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LOW–ENERGY BEHAVIOR OF TWO–POINT FUNCTIONS

arXiv:hep-ph/9306323v1 28 Jun 1993CERN TH 6924/93CPT-93 / P.2917NORDITA-93/43 N,PLOW–ENERGY BEHAVIOR OF TWO–POINT FUNCTIONSOF QUARK CURRENTSJohan BijnensNORDITA, Blegdamsvej 17DK–2100 Copenhagen ø, DenmarkEduardo de RafaelCentre de Physique Th´eoriqueCNRS - Luminy, Case 907F 13288 Marseille Cedex 9, FranceandHanqing ZhengCERN, CH–1211 Geneva 23, SwitzerlandNORDITA-93/43 N,PCPT-93 / P.2917CERN-TH 6924/93May 1993

AbstractWe discuss vector, axial-vector, scalar and pseudoscalar two-point func-tions at low and intermediate energies. We first review what is known fromchiral perturbation theory, as well as from a heat kernel expansion withinthe context of the extended Nambu-Jona-Lasinio (ENJL) model of ref.

[13].In this work we derive then these two-point functions to all orders in themomenta and to leading order in 1/Nc. We find an improved high-energybehaviour and a general way of parametrizing them that shows relations be-tween some of the two-point functions, which are also valid in the presence ofgluonic interactions.

The similarity between the shape of the experimentallyknown spectral functions and the ones we derive, is greatly improved with re-spect to those predicted by the usual constituent quark like models. We alsoobtain the scalar mass MS = 2MQ independent of the regularization scheme.In the end, we calculate fully an example of a nonleptonic matrix element inthe ENJL–model, the π+ −π0 electromagnetic mass difference and find goodagreement with the measured value.

1INTRODUCTION AND REVIEW OF KNOWN RESULTS IN QCDWe shall be concerned with two–point functions of the vector, axial–vector, scalarand pseudoscalar quark currents:V (a)µ (x)≡¯q(x)γµλ(a)√2 q(x)(1)A(a)µ (x)≡¯q(x)γµγ5λ(a)√2 q(x)(2)S(a)(x)≡−¯q(x)λ(a)√2 q(x)(3)P (a)(x)≡¯q(x)iγ5λ(a)√2 q(x)(4)where λ(a) are Gell-Mann SU(3)–matrices acting on the flavour triplets of Diracspinors: ¯q ≡(¯u(x), ¯d(x), ¯s(x)). Summation over the colour degrees of freedom of thequark fields is understood.

These are the quark currents of the QCD Lagrangianwith three light flavours u, d, s, in the presence of external vector vµ(x), axial vectoraµ(x), scalar s(x) and pseudoscalar p(x) field matrix sources; i.e.,LQCD(x) = L0QCD + ¯qγµ(vµ + γ5aµ)q −¯q(s −iγ5p)q ,(5)withL0QCD = −148XA=1G(A)µν G(A)µν + i¯qγµ(∂µ + iGµ)q(6)andGµ ≡gsN2C−1XA=1λ(A)2 G(A)µ (x)(7)the gluon field matrix in the fundamental SU(NC = 3)colour representation, withG(A)µν the gluon field strength tensorG(A)µν (x) = ∂µG(A)ν−∂νG(A)µ−gsfABCG(B)µ G(C)ν,(8)and gs the colour coupling constant (αs = g2s/4π).We want to consider the set of two–point functions:ΠVµν(q)ab=iZd4xeiq·x < 0|TV (a)µ (x)V (b)ν (0)|0 >(9)ΠAµν(q)ab=iZd4xeiq·x < 0|TA(a)µ (x)A(b)ν (0)|0 >(10)ΠSµ(q)ab=iZd4xeiq·x < 0|TV (a)µ (x)S(b)(0)|0 >(11)1

ΠPµ (q)ab=iZd4xeiq·x < 0|TA(a)µ (x)P (b)(0)|0 >(12)ΠS(q)ab=iZd4xeiq·x < 0|TS(a)(x)S(b)(0)|0 >(13)ΠP(q)ab=iZd4xeiq·x < 0|TP (a)(x)P (b)(0)|0 > . (14)The relevance of these two–point functions to hadronic physics, within the frame-work of current algebra, was emphasized a long time ago (See refs.

[1] to [4].). Withthe advent of QCD it became possible to compute the short–distance behavior ofthese two point functions using perturbation theory in the colour coupling con-stant because of the property of asymptotic freedom: αs(Q2 ≡−q2 >> Λ2QCD) ∼log−1(Q2/Λ2QCD), (See refs.

[5] to [7].). Non–perturbative corrections at large Q2 ap-pear as inverse powers in Q2 [8].

The inclusion of these corrections, combined witha phenomenological ansatz for the corresponding hadronic spectral functions at lowenergies, has developped into the active field of QCD sum rules. (See e.g.

ref. [9] fora review and references therein.) The QCD behaviour of the two point functionsabove at very low Q2 values, is controlled by chiral perturbation theory (χPT).

Itis important for our later purpose to review here what is known in this respect.From Lorentz–covariance and SU(3) invariance, the two–point functions abovecan be decomposed in invariant functions as follows:ΠVµν(q)ab = (qµqν −q2gµν)Π(1)V (Q2)δab + qµqνΠ(0)V (Q2)δab(15)ΠAµν(q)ab = (qµqν −q2gµν)Π(1)A (Q2)δab + qµqνΠ(0)A (Q2)δab(16)ΠSµ(q)ab=ΠSM(Q2)qµδab(17)ΠPµ (q)ab=ΠPM(Q2)iqµδab(18)ΠS(q)ab=ΠS(Q2)δab(19)ΠP(q)ab=ΠP(Q2)δab . (20)The low energy behaviour of these invariant functions in SU(2)L ×SU(2)R χPT hasbeen worked out in ref.[10].

It is easy to extend their analysis to SU(3)L × SU(3)RχPT. In the chiral limit, where the quark mass matrix M →0; and with neglectof chiral loops, which are non leading in the 1/NC–expansion [11], the results are asfollows:Π(1)V (Q2) = −4(2H1 + L10) + O(Q2)(21)Π(0)V (Q2) = 0(22)2

Π(1)A (Q2) = 2f 20Q2 −4(2H1 −L10) + O(Q2)(23)Π(0)A (Q2) = 0(24)ΠSM(Q2) = 0(25)ΠPM(Q2) = 2B0f 20Q2(26)ΠS(Q2) = 8B20(2L8 + H2) + O(Q2)(27)ΠP(Q2) = 2B20f 20Q2+ 8B20(−2L8 + H2) + O(Q2) . (28)The functions Π(1)A , ΠPM and ΠP get their leading behaviour from the pseudoscalarGoldstone pole.

The residue of the pole is proportional to the pion decay constantf0 (f0 ≃fπ = 93.3MeV ). The constant B0 is related to the vacuum expectationvalue< 0|¯qq|0 >|q=u,d,s= −f 20B0 (1 + O(M)) .

(29)The constants L8, L10, H1 and H2 are coupling constants of the O(p4) effectivechiral Lagrangian in the notation of Gasser and Leutwyler [12].The constantsL8 and L10 are known from the comparison between χPT and low energy hadronphenomenology. At the scale of the ρ mass:L8 = (0.9 ± 0.3) × 10−3(30)andL10 = (−5.5 ± 0.7) × 10−3 .

(31)The constant H1 and H2 correspond to couplings which involve external source fieldsonly and therefore cannot be extracted from experiment unambiguously.It has been recently shown [13] that the extended Nambu Jona–Lasinio model(ENJL–model) describes the values of the low energy parameters rather well. Inref.

[13], various relations between the parameters of a low energy effective La-grangian were obtained that were independant of the input parameters and possiblelow energy gluonic corrections. Among these was the relation3

f 2V M2V −f 2AM2A = f 2π(32)between couplings and masses of the lowest vector and axial–vector mesons and thepion–decay constant. This corresponds to what is usually called the first Weinbergsum rule [1].

The second Weinberg sum rule, in the lowest resonance saturationform i.e., f 2V M4V = f 2AM4A, was, however, not satisfied exactly though the deviationwas numerically small. Part of the motivation which triggered our interest in thepresent work has been to understand the origin of this unsatisfactory result.

Aswe shall show, the low energy expansion method used in ref. [13] is inappropriate todraw conclusions about the intermediate Q2–behaviour of two–point functions.

Therelevant contributions can however be easily summed using a Feynman–diagramexpansion of the four fermion couplings in the ENJL–model without introducingcollective fields. Section 3 gives the details of this summation method.

The sameresults could be obtained of course using the collective field method, provided thoughthat all higher orders in the Q2–expansion were kept.It is well known that the Weinberg sum rules play a crucial role in the calculationof the electromagnetic π+–π0 mass difference in the chiral limit. In a previous cal-culation of this mass difference [14] within the framework of the effective action ap-proach [15], it was shown that the quality of the matching between long–distance andshort–distance contributions to the photon–loop integral is still rather poor whenthe vector and axial–vector spectral functions are replaced by constituent quarkspectral functions alone.

We shall (see also ref. [16]) discuss this problem again insection 4, within the framework of the ENJL–model and with the full Q2/M2Q depen-dence summed.

This calculation, which as we shall see is rather succesful, providesthe first non–trivial example of a genuine one loop calculation in the ENJL–model.There are other applications one can now think of doing; in particular calculationsat the next to leading order in the 1/NC–expansion. We plan to investigate this inthe future.Throughout this paper we shall work in the chiral limit.

The inclusion of cor-rections, whenever neccesary, due to nonzero current quark masses can however bedone with the present technology.The rest of the paper is organized as follows. In section 2 we give an overview ofthe ENJL–model and the two-point functions in this model in the low-energy limit.We also describe here the parametrization usually used to go beyond the low-energyexpansion in this model.

Section 3 is the mainstay of this work. Here we derivethe two-point functions to all orders in the momenta.

We also discuss how gluoniccorrections can be included and present numerical results. In the next section we usethese two-point functions to calculate completely in the ENJL–model a non-leptonic4

matrix-element, the π+ −π0 electromagnetic mass difference. In the last section wepresent our main conclusions.

In the appendix we derive the underlying relationsthat allow for the same type of results for the two-point functions, as those foundin ref. [13] for the low-energy constants.2TWO–POINT FUNCTIONS IN THE ENJL–MODEL2.1A brief review of the ENJL–modelThe scenario suggested in ref.

[13], assumes that at intermediate energies below orof the order of the spontaneous chiral symmetry breaking scale Λχ, the Lagrangianof the ENJL–model is a good effective realization of the standard QCD Lagrangiani.e.,LQCD →L∗χQCD + LS,PNJL + LV,ANJL + O( 1Λ4χ)(33)withLS,PNJL(x) = 8π2GS(Λχ)NcΛ2χXa,b(¯qaRqbL)(¯qbLqaR)(34)andLV,ANJL(x) = −8π2GV (Λχ)NcΛ2χXa,bh(¯qaLγµqbL)(¯qbLγµqaL) + (L →R)i. (35)Here a and b are u, d, s flavour indices and colour summation within each quarkbilinear bracket is implicit; qL = 12(1−γ5)q(x) and qR = 12(1+γ5)q(x).

The couplingsGS and GV are dimensionless quantities. In principle they are calculable functionsof the ratio of the cut–offscale Λχ to the renormalization scale ΛMS.

In practice thecalculation requires knowledge of the non-perturbative behaviour of QCD and GSand GV will be taken as independent unknown constants. In choosing the forms (34)and (35) of these four quark operators we have only kept those couplings that areallowed by the symmetries of the original QCD Lagrangian, and which are leading inthe 1/NC–expansion [11].

With one inverse power of NC pulled–out, both couplingsGS and GV are O(1) in the large NC limit. The Λχ index in LΛχQCD means that onlythe low frequency modes of the quark and gluon fields are to be included.The basic assumption in considering the ENJL–model as a good effective La-grangian of QCD is that at intermediate energies below or of the order of the spon-taneous chiral symmetry breaking scale, the operators LS,PNJL and LV,ANJL are the lead-ing operators of higher dimension which, due to the growing of their couplings GSand GV as the ultraviolet cut-offapproaches its critical value from above, becomerelevant or marginal.As is well known in the Nambu Jona-Lasinio model [17], the LS,PNJL operator, forvalues of GS > 1, is at the origin of the the spontaneous chiral symmetry breaking.5

It is this operator which generates a constituent chiral quark mass term (U is aunitary 3×3 matrix which collects the pseudoscalar Goldstone field modes):−MQ(¯qLU†qR + ¯qRUqL) = −MQ ¯QQ ,(36)like the one which appears in the Georgi–Manohar model [18]; as well as in theeffective action approach of ref. [15].As discussed in ref.

[13], the LV,ANJL operator is at the origin of an effective axialcoupling of the constituent quark fields Q(x) with the Goldstone modesi2gA ¯Qγµγ5ξµQ ,(37)withgA =11 + 4GV ǫΓ(0, ǫ) ,ǫ ≡M2Q/Λ2χ(38)and Γ(0, ǫ) the incomplete gamma function:Γ(0, ǫ) =Z ∞ǫdzz e−z . (39)For ǫ small,Γ(0, ǫ) = −log ǫ −γE + O(ǫ) ,(40)with γE = 0.5772 · ··, Euler’s constant.

We recall that in eq. (37)ξµ = inξ† [∂µ −i(vµ + aµ)] ξ −ξ [∂µ −i(vµ −aµ)] ξ†o;(41)withU = ξξ(42)andQ = QL + QR(43)QL = ξqL ,QR = ξ†qR .

(44)The appearance of incomplete gamma functions is due to the proper time regu-larization which is used in calculating the fermion determinant via the Seeley–DeWitt expansion. The type of integrals which appear are (ǫ = M2Q/Λ2χ)Z ∞1/Λ2χdττ116π2τ 2τ ne−τM2Q=116π21(M2Q)n−2Z ∞ǫdzz e−zzn−2=116π21(M2Q)n−2Γ(n −2, ǫ) ;n = 1, 2, 3, · · ·(45)6

The result for the axial coupling gA in eq. (38) is the result to leading order inthe 1/NC–expansion.

In terms of Feynman diagrams it can be understood as theinfinite sum of constituent quark bubbles shown in Fig.1a, where the cross at the endrepresents the pion field. These are the diagrams generated by the GV – four fermioncoupling to leading order in the 1/NC–expansion [19].

The quark propagators inFig.1a are constituent quark propagators, solutions of the Schwinger–Dyson equationin the large NC approximation, which diagrammatically is represented in Fig.1b.In its simplest version where one assumes furthermore that all the relevant gluoniceffects for low energy physics can be absorbed in the couplings GS and GV , theENJL–model has only three free parameters: GS, GV and Λχ. We find it usefulto specify these in terms of MQ, Λχ and gA instead.

For this purpose one shouldremember that (ǫ = M2Q/Λ2χ)G−1S = ǫΓ(−1, ǫ) = e−ǫ −ǫΓ(0, ǫ) . (46)2.2Low Q2–behaviour of two–point functions in the ENJL–model.As we discussed in the previous section, the low–energy behaviour of two pointfunction quark currents, is governed by the constants f 2π, B0, L8, L10, H1 and H2.In the ENJL–model, these constants have been calculated in ref.

[13] with the results(ǫ = M2Q/Λ2χ):f 2π = NC16π24M2QgAΓ(0, ǫ)(47)2H1 + L10 = −NC16π213Γ(0, ǫ)(48)2H1 −L10 = −NC16π2g2A13 [Γ(0, ǫ) −Γ(1, ǫ)](49)L8 = NC16π2116g2AΓ(0, ǫ) −23Γ(1, ǫ)(50)H2 = NC16π218g2A" 1 −4 Γ(0, ǫ)Γ(−1, ǫ)!Γ(0, ǫ) + 23Γ(1, ǫ)#. (51)In fact, the calculations made in ref.

[13] suggest a possible improvement of theinvariant fuctions in eqs. (21) to (28) when the contribution from the correspondingresonance propagators is also taken into account, with the resultsΠ(1)V (Q2) = −4(2H1 + L10) −2f 2V Q2M2V + Q2 ,(52)Π(1)A (Q2) = 2f 2πQ2 −4(2H1 −L10) −2f 2AQ2M2A + Q2 ,(53)7

andΠS(Q2) = 8B20(2˜L8 + ˜H2 +2c2mQ2 + M2S). (54)For the other invariant functions, the results are the same as in eqs.

(21) to (28)with the parameter values given in eqs. (47) to (51).

Several comments on theseresults are in order:i) Two relations which follow from the ENJL–model [13] are2H1 −L10 = −f 2A/2and2H1 + L10 = −f 2V /2 . (55)These relations are of the same type as the first Weinberg sum rule relation ineq.

(32) i.e., they are independent of the input parameters and possible low energygluonic corrections. As first shown in ref.

[20], they are crucial to ensure that the lowenergy effective theory is compatible with the known short–distance properties ofthe underlying theory – QCD. It is reassuring that the ENJL–model indeed respectsthese constraints.

Using these relations, we can also write Π(1)Vand Π(1)A in the formΠ(1)V= 2f 2V M2VM2V + Q2 ,(56)Π(1)A = 2f 2πQ2 + 2f 2AM2AM2A + Q2 ;(57)a form much more similar to the usual vector meson dominance (VMD) phenomeno-logical parametrizations found in the literature.ii) The constant cm in eq. (54) denotes the couplingcmtrS(x)hξ†χξ† + ξχ†ξi(58)in the effective scalar Lagrangian.

Hereχ = 2B0 [s(x) + ip(x)] ,(59)with s(x) and p(x) the external scalar and pseudoscalar matrix field sources. Asdiscussed in ref.

[13], the couplings L8 and H2 receive contributions both from thequark loop – which we denote ˜L8 and ˜H2 – and from the integration of scalar fields– which we denote LS8 and HS2 ; i.e.,L8 = ˜L8 + LS8andH2 = ˜H2 + HS2 . (60)In fact2c2mM2S= 2LS8 + HS2 = 2HS2 = NC16π214g2AΓ(0, ǫ)Γ(−1, ǫ)2 [Γ(−1, ǫ) −2Γ(0, ǫ)]2 .

(61)8

iii) Equations (52), (53) and (54) imply a specific form of the resummation to allorders in an expansion in powers of Q2. As we shall see, this is not however thecorrect form which follows from the exact resummation of Feynman diagrams, toleading order in the 1/NC–expansion.We shall finally give the expressions for the masses MS, MV and MA which areobtained in the ENJL–model in ref.

[13]:M2S = 4M2Q11 −23Γ(1,ǫ)Γ(0,ǫ),(62)M2V = 32Λ2χGV1Γ(0, ǫ) = 6M2QgA1 −gA,(63)M2A = 6M2Q11 −gA11 −Γ(1,ǫ)Γ(0,ǫ). (64)3FULLQ2–DEPENDENTTWO–POINTFUNCTIONSINTHEENJL–MODEL3.1The Vector Two–Point FunctionTo illustrate the method, we shall first discuss with quite a lot of detail the vectorinvariant function Π(1)V (Q2).

In the ENJL–model, and to leading order in the 1/NC–expansion, we have to sum over the infinite class of bubble diagrams shown in Fig.2a. Algebraically, this corresponds to the sum(qµqν −q2gµν)Π(1)V +(qµqα −q2gµα)Π(1)V −4π2GVNCΛ2χ!×2(qαqν −q2gνα)Π(1)V +··· , (65)where the explicit factor of 2 in the second term comes from the two possible con-tractions between the fermion fields of the vector four–quark operator.

The overallresult is(qµqν −q2gµν)(Π(1)V + Π(1)V q28π2GVNCΛ2χΠ(1)V + · · ·). (66)Notice that for this two-point function, only the vector four quark interaction withcoupling GV can contribute.

No mixing between different operators can occur in thiscase. The one loop bubble in Fig.2b corresponds to the bare fermion–loop diagramof the mean field approximation defined in ref.

[13], which in what follows we shalldenote with an overlined expression1Π(1)V (Q2) = Π(1)V (Q2) . (67)It is easy to see that at the n–loop bubble level, the corresponding expression fornΠ(1)V (Q2) will be given by the n −1 – loop bubble result multiplied by the coupling9

GV and one more factor of the one-loop result, i.e.,nΠ(1)V (Q2) = n−1Π(1)V (Q2)8π2GVNCΛ2χ(−Q2)Π(1)V (Q2) . (68)This series can then be summed with the resultΠ(1)V=Π(1)V (Q2)1 + Q2 8π2GVNCΛ2χ Π(1)V (Q2).

(69)We discuss next the calculation of Π(1)V (Q2) in some detail. The spectral functionassociated to Π(1)V (Q2) is the one corresponding to the Q ¯Q intermediate state in aP–wave and can be calculated unambiguously with the well known result1πImΠ(1)V (t) = NC16π243 1 + 2M2Qt!

s1 −4M2Qtθ(t −4M2Q) . (70)The function Π(1)V (Q2) we seek for has to obey three criteria:i) It must obey the relevant Ward identities.ii) Its discontinuity should coincide with the spectral function in eq.

(70).iii) When expanded in powers of Q2 it must reproduce the heat kernel calculationof the effective action approach, with the same proper time regularization results.These three criteria will in fact apply to all the two–point functions we discuss.To proceed with the calculation of Π(1)V (Q2), we write a once subtracted dispersionrelation for this functionΠ(1)V (Q2) = Π(1)V (0) −Q2Z ∞0dtt1t + Q21πImΠ(1)V (t) . (71)Requirement iii) implies that Π(1)V (0) is fixed, with the result (ǫ = M2Q/Λ2χ)Π(1)V (0) = NC16π243Γ(0, ǫ) .

(72)With 1πImΠ(1)V (t) in eq. (70) inserted in the integrand of the r.h.s.

of eq. (71); andwith the successive change of variables4M2Qt= 1 −y2 and y = 1 −2x ,(73)we have thatZ ∞0dttQ2t + Q21πImΠ(1)V (t) =NC16π223Z 10 dx(1 −2x)2 [1 + 2x(1 −x)]Q2M2Q + Q2x(1 −x) .

(74)10

In order to match with the proper time regularization which has been used in thecalculation of Π(1)V (0), we next replace the denominator in the r.h.s. of eq.

(74) asfollows1M2Q + Q2x(1 −x) →Z ∞1/Λ2χdτe−τ[M2Q+Q2x(1−x)] . (75)Performing an integration by parts in the x–variable we then finally get the result(ǫ = M2Q/Λ2χ)Z ∞0dttQ2t + Q21πImΠ(1)V (t) = NC16π243Γ(0, ǫ) −6Z 10 dxx(1 −x)Γ(0, xQ)(76)where xQ is a short–hand notation, which we shall use from here onwards, forxQ = M2Q + Q2x(1 −x)Λ2χ.

(77)Combining eqs. (71), (72) and (76), we obtainΠ(1)V (Q2) = NC16π28Z 10 dxx(1 −x)Γ(0, xQ) .

(78)The first few terms in a Q2–expansion of this expression areΠ(1)V (Q2) = NC16π2(43Γ(0, ǫ) −415Γ(1, ǫ) Q2M2Q+ 135Γ(2, ǫ) Q4M4Q+ O(Q6)),(79)in agreement with the proper time regularized heat kernel effective action result.The imaginary part of Π(1)V (Q2) evaluated from eq. (78) using the iǫ prescriptionxQ →M2Q + Q2x(1 −x) −iǫ/Λ2χ in the log term: Γ(0, xQ) = −log xQ −γE +O(xQ), reproduces the spectral function in eq.

(70).We shall now try to cast the result for Π(1)V (Q2) in eq. (69) in the simple VMD–form of eq.

(56):Π(1)V (Q2) =NCΛ2χ/8π2GV(Π(1)V )−1 NCΛ2χ8π2GV + Q2= 2f 2V (Q2)M2V (Q2)M2V (Q2) + Q2,(80)where we have set2f 2V (Q2)M2V (Q2) = NCΛ2χ8π2GV,(81)andM2V (Q2) = Λ2χ4GV1R 10 dxx(1 −x)Γ(0, xQ) . (82)We find that the full Q2–dependent vector two–point function can indeed be castin the VMD–form of eq.

(56) provided that the coupling parameters fV (Q2) and11

MV (Q2) become Q2 dependent. Their value at Q2 = 0 happen to coincide, in thiscase, with the couplings in the low energy effective Lagrangian i.e.,M2V (Q2 = 0) = M2V = 32Λ2χGV1Γ(0, ǫ) .

(83)andf 2V (Q2 = 0) = f 2V = NC16π223Γ(0, ǫ) . (84)The product f 2V (Q2)M2V (Q2) is scale–invariant.In order to see the hadronic content of the full vector two–point function wepropose to examine the complete spectral function1πImΠ(1)V (t) =1πImΠ(1)V (t)1 −t8π2GVNCΛ2χ ReΠ(1)V (t)2+t8π2GVNCΛ2χ ImΠ(1)V (t)2(85)and plot it as a function of t for the input parameter valuesMQ = 265MeV,Λχ = 1165MeV(86)andgA = 0.61 .

(87)These are the values corresponding to fit #1 in ref.[13]. The plot is the one shownin Fig.

3 (the full line). For the sake of comparison, we have also ploted in the samefigure the spectral function 1πImΠ(1)V (t) corresponding to the mean field approxima-tion (the dashed line).

The improvement towards a reasonable simulation of thewell known experimental shape of the JP = 1−, I = 1 hadronic spectral function israther notorious.3.2The Axial–Vector Two–Point FunctionThe infinite series of bubble diagrams we have to sum in this case is formallyvery similar to the one already discussed in the previous subsection. Again, onlythe vector four–quark interaction with coupling GV contributes in this case with theresultΠ(1)A (Q2) =Π(1)A (Q2)1 + Q2 8π2GVNCΛ2χ Π(1)A (Q2).

(88)The axial two–point function ΠµνA (q) in the mean field approximation, has howevermore structure than the corresponding vector two–point function because now, dueto the presence of a constituent quark mass, the axial invariant function Π(0)A in thedecomposition corresponding to eq. (16) doesn’t vanish.

Both Π(1)A (Q2) and Π(0)A (Q2)12

have associated spectral functions which can be calculated unambiguously with theresults:1πImΠ(1)A (t) = NC16π243 1 −4M2Qt! s1 −4M2Qtθ(t −4M2Q)(89)1πImΠ(0)A (t) = NC16π28M2Qts1 −4M2Qtθ(t −4M2Q)(90)Notice that the three spectral functions1πImΠ(1)V (t),1πImΠ(1)A (t) and1πImΠ(0)A (t)satisfy the identity:1πImΠ(1)V (t) −1πImΠ(1)A (t) −1πImΠ(0)A (t) = 0 .

(91)In fact this is nothing but a particular case of a general Ward identity which two–point functions in the mean field approximation must obey:Π(1)V (Q2) −Π(1)A (Q2) −Π(0)A (Q2) = 0 . (92)The proof of this identity can be found in the Appendix.

It is precisely this identitywhich guarantees that the first Weinberg sum rule in the mean field approximationis automatically satisfied.From the asympotic behaviour of ImΠ(0)A (t) in eq. (90) we conclude that thedispersive part of the function Π(0)A (Q2) obeys an unsubtracted dispersion relation.To this we have to add the pole term calculated in the effective action approach i.e.,Π(0)A (Q2) = −2 ¯f 2πQ2+Z ∞0dtt + Q21πImΠ(0)A (t) ,(93)with (ǫ = M2Q/Λ2χ)¯f 2π = NC16π24M2QΓ(0, ǫ) .

(94)Using the same change of variables as in eqs. (73), we obtainZ ∞0dtt + Q21πImΠ(0)A (t) = NC16π24M2QZ 10 dx(1 −2x)21M2Q + Q2x(1 −x) .

(95)Next, we use the same proper time representation for the denominator in the righthand side as in eq. (75), and perform an integration by parts in the x–variable, withthe resultZ ∞0dtt + Q21πImΠ(0)A (t) = NC16π28M2QQ2Γ(0, ǫ) −Z 10 dxΓ(0, xQ),(96)with xQ defined in eq.

(77). Inserting this result in the r.h.s.

of eq. (93) leads to thefinal resultΠ(0)A (Q2) = −NC16π28M2QQ2Z 10 dxΓ(0, xQ) ≡−2 ¯f 2π(Q2)Q2,(97)13

which defines a running ¯fπ(Q2) in the mean field approximation. The first few termsin a Q2–expansion of ¯f 2π(Q2) are¯f 2π(Q2) = NC16π24M2Q(Γ(0, ǫ) −16Γ(1, ǫ) Q2M2Q+ O(Q4)),(98)in agreement with the proper time regularized heat kernel effective action result.Once we have calculated Π(0)A (Q2) and Π(1)V (Q2), the function Π(1)A (Q2) followsfrom the Ward identity in eq.

(92) with the resultΠ(1)A (Q2) = NC16π28(M2QQ2Z 10 dxΓ(0, xQ) +Z 10 dxx(1 −x)Γ(0, xQ)). (99)It is nevertheless instructive to calculate Π(1)A (Q2) independently as an illustration ofthe method we are using.

First we observe that the dispersive part of Π(1)A (Q2) needsa subtraction. On the other hand, we know from the effective action calculation thatΠ(1)A (Q2) has a pole term (see eqs.

(53) and (49))Π(1)A (Q2) = 2 ¯f 2πQ2 −NC16π243Γ(1, ǫ) + NC16π243Γ(0, ǫ) + O(Q2) . (100)We recognize the second term in the r.h.s.

of this expression as the constant term inthe Q2–expansion of ¯f 2π(Q2) in eq. (98), which means that the subtraction constantneeded for the dispersion relation is only the third term.However this term isprecisely the same as the one corresponding to the vector function Π(1)V (0) in eq.

(72).We must therefore separate the spectral function1πImΠ(1)A (t) in two pieces: onewhich reproduces the vector spectral function 1πImΠ(1)V (t), for which we shall writea once subtracted dispersion and the rest. But this is precisely the Ward identityseparation we already pointed out in eq.

(91). We then haveΠ(1)A (Q2)=2 ¯f 2πQ2 −NC16π2Z ∞0dtt + Q28M2Qts1 −4M2Qtθ(t −4M2Q) + NC16π243Γ(0, ǫ)−NC16π243Z ∞0dttQ2t + Q2 1 + 2M2Qt!

s1 −4M2Qtθ(t −4M2Q) . (101)the two dispersive integrals are those calculated before in eq.

(96) for the unsub-tracted piece. Putting these results together leads to the result in eq.

(99).With these results in hand, we shall now try to cast Π(1)A (Q2) in eq. (88) as closeas possible to the VMD–form of eq.

(57):Π(1)A (Q2) =Π(1)V −Π(0)A1 −Q2 8π2GVNCΛ2χ Π(0)A + Q2 8π2GVNCΛ2χ Π(1)V.(102)14

From the calculated expression for Π(0)A (Q2) in eq. (97), it follows that1 −Q28π2GVNCΛ2χΠ(0)A (Q2) = 1 + 4GVM2QΛ2χZ 10 dxΓ(0, xQ) .

(103)At Q2 = 0, the r.h.s. is precisely g−1A (see eq.

(38)). This is not a surprising result.The evaluation of the axial vector form factor of a constituent chiral quark from theinfinite series of bubble graphs in Fig.

1a, leads to the resultgA(Q2) =11 + (GV /Λ2χ)4M2QR 10 dxΓ(0, xQ) ,(104)which at Q2 = 0 coincides with the axial coupling constant gA obtained in thecalculation of the low energy effective action in ref.[13]. Using this result, we cannow rewrite the r.h.s.

of eq. (102) in the following simple form:Π(1)A (Q2) = 2f 2π(Q2)Q2+ 2f 2A(Q2)M2A(Q2)M2A(Q2) + Q2,(105)wheref 2π(Q2) = gA(Q2) ¯f 2π(Q2) ;(106)M2A(Q2) =1gA(Q2)M2V (Q2) ;(107)andf 2A(Q2) = g2A(Q2)f 2V (Q2) ,(108)with ¯f 2π(Q2), M2V (Q2) and f 2V (Q2) as given in eqs.

(97), (82) and (81), respectively.Notice that at Q2 = 0f 2π(Q2 = 0) = f 2π = NC16π24M2QgAΓ(0, ǫ) ;(109)i.e., the same expression which appears in the low energy effective action; howeverM2A(Q2 = 0) and f 2A(Q2 = 0) do not coincide with the expressions for the couplingsM2A and f 2A as given in eqs. (64) and (55), (49).

We now have insteadM2A(Q2 = 0) = 1gAM2V and f 2A(Q2 = 0) = g2Af 2V . (110)The remarkable new result is that now, in terms of the running couplings and runningmasses, both the first and second Weinberg sum rule are satisfiedf 2V (Q2)M2V (Q2) = f 2A(Q2)M2A(Q2) + f 2π(Q2)(111)andf 2V (Q2)M4V (Q2) = f 2A(Q2)M4A(Q2) .

(112)15

The fact that f 2A(Q2 = 0) does not coincide with the f 2A defined in the effec-tive action approach can be understood from a comparison between eqs. (57) andeqs.

(105). Both expressions coincide to O(Q2):Π(1)A |eq.

(57) = 2f 2πQ2 + 2f 2A + O(Q2)(113)Π(1)A |eq. (105) = 2f 2πQ2 −NC16π243g2AΓ(1, ǫ) + 2f 2A(Q2 = 0) + O(Q2)(114)It is the fact that part of the constant term is reabsorbed in the Q2–dependence off 2π, that is at the origin of this difference.

The precise relation isf 2A = f 2A(Q2 = 0) −NC16π243g2AΓ(1, ǫ)(115)3.3The Scalar Two–Point FunctionThe full scalar invariant function from the sum of the infinite series of bubble dia-grams has the formΠS(Q2) =ΠS(Q2)1 −4π2GSNCΛ2χ ΠS(Q2) . (116)It only involves the four–quark operator with GS–coupling.

As in the previous vectorand axial–vector discussion, we now proceed to the calculation of the scalar two–point function ΠS(Q2) in the mean field approximation. The associated spectralfunction can be calculated unambiguously, with the result1πImΠS(t) = NC16π28M2Q t4M2Q−1!

s1 −4M2Qtθ(t −4M2Q) . (117)From its assymptotic behaviour at large t, it follows that the function ΠS(Q2) obeys adispersion relation with two–subtractions for the term proportional to tq1 −4M2Q/tand one subtraction for the term M2Qq1 −4M2Q/t.

Accordingly, we write the dis-persion relationΠS(Q2) = ΠS(0) + Q2Π′S(0) −NC16π243Γ(1, ǫ)−Q2Z ∞0dtt1t + Q2NC16π2(−8M2Q)s1 −4M2Qtθ(t −4M2Q)+ Q4Z ∞0dtt21t + Q2NC16π2(2t)s1 −4M2Qtθ(t −4M2Q) ;(118)where ΠS(0) and Π′S(0) are already known from the effective action calculation i.e.,ΠS(0) = NC16π24M2Q [Γ(−1, ǫ) −2Γ(0, ǫ)] ;(119)16

andΠ′S(0) = −NC16π22Γ(0, ǫ) −23Γ(1, ǫ). (120)Notice that only the divergent pieces (i.e., terms proportional to Γ(−1, ǫ) and Γ(0, ǫ);but not Γ(n, ǫ), n ≃1) are retained in the subtraction constant.

The integral wehave to compute isZ ∞4M2Qdtt1t + Q2s1 −4M2Qt= 12Z 10 dx(1 −2x)21M2Q + Q2x(1 −x) ,(121)where we have made the standard change of variables of eq. (73).

Using the rep-resentation of eq. (75), and doing one integration by parts in the variable x, andreplacing eqs.

(119) and (120), we obtain the result (ǫ =M2QΛ2χ , xQ =M2Q+Q2x(1−x)Λ2χ)ΠS(Q2) = NC16π24M2Q(Γ(−1, ǫ) −2 Q24M2Q+ 1! Z 10 dxΓ(0, xQ)).

(122)We shall next evaluate the full ΠS(Q2) function in eq. (116), and try to cast it ina form as close as possible to eq.

(54):ΠS(Q2) =NCΛ2χ4π2GS ΠS(Q2)NCΛ2χ4π2GS −ΠS(Q2). (123)Using the fact that G−1S= ǫΓ(−1, ǫ), we can cancel the terms proportional toΓ(−1, ǫ) in the denominator; and write ΠS(Q2) in the simple formΠS(Q2) = 8B20(2 ˜˜L8 + ˜˜H2 + 2c2m(Q2)Q2 + M2S),(124)whereM2S = 4M2Q ,(125)and (xQ =M2Q+Q2x(1−x)Λ2χ)8B20 × 2c2m(Q2)M2S= NC16π24M2Q(Γ(−1, ǫ))22R 10 dxΓ(0, xQ) =⟨¯QQ⟩22M2Q ¯f 2π(Q2) .

(126)The term8B20(2 ˜˜L8 + ˜˜H2) = −NC16π24M2QΓ(−1, ǫ) = ⟨¯QQ⟩MQ(127)is not quite the same as in the effective action calculation. The value of ΠS(Q2 = 0)however, coincides with the same result as in the effective action:ΠS(0) = 8B20(2L8 + H2) = NC16π24M2QΓ(−1, ǫ)2Γ(0, ǫ) [Γ(−1, ǫ) −2Γ(0, ǫ)] .

(128)17

As in the previous subsection, it was an advantage to reabsorb some of the constantterms into the pole term, compared with the expression obtained from the effectiveaction calculation, to obtain a simple expression to all orders in Q2.The striking new feature of the summed scalar propagator is that the scalar massis constant and MS = 2MQ. This is to be contrasted with results of previous workin the literature, see e.g.

ref. [21] and references therein.

Eqs. (125) and (127) arealso true in the presence of gluonic corrections.

They, and the fact that MS = 2MQ,are a consequence of the identities derived in the appendix. It is only the specificform of the functions that depends on the inclusion of gluons or not.3.4Two–Point Functions with MixingThe case of the other two–point functions Π(0)A (Q2), ΠPM(Q2) and ΠP(Q2) issomewhat more involved because they mix through the Nambu–Jona-Lasinio four–fermion interaction terms in eqs.

(34) and (35). Therefore, for this case the resultat the n bubble level is a matrix equation in terms of the two–point functions at then-1 bubble level:nΠ ≡nΠ(0)AnΠPMnΠP=gV Π(0)AgSΠPM00gV Π(0)AgSΠPM0gV ΠPMgSΠP×n−1Π(0)An−1ΠPMn−1ΠP,(129)where Π(0)V (Q2), ΠPM(Q2) and ΠP(Q2) denote, as usual the two–point functions at theone loop level calculated in the mean field approximation; and gV , gS are short–handnotation forgV = 8π2GVNCΛ2χQ2 and gS = 4π2GSNCΛ2χ.

(130)The series we have to sum, in a vector–like notation, isΠ(Q2) =∞Xn=1nΠ =∞Xn=1Bn−1(Q2) Π(Q2) ,(131)where B(Q2) denotes the 3 × 3 two–point function matrix in eq. (129); and theΠ(Q2)’s three component two–point function vectors.

This series can be summed,Π(Q2) =11 −B(Q2)1Π(Q2) . (132)The matrix (1 −B) can be inverted, and after some algebra we get the resultΠ(0)A (Q2) =1∆(Q2)(1 −gSΠP)Π(0)A + gS(ΠPM)2;(133)ΠPM(Q2) =1∆(Q2)ΠPM(Q2) ;(134)18

andΠP(Q2) =1∆(Q2)(1 −gV Π(0)A )ΠP + gV (ΠPM)2,(135)with ∆(Q2) the function∆(Q2) =1 −gV Π(0)A 1 −gSΠP−gSgVΠPM2 . (136)It is illustrative to see what the result would be for the scalar two–point functionΠS(Q2) if we had carried the analysis keeping the functions Π(0)Vand ΠSM:ΠS(Q2) =−gV (ΠSM)2 + (1 −gV Π(0)V )ΠS(1 −gV Π(0)V )(1 −gSΠS) + gSgV (ΠSM)2 .

(137)If we now set Π(0)V= ΠSM = 0, we recover the result of eq. (116).We need now to calculate ΠP(Q2) and ΠPM(Q2).

To calculate ΠP(Q2) we proceedas for the other two–point functions we have already calculated. The correspondingspectral function is1πImΠP(t) = NC16π28M2Qt4M2Qs1 −4M2Qtθ(t −4M2Q) .

(138)For the large t–behaviour we conclude that ΠP(Q2) obeys a dispersion relationsubtracted twice:ΠP(Q2) = ΠP(0) + Q2Π′P(0) + Q4Z ∞0dtt21t + Q21πImΠP(t) ,(139)with the divergent pieces (divergent when ǫ = M2Q/Λ2χ →0) of ΠP(0) and Π′P(0) asknown from the effective action calculation i.e.,ΠP(0) = NC16π24M2QΓ(−1, ǫ) ;(140)andΠ′P(0) = −NC16π22Γ(0, ǫ) . (141)The integral in the r.h.s.

of eq. (139) has already been calculated in the sectionabout the scalar two–point function.

We then find the result (xQ =M2Q+Q2x(1−x)Λ2χ)ΠP(Q2) = NC16π24M2QΓ(−1, ǫ) −2Q2Z 10 dxΓ(0, xQ). (142)Let us check that this result satisfies the three criteria we discussed in section3.1.

First, there is a Ward identity which relates ΠP(Q2) to Π(0)A (Q2), because ofthe axial current divergence condition∂µ( ¯Qγµγ5Q) = 2MQ ¯Qiγ5Q(143)19

in the mean effective field theory. The Ward identity in question, which we proof inthe appendix, is4M2QΠP(Q2) + 4MQ < ¯QQ >=Q22Π(0)A (Q2) ;(144)and [13]< ¯QQ >= −NC16π24M3QΓ(−1, ǫ) .

(145)Equations (97) and (142) indeed satisfy this Ward identity. Second, the spectralfunction calculated from eq.

(142) via the iǫ prescription:log M2Q −tx(1 −x) −iǫΛ2χ= log |M2Q −tx(1 −x)Λ2χ| + iπθ tx(1 −x) −M2QΛ2χ!,is the same as in eq. (138).

Finally, the first few terms in the Q2–expansion ofΠP(Q2) coincide with those calculated in the effective action approach with theproper time heat kernel regularization.The last two–point function to calculate is ΠPM(Q2). The corresponding spectralfunction is1πImΠPM(t) = NC16π24MQs1 −4M2Qtθ(t −4M2Q) .

(146)The function ΠPM(Q2) obeys a once subtracted dispersion relation.We alreadyhave encountered the same situation with one of the terms in the scalar two–pointfunction, which we have discussed in detail. Therefore, we give the final result onlyΠPM(Q2) = NC16π24MQZ 10 dxΓ(0, xQ) .

(147)We can now proceed to the explicit calculation of the functions Π(0)A (Q2), ΠPM(Q2)and ΠP(Q2) in eqs. (133), (134) and (135).

We find thath1 −gSΠP(Q2)iΠ(0)A (Q2) + gSΠPM(Q2)2 = 0 ,(148)and henceΠ(0)A (Q2) = 0 ,(149)a result which must hold in QCD at the chiral limit. Equation (148) also impliesthat∆(Q2) = 1 −gSΠP(Q2) = Q2M2Q12Γ(−1, ǫ)Z 10 dxΓ(0, xQ) ;(150)and thereforeΠPM(Q2) = −2< ¯QQ >Q2.

(151)20

As for ΠP(Q2), from eq. (135) and using the relations (148) and (150) above, wefindΠP(Q2) = ΠP(Q2) −gV /gSΠ(0)A (Q2)1 −gSΠP(Q2)= ⟨¯QQ⟩MQ+2⟨¯QQ⟩2f 2π(Q2) Q2 .

(152)We see from this result that mixing between the pseudoscalar and the longitudi-nal axial degrees of freedom occurs at all orders in the Q2–expansion. Because ofeq.

(150), ΠP(Q2) has now a pole at Q2 = 0. We want to stress the fact that thefinal full results in eqs.

(149), (151) and (152) are very different to those at theone–loop level approximation. It is only when all the contribution to leading orderin 1/NC are summed that these relations, which are expected features of QCD, ap-pear.

These results are a big improvement with respect to the QCD effective actionapproach at the mean field approximation.3.5Inclusion of Gluonic CorrectionsIn Ref. [13] the low-energy corrections due to the lowest dimensional gluonic con-densate were also explicitly included.

A general analysis based on the possible typesof terms here corresponds essentially to keeping all the overlined functions as un-determined parameters but satisfying the relations derived in the appendix. Thisfollows from the fact that gluonic lines connecting different fermion loops in Fig.

2aare suppressed by extra factors of 1/Nc compared to the leading contribution.The correction due to the leading gluonic vacuum expectation values can in factbe easily included by using the results for two-point functions calculated for use inQCD sum rules [9]. These corrections can be rewritten in terms of the dimensionlessparameterg =π26Ncm4Q⟨αsπ G2⟩(153)and the set of functionsJN =Z 10 dx11 + Q2m2Qx(1 −x)N .

(154)The corrections needed for the spin-1 parts are (for Nc = 3)Π(1)V (Q2)=3gM4Q2π2Q4 (−1 + 3J2 −2J3) ,Π(1)A (Q2)=9gM4Q2π2Q4 (1 −J2) . (155)For the scalar two-point function we needΠS(Q2) = 9gM4Q4π2Q4 (−1 −2J1 + 3J2) .

(156)21

The spin-0 axial-vector, pseudo-scalar and mixed two-point functions have very largecancellations in the denominator and are numerically very unstable when gluoniccorrections are included. They can be handled similarly in principle.

The gluoniccorrection terms do also satisfy the relations derived in the appendix as required.We have checked that using the above formulas the two-point functions includingnon-zero gluonic vacuum expectation values converge for small values of Q2 to thelow energy expansion with these corrections included; i.e. those of eqs.

(21)-(28)with the values of the parameters calculated including gluonic corrections. All thenice features of the two-point functions as given in the previous subsections are stillvalid since the underlying cause for these properties were the relations derived inthe appendix and the gluonic corrections have to satisfy those as well.3.6Numerical ResultsIn this section we plot the two-point functions as calculated in the previous sub-sections.

As described in subsection 3.5 we can also include gluonic corrections. Inview of the result of ref.

[13] that a very good fit to the low-energy parameters couldalso be obtained without gluonic corrections, we only show the effect of the gluoniccorrections in the vector two-point functions. The input values used for all of theplots are those of fit 1 in ref.

[13]. They are MQ = 0.265 GeV , Λχ = 1.165 GeVand gA = 0.61.

The variation with the input parameters can be judged from table2 in ref. [13].

The size of the changes here is similar to the ones obtained there atvalues of momentum transfer Q2 = 0.In fig. 4 we have plotted the vector-two-point function for positive values of Q2for √Q2 from 0 to 1.5 GeV .

The full line corresponds to Π(1)V (q2) in the full ENJL–model. The dashed line is the corresponding result using the effective approximation,eq.

(56). The vector meson mass for the values of the parameters MQ, Λχ and gAwhich we have fixed, is about 0.81 GeV .

The short-dashed line corresponds to thevector-two-point function in the QCD effective action model of ref. [15].

As canbe seen in the figure, the full resummation leads to lower values for the two pointfunctions than those of the low-energy formulas when extended to higher Q2. Inthe same figure 4 we also show the effect of the gluonic corrections.

The dotted lineuses the same input values as given above but has a non-zero value for the gluonicbackground, we have set g = 0.5 (see eq. (153)).

It can be seen that the effect ofgluonic corrections is large at small values of Q2 but grows smaller at higher valuesof Q2.In fig. 5 we have plotted similarly to the vector case, the result for Q2ΠA1 (Q2).

(The extra factor of Q2 is included to remove the pole at Q2 = 0 due to pion ex-change.) In contrast to the vector two-point function, there is also a significant22

difference between the one-loop result of eq. (57) (dashed line) and the full resum-mation of eq.

(105) at low Q2(the full line). This is due to what is usually calledthe pseudoscalar-axial-vector mixing and was in our previous work described by thecoupling gA.

The value of this two-point function at Q2 = 0 determines f 2π. Thedashed line corresponds to the two-point function using the effective approximationof eq.

(57). The axial-vector mass here is about 1.3 GeV .Fig.

6 shows how the summation of the whole series of diagrams has produced thepole at Q2 = 0 that is required by the spontaneous breaking of the chiral symmetryfor the pseudoscalar two-point function. The one-loop result (dashed line) does nothave this behaviour, but the full resummed version (full line) of eq.

(152) does. Theshort-dashed line is the low-energy extrapolation of eq.

(28). Here we see how thefull resummation correctly reproduces the low-energy behaviour as derived in ref.

[13], but for larger values of momentum transfer, it starts to differ appreciably.We have not plotted the scalar two-point function. The pole is, as we have provenabove, always at MS = 2MQ.This pole is generated by the full resummation.Neither have we plotted the mixed pseudoscalar–axial-vector two-point functionsince this has the very simple behaviour of Eq.

(151).The overall picture of the high energy behaviour of the ENJL–model that emergesafter the resummation, is improved compared to the behaviour obtained from thelow-energy expansion. This is illustrated by the fact that now it also satisfies thesecond Weinberg sum rule.

The ENJL–model has another advantage over the simpleQCD effective action model of ref. [15].

By virtue of the extra 4-quark interactionspresent, this model naturally contains more or less correct meson poles while thesimple quark version, that corresponds to the one-loop result (or essentially theuse of the overlined two-point functions), does not. This means that for positivevalues of q2 the two-point functions are considerably enhanced both in the real andimaginary parts as compared to the one loop result.

The importance of this type ofbehaviour can be seen, e.g. in the determination of some low-energy constants usingdispersion relations.

As an example we show in fig. 3 how the imaginary part of thevector two-point function gets enhanced considerably over the one-loop result.The advantage of using the full ENJL–model over a parametrization with mesonresonances is that the number of free parameters remains within limits.

For instance,if one tries to extend the analysis of ref. [16] to non-leptonic matrix elements usinga parametrization with vector mesons it requires the knowledge of weak decays ofvector mesons which have not been observed experimentally.In ref.

[13] the masses and couplings of the mesons were determined from the low-energy expansion. These are essentially given by various combinations of derivativesof the two-point functions at q2 = 0.

An alternative way of determining the meson23

masses is to determine them by looking for the poles and residues of the full two-point functions. This procedure can be questioned on the grounds that for euclideanmomenta quark confinement is not so important but for momenta of q2 ≥4M2Q weget effects of free quarks included.

The two-point functions still have poles though.As an example we give the position of the poles for the vector, axial-vector andscalar for the parameters used above, in table 1. The pion mass is of course exactlyzero in both cases since we work in the chiral limit.

In general the masses are lowerthan those derived from the low energy approximation.4THE π+–π0 ELECTROMAGNETIC MASS DIFFERENCETo the lowest order in the chiral expansion, the effect of virtual electromagneticinteractions to lowest order in the fine structure constant αem = e2/4π, generates aterm in the effective action without derivatives [22]:Zd4xne2C1trQU(x)QU+(x)o,(157)where U is the unitary matrix which collects the pseudoscalar Goldstone fields andQ the quark electric charge matrix Q = 1/3diag(2, −1, −1). Expanding eq.

(157) inpowers of pseudoscalar fieldse2C1trQUQU+ = −2e2C1f 2π(π+π−+ K+K−) + O(φ4) ,(158)one sees explicitly that this term leads to a π+–π0 (and K+–K0) mass splitting∆m2π =m2π+ −m2π0EM = 2e2C1f 2π. (159)The constant C1, like f 2π, is not fixed by symmetry requirement alone.

It is deter-mined by the dynamics of the underlying theory. Formally, it is given by the integralrepresentation [14]2e2C1 = −ie2Zd4q(2π)4gµν −qµqνq2q2 −iǫ (qµqν −q2gµν)Π(1)LR(q2) ,(160)whereΠ(1)LR = 12Π(1)V −Π(1)A,(161)with Π(1)Vand Π(1)A the vector and axial vector invariant functions we have discussed.Performing a Wick rotation in eq.

(160) leads to the sum rule [2]∆m2π = αemπ116π2f 2π(−6π2)Z ∞0dQ2Q2 Q4 hΠ(1)V (Q2) −Π(1)A (Q2)i. (162)24

The purpose of this section is to discuss the evaluation of the ∆m2π sum ruleabove based on the two–point function results discussed in the previous sections. Itis then convenient to split the Q2–integral into long–distance (0 ≤Q2 ≤µ2) andshort–distance (µ2 ≤Q2 ≤∞) parts:Z ∞0dQ2 · ·· =Z µ20dQ2 · · · +Z ∞µ2 Q2 · ··(163)We shall concentrate first on the long–distance part calculation.a) Long–distance contribution.

Phenomenological approachThe very low Q2 contribution to the integral∆m2πLD = αemπ −38f 2π! Z µ20dQ2Q2 Π(1)V −Π(1)A(164)is fixed by chiral perturbation theory (see eqs.

(21) and (23)):Π(1)V (Q2) −Π(1)A (Q2) = −2f 2πQ2−8L10 + O(Q2) ,(165)from which it follows that [14]∆m2πχP T = αemπ34µ2(1 + 2L10f 2πµ2 + O(µ4)). (166)The known correction term O(µ2) in the parenthesis of the r.h.s.

can be used toestimate the value of the µ2–scale at which we can trust the validity of the χPT–contribution. From the fact that [20]4L10f 2π≃−1M2ρ,(167)we conclude that the χPT–result in eq.

(166) can only represent correctly the long–distance contribution to ∆m2π up to scalesµ2χP T < M2ρ . (168)Obviously, this is too small a scale to trust numerically a direct matching with theshort–distance contribution, which as we shall see later, it is expected to be valid forµ2–scales larger than a few GeV 2 at least.

(See however the first paper in ref. [22]and ref.

[14]. )Since the early work of Das et.

al. [2], the traditional phenomenological approachto the calculation of (∆m2π)LD has been to include the effect of vector and axial–vector particle states in the Q2–integral, using a parametrization that is constrained25

to satisfy the first and second Weinberg sum rules. The usual phenomenologicalVMD–model parametrization isΠ(1)V= 2f 2V M2VM2V + Q2(169)andΠ(1)A = 2f 2πQ2 + 2f 2AM2AM2A + Q2 ,(170)with the constants f 2π, f 2V , f 2A, M2V and M2A constrained by the relationsf 2π + f 2AM2A = f 2V M2V(171)andf 2V M4V = f 2AM4V ,(172)which ensure the convergence of the limitslimQ2→∞Q2(Π(1)V −Π(1)A ) →0 andlimQ2→∞Q4(Π(1)V −Π(1)A ) →0 ;(173)i.e., the superconvergence relations which lead to the first and second Weinberg sumrules.

One then has∆m2πV MD = αemπ34Z µ20dQ2M2AM2V(Q2 + M2A)(Q2 + M2V )(174)For MA, MV →∞, with µ2 fixed we recover the first term of the χPT calculationin eq. (166).

If we let the scale µ2 go to infinity; then, for MA =√2MV , one findsthe early result of Das et al. :∆m2π[2] = αemπ32M2ρlog2 = 1.4 × 103MeV 2 .

(175)Experimentally,(mπ+ −mπ0)Exp. = (4.5936 ± 0.0005)MeV ,(176)while the phenomenological result of Das et al.

corresponds to(mπ+ −mπ0)[2] = 5.2MeV ,(177)Recent phenomenological evaluations of the ∆m2π sum rule, which include explicitchiral symmetry breaking effects, can be found in refs. [23] to [25].b) Long–distance contribution in the ENJL–modelThe calculation of (∆m2π)LD in the QCD effective action approach of ref.

[15],which corresponds to the mean field approximation of the Nambu Jona-Lasiniomodel, was reported in ref.[14]. It is the approximation whereΠ(1)V −Π(1)A →Π(1)V −Π(1)A = −NC16π28M2QQ2Z 10 dxΓ(0, xQ) ,(178)26

which is the result obtained in eqs. (92) and (97).

This leads to the result (ǫ =M2Q/Λ2χ, xQ =Q2x(1−x)+M2QΛ2χ)∆m2π[14] = αemπ34Z µ20dQ21Γ(0, ǫ)Z 10 dxΓ(0, xQ) . (179)In ref.

[14] a proper time regularization for the photon propagator was used; andfor simplicity, the µ2–scale was identified with Λ2χ. The shape of this mean fieldapproximation evaluation versus µ2, for the input value of M2Q and Λ2χ which we havebeen considering (i.e., the value corresponding to fit 1 in ref.

[13]: MQ = 265MeVand Λχ = 1165MeV ) is plotted in Fig. 7.The evaluation of (∆m2π)LD in the full ENJL–model, with the expressions of thetwo–point functions Π(1)V (Q2) and Π(1)A (Q2) obtained in the previous section leads tothe result∆m2πENJL = αemπ34Z µ20dQ2f 2π(Q2)f 2πM2A(Q2)M2V (Q2)(Q2 + M2A(Q2)) (Q2 + M2V (Q2)) ,(180)with the Q2–dependent functions M2V (Q2), M2A(Q2) and f 2π(Q2) as given by eqs.

(82),(107), (104) and (106).The shape of (∆m2π)ENJL versus µ2 is shown in Fig. 7.

We expect the integrandin eq. (180) to be a good representation of the low and intermediate energy scales;and therefore, the matching with short–distance evaluation should now be muchsmoother than in the case of the mean field approximation.

This we discuss in thenext subsection.c) Short–distance contribution and numerical results.In QCD perturbation theory Π(1)V (Q2) = Π(1)A (Q2) . Spontaneous symmetrybreaking induces a deviation from this result which, at large Q2 and to leading orderin the 1/NC–expansion, can be calculated using the operator product expansion,with the result ([8] and first ref.

in [22])Π(1)V −Π(1)A= −1Q63π22NCαs(Q2)π(< ¯ψψ >)2 ,(181)where (NC →∞):NCαs(Q2)π→611 log(QΛQCD ) ;(182)and< ¯ψψ(Q2) >=ˆ< ¯ψψ >(log(Q/ΛQCD))922 . (183)Inserting this asymptotic estimate in the short–distance expression for ∆m2π, leadsto the result∆m2πSD = αemπ27π288f 2πˆ< ¯ψψ >µ22 Z ∞1dzz2 12 log µ2Λ2QCDz!!−211.

(184)27

Fig.7 also shows the shape of (∆m2π)SD versus µ2 for various values of the invariantquark condensateˆ< ¯ψψ >.∗Obviously, as the scale µ2 becomes small (∆m2π)SDdiverges. The matching between (∆m2π)SD and (∆m2π)LD is defined by the optimalchoice of µ2 which minimizes the variation of the total ∆m2π.

As seen in fig. 8 thisoccurs at value µ ≈950 MeV ; and in fact around the value, the stability is rathergood.

The corresponding value of ∆m2π in this range, is∆m2π ≈1.3 · 10−3 GeV 2 ,(185)and agree well with the experimental value, the horizontal dashed line in fig. 8.5CONCLUSIONSIn this paper we have extended the general analysis of the ENJL–model as donein ref.

[13] beyond the low-energy expansion. We have calculated directly the two-point functions within the ENJL–model to all orders in momenta.

The relationsthat the one-loop results have to satisfy lead after the full resummation to a set ofrather simple forms for the two-point functions. It should be stressed once morethat these are satisfied independent of the gluonic interactions and are thus valid ina wide class of ENJL-like models.The resulting expressions are, for the vector-axial-vector cases, very similar to theones usually obtained assuming some kind of vector, axial-vector meson dominance.The full resummations have a well behaved high-energy behaviour.

They satisfyboth the first and the second Weinberg sum rules. The resummation also obeys theWard identities of the full theory.Simple expressions were also found for the other two-point functions.

A byproductwas a proof that within this class of models the scalar two-point function always hasa pole corresponding to a mass of twice the constituent quark mass. Our derivationonly depends on the underlying symmetry properties of the Lagrangian and is henceregularization scheme independent.

The full resummation also reproduced the poleat Q2 = 0 in the pseudo-scalar two-point function explicitly showing how this modelobeys the Goldstone theorem.Finally, the two-point functions derived were used to start evaluating nonleptonicmatrix elements within the class of ENJL-like models. We have estimated the elec-tromagnetic π+ −π0 mass difference and found good agreement with the measuredvalue.∗The continuous curve is the one corresponding to the choice | < ¯ψψ > | = (281MeV )3, which isthe value predicted in the ENJL–model for the input values MQ = 265MeV and Λχ = 1162MeV .28

ACKNOWLEDGEMENTSWe would like to thank Ch. Bruno for collaboration in the early stages of this workand helpful comments.H.Z.

would like to thank the ICSC world laboratory forfinancial support. J.B. thanks CPT Marseille for hospitality.APPENDIXIn this appendix we derive the Ward identities that the one-loop two-point func-tions have to satisfy.

We first give a derivation based on the heat-kernel expansionand a general analysis of the type of terms that can contribute to the two-point func-tions. This method allows for explicit contact to be made with the regularizationchosen in the heat-kernel expansion.

A second method is essentially the traditionalway of deriving Ward identities but we have to take into account that ⟨qq⟩̸= 0.The second method can also be used to derive some of the identities that the fulltwo-point functions have to satisfy.The one-loop two-point functions are calculated using the Lagrangian (U = 1)L = qiD/ q −MQqq −q(s −ipγ5)q = qDq . (186)The last equality is the definition of D and the covariant derivative D/containsthe vector and axial-vector external fields.

The real part of the effective action inEuclidean space using the heat kernel expansion is then given by (ǫ = M2Q/Λ2χ):Seff = −132π2Xn≥1Γ(n −2, ǫ)(M2Q)2−nZd4xtrHn(x) . (187)The Hn(x) are the Seeley-DeWitt coefficients and these are constructed out ofE, Rµν and their covariant derivatives.

These are defined byD†EDE = −DµDµ + E + M2Qand[Dµ, Dν] = Rµν . (188)In terms of the external fields s, p, lµ and rν they are (only terms that can contributeto two-point functions are given):E=iγµγ5MQ (rµ −lµ) −i2σµνRµν+s2 + MQs + p2 + γµ∂µs −iγµγ5∂µp ,(189)Rµν=−i2 (lµν + rµν −γ5 (lµν −rµν)) .

(190)Here we see that E and Rµν vanish for vanishing external fields so only terms con-taining at most two factors of E and Rµν can contribute to the two-point functions.29

The first two coefficients are:H0 = 1andH1 = −E . (191)H1 thus contributes to the scalar and pseudoscalar two-point function.

These arethe only two-point functions that contain a quadratic divergence.The Hn≥2 only contain two types of terms that can contribute to two-point func-tions. Let us look at all possibilities.Terms with a single E. These are of the form D2(n−1)E and are total derivatives,so they do not contribute to the two-point functions.

The same argument appliesto terms with a single Rµν.Terms with one E and one Rµν. Extra derivatives acting on these can alwaysbe commuted, the commutator introduces an extra factor of Rµν and then onlycontributes earliest to a three point function.

We can also use partial integration.All this type of terms can thus be brought into the formDµDνED2(n−3)Rµν = 12 [Dµ, Dν] ED2(n−3)Rµν . (192)The commutator becomes an extra factor of Rµν so this type of terms does notcontribute to two-point functions.

We conclude that the Hn≥2 only contribute totwo-point functions through terms likeED2(n−2)E ,DαRαβD2(n−3)DµRµβandRαβD2(n−2)Rαβ . (193)Using Eq.

(190) the last two terms are of the formDαvαβD2(n−3)Dµvµβ + DαaαβD2(n−3)Dµaµβ(194)andvαβD2(n−2)vαβ + aαβD2(n−2)aαβ . (195)So these contribute only to the transverse part and equally for the vector and theaxial-vector two-point function.

The first term has a part containing Rµν as well. Itcontributes only to the transverse part, and equally for the vector and axial-vectorcase.The remaining type of terms can be rewritten using the explicit form of E.Zd4xtrED2(n−2)E=NcZd4xtr[16M2QAµ∂2(n−2)Aµ + 16MQAµ∂2(n−2)∂µP+16M2QP∂2(n−1)P+16M2QS 1 +∂24M2Q!∂2(n−2)S] .

(196)The axial-vector terms contribute only proportionally to gµν. This together withthe above contribution leads to:Π(0)V (Q2)=ΠS(M)(Q2) = 0 ,(197)Π(1)V (Q2)=Π(1)A (Q2) + Π(0)A (Q2) .

(198)30

The first two of these equations appear because the vector current in the LagrangianEq. (186) is conserved.

The third one is the reason why the first Weinberg sum ruleis satisfied even at the one-loop level. It also guarantees both Weinberg sum rulesafter the resummation.

Including the contributions from H1 we also have−2MQΠPM(Q2)=Q2Π(0)A (Q2) ,(199)2MQΠP(Q2)=−2⟨QQ⟩−Q2ΠPM(Q2) ,(200)ΠS(Q2)=ΠP(Q2) + Q2Π(0)A (Q2) . (201)In Eq.

(200) we have used the relation between the coefficient of H1 and the quarkvacuum expectation value.In the chiral limit this vacuum expectation value isdetermined uniquely by the contribution of H1 = −E. This derivation is also validin the presence of low-frequency gluons.

The effective action after including thelow-energy gluonic effects through gluonic vacuum expectation values, still has tobe constructed out of E and Rµν. This was precisely the argument used in Ref.

[13]to obtain relations between the low-energy coupling constants that are independentof the gluonic corrections. The results following from the relations (197-201) afterresummation are the equivalent relations for the two-point functions.

This is whatwe used in Sect. 3 to rewrite all the two-point functions in terms of essentially twofunctions and one constant.The preceding derivation was obtained using the Seeley-DeWitt expansion to allorders.

Let us now show how several results can also be obtained from the underlyingrelations in the Lagrangian (186). These relations are:∂µ (qγµq)=0 ,(202)∂µ (qγµγ5q)=2iMQqγ5q ,(203){qa†α (x), qbβ(0)}=δabδαβδ3(x) .

(204)Eq. (204) is valid at equal times.

a, b are colour-flavour indices and α, β are Diracspinor indices.We start fromqµΠAµν=Zd4x∂µeiq·x⟨0|T (Aµ(x)Aν(0)) |0⟩(205)=−2MQZd4x⟨0|T (Pµ(x)Aν(0)) |0⟩−Zd4xδµ0δ(x0)⟨0| [Aµ(x), Aν(0)] |0⟩(206)=−2iMQΠ(P )ν. (207)The matrix element of the equal time commutator vanishes for two identical currents.This follows from Eq.

(204). Putting in the form of the two-point functions this31

leads toq2qνΠ(0)A = 2MQqνΠP(M)(208)or the same as equation (199). In the full theory we have ∂µAµ = 0 so the identicalderivation leads to:Π(0)A (Q2) = 0 .

(209)This equation is satisfied by the fully resummed two-point function.A similar derivation leads toqµΠPµ = 2iMQΠP −Zd4xδµ0δ(x0)⟨0| [Aµ(x), P(0)] |0⟩. (210)Here the equal time commutator worked out using Eq.

(204) does not vanish. Aterm proportional to the quark vacuum expectation value remains and leads to Eq.(200).

In the full theory ∂µAµ = 0 so we obtainΠP(M)(Q2) = −2⟨QQ⟩Q2. (211)This equation is also satisfied by the fully resummed two-point function.References[1] S. Weinberg, Phys.

Rev. Lett.

18 (1967) 507. [2] T. Dass, G.S.

Guralnik, V.S. Mathur, F.E.

Low and J.E. Young, Phys.

Rev.Lett. 18 (1967) 759.

[3] S.L. Glashow and S. Weinberg, Phys.

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20 (1968) 224. [4] M. Gell-Mann, R.J. Oakes and B. Renner, Phys.

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[5] E.G. Floratos, S. Narison and E. de Rafael, Nucl.

Phys. B155 (1979) 115.

[6] C. Becchi, S. Narison, E. de Rafael and F.J. Yndur`ain, Z. Phys. C8 (1981) 335.

[7] D. Broadhurst, Phys. Lett.

101B (1981) 423. [8] M.A.

Shifman, A.I. Vainshtein and V.I.

Zakharov, Nucl. Phys.

B147 (1979) 385,447. [9] S. Narison, QCD Spectral Sum Rules, World Scientific Lecture Notes in Physics,Vol.

26. [10] J. Gasser and H. Leutwyler, Ann.

of Phys.(N.Y.) 158 (1984) 142.

[11] G. ’t Hooft, Nucl. Phys.

B72 (1974) 461.32

[12] J. Gasser and H. Leutwyler, Nucl. Phys.

B250 (1985) 465. [13] J. Bijnens, Ch.

Bruno and E. de Rafael, Nucl. Phys.

B390 (1993) 501. [14] J. Bijnens and E. de Rafael, Phys.

Lett. B273 (1991) 483.

[15] D. Espriu, E. de Rafael and J. Taron, Nucl. Phys.

B345 (1990) 22, erratumibid. B355 (1991) 278.

[16] J. Donoghue, E. Golowich and B. Holstein, Phys. Rev.

D46 (1992) 4076. [17] Y. Nambu and G. Jona-Lasinio, Phys.

Rev. 122 (1961) 345.

[18] A. Manohar and H. Georgi, Nucl. Phys.

B234 (1984) 189. [19] S. Peris and E. de Rafael, Constituent Quark Couplings and QCD in the largeNc limit, preprint CPT-93/P.2883, UAB-FT-310, to be published in Phys.

Lett.B. [20] G. Ecker, J. Gasser, H. Leutwyler, A. Pich and E. de Rafael, Phys.

lett. B223(1989) 425.

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62 (1989) 1343;G. Ecker, J. Gasser, A. Pich and E. de Rafael, Nucl. Phys.

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Peccei and J.Sola, Nucl. Phys.

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B.33

Table 1: Values of the masses determined from the poles in the two-point functionsand from the low-energy expansion of ref. [13].Mesonref.

[13]PoleMV0.81 GeV0.70 GeVMA1.3 GeV0.9 GeV †MS0.62 GeV0.53 GeV = 2MQ† In the resummed version there is an strong enhancement around this value of thetwo-point function. It does not become a pole with the values of the parameterschosen here.Figure CaptionsFig.

1: (a): The set of diagrams summed to obtain gA(Q2). X is the insertionof the pion field and the other lines are fermions.

(b): The gap equation. The thick line is the full fermion propagator.The thin line is the bare fermion propagator.Fig.

2: (a): The set of diagrams to be summed for the two-point functions. (b): The one loop fermion bubble.Fig.

3: The spectral function of the vector two-point function. The full line is thefull ENJL result.

The dashed line is the result at one-loop. Here Q =√t.Fig.

4: The real part of the vector two-point function. Plotted are the full resultwith (labelled gluon) and without (labelled full) gluonic corrections.

TheVMD parametrization (eff) and the one loop result (1-loop).Fig. 5: The real part of the axial-vector two-point function multiplied by Q2.

Theeffect of axial-pseudoscalar mixing that the full resummation reproduces isvisible at all Q’s. The labels have the same meaning as in Fig.

4.Fig. 6: The real part of the pseudoscalar two-point function multiplied by Q2.

No-tice how the resummed version produces the pole at Q = 0. The labels havethe same meaning as in Fig.

4.Fig. 7: Curves for ∆m2π in terms of the scale µ.Plotted are the long-distancepart for the pion exchange term only (LD-CHPT), the result of ref.

[14] (LD-mean) and the ENJL-result after the resummation (LD-ENJL).The short distance contributions are plotted for three values of ⟨¯QQ⟩=−(194 MeV )3(SD194), −(220 MeV )3 and −(281 MeV )3.Fig. 8: The full result ∆m2π versus µ corresponding to the sum of the long distanceENJL result with the short distance evaluation using ⟨¯QQ⟩as given by theENJL–model (see text).34

✲✲✫✪✬✩✫✪✬✩✫✪✬✩✈✈✈XFig. 1a✲=✫✪✬✩✉+✲Fig.

1b✛✚✘✙✉✛✚✘✙✉✛✚✘✙✉✛✚✘✙✉✛✚✘✙✉✉✉Fig. 2aFig.

2b✲✲✲✲✲✛✛✛✛✛35

00.10.20.30.40.50.60.70.50.550.60.650.70.750.80.850.90.951Q GeVImaginary part of vectorfull1-loop

00.010.020.030.040.050.060.0700.20.40.60.811.21.4Q GeVvectorfull1-loopeffgluon

00.010.020.030.040.050.0600.20.40.60.811.21.4GeV**2Q GeVQ**2 * axial-vectorfull1-loopeff

00.020.040.060.080.10.120.1400.20.40.60.811.21.4GeV**4Q GeVQ**2 * pseudo-scalarfull1-loopeff

00.00050.0010.00150.0020.00250.0030.20.40.60.811.2GeV**2mu GeVpi+-pi0 mass differenceLD-CHPTLD-meanLD-ENJLSD281SD194SD220

00.00050.0010.00150.0020.00250.0030.20.40.60.811.2GeV**2mu GeVpi+-pi0 mass differencefullexp.

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