# Correlated Electric and Magnetic Dipole Moment in Natural Supersymmetry

Sibo Zheng
Department of Physics, Chongqing University, Chongqing 401331, China

###### Abstract

In the context of supersymmetry, the two-loop Barr-Zee diagrams which induce
CP-violating electric dipole moment of electron due to superpartners
simultaneously yield CP-conserving magnetic dipole moment of muon. In this
paper, we derive the coherence between the electric and magnetic dipole
moments at two-loop level due to stops, charginos or neutralinos-charginos. We
also use the coherence to constrain superpartner masses and their CP-violating
phases, in the light of recent ACME limit on the electric dipole moment of
electron and future experiments about magnetic dipole moment of muon such as
Fermilab E989 experiment.

##  1 Introduction

In the context of new physics, which attempts to address the origin of Higgs,
the dark matter with mass of order weak scale, the puzzle of CP violation, and
even the grand unification of standard model (SM) gauge coupling constants,
natural supersymmetry (SUSY) [1] is one of theoretically well motivated
scenarios. In the minimal version of natural SUSY, there are quite a few
sfermions and gauginos with masses of order $\sim 1$ TeV, which once earned
this scenario a promising prospect at experimental facilities such as LHC.
However, the LHC Run I and II have yet observed any new physics except a SM
Higgs [2], which leaves us the only chance at HL-LHC [3].

While waiting for the era of HL-LHC, we may examine other experimental
results, especially the latest ones. Recently, the ACME Collaboration [4] has
updated the measurement on the electric dipole moment (EDM) of electron
$d_{e}$ in

$\displaystyle{}\mathcal{L}_{\text{EDM}}=-i\frac{d_{f}}{2}\bar{f}\sigma_{\rho\lambda}\gamma^{5}fF^{\rho\lambda},$
(1)

which is about an order improvement over the earlier limit [5]. The
implications of the new limit to natural SUSY were discussed in [6]. For
earlier discussions on EDM in the SM and beyond SM, see e.g. [7] and [8],
respectively. These studies reveal that indirect detections such as the ACME
limit on $d_{e}$ can actually reach mass range of several TeVs, which impose
stronger constraint on the natural SUSY than the direct detection at LHC. In
comparison with direct detection, the main drawback in such indirect detection
is that it usually sets a bound on multiple SUSY mass parameters, which reduce
the prediction.

The situation changes when the same SUSY particle yields multiple independent
observables. As is well known, the one-loop effect [9] due to light slepton
masses [10] can explain the observed anomaly of magnetic dipole moment (MDM)
of $\mu$ at Brookhaven E821 experiment [11]. When the stops [12], charginos
and neutralinos [13] running in the two-loop Barr-Zee diagram [14] contribute
to $d_{e}$, they simultaneously deliver correlated magnetic dipole moment
(MDM) of muon $a_{\mu}$ (for a review on MDM of muon, see e.g. [15]) in,

$\displaystyle{}\mathcal{L}_{\text{MDM}}=\frac{ea_{f}}{4m_{f}}\bar{f}\sigma_{\rho\lambda}fF^{\rho\lambda}.$
(2)

In the literature, earlier discussions on SUSY correlation between $d_{e}$ and
$a_{\mu}$ can be found in [16], and two-loop account of $a_{\mu}$ without
attention to the coherence in various contexts were presented in [17].

In the light of future improvement on the measurements of $d_{e}$ such as
ACME-III limit [18] and of $a_{\mu}$ such as the Fermilab E989 experiment
[19], the correlation between $d_{e}$ and $a_{\mu}$ offers a complementary way
to constrain the same SUSY particle in the inner loop, which motivates the
materials as follows. In Sec.2, we present the theoretic framework for the
calculation of two-loop $d_{e}$ and $a_{\mu}$. Sec.3 is devoted to explore the
numerical analysis of natural SUSY by utilizing the analytic results of Sec.2.
Finally, we conclude in Sec.4.

##  2 Correlated EDM and MDM

The two-loop Barr-Zee diagram can be effectively divided into an inner and an
outer loop. In the inner loop, various SUSY particles such as stops, charginos
or neutralinos-charginos run therein. In the outer loop, there are different
mediators such as SM Higgs scalar $h$ or gauge boson $V$. For calculating
$d_{e}$, CP-violation is required, and chirality flip always occurs in the
Feynman diagram, which implies that the mediators in the outer loop are
identified as one scalar $S=\\{h,A\\}$ etc and a vector boson
$V=\\{\gamma,Z\\}$ or alternatively two $W$ bosons. Furthermore, the external
photon in either Eq.(1) or Eq.(2) is only attached to the inner loop, which
together with the mediators ($S$, $V$) constitutes an effective vertex in the
outer loop.

What is of interest is that the effective vertex induced by the SUSY
particle(s) in the inner loop always contains both CP-violating and CP-
conserving parts. In this sense, the CP-violating contributions to $d_{e}$ are
tied to the CP-conserving ones to $a_{\mu}$. The coherence provides us a new
window to constrain the same SUSY particle(s) in the inner loop. In what
follows, we will consider the effective vertex $\Gamma_{\gamma...}$ arises
from a neutral scalar $S$, a charged scalar $S$ and two $W$ bosons, which will
be applied to the cases of stops, charginos, and neutralino-charginos
respectively in the next section.

###  2.1 Neutral Scalar

Firstly, we consider the outer loop composed of a neutral scalar $S$, gauge
boson $V$ and photon $\gamma$. The vertex $\Gamma_{\gamma SV}$ which is
induced by superpartner running in the inner loop takes the form at the
leading order of photon momentum $q$ [20]:

$\displaystyle{}\Gamma_{\gamma
SV}^{\mu\nu}(q,k)=\int^{1}_{0}dx\frac{1}{x(1-x)}\frac{1}{k^{2}-\Delta(x)}\left[c_{E}(q^{\nu}k^{\mu}-g^{\mu\nu}q\cdot
k)+c_{O}\epsilon^{\mu\nu\lambda\rho}q_{\lambda}k_{\rho}\right]$ (3)

where the CP-odd (even) coefficient $c_{O}$ ($c_{E}$) is uniquely determined
by the masses and couplings of superpartners running in the inner loop, and
$\Delta(x)$ in general depends on $x$ and superpartner mass parameters but not
$q$ and $k$.

Given the notation on interaction between scalar $S$, vector boson $V$ and SM
lepton $f=\\{e,\mu\\}$ in the outer loop as

$\displaystyle{}\mathcal{L}\supset
V_{\mu}\bar{f}(g^{V}_{f}\gamma^{\mu}+g^{A}_{f}\gamma^{\mu}\gamma^{5})f+S\bar{f}(g^{S}_{f}+ig^{P}_{f}\gamma^{5})f$
(4)

where all of coupling $g^{S,P}$ and $g^{V,A}$ are real numbers, we obtain the
contribution to EDM of $e$ and MDM of $\mu$ from the same inner loop:

$\displaystyle{}d_{e}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{2}m^{2}_{S}}\int^{1}_{0}dx\frac{1}{x(1-x)}j\left(\frac{m^{2}_{V}}{m^{2}_{S}},\frac{\Delta}{m^{2}_{S}}\right)\cdot\text{Re}(g_{e}^{V}(c_{O}g_{e}^{S}-c_{E}g_{e}^{P})),$
$\displaystyle a_{\mu}$ $\displaystyle=$
$\displaystyle-\left(\frac{e}{2m_{\mu}}\right)^{-1}\frac{1}{16\pi^{2}m^{2}_{S}}\int^{1}_{0}dx\frac{1}{x(1-x)}j\left(\frac{m^{2}_{V}}{m^{2}_{S}},\frac{\Delta}{m^{2}_{S}}\right)\cdot\text{Re}(g_{\mu}^{V}(c_{O}g_{\mu}^{P}+c_{E}g_{\mu}^{S})),$
(5)

with function $j(r,s)$ given as,

$\displaystyle{}j(r,s)=\frac{1}{r-s}\left(\frac{r\log{r}}{r-1}-\frac{s\log{s}}{s-1}\right).$
(6)

###  2.2 Charged Scalar

For a complex scalar $S$, the interaction is defined as

$\displaystyle{}\mathcal{L}\supset
V_{\mu}\bar{f}(g^{V}_{f}\gamma^{\mu}+g^{A}_{f}\gamma^{\mu}\gamma^{5})f^{\prime}+S\bar{f}(g^{S}_{f}+ig^{P}_{f}\gamma^{5})f^{\prime}+\text{H.c},$
(7)

where couplings $g^{V,A}$ and $g^{S,P}$ are not real in general. In this
situation, the contribution to EDM of $e$ and MDM of $\mu$ from the same inner
loop is given by respectively,

$\displaystyle{}d_{e}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{2}m^{2}_{S}}\int^{1}_{0}dx\frac{1}{x(1-x)}j\left(\frac{m^{2}_{V}}{m^{2}_{S}},\frac{\Delta}{m^{2}_{S}}\right)$
$\displaystyle\times$
$\displaystyle[\text{Re}(c_{O}g^{V}_{e}g_{e}^{S*})+\text{Im}(c_{O}g_{e}^{A}g^{P*}_{e})-\text{Re}(c_{E}g^{V}_{e}g_{e}^{P*})+\text{Im}(c_{E}g^{A}_{e}g^{S*}_{e})],$
$\displaystyle a_{\mu}$ $\displaystyle=$
$\displaystyle-\left(\frac{e}{2m_{\mu}}\right)^{-1}\frac{1}{16\pi^{2}m^{2}_{S}}\int^{1}_{0}dx\frac{1}{x(1-x)}j\left(\frac{m^{2}_{V}}{m^{2}_{S}},\frac{\Delta}{m^{2}_{S}}\right)$
(8) $\displaystyle\times$
$\displaystyle[\text{Re}(c_{O}g^{V}_{\mu}g_{\mu}^{P*})-\text{Im}(c_{O}g_{\mu}^{A}g^{S*}_{\mu})+\text{Re}(c_{E}g^{V}_{\mu}g_{\mu}^{S*})+\text{Im}(c_{E}g^{A}_{\mu}g^{P*}_{\mu})],$

###  2.3 W Boson Pair

Unlike in Eq.(3) the vertex for $\gamma W^{+}W^{-}$ is of form [21]

$\displaystyle{}\Gamma_{\gamma
WW}^{\mu\nu\rho}(q,k)=\int^{1}_{0}dx\frac{1}{x(1-x)}\frac{c_{O}}{k^{2}-\Delta(x)}\epsilon^{\mu\nu\rho\sigma}q_{\sigma}$
(9)

In this case the contribution to EDM of electron and MDM of moun from the same
inner loop described by $c_{O}$ and $\Delta$ reads as respectively,

$\displaystyle{}d_{e}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}m^{2}_{W}}\frac{m_{e}}{m^{2}_{W}}\int^{1}_{0}dx\frac{1}{x(1-x)}j\left(\frac{m^{2}_{f^{\prime}}}{m^{2}_{W}},\frac{\Delta}{m^{2}_{W}}\right)\times\text{Im}(c_{O})(\mid
g_{\mu}^{V}\mid^{2}+\mid g_{\mu}^{A}\mid^{2}),$ $\displaystyle a_{\mu}$
$\displaystyle=$
$\displaystyle-\left(\frac{e}{2m_{\mu}}\right)^{-1}\frac{1}{16\pi^{2}m^{2}_{W}}\frac{m_{\mu}}{m^{2}_{W}}\int^{1}_{0}dx\frac{1}{x(1-x)}j\left(\frac{m^{2}_{f^{\prime}}}{m^{2}_{W}},\frac{\Delta}{m^{2}_{W}}\right)\times\text{Re}(c_{O})(g_{e}^{V}g_{e}^{A*}+g_{e}^{A}g_{e}^{V*})$

##  3 Implications to Natural Supersymmetry

In the case of stop inner loop, the intermediator scalar $S$ in the outer loop
can be either the CP-even SM Higgs $h$ or CP-odd scalar $A$. As previously
shown in [20], the stop inner loop yields $c_{O}=0$ and
$c_{E}=\frac{3e^{2}}{4\pi^{2}}Q^{2}_{\tilde{t}_{i}}g^{S}_{\tilde{t}_{i}\tilde{t}_{i}}x(1-x)$,
which implies the intermediator scalar is $h$ ($A$) in the estimate of
$a_{\mu}$ ($d_{e}$). Substituting $g^{V}_{f}=e$, $g^{A}_{f}=0$,
$g^{S=h}_{f}=m_{f}/\upsilon$ ($\upsilon=$ 246 GeV), and $g^{P=A}_{f}=\tan\beta
m_{f}/\upsilon$ into Eq.(2.1), we obtain the the dominant contribution to
$a_{\mu}$ and $d_{e}$ from the same stop inner loop,

$\displaystyle{}\frac{d_{e}/e}{1.1\times 10^{-29}\text{cm}}$ $\displaystyle=$
$\displaystyle 1.79\times
10^{4}\left(\frac{\tan\beta}{10}\right)\left(\frac{100~{}\text{GeV}}{m_{A}}\right)^{2}\sum_{i}\int^{1}_{0}dx(1-x)j(0,\frac{m^{2}_{\tilde{t}_{i}}}{m^{2}_{A}}\frac{1}{x(1-x)})\frac{\text{Re}(g^{S=A}_{\tilde{t}_{i}\tilde{t}_{i}})}{\upsilon}$
$\displaystyle\frac{a_{\mu}}{43\times 10^{-10}}$ $\displaystyle=$
$\displaystyle-6.52\times
10^{-3}\sum_{i}\int^{1}_{0}dx(1-x)j(0,\frac{m^{2}_{\tilde{t}_{i}}}{m^{2}_{h}}\frac{1}{x(1-x)}),$
(11)

where in the small mixing effect

$\displaystyle{}\frac{\text{Re}(g^{S=A}_{\tilde{t}_{i}\tilde{t}_{i}})}{\upsilon}\simeq(-1)^{i+1}y^{2}_{t}\frac{\mid
A_{t}\mu\mid}{m^{2}_{\tilde{t}_{1}}-m^{2}_{\tilde{t}_{2}}}\sin(\text{arg}(A_{t}\mu))$
(12)

Fig.1 shows the correlation between $a_{\mu}$ and $d_{e}$ from the TeV-scale
stops. In the top plot one finds that $a_{\mu}$ is typically of order
$10^{-13}$ for stop masses $m_{\tilde{t}_{1}}=1$ TeV and $m_{\tilde{t}_{2}}$
between $1$ TeV and $2$ TeV, which indicates that the observed anomaly at the
Brookhaven E821 experiment has no relevance on stop scalars. Meanwhile, in the
bottom plot, we observe that the dependence of $d_{e}/e$ (in unit of
$1.1\times 10^{-29}$ cm) on heavier stop mass $m_{\tilde{t}_{2}}$ for
$\tan\beta=10$, small mixing $\mid A_{t}\mu\mid=(100~{}\text{GeV})^{2}$ and
different magnitudes of CP-violating phase
$\sin(\text{arg}(A_{t}\mu))=\\{0.1,0.01,0.001\\}$. This plot shows that the
magnitude of $d_{e}/e$ is about two orders below the ACME-II limit. In
contrast, it can excess the ACME-II limit in the case of large mixing effect
and large CP violation [6] in the stop sector. Because there is an enhancement
on $d_{e}/e$ about two orders of magnitude for large $A_{t}\sim 1$ TeV and
large CP-violating phase $\sin(\text{arg}(A_{t}\mu))\sim 1$ as shown by
Eq.(12). Therefore, the ACME-II limit imposes strong constraint on stop masses
when the mixing effect is large but still weak if the mixing effect is small.

There seems contradiction on the predictions of $A_{t}$ and $\tan\beta$ among
different experimental bounds on natural SUSY. Firstly, both large mixing and
large $\tan\beta$ are required by the observed Higgs mass in the context of
MSSM. Conversely, the ACME experiment strongly favors small $A_{t}$ and small
$\tan\beta$ [23], similar to the flavor violation experiment from
$b\rightarrow s\gamma$ [22]. Here we mention a simple way to reconcile the
situation. The Higgs mass can be uplifted by new sources [24] such as in the
MSSM extended by $5+\bar{5}$ vectorlike fermions [25] with mass of order TeV
scale, which retain the grand unification of SM gauge coupling constants
similar to MSSM. In these scenarios, the stop masses can be of order $\sim 1$
TeV as in natural SUSY, and the mixing effect and $\tan\beta$ can be small.

Figure 1: Correlated $a_{\mu}$ and $d_{e}$ from stop scalars. Top: the
dependence of $a_{\mu}$ on stop scalar mass $m_{\tilde{t}_{2}}$ for
$m_{\tilde{t}_{1}}=1$ TeV. Bottom: the dependence of $a_{\mu}$ on stop scalar
mass $m_{\tilde{t}_{2}}$ for $\tan\beta=10$, $m_{\tilde{t}_{1}}=m_{A}=1$ TeV,
$\mid A_{t}\mu\mid=(100~{}\text{GeV})^{2}$ and different values of
$\sin(\text{arg}(A_{t}\mu))=\\{0.1,0.01,0.001\\}$.

Unlike the stop inner loop, neither $c_{O}$ nor $c_{E}$ in Eq.(2.2) vanishes
in the situation of chargino inner loop [20]. In this case, the intermediator
scalar $S$ can be either $h$ or $A$. Substituting
$c_{O}=-\frac{e^{2}}{2\pi^{2}}g^{P=A}_{\tilde{\chi}^{+}_{i}\tilde{\chi}^{-}_{j}}(1-x)m_{\tilde{\chi}^{+}}$
and
$c_{E}=\frac{e^{2}}{2\pi^{2}}g^{S=h}_{\tilde{\chi}^{+}_{i}\tilde{\chi}^{-}_{j}}(1-x)(2x^{2}-2x+1)m_{\tilde{\chi}^{+}}$
[20] into Eq.(2.2) results in the EDM of electron and MDM of muon as follows,

$\displaystyle{}\frac{d_{e}/e}{1.1\times 10^{-29}\text{cm}}$ $\displaystyle=$
$\displaystyle 1.09\times
10^{3}\sum_{i=1,2}\frac{m_{\tilde{\chi}_{i}^{+}}}{100~{}\text{GeV}}\int^{1}_{0}\frac{dx}{x}$
$\displaystyle\times$
$\displaystyle\left[0.64j(0,\frac{\Delta}{m^{2}_{h}})\text{Re}g^{P=h}_{\tilde{\chi}^{+}_{i}\tilde{\chi}^{-}_{j}}+(2x^{2}-2x+1)\tan\beta\left(\frac{100~{}\text{GeV}}{m_{A}}\right)^{2}j(0,\frac{\Delta}{m^{2}_{A}})\text{Re}g^{S=A}_{\tilde{\chi}^{+}_{i}\tilde{\chi}^{-}_{j}}\right]$
$\displaystyle\frac{a_{\mu}}{43\times 10^{-10}}$ $\displaystyle=$
$\displaystyle 6.2\times
10^{-3}\sum_{i=1,2}\frac{m_{\tilde{\chi}_{i}^{+}}}{100~{}\text{GeV}}\int^{1}_{0}\frac{dx}{x}$
$\displaystyle\times$
$\displaystyle\left[0.64(2x^{2}-2x+1)j(0,\frac{\Delta}{m^{2}_{h}})\text{Re}g^{S=h}_{\tilde{\chi}^{+}_{i}\tilde{\chi}^{-}_{j}}+\tan\beta\left(\frac{100~{}\text{GeV}}{m_{A}}\right)^{2}j(0,\frac{\Delta}{m^{2}_{A}})\text{Re}g^{P=A}_{\tilde{\chi}^{+}_{i}\tilde{\chi}^{-}_{j}}\right]$

where $\Delta=m^{2}_{\tilde{\chi}_{i}^{+}}/x(1-x)$; the CP-violating Yukawa
couplings in $d_{e}$ are given by
$\text{Re}(g^{P=h}_{\tilde{\chi}^{+}_{i}\tilde{\chi}^{-}_{j}})\simeq\frac{g}{\sqrt{2}}\sin\beta\mid
C^{R*}_{1i}C^{L}_{2j}\mid\sin(\text{arg}(M_{2}\mu))$ and
$\text{Re}(g^{S=A}_{\tilde{\chi}^{+}_{i}\tilde{\chi}^{-}_{j}})\simeq-\frac{g}{\sqrt{2}}\sin\beta\mid
C^{R*}_{2i}C^{L}_{1j}\mid\sin(\text{arg}(M_{2}\mu)$ ($C^{L,R}$ refer to
unitary matrixes which diagonalize the chargino mass matrix); whereas the CP-
conserving Yukawa couplings in $a_{\mu}$ read as
$\text{Re}(g^{P=A}_{\tilde{\chi}^{+}_{i}\tilde{\chi}^{-}_{j}})=-\frac{g}{\sqrt{2}}\sin\beta\mid
C^{R*}_{2i}C^{L}_{1j}\mid$ and
$\text{Re}(g^{S=h}_{\tilde{\chi}^{+}_{i}\tilde{\chi}^{-}_{j}})=-\frac{g}{\sqrt{2}}\sin\beta\mid
C^{R*}_{1i}C^{L}_{2j}\mid$.

In Fig.2 we show the correlation between $a_{\mu}$ and $d_{e}$ from charginos.
For simplicity, we take nearly degenerate chargino mass spectrum
$m_{\tilde{\chi}_{1}}\simeq m_{\tilde{\chi}_{2}}$. In the top plot $a_{\mu}$
is typically of order $\sim 10^{-12}$, which is about one order of magnitude
larger than the result of stops. Similar results hold in the bottom plot,
where the value of $d_{e}/e$ due to charginos are larger than those due to
stops. As previously shown in [20], the ACME-II limits can exclude the
chargino mass below $10$ TeV for $\sin(\text{arg}(M_{2}\mu))$ of order unity.
Furthermore, the bottom plot tells that the excursion limit is $\sim 1$ TeV
for $\sin(\text{arg}(M_{2}\mu))$ of order $\sim 10^{-2}$, and the magnitude of
CP-violating phase $\sin(\text{arg}(M_{2}\mu))$ has to be of order $\sim
10^{-3}$ if sub-TeV charginos still survive in the ACME-II limit. Note, we
have taken $\text{sign}(\mu)=\text{sign}(M_{2})$ for our analysis. In the
situation with an opposite assignment of signs, the numerical values of
$a_{\mu}$ and $d_{e}$ both change, but their orders are still the same as
those of identical sign assignment.

Figure 2: Correlated $a_{\mu}$ and $d_{e}$ from charginos. Top: the dependence
of $a_{\mu}$ on nearly degenerate chargino mass $m_{\tilde{\chi}^{+}_{i}}$ for
$m_{A}=1$ TeV. Bottom: the dependence of $d_{e}$ on chargino mass
$m_{\tilde{\chi}}$ for $\tan\beta=10$, $m_{A}=1$ TeV and different values of
$\sin(\text{arg}(M_{2}\mu))=\\{0.1,0.01,0.001\\}$. The red line denotes the
ACME-II limit.

Finally, we discuss the contributions to $d_{e}$ and $a_{\mu}$ due to the
neutralino-chargino inner loop. According to the interaction

$\displaystyle{}\mathcal{L}\supset
g\bar{\chi}^{-}_{i}\gamma^{\mu}[G_{ij}^{L}P_{L}+G_{ij}^{R}P_{R}]\chi^{0}_{j}W^{+}_{\mu}+\text{H.c},$
(14)

where $G^{L}_{ij}=-C^{L*}_{1i}N_{2j}+\frac{1}{\sqrt{2}}C^{L*}_{2i}N_{4j}$ and
$G^{R}_{ij}=-C^{R*}_{1i}N^{*}_{2j}-\frac{1}{\sqrt{2}}C^{R*}_{2i}N^{*}_{3j}$,
we have

$\displaystyle{}\text{Im}(c_{O})$ $\displaystyle=$
$\displaystyle-\frac{e\alpha}{\pi
s^{2}_{W}}\text{Im}(G^{L}_{ij}G_{ij}^{R*})m_{\tilde{\chi}^{+}_{i}}m_{\tilde{\chi}^{0}_{j}}(1-x),$
$\displaystyle\text{Re}(c_{O})$ $\displaystyle=$
$\displaystyle\frac{e\alpha}{\pi s^{2}_{W}}(\mid G_{ij}^{L}\mid^{2}-\mid
G_{ij}^{R}\mid^{2})m^{2}_{\tilde{\chi}^{+}_{i}}(1-x)$ (15)

and
$\Delta=[(1-x)m^{2}_{\tilde{\chi}^{+}_{i}}+xm^{2}_{\tilde{\chi}^{0}_{j}}]/x(1-x)$
[26] in Eq.(2.3). Substituting them into Eq.(2.3) results in contributions to
$d_{e}$ and $a_{\mu}$ as follows,

$\displaystyle{}\frac{d_{e}/e}{1.1\times 10^{-29}\text{cm}}$ $\displaystyle=$
$\displaystyle-1.4\times
10^{3}\sum_{i,j}\int^{1}_{0}\frac{dx}{x}\text{Im}(G^{L}_{ij}G^{R*}_{ij})\frac{m_{\tilde{x}^{+}_{i}}}{100~{}\text{GeV}}\frac{m_{\tilde{x}^{0}_{j}}}{100~{}\text{GeV}}j(0,\frac{1}{x}\frac{m^{2}_{\tilde{\chi}^{+}_{i}}}{m^{2}_{W}}+\frac{1}{1-x}\frac{m^{2}_{\tilde{\chi}^{0}_{j}}}{m^{2}_{W}})$
$\displaystyle\frac{a_{\mu}}{43\times 10^{-10}}$ $\displaystyle=$
$\displaystyle-7.9\times 10^{-3}\sum_{i,j}\int^{1}_{0}\frac{dx}{x}(\mid
G_{ij}^{L}\mid^{2}-\mid
G_{ij}^{R}\mid^{2})\frac{m^{2}_{\tilde{x}^{+}_{i}}}{\left(100~{}\text{GeV}\right)^{2}}j(0,\frac{1}{x}\frac{m^{2}_{\tilde{\chi}^{+}_{i}}}{m^{2}_{W}}+\frac{1}{1-x}\frac{m^{2}_{\tilde{\chi}^{0}_{j}}}{m^{2}_{W}})$

wheres sum over $i=1-2$ and $j=1-4$. The result about $d_{e}$ in Eq.(3) agrees
with that of the 3rd reference in [13].

Figure 3: Correlated $a_{\mu}$ and $d_{e}$ from neutralinos and charginos with
universal mass $M_{1,2}=\mu=M$. Top: the dependence of $a_{\mu}$ on $M$.
Bottom: the dependence of $d_{e}$ on $M$ for different values of
$\sin(\text{arg}(M_{2}\mu))=\\{0.1,0.01,0.001\\}$.

From Eq.(3) it is clear that $a_{\mu}$ is sensitive to the mass spectrum of
neutralinos and charginos. When the mass splittings between neutralinos and
charginos are small, small $a_{\mu}$ is expected due to small $\mid
G_{ij}^{L}\mid^{2}-\mid G_{ij}^{R}\mid^{2}$ of order $\sim
m^{2}_{Z}s^{2}_{W}/M^{2}$, with $M$ referring to the universal neutralino and
chargino mass. Fig.3 shows the magnitudes of $a_{\mu}$ and $d_{e}$ in this
situation. The top plot shows that $a_{\mu}$, which decreases as $M$
increases, is typically of order $\sim 10^{-13}$. The bottom plot therein
reveals that $d_{e}$, which is $\log$-dependent on $M$, excesses the ACME-II
limit unless the magnitude of $\sin(\text{arg}(M_{2}\mu))$ is smaller than
$\sim 10^{-3}$. Otherwise, $M$ beneath $1$ TeV has been excluded for large or
moderate $\sin(\text{arg}(M_{2}\mu))$. The reason for the $\log$-dependence of
$d_{e}$ on $M$ is that the amplitude of $G^{L}_{ij}G^{R*}_{ij}$ is of order
unity, which is not sensitive to the universal mass $M$ much heavier than
$m_{Z}$. Rather, a relatively large $a_{\mu}$ appears when there is large mass
splittings between the masses of neutralino and charginos. An optimistic
estimate is that the enhancements on $a_{\mu}$ in Fig.3 is about two orders of
magnitude for $\mid G_{ij}^{L}\mid^{2}-\mid G_{ij}^{R}\mid^{2}$ of order
unity, which makes the contribution to $a_{\mu}$ due to neutralino-chargino
loop dominates over the other two-loop effects. However, we would like to
mention that this case is strongly constrained by the limits from LHC Run 1
and 2 [27], since there is large missing energy with large mass splitting
during the productions of either neutralinos or charginos at the LHC.

##  4 Conclusions

In this paper, we have explored the two-loop correlations between $a_{\mu}$
and $d_{e}$ due to either stops, charginos or neutralinos in natural SUSY. In
the individual situation, $a_{\mu}$ can probe the superparticle masses,
whereas $d_{e}$ can determine the CP-violating phase tied to the same
superparticle mass. Recent ACME-II limit has revealed that the magnitudes of
CP-violating phases should be small, with the range of order $0.1$ to $0.001$
for sub-TeV stops and neutralinos/charginos, respectively. On the other hand,
the magnitudes of the correlated $a_{\mu}$ vary from $\sim 10^{-13}$ to $\sim
10^{-12}$ for stops and charginos, respectively, which are beyond the scope of
Brookhaven E821 ($\sim 10^{-9}$) but may be examined by future experiments on
$a_{\mu}$ such as Fermilab E989 experiment. Together with constraints such as
flavor violation, the EDM and MDM studied here offer us a complementary way to
the direct detection on natural SUSY at future HL-LHC.

$\mathbf{Acknowledgments}$. This work is supported by the National Natural
Science Foundation of China (No. 11775039 and No. 11847301) and the
Fundamental Research Funds for the Central Universities at Chongqing
University (No. cqu2017hbrc1B05).

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