Axion cosmology from lattice QCDbodri.elte.hu/seminar/katz_20161005.pdf · IntroductionPeccei-Quinn mechanismEquation of stateTopological susceptibilityAxion mass Axion cosmology

Introduction Peccei-Quinn mechanism Equation of state Topological susceptibility Axion mass

Axion cosmology from lattice QCD

Sándor Katz

Institute for Theoretical Physics, Eötvös University

Sz. Borsanyi, Z. Fodor, J. Günther, K-H. Kampert, T. Kawanai, T.G. Kovacs, S.W. Mages,A. Pasztor, F. Pittler, J. Redondo, A. Ringwald, K.K Szabo

arXiv:1606.07494

ELTE Theory Seminar 2016

Sándor Katz Axion cosmology


Outline

1 Introduction

2 Peccei-Quinn mechanism

3 Equation of state

4 Topological susceptibility

5 Axion mass



Introduction

Strong CP problem→ axion→ dark matter candidateTwo important inputs for axion production: equation of state &topological susceptibility at high TDetermine topological susceptibility at high T in the physicalpoint using fixed Q integralExact zero modes of the Dirac operator for Q 6= 0 are crucial→ large discretization effects with staggered fermionsPossible solutions1) eigenvalue reweighting2) using chiral fermionsWe use 1) for the 3 flavor theory and 2) for going down to thephysical point



Strong CP problem

CP is violated in the Standard ModelQCD may have a CP violating Θ

32π2 Fµν F̃µν = Θ · q term

Experiments constrain |Θ|<∼10−10

Possible solutionsmu = 0 (ruled out by lattice results)Spontaneous CP violationAxion

Observation: −T log Z (Θ) has a minimum at Θ = 0→ promote Θ to a dynamical fieldElegant solution: Peccei-Quinn mechanism. Complex scalar fieldwith a U(1) symmetric Mexican hat potential.Spontaneous symmetry breaking→ axion is the(pseudo) Goldstone-boson



Peccei-Quinn mechanism

example: The KSVZ axion (Kim 1979, Shifman, Vainshtein, Zakharov 1980)

L = ∂µΦ∗∂µΦ + V (Φ∗Φ) + ΦΨ̄LΨR + Φ∗Ψ̄RΨL + Ψ̄D(A)Ψ +LQCDwhere Ψ is a heavy (m ∼ fA) fermion with color charge

Φ has a vev of fA; Φ = (fA + r)eiΘ

Simultaneous (chiral) U(1) rotations of Φ and ΨL = (fA + r)2∂µΘ∂µΘ + ∂µr∂µr + V ((fA + r)2) + (fA + r)Ψ̄Ψ ++Ψ̄D(A)Ψ + iΘq + ∂µΘΨ̄γ5γµΨ + LQCD

Integrating out Ψ and radial component leads toL = f 2

A∂µΘ∂µΘ + iΘq + ∂µΘ · (. . . ) + LQCD

The axion is a = fAΘ

If ΘQCD 6= 0 then Θ→ Θ + ΘQCD



One can find the effective potential for Θ by integrating out allother fields

For constant Θ this only contains moments of the topologicalcharge Q =

∫q(x)d4x

e−VT Veff = 〈eiQΘ〉

At high T only Q = 0,±1 contributes which leads to

Veff = χ(T )(1− cos(Θ)) = f 2Am2

A(T )(1− cos(Θ))

where χ = TV < Q2 > is the topological susceptibility

Mexican hat potential for Θ is slightly tilted by Veff , minimum is atΘ = 0→ strong CP solved

Solve classical equations for Θ in an expanding universe→axion production



Misalignment mechanism

We need to solve

d2θ

dt2 + 3Hdθdt

+dVeff (θ)

dθ= 0,

where H(t) is given by the Friedmann equations

H2 =8π

3M2Plρ

dρdt

= −3H(ρ+ p) = −3HsT

Two important inputs needed: equation of state (ρ, s) & χ(T )



Misalignment mechanism

Θ is initially frozen to Θ0 & starts to oscillate when 3H ≈ mAaxions are produced then n/s is conservedcurrent axion density: ρA;today = mAstoday · n/spre-inflation scenario: Θ0 uniformpost-inflation: all Θ0 present→ averagingalso (possibly large) string contribution

-0.5

0

0.5

1

1.5

0 0.5 1 1.5 2 0

100

200

300

400

500

600

700

800

900

1000

Θ

n/s

mA=50 µeVΘ0=1.0

Θ n/s

1/T [1/GeV]



The equation of state

Photons, neutrinos, charged leptons: free bosons/fermionsBelow the QCD transition: hadron resonance gas modelFor 100 MeV < T < 1 GeV lattice calculationO(g6) perturbation theory with fitted g6 coefficient agrees withthe lattice result above 500 MeVUse perturbation theoryabove 1 GeV and includeb quark above 500 MeVsimilarly to charmW±,Z ,H, t quark + EWphase transition fromLaine & Schroder 2006,Laine & Meyer 2015

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

200 400 600 800 1000 1200 1400 1600 1800 2000

(ρ-3

p)/

T4

T [MeV]

O(g6) Nf=3+1 qc=-3000

O(g6) Nf=3+1+1 qc=-3000

2+1+1 flavor EoS from lattice



The equation of state

Effective number of degrees of freedom including all SM particles

ρ =π2

30gρT 4 s =

2π2

45gsT 3 c =

2π2

15gcT 3

0.6

0.7

0.8

0.9

1ratios

gs(T)/gρ(T)

gs(T)/gc(T)

10

30

50

70

90

110

100

101

102

103

104

105

g(T)

T[MeV]

gρ(T)

gs(T)

gc(T)



The challenge of computing the susceptibility

large autocorrelation of Q on fine lattices (algorithmic problem)χ(T ) decreases strongly with temperature→ very few Q 6= 0configurations (physical problem)E.g. 〈Q2〉 = 10−6 means one Q = ±1 configuration per million.Even O(million) configurations can lead to large statistical errorsχ(T ) has large lattice artefacts

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014

χ [

fm-4

]

a2[fm

2]

standard(m

π,ph/mπ,ts)

2



Fixed sector integral

See also Frison,Kitano,Matsufuru,Mori,Yamada 1606.07175.

Instead of waiting for tunneling events, we make simulations infixed Q sectors and calculate the weight of sectors Zi from theaction difference between sectors.

First calculate derivative of log Z1/Z0:

b1(T ) ≡ −d log Z1/Z0

d log T

Use fixed Nt -approach, ie. T = (aNt )−1 is changed by β:

b1(T ) = − dβd log a

(〈Sg〉1 − 〈Sg〉0

)Sándor Katz Axion cosmology


Fixed sector integral

Integration gives the relative ratio:

Z1/Z0|T = exp

(−∫ T

T0

d log T ′ b1(T ′)

)Z1/Z0|T0

Start from a temperature T0, where standard approach works.For high temperatures, where only Q = 0 and 1 are contributing

〈Q2〉 ' 2Z1

Z0NtN3s a4

then b1 is directly related to the fall-off exponent:

b(T ) = − d logχd log T

' b1(T )− 4



Fixed Q integral - quenched

Fixed Q simulation: extra acc/rej step at the end of each update, aslattice spacing decreased the acceptance gets better.

Test in quenched case: pure Wilson action upto 7 · Tc and 8× 643

χ/T

c4

T/Tc

[1508.06917]DIGA

integral 8x32

10-8

10-6

10-4

10-2

1 2 3 4 5 6 7 8 9 10

b(T

)

8x328x645

6

7

8

standard method: extrapolation using a fit; integral method; DiluteInstanton Gas Approximation: exponent agrees nicely, but order of

magnitude difference in χ



Fixed Q integral with fermions

At high temperatures χ(T ) ∼ T−b only Q = 0,1 contribute

b1 = − dβd log a

〈Sg〉1−0 −∑

f

d log mf

d log amf 〈ψψf 〉1−0

Sg : small cutoff effects, huge statistics→ staggered Nf = 3mf 〈ψψf 〉1−0: large cutoff effects→staggered reweighting for Nf = 3, overlap for Nf = 2 + 1

0

10

20

30

100 200 500 1000 2000

nf=3+1

nf=2+1+1 phys

mud/m

udphys

T[MeV]

staggeredstaggered fix Q

overlap fix Q



Reweighting

Strong cut-off effects are related to the lack of exact zero-modes.

In the continuum non-trivial sectors are suppressed by thecontribution of zero-modes to the fermion determinant,ie. by the quark mass.

On the lattice the suppression is altered:m→ m + iλ0, where λ0 is a would be zero-mode.Weaker suppression→ χ(T ) overestimated.

To improve 1. identify would be zero-modes2. restore the continuum weight→ reweight

w [U] ∼ mm + iλ0



Reweighting - example

A direct comparison of methods @ T=300 MeV mud = mphysud .

10-5

10-4

10-3

10-2

10-1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

χ[f

m-4

]

(10/Nt)2

standardratio

reweightintegral

standard: huge lattice artefacts; ratio: χ(T )/χ(T = 0) apparentscaling is misleading; reweight: orders of magnitude smaller;

integral: calculate @ mud = ms directly and integrate down in mass,consistent with reweighting



Fixed Q integral - chiral condensate

In the high temperature (SB) limit only the zero mode contributesmf 〈ψψf 〉1−0 = Nf .

0

0.5

1

1.5

2

2.5

3

300 600 1200 2400

3 m

− ψψ

1−

0

T[MeV]

Nt=10Nt=8Nt=6

std Nt=4

Nt=10Nt=8Nt=6

rew+zm Nt=4

For large T goes to 0 instead of approaching the SB limit, this is alattice artefact. Non-chiral fermions fail spectacularly for large

temperatures! Solutions: reweighting or chiral fermions.



Mass integration with chiral fermions

There is an exact index theorem on the lattice for fermions whichsatisfy the Ginsparg-Wilson relation

{D, γ5} = aDγ5Dlattice artifacts can be largely reducedoverlap construction:

D =(m0 − m

2

)(1 + γ5sgn (HW )) + m,

much more expensive than staggered fermions, but condensatedifference can be computedwe find that mf 〈ψψf 〉1−0 = Nf for T ≥ 300 MeV within errorsthis gives the mass exponent required to integrate down to thephysical point



Topological susceptibility at the physical point

at T < Tc : χ ∼ mumdmu+md

while at T > Tc : χ ∼ mumd

isospin splitting in both cases results in a factor of 4mumd(mu+md )2 ≈ 0.88

10-12

10-10

10-8

10-6

10-4

10-2

100

102

100 200 500 1000 2000

χ[fm

-4]

T[MeV]

10-4

10-3

10-2

10-1

100 150 200 250



Comparison with other work

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100 200 500

χ[fm

-4]

T[MeV]

1512.06746

1606.03145 χt

1606.03145 m2χdisc

this work

Bonati et.al (1512.06746): smaller exponent, probably largelattice artefactsPetreczky, Schadler, Sharma (1606.03145): bosonic andfermionic definitions, consistent results with large errors



Constraints on the axion mass

10-4

10-3

10-2

10-1

100

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103

109

1010

1011

1012

1013

1014

1015

1016

1017

1018

1019

post-inflationmisalignmentrange

pre-inflationinitial angle

Θ0

mA [µeV]

fA [GeV]

Pre-inflation scenario: mA unambiguously determines the Θ0initial condition of our UniversePost-inflation: Θ0 average equivalent to Θ ≈ 2.15absolute lower limit (all DM from misalignment): mA>∼28(2) µeVassuming 50-99% other (e.g. strings): mA = 50− 1500 µeV



Summary

The axion offers a solution to both the strong CP and dark matterproblems

Calculating axion production in the early universe requires theEoS and χ(T )

Both were determined using lattice calculations up to hightemperatures

New techniques: fixed Q integral + eigenvalue reweighting

Axion mass in the post-inflation scenario: mA = 50− 1500 µeV


Documents

Axion cosmology from lattice QCDbodri.elte.hu/seminar/katz_20161005.pdf · IntroductionPeccei-Quinn mechanismEquation of stateTopological susceptibilityAxion mass Axion cosmology