Gauge Technique approximation to the $\pi \gamma$ production and the pion transition form factor

The $2\gamma^*\rightarrow \pi$ transition form factor , $G(q^2)$ is computed on the entire domain of spacelike and timelike momenta using a quantum field theory continuum approach. The analytical continuation of the function $G(Q^2)$ is based on the utilization of the Gauge Technique with the entry of QCD Green's functions determined from Minkowski space solution of QCD Dyson-Schwinger equations. The scale is set up by the phenomena of dynamical chiral symmetry breaking, which is a striking feature of low energy QCD.


I. INTRODUCTION
Transitions including single or few mesons are key elements for understanding exclusive QCD processes. For such processes the associated meson form factors, which carry nontrivial information about the interaction of quark constituents, depends on a few Lorentz invariant variables only. The neutral pion transition form factor G(Q 2 ) represents such process and even more it belongs to a rare exceptions, which has been measured on relatively large domain of timelike as well as spacelike Minkowski momenta. Kinematically selected events in more involved process e + e − → e + e − π 0 provide the transition form factor G(Q 2 ) from the cross sections which involves double variable function G γ * γ * π 0 (Q 2 , Q ,2 ) where Q, Q ′ are the momenta of the virtual photons. Both are the smooth function in the spacelike domain Q 2 = −q 2 , the first is the limit of the second G(x) = lim y→0 G γ * γ * π 0 (x, y) for one photon connecting the pion and the lepton pair being nearly on-shell y = Q '2 = 0. Such pion transition form factor for he spacelike Q 2 > 0 has been recently extracted from the data by the Belle Collaboration [1], and earlier by the BABAR [2], CELLO [3] and CLEO [4] collaborations. On the other side of q 2 variable axis, the transition form factor G(q 2 ) where q 2 > 0 is in the timelike domain of the four momentum, the form factor can be extracted from the annihilation process e + e − → π o γ with now the outgoing photon is perfectly on-shell. Ignoring the interference with other sources 3γ the cross section σ πγ which is linearly proportional to G 2 (q 2 ) has been extracted form the data collected by CMD2/SND collaboration [5].
Using predictions of perturbative QCD [10] the transition pion form factor is absolutely normalized at asymptotic momentum transfer with simple functional form: (1.1) The validity of perturbative QCD represented here by Eq. (1.1) has been experimentally tested at momenta Q < 6GeV . Owing the known discrepancy between the data collected by BaBaR and Belle Collaborations, only the later being with an agreement with the Eq. (1.1) at Q 2 0 ≃ 20GeV 2 , where Q 0 is the energy scale of perturbative QCD limit. The BaBar data overestimated the perturbative tail of (1.1) in a large region, provoking excited discussion where many were arguing that the BaBar sample of data does not accurately represent the pion transition from factor [6], [7], including also arguments coming from the prompt use of Dyson-Schwinger equations [8].
In this paper we contribute toward understanding of the pion transition form factor by analyzing this process using the existing Dyson-Schwinger equations (DSEs) calculation [9], which has the capacity to provide nonperturbative results not only for the Euclidean momentum (spacelike) , but on the entire domain of Minkowski space momentum (this includes both, spacelike as well as the timelike domain of momentum; herein we use the Minkowski metric convention g µν = diag(1, −1, −1, −1), thus q 2 is positive for the timelike momentum , while q 2 < 0; Q 2 = −q 2 in the spacelike domain of momentum). The DSEs are traditional tool for such calculations, they have potential to provide nonperturbative results in infrared, where the QCD coupling is strong, while smoothly approaching the perturbative limit when momenta are asymptotically large Q 2 >> Λ 2 QCD . Using the formalism of QCD DSEs , chiral symmetry breaking and its connection with meson spectra and continuous electromagnetic form factors are understood [17], [18], [19] , as well as Abelian anomaly and more general transition form factors were studied in unified picture of Euclidean space QCD DSE.
The set up of the paper is the following: In the next section a minimal symmetry preserving approximation of QCD Dyson-Schwinger equations model necessary for obtaining the pion transition form factor is presented. In the Section III. the Gauge Technique is reviewed for the quark propagator satisfying generalized spectral (Hilbert) representation. Results are discussed and presented in the Section IV and we conclude in the last section V.

II. EQUATIONS FOR TRANSITION FORM FACTOR
The perturbative QCD factorization formula for the transition form factor has the structure, which led to the Eq.(1.1). The evolution of the soft part -the pion distribution amplitude φ-was described in [10] and as was shown in light-cone quantized QCD it softly depends on Q as it receives logarithmic corrections. At large Q the function φ can be expressed as the light cone projection of Bethe-Salpeter amplitude and as recently elaborated in practice [11] such definition can be actually exploited to avoid problematic numerical integration in DSE formalism as well. The hard part of the parton scattering T is evaluated perturbatively within the use of on-shell quark spinors. At the other side, the phenomena of confinement in QCD can be mathematically formulated to the statement that there are no free on-shell quarks in the Nature, which has a crucial consequence since the on-shell spinors could not be an adequate description for the confined quark modes at all. The development of nonperturbative tools which incorporates confinement and naturally avoid the use of perturbation theory on-shell quarks is the aim of presented paper.
As a part of this analysis, we will elucidate the (in)sensitivity of the γ * γ → π 0 to the analytical behavior of the quark propagator S. We will use DSE solution [9], which has no on-shell pole in the quark propagator and calculate the transition form factor. The rainbow-ladder truncation skeleton diagram [11,20] which determines the function G(q) is computed from where the pion's momentum P = k + l, k and l are the photon momenta and two variable form factor is defined as where Γ π (q 1 , q 2 ) is the pion Bethe-Salpeter vertex function and G µν (q 1 , q 2 ; k, l) is non-amputated fully dressed quarkantiquark-two gamma vertex function in notation where variable q 1 and q 2 stands for the constituent (anti)quark momenta.
To get a single variable pion transition form factor, the kinematics constrains read l 2 = 0, P 2 = k 2 + 2k.l = m 2 π , (2.4) albeit we will keep both photons off-shell during the derivation here. The four-point function G µν consistent with rainbow-ladder approximation can be written uniquely as the decomposition of the quark propagators S and the proper quark-photon vertices Γ with momenta labeling that keeps q 1 = q 2 + k + l. The proper vertex Γ µ involves rich tensorial structure with 12 different form factor function included in and as the first approximation we instead of (2.5) exploit the Gauge Technique approximation [25], which allows to write where the weight function ρ was obtained from the solution of QCD Dyson-Schwinger equations in a simple AV symmetry preserving rainbow-ladder approximation, which we are going to describe in the next. Narrow meson width approximation gives rise the homogeneous Bethe-Salpeter equation, which is the traditional tool for calculation of meson masses [12][13][14][15][16]. Within the formalism the electromagnetic form-factors of pion and kaon [17][18][19] and meson transition form factors [11,20] were predicted or calculated in the spacelike region of momenta.
where the inverse of A is the renormalization wave function, while the (renormalization invariant) quark dynamical mass function is conventionally defined as M = B/A. The quark propagator is obtained from the solution of Dyson-Schwinger equation where following [9] we used the interaction kernel where L µν stands for the longitudinal projector L µν (q) = q µ q ν /q 2 . Stress here that the presence of nontrivial interaction proportional to longitudinal projector appears crucial for obtaining meaningful analytical results in the ladder rainbow approximation. We could mentioned here the empirical fact, that the solution with desired analytical properties is not automatically guaranteed in arbitrary gauge and its details (e.g. the branch points and the depth of the cut in between) does depend on the gauge parameters ξ. The origin of this numerical fail is not yet completely understood in general , but it lies beyond the success of the solution presented in [9] and a certain fail of strong coupling Landau gauge study presented in [23]. Here this the absence of the on-shell pole which is the main feature of optimized solution, showing up the confinement is realized in coexistence with the approximate existence of generalized (non-positive) spectral representation. The resulting propagator is shown in the Fig. 1, recalling that absence of ImS for the spacelike momenta was the aim of optimization strategy in [9].
For the sake of consistency, the Bethe-Salpeter and the quark Dyson-Schwinger equation must use the identical kernel. and to get the solution for the pion vertex , the following Bethe-Salpeter equation: has been solved. In the Eq. (2.11) P is the total momentum of meson satisfying P 2 = M 2 , M = 140M eV for the ground state and the arguments in the quark propagator are k ± = k ± P/2. The pion BSE vertex function reads where all Γ X was used to determine the pion mass, however for purpose of reduction computational difficulty associated with evaluation of 2.3 the single component approximation Γ(P, p) = γ 5 Γ A (P, p) was used through this paper. Note, we skip the isospin matrix notation, which is not necessary in given truncation of DSE system.

III. GAUGE TECHNIQUE ENTRY
One possibility to evaluate Eq. (2.3) is to know the solution for the Abelian gauge vertex Γ µ , which appears in the term (2.5). This can be achieved by solving the Dyson-Schwinger equation for this vertex as well, however to perform this task in the entire Minkowski space would require the knowledge of Integral Representation for all its components. Although, proliferation of Nakanishi's Perturbation Theory Integral Representation [24] is a logical (and likely correct) offer, the numerical realization of such a program is not yet recently in our hand. To accomplish this, we appreciate the fact that U (1) electromagnetic symmetry is unbroken in the Nature and employ the Gauge Technique [25] for the double vertex function here. It consist of writing a solution for the double vector un-amputated vertex in the form (2.6), where one needs that there exists a generalized spectral representation for the quark propagator .
The double vertex (2.6) satisfies the Ward-Takahashi identity: where untructated vertex takes familiar one body form of Gauge Technique [26], [27] It worthwhile to mention that within the use of the Gauge Technique we miss some important transverse pieces unavoidable for description of resonances. Future improvements, which will take into account transverse vertices as well, will always include the contribution already caught by the Gauge Technique. For general requirements needed for any acceptable Ansatz see for instance [11], where similar philosophy was used.
From now we will restrict to two flavors f = u, d in isospin (equal mass) limit. It is the experimental fact that the φ meson peak is suppressed by two orders when compared the ω/ρ peak in πγ cross section [5] . In the chiral limit , which we subsequently employ exclusively, the following Goldberger-Treiman-like identity [22]: exhibits the equivalence between the one-body quark and pseudoscalar two-body problem in QCD. Further, the renormalized quark function B obtained from the Eq. (2.8) satisfies the dispersion relation which we employ as well in this paper and we replace the pion vertex function by what B would be in the chiral limit 6) where N is the normalization of the BSE vertex, satisfying approximately N = f π , but with its exact value dictated by the canonical normalization of the BSE vertex [21]. Putting all together on gets the expression for the double variable pion transition form factor G(k 2 , l 2 ) ≃ I(q, l) the following integral where the integral over the variable ω is due to the Rel. (3.6) and the integral over the variable √ s is due to the Eq. (2.6) . Two weight functions ρ and ρ B are not independent but related through the definition (2.7).
In what follow we integrate over the momentum analytically and within the help (3.1) we rewrite the integral (3.7) equivalently into the expression where we integrate over the known and regular function: the quark propagator.
Let us sketch the derivation performance more explicitly here: First, we match all denominators by using the Feynman paramaterization. Lets use the variable x, y to match the free-like propagators in spectral s-space as a first steps. Then the obtained result we further match together with the denominator p 2 − ω stemming from the expression (3.6) used in (3.7). Then, after a usual shift of the momentum and Wick rotation one gets for (3.7), again in Minkowski space: where we have labeled r = k − l. After factorization of the variable z 2 out of the numerator, according to (3.1) one can recognize that the integrand together with the factor d √ ss is nothing else but the once differentiated quark propagator, which is further integrated over the auxiliary variables x, y, z. The expression (3.8) is exact and if needed , it can be straightforwardly used in order to get the result for the pion transition form factor. To avoid differentiation and in order to reduce number of auxiliary integrals, one can get very accurate expression by neglecting of the small term, which includes the pion mass z 2 m 2 π (1 − 2x) 2 y 2 /4 in the denominator of Eq. (3.8). Doing this explicitly and integrating over the variable x one gets where the variable √ s was absorbed into the usual definition of the scalar part of the quark propagator (2.7). Note, the derivation is independent on a given choice of the contour Γ, however the validity of the Wick rotation is assumed for. Recall for completeness that the quark propagator obtained in [9] belong to a class of solutions where the Wick rotation remains completely valid, according to the choice of Γ there.
Hence taking l 2 = 0 , the pion transition form factor finally reads where one has to take P.r = k 2 and r 2 = k 2 in the expression (3.10) and should take in mind that the first product turns as P.r = −k 2 when evaluating the function F (l, k).

IV. SOLUTION
We begin with the discussion of the spacelike asymptotic, which is often addressed in more common Euclidean space Dyson-Schwinger models. The asymptotic behavior of the pion transition form factor G(Q) has curious dependence on the interaction between quarks. For instance, the contact interaction typical for quarks in Nambu-Jona-Lasinio models provides non-QCD result with the asymptotic different from (1.1). It was shown in [8] that in this case (G c means just G but for this specific model) where M is constant constituent quark mass obtained within contact interaction.
Interesting asymptotic was obtained [28] for the double variable transition form factor in symmetric limit which tells us that with slightly smaller prefactor the same asymptotic behavior persist for the double variable transition form factor as well.
From the equations (1.1), (4.1) a specific logarithmic dependence of Q 2 G(Q) seems to be ruled by modeled interaction: a weaker interaction between the quarks at large relative momentum a smaller power of log (anomalous dimension) in G(q) one gets.
Herein, to get the quark propagator involved in the expression for the form factor (3.10) we employ the DSE kernel, which consists from the admixture of the softer interaction then given by tree level QCD, remind we have V ≃ 1/(k 2 ) 2 in the Eq. (2.9), noting the only longitudinal gauge fixing term behaves as usually 1/k 2 . As we can see, the presence of longitudinal mode is ignored and the expected log rule is confirmed here. The solution is shown in the Fig.1 2, where the function Q 2 G(Q 2 ) is displayed. Actually one can see that the pion transition form factor does not reach its QCD perturbative asymptotic, but instead the function Q 2 G(Q 2 ) shows up the maximum at few GeV, where it starts to decrease with negative anomalous dimension.
Very interesting is that the similar asymptotic is observed for the large timelike momenta as well, however it happens for a much larger asymptotic momentum q o ≃ 10GeV as is shown in the Fig. 3. Before that the function q 2 G(q) → const reach the flat plato, due to the interfrence, the details of its shape depends on the all components which enter into the expression for G.
We also show the graph of square of the pion transition form factor against the energy √ s in the Fig. 4, and compare to the extracted data from the SND cross section σ π,γ . We use the fact that SND data are far from the threshold E th = m π and the properly normalized ratio G(q)/(0) is extracted from the experiments by using: with α QED being the fine structure constant and σ π,γ is the total cross section of the process e + e − → π 0 γ. In the same Fig. 4 we also show the imaginary and the real parts of form factors. To distinguish an to guide eyes note that those lines optically vanishing at zero momenta stand for the imaginary parts. In all figures the single line labeled by I corresponds to the form factor as obtained via the Eq. (3.10). In the timelike domain we can see the cusp somewhere bellow vector meson ω energy. In fact for m π = 140M eV the cusp would be located at ≃ 500M eV , which likely contradicts a possible future near threshold measurements. (It is plain to say the SND data starts at 600 MeV, however in a good belief in an approximate universality of resonances observed in electron-positron annihilation together and within simultaneous knowledge of 3π and 2π production cross sections, we exclude any naive new bump bellow 1 GeV energy). On the other side, albeit several times smaller then the Breight-Wigner shape of transverse omega meson, the cusp is here and must be located at ρ − ω meson energy since being effectively a part of it. Obviously the cusp is general feature for all flavors, likely it is even more pronounced for heavier mesons then for u, d quarks considered here and as there is no other way to hide such structure they must be a part of structural shapes already known experimentally. Contrary to the case of the light quarks, the suspicious regions of different flavors cusps need not to be associated with the ground state quarkonia, but they can contribute to shape usually dedicated to the first or the second excitations ( the dip associated with locally vanishing cross section are a good candidates as well).
In the light of preceding section, the results obtained here, suggest that our ladder-rainbow model has not capacity to provide both in once -the right amount of chiral symmetry breaking and cusp position near the pole of vector meson mass. For the purpose of the graph, we rescale and take the pion mass to be m π = 200M eV rather the physical one. A convergence is expected only after a significant improvements of the model, which is recently beyond our reach. Whether the Gauge Technique alone can provide more significant portion of the ρ − ω peak has been raised by its phenomenological use in [31] , but seems to be unsupported by more self-consistent approach presented here. Not only for this purpose, within the use of the Gauge Technique employed here, we have checked the changes in G(q) which stem from different extrapolation form used for the pion Bethe-Salpeter vertex function.
Using basically the same previous strategy, we also derive the formula for the pion transition form factor, which comes from the interpolator suggested in [29]. The interpolator takes the form Γ π (p, P ) = γ 5 1 Following the same steps as above, leads to the following formula for the pion transition form factor: In the expression (4.5) we have used more general two variable integral representation for the pion vertex function: which arises when one study the integral representation for the pseudoscalar BSE in the Minkowski space more selfconsistently [30] . The Eq. (4.4) is obviously special and simplified choice of expression (4.7) where the Bethe-Salpeter weight function includes the delta function δ(u − Λ π ).
The results for the choice ρ 5 (z) = (1 − z 2 ) labeled as the model II. are added in figures for comparison, noting that the large enhancement of the cusp in the timelike region is the main feature of presented approximation. On the other hand as seen from figures the replacement changes the position of maximum of Q 2 G(Q 2 ) but does not affect the decreasing character of the asymptotic behavior in the ultraviolet spacelike (figs 2 5) as well as in the timelike domain of momenta (see the Fig. 3).
Interestingly, almost the same pion form factor G(Q) is achieved with even more primitive interpolator: for which the formula (4.5) applies as well. The pion transition form factor evaluated for the interpolator (4.8) is labeled as the model III in all figures presented here. For both models the constant Λ 2 π = 0.5(m π /0.37) 2 GeV 2 where we set up m π = 0.2GeV for purpose of figures.
As seen in the Fig.4 the cusp is similarly pronounced for interpolators II (4.7) and III (4.8) , mainly due to the fact that the pion vertex has similar singularity in both cases. On the other hand, the results are far away to remind the structure of vector resonances observed in the experiment. For the spacelike region studied experimentally by Belle and others, our model I. is overestimates, while II and III are in a better accordance with the experiments (not shown). This in-consistence is better visible when G(Q 2 ) of the model I. is compared to the model II. at log scale as done in the Fig. (5). The difference is attributed to a different asymptotic of approximated Bethe-Salpeter pion vertex function as well as to different sensitivity to neglections made when going from the Eq. (3.8) to simplified Eq. (3.10) (a similar step was made in order to get to the expression for the form factor (4.5)). Note, we do not optimize to higher scale since the asymptotic is already not in a complete accordance with the QCD expectation and we leave the error freely propagate to higher Q. Again, we expect that a mutual convergence of both expressions (3.10) and (4.5) could appear in the spacelike domain, albeit it persist on the peak position, where the quark function B is more smooth then a more naive interpolators II and III.
Recall here also some other important result obtained with further simplification made elsewhere. For instance taking ρ 5 (z) = δ(1 + z) + δ(1 − z) in Eq. (4.4) corresponds with the flat pion distribution amplitude, noting one also needs to take the constituent quark propagator for this purpose, where M is the quark constant mass. Recall this result corresponds with the aforementioned NJL constant interaction studied in [8] giving the non-QCD form factor (4.1).

V. CONCLUSIONS AND PROSPECTS
We completed a computation of the pion transition form factor G(q 2 ) in two flavor QCD. The Gauge Technique we employed made possible to compute G(q 2 ) on the entire domain of spacelike as well as of timelike momenta in Minkowski space for the first time. All required elements are determined by the solution of QCD's Dyson-Schwinger equations obtained in the rainbow-ladder truncation, the leading order in a systematic and symmetry preserving approximation scheme. The model provides correct pion properties at prize G(q 2 ) shows up a cusp 250 MeV bellow FIG. 5: Pion transition form factor G(Q) for spacelike Q as described in the text: the quantity Γγγ (Q) = (π/4)αQEDm 3 π G 2 (Q 2 ) is shown. Form factors are normalized such that the pion width is reproduced Γγγ (0) = 7.23eV .
----------------------------ω/ρ mass. Relaxing with an accurate determination of the pion mass then approximately 200M eV heavy pion is needed to get the cusp located at the left shoulder of vector mesons peak observed in experiments. Furthermore, according to UV weakness of modeled interaction between the quarks , the pion transition form factor shows up non-QCD non-zero negative anomalous behavior for an asymptotically large spacelike momentum. These two failures could be absent after the model [9] gets improvement, here the numerical results represent an instructive exhibition of how observable color singlet form factor behavior is affected by modeled QCD interaction. The inclusion of correct QCD running coupling, albeit numerically more demanding in Minkowski space, is underway.
The cusp in the square of the pion transition form factor is about one or two magnitudes smaller then ω/ρ peak observed in the experiment [5]. A missing amount of the function G(q 2 ) there is an expected weakness of the Gauge technique approximation we employ here, since it does not incorporate an important transverse components form factors in the quark-photon vertex. To get the solution, it was important to employ integral representation which allows to make analytical continuation at the level of QCD Dyson-Schwinger equations. In order to claim an understanding of the confinement of quarks in QCD, it is critical to obtain new schemes and methods, such as that herein, which provide reliably result in the entire Minkowski space. The formalism developed herein represents steps towards our better understanding of how confinement mechanism is incorporating in unified picture of hadron