35)
corresponding to the Mmeasured jets and NMunmeasured jets. The scales associated
with soft gluons entering the Mmeasured jets whose shapes are measured to be 1;:::;M
are1
S;:::;M
S, given by Eq. ( 5.23). The scale associated with soft gluons outside of
measured jets is 
Sgiven by Eq. ( 5.21). We have illustrated the ladder of these scales
in Fig. 7. Each of these soft scales can be associated with dierent soft elds Ai
swhose
momenta scale as 2
i!iwhereiis associated with the typical transverse momentum i!i
of the collinear modes for the ith jet. For measured jets, iis determined by i
a, while for
unmeasured jets itan(R=2). For soft gluons outside jets, however, the soft momentum
is set by the cuto scale , which is why 
Sappears in the ladder of Fig. 7.
{ 50 {At a scalelarger than all i
Sand
S, the soft function is calculated as we presented
in Sec. 5. In particular, we take the quantities i
aand  to be nonzero and nite. At a scale
below1
S, we integrate out soft gluons of virtuality 1
Sand match onto a theory with soft
gluons of virtuality 2
S. The scale 1
Sassociated with 1
ais taken to innity, and phase space
integrals for soft gluons entering the measured jet 1 become zero (see, e.g., Eq. ( B.17)).
Therefore, the matching coecient from the theory above 1
Sto the theory below is just
the measured jet 1 contribution Smeas(1
a) to the soft function given by Eq. ( 5.22). The
same occurs when matching from the theory above each scale i
Sfor soft gluons entering
measured jet ito the scale below i
S, giving a matching coecient Smeas(i
a).
When going through the scale 
S, in the theory above this scale, the calculation of
unmeasured contributions to the soft function gives the result Eq. ( 5.20), by treating  as a
nonzero, nite cuto. In the theory below 
S, we take  to innity, making all phase space
integrals originally cuto by  to be scaleless and thus zero. So the matching coecient
between the theory above and below 
Sis justSunmeas.
After performing the above matchings all the way down to the lowest soft scale in
Fig. 7, we nd that the original soft function S(1
a;:::;M
a;) can be expressed to all
orders as
S(1
a;:::;M
a;) =Sunmeas()MY
i=1Smeas(i
a;)h0jOy
SOSj0i; (6.36)
where to next-to-leading order SmeasandSunmeasare given by
Sunmeas() = 1 +Sunmeas
(1) () (6.37)
Smeas(i
a;) =(i
a) +Smeas
(1)(i
a;); (6.38)
whereSunmeas
(1)is given by Eq. ( 5.20) andSmeas
(1)is given by Eq. ( 5.22). Note that no
operators restricting the jet shape or the phase space appear in the nal matrix element of
soft elds living at the lowest scale on the ladder in Fig. 7. If all the scales on the ladder
are at a perturbative scale, we can now just use hOy
SOSi= 1 to eliminate the nal matrix
element. If any scale is nonperturbative, we should have stopped the matching procedure
before that scale, and dened the surviving soft matrix element still containing additional
delta function operators as a nonperturbative shape function.
Since the factors Sunmeas() andSmeas(i
a;) are now matching coecients between
two theories above and below the respective scales 
Sandi
S, we can run each of the
individual factors separately from their natural scale, instead of from a single soft scale 0
as in Eq. ( 6.32). The result for the RG-evolved soft function is then Eq. ( 6.36) where each
factor at NLO is given by the solution of its RGE,
Sunmeas() =Uunmeas
S (;
S)Sunmeas(
S) (6.39a)
Smeas(i
a;) =Z
d0Ui
S(i
a0;;i
S)Smeas(0;i
S): (6.39b)
These solutions allow us now to resum logarithms of all of the scales appearing in the
ladder in Fig. 7when these scales are widely disparate. However, the result we obtained
in Eq. ( 6.28) when we took all scales to be of the same order and had a single soft scale
{ 51 {has the form Eq. ( 6.39) at NLL accuracy. We will use equation Eq. ( 6.39) in all cases to
interpolate between these two extremes.
6.5 Total Resummed Distribution
Collecting together the above results for the running of hard, jet, and soft functions in
the factorized cross section Eq. ( 3.34), we obtain the RG-improved N-jet cross section
dierential in Mjet shapes,
1
(0)dN
d1a1


dMaM=H(H)H
!H!H(;H)NY
k=M+1Jk
!k(k
J) 
k
J
!ktanR
2!!k
J(;k
J)
Sunmeas(
S)
MY
i=1(

1 +fi
J(i
a;i
J) +fi
S(i
a;i
S) 
i
Stan1aR
2
!i!!i
S(;i
S)
i
J
!i(2a)!i
J(;i
J)1
[!i
J(;i
J)!i
S(;i
S)]1
(ia)1+!i
J(;i
J)+!i
S(;i
S))
+
exph
K(;H;1;:::;N
J;1;:::;M
S;
S) +
E
(;1;:::;M
J;1;:::;M
S)i
;
(6.40)
where !His dened by Eq. ( 6.25), the evolution parameters !F(;F) andKF(;F) are
dened in Eq. ( 6.15), and we have dened the collective parameters,
K(;H;1;:::;N
J;1;:::;M
S;
S) =KH(;H) +NX
i=1Ki
J(;i
J) +MX
j=1Kj
S(;j
S)
+Kunmeas
S (;
S)(6.41a)

(;1;:::;M
J;1;:::;M
S) =MX
i=1
i(;i
J;i
S)MX
i=1[!i
J(;i
J) +!i
S(;i
S)]:(6.41b)
Using results from Appendix C, we obtain the functions fi
J;Sgenerated by the nite pieces
of the measured jet and soft functions,
fi
J(i
a;i
J) =s(i
J)T2
i
2(max
ai
a)42a
1aln2i
J
!i(ia)1
2a+1
1a1
1a
22
6 (1)(
i)
+
ci+2
1aH(1
i)"
2 lni
J
!i(ia)1
2a+1
2aH(1
i)#
+ (42a) ln2tanR
22ciln tanR
2
+s(i
J)
2dJ(i
a) (6.42a)
fi
S(i
a;i
S) =s(i
S)T2
i
1
1a("
lni
Stan1aR
2
!iia+H(1
i)#2
+2
6 (1)(
i)+dS)
;
(6.42b)
{ 52 {whereci= 3=2 for quark jets and 0=(2CA) for gluon jets. max
ais the upper limit on i
a
found in the nite part of the na ve jet function, given in Appendix A.H(1
i) is the
harmonic number function, with 
 igiven by Eq. ( 6.41b ). (1)is the rst derivative of the
digamma function,  (1)(z) = (d=dz)[0(z)=(z)]. The terms dJ;Sare additional contribu-
tions from the nite parts of jet and soft functions that do not contain any logarithms,
wheredS=2=24, anddJis given in Eq. ( C.6) in the Appendix. dJ;Sare not needed
at NLL accuracy. Similarly, the terms containing large logarithms in the unmeasured jet
functions and unmeasured contribution to the soft function are
Ji
!(J) = 1 +
(s(J))T2
iln2J
!tanR
2+
k[s(J)] lnJ
!tanR
2+di
J
(6.43a)
Sunmeas(
S) = 1 + (s(
S))X
iT2
i
ln
S
2 tan1=2R
2
ln tan2R
22
8
+ (s(
S))X
i6=jTi
Tj
ln
S
2lnni
nj
2+ Li 2
12
ni
nj
; (6.43b)
wheredi
Jis the part of the unmeasured jet function containing no large logarithms (given
in Eqs. ( A.19) and ( A.30) in the Appendix).
The nite parts of the measured and unmeasured jet and soft functions given in
Eqs. ( 6.42) and ( 6.43) show that to minimize large logarithms in the O(s) nite parts in
the resummed distribution Eq. ( 6.40), we should choose initial scales for the running to be
H= !H (6.44a)
i
J=!i(i
a)1
2a; k
J=!ktanR
2(6.44b)
i
S=!ii
a
tan1aR
2; 
S= 2 tan1=2R
2: (6.44c)
These choices eliminate all large logarithms in the O(s) hard, jet, and soft functions.
They still leave logs of tanR
2andni
njin the unmeasured part of the soft function, and
logs of tanR
2in the measured jet function, but we already take Rnumerically ofO(1)14
to minimize power corrections from our implementation of the jet algorithm as discussed
in Sec. 3.4, andni
nj1 since the jet separation parameter tijis large compared to 1.
All logs ofR, , andi
aare eliminated in the unmeasured jet function and measured part
of the soft function.
14We still consider tan( R=2) to be of order kin the collinear sectors describing unmeasured jets, as
implied by Eq. ( 6.44). This means kis eectively much larger than the parameter iin a measured
jet sector. In fact, note that Eq. ( 6.44) tells us that tanR
2must be parametrically larger than ( i
a)1
2a;
otherwise, the jet scale falls below the soft scale in the measured jet sectors, invalidating the use of SCET
and, thus, the validity of the factorization theorem.
{ 53 {7. Plots of Distributions and Comparisons to Monte Carlo
Having resummed the jet shape distributions in ato NLL accuracy, in this section we
plot the distributions for various values of aandR, compare to Monte Carlo simulated
events, and perform scale variation on the resummed distribution. We use the process
e+e!3 jets to study our predictions of jet shapes, where the jets arise from partons
in the \Mercedes-Benz" conguration, with each parton having equal energy. In these
congurations, the partons lie in a plane and are equally separated with a pairwise angle
of 2=3. This allows us to study event shape distributions of well-separated jets where tis
reasonably large. We choose three values of Rto study,R= 1:0, 0.7, and 0.4. With these
values of R, the 1 =t2suppression factor for corrections to the large- tlimit are 0.10, 0.044,
and 0.014 respectively. We will measure the angularity of only one of the three jets; the
other two jets will be unmeasured.
In general, the Ti
Tjcolor correlations in the soft and hard functions lead to operator
mixing in color space under RG evolution. This implies that the RG kernels USandUH
are matrices in color space and must be studied on a process-by-process basis (see, e.g.,
[89,90,91,92,93,94]). For the case of N= 2;3 jets there is only one color-singlet operator
and hence no mixing. This can be seen, for example, by noting that all color correlations
reduce to the Casimir invariants ( CFandCA) in this case (cf. Appendix D). We have
restricted the example process we use in this work to N= 3 jets, avoiding the additional
complication of color-correlations that comes with a larger number of jets.
The NLL resummed distribution for one quark or gluon jet shape (jet 1) in a three-jet
nal state, written as the derivative of the radiator Eq. ( 1.5), is
1
(0)d3
d1a=H
!H!H(;H)1
J
!1(2a)!1
J(;1
J) 
2
J
!2tanR
2!!2
J(;2
J) 
3
J
!3tanR
2!!3
J(;3
J)
 
1
Stan1aR
2
!1!!1
S(;1
S)
exp
K(;H;1
J;2
J;3
J;1
S;
S) +
E
(;1
J;1
S)
[1 + ^fJ(1
a) +^fS(1
a)]1
[
(;1
J;1
S)]"
1
(1a)1+
(;1
J;1
S)#
+; (7.1)
where the various evolution parameters !i
J;S;
;Kare all dened in Eqs. ( 6.15) and ( 6.41),
and ^fJ;Sare given by fJ;Sin Eq. ( 6.42) with thedJ;Sterms set to zero (accurate to NLL).
The best scale choices Eq. ( 6.44) for this case are
H=
!T2
1
1!T2
2
2!T2
3
3 1
2CF+CA(7.2a)
1
J=!1(1
a)1
2a; 2;3
J=!2;3tanR
2(7.2b)
1
S=!11
a
tan1aR
2; 
S= 2 tan1=2R
2: (7.2c)
In Eq. ( 7.1) we have used tree-level initial conditions for the hard, jet, and soft functions
at these scales. Eq. ( 7.1) evolves these functions to the arbitrary scale at NLL accuracy.
{ 54 {τaq jet
R=1g jet
R=1
q jet
R=0.7g jet
R=0.7
q jet
R=0.4g jet
R=0.41
σ(0)dσ
dτa
a=0a=-1/2
a=-1/4
a=1/4a=1/2
τa0.000 0.002 0.004 0.006 0.008 0.010050100150200250300
0.000 0.005 0.010 0.015 0.020010203040506070
0.000 0.002 0.004 0.006 0.008 0.0100100200300400500
0.000 0.005 0.010 0.015 0.020020406080100120
0.000 0.002 0.004 0.006 0.008 0.0100200400600800
0.000 0.005 0.010 0.015 0.020050100150200Figure 8: Quark and gluon jet shapes for several values of aandR. The NLL resummed distribu-
tion in Eq. ( 7.1) is plotted for a=1
2;1
4;0;1
4;1
2for quark and gluon jets with R= 1;0:7;0:4. The
plots are for jets produced in e+eannihilation at center-of-mass energy Q= 500 GeV, with three
jets produced in a Mercedes-Benz conguration with equal energies EJ= 150 GeV, and minimum
threshold  = 15 GeV to produce a jet.
With these choices, we plot Eq. ( 7.1) in Fig. 8for several values of aandRfor a quark
or gluon jet shape in a three-jet nal state in e+eannihilation at center-of-mass energy
Q= 500 GeV.15The jets are chosen to be in a Mercedes-Benz conguration, where all
jets have equal energies of 150 GeV. We choose the jet energy cuto  to be 15 GeV. We
choose the factorization scale to be =H.
15The distributions plotted with the ^fJ;Sterms included in Eq. ( 7.1) exhibit a small negative dip near a=
0 (not shown) that can be cured by convolving with a nonperturbative shape function with a renormalon-
free gap parameter [ 38,54]. This is beyond the scope of the present work, so we only plot the perturbative
distributions where they are positive.
{ 55 {τa τa1
σ(0)dσ
dτa
a=1/2, R=0.4
τaa=0, R=0.4
a=-1/2, R=0.4a=1/2, R=0.7
a=0, R=0.7
a=-1/2, R=0.7a=1/2, R=1
a=0, R=1
a=-1/2, R=1Legend
Quark jets 
(blue)
Gluon jets 
(red)Theory NLLMonte Carlo
hadronization offMonte Carlo
hadronization on
0.000 0.002 0.004 0.006 0.008 0.0100501001502002500.000 0.002 0.004 0.006 0.008 0.010010203040506070
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.01401002003004005000.000 0.002 0.004 0.006 0.008 0.010 0.012 0.0140501001500.000 0.002 0.004 0.006 0.008 0.010 0.012 0.0140102030405060
0.000 0.005 0.010 0.015 0.0200501001502002503000.000 0.005 0.010 0.015 0.0200204060801001201400.000 0.005 0.010 0.015 0.0200102030405060
0.000 0.002 0.004 0.006 0.008 0.0100200400600800Figure 9: Quark vs. gluon jet shapes with comparison to Monte Carlo. Solid, straight curves
represent the resummed jet shape distribution in Eq. ( 7.1), and jagged curves are histograms from
the Monte Carlo, normalized as described in the text. The solid histogram has no hadronization,
while the dashed histogram includes the eects of hadronization. The distributions are plotted
fora=1
2;0;1
2with quark (blue) and gluon (red) jets compared on the same plot, for jets of
sizeR= 1:0;0:7;0:4. Gluon jet shape distributions peak at larger values of athan quark jets,
indicative of their broader shape. The plots are for jets in e+eannihilation at center-of-mass energy
Q= 500 GeV, with three jets produced with equal energies EJ= 150 GeV, angular separation
 = 2=3 between all pairs of jets, and minimum threshold  = 15 GeV to produce a jet.
We compare the results of a jet algorithm implemented on Monte Carlo simulated
events with our NLL resummed predictions for a variety of aandRvalues in Fig. 9. Because
the resummed NLL distribution we choose to study applies to an exclusive process, three-
jet events in the Mercedes-Benz conguration, we implement cuts on the simulated events
to obtain a sample that matches onto this conguration. We use MadGraph/MadEvent
v.4.4.21 [ 95] to generate parton-level e+e!qqgevents at a center-of-mass energy Q=
500 GeV, with cuts imposed to obtain partons in the Mercedes-Benz conguration. We
shower and hadronize the events with Pythia v.6.414 [ 96] usingpT-ordered parton showers.
The process of hadronization will induce a shift in the angularity distribution, which we do
{ 56 {not model in our resummed distribution. Therefore, we produce two samples: one sample
with only QCD nal-state showering, no initial-state radiation, and no hadronization, and
another sample with complete showering, initial-state radiation, and hadronization. The
anti-k Tjet algorithm is run on the nal state particles from Pythia, and we use FastJet [ 97]
to implement the jet algorithm. The anti-k Talgorithm is particularly well suited for this
comparison, as very few particles at an angle >R to the jet axis are included in the jet.
With anti-k T, the phase space cut on an individual particle matches well with the phase
space cuts in the next-to-leading order calculation.
To select a sample of events to compare to our resummed distributions, we make cuts on
the nal state jets, requiring each of the three hard, well-separated partons from MadGraph
to be associated with a jet. This involves a cut on the jet direction and momentum:
pparton
pjet
jppartonjjpjetj>0:9 andjjppartonjjpjetjj
jppartonj<0:15: (7.3)
We analyze events passing these cuts, and tag each associated jet as coming from a quark
or a gluon based on which parton it matches onto. The angularity value for each jet is
computed from the constituent particles of the jet, using the matching parton direction as
the jet axis. The jet direction only diers from the parton direction by a power correction
(see Sec. 3.2). In Fig. 9, we isolate some of the quark and gluon jet shapes in Fig. 8and
compare to Monte Carlo events.
The relative normalization between the distribution of Monte Carlo events and the
NLL resummed angularity distribution requires some explanation. Both our calculation
and the Monte Carlo simulation are most accurate in the region that includes the peak
of the distribution and the larger- side of the peak, but both are inaccurate as !0
and in the tail region. Therefore, each will dier in the relative normalization between
the peak region and the tail region. An accurate prediction of the tail region requires
matching onto a calculation at xed-order in sin full QCD as in [ 43,53,54]. In Fig. 9, we
choose to normalize the area of the Monte Carlo distribution to the total area of the NLL
aτpeak
a
gluon jets
quark jetsR=1
R=0.7
R=0.4
R=1
R=0.7
R=0.4
/Minus1.0 /Minus0.5 0.0 0.50.0000.0020.0040.0060.0080.0100.0120.014
Figure 10: Location of peak of jet shape distribution as a function of afor quark and gluon jets.
We plot the value of aat the peak of the jet shape distribution for abetween -1.0 and 0.8 for
quark vs. gluon jets, using R= 1;0:7;0:4.
{ 57 {quark jets1
σ(0)dσ
dτagluon jets
hard scale variation hard scale variation
    scale variation     scale variation
unmeasured jet scale variation unmeasured jet scale variation
measured jet scale variation measured jet scale variation
measured soft scale variation measured soft scale variationµΛ µΛ
τa τafactorization scale variation factorization scale variation
0.000 0.002 0.004 0.006 0.008 0.010050100150200
0.000 0.005 0.010 0.015 0.020010203040500.000 0.005 0.010 0.015 0.02001020304050
0.002 0.004 0.006 0.008 0.0100501001502000.000 0.005 0.010 0.015 0.02001020304050
0.002 0.004 0.006 0.008 0.0100501001502000.000 0.005 0.010 0.015 0.020010203040506070
0.002 0.004 0.006 0.008 0.0100501001502002500.000 0.005 0.010 0.015 0.0200102030405060
0.002 0.004 0.006 0.008 0.0100501001502002500.000 0.005 0.010 0.015 0.02001020304050
0.002 0.004 0.006 0.008 0.010050100150Figure 11: Scale variation of quark and gluon jet shapes. For a= 0 andR= 0:7, we display
the variation of the NLL resummed jet shape distributions with the hard scale H, the jet cuto
scale
S, the unmeasured jet scales 2;3
J, the measured jet scale 1
J(a), and the measured soft scale
S(a). In each case we vary the scale between 1 =2 and 2 times the natural choices in Eq. ( 6.44),
except for the measured soft scale, which we varied between 1 and 2 times the choice in Eq. ( 6.44).
We keep the factorization scale xed at the default hard scale given by Eq. ( 7.2),=!i.
{ 58 {resummed theory distribution. We nd the area under the theory curves for quark and
gluon jets to be approximately 0.3 for R= 0:4, 0.5 forR= 0:7, and 0.7 for R= 1. A more
accurate prediction of the normalizations may require summing remaining unsummed logs
of the phase space cuts in the theory and Monte Carlo predictions. These plots should be
interpreted as comparisons of the predictions of the shapes in aand these shapes' scaling
as we vary aandR, rather than the overall normalization.
The shapes of the theory and Monte Carlo distributions are largely similar, though they
display noticeable dierences at the leftmost endpoint near a= 0 and in the \sharpness" of
the peak. These may be due to the dierent ways the two approaches deal with the growth
of the strong coupling for small a, the dierent orders of log resummation (LL vs. NLL)
and the need to match the tails onto xed-order QCD predictions. Since neither the Monte
Carlo nor theory partonic predictions without inclusion of hadronization eects is yet a
prediction of a physically observable quantity, we use this comparison as an intermediate
diagnostic rather than a conclusive test of either method. Nevertheless, comparing the way
the shapes of the distributions and locations of the peaks vary over the range of values of
aandRthat we sample, the behavior agrees very well between the theory distributions
and the Monte Carlo distributions without hadronization for both quark and gluon jets.
In Fig. 10we plot the location of the peak of the jet shape distributions as a function
ofafor three values of R, displaying the dierent variation of the peak of quark and
gluon jet shape distributions. The peak value increases with increasing Randa, as wide
angle radiation is included (increasing R) and less suppressed (increasing a). Although
the dierence in the peak value between the quark and gluon jet angularity distributions is
large, the width of each distribution creates substantial overlap in angularity values between
quark and gluon jets. Distinguishing between quark and gluon jets using jet angularities is
a complex task which we will explore in future work; for now, we note only that the NLL
resummed distributions indicate that discrimination between quark and gluon jets using
jet angularities is possible.
As a rough estimate of the theoretical uncertainty in our NLL resummed predictions,
we show in Fig. 11the change in the a= 0 quark and gluon ajet shape distributions for
R= 0:7 when the various scales that appear in the resummed cross section Eq. ( 7.1) are
varied. These are the initial scales at which the hard, jet, and soft functions are evaluated
to minimize logarithms in the NLO xed-order part, from which the evolution kernels run
them to the common factorization scale . In the top row of Fig. 11, we varybetween
!H=2 and 2!H. The tiny variation is a sign of the consistency condition satised by the
anomalous dimensions in Eq. ( 6.33). In the next four rows, we vary the hard scale H,
the soft jet energy cuto scale 
S, the unmeasured jet scales 2;3
J, and the measured jet
scale1
J(1
a) between half and twice the natural values given in Eq. ( 7.2). In the last row,
we vary the measured soft scale 1
S(1
a) between one and two times the value in Eq. ( 7.2).
This is because too low a value of 1
S(1
a) asa!0 brings it into the nonperturbative
region where s(1
S) blows up, so that the perturbative estimate of uncertainty is not so
meaningful. We note that, while the uncertainty in the vertical scale of the distributions
is considerable in some cases, the location of the peak is much more stable.
Finally, in Fig. 12we give a sense for how robust our theoretical predictions are for other
{ 59 {0.000 0.001 0.002 0.003 0.004 0.0050100200300400
0.000 0.001 0.002 0.003 0.004 0.0050501001502002503001
σ0dσ
dτ1
0
τ1
0 τ1
0ψnear=π/2 ψnear=π/3
Measured 
“near” jets
Measured 
“far” jetsτ1
0
τ1
0ψnearblue   q near g
green  q near q
red   g near q
blue   q far
red   g far0.000 0.001 0.002 0.003 0.004 0.005050100150200250300
0.000 0.001 0.002 0.003 0.004 0.005050100150200250Figure 12: Jet shapes for other kinematic congurations. We compare our theoretical predictions
to Monte Carlo simulations for the shape 1
0(a= 0) for a quark or gluon jet found in a three-jet
conguration where the two jets with narrowest separation angle  nearhave equal energy. We
consider the two cases  near==2 and=3. In the rst row, we plot shapes of one of the jets in
the \near" pair. The blue solid curve is the shape of quark jet found near a gluon jet, the green
dotted curve is a quark found near an antiquark, and the red solid curve is a gluon found near a
quark. In the second row, we compare shapes of a quark or gluon jet found far from the near pair.
kinematic congurations. We consider e+e!qqgevents where the angle  nearbetween
two partons is either =2 or=3, and these partons have equal energy. We nd jets using
the anti-kT algorithm with R= 0:4, and plot jet shapes for a= 0. The selection cuts to
choose events from the Monte Carlo are the same as the Mercedes-Benz conguration. In
these events there are ve distinct characterizations for a single parton. If the event has
the quark (or antiquark) as the \far" (most well separated) parton, then each parton in
the event is distinct: there is the far quark, the near quark, and the near gluon. If the
event has the gluon as the far parton, then there are only two distinct types of partons:
the far gluon and the near quark (antiquark). In Fig. 12, we plot all these congurations
for both near==2 and near==3. The agreement between the theory predictions
and the Monte Carlo are as good as in the Mercedes-Benz case, a good indication that
our calculation applies to a broad range of kinematic congurations of multijet events.
Additionally, we observe features consistent with our intuition about the relative dierences
between the jet shape distributions between dierent jets in these congurations. As one
would expect, the distribution of near jet shapes is weighted more heavily towards larger
athan the far jet shapes, due to the enhanced soft radiation in the near jet system. When
the near quark is near a gluon, the distribution is weighted more heavily towards larger
athan when the near quark is near an antiquark, due to the enhanced radiation coming
from a gluon rather than a quark. These distributions serve as further evidence that jet
shapes may be an eective discriminant between quark and gluon jets.
{ 60 {8. Conclusions
In this work, we have factorized an N-jet exclusive cross section dierential in MN
jet observables and resummed global logarithms of the jet observable ato NLL accuracy,
leaving summation of non-global logarithms to future work. We demonstrated that the
anomalous dimensions of the hard, jet, and soft functions in the factorization theorem
satisfy the nontrivial consistency condition Eq. ( 6.33) toO(s), for any number of quark
and gluon jets, any number of jets whose shapes are measured, and any size Rof the jets,
as long as the jets are well-separated, meaning t1. This condition ensures the validity
of an eective theory with Ncollinear directions that are assumed to be distinct. We
identied and estimated important power corrections to the factorized form of the cross
section. We also illustrated that zero-bin subtractions give nonzero contributions to the
anomalous dimensions crucial for consistency.
Armed with consistent factorization and the xed-order jet and soft functions, we
resummed large logarithms in the jet shape distribution by running each individual function
from the scale where logs in it are minimized to the common factorization scale . We
thereby resummed, to NLL accuracy, global logs of the jet shape aand logs of the scale
=EJof soft radiation outside of jets, but leaving some non-global logs and logs of the
angular cut R(but we took Rto be numerically of order 1). This is the rst such calculation
of a resummed jet shape distribution in an exclusive multijet cross section.
We constructed a framework to deal with all the scales that appear in the multijet
soft function which depends on the values i
aof allMjet shapes and the phase space cuts
;R. By refactorizing the full soft function into individual pieces depending on one of
these scales at a time, we were able to sum logs of ratios of these scales.
We demonstrated the accuracy of our results by comparing our resummed prediction
for quark and gluon jet shapes in e+e!3 jets to the output of Monte Carlo event
generators, MadGraph/MadEvent and Pythia. We compared our predictions with the
Monte Carlo output without hadronization. The changes in shape and location of the peak
value as functions of aandRmatch quite well between the theory and Monte Carlo.
Our results provide a basis for future studies of other jet observables at both e+eand
hadron colliders, requiring recalculation of those parts of our jet and soft functions that
depend on the choice of observable. Studying jets at hadron colliders requires constructing
observables appropriate for that environment and the switching of two of our outgoing jets
to incoming beams, which can be described by beam functions in SCET [ 62].
Precision calculations of jet shapes will allow improved discrimination of jets of dierent
origins. We are applying the results of our predictions of light quark and gluon jet shapes
to distinguish quark and gluon jets with greater eciency than achieved before. Extensions
to the shapes of heavy jets and calculations of other types of jet shapes such as the 	( r=R)
shape introduced in [ 14,15,16] can also be performed.
Note added in nal preparation: As this paper was being completed, Ref. [ 98] appeared
reporting the calculation of a quark jet function for a jet dened with a Sterman-Weinberg
algorithm and whose invariant mass sis measured. This jet function is related to our
{ 61 {measured jet function Jq
!(a) for a cone jet at a= 0 given in Eq. ( 4.11), sinces=!20.
We have checked that the corresponding results between the two papers agree.
Acknowledgments
We are grateful to C. Bauer for valuable discussions and review of the draft. The authors at
the Berkeley CTP and in the Particle Theory Group at the University of Washington thank
one another's groups for hospitality during portions of this work. AH was supported in
part by an LHC Theory Initiative Graduate Fellowship, NSF grant number PHY-0705682.
The work of AH and CL was supported in part by the U.S. Department of Energy under
Contract DE-AC02-05CH11231, and in part by the National Science Foundation under
grant PHY-0457315. The work of SDE, CKV, and JRW was supported in part by the U.S.
Department of Energy under Grants DE-FG02-96ER40956.
A. Jet Function Calculations
A.1 Finite Pieces of the Quark Jet Function
Measured Quark Jet Function The nite pieces the jet functions, which depend on
the jet algorithm, share common features. For cone-type algorithms, the nite piece of the
naive part of the quark jet function, ~Jq
alg(a), is given by
~Jq
cone(a) =CF7
2+ 3 ln 22
3
(a) +CF
1a
2
Iq
cone(a)(max
aa)
a
+(A.1)
where in this Appendix, plus distributions are dened by [ 62]
[(x)g(x)]+= lim
!0d
dx[(x)G(x)]; withG(x) =Zx
1dx0g(x0); (A.2)
dened so as to satisfy the boundary conditionR1
0dx[(x)g(x)]+= 0. The quantity Iq
cone
depends implicitly on aandRand is given by
Iq
cone=Z1xcone
xconedx2(1x) +x2
x= 2 log1xcone
xcone3
2+ 3xcone: (A.3)
The parameter xcone=xcone(a) is the lower limit on the x=q=!scaled gluon momentum
integral imposed by the cone restriction. It is given by the solution of the equation
fcone(xcone) =a
tan2aR
2; (A.4)
wherefcone(x) is dened as
fcone(x)x2a[x1+a+ (1x)1+a] (A.5)
in the range 0 < x < 1=2, which is plotted in Fig. 13A. The limit max
ais given by the
maximum value over xof Eq. ( A.5). Similarly, for k T-type algorithms, ~Jq
kT(a) is given by
~Jq
kT(a) =CF13
222
3
(a) +CF
1a
2
Iq
kT(a)(max
aa)
a
+: (A.6)
{ 62 {x(A) (B)x2a−22a−2
τa
tan(2−a)R/2x
fcone (x) fkT(x) fkT(x)
0.2 0.4 0.6 0.8 1.00.050.100.150.200.25
x1 1−x1 x2 x1 1−x1 1−x20.2 0.4 0.6 0.8 1.00.010.020.030.040.050.06
0.2 0.4 0.6 0.8 1.00.20.40.60.81.0
xcone 1−xcone
(C)x xFigure 13: Regions of integration for the (A) cone and k T-type algorithms for (B) a >1 and
(C)a <1. The allowed region of xis when the (blue) functions fcone;kT(x) lie above the (red)
lines of constant a=tan(2a)R=2. Whena <1 for the k Talgorithm, there are two regions of
integration when a>2a2tan(2a)R=2.
Iq
kTis given by
Iq
kT=Z
Rdx2(1x) +x2
x(A.7)
whereRis the region in xwhere the constraint
fkT(x)x2a(1x)2a[x1+a+ (1x)1+a]a
tan2aR
2(A.8)
is satised. We plot this region in Fig. 13B and C for the cases a >1 anda <1,
repsectively. The boundaries of this region are the points x1;2illustrated in the gure, and
are given by the equation
fkT(x1;2) =a
tan2aR
2; (A.9)
where we take x2>x 1ifx2exists. The upper limit max
ais given by the maximum value
overxof the right-hand side of Eq. ( A.8). In general, the constraint Eq. ( A.8) is symmetric
aboutx=1
2, and so the region Ris symmetric about the same point. In general, if a>1
ora<2a2tan(2a)R
2, thenRis a single range in x. Otherwise,Ris two disjoint ranges
inx. Sincea2a2tan(2a)R
2can only occur for a<1, we can writeIq
kTas
Iq
kT=Z1x1
x1dx2(1x) +x2
x
a>2a2tan(2a)R
2Z1x2
x2dx2(1x) +x2
x
(A.10)
Note thatIq
coneandIq
kTinvolve the same integrand, but for each algorithm the integral
is over dierent ranges. In addition, both xconeandx1approach the same limiting value
for smalla,
xa!0!a
tan(2a)R
2: (A.11)
Thus, we can extract the small abehavior of both distributions by writing
1
aln1x
x
+="
1
aln 
a
tan(2a)R
21x
x!#
+"
1
aln 
a
tan(2a)R
2!#
+;(A.12)
{ 63 {wherex=xconeorx1for the cone and k Talgorithms, respectively. Dening
rq(x) = 3x+ 2 ln1x
x; (A.13)
using Eq. ( A.12), and including the zero-bin subtraction in Eq. ( 4.10), we nd that the
nite distributions of the full measured quark jet functions are
Jq
cone(a) =CF"
3
2ln2
!2tan2R
2+1a
2
1aln22
!2+
1a
2
ln2tan2R
2+7
2+ 3 ln 2
2
6
2 +1a
2
1a#
(a)CF" 
4
1alntan1aR
2
!a!
(amax
a)
a#
+
CF
1a
2"
(a)(max
aa)
a3
2+2a
1aln2
!21
1a=2
a
rq(xcone)2 lna
tan2aR
2#
+(A.14a)
and
Jq
kT(a) =CF"
3
2ln2
!2tan2R
2+1a
2
1aln22
!2+
1a
2
ln2tan2R
2+13
2
2
6
4 +1a
2
1a#
(a)CF" 
4
1alntan1aR
2
!a!
(amax
a)
a#
+
CF
1a
2(a)(max
aa)
a3
2+2a
1aln2
!21
1a=2
rq(x1)2 lna
tan2aR
2+ 
1
2aa>2 tanR
2
rq(x2)3
2
+:
(A.14b)
Fora= 0, these expressions for the jet functions can be simplied further to give
Jq
cone(0) =Jq
incl(0) +CF"
3(0)
tan2R
20

0+ tan2R
2+
0tan2R
2

0 
2 ln0
tan2R
2+3
2!#
;
(A.15a)
for the cone jet function, and
Jq
kT(0) =Jq
incl(0) +CF(
(0)1
4tan2R
20

0"
3x1+ 2 ln 
1x1
x10
tan2R
2!#
+
01
4tan2R
2

0 
2 ln0
tan2R
2+3
2!)
; (A.15b)
for the k Tjet function. In Eq. ( A.15b ),x1is given by its value for a= 0,
x1=1
2 
1s
140
tan2R
2!
: (A.16)
{ 64 {In Eq. ( A.15), we have divided the cone and k Tjet functions into the contribution Jq
incl(0)
to the inclusive jet function [ 75,76], given by
Jq
incl(0) =CF
(0)3
2ln2
!2+ ln22
!2+7
22
2
(0)
03
2+ 2 ln2
!2
+
;
(A.17)
and algorithm-dependent parts. The algorithm-dependent part of the a= 0 cone jet
function Eq. ( A.15a ) agrees with [ 98]. Note that if one takes Rto be parametrically larger
than0(cf. Sec. 3.4and Eq. ( 6.44)), the algorithm-dependent parts of Eq. ( A.15) are
power suppressed, and the cone and k Tjet functions reduce to the inclusive jet function.
Unmeasured Quark Jet Function The nite pieces for the unmeasured quark jet
function are
Jq
alg=3CF
2ln 
2
!2tan2R
2!
+CF
2ln2 
2
!2tan2R
2!
+dq;alg
J; (A.18)
where the constant term dq;alg
Jis given by
dq;cone
J=CF7
2+ 3 ln 252
12
; dq;kT
J=CF13
232
4
; (A.19)
for the cone and k Talgorithms, respectively.
A.2 Finite Pieces of the Gluon Jet Function
Measured Gluon Jet Function The nite distributions of the naive gluon jet function
are given by
~Jg
cone(a) =(a)
CA137
36+11
3ln 22
3
TRNf23
18+4
3ln 2
+1
1a
2
Ig
cone(a)(max
aa)
a
+; (A.20)
and
~Jg
kT(a) =(a)
CA67
922
3
TRNf23
9
+1
1a
2
Ig
kT(a)(max
aa)
a
+;
(A.21)
where the integrals Ig
algare given by
Ig
alg=Z
dx
CA1
x(1x)+x(1x)2
+TRNf(12x(1x))
; (A.22)
with the cone and k Tregions of integration the same as for the quark jet functions. The
valuemax
ais the same as in the measured quark jet function, for the respective jet algo-
rithm.
{ 65 {Going through similar steps as for the quark jet function, dening
rg(x) = 2CAln1x
x
+CAx2
3x2x+ 4
TRNfx4
3x22x+ 2
;(A.23)
and using Eq. ( A.12) to make all logarithmic dependence on aexplicit, we nd for the
cone and k T-type jet function nite distributions
Jg
cone(a) =(a)"
0
2ln2
!2tan2R
2+CA1a
2
1aln22
!2+CA
1a
2
ln2tan2R
2(A.24a)
+CA137
36+11
3ln 22
6
2 +1a
2
1a
TRNf23
18+4
3ln 2#
" 
4CA
1alntan1aR
2
!a!
(a)(amax
a)
a#
+1
1a
2"
(a)(max
aa)
a
 
0
2+2a
1aCAln2
!21
1a=2rg(xcone)2CAlna
tan2aR
2!#
+;
and
Jg
kT(a) =(a)"
0
2ln2
!2tan2R
2+CA1a
2
1aln22
!2+CA
1a
2
ln2tan2R
2(A.24b)
+CA67
92
6
4 +1a
2
1a
TRNf23
9
" 
4CA
1alntan1aR
2
!a!
(a)(amax
a)
a#
+
1
1a
2(
(a)(max
aa)
a"
0
2+2a
1aCAln2
!21
1a=2
rg(x1)2CAlna
tan2aR
2+ 
1
2aa>2 tanR
2
rg(x2)0
2#)
+;
wherexconeandx1;2are given in Eqs. ( A.3) and ( A.8).
Fora= 0, the simplied result for the gluon cone jet function is
Jg
cone(0) =Jg
incl(0) +(0)
tan2R
20

0+ tan2R
2f 
0
0+ tan2R
2!
+
0tan2R
2

0 
2CAln0
tan2R
2+0
2!
;(A.25)
where
f(x)CA2
3x2x+ 4
TRNf4
3x22x+ 2
: (A.26)
{ 66 {Fora= 0, the gluon k Tjet function is given by,
Jg
kT(0) =Jg
incl(0) +(0)1
4tan2R
20

0"
rg(x1) + 2CAln0
tan2R
2#
+
01
4tan2R
2

0 
2CAln0
tan2R
2+0
2!
;(A.27)
wherex1is given by Eq. ( A.16), andrg(x1) is given by Eq. ( A.23). In Eqs. ( A.25) and
(A.27), the contribution Jg
incl(0) to the inclusive gluon jet function [ 34,35,78,79] is
Jg
incl(0) =(0)"
0
2ln2
!2+CAln22
!2+CA67
182
2
10TRNf
9#
" 
0
2+ 2CAln2
!20!
(0)
0#
+(A.28)
As for the quark jet functions Eq. ( A.15), the gluon jet functions split up into the inclusive
jet function and algorithm-dependent pieces that are power suppressed for 0R.
Unmeasured Gluon Jet Function For the unmeasured gluon jet functions, the nite
pieces are given by
Jg
alg=CA
2ln22
!2tan2R
2+0
2ln2
!2tan2R
2+dg;alg
J(A.29)
where the constant part dg;alg
Jfor the cone and k Talgorithms is given by, respectively,
dg;cone
J=CA137
36+11
3ln 252
12
TRNf23
18+4
3ln 2
(A.30a)
and
dg;kT
J=CA67
932
4
TRNf23
9
: (A.30b)
B. Soft function calculations
B.1Sincl
ij
To evaluate the expression Eq. ( 5.11), we rst dene
Sincl
ij1
1
(1)s
242
42
Ti
TjIincl(ni
nj): (B.1)
We needIincltoO(). Working in a coordinate system with ~ nialigned along the z-axis
and~ njin thexz-plane and dening n1ni
nj=nz
j, we have
Iincl(ni
nj) =ni
nj4(1)
2p(1
2)Z
0dsin2Z
0dsin121
1cos
1
1nx
jsincosnz
jcos
=4
2(1)Z+1
1du(1u)1(1 +u)1n
1un2~F11
2;1; 1;z
(B.2)
{ 67 {wherez=(1n2)(1u2)
(1un)2. The integration over u= coshas singularities at the points u= 1
andu=nwhich correspond to z= 1 andz= 0, respectively. To isolate these singularities,
we split the integration over uinto the ranges (1;) and (;1) wheren<< 1,
Iincl(ni
nj) =Iincl
1(ni
nj) +Iincl
2(ni
nj); (B.3)
where
Iincl
1(ni
nj)4
2(1)Z
1du(1u)1(1 +u)1n
1un2~F11
2;1; 1;z
Iincl
2(ni
nj)4
2(1)Z1
du(1u)1(1 +u)1n
1un2~F11
2;1; 1;z
:(B.4)
Over the range of integration of uinIincl
1,z2[0;1) for<1. ForIincl
2,z2(0;1].
Furthermore, the singularity at u=ninIincl
1is made more explicit through the use
of the identity
2~F11
2;1; 1;z
=fa(z) +fb(z)
fa(z) =p
cos ()1nu
junj1+2
2~F11
2;;1
2; 1z
fb(z) =
cos ()
(1=2)(1)2~F11
2;1;3
2; 1z
: (B.5)
fa(z) gives anO(1=) contribution and we proceed by using the following trick that we
exploit multiple times throughout the Appendix.
To integrate a product of functions f(x;)g(x;) wherefdiverges at the point x0as
(xx0)1+O(), we write the integation as
Z
dxf(x;)g(x;) =Z
dxf(x;)g(x0;) +Z
dxf(x;)
g(x;)g(x0;)
: (B.6)
The rst integral has relatively simple xdependence since g(x0;) does not depend on x.
The term in parenthesis in the second integral vanishes as xx0for regular functions g
and so the entire integrand can be expanded in .
We can now evaluate fa(z) by adding and subtracting the non-singular part of the
integrand (which is the hypergeometric function) evaluated at u=nas in Eq. ( B.6),
whereasfb(z) isO() and so we can simply expand about = 0. Adding these contributions,
we nd that
Iincl
1(ni
nj) =4
2"p(1)
1n2

cos()1
2
Z
1du
junj1+2
Z
1dusgn(nu)
1u 
1ln4(nu)2
1n2!
+Z
1du
1u2
junjtanh1junj
1nu#
: (B.7)
{ 68 {ForIincl
2, the part of the integrand that is not singular at u= 1 is everything that
multiplies (1u)1, and so we add and subtract this part as in Eq. ( B.6). This gives
Iincl
2(ni
nj) =4
2"
1
2(1)+Z1
du
un 
1 +1n
1ulog(n1)2(u+ 1)
4(nu)2
log(1u) +un
1ulog(2)!#
: (B.8)
The integrals in Eqs. ( B.7) and ( B.8) give rise to many terms. However, we nd that,
after some lengthy algebra, the dependence on cancels in the sum as it must and that
the result can be simplied to
Iincl(ni
nj) =1
+ lnni
nj
2
+2
6+ Li 2
12
ni
nj
: (B.9)
B.2Si
ijandSmeas
ij(i
a)
B.2.1 Common Integrals
In evaluating the soft contributions Si
ijandSmeas
ij(i
a), we nd an integral of the following
form:
I(;;t ) = 2t2Z1
0du
uu2f(u;;t); (B.10)
wheret>1 and
f(u;;t) =(tan2R
2+u2)2
(u+t)22F1
1;1
2; 12;4tu
(u+t)2
: (B.11)
To evaluate this integral, we add and subtract the part of the integrand that is not singular
atu= 0, namely f(u;;t), as in Eq. ( B.6). This allows us to write
I(;;t ) = 2 tan4R
2Z1
0du
u1+2
+1
t2u2
u+ 2
ulnu+t2
ulnt2
t2u2+t2
uln
1 + tan2R
2u2
;
(B.12)
where we used that
f(0;;t) =1
t2tan4R
2; (B.13)
and that the expansion of the hypergeometric about = 0 fort>u is
2F1
1;1
2; 12;4tu
(u+t)2
=t+u
tu
1 + 2lnt2
t2u2+O(2)
: (B.14)
Evaluating the integals, we obtain
I(;;t ) =tan4R
2
t2
t21
+
+ 22

Li21
t21
2Li2 
1 +t2tan2R
2
t21!
+O(2): (B.15)
{ 69 {B.2.2Smeas
ij(i
a)
To evaluate Eq. ( 5.17) for the case that k=i, we use light cone coordinates in the frame
of jet i,k+=ni
kandk= ni
k. In terms of these variables, the on-shell condition can
be used to give
nj
k=k+cos2 ij
2+ksin2 ij
2p
k+ksin ijcos; (B.16)
with cos ij= 1ni
nj, andthe angle in k?-space (the azimuthal angle about ~ ni). We
can do the k?andk+integrals using the on-shell and adelta functions respectively. The
resultingSmeas
ij(i
a) has non-trivial integrals over kand:
Smeas
ij(i
a) =s
442
!2
(ni
nj)(Ti
Tj)1p(1
2)2!
2a1
(ia)2Z
0dsin2
Z1
0dk
(k)2!i
a
k1
 
tan2R
2!i
a
k2
2a!!i
a
k21a
2a
"!i
a
k2
2a
cos2 ij
2+ sin2 ij
2!i
a
k1
2a
sin ijcos#1
:
(B.17)
Making the change of variables u= cotR
2p
k+=k= cotR
2
!i
a
k1
2a, we nd that Smeas
ij(i
a)
can be written as
Smeas
ij(i
a) =s
2Ti
Tj1
(1)42
!2tan2(1a)R
21
ia1+2
I(1a;0;tij);(B.18)
whereI(;;t ) is dened in Eq. ( B.10). Using Eq. ( B.15) we nd the result given in
Eq. ( 5.18).
B.2.3Si
ij
The -functions in Eq. ( 5.14) are easiest to deal with if we shift to variables where each
-function is in a dierent variable. The simplest choices are just the arguments of the 
functions  and i
R,k0andu= cotR
2p
k+=k, respectively, where kare dened with
respect to direction ni. This gives a form similar to the integral in Smeas
ij(i
a),
Sj
ij=1
s
4Ti
Tj1
(1)42
42tan2R
2
I(1;1;tij): (B.19)
whereI(;;t ) is dened in Eq. ( B.10) and evaluates to Eq. ( B.15). This gives Eq. ( 5.15).
B.3Smeas
ij(k
a)andSk
ijfork6=i;j
We again use light cone coordinates centered on jet k. The integrations involved in
Smeas
ij(k
a) andSk
ijonly give rise to a 1 =pole as explained in the text, but integrating
the eikonal factor 1 =(ni
k)(nj
k) is more complicated than for the other cases since there
is a third direction, nk, involved.
{ 70 {For unmeasured jets when there are n3 total nal state jets, Sk
ijis needed. However,
as we explain in the text, measured jets violate consistency at O(1=t2) even forn= 2 (non
back-to-back) jets and the contribution of Sk
ijdoes not ameliorate this fact when n3.
To show this, we need to evaluate the divergent contribution of Sk
ij. In addition, we give
the form of the nite pieces which are O(1=t2).
For each measured jet when there are n3, the sum Smeas
ij(k
a) +Sk
ij(k
a) is needed.
However, in this case the 1 =pole cancels in this sum and we are left with only a single,
nite integral to evaluate. This is clear from the expressions for Smeas
ij(k
a) andSk
ijwhich
we derive in Sec. B.3.2 and Sec. B.3.3 , respectively. We evaluate the sum explicitly in
Sec.B.3.4 .
B.3.1 Common Integrals
We nd the following integral arising in both Smeas
ij(k
a) andSk
ij:
I(u;ta;tb;)2
Z
0d1sin21Z
0d2sin122t2
a+t2
b2tatbcos
u2+t2a2utacos1
1
u2+t2
b2utb(coscos1+ sinsin1cos2)
=I(0)(u;ta;tb;) +I(1)(u;ta;tb;) +O(2); (B.20)
where theO(0) andO(1) parts ofIare
I(0)(u;ta;tb;) =2
Z
0dA
A2B2t2
a+t2
b2tatbcos
u2+t2a2utacos(B.21)
I(1)(u;ta;tb;) =2
Z
0d2 ln (sin)A
A2B2+B
A2B2logAB
A+Bt2
a+t2
b2tatbcos
u2+t2a2utacos;
where we dened
A=u2+t2
b2utbcoscos
B= 2utbsinsin: (B.22)
We can evaluate I(0)straightforwardly. For the range of our interest, ta;b>1 and
0<u< 1, it gives
I(0)(u;ta;tb;) =2(t2
a+t2
b2tatbcos)(t2
at2
bu4)
(t2au2)(t2
bu2)(t2at2
b2tatbu2cos+u4): (B.23)
In addition, we will need the following integrals over I(0)(u):
f1(ta;tb;)Z1
0duuI(0)(u;ta;tb;) = lnt2
at2
b2tatbcos+ 1
(t2a1)(t2
b1)
; (B.24)
{ 71 {and
f2(ta;tb;;r)Z1
0duuI(0)(u;ta;tb;) ln(r+u2)
=(
g(ta;r) +g(tb;r) + 2 ln(r+ 1) ln(tatb)
+ 2 Re"
Li2tatbei
rei+tatb
Li2tatb
rei+tatb
+ lntatb
tatbei
ln(r+tatbei)#)
; (B.25)
where
g(t;r)Li2t2
t2+r
Li2t21
t2+r
+ ln(t21) ln(r+t2)ln(t2) ln((r+ 1)(r+t2)):
(B.26)
Forr= 0, this simplies to
f2(ta;tb;;0) =Li21
t2a
Li21
t2
b
+ 2 Re
Li2ei
tatb
: (B.27)
Notice that both f1andf2areO(1=t2).
TheO(1) piece,I(1), is less trivial. However, the only property of I(1)that we need
is that
f3(ta;tb;)Z1
0duuI(1)(u;ta;tb;) =O(1=t2); (B.28)
which can be seen by taking the large- tlimit ofI(1)in Eq. ( B.21). The integral is nite an
suppressed by 1 =t2.
B.3.2Sk
ij
To compute Sk
ij, we choose a coordinate system such the ~ nkis in thez-direction and ~ ni
lies in the xz-plane. In terms of the light-cone coordinates about nkand the variable
u= cotR
2p
k+=k, we have
ni
k=k+cos2 ik
2+ksin2 ik
2p
k+ksin ikcos1
=kcos2 ik
2tan2R
2h
u2+t2
ik2utikcos1i
nj
k=k+cos2 jk
2+ksin2 jk
2p
k+k(nx
jcos1+ny
jsin1cos2)
=kcos2 jk
2tan2R
2h
u2+t2
jk2utjk
cosijcos1+ sinijsin1cos2
i
;(B.29)
whereijis dened as the angle between the ik- andjk-planes. Using the relation
ni
nj
cos2 ik
2cos2 jk
2tan2R
2= 2(t2
ik+t2
jk2tiktjkcosij); (B.30)
{ 72 {we nd that Sk
ijcan be written as
Sk
ij=1
s
4Ti
Tj1
(1)42
42
tan2R
2Z1
0duu12
tan2R
2+u22
I(u;tik;tjk;ij);
(B.31)
whereI(u;ta;tb;ij) is dened in Eq. ( B.20). Expanding in , we nd
Sk
ij=s
4Ti
Tj"
1
f1(tik;tjk;ij) +F(tik;tjk;ij)#
; (B.32)
where the nite part is given by
F(ta;tb;)
f1(ta;tb;) ln2
42tan2R
2
f2(ta;tb;;0)
+ 2f2
ta;tb;;tan2R
2
+f3(ta;tb;)
; (B.33)
andf1,f2, andf3are given in Eqs. ( B.24), (B.25), and ( B.28), respectively.
B.3.3Smeas
ij(k
a)
Using the same coordinate system as for Sk
ij, we nd that Smeas
ij(k
a) can be written as
Smeas
ij(k
a) =s
2Ti
Tj1
(1)42
!2tan2(1a)R
21
ka1+2
Z1
0duu1+2(1a)I(u;t12;tik;ij): (B.34)
Expanding in gives
Smeas
ij(k
a) =s
2Ti
Tj"1
ka1+2
f1(tik;tjk;ij) +(k
a)G(tik;tjk;ij)#
; (B.35)
where
G(ta;tb;)1
2"
f1(ta;tb;) ln2
!2tan2(1a)R
2
+ (1a)f2(ta;tb;;0) +f3(ta;tb;)#
;
(B.36)
andf1,f2, andf3are given in Eqs. ( B.24), (B.25), and ( B.28), respectively.
B.3.4Sk
ij+Smeas
ij(k
a)
The sum of Eqs. ( B.32) and ( B.35) is nite. We nd
Smeas
ij(k
a) +Sk
ij(k
a) =s
4Ti
Tj"
(k
a) 
f1(tik;tjk;ij) ln42
!2tan2aR
2
+ (2a)f2(tik;tjk;ij;0)2f2
t12;tik;ij;tan2R
2!
21
ka
+f1(tik;tjk;ij)
: (B.37)
wheref1andf2are given in Eqs. ( B.24) and ( B.25), respectively.
{ 73 {C. Convolutions and Finite Terms in the Resummed Distribution
In evaluating the nal resummed distribution Eq. ( 6.40), each measured jet function must
be convolved against a corresponding soft function piece Smeas. These convolutions take
the form.
Z
dJdSd0
Jd0
SJ(0
J;J)Smeas(0
S;S)"
(J0
J)
(J0
J)1+!i
J#
+"
(S0
S)
(S0
S)1+!i
S#
+(JS):
(C.1)
For the class of functions of the form x1!with!6= 0 and! < 1, we dene the plus
distribution by
(x)
x1+!
+lim
!0(x)
x1+!!
!(x)
=(x)
!+1X
n=0(!)n(x) lnnx
x
+;(C.2)
where the plus functions on the second line are given by Eq. ( A.2),
(x) lnn(x)
x
+lim
!0(x) lnn(x)
x+lnn+1
n+ 1(x)
: (C.3)
From these denitions, we can derive the identities (see, e.g., Appendix B of [ 54])
Z
d00(00)
(00)1+!1
+(000)
(000)1+!2
+=(!1)(!2)
(!1!2)(0)
(0)1+!1+!2
+;(C.4)
and
Z
d0(0)
(0)1+!
+(0) =()
1+!
+(C.5a)
Z
d0(0)
(0)1+!
+(0)
0
+=()
1+!
+h
lnH(1!)i
(C.5b)
Z
d0(0)
(0)1+!
+(0) ln0
0
+=()
1+!
+[lnH(1!)]2+2
6 (1)(!)
2
(C.5c)
Using the above identities, we nd that the nal result can be written in the form
{ 74 {Eqs. ( 6.40) and ( 6.42) with the functions dJ(i
a) given by
dq;cone
J(i
a) =CF"
7
2+ 3 ln 22
6
2 +1a
2
1a#
+CF(max
ai
a)
1a
2(i
a)1+
i
Z
dJ(i
aJ)
(iaJ)1+
i
+"
(J)
J 
3xcone+ 2 ln1xcone
xconeJ
tan2aR
2!#
+
+ (i
amax
a)CF(
3
2ln2
!2tan2R
2+1a
2
1aln22
!2+
1a
2
ln2tan2R
2
4
1a
lni
aH(1
i)

lntan1aR
2
!i1
2
lni
aH(1
i)
2
2
12+1
2 (1)(
i)
+(i
a)1+
i
1a
2Zmax
a
0dJ1
(iaJ)1+
i
+1
J
rq(xcone)3
2
+)
(C.6a)
and
dq;kT
J(i
a) =CF"
13
22
6
4 +1a
2
1a#
+CF(max
ai
a)
1a
2(i
a)1+
i
Z
dJ(i
aJ)
(iaJ)1+
i
+"
(J)
J 
3x1+ 2 ln1x1
x1J
tan2aR
2

1
2aa2 tanR
2
rq(x2)3
2!#
+)
+ (i
amax
a)CF(
3
2ln2
!2tan2R
2+1a
2
1aln22
!2+
1a
2
ln2tan2R
2
4
1a
lni
aH(1
i)

lntan1aR
2
!i1
2
lni
aH(1
i)
2
2
12+1
2 (1)(
i)
+(i
a)1+
i
1a
2Zmax
a
0dJ1
(iaJ)1+
i
+1
J 
rq(x1)3
2

1
2aa2 tanR
2
rq(x2)3
2!#
+)
(C.6b)
{ 75 {for quarks and by
dg;cone
J(i
a) =CA137
36+11
3ln 22
6
2 +1a
2
1a
TRNf23
18+4
3ln 2
+(max
ai
a)
1a
2(i
a)1+
iZ
dJ(i
aJ)
(iaJ)1+
i
+
"
(J)
J 
rg(xcone) + 2CAlnJ
tan2aR
2!#
+
+ (i
amax
a)(
0
2ln2
!2tan2R
2+CA1a
2
1aln22
!2+CA
1a
2
ln2tan2R
2
4CA
1a
lni
aH(1
i)

lntan1aR
2
!i1
2
lni
aH(1
i)
2
2
12+1
2 (1)(
i)
+(i
a)1+
i
1a
2Zmax
a
0dJ1
(iaJ)1+
i
+(J)
J
rg(xcone)0
2
+)
(C.6c)
and
dg;kT
J(i
a) =CA67
92
6
4 +1a
2
1a
TRNf23
9
+(max
ai
a)
1a
2(i
a)1+
iZ
dJ(i
aJ)
(iaJ)1+
i
+
"
(J)
J 
rg(x1) + 2CAlnJ
tan2aR
2

1
2aa2 tanR
2
rg(x2)0
2!#
+)
+ (i
amax
a)(
0
2ln2
!2tan2R
2+CA1a
2
1aln22
!2+CA
1a
2
ln2tan2R
2
4CA
1a
lni
aH(1
i)

lntan1aR
2
!i1
2
lni
aH(1
i)
2
2
12+1
2 (1)(
i)
+(i
a)1+
i
1a
2Zmax
a
0dJ1
(iaJ)1+
i
+1
J
rg(x1)0
2

1
2aa2 tanR
2
rg(x2)0
2#
+)
(C.6d)
for gluons, where in all cases xcone,x1;2(dened in Eqs. ( A.3) and ( A.8)) are evaluated at
=0
Jandrq;gare dened in Eqs. ( A.13) and ( A.23).
{ 76 {D. Color Algebra for n= 2;3Jets
For the two and three jet cases, there are no color correlations since all color generator
inner products Ti
Tjcan be expressed in terms of the Casimir invariants CAandCF. For
n= 2, there is a quark jet with charge Tqand an anti-quark jet with charge Tqthat each
square toCF. There is only one inner-product in this case and using color conservation
(P
iTi= 0), we have that
Tq
Tq=T2
q=T2
q=CF: (D.1)
Forn= 3 jets color conservation gives that, for example,
T1
T2=1
2
(T1+T2)2T2
1T2
2
=1
2
T2
3T2
1T2
2
: (D.2)
Referring to the quark, anti-quark, and gluon generators as Tq,Tq, and Tg, respectively,
using T2
q=T2
q=CFandT2
g=CAin Eq. ( D.2) gives
Tq
Tq=CA
2CF
Tq
Tg=Tq
Tg=CA
2: (D.3)
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