Probing the Early Universewith Baryogenesis & Inflation

Wilfried BuchmüllerDESY, Hamburg

ICTP Summer School,Trieste, June 2015

• When and how was the baryon asymmetry generated?• What caused inflation, and at which energy scale?• How are inflation, baryogenesis and dark matter related?

[adapted from Schmitz ’12]

Outline

• BARYOGENESIS

1. Electroweak baryogenesis 2. Leptogenesis 3. Other models

• INFLATION

1. The basic picture 2. Recent developments

BARYOGENESIS

What is the origin of matter, i.e., the baryon-to-photon ratio

⌘B = nBn�

= (6.1± 0.1)⇥ 10�10

? ? ? ?

Key references

A. D. Sakharov, JETP Lett. 5 (1967) 24 G. ‘t Hooft, Phys. Rev. Lett. 37 (1976) 8 V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov, Phys. Lett. B 155 (1985) 36 J. A. Harvey and M. S. Turner, Phys. Rev. D 42 (1990) 3344

Sakharov’s conditions

Necessary conditions for generating a matter-antimatter asymmetry:

• baryon number violation

• C and CP violation

• deviation from thermal equilibrium

Alternative mechanisms: dynamics of scalar fields, e.g. Affleck-Dine baryogenesis, heavy moduli decay, ...

Check of 3rd condition:

hBi = Tr(e��HB) = Tr(⇥⇥�1e��HB)

= �Tr(e��HB) , ⇥ = CPT

Sphaleron processes

Baryon and lepton number not conserved in Standard Model,

JB

µ

=

1

3

X

generations

�qL

�µ

qL

+ uR

�µ

uR

+ dR

�µ

dR

�,

JL

µ

=

X

generations

�lL

�µ

lL

+ eR

�µ

eR

�,

divergence given by triangle anomaly,

@µJB

µ

= @µJL

µ

=

Nf

32⇡2

⇣�g2W I

µ⌫

fW Iµ⌫

+ g02Bµ⌫

eBµ⌫

⌘;

Nf

: number of generations; W I

µ

, Bµ

: SU(2) and U(1) gauge fields,

gauge couplings g and g0.

Change in baryon and lepton number related to the change in topological charge

the gauge field,

B(tf )�B(ti) =

Z tf

ti

dt

Zd

3x@

µJ

Bµ

= Nf [Ncs(tf )�Ncs(ti)] , with

Ncs(t) =g

3

96⇡

2

Zd

3x✏ijk✏

IJKW

IiW

JjW

Kk

−1 0 1

T=0, sphaleron

T=0, instanton

E/

W

Esph

Non-abelian gauge theory: �Ncs = ±1,±2, . . .. Jumps in Chern-Simons asso-

ciated with changes of baryon and lepton number,

�B = �L = Nf�Ncs .

Sphaleron

cL

τ

L

e

µ

LsLs t

b

b

ν

ν

ν

u

d

d

L

L

LL

L

In SM, e↵ective 12-fermion interaction

OB+L =

Y

i

(qLiqLiqLilLi) , �B = �L = 3

uc+dc + cc ! d+ 2s+ 2b+ t+ ⌫e + ⌫µ + ⌫⌧

Sphaleron rate crucially depends on temperature, only relevant in early universe:

zero temperature: � ⇠ e�Sinst= e�

4⇡↵

= O �10

�165�

EW phase transition:

�B+L

V=

M7W

(↵T )3exp (��Esph(T )) ,

with Esph(T ) ' 8⇡

gv(T )

high temperature phase:

�B+L

V= s↵

5T 4 ⇠ 10

�6

“consensus” among theorists: B+L violating processes in thermalequilibrium in temperature range:

TEW ⇠ 100 GeV < T < Tsph ⇠ 1012 GeV

... but no direct experimental evidence for instanton or sphaleron processes ...

Chemical potentials

In SM, with Higgs doublet H and Nf generations, there are 5Nf + 1 chemicalpotentials (qi, `i, ui, di, ei); for non-interacting gas of massless particles µi giveasymmetries in particle and antiparticle number densities,

ni � ni =gT 3

6

8<

:�µi +O

⇣(�µi)

3⌘

, fermions

2�µi +O⇣(�µi)

3⌘

, bosons

Relations between the various chemical potentials:

SU(2) instantons:X

i

(3µqi + µli) = 0

QCD instantons:

X

i

(2µqi � µui � µdi) = 0

vanishing hypercharge of plasma:

X

i

✓µqi + 2µui � µdi � µli � µei +

2

NfµH

◆= 0

Yukawa interactions:

µqi � µH � µdj = 0 , µqi + µH � µuj = 0 ,

µli � µH � µej = 0

Relations determine B and L in terms of B-L :

B =X

i

(2µqi + µui + µdi) , L =X

i

(2µli + µei)

B = cs(B � L) , L = (cs � 1)(B � L) , cs =8Nf + 4

22Nf + 13

Electroweak baryogenesis and leptogenesis fundamentally different!In EWBG B-L conserved, generation of B in strong phase transition;in LG generation of B from initial generation of B-L (then unaffected by sphaleron processes)

I. Electroweak Baryogenesis

Key references

D. A. Kirzhnits and A. D. Linde, Phys. Lett. B 42 (1972) 471 A. G. Cohen, D. B. Kaplan and A. E. Nelson, Phys. Lett. B 336 (1994) 41

Reviews

W. Bernreuther, Lect. Notes Phys. 591 (2002) 237 D. E. Morrissey and M. J. Ramsey-Musolf, New J. Phys. 14 (2012) 125003T. Konstandin, Physics - Uspekhi 56 (8) 747 (2013)

0 φcritφ

T = T C

Veff

T = T < TC1

T = T > TC2

0 φ

Veff

T1

TC

T2

2nd order vs 1st order (electroweak) phase transition, as universe cools down. What determines the shape of the effective potential? How does the phasetransition proceed in an expanding universe? How can the complicated nonequilibrium process be calculated, where all masses are generated?

Finite-temperature effective potential

Massive scalar field:

Euclidean field theory with finite time range β; add source term,calculate free energy (constant source, volume Ω):

S� =

Z

�

✓1

2(@⌧�)

2 +1

2(@i�)

2 + V (�)

◆

V (�) =1

2µ

2 +�

4�

4,

Z

�=

Z �

0d⌧

Zd

3x , � =

1

T

Z� [j] =

Z

�D� exp (�S� [�]�

Z

�j�) = exp (��⌦W�(j)) ,

@W�

@j

=

1

�⌦

hZ

��(x)i ⌘ '

Legendre transformation yields effective potential:

explicit calculation, high temperature expansion:

2nd term is free energy of massless boson; thermal bath generates “thermal mass” of boson; usefull concept to understand some effects in thermal field theory qualitatively, but different from kinematic mass; in gauge theories problem of gauge invariance ...

V�(') = VT=0(')�⇡2

90T 4 +

1

24m2(')T 2 � 1

12⇡m3(')T + . . .

m2(') = m2 + 3�'2 ,m(')

T⌧ 1

=1

2(m2 +

�

4T 2)'2 +

�

4'2 + . . .

V�(') = W�(j)� 'j , j = �@V�

@�

Higgs model & symmetry breaking

In (Abelian) Higgs symmetry “broken” in ground state:

finite-temperature potential (with “barrier temperature”, where the barrierdissappears):

S� =

Z

�

�(Dµ�)

⇤Dµ�+ µ2|�|2 + �|�|4�,

Dµ = @µ + igAµ , µ2 < 0 , Re �0 =��µ2/�

�1/2 ⌘ '0/p2 ,

mA = g'0 , mH =p2� '0

V�(') =a

2(T 2�T 2

b )'2 � b

3T'3 +

�

4'4 + . . .

@2V�

@2'

���'=0

= 0 : T 2b = �µ2

a, a =

3g2

16+

�

2

Cooling down, at critical temperature, Higgs vev jumps to critical vev:

phase transition weak for large Higgs mass (small coupling λ); Standard model and extensions: “a” and “b” in effective potential more complicated functions of gauge and Yukawa couplings.

Much work on effective potential (mostly mid-nineties): loop corrections,gauge dependence, infrared divergencies, treatment of Goldstone bosons,resummations, nonperturbative effects (gap equations, rigorous lattice studies!), beautiful work ... Baryogenesis needs strong phase transition:

not possible in SM, but possible in extensions ...

V�c('c) = V�c(0) :T 2c � T 2

b

T 2c

' b2

a�> 0 ,

'c

Tc=

2b

3�

'c

Tc> 1

Nonperturbative effects change 1st order transition to crossover atcritical Higgs mass:

[Csikor, Fodor, Heitger ’98][Jansen ’96]

critical endpoint, lattice: RHW =mH

mW, mc

H = 72.1± 1.4 GeV

gap equations, magnetic mass: mcH =

✓3

4⇡C

◆1/2

' 74 GeV ,

mSM = Cg2T , C ' 0.35

Phase diagram

Bubble nucleation & growth

bubbles formand expand

liquidT < TC

~_T TC

t = t0

t > t0

nucleation rate per volume:

�

V= A exp (��eff [�]) ,

� : saddle point of e↵ective

action, interpolating

between the two phases ,

Langer’s theory, ...

no 1st-order phase transition in SM, but in extensions(singlet model, 2HDM,...)

broken phase

becomes our world

unbroken phase

vWall

= 0φ

>> H

0CP CP_~ΓB + L

Sph

ΓB + L

Sph

q_q

qq

_

q_q

qq

_

CP violating scatterings at bubble wall (one-dimensional approximation):

Lf = �X

y ̄L R� , �(z) =⇢(z)p

2ei✓(z) , ⇢(z) =

vc2

✓1� tanh

z

Lw

◆

Calculating the baryon asymmetry very difficult, series of approximations: Schwinger-Keldysh → Boltzmann equations → diffusion equations ... ; CP violating interactions with bubble wall generate in front of wall excess of left-handed “tops”, converted to baryon asymmetry by sphaleron processes; in frame of wall chemical potentials only depend on distance from wall:

important parameters: critical Higgs vev, bubble wall velocity, bubble wall width, diffusion parameters, ...

chemical potentials: µqL(z) = 3(µq1(z) + µq2(z) + µq3(z))

baryon number density:

@nB

@t=

3

2

�sph

T

⇣µqL � cs

nB

T 2

⌘

di↵usion equations: vw@zµi �X

j

�ijµj + · · · = Si

final result: nB =3

2

�sph

vwT

Z 1

0µqL(z)e

�kBz , kB =3cs

2vw

�sph

T 3

Example 1: 2 Higgs-doublet model (2HDM)

V (H1, H2) = �µ21H

†1H1 � µ2

2H†2H2 � µ2

3(ei↵H†

1H2 + h.c.)

+

�1

2

(H†1H1)

2+

�2

2

(H†2H2)

2+ �3(H

†1H1)(H

†2H2)

+ �4|H†1H2|2 +

�5

2

⇣(H†

1H2)2+ h.c.

⌘

LY = yH2q3tc+ h.c. + . . .

approximate Z2 symmetry, broken by µ3; measure for size of couplings, one-loop

corrections:

� = max|��i/�i|Recent work:

Dorsch, Huber, No ’13: mH± . mH0 < mA0 , mA0 & 400 GeV ,

R�� = �(h0 ! ��)/�(h0 ! ��)SM 6= 1

Blinov, Profumo, Stefanik ’15: Inert Doublet Model OK, predictions:

(MH ,MA,MH± , R��) = (66, 300, 300, 0.90), (200, 400, 400, 0.93), (5, 265, 265, 0.90)

300

320

340

360

380

400

120 130 140 150 160 170 180 190

ηB=5

ηB=10

ηB=20

ηB=40

ηB=60

mH

mh

ξ=1

[Fromme, Huber, Seniuch ’06]

⌘B in units of 10�11; large enough baryon asymmetry requires⇠ = vc/Tc & 1.5 and � ⇠ 0.5, i.e. 2HDM is strongly coupled!Watch electric dipole moments!

Example 2: Standard Model with singlet Motivation: non-minimal composite Higgs models, additional singlet in low energy effective Lagrangian; strong first-order phase transition from tree-level potential, thermal instability for singlet and Higgs; CP violation via additional dim5-operator:

V (h, s, T ) =�h

4

h2 � v2c +

v2cw2

c

s2�2

+

4s2h2

+1

2(T 2 � T 2

c )(ch h2 + cs s

2)

mh ms vc f/b Lwvc vc/Tc

S1 120 GeV 81 GeV 188 GeV 1.88 TeV 7.1 2.0S2 140 GeV 139.2 GeV 177.8 GeV 1.185 TeV 3.5 1.5

Light singlet scalar in principle observable at LHC; problem: coupling onlyvia Higgs! Dedicated searches needed; also modification of Higgs properties, electroweak precision observables, ...

[Espinosa, Gripaios, Konstandin, Riva ’12]

sh

t tt

e e e

LtHs =s

fHq̄L3(a+ ib�5)tR + h.c.

CP violation for baryogenesis implies EDMs for electron and neutron;predictions close to upper experimental bound! Energy scale f/b ~ 1 TeV,i.e. low compositeness scale - other resonances due to compositeness should be in LHC range!

Note: baryogenesis in MSSM popular for many years, but stop lighter than stop required ... ! Possibility in NMSSM ... ?

Summary: electroweak baryogenesis

• Very interesting topic in nonequilibrium QFT, huge activity during the past past 30 years since closely related to Higgs mechanism

• Theoretical uncertainty considerable:

• Strong interactions and new particles are unavoidable; 2 HDM: charged Higgs bosons,... ; SM with singlet: light neutral scalar; ... • Dedicated searches at the LHC, stronger bounds on dipole moments and electroweak precision tests should settle the issue of EWBG

nB

s⇠ �sph

T 4

1

LwTc

�CP

4⇡e�m /Tc

✓vcTc

◆�

d . 10�9 ,

�sph/T4 ⇠ 10�6 , � = 3 . . . 4 , d = 10�2 . . . 10�1

' 6.2 · 10�10