Vector and axialvector correlators in a nonlocal chiral quark model
Alexander E. Dorokhov?
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980, Dubna, Russia, and Instituto de Fisica Teorica, Universidade Estudial Paulista, 01405900, Sao Paulo, SP, Brazil
Wojciech Broniowski?
The H. Niewodnicza? nski Institute of Nuclear Physics, PL31342 Cracow, Poland The behavior of nonperturbative parts of the isovectorvector and isovector and isosinglet axialvector correlators at Euclidean momenta is studied in the framework of a covariant chiral quark model with nonlocal quarkquark interactions. The gauge covariance is ensured with the help of the P exponents, with the corresponding modi?cation of the quarkcurrent interaction vertices taken into account. The low and highmomentum behavior of the correlators is compared with the chiral perturbation theory and with the QCD operator product expansion, respectively. The V ? A combination of the correlators obtained in the model reproduces quantitatively the ALEPH data on hadronic τ decays, transformed into the Euclidean domain via dispersion relations. The predictions for the electromagnetic π ± ? π 0 mass di?erence and for the pion electric polarizability are also in agreement with the experimental values. The topological susceptibility of the vacuum is evaluated as a function of the momentum, and its ?rst moment is predicted to be χ′ (0) ≈ (50 MeV)2 . In addition, the ful?llment of the Crewther theorem is demonstrated.
PACS numbers: 12.38.Aw, 12.38.Lg, 14.40.Aq, 11.10.Lm Keywords: Nonperturbative calculations for strong interactions, nonlocal theories and models, chiral symmetry
arXiv:hepph/0305037v2 30 Sep 2003
I.
INTRODUCTION
The vector (V ) and axialvector (A) currentcurrent correlators are fundamental quantities of the stronginteraction physics, sensitive to small and largedistance dynamics. They serve as an important testing ground for QCD as well as for e?ective models of strong interactions. In the limit of the exact isospin symmetry the V and A correlators in the momentum space (with ?q 2 ≡ Q2 ≥ 0) are de?ned as ΠV,ab ?ν (q ) = i ΠA,ab ?ν (q ) = i
2 ab 2 ΠV d4 x eiqx ΠV,ab T (Q )δ , ?ν (x) = q? qν ? g?ν q 2 2 ab 2 ab d4 x eiqx ΠA,ab ΠA + q? qν ΠA ?ν (x) = q? qν ? g?ν q T (Q )δ L (Q )δ ,
(1.1) (1.2)
a b ? ΠJ,ab ?ν (x) = 0 T J? (x)Jν (0)
0,
where the QCD currents are
a = qγ? T a q, J? 5a J? = qγ? γ5 T a q, 1 ab 2δ .
(1.3) The momentumspace
and T a denote the generators of the SUF (2) ?avor group, normalized to trT a T b = twopoint correlation functions obey a (suitably subtracted) dispersion relation, Π(Q2 ) = 1 π
∞ 0
ds ImΠ(s), s + Q2
(1.4)
where the imaginary parts of the correlators determine the spectral functions v1 (s) = 4π ImΠV T (s + i0), a1 (s) = 4π ImΠA T (s + i0).
(1.5)
? Electronic ? Electronic
address: dorokhov@thsun1.jinr.ru address: b4bronio@cyfkr.edu.pl
2 Recently, the inclusive nonstrange V and A spectral functions have been determined separately and with high precision by the ALEPH [1] and OPAL [2] collaborations from the hadronic τ lepton decays (τ → ντ + hadrons) in the interval of invariant masses up to the τ mass, 0 ≤ s ≤ m2 τ. The di?erence of the V and A correlation functions is very sensitive to the details of the spontaneous breaking of the chiral symmetry. In particular, the behavior of this combination is constrained by the chiral symmetry in the form of sum rules obtained through the use of the optical theorem [3, 4, 5, 6]. The experimental separation of the V and A spectral functions allows us to accurately test the chiral sum rules in the measured interval [1, 2]. The coe?cients of the Taylor expansion of the correlators at Q2 = 0 are expressed in terms of the lowenergy constants of the chiral perturbation theory(χPT)[6]. On the other hand, the larges behavior of the correlators can be confronted with perturbative QCD thanks to the su?ciently large value of the τ mass. In the highs limit ΠV (Q2 ) and ΠA (Q2 ) are dominated by the free?eld correlator, corrected by nonperturbative terms with inverse powers of Q2 . This follows from the fact that the correlators can be represented by an operator product expansion (OPE) series and thus are sensitive to the nonperturbative physics at smaller energy scales [7]. Recently, the interest in the OPE expansion has been revived due to a possible appearance of unconventional quadratic power corrections, ? 1/Q2 , found in [8], and also observed in lattice simulations [9]. The ALEPH and OPAL data have been intensely used in the literature in order to place limits on the leading coe?cients of the χPT and OPE expansions (see, e.g, [10, 11, 12, 13, 14, 15]). The aim of this work is to calculate the nonperturbative parts of the V and A currentcurrent correlators in the kinematic region reaching up to moderately large Euclidean Q2 and to extract experimentally observed characteristics. The calculations are carried out in the e?ective chiral model with nonlocal quarkquark interactions, which is made covariant by the inclusion of the P exponents in the nonlocal interaction vertex. A speci?c prescription for the Wilson lines and their di?erentiation, described in Sec. III, follows exactly Ref. [16]. That way the model is made consistent with the gauge invariance and can be used to analyze the V and A correlators. The model is a nonlocal extension of the wellknown NambuJona Lasinio model. Moreover, its nonlocal structure may be motivated by fundamental QCD interactions induced by the instanton and gluon exchanges, which induce the spontaneous breaking of the chiral symmetry and generate dynamically a momentumdependent quark mass. From the point of view of the standard OPE, the whole series of power corrections characterizes nonlocal properties of the QCD vacuum and may be described in terms of the nonlocal vacuum condensates [17, 18]. The use of a covariant nonlocal lowenergy quark model based on the selfconsistent approach to the dynamics of quarks has many attractive features, as it preserves the gauge invariance, is consistent with the lowenergy theorems, as well as takes into account the largedistance dynamics controlled by the bound states. Similar models with nonlocal fourquark interaction have been considered earlier in, e.g., [19, 20, 21, 22, 23, 24, 25, 26, 27, 28] and applied to describe various lowenergy phenomena. Nonlocal models have an important feature which makes them advantageous over the local models, such as the original Nambu–JonaLasinio model. At high virtualities the quark propagator and the vertex functions of the quark coupled to external ?elds reduce down to the free quark propagator and to local, pointlike couplings. This property allows us to straightforwardly reproduce the leading terms of the operator product expansion. For instance, the second Weinberg sum rule is reproduced in the model [25], which has not been the case of the local approaches. In addition, the intrinsic nonlocalities, inherent to the model, generate unconventional power and exponential corrections which have the same character as found in [8] and in the instanton model (see e.g. [13]). Recently, the nonlocal e?ective model was successively applied to the description of the data from the CLEO collaboration on the pion transition form factor in the interval of the spacelike momentum transfer squared up to 8 GeV2 [29, 30]. Importantly, in that study at zero photon virtualities the chiral anomaly result were reproduced, while at high photon virtualities the factorization of short and long distances occurs at a scale of the order of 1 GeV2 . This allowed for the extraction of the pion distribution amplitudes of leading and nexttoleading twists. There are several further advantages in using the nonlocal models compared to the local approaches. The nonlocal interactions regularize the theory in such a way that the anomalies are preserved [31, 32]. In other regularization methods in the local models [33, 34, 35] the preservation of the anomalies can only be achieved if the (?nite) anomalous part of the action is left unregularized, and only the nonanomalous (in?nite) part is regularized. Next, with nonlocal interactions the model is ?nite to all orders in the 1/Nc (loop) expansion. Finally, as shown in [27], stable solitons exist in a chiral quark model with nonlocal interactions without the extra constraint that forces the chiral ?elds to lie on the chiral circle. In the present paper we further test the nonlocal quark model by carrying out an analysis of the momentum dependence of the currentcurrent correlators. A transformation of the spectral functions measured by the ALEPH collaboration into the Euclidean momentum space allows us for a precise an unambiguous comparison of the experimental data with the model calculations. The paper is organized as follows. In Sect. II, we brie?y recall the results of the chiral perturbation theory and operator product expansion concerning the V and A correlators. In Sect. III and IV, we outline the gauged nonlocal quark model and introduce the quarkcurrent vertices. Then, we derive the expressions for the nonperturbative parts of transverse V and A correlators (Sect. V) and, after ?xing the model parameters in Sect. VI, confront the results with the available experimental data at large (Sect. VII) and low (Sect. VIII) Euclidean momenta. We explicitly demonstrate the transverse character of the V and nonsinglet A correlators
3 in Sect. IX. In Sect. X the contribution of the UA (1) axial anomaly to the ?avorsinglet longitudinal A correlator is displayed and the topological susceptibility is calculated as a function of the momentum. Below, in all cases we use the strict chiral limit, with current quark mass equal to zero. Numerically current quark mass corrections to observables discussed in the paper do not exceed more than few percents.
II. CHIRAL SUM RULES AND THE OPERATOR PRODUCT EXPANSION
Chiral sum rules are dispersion relations between the real and absorptive parts of a twopoint correlation function that transforms symmetrically under SU (2)L × SU (2)R (for the case of nonstrange currents). Through the use of the dispersion relations the sum rules are directly expressed in terms of the di?erence of the V and A spectral densities. Here is the list of sum rules, given in the strict chiral limit, which are investigated in this paper: The ?rst Weinberg sum rule (WSR I) [3], 1 4π 2
s0 →∞ 0 ?A ds [v1 (s) ? a1 (s)] = ?Q2 ΠV Q2 T Q2 →0 2 = fπ ,
(2.1)
the second Weinberg sum rule (WSR II) [3], 1 4π 2
s0 →∞ 0 ?A dss [v1 (s) ? a1 (s)] = Q2 ?Q2 ΠV Q2 T Q2 →∞
= 0,
(2.2)
the DasMathurOkubo (DMO) sum rule [4], 1 4π 2
s0 →∞ 0
? 1 ?A Q2 ΠV Q2 ds [v1 (s) ? a1 (s)] = T s ?Q2
2 = fπ Q2 →0
2 rπ ? FA , 3
(2.3)
and, ?nally, the DasGuralnikMathurLowYuong (DGMLY) sum rule [5], ? 1 4π 2
s0 →∞
dss ln
0
s [v1 (s) ? a1 (s)] = ?2
∞ 0 ?A dQ2 ?Q2 ΠV Q2 T
=
2 4πfπ 2 m2 π ± ? mπ 0 , 3α
(2.4)
where in the last equation α ≈ 1/137 is the ?ne structure constant. In the chiral limit of massless quarks the DGMLY sum rule is independent of the arbitrary normalization scale, ?2 , thanks to WSR II. It was shown by Witten [36] that the positive electromagnetic mass shift of the charged pions is a consequence of the DGMLY sum rule combined with the positivity property of the V ? A correlator,
?A ? Q2 ΠV Q2 T
0,
for
0
Q2
∞.
(2.5)
According to Witten, if the bare u and d quarks were massless and the mass shift were negative, the charged pions would become tachyons destabilizing the QCD vacuum. Whereas WSR I and DMO are lowenergy sum rules (in the sense that the righthand sides involve correlators at low momenta), and are reproduced in most lowenergy e?ective quark models, WSR II is a highmomentum sum rule. In local models it is not reproduced, as discussed shortly. The DMGLY sum rule collects contributions from the whole range of Q2 , both soft and hard. The left hand sides of the sum rules (2.1)(2.4) have been determined with the experimental data from [1] and [2], with s0 taken as the upper limit of the interval of the invariant mass covered by the experiment. The righthand sides of the sum rules are the theoretical predictions as s0 → ∞. The DMO sum rule relates the derivative of Q2 times the correlator to the square of the pion decay constant fπ = (92.4 ± 0.3) MeV [37] obtained from the decays π ? → ?? ν ? and π ? → ?? ν ? γ , to the pion axialvector form factor FA = 0.0058 ± 0.0008 for the radiative decays π ? → l? ν l γ , 2 = (0.439 ± 0.008) fm2 obtained from a oneparameter ?t to the spacelike and to the pion charge radiussquared rπ data [38]. The listed chiral sum rules provide important restrictions on the correlators at low and high energies. The ?rst Weinberg sum rule (2.1) ?xes the normalization of correlators and holds in all variants of the Nambu–JonaLasinio models, local or nonlocal. In general, the coe?cients of the Taylor expansion of the correlators at low Euclidean momenta are given by the low energy constants of the strong chiral Lagrangian. The second Weinberg sum rule (2.2) signals that the leading asymptotics of the high Q2 power expansion of the V ? A correlator essentially starts from dimension d = 6 term, and as such is valid in the nonlocal versions of the e?ective chiral quark models [25]. In local
4 models WSR II involves on the righthand side the large constituent quark mass times quark condensate, Mq < q ?q >, thus is violated badly. In this regard the nonlocal models are highly rewarding. More detailed shortdistance, or large Q2 , properties of the correlators are represented by the QCD operator product expansion [7]. For the V ? A and V + A combinations the OPE provides the following leadingtwist terms in the chiral limit:
?A ΠV (Q2 ) = T d=6,8... V +A ΠT (Q2 ) = d=0,2,4... 6 d OV OV 1 ?A ?A = + O( 8 ), d 6 Q Q Q d OV +A = Qd αs π
(2.6) (2.7) Ga ?ν Q4
2 6 OV 1 +A + O( 8 ), Q6 Q
1 αs λ2 1 Q2 αs = ? 2 1+ ln 2 ? 3 2 + 4π π ? 4π Q 12 where the vacuum matrix elements of dimension d = 6 operators are
6 OV ?A = παs
+
with λa being the color SU (3) matrices. The V ? A correlator does not acquire any perturbative contribution in the limit of massless quarks, hence it is sensitive entirely to the chiral symmetry breaking parameters. Already at relatively small Q2 the d = 6 term dominates in the expansion of ΠV ?A . On other hand, the sum of the correlators, ΠV +A , supplied with small power corrections, is close to the free?eld result for distances up to 1 fm [13]. In the expansion of ΠV +A we also included d = 2 term which violates the original OPE expansion of [7]. Motivation for inclusion this term into consideration was given in [8]. In (2.6) and (2.7) we do not explicitly show exponentially suppressed terms that may be induced by existence of instantons [7]. The magnitudes of the vacuum matrix elements which appear in the OPE can not be ?xed from the ?rst principles and are ?tted to various hadronic observables. Clearly, their determination is bound to carry experimental and theoretical uncertainties. Only a sign of the d = 6 term in the V ? A correlator is ?xed by 6 the Witten inequality: OV 0. ?A Di?erent models are used to estimate the vacuum expectation values. The standard approach in the calculation of the dimension d = 6 matrix elements suggested and used in original work [7] was to explore the factorization hypothesis, i.e. the saturation of the fourquark matrix elements with the intermediate vacuum state. Under this assumptions for the dimension d = 6 matrix elements one gets
6 OV ?A factor
6 ? (uγ? γ5 λa d) dγ? γ5 λa u OV +A = ?παs
(uγ? λa d) dγ? λa u ?
? (uγ? γ5 λa d) dγ? γ5 λa u + (uγ? λa d) dγ? λa u
, + 2 9
i=u,d j =u,d,s,...
(qi γ? λa qi ) (qj γ? λa qj ) ? ,
(2.8) ?
= ?32παs qq
2
/9,
6 OV +A
factor
= 64παs qq
2
/81.
(2.9)
However, some authors conclude that probably the factorization hypothesis is violated by a factor of 23 [39]. Moreover, quite di?erent result appears if one uses the instanton liquid model to calculate these matrix elements [40, 41]. In this work, for comparison with other model results, we use the following typical values of the condensates found via standard QCD sum rules without and with the inclusion of the d = 2 term: αs 2 λ = 0, π αs 2 λ = ?0.12 GeV2 , π αs Ga ?ν π
2
= 0.012 GeV4 ,
2
αs qq
2
= 2.4 · 10?4 GeV6
[7, 11],
(2.10)
αs Ga ?ν π
= 0.022 GeV4 ,
αs qq
2
= 5.8 · 10?4 GeV6
[42].
(2.11)
The above values hold at a typical renormalization scale of about 1 GeV.
III. GAUGING NONLOCAL MODELS
In local theories, the gauge principle of the minimum action uniquely determines the interaction of the matter ?elds with the gauge ?elds. However, in nonlocal theories such an interaction may be introduced in various ways, and its transverse part cannot be uniquely de?ned [43]. In order to obtain the nonlocal action in a gaugeinvariant form with
5
FIG. 1: Diagrammatic representation of the e?ective nonlocal fourquark interaction of Eq. (3.2). The hatched blobs represent the nonlocal interactions, with the Γi matrices present, and the dotted line indicates that the diagram can be cut across this line without crossing the quark lines going across the blobs.
respect to external ?elds V and A, we de?ne the delocalized quark ?eld, Q, with the help of the Schwinger gauge phase factor, a.k.a. the Wilson line or the link operator,
y
Q(x, y ) = P exp i
x
a a q (y ), (z ) + Aa dz? V? ? (z )γ5 T
Q(x, y ) = Q? (x, y )γ 0 .
(3.1)
a Here V? (z ) and Aa ? (z ) are the external gauge vector and axialvector ?elds, respectively, and P is the operator of ordering along the integration path, with y denoting the position of the quark and x being an arbitrary reference point. The P operator arranges the matrices in each term of the expansion of the exponent from the left to the right in the order determined by the point z moving along the path from x to y . We start with the nonlocal chirally invariant action which describes the interaction of soft quark ?elds. The nonlocal fourquark interaction is depicted in Fig. 1. The soft gluon ?elds have been integrated out. The corresponding gaugeinvariant action for quarks interacting through nonperturbative exchanges can be expressed in a form similar to the Nambu–JonaLasinio model [16]
S = +
d4 x q (x)γ ? [i?? ? V? (x) ? γ5 A? (x)] q (x) + 1 G 2
4
d4 X
n=1
d4 xn f (xn ) Q(X ? x1 , X )Γi Q(X, X + x3 )Q(X ? x2 , X )Γi Q(X, X + x4 ) ,
(3.2)
where in the simplest version of the model the spin?avor structure of the interaction is given by matrix product (Γi ? Γi ) = (1 ? 1 + iγ5 τ a ? iγ5 τ a ) . (3.3)
In Eq. (3.2) q = (u, d) denotes the quark ?avor doublet ?eld, G is the fourquark coupling constant, and τ a are the Pauli isospin matrices. The delocalization of the quark ?elds with the inclusion of the pathordered Schwinger phase factors, Eq. (3.1), ensures the gauge invariance of the nonlocal action (3.2). However, the presence of these factors modi?es the quarkcurrent interaction, as shown graphically in Fig. (2). The modi?cation of interaction, required by the gauge principle, poses a technical di?culty in dealing with nonlocal models, as many diagrams appear in the analysis of physical processes. The ambiguities in making the nonlocal 4quark interaction gauge invariant are manifest in the pathdependence in the de?nition (3.1), as well as in the choice of the junction of the quark sources with the gauge strings. In general, the Noether currents consist of two components: the pathindependent longitudinal part and the pathdependent transverse part. The dependence of the transverse component on the choice of the path is a feature of any method employed in constructing the Noether currents corresponding to a nonlocal action, and this freedom is immanent to the formulation of the model. We should recall here that the discussed ambiguities in the construction of the transverse parts of the Noether currents are by no means speci?c to chiral quark models. They also appear, e.g., in nuclear physics when one considers mesonexchange processes. To summarize, the choice of the path in Eq. (3.1) is a part of the model building. In what follows, we use the formalism [20, 43] based on the pathindependent de?nition of the derivative of the integral over a line for an arbitrary function F? (z ): ? ?y ?
y y
dz ν Fν (z ) = F? (y ),
x
δ (4) (x ? y )
x
dz ν Fν (z ) = 0.
(3.4)
This choice e?ectively means that the di?erentiation involves moving the endpoint of the line only, with the other part of the line kept ?xed. As a result, the terms with nonminimal coupling, which are induced by the kinetic term of the action, are omitted.
6
FIG. 2: Diagrammatic representation of gauging of the e?ective nonlocal fourquark interaction (3.2), shown up to the second order in the expansion in external ?elds, represented by the wavy lines. Note that vertices involving one quark line and multiple gauge ?elds appear.
In general, external ?elds entering into Eq. (3.2) are arbitrary auxiliary ?elds; however, some of them can be associated with electromagnetic, weak, or strong interactions. In the case of the electromagnetic interactions, the gauge factor takes into account the e?ects of the radiation of the photon ?eld when the two quarks are moving apart. This formalism was used in [20, 21, 23, 24, 25] to represent the nonlocal interaction in a gaugeinvariant form. The incorporation of a gaugeinvariant interaction with gauge ?elds is of principal importance if one desires to treat correctly the hadron characteristics probed by external currents, such as hadron electromagnetic and weak form factors, structure functions, distribution amplitudes, etc. In Eq. (3.2) the functions f (xn ), normalized to f (0) = 1, form the kernel of the fourquark interaction and characterize the spacedependence of the order parameter of the spontaneous chiralsymmetry breaking. Thus, the interaction is treated in the separable approximation. The choice of the nonlocal kernel in the form of (3.2) is motivated by the instantoninduced nonlocal quarkquark interaction [19], where the nonlocal function f (xn ) is related to the quark zero mode emerging in the instanton ?eld [18, 19]. To have the same ?avor symmetry as in the original instantoninduced ’t Hooft determinant interaction one needs to add yet another piece of the form G′ (τ a ? τ a + iγ5 ? iγ5 ) , (3.5)
with the coupling G′ = ?G. This term will be important in the discussion of the isosinglet axial currents in Sect. X. In the present work we do not consider an extended version of the model that explicitly includes vector and axialvector 2 2 degrees of freedom [23] (we take GV = 0, therefore gA = 1 and MV , MA → ∞). In order to compute physical quantities we must ?rst determine the full quark propagator and the full vertices for the vector and axialvector currents. All calculations will be done in the leading order of the 1/Nc expansion, also referred to as the onequarkloop level or the ladder approximation. In the nonlocal model the dressed quark propagator, S (p), with the momentumdependent quark scalar selfenergy (mass), M (p), is de?ned as S ?1 (p) = p ? M (p). (3.6)
Note that the considered model involves a constant quark wavefunction renormalization function, Z (p) = 1. The equation for the quark propagator in the ladder approximation, also known as the gap equation, M (p) = 4iGNf Nc f 2 (p) d4 k (2π )
4f 2
(k )
k2
M (k ) ? M 2 (k )
(3.7)
7 has the formal solution [23] of the form M (p) = Mq f 2 (p), (3.8)
with constant Mq ≡ M (0) determined dynamically from Eq. (3.7). The quark selfenergy is depicted in Fig. (3). Note that the functions f (p) are treated nondynamically, i.e. their dependence on p is ?xed, while M (p) is dynamical. Furthermore, the integrals over the momentum are calculated by transforming the integration variables into the Euclidean space, (k 0 → ik4 , k 2 → ?k 2 ).
IV. CONSERVED VECTOR AND AXIALVECTOR CURRENTS
The Noether currents and the corresponding vertices are formally obtained as functional derivatives of the action (3.2) with respect to the external ?elds at the zero value of the ?elds. For our purpose, it is necessary to construct the quarkcurrent vertices that involve one or two currents (contact terms). In the presence of the nonlocal interaction the conserved currents include both local and nonlocal terms. In order to expand the pathordered exponent in the external ?elds, we use the technique described in [20] (see also [16, 23]). Brie?y, this method is as follows. First, the Fourier transform of the interaction kernel in Eq. (3.2) is obtained and expanded in the Taylor series in momemta. Next, the momentum powers are replaced by the derivatives acting on both the pathordered exponent and the quark ?elds. Finally, the inverse Fourier transform is performed and summation is carried out again. The relations (3.4) and d4 xF x2 e?ipx
y λx+a
dz ? e?iqz = iλ (2p + qλ)
?
F (p + λq ) ? F (p) (p + λq ) ? p2
2
a
e?iqa + F (p)
y
dz ? e?iqz ,
(4.1)
where F (z 2 ) is an arbitrary function, are frequently used in the procedure described above [72]. The longitudinal projection of the above relation is q? d4 xF x2 e?ipx
y λx+a
dz ? e?iqz = i F (p + λq ) e?iqa ? F (p) e?iqy .
(4.2)
The algebra needed to obtain the vertices with this method is straightforward but somewhat tedious, hence below we present only the ?nal results. The vector vertex following from the model (3.2) is (Fig. 4)
′ a γ? ? (k + k ′ )? M (1) (k, k ′ ) , Γa ? (k, q, k = k + q ) = T
(4.3)
where M (1) (k, k ′ ) is the ?nitedi?erence derivative of the dynamical quark mass, q is the momentum corresponding to the current, and k (k ′ ) is the incoming (outgoing) momentum of the quark, k ′ = k + q . The ?nitedi?erence derivative of an arbitrary function F is de?ned as F (1) (k, k ′ ) = F (k ′ ) ? F (k ) . k ′2 ? k 2 (4.4)
FIG. 3: Diagrammatic representation of the quark scalar selfenergy of Eq. (3.7).
8
FIG. 4: Diagrammatic representation of the bare quarkcurrent vertices (4.3) and (4.6).
Thus, with the gauging prescription given by (3.2) and (3.4), one gets the minimumcoupling vector vertex without extra transverse pieces. The form of the vertex is the same as the longitudinal vector vertex resulting from the PagelsStokar construction [48]. The vertex satis?es the proper WardTakahashi identity:
?1 ′ ′ a a ?1 q? Γa ? (k, q, k ) = SF (k ) T ? T SF (k ) .
(4.5)
The vector vertex (4.3) is free of kinematic singularities and for this reason was advocated long ago in [48, 49]. For the case of the momentumindependent mass, as in local models, the vertex (4.3) reduces to the usual local form, a Γa ? = T γ? . The bare axialvector vertex obtained from the action (3.2) by the di?erentiation with respect to the ?elds is given by the formula (cf. Fig. 4) ? 2 M (k ′ ) ? M (k ) ? a ′ a ′ Γ5 + (4.6) ? (k, q, k = k + q ) = T ?γ? ? (k + k )? k ′2 ? k 2 + q? 2 q2 M (k ′ )M (k ) G J (q 2 ) ? 1 2 AP Mq γ5 ,
where we have introduced the notation JAP (q 2 ) = 4Nc Nf JAP (q 2 → 0) = with
′ JAP (0) =
d4 l M (l) 4 (2π ) D (l)
M (l + q ) M (l),
(4.7) (4.8)
2 Mq ′ ? q 2 JAP (0) + O Q4 , G
Nc Nf 32π 2
du
uM (u) [4M ′ (u) + 2uM ′′ (u)] ? u (M ′ (u)) , D (u)
2
(4.9)
where u = k 2 and (in the Euclidean space) D (k ) = k 2 + M 2 (k ). (4.10)
It satis?es the axial WardTakahashi identity,
In Refs. [16, 45] it was demonstrated that in order to obtain the full vertex corresponding to the conserved axialvector current it is necessary to add the term which contains the pion propagator. The presence of this term is associated with the wellknown pion–axial vector mixing. The addition of the term with the pion propagator exactly cancels the third term in Eq. (4.6), and the full conserved vertex acquires the form (cf. Figs. (5) and (6)) ? ? 2 ′) ? ′ M ( k M ( k ) ′2 2 M (k ) + M (k ) k ?k ? a ′ a? Γ5 (4.11) ? (k + k ′ ? q )? ? γ5 . ? (k, q, k = k + q ) = T ?γ? ? q? q2 q2 k ′2 ? k 2
?1 a ′ a a ?1 q? Γ5 ? (k, q, k ) = γ5 SF (k+ ) T + T SF (k? ) γ5 .
(4.12)
9 The axialvector vertex has a kinematic pole at q 2 = 0, a property that follows from the spontaneous breaking of the chiral symmetry in the limit of massless u and d quarks. Evidently, this pole corresponds to the massless Goldstone pion. We also need the vertices that couple two currents to the quark (cf. Fig. 2). In this regard it is convenient to introduce the notation
a (1) Ga (k, k + q ), ? (k, q ) = iT (2k + q )? f
(4.13)
and
′ ′ ′ a b g?ν f (1) (k, k + q + q ′ )+ Gab ?ν (k, q, q , k ) = ?f (k ) T T
(4.14)
+ [2 (k + q ′ ) + q ]? (2k + q ′ )ν f (2) (k, k + q ′ , k + q + q ′ ) + [(q, a, ?) ←→ (q ′ , b, ν )] , where the second ?nitedi?erence derivative is de?ned by F (2) (k, k ′ , k ′′ ) = Further, we need to introduce
(±)a a F? (k, q ) = Ga ? (k, q ) ± G? (k ? q, q ) , (±)ab F?ν
F (1) (k, k ′′ ) ? F (1) (k, k ′ ) . k ′′2 ? k ′2
(4.15)
(4.16) (k, q ′ ) . (4.17)
(k, q, q , k ) = ±
′
′
′ ′ ab ′ ′ ′ Gab ?ν (k, q, q , k ) + G?ν (k ? q ? q , q, q , k ) b ′ ′ ′ a ′ b Ga ? (k, q ) Gν (k ? q , q ) ± G? (k ? q, q ) Gν
With the above de?nitions one gets for the V V contact term
′ ′ ′ (+)ab Γab (k, q, q ′ , k ′ = k + q + q ′ ) ?ν (k, q, q , k = k + q + q ) = Mq F?ν
(4.18)
+ f (k )f (k ′ )G For the AA contact term one ?nds
d l
4
(2π )
4Tr
(+)ab S (l)F?ν (l, q, q ′ , l) .
ab ′ ′ ′ 5ab ′ ′ ′ 5ab ′ ′ ′ Γ5 ?ν (k, q, q , k = k + q + q ) = Γ(1)?ν (k, q, q , k = k + q + q ) + ?Γ?ν (k, q, q , k = k + q + q ),
(4.19)
where
′ ′ ′ (?)ab ab (k, q, q ′ , k ′ = k + q + q ′ ) Γ5 (1)?ν (k, q, q , k = k + q + q ) = Mq F?ν
(4.20) (4.21)
+ f (k )f (k ′ )G
d l (2π )4
4
(?)ab T r S (l)F?ν (l, q, q ′ , l) .
An additional contribution appears for the AA isotriplet contact term
ab ′ ′ ′ c b ′ b ′ ′ c ?Γ5 ?ν (k, q, q , k = k + q + q ) = ?G τ Gν (k, q ) ? Gν (k ? q , q ) τ
(4.22)
×
d4 l
(2π )
4Tr
(?)a S (l)τ c F? (l, q )
+ [(q, ?, a) ←→ (q ′ , ν, b)] .
FIG. 5: Diagrammatic representation of the full axialvector vertex obtained from the bare axial vertex supplied with rescat?V . tering process of Fig. 6. In the present model there is no rescattering in the vector channel, and ΓV = Γ
10
FIG. 6: Diagrammatic representation of the quarkquark scattering matrix, T .
In the above expressions T r denotes the trace over ?avor, color, and Dirac indices. In the following we also need to introduce the polarization operator in the pseudoscalar channel (cf. Fig. 7), JP P (q 2 )δab = ? i 2 Mq d4 k (2π )
4M
(k ) M (k + q ) T r S (k )γ5 τ a S (k + q ) γ5 τ b
(4.23)
and the correlator of the axial current vertex (4.6) and the pion vertex (cf. Fig. 8)
′ ′ a Γa π (k, k ) = igπ γ5 f (k )f (k )τ ,
(4.24)
de?ned by JπA q 2 δab = q? q2 d4 k (2π )
4Tr a a S (k )Γ5 ? (k, q, k + q )S (k + q ) Γπ (k + q, k ) .
(4.25)
Through the use of the gap equation (3.7) and the expression for the pion decay constant, fπ , given by [19, 23]
2 fπ
Nc = 4π 2
∞
du u
0
M (u)2 ? uM (u)M ′ (u) + u2 M ′ (u)2 , D2 (u)
(4.26)
these correlators have the following expansion at zero momentum: JP P (q 2 ) = 1 f2 + π2 q 2 + O q 4 , G Mq
2 + O q2 . JπA q 2 = fπ
(4.27)
In (4.26) we have used the notation u = k 2 and M ′ (u) = dM (u)/du. In Eq. (4.24) the quarkpion coupling, ?1 2 ′ gπ = [JP , and the pion decay constant, fπ , are connected by the GoldbergerTreiman relation, P (0)] gπ = Mq , fπ (4.28)
which is veri?ed to be valid in the nonlocal model [23], as required by the chiral symmetry.
V. CURRENTCURRENT CORRELATORS (TRANSVERSE PARTS)
Our goal is to obtain the nonperturbative parts of the currentcurrent correlators from the e?ective model and to compare them with the existing τ decay data. The currentcurrent correlators may be represented as a sum of two terms, dispersive (Fig. 9) and contact (Fig. 10), namely
2 ? Q2 ΠJ ?ν Q J J Q2 , Q2 + S?ν = K?ν
(5.1)
FIG. 7: The polarization operator of Eq. (4.23)
11
FIG. 8: The correlator of the bare axial current vertex (4.6) and the pion vertex (4.24).
V Q2 K?ν A K?ν Q2 J Q2 S?ν
= =
d4 k (2π ) d4 k (2π )
4Tr 4Tr
V ΓV ? (k, Q, k + Q) S (k + Q) Γν (k + Q, ?Q, k ) S (k ) , A ΓA ? (k, Q, k + Q) S (k + Q) Γν (k + Q, ?Q, k ) S (k ) ,
(5.2) (5.3) (5.4)
= 2Mq
d4 k (2π )
4Tr
S (k ) ΓJ ?ν (k, Q, ?Q, k + Q) .
′ J ′ ′ A ′ ′ The vertices ΓJ ? (k, q, k ) are given in Eqs. (4.3,4.11), Γ?ν (k, q, q , k ) in Eq. (4.18,4.19), and Γ?ν (k, q, q , k ) in Eq. 2 (4.6). The di?erence in the de?nitions of K?ν Q in (5.2) and (5.3) results from the necessity of taking into account the rescattering diagrams in the axial channel of the pseudoscalar (π or η ′ ) mesons (Fig. 9). The correlators (5.1) are de?ned in such a way that the perturbative contributions are subtracted,
Πnp Q2 = Πtot Q2 ? Πpert Q2 .
(5.5)
The perturbative part is obtained from the nonperturbative part by simply setting the dynamical quark mass M (k ) to zero. We extract the longitudinal and transverse parts of the correlators through the use of the projectors
L P?ν =
q? qν , q2
T P?ν =
1 3
g?ν ?
q?qν q2
.
(5.6)
We ?rst consider the transverse part of the V correlator, with the result
V KT Q2
= 2 Nc + 4 2 k 3 ⊥
1 (2π ) D+ D?
4
d4 k
2 2 M+ M? + k+ k? ? k⊥ 3
2
(5.7)
np
M+ ? M? 2 ? k2 k+ ? d4 k M (k ) 4 (2π ) D (k )
(k+ k? ? M+ M? ) ? M ′ (k ) +
2 2 M+ ? M? 2 2 k+ ? k?
, M (k + Q) ? M (k ) ( k + Q) ? k 2
2
V Q2 ST
= 4 Nc
2 4 k⊥ 3 k 2 ? (k + Q)2
M ′ (k ) ?
,
(5.8)
? where the notations k⊥ = k ? ? Q? (kQ)/Q2 , k± = k ± Q/2,
M± = M (k± ),
D± = D(k± )
(5.9)
FIG. 9: The resummation of the quark rescattering in the currentcurrent correlator in the axialvector channel.
12
FIG. 10: The contact terms in the currentcurrent correlators.
have been introduced. The subtraction of the perturbative part amounts to the replacement 2 2 1 k+ k? ? k⊥ D+ D? 3 2 2 =? k+ k? ? k⊥ 3 1 1 ? 2 2 . D+ D? k+ k? (5.10)
np
Further, we take the nonsinglet transverse projection of the A correlator and obtain
A Q2 KT
= 2 Nc + 4 2 k 3 ⊥
1 D + D? (2π )
4
d4 k
2 2 ?M+ M? + k+ k? ? k⊥ 3
4 2
+
np
(5.11) M?
2
M+ ?
M?
2 ? k2 k+ ?
(k+ k? + M+ M? ) ?
(M+ ? M? ) M+ ? 2 ? k2 k+ ? M (k )
2 2
,
A Q2 ST
V Q2 ? 4 Nc = ST
d4 k M (k ) 8 2 k⊥ (2π )4 D (k ) 3
M (k + Q) ?
2
.
(5.12)
(k + Q) ? k 2
Let us consider the di?erence of the V and A correlators, where a number of cancellations takes place and the ?nal result is quite simple, ? Q2 ΠV ?A Q2 = 4 Nc + d4 k
2 (2π ) D k+ 2 M+ ? M? 2 ? k2 2 k+ ? 4
1 2 D k?
M+ ? M? 4 2 M+ M? + k⊥ ? M+ M? 2 2 3 k+ ? k? M? k?
2
(5.13)
M+ k+ +
.
One may explicitly verify that the integrand of the above expression is indeed positivede?nite, irrespectively of the choice of the mass function M (p). Thus the Witten inequality (2.5) is indeed ful?lled. At Q2 = 0 one gets the results consistent with the ?rst Weinberg sum rule, ? Q2 ΠV,T Q2 = 0 = 0,
2 , ?Q2 ΠA,T Q2 = 0 = ?fπ 2 , ?Q2 ΠV ?A,T Q2 = 0 = fπ
(5.14)
where the explicit de?nition of the pion decay constant (4.26) is used. This serves as a useful algebraic check.
VI. MODEL PARAMETERS
The parameters of the model are ?xed in a way typical for e?ective lowenergy quark models. We demand that the pion decay constant fπ , (4.26), and the quark condensate (for a single ?avor), < q ?q >, given by q ?q = ? Nc 4π 2 du u M (u) , D (u) (6.1)
13 acquire their physical values. For simplicity, we take pro?le for the dynamical quark mass in a Gaussian form M (u) = Mq exp ?2u/Λ2 . With the model parameters Mq = 0.3 GeV, one obtains fπ = 93 MeV, q ?q = ? (224 MeV)3 , (6.4) Λ = 1.085 GeV, (6.3) (6.2)
where the quark condensate is supposed to be normalized at the scale of a few hundred MeV.
VII. LARGEQ2 EXPANSION
At large Q2 one ?nds the following asymptotic expansion for the di?erence and sum of the correlation functions in the inverse powers of Q2 (we do not display here the exponentiallysuppressed terms coming from powers of the dynamical quarkmass form factor): ?Q2 ΠV ?A Q2
Q2 →∞
=
+A ?Q2 ΠV Q2 → ∞ T
1 u2 M 2 (u) 2 Nc , +O du Q4 4 π 2 D ( u) Q6 Nc u 2 u2 u = 2 2 du M (u) M ′ (u) 1 ? 2 ? 4π D (u) Q 3 Q4
(7.1) ? 7 uM 2 (u) +O 6 Q4 1 Q6 . (7.2)
The e?ective model considered here is designed to describe low energy physics. At high energies it is certainly not expected to reproduce all the details of the asymptotic standard operator product expansion of QCD. On other hand, it is possible that the OPE works well only at very short distances while the e?ective model is applicable at large and intermediate distances. With this hope in mind we proceed to analyzing the largeQ2 expansions of the correlators, comparing them numerically to the OPE results, and trying to match the two approaches. It is important to note that the power corrections in the expansions (7.1) and (7.2) have the same inverse powers of Q2 as the OPE. We may now compare the expansion of the model correlators to the OPE , Eqs. (2.6,2.7). In Eq. (7.1) the formally leading d = 4 term absent in the chiral limit in accordance with the second Weinberg sum rule and the OPE QCD. The second term in Eq. (7.1) (and the last term in Eq. (7.2)) is proportional to the derivative of the gluon condensate, and via equations of motion it reduces to the fourquark condensate term appearing in the OPE, Eqs. (2.6,2.7). Let us compare the numerical estimates for the local d = 6 terms obtained from the QCD sum rules and from the nonlocal chiral quark model, labeled as NχQM:
V ?A O6 QCDsr
≈ ?(1 ÷ 2) · 10?3 GeV6 , = ?(3.4 ± 1.1) · 10?3 GeV6 , = ?1.1 · 10?3 GeV6 .
(7.3)
V ?A OPEτ O6 V ?A N χQM O6
The ?rst estimate is found on the basis of low energy theorems and QCD sum rules [7], while the second estimate is made with the help of the τ decay data [11]. The result of the present model is closer to the standard estimate obtained from the lowenergy phenomenology. Similar features of the shortdistance behavior of the correlators were found in the instanton model [13]. In the V + A correlator (7.2) the shortdistance expansion contains, in addition to the contributions coming from the local operators, the unconventional terms originating from the nonlocal operators of dimension d = 2, 4 and 6 (the ?rst terms in Eq. (7.2)). This kind of unconventional terms has recently attracted attention due to the revision of the standard OPE [8], as well as the lattice ?ndings, where the unconventional power corrections in the vector correlators were reported [9]. The appearance of this correction is usually related to the existence of the lowest d = 2 2 , which is due to an apparent gauge noninvariance, absent in the standard OPE. However, in condensate Aa ? a series of papers ([50, 51], and references therein) it was argued that it is possible to de?ne the nonlocal operator with the lowest dimension in a gaugeinvariant way. This situation is very similar to the famous spincrisis problem (cf. [52]). Analogously, in that case there is no twisttwo gluonic operator that may contribute to the singlet axial current matrix element, yet, it is possible to construct the matrix element from nonlocal operators [53]. We thus
14 see that our e?ective nonlocal model shares these unusual e?ects generated by the internal nonlocalities of the quark interaction. Furthermore, the lowestdimension power corrections are naturally present in the approaches similar to the analytic perturbative theory [55]. In that case in order to compensate the e?ects of the ghost pole in the strong coupling constant, the d = 2 power term is added. As discussed in Ref. [42], the justi?cation of the appearance of the unconventional power corrections at the same time means that the standard OPE is valid only at very large momenta. We also wish to comment that in the model expansion of the V + A correlator there are no explicit terms with the gluon condensate of dimension d = 4. The appearance of this term in the nonlocal chiral quark model would corresponded to the local matrix element Nc 4π 2 du uM 2 (u) , D (u)
that is related to the gluon condensate through the gap equation (3.7) [44]. However, the coe?cient of this term is equal to zero. This is due to the simple form of the quark propagator (3.6), which does not allow gluonic correlations between di?erent quark lines. The similar situation occurs in the QCD sum rules calculations (in the ?xed point gauge), where nonzero contribution comes from the diagram with quark lines correlated by soft gluon exchange. These (numerically small) correlation terms may be reconstructed in the e?ective model by introducing the gluonic ?eld in the e?ective action (3.2) by gauging kinetic and interaction terms (see also cf. [34]). From other hand the d = 4 term appears in (7.2) as a nonlocal matrix element. We end the discussion of the shortrange behavior of the correlators by giving the numerical estimates of the additional terms appearing in Eq. (7.2):
V +A O2 V +A O4 V +A O6 N χQM nonloc N χQM nonloc N χQM nonloc
= ?
Nc u M (u) M ′ (u) = 5.0 · 10?3 GeV2 , du 2π 2 D (u) u2 Nc M (u) M ′ (u) = ?1.8 · 10?3 GeV4 , du = 2π 2 D ( u) 4 Nc u3 M (u) M ′ (u) = ?7.6 · 10?4 GeV6 . = du 3 4π 2 D (u)
(7.4)
The sum of these terms, taken in the interval of momenta Q2 ? (1 ÷ 2) GeV2 where the model largeQ2 expansion is expected to be valid, agrees reasonably well with the coe?cient of the leading power correction in Eq. (2.7)
V +A O2 QCDsr
= 3.0 · 10?3 GeV2 ,
(7.5)
where we have taken the estimate (αs /π ) λ2 = ?0.12 GeV2 from Ref. [42]. V ?A Through the use of the factorization hypothesis (2.9) it is predicted that the chirality ?ip matrix element O6 is V +A strongly enhanced in absolute value over the chirality non?ip one O6
V ?A V +A O6 /O6 factor
= ?4.5.
(7.6)
V ?A In the nonlocal chiral quark model the chirality ?ip matrix element O6 is given by the local matrix element, but V +A the chirality non?ip one O6 is a mixture of the local and nonlocal matrix elements which transform to each other by integration by parts. We ?nd that their ratio V ?A V +A O6 /O6 N χQM
≈ ?3.2
(7.7)
has the same tendency as predicted in (7.6). It happens due to partial compensation of contributions of the local and V +A nonlocal matrix elements into O6 .
VIII. LOWENERGY OBSERVABLES AND THE ALEPH DATA
Let us now consider the lowenergy region where the e?ective model (3.2) should be fully predictive. From (5.13) and the DGMLY sum rule (2.4) we estimate the electromagnetic pion mass di?erence to be [mπ± ? mπ0 ]N χQM = 4.2 MeV, (8.1)
15 which is in remarkable agreement with the experimental value (after subtracting the md ? mu e?ect) [6] [mπ± ? mπ0 ]exp = 4.43 ± 0.03 MeV. (8.2)
It is interesting to estimate the electric polarizability of the charged pions, [57, 58]. With the help of the DMO sum rule [73], as done by Gerasimov in [56], we ?nd αE π± = α mπ
2 rπ IDMO , ? 2 3 fπ
(8.3)
where IDMO is the lefthand side of the DMO sum rule (2.3) IDMO (s0 ) = 1 4π 2
s0 0
ds [v1 (s) ? a1 (s)] . s
(8.4)
Equation (8.3) can be interpreted as a sum of the centerofmass recoil contribution and the intrinsic pion polarizability. In [56] it was demonstrated that in model calculations there occurs a delicate cancellation between the two contributions of Eq. (8.3). This requires the calculation of both terms consistently at the same level of approximations. With the experimental value for the pion mean squared radius [38] and the value of the IDMO integral estimated from the ALEPH and OPAL data [2] IDMO m2 τ one gets from Eq. (8.3) the result [2] αE π±
exp exp
= (26.3 ± 1.8) · 10?3
(8.5)
= (2.71 ± 0.88) · 10?4 fm3 .
(8.6)
From Eq. (2.3) also follows the relation obtained by Terentyev [59], which relates the pion polarizability and the pion axialvector form factor, αE π± = αFA . 2 mπ f π (8.7)
The last relation, used with the known values for FA , yields
?4 fm3 , αE π ± = (2.69 ± 0.37) · 10
(8.8)
which is very close to (8.6). Let us estimate the electric polarizability of the charged pions within the nonlocal chiral quark model. By calculating the derivative of ΠV ?A Q2 at zero momentum we estimate the left hand side of the DMO sum rule as [IDMO (s0 → ∞)]N χQM = 18.2 · 10?3 . The value of the pion charge radius squared,
2 rπ N χQM
(8.9)
= 0.33 fm2 ,
(8.10)
obtained in our model from the derivative of the charged pion form factor, is close to its limit of the local chiral model, found by Gerasimov long ago [56],
2 rπ χP T
=
Nc = 0.34 fm2 . 2 4 π 2 fπ
(8.11)
2 and IDMO are somewhat smaller than the experimental values given above. The The model predictions for rπ reason for this discrepancy may be attributed to vector meson degrees of freedom, neglected in our treatment, and to pion loops absent in the largeNc limit. However, these unconsidered contributions are essentially canceled in the combination (1.4) de?ning the electric pion polarizability (see, e.g. [14] for discussions). From (8.3) we ?nd with values given in Eqs. (8.9) and (8.10) the value
αE π±
N χQM
= 2.9 · 10?4 fm3 ,
(8.12)
16
υ1 – a1
2.5
–
ALEPH
τ → (V,A, I=1) ντ parton model/perturbative QCD
2
1.5
1
0.5
0
0.5
1
0
0.5
1
1.5
2
2.5
2
3
3.5
2 2
Mass (GeV/c )
FIG. 11: Inclusive vector minus axial vector spectral function, v1 ? a1 , measured by the ALEPH collaboration [1].
which is close to experimental number (8.6) and also to the prediction of the chiral perturbation theory at the oneloop level [60], αE π±
χPT
= 2.7 · 10?4 fm3 .
(8.13)
Let us note also that (8.12) is a factor of 2 smaller from the estimates obtained in a local chiral quark model [61], ?4 αE fm3 . We thus see that the model prediction for the pion polarizability, Eq. (8.12), is in a very π ± = 5.8 · 10 reasonable agreement with the experimental data. Next, we compare the model correlators with the ALEPH data, presented in Fig. 11. The ALEPH and OPAL data integrated up to the τ mass satisfy all chiral sum rules within the experimental uncertainty, but the central values di?er signi?cantly from the chiral model predictions. Following Ref. [13] we use s0 = 2.5 GeV2 as an upper integration limit, the value at which all chiral sum rules are satis?ed assuming that v1 (s) ? a1 (s) = 0 above s0 . Finally, a kinematic pole at q 2 = 0 is added to the axialvector spectral function. The resulting unsubtracted dispersion relation between the measured spectral densities and the correlation functions becomes
2 2 ΠT ? ΠT = V Q A Q 2 where fπ is given by the WSR I, 2 fπ =
1 4π 2
s0
ds
0
2 fπ v1 (s) ? a1 (s) ? , s + Q2 Q2
(8.14)
1 4π 2
s0
ds [v1 (s) ? a1 (s)] .
0
(8.15)
Having transformed the data into the Euclidean space, we may now proceed with the comparison to the model, which obviously applies to the Euclidean domain only. Admittedly, in the Euclidean presentation of the data the detailed resonance structure corresponding to the ρ and a1 mesons seen in the Minkowski region is smoothed out, hence the veri?cation of model results is not as stringent as would be directly in the Minkowski space. In Fig. 12 we show the normalized curves corresponding to the experimental data and the model prediction. We also show the prediction of the model of Ref. [15] (minimal hadronic approximation, MHA [62, 63])
2 ΠT V ?A Q MHA
=
2 2 fρ Mρ f 2M 2 f2 ? 2a a 2 ? π , 2 2 Q + Mρ Q + Ma Q2
(8.16)
17
1,0

QP Q
2 VA
(
2
)/f
2
p
0,8
0,6
0,4
0,2
0,0 0,0 0,5 1,0 1,5 2,0
Q
2
(GeV )
2
2 FIG. 12: Euclideanmomentum correlation function, ?Q2 ΠV ?A (Q2 )/fπ , constructed in the present model (solid line), in the model of Ref. [15] (dashed line), and reconstructed via Eq. (8.14) from the ALEPH experimental spectral function of Fig. 11 (dashdotted line).
where the contributions of the ρ and a1 mesons are taken into account with the model parameters Mρ = 0.750 GeV 2 2 and fa = Mρ /Ma = 0.5. Other parameters are constrained by the Weinberg sum rules. As demonstrated in Ref. [15], the good agreement between the data and the model predictions is far from trivial, since many analytic approaches discussed in the literature meet de?nite di?culties in the description of the ALEPH data in the region of moderately large Q2 . In Fig. (13) we also present the ratio of the nonperturbative parts of the V ? A (5.13) to V + A correlators in the nonsinglet channel. To conclude this Section we wish to recall that quite similar calculations of the vector and isovector axialvector correlators within a nonlocal model were done some time ago by Holdom and Lewis [45]. There are certain di?erences in the form of the nonlocal interaction and, as a consequence, the form of quarkcurrent vertices is di?erent. A more principal di?erence is that the authors of [45] have chosen a“two phase” strategy, describing the lowenergy part of correlators by full nonperturbative vertices and propagators, while the high energy parts were computed in the approximation when one of the vertices is local. In this case the problem of matching of two regimes occurs already at rather low energy scales. In the present calculations one prolongs the applicability of the model up to moderately large energies, which inter alias results in a good description of the ALEPH data. On other side, we have to admit that one of the goals of both approaches, namely the ?nding of a direct correspondence between the e?ective model calculations and the OPE in QCD has not yet been reached (see Sect. VII). To make correspondence closer it is necessary to supply the model with a more detailed information on the soft quarkgluon interaction[74].
IX. CURRENTCURRENT CORRELATORS (LONGITUDINAL PARTS)
In this Section we demonstrate explicitly the transverse character of the V and isovector (IV) A correlators (Figs. (9) and (10)). For the longitudinal component of the V correlator we get
V Q2 = KL
4 Nc Q2
d4 k M (k ) [M (k + Q) ? M (k )] , 4 (2π ) D (k )
V V Q2 , Q2 = ?K L SL
(9.1)
and therefore
2 ? Q2 ΠV = 0, L Q
(9.2)
as it certainly should be by the requirement of the vector current conservation.
18
1,6
P Q P Q
VA
(
2
)/
V+A
(
2
)
1,4
1,2
1,0
0,8
0,6
0,4
0,2
0,0 0,0 0,5 1,0 1,5 2,0
Q
2
(GeV )
2
FIG. 13: The ratio of the nonperturbative parts of the V ? A to V + A correlators in the nonsinglet channel.
Further, we consider the longitudinal projection of the A correlator. Then, we get contributions from the onequarkloop diagram
2 A =? KL, 1 Q
4 Nc Q2
d4 k M (k ) [M (k + Q) + M (k )] , (2π )4 D (k )
(9.3)
the twoloop diagram in the isovector channel
A KL, 2
Q
2
8 Nc = 2 Q
d4 k M (k ) (2π )4 D (k )
2
M (k + Q) M (k )
d4 k M 2 (k ) (2π )4 D (k )
?1
,
(9.4)
the oneloop contact diagram
2 A = SL, 1 Q
4 Nc Q2
d4 k M (k ) 3M (k ) + M (k + Q) ? 4 4 (2π ) D (k )
M (k ) M (k + Q) ,
(9.5)
and, ?nally, from the twoloop contact diagram in the isovector channel
2 A =? SL, 2 Q
8 Nc Q2
d4 k M (k ) M (k ) ? 4 (2π ) D (k )
2
M (k + Q) M (k )
d4 k M 2 (k ) 4 (2π ) D (k )
?1
.
(9.6)
The sum of all these contributions leads to the desired result
IV ? Q2 ΠA, Q2 = 0 L
(9.7)
consistent with the isovector axial current conservation in the strict chiral limit.
X. SINGLET AXIAL VECTOR CURRENT CORRELATOR AND THE TOPOLOGICAL SUSCEPTIBILITY
The cancellations in the longitudinal channels are consequences of the current conservation and follow simply from the application of the nonanomalous WardTakahashi identities. We have explicitly demonstrated this in the previous
19 Section in order to show the consistency of our calculations. The issue becomes important when we consider the longitudinal part of the isosinglet axialvector current correlator which is not conserved due to the UA (1) axial AdlerBellJackiw anomaly. This channel is dominated not by the pion, but by the η ′ meson intermediate state. Thus, in addition to the nonlocal interaction present in Eq. (3.2) we also need to include the interaction (3.5), where an exchange of the “η ” singlet meson, the SU (2) analog of the η ′ meson, occurs. It is well known that due to the anomaly the singlet axialvector current is not conserved even in the chiral limit,
50 ?? J? (x) = 2Nf Q5 (x) ,
(10.1)
a ?a where Q5 (x) is the topological charge density. In QCD it is de?ned as Q5 (x) = (αs /8π )Ga ?ν (x)G?ν (x), where G?ν is a a a ? is its dual, G ? = ε?νλσ G . The correlator of the singlet axialvector currents the gluonic ?eld strength, and G ?ν ?ν λσ has the same Lorentz structure as in Eq. (1.2), but without ?avor indices and with T a ≡ 1. In the chiral limit its longitudinal part is related to the topological susceptibility, the correlator of the topological charge densities Q5 (x),
χ Q2 = i by the relation (see, e.g., [64])
d4 x eiqx 0 T {Q5 (x)Q5 (0)} 0 ,
(10.2)
0 ΠA, Q2 = L
(2Nf )2 χ Q2 . Q2
(10.3)
At high Q2 the OPE for χ Q2 predicts [65] χ Q2 → ∞ = ? αs 16π αs Ga ?ν π
2
+ O(Q?2 ) + O(e?Qρ ),
(10.4)
where the perturbative contributions have been subtracted, and the exponential corrections are due to nonlocal instanton interactions [64]. At low Q2 the topological susceptibility χ Q2 can be represented as a sum of contributions coming purely from QCD and from hadronic resonances, [64]. Crewther proved the theorem [67] that χ (0) = 0 in any theory where at least one massless quark exists (the dependence of χ (0) on current quark masses has been found in [66][75]). Also, the contributions of nonsinglet hadron resonances are absent in the chiral limit. Thus, in the lowQ2 limit for massless current quarks one has χ Q2 → 0
χQCD
= ?Q2 χ′ (0) + O(Q4 ).
(10.5)
The estimates of χ′ (0) existing in the literature are rather controversial [76]: χ′ (0) = (48 ± 6 MeV)
2
[69],
χ′ (0) = (26 ± 4 MeV)
2
[70].
(10.6)
This makes further model estimates valuable. Now we turn to the model calculations. The bare isosinglet axialvector current obtained from the interaction terms (3.3) and (3.5) by the rules described in Sect. IV becomes M (k ′ ) ? M (k )
2
′ ′ Γ50 ? (k, q, k = k + q ) = γ? γ5 ? γ5 (k + k )?
k ′2 ? k 2
+
(10.7)
+ γ5
G′ q? G′ 2 ′ )M (k ) , 2 J ( q ) ? M ( k AP 2 q2 Mq G
where JAP (q 2 )is de?ned in Eq. (4.7). In order to get the full current we have to consider rescattering in the channel with the quantum numbers of the singlet pseudoscalar meson, “η ”, which results in M (k ′ ) ? M (k )
2
′ ′ Γ50 ? (k, q, k = k + q ) = γ? γ5 ? γ5 (k + k )?
k ′2 ? k 2 ′ q? G 1 ? GJP P (q 2 ) ? γ5 2 2 M (k ′ )M (k ) . q G 1 ? G′ JP P (q 2 )
(10.8)
20 Because of the singlet axial anomaly this current does not contain the massless pole anymore, since according to Eq. (4.27) one has at zero momentum: 1 ? GJP P (q 2 ) f2 = G π2 . 2 ?q Mq (10.9)
Instead, it develops a pole at the “η ” meson mass. So, within the model considered the same mechanism is responsible for the sponteneous breaking of chiral symmetry and violation of the UA (1) symmetry. The vertices satisfy the anomalous WardTakahashi identities:
′ ′ ′ q? Γ50 ? (k, q, k = k + q ) = qγ5 ? γ5 [M (k ) + M (k )] + γ5 2 M (k )M (k ) 1 ?
G′ JAP (q 2 ) + G′ 2 G Mq G′ G
,
(10.10)
and
′ ′ q? Γ50 ? (k, q, k = k + q ) = qγ5 ? γ5 [M (k ) + M (k )] + γ5
2 M (k ′ )M (k ) 1 ? G′ JP P (q 2 )
1?
,
(10.11)
where the last term is due to the anomaly. Thus the QCD pseudoscalar gluonium operator is interpolated by the pseudoscalar e?ective quark ?eld operator with coe?cient expressed in terms of dynamical quark mass. In the e?ective quark model the connection between quark and integrated gluon degrees of freedom is ?xed by the gap equation (3.7)[44]. By considering the forward matrix element (q = 0) one deduces that the singlet axial constant (0) is not renormalized within our scheme: GA (0) = 1. In order to get reduction of the singlet axial constant (“spin crisis”) we need to consider the e?ects of polarization of topologically neutral vacuum (see, e.g., [52]). It is instructive ?rst to consider the longitudinal part of the correlator of the local vertex, γ? γ5 , and the nonlocal vertex of Eq. (10.8), which is the construction of Pagels and Stokar [48]. In this model the decay constant is de?ned by
2 fπ,P S
Nc = 2 8π
∞
du u
0
2M (u)2 ? uM (u)M ′ (u) . D2 (u)
(10.12)
Then, through the use of Eq. (10.3) we get for the topological susceptibility the result (2Nf )2 χP S Q2 = ? Nc Nf G ? G′ G [1 ? G′ JP P (Q2 )] 4π 2 d4 k M+ M? [M+ (k? q ) ? M? (k+ q )] , D+ D?
2 fπ,P S . 2 Nf
(10.13)
with the coe?cients of the lowQ2 expansion given by χP S (0) = 0, χ′ P S (0) = (10.14)
2 Hence, the result is consistent with the Crewther theorem and it provides the estimate of χ′ P S (0) ≈ (39 MeV) obtained for Nf = 3. The second relation may be rewritten in the form resembling the generalized GoldbergerTreiman relation, as advocated by Veneziano and Shore [70]. Indeed, by using the standard GoldbergerTreiman relation, (4.24), valid in a given model, one ?nds PS Mq = gπ
2 N f χ′ P S (0),
(10.15)
which is just the quarklevel relation from Ref. [70]. At large Q2 the quantity χP S Q2 decreases according to the power of the dynamical quark form factor. Now let us turn to the full model calculation. Proceeding in a similar manner as in the previous Sections we get the topological susceptibility in the form ? (2Nf ) χ Q2
2
= 2 Nf
1?
G′ G
Q2 JπA Q2 1?
1?
G′ JAP (Q2 ) 1 + 2 ′ Mq 1 ? G JP P (Q2 )
(10.16)
2 + Mq JP P Q2
+
G 4 Nc Nf 2 Mq
G ? G′ GJAP (Q2 ) ? 2 Mq G [1 ? G′ JP P (Q2 )] ? 2? d4 k M (k ) , M (k ) ? M (k + Q) M (k ) 4 ? (2π ) D (k ) G′ J (Q2 ) 2 AP Mq
21 where we have used the relation between the singlet current correlator and the topological susceptibility of Eq. (10.3). At large Q2 one obtains the powerlike behavior consistent with the OPE prediction (10.4), namely ? (2Nf )2 χ Q2 → ∞ = At zero momentum the topological susceptibility is zero χ (0) = 0, in accordance with the Crewther theorem. For the ?rst moment of the topological susceptibility we obtain χ′ (0) = 1 2 Nf
2 fπ 2? 2 2 Nf M q G
1?
G′ G
.
(10.17)
(10.18)
G′ G
+ 1?
G′ G
2 ′ JAP (0) ,
(10.19)
2 ′ where fπ and JAP (0) are de?ned in Eqs. (4.26) and (4.9), respectively. In order to get numerical results we need to specify further the details of the model. We consider two possibilities. One involves the interaction with the exact symmetry as provided by the ’t Hooft determinant, G′ = ?G. For the second possibility the constants G and G′ are considered as independent of each other, and their values are ?xed with the help of the meson spectrum. In this more realistic scenario one has approximately the relation G′ = 0.1 G (for typical sets of parameters, c.f. Ref. [23]). Then we get the following estimates for the ?rst moment of the topological susceptibility:
χ′ (0) = (55 MeV)2 χ (0) = (50 MeV)
′ 2
(G′ = ?G), (G = 0.1 G).
′
(10.20) (10.21)
For the above estimates we have taken Nf = 3. Since the ?avor number enters only through the factor of 2Nf present in the de?nition (10.3), in this sense it is external to the model and its inclusion is very simple. We can see that the model gives the values of χ′ (0) which are close to the estimate of Ref. [69]. In Fig. 3 we present the model predictions for the topological susceptibility at low and moderate values of Q2 for the cases of the full (10.16) and PagelsStokar (10.13) model calculations. We should note that the predictions of our model have a limited character because we have used the SUf (2) model in the chiral limit and have not considered mixing e?ects. However, our ?nal result is formulated in terms of a physical observable, fπ , and thus we believe that the presented predictions may be not far from more realistic model calculations. In the region of small and intermediate momenta our results are quantitatively close to the predictions of the QCD sum rules with the instanton e?ects included [64]. In our opinion, both the instantonbased calculations (our model with (G′ = ?G) and the interpolation of the model [64]) overestimate the instanton contributions in the region Q2 ? 0.5 ? 2 GeV2 . It would be interesting to verify the predictions given in Fig. 14 by the modern lattice simulations.
XI. CONCLUSIONS
In this work we have analyzed the nonperturbative parts of the Euclideanmomentum correlation functions of the vector and axialvector currents within an e?ective nonlocal chiral quark model. To this end, we have derived the conserved vector and isotriplet axialvector currents and demonstrated explicitly the absence of longitudinal parts in the V and nonsinglet A correlators, which is consequence of the gauge invariance of the present approach. On the other hand, the singlet A correlator gains an anomalous contribution. From the properties of the V ? A correlator we have shown the ful?llment of the lowenergy relations. The values of the π ± ? π 0 electromagnetic mass di?erence and the electric pion polarizability are estimated and found to be remarkably close to the experimental values. In the highenergy region the relation to OPE has been discussed. In particular, the estimate of the 1/Q2 coe?cient is in agreement with the recent lattice ?ndings and the modi?ed OPE phenomenology. We stress that the momentum dependence of the dynamical quark mass is crucial for the ful?llment of the second Weinberg sum rule. The combination V ? A receives no contribution from perturbative e?ects and provides a clean probe for chiral symmetry breaking and a test ground for model veri?cation. We have found that our model describes well the transformed data of the ALEPH collaboration on the hadronic τ decay. The combination V + A, on the other hand, is dominated by perturbative contributions which are subtracted from our analysis. By considering the correlator of the singlet axialvector currents the topological susceptibility has been found as a function of the momentum, and its ?rst moment is predicted. In addition, the ful?llment of the Crewther theorem has been demonstrated.
22

c(Q )(10 GeV )
2 3 2
0,4
0,3
0,2
0,1
0,0 0 1 2 3
2 2
Q (GeV )
FIG. 14: Topological susceptibility versus Q2 predicted by the model with G′ = 0.1 G, Eq. (10.16), (solid line), and by the PagelsStokar construction, Eq. (10.13) ,(dashed line).
Acknowledgments
AED is grateful to S. B. Gerasimov, S.V. Mikhailov, N. I. Kochelev, M. K. Volkov, A. E. Radzhabov, L. Tomio, and H. Forkel for useful discussions of the subject of the present work. WB thanks M. Polyakov and E. Ruiz Arriola for discussions on certain topics at an early stage of this research. AED thanks for partial support from RFBR (Grants nos. 010216431, 020216194, 030217291), INTAS (Grant no. 0000366), and the “Fundacao de Amparo Pesquisa do Estado de Sao Paulo (FAPESP)”. We are grateful to the BogoliubovInfeld program for support.
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[26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72]
24
[73] In χPT in the oneloop approximation the righthandside of the DMO sum rule is expressed through the lowenergy constant L10 . The extraction of this constant from the experiment was considered in [10]. [74] Let us also refer to other important works [46], [47], where the problem of connection between the e?ective 4fermion models and QCD has been discussed. [75] Consistencies of the axialvector current conservation and UA (1) problem is also discussed in [68]. [76] The results obtained in Ref. [71] concerning χ(0) and χ′ (0) contradict to the lowenergy theorems.