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Tomohiro Abe

Institute for Advanced
Research, KMI, Nagoya U.

共同研究者
Motomo Fujiwara

Junji Hisano

Nagoya U.

Nagoya U, KMI, Kavli IPMU$#4'&10->*5B
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<latexit sha1_base64="7mmnHEhmLxiklI0ZU4ZxmNNhuWk=">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</latexit>

L

⇣

†
¯
g i 5 a0 +  ia0 H1 H2 + (h.c.)+07#3+#05

only pseudo-scalar interaction
thanks to CP invariance,⌘Note that other cubic terms
breaks CP invariancea0 H1† H1 , a0 H2† H2<latexit sha1_base64="Cnf1EDn2s9nLLW6A0X+TM8NbeAs=">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</latexit>

5

a0
A0

✓ ◆ ✓
A
cos ✓
=
a
sin ✓
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sin ✓
cos ✓

◆✓

0

A
a0

◆

q


loop calculations were calculated

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a, A

q

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¯ Gaµ⌫ Gaµ⌫

¯ q̄q
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5

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5<latexit sha1_base64="Jgbd7ozt+acrKRgrgjGfbtpC7Oo=">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</latexit>a, A<latexit sha1_base64="9kBS++2McKsR6zdfm1GZkoAbrLc=">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</latexit>'45+/#5'&$:64+0)5<latexit sha1_base64="Jgbd7ozt+acrKRgrgjGfbtpC7Oo=">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</latexit>↵s a aµ⌫Gµ⌫ G
12⇡mQ Q̄Q =a, A<latexit sha1_base64="9kBS++2McKsR6zdfm1GZkoAbrLc=">AAACjHichVG7SgNRED2u7/iK2gg2wRCxkDArgqIIEUEs8zAaUJHd9SYu7ovdTSCG/IBgaworBQvxD2xt/AGLfIJYRrCxcLJZEBV1lr333DNz5p7LqI6hez5Rs0vq7unt6x8YjAwNj4yORccndjy77Goir9mG7RZUxROGbom8r/uGKDiuUEzVELvqyUY7v1sRrqfb1rZfdcSBqZQsvahris9UTplfP4zGKUlBxH4COQRxhJG2o/fYxxFsaCjDhIAFn7EBBR5/e5BBcJg7QI05l5Ee5AXqiLC2zFWCKxRmT3gt8WkvZC0+t3t6gVrjWwz+XVbGkKAnuqUWPdIdPdP7r71qQY+2lyrvakcrnMOxs6nc278qk3cfx5+qPz37KGI58Kqzdydg2q/QOvrKaaOVW8kmarN0TS/s/4qa9MAvsCqv2k1GZC//8FPktcq9TM7UeUzy96H8BDsLSZmScmYxnloOBzaAacxgjqeyhBS2kEaebynhHBdoSKPSorQqrXVKpa5QM4kvIW1+AMIVkiY=</latexit>

Q = t, b, c(heavy quarks)

a, A

a, A

h/H
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q

5

q

<latexit sha1_base64="Jgbd7ozt+acrKRgrgjGfbtpC7Oo=">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</latexit>

a/A<latexit sha1_base64="csncdjiOFmho4lbqvg1ulEfAm1w=">AAACjHichVG7SgNRED2u7/iK2gg2wRCxirMiKIoQEcQyD6MBFdldb+LivtjdBGLIDwi2prBSsBD/wNbGH7DIJ4hlBBsLJ5sFUVFn2XvPPTNn7rmM6hi65xM1u6Tunt6+/oHByNDwyOhYdHxix7PLribymm3YbkFVPGHolsj7um+IguMKxVQNsauebLTzuxXherptbftVRxyYSsnSi7qm+EzllPn1w2ickhRE7CeQQxBHGGk7eo99HMGGhjJMCFjwGRtQ4PG3BxkEh7kD1JhzGelBXqCOCGvLXCW4QmH2hNcSn/ZC1uJzu6cXqDW+xeDfZWUMCXqiW2rRI93RM73/2qsW9Gh7qfKudrTCORw7m8q9/asyefdx/Kn607OPIpYDrzp7dwKm/Qqto6+cNlq5lWyiNkvX9ML+r6hJD/wCq/Kq3WRE9vIPP0Veq9zL5EydxyR/H8pPsLOQlCkpZxbjqeVwYAOYxg
sha1_base64="csncdjiOFmho4lbqvg1ulEfAm1w=">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</latexit>

5<latexit sha1_base64="Jgbd7ozt+acrKRgrgjGfbtpC7Oo=">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</latexit>

a/A<latexit sha1_base64="csncdjiOFmho4lbqvg1ulEfAm1w=">AAACjHichVG7SgNRED2u7/iK2gg2wRCxirMiKIoQEcQyD6MBFdldb+LivtjdBGLIDwi2prBSsBD/wNbGH7DIJ4hlBBsLJ5sFUVFn2XvPPTNn7rmM6hi65xM1u6Tunt6+/oHByNDwyOhYdHxix7PLribymm3YbkFVPGHolsj7um+IguMKxVQNsauebLTzuxXherptbftVRxyYSsnSi7qm+EzllPn1w2ickhRE7CeQQxBHGGk7eo99HMGGhjJMCFjwGRtQ4PG3BxkEh7kD1JhzGelBXqCOCGvLXCW4QmH2hNcSn/ZC1uJzu6cXqDW+xeDfZWUMCXqiW2rRI93RM73/2qsW9Gh7qfKudrTCORw7m8q9/asyefdx/Kn607OPIpYDrzp7dwKm/Qqto6+cNlq5lWyiNkvX9ML+r6hJD/wCq/Kq3WRE9vIPP0Veq9zL5EydxyR/H8pPsLOQlCkpZxbjqeVwYAOYxgzmeCpLSGELaeT5lhLOcYGGNCotSqvSWqdU6go1k/gS0uYHyIqSKQ==</latexit>h/HQ<latexit sha1_base64="svx+WB38pKXXtQsf9UWsPjq3h34=">AAAElHichVFNa9RAGH67rlrXj7YKIngJdleqDO1M/CpioVCEnqQf9gPaWmayk92hSSYk09Q25A94Fw+CouBB/AdevfgHPPQniMcKXjz4ZnZtrXXrhCTvPB9vnskr4kClhtLdvsqJ6slTp/vP1M6eO39hYHDo4mKqNxNPLng60Mmy4KkMVCQXjDKBXI4TyUMRyCWxMVXyS5lMUqWjx2Y7lmshb0XKVx43CK0PVfpXhWypKPdD31eBLPKm4q2Eh4VTO2AQiNs3ixHmUnKX3kAKwUD6JleMKLew+0S12ibXjGgEGiUitDE6zIUighHhEnGLiNtE+Ni7YS15HHAVFblySeaSLddaLdHkaVumxMiojD7BxtwiR0mWoaqHhARcyGCi3h6brqM4IxkrDn+FIUQ066bzEyl3pFU0tcltAKQc2yZ1fJ048qmRScQDp/y9aefUJZvXV722qpfBi3+g+jA6WwrZEWg/yG9zi4chf3KnDH8M5faktnq77E+TUfOPYTr7gJ17bX1wmI5Su5yjBesWw9BdM3rwI6xCEzR4sAkhSIjAYB0AhxSvFWBAIUZsDXLEEqyU5SUUUEPvJqokKjiiG/hs4W6li0a4L3um1u3hVwK8E3Q60KBf6Hu6Rz/TD/Qr/dmzV257lFm28S06XhmvDzy7Mv/jv64Q3wbaB65jMxvwYdxmVZg9tkh5Cq/jz3Ze7M3fn2vk1+lb+g3zv6G79BOeIMq+e+9m5dzLY/L4+NzGXiEyBY6J/T2Uo8WiO8roKJt1hyfHuwPrh6twDUZwKvdgEqZhBhbAq+jK88qryuvq5eqD6lT1YUda6et6LsGhVX30C12QScU=</latexit>.1E;:0@41%&;9;45>;.1$ !-3;E-'

Q

Q

Q



loop calculations were calculated!3%#&+'5#. 

"

but
this
replacement
isNOTaaµ⌫ justified!
¯ Gµ⌫ G¯ q̄q
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a, A

q

5

5

a, A

h/H

a, A

q
<latexit

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5<latexit sha1_base64="Jgbd7ozt+acrKRgrgjGfbtpC7Oo=">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</latexit>a, A<latexit sha1_base64="9kBS++2McKsR6zdfm1GZkoAbrLc=">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</latexit>'45+/#5'&$:64+0)↵5<latexit sha1_base64="Jgbd7ozt+acrKRgrgjGfbtpC7Oo=">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</latexit>a, A<latexit sha1_base64="9kBS++2McKsR6zdfm1GZkoAbrLc=">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</latexit><latexit sha1_base64="Smk667q2er+/LJ97SnGFkCM1MGo=">AAAC0XichVFLaxRBEP4yvuL6yJpcBC+NS8SLS00QDIIQEGKOmaybBDJx6Bl7kybzYh4Lm2Eg5Kg/wENOCh5Ef4FXL54FD0H/gHiM4MWDNbMjosFYTXdXfVVf9deUG/s6zYgOJ4xTp8+cPTd5vnXh4qXLU+0r06tplCee6nuRHyXrrkyVr0PVz3Tmq/U4UTJwfbXm7tyv8mtDlaQ6Ch9mo1htBnIr1APtyYwhp70YOJawXZkUVims1j1xS9iDRHqFLf14WzppWZhzdqxL8eBRIUunsI
sha1_base64="Smk667q2er+/LJ97SnGFkCM1MGo=">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</latexit>

s
a
aµ⌫
↵
s
a
aµ⌫
G
G
m
Q̄Q
=
Q=
Gµ⌫ G
mQ Q̄QQ = u, b, c (heavy quarks)
µ⌫
12⇡
12⇡

h/H
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q

5

q

<latexit sha1_base64="Jgbd7ozt+acrKRgrgjGfbtpC7Oo=">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</latexit>

a/A<latexit sha1_base64="csncdjiOFmho4lbqvg1ulEfAm1w=">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</latexit>

5<latexit sha1_base64="Jgbd7ozt+acrKRgrgjGfbtpC7Oo=">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</latexit>

a/A

a, A

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Q

Q

a, A
Q



loop calculations were calculated!3%#&+'5#. "

werevisitthetriple-scalar
couplings!
aaµ⌫¯ Gµ⌫ G¯ q̄q
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a, A

q

5

5

a, A

h/H

a, A

q
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5<latexit sha1_base64="Jgbd7ozt+acrKRgrgjGfbtpC7Oo=">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</latexit>a, A<latexit sha1_base64="9kBS++2McKsR6zdfm1GZkoAbrLc=">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</latexit>'45+/#5'&$:64+0)↵5<latexit sha1_base64="Jgbd7ozt+acrKRgrgjGfbtpC7Oo=">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</latexit>a, A<latexit sha1_base64="9kBS++2McKsR6zdfm1GZkoAbrLc=">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</latexit><latexit sha1_base64="Smk667q2er+/LJ97SnGFkCM1MGo=">AAAC0XichVFLaxRBEP4yvuL6yJpcBC+NS8SLS00QDIIQEGKOmaybBDJx6Bl7kybzYh4Lm2Eg5Kg/wENOCh5Ef4FXL54FD0H/gHiM4MWDNbMjosFYTXdXfVVf9deUG/s6zYgOJ4xTp8+cPTd5vnXh4qXLU+0r06tplCee6nuRHyXrrkyVr0PVz3Tmq/U4UTJwfbXm7tyv8mtDlaQ6Ch9mo1htBnIr1APtyYwhp70YOJawXZkUVims1j1xS9iDRHqFLf14WzppWZhzdqxL8eBRIUunsI
sha1_base64="Smk667q2er+/LJ97SnGFkCM1MGo=">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</latexit>

s
a
aµ⌫
↵
s
a
aµ⌫
G
G
m
Q̄Q
=
Q=
Gµ⌫ G
mQ Q̄QQ = u, b, c (heavy quarks)
µ⌫
12⇡
12⇡

h/H
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q

5

q

<latexit sha1_base64="Jgbd7ozt+acrKRgrgjGfbtpC7Oo=">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</latexit>

a/A<latexit sha1_base64="csncdjiOFmho4lbqvg1ulEfAm1w=">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</latexit>

5<latexit sha1_base64="Jgbd7ozt+acrKRgrgjGfbtpC7Oo=">AAACkXichVG7TgJREB3WF+IDlMbEhkgwVmTWaEQroo2JDQ95JIBkd73ghn1ldyFBwg9Y2lhgo4mF8Q9sbfwBCz7BWGJiY+HssolRIs7N3jv3zJzZc3NEQ5EtG7Hv4yYmp6Zn/LOBufmFxWBoaTlv6U1TYjlJV3SzKAoWU2SN5WzZVljRMJmgigoriI0Dp15oMdOSde3Ybhusogp1Ta7JkmATVCzXBVUVTraroSjG0Y3IaMJ7SRS8SOmhRyjDKeggQRNUYKCBTbkCAli0SsADgkFYBTqEmZTJbp1BFwLEbVIXow6B0AbtdbqVPFSjuzPTctkS/UWhzyRmBGL4gvc4wGd8wFf8/HNWx53haGnTKQ65zKgGL1ayH/+yVDptOPtmjdVsQw0SrlaZtBsu4rxCGvJb51eD7F4m1lnHW3wj/TfYxyd6gdZ6l+7SLNMbo6dGe5tmqVTpkk38b1NGk/xmnMc4n96KJhOeYX5YhTXYIFd2IAmHkIKc68Yl9OCaC3O7XJLbH7ZyPo8Thh/BHX0BIY2UfA==</latexit>

a/A

a, A

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Q

Q

a, A
Q

a

a

XENON1T. The light green (dark green) region will be probed by LZ and XENONnT (DARWIN).
@45?>1<8-/191:@/-::;@.16A?@5G10.A@A?105:!3%#&+'5#.
 " onThe
yellow region corresponds to the sensitivity to coherent scattering processesof neutrinos
nucleons.

mQ Q̄Q =↵s a aµ⌫G G
12⇡ µ⌫a, Asimplified model
a, A
a, A
a, A
Two comments
about
the reliability of this result are in order. First, the<latexit sha1_base64="Smk667q2er+/LJ97SnGFkCM1MGo=">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</latexit>

in which this computation has been made is not gauge invariant. Generically we expect that
an UV-completionof the simplified
model will introduce new degrees of freedom to restore
Q
Q
gauge-invariance. These new fields could allow for additional diagrams and therefore theamplitude considered here cannot be expected to be the full result. We will comment on
this8;;<05-3>-9??4;A80.1/-8/A8-@10
in more detail in section 3 where we analyze a representative example of such an UV!3%#&+'5#.  "
completion. In addition, there is a further complication which is related to the relation we
employed to replace the heavy quarks with the gluons in eq. (2.8). This procedure is justified
if the loop that generates the four-fermion interaction and the loop that relates the quarks to
the gluon-condensate factorize. While this assumption is reasonable for heavy new physics
which can be integrated out at energies above the top mass, it is not fully appropriate in the
scenario under scrutiny here since we are interested in ma < mt . In this case, the correct
top mass dependence of the e↵ective dark matter gluon interaction is only recovered by a
two-loop computation of the e↵ective dark matter gluon interaction [57] which is beyond thescope of this work. In the following, we will rely on eq. (2.8) while keeping in mind that theresult is only approximate.
<latexit sha1_base64="RnGINlBqQl4k4z3MFCIq2z1epPo=">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</latexit><latexit

sha1_base64="5bHd0p+6wmac/Lrmpk/dNDHatRQ=">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</latexit><latexit sha1_base64="iAss0Czht4qGAqiU9nSsXheC4mw=">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</latexit>

The behavior of the scattering cross section, as a function of the DM mass, for g = 0.5
and for some di↵erent assignations of ca and ma , is reported in figure 2. The predictions.1E;:0@41%!-3;E-'of thescattering cross sections &;9;45>;.1$are compared withthe current exclusion limit, as set by

AP03(2018)042

Figure 2.work
SI cross section
inducedat one-loop as functionof theDM mass
m , for g = 0.5 and
Our
(1)
:
two-loop
calc
forbox-gluon
for four assignations of (m , c ), as reported on the plot. The blue region is currently excluded byaa

XENON1T. The light green (dark green) region will be probed by LZ and XENONnT (DARWIN).
@45?>1<8-/191:@/-::;@.16A?@5G10.A@A?105:!3%#&+'5#.
 " onThe
yellow region corresponds to the sensitivity to coherent scattering processesof neutrinos
nucleons.

mQ Q̄Q =↵s a aµ⌫G G
12⇡ µ⌫a, Asimplified model
a, A
a, A
a, A
Two comments
about
the reliability of this result are in order. First, the<latexit sha1_base64="Smk667q2er+/LJ97SnGFkCM1MGo=">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</latexit>

in which this computation has been made is not gauge invariant. Generically we expect that
an UV-completionof the simplified
model will introduce new degrees of freedom to restore
Q
Q
gauge-invariance. These new fields could allow for additional diagrams and therefore theamplitude considered here cannot be expected to be the full result. We will comment on
this8;;<05-3>-9??4;A80.1/-8/A8-@10
in more detail in section 3 where we analyze a representative example of such an UV!3%#&+'5#.  "
completion. In addition, there is a further complication which is related to the relation we
employed to replace the heavy quarks with the gluons in eq. (2.8). This procedure is justified
if the loop that generates the four-fermion interaction and the loop that relates the quarks to
the gluon-condensate factorize. While this assumption is reasonable for heavy new physics
which can be integrated out at energies above the top mass, it is not fully appropriate in the
scenario under scrutiny here since we are interested in ma < mt . In this case, the correct
top mass dependence of the e↵ective dark matter gluon interaction is only recovered by a
two-loop computation of the e↵ective dark matter gluon interaction [57] which is beyond thescope of this work. In the following, we will rely on eq. (2.8) while keeping in mind that theresult is only approximate.
<latexit sha1_base64="RnGINlBqQl4k4z3MFCIq2z1epPo=">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</latexit><latexit

sha1_base64="5bHd0p+6wmac/Lrmpk/dNDHatRQ=">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</latexit><latexit sha1_base64="iAss0Czht4qGAqiU9nSsXheC4mw=">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</latexit>@C;8;;</;9<A@-@5;:5?@41?/;<1;2;A>C;>7
The behavior of the scattering cross section, as a function of the DM mass, for g = 0.5
and for some di↵erent assignations of ca and ma , is reported in figure 2. The predictions.1E;:0@41%!-3;E-'of thescattering cross sections &;9;45>;.1$are compared withthe current exclusion limit, as set by

AP03(2018)042

Figure 2.work
SI cross section
inducedat one-loop as functionof theDM mass
m , for g = 0.5 and
Our
(1)
:
two-loop
calc
forbox-gluon
for four assignations of (m , c ), as reported on the plot. The blue region is currently excluded by

Our work (1) : two-loop calc for box-gluon@45?>1<8-/191:@/-::;@.16A?@5G10.A@A?105:!3%#&+'5#. 
mQ Q̄Q =

a, A

a, A

Q

<latexit sha1_base64="Smk667q2er+/LJ97SnGFkCM1MGo=">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</latexit>↵s a aµ⌫G G
12⇡ µ⌫

a, A

"

a, A

Q

<latexit sha1_base64="RnGINlBqQl4k4z3MFCIq2z1epPo=">AAAEpnicjVK/b9QwFH4NB5TjR1tYECwRvUMHsio7FFEhVSpiYUL9wbWV2lLsnHNnNYmjxL3SRhlY+QcYmEBCCLHAysrCP8DQPwExFomFgRff0VJKC45iP3/f+14++0UkocoMpdsDzrHK8RMnB09VT585e25oeOT8fKbXU182fR3qdFHwTIYqlk2jTCgXk1TySIRyQazdLfmFrkwzpeMHZjORKxFvxypQPjcIrY44l5aFbKs4D6IgUKEs8pbi7ZRHhVvdYxBIOteLBqOUjNNrSCEYysDkihHlFXafqnbH5JoRjUC9RIQ2Rke5UEQwIjwibhAxTkSAtetWkichV3GRK490PbLhWaklWjzryIwYGZfWJykJuZDhZI0TcqdW5JjdZf+XusGwcLH/cwzVBAnN+k6DVMotaZNa2uTIln56yrprC2ZuoFNXPjYyjXnolhee9e6hZPPast9RtfIoxV9QvR+dKRPZAWjXzi9xm0cRf3izPPARlHcotXG4yt6JjFu/tdfdBeyfUF0dHqVj1A73YMD6wSj0x7Qe/gDL0AINPqxDBBJiMBiHwCHDZwkYUEgQW4EcsRQjZXkJBVRRu45ZEjM4oms4t3G31Edj3Jc1M6v28SshvikqXajTz/QN3aGf6Fv6hf44tFZua5ReNnEVPa1MVoeeXpz7/k9VhKuBzp7qSM8GApiwXhV6TyxSnsLv6btbz3bmbs/W86v0Jf2K/l/QbfoRTxB3v/mvZuTs8yP8BDhvYq0ImQLbxP5sysFg3htjdIzNeKNTE/2GDcJluAIN7MotmIJ7MA1N8J0nzmvnnfO+0qjcrzQrC71UZ6CvuQD7RuXRT5UFTvc=</latexit><latexit

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t b =40, g c =1, q=0.1

ma =100GeV, mA =600GeV
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Here we assume CP invariance in the Eq. (2.2). Thus all the parameters in Eq. (2.2)
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shows
Figure
The thecomparison about t
5, and theSI
lower for |c2 | = 1. The left column shows the casem2a| = 0.5,70 and the lower
case for |c
ght for ma = 100 GeV. The other parameters are fixed m
600theGeV,
GeV,
right for ma = 10
A =andmag=70= 0.1,= 10,
c1 = 0 for all the figure. The region where
 GeV
1 is the ✓left
handt sideof c1 = 0 for all th
the dashed vertical line. The expe
al line. The experimental bounds are the sameas in Fig. 6.
 

σ is large if C2 ≠ 0

 SI [cm2 ]10-46)"
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200

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400

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Summary
2 †

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2
VTHDM =m1 H1 H1 + m2 H2 H2 m3 H1 H2 + h.c.
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V)14-B1A?10=A->@5//;A<85:3?
port =(ia0 H1 H2 + h.c.) + c1 a0 H1 H1 + c2 a0 H2 H2 .
A 8-*=&6*3+8*2.,236*).28-*0.8*6&896*7
A 8-*=7.,2.?(&280=&B*(878-*(9'.(.28*6&(8.328*61&&-
e we assume
CP invariance in the Eq. (2.2). Thus all the param
A Da(&2'*;.8-.28-*+9896*46374*(873+"*232&2)#also assume
softly broken Z4 symmetry to avoid flavor changinextension of the softly broken Z2 symmetry often assumed in stud3
avoid flavor changing scalar couplings at the tree level [?]. Unde
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triangle diagrams with gluons@>5-:38105-3>-9C5@438A;:?
A

8;30334

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mQ Q̄Q =Q = u, b, c (heavy quarks)
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5

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a/A

a/A

<latexit sha1_base64="csncdjiOFmho4lbqvg1ulEfAm1w=">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</latexit>

mQ Q̄Q =<latexit sha1_base64="csncdjiOFmho4lbqvg1ulEfAm1w=">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</latexit><latexit sha1_base64="Smk667q2er+/LJ97SnGFkCM1MGo=">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</latexit>↵s a aµ⌫G G
12⇡ µ⌫<latexit sha1_base64="Jgbd7ozt+acrKRgrgjGfbtpC7Oo=">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</latexit>

a/A<latexit sha1_base64="csncdjiOFmho4lbqvg1ulEfAm1w=">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</latexit><latexit sha1_base64="csncdjiOFmho4lbqvg1ulEfAm1w=">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</latexit>

h/H
Q

a/A
h/H

Q

Q

Q

Q

<latexit sha1_base64="svx+WB38pKXXtQsf9UWsPjq3h34=">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</latexit><latexit sha1_base64="nOQiWPkMTBgKFQ6lDSCWZC2yVrA=">AAAFAnichVI7b9RAEJ7cHRCOhCTQINFY5A4FZB27BkSEFCkSTco8yEPKhcj2rX2rrL2WveeQWO6o+AMUVCBRABWiQ3Q0/AGKFPwARBkkGgrGeyYhCRfWsj37PcazM3YiwRNFyN5QpVo7c/bc8Pn6hZHRi2PjE5dWEtmLXbbsSiHjNcdOmOAhW1ZcCbYWxcwOHMFWna0HBb+asjjhMnyodiK2Edh+yD3u2gqhzYlK2naYz8PMCzyPC5ZnHW77sR3kRv2QQSDq3synqEVMSsgN5BAVzFMZpya3cr2Pud9VmaSmRKBZII5USgaZw02Hmo5lOrdN547peJi8qS1ZJGwemoqFRYUzVp5xy0wtc9vSSbSkYyddlhxoaJ6hIE1R82+BZQrbYWKm0b0110AxRXGp9FgcoOQw1x/pQqMftRLeYTPFwbTRt0wf7bS0+6InjxaLkoEkmiUtG+HFjO0yLexIlekTImXojyaGJ2ODPVYsDm1hFKNM+g0u2KzRdru8UXQm/wd60OsS9gslPYnJY1jbt4PAfnRXd2gwZQ2ktge79GRY2Pnr1zEOAP2X1TfHJ0mL6GWcDGgZTEK55uX4B2hDByS40IMAGISgMBZgQ4LXOlAgECG2ARliMUZc8wxyqKO3hyqGChvRLXz6uFsv0RD3Rc5Eu138isA7RqcBTfKFvCb75DN5S76RXwNzZTpHUcsOvp2+l0WbY0+vLP38ryvAt4LuoevUmhV4MK1r5Vh7pJHiFG7fn+4+21+6v9jMrpOX5DvW/4LskU94gjD94b5aYIvPT6nHw+cO5gqQyXFM9PhQTgYrVouSFl2wJmeny4ENw1W4BlM4lXswC3MwD8vgVr5Wq9WR6mjtSe1N7V3tfV9aGSo9l+HIqn38DV8/cno=</latexit>¯ Gaµ⌫ Gaµ⌫<latexit sha1_base64="sAwx4jBe/cxmwCmm9YA15JGt9lk=">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</latexit><latexit sha1_base64="QZx5PM5Ijf5ZIS8fRm9DIzH+uUo=">AAACtnichVHLShxRED3TSXyMUce4EbJpMhgCwlAtgoMrIYu49DUzgq3D7c6d8TL9oh8DYzM/kF1WWWSVgAvJBwSySnDjD7jwE0KWCm6ySHVPS0gkppquW3WqTt1zKStwVBQTXZa0Bw8fjY1PTJanHk/PzFbmnjQjPwlt2bB9xw/3LBFJR3myEavYkXtBKIVrObJl9V5m9VZfhpHyvd14EMgDV3Q91VG2iBlqV5ZMS4SpaR+poZ55/dWhaKemm+imlww5S4V+m7UrVapRbvrdwCiCKgrb9CtfYOI1fNhI4ELCQ8yxA4GIv30YIASMHSBlLORI5XWJIcrMTbhLcodgtMe+y9l+gXqcZzOjnG3zLQ7/ITN1LNIFndIVndMn+k4//zkrzWdkWgZ8WiOuDNqzbxZ2bv7LcvmMcfSbda/mGB3Uc62KtQc5kr3CHvH7x++udta2F9Pn9JF+sP4PdEln/AKvf22fbMnt9/fo6bAf8CyXK9majL+XcjdoLtcMqhlbK9X1erGwCTzFM7zgraxiHRvYRINveYvP+IpvWl071KTWHbVqpYIzjz9MC34Bmqmi1w==</latexit>.1E;:0@41%&;9;45>;.1$ !-3;E-'¯ Q̄QTable 2: Numerical values of form factors which is ta
form factors
micrOMEGAs[16]. Right panel shows the value for prot-@>5D18191:@?
A+361+&(8367+3659&6/7&6*

We useA +361+&(8367+36,0932.7
the following relations
to evaluate the spin-i
A
0334+&(836.28-*,09328*61.7&'736'*).2838-*+361+&(836operators,[ref](N )hN | mq qq | N i = mN fTq ,

↵
9↵s ⌦
(N )
a
aµ⌫
N Gµ⌫ GN = mN f TG ,
8⇡
✓
⌦
↵
1
q
N Oµ⌫ N =
pµ p⌫
mN

1 2
mN g µ4

where N stands for a nucleon, and mN is the nucleon m(N )factors (f ) and the second moments of the PDFs f.1E;:0@41%&;9;45>;.1$ !-3;E-'