dynamical mass generation in hadrons and leptons · dynamical mass generation in ! hadrons and...

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Christian S. Fischer Justus Liebig Universität Gießen Christian Fischer (University of Gießen) Dynamical mass generation / 46 Dynamical Mass Generation in Hadrons and Leptons 1 with Gernot Eichmann, Helios Sanchis-Alepuz, Richard Williams and Walter Heupel, Stanislav Kubrak

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Page 1: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian S. Fischer

Justus Liebig Universität Gießen

Christian Fischer (University of Gießen) Dynamical mass generation / 46

Dynamical Mass Generation in Hadrons and Leptons

1

with Gernot Eichmann, Helios Sanchis-Alepuz, Richard Williams and Walter Heupel, Stanislav Kubrak

Page 2: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

1.Introduction

2.Gluons, quarks and mesons

3.Pion cloud effects in baryons

4.Hadronic effects in (g-2)μ

5.Tetraquarks

−1=

−1−

+ pion cloud

+ pion cloud

Overview

2

Page 3: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

PANDA: Hadron physics CBM: QCD phase diagram

Facility for Antiproton Ion Reactions: FAIR

3

Nonperturbative QCD}Universities of Darmstadt, Frankfurt, Giessen + FIAS Continued funding by State of Hesse approved on 06/20/2014

Page 4: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

Properties of QCD: Dynamical mass generation

4

Dynamical quark masses via weak and strong force Yoichiro Nambu,

Nobel prize 2008

u d s c b tMweak 3 5 80 1200 4500 176000Mstrong 350 350 350 350 350 350Mtotal 360 360 450 1600 4900 176000[MeV/c2]

[MeV/c2]

[MeV/c2]

S�1(p) = [ip/ +M(p2)]/Zf (p2)

Page 5: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

Properties of QCD: Dynamical mass generation

4

Dynamical quark masses via weak and strong force Yoichiro Nambu,

Nobel prize 2008

u d s c b tMweak 3 5 80 1200 4500 176000Mstrong 350 350 350 350 350 350Mtotal 360 360 450 1600 4900 176000[MeV/c2]

[MeV/c2]

[MeV/c2]

S�1(p) = [ip/ +M(p2)]/Zf (p2)

Input parameters in Nf=2+1 QCD

Page 6: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

Effective theories in terms of hadrons

Nonperturbative QCD: Lattice, Functional methods

Properties of QCD: Confinement

5

Color confinement:

We are not detecting quarks and gluons, but baryons, mesons, tetraquarks, glueballs, hybrids...

Strategies to deal with this situation:

Page 7: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

QCD in covariant gauge

6

Quarks and Gluons

The Goal: gauge invariant information in a gauge fixed approach.

Landau gauge propagators in momentum space,

ZQCD =

Z

D[�, A] exp

�Z

d

4x

� (iD/ �m)�� 1

4

F

aµ�

�2

+gauge fixing⌘o

DGluonµ� (p) =

✓�µ� � pµp�

p2

◆Z(p2)

p2

SQuark(p) = Zf (p2)[�ip/ +M(p2)]�1

Page 8: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

Functional approaches (DSE, FRG, Hamilton): Chiral symmetry: physical quark masses Infinite volume and continuum limit Multi-scale problems feasible (e.g. (g-2)μ) Chemical potential: no sign problem

Nonperturbative QCD: Complementary approach

7

Quarks and gluons Hadrons

CF, Luecker, PLB 718 (2013) 1036

Effective theories and models (χPT, chiral models,.…) Physical degrees of freedom

Lattice simulations Ab initio

Page 9: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

1.Introduction

2.Gluons, quarks and mesons

3.Pion cloud effects in baryons

4.Hadronic effects in (g-2)μ

5.Tetraquarks

−1=

−1−

+ pion cloud

+ pion cloud

Overview

8

Page 10: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

DSEs and Bethe-Salpeter equation

9

π, ρ,...

Kernel K uniquely related to quark-DSE via axWTI

→Pion is bound state and Goldstone bosonMaris, Roberts, Tandy, PLB 420 (1998) 267

Two strategies:

I. use rainbow-ladder model for quark-gluon interaction →ok for some phenomenological applications II. calculate gluon and vertex from their DSEs

Page 11: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

Strategie I: Model for quark-gluon interaction

10

Maris, Tandy, 1999

�(k2) = ⇤⇥7✓k2

�2

◆e��2

⇣k2

�2

+ �UV (k2)

two (related) parameters and from fπ from perturbation theory

masses mu=md from mπ or mρ Renormalizable and momentum dependent -

in contrast to

↵UV

⌘ ⇤

(cf. (g-2)μ)

Page 12: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

Strategie I: Model for quark-gluon interaction

10

Maris, Tandy, 1999

�(k2) = ⇤⇥7✓k2

�2

◆e��2

⇣k2

�2

+ �UV (k2)

two (related) parameters and from fπ from perturbation theory

masses mu=md from mπ or mρ Renormalizable and momentum dependent -

in contrast to

↵UV

⌘ ⇤

NJL-model�(k2) = � �(⇥2 � k2) (cf. (g-2)μ)

Page 13: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

Quark mass: flavor dependence

11

M(p2): momentum dependent!

Dynamical mass: Mstrong≈350 MeV

Flavour dependence because of Mweak

Chiral condensate:

10-2 10-1 100 101 102 103

p2 [GeV2]

10-4

10-3

10-2

10-1

100

101

Qua

rk M

ass F

unct

ion:

M(p

2 ) [G

eV]

Bottom quarkCharm quarkStrange quarkUp/Down quarkChiral limit

Typical solution:

h��i ⇡ (250MeV)3

CF,  JPG 32 (2006) R253

[S(p)]�1 = [�ip/ +M(p2)]/Zf (p2)

Page 14: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

Quark mass: flavor dependence

11

M(p2): momentum dependent!

Dynamical mass: Mstrong≈350 MeV

Flavour dependence because of Mweak

Chiral condensate:

10-2 10-1 100 101 102 103

p2 [GeV2]

10-4

10-3

10-2

10-1

100

101

Qua

rk M

ass F

unct

ion:

M(p

2 ) [G

eV]

Bottom quarkCharm quarkStrange quarkUp/Down quarkChiral limit

Typical solution:

h��i ⇡ (250MeV)3

CF,  JPG 32 (2006) R253

Not directly measu

rable !

[S(p)]�1 = [�ip/ +M(p2)]/Zf (p2)

Page 15: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

Phenomenology from Maris-Tandy interaction

12

Slide from Pieter Maris

Page 16: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

good channels (ground state): 0-+, 1-- acceptable channels (ground state) : 2++, 3--,... clear deficiencies in other channels deficiencies for excited states

Rainbow-ladder: spectrum

13

��

��������

��������

������

������

�����

�������

�������

���� ������� ��

� ��� ��

� ������ ��������

������

��

������ ��� ��� ��� ��� �� �� �� ��� ��� ��� ��� ��� ��� ��

� �

������

�������

CF, Kubrak, Williams, arXiv:1406.4370

Page 17: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

naive and intuitive explanation for Regge (spinning flux tube) is not the mechanism at work here...

Rainbow-ladder: Regge-trajectory

14

1--2++ 3--

JPC

0,0

1,0

2,0

3,0

4,0

M2 [G

eV2 ]

isovector nnρ(770)

a2(1320)

ρ3(1690)

CF, Kubrak, Williams, arXiv:1406.4370

Page 18: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

treat Yang-Mills sector explicitly include gauge effects in vertex anomalous quark magnetic moment include gluon self-interaction effects include pion cloud effects

Strategy II: Beyond rainbow ladder

15

π, ρ,...

Chang, Roberts, PRL 103 (2009) Heupel, Goecke, CF, EPJA 50 (2014), [1402.5042]

CF, Nickel, Wambach PRD 76 (2007)

Recent improvements beyond rainbow-ladder:

CF, Williams, PRL 103 (2009)

CF, Watson, Cassing, PRD 72 (2005)

Chang, Liu, Roberts, PRL 106 (2011)

Page 19: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

spacelike momenta: excellent agreement with lattice

spectral function: positivity violations

Strategy II: Landau gauge gluon propagator

16

0 1 2 3 4 5p [GeV]

0

0.5

1

1.5

Z(p2 )

Sternbeck et. al. (2006)Funktionalzugang

spectral function

Strauss, CF, Kellermann, Phys. Rev. Lett. 109, (2012) 252001Gluon cannot appear in detector!

−1=

−1− 1

2

− 12 − 1

6

+ − 12

−1=

−1−

CF, Maas, Pawlowski, Annals Phys. 324 (2009) 2408. Huber and von Smekal, JHEP 1304 (2013) 149

Dµ⌫(p) =

✓�µ⌫ � pµp⌫

p2

◆Z(p2)

p2

Cornwall, Papavassiliou,...

Page 20: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

Strategy II: Pion effects in quark-gluon interaction

17

quark-gluon vertex:

quark:

= + + + +

Page 21: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

Strategy II: Pion effects in quark-gluon interaction

17

quark-gluon vertex:

quark:

= + Nc2

− 2Nc

+

π

+...

Page 22: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

Strategy II: Pion effects in quark-gluon interaction

17

quark-gluon vertex:

quark:

= + Nc2

− 2Nc

+

π

+...

−1

=

−1

Y M

π

Page 23: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

Strategy II: Pion effects in quark-gluon interaction

17

quark-gluon vertex:

= + Nc2

− 2Nc

+

π

+...

−1

=

−1

Y M

π

π+

π

π

Bethe-Salpeter equation:

Page 24: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

Attractive effects of pion cloud Scalar too large or ... too low!

Unquenching effects: Light mesons

18

CF, Williams, PRL 103 (2009), PRD 78 (2008)

Ebert, Faustov, Galkin, PRD 79 (2009) 114029 Parganlija, Kovacs, Wolf, Giacosa and Rischke,  PRD 87 (2013) 014011

→ tetraquarks !? → see later

Page 25: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

Attractive effects of pion cloud Scalar too large or ... too low!

Unquenching effects: Light mesons

18

CF, Williams, PRL 103 (2009), PRD 78 (2008)

Ebert, Faustov, Galkin, PRD 79 (2009) 114029 Parganlija, Kovacs, Wolf, Giacosa and Rischke,  PRD 87 (2013) 014011

→ tetraquarks !? → see later

Page 26: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

Attractive effects of pion cloud Scalar too large or ... too low!

Unquenching effects: Light mesons

18

CF, Williams, PRL 103 (2009), PRD 78 (2008)

Ebert, Faustov, Galkin, PRD 79 (2009) 114029 Parganlija, Kovacs, Wolf, Giacosa and Rischke,  PRD 87 (2013) 014011

→ tetraquarks !? → see later

Page 27: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

1.Introduction

2.Gluons, quarks and mesons

3.Pion cloud effects in baryons

4.Hadronic effects in (g-2)μ

5.Tetraquarks

−1=

−1−

+ pion cloud

+ pion cloud

Overview

19

Page 28: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

neglect irreducible three-body forces (three-gluon interaction !) approximate two-body interactions by RL-gluon exchange

one-parameter-model (MT) 64 tensor structures for nucleon: s, p, d - wave numerically expensive but manageable !

Eichmann, Alkofer, Krassnigg, Nicmorus, PRL 104 (2010)

Eichmann, PRD 84 (2011)

Sanchis-Alepuz ,Eichmann, Villalba-Chavez, Alkofer, PRD (2012)

Faddeev - equation

20

++= +

Page 29: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

neglect irreducible three-body forces (three-gluon interaction !) approximate two-body interactions by RL-gluon exchange

one-parameter-model (MT) 64 tensor structures for nucleon: s, p, d - wave numerically expensive but manageable !

Eichmann, Alkofer, Krassnigg, Nicmorus, PRL 104 (2010)

Eichmann, PRD 84 (2011)

Sanchis-Alepuz ,Eichmann, Villalba-Chavez, Alkofer, PRD (2012)

Faddeev - equation

20

++= +

Page 30: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

first covariant three-body calculations !

grosso modo: consistent description of mesons and baryons

masses dominated by s-waves

Baryon masses

21

Eichmann, Alkofer, Krassnigg, Nicmorus, PRL 104 (2010) Eichmann, PRD 84 (2011) Sanchis-Alepuz ,Eichmann, Villalba-Chavez, Alkofer, PRD (2012)

Page 31: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

missing pion cloud effects Eichmann, PRD 84 (2011) similar for axial form factors Eichmann and CF, Eur. Phys. J. A48 (2012) 9

Nucleon EM form factors

22

Passchier, Herberg, Zhu, Bermuth, Warren

Geis

Plaster/Madey

Riordan

Glazier

0

1

2

0 1 2 3

ҍ/ܯҍ

ҍ/ܯҍ

ݾ|(ҍ) ޗܩ| Anderson

Lachniet

Anklin

Lung

Rock

Kubon

00

1

2

3

1 2 3

ҍ/ܯҍ

ހ(ҍ) ޏܩ

ހ(ҍ) ޗܩ

Crawfordݾ(ҍ) ޏܩ

Paolone

Zhan

Punjabi

Ron

0.0

1.0

0.4

0.2

0.6

0.8

0.00

0.10

0.04

0.02

0.06

0.08

0 1 2 3 0 1 2 3 64 5ҍ/ܯҍ

Page 32: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

missing pion cloud effects in isovector moment κv no pion cloud effects in isoscalar moment κs

Eichmann, PRD 84 (2011)

Magnetic moments

23

Page 33: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

fix Λ by fπ, vary η s.t. fπ still ok effects of the order of 50-100 MeV missing: gluon self-interaction effects work in progress: mass splitting of Λ and Σ

Pion cloud effects in baryons

24

�(k2) = ⇤⇥7✓k2

�2

◆e��2

⇣k2

�2

+ �UV (k2)

M (G

eV)

0 0.1 0.2 0.3 0.4 0.5

Sanchis-Alepuz, CF, Kubrak, PLB 733 (2014) [1401.3183]

Page 34: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

fix Λ by fπ, vary η s.t. fπ still ok effects of the order of 50-100 MeV missing: gluon self-interaction effects work in progress: mass splitting of Λ and Σ

Pion cloud effects in baryons

24

�(k2) = ⇤⇥7✓k2

�2

◆e��2

⇣k2

�2

+ �UV (k2)

−1

=

−1

Y M

π

M (G

eV)

0 0.1 0.2 0.3 0.4 0.5

Sanchis-Alepuz, CF, Kubrak, PLB 733 (2014) [1401.3183]

Page 35: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

pion cloud does not change shape of nucleon: uniform skin pion cloud does affect sigma-term but still too small...

Pion cloud effects in baryons: structure

25

Nucleon Delta

s-wave p-wave d-wave s-wave p-wave d-wave f-wave

pure rainbow ladder

66 33 1 57 40 3 0,2

quark core 75 24 1 61 31 7 0,2

quark core plus

pion cloud 75 24 1 60 31 8 0,2

Page 36: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

Derive fermion-two-photon vertex consistent with gauge invariance free of kinematic singularities transverse part: on-shell nucleon Compton amplitude

Reproduce πΥΥ transition form factor on t-channel pole

Nucleon Compton scattering

26

Nonperturbative description of hadron-photon and hadron-meson scattering on quark-gluon basis

Eichmann and CF, PRD 85 (2012) 034015

Eichmann and CF, PRD 87 (2013) 036006

Technical/conceptual progress:

Next steps:Two-photon contributions to EM form factor Polarisabilities PANDA: pp ! ��

Page 37: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

1.Introduction

2.Gluons, quarks and mesons

3.Pion cloud effects in baryons

4.Hadronic effects in (g-2)μ

5.Tetraquarks

−1=

−1−

+ pion cloud

+ pion cloud

Overview

27

Page 38: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

Physics beyond the standard model (SM) ?

28

‘Anomalous’ magnetic moment of muon

Experiment:

aµ = (11 659 182.8± 4.9) · 10�10Theorie (SM):

Bennet et. al., Phys.Rev.D 73, 072003 (2006)

Hagiwara et.al., J.Phys.G G38 085003 (2011)

aµ = (11 659 208.9± 6.3) · 10�10

Discrepancy of three standard deviations...

al =gl � 2

2

�µ = gle~2mc

�S

Page 39: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

Hadronic contributions to aμ

29

Hadronic vacuum polarisation:

→ dominate theory error !

Light-by-light scattering:

aHV P (LO)µ = (692.1± 5.6) · 10�10

aLBLµ = (10.5± 2.6) · 10�10

aµ = (11 659 182.8± 4.9) · 10�10

Extracted from e+e- and τ-decay:

Model calculation:

Hagiwara et.al., J.Phys.G G38 085003 (2011)

J. Prades, E. de Rafael and A. Vainshtein, arXiv:0901.0306

Page 40: Dynamical Mass Generation in Hadrons and Leptons · Dynamical Mass Generation in ! Hadrons and Leptons 1 ... Quarks and gluons Hadrons CF, Luecker, ... Typical solution: h¯ i⇡(250MeV)3

Christian Fischer (University of Gießen) Dynamical mass generation / 46

Hadronic vacuum polarization

30

Need to determine quark propagator and quark-photon vertex:

Maris-Tandy model

Kernel K uniquely related to quark-DSE via axWTI

Bethe-Salpeter equation: meson mass and wave function

π, ρ,...

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

Quark-photon vertex and pion form factors

31

Q2

π π

Pion form factor:

Krassnigg, Schladming 2011; Maris, Tandy NPPS 161, 2006

Vector meson poles dynamically generated!

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

Results: Hadronic vacuum polarisation

32

Experiment:

DSE: (mπ)

(mρ)Goecke, CF, Williams, PLB 704 (2011)

ahad.(1)µ = (744.0± 2) · 10�10

ahad.(1)µ = (676.0± 2) · 10�10

ahad.(1)µ = (692.1± 5.6) · 10�10

Very reasonable agreement !

0.8 1 1.2 1.4 1.6M

ρ [GeV]

100

200

300

400

500

600

700

a µ [1

0-10 ]

RBC-UKQCD (Nf=2+1)Mainz CLS (Nf=2+1)Mainz CLS (Nf=2)ETMC (Nf=2)DSE (Nf=2+1)DSE (Nf=2)

10�10 u/d s c baµ 600 60 15 1

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

Hadronic contributions to aμ: LBL

33

Multi-scale problem NJL-model, Vector-meson dominance, largeNc

Lattice gauge theory: exploratory...

Dispersion theory: just started...

Bijnens, Knecht, Melnikov, Nyffeler, Pallante, Prades, de Rafael, Vainshtein...

Experiment:

Theorie (LBL):

Theorie (SM): aµ = (11 659 182.8± 4.9) · 10�10

aLBLµ = ( 10.5± 2.6) · 10�10

aµ = (11 659 208.9± 6.3) · 10�10

Challenge for non-perturbative QCD-calculation !

Blum, Izubuchi, ...

Colangelo, Vanderhaeghen,...

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

Hadronic Light-by-Light scattering

34

=

=

=

q

μμ

Dictated by gauge invariance!

Eichmann and CF, PRD 87 (2013) 3, 036006Goecke, CF, Williams, PRD 83 (2011) 094006

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

Hadronic Light-by-Light scattering

34

=

=

+...

. . .

. . .

...

... +...

. . . +...=

q

μμ

Dictated by gauge invariance!

Eichmann and CF, PRD 87 (2013) 3, 036006Goecke, CF, Williams, PRD 83 (2011) 094006

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

Hadronic Light-by-Light scattering

34

=

=

+...

. . .

. . .

...

... +...

. . . +...=

q

μμ

Dictated by gauge invariance!

Eichmann and CF, PRD 87 (2013) 3, 036006

No Double-Counting!

Goecke, CF, Williams, PRD 83 (2011) 094006

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

Hadronic Light-by-Light scattering

34

=

=

=

q

μμ

+...

+/ï +...

Quark-loop

Off-shell meson-exchange

Dictated by gauge invariance!

Eichmann and CF, PRD 87 (2013) 3, 036006Goecke, CF, Williams, PRD 83 (2011) 094006

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

Meson-exchange contribution

35

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

Meson-exchange contribution

35

12 Components

4 Components

2 Components

full momentum dependencies no free parameters

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

Fπγγ overall similar to VMD-models our value:

comparable with model calculations short distance constraints satisfied!

Meson exchange contribution to LBL

36

numerical: 0.2 systematic: 1.1

(a⇥,�,�0

µ )DSE = 8.1(1.3) · 10�10

Goecke, CF, Williams, PRD 83 (2011) 094006

⇥, �, �0

F(⇤,⇥,⇥0)��

F⇡��

see also Dorokhov, Broniowski, PRD 78 (2008) 073011

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

Quark-loop contribution to LBL

37

q

Numerically demanding due to superficial divergency Vertex contains 12 tensor structures

�µ =X

i=1,4

BCi Lµi +

X

i=1,8

Fi Tµi

gauge part WTI

transverse part vector-mesons

ENJL-model: (2.1± 0.3) · 10�10Bijnens, Pallante and Prades PRL 75, (1995)

DSE: bare vertex BC1 only BC1 + F1

(6.1± 0.2) · 10�10

(10.7± 0.3) · 10�10

CF, Goecke, Williams, EPJA 47 (2011) 28

Goecke, CF, Williams, PRD 87 (2013) 034013

(11.1± 0.2) · 10�10

Calculation not yet completed !

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

Hadronic LBL contributions - DSE + Models

38

Group Tool π quark-loop π SUM

BPP ENJL 8.5(1.3) 2.1(0.3) -1.9(1.3) 8.3(3.2)

HKS HLS 8.3(0.6) 1.0(1.1) -0.4(0.8) 8.9(1.6)

PdRV 11.4(1.3) 0,2 -1.9(1.3) 10.5(2.6)

GdR QM 6.8(0.3) 8.2 (0.7) 12.2 (1.8)(

GFW DSE 8.1(0.2) 10.7(0.3) 18.8(0.5)

BPP J. Bijnens, E. Pallante and J. Prades Phys. Rev. Lett. 75, 1447 (1995)

HKS M. Hayakawa, T. Kinoshita and A. I. Sanda, Phys. Rev. Lett. 75, 790 (1995)

PdRV J. Prades, E. de Rafael and A. Vainshtein, arXiv:0901.0306

GdR D. Greynat, E. de Rafael, JHEP 1207 (2012) 020

GFW T. Goecke, CF, R. Williams, Phys. Rev. D 87 (2013) 034013

numerical !systematic ?

preliminary!

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

Reliable theoretical error Reduce theoretical error down to 1-2 x 10-10 - level

Prospects for the future…

39

Goals:

Lattice gauge theory: increasing activities (cf. Lattice2014)

Dispersive approach: formalism outlined, experimental input needed

DSE/BSE: aim at LBL on 10-20% level

Tools:

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

Orthogonal basis: 136 elements (41 after gauge inv.) four-photon amplitude satisfies (exact within RL):

The four-photon amplitude

40

Quark-Compton-vertex

= +

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

basis constructed (128 elements) meson exchange contributions isolated and recovered

Quark-Compton-vertex

41

ߑΫ

ߑΪ

ߑ

߂

’’ ’’

= + +

Compton-vertex:

Quark-Loop

1PI

incl. “meson-exchange”

F⇡��

Next steps:calc. full quark-loop contribution calc. full 1PI contribution and systematically compare with

“off-shell meson approximation”

Eichmann and CF, PRD 87 (2013) 3, 036006

Eichmann, CF, Williams, work in progress

= +

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

1.Introduction

2.Gluons, quarks and mesons

3.Pion cloud effects in baryons

4.Hadronic effects in (g-2)μ

5.Tetraquarks

−1=

−1−

+ pion cloud

+ pion cloud

Overview

42

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

Diquark-Antidiquark Meson-Meson

Tetraquarks from DSEs/BSEs

Quark configurations:Hadrons

43

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

Diquark-Antidiquark Meson-Meson

Tetraquarks from DSEs/BSEs

Light meson sector: Scalars!

Quark configurations:Hadrons

43

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

Diquark-Antidiquark Meson-Meson

Tetraquarks from DSEs/BSEs

Light meson sector: Scalars!

Quark configurations:Hadrons

43

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

Input: Covariant Quark-Gluon interaction - Maris-Tandy model

Mesons and Diquarks via Bethe-Salpeter equation

Strategie II: Tetraquark-BSEs

44

Meson Diquark

Dynamical decision between Meson- and Diquark-configurations

−1=

−1− �(k2) = ⇤⇥7

✓k2

�2

◆e��2

⇣k2

�2

+ �UV (k2)

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

Results: scalar tetraquarks

Pion-Pion-contribution dominates ! m(0++ ) = 403 MeV

Narrow scalar cccc: m(0++) = 5.3 ± (0.5) GeV__

Heupel, Eichman, CF, PLB 718 (2012) 545-549

} f0(500)

45

up/down charm

see also Caprini, Colangelo and Leutwyler, PRL. 96 (2006) 132001 Parganlija, Kovacs, Wolf, Giacosa and Rischke,  PRD 87 (2013) 014011

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

Light and heavy mesons Explore quark-gluon interaction: consequences for pheno current work: masses and EM-properties of charmonia

Baryons form factors in rainbow-ladder masses including pion cloud effects current work: multipletts, excited states and Compton scattering

Hadronic contributions to (g-2)μ HVP: in agreement with lattice and experiment current work: determine LBL beyond simple models

Tetraquarks σ as tetraquark in π-π-configuration current work: include other channels; four body equation

Summary and outlook

46

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

Backup Slides

47

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

technically demanding

natural scale: MΔ

compare two RL models

no pion cloud yet...

Delta and Omega form factors

48

Sanchis-Alepuz, Williams and Alkofer, PRD 87 (2013) 095015.

Lattice: Alexandrou et al. NPA 825 (2009) 115, PoS CD 09 (2009) 092

Oblate shape of Δ and Ω agreement with lattice

Results:

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

Eichmann, PRD 84 (2011)

Form factor ratios

49

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

GT-relation satisfied missing pion cloud mA = 1.28 (6) GeV sizeable deviations from dipole form

Axial and pseudoscalar form factors

50

ҍ/ܯҍ0.0 0.5 1.0 1.5 2.0

0

5

10

15

(ҍ) ޘޘܩ Phenomenology

DSE/Faddeev

Lattice: Alexandrou ‘07(MeV 411 = )

ҍ/ܯҍ0.0 0.5 1.0 1.5 2.0

0.0

0.4

0.6

0.2

0.8

1.0

1.2

1.4

(ҍ) ދܩETMC (Alexandrou ‘11)

RBC/UKQCD (Yamazaki ‘09)

Lattice:

MeV 329 = MeV 416 =

MeV 298 =

0.0 0.1 0.2 0.3 0.4 0.5

ҍ[ҍܩ] 0.0 0.1 0.2 0.3 0.4 0.5

ҍ[ҍܩ] 0.8

1.0

1.2

1.4

0.0

1.0

0.2

0.4

0.6

0.8ދ [] ދݎ

RBC/UKQCD (Yamazaki ‘09)

ETMC (Alexandrou ‘11)QCDSF (Ali Khan ‘06)

Lattice:

LHPC (Bratt ‘10)

ChPT (Procura ‘07)DSE/Faddeev

Eichmann, CF, EPJA 48 (2012) 9

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Christian Fischer (University of Gießen) Dynamical mass generation / 46

Comparison ENJL vs. DSE

51

NJL: no momentum dependence in quark

NJL: no relative momenta in quark-photon vertex

artificial suppression of quark-loop contribution

�µ(P, p, p.P )

DSE (BC1 + F1): Nf = 2 Nf = 4

ENJL-model: 6.0 · 10�10 � 2.1 · 10�10

10.0 · 10�10 � 9.6 · 10�10

11.1 · 10�10 � 10.7 · 10�10

Goecke, CF, Williams, Phys. Rev. D 87 (2013) 034013