Axial vector transition form factors in holographic QCD and their contribution to the muon g−2
AAxial vector transition form factors in holographic QCDand their contribution to the muon 𝒈 − Josef Leutgeb and Anton Rebhan ∗ Institut für Theoretische Physik, Technische Universität Wien,Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria
E-mail: [email protected], [email protected]
Whereas the theoretical results for the dominant contributions to hadronic light-by-light scatteringcoming from pseudoscalar meson exchange have converged over the past years, the variouspublished estimates of the contribution due to axial vector meson exchange differ wildly. Sinceholographic AdS/QCD models have proved to provide rather good models of singly and doublyvirtual pion transition form factors, which reproduce remarkably well the known low-energy dataand also the asymptotic leading-order pQCD behavior, it is of interest to consider their predictionsfor axial vector transition form factors. Indeed, we find that these reproduce also very wellexisting data from the L3 experiment for 𝑓 → 𝛾𝛾 ∗ . Including the full infinite tower of axialvector mesons of the AdS/QCD models we moreover show that the Melnikov-Vainshtein short-distance constraint can be satisfied, while almost all existing models for hadronic light-by-lightscattering fail to incorporate it. The contribution to 𝑔 −
2, which is dominated by the first fewresonances, turns out to be significantly larger than estimated previously. ∗ Speaker © Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - ph ] D ec xial vector TFF in holographic QCD and their contribution to the muon 𝑔 − Anton Rebhan
1. Introduction
The long-standing discrepancy between the experimental and the theoretical Standard Model(SM) result for the anomalous magnetic moment of the muon 𝑎 𝜇 ≡ ( 𝑔 𝜇 − )/ Δ 𝑎 𝜇 : = 𝑎 exp 𝜇 − 𝑎 SM 𝜇 = ( ) × − . The estimatederror of the SM result, which comes almost exclusively from hadronic contributions, is dominatedby the error in the hadronic vacuum polarization (HVP) part, 40 × − according to [1]. While thehadronic light-by-light (HLBL) scattering contribution is about two orders of magnitude smaller,its uncertainties contribute an error that is about half of the HVP one, 19 × − . There has beenquite some progress in determining the leading HLBL contributions due to the exchange of neutralpseudoscalars with a convergence of refined phenomenological models, the dispersive approach,and lattice calculations (cf. the recent review [1]). However, there is still a great uncertaintyconcerning the size of the contributions of axial vector meson exchanges and the as we shall seerelated problem of a failure of most phenomenological models to satisfy the known short-distanceconstraints (SDC) of the HLBL tensor, in particular the one implied by the axial anomaly, whichhas been derived by Melnikov and Vainshtein [2] and which the latter have estimated to require asubstantial increase of previous HLBL results.In [3], we have recently shown that three of the simplest and most popular holographic QCD(hQCD) models, the two hard-wall models HW1 [5, 6], HW2 [7], and a simple soft-wall model (SW)[8–10], see Fig. 1, reproduce remarkably well the available low-energy data for the single virtual pion SSHW1HW2SW DRV4 Q [ GeV ] ( Q ,0 ) [ GeV - ] Figure 1:
Experimental data for the single-virtual pion TFF compared to the predictions of two hard-wall models HW1, HW2, and a simple soft-wall model (SW) [3]. Also included is the prediction of theholographic Sakai-Sugimoto (SS) model, which is a top-down string theoretic construction meant to coveronly the low-energy limit of QCD and which misses the short-distance constraints of QCD. The dashed lineis a recent fit by Danilkin et al. [4]. (Figure partially taken from Ref. [4].) xial vector TFF in holographic QCD and their contribution to the muon 𝑔 − Anton Rebhan [ GeV ] A ( Q ,0 ) / A ( ) Q [ GeV ] Q A ( Q , Q )/ A ( ) Figure 2:
Left: Single-virtual axial vector TFFs from holographic models (SS: blue, HW1: orange, HW2:red, HW2(UV-fit): red dotted) compared with dipole fit of L3 data for 𝑓 ( ) (grey band). The parametersof all models except HW2(UV-fit) are fixed by matching 𝑓 𝜋 and 𝑚 𝜌 . (The results for HW1 and HW2 almostcoincide here, but 𝐴 ( , ) differs; see [11].) Right: Double-virtual case with equal virtualities. The blackdashed lines correspond to the dipole model used in [12]. transition form factor (TFF) while also reproducing (partially or fully) the short-distance constraintsfor doubly virtual TFFs due to Brodsky and Lepage (which conventional phenomenological modelsfail to do, except for the interpolator proposed recently in [4]). In this paper we report the resultsobtained in [11], where we obtained axial vector TFFs from these hQCD models and evaluated theircontribution to 𝑎 𝜇 .
2. Axial vector TFFs and 𝑎 𝜇 In hQCD models, the axial anomaly is included by a five-dimensional Chern-Simons actioninvolving five-dimensional flavor gauge fields whose normalizable modes correspond to the pseu-doscalar Goldstone bosons and infinite towers of vector and axial vector fields; photons are includedthrough non-normalizable modes. Pion and axial vector TFFs are therefore unambiguously deter-mined once a hQCD model has been constructed. In hard-wall models where the holographiccoordinate 𝑧 is cut off at 𝑧 , the axial vector TFF 𝐴 , corresponding to the (symmetric) function 𝐴 in [12] has the form [11] (nicely realizing the Lee-Yang theorem by 𝑑 J ( , 𝑧 )/ 𝑑𝑧 = 𝐴 ( 𝑄 , 𝑄 ) ∝ 𝑄 ∫ 𝑧 𝑑𝑧 (cid:20) 𝑑𝑑𝑧 J ( 𝑄 , 𝑧 ) (cid:21) J ( 𝑄 , 𝑧 ) 𝜓 𝐴 ( 𝑧 ) , (1)where J is the bulk-to-boundary propagator of photons and 𝜓 𝐴 is the holographic wave functionof one of the axial vector mesons. In contrast to the usual simple model for the axial vector TFFinvolving a factorized dipole ansatz [12], the holographic result (1) is asymmetric in the photonvirtualities 𝑄 , 𝑄 . Its high-energy limit coincides with the asymmetric asymptotic form followingfrom perturbative QCD, which has only most recently been derived in [13].Experimental data for the singly virtual axial vector 𝑓 ( ) TFF have been obtained by theL3 experiment [14] in the form of a dipole fit, which is represented by the grey band in the left panelof Fig. 2 and compared with the results from the HW1 and HW2 models as well as the top-downSS model. The latter are right on top of the experimental results within errors. As shown in rightpanel of Fig. 2, for the doubly virtual case (where no experimental data are so far available) the3 xial vector TFF in holographic QCD and their contribution to the muon 𝑔 − Anton Rebhan [ GeV ] -( π / ) Q Q Π ( Q,Q, Q ) [ GeV ] × - × - × - × - × - × - ρ a ( Q,Q,0 ) Figure 3:
Axial-vector contribution to 𝑄 𝑄 ¯ Π ( 𝑄, 𝑄, 𝑄 ) as a function of 𝑄 at 𝑄 =
50 GeV in the HW2model and the integrand of 𝑎 AV 𝜇 [11]. The black line corresponds to the infinite sum over the tower of axialvector mesons, and the other lines give the contributions of the 1st to 5th lightest axial vector mesons. 𝑗 = 𝑗 ≤ 𝑗 ≤ 𝑗 ≤ 𝑗 ≤ 𝑎 AV 𝜇 HW1 31.4 36.2 37.9 39.1 39.6 40.6 × − HW2 23.0 26.2 27.4 27.9 28.2 28.7 × − Table 1:
The contribution of the infinite tower of axial vector mesons to 𝑎 AV 𝜇 in the two hard-wall AdS/QCDmodels considered in [11].) holographic prediction deviates substantially from the simple ansatz used in [12]. As a result, theholographic result for the axial-vector 𝑓 ( ) contribution to 𝑎 𝜇 deviates significantly from [12].A theoretically even more important aspect of the holographic result for the axial-vectorcontributions to 𝑎 𝜇 arises when the whole tower of axial vector mesons present in hQCD models issummed up. In the left panel of Fig. 3 the component of the HLBL tensor involved in the Melnikov-Vainshtein SDC is considered, showing that each individual axial vector mode gives a vanishingcontribution at infinite momenta, but the infinite sum does approach a finite value as required by theSDC. However, as shown in the right panel of Fig. 3, the integrand 𝜌 𝑎 of the two-loop integral for 𝑎 𝜇 receives significant contributions only from the first few axial vector modes. The correspondingnumerical results for 𝑎 𝜇 are given in Table 1.
3. Discussion
In Table 2 the results for 𝑎 PS 𝜇 obtained in [3] and the new results for 𝑎 AV 𝜇 are combined for the twohard-wall models considered by us. The parameters of these models have been fixed to reproducethe right values for the 𝜌 meson mass and the pion decay constant which led to the results shownin Fig. 1. Doing so, the model HW2 cannot be matched to the full value of the various asymptoticconstraints, but instead satisfies them at the level of only 62%. Model HW1 contains one more freeparameter so that the SDCs can be satisfied fully. However, neither has a running coupling constant.Therefore at large but finite momenta where NLO effects are important, the actual behavior in realQCD may well be in between that of HW1 and HW2, so that the two holographic results may betaken to delimit a plausible range of predictions. In Ref. [15] results similar to ours have been obtained by using only the HW2 model, where two sets of parametershave been used that also span the range of 62% and 100% SDC saturation. For the latter the mass of the rho meson can xial vector TFF in holographic QCD and their contribution to the muon 𝑔 − Anton RebhanHW1 (100% MV-SDC) HW2 (62% MV-SDC) 𝑎 PS 𝜇 [ 𝜋 + 𝜂 + 𝜂 (cid:48) ] × 𝑎 AV 𝜇 [ 𝐿 + 𝑇 ] × 𝑎 PS + AV 𝜇 ×
133 112
Table 2:
Summary of the results of our previous calculation of the pseudoscalar pole contribution of Ref. [3]and the results obtained in [11] for the contribution of axial vector mesons in the models HW1 and HW2.
In the White Paper [1] the contribution from axial vector mesons and the SDC have beenestimated as 𝑎 SDC 𝜇 = ( ) × − , 𝑎 axials 𝜇 = ( ) × − , and it was suggested to add their errorslinearly. The sum, 21 ( ) × − , may be compared to our combined HW1-HW2 results [11] 𝑎 AV 𝜇 [ 𝐿 + 𝑇 ] = ( ) [ ( ) + ( )] × − , (2)where the longitudinal (L) part is responsible for the saturation of the Melnikov-Vainshtein SDC.The holographic results thus overlap those of the White Paper [1] within errors, but are significantly larger, in particular as concerns the contribution from (transverse) axial vector mesons. In ouropinion, the hQCD models provide a more plausible model for axial vector TFFs than previousphenomenological models, because they reproduce rather well the experimental results at lowenergies while matching the very nontrivial doubly virtual form [13] of leading-order perturbativeQCD at high energies. It would be very interested to have them further tested by experiment.
Acknowledgments
J. L. was supported by the FWF doctoral program Particles & Interactions, project no. W1252.
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