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

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The Muon Anomalous Magnetic Moment and the Pion Polarizability ∗ Kevin T Physics Letters B 738 (2014) 123–127 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb 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 4 the O(p ) 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 r r the terms corresponding to α + α 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. r r μ-independent combination α + α . An experimental value has 9 10 It is natural to ask whether this disagreement has significant been obtained from the measurement of the rate for radiative pion HLBL r r −3 implications for aμ .In an initial exploration of this question, the decay [30], yielding (α + α )rad = (1.32 ± 0.14) × 10 (see also 9 10 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 μ . (5) + − ∂q = [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 aμ pro- been has been approved for the GlueX detector in Hall D at Jeffer- r r portional to α + α is divergent.
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