quarkonia and heavy-light mesons in a covariant quark model sofia leitão cftp, university of...
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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?
HUGS, JLab, USA June, 2015 Sofia Leitão 3
Example of the
day
interaction kernel
Covariant two-body bound-state equationStart from the Bethe-Salpeter (BS) equation
HUGS, JLab, USA June, 2015 Sofia Leitão 4
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)
HUGS, JLab, USA June, 2015 Sofia Leitão 5
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
HUGS, JLab, USA June, 2015 Sofia Leitão 6
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!
HUGS, JLab, USA June, 2015 Sofia Leitão 7
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
HUGS, JLab, USA June, 2015 Sofia Leitão 8
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)
HUGS, JLab, USA June, 2015 Sofia Leitão 9
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
HUGS, JLab, USA June, 2015 Sofia Leitão 10
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
HUGS, JLab, USA June, 2015 Sofia Leitão 11
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
HUGS, JLab, USA June, 2015 Sofia Leitão 12
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.
HUGS, JLab, USA June, 2015Sofia Leitão
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)
HUGS, JLab, USA June, 2015Sofia Leitão
With retardation we need to include a Pauli-Villars regularization, cut-off
light: large retardation effects
nonrelativistic limit: retardation vanishes
with retardationwithout retardation
with retardationwithout retardation
Heavy-heavy scenario
Light-light scenario
Heavy-light scenario
with retardationwithout retardation
HUGS, JLab, USA June, 2015Sofia Leitão
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
HUGS, JLab, USA June, 2015Sofia Leitão
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)
HUGS, JLab, USA June, 2015Sofia Leitão