Quarkonia and heavy-light mesons in a covariant
quark model
Sofia Leitão
CFTP, University of Lisbon, Portugal
in collaboration with:
Alfred Stadler, M. T. Peña and Elmar P. Biernat
HUGS, JLab, USA
June, 2015
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A unified model for all 𝑞 𝑞 mesons
Much important work was done on meson structure:
 Cornell-type potential models (Isgur and Godfrey, Spence and Vary, etc.)
But: nonrelativistic (or “relativized”); structure of constituent quark and relation to
existence of zero-mass pion in chiral limit not addressed
 Dyson-Schwinger approach (C. Roberts et al.)
But: Euclidean space; only Lorentz vector confining interaction
 Lattice QCD (also Euclidean space), EFT, Bethe-Salpeter, Light-front, Point-form, …
Our objectives:
 Construct a model to describe all 𝑞 𝑞-type mesons
 Covariant framework (CST) - light quarks require relativistic treatment
Work in Minkowski space (physical momenta)
 Quark self-energy from 𝑞 𝑞 interaction kernel (consistent quark mass function)
 Chiral symmetry: massless pion in chiral limit of vanishing bare quark mass
 Calculate meson spectrum and bound state vertex functions (wave functions)
 Pion elastic and transition form factors
 Learn about confining interaction (scalar vs. vector, etc.)
HUGS, JLab, USA
June, 2015
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2
CST main idea
interaction kernel
𝑀
scattering problem
𝜈
=
+ 𝜈
scattering amplitude
𝑀
 If kernel 𝒱 = ∑ (all 1PI diagrams) →
exact result for 𝑀
 Usual truncation: ladder approximation
 Cancelation theorem: 𝜙 3 - theory
2
1
+
=
[F.Gross, PR186, 1969]
[F.Gross, Relativistic Quantum Mechanics and Field Theory, 2004]
cancellation in all orders and exact in heavy-mass limit !
x
particle 1 on mass-shell: 𝑘12 = 𝑚12
A “good way” to sum the contribution of all ladder + crossed ladder diagrams is to use the
approximation of just 1 ladder diagram with the one particle on its mass-shell.
HUGS, JLab, USA
June, 2015
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3
Covariant two-body bound-state equation
Start from the Bethe-Salpeter (BS) equation
𝑑4𝑘
Γ𝐵𝑆 𝑝, 𝑃 = 𝑖
𝒱 𝑝, 𝑘; 𝑃 𝑆1 (𝑘1 )Γ𝐵𝑆 𝑘, 𝑃 𝑆2 (𝑘2 )
4
2𝜋
1
𝛴𝑖 𝑘𝑖 = 𝐴𝑖 𝑘𝑖2 + 𝑘𝑖 𝐵𝑖 (𝑘𝑖2 )
𝑆𝑖 𝑘𝑖 =
𝑚0𝑖 − 𝑘𝑖 + Σ𝑖 𝑘𝑖 − 𝑖𝜖
𝑃 = 𝑘1 − 𝑘2
𝑘 = 𝑘1 + 𝑘2 /2
Γ 𝑘, 𝑃 or Γ(𝑘1 , 𝑘2 )
𝒱 𝑝, 𝑘
total momentum
relative momentum
vertex function
kernel
Kernel contains confining interaction + color Coulomb +/or constant
In the BS equation it is effectively iterated to all orders
But the complete kernel is a sum of an infinite number of irreducible diagrams
→ has to be truncated (most often: ladder approximation)
Now we take a closer look at the loop integration over 𝑘0
HUGS, JLab, USA
June, 2015
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4
From Bethe-Salpeter to CST
Covariant Spectator Theory (CST)
Mini-review: A.Stadler, F. Gross, Few-Body Syst. 49, 91 (2010)
Integration over relative energy 𝑘0 :
 Keep only pole contributions from propagators
 Cancellations between ladder and crossed
ladder diagrams can occur
 Reduction to 3D loop integrations, but covariant
 Works very well in few-nucleon systems
If bound-state mass 𝜇 is small:
both poles are close together (both important)
Symmetrize pole contributions from both half planes:
resulting equation is symmetric under charge conjugation
HUGS, JLab, USA
June, 2015
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Four-channel CST equation
Closed set of equations when external legs are systematically placed on-shell
Approximations can be made for special cases:
 mesons with different quark constituent masses: 2 channels
 large bound-state mass: 1 channel
Nonrelativistic limit: Schrödinger equation
HUGS, JLab, USA
June, 2015
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6
CST bound-state (cont.)
2CS
Dominant pole!
1CS
Why to study such an equation?
𝒱
 prepare and test the numerics
 test already our phenomenological choice for 𝒱
 should be a good approach to large 𝜇 systems:
Quarkonia (𝑄𝑄) and heavy-light (𝑄𝑞)
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June, 2015
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7
Interaction kernel
 The 1CS eq. reads:
Γ1𝐶𝑆
𝑑3 𝑘
𝑝, 𝑃 = −∫
2𝐸𝑘1 2𝜋
3𝒱
𝑝, 𝑘; 𝑃 Λ1 𝑘1 Γ1𝐶𝑆 𝑘, 𝑃 𝑆2 𝑘2 ,
𝑆2 𝑘2 =
 Vertex function for a pseudoscalar meson:
1
𝑚02 + Σ2 𝑘2 − 𝑘2 + −𝑖𝜖
,
𝒎𝟐
Γ 𝑝 = Γ1 𝑝 𝛾 5 + Γ2 𝑝 𝛾 5 𝑚2 − 𝑝2 , the most general form for CST.
 Interquark interaction (phenomenological)
𝒱 𝑝, 𝑘 = 𝝈 𝑉𝐿 𝑝, 𝑘
1 − 𝒚 𝕝1 ⊗ 𝕝2 − 𝒚
𝜇
𝛾1
⊗
𝜇
𝛾2
+ 𝜶 𝑉𝑂𝐺𝐸 𝑝, 𝑘
correct nonrelativictic limit for arbitrary 𝑦
𝜇
𝜇
1 − 𝒚 𝕝1 ⊗ 𝕝2 + 𝛾15 ⊗ 𝛾25 − 𝒚 𝛾1 ⊗ 𝛾2
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June, 2015
𝜇
⊗ 𝛾2
𝛿 3 (𝒑 − 𝒌) 𝜇
𝜇
+ 𝑪
𝛾1 ⊗ 𝛾2
2𝐸𝑘1
One-gluon-exchange (OGE)
Linear confinement
𝑉𝐿 𝑝, 𝑘
𝜇
𝛾1
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Constant
CST 𝜋𝜋- scattering studies constraints - Lorentz structure
8
Linear confinement in momentum space
SL, A. Stadler, E. Biernat, M.T. Peña; PRD90, 096003 (2014)
Nonrelativistic case
𝑉𝐿 𝒓 = 𝜎𝑟
𝑉𝐿,𝜖 𝒓 =
𝜎
1 − 𝑒 −𝜖𝑟
𝜖
equiv.
𝜎
𝑉𝐿,𝜖 𝒓 = 𝑉𝐴,𝜖 𝒓 − 𝑉𝐴,𝜖 (0), with 𝑉𝐴,𝜖 𝒓 ≡ − 𝑒 −𝜖𝑟
𝜖
Fourier Transform
𝑉𝐿,𝜖 𝒑, 𝒌 = 𝑉𝐴,𝜖 𝒒 − 2𝜋 3 𝛿 (3)
𝑑 3 𝑞′
𝒒 ∫
𝑉𝐴,𝜖 𝒒′ lim 𝜖 → 0 𝑉𝐿 𝒑, 𝒌 = 𝑉𝐴 𝒒 − 2𝜋 3 𝛿
3
2𝜋
3
𝑑3𝑞′
𝒒 ∫
𝑉𝐴 𝒒′ ,
3
2𝜋
𝑉𝐴 𝒒 ≡ −
Relativistic case
𝑉𝐿 𝑝, 𝑘 = 𝑉𝐴 𝑝, 𝑘 − 2𝐸𝑝1 2𝜋 3 𝛿
3
𝒑−𝒌
𝑑 3 𝒌′
′
𝑉
𝑝,
𝑘
𝐴
2𝜋 3 2𝐸𝑘1′
𝑉𝐴 𝑝, 𝑘 = −
8𝜋𝜎
𝒒𝟒
8𝜋𝜎
𝑝−𝑘
4
 Covariant;

𝑑3𝒌
∫ 2𝐸 𝑉𝐿
𝑘1
𝑝, 𝑘 = 0;
 nonrelativistic limit:
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necessary to prove chiral symmetry
3
lim ∫ 𝑑 𝒌 𝑉𝐿,𝜖 𝒑 − 𝒌 𝜓 𝒌 = 𝑃∫
𝜖→0
June, 2015
Sofia Leitão
𝑑3𝑘
𝒑−𝒌 4
𝜓 𝒌 −𝜓 𝒑
well-defined
integral as a
Cauchy
Principal Value
integral
9
Input:
𝒎𝒃 = 𝟒. 𝟕𝟗𝟑𝟏
𝒎𝒄 = 𝟏. 𝟓𝟑𝟎𝟎
𝒎𝒔 = 𝟎. 𝟒𝟎𝟎𝟎
𝒎𝒖 = 𝒎𝒅 = 𝟎. 𝟐𝟓𝟖𝟎
𝚲 = 𝟏. 𝟕 𝒎
Free-parameters:
, , ,
𝜇
𝜇
Model 11
𝑉𝐿 𝑝, 𝑘
1 − 𝑦 𝕝1 ⊗ 𝕝2 − 𝑦 𝛾1 ⊗ 𝛾2
Model 2
𝑉𝐿 𝑝, 𝑘
1 − 𝑦 𝕝1 ⊗ 𝕝2 + 𝛾15 ⊗ 𝛾25 − 𝑦 𝛾1 ⊗ 𝛾2
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𝜇
June, 2015
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𝜇
10
1CSE fit to quarkonia and heavy-light pseudoscalar states
Model 11 𝝈 = 𝟎. 𝟐𝟐 𝑮𝒆𝑽𝟐 ,
𝜶 = 𝟎. 𝟑𝟖,
𝑪 = 𝟎. 𝟑𝟑𝟕𝑮𝒆𝑽,
𝒚 = 𝟖. 𝟔𝟎 × 𝟏𝟎−𝟕
𝝈 = 𝟎. 𝟐𝟏 𝑮𝒆𝑽𝟐 ,
𝜶 = 𝟎. 𝟑𝟕,
𝑪 = 𝟎. 𝟑𝟏𝟏 𝑮𝒆𝑽,
𝒚 = 𝟎. 𝟑𝟖 × 𝟏𝟎−𝟕
Model 2
 the linear confining interaction is compatible with y = 0
→ suggests vector component suppressed
 not very sensitive to the choice scalar vs scalar-plus-pseudoscalar structure
 for systems with larger 𝜇 the predictions are worse
→ can be explained by the pole behavior
HUGS, JLab, USA
June, 2015
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11
Summary and Outlook
1.
2.
We have solved the 1CS equation
Based on these early results, we can already state
that:
CST is a promising covariant, Minkowski space
approach to study the mesonic also
the bound-state problem.
 Extend the fit to all other mesons (vector mesons, etc...)
 Remake the fits using full, self-consistent 𝒒𝒒 1CS
𝒎𝟐 → 𝑴𝟐 𝒌𝟐 = 𝒎𝟎𝟐 + 𝜮𝟐 𝒌𝟐
 Solve the 2CS and 4CS using the numerical techniques developed – light sector (pion)
 Recalculate the pion form factor
E. Biernat, F. Gross, M.T. Peña, A. Stadler, PRD89, 016005 (2014), PRD89, 016006 (2014)
 Calculate quark-photon vertex dynamically
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June, 2015
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12
Backup slides
Subtraction Technique
 Kernel in momentum space - singularities both in linear and OGE pieces
→ First treatment in the nonrelativistic limit because singularities have the same nature
Example
Nonrelativistic, unscreened limit of 1CSE with just a linear potential:
s-wave
𝑝2
𝑘2
𝜓
𝑝 + 𝒫∫ 𝑑𝑘
𝑝ℓ𝑚 𝑉𝐴 𝑘ℓ𝑚 𝜓ℓ𝑚 𝑘 − 𝑝00 𝑉𝐴 𝑘00 𝜓ℓ𝑚 𝑝
2𝑚𝑅 ℓ𝑚
2𝜋 3
2𝑃ℓ (𝑦)
𝑝ℓ𝑚 𝑉𝐴 𝑘ℓ𝑚 = 2𝜋(−8𝜋𝜎)
𝑝2 − 𝑘 2
2
 For ℓ > 1:
2
′
𝑃′ℓ 𝑦
𝑝+𝑘
2𝑤ℓ−1
(𝑦)
+
2 − 2𝑝 𝑘 2 𝑙𝑛 𝑝 − 𝑘
2𝑝𝑘 2
1 singularities
𝒑=𝒌
𝑝2 + 𝑘 2
𝑦=
2𝑝𝑘
ℓ
′
𝑤ℓ−1
𝑦 =
𝑚=1
1
𝑃
𝑦 𝑃𝑚−1 (𝑦)
𝑚 ℓ−𝑚
∞
𝑄0 (𝑦) 𝜋 2
1 Add and subtract a term proportional to 𝒫 𝑑𝑘
=
,
𝑘
2
0
we can get rid of the logarithmic singularity.
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= 𝐸𝜓ℓ𝑚 𝑝
June, 2015
Spence, Vary, PRD35, 2191 (1987)
Gross, Milana, PRD43, 2401 (1991)
Maung, Kahana, Norbury, PRD47,1182 (1993)
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∞
2 Apply now a second subtraction based on we
can also remove the principal value singularity.
𝒫
0
𝑑𝑘
= 0,
2
2
𝑘 −𝑝
Singularity-free two-body equation
New technique
SL, A.Stadler, E.Biernat, M.T. Peña;
PRD90, 096003 (2014)
Before subtraction
After subtraction
 Cubic B-splines
 More stable results than the un-subtracted version for any partial wave;
 Less computational time
→ Back to the 1CSE:
This technique was very important for stability
purposes!
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June, 2015
All singularities are
eliminated from the
kernel!
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Heavy-light scenario
 With retardation we need to include a PauliVillars regularization, cut-off 𝚲
Light-light scenario
𝒎𝟏 = 𝟎. 𝟑𝟐𝟓 𝑮𝒆𝑽
𝒎𝟐 = 𝟎. 𝟑𝟐𝟓 𝑮𝒆𝑽
𝚲 = 𝟑. 𝟎 𝑮𝒆𝑽
𝒎𝟏 = 𝟓 𝒎𝟏
𝒎𝟐 = 𝟎. 𝟑𝟐𝟓 𝑮𝒆𝑽
𝚲 = 𝟑. 𝟎 𝑮𝒆𝑽
without retardation
with retardation
Heavy-heavy scenario
without retardation
with retardation
𝒎𝟏 = 𝟒. 𝟖 𝑮𝒆𝑽
𝒎𝟐 = 𝟒. 𝟖 𝑮𝒆𝑽
𝚲 = 𝟏𝟎. 𝟎 𝑮𝒆𝑽
 𝑞𝑞 light: large retardation effects
 nonrelativistic limit: retardation vanishes
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June, 2015
without retardation
with retardation
Sofia Leitão
SL, A.Stadler, E.Biernat, M.T. Peña; Phys. Rev. D90, 096003 (2014)
NR 1CSE
𝝈 = 𝟎. 𝟏𝟔𝟕 𝑮𝒆𝑽𝟐 ,
𝜶 = 𝟎. 𝟓𝟏𝟔𝟕,
𝑪 = 𝟎. 𝟎 𝑮𝒆𝑽,
Model 11
𝝈 = 𝟎. 𝟐𝟐 𝑮𝒆𝑽𝟐 ,
𝜶 = 𝟎. 𝟑𝟖,
𝑪 = 𝟎. 𝟑𝟑𝟕𝑮𝒆𝑽,
𝒚 = 𝟖. 𝟔𝟎 × 𝟏𝟎−𝟕
Model 2
𝝈 = 𝟎. 𝟐𝟏 𝑮𝒆𝑽𝟐 ,
𝜶 = 𝟎. 𝟑𝟕,
𝑪 = 𝟎. 𝟑𝟏𝟏 𝑮𝒆𝑽,
𝒚 = 𝟎. 𝟑𝟖 × 𝟏𝟎−𝟕
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June, 2015
𝒎𝒃 = 𝟒. 𝟕𝟗𝟑𝟏𝑮𝒆𝑽
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First energy state (positive component)
1CSE Results
𝜓1− (𝑝)
𝑬𝟏+ = 𝟎. 𝟗𝟑𝟖𝟕 𝑮𝒆𝑽
𝜓1+ (𝑝)
First energy state (negative component)
𝜓1+ (𝑝)
𝑬𝟏− = −𝟎. 𝟗𝟑𝟔𝟑 𝑮𝒆𝑽
𝜓1− (𝑝)
Parameters used: y = 0 (pure scalar) 𝜎 = 0.2𝐺𝑒𝑉 2 (linear piece)
𝑚2 = 0.325 𝐺𝑒𝑉
𝑚1 / 𝑚2 =5
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June, 2015
 Perfect agreement with previous
results – faster convergence
M. Uzzo, F. Gross, PRC59, 1009 (1999)
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