Topology and QCD axion’s properties

from lattice hot QCD with a dynamical charm

M.P. Lombardo

in collaboration with A. Trunin, F. Burger, E.-M. Ilgenfritz, M. Muller-Preussker ¨ †

February 12-15, 2017

Slide from Hanna Petersen QM2017 Plenary talk

Temperatures:

Quark Gluon Plasma: Topology

150 MeV < T < 500 MeV

..and beyond

Temperatures Time from Big Bang

Quark Gluon Plasma: Topology

150 MeV < T < 500 MeV

..and beyond

N(T)

Temperature and Time from BigBangare linked by the Equation of State

TemperatureTime from Big Bang

Axions’s freezout

Quark Gluon Plasma:Topology

Axions’ mass and density

today

QCD Topology,Strong CP Problem Dark Matter Candidate

The two faces of the axion

Outline

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- Gluonic operator and gradient flow - Fermionic operator

-Hot QCD& the lattice

-Topology:

-A window on the QCD axion

-Comments/outlook

✓ vacuum, strong CP problem

Hot QCD and Nf 2+1+1 twisted mass

up,down in the isospin limit

Physical strangePhysical charm

Wilson fermions

Our setup at a glance

Table 1. Number and parameters of used configurations

T = 0 (ETMC)nomenclature

� a [fm] [6] N

3

�

N

⌧

T [MeV] # confs.

A60.24 1.90 0.0936(38)243

323

567891011121314

422(17)351(14)301(12)263(11)234(10)211(9)192(8)176(7)162(7)151(6)

585137034197057752522710522941988

B55.32 1.95 0.0823(37) 323

5678910111213141516

479(22)400(18)342(15)300(13)266(12)240(11)218(10)200(9)184(8)171(8)160(7)150(7)

59534532723345329566711023081304456823

D45.32 2.10 0.0646(26)323

403

483

678101214161820

509(20)436(18)382(15)305(12)255(10)218(9)191(8)170(7)153(6)

403412416420380793626599582

7

Fixedvaryingscale

Four pionmasses

For each lattice spacing we explore a range of temperatures 150MeV — 500 MeV by varying Nt

We repeat this for three different lattice spacings following ETMC T=0 simulations.

Advantages: we rely on the setup of ETMC T=0 simulations. Scale is set once for all.

Disadvantages: mismatch of temperatures - need interpolation before taking the continuum limit

Setup

The pseudo critical temperatures

From < ̄ >�discFrom

Slide from M. Ilgenfritz,

Dubna Nov. 2016

The pseudo critical temperature

Topology

QCD topology, long standing focus of strong interaction:

-learning about the structure of the (s)QGP -fundamental symmetries, strongCP problem —> axions -hampered by technical difficulties

Recent developments:

-methodological progress: gradient flow, chiral fermions -first results for dynamical fermions at high temperature:

Trunin et al. J.Phys.Conf.Ser. 668 (2016) no.1, 012123 Bonati et al. JHEP 1603 (2016) 155 Borsany et al. Nature 539 (2016) no.7627, 69-71 Petreczky et al. Phys.Lett. B762 (2016) 498-505

= (< Q2 > � < Q >2)/V

✓ term, strong CP problem and topology

e�F (✓) = hei✓Qi

P⌫

=

Z⇡

�⇡

d✓

2⇡e�i✓⌫e�F (✓)

Cn

= (�1)n+1 1

V

d2n

d✓2nF (✓)

���✓=0

⌘< Q2n >conn

F (✓) = V1X

n=1

(�1)n+1 ✓2n

(2n)!C

n

P⌫

=e�

⌫2

2�2

p2⇡�2

1 +

1

4!

⌧

�2He4 (⌫/�)

�

�2 = V C1 and ⌧ = C2/C1

Taylor expansion, and cumulants of the topological charge distributionP (Q)

Q

Q = ⌫

P(Q) is Gaussian for V ! 1Details on F (✓) hidden in cumulants’ ratio.

In practice:

F (✓, T ) = 1/2�(T )✓2s(✓, t)

s(✓, T )1 + b2(T )✓2 + ...

b2 = �<Q4>�3<Q2>2

12<Q2>

DIGA — at very high temperature — predicts

F (✓, T )� F (0, T ) = �(T )(1� cos(✓)) b2 = �1/12

Q

Cumulants ofP(Q)

Taylor coefficients of F (✓, T )

Gluonic (butterfly) operator +

Gradient Flow Method

Results I

Gradient flow

Stop flowing when

Continuum limit of is independent on thechosen reference value

Monitor as a function of t

Evolve the link variables in a fictitious flow time:

< O(t) >Observables renormalized at µ = 1/p8t

< O(t) >

Luscher, Luscher Weisz¨ ¨

t2 < E > |t=t

x

,x=0�6 = (0.3, 0.66, 0.1, 0.15, 0.2, 0.25, 0.45)

Flowing towards the plateau

�0.25[MeV ]

T

We focuson

t(0.3) = t0and

t(0.66)= t1

Reference value

a = 0.082 fm

t0 t1

a = 0.082 fm a = 0.065 fm

On finer lattices, plateau is almost reached: �0.25[M

eV]

Reference value Reference value

Gradient method coincides with cooling

Continuum limit: - in principle independent on flow limit - we need to interpolate results at fixed scale to match T

Results for the topological susceptibility for M⇡ = 270 MeV

�(T,m⇡

) = lima!0 �1/4(T, a,m

⇡

, tx

)

�1/4(T, a,m⇡

, tx

) = �1/4(T,m⇡

) + a2k(T, tx

)

Very slowdecay:Nearly linear, or power-like with a tiny exponent: 60

80

100

120

140

160

180

200

150 200 250 300 350 400 450

χ0.25

T�(T )0.25 ' aT�0.26 ' T, T > 200 MeV

Continuum results for

Used: Bezier interp.

m⇡ = 370 MeV

(In)dependence of continuum limit on flow’s limit: 0.3 OK

Detailed analysis for T > 200 MeV (use approx. linearity) - 1

Detailed analysis for T > 200 MeV - 2

Interpolation ok.

1.0 1.5 2.0 2.5

50

100

150

200

c1ê4 @M

eVD

T êTcNf = 2+1+1, mp ~ 370 MeV

Nf =2+1, mp ~135 MeVrescaled as c~mp4

rescaled as c~mp2

Bonati et al. 2016

Bonati et al. 2016

A mass rescaling appears to work nicely

Detailed analysis for T > 200 MeV - 3

150 200 250 300 350 400 450 500

80

100

120

140

160

180

200 t2XE\= .3b = 1.95

mp~370mp~470mp~260

ËÁ

¥

c1ê4 @M

eVD

T @MeVD 1.0 1.5 2.0 2.5

80

100

120

140

160

180

200 t2XE\= .3b = 1.95

mp~370mp~470mp~260

ËÁ

¥

c1ê4 @M

eVD

T êTc

200 300 400 500

80

100

120

140

160

180

200 t2XE\= .3b = 2.10

mp~370mp~220

ËÁ

¥

c1ê4 @M

eVD

T @MeVD 1.0 1.5 2.0 2.5

80

100

120

140

160

180

200 t2XE\= .3b = 2.10

mp~370mp~220

ËÁ

¥

c1ê4 @M

eVD

T êTc

150 200 250 300 350 400

80

100

120

140

160

180

200 t2XE\= .3b = 1.90

mp~370mp~470

ËÁ

¥

c1ê4 @M

eVD

T @MeVD 1.0 1.5 2.0

80

100

120

140

160

180

200 t2XE\= .3b = 1.90

mp~370mp~470

ËÁ

¥

c1ê4 @M

eVD

T êTc

150 200 250 300 350 400 450 500

80

100

120

140

160

180

200 t2XE\= .3b = 1.95

mp~370mp~470

ËÁ

¥

c1ê4 @M

eVD

T @MeVD 1.0 1.5 2.0 2.5

80

100

120

140

160

180

200 t2XE\= .3b = 1.95

mp~370mp~470

ËÁ

¥

c1ê4 @M

eVD

T êTc

However: there is no mass

dependence..

Instanton potential - cumulants’ ratio b2

Consistent with Bonati et al.

-2

-1

0

1

2

3

4

5

6

150 200 250 300 350 400 450 500 550

(<Q

4 > -3

<Q2 >2 )/<

Q2 >)

= -1

2 b2

T

a=0.082 fm,t0a=0.082 fm,t1a=0.065 fm,t0a=0.065 fm,t1

DIGAChPT

Gaussian

DIGA limit for T > 350 MeV

b2 = -1/12

Fermionic operator

Results II

Topological and chiral susceptibility

�top

=< Q2top

> /V = m2l

�5,disc

�top

=< Q2top

> /V = m2l

�disc

T > TU(1)A ' Tc

HotQCD, 2012

Chiral susceptibility

0.001

0.01

0.1

1

10

100

150 200 250 300 350 400 450 500 550

χ/T2

T

Pion mass 470 MeV

a = 0.094 fma = 0.082 fm

0.0001

0.001

0.01

0.1

1

10

100

100 150 200 250 300 350 400 450 500 550

χ/T2

T

Pion mass 370 MeV

a = 0.094 fma = 0.082 fma = 0.065 fm

�disc/T

2

Within errors, no discernable spacing dependence

[Mev] [MeV]

10

100

150 200 250 300 350 400 450 500

(ml2 χ

disc

)0.25

T

pion 260 MeVpion 370 MeVpion 470 MeV

�1/4topT > Tc:

[MeV]

[MeV

]

10

100

150 200 250 300 350 400 450 500

(ml2 χ

disc

)0.25

T

pion 260 MeVpion 370 MeVpion 470 MeV

χtop

0.25 Borsanyi et al, continuumχtop

0.25 Bonati et al, a = 0.057 fmχtop

0.25 , Bonati et al, continuum

�1/4top

T > Tc:

[MeV]

[MeV

]

10

100

100 200 300 400 500 600 700 800

((ml2 χ

dis

c)0.

25) m

π ph

ys/m

π

T

pion 260 MeVpion 370 MeVpion 470 MeV

χ top0.25 Borsanyi et al, pion 135 MeV

Results for physical pion mass

Scaled

[MeV

]

[MeV]

0

1

2

3

4

5

6

250 300 350 400 450

d(T)

T

Fermionic, all massesGluonic (nearly linear)

DIGA, Nf = 2DIGA, Nf = 3

Effective exponent :

�0.25(T ) = aT�d(T )

d(T ) = �T ddT ln�0.25(T )

Possibly consistent with instant -dyon?

Shuryak 2017

Faster decrease before DIGA sets in

[MeV]

0

1

2

3

4

5

6

250 300 350 400 450

d(T)

T

Fermionic, all massesGluonic (nearly linear)

DIGA, Nf = 2DIGA, Nf = 3

Effective exponent : Same DIGA onset seen in b2 ≈ 350 MeV

-2

-1

0

1

2

3

4

5

6

150 200 250 300 350 400 450 500 550

(<Q

4 > -3

<Q2 >2 )/<

Q2 >)

= -1

2 b2

T

a=0.082 fm,t0a=0.082 fm,t1a=0.065 fm,t0a=0.065 fm,t1

DIGAChPT

Gaussian

[MeV] [MeV]

�1/4top

= aT�d(T )

0

10

20

30

40

50

60

70

80

250 300 350 400 450

χto

p0.25

T

Results for physical pion mass

Fermionic, rescaled from mπ 260 MeVFermionic, rescaled from mπ 370 MeVFermionic, rescaled from mπ 470 MeV

Gluonic, rescaled from mπ 370 MeVBorsanyi et al

Bonati et al.

Comparison with other results

Petreczky et al.Qualitative agreement

???

[MeV]

0

1

2

3

4

5

6

250 300 350 400 450

d(T)

T

Fermionic, all massesGluonic (nearly linear)

Borsanyi et al.Bonati et al.

Petreczky et al.DIGA, Nf = 2DIGA, Nf = 3

Comparisons with other results : �1/4top

= aT�d(T )Effective exponent d(T):

[MeV]

A window on the axions

Inspiring paper: Berkowitz, Buchoff, Rinaldi Phys.Rev. D92 (2015) no.3, 034507

Axion potential

p�(T )/fa=

PhD Thesis, G. Grilli di Cortona, Sissa 2016(advisor G. Villadoro)

Petreczky et al.

This work, fermionic

This work, gluonic

�(T ) / 1/T↵

Assume:

T > 1 GeV

Axions make all of Dark MatterNeeded assumption on fraction of DM made of axions

Borsanyi et al

B

T

D

Updated from Nature N&V

PBonTg

Lower limits on the axion mass assuming that axions make 100% of DM:

Tg: This work, gluonic; Bon: Bonati et al.; D: DIGA, B: Borsanyi et al., P: Petreczky et al., T: this work, fermionic

Assuming that axionscontribute from 50% to 1% to DM

Summary

-Gluonic operator with gradient flow method:Puzzling features : decreases slowly, no mass dependence.However, b2 is approaching DIGA value for T > 300 MeV.

-Fermionic operator: DIGA exponent for T > 300, apparently reached from above. The results for T> 300 are consistent with others once rescaled to the physical pion mass. Some differences around and slightly above Tc

-In general:There is an emerging consensus on the results among different groups. Remaining differences still not understood (so surprises may be possible). FRG approach??

-Axions:The results on topological susceptibility help constraining the axions mass.Slower decays imply lower bounds for the axion mass, while exponents in the DIGA range make plausible a region of largish masses not yet explored, well within the scope of MADMAX.

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