Nucleon Axial Radius and Muonic Hydrogen

# Nucleon Axial Radius and Muonic Hydrogen

Richard J. Hill Peter Kammel Center for Experimental Nuclear Physics and Astrophysics and Department of Physics, University of Washington, Seattle, WA 98195, USA William J. Marciano Department of Physics, Brookhaven National Laboratory, Upton, NY 11973, USA Alberto Sirlin Department of Physics, New York University, New York, NY 10003 USA
August 28, 2017
###### Abstract

Weak capture in muonic hydrogen (H) as a probe of the chiral properties and nucleon structure predictions of Quantum Chromodynamics (QCD) is reviewed. A recent determination of the axial-vector charge radius squared, , from a model independent expansion analysis of neutrino-nucleon scattering data is employed in conjunction with the MuCap measurement of the singlet muonic hydrogen capture rate, , to update the induced pseudoscalar nucleon coupling: derived from experiment, and predicted by chiral perturbation theory. Accounting for correlated errors this implies , confirming theory at the 8% level. If instead, the predicted expression for is employed as input, then the capture rate alone determines , or together with the independent expansion neutrino scattering results, a weighted average . Sources of theoretical uncertainty are critically examined and potential experimental improvements are described that can reduce the capture rate error by about a factor of 3. Muonic hydrogen can thus provide a precise and independent value which may be compared with other determinations, such as ongoing lattice gauge theory calculations. The importance of an improved determination for phenomenology is illustrated by considering the impact on critical neutrino-nucleus cross sections at neutrino oscillation experiments.

## 1 Introduction

Muonic hydrogen, the electromagnetic bound state of a muon and proton, is a theoretically pristine atomic system. As far as we know, it is governed by the same interactions as ordinary hydrogen, but with the electron of mass 0.511 MeV replaced by the heavier muon of mass 106 MeV, an example of electron-muon universality. That mass enhancement (207) manifests itself in much larger atomic energy spacings and a smaller Bohr radius of Å. This places the muonic hydrogen size about halfway (logarithmically) between the atomic angstrom and the nuclear fermi (1 fm Å) scale.

Those differences make muonic hydrogen very sensitive to otherwise tiny effects such as those due to proton size and nucleon structure parameters governing weak interaction phenomenology. Indeed, muonic hydrogen Lamb shift spectroscopy [1, 2] has provided a spectacularly improved measurement of the proton charge radius that differs by about 7 standard deviations from the previously accepted value inferred from ordinary hydrogen and electron-proton scattering [3]. (That so called Proton Radius Puzzle is currently unresolved [4, 5, 6]). Similarly, the larger muon mass kinematically allows the weak muon capture process depicted in Fig. 1,

 μ−+p→νμ+n, (1)

to proceed, while ordinary hydrogen is (fortunately for our existence) stable.

Weak muon capture in nuclei has provided a historically important probe of weak interactions and a window for studying nuclear structure. In particular, weak capture in muonic hydrogen is a sensitive probe of the induced pseudoscalar component of the axial current matrix element which is well predicted from the chiral properties of QCD. However, early experimental determinations of that pseudoscalar coupling, ,111The quantity  is defined at the characteristic momentum for muon capture, see Eqs. (8),(23) below. had, for some time, appeared problematic [7]. All  extractions from ordinary muon capture in hydrogen suffered from limited precision, while the more sensitive extraction from radiative muon capture [8] disagreed with ordinary muon capture and the solid prediction of Chiral Perturbation Theory (PT) [9, 10, 11, 12, 13]. An important underlying contribution to this problem was the chemical activity of muonic hydrogen, which like its electronic sibling, can form molecular ions, . The highly spin dependent weak interaction leads to very different capture rates from various muonic atomic and molecular states. Thus, atomic physics processes like ortho-para transitions in the muonic molecule, which flip the proton spins, significantly change the observed weak capture rates and often clouded the interpretation of experimental results in the 55-year history of this field. Unfortunately, the uncertainty induced by molecular transitions was particularly severe for the most precise measurements which were performed with high density liquid hydrogen targets, where, because of rapid formation, essentially capture from the molecule, not the atom, is observed. This problem was resolved by the MuCap Collaboration at the Paul Scherrer Institute (PSI) which introduced an active, in situ, target, where ultra-pure hydrogen gas served both as the target as well as the muon detector, thus enabling a measurement of the muonic hydrogen capture rate at low density, where  formation is suppressed. MuCap unambiguously determined the spin singlet muonic hydrogen capture rate  [14, 15] to 1% accuracy which, when corrected for an enhancement from radiative corrections [16], and using prevailing form factor values at the time implied , in excellent agreement with , the predicted value.

We note, however, that the determination of from both experiment and theory required the input of the axial charge radius squared, traditionally taken from dipole form factor fits to neutrino-nucleon quasielastic charged current scattering () and pion electroproduction () data, which at the time implied the very precise [17] . Recently, that small () uncertainty in has been called into question, since it derives from the highly model dependent dipole form factor assumption.222The dipole ansatz corresponds to with fit mass parameter . The axial radius, which is central to this paper, governs the momentum dependence of the axial-vector form factor, by means of the expansion at small , {linenomath*}

 FA(q2)=FA(0)(1+16r2Aq2+…). (2)

In the one-parameter dipole model, the terms denoted by the ellipsis in Eq. (2) are completely specified in terms of . However, the true functional form of is unknown, and the dipole constraint represents an uncontrolled systematic error. We may instead employ the expansion formalism, a convenient method for enforcing the known complex-analytic structure of the form factor inherited from QCD, while avoiding poorly controlled model assumptions. This method replaces the dipole with , which in terms of the conformal mapping variable , has a convergent Taylor expansion for all spacelike . The size of the expansion parameter, and the truncation order of the expansion necessary to describe data of a given precision in a specified kinematic range, are determined a priori. This representation helps ensure that observables extracted from data are not influenced by implicit form factor shape assumptions. Using the expansion [18] to fit the neutrino data alone leads to [19] with a larger (), more conservative but better justified error. As we will discuss below, traditional analyses of pion electroproduction data have also used a dipole assumption to extract from , and in addition required the a priori step of phenomenological modeling to extract from data. Since these model uncertainties have not been quantified, we refrain from including pion electroproduction determinations of in our analysis. Similarly, we do not include extractions from neutrino-nucleus scattering on nuclei larger than the deuteron, in order to avoid poorly quantified nuclear model uncertainties. In this context, we note that dipole fits to recent -C scattering data suggest a smaller  [20], compared to historical dipole values  [17]. This discrepancy may be due to form factor shape biases (i.e., the dipole assumption), mismodeling of nuclear effects, or something more interesting. Independent determination of is a necessary ingredient for resolving this discrepancy. Finally, we do not include recent interesting lattice QCD results [21, 22, 23, 24], some of which suggest considerably smaller values. As we shall discuss below in Sec. 5, future improvements on these lattice QCD results could provide an independent value with controlled systematics, that would open new opportunities for interpreting muon capture. To illustrate the broad range of possible values, we provide in Table 1 some representative values considered in the recent literature.

Accepting the larger uncertainty from the expansion fit to neutrino data, leads to renewed thinking about the utility of precision measurements of muonic hydrogen capture rates for probing QCD chiral properties. As we shall see, the determination of  becomes and , which are still in good agreement, but with errors enlarged by factors of 1.7 and 3.5, respectively, compared to results using [cf. Eqs. (31),(32) below]. However, taking into account the correlated uncertainties, the comparison can be sharpened to .

Instead of determining , one can use the functional dependence of this quantity, , predicted from PT to extract from the singlet capture rate. As we shall show, that prescription currently gives a sensitivity to comparable to -expansion fits to neutrino-nucleon scattering. We use the resulting value from muon capture to derive a combined weighted average. We also examine how such a method can be further improved by better theory and experiment, and demonstrate that a factor of 3 improvement in the experimental precision appears feasible and commensurate with our updated theoretical precision.

The axial radius is indispensable for ab-initio calculations of nucleon-level charged current quasielastic cross sections needed for the interpretation of long baseline neutrino oscillation experiments at . Its current uncertainty is a serious impediment to the extraction of neutrino properties from such measurements. We quantify the impact that an improved muon capture determination of would have on neutrino-nucleon cross sections, and discuss the status and potential for other determinations, particularly from the promising lattice QCD approach.

The remainder of this paper is organized as follows: In Sec. 2 we give an overview and update regarding the theory of - capture in muonic hydrogen. We first discuss the lowest order formalism. We then outline the status of radiative corrections and argue for a reduction in the overall theory error on the capture rate to about based on new considerations. Uncertainties in the input parameters are described, with particular emphasis on a numerical analysis of the axial charge radius squared and its potential extraction from the singlet 1S capture rate in . Then, in Sec. 3, we describe the experimental situation. After reviewing the MuCap result, we discuss possible improvements for a next generation experiment that would aim for a further factor of 3 error reduction. In Sec. 4, we discuss what can be learned from the present MuCap result and an improved experiment. We update the determination of  from the MuCap measurement using the more conservative expansion value of obtained from neutrino-nucleon scattering. Then, as a change in strategy, using the theoretical expression for  obtained from PT as input, is extracted from the MuCap capture rate and averaged with the expansion value. Other utilizations of MuCap results are also discussed. In Sec. 5, we illustrate the impact of an improved determination on quasielastic neutrino scattering cross sections and discuss the status of, and prospects for, improving alternative determinations. Section 6 concludes with a summary of our results and an outlook for the future.

## 2 Muon capture theory update

The weak capture process, Eq. (1), from a muonic hydrogen bound state is a multi-scale field theory calculational problem, involving electroweak, hadronic and atomic mass scales. In this section, we review the essential ingredients of this problem before discussing the status of phenomenological inputs and the numerical evaluation of the capture rate.

### 2.1 Preliminaries

For processes at low energy, , where is the weak charged vector boson mass, the influence of heavy particles and other physics at the weak scale is rigorously encoded in the parameters of an effective Lagrangian containing four-fermion operators. For muon capture the relevant effective Lagrangian is {linenomath*}

 L=−GFVud√2¯νμγμ(1−γ5)μ¯dγμ(1−γ5)u+H.c.+…, (3)

where and are the Fermi constant and the CKM up-down quark mixing parameter respectively (cf. Table 2), and the ellipsis denotes effects of radiative corrections. Atomic physics of the muonic hydrogen system is described by the effective Hamiltonian, {linenomath*}

 H=p22mr−αr+δVVP−iG2F|Vud|22[c0+c1(sμ+sp)2]δ3(r), (4)

where is the reduced mass, accounts for electron vacuum polarization as discussed below, and , are muon and proton spins. The annihilation process is described by an anti-Hermitian component of  [25]. Since the weak annihilation is a short-distance process compared to atomic length scales, this anti-Hermitian component can be expanded as a series of local operators. At the current level of precision terms beyond the leading one, , are irrelevant [25]. Relativistic corrections to the Coulomb interaction in Eq. (4) are similarly irrelevant [26]. In both cases, neglected operators contribute at relative order , where is the nonrelativistic bound state velocity. Electron vacuum polarization enters formally at order , but is enhanced by a factor making it effectively a first order correction [27, 28].

Having determined the structure of the effective Hamiltonian (4), the numbers are determined by a matching condition with the quark level theory (3). The annihilation rate in the 1S state is then computed from to be {linenomath*}

 Λ =G2F|Vud|2×[c0+c1F(F+1)]×|ψ1S(0)|2+…, (5)

where is the ground state wavefunction at the origin squared and is the total spin ( for singlet, for triplet). Equation (5), with expressed in terms of hadronic form factors (cf. Eq. (2.2) below), exhibits the factorization of the process into effects arising from weak, hadronic and atomic scales.

### 2.2 Tree level calculation

Hadronic physics in the nucleon matrix elements of the vector and axial-vector quark currents of Eq. (3) is parameterized as:333We choose a convention for the pseudoscalar form factor that is independent of lepton mass: , in terms of used in Ref. [29]. Our sign conventions for and are such that and all other form factors are positive. {linenomath*}

 ⟨n|(Vμ−Aμ)|p⟩ =¯un[F1(q2)γμ+iF2(q2)2mNσμνqν−\mbox{FA}(q2)γμγ5−\mbox{FP}(q2)mNqμγ5 +FS(q2)mNqμ−iFT(q2)2mNσμνqνγ5]up+…, (6)

where , and the ellipsis again denotes effects of radiative corrections. For definiteness we employ the average nucleon mass . The form factors and are so-called second class amplitudes that violate parity and are suppressed by isospin violating quark masses or electromagnetic couplings [30, 31, 32, 33]. They would appear in the capture rate, Eq. (2.2) below, accompanied by an additional factor relative to and . Similar to isospin violating effects in , discussed below in Sec. 2.4, power counting predicts negligible impact of and at the permille level; we thus ignore them in the following discussion.

The in Eq. (5) are determined by matching the quark level theory (3) to the nucleon level theory (4), using the hadronic matrix elements (2.2). This matching is accomplished by enforcing, e.g., equality of the annihilation rate for computed in both theories for the limit of free particles, with the proton and muon at rest. For the coefficients corresponding to singlet and triplet decay rates, this yields [34, 16] {linenomath*}

 c0 =E2ν2πM2(M−mn)2[2M−mnM−mnF1(q20)+2M+mnM−mnFA(q20)−mμ2mNFP(q20) +(2M+2mn−3mμ)F2(q20)4mN]2, c0+2c1 =E2ν24πM2(M−mn)2{[mμmNFP(q20)−2mnM−mn(F1(q20)−FA(q20)) (7)

where the initial state mass is , the neutrino energy is , and the invariant momentum transfer is {linenomath*}

 q20≡m2μ−2mμEν=−0.8768m2μ. (8)

Since the matching is performed with free particle states, the quantities , and are defined independent of the atomic binding energy, as necessary for determination of the state-independent coefficients of the effective Hamiltonian (4).444In particular, a binding energy is not included in the initial-state mass , but would anyways correspond to a relative order correction that is beyond the current level of precision.

The amplitudes (2.2) can also be expressed as an expansion in PT [12, 35, 36, 37]. However, the general formulas in Eq. (2.2) allow us to more directly implement and interpret experimental constraints on the form factors and do not carry the intrinsic truncation error of NNLO PT derivations (estimated in Ref. [37] as ). For example, we may take the vector form factors , directly from experimental data, rather than attempting to compute them as part of an expansion in PT. No approximation is yet made in Eq. (2.2), except for neglect of second class currents, as justified above. We investigate below the restricted application of PT to express in terms of and other experimentally measured quantities.

The electroweak radiative corrections to muon capture in muonic hydrogen, depicted in Fig. 2, were first calculated in Ref. [16]. Here, we briefly describe the origin of such quantum loop effects and take this opportunity to update and reduce their estimated uncertainty. The computational strategy relies on the well known electroweak corrections to (i) the muon lifetime [38, 39], (ii) super-allowed decays [38, 40, 41], and (iii) the neutron lifetime [42, 43].

Radiative corrections to weak decay processes in the Standard Model involve ultraviolet divergences that can be renormalized, yielding finite phenomenological parameters such as the Fermi constant obtained from the measured muon lifetime [39] and the CKM matrix element obtained from super-allowed decays (see Table 2). In terms of those parameters, the radiative corrections to the neutron lifetime and the muon capture rate are rendered finite and calculable. We note that the matrix element of the vector current is absolutely normalized at , corresponding to a Conserved Vector Current (CVC): , up to second order corrections in small isospin violating parameters [44, 45, 46]. On the other hand, the normalization of the remaining form factors appearing in Eq. (2.2) requires a conventional definition in the presence of radiative corrections. This definition is specified at by a factorization requirement that expresses the total process as a tree level expression times an overall radiative correction. For example, the neutron decay rate in this scheme involves the factor , where is the tree level expression with , and RC denotes the radiative corrections. By the definition of , these corrections are the same for vector and axial-vector amplitudes, but are actually computed for the vector amplitude. In that way, can be obtained from the neutron lifetime, used in conjunction with via the relationship [41, 42]

 (1+3g2A)|Vud|2τn=4908.7(1.9)s. (9)

Alternatively, can be directly obtained from neutron final state decay asymmetries. We employ the lifetime method here, because it is currently more precise.

In the case of muon capture, we have four form factors all evaluated at : vector (), induced weak magnetism (), axial-vector () and induced pseudoscalar (). We define these form factors to all have the same electroweak radiative corrections and explicitly compute those corrections for . Short-distance corrections (which dominate) correspond to a renormalization of the relevant four-fermion operator, and are automatically the same for all form factors. Long distance corrections, although not as important, are incorporated through the form factor definitions in much the same way as is renormalized by definition in neutron decay.

Given the above form factor definitions, their common total radiative correction is conventionally written as the sum of three terms, {linenomath*}

 RC =RC(electroweak)+RC(finitesize)+RC(electronVP), (10)

which we now specify. Neglecting terms of , where is the charged lepton energy and the momentum transfer,555For the kinematics of muon capture, . the radiative corrections to the vector parts of neutron decay and muon capture are of the same form, but evaluated at different and with different lepton mass. The RC (electroweak) radiative corrections to muon capture [16] were obtained from the original neutron decay calculation, but including higher-order leading log effects denoted by ellipsis in the following Eq. (11): {linenomath*}

 RC(electroweak) =α2π[4logmZmp−0.595+2C+g(mμ,βμ=0)]+⋯=+0.0237(10), (11)

where , ,  [41], and was obtained from Eq. (20b) in Ref. [47] by replacing , ignoring bremsstrahlung and taking the and limits. The ellipsis in Eq. (11) denotes higher order (in ) corrections enhanced by large logarithms [42]. These effects have been added to the order correction to obtain the total electroweak radiative correction. The uncertainty has been reduced from 0.4% in Ref. [16] to 0.1%. That reduction is justified by two improvements in the analysis. First, the radiative corrections to (such as ) are correlated with similar corrections in Eq. (11), and their uncertainties largely cancel. Second, (ignoring nuclear structure), direct calculation of corrections to muon capture (that were ignored in Ref. [16]) were found to cancel and not contribute to the uncertainty in Eq. (11).

Here, we assume that corrections of due to nuclear structure are parametrized by the nucleon finite size reduction factor [48] {linenomath*}

 |ψ1S(0)|2→m3rα3π(1−2αmr⟨r⟩), (12)

where denotes the first moment of the proton charge distribution. Based on a range of model forms for this distribution, the correction (12) evaluates to {linenomath*}

 RC(finitesize) =−0.005(1), (13)

where the error spans the central values  [31],  [16], and  [35] given in the literature. We note that the quoted uncertainty may not fully account for possible additional effects of nuclear structure which could be estimated using a relativistic evaluation of the - box diagrams, but are beyond the scope of this article.666The finite size ansatz (12) becomes exact in the large-nucleus limit, , where is the nuclear (proton) charge radius and denotes a weak vector or axial radius.

The corrections RC(electroweak) and RC(finite size) modify the coefficients of the effective Hamiltonian (4). The remaining radiative correction, from the electron vacuum polarization modification to the muonic atom Coulomb potential, is described by . This contribution amounts to {linenomath*}

 RC(electronVP) =+0.0040(2), (14)

where the very small uncertainty 0.02% is estimated by the difference between of Ref. [16, 27] and of Ref. [35].

In Eq. (10), we have defined the total radiative correction to include electroweak, finite size and electron vacuum polarization contributions. In Ref. [16], the finite size correction was treated separately, and “radiative correction” referred to the sum of our RC(electroweak) and RC(electron VP), amounting to .

### 2.4 Inputs

The relevant inputs used to compute the capture rate are displayed in Table 2. The Fermi constant is determined from the muon lifetime [39] and its uncertainty is negligible in determining the muon capture rate. The CKM matrix element is determined from superallowed decays [40]. The uncertainty in Table 2 is divided into a nucleus-independent radiative correction term, , and a second term representing the sum in quadrature of other theoretical-nuclear and experimental uncertainties. The former radiative correction is strongly correlated with RC(electroweak) in Eq. (11), and the corresponding uncertainty largely cancels when the muon capture rate is expressed in terms of . This cancellation has been accounted for in our discussion of radiative corrections; in the numerical analysis the uncertainty contribution to is dropped.777The muon capture rate could be expressed directly in terms of decay observables, such as the neutron lifetime, where does not appear explicitly.

The charged current isovector vector form factors are obtained from the isovector combination of electromagnetic form factors. Deviations from occur at second order in small isospin violating quantities. At the quark level these quantities may be identified with the quark mass difference and the electromagnetic coupling . At the hadron level, isospin violation manifests itself as mass splittings within multiplets, such as isodoublet and isotriplet  [44, 45, 46]. As shown in Ref.[44], first-order isodoublet mass splitting corrections vanish in and , for general , while first order isotriplet ones cancel in but contribute in for and in for all values of . Estimating these corrections to be of , where is the meson mass (representing a typical hadronic mass scale), we note that in they are accompanied by the further suppression factor , so they amount to . Corrections to the isospin limit in are thus negligible at the required permille level. In the case of , we note that in the expression for the singlet capture rate [Eq. (20) below], a correction to the term within square brackets amounts to , while the total contribution from the four form factors is 4.217.888The additional suppression may be traced to a factor appearing in the coefficients of relative to in Eq. (2.2). A similar power counting applies to the second class form factors, and in Eq. (2.2), that we have neglected in our analysis. Thus, a isotriplet mass splitting correction to induces a correction to the singlet capture rate, which is also negligible at the permille level.

Neglecting these small corrections, the Dirac form factor is thus normalized to . The Pauli form factor at zero momentum transfer is given by the difference of the proton and neutron anomalous magnetic moments: , where and are measured in units of . This leads to . Note that since the PDG [43] expresses both proton and neutron magnetic moments in units of , our value for differs from a simple difference of magnetic moments quoted there by a factor .

The dependence of the form factors is encoded by the corresponding radii, defined in terms of the form factor slopes: {linenomath*}

 1Fi(0)dFidq2∣∣∣q2=0≡16r2i. (15)

Curvature and higher-order corrections to this linear approximation enter at second order in small parameters , where is a hadronic scale characterizing the form factor. These corrections may be safely neglected at the permille level. Isospin violating effects in the determination of the radii may be similarly neglected. The Dirac-Pauli basis , is related to the Sachs electric-magnetic basis , by , . In terms of the corresponding electric and magnetic radii,999The isovector form factors can be written in the form , , , where the subscripts and refer to the proton and neutron contributions. The electric and magnetic radii are defined analogously to Eq. (15) in terms of the slopes of , , and . For the neutron, with , .

 r21=r2E,p−r2E,n−32m2NF2(0),r22=1F2(0)(κpr2M,p−κnr2M,n−r21). (16)

The neutron electric radius is determined from neutron-electron scattering length measurements,  [43]. The proton electric radius is precisely determined from muonic hydrogen spectroscopy,  [2]; this result remains controversial, and is discrepant with the value obtained in the CODATA 2014 adjustment [51] of constants using electron scattering and ordinary hydrogen spectroscopy. We take as default the more precise muonic hydrogen value, but verify that this puzzle does not impact the capture rate at the projected 0.33% level. The magnetic radii are less well constrained. We adopt the values  [43] and  [52].101010This PDG value for represents the expansion reanalysis [53] of A1 collaboration electron-proton scattering data [54]. A similar reanalysis of other world data in Ref. [53] obtained . We verify that this discrepancy does not impact the capture rate at the projected 0.33% level. For , we adopt the value from the expansion reanalysis [52] of extractions, combined with dispersive constraints (see also Ref. [55]). The larger uncertainty encompasses the PDG value, , obtained by averaging with the dispersion analysis of Ref. [56].

Currently, the most precise determination of comes indirectly via the neutron lifetime, , used in conjunction with obtained from super-allowed nuclear decays [40, 41, 42]. Correlating theoretical uncertainties in the electroweak radiative corrections to and ,111111The first, , uncertainty on in Table 2 is correlated with the uncertainty on the right hand side of Eq. (9). These uncertainties cancel. reduces the uncertainty in Eq. (9) to .121212The formula relating and  [42] is based on a neutron decay phase space factor (cf. Ref. [57]). For the PDG average [43], , one finds , the value we use throughout this text. That average includes a PDG scale factor of 1.9 which primarily reflects a lifetime disagreement between trapped neutron decays and free neutron beam decay in flight [58]. The direct neutron decay asymmetry PDG average [43], , is lower; but most more recent experiments find values close to 1.276.

Our knowledge about the functional form of relies primarily on neutrino-deuteron scattering data from bubble chamber experiments in the 1970’s and 1980’s: the ANL 12-foot deuterium bubble chamber experiment [59, 60, 61], the BNL 7-foot deuterium bubble chamber experiment [62], and the FNAL 15-foot deuterium bubble chamber experiment [63, 64]. As mentioned in the Introduction, the original analyses and most follow-up analyses employed the one-parameter dipole model of the axial form factor. A more realistic assessment of uncertainty allows for a more general functional form. Using a expansion analysis [19], the uncertainty on the axial radius is found to be significantly larger than from dipole fits, {linenomath*}

 r2A(zexp.,ν) =0.46(22)fm2. (17)

This value may be compared to a fit of scattering data to the dipole form,  [17]. Note that the value quoted in the Introduction is obtained by averaging this neutrino scattering result with an extraction from pion electroproduction [17], . As observed in Ref. [18], the electroproduction extraction is also strongly influenced by the dipole assumption. A more detailed discussion of the electroproduction constraints is given in Sec. 5, with the conclusion that further control over systematics is required in order to provide a reliable extraction. The pion decay constant and pion nucleon coupling , along with , are used to determine the induced pseudoscalar form factor [11] {linenomath*}

 \mbox{FP}(q20)=2mNgπNNfπm2π−q20−13gAm2Nr2A+…, (18)

where is the charged pion mass. Two loop PT corrections, indicated by the ellipsis in Eq. (18), were estimated to be negligible, as long as the low energy constants involved remain at natural size [13]. is determined from the measured rate for , and its uncertainty is dominated by hadronic structure dependent radiative corrections. For we take as default the value  [49, 50], where the first two errors are attributed to pion-nucleon scattering phase shifts and integrated cross sections, respectively, entering the Goldberger-Miyazawa-Oehme (GMO) sum rule for . The third error is designed to account for isospin violation and was motivated by evaluating a subset of PT diagrams. Other values include from partial wave analysis of nucleon-nucleon scattering data [65]; and  [66],  [67] from partial wave analysis of pion-nucleon scattering data. That range of values is covered by the error given in Table 2.

### 2.5 Numerical results

Employing the radiative corrections given above, the full capture rates become {linenomath*}

 Λ=[1+RC]Λtree=[1+0.0277(10)(2)−0.005(1)]Λtree, (19)

where is the tree level expression for the chosen spin state. We have displayed a conventional separation of the radiative corrections in Eq. (19), where the first  includes the electroweak and electron vacuum polarization corrections, and the second is the finite size correction. Inserting the relevant quantities from Table 2, the singlet 1S capture rate is given by {linenomath*}

 Λsinglet=40.226(56)[F1(q20)+0.08833F2(q20)+2.63645¯gA−0.04544¯gP]2s−1, (20)

where the quantities and are defined below and the relative uncertainty  = 1.4 in the prefactor of Eq. (20) quadratically sums the relative uncertainties (RC) = 1.40 and () = 0.20 .131313As discussed above, a large part of the uncertainty in cancels with the corresponding uncertainty in radiative corrections to muon capture. In the discussion above, we define as the relative uncertainty in the considered quantity having an uncertainty . The relative uncertainty in induced by parameter with uncertainty is .

As a next step, we evaluate the form factors at the momentum transfer relevant for muon capture. For the vector form factors, we expand to linear order using Eq. (15), {linenomath*}

 F1(q20)=0.97578(8),%$F2$(q20)=3.5986(82). (21)

For the axial form factor we have

 ¯gA≡FA(q20)=1.2503(118)r2A(9)gA=1.2503(118), (22)

with the uncertainty dominated by () = . Finally, the pseudoscalar form factor predicted by PT is {linenomath*}

 ¯gP≡mμmNFP(q20% )=8.743(67)gπNN(9)fπ−0.498(238)r2A=8.25(25), (23)

where the contribution from the pole and higher order term in Eq.(18) are shown separately. While the pole term dominates the value for , the uncertainty is actually dominated by the non-pole term, due to the rather dramatically increased uncertainty in .

We exhibit the sensitivity to the axial form factors by inserting the relatively well known vector form factors in Eq. (20) to obtain {linenomath*}

 Λsinglet=67.318(94)[1.00000(56)+2.03801¯gA−0.03513¯gP]2s−1. (24)

At the central values for and , the uncertainty in this equation from the remaining inputs is , corresponding to a relative error =1.44, which is still dominated by RC, with a minor contribution from ()=0.3. At this point the traditional approach would be to insert  and  in the equation above and to specify the uncertainties in  arising from these two axial form factors. However, as both  and  depend on the axial radius squared , which is not well known, they cannot be treated as independent input quantities. To avoid their correlation, we express  in terms of the independent input parameters (, , ): {linenomath*}

 Λsinglet=67.318(94)[1.00000(56)−0.02341(3)gπNN+(2.03801−0.05556r2A)gA]2s−1, (25)

with in units of fm. Using the current knowledge of these independent input quantities from Table 2, we obtain our best prediction for the muon capture rate in the singlet and triplet hyperfine states of muonic hydrogen as {linenomath*}

 Λsinglet =714.8(7.0)s−1, (26) Λtriplet =12.09(52)s−1. (27)

We have employed the same methodology as above for  to obtain . The total relative uncertainty for , () = 9.8, is calculated as the quadratic sum of (RC) = 1.4, () = 1.4, () = 1.1, () = 9.5 and a negligible uncertainty from . Assuming no uncertainty in , the prediction for  would have a more than 4 times smaller error of .

## 3 Muon capture experiment update

Precise measurements of muon capture in hydrogen are challenging, for the following reasons [7, 68]. (i) Nuclear capture takes place after muons come to rest in matter and have cascaded down to the ground state of muonic atoms. As exemplified for the case of H in Eq. (5), the capture rate is proportional to the square of the muonic wavefunction at the origin,141414Compare Ref. [69] for corrections relevant for capture and muon-electron conversion in heavy nuclei. , which, after summing over the number of protons in a nucleus of charge , leads to a steep increase of the capture rate with , such that the muon capture and decay rates are comparable for 13. For H, where , this amounts to a small capture rate of order 10 compared to muon decay, as well as dangerous background from muon stops in other higher materials, where the capture rate far exceeds the one in H. (ii) On the normal atomic scale H atoms are small and can easily penetrate the electronic cloud to transfer to impurities in the hydrogen target gas, or to form muonic molecular ions . The former issue requires target purities at the part-per-billion level. The latter problem, depicted in Fig. 3, has been a primary source of confusion in the past, as the helicity dependence of weak interactions implies large differences in the capture rates from the possible states. The rates for the two atomic hyperfine H states are given in Eqs. (26,27), while the molecular rates can be calculated as {linenomath*}

 Λortho =544s−1, (28) Λpara =215s−1, (29)

using the molecular overlap factors given in Eq. (11) of Ref. [7].151515We do not estimate uncertainties for the molecular rates, as a reliable error evaluation at the permille level should include a modern confirmation of the original calculation of the space and spin structure [70]. To interpret a specific experimental capture rate, the fractional population of states for the given experimental conditions has to be precisely known, which is especially problematic for high density targets. (iii) Finally, muon capture in hydrogen leads to an all neutral final state, , where the 5.2 MeV neutron is hard to detect with well-determined efficiency.

### 3.1 MuCap experiment: strategy and results

Over the past two decades, the He experiment, the MuCap and later the MuSun collaboration have developed a novel active target method based on high pressure time projection chambers (TPC) filled with pure He, ultra-pure hydrogen (1% of liquid hydrogen (LH) density) or cryogenic deuterium gas (6% of LH density), respectively, to overcome the above challenges. The first experiment [71], benefiting from the charged final state, determined the rate for with an unprecedented precision of 0.3% as . The most recent extraction [72] of  from this result gives , with uncertainties due to nuclear structure theory. Additional uncertainties would enter if the new is taken into account.161616We refrain from updating this result with new form factors. As the calculation uses tritium decay as input, changes in form factors at =0 are expected to cancel, but the uncertainty in the momentum dependence enters via .

MuCap measured  in the theoretically clean H system to extract  more directly. The original publication [14] gave , which was slightly updated based on an improved determination of the () molecular formation rate  [15] to its final value

 ΛMuCapsinglet=(715.6±5.4stat±5.1syst) s−1. (30)

The scientific goal of MuSun [73, 7] is the determination of an important low energy constant (LEC), which characterizes the strength of the axial-vector coupling to the two-nucleon system and enters the calculation of fundamental neutrino astrophysics reactions, like fusion in the sun and scattering in the Sudbury Neutrino Observatory [74].

As muon capture involves a characteristic momentum transfer of the order of the muon mass, extractions of form factors and LECs from all of these experiments are sensitive to the modified theoretical capture rate predictions or uncertainties implied by the use of the new .

In view of potential further improvements, let us analyze in some detail how MuCap achieved its high precision 1% measurement. Fig. 4 illustrates the basic concept. Muons are detected by entrance detectors, a 500- thick scintillator (SC) and a wire chamber (PC), and pass through a 500--thick hemispherical beryllium pressure window to stop in the TPC, which is filled with ultrapure, deuterium-depleted hydrogen gas at a pressure of 1.00 MPa and at ambient room temperature. Electrons from muon decay are tracked in two cylindrical wire chambers (PC, green) and a 16-fold segmented scintillator array (SC, blue). The experimental strategy involves the following key features.

Low density and suppressed  formation: As the target has only 1% of liquid hydrogen density, molecule formation is suppressed and 97% of muon capture occurs in the H singlet atom, providing unambiguous interpretation of the signal.

Lifetime method [75]: The observable is the disappearance rate of negative muons in hydrogen, given by the time between muon entrance and decay electron signal. The capture rate is extracted as the difference , where is the precisely known positive muon decay rate [39]. Contrary to the traditional method of detecting capture neutrons from process (1) which requires absolute efficiencies, only precise time measurements are needed, albeit at large statistics.

Selection of muon stops in hydrogen by tracking: The TPC [76] tracks the incident muons in three dimensions to accept only hydrogen stops sufficiently far away from wall materials with higher capture rate. Its sensitive volume is , with an electron drift velocity of 5.5 mm/s at a field of 2 kV/cm in vertical -direction. The proportional region at the bottom of the chamber was operated at a gas gain of 125 – 320, with anode (in -direction) and cathode (in -direction) wires read out by time-to-digital converters using three different discriminator levels.

Ultra-pure target gas: Target purity of  ppb was maintained with a continuous circulation and filter system [77]. The TPC allowed in-situ monitoring of impurities by observing charged nuclear recoils from , in the rare (10 cases of muon transfer to impurities. Isotopically pure protium was produced onsite [78] and verified by accelerator mass spectroscopy [79]. In total, decay events were accepted with muons stopping in the selected restricted fiducial volume.

### 3.2 Conceptual ideas towards a 3-fold improved muon capture experiment

The 3-fold uncertainty reduction over MuCap implies a precision goal of