Hadronic light-by-light contribution to the muon g-2
aa r X i v : . [ h e p - ph ] D ec Hadronic light-by-light contribution to the muon g − Adolfo Guevara
Departamento de F´ısica, Cinvestav del IPN, Apdo. Postal 14-740, 07000 Mexico DF. MexicoE-mail: [email protected]
Abstract.
We have computed the hadronic light-by-light (LbL) contribution to the muonanomalous magnetic moment a µ in the frame of Chiral Perturbation Theory with the inclusionof the lightest resonance multiplets as dynamical fields (R χ T). It is essential to give a moreaccurate prediction of this hadronic contribution due to the future projects of J-Parc and FNALon reducing the uncertainty in this observable. We, therefore, computed the pseudoscalartransition form factor and proposed the measurement of the e + e − → µ + µ − π cross sectionand dimuon invariant mass spectrum to determine more accurately its parameters. Then, weevaluated the pion exchange contribution to a µ , obtaining (6 . ± . · − . By comparingthe pion exchange contribution and the pion-pole approximation to the corresponding transitionform factor ( π TFF) we recalled that the latter underestimates the complete π TFF by (15-20)%.Then, we obtained the η ( ′ ) TFF, obtaining a total contribution of the lightest pseudoscalarexchanges of (10 . ± . · − , in agreement with previous results and with smaller error.
1. Introduction
Ever since the measurement of the electron magnetic moment in the splitting of the groundstates of deuterium and molecular hydrogen [1], the anomalous magnetic moment has been anever more stringent test of the underlying theory governing the interactions among elementalparticles; giving us the lead from a way of renormalizing QED [2] to an outstanding confirmationof QFT with QED contributions [3] up to order (cid:0) απ (cid:1) . In this spirit, the a µ has been seen asa very stringent test of beyond standard model physics (BSM). With the most recent measure-ments [4], a deviation from the Standard Model (SM) results would imply a contribution fromBSM with a scale ∼
100 TeV (assuming an interaction ∼ σ and future plans on measuring more accurately this observable force theorists to make moreprecise predictions of SM contributions to the µ anomalous magnetic moment.Within the SM, the contributions to the a µ that have a greater uncertainty are the hadronicones [5]. This is due to the fact that the underlying theory cannot be taken perturbatively in thewhole energy range of the quark loop integrals, forcing theorists to compute these contributionsusing effective field theories (EFT) based on symmetries of Quantum Chromodynamics (QCD).This hadronic contribution can be splitted into two sub-contributions, the Hadron VacuumPolarization (HVP) and the Hadronic Light-by-Light (HLbL), shown in Fig. 1. We analize thelatter one by studying the P γ ∗ γ ∗ interaction through its form factor (also called P transitionform factor, PTFF), which gives the leading contribution to the HLbL through a pseudoscalarexchange diagram shown in Fig. 2. At low energies ( i.e. in the chiral limit), the prediction forthe π TFF has been confirmed by the measured rate of π → γγ decays [5]. On the other hand, adrons Hadrons
Figure 1.
The two hadronic contribution to a µ : Hadronic Light-by-Light (left) and HadronVacuum Polarization (right).the prediction for a nearly on-shell photon and one with very large virtuality seems to be at oddswith measurements at B-factories [6, 7]. These two limits have ruled the way of constructing theform factor to describe interactions in the intermediate energy region, where hadronic degrees offreedom play a crucial role. The EFT we use to compute the TFF is Resonance Chiral Theory(R χ T) [8, 9], which makes use of short-distance QCD predictions to obtain the parameters ofthe theory in terms of known constants. In this work, we fit one of the parameters in the π TFFwith the B-factories data and, using this information, we predict the η ( ′ ) TFF and then obtainthe contribution to the a µ using these form factors. π , η, η ′ Figure 2.
Pseudoscalar exchange contribution to the HLbL.
2. Theoretical Framework
Chiral Perturbation Theory ( χ PT) [10] is the EFT dual to QCD at low energies [11]. It isbased on an expansion in powers of momenta and masses of the lightest pseudoscalar mesonsover the chiral symmetry breaking scale (Λ ∼ & M ρ ) the theory is no longer applicable. A generalization of χ PT is obtained using 1 /N C asan expansion parameter [12] to include resonances as dynamical degrees of freedom. The theorythat incorporates these elements is Resonance Chiral Theory (R χ T) [8, 9], which requires unitarysymmetry for the resonance multiplets. No a priori assumptions are made with respect to therole of resonances in this theory, therefore one obtains naturally Vector Meson Dominance [13]as a dynamical result of the theory [8]. The final ingredient of the theory comes from QCDbehavior at short distances, which constraints a great amount of free parameters in the theory.
3. The πγ ∗ γ ∗ form factor in R χ T In this framework, the form factor we obtain [14] is π γ ∗ γ ∗ ( p , q , r ) = 2 r F (cid:20) − N C π r + 4 F V d ( p + q )( M V − p )( M V − q ) r + 4 F V d ( M V − p )( M V − q )+ 16 F V P ( M V − p )( M V − q )( M P − r ) − √ M V − p (cid:18) F V M V r c − p c + q c r + 8 P F V ( M P − r ) (cid:19) +( q ↔ p ) (cid:3) . (1)It contains contributions from the pseudoscalar resonances (as can be seen through thecouplings P and P ) which need to be taken into account to obtain consistent short distanceconstraints [9, 15]. All the parameters, but one, can be obtained through these constraints. P cannot be obtained requiring high energy constraints, therefore it is fitted using the combinedanalyses of π (1300) → γγ and π (1300) → ργ decays as given in references [9, 14]. The consistentshort distance constraints on the resonance couplings in the odd-intrinsic parity sector can beseen in refs [14, 15]. Thus, we obtain P = ( − . ± . · − GeV . (2) (GeV )00,050,10,150,20,250,3 Q F π γγ ∗ ( Q ) ( G e V ) CELLOCLEOBaBarBelleB-L asymptotic behaviourOur fit
Figure 3.
Our best fit compared to CELLO, CLEO, BaBar and Belle data for the π TFF.On the other hand, the π TFF does not fit very well experimental data [6, 7] when P isconstrained by the short distance prediction. Therefore we allowed for it a small variation ina fit to Babar and Belle data of this form factor, where they measure the π TFF spectrum ina kinematical configuration that ensures that one of the photons is on-shell and the other isvirtual. The form factor for such a configuration is given by taking p → Q = − q ineq. (1) F π γ ∗ γ ( Q ) = − F Q (1 + 32 √ P F V F ) + N C π M V F M V ( M V + Q ) . (3)We keep a very conservative 10% uncertainty from the asymptotic value of F V around itspredicted value [15] of √ F . Fig. 3 shows our best fit, with which we obtain P = ( − . ± . · − GeV, χ /dof = 1 . . (4) By the kinematical configuration in which the process is chosen to be measured, the momenta of both photonsare space-like.
10 20 30 40Q (GeV )00,050,10,150,20,25 Q F η γγ ( G e V ) BaBar dataCELLO dataOur upper limitOur lower limit 0 10 20 30 40Q (GeV )00,10,20,3 Q F η ’ γγ ( G e V ) BaBar dataCELLO dataCLEO dataOur upper limitOur lower limit
Figure 4.
Our prediction of the η TFF (left) and η ′ TFF (right) compared to CELLO, CLEOand BaBar data and using the parameters found with the π TFF.
4. The pseudoscalar exchange contribution to the a HLbLµ
Once all the parameters in the π TFF are determined, we insert the full off-shell TFF in therelations given in [16] obtaining thus a π LbLµ = (5 . ± . · − on-shell π a π LbLµ = (6 . ± . · − whole π TFF . (5) Table 1.
Our result compared with other results obtained through different methods. a π LbLµ ·
10 Model and Reference . ± .
05 Extended Nambu-Jona-Lasinio [17]5 . ± .
01 Naive VMD [18]5 . ± . N C with two vector multiplets π -pole [16]7 . ± . π -exchange contribution [19]6 . ± .
25 Holographic models of QCD [20]6 . ± .
12 Lightest pseudoscalar and vector resonance saturation [9]6 . ± .
56 Rational approximants [21]5 . ± . . ± .
06 Our result with on-shell π [14] . ± .
21 Our result whole π TFF [14]This clearly shows that assuming an on-shell pion in the a HLbLµ underestimates thecontribution in ∼ F V , P and in a chiral correction from very-low energy physics. We compare ourresult with previous results in table 1. The form factor for the η and η ′ can be obtained withthe π TFF through eq. (6) with the minus sign for the case of the η . F η ( ′ ) γ ∗ γ ∗ = C ( q ′ ) ∓ √ C ( s ′ ) ! F π γ ∗ γ ∗ (6)With this, we can compute the whole pseudoscalar exchange contribution to the a HLbLµ , shownin table 2 which includes other sub-leading HLbL contributions. able 2.
Comparison of our contributions of the full a HLbLµ to previous determinations. a HLbLµ · Contributions11 . ± . . ± . [23]11 . ± .
5. Genuine probe of π TFF
All the experimental observables available to fit the parameters in the π TFF so far need anon shell photon and/or have photons with space-like momenta, while the HLbL contribution tothe a µ has both photons with time-like momenta. The photons in the process we study in thissection, namely σ ( e + e − → γ ∗ → π γ ∗ → µ + µ − π ), both have time-like momenta and can bemeasured at very high photon virtualities by KLOE collab. for q ∼ and Belle-II collab.for q ∼ . . With the π TFF parameters fully determined, the prediction we obtain isshown in Fig. 5 )00,00050,0010,0015 σ ( nb ) (GeV )00,00050,001 d σ / d s ( nb G e V - ) Figure 5.
Our prediction for σ ( s ) (left) and for sσds with s = 1 .
02 GeV , the error bands cannotbe appreciated in these plots.
6. Conclusion
We found the pseudoscalar exchange contribution to the a HLbLµ with a very competitive uncer-tainty and consistent with other theoretical models; improving the analysis by including highenergy constraints not realized in the reference [9] and also using Belle data released after thereference was published. Our error estimate is also more robust, since in addition to the errorsof the resonance couplings, we have also included the uncertainty due to the value of the π TFFat very low energies.We also obtained the first prediction for the cross section σ ( e + e − → µ + µ − π ), which might bemeasured in KLOE-2 and Belle-II. The measurement of this observable would be an interestingway of trying to reduce the error in the parameters of the π TFF. This, may also help to reducethe uncertainty on the mixing parameters between the η and η ′ mesons. Acknowledgments
The author wishes to thank the DPyC and Cinvestav for financial support. eferences [1] Nafe J E, Nelson E B and Rabi I I 1947 Phys. Rev. et al. [Muon g-2 Collab.] 2006 Phys. Rev. D et al. [PDG] 2014 Chin. Phys. C .[6] Aubert B et al. [BABAR Collab.] 2009 Phys. Rev. D et al. [Belle Collab.] 2013 Phys. Rev. D B321
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