ACFI-T13-01, CALT CALT 68-2859

The Muon Anomalous Magnetic Moment and the Pion Polarizability

Kevin T. Engel1 and Michael J. Ramsey-Musolf2 1California Institute of Technology, Pasadena, CA 91125 USA 2Physics Department, University of Massachusetts Amherst, Amherst, MA 01003, USA and Kellogg Radiation Laboratory, California Institute of Technology, Pasadena, CA 91125

We compute the charged pion loop contribution to the muon anomalous magnetic moment aµ, taking into account the effect of the charged pion polarizability, (α1 − β1)π+ . We evaluate this contribution using two different models that are consistent with the requirements of chiral sym- metry in the low-momentum regime and perturbative quantum chromodynamics in the asymptotic region. The result increases the disagreement between the present experimental value for aµ and the theoretical, Standard Model prediction by as much as ∼ 60 × 10−11, depending on the value of (α1 − β1)π+ and the choice of the model. The planned determination of (α1 − β1)π+ at Jefferson Laboratory will eliminate the dominant parametric error, leaving a theoretical model uncertainty commensurate with the error expected from planned Fermilab measurement of aµ.

The measurement of the muon anomalous magnetic Both uncertainties are similar in magnitude to the goal moment, aµ, provides one of the most powerful tests experimental error for the proposed Fermilab measure- of the Standard Model of particle physics and probes ment. Thus, it is of interest to revisit previous compu- of physics that may lie beyond it. The present exper- tations of the charged pion loop contribution, scrutinize imental value obtained by the E821 Collaboration[1– the presently quoted error, and determine how it might exp −11 3] aµ = 116592089(63) × 10 disagrees with the be reduced. most widely quoted theoretical SM predictions by 3.6σ: In previously reported work[28], we completed a step SM −11 µναβ aµ = 116591802(49) × 10 (for recent reviews, see in this direction by computing the amplitude Π for Ref. [4, 5] as well as references therein). This differ- light-by-light scattering for low-momentum off-shell pho- ence may point to physics beyond the Standard Model tons. In this regime, Chiral Perturbation Theory (χPT ) (BSM) such as weak scale supersymmetry or very light, provides a first principles, effective field theory descrip- weakly coupled neutral gauge bosons[6–9] . A next gen- tion of strong interaction dynamics that incorporates the eration experiment planned for Fermilab would reduce approximate chiral symmetry of quantum chromodynam- the experimental uncertainty by a factor of four[10]. If ics (QCD) for light quarks. Long-distance hadronic ef- a corresponding reduction in the theoretical, SM uncer- fects can be computed order-by-order in an expansion of tainty were achieved, the muon anomalous moment could p/Λχ, where p is a typical energy scale (such as the pion provide an even more powerful indirect probe of BSM mass mπ or momentum) and Λχ = 4πFπ ∼ 1 GeV is the physics. hadronic scale with Fπ = 93.4 MeV being the pion decay The dominant sources of theoretical uncertainty are as- constant. At each order in the expansion, presently incal- sociated with non-perturbative strong interaction effects culable strong interaction effects associated with energy that enter the leading order hadronic vacuum polariza- scales of order Λχ are parameterized by a set of effec- tion (HVP) and the hadronic light-by-light (HLBL) con- tive operators with a priori unknown coefficients. After HVP −11 HLBL tributions: δaµ (LO) = ±42 × 10 and δaµ = renormalization, the finite parts of these coefficients – ±26 × 10−11 [11] (other authors give somewhat different “low energy constants” (LECs) – are fit to experimental error estimates for the latter [12–26] , but we will refer results and then used to predict other low-energy observ-

arXiv:1309.2225v1 [hep-ph] 9 Sep 2013 to these numbers as points of reference; see [27] for a ables. review). In recent years, considerable scrutiny has been Working to next-to-next-to leading order (NNLO) in HVP applied to the determination of aµ (LO) from data on this expansion, we showed that models used to date in + − HLBL σ(e e → hadrons) and hadronic τ decays. A significant computing the full charged pion contribution to aµ reduction in this HVP error will be needed if the levels do not reproduce the structure of the low-momentum off- of theoretical and future experimental precision are to be shell HLBL scattering amplitude implied by the approxi- comparable. mate chiral symmetry of QCD. In particular, these mod- In this Letter, we concentrate on the more els fail to generate terms in the amplitude proportional HLBL theoretically-challenging aµ . At leading order in the to the pion polarizability, a ππγγ interaction arising from HLBL two terms in the O(p4) chiral Lagrangian: expansion of the number of colors NC , aµ is gener-  µ ν †
ated by the pseudoscalar pole contributions that in prac- L ⊃ ieα9 Fµν Tr Q D Σ,D Σ (1) tice turn out to be numerically largest. The contribution 2 2 †
+e α10 F Tr QΣQΣ , arising from charged pion loops is subleading in NC , yet the associated error is now commensurate with the uncer- where Q = diag(2/3, −1/3) is the electric charge ma- a a tainty typically quoted for the pseudoscalar pole terms. trix and Σ = exp(iτ π /Fπ) with a = 1, 2, 3 giving the 2 non-linear realization of the spontaneously broken chiral low-momentum off-shell HLBL amplitude match onto the r symmetry. The finite parts of the coefficients, α9 and χPT results for the charge radius contributions when one r 2 2 α10, depend on the renormalization scale, µ. identifies rπ = 6/mρ. In contrast, the terms correspond- r r The first term in Eq. (1) gives the dominant contribu- ing to α9 + α10 are absent. Moreover, the coefficients of tion to the pion charge radius for µ = mρ, the ρ-meson the polarizability contributions are comparable to those mass. The polarizability amplitude arises from both involving the charge radius, implying that the ENJL and terms and is proportional to the µ-independent combina- HLS model results for the low-momentum regime are in r r tion α9 + α10. An experimental value has been obtained disagreement with the requirements of QCD. from the measurement of the rate for radiative pion de- It is natural to ask whether this disagreement has sig- r r −3 HLBL cay [29], yielding (α9 + α10)rad = (1.32 ± 0.14) × 10 nificant implications for aµ . In an initial exploration (see also Refs. [30, 31]). On the other hand, direct deter- of this question, the authors of Ref. [38, 39] (see also [40]) µναβ minations of the polarizability (α1 −β1)π+ have been ob- included the operators in Eq. (1) in Π (q, k2, k3, k4), 0 + tained from radiative pion photoproduction γp → γ π n where q and the kj are the real and virtual photon mo- 0 and the hadronic Primakov process πA → π γA where A menta, respectively, with k4 = −(q + k2 + k3). Since is a heavy nucleus. Using the anomalous magnetic moment amplitude is linear in qµ, one need retain only the first non-trivial term in an r r 2 (α1 − β1)π+ = 8α(α9 + α10)/(Fπ mπ) + ··· , (2) expansion in the external photon momentum. Differenti- λναβ ating the QED Ward identity qλΠ = 0 with respect where the “+ ··· ” indicate corrections that vanish in the to qµ implies that one may then express the HLBL am- chiral limit (see e.g., Refs. [32, 33]), these direct mea- plitude entering the full integral for a as[12] r r −3 µ surements yield (α9 +α10)γp = (3.1±0.9)×10 [34] and r r −3 λναβ (α9 + α10)πA = (3.6 ± 1.0) × 10 [35], respectively. The µναβ ∂Π (q, k2, k3, −q − k2 − k3) Π = −qλ . (5) COMPASS experiment has undertaken a new determi- ∂qµ q=0 nation using the hadronic Primakov process[36], while a Using this procedure, one finds that the contribution to measurement of the process γγπ+π− is underway at Fras- a proportional to αr + αr is divergent. The authors cati. A determination using the reaction γA → π+π−A µ 9 10 of Refs. [38, 39] thus regulated the integral by imposing has been has been approved for the GlueX detector in a cutoff K2 ≡ (k + k )2 < (500 MeV)2. The resulting Hall D at Jefferson Laboratory (JLab), with an antici- 2 3 impact on aHLBL amounts to a ∼ 10% increase in the pated absolute uncertainty of 0.16 × 10−4[37]. µ magnitude of the overall charged pion loop contribution In terms of the off-shell HLBL amplitude, the LO con- compared to the simplest VMD model prediction. tribution arises solely from interactions appearing in the Here, we report on a computation of aHLBL that is O(p2) Lagrangian, corresponding to scalar quantum elec- µ consistent with the low-momentum requirements of QCD trodynamics [see Fig. 1(a)]. The associated contribution and that does not rely on an ad hoc cut off when extrap- to aHLBL is finite and was first computed in Ref. [12]. At µ olating to the higher momentum regime where χPT is NNLO in χPT , one encounters distinct contributions to not applicable. Instead, we employ two different mod- the low-momentum off-shell HLBL amplitude associated els for the high-momentum behavior of the pion virtual with the square of the pion charge radius Compton amplitude that are consistent with both the 12 1   µ2   strictures of chiral symmetry in the low momentum re- r2 = αr(µ) + ln − 1 (3) π F 2 9 Λ2 m2 gion and the requirements of perturbative QCD in the π χ π domain of large photon virtuality. Compared with the r r conclusions of Refs. [38, 39], we find that the impact and the polarizability LECs α9+α10. The resulting γγππ µν 2 2 on a may be significant, leading to an increase in the vertex V εµ(k1)εν (k2) for |kj | << mπ is given by µ discrepancy with the experimental result by as much as 2 −11 µν 2n µν rπ  µν 2 2 µ ν µ ν  ∼ 60 × 10 , depending on which whether one takes the VχPT = 2ie g + g (k1 + k2) − k1 k1 − k2 k2 r r 6 value of α9 + α10 from radiative pion decays or direct determinations of the polarizability. The planned deter- 4(α9 + α10) µν µ ν o + [k1 · k2g − k k ] . (4) r r 2 2 1 mination of α9 + α10 at JLab could significantly reduce Fπ the spread of polarizability contributions to aµ. In the From the model standpoint, efforts to incorporate the longer term, studies of the off-shell Compton amplitude effects of pion substructure in electromagnetic interac- could help reduce the theoretical uncertainty associated tions have generally followed a vector meson dominance with interpolating between the chiral and asymptotic do- (VMD) type of approach [see Fig. 1(b)]. The first efforts mains. with the simplest VMD implementation [12] were fol- In modeling the higher momentum behavior of the po- lowed by use an extended Nambu-Jona-Lasinio (ENJL) larizability, we are guided by several considerations: model[15] the Hidden Local Symmetry (HLS) approach Chiral symmetry. In the low-momentum regime, any [16, 17]. In all cases, the associated contributions to the model should reproduce the γππ and γγππ interactions 3 implied by the O(p4) operators in Eq. (1). As indicated earlier, neither the HLS nor the ENJL prescriptions are fully consistent with this requirement. While they incor- r porate the α9 (charge radius) contribution to the γππ interaction, they omit the contribution to the γγππ in- r r teraction proportional to α9 + α10. Asymptotic behavior. By using the operator product ex- (a) (b) (c) pansion, it is possible to show that the virtual Compton 2 amplitude Tµν (k, −k) must vanish as 1/k in the large 2 FIG. 1: Representative diagrams for charged pion loop con- k regime. Neither the HLS nor the ENJL models sat- HLBL tributions to aµ : (a) LO; (b) VMD (ρ-meson) model for isfy this requirement. The HLS approach gives a non- the γπ+π− vertex; (c) Model I and II γγπ+π− form factor. vanishing Tµν in the asymptotic limit, while in the ENJL framework the Compton amplitude falls off as 1/(k2)2. Resonance saturation. The LECs of the O(p4) chiral La- Unfortunately, the a1 AT model does not yield a finite HLBL grangian are known to be saturated by spin-one meson result for aµ . We are, thus, motivated to consider resonances for µ ' mρ. The ENJL and HLS approaches alternative models that incorporate as many features of incorporate these “resonance saturation” dynamics for the a1 dynamics as possible while satisfying the require- the ρ-meson, thereby obtaining the well-established re- ments of chiral symmetry, asymptotic scaling, and finite 2 2 aHLBL. Our strategy is to modify the γγππ polarizabil- lation rπ = 6/mρ. By itself, however, inclusion of the ρ µ does not lead to a correct description of the polarizability, ity vertex by the introduction of vector meson-like form as our study of the off-shell LBL amplitude demonstrates. factors. We consider two models: Pion mass splitting. The degeneracy between charged e2  1  L = − F π+ F µν π− + h.c. +··· , (8) and neutral pion masses is broken by the light quark I 4 µν D2 + M 2 mass difference and by the electromagnetic interaction. A The pion polarizability and charge radius contribute to where Dµ = ∂µ + ieQAµ is the covariant derivative and the latter when one embeds Tµν in the one-loop pion self the + ··· are higher order terms in pion fields as dictated energy. Retaining only the O(p4) interactions in Eq. (1) by chiral symmetry; and yields a divergent result that is rendered finite by inclu- e2  M 2  2 sion of the operator[41] L = − π+π− V F µν + ··· , (9) II 2M 2 ∂2 + M 2 2 A V e C †
L ⊃ Tr QΣQΣ (6) with the partial derivatives acting only on the field F 2 π strength tensors immediately to the right. In order to 2 2 2 2 that contributes 2e C/Fπ to mπ+ but nothing to mπ0 . obtain the appropriate asymptotic behavior for Tµν , one Note that this contribution does not vanish in the chiral must combine the Model I Lagrangian (8) with either the limit. The HLS approach also does not give a finite con- AT or HLS formulation for the ρ-meson contributions, tribution. While it would be desirable that any model whereas in using the Model II Lagrangian (9) it is neces- used to interpolate to the higher-momentum regime also sary to employ the full VMD prescription for the ρ (simi- 2 reproduce the known value of (∆mπ)EM, this mass split- lar to the ENJL case, but with a momentum-independent ting does not enter directly into the HLBL amplitude to MV ). the order of interest here. By construction, both models reproduce the correct One approach that satisfies the aforementioned crite- polarizability and charge radius interactions that appear 4 ria is to include the axial vector a1 meson as well as at O(p ) and yield a Compton amplitude Tµν (k, −k) the ρ meson in the low-energy effective Lagrangian us- that falls off as 1/k2. Both also generate a finite con- HLBL ing the anti-symmetric tensor (AT) formulation[42]. De- tribution to aµ . When the Kawarabayashi-Suzuki- tailed application of this approach to the pion polarizabil- √Fayyazuddin-Riazuddin (KSFR) relation[45, 46] MA = 2 ity and (∆mπ)EM have been reported in Refs. [43, 44]. 2MV is imposed, Model I gives rise to a finite con- 2 The polarizability term in Eq. (4) is given by tribution to (∆mπ)EM, whereas Model II requires the r r 2 2 additional counterterm in Eq. (6). Since, however, 4(α9 + α10)a1 = FA/MA , (7) r r 2 (α9 + α10) ∼ 1/MA in these models [see Eq. (7)], choos- where MA and FA are the a1 mass and electromagnetic ing MA and MV to reproduce the experimental results coupling, respectively. (Consistency with a variety of for the polarizability and charge radius (MV = mρ) may theoretical and empirical considerations suggests taking lead to a violation of the KSFR relation, thus implying 2 FA = Fπ, which we follow below). Introduction of ad- for Model I both a divergent (∆mπ)EM as well as incor- ditional form factors leads to a finite contribution to rect asymptotic behavior for Tµν . 2 (∆mπ)EM. Numerically, one finds that the experimen- An example of the additional diagrams needed for HLBL tal value for the pion mass splitting is well-reproduced. complete evaluation the new contributions to aµ are 4 shown in Fig. 1(c). Note that in Model I, one encounters the polarizability obtained from radiative pion photopro- additional vertices associated with the action of the co- duction. Note that only the results in the last three lines variant derivative on the pion and field strength tensors. of Table I depend on this choice. For case (a), the value The form of the interactions in Models I and II facili- of MA implied by Eq. (7) is about 20 % larger than given HLBL tates numerical evaluation of the full aµ integral. The by the KSFR relation; consequently, Model I no longer 2 momentum space structures are propagator-like, thereby yields a finite value for (∆mπ)EM for this choice. allowing us to employ conventional Feynman parameter- Several features emerge from Table I: ization. In doing so, we follow the procedure described in (i) Inclusion of the polarizability tends to decrease the HLBL Ref. [12], wherein evaluation of the loop integrals yields Standard Model prediction for aµ , regardless of which Feynman parameter integrals of the form procedure one follows in treating its high momentum be- Z havior, thereby increasing the discrepancy with the ex- X N(x) perimental result. M = Πjdxjδ(1 − xj) α β . (10) U(x) V (x) (ii)Use of a model that interpolates to high momentum Here, U(x) and V (x) arise from the denominator struc- and that is consistent with the required asymptotic be- ture of the diagram, with U(x) encoding the self-coupling havior of the virtual Compton amplitude leads to a sub- of the loop momenta and V (x) containing mass terms stantially larger shift than does the imposition of a cut- that govern the infrared behavior the integrand. The nu- off. When compared to the Standard Model prediction obtained using the ENJL model, this shift can be has merator N(x) follows from the detailed structure of the −11 −11 interaction vertices. We have written a separate Monte much as ∼ 30×10 [case (a)] or ∼ 60×10 [case (b)]. (iii) The uncertainties in the polarizability contribution Carlo routine for evaluating these Feynman parameter r r integrals, details of which will appear elsewhere. associated with both the experimental value of (α9 +α10) and the choice of a model for interpolating to the asymp- totic domain are significant, particularly compared with HLBL TABLE I: Charged pion loop contributions to aµ in differ- the anticipated experimental error for the future FNAL −11 ent approaches discussed in text. Second and third columns measurement of δaµ = ±16 × 10 . correspond to different values for the polarizability LECs, r r −3 −3 Clearly, it will be desirable to reduce the uncertainties (α9 + α10): (a) (1.32 ± 1.4) × 10 and (b) (3.1 ± 0.9) × 10 . associated with polarizability contribution. The planned Note that only the NLO/cut-off and Models I and II depend JLab experiment will determine the polarizability LECs on these LECs. r r −3 + − + − with an uncertainty of δ(α +α ) = 0.16×10 , thereby Approach aπ π × 1011 (a) aπ π × 1011 (b) 9 10 µ µ reducing the parametric error well below the level of LO -44 -44 the expected FNAL uncertainty in aµ. Reducing the HLS -4.4 (2) -4.4 (2) model-dependent uncertainty will require additional in- ENJL -19 (13) -19(13) put. To be on the conservative side, one would like to NLO/cut-off -20 (5) -24 (5) have in hand an independent, experimental test of the Model I -11 -34 momentum-dependence of the polarizability that could Model II -40 -71 help discriminate between Models I and II and any other prescriptions for interpolating to the asymptotic domain. Results are shown in Table I. For comparison, we also The possibilities for doing so will be the subject of forth- give the charged pion loop results obtained in the leading coming work. order calculation, HLS and ENJL approaches, and using Acknowledgements We thank J. Donoghue, B. Holstein, the NLO operators in Eq. (1) but imposing the cut-off and R. Miskimen for useful conversations; J. Bijnens K2 < (500 MeV)2 discussed earlier. As a cross check on for providing input on the NLO/cut-off evaluation; and our evaluation of the integrals, we have reproduced the D. Hertzog and D. Kawall for critical reading of this LO and HLS results reported in Refs. [12, 16, 17]. In the manuscript. The work is partially supported by U.S. case of the ENJL model, one must include a momentum- Department of Energy contracts DE-FG02-92-ER40701 dependence for the vector meson mass, an effect we (KTE) and DE-FG02-08ER41531 (MJRM) and the Wis- are not able to implement using the integration proce- consin Alumni Research Foundation (MJRM). dure described above. However, taking a momentum- independent mass yields −16×10−11, in good agreement with the full ENJL result reported in Ref. [15] . The last three lines in Table I include the results from Refs. [38, 39] and the two models adopted in this work. [1] G. W. Bennett et al. [ Muon G-2 Collaboration ], Phys. Rev. D73, 072003 (2006). The second and third columns give the results for two r r −3 [2] G. W. Bennett et al. [ Muon g-2 Collaboration ], Phys. different values for (α9 + α10): (a) (1.32 ± 1.4) × 10 , Rev. Lett. 92, 161802 (2004). obtained using the results of pion radiative decay and (b) [3] G. W. Bennett et al. [ Muon g-2 Collaboration ], Phys. (3.1 ± 0.9) × 10−3, corresponding to the determination of Rev. Lett. 89, 101804 (2002). 5

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