Hadronic light-by-light scattering in the muon : a new short-distance constraint on pion exchange
We summarize our recent new evaluation of the pion-exchange contribution to hadronic light-by-light scattering in the muon . We first derive a new short-distance constraint on the off-shell pion-photon-photon form factor at the external vertex in which relates the form factor to the quark condensate magnetic susceptibility in QCD. We then evaluate the pion-exchange contribution in the framework of large- QCD using an off-shell form factor which fulfills all short-distance constraints and obtain the new estimate . Updating our earlier results for the contributions from the exchanges of the and using simple vector-meson dominance form factors, we get for the sum of all light pseudoscalars. Combined with available evaluations for the other contributions to hadronic light-by-light scattering this leads to the estimate . The corresponding contributions to the anomalous magnetic moment of the electron are also given.
Hadronic light-by-light scattering in the muon : a new short-distance constraint on pion exchange
Regional Centre for Accelerator-based Particle Physics
Harish-Chandra Research Institute
Chhatnag Road, Jhusi
Allahabad - 211 019, India
The muon has served over many decades as an important test of the Standard Model (SM). It is also sensitive to contributions from New Physics slightly above the electroweak scale. In fact, for several years now a discrepancy of more than three standard deviations has existed between the SM prediction and the experimental value, see the recent reviews Refs. [1, 2, 3] on the muon . The main error in the theoretical SM prediction comes from hadronic contributions, i.e. hadronic vacuum polarization and hadronic light-by-light (had. LbyL) scattering. Whereas the hadronic vacuum polarization contribution can be related to the cross section , no direct experimental information is available for had. LbyL scattering. One therefore has to rely on hadronic models to describe the strongly interacting, nonperturbative dynamics at the relevant scales from the muon mass up to about 2 GeV. This leads to large uncertainties, see Refs. [4, 5, 3] for recent reviews on had. LbyL scattering, largely based on the original works [6, 7, 8, 9].
Essentially, these models describe the interactions of hadrons with photons, usually with the help of some form factors. One can reduce this model dependence and the corresponding uncertainties by relating the form factors at low energies to results from chiral perturbation theory (ChPT)  and at high energies (short distances) to the operator product expansion (OPE) . In this way, one connects the form factors to the underlying theory of QCD. In particular, this has been done in Refs. [6, 7, 12, 8, 9] for the numerically dominant contribution from the exchange of light pseudoscalars .
The pseudoscalar-exchange contributions to had. LbyL scattering are given by the diagrams shown in Fig. 1.
It was pointed out recently in Ref. , that one should use fully off-shell form factors for the evaluation of the LbyL scattering contribution. This seems to have been overlooked in the recent literature, in particular, in Refs. [12, 8, 9, 4, 5]. The on-shell form factors as used in Refs. [8, 12] actually violate four-momentum conservation at the external vertex, as observed already in Ref. .
The exchange of the lightest state yields the largest contribution and therefore warrants special attention. In these proceedings, based on the results obtained in Ref. , we present a new QCD short-distance constraint on the off-shell pion-photon-photon form factor at the external vertex by relating it to the quark condensate magnetic susceptibility of QCD. We then evaluate this contribution in the framework of large- QCD , using a form factor which fulfills this new and other relevant short-distance constraints.
2 On-shell versus off-shell form factors
up to small mixing effects with the states and and neglecting exchanges of heavier states like . Here is the light quark part of the electromagnetic current and .
The corresponding contribution to the muon may be worked out with the result 
where the external photon has now zero four-momentum. See Ref.  for the expressions for .
Although pole dominance might be expected to give a reasonable approximation, it is not correct as it was used in those references, as stressed in Refs. [9, 2]. The point is that the form factor sitting at the external photon vertex in the pole approximation for violates four-momentum conservation, since the momentum of the external (soft) photon vanishes. The latter requires . Ref.  then proposed to use instead
Note that putting the pion on-shell at the external vertex automatically leads to a constant form factor, given by the Wess-Zumino-Witten (WZW) term . However, this prescription does not yield the pion-exchange contribution with off-shell form factors, which we calculate with Eq. (2.0).
Strictly speaking, the identification of the pion-exchange contribution is only possible, if the pion is on-shell. If one is off the mass shell of the exchanged particle, it is not possible to separate different contributions to the , unless one uses some particular model where elementary pions can propagate. In this sense, only the pion-pole contribution with on-shell form factors can be defined, at least in principle, in a model-independent way. On the other hand, the pion-pole contribution is only a part of the full result, since in general, e.g. using some resonance Lagrangian, the form factors will enter the calculation with off-shell momenta. In this respect, we view our evaluation as being a part of a full calculation of had. LbyL scattering using a resonance Lagrangian whose coefficients are tuned in such a way as to systematically reproduce the relevant QCD short-distance constraints, along the lines of the resonance chiral theory developed in Ref. .
3 A new short-distance constraint on the off-shell pion-photon-photon form factor
The form factor defined in Eq. (2.0) is determined by nonperturbative physics of QCD and cannot (yet) be calculated from first principles. Therefore, various hadronic models have been used in the literature. At low energies, the form factor is normalized by the decay amplitude, . To a good approximation, all hadronic models thus have to satisfy the constraint .111We note that in our work  and in Refs. [6, 7, 8, 9] simply is used, without any error attached. Maybe this could be an additional source of uncertainty in , in particular in view of the new value presented in Ref. ; see also the discussion in Ref.  and references therein.
For an on-shell pion, there is also experimental data available for one on-shell and one off-shell photon, from the process . Several experiments  thereby fairly well confirm the Brodsky-Lepage  behavior for large Euclidean momentum and any model should reproduce this behavior, maybe with a different prefactor.222Note, however, that a recent measurement of the form factor by the BABAR collaboration  at momentum transfers between and does not show such a falloff. We will come back to this issue in Section 4.
Apart from these experimental constraints, any consistent hadronic model for the off-shell form factor should match at large momentum with short-distance constraints from QCD that can be calculated using the OPE. In Ref.  the short-distance properties for the three-point function in Eq. (2.0) in the chiral limit and assuming octet symmetry have been worked out in detail. Two limits are of interest. In the first case, the two momenta become simultaneously large, which describes the situation where the space-time arguments of all three operators tend towards the same point at the same rate. The second situation corresponds to the case where the relative distance between only two of the three operators in becomes small. When the space-time arguments of the two vector currents in approach each other, the leading term in the OPE leads to the Green’s function . The explicit results for both these cases can be found in Refs. [22, 13].
The new short-distance constraint on the off-shell form factor at the external vertex in had. LbyL scattering arises when the space-time argument of one of the vector currents in approaches the argument of the pseudoscalar density. This leads to the two-point function of the vector current and the antisymmetric tensor density
In particular, at the external vertex in LbyL scattering in Eq. (2.0), the limit is relevant.
As pointed out in Ref. , the value of at zero momentum is related to the quark condensate magnetic susceptibility in QCD in the presence of a constant external electromagnetic field, introduced in Ref. : , with and . With our definition of in Eq. (3.0) one obtains the relation (see also Ref. ) and the new short-distance constraint at the external vertex can be written as 
Note that there is no falloff in this limit, unless vanishes.
Unfortunately there is no agreement in the literature what the actual value of should be. In comparing different results one has to keep in mind that actually depends on the renormalization scale . In Ref.  the estimate was given in a QCD sum rule evaluation of nucleon magnetic moments. A similar value was obtained in Ref. , probably again for a low scale as argued in Ref. .
On the other hand, saturating the leading short-distance behavior of the two-point function  with one multiplet of lowest-lying vector mesons (LMD) [28, 23, 22] leads to the estimate . Again, it is not obvious at which scale this relation holds, it might be at . This LMD estimate was soon afterwards improved by taking into account higher resonance states () in the framework of QCD sum rules, with the results  and . A more recent analysis  yields, however, a smaller absolute value , close to the original LMD estimate.333After the publication of our paper Ref. , two new estimates for appeared, both based on the analysis of the zero-modes of the Dirac operator. Ref.  presents an analytical approach which yields with an estimated error of . A quenched lattice calculation  for gives a very small absolute value . No scale dependence is given, the lattice spacing corresponds to 2 GeV. For a quantitative comparison of all these estimates for we would have to run them to a common scale, for instance, 1 GeV or 2 GeV, which can obviously not be done within perturbation theory starting from such low scales as .
4 New evaluation of the pseudoscalar-exchange contribution in large- Qcd
In the spirit of the minimal hadronic Ansatz  for Green’s functions in large- QCD, an off-shell form factor has been constructed in Ref. . It contains the two lightest multiplets of vector resonances, the and the (LMD+V), and fulfills all the OPE constraints discussed earlier:
Below we reevaluate the pion-exchange contribution using off-shell LMD+V form factors at both vertices. The constants in the Ansatz for in Eq. (4.0) are determined as follows. The normalization with the pion decay amplitude yields , where we used and . The Brodsky-Lepage behavior can be reproduced by choosing . In Ref.  a fit to the CLEO data  for the on-shell form factor was performed, with the result . The constant can be obtained from higher-twist corrections in the OPE with the result .
The LMD estimate is close to obtained in Ref.  using QCD sum rules with several vector resonances , and . Assuming that the LMD/LMD+V framework is self-consistent, we will therefore take in our numerical evaluation, with a typical large- uncertainty of about 30%. We will vary in the range and determine from Eq. (4.0) and vice versa.
The coefficient in the LMD+V Ansatz is undetermined as well. It enters at order in the low-energy expansion of in one combination of low-energy constants from the chiral Lagrangian of odd intrinsic parity, . The LMD ansatz with only one multiplet of vector resonances yields . If the LMD/LMD+V framework is self-consistent, the change in these estimates, while going from LMD to LMD+V, should not be too big. Since the size of this low-energy constant seems to be small compared to another combination of low-energy constants which enters at order , we allow for a 100% uncertainty of and get the range , see Ref.  for details.
Varying by changes the result for by at most. The uncertainty in affects the result by up to . The variation of with [with determined from the constraint in Eq. (4.0) or vice versa] is much smaller, at most . In the absence of more information on the values of the constants and , we take the average of the results obtained with for and for as our central value: . Adding all uncertainties from the variations of , (or ), and linearly to cover the full range of values obtained with our scan of parameters, we get [13, 3]
This value replaces the result obtained in Ref.  with on-shell LMD+V form factors at both vertices. We think the 16% error should fairly well describe the inherent model uncertainty using the off-shell LMD+V form factor. In order to facilitate updates of our result in case some of the parameters in the LMD+V Ansatz in Eq. (4.0) will be known more precisely, we have given in the Appendix of Ref.  a parametrization of for arbitrary coefficients .444A fit of the on-shell LMD+V form factor to the recent BABAR data  yields and with . In this way we would get the new average value , i.e. the result is essentially unchanged from Eq. (4.0).
As far as the contribution to from the exchanges of the other light pseudoscalars and is concerned, it is not so straightforward to apply the above analysis within the LMD+V framework to these resonances. In particular, the short-distance analysis in Ref.  was performed in the chiral limit and assumed octet symmetry. We therefore resort to a simplified approach which was also adopted in other works [6, 7, 8, 9] and take a simple VMD form factor normalized to the experimental decay width . In this way we obtain the results and , which update the values given in Ref. . Adding up the contributions from all the light pseudoscalar exchanges, we obtain the estimate [13, 3]
where we have assumed a 16% error, as inferred above for the pion-exchange contribution.555Applying the same procedure to the electron, we get . This number supersedes the value given in Ref. . Note that the naive rescaling yields a value which is almost a factor of 2 too small. Our estimates for the other pseudoscalars contributions using VMD form factors at both vertices are and . Therefore we get , where the relative error of about 12% is again taken over from the pion-exchange contribution. Assuming that the pseudoscalar contribution yields the bulk of the result of the total had. LbyL scattering correction, we obtain , with a conservative error of about 30%, see Ref. . This value was later confirmed in the published version of Ref.  where a leading logs estimate yielded .
5 Discussion and conclusions
We would like to stress that although our result for the pion-exchange contribution is not too far from the value given in Ref. , this is pure coincidence. We have used off-shell LMD+V form factors at both vertices, whereas the authors of Ref.  evaluated the pion-pole contribution using the on-shell LMD+V form factor at the internal vertex and a constant WZW form factor at the external vertex, see for instance Eq. (18) in Ref. . Since only the pion-pole contribution is considered in Ref. , their short-distance constraint cannot be applied to our approach either. However, our ansatz for the pion-exchange contribution agrees qualitatively with the short-distance behavior of the quark-loop derived in Ref. , see the discussion in Refs. [13, 3].
Our results for the pion and the sum of all pseudoscalars are about 20% larger than the values in Refs. [6, 7] which used other hadronic models. An evaluation of the pion-exchange contribution using an off-shell form factor based on a nonlocal chiral quark model yielded . In that model, off-shell effects of the pion always lead to a strong damping in the form factor and the result is therefore smaller than the pion-pole contribution obtained in Ref. . In our model, there are some corners of the parameter space where the result is larger than the pion-pole contribution, for instance, we get a maximal value of in the scanned region. Very recently, a value of with an estimated error of at most 30% was obtained in Ref.  within an AdS/QCD approach.
for the total had. LbyL scattering contribution to the anomalous magnetic moment of the muon. To be conservative, we have added all the errors linearly, as has become customary in recent years. In the very recent review  the central values of some of the individual contributions to had. LbyL scattering are adjusted and some errors are enlarged to cover the results obtained by various groups which used different models. The errors are finally added in quadrature to yield the estimate . Note that the dressed light quark loops are not included as a separate contribution in Ref. . They are assumed to be already covered by using the short-distance constraint from Ref.  on the pseudoscalar-pole contribution. Certainly, more work on the had. LbyL scattering contribution is needed to fully control all the uncertainties.
I would like to thank the organizers of Chiral Dynamics 2009 for their financial support and for providing such a stimulating atmosphere. I am grateful to F. Jegerlehner for pointing out that fully off-shell form factors should be used to evaluate the pion-exchange contribution, for helpful discussions and for numerous correspondences. Furthermore, I would like to thank G. Colangelo, J. Gasser, M. Knecht, H. Leutwyler, P. Minkowski, B. Moussallam, M. Perrottet, A. Pich, J. Portoles, J. Prades, E. de Rafael and A. Vainshtein for illuminating discussions. This work was supported by the Department of Atomic Energy, Government of India, under a 5-Years Plan Project.
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