Coherent elastic neutrino-nucleus scattering:
EFT analysis and nuclear responses
Martin Hoferichter ,1,2,* Javier Mene´ndez ,3,4,† and Achim Schwenk 5,6,7,‡
1Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics,
University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
2Institute for Nuclear Theory, University of Washington, Seattle, Washington 98195-1550, USA
3Department of Quantum Physics and Astrophysics and Institute of Cosmos Sciences,
University of Barcelona, 08028 Barcelona, Spain
4Center for Nuclear Study, The University of Tokyo, 113-0033 Tokyo, Japan
5Institut für Kernphysik, Technische Universität Darmstadt, 64289 Darmstadt, Germany
6ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH,
64291 Darmstadt, Germany
7Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany
(Received 21 July 2020; accepted 10 September 2020; published 23 October 2020)
The cross section for coherent elastic neutrino-nucleus scattering (CEνNS) depends on the response of
the target nucleus to the external current, in the Standard Model (SM) mediated by the exchange of a
Z boson. This is typically subsumed into an object called the weak form factor of the nucleus. Here, we
provide results for this form factor calculated using the large-scale nuclear shell model for a wide range of
nuclei of relevance for current CEνNS experiments, including cesium, iodine, argon, fluorine, sodium,
germanium, and xenon. In addition, we provide the responses needed to capture the axial-vector part of the
cross section, which does not scale coherently with the number of neutrons, but may become relevant for
the SM prediction of CEνNS on target nuclei with nonzero spin. We then generalize the formalism allowing
for contributions beyond the SM. In particular, we stress that in this case, even for vector and axial-vector
operators, the standard weak form factor does not apply anymore, but needs to be replaced by the
appropriate combination of the underlying nuclear structure factors. We provide the corresponding
expressions for vector, axial-vector, but also (pseudo)scalar, tensor, and dipole effective operators,
including two-body-current effects as predicted from chiral effective field theory (EFT). Finally, we update
the spin-dependent structure factors for dark matter scattering off nuclei according to our improved
treatment of the axial-vector responses.
DOI: 10.1103/PhysRevD.102.074018
I. INTRODUCTION
CEνNS, suggested as a probe of the weak current as
early as 1974 [1], was finally observed by the COHERENT
collaboration in 2017 [2]. After the initial detection in CsI,
also the scattering off argon has just been observed [3,4].
With future advances in COHERENT and other experi-
ments [5–13], the CEνNS process will soon develop into
another sensitive low-energy probe of physics beyond the
Standard Model (BSM) [14].
A crucial input in interpreting the measured cross section
is the response of the nucleus. If BSM constraints are to be
extracted, the nuclear structure has to be provided from
elsewhere. In fact, since the weak charge of the proton is
small, the SM CEνNS process mainly probes the nuclear
neutron distribution, which is significantly less constrained
experimentally than the electromagnetic charge distribution.
Apart from CEνNS, the only direct information of the
neutron distribution comes from parity-violating electron
scattering (PVES) off lead [15,16]. Accordingly, assuming
the absence of a significant BSM signal, the measured
CEνNS cross section could be used to constrain the neutron
distribution instead [17–21]. Currently, the nuclear input
used in the interpretation of CEνNS experiments is mainly
derived from relativistic mean-field methods (RMF)
[22,23], even though results based on nonrelativistic
energy-density functionals are also available [24–26]. For
argon, there is a recent first-principles calculation based on
coupled-cluster theory [27].
*hoferichter@itp.unibe.ch
†menendez@fqa.ub.edu
‡schwenk@physik.tu-darmstadt.de
Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article’s title, journal citation,
and DOI. Funded by SCOAP3.
PHYSICAL REVIEW D 102, 074018 (2020)
2470-0010=2020=102(7)=074018(30) 074018-1 Published by the American Physical Society
Here, we provide nuclear structure results for CEνNS,
extending large-scale nuclear shell model calculations
developed in the context of nuclear structure factors in
direct-detection searches for dark matter [28–36]. First, the
level of agreement between the shell-model, the RMF, and
coupled-cluster results suggests that the form factor uncer-
tainties are not as severe as claimed in Ref. [37], but in
addition the shell-model approach also allows us to address
the spin-dependent (SD) responses, which are similar, but
somewhat different to the ones in SD dark matter searches.
To this end, we first derive the decomposition of the
cross section into Wilson coefficients of effective operators,
hadronic matrix elements, and nuclear structure factors.
In the SM the effective operators just parametrize the
Z-boson exchange, but this approach can be conveniently
extended to include BSM effects. The hadronic matrix
elements determine the hadronization of these operators
at the single-nucleon level, and finally the nuclear structure
factors take into account the many-body nuclear matrix
element of these single-nucleon currents. As a first step,
we demonstrate how the standard weak form factor
emerges when combining all these ingredients into a single
object. However, this analysis shows that even for the
coherent part of the nuclear response four different under-
lying structure factors contribute to the cross section.
Therefore, in principle the weak form factor needs to be
modified as well when allowing for BSM effects in the
Wilson coefficients, since their contribution does not
factorize.
While for the dominant vector operators corrections
beyond the single-nucleon currents are small [38,39],
since the magnetic-moment form factors happen to be
kinematically suppressed, this is no longer true for the
axial-vector [29] or for scalar currents, see [32–34,36,
40–45] for the analogous effects in the case of dark matter
scattering off nuclei. As long as the SM dominates, such
effects will only become relevant for CEνNS once experi-
ments become sensitive to SD responses. Otherwise,
mainly the limits on scalar operators would be affected,
but in CEνNS such contributions are suppressed due to the
need for right-handed neutrinos and lack of interference
with the SM.
The paper is organized as follows: in Sec. II we first
review the necessary formalism, regarding effective oper-
ators, hadronic matrix elements, and nuclear responses,
with some details of the multipole expansion and nuclear-
structure calculations summarized in the Appendices. In
Sec. III we first introduce the charge and weak form factors
in the context of electron scattering, before discussing
the application to CEνNS. In particular, we present an
improved treatment of the axial-vector responses both for
CEνNS and dark matter. In Sec. IV we discuss how the
nuclear responses need to be adapted when considering
SM extensions, before concluding with a summary in
Sec. V.
II. FORMALISM
A. Effective Lagrangians
As a first step, we review the operator basis for CEνNS
[36,46]1
Lð5Þ ¼ CFν¯σμνPLνFμν;
Lð6Þ ¼
X
q
ðCVq ν¯γμPLνq¯γμqþ CAq ν¯γμPLνq¯γμγ5q
þ CTq ν¯σμνPLνq¯σμνqÞ;
Lð7Þ ¼
X
q

CSq þ
8π
9
C0gS

ν¯PLνmqq¯q
þ
X
q
CPq ν¯PLνmqq¯iγ5q −
8π
9
C0gSν¯PLνθ
μ
μ; ð1Þ
where we adopted the following conventions: neutrino
indices are suppressed throughout, indicating that oscil-
lation effects are usually negligible at the scale of CEνNS
experiments, so that incoming and outgoing flavors are
understood to be identical. In comparison to the case of a
dark-matter spin-1=2 particle [36] the number of operators
is reduced by a factor of 2 when assuming that the neutrino
is left handed. This is implemented in Eq. (1) in terms of
projectors PL ¼ ð1 − γ5Þ=2, and given that observing
chirality-violating effects would require right-handed neu-
trino beams (suppressed by tiny neutrino masses), we will
not consider the opposite chirality in the following.
With these conventions the set of operators is then
similar to the dark-matter case: at dimension-5 level there
is a single (dipole) operator involving the photon field
strength tensor Fμν. At dimension-6 we have the vector and
axial-vector operators already present in the SM, as well as
the tensor operator. Introducing quark masses for renorm-
alization-group invariance, the scalar and pseudoscalar
operators would be counted as dimension-7. The operator
involving the QCD trace anomaly θμμ would also be of
dimension-7. We have already rewritten the gluon term
GaμνG
μν
a in terms of this operator (we will not consider the
operators involving the dual field strength tensor G˜aμν or the
photon field strength). In particular, we already integrated
out the heavy quarks [47] and absorbed their effect into
C0gS ¼ CSg −
1
12π
X
Q¼c;b;t
CSQ; ð2Þ
where CSg is the original coefficient of the ν¯PLναsGaμνG
μν
a
gluon operator and we used the relation
1This definition strictly applies to Dirac neutrinos, while in the
Majorana case a symmetry factor of 2 would arise in the
amplitudes. To avoid this complication, an additional factor of
1=2 is implied in the definition of the effective operators for
Majorana neutrinos, in which case also the diagonal vector and
tensor currents vanish.
HOFERICHTER, MENE´NDEZ, and SCHWENK PHYS. REV. D 102, 074018 (2020)
074018-2
θμμ ¼
X
q
mqq¯q −
9
8π
αsGaμνG
μν
a þOðα2sÞ ð3Þ
in the transition. Elsewhere, the sum over q in principle refers
to all quark species, but in practicewewill restrict the analysis
to the light quarksq ¼ u,d, s. Finally, there areoperatorswith
derivatives acting on the neutrino fields (in analogy to the
spin-2 operator for dark matter), but we will concentrate on
the more frequently considered operators in Eq. (1). We note
that the dimensional counting is not unambiguous regarding
the quark mass mq, e.g., sometimes the tensor operator is
introduced at dimension-7 by adding a factor mq in this
operator as well [46] [the notation in Eq. (1) follows
Ref. [48] ]. Finally, we stress that the chirality-flipping
operators, with scalar and tensor operators on the neutrino
bilinear, require the presence of (final-state) right-handed
neutrinos. In SMEFT [49] such operators are suppressed
beyond dimension-6 level. In addition, the dipole operator
leads to a new long-range interaction, and therefore CF is
strongly constrained by astrophysical observations [50,51].
However, such operators have been suggested as a potential
BSM explanation of the excess of electronic recoil events
observed by XENON1T [52].
In the SM all Wilson coefficients except for CVq and CAq
vanish, with Z exchange leading to the matching relations
CVu ¼ −
GFffiffiffi
2
p

1 −
8
3
sin2 θW

;
CVd ¼ CVs ¼
GFffiffiffi
2
p

1 −
4
3
sin2 θW

;
CAu ¼ −CAd ¼ −CAs ¼
GFffiffiffi
2
p ; ð4Þ
with Fermi constant GF ¼ 1.1663787ð6Þ × 10−5 GeV−2
[53,54]. In the notation of Refs. [2,3,14], the deviations
from these SM values are often expressed as
CVq − CVq jSM ¼ −
ffiffiffi
2
p
GFϵ
qV
ee ¼ −
ffiffiffi
2
p
GFðϵqLee þ ϵqRee Þ;
CAq − CAq jSM ¼
ffiffiffi
2
p
GFϵ
qA
ee ¼
ffiffiffi
2
p
GFðϵqLee − ϵqRee Þ; ð5Þ
where the sign of ϵqAee has been chosen in accordance with
Ref. [14]. Finally, we can define dimensionless Wilson
coefficients C˜i ¼ Ci=Λn, where Λ either corresponds to the
respective BSM scale, or, in the SM, to the Higgs vacuum
expectation value Λ ¼ ð ffiffiffi2p GFÞ−1=2 ¼ v ¼ 246 GeV.
B. Dimension-5 matrix elements
Having defined the operator basis (1), the second step
concerns the nonperturbative input required to define
amplitudes at the hadronic level. We will largely follow
the conventions of Refs. [32,36], but for completeness
review here the respective hadronic matrix elements.
For the dimension-5 operator only the electromagnetic
form factors of the nucleon are required, without the need
for a flavor decomposition. With N ¼ fp; ng, we thus have
the usual Dirac and Pauli form factors F1 and F2,
hNðp0ÞjjμemjNðpÞi ¼ u¯ðp0Þ

FN1 ðtÞγμ − FN2 ðtÞ
iσμνqν
2mN

uðpÞ;
ð6Þ
where jμem¼
P
q¼u;d;s q¯Qqγμq, Q ¼ diagð2;−1;−1Þ=3, and
q ¼ p − p0. For smallmomentum transfer t ¼ ðp0 − pÞ2, it is
sufficient to consider the expansion around t ¼ 0:
FN1 ðtÞ ¼ QN þ
hr21iN
6
tþOðt2Þ;
FN2 ðtÞ ¼ κN þOðtÞ; ð7Þ
with charge QN , anomalous magnetic moment κN , and
hr21iN ¼ hr2EiN −
3κN
2m2N
; ð8Þ
in terms of the charge radius hr2EiN .Wewill use the numerical
values given in Table I. In particular, we will use the proton
TABLE I. Values of the hadronic matrix elements.
κp 1.79284734462(82) [54,55]
κn −1.91304273ð45Þ [54,56]
hr2Eip ½fm2 0.7071(7) [57]
hr2Ein ½fm2 −0.1161ð22Þ [54,58,59]
κNs −0.017ð4Þ [60,61]
hr2E;siN ½fm2 −0.0048ð6Þ [60,61]
gA 1.27641(56) [62]
gu;pA 0.842(12) [54,63]
gd;pA −0.427ð13Þ [54,63]
gs;NA −0.085ð18Þ [54,63]
hr2Ai ½fm2 0.46(16) [64]
Fu;p1;T ð0Þ 0.784(28) [65]
Fd;p1;T ð0Þ −0.204ð11Þ [65]
Fs;N1;T ð0Þ −0.0027ð16Þ [65]
Fu;p2;T ð0Þ −1.5ð1.0Þ [66]
Fd;p2;T ð0Þ 0.5(3) [66]
Fs;N2;T ð0Þ 0.009(5) [66]
Fu;p3;T ð0Þ 0.1(2) [66]
Fd;p3;T ð0Þ −0.6ð3Þ [66]
Fs;N3;T ð0Þ −0.004ð3Þ [66]
fpu ½10−3 20.8(1.5) [67]
fpd ½10−3 41.1(2.8) [67]
fnu ½10−3 18.9(1.4) [67]
fnd ½10−3 45.1(2.7) [67]
fNs ½10−3 43(20) [68–71]
fNQ ½10−3 68(1) [34,72]
_σ ½GeV−1 0.27(1) [33,73,74]
_σs ½GeV−1 0.3(2) [33,73,74]
fπu 0.315(14) [32,75]
fπd 0.685(14) [32,75]
COHERENT ELASTIC NEUTRINO-NUCLEUS SCATTERING: … PHYS. REV. D 102, 074018 (2020)
074018-3
charge radius from muonic atoms
ffiffiffiffiffiffiffiffiffiffiffi
hr2Eip
p
¼
0.84087ð39Þ fm [57,76], in line with most recent electron
spectroscopy measurements [77–79], the PRad electron
scattering data [80], and the expectation from dispersion
relations [81,82]. For the neutron, we use the charge radius
from Ref. [54], but note that a recent extraction from the
deuteron points to a slightly smaller value [83].
C. Dimension-6 matrix elements
At dimension-6 we first need the vector matrix elements
for each quark flavor separately:
hNðp0Þjq¯γμqjNðpÞi
¼ u¯ðp0Þ

Fq;N1 ðtÞγμ − Fq;N2 ðtÞ
iσμνqν
2mN

uðpÞ: ð9Þ
To perform the flavor decomposition, we will assume
isospin symmetry (see Ref. [84] for corrections), which
leads to
Fu;pi ðtÞ ¼ Fd;ni ðtÞ ¼ 2Fpi ðtÞ þ Fni ðtÞ þ Fs;Ni ðtÞ;
Fu;di ðtÞ ¼ Fu;ni ðtÞ ¼ Fpi ðtÞ þ 2Fni ðtÞ þ Fs;Ni ðtÞ;
Fs;pi ðtÞ ¼ Fs;ni ðtÞ≡ Fs;Ni ðtÞ: ð10Þ
Information on the strangeness form factors has tradition-
ally been extracted from PVES, but the uncertainties are
sizable [85]. More recently, it has been shown in lattice
QCD that the strangeness contribution is very small; in
Table I we quote the naive average of Refs. [60,61].
The second dimension-6 operator requires input on the
axial-vector form factors, as they appear in the decom-
position
hNðp0Þjq¯γμγ5qjNðpÞi
¼ u¯ðp0Þ

γμγ5G
q;N
A ðtÞ − γ5
qμ
2mN
Gq;NP ðtÞ
−
iσμν
2mN
qνγ5G
q;N
T ðtÞ

uðpÞ; ð11Þ
where, for completeness, we included the second-class
current Gq;NT ðtÞ [86], but will not further consider its
contribution in the following. The normalization is deter-
mined by the axial-vector charges
Gq;NA ð0Þ≡ gq;NA ≡ ΔqN: ð12Þ
Assuming again isospin symmetry
gu;pA ¼ gd;nA ; gd;pA ¼ gu;nA ; gs;pA ¼ gs;nA ≡ gs;NA ; ð13Þ
these couplings are constrained by
gu;pA − g
d;p
A ¼ gA; gu;NA þ gd;NA − 2gs;NA ¼ 3F −D;
ð14Þ
in terms of the axial-vector coupling of the nucleon gA ¼
1.27641ð56Þ [62] and the SUð3Þ couplings D and F that
can be extracted from semileptonic hyperon decays. In
combination with the singlet combination from Ref. [63],
this leads to the couplings listed in Table I. These values are
in reasonable agreement with lattice QCD [87],
Nf ¼ 2þ 1þ 1 ½88∶ gu;pA ¼ 0.777ð39Þ;
gd;pA ¼ −0.438ð35Þ; gs;NA ¼ −0.053ð8Þ;
Nf ¼ 2þ 1 ½89∶ gu;pA ¼ 0.847ð37Þ;
gd;pA ¼ −0.407ð24Þ; gs;NA ¼ −0.035ð9Þ; ð15Þ
but in view of the present uncertainties we adopt the
phenomenological determination. However, while part of
the difference to phenomenology could be due to the scale
dependence of the singlet combination, both lattice calcu-
lations point to a smaller strangeness coupling than
extracted from the spin structure functions.
The triplet and octet components of the induced pseu-
doscalar form factor GPðtÞ are constrained by Ward
identities, whose manifestation at leading order in the
chiral expansion becomes
G3AðtÞ ¼ gA; G8AðtÞ ¼
3F −Dffiffiffi
3
p ≡ g8A;
G3PðtÞ ¼ −
4m2NgA
t −M2π
; G8PðtÞ ¼ −
4m2Ng
8
A
t −M2η
: ð16Þ
Finally, for the triplet component there is also experimental
information on the momentum dependence. Defining the
axial radius by
G3AðtÞ ¼ gA

1þ hr
2
Ai
6
tþOðt2Þ

; ð17Þ
a simple dipole ansatz
G3AðtÞ ¼
gA
ð1 − t=M2AÞ2
; ð18Þ
with mass scale MA around 1 GeV [90], implies
hr2Ai ¼ 12=M2A ∼ 0.47 fm2. The central value agrees with
Ref. [64], a global analysis of muon capture and neutrino
scattering, but the uncertainties are substantial, see Table I.
To ensure that the Ward identity is satisfied up to higher
HOFERICHTER, MENE´NDEZ, and SCHWENK PHYS. REV. D 102, 074018 (2020)
074018-4
orders, the pseudoscalar form factor needs to be modified
according to [91]
G3PðtÞ ¼ −
4mNgπNNFπ
t −M2π
−
2
3
gAm2Nhr2Ai þOðt;M2πÞ ð19Þ
when including the radius corrections (17). The full πN
coupling constant gπNN has been introduced as a convenient
way to capture all chiral corrections at Oð1Þ. In the
numerical analysis we will use Fπ ¼ 92.28ð9Þ MeV [54]
and g2πNN=ð4πÞ ¼ 13.7ð2Þ [92–96]. With this input, the
Goldberger–Treiman discrepancy becomes
ΔGT ¼
gπNNFπ
mNgA
− 1 ¼ 1.0ð7Þ%; ð20Þ
demonstrating that the chiral corrections are rather small.
The final matrix elements in the dimension-6 Lagrangian
concern the tensor operator q¯σμνq, for which we use the
decomposition
hNðp0Þjq¯σμνqjNðpÞi
¼ u¯ðp0Þ

σμνFq;N1;T ðtÞ −
i
mN
ðγμqν − γνqμÞFq;N2;T ðtÞ
−
i
m2N
ðPμqν − PνqμÞFq;N3;T ðtÞ

uðpÞ: ð21Þ
The tensor charges gq;pT ¼ Fq;p1;Tð0Þ, as given in Table I,
are taken from lattice QCD [65]. The other form-factor
normalizations come from Ref. [66] (with strangeness input
updated to Ref. [61]).
D. Dimension-7 matrix elements
At dimension-7 we first need the scalar matrix elements
of the nucleon:
hNðp0Þjmqq¯qjNðpÞi ¼ mNfNq ðtÞu¯ðp0ÞuðpÞ: ð22Þ
To separate the momentum dependence we define
fNu ðtÞ ¼ fNu þ
1 − ξud
2mN
_σtþOðt2Þ;
fNd ðtÞ ¼ fNd þ
1þ ξud
2mN
_σtþOðt2Þ;
fNs ðtÞ ¼ fNs þ
_σs
mN
tþOðt2Þ: ð23Þ
The scalar couplings fNu;d are largely determined by the
pion-nucleon σ-term σπN [97], up to isospin-breaking
corrections that can be extracted from the proton-neutron
mass difference [98–102]. The numbers given in Table I
follow from σπN as extracted from data on pionic atoms
[93,94,103–105] when used as input for a dispersive
analysis of pion-nucleon scattering [67,106]. This result
has been confirmed using independent input from scatter-
ing data [107], but there is a persistent tension with lattice
QCD that still has not been resolved [68–71,108,109].
Accordingly, we have increased the error in fNs given that in
this case all phenomenological extractions are subject to
large SUð3Þ uncertainties. The heavy-quark couplings fQ
effectively describe the matrix element of the trace anomaly
at OðαsÞ:
fNQðtÞ ¼
2
27

θN0 ðtÞ
mN
−
X
q¼u;d;s
fNq ðtÞ

;
θN0 ðtÞ ¼ hNðp0ÞjθμμjNðpÞi; ð24Þ
with normalization θN0 ð0Þ ¼ mN , while perturbative cor-
rections especially for the charm quark lead to additional
uncertainties [72]. For the momentum dependence, _σ and
_σs are taken from Refs. [33,73,74], and [75,87],
ξud ¼
md −mu
md þmu
¼ 0.35ð2Þ; ð25Þ
which also determines the scalar couplings of the pion
hπjmqq¯qjπi ¼ fπqM2π; ð26Þ
according to
fπu ¼
mu
mu þmd
¼ 1
2
ð1 − ξudÞ ¼ 0.315ð14Þ;
fπd ¼
md
mu þmd
¼ 1
2
ð1þ ξudÞ ¼ 0.685ð14Þ: ð27Þ
These matrix elements arise in two-body corrections to
scalar currents and are also included in Table I.
Finally, we parametrize the pseudoscalar matrix ele-
ment as
hNðp0Þjmqq¯iγ5qjNðpÞi ¼ mNGq;N5 ðtÞu¯ðp0Þiγ5uðpÞ: ð28Þ
For the nonsinglet component the new form factor is related
to the axial-vector ones by the Ward identity
Gq;N5 ðtÞ ¼ Gq;NA ðtÞ þ
t
4m2N
Gq;NP ðtÞ; ð29Þ
but in the singlet case this relation is broken by the anomaly
contribution from GaμνG˜
μν
a , similarly to the gluonic con-
tribution to the trace anomaly. For a consistent treatment
of singlet effects one would thus have to extend the
operator basis in Eq. (1) accordingly. In the past, the
singlet pseudoscalar matrix element has often been esti-
mated by assuming [110,111]
COHERENT ELASTIC NEUTRINO-NUCLEUS SCATTERING: … PHYS. REV. D 102, 074018 (2020)
074018-5
hNj
X
q¼u;d;s
q¯ iγ5qjNi ¼ 0; ð30Þ
but the analogous relation for the axial-vector singlet
combination
P
q¼u;d;s Δq does not display the expected
1=Nc suppression. The matrix element of the gluon
anomaly could be extracted with similar techniques as
used for lattice calculations of the QCD θ term [112].
E. Nuclear responses
As a final step, the nucleon-level matrix elements need to
be convolved with the nuclear states. Formally, the decom-
position into distinct nuclear responses requires a multipole
decomposition, see Refs. [113–118], which in full general-
ity becomes very complex. Here, we concentrate on the
most relevant contributions, with the main features sum-
marized in Table II, and review some of the details needed
later in Appendix A.
By far the most important response is related to the
charge operator, it is denoted by the structure factors
FM ðq2Þ normalized by
FMþ ð0Þ ¼ N þ Z ¼ A; FM− ð0Þ ¼ Z − N: ð31Þ
This is the only response that is fully coherent. In addition
to FM , we also need the so-called F
Φ00
 structure factor,
which can be interpreted in terms of spin-orbit corrections.
This response vanishes for q2 ¼ 0, but it interferes withFM
and receives some coherent enhancement, especially for
heavy nuclei. This is because in the relevant nuclei
nucleons tend to occupy orbitals with spin parallel to the
angular orbital momentum (lowered in energy by the
nuclear spin-orbit interaction) and high-energy orbitals
with antiparallel spin, which would cancel FΦ
00
 , remain
mostly empty. The interference with FM and partial
coherence make the Φ00 response the most relevant cor-
rection. In principle, both FM , F
Φ00
 may contribute beyond
the leading L ¼ 0 multipole, but such effects are not
coherent, vanish at q2 ¼ 0, and without interference with
the leading multipole effectively become negligible. Due to
this we will continue to identify both responses with their
L ¼ 0 multipole.
Finally, there are several responses that emerge from the
axial-vector operator. Their contribution again is not
coherent, but remains finite at q2 ¼ 0. In these cases,
several multipoles and responses become relevant, but we
will continue to use a notation in which these effects are
subsumed into structure factors Sij (with indices i, j ¼ 0, 1
corresponding to isoscalar/isovector combinations). We
keep the induced pseudoscalar form factors Gq;NP , whose
contribution is enhanced by the presence of the pion pole,
but do not consider any other subleading noncoherent
responses. Further aspects of the multipole decomposition
are discussed in Sec. III whenever necessary to introduce
the nuclear responses for a given process.
III. NUCLEAR RESPONSES
IN THE STANDARD MODEL
In this section we will collect the nuclear responses as
they appear in electron-nucleus and neutrino-nucleus scat-
tering. In particular, we demonstrate how the traditional
charge and weak form factors emerge in the formalism
established in Sec. II. In either case, the kinematics are
defined by
lðkÞ þN ðpÞ → lðk0Þ þN ðp0Þ; l ∈ fe−; νg; ð32Þ
with
q ¼ k0 − k ¼ p − p0; ð33Þ
and invariants
s ¼ ðpþ kÞ2; t ¼ ðp− p0Þ2; u ¼ ðp− k0Þ2; ð34Þ
fulfilling sþ tþ u ¼ 2m2A. Here, mA refers to the mass of
the nucleus and lepton masses are neglected throughout.
We also define P ¼ pþ p0 and write t ¼ q2 ¼ −Q2.
A. Parity-conserving electron scattering
For electron scattering, the invariants (34) are conven-
tionally replaced in favor of
η ¼ − t
4m2A
; z ¼ cos θ ¼ 1− 2m
2
At
ðs−m2AÞðu−m2AÞ
; ð35Þ
where θ is the scattering angle in the laboratory frame. In
this frame the relation of the spin-averaged scattering
amplitude jM¯j2 to the cross section becomes
TABLE II. Nomenclature for the nuclear structure factors. The
second column gives the leading operators on the single-nucleon
level, the third one indicates the extent to which the response
scales coherently with nucleon number, and the fourth column
gives its physical interpretation. The axial responses include
longitudinal, transverse electric, and transverse magnetic multi-
poles. SN ¼ σN=2 denotes the nucleon spin operator and the
momenta are defined as in Sec. III.
Responses Operator Coherence Interpretation
FM 1 Coherent Charge
FΦ
00
 SN · ðq × PÞ Semicoherent Spin orbit
Sij SN Not coherent Axial
HOFERICHTER, MENE´NDEZ, and SCHWENK PHYS. REV. D 102, 074018 (2020)
074018-6
dσ
dΩ
¼ jM¯j
2
64π2mAðmA þ Eð1 − zÞÞ
E0
E
¼

dσ
dΩ

Mott
×
E0
E
×
t2jM¯j2
4e4ðm4A − suÞ
; ð36Þ
where the last relation defines the Mott cross section

dσ
dΩ

Mott
¼ e
4ðm4A − suÞ
16π2t2mAðmA þ Eð1 − zÞÞ
¼ α
2
4E2
cos2 θ
2
sin4 θ
2
: ð37Þ
The incoming and outgoing electron energies are given by
E ¼ s −m
2
A
2mA
; E0 ¼ m
2
A − u
2mA
: ð38Þ
For the parity-conserving case, the amplitude becomes
jM¯j2 ¼ 1
2ð2J þ 1Þ
X
spins
jMj2;
M ¼ − e
2
t
u¯ðk0ÞγμuðkÞhN ðp0ÞjjμemjN ðpÞi; ð39Þ
and at the single-nucleon level the hadronization follows
from Eq. (6). The leptonic trace
Lμν ¼ Trð=k0γμ=kγνÞ ¼ 4ðkμk0ν þ k0μkν − gμνk · k0Þ; ð40Þ
fulfilling kμLμν ¼ k0μLμν ¼ 0, needs to be contracted with
the nuclear amplitude, which we express in terms of
multipoles according to Sec. II E, see Ref. [117] and
Appendix A. The leading terms can be read off from the
nonrelativistic expansion of the single-nucleon current,
hNðp0Þjj0emjNðpÞi¼FN1 ðtÞþ
FN1 ðtÞþ2FN2 ðtÞ
8m2N
t
−iSN ·ðq×PÞ
FN1 ðtÞþ2FN2 ðtÞ
4m2N
; ð41Þ
where we dropped the remaining two-component spinors
and the spacelike components do not contribute to the M
and Φ00 responses. After the multipole decomposition, the
first line of Eq. (41) will contribute to FM, the second to
FΦ
00
, and the combination to an interference term between
the two responses. Concentrating on the L ¼ 0 multipole,
the result takes a very compact form and is typically
expressed as
dσ
dΩ
¼

dσ
dΩ

Mott
×
E0
E
× Z2 × ½Fchðq2Þ2; ð42Þ
with the charge form factor defined by
Fchðq2Þ ¼
1
Z

1þ hr
2
Eip
6
tþ 1
8m2N
t

FMp ðq2Þ
þ hr
2
Ein
6
tFMn ðq2Þ
−
1þ 2κp
4m2N
tFΦ
00
p ðq2Þ −
2κn
4m2N
tFΦ
00
n ðq2Þ

: ð43Þ
The proton/neutron combinations are related to the isospin
components by
FM ðq2Þ ¼ FMp ðq2Þ  FMn ðq2Þ;
FΦ
00
 ðq2Þ ¼ FΦ
00
p ðq2Þ  FΦ00n ðq2Þ; ð44Þ
and we have replaced the full form factors in Eq. (41) by the
first terms in the expansion (7). The charge form factor
fulfills the normalization Fchð0Þ ¼ 1, and the correspond-
ing representation (42) is exact for spin-0 nuclei. For
nonvanishing spin, there are further form factors, e.g.,
the magnetic form factor for spin-1=2 in analogy to the
nucleon, but for the reasons given in Sec. II E these
contributions are small in heavy nuclei. In addition, two-
body effects only enter at loop level, so that in contrast to
the magnetic form factor two-body modifications of the
charge form factor are also small. Finally, we give the
corresponding expansion for the charge radius
R2ch ¼ R2p þ hr2Eip þ
N
Z
hr2Ein þ
3
4m2N
þ hr2iso;
hr2iso ¼ −
3
2m2NZ
ðð1þ 2κpÞFΦ00p ð0Þ þ 2κnFΦ00n ð0ÞÞ; ð45Þ
where R2p is the so-called point-proton radius defined as
R2p ¼ −
6
Z
dFMp ðq2Þ
dq2

q2¼0
; ð46Þ
and hr2iso represents the spin-orbit contribution encoded in
Φ00 [119]. In the case of Eq. (43) the matching of matrix
elements and Wilson coefficients is trivial, since so far only
the long-range contribution in the SM has been taken into
account. A potential modification would be provided by the
electron analog of Lð5Þ given in Eq. (1). In the next step, we
extend the discussion to short-range contributions from Z
exchange, which are responsible for PVES in the SM.
B. Parity-violating electron scattering
The central observable in PVES is the asymmetry
APVES ¼
ðdσdΩÞR − ðdσdΩÞL
ðdσdΩÞR þ ðdσdΩÞL
; ð47Þ
COHERENT ELASTIC NEUTRINO-NUCLEUS SCATTERING: … PHYS. REV. D 102, 074018 (2020)
074018-7
where the cross sections involve left- or right-handed
initial-state electrons, respectively. The corresponding
lepton traces are
LμνL=R ¼ Trðk 0γμPL=RkγνðgeV − geAγ5ÞÞ
¼ 2ðgeV  geAÞ
× ðkμk0ν þ k0μkν − gμνk · k0  iϵμναβkαk0βÞ; ð48Þ
where
geV ¼ −
1
2
þ 2sin2θW; geA ¼ −
1
2
ð49Þ
are the vector and axial-vector weak charges of the electron
in the normalization of Ref. [54]. The terms in Eq. (48)
involving an ϵ tensor will lead to SD corrections, which we
will study below in the context of CEνNS, while the
remainder follows in close analogy to the parity-conserving
case, the only difference being that the electromagnetic
form factor needs to be replaced by its weak analog. With
quark-level Wilson coefficients as in the SM and matrix
elements from Eq. (10), the result takes the simple form
APVES ¼
GFt
4πα
ffiffiffi
2
p QwFwðq
2Þ
ZFchðq2Þ
; ð50Þ
where the weak charge
Qw ¼ ZQpw þ NQnw;
Qpw ¼ 1 − 4sin2θW; Qnw ¼ −1; ð51Þ
has been separated from the weak form factor Fwðq2Þ.
However, we note that, in contrast to the electromagnetic
charge and Fchðq2Þ, Qw does not actually factorize. The
explicit definition reads
Fwðq2Þ¼
1
Qw

Qpw

1þhr
2
Eip
6
tþ 1
8m2N
t

þQnw
hr2Einþhr2E;siN
6
t

FMp ðq2Þ
þ

Qnw

1þhr
2
Eipþhr2E;siN
6
tþ 1
8m2N
t

þQpw hr
2
Ein
6
t

FMn ðq2Þ
−
Qpwð1þ2κpÞþ2QnwðκnþκNs Þ
4m2N
tFΦ
00
p ðq2Þ
−
Qnwð1þ2κpþ2κNs Þþ2Qpwκn
4m2N
tFΦ
00
n ðq2Þ
#
; ð52Þ
where we have used that in the SM the Wilson coefficients
for d- and s-quarks are identical to write the strangeness
contribution in terms of Qnw. The corresponding weak
radius reads
R2w ¼
ZQpw
Qw

R2p þ hr2Eip þ
Qnw
Qpw
ðhr2Ein þ hr2E;siNÞ

þ NQ
n
w
Qw

R2n þ hr2Eip þ hr2E;siN þ
Qpw
Qnw
hr2Ein

þ 3
4m2N
þ hr˜2iso; ð53Þ
with spin-orbit contribution [120]
hr˜2iso ¼ −
3Qpw
2m2NQw

1þ 2κp þ 2Q
n
w
Qpw
ðκn þ κNs Þ

FΦ
00
p ð0Þ
−
3Qnw
2m2NQw

1þ 2κp þ 2κNs þ 2
Qpw
Qnw
κn

FΦ
00
n ð0Þ:
ð54Þ
Numerically, we use the values [54]
Qpw ¼ −
ffiffiffi
2
p
GF
ð2CVu þ CVd Þ ¼ 0.0714;
Qnw ¼ −
ffiffiffi
2
p
GF
ðCVu þ 2CVd Þ ¼ −0.9900; ð55Þ
and accordingly for Qw, because the process-dependent
radiative corrections [54,121,122] as for atomic parity
violation or PVES are not yet available.
We have calculated the nuclear responses FMN ðqÞ,
FΦ
00
N ðqÞ, and the corresponding nuclear radii for isotopes
relevant for experiment with the nuclear shell model. The
calculations use the same configuration spaces and nuclear
interactions as in previous works [29,33,36]. In particular,
the shell-model interactions used are USDB for 19F and
23Na [123] (with 0d5=2, 1s1=2, and 0d3=2 single-particle
orbitals), SDPF.SM [124] for 40Ar (0d5=2, 1s1=2, 0d3=2,
0f7=2, 1p3=2, 1p1=2, and 0f5=2 space), RG [125] for 73Ge
(1p3=2, 0f5=2, 1p3=2, and g9=2 orbitals), and GCN5082
[126] for 127I, 133Cs, and 129;131Xe (0g7=2, 1d5=2, 1s1=2,
0d3=2, and h11=2 space). The notation for harmonic oscil-
lator orbitals is nlj, where n is the principal quantum
number, and l, j denote the orbital and total angular
momentum. For additional details on the calculations,
see Refs. [29,33,36]. The nuclear-structure calculations
have been performed with the shell-model code ANTOINE
[127,128].
While the phenomenological character of the nuclear
interactions used in our work prevents the assessment of
reliable nuclear-structure uncertainties, the shell-model
results agree very well with experiment. For instance,
our calculations reproduce well the energies of the
HOFERICHTER, MENE´NDEZ, and SCHWENK PHYS. REV. D 102, 074018 (2020)
074018-8
lowest-lying excited states of these nuclei [29,31,36] and
the electromagnetic transitions between them, including for
the ground states involved in CEνNS [36]. The magnetic
moments of the ground states of odd-mass nuclei, and the
quadrupole moments of first excited states are also in good
agreement with experiment [36]. From all these isotopes,
only 133Cs is presented here for the first time, compared to
our previous works. This calculation was carried out
without truncations in the configuration space, and the
quality of the 133Cs results is illustrated by the energy
spectrum discussed in Appendix C.
As an example, Fig. 1 shows theM andΦ00 responses for
133Cs. The coherent and partially coherent characters of M
and Φ00, respectively, are clearly observed at q ¼ 0, where
about 20%–25% of the nucleons contribute coherently for
FΦ
00
N ð0Þ. The minimum of FMn at lower jqj compared toFMp
indicates a larger neutron than proton radius. Explicit
parametrizations of all nuclear structure factors are pro-
vided in Appendix E.
We obtain the charge and weak radii given in Table III. In
addition, Table III also shows the so-called neutron skin,
defined as the difference between neutron and proton point
radii,Rn − Rp. Calculated charge radii are in goodagreement
with experiment, similar to other approaches [17,25,27]. The
disagreement between calculations increases for predictions
of theweak radii and neutron skin. The shell model generally
predicts larger weak radii and especially larger neutron skins
than other many-body approaches [17,23–25,27,129,130].
FIG. 1. M and Φ00 responses for cesium.
TABLE III. Shell-model charge and weak radii (in fm). The experimental data for the charge radii are from
Ref. [131]. The table also includes our results for the neutron skin Rn − Rp.
19F 23Na 40Ar 70Ge
Rch Th 2.83 3.01 3.43 4.06
Rch Exp 2.8976(25) 2.9936(21) 3.4274(26) 4.0414(12)
Rw Th 2.90 3.06 3.55 4.14
Rn − Rp Th 0.06 0.04 0.11 0.08
72Ge 73Ge 74Ge 76Ge
Rch Th 4.07 4.08 4.08 4.08
Rch Exp 4.0576(13) 4.0632(14) 4.0742(12) 4.0811(12)
Rw Th 4.20 4.23 4.26 4.31
Rn − Rp Th 0.13 0.14 0.17 0.21
127I 133Cs
Rch Th 4.73 4.78
Rch Exp 4.7500(81) 4.8041(46)
Rw Th 5.00 5.08
Rn − Rp Th 0.26 0.27
128Xe 129Xe 130Xe 131Xe
Rch Th 4.75 4.75 4.76 4.77
Rch Exp 4.7774(50) 4.7775(50) 4.7818(49) 4.7808(49)
Rw Th 5.01 5.03 5.04 5.06
Rn − Rp Th 0.24 0.26 0.26 0.27
132Xe 134Xe 136Xe
Rch Th 4.77 4.78 4.79
Rch Exp 4.7859(48) 4.7899(47) 4.7964(47)
Rw Th 5.08 5.10 5.13
Rn − Rp Th 0.28 0.30 0.32
COHERENT ELASTIC NEUTRINO-NUCLEUS SCATTERING: … PHYS. REV. D 102, 074018 (2020)
074018-9
The corresponding results for the weak form factors are
shown in Figs. 2–5. In each case, we show the shell-model
results for the modulus of the weak form factor including
all corrections given in Eq. (52). Coherence is kept until
larger momentum transfers in lighter nuclei with smaller
neutron radius, see Fig. 2. For germanium and xenon
isotopes, Figs. 4 and 5 show the difference between the
weak form factors of stable isotopes.
In the case of 40Ar, Fig. 3 compares our results to the
RMF calculation of Ref. [23] as well as ab initio results
from coupled-cluster theory [27]. All calculated weak form
factors give similar results, within the uncertainty band
estimated in Ref. [27]. This suggests that uncertainties in
the neutron distribution are relatively small, in contrast to
the assumptions in Ref. [37]. We stress that apart from the
nuclear structure, minor differences in the weak form factor
arise from the precise input for the hadronic matrix
elements and weak charges, primarily the proton charge
radius, for which Refs. [23,27] use hr2Eip ≃ 0.77 fm2.
C. Neutrino scattering
The dominant contribution to the CEνNS cross section in
the SM involves the same nuclear form factor as in the case
of PVES, since apart from overall prefactors the combi-
nation of Wilson coefficients, hadronic matrix elements,
and nuclear structure factors remains unchanged. This
dominant piece of the differential cross section takes the
form
FIG. 2. Shell-model results for the weak form factor of 19F,
23Na, 127I, and 133Cs.
FIG. 3. Shell-model results for the weak form factor of 40Ar, in
comparisontoRMF[23]andcoupled-cluster [27]results.Thecurves/
bands labeled (EM)-(PWA),NNLOsat, andΔNNLOGOð450Þ refer to
the chiral interactions considered in Ref. [27].
FIG. 4. Shell-model results for the weak form factor of
germanium.
FIG. 5. Shell-model results for the weak form factor of xenon
(we only show selected isotopes for better visibility).
HOFERICHTER, MENE´NDEZ, and SCHWENK PHYS. REV. D 102, 074018 (2020)
074018-10
dσA
dT

coherent
¼ G
2
FmA
4π

1 −
mAT
2E2ν

Q2wjFwðq2Þj2; ð56Þ
where Eν is the energy of the incoming neutrino and the
nuclear recoil,
T ¼ Eν − E0ν ¼ −
t
2mA
; ð57Þ
takes values in ½0; 2E2ν=ðmA þ 2EνÞ. Terms of orderT=Eν ≲
2Eν=mA are usually neglected due to typical neutrino
energies Eν ≲ 50 MeV. The cross section in Eq. (56) rep-
resents the truly “coherent” contribution, in the sense that the
nuclear structure factors that enter the definition of Fw, see
Eq. (52), indeed scale with Z and N (FM) or at least can
receive some partial coherent enhancement with respect to
closed shells (FΦ
00
). Two-body corrections to Eq. (56) again
only arise at the loop level, and are thus significantly
suppressed in the chiral expansion.
Before extending Eq. (56) to the axial-vector responses,
we comment on some details of the derivation as well as
subleading kinematic effects. The starting point is the
leptonic trace,
Lμν ¼ Trðk 0γμPL=kγνPLÞ
¼ 2ðkμk0ν þ k0μkν − gμνk · k0 þ iϵμναβkαk0βÞ; ð58Þ
whose components determine the spin sums:
X
spins
l0l0 ¼ L00 ¼ 2E2ν

2 −
mAT
E2ν
−
2T
Eν

þOðT2Þ;
X
spins
l3l3 ¼ L33 ¼ 2E2ν
T
mA
þOðT2Þ;
X
spins
l0l3 ¼ L03 ¼ 2E2ν
ffiffiffiffiffiffi
2T
mA
s
þOðT3=2Þ;
X
spins
l · l ¼ Lii ¼ 2E2ν

2þmAT
E2ν
−
2T
Eν

þOðT2Þ;
X
spins
ðl × lÞ3 ¼ ϵ3ijLij ¼ −4iEν
ffiffiffiffiffiffiffiffiffiffiffiffi
2mAT
p
þOðT3=2Þ: ð59Þ
The spherical components are defined with respect to the
direction of q ¼ k0 − k, e.g.,
k3 ¼
k · q
jqj ¼ −
TðmA þ EνÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Tð2mA þ TÞ
p ;
k03 ¼
k0 · q
jqj ¼
TðmA þ T − EνÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Tð2mA þ TÞ
p : ð60Þ
In particular, the combination L33 is strongly suppressed by
T=mA ≲ 2E2ν=m2A, while L03 or the additional terms in L00
and Lii are only suppressed by T=Eν ≲ 2Eν=mA. In
consequence, the longitudinal multipoles in Eq. (A1) can
be safely neglected. The interference with the Coulomb
multipoles could in principle become relevant, but the
longitudinal multipoles involve an additional suppression
by q0=jqj ¼ − ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT=ð2mAÞp ≲ −Eν=mA from the application
of current conservation, see Eq. (A3). Accordingly, all
potentially relevant subleading kinematic effects can be
taken into account by

1 −
mAT
2E2ν

→

1 −
mAT
2E2ν
−
T
Eν

ð61Þ
in Eq. (56).
Next, there could be interference terms between the
vector and axial-vector responses. The vector contributions
to the transverse multipoles vanish for T → 0 and are not
coherent, so the only potentially relevant interferences arise
from the longitudinal and Coulomb multipoles. However,
all such interferences vanish due to Eq. (A3).
Therefore, the dominant correction to Eq. (56) comes
solely from the axial-vector part of the interaction. This
contribution becomes relevant for precision studies of
nuclei with nonvanishing spin, especially, because in
contrast to other less relevant corrections their contribution
remains finite in the limit T → 0. The SD structure factors
are obtained by adapting the formalism from Ref. [29],
most notably, by only keeping the transverse electric
multipoles, due to the strong suppression of the longi-
tudinal ones (transverse magnetic multipoles do not con-
tribute to elastic scattering due to time reversal). Collecting
the kinematic factors, the resulting contribution to the
CEνNS cross section takes the form
dσA
dT

SD
¼ 2mA
2Jþ 1

2þmAT
E2ν
−
2T
Eν

× ððg0AÞ2ST00ðq2Þ þ g0Ag1AST01ðq2Þ þ ðg1AÞ2ST11ðq2ÞÞ;
ð62Þ
where the structure factors STijðq2Þ are the same as for dark
matter except that longitudinal multipoles need to be
omitted, see Sec. III E as well as Appendices A and B
for the precise definitions. In particular, the normalizations
are related to hSNi, the nucleon (proton and neutron) spin
expectation values2:
F
Σ0
1
N ð0Þ ¼
ffiffiffi
2
p
F
Σ00
1
N ð0Þ ¼
ffiffiffi
2
3
r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð2J þ 1ÞðJ þ 1Þ
4πJ
r
hSNi: ð63Þ
2Note that Eq. (63) includes an additional factor 1=2 compared
to the q ¼ 0 limit of the standard definitions of the Σ0, Σ00
operators in Eqs. (B2) and (B3) [the same is true for the full
F
Σ0L
N ðq2Þ, FΣ
00
L
N ðq2Þ]. This factor is compensated by the factor 2 in
Eqs. (83) and (84), which is needed for consistency with the
definition of Sij in Eqs. (80)–(82) in the literature.
COHERENT ELASTIC NEUTRINO-NUCLEUS SCATTERING: … PHYS. REV. D 102, 074018 (2020)
074018-11
We have obtained the nuclear responses F
Σ0
1
N ðqÞ and
F
Σ00
1
N ðqÞ and the corresponding spin expectation values
with the nuclear shell model calculations described in
Sec. III B. The results for the spin expectation values are
given in Table IV, see also Refs. [28,29]. The isoscalar/
isovector coefficients are
g0A ¼
gpA þ gnA
2
; g1A ¼
gpA − gnA
2
; ð64Þ
where gNA ¼
P
q C
A
qg
q;N
A . In the SMwe have, using Eqs. (4),
(13), and (14),
gpA ¼
GFffiffiffi
2
p ðgA − gs;NA Þ; gnA ¼ −
GFffiffiffi
2
p ðgA þ gs;NA Þ;
g0A ¼ −
GFffiffiffi
2
p gs;NA ; g1A ¼
GFffiffiffi
2
p gA; ð65Þ
so that the full expression for the cross section becomes
dσA
dT
¼ G
2
FmA
4π

1 −
mAT
2E2ν
−
T
Eν

Q2wjFwðq2Þj2
þG
2
FmA
4π

1þmAT
2E2ν
−
T
Eν

FAðq2Þ; ð66Þ
where
FAðq2Þ ¼
8π
2J þ 1 ððg
s;N
A Þ2ST00ðq2Þ − gAgs;NA ST01ðq2Þ
þ ðgAÞ2ST11ðq2ÞÞ ð67Þ
is the axial-vector analog of jFwðq2Þj2. As expected, the
dominant SD correction arises from the isovector compo-
nent, with the normalization
FAð0Þ ¼
4
3
g2A
J þ 1
J
ðhSpi − hSniÞ2; ð68Þ
when strangeness and two-body corrections are neglected.
The induced pseudoscalar form factor GPðtÞ only contrib-
utes to the longitudinal multipoles, see Eq. (A3). Since gA
factorizes, the radius corrections from Eq. (17) are usually
absorbed into the structure factors, as are corrections from
two-body currents, to which we will turn in the next
subsection.
D. Improved treatment of axial-vector
two-body currents
Axial-vector currents are responsible for SD scattering.
In the nonrelativistic limit the leading one-body (1b)
currents are given by
J3i;1b ¼
1
2
τ3i

G3Aðq2Þσi −
G3Pðq2Þ
4m2N
ðq · σiÞq

; ð69Þ
so that axial responses are driven by the nucleon spin
Si ¼ σi=2, as indicated by Table II.
A sizable correction to the leading one-body terms
comes from subleading axial-vector two-body currents
[32]. In medium-mass and heavy nuclei, these contribu-
tions have been evaluated in previous studies of β decays
[132,133] and WIMP-nucleus scattering [28,29]. However,
the studies of SDWIMP scattering off nuclei focus on pion-
exchange two-body currents proportional to the low-energy
couplings c3, c4, and c6 [28,29] and neglected the contact
two-body axial-vector current proportional to the couplings
d1, d2 [32], which is only included in the jqj ¼ 0 limit in β
decay [132,133].
Here we improve previous studies by including all pion-
exchange, pion-pole, and contact terms derived in Ref. [32]:
J312 ¼ −
gA
2F2π
½τ1 × τ23

c4

1 −
q
q2 þM2π
q·

ðσ1 × k2Þ þ
c6
4
ðσ1 × qÞ þ i
p1 þ p01
4mN

σ2 · k2
M2π þ k22
−
gA
F2π
τ32

c3

1 −
q
q2 þM2π
q·

k2 þ 2c1M2π
q
q2 þM2π

σ2 · k2
M2π þ k22
− d1τ31

1 −
q
q2 þM2π
q·

σ1 þ ð1↔ 2Þ − d2ðτ1 × τ2Þ3ðσ1 × σ2Þ

1 − ·q
q
q2 þM2π

; ð70Þ
TABLE IV. Shell-model proton (hSpi) and neutron (hSni) spin
expectation values for the odd-mass isotopes considered in this
work.
19F 23Na 73Ge 127I
hSpi 0.478 0.224 0.032 0.346
hSni −0.002 0.024 0.439 0.031
129Xe 131Xe 133Cs
hSpi 0.010 −0.009 −0.343
hSni 0.329 −0.272 0.001
HOFERICHTER, MENE´NDEZ, and SCHWENK PHYS. REV. D 102, 074018 (2020)
074018-12
where ki ¼ p0i − pi, q ¼ −k1 − k2, and ð1↔ 2Þ applies to
the entire expression except for the last term. Relativistic
1=mN corrections to Eq. (70), besides the term proportional
to p1 þ p01, can be absorbed into c4 → c4 þ 1=ð4mNÞ,
c6 → c6 þ 1=mN , where we use a dimensionful c6 for
consistency with the previous literature on the axial current
(note that our choice of c6 corresponds to c6=mN in the
conventions of Ref. [134]). In the counting of Refs. [32,135]
these relativistic corrections are formally of higher order,
but we keep them both for consistency with Ref. [133] and
in analogy to our treatment of higher-order effects in the ci,
see below.
Following Refs. [28,29] we approximate the two-
nucleon currents by a normal-ordering approximation
with respect to spin-isospin symmetric reference state with
density ρ ¼ 2k3F=ð3π2Þ (kF is the Fermi momentum)
Jeffi;2b ¼
X
j
ð1 − PijÞJ3ij; ð71Þ
where Pij is the exchange operator and the sum is
performed over the second nucleon j.
As a result, axial-vector two-body currents transform
into effective one-body currents [29,136]:
Jeff;σi;2b ðρ;q;PÞ ¼ −gAσi
τ3i
2
ρ
2F2π

−
1
3

c3 −
1
4mN

½Iσ1ðρ; jP − qjÞ þ Iσ1ðρ; jPþ qjÞ þ
c4
3
½3Iσ2ðρ; jP − qjÞ − Iσ1ðρ; jP − qjÞ
þ 3Iσ2ðρ; jPþ qjÞ − Iσ1ðρ; jPþ qjÞ þ
c6
12

Ic6ðρ; jP − qjÞ
q · ðP − qÞ
ðP − qÞ2
− Ic6ðρ; jPþ qjÞ
q · ðPþ qÞ
ðPþ qÞ2

−
cD
2gAΛχ

; ð72Þ
Jeff;Pi;2b ðρ;q;PÞ ¼ −gA
τ3i
2
ðq · σiÞq
ρ
2F2π

4ðc3 − 2c1Þ
M2π
ðM2π þ q2Þ2
−
1
3

c3 þ c4 −
1
4mN

IPðρ; jP − qjÞ þ IPðρ; jPþ qjÞ
q2
þ 1
3
ðc3 þ c4Þ
1
M2π þ q2

Iσ1ðρ; jP − qjÞ þ Iσ1ðρ; jPþ qjÞ þ
q2IPðρ; jP − qjÞ
ðP − qÞ2 þ
q2IPðρ; jPþ qjÞ
ðPþ qÞ2

− c4
1
M2π þ q2
½Iσ2ðρ; jP − qjÞ þ Iσ2ðρ; jPþ qjÞ
þ

c6
12
−
2
3
c1M2π
M2π þ q2

Ic6ðρ; jP − qjÞ
ðP − qÞ2 þ
Ic6ðρ; jPþ qjÞ
ðPþ qÞ2

þ cD
2gAΛχ
1
M2π þ q2

: ð73Þ
These two effective currents have the same structure as the
two terms in the leading one-body current, Eq. (69), so they
can be treated in the same way.
The currents in Eqs. (72) and (73) depend on the nuclear
density ρ, the momentum transfer q, and the combined
momentum P. Because the dependence on P is small [29],
in practice we evaluate the expressions taking P ¼ 0.
Likewise, we neglect additional effective one-body currents
proportional to P and P · σi. The functions Iσ1ðρ; KÞ,
Iσ2ðρ; KÞ, IPðρ; KÞ, and Ic6ðρ; KÞ appear due to the
summation over occupied states in the exchange terms
in Eq. (71). They can be expressed as integrals, with
analytical expressions given in Ref. [29].
In the P ¼ 0 approximation, the combined effective
currents can be written in analogy to Eq. (69):
Jeffi;2bðρ;qÞ ¼ gA
τ3i
2

δaðq2Þσi þ
δaPðq2Þ
q2
ðq · σiÞq

; ð74Þ
where
δaðq2Þ ¼ − ρ
F2π

c4
3
½3Iσ2ðρ; jqjÞ − Iσ1ðρ; jqjÞ −
1
3

c3 −
1
4mN

Iσ1ðρ; jqjÞ −
c6
12
Ic6ðρ; jqjÞ −
cD
4gAΛχ

; ð75Þ
δaPðq2Þ ¼ ρ
F2π

−2ðc3 − 2c1Þ
M2πq2
ðM2π þ q2Þ2
þ 1
3

c3 þ c4 −
1
4mN

IPðρ; jqjÞ −

c6
12
−
2
3
c1M2π
M2π þ q2

Ic6ðρ; jqjÞ
−
q2
M2π þ q2

c3
3
½Iσ1ðρ; jqjÞ þ IPðρ; jqjÞ þ
c4
3
½Iσ1ðρ; jqjÞ þ IPðρ; jqjÞ − 3Iσ2ðρ; jqjÞ

−
cD
4gAΛχ
q2
M2π þ q2

: ð76Þ
COHERENT ELASTIC NEUTRINO-NUCLEUS SCATTERING: … PHYS. REV. D 102, 074018 (2020)
074018-13
For β decays q ≃ 0, and axial-vector two-body currents
have been studied beyond the normal-ordering approxi-
mation in Eq. (71) [133]. The approximation for Jeffi;2b was
found to be very good when taking ρ ∼ 0.10 fm−3, which is
a typical value for the density of the nuclear surface. Based
on this, for our evaluation of the nuclear structure factors
we consider the density range ρ ¼ 0.09…0.11 fm−3. This
range includes slightly lower densities, but is consistent
with the one considered in Refs. [28,29].
The contributions from two-body currents in Eqs. (72)
and (73) depend on the low-energy couplings c1, c3,
c4, c6, and cD. Due to antisymmetrization of the currents,
the two couplings of the contact two-body term com-
bine into a single contribution proportional to
cD ¼ −4ðd1 þ 2d2Þ=ðF2πΛχÞ. The values of ci, cD to be
used should in principle be given by the nuclear interaction
used to solve the many-body problem for the nucleus of
interest. However, accurate many-body calculations using
chiral interactions that depend explicitly on ci, cD are still not
available for all nuclei discussed in this work. Instead, our
results are based on many-body calculations that use shell-
modelHamiltonians, which, despite being based on nucleon-
nucleon interactions, are modified by phenomenological
adjustments in order to improve their description of the
nuclear structure of selected regions of nuclei. Therefore,
we cannot use consistent ci, cD couplings between the
nuclear interactions and the two-nucleon currents given in
Eqs. (72) and (73).
Our strategy is as follows. First, we use the values for c1,
c3, and c4 determined in the Roy-Steiner equation analysis
of πN scattering [106,137]. This improved determination of
the ci values allows us to obtain results with reduced
theoretical uncertainties compared to Refs. [28,29], which
considered a broad range of c3 and c4 (the smaller c1
contributions are included for the first time in this work). In
fact, at a given chiral order the uncertainties in the ci are
now negligible, with the main uncertainty arising from the
chiral expansion. Strictly speaking, one should use the
next-to-leading-order values from Refs. [106,137] to be
consistent with the chiral order we use for the axial-vector
current, but this assumes that the latter is affected by large
loop corrections in the same way as πN scattering, which is
known not to be the case. Instead, we make use of the fact
that the two-nucleon axial-vector current is matched to the
three-nucleon force [135], in such a way that the leading
loop corrections in the axial-vector current coincide with
the ones in the three-nucleon force [138,139]. These
corrections can be represented by a simple shift δci [140]:
δc1 ¼ −
g2AMπ
64πF2π
; δc3 ¼ −δc4 ¼
g4AMπ
16πF2π
: ð77Þ
The values shown in Table V are then obtained as the
combination of the next-to-next-to-leading-order values
from Refs. [106,137] in combination with these δci
(as well as the relativistic correction for c4), and the
uncertainties represent the shifts between the two chiral
orders. The value of c6 is related to the isovector magnetic
moments via [134]
c6 ¼
κp − κn
mN
þ g
2
AMπ
4πF2π
; ð78Þ
where we have indicated the leading loop correction.
Similarly to the other ci, this correction is large despite
being formally of higher order (in part due to the enhance-
ment by a factor of π [141]). However, similar corrections
arise from chiral loops in the axial-vector current
[135,142], the dominant of which can again be represented
as a shift in c6,
δc6 ¼ −
g2AMπ
4πF2π
; ð79Þ
and cancels thematching correction inEq. (78). Including the
relativistic corrections discussed before, we will thus equate
c6 ¼ ðκp − κn þ 1Þ=mN ¼ 5.01, as given in Table V.
We then fix the value of the contact coupling cD, while at
the same time correcting for the shortcomings of our
phenomenological calculations. Shell-model nuclear
matrix elements involving the axial-vector current typically
overestimate experiment [143] by about 20% to 30%.
Recently, Ref. [133] showed for β decay (where it is
sufficient to take jqj ¼ 0) that this is because of a
combination of missing two-body axial-vector currents,
see Eq. (70), and additional nuclear correlations that are
beyond the standard shell-model approach. In order to
account for this, we adjust the value of cD so that our shell-
model calculations receive a contribution from two-nucleon
currents such that, at jqj ¼ 0, Eq. (72) reduces the leading
term in Eq. (69) in the range 20% to 30%. The q
dependence of the effective two-body currents is the one
predicted by Eqs. (72) and (73). Since the leading con-
tribution from two-body axial-vector currents comes from
the pion-exchange part proportional to c3 and c4, the part
TABLE V. Nuclear density ρ and low-energy couplings ci and
cD used in this work. The smallest (largest) value of cD is only
reached for the lowest (highest) density ρ ¼ 0.09 fm−3
(ρ ¼ 0.11 fm−3) and 30% (20%) contribution of two-body axial
currents at jqj ¼ 0. The values for c1;3;4;6 include the leading-loop
effects and relativistic corrections as described in the main text.
The chiral scale in the definition of cD is set to Λχ ¼ 700 MeV.
c1 [GeV−1] −1.20ð17Þ
c3 [GeV−1] −4.45ð86Þ
c4 [GeV−1] 2.96(70)
c6 [GeV−1] 5.01(1.06)
cD −6.08…0.30
ρ [fm−3] 0.09…0.11
HOFERICHTER, MENE´NDEZ, and SCHWENK PHYS. REV. D 102, 074018 (2020)
074018-14
considered in Refs. [28,29], our results are consistent with
these previous calculations.
The values of ci and cD used in this work are summa-
rized in Table V, where the extreme values cD ¼ −6.08
(cD ¼ 0.30) only correspond to the low density ρ ¼
0.09 fm−3 (high density ρ ¼ 0.11 fm−3). In practice, we
neglect the remaining uncertainties in the ci due to effects
from higher chiral orders not captured here, as those
are subleading compared to the uncertainty in the
range of cD values, which also depend on the nuclear
density ρ. Ultimately, our uncertainty depends on the range
imposed on the impact of the two-body currents at jqj ¼ 0,
20%–30%, as estimated from β decay [133,143].
E. Spin-dependent responses for CEνNS
and dark matter
The nuclear responses for CEνNS and SD dark matter
scattering off nuclei can be expressed in terms of the
transverse and longitudinal SD structure factors F
Σ0L
 ðq2Þ
and F
Σ00L
 ðq2Þ, respectively. For CEνNS, only the transverse
component contributes, while for dark matter scattering
both longitudinal and transverse parts need to be taken into
account.
The expressions are given by
S00 ¼ ST00 þ SL00
¼
X
L
½FΣ0Lþ ðq2Þ2 þ
X
L
½FΣ00Lþ ðq2Þ2; ð80Þ
S11 ¼ ST11 þ SL11
¼
X
L
½½1þ δ0ðq2ÞFΣ0L− ðq2Þ2
þ
X
L
½½1þ δ00ðq2ÞFΣ00L− ðq2Þ2; ð81Þ
S01 ¼ ST01 þ SL01
¼
X
L
2½1þ δ0ðq2ÞFΣ0Lþ ðq2ÞFΣ
0
L− ðq2Þ
þ
X
L
2½1þ δ00ðq2ÞFΣ00Lþ ðq2ÞFΣ
00
L− ðq2Þ; ð82Þ
which can be expressed in terms of the proton/neutron
instead of the isoscalar/isovector basis as
Sp ¼ STp þ SLp
¼
X
L
½2FΣ0Lp ðq2Þ þ δ0ðq2ÞðFΣ
0
L
p ðq2Þ − FΣ
0
L
n ðq2ÞÞ2
þ
X
L
½2FΣ00Lp ðq2Þ þ δ00ðq2ÞðFΣ
00
L
p ðq2Þ − FΣ
00
L
n ðq2ÞÞ2;
ð83Þ
Sn ¼ STn þ SLn
¼
X
L
½2FΣ0Ln ðq2Þ − δ0ðq2ÞðFΣ
0
L
p ðq2Þ − FΣ
0
L
n ðq2ÞÞ2
þ
X
L
½2FΣ00Ln ðq2Þ − δ00ðq2ÞðFΣ
00
L
p ðq2Þ − FΣ
00
L
n ðq2ÞÞ2;
ð84Þ
where the proton/neutron combinations are related to the
isospin ones analogously to Eq. (44),
F
Σ0L
 ðq2Þ ¼ FΣ
0
L
p ðq2Þ  FΣ
0
L
n ðq2Þ;
F
Σ00L
 ðq2Þ ¼ FΣ
00
L
p ðq2Þ  FΣ
00
L
n ðq2Þ: ð85Þ
The terms δ0ðq2Þ; δ00ðq2Þ encode the corrections beyond
the leading SD coupling to the transverse and longitudinal
SD responses, respectively. They capture the combined
effect of the pseudoscalar form factor, radius corrections,
and two-body currents. They are given by
δ0ðq2Þ ¼ −q
2hr2Ai
6
þ δaðq2Þ;
δ00ðq2Þ ¼ − gπNNFπ
gAmN
q2
q2 þM2π
þ δaðq2Þ þ δaPðq2Þ; ð86Þ
where the two-body current contributions δaðq2Þ and
δaPðq2Þ are defined in Eqs. (75) and (76).
Note that currents proportional to ðq · σiÞq only con-
tribute to the longitudinal multipoles. Moreover, their
contribution can be treated similarly to terms proportional
to σi because
ðq · σiÞq ¼ q2σi þ q × ðq × σiÞ; ð87Þ
where the second term is perpendicular to q and vanishes
for longitudinal multipoles.
As a first application we show the results for the structure
factors SNðq2Þ for xenon, in comparison to our previous
work from Ref. [29], see Fig. 6. There is good consistency
within the earlier theoretical band. As expected, recent
progress in the understanding of low-energy constants and
two-body currents in β decays allows us to reduce the
theoretical uncertainties. Figure 6 shows that for xenon this
is especially the case for Sp, as this response is dominated
by two-body contributions. In general, uncertainty bands
are reduced most for the smaller structure factors corre-
sponding to the species with an even number of nucleons.
Second, we show the variant of the SD structure factors
required for CEνNS, see Figs. 7 and 8. As discussed in
Sec. III C, only the transverse multipoles contribute to the
final expression in Eq. (66), but unless the strangeness
COHERENT ELASTIC NEUTRINO-NUCLEUS SCATTERING: … PHYS. REV. D 102, 074018 (2020)
074018-15
contribution is neglected all isospin components enter. The
figures show our shell-model results, including two-body
currents and form factor corrections represented by δ0ðq2Þ,
δ00ðq2Þ in Eq. (86). For a given nucleus, the shape of the
isovector and isoscalar responses is similar because all of
them are ultimately dominated by either Spðq2Þ, if the
nucleus has an unpaired proton, or Snðq2Þ, for nuclei with
odd number of neutrons. A comparison between the 131Xe
structure factors in Figs. 6 and 8 shows that the shape of the
transverse component may differ significantly from the
total structure factor (dominated by the longitudinal com-
ponent in that case, see Ref. [29]). According to Eq. (63),
the normalization of the transverse contribution differs
by 2=3 from the sum. Moreover, as can be seen from
Figs. 7 and 8, the isovector combination ST11, which is most
relevant for Eq. (66), is the smallest of the isospin
components. This is partly because of the reduction caused
by axial-vector two-body currents, which are isovector, as
one-body S11 and S00 structure factors are of similar size.
IV. NUCLEAR RESPONSES BEYOND THE
STANDARD MODEL
A. Vector and axial-vector operators
As a first step, we generalize Eq. (66) to include
scenarios in which still only vector and axial-vector
operators are present, but whose Wilson coefficients are
allowed to deviate from the SM. Especially the case with
BSM contributions only to the vector operators is a
frequently studied scenario [2,3].
To collect the combination of Wilson coefficients and
hadronic matrix elements, we define
FIG. 6. Structure factors SNðq2Þ, as defined in Eqs. (83) and
(84), for xenon. The dark bands refer to the results from this
work, the light bands to the ones from Ref. [29].
FIG. 7. Transverse SD structure factors for CEνNS, as required
for Eq. (66). The figure includes all isospin channels, for sodium
and germanium (top) and cesium and iodine (bottom).
HOFERICHTER, MENE´NDEZ, and SCHWENK PHYS. REV. D 102, 074018 (2020)
074018-16
gNV;iðtÞ ¼
X
q¼u;d;s
CVqF
q;N
1 ðtÞ; i ∈ f1; 2g;
gNA ðtÞ ¼
X
q¼u;d;s
CAqG
q;N
A ðtÞ; ð88Þ
as well as the short-hand notation
gNV ≡ gNV;1ð0Þ; gNV;2 ≡ gNV;2ð0Þ; gNA ¼ gNA ð0Þ;
gNV;1ðtÞ ¼ gNV þ _gNV tþOðt2Þ; ð89Þ
where
_gpV ¼ gpV
hr2Eip
6
−
κp
4m2N

þ gnV
hr2Ein
6
−
κn
4m2N

þ gBV
hr2E;siN
6
−
κNs
4m2N

;
gpV;2 ¼ gpVκp þ gnVκn þ gBVκNs ; ð90Þ
and the neutron equations follow by gpV ¼ 2CVuþ
CVd ↔ g
n
V ¼ CVu þ 2CVd . For the strangeness contribution
we have introduced the “baryon-number” coupling
gBV ¼
X
q¼u;d;s
CVq : ð91Þ
In the SM, where CVd ¼ CVs , this new coupling coincides
with gnV and was therefore not needed in Eq. (52).
Collecting all terms, the generalization of Eq. (66) becomes
dσA
dT
¼ mA
2π

1 −
mAT
2E2ν
−
T
Eν

Q˜2wjF˜wðq2Þj2
þmA
2π

1þmAT
2E2ν
−
T
Eν

F˜Aðq2Þ; ð92Þ
where
F˜Aðq2Þ¼
8π
2Jþ1
× ððg0AÞ2ST00ðq2Þþg0Ag1AST01ðq2Þþðg1AÞ2ST11ðq2ÞÞ:
ð93Þ
The isoscalar and isovector couplings for the axial-vector
part are defined as in Eq. (64), so that F˜A → G2F=2FA in the
SM. Similarly, the new “weak charge,”
Q˜w ¼ ZgpV þ NgnV; ð94Þ
reduces to −GF=
ffiffiffi
2
p
Qw in the SM, see Eq. (51), and the
new “weak form factor” becomes
F˜wðq2Þ ¼
1
Q˜w

gpV þ _gpVtþ
gpV þ 2gpV;2
8m2N
t

FMp ðq2Þ
þ

gnV þ _gnVtþ
gnV þ 2gnV;2
8m2N
t

FMn ðq2Þ
−
gpV þ 2gpV;2
4m2N
tFΦ
00
p ðq2Þ −
gnV þ 2gnV;2
4m2N
tFΦ
00
n ðq2Þ

:
ð95Þ
Modifications due to BSM physics thus affect the CEνNS
cross section in two ways: the normalization at q2 ¼ 0
changes, visible as the change in the weak charge, but in
addition the weak form factor changes as well, which is due
to the fact thatQw does not actually factorize, but emerges as
a sum of different underlying nuclear responses. Only in
special cases inwhich the shifts in theWilson coefficients are
aligned with the SM, i.e., all coefficients are modified by the
same relative factor, would Fwðq2Þ remain unaltered.
To quantify the changes with respect to Fwðq2Þ, the new
form factor is shown in Fig. 9 for several points in the BSM
parameter space. These contributions to the u- and d-quark
vector Wilson coefficients, defined as in Eq. (5), are large
but realistic in view of current bounds from CEνNS [2,3].
By definition, the deviations vanish at jqj ¼ 0, and they
become most visible in the vicinity of the zeros. The second
point is illustrated in Fig. 10, which shows that sufficiently
far away from the zeros the changes are at the few-percent
level, while the relative deviations are enhanced once the
process becomes less coherent. The relative changes to
FIG. 8. Same as Fig. 7, for the two odd-mass xenon isotopes.
COHERENT ELASTIC NEUTRINO-NUCLEUS SCATTERING: … PHYS. REV. D 102, 074018 (2020)
074018-17
Fwðq2Þ in Fig. 10 are comparable to the current nuclear-
structure uncertainties suggested by Fig. 3.
B. Operators not present in the Standard Model
Next, we turn to the operators in Eq. (1) not present in
the SM. At dimension-5 there is only the dipole operator,
leading to the lepton trace
Lμν ¼ −Trðk 0½γα; γμPL=k½γβ; γνPRÞqαqβ
¼ −8tðkμk0ν þ k0μkνÞ; ð96Þ
where we dropped terms that vanish upon contraction with
the nuclear matrix element due to gauge invariance. Since
the interference terms with the SM contribution vanish, the
presence of a dipole contribution would manifest itself as a
new, long-range interaction,
dσA
dT

dipole
¼ 4αC
2
F
T
Z2jFchðq2Þj2 þOðT0Þ: ð97Þ
One power of 1=t from the photon propagator in the
squared matrix element cancels with the lepton trace in
Eq. (96), but the second remains and leads to the divergence
for T → 0, due to the relation between momentum transfer
and nuclear recoil given in Eq. (57).
Next, the lepton trace for the scalar operator is
L ¼ Trðk 0PLkPRÞ ¼ 2k · k0 ¼ −t: ð98Þ
The diagonal term in the cross section can be expressed as
dσA
dT

scalar
¼ m
2
AT
4πE2ν
jFSðq2Þj2: ð99Þ
This expression vanishes for T → 0, but otherwise there is
no kinematic suppression compared to the vector contri-
bution due to the scaling mAT=ð2E2νÞ ≲ 1. We have
collected all the relevant couplings and form factors in
the scalar combination FS, which is defined as
FSðq2Þ ¼
X
N¼n;p

fN þ
t
m2N
_fN

FMN ðq2Þ
þ ðfπ þ 2fθπÞF πðq2Þ þ fθπF bðq2Þ; ð100Þ
with FMN given in Eq. (44), the two-body contributions
F πðq2Þ, F bðq2Þ from Ref. [36], and the following combi-
nations of Wilson coefficients and hadronic couplings:
fN ¼ mN
 X
q¼u;d;s
CSqfNq − 12πfNQC0Sg

;
_fN ¼ CSu
1 − ξud
2
_σ þ CSd
1þ ξud
2
_σ þ CSs _σs;
fπ ¼ Mπ
X
q¼u;d

CSq þ
8π
9
C0Sg

fπq;
fθπ ¼ −Mπ
8π
9
C0Sg : ð101Þ
Again, there is no interference with the SM, but the scalar
contribution does interfere with the dipole, leading to
FIG. 10. Relative changes in the weak form factor for 133Cs, for
the same scenarios shown in Fig. 9.
FIG. 9. Changes in the weak form factor for 133Cs in the
presence of BSM contributions to the u- and d-quark vector
Wilson coefficients (5).
HOFERICHTER, MENE´NDEZ, and SCHWENK PHYS. REV. D 102, 074018 (2020)
074018-18
dσA
dT

dipoleþscalar
¼ m
2
AT
4πE2ν
×
FSðq2Þ þ 2Eν − TmAT ZeCFFchðq2Þ
2:
ð102Þ
For the pseudoscalar operator there is also no interfer-
ence with the SM, and due to the SD nature of the nucleon
matrix elements such a response should be even further
suppressed than in the scalar case. To corroborate that
expectation we rewrite the operator by means of the axial
Ward identity,
ν¯PLνmqq¯iγ5q ¼ −
i
2
qμν¯PLνq¯γμγ5q; ð103Þ
so that we can define a leptonic trace,
Lμν ¼ 1
4
qμqνTrð=kPL=kPRÞ ¼ −
t
4
qμqν; ð104Þ
to be contracted with the same nuclear responses already
studied for the axial-vector case. The relevant spin sums are
given by L33 ¼ Lii ¼ t2=4, leading to a kinematic sup-
pression with respect to the axial-vector contribution that
scales as
t2
16E2νm2N
¼ m
2
AT
2
4E2νm2N
≲ E
2
ν
m2N
: ð105Þ
The scale mN emerges assuming that the formal difference
between the dimension-7 and dimension-6 operators is
mainly due to hadronic scales [as is manifest for the matrix
elements of the scalar operator, see Eq. (22)], and for higher
scales the suppression would be even stronger. In either
case we conclude that pseudoscalar contributions to
CEνNS are negligible.
For the tensor operator, the most relevant contributions
are expected from the spacelike components σij, because
only those are momentum independent and not suppressed
by 1=mN in the nonrelativistic expansion. For the same
reason, the induced terms in Eq. (21) are subleading. The
result of the multipole decomposition for tensor currents,
see Appendix D, then leads to the following expressions:
defining the couplings via
gNT;1ðtÞ ¼
X
q¼u;d;s
CTqF
q;N
1;T ðtÞ; gNT;1 ≡ gNT;1ð0Þ; ð106Þ
and
g0T;1 ¼
gpT;1 þ gnT;1
2
; g1T;1 ¼
gpT;1 − gnT;1
2
; ð107Þ
the cross section becomes
dσA
dT

tensor
¼ 8mA
2J þ 1

2 −
mAT
E2ν
−
2T
Eν

½ðg0T;1Þ2S¯T00ðq2Þ
þ g0T;1g1T;1S¯T01ðq2Þ þ ðg1T;1Þ2S¯T11ðq2Þ
þ 32mA
2J þ 1

1 −
T
Eν

½ðg0T;1Þ2S¯L00ðq2Þ
þ g0T;1g1T;1S¯L01ðq2Þ þ ðg1T;1Þ2S¯L11ðq2Þ: ð108Þ
Contrary to the axial-vector response, there is now also a
contribution from the longitudinal multipoles, S¯Lijðq2Þ.
These response functions are identical to the ones derived
for the axial-vector case only at leading order, i.e., the two-
body corrections for the tensor current would take a
different form and likewise the corrections from the
induced pseudoscalar and the axial-vector radius need to
be removed:
S¯Tijðq2Þ¼STijðq2Þjδ0ðq2Þ¼0; S¯Lijðq2Þ¼SLijðq2Þjδ00ðq2Þ¼0:
ð109Þ
There are again no interference terms with the SM, but
the lepton traces do allow for potential interference terms
with scalar, pseudoscalar, and dipole operators. In addition,
there would be additional contributions from the σ0i
components of the tensor current as well as the induced
form factors in Eq. (21). In case such contributions became
relevant, the formalism could be extended accordingly.
V. SUMMARY
In this paper we have provided a detailed account of the
CEνNS cross section both within the SM and beyond. To
this end, we started from a decomposition into effective
operators, hadronic matrix elements, and nuclear structure
factors, including both the vector and axial-vector operators
already present in the SM, but also considering the effects
of (pseudo)scalar, tensor, and dipole operators. Light BSM
degrees of freedom could be included along similar lines.
As a first step, we introduced the charge and weak form
factors as typically defined in electron scattering, to
exemplify their decomposition in terms of underlying
nuclear structure factors, but also hadronic matrix elements
and Wilson coefficients. The analogous decomposition for
CEνNS is then used to address the question how, e.g., the
weak form factor needs to be modified once BSM con-
tributions are permitted, and to derive master formulas for
the cross section in the various cases.
Our results for the nuclear structure factors are based on
the large-scale nuclear shell model. In addition to the
coherent part of the response, which is largely determined
by charge operators, radius and relativistic corrections, as
well as spin-orbit contributions, we have also performed a
detailed study of the typically neglected axial-vector
responses. While the general formalism is similar to the
COHERENT ELASTIC NEUTRINO-NUCLEUS SCATTERING: … PHYS. REV. D 102, 074018 (2020)
074018-19
spin-dependent responses for dark matter scattering off
nuclei, there are key differences. Most notably, only the
transverse multipoles contribute to CEνNS due to the lepton
trace. We have also calculated updates for the structure
factors relevant for spin-dependent dark matter scattering.3
Our calculation of the spin-dependent responses takes
advantage of several developments in recent years that allow
us to improve the treatment of two-body currents as
predicted from chiral EFT. These include improved deter-
minations of the relevant low-energy constants from pion-
nucleon scattering, the calculation of one-loop corrections to
the nuclear axial-vector current, and insights from ab initio
studies of two-body effects in medium-mass and heavy
nuclei. While the nuclear interactions used in this work are
still phenomenological, this strategy allows us to incorporate
as many constraints from chiral EFT as possible, including,
for the first time, the effect of contact operators and pion-
pole contributions to the two-body currents.
Finally, we provide further details of the multipole
expansion of the nuclear responses, tailored towards the
aspects relevant for the CEνNS application and making the
connection to the notation in the nuclear-physics literature.
Together with the fits of the resulting nuclear responses as
well as the EFT decomposition of the cross section, this
defines general CEνNS responses for a wide range of
isotopes and effective operators.
Future precision studies of CEνNS will require improved
nuclear responses, especially those involving neutrons. As
CEνNS may, in fact, be the most promising probe of the
neutron responses of atomic nuclei, a global analysis of
multiple targets will be required to disentangle nuclear-
structure and potential BSM effects.
ACKNOWLEDGMENTS
We thank C. Alexandrou, S. Bacca, A. Crivellin, J.
Detwiler, J. Erler, D. Gazit, H. Krebs, S. Pastore, J.
Piekarewicz, K. Scholberg, A. Shindler, and J. de Vries
for valuable discussions. This work was supported in part
by the Swiss National Science Foundation (Project
No. PCEFP2_181117), the United States Department of
Energy (Grant No. DE-FG02-00ER41132), the Spanish
Ministerio de Ciencia e Innovación through the “Ramón y
Cajal” program with Grant No. RYC-2017-22781, the
Spanish Ministerio de Economía y Competitividad Grant
No. FIS2017-87534-P, the Japanese Society for the
Promotion of Science through Grant No. 18K03639,
MEXT as Priority Issue on Post-K Computer
(Elucidation of the Fundamental Laws and Evolution of
the Universe), the Joint Institute for Computational
Fundamental Science, the CNS-RIKEN joint project for
large-scale nuclear structure calculations, the Deutsche
Forschungsgemeinschaft (DFG, German Research
Foundation)—Project-ID 279384907—SFB 1245, and
the Max Planck Society.
APPENDIX A: MULTIPOLE EXPANSION
In this Appendix we review the main features of the
multipole expansion, following closely Refs. [113,118].
The starting point is the leptonic current lμ, which is
decomposed into the temporal component l0 and the
spatial, spherical components lλ, λ ¼ ; 3 with respect to
the reference vector q, where the latter index is chosen to
avoid confusion with the temporal component. The spin
sum takes the form
X
spins
jhfjLjiij2 ¼ 4π
X
spins
X
L≥0
½l3l3jhJfkLL þ L5LkJiij2 þ l0l0jhJfkML þM5LkJiij2
− 2Reðl3l0hJfkLL þ L5LkJiihJfkML þM5LkJiiÞ
þ 1
2
X
λ¼1
lλlλ
X
L≥1
jhJfkT elL þ T el5L þ λðT magL þ T mag5L ÞkJiij2

; ðA1Þ
where the reduced matrix elements refer to the longitudinal (L), Coulomb (M), transverse electric (T el), and transverse
magnetic (T mag) multipoles. The latter can be simplified to
X
spins
jhfjLjiij2jT ¼ 2π
X
spins
X
L≥1
½ðl · l − l3l3ÞðjhJfkT elL þ T el5L kJiij2 þ jhJfkT magL þ T mag5L kJiij2Þ
− 2iðl × lÞ3ReðhJfkT elL þ T el5L kJiihJfkT magL þ T mag5L kJiiÞ: ðA2Þ
The single-nucleon contributions, obtained by nonrelativistic expansion of Eqs. (9) and (11), can then be expressed in terms
of fundamental multipole operators according to
3Our results for the nuclear structure factors, as can be reconstructed from the fits for the nuclear responses in Appendix E, are also
available as text files upon request.
HOFERICHTER, MENE´NDEZ, and SCHWENK PHYS. REV. D 102, 074018 (2020)
074018-20
MLM ¼ FN1MML þ
q2
4m2N
ðFN1 þ 2FN2 Þ

Φ00ML −
1
2
MLM

;
LLM ¼
q0
jqjMLM;
T elLM ¼
jqj
mN

FN1 Δ0ML þ
FN1 þ FN2
2
ΣML

;
T magLM ¼ −i
jqj
mN

FN1 ΔML −
FN1 þ FN2
2
Σ0ML

;
M5LM ¼ −i
jqj
mN
GNA

ΩML þ
1
2
Σ00ML

;
L5LM ¼ i

GNA

1 −
q2
8m2N

−
q2
4m2N
GNP

Σ00ML ;
T el5LM ¼ iGNA

1 −
q2
8m2N

Σ0ML ;
T mag5LM ¼ GNA

1 −
q2
8m2N

ΣML ; ðA3Þ
where we dropped the quark labels for the form
factors, terms suppressed by q0=mN, and several subleading
multipoles in the axial-vector contribution. The explicit
expressions for the multipoles in harmonic oscillator
basis are given in Ref. [113], where an additional
operator Ω0L ¼ Δ00L −Φ00L is introduced. Not all multipoles
will be needed in the analysis, the most important ones are
M and Φ00 for the vector responses and Σ0, Σ00 for the SD
ones. The nuclear responses ΣL, Δ0L, as well as the
combinations ðΔ00L − 12MLÞ, ðΩL þ 12Σ00LÞ vanish for elastic
scattering.
APPENDIX B: NUCLEAR RESPONSES
The nuclear responses associated to the M, Σ0, Σ00, and
Φ00 operators are defined as
MJ ¼
X
i
jJðqriÞYJðrˆiÞ; ðB1Þ
Σ0J ¼
X
i
1ffiffiffiffiffiffiffiffiffiffiffiffiffi
2J þ 1p ½−
ffiffiffi
J
p
jJþ1ðqriÞ½YJþ1ðrˆiÞσiJ þ
ffiffiffiffiffiffiffiffiffiffiffi
J þ 1p jJ−1ðqriÞ½YJ−1ðrˆiÞσiJ; ðB2Þ
Σ00J ¼
X
i
1ffiffiffiffiffiffiffiffiffiffiffiffiffi
2J þ 1p ½
ffiffiffiffiffiffiffiffiffiffiffi
J þ 1p jJþ1ðqriÞ½YJþ1ðrˆiÞσiJ þ
ffiffiffi
J
p
jJ−1ðqriÞ½YJ−1ðrˆiÞσiJ; ðB3Þ
Φ00J ¼ i
X
i
1
q
∇iðjJðqriÞYJðrˆiÞÞ ·

σi ×
1
q
∇i

¼ i
X
i
ffiffiffiffiffiffiffiffiffiffiffi
J þ 1pffiffiffiffiffiffiffiffiffiffiffiffiffi
2J þ 1p

jJþ1ðqrÞYJþ1ðrˆiÞ

σi ×
1
q
∇i

J
þ
ffiffiffi
J
pffiffiffiffiffiffiffiffiffiffiffiffiffi
2J þ 1p

jJ−1ðqrÞYJ−1ðrˆiÞ

σi ×
1
q
∇i

J
; ðB4Þ
where ½O1O2J indicates the coupling of operators O1 and O2 to a tensor of rank J, and tensor projections are omitted. The
single-particle harmonic-oscillator matrix elements needed for the calculation of the nuclear responses are

n0l0
1
2
j0
				MJ
				nl 12 j


¼ hn0l0jjJðqriÞjnlið−1Þjþ1=2þJ
ffiffiffiffiffi
1
4π
r
½ð2j0 þ 1Þð2jþ 1Þ12½ð2J þ 1Þð2lþ 1Þð2l0 þ 1Þ12
×

l0 J l
0 0 0

l0 j0 ½
j l J

; ðB5Þ

n0l0
1
2
j0
				jJ0 ðpriÞ½YJ0 ðrˆiÞσiJ
				nl 12 j


¼ hn0l0jjJ0 ðqriÞjnlið−1Þl0
ffiffiffiffiffi
6
4π
r
½ð2l0 þ 1Þð2lþ 1Þð2j0 þ 1Þð2jþ 1Þ12
× ½ð2J0 þ 1Þð2J þ 1Þ12

l0 J0 l
0 0 0
8><
>:
l0 l J0
½ ½ 1
j0 j J
9>=
>;; ðB6Þ
COHERENT ELASTIC NEUTRINO-NUCLEUS SCATTERING: … PHYS. REV. D 102, 074018 (2020)
074018-21
hn0l0j0kΦ00Jknlji ¼ ð−1Þl0
6ffiffiffiffiffi
4π
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð2j0 þ 1Þð2jþ 1Þð2l0 þ 1Þ
p
×
 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðJ þ 1Þð2J þ 3Þ
p XJþ1
L¼J
ð−1ÞJþLð2Lþ 1Þ

J þ 1 1 L
1 J 1
8><
>:
l0 l L
½ ½ 1
j0 j J
9>=
>;
×
 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðlþ 1Þð2lþ 3Þ
p  J þ 1 1 L
l l0 lþ 1

l0 J þ 1 lþ 1
0 0 0

hn0l0jjJþ1ðqriÞ
 ∂
∂ðqriÞ −
l
qri

jnli
−
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
lð2l − 1Þ
p  J þ 1 1 L
l l0 l − 1

l0 J þ 1 l − 1
0 0 0

hn0l0jjJþ1ðqriÞ
 ∂
∂ðqriÞ þ
lþ 1
qri

jnli

þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Jð2J − 1Þ
p XJ
L¼J−1
ð−1ÞJþLð2Lþ 1Þ

J − 1 1 L
1 J 1
8><
>:
l0 l L
½ ½ 1
j0 j J
9>=
>;
×
 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðlþ 1Þð2lþ 3Þ
p  J − 1 1 L
l l0 lþ 1

l0 J − 1 lþ 1
0 0 0

hn0l0jjJ−1ðqriÞ
 ∂
∂ðqriÞ −
l
qri

jnli
−
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
lð2l − 1Þ
p  J − 1 1 L
l l0 l − 1

l0 J − 1 l − 1
0 0 0

hn0l0jjJ−1ðqriÞ
 ∂
∂ðqriÞ þ
lþ 1
qri

jnli

: ðB7Þ
APPENDIX C: NUCLEAR STRUCTURE
CALCULATION OF 133Cs
In order to illustrate the quality of the shell-model
calculations for 133Cs, Fig. 11 compares the calculated
and experimental low-energy excitation spectrum of 133Cs.
Even though our calculation incorrectly predicts a ground
state with angular momentum and parity JP ¼ 5=2þ, the
difference with the 7=2þ state is only 10 keV. The angular
momentum and parity of the lowest energy levels is
predicted well, even though the energy of the calculated
second 3=2þ state is lower than in experiment. Overall the
agreement with experiment is similar as in other odd-mass
nuclei with similar mass number.
APPENDIX D: Multipole decomposition
for tensor currents
Including the tensor operator from Eq. (1) into the
analysis requires a generalization of the multipole decom-
position reviewed in Appendix A. Here we follow closely
the original derivation in Refs. [144–146], including the
lepton trace
Lμνλσ ¼ Trð=kσμνPL=kσλσPRÞ
¼ 2½ðgμλgνσ − gμσgνλÞk · k0 þ iϵμνλαkσk0α
− iϵμνσαkλk0α − iϵμλσαk0νkα þ iϵνλσαk0μkα
− gμλðkνk0σ þ k0νkσÞ þ gμσðkνk0λ þ k0νkλÞ
þ gνλðkμk0σ þ k0μkσÞ − gνσðkμk0λ þ k0μkλÞ; ðD1Þ
and then specify the spin sums relevant for CEνNS. The key
idea in the generalized multipole expansion is then that the
antisymmetric tensor current jμν essentially admits two
vectorial components, jð0Þi ¼ j0i and jð1Þi ¼ − iffiffi2p ϵijkjjk, in
terms of which the analog of Eqs. (A1) and (A2) becomes
[144,146]
Theory Exp.
0
100
200
300
400
500
600
700
800
900
Ex
ci
ta
tio
n 
en
er
gy
 (k
eV
)
133Cs
5/2+
5/2+
7/2+
1/2+
(3/2)+
(9/2)+
(7/2)+
11/2+
9/2+
(5/2,7/2,9/2+)
3/2+
5/2+
5/2+
9/2+
1/2+
3/2+
9/2+
3/2+
7/2+
5/2+
11/2+
7/2+
FIG. 11. Calculated 133Cs spectrum compared to experiment.
HOFERICHTER, MENE´NDEZ, and SCHWENK PHYS. REV. D 102, 074018 (2020)
074018-22
X
spins
jhfjLjiij2 ¼ 4π
X
spins
X
L≥0
½lð1Þ3 lð1Þ3 jhJfkLð1ÞL kJiij2 þ 4lð0Þ3 lð0Þ3 jhJfkLð0ÞL kJiij2
þ 4Reðlð1Þ3 lð0Þ3 hJfkLð1ÞL kJiihJfkLð0ÞL kJiiÞ
þ 2π
X
spins
X
L≥1
½ðlð1Þ · lð1Þ − lð1Þ3 lð1Þ3 ÞðjhJfkT elð1ÞL kJiij2 þ jhJfkT magð1ÞL kJiij2Þ
þ 4ðlð0Þ · lð0Þ − lð0Þ3 lð0Þ3 ÞðjhJfkT elð0ÞL kJiij2 þ jhJfkT magð0ÞL kJiij2Þ
þ 4ðlð1Þ · lð0Þ − lð1Þ3 lð0Þ3 ÞðhJfkT elð1ÞL kJiihJfkT elð0ÞL kJii þ hJfkT magð1ÞL kJiihJfkT magð0ÞL kJiiÞ
− 2iðlð1Þ × lð1ÞÞ3ReðhJfkT elð1ÞL kJiihJfkT magð1ÞL kJiiÞ
− 8iðlð0Þ × lð0ÞÞ3ReðhJfkT elð0ÞL kJiihJfkT magð0ÞL kJiiÞ
− 4iðlð1Þ × lð0ÞÞ3ReðhJfkT elð1ÞL kJiihJfkT magð0ÞL kJii þ hJfkT magð1ÞL kJiihJfkT elð0ÞL kJiiÞ; ðD2Þ
where we dropped the distinction between the two parities in each multipole. Since the nonrelativistic reduction of σ0i only
starts at Oð1=mNÞ and depends on momenta, the most interesting tensor contribution originates from the σij → ϵijkσk
components, contained in jð1Þ. The relevant spin sum reads
X
spins
lð1Þi l
ð1Þ
j ¼
1
2
ϵiklϵjmnLklmn ¼ −2tδij þ 4ðδijk · k0 − kik0j − k0ikjÞ þ 4iϵijkðkkE0ν þ k0kEνÞ; ðD3Þ
with projections
X
spins
lð1Þ3 l
ð1Þ
3 ¼ 8E2ν

1 −
T
Eν

;
X
spins
ðlð1Þ · lð1Þ − lð1Þ3 lð1Þ3 Þ ¼ 4E2ν

2 −
mAT
E2ν
−
2T
Eν

;
X
spins
ðlð1Þ × lð1ÞÞ3 ¼ −8iE2ν
ffiffiffiffiffiffi
2T
mA
s
: ðD4Þ
In contrast to Eq. (59), the longitudinal multipole is no
longer kinematically suppressed, but instead the interfer-
ence term between electric and magnetic multipoles can be
dropped. In our normalization the hadronic current starts
with − iffiffi
2
p ϵijkσjk → −i
ffiffiffi
2
p
σi, so that, up to the prefactor and
the different lepton traces, the remainder of the calculation
follows along the same lines as for the axial-vector
response.
APPENDIX E: PARAMETRIZATIONS OF THE
NUCLEAR RESPONSES
In this Appendix we provide explicit parametrizations
for the M and Φ00 responses not already given in previous
work [33], see Tables VI and VII. The parametrizations for
the Σ0 and Σ00 responses are given in Tables VIII–XI.
COHERENT ELASTIC NEUTRINO-NUCLEUS SCATTERING: … PHYS. REV. D 102, 074018 (2020)
074018-23
TABLE VI. Spin/parity JP of the nuclear ground states, harmonic-oscillator length b, and fit coefficients for the
nuclear response functions FM and F
Φ00
 . The fit functions are F
M
 ðuÞ ¼ e−
u
2
PnM
i¼0 c
M
i u
i (with c0 ¼ A and
c0 ¼ Z − N, respectively) and FΦ00 ðuÞ ¼ e−
u
2
PnΦ00
i¼0 c
Φ00
i u
i, with u ¼ q2b2=2. These forms correspond to the
analytical solution in the harmonic-oscillator basis [115,147], with nM and nΦ00 as implied by the table. Our results
for xenon are given in Ref. [33], the ones for germanium in Table VII.
Isotope 19F 23Na 40Ar 127I 133Cs
JP 1=2þ 3=2þ 0þ 5=2þ 7=2þ
b [fm] 1.7623 1.8048 1.9399 2.2821 2.2976
cMþ1 −6.00039 −8.66651 −20.9778 −125.164 −134.2
cMþ2 0.317846 0.555305 2.41486 35.3993 38.9577
cMþ3       −0.0368597 −3.62687 −4.12938
cMþ4          0.125083 0.151119
cMþ5          −0.000670162 −0.00103353
cM−1 0.666687 0.666658 3.42422 30.4307 33.9495
cM−2 −0.102251 −0.0655647 −0.618209 −12.321 −13.9502
cM−3       0.0268957 1.78131 2.04567
cM−4          −0.0870947 −0.102733
cM−5          0.000697815 0.000944352
cΦ
00þ
0
−0.764186 −2.89325 −4.79093 −26.1218 −28.2527
cΦ
00þ
1
0.152842 0.578667 1.4068 18.1692 20.4868
cΦ
00þ
2
      −0.0683192 −3.50413 −4.09303
cΦ
00þ
3
         0.223523 0.275572
cΦ
00þ
4
         −0.00360552 −0.0051254
cΦ
00−
0
0.36285 0.336942 0.326509 3.58476 8.98993
cΦ
00−
1
−0.0725723 −0.0673903 −0.452519 −4.58091 −8.67714
cΦ
00−
2
      0.0589909 1.46191 2.21868
cΦ
00−
3
         −0.139708 −0.189453
cΦ
00−
4
         0.0035109 0.00473947
TABLE VII. Same as Table VI, for germanium isotopes.
Isotope 70Ge 72Ge 73Ge 74Ge 76Ge
JP 0þ 0þ 9=2þ 0þ 0þ
b [fm] 2.0952 2.1035 2.1076 2.1117 2.1120
cMþ1 −51.2373 −53.5901 −54.7404 −55.9913 −58.3541
cMþ2 9.61013 10.2948 10.6249 10.9743 11.6381
cMþ3 −0.515768 −0.57547 −0.603598 −0.634449 −0.691196
cMþ4 0.0039318 0.0050503 0.00552928 0.00632403 0.00747821
cM−1 6.06953 8.67126 9.80348 11.356 13.9175
cM−2 −1.71276 −2.51496 −2.84183 −3.34586 −4.13067
cM−3 0.130409 0.20692 0.234571 0.287529 0.361556
cM−4 −2.22453 × 10−4 −0.00213335 −0.00255345 −0.0043077 −0.00609108
cΦ
00þ
0
−14.7388 −15.3806 −15.5467 −16.2171 −16.7737
cΦ
00þ
1
7.10953 7.53352 7.6085 8.07754 8.59006
cΦ
00þ
2
−0.811295 −0.869702 −0.875102 −0.951994 −1.04772
cΦ
00þ
3
0.0193996 0.0219601 0.0220616 0.0252548 0.02986
cΦ
00−
0
−3.27309 −0.924438 −0.848625 2.04591 4.22205
cΦ
00−
1
1.25408 −0.166778 −0.302814 −1.90271 −3.22233
cΦ
00−
2
−0.0487671 0.146851 0.177212 0.388221 0.576533
cΦ
00−
3
−8.74439 × 10−4 −0.00851802 −0.0101962 −0.017214 −0.0243454
HOFERICHTER, MENE´NDEZ, and SCHWENK PHYS. REV. D 102, 074018 (2020)
074018-24
TABLE VIII. Fit coefficients for the nuclear response functionsF
Σ0L
p;n andF
Σ00L
p;n for the relevant isotopes of fluorine, sodium, and xenon.
In analogy to Table VI, the fit functions areF ðuÞ ¼ e−u2Pi ciui, with nonzero coefficients as indicated. The results for the other isotopes
considered in this work are listed in Tables IX–XI.
Isotope 19F 23Na 129Xe 131Xe
L 1 1 3 1 1 3
cΣ
0p
0
0.269513 0.132973    0.00576416 −0.00511011   
cΣ
0p
1
−0.18098 −0.104393 0.0899535 −0.0069211 0.00702863 −0.0000968882
cΣ
0p
2
0.0296873 0.00909271 −0.0142746 0.00450247 −0.00156217 0.000171958
cΣ
0p
3
         −0.000867868 0.0000331178 −0.0000934431
cΣ
0p
4
         0.000038544 3.08471 × 10−6 7.87133 × 10−6
cΣ
0p
5
         9.80727 × 10−9 −1.94585 × 10−8 −1.56561 × 10−8
cΣ
0n
0
−0.00113172 0.0141201    0.185828 −0.161697   
cΣ
0n
1
0.00038188 −0.00774151 −0.000878018 −0.267263 0.334948 0.0364067
cΣ
0n
2
0.000744991 0.000326936 −0.000231297 0.149565 −0.174187 −0.079646
cΣ
0n
3
         −0.0274886 0.0310707 0.022489
cΣ
0n
4
         0.00173304 −0.00151254 −0.00171746
cΣ
0n
5
         −3.87392 × 10−7 −3.84408 × 10−7 −4.0527 × 10−7
cΣ
00p
0
0.190574 0.0940265    0.00407586 −0.00361339   
cΣ
00p
1
−0.125204 −0.0404172 0.0779019 −0.00646161 0.00442108 −0.0000839117
cΣ
00p
2
0.0206132 −0.000254736 −0.00592251 0.00321675 −0.00205213 0.000213614
cΣ
00p
3
         −0.000582408 0.000349931 −0.0000258884
cΣ
00p
4
         0.0000294951 −0.0000169039 2.73765 × 10−7
cΣ
00p
5
         3.82107 × 10−9 −3.20028 × 10−9 −4.49323 × 10−9
cΣ
00n
0
−0.000800244 0.00998438    0.131401 −0.114337   
cΣ
00n
1
0.00106046 −0.00902057 −0.000760388 −0.150054 −0.0175951 0.0315279
cΣ
00n
2
−0.000167277 0.00180209 −0.000223599 0.0820897 0.0321689 0.0476438
cΣ
00n
3
         −0.0148368 −0.00881948 −0.0170447
cΣ
00n
4
         0.000990728 0.000540511 0.00152533
cΣ
00n
5
         −1.50839 × 10−8 −3.05396 × 10−8 −1.37901 × 10−7
COHERENT ELASTIC NEUTRINO-NUCLEUS SCATTERING: … PHYS. REV. D 102, 074018 (2020)
074018-25
TABLE IX. Same as Table VIII, for cesium.
Isotope 133Cs
L 1 3 5 7
cΣ
0p
0
−0.253012         
cΣ
0p
1
0.483027 0.104388      
cΣ
0p
2
−0.164531 −0.08238 −0.0150628   
cΣ
0p
3
0.0168134 0.0118925 0.00856552 0.000657954
cΣ
0p
4
−0.00048879 −0.000423071 −0.000519134 −0.000651735
cΣ
0p
5
−5.62349 × 10−8 −3.52071 × 10−8 2.07474 × 10−8 4.02019 × 10−8
cΣ
0n
0
0.00070445         
cΣ
0n
1
−0.00520619 −0.00507773      
cΣ
0n
2
0.00351738 0.00295876 0.000728257   
cΣ
0n
3
−0.00069372 −0.000444073 −0.000228224 −0.0000513882
cΣ
0n
4
0.000060668 0.0000235555 0.000018572 6.60564 × 10−6
cΣ
0n
5
−1.0888 × 10−6 −9.09827 × 10−7 −4.68954 × 10−7 −2.43844 × 10−7
cΣ
00p
0
−0.178908         
cΣ
00p
1
0.0320074 0.0904055      
cΣ
00p
2
0.0211378 0.0034629 −0.0137503   
cΣ
00p
3
−0.00419937 −0.00308878 −0.00344057 0.000615352
cΣ
00p
4
0.000173592 0.000141839 0.000290862 0.000607814
cΣ
00p
5
−6.77831 × 10−8 −2.95736 × 10−8 1.61033 × 10−8 1.93527 × 10−8
cΣ
00n
0
0.000498115         
cΣ
00n
1
−0.000408223 −0.00439751      
cΣ
00n
2
−0.000741592 0.00230722 0.000664811   
cΣ
00n
3
0.000215744 −0.000355182 −0.000138555 −0.0000480682
cΣ
00n
4
−0.0000124709 0.0000227478 8.67291 × 10−6 6.82111 × 10−6
cΣ
00n
5
−1.28214 × 10−7 −2.62241 × 10−7 −2.143 × 10−7 −1.52006 × 10−7
HOFERICHTER, MENE´NDEZ, and SCHWENK PHYS. REV. D 102, 074018 (2020)
074018-26
TABLE X. Same as Table VIII, for iodine.
Isotope 127I
L 1 3 5
cΣ
0p
0
0.231258      
cΣ
0p
1
−0.374391 −0.153173   
cΣ
0p
2
0.195962 0.105378 0.0743581
cΣ
0p
3
−0.0342014 −0.0228849 −0.0234546
cΣ
0p
4
0.00162438 0.00130854 0.00188104
cΣ
0p
5
−3.37595 × 10−7 −2.56507 × 10−8 −9.24252 × 10−8
cΣ
0n
0
0.0205005      
cΣ
0n
1
−0.0362175 −0.00369561   
cΣ
0n
2
0.0174239 0.00235829 0.0000803278
cΣ
0n
3
−0.00285902 −0.000383903 −0.0000110023
cΣ
0n
4
0.000174649 0.0000204291 2.57961 × 10−8
cΣ
0n
5
−2.13335 × 10−6 −6.28715 × 10−7 −4.55316 × 10−8
cΣ
00p
0
0.163523      
cΣ
00p
1
−0.125749 −0.132651   
cΣ
00p
2
0.0450115 0.0668207 0.0678788
cΣ
00p
3
−0.00624361 −0.0124938 −0.0207994
cΣ
00p
4
0.000245811 0.000614695 0.00171886
cΣ
00p
5
−1.85038 × 10−7 −1.48556 × 10−8 −3.6698 × 10−8
cΣ
00n
0
0.0144959      
cΣ
00n
1
−0.017996 −0.00320049   
cΣ
00n
2
0.00679698 0.00161307 0.0000733265
cΣ
00n
3
−0.000985306 −0.000240205 −0.0000296856
cΣ
00n
4
0.0000461489 0.0000177136 2.66146 × 10−6
cΣ
00n
5
−2.6458 × 10−7 −1.81495 × 10−7 −2.07467 × 10−8
COHERENT ELASTIC NEUTRINO-NUCLEUS SCATTERING: … PHYS. REV. D 102, 074018 (2020)
074018-27
[1] D. Z. Freedman, Phys. Rev. D 9, 1389 (1974).
[2] D. Akimov et al. (COHERENT Collaboration), Science
357, 1123 (2017).
[3] D. Akimov et al. (COHERENT Collaboration), arXiv:
2003.10630.
[4] D. Akimov et al. (COHERENT Collaboration), Phys. Rev.
D 100, 115020 (2019).
[5] V. Belov et al., J. Instrum. 10, P12011 (2015).
[6] S. Kerman, V. Sharma, M. Deniz, H. T. Wong, J.-W. Chen,
H. B. Li, S. T. Lin, C.-P. Liu, and Q. Yue (TEXONO
Collaboration), Phys. Rev. D 93, 113006 (2016).
[7] G. Agnolet et al. (MINER Collaboration), Nucl. Instrum.
Methods Phys. Res., Sect. A 853, 53 (2017).
[8] D. Y. Akimov et al., J. Instrum. 12, C06018 (2017).
[9] J. Hakenmüller et al., Eur. Phys. J. C 79, 699 (2019).
[10] G. Angloher et al. (NUCLEUS Collaboration), Eur. Phys.
J. C 79, 1018 (2019).
[11] A. Aguilar-Arevalo et al. (CONNIE Collaboration), Phys.
Rev. D 100, 092005 (2019).
[12] A. Juillard et al., J. Low Temp. Phys. 199, 798 (2020).
[13] D. Baxter et al., J. High Energy Phys. 02 (2020) 123.
[14] J. Barranco, O. G. Miranda, and T. I. Rashba, J. High
Energy Phys. 12 (2005) 021.
[15] S. Abrahamyan et al., Phys. Rev. Lett. 108, 112502
(2012).
[16] C. J. Horowitz et al., Phys. Rev. C 85, 032501 (2012).
[17] M. Cadeddu, C. Giunti, Y. F. Li, and Y. Y. Zhang, Phys.
Rev. Lett. 120, 072501 (2018).
[18] X. R. Huang and L.W. Chen, Phys. Rev. D 100, 071301
(2019).
[19] A. N. Khan andW. Rodejohann, Phys. Rev. D 100, 113003
(2019).
[20] M. Cadeddu, F. Dordei, C. Giunti, Y. F. Li, and Y. Y.
Zhang, Phys. Rev. D 101, 033004 (2020).
[21] P. Coloma, I. Esteban, M. C. Gonzalez-Garcia, and J.
Mene´ndez, J. High Energy Phys. 08 (2020) 030.
[22] C. J. Horowitz, K. J. Coakley, and D. N. McKinsey, Phys.
Rev. D 68, 023005 (2003).
[23] J. Yang, J. A. Hernandez, and J. Piekarewicz, Phys. Rev. C
100, 054301 (2019).
[24] K. Patton, J. Engel, G. C. McLaughlin, and N. Schunck,
Phys. Rev. C 86, 024612 (2012).
[25] G. Co’, M. Anguiano, and A. M. Lallena, J. Cosmol.
Astropart. Phys. 04 (2020) 044.
[26] N. Van Dessel, V. Pandey, H. Ray, and N. Jachowicz,
arXiv:2007.03658.
TABLE XI. Same as Table VIII, for germanium.
Isotope 73Ge
L 1 3 5 7 9
cΣ
0p
0
0.0257064            
cΣ
0p
1
−0.0418759 −0.00920991         
cΣ
0p
2
0.015169 0.00417235 0.000347229      
cΣ
0p
3
−0.00163883 −0.000562315 −0.0000416612 −0.0000106207   
cΣ
0p
4
0.000045204 0.0000198897 3.28572 × 10−6 7.80947 × 10−7 0.0000282336
cΣ
0n
0
0.353305            
cΣ
0n
1
−0.562061 −0.233908         
cΣ
0n
2
0.181791 0.117078 0.0549077      
cΣ
0n
3
−0.0180905 −0.0142114 −0.0112771 −0.00718702   
cΣ
0n
4
0.00051239 0.000451122 0.000469159 0.000528465 0.000752534
cΣ
00p
0
0.0181769            
cΣ
00p
1
−0.0174535 −0.00797608         
cΣ
00p
2
0.00405522 0.00361834 0.000316977      
cΣ
00p
3
−0.00028759 −0.000359967 −0.0000879456 −9.93462 × 10−6   
cΣ
00p
4
6.37115 × 10−6 6.89172 × 10−6 1.79986 × 10−6 5.84383 × 10−7 0.0000267848
cΣ
00n
0
0.249824            
cΣ
00n
1
−0.205762 −0.202568         
cΣ
00n
2
0.045436 0.0675417 0.0501235      
cΣ
00n
3
−0.00335668 −0.00613658 −0.00769287 −0.00672282   
cΣ
00n
4
0.0000723721 0.00015611 0.000256944 0.000395465 0.000713918
HOFERICHTER, MENE´NDEZ, and SCHWENK PHYS. REV. D 102, 074018 (2020)
074018-28
[27] C. G. Payne, S. Bacca, G. Hagen, W. Jiang, and T.
Papenbrock, Phys. Rev. C 100, 061304 (2019).
[28] J. Mene´ndez, D. Gazit, and A. Schwenk, Phys. Rev. D 86,
103511 (2012).
[29] P. Klos, J. Mene´ndez, D. Gazit, and A. Schwenk, Phys.
Rev. D 88, 083516 (2013); 89, 029901(E) (2014).
[30] L. Baudis, G. Kessler, P. Klos, R. F. Lang, J. Mene´ndez, S.
Reichard, and A. Schwenk, Phys. Rev. D 88, 115014
(2013).
[31] L. Vietze, P. Klos, J. Mene´ndez, W. C. Haxton, and A.
Schwenk, Phys. Rev. D 91, 043520 (2015).
[32] M. Hoferichter, P. Klos, and A. Schwenk, Phys. Lett. B
746, 410 (2015).
[33] M. Hoferichter, P. Klos, J. Mene´ndez, and A. Schwenk,
Phys. Rev. D 94, 063505 (2016).
[34] M. Hoferichter, P. Klos, J. Mene´ndez, and A. Schwenk,
Phys. Rev. Lett. 119, 181803 (2017).
[35] A. Fieguth, M. Hoferichter, P. Klos, J. Mene´ndez, A.
Schwenk, and C. Weinheimer, Phys. Rev. D 97, 103532
(2018).
[36] M. Hoferichter, P. Klos, J. Mene´ndez, and A. Schwenk,
Phys. Rev. D 99, 055031 (2019).
[37] D. Aristizabal Sierra, J. Liao, and D. Marfatia, J. High
Energy Phys. 06 (2019) 141.
[38] S. Bacca and S. Pastore, J. Phys. G 41, 123002 (2014).
[39] N. Nevo Dinur, O. J. Hernandez, S. Bacca, N. Barnea, C.
Ji, S. Pastore, M. Piarulli, and R. B. Wiringa, Phys. Rev. C
99, 034004 (2019).
[40] G. Pre´zeau, A. Kurylov, M. Kamionkowski, and P. Vogel,
Phys. Rev. Lett. 91, 231301 (2003).
[41] V. Cirigliano, M. L. Graesser, and G. Ovanesyan, J. High
Energy Phys. 10 (2012) 025.
[42] V. Cirigliano, M. L. Graesser, G. Ovanesyan, and I. M.
Shoemaker, Phys. Lett. B 739, 293 (2014).
[43] C. Körber, A. Nogga, and J. de Vries, Phys. Rev. C 96,
035805 (2017).
[44] L. Andreoli, V. Cirigliano, S. Gandolfi, and F. Pederiva,
Phys. Rev. C 99, 025501 (2019).
[45] E. Aprile et al. (XENON Collaboration), Phys. Rev. Lett.
122, 071301 (2019).
[46] W. Altmannshofer, M. Tammaro, and J. Zupan, J. High
Energy Phys. 09 (2019) 083.
[47] M. A. Shifman, A. Vainshtein, and V. I. Zakharov, Phys.
Lett. 78B, 443 (1978).
[48] J. Goodman, M. Ibe, A. Rajaraman, W. Shepherd, T. M. P.
Tait, and H. B. Yu, Phys. Rev. D 82, 116010 (2010).
[49] B. Grzadkowski, M. Iskrzyński, M. Misiak, and J. Rosiek,
J. High Energy Phys. 10 (2010) 085.
[50] R. N. Mohapatra and P. B. Pal, World Sci. Lect. Notes
Phys. 60, 1 (1998); 72, 1 (2004).
[51] C. Giunti and A. Studenikin, Rev. Mod. Phys. 87, 531
(2015).
[52] E. Aprile et al., arXiv:2006.09721.
[53] V. Tishchenko et al. (MuLan Collaboration), Phys. Rev. D
87, 052003 (2013).
[54] M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98,
030001 (2018).
[55] G. Schneider et al., Science 358, 1081 (2017).
[56] P. J. Mohr, D. B. Newell, and B. N. Taylor, Rev. Mod.
Phys. 88, 035009 (2016).
[57] A. Antognini et al., Science 339, 417 (2013).
[58] L. Koester, W. Waschkowski, L. V. Mitsyna, G. S.
Samosvat, P. Prokofevs, and J. Tambergs, Phys. Rev. C
51, 3363 (1995).
[59] S. Kopecky, M. Krenn, P. Riehs, S. Steiner, J. A. Harvey,
N. W. Hill, and M. Pernicka, Phys. Rev. C 56, 2229 (1997).
[60] D. Djukanovic, K. Ottnad, J. Wilhelm, and H. Wittig,
Phys. Rev. Lett. 123, 212001 (2019).
[61] C. Alexandrou, S. Bacchio, M. Constantinou, J. Finkenrath,
K. Hadjiyiannakou, K. Jansen, and G. Koutsou, Phys. Rev.
D 101, 031501 (2020).
[62] B. Märkisch et al., Phys. Rev. Lett. 122, 242501 (2019).
[63] A. Airapetian et al. (HERMES Collaboration), Phys. Rev.
D 75, 012007 (2007).
[64] R. J. Hill, P. Kammel, W. J. Marciano, and A. Sirlin, Rep.
Prog. Phys. 81, 096301 (2018).
[65] R. Gupta, B. Yoon, T. Bhattacharya, V. Cirigliano, Y. C.
Jang, and H.W. Lin, Phys. Rev. D 98, 091501 (2018).
[66] M. Hoferichter, B. Kubis, J. Ruiz de Elvira, and P. Stoffer,
Phys. Rev. Lett. 122, 122001 (2019); 124, 199901(E)
(2020).
[67] M. Hoferichter, J. Ruiz de Elvira, B. Kubis, and U.-G.
Meißner, Phys. Rev. Lett. 115, 092301 (2015).
[68] S. Dürr et al. (BMW Collaboration), Phys. Rev. Lett. 116,
172001 (2016).
[69] Y. B. Yang, A. Alexandru, T. Draper, J. Liang, and K.-F.
Liu (χQCD Collaboration), Phys. Rev. D 94, 054503
(2016).
[70] A. Abdel-Rehim, C. Alexandrou, M. Constantinou, K.
Hadjiyiannakou, K. Jansen, Ch. Kallidonis, G. Koutsou,
and A. V. Avile´s-Casco (ETM Collaboration), Phys. Rev.
Lett. 116, 252001 (2016).
[71] G. S. Bali, S. Collins, D. Richtmann, A. Schäfer, W.
Söldner, and A. Sternbeck (RQCD Collaboration), Phys.
Rev. D 93, 094504 (2016).
[72] R. J. Hill and M. P. Solon, Phys. Rev. D 91, 043505 (2015).
[73] M. Hoferichter, C. Ditsche, B. Kubis, and U.-G. Meißner,
J. High Energy Phys. 06 (2012) 063.
[74] C. Ditsche, M. Hoferichter, B. Kubis, and U.-G. Meißner,
J. High Energy Phys. 06 (2012) 043.
[75] Z. Fodor, C. Hoelbling, S. Krieg, L. Lellouch, Th. Lippert,
A. Portelli, A. Sastre, K. K. Szabo, and L. Varnhorst, Phys.
Rev. Lett. 117, 082001 (2016).
[76] R. Pohl et al., Nature (London) 466, 213 (2010).
[77] A. Beyer et al., Science 358, 79 (2017).
[78] H. Fleurbaey, S. Galtier, S. Thomas, M. Bonnaud, L.
Julien, F. Biraben, F. Nez, M. Abgrall, and J. Gue´na, Phys.
Rev. Lett. 120, 183001 (2018).
[79] N. Bezginov, T. Valdez, M. Horbatsch, A. Marsman, A. C.
Vutha, and E. A. Hessels, Science 365, 1007 (2019).
[80] W. Xiong et al. (PRad Collaboration), Nature (London)
575, 147 (2019).
[81] I. T. Lorenz, H.-W. Hammer, and U.-G. Meißner, Eur.
Phys. J. A 48, 151 (2012).
[82] M. Hoferichter, B. Kubis, J. Ruiz de Elvira, H.-W.Hammer,
and U.-G. Meißner, Eur. Phys. J. A 52, 331 (2016).
[83] A. A. Filin, V. Baru, E. Epelbaum, H. Krebs, D. Möller,
and P. Reinert, Phys. Rev. Lett. 124, 082501 (2020).
[84] B. Kubis and R. Lewis, Phys. Rev. C 74, 015204
(2006).
COHERENT ELASTIC NEUTRINO-NUCLEUS SCATTERING: … PHYS. REV. D 102, 074018 (2020)
074018-29
[85] R. González-Jime´nez, J. A. Caballero, and T.W. Donnelly,
Phys. Rev. D 90, 033002 (2014).
[86] S. Weinberg, Phys. Rev. 112, 1375 (1958).
[87] S. Aoki et al. (Flavour Lattice Averaging Group), Eur.
Phys. J. C 80, 113 (2020).
[88] H.W. Lin, R. Gupta, B. Yoon, Y. C. Jang, and T.
Bhattacharya, Phys. Rev. D 98, 094512 (2018).
[89] J. Liang, Y. B. Yang, T. Draper, M. Gong, and K. F. Liu,
Phys. Rev. D 98, 074505 (2018).
[90] A. Bodek, S. Avvakumov, R. Bradford, and H. S. Budd,
Eur. Phys. J. C 53, 349 (2008).
[91] V. Bernard, L. Elouadrhiri, and U.-G. Meißner, J. Phys. G
28, R1 (2002).
[92] J. J. de Swart, M. C. M. Rentmeester, and R. G. E. Timmer-
mans, PiN Newslett. 13, 96 (1997), https://arxiv.org/abs/
nucl-th/9802084.
[93] V. Baru, C. Hanhart, M. Hoferichter, B. Kubis, A. Nogga,
and D. R. Phillips, Phys. Lett. B 694, 473 (2011).
[94] V. Baru, C. Hanhart, M. Hoferichter, B. Kubis, A. Nogga,
and D. R. Phillips, Nucl. Phys. A872, 69 (2011).
[95] R. Navarro Pe´rez, J. E. Amaro, and E. Ruiz Arriola, Phys.
Rev. C 95, 064001 (2017).
[96] P. Reinert, H. Krebs, and E. Epelbaum, arXiv:2006.15360.
[97] A. Crivellin, M. Hoferichter, and M. Procura, Phys. Rev. D
89, 054021 (2014).
[98] J. Gasser and H. Leutwyler, Nucl. Phys. B94, 269 (1975).
[99] S. Borsanyi et al., Science 347, 1452 (2015).
[100] J. Gasser, M. Hoferichter, H. Leutwyler, and A. Rusetsky,
Eur. Phys. J. C 75, 375 (2015); 80, 353(E) (2020).
[101] D. A. Brantley, B. Joó, E. V. Mastropas, E. Mereghetti, H.
Monge-Camacho, B. C. Tiburzi, and A. Walker-Loud,
arXiv:1612.07733.
[102] J. Gasser, H. Leutwyler, and A. Rusetsky, arXiv:
2003.13612.
[103] D. Gotta et al., Lect. Notes Phys. 745, 165 (2008).
[104] T. Strauch et al., Eur. Phys. J. A 47, 88 (2011).
[105] M. Hennebach et al., Eur. Phys. J. A 50, 190 (2014).
[106] M. Hoferichter, J. Ruiz de Elvira, B. Kubis, and U.-G.
Meißner, Phys. Rep. 625, 1 (2016).
[107] J. Ruiz de Elvira, M. Hoferichter, B. Kubis, and U.-G.
Meißner, J. Phys. G 45, 024001 (2018).
[108] M. Hoferichter, J. Ruiz de Elvira, B. Kubis, and U.-G.
Meißner, Phys. Lett. B 760, 74 (2016).
[109] S. Borsanyi, Z. Fodor, C. Hoelbling, L. Lellouch, K. K.
Szabo, C. Torrero, and L. Varnhorst, arXiv:2007.03319.
[110] H. Y. Cheng, Phys. Lett. B 219, 347 (1989).
[111] H. Y. Cheng and C.W. Chiang, J. High Energy Phys. 07
(2012) 009.
[112] J. Dragos, T. Luu, A. Shindler, J. de Vries, and A. Yousif,
arXiv:1902.03254.
[113] B. D. Serot, Nucl. Phys. A308, 457 (1978).
[114] T. W. Donnelly and R. D. Peccei, Phys. Rep. 50, 1 (1979).
[115] T. W. Donnelly and W. C. Haxton, At. Data Nucl. Data
Tables 23, 103 (1979).
[116] B. D. Serot, Nucl. Phys. A322, 408 (1979).
[117] T.W. Donnelly and I. Sick, Rev.Mod. Phys. 56, 461 (1984).
[118] J. D. Walecka, Oxford Stud. Nucl. Phys. 16, 1 (1995).
[119] W. Bertozzi, J. Friar, J. Heisenberg, and J. Negele, Phys.
Lett. B 41, 408 (1972).
[120] C. Horowitz and J. Piekarewicz, Phys. Rev. C 86, 045503
(2012).
[121] J. Erler and S. Su, Prog. Part. Nucl. Phys. 71, 119 (2013).
[122] P. G. Blunden, W. Melnitchouk, and A.W. Thomas, Phys.
Rev. Lett. 109, 262301 (2012).
[123] B. A. Brown and W. A. Richter, Phys. Rev. C 74, 034315
(2006).
[124] E. Caurier, J. Mene´ndez, F. Nowacki, and A. Poves, Phys.
Rev. C 75, 054317 (2007); 76, 049901(E) (2007).
[125] J. Mene´ndez, A. Poves, E. Caurier, and F. Nowacki, Phys.
Rev. C 80, 048501 (2009).
[126] J. Mene´ndez, A. Poves, E. Caurier, and F. Nowacki, Nucl.
Phys. A818, 139 (2009).
[127] E. Caurier and F. Nowacki, Acta Phys. Pol. B 30, 705
(1999), https://www.actaphys.uj.edu.pl/fulltext?series=
Reg&vol=30&page=705.
[128] E. Caurier, G. Martínez-Pinedo, F. Nowacki, A. Poves, and
A. P. Zuker, Rev. Mod. Phys. 77, 427 (2005).
[129] M. Centelles, X. Roca-Maza, X. Vinas, and M. Warda,
Phys. Rev. Lett. 102, 122502 (2009).
[130] B. A. Brown, A. Derevianko, and V. V. Flambaum, Phys.
Rev. C 79, 035501 (2009).
[131] I. Angeli and K. P. Marinova, At. Data Nucl. Data Tables
99, 69 (2013).
[132] J. Mene´ndez, D. Gazit, and A. Schwenk, Phys. Rev. Lett.
107, 062501 (2011).
[133] P. Gysbers, G. Hagen, J. Holt, G. Jansen, T. Morris, P.
Navrátil, T. Papenbrock, S. Quaglioni, A. Schwenk, S.
Stroberg, and K. Wendt, Nat. Phys. 15, 428 (2019).
[134] V. Bernard, N. Kaiser, J. Kambor, and U.-G. Meißner,
Nucl. Phys. B388, 315 (1992).
[135] H. Krebs, E. Epelbaum, and U.-G. Meißner, Ann. Phys.
(Amsterdam) 378, 317 (2017).
[136] P. Klos, Master’s thesis, Technical University Darmstadt,
2014.
[137] M. Hoferichter, J. Ruiz de Elvira, B. Kubis, and U.-G.
Meißner, Phys. Rev. Lett. 115, 192301 (2015).
[138] V. Bernard, E. Epelbaum, H. Krebs, and U.-G. Meißner,
Phys. Rev. C 77, 064004 (2008).
[139] S. Ishikawa and M. R. Robilotta, Phys. Rev. C 76, 014006
(2007).
[140] E. Epelbaum, H.W. Hammer, and U.-G. Meißner, Rev.
Mod. Phys. 81, 1773 (2009).
[141] V. Baru, E. Epelbaum, C. Hanhart, M. Hoferichter, A. E.
Kudryavtsev, and D. R. Phillips, Eur. Phys. J. A 48, 69
(2012).
[142] A. Baroni, L. Girlanda, S. Pastore, R. Schiavilla, and M.
Viviani, Phys. Rev. C 93, 015501 (2016); 93, 049902(E)
(2016); 95, 059901(E) (2017).
[143] G. Martínez-Pinedo, A. Poves, E. Caurier, and A. Zuker,
Phys. Rev. C 53, R2602 (1996).
[144] A. Glick-Magid, Master’s thesis, Hebrew University of
Jerusalem, 2016.
[145] A. Glick-Magid, Y. Mishnayot, I. Mukul, M. Hass, S.
Vaintraub, G. Ron, and D. Gazit, Phys. Lett. B 767, 285
(2017).
[146] A. Glick-Magid and D. Gazit (to be published).
[147] W. Haxton and C. Lunardini, Comput. Phys. Commun.
179, 345 (2008).
HOFERICHTER, MENE´NDEZ, and SCHWENK PHYS. REV. D 102, 074018 (2020)
074018-30