RECENT ADVANCESINTHECALCULATIONOF … · RECENT ADVANCESINTHECALCULATIONOF HADRONFORMFACTORSUSING DYSON-SCHWINGER EQUATIONSOFQCD ... Jorge Segovia (Argonne National ... symmetry pattern

RECENT ADVANCES IN THE CALCULATION OF

HADRON FORM FACTORS USING

DYSON-SCHWINGER EQUATIONS OF QCD

Jorge Segovia

Argonne National Laboratory

Thomas Jefferson National Accelerator Facility

Newport News (Virginia)

December 4th, 2013

Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 1/45

Collaborators

Argonne National LaboratoryCraig D. Roberts (supervisor)

Ian C. Cloet

University of Science and Technology of ChinaChen Chen

Shaolong Wan

Forschungszentrum JulichSebastian M. Schmidt

University of AdelaideLei Chang

Nanjing UniversityHong-Shi Zong


The challenge of QCD

Quantum Chromodynamics is the only known example in nature of a nonperturvativefundamental quantum field theory

QCD have profound implications for our understanding of the real-world:

Explain how quarks and gluons bind together to form hadrons.

Origin of the 98% of the mass in the visible universe.

Given QCD’s complexity:

The best promise for progress is a strong interplay between experiment and theory.

Emergent phenomena

ւ ցQuark and gluon confinement Dynamical chiral symmetry breaking

↓ ↓Colored particles

have never been seenisolated

Hadrons do notfollow the chiralsymmetry pattern

Neither of these phenomena is apparent in QCD’s Lagrangianyet!

They play a dominant role determining characteristics of real-world QCD


Emergent phenomena: Confinement

Confinement is associated with dramatic, dynamically-driven changes in the analyticstructure of QCD’s propagators and vertices (QCD’s Schwinger functions)

Dressed-propagator for a colored state:

An observable particle is associated with a poleat timelike-P2.

When the dressing interaction is confining:

Real-axis mass-pole splits, moving into a pairof complex conjugate singularities.

No mass-shell can be associated with a particlewhose propagator exhibits such singularity.

Dressed-gluon propagator:

Confined gluon.

IR-massive but UV-massless.

mG ∼ 2− 4ΛQCD (ΛQCD ≃ 200MeV).

Any 2-point Schwinger function with an inflexionpoint at p2 > 0:→ Breaks the axiom of reflexion positivity→ No physical observable related with


Emergent phenomena: Dynamical chiral symmetry breaking

Spectrum of a theory invariant under chiral transformations should exhibit degenerateparity doublets

π JP = 0− m = 140MeV cf. σ JP = 0+ m = 500MeV

ρ JP = 1− m = 775MeV cf. a1 JP = 1+ m = 1260MeV

N JP = 1/2+ m = 938MeV cf. N(1535) JP = 1/2− m = 1535MeV

Splittings between parity partners are greater than 100-times the light quark massscale: mu/md ∼ 0.5, md = 4MeV

Dynamical chiral symmetry breaking

Mass generated from the interaction of quarks withthe gluon-medium.

Quarks acquire a HUGE constituent mass.

Responsible of the 98% of the mass of the proton.

(Not) spontaneous chiral symmetry breaking

Higgs mechanism.

Quarks acquire a TINY current mass.

Responsible of the 2% of the mass of the proton. 0 1 2 3

p [GeV]

0

0.1

0.2

0.3

0.4

M(p

) [G

eV

] m = 0 (Chiral limit)m = 30 MeVm = 70 MeV

effect of gluon cloudRapid acquisition of mass is


Theory tool: Dyson-Schwinger equations

Confinement and Dynamical Chiral Symmetry Breaking (DCSB) can be identified withproperties of dressed-quark and -gluon propagators and vertices (Schwinger functions)

Dyson-Schwinger equations (DSEs)

The quantum equations of motion of QCD whosesolutions are the Schwinger functions.

→ Propagators and vertices.

Generating tool for perturbation theory.

→ No model-dependence.

Nonperturbative tool for the study of continuum strongQCD.

→ Any model-dependence should be incorporated here.

Allows the study of the interaction between light quarksin the whole range of momenta.

→ Analysis of the infrared behaviour is crucial todisentangle confinement and DCSB.

Connect quark-quark interaction with experimentalobservables.

→ Elastic and transition form factors can be used toilluminate QCD (at infrared momenta).

0

0.1

0.2

0.3

0.4

M(p)

(GeV)

0 1 2 3 4

p (GeV)

α = 2.0

α = 1.8

α = 1.4

α = 1.0

−0.2

0

0.2

0.4

0.6

0.8

1.0

µpG

Ep/G

Mp

0 2 4 6 8 10

Q2 (GeV2)

α = 2.0α = 1.8α = 1.4α = 1.0

Ian C. Cloet and Craig D. RobertsarXiv:nul-th/1310.2651


The simplest example of DSEs: The gap equation

The quark propagator is given by the gap equation:

S−1(p) = Z2(iγ · p +mbm) + Σ(p)

Σ(p) = Z1

∫ Λ

qg2Dµν(p − q)

λa

2γµS(q)

λa

2Γν(q, p)

General solution:

S(p) =Z(p2)

iγ · p +M(p2)

Kernel involves:Dµν (p − q) - dressed gluon propagatorΓν(q, p) - dressed-quark-gluon vertex

0 1 2 3

p [GeV]

0

0.1

0.2

0.3

0.4

M(p

) [G

eV

]

m = 0 (Chiral limit)m = 30 MeVm = 70 MeV

effect of gluon cloudRapid acquisition of mass is

M(p2) exhibits dynamicalmass generation

Each of which satisfies its own Dyson-Schwinger equation

↓Infinitely many coupled equations

↓Coupling between equations necessitates truncation


Ward-Takahashi identities

Symmetries should be preserved by any truncation

↓Highly nontrivial constraint → failure implies loss of any connection with QCD

↓Symmetries in QFT are implemented by WTIs which relate different Schwinger functions

For instance, axial-vector Ward-Takahashi identity:

These observations show that symmetries relate the kernel of the gap equation – aone-body problem – with that of the Bethe-Salpeter equation – a two-body problem –


Bethe-Salpeter and Faddeev equations

Hadrons are studied via Poincare covariant bound-state equations

Mesons

A 2-body bound state problem in quantumfield theory.

Properties emerge from solutions ofBethe-Salpeter equation:

Γ(k;P) =

∫

d4q

(2π)4K(q, k;P)S(q+P) Γ(q;P) S(q)

The kernel is that of the gap equation.

=

iΓ

iS

iΓ

K

iS

Baryons

A 3-body bound state problem in quantumfield theory.

Structure comes from solving the Faddeevequation.

Faddeev equation: Sums all possible quantumfield theoretical interactions that can takeplace between the three quarks that define itsvalence quark content.

=aΨ

P

pq

pd Γb

Γ−a

pd

pq

bΨP

q


Diquarks inside baryons

The attractive nature of quark-antiquark correlations in a color-singlet meson is alsoattractive for 3c quark-quark correlations within a color-singlet baryon

Diquark correlations:

Empirical evidence in support of strong diquarkcorrelations inside the nucleon.

A dynamical prediction of Faddeev equationstudies.

In our approach: Non-pointlike color-antitripletand fully interacting.

Diquark composition of the nucleon and ∆

Positive parity states

ւ ցpseudoscalar and vector diquarks scalar and axial-vector diquarks

↓ ↓Ignored

wrong paritylarger mass-scales

Dominantright parity

shorter mass-scales

→ N ⇒ 0+, 1+ diquarks∆ ⇒ only 1+ diquark


Study of electromagnetic form factors - Motivation

A central goal of Nuclear Physics: understand the structure and properties of protonsand neutrons, and ultimately atomic nuclei, in terms of the quarks and gluons of QCD.

Elastic and transition form factors

ւ ցUnique window into its

quark and gluon structureHigh-Q2 reach by

experiments

↓ ↓Distinctive information on theroles played by confinement

and DCSB in QCD

Probe the excited nucleonstructures at perturbative andnon-perturbative QCD scales

CEBAF Large Acceptance Spectrometer (CLAS)

Most accurate results for the electroexcitation amplitudesof the four lowest excited states.

They have been measured in a range of Q2 up to:

8.0GeV2 for ∆(1232)P33 and N(1535)S11 .

4.5GeV2 for N(1440)P11 and N(1520)D13 .

The majority of new data was obtained at JLab.

Upgrade of CLAS up to 12GeV → CLAS12 (New generation experiments in 2015)


The γ∗N → ∆ reaction

Two ways in order to analyze the structure of the ∆-resonances

ւ ցπ-mesons as a probe photons as a probe

↓ ↓complex relatively simple

BUT: B(∆ → γN) . 1%

This became possible with the advent of intense, energetic electron-beam facilities

Reliable data on the γ∗p → ∆+ transition:

Available on the entire domain 0 ≤ Q2 . 8GeV2.

Isospin symmetry implies γ∗n → ∆0 is simply related with γ∗p → ∆+.

γ∗p → ∆+ data has stimulated a great deal of theoretical analysis:

Deformation of hadrons.

The relevance of pQCD to processes involving moderate momentum transfers.

The role that experiments on resonance electroproduction can play in exposingnon-perturbative phenomena in QCD:

The nature of confinement and Dynamical Chiral Symmetry Breaking.


The electromagnetic current

The electromagnetic current can be generally written as:

Jµλ(K ,Q) = Λ+(Pf )Rλα(Pf ) iγ5 Γαµ(K ,Q) Λ+(Pi )

Incoming nucleon: P2i = −m2

N , and outgoing delta: P2f = −m2

∆.

Photon momentum: Q = Pf − Pi , and total momentum: K = (Pi + Pf )/2.

The on-shell structure is ensured by the N- and ∆-baryon projection operators.

The composition of the 4-point function Γαµ is determined by Poincare covariance:

Convenient to work with orthogonal momenta ↔ Simplify its structure considerably

↓Not yet the case for K and Q ↔ ∆(m∆ −mN) 6= 0 ⇒ K · Q 6= 0

↓We take instead K⊥

µ = TQµν Kν and Q

Vertex decomposes in terms of three (Jones-Scadron) form factors:

Γαµ = k

[

λm

2λ+(G∗

M − G∗E )γ5εαµγδK

⊥γ Qδ − G∗

ETQαγT

Kγµ − iς

λmG∗C QαK

⊥µ

]

,

Magnetic dipole ⇒ G∗M Electric quadrupole ⇒ G∗

E Coulomb quadrupole ⇒ G∗C


Extraction of the form factors

The Jones-Scadron form factors are:

G ∗M = 3(s2 + s1),

G ∗E = s2 − s1,

G ∗C = s3.

G∗M,Ash

vs G∗M,J−S

G∗M,Ash = G∗

M,J−S

(

1 +Q2

(m∆ +mN)2

)− 12

The scalars are obtained from the following Dirac traces and momentum contractions:

s1 = n

√

ς(1 + 2d )

d − ςTKµν K

⊥λ Tr[γ5Jµλγν ],

s2 = nλ+

λm

TKµλTr[γ5Jµλ],

s3 = 3nλ+

λm

(1 + 2d )

d − ςK⊥

µ K⊥λ Tr[γ5Jµλ].

We have used the following notation:

n =

√

1 − 4d 2

4ik λm

, λ± =(m∆±mN )2+Q2

2(m2∆+m2

N)

, ς =Q2

2(m2∆ + m2

N),

d =m2

∆ − m2N

2(m2∆ + m2

N ), λm =

√

λ+λ−, k =

√

3

2

(

1 +m∆

mN

)

.

G. Eichman et al., Phys. Rev. D 85, 093004 (2012).


Experimental results and theoretical expectations

I.G. Aznauryan and V.D. Burkert Prog. Part. Nucl Phys. 67, 1-54 (2012)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

10-1

1Q2 (GeV2)

G* M

,Ash

/3G

D

-7-6-5-4-3-2-1012

RE

M (

%)

-35

-30

-25

-20

-15

-10

-5

0

10-1

1Q2 (GeV2)

RS

M (

%)

The REM ratio is measured to beminus a few percent.

The RSM ratio does not seem to

settle to a constant at large Q2.

pQCD predictions

For Q2 → ∞G∗M → 1/Q4.

REM → +100%.

RSM → constant.

CQM predictions

Without quark orbital angularmomentum:

REM → 0.

RSM → 0.

SU(6) predictions

〈p|µ|∆+〉 = 〈n|µ|∆0〉〈p|µ|∆+〉 = −

√2 〈n|µ|n〉

Data do not support these predictions

↓Our aim: try to understand this longstanding puzzle


Electromagnetic current description in the quark-diquark picture

To compute the electromagnetic properties of the γ∗N∆ reaction in a givenframework, one must specify how the photon couples to its constituents.

There are six contributions to the current.

The picture shows the one-loop diagrams

1 Coupling of the photon to the dressed quark.

2 Coupling of the photon to the dressed diquark:

Elastic transition.

Induced transition.

scalar diquark correlations are absent from the ∆-resonance

↓Only axial-vector diquark correlations contribute in the top

and middle diagrams

Each diagram can be expressed like the electromagneticcurrent:

Γµλ = Λ+(Pf )Rλα(Pf )Jµα(K ,Q)Λ+(Pi )

i

iΨ ΨPf

f

P

Q

i

iΨ ΨPf

f

P

Q

scalaraxial vector

i

iΨ ΨPf

f

P

Q


Elastic form factor of the proton in the quark-diquark picture (I)

Each diagram can be expressed in a similar way:

Γµ = Λ+(Pf )Jµ(K ,Q)Λ+(Pi )

Photon coupling directly to a dressed-quark with the diquark acting as a bystander

Initial state and final state: Proton

Two axial-vector diquark isospin states:

(I , Iz ) = (1, 1) → flavor content: uu(I , Iz ) = (1, 0) → flavor content: ud

In the isospin limit, they appear with relative

weighting: (−√

2/3) : (√

1/3)

Therefore

Jscalarµ =

√

1

3

√

1

3eu I

udµ 6= 0

Jaxialµ =

√

2

3

√

2

3ed I

uuµ +

√

1

3

√

1

3eu I

udµ = 0

ΨiΨfPf Pi

Q

Pf

Pi

Q

ΨiΨf

axial − vector

scalar

Hard contributions appear in the microscopic description of the elastic form factor ofthe proton


Elastic form factor of the proton in the quark-diquark picture (II)

Remaining diagrams: Photon interacting with diquarks

H.L.L. Roberts et al. Phys. Rev. C 83, 065206 (2011)

ΨiΨf PiPf

Q

Q

ΨiΨfPf Pi

axial scalar 0 2 4 6 8 10-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Q2HGeV2

L

GE

,M,Q

1+

,G

M1+

GQ1+

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

Q2HGeV2

L

G0+Γ

1+

,GΠΓΡ

Composite object

↓Electromagnetic radius is nonzero (rqq & rπ)

↓Softer contribution to the form factors

Soft contributions appear in the microscopic description of the elastic form factor ofthe proton


Transition form factor of γ∗N∆ in the quark-diquark picture

Diagrams in which the photon interact with diquarks appear

Photon coupling directly to a dressed-quark with the diquark acting as a bystander

Initial state: Proton

Two axial-vector diquark isospin states:

(I , Iz ) = (1, 1) → flavor content: uu(I , Iz ) = (1, 0) → flavor content: ud

In the isospin limit, they appear with relative

weighting: (−√

2/3) : (√

1/3)

Final state: ∆+

Same isospin states of axial-vector diquark.

Different weighting due to I∆ = 3/2:

(√

1/3) : (√

2/3)

Therefore

J1,axialµα = −

√

2

3

√

1

3ed I

1uuµα +

√

1

3

√

2

3eu I

1udµα =

√2

3I1qqµα (K ,Q)

ΨiΨfPf Pi

Q

Pf

Pi

Q

ΨiΨf

axial − vector

scalar

Soft and still hard contributions appear in the microscopic description of the γ∗N∆electromagnetic reaction


General observation

GpM vs G∗p

M

Similar contributions in both cases:

G∗pM should fall asymptotically at the same rate as Gp

M .

By isospin considerations:

G∗nM should fall asymptotically at the same rate as G∗p

M .

Hold SU(6):

〈p|µ|∆+〉 ∝ 〈n|µ|∆0〉 ∝ 〈p|µ|p〉 .


Simple framework

Symmetry preserving Dyson-Schwinger equation treatment of a vector × vectorcontact interaction

Gluon propagator: Contact interaction.

g2Dµν(p − q) = δµν4παIR

m2G

Truncation scheme: Rainbow-ladder.

Γaν(q, p) = (λa/2)γν

Quark propagator: Gap equation.

S−1(p) = iγ · p +m+ Σ(p)

= iγ · p +M

M ∼ 0.4GeV = constant.

Implies momentum independent BSAs.

Baryons: Faddeev equation.

mN = 1.14GeV m∆ = 1.39GeV

(masses reduced by meson-cloud effects)

Gap equation

Σ

γ

S Γ

D

=

Bethe-Salpeter equation

=

iΓ

iS

iΓ

K

iS

Faddeev equation

=aΨ

P

pq

pd Γb

Γ−a

pd

pq

bΨP

q


Series of papers establishes strengths and limitations

Used judiciously, produces results indistinguishable from most-sophisticatedinteractions employed in the rainbow ladder truncation of QCDs DSEs

1 Features and flaws of a contact interaction treatment of the kaonC. Chen, L. Chang, C.D. Roberts, S.M. Schmidt S. Wan and D.J. WilsonPhys. Rev. C 87 045207 (2013). arXiv:1212.2212 [nucl-th]

2 Spectrum of hadrons with strangenessC. Chen, L. Chang, C.D. Roberts, S. Wan and D.J. WilsonFew Body Syst. 53 293-326 (2012). arXiv:1204.2553 [nucl-th]

3 Nucleon and Roper electromagnetic elastic and transition form factorsD.J. Wilson, I.C. Cloet, L. Chang and C.D. RobertsPhys. Rev. C 85, 025205 (2012). arXiv:1112.2212 [nucl-th]

4 π- and ρ-mesons, and their diquark partners, from a contact interactionH.L.L. Roberts, A. Bashir, L.X. Gutierrez-Guerrero, C.D. Roberts and D.J. WilsonPhys. Rev. C 83, 065206 (2011). arXiv:1102.4376 [nucl-th]

5 Masses of ground and excited-state hadronsH.L.L. Roberts, L. Chang, I.C. Cloet and C.D. RobertsFew Body Syst. 51, 1-25 (2011). arXiv:1101.4244 [nucl-th]

6 Abelian anomaly and neutral pion productionH.L.L. Roberts, C.D. Roberts, A. Bashir, L.X. Gutierrez-Guerrero and P.C. TandyPhys. Rev. C 82, 065202 (2010). arXiv:1009.0067 [nucl-th]

7 Pion form factor from a contact interactionL.X. Gutierrez-Guerrero, A. Bashir, I.C. Cloet and C.D. RobertsPhys. Rev. C 81, 065202 (2010). arXiv:1002.1968 [nucl-th]


Weakness of contact-interaction

A truncation which produces Faddeev amplitudes that are independent of relativemomentum:

Underestimates the quark orbital angular momentum content of the bound-state.

Suppresses the two-loop diagrams.

G∗M (Q2 = 0) ind.-p DSE kernels dep.-p DSE kernels

axial-diquark(∆) − axial-diquark(p) 0.85 0.96axial-diquark(∆) − scalar-diquark(p) 0.18 1.27

Two sets of results

1 Original result.

2 Improved version:Rescale the axial-diquark(∆) − scalar-diquark(p) diagram using:

1 +gas/aa

1 + Q2/m2ρ

axial(∆)-scalar(p) = axial(∆)-axial(p) for G∗M (Q2 = 0)

Incorporate dressed quark-anomalous magnetic moment ⇔ Consequence of the DCSB.


Q2-behaviour of G ∗M,Jones−Scadron

(I)

G∗M,J−S cf. Experimental data and dynamical models

Solid-black: Original result.Dashed-blue: Improved version.Dot-Dashed-green: Dynamical model without meson-cloud effects.Dotted-brown: Estimation with a sophisticated interaction.

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0 1 2 30

1

2

3

x=Q 2m Ρ

2

GM*

Both computed curves are consistent with data for Q2 & 2m2ρ.

They are in marked disagreement at infrared momenta.

Similarity between Dashed-blue and Dot-Dashed-green.

The discrepancy results from the omission of meson-cloud effects.


Q2-behaviour of G ∗M,Jones−Scadron

(II)

Transition cf. elastic magnetic form factors

æ æ æ æ æ æ æ æ æ æ

0 2 4 6 8 100.8

0.9

1.0

1.1

x=Q 2m Ρ

2

HΜ

nΜ

N*LH

GM*

NG

MnL

Solid-black: Proton.

Red-points: Neutron.

Fall-off rate of G∗M,J−S (Q

2) in the γ∗p → ∆+ must much that of GM(Q2).

With isospin symmetry:〈p|µ|∆+〉 = −〈n|µ|∆0〉

so same is true of the γ∗n → ∆0 magnetic form factor.

These are statements about the dressed quark core contributions

→ Outside the domain of meson-cloud effects, Q2 & 2GeV2


Q2-behaviour of G ∗M,Ash

Presentations of experimental data typically use the Ash convention

G∗M,Ash

(Q2) falls faster than a dipole.

There is no sound reason to expect:

G∗M,Ash/G

nM ∼ constant

Jones-Scadron should exhibit:

G∗M,J−S/G

nM ∼ constant

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0 2 4 6 8 10

1

2

3

x=Q 2m Ρ

2

GM

,Ash

*G

D

Two main reasons

Meson-cloud effects

↓Provide more than 30% for Q2 . 2m2

ρ

↓These contributions are very soft

↓They disappear rapidly

G∗M,Ash

vs G∗pM,J−S

G∗M,Ash = G∗

M,J−S

(

1 +Q2

(m∆ +mN )2

)− 12

↓A factor 1/Q of difference

↓Provides material damping for Q2 & 4m2

ρ


Electric and coulomb quadrupoles

REM = RSM = 0 in SU(6)-symmetric CQM.

Deformation of the hadrons involved.

Modification of the structure of the transitioncurrent. ⇔

RSM : Good description of the rapid fallat large momentum transfer.

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0.0 0.5 1.0 1.5 2.0

0

-4

-8

-12

x=Q 2m Ρ

2

RS

MH%L

REM : A particularly sensitive measure oforbital angular momentum correlations.

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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

-2

-4

-6

-8

x=Q 2m Ρ

2R

EMH%L

Solid-black: Original

Dashed-blue: Improved

Dotted-brown: Sophisticated

REM = 0 at x ∼ 1.50 (contact)at x ∼ 3.25 (sophisticated).

Even a contact interaction produces correlationsthat are comparable with those observed empirically.


Large Q2-behaviour of the quadrupole ratios

Helicity conservation arguments in pQCD should apply equally to aninternally-consistent symmetry-preserving treatment of a contact interaction

REMQ2→∞

= 1, RSMQ2→∞

= constant

0 20 40 60 80 100-0.5

0.0

0.5

1.0

x=Q 2m Ρ

2

RS

M,R

EM

Observations:

Truly asymptotic Q2 is required before predictions are realized.

REM = 0 at an empirical accessible momentum and then REM → 1.

RSM → constant. Curve contains the logarithmic corrections expected in QCD.


The ∆ elastic form factors

The small-Q2 behavior of the ∆ elastic form factors is a necessary element incomputing the γ∗N → ∆ transition form factors.

The electromagnetic current can be generally written as:

Jµ,λω(K ,Q) = Λ+(Pf )Rλα(Pf ) Γµ,αβ(K ,Q) Λ+(Pi )Rβω(Pi )

Vertex decomposes in terms of four form factors:

Γµ,αβ(K ,Q) =

[

(F∗1 + F∗

2 )iγµ − F∗2

m∆Kµ

]

δαβ −[

(F∗3 + F∗

4 )iγµ − F∗4

m∆Kµ

]

QαQβ

4m2∆

The multipole form factors:

GE0(Q2), GM1(Q

2), GE2(Q2), GM3(Q

2),

are functions of F∗1 , F

∗2 , F

∗3 and F∗

4 .

They are obtained by any four sensible projection operators. Physical interpretation:

GE0 and GM1 → Momentum space distribution of ∆’s charge and magnetization.

GE2 and GM3 → Shape deformation of the ∆-baryon.


No experimental data

Since one must deal with the very short ∆-lifetime (τ∆ ∼ 10−16τπ+ ):

Little is experimentally known about the elastic form factors.

Lattice-regularised QCD results are usually used as a guide.

Lattice-regularised QCD produce ∆-resonance masses that are very large:

Approach mπ mρ m∆

Unquenched I 0.691 0.986 1.687Unquenched II 0.509 0.899 1.559Unquenched III 0.384 0.848 1.395Hybrid 0.353 0.959 1.533Quenched I 0.563 0.873 1.470Quenched II 0.490 0.835 1.425Quenched III 0.411 0.817 1.382

We artificially inflate the mass of the ∆-baryon

Black solid line: DSEs + contact interaction + m∆ = 1.8GeV

Blue dashed line: DSEs + sophisticated interaction (+ m∆ = 1.8GeV)


GE0 and GM1 with(out) an inflated quark core mass

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Observation

The ∆ elastic form factors are very sensitive to m∆

Lattice regularised QCD produce ∆-resonance masses that are very large.

Form factors obtained therewith are a poor guide to the ∆’s properties.


The Ω− elastic form factors - General remarks

The Ω−-baryon’s electromagnetic properties must be more amenable to measurement

Magnetic dipole moment has measured with some precision:

µΩ− = −(2.02± 0.05)µN

The Ω− consists of three valence s-quarks:

Only decays via the weak interaction → Significantly more stable.

Ω’s Lifetime: τΩ− ∼ 1013τ∆ ∼ 10−3τπ+ .

kaon-cloud effects are smaller than pion-cloud effects:

OZI rule: Meson-loops dominated by kaons in Ω−.

Kaons are heavier than pions → Meson-loop effects are smaller than in the ∆baryon.

Dressed-quark-core should be a good approximation for the Ω−

mthe = 1.76GeV ∼ 1.67GeV = mexp

mthe,∆ = 1.39GeV ∼ 1.23GeV = mexp,∆

The value mπ should affect less to masses of higher excited states inLattice-regularised QCD:

Therefore one expects results in better agreement.


The Ω− elastic form factors - Results

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Bound state amplitudes (BSAs)

Exclusive reactions can be described in terms of Poincare-covariant hadronbound-state amplitudes (BSAs)

Bethe-Salpeter amplitudes → Mesons

Faddeev amplitudes → Baryons

This approach has been used widely:

Elastic and transition electromagnetic form factors.

Strong decays of hadrons.

Semileptonic and nonleptonic weak decays of heavy mesons.

The BSAs are predictions of the framework and the associated computationalscheme is applicable on the entire domain of accessible momentum transfers.

Truncations must be employed in formulating the problem.

Issues related to the construction of veracious truncation schemes.

In equal-time quantization, the wave function for a hadron is a frame dependentconcept:

As it is defined by observations of different space points at a fixed time.

Boost operators are interaction dependent, i.e. are dynamical.


Parton distribution amplitudes (PDAs)

In high energy scattering experiments particles move at near speed of light.

Natural to quantize a theory at equal light-front time: τ = (t + z)/√2.

Light-front wave functions, ψ(xi , ~k⊥i ), have many remarkable properties:

Provide a frame-independent representation of hadrons.

Have a probability interpretation - as close as QFT gets to QM.

Do not depend on the hadrons 4-momentum; only internal variables: xi and ~k⊥i .

Boosts are kinematical - not dynamical!!

PDAs are (almost) observables and are related to light-front wave functions:

ϕ(xi ) =

∫

d2~k⊥i ψ(xi , ~k⊥i )

xi ≡ L-F fraction of the bound-state total-momentum carried by the quark.

Hard exclusive hadronic reactions can be expressed in terms of the PDAs.

Example: Electromagnetic form factor of light pseudoscalar meson:

∃Q0 >ΛQCD | Q2Fπ(Q2)

Q2>Q20≈ 16παs (Q

2)f 2πw2ϕ,

wϕ =1

3

∫ 1

0dx

1

xϕπ(x) ,

The scale Q0 and PDAs are not determined by the analysis framework.


Connection between BSAs and PDAs

PDAs may be obtained as light-front projections of the hadron BSAs.

Lei Chang et al., Phys. Rev. Lett. 110, 132001 (2013); 111, 141802 (2013).

Ian C. Cloet et al., Phys. Rev. Lett. 111, 092001 (2013).

Example: The pion’s PDA:

fπ ϕπ(x) = Z2

∫

d4k

(2π)2δ(k+ − xp+)Tr

[

γ+γ5 S(k)Γπ (k, p)S(k − p)]

S(k)Γπ (k, p)S(k − p) is the pion’s Bethe-Salpeter amplitude.

ϕπ(x) is the axial-vector projection of the pion’s BSA onto the light-front.

Two features emerged in developing the connection between BSAs and PDAs:

The so-called asymptotic PDA:

ϕasy(x) = 6x(1− x)

provides an unacceptably poor description of meson internal structure at all scaleswhich are either currently accessible or foreseeable in experiments.

→ Expected!! evolution with energy scale in QCD is logarithmic.

The leading twist PDAs for light-quark mesons are concave functions at allenergy scales:

Eliminates the possibility of “humped” distributions.Enables one to obtain the meson’s PDA from a very limited number of moments.


Computing PDAs from moments (I)

Energy scale dependence of PDA (c.f. DGLAP equations for PDFs):

τd

dτϕ(x , τ) =

∫ 1

0dy V (x , y)ϕ(y , τ)

This evolution equation has a solution of the form:

ϕ(x ; τ) = ϕasy(x)

[

1 +∞∑

j=1,2,...

a3/2j (τ)C

(3/2)j (2x − 1)

]

QCD is invariant under the collinear conformal group SL(2;R) in Q2 → ∞ limit.

Gegenbauer-α = 3/2 polynomials are the irreducible representations of this group.

Nonperturbative methods in QCD typically provide access to moments of the PDA:

〈(x − x)m〉τϕ =

∫ 1

0dx (x − x)m ϕ(x ; τ), x = 1− x

Accurate approx. to ϕ(x ; τ) is obtained by usingjust first few terms of expansion.

Leads to ϕ(x) whose behaviour is not concave→ “humped” distributions.

Slow convergence and spurious oscillations 0.0 0.25 0.50 0.75 1.00.0

0.5

1.0

1.5

x

ΦΠHxL


Computing PDAs from moments (II)

ALTERNATIVE

Accept that, at all accessible scales, PDAs determined by nonperturbative dynamics

PDAs should be reconstructed from moments by using Gegenbauer polynomials oforder α, with this order determined by the moments themselves, not fixed beforehand.

ϕ(x ; τ) = Nα [x(1− x)]α−1/2

[

1 +

js∑

j=1,2,...

aαj (τ)C(α)j (2x − 1)

]

Quick convergence → only js = 2 needed for the pion (there is no js = 1)

One may project ϕ(x ; τ) onto the asymptotic form

a3/2j (τ) =

2

3

2 j + 3

(j + 2) (j + 1)

∫ 1

0dx C

(3/2)j (2x − 1)ϕ(x ; τ),

Therewith obtaining all coefficients necessary to represent any computeddistribution in the conformal form without ambiguity or difficulty.

One may determine the distribution at any τ ′ 6= τ using the ERBL evolution

equations for the coefficients a3/2j (τ), i = 1, 2, . . ..


Meson PDA moments from Lattice

At most two nontrivial moments of ϕ(x) can be computed using numerical simulationsof lattice-regularised QCD.

Proposed procedure:

Reconstruct PDAs from lattice-QCD moments using:

ϕ(x) = xα (1 − x)β/B(α, β).

Valid also for mesons comprised from quarks with nondegenerated masses.

Evolving the distribution obtained to another momentum scale:Project the formula onto the asymptotic form.Employ the ERBL evolution equations.

Lattice-regularised QCD data:

Indistinguishable:

ϕV|| ∼ ϕV

⊥ ∼ ϕP

The appearance of precision ismisleading: 23% for pion!!

1/2 = 〈(x − x)2〉ϕasy ≤ 〈(x − x)2〉ϕ(x − x)2〉ϕ ≤ 〈(x − x)2〉ϕpoint = 1/3

meson 〈(x − x)n〉 163 × 32 243 × 64

π n = 2 0.25(1)(2) 0.28(1)(2)ρ‖ n = 2 0.25(2)(2) 0.27(1)(2)K n = 1 0.035(2)(2) 0.036(1)(2)K∗‖

n = 1 0.037(1)(2) 0.043(2)(3)

K n = 2 0.25(1)(2) 0.26(1)(2)K∗‖

n = 2 0.25(1)(2) 0.25(2)(2)


Light pseudoscalar and vector mesons with equal-mass valence-quarks

0.0 0.25 0.50 0.75 1.00.0

0.5

1.0

x

ΦduHxLH1

63´

32L

A B

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ΦduHxLH2

43´

64L

A

C

• Curve A: DSE prediction for the chiral pion

• Region B: 163 × 32, αud = βud = 0.50+0.20−0.16

• Region C: 243 × 64, αud = βud = 0.29+0.15−0.13

Observations:

PDAs obtained from the two different lattices have overlapping error bands.However, the differences are material.

It appears that the 243 × 64 lattice configurations produce a form of φud (x) thatis too broad.


Light pseudoscalar and vector mesons with no equal-mass valence-quarks

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D

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F E

D

• Curve A: DSE prediction for the chiral pion

• Region D: 243 × 64, αus = 0.48+0.19−0.16 βus = 0.38+0.23

−0.15

• Region E: 243 × 64, αus = 0.62+0.15−0.13 βus = 0.53+0.14

−0.12

Observations:

Shift in position relative to the peak in the pion’s PDA.

SU(3)-flavor symmetry is broken nonperturbatively at the level of 10%.

PDA shift toward φasy when increasing energy scale.

Region E: 2GeV → 10GeV. PDA evolution is extremely slow.


ERBL evolution

Evolution with energy scale in QCD is LOGARITHMIC

Example: first moment:

Vertical dashed line: Marks τ = 100GeV.

Horizontal dotted line: Marks 0.5× 〈x − x〉 → not reach until τ = 7.3TeV.

0.0 2 4 6 80.0

0.02

0.04

Ln@ΖΖ2D

<x-

x>

suH2

43´

64L

Positive value: s-quark carries more momentum than the u-quark.

It is responsible for the shift in position.


Conclusions The γ∗N → ∆ transition form factors

Jones-Scadron G∗pM :

G∗pM falls asymptotically at the same rate as Gp

M .

Compatible with isospin symmetry and pQCD predictions.

Data do not fall unexpectedly rapid once the kinematic relation betweenJones-Scadron and Ash conventions is properly account for.

REM and RSM :

Limits of pQCD, REM → 1 and RSM → constant, are apparent in ourcalculation but truly asymptotic Q2 is required before the predictions are realized.

Strong diquark correlations within baryons produce a zero in the transitionelectric quadrupole at Q2 ∼ 2GeV

2.

The ∆ elastic form factors

The ∆ elastic form factors are very sensitive to m∆.

Lattice-regularised QCD produce ∆-resonance masses that are very large, theform factors obtained therewith should be interpreted carefully.

Parton distribution amplitudes (PDAs)

The method introduced enables one to obtain a pointwise accurate approximationto meson PDAs from limited moments (information).

PDAs of pseudoscalar and vector mesons with (non)-equal-mass valence-quarksare presented.

At all energy scales, the PDAs are concave functions whose dilation andasymmetry express the strength of dynamical chiral symmetry breaking.


RECENT ADVANCESINTHECALCULATIONOF … · RECENT ADVANCESINTHECALCULATIONOF HADRONFORMFACTORSUSING DYSON-SCHWINGER EQUATIONSOFQCD ... Jorge Segovia (Argonne National ... symmetry pattern

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