KCL-PH-TH/2023-27, CERN-PH-TH-2023-083
FTPI-MINN-23/08, UMN-TH-4214/23

Electroweak Loop Contributions to the Direct Detection of Wino Dark Matter

John Ellisa, Natsumi Nagatab, Keith A. Olivec, and Jiaming Zhengd

aTheoretical Particle Physics and Cosmology Group, Department of Physics,
King’s College London, London WC2R 2LS, United Kingdom;
Theoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland
bDepartment of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113–0033, Japan

cWilliam I. Fine Theoretical Physics Institute, School of Physics and
Astronomy,
University of Minnesota, Minneapolis, Minnesota 55455, USA

dTsung-Dao Lee Institute & School of Physics and Astronomy,
Shanghai Jiao Tong University, Shanghai 200240, China

Electroweak loop corrections to the matrix elements for the spin-independent
scattering of cold dark matter particles on nuclei are generally small,
typically below the uncertainty in the local density of cold dark matter.
However, as shown in this paper, there are instances in which the electroweak
loop corrections are relatively large, and change significantly the spin-
independent dark matter scattering rate. An important example occurs when the
dark matter particle is a wino, e.g., in anomaly-mediated supersymmetry
breaking (AMSB) and pure gravity mediation (PGM) models. We find that the one-
loop electroweak corrections to the spin-independent wino LSP scattering cross
section generally interfere constructively with the tree-level contribution
for AMSB models with negative Higgsino mixing, $\mu<0$, and in PGM-like models
for both signs of $\mu$, lifting the cross section out of the neutrino fog and
into a range that is potentially detectable in the next generation of direct
searches for cold dark matter scattering.

##  1 Introduction

As a general rule, electroweak loop corrections to the matrix elements for the
scattering of dark matter particles on nuclei are expected to be small
compared to other uncertainties such as those in the local density of dark
matter, its velocity spectrum and the distributions of quark and gluon
constituents in nuclear matter. However, there are instances in which
electroweak loops can make non-negligible contributions to the dark matter
scattering matrix elements, as we discuss in this paper, focusing on the case
of the scalar matrix elements that dominate spin-independent dark matter
scattering.

These instances arise when the tree-level scattering matrix element is
suppressed, e.g., because scalar exchange is parametrically reduced as in the
case of a wino-like dark matter particle that couples weakly to an
intermediate Higgs boson, or because of an accidental cancellation for
specific values of the contributions to the hadronic scattering matrix element
that causes a ‘blind spot’ in the model parameter space. The former case is
relevant, in particular, in models with anomaly-mediated supersymmetry
breaking (AMSB) [1, 2, 3, 4] or pure gravity mediation (PGM) [5, 6, 7], which
predict that the lightest supersymmetric particle (LSP) is a wino-like
neutralino. A first example of a ‘blind spot’ was given in the MSSM with
supersymmetry-breaking parameters constrained to be universal at the input
grand unification scale (the CMSSM) for a negative sign of the Higgsino mixing
term $\mu$ [8, 9, 10].

A lot of experimental water has passed under the supersymmetric bridge since
the CMSSM, AMSB and PGM were initially formulated, with the LHC and direct
searches for dark matter via scattering experiments taking their tolls on the
respective supersymmetric model parameter spaces [11]. For example, a recent
survey of the CMSSM [12] found allowed strips of parameter space in which the
dark matter density was brought into the allowed range by either the focus-
point mechanism [13] or stop coannihilation [14], for Higgsino or bino LSP
with Higgs mass calculations enforcing a heavy spectrum. A global analysis of
the AMSB model that took into account LHC and other experimental constraints
[3] found them to be consistent with a wino LSP weighing $\sim 3$ TeV [15, 16]
that had a very low spin-independent dark matter scattering cross section that
could descend into the neutrino background ‘fog’ [17] 111We note that several
studies have argued that wino dark matter may be in conflict with Fermi-LAT
and H.E.S.S. observations of the gamma-rays coming from the Galactic centre
and dwarf spheroidal galaxies [18]. The degree of this tension depends,
however, on the uncertainties in the dark matter density profiles in these
objects.. However, this analysis did not take into account electroweak loop
corrections to the spin-independent scattering matrix element, which can
become important in regions where the tree-level scattering matrix element is
suppressed, e.g., because of large sparticle masses and/or a chance
cancellation.

We recall that in mAMSB models [2] the number of free parameters is reduced
relative to the CMSSM, from four to three. The gravitino mass, $m_{3/2}$
provides a seed for gaugino masses and $A$-terms, which are generated by
radiative corrections. For example, at one loop, the gaugino masses at some
high-energy scale (often taken to be the GUT scale as in the CMSSM) are given
by [1]:

$\displaystyle M_{1}$ $\displaystyle=$
$\displaystyle\frac{33}{5}\frac{g_{1}^{2}}{16\pi^{2}}m_{3/2}\ ,$ (1)
$\displaystyle M_{2}$ $\displaystyle=$
$\displaystyle\frac{g_{2}^{2}}{16\pi^{2}}m_{3/2}\ ,$ (2) $\displaystyle M_{3}$
$\displaystyle=$ $\displaystyle-3\frac{g_{3}^{2}}{16\pi^{2}}m_{3/2}\ ,$ (3)

with $M_{i}\ll m_{3/2}$. This results in a mass spectrum with
$|M_{1}|:|M_{2}|:|M_{3}|\approx 2.8:1:7.1$, resulting in a wino LSP if the
scalar masses are sufficiently heavy. Similar one-loop expressions determine
the trilinear terms at the same high-energy input scale. Requiring that scalar
masses are also determined radiatively results in an unrealistic model,
because renormalization leads to negative squared masses for sleptons. Thus
the minimal AMSB scenario (mAMSB) adds a constant $m_{0}^{2}$ to all squared
scalar masses [2]. Thus the mAMSB model has three continuous free parameters:
$m_{3/2}$, $m_{0}$ and the ratio of Higgs vevs, $\tan\beta$. In the limit that
$m_{0}=m_{3/2}$, one recovers the set of two-parameter PGM models [6],
specified by $m_{3/2}$ and $\tan\beta$. For these models to be viable,
$\tan\beta\approx 2$ is required. As in the CMSSM, the $\mu$-term and the
Higgs pseudoscalar mass (or $B_{0}\mu$) are determined from the minimization
of the Higgs potential, and the sign of $\mu$ is free.

In both the mAMSB [2] and PGM-like [7] models the LSP is typically a wino-like
neutralino 222Although there are limiting cases in both models where the LSP
becomes Higgsino-like.. In such scenarios, if the Higgsino mixing parameter
$\mu\gg M_{2}$ and $m_{Z}$, the tree-level Higgs exchange contribution to
spin-independent dark matter scattering is strongly suppressed, which is why
the electroweak loop corrections can be important. This is the case, in
particular, when there is a cancellation in the tree-level scattering matrix
element, as can occur in both the mAMSB and PGM-like models, as we discuss
here.

The outline of this paper is as follows. In Section 2 we discuss calculations
of the spin-independent dark matter scattering cross section, first reviewing
the relevant effective interactions and then the tree-level contribution of
Higgs exchange and then the form of the one-loop electroweak radiative
corrections. In Section 3 we discuss calculations of two auxiliary quantities,
namely the relic wino dark matter density and the Higgs mass, paying
particular attention to the requirements of radiative electroweak symmetry
breaking and a reliable calculation of the Higgs mass in supersymmetric models
with a very heavy spectrum. We present our results for the spin-independent
wino dark matter scattering cross section in Section 4, exhibiting the regions
of AMSB and PGM-like parameter space where the one-loop effects enhance the
cross section, as well as regions where they suppress it. For reasons that we
explain, both effects are possible in the AMSB model, whereas the cross
section is generally enhanced in the PGM-like model. As we discuss in Section
5, there are generic regions of the models’ parameter spaces where the
enhanced cross section rises out of the neutrino fog and may be detectable in
the next generation of direct searches for cold dark matter scattering.

##  2 Spin-Independent Scattering Cross Sections

###  2.1 Effective interactions

We first review the calculation of the cross section for the elastic
scattering of wino-like dark matter on a nucleus. We focus on the spin-
independent elastic scattering, as the spin-dependent scattering cross section
turns out to be negligibly small for the cases considered here. The spin-
independent scattering cross section for generic Majorana fermion dark matter
is given by [19, 20, 21, 22, 23, 24, 8]

$\sigma_{\text{SI}}^{Z,A}=\frac{4}{\pi}\biggl{(}\frac{m_{\chi}m_{T}}{m_{\chi}+m_{T}}\biggr{)}^{2}\left[Zf_{p}+(A-Z)f_{n}\right]^{2}~{},$
(4)

where $m_{\chi}$ and $m_{T}$ are the masses of the Majorana dark matter and
the target nucleus, respectively, $A$ and $Z$ are the mass and atomic numbers
of the nucleus, and $f_{N}$ $(N=p,n)$ are the effective dark matter-nucleon
couplings.

The effective couplings $f_{N}$ are obtained as a sum of the Wilson
coefficients of dark matter-quark/gluon effective operators multiplied by
their nucleon matrix elements. The operators relevant to our discussions are
[23, 8, 25, 26, 27, 28]

$\displaystyle{\cal L}_{\text{eff}}$
$\displaystyle=\sum_{q}\alpha_{3q}\,\overline{\tilde{\chi}^{0}}\tilde{\chi}^{0}\bar{q}q+\alpha_{G}\,\frac{\alpha_{s}}{\pi}\overline{\tilde{\chi}^{0}}\tilde{\chi}^{0}G^{a}_{\mu\nu}G^{a\mu\nu}$
$\displaystyle+\sum_{q}\frac{\beta_{1q}}{m_{\chi}}\,\overline{\tilde{\chi}^{0}}i\partial^{\mu}\gamma^{\nu}\tilde{\chi}^{0}\,{\cal
O}^{q}_{\mu\nu}+\sum_{q}\frac{\beta_{2q}}{m_{\chi}^{2}}\,\overline{\tilde{\chi}^{0}}i\partial^{\mu}i\partial^{\nu}\tilde{\chi}^{0}\,{\cal
O}^{q}_{\mu\nu}~{},$ (5)

where $\tilde{\chi}^{0}$ denotes the dark matter field, the $q$ are quark
fields, $G^{a}_{\mu\nu}$ is the field strength tensor of the gluon,
$\alpha_{s}\equiv g_{s}^{2}/(4\pi)$ is the strong gauge coupling constant, and
the ${\cal O}^{q}_{\mu\nu}$ are the so-called twist-2 operators of the quark
fields 333One could also consider the interaction described by the gluonic
twist-2 operator. However, as we see below, this interaction is induced at
higher order in $\alpha_{s}$ compared with the other interactions, and thus
can be neglected for the leading order computation. defined by [23, 24]

${\cal
O}^{q}_{\mu\nu}\equiv\frac{1}{2}\bar{q}i\Bigl{(}D_{\mu}\gamma_{\nu}+D_{\nu}\gamma_{\mu}-\frac{1}{2}\eta_{\mu\nu}{\ooalign{\hfil/\hfil\crcr$D$}}\Bigr{)}q~{},$
(6)

with $D_{\mu}$ denoting the covariant derivative.

###  2.2 Tree-level Higgs exchange

Figure 1: The Higgs-boson exchange processes that contribute at leading order
to the spin-independent scattering of wino-like neutralino with nucleons.

The effective four-fermion coupling, $\alpha_{3q}$ in Eq. (5) contains
contributions from both squark and Higgs exchange. However, in the region of
parameter space of interest here, the squark masses and the heavy Higgs scalar
mass, $m_{H}$, are significantly larger than the light Higgs scalar mass,
$m_{h}$. Thus, at leading order, wino-like neutralino dark matter-nucleon
scattering is induced by the light Higgs-boson exchange processes shown in
Fig. 1. By integrating out the Higgs boson, we can readily obtain the
effective quark couplings $\alpha_{3q}$ [8, 25]

$\alpha_{3q}^{(\text{tree})}\simeq\frac{gm_{q}}{4m_{W}m_{h}^{2}}\text{Re}\left[\left(Z_{\chi
3}\cos\beta-Z_{\chi 4}\sin\beta\right)\left(gZ_{\chi 2}-g^{\prime}Z_{\chi
1}\right)\right]~{},$ (7)

where $g$ and $g^{\prime}$ denote the SU(2)L and U(1)Y gauge couplings,
respectively, $m_{W}$ is the $W$-boson mass, the $m_{q}$ are the quark masses,
and $\tan\beta$ is the ratio of the Higgs VEVs defined by
$\tan\beta\equiv\langle H_{u}^{0}\rangle/\langle H_{d}^{0}\rangle$. Note that
in the assumed limit, $m_{H}\gg m_{h}\simeq 125$ GeV, the Higgs mixing angle,
$\alpha$ can be approximated by the decoupling limit $\alpha\to\beta-\pi/2$,
so that $\cos\alpha\to\sin\beta$ and $\sin\alpha\to-\cos\beta$. The $Z_{\chi
i}$ ($i=1,2,3,4$) in Eq. (7) are defined by

$\tilde{\chi}^{0}=Z_{\chi 1}\tilde{B}+Z_{\chi 2}\tilde{W}^{0}+Z_{\chi
3}\tilde{H}_{d}^{0}+Z_{\chi 4}\tilde{H}^{0}_{u}~{},$ (8)

where $\tilde{B}$, $\tilde{W}^{0}$, $\tilde{H}^{0}_{d}$, and
$\tilde{H}^{0}_{u}$, denote the bino, wino, and Higgsino fields, respectively.
Focusing on the region of parameter space in which $\tilde{\chi}^{0}$ is wino-
like and $m_{W}\ll|M_{i}-\mu|$ ($i=1,2$), where $M_{1}$, $M_{2}$ and $\mu$ are
the bino mass, wino mass and Higgsino mixing parameters, respectively, we have

$\displaystyle Z_{\chi 1}\simeq 0,\quad Z_{\chi 2}\simeq 1,$ $\displaystyle
Z_{\chi 3}\simeq\frac{m_{W}}{M_{2}^{2}-\mu^{2}}(M_{2}\cos\beta+\mu\sin\beta),$
$\displaystyle Z_{\chi 4}$ $\displaystyle\simeq$
$\displaystyle-\frac{m_{W}}{M_{2}^{2}-\mu^{2}}(M_{2}\sin\beta+\mu\cos\beta)\,,$
(9)

and we can further approximate Eq. (7) as

$\alpha_{3q}^{(\text{tree})}\simeq\frac{g^{2}m_{q}(M_{2}+\mu\sin
2\beta)}{4m_{h}^{2}(M_{2}^{2}-\mu^{2})}~{}.$ (10)

From this expression, we see that $\alpha_{3q}^{(\text{tree})}$ decreases as
$\propto\mu^{-1}$ when $|\mu|\gg M_{2}$, which is the case for the AMSB so
long as $m_{0}\ll m_{3/2}$. These approximations also hold in the PGM model
for values of $\tan\beta$ that maintain $|\mu|\gg M_{2}$.

The contribution of the diagrams in Fig. 1 to the effective nucleon couplings
is obtained by using the nucleon matrix elements of the quark-antiquark scalar
operators, $\bar{q}q$. Their values are often described by the mass fractions
$f_{T_{q}}^{(N)}$ defined by

$m_{N}f_{T_{q}}^{(N)}\equiv\langle
N|m_{q}\bar{q}q|N\rangle\equiv\sigma_{q}\equiv m_{q}B_{q}^{N}\,,$ (11)

with $m_{N}$ the nucleon mass. For the light quarks, combinations of the
matrix elements can be related to $\sigma$ terms such as the pion-nucleon
$\sigma$ term

$\Sigma_{\pi
N}\;=\;\frac{1}{2}(m_{u}+m_{d})\left(B_{u}^{p}+B_{d}^{p}\right)\,,$ (12)

which may be computed directly with lattice simulations [29] or extracted
phenomenologically from data on low-energy $\pi$-nucleon scattering or on
pionic atoms [30]. Another combination can be extracted phenomenologically
from the octet baryon mass splittings,

$\sigma_{0}\;=\;\frac{1}{2}(m_{u}+m_{d})\left(B_{u}^{p}+B_{d}^{p}-2B_{s}^{p}\right)\,,$
(13)

which gives

$\sigma_{s}=m_{s}B^{p}_{s}=\frac{m_{s}}{m_{u}+m_{d}}(\Sigma_{\pi
N}-\sigma_{0})\,.$ (14)

A third relation is obtained from baryon octet masses

$z\;\equiv\;\frac{B_{u}^{p}-B_{s}^{p}}{B_{d}^{p}-B_{s}^{p}}\;=\;\frac{m_{\Xi^{0}}+m_{\Xi^{-}}-m_{p}-m_{n}}{m_{\Sigma^{+}}+m_{\Sigma^{-}}-m_{p}-m_{n}}\;=\;1.49\,.$
(15)

From a recent compilation in Ref. [31], we use

$\Sigma_{\pi N}\;=\;46\pm 11\;{\rm MeV}\qquad\sigma_{s}\;=\;35\pm 16\;{\rm
MeV}\,,$ (16)

which determines the matrix elements of the three light quarks and we list
them in the first three columns of Table 1 for the reader’s convenience.

Table 1: Values of the mass fractions $f^{(N)}_{T_{q,G}}$ obtained in Ref. [31]. | $f^{(N)}_{T_{u}}$ | $f^{(N)}_{T_{d}}$ | $f^{(N)}_{T_{s}}$ | $f^{(N)}_{T_{G}}$ | $f^{(N)}_{T_{c}}$ | $f^{(N)}_{T_{b}}$ | $f^{(N)}_{T_{t}}$
---|---|---|---|---|---|---|---
Proton | 0.018(5) | 0.027(7) | 0.037(17) | 0.917(19) | 0.078(2) | 0.072(2) | 0.069(1)
Neutron | 0.013(3) | 0.040(10) | 0.037(17) | 0.910(20) | 0.078(2) | 0.071(2) | 0.068(2)

The contributions of the heavy quarks to the scattering cross section are
often calculated by integrating them out and replacing them by the one-loop
gluon contributions so that $f_{T_{Q}}^{(N)}=\frac{2}{27}f_{T_{G}}^{(N)}$ for
$Q=c,b,t$, where the nucleon matrix element of the gluon scalar operator is
evaluated using the trace anomaly formula given in [32]:

$\langle
N|\frac{\alpha_{s}}{\pi}G^{a}_{\mu\nu}G^{a\mu\nu}|N\rangle=-\frac{8}{9}m_{N}f_{T_{G}}^{(N)}~{},$
(17)

which holds at the leading order in $\alpha_{s}$, and we note in addition that
$f_{T_{G}}^{(N)}\equiv 1-\sum_{q=u,d,s}f_{T_{q}}^{(N)}$.

However, momenta around the mass scale of the quark running in the loop make
the most important contributions to the integral in the loop diagram shown in
Fig. 1. These contributions are often referred to as long-distance
contributions as the relevant energy scales are much lower than the
electroweak scale in the cases of the charm and bottom quarks. Since
$\alpha_{s}(m_{Q})$ is rather large at the scales of these masses, higher-
order QCD corrections are significant. We take these corrections into account
to ${\cal O}(\alpha_{s}^{3})$ in perturbative QCD, following Ref. [33],
finding

$\displaystyle\sigma_{c}$ $\displaystyle=$
$\displaystyle\frac{2}{27}\left(-0.3+1.48f^{(N)}_{T_{G}}\right)M_{N}\;=\;73.4\pm
1.9\;{\rm MeV}\,,$ (18) $\displaystyle\sigma_{b}$ $\displaystyle=$
$\displaystyle\frac{2}{27}\left(-0.16+1.23f^{(N)}_{T_{G}}\right)M_{N}\;=\;67.3\pm
1.6\;{\rm MeV}\,,$ (19) $\displaystyle\sigma_{t}$ $\displaystyle=$
$\displaystyle\frac{2}{27}\left(-0.05+1.07f^{(N)}_{T_{G}}\right)M_{N}\;=\;64.7\pm
1.4\;{\rm MeV}\,.$ (20)

Using (11), the corrected contributions for all quarks can then be expressed
in terms of the effective mass fractions $f^{(N)}_{T_{Q}}$, whose values we
show in the last three columns of Table 1. The resulting effective coupling is
given by

$f_{N}^{(\text{tree})}=m_{N}\sum_{q=u,d,s,c,b,t}\frac{\alpha_{3q}^{(\text{tree})}}{m_{q}}f_{T_{q}}^{(N)}~{}.$
(21)

This treatment increases the resulting value of $f_{N}$ by ${\cal O}(10)$%
[31] relative to computing $f_{N}$ using a common $f_{T_{Q}}^{(N)}$ for the
heavy quarks. We note that even though the contributions from heavy quarks are
induced by QCD loop diagrams, the resultant contributions are similar in
magnitude to those from the light quarks [23, 26, 27]. This is because of the
large contribution of the gluons to the mass of nucleon: $f_{T_{G}}^{(N)}\gg
f_{T_{q}}^{(N)}$.

###  2.3 Electroweak loop contributions

As seen in Eq. (10), the tree-level Higgs exchange contribution is suppressed
when the Higgsino mass is very large. In this case, the contributions from the
loop processes shown in Figs. 2 and 3 may dominate over the tree-level
contribution [34]. These loop contributions have been computed in the
literature [26, 27, 35, 36, 37], and we include them in our analysis with the
following approximations:

  * •

We use the results obtained for a pure wino, though the wino-like neutralino
LSP in our case also contains admixtures of the bino and Higgsinos. This
approximation is valid since the loop corrections can be significant only when
the Higgsino mass is quite large so that the tree-level bino contribution
becomes small—in this case, $|Z_{\chi 1}|,|Z_{\chi 3}|,|Z_{\chi 4}|\ll 1$ as
can be seen from Eq. (9), i.e., the LSP is almost pure wino.

  * •

Although the electroweak loop contributions have been computed at NLO in QCD
[37], we use the LO result, since the difference between the two
approximations can be neglected for our purposes.

In the rest of this subsection, we summarize our results for the electroweak
loop contributions.

The left diagram in Fig. 2 gives rise to scalar light-quark operators, with
coefficients given by

$\alpha_{3q}^{(\text{loop})}=\frac{\alpha_{2}^{2}m_{q}}{4m_{W}m_{h}^{2}}g_{\rm
H}(\omega)~{},$ (22)

where $\alpha_{2}\equiv g^{2}/(4\pi)$ is the SU(2)L coupling strength and
$\omega\equiv m_{W}^{2}/m_{\chi}^{2}$. The mass function $g_{\text{H}}(x)$ is
given in the Appendix. In addition, heavy quarks provide scalar gluon operator
contributions via the two-loop diagrams in Fig. 3. As in the case of the tree-
level Higgs exchange processes, these two-loop contributions can be comparable
to the one-loop contribution to the scalar quark operators, even though they
are induced at a higher order in ${\cal O}(\alpha_{s})$.

Figure 2: One-loop contributions to wino-nucleon scattering.  Figure 3: Two-
loop contributions to wino-nucleon scattering.

We again take account of the long-distance QCD corrections to the left diagram
in Fig. 3 by using the $f^{(N)}_{T_{Q}}$ $(Q=c,b,t)$ values given in Table 1.
For each heavy quark we have

$\alpha_{3Q}^{(\text{loop})}=\frac{\alpha_{2}^{2}m_{Q}}{4m_{W}m_{h}^{2}}g_{\rm
H}(\omega)~{},$ (23)

which is the same as in Eq. (22).

The contribution from the right diagram in Fig. 3 is included in the
coefficient $\alpha_{G}$ of the scalar gluon operator:

$\alpha_{G}=2\times\frac{\alpha_{2}^{2}}{4m_{W}^{3}}g_{\rm
B1}(\omega)+\frac{\alpha_{2}^{2}}{4m_{W}^{3}}g_{\rm B3}(\omega,\tau)\ ,$ (24)

where $\tau\equiv m_{t}^{2}/m_{\chi}^{2}$ and the mass functions
$g_{\text{B1}}$ and $g_{\text{B3}}$ are given in the Appendix. The first
(second) term in the right-hand side of the above expression corresponds to
the first- and second-generation (third-generation) contribution. We use the
LO formula (17) to evaluate the nucleon matrix element of the gluon scalar
operator. A more systematic treatment for the inclusion of the higher-order
QCD effects as well as the separation between the long- and short-distance
contributions using the matching procedure is discussed in Ref. [37].

The right diagram in Fig. 2 induces interactions described by the twist-2
operators in Eq. (5). The Wilson coefficients of these operators are found to
be

$\displaystyle\beta_{1q}$
$\displaystyle=\frac{\alpha_{2}^{2}}{m_{W}^{3}}g_{\rm T1}(\omega)\ ,$ (25)
$\displaystyle\beta_{2q}$
$\displaystyle=\frac{\alpha_{2}^{2}}{m_{W}^{3}}g_{\rm T2}(\omega)\ ,$ (26)

for $q=u,d,s,c$ and

$\displaystyle\beta_{1b}$
$\displaystyle=\frac{\alpha_{2}^{2}}{m_{W}^{3}}h_{\rm T1}(\omega,\tau)\ ,$
(27) $\displaystyle\beta_{2b}$
$\displaystyle=\frac{\alpha_{2}^{2}}{m_{W}^{3}}h_{\rm T2}(\omega,\tau)\ ,$
(28)

for the $b$ quark. The nucleon matrix elements of the twist-2 operators are
given by the second moments of the parton distribution functions (PDFs) [23,
24]:

$\displaystyle\langle N(p)|{\cal O}_{\mu\nu}^{q}|N(p)\rangle$
$\displaystyle=m_{N}\left(\frac{p_{\mu}p_{\nu}}{m_{N}^{2}}-\frac{1}{4}\eta_{\mu\nu}\right)\left[q_{N}(2)+\bar{q}_{N}(2)\right]~{},$
(29)

with

$\displaystyle q_{N}(2)$ $\displaystyle=\int^{1}_{0}dx~{}x\ q_{N}(x)~{},$ (30)
$\displaystyle\bar{q}_{N}(2)$ $\displaystyle=\int^{1}_{0}dx~{}x\
\bar{q}_{N}(x)~{},$ (31)

where $q_{N}(x)$ and $\bar{q}_{N}(x)$ are the PDFs of the quark and antiquark,
respectively. We present in Table 2 the values of the second moments at the
scale $\mu=m_{Z}$ for the proton, where we have used the CJ12 NLO PDFs given
by the CTEQ-Jefferson Lab collaboration [38]. As mentioned above, there is
also a gluon twist-2 contribution, but this can be neglected, as it is higher
order in $\alpha_{s}/\pi$, and the nucleon matrix element of the gluon twist-2
operator, $g(2)$, is not so much larger than the light-quark operators, $u(2)$
and $d(2)$, in contrast to the cases for the scalar operators. Thus the gluon
twist-2 contribution is always suppressed by an extra $\alpha_{s}/\pi$ factor
compared to the light-quark twist-2 contributions.

Table 2: Second moments of the PDFs of partons in the proton evaluated at $\mu=m_{Z}$. We use the CJ12 next-to-leading order PDFs given by the CTEQ-Jefferson Lab collaboration [38].  $g(2)$ | 0.464(2) |  |
---|---|---|---
$u(2)$ | 0.223(3) | $\bar{u}(2)$ | 0.036(2)
$d(2)$ | 0.118(3) | $\bar{d}(2)$ | 0.037(3)
$s(2)$ | 0.0258(4) | $\bar{s}(2)$ | 0.0258(4)
$c(2)$ | 0.0187(2) | $\bar{c}(2)$ | 0.0187(2)
$b(2)$ | 0.0117(1) | $\bar{b}(2)$ | 0.0117(1)

###  2.4 Summary

Combining the results above, the effective coupling $f_{N}$ is evaluated as

$\displaystyle\frac{f_{N}}{m_{N}}$
$\displaystyle=\frac{f_{N}^{(\text{tree})}}{m_{N}}+\sum_{q=u,d,s,c,b,t}f_{T_{q}}^{(N)}\frac{\alpha_{3q}^{(\text{loop})}}{m_{q}}-\frac{8}{9}f_{T_{G}}^{(N)}\alpha_{G}+\sum_{q=u,d,s,c,b}\frac{3}{4}(q_{N}(2)+\bar{q}_{N}(2))(\beta_{1q}+\beta_{2q})~{},$
(32)

where $f_{N}^{(\text{tree})}$ is given in Eq. (21), the $f_{T_{q}}^{(N)}$ are
given in Table 1, $\alpha_{3q}^{(\text{loop})}$ is given in Eq. (22),
$\alpha_{G}$ is given in Eq. (24), $\beta_{1q}$ and $\beta_{2q}$ are shown in
Eqs. (26) and (28), respectively, the second moments $q_{N}(2)$ and
$\bar{q}_{N}(2)$ are given in Table 2, and the mass functions in the
coefficients are summarized in the Appendix.

##  3 Calculation of the Relic Density and Higgs Mass

###  3.1 Relic Wino LSP Density

As discussed earlier, the mAMSB model is characterized by three continuous
parameters, $m_{3/2},m_{0}$ and $\tan\beta$, and the sign of $\mu$. Examples
of $(m_{0},m_{3/2})$ planes with $\tan\beta=5$ and 2 and both signs of $\mu$
are shown in Fig. 4, which updates a similar plot given in [4]. In the pink
shaded region to the right of each panel, there are no solutions for the
minimization of the Higgs potential, and therefore radiative electroweak
symmetry breaking is not possible in this region. The red lines are contours
of $m_{h}$ in GeV. (Our calculation of the Higgs mass is discussed in more
detail below.) In the largely horizontal blue-shaded region the LSP relic
density is $\Omega_{\chi}h^{2}=0.12\pm 0.01$.

Figure 4: The $(m_{0},m_{3/2})$ plane in the mAMSB/PGM models for
$\tan\beta=5$ (upper panels) and $\tan\beta=2$ (lower panels) for $\mu>0$
(left panels) and $\mu<0$ (right panels). Electroweak symmetry breaking does
not occur in the region shaded pink. Contours of $m_{h}$ are shown as red
lines labeled in units of GeV. The LSP relic density is
$\Omega_{\chi}h^{2}=0.12\pm 0.01$ in the region shaded blue.

The wino mass increases monotonically with $m_{3/2}$ as seen in Eq. (2). For
$\tan\beta=5$ and $m_{3/2}\approx 0.9-1$ PeV, the wino mass is roughly 3 TeV
and, as seen in the upper panels of Fig. 4, it is able to provide the correct
relic density when the Sommerfeld enhancements are included. For relatively
low values of $\tan\beta$ such as those chosen in Fig. 4, the Higgs mass is
quite sensitive to parameter choices and increases with $\tan\beta$, and we
find that for $\tan\beta\simeq 5$ the relic density is satisfied for an
acceptable value of the Higgs mass.

We note that, as $m_{0}$ increases, the value of the $\mu$ eventually starts
to decrease so that the LSP becomes more Higgsino-like close to the region
with no electroweak symmetry breaking. When $\mu\sim 1$ TeV and $m_{3/2}$ is
sufficiently large the LSP is almost a pure Higgsino and the relic density is
acceptable. This region is visible as the diagonal blue strip running close to
the boundary of the pink-shaded region in the upper panels of Fig. 4. However,
we do not consider this strip, as our approximations for the elastic
scattering cross section only apply in the wino-like case represented by the
horizontal blue-shaded band 444Electroweak corrections for the Higgsino-like
LSP are found to be negligibly small [35, 36, 37, 39]. .

As one can see in the upper panels of Fig. 4, for $\tan\beta=5$ the
requirements of electroweak symmetry breaking and the cold dark matter density
enforce $m_{0}\ll m_{3/2}$. PGM boundary conditions with $m_{0}\simeq m_{3/2}$
are only possible at lower values of $\tan\beta\simeq 2$ [6]. In the lower two
panels of Fig. 4, we show the $(m_{0},m_{3/2})$ plane with $\tan\beta=2$. In
the horizontal blue-shaded region, the LSP is again wino-like with the
required relic density. The Higgs mass is generally too low, except for the
largest values of $m_{0}$ allowed by radiative electroweak symmetry breaking.
We note that higher values of $m_{h}$ can be attained for slightly higher
values of $\tan\beta$. For example, $m_{h}=125$ GeV is possible for
$\tan\beta=2.05$.

The neutralino mass spectra for $\tan\beta=2$ and 5 are shown in Fig. 5 for
both signs of $\mu$. Along each of the curves, the value of $m_{3/2}$ is
chosen so that the relic density is $\Omega_{\chi}h^{2}=0.12$. For
$\tan\beta=5$, the range of gravitino masses is 0.87–0.95 PeV for $\mu>0$ and
0.86–0.94 PeV for $\mu<0$. Similarly, the range for the gravitino mass when
$\tan\beta=2$ is 1.0–1.3 PeV for $\mu>0$ and 0.49–0.70 PeV for $\mu<0$. In all
of the cases shown, we see that both the bino and wino masses are relatively
independent of $m_{0}$ when $m_{0}\ll m_{3/2}$. On the other hand, the
Higgsino mass is quite sensitive to $m_{0}$, as it is essentially determined
by $\mu$, which is in turn fixed by the electroweak symmetry breaking
conditions. At very large $m_{0}$, $|\mu|$ begins to drop, and at sufficiently
large $m_{0}$ the focus-point region [13] is reached and the Higgsino becomes
the LSP.

Figure 5: Calculations of the neutralino and Higgs masses as functions of
$m_{0}$ for $\tan\beta=5$ (upper panels) and $\tan\beta=2$ (lower panels) and
$\mu>0$ (left panels) and $\mu<0$ (right panels). The (Higgsino-) (bino-)
wino-like mass contours are (blue) (red) black. For each value of $m_{0}$, the
value of $m_{3/2}$ is chosen so that the relic LSP density is
$\Omega_{\chi}h^{2}=0.12$. Also shown is the Higgs mass given by the scale on
the right.

It is instructive to consider the dependences of the neutralino masses on
$\tan\beta$ as shown in the left panels of Fig. 6 for the cases $m_{3/2}=950$
TeV, $m_{0}=15$ TeV and both signs of $\mu$. We see that the lightest
neutralino is always wino-like with a mass around 3 TeV, and the bino-like
neutralino has a mass $\sim 9$ TeV. The second lightest neutralino is bino-
like for $\tan\beta\lesssim 5$ and Higgsino-like for larger $\tan\beta$. Also
shown in the left panels of Fig. 6 is the Higgs mass plotted as a function of
$\tan\beta$. Here we see clearly the strong dependence of $m_{h}$ on
$\tan\beta$ for $\tan\beta\lesssim 10$. All of the masses are relatively
independent of $\tan\beta$ when $\tan\beta\gtrsim 10$.

Figure 6: Neutralino and Higgs masses (left panels) and the cold dark matter
density (right panels) as functions of $\tan\beta$ for $m_{3/2}=950$ TeV,
$m_{0}=15$ TeV with $\mu>0$ (upper panels) and $\mu<0$ (lower panels). The
lightest neutralino is always wino-like (black line). The bino-like neutralino
mass is shown as a green line and the Higgsino-like neutralino masses as a
blue line. (The two Higgsinos are nearly degenerate.) The red dashed line
shows the Higgs mass.

For the same choices of parameters, we show the relic LSP density as a
function of $\tan\beta$ in the right panels of Fig. 6. While it might appear
that there is a strong dependence on $\tan\beta$, this is largely due to the
choice of scale. The relic density is acceptable (particularly when
calculational uncertainties are taken into account) throughout the range of
$\tan\beta$ shown.

In the limit that $m_{0}=m_{3/2}$ as in PGM models, only a relatively narrow
range of $\tan\beta$ is allowed, as seen in Figs. 7 where the neutralino and
Higgs masses and relic density are plotted as functions of $\tan\beta$
assuming $m_{0}=m_{3/2}=1.15$ PeV for $\mu>0$ (upper panels) and
$m_{0}=m_{3/2}=0.7$ PeV for $\mu<0$ (lower panels). The mass spectra plotted
in the left panels show again a wino-like LSP with mass around 3 TeV. Because
the $\mu$ parameter typically takes values of order $m_{0}$, the Higgsino
masses are very large in this case until $\tan\beta$ is sufficiently large
that the focus-point region is approached. At this point, the Higgs mass is
increased toward its experimental value. At still larger $\tan\beta$ radiative
electroweak symmetry breaking is no longer possible.

Figure 7: Neutralino and Higgs masses (left panels) and the relic LSP density
(right panels) as functions of $\tan\beta$ for the PGM model when
$m_{3/2}=m_{0}=1.15$ PeV with $\mu>0$ (upper panels) and $\mu<0$ (lower
panels).

The relic LSP density for the same sets of parameters is shown in right panels
of Fig. 7. Depending on the sign of $\mu$, the relic LSP density (which is
always close to the observationally determined values) increases ($\mu>0$) or
decreases ($\mu<0$), tracking the behavior of the wino-like LSP mass.

###  3.2 Calculation of the Higgs Boson Mass

Reliable calculations of the light Higgs mass are challenging when the
supersymmetry-breaking scale is ${\cal O}({\rm PeV})$, as is the case for the
AMSB and PGM models considered here. Accordingly, we have compared results
from two codes: FeynHiggs 2.18.1 [40] and mhsplit, a simplified code devised
specifically for large supersymmetry-breaking scales. Following [41, 42], the
2-loop RGE evolution of the couplings of the effective theory below the
supersymmetry breaking scale are used to obtain the Higgs mass at full next-
to-leading order accuracy as described in more detail in [43] for a model with
an AMSB-like spectrum. The same code was also adapted to PGM models [6]. A
comparison of the results of the two codes is shown in Fig. 8, where we plot
the Higgs mass as a function of $m_{0}$ given by each code for $m_{3/2}=950$
TeV and $\tan\beta=2$ with $\mu>0$ (left) and $\mu<0$ (right). The Higgs mass
from mhsplit is shown by the (smooth) solid curve, and that from FeynHiggs
2.18.1 by the dashed curve. For $m_{0}=0.1-0.6$ PeV, both codes are in good
agreement and appear well-behaved. At lower values of $m_{0}$, the
approximations used in mhsplit break down. At $m_{0}\gtrsim 0.6$ PeV,
FeynHiggs 2.18.1 starts to exhibit irregularities that are absent in the
simplified calculation.

(a) $\mu>0$ (b) $\mu<0$

Figure 8: Comparisons of the calculations of $m_{h}$ made using FeynHiggs
2.18.1 (dashed lines) and mhsplit, a simplified code devised specifically for
large supersymmetry-breaking scales (solid lines). We see that the simplified
calculation of $m_{h}$ is very stable at large values of $m_{0}$ for
$m_{3/2}=950$ TeV, $\tan\beta=2$ and both signs of $\mu$, and agrees very well
with the FeynHiggs results for values of $m_{0}\simeq 0.1-0.6$ PeV.

Motivated by the above comparison, the Higgs mass contours for the mAMSB
models shown in the upper panels of Fig. 4 are run using FeynHiggs 2.18.1,
whereas those in the lower panels for the PGM-like models with significantly
higher supersymmetry-breaking mass parameters are run using mhsplit. The Higgs
mass as a function of $m_{0}$ for fixed $\tan\beta$ and $m_{3/2}$ fixed to
yield $\Omega_{\chi}h^{2}=0.12$ is shown in Fig. 5. The upper panels with
$\tan\beta=5$ are calculated using FeynHiggs 2.18.1 while the lower panels
with $\tan\beta=2$ are calculated using mhsplit. As one can see, the scalar
mass range there is highly sensitive to $\tan\beta$. It is relatively easy in
the mAMSB models with $\tan\beta=5$ to obtain acceptable Higgs masses with
$m_{0}\sim\mathcal{O}(10)$ TeV, whereas $m_{0}\gtrsim 1$ PeV is required in
the PGM model with $\tan\beta=2$, and an acceptable Higgs mass is only
possible when the Higgsino mass drops precipitously as one approaches the
focus-point region. The dependence of the Higgs mass on $\tan\beta$ is shown
in Fig. 6 for the mAMSB models and in the left panels of Fig. 7 for the PGM
models.

##  4 Results for the Spin-Independent Wino Scattering Cross Section

We now present our results for the spin-independent scattering cross section
of a wino-like neutralino, $\sigma_{p}$, obtained using the results presented
in Section 2. Explicit expressions for the mass functions introduced there are
given in the Appendix. We also provide in the Appendix simple approximate
formulae that apply when the wino-like neutralino mass $m_{\chi}$ is much
larger than the electroweak scale, which is the case for the parameter regions
of interest in this paper.

We present in Fig. 9 a comparison of calculations of $\sigma_{p}$ made using
tree-level matrix elements (solid black lines) and one-loop calculations
(solid blue lines). The spin-independent cross section is plotted as a
function of $m_{0}$ for the AMSB model with $\tan\beta=5$ (upper panels) for
both $\mu>0$ (left panels) and $\mu<0$ (right panels). In the lower panels we
show the cross section in the PGM-like model with $\tan\beta=2$. The orange-
shaded region in each plot corresponds to cross sections in the neutrino fog
[17]. For comparison, we note that for $m_{\chi}\simeq 3$ TeV the current best
experimental limit, from the LUX-ZEPLIN (LZ) experiment [44], is
$\sigma_{p}\lesssim 10^{-9}$ pb.

Figure 9: Comparison of calculations of $\sigma_{p}$ as a function of $m_{0}$
made using tree-level matrix elements (black lines) and one-loop calculations
made using the full expressions in the Appendix (solid blue lines) and the
simplified expressions in Eq. (42) that become exact when the wino neutralino
mass $m_{\chi}\gg m_{W}$ (dashed blue lines). The calculations are for the
AMSB model with $m_{3/2}=950$ TeV, $m_{0}=15$ TeV and both signs of $\mu$
(upper panels) and the PGM-like model with $m_{3/2}=m_{0}=1.15$ PeV for
$\mu>0$ (lower left panel) and with $m_{3/2}=m_{0}=0.7$ PeV for $\mu<0$ (lower
left panel). The orange-shaded region corresponds to cross section values in
the neutrino fog.

In the mAMSB model with $\tan\beta=5$ and $\mu>0$, we have the discouraging
result that although the tree level result is above the neutrino fog (and
hence in principle observable, though it is still two orders of magnitude
below the current LZ limit), the one-loop correction induces a cancellation
and the cross section drops into the neutrino fog unless $m_{0}\gtrsim 23$
TeV. Indeed, the cancellation is near-total when $m_{0}=16$ TeV. This
cancellation can be understood by looking at the tree and loop contributions
to the cross section separately. From Eqs. (10) and (21), we see that
$\alpha_{3q}/m_{q}$ is independent of $m_{q}$ for a wino LSP in mAMSB, and we
can write

$f_{p}^{(\rm tree)}\simeq 2\times 10^{-6}{\rm
GeV}^{-1}\times\frac{(M_{2}+\mu\sin 2\beta)}{(M_{2}^{2}-\mu^{2})}\,,$ (33)

when taking $m_{h}=125$ GeV. When the relic density is fixed, requiring the
wino mass, $M_{2}\simeq 3$ TeV, and noting that $\mu$ (nearly equal to the
Higgsino mass shown in Fig. 5) varies with $m_{0}$, we see that the tree-level
cross section varies little at low $m_{0}$ (as does $\mu$) and increases as
$\mu$ begins to drop. This is the behaviour seen in the upper left panel of
Fig. 9. When $M_{2}\simeq 3$ TeV for wino DM and $m_{h}$ is restricted to the
observed value, the loop correction is roughly constant and takes the value

$f_{p}^{(\rm loop)}\simeq 1.6\times 10^{-10}\,{\rm GeV}^{-2}\,.$ (34)

However, when $\mu>0$ the tree level amplitude is negative and for a certain
value of $\mu$ ($m_{0}$) will cancel the one-loop contribution as seen in Fig.
9. It is easy to see that a complete cancellation, $f_{p}^{(\rm
tree)}+f_{p}^{(\rm loop)}=0$, occurs when

$\sin 2\beta\simeq\frac{\mu}{12{\rm TeV}}-\frac{3.7{\rm TeV}}{\mu}\,.$ (35)

This occurs when $m_{0}\sim 17$ TeV, when $\tan\beta=5$ and $\mu\sim 9$ TeV as
seen in Fig. 5. However, we note that the amplitude will not vanish if
$|\mu|\gtrsim 15$ TeV. This same cancellation can be seen in the upper panels
of Fig. 10, where we show the cross section as a function of $\tan\beta$ for
$m_{3/2}=950$ TeV and $m_{0}=15$ TeV. 555We note in passing that the results
obtained using the approximate expressions for the one-loop mass functions
given in Eq. (42) (dashed blue lines) are very similar to the exact results,
as was to be expected since $m_{\chi}\gg m_{W}$.

Figure 10: Comparison of calculations of $\sigma_{p}$ as a function of
$\tan\beta$ made using tree-level matrix elements (black lines) and one-loop
calculations made using the full expressions in the Appendix (solid blue
lines) and the simplified expressions in Eq. (42) that become exact when the
wino neutralino mass $m_{\chi}\gg m_{W}$ (dashed blue lines). The calculations
are for the AMSB model with $m_{3/2}=950$ TeV, $m_{0}=15$ TeV and both signs
of $\mu$ (upper panels) and the PGM-like models with $m_{3/2}=m_{0}=1.15$ PeV
for $\mu>0$ (lower left panel) and with $m_{3/2}=m_{0}=0.7$ PeV for $\mu<0$
(lower right panel). The orange-shaded region corresponds to cross sections in
the neutrino fog. In all four panels the value of $m_{3/2}$ is chosen to give
$\Omega h^{2}\simeq 0.12$.

On the other hand, the tree-level scattering amplitude vanishes when
$M_{2}+\mu\sin 2\beta=0$. For $\tan\beta=5$, this occurs when $\mu\sim-8$ TeV,
corresponding to $m_{0}\sim 21$ TeV. However, when $\mu<0$ the one-loop
correction can enhances the total cross section, lifting it out of the
neutrino fog and into the domain of possible experimental observation (though
still unobservable at present). The large loop correction can significantly
shift this point of vanishing amplitude. In the case of the upper right panel
of Fig. 9, the cancellation would occur at $m_{0}>25$ TeV, past the point
where radiative electroweak symmetry breaking is possible.

In the PGM-like model with $\tan\beta=2$ ($\sin 2\beta=0.8$), $\mu\gg M_{2}$,
and the tree-level amplitude is approximately proportional to $1/\mu$. So long
as $\mu$ is very large, the one-loop correction dominates the cross section
substantially, lifting it out the neutrino fog unless $m_{0}\gtrsim 1.2$ PeV
when $\mu>0$, as seen in the lower panels of Figs. 9 and 10.

Before concluding, we note that the tree-level cross section in the mAMSB
model was explored in a frequentist analysis using constraints from cosmology
and accelerator experiments in [3]. The best-fit cross section for wino dark
matter was found to be slightly above (below) the neutrino fog for $\mu>0$
($\mu<0$), and values for $\mu<0$ extended far into the neutrino fog. It was
also found that values of the cross section extend to much higher values. This
occurs when $m_{0}$ takes large values and approaches the region with no
electroweak symmetry breaking. This behaviour is seen, for example, in the
upper left panel of Fig. 9 where the cross section increases by over an order
of magnitude as $m_{0}$ is increased.

##  5 Conclusion and Discussion

The results in Figs. 9 and 10 illustrate clearly the potential importance of
one-loop electroweak corrections to the spin-independent scattering cross
section for a wino-like LSP. In the case of the AMSB model, the results in the
upper panels of these two figures show sharp difference between the two signs
of $\mu$. In the case of positive $\mu$ (upper left panels), the one-loop
correction interferes negatively with the tree-level contribution and has
similar magnitude, potentially causing a total cancellation. On the other
hand, in the case of negative $\mu$ (upper right panels), the one-loop
correction has the same sign as the tree-level contribution for $m_{0}\lesssim
20$ TeV and $\tan\beta\lesssim 5$, enhancing the spin-independent scattering
cross section. Furthermore, in the case of the PGM-like model shown in the
lower panels of Figs. 9 and 10, the one-loop corrections enhance the cross
section for both signs of $\mu$.

The consequences of these effects is to invert the conclusions about
experimental observability that would be drawn from considering naively the
tree-level cross section alone. In the case of the AMSB model shown in the
upper panels of Figs. 9 and 10, whereas the tree-level calculation gives an
observable cross section for $\mu>0$ and a cross section lost in the neutrino
fog for $\mu<0$, the one-loop correction pushes the cross section down into
the fog for most values of $m_{0}$ and $\tan\beta$ when $\mu>0$ and lifts it
out of the fog when $\mu<0$. On the other hand, in the PGM-like model shown in
the lower panels of Figs. 9 and 10, the effect of the one-loop electroweak
correction is generally positive, lifting the spin-independent cross section
out of the fog for most of the displayed ranges of $m_{0}$ and $\tan\beta$.

The one-loop electroweak corrections must therefore be taken into account when
assessing the implications of direct searches for the scattering of cold dark
matter on nuclear targets for the viability of AMSB and PGM-like models. 666As
discussed in the Appendix, the full form of the one-loop correction is
relatively complicated. However, the large mass of the wino dark matter
implies that it has a simple analytic approximation, see Eq. (42). The good
news is that the cross section is likely to be above the neutrino fog for the
AMSB model if $\mu<0$ and for PGM-like models with either sign of $\mu$. On
the other hand, even in these encouraging cases the cross section is not much
higher than the neutrino fog, so a strenuous experimental effort [45] will be
necessary to explore these models. However, the results of this paper indicate
that this effort would have a good chance of being rewarded if it is able to
reach down to the level of the neutrino fog.

## Acknowledgments

The work of J.E. was supported by the United Kingdom STFC Grant ST/T000759/1.
The work of N.N. was supported by Grants-in-Aid for Scientific Research B (No.
20H01897) and Young Scientists (No.21K13916). K.A.O. was supported in part by
DOE grant DE-SC0011842 at the University of Minnesota. J.Z. was supported in
part by the NSF of China grants 11675086 and 11835005.

## Appendix

We list here explicit expressions for the mass functions introduced above:

$\displaystyle g_{\rm H}(x)$ $\displaystyle=2\sqrt{x}(2-x\ln
x)-\frac{2}{b_{x}}(2+2x-x^{2})\tan^{-1}\biggl{(}\frac{2b_{x}}{\sqrt{x}}\biggr{)}~{},$
(36) $\displaystyle g_{\rm B1}(x)$ $\displaystyle=\frac{1}{24}\sqrt{x}(2-x\ln
x)+\frac{1}{24b_{x}}(4-2x+x^{2}){\rm
tan}^{-1}\biggl{(}\frac{2b_{x}}{\sqrt{x}}\biggr{)}~{},$ (37) $\displaystyle
g_{\rm B3}(x,y)$
$\displaystyle=-\frac{x^{\frac{3}{2}}(2y-x)}{12(y-x)^{2}}-\frac{x^{\frac{3}{2}}y^{3}\ln
y}{24(y-x)^{3}}+\frac{x^{\frac{5}{2}}(3y^{2}-3xy+x^{2})\ln x}{24(y-x)^{3}}$
$\displaystyle+\frac{x^{\frac{3}{2}}\sqrt{y}(y^{3}-2y^{2}-14y+6x)\tan^{-1}\left(\frac{2b_{y}}{\sqrt{y}}\right)}{24b_{y}(y-x)^{3}}$
$\displaystyle-\frac{x\left(x^{4}-3yx^{3}-2x^{3}+3y^{2}x^{2}+6yx^{2}+4x^{2}-6y^{2}x-6yx-6y^{2}\right)\tan^{-1}\left(\frac{2b_{x}}{\sqrt{x}}\right)}{24b_{x}(y-x)^{3}}\
,$ $\displaystyle g_{\rm T_{1}}(x)$
$\displaystyle=\frac{1}{12}\sqrt{x}\\{1-2x-x(2-x)\ln
x\\}+\frac{1}{3}b_{x}(2+x^{2})\tan^{-1}\biggl{(}\frac{2b_{x}}{\sqrt{x}}\biggr{)}~{},$
(38) $\displaystyle g_{\rm T_{2}}(x)$
$\displaystyle=-\frac{1}{4}\sqrt{x}\\{1-2x-x(2-x)\ln
x\\}+\frac{1}{4b_{x}}x(2-4x+x^{2})\tan^{-1}\biggl{(}\frac{2b_{x}}{\sqrt{x}}\biggr{)}~{},$
(39) $\displaystyle h_{{\rm T_{1}}}(x,y)$
$\displaystyle=\frac{x^{\frac{3}{2}}\\{x(1-2x)+y(13+2x)-2y^{2}\\}}{12(x-y)^{2}}$
$\displaystyle-\frac{x^{\frac{3}{2}}\\{x^{3}(2-x)+2xy(3-3x+x^{2})+6y^{2}(2-x)\\}}{12(x-y)^{3}}\ln
x$
$\displaystyle+\frac{x^{\frac{3}{2}}y\\{2x(3-6y+y^{2})+y(12+2y-y^{2})\\}}{12(x-y)^{3}}\ln
y$
$\displaystyle+\frac{x\\{4x^{2}b_{x}^{2}(2+x^{2})-2xy(6-7x+5x^{2}-x^{3})-6y^{2}(2-4x+x^{2})\\}}{12b_{x}(x-y)^{3}}\tan^{-1}\left(\frac{2b_{x}}{\sqrt{x}}\right)$
$\displaystyle-\frac{x^{\frac{3}{2}}y^{\frac{1}{2}}\\{2x(3-y)(2+5y-y^{2})-y(2-y)(14+2y-y^{2})\\}}{12b_{y}(x-y)^{3}}\tan^{-1}\left(\frac{2b_{y}}{\sqrt{y}}\right),$
(40) $\displaystyle h_{{\rm T_{2}}}(x,y)$
$\displaystyle=\frac{x^{\frac{3}{2}}\\{x(-1+2x)-(1+2x)y+2y^{2}\\}}{4(x-y)^{2}}$
$\displaystyle+\frac{x^{\frac{5}{2}}\\{(2-x)x^{2}+2y(1-3x+x^{2})\\}}{4(x-y)^{3}}\ln
x$
$\displaystyle+\frac{x^{\frac{3}{2}}y\\{y^{2}(y-2)-2x(1-3y+y^{2})\\}}{4(x-y)^{3}}\ln
y$
$\displaystyle+\frac{x^{3}\\{x(2-4x+x^{2})-2y(5-5x+x^{2})\\}}{4b_{x}(x-y)^{3}}\tan^{-1}\left(\frac{2b_{x}}{\sqrt{x}}\right)$
$\displaystyle+\frac{x^{\frac{3}{2}}y^{\frac{3}{2}}(2x(5-5y+y^{2})-y(2-4y+y^{2}))}{4b_{y}(x-y)^{3}}\tan^{-1}\left(\frac{2b_{y}}{\sqrt{y}}\right)~{},$
(41)

where we have defined $b_{x}\equiv\sqrt{1-x/4}$.

When the mass of the wino-like neutralino $m_{\chi}$ is much larger than the
electroweak scale; i.e., $\omega,\tau\to 0$, the above mass functions can be
approximated as

$\displaystyle g_{\rm H}(\omega)$ $\displaystyle\simeq-2\pi\ ,$ $\displaystyle
g_{\rm B1}(\omega)$ $\displaystyle\simeq\frac{\pi}{12}\ ,$ $\displaystyle
g_{\rm B3}(\omega,\tau)$
$\displaystyle\simeq\frac{\pi(2+3r)}{24(1+r)^{3}}~{},$ $\displaystyle g_{\rm
T1}(\omega)$ $\displaystyle\simeq\frac{\pi}{3}\ ,$ $\displaystyle g_{\rm
T2}(\omega)$ $\displaystyle\simeq 0\ ,$ $\displaystyle h_{\rm
T1}(\omega,\tau)$ $\displaystyle\simeq\frac{\pi(2+3r)}{6(1+r)^{3}}~{},$
$\displaystyle h_{\rm T2}(\omega,\tau)$ $\displaystyle\simeq 0\ ,$ (42)

with $r\equiv\sqrt{\tau/\omega}=m_{t}/m_{W}$.

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