Dyson Schwinger 101 Research in the DSE approach Esther Weil - - PowerPoint PPT Presentation

December, 12th 2017

Justus-Liebig-Universität Gießen

Institut für Theoretische Physik

Cold Quantum Coffee-Heidelberg

Esther Weil

γµ

Dyson Schwinger 101

Research in the DSE approach

Esther Weil JLU Giessen

Dyson-Schwinger 101

/18 Tues, 12.12.17 2

Goal:

Compute Hadron Properties (such as spectra, form factors, scattering amplitudes …) using QCD’s Greens functions on a quark-gluon level.

QCD effective action Dyson-Schwinger equations (DSE)

derive “The quantum equation of motion“

Intro

+ Bethe Salpeter equations

describing bound states But for numerical evaluation a truncation is needed. non-perturbative, at all momentum scales, light and heavy quarks.

Esther Weil JLU Giessen

Dyson-Schwinger 101

/18 Tues, 12.12.17 3

Christian Fischers group, JLU Gießen

Group topics include:

• Meson and Bayron spectroscopy
• Nucleon Compton scattering
• Muons g-2
• Elastic & transition form factors
• Tetraquarks
• Exotic & Hybrid mesons
• QCD phase diagram

Wallbott, Eichmann, Fischer Eichmann, Fischer, Weil, Williams Gunkel, Isserstedt, Haque Eichmann, Sanchis-Alepuz, Williams, Fischer - recent Review (2016) Eichmann, Fischer, PRD 87 (2014) Bonnet, Eichmann, Fischer, Williams eg.

Intro

Söhngen, Eshraim, Williams, Huber

• Phys. Rev. D 86, 099901 (2012)

PLB, 753, 1508.07178 (2016) Eichmann, G. Few-Body Syst (2016) 57: 965

eg.

Esther Weil JLU Giessen

Dyson-Schwinger 101

/18 Tues, 12.12.17 4

QCD Lagrangian

quarks, gluons

mesons baryons glueballs? hybrids? tetraquarks?

degrees of freedom in QCD

L = ¯ ψ(x)(i/ ∂ + g / A − M)ψ(x) − 1 4F a

µνF µν a but those can not be seen in nature. We observe only hadrons asymptotic freedom

webific.ific.uv.es/web/en

Intro

calls for non-perturbative methods quark and gluons are confined such as the Dyson-Schwinger approach !

Esther Weil JLU Giessen

Dyson-Schwinger 101

/18 Tues, 12.12.17 5

quark propagator

Theory stuff

Are these included? The so called self-energy includes all possible quark self interactions. Deriving the quark Dyson-Schwinger equation diagrammatically (i) QCD’s action Origin of the equation Starting from we obtain the following action diagrammatically (ii) Take the functional derivative with respect to the quark fields

S = ⁄ d4x 5 ¯ ψ(x)(i/ ∂ + g / A − M)ψ(x) − 1 4F a

µνF µν a

6

Esther Weil JLU Giessen

Dyson-Schwinger 101

/18 Tues, 12.12.17 5

quark propagator

Theory stuff

Are these included? The so called self-energy includes all possible quark self interactions. Deriving the quark Dyson-Schwinger equation diagrammatically (i) QCD’s action Origin of the equation Starting from we obtain the following action diagrammatically (ii) Take the functional derivative with respect to the quark fields

S = ⁄ d4x 5 ¯ ψ(x)(i/ ∂ + g / A − M)ψ(x) − 1 4F a

µνF µν a

6

Esther Weil JLU Giessen

Dyson-Schwinger 101

/18 Tues, 12.12.17 5

quark propagator

Theory stuff

Are these included? The so called self-energy includes all possible quark self interactions. Deriving the quark Dyson-Schwinger equation diagrammatically (i) QCD’s action Origin of the equation Starting from we obtain the following action diagrammatically (ii) Take the functional derivative with respect to the quark fields

S = ⁄ d4x 5 ¯ ψ(x)(i/ ∂ + g / A − M)ψ(x) − 1 4F a

µνF µν a

6

Esther Weil JLU Giessen

Dyson-Schwinger 101

/18 Tues, 12.12.17 6

Theory stuff

(ii) Take the functional derivative with respect to the quark fields (iii) Replace the action by the effective action (iv) Take another derivative and set the field to zero

quark Dyson-Schwinger equation

and there is more! - equations for the different Greens functions

derivation inspired by Gernot Eichmann, Marcus Huber, Reinhard Alkhofer and Kai Schwenzer.

connecting the dots in all possible ways lead to the quantum equation

• f motion

and are related thru a Legendre-Trafo.

Γ S

Esther Weil JLU Giessen

Dyson-Schwinger 101

/18 Tues, 12.12.17

+ =

Γµ(p, P)

γµ

K

P

7

Theory stuff

quark propagator

= Bethe-Salpeter equation (BSE)

gluon propagator

Γ

p

=

Γ

q

q+

q−

K

P

quark propagator

• eg. for a meson

ghost propagator

vertices

• eg. quark-gluon vertex, quark-photon vertex

bound states equation for bound states and vertices + + + = Dyson-Schwinger equation (DSE)

Esther Weil JLU Giessen

Dyson-Schwinger 101

/18 Tues, 12.12.17

but but we have an infinite set of coupled integral equations!

8

all these equations are rigorous and non-perturbative quark-gluon vertex

gluon propagator

includes higher order Greens functions

Need to introduce a truncation for numerical evaluations !

Theory stuff

quark propagator

vertices bound states + +

gluon propagator ghost propagator

+

Esther Weil JLU Giessen

Dyson-Schwinger 101

/18 Tues, 12.12.17 9

Numerical evaluations

Truncate, but how?

To preserve chiral symmetry we use a consistent truncation in each equation. We choose an Ansatz for higher order Greens function: In the case of the rainbow-ladder approximation we combine the quark-gluon vertex and gluon propagator to a tree-level vertex + effective coupling

K = ∂Σ ∂S =

Σ = ∂Γ ∂S =

Rainbow-ladder truncation

DSE self-energy term BSE kernel Ansatz = tree-level + effective coupling

α(k2) α(k2)

self-energy and kernel are related through a symmetry relation, Axial-Vector Ward-Takehashi identity

Esther Weil JLU Giessen

Dyson-Schwinger 101

/27 Tues, 12.12.17 10

α(k2) = αIR(k2/

Λ2, η) + αUV(k2)

scale is adjusted to observables, like .

masses mu=md, ms from m , mK

from perturbation theory - UV: rigorous field theory part parameter : band of results - IR: modeled part

fπ π

Λ η αUV

α(k2)

Combine gluon with quark-gluon vertex to an effective coupling α(k2) + tree-level vertex

Maris, Roberts,Tandy, PRC 56 (1997), PRC 60 (1999)

Ansatz effective coupling:

Numerical evaluations

Esther Weil JLU Giessen

Dyson-Schwinger 101

/27 Tues, 12.12.17

limitations in kinematic regim due to poles in the calculations, e.g.. quark poles.

11

Numerical evaluations

350 MeV 3 MeV

Bottom Charm Strange Up/down Chiral limit

Quark mass function [GeV]: Fischer J. Phys G. 32 (2006)

DSE results for the quark propagator, for different quark masses: Using the rainbow-ladder approximation: works well for not as well for

pseudo-scalar and vector mesons, also form- factors & decays excited scalar, axial- vector mesons

• too light.

quark mass generation through dynamical chiral symmetry breaking.

excited state are problematic! Calculations beyond rainbow-ladder similar for the baryons. More complicated, two different approaches. excited bayrons Using RL, we are ready to calculate various QCD observables.

the kinematic regime where we can calculate is limited due to poles in the quark propagators.

Eichmann, Fischer, Williams, Alkhofer PPN 91 (2016)

• Phys. Rev. D 93, 034026

Esther Weil JLU Giessen

Dyson-Schwinger 101

/18 Tues, 12.12.17 12

Recent Research

Pion transition form factor

Next:

Λµν(Q, Q0) = e2 F(Q2, Q02) 4π2fπ εµναβQ0αQβ

π0

+

π0 → γγ

In the chiral limit ( )

mπ = 0

SU(3) flavor anomaly

ADLER, BELL, JACKIW, 1969

From the anomaly we know F(0, 0) = 1

PLB V 774, 425-429 arXiv:1704.05774v3 [hep-ph]

Esther Weil JLU Giessen

Dyson-Schwinger 101

/18 Tues, 12.12.17 12

Recent Research

Pion transition form factor

Next:

Λµν(Q, Q0) = e2 F(Q2, Q02) 4π2fπ εµναβQ0αQβ

We need these building blocks:

quark propagator

Pion amplitude Quark-photon vertex

π0 → γγ

π0

γµ

In the chiral limit ( )

mπ = 0

SU(3) flavor anomaly

ADLER, BELL, JACKIW, 1969

From the anomaly we know F(0, 0) = 1

PLB V 774, 425-429 arXiv:1704.05774v3 [hep-ph]

Esther Weil JLU Giessen

Dyson-Schwinger 101

/18 Tues, 12.12.17

Result for the form factor in the asymmetric and symmetric ( ) limit

13

Recent Research

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

cone asym sym

(, ´)

[]

0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

[]

Q2 = QÕ2

Dispersion relation Lattice estimate of the continuum limit

singly-virtual up to

• M. Hoferichter et al.(2014), arXiv:1410.4691[hep-ph]

2 ∼ 2GeV2

• A. Geradin et al. (2016), arXiv:1508.07178 [hep-lat]

symmetric limit for all asymmetric kinematic restriction due to the limitations

• f the building blocks, e.g quark

propagator in the complex plane

Q2

Q2 < 4GeV2

In comparison Non-perturbative input to further electro-magnetic pion decays combined effort of different non- perturbative methods.

DSE (asym) DSE (sym) Belle BaBar CLEO CELLO

2.0 1.5 1.0 0.5 0.0 5 10 15 20

[]

)

2 ′

, Q

2

Q ( F

• Â

F(Q2, QÕ2) = η+ F(Q2, QÕ2) 4π2f 2

π

Q2

Esther Weil JLU Giessen

Dyson-Schwinger 101

/18 Tues, 12.12.17 14

Recent Research

Rare decay of the pion

Next:

π0 → e+e−

Discrepancy between theory and experiment of 2σ with t0 = −m2

π/4

The branching ratio is given by and the scalar amplitude

BRexp(π0

e+e−) ≈ 6.87(36) × 10−8

theory calculations

A(t) ∼

∆(π0)

• Phys. Rev. D 96, 014021

arXiv:1704.06046v2 [hep-ph]

Esther Weil JLU Giessen

Dyson-Schwinger 101

/18 Tues, 12.12.17 14

Recent Research

Rare decay of the pion

Next:

π0 → e+e−

Discrepancy between theory and experiment of 2σ with t0 = −m2

π/4

The branching ratio is given by and the scalar amplitude

BRexp(π0

e+e−) ≈ 6.87(36) × 10−8

theory calculations

Integration domain includes poles

A(t) ∼

∆(π0)

• Phys. Rev. D 96, 014021

arXiv:1704.06046v2 [hep-ph]

Esther Weil JLU Giessen

Dyson-Schwinger 101

/18 Tues, 12.12.17 15

Recent Research

Rare decay of the pion

Next:

π0 → e+e−

BRexp(π0

e+e−) ≈ 6.87(36) × 10−8

A(t) ∼

∆(π0)

Integration domain includes poles!

poles

(i) dispersion relation (ii) “direct“ method Two ways to calculate the integral:

L.Bergström al.(2014), Phys Let. B126, 8117(19839)

• Phys. Rev. D 96, 014021

arXiv:1704.06046v2 [hep-ph]

Esther Weil JLU Giessen

Dyson-Schwinger 101

/18 Tues, 12.12.17 16

Recent Research

Rare decay of the pion

Next:

π0 → e+e−

(ii) “direct“ method

integration path in the complex plane

Our quantities are defined in Euclidean space, from the Minkowski description there is only one rigorous path that corresponds to the physical decay rate, but which?

t = (−1 + i)m2

π/4

• Phys. Rev. D 96, 014021

arXiv:1704.06046v2 [hep-ph]

A(t) = 1 2π2t Z

γ

d Σ2 Z d cos θ Z dφ (Σ · ∆)2 − Σ2∆2 (p + Σ)2 + m2

e

F(Q2, Q02) Q2 Q02 .

Esther Weil JLU Giessen

Dyson-Schwinger 101

/18 Tues, 12.12.17 16

Recent Research

Rare decay of the pion

Next:

π0 → e+e−

(ii) “direct“ method

Photon cuts

(opens at )

integration path in the complex plane

Our quantities are defined in Euclidean space, from the Minkowski description there is only one rigorous path that corresponds to the physical decay rate, but which?

t = (−1 + i)m2

π/4

• Phys. Rev. D 96, 014021

arXiv:1704.06046v2 [hep-ph]

A(t) = 1 2π2t Z

γ

d Σ2 Z d cos θ Z dφ (Σ · ∆)2 − Σ2∆2 (p + Σ)2 + m2

e

F(Q2, Q02) Q2 Q02 .

Esther Weil JLU Giessen

Dyson-Schwinger 101

/18 Tues, 12.12.17 16

Recent Research

Rare decay of the pion

Next:

π0 → e+e−

(ii) “direct“ method

Photon cuts

(opens at )

integration path in the complex plane

Our quantities are defined in Euclidean space, from the Minkowski description there is only one rigorous path that corresponds to the physical decay rate, but which?

t = (−1 + i)m2

π/4

• Phys. Rev. D 96, 014021

arXiv:1704.06046v2 [hep-ph]

A(t) = 1 2π2t Z

γ

d Σ2 Z d cos θ Z dφ (Σ · ∆)2 − Σ2∆2 (p + Σ)2 + m2

e

F(Q2, Q02) Q2 Q02 .

Esther Weil JLU Giessen

Dyson-Schwinger 101

/18 Tues, 12.12.17 16

Recent Research

Rare decay of the pion

Next:

π0 → e+e−

(ii) “direct“ method

Photon cuts

(opens at )

Lepton cuts

(open at )

integration path in the complex plane

Our quantities are defined in Euclidean space, from the Minkowski description there is only one rigorous path that corresponds to the physical decay rate, but which?

t = (−1 + i)m2

π/4

• Phys. Rev. D 96, 014021

arXiv:1704.06046v2 [hep-ph]

A(t) = 1 2π2t Z

γ

d Σ2 Z d cos θ Z dφ (Σ · ∆)2 − Σ2∆2 (p + Σ)2 + m2

e

F(Q2, Q02) Q2 Q02 .

Esther Weil JLU Giessen

Dyson-Schwinger 101

/18 Tues, 12.12.17 16

Recent Research

Rare decay of the pion

Next:

π0 → e+e−

(ii) “direct“ method

• Re

Im

(1) (2) (3) (4)

0.01 0.00

• 0.01
• 0.01

0.00 0.01 0.02 0.03

Photon cuts

(opens at )

Lepton cuts

(open at )

integration path in the complex plane

Our quantities are defined in Euclidean space, from the Minkowski description there is only one rigorous path that corresponds to the physical decay rate, but which?

t = (−1 + i)m2

π/4

• Phys. Rev. D 96, 014021

arXiv:1704.06046v2 [hep-ph]

A(t) = 1 2π2t Z

γ

d Σ2 Z d cos θ Z dφ (Σ · ∆)2 − Σ2∆2 (p + Σ)2 + m2

e

F(Q2, Q02) Q2 Q02 .

Esther Weil JLU Giessen

Dyson-Schwinger 101

/18 Tues, 12.12.17 16

Recent Research

Rare decay of the pion

Next:

π0 → e+e−

(ii) “direct“ method

Photon cuts

(opens at )

Lepton cuts

(open at )

integration path in the complex plane

technical challenge: cuts are almost congruent, due to the small electron mass Our quantities are defined in Euclidean space, from the Minkowski description there is only one rigorous path that corresponds to the physical decay rate, but which?

t = (−1 + i)m2

π/4

• Phys. Rev. D 96, 014021

arXiv:1704.06046v2 [hep-ph]

A(t) = 1 2π2t Z

γ

d Σ2 Z d cos θ Z dφ (Σ · ∆)2 − Σ2∆2 (p + Σ)2 + m2

e

F(Q2, Q02) Q2 Q02 .

Esther Weil JLU Giessen

Dyson-Schwinger 101

/18 Tues, 12.12.17 17

Recent Research

Rare decay of the pion

Next:

π0 → e+e−

(ii) “direct“ method in the whole complex plane

A(t)

• direct and dispersive calculation agree
• discrepancy: theory and experiment

Esther Weil JLU Giessen

Dyson-Schwinger 101

/18 Tues, 12.12.17 18

Summary

Summary

T h a n k you !

• how to calculate physical observables in the DSE approach.
• recent publications on the pion form factor.
• interesting contour deformation to solve a 1-loop diagram in

Euclidean space.

• Sparked your interest in the DSE approach.