Quarkonia and heavy-light mesons in a covariant quark model Sofia Leitão CFTP, University of Lisbon, Portugal in collaboration with: Alfred Stadler, M

Quarkonia and heavy-light mesons in a covariant quark model

Sofia LeitãoCFTP, University of Lisbon, Portugal

in collaboration with:Alfred Stadler, M. T. Peña and Elmar P. Biernat

HUGS, JLab, USA June, 2015Sofia Leitão

A unified model for all mesonsMuch 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, …

HUGS, JLab, USA June, 2015 Sofia Leitão 2

Our objectives: Construct a model to describe all -type mesons Covariant framework (CST) - light quarks require relativistic treatment Work in Minkowski space (physical momenta) Quark self-energy from 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.)

We have just learned about it :)

particle 1 on mass-shell:

+¿ ¿

+¿¿𝑀 𝜈𝜈 𝑀

[F.Gross, PR186, 1969][F.Gross, Relativistic Quantum Mechanics and Field Theory, 2004]

cancellation in all orders and exact in heavy-mass limit !

CST main idea

If kernel (all 1PI diagrams) exact result for

scattering problem

Cancelation theorem: - theory

Usual truncation: ladder approximation

x

2

1

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.

scattering amplitude

How to truncate?


Example of the

day

interaction kernel

Covariant two-body bound-state equationStart from the Bethe-Salpeter (BS) equation


Γ𝐵𝑆 (𝑝 ,𝑃 )=𝑖∫ 𝑑4𝑘(2𝜋 )4

𝒱 (𝑝 ,𝑘;𝑃 )𝑆1(𝑘1)Γ𝐵𝑆 (𝑘 ,𝑃 )𝑆2(𝑘2)

𝑆 𝑖 (𝑘𝑖 )=1

𝑚0 𝑖−𝑘𝑖+Σ𝑖 (𝑘𝑖 )− 𝑖𝜖𝛴 𝑖 (𝑘𝑖 )=𝐴𝑖(𝑘𝑖2 )+𝑘𝑖𝐵𝑖 (𝑘𝑖2)

𝑃=𝑘1−𝑘2𝑘=𝑘1+𝑘2/2Γ (𝑘 ,𝑃 )∨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

Back to mesons!

From Bethe-Salpeter to CSTCovariant Spectator Theory (CST)


Integration over relative energy :

If bound-state mass is small: both poles are close together (both important)

Symmetrize pole contributions from both half planes:resulting equation is symmetric under charge conjugation

Mini-review: A.Stadler, F. Gross, Few-Body Syst. 49, 91 (2010)

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

Four-channel CST equationClosed 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

CST bound-state (cont.)2CS

1CS

Why to study such an equation?

prepare and test the numerics

test already our phenomenological choice for

should be a good approach to large systems:

𝒱Quarkonia ( and heavy-light (

Dominant pole!


𝒱

Interaction kernel The 1CS eq. reads:

Γ 1𝐶𝑆 (𝑝 ,𝑃 )=− ∫ 𝑑3𝑘2𝐸𝑘1

(2𝜋 )3𝒱 (𝑝 ,𝑘 ;𝑃 ) Λ1 (�̂�1 )Γ1𝐶𝑆 (𝑘 ,𝑃 )𝑆2 (𝑘2) ,

𝑆2 (𝑘2 )= 1𝑚02+Σ2 (𝑘2 )−𝑘2+−𝑖𝜖

,

Vertex function for a pseudoscalar meson:

Γ (𝑝 )=Γ1 (𝑝)𝛾5+Γ 2 (𝑝 )𝛾5 (𝑚2−𝑝2) , the most general form for CST.

Interquark interaction (phenomenological)

Linear confinement One-gluon-exchange (OGE) Constant

𝒱 (𝑝 ,𝑘)=𝝈𝑉 𝐿 (𝑝 , �̂� ) [ (1−𝒚 )𝕝1⊗ 𝕝2−𝒚 𝛾1𝜇⊗𝛾 2

𝜇]+𝜶𝑉 𝑂𝐺𝐸 (𝑝 , �̂� )𝛾1𝜇⊗𝛾 2𝜇+𝑪𝛿3(𝒑−𝒌)2𝐸𝑘1

𝛾1𝜇⊗𝛾 2

𝜇

correct nonrelativictic limit for arbitrary

𝑉 𝐿 (𝑝 ,�̂� ) [ (1−𝒚 ) (𝕝1⊗𝕝2+𝛾15⊗𝛾25 )− 𝒚 𝛾1𝜇⊗𝛾2𝜇] CST - scattering studies - constraints - Lorentz structure

𝒎𝟐

for now,fixed masses


Linear confinement in momentum space

~𝑉 𝐿 ,𝜖 (𝒓 )=𝜎

𝜖(1−𝑒−𝜖𝑟 )~

𝑉 𝐿 (𝒓 )=𝜎 𝑟

Nonrelativistic case~𝑉 𝐴 , 𝜖 (𝒓 )≡− 𝜎

𝜖𝑒−𝜖 𝑟,

𝑉 𝐿 ,𝜖 (𝒑 ,𝒌 )=𝑉 𝐴 ,𝜖 (𝒒 )− (2𝜋 )3 𝛿(3) (𝒒 ) ∫ 𝑑3𝑞 ′

(2𝜋 )3𝑉 𝐴 ,𝜖 (𝒒 ′ ) 𝑉 𝐿 (𝒑 ,𝒌 )=𝑉 𝐴 (𝒒 )− (2𝜋 )3 𝛿(3 ) (𝒒 ) ∫ 𝑑

3𝑞 ′

(2𝜋 )3𝑉 𝐴 (𝒒 ′ ) ,

𝑉 𝐴 (𝒒 )≡− 8𝜋𝜎𝒒𝟒

Fourier Transform

lim

with

𝑉 𝐿 (𝑝 ,�̂� )=𝑉 𝐴 (𝑝 , �̂� )−2𝐸𝑝 1 (2𝜋 )3𝛿( 3) (𝒑−𝒌 )∫ 𝑑3𝒌 ′(2𝜋 )32𝐸𝑘1 ′

𝑉 𝐴 (𝑝 ,�̂�′ )

Relativistic case

Covariant; necessary to prove chiral symmetry nonrelativistic limit:

well-defined integral as a

Cauchy Principal Value

integral

𝑉 𝐴 (𝑝 , �̂�)=− 8𝜋𝜎

(𝑝− �̂�)4

equiv.

SL, A. Stadler, E. Biernat, M.T. Peña; PRD90, 096003 (2014)


subtle detail

Model 11

Model 2

𝒎𝒃=𝟒 .𝟕𝟗𝟑𝟏

𝒎𝒄=𝟏 .𝟓𝟑𝟎𝟎𝒎𝒔=𝟎 .𝟒𝟎𝟎𝟎

𝒎𝒖=𝒎𝒅=𝟎 .𝟐𝟓𝟖𝟎

𝚲=𝟏 .𝟕𝒎

𝑉 𝐿 (𝑝 ,𝑘 ) [ (1− 𝑦 ) 𝕝1⊗𝕝2− 𝑦 𝛾1𝜇⊗𝛾 2

𝜇 ]𝑉 𝐿 (𝑝 ,𝑘 ) [ (1− 𝑦 ) (𝕝1⊗ 𝕝2+𝛾 1

5⊗𝛾25 )− 𝑦𝛾 1𝜇⊗𝛾 2𝜇]

𝜎 ,𝛼 ,𝐶 , 𝑦

Input:

Free-parameters:

some results


Model 11

1CSE fit to quarkonia and heavy-light pseudoscalar states

Model 2

𝝈=𝟎 .𝟐𝟐𝑮𝒆𝑽 𝟐 , 𝜶=𝟎 .𝟑𝟖 ,𝑪=𝟎 .𝟑𝟑𝟕𝑮𝒆𝑽 , 𝒚=𝟖 .𝟔𝟎×𝟏𝟎−𝟕

𝝈=𝟎 .𝟐𝟏𝑮𝒆𝑽 𝟐 , 𝜶=𝟎 .𝟑𝟕 ,𝑪=𝟎 .𝟑𝟏𝟏𝑮𝒆𝑽 , 𝒚=𝟎 .𝟑𝟖×𝟏𝟎−𝟕

the linear confining interaction is compatible with suggests vector component suppressed not very sensitive to the choice scalar vs scalar-plus-pseudoscalar structure for systems with larger the predictions are worse can be explained by the pole behavior


Extend the fit to all other mesons (vector mesons, etc...) Remake the fits using full, self-consistent 1CS

Solve the 2CS and 4CS using the numerical techniques developed – light sector (pion)

Recalculate the pion form factor

Calculate quark-photon vertex dynamically

E. Biernat, F. Gross, M.T. Peña, A. Stadler, PRD89, 016005 (2014), PRD89, 016006 (2014)

1. We have solved the 1CS equation2. 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.

Summary and Outlook


soon, we hope!

Backup slides

Subtraction Technique

singularities

𝑝2

2𝑚𝑅

𝜓 ℓ𝑚 (𝑝 )+𝒫 ∫ 𝑑𝑘 𝑘2

(2𝜋 )3[ ⟨𝑝 ℓ𝑚|𝑉 𝐴|𝑘 ℓ𝑚 ⟩𝜓 ℓ𝑚 (𝑘 )− ⟨𝑝 00|𝑉 𝐴|𝑘00 ⟩𝜓 ℓ𝑚 (𝑝 ) ]=𝐸𝜓 ℓ𝑚 (𝑝 )

⟨𝑝 ℓ𝑚|𝑉 𝐴|𝑘 ℓ𝑚⟩=2𝜋 (−8𝜋𝜎)[ 2𝑃 ℓ (𝑦 )

(𝑝2−𝑘2 )2−𝑃 ′ ℓ (𝑦 )

(2𝑝𝑘)2𝑙𝑛( 𝑝+𝑘𝑝−𝑘 )

2

+2𝑤 ℓ− 1

′ (𝑦 )

(2𝑝𝑘 )2 ] 𝑦=𝑝2+𝑘2

2𝑝𝑘

𝑤 ℓ−1′ (𝑦 )=∑

𝑚=1

ℓ1𝑚𝑃 ℓ −𝑚 (𝑦 )𝑃𝑚−1(𝑦 )𝒑=𝒌 1

1 Add and subtract a term proportional to 𝒫∫0

∞

𝑑𝑘𝑄0(𝑦 )𝑘

= 𝜋2

2 ,

s-wave

Kernel in momentum space - singularities both in linear and OGE pieces

→ First treatment in the nonrelativistic limit because singularities have the same nature

ExampleNonrelativistic, unscreened limit of 1CSE with just a linear potential:

Spence, Vary, PRD35, 2191 (1987)Gross, Milana, PRD43, 2401 (1991)

Maung, Kahana, Norbury, PRD47,1182 (1993)

2

For

we can get rid of the logarithmic singularity.


Singularity-free two-body equationBefore 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!

Apply now a second subtraction based on we 𝒫∫0

∞𝑑𝑘𝑘2−𝑝2

=0 , New technique

All singularities are eliminated from the

kernel!

2

can also remove the principal value singularity. SL, A.Stadler, E.Biernat, M.T. Peña; PRD90, 096003 (2014)


With retardation we need to include a Pauli-Villars regularization, cut-off

light: large retardation effects

nonrelativistic limit: retardation vanishes

with retardationwithout retardation


Heavy-heavy scenario

Light-light scenario

Heavy-light scenario



SL, A.Stadler, E.Biernat, M.T. Peña; Phys. Rev. D90, 096003 (2014)

PRELIMINARY

Model 11

Model 2

𝝈=𝟎 .𝟐𝟐𝑮𝒆𝑽 𝟐 , 𝜶=𝟎 .𝟑𝟖 ,𝑪=𝟎 .𝟑𝟑𝟕𝑮𝒆𝑽 , 𝒚=𝟖 .𝟔𝟎×𝟏𝟎−𝟕

𝝈=𝟎 .𝟐𝟏𝑮𝒆𝑽 𝟐 , 𝜶=𝟎 .𝟑𝟕 ,𝑪=𝟎 .𝟑𝟏𝟏𝑮𝒆𝑽 , 𝒚=𝟎 .𝟑𝟖×𝟏𝟎−𝟕

NR 1CSE 𝝈=𝟎 .𝟏𝟔𝟕𝑮𝒆𝑽 𝟐 ,𝜶=𝟎 .𝟓𝟏𝟔𝟕 ,𝑪=𝟎 .𝟎𝑮𝒆𝑽 ,𝒎𝒃=𝟒 .𝟕𝟗𝟑𝟏𝑮𝒆𝑽


First energy state (positive component)

𝜓 1+¿(𝑝)¿

𝜓1−(𝑝 )

𝜓 1−(𝑝 )

𝜓 1+¿(𝑝)¿

𝑬𝟏+¿=𝟎 .𝟗𝟑𝟖𝟕𝑮𝒆𝑽 ¿

𝑬𝟏−=−𝟎 .𝟗𝟑𝟔𝟑𝑮𝒆𝑽

First energy state (negative component)

Parameters used: y=0𝑚2=0.325𝐺𝑒𝑉=5𝜎=0.2𝐺𝑒𝑉 2(pure scalar) (linear piece) Perfect agreement with previous

results – faster convergence

1CSE Results

M. Uzzo, F. Gross, PRC59, 1009 (1999)


Documents

Quarkonia and heavy-light mesons in a covariant quark model Sofia Leitão CFTP, University of Lisbon, Portugal in collaboration with: Alfred Stadler, M