Determination of axion
abundance using lattice QCD
Norikazu Yamada(KEK Theory Center/SOKENDAI)

基研研究会「熱場の量子論とその応用2017」@京大基研 2017/08/30

Collaboration withJulien Frison,

Ryuichiro Kitano,

Hideo Matsufuru,

Shingo Mori

Goal
QCD θ-vacuum(spontaneous CPV),
role of instantons [poster by Mori],…

axion abundanceStrong CP problem
Symmetries of the SM do allow the θ term,

iµL =
232i
Tr(Gµ G ) =GG̃232
Gµ : gluon field strengthθ = θQCD + θYukawa−10NEDM exp ➡ θ ≲ 10 !
➡ Why is θ so small?

Two possible solutions✓mu = 0θ-term ⇒ unphysical✓Peccei-Quinn mechanism
θ-term dynamically vanishesume Isotropy
caying neutrinos
generate v

mu = 0?

iabatic

iabatic+decaying
thermalthermalferent from the heavy quark masses obtained in potential models.
For this edition we have attempted to give a conservative rangeof masses. In the next edition we will provide a more extensive

In 1992, “quark masses” appears
treatment.
for the ﬁrst time.

v'sMass m = 5 to 15 MeV
md/ms ——0.04 to 0.06

MMENTCharge= —~ eiz = —g1

The d-, u-, and s-quark masses are estimates of so-called "current"
quark masses, with ratios mu/md and md/ms extracted from pionand kaon masses using chiral symmetry. The estimates of d and umasses are not without controversy and remain under active investigation. Within the literature there are even suggestions that theu quark could be essentially massless. The s-qIJark mass is estimatedfrom SU(3) splitting in hadron masses.

~ ~ ~

ac v
jorana v

MMENT~ ~ ~(0.1/ GF (0.01/GFizedbythe equation:

Mass m = 2 to 8 MeV—0.25 to 0.70
mu/md

Charge=p e

See the comment for the d quark above.
and Cosmology

I, =+&1mu = 0?

Citation: C. Patrignani et al. (Particle Data Group), Chin. Phys. C, 40, 100001 (2016) and 2017 update
LatestPDG (2016)

Light Quarks (u, d, s)
OMITTED FROM SUMMARY TABLE
u-QUARK MASS
The u-, d-, and s-quark masses are estimates of so-called “current-quark
masses,” in a mass- independent subtraction scheme such as MS. The
ratios m u /m d and m s /m d are extracted from pion and kaon masses
using chiral symmetry. The estimates of d and u masses are not without
controversy and remain under active investigation. Within the literature
there are even suggestions that the u quark could be essentially massless.
The s-quark mass is estimated from SU(3) splittings in hadron masses.
We have normalized the MS masses at a renormalization scale of µ = 2
GeV. Results quoted in the literature at µ = 1 GeV have been rescaled bydividing by 1.35. The values of “Our Evaluation” were determined in part
via Figures 1 and 2.
MS MASS (MeV)

DOCUMENT ID

TECN+ 0.6 OUR EVALUATION
2.2 −0.4

See the ideogram below.

2.27± 0.06 ± 0.06
2.36± 0.24
2.57± 0.26 ± 0.07
2.15± 0.03 ± 0.10
1.9 ± 0.2
2.24± 0.10 ± 0.34

1 FODOR
2 CARRASCO
3 AOKI
4 DURR
5 BAZAVOV
6 BLUM
7161412111010LATT
LATT
LATT
LATT
LATT
LATT

(4)
leads to higher χt values for those actions (as seen in next plot). At finer lattice

mu = 0?

spacings (bottom half, β = 3.7), all actions tend to be more similar, qualitatively
and quantitatively

Topological susceptibility in Nf=1+20.003

are combined as

Nf=1+2 (5)m1 m2 = m3
blem.

0.0025β=3.31, Wilson flow
β=3.31, Symanzik flow
β=3.31, Iwasaki flow
β=3.31, DBW2 flow
β=3.5, Wilson flow
β=3.5, Iwasaki flow
β=3.61, Wilson flow
β=3.61, Symanzik flow
β=3.7, Wilson flow
β=3.7, Symanzik flow
β=3.7, Iwasaki flow
β=3.8, Symanzik flow

0.002

t is, in fact, not guaranχt : topologicalre, it is highly nontrivialsusceptibilityof non-perturbative
andχtχtw0.54

physical meaning of the0.0015

0.001

m1

tive renormalisation, we

0.0005

0-0.2

00.2

0.4
PCACPCAC
mu/ms

0.6

0.8

1

J. Frison, et al., in preparation

Figure 2: Smaller mPu CAC values correspond to smaller χt as expected, but a large
discretisation eﬀect remains which prevents an exact cancellation of χt at finite

[Peccei and Quinn

PQ mechanism [Peccei andQuinn (77)]Axion[Peccei and Quinn ’77]

Axion
Introduce a complex scalar field φ(x)=|φ(x)|eia(x)/fa

OK, maybe mu is non zero and θ is physical.

is non zero and θ is physical.
2 2fa Then,
why is θ soismall?
SSBa/fa
2V( )=| |e2n, why is θ so small?
The axion provides a nice solution.

the coupling to quarks
provides a Throughnice solution.

Minimizing axion potential(dynamically sele(dynamically selected)⇒ Strong CP problem gone.

term really physical?
Axion mass

oes the partition function Z depend on θ?

Axion mass:
22ma (T) = χt(T) (CP)/ 2 fa(topologicalcharge=integers!)
χ : topological susceptibilityt

Q: topological charge
1Q = 32 24At T=0,χt ≈ [80 MeV]
(topologicalsusceptibility)⇒ ma

61210GeV
610 eV
faf/Nad4 x GG̃

V(a)axion abundance
Evolution of coherent component:2¨˙ä + 3H(T ) ȧ + ma (T ) a = 03H(T)2 Mpl2 ∝ T 4
ma2(T) = χt(T) / 2 fa2
Initially, 3H(T) ≫ ma(T) ⇒ ȧ =0T≫T*ma ~ 0 ≪ 3HT>T*ma ≠0 < 3H

When 3H(T*) ∼ ma(T*) ⇒ a starts to oscillate.
na(T0) ≈ na(T*)(T0/T*)3~ ma(T*) fa2 θa,i2 (T0/T*)3(θa,i = θ + aini / fa)
Need to know T dependence of χt(T)T≈T*
ma* ≈ 3H

T=T0
ma0 > 3H0a

Constraints on axion mass

Current Axion Constraints
Gen 2 ADMX Projected Sensitivity
kmodel / fa

Cavity Frequency (GHz)
101

100

10-10

10

10

10-13-14-15

ADMX Published Limits

White Dwarf and Supernova Bounds
Too Much Dark Matter

Axion Coupling |gaaa | (GeV-1)

Non RF-cavity Techniquesterdro"Hanicg
plinuo"Clingrark
Ddoln CoAxi

teMatoupmCimuMin2019
2018

2017

2016
2015

mWaratk M

Darvery small 1/ fa⇒ long life time
⇒ DM candidate
ma10-16
10100
Axion Mass (µeV)

1000
Courtesy GP Carosi, ADMX

Axion abundance ≤ DM abundanceOver-closure bound depending on χt(T)he standard way to calculate the tempera
χt(T) frominstanton
Instanton calculus paradigm[Gross, Pisarski, Yaﬀe (1981)]
ndence of ma is based on the dilute instan
EstimateTheof axion
abundanceχt(T) in theTc<T<10 Tc.
standardwayrequires
to calculate
temperatureapproximation.
dependenceof mthe diluteinstanton
a is based on
DiluteInstanton Gas
Approximation(DIGA)
is often
used. ga

[Pisarsapproximation.

[Pisarsky, Yaffe!tAxion Dark
Matt
instanton action
2/g2−8πInstanton action = einstanton
actionb: beta function

TcT

Over-closure
boundAxion Dark Matter
Axion abundance ≤ DM abundance

Current Axion Constraints−8Gen 2 ADMX Projected Sensitivity

Instanton: χt ~Tkmodel / fa

⇒Cavity Frequency (GHz)110100[P10-101010-13-1410-15

ADMX Published Limits

White Dwarf and Supernova Bounds
Too Much Dark Matter

Axion Coupling |gaaa | (GeV-1)

Non RF-cavity Techniques

tertaM
ark
D2019
dColn
2018
oAxi
2017

W

oumC

imu

Min

axion window

2015

-16

10100
Axion Mass (µeV)

Mild T-dependence

Darmgplin

2016

10"croni

d"Ha

arkg
plin
Cou

tertaMaxion window
1000
Courtesy GP Carosi, ADMX

Steep T-dependenceOver-closure boundgood DM abundanceIs DIGA reliable?
At low T (~ΛQCD), validity is questionable.
The following extreme behaviors are suggested or yet-allowed.

✓ Step function-like behavior
𝜒t(T) ~ 𝜒t(T=0) θ(Tc−T) if mu,d < mqcrit✓ A bit milder case𝜒t(T) ~ 𝜒t(T=0)
for T ≲ Tc
~ 𝜒t(T=0) exp[ −2c(mq) T2/Tc2 ] for T > Tc

In extreme casesT≫T*ma ~ 0 ≪ 3HT≈T*≈Tc≈200 MeV

V(a)[Kitano, Yamada (2015)]

ma(T) rapidly grows

T≈T*≈Tc
ma0 > 3H

T=T0
ma0 > 3H0

V(a)

V(a)

V(a)

ρa(T*)

aa

scale factor a/ai (or t)aa
ρa(T0)=ρa(T*)×(a*/a0)3

Oscillation begins.
ρa(T*) is larger than the adiabatic case.

Goal2
χt(T) = 〈Q 〉/V4in Tc < T < O(10×Tc)

Lattice calculations of χt(T)!t on the lattice MC history of Q•

Generate configurations

Measure Q at eachweconfiguration
just need to measure Q in each configuration.

•Lattice calculationsof χt(T)𝜒tDistribution of Qwidth〈Q2〉!t on the lattice-3-2-10123!t measures how often instantons appear in the path integral.
If !t is nonzero, θ is physical.

we just need to measure Q in each configur

χt in Somehow,
pure Yang-Milles
in 2015, three independent calculations appeared.
(in the SU(3) Yang-Milles theory, no quarks yet)
E. Berkowiz, M. Buchoff, E. Rinaldi (LLNL)
Bosonic (cooling)
RKand N. Yamada(overlap)
R. Kitanoand NY (KEK)(KEK) FermionicIndex theorem
S. Mages et al (BMW)Bosonic (Wilson Flow)

All look consistent

Quenched approximation (no dynamical quark)(at least qualitatively)[Mages (Lattice 2015 conf.)]

We see a clear power law even at a very low temperature.

Standard method fails at T > 4 Tc .

Problem in the standard method at high T𝜒2〉: width of Q-distribution〈QtLow THigh T

Q-3-2

-1

0123

­3 ­2 ­1 0 1 2 3Aboveacertain
highT,
distribution⇒
δ(Q)measures how often instantons appear in the path integral.
If !t is nonzero,
physical. χt ( =〈Q2〉/V4 ) when 〈Q2〉≪1Fail toθ isestimate✓ New MethodJ. Frison, R. Kitano, H. Matsufuru, S. Mori, NY, JHEP 1609 (2016) 021 [arXiv:1606.07175 [hep-lat]]New Method
𝜒tJ. Frison, R. Kitano, H. Matsufuru, S. Mori, NY, JHEP 1609 (2016) 021 [arXiv:1606.07175 [hep-lat]]

Low THigh T

Q-3-2-10

12

3measures how often instantons appear in the path integral.

­3 ­2 ­1 0 1 2 3

Giveupχ(T)t
If ! is nonzero, θ is physical.
Focus on d ln χt(T) / d ln Tt

[9].

New Method
J. Frison, R. Kitano, H. Matsufuru, S. Mori, NY, JHEP 1609 (2016) 021 [arXiv:1606.07175 [hep-lat]]

ANG ZEROS ASSOCIATED WITH DISCONTINUITY OF F (✓)

partition function is given by the summation over all topological sectω(T) = χt(T) V4 (mπ and Nsite are fixed)
1Xi✓QF (✓)Z(✓) =ZQ e = e,
Q= 1this note, we call F (✓) free energy. The topological susceptibility!=t V4 =d2 ln Z(✓)=2d✓✓=0d 1 Xi✓Qi
QZQ ed✓ Z(✓) Q✓=0!

t

To experts

ω(T) = χt(T) V4 (mq and Nsite are fixed)(Q2)

where(Q1)To experts

ω(T) = χt(T) V4 (mq and Nsite are fixed)
When ω ≫ 1,When ω ≪ 1,(1+…)

New Method
J. Frison, R. Kitano, H. Matsufuru, S. Mori, NY, JHEP 1609 (2016) 021 [arXiv:1606.07175 [hep-lat]](Q2)

Looks complicated, but calculable!

(Q1)

High T Limit (ɡ2→0 limit)
Hight T Limit ⇒28(Q)
Sg |BPST = 2 |Q|
gcf.

≈ −11 + 4
With |Q|=1, DIGA results χt(T) ~ T −7 is reproduced.
The above is the quenched case.
Similar argument in dynamical case.

✓ Test in quench

J. Frison, R. Kitano, H. Matsufuru, S. Mori, NY,JHEP 1609 (2016) 021 [arXiv:1606.07175 [hep-lat]]•••

V=163×4 (and 243×4)
Iwasaki gauge action w/ Q=0, 1, 2Tc < T < 104 Tc

Test in the quenched approximation[Kitano, Frison, Matsufuru, Mori, NY (2016)]

(Q2)

(Q1)

Consistent with DIGA,χt ∝ T −7 at T > 2 Tc

16^3 x 4

But
poor accuracy
behavior around Tc
unclear.

✓ Improved calculation
J. Frison, R. Kitano, H. Matsufuru, S. Mori, NY, in preparation•••

V=243×6Iwasaki gauge action w/ Q=0, 5, 19, 36, 45
Tc < T < 6 Tc

Test in the quenched approximation[Kitano, Frison, Matsufuru, Mori, NY, in preparation](Q2)

424^3 x 6d ln χt /d ln T

2

Stat. error significantly
reduced !

Q=1936450χt ∝ T −8 rather than T −7around T ~ 6 Tc

-2-4-6

Instanton calculus

-8-10
0

1

2

3

4T/Tc

5

(Q1)

67

Summary✓ Lattice QCD can explore axion physics!
✓ New method allows us to calculate χt(T) atvery high T.
✓ The instanton calculus will be replaced bylattice determination of χt(T) in the estimateof the axion abundance.