Towards a uniﬁed treatment of S0parity violation in low-energy nuclear processes

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Towards a uniﬁed treatment of ∆S=0parity violation in low-energy nuclear processes

Susan Gardner∗and Girish Muralidhara†

Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506-0055, USA

We revisit the uniﬁed treatment of low-energy hadronic parity violation espoused by Desplanques, Donoghue,

and Holstein to the end of an ab initio treatment of parity violation in low-energy nuclear processes within the

Standard Model. We use our improved eﬀective Hamiltonian and precise non-perturbative assessments of the

quark charges of the nucleon within lattice QCD to make new assessments of the parity-violating meson-nucleon

coupling constants. Comparing with recent, precise measurements of hadronic parity violation in few-body

nuclear reactions, we ﬁnd improved agreement with these experimental results, though some tensions remain.

We thus note the broader problem of comparing low-energy constants from nuclear and few-nucleon systems,

considering, too, unresolved theoretical issues in connecting an ab initio, eﬀective Hamiltonian approach to

chiral eﬀective theories. We note how future experiments and lattice QCD studies could sharpen the emerging

picture, promoting the study of hadronic parity violation as a laboratory for testing “end-to-end” theoretical

descriptions of weak processes in hadrons and nuclei at low energies.

I. INTRODUCTION

In spite of decades of research, hadronic parity violation in ﬂavor non-changing processes remains poorly understood [1–6].

The pertinent body of experimental work involves the low-energy interactions of hadrons and nuclei, so that we are compelled to

address the interplay of the physics of the weak interaction and of nonperturbative strong dynamics. Ultimately we hope that this

problem can be largely conquered once the direct computation of two-nucleon matrix elements of a suitable eﬀective Hamiltonian

within lattice QCD (LQCD) becomes possible [7], though, as we shall see, there are further issues to address. As an interim

step, we revisit the uniﬁed treatment of hadronic parity violation by Desplanques, Donoghue, and Holstein (DDH) [1]. There,

the description of low-energy hadronic parity violation is framed within an one-meson-exchange model, and DDH show that it

is possible to compute the appropriate meson-nucleon coupling constants starting from the Standard Model (SM) Lagrangian.

Since that early work, powerful ﬁeld theoretic treatments exploiting the low-energy symmetries of QCD have been developed

and applied to the analysis of hadronic parity violation [4, 6, 8–13]. Yet in these chiral eﬀective ﬁeld theory treatments, organized

in terms of hadronic degrees of freedom, the eﬀective couplings are determined from experiment, and the underlying theoretical

connection to QCD and the SM is lost. We note, however, nascent work that would compute the parity-violating pion-nucleon

constant in an ab initio way [14–16]. Here we assess the current status of this problem by revisiting and updating the treatment

of DDH. Namely, we employ our improved eﬀective Hamiltonian [17] to compute the parity-violating meson-nucleon coupling

constants, using the factorization approximation (as it is now employed [18]) and LQCD assessments of the quark ﬂavor charges

of the nucleon [19]. Our particular purpose is to see how these updated assessments combine to confront the constraints on these

parameters from precise experimental measurements of hadronic parity violation in few-body nuclear systems, namely, from the

NPDGamma [20] and n3He [21] collaborations that measure the parity-violating asymmetry from neutron-spin reversal in the

n+p→d+γand in ~

n+3He →t+preactions, respectively.

The NPDGamma measurement is particularly sensitive to the parity-violating pion-nucleon coupling, whereas that made by

the n3He collaboration also probes four-nucleon contact interactions of isoscalar and isovector character, which we interpret in

terms of contributions from vector-meson exchanges between nucleons. Much of the past theoretical eﬀort has concentrated on

studying charged pion-nucleon interactions, due to a longstanding notion of its dominance in hadronic-parity-violating observ-

ables [1]. However, noting the non-observation of parity violation in 18F radiative decay [22–24], and thus ﬁnding no clear sign

of this dominance, and with the direct theoretical analysis of nucleon-nucleon (NN) amplitudes in pionless eﬀective ﬁeld theory

(EFT) in the large number of colors (Nc) limit showing that isoscalar and isotensor interactions should play driving phenomeno-

logical roles [5, 20, 21, 25, 26], we believe the contributions from all isosectors should be computed. Earlier studies of QCD

evolution eﬀects have either made calculational approximations [1, 27, 28], or focused on the isovector case [29–31]. We note,

for example, that the original estimates of parity-violating meson-nucleon couplings were performed with a low-energy Hamil-

tonian built using phenomenological K-factors to account for QCD evolution eﬀects on weak processes [1]. In this work, we

employ our low-energy eﬀective Hamiltonian [17], which makes a complete renormalization group evolution in leading-order

QCD, with matching across heavy-ﬂavor thresholds, to give a uniﬁed treatment of all three isosectors in order to compare their

contributions to recent experimental measurements.

∗gardner@pa.uky.edu

†girish.muralidhara@uky.edu

arXiv:2210.03567v2 [nucl-th] 19 Feb 2023

Powerful searches for physics beyond the SM can be made through low-energy, precision measurements of symmetry-

breaking eﬀects in nucleons and nuclei [32, 33]. For example, in the case of searches for permanent electric dipole moments,

for neutrinoless double βdecay, or for µ→econversion on nuclear targets, the expected SM contribution is either negligibly

small with respect to current experimental sensitivities or altogether absent. Thus the discovery of signiﬁcantly non-zero results

in these systems would signal the existence of physics beyond the SM. Here theory is key to assessing the relative sensitivity

of diﬀerent nuclear systems to the eﬀects of interest. Theory is also essential to the interpretation of a non-zero experimental

result or limit in terms of the parameters of an underlying new physics model — and, more broadly, to using the experimental

limit to estimate a lower bound on the energy scale of new physics, assuming that it lies beyond the weak scale. A theoretical

analysis that connects the scale of new physics to that of the pertinent low-energy experiments requires the consideration of

multiple physical scales, and “end-to-end” eﬀective-ﬁeld-theory treatments are being developed to accomplish that [34–36]. In

this context, we believe QCD studies of hadronic parity violation have a crucial role to play in the benchmarking of these treat-

ments, because its observables are not only nonzero within the SM but also, given the success of the SM in describing ultra-low

energy, parity-violating electron-nucleon interactions [37], new-physics eﬀects presumably play a subdominant role. Thus the

comparison of theory and experiment in hadronic parity violation provides a welcome test of the overall theoretical framework,

as such tests possess aspects common to new-physics searches as well.

In this paper, we embark on this program by determining the parity-violating meson-nucleon coupling constants at a renor-

malization scale of 2 GeV, and, as we shall detail, we ﬁnd improved agreement with the experimental results. Although better

agreement speaks to progress, our longer-term goals are to reﬁne our results to higher precision and also to evolve our descrip-

tion to still smaller scales. We note that the parity-violating meson-nucleon couplings, and, generally, the low-energy constants

associated with the operators of an eﬀective ﬁeld theory are not in themselves observables and can be expected to depend on the

renormalization scale. In this paper we discuss various assessments of the parity-violating pion-nucleon coupling constant from

this perspective, as it is the most precisely determined. Generally, we anticipate diﬃculties can arise both from the gap between

the lowest scale to which we can potentially apply perturbative QCD accurately and the highest scale to which we can employ

chiral perturbation theory, as well as from the eﬀects of the massive charm quark. The latter aﬀects the splay of operators that

can appear, even if the charm quark is still active [38], as we have developed explicitly in the ∆S=0 case [17]. Moreover, the

truncation error from matching a four to three-ﬂavor theory, at a ﬁxed order of perturbation theory, at charm threshold can be

signiﬁcant, as studied in K→ππ decay [39, 40], where we refer to Ref. [41] for a broader discussion. In this paper we comment

on how some of these eﬀects can impact our results.

We conclude this section with an outline of the rest of the paper. In Sec.II, we recapitulate the outcomes of our eﬀective weak

Hamiltonian computation [17] that are pertinent here. In Sec.III we discuss the factorization approximation and its validity. In

Sec.IV we employ these results to compute the parity-violating meson-NN coupling constants. In Sec. V we compare our results

with the parameters extracted from experiments and discuss the perspectives they oﬀer, and we oﬀer a concluding summary and

outlook in Sec. VI.

II. EFFECTIVE HAMILTONIAN AND WILSON COEFFICIENTS

The eﬀective Hamiltonian for hadronic parity violation at a particular energy scale is deﬁned in terms of four-quark operators

and corresponding Wilson coeﬃcients. In evolving the theory from one energy scale to another, such as from the Wmass to

scale µ, the Wilson coeﬃcients follow the relation:

C(µ)=exp "Zgs(µ)

gs(MW)

dg γT(µ)

β(gs)#~

C(MW) with β(gs)=−g3

48π2(33 −2nf),(1)

where we work in leading-order (LO) QCD, noting that the anomalous dimension matrix γarises from the LO QCD mixing

of the operators. We refer to Ref.[17] for all details. Allowing the eﬀective theory to ﬂow from the Wmass scale to hadronic

energy scales, with µ=2 GeV, results in the Hamiltonian:

HPV

eﬀ(2 GeV) =GFs2

3√2

i=1

Ci(2 GeV) Θi,(2)

where Θiare four-quark operators. Twelve such operators form a closed set under LO QCD mixing and thus describe the theory

of hadronic parity nonconservation, for all three isosectors, where we refer to App. A1 for a complete list. Moreover, we use

the weak-mixing angle θWwith sin2θW≡s2

w=0.231 and the Fermi constant GF=1.166 ×10−5GeV−2[42]. Just below the W

mass scale, the eﬀects of QCD are negligible, and we can collect the Wilson coeﬃcients by summing the tree-level W and Z0

exchange contributions in ∆S=0 quark interactions, giving ~

C(MW)=(1,0,0,0,−3.49,0,0,0,−13.0 cos2θc,0,−13.0 sin2θc,0),

with the Cabibbo angle given by sin θc=0.2253. Upon performing the RG ﬂow to 2 GeV using Eq.1, we have [17]

C(2 GeV)=







1.09 [1.17 . . . 1.06][1.08 . . . 1.04] [1.07][1.06]

0.018 [0.014 . . . 0.021][0.033 . . . 0.006] [−0.006][−0.006]

0.199 [0.321 . . . 0.133][0.193 . . . 0.127] [0.158][0.153]

−0.583 [−0.990 ··· − 0.385][−0.571 ··· − 0.374] [−0.460][−0.456]

−4.36 [−4.99 ··· − 4.05][−4.34 ··· − 4.03] [−4.16][−4.14]

1.72 [2.63 . . . 1.19][1.67 . . . 1.16] [1.40][1.36]

−0.170 [−0.288 ··· − 0.110][−0.165 ··· − 0.105] [−0.134][−0.129]

0.332 [0.496 . . . 0.235][0.322 . . . 0.225] [0.275][0.268]

−16.2 [−18.6··· − 15.0][−16.1··· − 15.0] [−15.48][−15.4]

6.38 [9.76 . . . 4.44][6.22 . . . 4.30] [5.19][5.05]

−16.2 [−18.6··· − 15.0][−16.1··· − 15.0] [−15.48][−15.4]

6.38 [9.76 . . . 4.44][6.22 . . . 4.30] [5.19][5.05]







,(3)

where the last four entries should be multiplied by factors of cos2θc,cos2θc,sin2θc, and sin2θc, respectively. The primary result

is given by the leftmost column of numbers. The other columns illustrate the uncertainties in the computation. In the central

column, the left set shows the ranges of Wilson coeﬃcients that result in the Nf=2+1 theory for renormalization scales of

µ=1−4 GeV and the right set shows them in the Nf=2+1+1 theory with µ=2−4 GeV. The rightmost column gives Wilson

coeﬃcients if the αsrunning and matching is computed at NLO (left) and NNLO (right).

For the present work, it is useful to make the diﬀerent isosector contributions explicit and separated as

HPV

eﬀ(2 GeV) =HI=1

eﬀ(2 GeV) +HI=0⊕2

eﬀ(2 GeV).(4)

Isovector (I=1) Wilson coeﬃcients at high and low energies are: ~

CI=1(MW)=(1,0,0,0,3.49,0,3.49,0,−13.0 cos2θc,0) and

CI=1(2 GeV)=







1.09 [1.17 . . . 1.06][1.08 . . . 1.04] [1.07][1.06]

0.018 [0.014 . . . 0.021][0.033 . . . 0.006] [−0.006][−0.006]

0.199 [0.321 . . . 0.133][0.193 . . . 0.127] [0.158][0.153]

−0.583 [−0.990 ··· − 0.385][−0.571 ··· − 0.374] [−0.460][−0.456]

4.36 [4.99 . . . 4.05][4.34 . . . 4.03] [4.16][4.14]

−1.72 [−2.63 ··· − 1.19][−1.67 ··· − 1.16] [−1.40][−1.36]

4.36 [4.99 . . . 4.05][4.34 . . . 4.03] [4.16][4.14]

−1.72 [−2.63 ··· − 1.19][−1.67 ··· − 1.16] [−1.40][−1.36]

−16.2 [−18.6··· − 15.0][−16.1··· − 15.0] [−15.48][−15.4]

6.38 [9.76 . . . 4.44][6.22 . . . 4.30] [5.19][5.05]







,(5)

where the last two entries should be multiplied by a factor sin2θcand the error estimates are deﬁned as in Eq. (3). Wilson

coeﬃcients for the I=0⊕2 sector at high and low energies are: ~

CI=0⊕2(MW)=(−1,0,0,0,−3.49,0,0,0,−13.0 cos2θc,0) and

CI=0⊕2(2 GeV)=







−1.09 [−1.17 ··· − 1.06][−1.08 ··· − 1.04] [−1.07][−1.06]

−0.018 [−0.014 ··· − 0.021][−0.033 ··· − 0.006] [0.006][0.006]

−0.199 [−0.321 ··· − 0.133][−0.193 ··· − 0.127] [−0.158][−0.153]

0.583 [0.990 . . . 0.385][0.571 . . . 0.374] [0.460][0.456]

−4.36 [−4.99 ··· − 4.05][−4.34 ··· − 4.03] [−4.16][−4.14]

1.72 [2.63 . . . 1.19][1.67 . . . 1.16] [1.40][1.36]

−0.170 [−0.288 ··· − 0.110][−0.165 ··· − 0.105] [−0.134][−0.129]

0.332 [0.496 . . . 0.235][0.322 . . . 0.225] [0.275][0.268]

−16.2 [−18.6··· − 15.0][−16.1··· − 15.0] [−15.48][−15.4]

6.38 [9.76 . . . 4.44][6.22 . . . 4.30] [5.19][5.05]







(6)

where the last two entries should be multiplied by a factor cos2θcand the error estimates are deﬁned as in Eq. (3). Although

Eqs. (3, 5, 6) appeared in our earlier paper [17], we have included them here to make our presentation self-contained.

III. FACTORIZATION APPROXIMATION

The eﬀective Hamiltonian presented in the previous section can be used in the computation of various parity-violating meson-

nucleon coupling constants of isospin I:hI

M. These parameters are introduced to quantify the observable eﬀects of hadronic

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TowardsauniedtreatmentofS=0parityviolationinlow-energynuclearprocessesSusanGardnerandGirishMuralidharayDepartmentofPhysicsandAstronomy,UniversityofKentucky,Lexington,KY40506-0055,USAWerevisittheuniedtreatmentoflow-energyhadronicparityviolationespousedbyDesplanques,Donoghue,andHolsteintotheendofa...

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Towards a uniﬁed treatment of S0parity violation in low-energy nuclear processes

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