# Tritium -decay in chiral effective field theory

###### Abstract

We evaluate the Fermi and Gamow-Teller (GT) matrix elements in tritium -decay by including in the charge-changing weak current the corrections up to one loop recently derived in nuclear chiral effective field theory (EFT). The trinucleon wave functions are obtained from hyperspherical-harmonics solutions of the Schrödinger equation with two- and three-nucleon potentials corresponding to either EFT (the N3LO/N2LO combination) or meson-exchange phenomenology (the AV18/UIX combination). We find that contributions due to loop corrections in the axial current are, in relative terms, as large as (and in some cases, dominate) those from one-pion exchange, which nominally occur at lower order in the power counting. We also provide values for the low-energy constants multiplying the contact axial current and three-nucleon potential, required to reproduce the experimental GT matrix element and trinucleon binding energies in the N3LO/N2LO and AV18/UIX calculations.

###### pacs:

21.45.-v, 23.40-s## I Introduction

Recently, nuclear axial current and charge operators have been derived in chiral effective field theory (EFT) up to one loop in a formalism based on time-ordered perturbation theory, in which, along with irreducible contributions, non-iterative terms in reducible contributions were identified and accounted for order-by-order in the power counting Baroni16 (). Ultraviolet divergencies associated with the loop corrections were isolated in dimensional regularization. The resulting axial current was found to be finite and conserved in the chiral limit, while the axial charge required renormalization. In particular, the divergencies in the loop corrections to the one-pion exchange axial charge were reabsorbed by renormalization of some of the low-energy constants (LECs) characterizing the sub-leading Lagrangian Fettes00 (). For a detailed discussion of these issues (formalism, renormalization, etc.) we defer to Ref. Baroni16 (). However, a brief summary is provided in the next section.

In the present paper, the focus is on the axial current, whose contributions up to one loop are illustrated diagrammatically in Fig. 1. Pion-pole terms are crucial for the current to be conserved in the chiral limit Baroni16 ()—these terms were ignored in the earlier studies of Park et al. Park93 (); Park03 (); of course, they are suppressed in low momentum transfer processes such as the tritium -decay under consideration here. Vertices involving three or four pions, such as those, for example, occurring in panels (l), (p), (q), and (r) of Fig. 1, depend on the pion field parametrization. This dependence must cancel out after summing the individual contributions associated with these diagrams, as indeed it does Baroni16 () (this and the requirement that the axial current be conserved in the chiral limit provide useful checks of the calculation).

In Fig. 1 the labeling NLO corresponds to the power counting , where denotes generically the low momentum scale and is for the axial current Baroni16 (). The LO and N2LO currents consist of single-nucleon terms; the N2LO current includes relativistic corrections proportional to ( is the nucleon mass), suppressed by two powers of relative to the LO. Pion-range currents contribute at N3LO, panels (e) and (f) of Fig. 1, and involve vertices from the sub-leading chiral Lagrangian Fettes00 (), proportional to the LECs , , and . At this order (N3LO) there is also a contact current proportional to a single LEC, which we denote as following Ref. Baroni16 (). This LEC is related to the LEC (in standard notation), which enters the three-nucleon chiral potential at leading order. The two LECs and which fully characterize this potential have recently been constrained by reproducing the empirical value of the Gamow-Teller (GT) matrix element in tritium decay and the binding energies of the trinucleons Gazit09 (); Marcucci12 (). However, the value determined for in those earlier studies was based on calculations which retained only terms up to N3LO in the axial current. As a matter of fact, one of the goals of the present work is to provide a determination of by also accounting for the N4LO corrections, represented by diagrams (i)-(x) in Fig. 1.

Most calculations of nuclear axial current matrix elements, such as those reported for the and He weak fusions of interest in solar physics in Refs. Park03 (); Marcucci13 (), and for muon capture on H and He in Ref. Marcucci12 (), have ignored these N4LO corrections. One exception is Ref. Klos13 (), which included effective one-body reductions, for use in a shell-model study, of some of the two-pion exchange terms derived in Ref. Park03 (). However, a systematic study of axial current contributions at N4LO is still lacking. The other goal of the present work is to provide a numerically exact estimate of these contributions in the H GT matrix element.

## Ii Formalism

The starting point of the derivation of nuclear current operators is the chiral Lagrangian for interacting pions and nucleons. This defines a quantum field theory which satisfies, besides all common general properties, like unitarity, analiticity, crossing symmetry and cluster decomposition, all constraints from chiral symmetry, in the form of chiral Ward identities, e.g., (partial) current conservation. Due to the (pseudo-) Goldstone boson character of the pions, their interactions can be organized according to increasing powers of their momenta, whose magnitude is generically denoted , much smaller than the hadronic scale GeV. From the chiral Lagrangian one can derive, in the canonical formalism, the chiral Hamiltonian, divided into a free part and an interacting part , which allows one to calculate transition amplitudes by applying the rules of time-ordered perturbation theory (TOPT),

(1) |

The evaluation of this amplitude is in practice carried out by inserting complete sets of eigenstates between successive terms of . Power counting is then used to organize the diagrammatic expansion (which in general will involve reducible—i.e., with purely nucleonic intermediate states—and irreducible contributions) in powers of . In this expansion we also take into account non-static contributions which represent nucleon-recoil corrections, by expanding a generic energy denominator as

(2) |

where denotes the kinetic energy of the intermediate purely-nucleonic state, the pion energy (or energies, as the case may be), and the ratio is of order . As a result the scattering amplitude admits the following expansion:

(3) |

where , and chiral symmetry ensures that is finite. In the case of the two-nucleon amplitude . Obviously, an infinite set of contributions to the TOPT expansion must be resummed in order to describe nuclear bound states. This is achieved by definining a kernel that satisfies a Lippmann-Schwinger (LS) equation and generates the above perturbative expansion of the scattering amplitude. Thus, a two-nucleon potential can be derived, assumed to admit the same kind of low-energy expansion as in Eq. (3), which when iterated in the LS equation,

(4) |

where denotes the free two-nucleon propagator , leads to the on-the-energy-shell () -matrix in Eq. (3), up to any specified order in the power counting. In this way one obtains

(5) | |||||

(6) | |||||

(7) | |||||

Notice that a term like is of order , since is of order and the implicit loop integration brings in a factor . The leading-order (LO) term, , consists of two (non-derivative) contact interactions and (static) one-pion exchange (OPE) (respectively displayed in panels (a’) and (b’), of Fig. 2), while the next-to-leading (NLO) term, , is easily seen to vanish Pastore11 (), since the leading non-static corrections to the (static) OPE amplitude add up to zero on the energy shell, while the remaining diagrams in represent iterations of , whose contributions are exactly canceled by (complete or partial cancellations of this type persist at higher orders). The next-to-next-to-leading (N2LO) term, which follows from Eq. (7), contains contact (involving two gradients of the nucleon fields) interactions, two-pion-exchange (TPE), loop corrections to LO contact interactions, and loop corrections to OPE potential (respectively displayed in panels (c’), (d’)-(f’), (g’) and (h’), and (i’), of Fig. 2). However, the procedure outlined above does not specify the potential uniquely, being affected by well known off energy-shell ambiguities. Indeed, at N2LO there is also a recoil correction to the OPE, which we write as Friar77 ()

(8) |

where is the leading order OPE potential, defined as

(9) |

() and () are the initial and final energies (momenta) of nucleon , and . There is an infinite class of corrections , labeled by the parameter , which, while equivalent on the energy shell () and hence independent of , are different off the energy shell. Friar Friar77 () has in fact shown that these different off-the-energy-shell extrapolations are unitarily equivalent, and thus do not affect physical observables. The off-shell ambiguity propagates to the next-order , but the unitary equivalence persists also at this order, i.e., at the two-pion exchange level Pastore11 ().

The inclusion (in first order) of electroweak interactions in the perturbative expansion of Eq. (1) is in principle straightforward. The weak transition operator can be expanded as Pastore11 (); Baroni16 ():

(10) |

where is of order and in this case. The nuclear weak axial charge, , and current, , operators follow from , where is the weak axial field, and it is assumed that has a similar expansion as . The requirement that, in the context of the LS equation, matches order by order in the power counting implies relations for , which can be found in Refs. Pastore11 (); Baroni16 (), similar to those derived above for , the strong-interaction potential. The lowest order terms that contribute to the axial current operators have , while for the axial charge. This implies that the off-shell ambiguity affects the axial current already at N3LO and the axial charge at N4LO. In the case of the electromagnetic operators the same was true with inverted roles of the charge and current Pastore11 (). There it was shown that different choices for the parameter for both the potential and the electromagnetic charge operator were unitarily equivalent. We expect the same to occur for the axial current, although this has not been verified explicitly. The specific form of the axial current we use corresponds to the choice for and , specifically Eq. (8) above and Eq. (19) of Ref. Pastore11 (). The remaining non-static corrections in the potential are as given in Eqs. (B8), (B10), and (B12) of that work.

We notice that at N4LO there are several one loop diagrams that contribute to the nuclear axial current. Diagrams (k), (l), (p), (q), and (r) of Fig. 1 are irreducible and in Ref. Baroni16 () they were shown to give the same contribution both in TOPT and HBPT. The remaining topologies contain reducible diagrams and require the subtraction of the iterations generated by the LS equation Pastore11 (); Piarulli13 (); Baroni16 (). The partially conserved axial current (PCAC) relation implies the conservation of the weak axial current in the chiral limit with the two-nucleon Hamiltonian given by and where the (two-nucleon) kinetic energy is counted as . This requirement, order by order in the power counting, translates into a set of non-trivial relations between the and the , , and (note that commutators implicitly bring in factors of ), see Eqs. (7.9)–(7.12) of Ref. Baroni16 (). These relations couple contributions of different orders in the power counting of the operators, and can only be satisfied up to a truncation of the low-energy expansion. In Ref. Baroni16 () it has been shown that the axial current, up to order , is conserved in the chiral limit. In particular we note that the sum of the loop corrections at order displayed in Fig. 1, when contracted with the three momentum of the external axial field, is equal to the following commutator

(11) |

where is the OPE potential, panel (b’) of Fig. 2, and is the LO two-body axial charge. Finally we note that the verification of PCAC, for nonvanishing pion mass, should come out as a natural consquence of the fact that we used chiral Lagrangians without making any approximations (besides neglecting some corrections at order , for further details we defer to Sec IV.B of Ref. Baroni16 ()). However an explicit verification of PCAC for tree level diagrams as well as loop corrections at order of Fig. 1 has not yet been performed.

## Iii Nuclear axial currents in Eft

In this section we report the expressions for the nuclear axial current in the limit of vanishing external field momentum (denoted as ) Baroni16 (). Of course, pion-pole contributions in Fig. 1 vanish in this limit. The expressions at LO and N2LO read

(12) | |||||

(13) |

while those at N3LO are separated into one-pion exchange (OPE) and contact (CT) terms corresponding respectively to panels (e) and (g) of Fig. 1,

(14) | |||||

(15) |

The LECs and in the OPE current effectively include the contributions associated with -isobar excitations ( degrees of freedom are integrated out in the EFT formulation adopted here) as well as short-range contributions involving vector meson exchanges, such as axial - transition mechanisms Park03 ().

Lastly, the expressions at N4LO are separated into terms originating from OPE, panel (s), and multi-pion exchange (MPE), panels (i), (k), (m), and (p),

(16) | |||||

(17) | |||||

where the loop functions are given by

(18) | |||||

(19) | |||||

(20) | |||||

(21) | |||||

(22) |

and

(23) |

In the equations above, and are the nucleon axial coupling constant and pion decay amplitude, and are the nucleon and pion mass, is the pion energy, and , , and are LECs, and entering the Lagrangian and multiplying the contact axial current (these LECs are discussed in Sec. IV). The nucleon spin and isospin operators are denoted by and , respectively, and the following charge-raising () and charge-lowering () combinations have been defined:

(24) |

The momenta and are

(25) |

where () is the nucleon initial (final) momentum and, in the limit of vanishing external field momentum, and are related via

(26) |

In Ref. Baroni16 () diagrams (w) and (x) of Fig. 1 were inadvertently omitted, only diagrams (u) and (v) were considered. We have evaluated them here, and obtained for the combined contribution of (u) and (w) the N4LO contact current

(27) |

where (in standard notation) is one of the two LECs in the four-nucleon contact interaction at LO. The pion-pole contribution from diagrams (v)+(x) follows as

(28) |

However, use of Fierz identities shows that the contact current in Eq. (27) vanishes identically Baroni16 ().

In a three-nucleon system the two-body loop corrections to the axial current enter at order , owing to the presence of a momentum-conserving -function . These loop corrections turn out to be of the same order as the three-body axial current, illustrated in Fig. 3 and first derived in Ref. Park03 (),

(29) | |||||

where the sum is over the cyclic permutations of the three nucleons, and in the limit .

Configuration-space expressions for these two- and three-body operators (denoted generically as 2B and 3B, respectively) follow from

(30) | |||||

(31) |

where the relative positions are defined as , and is the momentum cutoff, which we take as

(32) |

This cutoff does not modify the power counting of the various terms, as it is easily seen by expanding in powers of . In particular, the conservation of the vector current and axial current (in the chiral limit) is preserved up to the order considered in the present work.

Lastly, terms proportional to in the N2LO and N3LO currents are obtained by replacing with in configuration space (the momentum operator), and need to be symmetrized accordingly to preserve hermiticity. Explicit expressions for these Fourier transforms are listed in Appendix A.

## Iv Gamow-Teller matrix element in tritium -decay

The Gamow-Teller (GT) matrix element is obtained from the tritium half-life via (see Schiavilla98 () and references therein)

(33) |

where is the current experimental value g_A () for the nucleon axial coupling constant, is the outer radiative correction d_R (), is the half-life of , and and are Fermi functions reported in Ref. Simpson87 () to have the values and , respectively. The experimental value used for is s as obtained from Ref. Hardy15 (), and that used for is s as reported in Ref. Simpson87 (). Finally, and denote the reduced matrix element of the Fermi (F) and GT operators. The GT operator is the axial current constructed in Sec. III. The F operator is the vector charge and, while it too includes one- and two-body terms derived in Ref. Pastore11 (), the latter vanish in the limit of vanishing external field momentum, and only the one-body term at LO contributes in this limit.

The F and GT matrix elements are calculated with and wave functions
obtained with the hyperspherical-harmonics (HH) expansion method (see review Kievsky08 ())
from two- and three-nucleon potentials derived from either EFT or the conventional approach.
The combination of chiral potentials is denoted as N3LO/N2LO(500) [N3LO/N2LO(600)] corresponding
to cutoff MeV ( MeV), and consists of two-nucleon potentials at N3LO
from Refs. Entem03 (); Machleidt11 () and three-nucleon potentials at N2LO
from Refs. Epelbaum02 (); Nav07 (). ^{1}^{1}1Note that for consistency with the convention adopted
in Fig. 1, it would be more appropriate to label these two- and three-nucleon
potentials, respectively, as N4LO and N3LO. However, this is not the standard notation
used in the literature.
The combination of conventional potentials is denoted as AV18/UIX and consists of the Argonne
(AV18) two-nucleon potential Wiringa95 () and Urbana-IX (UIX) three-nucleon
potential Pudliner95 (). In all cases we obtain .
From this value we extract via Eq. (33) the experimental GT matrix element
as

(34) |

Contributions to the GT matrix element corresponding to the LO, N2LO, N3LO, N4LO, and N4LO(3Ba) axial operators are reported in Table 1, where the LEC in the N3LO(CT) operator is taken as in units of . The LECs and in the N3LO(OPE) operators are constrained by fits to scattering data, and two different sets of values (listed in the table caption) have been used in the present study, one from Refs. Entem03 (); Machleidt11 () and the other from a recent analysis of these data based on Roy-Steiner equations Hoferichter15 (), specifically the values corresponding to the column labeled N3LO in Table II of that work. The first set of and values (from Refs. Entem03 (); Machleidt11 ()) enters the chiral two- and three-nucleon potentials, used here to generate the H and He wave functions. Clearly, use of the second set from Ref. Hoferichter15 () in the N3LO(OPE) axial current is not consistent with these potentials; results for the GT matrix element are provided in that case only to give an estimate of the their sensitivity to the and values. As per the additional LECs in the three-nucleon potential, these have been obtained by the fitting procedure described below. In particular, we note that the LEC in the N3LO(CT) operator is related to via Eq. (37).

In the N4LO(3Ba) current we have only considered the term of Eq. (48) and neglected the term of Eq. (54) for reasons explained in Appendix A. The GT (and F) matrix elements are computed exactly, without approximation, with quantum Monte Carlo methods. The spin-isospin algebra is carried out with techniques similar to those developed in Ref. Schiavilla89 () for the electromagnetic current operator. The results reported in the tables below are based on random walks consisting of configurations. Statistical errors are not listed, but are typically at the few parts in , except in the special case of the N3LO(OPE) results, for which they are at the few % level (see below).

In Table 1 we report the results for the N3LO/N2LO(500) and N3LO/N2LO(600) models, and in parentheses those for the AV18/UIX model. The LO and N2LO axial operators do not need to be regularized, and hence the corresponding contributions for the AV18/UIX are the same for MeV and 600 MeV. However, the N3LO/N2LO contributions change (rather significantly at N2LO) as varies in this range due to the intrinsic cutoff dependence of the potentials. In the N3LO axial current of Eq. (14) the terms proportional to and have opposite signs and tend to cancel each other. This cancellation depends crucially on the values of the LECs and Hamiltonian model. In particular, when and are taken from Refs. Entem03 (); Machleidt11 (), the sum of their contributions for the N3LO/N2LO model is (in magnitude) comparable to the contribution from the non-local terms proportional to in Eq. (14).

The contributions from loop corrections, row labeled N4LO(MPE), are relatively large and comparable to those at N3LO(OPE). As a matter of fact, when the values for the and LECs are from Refs. Entem03 (); Machleidt11 (), the N3LO(OPE) contributions are an order of magnitude smaller than the N4LO(MPE) in the case of the chiral potentials. The origin of this large contribution can be traced back to the term proportional to the loop function in Eq. (17), specifically to the term with the factor in Eq. (18). It originates from box diagrams, panel (m) of Fig. 1 (see Ref. Baroni16 ()). All the N4LO corrections have opposite signs relative to the LO and N3LO(OPE).

500 MeV | 600 MeV | |
---|---|---|

LO | 0.9363(0.9224) | 0.9322 (0.9224) |

N2LO | –0.569(–0.844) | –0.457(–0.844) |

N3LO(OPE) | 0.825(1.304) | 0.043(7.517) |

N3LO(OPE) | 0.579(0.812) | 0.652(1.413) |

N3LO(CT) | –0.586(–0.721) | –0.717(–0.644) |

N4LO(OPE) | –0.697(–0.964) | –0.867(–1.216) |

N4LO(MPE) | –0.430(–0.565) | –0.532(–0.775) |

N4LO(3Ba) | –0.143(–0.183) | –0.153(–0.205) |

Next, we discuss the determination of the value for the LEC required to reproduce GT for the various Hamiltonian models we consider, by retaining corrections in the axial current up to either N3LO or N4LO. In order to compare with previous determinations of this LEC Park03 (); Gazit09 (); Marcucci12 (), we define an adimensional by rescaling as

(35) |

This is simply given by in terms of the LECs and introduced in Ref. Park03 () (in Park03 () these LECs multiply contact axial currents related to each other by a Fierz rearrangement, and are not therefore independent). We also note the relation

(36) |

where are adimensional, and was fixed in Ref. Park03 () by fitting GT in a hybrid calculation based on the AV18/UIX model and including N3LO corrections in the axial current.

N3LO | N4LO | |||
---|---|---|---|---|

500 | 600 | 500 | 600 | |

–0.421 | 0.742 | –1.607 | –1.048 | |

2.122 | 3.285 | 0.936 | 1.495 | |

–0.571 | 1.007 | –2.180 | –1.421 | |

0.769 | 2.038 | –0.417 | 0.235 | |

1.850 | 3.115 | 0.660 | 1.311 | |

1.043 | 2.764 | –0.566 | 0.318 |

Lastly, the LEC in the three-nucleon potential at N2LO is related to via Park03 (); Gazit09 (); Marcucci12 ()

(37) |

where is taken as 1 GeV here, while in Refs. Gazit09 (); Marcucci12 () GeV was adopted ( is not to be confused with the cutoff which regularizes the configuration-space expressions of the axial operators).

Values for the LECs are reported in Table 2 for the hybrid calculation based on the AV18/UIX Hamiltonian model, and in Table 3 for the chiral Hamiltonian model. In Table 2 the values for the various combinations considered above are listed, so that they can be compared with previous determinations Park03 (); Marcucci12 (); Marcucci11 (): they follow simply from reproducing the central value of GT in Eq. (34). In order to determine the values corresponding to the chiral potentials, we proceed as in Ref. Marcucci12 (). The H and He ground state wave functions are calculated using these potentials for = MeV and 600 MeV. We span the range , and, in correspondence to each in this range, determine so as to reproduce the binding energies of either H or He. The resulting trajectories are essentially indistinguishable, as shown in Fig. 4 for = MeV and in Fig. 5 for = MeV, and as already obtained in Ref. Marcucci12 (). Then, for each set of , the triton and He wave functions are calculated and the Gamow-Teller matrix element, denoted as GT, is determined, by including in the axial current corrections up to N3LO or N4LO. The ratio GT/GT for both values of the cutoff is shown in Fig. 6 for the N3LO case and Fig. 7 for the N4LO one. The LECs that reproduce GT (its central value) and the trinucleon binding energies are given in Table 3. The values for at N3LO are found to be consistent with those listed in Marcucci12 (), after allowance is made for the different (0.7 GeV in that work versus 1 GeV above) and for the fact that GT as determined here is slightly smaller than adopted in Marcucci12 ().

N3LO | N4LO | |||
---|---|---|---|---|

500 | 600 | 500 | 600 | |

–0.353 | –0.443 | –1.847 | –2.030 | |

–0.305 | –1.224 | –0.548 | –1.553 |

Alternatively, we could choose a different set of three-nucleon observables to fit these LECs. We consider here, together with the = binding energy, the doublet scattering length , for which we take the experimental value fm, obtained in Ref. Schoen03 (). In the range the resulting trajectories are displayed in Figs. 4 and 5 for MeV and 600 MeV, respectively. The experimental uncertainty in has been taken into account, and therefore the results of Figs. 4 and 5 are presented as a band. The trajectories originating from the = binding energies and scattering length are quite close to each other, but do not overlap. In the MeV case, there is a crossing point at =, while for MeV there is no crossing. In particular, using the in Table 3, we obtain fm for = MeV and fm for = MeV, when the N4LO (N3LO) contributions in the axial current are retained. The present calculations of the scattering wave functions ignore higher order electromagnetic interaction terms, such as those associated with the nucleons’ magnetic moments. These terms are known to reduce the value of about 3 % Kievsky08 (), when the AV18/UIX Hamiltonian model is used. Thus, the present analysis seems to indicate that the three = observables (= binding energies, GT, and ) are simultaneously reproduced, at least for MeV, when the nuclear axial current retains corrections up to N4LO.

## V Conclusions

To summarize, in the present work we have carried out a calculation of the F and GT matrix elements in H -decay with the charge-changing weak current recently derived in EFT up to N4LO (one loop). The trinucleon wave functions have been obtained from accurate hyperspherical harmonics solutions of the Schrödinger equation corresponding to either chiral (N3LO/N2LO) or conventional (AV18/UIX) nuclear potentials, and the relevant matrix elements have been computed by Monte Carlo integration methods without any approximations (statistical errors are typically at the level of a few parts in ).

We find that the OPE contributions at N3LO proportional to and interfere destructively and therefore depend strongly on the values of these LECs. As a consequence, the N4LO contributions turn out to be comparable (in magnitude) to the N3LO ones, even though nominally they are suppressed by a factor of relative to N3LO. This leads to a strong variation of the LEC as determined respectively at N3LO or at N4LO. It is possible that the convergence of the chiral series is not satisfactory for this observable and that the effective theory should be enlarged to include explicit ’s. An additional caveat is that, strictly speaking, the N4LO axial current calculations reported here should have involved the three-nucleon interaction at N3LO, whereas only the N2LO component has been considered in this work. Furthermore, the definition of the current operator is closely related to the prescription adopted for defining the nuclear potential off the energy-shell Baroni16 (). Whether different prescriptions lead to the same convergence pattern is a question that would require further investigation.

Finally, the LEC multiplying the contact axial current is related to the LEC in the three-nucleon potential. This and the other LEC which fully characterize this (contact) potential have been constrained by a simultaneous fit to the empirical values of the three-nucleon binding energies and GT matrix element. When the fit is carried out in a calculation including the axial current at N4LO, the resulting and also lead to a doublet scattering length in reasonable agreement with the experimental value for = MeV.

## Acknowledgments

An email exchange with B. Kubis in reference to the LECs is gratefully acknowledged. This research is supported by the U.S. Department of Energy, Office of Nuclear Physics, under contract DE-AC05-06OR23177 (A.B. and R.S.). A.B. was supported by a Jefferson Science Associates Theory Fellowship.

## Appendix A Configuration-space expressions

The Fourier transforms of two-body operators are easily reduced to one-dimensional integrals [or two-dimensional ones in the case of the N4LO(MPE) operator], which are then evaluated by Gaussian quadrature formulae. For example, the N3LO(OPE) current is given by

(38) |

where

(39) | |||||

(40) | |||||

(41) |

Here we have defined , the adimensional variable , as the adimensional momentum operator, and the radial functions

(42) | |||||

(43) |

where are spherical Bessel functions. We have also introduced adimensional constants (denoted with the overline) expressing them units of the cutoff . They are given by

(44) |

The functions and , and follow from those above by the replacement of the pre-factor as

(45) | |||

(46) |

The Fourier transform of the three-body operator is more involved. We express it as

(47) |

where

(48) | |||||

and the function is obtained from by replacing

(49) |