Thermal dark matter co-annihilating with a strongly interacting scalar

19 Pages·2017·0.69 MB·English

Published for SISSA by Springer
Received: January 25, 2018
Accepted: April 4, 2018
Published: April 13, 2018
Thermal dark matter co-annihilating with a strongly J
H
interacting scalar
E
P
0
4
S. Biondini and M. Laine
(
AEC, Institute for Theoretical Physics, University of Bern,
2
Sidlerstrasse 5, CH-3012 Bern, Switzerland
0
E-mail: [email protected], [email protected]
1
Abstract: Recently many investigations have considered Majorana dark matter co- 8
annihilating with bound states formed by a strongly interacting scalar field. However )
only the gluon radiation contribution to bound state formation and dissociation, which at
0
high temperatures is subleading to soft 2 2 scatterings, has been included. Making use
→ 7
of a non-relativistic effective theory framework and solving a plasma-modified Schr¨odinger
2
equation, we address the effect of soft 2 2 scatterings as well as the thermal dissociation
→
of bound states. We argue that the mass splitting between the Majorana and scalar field
has in general both a lower and an upper bound, and that the dark matter mass scale can
be pushed at least up to 5...6TeV.
Keywords: ThermalFieldTheory,CosmologyofTheoriesbeyondtheSM,EffectiveField
Theories, Nonperturbative Effects
ArXiv ePrint: 1801.05821
Open Access, (cid:13)c The Authors.
https://doi.org/10.1007/JHEP04(2018)072
Article funded by SCOAP3.
Contents
1 Introduction 1
2 Model 2
3 Parametric forms of thermal masses and interaction rates 3
4 Quantitative framework for estimating the annihilation rate 5
J
4.1 Non-relativistic fields 5
H
4.2 Imaginary parts of 4-particle operators 6
4.3 Number density, effective cross section, evolution equation 7 E
4.4 Plasma-modified Schr¨odinger equation and generalized Sommerfeld factors 8 P
4.5 Thermal potentials 9
0
5 Numerical evaluations 10 4
(
6 Conclusions 12
2
A Fixing of vacuum and thermal couplings 13 0
1
8
)
1 Introduction
0
7
Negative results from direct and indirect detection experiments and collider searches pose
a challenge for many minimal dark matter models. This has led to the construction of less 2
minimal models. In the latter, the cross sections probed experimentally are not directly
related to the cross section affecting freeze-out dynamics in the early universe. Therefore
experimental bounds might be respected while at the same time maintaining the correct
cosmological abundance.
IfthedarkmatterparticlesaremassiveandinteractstronglyenoughwiththeStandard
Model to have been in equilibrium with it at some time in the early universe, the basic
feature that is needed for the above task is a strongly temperature-dependent annihilation
cross section. At low temperatures, the cross section needs to be very small, to satisfy the
non-observation bounds from indirect detection. In the early universe, the cross section
needs to be large enough to keep dark matter in chemical equilibrium for a long while,
reducing its number density and thereby evading overclosure of the universe.
An example of a possible scenario along these lines is to postulate a model in which
the dark sector consists of two particle species. The lighter one is the true dark matter,
long-lived and interacting very weakly. In contrast, the heavier one could interact strongly
and act as an efficient dilution channel for the overall abundance at high temperatures (cf.,
e.g., ref. [1]).
– 1 –
If the heavy species interacts strongly, as in QCD, this scenario could lead to rather
rich phenomenology. Strongly interacting particles form generally bound states. Because
of the associated binding energy, their thermal abundance is larger than that for the same
particles in a scattering state. Bound states may annihilate efficiently because the two
particles are close to each other. Though often alluded to previously, a more intensive
study of bound-state effects on the freeze-out dynamics has only started a few years ago
(cf., e.g., refs. [2, 3]).
Recently, wehaveparticipatedindevelopinganon-perturbativeformalismforaddress-
ing the thermal annihilation of non-relativistic particles [4, 5]. The formalism was already
applied to a first full model, which did not include strongly interacting particles but never- J
theless displayed weakly bound states [6]. The purpose of the current paper is to apply the
H
formalismtoastronglyinteractingmodelthathasbeenmuchdiscussedinrecentliterature.
E
Our plan is as follows. Having introduced the model in section 2, we review some
P
salient features concerning its thermal behaviour in section 3. The main technical ingre-
dients of our analysis are specified in section 4: the operators responsible for the hard 0
annihilation event; the spectral functions describing the soft initial-state effects that in- 4
fluence the annihilation; as well as generalized “Sommerfeld factors” which capture the
(
effect both of bound and scattering states on the thermal annihilation cross sections. The
2
cosmological evolution equations are integrated in section 5, whereas conclusions and an
0
outlook are offered in section 6.
1
2 Model 8
)
ThemodelconsideredconsistsoftheStandardModelextendedbyagaugesingletMajorana
0
fermion (χ) as well as a scalar field (η) which is singlet in SU (2) but carries non-trivial
L
7
QCD and hypercharge quantum numbers.1 The Majorana fermion is chosen as the dark
2
matter particle, given that its low-energy scattering cross section is naturally suppressed,
being p-wave at tree level [8]. In the MSSM language, the Majorana fermion could be a
bino-like neutralino and the scalar a right-handed stop or sbottom. However, for generality
we do not fix couplings to their MSSM values. The hypercharge coupling of the scalar is
generally omitted, as its effects are subleading compared with QCD effects.
The Lagrangian for this extension of the Standard Model can be expressed as
1
= + χ¯ i/∂ M χ+(D η)†Dµη M2η†η λ (η†η)2
L LSM 2 − µ − η − 2
λ η†ηH(cid:0)†H yη†(cid:1)χ¯a q y∗q¯a χη. (2.1)
− 3 − R − L
The notation λ is reserved for the self-coupling of the Higgs doublet (H). The chiral
1
projectors a = (1 γ )/2, a = (1+γ )/2 imply that χ only interacts with SU (2) singlet
L − 5 R 5 L
projections of quarks. We assume that the Yukawa coupling y couples dominantly to one
quark flavour only. The Yukawa coupling determining the mass of that flavour is denoted
by h, and the strong gauge coupling by g . The free parameters of the model are then the
s
1An SUL(2) doublet η would lead to similar results but a somewhat more complicated analysis [7].
– 2 –
two mass scales2 M and ∆M M M as well as the two “portal” couplings λ and y 2
≡ η− 3 | |
that are assumed to be small at the MS scale µ¯ 2M.
∼
In the MSSM context, the importance of co-annihilations in such a model was stressed
longago[1]. SommerfeldenhancementsfromQCDinteractionswereincludedinrefs.[9,10],
however without the consideration of bound states. Similar theoretical ingredients were
appliedtothegeneralizedmodelinref.[11]. Direct, indirectandcolliderconstraintsonthe
generalized model were reviewed in ref. [7]. More recently, bound-state effects have been
approximately included in this model [12–14], as a single additional degree of freedom in a
set of Boltzmann equations, a treatment which we aim to improve upon in the following.
J
3 Parametric forms of thermal masses and interaction rates H
E
Thecolouredscalarsareresponsibleformostoftheannihilationsduringthermalfreeze-out.
P
We start by reviewing the thermal mass corrections and interaction rates that they experi-
0
ence. The important point is that, because of Bose enhancement, the gluonic contributions
are infrared (IR) sensitive, and need to be properly resummed for a correct result. 4
As a first step, consider a naive (i.e. unresummed) computation of the self-energy of (
the coloured scalar. Evaluating the (retarded) self-energy at the on-shell point yields (the
2
line “ ” stands for η and the wiggly line for a gluon)
0
ReΠ g2C T2 1
+ R = s F , (3.1)
⇒ 2Mη 12Mη 8
ImΠ
2 R = 0, (3.2) )
∼ ⇒ 2Mη 0
where C (N2 1)/(2N ). The real part is analogous to that for a heavy fermion [15]. 7
F ≡ c − c
The imaginary part vanishes because there is no phase space for the 1 2 process. 2
↔
However,athightemperaturesthese naive results are misleading. Perhapsthesimplest
way to see this is to replace the scalar in the loop by a particle with a different mass,
M +∆M, andconsiderthecase∆M πT M . ThenitcanbeverifiedthatReΠ /M
η ≪ ≪ η R η
is modified by a correction of order g2C ∆M, and ImΠ /M by a correction of order
∼ s F R η
g2C ∆M n ( ∆M ) g2C T, where n is the Bose distribution. In other words, the
∼ s F| | B | | ≈ s F B
result in eq. (3.2) seems to change qualitatively because Bose enhancement of the soft
contribution compensates against the phase-space suppression.
The correct treatment of the Bose-enhanced IR contribution requires resummation.
The heavy scalars are almost static, and interact mostly with colour-electric fields (Aa).
0
In a plasma, colour-electric fields get Debye screened. We denote the Debye mass by m .
D
Parametrically, m g T, where g √4πα . The proper inclusion of Debye screening in
D ∼ s s ≡ s
a gauge theory requires Hard Thermal Loop (HTL) resummation [16–19]. Recomputing
the 1-loop self-energy with HTL propagators, and setting ∆M 0 since IR sensitivity
→
2Moreprecisely,M andM refertotherenormalizedpartsofthemassesappearinginthenon-relativistic
η
effectivetheoryforχandη. Thenon-perturbativeQCDcontributiontoM isoftheorderO(GeV/TeV)∼
η
10−3 which is smaller than the effects that we discuss below.
– 3 –
has now been regulated, we get (here p p and a blob stands for a HTL-resummed
≡ | |
propagator)
ReΠ∗ g2C T2 g2C 1
+ R = s F + s F
⇒ 2M 12M 2 p2+m2
η η Zp D
g2C T2 g2C m
= s F s F D , (3.3)
12M − 8π
η
ImΠ∗ g2C πTm2
2 R = s F D
∼ ⇒ 2Mη − 2 Zp p(p2+m2D)2
g2C T J
= s F . (3.4)
− 8π H
Thenewcontributionineq.(3.3),originatingfromtheDebye-screenedCoulombself-energy, E
is known as the Salpeter correction (cf. ref. [20] for a review). It dominates over the other P
mass correction if T . g M , which is generally the case. The imaginary part in eq. (3.4),
s η 0
i.e. the interaction rate, reflects fast colour and phase-changing 2 2 scatterings off light
→ 4
medium particles. It was first derived for the case of a heavy quark [16].
We finally replace the coloured scalar by a pair of heavy scalars, separated by a dis- (
tance r. The HTL-resummed computation of the thermal mass correction (“static poten- 2
tial”) and interaction rate as a function of r was carried out in refs. [21–23]. At leading
0
non-trivial order the result can be expressed as
1
G(r,t) t→+∞ G(r,0) exp i[V(r) iΓ(r)]t , (3.5) 8
∼ − −
)
(cid:8) (cid:9)
g2C exp( m r)
r V(r) = s F m + − D , (3.6) 0
− 4π D r
(cid:20) (cid:21) 7
g2C T ∞ dzz sin(zm r)
Γ(r) = s F 1 D . (3.7) 2
2π (z2+1)2 − zm r
Z0 (cid:20) D (cid:21)
As a crosscheck, for r twice the results of eqs. (3.3) and (3.4) are reproduced.
→ ∞
The interaction rate in eq. (3.7) can again be traced back to 2 2 scatterings. At
→
short distances, up to logarithms, Γ(r) g2C Tm2r2. This can be compared with the
∼ s F D
1 2 gluon radiation contribution, g2C (∆E)3r2n (∆E) [23], where ∆E is the energy
↔ ∼ s F B
difference between the singlet and octet potentials. At high temperatures, when m ,πT
D ≫
∆E, the 2 2 contribution dominates over the 1 2 one.
→ ↔
In order to determine the spectral function of the scalar pair, characterizing the states
that appear in the scalar-antiscalar sector of the Fock space, V(r) and Γ(r) can be in-
serted into a time-dependent Schr¨odinger equation satisfied by the appropriate Green’s
function [24]. More details are given in section 4. We have checked numerically that,
in accordance with theoretical expectations [25], the states originating from this solution
respect the qualitative pattern seen above for Γ(r), namely that at high temperatures the
width from 2 2 reactions dominates over the gluo-dissociation contribution.
→
We close this section by considering another essential ingredient of the framework,
namely the rate at which Majorana dark matter particles convert into the coloured scalars.
– 4 –
Once more, this rate is dominated by 2 2 scatterings, and obtaining the correct result
→
requires HTL resummation. Setting for simplicity the external momentum to zero, we find
(the thick line is the Majorana fermion and the arrowed line the quark flavour with which
it interacts, treated for simplicity as massless in vacuum which is a good approximation if
m . πT)
vac
∼ 2+ 2 ⇒ ImΣ∗R = −|y8|M2Nc Zp πm2qpn(Fp(cid:0)2∆+Mm+2q)2pM2 (cid:1) (3.8)
y 2N m2 1.76388MT
| | c q ln , (3.9)
≈ − 64πM m2 J
(cid:18) q (cid:19)
H
where the last line applies under the assumption ∆M m ,πT √TM. The thermal
≪ q ≪ E
quark mass m , originating from the phase space integral of the light plasma particles off
q
which the 2 2 scattering takes place, is P
→
0
n (q)+n (q) g2T2C
m2q = 2gs2CF B q F = s 4 F . (3.10) 4
q
Z
(
The rate in eqs. (3.8) and (3.9) is faster than the Hubble rate in a broad temperature
2
range, e.g. down to M/T & 3000 for y = 0.3 and ∆M/M . 0.01. It does fall out of
0
equilibrium when T ∆M, however transitions to virtual bound-state constituents may
≪
continue and form presumably the relevant concern. Non-equilibrium effects have been 1
discussed in ref. [26]. 8
)
4 Quantitative framework for estimating the annihilation rate
0
Wenowpresentaframeworkforcomputing(co-)annihilationratesinthemodelofsection2. 7
2
4.1 Non-relativistic fields
The basic premise of our framework is to make use of the non-relativistic approximation,
assumingthat πT, m , ∆M M, whereM isthedarkmattermassand∆M = M M
top ≪ η−
is the mass splitting within the dark sector. This simplification opens up the avenue to a
non-relativistic effective field theory investigation of soft initial-state effects.
Inthenon-relativisticlimit, theinteractionpicturefieldoperatorofthecolouredscalar
is expressed as
1 1
η = φe−iMηt+ϕ†eiMηt , η† = ϕe−iMηt+φ†eiMηt . (4.1)
2M 2M
η η
(cid:16) (cid:17) (cid:16) (cid:17)
Thenon-relpativisticfieldsφandϕ† transforminthpefundamentalrepresentationofSU(N ),
c
with colour indices denoted by α,β,γ,δ,.... The Majorana spinor χ is simplest to handle
bychoosingthestandardrepresentationfortheDiracmatrices,i.e.γ0 = diag(1, 1). Then
−
ψe−iMt
χ = , χ¯ = ψ†eiMt, ψTiσ e−iMt , (4.2)
iσ ψ∗eiMt − 2
− 2 ! (cid:16) (cid:17)
– 5 –
= + , = , = ,
Ma Mb Mc
= + , = + ,
Md Me
= + + + ,
Mf
= , = + + ,
Mg Mh
J
= + , = + . H
Mi Mj
E
P
Figure 1. 2 2 annihilation processes leading to the coefficients in eq. (4.4). Thick solid lines
→
standforMajoranaparticles,thickdashedlinesforcolouredscalars,wigglylinesforgluons,arrowed 0
lines for quarks, and thin dashed lines for Higgs bosons and longitudinal polarizations of W± and 4
Z0 bosons.
(
2
where the Grassmannian spinor ψ has two spin components, labelled by p,q,r,s,.... Only
the left-chiral projection of χ participates in interactions according to eq. (2.1). 0
In the following, we generally set M M whenever possible. The influence of
η → 1
∆M = 0 (and its thermal modification) is discussed in section 4.3.
6 8
4.2 Imaginary parts of 4-particle operators )
The first step is to determine annihilation cross sections for all possible processes with 0
dark matter initial states. The leading order Feynman diagrams are shown in figure 1. 7
According to the optical theorem, the amplitudes squared 2 can be expressed as an
2
|M|
imaginary (or “absorptive”) contribution to an effective Lagrangian [27].
An important simplification in the Majorana case follows from the identity satisfied
by Pauli matrices, σk σk = 2δ δ δ δ . Therefore a possible spin-dependent operator
pq rs ps qr− pq rs
can be reduced to a spin-independent one: ψ†ψ†ψ ψ σk σk = 3ψ†ψ†ψ ψ .
p r s q pq rs − p q q p
At leading order in an expansion in 1/M2, the absorptive operators read
= i c ψ†ψ†ψ ψ +c ψ†φ†ψ φ +ψ†ϕ†ψ ϕ
Labs 1 p q q p 2 p α p α p α p α
n
+c φ†ϕ†ϕ φ +c(cid:0) φ†ϕ† ϕ φ Ta Ta +c(cid:1) φ†φ†φ φ +ϕ†ϕ†ϕ ϕ . (4.3)
3 α α β β 4 α β γ δ αβ γδ 5 α β β α α β β α
o
Here Ta are Hermitean generators of SU(N ). In the p(cid:0)artial wave language, the(cid:1)operators
c
in eq. (4.3) correspond to s-wave annihilations, whereas p-wave annihilations would lead
to operators of (1/M4). At leading order in couplings, the coefficients read
O
y 2( h 2+g2C )
c = 0, c = | | | | s F ,
1 2 128πM2
1 g4C g4(N2 4) y 4
c = λ2+ s F , c = s c − , c = | | . (4.4)
3 32πM2 3 N 4 64πM2N 5 128πM2
(cid:18) c (cid:19) c
– 6 –
A non-zero value of c may be generated at higher orders. To minimize the magnitude
1
of higher-order effects, the couplings should be evaluated at the MS renormalization scale
µ¯ 2M. We note that c gets contributions from the “Majorana-like” processes and
∼ 5 Mi
in figure 1, but not from the “Dirac-like” amplitude .
Mj Mh
4.3 Number density, effective cross section, evolution equation
Within Boltzmann equations the overall dark matter abundance evolves as [28–30]
n˙ = σ v n2 n2 , (4.5)
−h eff i − eq J
(cid:0) (cid:1)
where n˙ is the covariant time derivative in an expanding background. To go beyond H
the quasiparticle approximation underlying the Boltzmann approach, the effective cross E
section can be re-interpreted as a chemical equilibration rate, Γ , and then defined on
chem P
thenon-perturbativelevelwithinlinearresponsetheory[31]. Furthermore, withinthenon-
0
relativistic effective theory, Γ can be related to the thermal expectation value of
chem Labs 4
from eq. (4.3) [4]. These relations can be expressed as
(
Γ 4
σ v = chem = Im . (4.6) 2
h eff i 2n n2 h Labsi
eq eq
0
In our model the number density amounts to 1
8
p2
neq = e−Ep/T 2+2Nce−∆MT/T , Ep ≡ M + 2M . (4.7) )
p
Z n o
0
The mass difference ∆M gets a vacuum contribution, ∆M = M M, and a thermal
T η − 7
correction from eq. (3.3) as well as from a similar tadpole involving λ ,
3
2
(g2C +λ )T2 g2C m
∆M ∆M + s F 3 s F D . (4.8)
T ≡ 12M − 8π
Note that the negative Salpeter correction may cancel against the positive terms. At
leading order the Debye mass parameter amounts to
N N
c f
m = g T + , (4.9)
D s 3 6
r
where N is the number of quark flavours (cf. ref. [32] for higher orders). The effective
f
values of g and N are changed with the temperature, as reviewed in appendix A.
s f
For future reference we define a “tree-level” effective cross section, σ v (0), by eval-
h eff i
uating the thermal expectation value Im at leading order and then making use of
h Labsi
eqs. (4.6) and (4.7). Wick contracting the indices in eq. (4.3) leads to
σ v (0) = 2c1+4c2Nce−∆MT/T +[c3+c4CF+2c5(Nc+1)]Nce−2∆MT/T . (4.10)
eff 1+Nce−∆MT/T 2
(cid:10) (cid:11)
(cid:0) (cid:1)
– 7 –
4.4 Plasma-modified Schro¨dinger equation and generalized Sommerfeld
factors
Goingbeyondleadingorder,weevaluate Im aselaborateduponinref.[5],expressing
h Labsi
it as a Laplace transform of a spectral function characterizing the dynamics of the dark
matterparticlesbeforetheirannihilation. DenotingbyE′ theenergyoftherelativemotion
and by k the momentum of the center-of-mass motion, this implies
ImLabs ≈ e−2TM−4Mk2T ∞dπE′ e−E′/T ciρi(E′)
Zk Z−Λ i
(cid:10) (cid:11) = MT 3/2e−2M/T ∞dE′ e−EX′/T c ρ (E′), (4.11) J
π π i i H
(cid:16) (cid:17) Z−Λ Xi
E
where α2M Λ M restricts the average to the non-relativistic regime.3 The spectral
P
≪ ≪
functions are obtained as imaginary parts of Green’s functions,4
0
2 4
∇r + (r) E′ G (E′;r,r′) = N δ(3)(r r′), (4.12)
−M Vi − i i −
(cid:20) (cid:21) (
lim ImG (E′;r,r′) = ρ (E′). (4.13)
r,r′→0 i i 2
0
Here contains a negative imaginary part, and N is a normalization factor giving the
Vi i 1
number of contractions related to the operator that c multiplies in eq. (4.3):
i
8
N 2, N 4N , N N , N N C , N 2N (N +1). (4.14)
1 ≡ 2 ≡ c 3 ≡ c 4 ≡ c F 5 ≡ c c )
0
If the potentials (r) were r-independent and with an infinitesimal imaginary part,
Vi
i.e. (r) = Re ( ) i0+, they would only induce mass shifts. In this case the spectral 7
Vi Vi ∞ −
functions can be determined analytically, 2
3
ρ(0)(E′) = NiM2 θ E′ Re ( ) E′ Re ( ). (4.15)
i 4π − Vi ∞ − Vi ∞
q
(cid:0) (cid:1)
This form can be used for defining generalized Sommerfeld factors:
S¯ −∞ΛdπE′ e−E′/T ρi(E′) = 4π 23 ∞dE′ e[ReVi(∞)−E′]/T ρi(E′) . (4.16)
i ≡ R−∞ΛdπE′ e−E′/T ρ(i0)(E′) (cid:16)MT(cid:17) Z−Λ π Ni
R
Then eq. (4.11) combined with eqs. (4.6) and (4.7) leads to a generalization of eq. (4.10),
σ v = 2c1+4c2Nce−∆MT/T+[c3S¯3+c4S¯4CF+2c5S¯5(Nc+1)]Nce−2∆MT/T . (4.17)
eff 1+Nce−∆MT/T 2
(cid:10) (cid:11)
3Someelaborationabouttheneedtointro(cid:0)ducesuchacutoffca(cid:1)nbefoundinref.[5]. Inpractice,wechoose
Λ≃2α2M,andhaveverifiedthatmakingite.g.2-3timeslargerplaysnoroleonournumericalresolution.
4At higher orders in the non-relativistic expansion, kinetic terms and potentials suppressed by powers
of 1/M2 could be added. In addition, operators suppressed by 1/M4 should be added in eq. (4.3).
– 8 –
If a potential (r) leads to a bound state, whose width is much smaller than the bind-
Vi
ing energy, the corresponding generalized Sommerfeld factor can be computed in analytic
form. In this case eq. (4.12) can be solved in a spectral representation, resulting in
ψ (0) 2δ(E′ E′)
ρ (E′) = πN | j | − j , (4.18)
i i d3r ψ (r) 2
j | j |
X
R
where ψ are the bound state wave functions. Inserting into eq. (4.16), the contribution of
j
the jth bound state to S¯ reads
i
4π 3 ψ (0) 2e[ReVi(∞)−Ej′]/T J
∆ S¯ = 2 | j | . (4.19)
j i MT d3r ψ (r) 2 H
(cid:16) (cid:17) | j |
E
R
This becomes (exponentially) large when T α2M, however chemical equilibrium is lost
≪ s P
in the dark sector at low T, which imposes an effective cutoff on the growth (cf. sections 5
0
and 6).
4
4.5 Thermal potentials
(
In order to write down the potentials (r) appearing in eq. (4.12), let us define 2
Vi
0
g2 1 iπT m2
v(r) s eik·r D (4.20)
≡ 2 k2+m2 − k (k2+m2)2 1
Zk (cid:26) D D (cid:27)
8
exp( m r) iT ∞dz sin(zm r)
= gs2  4−πr D − 2πmDr Z0 (z2+1)D2 , r > 0 . (4.21) )
2 × mD iT , r = 0 0
−4π − 4π 7


The integrand in eq. (4.20) corresponds to the static limit of the time-ordered HTL- 2
resummed temporal gluon propagator. Then we find
(r) = 0, (r) = C v(0), (r) = 2C v(0) v(r) , (4.22)
V1 V2 F V3 F −
(cid:2) (cid:3)
v(r) (N 1)v(r)
(r) = 2C v(0)+ , (r) = 2C v(0)+ c− . (4.23)
V4 F N V5 F N
c c
The structure (r) equals the combination V(r) iΓ(r) shown in eqs. (3.5)–(3.7), whereas
V3 −
C Re[v(0)] yields the Salpeter part of ∆M in eq. (4.8). The potential (r) corresponds
F T V3
to a singlet potential, (r) to an octet potential, and (r) to a particle-particle potential,
V4 V5
relevant because of the presence of a particle-particle annihilation channel generated by
Majorana exchange (cf. the discussion around the end of section 4.2).
We note in passing that at T < 160GeV, when the Higgs mechanism is operative,
additional potentials can be generated, particularly through the Higgs portal coupling λ
3
in eq. (2.1) (cf. e.g. ref. [33]). However the coefficients of these potentials are suppressed by
λ2v2/M2, where v is the Higgs expectation value. Given that we consider M 2TeV,
∼ 3 ≥
we expect their contributions to be negligible compared with QCD effects and have not
– 9 –