Properties of bound states from the FRG and the DSE-BSE approaches - - PowerPoint PPT Presentation

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Properties of bound states from the FRG and the DSE-BSE approaches - - PowerPoint PPT Presentation

Aug 13, 2022 196 likes •531 views

Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary Properties of bound states from the FRG and the DSE-BSE approaches Jordi Par s L opez Advisors: R. Alkofer and H. Sanchis-Alepuz

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Properties of bound states from the FRG and the DSE-BSE approaches

Jordi Par´ ıs L´

  • pez

Advisors: R. Alkofer and H. Sanchis-Alepuz Karl-Franzens-Universit¨ at Graz, Austria

Jordi Par´ ıs L´

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Content

Motivation Brief introduction to the FRG FRG techniques Results in the Quark-Meson model Preliminary comparison with the DSE-BSE approach Summary and Outlook

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Motivation

Observable properties of hadrons are difficult to extract from QCD’s degrees of freedom: Need theoretical assumptions: Bound states and QCD (at hadronic energies) are not perturbative. Many approaches and models are built to solve these problems. Lattice QCD. Functional methods.

Dyson-Schwinger equations (DSE). Bethe-Salpeter equations (BSE).

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Both equations need to be truncated to be numerically solved. Rainbow-Ladder truncation:

Bethe-Salpeter equation:

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Results for the pion and other ground state mesons are well understood. However: The solution relies strongly on the truncation. For more complex systems:

Other terms appear in the DSE. Rainbow-Ladder truncation is not good enough. New technical issues appear.

Work in a different approach: Use of the Functional Renormalization Group (FRG) to find properties of mesons. The FRG approach is consistent with BSE.

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Introduction to the FRG

Generating Functional in Euclidean space as starting point: Z[J] = eW [J] =

  • Dψ e−S[ψ]+
  • x Jψ

Effective Action Γ[φ] from W [J] Legendre transformation: e−Γ[φ] =

  • Dψ exp
  • −S[φ + ψ] +
  • x

dΓ[φ] dφ ψ

  • with δΓ

δψ ≡ J , φ ≡ δW [J] δJ

= ψJ. Γ[φ] expressed as sum of 1PI diagrams. Introduction of scale k and regulator ∆Sk: Γk[φ] = Γ[φ] − ∆Sk[φ]

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Initial and final conditions are fixed in theory space:

The choice of the regulator is not unique.

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

With the choice of quadratic regulators ∆Sk[φ] =

  • p φRkφ, the properties
  • f the scale-dependent effective action lead to the 1-loop

integral-differential equation: ∂tΓk = 1 2 Tr

  • ∂tRk
  • Γ(2)

k

+ Rk −1 Wetterich’s Flow Equation with t = ln k ΛUV

  • ∂t = k d

dk The properties of this exact flow equation are very convenient for physical calculations since it is an Euclidean 1-loop integral-differential equation.

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Wetterich’s equation can be solved applying vertex expansion: Γk[φ] =

∞

  • n=0

1 n!

  • p1...pn

Γ(n)

k (p1, ...pn)φ(p1)...φ(pn)

Applying n−derivatives and averaging the fields one obtains the flow of the momentum dependent vertex functions. Remark: a truncation/approximation is needed.

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Dynamical Hadronization

Macroscopic QCD degrees of freedom are mesons and baryons. Introduced in the effective action through 4-Fermi Hubbard-Stratonovich transf.: Problem: 4-Fermi interaction flow non-zero, H-S transfomation must be applied in every RG-step = ⇒ Solved by Dynamical Hadrnization.

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Introduction of scale dependent bosonic field: ∂tφk(p) = ∂tAk(p)( ¯ ψτψ)(p) + ∂tBk(p)φk(p) with ∂tAk and ∂tBk defined such that 4-Fermi flow is cancelled: This generalizes Hubbard-Stratonovich transf. for every RG-step. Green’s functions computed with meson exchange diagrams.

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Analytical Continuation

Search for bound state properties through real-time Green’s functions: analytical continuation must be performed. Since the inverse propagator of 2-point function is proportional to (p2 + M2) in Euclidean Space, goal is to continue to purely imaginary p0. Extrapolation by fitting Euclidean momenta p2 data to a parametrized function and evaluating it at Minkowski momenta −p2. Direct calculation using the properties of the regulators.

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Analytical continuation by extrapolation: Pad´ e approximant: R(m n)(x) =

m

  • i=0

cixi 1 +

n

  • j=1

djxj Schlessinger point method for a set of M data points: C(x) = F(x1) 1 +

z1(x−x1) 1+

z2(x−x2)

. . .

zM−1(x−xM−1) Jordi Par´ ıs L´

  • pez

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Analytical continuation by direct calculation: Complex momenta accessed by direct calculation, not extrapolation. Already achieved using 3D-regulators [R-A Tripolt et al, arXiv:1311.0630v2]. Successfully performed with a 4D modified regulator for zero temperature O(N) model [J. M. Pawlowski and N. Strodthoff, arXiv:1508.01160v3]. Rk;∆m2

r (p2) =

  • ∆Γ(2)

k (p2)|φ=φ0 + ∆m2 r

  • r

p2 + ∆m2

r

k2

  • Jordi Par´

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Modified regulator moves poles out the integrating region = ⇒ physical implications induced. [J. M. Pawlowski and N. Strodthoff, arXiv:1508.01160v3]

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Preliminary results in the Quark-Meson model

Why using QM as starting point? A.Cyrol et al, arXiv:1605.01856 Low-energy effective theory with gluons decoupled.

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Free massless quark effective action + 4-Fermi interaction: Γk[ ¯ ψ, ψ] =

  • d4p

(2π)4 Zk,ψ ¯ ψA

a i/

p ψA

a

+ Γ4−int

k

[ ¯ ψ, ψ] Applying Hubbard-Stratonovich transformation introducing φ = (σ, π) fields: Γk ¯ ψ, ψ, σ, π

  • =
  • p
  • Zψ,k ¯

ψA

a i/

p ψA

a +

+ 1 2

  • Zk,φp2 + m2

k,φ

σ2 + πzπz

  • − cσ +

+

  • q

hk ¯ ψA

a

σ 2 δab + iγ5(τz)abπz ψA

b

  • Jordi Par´

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Results using the LPA’+DH Wave-function renormalizations Zk,i = 1 and non-kinetic bosonic terms rewritten in the same O(N) potential: Vk(ρ) =

∞

  • n=0

V (n)

k

n! (ρ − ρ0)n with ρ = 1

2

  • σ2 +

π2 and ρ0 scale independent expansion point. For non-zero renormalization wavefunctions the definition of the renormalized couplings/fields is needed: ρ = Zk,φρ ˆ ψ = Z

1 2

k,ψψ

hk = hk Zk,ψZ

1 2

k,φ

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Consequences of rewriting effective action: Renormalized potential Vk(ρ) =

∞

  • n=0

V (n)

k

n! (ρ − ρ0)n With V (n)

k

= V (n)

k

Z n

k,ρ and ρ0 = Zk,φρ0 =

⇒ running expansion point. Terms proportional to (momentum dependent) anomalous dimensions ηi appear explicitly in the flow equation: ηk,i = −∂tZk,i Zk,i

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Two ways to proceed: Solving for renormalized couplings.

Need to rewrite flow equations with renormalized parameters. Extra terms appear in the flows: ∂khk = FLOW + (ηψ + 1 2ηρ)hk Applied at running ρ0.

Solving for unrenormalized couplings with constant ρ0 and then translate into renormalized parameters.

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Definition of momentum dependent correlators: Γ(2)

k,φi(p2) = Zk,φ(p2)(p2 + m2 k,i(p2))

Γ(2)

k,ψ(p2) = Zk,ψ(p2)(i/

p + m2

k,ψ(p2))

Momentum dependence introduced from functional derivative. m2

k,π(p2) = V (1) k

(p2) = m2

k,π(0)

Zk,φ(p2) m2

k,σ(p2) = V (1) k

(p2) + 2ρ(p2)V (2)

k

(p2) = m2

k,σ(0)

Zk,φ(p2) mψ(p2) = hk(p2) σ √2Nf

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Mass (MeV)

200 400 600 800 1.000

k (MeV)

200 400 600 800 1.000

Sigma Pion

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Renormalized Yukawa

4,8 5 5,2 5,4 5,6 5,8 6 6,2 6,4

k (MeV)

200 400 600 800 1.000

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Quark Mass (MeV)

140 160 180 200 220 240 260 280 300

p2 (MeV2)

10-4 10-2 100 102 104 106

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Renormalized pion 2-point function (MeV2) 106 2x106 3x106 4x106 p2 (MeV2) 106 2x106 3x106 4x106

Pion Propagator (MeV-2) 10-5 2x10-5 3x10-5 4x10-5 5x10-5 6x10-5 p2 (MeV2) 10-2 100 102 104 106 Sigma 2-point function (MeV2) 106 2x106 3x106 4x106 p2 (MeV2) 106 2x106 3x106 4x106 Sigma Propagator (MeV-2) 10-6 2x10-6 3x10-6 4x10-6 5x10-6 6x10-6 p2 (MeV2) 10-2 100 102 104 106

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Applying analytical continuation and comparing with curvature masses mk,π(0) = 138.5 MeV and mk,σ(0) = 441.7 MeV, we obtained: Pole mπ ≈ 137.3 ± 0.4 MeV Pole mσ ≈ 395 ± 10 MeV Results compatible with Schlessinger with the number of points used. Pole mπ ≈ 137.953 MeV Pole mσ ≈ 382.961 MeV

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Conclusions FRG results: Calculation of momentum dependent anomalous dimensions and solving the system requires a large numerical effort. In terms of pole masses, Pad´ e approximant and Schlessinger point method works fairly well for the pion, but differs for the sigma meson due to the larger mass. Schlessinger point method stability requires large number of data points. Quark mass decreases with larger momentum but does not reach smaller values in the Quark-Meson model validity range. Behaviour agrees in general with QCD results from DSE-BSE at low energies.

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Preliminary comparison with the DSE-BSE approach

The good behaviour of the results obtained from the FRG in the LPA’+DH approximation agrees quite well with DSE-BSE results in QCD with the Rainbow-Ladder approximation. Hence, we ought to: Reproduce the same results for the Quark-Meson model in the DSE-BSE approach using a Rainbow-Ladder-like approximation. Establish relation between FRG and DSE-BSE approximations.

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Diagrammatic expression for the quark propagator DSE:

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

The parameter α(k2) needs to be modeled. Requiring: Cancellation of the meson loop contribution for momenta larger than the validity range of the Quark-Meson model. Convergence of the system given the same boundary conditions as in the FRG. Several models are being tried, for instance: α(k2) = k2 η 2 exp

  • −

k2 λ 4

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Quark Mass (MeV) Quark Mass (MeV)

140 140 160 160 180 180 200 200 220 220 240 240 260 260 280 280 300 300

p p2

2 (MeV

(MeV2

2)

)

10 100 10 102 10 104 10 106

DSE DSE FRG FRG

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Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary

Summary and Outlook

The FRG provides an alternative procedure to the BSE/Faddeev equation to obtain resonance masses and decay widths. Analytical continuation of the 4-fermi interaction to timelike momenta are to be performed in the Quark-Meson model. Big numerical effort and tools are needed to obtain accurate results. Systematic comparison of DSE/BSE vs FRG results for QCD. THANK YOU FOR YOUR ATTENTION

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