In-Medium Vector-Meson Spectral Functions with the

Ralf-Arno Tripolt, ECT*, Trento, Italy
Based on:
C. Jung, F. Rennecke, R.-A. T., L. von Smekal and Jochen Wambach,
Phys. Rev. D 95, 036020 (2017)
Workshop: Space-like and time-like electromagnetic baryonic transitions
Trento, May 11th, 2017
I Flow equations for 2-point functions
I Analytic continuation
IV) Summary and Outlook
[B. Mohanty, arXiv: 1308.3328 [nucl-ex]]
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0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4
In-In→ µ + µ
NA60 Data hadronic
QGP total thermal
[R. Rapp, H. van Hees, Phys.Lett. B 753 (2016)]
I Exploration of phase structure of QCD-matter through heavy ion collisions
I Useful probes: photons and dileptons, negligible interaction with hadronic medium
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I Vector mesons can decay directly into lepton pairs
I Interesting: light vector mesons ρ, ω and φ
I Lifetime τρ ≈ 1.3 fm/c, smaller than lifetime of fireball (≈ 10 fm/c)
Dilepton rate:
dNll d4xd4q
ImΠµν em ∼ ImDµν
[A. Drees, Nuclear Physics A 830 (2009)]
⇒ Aim: In-medium spectral functions of vector mesons and order parameter for chiral phase transition within the same framework
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∂kΓk = 1
2 STr
¶kGk 1 2
[wikipedia.org/wiki/Functional renormalization group]
I Γk interpolates between bare action S at k = Λ and effective action Γ at k = 0
I regulator Rk acts as a mass term and suppresses fluctuations with momenta smaller than k
I the use of 3D regulators allows for a simple analytic continuation procedure
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Modelling Vector Mesons
I Sakurai (1960): vector mesons as gauge bosons of local gauge symmetry SU(2)
→ Electromagnetic-hadronic interaction via exchange of vector mesons
→ Current field identity (CFI):
[Berghaeuser, www.staff.uni-giessen.de (2016)]
I Lee and Nieh (1960s): Gauged linear sigma model, local gauge symmetry SU(2)L × SU(2)R
⇒ ρ meson and chiral partner a1 meson as gauge bosons
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Gauged Linear Sigma Model with Quarks
I Local gauge symmetry SU(2)L × SU(2)R: Low-energy model of two-flavor QCD
I Ansatz for the effective average action Γk ≡ Γk[σ, π, ρ, a1, ψ, ψ] (LPA):
Γk =
∫ d4x { ψ ( /∂ − µγ0 + hS (σ + i~τ~πγ5) + ihV (γµ~τ~ρ
µ + γµγ5~τ~a µ 1 ) ) ψ + Uk(φ2)− cσ
+ 1
− 1
4 g2TrVµVν [Vµ, Vν ]
}
I Mesonic fields: φ ≡ (~π, σ) and Vµ ≡ ~ρµ ~T + ~a1,µ ~T 5
I Scale dependent quantities: Uk,m 2 k,V
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∂kΓ (2) ρ,k =
ψ
ψ
I Neglect vector mesons inside the loops
I Vertices extracted from ansatz of the effective average action Γk
I Tadpole diagrams give ω-independent contributions
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nB,F (E + ip0)→ nB,F (E)
2) Substitute p0 by continuous real frequency ω:
Γ(2),R(ω) = − lim ε→0
ρ(ω) = −Im(1/Γ(2),R(ω))/π
[R.-A. T., N. Strodthoff, L. v. Smekal, and J. Wambach, Phys. Rev. D 89, 034010 (2014)]
[J. M. Pawlowski, N. Strodthoff, Phys. Rev. D 92, 094009 (2015)]
[N. Landsman and C. v. Weert, Physics Reports 145, 3&4 (1987) 141]
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III) Results
Phase Diagram of the Model
I chiral order parameter σ0 decreases towards higher T and µ
I a crossover is observed at T ≈ 175 MeV and µ = 0
I critical endpoint (CEP) at µ ≈ 292 MeV and T ≈ 10 MeV
I we will study spectral functions along µ = 0 and T ≈ 10 MeV 100 200 300 400
Μ @MeVD
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Phase Diagram of the Model
I chiral order parameter σ0 decreases towards higher T and µ
I a crossover is observed at T ≈ 175 MeV and µ = 0
I critical endpoint (CEP) at µ ≈ 292 MeV and T ≈ 10 MeV
I we will study spectral functions along µ = 0 and T ≈ 10 MeV 100 200 300 400
Μ @MeVD
5
10
15
20
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Flow of Euclidean Masses
I Starting at cutoff scale Λ = 1500 MeV in chirally symmetric phase
I Chiral symmetry breaking sets in at kχSB
≈ 700 MeV
mσ = 557 MeV
mπ = 140 MeV
mρ = 1298 MeV
ma1 = 1676 MeV
mψ = 300 MeV
500
1000
1500
mσ
mπ
mψ
mρ
ma1
[C. Jung, F. Rennecke, R.-A. T., L. von Smekal, and J. Wambach, Phys. Rev. D 95, 036020 (2017)]
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I Euclidean (curvature) masses determine thresholds for the decay processes
I Euclidean masses in the vacuum:
mσ = 557 MeV
mπ = 140 MeV
mρ = 1298 MeV
ma1 = 1676 MeV
mψ = 300 MeV
I Mass degeneration of chiral partners for high T and µ
0 50 100 150 200 250 300 0
500
1000
1500
500
1000
1500
mσ
mπ
mψ
mρ
ma1
[C. Jung, F. Rennecke, R.-A. T., L. von Smekal, and J. Wambach, Phys. Rev. D 95, 036020 (2017)] 16 / 34
Decay Channels of the ρ Meson
ρ∗ → π + π
(2mψ)2 + ~p 2
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a∗1 → σ + π
(2mψ)2 + ~p 2
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0 200 400 600 800 1000 1200 1400
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-2
-1
0
1
e V
0 200 400 600 800 1000 1200 1400 0
1
2
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4
I Pole mass ∼ zero-crossing of the real part:
mp ρ = 789 MeV, mp
a1 = 1275 MeV
I Decay channels accessible when imaginary part is non-zero
I ρ and a1 meson unstable due to mesonic and quark-antiquark decay channels
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0 200 400 600 800 1000 1200 1400
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-2
-1
0
1
e V
0 200 400 600 800 1000 1200 1400 0
1
2
3
4
ψψ
σ/π
mp ρ = 789 MeV, mp
a1 = 1275 MeV
I Decay channels accessible when imaginary part is non-zero
I ρ and a1 meson unstable due to mesonic and quark-antiquark decay channels
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10-4
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1
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5
3 : a∗1 + π → σ
4 : a∗1 → π + σ
5 : a∗1 → ψ + ψ
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T = 300 MeV, μ = 0 MeV
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Discussion and T -dependent Pole Masses
I Degeneration of ρ and a1 spectral functions in chirally symmetric phase
I Broadening of spectral functions with increasing T
I Pole masses do not vary much, no dropping ρ mass
⇒ Consistent with broadening/melting-ρ-scenario
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1500
p
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ρ∗ + π → π
0 ≤ ω ≤ |~p|
ρ∗ + ψ → ψ
ρ∗ + ψ → ψ
0 ≤ ω ≤ |~p|
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a∗1 + σ → π
0 ≤ ω ≤ |~p|
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I shown for µ = 0 and T = 100 MeV
I time-like region (ω > ~p) is Lorentz-boosted to higher energies
I space-like region (ω < ~p) is non-zero at finite T due to space-like processes
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I shown for µ = 0 and T = 100 MeV
I time-like region (ω > ~p) is Lorentz-boosted to higher energies
I space-like region (ω < ~p) is non-zero at finite T due to space-like processes
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I Introduce wave function renormalization to include dressing of momentum dependence of meson propagators:
Γ (meson,kin) k =
m2 α 7−→ m2
Truncation mcurv ρ mpole
ZS,k = 1, ZV,k 914 MeV 786 MeV
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Summary and Outlook
I Analytic continued flow equations for vector meson two-point functions based on the gauged linear sigma model within the FRG
I Chiral order parameter and in-medium spectral functions obtained by the same theoretical framework
I Degeneration of ρ and a1 spectral functions indicating restoration of chiral symmetry, consistent with broadening/melting-ρ-scenario
Work in progress:
I Improve truncation (e.g. include wave function renormalization for all mesons)
I Improve phenomenology (e.g. include vector mesons inside the loops)
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∂kΓ (2) ρ,k = − 1
ψ
Summary and Outlook
I Analytic continued flow equations for vector meson two-point functions based on the gauged linear sigma model within the FRG
I Chiral order parameter and in-medium spectral functions obtained by the same theoretical framework
I Degeneration of ρ and a1 spectral functions indicating restoration of chiral symmetry, consistent with broadening/melting-ρ-scenario
Work in progress:
I Improve phenomenology (e.g. include vector mesons inside loops)
Aim: Provide spectral functions for transport simulations to compute dilepton spectra!
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In-Medium Vector-Meson Spectral Functions with the