Constraining Spectral Functions at Finite Temperature and Chemical Potential with Exact Sum

a r X i v :h e p -p h /9412246v 1 6 D e c 1994MIT-CTP#2357

hep-ph/9412246

INFNCA-TH-94-16

Constraining Spectral Functions

at Finite Temperature and Chemical Potential

with Exact Sum Rules in Asymptotically Free Theories

Suzhou Huang (1,2)?and Marcello Lissia (1,3)?(1)Center for Theoretical Physics,Laboratory for Nuclear Science and Department of Physics,Massachusetts Institute of Technology,Cambridge,Massachusetts 02139(2)Department of Physics,FM-15,University of Washington,Seattle,Washington 98195?(3)Istituto Nazionale di Fisica Nucleare,via Ada Negri 18,I-09127Cagliari,Italy ?and Dipartimento di Fisica dell’Universit`a di Cagliari,I-09124Cagliari,Italy (February 1,2008)Abstract Within the framework of the operator product expansion (OPE)and the renormalization group equation (RGE),we show that the temperature and chemical potential dependence of the zeroth moment of a spectral function (SF)is completely determined by the one-loop structure in an asymptotically free theory,and in particular in QCD.Logarithmic corrections are found to play an essential role in the derivation.This exact result constrains the shape of SF’s,and implies striking e?ects near phase transitions.Phenomenological parameterizations of the SF,often used in applications such as the analysis of lattice QCD data or QCD sum rule calculations at ?nite temperature and

baryon density must satisfy these constraints.We also explicitly illustrate in

detail the exact sum rule in the Gross-Neveu model.

Typeset using REVT E X

I.INTRODUCTION

Hadronic properties at?nite temperature and baryon density are of great importance in the phenomenology of heavy ions collisions,star interior and the early universe.Moreover, the theoretical expectation of transitions to a chirally symmetric phase and,perhaps,to a quark-gluon plasma phase contributes to the interest in studying the e?ect of matter and temperature on the quantum chromodynamics(QCD)vacuum.

Our present understanding of QCD at?nite temperature(T)and baryon density(or chemical potentialμ)is mainly limited in the Euclidean realm,due to the lack of non-perturbative and systematic calculating tools directly in the Minkowski space.Typical methods,with QCD Lagrangian as the starting point,are the OPE and lattice simula-tions.Because these two formulations are intrinsically Euclidean,only static quantities are conveniently studied.In order to gain dynamical informations,which are more accessible experimentally,the analytic structure implemented through dispersion relations often have to be invoked within the theory of linear response.

In principle,dispersion relations allow the determination of the spectral function(SF), which carries all the real-time information,from the corresponding correlator in the Eu-clidean region.In practice,realistic calculations,e.g.OPE or lattice simulations,yield only partial information on the correlators,making impossible a direct inversion of the dispersion relation.

Therefore,the standard approach assumes a phenomenological motivated functional form with several parameters for the SF,and uses the information from the approximate corre-lator,mediated by the dispersion relation,only to determine the value of parameters by a?t.This approach has been quite successful at zero temperature and density,thanks to the fortuitous situation that we roughly know how to parameterize the SF’s in many cases. Two important examples are the QCD sum rules pioneered by Shifman,Vainshtein and Zakharov[1,2],and the analysis of lattice QCD data[3].So far,standard parameterizations have included poles plus perturbative continuum[4].

The success of such approaches heavily rests on our good understanding of the qualitative behavior of SF’s at zero(T,μ).We can?nd other such favorable examples in the low-T regime[5],where the shape of the SF has the same qualitatively features of the zero(T,μ) case,or even in the high-T regime for simple models[6],for which the functional form of the SF is known.

The QCD sum rules approach has been extended also to systems at?nite tempera-ture[7,8,5].The lack of experimental data,and of reliable nonperturbative calculations has prompted people to use the same kind of parameterizations that have worked so well at zero temperature with,at most,perturbative corrections.We believe that physical results at?nite(T,μ)can be strongly biased by this assumption.In fact,recent interpretations of lattice simulation data[9–11]appear to indicate the existence of such problems.

The purpose of this work is to derive exact sum rules that constrain the variation of SF’s with(T,μ).In addition,we apply these sum rules to the chiral phase transition,and demonstrate that SF’s in some channels are drastically modi?ed compared to both their zero(T,μ)and their perturbative shapes.This result con?rm our worries about non-trivial e?ect of?nite T or baryon density on the shape of the SF.

Our derivation of these exact sum rules,based on the OPE and the RGE,has a closer

relation with the derivation of sum rules for deep inelastic scatterings than with the QCD sum rule approach of SVZ[1].In fact,we establish relationships between moments of the SF and corresponding condensates as functions of(T,μ),without assuming any functional form of the SF.

In the derivation process,we?nd that the logarithmic corrections are essential to estab-lish the exact sum rules.In contrast,the QCD logarithmic corrections are only marginally relevant in the?nite energy sum rules,and hence are rarely discussed in the literature.To properly take into account the logarithmic corrections,a repeated partial integration method is used to match the relevant asymptotic expansions.

Since no further assumptions other than the validity of the OPE and RGE are involved in the derivation,our sum rules are very general and can be applied anywhere in the(T,μ)-plane,even near or at the boundary of a phase transition.

The paper is organized as follows.In section II we present the general derivation of exact sum rules in asymptotically free theories.The matching of asymptotic behaviors of the correlator and the dispersion integral,including explicitly their logarithmic corrections,are carefully discussed.In section III we illustrate each single step of the derivation in a soluble model,the Gross-Neveu model in the large-N limit.In this model we can calculate exactly all the relevant quantities(spectral functions,Wilson coe?cients,condensates,anomalous dimensions and correlators in space-like region,etc.),and,therefore,give a concrete example of how our method works.The application of our method to the derivation of exact sum rules for the mesonic channels in QCD is presented in section IV.In the same section,we also discuss the phenomenological consequences of the exact sum rules near chiral restoration phase transitions.Finally,in section V we summarize our work,draw some conclusions,and discuss possible future directions.

II.GENERAL DERIV ATION

We start this section with a short review of the linear response theory,the OPE and the RGE.Next we introduce a convenient subtraction in the dispersion relation for studying the dependence of SF’s on T andμ.

Then,we present in detail a crucial part of our method:how to match the asymptotic OPE expansion with a corresponding asymptotic expansion of the SF and its dispersion integral.This approach is necessary for properly taking into account the logarithmic cor-rections,and studying the convergence properties of the relevant moments of the SF.More naive approaches not only yield,in general,incorrect sum rules,but might also fail to rec-ognize that a given sum rule does not exist in the?rst place,since the integral of the SF involved is divergent.

Finally,a comparison of the two asymptotic expansions leads to the desired exact sum rules.We end this section with some general comments on the derivation and meaning of these sum rules.

A.Linear Response

The real-time linear response[12]to an external source,S(x),coupled to a renormalized current J(x)in the form of?L=S(x)J(x)is given by the retarded correlator:

K(x;T,μ)≡θ(x0) [J(x),J(0)] T,μ,(1) where the average is on the grand canonical ensemble speci?ed by temperature T and chemi-cal potentialμ.Disregarding possible subtraction terms,which are(T,μ)-independent since they are related to short distance properties of the theory,we can write the following dis-persion relation for the frequency dependence of the retarded correlation function:

?K(ω,k;T,μ)= ∞0du2ρ(u,k;T,μ)

+2ΓJ?Γn C n(Q2,m2(κ),g2(κ),κ)=0,(4a)

dκ

where

κd

?κ

+β

?

?m

,(4b)

andΓJ,Γn andΓm are the anomalous dimensions for the current J,the operator O n and the current quark mass m respectively.For the purpose of illustration,we are only considering operators that do not mix under renormalization,but the mixing will be properly taken into account when necessary.The standard approach[13]to the renormalization group equation is the introduction of a running coupling g2(Q)and a running mass m(Q). In asymptotically free theories g2(Q)vanishes logarithmically at large Q2.It is therefore meaningful to consider,in this limit,a perturbative expansion of the renormalization group equation functions

Γi=?γi g2+O(g4),andβ=?bg4+O(g6),(5) where i=J,n,m,whileγi,b are pure numbers determined by a one-loop calculation. Within this perturbative context,Eq.(4)can be solved[13]:

C n(Q2,m(κ),g2(κ),κ)= 12 g2(Q)2b×c n(m2(Q)

u2+Q2

,(7) where?ρ(u)≡ρ(u;T,μ)?ρ(u;T′,μ′).This subtraction is crucial to remove?K0(iQ,κ), which contains terms not suppressed by powers of1/Q2and is explicitly dependent on the renormalization point.The OPE asymptotic expansion of??K(iQ)is then

??K(iQ)～ n 12 g2(Q)2b×c n(m2(Q)

D.Asymptotic Expansion

Since c n(m2(Q)/Q2,g2(Q))is perturbative and hence can be expanded in power of g2(Q)

(and denoting the?rst non-vanishing power asν(0)n),the left-hand side of Eq.(7)can be expressed as a double(in1/Q2and g2(Q))asymptotic expansion of the form:

∞ n,ν=0c(ν)n(κ)? [O n]κ

??K(iQ)～

∞ ν=0a(ν)n(T,μ)[ln(u2)]?ν,(10b)

u2(n+1)

where we have explicitly isolated in?ρexp(u)all terms that vanish exponentially when u2→∞,such as the pole contributions to?ρ(u)or terms containing the factor exp(?u/T). To simplify the notation we have chosen the units such that the running coupling has the form at one-loop level g2(u)=1/(b ln u2)(or equivalently,energy scales are measured in units of the relevantΛ-parameter).

This ansatz is su?cient to produce an asymptotic series of the form of Eq.(9)to one-loop level.More generally,one could replace1/ln u2with the full running coupling g2(u) in the asymptotic sequence and generalize the method we are going to describe;the general strategy involved in this generalization can be found,for example,in Ref.[14].Since we are presently only interested in one-loop calculations,we can regard g2(Q)and1/ln Q2to be proportional.It is easy to recognize that the sum over“n”is meant to match the sum over mass dimension in Eq.(9),while the sum over“ν”will match the sum over the order in perturbation series.The existence of anomalous dimensions in the OPE makes it necessary to introduceξn in the expansion for the spectral function.In the following we ignore the dependence ofξn on“n”,as we have already ignored the fact that there generally exist more than one operator at a given dimension,and writeξto avoid a too cumbersome notation. The complete notation will be restored when necessary.

Before we proceed further,we wish to discuss whether it is possible that additional terms might appear in the expansion of?ρpower(u)in Eq.(10b).In general,our asymptotic expansion procedure is powerful enough to exclude this possibility within the framework of the OPE.In fact,it allows to verify unambiguously that terms di?erent from the ones already present generate,when substituted in the dispersion integral,terms that are missing in the

OPE.Let us examine two speci?c examples that might be suspected to exist otherwise.First,

√

dimensional arguments could suggest terms like g2T/

1/√Q 2term in ??K (iQ ),which does not correspond to any known condensate,and it is therefore excluded.The second example is given by terms such as ln(T 2/u 2),which are naturally produced by elementary perturbative calculations of the spectral function:are such terms present in Eq.(10b)?It is indeed true that such terms appear in the high-T expansion of the spectral function at ?xed u 2.However,the expansion of the spectral function that is relevant for comparing to the OPE is a high-u 2expansion at ?xed T .In the next section an explicit calculation in the Gross-Neveu model will illustrate the general fact that,contrary to the high-T expansion,the high-u 2expansion does not generate terms like ln(T 2/u 2).

At this stage,we can already recognize a fundamental,and often overlooked,character-istic of the spectral function.If we insert the term ?ρexp (u )in the dispersion integral,we only obtain pure powers of 1/Q 2,since the exponential convergence allows a naive expan-sion of the factor 1/(u 2+Q 2)(Watson’s Lemma [15]).Therefore logarithmic corrections,i.e.powers of g 2(Q ),come solely from the ?ρpower (u )term.Because we know that the running coupling g 2(Q )is always present in the OPE series,the term ?ρpower (u )must be present in the subtracted SF,and it obviously dominates the asymptotic regime (u 2

→∞).This fact immediately implies that only a ?nite number of moments of the subtracted SF can possibly be ?nite,i.e.the naive expansion of the factor 1/(u 2+Q 2)is generally wrong,and that logarithmic corrections play a important role.

A standard method to tackle the dispersion integral in the large-Q 2limit is the Mellin transform.However,the use of Mellin transform methods [15]is extremely cumbersome when logarithms appear in the denominator.Since inverse logarithms cannot be avoided in the spectral function,we need to resort to other means.

We carry out the dispersion integral of the ?ρpower (u )term by splitting the integral in Eq.(7)into three intervals:(0,λ2),(λ2,Q 2)and (Q 2,∞).

The integral over the ?rst interval can be naively expanded in powers of 1/Q 2,since there are no convergence problems:

??K 1(iQ )≡ λ20du 2?ρpower (u )

Q 2∞ n =0

(?1)n Q 2 λ2

0du 2?ρpower (u )+O (1/Q 4).(12)

In the second interval,we use the asymptotic form of ?ρpower given in Eq.(10b)and obtain (for instance by repeatedly integrating by parts)

??K 2(iQ )≡ Q 2

λ2du 2?ρpower (u )

Q 2∞

n,ν=0a (ν)

n (?1)n α

+∞ l =0l =n (?1)l (n ?l )α

(13)

where α≡1?ν?ξ,and Γ[α,z ]is the incomplete Gamma function.Notice that when α≡1?ν?ξ=0the correct result for the term with l =n is ln ln Q 2

its limiting value:lim α→0[(ln Q 2)α?(ln λ2)α]/α=ln ln Q 2

z m Γ(1?α)

.(14)The resulting expression for ??K

2(iQ )is ??K

2(iQ )～1Q 2l ∞ n =0n =l a (ν)n Γ(α,(n ?l )ln λ2)α (15)?1

Q 2n ∞ l =0

l =n ∞

m =0

(?1)l +m Γ(1?α)?(?1)n [ln Q 2]α(n ?l )α?a (ν)l (ln λ2)

αQ 2

∞ ν=0 ∞ n =1a (ν)n Γ(α,n ln λ2)α (17)

+1l m +1[ln Q 2]m +1?α

Γ(m +1?α)α +O 1

u 2+Q 2

～?1(n +l )α～?

1Q 2n ∞ l =1∞ m =0(?1)l +m Γ(1?α)～?1l m +1[ln Q 2]m +1?αΓ(m +1?α)

Q 4 .(18)

In the end we add the leading 1/Q 2contributions from Eqs.(12),(18)and (18)to the corresponding contribution from the naive expansion of ?ρexp and obtain

??K (iQ )≡ ∞

0du 2?ρ(u )

?ρQ 2

∞ ν=0a (ν)01?ξ?ν+2

∞ l =1

m =0(?1)l Γ(ν+ξ) +O 1

where

?ρ≡

λ20du 2?ρpower (u )+ ∞0du 2?ρexp (u )+

∞ n =1ν=0a (ν)n Γ(1?ν?ξ,n ln λ2)1?ν?ξ.(20)Since the integral ∞dxx ?1(ln x )?ξcan be ?nite only if ξ>1,we can use Eq.(16)to identify

?ρas de?ned by Eq.(20)still exists.

E.Exact Sum Rules

We derive the sum rules by comparing the coe?cient of 1/Q 2in Eq.(19)

Q 2

∞0du 2?ρ(u )?ρ+a (0)0× [ln Q 2]1?ξ

Q 2,1

Q 2,[g 2(Q )]η+1 ,(22)

where we are only considering cases with n =1and d 1=2.Furthermore,we are presently interested in exact sum rules that can be derived with one-loop calculations and,therefore,we only compare leading orders in 1/ln Q 2.Then there exist three possibilities,depending on the value of η,which can be calculated using the OPE.

(1)If η=0,then ξ=2(or a bigger integer)and

?ρ= ∞

0du 2?ρ(u )=0.(24)

(3)If η<0(positive powers of ln Q 2in Eq.(8))or if the term ln ln Q 2appears in Eq.(8),then ξ<1or ξ=1,respectively.This implies that

We remark that,even in this case when the moment is not?nite,the asymptotic expansion is still well de?ned.It is nice to see that whether the zeroth moment of the subtracted SF exists is re?ected directly through the leading power of g2(Q)in the OPE series.

Our main results,Eq.(23)and Eq.(24),can be expressed in physical terms as follows. The zeroth moment of a SF for a current J whose OPE series yieldsη>0is independent of T andμ,while the same moment for a current withη=0changes with T andμproportionally to the corresponding change(s)of the condensate(s)of the leading operator(s).Although c(0)1(κ)and? [Q1]κ can separately depend onκ,their product must be independent ofκ, since the zeroth moment is independent ofκ.

F.Discussion

At this point several general comments are appropriate:

(1)Our derivation relies on the fact that an asymptotically free theory allows a perturbative expansion at short distances,making practical the use of the OPE and of the RGE.We understand why only short distance physics is involved if we realize that the integral over frequencies reduces Eq.(1)to the ensemble average of the equal-time commutator of the currents.Therefore,results such as Eq.(23)and Eq.(24)are completely determined by the one-loop structure of the theory and the particular current under exam.

(2)Flavor,or other non-dynamical quantum numbers,does not change the expansions at the one-loop level in an essential way.Therefore,one can derive analogous sum rules by using other kind of subtractions,instead of the one we adopted.One such example is given by the exact Weinberg sum rules(T=0)in the chiral limit[16].

(3)The derivation of sum rules for higher moments of the SF requires the complete cance-lation of all the lower dimensional operator terms,not just the leading g2(Q).In particular, one also needs current quark mass corrections to the Wilson coe?cients.Without appropri-ate subtractions,higher moments do not even converge[17].

(4)It is essential to properly take into account the logarithmic corrections when deriving ex-act sum rules,since the logarithmic corrections not only dictate whether

?ρ[17].This procedure is in sharp contrast with the usual QCD sum rule approach,where the convergence issue is by-passed by applying the Borel improvement by explicitly introducing a cut-o?parameter(the Borel mass).

(5)We believe that the(T,μ)-dependent part of the leading condensate appearing in Eq.(23) does not su?er from the infrared renormalon ambiguity.In fact,only the perturbative term?K0can generate contributions to the leading condensate that are dependent on the prescription used to regularize these renormalons.But?K0is independent of T andμ:any prescription dependence cancels out when we make the subtraction in Eq.(7).On the contrary,unless we generalize Eq.(7)and make other subtractions,sum rules that involve non-leading condensates are,in principle,ambiguous.

(6)It is well-known that conserved operators are not renormalized and,barring anomalous violations,verify the same“classical”identities that can be derived at the tree level.This fact is also veri?ed in the sum rules.In fact when both the currents and the operators are conserved(η=0)and we obtain the result of Eq.(23),i.e.one can use the“naive

asymptotic expansion”to derive the sum rule.Conservation of the current alone is not enough to warrant a“classical”identity.

III.GROSS-NEVEU MODEL

In the preceding section we have derived sum rules valid for any asymptotically free theory.In this section we illustrate the procedure in the1+1dimensional Gross-Neveu model[18]in the in?nite-N limit.On one hand,we can derive the sum rules in Eqs.(23) and(24)by explicitly calculating Wilson coe?cients,β-andΓ-functions in the vector and pseudoscalar channels,following the general procedure discussed in section II.On the other hand,since this model is soluble,we can obtain the exact spectral function at arbitrary (T,μ),and then explicitly verify both that the sum rules are satis?ed and that the asymptotic expansion of the spectral function has the form given in Eq.(10b).Moreover,we can also calculate the condensates,and therefore explicitly check that the OPE really matches the asymptotic expansion of the exact spectral integral.

The Lagrangian of the Gross-Neveu model is

L=ˉψiγ·?ψ+g2

(σ2+π2),(27)

2g2

whereσandπare auxiliary?elds.The coupling constant g2is independent of N and held?xed in the limit N→∞.This model is asymptotically free when D=1+1,and the chiral symmetry is dynamically broken at T=0andμ=0to the leading order in 1/N[18].In the following we give the exact solution at N→∞with?nite(T,μ).Although formulas are explicitly considered in the symmetry breaking phase,they are also valid in the symmetric phase provided that the vanishing limit of certain condensates(such as the dynamical fermion mass)is properly taken.

We wish to remark on the well-known fact that the limit N→∞here should not be interpreted as a starting point for an expansion of the model at?nite N,but rather as di?erent model in itself,which is in fact the model we decided to use for the purpose of illustration.Moreover,there exist arguments[19,20]suggesting that this model in the limit N→∞is actually more relevant to3+1phenomenology than the model with?nite N.

A.Exact solution in the large-N limit

Since the procedure to obtain the exact solution in the large-N limit is rather stan-dard and some of the intermediate steps can be found in the literature,see for instance Refs.[18,21,19,6,22],we only give the de?nitions and?nal results.

1.Gap equation and phase diagram

The gap equation and the phase diagram can be derived from the ?rst derivative of the e?ective potential

?V (σ2)

2π π(iωn +μ)2?(k 2+σ2)

ωn =(2n +1)πT .

(28)

We introduce a momentum cuto?Λand add the counterterm N ?σ2=N g 2+1κ2+ ∞

0dk f (√

√

2π 1M 2+ ∞0dk f (√

√

e (x +μ)/T +1+

1

4π σ2 ln σ2

(1+e ?(√

k 2+σ2?μ)/T ) .(31)

The gap equation is simply given by

?V (σ2)

2 M 0dk

cosh[(k +μ)/2T ]cosh[(k ?μ)/2T ]? ∞M dk

the minimum given by Eq.(33).Since

the mass gap is?nite on the dash line(given by

the solution of V(σ2)=V′(σ2)=0withσ=0,where prime denote a derivative with respect toσ2),the dash line is a?rst order phase boundary.The critical temperature at

μ=0is given by T c=(eγE/π)M≈0.566933M.The“tricritical point”(the heavy dot in Fig.1)can be found by imposing that V′(σ2→0)=V′′(σ2→0)=0:T3≈0.318328M,μ3≈0.608221M[19].When T=0the chemical potential at which the?rst order phase transition

takes place isμc=M/

√

πln(

κ

2π

1

1+4m2/Q2

ln √Q2

Q2+4m2?√

π ∞0dk f(E k;T,μ)E k

m2+k2.We shall also use the

short-hand notation

A ≡1E

k

A(k)f(E k;T,μ).(35) For instance,the use of this notation makes possible to write the gap equation as

ln κ

g2

= ∞0dk

N

Π(ω) ?1.(37)

In the Euclidean region,we can then expand this correlator in the Q2→∞limit as

1

1+g2ln(Q2/κ2)/2π?2g2(Q)m2/g4

2π

+ g2(Q)Q2 E2k ?m2Q4 ,(38)

where we have introduced the running coupling constant

g 2(Q )≡g 2

2

m 2 E 0k = [θ00] T,μis the condensate of the energy-momentum tensor.We shall see that the perturbative contribution and the coe?cients of the condensates are indeed the ones

obtained by the OPE of J 5J 5,and therefore the expansion of the correlator shown in Eq.(38)is of the form of the OPE.

Note that in de?ning the condensates we have absorbed factors of N in order to simplify the notation. 3.Spectral function

To obtain the spectral function we need the bubble graph in the time-like region ω2>4m 2:

1

4 2T +tanh |ω|/2?μ

N Re Π(ω) =?11?4m 2/ω2

1?4m 2/ω2+ω2E k f (E k ;T,μ)

g 4+θ(ω2?4m 2)ρcont (ω).(42)

The pole contribution comes from the bound state (the pion),whose mass is the solution to the equation

N

?ω2Π(ω)/N

?1ω2=m 2π=0.(44)

The continuum part of the spectral function is related to the bubbleΠ(ω)in the time-like region through

ρcont(ω)=

ImΠ(ω)

N

Im(Π(ω))～g2?(ω) 1ω2+O(1

N

Re Π(ω) ～g2ω2 2m2(1?g2(ω)ω4) ,(47) which leads to the following asymptotic form for the continuum part of the spectral function

1

2(g2(ω)/g2)2

ω2 [(

ˉψψ)2]

T,μ

+4g2(ω)

g2 2 [θ00] T,μ 1+O(g2(Q)) .(48)

Here we only concern with the g2(Q)-dependent terms and postpone the pure1/Q2terms (related to?ρexp)to a later subsection.We recognize again the?rst term in Eq.(48as the asymptotic perturbative spectral function.Moreover,the(T,μ)-dependent part of Eq.(48) has indeed the general form of Eq.(10b).

In connection with the comments we made after Eq.(10b),we point out that Eq.(48) has been obtained by expanding the spectral function in the limit Q2→∞at?xed T.Had we made instead a high-T expansion(T→∞)at?xed Q2,we would have obtained,for instance,ln(T2/ω2)from the third term of Eq.(41).

Upon substituting Eq.(48)in the dispersion integral,and using the identity

dω2ω2～2πQ2 1+O(g2(Q))

whereν>1and we have kept only the g2(Q)-dependent part,we?nd the following series to the leading order in g2(Q),

dω2N～ J5J5 (0)(Q)Q2+4 g2(Q)Q2+ (49)

This result is in agreement with the OPE series of Eq.(38),which has been directly expanded from the correlator in the Euclidean region.

The fact that no pure1/Q2term appears in Eq.(49)implies that the zeroth moment of the subtracted spectral function vanishes:

ρ=0.

B.Direct solution in terms of the OPE

We have just veri?ed that in the Gross-Neveu model the exact correlation function can indeed be expanded in an asymptotic series precisely in the form of the OPE.Now we shall calculate the Wilson coe?cients,β-function and appropriate anomalous dimensions for the OPE,and verify that they match the coe?cients and exponents of this asymptotic series. Furthermore,though it is beyond the scope of the OPE and RGE,we shall also calculate the condensates.In this way,we explicitly verify that the asymptotic expansion of the exact correlation function is identical to the OPE.

1.One-loop calculation

The one-loopβ-function of the1+1dimensional Gross-Neveu model has already been calculated in the original paper[18]:β=?g4/2π,i.e.according to our notation b=1/2π. The anomalous dimensions for pseudoscalar current J5≡ˉψiγ5ψand four-quark operator O4=(ˉψψ)2can be calculated using the Feynman diagrams shown in Fig.4a and Fig.4b,

respectively.According to the notation of Eq.(5)we?ndγJ

5=1/π,andγO

4

=2/π.The

energy-momentum conservation makes the anomalous dimension forθ00vanish,i.e.γθ

00

=0.

The relevant Wilson coe?cients corresponding to operators O4andθ00can be calculated, to leading order in g2(Q),using the Feynman diagrams shown in Fig.5a and Fig.5b,

respectively.The resulting coe?cient are c O

4=?2g2(Q)(ν(0)O

4

=1according to notation in

Eq.(9))and cθ

00=4(ν(0)θ

00

=0).

These results lead to the exponentsηO

4=1+(2γJ

5?γO4)/2b=1andηθ00=(2γJ5?

γθ

00

)/2b=2,which are precisely what we found in Eq.(49).

These explicit one-loop calculations exactly match the solution in Eq.(49).

At last a comment on the one-loop calculation:this result is exact to the leading order in 1/N and only involves insertion of the quark-bubble chain.Since the quark-bubble behaves like∝ln Q2,this insertion is equivalent to substituting the coupling constant g2with the running coupling constant g2(Q).So one can easily identify those sets of Feynman graphs whose sum leads to the solution of the RGE in Eq.(4).

2.Condensates

The OPE itself does not specify how the relevant condensates are calculated,since the OPE is only a RGE-improved perturbative procedure.The calculation of the condensates can be done only in a non-perturbative context.Here,we carry out this calculation using the1/N expansion.

The bare quark condensate ˉψψ T,μis given by the dynamical quark tadpole graph ˉψψ T,μ=(?iT) Λdk k.γ?m=2mT Λdk(iωn+μ)2?(k2+m2).(50)

Using the standard contour integral technique to carry out the Matsubara frequency sum, we obtain

ˉψψ T,μ=m E0k ?m m .(51) The renormalized quark condensate is obtained by replacing2Λwith the subtraction point κin the above equation

[ˉψψ] T,μ=m E0k ?m m =?m

2π n Trγ0k0i2π n(iωn+μ)2

After removing the(T,μ)-independent volume and quadratic divergences,and introduced a subtraction point for the logarithmic divergence the renormalized kinetic energy becomes

[ˉψiγ0?0ψ] T,μ= E2k ?m2m .(54)

After subtracting the trace term from the kinetic energy,we?nally obtain the expectation value of the traceless energy-momentum tensor

[θ00] T,μ≡ [ˉψiγ0?0ψ?12 E0k .(55)

In the symmetric phase(m=0),the expectation value of the energy-momentum tensor has the very simple form atμ=0

[θ00] T,μ=0=π

ω4

),(57) where

?ρ2(ω)=4

g4 ??m2π?(m2+4π [θ00] T,μ)

A2

),(59)

where the integral of?ρ2(ω)can be done analytically and we obtain

A20dω2?ρ(ω)=ππtan?12ln A(ln A)2+π2/4 +O(1

π 1?8m2

Q2(Q2+4m2)

tanh?1 Q2+4m2+2m2 ∞0dk E2k+Q2/4 .

(61) We de?ne the spectral function in vector channel as

JαJα T,μ(iQ)=1ω2+Q2 .(62) Then we?nd the following explicit expression for?ρ

?ρ(ω)=4m2θ(ω2?4m2)

ω2(ω2?4m2)

1?2

the section II we have shown that,corresponding to this divergence,we must?nd negative powers of g2(Q)(positive power of ln(Q2))in the OPE series.

When there are no explicit logarithms in?ρ(ω),it is easier to make the asymptotic ex-pansion by using the Mellin transform method,and in particular the convolution property of the Mellin transform[15],

∞0dt f(t)h(t)=1

πΓ(1?z)/Γ(3/2?z).(67) Using the convolution,Eq.(64),and the Mellin transform of1/(u2+Q2)

M[(u2+Q2)?1;1?z]=π(Q2)?z/sinπ(1?z),(68) we?nd(in this case c in Eq.(64)obeys0

∞0dω2?ρpower Q2 z√Γ(3/2?z)π

ln Q2Q4 1+ln Q2Q6 .(69)

Q2

Similarly to the pseudoscalar correlator,it is possible to check this result in two ways.We can directly expand the power part(temperature independent part)of Eq.(61),and we can

=8g2(Q)for the1/Q2term).

derive the OPE(γV=0and c O

4

As promised,there indeed appears a positive power of ln(Q2)in the OPE series.At the same time,the singularities of M[?ρpower;z]at z=positive integer hints the fact that all the non-negative integer moments of?ρpower do not exist.On the other hand,the asymptotic expansion in Eq.(69)is well de?ned.Even though the vector current is conserved,due to the fact that the anomalous dimension of O4is not zero,the correct result can not be obtained by a naive asymptotic expansion in this case,in accordance with the remark number(6)at the end of section II.

This same result could have been obtained also by the general method developed in section II,but it should be clear by now that the use of the Mellin transform,when possible, is more straightforward and require less labour.

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WRFChem安装及使用说明手册

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