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The Standard Model prediction for R_{e/mu}^{(pi,K)}

We study the ratios R_{e/mu}^{(P)} = Gamma(P ->e nu [gamma])/Gamma(P ->mu nu [gamma]) (P=pi,K) in Chiral Perturbation Theory to order e^2 p^4. We complement the two-loop effective theory results with a matching calculation of the counterterm, finding R_{e/mu}^{(pi)} = (1.2352 \pm 0.0001)*10^(-4) and R_{e/mu}^{(K)} = (2.477 \pm 0.001)*10^(-5).

Introduction -The ratio R (P ) e/µ ≡ Γ(P → eν e [γ])/Γ(P → µν µ [γ]) (P = π, K) is helicity-suppressed in the Standard Model (SM), due to the V − A structure of charged current couplings. It is therefore a sensitive probe of all SM extensions that induce pseudoscalar currents and non-universal corrections to the lepton couplings [1], such as the minimal supersymmetric SM [2]. Effects from weak-scale new physics are expected in the range (∆R e/µ )/R e/µ ∼ 10 −4 − 10 −2 and there is a realistic chance to detect or constrain them because: (i) ongoing experimental searches plan to reach a fractional uncertainty of (∆R (π) e/µ )/R (π) e/µ < ∼ 5 × 10 −4 [3] and (∆R [4], which represent respectively a factor of 5 and 10 improvement over current errors [5]. (ii) The SM theoretical uncertainty can be pushed below this level, since to a first approximation the strong interaction dynamics cancels out in the ratio R e/µ and hadronic structure dependence appears only through electroweak corrections. Indeed, the most recent theoretical predictions read R (π) e/µ = (1.2352 ± 0.0005) × 10 −4 [6], R (π) e/µ = (1.2354 ± 0.0002)× 10 −4 [7], and R (K) e/µ = (2.472 ± 0.001) × 10 −5 [7]. The authors of Ref. [6] provide a general parameterization of the hadronic effects and estimate the induced uncertainty via dimensional analysis. On the other hand, in Ref. [7] the hadronic component is calculated by modeling the low-and intermediate-momentum region of the loops involving virtual photons.
With the aim to improve the existing theoretical status, we have analyzed R e/µ within Chiral Perturbation Theory (ChPT), the low-energy effective field theory (EFT) of QCD. The key feature of this framework is that it provides a controlled expansion of the amplitudes in terms of the masses of pseudoscalar mesons and charged leptons (p ∼ m π,K,ℓ /Λ χ , with Λ χ ∼ 4πF π ∼ 1.2 GeV), and the electromagnetic coupling (e). Electromagnetic corrections to (semi)-leptonic decays of K and π have been worked out to O(e 2 p 2 ) [8,9], but had never been pushed to O(e 2 p 4 ), as required for R e/µ . In this letter we report the results of our analysis of R e/µ to O(e 2 p 4 ), deferring the full details to a separate publication [10]. To the order we work, R e/µ features both model independent double chiral logarithms (previously neglected) and an a priori unknown low-energy coupling (LEC), which we estimate by means of a matching calculation in large-N C QCD. The inclusion of both effects allows us to further reduce the theoretical uncertainty and to put its estimate on more solid ground.
Within the chiral power counting, R e/µ is written as: The leading electromagnetic correction ∆ (P ) e 2 p 2 corresponds to the point-like approximation for pion and kaon, and its expression is well known [6,11]. Neglecting terms of order (m e /m ρ ) 2 , the most general parameterization of the NLO ChPT contribution can be written in the form ∆ (P ) which highlights the dependence on lepton masses. The dimensionless constants c  do not coincide with C 2,3 of Ref. [6], because their C 3 is not constrained to be m ℓ -independent.) Finally, depending on the treatment of real photon emission, one has to include in R e/µ terms arising from the structure dependent contribution to P → eν e γ [12], that are formally of O(e 2 p 6 ), but are not helicity suppressed and behave as The calculation -In order to calculate the various coefficients c (P ) i within ChPT to O(e 2 p 4 ), one has to consider (i) two-loop graphs with vertices from the lowest order effective lagrangian (O(p 2 )); (ii) one-loop graphs with one insertion from the NLO lagrangian [13] (O(p 4 )); (iii) tree-level diagrams with insertion of a local counterterm of O(e 2 p 4 ). In Fig. 1 we show all the relevant one-and two-loop 1PI topologies contributing to R e/µ . Note that all diagrams in which the virtual photon does not connect to the charged lepton line have a trivial dependence on the lepton mass and drop when taking the ratio of e and µ rates. We work in Feynman gauge and use dimensional regularization to deal with ultraviolet (UV) divergences.
By suitably grouping the 1PI graphs of Fig. 1 with external leg corrections, it is possible to show [10] that the effect of the O(e 2 p 4 ) diagrams amounts to: (i) a renormalization of the meson mass m P and decay constant F P in the one-loop result ∆ (P ) ). This correction can be expressed as the convolution of a known kernel with the vertex func- , once the Born term has been subtracted from the latter. Explicitly, in the case of pion decay one has (W = p − q, ǫ 0123 = +1) To the order we work, the form factors V 1 (q 2 , W 2 ), A i (q 2 , W 2 ) and F ππ V (q 2 ) have to be evaluated to O(p 4 ) in ChPT in d-dimensions. Their expressions are well known for d = 4 [12] and have been generalized to any d [10]. are parameter-free predictions of ChPT (they depend only on m π,K , F π , and the LECs L 9,10 determined in other processes [13]), c ). While this representation is valid at scales below m ρ (and gener-ates the correct single-and double-logs upon integration in d d q) it leads to the incorrect UV behavior of the integrand in Eq. 4, which is instead dictated by the Operator Product Expansion (OPE) for the V V P and V AP correlators. So in order to estimate the finite local contribution (dominated by the UV region) we need a QCD representation of the correlators valid for momenta beyond the chiral regime (T µν → T QCD µν ) . This program is feasible only within an approximation scheme to QCD. We have used a truncated version of large-N C QCD, in which the correlators are approximated by meromorphic functions, representing the exchange of a finite number of narrow resonances, whose couplings are fixed by requiring that the vertex functions π|V A|0 and π|V V |0 obey the leading and next-to-leading OPE behavior at large q [14]. This procedure allows us to obtain a simple analytic form for the local coupling (see Eq. 10).
Results -The results for c e/µ is fully inclusive on the radiative mode, so that ∆ (π) e 2 p 4 receives a contribution from the interference of T IB and T SD , and one also has to include the effect of ∆
As a check on our calculation, we have verified that if we neglect c CT 3 and pure two-loop effects, and if we use L 9 = F 2 /(2m 2 ρ ) (vector meson dominance), our results for c (π) 2,3,4 are fully consistent with previous analyses of the leading structure dependent corrections based on current algebra [6,17]. Moreover, our numerical value of ∆ (π) e 2 p 4 reported in Table II is very close to the corresponding result in Ref. [6], ∆ (π) e 2 p 4 = (0.054 ± 0.044) × 10 −2 . For completeness we report here the contribution to ∆ (π) e 2 p 6 induced by structure dependent radiation: Results for R (K) e/µ -In this case we have: where r 2 (K) V is the O(p 4 ) kaon charge radius. c  5.2 ± 0.4L 9 ± 0.01γ 4.3 ± 0.4L 9 ± 0.01γ c (P ) 3 −10.5 ± 2.3m ± 0.53L 9 −4.73 ± 2.3m ± 0.28L 9 c (P ) 4 (mµ) 1.69 ± 0.07L 9 0.22 ± 0.01L 9 of Eq. 3 (P = π, K). The uncertainties correspond to the input values L r 9 (µ = mρ) = (6.9 ± 0.7) × 10 −3 , γ = 0.465 ± 0.005 [16], and to the matching procedure (m), affecting only c  13/9 γ, by dropping the term proportional to log m 2 ρ /m 2 π , and by inter-changing everywhere else the label π with K (masses, ℓ π → ℓ K , etc.). c   Table I. Resumming leading logarithms -At the level of uncertainty considered, one needs to include higher order long distance corrections to the leading contribution ∆ e 2 p 2 ∼ −3α/π log m µ /m e ∼ −3.7%. The leading logarithms can be summed via the renormalization group and their effect amounts to multiplying R Conclusions -In Table II we summarize the various corrections to R (π,K) e/µ , which lead to our final results: In the case of R (K) e/µ we have inflated the nominal uncertainty arising from matching by a factor of four, to account for higher order chiral corrections of expected size ∆ e 2 p 4 × m 2 K /(4πF ) 2 . Our results have to be compared with the ones of Refs. [6] and [7] reported in the introduction. While R (π) e/µ is in good agreement with both previous results, there is a discrepancy in R (K) e/µ that goes well outside the estimated theoretical uncertainties. We have traced back this difference to the following problems in Ref. [7]: (i) the leading log correction ∆ LL is included with the wrong sign (this accounts for half of the discrepancy); (ii) the NLO virtual correction ∆ (K) e 2 p 4 = 0.058% is not reliable because the hadronic form factors modeled in Ref. [7] do not satisfy the QCD short-distance behavior.
In conclusion, by performing the first ever ChPT calculation to O(e 2 p 4 ), we have improved the reliability of both the central value and the uncertainty of the ratios R (π,K) e/µ . Our final result for R (π) e/µ is consistent with the previous literature, while we find a discrepancy in R (K) e/µ , which we have traced back to inconsistencies in the analysis of Ref. [7]. Our results provide a clean basis to detect or constrain non-standard physics in these channels by comparison with upcoming measurements.