Thermodynamic properties along the QCD crossover from Lattice Peter - - PowerPoint PPT Presentation

thermodynamic properties along the qcd crossover from
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Thermodynamic properties along the QCD crossover from Lattice Peter - - PowerPoint PPT Presentation

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SLIDE 1

Thermodynamic properties along the QCD crossover from Lattice

Peter Petreczky

GHP19, 8th Workshop of the APS Topical Group on Hadronic Physics, Denver, April 10-12, 2019

searches for criticality in QCD phase-diagram and thermodynamics along the phase boundary: phase boundary @ µB > 0 & critical point RHIC, NA61, FAIR, NICA, J-PARC

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Remnants of criticality @ µB = 0 RHIC, LHC

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Thermodynamis along the QCD crossover

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SLIDE 2

Summit (TOP500: 1) Titan (TOP500: 9) Cori (TOP500:12) Piz Daint (TOP500: 5) NSC3 (CCNU, Wuhan, China)

Large scale numerical calculations by HotQCD Collaboration

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SLIDE 3

O(N) scaling and the chiral transition temperature

For sufficiently small ml and in the vicinity of the transition temperature:

governed by universal O(4) scaling

Tc0 is critical temperature in the mass-less limit, h0 and t0 are scale parameters Pseudo-critical temperatures for non-zero quark mass are defined as peaks in the susceptibilities = Tc0

in the zero quark mass limit

universal scaling function has a peak at z=zp

hq¯ qi = T(∂ln Z)/∂mf

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SU(2)V ⊗ SU(2)A ∼ O(4)

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slide-4
SLIDE 4

The chiral cross-over temperature for physical masses

hq¯ qi = T(∂ln Z)/∂mf

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Σ = 1 f 4

K

⇥ msh¯ uu + ¯ ddi (mu + md)h¯ ssi ⇤

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χΣ = ms ✓ ∂ ∂mu + ∂ ∂md ◆ Σ

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χ = m2

s

f 4

K

h h

  • ¯

uu + ¯ dd 2i

uui + h ¯ ddi 2i

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Chiral order parameter: and the corresponding susceptibilities: For non-zero chemical potential we use Taylor expansion Σ(T, µX) =

X

n=0

2n(T)

(2n)! ⇣µX T ⌘2n

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χ(T, µX) =

X

n=0

2n(T)

(2n)! ⇣µX T ⌘2n

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χΣ, Cχ

0 (T) ∼ χl,m

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∂T CΣ

0 , CΣ 2 (T) ∼ χt,m

<latexit sha1_base64="iqiIljVcWy6t17mtVOKXOSAnRN0=">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</latexit><latexit sha1_base64="iqiIljVcWy6t17mtVOKXOSAnRN0=">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</latexit><latexit sha1_base64="iqiIljVcWy6t17mtVOKXOSAnRN0=">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</latexit><latexit sha1_base64="iqiIljVcWy6t17mtVOKXOSAnRN0=">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</latexit>

0 = Σ

<latexit sha1_base64="FzkuUp3kqDoVhQFem8iDiEptcY8=">AB/HicdZDLSgMxFIYzXmu9jXbpJlgEVyUpYtuFUOjGZUV7gXYcMmDc1cSDLCMNRXceNCEbc+iDvfxkxbQUPhHz8/znk5PdiwZVG6MNaWV1b39gsbBW3d3b39u2Dw6KEklZh0Yikn2PKCZ4yDqa8H6sWQk8ATredNW7vfumFQ8Cm90GjMnIOQ+5wSbSTXLrVcdJsNr/k4ILOLxe3aZVRBCGMYQ64do4MNBr1Kq5DnFumymBZbd+H4imgQs1FQpQYxdrJiNScCjYrDhPFYkKnZMwGBkMSMOVk8+Vn8MQoI+hH0pxQw7n6fSIjgVJp4JnOgOiJ+u3l4l/eINF+3cl4GCeahXTxkJ8IqCOYJwFHXDKqRWqAUMnNrpBOiCRUm7yKJoSvn8L/oVutYMNXZ+VmdRlHARyBY3AKMKiBJrgEbdABFKTgATyBZ+verRerNdF64q1nCmBH2W9fQJh2JSO</latexit><latexit sha1_base64="FzkuUp3kqDoVhQFem8iDiEptcY8=">AB/HicdZDLSgMxFIYzXmu9jXbpJlgEVyUpYtuFUOjGZUV7gXYcMmDc1cSDLCMNRXceNCEbc+iDvfxkxbQUPhHz8/znk5PdiwZVG6MNaWV1b39gsbBW3d3b39u2Dw6KEklZh0Yikn2PKCZ4yDqa8H6sWQk8ATredNW7vfumFQ8Cm90GjMnIOQ+5wSbSTXLrVcdJsNr/k4ILOLxe3aZVRBCGMYQ64do4MNBr1Kq5DnFumymBZbd+H4imgQs1FQpQYxdrJiNScCjYrDhPFYkKnZMwGBkMSMOVk8+Vn8MQoI+hH0pxQw7n6fSIjgVJp4JnOgOiJ+u3l4l/eINF+3cl4GCeahXTxkJ8IqCOYJwFHXDKqRWqAUMnNrpBOiCRUm7yKJoSvn8L/oVutYMNXZ+VmdRlHARyBY3AKMKiBJrgEbdABFKTgATyBZ+verRerNdF64q1nCmBH2W9fQJh2JSO</latexit><latexit sha1_base64="FzkuUp3kqDoVhQFem8iDiEptcY8=">AB/HicdZDLSgMxFIYzXmu9jXbpJlgEVyUpYtuFUOjGZUV7gXYcMmDc1cSDLCMNRXceNCEbc+iDvfxkxbQUPhHz8/znk5PdiwZVG6MNaWV1b39gsbBW3d3b39u2Dw6KEklZh0Yikn2PKCZ4yDqa8H6sWQk8ATredNW7vfumFQ8Cm90GjMnIOQ+5wSbSTXLrVcdJsNr/k4ILOLxe3aZVRBCGMYQ64do4MNBr1Kq5DnFumymBZbd+H4imgQs1FQpQYxdrJiNScCjYrDhPFYkKnZMwGBkMSMOVk8+Vn8MQoI+hH0pxQw7n6fSIjgVJp4JnOgOiJ+u3l4l/eINF+3cl4GCeahXTxkJ8IqCOYJwFHXDKqRWqAUMnNrpBOiCRUm7yKJoSvn8L/oVutYMNXZ+VmdRlHARyBY3AKMKiBJrgEbdABFKTgATyBZ+verRerNdF64q1nCmBH2W9fQJh2JSO</latexit><latexit sha1_base64="FzkuUp3kqDoVhQFem8iDiEptcY8=">AB/HicdZDLSgMxFIYzXmu9jXbpJlgEVyUpYtuFUOjGZUV7gXYcMmDc1cSDLCMNRXceNCEbc+iDvfxkxbQUPhHz8/znk5PdiwZVG6MNaWV1b39gsbBW3d3b39u2Dw6KEklZh0Yikn2PKCZ4yDqa8H6sWQk8ATredNW7vfumFQ8Cm90GjMnIOQ+5wSbSTXLrVcdJsNr/k4ILOLxe3aZVRBCGMYQ64do4MNBr1Kq5DnFumymBZbd+H4imgQs1FQpQYxdrJiNScCjYrDhPFYkKnZMwGBkMSMOVk8+Vn8MQoI+hH0pxQw7n6fSIjgVJp4JnOgOiJ+u3l4l/eINF+3cl4GCeahXTxkJ8IqCOYJwFHXDKqRWqAUMnNrpBOiCRUm7yKJoSvn8L/oVutYMNXZ+VmdRlHARyBY3AKMKiBJrgEbdABFKTgATyBZ+verRerNdF64q1nCmBH2W9fQJh2JSO</latexit>

0 = χ

<latexit sha1_base64="SjbY7AzhEWryjso6wQXZ15fRdpc=">AB+HicdZDLSgMxFIYz9VbrpaMu3QSL4KokRWy7EArduKxgL9COQyZN29BMZkgyQh36JG5cKOLWR3Hn25hpK6jogYSP/z+HnPxBLg2CH04ubX1jc2t/HZhZ3dv+geHZ0lCjK2jQSkeoFRDPBJWsbgTrxYqRMBCsG0ybmd+9Y0rzSN6YWcy8kIwlH3FKjJV8t9j0W06oBM+v8xu3y2hMkIYwzwNULZKFer1VwDeLMslUCq2r57vtgGNEkZNJQbTuYxQbLyXKcCrYvDBINIsJnZIx61uUJGTaSxeLz+GpVYZwFCl7pIEL9ftESkKtZ2FgO0NiJvq3l4l/ef3EjGpeymWcGCbp8qFRIqCJYJYCHLFqBEzC4QqbneFdEIUocZmVbAhfP0U/g+dShlbvj4vNSqrOPLgGJyAM4BFTAFWiBNqAgAQ/gCTw7986j8+K8LltzmrmCPwo5+0TeHuS7A=</latexit><latexit sha1_base64="SjbY7AzhEWryjso6wQXZ15fRdpc=">AB+HicdZDLSgMxFIYz9VbrpaMu3QSL4KokRWy7EArduKxgL9COQyZN29BMZkgyQh36JG5cKOLWR3Hn25hpK6jogYSP/z+HnPxBLg2CH04ubX1jc2t/HZhZ3dv+geHZ0lCjK2jQSkeoFRDPBJWsbgTrxYqRMBCsG0ybmd+9Y0rzSN6YWcy8kIwlH3FKjJV8t9j0W06oBM+v8xu3y2hMkIYwzwNULZKFer1VwDeLMslUCq2r57vtgGNEkZNJQbTuYxQbLyXKcCrYvDBINIsJnZIx61uUJGTaSxeLz+GpVYZwFCl7pIEL9ftESkKtZ2FgO0NiJvq3l4l/ef3EjGpeymWcGCbp8qFRIqCJYJYCHLFqBEzC4QqbneFdEIUocZmVbAhfP0U/g+dShlbvj4vNSqrOPLgGJyAM4BFTAFWiBNqAgAQ/gCTw7986j8+K8LltzmrmCPwo5+0TeHuS7A=</latexit><latexit sha1_base64="SjbY7AzhEWryjso6wQXZ15fRdpc=">AB+HicdZDLSgMxFIYz9VbrpaMu3QSL4KokRWy7EArduKxgL9COQyZN29BMZkgyQh36JG5cKOLWR3Hn25hpK6jogYSP/z+HnPxBLg2CH04ubX1jc2t/HZhZ3dv+geHZ0lCjK2jQSkeoFRDPBJWsbgTrxYqRMBCsG0ybmd+9Y0rzSN6YWcy8kIwlH3FKjJV8t9j0W06oBM+v8xu3y2hMkIYwzwNULZKFer1VwDeLMslUCq2r57vtgGNEkZNJQbTuYxQbLyXKcCrYvDBINIsJnZIx61uUJGTaSxeLz+GpVYZwFCl7pIEL9ftESkKtZ2FgO0NiJvq3l4l/ef3EjGpeymWcGCbp8qFRIqCJYJYCHLFqBEzC4QqbneFdEIUocZmVbAhfP0U/g+dShlbvj4vNSqrOPLgGJyAM4BFTAFWiBNqAgAQ/gCTw7986j8+K8LltzmrmCPwo5+0TeHuS7A=</latexit><latexit sha1_base64="SjbY7AzhEWryjso6wQXZ15fRdpc=">AB+HicdZDLSgMxFIYz9VbrpaMu3QSL4KokRWy7EArduKxgL9COQyZN29BMZkgyQh36JG5cKOLWR3Hn25hpK6jogYSP/z+HnPxBLg2CH04ubX1jc2t/HZhZ3dv+geHZ0lCjK2jQSkeoFRDPBJWsbgTrxYqRMBCsG0ybmd+9Y0rzSN6YWcy8kIwlH3FKjJV8t9j0W06oBM+v8xu3y2hMkIYwzwNULZKFer1VwDeLMslUCq2r57vtgGNEkZNJQbTuYxQbLyXKcCrYvDBINIsJnZIx61uUJGTaSxeLz+GpVYZwFCl7pIEL9ftESkKtZ2FgO0NiJvq3l4l/ef3EjGpeymWcGCbp8qFRIqCJYJYCHLFqBEzC4QqbneFdEIUocZmVbAhfP0U/g+dShlbvj4vNSqrOPLgGJyAM4BFTAFWiBNqAgAQ/gCTw7986j8+K8LltzmrmCPwo5+0TeHuS7A=</latexit>

e.g. ∂T Cχ

0 ∼ Cχ 2

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5 different definitions of T pc:

<latexit sha1_base64="oVvL8p4bRCxvO+E+52VdNOsPcw4=">ACDXicdZDLSgMxFIYzXmu9jbp0E2wFV2WmUBRXBTcuK/QGbS2ZzJk2NDMZkoxQhr6AG1/FjQtF3Lp359uYaegogcCP/9/TpLzeTFnSjvOp7Wyura+sVnYKm7v7O7t2weHbSUSaFBRey6xEFnEXQ0kxz6MYSOhx6HiTqyzv3IFUTERNPY1hEJRxAJGiTbW0C7XsM+CACREGvsQsIhlgcIiwOXmbRrTWflyaJecSs3JCjsVZylyx82dEsqrMbQ/+r6gSWgupZwo1XOdWA9SIjWjHGbFfqIgJnRCRtAzMiIhqE632aGT43j40BIc8yn5u73iZSESk1Dz3SGRI/V7ywz/8p6iQ4uBimL4kRDRBcPBQnHWuAMjQEhgWo+NYJQaThQTMdEqoNwKBsNwU/y/a1Ypr9E21VK/mOAroGJ2gM+Sic1RH16iBWoie/SIntGL9WA9Wa/W26J1xcpnjtCPst6/ACY9muc=</latexit><latexit sha1_base64="oVvL8p4bRCxvO+E+52VdNOsPcw4=">ACDXicdZDLSgMxFIYzXmu9jbp0E2wFV2WmUBRXBTcuK/QGbS2ZzJk2NDMZkoxQhr6AG1/FjQtF3Lp359uYaegogcCP/9/TpLzeTFnSjvOp7Wyura+sVnYKm7v7O7t2weHbSUSaFBRey6xEFnEXQ0kxz6MYSOhx6HiTqyzv3IFUTERNPY1hEJRxAJGiTbW0C7XsM+CACREGvsQsIhlgcIiwOXmbRrTWflyaJecSs3JCjsVZylyx82dEsqrMbQ/+r6gSWgupZwo1XOdWA9SIjWjHGbFfqIgJnRCRtAzMiIhqE632aGT43j40BIc8yn5u73iZSESk1Dz3SGRI/V7ywz/8p6iQ4uBimL4kRDRBcPBQnHWuAMjQEhgWo+NYJQaThQTMdEqoNwKBsNwU/y/a1Ypr9E21VK/mOAroGJ2gM+Sic1RH16iBWoie/SIntGL9WA9Wa/W26J1xcpnjtCPst6/ACY9muc=</latexit><latexit sha1_base64="oVvL8p4bRCxvO+E+52VdNOsPcw4=">ACDXicdZDLSgMxFIYzXmu9jbp0E2wFV2WmUBRXBTcuK/QGbS2ZzJk2NDMZkoxQhr6AG1/FjQtF3Lp359uYaegogcCP/9/TpLzeTFnSjvOp7Wyura+sVnYKm7v7O7t2weHbSUSaFBRey6xEFnEXQ0kxz6MYSOhx6HiTqyzv3IFUTERNPY1hEJRxAJGiTbW0C7XsM+CACREGvsQsIhlgcIiwOXmbRrTWflyaJecSs3JCjsVZylyx82dEsqrMbQ/+r6gSWgupZwo1XOdWA9SIjWjHGbFfqIgJnRCRtAzMiIhqE632aGT43j40BIc8yn5u73iZSESk1Dz3SGRI/V7ywz/8p6iQ4uBimL4kRDRBcPBQnHWuAMjQEhgWo+NYJQaThQTMdEqoNwKBsNwU/y/a1Ypr9E21VK/mOAroGJ2gM+Sic1RH16iBWoie/SIntGL9WA9Wa/W26J1xcpnjtCPst6/ACY9muc=</latexit><latexit sha1_base64="oVvL8p4bRCxvO+E+52VdNOsPcw4=">ACDXicdZDLSgMxFIYzXmu9jbp0E2wFV2WmUBRXBTcuK/QGbS2ZzJk2NDMZkoxQhr6AG1/FjQtF3Lp359uYaegogcCP/9/TpLzeTFnSjvOp7Wyura+sVnYKm7v7O7t2weHbSUSaFBRey6xEFnEXQ0kxz6MYSOhx6HiTqyzv3IFUTERNPY1hEJRxAJGiTbW0C7XsM+CACREGvsQsIhlgcIiwOXmbRrTWflyaJecSs3JCjsVZylyx82dEsqrMbQ/+r6gSWgupZwo1XOdWA9SIjWjHGbFfqIgJnRCRtAzMiIhqE632aGT43j40BIc8yn5u73iZSESk1Dz3SGRI/V7ywz/8p6iQ4uBimL4kRDRBcPBQnHWuAMjQEhgWo+NYJQaThQTMdEqoNwKBsNwU/y/a1Ypr9E21VK/mOAroGJ2gM+Sic1RH16iBWoie/SIntGL9WA9Wa/W26J1xcpnjtCPst6/ACY9muc=</latexit>

∂T Cχ

0 = 0, ∂T CΣ 0 = 0 , Cχ 2 = 0

<latexit sha1_base64="CpmcjZKYDZyEZNt7hymk9k8VirQ=">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</latexit><latexit sha1_base64="CpmcjZKYDZyEZNt7hymk9k8VirQ=">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</latexit><latexit sha1_base64="CpmcjZKYDZyEZNt7hymk9k8VirQ=">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</latexit><latexit sha1_base64="CpmcjZKYDZyEZNt7hymk9k8VirQ=">ACKXicdVDLSgMxFM3UV62vqks3wSK4KCUzlrYuhEI3Liv2BW0dMmnahmYeJBmhDO3nuPFX3Cgo6tYfMdNW8XkgcDjnXG7ucQLOpELoxUgsLa+sriXUxubW9s76d29hvRDQWid+NwXLQdLyplH64opTluBoNh1OG06o0rsN6+pkMz3amoc0K6LBx7rM4KVlux0uRNgoRjmdg1WbHQVdciQTc5QdvrLuGQDF2trmq3Y1mfQTmdQ7rRUsPIFiHIFU3LjIlVzJ/koamVGBmwQNVOP3R6Pgld6inCsZRtEwWqG8XLCKeTVCeUNMBkhAe0ramHXSq70ezSCTzSg/2faGfp+BM/ToRYVfKsevopIvVUP70YvEvrx2qfqkbMS8IFfXIfFE/5FD5MK4N9pigRPGxJpgIpv8KyRALTJQuN6VL+LgU/k8aVs7U/CKfKVuLOpLgAByCY2CIiDc1AFdUDADbgDj+DJuDXujWfjdR5NGIuZfANxts7eK6mFw=</latexit>

∂2

T CΣ 0 = 0, ∂T CΣ 2 = 0

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Derivatives in µ2

X are similar to derivatives in T

⇒ the following quantities will peak at Tc

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HotQCD, arXiv:1812.08235 The 5 different Tc values reduce to Tl,m and Tl,t if regular part is zero

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slide-5
SLIDE 5

5 10 15 20 25 30 135 145 155 165 175 CΣ T [MeV]

20 40 60 80 100 135 145 155 165 175 −TdCΣ

0 /dT

T [MeV]

Nτ = 16 12 8 6

  • 2.0
  • 1.5
  • 1.0
  • 0.5

0.0 0.5 135 145 155 165 175 µQ = µS = 0 CΣ

2 /2

T [MeV] Nτ = 12 8 6

Lattice calculations based on 100K - 500 K configurations, Nτ = 6 − 12, and 4K configurations for Nτ = 16

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50 100 150 200 135 145 155 165 175 Cχ T [MeV] Nτ = 16 12 8 6

  • 20
  • 15
  • 10
  • 5

5 10 15 20 135 145 155 165 175 µQ = µS = 0 Cχ

2 /2

T [MeV] Nτ = 12 8 6

HotQCD, arXiv:1812.08235

slide-6
SLIDE 6

150 155 160 165 170 N

τ

= ∞ N

τ

= 1 6 N

τ

= 1 2 N

τ

= 8 N

τ

= 6 Tc(0) [MeV] 1/N2

τ

χΣ Cχ CΣ CΣ

2

2

(156.5 ± 1.5) MeV

Different definitions of Tc surprisingly agree in the continuum limit and we for zero chemical potential we get Tc = 156 ± 1.5 MeV

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HotQCD, arXiv:1812.08235

slide-7
SLIDE 7

little change in peak-height & width with increasing baryon chemical potential: no indication of a stronger transition becoming stronger

The chiral susceptibility at baryon density non-zero density

Conditions in heavy ion collisions: nB > 0, nS = 0, nQ = 0.4nB (for Au, Pb)

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slide-8
SLIDE 8

The chiral cross-over temperature at non-zero density

Tc(µB) = Tc(0) " 1 − κB

2

✓ µB Tc(0) ◆2 − κB

4

✓ µB Tc(0) ◆4#

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  • 0.01

0.00 0.01 0.02 0.03 0.04 Nτ = ∞ Nτ = 12 Nτ = 8 Nτ = 6 µQ = µS = 0 1/N2

τ

χ: κB

2

κB

4

135 140 145 150 155 160 165 170 175 50 100 150 200 250 300 350 400 Tc [MeV] µB [MeV] nS = 0, nQ

nB = 0.4

crossover line: O(µ4

B)

constant: ✏ s freeze-out: STAR ALICE

The µB dependence of Tc is small

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The freeze-out condition corresponding to constant energy density or constant entropy density agrees with the crossover line within errors

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κB,χ

2

' κB,Σ

2

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HotQCD, arXiv:1812.08235

slide-9
SLIDE 9

Chiral phase transition in 2+1 flavor QCD

What is the nature of the chiral transition in 2+1 flavor QCD for fixed ms and ml → 0 ?

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100 200 300 400 500 600 700 130 135 140 145 150 155 160 165 170 175 180 χM T [MeV]

mπ [MeV] ms/ml Nτ=8 20 160 27 140 40 110 80 80 160 55

150 200 250 300 350 400 450 135 140 145 150 155 160 165 170 χM T [MeV]

Nτ=8 ms/ml=80 Nσ=32 Nσ=40 Nσ=56

Tp(H, L) = T 0

c

✓ 1 + zp(zL) z0 H1/βδ ◆ + sub leading

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M = h1/δfG(z, zL) + fsub(T, H, L) , χM = h−1

0 h1/δ−1fχ(z, zL) + ˜

fsub(T, H, L) .

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1903.04801 HotQCD, arXiv:1903.04801 TX(H, L) = T 0

c

✓ 1 + ✓zX(zL) z0 ◆ H1/βδ ◆

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zL = l0/(Lhν/βδ)

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Finite volume: New estimators:

slide-10
SLIDE 10

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 125 130 135 140 145 150 155 160 165 170 H⋅χM/M T [MeV]

Nτ=12 ms/ml=80

1/δ

Nσ=48 Nσ=60 Nσ=72

HχM(Tδ, H, L) M(Tδ, H, L) = 1 δ , χM(T60, H) = 0.6χmax

M

.

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TX(H, L) = T 0

c

✓ 1 + ✓zX(zL) z0 ◆ H1/βδ ◆ +cXH1−1/δ+1/βδ , X = δ, 60

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125 130 135 140 145 150 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Tδ [MeV] [Nτ/Nσ(ms/ml)ν/βδ]3

ms/ml=80 : filled

1/27 1/40 1/40 1/40 1/160 1/27 1/40 1/40 1/27 1/40 1/60

Nτ=6, 8, 12 Nτ=8, 12

Nτ=6 8 12

T 0

c = 132+3 −6 MeV

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Use O(4) fits for ml and volume dependence

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Continuum extrapolations:

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HotQCD, arXiv:1903.04801 z60 ' zδ ' 0

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slide-11
SLIDE 11

Equation of state at non-zero net baryon density

0.2 0.4 0.6 0.8 1 140 160 180 200 220 240 260 280

µB/T=2 µB/T=2.5 µB/T=1 HRG µQ=µS=0

[P(T,µB)-P(T,0)]/T4 T [MeV]

O(µB

6)

O(µB

4)

O(µB

2)

6th order Taylor expansion, HotQCD, PRD 95 (2017) 054504 Truncation errors of the 6th order Taylor expansions are small for μB/T<2.5

slide-12
SLIDE 12

S/NB (nQ=0.4nB, nS=0)

50 100 150 200 250 300 350 µB [MeV] 135 155 175 195 215 235 255 275 T [MeV] 100 200 300 400 500 600 700 800

√s=200 GeV (S/NB=386) 62.4 GeV (124) 3 9 G e V ( 7 9 ) 27 GeV (56) 19.6 GeV (42) 14.5 GeV (31)

approximate trajectories in heavy-ion collisions follow constant entropy to baryon number ratio

Iso-entropic Equation of state at non-zero net baryon density

slide-13
SLIDE 13

Equation of state along the “phase boundary”

15.0 20.0 25.0 30.0 35.0 40.0 45.0 50 100 150 200 250 cV (Tc(µB), µB)/Tc(µB)3 µB [MeV] µQ = µS = 0 O(µ4

B)

O(µ2

B)

HRG

specific heat @ constant volume: (inverse) specific heat@ constant pressure:

0.000 0.002 0.004 0.006 0.008 0.010 0.012 50 100 150 200 250 Tc(µB)3/cP(Tc(µB), µB) µB [MeV] µQ = µS = 0 O(µ6

B)

O(µ4

B)

HRG

isentropic speed of sound:

0.10 0.12 0.14 0.16 0.18 0.20 50 100 150 200 250 c2

s(Tc(µB), µB)

µB [MeV] µQ = µS = 0 O(µ4

B)

O(µ2

B)

HRG

cV = T ✓ ∂s ∂T ◆

nB

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cp = T (s/nB) ✓∂(s/nB) ∂T ◆

p

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c2

s =

✓@p @✏ ◆

s/nB

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slide-14
SLIDE 14

increase remains less than (ideal) hadron gas resonance gas model (HRG)

Net baryon number fluctuations along the “phase boundary”

σ2

B

V f 3

K

= 1 V f 3

K

∂ ln Z ∂(µB/T)2 =

X

n=1

χB

2n

⇣µB T ⌘2n

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Baryon number fluctuations are expected to increase as we approach the critical point

slide-15
SLIDE 15

Radius of convergence of Taylor series and critical point

P(T, µB) − P(T, 0) T 4 =

X

n=1

χB

2n(T)

(2n)! ⇣µB T ⌘2n χB

2 (T, µB) = ∞

X

n=1

χB

2n+2(T)

(2n)! ⇣µB T ⌘2n T < Tc: χB

n > 0 ⇒ convergence of the Taylor expansion

is limited by a similarity on the real axis µB = µc

B (critical point).

Estimator for radius of convergence: rχ

n =

  • 2n(2n − 1)χB

2n

χB

2n+2

  • 2

Critical point is disfavored for µB < 300 MeV

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slide-16
SLIDE 16

Summary

  • The chiral transition temperature at the physical quark mass can be pre-

cisely determined: Tc = 156 ± 1.5 MeV and can be related to chiral phase transition temperature For mu,d = 0: Tc = 132+3

−6 MeV

  • Equation of state at non-zero baryon density can be obtained from the

Taylor expansion and no indication of limited convergence radius for µB < 300 MeV

  • The dependence of Tc on µB is very small and is consistent with freeze-out

curve in HI, the width of the chiral susceptibility and the peak height does not change with µB

  • The µB dependence of EoS and baryon number fluctuations follows the

HRG expectation

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