beyond rainbow-ladder in a covariant three-body bethe...

Beyond Rainbow-Ladder in a covariant three-body

Bethe-Salpeter approach: Baryons

Helios Sanchis AlepuzJustus-Liebig University - Giessen

In collaboration with:Christian FischerStas Kubrak

Menu 2013 — Sept. 30 - Oct. 4, 2013

Introduction. Covariant BSEs.

Rainbow-Ladder truncation.

Technicalities.

Selection of results in RL.

First steps beyond-RL.

Helios Sanchis Alepuz (JLU - Giessen) Menu 2013 — Sept. 30 - Oct. 4, 2013 2 / 16

Goals

Continuum functional methods are currently one more tool in the toolbox tounderstand hadron physics from QCD.

What features of QCD (quarks, gluons and ghosts) are relevant to understandhadrons properties?

In principle, hadron properties are encoded in QCD’s Green’s functions ↪→Dyson-Schwinger equations (DSEs)

Bound states appear as poles of the Green functions ↪→ Bethe-Salpeter equations(BSEs)

Some applications of DSE/BSE in hadron phenomenology:

Baryon spectrum and form factors (Alkofer, Eichmann, Nicmorus, Roberts, HSA, ...)

Hadronic contributions to g − 2 (Fischer, Gocke, Williams).

Sigma properties/decay (Alkofer, Williams).

Nucleon Compton scattering (Eichmann).

. . .


Goals

Most current DSE/BSE calculations are done using the simplest possible interactionkernels (Rainbow-Ladder).

This works quite well for several ground-state properties.

What are the most important terms beyond-RL?


Goals

Most current DSE/BSE calculations are done using the simplest possible interactionkernels (Rainbow-Ladder).

This works quite well for several ground-state properties.

What are the most important terms beyond-RL?

ROADMAP

1 Form factors give details about the structure of the baryon. Give hints of missingdynamics.

2 Good (and relatively easy) laboratories to check beyond-RL: excited states, octetand decuplet

3 Excited states within Rainbow-Ladder

4 Excited states with pionic corrections We’re almost there!

5 Masses of the baryon octet and decuplet Hopefully soon


Baryon spectrum from BSE/DSEs:��

�



�

Rainbow-Ladder

Quark DSE: Dtree−levelµν (q)γναeff (q2)

Two-body kernel: K = 4πZ 22 Dtree−level

µν αeff (q2)γµ ⊗ γν

This truncation preserves Axial-vector WTI.

Thus, it preserves Chiral symmetry (in the chiral limit) and allows its dynamicalbreaking.

Also, for the 3-body BSE: Neglect 3-body irreducible contributions.

We need a model for the effective interaction αeff (q2).



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Maris-Tandy modelP. Maris and C. D. Roberts, PRC56 (1997)

P. Maris, P. Tandy PRC60 (1999)

α(k2) = αIR (k2; Λ, η) + αUV (k2)

Purely phenomenological model.

Λ fitted to fπ.

Current quark masses fitted to π, K .

Ground-state pseudoscalar propertiesalmost insensitive to η around 1.8

Describes succesfully ground-state properties:Chang, Roberts,Tandy arXiv:1107.4003 [nucl-th]Eichmann et al. Ann Phys 323 (2008)...

0

2

4

6

8

10

12

14

k2 (GeV2)0 0,5 1 1,5 2

α(k2): MT model

Λ=0.72 GeV

η=1.6 η=1.8 η=2.0


Technicalities I. Choice of the quantum numbers:

⇒ Expand in a basis: ΨαβγI =P

i f(i)(p, q,P) τ

(i)αβγI(p, q,P) ⊗ Flavor ⊗ Color




i f(i)(p, q,P) τ


Facts about the basis:

It is independent of any approximation in the BSE equation.

Only Poincare covariance and parity invariance are needed to construct the basis.

The basis includes all possible internal spin and orbital angular momentum values.

Color part: we want a color singlet. Flavor part: SU(2) flavor symmetry.

For fixed-parity, positive-energy (particle) baryons the basis consists of:




i f(i)(p, q,P) τ


Facts about the basis:

It is independent of any approximation in the BSE equation.

Only Poincare covariance and parity invariance are needed to construct the basis.

The basis includes all possible internal spin and orbital angular momentum values.

Color part: we want a color singlet. Flavor part: SU(2) flavor symmetry.

For fixed-parity, positive-energy (particle) baryons the basis consists of:

spin- 12

particle: 64 elements

# elementss-wave 8p-wave 36d-wave 20

Eichmann et al. PRL. 104 (2010) 201601

spin- 32

particle: 128 elements

# elementss-wave 4p-wave 36d-wave 60f-wave 28

HSA et al. PRD 84 (2011) 096003


Masses:

Good agreement withexperimental values, except forvector mesons.

Reasonably good agreementwith lattice.

Unclear from here what isneeded to improve the results.


Form Factors in a covariant BSE approach

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†A. N. Kvinikhidze and B. Blankleider, Phys. Rev. C60, 044004 (1999)

A. N. Kvinikhidze and B. Blankleider, Phys. Rev. C60, 044003 (1999)


Form Factors in a covariant BSE approach

If we use the Rainbow-Ladder truncation:

��

��

And from the current we extract the corresponding Form Factors.


Nucleon EM form factors

G. Eichmann, Phys. Rev. D84, 014014 (2011)


Delta EM form factors

HSA, R. Alkofer and R. Williams, Phys. Rev. D87, 096015 (2013)Helios Sanchis Alepuz (JLU - Giessen) Menu 2013 — Sept. 30 - Oct. 4, 2013 10 / 16

Omega EM form factors

HSA, R. Alkofer and R. Williams, Phys. Rev. D87, 096015 (2013)


N∆γ transition form factors

PRELIMINARY!

HSA, R. Alkofer and R. Williams, in preparation.


First steps beyond Rainbow-Ladder

Pionic corrections:

Form factor calculations ask for Pion cloudPion corrections can be justified as the dominant corrections.


Baryon masses with pion corrections

Calculation in progress!

When pion exchange is included, the effective interaction must be refitted.

As before, it is refitted in the pseudoscalar meson sector.

There are no additional parameters.


Technicalities II. Excited states with BSEs

T = −iK − iKG0TPole assumption→ T ∼ ΨΨ

P2+M2 ⇒�� Ψ = −iKG0Ψ Homogenous BSE

Algorithm:

Introduce fictitous eigenvalue: λΨ = −iKG0Ψ

Solve the eigenvalue problem for λ(M), for fixed M.

When λ(M) = 1 we recover the original equation and M isthe mass of the bound state.

No problem for ground states (largest eigenvalue).

Beyond the largest eigenvalue, we encounter many complex(unphysical) eigenvalues.





Algorithm:






Inhomogeneous BSE: Γ = Γ0 − iKG0Γ for Γ0 with the appropriate quantum numbers.

Algorithm:

Solve as a linear equation: (1 + iKG0)Γ = Γ0

Look for values of M for which Γ has a pole.

We don’t care about unphysical eigenvalues: easier to solve(for excited states).

Less accurate determination of bound-state masses.Helios Sanchis Alepuz (JLU - Giessen) Menu 2013 — Sept. 30 - Oct. 4, 2013 15 / 16




Algorithm:






Spectral transformations (Shift, or Shift-Invert):

Solve initial eigenvalue problem applying some spectraltransformation.

These allows to look for one or two eigenvalues close to 1.

Still more expensive than calculating ground states.


Outlook

1 Complete calculation of baryon masses with pion exchangeCalculate negative parity excitations. Do we get the correct level ordering?How far are we from the physical masses? How far should we be?

2 Recalculate form factors with these new interactionIs a pion exchange enouch to reconcile with experiment?

3 Calculate octet and decuplet statesThese states will certainly require a quark-mass dependent interactionAgain, will a pion exchange be enough?


Masses Rho, Nucleon, Delta and Omega

Nucleon σ-term (MeV) MT AFW30 47.8

PoS QCD-TNT-II (2011) 041


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