# General Heavy WIMP Nucleon Elastic Scattering

Qing Chen<EMAIL_ADDRESS>Interdisciplinary Center for Theoretical Study,
University of Science and Technology of China, Hefei, Anhui 230026, China
Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, China  Gui-Jun
Ding<EMAIL_ADDRESS>Interdisciplinary Center for Theoretical Study,
University of Science and Technology of China, Hefei, Anhui 230026, China
Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, China  Richard
J. Hill<EMAIL_ADDRESS>Department of Physics and Astronomy, University
of Kentucky, Lexington, KY 40506, USA  Theoretical Physics Department,
Fermilab, Batavia, IL 60510, USA

###### Abstract

Heavy WIMP (weakly-interacting-massive-particle) effective field theory is
used to compute the WIMP-nucleon scattering rate for general heavy electroweak
multiplets through order $m_{W}/M$, where $m_{W}$ and $M$ denote the
electroweak and WIMP mass scales. The lightest neutral component of such an
electroweak multiplet is a candidate dark matter particle, either elementary
or composite. Existing computations for certain representations of electroweak
$\mathrm{SU(2)}_{W}\times\mathrm{U(1)}_{Y}$ reveal a cancellation of
amplitudes from different effective operators at leading and subleading orders
in $1/M$, yielding small cross sections that are below current dark matter
direct detection experimental sensitivities. We extend those computations and
consider all low-spin (spin-0, spin-1/2, spin-1, spin-3/2) heavy electroweak
multiplets with arbitrary $\mathrm{SU(2)}_{W}\times\mathrm{U(1)}_{Y}$
representations and provide benchmark cross section results for dark matter
direct detection experiments. For most self-conjugate TeV WIMPs with isospin
$\leq 3$, the cross sections are below current experimental limits but within
reach of next-generation experiments. An exception is the case of pure
electroweak doublet, where WIMPs are hidden below the neutrino floor.

##  1 Introduction

The field of dark matter direct detection [1, 2, 3, 4] comprises a large class
of experiments mainly designed to detect WIMPs (Weakly-Interacting-Massive-
Particles) [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. WIMPs can naturally
explain the astronomically observed relic abundance of dark matter mass
density, created in thermal equilibrium with other particles in the early
Universe [16]. The primary signal process for these experiments is elastic
scattering of WIMPs from atomic nuclei, detected by observing the recoiling
nucleus. Since the precise nature of the dark matter particle is unknown, the
WIMP-nucleus cross section is a priori unknown. In order to make predictions,
the problem can be approached from the “top-down” or the “bottom-up”
perspective. In the “top-down” approach, a specific UV complete theory
determines all possible couplings between the new particle and Standard Model
(SM) particles [17]; however, the parameter space of all new physics models is
large and predictions rely on model assumptions. In the “bottom-up” approach,
a non-relativistic expansion enforcing only the constraints of spacetime
symmetries can be employed [18, 19]. While this approach does not depend on
the underlying ultra-violet (UV) theory of the dark matter, the coefficients
of the associated effective operators, and hence the dark matter-nucleus
scattering rate, are undetermined.

The null results of collider searches up to a few hundred GeV [20] and thermal
relic abundance estimates [21, 22] suggest WIMP masses greater than the
electroweak scale, $M\gtrsim{\rm few}\times 100\,{\rm GeV}\gg m_{W}$. Heavy
WIMP effective theory (HWET) is operative in this regime and has advantages of
both the “top-down” and “bottom-up” approaches. By using the scale separation
between $m_{W}$ and $M$, HWET describes large classes of UV theories, and
predicts absolute WIMP-nucleus cross sections.

The interactions of electroweak-charged WIMPS with quarks and gluons involve
two classes of quark and gluon operators, transforming as spin-0 and spin-2,
that largely cancel at the amplitude level, resulting in an “accidentally”
suppressed cross section. Such cancellations have been found using
relativistic WIMP effective theory in specific UV completions [23], and in
HWET at both leading and subleading power [24, 25, 26]. For example, for
benchmark Wino-like or Higgsino-like particles, the leading order of HWET
predicts a cross section one or a few orders of magnitude smaller [27] than
the current experimental limits. In fact, the cancellation essentially remains
after including $1/M$ power corrections, and accounting for potential
differences in nuclear responses for spin-0 and spin-2 channels [28, 29]:
subleading contributions do not lift the cross sections up to the discovery
limits of current direct detection experiments.

The next generation dark matter direct detection experimental sensitivities
will be improved by orders of magnitude [30, 31, 32], and more stringent
constraints on the supersymmetric electroweak multiplets will be placed by
collider experiments [33, 34]. To shed further light on the above-mentioned
amplitude cancellation and to determine sensitivity targets for next
generation experiments, we consider more general WIMPs with arbitrary
electroweak representation and spin. We aim to carry out a thorough survey of
general electroweak-charged heavy WIMPs and compute their cross sections for
scattering on a nucleon utilizing heavy WIMP effective theory through first
subleading power, providing benchmark theoretical results for future direct
detection phenomenology. Our computations will show that, for most of the
self-conjugate WIMPs with low isospins, the WIMP and nucleon elastic
scattering spin-independent cross sections naturally lie close to the neutrino
floor of direct detection experiments, and are within striking range of next
generation experiments at or below the neutrino floor.

The remainder of the paper is organized as follows. Section.2 constructs the
heavy WIMP effective theory at the electroweak scale including order $1/M$
power corrections. Section.3 constructs the low energy effective theory
containing the WIMP and low-energy QCD. Section.4 matches the electroweak
scale heavy WIMP effective theory onto this low energy effective theory.
Section.5 illustrates minimal UV completions of the electroweak scale heavy
WIMP effective theory. Section.6 computes the cross sections for WIMP-proton
elastic scattering and provides comparisons with experimental sensitivities.
Section.7 is a summary.

##  2 Subleading Power Heavy WIMP Effective Theory

Let us consider an electroweak multiplet with mass $M$ large compared to the
weak scale, and construct the effective theory for this heavy particle in
powers of $1/M$. We restrict attention to the case of a self-conjugate heavy
particle; this forbids tree level $Z^{0}$ boson interactions, enabling the
particle to survive current experimental exclusion limits. Universal behavior
is shared by heavy WIMPs of different spin at the leading order of the heavy
WIMP effective theory [24]. We here investigate effects through subleading
order $1/M$ and will consider Lorentz spin-0, spin-1/2, spin-1 and spin-3/2
WIMPs.

###  2.1 Effective lagrangians

The effective Lagrangian in the one-heavy-particle sector takes the following
form for spin-0, spin-1/2, spin-1 and spin-3/2:

$\displaystyle\mathcal{L}_{\rm HWET}^{\rm spin-0}$ $\displaystyle=$
$\displaystyle{\phi}^{\dagger}_{v}\bigg{[}iv\cdot D-\delta
m-\frac{{D}_{\perp}^{2}}{2M}-\frac{f(H)}{M}-\frac{g(W,B)}{M}+\dots\bigg{]}\phi_{v}\,,$
(1) $\displaystyle\mathcal{L}_{\rm HWET}^{\rm spin-1/2}$ $\displaystyle=$
$\displaystyle\bar{\chi}_{v}\bigg{[}iv\cdot D-\delta
m-\frac{{D}_{\perp}^{2}}{2M}-\frac{f(H)}{M}-\frac{g(W,B)}{M}+\dots\bigg{]}\chi_{v}\,,$
(2) $\displaystyle\mathcal{L}_{\rm HWET}^{\rm spin-1}$ $\displaystyle=$
$\displaystyle\mathcal{V}^{\mu\dagger}_{v}\bigg{[}\left(iv\cdot D-\delta
m-\frac{{D}_{\perp}^{2}}{2M}-\frac{f(H)}{M}\right)\left(-g_{\mu\nu}\right)+\frac{g(W,B)_{\mu\nu}}{M}+\dots\bigg{]}\mathcal{V}^{\nu}_{v}\,,$
(3) $\displaystyle\mathcal{L}_{\rm HWET}^{\rm spin-3/2}$ $\displaystyle=$
$\displaystyle\bar{\xi}^{\mu}_{v}\bigg{[}\left(iv\cdot D-\delta
m-\frac{{D}_{\perp}^{2}}{2M}-\frac{f(H)}{M}\right)\left(-g_{\mu\nu}\right)+\frac{g(W,B)_{\mu\nu}}{M}+\dots\bigg{]}\xi^{\nu}_{v}\,.$
(4)

Here the ellipses denote terms of order $1/M^{2}$, $v^{\mu}$ is the heavy WIMP
velocity with $v^{2}=1$, and $\delta m$ is a residual mass matrix after
integrating out the heavy particle. The covariant derivative is
$D_{\mu}=\partial_{\mu}-ig_{1}YB_{\mu}-ig_{2}W_{\mu}^{a}T^{a}$ where $Y$ is
the $\mathrm{U(1)}_{Y}$ hypercharge and $T^{a}$ are $\mathrm{SU(2)}_{W}$
generators, with $a=1,2,3$. Perpendicular components are projected using
$g_{\perp}^{\mu\nu}\equiv g^{\mu\nu}-v^{\mu}v^{\nu}$ as usual (thus e.g.
$\sigma^{\mu\nu}_{\perp}=g_{\perp}^{\alpha\mu}g_{\perp}^{\beta\nu}\sigma_{\alpha\beta}$,
$D_{\perp}^{\mu}=D^{\mu}-v^{\mu}v\cdot D$). We have applied field
redefinitions to remove redundant operators, and enforced the constraints for
expressing heavy particles in terms of four-component Dirac spinors with
vector indices (e.g. $v_{\mu}\mathcal{V}^{\mu}_{v}=0$,
$\not{v}\xi^{\mu}_{v}=\xi^{\mu}_{v}$ and $\gamma_{\mu}\xi^{\mu}_{v}=0$) [35].
The Higgs-WIMP interaction $f(H)$ will be discussed below. Terms contained in
$g(W,B)$ give rise to suppressed spin-dependent scattering rates and will not
be considered further in this work [29].

###  2.2 Higgs interactions

The Higgs-WIMP interaction, $f(H)$, depends on the specific spin and
electroweak representation of the WIMP. Let us construct gauge- and Lorentz-
invariant operators containing WIMP and Higgs fields, for a general $(J,\,Y)$
representation of ${\rm SU}(2)_{W}\times{\rm U}(1)_{Y}$, with $J$ being ${\rm
SU}(2)$ isospin and $Y$ being ${\rm U}(1)$ hypercharge. Since we are concerned
with the one-WIMP sector, there must be two WIMP fields in the Lagrangian
interaction. Since the Standard Model Higgs $H$ is a $(1/2,1/2)$
representation under ${\mathrm{SU}}(2)_{W}\times{\mathrm{U}}(1)_{\mathrm{Y}}$,
there is no three-point gauge-invariant interaction. The leading WIMP-Higgs
interaction arises from four-point interactions with two WIMP fields and two
Higgs fields. The Higgs bilinear $H^{\dagger}H$ transforms as a singlet, and
$H^{\dagger}\tau^{a}H$ transforms as a triplet with $\tau^{a}$ the isospin
Pauli matrices. It is convenient to introduce $\tilde{H}\equiv
i\tau^{2}H^{*}$, which transforms identically to $H$ under ${\mathrm{SU}}(2)$
but has opposite hypercharge.

The possible forms for $f(H)$ can be tabulated by first considering manifestly
Lorentz-invariant Lagrangians and then making the identifications

$\displaystyle\mathrm{spin-0:}\hskip
10.0pt\Phi(x)=\sqrt{\frac{1}{M}}e^{-iMv\cdot x}\phi_{v}(x)\,,$ (5)
$\displaystyle\mathrm{spin-1/2:}\hskip 10.0pt\chi=\sqrt{2}e^{-iMv\cdot
x}\left(\chi_{v}+X_{v}\right)\,,$ (6) $\displaystyle\mathrm{spin-1:}\hskip
10.0pt\mathcal{V}^{\mu}(x)=\sqrt{\frac{1}{M}}e^{-iMv\cdot
x}\mathcal{V}^{\mu}_{v}(x)\,,$ (7) $\displaystyle\mathrm{spin-3/2:}\hskip
10.0pt\xi^{\mu}=\sqrt{2}e^{-iMv\cdot
x}\left(\xi_{v}^{\mu}+\Xi_{v}^{\mu}\right)\,,$ (8)

where for $\chi$ and $\xi^{\mu}$, the second terms in parentheses denote anti-
particle degrees of freedom that are integrated out (the components satisfy
$\not{v}\chi_{v}=\chi_{v}$, $\not{v}\xi^{\mu}=\xi^{\mu}$, and
$\not{v}X_{v}=-X_{v}$, $\not{v}\Xi^{\mu}=-\Xi^{\mu}$). Let us consider
separately the cases of spin 0, $1/2$, $1$ and $3/2$.

####  2.2.1 Spin 0

The relativistic spin-0 electroweak multiplet and Higgs interaction takes the
form

$\displaystyle\mathcal{L}_{H}^{\rm spin-0}$
$\displaystyle=c_{1,0}\phi^{\dagger}\phi
H^{\dagger}H+c_{2,0}\phi^{\dagger}t^{a}\phi H^{\dagger}\tau^{a}H$
$\displaystyle+\left(c_{3,0}\phi^{\dagger}t^{a}\tilde{\phi}\tilde{H}^{\dagger}\tau^{a}H+\rm{h.c.}\right)\delta_{Y,\,1/2}$
$\displaystyle+\left(c_{4,0}\phi^{\dagger}t^{a}\tilde{\phi}H^{\dagger}\tau^{a}\tilde{H}+\rm{h.c.}\right)\delta_{Y,\,-1/2}\,,$
(9)

where $t^{a}$ is an ${\mathrm{SU}}(2)$ generator, $\tilde{\phi}\equiv
U\phi^{*}$ and $U$ is a $(2J+1)\times(2J+1)$ matrix acting as a similarity
transformation for isospin-$J$ representation [36],

$U\left(e^{i\boldsymbol{\alpha\cdot
t}}\right)^{*}U^{-1}=e^{i\boldsymbol{\alpha\cdot t}}\,,$ (10)

where $\boldsymbol{t}=(t^{1},t^{2},t^{3})$ are ${\rm SU}(2)$ generators and
$\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2},\alpha_{3})$ are real parameters.
The explicit matrix elements are

$\displaystyle
t^{1}_{mn}=\left(\sqrt{m(2J+1-m)}\delta_{n,m+1}+\sqrt{n(2J+1-n)}\delta_{n,m-1}\right)/2\,,$
$\displaystyle
t^{2}_{mn}=-i\left(\sqrt{m(2J+1-m)}\delta_{n,m+1}-\sqrt{n(2J+1-n)}\delta_{n,m-1}\right)/2\,,$
$\displaystyle t^{3}_{mn}=(J+1-m)\delta_{mn}\,,$ $\displaystyle
U_{mn}=(-1)^{m+1}\delta_{m+n,2J+2}\,,$ (11)

where $m,\,n=1,\,2,\,...,\,2J+1$. In particular, when $J=1/2$, $U$ is the
matrix $i\tau^{2}$ that we have introduced above in the construction of
$\tilde{H}$. The coefficient $c_{i,0}$ has a subscript $0$ standing for
spin-0. The notation h.c. denotes hermitian conjugate. Gauge-invariant
interactions among electroweak multiplets including Higgs field can also be
obtained by brute-force construction of gauge-singlets using Clebsch-Gordan
coefficients [36], and we have checked the equivalency of the two methods.

When the hypercharge $Y=0$, we restrict attention to integer isospin, for
which there is an electrically neutral dark matter candidate. For integer
isospin, the ${\mathrm{SU}}(2)$ representation is real, and the scalar field
$\phi$ may be chosen real, and identified with $\Phi$ in Eq. (5). Interaction
$c_{2,0}$ vanishes in this case, and only $c_{1,0}$ appears; according to Eq.
(5), $f(H)=c_{1,0}H^{\dagger}H\equiv-c_{H}H^{\dagger}H$ [24, 28]. When $Y\neq
0$, the field $\Phi$ in Eq. (5) is identified with the column vector of two
real scalar fields,

$\displaystyle\Phi$
$\displaystyle=\left(\begin{array}[]{c}\frac{1}{\sqrt{2}}(\phi+\phi^{*})\\\
\frac{i}{\sqrt{2}}(\phi-\phi^{*})\end{array}\right)\,,$ (14)

and $f(H)$ may be read off according to Eq. (5).

####  2.2.2 Spin 1/2

For a spin-1/2 electroweak multiplet, let us construct the self-conjugate
fields from Weyl spinors, $\psi_{L}$ and $\psi_{L}^{\prime}$, transforming
under ${\mathrm{SU}}(2)_{W}\times{\mathrm{U}}(1)_{Y}$ as

$\displaystyle\psi_{L}\rightarrow e^{i\boldsymbol{\alpha\cdot t}}e^{i\beta
Y}\psi_{L}\,,$ $\displaystyle\psi_{L}^{\prime}\rightarrow
e^{-i\boldsymbol{\alpha\cdot t^{*}}}e^{-i\beta Y}\psi_{L}^{\prime}\,.$ (15)

The general spin-1/2 Higgs interaction is then

$\displaystyle\mathcal{L}_{H}^{\rm spin-1/2}$
$\displaystyle=-\frac{1}{M}\Bigg{[}c_{1,\frac{1}{2}}H^{\dagger}H\left({\psi}^{\prime
T}_{L}i\sigma^{2}\psi_{L}\right)+c_{2,\frac{1}{2}}H^{\dagger}\tau^{a}H\left({\psi}^{\prime
T}_{L}i\sigma^{2}t^{a}\psi_{L}\right)$
$\displaystyle+c_{3,\frac{1}{2}}{H}^{\dagger}\tau^{a}\tilde{H}\left({\psi}^{T}_{L}i\sigma^{2}U^{\dagger}t^{a}\psi_{L}\right)\delta_{Y,\,1/2}+c_{4,\frac{1}{2}}\tilde{H}^{\dagger}\tau^{a}{H}\left({\psi}^{\prime
T}_{L}i\sigma^{2}t^{a}U\psi_{L}^{\prime}\right)\delta_{Y,\,1/2}$
$\displaystyle+c_{5,\frac{1}{2}}\tilde{H}^{\dagger}\tau^{a}{H}\left({\psi}^{T}_{L}i\sigma^{2}U^{\dagger}t^{a}\psi_{L}\right)\delta_{Y,\,-1/2}+c_{6,\frac{1}{2}}H^{\dagger}\tau^{a}\tilde{H}\left({\psi}^{\prime
T}_{L}i\sigma^{2}t^{a}U\psi_{L}^{\prime}\right)\delta_{Y,\,-1/2}+\rm{h.c.}\Bigg{]}\,.$
(16)

To connect with Eq. (6), let us embed $\psi_{L}$ and $\psi_{L}^{\prime}$ into
the Dirac field $\psi$, with its conjugate $\psi^{c}$:

$\displaystyle\psi=\begin{pmatrix}\psi_{L}\\\ i\sigma^{2}\psi_{L}^{\prime*}\\\
\end{pmatrix},\hskip 28.45274pt\psi^{c}=\begin{pmatrix}\psi_{L}^{\prime}\\\
i\sigma^{2}\psi_{L}^{*}\\\ \end{pmatrix}\,,$ (17)

where the Pauli matrix $\sigma^{2}$ acts as a generator in the Lorentz group.
The self-conjugate (Majorana) fermion field $\chi$ in Eq. (6) is then
identified with

$\displaystyle\chi$ $\displaystyle=\left(\begin{array}[]{c}\chi_{1}\\\
\chi_{2}\end{array}\right)=\left(\begin{array}[]{c}\frac{1}{\sqrt{2}}(\psi+\psi^{c})\\\
\frac{i}{\sqrt{2}}(\psi-\psi^{c})\end{array}\right)\,.$ (22)

For hypercharge $Y=0$ and integer isospin, we may choose irreducible
representations involving a single Weyl fermion, i.e.
$\psi_{L}^{\prime}=\psi_{L}$. For this case, all interactions except
$c_{1,\frac{1}{2}}$ vanish. Coefficient $c_{1,\frac{1}{2}}$ may be chosen
real111 This may be obtained by field redefinition
$\delta\psi_{L}\sim(H^{\dagger}H/M^{2})\psi_{L}$.  and according to Eq. (6),
$f(H)=c_{1,\frac{1}{2}}H^{\dagger}H\equiv-2c_{H}H^{\dagger}H$ [29]. For the
Dirac fermion (“Higgsino”) case, $J=Y=1/2$, the gauge-invariant interaction
(16) may be simplified. After expressing $\psi_{L}$ and $\psi_{L}^{\prime}$ in
terms of $\chi_{v}$ via Eqs. (17), (22) and (6), $f(H)$ in Eq. (2) is
expressed as a matrix with four real parameters [29]. 222The correspondence
with Eq. (2) of Ref. [29] is $a=-{\rm Re}(c_{2})/2$, $b=-(c_{3}^{*}+c_{4})/2$
and $c=-{\rm Re}(c_{1}-c_{2}/2)/2$.

####  2.2.3 Spin 1

Similar to the spin-0 case, the spin-1 electroweak multiplet and Higgs
bilinear interaction takes the form

$\displaystyle\mathcal{L}_{H}^{\rm spin-1}$
$\displaystyle=c_{1,1}V^{\mu\dagger}V_{\mu}H^{\dagger}H+c_{2,1}V^{\mu\dagger}t^{a}V_{\mu}H^{\dagger}\tau^{a}H+\left(c_{3,1}V^{\mu\dagger}t^{a}\tilde{V}_{\mu}\tilde{H}^{\dagger}\tau^{a}H+\rm{h.c.}\right)\delta_{Y,1/2}$
$\displaystyle+\left(c_{4,1}V^{\mu\dagger}t^{a}\tilde{V}_{\mu}H^{\dagger}\tau^{a}\tilde{H}+\rm{h.c.}\right)\delta_{Y,-1/2}\,,$
(23)

where $\tilde{V}_{\mu}=UV_{\mu}^{*}$, and the self-conjugate basis consists of
two real vectors,

$\displaystyle V^{\mu}_{1}=\frac{1}{\sqrt{2}}(V^{\mu}+V^{\mu\dagger}),\hskip
28.45274ptV^{\mu}_{2}=\frac{i}{\sqrt{2}}(V^{\mu}-V^{\mu\dagger})\,,$ (24)

where $\mathcal{V}^{\mu}=\left(V^{\mu}_{1},V^{\mu}_{2}\right)^{T}$ is the
relativistic field mapping onto the heavy vector $\mathcal{V}_{v}^{\mu}$ in
Eq. (7).

####  2.2.4 Spin 3/2

Similar to the spin-1/2 case, the spin-3/2 electroweak multiplet and Higgs
bilinear interaction takes the form

$\displaystyle\mathcal{L}_{H}^{\rm spin-3/2}$
$\displaystyle=\frac{1}{M}\Bigg{[}c_{1,\frac{3}{2}}H^{\dagger}H\left({\varPsi}^{\prime\mu
T}_{L}i\sigma^{2}\varPsi_{\mu
L}\right)+c_{2,\frac{3}{2}}H^{\dagger}\tau^{a}H\left({\varPsi}^{\prime\mu
T}_{L}i\sigma^{2}t^{a}\varPsi_{\mu L}\right)$
$\displaystyle+c_{3,\frac{3}{2}}{H}^{\dagger}\tau^{a}\tilde{H}\left({\varPsi}^{\mu
T}_{L}i\sigma^{2}U^{\dagger}t^{a}\varPsi_{\mu
L}\right)\delta_{Y,\,1/2}+c_{4,\frac{3}{2}}\tilde{H}^{\dagger}\tau^{a}{H}\left({\varPsi}^{\prime\mu
T}_{L}i\sigma^{2}t^{a}U\varPsi_{\mu L}^{\prime}\right)\delta_{Y,\,1/2}$
$\displaystyle+c_{5,\frac{3}{2}}\tilde{H}^{\dagger}\tau^{a}{H}\left({\varPsi}^{\mu
T}_{L}i\sigma^{2}U^{\dagger}t^{a}\varPsi_{\mu
L}\right)\delta_{Y,\,-1/2}+c_{6,\frac{3}{2}}H^{\dagger}\tau^{a}\tilde{H}\left({\varPsi}^{\prime\mu
T}_{L}i\sigma^{2}t^{a}U\varPsi_{\mu
L}^{\prime}\right)\delta_{Y,\,-1/2}+\rm{h.c.}\Bigg{]}\,,$ (25)

where we have the Rarita-Schwinger field $\varPsi^{\mu}$ and its conjugate
$\varPsi^{c\,\mu}$,

$\displaystyle\varPsi^{\mu}=\begin{pmatrix}\varPsi_{L}^{\mu}\\\
i\sigma^{2}\varPsi_{L}^{\prime\mu*}\\\ \end{pmatrix},\hskip
28.45274pt\varPsi^{\mu\,c}=\begin{pmatrix}\varPsi_{L}^{\prime\mu}\\\
i\sigma^{2}\varPsi_{L}^{\mu*}\\\ \end{pmatrix}\,.$ (26)

Constructing self-conjugate fermions from $\varPsi^{\mu}$ and
$\varPsi^{c\,\mu}$,

$\displaystyle\xi^{\mu}_{1}=\frac{1}{\sqrt{2}}(\varPsi^{\mu}+\varPsi^{\mu\,c}),\hskip
28.45274pt\xi^{\mu}_{2}=\frac{i}{\sqrt{2}}(\varPsi^{\mu}-\varPsi^{\mu\,c})\,,$
(27)

we identify
$\xi^{\mu}\equiv\begin{pmatrix}\xi^{\mu}_{1},&\xi^{\mu}_{2}\end{pmatrix}^{T}$
as the relativistic field in Eq. (8).

####  2.2.5 EWSB and Feynman rules

After electroweak symmetry breaking, the Higgs field acquires its vacuum
expectation value

$\displaystyle\langle H\rangle=\frac{v}{\sqrt{2}}\begin{pmatrix}0\\\
1\end{pmatrix}\,,$ (28)

and in the self-conjugate basis, the mass matrix becomes

$\displaystyle\delta M(v)=\delta
m+\frac{v^{2}}{2M}\begin{pmatrix}M_{1}+\mathbf{Re}\left(M_{2}\right)&-\mathbf{Im}\left(M_{2}\right)\\\
-\mathbf{Im}\left(M_{2}\right)&M_{1}-\mathbf{Re}\left(M_{2}\right)\end{pmatrix}\,,$
(29)

where the matrices $M_{1}$ and $M_{2}$ are

$\displaystyle\left(M_{1}\right)_{kl}$ $\displaystyle=(A+Bk)\delta_{kl}\,,$
$\displaystyle\left(M_{2}\right)_{kl}$
$\displaystyle=\delta_{Y,\,1/2}C(-1)^{k}\sqrt{(k-1)(2J+2-k)}\delta_{k+l,\,2J+3}+\delta_{Y,\,-1/2}C^{\prime}(-1)^{k}\sqrt{k(2J+1-k)}\delta_{k+l,\,2J+1}\,,$
(30)

with $k,\,l=1,\,2,\,...,\,2J+1$ and

$\displaystyle A$
$\displaystyle=\frac{1}{2}\left[\mathbf{Re}(c_{1,s})-\mathbf{Re}(c_{2,s})(J+1)\right]\,,$
$\displaystyle B$ $\displaystyle=\frac{1}{2}\mathbf{Re}(c_{2,s})\,,$
$\displaystyle C$
$\displaystyle=\frac{c^{*}_{3,s}+c_{4,s}}{2}\left(\delta_{s,\frac{1}{2}}+\delta_{s,\frac{3}{2}}\right)+c_{3,s}\left(\delta_{s,0}+\delta_{s,1}\right)\,,$
$\displaystyle C^{\prime}$
$\displaystyle=\frac{c^{*}_{5,s}+c_{6,s}}{2}\left(\delta_{s,\frac{1}{2}}+\delta_{s,\frac{3}{2}}\right)+c_{4,s}\left(\delta_{s,0}+\delta_{s,1}\right)\,,$
(31)

where $s=0,\,\frac{1}{2},\,1,\,\frac{3}{2}$ is the spin.

Diagonalizing the mass matrix Eq. (29), when isospin $J$ is half-integer
(even-dimension representation) and hypercharge $Y=1/2$, the eigenvalues are

$\displaystyle\frac{v^{2}}{2M}\left(A+B\right),\,\frac{v^{2}}{2M}\left(A+B\right)\,,$
$\displaystyle\frac{v^{2}}{2M}\left[A+\left(J+\frac{3}{2}\right)B\pm\left(J+\frac{1}{2}\right)|C|\right]\,,$
$\displaystyle\frac{v^{2}}{2M}\left[A+\left(J+\frac{3}{2}\right)B\pm\sqrt{n^{2}B^{2}+\left[\left(J+\frac{1}{2}\right)^{2}-n^{2}\right]|C|^{2}}\right]\,,$
$\displaystyle\frac{v^{2}}{2M}\left[A+\left(J+\frac{3}{2}\right)B\pm\sqrt{n^{2}B^{2}+\left[\left(J+\frac{1}{2}\right)^{2}-n^{2}\right]|C|^{2}}\right]\,,$
(32)

for integer $n=1,...,J-1/2$. The two mass eigenvalues in the first row
correspond to the components with the largest magnitudes of charge,
$Q=\pm(J+1/2)$. The non-vanishing $C$ term acts to split the degenerate mass
of two neutral components and
$\frac{v^{2}}{2M}\left[A+\left(J+\frac{3}{2}\right)B-\left(J+\frac{1}{2}\right)|C|\right]$
in the second row is the mass of the lightest neutral component, the WIMP. The
four mass eigenvalues in the last two rows are for the two pairs of components
with opposite charges $Q=\pm n$.

When isospin $J$ is a half-integer and hypercharge $Y=-1/2$, the mass
eigenvalues are

$\displaystyle\frac{v^{2}}{2M}\left(A+(2J+1)B\right),\,\frac{v^{2}}{2M}\left(A+(2J+1)B\right)\,,$
$\displaystyle\frac{v^{2}}{2M}\left[A+\left(J+\frac{1}{2}\right)B\pm\left(J+\frac{1}{2}\right)|C^{\prime}|\right]\,,$
$\displaystyle\frac{v^{2}}{2M}\left[A+\left(J+\frac{1}{2}\right)B\pm\sqrt{n^{2}B^{2}+\left[\left(J+\frac{1}{2}\right)^{2}-n^{2}\right]|C^{\prime}|^{2}}\right]\,,$
$\displaystyle\frac{v^{2}}{2M}\left[A+\left(J+\frac{1}{2}\right)B\pm\sqrt{n^{2}B^{2}+\left[\left(J+\frac{1}{2}\right)^{2}-n^{2}\right]|C^{\prime}|^{2}}\right]\,,$
(33)

with similar notation as for the $Y=1/2$ case. Here
$\frac{v^{2}}{2M}\left[A+\left(J+\frac{1}{2}\right)B-\left(J+\frac{1}{2}\right)|C^{\prime}|\right]$
is the mass of the lightest neutral component, the WIMP.

When $J$ is an integer (odd-dimension representation), the hypercharge must be
an integer to provide a neutral component for the WIMP. Further, if the
electroweak multiplet is self-conjugate, hypercharge must be zero. After
electroweak symmetry breaking, the mass matrix becomes

$\displaystyle\delta M(v)=\delta m+\frac{v^{2}}{2M}M_{1}^{\prime}\,,$ (34)

where

$\displaystyle\left(M_{1}^{\prime}\right)_{kl}=(A^{\prime}+B^{\prime}k)\delta_{kl},$
(35)

for $k,l=1,\,2,...,\,2J+1$, with

$\displaystyle A^{\prime}$
$\displaystyle=\mathbf{Re}(c_{1,s})-\mathbf{Re}(c_{2,s})(J+1)\,,$
$\displaystyle B^{\prime}$ $\displaystyle=\mathbf{Re}(c_{2,s}).$ (36)

The eigenvalue for the neutral state is
$v^{2}[A^{\prime}+B^{\prime}(J+1)]/(2M)$.

Figure 1:  WIMP-WIMP-Higgs boson effective interaction at order $1/M$. The
double line denotes the heavy WIMP field and the dashed line denotes the Higgs
boson.

After electroweak symmetry breaking, the WIMP interacts with the dynamical
components of the Higgs field, $\phi_{1}$, $\phi_{2}$, $\phi_{3}$ and $h$ as
follows,

$\displaystyle H=\langle
H\rangle+\frac{1}{\sqrt{2}}\begin{pmatrix}\phi_{1}+i\phi_{2}\\\
h+i\phi_{3}\end{pmatrix}\,,$ (37)

and the Lagrangian of the heavy field after electroweak symmetry breaking in
the charge and mass eigenstate basis $\chi_{0}^{(\mu)}$ is

$\displaystyle\mathcal{L}$
$\displaystyle=\bar{\chi}_{0}^{(\mu)}\Big{[}iv\cdot\partial+eQv\cdot
A+\frac{g_{2}}{\cos{\theta_{W}}}v\cdot
Z(\tilde{T}^{3}-\sin^{2}{\theta_{W}}Q)+\frac{g_{2}}{\sqrt{2}}v\cdot(W^{+}\tilde{T}^{+}+W^{-}\tilde{T}^{-})$
$\displaystyle\quad-\delta
M-\frac{{D}_{\perp}^{2}}{2M}+\frac{f(h)}{M}+\frac{g(W,\,Z)}{M}+...\Big{]}\chi_{0{(\mu)}}\,,$
(38)

where the WIMP is the lightest neutral state of the electroweak multiplet
which we denote as $\chi_{0}^{(\mu)}$ (with or without the vector index $\mu$
depending on its spin), and the tilded gauge generator matrices refer to this
basis. Here $f(h)$ describes the WIMP interaction with the Higgs boson, cf.
Fig. 1. The Feynman rule for this vertex, $ivc_{H}(g_{\mu\nu})/M$, is related
to the mass eigenvalue of the lightest neutral state. When hypercharge
$Y=1/2$, it is

$\displaystyle-i\frac{v}{M}\left[A+\left(J+\frac{3}{2}\right)B-\left(J+\frac{1}{2}\right)|C|\right]\,.$
(39)

When hypercharge $Y=-1/2$, it is

$\displaystyle-i\frac{v}{M}\left[A+\left(J+\frac{1}{2}\right)B-\left(J+\frac{1}{2}\right)|C^{\prime}|\right]\,.$
(40)

When hypercharge $Y=0$, it is

$\displaystyle-i\frac{v}{M}\left[A^{\prime}+\left(J+1\right)B^{\prime}\right]\,.$
(41)

If inelastic scattering is considered, the relevant operators will involve
components in the electroweak multiplet other than the WIMP, as well as
$\phi_{1}$, $\phi_{2}$, $\phi_{3}$ components in the Higgs field, which can be
found by inserting Eq. (37) into the $f(H)$ term in the Lagrangians, Eq. (1),
Eq. (2), Eq. (3) and Eq. (4), then diagonalizing to the charge and mass
eigenstate basis as in the above procedure. We focus on elastic scattering in
this paper and leave the inelastic case to future work.

##  3 Low Energy Effective Theory

After electroweak symmetry breaking, the dark matter particle
$\chi_{0}^{(\mu)}$ is a singlet under
$\mathrm{SU}(3)_{c}\times\mathrm{U}(1)_{\rm e.m.}$. The low energy effective
operators for dark matter (DM) and nucleon scattering at the quark level can
be constructed from heavy WIMP bilinears, and quark and gluon bilinears.

We focus on WIMP-nucleon spin-independent elastic scattering, which is the
dominant process for many dark matter direct detection experiments. The
relevant low energy effective theory for the spin-independent interaction of
spin-0 and spin-1/2 heavy WIMP with quarks (top-quark been integrated out) and
gluons is [26]

$\displaystyle\mathcal{L}=\bar{\chi}_{0}\chi_{0}\bigg{\\{}\sum_{q=u,d,s,c,b}\left[c_{q}^{(0)}O_{q}^{(0)}+c_{q}^{(2)}v_{\mu}v_{\nu}O_{q}^{(2)\mu\nu}\right]+c_{g}^{(0)}O_{g}^{(0)}+c_{g}^{(2)}v_{\mu}v_{\nu}O_{g}^{(2)\mu\nu}\bigg{\\}}\,,$
(42)

where

$\displaystyle O_{q}^{(0)}=m_{q}\bar{q}{q},\quad
O^{(2)\mu\nu}_{q}=\frac{1}{2}\bar{q}\left(\gamma^{\\{\mu}iD_{-}^{\nu\\}}-\frac{g^{\mu\nu}}{d}i\not{D}_{-}\right)q\,,$
$\displaystyle O^{(0)}_{g}=G^{A\,\mu\nu}G^{A}_{\mu\nu},\quad
O^{(2)\mu\nu}_{g}=-G^{A\mu\lambda}G^{A\nu}_{~{}~{}~{}\lambda}+\frac{1}{d}g^{\mu\nu}(G^{A}_{\alpha\beta})^{2}\,,$
(43)

with $D^{\mu}_{-}\equiv D^{\mu}-\overleftarrow{D}^{\mu}$.

For higher-spin particles, no essentially new operators appear in the spin-
independent sector of the low energy effective theory. 333This may be seen by
constructing an explicit basis, and enforcing the constraints on higher-spin
representations for heavy particles [35], e.g. $v_{\mu}\chi_{0}^{\mu}=0$,
$\not{v}\chi_{0}^{\mu}=\chi_{0}^{\mu}$,
$\epsilon_{\nu\alpha\beta\mu}v^{\nu}\sigma^{\alpha\beta}\chi_{0}^{\mu}=0$.
For spin-1 and spin-3/2 heavy WIMPs interacting with quarks and gluons, we
have

$\displaystyle\mathcal{L}=\bar{\chi}_{0}^{\mu}\chi_{0}^{\nu}\bigg{\\{}\sum_{q=u,d,s,c,b}\left[c_{q}^{(0)}O_{q}^{(0)}+c_{q}^{(2)}v_{\alpha}v_{\beta}O_{q}^{(2)\alpha\beta}\right]+c_{g}^{(0)}O_{g}^{(0)}+c_{g}^{(2)}v_{\alpha}v_{\beta}O_{g}^{(2)\alpha\beta}\bigg{\\}}g_{\mu\nu}\,.$
(44)

##  4 Weak Matching

To determine the Wilson coefficients in the effective theories Eq. (42) and
Eq. (44) and obtain results for WIMP-nucleon elastic scattering, we match the
low energy Lagrangian to the electroweak scale effective theory (38) by
integrating out the weak scale particles.

The matching diagrams for WIMP and quark operators are shown in Fig. 2. Note
that all diagrams involving Nambu-Goldstone bosons are suppressed compared to
the diagrams present in Fig. 2. All Standard Model particles are treated as
massless except the weak scale particles $W^{\pm}$, $Z^{0}$, $h$, $t$-quark,
which will be integrated out. The matching for WIMP and gluon operators are
shown in Fig. 3. The details of the matching can be found in [37].

Figure 2:  Diagrams contributing through $1/M$ order to quark matching. Double
lines denote the WIMP, solid lines denote quarks, zigzag lines denote weak
gauge bosons, and dashed lines denote the Higgs boson. The encircled cross
denotes an insertion of $1/M$ order effective operators.  Figure 3:  Diagrams
contributing through $1/M$ order to gluon matching, with the same notation as
in Fig. 2. Curly lines denote gluons. Diagrams with both gluons attached to
the upper quark line or with one gluon attached to each of the upper and lower
quark lines are not shown in the second row.

The renormalized Wilson coefficients in Eq. (42) and Eq. (44) for the low
energy five-flavor quarks and gluons effective theory are

$\displaystyle\hat{c}_{U}^{(0)}(\mu)$
$\displaystyle=-\frac{1}{2x_{h}^{2}}\left(f_{W}+\frac{f_{Z}}{c_{W}^{3}}\right)+\frac{f_{Z}}{8c_{W}}(c_{V}^{(U)2}-c_{A}^{(U)2})-\frac{m_{W}}{\pi
M}\frac{{c}_{H}}{\alpha_{2}^{2}x_{h}^{2}}\,,$
$\displaystyle\hat{c}_{D}^{(0)}(\mu)$
$\displaystyle=-\frac{1}{2x_{h}^{2}}\left(f_{W}+\frac{f_{Z}}{c_{W}^{3}}\right)+\frac{f_{Z}}{8c_{W}}(c_{V}^{(D)2}-c_{A}^{(D)2})-\delta_{Db}f_{W}\frac{x_{t}}{8(x_{t}+1)^{3}}-\frac{m_{W}}{\pi
M}\frac{{c}_{H}}{\alpha_{2}^{2}x_{h}^{2}}\,,$
$\displaystyle\hat{c}_{g}^{(0)}(\mu)$
$\displaystyle=\frac{\alpha_{s}(\mu)}{4\pi}\bigg{\\{}\frac{1}{6}\bigg{[}\frac{1}{x_{h}^{2}}\Big{(}f_{W}+\frac{f_{Z}}{c_{W}^{3}}\Big{)}+f_{W}\left[\frac{N_{l}}{2}+\frac{1}{2(x_{t}+1)^{2}}\right]\bigg{]}+\frac{m_{W}}{\pi
M}\frac{{c}_{H}}{3\alpha_{2}^{2}x_{h}^{2}}$
$\displaystyle\quad+\frac{f_{Z}}{16c_{W}}\bigg{[}(c_{V}^{(D)2}+c_{A}^{(D)2})+\frac{1}{4}(c_{V}^{(U)2}+c_{A}^{(U)2})\Big{[}\frac{4(48y_{t}^{6}-2y_{t}^{4}+9y_{t}^{2}-1)}{3(4y_{t}^{2}-1)^{3}}$
$\displaystyle\quad+\frac{8}{3}-\pi
y_{t}+\frac{32y_{t}^{6}(8y_{t}^{2}-7)}{(4y_{t}^{2}-1)^{7/2}}\arctan(\sqrt{4y_{t}^{2}-1})\Big{]}+\frac{1}{4}(c_{V}^{(U)2}-c_{A}^{(U)2})\Big{[}3\pi
y_{t}$
$\displaystyle\quad-\frac{32y_{t}^{4}(24y_{t}^{4}-21y_{t}^{2}+5)}{(4y_{t}^{2}-1)^{7/2}}\arctan(\sqrt{4y_{t}^{2}-1})-\frac{4(144y_{t}^{6}-70y_{t}^{4}+9y_{t}^{2}-2)}{3(4y_{t}^{2}-1)^{3}}\Big{]}\bigg{]}\bigg{\\}}\,,$
$\displaystyle\hat{c}_{U}^{(2)}(\mu)$
$\displaystyle=\frac{f_{W}}{3}+\frac{f_{Z}}{6c_{W}}(c_{V}^{(U)2}+c_{A}^{(U)2})-\frac{f_{Z}}{4c_{W}^{2}}(c_{V}^{(U)2}+c_{A}^{(U)2})\frac{m_{W}}{\pi
M}-f_{W}\frac{m_{W}}{2\pi M}\,,$ $\displaystyle\hat{c}_{D}^{(2)}(\mu)$
$\displaystyle=\frac{f_{W}}{3}+\frac{f_{Z}}{6c_{W}}(c_{V}^{(D)2}+c_{A}^{(D)2})-\frac{f_{Z}}{4c_{W}^{2}}(c_{V}^{(D)2}+c_{A}^{(D)2})\frac{m_{W}}{\pi
M}-f_{W}\frac{m_{W}}{2\pi M}$
$\displaystyle\quad+f_{W}\frac{\delta_{Db}}{2}\Big{[}\frac{3x_{t}+2}{3(x_{t}+1)^{3}}-\frac{2}{3}-\frac{m_{W}}{\pi
M}\frac{x_{t}^{2}(1-x_{t}^{4}+4x_{t}^{2}\log{x_{t}})}{(x_{t}^{2}-1)^{3}}\Big{]}\,,$
$\displaystyle\hat{c}_{g}^{(2)}(\mu)$
$\displaystyle=\frac{\alpha_{s}(\mu)}{4\pi}\bigg{\\{}2f_{W}\bigg{[}N_{\ell}\left(-{4\over
9}\log{\mu\over m_{W}}-{1\over 2}\right)-{(2+3x_{t})\over
9(1+x_{t})^{3}}\log{\mu\over m_{W}(1+x_{t})}$
$\displaystyle\quad-{(12x_{t}^{5}-36x_{t}^{4}+36x_{t}^{3}-12x_{t}^{2}+3x_{t}-2)\over
9(x_{t}-1)^{3}}\log{x_{t}\over 1+x_{t}}$
$\displaystyle\quad-{2x_{t}(-3+7x_{t}^{2})\over 9(x_{t}^{2}-1)^{3}}\log
2-{48x_{t}^{6}+24x_{t}^{5}-104x_{t}^{4}-35x_{t}^{3}+20x_{t}^{2}+13x_{t}+18\over
36(x_{t}^{2}-1)^{2}(1+x_{t})}\bigg{]}$
$\displaystyle\quad+f_{W}\frac{m_{W}}{2\pi
M}\bigg{[}N_{\ell}\left(\frac{8}{3}\log{\mu\over
m_{W}}-\frac{1}{3}\right)+\frac{16x_{t}^{4}}{3(x_{t}^{2}-1)^{3}}\log{x_{t}}\log{\mu\over
m_{W}}-\frac{4(3x_{t}^{2}-1)}{3(x_{t}^{2}-1)^{2}}\log{\mu\over m_{W}}$
$\displaystyle\quad+\frac{16x_{t}^{2}}{3}\log^{2}{x_{t}}-\frac{4(4x_{t}^{6}-16x_{t}^{4}+6x_{t}^{2}+1)}{3(x_{t}^{2}-1)^{3}}\log{x_{t}}+\frac{8x_{t}^{2}(x_{t}^{6}-3x_{t}^{4}+4x_{t}^{2}-1)}{3(x_{t}^{2}-1)^{3}}{\mathrm{Li}}_{2}(1-x_{t}^{2})$
$\displaystyle\quad+\frac{4\pi^{2}x_{t}^{2}}{9}-\frac{8x_{t}^{4}-7x_{t}^{2}+1}{3(x_{t}^{2}-1)^{2}}\bigg{]}+\frac{f_{Z}}{16c_{W}}\bigg{[}\Big{[}2(c_{V}^{(U)2}+c_{A}^{(U)2})+3(c_{V}^{(D)2}+c_{A}^{(D)2})\Big{]}\Big{[}-\frac{32}{9}\log{{\mu\over
m_{Z}}}-4\Big{]}$
$\displaystyle\quad+(c_{V}^{(U)2}+c_{A}^{(U)2})\Big{[}\frac{32(24y_{t}^{8}-21y_{t}^{6}-4y_{t}^{4}+5y_{t}^{2}-1)}{9(4y_{t}^{2}-1)^{7/2}}\arctan(\sqrt{4y_{t}^{2}-1})-\frac{\pi
y_{t}}{3}$
$\displaystyle\quad+\frac{4(48y_{t}^{6}+62y_{t}^{4}-47y_{t}^{2}+9)}{9(4y_{t}^{2}-1)^{3}}\Big{]}+(c_{V}^{(U)2}-c_{A}^{(U)2})\Big{[}\frac{4y_{t}^{2}(624y_{t}^{4}-538y_{t}^{2}+103)}{9(4y_{t}^{2}-1)^{3}}-\frac{13\pi
y_{t}}{3}$
$\displaystyle\quad+\frac{32y_{t}^{2}(104y_{t}^{6}-91y_{t}^{4}+35y_{t}^{2}-5)}{3(4y_{t}^{2}-1)^{7/2}}\arctan(\sqrt{4y_{t}^{2}-1})\Big{]}\bigg{]}$
$\displaystyle\quad+\frac{f_{Z}}{24c_{W}^{2}}\frac{m_{W}}{\pi
M}\bigg{[}\Big{[}2(c_{V}^{(U)2}+c_{A}^{(U)2})+3(c_{V}^{(D)2}+c_{A}^{(D)2})\Big{]}\Big{(}8\log{{\mu\over
m_{Z}}}-1\Big{)}$
$\displaystyle\quad-(c_{V}^{(U)2}+c_{A}^{(U)2})\Big{[}\frac{1-18y_{t}^{2}+36y_{t}^{4}}{(4y_{t}^{2}-1)^{2}}+\frac{8(1-4y_{t}^{2}+3y_{t}^{4}+18y_{t}^{6})\log{y_{t}}}{(4y_{t}^{2}-1)^{3}}$
$\displaystyle\quad+\frac{16y_{t}^{2}(2-13y_{t}^{2}+32y_{t}^{4}-18y_{t}^{6})}{(4y_{t}^{2}-1)^{7/2}}\big{[}2\arctan{\Big{(}\frac{1}{\sqrt{4y_{t}^{2}-1}}\Big{)}}\log{y_{t}}-{\mathrm{Im}}\,{{\mathrm{Li}}_{2}\Big{(}\frac{1-i\sqrt{4y_{t}^{2}-1}}{2y_{t}^{2}}\Big{)}}\big{]}\Big{]}$
$\displaystyle\quad+4y_{t}^{2}(c_{V}^{(U)2}-c_{A}^{(U)2})\Big{[}-\frac{8-59y_{t}^{2}+108y_{t}^{4}}{(4y_{t}^{2}-1)^{3}}-\frac{(29-128y_{t}^{2}+108y_{t}^{4})\log{y_{t}}}{(4y_{t}^{2}-1)^{3}}$
$\displaystyle\quad+\frac{2(-7+38y_{t}^{2}-82y_{t}^{4}+108y_{t}^{6})}{(4y_{t}^{2}-1)^{7/2}}\big{[}2\arctan{\Big{(}\frac{1}{\sqrt{4y_{t}^{2}-1}}\Big{)}}\log{y_{t}}-{\mathrm{Im}}\,{{\mathrm{Li}}_{2}\Big{(}\frac{1-i\sqrt{4y_{t}^{2}-1}}{2y_{t}^{2}}\Big{)}}\big{]}\Big{]}\bigg{]}\Bigg{\\}}\,,$
(45)

where the reduced coefficients $\hat{c}_{i}^{(S)}$ are given in terms of the
original Wilson coefficients by
$c_{i}^{(S)}\equiv(\pi\alpha_{2}^{2}/m_{W}^{3})\hat{c}^{(S)}_{i}$ with
$\alpha_{2}=g_{2}^{2}/(4\pi)$, where $i=u,d,s,c,b,g$ is the index for quark or
gluon and $U$ denotes up-type while $D$ denotes down-type and we have
neglected small corrections from $|V_{td}|^{2}$ and $|V_{ts}|^{2}$, The $u$
and $c$ quarks have the same coefficients, as do $d$ and $s$ quarks through
all the weak matching calculations. The group theory factors are
$f_{W}=J(J+1)-Y^{2}$, $f_{Z}=Y^{2}$. The strong coupling is denoted by
$\alpha_{s}(\mu)$. The mass ratios are defined as $x_{j}\equiv m_{j}/m_{W}$
and $y_{j}\equiv m_{j}/m_{Z}$ where $m_{Z}$ is the mass of $Z^{0}$ boson, and
$j$ is the index of the specific particle, e.g. $j=t$ stands for top quark,
$j=h$ for Higgs boson.
$\mathrm{Li}_{2}(z)\equiv\sum_{k=1}^{\infty}{z^{k}}/{k^{2}}$ is the
dilogarithm function. $N_{\ell}=2$ is the number of massless Standard Model
generations.

##  5 Illustrative UV Completions

At subleading order $1/M$, the underlying UV completion impacts spin-
independent direct detection cross sections via the single parameter $c_{H}$,
cf. Eq. (4). This parameter is in turn determined by coefficients $A$, $B$,
$C$, $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$ in Eqs. (39), (40), and (41)
We illustrate the determination of $c_{H}$ by considering minimal Standard
Model extensions with a new electroweak multiplet containing our dark matter
WIMP.

###  5.1 Real bosons

Figure 4:  Matching condition for the coefficients in the EFT for UV theory
consisting of the Standard Model plus an electroweak multiplet real boson.
Solid lines denote the boson, dashed lines denote Higgs doublet, zigzag lines
denote $\mathrm{SU(2)}_{W}\times\mathrm{U(1)}_{Y}$ gauge fields. The black
square denotes a counter term contact interaction. Matching is performed in
the electroweak symmetric theory. Double lines on the R.H.S. denote heavy
WIMPs and the encircled cross denotes insertion of a $1/M$ effective theory
vertex.

For a real boson electroweak multiplet, the matching between the UV theory and
the effective theory to determine WIMP-Higgs interaction operators is shown in
Fig. 4. For a real scalar, we take the Lagrangian

$\mathcal{L}^{\mathrm{spin-0}}_{\mathrm{UV}}=\frac{1}{2}D_{\mu}\Phi
D^{\mu}\Phi-\frac{1}{2}M^{2}\Phi^{2}\,,$ (46)

and for a real vector, we take the Lagrangian

$\mathcal{L}^{\mathrm{spin-1}}_{\mathrm{UV}}=-\frac{1}{4}\left(D_{\mu}\mathcal{V}_{\nu}-D_{\nu}\mathcal{V}_{\mu}\right)\left(D^{\mu}\mathcal{V}^{\nu}-D^{\nu}\mathcal{V}^{\mu}\right)+\frac{1}{2}M^{2}\mathcal{V}_{\mu}\mathcal{V}^{\mu}\,.$
(47)

For a real scalar, a generalization of the results in Ref. [28] yields

$c_{H}^{\mathrm{spin-0}}=\frac{J(J+1)}{2}\alpha_{2}^{2}\log{\frac{\Lambda_{\rm
UV}^{2}}{M^{2}}}+\dots\,,$ (48)

where $\Lambda_{\rm UV}$ is a scale intrinsic to the UV theory (the “UV theory
of the UV theory” scale) and the ellipsis denotes terms that are not
logarithmically enhanced in the limit $\Lambda_{\rm UV}\gg M$. In models such
as a “weakly interacting stable pion” [38], this scale is $\Lambda_{\rm
UV}^{2}\sim M^{2}/\alpha_{2}$, and we consider this case in Sec. 6. Similarly,
for a real vector we find

$c_{H}^{\mathrm{spin-1}}=\frac{J(J+1)}{6}\alpha_{2}^{2}\log{\frac{\Lambda_{\rm
UV}^{2}}{M^{2}}}\,.$ (49)

###  5.2 Fermions

For fermionic electroweak multiplets with hypercharge $Y=\pm 1/2$, we consider
a UV theory containing a Dirac field $\psi$, related to the self-conjugate
field $\chi$ as in Eq. (22). We include another electroweak multiplet Majorana
fermion $\chi^{\prime}$, in a $(J^{\prime},\,Y^{\prime})$ representation of
$\mathrm{SU}(2)_{W}\times\mathrm{U}(1)_{Y}$, with mass $M^{\prime}\gg M$. The
field $\chi^{\prime}$ serves to split the degenerate mass of the neutral
components in $\chi$ (for $Y^{\prime}\neq 0$, the Majorana fermion
$\chi^{\prime}$ is a reducible representation of
$\mathrm{SU}(2)_{W}\times\mathrm{U}(1)_{Y}$ constructed from a Dirac fermion
$\lambda$, similar to the construction of $\chi$ from $\psi$). We include the
general renormalizable interaction $F(\psi,\,\chi^{\prime},\,H)$ allowed by
gauge invariance,

$\displaystyle\mathcal{L}_{\mathrm{UV}}^{\mathrm{spin-1/2}}$
$\displaystyle=\bar{\psi}(i\not{D}-M)\psi+\frac{1}{2}\bar{\chi}^{\prime}(i\not{D}-M^{\prime})\chi^{\prime}+F(\psi,\,\chi^{\prime},\,H)\,,$
(50)

where the detailed expression of $F(\psi,\,\chi^{\prime},\,H)$ can be found in
Appendix A.

Figure 5:  Matching condition for the coefficients in the EFT for UV theory
consisting of the Standard Model plus electroweak multiplet self-conjugate
fermion. Solid lines denote the fermions, with the blue line denoting the
heavier fermion with mass $M^{\prime}$. Double lines denote the WIMPs. Other
notations are the same as in the bosonic matching, Fig. 4. The third diagram
on the L.H.S only appears when $Y=\pm 1/2$ for even-dimension electroweak
multiplet. The last diagram on the L.H.S appears when the fermion is spin-3/2.

The matching is shown in Fig. 5. The Higgs-WIMP-WIMP coupling is determined by
the quantities $A$, $B$, $C$ in Eq. (31), which are given by explicit
computation as in Appendix A. In the pure WIMP limit, i.e. $M^{\prime}\gg M$,
the Wilson coefficient in the Feynman rules Eq. (39) and Eq. (40) reduce to

$\displaystyle
c_{H}=-\frac{3}{2}\alpha_{2}^{2}\left[J(J+1)+\tan^{2}{\theta_{W}}|Y|+\tan^{4}{\theta_{W}}Y^{2}\right]\,,$
(51)

where $\theta_{W}$ is the weak mixing angle, and $Y=\pm 1/2$ for even-
dimension electroweak multiplets.

For the $Y=0$ case, we consider the limit $M^{\prime}\to\infty$ and the
renormalizable Lagrangian containing a single electrically neutral component
reduces to

$\displaystyle\mathcal{L}_{\mathrm{UV}}^{\mathrm{spin-1/2}}$
$\displaystyle=\frac{1}{2}\bar{\chi}(i\not{D}-M)\chi\,.$ (52)

The matching is again described by the diagrams in Fig. 5. In the pure WIMP
limit, we have $A^{\prime}=\frac{3}{2(4\pi)^{2}}g_{2}^{4}J(J+1)$ and
$B^{\prime}=0$, and the Wilson coefficient from Eq. (41) reduces to

$\displaystyle c_{H}=-\frac{3}{2}\alpha_{2}^{2}J(J+1)\,.$ (53)

For a Rarita-Schwinger, spin-3/2, WIMP, we take the effective UV Lagrangian to
be

$\mathcal{L}_{\mathrm{UV}}^{\mathrm{spin-3/2}}=-\bar{\varPsi}^{\mu}\left[\left(i\not{D}-M\right)g_{\mu\nu}-\left(i\gamma_{\mu}D_{\nu}+i\gamma_{\nu}D_{\mu}\right)+\gamma_{\mu}\left(i\not{D}+M\right)\gamma_{\nu}\right]{\varPsi}^{\nu}\,,$
(54)

and do the matching as in Fig. 5, similar to the procedures for the spin-1/2
case. We obtain the coefficient

$\displaystyle
c_{H}=-\frac{2}{3}\alpha_{2}^{2}\left[J(J+1)+\tan^{2}{\theta_{W}}|Y|+\tan^{4}{\theta_{W}}Y^{2}\right]\log{\frac{\Lambda_{\rm
UV}^{2}}{M^{2}}}+\dots\,,$ (55)

where $Y=0$ for odd-dimension multiplets and $Y=\pm 1/2$ for even-dimension
multiplets.

##  6 Cross Sections

The benchmark WIMP-nucleon elastic scattering spin-independent cross section
is

$\displaystyle\sigma_{N}=\frac{m_{r}^{2}}{\pi}|\mathcal{M}^{(0)}_{N}+\mathcal{M}^{(2)}_{N}|^{2}\,,$
(56)

where $N=n,p$ is a nucleon, $m_{r}=m_{N}M/(m_{N}+M)\approx m_{N}$ is the
reduced mass of the WIMP-nucleon system, and the scattering amplitude is

$\mathcal{M}^{(S)}_{N}=\sum_{i=q,g}c^{(S)}_{i}(\mu_{0})\langle
N|O^{(S)}_{i}(\mu_{0})|N\rangle\,,$ (57)

where $S=0,2$ for operators with different spins. The nucleon states
$|N\rangle$ are non-perturbative and we use Lattice QCD to evaluate the
nucleon matrix elements at energy scale $\mu_{0}\sim$ GeV. So the heavy
quarks, bottom and charm need to be integrated out from the 5-flavor QCD
theories, Eq. (42). Renormalization group evolution from the 5-flavor
effective QCD theory at the weak scale $\mu_{t}$ to the bottom quark mass
scale $\mu_{b}\sim m_{b}$, threshold matching at $\mu_{b}$, running from
$\mu_{b}$ to charm quark mass scale $\mu_{c}\sim m_{c}$, threshold matching at
$\mu_{c}$, further running from $\mu_{c}$ to $\mu_{0}$ are performed. Details
can be found in Ref. [26]. Specifically, we take
$\mu_{t}=(m_{t}+m_{W})/2=126\,{\rm GeV}$, $\mu_{b}=4.75\,{\rm GeV}$,
$\mu_{c}=1.4\,{\rm GeV}$, and $\mu_{0}=1.2\,{\rm GeV}$. For the spin-0
coefficients, renormalization group evolution and threshold matching are
performed at NNNLO. For spin-2 coefficients, the running and matching are at
NLO. In the end, we obtain the 3-flavor effective QCD theory, with $q=u,d,s$
in Eq. (57) being the three light flavors and $g$ denoting the gluon. We take
the same Lattice QCD data for nucleon matrix elements as in Ref. [29].

Figure 6: Constraints on the dimensionless parameter $c_{H}$ for WIMPS with
different isospins, versus WIMP mass. Zoomed bounds for WIMP mass smaller than
2 TeV are exhibited at the upper left corners.

For our default matching scales $\mu_{t}$, $\mu_{b}$, $\mu_{c}$ and $\mu_{0}$,
and with the central values of all nucleon matrix elements at scale $\mu_{0}$,
we find that the spin-0 and spin-2 amplitudes for WIMP and proton scattering
are (normalized by spin-2 amplitude
${\mathcal{M}}^{(2)}_{p}|_{M\to\infty}=J(J+1)$ when $Y=0$)

$\displaystyle{\mathcal{M}}_{p}^{(0)}$
$\displaystyle=-0.82J(J+1)-0.42Y^{2}-299.50c_{H}{m_{W}\over M}\,,$
$\displaystyle{\mathcal{M}}_{p}^{(2)}$
$\displaystyle=J(J+1)-0.25Y^{2}-\left[0.51J(J+1)-0.072Y^{2}\right]{m_{W}\over
M}\,,$ (58)

where the low energy effective theory of WIMP and 3-flavor QCD operators at
$1/M$ order is yet to be determined by one parameter $c_{H}$. We may constrain
$c_{H}$ by current direct detection experimental limits [1]. We plot the
allowed region of $c_{H}$ for different isospins of a WIMP in Fig. 6.

We may match onto the minimal UV theories in Sec. 5 to obtain concrete values
for $c_{H}$ and predict benchmark results for general WIMP and nucleon spin-
independent scattering cross sections. For a real bosonic heavy WIMP, the
central values for the amplitudes are

$\displaystyle{\mathcal{M}}_{p}^{(0)}$
$\displaystyle=-J(J+1)\left(0.824-0.342\eta{m_{W}\over M}\log{\Lambda_{\rm
UV}\over M}\right)\,,$ $\displaystyle{\mathcal{M}}_{p}^{(2)}$
$\displaystyle=J(J+1)\left(1-0.515{m_{W}\over M}\right)\,,$ (59)

where $\eta=1$ for a spin-0 WIMP, $\eta=1/3$ for a spin-1 WIMP, and
$\Lambda_{\rm UV}$ is a UV scale. We take $\Lambda_{\rm UV}\sim
M/\sqrt{\alpha_{2}}\sim 10\,M$ for illustration, as discussed in Sec. 5. For a
self-conjugate spin-1/2 heavy WIMP, the amplitudes are

$\displaystyle{\mathcal{M}}_{p}^{(0)}$
$\displaystyle=-0.824J(J+1)-0.417Y^{2}+\left[0.513J(J+1)+0.153|Y|+0.0457Y^{2}\right]{m_{W}\over
M}\,,$ $\displaystyle{\mathcal{M}}_{p}^{(2)}$
$\displaystyle=J(J+1)-0.247Y^{2}-\left[0.515J(J+1)-0.0716Y^{2}\right]{m_{W}\over
M}\,.$ (60)

For a self-conjugate spin-3/2 heavy WIMP, the amplitudes are

$\displaystyle{\mathcal{M}}_{p}^{(0)}$
$\displaystyle=-0.824J(J+1)-0.417Y^{2}+\left[0.456J(J+1)+0.136|Y|+0.0407Y^{2}\right]{m_{W}\over
M}\log{\Lambda_{\rm UV}\over M}\,,$ $\displaystyle{\mathcal{M}}_{p}^{(2)}$
$\displaystyle=J(J+1)-0.247Y^{2}-\left[0.515J(J+1)-0.0716Y^{2}\right]{m_{W}\over
M}\,,$ (61)

where again $\Lambda_{\rm UV}$ is a UV scale and we will take it to be
$M/\sqrt{\alpha_{2}}$. From expressions (59), (60) and (61) we see clearly the
cancellation between spin-0 and spin-2 amplitudes. For all values of spin and
for all electroweak quantum numbers with $J(J+1)\geq Y^{2}$ (such that the
multiplet contains an electrically neutral component), ${\cal M}^{(0)}$ is
negative at leading power and ${\cal M}^{(2)}$ is positive. Similarly at $1/M$
order, the contributions to ${\cal M}^{(0)}$ and ${\cal M}^{(2)}$ have
opposite sign. The cancellation is especially severe for the Higgsino-like
case $J=Y=1/2$.

Figure 7: Spin-independent scattering cross section for different bosonic WIMP
multiplets on proton, versus the WIMP mass.

Figure 8: Spin-independent scattering cross section for different fermionic
WIMP multiplets on proton, versus the WIMP mass.

We plot the spin-independent cross sections for different heavy WIMPs and
proton scattering in Fig. 7 and Fig. 8, versus the mass of the WIMP. When
evaluating the nucleon-level amplitude, we have perturbative uncertainties
from Wilson coefficients and non-perturbative uncertainties from hadronic
matrix elements. Uncertainty for perturbative computation of the matching
coefficients is estimated by varying the matching scales within the ranges
$m_{W}^{2}/2\leq\mu_{t}^{2}\leq 2m_{t}^{2}$, $m_{b}^{2}/2\leq\mu_{b}^{2}\leq
2m_{b}^{2}$, $m_{c}^{2}/2\leq\mu_{c}^{2}\leq 2m_{c}^{2}$, and $1.0\,{\rm
GeV}\leq\mu_{0}\leq 1.4\,{\rm GeV}$. Uncertainties from neglect of higher
order (starting from $1/M^{2}$ order) power corrections are estimated by
shifting ${\cal M}^{(2)}_{p}\to{\cal
M}^{(2)}_{p}|_{M\to\infty}[1\pm(m_{W}/M)^{2}]$. Uncertainties from nucleon
matrix elements are propagated to the observable cross section [39, 40, 41,
42, 26]. We add the errors in quadrature from different sources mentioned
above, for spin-0 and spin-2 amplitudes separately. The maximum and minimum of
all possible values of the combination
$|\mathcal{M}^{(0)}_{p}+\mathcal{M}^{(2)}_{p}|$ set the bounds of the colored
cross section bands for each WIMP in Fig. 7 and Fig. 8. The cross section
increases as the isospin increases, and the central value varies from
$10^{-50}\,\rm{cm}^{2}$ order to $10^{-46}\,\rm{cm}^{2}$ order from lowest
isospin $J=1/2$ to highest isospin $J=3$ for WIMP mass at 1 TeV.

In Fig. 7 and Fig. 8, also shown are the recent dark matter direct detection
experimental exclusion ($90\%$ confidence) limits [1, 2, 3, 4] for the
relevant TeV mass range. Above the TeV scale, for isospin $J$ smaller than 3,
the WIMP proton cross section is below current experimental sensitivity. For
fermionic WIMPs, when $J=3$, the cross section overlaps with LZ’s limit in the
1 TeV mass region. For bosonic triplet ($J=1$) WIMPs, the cross section is
close to the neutrino floor [43]. For fermionic doublet ($J=1/2$) WIMPs, the
cross section upper bound is much lower than the neutrino floor. Other low-
isospin WIMPs lie between current experimental limit and the neutrino floor.

##  7 Summary

We have used heavy particle effective theory to study general heavy WIMP and
nucleon scattering at subleading $1/M$ order, and to compute cross sections
for arbitrary electroweak representations and low-spin particles. We focused
on the elastic and spin-independent process which is a primary target for dark
matter direct detection experiments.

The spin-independent cross section is universal at leading power, determined
by Standard Model parameters once the WIMP spin and electroweak representation
are specified. At subleading $1/M$ order, dependence on UV structure is
encoded by a single coefficient $c_{H}$ describing the WIMP-WIMP-Higgs boson
coupling. We constrain this coefficient using current experimental exclusion
limits [1] and find $-0.2\lesssim c_{H}\lesssim 0.1$ (90% CL) at 1 TeV WIMP
mass, with a slight dependence on the isospin of the WIMP, cf. Fig. 6. These
model-independent results can be interpreted as constraints on the parameter
space of specific UV completions, cf. e.g. Ref. [44].

We also predict benchmark cross sections in dark matter direct detection
experiments through $1/M$ order, by matching our heavy WIMP effective theory
to minimal UV extensions of Standard Model to obtain $c_{H}$. The
corresponding cross sections are below current experimental limits for low
isospin electroweak multiplets, either bosonic or fermionic, mostly lying
between the experimental limit and the neutrino floor. Central values vary
between $\sim 10^{-50}\,\rm{cm}^{2}$ and $\sim 10^{-46}\,\rm{cm}^{2}$ from
lowest isospin $J=1/2$ to highest isospin $J=3$ for WIMP mass at 1 TeV. These
cross sections are within striking range of next-generation experiments, with
the exception of electroweak doublets, which hide below the neutrino floor. In
general, higher-isospin WIMPs have larger cross section and will be discovered
or excluded first. Uncertainties due to nuclear effects of the heavy element
experimental target, e.g. Xenon, are similar in magnitude to uncertainties of
the cross section which have been computed here, and are not expected to
change the predicted discovery range for these WIMPs [29]. For most cases
these heavy WIMPs can be discovered or excluded with next-generation direct
detection experiments. An exceptional case is the electroweak doublet, whose
cross section is impacted by a severe amplitude cancellation. Experimental
methods such as directional discrimination, annual modulation and improved
flux measurements [45, 46, 47] may allow access to cross sections below the
neutrino floor, complementing indirect searches [48, 49].

Acknowledgements. QC acknowledges postdoctorate fellowship supported by
University of Science and Technology of China and Peng Huanwu Center for
Fundamental Theory (PCFT), Hefei. PCFT is supported by National Natural
Science Foundation of China under grant No. 12247103. GJD is supported by the
National Natural Science Foundation of China under Grant Nos. 11975224,
11835013. Research of RJH supported by the U.S. Department of Energy, Office
of Science, Office of High Energy Physics, under Award Number DE-SC0019095.
This manuscript has been authored by Fermi Research Alliance, LLC under
Contract No. DE- AC02-07CH11359 with the U.S. Department of Energy, Office of
Science, Office of High Energy Physics.

##  Appendix A Details of UV matching onto HWET for fermions

Consider the Lagrangian,

$\displaystyle\mathcal{L}_{\mathrm{UV}}^{\mathrm{spin-1/2}}$
$\displaystyle=\bar{\psi}(i\not{D}-M)\psi+\frac{1}{2}\bar{\chi}^{\prime}(i\not{D}-M^{\prime})\chi^{\prime}$
$\displaystyle+$
$\displaystyle\delta_{Y\prime,\,Y-1/2}\Bigg{\\{}\bar{\psi}_{k}\bigg{[}A_{1}\sqrt{k-1}\delta_{l,\,k-1}\delta_{J^{\prime},\,J-1/2}+B_{1}\sqrt{2J+2-k}\delta_{lk}\delta_{J^{\prime},\,J+1/2}\bigg{]}\lambda_{l}H_{2}$
$\displaystyle\quad-\bar{\psi}_{k}\bigg{[}-A_{1}\sqrt{2J+1-k}\delta_{l,\,k-1}\delta_{J^{\prime},\,J-1/2}+B_{1}\sqrt{k}\delta_{l,\,k+1}\delta_{J^{\prime},\,J+1/2}\bigg{]}\lambda_{l}H_{1}$
$\displaystyle\quad+\bar{\psi}_{k}\bigg{[}A_{1}^{\prime}\sqrt{k-1}\delta_{l,\,k-1}\delta_{J^{\prime},\,J-1/2}+B_{1}^{\prime}\sqrt{2J+2-k}\delta_{lk}\delta_{J^{\prime},\,J+1/2}\bigg{]}\gamma^{5}\lambda_{l}H_{2}$
$\displaystyle\quad-\bar{\psi}_{k}\bigg{[}-A_{1}^{\prime}\sqrt{2J+1-k}\delta_{l,\,k-1}\delta_{J^{\prime},\,J-1/2}+B_{1}^{\prime}\sqrt{k}\delta_{l,\,k+1}\delta_{J^{\prime},\,J+1/2}\bigg{]}\gamma^{5}\lambda_{l}H_{1}$
$\displaystyle\quad-\bar{\psi_{k}^{c}}\bigg{[}A_{8}\sqrt{2J+1-k}\delta_{kl}\delta_{J^{\prime},\,J-1/2}-B_{8}\sqrt{k}\delta_{l,\,k+1}\delta_{J^{\prime},\,J+1/2}\bigg{]}\lambda^{c}_{l}H_{1}^{*}$
$\displaystyle\quad-\bar{\psi_{k}^{c}}\bigg{[}A_{8}\sqrt{k-1}\delta_{l,\,k-1}\delta_{J^{\prime},\,J-1/2}+B_{8}\sqrt{2J+2-k}\delta_{lk}\delta_{J^{\prime},\,J+1/2}\bigg{]}\lambda^{c}_{l}H_{2}^{*}$
$\displaystyle\quad-\bar{\psi_{k}^{c}}\bigg{[}A_{8}^{\prime}\sqrt{2J+1-k}\delta_{kl}\delta_{J^{\prime},\,J-1/2}-B_{8}^{\prime}\sqrt{k}\delta_{l,\,k+1}\delta_{J^{\prime},\,J+1/2}\bigg{]}\gamma^{5}\lambda^{c}_{l}H_{1}^{*}$
$\displaystyle\quad-\bar{\psi_{k}^{c}}\bigg{[}A_{8}^{\prime}\sqrt{k-1}\delta_{l,\,k-1}\delta_{J^{\prime},\,J-1/2}+B_{8}^{\prime}\sqrt{2J+2-k}\delta_{lk}\delta_{J^{\prime},\,J+1/2}\bigg{]}\gamma^{5}\lambda^{c}_{l}H_{2}^{*}\Bigg{\\}}$
$\displaystyle\quad+\delta_{Y\prime,\,Y+1/2}\Bigg{\\{}-\bar{\psi}_{k}\bigg{[}A_{2}\sqrt{k-1}\delta_{l,\,k-1}\delta_{J^{\prime},\,J-1/2}+B_{2}\sqrt{2J+2-k}\delta_{lk}\delta_{J^{\prime},\,J+1/2}\bigg{]}\lambda_{l}H_{1}^{*}$
$\displaystyle\quad-\bar{\psi}_{k}\bigg{[}-A_{2}\sqrt{2J+1-k}\delta_{l,\,k-1}\delta_{J^{\prime},\,J-1/2}+B_{2}\sqrt{k}\delta_{l,\,k+1}\delta_{J^{\prime},\,J+1/2}\bigg{]}\lambda_{l}H_{2}^{*}$
$\displaystyle\quad-\bar{\psi}_{k}\bigg{[}A_{2}^{\prime}\sqrt{k-1}\delta_{l,\,k-1}\delta_{J^{\prime},\,J-1/2}+B_{2}^{\prime}\sqrt{2J+2-k}\delta_{lk}\delta_{J^{\prime},\,J+1/2}\bigg{]}\gamma^{5}\lambda_{l}H_{1}^{*}$
$\displaystyle\quad-\bar{\psi}_{k}\bigg{[}-A_{2}^{\prime}\sqrt{2J+1-k}\delta_{l,\,k-1}\delta_{J^{\prime},\,J-1/2}+B_{2}^{\prime}\sqrt{k}\delta_{l,\,k+1}\delta_{J^{\prime},\,J+1/2}\bigg{]}\gamma^{5}\lambda_{l}H_{2}^{*}$
$\displaystyle\quad+\bar{\psi_{k}^{c}}\bigg{[}A_{6}\sqrt{2J+1-k}\delta_{kl}\delta_{J^{\prime},\,J-1/2}-B_{6}\sqrt{k}\delta_{l,\,k+1}\delta_{J^{\prime},\,J+1/2}\bigg{]}\lambda^{c}_{l}H_{2}$
$\displaystyle\quad-\bar{\psi_{k}^{c}}\bigg{[}A_{6}\sqrt{k-1}\delta_{l,\,k-1}\delta_{J^{\prime},\,J-1/2}+B_{6}\sqrt{2J+2-k}\delta_{lk}\delta_{J^{\prime},\,J+1/2}\bigg{]}\lambda^{c}_{l}H_{1}$
$\displaystyle\quad+\bar{\psi_{k}^{c}}\bigg{[}A_{6}^{\prime}\sqrt{2J+1-k}\delta_{kl}\delta_{J^{\prime},\,J-1/2}-B_{6}^{\prime}\sqrt{k}\delta_{l,\,k+1}\delta_{J^{\prime},\,J+1/2}\bigg{]}\gamma^{5}\lambda^{c}_{l}H_{2}$
$\displaystyle\quad-\bar{\psi_{k}^{c}}\bigg{[}A_{6}^{\prime}\sqrt{k-1}\delta_{l,\,k-1}\delta_{J^{\prime},\,J-1/2}+B_{6}^{\prime}\sqrt{2J+2-k}\delta_{lk}\delta_{J^{\prime},\,J+1/2}\bigg{]}\gamma^{5}\lambda^{c}_{l}H_{1}\Bigg{\\}}$
$\displaystyle\quad+\delta_{Y\prime,\,-Y+1/2}\Bigg{\\{}\bar{\psi}_{k}(-1)^{k}\bigg{[}A_{3}\sqrt{k-1}\delta_{l+k,\,2J+2}\delta_{J^{\prime},\,J-1/2}$
$\displaystyle\quad-
B_{3}\sqrt{2J+2-k}\delta_{k+l,\,2J+3}\delta_{J^{\prime},\,J+1/2}\bigg{]}\lambda^{c}_{l}H_{2}$
$\displaystyle\quad-\bar{\psi}_{k}(-1)^{k}\bigg{[}A_{3}\sqrt{2J+1-k}\delta_{l+k,\,2J+1}\delta_{J^{\prime},\,J-1/2}+B_{3}\sqrt{k}\delta_{l+k,\,2J+2}\delta_{J^{\prime},\,J+1/2}\bigg{]}\lambda^{c}_{l}H_{1}$
$\displaystyle\quad+\bar{\psi}_{k}(-1)^{k}\bigg{[}A_{3}^{\prime}\sqrt{k-1}\delta_{l+k,\,2J+2}\delta_{J^{\prime},\,J-1/2}-B_{3}^{\prime}\sqrt{2J+2-k}\delta_{k+l,\,2J+3}\delta_{J^{\prime},\,J+1/2}\bigg{]}\gamma^{5}\lambda^{c}_{l}H_{2}$
$\displaystyle\quad-\bar{\psi}_{k}(-1)^{k}\bigg{[}A_{3}^{\prime}\sqrt{2J+1-k}\delta_{l+k,\,2J+1}\delta_{J^{\prime},\,J-1/2}+B_{3}^{\prime}\sqrt{k}\delta_{l+k,\,2J+2}\delta_{J^{\prime},\,J+1/2}\bigg{]}\gamma^{5}\lambda^{c}_{l}H_{1}$
$\displaystyle\quad-\bar{\psi_{k}^{c}}(-1)^{k}\bigg{[}A_{7}\sqrt{2J+1-k}\delta_{k+l,\,2J+1}\delta_{J^{\prime},\,J-1/2}+B_{7}\sqrt{k}\delta_{k+l,\,2J+2}\delta_{J^{\prime},\,J+1/2}\bigg{]}\lambda_{l}H_{1}^{*}$
$\displaystyle\quad-\bar{\psi_{k}^{c}}(-1)^{k}\bigg{[}-A_{7}\sqrt{k-1}\delta_{k+l,\,2J+2}\delta_{J^{\prime},\,J-1/2}+B_{7}\sqrt{2J+2-k}\delta_{k+l,\,2J+3}\delta_{J^{\prime},\,J+1/2}\bigg{]}\lambda_{l}H_{2}^{*}$
$\displaystyle\quad-\bar{\psi_{k}^{c}}(-1)^{k}\bigg{[}A_{7}^{\prime}\sqrt{2J+1-k}\delta_{k+l,\,2J+1}\delta_{J^{\prime},\,J-1/2}+B_{7}^{\prime}\sqrt{k}\delta_{k+l,\,2J+2}\delta_{J^{\prime},\,J+1/2}\bigg{]}\gamma^{5}\lambda_{l}H_{1}^{*}$
$\displaystyle\quad-\bar{\psi_{k}^{c}}(-1)^{k}\bigg{[}-A_{7}^{\prime}\sqrt{k-1}\delta_{k+l,\,2J+2}\delta_{J^{\prime},\,J-1/2}+B_{7}^{\prime}\sqrt{2J+2-k}\delta_{k+l,\,2J+3}\delta_{J^{\prime},\,J+1/2}\bigg{]}\gamma^{5}\lambda_{l}H_{2}^{*}\Bigg{\\}}$
$\displaystyle\quad+\delta_{Y\prime,\,-Y-1/2}\Bigg{\\{}-\bar{\psi}_{k}(-1)^{k}\bigg{[}A_{4}\sqrt{k-1}\delta_{l+k,\,2J+2}\delta_{J^{\prime},\,J-1/2}$
$\displaystyle\quad-
B_{4}\sqrt{2J+2-k}\delta_{k+l,\,2J+3}\delta_{J^{\prime},\,J+1/2}\bigg{]}\lambda^{c}_{l}H_{1}^{*}$
$\displaystyle\quad-\bar{\psi}_{k}(-1)^{k}\bigg{[}-A_{4}\sqrt{2J+1-k}\delta_{l+k,\,2J+1}\delta_{J^{\prime},\,J-1/2}+B_{4}\sqrt{k}\delta_{l+k,\,2J+2}\delta_{J^{\prime},\,J+1/2}\bigg{]}\lambda^{c}_{l}H_{2}^{*}$
$\displaystyle\quad-\bar{\psi}_{k}(-1)^{k}\bigg{[}A_{4}^{\prime}\sqrt{k-1}\delta_{l+k,\,2J+2}\delta_{J^{\prime},\,J-1/2}-B_{4}^{\prime}\sqrt{2J+2-k}\delta_{k+l,\,2J+3}\delta_{J^{\prime},\,J+1/2}\bigg{]}\gamma^{5}\lambda^{c}_{l}H_{1}^{*}$
$\displaystyle\quad-\bar{\psi}_{k}(-1)^{k}\bigg{[}-A_{4}^{\prime}\sqrt{2J+1-k}\delta_{l+k,\,2J+1}\delta_{J^{\prime},\,J-1/2}+B_{4}^{\prime}\sqrt{k}\delta_{l+k,\,2J+2}\delta_{J^{\prime},\,J+1/2}\bigg{]}\gamma^{5}\lambda^{c}_{l}H_{2}^{*}$
$\displaystyle\quad+\bar{\psi_{k}^{c}}(-1)^{k}\bigg{[}A_{5}\sqrt{2J+1-k}\delta_{k+l,\,2J+1}\delta_{J^{\prime},\,J-1/2}+B_{5}\sqrt{k}\delta_{k+l,\,2J+2}\delta_{J^{\prime},\,J+1/2}\bigg{]}\lambda_{l}H_{2}$
$\displaystyle\quad-\bar{\psi_{k}^{c}}(-1)^{k}\bigg{[}-A_{5}\sqrt{k-1}\delta_{k+l,\,2J+2}\delta_{J^{\prime},\,J-1/2}+B_{5}\sqrt{2J+2-k}\delta_{k+l,\,2J+3}\delta_{J^{\prime},\,J+1/2}\bigg{]}\lambda_{l}H_{1}$
$\displaystyle\quad+\bar{\psi_{k}^{c}}(-1)^{k}\bigg{[}A_{5}^{\prime}\sqrt{2J+1-k}\delta_{k+l,\,2J+1}\delta_{J^{\prime},\,J-1/2}+B_{5}^{\prime}\sqrt{k}\delta_{k+l,\,2J+2}\delta_{J^{\prime},\,J+1/2}\bigg{]}\gamma^{5}\lambda_{l}H_{2}$
$\displaystyle\quad-\bar{\psi_{k}^{c}}(-1)^{k}\bigg{[}-A_{5}^{\prime}\sqrt{k-1}\delta_{k+l,\,2J+2}\delta_{J^{\prime},\,J-1/2}+B_{5}^{\prime}\sqrt{2J+2-k}\delta_{k+l,\,2J+3}\delta_{J^{\prime},\,J+1/2}\bigg{]}\gamma^{5}\lambda_{l}H_{1}\Bigg{\\}}$
$\displaystyle\quad+\mathrm{h.c.\,terms}\,,$ (62)

where $\lambda$ is a Dirac fermion to construct $\chi^{\prime}$, similar to
the role of $\psi$ for $\chi$, $A_{i}$, $A^{\prime}_{i}$ and $B_{i}$,
$B^{\prime}_{i}$ are coupling constants in the UV theory. We match the UV
theory Eq. (62) onto the effective theory Eq. (16), and determine the
effective theory coefficients from the UV couplings.

Let us do the matching for the operator
$\bar{\chi}_{\alpha}^{m}\chi_{\beta}^{l}H_{i}^{\dagger}H_{j}$, where
$\alpha,\,\beta=1,\,2$ are indices for two Majorana fermions $\chi_{1}$ or
$\chi_{2}$, $m,\,l=1,\,2$ are indices for two components of each Majorana
fermion $\chi_{\alpha}$, and $i,\,j=1,\,2$ are indices for the two components
of Higgs doublet. The first diagram on the L.H.S with exchange of two $W$
fields yields a group factor

$\left(\tilde{T}^{a}\tilde{T}^{b}\right)_{\alpha\beta}^{ml}\left(\tau^{a}\tau^{b}+\tau^{b}\tau^{a}\right)_{ij}=\frac{1}{2}J(J+1)\delta_{\alpha\beta}\delta_{ml}\delta_{ij}\,,$
(63)

When the first diagram of L.H.S contains one $W$ and one $B$ exchange, it
gives a group factor

$\displaystyle\left[\left(\tilde{T}^{a}\tilde{T}^{0}\right)^{ml}_{\alpha\beta}+\left(\tilde{T}^{0}\tilde{T}^{a}\right)^{ml}_{\alpha\beta}\right]\left(\tau^{a}\tau^{0}+\tau^{0}\tau^{a}\right)_{ij}$
$\displaystyle=Y\delta_{\alpha\beta}(J_{ml}^{1}\sigma_{ij}^{1}+J_{ml}^{3}\sigma_{ij}^{3})+iY\sigma_{ij}^{2}J_{ml}^{2}\left(-\delta_{\alpha
1}\delta_{\beta 2}+\delta_{\alpha 2}\delta_{\beta 1}\right)\,,$ (64)

where $a,\,b=1,\,2,\,3$.

When the first diagram of L.H.S contains two $B$ fields exchange, it gives a
group factor

$\displaystyle\left(\tilde{T}^{0}\tilde{T}^{0}\right)_{\alpha\beta}^{ml}\left(\tau^{0}\tau^{0}+\tau^{0}\tau^{0}\right)_{ij}=\frac{Y^{2}}{2}\delta_{\alpha\beta}\delta_{ml}\delta_{ij}\,.$
(65)

Thus, working out all the Feynman rules and the first diagram of L.H.S
contributes to the operator
$\bar{\chi}_{\alpha}^{m}\chi_{\beta}^{l}H_{i}^{\dagger}H_{j}$ a coefficient

$\displaystyle-\left[\frac{1}{2}J(J+1)g_{2}^{4}\delta_{ml}\delta_{ij}+Yg_{1}^{2}g_{2}^{2}(J_{ml}^{1}\sigma_{ij}^{1}+J_{ml}^{3}\sigma_{ij}^{3})+\frac{Y^{2}}{2}g_{1}^{4}\delta_{ml}\delta_{ij}\right]\delta_{\alpha\beta}I_{\mathrm{loop}}$
$\displaystyle-
iYg_{1}^{2}g_{2}^{2}\sigma_{ij}^{2}J_{ml}^{2}\left(-\delta_{\alpha
1}\delta_{\beta 2}+\delta_{\alpha 2}\delta_{\beta
1}\right)I_{\mathrm{loop}}\,,$ (66)

where $I_{\mathrm{loop}}$ is the loop integral for the first diagram on the
L.H.S and

$\displaystyle
I_{\mathrm{loop}}=\int\frac{d^{d}p}{(2\pi)^{d}}\frac{\gamma^{\mu}(\not{p}+\not{q}+M)\gamma_{\mu}}{\left[(p+q)^{2}-M^{2}+i0\right](p^{2}+i0)^{2}}=\frac{i}{(4\pi)^{2-\epsilon}}\frac{\Gamma(1+\epsilon)}{M^{1+2\epsilon}}(3-2\epsilon)\,,$
(67)

with $d=4-2\epsilon$.

The second diagram vanishes by straightforward computation in dimensional
regularization. The only surviving diagram on the R.H.S is the last diagram
and other diagrams vanish since the loop integrals are scaleless but
dimensionful.

Specifically, let us consider the matching when $J^{\prime}=J-1/2$ and
$Y^{\prime}=Y-1/2=-Y+1/2=0$. Other choices will bring us similar results and
the heavy-limit result is the same as we will see in Eq. (51). When we choose
$\alpha=\beta=1$, $i=j=1$ and $m=l$, we obtain

$\displaystyle\mathrm{R.H.S}$
$\displaystyle=-\frac{i}{2M}\left[\mathbf{Re}(c_{1,1/2})-\mathbf{Re}(c_{2,1/2})(l-J-1)\right]\delta_{ml}\,,$
$\displaystyle\mathrm{L.H.S}$
$\displaystyle=-\left[\frac{1}{2}J(J+1)g_{2}^{4}\delta_{ml}\delta_{ij}+Yg_{1}^{2}g_{2}^{2}(J_{ml}^{1}\sigma_{ij}^{1}+J_{ml}^{3}\sigma_{ij}^{3})+\frac{Y^{2}}{2}g_{1}^{4}\delta_{ml}\delta_{ij}\right]I_{\mathrm{loop}}$
$\displaystyle\quad+\frac{i}{4M^{\prime}}\left(|A_{1}|^{2}+|A_{3}|^{2}+|A_{7}|^{2}+|A_{8}|^{2}\right)(2J+1-l)\delta_{ml}\,,$
(68)

and

$\displaystyle\mathbf{Re}(c_{1,1/2})$
$\displaystyle=\frac{3}{(4\pi)^{2}}\left[J(J+1)g_{2}^{4}+Y^{2}g_{1}^{4}\right]-\frac{M}{2M^{\prime}}\left(|A_{1}|^{2}+|A_{3}|^{2}+|A_{7}|^{2}+|A_{8}|^{2}\right)J\,,$
$\displaystyle\mathbf{Re}(c_{2,1/2})$
$\displaystyle=\frac{6g_{1}^{2}g_{2}^{2}}{(4\pi)^{2}}Y-\frac{M}{2M^{\prime}}\left(|A_{1}|^{2}+|A_{3}|^{2}+|A_{7}|^{2}+|A_{8}|^{2}\right)\,.$
(69)

Then we have the coefficients in Eq. (31),

$\displaystyle A$
$\displaystyle=\frac{1}{2}\left[\mathbf{Re}(c_{1,1/2})-\mathbf{Re}(c_{2,1/2})(J+1)\right]$
$\displaystyle=\frac{3}{2(4\pi)^{2}}\left[g_{2}^{4}J(J+1)-2g_{1}^{2}g_{2}^{2}Y(J+1)+Y^{2}g_{1}^{4}\right]+\frac{M}{4M^{\prime}}\left(|A_{1}|^{2}+|A_{3}|^{2}+|A_{7}|^{2}+|A_{8}|^{2}\right)\,,$
$\displaystyle B$
$\displaystyle=\frac{1}{2}\mathbf{Re}(c_{2,1/2})=\frac{3g_{1}^{2}g_{2}^{2}}{(4\pi)^{2}}Y-\frac{M}{4M^{\prime}}\left(|A_{1}|^{2}+|A_{3}|^{2}+|A_{7}|^{2}+|A_{8}|^{2}\right)\,.$
(70)

To obtain the coefficient $C$ in Eq. (31), let us consider the matching for
operator $\bar{\chi}_{\alpha}^{m}\chi_{\beta}^{l}H_{i}H_{j}$ and we choose
$\alpha=2$, $\beta=1$, $i=j=1$ and $m+l=2J+1$,

$\displaystyle\mathrm{R.H.S}$
$\displaystyle=-\frac{c_{4,1/2}+c_{3,1/2}^{*}}{4M}(-1)^{l}\sqrt{l(2J+1-l)}\delta_{l+m,\,2J+1}\,,$
$\displaystyle\mathrm{L.H.S}$
$\displaystyle=\frac{i}{4M^{\prime}}A_{3}A_{8}^{*}(-1)^{l}\sqrt{l(2J+1-l)}\delta_{l+m,\,2J+1}\,,$
(71)

yielding

$\displaystyle
C=\frac{c_{4,1/2}+c_{3,1/2}^{*}}{2}=-i\frac{A_{3}A_{8}^{*}}{2}\frac{M}{M^{\prime}}\,.$
(72)

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