revealing diquarks in the proton

Revealing Diquarks in the Proton

Jorge Segovia

Institut de Fısica d’Altes Energies

Universitat Autonoma de Barcelona

Juan de la Cierva

Theory Seminar at IFAEMarch 2nd, 2018

Main collaborators (in this research line):

C. Chen (IFT-UNESP), Y. Lu (NJU), C. Mezrag (INFN) and C.D. Roberts (ANL).

Jorge Segovia ([email protected]) Revealing diquarks in the proton 1/37

The Nucleon’s electromagnetic current

+ The electromagnetic current can be generally written as:

Jµ(K ,Q) = ie Λ+(Pf ) Γµ(K ,Q) Λ+(Pi )

Incoming/outgoing nucleon momenta: P2i = P2

f = −m2N .

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

The on-shell structure is ensured by the Nucleon projection operators.

+ Vertex decomposes in terms of two form factors:

Γµ(K ,Q) = γµF1(Q2) +1

2mNσµνQνF2(Q2)

+ The electric and magnetic (Sachs) form factors are a linear combination of the

Dirac and Pauli form factors:

GE (Q2) = F1(Q2)−Q2

4m2N

F2(Q2)

GM(Q2) = F1(Q2) + F2(Q2)

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

GE ⇒ Momentum space distribution of nucleon’s charge.

GM ⇒Momentum space distribution of nucleon’s magnetization.


Phenomenological aspects (I)

+ Perturbative QCD predictions for the Dirac and Pauli form factors:

F p1 ∼ 1/Q4 and F p

2 ∼ 1/Q6 ⇒ Q2F p2 /F

p1 ∼ constant

+ Consequently, the Sachs form factors scale as:

GpE ∼ 1/Q4 and Gp

M ∼ 1/Q4 ⇒ GpE/G

pM ∼ constant

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• Jones et al., Phys. Rev. Lett. 84 (2000) 1398.

• Gayou et al., Phys. Rev. Lett. 88 (2002) 092301.

• Punjabi et al., Phys. Rev. C71 (2005) 055202.

• Puckett et al., Phys. Rev. Lett. 104 (2010) 242301.

• Puckett et al., Phys. Rev. C85 (2012) 045203.


Phenomenological aspects (II)

Updated perturbative QCD prediction

Q2F p2 /F

p1 ∼ constant ê ì ë ì ê Q2F p

2 /Fp1 ∼ ln2

[Q2/Λ2

]+ The prediction has the important feature that it includes components of the

quark wave function with nonzero orbital angular momentum.

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0 1 2 3 4 5 6 7 8 90.0

1.0

2.0

3.0

4.0

Q2@GeV

2D

Q2

F2p�F

1p

Curve: ln2(Q2/Λ2) for Λ = 0.3 GeV which is normalized to the data at 2.5 GeV 2

→ Λ is a soft scale parameter related to the size of the nucleon.


Phenomenological aspects (III)


Our goal

In view of these facts it is of significant interest to look for the (non-perturbative)

origin of the observed Q2-dependence of the Dirac and Pauli form factors


Non-perturbative QCD:Confinement and dynamical chiral symmetry breaking (I)

Hadrons, as bound states, are dominated by non-perturbative QCD dynamics

Explain how quarks and gluons bind together ⇒ Confinement

Origin of the 98% of the mass of the proton ⇒ DCSB

Emergent phenomena

↙ ↘Confinement DCSB

↓ ↓Coloredparticles

have neverbeen seenisolated

Hadrons donot followthe chiralsymmetry

pattern

Neither of these phenomena is apparent in QCD’s Lagrangian

however!

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


Non-perturbative QCD:Confinement and dynamical chiral symmetry breaking (II)

From a quantum field theoretical point of view

Emergent phenomena could be associated with dramatic,

dynamically driven changes in the analytic structure of

QCD’s Schwinger functions (propagators and vertices).

+ Dressed-quark propagator in Landau gauge:

S−1(p) = Z2(iγ·p+mbm)+Σ(p) =

(Z(p2)

iγ · p + M(p2)

)−1

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

Light quarks acquire a HUGE constituent mass.

Responsible of the 98% of the mass of the proton andthe large splitting between parity partners.

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 cloud

Rapid acquisition of mass is

+ Dressed-gluon propagator in Landau gauge:

i∆µν = −iPµν∆(q2), Pµν = gµν − qµqν/q2

An inflexion point at p2 > 0.

Breaks the axiom of reflexion positivity.

No physical observable related with.


Dyson-Schwinger equations

The Schwinger functions are solutions of the quantum equations of motion (≡ DSEs)


Theory tool based on Dyson-Schwinger equations

+ Advantages:

Continuum quantum field theory approach that provides access to:

Perturbative regime of QCD → No model-dependence.

Nonperturbative regime of QCD → Model-dependence should be incorporated here.

Poincare covariant formulation → Valid for all momentum scales.

Cover full quark mass range → between chiral limit and the heavy quark domain.

+ The main caveat concerns the complexity of this framework:

Truncation of the infinite set of equations.

Modelling in the nonperturbative regime.

Ansatze are constrained by symmetry properties, multiplicative renormalizability,perturbative limits, few model parameters related with fundamental quantities


Infinite system of coupled integral equations: Truncation

α(k2) = πη7(

k2

Λ2

)2e− η

2k2

Λ2 + 12π11NC−2Nf

1−e−k2/Λ2

12

ln[e2−1+(1+k2/Λ2QCD

)2]


The bound-state problem in quantum field theory

Extraction of hadron properties from poles in qq, qqq, qqqq... scattering matrices

Use scattering equation (inhomogeneous BSE) toobtain T in the first place: T = K + KG0T

Homogeneous BSE forBS amplitude:

+ Baryons. A 3-body bound state problem in quantum field theory:

Faddeev equation in rainbow-ladder truncation

Faddeev equation: Sums all possible quantum field theoretical exchanges andinteractions that can take place between the three dressed-quarks that define itsvalence quark content.


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:

A tractable truncation of the Faddeevequation.

In Nc = 2 QCD: diquarks can form colorsinglets with are the baryons of the theory.

In our approach: Non-pointlike color-antitripletand fully interacting. Thanks to G. Eichmann.

Diquark composition of the Nucleon

Positive parity state

↙ ↘pseudoscalar and vector diquarks scalar and axial-vector diquarks

↓ ↓Ignored

wrong paritylarger mass-scales

Dominantright parity

shorter mass-scales


Diquark properties

Meson BSE Diquark BSE

+ Owing to properties of charge-conjugation, a diquark with spin-parity JP may beviewed as a partner to the analogous J−P meson:

Γqq(p;P) = −∫

d4q

(2π)4g2Dµν(p − q)

λa

2γµ S(q + P)Γqq(q;P)S(q)

λa

2γν

Γqq(p;P)C† = −1

2

∫d4q

(2π)4g2Dµν(p − q)

λa

2γµ S(q + P)Γqq(q;P)C†S(q)

λa

2γν

+ Whilst no pole-mass exists, the following mass-scales express the strength andrange of the correlation:

m[ud ]0+= 0.7−0.8 GeV, m{uu}1+

= 0.9−1.1 GeV, m{dd}1+= m{ud}1+

= m{uu}1+

+ Diquark correlations are soft, they possess an electromagnetic size:

r[ud ]0+& rπ , r{uu}1+

& rρ, r{uu}1+> r[ud ]0+


Remark about the 3-gluon vertex

+ A Y-junction flux-tube picture of nucleon structure isproduced in quenched lattice QCD simulations that usestatic sources to represent the proton’s valence-quarks.

F. Bissey et al. PRD 76 (2007) 114512.

+ This might be viewed as originating in the 3-gluonvertex which signals the non-Abelian character of QCD.

+ These suggest a key role for the three-gluon vertex in nucleon structure if they wereequally valid in real-world QCD: finite quark masses and light dynamical/sea quarks.

G.S. Bali, PRD 71 (2005) 114513.

The dominant effect of non-Abelian multi-gluon vertices is expressed in the formationof diquark correlations through Dynamical Chiral Symmetry Breaking.


The quark+diquark structure of the nucleon (I)

Faddeev equation in the quark-diquark picture

Ppd

pq

Ψa =P

pq

pd

ΨbΓa

Γb

Dominant piece in nucleon’s eight-componentPoincare-covariant Faddeev amplitude: s1(|p|, cos θ)

There is strong variation with respect to botharguments in the quark+scalar-diquark relativemomentum correlation.

Support is concentrated in the forward direction,cos θ > 0. Alignment of p and P is favoured.

Amplitude peaks at (|p| ∼ MN/6, cos θ = 1),whereat pq ∼ pd ∼ P/2 and hence the naturalrelative momentum is zero.

In the anti-parallel direction, cos θ < 0, support isconcentrated at |p| = 0, i.e. pq ∼ P/3, pd ∼ 2P/3.


The quark+diquark structure of the nucleon (II)

+ A nucleon (and kindred baryons) can be viewedas a Borromean bound-state, the binding withinwhich has two contributions:

Formation of tight diquark correlations.

Quark exchange depicted in the shaded area.

Ppd

pq

Ψa =P

pq

pd

ΨbΓa

Γb

+ The exchange ensures that diquark correlations within the nucleon are fullydynamical: no quark holds a special place.

+ The rearrangement of the quarks guarantees that the nucleon’s wave functioncomplies with Pauli statistics.

+ Modern diquarks are different from the old static, point-like diquarks whichfeatured in early attempts to explain the so-called missing resonance problem.

+ The number of states in the spectrum of baryons obtained is similar to that foundin the three-constituent quark model, just as it is in today’s LQCD calculations.

+ Modern diquarks enforce certain distinct interaction patterns for the singly- anddoubly-represented valence-quarks within the proton.


Baryon-photon vertex

One must specify how the photoncouples to the constituents within

the baryon.

⇓

Six contributions to the current inthe quark-diquark picture

⇓

1 Coupling of the photon to thedressed quark.

2 Coupling of the photon to thedressed diquark:

å Elastic transition.

å Induced transition.

3 Exchange and seagull terms.

One-loop diagrams

i

iΨ ΨPf

f

P

Q

i

iΨ ΨPf

f

P

Q

scalaraxial vector

i

iΨ ΨPf

f

P

Q

Two-loop diagrams

i

iΨ ΨPPf

f

Q

Γ−

Γ

µ

i

i

X

Ψ ΨPf

f

Q

P Γ−

µi

i

X−

Ψ ΨPf

f

P

Q

Γ


Sachs electric and magnetic form factors

+ Q2-dependence of proton form factors:

0 1 2 3 4

0.0

0.5

1.0

x=Q2�mN

2

GEp

0 1 2 3 4

0.0

1.0

2.0

3.0

x=Q2�mN

2

GMp

+ Q2-dependence of neutron form factors:

0 1 2 3 40.00

0.04

0.08

x=Q2�mN

2

GEn

0 1 2 3 4

0.0

1.0

2.0

x=Q2�mN

2

GMn

QCD-based

NJL-model

Experiment


Unit-normalized ratio of Sachs electric and magnetic form factors

Both CI and QCD-kindred frameworks predict a zero crossing in µpGpE/G

pM

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2D

ΜnG

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QCD-based

NJL-model

The possible existence and location of the zero in µpGpE/G

pM is a fairly direct measure

of the nature of the quark-quark interaction


A world with only scalar diquarks (I)

The singly-represented d-quark in the proton≡ u[ud ]0+

is sequestered inside a soft scalar diquark correlation.

+ Observation:

diquark-diagram ∝ 1/Q2 × quark-diagram

Contributions coming from u-quark

Ψi

Ψi

Ψf

ΨfPf Pi

PiPf

Q

Q

Contributions coming from d-quark

Ψi

Ψi

Ψf

ΨfPf Pi

PiPf

Q

Q


A world with only scalar diquarks (II)

The d-quark contributions to the proton form factors should be suppressedrespect the u-quark contributions

+ Remind the experimental data...


A world with scalar and axial-vector diquarks (I)

The singly-represented d-quark in the proton isnot always (but often) sequestered inside a softscalar diquark correlation.

+ Observation:

P scalar ∼ 0.621, Paxial ∼ 0.287, Pmix = 0.092

Contributions coming from u-quark

Ψi

Ψi

Ψf

ΨfPf Pi

PiPf

Q

Q

Contributions coming from d-quark

Ψi

Ψi

Ψf

ΨfPf Pi

PiPf

Q

Q


A world with scalar and axial-vector diquarks (II)

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0.0

0.5

1.0

1.5

2.0

x=Q2�MN

2

x2F

1p

d,

x2F

1p

u

u-quark

d-quarkæ

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0.0

0.2

0.4

0.6

x=Q2�MN

2

HΚ

pdL-

1x

2F

2p

d,HΚ

puL-

1x

2F

2p

u

u-quark

d-quark

+ Observations:

F d1p is suppressed with respect F u

1p in the whole range of momentum transfer.

The location of the zero in F d1p depends on the relative probability of finding 1+

and 0+ diquarks in the proton.

F d2p is suppressed with respect F u

2p but only at large momentum transfer.

There are contributions playing an important role in F2, like the anomalousmagnetic moment of dressed-quarks or meson-baryon final-state interactions.


Comparison between worlds (I)

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2F

2d

Only scalar

Scalar and axial

Only axial


Comparison between worlds (II)

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1.0

2.0

3.0

4.0

x=Q2�MN

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@x

F2pD�F

1p

Total

Scalar all-wave

Scalar S-wave

Axial all-wave

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Total

Scalar all-wave

Scalar S-wave

Axial all-wave

+ Observations:

Axial-vector diquark contribution is not enough in order to explain the proton’selectromagnetic ratios.

Scalar diquark contribution is dominant and responsible of the Q2-behaviour ofthe the proton’s electromagnetic ratios.

Higher quark-diquark orbital angular momentum components of the nucleon arecritical in explaining the data.

The presence of higher orbital angular momentum components in the nucleon is aninescapable consequence of solving a realistic Poincare-covariant Faddeev equation


Hard (semi-)inclusive and (semi-)exclusive reactions are coming to the forefront ofhigh energy nuclear and particle physics


Multidimensional view of the nucleon structure


Hard exclusive hadronic reactions

+ Hard exclusive hadronic reactions can be expressed in terms of hadron distributionamplitudes (DAs).

Example: Electromagnetic form factor of light pseudoscalar meson:

∃Q0 >ΛQCD | Q2Fπ(Q2)Q2>Q2

0≈ 16παs(Q2)f 2πw 2

ϕ,

wϕ =1

3

∫ 1

0dx

1

xϕπ(x) ,

x ≡ Light-front fraction of the bound-state total-momentum carried by the quark.

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

In the example above

ϕπ(x) =

∫d2~k⊥ ψπ(x , ~k⊥)

~k⊥ ≡ Transverse coordinates of the light-front relative momentum.

+ Light-front wave functions, ψπ(x , ~k⊥), have many remarkable properties:

Do not depend on the hadron’s 4-momentum; only internal variables: x and ~k⊥.

Provide a frame-independent representation of hadrons.

Have a probability interpretation.


Proton’s distribution amplitude(s)

+ Light-front quantization allows to expand hadrons on a Fock basis; for a baryon:

|P,N〉 ∝∑β

Ψqqqβ |qqq〉+

∑β

Ψqqq,qqβ |qqq, qq〉+ . . .

where ΨN is the N-particles light-front wave functions → Non-perturbative physics.

+ The notion of hadron DAs refers to hadron-to-vacuum matrix elements ofnonlocal operators built of quark and gluon fields at light-like separations:

〈0|εijkui′α(a1z) [a1z, a0z]i′,i u

j′

β (a2z) [a2z, a0z]j′,j dk′γ (a3z) [a3z, a0z]k′,k |P, λ〉

|P, λ〉 denotes the proton state with momentum P, P2 = M2 and helicity λ.

u, d are the quark-field operators.

Greek letters α, β, γ stand for Dirac indices. Latin letters i , j , k refer to color.

z is an arbitrary light-like vector, z2 = 0, the ai are real numbers.

The gauge-factors [x , y ] are defined as

[x , y ] = P exp

[ig

∫ 1

0dt (x − y)µA

µ(tx + (1− t)y)

],

and render the matrix element in gauge-invariant.

V. Braun et al. Nucl. Phys. B589 (2000) 381-409


Proton’s distribution amplitude(s)

+ Taking into account Lorentz covariance, spin and parity of the nucleon, the mostgeneral decomposition of the matrix element involves 24 invariant functions:

4〈0|εijkuiα(a1z)ujβ(a2z)dkγ(a3z)|P〉 =

= S1MCαβ (γ5N)γ + S2M2Cαβ (6zγ5N)γ + P1M (γ5C)αβ Nγ + P2M

2 (γ5C)αβ (6zN)γ

+ V1 (6PC)αβ (γ5N)γ + V2M (6PC)αβ (6zγ5N)γ + V3M (γµC)αβ (γµγ5N)γ

+ V4M2 (6zC)αβ (γ5N)γ + V5M

2 (γµC)αβ (iσµνzνγ5N)γ + V6M3 (6zC)αβ (6zγ5N)γ

+ A1 (6Pγ5C)αβ Nγ + A2M (6Pγ5C)αβ (6zN)γ + A3M (γµγ5C)αβ (γµN)γ

+ A4M2 (6zγ5C)αβ Nγ + A5M

2 (γµγ5C)αβ (iσµνzνN)γ + A6M3 (6zγ5C)αβ (6zN)γ

+ T1 (Pν iσµνC)αβ (γµγ5N)γ + T2M (zµPν iσµνC)αβ (γ5N)γ + T3M (σµνC)αβ (σµνγ5N)γ

+ T4M (PνσµνC)αβ (σµρzργ5N)γ + T5M2 (zν iσµνC)αβ (γµγ5N)γ

+ T6M2 (zµPν iσµνC)αβ (6zγ5N)γ + T7M

2 (σµνC)αβ (σµν 6zγ5N)γ

+ T8M3 (zνσµνC)αβ (σµρzργ5N)γ ,

V. Braun et al. Nucl. Phys. B589 (2000) 381-409


Proton’s DAs: Leading twist

+ Light-front quantization allows to expand hadrons on aFock basis; for a baryon:

|P,N〉 ∝∑β

Ψqqqβ |qqq〉+

∑β

Ψqqq,qqβ |qqq, qq〉+ . . .

Non-perturbative physics is contained in the N-particleslight-front wave functions, ΨN .

+ At leading twist: the matrix element defining the proton3-body wave function is written in terms of three Lorentzinvariant functions:

ϕas ≡ 120x1x2x3

〈0|εijkuiα(z1)ujβ(z2)dkγ(z3)|P, λ〉 =

1

4fN

[(/pC)αβ

(γ5N

+)γV (zip · n)

+(/pγ5C

)αβ

(N+)γA(zip · n)− (ipµσµνC)αβ

(γνγ5N

+)γT (zip · n)

]+ Two leading-twist distribution amplitudes can be defined:

ϕ(xi ) = V (xi )− A(xi ), T (xi ) ;[F (zip · n) =

∫Dx e−ip·n

∑i xi zi F (xi )

]+ Proton case at leading twist:

|P, ↑〉 =

∫[dx]

8√

6x1x2x3|uud〉⊗ [ϕ(x1, x2, x3)| ↑↓↑〉+ ϕ(x2, x1, x3)| ↓↑↑〉 − 2T (x1, x2, x2)| ↑↑↓〉]


Proton’s DAs: Lattice vs DSEs (I)

ϕNfN− ϕas

G.S. Bali et al. JHEP 1602 (2016) 070

The ϕ-DA is quite different withrespect the asymptotic one.

The peak of the ϕ-DA is shifted withrespect to the barycenter.

Similar features are found inLattice-QCD and DSEs.

The peak of the ϕ-distribution isshifted toward the region where thesingle quark carries most of thenucleon light-cone momentumfraction.

There is an asymmetry due to thepresence of different contributions, inparticular, scalar and axial-vectordiquarks.

ϕNfN

C. Mezrag et al. arXiv:1711.09101 [nucl-th]


Proton’s DAs: Lattice vs DSEs (II)

TfN− ϕas

G.S. Bali et al. JHEP 1602 (2016) 070

The T -DA is quite different withrespect the asymptotic one.

The peak of the T -DA is shifted withrespect to the barycenter.

Similar features are found inLattice-QCD and DSEs.

Our framework is fully consistent.Independent computation of ϕ and T ,and symmetries are fulfilled:

2T (x1, x2, x3) = ϕ(x1, x2, x3)+ϕ(x2, x3, x1)

TNfN

C. Mezrag et al. arXiv:1711.09101 [nucl-th]


Average values of the momentum fraction

Scalar S+AV Evolved Braun:2014wpa Bali:2015ykx

〈x1〉ϕ 0.434 0.392(5) 0.379(4) 0.372(7) 0.358(6)〈x2〉ϕ 0.283 0.291(2) 0.302(1) 0.314(3) 0.319(4)〈x3〉ϕ 0.283 0.316(4) 0.319(3) 0.314(7) 0.323(6)

〈x1〉T 0.358 0.354 0.349 0.343 0.34〈x2〉T 0.358 0.354 0.349 0.343 0.34〈x3〉T 0.283 0.291 0.302 0.314 0.319


Nucleon’s normalization/decay constant


Summary and conclusions

Quantum Field Theory vision of the proton:

+ Poincare covariance demands the presence of dressed-quark orbital angularmomentum in the nucleon.

+ Dynamical chiral symmetry breaking and its correct implementation produces pionsas well as strong electromagnetically-active diquark correlations.

Proton’s electromagnetic form factors:

+ The presence of strong diquark correlations within the nucleon is sufficient tounderstand empirical extractions of the flavor-separated form factors.

+ Scalar diquark dominance and the presence of higher orbital angular momentumcomponents are responsible of the Q2-behaviour of Gp

E/GpM and F p

2 /Fp1 .

Proton’s distribution amplitude:

+ It is a broad, concave function, whose maximum is shifted relative to the peak inQCD’s conformal limit.

+ The shift and asymmetry signal the presence of both scalar and pseudovectordiquark correlations in the nucleon.


revealing diquarks in the proton

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