Upload
others
View
18
Download
0
Embed Size (px)
Citation preview
Christian S. Fischer
Justus Liebig Universität Gießen
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
Glueballs and tetraquarks from Dyson-Schwinger equations
1
with Gernot Eichmann, Walter Heupel and Helios Sanchis-Alepuz
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
1.Introduction
2.Gluons and glueballs
3.Quarks and mesons
4.Tetraquarks
−1=
−1−
Overview
2
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
States in the light and heavy quark energy regions
Most calculations quenched
Preliminary unquenched results: larger masses
Glueballs
3
Morningstar and Peardon, PRD 60 (1999) 034509 Y.~Chen et al., PRD 73 (2006) 014516
Gregory et al., JHEP 1210 (2012) 170
DSE:
Meyers, Swanson, PRD 87 (2013) 3, 036009 Sanchis-Alepuz, CF, Kellermann and von Smekal, PRD 92 (2015) 3, 034001
Lattice:
structural information
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
Tetraquarks in the light meson sector
4
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
Tetraquarks in the light meson sector
Light meson sector: Scalars!
4
f0(980) ! ��,KK
a0(980) ! ��,KK
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
Tetraquarks in the light meson sector
Light meson sector: Scalars!
4
f0(980) ! ��,KK
a0(980) ! ��,KK
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
Related to details of underlying QCD forces between quarks
Tetraquark candidates in charmonium region
5
Internal structure ??
Wolfgang Gradl, BESIII, St Goar 2015
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
Basic idea: solve four-body equation without any assumption on internal clustering
Key elements: quark propagator and interaction kernels
Tetraquarks from the four-body interaction
6
Exact equation:
Two-body interactions Three- and four-body interactions
Kvinikhidze & Khvedelidze, Theor. Math. Phys. 90 (1992)
Heupel, Eichman, CF, PLB 718 (2012) 545-549
Eichman, CF, Heupel, 1508.07178
+ perm.
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
1.Introduction
2.Gluons and glueballs
3.Quarks and mesons
4.Tetraquarks
−1=
−1−
Overview
7
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
QCD in covariant gauge
8
Quarks, gluons and ghosts
The Goal: gauge invariant information in a gauge fixed approach.
Landau gauge propagators in momentum space,
DGluonµ� (p) =
✓�µ� � pµp�
p2
◆Z(p2)
p2
SQuark(p) = Zf (p2)[�ip/ +M(p2)]�1
ZQCD =
ZD[ , A, c] exp
(�Z
d
4x
✓¯
(iD/ �m) � 1
4
(F
aµ⌫)
2
+gauge term + c(�@D)c)
)
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
spacelike momenta: excellent agreement with lattice
spectral function: positivity violations
Landau gauge gluon propagator
9
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,...
0 1 2 3 4 5p [GeV]
0
0.5
1
1.5
Z(p2 )
Sternbeck et. al. (2006)DSE
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
Mixing of two-gluon amplitudes with ghost-antighost Probes analytical structure of gluons and ghosts
Glueballs from DSE/BSEs
10
Results: M(0++) = 1.64GeV
M(0�+) = 4.53GeV
Sanchis-Alepuz, CF, Kellermann and von Smekal, PRD 92 (2015) 3, 034001
ghost do not contribute !
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
1.Introduction
2.Gluons and glueballs
3.Quarks and mesons
4.Tetraquarks
−1=
−1−
Overview
11
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
DSEs and Bethe-Salpeter equation
12
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) Glueballs and tetraquarks from DSEs / 21
Strategie I: Model for quark-gluon interaction
13
Maris, Roberts,Tandy, PRC 56 (1997), PRC 60 (1999)�(k2) = ⇤⇥7✓k2
�2
◆e��2
⇣k2
�2
⌘
+ �UV (k2)
fix from fπ; small dependence of many results on
masses mu=md, ms, mc, from π, Κ, J/ψ Renormalizable and momentum dependent ! Qualitatively similar to results from explicit calculation
⌘⇤
CF, Maas, Pawlowski, Annals Phys. 324 (2009) 2408. Williams, EPJA 51 (2015) 5, 57.
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
Quark mass: flavor dependence
14
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)30.0 1.0 2.0 3.0 4.0p [GeV]
0.0
0.1
0.2
0.3
0.4
M(p
) [G
eV]
Lattice: quenched Lattice: unquenched (N
f=2+1)
DSE: quenchedDSE: unquenched (N
f=2)
CF, Nickel, Williams, EPJ C 60 (2009) 47
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
Quark mass: flavor dependence
14
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
Not directly measu
rable !
0.0 1.0 2.0 3.0 4.0p [GeV]
0.0
0.1
0.2
0.3
0.4
M(p
) [G
eV]
Lattice: quenched Lattice: unquenched (N
f=2+1)
DSE: quenchedDSE: unquenched (N
f=2)
CF, Nickel, Williams, EPJ C 60 (2009) 47
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
good channels: 1--,2++, 3--,... acceptable channels: 0-+ clear deficiencies in other channels: missing spin-structure
Charmonium spectrum
15
CF, Kubrak, Williams, EPJA 51 (2015) Hilger et al. PRD 91 (2015)
��������
���������
�� �����
�������
����� ��
���������
������
���� ���������
�������
�������������
�������
��������
�������
���������������
�������
������
��
��
��
��
�
���
�� ��� �� ��� ��� ��� ��� ��� �� �� �� �� �� ���
� ��!"!�����
#$%&$'(!)$(*�(
DSE/BSE
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
good channels: 1--,2++, 3--,... acceptable channels: 0-+ clear deficiencies in other channels: missing spin-structure
Charmonium spectrum
15
CF, Kubrak, Williams, EPJA 51 (2015) Hilger et al. PRD 91 (2015)
��������
���������
�� �����
�������
����� ��
���������
������
���� ���������
�������
�������������
�������
��������
�������
���������������
�������
������
��
��
��
��
�
���
�� ��� �� ��� ��� ��� ��� ��� �� �� �� �� �� ���
� ��!"!�����
#$%&$'(!)$(*�(
DSE/BSE
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
good channels: 1--,2++, 3--,... acceptable channels: 0-+ clear deficiencies in other channels: missing spin-structure
Charmonium spectrum
15
CF, Kubrak, Williams, EPJA 51 (2015) Hilger et al. PRD 91 (2015)
��������
���������
�� �����
�������
����� ��
���������
������
���� ���������
�������
�������������
�������
��������
�������
���������������
�������
������
��
��
��
��
�
���
�� ��� �� ��� ��� ��� ��� ��� �� �� �� �� �� ���
� ��!"!�����
#$%&$'(!)$(*�(
DSE/BSE
see talk of T. Hilger
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
1.Introduction
2.Gluons and glueballs
3.Quarks and mesons
4.Tetraquarks
−1=
−1−
Overview
16
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
Solving the four-body equation
17
Input: Non-perturbative quark, quark-gluon interaction
�(k2) = ⇤⇥7✓k2
�2
◆e��2
⇣k2
�2
⌘
+ �UV (k2)
+ perm.
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
good approximation: keep s-waves only; 16 tensor structures
Structure of the amplitude
18
Scalar tetraquark:
9 Lorentz scalars (built from P,p,q,k)
256 tensor structures
(scalar tetra)
�(P, p, q, k) =X
i
fi(s1, ..., s9)⇥ �i(P, p, q, k)⇥ color ⇥ flavor
3⌦ ¯
3, 6⌦ ¯
6 or
1⌦ 1, 8⌦ 8
p q k P
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
Singlet: , carries overall scale Doublet:
Four-body equation:
19
Organise Dirac-Lorentz-tensors into multiplets of S4
S0 = (p2 + q2 + k2)/4
a =p3(q2 � p2)/(4S0); s = (p2 + q2 � 2k2)/(4S0)
meson poles
diquark pole
Two tripletsEichman, CF, Heupel, arXiv: 1505.06336
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
Different levels of approximations:
Bound state masses
20
MTetra[MeV ]0 1500
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
Different levels of approximations:
Bound state masses
20
Singlet only
MTetra[MeV ]0 1500
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
Different levels of approximations:
Bound state masses
20
Singlet onlySinglet +Triplet I
MTetra[MeV ]0 1500
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
Different levels of approximations:
Bound state masses
20
Singlet onlySinglet +Triplet I
MTetra[MeV ]0 15001190
Singlet +Triplet II
Bound state of four massive quarks
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
Different levels of approximations:
Bound state masses
20
Singlet onlySinglet +Triplet I
MTetra[MeV ]0 15001190
Singlet +Triplet II
Bound state of four massive quarks
Singlet + Doublet
350
Two-pion resonance
4m2⇡ �M2
S0
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
Resonance becomes bound state for large mq Dynamical decision: meson clusters, not diquarks Results:
Mass evolution of tetraquark
21
qualitatively similar to two-body frameworkHeupel, Eichman, CF, PLB 718 (2012) 545-549
Eichman, CF, Heupel, 1508.07178
mssss ⇠ 1.5GeV
mcccc ⇠ 5.7GeV
m� ⇠ 350MeV
m ⇠ 750MeV
ma0,f0 ⇠ 1080MeV
3
4
6
5
2
1
00.0
0.5
1.0
1.5
0.02 4 6 8 10 0.2 0.4 0.6 0.8
[ܯ] [ܩ]
[ܩ] ܯ
ߪߢ
ҝ/ҝ
strange
4-body BSE
2-body BSE
22
charm
MTetra[GeV]
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
Outlook: heavy-light systems
22
Dynamical decision of most important clustering!
heavy-light diquarks
hadro-charmoniummeson-molecules
Dynamical situation in S4-doublet:
cq � cq
cq � cq cc� qq
Christian Fischer (University of Gießen) Glueballs and tetraquarks from DSEs / 21
Mass gap in YM-theory: scalar glueball mass
Tetraquarks dominated by internal meson-meson configurations
Dynamical description of σ as π-π resonance
Bound cccc-tetraquark....
Summary and outlook
23
Improve numerical framework: precision, systematics
Implement quark mass dependence of quark-gluon interaction
Explore heavy-light systems
Summary
Outlook
Williams, EPJA 51 (2015)