Lattice QCD Input to Axion Cosmology · Lattice QCD Input to Axion Cosmology Evan Berkowitz...
Transcript of Lattice QCD Input to Axion Cosmology · Lattice QCD Input to Axion Cosmology Evan Berkowitz...
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→∞
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
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→∞
��� �=���=��� ��
���� ��
��
��
��
��
��
��
��
��
��
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��
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�
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���� ��&�'(�)��444 (��=�++ ��=01� �� ) 64
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144
26
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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→∞
27