Spectral Functions from the FunctionalRenormalization Group

Jochen Wambach

ECT*, Trento, Italy

Seminar

Lisbon, November 16, 2017

in collaboration with:C. Jung, F. Rennecke, R.-A. Tripolt and L. von Smekal

Outline

I Motivation

I Functional Renormalization Group

I Spectral functions

I Vector mesons in the FRG

I Summary and outlook

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Mass without mass

QCD Lite:

LQCD = q̄i(i(γµDµ)ij −mi δij)qj − 14GaµνGµνa

mu = 2.3± 0.7± 0.5 md = 4.8± 0.5± 0.3I only apparent scale in QCD is the quark massI quark mass generated by the Higgs mechanismI for the proton this ∼ 1% of the massI nuclear physics mass scale - 1 GeV - is emergent

- no amount of staring at LQCD can reveal that scaleremove the quark mass:

I there is no scale left at the classical level

x→ λx LQCD(λx) = LQCD(x)I QCD becomes ’chirally symmetric’

(left- and right-handed quarks do not interact)

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Mass without mass

I as a consequence of scale invariance the energy-momentum tensor istraceless at the classical level:

Tµµ = 0

I due to quantum fluctuations and finite quark masses Tµµ receivesextra contributions

Tµµ =β(g)

2gGaµνG

µνa +muūu+mdd̄d+ · · ·

I and the proton mass becomes

〈p(P )|Tµµ |p(P )〉 = −P 2 = M20 + Σ2

M0 ∼ 880 MeV, Σ ∼ 60 MeV

I in QCD matter scale invariance and broken chiral symmetry arerestored at high temperature

I what does this imply for hadron masses in the medium?

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Why study hadrons in QCD matter?

Trace anomaly in QCD Matter〈Tµµ〉

= �− 3p

Chiral condensate in QCD Matter

〈q̄q〉 = −∂p/∂m

I scale invariance and broken chiral symmetry get restored in the mediumI hadron masses drop or new dgrees of freedom?I doubling of parity partners?

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Why study hadrons in QCD matter?

parity doubling

I scale invariance and broken chiral symmetry get restored in the mediumI hadron masses drop or new degrees of freedom?I doubling of parity partners

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Consistent theoretical framework

want a theoretical framework for computing the thermodynamic and thespectral properties of QCD matter on the same footing!

Requirements:

I thermodynamic consistencyI preservation of symmetries and their breaking pattern

Candidates:

I mean-field theoryI Functional Renormalization Group (FRG)

FRG superior since it includes both thermal and quantum fluctuationsand hence properly deals with phase transitions!

Method widely used in condensed matter physics, particle- and nuclearphysics, statistical physics and quantum gravity

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Functional Renormalization Group

partition function: (scalar field φ(x))

Z[j] = eW [j] =

∫[Dφ] e−S[φ]+

∫d4x φ(x)j(x)

generating functional:

δW [j]

δj(x)

∣∣∣∣j=0

=1

Z[0]

∫[Dφ]φe−S[φ] = 〈φ(x)〉 ≡ ϕ(x)

two-point correlation function: (Euclidean)

δ2W [j]

δj(x)δj(y)

∣∣∣∣j=0

= 〈φ(x)φ(y)〉 − 〈φ(x)〉 〈φ(y)〉 ≡ G(x, y)

effective action: (Legendre transform of W)

Γ[ϕ] = −W [j] +∫d4x ϕ(x)j(x)

stationarity condition and thermodynamic potential:

δΓ[ϕ]

δϕ

∣∣∣∣ϕ=ϕ0

= 0; → Ω(T ) = TV

Γ[ϕ0]8 / 31

Functional Renormalization Group

Wilsonian coarse graining:

at given resolution scale k split φ into low- and high-frequency modes:

φ(x) = φq≤k(x) + φq>k(x)

→ Z[j] =∫

[Dφ]q≤k∫

[Dφ]q>k e−S[φ]+∫d4x φj︸ ︷︷ ︸

=Zk[j]

; limk→0

Zk[j] = Z[j]

regulator Rk(q):

limk→0

Rk(q) = 0

limk→Λ

Rk(q) = ∞

Zk[j] =

∫[Dφ] e−S[φ]−∆Sk[φ]+

∫d4x φj

∆Sk[φ] =1

2

∫d4q

(2π)4φ(−q)Rk(q)φ(q)︸ ︷︷ ︸

acts like a mass term mkscale-dependent effective action:

→ Γk[ϕ] = − lnZk[j] +∫d4x ϕ(x)j(x)−∆Sk[ϕ]

Γk interpolates between k = Λ (no fluct.) and k = 0 (full quantum action)

limk→Λ

Γk[ϕ] = S[ϕ]; limk→0

Γk[ϕ] = Γ[ϕ]9 / 31

Functional Renormalization Group

flow equation for the effective action:

∂kΓk[ϕ] =1

2Tr

(∂kRk

[Γ

(2)k +Rk

]−1)

Γ(2)k (q) =

δ2Γk[ϕ]

δϕ(−q)δϕ(q)

[C. Wetterich, Phys. Lett. B301 (1993) 90]

¶kGk12

The average action Γk corresponds to an integration over all modes of the quantum fields withEuclidean momenta larger than the infrared cutoff scale, i.e., q2 > k2. The modified Legendretransform guarantees that the only difference between Γk and Γ is the effective IR cutoff ∆kSand thus only quantum fluctuations with momenta larger than k are included.

Figure 4.33: The effective average action Γk as an interpolation between the bare action in theUV and the full effective action Γ in the IR.

In the limit k → 0, the infrared cutoff is removed and the effective average action becomesthe full quantum effective action Γ containing all quantum fluctuations. Thus, for any finiteinfrared cutoff k the integration of quantum fluctuations is only partially done. The influence ofmodes with momenta q2 < k2 is not considered. This scenario is visualized in Fig. 4.33 wherethe k-dependent effective average action Γk as an interpolation between the bare action in theultraviolet and the full effective action in the infrared is shown.

In the limit k → ∞ the effective average action matches the bare or classical action. In atheory with a physical UV cutoff Λ, we therefore associate Γk=Λ with the bare action because nofluctuations are effectively taken into account. As the scale k is lowered, more and more quantumfluctuations are taken into account. As a consequence, Γk can be viewed as a microscope with avarying resolution whose length scale is proportional to 1/k. It averages the pertinent fields overa d-dimensional volume with size 1/kd and permits to explore the system on larger and largerlength scales. In this sense, it is closely related to an effective action for averages of fields, henceits denotation as effective average action becomes manifest. Thus, for large scale k one has avery precise spatial resolution, but one also investigates effectively only a small volume 1/kd.For lower k the resolution is smeared out and the detailed information of the short distancephysics is lost. However, since the observable volume is increased, long distance effects such ascollective phenomena which play an important role in statistical physics become more and morevisible.

The ’decimation’ idea, presented above, is in close analogy to a repeated application of theso-called block-spin transformation on a lattice invented by Kadanoff et al. [649]. This trans-formation is based on integrating out the fluctuations with short wavelengths and a subsequentrescaling of the parameters which govern the remaining long-range fluctuations such as the mass,coupling constant etc. On the sites of a coarse lattice more and more spin-blocks are averagedover. Hence, in the language of statistical physics, the effective average action can also beinterpreted as a coarse grained free energy with a coarse graining scale k.

189

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Functional Renormalization Group

flow equation for Γk including bosons and fermions:

∂kΓk[ϕ,ψ] = Tr

∫q

(1

2Gϕ,k(q)Rϕ,k(q)−Gψ,k(q)Rψ,k(q)

)

Gϕ,k(q) =[Γ

(2)k [ϕ] +Rϕ,k(q)

]−1Gψ,k(q) =

[Γ

(2)k [ψ] +Rψ,k(q)

]−1Γ

(2)k [ϕ] =

δ2Γk[ϕ,ψ]

δϕ2; Γ

(2)k [ψ] =

δ2Γk[ϕ,ψ]

δψδψ̄

¶kGk12

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Quark-meson model

I effective low-energy model for QCD (Nf = 2)I pion and sigma fields φ = σ + i~τ~π and quarks q̄, qI describes spontaneous and explicit chiral symmetry breaking

Scale-dependent effective action: (gradient expansion)

Γk[q, q̄, φ] =

∫x

{q̄Zq,k (∂/− µγ0) q + hq,kq̄ (σ + i~τ~πγ5) q

+1

2Zφ,k(∂µφ)

2 +1

8Yφ,k

(∂µφ

2)2

+ Uk(φ2)− cσ + · · ·

}∫x≡∫ 1/T0

dx0

∫Vd3x

’Local potential approximation’: (only Uk flows)

UΛ(φ2) = ~π2 + σ2; 〈σ〉 = 0

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Momentum flow of the effective action

(Loading movie...)

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Lavf55.44.100

pot_broken_k.mp4Media File (video/mp4)

Phase diagram of the Quark-Meson Model

I chiral order parameter σ0decreases towards higher T and µ

I a crossover is observed atT ≈ 175 MeV and µ = 0

I critical endpoint (CEP) atµ ≈ 292 MeV and T ≈ 10 MeV

I vacuum: σ0 = 93.5 MeV,mπ = 138 MeV, mσ = 509 MeV,mq = 299 MeV 100 200 300 400

Μ @MeVD

50

100

150

200

250T @MeVD

270 290 310

5

10

15

20

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In-medium spectral functions from the FRG

How are in-medium modifications of hadrons related to the change of thevacuum structure of QCD? (chiral symmetry restoration ..)

I need to calculate equilibrium properties and spectral function on thesame footing → FRG

equilibrium FRG formulated and solved in Euclidean space-time

flow equations for Euclidean two-point functions:

∂kΓ(2)k =

1

2

δ2

δϕ2Trq

{Gϕ,kRϕ,k)

}−

δ2

δψψ̄Trq

{Gψ,kRψ,k

}= Trq

{∂kRϕ,k

(Gϕ,kΓ

(3)k Gϕ,kΓ

(3)k Gϕ,k

)}−

1

2Trq

{∂kRϕ,k

(Gϕ,kΓ

(4)k Gϕ,k

)}+· · ·

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Flow equations for two-point functions

I mesonic vertices from scale-dependent effective potential, U (3)k , U(4)k

I quark-meson vertices are givn by Γ(3)ψ̄ψσ

= h, Γ(3)

ψ̄ψ~π= ihγ5~τ

I one-loop structure preservedI thermodynamically consistent and symmetry preserving

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In-medium spectral functions from the FRG

How are in-medium modifications of hadrons related to the change of thevacuum structure of QCD? (deconfinement and chiral symmetry restoration ..)

I need to calculate equilibrium properties and spectral function on thesame footing → FRG

equilibrium FRG formulated and solved in Euclidean space-time

flow equations for Euclidan two-point functions:

∂kΓ(2)k =

1

2

δ2

δϕ2Trq

{Gϕ,kRϕ,k)

}−

δ2

δψψ̄Trq

{Gψ,kRψ,k

}= Trq

{∂kRϕ,k

(Gϕ,kΓ

(3)k Gϕ,kΓ

(3)k Gϕ,k

)}−

1

2Trq

{∂kRϕ,k

(Gϕ,kΓ

(4)k Gϕ,k

)}+· · ·

spectral functions are real-time quantities!

analytic continuation procedure is needed

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Analytic continuation

I use periodicity of occupation numbers with discrete Euclidean energyip0 = i2nπT

nB,F (E + ip0)→ nB,F (E)

I substitute p0 by continuous real frequency ω

Γ(2),Rk,j (ω, ~p) = − lim�→0 Γ

(2),Ek,j (p0 = −i(ω + i�), ~p); for j = π, σ

I solve flow equations Re ∂kΓ(2),Rk,j , Im ∂kΓ

(2),Rk,j at global minimum of Uk→0

I finally obtain spectral functions as

ρj(ω, ~p) =1

π

Im Γ(2),Rk→0,j(ω, ~p)(

Re Γ(2),Rk→0,j(ω, ~p)

)2+(

Im Γ(2),Rk→0,j(ω, ~p)

)218 / 31

Flow of the σ and π spectral functions in vaccum

(Loading movie...)

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Lavf56.40.101

spectral_flow_vac.mp4Media File (video/mp4)

σ- and π spectral function for finite T at µ = 0

(Loading movie...)

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Lavf57.73.100

spectral_T_eps0_2.mp4Media File (video/mp4)

σ spectral function with increasing T at µ = 0

(Loading movie...)

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Lavf57.11.100

sigma_mu0_T_3D.mp4Media File (video/mp4)

π spectral function with increasing T at µ = 0

(Loading movie...)

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Lavf57.26.100

pion_mu0_T_3D.mp4Media File (video/mp4)

Modelling vector mesons

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Gauged Linear Sigma Model with quarks

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Flow equations for two-point functions

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ρ- and a1 spectral function for finite T at µ = 0

(Loading movie...)

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Lavf57.57.100

vector_T_2.mp4Media File (video/mp4)

T-dependence of V-A spectral functions

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T-dependent pole masses

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ρ spectral function with increasing T at µ = 0

(Loading movie...)

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Lavf57.46.100

rho_T_3D.mp4Media File (video/mp4)

Summary and Outlook

I analytically continued flow equations for scalar- and vector-meson two-pointfunctions including parity partners within the FRG

I chiral order parameter and in-medium spectral functions obtained within thesame theoretical framework

I complete degeneration of the (ω, ~p) - dependent spectral functions of paritypartners in the chirally restored phase

I degeneracy of ρ and a1 spectral functions consistent with broadingρ-scenario

work in progress:

I improve truncation (e.g. include wave function renormalization)

I improve phenomenology (e.g. include vector mesons inside loops)

aim: provide spectral functions for hydrodynamicsand/or coarse-grained transport simulations

to compute dilepton spectra

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