Minimal In ation Luis Alvarez-Gaum ea, C esar G omezb,a, Raul Jimenezc,a aTheory Group, Physics Department, CERN, CH-1211, Geneva 23, Switzerland. bInstituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, E-28049 Madrid, Spain. cICREA & Institute of Sciences of the Cosmos (ICC), University of Barcelona, 08028 Barcelona, Spain. Abstract Using the universal Xsuper eld that measures in the UV the violation of conformal invariance we build up a model of multi eld in ation. The underlying dynamics is the one controlling the natural ow of this eld in the IR to the Goldstino super eld once SUSY is broken. We show that at directions satisfying the slow roll conditions exist only if R-symmetry is broken. Naturalness of our model leads to scales of SUSY breaking of the order of 101113Gev, a nearly scale-invariant spectrum of the initial perturbations and negligible gravitational waves. We obtain that the in aton eld is lighter than the gravitino by an amount determined by the slow roll parameter . The existence of slow-roll conditions is directly linked to the values of supersymmetry and R-symmetry breaking scales. We make cosmological predictions of our model and compare them to current data. Key words: SUSY; cosmology; in ation 1. Introduction In spite of the enormous success of in ationary cosmol- ogy [1, 2, 3, 4, 5, 6, 7, 8, 9] at describing the observed properties of the Universe, we are still missing a deriva- tion from rst principles where the in aton eld is iden- ti ed with one, or several, fundamental elds in particle physics. This manifests itself the in the fact that we still do not count with a natural way of identifying the in aton eld and the properties of its potential required to satisfy experimental constraints [10, 11]. It was quickly realized after the in ationary scenario was proposed more than 30 years ago, that supersymmetry could provide a natural scenario with plenty of at direc- tions which could lead to in ation [18, 19, 20, 21, 22, 23]. When the theory couples to supergravity, there are a num- Email addresses: Luis.Alvarez-Gaume@cern.ch (Luis Alvarez-Gaum e), cesar.gomez@uam.es (C esar G omez), jimenez@icc.ub.edu (Raul Jimenez)ber of new problems that appear [24], and we will discuss some of them later on. Current observational constraints from CMB tempera- ture and polarization experiments and large-scale struc- ture limit the amount the in aton eld has moved to ap- proximately <2Mpl[14], where Mplis the reduced Planck mass. Therefore, in ationary models that search for the in aton at very large energies, like for example chaotic in ation, are severely constrained already by current ob- servations. With the current new generation of CMB ex- periments (Planck, EBEX, Spider, SPUDS etc...) it will be possible to further constraint how much the in aton eld has displaced during the in ationary period that gave rise to our current casual horizon. It is therefore useful to revisit again the problem of steep directions in SUGRA models to understand if a at direction can be obtained at all. In this paper we will suggest a natural embedding of in- ationary dynamics in the e ective low-energy Lagrangian Preprint submitted to Physics Letters B October 26, 2018arXiv:1001.0010v1 [hep-th] 30 Dec 2009describing supersymmetry breaking. Our approach will be quite independent of the microphysics underlying su- persymmetry breaking, and will only rely on universal properties of this symmetry. Since we are not commit- ting ourselves to any particular microscopic realization of supersymmetry breaking, some of our comments about re- heating for instance will be rather sketchy. A more de- tailed and precise presentations of our ideas will appear elsewhere [25]. Like most in ationary theories containing supersymmetry, we present a simple model of multi eld in ation (sometimes called hybrid) [26], identify naturally the in aton eld and its potential, and then t a few obser- vational data to estimate the few parameters of our model. We compute, in particular, the number of e-folding and the amplitude of density uctuations at horizon crossing. It is surprising to nd that the scale of supersymmetry break- ing indicated by this analysis is between 10111014GeV. An interesting spin-o of our model is that the in aton is lighter than the gravitino by an amountp, whereis one of the slow roll parameters (see below). We would like to stress that in this paper we are always assuming F-breaking of supersymmetry. In D-breaking scenarios our arguments do not apply, at least as presented here1. 2. General framework Supersymmetry is a natural framework to de ne in- ationary scenarios for two main reasons. First of all, SUSY naturally leads to the existence of at, or nearly at directions (pseudomoduli), allowing for slow roll sce- narios. Second, and more important, the order parameter of supersymmetry breaking is the vacuum energy density. Hence, naturally associated with its breaking, supersym- metry contains two main ingredients necessary in in a- tionary scenarios: vacuum energy and reasonably at di- rections. 1We thank Gia Dvali for raising this point. See for instance the last entry in [21]In a remarkable recent work, Komargodski and Seiberg [27] have presented a new formalism to understand super- symmetry breaking, its general properties, its non-linear realizations [28], and a systematic way to understand the low-energy couplings of goldstinos to other elds. Al- though many things were known before (see references in [27]) this work, the presentation is quite insightful, and it played a major part in the inspiration of this work. The basic starting point in [27] is the Ferrara-Zumino multiplets of currents [29]. A vector super eld composed of the R-symmetry current, the supercurrent, and the en- ergy momentum tensor. This vector super eld satis es the general relation: D_ J ;_ =D X: (1) The chiral super eld Xis essentially de ned uniquely2 in the ultraviolet. Following [27] the super eld Xhas the following properties: In the UV description of the theory, it appears in the right hand side of 1, where it represents a measure of the violation of conformal invariance. The expectation value of its 2component is the or- der parameter of supersymmetry breaking. In this work we are only considering F-breaking of super- symmetry. We denote by fthe expectation value of theF-component of X. It will sometimes be useful to writef=2, whereis the microscopic scale of supersymmetry breaking. When supersymmetry is spontaneously broken, we can follow the ow of Xto the infrared (IR). In the IR this eld satis es a non-linear constraint and becomes 2The ambiguities in the supercurrent multiplet and Xare related to improvement terms in the various currents. 2the \goldstino" super eld3. X2 NL= 0; (2) XNL=G2 2F+p 2G+2F: (3) The scalar component xofXbecomes a goldstino bilinear. Its fermionic component is the goldstino fermionG, andFis the auxiliary eld that gets the vacuum expectation value. A major part in the anal- ysis in [27] is based on this novel nonlinear constraint satis ed by the super eld Xin the IR. As shown there, the correct normalization of the goldstino su- per eld to derive all relevant low-energy theorems of broken supersymmetry is XNL=3 8fX. Finally,Xgeneralizes the usual spurion couplings ap- pearing in the description of low-energy supersymmet- ric lagrangians. If msoftdescribes the soft supersym- metry breaking masses at low energies, the standard spurion in the lagrangian is replaced bymsoft fXNL. This allows one to write the leading low-energy cou- plings of the goldtino to other matter elds. Since we are going to consider goldstino couplings, we will work with a eld whose expectation values are well below the Planck scale. Our proposal is to identify in the UV the in aton eld with the scalar component of the super eld X. SinceXis de ned uniquely (up to the ambiguity mentioned in foot- note one) in the UV, this provides a well de ned prescrip- tion. Furthermore, we will identify the in ationary period precisely with the ow of Xfrom the UV to the IR i.e. X!XNL. Note that by making this assumption we do not need to think of the in aton as any extra fundamental eld. In fact, independently of how SUSY is broken, and 3A modi ed version of the nonlinear constraint (2) appears when one considers spontaneous R-symmetry breaking. In that case, the goldstino and the corresponding axion will be part of the same mul- tiplet.what is the underlying fundamental theory we can always identify the Xsuper eld as well as its scalar component x. More importantly, by making this assumption we are identifying the vacuum energy driven in ation with the actual SUSY breaking order parameter. In the supergravity context, once we have the K ahler potentialK(X;X) and the superpotential W(X), the full scalar potential is given by [30]: V=eK M2(K1 X;XDW DW3 M2jWj2) (4) with DW =@XW+1 M2@XKW: (5) Mis the high energy scale below which we can write the ef- fective action describing the dynamics of the X-super eld. It could be the Planck scale, or a GUT scale depending on the microscopic theory. We will work well below the scale M, and for simplicity take M=MplIn equation (4) we can see one of the basic problems in supergravity in a- tion [24]. As we will see later on, to satisfy the slow roll conditions, a necessary condition is that the -parameter, de ned by: =M2 plV00 V; (6) be much smaller than one. If we choose a K ahler potential K(X;X) with R-symmetry, for instance the canonical one K(X;X) =XX+:::, where the :::represents a function ofXX, it is easy to see that from the exponent of (4) we always get a contribution to equal to 1: = 1 +:::, no matter which component of Xis taken as the in aton eld. This of course violates the slow roll conditions. Since we are considering a situation with supersymmetry breaking and gravity (early universe), we cannot exclude supergrav- ity from the picture, and this leads to the -problem in these theories. The simplest way out of this problem without unreason- able ne tuning, is to have explicit R-symmetry breaking 3in the K ahler potential4. If we have explicit R-breaking, the expansion of Vfor small elds takes the form: X=M( +i ) (7) V=f2(1 +A1( 2+ 2) +B1( 2 2) +:::)(8) fis the supersymmetry breaking parameter representing the expectation value of an F-term, and hence with square mass dimensions. We assume that Vis locally stable at least during in ation. Hence A1B1>0. We express the potential in terms of the dimensionless elds ; . Their masses can be read o from (8): m2 =2f2 M2(A1+B1); m2 =2f2 M2(A1B1):(9) The numbers A1;B1are taken to be O(1). One could be more explicit, and choose some super- symmetry breaking superpotential, like W=fX, and K ahler potential explicitly breaking R-symmetry, like: K=XX+ (c=M2)(X3X+XX3) +:::as in [27] lead- ing to an e ective action description of Xfor scales well belowM. At this stage, we prefer not to consider explicit examples of UV-completions of the theory. We consider the beginning of in ation well below M, hence the initial conditions are such that ; << 1. In fact, since is the lighter eld, we take this one to be the in aton, and consider that initially ; pf=M . For us the in ationary period goes from this scale until the value of the eld is close to the typical soft breaking scale of the problemmsoft, where the eld X!XNL(2), at this scale XNLbehaves like a spurion [27] and as shown in Ref. [27], the leading couplings to low-energy supersymmetric mat- ter can be computed as spurion couplings, for instance5, ifQ;V represent respectively low energy chiral and vector 4R-symmetry is a well-known problem in phenological applica- tions of supersymmetry. R-symmetry does not allow soft breaking masses for the gauginos; and spontaneous breaking of the symmetry may lead to axions with unacceptable couplings. Often one wants to preserve R-parity to avoid other possible phenomenological disasters. 5The details can be found in[27] section 4, in particular around equations (4.3,4).super elds, we can have the couplings: L=Z d4 XNL f 2 m2QeVQ (10) +Z d2XNL f1 2BijQiQj+:::+h:c: plus gauge couplings. Once we reach the end of in ation, the eld Xbecomes nonlinear, its scalar component is a goldstino bilinear and the period of reheating begins. The details of reheating de- pend very much on the microscopic model. At this stage one should provide details of the \waterfall" that turns the huge amount of energy f2into low energy particles. Part of this energy will be depleted and converted into low energy particles through the soft couplings in (10), and hence we can in principle compute a lower bound on the reheating temperature. Before making some comments on the re- heating period, we analyze the cosmological consequences of a potential as simple as (8), as well as the assumptions we have made earlier about the in aton and its range as in ation takes place. 3. The In aton Potential and Slow Roll Conditions To study the conditions under which our potential pro- vides in ation consistent with the latest cosmological con- straints, we examine the slow-roll parameters, de ned as [13]: =M2 pl 2V0 V2 ; (11) =M2 plV00 V; (12) whereMplis the reduced Planck mass and ' denotes deriva- tive with respect to the in aton eld. The observables are then expressed in terms of the above slow roll parameters as: nS= 16+ 2; (13) r= 16 (14) 4nt=2; (15) 2 R=VM4 pl 242: (16) nSis the slope of the scalar primordial power spectrum, ntis the corresponding tensor one, ris the scalar to tensor ratio and 2 Ris the amplitude of the initial perturbations. All these numbers are constrained by current cosmological observations [10, 11, 12]. We will use their constraints to explore the naturalness of our in ationary trajectories. In- ation takes place when the slow-roll parameters are much smaller than 1. We will use the amplitude of initial perturbations and the number of efoldings to t some of the paramenters of the toy model in the previous section. Recall that the potential in the range of interest is: V=f2(1 +A1( 2+ 2) +B1( 2 2) +:::);(17) which appears in gure 1. We can compute ;while rolling in the direction: = 2 ( (A1B1) )2+::: (18) = 2 (A1B1) +:::; (19) since << 1,is naturally small. We can make small by a slight ne tuning of the di erence A1B1. We will writelater as a ratio of the in aton and gravitino masses. Once the slow roll conditions are satis ed, we can compute the number of efoldings (see for instance [16, 17]): N=1 M Zdxp 2 = Z f id 2p (20) From (19) we get: N=1p 2jA1B1j log f i : (21) In most models of supersymmetry breaking, the gravitino mass is given by: m3=2=f M; (22) hence, we can rewrite the parameters and masses in (9) as: jA1B1j=1 2m2 m2 3=2;jA1+B1j=1 2m2 m2 3=2;(23)thus: N=p 2m3=2 m 2 log f i (24) The number of efoldings is considered normally to be be- tween 50100. Finally we will use the amplitude of initial perturbations to get one extra condition in the parameters of our potential. Using [11] (16) can be written as: V 1=4 =f1=2 21=4(jA1B1j )1=2=:027M; (25) where is taken atN-efoldings before the end of in ation. Summarizing, the two cosmological constraints we get on the parameters of our potential can be written as: N=p 2m3=2 m 2 log f i ; (26) 21=4m3=2 m pf M1=2 = 0:027; (27) and theparameter can be written as: =m m3=22 : (28) We take iabove the supersymmetry breaking scale pf=M ==M , and fclose tomsoft=M, therefore we can easily get values for Nbetween 50100 for moderate values of, which is expressed here as the square of the ratio of the in aton to the gravitino mass. It is interesting to notice that from (27), we can write the supersymmetry breaking scale in terms of the -parameter:  M5:2 104: (29) Hence for a value of :1 we can get 1013GeV. Lower values of the supersymmetry breaking scale can be obtained by reducing . However, since the in aton mass is m =m3=2p; (30) we may end up with an in aton whose mass is substantially lighter than the gravitino. For these values of ;, we have that i1013=M; f103=M, and the number of efoldings is110. 5We conclude then that with moderate values of be- tween:1:01 we can get supersymmetry breaking scales between 10111013without major ne tunings. We eas- ily get enough efoldings, and furthermore, the in aton is lighter than the gravitino by an amount given byp. For the above range of parameters we can compare the predicted value of nSin our model with observational con- straints. This is shown in the right panel of Fig. 1. The yellow region is the current cosmological constraints from WMAP5 [11] and the other colored areas are the predic- tions for our model with minimal ne tuning for an stable (unstable)Xpotential, i.e. the eld is concave (convex) re- spectively. The constraints will improve greatly when the Planck satellite releases its results next year, and therefore our model can be tested much more accurately. Reheating can proceed in many ways, since we have not provided a detailed microscopic model. Once in the non- linear regime, the XNL eld (whose scalar component is made of a goldstino bilinear) could eciently convert the f2-energy density into radiation. We can calculate the amount of entropy and particle density by using the Boltz- man equation and assuming that the pair of Goldstinos will have an out-of-equilibrium decay[16]. Using that TRH= 1010p f=GeV3=2 GeV (31) we obtain a range 107< TRH<109. This produces a particle abundance of n107090which are standard values. We can also compute the amount of entropy gen- erated by the out-of-equilibrium decay as Sf=Si= 107(p f=GeV )1=2(32) which yields values in the range 10 to 1, and assures that there is no entropy overproduction. We could also compute the depletion of this energy through the soft couplings (10) yielding very similar values as above. In both cases, we can get sucient reheating with temperatures betweenpf and a fraction of m3=2. The true value depends very much on the details of the microscopic model. However, thereseems to be no obstruction to reheating the universe to and acceptable value of temperature, particle abundances and entropy. We are currently working in a more detailed theory incorporating our scenario [25]. 4. Conclusions In this short note we have studied the possibility of hav- ing supersymmetry breaking as the driving force of in a- tion. We have used the unique chiral super eld Xwhich represents the breaking of conformal invariance in the UV, and whose fermionic component becomes the goldstino at low energies. Its auxiliary eld is the F-term which gets the vacuum expectation value breaking supersymmetry. It is crucial in our analysis to have explicit R-symmetry breaking along with supersymmetry breaking. This allows us to avoid the problem in supergravity and to take the supersymmetric limit. The simplest model we obtain de- scribes the components of Xwell below the Planck scale. It is written in terms of three parameters: the supersym- metry breaking parameter fand the masses of the real and imaginary components of the eld x(the scalar component of X). In our analysis the imaginary part of xplays the role of the in aton, and its mass was shown to be smaller than the gravitino mass by an amount given byp. This imag- inary component represents a pseudo-goldstone boson, or rather, a pseudomoduli. In supersymmetric theories such elds abound, and any of them could be used to construct some form of hybrid in ation. In our case, however, we want to use the minimal choice that is naturally provided by the universal super eld Xthat must exist in any su- persymmetric theory. Since we have not presented any detailed model, the cos- mological consequences are a bit rudimentary, especially concerning reheating at the end of in ation. However, the comparison of the simplest model with present data, yields very interesting values for the supersymmetry breaking scale, and the ratio of the in aton and gravitino masses. 6Figure 1: Left panel: The potential as a function of ( ) and ( ) components of the eld X. Note the nearly at direction ( ) that we use for our in ationary trajectories. Graceful exit and particle creation occurs in the non-linear part of the X eld. Right panel: WMAP5 cosmological constraints (yellow region) in the rnSplane. For no- ne-tuned minimal in ation models the green and red area show our predictions for both cases of a stable (concave) potential and unstable (convex) potential. The Planck satellite will be able to provide signi cantly tigther constraints on rand especially nS(at the<0:5% level) thus further constraining our model. The dashed line is the limit in rthat can be achieved with an ideal CMB polarization experiment [14] These are bonuses which come directly from the observa- tions of the initial density perturbations from WMAP data [11]. The fact that the in aton is lighter than the gravitino may have interesting low-energy phenomenological impli- cations. Furthermore in this simple model it is easy to obtain sucient number of efoldings with moderate values of theparameter. To explore our proposal in more detail, it is important to construct an explicit model, even if not very realistic, in order to understand in more detail the end of in ation, the reheating mechanisms, and also the ne structure of the in aton potential. We hope to report on this in the near future [25]. Acknowledgements We would like to thank G. Dvali, G. Giudice, J. Les- gourgues, S. Matarrese, G. Ross, Nathan Seiberg, M.A. V azquez Mozo, and L. Verde for useful discussion. 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