Reduced hadronic uncertainty in the determination of $V_{ud}$
Chien-Yeah Seng Mikhail Gorchtein Hiren H. Patel Michael J. Ramsey-Musolf

ACFI-T18-12
MITP/18-070






gorshtey@uni-mainz.de

 

^aINPAC, Shanghai Key Laboratory for Particle Physics and Cosmology, 

	MOE Key Laboratory for Particle Physics, Astrophysics and Cosmology,  

	School of Physics and Astronomy, Shanghai Jiao-Tong University, Shanghai 200240, China
^bInstitut für Kernphysik, PRISMA Cluster of Excellence

	Johannes Gutenberg-Universität, Mainz, Germany
^cAmherst Center for Fundamental Interactions, Department of Physics, University of Massachusetts, Amherst, MA 01003
^dKellogg Radiation Laboratory, California Institute of Technology,
	Pasadena, CA 91125 USA





We analyze the universal radiative correction Δ_R^V to neutron and superallowed nuclear β decay by expressing the hadronic γ W-box contribution in terms of a dispersion relation, which we identify as an integral over the first Nachtmann moment of the γ W interference structure function F_3^(0).  By connecting the needed input to existing data on neutrino and antineutrino scattering, we obtain an updated value of Δ_R^V = 0.02467(22), wherein the hadronic uncertainty is reduced.  Assuming other Standard Model theoretical calculations and experimental measurements remain unchanged, we obtain an updated value of |V_ud| = 0.97366(15), raising tension with the first row CKM unitarity constraint.  We comment on ways current and future experiments can provide input to our dispersive analysis.


Reduced hadronic uncertainty in the determination of V_ud
    Michael J. Ramsey-Musolf^c,d
    December 30, 2023
=========================================================



The unitarity test of the Cabibbo-Kobayashi-Maskawa (CKM) matrix serves as one of the most important precision tests of the Standard Model.  In particular, tests of first-row CKM unitarity |V_ud|^2+|V_us|^2+|V_ub|^2=1 receive the most attention since these matrix elements are known with highest precision, all with comparable uncertainties.   The good agreement with unitarity  <cit.> serves as a powerful tool to constrain New Physics scenarios.

Currently, the most precise determination of |V_ud| comes from measurements of half-lives of superallowed 0^+→0^+ nuclear β decays with a precision of 10^-4 <cit.>.  At tree-level, these decays are mediated by the vector part of the weak charged current only, which is protected against renormalization by strong interactions due to conserved vector current (CVC), making the extraction of |V_ud| relatively clean.  Beyond tree-level, however, electroweak radiative corrections (EWRC) involving the axial current are not protected, and lead to a hadronic uncertainty that dominates the error in the determination of |V_ud|. 

The master formula relating the CKM matrix element |V_ud| to the superallowed nuclear β decay half-life is <cit.>:

    |V_ud|^2=2984.432(3) s/ℱt(1+Δ_R^V) ,

where the nucleus-independent ℱt-value is obtained from the experimentally measured ft-value by absorbing all nuclear-dependent corrections, and where Δ_R^V represents the nucleus-independent EWRC.  Currently, an average of the 14 best measured half-lives yields an extraordinarily precise value of ℱt=3072.27(72) s. A similar master formula exists for free neutron β decay <cit.> depending additionally on the axial-to-vector nucleon coupling ratio λ=g_A/g_V, and is free of nuclear-structure uncertainties.  But the much larger experimental errors in the measurement of its lifetime and the ratio λ <cit.> makes it less competitive in the extraction of |V_ud|.  Regardless, if first-row CKM unitarity is to be tested at a higher level of precision, improvement in the theoretical estimate of Δ_R^V by reducing hadronic uncertainties is essential.

The best determination of Δ_R^V = 0.02361(38) was obtained in 2006 by Marciano and Sirlin <cit.> (in the following, we refer to their work as [MS]). They were able to reduce the hadronic uncertainty by a factor of 2 over their earlier calculation <cit.> by using high order perturbative QCD corrections originally derived for the polarized Bjorken sum rule to precisely estimate the short distance contribution. At intermediate distances, an interpolating function motivated by vector meson dominance (VMD) was used to connect the long and short distances and was identified as the dominant source of theoretical uncertainty. This result leads to the current value of |V_ud| = 0.97420(21) <cit.>. 


In this Letter, we introduce a new approach for evaluating Δ_R^V based on dispersion relations which relate it to directly measurable inclusive lepton-hadron and neutrino-hadron structure functions. 
Dispersion relations have proved crucial for evaluating the γ Z-box correction to parity violating electron-hadron interaction in atoms and in scattering processes <cit.>. It led to a significant shift in the 1-loop SM prediction for the hadronic weak charges, and ensured a correct extraction of the weak mixing angle at low energy <cit.>. 
Using existing data on neutrino and anti-neutrino scattering, we obtain a more precise value of the nucleus-independent EWRC,

    Δ_R^V = 0.02467(22) ,
 and therefore a new determination of |V_ud|,

    |V_ud| = 0.97366(15).

We summarize in this Letter the essential features of our analysis that lead us to these values, and defer details to an upcoming longer paper <cit.>. 




Among the various contributions at 𝒪(α/π) to the neutron β decay amplitude, Sirlin established <cit.> that the only one sensitive to the hadronic scale is the part in the γ W box amplitude (Fig. <ref>),

    ℳ_VA=2√(2)e^2G_FV_ud∫d^4q/(2π)^4[
    u̅_e(𝐤)γ_μ(k-q+m_e)γ_ν P_L v_ν(𝐤)/q^2[(k-q)^2-m_e^2]M_W^2/q^2-M_W^2T^μν_VA],

involving the nucleon matrix element of the product of the electromagnetic (EM) and the axial part of the weak charged current

    T^μν_VA = 1/2∫ d^4x   e^iqx⟨ p(𝐩)|T[J^μ_em(x)J^ν_W,A(0)]|n(𝐩)⟩ .

After inserting the nucleon matrix element parametrized in terms of the P-odd invariant function T^μν_VA = iϵ^μναβp_α q_β/2 p· q T_3 into the amplitude (<ref>), the correction to the tree level amplitude is expressed as <cit.> 

    ^VA_γ W = α/8π∫_0^∞ dQ^2 M_W^2/M_W^2+Q^2×
    ∫_-i√(Q^2)^i√(Q^2)dν/ν4(Q^2+ν^2)^3/2/π M Q^4T_3(ν,Q^2)

where after Wick rotation the azimuthal angles of the loop momentum have been integrated over and the remaining integrals have been expressed in terms of Q^2 = -q^2 and ν=(p· q)/M.  With negligible error, we assume a common nucleon mass M in the isospin symmetric limit and we work in the recoil-free approximation.  This contributes to the nucleus-independent EWRC as

    Δ_R^V=2_γ W^VA+… ,

where the ellipses denote all other corrections insensitive to the hadronic scale.

Marciano and Sirlin estimate ^VA_γ W by phenomenologically treating the ν-integral F_M.S.(Q^2) ≡∫ dν… in the second line of (<ref>) as a function of Q^2, and parametrizing it piecewise over three domains: in the short distance domain Q^2>(1.5 GeV)^2, the leading term in the OPE corrected by high order perturbative QCD is used; in the long distance domain Q^2<(0.823 GeV)^2, the elastic nucleon with dipole form factors is used with a 10% uncertainty; and at intermediate scales (0.823 GeV)^2<Q^2<(1.5 GeV)^2, an interpolating function inspired by VMD is used and is assigned a generous 100% uncertainty.  Performing the integration over Q^2 in (<ref>) yields their value of Δ_R^V quoted above.

Our evaluation of ^VA_γ W begins by first separating the invariant amplitude T_3 with respect to isosinglet and isotriplet components of the EM current T_3=T_3^(0)+T_3^(3).  Crossing symmetry implies T_3^(0) is odd under ν→-ν while T_3^(3) is even.  Since the ν integration measure in (<ref>) is odd, only T_3^(0) contributes to ^VA_γ W.  We then write a dispersion relation in ν for T_3^(0), taking into account the physical sheet singularities.  Poles at ν_B=± Q^2/(2M) correspond to the elastic single-nucleon intermediate state, and branch points at ν_π=±(m_π^2+2Mm_π+Q^2)/(2M) correspond to single pion production thresholds.  We identify the discontinuity of T_3^(0) across the cut as the γ W-interference structure function,  
4π F^(0)_3(ν,Q^2) = T_3^(0)(ν+iϵ,Q^2)-T_3^(0)(ν-iϵ,Q^2), so that the dispersion relation reads

    T_3^(0)(ν,Q^2)=-4iν∫_0^∞ dν'F_3^(0)(ν',Q^2)/ν'^2-ν^2.

where F_3^(0) contains both the elastic and inelastic contributions.  No subtraction constant appears since T_3^(0) is an odd function of ν.  Only I=1/2 intermediate states contribute because the EM current is isoscalar.  After inserting (<ref>) into (<ref>), performing the ν-integral, and changing the integration variable ν' → Q^2/(2Mx) we obtain

    ^VA_γ W=3α/2π∫_0^∞dQ^2/Q^2M_W^2/M_W^2+Q^2M_3^(0)(1,Q^2),

where M_3^(0)(1,Q^2) is the first Nachtmann moment of the structure function F_3^(0) <cit.>

    M_3^(0)(1,Q^2)=4/3∫_0^1 dx 1+2r/(1+r)^2F_3^(0)(x,Q^2),

and r=√(1+4M^2x^2/Q^2).  To estimate ^VA_γ W, we require the functional form of F_3^(0) depending on x and Q^2, or equivalently, W^2=M^2+(1-x)Q^2/x and Q^2.



We draw attention to the fact that (<ref>) relates [MS]'s phenomenological function to the first Nachtmann moment

    F_M.S.(Q^2)=12/Q^2M_3^(0)(1,Q^2) ,

which will prove useful when comparing their results with ours.  Furthermore, since F_3^(0) depends directly on on-shell intermediate hadronic states, it provides better handle on the physics that may enter at various scales.  Fig. <ref> depicts the domain in the W^2–Q^2 plane over which F_3^(0) has support: the single-nucleon elastic pole is at W^2=M^2, and the inelastic continuum covers the region above W^2 > (M + m_π)^2.



Our parameterization of F_3^(0) is as follows:

    F_3^(0)=F_Born+
    
    F_pQCD,     Q^2 ≳ 2 GeV^2 
    F_π N+F_res+F_ℝ,    Q^2 ≲ 2 GeV^2 ,

where each component is given by

    F_Born=-1/4(G_M^p+G_M^n)G_Aδ(1-x) 
    ∫_0^1 dx F_pQCD = 1/12[1+pQCD] 
    F_π N = F_χPT×(F_1^p+F_1^n)|G_A|/g_A
    
    F_res = negligible
    F_ℝ = C_γ Wf_thm_ω^2/m_ω^2+Q^2m_a_1^2/m_a_1^2+Q^2(ν/ν_0)^α_0^ρ ,

and supplies the dominant contribution to F_3^(0) in various regions indicated in Fig. <ref> which we describe next.

We obtain the elastic Born contribution at W^2 = M^2 in (<ref>) by using the updated values of the magnetic Sachs form factor G_M and the axial form factor G_A for the nucleon <cit.>.  Above threshold, W^2 ≥ (M+m_π)^2, we consider the dominant physics operating in various of domains in the Q^2–W^2 plane separately. At large Q^2 ≳ 2 GeV^2, the Nachtmann moment M_3^(0) reduces to the Mellin moment and is fixed by the sum rule corrected by pQCD in Eq. (<ref>) by analogy with that of the polarized Bjorken sum rule [MS].  At small Q^2 ≲ 2 GeV^2, we estimate the contribution (<ref>) near the inelastic threshold by computing the single pion production contribution  F_χPT in Chiral Perturbation Theory (χPT) at leading order.  To improve the behavior of F_χPT at larger Q^2, we replace the point-like nucleon vertices with measured Dirac and axial nucleon form factors, F_1 and G_A.  At higher W^2, we investigated the impact of several low-lying I=1/2 resonances based on a few models <cit.>, and found their contributions to _γ W^VA to be negligible.



Finally, at large W^2, we use the form in Eq. (<ref>) inspired by Regge phenomenology together with VMD <cit.> as illustrated in Fig. <ref>a.  In this picture, the Regge behavior (ν/ν_0)^α_0^ρ arises from the exchange of the ρ trajectory with intercept α_0^ρ = 0.477 <cit.>, and is coupled to the external currents via a_1 and ω mesons encoded by the VMD factors m^2_V/(m_V^2 + Q^2).  We include a threshold function f_th=Θ(W^2-W_th^2)(1-exp[(W_th^2-W^2)/Λ_th^2]) which smoothly vanishes at the two-pion threshold point W_th^2=(M+2m_π)^2 to model the smooth background in the resonance region <cit.>.  We choose equal values for the Regge and threshold scales of ν_0 = Λ_th = 1 GeV, to ensure that Regge behavior sets in around W^2∼(2.5 GeV)^2.  The function C_γ W (Q^2) accounts for residual Q^2-dependence beyond that of the VMD, which we infer from experimental data as explained below.




Since the isospin structure of F_3^(0) is , it is not directly accessible experimentally.  However, information about the P-odd structure function with a different isospin structure  is available from ν- and ν̅-scattering.  In particular, data exists on the first Nachtmann moment M_3^ν p+ν̅p for the combination F_3^ν p+ν̅p=(F_3^W^-+F_3^W^+)/2 derived from the difference of ν p and ν̅p differential cross sections.  The data by CCFR <cit.>, BEBC/Gargamelle <cit.> and WA25 <cit.> cover a wide region of Q^2 from 0.15 to 600 GeV^2 (see Fig. <ref>).  Although the precision below Q^2≈1.4 GeV^2 is less satisfactory, we are able to use it to collect information about the form of the analogous Regge coefficient function C_WW(Q^2) for this structure function, and thereby infer the form of the required C_γ W(Q^2) as follows. 

We parametrize the structure function F_3^ν p+ν̅p in precisely the same way as in (<ref>) for F_3^(0), and establish F_Born^ν p+ν̅p, F_pQCD^ν p+ν̅p, F_π N^ν p+ν̅p, F_res^ν p+ν̅p and F_ℝ^ν p+ν̅p along similar lines.  In this case, ∫_0^1 dx  F_pQCD^ν p+ν̅p satisfies the Gross-Llewellyn-Smith sum rule <cit.> corrected by pQCD <cit.>, while at low Q^2, the Δ-resonance and the Born contribution saturate the Nachtmann moment <cit.>.  At large W^2, the ω trajectory controls the leading behavior, and couples to the external currents by the a_1 and ρ mesons (see Fig. <ref>b), leading to

    F_ℝ^ν p+ν̅p=C_WWf_thm_ρ^2/m_ρ^2+Q^2m_a_1^2/m_a_1^2+Q^2(ν/ν_0)^α_0^ω .

We then fit the unknown function C_WW(Q^2) to the data for M_3^ν p+ν̅p(1,Q^2) in the range Q^2≤ 2 GeV^2. Due to the quality of the data, we choose the simple linear form

    C_WW(Q^2)=A_WW(1+B_WWQ^2)

and obtain A_WW=5.2± 1.5, B_WW=1.08^+0.48_-0.28 GeV^-2.  The result of the fit is shown by the blue curve in Fig. <ref>.  The solid curve corresponds to the central value of the fit, and the dotted curve indicates the maximum variation in M_3^ν p+ν̅p allowed by the errors in the fit. We do not fit the three data points below Q^2=0.1 GeV^2 where Born and resonance contributions dominate the GLS sum rule: rather, we use the resonance parameters obtained in <cit.> from a fit to modern neutrino data.

Finally, to obtain C_γ W(Q^2), we require the ratio of Nachtmann moments M_3,ℝ^(0)(1,Q^2)/M_3,ℝ^ν p+ν̅p(1,Q^2) to agree with the value predicted by VMD at Q^2=0, and the QCD-corrected parton model at Q^2=2  GeV^2. Since the ρ and ω Regge trajectories are nearly degenerate <cit.>, the two conditions predict the same ratio <cit.>

    M_3,ℝ^(0)(1,0)/M_3,ℝ^ν p+ν̅p(1,0)≈M_3,ℝ^(0)(1,2 GeV^2)/M_3,ℝ^ν p+ν̅p(1,2 GeV^2)≈1/36.

For the linear parametrization in Eq. (<ref>), this implies 

    C_γ W(Q^2)=1/36C_WW(Q^2) ,

providing us with the final piece of F_ℝ in (<ref>).

Upon inserting our parameterization (<ref>) for the structure function F_3^(0) into (<ref>, <ref>) and performing the integrations, we obtain the following contributions to _γ W^VA in units of 10^-3: 2.17(0) from parton+pQCD, 1.06(6) from Born and 0.56(8) from Regge+resonance+π N, the digit in parentheses indicating the uncertainty. Combining them with the remaining known contributions [MS] gives our new values, 
Δ_R^V=0.02467(22) and |V_ud|=0.97366(15). Our reevaluation of Δ_R^V represents a reduction in theoretical uncertainty over the previous [MS] result by nearly a factor of 2.  However, it also leads to a substantial upward shift in the central value of Δ_R^V and a corresponding downward shift of |V_ud| by nearly three times their quoted error, now raising tension with the first-row CKM unitarity constraint: |V_ud|^2+|V_us|^2+|V_ub|^2=0.9983(4).





We pause to comment on the origin of the large shift in the central value for Δ_R^V with respect to [MS].  In Fig. <ref> we plot the integrand M_W^2/M_W^2 + Q^2 M_3^(0)(1,Q^2) of Eq. (<ref>) as a function of Q^2.  In solid blue, we show the result of our parametrization (<ref>) after integrating over x.  In dashed red, we show the piecewise parametrization by [MS] obtained with the help of (<ref>). 
The discontinuity in their parametrization at Q^2 = (1.5 GeV)^2 arises from their choice of matching the Q^2 integrals of pQCD and the interpolating function over the short distance domain, rather than matching the functions themselves.
The log-linear scale conveniently accounts for the integration measure dQ^2/Q^2 in (<ref>) so that the correction ^VA_γ W is directly proportional to the area under the curve.  Although the line shapes are in agreement above Q^2≳2 GeV^2 and below Q^2≲0.001 GeV^2, ours lies significantly above that of [MS] for intermediate Q^2. This difference is the origin of the discrepancy between our central values for Δ_R^V.  By working with the two-variable structure function F_3^(0)(x, Q^2), we were able to capture a broad variety of physics (Born, Nπ, Regge) operating at intermediate Q^2 in contrast with the one-variable analysis of F_M.S.(Q^2) by [MS]. 
We therefore believe our updated result provides a more realistic assessment of Δ_R^V, even though the difference with them is larger than their quoted theoretical uncertainty.



We conclude by discussing how new measurements could provide tests of our parameterization of F_3^(0) and further reduce the uncertainty in Δ_R^V. In view of the upcoming high-intensity neutrino beam program at Fermilab, we wish to point out the potential impact which new, more precise measurements of M_3^ν p+ν̅p(1,Q^2) at low Q^2 can have on our fit, as evidenced by Fig. <ref>. That said, we have related F_3^(0) and F_3^ν p+ν̅p within a model. However, by making use of isospin symmetry, we can establish a more robust relationship between F_3^(0) and the P-odd structure function F_3,γ Z^N. The latter is accessible with parity-violating deep inelastic (inclusive) electron scattering. Since the axial component of the weak neutral current is predominantly isovector, we obtain

    4F_3^(0)≈ F_3,γ Z^p-F_3,γ Z^n≈ 2F_3,γ Z^p-F_3,γ Z^d.

Thus, fixed target measurements using hydrogen and deuterium can in principle provide a more direct way to determine _γ W^VA from data.  High quality data in the range 0.1 GeV^2 ≲ Q^2 ≲ 1 GeV^2 and W^2 ≳ 5 GeV^2 would be particularly advantageous, as our parametrization of F_3^(0) admits the greatest model-dependence and exhibits the largest difference from that of [MS] in this domain.  Such an experimental program will however require a dedicated feasibility study, as the contribution of F_3,γ Z to the parity-violating asymmetry with a polarized electron beam is suppressed by the small weak charge of the electron.  Finally, with the  reduction in the uncertainty of |V_ud|, the error in the first-row CKM unitarity constraint is dominated by the uncertainty in |V_us| = 0.2243(5). Combined with our results presented here, a commensurate reduction in the latter uncertainty would enhance the impact of first row CKM unitarity tests.

We acknowledge helpful discussions with Bill Marciano, John Hardy, Vincenzo Cirigliano, Peter Blunden, Emmanuel Paschos, Cheng-Pang Liu, Chung-Wen Kao, Hubert Spiesberger and Jens Erler.  Significant progress was made during the scientific program “Bridging the Standard Model to New Physics with the Parity Violation Program at MESA" hosted by MITP Mainz. 
M.G.'s work was supported by the Deutsche Forschungsgemeinschaft under the personal grant GO 2604/2-1. CYS's work is supported in part by the National Natural Science Foundation of China (NSFC) under Grant Nos.11575110, 11655002, 11735010, Natural Science Foundation of Shanghai under Grant No. 15DZ2272100 and No. 15ZR1423100, by Shanghai Key Laboratory for Particle Physics and Cosmology, and by Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education, and also appreciates the support through the Recruitment Program of Foreign Young Talents from the State Administration of Foreign Expert Affairs, China.  HHP and MJRM were supported in part by US Department of Energy Contract DE-SC0011095.


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