the effect of the early kinetic decoupling in a fermionic ... · the e↵ect of the early kinetic...

The effect of the early kinetic decoupling in a fermionic dark matter model

Tomohiro Abe (IAR, KMI Nagoya U)

arXiv:2004.10041

Tomohiro Abe Institute for Advanced Research,

KMI, Nagoya U.

http://www.apple.com

Tomohiro Abe (IAR, KMI Nagoya U) 2

Weakly Interacting Massive ParticleWIMP (Weakly Interacting Massive Particle)

• has short range interactions with the standard model particles• energy density is explained by the freeze-out mechanism • correlation btw. various processes

DM

DM SM

SM

annihilation（thermal relic, indirect detection）

pair production（LHC)

scattering（

direct detection, kinetic equilibrium)


Constraints from direct detection

• WIMP models have been severely constrained today• We need ideas to avoid this strong constraint

101 102 103

WIMP mass [GeV/c2]

10°47

10°46

10°45

10°44

10°43

WIM

P-nu

cleo

ns S

I[c

m2 ]

LUX (2017)PandaX-II (2017)

XENON1T (1 t£yr, this work)

101 102 103

WIMP mass [GeV/c2]

10°1

100

101

Nor

mal

ized

[XENON1T (2018)]


Small coupling

• suppress the scattering cross section (σSI O(10-46) cm2)• avoid the constraint from XENON1T experiment• keep annihilation cross section ( = 10-26 cm3/s) by resonance

≪

⟨σv⟩

DM

DM SM

SM

annihilation（Keep this for Ωh2）

scattering（

suppress this to avoid Xenon1T)(example)

hλhsmake this small


DM temperature• temperature of DM is assumed to be the same as the temperature of

the thermal bath in WIMP models ( )• This assumption is valid if the scattering processes are frequent• Now the scattering is suppressed to avoid the XENON1T• We cannot assume

• We have to calculate by solving the Boltzmann equation

Tχ = T

Tχ = TTχ

DM

DM SM

SM

hλhssmall coupling


withOUT assuming Tχ = T

6

Boltzmann equation

these two higher-dimensional operators. This fermionic DM interacts with the SM particles only

through the exchange of the Higgs boson. The di↵erence between the two types of interactions is

important. For elastic scatterings of DM o↵ SM particles, the scattering amplitudes induced by

the CP-violating operator are suppressed by the momentum transfer in addition to the small DM-

Higgs coupling due to the Higgs resonance. The momentum transfer is very small because the DM

is non-relativistic in the scattering processes due to the Boltzmann suppression. Consequently, the

scattering is less e�cient if the CP-violating operator mainly induces the interaction. Therefore,

the e↵ect of the early kinetic decoupling is more important in the fermionic DM model with the

CP-violating coupling.

The rest of this paper is organized as follows. In Sec. 2, we briefly review the early kinetic

decoupling. The zeroth and second moments of the Boltzmann equation are discussed, which have

information on the number density and the temperature of DM, respectively. The coupled equations

to be solved are summarized. In Sec. 3, the fermionic DM model is described. The result with

the early kinetic decoupling is discussed in Sec. 4. We show the CP-violating interaction certainly

requires larger coupling compared to the one in the standard calculation to obtain the measured

value of the DM energy density. We vary the ratio of the CP-conserving and CP-violating couplings

and show that it a↵ects the kinetic decoupling. Using the values of the couplings required for the

right amount of the DM relic abundance, we discuss the Higgs invisible decay and prospects of its

measurements at collider experiments. We find that the branching ratio of the Higgs decaying into

two DM particles can be larger than the value predicted in the standard calculation. Section 5 is

devoted to our conclusion.

2 The early kinetic decoupling

We briefly review how to calculate the DM number density with taking into account the e↵ect

of the early kinetic decoupling based on the discussion in Ref. [7].

The Boltzmann equation for our universe is given by

E

✓@

@t�H~p ·

@

@~p

◆f�(t, ~p) = Cann.[f�] + Cel.[f�], (2.1)

where E is the energy of the DM, H is the Hubble parameter, ~p is the momentum of DM, and f� is

the phase-space density of DM. The collision term is divided into two parts. One is for annihilation

of pairs of DM particles (Cann.), and the other is for elastic scatterings of a DM particle o↵ a SM

3

n�(T�) = g�

Zd3p

(2⇡)3f�(~p, T�) = sY

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DM number density

particle in the thermal bath (Cel.). For two-to-two processes, they are written as

Cann. =1

2g�

XZd3p0

(2⇡)32Ep0

Zd3k

(2⇡)32Ek

Zd3k0

(2⇡)32Ek0(2⇡)4�4(p+ p

0� k � k

0)

⇥

⇣�|M��!BB0 |

2f�(~p)f�(~p0)(1± f

eqB (~k))(1± f

eqB0 (~k0))

+ |MBB0!��|2feqB (~k)f eq

B0 (~k0)(1± f�(~p))(1± f�(~p0))⌘, (2.2)

Cel. =1

2g�

XZd3p0

(2⇡)32Ep0

Zd3k

(2⇡)32Ek

Zd3k0

(2⇡)32Ek0(2⇡)4�4(p+ p

0� k � k

0)

⇥

⇣�|M�B!�B|

2f�(~p)f

eqB (~k)(1± f�(~p0))(1± f

eq.B (~k0))

+ |M�B!�B|2f�(~p0)f

eq.B (~k0)(1± f�(~p))(1± f

eq.B (~k))

⌘, (2.3)

where B and B0 stand for particles in the thermal bath such as quarks, g� is the number of internal

degrees of freedom of DM, and feqB is given by the Fermi-Dirac or Bose-Einstein distribution

depending on the spin of B. The summation should be taken for all the internal degrees of freedom

for all the particles. For the non-relativistic DM, Cel. is simplified as1 [20]

Cel. '1

2g�E

@

@~p·

(1

384⇡3m3�T

ZdEkf

eqB (Ek)(1± f

eqB (Ek))

⇥

Z 0

�4k2cm

dt(�t)X

|M�B!�B|2

✓m�T

@

@~pf� + ~pf�

◆), (2.4)

where k2cm is given by

k2cm =

m2�(E

2k �m

2B)

m2� +m

2B + 2m�Ek

. (2.5)

Here Ek is the energy of B. Note that k2cm 6= E2k �m

2B = |~k|

2.

The temperature of the DM, T�, and a related variable y are defined by

T� =g�

3n�

Zd3p

(2⇡)3~p2

Ef�(~p) =

s2/3

m�y, (2.6)

where n� is the number density of the DM, and s is the entropy density. Here s is a function of

the temperature of the thermal bath, T . From this definition, T� and y are the function of T . The

yield and x are defined as usual,

Y =n�

s, x =

m�

T. (2.7)

1 Eq. (2.4) is the same as Eq. (5) in [7]. The expression here makes it clear that Cel. does not contribute to thezeroth moment of the Boltzmann equation.

4

DM temperature

dn�

dt=(complicated equations),

dT�

dt=(complicated equations).

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Fermion dark matter model

7

• : dark matter (Majorana fermion, gauge singlet)• H : Higgs boson

χ

Two types of interactions• scalar-type : • pseudo scalar-type :

χχH†Hχiγ5χH†H

During the QCD phase transition, we cannot treat particles as free particles. Dedicated studies

are required for that regime. In Ref. [21], the table is provided for g⇤ and gs for 0.036 MeV

. T . 8.6 TeV. Since the values of g⇤ and gs do not change for T . 0.036 MeV, we can regard the

values of g⇤ and gs at T = 0.036 MeV as the values at the temperature today.

We solve Eqs. (2.8) and (2.9) numerically with the following initial condition

Y (xini.) =Yeq(xini.), (2.17)

y(xini.) =yeq(xini.), (2.18)

where xini. ' 10. After solving the coupled equations and obtain Y (x0), where x0 is defined by the

temperature of the current universe T0 as x0 = m�/T0, we convert Y (x0) into ⌦h2 that is given by

⌦h2 =m�s0Y (x0)

⇢cr.h�2

, (2.19)

where [22]

s0 =2⇡2

45gs(x0)T

30 , (2.20)

⇢cr.h�2 =1.05371⇥ 10�5 [GeV cm�3], (2.21)

T0 =2.35⇥ 10�13 [GeV]. (2.22)

The measured value of ⌦h2 by the Planck Collaboration is ⌦h2 = 0.120± 0.001 [23]. We can use

this value to determine a model parameter.

3 Model

We describe a model that we investigate in the following. We consider a gauge singlet Majorana

fermion DM. A discrete symmetry Z2 is assumed to stabilize the DM particle. Under the Z2

symmetry, the DM is odd while all the other particles, namely the SM particles, are even. Then,

renormalizable operators composed of the DM and SM fields are forbidden. The DM particle

interacts with the SM particles through higher-dimensional operators. Therefore, the model is

regarded as an e↵ective theory of fermionic DM models. Up to dimension-five operators, the

Lagrangian is given by

L =LSM +1

2� (i�µ@µ �m�)�+

cs

2��

✓H

†H �

v2

2

◆+

cp

2�i�5�

✓H

†H �

v2

2

◆, (3.1)

where � is the DM candidate, H is the SM Higgs field, and v is the vacuum expectation value

of the Higgs field, v ' 246 GeV. The three parameters (m�, cs, and cp) are real. There are two

6


Scattering process

8

q

• suppressed by • For non-relativistic DM, is typically small• elastic scattering is more suppressed by pseudo-

scalar coupling

qq

Two types of interactions• scalar-type : • pseudo scalar-type :

χχH†Hχiγ5χH†H

During the QCD phase transition, we cannot treat particles as free particles. Dedicated studies

are required for that regime. In Ref. [21], the table is provided for g⇤ and gs for 0.036 MeV

. T . 8.6 TeV. Since the values of g⇤ and gs do not change for T . 0.036 MeV, we can regard the

values of g⇤ and gs at T = 0.036 MeV as the values at the temperature today.

We solve Eqs. (2.8) and (2.9) numerically with the following initial condition

Y (xini.) =Yeq(xini.), (2.17)

y(xini.) =yeq(xini.), (2.18)

where xini. ' 10. After solving the coupled equations and obtain Y (x0), where x0 is defined by the

temperature of the current universe T0 as x0 = m�/T0, we convert Y (x0) into ⌦h2 that is given by

⌦h2 =m�s0Y (x0)

⇢cr.h�2

, (2.19)

where [22]

s0 =2⇡2

45gs(x0)T

30 , (2.20)

⇢cr.h�2 =1.05371⇥ 10�5 [GeV cm�3], (2.21)

T0 =2.35⇥ 10�13 [GeV]. (2.22)

The measured value of ⌦h2 by the Planck Collaboration is ⌦h2 = 0.120± 0.001 [23]. We can use

this value to determine a model parameter.

3 Model

We describe a model that we investigate in the following. We consider a gauge singlet Majorana

fermion DM. A discrete symmetry Z2 is assumed to stabilize the DM particle. Under the Z2

symmetry, the DM is odd while all the other particles, namely the SM particles, are even. Then,

renormalizable operators composed of the DM and SM fields are forbidden. The DM particle

interacts with the SM particles through higher-dimensional operators. Therefore, the model is

regarded as an e↵ective theory of fermionic DM models. Up to dimension-five operators, the

Lagrangian is given by

L =LSM +1

2� (i�µ@µ �m�)�+

cs

2��

✓H

†H �

v2

2

◆+

cp

2�i�5�

✓H

†H �

v2

2

◆, (3.1)

where � is the DM candidate, H is the SM Higgs field, and v is the vacuum expectation value

of the Higgs field, v ' 246 GeV. The three parameters (m�, cs, and cp) are real. There are two

6

SM

�

SM

�

' u(p)(cs + icp�5)u(p) ⇠ cs + cp~q·~s

mDM

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Result : only pseudo-scalar case

9

Brinv>0.13

Br inv=0.019

Br inv=0.0026

Br inv=0.00024

QCD-BQCD-AT

�=T

50 52 54 56 58 60 620.001

0.005

0.010

0.050

0.100

0.500

1

mDM [GeV]

c p[TeV

-1]

cs=0

Figure 1: Left: The values of cp that explain the measured value of the DM energy density in the maximally

CP-violating case. The black-solid curve is for the standard calculation without taking into account the e↵ect

of the early kinetic decoupling. The blue-dashed and blue-dotted curves are the results with the e↵ect of

the early kinetic decoupling in the QCD-A and QCD-B scenario, respectively. The constraint and prospects

from the Higgs invisible decay search are also shown. The gray shaded region is already excluded by the

ATLAS and CMS experiments. The black dashed curves show the prospects of the HL-LHC, ILC, and FCC

experiments. Right: The branching ratio of the Higgs invisible decay for cs = 0. The color notations are

the same as in the left panel.

at 95% CL. The prospects of various experiments are summarized in [29],

BRinv <

8>>>>><

>>>>>:

0.019 (HL-LHC)

0.0026 (ILC(250))

0.00024 (FCC)

(4.2)

at 95% CL, where FCC corresponds to the combined performance of FCC-ee240, FCC-ee365, FCC-

eh, and FCC-hh. The prospects for the ILC, and FCC are obtained by combining with the HL-LHC.

We show the model prediction of the branching ratio of the Higgs invisible decay in the right panel

in Fig. 1 with these prospects and the current bound. Due to the large enhancement of cp by

the early kinetic decoupling, the bound on the mass of the DM is stringent. The current lower

mass bound on the DM is obtained as 58.1 GeV in QCD-B, while it is 55.2 GeV in the standard

treatment where the e↵ect of the kinetic decoupling is ignored. The constraint and prospects are

also shown in the left panel in Fig. 1.

10

BRinv <

(0.13 (ATLAS)

0.19 (CMS)

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BRinv <

8>>>>>>>><

>>>>>>>>:

0.019 (HL-LHC)

0.0026 (ILC(250))

0.0023 ILC500

0.0022 ILC1000

0.0027 (CEPC)

0.00024 (FCC)

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current bound

prospects


Summary

10

WIMP models are constrained by direct detection experiments • small coupling with the Higgs resonance is a possible way to avoid the

constraints

is not a good assumption • though it is assumed in standard calculations• should be calculated from the Boltzmann equation

Larger coupling is required • to explain the energy density of DM by the freeze-out mechanism• the coupling enhancement is really large in a fermionic DM model• more chance to see DM signals at experiments

Tχ = T

Tχ

Backup

12


Freeze-out mechanism

13

dn

dt+ 3Hn = �h�vi

�n2 � n

2eq

�scale factor

a3

na3

H =1

a

da

dt

a

volume：

Hubble const.particle number

DM

DM SM

SM

annihilation stop if the expansion ratio is larger than the

annihilation rate

creation stop if temperature of the universe decrees and creation process is kinematically forbidden

Eur. Phys. J. C (2018) 78:203 Page 3 of 57 203

being the entropy density (heff(T ) is the effective number ofentropy degrees of freedom at the temperature T ), so that itis possible to get rid of the term dependent on the Hubbleexpansion rate on the left-hand side of Eq. (1), giving:

dY!

dt= ds

dt!"v"3H

Y 2!

!

1 #Y 2

! ,eq

Y 2!

"

. (7)

To obtain the last equation we have used the entropyconservation relation ds

dt = #3Hs. Qualitatively, Eq. (7),describes the following picture: If DM interactions areenough efficient, as in the case of WIMPs, at early timesthe annihilation rate #ann = !"v"Y! s exceeds the Hubbleexpansion rate and Eq. (7) is solved for Y! = Y! ,eq , mean-ing that the DM is in thermal equilibrium with the primordialthermal bath. At later times, when the temperature eventu-ally drops below the DM mass, the DM yield becomes Boltz-mann suppressed, Y! ,eq $ exp(#m!/T ), so that the annihi-lation rate falls below the Hubble expansion rate leading tothe thermal freeze-out of this “cold” relic, i.e., thereafter Y!

is approximately constant with time.1 Equation (7) can besolved by adopting the temperature T 2 of the thermal bathor x = m!/T as independent variable. A good approxi-mate solution is represented by the following semi-analyticalexpression [23]:

Y (T0) % Y0 &#

$

45MPl

$% T f

T0

g1/2' !"v"dT

&#1

, (9)

where:

g1/2' = heff

g1/2eff

'1 + 1

3Theff

dheff

dT

(, (10)

with T0 as the present time temperature while T f representsthe freeze-out temperature which can be determined by solv-ing the equation:

#$

45MPl

g1/2' m!

x2 !"v"Y! ,eq%(% + 2) = #d log Y! ,eq

dx, (11)

where % = (Y! # Y! ,eq)/Y! ,eq is conventionally set to 1.5while x = m!/T .

The DM relic abundance is usually expressed in terms ofthe parameter &DMh2 where h ( 0.7 is the value Hubbleexpansion rate at present times in units of 100 (km/s)/Mpc

1 See Ref. [22] for an exception (“relentless” DM) for modified expan-sion histories.2 We make also use of: ds

dt =)

3sT

*1 + T

3heff

dheffdT

+,dTdt . (8)

Fig. 2 Comoving number density evolution as a function of the ratiom!/T in the context of the thermal freeze-out. Notice that the sizeof the annihilation cross-section determines the DM abundance since&DMh2 $ 1/!"v"

while &DM represents the ratio between the DM energy den-sity 'DM and the so called critical energy density 'cr , namely:

&DM = 'DM/'cr(T0), 'DM = m! s0Y0,

'cr(T ) = 3H(T )2M2PL/8$, 'cr(T0) & 10#5 GeV cm#3,

(12)

where s0 = s(T0) is the entropy density at present times.By combining the expressions above, the DM relic density

can be numerically estimated as:

&DMh2 ) 8.76 * 10#11 GeV#2$% T f

T0

g1/2' !"v"dT

m!

&#1

.

(13)

The behavior of the solution of the Boltzmann equationis illustrated in Fig. 2. As expected, the DM relic density isbasically set by the inverse value of the thermally averagedcross-section (calculated at the freeze-out temperature), witha logarithmic dependence on m! . It can be straightforwardlyverified that the experimental determination of &DMh2 )0.12 [1] is matched by a value of the cross-section of the orderof 10#9 GeV#2 corresponding to !"v" ( 10#26 cm3 s#1.

The WIMP paradigm hence reduces, under the hypothe-sis of standard cosmological evolution of the Universe, thesolution of the DM problem to the determination of a sin-gle particle physics input, i.e., the thermally averaged pairannihilation cross-section of the DM.

Its formal definition reads [23]:3

3 In scenarios where the DM is not the only new particle state, otherprocesses like co-annihilations, might also be relevant for the DM relicdensity. A more general definition of !"v", including such processes,can be found e.g., in Ref. [24].

123

[Arcadi+ (1703.07364)]

thermal average of the annihilation cross section


This constraint is very strong(e.g.) SM + singlet scalar

• introduce a gauge singlet scalar “S” that is Z2 odd

• two free parameters: DM mass and λsH

• determine λsH to obtain measured value of the DM energy density• then we can predict σSI uniquely

L = LSM +1

2@µS@µS � m

2

2S2 � �sH

2S2H

†H � �s

4!S4

<latexit sha1_base64="1qa/4esHeJJbEVqo9hEO9lZCI2Q=">AAADfnichVHLbtNAFL2ueZTwSCgbJDYDUSsk1DAOrYiQgEpssgCprUlbqW6s8WSSWh0/ZE8iwsg/wA+wYAUSi4rPYMMPsOiKNWJZJDYsuHbcIiikY3nmzrnn3LkPL5Z+qig9MGbMM2fPnZ+9ULl46fKVau3q3EYaDRMuOjySUbLlsVRIPxQd5SsptuJEsMCTYtPbe5L7N0ciSf0ofK7GsdgJ2CD0+z5nCiG3tq8dziR5mpGH5Mh0taPEC6XtZ1lWuUOcfsK4tjLdzIgTs0T5THa1EwwzYh8DbglUFkt+0G0WCrvbPMYciYn1mKvTdnbkJO2u02ODgUhI+x9E5C3dzIlLbq1OG7RY5KRhlUYdyrUa1b6AAz2IgMMQAhAQgkJbAoMUv22wgEKM2A5oxBK0/MIvIIMKaofIEshgiO7hPsDbdomGeM9jpoWa4ysS/wSVBObpZ7pPD+kn+oF+pT//G0sXMfJcxnh6E62I3eqr6/aPU1UBngp2f6umKNKy6l289aZUpzGL4JTqFfShVVTtYxfiAsn7wSeZjF6+PrQfrM/rBfqOfsNOvKUH9CP2Ihx95+/XxPqbKa/3cR9jrAA9GQ7c+nu8J42NZsO616BrtL7SKkc/CzfgFtzG+d6HFWjDKnSAG1Vj2XhkPDbBXDAXzbsT6oxRaq7BH8ts/QKk8tr6</latexit>

[Silveria et.al. (‘85), McDonald (’94), Burgess (’01), …,Cline et.al. (’13), TA Kitano Sato (’15), …GAMBIT collaboration (’17, ‘19) ]


How strong is the Xenon1T constraint?singlet scalar DM vs direct detection

• mDM > 1TeV is allowed• mDM ~ mh/2 is also allowed (Higgs funnel/resonance)

10-50

10-49

10-48

10-47

10-46

10-45

10-44

10-43

10-42

10-41

101 102 103 104

� SI[cm

2 ]

mDM [GeV]

scalar singlet

Neutrino

FloorXEN

ONnT/LZ

(prospect)

Xenon1T

(2018)

[TA Kitano Sato (’15) w/ update]


Early kinetic decouplingCalculation in the Higgs funnel region required dedicated study

• coupling is very small• kinetic decoupling happens before chemical decoupling• ( is assumed in standard treatment)• coupling should be larger ==> stronger constraint from Xenon1T

Tχ ≠ T Tχ = T

[Duch, Grzadkowski (’17);Binder, Bringmann, Gustafsson, Hryczuk (’17)]

7

FIG. 1. The required value of the Singlet-Higgs coupling �S ,as a function of the Scalar Singlet mass mS , in order to obtaina relic density of ⌦h2 = 0.1188. The blue dashed line showsthe standard result as established by Gondolo & Gelmini [9],based on the assumption of local thermal equilibrium duringfreeze-out. For comparison, we also plot the result of solvinginstead the coupled system of Boltzmann equations (27) and(28) for the maximal (‘B’) and minimal (‘A’) quark scatter-ing scenarios defined in the main text (red solid and dashedlines, respectively). Finally, we show the result of fully solvingthe Boltzmann equation numerically, for the maximal quarkscattering scenario and with no DM self-interactions included(‘full BE’).

B: only light quarks (u, d, s) contribute to the scat-tering, and only for temperatures above 4Tc ⇠

600MeV, below which hadronization e↵ects start tobecome sizeable [63] (smallest scattering scenario,as adopted in [12]).

Finally, we adopt the recent results from Drees etal. [64] for the e↵ective number of relativistic de-grees of freedom ge↵(T ) that enter the calculation ofthe Hubble rate during radiation domination, H =p

4⇡3ge↵/45T 2/mPl, as well as the entropy degrees of

freedom entering for example in the calculation of g(T )as defined in Eq. (18).

B. Relic density of scalar singlet dark matter

Let us first compute the relic density following thestandard treatment adopted in the literature. To thisend, we numerically solve Eq. (17) for a given set of pa-rameters (mS ,�S) and determine the resulting asymp-totic value of Y0. The blue dashed line in Fig. 1 showsthe contour in this plane that results in Y0 correspond-ing to a relic density of ⌦h2 = 0.1188, c.f. Eq. (19). Werestrict our discussion to values of mS in the kinematicrange where h�vi is enhanced due to the Higgs propaga-tor given in Eq. (41), and the coupling �S that resultsin the correct relic density is hence correspondingly de-

FIG. 2. Temperatures at which DM number density and ve-locity dispersion (‘temperature’) start to deviate from theirequlibrium values, defined for the purpose of this figure as|Y �Yeq| = 0.1Yeq and |y�yeq| = 0.1 yeq, respectively. Thesecurves are based on solving the coupled system of Boltzmannequations (27) and (28), for the same parameter combinationsas in Fig. 1 (resulting thus in the correct relic density).

creased. This curve agrees with the corresponding resultobtained in Ref. [49].

For comparison, we show in the same figure the re-quired value of �S that results when instead solving thecoupled system of Boltzmann equations (27) and (28), orwhen numerically solving the full Boltzmann equation asdescribed in Section IIC . Here, the solid (dashed) lineshows the situation for the ‘B’ (‘A’) scenario for scat-terings on quarks. Outside the resonance region, thecoupled Boltzmann equations lead to identical resultscompared to the standard approach, indicating that ki-netic decoupling indeed happens much later than chemi-cal decoupling and that the assumption of local thermalequilibrium during chemical freeze-out thus is satisfied.For DM masses inside the resonance region, on the otherhand, we can see that the two methods can give signif-icantly di↵erent results, implying that this assumptionmust be violated. For the same reason, a smaller scat-tering rate (as in scenario ‘B’) leads to an even largerdeviation from the standard scenario than the maximalscattering rate adopted in scenario ‘A’.

This interpretation is explicitly confirmed in Fig. 2,where we plot the temperatures at which the DM num-ber density and temperature start to deviate from theequilibrium values: in the parameter range that we focuson here, kinetic decoupling happens indeed very closeto chemical decoupling. The reason for this very earlykinetic decoupling is straight-forward to understand asthe result of a strongly suppressed momentum transferrate �(T ), compared to the annihilation rate, due totwo independent e↵ects: i) the small coupling �S neededto satisfy the relic density requirement, without a cor-responding resonant enhancement of �(T ), and ii) the


withOUT assuming (cont’d)Tχ = T

17

Note that x is defined by T not T�. Di↵erential equations for Y and y are obtained from the zeroth

and second moments of the Boltzmann equation, namely g�R d3p

(2⇡)31E⇥Eq. (2.1) and g�

R d3p(2⇡)3

1E

~p2

E2⇥

Eq. (2.1). Note that the elastic scattering term given in Eq. (2.4) does not contribute to the zeroth

moment term. This is a natural consequence because the elastic scattering processes do not change

the number density of DM. After some algebra, the following coupled equations are obtained.

dY

dx=

s8m2

pl⇡2

45

m�

x2

pg⇤(T )

⇣�h�viT�

Y2 + h�viT Y

2eq

⌘, (2.8)

1

y

dy

dx=

s8m2

pl⇡2

45

m�

x2

pg⇤(T )

(Y

⇣h�viT�

� h�vi2,T�

⌘+

Y2eq

Y

✓yeq

yh�vi2,T � h�viT

◆)

+pg⇤(T )

x2

gs(T )�

✓yeq

y� 1

◆+

✓1 +

T

3gs(T )

dgs(T )

dT

◆1

3m�

yeq

y

⌧p4

E3

�, (2.9)

where

pg⇤(T ) =

gs(T )pg(T )

✓1 +

T

3gs(T )

dgs(T )

dT

◆, (2.10)

� =

s8m2

pl⇡2

45

15

256⇡5m6�g�

X

B

Z 1

mB

dEkfeqB (Ek)(1± f

eqB (Ek))

Z 0

�4k2cm

dt(�t)X

|M�B!�B|2

=

s8m2

pl⇡2

45

15T

16⇡5m6�g�

X

B

Z 1

mB

dEkfeqB (Ek)

@k2cm

@Ekk2cm

X|M�B!�B|

2��t=�4k2cm

, (2.11)

h�viT�=

g2�

(neq� )2

Zd3p

(2⇡)3

Zd3q

(2⇡)3(�v)��!BB0 f

eq� (~p, T�)f

eq� (~q, T�), (2.12)

h�vi2,T�=

g2�

(neq� )2T�

Zd3p

(2⇡)3

Zd3q

(2⇡)3~p · ~p

3E(�v)��!BB0 f

eq� (~p, T�)f

eq� (~q, T�), (2.13)

⌧p4

E3

�=

g�

neq� (T�)

Zd3p

(2⇡)3(~p · ~p)2

E3e� E

T� . (2.14)

Here g and gs are the e↵ective degrees of freedom for the energy and entropy densities respectively,

feq� is given by the Boltzmann distribution, and mpl is the reduced Plank mass, mpl = 1.220910⇥

1019(8⇡)�1/2 GeV. For h�viT and h�vi2,T , replace T� by T in h�viT�and h�vi2,T�

, respectively.

neq� (T�) is given by

neq� (T�) = g�

Zd3p

(2⇡)3feq� (~p, T�) = g�

Zd3p

(2⇡)3e�Ep

T� . (2.15)

From the first to the second line in Eq. (2.11), we used the following relation,

feqB (Ek)(1± f

eqB (Ek)) = �T

@

@EkfeqB (Ek), (2.16)

and integration by parts.

5


withOUT assuming (cont’d)Tχ = T

18



(2⇡)31E⇥Eq. (2.1) and g�

R d3p(2⇡)3

1E

~p2

E2⇥




dY

dx=

s8m2

pl⇡2

45

m�

x2

pg⇤(T )

⇣�h�viT�

Y2 + h�viT Y

2eq

⌘, (2.8)

1

y

dy

dx=

s8m2

pl⇡2

45

m�

x2

pg⇤(T )

(Y

⇣h�viT�

� h�vi2,T�

⌘+

Y2eq

Y

✓yeq


◆)

+pg⇤(T )

x2

gs(T )�

✓yeq

y� 1

◆+

✓1 +

T

3gs(T )

dgs(T )

dT

◆1

3m�

yeq

y

⌧p4

E3

�, (2.9)

where

pg⇤(T ) =

gs(T )pg(T )

✓1 +

T

3gs(T )

dgs(T )

dT

◆, (2.10)

� =

s8m2

pl⇡2

45

15

256⇡5m6�g�

X

B

Z 1

mB

dEkfeqB (Ek)(1± f

eqB (Ek))

Z 0

�4k2cm

dt(�t)X

|M�B!�B|2

=

s8m2

pl⇡2

45

15T

16⇡5m6�g�

X

B

Z 1

mB

dEkfeqB (Ek)

@k2cm

@Ekk2cm

X|M�B!�B|

2��t=�4k2cm

, (2.11)

h�viT�=

g2�

(neq� )2

Zd3p

(2⇡)3

Zd3q

(2⇡)3(�v)��!BB0 f

eq� (~p, T�)f

eq� (~q, T�), (2.12)

h�vi2,T�=

g2�

(neq� )2T�

Zd3p

(2⇡)3

Zd3q

(2⇡)3~p · ~p

3E(�v)��!BB0 f

eq� (~p, T�)f

eq� (~q, T�), (2.13)

⌧p4

E3

�=

g�

neq� (T�)

Zd3p

(2⇡)3(~p · ~p)2

E3e� E

T� . (2.14)




, respectively.


neq� (T�) = g�

Zd3p

(2⇡)3feq� (~p, T�) = g�

Zd3p

(2⇡)3e�Ep

T� . (2.15)


feqB (Ek)(1± f

eqB (Ek)) = �T

@



5



(2⇡)31E⇥Eq. (2.1) and g�

R d3p(2⇡)3

1E

~p2

E2⇥




dY

dx=

s8m2

pl⇡2

45

m�

x2

pg⇤(T )

⇣�h�viT�

Y2 + h�viT Y

2eq

⌘, (2.8)

1

y

dy

dx=

s8m2

pl⇡2

45

m�

x2

pg⇤(T )

(Y

⇣h�viT�

� h�vi2,T�

⌘+

Y2eq

Y

✓yeq


◆)

+p

g⇤(T )x2

gs(T )�

✓yeq

y� 1

◆+

✓1 +

T

3gs(T )

dgs(T )

dT

◆1

3m�

yeq

y

⌧p4

E3

�, (2.9)

where

pg⇤(T ) =

gs(T )pg(T )

✓1 +

T

3gs(T )

dgs(T )

dT

◆, (2.10)

� =

s8m2

pl⇡2

45

15

256⇡5m6�g�

X

B

Z 1

mB

dEkfeqB (Ek)(1± f

eqB (Ek))

Z 0

�4k2cm

dt(�t)X

|M�B!�B|2

=

s8m2

pl⇡2

45

15T

16⇡5m6�g�

X

B

Z 1

mB

dEkfeqB (Ek)

@k2cm

@Ekk2cm

X|M�B!�B|

2��t=�4k2cm

, (2.11)

h�viT�=

g2�

(neq� )2

Zd3p

(2⇡)3

Zd3q

(2⇡)3(�v)��!BB0 f

eq� (~p, T�)f

eq� (~q, T�), (2.12)

h�vi2,T�=

g2�

(neq� )2T�

Zd3p

(2⇡)3

Zd3q

(2⇡)3~p · ~p

3E(�v)��!BB0 f

eq� (~p, T�)f

eq� (~q, T�), (2.13)

⌧p4

E3

�=

g�

neq� (T�)

Zd3p

(2⇡)3(~p · ~p)2

E3e� E

T� . (2.14)




, respectively.


neq� (T�) = g�

Zd3p

(2⇡)3feq� (~p, T�) = g�

Zd3p

(2⇡)3e�Ep

T� . (2.15)


feqB (Ek)(1± f

eqB (Ek)) = �T

@



5

elastic scatteringB: particles in the thermal bath

<latexit sha1_base64="7rX/uTf4cac5M6bemSW9wX+2svA=">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</latexit>


Result3: with scalar-couplings

19

XENON1T

XENONnT/LZ

� floor

Brinv>0.13

Br inv=0.019

Br inv=0.0026

Br inv=0.00024

QCD-BQCD-A

T�=T

50 52 54 56 58 60 620.000

0.005

0.010

0.050

0.100

0.500

1

mDM [GeV]

c p[TeV

-1]

cs=0.1cp

[-1]

Figure 3: The values of the couplings that explain the measured value of the DM energy density. The

blue-hatched region (\\\) is excluded by the XENON1T experiment [4]. The red-dashed line shows the

prospect of the XENONnT and LZ experiments [32, 33]. The orange-hatched region (///) is below the

neutrino floor and cannot be accessed by the direct detection experiments. The other color notation is the

same as in Fig. 1.

the Higgs invisible decay is essential to test the model.

13

XENON1T

XENONnT/LZ

� floor

Brinv>0.13

Br inv=0.019

Br inv=0.0026

Br inv=0.00024

50 52 54 56 58 60 620.001

0.010

0.100

1

10

mDM [GeV]

c s[TeV

-1]

cp=0

Figure 3: The values of the couplings that explain the measured value of the DM energy density. The

blue-hatched region (\\\) is excluded by the XENON1T experiment [4]. The red-dashed line shows the

prospect of the XENONnT and LZ experiments [32, 33]. The orange-hatched region (///) is below the

neutrino floor and cannot be accessed by the direct detection experiments. The other color notation is the

same as in Fig. 1.

the Higgs invisible decay is essential to test the model.

13

the effect of the early kinetic decoupling in a fermionic ... · the e↵ect of the early kinetic...

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