Analytic continuation and physical content of the gluon propagator Fabio Siringoand Giorgio Comitiniy Dipartimento di Fisica e Astronomia dellUniversità di Catania

2025-04-26 0 0 622.33KB 42 页 10玖币

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Analytic continuation and physical content of the gluon propagator

Fabio Siringo∗and Giorgio Comitini†

Dipartimento di Fisica e Astronomia dell’Università di Catania,

INFN Sezione di Catania, Via S.Soﬁa 64, I-95123 Catania, Italy

(Dated: April 24, 2023)

Abstract

The analytic continuation of the gluon propagator is revised in the light of recent ﬁndings on the

possible existence of complex conjugated poles. The contribution of the anomalous pole must be

added when Wick rotating, leading to an eﬀective Minkowskian propagator which is not given by

the trivial analytic continuation of the Euclidean function. The eﬀective propagator has an integral

representation in terms of a spectral function which is naturally related to a set of elementary

(complex) eigenvalues of the Hamiltonian, thus generalizing the usual Källén-Lehmann description.

A simple toy model shows how the elementary eigenvalues might be related to actual physical

quasiparticles of the non-perturbative vacuum.

∗fabio.siringo@ct.infn.it

†giorgio.comitini@dfa.unict.it

arXiv:2210.11541v2 [hep-th] 21 Apr 2023

I. INTRODUCTION

The gluon and quark propagators play a very important role in the study of strong

interactions and a detailed knowledge of the real-time correlators would provide the basic

blocks for a study of heavy-ion collisions from ﬁrst principles. However, in the low energy

nonperturbative regime of strong interactions, our knowledge of the propagators is very

limited and usually based on numerical calculations in the Euclidean space, including lattice

simulations[1–12] and continuum studies[13–35]. Thus, the problem of analytic continuation

from Euclidean to Minkowski space is still under intense debate[36–47].

For a generic ﬁeld theory which describes physical particles, many exact results have been

developed in the past and some of them have been even extended to N-point functions[48–52].

If we did not know about conﬁnement, then the non-Abelian gauge theory would be expected

to satisfy the same general conditions which hold for all physical particles: the propagators

should be characterized by the usual analytic properties and could be written by the standard

Källén-Lehmann integral representation in terms of a positive deﬁned spectral function.

Then, the knowledge of the spectral function would allow a trivial analytic continuation from

Euclidean to Minkowski space. Actually, from the formal point of view, there is nothing in

the Lagrangian which might foreshadow a diﬀerent behavior for the correlators of QCD in

comparison to, for instance, QED. For the same reason, we still miss a full understanding

of conﬁnement. On the other hand, the interacting compact QED seems to follow the same

anomalous features of Yang-Mills theory[53].

Because of color conﬁnement, gluon and quarks are usually regarded as internal degrees

of freedom of the theory. More precisely, they do not occur in the asymptotic states, but

they do exist as quasiparticles in a very hot quark-gluon plasma above the deconﬁnement

transition. Thus, they cannot be regarded as totally unphysical mathematical degrees of

freedom like a ghost. But, since we cannot detect a free gluon or a free quark, some unitarity

constraints might be relaxed for these particles and there is no reason to believe that the

same positivity conditions should still hold for their spectral functions. Moreover, we don’t

have any formal proof that there is any spectral representation at all, so that the usual

analytic properties of the propagators might be questioned. That explains why the problem

of analytic continuation is still so strongly debated.

On the other hand, we believe by now that QCD is a complete consistent theory which

generates its IR cutoﬀ dynamically and we expect that the conﬁnement must arise from the

same Lagrangian, as it actually happens in the lattice, without adding spurious eﬀects by

hand. Thus, it is also very reasonable to expect that the exact propagators of the theory

should be substantially diﬀerent than the other propagators of the standard model. Some-

how, some sign of conﬁnement must appear dynamically in the structure of the propagators

and must be buried in their analytic properties, in the complex plane. But, since our most

accurate information on the propagators is found numerically in the Euclidean space, we

have no direct knowledge of the analytic properties in the complex plane, and the continu-

ation has the nature of a guessing work. Moreover, there are many clues that the analytic

structure is untrivial. For instance, from lattice and continuous calculations we know that

the curvature of the propagator changes sign and the Schwinger function crosses the zero,

becoming negative at a length of some Fermi units[54, 55]. These are all signs of a spectral

function which is not positive deﬁned, if there is a spectral function. What is even more dis-

turbing, there are independent predictions of complex conjugated poles which invalidate the

Källén-Lehmann spectral representation, even if the spectral density were negative[45, 46].

Complex poles were predicted by eﬀective models[56–58] for the gluon propagator in the

past. From ﬁrst principles, their existence arises from a one-loop screened expansion of the

exact Lagrangian[55, 59–67]. But they also occur in one-loop approximations[46] of eﬀective

models like Curci-Ferrari[69–76].

Many numerical attempts at reconstructing the gluon propagator and its spectral function

have shown a better agreement with the data if a pole part, with complex conjugated

poles, is added to the usual spectral integral[43, 77]. Even the outcome of Schwinger-Dyson

euqations in the complex plane seems to suggest the existence of singularities outside the

real axis[82]. The quark propagator has also been reported to show complex conjugated

poles by the one-loop screened expansion[78] and a general study of the pole structure in

one-loop approximations has been discussed in Ref.[46].

While there are reconstruction methods which describe the lattice data without requiring

the existence of complex poles[41, 42], the quality of the reconstruction seems to improve

when the poles are added[77]. Then the issue of the existence and dynamical meaning of

the complex poles becomes of paramount importance.

In this paper, we discuss how a consistent quantum theory can be recovered when there

are complex poles in the Euclidean propagator. Assuming that complex conjugated poles

do exist in the exact propagator and that they might play a physical role in the conﬁnement

mechanism[79], we show how well deﬁned propagators can be actually derived in real time by

a modiﬁed Wick rotation. Then, we see how a modiﬁed Källén-Lehmann spectral represen-

tation, including the anomalous pole part, can be derived from ﬁrst principles in presence of

zero-norm states, with complex energies. Finally we speculate on a direct relation between

the complex energies and a set of observable glueball physical states.

While we cannot say if the complex poles are genuine and if they do exist at all in the exact

gluon propagator, here we show how their existence would lead to untrivial consequences in

the analytic continuation to real time.

The paper is organized as follows: the problem of analytic continuation is discussed in

Sec. II and the standard Wick rotation is recovered in Sec. III in order to ﬁx the notation;

in Sec. IV a modiﬁed analytic continuation is derived by two diﬀerent methods, by residue

subtraction and by convergence arguments, yielding an interesting spectral representation;

in Sec. V the same anomalous spectral density is derived from ﬁrst principles as a modiﬁed

Källén-Lehmann representation in presence of complex eigenvalues; in Sec. VI a Hermitian

toy model is discussed which leads to a speculative physical interpretation of the anoma-

lous spectral density in terms of physical states; ﬁnally, in Sec. VII, the main results are

summarized and discussed.

II. ANALYTIC CONTINUATION OF THE GLUON PROPAGATOR

While most of the rigorous results in quantum ﬁeld theory have been established in

the Euclidean space, the physical content of a theory is usually extracted in Minkowski

space. However, if there are complex conjugated poles, a general rigorous connection between

amplitudes in Euclidean and Minkowski spaces is missing because the singularities do not

allow the usual Wick rotation and the standard Källén-Lehmann spectral representation

does not hold[45–47]. Thus, the extraction of the physical content from the theory might be

quite tricky and might rely on some guessing work. Moreover, the numerical knowledge of

an amplitude on the real axis of the Euclidean space is usually not enough for reconstructing

its analytic continuation to Minkowski space[41–44, 52].

In perturbation theory, it is assumed and generally found that the Fourier Transform

(F.T.) of the physical amplitudes have poles in the second and fourth quadrant of the

complex-energy plane and a branch cut on the real axis. Then, Wick rotation is allowed and

gives a well deﬁned connection between the physics which occurs in Minkowski space and

the amplitudes which are evaluated in the Euclidean space. In that case, we ﬁnd a circular

path going: (i) from real time to real energy (by a F.T.); (ii) to the Euclidean space through

Wick rotation in the complex-energy plane; (iii) to imaginary time by an inverse F.T.; and

as shown in the following line,

tF.T.

⇐⇒ p0

Wick

⇐⇒ ip4

F.T.

⇐⇒ −i x4

τorder

⇐⇒ t(1)

(iv) the circle closes if a well deﬁned prescription is given for the analytic continuation from

imaginary time to real time. Time-ordered functions are not analytic in time, because of

the functions θ(±t), then the relation between real-time and imaginary-time is not unique,

in principle. The position t=−iτ, where t=x0in Minkowski space and τ=x4in the

Euclidean space, can be explained by the physical motivation of mapping the time-evolution

operator U(t) = exp(−iHt)on a thermal average by U(−iτ) = exp(−τH)where 0≤τ≤β.

If we look at the general structure of a time-ordered correlator

h0|T[A(t)B(0)|0i=θ(t)X

ρne−iEnt+θ(−t)X

ρ0

neiEnt(2)

we ﬁnd positive frequencies for t > 0and negative frequencies for t < 0, which can be seen as

antiparticle states going backwards in time. The position t=−iτ gives a weight exp(−Enτ)

for positive frequencies and a weight exp(Enτ)for negative frequencies. The correct thermal

weight exp(−En|τ|)is obtained in all cases if τ < 0when t < 0and vice versa. Thus, we

generally assume that the generic time-ordered average transforms according to

θ(t)hA(t)B(0)i+θ(−t)hB(0)A(t)i=⇒θ(τ)hA(−iτ)B(0)i+θ(−τ)hB(0)A(−iτ)i(3)

when going to the Euclidean space. Then, if the functions hA(t)B(0)iare analytic functions,

there is a well deﬁned and unique way to connect real-time amplitudes and imaginary-time

averages. With the imaginary-time order understood in the analytic continuation, the circle

is closed and we have a well deﬁned connection among the diﬀerent representations of the

same theory as shown in Eq.(1). For a physical particle, which is present in the asymptotic

states, causality and unitarity determine the Källén-Lehmann spectral representation, giving

a formal proof of the relation between time order and imaginary-time order. Thus our

physical motivation is based on a solid formal background[48–51].

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AnalyticcontinuationandphysicalcontentofthegluonpropagatorFabioSiringoandGiorgioComitiniyDipartimentodiFisicaeAstronomiadell'UniversitàdiCatania,INFNSezionediCatania,ViaS.Soa64,I-95123Catania,Italy(Dated:April24,2023)AbstractTheanalyticcontinuationofthegluonpropagatorisrevisedinthelightofrecentnd...

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Analytic continuation and physical content of the gluon propagator Fabio Siringoand Giorgio Comitiniy Dipartimento di Fisica e Astronomia dellUniversità di Catania

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