Wigner distributions of kaon using light-cone quark …...Satvir Kaur and Harleen Dahiya Dr. B. R....

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Wigner distributions of kaon using light-cone quarkmodel

Satvir Kaur and Harleen Dahiya

Dr. B. R. Ambedkar National Institute of Technology, Jalandhar (INDIA).MENU-2019, Pittsburgh, USA

dahiyah@nitj.ac.in

June 02-07, 2019

Satvir Kaur and Harleen Dahiya (NITJ) Wigner distributions of kaon June 02-07, 2019 1 / 40

Overview

1 Light-Front Dynamics

2 PDFs, GPDs and TMDs

3 Wigner Distributions

4 Generalized Transverse Momentum-Dependent Parton Distributions (GTMDs)

5 Light-Cone Quark Model

6 Results

7 Conclusions

Light-Front Dynamics

Light-front framework was first introduced by Dirac in 1949.

-P. A. M. Dirac, Rev. Mod. Phys. 21, 392 (1949).

The dynamical framework is formulated in Minkowski space.

One of the most outstanding problem of particle physics is to unravel theinternal structure of hadrons.

QCD provides a fundamental description of hadronic and nuclear structureand dynamics in terms of their quark and gluon degrees of freedom.

The front-form was the most interesting achievement, applied to examine the”constituent picture of the hadron”.

Instant form v/s Front form

Figure: The instant form Figure: The front form

All measurements are made atfixed t i.e. at x0 = 0.

All measurements are made atfixed light-cone time x+ i.e. atx+ = x0 + x3 = 0.

Energy-momentum dispersion relation:

In the instant form,p0 =

√~p2 + m2.

In the front form,

p− =~p2⊥+m2

No square-root for the Hamiltonian in light front form.Therefore, simplifes the dynamical structure.

Instant-form vacuum is infinitelycomplex.

Light-front vacuum issimple, as all the massive fluctua-tions in the ground state are absent.

Light-front provides the wavefunctions (LFWFs) required to describe the structureand dynamics of hadrons in terms of their constituents (quarks and gluons).

- S. J. Brodsky, G. F. de Teramond, Phys. Rev. D 77, 056007 (2008).

PDFs, GPDs and TMDs

Parton Distribution Functions (PDFs)Generalized Parton Distributions (GPDs)Transverse Momentum-Dependent Parton Distributions (TMDs)

PDFs, GPDs and TMDs

Parton Distribution Functions (PDFs)

To understand the structure of the hadron in terms of quarks andgluons, different categories of parton distributions are present.

PDFs were introduced by Feynman in 1969.

PDF f (x) imparts an information about the probability of finding a partoncarrying a longitudinal momentum fraction x inside the hadron.

But how partons are distributed in the plane transverse to the motionof hadron?

This missing information was then compensated in Generalized PartonDistributions (GPDs).

PDFs, GPDs and TMDs

Generalized Parton Distributions (GPDs)

GPDs f (x , ζ, t), are functions of longitudinal momentum fraction x = p+

carried by the active quark, longitudinal momentum transferred ζ = − ∆+

(skewness) and squared of total momentum transferred to the hadron t = ∆2.

For zero skewness i.e. ζ = 0, one is left with GPDs f (x , 0, t), where the

momentum transferred is only in the transverse direction i.e. t = −~∆2⊥.

There is only one quark(anti-quark) GPD at the leading twist in case ofpseudoscalar mesons (spin-0). However, 4 quark GPDs are present forspin-1/2 composite systems.

PDFs, GPDs and TMDs

Transverse Momentum-Dependent PartonDistributions (TMDs)

To get the information of hadron structure in momentum space, transversemomentum-dependent parton distributions (TMDs) were introduced.

TMDs f (x , ~k⊥), are function of longitudinal momentum fraction carried by

the active quark x = k+

P+ and the quark transverse momentum ~k⊥.

TMDs represent three-dimensional hadron picture in momentum space.

PDFs, GPDs and TMDs

-Image courtesy: A. Sig-nori

There is one quark TMD atleading-twist in case of kaon,

while 8 quarkand gluon TMDs atthe leading twist in

case of nucleon(spin-1/2).

In the figure,partons

(quarks and gluons)are like fishes

confined inside afishbowl (the proton).

Each parton has its owncollinear and transverse

velocity, indicated by blackand colored arrows

respectively.

Wigner Distributions

To understand the hadron structure more precisely,the joint position and momentum distributions (quantum analog to theclassical phase-space distributions) Wigner distributions were introduced.

Wigner distributions were first introduced by E. Wigner in 1932.

-E. Wigner Phys. Rev. 70, 749 (1932)

These distributions are the quasi-probabilistic distributions.

In QCD, Wigner distributions were first introduced by Xiangdong Ji

-X. -d. Ji, Phys. Rev. Lett. 91, 062001 (2003).

ρ[Γ](b⊥, k⊥, x) =

∫d2∆⊥(2π)2

e−i∆⊥·b⊥W [Γ](∆⊥, k⊥, x),

where W [Γ](∆⊥, k⊥, x) in the meson state |M(P)〉 at fixed light-cone timez+ = 0 is defined as

W [Γ](~∆⊥, ~p⊥, x) =

∫dz−d2z⊥

(2π)3e ip·z 〈M(P ′′)| ψ(−z/2)ΓW[− z

2 ,z2 ]ψ(z/2) |M(P ′)〉 .

Here, Γ indicates the Dirac γ-matrix, specifically

γ+ : corresponding to unpolarized quark,

γ+γ5 : corresponding to longitudinally-polarized quark,

iσj+γ5: corresponding to transversely-polarized parton, where j = 1 or 2,depending upon the polarization direction of quark.

For meson, the two-particle Fock state expansion is defined as

|M(P)〉 =∑λ1,λ2

∫dxd2k⊥√

x(1− x)16π3|x , k⊥, λ1, λ2〉ψSz (x , k⊥, λ1, λ2),

where λ1 and λ2 describes the helicities of quark and anti-quark in mesonrespectively.

-W. Qian and B. -Q. Ma, Phys. Rev. D 78, 074002 (2008).

There are 4 independent twist-2 quark (anti-quark) Wigner distributions,depending on various polarization configurations for spin-0 hadron(pseudoscalar mesons).

There are 16 independent Wigner distributions for the case of proton.

-Z. -L. Ma and Z. Lu, Phys. Rev. D 98, 054024 (2018).

For unpolarized quark in unpolarized hadron, we have

ρUU(b⊥, k⊥, x) = ρ[γ+](b⊥, k⊥, x)

For the longitudinally-polarized quark in the unpolarized hadron, we have

ρUL(b⊥, k⊥, x) = ρ[γ+γ5](b⊥, k⊥, x)

For the transversely-polarized quark in the unpolarized hadron, we have

ρjUT (b⊥, k⊥, x) = ρ[iσj+γ5](b⊥, k⊥, x).

Wigner distributions are designed as

where x : polarization state of kaon, which always remain unpolarized.and y : polarization state of quark or anti-quark.

Generalized Transverse Momentum-Dependent Parton Distributions (GTMDs)

Generalized Transverse Momentum-DependentParton Distributions (GTMDs)

The twist-2 GTMDs related to unpolarized pseudoscalar meson with spin-0 isconnected with Wigner correlator or operator as

W [γ+] = F1,

W [γ+γ5] =iεij⊥k

i⊥∆j⊥

W [iσj+γ5] =iεij⊥k

1 +iεij⊥∆i

H∆1 ,

with the anti-symmetric tensor εij⊥ = ε−+ij , ε0123 = 1 and σab = i2 [γa, γb].

All the leading-twist GTMDs are the function of six variables, we have(x , ζ, k2

⊥, k⊥.∆⊥,∆2⊥).

Generalized Transverse Momentum-Dependent Parton Distributions (GTMDs)

GTMDs are known asmother distributions.

One can obtain GPDsand TMDs from GTMDs

under certain kinematiclimits.

Light-Cone Quark Model

The light-cone quark model is described for the bound states of mesons. TheFock state expansion of kaon is considered to be |k〉 = |qq〉ψqq.The light-cone wavefunctions ψSz=0(x , k⊥, λ1, λ2), by considering differentcombinations of helicities of active quark and spectator anti-quark in kaon,are given as

ψ0(x , k⊥, ↑, ↑) = − 1√2

k1 − ik2√k2⊥ + l2

ϕ(x , k⊥),

ψ0(x , k⊥, ↑, ↓) =1√2

(1− x)m1 + xm2√k2⊥ + l2

ϕ(x , k⊥),

ψ0(x , k⊥, ↓, ↑) = − 1√2

(1− x)m1 + xm2√k2⊥ + l2

ϕ(x , k⊥),

ψ0(x , k⊥, ↓, ↓) = − 1√2

k1 + ik2√k2⊥ + l2

ϕ(x , k⊥),

l2 = (1− x)m21 + xm2

2 − x(1− x)(m1 −m2)2.

The momentum-space wavefunction ϕ(x , k⊥) is described by using theBrodsky-Hwang-Lepage prescreption, we have

-B. -W. Xiao, et al., Eur. Phys. J. A 15, 523 (2002).

ϕ(x , k⊥) = A exp

k2⊥+m2

x +k2⊥+m2

1−x8β2

− (m21 −m2

(k2⊥+m2

x +k2⊥+m2

)],The parameters are taken asmass of u-quark : m1 = 0.25 GeV ,mass of s-quark : m2 = 0.5 GeV ,β = 0.393 GeVA = 74.2.

Since kaon is considered as a two particle constituent, so both u-quark andanti-s quark Wigner distributions and GTMDs will contribute.

The distributions have a support interval x ∈ [−1, 1], in which DGLAP andERBL regions are included.

-Image Courtesy : M. Diehl, Phys. Rept. 388, 41 (2003).

We restrict our calculations in DGLAP regions. For anti-quark and quark, therespective regions −1 < x < 0 and 0 < x < 1, as we take ζ = 0 in our work.

The Wigner correlation operator W [Γ](∆⊥, k⊥, x) for Γ = γ+, γ+γ5, iσj+γ5 in

overlap form as

W [γ+] =1

[ψ†0(x , k′′, ↑, ↑)ψ0(x , k′, ↑, ↑)

+ψ†0(x , k′′, ↓, ↑)ψ0(x , k′, ↓, ↑)+ψ†0(x , k′′, ↑, ↓)ψ0(x , k′, ↑, ↓)+ψ†0(x , k′′, ↓, ↓)ψ0(x , k′, ↓, ↓)

]W [γ+γ5] =

[ψ†0(x , k′′, ↑, ↑)ψ0(x , k′, ↑, ↑)

−ψ†0(x , k′′, ↓, ↑)ψ0(x , k′, ↓, ↑)+ψ†0(x , k′′, ↑, ↓)ψ0(x , k′, ↑, ↓)−ψ†0(x , k′′, ↓, ↓)ψ0(x , k′, ↓, ↓)

]W [iσj+γ5] =

16π3εij⊥[(−i)iψ0(x , k′′, ↑, ↑)ψ0(x , k′, ↓, ↑)

+(i)iψ0(x , k′′, ↓, ↑)ψ0(x , k′, ↑, ↑)+(−i)iψ0(x , k′′, ↑, ↓)ψ0(x , k′, ↓, ↓)+(i)iψ0(x , k′′, ↓, ↓)ψ0(x , k′, ↑, ↓)

]Satvir Kaur and Harleen Dahiya (NITJ) Wigner distributions of kaon June 02-07, 2019 21 / 40

The initial and final momenta of the struck quark(anti-quark) are describedby

k′⊥ = k⊥1 + (1− x1)∆⊥

and k′′⊥ = k⊥1 − (1− x1)∆⊥

For the anti-quark(quark) spectator, the initial and final states of momentaare defined as

k′⊥s = k⊥2 − x2∆⊥

and k′′⊥s = k⊥2 + x2∆⊥

respectively.

The required conditions which should be satisfied for initial and final statesare

2∑i=1

xi = 1,2∑

k′⊥i = 0⊥, and

2∑i=1

k′′⊥i = 0⊥.

For the struck quark and spectator anti-quark :Struck quark transverse momentum : k⊥1 = k⊥.Longitudinal momentum fraction carried by struck quark : x1 = x .Spectator anti-quark transverse momentum : k⊥2 = −k⊥.Longitudinal momentum fraction carried by spectator anti-quark : x2 = 1− x .

For the struck anti-quark and spectator quark :Struck anti-quark transverse momentum : k⊥1 = −k⊥.Longitudinal momentum fraction carried by struck anti-quark : x1 = −x .Spectator anti-quark transverse momentum : k⊥2 = k⊥.Longitudinal momentum fraction carried by spectator anti-quark : x2 = 1 + x .

The anti-s quark distributions are defined from u-quark distributions as

ρu(x , k⊥,b⊥,m1,m2) = −ρs(−x ,−k⊥,b⊥,m2,m1),

F u1 (x , ζ, k2

⊥, k⊥.∆⊥,∆2⊥,m1,m2) = −F s

1 (−x , ζ, k2⊥,−k⊥.∆⊥,∆

2⊥,m2,m1),

In the present work, the GTMDs results are restricted to ζ = 0.

We evaluate the Wigner distributions ρUU , ρUL and ρjUTin impact-parameter plane, by taking the fixed transverse momentum ask⊥ = k⊥ey , where k⊥ = 0.2 GeV ,in momentum plane, by taking the fixed impact-parameter co-ordinate

b⊥ = b⊥ey , where b⊥ = 0.4 GeV−1.

Results

Wigner distribution for unpolarized u-quark in unpolarized kaon, we have

ρuUU(b⊥, k⊥) =1

∫d∆xd∆y

(2π)2

∫dx cos(∆xbx + ∆yby )

×[k2⊥ − (1− x)2 ∆2

+ ((1− x)m1 + xm2)2]

× ϕ†u(x , k′′⊥)ϕu(x , k′⊥)√k′′2 + l2u

√k′2 + l2u

for unpolarized s-quark in unpolarized kaon, we have

ρsUU(b⊥, k⊥) = − 1

∫d∆xd∆y

(2π)2

∫dx cos(∆xbx + ∆yby )

×[k2⊥ − (1 + x)2 ∆2

+ ((1 + x)m2 − xm1)2]

× ϕ†s (x , k′′⊥)ϕs(x , k′⊥)√k′′2 + l2s

√k′2 + l2s

Results

(a)-2 -1 0 1 2

bx (GeV-1 )

by(GeV

ρUUu (bx,by)

(b)-2 -1 0 1 2

bx (GeV-1 )

by(GeV

ρUUs_

(bx ,by )

-0.175

-0.125

-0.075

-0.025

(c)-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

kx (GeV)

ky(GeV

ρUUu (kx ,ky )

(d)-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

kx (GeV)ky(GeV

ρUUs_

(kx ,ky )

-0.225

-0.175

-0.125

-0.075

-0.025

Figure: Wigner distribution ρUU for u-quark and s-quark in impact-parameter plane andmomentum plane.

Results

We observe that in case of s-quark distribution is more concentrated at thecenter as compared to u-quark in the impact-parameter plane (b⊥-plane) andmomentum plane (k⊥-plane).

The distributions in both cases are opposite in direction.

The distribution ρUU is related to the unpolarized TMD f1 and unpolarizedGPD H, when integrated upon certain limits.

Results

Wigner distribution for longitudinally-polarized u-quark in unpolarized kaon

ρuUL(b⊥, k⊥) =1

∫d∆xd∆y

(2π)2

∫dx sin(∆xbx + ∆yby )

×(1− x)(ky∆x − kx∆y )ϕ†u(x , k′′⊥)ϕu(x , k′⊥)√

k′′2 + l2u√

k′2 + l2u

For longitudinally-polarized s-quark in unpolarized kaon, we have

ρsUL(b⊥, k⊥) =1

∫d∆xd∆y

(2π)2

×(1 + x)(kx∆y − ky∆x)ϕ†s (x , k′′⊥)ϕs(x , k′⊥)√

k′′2 + l2s√

k′2 + l2s

Results

(a)-2 -1 0 1 2

bx (GeV-1 )

by(GeV

ρULu (bx,by)

(b)-2 -1 0 1 2

bx (GeV-1 )

by(GeV

ρULs_

(bx ,by )

(c)-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

kx (GeV)

ky(GeV

ρULu (kx ,ky )

(d)-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

kx (GeV)ky(GeV

ρULs_

(kx ,ky )

-1. × 10-2

1. × 10-2

Figure: Wigner distribution ρUL for u-quark and s-quark in impact-parameter plane andmomentum plane.

Results

In impact-parameter plane, the distribution ρUL shows a dipole behaviour,whereas it shows the positive behaviour at bx > 0 for both quark andanti-quark.

However, in transverse momentum plane, it reverses the direction, ρUL ispositive at bx < 0.

No TMD or GPD is present corresponding to ρUL. Generally, this Wignerdistribution is related to orbital angular momentum (OAM), but for thepseudoscalar meson, the net quark and anti-quark spin and OAM are zero.

Results

Wigner distribution for transversely-polarized u-quark in unpolarized kaon

ρuUT (b⊥, k⊥) =1

∫d∆xd∆y

(2π)2

×((1− x)m1 + xm2)(1− x)∆y

× ϕ†u(x , k′′⊥)ϕu(x , k′⊥)√k′′2 + l2u

√k′2 + l2u

For transversely-polarized s-quark in unpolarized kaon, we have

ρsUT (b⊥, k⊥) = − 1

∫d∆xd∆y

(2π)2

×((1 + x)m2 − xm1)(1 + x)∆y

× ϕ†s (x , k′′⊥)ϕs(x , k′⊥)√k′′2 + l2s

√k′2 + l2s

Results

(a)-2 -1 0 1 2

bx (GeV-1 )

by(GeV

ρUTu (bx,by)

(b)-2 -1 0 1 2

bx (GeV-1 )

by(GeV

ρUTs_

(bx ,by )

(c)-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

kx (GeV)

ky(GeV

ρUTu (kx ,ky )

(d)-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

kx (GeV)ky(GeV

ρUTs_

(kx ,ky )

Figure: Wigner distribution ρUT for u-quark and s-quark in impact-parameter plane andmomentum plane.

Results

We observe that the distribution vanishes by considering the quark(antiquark)spin direction along the direction of quark(anti-quark) transverse co-ordinate.

A dipolar distribution is observed for both quark and anti-quark with theopposite polarities in impact-parameter plane.

In transverse momentum plane ρUT is more focused at the center(px = py = 0) in case of s-quark, while it is extended to the periphery incase of u-quark.

ρUT relates to the T-odd Boer-Mulder TMD h1⊥ and T-odd GPD ET at TMD

limit and IPD limit respectively. But here, we are not able to extract anyTMD or GPD, because gluon contribution is not considered in this work.

Results

We evaluate the mother distributions viz. GTMDs F1, G1 and H∆1 . For

ζ = 0, the GTMD corresponding to transversely-polarized quark(anti-quark)in unpolarized kaon, Hk

1 = 0.

We plot the variation of GTMDs w.r.t the longitudinal momentum fractioncarried by quark (x) or anti-quark (−x),at constant value of ∆⊥ (= 1 GeV ) and at different values of (k⊥),at constant value of k⊥ (= 0.2 GeV ) and at different values of (∆⊥).

The GTMD F1 is related to the unpolarized Wigner distribution ρUU and isevaluated as

F(u)1 =

[k2⊥ − (1− x)2 ∆2

+((1− x)m1 + xm2

× ϕ†u(x , k′′⊥)ϕu(x , k′⊥)√k′′2 + l2u

√k′2 + l2u

F(s)1 = − 1

[k2⊥ − (1 + x)2 ∆2

+((1 + x)m2 − xm1

× ϕ†s (x , k′′⊥)ϕs(x , k′⊥)√k′′2 + l2s

√k′2 + l2s

Results

k⊥=0.05 GeV

k⊥=0.2 GeV

k⊥=0.3 GeV

u-quark

s--quark

-1.0 -0.5 0.0 0.5 1.0

Δ⊥=0.3 GeV

Δ⊥=0.5 GeV

Δ⊥=1.0 GeV

u-quark

s--quark

-1.0 -0.5 0.0 0.5 1.0

Figure: The plot of GTMD F1(x , k2⊥, k⊥.∆⊥,∆

2⊥) w.r.t x for u and s-quarks (i) at

differnt values of k⊥ with fixed ∆⊥ = 1 GeV (left panel) and (ii) at differnt values of∆⊥ with fixed k⊥ = 0.2 GeV (right panel).

Results

The GTMD G1 is related to Wigner distribution ρUL. They both are relatedto quark spin and orbital angular momentum correlation.

G(u)1 = − M2

16π3(1− x)

ϕ†u(x , k′′⊥)ϕu(x , k′⊥)√k′′2 + l2u

√k′2 + l2u

G(s)1 =

16π3(1 + x)

ϕ†s (x , k′′⊥)ϕs(x , k′⊥)√k′′2 + l2s

√k′2 + l2s

k⊥=0.05 GeV

k⊥=0.2 GeV

k⊥=0.3 GeV

u-quark

s--quark

-1.0 -0.5 0.0 0.5 1.0

Δ⊥=0.3 GeV

Δ⊥=0.5 GeV

Δ⊥=1.0 GeV

u-quark

s--quark

-1.0 -0.5 0.0 0.5 1.0

Figure: The plot of GTMD G1(x , k2⊥, k⊥.∆⊥,∆

2⊥) w.r.t x for u and s-quarks (i) at

differnt values of k⊥ with fixed ∆⊥ = 1 GeV (left panel) and (ii) at differnt valuesof ∆⊥ with fixed k⊥ = 0.2 GeV (right panel).

Results

The GTMD G1 is related to Wigner distribution ρUL. They both are relatedto quark spin and orbital angular momentum correlation.

H∆(u)1 = − M

((1− x)m1 + xm2

)(1− x)

ϕ†u(x , k′′⊥)ϕu(x , k′⊥)√k′′2 + l2u

√k′2 + l2u

H∆(s)1 =

((1 + x)m2 − xm1

)(1 + x)

ϕ†s (x , k′′⊥)ϕs(x , k′⊥)√k′′2 + l2s

√k′2 + l2s

k⊥=0.05 GeV

k⊥=0.2 GeV

k⊥=0.3 GeV

u-quark

s--quark

-1.0 -0.5 0.0 0.5 1.0

Δ⊥=0.3 GeV

Δ⊥=0.5 GeV

Δ⊥=1.0 GeV

u-quark

s--quark

-1.0 -0.5 0.0 0.5 1.0

Figure: The plot of GTMD H∆1 (x , k2

⊥, k⊥.∆⊥,∆2⊥) w.r.t x for u and s-quarks (i) at

differnt values of k⊥ with fixed ∆⊥ = 1 GeV (left panel) and (ii) at differnt valuesof ∆⊥ with fixed k⊥ = 0.2 GeV (right panel).

Conclusions

We evaluate the Wigner distributions and GTMDs by considering kaon to bea composite of quark and anti-quark.

The overlap representation of light-cone wavefunctions is used for thecalculations.

We take different polarization configurations of quark (anti-quark) inunpolarized kaon.

The GTMDs are calculated, when there is no longitudinal momentumtransfer (ζ = 0).

The results provide rich and interesting information on the internal structureof the kaon.

Conclusions

-By S. J. Brodsky

Wigner distributions of kaon using light-cone quark …...Satvir Kaur and Harleen Dahiya Dr. B. R....

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