On the quantum statistical physics

of dark matter freeze-out1,2

Mikko Laine

University of Bern, Switzerland

1Based partly on collaborations with Simone Biondini and Seyong Kim.

2Supported by the SNF under grant 200020-168988.

1

There is “dark matter”

2

3D. Clowe et al, A direct empirical proof of the existence of DM, astro-ph/0608407.

3

What is dark matter?

4

Yet to be discovered particles? Basic requirements:

• not visible ⇒ electrically neutral

• around long ago & still today ⇒ stable or very long-lived

• correct structure formation long ago ⇒ rather heavy

Known particles (well described by the Glashow-Weinberg-Salam

“Standard Model”) fail to satisfy these requirements.

(An extension by keV sterile neutrinos might help. An extension

by µeV axions is quite popular. Primordial black holes may help,

but they cannot be generated within the Standard Model.)

5

In this talk: “WIMP paradigm”

Postulate the existence of Weakly Interacting Massive Particles

(“heavy neutrinos”) which cannot decay and are thus stable.

“Indirect detection” from galactic center:

DM

DM

DM′

“Direct detection” by nuclear recoil:

DM DM′

“Collider search” through missing energy:

DM′

DM

6

Text-book WIMP is in trouble

7

Lee-Weinberg equation4 (n=number density, H=Hubble rate)

(∂t + 3H)n = −〈σv rel〉 (n2 − n

2eq) .

DM

DM

Entropy conservation (part of Einstein equations)

(∂t + 3H)s = 0 .

Combining the two we get

∂t

(n

s

)

= −〈σv rel〉(n2 − n2

eq)

s.

4B.W. Lee and S. Weinberg, Cosmological Lower Bound on Heavy Neutrino Masses,

Phys. Rev. Lett. 39 (1977) 165.8

Insert relation of time and temperature (c = heat capacity):

dT

dt= −3Hs

c, ∂t =

dT

dt∂T .

Defining a “yield parameter” through Y ≡ n/s we get

∂TY =c〈σv rel〉3H

(Y2 − Y

2eq) .

Inserting c ∼ T 2, 〈σv rel〉 ∼ α2/M2, H ∼ T 2/mPl, where M

and α denote the WIMP mass and coupling, we get

∂TY ∼ mPlα2

M2(Y

2 − Y2eq) .

9

The equilibrium number density is a known function of T,M :

n eq ∝∫

d3p

(2π)31

e√

p2+M2/T ± 1≈

(MT

2π

)3/2

e−M/T

.

The entropy density is dominated by massless modes: s ∝ T 3.

Linearize around equilibrium (Y = Y eq + δY , δY = Y − Y eq):

z ≡ M

T⇒ Y

′(z) ∼ − mPlα

2

M

Y eq(z)[Y (z) − Y eq(z)]

z2.

The differential equation has a “thermal fixed point” at Y (z) =

Y eq(z) but cannot keep close to it for Y eq(z) ≪ 1. 10

Numerical solution shows a “freeze-out”:

0 5 10 15 20 25 30z = M / T

1e-12

1e-08

1e-04

1e+00

Y

Yeq

M = 7 TeV

M = 5 TeV

M = 3 TeV

M = 1 TeV

α = 0.01, mPl

= 1.2 ∗ 1016

TeV

11

Final energy density (e ≡ Mn) compared to radiation ∼ T 4:

1 2 3 4 5 6 7 8

M / TeV

0e+00

1e-05

2e-05

3e-05

e /

T4 @

T =

1 G

eV

α = 0.01, mPl

= 1.2 ∗ 1016

TeV

WIMP miracle

overclosure

LHC pushes up M , so there is a danger “overclosure”!12

Could increased 〈σvrel〉 help?

13

Much discussed:5 “Sommerfeld effect”6:

〈σv rel〉 −→ 〈σ tree v rel S(v rel)〉 .

For attractive Coulomb-like interaction,

S(v rel) ∼ α

v rel

for v rel<∼α .

5e.g. J. Hisano, S. Matsumoto, M. Nagai, O. Saito and M. Senami, Non-perturbative

effect on thermal relic abundance of dark matter, hep-ph/0610249; J.L. Feng, M. Kaplinghatand H.-B. Yu, Sommerfeld Enhancements for Thermal Relic Dark Matter, 1005.4678.

6L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Non-Relativistic Theory, Third

Edition, §136; V. Fadin, V. Khoze and T. Sjostrand, On the threshold behavior of heavy top

production, Z. Phys. C 48 (1990) 613.14

More recent:7 bound states (“WIMPonium”):

Mbound = 2M − ∆E ⇒ e−Mbound/T > e

−2M/T.

This is quantum mechanics in a statistical background.

7e.g. B. von Harling and K. Petraki, Bound-state formation for thermal relic dark

matter and unitarity, 1407.7874.15

Some quantum statistical physics

16

Particles in the initial state: most energy is carried by mass.

E rest ∼ 2M , Ekin ∼k2

2M∼ T .

Particles in the final state: all energy is carried by momentum.

Ekin ∼ 2k ∼ 2M ⇒ ∆x ∼ 1

k∼ 1

M≪ 1

T.

DM

DM

Therefore the “hard” annihilation process is local.8

8e.g. G.T. Bodwin, E. Braaten and G.P. Lepage, Rigorous QCD analysis of inclusive

annihilation and production of heavy quarkonium, hep-ph/9407339; L.S. Brown andR.F. Sawyer, Nuclear reaction rates in a plasma, astro-ph/9610256.

17

But before the annihilation there are “soft” initial-state effects:

. . .

soft hard

18

A “linear response” analysis shows that

〈σv rel〉 ∼ α2

M2n2eq

1

Z

∑

m,n

e−Em/T〈m|φ†

φ†

⇒1

︷ ︸︸ ︷|n〉〈n|φφ|m〉

︸ ︷︷ ︸

=:〈O†(0)O(0)〉T

.

Here |m〉 are eigenstates containing a DM-DM pair,

and φφ annihilates the DM-DM pair.

19

The DM-DM problem can be reduced to a 1-body problem:

Em =: E′+

[

2M +k2

4M

]

︸ ︷︷ ︸

center-of-mass energy

.

Carrying out the integral over k we are left with

〈O†(0)O(0)〉T = e

−2M/T

(MT

π

)3/2 ∫ ∞

−Λ

dE′

πe−E′/T

ρ(E′) .

20

The “spectral function” ρ(E′) represents the solution of a

Schrodinger equation for a Green’s function.

[HT − iΓT (r) − E

′]G(E

′; r, r

′) = δ

(3)(r − r

′) ,

limr,r′→0

ImG(E′; r, r

′) = ρ(E

′) .

Here the Hamiltonian has a standard from

HT = −∇2r

M+ VT (r) , r = |r| ,

whereas −iΓT (r) accounts for real scatterings with the plasma.

21

Procedure in a nutshell

• Compute thermal self-energy of the exchanged particle

• Determine the corresponding time-ordered propagator

• Fourier-transform for potential VT (r) and width ΓT (r)

• Solve for ρ(E′) = ImG(E′; 0, 0)

• Laplace-transform with weight e−E′/T for 〈O†(0)O(0)〉T

• Integrate Lee-Weinberg equation numerically

22

Results for simple cases

23

Spectral function for typical WIMPs

-1×10-2 0 1×10

-22×10

-23×10

-2

E’ / M

0

2×10-3

4×10-3

ρ 1 /

(ω

2 Ν1)

free

free ∗ S1

M = 4 TeV

T = 200 GeV

24

It becomes interesting if M is large and T low

-4×10-4

-2×10-4 0 2×10

-4

E’ / M

0

2×10-3

4×10-3

6×10-3

ρ 1 /

(ω

2 Ν1)

T = 600 GeV

T = 100 GeV

T = 10 GeV

M = 12 TeV

25

Even more interesting if we decrease T further

-4e-06 -2e-06 0e+00 2e-06E’ / M

0e+00

2e-03

4e-03

6e-03ρ 1

/ (

ω2 Ν

1)

T = 600 GeV

T = 100 GeV

T = 10 GeV

T = 1 GeV

M = 12 TeV

26

A nice relation to heavy ion collision experiments

S. Chatrchyan et al. [CMS Collaboration], Suppression of excited Υ states in PbPb collisions

at√sNN = 2.76 TeV, Phys. Rev. Lett. 107 (2011) 052302 [1105.4894].

This follows a general pattern previously predicted theoretically.9

9e.g. F. Karsch, D. Kharzeev and H. Satz, Sequential charmonium dissociation, Phys.

Lett. B 637 (2006) 75 [hep-ph/0512239].27

Dark matter density evolution in the Early Universe

101

102

103

z = M / T

10-16

10-14

10-12

10-10

10-8

Y

λ3,4,5

= 0

λ3,4,5

= 1

λ3,4,5

= π

M = 12.0 TeV, ∆M = 10-3

M

Yeq

28

Examples of overclosure bounds

0 2 4 6 8 10 12 14M / TeV

0.0

0.1

0.2

0.3

Ωdm

h2

λ3,4,5

= 0

λ3,4,5

= 1

λ3,4,5

= π

overclosure

viable

29

Relation to indirect non-detection

30

Why is Early Universe not much constrained by present day?

Long ago: t ∼ 10−12 s, T ∼ 100 GeV.

DM annihilation:

DM

DM

DM’ annihilation:

DM′

DM′

DM ↔ DM’ is in thermal equilibrium ⇒ annihilation can proceed

through the heavier DM’ channel if this is more efficient.

31

Today: t ∼ 1017 s, T ≪ eV.

DM annihilation is active:

DM

DM

DM’ decayed long ago.

Therefore present day indirect detection constraints cannot

directly fix 〈σv rel〉 relevant for Early Universe cosmology.

32

If DM’ interacts strongly, there are large effects.10

-4e-02 -2e-02 0e+00 2e-02E’ / M

0e+00

2e-02

4e-02

6e-02

8e-02

ρ / ω

2

free ∗ S1

free

T = 100 GeV

T = 150 GeV

T = 200 GeV

T = 300 GeV

T = 500 GeV

Gluon exchange, M = 3 TeV

10e.g. J. Ellis, F. Luo and K.A. Olive, Gluino Coannihilation Revisited, 1503.07142.

33

Summary

• Apart from model uncertainties, generic dark matter studies

also contain theoretical uncertainties.

• In general, both quantum-mechanical effects (bound states,

multiple interactions) and statistical physics phenomena (Debye

screening, frequent scatterings on plasma particles) play a role.

• Strong interactions may enhance the annihilation rate because

of bound states, and this may help to avoid overclosure.

• Model-specific studies are needed for definite conclusions.

34