(Light) tetraquarks in a Dyson-Schwinger, Bethe-Salpeter approach - - PowerPoint PPT Presentation

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(Light) tetraquarks in a Dyson-Schwinger, Bethe-Salpeter approach 1 P. C.WallboA, G. Eichmann, C. S. Fischer, W. Heupel 02.06.17 P.C. WallboA, FAIRNESS 1 Contents Physics: The scalar mesons CalculaRons: How the method works

SLIDE 1

(Light) tetraquarks in a Dyson-Schwinger, Bethe-Salpeter approach

P. C.WallboA, G. Eichmann, C. S. Fischer, W. Heupel

02.06.17 P.C. WallboA, FAIRNESS 1

SLIDE 2

Physics: The scalar mesons
CalculaRons: How the method works
Outlook

02.06.17 P.C. WallboA, FAIRNESS 2

SLIDE 3

Physics: The scalar mesons
CalculaRons: How the method works
Outlook

02.06.17 P.C. WallboA, FAIRNESS 3

SLIDE 4

Light mesons: MulRplet order

PDG

JP C

u-, d-, s-quarks

02.06.17 P.C. WallboA, FAIRNESS 4

SLIDE 5

MulRplet order

PDG

Quark models

(−1)L+1

02.06.17 P.C. WallboA, FAIRNESS 5

SLIDE 6

MulRplet order

PDG

02.06.17 P.C. WallboA, FAIRNESS 6

SLIDE 7

Masses ...

M [GeV] 1 1 1 1 2

02.06.17 P.C. WallboA, FAIRNESS 7

SLIDE 8

... and widths

M [GeV] ??? OZI? S-quarks

02.06.17 P.C. WallboA, FAIRNESS 8

SLIDE 9

Are they tetraquarks?

M [GeV]

Jaffe, RJ: Physical Review D, 15(1):267, 1977

S-quarks

02.06.17 P.C. WallboA, FAIRNESS 9

SLIDE 10

Results in BSE/DSE

Eichmann, Fischer, Heupel Phys. LeF. B753:282-287, 2016

02.06.17 P.C. WallboA, FAIRNESS 10

SLIDE 11

CalculaRon roadmap

Quark propagator (DSE)
Bound states (BSE) for mesons, baryons,

tetraquarks…

02.06.17 P.C. WallboA, FAIRNESS 11

SLIDE 12

Quark DSE

p q p p k

Self energy Non interacRng Full propagator

S−1(p) = S−1

0 (p) +

Z

Dµν(k)γµS(q)Γν(p, q)

02.06.17 P.C. WallboA, FAIRNESS 12

SLIDE 13

Quark DSE

S−1

0 (p) = −ipµγµ + m

S(q)

S−1(p) = −ipµγµA(p2) + B(p2)

?? ??

Γµ(q, k)

Dµν = ✓ δµν − kµkν k2 ◆ Z(k2) k2

?? ??

S−1(p) = S−1

0 (p) +

Z

Dµν(k)γµS(q)Γν(p, q)

02.06.17 P.C. WallboA, FAIRNESS 13

SLIDE 14

Quark DSE

S−1

0 (p) = −ipµγµ + m

S(q)

S−1(p) = −ipµγµA(p2) + B(p2)

?? ?? Rainbow-Ladder + Maris-Tandy In RL + MT only A,B unknown

> solve for A,B
> quark propagator

∝ γµαeff(k2) Γµ(q, k)

Dµν = ✓ δµν − kµkν k2 ◆ Z(k2) k2

02.06.17

P.C. WallboA, FAIRNESS 14

SLIDE 15

Quark DSE, results:

M = B A

b c s u/d chiral 1 1000 1 10 [GeV] [GeV]

02.06.17 P.C. WallboA, FAIRNESS 15

SLIDE 16

Meson BSE setup

Quark propagators from DSE cut Self energy -> scaAering kernel

Γ = K · Γ

Γab(p, P) = Z

Kad,cb(p, q, P) [S(q+)Γ(q, P)S(−q−]cd

02.06.17 P.C. WallboA, FAIRNESS 16

SLIDE 17

Meson BSE solve it

Γ = K · Γ λ(M)

Γab(p, P) = Z

Kad,cb(p, q, P) [S(q+)Γ(q, P)S(−q−]cd

Γ =

i=1

fi(Ω)τi ⊗ color ⊗ flavor

Ω(p2, p · P)

02.06.17 P.C. WallboA, FAIRNESS 17

SLIDE 18

Tetraquark BSE(s)

2-body approximaRon full equaRon

Heupel, Eichmann Fischer

Phys. LeF. B718:545-549, 2012

Eichmann, Fischer, Heupel

Phys. LeF. B753:282-287, 2016

WallboF, tetraquarks in a DSE, BSE approach

A. Khvedelidze, A. Kvinikhidze, Theor. Math.
Phys. 90, 62 (1992)

02.06.17 P.C. WallboA, FAIRNESS 18

SLIDE 19

+ perm.

Tetraquark BSE, setup

cut Quark propagators from DSE

Γ = K · Γ λ(M)

02.06.17 P.C. WallboA, FAIRNESS 19

SLIDE 20

+ perm.

Tetraquark BSE, setup

cut Quark propagators from DSE S-waves only! S4

Ω(p2, q2, k2, p · q, ...) → Ω(S0, D, T1, T2)

02.06.17 P.C. WallboA, FAIRNESS 20

SLIDE 21

Importance of mulRplets

:= set to constant value

D important, T not!

02.06.17 P.C. WallboA, FAIRNESS 21

SLIDE 22

Why is D important?

Phase space restricted to triangle!

√ 3q2 − p2

p2 + q2 − 2k2

02.06.17 P.C. WallboA, FAIRNESS 22

SLIDE 23

Poles!

f1( √ 3q2 − p2, p2 + q2 − 2k2) = f1(D)

Γ =

i=1

fi(Ω)τi ⊗ color ⊗ flavor

τ1 = γ5 ⊗ γ5

02.06.17 P.C. WallboA, FAIRNESS 23

SLIDE 24

Poles!

f1( √ 3q2 − p2, p2 + q2 − 2k2) = f1(D)

Moving poles! Exhibits poles! π π ??

σ

f1 = 1 m2

π + (p1 + p3)2 ·

1 m2

π + (p2 + p4)2

02.06.17 P.C. WallboA, FAIRNESS 24

SLIDE 25

Summary of results

Light scalars, because

dynamically generated meson poles dominate

Tetraquark is

resonance above two pion threshold

02.06.17 P.C. WallboA, FAIRNESS 25

SLIDE 26

Outlook

Technical improvements

– Analysis of higher parRal waves – More systemaRc studies of results – Rigorous calculaRon of decay properRes

Establish method for heavy-light systems

σ → ππ

02.06.17 P.C. WallboA, FAIRNESS 26

SLIDE 27

Outlook

02.06.17 P.C. WallboA, FAIRNESS 27

SLIDE 28

Outlook

Establish method for

heavy-light systems

And other quantum

numbers

Esposito et al., I Journal Mod Phys A 30, 2014

02.06.17 P.C. WallboA, FAIRNESS 28

SLIDE 29

Outlook (approach)

Hard calculaRon, limited to (almost) equal quark masses

02.06.17 P.C. WallboA, FAIRNESS 29

SLIDE 30

Outlook (approach)

“Physical construcRon” + poles Now back to s-wave truncated amplitude

Γ =

i=1

fi(Ω)τi ⊗ color ⊗ flavor

Γq¯

qq¯ q = Γπ ⊗ Γπ + Γρ ⊗ Γρ + ...

1− ⊗ 1− = 0+

PW, Tetraquarks in a DSE/BSE approach

02.06.17 P.C. WallboA, FAIRNESS 30

SLIDE 31

Summary

Strength:

– Once truncaRon is set, no further input necessary – System chooses configuraRon dynamically (π-π) – Growing computaRon power -> full soluRon possible

Challenges:

– heavy-light systems – Decays (extrapolaRon) – Mixing – Redo lots of coding

02.06.17 P.C. WallboA, FAIRNESS 31

SLIDE 32

02.06.17 P.C. WallboA, FAIRNESS 32

SLIDE 33

Also interesRng

02.06.17 P.C. WallboA, FAIRNESS 33

SLIDE 34

Why D? Poles!

02.06.17 P.C. WallboA, FAIRNESS 34

SLIDE 35

Tetraquarks in a DSE BSE approach

WallboF, tetraquarks in a BSE/DSE approach

02.06.17 P.C. WallboA, FAIRNESS 35

SLIDE 36

Tetraquarks in a DSE BSE approach

sRll strong doublet D

dependence of results

Problem: Symmetries!

WallboF, tetraquarks in a BSE/DSE approach

02.06.17 P.C. WallboA, FAIRNESS 36

SLIDE 37

BSE eigenvalue/ extrapolaRon

02.06.17 P.C. WallboA, FAIRNESS 37

SLIDE 38

BSE eigenvalue/ extrapolaRon

02.06.17 P.C. WallboA, FAIRNESS 38

SLIDE 39

BSE eigenvalue/ extrapolaRon

Treshold !!

02.06.17 P.C. WallboA, FAIRNESS 39