Radiative corrections to nucleon weak charges and Beyond Standard Model impact
RRadiative corrections to nucleon weak charges and Beyond Standard Model impact
Leendert Hayen
1, 2, 3, ∗ Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA Triangle Universities Nuclear Laboratory, Durham, North Carolina 27708, USA Instituut voor Kern- en Stralingsfysica, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium (Dated: February 9, 2021)The nucleon axial charge is a central ingredient in nuclear and particle physics, and a key ob-servable in precision tests of the electroweak Standard Model sector and beyond. We report onthe first complete calculation of its electroweak quantum corrections up to O ( α ), using a combina-tion of current algebra techniques and QCD sum rules. We find a substantial enhancement due tothe weak magnetism contribution in the elastic channel, and include higher-twist and target masscorrections at low Q to find ∆ AR = 0 . VR = 0 . g A = 1 . > σ shift away from the commonly quoted value. We use this newresult to set constraints on exotic right-handed currents by comparing to lattice QCD results, andresolve a double-counting issue in the | V ud | extraction from mirror decays. Precision studies of neutron β decay put stringent testson extensions of the charged weak sector at the TeVscale with precision equal to or exceeding that of collid-ers [1–6]. In particular, studies of the unitarity require-ment of the Cabibbo-Kobayashi-Maskawa (CKM) matrixand comparison to lattice QCD calculations have showntremendous reach in several Beyond Standard Model(BSM) channels [7, 8]. This is possible only throughexquisite control of strong and electroweak quantum cor-rections to the tree-level vector-axial vector ( V - A ) weakinteraction [9, 10].Even though the vector current is protected from thestrong interaction [11], QCD effects shift the nucleonaxial-vector coupling constant, g A , away from unity. Intypical charged current processes such as (nuclear) β de-cay, however, these bare quantities are not observableexperimentally as both g V and g A contain percent-levelcorrections due to electroweak radiative processes g L → g L (1 + ∆ LR ) (1)with L = V, A . Up to now, however, only ∆ VR has re-ceived much attention due to its importance in the | V ud | extraction from pure vector transitions, with V ud the up-down CKM matrix element [12–15]. Traditionally, formixed transitions such as the neutron one ignores thisdifference and defines the neutron lifetime, τ n , by ex-tracting the vector radiative corrections as follows(1 + ∆ VR ) (cid:2) g V f V + 3( g eff A ) f A (cid:3) = K | V ud | G F τ n (2)with K a collection of constants, G F the Fermi couplingconstant and f V,A so-called phase space factors. Equa-tion 2 implies an ‘operational’ definition of g A [10] g eff A = g QCD A (cid:2) (cid:0) ∆ AR − ∆ VR (cid:1) / (cid:3) [1 − (cid:15) R ] , (3)where we added a possible contamination from a BSMright-handed coupling, (cid:15) R ∝ M W / Λ BSM [16]. This def-inition can be made in the tree-level Lagrangian so that all experimental observables probe g eff A instead. Withthe rise of precision lattice QCD (LQCD) calculationsfor g QCD A [17–19], comparison with experimental g eff A re-sults was recognized as an extremely clean way of probing (cid:15) R [8]. A correct assessment, however, relies on a cleanseparation of electroweak and strong effects.In this Letter, we present a first complete StandardModel calculation of electroweak radiative corrections(EWRC) to the neutron axial vector charge, g A , and anew result for ∆ VR . Using a combination of current alge-bra techniques with QCD sum rules, we find∆ AR = 0 . , (4a)∆ VR = 0 . VR results can bemade in agreement [14, 15]. As a consequence, we canfor the first time extract g A ≡ g eff A AR − ∆ VR ) / . , (5)which is a > σ departure from its quoted value, g eff A =1 . | V ud | ex-traction from T = 1 / | V ud | value from superallowed 0 + → + decays [21] fromEq. (4b), | V ud | mirror = 0 . , (6a) | V ud | super = 0 . . (6b)We summarize in this Letter the essential features of ouranalysis, and defer details to a longer paper [22].The small momentum exchange in β decay ( q ∼ MeV)relative to the W -boson mass limits the number of dia-grams that contribute to order O ( G F α ) with α the fine-structure constant, summarized in the seminal work by a r X i v : . [ h e p - ph ] F e b γ Z W e νWγ Z Figure 1. O ( α ) radiative corrections that give rise to differ-ences in vector and axial vector transitions. Sirlin [12]. Further, when evaluating Eq. (3) we needto worry only about those diagrams which depend onthe vector or axial-vector nature of the transition. Dif-ferences are anticipated a priori in the vertex and boxcorrection diagrams, shown in Fig. 1. It is straightfor-ward, however, to show that only diagrams containingvirtual photons contribute to first order, as those witheither Z or W bosons are infrared convergent and are assuch either higher-order or common to both ∆ LR .The total EWRC can be parametrized according to[15, 23] ∆ LR = 0 . . L NP + 1 . L P (7)where ( N ) P designates (non-)perturbative contributions,respectively. The first term combines large-log termsoriginating from high-momentum behaviour common tovector and axial transitions [24], and includes the asymp-totic ZW box contribution [15].Starting with the photonic vertex correction, wecan use the on mass-shell renormalization theoremwhich states that the modification of a vertex function (cid:104) p f | Γ µ | p i (cid:105) = F µ ( p f , p i ) due to an interaction term δ L inthe Lagrangian is [12, 14] δF µ ( p f , p i ) = lim ¯ q → q iT µ (¯ q, p i , p f ) , (8)where q = p i − p f and the tensor T µ = T µλλ (¯ q, p f , p i ) is T µλλ = (cid:90) d x e i ¯ qx (cid:104) p f | T { J µW ( x ) δ L (0) }| p i (cid:105) − B µ , (9)where J µW is the weak current and the final term subtractsthe poles from the mass renormalization to make T µ pole-free by construction. In Fig. 1 the interaction is δ L (0) = α (2 π ) (cid:90) d kk (cid:90) d y e iky T { J λγ ( y ) J γλ (0) } , (10)where J γλ is the photonic current. Using the identity iT µ = − ¯ q ν ∂∂ ¯ q µ iT ν + ∂∂ ¯ q µ ( i ¯ q ν T ν ) , (11)a partial integration of ¯ q ν T ν unlocks equal-time current commutators through ∂∂x ν T (cid:8) J νW ( x ) J λγ ( y ) J γλ (0) (cid:9) = T (cid:26) ∂ ν J νW ( x ) J λγ ( y ) J γλ (0) + δ ( x ) (cid:2) J W ( x ) , J λγ (0) (cid:3) J γλ ( y )+ δ ( x − y ) (cid:2) J W ( x ) , J λγ ( y ) (cid:3) J γλ (0) (cid:27) . (12)The commutators can be evaluated immediately usingthe current algebra relationship (cid:2) J W ( t, x ) , J µγ ( t, y ) (cid:3) = J µW ( x ) δ (3) ( x − y ) , (13)which is conserved even in the presence of strong inter-action effects [25, 26]. From Eq. (12) we obtain a three-current, denoted D γ , and two-current piece. The lattercan be shown to cancel with an opposite contributionfrom the γW box diagram and we continue with the for-mer. Since it depends on the divergence of the weakcurrent, the vector part of which is conserved, it van-ishes for vector transitions and we anticipate a non-zerocontribution for an axial transition.In the high-momentum region ( k (cid:29) ) the di-vergence vanishes also for an axial transition due to chiralsymmetry - or equivalently using the partially conservedaxial current. This can also be checked explicitly [22]through, e.g., an operator product expansion or, in thiscase equivalently, a Bjorken-Johnson-Low limit [27–29].In the low energy limit we can describe the photon andweak coupling for on-shell nucleons, N , in the isospinformalism J γ,Iµ = i ¯ N (cid:20) F I γ µ + i F I M σ µν q ν (cid:21) τ I N, (14a) J Wµ = i ¯ N (cid:104) g V γ µ + i g M M σ µν q ν + g A γ µ γ (cid:105) τ ± N (14b)with I = 0 , F , ( q ) are the charge and magnetic form fac-tors, g i ( q ) are weak form factors, g M (0) = κ p − κ n =3 .
706 is the so-called weak magnetism contribution and τ are the SU (2) generators. By enforcing G -parity [30],the photon currents in D γ are restricted to be either bothisoscalar or both isovector. Using the fact that ∂ µ J Wµ must be proportional to τ ± it follows after some algebrathat D Born γ ≈ (cid:20) ( F ) − ( F ) (cid:21) g A M [ ¯ N (cid:48) γ τ ± N ] × (cid:90) d kk Λ Λ − k k + i(cid:15) . (15)where Λ is a Pauli-Villars mass regulator. In the isospinlimit and for small momenta we have F (0) = F (0) = 1and the contribution vanishes also at low energy.Moving on, the matrix element for the photonic boxdiagram in Fig. 1 can be written down as [12] M γW = αG F V ud / π (cid:90) d kk ¯ e (2 l µ − γ µ (cid:54) k ) γ ν (1 − γ ) ν [ k − l · k ][1 − k /M W ] T γWµν (16)where k and l are virtual loop and outgoing lepton four-momenta, respectively, and T γWµν describes the blob inFig. 1 T γµν ( k ) = (cid:90) d xe ikx (cid:104) p f | T { J γµ ( x ) J Wν (0) }| p i (cid:105) . (17)Equation (16) can be reduced by expanding the γ matrixproduct and using Ward-Takahashi identities k µ T γWµν = i (cid:104) p f | J Wν | p i (cid:105) (18a) k ν T γWµν = i (cid:104) p f | J Wµ | p i (cid:105) + q ν T µν + i (cid:90) d xe i ( k − q ) · x (cid:104) p f | T { ∂ ν J Wν J γµ }| p i (cid:105) . (18b)where we used the conservation of the QED current.Terms proportional to the tree-level interaction drop outin Eq. (3), and after neglecting terms of O ( αq ) and com-bining other finite terms with other diagrams, contribu-tions arise from two terms M γW = αG F V ud π L α (cid:90) d kk [ i(cid:15) µλνα k λ T γWµν − D α ]1 − k /M W (19)where we have taken l, m e → D α , cor-responds to the second line in Eq. (18b) and dependson the divergence of the weak current responsible for thetransition. This resembles the three-current correlatordiscussed above and so in principle introduces a differ-ence in ∆ LR . Analogous to our discussion of the vertexcorrection above, however, the high-energy behaviour of D α can be shown to vanish due to chiral symmetry.The low-energy behaviour, on the other hand, can betreated similarly to Eq. (15) by once more invoking G -parity meaning only the isovector photons contribute andwe find D Born µ ≈ − iπδ ( k ) F ¯ N [ τ z ∂ ν J νW + ∂ ν J νW τ z ] N, (20)neglecting the O (1 /M ) magnetic term. Since ∂ ν J Wν ∝ τ ± the contribution vanishes by crossing symmetry.This leaves the analogous contribution of the γW boxto axial transitions as the one that has occupied researchfor vector transitions over the past half-century. Recentwork [13–15] has shown that contributions from physicsat intermediate scales are significant and extend into theelastic regime. Going forward, we simplify the notation of the remaining term in the box contribution of Eq. (19)by introducing a general function F A,V ( Q ) [31] (cid:3) γW = α π (cid:90) ∞ dQ M W Q + M W F V,A ( Q ) (21)where Q = − k is the virtuality. The Born contributioncan be written down easily using Eqs. (14a)-(14b) andsome algebra results in F V Born = 1 Q g A ( F + F ) P (22a) F A Born = 1 Q (cid:20) g V ( F + F ) + g M F (cid:21) P (22b)neglecting O (1 /M ) terms and where P is the kinematiccontribution from the nucleon propagators P = 1 + 2 (cid:112) M /Q [1 + (cid:112) M /Q ] (23)We can once more use, e.g., G -parity to show that onlyisoscalar photons contribute, which is a well-known result[32]. We perform the integral in Eq. (21) numericallyover the full range using experimental data for g V ( Q )and g M ( Q ) [33] and input from Ref. [34] for g A ( Q ) tofind V NP = 0 . απ (24a) A NP = [0 . . απ (24b)where we explicitly separated contributions due to g V and g M in the second line, and assign uncertainties as inRef. [14]. Interestingly, the weak magnetism contribu-tion clearly dominates for axial transitions and providesa much larger enhancement than anticipated.For the remaining momentum range we take inspira-tion from the idea originally by Marciano and Sirlin [31]and its recent extension [15]. We may decompose Eq.(17) into structure functions after spin summation (cid:88) spins T µν asy −→ i(cid:15) µναβ k α p β p · k ) F ( ν, Q )+ i(cid:15) µναβ q α p · k (cid:20) S β g ( ν, Q ) + (cid:18) S β − p β S · kp · k (cid:19) g ( ν, Q ) (cid:21) , (25)with S β the four-polarization, ν = p · k/M and writingonly terms that survive the contraction with the Levi-Cevita tensor in Eq. (19). For an axial transition, wemay relate Eq. (17) to the Bjorken (Bj) sum rule ofpolarized electron scattering through an isospin rotation,i.e. γW → γγ , specifically (cid:90) dx [ g p ( x, Q ) − g n ( x, Q )] = g A (cid:20) − α g ( Q ) π (cid:21) , (26) Q [GeV ]0.000.050.100.150.20 p n BjSR FitData g / Figure 2. Parametrization of the PBjSR and running coupling α g ( Q ) /π using the pQCD MS parametrization, Eq. (26),for Q > Q = 0 .
910 GeV and the LFH result of Eq. (28)for Q ≤ Q , together with experimental data between 0.05GeV and 3 GeV . Figure adopted from Ref. [35]. where x = Q / M ν and α g ( Q ) is an effective QCDcoupling constant. In the isospin limit, the running ofEq. (26) is identical to that of T µνγ for axial transitions,and so F A DIS ( Q ) ≈ Q (cid:20) − α g ( Q ) π (cid:21) . (27)We can take advantage of the substantial amount of ex-perimental data of Eq. (26) down to Q ≈ . [35], thereby also neatly taking into account low energycontributions shown in Fig. 2.Similar to Refs. [15, 35] we stitch together expres-sions from perturbative QCD (pQCD) at high Q [36]and a light-front holography (LFH) motivated expressionat low Q [37]1 − α g ( Q ) π Q Cur-rent and Future Status of the First-Row CKM Unitar-ity workshops for productive discussions related to thismanuscript. I acknowledge support by the U.S. NationalScience Foundation (PHY-1914133), U.S. Department ofEnergy (DE-FG02-ER41042), the Belgian Federal Sci-ence Policy Office (IUAP EP/12-c) and the Fund for Sci-entific Research Flanders (FWO). Q [GeV ]0.00.51.01.52.02.5 Q F ( Q ) / ( + Q / M W ) Gamow-TellerBorn GTFermiBorn F Figure 3. Summary of the results for vector (Fermi) andaxial (Gamow-Teller) transitions including target mass cor-rections, shown as in Ref. [43], using the relationship F ( Q ) = 12 M (1 , Q ) /Q , where M is the Nachtmann mo-ment. Dashed lines show the contribution of the Born ampli-tude. R L WH , S = 14 TeV ( v ) L Z pole g A FLAG'19Most precise g QCDA @ 0.1%( AR VR )/2 Figure 4. Current limits (68% C.L.) on left and right-handedcouplings interpreted in the SMEFT, showing Z -pole (blue)[46, 47], LHC (black) [48], LQCD results from FLAG’19(green) [19] and Ref. [45] (orange). In red we show antic-ipated limits when g A reaches 0.1% on the lattice. 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