Weak mixing below the weak scale in darkmatter direct detection
HJE
mixing below the weak scale in darkmatter
Joachim Brod 0 1 3 6 7
Benjamin Grinstein 0 1 3 4 7
Emmanuel Stamou 0 1 3 5 7
Jure Zupan 0 1 2 3 7
0 5640 S Ellis Ave , Chicago, IL, 60637 U.S.A
1 9500 Gilman Dr. , La Jolla, CA, 92093 U.S.A
2 Department of Physics, University of Cincinnati
3 OttoHahnStr. 4, Dortmund, D44221 Germany
4 Department of Physics, University of CaliforniaSan Diego , USA
5 Enrico Fermi Institute, University of Chicago
6 Fakultat fur Physik, TU Dortmund
7 400 Geology/Physics Bldg. , Cincinnati, Ohio, 45221 U.S.A
If dark matter couples predominantly to the axialvector currents with heavy quarks, the leading contribution to darkmatter scattering on nuclei is either due to oneloop weak corrections or due to the heavyquark axial charges of the nucleons. We calculate the e ects of Higgs and weak gaugeboson exchanges for dark matter coupling to heavyquark axialvector currents in an e ective theory below the weak scale. By explicit computation, we show that the leadinglogarithmic QCD corrections are important, and thus resum them to all orders using the renormalization group.
Beyond Standard Model; Perturbative QCD; Renormalization Group; Re

Weak
1 Introduction
2
3
4
5
6
7
8
Majorana and scalar dark matter
7.1
7.2
Majorana dark matter
Scalar dark matter
Conclusions
A Unphysical operators
A.1 Evanescent operators A.2 E.o.m.vanishing operators
Standard Model weak e ective Lagrangian
Operator mixing and anomalous dimensions
Renormalization group evolution
5.1
Numerical analysis and the impact of resummation
Connecting to the physics above the weak scale
1
Introduction
A useful approach to describe the results of Dark Matter (DM) directdetection experiments
is to relate them to an E ective Field Theory (EFT) of DM coupling to quarks, gluons,
leptons, and photons [1{17]. In this EFT, the level of suppression of DM interactions with
the Standard Model (SM) depends on the mass dimension of the interaction operators, i.e.,
the higher the mass dimension the more suppressed the operator is. The mass dimension
of operators is thus the organizing principle in capturing the phenomenologically most
relevant e ects, which is why in phenomenological analyses one keeps all relevant terms up
to some mass dimension, d. An important question is, at which value of d one can truncate
the expansion. The obvious choice would be to keep all operators of dimension
ve and
refs. [10, 11], with the phenomenological implications further discussed in [17]. We
improve on the analysis of ref. [11] in two ways: i) we clarify how to consistently include the
doubleinsertion contributions in the EFT framework, ii) we also perform the resummation
of the QCD corrections at leadinglogarithmic accuracy. Moreover, the generality of our
approach covers also the case of nonsinglet DM in the theory above the electroweak scale.
The paper is structured as follows. In sections 2{6 we derive our results for the case of
Diracfermion DM. These are then extended to the case of Majoranafermion DM and to
the case of scalar DM in section 7. In section 2 we rst show that the electroweak
correcoperators entering in intermediate steps of our calculation.
2
The importance of weak corrections for axial currents
We start by considering the DM EFT valid below the electroweak scale, b <
< ew, for
Diracfermion DM when ve quark avors are active,
L
=
X Ca
a;d
(d)
d 4 Q(ad) :
(2.1)
(2.2)
(2.3)
The sums run over the dimensions of the operators, d, and the operator labels, a. The
operators are multiplied by dimensionless Wilson coe cients, Ca(d), and the appropriate
powers of the mediator mass scale,
. Since we are interested in the theory below the
electroweak scale, any interactions with the top quark, W , Z bosons, and the Higgs are
integrated out and are part of the Wilson coe cients Ca(d). In this work, we focus on
dimensionsix operators, namely
Q(16;f) = (
Q(36;f) = (
)(f
)(f
f ) ;
tude, the nuclear response functions [13, 14, 20{23], and all the relevant kinematic factors.
We estimate A[Q(ad)] in three di erent limits: i) in the limit of only strong interactions, ii)
including QED corrections, and iii) also including corrections from weak interactions.
HJEP02(18)74
i) Switching o
QED and weak interactions, the e ective scattering amplitudes for
dimensionsix operators have the following parametric sizes (see ref. [18]):
A[Q(16;u)(d)]
A[Q(26;u)(d)]
A[Q(36;u)(d)]
A ;
10 3 is the typical DM velocity in the
laboratory frame, q is the typical momentum exchange, q=mN . 0:1, where mN is
the nucleon mass, and A is the nuclear mass number (for heavy nuclei A
102). The
approximate expressions for the e ective scattering amplitudes in eqs. (2.5){(2.8)
include the parametric O(A) coherent enhancement of the spinindependent nuclear
response function, WM (q), while all the other response functions were counted as
O(
1
). The vector and axial form factors at zero recoil are O(
1
) for u; d quarks. For
the strange, charm and bottom quarks the vector form factors vanish. The axial
charge for the strange quark is reasonably well known,
s =
factor of two uncertainty on these estimates.
Due to the nonrelativistic nature of the problem and the sizes of the nuclear
matrix elements, there are large hierarchies between the e ective scattering
amplitudes. For light quarks this hierarchy can be as large as vT =A
χ
Q(a6,q)′
χ
χ
f
γ
Q(a6,q)′
q
′
Ob(,6q)q′
heavy quarks into DM vector interactions with light quarks. Here, f = u; d; s; c; b; e; ; can denote
any of the quarks or charged leptons.
see eq. (3.1) in the next section. A double insertion of one fourfermion operator from
LeSM and one from L , see gure 2, induces the additional contributions to
A[Q(36;c)(b)]
4 s2w mmc2(2Zb) A ;
4 s2w mmc2(2Zb) max vT A;
q
mN
; (2.11)
(6)
where sw is the sine of the weak mixing angle. The proportionality to the square of
the heavyquark mass mc(b)  necessary for dimensional reasons  can be deduced
from the fact that it is the only relevant mass scale in the regime c(b) <
<
ew.
For Q3;c(b) these contributions dominate over the axial charge contribution, eq. (2.7),
{ 4 {
(6)
by several orders of magnitude, while for Q4;c(b) the electroweak corrections are either
comparable or smaller than in eq. (2.8). More details follow in the next sections.
The above estimates show that QED and weak corrections are essential to
capture the leading contributions for the dimensionsix operators in eqs. (2.2){(2.3) that
involve heavy quarks. The same type of QED and weak radiative corrections also
induce the leading e ective amplitudes for the scattering on nucleons when the DM
couples, at tree level, only to leptons. The logarithmically enhanced QED
contributions are known, see for instance refs. [9, 11, 18]. In the present work, we calculate
the logarithmically enhanced contributions due to the weak interactions. They arise,
via double insertions, at second order in the dimensionsix e ective interactions,
Accordingly, they can mix into dimensioneight operators, which,
therefore, also have to be included.
It turns out that the weak corrections are numerically irrelevant for operators
coupling DM to light quarks at tree level. Since the weak interactions do not conserve
parity, they can lift the velocity suppression in the matrix elements of Q(36;q) through the
mixing into the coherently enhanced operator Q(16;q). However, the resulting relative
enhancement of order A=vT
sion of the weak corrections, of order
105 is not enough to compensate for the large
suppres
=(4 s2w)(mq=mZ )2 . 10 9(mq=100 MeV)2.
The weak corrections are also much less important for the dimension ve and
dimensionseven operators coupling DM to the SM
elds [18, 19]. Most of these
operators have a nonzero nucleon matrix element already without including electroweak
corrections, in which case the latter only give subleading corrections. This is the
case for the operators coupling DM to gluons or photons, for pseudoscalar currents
with light quarks, and for scalar quark currents, including the ones with heavy
bottom and charm quarks. In the special case where DM couples only to pseudoscalar
heavyquark currents the nuclear matrix elements vanish. This remains true also
after oneloop electroweak corrections are included.
In the next two sections, we will obtain the leadinglogarithmic expressions for
the electroweak contributions in eq. (2.11) and also resum the QCD corrections by
performing the RG running from the weak scale, ew
O(mZ ), to the hadronic scale,
had
O(2 GeV), where we match to the nonrelativistic theory.
3
Standard Model weak e ective Lagrangian
The SM interactions below the weak scale are described by an e ective Lagrangian,
obtained by integrating out the top quark and the Z, W , and Higgs bosons at the scale
ew
mZ . In this section we focus on quark interactions. We discuss leptons in
section 6. We can neglect any operators involving avorchanging neutral currents as well as
terms suppressed by o diagonal CabibboKobayashiMaskawa (CKM) matrix elements.
The only necessary operators are
LeSM
p
2GF
8
<X
:q6=q0
2
1
X
4 2 i=1;2;4;5
Di;qq0 Oi(;6q)q0 +
(6)
X
i=3;6
3
Di;qq0 Oi;qq0 5 + X
(6) (6)
X
q i=1;:::;4
(6)
Di;q Oi;q
(6)=
;
9
;
(3.1)
{ 5 {
Here, T a are the SU(3)c generators normalised as Tr(T aT b) = 12 ab. As seen from the
above operator basis, there are fewer linearly independent operators with a single quark
than with two di erent quarks. The reason is that Fierz identities relate operators, like
for instance the counterpart of Oq(6q)0 with four equal quark
elds, to the operators Oi;q
(6)
with i = 1; : : : ; 4.
One way of implementing the Fierz relations is to project Green's
functions onto the basis that includes socalled Fierzevanescent operators, like E7q and E8
q
in eq. (A.2) of appendix A, that vanish due to Fierz identities. SM operators with scalar or
tensor currents do not contribute in our calculation. This is most easily seen by inspecting
their chiral and Lorentz structure, neglecting operators with derivatives (see below).
Integrating out the W and the Z bosons at tree level gives the following values for the
Wilson coe cients at ew
(6)
D1;qq0 = 4s2wc2wvqvq0 +
(6)
D3;qq0 = 4s2wc2waqvq0
I
j q
3
I
j q
3
6
6
3
Iq0 j
3
Iq0 j
jVqq0 j2 ;
jVqq0 j2 ;
and
(6)
D2;qq0 = 4s2wc2waqaq0 +
I
j q
3
3
Iq0 j
6
jVqq0 j2 ;
(6)
D4;qq0 = D5;qq0 =
(6)
(6)
3
D6;qq0 = jIq
Iq30 jjVqq0 j2 ;
where GF is the Fermi constant and Da
run over all light quarks, q; q0 = u; d; s; c; b, and the labels of the operators with two di erent
(6) are dimensionless Wilson coe cients. The sums
quark avors (q 6= q0)
and a single quark avor,
O1(6;q) = (q q) (q
q) ;
O3(6;q) = (q
D1(6;q) = 2s2wc2wvq2 ;
D2(6;q) = 2s2wc2waq2 ;
D3(6;q) = 4s2wc2wvqaq ;
D4(6;q) = 0 :
for q = u; c and Iq3 =
the quarks are encoded in
Here, sw
sin w, cw
cos w, with
w the weak mixing angle, while Iq3 is the third
component of the weak isospin for the corresponding lefthanded quark, i.e., I
q3 = 1=2
1=2 for q = d; s; b. The CKM matrix, Vqq0 , will be set to unity
unless speci ed otherwise, while the vector and axialvector couplings of the Z boson to
vq
I
3
q
2s2wQq
2swcw
;
aq
I
3
q
2swcw
;
where Qq is the electric charge of the corresponding quark. Note that Di;qq0
i = 1; 2; 4; 5, since the corresponding operators are symmetric under q $ q0.
(6)
{ 6 {
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
Di(;6q)0q for
HJEP02(18)74
We are now ready to derive the leading contributions to the DMnucleon scattering rates
for the case that, at the weak scale, DM interacts with the visible sector only through the
dimensionsix operators Q(36;q) or Q(46;q), with q = b; c. To properly describe all the leading
DM interactions we need to extend the dimensionsix e ective Lagrangian L , eq. (2.1),
to include the following dimensioneight operators
where
Q(18;q) =
Q(38;q) =
mq2
g2 (
s
mq2
g2 (
s
L
p
2GF
2
)(q
)(q
q) ;
5q) ;
X
For future convenience, we de ned the operators including two inverse powers of the strong
coupling constant. Even if the Wilson coe cient of the dimensioneight operators are zero
at ew, they are generated below the electroweak scale from a double insertion of one of
the dimensionsix operators in L in eq. (2.1) and one of the dimensionsix operator from
LeSM in eq. (3.1), see gure 2.1 The logarithmic part of the running from
ew to had gives
where we set ew = mZ . This equation shows that the operators with derivatives, for
instance, (
q), can be neglected because their e ect on the scattering rates is not
enhanced by the large ratio mb2;c=mq2. Furthermore, the set of operators in eqs. (4.2){(4.3)
is closed under RG running up to massdimension eight, if we keep only terms proportional
to two powers of the bottom or charmquark mass in the RG evolution. At higher orders
in QCD the purely electroweak expression eq. (4.4) gets corrected by terms of the order of
sn 1 logn(mb(c)=mZ ). Since mb(c)
mZ , these terms can amount to O(
1
) corrections. In
the following we resum these large QCD logarithms to leadinglogarithmic order.
The techniques for the calculation of leadinglogarithmic QCD corrections with
double insertions are standard [29{36].
We
rst replace the bare Wilson coe cients in
1The only exception occurs when the values of the Wilson coe cients at the weak scale conspire to
exactly cancel the divergence, so that the sum of the doubleinsertion diagrams is nite. This scenario is
the parameter K, where the GIM mechanism associated with the approximate avor symmetry of the SM
serves to cancel all divergences. We call the analogous mechanism for DM the \judicious operator equality
GIM ", in short \JoeGIM" mechanism. For JoeGIM DM there is no mixing of dimensionsix operators
into dimensioneight operators below the weak scale [29]. The leading contributions to the dimensioneight
operators are then obtained by a nite oneloop matching calculation at the heavyquark scales.
{ 7 {
The compound indices a, b, c, run over both the operator labels and quark avor indices.
In eq. (4.5), we have already made use of the fact that the QCD anomalous dimensions of
the operators Q(a6) in eqs. (2.2){(2.3) are zero, and have not introduced the corresponding
renormalization constants.
In dimension regularization around d = 4
2 spacetime dimensions, the
renormalization constants admit a double expansion in the strong coupling constant and
HJEP02(18)74
Zab = ab +
4
s
X
k=0;1
1k Za(1b;k) + O( s2) ;
and similarly for Z~ and Z^.
equation that is linear in the Wilson coe cients,
The RG evolution of the dimensionsix Wilson coe cients is determined by a RG
eqs. (2.1), (3.1), and (4.1) with their renormalized counterparts. The corresponding
effective Lagrangian reads
Le =
1
2
X
a
p
2GF
2
= (
1
) + (2) +
;
{ 8 {
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
(4.10)
d
d Da(6)( ) =
X
b
Db(6)( ) ba ;
with
=
d log Z
d log
:
On the other hand, the running of the dimensioneight Wilson coe cients receives two
contributions. In addition to the running of the mq2=gs2 prefactor, encoded in ~, there are
contributions from double insertions of dimensionsix operators, see gure 2. This leads to
a RG equation that is quadratic in dimensionsix Wilson coe cients [33, 34],
dd Ca(8)( ) =
X
b
Cb(8)( )~ba + X
Db(6)( )Cc(6)( )^bc;a :
bc
To leading order in the strong coupling constant the rankthree anomalous dimension tensor
^ab;c [33, 34] is given by
^ab;c =
2
s Z^a(1b;;c1) + O( s2) :
Next we provide the explicit values for the anomalous dimensions. In our notation, the
anomalous dimensions are expanded in powers of s
,
with (n)
/ ( s=4 )n, and similarly for ~ and ^.
We start by giving the results for the mixing of the dimensioneight operators coupling
DM to quarks, eqs. (4.2) and (4.3). This mixing is encoded in the ~ and ^ anomalous
q
′
q
q
′
q
q
′
q
q
′
q
q
of fourfermion operators. The poles of QCD penguin diagrams a ect the mixing via the
e.o.m.vanishing operators de ned in appendix A.
dimensions. We obtain the ^ from the poles of the double insertions, gure 2. The only
nonzero entries leading to mixing into operators with lightquark currents are
^O(13()6;q)0q;Q(36;q)0 ;Q1;q
(
1
)
(
1
)
(8) = ^O3(6;q)0q;Q(46;q)0 ;Q2;q
(8) = ^O2(6;q)q0 ;Q(36;q)0 ;Q3;q
(8) = ^O2(6;q)q0 ;Q(46;q)0 ;Q4;q
(8) =
(
1
)
The remaining contribution to the RG running of the dimensioneight operators is entirely
due to the mq2=gs2 prefactors in the de nition of the operators, namely
~ab = 2( m(0)
with CF = 4=3 for QCD, and Nf the number of active quark avors.
The RG running of the dimensionsix operators in the SM weak e ective Lagrangian is
due to oneloop gluon exchange diagrams, see gure 3. Since the corresponding anomalous
dimension matrix
has many entries, we split the result into several blocks.
The anomalous dimension matrix in the subsector spanned by the operators in
16Nc mq2 4
mq20 s
reads
into the operators
Note that, at oneloop, there is no mixing into operators with a di erent quark avor.
The anomalous dimensions describing the mixing of the same operators,
O1;q ; O2;q ; O3;q ; O4(6;q) ;
(6) (6) (6)
(
1
) =
4
0
B
B
B
C
C :
A
The anomalous dimension
matrix in the subsector spanned by the operators in
read
eqs. (3.2){(3.4),
reads
is
into the operators
has only two nonzero entries,
(
1
) =
4
s B
B
0
B
B
B
Finally, the part of the anomalous dimension matrix mixing the operators
0
12
0
0
5
0
0
19
3
(6)
(6)
(6)
(6)
8
3
8
3
9
12
0
0
0
5
0
0
9
0
0
0
0
0
12
23
3
5
(6)
(6)
4
3
4
3
(
1
) =
4
00 0 0 01
B
B
B0 0 0 0C
B0 0 0 0C
C
C
s BBB00 00 00 04CCC :
B
C
12
0
0
0
0
0
5
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
CC :
The part of the anomalous dimension matrix mixing the operators,
into the same operators, but with di erent quark
avor structure, q00 6= q0,
(6)
O1;qq0
(6)
(6)
; O2;qq0
; O3;qq0
; O3;q0q ; O4;qq0
; O5;qq0
; O6;qq0
; O6;q0q
;
(6)
(6)
(6)
(6)
;
(6)
(6)
(
1
) =
diag
0; 0; 0; 0; ; 0; ; 0
:
(6)
(6)
(6)
O1;qq0 ; O2;qq0 ; O3;qq0 ; O3;q0q ; O4;qq0 ; O5;qq0 ; O6;qq0 ; O6;q0q
(6)
(6)
(6)
(6)
(6)
O1;q ; O2;q ; O3;q ; O4;q
(6)
All the remaining entries in
vanish. We extracted the anomalous dimensions from
the o shell renormalization of Green's functions with appropriate external states. We
checked explicitly that our results are gaugeparameter independent. In appendix A, we
list the evanescent and e.o.m.vanishing operators that enter at intermediate stages of the
computation.
5
Renormalization group evolution
With the anomalous dimensions of section 4, we can now compute the Wilson coe cients at
had
not run, thus Cb(6)( had) = Cb(6)( ew). The RG running for the remaining Wilson coe cients
2 GeV in terms of initial conditions at the weak scale. The Wilson coe cients Cb(6) do
is controlled by the RG equations in eqs. (4.7) and (4.8), which we combine into a single
Here, we de ned a vector of Wilson coe cients as
d
d
Ca( ) =
X Cb( ) bea :
b
C ( )
D
C
(6)( )
(8)( )
!
;
(5.1)
(5.2)
(5.3)
and absorbed the (scaleindependent) Wilson coe cients Ca(6) into the e ective
anomalousdimension matrix
e
0
C
(6) ^
~
!
with (C(6) ^)ba
X
c
Cc(6)^bc;a :
Since the Ca(6) Wilson coe cients are RG invariant, the tensor product e ectively transforms
the rankthree tensor ^ab;c into an equivalent matrix, C(6) ^, with all its entries constant,
that is equivalent to the tensor for the purpose of RG running. This has the advantage
that one can use the standard methods for single insertions to solve the RG equations.
The RG evolution proceeds in multiple steps. The rst step is the matching of the
(6)
(complete or e ective) theory of DM interactions above the weak scale onto the ve avor
EFT. This matching computation yields the initial conditions for Ca(6)( ew) and Ca(8)( ew)
at the weak scale. At leadinglogarithmic order it su ces to perform the matching at
M
ew at treelevel. If the mediators have weakscale masses, we obtain Ca(6)( ew) 6= 0 and
Ca(8)( ew) = 0. If the mediators are much heavier than the weak scale, with masses of order
mZ , the RG running above the electroweak scale can induce nonzero Ca(8)( ew)
log( ew=M ). We discuss the latter case in section 6. For the RG evolution below the
electroweak scale one also needs the coe cients Da(6)( ew). The SM contributions to the
treelevel initial conditions for Da ( ew) are provided in eqs. (3.7){(3.9).
The second step is to evolve Ca ( ) and Da(6)( ) from the electroweak scale to lower
(8)
scales according to eq. (5.1). The RG evolution is in a theory with Nf = 5 quark avors,
when b
ew, with Nf = 4, when c
b, and with Nf = 3, when
had
c. Here the b(c) denote the threshold scale at which the bottom(charm)quark is removed
from the theory. In our numerical analysis we will use b = 4:18 GeV and c = 2 GeV.
At leadinglogarithmic order, there are no nontrivial matching corrections at the
bottomand charmquark thresholds, and we simply have
Ca(d)( b) Nf =4 = Ca(d)( b) Nf =5 ;
Ca(d)( c) Nf =3 = Ca(d)( c) Nf =4 :
(5.4)
This means that we can switch to the EFT with four active quark avors by simply
dropping all operators in eq. (4.5) that involve a bottomquark
eld, and to the EFT with
three active quark
avors by simply dropping all operators with charmquark
elds. The
leadingorder matching at q0
mq0 comes with a small uncertainty due to the choice
of matching scale that is of order log( q0 =mq0 ). This is formally of higher order in the
RGimproved perturbation theory. The uncertainty is canceled in a calculation at
nexttoleadinglogarithmic order by nite threshold corrections at the respective threshold scale.
This is a good point to pause and compare our results with the literature. The RG
evolution of the operators in eqs. (2.2){(2.3) below the electroweak scale has been studied
in ref. [11], which e ectively resummed the large logarithms log( had= ew) to all orders in
the Yukawa couplings. Such a resummation is problematic for two reasons. Firstly, it does
not take into account that the RG evolution stops at the heavyquark thresholds, below
which the Wilson coe cients are RG invariant. (Below the heavyquark thresholds, there
are no double insertions with heavy quarks and the running of the factor mq2=gs2 is precisely
canceled by the running of the Wilson coe cients of the dimensioneight operators.) This
introduces a spurious scale dependence of the Wilson coe cients in the three avor EFT, of
the order of jlog( had=mb(c))j . 50%, that is not canceled by the hadronic matrix elements.
Secondly, such a resummation is not consistent within the EFT framework. Since there
are no Higgsboson exchanges in the EFT below the weak scale, the schemedependence of
the anomalous dimensions and the residual matching scale dependence at the heavyquark
thresholds is not consistently canceled by higherorders, leading to unphysical results.
Continuing with our analysis, the nal step is to match the three avor EFT onto the
EFT with nonrelativistic neutrons and protons that is then used to predict the scattering
rates for DM on nuclei using nuclear response functions. The matching for the
dimensioneight contributions proceeds in exactly the same way as described in refs. [14, 16] for the
operators up to dimension seven. In practice, this means that we obtain the following
contributions to the nonrelativistic coe cients (see refs. [14, 16, 18, 20, 22]),
HJEP02(18)74
c1p =
c4p =
c6p =
c7p =
1
s
p
2
4
1
s
4
1
1
s
p
2GF
2
X
(5.5)
(5.6)
(5.7)
(5.8)
c8p =
c9p =
2
2
2GF
2
2
q=u;d;s
X
and similarly for neutrons, with p ! n. The quark masses and the strongcoupling
constant in these expressions should be evaluated in the three avor theory at the same scale
as the nuclear response functions, i.e.,
had = 2 GeV. The ellipses denote the
contributions from dimensionsix interactions proportional to Ca
to dimension ve and dimensionseven operators, which can be found in eqs. (17){(24)
(6) as well as the contributions due
s appears in eqs. (5.5){(5.10) as a consequence of the 1=gs2
prefactor in the de nition of the dimensioneight operators in eqs. (4.2){(4.3).
When
expanding the resummed results in the strong coupling constant, the
s cancels in the
leading expressions, and we nd
p
c1 '
p
c4 '
p
c6 '
p
c7 '
c8p =
c9p =
3 p
2
2GF
2
12
2
p
2GF
2
3 p
2
2GF
2
6 p
2
2GF
2
6 p
2GF
6 p
2GF
2
2
2
2
X
The quark masses in these expressions should be evaluated at the weak scale, mq0 =
mq0 (mZ ), while q0 is the scale at which the q0 quark is integrated out. We have provided the
SM Wilson coe cients, D2;qq0 and D3;q0q, in eqs. (3.7) and (3.8). The expanded equations
clearly illustrate that the leading terms are of electroweak origin, and thus of O( s0), while
the corrections due to QCD resummation start at O( s).
0−
0−
× − ×
0−
× − ×
the scattering of DM on protons, taking only one of the Wilson coe cients nonzero, setting it to
Ca
(d) = 1= 2, with
= 1 TeV. Hatched: the results without QCD resummation, eqs. (5.11){(5.16).
5.1
Numerical analysis and the impact of resummation
In gures 4 and 5 we show two numerical examples that illustrate the relative importance
of the above results.
We set
= 1 TeV and switch on a single dimensionsix Wilson
coe cient, C3;c , C4;c , C3(d;b), or C4(d;b) at a time, setting it to Ca
(d) (d)
(d) = 1. Figure 4 shows the
resulting nonrelativistic couplings for SM scattering on protons, cip. The magenta columns
are the full results, including QCD resummation. The hatched columns give the results
without the QCD resummation from eqs. (5.11){(5.16). Figure 5 shows the corresponding
results for DM couplings to neutrons, cin.
In these examples we set had =
c = 2 GeV and used the following quark masses
at
= mZ ,
mu(mZ ) = 1:4 MeV ;
mc(mZ ) = 0:78 GeV ;
md(mZ ) = 3:1 MeV ;
mb(mZ ) = 3:1 GeV :
ms(mZ ) = 63 MeV ;
These were obtained using the oneloop QCD running to evolve the MS quark masses
mu;d;s(2 GeV) and mc(b)(mc(b)), taken from ref. [37], to the common scale
= mZ . For
the nuclear coe cients that depend on the DM mass and/or the momentum transfer, we
choose m
= 100 GeV and a momentum transfer of q = 50 MeV.
As seen from
gures 4 and 5, the resummation of QCD logarithms enhances cip(n) by
approximately 10% to 50% depending on the speci c case. The typical enhancement is
O(30%). In the numerics we have set the CKM matrix element to unity, thus ignoring
all avor changing transitions. This is a very good approximation for operators with
10− 10−
× − ×
DM scattering on neutrons
cn
4
cn
6
cn
7
× − ×
0−
0−
8
× − ×
× − ×
0−
× 1 × − × − ×
bottom quarks. For charm quarks the e ect of avor o diagonal CKM matrix elements is
more important, yet still subleading. If we including the o diagonal terms in eqs. (3.7){
(3.8), then the largest correction reaches 16% for c1p induced from C3(6;c) (31% for the result
without resummation), as there is an up to 10% cancellation between the D3;cu and D3;cd
contributions with respect to the case of unit CKM matrix. For all other cases the error
due to setting the CKM matrix to unity is less than 10%.
Finally, we compare the contributions to DM scattering originating from electroweak
corrections as opposed to the intrinsic charm and bottom axial charges. For the case of
axialvectoraxialvector interactions (C4(6;c) 6= 0, C4(6;b) 6= 0) we have
weak:
intrinsic:
p
c4 '
p
c4 '
1
2 0:02 C4(6;c)
0:26 C4(6;b)
10 3;
4
2
c C4(6;c) +
b C4(6;b)
1
2 2 C4(6;c) + 0:2C4(6;b)
We see that for the bottom quarks the weak contribution, eq. (5.12), is comparable to the
contribution from the intrinsic bottom axial charge, while for charm quarks the contribution
due to the intrinsic charm axial charge dominates.
For vectoraxialvector interactions (C3(6;c) 6= 0, C3(6;b) 6= 0) we have
weak:
intrinsic:
p
c1 '
p
c7 '
p
c9 '
1
2 0:4 C3(6;c)
2
2
2 mN
2 m
c C3(6;c) +
b C3(6;b)
c C3(6;c) +
b C3(6;b)
1
2 C3(6;c) + 0:1C3(6;b)
10 3;
1
2 9 C3(6;c) + 0:9C3(6;b)
(5.17)
(5.18)
(5.19)
(5.20)
(5.21)
where in the last equality we set m
= 100 GeV. The e ective scattering amplitude is
parametrically given by
A
Ac1p + vT c7p +
q
mN 9
p
c ;
where vT
10 3, q=mN . 0:1, and A
100 for heavy nuclei. The loopinduced weak
contributions thus dominate the scattering rates of weak scale DM.
6
Connecting to the physics above the weak scale
We now describe how to apply and extend our results for the case in which there is a
separation between the mediator scale and the electroweak scale, i.e., if
mZ . In this
case, the e ective Lagrangian valid above the weak scale is
L
=
X
a;d
Ca(d)
d 4 Q(ad) ;
with the operators Q(ad) manifestly invariant under the full SM gauge group. We focus on the
dimensionsix e ective interactions involving DM and quarks currents, analogous to those
in eqs. (2.2){(2.3). For the case of Diracfermion DM in a generic SU(2)L representation
with generators ~a and hypercharge Y , the basis of dimensionsix operators is [38]
Q(16;i) = (
Q(26;i) = (
Q(36;i) = (
Q(46;i) = (
~a )(QiL
aQiL) ;
)(QiL
)(uiR
)(diR
QiL) ;
uiR) ;
diR) ;
Q(56;i) = (
Q(66;i) = (
Q(76;i) = (
Q(86;i) = (
5 ~a )(QiL
aQiL) ;
5 )(QiL
5 )(uiR
5 )(diR
QiL) ;
uiR) ;
diR) ;
where the index i = 1; 2; 3 labels the generation, and a =
a=2, with the Pauli matrices
a. If
is an electroweak singlet, the operators Q(16;i) and Q(56;i) do not exist. Below the
weak scale the above operators, Q(nd;i), match onto the operators Q(md;)f in eqs. (2.2){(2.3).
For certain patterns of Wilson coe cients, DM couples only to bottom and/or
charmquark axialvector currents. Such possibilities are the main focus of this work. For instance,
'
DM couples (at the mediator scale
mZ ) only to the axial bottomquark current
if the only nonzero Wilson coe cients are C5(6;3), C6(6;3), and C8(6;3), and such that they satisfy
the relation
Y C5(6;3) = 4C6(6;3) =
2C8(6;3) :
We rst derive the leading electroweak contribution to DMnucleon scattering rates for this
case and then discuss the case in which DM couples only to charm axial currents.
To this end, we st assume that the initial conditions at
satisfy eq. (6.6). At
scales ew <
<
, the operators Q(a6;i) mix at oneloop via the SM Yukawa interactions
'
into the Higgscurrent operators [10, 11, 38]
Q(166) = (
)(HyiD$ H) ;
Q(168) = (
5 )(HyiD$ H) ;
(5.22)
(6.1)
(6.2)
(6.3)
(6.4)
(6.5)
(6.6)
(6.7)
χ
Q(a6,i)
H †
Z
χ
χ
hH †i
Higgs operators, Q(166). Right: treelevel diagram that generates the dimensioneight operators from
the Higgs operators through matching at the weak scale.
and a similar set of operators with the ~a
a structure (above, D
D
D ). This mix
ing is generated by \electroweak sh" diagrams, see gure 6 (left), and induces at ew ' mZ
$
C1(66)(mZ )
y
2
16b 2 Ca(6;3)( ) log
mZ
:
Here, we only show the parametric dependence and suppress O(
1
) coe cients from the
actual value of the anomalous dimensions.
At energies close to the electroweak scale, at which the Higgs obtains its vacuum
expectation value, the two operators in eq. (6.7) result in couplings of DM currents to
the Z boson. Integrating out the Z boson at treelevel induces DM couplings to quarks,
see gure 6 (right). The Higgscurrent operators in eq. (6.7) therefore match, at ew, onto
fourfermion operators of the ve avor EFT that couple DM to quarks with an interaction
GF C1(66)(mZ ) v2= 2
. The factor v2 originates from the two
strength of parametric size
out the Z boson.
Higgs elds relaxing to their vacuum expectation values and the factor GF from integrating
Since the oneloop RG running from
to ew, eq. (6.8), followed by treelevel matching
at ew, induces interactions proportional to yb2v2, it is convenient to match such corrections
to initial conditions of the dimensioneight operators in eqs. (4.3).2 For the pattern of initial
conditions in eq. (6.6), we then
operator Q(48;b) at ew ' mZ is
nd that the Wilson coe cient of the dimensioneight
and the Wilson coe cient of the dimensionsix operator Q(46;b) is
C4(8;b)(mZ )
g
2
16s 2 Ca(6;3)( ) log
mZ
;
C4(6;b)(mZ )
Ca(6;3)( ) :
(6.8)
(6.9)
(6.10)
2Here, we have decided to ascribe the treelevel Z exchange contribution from the matching at ew to
dimensioneight fourfermion operators. Alternatively, we could have absorbed also this contribution into
the Wilson coe cients of dimensionsix operators. This choice would have the unattractive property of
having the parametric suppression of yb2v2GF = mb2GF hidden in the smallness of some of (the parts of) the
Wilson coe cients Ca(6;f) , thus making the ve avor EFT less transparent. With our choice, the parametric
suppression of mb2GF is factored out of the Wilson coe cients and is part of the massdimension counting
of the dimensioneight operator.
Again, we only show the parametric dependence, including loop factors, but omit O(
1
)
factors, e.g., from the actual values of anomalous dimensions (for details see ref. [38]). In
particular, Ca(6;3)( ) denotes a linear combination of the Wilson coe cients with a = 5; 6; 8.
The subsequent RG evolution from
ew to
had proceeds as described in section 5,
eqs. (5.1){(5.4).
nonzero. For instance, in the result for the nonrelativistic coe cient c4p in eq. (5.12),
there is an additional contribution from C4(8;b)(mZ ) / log(mZ = ). If one neglects the
QCD e ects, the two contributions amount to adding up two logarithmically enhanced
The only di erence is that the initial conditions C4(8;b)(mZ ) are now
terms with exactly the same prefactors. The net e ect is to replace log(mq0 =mZ ) with
log(mq0 = ) = log(mq0 =mZ ) + log(mZ = ) in eqs. (5.11){(5.16). This is equivalent to simply
calculating the electroweak sh diagram, with u; d; s fermions attached to the Z line, and
keeping only the log(mb= )enhanced part.
An analogous analysis applies if the only nonzero Wilson coe cients satisfy Y C5(6;2) =
4C6(6;2) = 2C7(6;2), so that just the (
5 )(c
5c) operator is generated (setting the CKM
matrix to unity for simplicity). Similarly, if the only nonzero Wilson coe cients satisfy
HJEP02(18)74
Y C1(6;3) = 4C2(6;3) =
the (
2C4(6;3) 6= 0 or Y C1(6;2) =
4C2(6;2) = 2C3(6;2) 6= 0 this means that just
essarily imply
netuning, as they can originate from the quantum number assignments
for the mediators, DM, and quark
elds in the UV theory. They do require the DM
hypercharge Y
to be nonzero.3
This conclusion changes, if at
'
we also include
dimensioneight operators of the form (
)(QLH
HQL) alongside the dimensionsix
bR) operators. In this case, it is possible to induce only the (
malizable interaction to the Z boson). This, however, requires netuning of dimensionsix
and dimensioneight contributions.
Note that the relation in eq. (6.6) also requires DM to be part of an electroweak
multiplet. For singlet DM there is no operator Q(56;i) and so C5(6;i) is trivially zero. Therefore,
for singlet DM a coupling to an axialvector bottomquark current is always
accompanied by couplings to top quarks. In this case our results get corrected by terms of order
yt2 log( ew= ) from the RG evolution above the electroweak scale due to topYukawa
interactions [38].
Another phenomenologically interesting case is the one of DM coupling only to leptons
at ew, i.e., through operators in eqs. (2.2){(2.3) with f = e; ; . We can readily adapt
our results to this case by replacing in eqs. (3.2){(3.6) the bottom and charmquarks
with leptons. The new operators are either colorsinglets or conserved QCD currents so
that their anomalous dimensions vanish. The hadronic functions cip(n), controlling DM
scattering on nuclei, are then given by eqs. (5.11){(5.16) after substituting q0 ! ` = ; ,
and dividing by the number of colors, Nc = 3, implicit in these equations.
3This may or may not lead to potentially dangerous renormalizable couplings of DM to the Zboson. An
example of the latter is a DM multiplet that is a pseudoDirac fermion (a Dirac fermion with an additional
small Majorana mass term), such as an almost pure Higgsino in the MSSM. In this case, the lightest mass
eigenstate, the Majoranafermion DM, does not couple diagonally to the Z boson at tree level.
cross section.
Strictly speaking, only the lepton can be integrated out when matching to the
threeavor theory, and the analogy to the heavyquark case breaks down for the muon, which
will appear in the lowenergy theory as an active degree of freedom. Since the muon mass
is larger than the typical momentum transfer in the direct detection experiments, one
might expect that keeping only the local contributions via eqs. (3.2){(3.6) is still a good
For couplings to electrons, ` = e, we expect nonlocal contributions as well as
dimensioneight operators with derivatives, which we have not considered here, to
contribute to the scattering on nuclei at approximately the same order. Hence, keeping only
the local contribution in eqs. (3.2){(3.6) is not expected to lead to a good estimate of the
Majorana and scalar dark matter
So far we focused on DM that is a Dirac fermion. However, the RG results discussed in this
work do not depend on the precise form of the DM current. We can, therefore, generalize
our results to the case of Majorana and scalar DM.
Majorana dark matter
Our results apply for Majorana DM with only small modi cations. It is su cient to drop
from the operator basis the operators Q(16;f) and Q(36;f) in eqs. (2.2){(2.3) and likewise the
operators Q(18;q) and Q(38;q) in eqs. (4.2){(4.3). The Lagrangian terms proportional to the
remaining operators, Q2;f , Q4;f , Q(28;f) , and Q(48;f) should be multiplied by a factor of 1=2 to
(6) (6)
account for the additional Wick contractions (see, for instance, ref. [18]). With these
modi cations, the coe cients of the nuclear e ective theory are still given by eqs. (5.5){(5.10).
7.2
Scalar dark matter
The relevant set of operators for scalar DM is
Lagrangian for scalar DM, cf., ref. [14],
$
Q(16;f) = ' i@ ' (f
f ) ;
$
Q(26;f) = ' i@ ' (f
(@ ' )'. These operators are part of the dimensionsix e ective
L' =
X Ca
(6)
2 Q(a6) ;
a
with Ca
(6) the dimensionless Wilson coe cients. Note that we adopt the same notation for
operators and Wilson coe cients in the case of scalar DM as we did for fermionic DM. No
confusion should arise as this abuse of notation is restricted to this subsection.
Apart from having a di erent DM current, nothing changes in our calculations.
Therefore, after de ning the dimensioneight e ective Lagrangian in the three avor theory as
L'(8) =
p
2GF
2
X
(7.1)
(7.2)
the additional contributions to the nuclear coe cients are given, for complex scalar DM,
by (cf. ref. [18])
c
1N =
c
7N =
1
s
p
2
1
s
p
2
2GF m'
2
2GF m'
X
For real scalar DM, the operators in eq. (7.1) vanish. For completeness, we display also
the dimensioneight contributions to the nuclear coe cients, expanded to leading order in
HJEP02(18)74
the strong coupling constant,
c
N
1 '
c
N
7 '
6m'
2
p
2GF
2
p
X
If DM couples only to bottom or charmquark axialvector currents, the dominant
contribution to DM scattering on nuclei is either due to oneloop electroweak corrections or
due to the intrinsic bottom and charm axial charges of the nucleons. Below the weak scale
the electroweak contributions are captured by double insertions of both the DM e ective
Lagrangian and the SM weak e ective Lagrangian. These convert the heavyquark currents
to the currents with u; d, and s quarks that have nonzero nuclear matrix elements. In this
paper we calculated the nonrelativistic couplings of DM to neutrons and protons that result
from such electroweak corrections, including the resummation of the leadinglogarithmic
QCD corrections. The latter are numerically important, as they lead to O(
1
) changes in
the scattering rates on nuclei. Our results can be readily included in the general framework
of EFT for DM direct detection, and will be implemented in a future version of the public
code DirectDM [18].
Acknowledgments
We thank Francesco D'Eramo for useful discussions, and especially Fady Bishara for
checking several equations. JZ acknowledges support in part by the DOE grant DESC0011784.
The research of BG was supported in part by the DOE Grant No. DESC0009919.
Unphysical operators A
4
We extract the anomalous dimensions by renormalizing o shell Green's functions in d =
2 dimensions. In some intermediate stages of the computation it is thus necessary to
introduce some unphysical operators.
The oneloop mixing among the \physical" operators is not a ected by the de nition of
evanescent operators, i.e., operators that are required to project oneloop Green's functions
in d = 4
2 dimensions but vanish in d = 4. Indeed, our oneloop results could also
have been obtained by performing the Dirac algebra in d = 4 instead o in noninteger
dimensions. Since i) this no longer possible at nexttoleading order computations and
ii) we use dimensional regularization to extract the poles of loop integrals, we
nd it
convenient to also perform the Dirac algebra in noninteger dimensions. To project the
d = 4
2 amplitudes we thus need to also include some evanescent operators in the
basis. For completeness and future reference, we list below the ones entering the oneloop
computations:
HJEP02(18)74
and
E1
qq0 = (q
E2
qq0 = (q
E3
E4
E5
E6
qq0 = (q
qq0 = (q
qq0 = (q
qq0 = (q
E1q = (q
E2q = (q
E3q = (q
E4q = (q
E5q = (q
E6q = (q
E7q = (q
E8q = (q
(A.1)
O1q + O2
q
4O4q ;
1
Nc
1
Nc
q
O3 ;
O1q + O2q + 4O4q ;
(A.2)
q) (q0
T aq)
operators, i.e., they vanish due to Fierz identities and not due to special d = 4 relations of
the Dirac algebra.
A.2
E.o.m.vanishing operators
In our conventions the equation of motion (e.o.m.) for the gluon eld reads
D G
a
gsf abcG ;c)Gb
X qT a
q ;
q
up to gauge xing and ghost terms. The sum is over all active quark elds. Hence the
following operators vanish via the e.o.m.
HJEP02(18)74
q
N1;e.o.m. =
N2;e.o.m. =
1
gs
1
gs
T aq)D G
a + O4(6;q) +
O4(6;q)q0 ;
5 T aq)D G
a +
1
1
2
X
q06=q
1
Nc
O3q +
X
q06=q
O6(6;q)q0 :
(A.3)
(A.4)
The fourfermion pieces of these e.o.m.vanishing operators contribute to the same
amplitudes as the physical fourfermion operators. Therefore, the mixing of physical operators
into the e.o.m.vanishing operators (computed from QCD penguin diagrams, gure 3)
affects the anomalous dimensions of fourfermion operators.
Open Access.
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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