Abelian anomaly and neutral pion production

# Abelian anomaly and neutral pion production

H. L. L. Roberts Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA Institut für Kernphysik, Forschungszentrum Jülich, D-52425 Jülich, Germany    C. D. Roberts Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA Institut für Kernphysik, Forschungszentrum Jülich, D-52425 Jülich, Germany Department of Physics, Peking University, Beijing 100871, China    A. Bashir Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Apartado Postal 2-82, Morelia, Michoacán 58040, Mexico    L. X. Gutiérrez-Guerrero Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Apartado Postal 2-82, Morelia, Michoacán 58040, Mexico    P. C. Tandy Center for Nuclear Research, Department of Physics, Kent State University, Kent OH 44242, USA
###### Abstract

We show that in fully-self-consistent treatments of the pion; namely, its static properties and elastic and transition form factors, the asymptotic limit of the product , determined a priori by the interaction employed, is not exceeded at any finite value of spacelike momentum transfer. Furthermore, in such a treatment of a vector-vector contact-interaction one obtains a transition form factor that disagrees markedly with all available data. We explain that the contact interaction produces a pion distribution amplitude which is flat and nonvanishing at the endpoints. This amplitude characterises a pointlike pion bound-state. Such a state has the hardest possible form factors; i.e., form factors which become constant at large momentum transfers and hence are in striking disagreement with completed experiments. On the other hand, interactions with QCD-like behaviour produce soft pions, a valence-quark distribution amplitude that vanishes as for , and results that agree with the bulk of existing data. Our analysis supports a view that the large- data obtained by the BaBar Collaboration is not an accurate measure of the form factor.

###### pacs:
13.25.Cq, 13.40.Gp, 11.10.St, 24.85.+p

## I Introduction

The process is fascinating because in order to explain the associated transition form factor within the Standard Model on the full domain of momentum transfer, one must combine, using a single internally-consistent framework, an explanation of the essentially nonperturbative Abelian anomaly with the features of perturbative QCD. The case for attempting this has received a significant boost with the publication of data from the BaBar Collaboration Aubert:2009mc () because, while they agree with earlier experiments on their common domain of squared-momentum-transfer Behrend:1990sr (); Gronberg:1997fj (), the BaBar data are unexpectedly far above the prediction of perturbative QCD at larger values of .

Herein we contribute toward understanding the discrepancy by analysing this process using the Dyson-Schwinger equations (DSEs) Roberts:1994dr (); Roberts:2000aa (); Maris:2003vk (); Fischer:2006ub (); RodriguezQuintero:2010wy (); Roberts:2007jh (), which are known to have the capacity to connect nonperturbative and perturbative phenomena in QCD. In particular, the connection between dynamical chiral symmetry breaking (DCSB) and the Abelian Roberts:1987xc (); Bando:1993qy (); Roberts:1994hh (); Alkofer:1995jx (); Maris:1998hc (); Bistrovic:1999dy (); Holl:2005vu () and non-Abelian Bhagwat:2007ha () anomalies is understood, as is the manner through which the perturbative QCD results for the large- behaviour of transition form factor can be obtained Kekez:1998rw (); Roberts:1998gs ().

As part of this analysis, we will elucidate the sensitivity of the transition form factor, , to the pointwise behaviour of the interaction between quarks. We will use existing DSE calculations Maris:2002mz () of this and the kindred form factor to characterise the -dependence of which is produced by a quark-quark interaction that is mediated by massless vector-bosons. For comparison, we will compute the behaviour obtained if quarks interact instead through a contact interaction. Such comparisons are important to achieving a goal of charting the long-range behaviour of the strong-interaction in the Standard Model Aznauryan:2009da ().

In Sec. II we describe a symmetry-preserving regularisation and DSE-formulation of the contact interaction, and explain how a dressed-quark comes simultaneously to have a nonzero charge radius and a hard form factor. In Sec. III we discuss the transition form factor in detail, describing: its connection with the Abelian anomaly; its asymptotic behaviour in QCD cf. that produced by a contact interaction; and how the nature of the interaction determines the pion’s distribution amplitude. Section IV places our results in context with extant data; and Sec. V expresses our conclusions.

## Ii Bound state pion

### ii.1 Bethe-Salpeter and gap equations

Poincaré covariance entails that the Bethe-Salpeter amplitude for an isovector pseudoscalar bound-state of a dressed-quark and -antiquark takes the form

 Γjπ(k;P)=τjγ5[iEπ(k;P)+γ⋅PFπ(k;P) (1) +γ⋅kGπ(k;P)+σμνkμPνHπ(k;P)],

where is the relative and the total momentum of the constituents, and are the Pauli matrices.111We employ a Euclidean metric with: ; ; ; and . A timelike four-vector, , has . Furthemore, we consider the isospin-symmetric limit. This amplitude is determined from the homogeneous Bethe-Salpeter equation (BSE):

 [Γjπ(k;P)]tu=∫d4q(2π)4[χjπ(q;P)]srKrstu(q,k;P), (2)

where , represent colour, flavour and spinor indices, and is the quark-antiquark scattering kernel. In Eq. (2), is the dressed-quark propagator; viz., the solution of the gap equation:

 S(p)−1=iγ⋅p+m (3) +∫d4q(2π)4g2Dμν(p−q)λa2γμS(q)λa2Γν(q,p),

wherein is the Lagrangian current-quark mass, is the gluon propagator and is the quark-gluon vertex.

### ii.2 Momentum-independent vector-boson exchange

Following Ref. GutierrezGuerrero:2010md () we define

 g2Dμν(p−q)=δμν1m2G, (4)

where is a gluon mass-scale (such a scale is generated dynamically in QCD, with a value GeV Bowman:2004jm ()) and proceed by embedding this interaction in a rainbow-ladder truncation of the DSEs. This means in both Eq. (3) and the construction of in Eq. (2). Rainbow-ladder is the leading-order in a nonperturbative, symmetry-preserving truncation Munczek:1994zz (); Bender:1996bb (). It is known and understood to be an accurate truncation for pseudoscalar mesons Bhagwat:2004hn (); Chang:2009zb ().

With this interaction the gap equation becomes

 S−1(p)=iγ⋅p+m+431m2G∫d4q(2π)4γμS(q)γμ. (5)

The integral possesses a quadratic divergence, even in the chiral limit. If the divergence is regularised in a Poincaré covariant manner, then the solution is

 S(p)−1=iγ⋅p+M, (6)

where is momentum-independent and determined by

 M=m+M3π2m2G∫∞0dss1s+M2. (7)

To proceed we must specify a regularisation procedure. We write Ebert:1996vx ()

 1s+M2 = ∫∞0dτe−τ(s+M2) (9) → ∫τ2irτ2uvdτe−τ(s+M2) =e−(s+M2)τ2uv−e−(s+M2)τ2irs+M2,

where are, respectively, infrared and ultraviolet regulators. It is apparent from Eq. (9) that a nonzero value of implements confinement by ensuring the absence of quark production thresholds Krein:1990sf (); Roberts:2007ji (). Furthermore, since Eq. (4) does not define a renormalisable theory, cannot be removed but instead plays a dynamical role and sets the scale of all dimensioned quantities.

The gap equation can now be written

 M=m+M3π2m2GCiu(M2), (10)

where , with being the incomplete gamma-function.

Using the interaction we’ve specified, the homogeneous BSE for the pseudoscalar meson is (

 Γπ(P)=−431m2G∫d4q(2π)4γμχπ(q+,q)γμ. (11)

With a symmetry-preserving regularisation of the interaction in Eq. (4), the Bethe-Salpeter amplitude cannot depend on relative momentum. Hence Eq. (1) reduces to

 Γπ(P)=γ5[iEπ(P)+1Mγ⋅PFπ(P)]. (12)

Crucially, , a component of pseudovector origin, remains. It is an essential component of the pion, which has very significant measurable consequences and thus cannot be neglected.

### ii.3 Ward-Takahashi identity

Preserving the vector and axial-vector Ward-Takahashi identities is essential when computing properties of the pion. The axial-vector identity states

 PμΓ5μ(k+,k)=S−1(k+)iγ5+iγ5S−1(k), (13)

where is the axial-vector vertex, which is determined by

 Γ5μ(k+,k)=γ5γμ−431m2G∫d4q(2π)4γαχ5μ(q+,q)γα. (14)

To achieve this, one must implement a regularisation that maintains Eq. (13). To see what this entails, contract Eq. (14) with and use Eq. (13) within the integrand. This yields the following two chiral limit identities:

 M = 83Mm2g∫d4q(2π)4[1q2+M2+1q2++M2], (15) 0 = ∫d4q(2π)4[P⋅q+q2++M2−P⋅qq2+M2], (16)

which must be satisfied after regularisation. Analysing the integrands using a Feynman parametrisation, one arrives at the follow identities for :

 M = 163Mm2G∫d4q(2π)41[q2+M2], (17) 0 = ∫d4q(2π)412q2+M2[q2+M2]2. (18)

Equation (17) is just the chiral-limit gap equation. Hence it requires nothing new of the regularisation scheme. On the other hand, Eq. (18) states that the axial-vector Ward-Takahashi identity is satisfied if, and only if, the model is regularised so as to ensure there are no quadratic or logarithmic divergences. Unsurprisingly, these are the just the circumstances under which a shift in integration variables is permitted, an operation required in order to prove Eq. (13).

We observe in addition that Eq. (13) is valid for arbitrary . In fact its corollary, Eq. (15), can be used to demonstrate that in the chiral limit the two-flavour scalar-meson rainbow-ladder truncation of the contact-interaction DSEs produces a bound-state with mass Roberts:2010gh (). The second corollary, Eq. (16) entails

 0=∫10dα[Ciu(ω(M2,α,P2))+Ciu1(ω(M2,α,P2)], (19)

with and .

### ii.4 Pion’s Bethe-Salpeter kernel

We are now in a position to write the explicit form of Eq. (11):

 [Eπ(P)Fπ(P)]=13π2m2G[KEEKEFKFEKFF][Eπ(P)Fπ(P)], (20)

where

 KEE = ∫10dα[Ciu(ω(M2,α,−m2π)) (21) +2α(1−α)m2π¯¯¯Ciu1(ω(M2,α,−m2π))], KEF = −m2π∫10dα¯¯¯Ciu1(ω(M2,α,−m2π)), (22) KFE = 12M2∫10dα¯¯¯Ciu1(ω(M2,α,−m2π)), (23) KFF = −2KFE, (24)

with . We used Eq. (19) to arrive at this form of .

In the computation of observables, one must use the canonically-normalised Bethe-Salpeter amplitude; i.e., is rescaled so that

 Pμ=Nctr∫d4q(2π)4Γπ(−P)∂∂PμS(q+P)Γπ(P)S(q). (25)

In the chiral limit, this means

 1=Nc4π21M2C1(M2;τ2ir,τ2uv)Eπ[Eπ−2Fπ]. (26)

With the parameter values (in GeV) GutierrezGuerrero:2010md ()

 mG=0.11,Λir=0.24,Λuv=0.823, (27)

one obtains the results presented in Table 1. We note that the leptonic decay constant and in-pion condensate are given by the following expressions Maris:1997hd (); Maris:1997tm ():

 fπ = 1M32π2[Eπ−2Fπ]KP2=−m2πFE, (28) κπ = fπ34π2[EπKP2=−m2πEE+FπKP2=−m2πEF]. (29)

In the chiral limit ; i.e., the so-called vacuum quark condensate Brodsky:2010xf (). Moreover, also in this limit, one may readily verify that GutierrezGuerrero:2010md ()

 Eπ\lx@stackrelm=0=Mfπ, (30)

which is a particular case of one of the Goldberger-Treiman relations proved in Ref. Maris:1997hd ().

### ii.5 Dressed-photon-quark vertex

In coupling photons to a bound-state constituted from dressed-quarks, it is important that the quark-photon vertex be dressed so that it satisfy the vector Ward-Takahashi identity Roberts:1994hh (). Indeed, where possible it should be dressed at a level consistent with the truncation used to compute the bound-state’s Bethe-Salpeter amplitude Maris:1999bh (). With our treatment of the interaction described in connection with Eq. (4), the bare vertex is sufficient to satisfy the Ward-Takahashi identity and ensure, e.g., a unit value for the charged pion’s electromagnetic form factor GutierrezGuerrero:2010md (). However, given the simplicity of the DSE kernels, one can readily do better.

A vertex dressed consistently with our rainbow-ladder pion is determined by the following inhomogeneous Bethe-Salpeter equation:

 Γμ(Q)=γμ−431m2G∫d4q(2π)4γαχμ(q+,q)γα, (31)

where . Owing to the momentum-independent nature of the interaction kernel, the general form of the solution is

 Γμ(Q)=γTμPT(Q2)+γLμPL(Q2), (32)

where and . This simplicity doesn’t survive with a more sophisticated interaction.

Upon insertion of Eq. (32) into Eq. (31), one can readily obtain

 PL(Q2)=1, (33)

owing to Eq. (16). Using this same identity, one finds

 PT(Q2)=11+Kγ(Q2) (34)

with

 Kγ(Q2)=13π2m2G (35) ×∫10dαα(1−α)Q2¯¯¯Ciu1(ω(M2,α,Q2)).

It is plain that

 PT(Q2=0)=1, (36)

so that at in the rainbow-ladder treatment of the interaction in Eq. (4) the dressed-quark-photon vertex is equal to the bare vertex.222Equations (33), (36) guarantee a massless photon and demonstrate that our regularisation also ensures preservation of the Ward-Takahashi identity for the photon vacuum polarisation Burden:1991uh (). However, this is not true for . Indeed, the transverse part of the dressed-quark-photon vertex will exhibit a pole at that for which

 1+Kγ(Q2)=0. (37)

This is just the model’s Bethe-Salpeter equation for the ground-state vector meson. The mass obtained therefrom is listed in Table 1.

In Fig. 1 we depict the function that dresses the transverse part of the quark-photon vertex. The pole associated with the ground-state vector meson is clear. Another important feature is the behaviour at large spacelike-; namely, as . This is the statement that a dressed-quark is pointlike to a large- probe. The same is true in QCD, up to the logarithmic corrections which are characteristic of an asymptotically free theory Maris:1999bh ().

One can define an electromagnetic radius for a dressed-quark; viz.,

 r2q=−6ddQ2PT(Q2)∣∣∣Q2=0. (38)

Our computed value is reported in Table 1. It is noteworthy that so that, although the ground-state vector-meson pole is a dominant feature of in the vicinity of , it does not completely determine the electromagnetic radius of the dressed-quark.

Nor, in fact, of anything else, as one can infer from the computed values of the pion charge radius reported in Table 1:333In computing , we follow Ref. GutierrezGuerrero:2010md (). Owing to the vector Ward-Takahashi identity, the longitudinal part of the vertex does not contribute to the pion’s elastic form factor. is obtained with ; is computed with ; and , which is just a consequence of the product rule. This emphasises again that single-pole vector-meson-dominance is a helpful phenomenology but not a hard truth Roberts:2000aa (); Maris:2003vk (); Maris:1999bh ().

We show in Fig. 2 that dressing the quark-photon vertex does not qualitatively alter the behaviour of at spacelike momenta. In particular, it does not change the fact that a momentum-independent interaction, Eq. (4), regularised in a symmetry-preserving manner, produces444The rainbow-ladder truncation omits a so-called meson-cloud component of but this, too, affects only the behaviour in a measurable neighbourhood of Alkofer:1993gu ().

 Femπ(Q2→∞)=constant. (39)

## Iii Transition form factor: γ∗π0γ

In the rainbow-ladder truncation this process is computed from Holl:2005vu ()

 Tμν(k1,k2)=Tμν(k1,k2)+Tνμ(k2,k1), (40)

where the pion’s momentum , and are the photon momenta and

 Tμν(k1,k2)=αemπfπϵμναβk1αk2βG(k21,k1⋅k2,k22) = tr∫d4ℓ(2π)4χπ(ℓ1,ℓ2)iQΓμ(ℓ2,ℓ12)S(ℓ12)iQΓν(ℓ12,ℓ1),

with , , , and , . The kinematic constraints are:

 k21=Q2,k22=0,2k1⋅k2=−(m2π+Q2). (43)

### iii.1 Anomaly

We first consider the chiral limit and , in which case Eq. (LABEL:anomalytriangle) describes the “triangle diagram” that produces the Abelian anomaly and one must compute . We have explained above that our regularisation of the interaction in Eq. (4) ensures that the non-anomalous vector and axial-vector Ward-Takahashi identities are satisfied. The outcome for the anomalous case is therefore very interesting.

Two contributions are obtained upon inserting Eq. (12) into Eq.(LABEL:anomalytriangle); viz., one associated with , which we’ll denominate , and the other with , to be called . We first examine the latter. To obtain one need only expand the integrand in Eq. (LABEL:anomalytriangle) around and keep the term linear in , a process which yields

 GF(0,0,0)=−fπM∫∞0dss2Fπ(P)σV(s) (44) ×[σV(s)2+sσV(s)σ′V(s)+σS(s)σ′S(s)],

where and we have written

 S(ℓ)=−iγ⋅ℓσV(ℓ2)+σS(ℓ2). (45)

Using Eq. (6), one readily finds that , . These identities, when inserted into Eq. (44), reveal that the integrand is identically zero, so that

 GF(0,0,0)=0. (46)

This is a particular case of the general result proved in Ref. Maris:1998hc (). As explained therein, since the integral in Eq. (44) is logarithmically divergent, the result is only transparent with the choice of momentum-partitioning that we have employed.

The remaining contribution is , which, following the methods of Sec. II, can be written

 GE(0,0,0)=Mfππ2∫d4ℓEπ(P)σV(ℓ212)σV(ℓ21)σV(ℓ22). (47)

The integral is convergent and therefore a shift in integration variables cannot affect the result. It follows that

 GE(0,0,0)=Eπ(P)fπM∫∞0dssM2(s+M2)3. (48)

If we employ Eq. (9), as with all other computations hitherto, this becomes

 GE(0,0,0)=1M2Ciu2(M2), (49)

where and we have used the Goldberger-Treiman relation in Eq. (30).

In what has long been a textbook result, the anomalous Ward-Takahashi identity states that : truly, just this simple fraction. Equation (49) is plainly inconsistent with this because it produces a number that depends on the values of the parameters , . Indeed, with the values in Eq. (27), our regularisation of Eq. (4) gives . What has gone wrong?

The answer lies in the observation that

 C∞02(M2)=C2(M2,τ2ir→∞,τ2uv→0)=12M2. (50)

One could have judged at the outset that no regularisation scheme which bounds the loop-integral can supply the correct result for the anomalous Ward-Takahashi identity because it blocks the crucial connection between the anomaly, topology and DCSB Witten:1983tw ().

To elucidate, return to Eq. (48). The integral is convergent and dimensionless. Hence, it cannot depend on . In a particular application of the procedure elucidated in Refs. Roberts:1987xc (); Roberts:1994hh (); Alkofer:1995jx (); Maris:1998hc (), the change of variables yields

 GE(0,0,0) = ∫∞0dC1(1+C)3 (51) = −12∫∞0dCddC1(1+C)2=12. (52)

The last line emphasises the connection between the simple rational-number result and the spacetime boundary: the anomaly is determined by the integral of a total derivative. The result in Eq. (52) is obtained if, and only if, chiral symmetry is dynamically broken, since in this instance alone can Eq. (30) be used to completely eliminate the pion structure-factor: , from the expression.

### iii.2 Asymptotic behaviour

#### iii.2.1 Massless vector-boson exchange

In Ref. Lepage:1980fj (), using the methods of light-front quantum field theory, it was shown that

 limQ2→∞Q2G(Q2,−\footnotesize12Q2,0)=4π2f2π. (53)

It is notable that this is a factor of bigger than the kindred limit of the elastic pion form factor Lepage:1980fj (); Farrar:1979aw (); Efremov:1979qk (); i.e., at GeV, more than an order-of-magnitude larger.

Our analysis of Eq. (LABEL:anomalytriangle) is performed within a Poincaré covariant formulation. In this case, as elucidated in Ref. Maris:2002mz (), the asymptotic limit of the doubly-off-shell process () is reliably computable in the rainbow-ladder truncation because both quark-legs in the dressed-quark-photon are sampled at the large momentum-scale , with the result Kekez:1998rw (); Roberts:1998gs ()

 limQ2→∞Q2G(Q2,−Q2−m2π/2,Q2)=234π2f2π, (54)

if the propagator of the exchanged vector-boson behaves as for large-. In order to obtain this result it is crucial that the pion’s Bethe-Salpeter amplitude depends on the magnitude of the relative momentum and behaves as for large-, as it does in QCD. (See also Sec. III.3.)

Equations (53) and (54) correspond to the asymptotic limits of different but related processes. Part of the mismatch owes to the fact that in the process not all quark-legs attached to vertices carry the large momentum-scale ; namely, in Eq. (LABEL:anomalytriangle), and hence some amount of vertex dressing contributes, even at large . This is consistent with a more general observation; namely, that in a covariant calculation any number of loops contribute to at leading order Lepage:1980fj (), and these provide a series of logarithmic corrections which should be summed. The same is true of the pion’s elastic form factor Maris:1998hc (). Nevertheless, the correct power-law behaviour is necessarily produced.

#### iii.2.2 Contact interaction

With the QCD-based expectation made clear, we now turn to the outcome produced by the contact interaction, Eq. (4). In this instance the arguments used to obtain Eq. (54) fail conspicuously because the pion’s Bethe-Salpeter amplitude is completely independent of the relative momentum: all values of the relative momenta are equally likely. This is why the interaction yields Eq. (39), a result in striking disagreement with experiment. A similar result is obtained in the present context. However, a decision must be made before that can be exhibited.

Recall Eqs. (44) and (48): the first is logarithmically divergent while the second is convergent even if the regularisation parameters are removed. Indeed, one needs to remove the regularisation scales if the anomaly value is to be recovered. However, the form factor is then ill-defined because the -term contributes the logarithmic divergence just noted. We proceed by removing the regularisation in computing but retaining it in calculating . Notably, as we will see, with Eq. (4) no internally consistent scheme can provide QCD-like ultraviolet behaviour but this prescription serves to preserve the infrared behaviour.

There is one more step in implementing this scheme. In arriving at expressions such as those defining the pion’s Bethe-Salpeter kernel (see Sec. II.4), we re-express a product of propagator-denominators via a Feynman parametrisation, then perform a change-of-variables, and thereafter rewrite the result using Eq. (9). This does not introduce any difficulties when the boundary at spacetime-infinity has no physical impact. However, as we have seen, that is not the case with the anomaly. The integral which defines is logarithmically divergent. A shift of integration variables changes its value, and in doing that one runs afoul of the fact that it is impossible to simultaneously preserve the vector and axial vector Ward-Takahashi identities for triangle diagrams in field theories with axial currents that are bilinear in fermion fields. Any shift in variables from that used in Eq. (LABEL:anomalytriangle) changes the value of . We compensate by an additional regularising subtraction; i.e., by redefining

 GF(Q2,−(m2π+Q2)/2,0) (55) → GF(Q2,−(m2π+Q2)/2,0)−GF(0,0,0).

In doing so we implement an anomaly-free electromagnetic current Jackiw ().

In Fig. 3 we depict the result produced from Eq. (4) using the regularisations just described. Comparing the solid- and dashed-curves, it is evident that the effect of dressing the quark-gluon vertex diminishes with increasing and therefore has no impact on the asymptotic behaviour of the transition form factor. It does, however, affect the neutral-pion interaction-radius, which can be defined via

 r∗2π0=−6ddQ2lnG(Q2,−(m2π+Q2)/2,0)∣∣∣Q2=0. (56)

This yields fm and fm, values that are not sensibly distinguishable from the charged-pion values listed in Table 1. Near equality of and is also found in the QCD-based calculations of Refs. Maris:2002mz (); Maris:1999bh ().555Choosing instead the form factor, one finds fm and fm, values which are larger because the momentum-scale enters into both quark-photon vertices.

More significantly, the solid- and dashed-curves in Fig. 3 show that, as with the elastic form factor GutierrezGuerrero:2010md (), the presence of the pion’s necessarily-nonzero pseudovector component, , leads to

 limQ2→∞G(Q2,−(m2π+Q2)/2,0)=constant. (57)

This is consistent with the picture developed in Ref. GutierrezGuerrero:2010md (); namely, it is possible to treat the contact interaction, Eq. (4), so that it yields static properties of the pion in agreement with experiment and computations based on well-defined and systematically improvable truncations of QCD’s DSEs. However, a marked deviation from experiment occurs in processes that probe the pion with and it is impossible to obtain results which agree with perturbative-QCD, even at the gross level of form-factor power-laws.

These observations are emphasised by the comparisons presented in Figs. 4 and 5. The form factor obtained using the symmetry-preserving, fully-self-consistent rainbow-ladder treatment of the contact interaction in Eq. (4) is in glaring disagreement with all existing data. This is what it means to have a pointlike-component in the pion: all form factors must asymptotically approach a constant. That limit rapidly becomes apparent with increasing momentum transfer because the dynamically generated mass-scale associated with low-energy hadron phenomena is GeV. No study that neglects the pion’s pseudovector component can provide a valid explanation or interpretation of the transition form factor, or any other of the pion’s form factors.

On the other hand, it is noteworthy that the DSE result Maris:2002mz (), which is based on an interaction that preserves the one-loop renormalisation group behavior of QCD, agrees with all but the large- BaBar data.

### iii.3 Pion distribution amplitude

It is worthwhile to consider a little further the nature of a pointlike pion. As explained in Ref. Holt:2010vj (), with the dressed-quark propagator and pion Bethe-Salpeter amplitude in hand, one can compute the pion’s valence-quark parton distribution function in rainbow-ladder truncation. For the contact interaction, the result is

 qV(x)=32itrD∫d4ℓ(2π)4δ(n⋅ℓ−xn⋅P) (58) × Γπ(−P)S(ℓ)n⋅γS(ℓ)Γπ(P)S(ℓ−P),

where , , and is the Bjorken variable.

It follows from this expression that

 (n⋅P)n+1∫10dxxnqV(x)=32itrD∫d4ℓ(2π)4(n⋅ℓ)n (59) × Γπ(−P)S(ℓ)n⋅γS(ℓ)Γπ(P)S(ℓ−P).

At this point we’ll specialise to the chiral limit and: evaluate the Dirac-trace; use a Feynman parametrisation to re-express the product which arises; shift variables , where is the Feynman parameter; use the invariance of the measure to evaluate the angular integrals; and thereby arrive at

 ∫10dxxnqV(x) = 1n+134π2¯¯¯Ciu1(M2)Eπ[Eπ−2Fπ] (60) = 1n+1,

where the last line follows because the pion’s Bethe-Salpeter amplitude is canonically normalised, Eq. (26).

The distribution function is readily reconstructed from Eq.(60); and one finds that even with inclusion of the pion’s necessarily-nonzero pseudovector component, the contact-interaction produces

 qV(x)=θ(x)θ(1−x), (61)

which corresponds to a pion distribution amplitude

 ϕπ(x)=constant. (62)

This outcome provides another way of understanding the inability of the contact interaction to reproduce the results of QCD.

As reviewed and explained in Ref. Holt:2010vj (), Goldstone’s theorem in QCD is expressed in a remarkable correspondence between the quark-propagator and the pion’s Bethe-Salpeter amplitude; i.e., between the one- and two-body problems Maris:1997hd (). The long-known fact that the dressed-quark mass function behaves as Lane:1974he (); Politzer:1976tv (); Bhagwat:2003vw (); Bhagwat:2006tu (); Bowman:2005vx ()

 M(p2)\lx@stackrellarge-p2∼1p2, (63)

entails that in QCD every scalar function in the pion’s Bethe-Salpeter amplitude, Eq. (1), depends on the relative momentum, , as for large- (with additional logarithmic suppression). It is impossible to find a kinematic arrangement of the dressed-quarks constituting the pion in which the Bethe-Salpeter amplitude remains nonzero in the limit of infinite relative momentum.

It follows that in QCD the pion’s valence-quark distribution behaves as , , for , at a renormalisation scale of GeV Holt:2010vj (); Hecht:2000xa (). Hence there is no renormalisation scale in the application of perturbative QCD at which Eq. (61) is a valid representation of nonperturbative QCD dynamics; namely, no scale at which it is tenable to employ constant, or even, more weakly, flat and nonzero at .

## Iv Reflections on extant data

We have shown that an internally consistent treatment of the contact interaction is incompatible with extant pion elastic (Fig. 2) and transition form factor data (Figs. 4, 5). Notwithstanding this, the results elucidated can be used in combination with QCD-based DSE studies in order to comment on available data for the process .

To begin we remark upon a similarity between the -dependence of the dash-dot- and dotted-curves in Fig. 3; i.e., the QCD-based DSE result and that obtained from the contact interaction if the pion’s pseudovector component is artificially eliminated. We emphasise that if is forced to zero, then one is no longer representing faithfully the features and consequences of a vector-vector contact interaction. Hitherto, this has nevertheless been a conventional mistreatment of Eq. (4). Its consequences were first elucidated in Ref.GutierrezGuerrero:2010md (). We describe results obtained through this intervention in order to clarify its real implications for the transition form factor.

An analysis of Eq. (LABEL:anomalytriangle) shows that the contact interaction yields

 GE(Q2,−(Q2+m2π)/2,0)\lx@stackrelQ2≫M2=\footnotesize12M2Q2[lnQ2M2]2, (64)

cf. Eq. (54). A similar result is obtained for the doubly-off-shell process, with the only difference being that the power of the logarithm is reduced “”.

We stress that in Eq. (64), is the dressed-quark mass. Table 1 emphasises that is a computed quantity, which is completely determined once the interaction and truncation are specified. The value of is tightly connected with those of all other measurable properties in the Table. Thus, in a well-constrained and internally-consistent analysis, one cannot significantly alter without materially changing all the other static properties which characterise the pion. No theoretical analysis is reliable if it allows itself to skirt these constraints.

In Fig. 6 we depict the contact-interaction result for normalised by the asymptotic form in Eq. (64). In addition, we plot a monopole with mass-scale , which bounds uniformly from above the QCD-based DSE calculation of the transition form factor reported in Ref. Maris:2002mz (); and a monopole with mass-scale , which is a fit to the transition form factor also computed therein. The origin of these mass-scales was discussed in Sec. III.2.1. It is striking that these curves all approach their asymptotic limits from below. Stated differently, each is a monotonically-increasing concave function. Indeed, this is even true of the solid curve in Fig. 5.

## V Conclusions

We have shown that in fully-self-consistent treatments of pion: static properties; and elastic and transition form factors, the asymptotic limit of the product , which is determined a priori by the interaction employed, is not exceeded at any finite value of spacelike momentum transfer: the product is a monotonically-increasing concave function. We understand a consistent approach to be one in which: a given quark-quark scattering kernel is specified and solved in a well-defined, symmetry-preserving truncation scheme; the interaction’s parameter(s) are fixed by requiring a uniformly good description of the pion’s static properties; and relationships between computed quantities are faithfully maintained.

Within such an approach it is nevertheless possible for , with being the elastic form factor, to exceed its asymptotic limit because the leading-order matrix-element involves two Bethe-Salpeter amplitudes. This permits an interference between dynamically-generated infrared mass-scales in the computation. Moreover, for the perturbative QCD limit is more than an order-of-magnitude smaller than . Owing to the proximity of the -meson pole to , the latter mass-scale must provide a fair first-estimate for the small- evolution of . A monopole based on this mass-scale exceeds the pQCD limit . For the transition form factor, however, the opposite is true because is less-than the pQCD limit, Eq. (53).

A vector current-current contact-interaction may be described as a vector-boson exchange theory with vector-field propagator , . We have shown (see Figs. 4, 5) that the consistent treatment of such an interaction produces a transition form factor that disagrees with all available data. On the other hand, precisely the same treatment of an interaction which preserves the one-loop renormalisation group behaviour of QCD, produces a form factor in good agreement with all but the large- data from the BaBar Collaboration Aubert:2009mc ().

Studies exist which interpret the BaBar data as an indication that the pion’s distribution amplitude, , deviates dramatically from its QCD asymptotic form, indeed, that constant, or is at least flat and nonvanishing at Radyushkin:2009zg (); Polyakov:2009je (). We have explained that such a distribution amplitude characterises an essentially-pointlike pion; and shown that, when used in a fully-consistent treatment, it produces results for pion elastic and transition form factors that are in striking disagreement with experiment. A bound-state pion with a pointlike component will produce the hardest possible form factors; i.e., form factors which become constant at large-.

On the other hand, QCD-based studies produce soft pions, a valence-quark distribution amplitude for the pion that vanishes as for , and results that agree well with the bulk of existing data.

Our analysis shows that the large- BaBar data is inconsistent with QCD and also inconsistent with a vector current-current contact interaction. It supports a conclusion that the large- data reported by BaBar is not a true representation of the transition form factor, a perspective also developed elsewhere Mikhailov:2009sa (). We are confirmed in this view by the fact that the and transition form factors have also been measured by the BaBar Collaboration Aubert:2006cy (), at GeV, and in these cases the results from CLEO Gronberg:1997fj () and BaBar are fully consistent with perturbative-QCD expectations.

###### Acknowledgements.
We acknowledge valuable: correspondence with S. J. Brodsky; and discussions with C. Hanhart, R. J. Holt and S. M. Schmidt. This work was supported by: Forschungszentrum Jülich GmbH; the U. S. Department of Energy, Office of Nuclear Physics, contract no. DE-AC02-06CH11357; the Department of Energy’s Science Undergraduate Laboratory Internship programme; CIC and CONACyT grants, under project nos. 4.10 and 46614-I; and the U. S. National Science Foundation under grant no. PHY-0903991 in conjunction with a CONACyT Mexico-USA collaboration grant.

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