Chiral effective field theory predictions for muon capture on deuteron and 3He

Abstract

The muon-capture reactions 2H(\mu^-,\nu_\mu)nn and 3He(\mu^-,\nu_\mu)3H are
studied with nuclear strong-interaction potentials and charge-changing weak
currents, derived in chiral effective field theory. The low-energy constants
(LEC's) c_D and c_E, present in the three-nucleon potential and (c_D)
axial-vector current, are constrained to reproduce the A=3 binding energies and
the triton Gamow-Teller matrix element. The vector weak current is related to
the isovector component of the electromagnetic current via the
conserved-vector-current constraint, and the two LEC's entering the contact
terms in the latter are constrained to reproduce the A=3 magnetic moments. The
muon capture rates on deuteron and 3He are predicted to be 399(3) sec^{-1} and
1494 (21) sec^{-1}, respectively, where the spread accounts for the cutoff
sensitivity as well as uncertainties in the LEC's and electroweak radiative
corrections. By comparing the calculated and precisely measured rates on 3He, a
value for the induced pseudoscalar form factor is obtained in good agreement
with the chiral perturbation theory prediction.

Full text

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arXiv:1109.5563v2 [nucl-th] 24 Nov 2011
Chiral effective field theory predictions for muon capture on deuteron and3He
L.E. Marcuccia,b, A. Kievskyb, S. Rosatia, R. Schiavillac,d, and M. Vivianib
aDepartment of Physics, University of Pisa, 56127 Pisa, Italy
bINFN-Pisa, 56127 Pisa, Italy
cDepartment of Physics, Old Dominion University, Norfolk, VA 23529, USA
dJefferson Lab, Newport News, VA 23606
(Dated: November 28, 2011)
The muon-capture reactions2H(µ−,νµ)nn and3He(µ−,νµ)3H are studied with nuclear potentials
and charge-changing weak currents, derived in chiral effective field theory. The low-energy constants
(LEC’s) cD and cE, present in the three-nucleon potential and (cD) axial-vector current, are con-
strained to reproduce the A = 3 binding energies and the triton Gamow-Teller matrix element. The
vector weak current is related to the isovector component of the electromagnetic current via the
conserved-vector-current constraint, and the two LEC’s entering the contact terms in the latter are
constrained to reproduce the A = 3 magnetic moments. The muon capture rates on deuteron and
3He are predicted to be 399 ± 3 sec−1and 1494 ± 21 sec−1, respectively. The spread accounts for
the cutoff sensitivity as well as uncertainties in the LEC’s and electroweak radiative corrections. By
comparing the calculated and precisely measured rates on3He, a value for the induced pseudoscalar
form factor is obtained in good agreement with the chiral perturbation theory prediction.
PACS numbers: 23.40.-s,21.45.-v,27.10.+h
When negative muons pass through matter, they can
be captured into high-lying atomic orbitals. They then
quickly cascade down into the 1S orbit, where two com-
peting processes occur: one is ordinary decay µ−→
e−νeνµ, and the other is (weak) capture by the nucleus
µ−A(Z,N) → νµA(Z − 1,N + 1). Apart from tiny cor-
rections due to bound-state effects (chief among which is
time-dilation) [1], the decay rate is essentially the same
as for a free muon and, in light nuclei, is much larger than
the rate for capture. The latter proceeds predominantly
through the basic process pµ−→ nνµinduced by the ex-
change of a W+boson, and its rate, which would naively
be expected to scale with the number Z of protons in the
nucleus, is enhanced by an additional flux factor of Z3,
originating from the square of the atomic wave function
(w.f.) evaluated at the origin [2]. Thus capture, with a
rate proportional to Z4, dominates decay at large Z.
Muon capture on hydrogen is, in principle, best suited
to obtain information on the nucleon matrix element of
the charge-changing quark current dγµ(1−γ5)u, respon-
sible for the process pµ−→ nνµ. Ignoring contributions
from second-class currents [3] for which there is presently
no firm experimental evidence [4], it is parametrized in
terms of four form factors (f.f.’s): two, F1(q2) and F2(q2),
from the polar-vector component of the weak current
are related to the (isovector) electromagnetic (EM) form
factors of the nucleon by the conserved-vector-current
(CVC) constraint; two, the axial and induced pseu-
doscalar f.f.’s GA(q2) and GPS(q2), go along with the
axial-vector part of the weak current. The F1(q2) and
F2(q2) f.f.’s are well known over a wide range of momen-
tum transfers q2from elastic electron scattering data on
the nucleon [5]. The f.f. GA(q2) is also quite well known:
its value at vanishing q2, gA= 1.2695(29), is from neu-
tron β-decay [6], while its q2-dependence is parametrized
as GA(q2) = gA/(1 − q2/Λ2
an analysis of pion electro-production data [7] and direct
measurements of the reaction pνµ→ nµ+[8].
Of the four f.f.’s, the induced pseudoscalar GPS(q2)
is the least known. The MuCap collaboration at PSI
has recently reported a precise measurement of the rate
for capture on hydrogen in the 1S singlet hyperfine
state:Γ(1H)|singlet = 725.0 ± 13.7(stat) ± 10.7(syst)
sec−1[9].Based on this value, an indirect “experi-
mental”determination of GPS at the momentum trans-
fer q2
µrelevant for µ−capture on hydrogen,
GEXP
0) = 7.3±1.2, has been obtained by using for the
remaining f.f.’s the values discussed above and by eval-
uating electroweak radiative corrections [10]. The latter
lead to a 2.8% increase in the rate on hydrogen, and are
crucial for bringing GEXP
PS
within less than 1σ of the most
recent theoretical prediction, GTH
obtained in chiral perturbation theory (χPT). For a re-
cent and comprehensive review of theoretical and exper-
imental efforts to determine GPS(q2) see Refs. [12, 13].
In the present letter, we focus on the reactions
2H(µ−,νµ)nn and3He(µ−,νµ)3H, hereafter referred to
as µ–2 and µ–3, respectively. There are a couple of rea-
sons for undertaking this study now: (i) the forthcoming
measurement of the µ–2 rate Γ(2H) in the doublet hy-
perfine state by the MuSun collaboration at PSI with a
projected 1% precision [13, 14]. This and the already
available, and remarkably precise, measurement of the
µ–3 rate, Γ(3He) = 1496±4 sec−1[15], will make it pos-
sible to put tight constraints on GEXP
χPT prediction for this f.f. far more sharply than up to
now. (ii) A number of low-energy weak processes of as-
trophysical interest, such as the weak captures on proton
A)2, with ΛA= 1 GeV from
0= −0.88m2
PS(q2
PS(q2
0) = 8.2 ± 0.2 [11],
PS(q2) and to test the

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and3He, and neutrino reactions on light nuclei, are not
accessible experimentally. In order to have some level
of confidence in the reliability of their cross section esti-
mates, it becomes crucial to study, within the same the-
oretical framework, related electroweak reactions, whose
rates are known experimentally, like muon captures [16].
Theoretical work on the µ–2 and µ–3 reactions is quite
extensive (see Refs. [12, 13, 17, 18]).
tions have been performed within two different schemes:
the “standard nuclear physics approach” (SNPA) and the
approach known as “hybrid” chiral effective field theory
(χEFT). In SNPA, Hamiltonians based on conventional
two-nucleon (NN) and three-nucleon (NNN) potentials
are used to calculate the nuclear w.f.’s, and the weak
transition operator includes, beyond the one-body contri-
bution (the impulse approximation—IA) associated with
the basic process pµ−→ nνµ, meson-exchange currents
as well as currents arising from the excitation of ∆-isobar
degrees of freedom [19]. In the hybrid χEFT approach,
the weak operators are derived in χEFT, but their ma-
trix elements are evaluated between w.f.’s obtained from
conventional potentials. Typically, the SNPA and hybrid
χEFT predictions are in good agreement with each other.
For example, for the µ–2 rate, the SNPA calculation of
Ref. [18] gives 391 sec−1, to be compared with the hy-
brid χEFT studies of Refs. [20] and [18], which report
386 sec−1and 393 ± 1 sec−1, respectively. The differ-
ences between Refs. [20] and [18] are due to contributions
of loop corrections and contact terms in the vector part
of the weak current, which were neglected in Ref. [20].
For the µ–3 rate, the SNPA calculation of Ref. [18] gives
1486 sec−1, while the hybrid χEFT studies of Refs. [21]
and [18] report, respectively, 1499±16 sec−1and 1484±4
sec−1. Here, the differences between Refs. [21] and [18]
arise mostly from the inclusion in Ref. [21] of vacuum
polarization effects on the muon bound state w.f. [10]—
these would increase the SNPA and hybrid χEFT results
of Ref. [18] quoted above for the µ–3 rate to, respectively,
1496 sec−1and 1494± 4 sec−1.
One of the objectives of the present work is to carry
out a χEFT calculation of the µ–2 and µ–3 rates. Chi-
ral EFT is a formulation of quantum chromodynamics
(QCD) in terms of effective degrees of freedom suitable
for low-energy nuclear physics: pions and nucleons. The
symmetries of QCD, in particular its (spontaneously bro-
ken) chiral symmetry, severely restrict the form of the in-
teractions of nucleons and pions among themselves and
with external electroweak fields, and make it possible
to expand the Lagrangian describing these interactions
in powers of Q/Λχ, where Q is pion momentum and
Λχ ∼ 700 MeV is the chiral-symmetry-breaking scale.
As a consequence, classes of Lagrangians emerge, each
characterized by a given power of Q/Λχand each involv-
ing a certain number of unknown coefficients, so called
low-energy constants (LEC’s). While these LEC’s could
in principle be determined by theory (for instance, in lat-
So far, calcula-
tice QCD calculations), they are in practice constrained
by fits to experimental data. Some of them (for exam-
ple, gA and the pion decay amplitude Fπ) characterize
the coupling (at lowest order) of pions to nucleons and,
in particular, the strength of one- and two-pion-exchange
terms (denoted, respectively, OPE and TPE) in the NN
potential [22, 23], i.e. its long-rangecomponents. Some of
the other LEC’s multiply NN (and multinucleon) contact
interactions, and therefore encode short-range physics,
which in a meson-exchange picture would, for example,
be associated with vector-meson exchanges or excitation
of baryon resonances, like the ∆ isobar.
The NN potential has been derived up to order
(Q/Λχ)4in the chiral expansion [22, 23].
of OPE and TPE with interaction vertices from lead-
ing, next-to-leading, and next-to-next-to-leading πN chi-
ral Lagrangians, and of contact terms. The LEC’s have
been constrained by accurate fits to the NN scattering
database at energies below the pion production thresh-
old (see Ref. [23] for a review).
which first contributes at order (Q/Λχ)3, includes S- and
P-wave TPE—its P-wave piece is the familiar Fujita-
Miyazawa NNN potential—a OPE plus NN contact term
with LEC cDand a NNN contact terms with LEC cE.
The vector and axial pieces of the weak current
have been derived up to order Q/Λχ in, respectively,
Refs. [24, 25] and [26]. The one-body operators are the
same as those obtained in the SNPA by retaining, in the
expansion of the covariant single-nucleon four-current,
corrections up to order (v/c)2relative to the leading-
order term [19]. Two-body operators in the axial cur-
rent (charge) first enter at order (Q/Λχ)0[(Q/Λχ)−1],
and are suppressed, in the power counting, by (Q/Λχ)3
[Q/Λχ] relative to the one-body term of order (Q/Λχ)−3
[(Q/Λχ)−2]. In the axial current, these terms include a
OPE contribution, involving the known LEC’s c3and c4
(determined by fits to the NN data [23]), and one contact
current, whose strength is parametrized by the LEC dR
(see below). In the axial charge, only OPE contributes,
and the associated operator is proportional to gA/F2
One-loop corrections to the axial charge and current from
TPE, which enter at Q/Λχ and are therefore strongly
suppressed relative to the leading-order one-body terms,
are ignored, since their contributions are expected to be
tiny.
The vector weak current is related (via the CVC con-
straint) to the EM current, which includes, up to or-
der Q/Λχ, OPE and TPE (i.e., one-loop corrections),
as well as isoscalar and isovector contact terms, whose
strengths are parametrized by the LEC’s denoted, re-
spectively, as g4Sand g4V in the following [24, 26]. It has
been shown [25] that such a current satisfies the continu-
ity equation with the NN potential at order (Q/Λχ)2. In
this regard, we note that the construction of a conserved
current with the (Q/Λχ)4NN potential used here would
require the inclusion of terms up to order (Q/Λχ)3, i.e.,
It consists
The NNN potential,
π.

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3
two-loop corrections. This is a daunting task, well be-
yond the present state of the art. In a more speculative
vein, it is also not obvious that such a theory could be
made predictive, given the presumably large number of
contact terms with unknown LEC’s that it would entail.
Finally, we notice that potentials and currents have
power-law behavior for large momenta, and need to be
regularized. This is accomplished in practice by introduc-
ing a momentum-cutoff function. In the present work,
the cutoff Λ is taken to be 500 MeV and 600 MeV.
We now turn our attention to the determination of
the LEC’s dR, cD, cE, g4S, and g4V. In the past, cD
and cE were fixed by fitting the triton binding energy
(BE) together with an additional strong-interaction ob-
servable, such as the nd doublet scattering length2and
or4He BE. However, this led to significant uncertain-
ties, due to strong correlations between these observ-
ables [27]. As the authors of Ref. [28] have observed,
the LEC’s dR and cD are related to each other via
dR =
cleon mass), and therefore one can fix cD (or dR) and
cE by fitting the triton BE and half-life (specifically,
the Gamow-Teller matrix element). Thus, we proceed
as follows. The3H and3He ground state w.f.’s are cal-
culated with the hyperspherical-harmonics method (see
Ref. [29] for a review) using the chiral NN+NNN po-
tentials of Refs. [22, 23, 30] for Λ = 500 and 600 MeV.
The corresponding set of LEC’s {cD,cE} is determined
by fitting the A = 3 experimental BE’s, BE(3H)=8.475
MeV and BE(3He)=7.725 MeV, corrected for small con-
tributions (+7 keV in3H and –7 keV in3He) due to
the n-p mass difference [31], since this effect is neglected
in the present calculations.
cD ∈ [−3,3], and, in correspondence to each cD in this
range, determine cEso as to reproduce either BE(3H) or
BE(3He). The resulting trajectories are shown in Fig. 1,
and are nearly indistinguishable. Their average, shown
by the red lines in Fig. 1, leads to A = 3 BE’s within 10
keV of the experimental values above. Then, for each set
of {cD,cE}, the triton and3He w.f.’s are calculated and,
using the χEFT weak axial current discussed above, the
Gamow-Teller matrix element of tritium β-decay (GTTH)
is determined.The ratio GTTH/GTEXPis shown in
Fig. 2, for both values of the cutoff Λ. We have used
GTEXP= 0.955 ± 0.004, as obtained in Ref. [18], ex-
cept that we have conservatively doubled the error, rep-
resented by the shadowed band in the figure. The range
of cD values, for which GTTH= GTEXPwithin the ex-
perimental error, are [−0.20,−0.04] for Λ = 500 MeV,
and [−0.32,−0.19] for Λ = 600 MeV. The corresponding
ranges for cEare [−0.208,−0.184] and [−0.857,−0.833],
respectively. We note that, for each pair of {cD,cE} in
the selected range, the scattering length2and is calcu-
lated to be2and = 0.666 ± 0.001 fm for Λ = 500 MeV
and2and = 0.696 ± 0.001 fm for Λ = 600 MeV, which
should be compared with2and= 0.675 fm [29], obtained
MN
ΛχgAcD+1
3MN(c3+ 2c4) +1
6(MN is the nu-
We then span the range
-3 -2
-1
0
cD
1
23
-2
-1.5
-1
-0.5
0
0.5
cE
3H
3He
average
Λ=500 MeV
Λ=600 MeV
FIG. 1: (Color online) cD-cE trajectories fitted to reproduce
the experimental3H and3He BE’s. See text for explanation.
-3-2
-1
0
cD
1
23
0.95
0.96
0.97
0.98
0.99
1
1.01
1.02
1.03
1.04
1.05
GT
TH/GT
EXP
Λ=500 MeV
Λ=600 MeV
FIG. 2: The ratio GTTH/GTEXPas function of the LEC cD.
with Λ = 500 MeV and {cD,cE} = {1.0,−0.029}, as
originally set in Ref. [30]. The most recent experimental
determination gives2and= 0.645 ± 0.010 fm [32].
For the minimum and maximum values of {cD,cE}
intheselected range,
i.e.,
{−0.04,−0.184} for Λ = 500 MeV, and {−0.32,−0.857}
and {−0.19,−0.833} for Λ = 600 MeV, we have deter-
mined the isoscalar and isovector LEC’s, g4S and g4V,
entering the NN contact terms of the EM current by re-
producing the A = 3 magnetic moments. These LEC’s
are listed in Table I.
{−0.20,−0.208} and
TABLE I: The LEC’s g4S and g4V associated with the
isoscalar and isovector NN contact terms in the EM current
for Λ = 500 and 600 MeV. See text for explanation.
{cD,cE}
Λ=500 MeV {−0.20,−0.208} 0.207 ± 0.007 0.765 ± 0.004
{−0.04,−0.184} 0.200 ± 0.007 0.771 ± 0.004
Λ=600 MeV {−0.32,−0.857} 0.146 ± 0.008 0.585 ± 0.004
{−0.19,−0.833} 0.145 ± 0.008 0.590 ± 0.004
g4S
g4V
Having fully constrained the NNN potential and weak
current, we present in Table II the χEFT predictions for
the µ–2 and µ–3 rates, Γ(2H) and Γ(3He). For Γ(2H),
we also show the individual contributions of nn chan-
nels with total angular momentum J ≤ 2 (1S0,3P0,3P1,
3P2,1D2 and3F2). Higher partial waves are known to
contribute less than 0.5 % to Γ(2H) [18]. The one-body
(IA) and (one+two)-body (FULL) results are listed sep-

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4
arately. Note that the IA results depend on the cutoff
Λ through the nuclear potentials. Theoretical errors in
the FULL results arise from the fitting procedure, and are
due primarily to the experimental error on GTEXP. They
are not indicated when less than 0.1 sec−1. Electroweak
radiative corrections have been included as estimated in
Ref. [10] for hydrogen and3He—we have assumed that
those for deuterium are the same as for hydrogen. The
cutoff dependence of the predictions is weak, at less than
1% level, thus suggesting that the mismatch between the
chiral order of the potentials and that of the currents
may be of little numerical import. If we also account
for uncertainties in the electroweak radiative corrections
of the order of 0.4% [10], we can conservatively quote
Γ(2H) = 399 ± 3 sec−1and Γ(3He) = 1494 ± 21 sec−1.
These predictions are in good agreement with available
experimental data (although those on Γ(2H) [12] have
large errors), as well as with results of recent theoreti-
cal studies [18, 20, 21]. Finally, a comparison between
the calculated and measured µ–3 rates makes it possi-
ble to put a constraint on the induced pseudoscalar f.f.
GPS(q2) at q2
µrelevant for the µ–3 reac-
tion. By varying GPS(q2
0) so as to match the theoretical
upper (lower) value with the experimental lower (upper)
value for the rate, we obtain GPS(q2
agreement with the χPT prediction of 7.99 ± 0.20 [11].
0= −0.954m2
0) = 8.2±0.7, in good
TABLE II: Total rates for muon capture on deuteron Γ(2H) and3He Γ(3He), in sec−1, corresponding to Λ = 500 and 600 MeV.
The one-body (IA) and (one+two)-body (FULL) contributions are listed, along with the individual partial-wave contributions
to Γ(2H). Theoretical uncertainties in the FULL results, not reported when below 0.1 sec−1, are due to the fitting procedure.
1S0
238.8
238.7
3P0
21.1 44.0 72.4 4.5 0.9
20.9 43.8 72.0 4.5 0.9
3P1
3P2
1D2
3F2
Γ(2H)
381.7
380.8
Γ(3He)
1362
1360
IA(Λ = 500 MeV)
IA(Λ = 600 MeV)
FULL(Λ = 500 MeV) 254.4±0.9 20.5 46.8 72.1 4.5 0.9 399.2±0.9 1488±9
FULL(Λ = 600 MeV) 255.2±1.0 20.3 46.6 71.6 4.5 0.9 399.1±1.0 1499±9
The authors would like to thank P. Kammel for encour-
aging us to carry out this study, and D. Gazit, P. Navr´ atil
and S. Quaglioni for useful discussions. The work of R.S.
is supported by the U.S. Department of Energy, Office of
Nuclear Physics under contract DE-AC05-06OR23177.
[1] A. Czarnecki, G.P. Lepage, and W.J. Marciano, Phys.
Rev. D 61, 073001 (2000).
[2] H. Primakoff, Rev. Mod. Phys. 31, 802 (1959).
[3] S. Weinberg, Phys. Rev. 112, 1375 (1958).
[4] N. Severijns, M. Beck, and O. Naviliat-Cuncic, Rev. Mod.
Phys. 78, 991 (2006).
[5] C.E. Hyde-Wright and K. de Jager, Ann. Rev. Nucl. Part.
Sci. 54, 217 (2004).
[6] C. Amsler et al. (Particle Data Group), Phys. Lett. B
667, 1 (2008).
[7] E. Amaldi, S. Fubini, and G. Furlan, Electroproduction at
Low Energy and Hadron Form Factors, (Springer Tracts
in Modern Physics No. 83, 1979), p.1.
[8] T. Kitagaki et al., Phys. Rev. D 28, 436 (1983).
[9] V.A. Andreev et al. (MuCap Collaboration), Phys. Rev.
Lett. 99, 032002 (2007).
[10] A. Czarnecki, W.J. Marciano, and A. Sirlin, Phys. Rev.
Lett. 99, 032003 (2007).
[11] V. Bernard, N. Kaiser, and Ulf-G. Meissner, Phys. Rev.
D 50, 6899 (1994); N. Kaiser, Phys. Rev. C 67, 027002
(2003).
[12] T. Gorringe and H.W. Fearing, Rev. Mod. Phys. 76, 31
(2004).
[13] P. Kammel and K. Kubodera, Ann. Rev. Nucl. Part. Sci.
60, 327 (2010).
[14] V.A.Andreev
et al.
arXiv:1004.1754.
[15] P. Ackerbauer et al., Phys. Lett. B 417, 224 (1998).
[16] E.G. Adelberger et al., Rev. Mod. Phys. 83, 195 (2011).
[17] D.F. Measday, Phys. Rep. 354, 243 (2001).
[18] L.E. Marcucci et al., Phys. Rev. C 83, 014002 (2011).
[19] L.E. Marcucci et al., Phys. Rev. C 63, 015801 (2000).
[20] S. Ando et al., Phys. Lett. B 533, 25 (2002).
[21] D. Gazit, Phys. Lett. B 666, 472 (2008).
[22] D.R. Entem and R. Machleidt, Phys. Rev. C 68, 041001
(2003).
[23] R. Machleidt and D.R. Entem, Phys. Rep. 503, 1 (2011).
[24] T.-S. Park, D.-P. Min, and M. Rho, Nucl. Phys. A 596,
515 (1996); Y.-H. Song, R. Lazauskas, and T.-S. Park,
Phys. Rev. C 79, 064002 (2009).
[25] S. Pastore et al., Phys. Rev. C 80, 034004 (2009); S.
K¨ olling et al., Phys. Rev. C 80, 045502 (2009).
[26] T.-S. Park et al., Phys. Rev. C 67, 055206 (2003).
[27] A. Kievsky et al., Phys. Rev. C 81, 044003 (2010).
[28] A. Gardestig and D.R. Phillips, Phys. Rev. Lett. 96,
232301 (2006); D. Gazit, S. Quaglioni, and P. Navr´ atil,
Phys. Rev. Lett. 103, 102502 (2009).
[29] A. Kievsky et al., J. Phys. G: Nucl. Part. Phys. 35,
063101 (2008).
[30] P. Navr´ atil, Few-Body Syst. 41, 117 (2007).
[31] A. Nogga et al., Phys. Rev. C 67, 034004 (2003).
[32] K. Schoen et al., Phys. Rev. C 67, 044005 (2003).
(MuSun Collaboration),