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Electromagnetic Form Factors Large-NC QCD Regge Theory and Duality Veneziano Dual-Resonance Model Results for the ∆(1232) Chiral Perturbation Parameters Conclusion, Further Work and References

ELECTROMAGNETIC FORM FACTORS INDUAL-LARGE NC QCD

Raoul Rontsch

Centre for Theoretical Physics and Astrophysics, University of Cape Town

International WorkshopStandard Model and Beyond in the LHC Era

Valparaiso, Chile10 January 2008

Raoul Rontsch ELECTROMAGNETIC FORM FACTORS IN DUAL-LARGE NC QCD

1 Electromagnetic Form Factors

2 Large-NC QCD

3 Regge Theory and Duality

4 Veneziano Dual-Resonance Model

5 Results for the ∆(1232)

6 Chiral Perturbation Parameters

7 Conclusion, Further Work and References

POINT PARTICLES

Electron has no internal structure, completelydescribed by Dirac equation.

Current is jµ = e0ψeγµψe

POINT PARTICLES

Electron has no internal structure, completelydescribed by Dirac equation.

Current is jµ = e0ψeγµψe

PARTICLES WITH INTERNAL STRUCTURE

Other particles, e.g. proton, have internalstructure.

virtual particlesmagnetic moment

Current is j(p)µ = e0ψpΓµψp

virtual particles

magnetic moment

INFLUENCE OF VIRTUAL PARTICLES

Virtual particle emitted by proton at x interactswith photon at x ′.

Described by Green’s function F (x − x ′).

Can accommodate influence of virtual particlesby writing F (q2)γµ instead of γµ.

FORM FACTORS FOR PROTONS

Magnetic moment effect can be accommodatedby a Pauli term iσµν

Thus Γµ = F1(q2)γµ + iF2(q

m σµν

F1 is charge form factor

F2 is magnetic moment form factor

m σµν

Electromagnetic form-factors are a measureof the internal electromagnetic structure ofa particle.

F1 is Fourier transform of charge density.

F2 is Fourier transform of magnetic momentdensity.

Leads to normalization of form factors:F1(0) = 1F2(0) = 1.79

Leads to normalization of form factors:

F1(0) = 1F2(0) = 1.79

ELECTRIC AND MAGNETIC FORM FACTORS

More common to combine F1 and F2 into electricand magnetic form factors

GM(q2) = F1(q2) + F2(q

GE (q2) = F1(q2) +

4m2F2(q

with normalization GM(0) = 2.79 and GE (0) = 1.

ELECTRIC AND MAGNETIC FORM FACTORS

More common to combine F1 and F2 into electricand magnetic form factors

GM(q2) = F1(q2) + F2(q

GE (q2) = F1(q2) +

4m2F2(q

with normalization GM(0) = 2.79 and GE (0) = 1.

MOTIVATION BEHIND LARGE-NC QCD

Use number of colours NC as an expansionparameter in QCD, and comparephenomenology with experiment.

In the limit NC →∞, QCD becomes solvable.

MOTIVATION BEHIND LARGE-NC QCD

Use number of colours NC as an expansionparameter in QCD, and comparephenomenology with experiment.

In the limit NC →∞, QCD becomes solvable.

In this limit, hadronic spectrum consists ofinfinitely many zero-width resonances.

Width ∼ 1√N→ 0 in the limit NC →∞.

Infinite number of resonance related to thetwo-point function〈J(k)J(−k)〉 =

k2−m2n

which is logarithmically divergent.

This means that RHS has to be an infinite sum,hence infinitely many resonances.

k2−m2n

RESULTS OF REGGE THEORY

Every stable particle or resonance can berepresented by a Regge trajectory α(t),whose real part is linear in t and whoseimaginary part depends on the stability ofthe particle.

HADRONIC DUALITY

Hadronic spectral function shows resonanceswhose width increases with q2.

At high enough q2, have QCD background inwhich no structure can be distinguised.

Can be described on average by Reggetrajectories.

HADRONIC DUALITY

IGI AND MATSUBA

VENEZIANO DUAL-RESONANCE MODEL

Dual model based on Regge phenomenologythat shares important similarities with large-NC

Allows predictions of large-NC QCD to beconfronted with experiment.

Write form factors as Euler beta-functions

B(x , y) =Γ(x)Γ(y)

Γ(x + y)

B(x , y) =Γ(x)Γ(y)

Γ(x + y)

B(x , y) =Γ(x)Γ(y)

Γ(x + y)

SIMILARITIES WITH LARGE-NC QCD

Euler beta-function has infinitely many poles:

1− α(t) = −nwhich reflects infinitely many zero-widthresonances.

Euler beta-function has infinitely many poles:1− α(t) = −n

which reflects infinitely many zero-widthresonances.

Euler beta-function has infinitely many poles:1− α(t) = −nwhich reflects infinitely many zero-widthresonances.

VENEZIANO EXPRESSION FOR FORM FACTORS

Expect the form factors to have Reggebehaviour F (t) ∝ t−n

Regge trajectories should be linear in t and givea pole at t = m2

F (s, t) = N√π

Γ(n− 12 )Γ( 1

2m2ρ)

Γ(n− 12−

2m2ρ)

F (s, t) = N√π

Γ(n− 12 )Γ( 1

2m2ρ)

Γ(n− 12−

2m2ρ)

F (s, t) = N√π

Γ(n− 12 )Γ( 1

2m2ρ)

Γ(n− 12−

2m2ρ)

VECTOR MESON DOMINANCE

Vector meson dominance modifies normal EMinteraction

Virtual hadron surrounding photon couples tonucleon

Strong coupling dominates electromagneticcoupling.

Hadron must have same quantum numbers as γ.

Most likely (lightest) is ρ(770).

Most likely (lightest) is ρ(770).Raoul Rontsch ELECTROMAGNETIC FORM FACTORS IN DUAL-LARGE NC QCD

THE ∆(1232)

First excited states of nucleon.

Spins of all three quarks aligned - hence spin 32

(Reason for introduction of colour d.o.f.)Three charge states:

∆∆−

corresponding to uuu, uud , udd , and ddd

THE ∆(1232)

(Reason for introduction of colour d.o.f.)

Three charge states:∆++

∆∆−

THE ∆(1232)

(Reason for introduction of colour d.o.f.)Three charge states:

∆∆−

THE ∆(1232) CREATION CHANNELS

Main creation and decay channel is via strong interaction:

πN → ∆ → πN

Secondary mechanism is via electromagnetic interaction

γN → ∆ → γN

Photon induces a spin-flip in one of the quarks.

Quark in pure s-state ⇒ magnetic dipole transition.

Electric and Coulombic transitions possible IF admixture ofs- and d-states

πN → ∆ → πN

γN → ∆ → γN

πN → ∆ → πN

γN → ∆ → γN

πN → ∆ → πN

γN → ∆ → γN

πN → ∆ → πN

γN → ∆ → γN

FIT FOR G ∗M

First wrote G ∗M,E ,C as

G ∗M,E ,C = N√

πΓ(βM,E ,C − 1

2)Γ( 1

2M2ρt)

Γ(βM,E ,C− 12+ 1

2M2ρt)

then fitted βM to data for GM .

with βM = 4.6− 4.8

FIT FOR G ∗M

First wrote G ∗M,E ,C as

G ∗M,E ,C = N√

πΓ(βM,E ,C − 1

2)Γ( 1

2M2ρt)

Γ(βM,E ,C− 12+ 1

2M2ρt)

then fitted βM to data for GM .

with βM = 4.6− 4.8Raoul Rontsch ELECTROMAGNETIC FORM FACTORS IN DUAL-LARGE NC QCD

FITS FOR G ∗E AND G ∗

Define ratios REM and RSM .

REM = −G ∗M

G ∗E

RSM = −Q+Q−

4M2∆

G ∗C

G ∗M

βE and βC fitted to data for these ratios, usingG ∗

M as obtained from previous fit.

FITS FOR G ∗E AND G ∗

Define ratios REM and RSM .

REM = −G ∗M

G ∗E

RSM = −Q+Q−

4M2∆

G ∗C

G ∗M

βE and βC fitted to data for these ratios, usingG ∗

M as obtained from previous fit.

FIT FOR G ∗E

Experimental errors too large for good fit to bemade.Because of this, set βE = βM = 4.6

FIT FOR G ∗C

Experimental errors still very large.

Decent fit with βC = 6.0− 6.2

CHIRAL PERTURBATION PARAMETERS

The form factors can be written in terms of chiralperturbation parameters gM , gE and gC

G ∗M = gM +

2(−M2

∆ + M2N + Q2)gE + Q2gC

]G ∗

2(−M2

∆ + M2N + Q2)gE + Q2gC

]G ∗

[(−M2

∆ + M2N + Q2)gC − 2M2

CHIRAL PERTURBATION PARAMETERS

The form factors can be written in terms of chiralperturbation parameters gM , gE and gC

G ∗M = gM +

2(−M2

∆ + M2N + Q2)gE + Q2gC

]G ∗

2(−M2

∆ + M2N + Q2)gE + Q2gC

]G ∗

[(−M2

∆ + M2N + Q2)gC − 2M2

CHIRAL PERTURBATION PARAMETERS ASFUNCTIONS OF FORM FACTORS

Chiral perturbation parameters are usually taken to beeither constant or described by dipole formula.

but. . .

gM = G ∗M − G ∗

gC = Q2+

G ∗C (−M2

∆ + M2N + Q2) + 4M2

∆G ∗E

(−M2∆ + M2

N + Q2)2+ 4M2

−M2∆ + M2

N + Q2

+G ∗E − Q2gC

but. . .

gM = G ∗M − G ∗

gC = Q2+

G ∗C (−M2

∆ + M2N + Q2) + 4M2

∆G ∗E

(−M2∆ + M2

N + Q2)2+ 4M2

−M2∆ + M2

N + Q2

+G ∗E − Q2gC

but. . .

gM = G ∗M − G ∗

gC = Q2+

G ∗C (−M2

∆ + M2N + Q2) + 4M2

∆G ∗E

(−M2∆ + M2

N + Q2)2+ 4M2

−M2∆ + M2

N + Q2

+G ∗E − Q2gC

but. . .

gM = G ∗M − G ∗

gC = Q2+

G ∗C (−M2

∆ + M2N + Q2) + 4M2

∆G ∗E

(−M2∆ + M2

N + Q2)2+ 4M2

−M2∆ + M2

N + Q2

+G ∗E − Q2gC

ESTIMATE OF FORM OF CHIRAL PERTURBATIONPARAMETERS

Clearly not constant! Our results give a way toestimate chiral perturbation parameters.

Definitely not constant.

gM and gE might be described by dipole formulaalthough falls off too fast.

gC crosses zero so can’t be described by dipoleformula.

gM and gE might be described by dipole formula

although falls off too fast.

CONCLUSION

Excellent agreement with data for G ∗M up to

−t = 10GeV 2 using parameter βM = 4.6− 4.8.

Data too inaccurate to give meaningful fit forG ∗

E (better data will improve situation).

Both of these form factors fall off faster thandipole (∼ t−4).

Decent agreement with data for G ∗C for

βC = 6.0− 6.2.

Estimate of chiral perturbation parametersobtained.

CONCLUSION

βC = 6.0− 6.2.

CONCLUSION

βC = 6.0− 6.2.

CONCLUSION

βC = 6.0− 6.2.

CONCLUSION

βC = 6.0− 6.2.

FURTHER WORK AND REFERENCES

Same model has already been successfully applied to pion andnucleon form factors(C.A. Dominguez, Phys. Lett. B 512 (2001) 331C.A. Dominguez, T. Thapedi, JHEP 10 (2004) 003)

Future work looks to include the proton in the time-like region.

Other references of interest:V. Pascalutsa, M. Vanderhaegen and S.N. Yang, Phys. Rept.437 (2007) 125P.H. Frampton, Dual resonance models, Benjamin (1974)

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