The Muon Anomalous Magnetic Moment and the Pion Polarizability ∗ Kevin T

Physics Letters B 738 (2014) 123–127

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Physics Letters B

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The muon anomalous magnetic moment and the pion polarizability ∗ Kevin T. Engel a, Michael J. Ramsey-Musolf b,c, a University of Maryland, College Park, MD 20742, USA b Physics Department, University of Massachusetts Amherst, Amherst, MA 01003, USA c Kellogg Radiation Laboratory, California Institute of Technology, Pasadena, CA 91125, USA a r t i c l e i n f o a b s t r a c t

Article history: We compute the charged pion loop contribution to the muon anomalous magnetic moment aμ, taking Received 4 April 2014 into account the previously omitted effect of the charged pion polarizability, (α1 − β1)π + . We evaluate Received in revised form 13 August 2014 this contribution using two different models that are consistent with the requirements of chiral symmetry Accepted 3 September 2014 in the low-momentum regime and perturbative quantum chromodynamics in the asymptotic region. The Available online 6 September 2014 result may increase the disagreement between the present experimental value for a and the theoretical, Editor: W. Haxton μ −11 Standard Model prediction by as much as ∼ 60 × 10 , 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. © 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.

The measurement of the muon anomalous magnetic moment, and hadronic τ decays. A significant reduction in this HVP error aμ, provides one of the most powerful tests of the Standard Model will be needed if the levels of theoretical and future experimental of particle physics and probes of physics that may lie beyond it. precision are to be comparable. The present experimental value obtained by the E821 Collabo- In this Letter, we concentrate on the more theoretically- exp −11 HLBL ration [1–3] aμ = 116592089(63) × 10 disagrees with the challenging aμ , computing the previously omitted contribution SM = from the charged pion polarizability and estimating the associated most widely quoted theoretical SM predictions by 3.6σ : aμ − parametric and model-dependent uncertainties. At leading order 116591802(49) × 10 11 (for recent reviews, see Refs. [4–6] as well in the expansion of the number of colors N , aHLBL is generated as references therein). This difference may point to physics beyond C μ by the pseudoscalar pole contributions that in practice turn out the Standard Model (BSM) such as weak scale supersymmetry or to be numerically largest. The contribution arising from charged very light, weakly coupled neutral gauge bosons [7–10]. A next pion loops is subleading in N , yet the associated error is now generation experiment planned for Fermilab would reduce the ex- C commensurate with the uncertainty typically quoted for the pseu- perimental uncertainty by a factor of four [11]. If a corresponding doscalar pole terms. Both uncertainties are similar in magnitude reduction in the theoretical, SM uncertainty were achieved, the to the goal experimental error for the proposed Fermilab measure- muon anomalous moment could provide an even more powerful ment. Thus, it is of interest to revisit previous computations of the indirect probe of BSM physics. charged pion loop contribution, scrutinize the presently quoted er- The dominant sources of theoretical uncertainty are associ- ror, and determine how it might be reduced. ated with non-perturbative strong interaction effects that enter In previously reported work [29], we completed a step in this the leading order hadronic vacuum polarization (HVP) and the direction by computing the amplitude Πμναβ for light-by-light hadronic light-by-light (HLBL) contributions: δaHVP(LO) =±42 × μ scattering for low-momentum off-shell photons. In this regime, −11 HLBL =± × −11 10 and δaμ 26 10 [12] (other authors give some- Chiral Perturbation Theory (χPT) provides a first principles, effec- what different error estimates for the latter [13–27], but we will tive field theory description of strong interaction dynamics that refer to these numbers as points of reference; see [28] for a re- incorporates the approximate chiral symmetry of quantum chro- view). In recent years, considerable scrutiny has been applied to modynamics (QCD) for light quarks. Long-distance hadronic effects HVP + − → the determination of aμ (LO) from data on σ (e e hadrons) can be computed order-by-order in an expansion of p/Λχ , where p is a typical energy scale (such as the pion mass mπ or mo- mentum) and Λχ = 4π Fπ ∼ 1GeVis the hadronic scale with * Corresponding author. Fπ = 92.4MeVbeing the pion decay constant. At each order in http://dx.doi.org/10.1016/j.physletb.2014.09.006 0370-2693/© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3. 124 K.T. Engel, M.J. Ramsey-Musolf / Physics Letters B 738 (2014) 123–127 the expansion, presently incalculable strong interaction effects as- sociated with energy scales of order Λχ are parameterized by a set of effective operators with a priori unknown coefficients. After renormalization, the finite parts of these coefficients – “low energy constants” (LECs) – are fit to experimental results and then used to predict other low-energy observables. Working to next-to-next-to leading order (NNLO) in this expan- sion, we showed that models used to date in computing the full HLBL charged pion contribution to aμ do not reproduce the structure HLBL of the low-momentum off-shell HLBL scattering amplitude implied Fig. 1. Representative diagrams for charged pion loop contributions to aμ : (a) LO; + − + − by the approximate chiral symmetry of QCD. In particular, these (b) VMD (ρ-meson) model for the γπ π vertex; (c) Models I and II γγπ π models fail to generate terms in the amplitude proportional to the form factor. pion polarizability, a ππγγ interaction arising from two terms in the O(p4) chiral Lagrangian: followed a vector meson dominance (VMD) type of approach [see      Fig. 1(b)]. The first efforts with the simplest VMD implementation μ ν † 2 2 † L ⊃ ieα9 Fμν Tr Q D Σ, D Σ + e α10 F Tr Q Σ Q Σ , [13] were followed by use an extended Nambu–Jona-Lasinio (ENJL) (1) model [16] the Hidden Local Symmetry (HLS) approach [17,18]. In all cases, the associated contributions to the low-momentum off- where Q = diag(2/3, −1/3) is the electric charge matrix and Σ = shell HLBL amplitude match onto the χPT results for the charge exp(i a a/F ) with a = 1, 2, 3givingthe non-linear realization of 2 = 2 τ π π radius contributions when one identifies rπ 6/mρ .In contrast, the spontaneously broken chiral symmetry. The finite parts of the the terms corresponding to αr + αr are absent. Moreover, the r r 9 10 coefficients, α9 and α10, depend on the renormalization scale, μ. coefficients of the polarizability contributions are comparable to The first term in Eq. (1) gives the dominant contribution to the those involving the charge radius, implying that the ENJL and HLS = pion charge radius for μ mρ , the ρ-meson mass. The polariz- model results for the low-momentum regime are in disagreement ability amplitude arises from both terms and is proportional to the with the requirements of QCD. -independent combination r + r . An experimental value has μ α9 α10 It is natural to ask whether this disagreement has significant been obtained from the measurement of the rate for radiative pion HLBL − implications for aμ .In an initial exploration of this question, the decay [30], yielding (αr + αr ) = (1.32 ± 0.14) × 10 3 (see also 9 10 rad authors of Refs. [39,40] (see also [41]) included the operators in Refs. [31,32]). On the other hand, direct determinations of the po- μναβ Eq. (1) in Π (q, k2, k3, k4), where q and the k j are the real and larizability (α1 − β1)π + have been obtained from radiative pion  + virtual photon momenta, respectively, with k =−(q + k + k ). photoproduction γ p → γ π n and the hadronic Primakov process 4 2 3  μ π A → π γ A where A is a heavy nucleus. Using Since the anomalous magnetic moment amplitude is linear in q ,     one need retain only the first non-trivial term in an expansion + r r 2 in the external photon momentum. Differentiating the QED Ward (α1 − β1)π = 8α α + α / F mπ +···, (2) 9 10 π λναβ μ identity qλΠ = 0with respect to q implies that one may where the “+ ···” indicate corrections that vanish in the chi- then express the HLBL amplitude entering the full integral for aμ ral limit (see e.g., Refs. [33,34]), these direct measurements yield as [13] r + r = ± × −3 r + r = ± (α9 α10)γ p (3.1 0.9) 10 [35] and (α9 α10)π A (3.6 − 1.0) ×10 3 [36], respectively. The COMPASS experiment has under- λναβ − − − μναβ ∂Π (q,k2,k3, q k2 k3) taken a new determination using the hadronic Primakov process Π =−qλ . (5) + − qμ [37], while a measurement of the process γγπ π is underway ∂ q=0 + − at Frascati. A determination using the reaction γ A → π π A has Using this procedure, one finds that the contribution to a pro- been has been approved for the GlueX detector in Hall D at Jeffer- μ portional to αr + αr is divergent. The authors of Refs. [39,40] son Laboratory (JLab), with an anticipated absolute uncertainty of 9 10 − 2 ≡ + 2 0.16 × 10 3 [38]. thus regulated the integral by imposing a cutoff K (k2 k3) < 2 HLBL ∼ In terms of the off-shell HLBL amplitude, the LO contribution (500 MeV) . The resulting impact on aμ amounts to a 10% arises solely from interactions appearing in the O(p2) Lagrangian, increase in the magnitude of the overall charged pion loop contri- corresponding to scalar quantum electrodynamics [see Fig. 1(a)]. bution compared to the simplest VMD model prediction. HLBL HLBL Here, we report on a computation of a that is consistent The associated contribution to aμ is finite and was first com- μ with the low-momentum requirements of QCD and that does not puted in Ref. [13]. At NNLO in χPT, one encounters distinct contri- rely on an ad hoc cut off when extrapolating to the higher mo- butions to the low-momentum off-shell HLBL amplitude associated mentum regime where χPT is not applicable. Instead, we employ with the square of the pion charge radius    two different models for the high-momentum behavior of the pion 12 1 μ2 virtual Compton amplitude that are consistent with both the stric- r2 = αr (μ) + ln − 1 (3) π 2 9 2 2 tures of chiral symmetry in the low momentum region and the Fπ Λχ mπ requirements of perturbative QCD in the domain of large pho- r + r and the polarizability LECs α9 α10. The resulting γγππ vertex ton virtuality. Compared with the conclusions of Refs. [39,40], we μν | 2|  2 V εμ(k1)εν(k2) for k j mπ is given by find that the impact on aμ may be significant, leading to an in- crease in the discrepancy with the experimental result by as much 2     − μν r μ μ as ∼ 60 × 10 11, depending on whether one takes the value of V = 2ie2 gμν + π gμν k2 + k2 − k kν − k kν χPT 6 1 2 1 1 2 2 αr + αr from radiative pion decays or direct determinations of 9 10 +   the polarizability. The planned determination of αr + αr at JLab 4(α9 α10) μν μ ν 9 10 + k1 · k2 g − k k . (4) could significantly reduce the spread of polarizability contributions F 2 2 1 π to aμ. In the longer term, studies of the off-shell Compton ampli- From the model standpoint, efforts to incorporate the effects tude could help reduce the theoretical uncertainty associated with of pion substructure in electromagnetic interactions have generally interpolating between the chiral and asymptotic domains. K.T. Engel, M.J. Ramsey-Musolf / Physics Letters B 738 (2014) 123–127 125

In modeling the higher momentum behavior of the polarizabil- where Dμ = ∂μ + ieQ Aμ is the covariant derivative and the + ··· ity, we are guided by several considerations: are higher order terms in pion fields as dictated by chiral symme- Chiral symmetry. In the low-momentum regime, any model try; and   should reproduce the γππ and γγππ interactions implied by the 2 4 2 2 O(p ) operators in Eq. (1). As indicated earlier, neither the HLS e + − MV μν LII =− π π F +···, (9) nor the ENJL prescriptions are fully consistent with this require- 2M2 ∂2 + M2 r A V ment. While they incorporate the α9 (charge radius) contribution to the γππ interaction, they omit the contribution to the γγππ with the partial derivatives acting only on the field strength r + r tensors immediately to the right. In order to obtain the ap- interaction proportional to α9 α10. Asymptotic behavior. By using the operator product expansion, it propriate asymptotic behavior for Tμν , one must combine the Model I Lagrangian (8) with either the AT or HLS formulation is possible to show that the virtual Compton amplitude Tμν (k, −k) for the ρ-meson contributions, whereas in using the Model II must vanish as 1/k2 in the large k2 regime. Neither the HLS nor Lagrangian (9) it is necessary to employ the full VMD prescrip- the ENJL models satisfy this requirement. The HLS approach gives tion for the ρ (similar to the ENJL case, but with a momentum- a non-vanishing Tμν in the asymptotic limit, while in the ENJL independent MV ). framework the Compton amplitude falls off as 1/(k2)2. By construction, both models reproduce the correct polariz- Resonance saturation. The LECs of the O(p4) chiral Lagrangian ability and charge radius interactions that appear at O(p4) and are known to be saturated by spin-one meson resonances for 2 yield a Compton amplitude Tμν(k, −k) that falls off as 1/k . μ m . The ENJL and HLS approaches incorporate these “reso- ρ Both also generate a finite contribution to aHLBL.When the nance saturation” dynamics for the ρ-meson, thereby obtaining μ Kawarabayashi–Suzuki–Fayyazuddin–Riazuddin (KSFR) relation [46, the well-established relation r2 = 6/m2 . By itself, however, inclu- √ π ρ 47] M = 2M is imposed, Model I gives rise to a finite con- sion of the ρ does not lead to a correct description of the polariz- A V tribution to (m2 ) , whereas Model II requires the additional ability, as our study of the off-shell LBL amplitude demonstrates. π EM counterterm in Eq. (6). Since, however, (αr + αr ) ∼ 1/M2 in these Pion mass splitting. The degeneracy between charged and neutral 9 10 A models [see Eq. (7)], choosing M and M to reproduce the exper- pion masses is broken by the light quark mass difference and by A V imental results for the polarizability and charge radius (M = m ) the electromagnetic interaction. The pion polarizability and charge V ρ may lead to a violation of the KSFR relation, thus implying for radius contribute to the latter when one embeds Tμν in the one- Model I both a divergent (m2 ) as well as incorrect asymptotic loop pion self-energy. Retaining only the O(p4) interactions in π EM behavior for Tμν . Eq. (1) yields a divergent result that is rendered finite by inclu- Although the chiral invariance associated with Models I and II is sion of the operator [42] not manifest, one may construct them from operators that respect L e2C   the chiral Ward identities. To illustrate, we first observe that I,II L ⊃ Tr Q Σ Q Σ † (6) correspond to a sums of an infinite tower of operators, starting off 2 Fπ with terms at O(p4) that match explicitly onto the combination 2 2 2 2 in Eq. (1) that give the polarizability. The higher order terms for that contributes 2e C/F to m + but nothing to m .Note that π π π 0 Model II are obtained in a straightforward matter from a tower of this contribution does not vanish in the chiral limit. The HLS ap- operators of the form proach also does not give a finite contribution. While it would      be desirable that any model used to interpolate to the higher- 2 n 2 † 2 ∂ F Tr Q Σ Q Σ (10) momentum regime also reproduce the known value of (mπ )EM, this mass splitting does not enter directly into the HLBL amplitude that have the same chiral transformation properties as the corre- 4 to the order of interest here. sponding O(p ) operator proportional to α10. With appropriately One approach that satisfies the aforementioned criteria is to in- chosen coefficients and resummation, the operators in Eq. (10) will 2 L clude the axial vector a1 meson as well as the ρ meson in the then yield the higher order (in p ) terms in II. The construction low-energy effective Lagrangian using the anti-symmetric tensor for Model I is less straightforward, though a resummed tower of (AT) formulation [43]. Detailed application of this approach to the operators of the form pion polarizability and (m2 ) have been reported in Refs. [44,      π EM 2 n μ ν † 45]. The polarizability term in Eq. (4) is given by Fμν Tr Q D D Σ, D Σ (11)

  with appropriately-chose coefficients will contain LI as well as 4 αr + αr = F 2 /M2 , (7) 9 10 a1 A A other pion–photon interactions. As our present purpose is to assess HLBL the sensitivity of aμ to the higher-order momentum dependence where M A and F A are the a1 mass and electromagnetic coupling, as needed to obtain the correct asymptotic behavior of Tμν (k, −k), respectively. (Consistency with a variety of theoretical and em- we defer a detailed discussion of the model construction and its pirical considerations suggests taking F = F , which we follow A π phenomenological consequences to future work. below.) Introduction of additional form factors leads to a finite An example of the additional diagrams needed for complete contribution to (m2 ) . Numerically, one finds that the exper- π EM evaluation the new contributions to aHLBL are shown in Fig. 1(c). imental value for the pion mass splitting is well-reproduced. μ Note that in Model I, one encounters additional vertices associated Unfortunately, the a AT model does not yield a finite result for 1 with the action of the covariant derivative on the pion and field aHLBL. We are, thus, motivated to consider alternative models that μ strength tensors. The form of the interactions in Models I and II incorporate as many features of the a dynamics as possible while 1 facilitates numerical evaluation of the full aHLBL integral. The mo- satisfying the requirements of chiral symmetry, asymptotic scaling, μ mentum space structures are propagator-like, thereby allowing us and finite aHLBL. Our strategy is to modify the γγππ polarizabil- μ to employ conventional Feynman parameterization. In doing so, we ity vertex by the introduction of vector meson-like form factors. follow the procedure described in Ref. [13], wherein evaluation of We consider two models: the loop integrals yields Feynman parameter integrals of the form   e2 1    L =− + μν − + +··· N(x) I Fμνπ F π h.c. , (8) M = Π dx δ 1 − x . (12) 4 2 + 2 j j j α β D M A U(x) V (x) 126 K.T. Engel, M.J. Ramsey-Musolf / Physics Letters B 738 (2014) 123–127

Table 1 Clearly, it will be desirable to reduce the uncertainties asso- HLBL Charged pion loop contributions to aμ in different approaches discussed in text. ciated with polarizability contribution. The planned JLab experi- Second and third columns correspond to different values for the polarizability LECs, − − ment will determine the polarizability LECs with an uncertainty (αr + αr ): (a) (1.32 ± 1.4) × 10 3 and (b) (3.1 ± 0.9) × 10 3. Note that only the 9 10 r + r = × −3 NLO/cut-off and Models I and II depend on these LECs. of δ(α9 α10) 0.16 10 , thereby reducing the parametric er- + − + − ror well below the level of the expected FNAL uncertainty in aμ. π π × 11 π π × 11 Approach aμ 10 (a) aμ 10 (b) Reducing the model-dependent uncertainty will require additional LO −44 −44 input. Measuring the momentum-dependence of the polarizability − − HLS 4.4 (2) 4.4 (2) could help discriminate between Models I and II and any other ENJL −19 (13) −19 (13) NLO/cut-off −20 (5) −24 (5) prescriptions for interpolating to the asymptotic domain. The pos- Model I −11 −34 sibilities for doing so will be the subject of forthcoming work. On ± Model II −40 −71 the theoretical side, exploring the impact of off-shell π momenta and other dynamical effects, as has been done for the pseudoscalar pole contribution in the context of Dyson Schwinger equations [48, Here, U (x) and V (x) arise from the denominator structure of the 49] and other approaches [5], remains a future task as well. As diagram, with U (x) encoding the self-coupling of the loop mo- discussed above, it would also be worthwhile to investigate the menta and V (x) containing mass terms that govern the infrared explicit construction of models yielding the correct asymptotic be- behavior the integrand. The numerator N(x) follows from the de- havior of Tμν , possibly utilizing explicit a1 degrees of freedom in tailed structure of the interaction vertices. We have written a sep- the AT formulation (see, e.g., Ref. [50]). Together with the results arate Monte Carlo routine for evaluating these Feynman parameter of this work, such studies should yield insights that complement integrals, details of which will appear elsewhere. ongoing lattice QCD efforts [27]. Results are shown in Table 1. For comparison, we also give the charged pion loop results obtained in the leading order calculation, Acknowledgements HLS and ENJL approaches, and using the NLO operators in Eq. (1) but imposing the cut-off K 2 <(500 MeV)2 discussed earlier. As a We thank V. Cirigliano, J. Donoghue, B. Holstein, and R. Miski- cross check on our evaluation of the integrals, we have reproduced men for useful conversations; J. Bijnens for providing input on the the LO and HLS results reported in Refs. [13,17,18]. In the case NLO/cut-off evaluation; and D. Hertzog and D. Kawall for critical of the ENJL model, one must include a momentum-dependence for reading of this manuscript. The work is partially supported by U.S. the vector meson mass, an effect we are not able to implement us- Department of Energy contracts DE-FG02-92-ER40701 (KTE) and ing the integration procedure described above. However, taking a − DE-FG02-08ER41531 and DE-SC0011095 (MJRM) and the Wiscon- momentum-independent mass yields −16 × 10 11, in good agree- sin Alumni Research Foundation (MJRM). ment with the full ENJL result reported in Ref. [16]. The last three lines in Table 1 include the results from Refs. [39, References 40] and the two models adopted in this work. The second and third columns give the results for two different values for [1] G.W. Bennett, et al., Muon G-2 Collaboration, Phys. Rev. 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