Dispersive analysis of the pion transition form factor

# Dispersive analysis of the pion transition form factor

###### Abstract

We analyze the pion transition form factor using dispersion theory. We calculate the singly-virtual form factor in the time-like region based on data for the cross section, generalizing previous studies on decays and scattering, and verify our result by comparing to data. We perform the analytic continuation to the space-like region, predicting the poorly-constrained space-like transition form factor below , and extract the slope of the form factor at vanishing momentum transfer . We derive the dispersive formalism necessary for the extension of these results to the doubly-virtual case, as required for the pion-pole contribution to hadronic light-by-light scattering in the anomalous magnetic moment of the muon.

###### Keywords:
Dispersion relations Meson–meson interactions Chiral Symmetries Electric and magnetic moments
###### pacs:
11.55.Fv 13.75.Lb 11.30.Rd 13.40.Em
journal: Eur. Phys. J. C

## 1 Introduction

One of the biggest challenges of contemporary particle physics is the unambiguous identification of signs of beyond-the-standard-model physics. While high-energy experiments are mainly devoted to the search for new particles, high-statistics low-energy experiments can provide such a high precision that standard-model predictions can be seriously scrutinized. A particularly promising candidate for such an enterprise is the gyro-magnetic ratio of the muon, for a review see Jegerlehner:2009ry (). Since the muon is an elementary spin-1/2 fermion, the decisive quantity is the deviation of its gyro-magnetic ratio from its classical value. This difference, caused by quantum effects, is denoted by .

From the theory side the potential to isolate effects of physics beyond the standard model is limited by the accuracy of the standard-model prediction. Typically the limiting factor is our incomplete understanding of the non-perturbative sector of the standard model, i.e. the low-energy sector of the strong interaction, which is governed by hadrons as the relevant degrees of freedom instead of the elementary quarks and gluons. In fact, for the hadronic contributions by far dominate the uncertainties for the standard-model prediction. The largest hadronic contribution, hadronic vacuum polarization (HVP), enters at order in the fine-structure constant and can be directly related to one observable quantity, the cross section of the reaction hadrons, by means of dispersion theory. In that way a reliable error estimate of HVP emerges from the knowledge of the experimental uncertainties in the measured cross section. At order there are next-to-leading-order iterations of HVP as well as a new topology, hadronic light-by-light scattering (HLbL) Calmet:1976kd (). It was recently shown in Kurz:2014wya () that even next-to-next-to-leading-order iterations of HVP are not negligible at the level of accuracy required for the next round of experiments planned at FNAL Welty-Rieger:2013saa () and J-PARC Saito:2012zz (), while an estimate of next-to-leading-order HLbL scattering indicated a larger suppression Colangelo:2014qya ().

With the increasing accuracy of the cross-section measurement for hadrons that can be expected in the near future Blum:2013xva (), the largest uncertainty for will then reside in the HLbL contribution. The key quantity here is the coupling of two (real or virtual) photons to any hadronic single- or many-body state. This quantity is not directly related to a single observable. However, it is conceivable to build up the hadronic states starting with the ones most dominant at low energies, in particular the light one- and two-body intermediate states. Based on a dispersive description of the HLbL tensor an initiative has recently been started to relate the one- and two-pion contributions for HLbL scattering to observable quantities Bern:lbl (); roadmap (); Bern:lbl-procs ().111A different approach, based on dispersion relations for the Pauli form factor instead of the HLbL tensor, was recently proposed in Pauk:2014rfa (). For a first calculation in lattice QCD, an alternative strategy to reduce the model dependence in the HLbL contribution, see Blum:2014oka (). The present work should be understood as an input for this initiative. We focus on the lowest hadronic state, the neutral pion, and its coupling to two (real or virtual) photons (a similar program is currently also being pursued for and , see Stollenwerk (); Hanhart_eta ()). Thus the central object of interest is the pion transition form factor. Its importance for the HLbL contribution to has been stressed early on, see e.g. Bijnens:2007pz (); Jegerlehner:2009ry (); Czerwinski:2012ry (), and triggered many studies of the transition from factor in this context Hayakawa:1997rq (); Bijnens:2001cq (); KN (); MV (); Nyffeler:2009tw (); Goecke (); Masjuan (); Terschlusen:2013iqa (); Roig:2014uja (). It is defined by

 ∫d4xeiq1⋅xi⟨0|Tjμ(x)jν(0)|π0(q1+q2)⟩ =−ϵμναβqα1qβ2Fπ0γ∗γ∗(q21,q22), (1)

where

 jμ=e∑fQf¯qfγμqf (2)

denotes the electromagnetic current carried by the quarks and the electric charge of the quark of flavor (in units of the proton charge ).

The normalization of the form factor is given by a low-energy theorem LET_pi0_1 (); LET_pi0_2 (); LET_pi0_3 (). In the chiral limit one finds

 Fπ0γ∗γ∗(0,0)→e24π2Fπ≡Fπγγ, (3)

which agrees with experiment to a remarkable accuracy, see Bernstein_review () for a recent review. In (3) denotes the pion decay constant PDG ().

For the dispersive treatment of the HLbL contribution to as envisaged in Bern:lbl (); roadmap (); Bern:lbl-procs () one needs the pion transition form factor for arbitrary space-like virtualities and of the two photons. We will approach this aim in a multi-step process. In the present work we will formulate the dispersive framework for the general doubly-virtual transition form factor, but restrict the numerical analysis to the singly-virtual case, both in the space- and time-like regions. We will use data on to fix the parameters and predict the cross section for as well as the space-like transition form factor to demonstrate the viability of the approach. While presently low-energy space-like data are scarce CELLO (); CLEO (), new high-statistics data can be expected in the near future from BESIII (see Benayoun:2014tra (); Amaryan:2013eja ()), which makes a calculation of the space-like singly-virtual form factor particularly timely. In a second step, the experimental information from both in space- and time-like kinematics will then serve as additional input for a full analysis of the doubly-virtual form factor.

The basic idea of the dispersive approach for the calculation of the pion transition form factor is its reconstruction from the most important intermediate states in the unitarity relation (see also Gorchtein (); Amaryan:2013eja ()). At low energies these are the two-pion and three-pion states with isospin and , respectively. Assuming perfect isospin symmetry one of the two photons of the amplitude must be in an isovector and one in an isoscalar state. We shall denote this assignment by the indices and , respectively. Then at low energies the unitarity relation for is dominated by , see the left diagram in Fig. 1. Additional inelasticities start contributing only at an invariant mass of the isovector photon above , predominantly in the form of four pions, cf. pionvff_Hanhart (). We will not consider such contributions explicitly in the present work, but estimate their potential impact by variations of the phase shifts in the inelastic region. The crucial building blocks of the dispersive treatment are the charged pion vector form factor , defined by

 ⟨0|jμ(0)| π+(p+)π−(p−)⟩= −e(pμ+−pμ−)FVπ((p++p−)2), (4)

and the amplitude for the reaction. The pion vector form factor with its normalization has been studied in great detail both from the theoretical and experimental side, see e.g. pionvff_BELLE (); pionvff_BaBar (); pionvff_KLOE (); pionvff_Anant (); pionvff_Hanhart (). It is closely related to the Omnès function to which we will come back in Sect. 2, see also omegaTFF (); g3pi () for more details.

In contrast, the structure of the amplitude for is much more involved. It will be discussed in detail in Sect. 2. Its two-body unitarity relation, illustrated by the right diagram in Fig. 1, involves the rescattering of pion pairs, which can be resummed in terms of the -wave phase shift within the dispersive approach. While two-body unitarity is exact, we do not consider full three-body unitarity as required by the intermediate states in , see left diagram in Fig. 2. However, with two-body unitarity fully implemented, the rescattering in generates topologies such as the one shown in the right diagram in Fig. 2, which manifestly contains three-pion cuts. The part of this diagram indicated by the dashed box can be interpreted as a special case of the full amplitude. Therefore, in our framework the structure of the left-hand cut in is approximated by pion pole terms.

The rest of the paper is organized as follows: in Sect. 2 we describe our framework for the determination of the amplitude. In Sect. 3 we formulate the general dispersion relation for the pion transition form factor with arbitrary virtualities for the two photons. In Sect. 4 we specialize the general framework to the case of one on-shell and one time-like photon. As a first application we will determine the cross section of the reaction and compare to the corresponding experimental results. Section 5 is devoted to the analytic continuation into the space-like region as well as the calculation of the slope of the form factor at zero momentum transfer. The Dalitz decay region is discussed in Sect. 6. We close with a summary and outlook in Sect. 7. An Appendix is added to discuss the comparison of our results to the simple vector-meson-dominance picture.

## 2 The γ∗→3π amplitude

### 2.1 Formalism

A key ingredient for the dispersive calculation of the pion transition form factor is the amplitude for the reaction . We define

 ⟨0|jμ(0)| π+(p+)π−(p−)π0(p0)⟩= −ϵμναβpν+pα−pβ0F(s,t,u;q2) (5)

with , , , , and .

The low-energy limit of is dictated by the chiral anomaly. In the chiral limit this leads to the identification WZW_1 (); WZW_2 (); WZW_3 (); WZW_4 (); WZW_5 (); g3pi ()

 F(0,0,0;0)→e4π2F3π≡F3π. (6)

A comment is in order to which extent the chiral predictions (3) and (6) have been confronted with experiment so far. has been tested up to in Primakoff measurements of  PrimEx () including chiral Bijnens:1988kx (); Goity:2002nn () and radiative Ananthanarayan:2002kj () corrections, the former up to two-loop order Kampf:2009tk (). Both the world average PDG () and the PrimEx result PrimEx () are fully consistent with the chiral tree-level prediction (3), the former even at accuracy, while chiral corrections predict an increase of up to mainly due to mixing Kampf:2009tk (), in slight tension with the world average. Here, we use (3) directly, given that apart from the very low-energy region the associated uncertainties are sub-dominant.

In contrast to this high accuracy the extractions of both from Primakoff measurements Antipov () (with chiral and radiative corrections from Bijnens90 (); Hannah (); Ametller ()) and  Scherer () presently allow a test at the level only. In g3pi () a dispersive framework (see also Hannah (); Holstein (); Truong () for earlier work in this direction) was presented that provides a two-parameter description of the cross section valid up to . This opens the possibility to profit from the high-statistics Primakoff data currently analyzed at COMPASS Friedrich:2014dna () concerning the extraction of to higher accuracy.

We decompose as

 F(s,t,u;q2)=F(s,q2)+F(t,q2)+F(u,q2). (7)

This decomposition neglects discontinuities in - and higher partial waves, see Hannah (). Using the (-channel) partial-wave decomposition

 F(s,t,u;q2) =∑ℓoddfℓ(s,q2)P′ℓ(cosθs), cosθs =t−uκ(s,q2), κ(s,q2) =σπ(s)λ1/2(q2,M2π,s), (8)

with the Källén function and , we find that the function in (7) is related to the -wave amplitude according to V3pi ()

 f1(s,q2) =F(s,q2)+^F(s,q2), ^F(s,q2) =32∫1−1dz(1−z2)F(t(s,q2,z),q2), (9)

with

 t(s,q2,z)=12(3M2π+q2−s)+12κ(s,q2)z. (10)

Note that for positive the evaluation of (2.1) is straightforward, while some care is needed for the proper analytic continuation of the square roots for negative . Therefore the framework presented here can be immediately applied for instance to the singly-virtual time-like transition form factor, as will be shown in Sect. 4. For the corresponding space-like form factor, to be tackled in Sect. 5, we will refrain from an analytic continuation of the formulae presented here but instead use a dispersion relation to determine the space-like transition form factor from the imaginary part of the time-like one.

For fixed , the quantity , given in (9), only has a right-hand cut starting at . The left-hand cut of the partial wave entirely resides in . Furthermore, the amplitude develops a three-pion cut for , i.e. in kinematics allowing for the physical decay . In this situation, the right- and left-hand cuts in begin to overlap, which leads to a significant complication of the analytic structure, see the corresponding discussion in V3pi ().

The discontinuity of the partial wave along the right-hand cut is given by

 discf1(s,q2)=2if1(s,q2)θ(s−4M2π)sinδ(s)e−iδ(s), (11)

where is the -wave phase shift. Noting that along the right-hand cut, we can recast this relation into the form

 discF(s,q2) ×θ(s−4M2π)sinδ(s)e−iδ(s). (12)

A once-subtracted dispersive representation solving (2.1) is given by V3pi ()

 F(s,q2) =Ω(s) (13) ×{a(q2)+sπ∫∞4M2πds′^F(s′,q2)sinδ(s′)s′(s′−s)|Ω(s′)|},

where

 Ω(s)=exp{sπ∫∞4M2πds′δ(s′)s′(s′−s)} (14)

is the Omnès function Omnes ().

An important property of (13) concerns its linearity in the subtraction function , which follows from the fact that is defined in terms of the angular average of itself (9). In this way, takes the role of a normalization, so that in practice (9) and (13) are solved by iteration for , while the full solution is recovered by multiplying with in the end. However, since as a function of implicitly depends on , the subtraction function is not the only source of dependence in the full solution.

For fixed virtualities the solutions of (9) and (13) have been studied in V3pi () to describe the vector-meson decays .222For a variant of this calculation see Danilkin:2014cra (). In this case the respective subtraction constant is fixed by the overall normalization of the Dalitz plot distribution and hence the corresponding partial decay width. The main complication when extending (13) to arbitrary virtualities of the incoming photon arises from the fact that depends on , a dependence that cannot be predicted within the dispersive framework itself, but has to be determined by different methods. Physically, contains the information how the isoscalar photon couples to hadrons. At low energies, this coupling is dominated by the three-pion state and can be accessed in . For the extraction of we need a representation that preserves analyticity and accounts for the phenomenological finding that the three-pion state is strongly correlated to the very narrow and resonances. We take

 a(q2)=α+βq2+q4π∫∞sthrds′ImA(s′)s′2(s′−q2), (15)

with modeled using two relativistic Breit–Wigner functions

 A(q2) =cωM2ω−q2−i√q2Γω(q2) +cϕM2ϕ−q2−i√q2Γϕ(q2). (16)

In the following we refer to as the spectral function. In (2.1) is the energy-dependent width of the meson, respectively. We take into account the main decay channels of and via

 Γω(q2) =γω→3π(q2)γω→3π(M2ω)Γω→3π+γω→π0γ(q2)γω→π0γ(M2ω)Γω→π0γ, Γϕ(q2) =γϕ→3π(q2)γϕ→3π(M2ϕ)Γϕ→3π +∑K=K+,K0γϕ→K¯K(q2)γϕ→K¯K(M2ϕ)Γϕ→K¯K, (17)

where denotes the measured partial decay width for the decay , while the energy-dependent coefficients are given by

 γω→π0γ(q2) =(q2−M2π)3(q2)3/2, γϕ→K¯K(q2) =(q2−4M2K)3/2q2, (18)

and the calculation of is performed along the lines described in V3pi (). For completeness we also include the decay channel of the , which strictly speaking corresponds to a radiative correction. As a consequence the threshold in (15) is actually instead of . However, we checked that as expected the impact of the channel is very small numerically.

The representation (15) can be understood as a dispersively improved Breit–Wigner parametrization Lomon (); Moussallam:gg*pipi (): the reconstruction of the real part via a dispersive integral ensures a reasonable behavior of the phase of despite the energy dependence of the widths. We decide to subtract (15) twice: the first subtraction constant is fixed by the chiral anomaly for at the real-photon point (corrected for quark-mass renormalization) Bijnens90 (); g3pi (),

 α=F3π3×(1.066±0.010)≡α3π. (19)

The second subtraction serves as an additional background term and is fitted to cross-section data, together with the residues and . Note that the precise form of the spectral function in (2.1) is irrelevant: the only requirement is to have an analytically rigorous representation of the cross section.

Finally, we give the explicit relation between the amplitude (7) and the cross section (neglecting the electron mass)

 σe+e−→3π=∫smaxsminds∫tmaxtmindtd2σdsdt, (20)

with

 d2σdsdt=e2P96(2π)3q6|F(s,t,u;q2)|2 (21)

and

 P ≡−gμμ′ϵμναβpν+pα−pβ0ϵμ′ν′α′β′pν′+pα′−pβ′0 =14(stu−M2π(q2−M2π)2) =116sκ(s,q2)2sin2θs, (22)

as well as integration boundaries

 smin=4M2π,smax=(√q2−Mπ)2, (23)

and

 tmin/max =(E∗−+E∗0)2 −(√E∗2−−M2π±√E∗20−M2π)2, E∗− =√s2,E∗0=q2−s−M2π2√s. (24)

We note in passing that for fixed, but arbitrary we can predict the shape of the two-fold differential distribution (21). The knowledge of is only needed for the overall normalization, not for the and dependence.

It has been noted in V3pi () that the amplitude representation (13) is not accurate enough to give a statistically valid description of the very precise Dalitz plot determination by the KLOE collaboration KLOE:phi (). For this purpose, a second subtraction was introduced, leading to the representation

 F(s,q2) =Ω(s){a(q2)+b(q2)s +s2π∫∞4M2πds′^F(s′,q2)sinδ(s′)s′2(s′−s)|Ω(s′)|} (25)

(only used for in V3pi ()). Similarly, for a twice-subtracted amplitude representation was envisaged theoretically in g3pi (). For general , the second subtraction will again be -dependent. Provided future measurements allow us to determine such a second subtraction both from cross-section data () and from an Dalitz plot (), the three data points—together with —should permit a smooth interpolation of in a representation similar to (15) (with only a single subtraction). In the absence of such additional high-precision data, we will utilize the singly-subtracted representation (13) of the partial wave for the purpose of this study.

### 2.2 Fits to e+e−→3π

Before turning to the fit results, we first summarize the various uncertainty estimates that we have performed in the context of our fits to . First of all, in the calculation of we used three different phase shifts, the phases from CCL (); Madrid () and a version of CCL () that includes the and the resonances in an elastic approximation to try to mimic the possible impact of inelasticities V3pi (). In addition, we varied the cutoff in the dispersive integral (13) above which asymptotic behavior is assumed between and , see omegaTFF ().

Next, our representation for is only adequate below , given that above this energy excited states of and may contribute. The isoscalar vector resonances listed in PDG () below with a sizable branching fraction are the and the , with masses and widths

 Mω′ =(1.425±0.025)GeV, Γω′ =(0.215±0.035)GeV, Mω′′ =(1.67±0.03)GeV, Γω′′ =(0.315±0.035)GeV. (26)

To estimate the effect of these states, we also consider a version of the fits where additional terms for and are included in (2.1), identical to the expression for the apart from the channel (we assume branching fraction to for and ). In total, we thus have a three- (five-)parameter representation to be fit to data, with free parameters , , (and , ).

The prime source of data below/above are the SND ee3pi_SND_1 (); ee3pi_SND_2 () and CMD2 ee3pi_CMD2_1 (); ee3pi_CMD2_2 ()/the BaBar data sets ee3pi_BaBar (), respectively. Restricting the fit (without and ) to the energy region below , we observed that the SND data set can be described with a reduced close to , while the CMD2 scans can only be accommodated with a significantly worse (around ). We also checked if the respective fit reproduced the correct chiral anomaly by including in (15) as another fit parameter. For SND we indeed obtain , while the fit to CMD2 even produces a negative value of .

One explanation for this apparent tension could be provided by the fact that radiative corrections were not treated in exactly the same way in both experiments. Moreover, the CMD2 scans were restricted to a relatively narrow region around the and masses, limiting the sensitivity to the low-energy region (and thus particularly to the chiral anomaly). Such inconsistencies in the data base were already observed in Hagiwara:2011af () in the context of the HVP contribution to , where the channel entered with a global reduced of . For the present study we will therefore consider two data sets: first, SND+BaBar and, second, the compilation from Hagiwara:2011af (), in the following denoted by HLMNT. It includes all data sets mentioned so far as well as some older experiments Cordier:1979qg (); Dolinsky:1991vq (); Antonelli:1992jx (); Akhmetshin:1995vz (); Akhmetshin:1998se (); ee3pi_CMD2_3 (). The rationale for doing so is that for the reasons explained above SND/BaBar appear to be the most comprehensive single data sets for low/high energies. Confronting the outcome of fits to the combination of both and to the comprehensive data compilation of Hagiwara:2011af () should allow for a reasonable estimate of the impact of the uncertainties in the cross section on the prediction for the pion transition form factor.

The result of the three-parameter fit to SND+BaBar below is shown in the left panel of Fig. 3, with fit parameters summarized in Table 1. Since the fits to are hardly distinguishable visually, we only show the curves for the phase shift from CCL () and , but give the ranges for the fit parameters found in the full calculation. For these data sets and energy region the reduced is very close to . As alluded to above, the deteriorates substantially when fitting to the full data base of Hagiwara:2011af (), but the central values of the fit parameters remain largely unaffected.

Extending the fit to higher energies by including and in the spectral function yields a reasonable fit up to , at the expense of a slight deterioration of the data description between the and , see the right panel of Fig. 3 and Table 1. Again, we observe that the fit result is relatively insensitive to the data set chosen, with larger differences evolving in the region. We will use the outcome of this extended fit to estimate the impact of the high-energy region on the analytic continuation of the transition form factor into the space-like region in Sect. 5.

## 3 Dispersion relations for the doubly-virtual π0 transition form factor

We decompose the pion transition form factor into definite isospin components according to

 Fπ0γ∗γ∗(q21,q22)=Fvs(q21,q22)+(q1↔q2), (27)

where the first/second index refers to isovector () and isoscalar () quantum numbers of the photon with momentum /. For fixed isoscalar virtuality we can write a once-subtracted dispersion relation in the isovector virtuality g3pi ()

 Fvs(s1,s2) =Fvs(0,s2) (28) +es112π2∫∞4M2πds′q3π(s′)FV∗π(s′)f1(s′,s2)s′3/2(s′−s1),

where , and is the pion vector form factor (4). Assuming both and to asymptotically fall off like  Lepage:1980fj (); Duncan:1979hi (); Froissart:1961ux (); V3pi (); omegaTFF () (for fixed ), there is a sum rule for the subtraction function in (28),

 Fvs(0,s2)=e12π2∫∞4M2πds′q3π(s′)s′3/2FV∗π(s′)f1(s′,s2). (29)

This sum rule formally converges only with a partial wave based on the singly-subtracted representation (13), with a second subtraction (25) it can at best be evaluated below a certain cutoff. The representation (28) as well as the sum rule (29) have been employed before: for , they yield the vector meson transition form factors for , including (from the sum rule) the normalization for the real-photon decays omegaTFF (). For , one obtains the isovector part of the singly-virtual transition form factor, with the sum rule yielding  g3pi (). Numerically, these sum rules were found to be saturated at the level omegaTFF (); g3pi ().

Taken together, (28) and (29) are equivalent to an unsubtracted dispersion relation

 Fvs(s1,s2)=e12π2∫∞4M2πds′q3π(s′)FV∗π(s′)f1(s′,s2)s′1/2(s′−s1). (30)

We can perform a (necessarily less explicit) subtraction of (28) in as well, defining a subtracted partial wave

 ¯f1(s,q2)=f1(s,q2)−f1(s,0)q2. (31)

The alternative formulation of the dispersive representation, making use of the sum rule (29), then reads

 Fvs(s1,s2) =Fvs(s1,0)+Fvs(0,s2)−Fπγγ2 (32) +es1s212π2∫∞4M2πds′q3π(s′)FV∗π(s′)¯f1(s′,s2)s′3/2(s′−s1).

## 4 Time-like form factor and e+e−→π0γ

We now specialize the general expressions (27) and (30) to the singly-virtual case for further phenomenological investigation. The transition form factor can be written out explicitly according to

 Fπ0γ∗γ(q2,0) =Fπγγ+e12π2∫∞4M2πds′q3π(s′)FV∗π(s′)s′3/2 ×{f1(s′,q2)−f1(s′,0)+q2s′−q2f1(s′,0)}. (33)

Here we have again made use of the sum rule (29) to fix the full transition form factor at to the chiral anomaly . Neglecting the mass of the electron for simplicity, the relation between the cross section and the pion transition form factor is given by

 σe+e−→π0γ=e2(q2−M2π0)396πq6|Fπ0γ∗γ(q2,0)|2. (34)

To ensure consistency with the calculation of the amplitude we assume asymptotic behavior of and in (4) above and use a twice-subtracted Omnès representation for (cf. Guo:2008nc ())

 FVπ(s)=exp{⟨r2⟩Vπ6s+s2π∫∞4M2πds′δ(s′)s′2(s′−s)}, (35)

with a radius and the same phase shift as in the respective version of . The isoscalar part, corresponding to the difference in (4), is then calculated by the same methods as in V3pi () with the normalization fixed from as described in Sect. 2. The isovector part, corresponding to the last term in (4), is completely determined by and can thus be measured in . Here, we use a finite matching point of and fix the normalization to the chiral anomaly g3pi (), but this representation can be improved once the COMPASS data for become available.

Our result for the cross section is shown in Fig. 4. We repeat the calculation for each set of phase shifts and , fitting the isoscalar part in each case both to SND+BaBar and HLMNT. The error band in Fig. 4 represents the uncertainty deduced from scanning over the input quantities in this way. Within uncertainties, the outcome agrees perfectly with the cross section measured by eepi0g_SND_1 (); eepi0g_SND_2 (); eepi0g_CMD2 (). We would like to stress that this result is a prediction solely based on the input quantities described above, most prominently, cross-section data, the -wave phase shift, the pion vector form factor, and the low-energy theorems for and .

To provide a quantitative measure of the agreement between our result and experiment, we first give the reduced of the mean of our band when comparing to the various data sets, see Table 2. However, the usual does not account for the theory uncertainty, so that it is not surprising that values significantly larger than are obtained. If one assumed the theory band to be statistically distributed with mean values and uncertainties , uncorrelated for each data point , one could consider the difference between theory and experiment with combined error and test the distribution for consistency with zero, leading to a modified ,

 χ2→~χ2=N∑i=1(yi−yth(qi))2σ2i+σ2th(qi). (36)

The corresponding values for this quantity are also summarized in Table 2. Given that in practice correlations between different points of the theory band are not negligible, the statistical interpretation of (36) is not obvious. However, taken together with the observation that curves within the theory band can be constructed with even smaller , it provides quantitative evidence for the consistency of our result with the data. In addition, the comparison of the and for the two fits reveals that, while the fit is deteriorated mostly in the energy region above the , including , improves the agreement with below .

## 5 Slope parameter and space-like form factor

We reconstruct the transition form factor in the space-like region again dispersively, making use of the imaginary part determined from the study of the time-like region in the previous sections

 Fπ0γ∗γ(q2,0)=Fπγγ+q2π∫∞sthrds′ImFπ0γ∗γ(s′,0)s′(s′−q2). (37)

If we assume the transition form factor to fulfill even an unsubtracted dispersion relation, this relation implies a sum rule for the chiral anomaly

 Fπγγ=1π∫∞sthrds′ImFπ0γ∗γ(s′,0)s′. (38)

The slope of the form factor obeys

 aπ =M2π0Fπγγ∂∂q2Fπ0γ∗γ(q2,0)∣∣∣q2=0 =M2π0Fπγγ1π∫∞sthrds′ImFπ0γ∗γ(s′,0)s′2. (39)

For the evaluation of these relations we need to specify how to treat the high-energy region of the integrals. Perturbative QCD in the factorization framework of Lepage:1980fj () predicts an asymptotic behavior

 Fπ0γ∗γ(−Q2,0)∼2e2FπQ2.