progress on the nature of the qcd thermal transition as a ...f. cuteri - goethe universität qcd...
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CRC - TR
Progress on the nature of the QCDthermal transition
as a function of quark flavors and masses
Francesca Cuteri, Owe Philipsen and Alessandro Sciarrapartially based on Phys.Rev. D93 (2016) no.5, 054507
36th Annual International Symposium on Lattice Field Theory @ MSU
July 26th, 2018
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Standard pms,mu,dq Columbia plotDependence of the order of the QCD thermal phase transition on the quark masses
Broken global Zp3q for mu,d,ms Ñ8 Svetitsky, Yaffe (1982)
Restored global SULpNfq ˆ SURpNfq for mu,d,ms Ñ 0 Pisarski, Wilczek (1984)
Aoki et al. (2006)
de Forcrand, Philipsen (2007)Bazavov et al. (2017)
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 2 / 15
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Standard pms,mu,dq Columbia plotDependence of the order of the QCD thermal phase transition on the quark masses
Broken global Zp3q for mu,d,ms Ñ8 Svetitsky, Yaffe (1982)
Restored global SULpNfq ˆ SURpNfq for mu,d,ms Ñ 0 Pisarski, Wilczek (1984)
Aoki et al. (2006)
de Forcrand, Philipsen (2007)Bazavov et al. (2017)
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 2 / 15
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Standard pms,mu,dq Columbia plotDependence of the order of the QCD thermal phase transition on the quark masses
Broken global Zp3q for mu,d,ms Ñ8 Svetitsky, Yaffe (1982)
Restored global SULpNfq ˆ SURpNfq for mu,d,ms Ñ 0 Pisarski, Wilczek (1984)
Aoki et al. (2006)
de Forcrand, Philipsen (2007)Bazavov et al. (2017)
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 2 / 15
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Standard pms,mu,dq Columbia plotDependence of the order of the QCD thermal phase transition on the quark masses
Broken global Zp3q for mu,d,ms Ñ8 Svetitsky, Yaffe (1982)
Restored global SULpNfq ˆ SURpNfq for mu,d,ms Ñ 0 Pisarski, Wilczek (1984)
Aoki et al. (2006)
de Forcrand, Philipsen (2007)Bazavov et al. (2017)
At the physical point, continuumextrapolated results
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 2 / 15
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Standard pms,mu,dq Columbia plotDependence of the order of the QCD thermal phase transition on the quark masses
Broken global Zp3q for mu,d,ms Ñ8 Svetitsky, Yaffe (1982)
Restored global SULpNfq ˆ SURpNfq for mu,d,ms Ñ 0 Pisarski, Wilczek (1984)
Aoki et al. (2006)
de Forcrand, Philipsen (2007)Bazavov et al. (2017)
At the physical point, continuumextrapolated resultsmu,d very small. Transition affected byremnants of the chiral universality class?
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 2 / 15
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Standard pms,mu,dq Columbia plotDependence of the order of the QCD thermal phase transition on the quark masses
Broken global Zp3q for mu,d,ms Ñ8 Svetitsky, Yaffe (1982)
Restored global SULpNfq ˆ SURpNfq for mu,d,ms Ñ 0 Pisarski, Wilczek (1984)
Aoki et al. (2006)
de Forcrand, Philipsen (2007)Bazavov et al. (2017)
At the physical point, continuumextrapolated resultsmu,d very small. Transition affected byremnants of the chiral universality class?Relevance of the strength of the Up1qAanomaly at Tc Pisarski, Wilczek (1984)
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 2 / 15
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Standard pms,mu,dq Columbia plotDependence of the order of the QCD thermal phase transition on the quark masses
Broken global Zp3q for mu,d,ms Ñ8 Svetitsky, Yaffe (1982)
Restored global SULpNfq ˆ SURpNfq for mu,d,ms Ñ 0 Pisarski, Wilczek (1984)
Aoki et al. (2006)
de Forcrand, Philipsen (2007)Bazavov et al. (2017)
At the physical point, continuumextrapolated resultsmu,d very small. Transition affected byremnants of the chiral universality class?Relevance of the strength of the Up1qAanomaly at Tc Pisarski, Wilczek (1984)
Chiral 1st order region wider for largerNf, until Nf “ 4 de Forcrand, D’Elia (2017)
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 2 / 15
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Standard pms,mu,dq Columbia plotDependence of the order of the QCD thermal phase transition on the quark masses
Broken global Zp3q for mu,d,ms Ñ8 Svetitsky, Yaffe (1982)
Restored global SULpNfq ˆ SURpNfq for mu,d,ms Ñ 0 Pisarski, Wilczek (1984)
Aoki et al. (2006)
de Forcrand, Philipsen (2007)Bazavov et al. (2017)
At the physical point, continuumextrapolated resultsmu,d very small. Transition affected byremnants of the chiral universality class?Relevance of the strength of the Up1qAanomaly at Tc Pisarski, Wilczek (1984)
Chiral 1st order region wider for largerNf, until Nf “ 4 de Forcrand, D’Elia (2017)
Strong cut-off and discretizationdependence of chiral Z2 boundaryI Critical quark masses unreachably
small for highly improved actions
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 2 / 15
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pms,mu,dq Columbia plot in the continuum
Columbia plot from the “unimproved viewpoint”. Focus on Nf “ 2.I Qualitative/Quantitative agreement between Wilson and staggered studies
I Chiral 1st order region shrinks towards the continuum limit
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 3 / 15
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pms,mu,dq Columbia plot in the continuum
Columbia plot from the “unimproved viewpoint”. Focus on Nf “ 2.I Qualitative/Quantitative agreement between Wilson and staggered studiesI Chiral 1st order region shrinks towards the continuum limit
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 3 / 15
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pms,mu,dq Columbia plot in the continuum
Columbia plot from the “unimproved viewpoint”. Focus on Nf “ 2.I Qualitative/Quantitative agreement between Wilson and staggered studiesI Chiral 1st order region shrinks towards the continuum limit
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 3 / 15
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pms,mu,dq Columbia plot in the continuum
Columbia plot from the “unimproved viewpoint”. Focus on Nf “ 2.I Qualitative/Quantitative agreement between Wilson and staggered studiesI Chiral 1st order region shrinks towards the continuum limit
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 3 / 15
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pms,mu,dq Columbia plot in the continuum
Columbia plot from the “unimproved viewpoint”. Focus on Nf “ 2.I Qualitative/Quantitative agreement between Wilson and staggered studiesI Chiral 1st order region shrinks towards the continuum limit
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 3 / 15
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pms,mu,dq Columbia plot in the continuumAt least two possible scenarios for the nature of Nf “ 2 chiral transition
“Direct approach" µ “ 0, Nf “ 2 and mu,d Ñ 0 proved to be too expensive
“Indirect approaches" exploit tricritical scaling laws for controlled extrapolationsto the chiral limitI From imaginary chemical potential µ “ iµiI At µ “ 0 and p2` 1qNf
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pms,mu,dq Columbia plot in the continuumAt least two possible scenarios for the nature of Nf “ 2 chiral transition
“Direct approach" µ “ 0, Nf “ 2 and mu,d Ñ 0 proved to be too expensive“Indirect approaches" exploit tricritical scaling laws for controlled extrapolationsto the chiral limitI From imaginary chemical potential µ “ iµiI At µ “ 0 and p2` 1qNf
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 3 / 15
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pms,mu,dq Columbia plot in the continuumAt least two possible scenarios for the nature of Nf “ 2 chiral transition
“Direct approach" µ “ 0, Nf “ 2 and mu,d Ñ 0 proved to be too expensive“Indirect approaches" exploit tricritical scaling laws for controlled extrapolationsto the chiral limitI From imaginary chemical potential µ “ iµiI At µ “ 0 and p2` 1qNf
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 3 / 15
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Alternative pm,Nfq Columbia plot
Yet another “indirect approach", promoting Nf to continuous real parametermZ2
pNfq according to the two considered scenarios
N tricf ă 2 N tric
f ą 2
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Continuous Nf in the path integral
Partition function describing Nf flavors of degenerate mass m
ZNfpmq “∫
DU [detMpU,mq]Nf e´SG
Being of 1st order for Nf ě 3, a 2nd order transition for Nf “ 2 requires atricritical point N tric
f in between8 many interpolations such that limNfÑ2,3 ZpNf “ 2.#q “ ZpNf “ 2, 3q
ZNf“NIf`1 “
∫DU [detMpU,m1q]N
If [detMpU,m2q] e´SG
ZN If`1pm,m1q “ ZNfpmq
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Continuous Nf in the path integral
Partition function describing Nf flavors of degenerate mass mBeing of 1st order for Nf ě 3, a 2nd order transition for Nf “ 2 requires atricritical point N tric
f in between
8 many interpolations such that limNfÑ2,3 ZpNf “ 2.#q “ ZpNf “ 2, 3q
ZNf“NIf`1 “
∫DU [detMpU,m1q]N
If [detMpU,m2q] e´SG
ZN If`1pm,m1q “ ZNfpmq
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Continuous Nf in the path integral
Partition function describing Nf flavors of degenerate mass mBeing of 1st order for Nf ě 3, a 2nd order transition for Nf “ 2 requires atricritical point N tric
f in between8 many interpolations such that limNfÑ2,3 ZpNf “ 2.#q “ ZpNf “ 2, 3q
ZNf“2.# “∫
DU [detMpU,mq]2.# e´SG
ZNf“NIf`1 “
∫DU [detMpU,m1q]N
If [detMpU,m2q] e´SG
ZN If`1pm,m1q “ ZNfpmq
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Continuous Nf in the path integral
Partition function describing Nf flavors of degenerate mass mBeing of 1st order for Nf ě 3, a 2nd order transition for Nf “ 2 requires atricritical point N tric
f in between8 many interpolations such that limNfÑ2,3 ZpNf “ 2.#q “ ZpNf “ 2, 3q
ZNf“2.# “∫
DU [detMpU,mq]2.# e´SG
ZNf“NIf`1 “
∫DU [detMpU,m1q]N
If [detMpU,m2q] e´SG
ZN If`1pm,m1q “ ZNfpmq
ZNf“2.# partition function of a statistical system that represents one particularinterpolation between QCD with integer N I
f
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Continuous Nf in the path integral
Partition function describing Nf flavors of degenerate mass mBeing of 1st order for Nf ě 3, a 2nd order transition for Nf “ 2 requires atricritical point N tric
f in between8 many interpolations such that limNfÑ2,3 ZpNf “ 2.#q “ ZpNf “ 2, 3q
ZNf“2.# “∫
DU [detMpU,mq]2.# e´SG
ZNf“NIf`1 “
∫DU [detMpU,m1q]N
If [detMpU,m2q] e´SG
ZN If`1pm,m1q “ ZNfpmq
ZNf“2.# partition function of a statistical system that represents one particularinterpolation between QCD with integer N I
f
Relative position of N tricf with respect to Nf “ 2 uniquely determined, its
precise value has no meaning other than being located between two integer N If
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Conclusion from the tricritical scaling
0.5 1 1.5 2 2.5 3 3.50
0.005
0.01
0.015
0.02
0.025 Tricritical scaling fitNf ∈ [2.0, 2.2]
m ∝ (Nf −Ntricf )5/2
Nf
m
N tricf < 2
Coarse Nτ “ 4 lattices explored with rooted staggered fermion discretizationWidth of the scaling window in m same as found in the extrapolation from µi
Cheapest extrapolation while changing Nτ?
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Outline
1 How could linearity help?
2 Strategy
3 Results
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Z2 boundary linear in some Nf range?
If it is reasonable to expect both linearity within some range in Nf and tricriticalscaling closer to the chiral limitI make use of a linear extrapolation to m “ 0 to get an upper bound for N tricr
f ,without the need to enter the tricritical scaling region
lower costno need for simulations at non integer number of flavors?!
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Z2 boundary linear in some Nf range?
If it is reasonable to expect both linearity within some range in Nf and tricriticalscaling closer to the chiral limitI make use of a linear extrapolation to m “ 0 to get an upper bound for N tricr
f ,without the need to enter the tricritical scaling region
lower costno need for simulations at non integer number of flavors?!
If N linf ă 2, while one simulates at
larger and larger Nτ towards thecontinuum limitI Transition in the Nf “ 2 chiral limit
keeps being a first order oneAs soon as N lin
f ě 2
I No conclusion can be drawn
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Outline
1 How could linearity help?
2 Strategy
3 Results
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Code & investigated parameter space
Simulations employ the OpenCL-based and publicly available CL2QCD codePhilipsen et al. (2014)
I Unimproved rooted staggered fermion discretization (RHMC algorithm)
No. of flavors ÝÑ Nf “ 3, 4, 5 on Nτ “ 4Nf “ 3.6, 4, 4.4 on Nτ “ 6
Chemical potential ÝÑ µ “ 0
Scan in mass ÝÑ m P r0.0075, 0.0900sFinite size scaling ÝÑ Nσ{Nτ “ 2, 3, 4
Scan in temperature ÝÑ p3´ 5q β values, then reweighting
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Strategy to map out the Z2 boundary
Sample the (approximate) order parameter O “ 〈ψ̄ψ〉 and extract centralmoments of the distribution, which gets shifted and deformed while β is varied
Bnpβ,m,Nσq “ 〈(O ´ 〈O〉)n〉⟨(O ´ 〈O〉)2
⟩n{2 .I On phase boundaries, i.e. at βc and in the thermodynamic limit
B3pβc,mq “ 0; B4pβc,mq “
1, 1st order1.5, 1st order triple1.604, 2nd order Z2
2, tricritical3, crossover
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Strategy to map out the Z2 boundary
Sample the (approximate) order parameter O “ 〈ψ̄ψ〉 and extract centralmoments of the distribution, which gets shifted and deformed while β is varied
F. C., O. Philipsen, A. Sciarra (2017)
β � βc βc β � βc
−0.2
0
0.2
0.4
0.6
0.8
1
β
µ(x)
β � βc βc β � βc
0
0.5
1
1.5
β
σ2(x)
0 0.2 0.4 0.6 0.80
2
4
6
8
x
Ppx
q
β ! βc
β � βc βc β � βc
−1
0
1
β
B3(x)
β � βc βc β � βc
2
4
6
β
B4(x)
B3pβcq “ 0 pinpoints βc & B4pβcq reveals the order of the transition
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 11 / 15
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Strategy to map out the Z2 boundary
Sample the (approximate) order parameter O “ 〈ψ̄ψ〉 and extract centralmoments of the distribution, which gets shifted and deformed while β is varied
F. C., O. Philipsen, A. Sciarra (2017)
β � βc βc β � βc
−0.2
0
0.2
0.4
0.6
0.8
1
β
µ(x)
β � βc βc β � βc
0
0.5
1
1.5
β
σ2(x)
0 0.2 0.4 0.6 0.80
2
4
6
8
x
Ppx
q
β ! βc β À βc
β � βc βc β � βc
−1
0
1
β
B3(x)
β � βc βc β � βc
2
4
6
β
B4(x)
B3pβcq “ 0 pinpoints βc & B4pβcq reveals the order of the transition
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 11 / 15
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Strategy to map out the Z2 boundary
Sample the (approximate) order parameter O “ 〈ψ̄ψ〉 and extract centralmoments of the distribution, which gets shifted and deformed while β is varied
F. C., O. Philipsen, A. Sciarra (2017)
β � βc βc β � βc
−0.2
0
0.2
0.4
0.6
0.8
1
β
µ(x)
β � βc βc β � βc
0
0.5
1
1.5
β
σ2(x)
0 0.2 0.4 0.6 0.80
2
4
6
8
x
Ppx
q
β ! βc β À βc β “ βc
β � βc βc β � βc
−1
0
1
β
B3(x)
β � βc βc β � βc
2
4
6
β
B4(x)
B3pβcq “ 0 pinpoints βc & B4pβcq reveals the order of the transition
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 11 / 15
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Strategy to map out the Z2 boundary
Sample the (approximate) order parameter O “ 〈ψ̄ψ〉 and extract centralmoments of the distribution, which gets shifted and deformed while β is varied
F. C., O. Philipsen, A. Sciarra (2017)
β � βc βc β � βc
−0.2
0
0.2
0.4
0.6
0.8
1
β
µ(x)
β � βc βc β � βc
0
0.5
1
1.5
β
σ2(x)
0 0.2 0.4 0.6 0.80
2
4
6
8
x
Ppx
q
β ! βc β À βc β “ βc β Á βc
β � βc βc β � βc
−1
0
1
β
B3(x)
β � βc βc β � βc
2
4
6
β
B4(x)
B3pβcq “ 0 pinpoints βc & B4pβcq reveals the order of the transition
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 11 / 15
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Strategy to map out the Z2 boundary
Sample the (approximate) order parameter O “ 〈ψ̄ψ〉 and extract centralmoments of the distribution, which gets shifted and deformed while β is varied
F. C., O. Philipsen, A. Sciarra (2017)
β � βc βc β � βc
−0.2
0
0.2
0.4
0.6
0.8
1
β
µ(x)
β � βc βc β � βc
0
0.5
1
1.5
β
σ2(x)
0 0.2 0.4 0.6 0.80
2
4
6
8
x
Ppx
q
β ! βc β À βc β “ βc β Á βc β " βc
β � βc βc β � βc
−1
0
1
β
B3(x)
β � βc βc β � βc
2
4
6
β
B4(x)
B3pβcq “ 0 pinpoints βc & B4pβcq reveals the order of the transitionF. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 11 / 15
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Critical parameters for Z2 transitions
Fitting B4pX,Nσq for sets at different spatial extent Nσ, as function of X
1.35
1.4
1.45
1.5
1.55
1.6
1.65
1.7
1.75
1.8
1.85
1.9
0.017 0.018 0.019 0.02 0.021 0.022 0.023 0.024 0.025 0.026
B4pβc,m
,Nσq
m
Nf “ 2.8
Nσ “ 8
Nσ “ 12
Nσ “ 16
B4pX,Nσq “ B4pXc,8q ` a1 x` a2 x2 `Opx3q
x ” pX ´XcqN1{νσ , X “ m
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 12 / 15
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Outline
1 How could linearity help?
2 Strategy
3 Results
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 13 / 15
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Results, so far...
1 2 3 4 5 60.000
0.020
0.040
0.060
0.080
Linear fitNf ∈ [2.4, 5]
Tricritical scaling fitNf ∈ [2.0, 2.2]
m ∝ (Nf −Ntricf )5/2
Pre
lim
inar
yNf
m
N tricf
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 14 / 15
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Results, so far...
1 1.5 2 2.5 30.000
0.005
0.010
0.015
0.020
0.025
Linear fitNf ∈ [2.4, 5]
Tricritical scaling fitNf ∈ [2.0, 2.2]
m ∝ (Nf −Ntricf )5/2
Pre
lim
inar
y
Nf
m
N tricf N lin
f
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 14 / 15
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Results, so far...
1 2 3 4 5 60.000
0.020
0.040
0.060
0.080
Linear fitNf ∈ [2.4, 5]
Linear fitNf ∈ [3.6, 4.4]
Tricritical scaling fitNf ∈ [2.0, 2.2]
m ∝ (Nf −Ntricf )5/2
Pre
lim
inar
y
Nτ = 4
Prelim
inar
yNτ = 6
Nf
m
N tricf
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 14 / 15
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Results, so far...
1 2 3 4 5 60.000
0.020
0.040
0.060
0.080
Linear fitNf ∈ [2.4, 5]
Linear fitNf ∈ [3.6, 4.4]
Tricritical scaling fitNf ∈ [2.0, 2.2]
m ∝ (Nf −Ntricf )5/2
Pre
lim
inar
y
Nτ = 4
Prelim
inar
yNτ = 6
Nf
m
N tricf
Nf “ 3, Nτ “ 6 from de Forcrand, Kim, Philipsen (2007)
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 14 / 15
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Conclusions, so far...
1 2 3 4 5 60.000
0.020
0.040
0.060
0.080
Linear fitNf ∈ [2.4, 5]
Linear fitNf ∈ [3.6, 4.4]
Tricritical scaling fitNf ∈ [2.0, 2.2]
m ∝ (Nf −Ntricf )5/2
Pre
lim
inar
y
Nτ = 4
Prelim
inar
yNτ = 6
Nf
m
N tricf
Found N linf ě 2 for Nτ “ 4, 6.
On the one handI Nature of the chiral phase transition
at Nf “ 2 remains elusive to ourlinear extrapolation already on coarselattices.
I While being interesting on its own,linearity does not seem to help inresolving the Nf “ 2 puzzle.
On the other handI If we think of the size of the shift in
the boundary, the first order scenariowould look more and more contrivedwith larger and larger Nτ values.
I It would be interesting to obtainresults for other fermiondiscretizations based on this method.
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 15 / 15
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Conclusions, so far...
1 2 3 4 5 60.000
0.020
0.040
0.060
0.080
Linear fitNf ∈ [2.4, 5]
Linear fitNf ∈ [3.6, 4.4]
Tricritical scaling fitNf ∈ [2.0, 2.2]
m ∝ (Nf −Ntricf )5/2
Pre
lim
inar
y
Nτ = 4
Prelim
inar
yNτ = 6
Nf
m
N tricf
Found N linf ě 2 for Nτ “ 4, 6.
On the one handI Nature of the chiral phase transition
at Nf “ 2 remains elusive to ourlinear extrapolation already on coarselattices.
I While being interesting on its own,linearity does not seem to help inresolving the Nf “ 2 puzzle.
On the other handI If we think of the size of the shift in
the boundary, the first order scenariowould look more and more contrivedwith larger and larger Nτ values.
I It would be interesting to obtainresults for other fermiondiscretizations based on this method.
Thank you for your attention!
F. Cuteri - Goethe Universität QCD thermal transition in the pm,Nfq plane July 26th, 2018 15 / 15