Lattice QCD Input to Axion CosmologyEvan BerkowitzLawrence Livermore National Laboratory

INT-15-3Intersections of BSM Phenomenology and QCD for New Physics SearchesInstitute for Nuclear Theory

Tuesday, October 13th, 2015PRD 92 034507 / arXiv:1505.07455 – E. Berkowitz, M. Buchoff, E. Rinaldi.

LLNL-PRES-669910This work was performed under the auspices of the U.S. Department of Energy by

Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344

Colgate-Palmolive

1

Outline

• Introduction

• Whence axions?

• What is the over-closure bound?

• Inputs to the over-closure bound from lattice QCD

• Outlook

2

Big Idea

• Axions were originally proposed to deal with the Strong CP Problem, also form a plausible DM candidate.• Calculating the axion energy density

requires nonperturbative QCD input.

• Being sought in ADMX (LLNL, UW) & CAST (CERN), and (soon) IAXO with large discovery potential in the next few years.

• Requiring Ωa ≤ ΩCDM yields a lower bound on the axion mass today.

The Economist, 19 Dec 2006

CDM

Λ

Β

Ωtot = 1.000(7)PDG 2014 via

P.A.R. Ade, et al., (Planck Collab. 2013 XVI), arXiv: 1303.5076v1.

Preskill, Wise & Wilczek, Phys Lett B 120 (1983) 127-132

3

QCD Theta Term

• QCD has a parameter, θ.

• Controls QCD CP violation.

• Topological.

• θ can take any value in (-π,π].

• Neutron EDM ≲ 3 10-26e•cm

• |θ| ≲ 10-10

CP Violating

Strong CP Problem:Why is θ so small?

Q =1

32⇡2

Zd

4x ✏

µ⌫⇢�Fµ⌫F⇢� 2 Z

LQCD 3 ✓1

32⇡2✏µ⌫⇢�Fµ⌫F⇢�

Baker et al., PRL 97, 131801 (2006) / hep-ex/0602020

eiS / eiQ✓

4

Colangelo et al. (FLAG) arXiv:1011.4408

(Some) Resolutions of the Strong CP Problem

• Just declare CP to be good in the strong sector• Why in the strong and not in the weak?

• mu = 0

• mu≠0

• Additional Peccei-Quinn symmetry & axions

• Fine tuning problem can be reintroduced via high-dimensional (11+) operators at Planck scale.

q̄⇣i /D �mei✓

0�5

⌘q

Gasser & Leutwyler PhysRept 87 77-169 (1982)

Peccei & Quinn: PRL 38 (1977) 1440, PR D16 (1977) 1791

Holman et al. arXiv:hep-ph/9203206Cheung arXiv:1003.0941

Kaplan & Manohar PRL 56 2004 (1986)

‘t Hooft PRL 37 8 (1976)Jackiw & Rebbi, PRL 37 127 (1976)Callan, Dashen & Gross PLB 63 335 (1976)

5

Axions

• Couple to topological charge

• Otherwise have shift symmetry.

• Amenable to effective theory treatment

• PQ symmetry can break before or after inflation.

Peccei & Quinn: PRL 38 (1977) 1440, PR D16 (1977) 1791

LQCD 3 ✓1

32⇡2✏µ⌫⇢�Fµ⌫F⇢�

6

Axions

• Couple to topological charge

• Otherwise have shift symmetry.

• Amenable to effective theory treatment

• PQ symmetry can break before or after inflation.

Peccei & Quinn: PRL 38 (1977) 1440, PR D16 (1977) 1791

a ! a+ ↵

Laxions

=1

2(@µa)

2 +

✓a

fa+ ✓

◆1

32⇡2

✏µ⌫⇢�Fµ⌫F⇢�

m2af

2a =

@2F

@✓2

����✓=0

Ve↵ ⇠ cos

✓✓ +

haifa

◆

6

Axions

• Couple to topological charge

• Otherwise have shift symmetry.

• Amenable to effective theory treatment

• PQ symmetry can break before or after inflation.

Peccei & Quinn: PRL 38 (1977) 1440, PR D16 (1977) 1791

a ! a+ ↵

m2af

2a = �

Laxions

=1

2(@µa)

2 +

✓a

fa+ ✓

◆1

32⇡2

✏µ⌫⇢�Fµ⌫F⇢�

Ve↵ ⇠ cos

✓✓ +

haifa

◆

6

Axions

• Couple to topological charge

• Otherwise have shift symmetry.

• Amenable to effective theory treatment

• PQ symmetry can break before or after inflation.

Peccei & Quinn: PRL 38 (1977) 1440, PR D16 (1977) 1791

a ! a+ ↵

m2af

2a = �

Axion mass

QCDTopological

Susceptibility

Laxions

=1

2(@µa)

2 +

✓a

fa+ ✓

◆1

32⇡2

✏µ⌫⇢�Fµ⌫F⇢�

Ve↵ ⇠ cos

✓✓ +

haifa

◆

6

10-10

10-16

10-15

10-14

10-13

10 100 1000

1 10 100

Axio

n C

oupl

ing

|gaaa |

(GeV

-1)

Axion Mass (µeV)

Gen 2 ADMX Projected Sensitivity

Cavity Frequency (GHz)

2015

2016

2017

20182019

"Hadronic" Coupling

Minimum Coupling

Axion

Cold D

ark Ma

tter

Warm D

ark Ma

tter

ADMX Published Limits

Non RF-cavity Techniques

Too Much Dark Matter

White Dwarf and Supernova Bounds

Current Axion Constraints

Courtesy GP Carosi, ADMX

7

10-10

10-16

10-15

10-14

10-13

10 100 1000

1 10 100

Axio

n C

oupl

ing

|gaaa |

(GeV

-1)

Axion Mass (µeV)

Gen 2 ADMX Projected Sensitivity

Cavity Frequency (GHz)

2015

2016

2017

20182019

"Hadronic" Coupling

Minimum Coupling

Axion

Cold D

ark Ma

tter

Warm D

ark Ma

tter

ADMX Published Limits

Non RF-cavity Techniques

Too Much Dark Matter

White Dwarf and Supernova Bounds

Current Axion Constraints

Over-closureconstraint onaxion mass

Courtesy GP Carosi, ADMX

7

The Over-Closure Bound

High temperature argumentsimply χ vanishes as T→∞

8

The Over-Closure Bound

High temperature argumentsimply χ vanishes as T→∞

Universe cools as it expands

8

The Over-Closure Bound

High temperature argumentsimply χ vanishes as T→∞

Universe cools as it expands

Axion number fixed when 3H ~ ma

T1 ≈ 5.5 Tc

H: Hubble constant

8

Axion number fixed when 3H ~ ma

T1 ≈ 5.5 Tc

The Over-Closure Bound

3H(T1) ⇠ ma(T1)

9H2(T1)f2a ⇠ �(T1)

T1 = T1(fa,�(T ))

9

Axion number fixed when 3H ~ ma

T1 ≈ 5.5 Tc

The Over-Closure Bound

Axions continue to get heavierafter production stops!⇢(t) 6=

✓a(t1)

a(t)

◆3

⇢(t1)

3H(T1) ⇠ ma(T1)

9H2(T1)f2a ⇠ �(T1)

T1 = T1(fa,�(T ))

9

Axion number fixed when 3H ~ ma

T1 ≈ 5.5 Tc

The Over-Closure Bound

⇢(t)R3

ma(t)= # axions in a fixed comoving volume

Axions continue to get heavierafter production stops!

3H(T1) ⇠ ma(T1)

9H2(T1)f2a ⇠ �(T1)

T1 = T1(fa,�(T ))

9

Axion Density ⇢(t)R3

ma(t)= # axions in a fixed comoving volume

Cosmological EOM:

⇢(T�) =1

2✓21

q�(T�)�(T1)

✓R(T1)

R(T�)

◆3

⇢(T1) =1

2✓21m

2af

2a =

1

2✓21�(T1)

10

Axion Density ⇢(t)R3

ma(t)= # axions in a fixed comoving volume

Cosmological EOM:

⇢(T�) =1

2✓21

q�(T�)�(T1)

✓R(T1)

R(T�)

◆3

⇢(T1) =1

2✓21m

2af

2a =

1

2✓21�(T1)

χPT

f2am

2a(T�) =

mumd

(mu +md)2f2⇡m

2⇡

10

Axion Density ⇢(t)R3

ma(t)= # axions in a fixed comoving volume

Cosmological EOM:

⇢(T�) =1

2✓21

q�(T�)�(T1)

✓R(T1)

R(T�)

◆3

⇢(T1) =1

2✓21m

2af

2a =

1

2✓21�(T1)

CosmologyStrong Dynamics

3H ~ ma

χPT

10

Axion Density ⇢(t)R3

ma(t)= # axions in a fixed comoving volume

Cosmological EOM:

⇢(T�) =1

2✓21

q�(T�)�(T1)

✓R(T1)

R(T�)

◆3

⇢(T1) =1

2✓21m

2af

2a =

1

2✓21�(T1)

CosmologyStrong Dynamics

3H ~ ma

Initial conditionsχPT

10

Axion Density ⇢(t)R3

ma(t)= # axions in a fixed comoving volume

Cosmological EOM:

⇢(T�) =1

2✓21

q�(T�)�(T1)

✓R(T1)

R(T�)

◆3

⇢(T1) =1

2✓21m

2af

2a =

1

2✓21�(T1)

CosmologyStrong Dynamics

3H ~ ma

Initial conditions

Early PQ breaking: constraints depend on θ1

Late PQ breaking: ⌦✓21↵=

⇡2

3constraints on ma, fa

χPT

10

9H2(T1)f2a ⇠ �(T1)

T1 goes down as fa goes up.If fa goes up, ρ(Τγ) goes up...

Axion Density ⇢(t)R3

ma(t)= # axions in a fixed comoving volume

⇢(T�) =1

2✓21

q�(T�)�(T1)

✓R(T1)

R(T�)

◆3

If T1 goes down, ρ(Τγ) goes up...

Small ma are excluded by observed dark matter abundances!

m2af

2a = �

11

Prior Over-Closure Bound

• ma(T) is provided by models.

⇢

⇢c< ⌦CDM = 0.12

Wantz & Shellard, arXiv:0910.1066

fa ≤ (2.8 ± 2) 1011 GeVma ≥ 21 ± 2 μeV( ± uncontrolled systematic)

12

10-10

10-16

10-15

10-14

10-13

10 100 1000

1 10 100

Axio

n C

oupl

ing

|gaaa |

(GeV

-1)

Axion Mass (µeV)

Gen 2 ADMX Projected Sensitivity

Cavity Frequency (GHz)

2015

2016

2017

20182019

"Hadronic" Coupling

Minimum Coupling

Axion

Cold D

ark Ma

tter

Warm D

ark Ma

tter

ADMX Published Limits

Non RF-cavity Techniques

Too Much Dark Matter

White Dwarf and Supernova Bounds

Current Axion Constraints

Courtesy GP Carosi, ADMX

Wantz & Shellard, arXiv:0910.1066

⌦a ⌦CDM

13

10-10

10-16

10-15

10-14

10-13

10 100 1000

1 10 100

Axio

n C

oupl

ing

|gaaa |

(GeV

-1)

Axion Mass (µeV)

Gen 2 ADMX Projected Sensitivity

Cavity Frequency (GHz)

2015

2016

2017

20182019

"Hadronic" Coupling

Minimum Coupling

Axion

Cold D

ark Ma

tter

Warm D

ark Ma

tter

ADMX Published Limits

Non RF-cavity Techniques

Too Much Dark Matter

White Dwarf and Supernova Bounds

Current Axion Constraints

Courtesy GP Carosi, ADMX

Wantz & ShellardIILM

Wantz & Shellard, arXiv:0910.1066

systematic uncertainty?

⌦a ⌦CDM

13

10-10

10-16

10-15

10-14

10-13

10 100 1000

1 10 100

Axio

n C

oupl

ing

|gaaa |

(GeV

-1)

Axion Mass (µeV)

Gen 2 ADMX Projected Sensitivity

Cavity Frequency (GHz)

2015

2016

2017

20182019

"Hadronic" Coupling

Minimum Coupling

Axion

Cold D

ark Ma

tter

Warm D

ark Ma

tter

ADMX Published Limits

Non RF-cavity Techniques

Too Much Dark Matter

White Dwarf and Supernova Bounds

Current Axion Constraints

Courtesy GP Carosi, ADMX

Wantz & ShellardIILM

Reliance on models is unnecessary:we can calculate χ from lattice QCD!

Wantz & Shellard, arXiv:0910.1066

systematic uncertainty?

⌦a ⌦CDM

13

We study pure Yang-Mills, and not yet full QCD.CAVEAT

• Dramatically more efficient algorithms enable huge statistics and volumes, shorter autocorrelation times.

• Tc is ~284 MeV, compared to 154 MeV in QCD.

• High temperature tends to suppress quark loops.

• What counts as high temperature?

• Unclear if this holds true for topological observables.

But: for getting a lower bound on ma it's not so bad!14

• SU(3) YM with Wilson plaquette action

• T between 1.2 and 2.5

• Nσ between 48 and 144 (larger at higher T)

• Nτ either 6 or 8

• Between 14000 and 52000 measurements• Combined hot & cold starts• Cut of 2000 cfg.s for thermalization• 10 compound sweeps of 1 heatbath step

and 8 over-relaxation steps

Overview of Lattice Ensembles & Measurements

1

32⇡2

X

x

✏µ⌫⇢�⇤µ⌫

⇤⇢�

Berkowitz, Buchoff, and Rinaldi, arXiv:1505.07455

QR

QZ

Qa

Qf

raw measurement

naïve rounding

artifact corrected

globally fit

Lucini & Teper, hep-lat/0103027

del Debbio et al., hep-th/0204125

QOV overlap

Wilson FlowQWF

15

• SU(3) YM with Wilson plaquette action

• T between 1.2 and 2.5

• Nσ between 48 and 144 (larger at higher T)

• Nτ either 6 or 8

• Between 14000 and 52000 measurements• Combined hot & cold starts• Cut of 2000 cfg.s for thermalization• 10 compound sweeps of 1 heatbath step

and 8 over-relaxation steps

Overview of Lattice Ensembles & Measurements

Essentially no discretization or finite volume corrections

1

32⇡2

X

x

✏µ⌫⇢�⇤µ⌫

⇤⇢�

Berkowitz, Buchoff, and Rinaldi, arXiv:1505.07455

QR

QZ

Qa

Qf

raw measurement

naïve rounding

artifact corrected

globally fit

Lucini & Teper, hep-lat/0103027

del Debbio et al., hep-th/0204125

15

• SU(3) YM with Wilson plaquette action

• T between 1.2 and 2.5

• Nσ between 48 and 144 (larger at higher T)

• Nτ either 6 or 8

• Between 14000 and 52000 measurements• Combined hot & cold starts• Cut of 2000 cfg.s for thermalization• 10 compound sweeps of 1 heatbath step

and 8 over-relaxation steps

Overview of Lattice Ensembles & MeasurementsBerkowitz, Buchoff, and Rinaldi, arXiv:1505.07455

QR

QZ

Qa

Qf

raw measurement

naïve rounding

artifact corrected

globally fit

Lucini & Teper, hep-lat/0103027

del Debbio et al., hep-th/0204125

15

0. 0.5 1. 1.5 2. 2.50.

0.2

0.4

0.6

0.8

T/Tc

�1/4 /T c

Measurement N�

���������� �� ��� 12

16

20

0. 0.5 1. 1.5 2. 2.50.

0.2

0.4

0.6

0.8

T/Tc

�1/4 /T c

Measurement N�

���������� �� ��� 12

� - ���� � 16

20

64

80

96

144

Best Lattice ResultsPure glue measurementsshow χ vanishes as T→∞

Gattringer et al., arXiv:hep-lat/0203013 up to T/Tc ~ 1.3

Berkowitz, Buchoff, and Rinaldi, arXiv:1505.07455

16

0.082 0.084 0.086 0.088 0.090

5.55

5.60

5.65

5.70

5.75

C

n

Best Fit1�2�3�

DIGM Best Fit & Extrapolation

�

T 4c

=C

(T/Tc)n

Pure glue measurementsshow χ vanishes as T→∞

C nBest Fit 0.0865 5.63

Covariance MatrixC 2.×10-6 5.×10-5

n 5.×10-5 0.0012

1. 2. 3. 4. 5. 6.0.

0.1

0.2

0.3

0.4

0.5

0.6

T/Tc

�1/4 /T c

Measurement N�

� - �������� �� 64

80

96

144

Berkowitz, Buchoff, and Rinaldi, arXiv:1505.07455

17

0 5 10 15 20 25 30

10-9

10-7

10-5

10-3

T/Tc

�/Tc4

DIGM fit9H2 fa2/Tc4, fa=1010 GeV9H2 fa2/Tc4, fa=1011 GeV9H2 fa2/Tc4, fa=1012 GeV

Axion Production Ceases Axions production stops when

3H ~ maT1 ≈ 5.5 Tc from models

fa [GeV]

1012

1011

1010

3H = ma

Berkowitz, Buchoff, and Rinaldi, arXiv:1505.07455

18

0 5 10 15 20 25 30

10-9

10-7

10-5

10-3

T/Tc

�/Tc4

DIGM fit9H2 fa2/Tc4, fa=1010 GeV9H2 fa2/Tc4, fa=1011 GeV9H2 fa2/Tc4, fa=1012 GeV

Axion Production Ceases Axions production stops when

3H ~ maT1 ≈ 5.5 Tc from models

fa [GeV]

1012

1011

1010

9H2f2a = m2

af2a = �

H2 =⇡2

90

1

m2P

g⇤R(T ) T4

Berkowitz, Buchoff, and Rinaldi, arXiv:1505.07455

18

Axion Density ⇢(t)R3

ma(t)= # axions in a fixed comoving volume

Cosmological EOM:

⇢(T�) =1

2✓21

q�(T�)�(T1)

✓R(T1)

R(T�)

◆3

⇢(T1) =1

2✓21m

2af

2a =

1

2✓21�(T1)

χPT Cosmology

3H ~ ma

Initial conditions Strong Dynamics

19

Axion Density ⇢(t)R3

ma(t)= # axions in a fixed comoving volume

Cosmological EOM:

⇢(T�) =1

2✓21

q�(T�)�(T1)

✓R(T1)

R(T�)

◆3

⇢(T1) =1

2✓21m

2af

2a =

1

2✓21�(T1)

χPT Cosmology

3H ~ ma

Initial conditions Lattice!

19

10-10

10-16

10-15

10-14

10-13

10 100 1000

1 10 100

Axio

n C

oupl

ing

|gaaa |

(GeV

-1)

Axion Mass (µeV)

Gen 2 ADMX Projected Sensitivity

Cavity Frequency (GHz)

2015

2016

2017

20182019

"Hadronic" Coupling

Minimum Coupling

Axion

Cold D

ark Ma

tter

Warm D

ark Ma

tter

ADMX Published Limits

Non RF-cavity Techniques

Too Much Dark Matter

White Dwarf and Supernova Bounds

The Over-Closure Bound As It Stands Today

Courtesy GP Carosi, ADMX

Wantz & ShellardIILM

Berkowitz, Buchoff, and Rinaldi, arXiv:1505.07455

⌦a ⌦CDM

20

10-10

10-16

10-15

10-14

10-13

10 100 1000

1 10 100

Axio

n C

oupl

ing

|gaaa |

(GeV

-1)

Axion Mass (µeV)

Gen 2 ADMX Projected Sensitivity

Cavity Frequency (GHz)

2015

2016

2017

20182019

"Hadronic" Coupling

Minimum Coupling

Axion

Cold D

ark Ma

tter

Warm D

ark Ma

tter

ADMX Published Limits

Non RF-cavity Techniques

Too Much Dark Matter

White Dwarf and Supernova Bounds

The Over-Closure Bound As It Stands Today

Courtesy GP Carosi, ADMX

Berkowitz, Buchoff, and Rinaldi, arXiv:1505.07455

Lattice SU(3)Pure Glue

Late PQ Breaking

fa < (4.10±0.04) 1011 GeVma > (14.6±0.1) μeV

⌦a ⌦CDM

20

10-10

10-16

10-15

10-14

10-13

10 100 1000

1 10 100

Axio

n C

oupl

ing

|gaaa |

(GeV

-1)

Axion Mass (µeV)

Gen 2 ADMX Projected Sensitivity

Cavity Frequency (GHz)

2015

2016

2017

20182019

"Hadronic" Coupling

Minimum Coupling

Axion

Cold D

ark Ma

tter

Warm D

ark Ma

tter

ADMX Published Limits

Non RF-cavity Techniques

Too Much Dark Matter

White Dwarf and Supernova Bounds

The Over-Closure Bound As It Stands Today

Courtesy GP Carosi, ADMX

Berkowitz, Buchoff, and Rinaldi, arXiv:1505.07455

Lattice SU(3)Pure Glue

Late PQ Breaking

fa < (4.10±0.04) 1011 GeVma > (14.6±0.1) μeV

axions < 100% of DM

smaller χ

⌦a ⌦CDM

20

Conclusions & Outlook

• PQ symmetry:• cleans up the Strong CP problem• provides a plausible, largely unconstrained DM candidate: the axion.

• Axion searches will search large swaths of interesting parameter space soon.

• Power law (DIGM-inspired) fits outstandingly to pure glue at high temperature.

The Economist, 19 Dec 2006

Lattice QCD can provide important nonperturbative input for calculating Ωa

21

Future Steps

• Measure higher moments? May be able to get χ4, χ6.

• Incorporate quarks

• Move to Wilson Flow definition

• Move to anisotropic lattices to alleviate finite-volume effects at high T.

• Fixed topology methods / open boundary conditions at high T.

• Finite θ:• Imaginary-θ has no sign problem• Real, finite θ may be amenable to Langevin methods

Lüscher & Schæfer, arXiv:1105.4749Aoki et al., arXiv:0707.0396v2

T=0: Cé, Consonni, Engle & Giusti, arXiv:1506.06052

22

Backup Slides

23

Finite Volume EffectsBerkowitz, Buchoff, and Rinaldi, arXiv:1505.07455

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2. 2.10.

0.1

0.2

0.3

0.4

T/Tc

�1/4 /T c

Measurement N�

� - ������� 48

� - ���� ���� �� 64

� - ����� 80

� - �������� � 96

24

Discretization EffectsBerkowitz, Buchoff, and Rinaldi (arXiv:1505.07455), Kitano & Yamada (arXiv:1506.00370)

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2. 2.1 2.2 2.30.

0.1

0.2

0.3

0.4

T/Tc

�1/4 /T c

Measurement N�

� - ������� 4

� - ���� ���� �� 6

� - ����� 8

� - �������� ����� ��� ������

25

ComparisonsPure glue measurementsshow χ vanishes as T→∞

��� �=���=��� ��

���� ��

��

��

��

��

��

��

��

��

��

��

��

0. 0.5 1. 1.5 2. 2.5 3. 3.5 4.0.

0.2

0.4

0.6

0.8

1.

1.2

1.4

1.6

T/Tc

�1/4 /T c

Measurement N�

������� �� �� �� ����� (���� ����) 12

������� �� �������� ���� � ��� ��=��� �� ! 16

"#�/"����$��% ��&�'(��)�*+� ,��- ��=*� �� ! 18

"#�/"����$��% ��&�'(��)�*+� ,��- ��=*� �� ,.� 20

/����� ��% .� �%� ��&�'(�*)��01� 24

� - ���2���� 3� 32

���� ��&�'(�)��444 (��=�++ ��=01� �� ) 64

���������� �� ��� ��5-���/���0�0 80

6�-�� (��=� ��=0*4 �� �=7� �� ) 96

144

26

��� �=���=��� ��

��

��

��

0. 0.5 1. 1.5 2.0.

0.2

0.4

0.6

0.8

T/Tc

�1/4 /T c

Measurement N�

������� �� �� �� ����� (���� ����) 12

������� �� �������� ���� � ��� ��=��� �� ! 16

� - ���"���� #� 20

���������� �� ��� ��$-���/���%�% 24

&�'�� (��=� ��=%() �� �=*� �� ) 32

64

80

96

144

ComparisonsPure glue measurementsshow χ vanishes as T→∞

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The Misalignment Mechanism

Axion begins to oscillate when Compton λfits inside the universe

At T ~ 1 GeV, potential develops and axion mass begins to grow.

At high temperature, the axion potential is flat.

3H ~ ma

28