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http://arxiv.org/abs/2409.03353v1 | 20240905085539 | Modelling the age distribution of longevity leaders | [
"Csaba Kiss",
"László Németh",
"Bálint Vető"
] | q-bio.PE | [
"q-bio.PE",
"math.PR",
"60J25, 60K20"
] |
Real-time diagnostics on a QKD link via
QBER Time Series Analysis
Dimitris Syvridis
MOX - Dipartimento di Matematica “F. Brioschi”, Politecnico di
Milano, via Bonardi 9, 20133 Milan, Italy
^1 arashandrea.roknian@mail.polimi.it
^2 anna.scotti@polimi.it
^3 alessio.fumagalli@polimi.it
September 9, 2024
===============================================================================================================================================================================================================================================
empty
§ INTRODUCTION
Exceptionally long human lifespans are one of the cornerstones of demography and mortality research.
Studying the group of record holders may reveal not only the underlying mortality mechanism of a population but also potentially shed some light on the future developments of human longevity.
As life expectancy increases <cit.> with deaths shifting to older ages, the distribution of deaths at the oldest-old ages <cit.> gains more interest of demographers, actuaries and decision makers of numerous disciplines.
The pattern of adult human mortality has already been described <cit.> but there is still a debate about the exact distribution of deaths at adult ages. More details on the necessity of a correctly specified model for the underlying mortality process and its impact on further research are discussed in <cit.>. Numerous publications review the ongoing discussion on the existence of a mortality plateau <cit.>, and the levelling-off of adult human death rates at the oldest ages is supported by the findings in <cit.> while others cast some doubt on this observation <cit.>. If there is a mortality plateau then the distribution of deaths at the oldest-old ages must be gamma-Gompertz and human lifespan can increase further without any maximum <cit.>. Further studies discuss the existence of a limit to human lifespan with more focus on the extreme value distribution aspect of the deaths at the oldest-old ages for various populations and different lifetime distributions <cit.>. These models can be helpful in determining the plausibility of longevity leaders as well.
We contribute to this discussion by proposing a stochastic model to describe the evolution of the age of the world's oldest person. Based on our estimates the model provides a good fit to the titleholder data since 1955, collected by the Gerontology Research Group <cit.>.
With the model results, it is possible to predict the age of the oldest person in the world in the future.
When should we expect to see the next Jeanne Calment, the supercentenarian with the longest human lifespan ever documented?
Will her record ever be surpassed?
Our results provide a prediction for the age distribution of the record holder in the coming decades to answer these questions.
§ RESULTS
Our model describes the evolution of the age of the oldest living person under the following assumptions.
We assume that the births of individuals follow a Poisson process with time-dependent intensity <cit.>.
The lifespans of individuals in the population are independent and their distribution may depend on the date of birth.
Then the age of the record holder in the population evolves in time as a Markov process with explicit transition probabilities.
As the first main result of this paper, we explicitly compute the distribution of the age of the record holder for any given birth rate parameter and lifespan distribution.
The detailed mathematical description and the properties of the general model are described in Section <ref>.
We apply our general result to the case which approximates the human birth rate over the world and the human lifespan.
We specify the intensity function of the Poisson process of births to have an exponential growth in time.
The underlying force of mortality is chosen so that it follows an extension of the Gompertz mortality model <cit.>
and the lifespan distribution of individuals is given by the gamma–Gompertz () distribution with time-dependent parameters.
This distribution adequately captures the slowing down of senescence mortality at the oldest old ages.
Given the growth parameters of the birth rate, we fit the model parameters to the statistics of the oldest person titleholder data using maximum likelihood method.
The optimal parameters of the model fit well to the data.
It shows in particular that the age of the oldest person alive increases over time, and it will most likely increase further in the future.
We compute the expected value and a confidence interval for the age of the world's oldest person using the fitted model parameters for each year between 1955 and 2019 shown by the green curves on Figure <ref>.
The detailed discussion of the model specification, likelihood calculations as well as the parameter fitting are given in Section <ref>.
Section <ref> contains calculations related to the gamma–Gompertz–Makeham generalization of the gamma–Gompertz distribution.
Our results enable us to predict the age distribution of the world's oldest person at future time points.
We compute the probability density of the age of the world's oldest person in different years not only in the past but also in the future.
These densities are shown in Figure <ref>.
When comparing the age distribution of the oldest person in the world in different years to the age of Jeanne Calment at her death, we find that on Jan 1st 2060 we can expect that the age of the world's oldest person will exceed her age with probability around 0.5.
This also means that with high probability her age record will already be broken by that time.
In Figure <ref> two extreme outliers with unexpectedly long lifetimes can be observed. Jeanne Calment died at the age of 122.45 years in 1997, and Sarah Knauss died at the age 119.27 in 2000.
In our model, the probability of observing an age greater than or equal to their actual age at the time of their death is 0.000286 for Calment and 0.0116 for Knauss.
See details in Subsection <ref>.
The fact that Calment and Knauss are outliers among the oldest old in the world became even more evident when we performed a backtesting of our model. We estimated the parameters based on the data on the world's oldest person between 1955 and 1988 where the ending date is the time when Calment became the world's oldest person. The model-estimated mean and confidence interval of the world's oldest person using the full data and the partial dataset (before Calment) are shown in Figure <ref> by green and red, respectively.
The estimate using the data until 1988 is less reliable after 2000 which is shown by the fact that the observed data is out of the confidence interval in the majority of the time after 2000.
When we compare the two confidence intervals, we can conclude that, based on the data before 1988, Calment and Knauss already had extremely high ages at their deaths.
Adding the remaining data set, the estimated mean age of the world's oldest person becomes lower.
Hence we can conclude that now we can consider the ages of the two outliers between 1988 and 2000 at their death to be more extreme than based on the information available in 1988.
The other important observable is the reign length of a record holder.
The numerical value of the expected reign length with our estimated model parameters is 1.195 in 1955 and it is 1.188 in 2019.
The empirical value of the reign length is 1.008 which is not much less than the model-based estimate.
Our approach to studying the age of the oldest old is completely novel because it takes into account jointly the age and the time of birth of individuals.
Although the age and the reign length of the world's oldest person depend in a complex non-linear way on the total lifespan and the time of birth of supercentenarians, we compute explicitly the probability distribution of the age of the oldest person.
Hence, the performance of our predictions cannot be directly compared to previous results in the literature because in the usual approach, the oldest person in each cohort is considered separately, and it is not relevant whether this person was ever the oldest in the population, see e.g. the extreme value method in <cit.>.
Our model contributes to the mathematical understanding of the evolution of the oldest individual, which is the extra benefit compared to a prediction using the trend in the data, e.g. in linear regression.
In this way, we not only observe but also prove mathematically that the dynamics of the birth process and that of the lifespan distribution which we consider in this paper necessarily imply the increase of the expected age of the world's oldest person.
§ MATHEMATICAL MODEL FOR THE AGE OF THE OLDEST PERSON
In this section, we provide the mathematical definition of a general model for the age of the world's oldest person, where the births of individuals follow a Poisson process and their lifespans are independent.
Under the assumptions of the general model, the age of the record holder in the population evolves in time as a Markov process with explicit transition probabilities.
In Subsections <ref>–<ref>, we describe the exact distribution of the age of the record holder in this generality
for any given birth rate parameter and lifespan distribution using the two-dimensional representation of the age process of the oldest person.
In the time-homogeneous case with constant birth rate and identical lifespan distributions the reign length distribution of a record holder is computed in Subsections <ref>–<ref>.
We explain the role of the entry age parameter in Subsection <ref>.
§.§ Model description and two-dimensional representation
The model is formally defined as follows.
Let λ(t) be the birth rate parameter which depends on time and let F_t and f_t be a family of cumulative distribution functions and density functions corresponding to non-negative random variables which are also time-dependent.
We assume that individuals are born according to a Poisson point process at rate λ(t) and that the lifespan of an individual born at time t is given by F_t so that lifespans are independent for different individuals.
Let Y_t denote the age of the oldest person in the population at time t.
The process (Y_t:t∈) is Markovian.
The Markov property holds because at any time t the history of the process (Y_s:s≤ t) provides information about the lifetime of individuals born before the current record holder
while any transition of (Y_s:s≥ t) depends only on the lifetime of the current record holder and of those born after them.
The evolution of the Markov process Y_t is the following.
It has a deterministic linear growth with slope 1 due to the ageing of the current record holder.
This happens until the death of the record holder.
Additionally, given that Y_t-=lim_s↑ tY_s=y for some t with y>0,
the process has a downward jump at time t at rate f_t(y)/(1-F_t(y)) which is the hazard rate of the distribution F_t at y.
This corresponds to the possibility that the record holder dies at time t which happens at rate f_t(y)/(1-F_t(y)).
The conditional distribution of the jump is given by
(Y_t<x | Y_t-=y,Y_t<y)
=exp(-∫_x^yλ(t-u)(1-F_t-u(u)) ụ)
for all x>0.
The jump distribution in (<ref>) has an absolutely continuous part supported on [0,y] with density
j_y,t(x)=exp(-∫_x^yλ(t-u)(1-F_t-u(u)) ụ)
λ(t-x)(1-F_t-x(x))
and a point mass at 0 with probability
a_y,t=exp(-∫_0^yλ(t-u)(1-F_t-u(u)) ụ).
As we shall see in the relevant parameter regime the probability a_y,t of the point mass at 0 is negligible.
The transition formula in (<ref>) can be proven using the description below.
We introduce a two-dimensional representation of the process Y_t as follows.
Let Λ={(t_i,x_i):i∈ I} be a marked Poisson process in ×_+
where {t_i:i∈ I} forms a Poisson point process on with intensity λ(t)
and x_i≥0 is sampled independently for each i∈ I according to the distribution F_t_i.
The point (t_i,x_i) represents an individual born at time t_i with lifespan x_i for all i∈ I, that is, the individual i is alive in the time interval [t_i,t_i+x_i) and their age at time t is t-t_i if t∈[t_i,t_i+x_i).
Hence the marked Poisson process Λ contains all relevant information about the age statistics of the population at any time.
In particular the age of the oldest person Y_t can be expressed in terms of Λ as
Y_t=max{(t-t_i)𝕀_{t∈[t_i,t_i+x_i)}:i∈ I}.
where the indicator 𝕀_{t∈[t_i,t_i+x_i)} is 1 exactly if the ith person is alive at time t.
The transition distribution formula (<ref>) can be seen using the two-dimensional representation as follows.
Given that the current record holder dies at time t at age y the event {Y_t<x} means that nobody with age between x and y can be alive at time t.
This event can be equivalently characterized in terms of the Poisson process of birth at rate λ(·) thinned by the probability that the person is still alive at time t.
Indeed the event {Y_t<x} can be expressed as a Poisson process of intensity at time t-u given by λ(t-u)(1-F_t-u(u)) for u∈[x,y] not having any point in the time interval [t-y,t-x].
This probability appears exactly on the right-hand side of (<ref>).
In other words, for any u∈[x,y] people are born at time t-u at rate λ(t-u).
On the other hand, the probability for a person born at time t-u to be alive at time t (that is, at age u) is 1-F_t-u(u).
See Figure <ref> for illustration.
§.§ Exact distribution of the oldest person's age process
We assume that all birth events are already sampled on (-∞,t] together with the corresponding lifespans.
Then the distribution of Y_t can be computed explicitly for all t∈ using the two-dimensional representation.
For all t∈ the density
h_t(x)=exp(-∫_x^∞λ(t-u)(1-F_t-u(u)) ụ)
λ(t-x)(1-F_t-x(x))
for all x>0 and the point mass at 0
m_t=exp(-∫_0^∞λ(t-u)(1-F_t-u(u)) ụ)
characterize the distribution of Y_t which can be seen as follows.
We mention that the point mass at 0 is negligible in the application.
Similarly to the proof of the transition formula in (<ref>) the event {Y_t<x} for any x>0 is the same as the event that nobody with age at least x is alive at time t.
We express this event in terms of the Poisson process of birth at rate λ(·) thinned by the probability that the person is still alive at time t.
The event {Y_t<x} means that a Poisson process of intensity at time t-u given by λ(t-u)(1-F_t-u(u)) for u≥ x does not have any point in (-∞,t-x] yielding
(Y_t<x)=exp(-∫_x^∞λ(t-u)(1-F_t-u(u)) ụ).
In other words for any u≥ x individuals are born at time t-u at rate λ(t-u).
A person born at time t-u is alive at time t at age u with probability 1-F_t-u(u).
(<ref>)–(<ref>) follow by differentiation in (<ref>) and by taking the x→0 limit.
See Figure <ref> for illustration.
§.§ Homogeneous model
The exact computation of the reign length distribution (see Subsection <ref>) can only be performed in a special case of our general model described in Subsection <ref>.
We introduce this special case as the homogeneous model where individuals are born at the times of a Poisson process of constant rate λ=λ(t)>0 for all t.
The lifespan of individuals are independent and identically distributed with a fixed density f=f_t and cumulative distribution function F=F_t for all t which does not depend on time.
In the homogeneous model the jump distribution given in (<ref>)–(<ref>) simplifies to
j_y(x) =j_y,t(x)=exp(-λ∫_x^y(1-F(u)) ụ)λ(1-F(x)),
a_y =a_y,t=exp(-λ∫_0^y(1-F(u)) ụ).
The distribution of Y_t does not depend on time in this case hence it is a stationary distribution as well.
The formulas for the density of Y_t and point mass at 0 reduce in the homogeneous case to
h(x) =exp(-λ∫_x^∞(1-F(u)) ụ)λ(1-F(x)),
m =exp(-λ∫_0^∞(1-F(u)) ụ)
where the integral ∫_0^∞(1-F(u)) ụ is equal to the expected lifespan.
The equilibrium condition for the homogeneous density h can be written as
h'(x)+h(x)f(x)/1-F(x)-∫_x^∞ h(y)f(y)/1-F(y)j_y(x) ỵ=0.
After differentiation and using the fact that
/x̣j_y(x)=j_y(x)(λ(1-F(x))-f(x)/1-F(x))
one can derive from (<ref>) the second order differential equation
h”(x)+(2f(x)/1-F(x)-λ(1-F(x)))h'(x)
+(2f(x)^2/(1-F(x))^2+f'(x)/1-F(x))h(x)=0.
The point mass m at 0 satisfies
mλ=∫_0^∞ h(y)f(y)/1-F(y)j_y(0) ỵ.
§.§ The peaks process
In the homogeneous model the sequence of peaks in Y_t forms a discrete time Markov chain.
By peak we mean a local maximum of Y_t with value being equal to the lifespan of the last record holder.
Each time the oldest person dies the process Y_t has a peak with a downward jump following it.
Let Z_n denote the age of record holders at which they die which are the values of the peaks of the process Y_t.
The sequence Z_n forms a discrete time Markov chain.
The Markov property follows by the fact that ages at death of previous record holders only give information on people born before the current record holder but transitions depend on the lifespan of the current record holder and that of people born after them.
The stationary density of Z_n is given by
z(x)=f(x)exp(-λ∫_x^∞(1-F(u)) ụ)/∫_0^∞ f(y)exp(-λ∫_y^∞(1-F(u)) ụ)ỵ.
The formula can be seen as follows.
To have a record holder who dies at age x there has to be a person who has lifespan x which gives the factor f(x) in the numerator on the right-hand side of (<ref>).
The exponential factor is by the two-dimensional representation equal to the probability that no people born before the record holder who just died can be alive at the time the record holder dies.
The denominator on the right-hand side of (<ref>) makes z(x) a probability density function.
The density of Z_n can also be characterized by the following description.
It satisfies the integral equation
z(x)=∫_0^∞ z(w)∫_0^min(x,y)j_w(y)f(x)/1-F(y) ỵ ẉ+∫_0^∞ z(w)a_wf(x) ẉ
which comes from the possible transitions of the peak process as follows.
If the previous record holder had a total lifetime w∈[0,∞) then at the death the process Y_t jumps down to some value y at rate j_w(y) or to 0 with probability a_w.
The density of the age at which a person dies who becomes a record holder at age y is f(x)/(1-F(y)).
From the integral equation in (<ref>) one can derive the second order differential equation for the function g(x)=z(s)/f(x) given by
g”(x)-λ(1-F(x))g'(x)+λ f(x)g(x)=0
which is satisfied by g(x)=cexp(-λ∫_x^∞(1-F(u)) ụ) in accordance with (<ref>).
§.§ Reign length distribution
In the homogeneous model, let W_n denote the reign length of the nth record holder, that is, the time length for which this person is the oldest person of the population.
The density of the random reign length is given by
r(w)=∫_0^∞ h(y)f(y+w) ỵ+m∫_0^wf(z)λ e^-λ(w-z)f(z) ẓ/∫_0^∞ h(y)(1-F(y)) ỵ+m.
The density formula in (<ref>) can be derived based on the stationary density of the peaks process given by (<ref>) as follows.
It holds for the density of the reign length that
r(w)=∫_0^∞ z(x)(∫_0^xj_x(y)f(y+w)/1-F(y) ỵ
+a_x∫_0^wf(z)λ e^-λ(w-z) ẓ)x̣
based on the decomposition with respect to the previous value of the peaks process Z_n.
The integral ∫_0^wf(z)λ e^-λ(w-z) ẓ is the density of the convolution of the density f with an independent exponential distribution of parameter λ.
On the right-hand side of (<ref>), one can use the definitions of the density z given by (<ref>) and the homogeneous jump distribution j_x and a_x given by (<ref>).
Then in the numerator after the exchange of the order of integrations in the first term and by using the formula for the stationary distribution given by (<ref>) one gets the numerator of (<ref>).
In the denominator one can use the equality
∫_0^∞ f(y)exp(-λ∫_y^∞(1-F(u)) ụ)ỵ
=∫_0^∞ h(y)(1-F(y)) ỵ+m.
which follows by integration by parts.
Note also that the density is not equal to the remaining reign length of Y_t under the stationary distribution because it would involve the integral ∫_0^∞ h(y)f(y+w)/(1-F(y)) ỵ in place of the first term in the numerator on the right-hand side of (<ref>).
§.§ Entry age parameter
Next we introduce another parameter which we call the entry age and we denote it by E.
As opposed to our original model we consider individuals as being born at age E at a modified birth rate and with a modified lifespan distribution.
As a result we obtain a model to the age of the world's oldest person all values of the entry age parameter E≥0 and we can fit the model parameters with different values of the entry age.
For any value E≥0 of the entry age we denote by λ_E(t) the rate at which people reach the age E at time t, that is,
λ_E(t)=λ(t-E)(1-F_t-E(E))
because the new birth process of rate λ_E(t) is obtained by an inhomogeneous thinning of the original Poisson process of the birth events.
The lifespan distribution of those born at time t with age E becomes the remaining lifetime distribution at age E.
The modified cumulative distribution function and density are given by
F_t^E(x)=F_t-E(x+E)-F_t-E(E)/1-F_t-E(E),
f_t^E(x)=f_t-E(x+E)/1-F_t-E(E).
§ MODEL SPECIFICATION AND PARAMETER FITTING
In Subsection <ref> we specify the general model introduced and discussed in Section <ref>, that is,
we assume that the birth rate parameter increases exponentially in time and that the lifespan distribution is given by the gamma–Gompertz–Makeham distribution with time-dependent parameters.
We provide the details of the computation of the likelihood as a function of the model parameters in Subsection <ref>.
We show the way to maximize the likelihood and how the optimal parameters can be found using the Nelder–Mead method in Subsection <ref>.
With these values of the parameters, the age of the world's oldest person and the reign length of the record holder can be computed as described in Subsection <ref>.
§.§ Model specification: birth rate parameter and lifespan distribution
For the rest of the paper we specifiy our general model described in Section <ref>
to the following choice of the birth rate parameter and of the lifespan distribution.
We choose the value of the entry age to be E=0,30,60 and we fit the model parameters for all three values of E separately.
First we specify the intensity function of the Poisson process of births with an exponential growth in time.
For any of the three values of the entry age E we assume that the birth rate at age E is given by
λ_E(t)=C_Ee^κ_Et
where the numerical values of the parameters C_E and κ_E are obtained
by linear regression of the logarithmic data of newborns, people at age 30 and 60 published by the United Nations since 1950.
We extrapolate the linear regression backwards in time and we use the numerical values shown in Table <ref>.
We assume that the underlying force of mortality is chosen so that the lifespan distribution of individuals follows the gamma–Gompertz () distribution with cumulative distribution function and density
F^_a,b,γ(x) =1-(1+aγ/b(e^bx-1))^-1/γ,
f^_a,b,γ(x) =ae^bx(1+aγ/b(e^bx-1))^-1-1/γ
for x≥0 where a,b,γ are positive parameters.
We mention that the gamma–Gompertz–Makeham () distribution differs from the distribution by the presence of a non-negative extrinsic mortality parameter c which appears as an additive term in the force of mortality.
See (<ref>) for the definition of the distribution.
In our model, we exclude the extrinsic mortality for the following two reasons.
Since the extrinsic mortality becomes irrelevant at high ages and we aim to model the front-end of the death distribution at the oldest-old ages,
we do not expect to obtain a reliable estimate on the extrinsic mortality using the data about the world's oldest person.
On the other hand, as explained later in Subsection <ref>, the likelihood maximization provides unrealistic lifespan distributions even for the model if one tries to optimize in all the parameters at the same time.
In order for the algorithm to result in a distribution close to the actual human lifespan distribution, the number of model parameters had to be decreased.
For our model we suppose that in the lifespan distribution, parameters b=b_E, the rate of aging and γ=γ_E, the magnitude of heterogeneity
are constants over time and that they only depend on the value of the entry age parameter E.
The parameter a, the initial level of mortality at the entry age for individuals born at time t, depends on time given by the exponentially decreasing function
a_E(t)=K_Ee^-α_E(t-2000)
where the exponent α_E and the constant K_E only depends on the entry age E.
The reason for subtracting 2000 in (<ref>) is only technical, the numerical values of the parameters do not become tiny with this definition.
In the model with entry age E, we assume that the birth rate λ_E(t) is given by (<ref>) and we fit the gamma–Gompetz distribution with parameters b_E,γ_E and a_E(t) given by (<ref>) for the modified distribution function F_t^E(x) and density f_t^E(x) in (<ref>).
This means that we search for the best fitting values of the parameters α_E,K_E,b_E,γ_E which results in an approximation of the remaining lifetime distribution at the age E.
§.§ Likelihood calculations
The aim of the maximum likelihood method is to give an estimate to the parameters α_E, K_E, b_E and γ_E for E=0,30,60 by finding those values for which the likelihood of the full sample is the largest.
The sample is obtained from the the historical data on the world's oldest person available in <cit.>.
We transform this information into a list of triples of the form (t_i,y_i,z_i) for i=1,…,n
where t_i is the ith time in the sample when the oldest person dies at age y_i and the new record holder has age z_i at time t_i.
Then the data has to satisfy the consistency relation t_i-z_i=t_i+1-y_i+1 since the two sides express the date of birth of the same person.
In the model with entry age E, the likelihood of the ith data point (t_i,y_i,z_i) given the previous data point is equal to
f_t_i-y_i+E^E(y_i-E)/1-F_t_i-y_i+E^E(z_i-1-E)j^E_y_i,t_i(z_i)
=f_t_i-y_i+E^E(y_i-E)/1-F_t_i-y_i+E^E(z_i-1-E)exp(-∫_z_i^y_iλ_E(t_i-u+E)(1-F_t_i-u+E^E(u-E)) ụ)
λ_E(t_i-z_i+E)(1-F_t_i-z_i+E^E(z_i-E))
for all i=2,3,…,n except for i=1 in which case the 1-F_y_1-t_1+E^E(z_0-E) factor in the denominator is missing.
In (<ref>) above we use the transition probabilities of the model with entry age E given by
j^E_y,t(x)=exp(-∫_x^yλ_E(t-u+E)(1-F_t-u+E^E(u-E)) ụ)λ_E(t-x+E)(1-F_t-x+E^E(x-E))
as a generalization of (<ref>).
The explanation of the left-hand side of (<ref>) is that the person died at time t_i at age y_i had age E at time t_i-y_i+E.
The previous data point ensures that this person has already reached age z_i-1 hence we condition their lifetime distribution on this fact.
The transition probabilities in (<ref>) are obtained similarly to (<ref>) with the difference that a person at age u with u∈[x,y] at time t had age E at time t-u+E.
Note that when computing the likelihood of the full data by multiplying the right-hand side of (<ref>) for different values of i the consistency relation of the data implies that the factor 1-F_t_i-z_i+E(z_i-E) of the ith term cancels with the factor 1-F_t_i+1-y_i+1+E(z_i-E) coming from the (i+1)st term.
Hence the log-likelihood of the full sample is given by
l(α,K,b,γ)
=∑_i=1^n(log f_t_i-y_i+E^E(y_i-E)
-∫_z_i^y_iλ_E(t_i-u+E)(1-F_t_i-u+E^E(u-E)) ụ+logλ_E(t_i-z_i+E))
+log(1-F_t_n-z_n-E^E(z_n+E))
=∑_i=1^n(log f^_Ke^-α(t_i-y_i+E-2000),b,γ(y_i-E)
-∫_z_i^y_iCe^κ(t_i-u+E)(1-F^_Ke^-α(t_i-u+E-2000),b,γ(u-E))ụ)
+log(1-F^_Ke^-α(t_n-z_n+E-2000),b,γ(z_n-E))
+nlog C+∑_i=1^nκ(t_i-z_i+E).
where we suppress the dependence of the parameters α,K,b,γ on the entry age.
Note that the last two terms do not depend on the parameters α,K,b,γ hence we can omit these terms in the maximization of the log-likelihood.
§.§ Likelihood maximization
We implemented the calculation of the log-likelihood function l(α,K,b,γ) given by (<ref>) in Python.
We used numerical integration to obtain the integrals on the right-hand side of (<ref>).
We mention that the general integral formula in (<ref>) could not be used because the parameter a of the gamma–Gompertz distribution in the integrand depends on the integration variable on the right-hand side of (<ref>).
In order to maximize the value of the log-likelihood function l(α,K,b,γ) we applied the Nelder–Mead method
<cit.> which is already implemented in Python.
We mention that initially we used the gamma–Gompertz–Makeham distribution as lifespan distribution, see (<ref>) for the definition, which contains the extra parameter c to be fitted but it turned out that the number of model parameters has to be reduced.
The behaviour of the optimization algorithm in the five parameters α,K,b,c,γ using the gamma–Gompretz–Makeham model was very similar the case of four parameters α,K,c,γ in the gamma–Gompertz model.
Running the optimization in the full set of parameters (α,K,b,c,γ in the gamma–Gompertz–Makeham model or α,K,b,γ in the gamma–Gompertz model), it turned out that after a few rounds the parameter K started to decrease dramatically and reached values below 10^-10.
The resulting lifespan distribution seemed very unrealistic with almost no mortality before the age of 100.
This happened for all values of the entry age E=0,30,60.
We explain this phenomenon by the fact that historical data about the oldest person in the world only gives information about the behaviour of the lifespan distribution between the ages 107 and 123.
The simple optimization in the four parameters α,K,b,γ simultaneously yields an excellent fit for the tail decay of the lifespan distribution with the historical data but the result may be very far from the actual human lifespan.
This would limit the practical relevance of our results.
The mathematical reason for the fact that the four-parameter optimization does not result in a satisfactory approximation to the human lifespan distribution is the following.
In these cases, the optimization procedure diverges to those regimes of the parameter space _+^4 where the corresponding gamma–Gompertz distribution is degenerate.
One can prevent reaching these unrealistic combinations of parameters by reducing the amount of freedom in the optimization.
Hence we specify some of the parameters a priori and we perform the optimization in the remaining ones so that it provides a good fit to the data on the age of the oldest old as well as a realistic lifespan distribution.
We believe that the most robust of the four parameters of the model is b which is the exponent in the time dependence of the mortality rate.
By setting the rate of aging b=0.09 the algorithm gives the optimal triple α,K,γ with the best likelihood which is very stable under changing the initial values of these parameters.
The running time is also very short.
The Nelder–Mead algorithm, being a numerical maximization method, heavily relies on the tolerance parameter, which determines the minimal improvement required for the algorithm to continue running. If this parameter is set too high, the algorithm might stop before reaching the optimum. Conversely, if set too low, the algorithm might take excessively long to converge. To address this, we drew inspiration from dynamic learning rate algorithms used in neural network training and developed the following meta-algorithm.
First, we run the Nelder–Mead optimization. Based on the improvement from the starting point, we dynamically adjust the tolerance factor, similar to how learning rates are modified during neural network training. We then run the optimization again, recalibrating the tolerance factor based on the observed improvement, and repeat the process. This iterative adjustment allows us to get closer to the optimum, a hypothesis supported by our practical experience with this meta-algorithm.
Following this meta-algorithm, only a few calls of the Nelder–Mead method is enough to reach the optimum.
The Python codes for the likelihood calculations as well as the Nelder–Mead optimization implemented to this problem are available in <cit.>.
The numerical values of the resulting parameters for the three choices of the entry age are shown in Table <ref>.
The survival probability functions with the parameters given in Table <ref> for individuals born in 2000 corresponding to the entry age E=0,30,60 are shown on Figure <ref> as a function of the age.
We also computed the optimal values of the parameters α,K,γ for other values of the rate of aging b as a sensitivity analysis.
The resulting parameter values for the choices b=0.11, b=0.13 and b=0.15 are shown in Table <ref>.
We mention as an alternative approach that scaling the parameters could enhance the optimization process, but this requires prior knowledge of the range within which the parameters vary. This range could be determined through our iterative application of the Nelder–Mead algorithm.
§.§ Computation of the oldest person's age and of the reign length
We observe that the model with the parameters in Table <ref> fits well to the titleholder data.
We focus on two statistics of the process in order to support this observation about the comparison:
the age of the world's oldest person and the reign length of the record holder.
In the case of both statistics exact formulas are only available for the homogeneous model introduced in Subsection <ref> where the birth rate is constant as well as the lifespan distribution does not depend on time.
Hence we apply an approximation where the error is negligible compared to the difference from the statistics computed using the data.
In the general model the distribution of the age of the world's oldest person at time t is given by the density h_t(x) in (<ref>) and by the point mass m_t at 0 in (<ref>).
For the numerical computations, we ignore the point mass m_t which is below the round-off error in the numerical results.
The difficulty in computing the mean age of the oldest person at time t is that parameter a of the gamma–Gompertz–Makeham distribution function F_t-u in the exponent of (<ref>) also depends on the integration variable u.
In our approximation we fix the value of the parameter a of the distribution in h_t(x) in (<ref>) to a value which is equal to a_0(t-d) in (<ref>) with some delay d.
The delay d is chosen so that the mean age of the oldest person computed using a_0(t-d) as parameter a for all times in the distribution function in (<ref>) is equal to the same value d.
For a given t, this value of d can be obtained as the fixed point of the contraction map
d↦∫_0^∞ xe^-∫_x^∞λ(t-u)(1-F_t-d(u)) ụλ(t-x)(1-F_t-d(x)) x̣
which provides a reasonable approximation for the mean age of the world's oldest person.
For the comparison with the data and for the prediction, we use the model with entry age E=0.
Hence in (<ref>), the function λ is given in (<ref>) with E=0 and F_t-d is the distribution function with parameters given by the E=0 values in Table <ref> and with a=a_0(t-d) in (<ref>).
This approximation is reasonable because the distribution of the age of the oldest person is highly concentrated.
The fixed point of the map in (<ref>) as the expected age of the world's oldest person can be found in a few steps of iterations.
We show the result on Figure <ref>.
We applied 10 iterations using the fitted model parameters for each year between 1955 and 2019.
By computing the standard deviation of the age of the oldest person as well we obtain the mean and a confidence interval for the age.
The predictions for the age distribution of the world's oldest person in the future shown on Figure <ref>.
We obtained them by using the exact age distribution formula in Subsection <ref> along with the numerical values of the parameters C,κ,α,K,γ given in Tables <ref> and <ref> for entry age E=0.
In our model, the distribution function of the age of the world's oldest person is given in (<ref>) where the distribution function can be substituted with the estimated parameter values at any time.
In this way, the probability of observing an age greater than or equal to Calment's or Knauss' actual age at the time of their death can be computed exactly.
The numerical values are 0.000286 for Calment and it is 0.0116 for Knauss.
The backtesting mentioned in the Results section is performed as follows.
We estimated the best parameter values with entry age 0 based on the reduced data on the world's oldest person between 1955 and 1988 where the ending date is the time when Calment became the world's oldest person.
The resulting parameters α=0.01516,K=0.00002064,γ=0.08413 are numerically not very far from the optimal parameters in Table <ref> but the difference is more visible on Figure <ref>.
The figure shows the model-based mean age and confidence interval for the age of the world's oldest person computed using the full data as well as the data until 1988.
For the reign length of record holders, we again used the expected age at a given time obtained as the fixed point of the iteration in (<ref>).
The numerical value of the expected reign length obtained from the iteration is 1.195 in 1955 and it is 1.188 in 2019.
The empirical value of the reign length is 1.008 computed from the data by dividing the total length of the time interval between 1955 and 2019 by the number of record holders.
§ METHODS
In this section, we provide supplementary information related to the main result of this paper.
We perform explicit computations with the gamma–Gompertz–Makeham model
and we express the integral of the survival function in terms of a hypergeometric function.
The cumulative distribution function and the density of the gamma–Gompertz–Makeham () distribution are given by
F^_a,b,c,γ(x) =1-e^-cx/(1+aγ/b(e^bx-1))^1/γ,
f^_a,b,c,γ(x) =e^-cx/(1+aγ/b(e^bx-1))^1+1/γc(b-aγ)+a(b+cγ)e^bx/b
for x≥0 where a,b,c,γ are positive parameters.
The positivity of parameters implies the finiteness of all moments and, in particular, the convergence of the integral of the survival function
∫_x^∞(1-F^_a,b,c,γ(u)) ụ.
In the homogeneous model, the integral of the survival function appears in the density of the distribution of Y_t in (<ref>) and in the stationary density of the peaks process in (<ref>).
We show below that in the gamma–Gompertz–Makeham model the integral of the survival function can be computed explicitly and it is given by
∫_x^∞(1-F^_a,b,c,γ(u))ụ
=( b/a γ)^1/γe^-(c+b/γ)x/b/γ+c_2F_1(1/γ,1/γ+c/b;1+1/γ+c/b;a γ-b/a γe^-bx)
where _2F_1(a,b;c;z) is the hypergeometric function.
See 15.1.1 in <cit.> for the definition and properties.
We prove (<ref>) based on the following integral representation 15.3.1 in <cit.> of the hypergeometric function
_2F_1(a,b,c,z)=Γ(c)/Γ(b) Γ(c-b)∫_0^1 t^b-1(1-t)^c-b-1(1-tz)^-a ṭ
which holds whenever (c)>(b)>0.
First we prove an identity for complex parameters α,β,δ which satisfy (α+β)>0 and we compute
∫_x^∞e^-β u/(1+δ e^u)^α ụ =1/δ^α∫_x^∞e^-(α+β)u/(1+e^-u/δ)^α ụ
=e^-(α+β)x/δ^α∫_0^1 y^α+β-1(1+e^-xy/δ)^-α ỵ
=e^-(α+β)x/(α+β)δ^α·_2F_1(α,α+β;1+α+β;-e^-x/δ)
where we applied a change of variables y=e^x-u in the second equality above and we applied the hypergeometric identity (<ref>) in the last equality with a=α, b=α+β, c=1+α+β, z=-e^-x/δ together with the observation that with these values of the parameters the prefactor of the integral on the right-hand side of (<ref>) simplifies to α+β.
Note that the condition (c)>(b)>0 for (<ref>) to hold is satisfied by our assumption (α+β)>0 which also makes the integrals in (<ref>) convergent.
Next we show (<ref>) using (<ref>) as follows.
We write
∫_x^∞(1-F^_a,b,c,γ(u)) ụ =1/(1-aγ/b)^1/γ∫_x^∞e^-cu/(1+aγ/b-aγe^bu)^1/γ ụ
=1/(1-aγ/b)^1/γb∫_bx^∞e^-cv/b/(1+aγ/b-aγe^v)^1/γ ṿ
=1/(1-aγ/b)^1/γbe^-(1/γ+c/b)bx/(1/γ+c/b)(aγ/b-aγ)^1/γ_2F_1(1/γ,1/γ+c/b;1+1/γ+c/b;a γ-b/a γe^-bx)
where we applied the change of variables v=bu in the second equality above and we used (<ref>) with α=1/γ, β=c/b, δ=aγ/(b-aγ) and with x replaced by bx.
The right-hand side of (<ref>) simplifies to that of (<ref>).
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§ ACKNOWLEDGEMENTS
We thank Katalin Kovács for some useful advice at an early stage of the project which led to this collaboration.
The work of Cs. Kiss and B. Vető was supported by the NKFI (National Research, Development and Innovation Office) grant FK142124.
B. Vető is also grateful for the support of the NKFI grant KKP144059 “Fractal geometry and applications” and for the Bolyai Research Scholarship of the Hungarian Academy of Sciences.
L. Németh was supported by MaRDI, funded by the Deutsche Forschungsgemeinschaft (DFG), project number 460135501, NFDI 29/1 “MaRDI – Mathematische Forschungsdateninitiative.
§ AUTHOR CONTRIBUTIONS
B. V. initiated the research. Cs. K. and B. V. derived the model, prepared the scripts and figures, and carried out the estimations. Cs. K., L. N. and B. V. analyzed the results and wrote the manuscript.
§ DATA AVAILABILITY
The titleholder data are freely available at https://grg.org/Adams/C.HTM
§ ADDITIONAL INFORMATION
The authors declare no competing interests.
§ TABLES
|
http://arxiv.org/abs/2409.03097v1 | 20240904215744 | Real-time operator evolution in two and three dimensions via sparse Pauli dynamics | [
"Tomislav Begušić",
"Garnet Kin-Lic Chan"
] | quant-ph | [
"quant-ph"
] |
gkc1000@gmail.com
Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA
§ ABSTRACT
We study real-time operator evolution using sparse Pauli dynamics, a recently developed method for simulating expectation values of quantum circuits. On the examples of energy and charge diffusion in 1D spin chains and sudden quench dynamics in the 2D transverse-field Ising model, it is shown that this approach can compete with state-of-the-art tensor network methods. We further demonstrate the flexibility of the approach by studying quench dynamics in the 3D transverse-field Ising model which is highly challenging for tensor network methods.
For the simulation of expectation value dynamics starting in a computational basis state, we introduce an extension of sparse Pauli dynamics that truncates the growing sum of Pauli operators by discarding terms with a large number of X and Y matrices. This is validated by our 2D and 3D simulations. Finally, we argue that sparse Pauli dynamics is not only capable of converging challenging observables to high accuracy, but can also serve as a reliable approximate approach even when given only limited computational resources.
Real-time operator evolution in two and three dimensions via sparse Pauli dynamics
Garnet Kin-Lic Chan
September 9, 2024
==================================================================================
./figures/
§ INTRODUCTION
Numerical simulations of quantum dynamics are essential for our understanding of strongly correlated many-body physics. For large systems and long-time dynamics, exact state-vector or tensor network contraction <cit.> methods become computationally intractable and approximate numerical methods are required. These can be formulated in different pictures, for example, the Schrödinger picture, where the state is evolved, or the Heisenberg picture, where the state is unchanged and time evolution is applied to the observable.
Operator time evolution appears naturally in the dynamics of high-temperature systems <cit.> and the theory of operator spreading <cit.>, and has proven useful in the computation of time-correlation functions <cit.> and out-of-time-order correlations <cit.>. Compared to the Schrödinger picture, working in the Heisenberg picture has the benefit that one can take advantage of the dynamical light-cone structure, i.e., the fact that the support of some relevant observables is initially local and grows in time. Operator time evolution has been the subject of tensor network studies, e.g., those based on matrix-product operators (MPO) <cit.> or projected entangled-pair operators (PEPO) <cit.>, whose performance depends on the degree of operator entanglement in the evolved observable.
Alternative Heisenberg-picture methods have been formulated to take advantage of the sparsity of the observable in the Pauli operator representation.
Under time evolution, local observables spread to a superposition of an increasing number of Pauli operators, and
various strategies to curb the (at worst) exponential growth of the number of Pauli operators have been employed, including stochastic sampling of Pauli paths <cit.> or discarding Pauli paths based on Pauli weights <cit.>, perturbation order with respect to the nearest Clifford transformation <cit.>, Fourier order <cit.>, or combinations thereof <cit.>. These methods have mainly been developed in the context of simulating quantum circuit expectation values, where the number of Pauli operators can be low due to the use of Clifford gates or noise <cit.>.
Here, we consider one of these methods, namely the sparse Pauli dynamics (SPD) <cit.>, which was applied successfully recently to simulate the kicked transverse-field Ising model quantum simulation of Ref. <cit.>. There, we showed that it can simulate the expectation values faster than the quantum device to reach comparable accuracy and that, given more time, can produce more accurate results than the quantum processor. These simulations also demonstrated that SPD performs well compared to other classical simulation strategies based on state <cit.> and operator evolution <cit.>. In this work, we analyze its performance in simulating real-time dynamics on 1D spin chains, where we compare it to MPO dynamics including its recent dissipation assisted variant <cit.>, as well as in the transverse-field Ising model on square (2D) and cubic (3D) lattices, where we use available 2D tensor network simulation benchmarks. For the latter, we introduce a modification of the original method that further reduces the computational cost of simulating time-dependent expectation values when the initial state is a computational basis state (e.g., | 0^⊗ n⟩).
§ REAL-TIME SPARSE PAULI DYNAMICS
In SPD we write the observable operator
O = ∑_P ∈𝒫 a_P P
as a sum of Pauli operators P with complex coefficients a_P, where 𝒫 is a set of Pauli operators that contribute to O. The central part of our algorithm is the action of a Pauli rotation operator U_σ(θ) = exp(-i θσ / 2) on the observable (<ref>):
Õ = U_σ(θ)^† O U_σ(θ) = ∑_P ∈𝒫_C a_P P + ∑_P ∈𝒫_A (a_P cos(θ) P + i a_P sin(θ) σ· P),
which follows from
U_σ(θ)^† P U_σ(θ) =
P, [P, σ] = 0,
cos(θ) P + i sin(θ) σ· P, {P, σ} = 0,
where [·, ·] denotes a commutator and {·, ·} an anticommutator. In Eq. (<ref>), 𝒫_C (𝒫_A) is a set of Pauli operators in 𝒫 that commute (anticommute) with σ. The right-hand side of Eq. (<ref>) can be brought into the form of Eq. (<ref>) by identifying which σ· P already exist in 𝒫 and which have to be added to represent the rotated observable Õ. In general, the number of Pauli operators that we have to store, N = |𝒫|, grows exponentially with the number of unitary Pauli rotation operators applied. To limit the growth of N, we replace the exactly rotated observable Õ by
Õ_δ = Π_δ (Õ),
where Π_δ acts by discarding all Pauli operators P with |a_P| < δ. Here, the threshold δ defines the approximation error, i.e., δ = 0 corresponds to exact dynamics. In practice, the goal is to converge the simulation with respect to this tunable parameter. Additional implementation details can be found in Appendix <ref>.
For real-time dynamics under a Hamiltonian
H = ∑_j c_j H_j,
where c_j are real coefficients and H_j are Pauli operators, we replace the exact time-evolution operator U_Δ t = exp(- i H Δ t), corresponding to a time step Δ t, by the first-order Trotter splitting formula
U_Δ t≈∏_j exp(- i c_j Δ t H_j).
Now the real-time evolution takes the form of applying multiple Pauli rotation gates, which allows us to use the SPD method as defined above. The size of the time step determines not only the Trotter error in Eq. (<ref>) but also the truncation error in SPD. Specifically, for small values of threshold δ and time step Δ t, the error is a function of their ratio δ / Δ t. To show this, let us consider one substep of the dynamics in which we apply one Pauli rotation operator exp(- i c_j Δ t H_j) to the observable (<ref>), the result of which is shown in Eq. (<ref>) with θ = 2 c_j Δ t. For a sufficiently small time step, we can expand Eq. (<ref>) up to first order in Δ t,
Õ≈∑_P ∈𝒫_C a_P P + ∑_P ∈𝒫_A (a_P P + 2 i a_P c_j Δ t σ· P) = O + 2 i c_j Δ t ∑_P ∈𝒫_A a_P H_j · P.
For small threshold δ, we can assume that the threshold-based truncation will mainly discard Pauli operators H_j · P that do not already exist in O. The condition under which they are discarded reads |2 c_j Δ t a_P| < δ, i.e., |a_P| < (δ / Δ t) / 2|c_j|. Consequently, the truncation error and the number of Pauli operators depend only on the ratio δ / Δ t. Since the number of Pauli operators determines the computational cost per time step, it is generally preferred to use a larger time step—at fixed δ / Δ t, the cost per time step is constant, but a larger time step requires less steps to reach a given total time. In turn, the size of the time step is limited by the Trotter error, which must be validated before converging the calculation with respect to truncation threshold δ.
§ RESULTS
§.§ Spin and energy diffusion constants in 1D chains
We begin with the computation of diffusion constants of conserved densities in spin chains of length L at infinite temperature. The diffusion constant <cit.>
D = 1/2∂/∂ t d^2(t)
is defined through the time-derivative of the mean-square displacement
d^2(t) = ∑_j C_j(t) j^2 - ( ∑_j C_j(t) j )^2,
where
C_j(t) = Tr[q_j q_L+1/2(t)]/{∑_j Tr[q_j q_L+1/2(0)] }
are the dynamical correlations between the operators q localized at the central site (L+1)/2 and sites j. These operators represent conserved densities in the sense that ∑_j q_j(t) = ∑_j q_j(0), which leads to ∑_j C_j(t) = ∑_j C_j(0) = 1.
This problem is naturally formulated in the Heisenberg picture and we employ SPD to numerically evolve q_L+1/2(t) in time. Within the sparse Pauli representation of the operator, it is also easy to evaluate the overlap with another Pauli operator (or a sum of Pauli operators) q_j (see Appendix <ref>). Since the conservation laws are not strictly obeyed by the non-unitary truncation scheme employed in SPD, we replace the denominator in Eq. (<ref>) by ∑_j Tr[q_j q_L+1/2(t)], i.e., we renormalize the correlations at post-processing.
§.§.§ Models
Below, we introduce the examples of Ref. <cit.>, which were studied using dissipation assisted operator evolution (DAOE) combined with a MPO representation of the operator. There, the authors introduced an artificial dissipator that reduces the entanglement of the said MPO and demonstrated that this computational strategy is well founded for the simulation of diffusive transport at high temperature.
The first example is the one-dimensional tilted-field Ising model <cit.>
H = ∑_j=1^L-1 Z_j Z_j+1 + ∑_j=1^L (1.4 X_j + 0.9045 Z_j)
with open boundary conditions, while the conserved densities are the local energies
q_j = 1/2 Z_j-1 Z_j + 1/2 Z_j Z_j+1 + 1.4 X_j + 0.9054 Z_j.
L=51 is the number of sites in the chain.
The second model is the XX-ladder Hamiltonian <cit.>
H = 1/4∑_j=1^L-1∑_a=1,2 (X_j,a X_j+1, a + Y_j,a Y_j+1,a) + 1/4∑_j=1^L (X_j,1 X_j, 2 + Y_j,1 Y_j,2),
where the total spin is a conserved property and we consider the diffusion of q_j = (Z_j,1 + Z_j,2)/2 along the chain of length L=41 (number of sites is n=2L=82).
For these two models, we simulated the mean-square displacement (<ref>) for a total time of t = 20, with a time step of Δ t = 0.02, unless stated otherwise. The diffusion constant was computed by linear regression of d^2(t) between t=10 and t=20.
§.§.§ Numerical results
Figure <ref> shows the mean-square displacements for the tilted-field Ising (panel A) and XX-ladder models (panel C), simulated using SPD with different values of the threshold δ ranging from 2^-18 to 2^-13. We can observe how d^2(t) converges onto a straight line at large t as we reduce the threshold and make the simulations more accurate. Figures <ref>B, D plot the corresponding diffusion constant as a function δ/Δ t for two sets of data, one using Δ t = 0.01 and the other using Δ t = 0.02. For this property, the results are amenable to an extrapolation in δ→ 0, and we find D ≈ 1.4 for the tilted-field Ising chain (same value was reported in Ref. <cit.>), while D ≈ 0.94 for the XX-ladder (compared to D ≈ 0.95 of Ref. <cit.> and D ≈ 0.96-0.98 reported in Ref. <cit.>). The plots of the diffusion constant also reveal the scaling relationship discussed in Sec. <ref>. Namely, as the threshold is reduced, the simulations using different δ and Δ t but the same δ / Δ t become closer to each other. In addition, we can numerically verify that using a larger time step reduces the computational cost. For example, the two points in Fig. <ref>D at fixed δ / Δ t = 2^-18/0.02 ≈ 0.00019 take around 84h (Δ t = 0.01, blue circle) and 43h (Δ t = 0.02, orange square) to simulate on 6 cores.
To illustrate that the above problems are non-trivial to simulate, we present a comparison between SPD and matrix-product operator (MPO) dynamics without dissipation assistance (Fig. <ref>, see details in Appendix <ref>), a standard method for 1D dynamics.
Using a MPO bond dimension up to χ=2^9 we find that the MPO simulations are still far from the exact results even for shorter chains (L=9-21) of the tilted-field Ising model, whereas SPD is more accurate already at rather large values of δ. Similarly, Ref. <cit.> reported that the MPO dynamics of the L=51 chain with bond dimension χ = 2^9 diverges from the exact result already at t=8. For the same system, SPD is visually well converged up to t=15 already at δ = 2^-15.
In the remainder of this Section, we analyze the operator evolution in terms of contributions from Pauli operators of different weights. Here, the Pauli weight is defined as the number of non-identity Pauli matrices in a Pauli operator. To this end, for any operator O=∑_P ∈𝒫 a_P P, we introduce F_m = ∑_P ∈𝒫_m |a_P|^2, where 𝒫_m ⊆𝒫 is a subset of Pauli operators with Pauli weight m. Then the sum F = ∑_m=1^n F_m is constant for unitary dynamics and equal to the square of the Frobenius norm, i.e., F = Tr(O^†O)/2^n. Figure <ref>A shows the breakdown of F into individual components F_m (up to m=12) for the example of tilted-field Ising model [Eqs. (<ref>) and (<ref>)], demonstrating how the dynamics evolves initially low-weight Pauli operators into a sum of operators with higher weights. After some time, F of the operator evolved with SPD deviates from its initial value because of threshold-based truncation. This truncation appears to affect high-weight Pauli operators more than the low-weight Paulis. This is confirmed in Fig. <ref>B, where we show F_m contributions for m up to 5. As shown in previous works <cit.>, Pauli operators corresponding to the local conserved quantity (here m=1 and m=2 for the local energy (<ref>)) obey the long-time scaling F_m ∼ t^-1/2, whereas other Pauli operators (here m>2) obey F_m ∼ t^-3/2 in the long-time limit.
This observation motivates methods that truncate the time-evolved operator based on Pauli weights, including the dissipation assisted operator evolution (DAOE) approach <cit.>, low-weight simulation algorithm (LOWESA) <cit.> and observable's back-propagation on Pauli paths (OBPPP) <cit.>, two approaches that were originally developed for the simulation of noisy quantum circuits, and the restricted state space approximation <cit.>, introduced in the context of simulating nuclear magnetic resonance experiments. Our results indicate that SPD can take advantage of such behavior without explicitly truncating the sum based on Pauli weights because high-weight Pauli operators appear with small coefficients that do not meet the threshold criterion.
§.§ Time-dependent expectation values
§.§.§ X-truncated sparse Pauli dynamics
In the examples above, we focused on infinite-temperature time-correlation functions of a time-evolved operator with a local, low-weight Pauli operator. Here, we consider expectation values of the form ⟨ O ⟩_t = ⟨ 0^⊗ n | O(t) | 0^⊗ n⟩, where high-weight Pauli operators can contribute as long as they are composed only of identity and Z matrices (i.e., if they are diagonal in the computational basis). Default Heisenberg evolution does not account for the information about the state over which we take the expectation value but rather treats all Pauli operators equally. Within the framework of SPD, this means that we keep a large number of Pauli operators, of which only a fraction contribute to the observable.
To further truncate the number of Pauli operators without introducing large errors in the expectation value ⟨ O ⟩_t, we propose to discard Pauli operators composed of more than M X or Y Pauli matrices. We refer to the number of X/Y matrices as the X-weight of a Pauli operator and we call this additional truncation scheme X-truncated SPD (xSPD). The truncation introduces the additional assumption that for certain (short) times, there is limited operator backflow from high X-weight Paulis to the manifold of Z-type Pauli operators. For each calculation, we test the value of M to ensure that the error introduced by the X-truncation scheme is sufficiently small (for example, smaller than the target convergence criterion). The X-truncation is applied only every T steps of the dynamics to limit the impact on the accuracy. In our calculations, we fixed T to 5 steps. Finally, we note here that alternatives to our hard M cutoff could also be considered, such as introducing an artificial dissipator based on the Pauli's X-weight that would be similar in spirit to DAOE <cit.>.
In the following, we apply this modification of the original SPD method to dynamics in the 2D (square lattice) and 3D (cubic lattice) transverse-field Ising models described by the Hamiltonian
H = -∑_⟨ j k ⟩ X_j X_k - h ∑_j=1^L Z_j,
where the first sum runs over nearest neighbors on the lattice with open boundary conditions and h controls the magnitude of the field. We consider the time-dependent magnetization ⟨ Z⟩_t = ⟨ 0^⊗ n| Z_0(t) |0^⊗ n⟩, where Z_0 denotes the Z Pauli operator on the central site. Physically this corresponds to the magnetization induced after a sudden quench from infinite h to a finite value of h.
In this setting, the first-order Trotter splitting
U^(1)(t) = [e^ i Δ t ∑_⟨ j k ⟩ X_j X_k e^ i h Δ t ∑_j=1^L Z_j]^K,
where K = t/Δ t is the number of time steps, is equivalent to the second-order splitting
U^(2)(t) = [e^ i 1/2 h Δ t ∑_j=1^L Z_j e^ i Δ t ∑_⟨ j k ⟩ X_j X_k e^ i 1/2 h Δ t ∑_j=1^L Z_j]^K
= e^ i 1/2 h Δ t ∑_j=1^L Z_j U^(1)(t) e^ - i 1/2 h Δ t ∑_j=1^L Z_j
because Z-Pauli rotations commute with the observable and apply only a phase to the initial state.
§.§.§ 2D transverse-field Ising model
The quantum quench dynamics of magnetization in the 2D transverse-field Ising model has been studied by means of infinite projected entangled pair state (iPEPS) <cit.> and neural network quantum state <cit.> calculations. While the iPEPS simulations correspond to dynamics in the thermodynamic limit, neural network simulations were performed on a 10 × 10 lattice, which was shown to be sufficiently large to exhibit negligible finite-size effects <cit.>. In our simulations, we used an 11 × 11 square lattice. We set h=h_c, where h_c = 3.04438(2) corresponds to the quantum critical point <cit.>, and simulate the magnetization up to t = 0.92, where we can compare our results to different update schemes used in iPEPS simulations, namely the full update (FU) <cit.>, neighborhood tensor update (NTU) <cit.>, and gradient tensor update (GTU) <cit.>.
Figure <ref> shows the convergence of xSPD with respect to the threshold δ. As expected, the method converges quickly at short times but requires small values of δ to converge the values at longer times. Our most accurate simulation agrees well with FU and GTU iPEPS results, and shows some deviation from the NTU scheme at the end of the simulation time.
We note that although the NTU iPEPS calculation corresponds to the largest bond dimension amongst the iPEPS data, the accuracy of truncation is also believed to be less than for the FU iPEPS simulation, thus the relative accuracy of the different reference iPEPS data is unclear.
The disagreement between the two smallest δ xSPD simulations at the end of the simulation time is only 0.002 (0.007 over the 3 smallest δs) which provides an estimate of the threshold error. This threshold error is comparable to the estimated Trotter and X-truncation errors (discussed below).
In this example, we employed a time step of Δ t = 0.04 and set the X-truncation parameter to M=5. To validate this choice of parameters, we analyze the associated errors in Fig. <ref>. Specifically, for the time step (Fig. <ref>A), we set δ / Δ t = 2^-19/0.01 = 1.90734 × 10^-4 and compute the observable using three different time steps. We estimate that the Trotter error is below 0.003 within the simulated time of t=0.92. Similarly, we perform SPD and xSPD simulations with varying values of M, using δ=2^-20. We estimate that the error due to M=5 X-truncation is about 0.003. In contrast, employing M=7 would lead to almost no error but also limited computational savings, while M=3 produces an error that is greater than our convergence target of less than 0.01. Note that due to symmetry, all Pauli operators appear with an even X-weight, which is why we only consider odd values of M.
Regarding the computational cost, the most accurate simulation at δ = 2^-23 generated up to 8.5 billion Pauli operators, used over 1TB of memory, and took around 36 hours on 16 CPU cores. For comparison, the least accurate simulation shown in Fig. <ref>, corresponding to δ = 2^-18, generated at most 84 million Pauli operators in 32 minutes on 16 CPU cores.
§.§.§ 3D transverse-field Ising model
As our final example, we present the quench dynamics of magnetization for spins on a simple cubic lattice with L=11 and n=L^3=1331 sites. Here we consider two values of h, h=1 (weak field) and h=5.15813(6) (critical point).
In the case of h=1, using Δ t = 0.04 and M=7, we could run the dynamics up to t=1 with thresholds as low as 2^-19 (see Fig. <ref>). The most accurate simulations (δ≤ 2^-17) agree to within ≈ 0.02, which is comparable to the estimated time step (Trotter) and X-truncation errors (see Fig. <ref>A, C in Appendix <ref>). Interestingly, even the fastest, least-accurate simulation (δ=2^-14) exhibits “qualitative accuracy”, i.e., recovers the general trend of the most accurate available result. We ascribe this to the fact that sparse Pauli representation can easily reproduce the dynamics dominated by few Pauli operators. However, the method struggles to include small contributions from many Pauli operators generated by the time evolution. For example, while only about 10^6 Pauli operators are generated at threshold δ=2^-14 after 1 time unit, around 7 × 10^8 Pauli operators are generated during the same dynamics with a threshold of δ=2^-19. Yet, the difference in the observable appears limited to around 0.04.
The system with h=5.158136 (Fig. <ref>B) was propagated up to t=0.6 using Δ t= 0.02 and M=5. For this choice of parameters, the Trotter and X-truncation errors are estimated to be below 0.005 (see Fig. <ref>B, D in Appendix <ref>). Because of the shorter dynamics we could use a smaller value of the X-truncation parameter M compared to the weak-field case. The results, using thresholds as low as δ=2^-20, are converged to below 0.01 for times t<0.5, after which our most accurate simulations begin to deviate from each other.
In general, these 3D calculations are expected to pose challenges for tensor network techniques, which for a fixed bond dimension show an exponential scaling with the connectivity of the lattice (assuming site tensors with a number of bonds equal to the number of neighbours). Although SPD does not show this exponential scaling, the 3D simulations for a given side-length L are still more costly than the 2D ones, primarily because the number of sites n=L^3 is a factor of L larger than in the 2D case.
For these reasons, we cannot simulate as many Pauli operators as in the 2D case. Specifically, our memory budget of about 1.5TB is reached already with less than 10^9 Pauli operators, an order of magnitude less than in our 2D square lattice simulations. With this number of Pauli operators, the computation with h=1 takes around 73 h on 16 CPU cores, about two times longer than our most accurate 2D calculation. Nonetheless, the relative feasibility of these simulations illustrate the kinds of systems that can be (approximately) studied by SPD dynamics that would otherwise be challenging to consider.
§.§.§ Computational savings from X-truncation
We now analyze the savings due to the X-truncation scheme. Figures <ref>A, B show the number of Pauli operators as a function of time for 2D and 3D simulations. While the number of Pauli operators in xSPD is suppressed by the X-truncation, the number of Z-type operators (Pauli operators composed only of Z and identity matrices) is almost the same as in SPD. Since the total cost of the computation is proportional to the number of all Pauli operators N, we can use the ratio N_ SPD/N_ xSPD to quantify the computational savings (Fig. <ref>C). We observe a factor of 6 decrease in the number of Pauli operators in the 2D case, and up to a factor of 5 for the 3D simulation. Such savings allow a factor of 2–4 lower thresholds compared to the original SPD. For example, within a budget of a few days and 1.5 TB of memory, the most accurate SPD calculation we could run for the 2D system would be limited to δ=2^-21, which is not converged with respect to threshold at the longest simulated times (see Fig. <ref>). In contrast, with xSPD we could afford to run the same simulation with δ=2^-23.
§ CONCLUSION
To conclude, we have presented numerical simulations of real-time operator evolution with SPD and its modified version, xSPD. We have shown that, in systems for which reference data is available, the performance of these methods is at the level of state-of-the-art tensor network approaches, while the flexibility and simplicity of SPD allows for applications to dynamics problems where tensor network approaches have yet to make an impact, such as in 3D lattices. For the studied systems, we found that we can obtain very accurate results either by converging the observables in the case of short-time, 2D and 3D transverse field Ising model dynamics, or by extrapolating to zero threshold for the long-time diffusion coefficients in 1D chains. Apart from reaching quantitatively converged results for these challenging examples of time-dependent observables, we also found that SPD could serve as a practical method for computing qualitatively accurate results. In this respect, our work motivates further research into using sparse Pauli representations for real-time quantum dynamics.
We thank Jacek Dziarmaga for sharing the iPEPS simulation data presented in Fig. <ref> and Huanchen Zhai for help with setting up MPO simulations presented in Fig. <ref>.
The authors were supported by the US Department of Energy, Office of Science, Office of Advanced Scientific Computing Research and Office of Basic Energy Sciences, Scientific Discovery through Advanced Computing (SciDAC) program under Award Number DE-SC0022088. TB acknowledges financial support from the Swiss National Science Foundation through the Postdoc Mobility Fellowship (grant number P500PN-214214). GKC is a Simons Investigator in Physics. Computations presented here were conducted in the Resnick High Performance Computing Center, a facility supported by Resnick Sustainability Institute at the California Institute of Technology.
§ ADDITIONAL DETAILS OF SPD IMPLEMENTATION AND WORKING EQUATIONS
SPD is implemented as described in Ref. <cit.>. Briefly, a sum of N Pauli operators is stored in the form of three arrays: an array of N complex coefficients a, an array of N integer phases φ, and a N × 2n_64 array of 64-bit unsigned integers that stores two bitstrings x and z for each Pauli operator:
O = ∑_j=0^N-1 a_j (-i)^φ∏_k=0^n-1 Z_k^z_jk X_k^x_jk.
The number of unsigned integers needed to store n bits is n_64 = ⌈ n ⌉. Pauli operators are sorted using lexicographic ordering on the bitstrings. In this way, we can find the position j of a given Pauli operator in the sum—or the position at which to insert a new Pauli operator so that the ordering is preserved—in 𝒪(log N) time. Similarly, deleting Pauli operators preserves the order trivially.
Apart from searching, inserting, and deleting Pauli operators, other key operations on this sparse representation of a sum of Pauli operators include identifying which Pauli operators in the sum anticommute with a given Pauli operator and multiplying the sum of Pauli operators by a Pauli. For the anticommutation of Pauli operators A and B, we have
A anticommutes with B = z_A · x_B - x_A · z_B.
Here the multiplications on the right-hand side correspond to the AND logical operator between bits and additions to the XOR logical operator (addition in ℤ_2). The product of two Pauli operators C = A B is given by
(z_C, x_C) = (z_A + z_B, x_A + x_B),
φ_C = φ_A + φ_B + 2 z_A · x_B,
a_C = a_A a_B.
In Sec. <ref>, we also introduced an overlap (inner product) between sums of Pauli operators Tr[O_1^† O_2]/2^n represented in the sparse Pauli format described above. Let us assume that N_1 < N_2, i.e., that O_1 has fewer Pauli operators than O_2. Then, we can search for all Pauli operators of O_1 in O_2 in 𝒪(N_1 log N_2) time and the overlap is
1/2^nTr[O_1^† O_2] = ∑_j (-i)^φ_2, k[j] - φ_1, j a_1, j^∗ a_2, k[j],
where the sum runs over Pauli operators in O_1 that were found in O_2 and k[j] is the index of the j-th found Pauli in O_2. The expectation over the all-zero state, as needed in Sec. <ref>, can be computed as
⟨ 0^⊗ n | O | 0^⊗ n⟩ = ∑_j a_j (-i)^φ_j,
where the sum runs over Z-type Pauli operators (for which all bits in x_j are 0). Finally, the X-weight of an operator is computed as the number of set bits in the corresponding x array.
For convenience, our implementation interfaces to Qiskit <cit.> for setting up the calculations. Specifically, it converts Qiskit's SparsePauliOp into our representation described above that is then used in the simulations.
§ DETAILS OF MPO SIMULATIONS
MPO simulations were performed using the time-dependent density matrix renormalization group (TD-DMRG) method, as implemented in the Block2 code <cit.>. After constructing the MPO of the observable at time zero, we convert it to an MPS | O with a doubled number of sites, i.e., ⟨ i_1 i_2 … | O | j_1 j_2 …⟩ / 2^n/2 = i_1 j_1 i_2 j_2 … | O. The sites of the MPS are ordered so that the two physical legs of a single site in the MPO appear on neighboring sites in the MPS. The Liouvillian superoperator L |O≡ [H, O] that governs the dynamics of the observable is then represented as an MPO in the extended Hilbert space with twice as many sites. Specifically, each Pauli operator in the Hamiltonian corresponds to a sum of two Pauli operators in the Liouvillian. For single-site Pauli operators σ_i ∈{I, X, Y, Z} at site i, we have σ_2i - σ_2i+1, while the nearest-neighbor interaction terms σ_i σ_i+1 correspond to σ_2iσ_2i+2 - σ_2i+1σ_2i+3 in the superoperator picture.
The initial observable MPS is propagated with the Liouvillian MPO using the time-step-targeting method <cit.> and a time step of Δ t = 0.04. The correlation functions of the form Tr[O_1^† O_2]/2^n, used in Eq. (<ref>), were evaluated as the inner product O_1 | O_2 of the two matrix product states.
§ TIME STEP AND X-TRUNCATION PARAMETERS IN 3D SIMULATIONS
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"Lei Zhao",
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"A. Zhemchugov",
"B. Zheng",
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"A. N. Zhu",
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"K. Zhu",
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"L. Zhu",
"L. X. Zhu",
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"J. H. Zou",
"J. Zu"
] | hep-ex | [
"hep-ex"
] |
§ ABSTRACT
In the effective field theory, the massless dark photon γ' can only couple with the Standard Model particle through operators of dimension higher than four, thereby offering a high sensitivity to the new physics energy scale.
Using 7.9 fb^-1 of e^+e^- collision data collected at √(s)=3.773 GeV with the BESIII detector at the BEPCII collider, we measure the effective flavor-changing neutral current coupling of cuγ' in D^0→ωγ' and D^0→γγ' processes to search for the massless dark photon.
No significant signals are observed, and the upper limits at the 90% confidence level on the massless dark photon branching fraction are set to be 1.1×10^-5 and 2.0×10^-6 for D^0→ωγ' and D^0→γγ', respectively. These results provide the most stringent constraint on the new physics energy scale associated with cuγ' coupling in the world, with the new physics energy scale related parameter |ℂ|^2+|ℂ_5|^2<8.2×10^-17^-2 at the 90% confidence level, playing a unique role in the dark sector search with the charm sector.
-0.2cm
-0.2cm
Searching for the massless dark photon in c→ uγ'
M. Ablikim^1, M. N. Achasov^4,c, P. Adlarson^76, O. Afedulidis^3, X. C. Ai^81, R. Aliberti^35, A. Amoroso^75A,75C, Y. Bai^57, O. Bakina^36, I. Balossino^29A, Y. Ban^46,h, H.-R. Bao^64, V. Batozskaya^1,44, K. Begzsuren^32, N. Berger^35, M. Berlowski^44, M. Bertani^28A, D. Bettoni^29A, F. Bianchi^75A,75C, E. Bianco^75A,75C, A. Bortone^75A,75C, I. Boyko^36, R. A. Briere^5, A. Brueggemann^69, H. Cai^77, X. Cai^1,58, A. Calcaterra^28A, G. F. Cao^1,64, N. Cao^1,64, S. A. Cetin^62A, X. Y. Chai^46,h, J. F. Chang^1,58, G. R. Che^43, Y. Z. Che^1,58,64, G. Chelkov^36,b, C. Chen^43, C. H. Chen^9, Chao Chen^55, G. Chen^1, H. S. Chen^1,64, H. Y. Chen^20, M. L. Chen^1,58,64, S. J. Chen^42, S. L. Chen^45, S. M. Chen^61, T. Chen^1,64, X. R. Chen^31,64, X. T. Chen^1,64, Y. B. Chen^1,58, Y. Q. Chen^34, Z. J. Chen^25,i, Z. Y. Chen^1,64, S. K. Choi^10, G. Cibinetto^29A, F. Cossio^75C, J. J. Cui^50, H. L. Dai^1,58, J. P. Dai^79, A. Dbeyssi^18, R. E. de Boer^3, D. Dedovich^36, C. Q. Deng^73, Z. Y. Deng^1, A. Denig^35, I. Denysenko^36, M. Destefanis^75A,75C, F. De Mori^75A,75C, B. Ding^67,1, X. X. Ding^46,h, Y. Ding^40, Y. Ding^34, J. Dong^1,58, L. Y. Dong^1,64, M. Y. Dong^1,58,64, X. Dong^77, M. C. Du^1, S. X. Du^81, Y. Y. Duan^55, Z. H. Duan^42, P. Egorov^36,b, Y. H. Fan^45, J. Fang^59, J. Fang^1,58, S. S. Fang^1,64, W. X. Fang^1, Y. Fang^1, Y. Q. Fang^1,58, R. Farinelli^29A, L. Fava^75B,75C, F. Feldbauer^3, G. Felici^28A, C. Q. Feng^72,58, J. H. Feng^59, Y. T. Feng^72,58, M. Fritsch^3, C. D. Fu^1, J. L. Fu^64, Y. W. Fu^1,64, H. Gao^64, X. B. Gao^41, Y. N. Gao^46,h, Yang Gao^72,58, S. Garbolino^75C, I. Garzia^29A,29B, L. Ge^81, P. T. Ge^19, Z. W. Ge^42, C. Geng^59, E. M. Gersabeck^68, A. Gilman^70, K. Goetzen^13, L. Gong^40, W. X. Gong^1,58, W. Gradl^35, S. Gramigna^29A,29B, M. Greco^75A,75C, M. H. Gu^1,58, Y. T. Gu^15, C. Y. Guan^1,64, A. Q. Guo^31,64, L. B. Guo^41, M. J. Guo^50, R. P. Guo^49, Y. P. Guo^12,g, A. Guskov^36,b, J. Gutierrez^27, K. L. Han^64, T. T. Han^1, F. Hanisch^3, X. Q. Hao^19, F. A. Harris^66, K. K. He^55, K. L. He^1,64, F. H. Heinsius^3, C. H. Heinz^35, Y. K. Heng^1,58,64, C. Herold^60, T. Holtmann^3, P. C. Hong^34, G. Y. Hou^1,64, X. T. Hou^1,64, Y. R. Hou^64, Z. L. Hou^1, B. Y. Hu^59, H. M. Hu^1,64, J. F. Hu^56,j, Q. P. Hu^72,58, S. L. Hu^12,g, T. Hu^1,58,64, Y. Hu^1, G. S. Huang^72,58, K. X. Huang^59, L. Q. Huang^31,64, X. T. Huang^50, Y. P. Huang^1, Y. S. Huang^59, T. Hussain^74, F. Hölzken^3, N. Hüsken^35, N. in der Wiesche^69, J. Jackson^27, S. Janchiv^32, J. H. Jeong^10, Q. Ji^1, Q. P. Ji^19, W. Ji^1,64, X. B. Ji^1,64, X. L. Ji^1,58, Y. Y. Ji^50, X. Q. Jia^50, Z. K. Jia^72,58, D. Jiang^1,64, H. B. Jiang^77, P. C. Jiang^46,h, S. S. Jiang^39, T. J. Jiang^16, X. S. Jiang^1,58,64, Y. Jiang^64, J. B. Jiao^50, J. K. Jiao^34, Z. Jiao^23, S. Jin^42, Y. Jin^67, M. Q. Jing^1,64, X. M. Jing^64, T. Johansson^76, S. Kabana^33, N. Kalantar-Nayestanaki^65, X. L. Kang^9, X. S. Kang^40, M. Kavatsyuk^65, B. C. Ke^81, V. Khachatryan^27, A. Khoukaz^69, R. Kiuchi^1, O. B. Kolcu^62A, B. Kopf^3, M. Kuessner^3, X. Kui^1,64, N. Kumar^26, A. Kupsc^44,76, W. Kühn^37, L. Lavezzi^75A,75C, T. T. Lei^72,58, Z. H. Lei^72,58, M. Lellmann^35, T. Lenz^35, C. Li^47, C. Li^43, C. H. Li^39, Cheng Li^72,58, D. M. Li^81, F. Li^1,58, G. Li^1, H. B. Li^1,64, H. J. Li^19, H. N. Li^56,j, Hui Li^43, J. R. Li^61, J. S. Li^59, K. Li^1, K. L. Li^19, L. J. Li^1,64, L. K. Li^1, Lei Li^48, M. H. Li^43, P. R. Li^38,k,l, Q. M. Li^1,64, Q. X. Li^50, R. Li^17,31, S. X. Li^12, T. Li^50, W. D. Li^1,64, W. G. Li^1,a, X. Li^1,64, X. H. Li^72,58, X. L. Li^50, X. Y. Li^1,64, X. Z. Li^59, Y. G. Li^46,h, Z. J. Li^59, Z. Y. Li^79, C. Liang^42, H. Liang^1,64, H. Liang^72,58, Y. F. Liang^54, Y. T. Liang^31,64, G. R. Liao^14, Y. P. Liao^1,64, J. Libby^26, A. Limphirat^60, C. C. Lin^55, C. X. Lin^64, D. X. Lin^31,64, T. Lin^1, B. J. Liu^1, B. X. Liu^77, C. Liu^34, C. X. Liu^1, F. Liu^1, F. H. Liu^53, Feng Liu^6, G. M. Liu^56,j, H. Liu^38,k,l, H. B. Liu^15, H. H. Liu^1, H. M. Liu^1,64, Huihui Liu^21, J. B. Liu^72,58, J. Y. Liu^1,64, K. Liu^38,k,l, K. Y. Liu^40, Ke Liu^22, L. Liu^72,58, L. C. Liu^43, Lu Liu^43, M. H. Liu^12,g, P. L. Liu^1, Q. Liu^64, S. B. Liu^72,58, T. Liu^12,g, W. K. Liu^43, W. M. Liu^72,58, X. Liu^38,k,l, X. Liu^39, Y. Liu^38,k,l, Y. Liu^81, Y. B. Liu^43, Z. A. Liu^1,58,64, Z. D. Liu^9, Z. Q. Liu^50, X. C. Lou^1,58,64, F. X. Lu^59, H. J. Lu^23, J. G. Lu^1,58, X. L. Lu^1, Y. Lu^7, Y. P. Lu^1,58, Z. H. Lu^1,64, C. L. Luo^41, J. R. Luo^59, M. X. Luo^80, T. Luo^12,g, X. L. Luo^1,58, X. R. Lyu^64, Y. F. Lyu^43, F. C. Ma^40, H. Ma^79, H. L. Ma^1, J. L. Ma^1,64, L. L. Ma^50, L. R. Ma^67, M. M. Ma^1,64, Q. M. Ma^1, R. Q. Ma^1,64, T. Ma^72,58, X. T. Ma^1,64, X. Y. Ma^1,58, Y. M. Ma^31, F. E. Maas^18, I. MacKay^70, M. Maggiora^75A,75C, S. Malde^70, Y. J. Mao^46,h, Z. P. Mao^1, S. Marcello^75A,75C, Z. X. Meng^67, J. G. Messchendorp^13,65, G. Mezzadri^29A, H. Miao^1,64, T. J. Min^42, R. E. Mitchell^27, X. H. Mo^1,58,64, B. Moses^27, N. Yu. Muchnoi^4,c, J. Muskalla^35, Y. Nefedov^36, F. Nerling^18,e, L. S. Nie^20, I. B. Nikolaev^4,c, Z. Ning^1,58, S. Nisar^11,m, Q. L. Niu^38,k,l, W. D. Niu^55, Y. Niu ^50, S. L. Olsen^10,64, S. L. Olsen^64, Q. Ouyang^1,58,64, S. Pacetti^28B,28C, X. Pan^55, Y. Pan^57, A. Pathak^34, Y. P. Pei^72,58, M. Pelizaeus^3, H. P. Peng^72,58, Y. Y. Peng^38,k,l, K. Peters^13,e, J. L. Ping^41, R. G. Ping^1,64, S. Plura^35, V. Prasad^33, F. Z. Qi^1, H. Qi^72,58, H. R. Qi^61, M. Qi^42, T. Y. Qi^12,g, S. Qian^1,58, W. B. Qian^64, C. F. Qiao^64, X. K. Qiao^81, J. J. Qin^73, L. Q. Qin^14, L. Y. Qin^72,58, X. P. Qin^12,g, X. S. Qin^50, Z. H. Qin^1,58, J. F. Qiu^1, Z. H. Qu^73, C. F. Redmer^35, K. J. Ren^39, A. Rivetti^75C, M. Rolo^75C, G. Rong^1,64, Ch. Rosner^18, M. Q. Ruan^1,58, S. N. Ruan^43, N. Salone^44, A. Sarantsev^36,d, Y. Schelhaas^35, K. Schoenning^76, M. Scodeggio^29A, K. Y. Shan^12,g, W. Shan^24, X. Y. Shan^72,58, Z. J. Shang^38,k,l, J. F. Shangguan^16, L. G. Shao^1,64, M. Shao^72,58, C. P. Shen^12,g, H. F. Shen^1,8, W. H. Shen^64, X. Y. Shen^1,64, B. A. Shi^64, H. Shi^72,58, H. C. Shi^72,58, J. L. Shi^12,g, J. Y. Shi^1, Q. Q. Shi^55, S. Y. Shi^73, X. Shi^1,58, J. J. Song^19, T. Z. Song^59, W. M. Song^34,1, Y. J. Song^12,g, Y. X. Song^46,h,n, S. Sosio^75A,75C, S. Spataro^75A,75C, F. Stieler^35, S. S Su^40, Y. J. Su^64, G. B. Sun^77, G. X. Sun^1, H. Sun^64, H. K. Sun^1, J. F. Sun^19, K. Sun^61, L. Sun^77, S. S. Sun^1,64, T. Sun^51,f, W. Y. Sun^34, Y. Sun^9, Y. J. Sun^72,58, Y. Z. Sun^1, Z. Q. Sun^1,64, Z. T. Sun^50, C. J. Tang^54, G. Y. Tang^1, J. Tang^59, M. Tang^72,58, Y. A. Tang^77, L. Y. Tao^73, Q. T. Tao^25,i, M. Tat^70, J. X. Teng^72,58, V. Thoren^76, W. H. Tian^59, Y. Tian^31,64, Z. F. Tian^77, I. Uman^62B, Y. Wan^55, S. J. Wang ^50, B. Wang^1, B. L. Wang^64, Bo Wang^72,58, D. Y. Wang^46,h, F. Wang^73, H. J. Wang^38,k,l, J. J. Wang^77, J. P. Wang ^50, K. Wang^1,58, L. L. Wang^1, M. Wang^50, N. Y. Wang^64, S. Wang^38,k,l, S. Wang^12,g, T. Wang^12,g, T. J. Wang^43, W. Wang^73, W. Wang^59, W. P. Wang^35,58,72,o, X. Wang^46,h, X. F. Wang^38,k,l, X. J. Wang^39, X. L. Wang^12,g, X. N. Wang^1, Y. Wang^61, Y. D. Wang^45, Y. F. Wang^1,58,64, Y. H. Wang^38,k,l, Y. L. Wang^19, Y. N. Wang^45, Y. Q. Wang^1, Yaqian Wang^17, Yi Wang^61, Z. Wang^1,58, Z. L. Wang^73, Z. Y. Wang^1,64, Ziyi Wang^64, D. H. Wei^14, F. Weidner^69, S. P. Wen^1, Y. R. Wen^39, U. Wiedner^3, G. Wilkinson^70, M. Wolke^76, L. Wollenberg^3, C. Wu^39, J. F. Wu^1,8, L. H. Wu^1, L. J. Wu^1,64, X. Wu^12,g, X. H. Wu^34, Y. Wu^72,58, Y. H. Wu^55, Y. J. Wu^31, Z. Wu^1,58, L. Xia^72,58, X. M. Xian^39, B. H. Xiang^1,64, T. Xiang^46,h, D. Xiao^38,k,l, G. Y. Xiao^42, S. Y. Xiao^1, Y. L. Xiao^12,g, Z. J. Xiao^41, C. Xie^42, X. H. Xie^46,h, Y. Xie^50, Y. G. Xie^1,58, Y. H. Xie^6, Z. P. Xie^72,58, T. Y. Xing^1,64, C. F. Xu^1,64, C. J. Xu^59, G. F. Xu^1, H. Y. Xu^67,2, M. Xu^72,58, Q. J. Xu^16, Q. N. Xu^30, W. Xu^1, W. L. Xu^67, X. P. Xu^55, Y. Xu^40, Y. C. Xu^78, Z. S. Xu^64, F. Yan^12,g, L. Yan^12,g, W. B. Yan^72,58, W. C. Yan^81, X. Q. Yan^1,64, H. J. Yang^51,f, H. L. Yang^34, H. X. Yang^1, J. H. Yang^42, T. Yang^1, Y. Yang^12,g, Y. F. Yang^1,64, Y. F. Yang^43, Y. X. Yang^1,64, Z. W. Yang^38,k,l, Z. P. Yao^50, M. Ye^1,58, M. H. Ye^8, J. H. Yin^1, Junhao Yin^43, Z. Y. You^59, B. X. Yu^1,58,64, C. X. Yu^43, G. Yu^1,64, J. S. Yu^25,i, M. C. Yu^40, T. Yu^73, X. D. Yu^46,h, Y. C. Yu^81, C. Z. Yuan^1,64, J. Yuan^34, J. Yuan^45, L. Yuan^2, S. C. Yuan^1,64, Y. Yuan^1,64, Z. Y. Yuan^59, C. X. Yue^39, A. A. Zafar^74, F. R. Zeng^50, S. H. Zeng^63A,63B,63C,63D, X. Zeng^12,g, Y. Zeng^25,i, Y. J. Zeng^59, Y. J. Zeng^1,64, X. Y. Zhai^34, Y. C. Zhai^50, Y. H. Zhan^59, A. Q. Zhang^1,64, B. L. Zhang^1,64, B. X. Zhang^1, D. H. Zhang^43, G. Y. Zhang^19, H. Zhang^81, H. Zhang^72,58, H. C. Zhang^1,58,64, H. H. Zhang^59, H. H. Zhang^34, H. Q. Zhang^1,58,64, H. R. Zhang^72,58, H. Y. Zhang^1,58, J. Zhang^59, J. Zhang^81, J. J. Zhang^52, J. L. Zhang^20, J. Q. Zhang^41, J. S. Zhang^12,g, J. W. Zhang^1,58,64, J. X. Zhang^38,k,l, J. Y. Zhang^1, J. Z. Zhang^1,64, Jianyu Zhang^64, L. M. Zhang^61, Lei Zhang^42, P. Zhang^1,64, Q. Y. Zhang^34, R. Y. Zhang^38,k,l, S. H. Zhang^1,64, Shulei Zhang^25,i, X. M. Zhang^1, X. Y Zhang^40, X. Y. Zhang^50, Y. Zhang^73, Y. Zhang^1, Y. T. Zhang^81, Y. H. Zhang^1,58, Y. M. Zhang^39, Yan Zhang^72,58, Z. D. Zhang^1, Z. H. Zhang^1, Z. L. Zhang^34, Z. Y. Zhang^77, Z. Y. Zhang^43, Z. Z. Zhang^45, G. Zhao^1, J. Y. Zhao^1,64, J. Z. Zhao^1,58, L. Zhao^1, Lei Zhao^72,58, M. G. Zhao^43, N. Zhao^79, R. P. Zhao^64, S. J. Zhao^81, Y. B. Zhao^1,58, Y. X. Zhao^31,64, Z. G. Zhao^72,58, A. Zhemchugov^36,b, B. Zheng^73, B. M. Zheng^34, J. P. Zheng^1,58, W. J. Zheng^1,64, Y. H. Zheng^64, B. Zhong^41, X. Zhong^59, H. Zhou^50, J. Y. Zhou^34, L. P. Zhou^1,64, S. Zhou^6, X. Zhou^77, X. K. Zhou^6, X. R. Zhou^72,58, X. Y. Zhou^39, Y. Z. Zhou^12,g, Z. C. Zhou^20, A. N. Zhu^64, J. Zhu^43, K. Zhu^1, K. J. Zhu^1,58,64, K. S. Zhu^12,g, L. Zhu^34, L. X. Zhu^64, S. H. Zhu^71, T. J. Zhu^12,g, W. D. Zhu^41, Y. C. Zhu^72,58, Z. A. Zhu^1,64, J. H. Zou^1, J. Zu^72,58
(BESIII Collaboration)
^1 Institute of High Energy Physics, Beijing 100049, People's Republic of China
^2 Beihang University, Beijing 100191, People's Republic of China
^3 Bochum Ruhr-University, D-44780 Bochum, Germany
^4 Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090, Russia
^5 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
^6 Central China Normal University, Wuhan 430079, People's Republic of China
^7 Central South University, Changsha 410083, People's Republic of China
^8 China Center of Advanced Science and Technology, Beijing 100190, People's Republic of China
^9 China University of Geosciences, Wuhan 430074, People's Republic of China
^10 Chung-Ang University, Seoul, 06974, Republic of Korea
^11 COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind Road, 54000 Lahore, Pakistan
^12 Fudan University, Shanghai 200433, People's Republic of China
^13 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
^14 Guangxi Normal University, Guilin 541004, People's Republic of China
^15 Guangxi University, Nanning 530004, People's Republic of China
^16 Hangzhou Normal University, Hangzhou 310036, People's Republic of China
^17 Hebei University, Baoding 071002, People's Republic of China
^18 Helmholtz Institute Mainz, Staudinger Weg 18, D-55099 Mainz, Germany
^19 Henan Normal University, Xinxiang 453007, People's Republic of China
^20 Henan University, Kaifeng 475004, People's Republic of China
^21 Henan University of Science and Technology, Luoyang 471003, People's Republic of China
^22 Henan University of Technology, Zhengzhou 450001, People's Republic of China
^23 Huangshan College, Huangshan 245000, People's Republic of China
^24 Hunan Normal University, Changsha 410081, People's Republic of China
^25 Hunan University, Changsha 410082, People's Republic of China
^26 Indian Institute of Technology Madras, Chennai 600036, India
^27 Indiana University, Bloomington, Indiana 47405, USA
^28 INFN Laboratori Nazionali di Frascati , (A)INFN Laboratori Nazionali di Frascati, I-00044, Frascati, Italy; (B)INFN Sezione di Perugia, I-06100, Perugia, Italy; (C)University of Perugia, I-06100, Perugia, Italy
^29 INFN Sezione di Ferrara, (A)INFN Sezione di Ferrara, I-44122, Ferrara, Italy; (B)University of Ferrara, I-44122, Ferrara, Italy
^30 Inner Mongolia University, Hohhot 010021, People's Republic of China
^31 Institute of Modern Physics, Lanzhou 730000, People's Republic of China
^32 Institute of Physics and Technology, Peace Avenue 54B, Ulaanbaatar 13330, Mongolia
^33 Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica 1000000, Chile
^34 Jilin University, Changchun 130012, People's Republic of China
^35 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz, Germany
^36 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
^37 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-Buff-Ring 16, D-35392 Giessen, Germany
^38 Lanzhou University, Lanzhou 730000, People's Republic of China
^39 Liaoning Normal University, Dalian 116029, People's Republic of China
^40 Liaoning University, Shenyang 110036, People's Republic of China
^41 Nanjing Normal University, Nanjing 210023, People's Republic of China
^42 Nanjing University, Nanjing 210093, People's Republic of China
^43 Nankai University, Tianjin 300071, People's Republic of China
^44 National Centre for Nuclear Research, Warsaw 02-093, Poland
^45 North China Electric Power University, Beijing 102206, People's Republic of China
^46 Peking University, Beijing 100871, People's Republic of China
^47 Qufu Normal University, Qufu 273165, People's Republic of China
^48 Renmin University of China, Beijing 100872, People's Republic of China
^49 Shandong Normal University, Jinan 250014, People's Republic of China
^50 Shandong University, Jinan 250100, People's Republic of China
^51 Shanghai Jiao Tong University, Shanghai 200240, People's Republic of China
^52 Shanxi Normal University, Linfen 041004, People's Republic of China
^53 Shanxi University, Taiyuan 030006, People's Republic of China
^54 Sichuan University, Chengdu 610064, People's Republic of China
^55 Soochow University, Suzhou 215006, People's Republic of China
^56 South China Normal University, Guangzhou 510006, People's Republic of China
^57 Southeast University, Nanjing 211100, People's Republic of China
^58 State Key Laboratory of Particle Detection and Electronics, Beijing 100049, Hefei 230026, People's Republic of China
^59 Sun Yat-Sen University, Guangzhou 510275, People's Republic of China
^60 Suranaree University of Technology, University Avenue 111, Nakhon Ratchasima 30000, Thailand
^61 Tsinghua University, Beijing 100084, People's Republic of China
^62 Turkish Accelerator Center Particle Factory Group, (A)Istinye University, 34010, Istanbul, Turkey; (B)Near East University, Nicosia, North Cyprus, 99138, Mersin 10, Turkey
^63 University of Bristol, (A)H H Wills Physics Laboratory; (B)Tyndall Avenue; (C)Bristol; (D)BS8 1TL
^64 University of Chinese Academy of Sciences, Beijing 100049, People's Republic of China
^65 University of Groningen, NL-9747 AA Groningen, The Netherlands
^66 University of Hawaii, Honolulu, Hawaii 96822, USA
^67 University of Jinan, Jinan 250022, People's Republic of China
^68 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
^69 University of Muenster, Wilhelm-Klemm-Strasse 9, 48149 Muenster, Germany
^70 University of Oxford, Keble Road, Oxford OX13RH, United Kingdom
^71 University of Science and Technology Liaoning, Anshan 114051, People's Republic of China
^72 University of Science and Technology of China, Hefei 230026, People's Republic of China
^73 University of South China, Hengyang 421001, People's Republic of China
^74 University of the Punjab, Lahore-54590, Pakistan
^75 University of Turin and INFN, (A)University of Turin, I-10125, Turin, Italy; (B)University of Eastern Piedmont, I-15121, Alessandria, Italy; (C)INFN, I-10125, Turin, Italy
^76 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
^77 Wuhan University, Wuhan 430072, People's Republic of China
^78 Yantai University, Yantai 264005, People's Republic of China
^79 Yunnan University, Kunming 650500, People's Republic of China
^80 Zhejiang University, Hangzhou 310027, People's Republic of China
^81 Zhengzhou University, Zhengzhou 450001, People's Republic of China
^a Deceased
^b Also at the Moscow Institute of Physics and Technology, Moscow 141700, Russia
^c Also at the Novosibirsk State University, Novosibirsk, 630090, Russia
^d Also at the NRC "Kurchatov Institute", PNPI, 188300, Gatchina, Russia
^e Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany
^f Also at Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education; Shanghai Key Laboratory for Particle Physics and Cosmology; Institute of Nuclear and Particle Physics, Shanghai 200240, People's Republic of China
^g Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and Institute of Modern Physics, Fudan University, Shanghai 200443, People's Republic of China
^h Also at State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, People's Republic of China
^i Also at School of Physics and Electronics, Hunan University, Changsha 410082, China
^j Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal University, Guangzhou 510006, China
^k Also at MOE Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou 730000, People's Republic of China
^l Also at Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou 730000, People's Republic of China
^m Also at the Department of Mathematical Sciences, IBA, Karachi 75270, Pakistan
^n Also at Ecole Polytechnique Federale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
^o Also at Helmholtz Institute Mainz, Staudinger Weg 18, D-55099 Mainz, Germany
September 9, 2024
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Although the standard model (SM) has achieved great success in high-energy physics, some questions like, e.g., dark matter, matter and anti-matter asymmetry, fermion mass hierarchy remain unresolved. So-called “dark" sectors have been theorized, named as such due to their assumed extremely faint interactions with the visible sector. Searching for these dark sectors offers a unique opportunity to uncover new physics (NP) beyond the SM, which would require extension to accommodate for these new particles and their interaction with SM matter. In the SM, the Abelian gauge group U(1)_Y (with hypercharge Y) describes the electromagnetic interaction and produces the associated gauge boson, the SM photon γ.
A minimum extension of the SM with an extra Abelian gauge group, U(1)_D (D denoting dark) produces another gauge boson, the dark photon <cit.>.
These two Abelian gauge fields could have a kinetic mixing because the field strengths can be multiplied to form a dimension-four operator <cit.>.
The dark photon may be a portal between the SM matter and the dark sector matter, and it could be massive or massless. If the symmetry of U(1)_D is broken spontaneously, the dark photon will acquire mass, which will be called A'. If the symmetry of U(1)_D remains unbroken, it will produce the massless dark photon, γ' <cit.>.
This massless dark photon could provide a natural explanation for the fermion mass hierarchy puzzle <cit.>.
While the massive dark photon couples to SM matter via the coupling strength eε through the kinetic mixing, where ε is the mixing parameter <cit.>, the massless dark photon has no direct interaction with the SM particle in the dimension-four operator <cit.>.
In the massless dark photons case, the SM photon can couple with the dark sector particle, which is usually called milli-charge particle <cit.>.
The massive dark photon can be produced in many processes by replacing the SM photon, such as e^+e^-→γ A', a process that has not been confirmed yet despite extensive search. A stringent constraint is set on the mixing parameter ε with different m_A' <cit.>, indicating that the value of ε must be small.
Since the massless dark photon has no interaction with SM particles within the dimension-four operator, the restrictions obtained from the massive dark photon do not apply to the massless dark photon.
A dimension-six operator has been proposed to provide a connection between SM matter and the massless dark photon <cit.>:
ℒ_NP= 1/Λ^2_NP ( C^U_jkq̅_j σ^μν u_k H̃ + C^D_jkq̅_j σ^μν d_k H
+ C^L_jkl̅_j σ^μν e_k H + h.c. ) F'_μν,
where Λ_NP is the effective mass, indicating the NP energy scale, C^U_jk, C^D_jk, and C^L_jk are the up-type, down-type, and charged-lepton-type dimensionless coefficients, respectively, depending on the NP and not necessarily related to one another, j(k)=1,2,3 is the generation tag of the SM particle. More details can be found in Ref. <cit.>.
The first three terms in this equations are the couplings between the massless dark photon and the up-type quarks, down-type quarks, and charged leptons, where the flavors of the two quarks or leptons could be identical or different, differing to flavor diagonal of the tree-level couplings of the massive dark photon.
This effective operator may cover some new dark-sector particles with very heavy mass in the NP energy scale Λ_NP <cit.>. Up to now, no new particles have been found up to the mass of ∼ 1 TeV, but some anomalies require an energy scale above the electroweak energy scale Λ_EW (∼ 100 GeV).
Similar to the β decay observed in low energy experiments <cit.>, where a missing neutrino within the four-fermion effective coupling <cit.> can predict the electroweak energy scale at about 100 GeV <cit.>, the massless dark photon could provide a portal for exploring the NP energy scale Λ_NP beyond the TeV magnitude.
In this Letter, we focus on the first item of the dimension-six operator in Eq. (<ref>), which causes the cuγ' coupling in the flavor-changing neutral current (FCNC) process of a charm quark with j=1,k=2.
In the SM, the FCNC processes are strongly suppressed by the Glashow-Iliopoulos-Maiani mechanism <cit.>, stating that these processes are forbidden at the tree level and can only happen through a loop diagram. In the SM, the branching fraction (BF) of the FCNC charm process is expected to be smaller than 10^-9 <cit.>. But for the cuγ' coupling, its FCNC process originates from the NP coupling in the NP energy scale, which is different from the SM.
In the charm sector, the massless dark photon can be searched for in D meson or Λ^+_c baryon decays, such as D→ Vγ', D→γγ', or Λ^+_c→ p γ', where V is a vector particle like ρ or ω. The BFs of these processes are directly related to |ℂ|^2+|ℂ_5|^2 <cit.>. Here ℂ=Λ_NP^-2(C^U_12+C^U*_12)ν/√(8) and ℂ_5=Λ_NP^-2(C^U_12-C^U*_12)ν/√(8) with the Higgs vacuum expectation value ν=246.2 GeV <cit.>, which are determined by the NP energy scale Λ_NP and the up-type dimensionless coefficient C^U_12. From the constraint of the dark matter (DM) and the vacuum stability (VS) in the universe <cit.>, the allowed BF of the massless dark photon in charm FCNC processes can be enhanced to 10^-7∼10^-5 <cit.>. Previously, BESIII searched for Λ^+_c→ p γ' and set the upper limit (UL) of its decay BF to be 8×10^-5 at the 90% confidence level (CL) <cit.>. This, however, is still above the allowed region obtained from DM and VS <cit.>. In this Letter, we search for the massless dark photon and probe the NP energy scale through the FCNC processes D^0→ωγ' and D^0→γγ' for the first time, which can be mediated via Feynman diagrams shown in FIG <ref>, by analyzing e^+e^- collision data of 7.9 fb^-1 at a center-of-mass energy of √(s)=3.773 GeV with the BESIII detector.
Details about the BESIII detector design and performance are provided elsewhere <cit.>. The simulated Monte Carlo (MC) samples, also described in Ref. <cit.>, are used to determine detection efficiencies and to estimate backgrounds.
The generator of the signal MC samples is parameterized by the helicity amplitudes same as the radiative decay of the D meson <cit.>.
At √(s)=3.773 GeV, the D^0D̅^0 meson pairs are produced from ψ(3770) decays without accompanying hadrons, which provide an ideal opportunity to study invisible massless dark photon decays of D mesons using the double tag (DT) method <cit.>. The D̅^0 mesons are first tagged with the main hadronic-decay modes D̅^0→ K^+π^-, D̅^0→ K^+π^-π^0 and D̅^0→ K^+π^-π^+π^-, and the selected candidates are referred to as the single tag (ST) sample. Here and throughout this letter, charge conjugations are always implied. Then, the signal processes D^0→ωγ' and D^0→γγ' are searched for in the system recoiling against the ST D̅^0 meson, and the selected candidates are denoted as the DT sample. Here, ω is reconstructed through its decay ω→π^+π^-π^0, π^0→γγ, and γ' is missing in the detector. The BFs of D^0→ωγ' and D^0→γγ' are calculated by
ℬ(D^0→ω(γ) γ')= N_sig/ϵ̂/ℬ_int∑_i N_i^ST,
with ∑_i N_i^ST=(6306.7±2.8)×10^3 <cit.> and the effective efficiency
ϵ̂=∑_i ϵ_i^DT/ϵ_i^ST×N_i^ST/∑_i N_i^ST,
where i indicates each mode of D̅^0→hadrons, N_sig is the signal yield of the massless dark photon in data, N_i^ST is the ST yield of D̅^0 meson samples in data, ϵ_i^ST is the ST efficiency of D̅^0→hadrons, ϵ_i^DT is the DT efficiency of D̅^0→hadrons, D^0→ω(γ)γ', ℬ_int=ℬ(ω→π^+π^-π^0)×ℬ(π^0→γγ) is obtained from Particle Data Group <cit.> for D^0→ωγ' and ℬ_int=1 for D^0→γγ'.
The selection criteria of ST samples, the ST yield N_i^ST, and the ST efficiency ϵ_i^ST can be found in Ref. <cit.>. The selection criteria of D^0→ωγ' and D^0→γγ', based on the tagged D̅^0 meson samples, are described below. To select D^0→γγ', no additional charged track is allowed.
The good charged track, particle identification (PID), and photon candidates are selected with the same strategy as outlined in Ref. <cit.>.
To select D^0→ωγ', only events with exactly two selected charged tracks, both identified as pions with zero net charge, are retained for further analysis.
There should be at least one photon with energy larger than 0.5 GeV for D^0→γγ' and at least two photons for D^0→ωγ', where the two photons with minimum χ^2 value of the kinematic fit <cit.> constraining M_γγ to the nominal π^0 mass are regarded as the correct photons from the π^0 meson. To select the ω meson in the data samples, the invariant mass of the two photons M_γγ before the kinematic fit must be in the region of [0.115, 0.150], and the invariant mass M_π^+π^-π^0 of the ω candidate is required to be in the region of [0.700, 0.850]. To further reduce the non-ω background, a kinematic fit <cit.> constraining M_γγ to the nominal π^0 mass and M_π^+π^-π^0 to the nominal ω mass is performed to obtain the χ^2_2C value which is required to be less than 44, optimized with the Punzi-optimization method <cit.>. To suppress the background with additional photons or π^0, the total energy of photon candidates other than those from the π^0 (γ) and the D̅^0 meson (E^tot_oth.γ) is required to be less than 0.1 GeV for D^0→ωγ' (D^0→γγ'). After these selections, there may still be some background particles flying to the endcap of the detector that cannot be effectively detected <cit.>, so the recoiling angle of D̅^0ω (D̅^0γ) is applied to veto these associated background events. The cosine of the recoiling angle is defined as cosθ^reccoil_D̅ω(γ)=|p⃗_cms - p⃗_D̅^0 - p⃗_ω(γ)|_z/|p⃗_cms - p⃗_D̅^0 - p⃗_ω(γ)|, where p⃗_cms is the momentum of the center-of-mass in e^+e^- collision, p⃗_D̅^0 is the reconstructed momentum of the D̅^0 meson, p⃗_ω(γ) is the reconstructed momentum of ω (γ), and the subscript z refers to the z-component. To suppress these background events, a requirement of |cosθ_D̅ω(γ)^recoil|<0.7 is applied.
With the above selection criteria,
the effective efficiency is estimated from the MC samples, which is ϵ̂=(15.98±0.02)% for D^0→ωγ' and ϵ̂=(52.18±0.05)% for D^0→γγ'.
The main background after the selections comes from the K^0_L associated background events, such as D^0→ω K^0_L for D^0→ωγ' and D^0→π^0 K^0_L for D^0→γγ'.
The signals of the massless dark photon are extracted from an unbinned extended maximum likelihood fit on the distribution of the square of the missing mass, M^2_miss, defined as
M^2_miss=|p_cms - p_D̅^0 - p_ω(γ)|^2/c^4,
where p_cms is the four-momentum of the e^+e^- center-of-mass system in the laboratory frame, p_ω(γ) is the kinematic fitted (reconstructed) four-momentum of ω(γ), p_D̅^0 is the four-momentum of the D̅^0 meson, achieved by the kinematic fit <cit.> constraining M_γγ to the nominal π^0 mass and M_K^+π^-, M_K^+π^-π^0, M_K^+π^-π^+π^- to the nominal D̅^0 meson mass. In the fit, the background is separated into the K^0_L-related and non-K^0_L backgrounds <cit.>.
The background shape is derived from the inclusive MC sample <cit.>, with the number of non-K^0_L background events assumed to follow a Gaussian distribution and constrained by the MC simulation (referred to as the Gaussian constraint), while the number of K^0_L-related background events is left as a floating parameter.
The signal is derived by the simulated shape convolved with a Gaussian function G(μ,σ), where μ and σ are restrained to the values obtained from the control samples D^0→ω K^0_S (D^0→π^0 K^0_S) for D^0→ωγ' (D^0→γγ'). The fit results are shown in FIG <ref>, with the massless dark photon signal yield N_sig=-15±8 for D^0→ωγ' and N_sig=-6±4 for D^0→γγ'.
The systematic uncertainty sources for the BF measurement include ST yield, intermediate BF, signal generator, DT signal efficiency, and signal extraction. With the DT method, several systematic uncertainties associated with the ST selection can be canceled without impacting the BF measurement. The uncertainty of ST yield is assigned as 0.1% <cit.>.
The uncertainty of the generator is estimated from the efficiency difference compared with a flat angular generation of γ' (phase space model), which is 1.3% (0.6%) for D^0→ωγ' (D^0→γγ').
The uncertainty of the BF of ω→π^+π^-π^0 is 0.8% and that of π^0→γγ is negligible <cit.>.
The uncertainty of photon detection is assigned as 1.0% per photon <cit.>. The uncertainties of pion tracking and PID are studied from the control sample J/ψ→π^+π^-π^0, which is assigned as 0.9% for tracking and 1.1% for PID of the two pions.
The uncertainties of other selections are estimated from the control sample D^0→ω K^0_S (D^0→π^0 K^0_S) for D^0→ωγ' (D^0→γγ'), where the K^0_S meson is regarded as a missing particle. For D^0→ωγ', the uncertainty is 0.2% for the M_γγ selection, negligible for the M_π^+π^-π^0 selection, 3.1% for the χ^2_2C selection, 5.9% for the E^tot_oth.γ selection and 1.1% for the cosθ^recoil_D̅ω selection, respectively. For D^0→γγ', the uncertainty is 2.4% for the E^tot_oth.γ selection and is 0.6% for the cosθ^recoil_D̅γ selection, respectively. The total systematic uncertainty is calculated by summing up all sources in quadrature, yielding 7.4% for D^0→ωγ' and 2.7% for D^0→γγ'.
For the uncertainty from the signal extraction, the signal shape is convolved with a Gaussian function to describe the difference where the parameter of the Gaussian function is in a Gaussian constraint within its uncertainty, the K^0_L background yield is floating in the fit, and the non-K^0_L background yield is also floating in a Gaussian constraint within its uncertainty. The uncertainty of signal extraction is negligible.
Since no significant excess of signal above the background is observed, a UL on the BF is set using a Bayesian approach following Ref. <cit.>, where the BF is calculated by Eq. (<ref>) and the systematic uncertainty is estimated with the method in Refs. <cit.>. The UL on the BF at the 90% CL is calculated by integrating the likelihood distribution with different signal assumptions to the 90% region, which is ℬ(D^0→ωγ')<1.1×10^-5 and ℬ(D^0→γγ')<2.0×10^-6.
Note that in the D^0→ωγ' measurement, the non-ω contribution can not be fully removed, and the current UL of ℬ(D^0→ωγ') is a conservative estimation.
Since the BF of massless dark photon production is related to |ℂ|^2+|ℂ_5|^2 and directly includes the NP energy scale <cit.>, the constraint on |ℂ|^2+|ℂ_5|^2 can be performed as well.
The UL of |ℂ|^2+|ℂ_5|^2 is shown in FIG <ref>. For D^0→ωγ', one sees that |ℂ|^2+|ℂ_5|^2<8.2×10^-17^-2, reaching the DM and VS allowed region <cit.> for the first time.
The channel D^0→γγ' has a better UL of the BF but a worse constraint on cuγ' coupling due to an additional electromagnetic vertex in FIG <ref>(b).
The two-dimensional constraint on the NP energy scale Λ_NP and the up-type dimensionless coefficient C^U_12 is given in FIG <ref> (b). Our result currently resemble the best constraint on the NP parameter space. Assuming |C^U_12|=1, our results can exclude NP energy scales below 138 TeV in the dark sector, which is approximately ten times the energy reached at the Large Hadron Collider <cit.>, suggesting a challenge of directly detecting superheavy particles at the NP energy scale associated with the cuγ' coupling in the present collider settings. Note that the value of |C^U_12| is model-dependent.
In the dimension-six operator of the massless dark photon (Eq. (<ref>)), there are three terms with the same NP energy scale but different types of dimensionless coefficients C^U_jk, C^D_jk, C^L_jk. For each type, the massless dark photon can couple with a pair of SM particles involving identical or different flavors. In principle, these dimensionless coefficients are not necessarily related to one another.
To compare the difference between different couplings, we follow the method from Ref. <cit.>, constructing a uniform variable F_NP=Λ_NP/√(|C^i_jk|ν/√(2m^2))
to perform the comparison on the first two generations, where i=U,D,L, |C^i_jk|ν/√(2m^2) indicates the strength of the massless dark photon coupled with the SM particle. Here, m is the mass of the heavier SM particle in the coupling, and F_NP represents the NP energy scale when |C^i_jk| equals to the Higgs-fermions coupling √(2)m/v <cit.>. The summary of the constraints in different couplings is shown in FIG <ref>, depicting only the best constraint on the coupling.
The constraints on the massless dark photon coupled with the same-flavor SM particles are mainly from astrophysics and cosmology <cit.>, while the constraints on the different flavors are mainly from laboratory physics <cit.>. This Letter provides the best constraint for the cuγ' coupling, which plays a unique role in the dark sector of the charm sector.
In summary,
we search for the massless dark photon and constraint the NP scale through the cuγ' coupling in D^0→ωγ' and D^0→γγ' for the first time. Based on 7.9 fb^-1 of e^+e^- collision data at √(s)=3.773 GeV, no significant signals are observed. The constraints on the BF and the NP energy scale of the massless dark photon production are given. The result of D^0→ωγ' gives the most stringent constraint on cuγ' coupling to date, exploring the DM and VS allowed space for the first time. The result of D^0→γγ' has a 5.5 times better UL of the BF than D^0→ωγ' but a worse constraint on cuγ' coupling.
Acknowledgement
The BESIII Collaboration thanks the staff of BEPCII and the IHEP computing center for their strong support. This work is supported in part by National Key R&D Program of China under Contracts Nos. 2023YFA1606000, 2020YFA0406400, 2020YFA0406300; National Natural Science Foundation of China (NSFC) under Contracts Nos. 11635010, 11735014, 11935015, 11935016, 11935018, 12025502, 12035009, 12035013, 12061131003, 12175321, 12192260, 12192261, 12192262, 12192263, 12192264, 12192265, 12221005, 12225509, 12235017, 12361141819; the Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; the CAS Center for Excellence in Particle Physics (CCEPP); Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under Contract No. U1832207, U1932101; 100 Talents Program of CAS; The Institute of Nuclear and Particle Physics (INPAC) and Shanghai Key Laboratory for Particle Physics and Cosmology; German Research Foundation DFG under Contracts Nos. FOR5327, GRK 2149; Istituto Nazionale di Fisica Nucleare, Italy; Knut and Alice Wallenberg Foundation under Contracts Nos. 2021.0174, 2021.0299; Ministry of Development of Turkey under Contract No. DPT2006K-120470; National Research Foundation of Korea under Contract No. NRF-2022R1A2C1092335; National Science and Technology fund of Mongolia; National Science Research and Innovation Fund (NSRF) via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation of Thailand under Contracts Nos. B16F640076, B50G670107; Polish National Science Centre under Contract No. 2019/35/O/ST2/02907; Swedish Research Council under Contract No. 2019.04595; The Swedish Foundation for International Cooperation in Research and Higher Education under Contract No. CH2018-7756; U. S. Department of Energy under Contract No. DE-FG02-05ER41374.
Other Fund Information
To be inserted with an additional sentence into papers that are relevant to the topic of special funding for specific topics. Authors can suggest which to Li Weiguo and/or the physics coordinator.
Example added sentence: This paper is also supported by the NSFC under Contract Nos. 10805053, 10979059, ....National Natural Science Foundation of China (NSFC), 10805053, PWANational Natural Science Foundation of China (NSFC), 10979059, Lund弦碎裂强子化模型及其通用强子产生器研究National Natural Science Foundation of China (NSFC), 10775075, National Natural Science Foundation of China (NSFC), 10979012, baryonsNational Natural Science Foundation of China (NSFC), 10979038, charmoniumNational Natural Science Foundation of China (NSFC), 10905034, psi(2S)->B BbarNational Natural Science Foundation of China (NSFC), 10975093, D 介子National Natural Science Foundation of China (NSFC), 10979033, psi(2S)->VPNational Natural Science Foundation of China (NSFC), 10979058, hcNational Natural Science Foundation of China (NSFC), 10975143, charmonium rare decays
apsrev4-1
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http://arxiv.org/abs/2409.02574v1 | 20240904094827 | Solving Video Inverse Problems Using Image Diffusion Models | [
"Taesung Kwon",
"Jong Chul Ye"
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"cs.CV",
"cs.AI",
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] | "\n\nBMI Prediction from Handwritten English Characters Using a Convolutional Neural Network\n\n\n\(...TRUNCATED) |
http://arxiv.org/abs/2409.02417v1 | 20240904034248 | "Generation of Scalable Genuine Multipartite Gaussian Entanglement with a Parametric Amplifier Netwo(...TRUNCATED) | [
"Saesun Kim",
"Sho Onoe",
"Alberto M. Marino"
] | quant-ph | [
"quant-ph"
] | "\n\n\n\nsaesun.kim@jpl.nasa.gov\n\n\nmarino@ou.edu, marinoa@ornl.gov\n^1Homer L. Dodge Department o(...TRUNCATED) |
http://arxiv.org/abs/2409.02790v1 | 20240904150444 | "Symmetry based efficient simulation of dissipative quantum many-body dynamics in subwavelength quan(...TRUNCATED) | [
"Raphael Holzinger",
"Oriol Rubies-Bigorda",
"Susanne F. Yelin",
"Helmut Ritsch"
] | quant-ph | [
"quant-ph",
"physics.atom-ph",
"physics.comp-ph"
] | "\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\(...TRUNCATED) |
http://arxiv.org/abs/2409.02178v1 | 20240903180002 | Modular flavored dark matter | [
"Alexander Baur",
"Mu-Chun Chen",
"V. Knapp-Perez",
"Saul Ramos-Sanchez"
] | hep-ph | [
"hep-ph",
"hep-th"
] | "\n\n\n\n\n \n\n\n\n\nUCI-TR-2024-12\n\n\nTUM-HEP 1520/24\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nAlexand(...TRUNCATED) |
http://arxiv.org/abs/2409.02853v1 | 20240904162335 | Hardy perturbations of subordinated Bessel heat kernels | [
"Krzysztof Bogdan",
"Tomasz Jakubowski",
"Konstantin Merz"
] | math.AP | [
"math.AP",
"math-ph",
"math.FA",
"math.MP",
"math.PR"
] | "\n\nHeat kernel bounds — ]Hardy perturbations of \n subordinated Bessel heat kernels\n\n\nK. Bogd(...TRUNCATED) |
http://arxiv.org/abs/2409.02761v1 | 20240904143808 | Sampling methods for recovering buried corroded boundaries from partial electrostatic Cauchy data | [
"Isaac Harris",
"Andreas Kleefeld",
"Heejin Lee"
] | math.AP | [
"math.AP"
] | "\n\n\n\n\n\nSampling methods for recovering buried corroded boundaries from partial electrostatic C(...TRUNCATED) |
http://arxiv.org/abs/2409.03064v1 | 20240904203302 | A posteriori error estimates for a bang-bang optimal control problem | [
"Francisco Fuica"
] | math.OC | [
"math.OC",
"cs.NA",
"math.NA",
"49M25, 65N15, 65N30"
] | "\n\n\n\n\nFacultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mack(...TRUNCATED) |
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