Pis'ma v ZhETF, vol.92, iss. 10, pp.777-789

© 2010 November 25


Photon-pion transition form factor at high photon virtualities within

the nonlocal chiral quark model

A. E. Dorokhov

Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics, 141980 Dubna, Moscow region, Russia Institute for Theoretical Problems of Microphysics, Moscow State University, RU-119899 Moscow, Russia

Submitted 20 September 2010

Recently, the BABAR collaboration reported the measurments of the photon-pion transition form factor -PV77* (Q2), which are in strong contradiction to the predictions of the standard factorization approach to perturbative QCD. In the present work, based on a nonperturbative approach to the QCD vacuum and on rather universal assumptions, we show that there exists two asymptotic regimes for the pion transition form factor. One regime with asymptotics (Q2) ~ 1 /Q2 corresponds to the result of the standard QCD

factorization approach, while other violates the standard factorization and leads to asymptotic behavior as F-ki'i (Q2) ~ ln (Q2) /Q2• Furthermore, considering specific nonlocal chiral quark models, we find the region of parameters, where the existing CELLO, CLEO and BABAR data for the pion transition form factor are successfully described.

I. Introduction. In the years 1977-1981, the theory of hard exclusive processes was formulated within the factorization approach to perturbative quantum chromo-dynamics (pQCD) [1-7]. The main ingredients of this approach are the operator product expansion (OPE), the factorization theorems, and the pQCD evolution equations. In this context, the form factor for the photon-pion transition 7*7* —¥ 7T°, with both photons being spacelike (with photon virtualities Q\,Q\ >0), was considered in [6, 7]. Since only one hadron is involved, the corresponding form factor (Qf, Q\) has the sim-

plest structure for the pQCD analysis among the hard exclusive processes. The nonperturbative information about the pion is accumulated in the pion distribution amplitude (DA) ip„ (x) for the fraction x of the longitudinal pion momenta p, carried by a quark. Another simplification is, that the short-distance amplitude for the 7*7* 7T° transition is, to leading order, just given by a single quark propagator. Finally, the photon-pion form factor is related to the axial anomaly [8, 9], when both photons are real.

Experimentally, the easiest situation is, when one photon virtuality is small and the other large. Under these conditions, the form factor F„j*j(Q2,0) was measured at e+e- colliders by CELLO [10], CLEO [11] Collaborations (Fig.5). In the region of large virtualities

Q2 1 GeV2, the pQCD factorization approach for exclusive processes predicts to leading order in the strong coupling constant [6, 7]

<,c7d(Q2,0) = |^J, (1)


J = Г dx(2)

Jo x

is the inverse moment of the pion DA, and f„ = 92.4 MeV. The factor 1/Q2 reflects the asymptotic property of the quark propagator connecting two quarkphoton vertices (Fig.l). The formula (1) is derived un-

n (p)


k - p

k - q


Fig.l. The triangle diagram in momentum and a-representation notation

der the assumption, that the QCD dynamics at large distances (the factor Jf„) and the QCD dynamics at

small distances (the factor 1/Q2) is factorized. Moreover, under this assumption, the asymptotics is reached already at the typical hadronic scale of a few GeV2. The pion DA ip„ (x), in addition, evolves in shape with the change of the renormalization scale [4, 6] and asymptotically equals [3] ip^ (x) = 6x (1 — a;). From this follows the famous asymptotic prediction (the straight line in Fig.5)

^D'As(g2,o) =




Recently, the BABAR collaboration published new data (Fig. 5) for the 77* 7r° transition form factor in the momentum transfer range from 4 to 40 GeV2 [12]. They found the following puzzling result: At Q2 > 10 GeV2 the measured form factor multiplied by the photon virtuality Q2F7t-y*-y{Q2,0) exceeds the predicted asymptotic limit (3) and, moreover, continues to grow with increasing Q2. This result is in strong contradiction to the predictions of the standard QCD factorization approach mentioned above. The BABAR data very well match the older data obtained by the CLEO collaboration in the smaller Q2 region, but extend to a much lager Q2 values.

There are several QCD based approaches to treat the nonperturbative aspects of strong interactions. They are the lattice QCD, QCD sum rules, Schwinger-Dyson approach, Nambu-Jona-Lasinio model, etc. In the present paper, we analyze the photon-pion transition form factor in the gauged nonlocal chiral quark model based on the picture of nontrivial QCD vacuum. The attractive feature of this model is, that it interpolates the physics at large and small distances. At low energy, it enjoys the spontaneous breaking of chiral symmetry, the generation of the dynamical quark mass, and it satisfies the basic low energy theorems. At energies much higher than the characteristic hadronic scale, it becomes the theory of free massless quarks (in chiral limit).

The paper is organized as follows: In Sec. II, we give the basic elements of the effective chiral quark model, the quark propagator and the quark-photon and quark-pion vertices. In Sec. Ill, we transform the expression for the pion transition form factor into the «-representation and analyze, under rather general requirements on the nonperturbative dynamics, the asymptotic behavior of the form factor for different kinematics. In Sec. IV, we specify two kinds of nonlocal chiral quark model implementing different asymptotic regimes and obtain the pion DA for various sets of parameters. In Sec. V, we are looking for the space of parameters that give a satisfactory fit of the CELLO, CLEO and BABAR data. Sec. VI contains our conclusions.

II. Nonlocal chiral quark model. Let us discuss the properties of the triangle diagram (Fig.l) within the effective approach to nonperturbative QCD dynamics. To consider the asymptotics of the photon-pion transition form factor, we do not need to completely specify the elements of the diagram technique, which are, in general, model dependent, but shall restrict ourselves to rather general requirements. All expressions will be treated in Euclidean space appropriate for the process under consideration and for the treatment of nonperturbative physics. The nonperturbative quark propagator, dressed by the interaction with the QCD vacuum, is

S(k) =

fc + to (fc2) D(k2)


The main requirement to the quark propagator is, that at large quark virtualities one has



We assume also, that the dynamical quark mass is a function of the quark virtuality k2 and normalized at zero as

to (0) = Mq, D (0) = M2.


At large virtualities, it drops to the current quark mass TOcurr faster than any power of k 2 (see the discussion in [13])

to (fc2) ~ Mqexp (- (fc2)a) + TOcurr, a > 0. (7)

This is, firstly, because the dynamical quark mass is directly related to the nonlocal quark condensate [14, 15] and, secondly, the quark propagators with powerlike dynamical mass induce false power corrections that are in contradiction to OPE. On the other hand, the dynamical quark mass (7) generates exponentially small corrections, invisible in the standard OPE. The direct instanton contributions provide a famous example of these exponential corrections in the QCD sum rules approach [16, 17]. The denominator in (4) at large virtualities is

k2 ) GO

D (fc2) k2 and the typical expression is

D (fc2) = fc2 + to2 (fc).


It is well known (see, e.g., [18, 19]), that the change of the quark propagator leads to a modification of the quark-photon vertex in order to preserve the Ward-Takahashi identity

r„(k,q,k'=k + q) = ^ieq [7" - ArM (k,q,k' = k + q)],


where the extra term guarantees the property

(k, q,k' = k + q) = S-1 (k') - S-1 (k). (10)

The term AFM (q) is not uniquely defined, even within a particular model, especially its transverse part. The importance of the full vertex FM is, that the axial anomaly is reproduced [20], and thus the photon-transition form factor correctly normalized. Fortunately, due to the fact, that AFM is not proportional to 7M matrix, the corresponding amplitude has no projection onto the leading twist operator. Thus, this term is suppressed, if a large photon virtuality passes through the vertex, and hence does not participate in the leading asymptotics of the form factor. Its leading asymptotics results exclusively from the local part of the photon vertex

FAs (k,q,k' = k + q) =


Furthermore, we need the quark-pion vertex,

Tl(p)=^TaF{k\,k2_), Jit



where k+ and k are the quark and antiquark momenta. It is important to note, that the quark-pion vertex function F k2_) plays a similar role in our consideration as the light-cone wave function (x,k±) in [1-7]. The vertex function F {k\,k2_) is symmetric in the quark virtualities k1 and k1. and rapidly decreases, when both virtualities are large. If it were a function of a linear combination of the quark momenta k+ and k . then it would led to a growing form factor with increasing spacelike photon momenta (see for discussions [21]). The spontaneous breaking of chiral symmetry ensures, that the vertex function F (fc^, k2_) is a functional of the dynamical mass m (k2). In particular, the vertex function is normalized via

F(k2,k2) =m(k2)


In the following, the important feature of the vertex function F {k\,k2_) will be its behavior in the limit, when one quark virtuality is asymptotically large and the other remains finite. There are two possibilities,

Ff (k2+,k2_) fe-4°°o


(k\,k2_) g {k\).



Finally, one needs the projection of the pion state onto the leading twist operator

lt'As (k,q,k' = k + q) = rI5


This projection is determined by the matrix element (0 \q/yIJ,'y5Taq\ na (p)) = —i'2f-.\>a, wh

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