Joint Lattice QCD–Dispersion Theory Analysis Confirms the Quark-Mixing Top-Row Unitarity Deficit

PHYSICAL REVIEW D 101, 111301(R) (2020) Rapid Communications

Joint lattice QCD–dispersion theory analysis confirms the quark-mixing top-row unitarity deficit

Chien-Yeah Seng ,1,* Xu Feng ,2,3,4,5 Mikhail Gorchtein,6,7,8 and Lu-Chang Jin 9,10 1Helmholtz-Institut für Strahlen- und Kernphysik and Bethe Center for Theoretical Physics, Universität Bonn, 53115 Bonn, Germany 2School of Physics, Peking University, Beijing 100871, China 3Collaborative Innovation Center of Quantum Matter, Beijing 100871, China 4Center for High Energy Physics, Peking University, Beijing 100871, China 5State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China 6Helmholtz Institute Mainz, Mainz D-55099, Germany 7GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt 64291, Germany 8Johannes Gutenberg University, Mainz D-55099, Germany 9RIKEN-BNL Research Center, Brookhaven National Lab, Upton, New York 11973, USA 10Physics Department, University of Connecticut, Storrs, Connecticut 06269-3046, USA

(Received 1 April 2020; accepted 26 May 2020; published 3 June 2020)

Recently, the first ever lattice computation of the γW-box radiative correction to the rate of the semileptonic pion decay allowed for a reduction of the theory uncertainty of that rate by a factor of ∼3. A recent dispersion evaluation of the γW-box correction on the neutron also led to a significant reduction of the theory uncertainty, but shifted the value of Vud extracted from the neutron and superallowed nuclear β decay, resulting in a deficit of the Cabibbo-Kobayashi-Maskawa (CKM) unitarity in the top row. A direct lattice computation of the γW-box correction for the neutron decay would provide an independent cross- check for this result but is very challenging. Before those challenges are overcome, we propose a hybrid analysis, converting the lattice calculation on the pion to that on the neutron by a combination of dispersion theory and phenomenological input. The new prediction for the universal radiative correction to free and β ΔV 0 02477 24 bound neutron -decay reads R ¼ . ð Þ, in excellent agreement with the dispersion theory result ΔV 0 02467 22 R ¼ . ð Þ. Combining with other relevant information, the top-row CKM unitarity deficit persists.

DOI: 10.1103/PhysRevD.101.111301

Universality of the weak interaction, conservation of quantum chromodynamics (QCD), affects the value of vector current and completeness of the Standard Model jVudj extracted from the free neutron and superallowed (SM) finds its exact mathematical expression in the require- nuclear β decays. In a series of recent papers, this RC was ment of unitarity of the Cabibbo-Kobayashi-Maskawa reevaluated within the dispersion relation technique [3–5]. (CKM) matrix. Of various combinations of the CKM matrix In particular, Ref. [3] observed that the universal, free- elements constrained by unitarity, the top-row constraint is neutron correction received a significant shift, later con- the best known both experimentally and theoretically. firmed qualitatively by Ref. [6]. This shift is the main cause Δu ≡ 2 2 2 − 1 Δu The 2018 value, CKM jVudj þjVusj þjVubj ¼ of the current apparent unitarity deficit, CKM ¼ −0.0006ð5Þ [1,2] is in good agreement with zero required −0.0016ð6Þ (using an average of Vus from Kl2 and Kl3 in the SM, putting severe constraints on beyond Standard decays [2]). The slight increase in the uncertainty is due to Model (BSM) physics. nuclear structure effects [4,5]. Δu β 10−4 Notably, the main source of the uncertainty in the CKM Since in superallowed decays one aims for a constraint is theoretical: the γW-box radiative correction precision, it is highly desirable to assess the uncertainty and (RC), prone to effects of the strong interaction described by possible, unaccounted for, systematic effects in the nonperturbative regime of QCD in a model-independent way. A common limitation of the studies above is the lack of *[email protected] experimental data to directly constrain the hadronic matrix element relevant to the RC. By means of isospin symmetry, Published by the American Physical Society under the terms of Ref. [3] relates the input to the dispersion integral at low the Creative Commons Attribution 4.0 International license. 2 Further distribution of this work must maintain attribution to photon virtuality Q to a very limited and imprecise set of the author(s) and the published article’s title, journal citation, data on neutrino scattering on light nuclei from the 1980s and DOI. Funded by SCOAP3. [7,8]. The analysis of Ref. [6] consists of pure model studies.

2470-0010=2020=101(11)=111301(6) 111301-1 Published by the American Physical Society SENG, FENG, GORCHTEIN, and JIN PHYS. REV. D 101, 111301 (2020)

A complete change of landscape is expected following resulting from the product between the axial charged weak the first direct application of the lattice QCD to RC in current and the isoscalar electromagnetic current: leptonic meson decays, K → μνμ and π → μνμ [9]. Very ϵμναβ recently, the first ever direct lattice calculation of the RC in i pαqβ ð0Þ 2 F ðx; Q Þ semileptonic β decay was presented, where the relevant 2p · q 3H hadronic matrix element responsible for the γW-box dia- 1 X 2π 4δð4Þ − gram in the pion is calculated to high precision as a function ¼ 8π ð Þ ðp þ q pXÞ of Q2 [10]. As a result, the theory uncertainty of the X π π− → π0 ν¯ ð0Þμ ν e3ð e eÞ decay rate is reduced by a factor of 3. × hHfðpÞjJem jXihXjðJWÞAjHiðpÞi: ð3Þ While theoretically very clean, πe3 is not the easiest avenue ∼10−8 2 − 2 to extract Vud due to its tiny branching ratio . Above, MH is the averagep massffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of Hi, Hf, Q ¼ q , Nonetheless, it provides useful information about the 2 2 1 4 2 2 2 x ¼ Q = p · q, and rH ¼ þ MHx =Q , and the fac- involved nonperturbative dynamics, especially its low-Q2 H tor Fþ defines the normalization of the tree-level hadronic behavior and its smooth transition to the perturbative matrix element of the vector charged weak current: regime. Using the same method or other approaches such μ H2 μ as Feynman-Hellmann theorem [11,12], a first-principle hHfðpÞjðJWÞVjHiðpÞi ¼ VudFþ p : ð4Þ calculation of the RC to the free neutron β decay, while very − pffiffiffi n 1 π 2 challenging, is expected to be performed in the near future. By isospin symmetry, Fþ ¼ and Fþ ¼ . — δVA In this paper, we perform a combined lattice QCD The quantity γW;H is the source of the largest theory phenomenological analysis. Making use of a body of uncertainty of the RC in the πe3, free neutron β decay, and hadron-hadron scattering data, known meson decay widths the universal RC in superallowed nuclear β decays, and has and the guidance of Regge theory and vector dominance, long been the limiting factor for the precise determination δVA along with constraints from isospin symmetry, analyticity, of Vud. To obtain γW;H we need to know the Nachtmann and unitarity, we are able to unambiguously relate the input ð0Þ 2 2 2 moment M ð1;Q Þ as a function of Q . At large Q , the into the dispersion integral for the γW-box RC on the pion 3H product of currents in Eq. (3) is given by the leading-order and on the neutron. Fixing the strength of the pion matrix (LO) operator product expansion (OPE) and the perturba- element from the lattice, we thus obtain an estimate of an tive QCD (pQCD) corrections. The LO OPE pQCD analogous matrix element on the neutron, in accord with all þ result is independent of the external state H and is known the aforementioned physics constraints. 4 up to order OðαsÞ [13,14], with αs the strong coupling We start by writing down the dispersive representation of 2 the contribution of the γW box diagram (see Fig. 1) to the constant. However, at low Q the structure function β ð0Þ 2 rate of the Fermi part of a semileptonic decay process of F3Hðx; Q Þ depends on details of different on-shell inter- Hi → Hfeν¯e [3,4]: mediate states jXi that dominate different regions of 2 Z 2 2 fx; Q g (see Fig. 2 of Ref. [3] for the explanation). 3α ∞ dQ M 0 δVA W ð Þ 1 2 Also, the transition point between perturbative and non- γW;H ¼ π 2 2 2 M3Hð ;Q Þ; ð1Þ 0 Q MW þ Q perturbative regime is a priori unknown, or uncertain. ð0Þ 1 2 where α is the fine-structure constant. The above definition The first calculation of M3π ð ;Q Þ on the lattice in 2 Ref. [10] serves as an important step in addressing the of the γW-box correction corresponds to a shift jVudj → 2 questions above. Its result is presented in Fig. 2 as a jV j ð1 þ δVA Þ, affecting the apparent value of V ud γW;H ud function of Q2.AtlowQ2 where the integral (1) is strongly extracted from an experiment. The function weighted, lattice provides an extremely precise description Z 0 4 1 1 2 ð0Þ 2 ð Þ 1 2 2 ð0Þ 2 þ rH F3Hðx; Q Þ of M3π ð ;Q Þ, but its uncertainty increases at large Q due M 1;Q dx 2 2 2 3Hð Þ¼3 1 2 H ð Þ to the discretization error. Fortunately, at Q > 2 GeV 0 ð þ rHÞ Fþ there exists very precise data for the first Nachtmann ν ν¯ stands for the first Nachtmann moment of the (spin- moment of the parity-odd structure function F pþ p mea- ð0Þ 2 3 independent) parity-odd structure function F3Hðx; Q Þ, sured in the ν=ν¯ scattering on light nuclei by the CCFR Collaboration [15,16]. Their good agreement with pQCD prediction indicates a smooth transition to the perturbative regime at Q2 > 2 GeV2, which also implies that these data, ð0Þ 2 1 upon simple rescaling, can be converted to M3π ð1;Q Þ.

1 νpþν¯p Strictly speaking, the pQCD correction to F3 differs from ð0Þ O α3 that of F3H at ð s Þ, but such a difference is numerically FIG. 1. The γW box diagram in free neutron decay. insignificant at Q2 > 2 GeV2.

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0.08 nonresonant part of the ππ continuum in the p-wave; however, this partial wave is known to be entirely domi- 0 0.06 nated by the ρ resonance up to the KK¯ threshold. Comparing the parametrizations of Eqs. (5), (6),we 0.04 make an important observation. Among the various contributions there are the process-specific ones that reside in the lower part of the spectrum (elastic, resonance and low- 0.02 energy continuum). They have to be explicitly calculated for the pion and for the nucleon and cannot be related to 0.00 each other. On the other hand, the asymptotic contributions 0 1 2 3 4 5 (Regge and pQCD) are universal. This is the central point of our analysis. Universality of the OPE is straightforward. The FIG. 2. Comparison between the lattice calculation of ð0Þ ð0Þ ð0Þ 1 2 only difference between F3N;pQCD and F3π;pQCD is in the M3π ð ;Q Þ (blue band), the prediction from LO OPE with ð0Þ 4 2 normalization of the isospin states, thus F3π ¼ Oðαs Þ pQCD corrections (red curve) and the low-Q CCFR data ;pQCD [15,16] (green points). π− n ð0Þ ðFþ =FþÞF3N;pQCD. Universality is among the central predictions of Regge On the other hand, below 2 GeV2 effects of generic theory. It dictates that the upper and lower vertices in the ρ ρ þ π− → γ π0 higher-twist terms start to show up, and the LO OPE Regge -exchange amplitudes T ðW þ þ Þ and þ ρ þ → γ pQCD prediction disagrees significantly with the lattice T ðW þ n þ pÞ in Fig. 3 factorize, so that, e.g., result. ρ ρ ρ T − 0 We shall describe how the lattice result for δVA on the Wþþπ →γþπ Tππ→ππ TπN→πN γW Rπ ¼ ρ ¼ ρ ¼ ρ ; ð7Þ δVA =N T T T pion can be used to improve our understanding of γW on Wþþn→γþp πN→πN NN→NN the neutron. First, for the neutron we parametrized the ρ ρ ρ ð0Þ ð0Þ where Tππ→ππ;T ;T stand for the amplitudes structure function F (hence, also M )as[3,4]: πN→πN NN→NN 3N 3N in elastic ππ; πN;NN scattering in the channel that corre- ( 0 0 0 sponds to an exchange of the quantum numbers of the ρ Fð Þ Fð Þ Fð Þ ;Q2 ≤ Q2; ð0Þ ð0Þ 3N;res þ 3N;πN þ 3N;R 0 meson in the t-channel. Regge factorization has been tested F3 ¼ F3 þ N N;el ð0Þ 2 2 on global data sets for elastic pion, pion-nucleon, and F3 ;Q≥ Q0; N;pQCD nucleon-nucleon scattering. ð5Þ This leads to a prediction based on Regge universality, 2 2 2 αρ where Q0 ≈ 2 GeV is the scale above which the LO 0 Q 0 Fð Þ ðx; Q2Þ¼R−1 Fn AðQ2ÞfNðW2Þ OPE þ pQCD description is valid. Above, we isolated 3N;R π=N þ th x the contributions from the elastic intermediate state (el) 2 ρ α0 fixed by the nucleon magnetic [17,18] and axial elastic ð0Þ 2 π− 2 π 2 Q F3π;Rðx; Q Þ¼Fþ AðQ ÞfthðW Þ ; ð8Þ form factor [19], from the nonresonance πN continuum x (πN) in the low-energy region, from the N resonances αρ 0 477 (res),2 and the Regge contribution (R) that allow to with 0 ¼ . [20]. Here we define the threshold H Θ 2 − 2 1 − 2 − 2 Λ2 economically describe the multihadron continuum. function fth ¼ ðW Wth;HÞð exp½ðWth;H W Þ= Þ, 2 2 2 1 − 1 Λ 1 2 In a similar way, we parametrize the pion structure where W ¼ MH þ Q ðx Þ and ¼ GeV [21]. The function as threshold parameter Wth;H characterizes the threshold for ( the multihadron contributions. In Ref. [3] we fixed ð0Þ ð0Þ 2 ≤ 2 ð0Þ F3π;res þ F3π;R;Q Q0; F3π ¼ ð6Þ ð0Þ 2 ≥ 2 F3π;pQCD;QQ0:

We note the absence of the elastic and the low-energy continuum contributions. The former is identically zero because the axial current does not couple to the spin-0 pion ground state. The latter would correspond to the ð0Þ FIG. 3. The Regge-exchange contribution to F3 for neutron 2Δ resonances do not contribute due to the isoscalar nature of and pion. The vertical propagator represents the exchange of the the photon. ρ-trajectory.

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2 Wth;N ¼ mN þ Mπ, such that the threshold function N ≈ 1 ≳ 2 5 fth for W . GeV. In the pion sector, one expects Wth;π to lie between Mρ and 1.2 GeV, the scale above which Regge description is valid [22]. In this work we choose ≈ 1 Wth;π GeV, and account for the uncertainty due to its variation between the two boundaries. The function AðQ2Þ describes the interaction at the upper half of Fig. 3 and is, within the Regge framework, common for neutron and pion. It is generally unknown but is now completely fixed by the lattice result plotted in Fig. 2— upon subtracting the resonance contribution. With these ingredients, the ratio of the first Nachtmann moments of the ð0Þ 1 2 Regge contributions reads, FIG. 5. The new determination of M3N;Rð ;Q Þ (blue band R with solid boundaries) is compared to the result of Ref. [3] 1 2 −αρ ð0Þ 2 1 þ rN N 2 0 (orange band with dashed boundaries), the pQCD prediction (red M3 Rð1;Q Þ 1 0 dx 1 2 fthðW Þx N; R ð þrN Þ curve), and the CCFR data [15,16] (green points). In the bottom- 0 ¼ 1 2 −αρ : ð9Þ ð Þ 2 1 þ rπ π 2 0 ð0Þ 1 Rπ=N 0 dx 2 f ðW Þx M3π;Rð ;Q Þ ð1þrπÞ th right subview, the resonance contribution to M3π;res (red dashed curves and band) is shown along with the full lattice calculation ð0Þ ð0Þ To fully specify the parametrization of F3π we turn M3π (blue solid curves and band). now to the resonance contribution depicted in Fig. 4. Its strength is derived from the following effective Lagrangian the full lattice calculation (blue curves and band). This densities [23], smallness guarantees that the removal of the nonuniversal egργπ resonance contribution does not introduce an uncontrolled L 2 a μν ˜ πa ργπ ¼ 2 FωðQ ÞðFρÞ Fμν systematic uncertainty in our analysis. Mρ ð0Þ 1 2 With Eq. (9), M3N;Rð ;Q Þ could now be directly ga1ρπ abc a μν b c L ρπ ¼ ε ðFρÞ ðF Þ π obtained from the lattice results and the rescaling factor a1 2M a1 μν a1 Rπ=N. A recent analysis of ππ scattering [22] made the 2 factorization test with respect to πN analysis and found gMa1 2 − μ − L þ π n Wa1 ¼ wa1 Fa1 ðQ ÞVudWμ a1 þ H:c:; ð10Þ (omitting the isospin factor F =F ), 2gρ þ þ ρ Tππ→ππ 1 35þ0.21 where we explicitly include the vector dominance form ρ ¼ . −0.26 : ð11Þ 2 1 2 2 −1 TπN→πN factors Fω;a1 ðQ Þ¼½ þ Q =Mω;a1 . The couplings are obtained as follows: jgργπj¼0.645ð43Þ from the ρ → γπ On the other hand, the OPE suggests that Rπ=N ¼ 1 in the decay width, jg ρπj is allowed to vary from 0 all the way to a1 perturbative regime (note also the ρ coupling universality 5.7(1.3) which saturates the full a1 decay width [2], and − − hypothesis in the hidden local symmetry [25]). Therefore, w =gρ 0.133 from the τ → a ντ decay width [24]. 2 j a1 j¼ 1 to ensure a continuous matching at all Q values we allow ð0Þ 2 Finally, the overall sign of M3π;res is fixed by requiring that Rπ=N to slightly depend on Q , it matches the sign of the ππ contribution calculated in 2 2 2 chiral perturbation theory at small Q . Numerically, the size Rπ=NðQ Þ¼Rπ=Nð0ÞþbQ ; ð12Þ ð0Þ ≤ 10% of M3π;res is rather small, of the total, as can be seen where Rπ 0 is fixed by Eq. (11), and b is fixed by in the bottom-right subview of Fig. 5 where the resonance =Nð Þ ð0Þ estimate (red dashed curves and band) is plotted along with requiring M3N;R to reproduce the CCFR datum at the 2 2 matching point Q0 ¼ 2 GeV ,

ð0Þ 1 2 0 0667 35 M3N;Rð ;Q0Þ¼ . ð Þ: ð13Þ

−0 076þ0.100 −2 The result reads b ¼ . −0.072 GeV . ð0Þ 1 2 With the prescription above we fully fix M3N;Rð ;Q Þ at 2 ð0Þ 2 low Q using the lattice curve of M3π ð1;Q Þ. The result is 2 ρ ð0Þ shown in Fig. 5, with the uncertainties from Rπ=NðQ Þ and FIG. 4. The -exchange contribution to F3π . The propagators 2 of ω and a1 mesons indicate the vector-meson-dominance form Wth;π added in quadrature. Integrating over Q gives an δVA factors. updated estimate of the Regge contribution to γW;N:

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δVA 1 12 16 9 3 10−3 Δu ð γW;NÞR ¼ . ð Það Þbð Þc × ; ð14Þ TABLE I. Summary of CKM for different cases. u u V Δ with Kl2 Δ with Kl3 where the uncertainties are from (a) the pion-nucleon j udj CKM CKM −0 0012 4 −0 0021 4 matching, including the rescale factor Rπ=N and the lattice w/o NNC 0.97365(15) . ð Þ . ð Þ uncertainty, (b) the Regge parameterization and (c) the w=NNC 0.97366(33) −0.0012ð7Þ −0.0021ð7Þ resonance subtraction. Our result is in excellent agreement δVA 1 02 16 10−3 with the previous determination ð γW;NÞR ¼ . ð Þ× Our new analysis implies ΔR ¼ 0.04002ð24Þ (ΔR is the [3]. One can also study the effect of varying the perturbative V 2 sum of Δ and the Sirlin’s function [29]), which leads to matching point by evaluating the Q -integral in Eq. (1) R 2 2 jV j¼0.97297ð58Þ given the neutron lifetime τ ¼ between 2 GeV and 3 GeV using the CCFR data instead ud n 879.7ð8Þ s [30–32] and the axial-vector ratio λ ¼ of the pQCD expression. That gives an insignificant extra −1.27641ð56Þ [33,34]. The result is consistent with that uncertainty of 1 × 10−5, confirming the robustness of our from the superallowed nuclear β decays. error analysis. Finally, we discuss the current situation of the top-row We next discuss the impact of this result on the extraction CKM unitarity. There are two different measurements of of Vud. From superallowed nuclear β decay, we have [26]: Vus, using Kl2 [2] and Kl3 [35] decay separately:

2 2984.43 s jVudj ¼ V ; ðsuperallowedÞð15Þ 0 2253 7 0 2233 6 F 1 Δ jVusjKl2 ¼ . ð Þ; jVusjKl3 ¼ . ð Þ: ð18Þ tð þ RÞ

F ΔV 2σ where t is the ft-value corrected by nuclear effects, R ¼ They disagree with each other at level, Kl3 giving a δVA … smaller jV j which leads to a larger unitarity violation. γW;N þ is the nucleus-independent RC that contains the us largest theoretical error. In this paper we update the Regge This, however, depends critically on the existing lattice 0π− δVA calculation of the Kπ vector form factor fK ð0Þ which is contribution to γW;N according to Eq. (14). Meanwhile, we þ also update the pQCD contribution above 2 GeV2 from recently questioned by theory [36] and a new lattice paper O α3 O α4 ΔV [37]. Another possible issue is the electromagnetic RC in ð s Þ to ð sÞ [13,14], which reduces R by mere −5 Kl3, which may be reanalyzed in a dispersive approach 1 × 10 . As a result we obtain a slight shift upward with [38]. We summarize the resulting Δu from different respect to the result of Ref. [3]: CKM combinations in Table I. In short, we observe a ð3 − 5Þσ 1 7 − 3 σ ΔV ¼ 0.02467ð22Þ → 0.02477ð24Þ: ð16Þ unitarity violation excluding the NNC, and ð . Þ R violation with the NNC. Our results can be tested with a ΔV future, direct lattice calculation of the γW-box on the The recent Ref. [6] estimated a lower value, R ¼ 0.02426ð32Þ, based on the assumption that the full neutron. After that, the emphasis should be shifted to a Nachtmann moment should follow the perturbative curve reassessment of the nuclear structure corrections that enter 2 2 the analysis of superallowed nuclear decay. down to as far as Q ¼ 1 GeV , and only afterwards higher-twist effects (estimated in a holographic QCD We appreciate Guido Martinelli and Ulf-G. Meißner for model) become important. The lattice calculation on the inspiring discussions. The work of C. Y. S. is supported in 2 2 pion [10] suggests that already at Q ≤ 2 GeV the higher part by the DFG (Grant No. TRR110) and the NSFC (Grant twist contributions are non-negligible. No. 11621131001) through the funds provided to the The implication of Eq. (16) on Vud is as follows. First, if Sino-German CRC 110 “Symmetries and the Emergence F 3072 07 63 0 97365 15 we take t¼ . ð Þ s [27], then jVudj¼ . ð Þ. of Structure in QCD”, and also by the Alexander von However, recent studies in Ref. [4,5] unveil two mutually Humboldt Foundation through the Humboldt Research competing new nuclear corrections (NNC) whose net effect Fellowship. M. G. is supported by EU Horizon 2020 is to enhance the uncertainty, Ft ¼ 3072ð2Þ s. Taking that research and innovation programme, STRONG-2020 0 97366 33 into account gives jVudj¼ . ð Þ. For completeness, project, under Grant Agreement No. 824093 and by the we also quote the impact of our result to neutron beta decay, German-Mexican research collaboration Grant No. 278017 where Vud is determined by [28]: (CONACyT) and No. SP 778/4-1 (DFG). X. F. was supported in part by NSFC of China under Grant 5099.34 s V 2 : No. 11775002. L. C. J. acknowledges support by DOE j udj ¼ τ 1 3λ2 1 Δ ðneutronÞð17Þ nð þ Þð þ RÞ Grant No. DE-SC0010339.

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