Consequences of a dressed quark-gluonvertex in heavy-light mesons

Marıa Gomez-Rocha

in collaboration withT. Hilger, C. Popovici, A. Krassnigg

P25121-N27

Jefferson Lab, May 21, 2014,

Motivation

• QCD and the structure of hadrons

? the role of the quark masses in hadron spectrum and structure

• Heavy-light mesons

? Dynamics of the heavy-light meson controlled by dynamics of theheavy quark

? For mesons with mQ ∼ mM andmq

mQ∼ 0: some conditions are

provided by “heavy-quark symmetry”

Neubert Phys. Rept. 245 (1994)

• Heavy-quark symmetry as a guidance

? Use constraints of heavy-quark symmetry to check the correctbehaviour of the mass

? Similar procedure within point-form Hamiltonian dynamics using aconstituent quark model: fully relativistic treatment of light quark isnecessary

Gomez-Rocha, Schweiger PRD 86 (2012)

• Limitations of BSE/DSE description of hadrons when using therainbow-ladder truncation approach

? In particular: difficulties to describe heavy-light mesons

The main goal of this work...Based on previous studies of consequences of a dressed quark-gluonvertex in qq mesons...

Bender, Detmold, Thomas, Roberts, PRC 65 (2002)Bhagwat, Holl, Krassnigg, Roberts, Tandy PRC 70 (2004)

Use the same interaction model:

Munczek and Nemirovsky PRD 28, 181 (1983)

• sufficiently simple → allow to compute an arbitrary number ofiterations

• sufficiently realistic → important features in common with QCD: e.g.confinement, dynamical chiral symmetry breaking

... attempt to generalize to qQ mesons

... ask:

Which kind of consequences do corrections to the bare vertex in heavy-lightmesons have?

or...

Which kind of heavy-light physics are we “truncating” using RLapproximation?

Dyson-Schwinger/Bethe-Salpeter approachThe interaction model

DSE:

S−1(p) = i/p+ 1mq +

∫d4q

(2π)4g2Dµν(p− q)λ

a

2γµS(q)Γaν(q; p)

Interaction model : Dµν(k) := 43δµν(2π)4G2δ4(k)

Munczek and Nemirovsky PRD 28, 181 (1983)

DSE becomes an algebraic equation

⇒ Solvable at every truncationorder: n = 0, 1, 2, 3...

⇒ Analytical solution in RL

⇒ Solvable in the limit n→∞(fully dressed)

figure adapted from Bender et al. PRC 65 2002

ΓCµ,n+1(p) = −CγνS(p)ΓCµ,n(p)S(p)γν

Solutions to the DSE for n loopsfor mu = 10 MeV: Bhagwat et al. PRC 70 (2004)

-2 -1 0 1 2

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

AHp2 L

-2 -1 0 1 20.0

0.5

1.0

1.5

MHp2 L

n=¥

n=4

n=3

n=2

n=1

n=0

for ms = 0.166 GeV: C = 0.51

-4 -2 0 2 4

1.0

1.5

2.0

2.5

3.0

3.5

4.0

AHp2 L

-4 -2 0 2 4

0.5

1.0

1.5

2.0

2.5

MHp2 L

n=¥

n=4

n=3

n=2

n=1

n=0

Quark propagator: S−1(p) = i/pA(p2) +B(p2); M(p2) := B(p2)/A(p2)

Solutions to the DSE for n loopsfor mc = 1.33 GeV: C = 0.51

-15 -10 -5 0 5 10

1.0

1.5

2.0

2.5

AHp2 L

-15 -10 -5 0 5 10

2.5

3.0

3.5

4.0

MHp2 L

n=¥

n=4

n=3

n=2

n=1

n=0

for mb = 4.62 GeV:

-80 -60 -40 -20 0

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6AHp2 L

-80 -60 -40 -20 0

7.0

7.5

8.0

8.5

9.0

9.5

10.0

MHp2 L

n=¥

n=4

n=3

n=2

n=1

n=0

Quark propagator: S−1(p) = i/pA(p2) +B(p2); M(p2) := B(p2)/A(p2)

Dyson-Schwinger/Bethe-Salpeter approachThe interaction model

BSE:

ΓM (p;P ) =

∫d4q

(2π)4g2[K(p; q;P )]S(q+)ΓM (q;P )S(q−)︸ ︷︷ ︸

χM (q;P )

Γaν(q; p)

p+ = ηP, p− = (η − 1)P

Use an appropriate kernelthat guarantees validity of Ward-Takahashi identities

Bender et al. PRC65(2002)

Generalize to mq 6= mq and use our MN model to simplify the calculation

Dyson-Schwinger/Bethe-Salpeter approachThe interaction model

Generalize to mq 6= mq

ΓM (p;P ) =

∫ Λ

q

Dµν(p− q) la 1

2

[γµχM (q;P ) la Γν(q−, p−) + ΛaMν(q, p;P )S(q−)γµ

+la Γν(q+, p+)χM (q;P )γµ + γµS(q+) ΛaMν(q, p;P )],

figure adapted from Bender et al. PRC 65 (2002)

Derive recursion relation for ΛaMν(q, p;P ) for mq 6= mq

⇒ Corrections are more involve than for mq = mq

Search for bound-state solutions...

• Pseudoscalar mesons: qq, QQ, qQ

• Vector mesons: qq, QQ, qQ

• Scalars → no solution

• Axial vectors → no solution

Solutions to the BSEFirst impression for qq systems

The pion

n=0 n=1 n=2 n=3 n=4 n=¥

0.13

0.14

0.15

0.16

m HGeVL

Π C=0.51

n = # of loops

Now: check η-dependence, since this is an issue for unequal-massconstituents(see also original MN-paper, η was used as fitting parameter).

η =p+

P, (η − 1) =

p−P

Why η-dependence?

Because MN omits 2 of 4 covariants in BS amplitude → this breaks Lorentzcovariance: systematic error < 0.8%

n=0 n=1 n=2 n=3 n=4 n=¥

0.13

0.14

0.15

0.16

m HGeVL

Π C=0.51

0.2 0.4 0.6 0.8 1.0Η

0.135

0.140

0.145

m HGeVL

Correction to RL ∼ 8% for η = 0.5

Heavy quarkonia QQ

Charmonium Bottomonium

n=0 n=1 n=2 n=3 n=4 n=¥

2.8

3.0

3.2

3.4

m HGeVL

Ηc c=0.51

n=0 n=1 n=2 n=3 n=4 n=¥9.2

9.3

9.4

9.5

9.6

9.7

9.8mHGeVL

Ηb C=0.51

Correction to RL ∼ 1.6% for η = 0.5Error due to η-dep < 6%

Correction to RL ∼ 0.2% for η = 0.5Error due to η-dep < 1.5%

η-dep increases with increasing n

First check for mesons with mq 6= mq:

Kaon

n=0 n=1 n=2 n=3 n=4 n=¥

0.35

0.40

0.45

0.50

0.55

m HGeVL

K C=0.51

0.2 0.4 0.6 0.8 1.0Η

0.40

0.42

0.44

0.46

0.48

m HGeVL

Correction to RL ∼ 7.2% for η = 0.5Error due to η-dep < 10%

qQ-mesons

n=0 n=1 n=2 n=3 n=4 n=¥

1.6

1.8

2.0

2.2

m HGeVL

D C=0.51

n=0 n=1 n=2 n=3 n=4 n=¥

4.8

5.0

5.2

5.4

m HGeVL

B C=0.51

Correction to RL ∼ 6.3% for η = 0.75Error due to η-dep < 15%

mfullD = 1.68 GeV

Correction to RL ∼ 3.6% for η = 0.95Error due to η-dep < 12%

mfullB = 5.08 GeV

η-dep decreases with increasing n

Summary

• Interest in qQ meson properties, phenomenology and structure fromQCD

• Showed you an approach to these problems within DSE/BSEformalism

• Based on previous works, used an interaction model where

? Very heavy quarks and mesons can be studied numerically (evenheavy-quark limit)

? Effects beyond the popular RL truncation of DSE/BSE can bestudied systematically and quantitatively

π ∼ 8%ηc ∼ 1.6%ηb ∼ 0.2%

K ∼ 7.2 %D ∼ 6.3 %B ∼ 3.6 %

BUT: η-dependence must be better understood

Conclusions and Outlook

• Setup established previously was adapted to investigate qq, QQ, qQmeson

• Corrections to the RL truncation are more complicated in qQ mesonsthan in qq mesons

• Model artifacts have to be taken into account and kept under control(η-dependence)

• Heavy-light meson masses computed from the given model andparameters are reasonable

• Check heavy-quark symmetry predictions: relations between vectorand pseudoscalar mesons in extreme case mQ � mq

• Decay constants

Thank you!

Conclusions and Outlook

• Setup established previously was adapted to investigate qq, QQ, qQmeson

• Corrections to the RL truncation are more complicated in qQ mesonsthan in qq mesons

• Model artifacts have to be taken into account and kept under control(η-dependence)

• Heavy-light meson masses computed from the given model andparameters are reasonable

• Check heavy-quark symmetry predictions: relations between vectorand pseudoscalar mesons in extreme case mQ � mq

• Decay constants

Thank you!