topology and qcd axion’s properties from lattice hot qcd
TRANSCRIPT
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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