θ physics and axion
Ryuichiro Kitano (KEK)

talk@Axion physics and dark matter cosmology, Osaka, Dec. 20-21, 2017θ term in QCD

ZQCD (✓) =ZSQCD¯[dA][d ][d ]einstanton number:(integer!)

How does θ affect physics?

It is actually physical, and even known to be problematic.

Strong CP problem
θ term breaks CP[’t Hooft ’76]

ɣπ????
nn[Crewther, Di Vecchia,
Veneziano, Witten ’79]θ dependence of free energy?
F (✓) =1log Z(✓)
V

topological susceptibility
200=hQi/V > 0t = F (✓)

⇡0⇡

2⇡✓

dilute instanton gas picture
F (✓) =1log Z(✓)
VF (✓) =⇡0⇡
t =F00t (12⇡(✓)

cos ✓)✓

[Witten ’98]

large N picture
F (✓) =1log Z(✓)
VF (✓) =

⇡0

⇡

2⇡

t 22

✓ +…

✓00
=F
(✓)
t

This seems to hold in YM theory with finite N. (maybe N>2)
[Gaiotto, Kapustin, Komargodski, Seiberg ’17] [RK, Suyama, Yamada ’17]

massless quarkF (✓) =1log Z(✓)
VF (✓) = 0

θ is unphysicalno strong CP problem⇡0⇡t =02⇡✓𝜒t and quark mass

If mu is non zero, θ is physical.
If mu=0, physics does not depend on θ.
—> no strong CP problem

mu=0?
[PDG]

Lattice QCD?
The quark mass is something you define
it’s not just a parameter in the Lagrangian.
One should define the quark mass so thatt / muin order to establish
mu 6= 0 and the strong CP problem is real.

𝜒t on the lattice-3-2-10123

Let’s examine if the “lattice quark mass”
satisfiest / mu1+2 flavor QCD
measure of the Strong CP problem(physical)[RK, Yamada, Frison ’16]

Preliminary

lattice quark mass

It seems that there is a large discretization effects..
We need more data to say something quantitative.

Hopefully we can say
something soon…
Sorry! No conclusion yet…

[Peccei, Quinn ’77][Weinberg ’78][Wilczek ’78]
[Kim ’79][Shifman, Vainstein, Zakharov ’80]

Axion
promote θ toa dynamical field[Zhitnitsky ’80]

[Dine, Fischler, Srednicki ’81]

minimumof the potential preserve CP!

F (✓)

⇡0⇡
t =F002⇡(✓) = m2a fa2✓

axion mass[Preskill, Wise, Wilczek ’83][Abbott, Sikivie ’83]
[Dine, Fischler ’83]

Axion Dark Matter
V(a)this potential changes asa function of temperatureawheretemperature dependence of the axion massis the essential information to estimate the abundance.

instanton paradigm
The standard way to calculate the temperaturedependence of ma is based on the dilute instanton gas
approximation.
[Pisarsky, Yaffe ’80]

𝜒tinstanton action

TcTif 𝜒t shuts off very quickly at Tc
the axion suddenly starts to oscillate at T~Tc

𝜒tTc

Tenhancement due to the non-adiabatic evolutionof the potential.

Dark Matter abundance is controlled by
How quickly the potential grows as temperature.

lattice results (SU(3) YM)
index

semi-classicalinstanton estimates(T^7)[RK, Yamada]

Wilson flow
Cooling[Berkowitz et al.][Borsanyi et al.]Power Law
We see a clear power law even at a very low temperature.
The dilute Instanton gas picture seems to be pretty good
above the critical temperature.

directly access the exponent[Frison, RK, Matsufuru, Mori, Yamada ’16]

wecanmeasurethisinstanton prediction is “-b+4-Nf”
by fixing the topology.

[Frison, RK, Matsufuru, Mori, Yamada ’16]

Fixed topology measurements(quenched)

163x4

This quantity is related to the difference of <S> for fixed Q.
Instanton looks good.

fun with instantons
high temperature

low temperature

instanton gas predictions

dynamical fermions
from [1606.07494 Borsanyi et. al.]Instanton picture looks essentially OK,
but normalization differs by O(10).

Instanton picture seems to work at high temperature.
(Actually, there are still active discussion on this point.)See Fukaya’s talk!

Summaryθ is physicalif quark is massive.
We have been considering• Is quark really massive? (under progress..)• How Axion mass behaves as a function oftemperature? (looks like dilute instanton gas is goodat least qualitatively)• What’s the actual θ dependence of the vacuum?
Especially, what happens at θ=π? (spontaneous CPviolation or deconfinement!)