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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
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
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
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)
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
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:
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
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
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
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
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)
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)
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)
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)
Christian Fischer (University of Gießen) Dynamical mass generation / 46
Phenomenology from Maris-Tandy interaction
12
Slide from Pieter Maris
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
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
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)
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,...
Christian Fischer (University of Gießen) Dynamical mass generation / 46
Strategy II: Pion effects in quark-gluon interaction
17
quark-gluon vertex:
quark:
= + + + +
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
+...
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
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
π
Bethe-Salpeter equation:
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
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
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
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
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
++= +
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
++= +
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)
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
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
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]
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]
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
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 ! ��
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
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
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
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,...
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!
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
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,...
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
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
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
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
Christian Fischer (University of Gießen) Dynamical mass generation / 46
Meson-exchange contribution
35
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
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
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 !
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!
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:
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
= +
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
= +
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
Christian Fischer (University of Gießen) Dynamical mass generation / 46
Diquark-Antidiquark Meson-Meson
Tetraquarks from DSEs/BSEs
Quark configurations:Hadrons
43
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
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
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)
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
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
Christian Fischer (University of Gießen) Dynamical mass generation / 46
Backup Slides
47
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:
Christian Fischer (University of Gießen) Dynamical mass generation / 46
Eichmann, PRD 84 (2011)
Form factor ratios
49
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
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