INHA-NTG-112022 Gravitational form factors of the baryon octet with avor SU3 symmetry breaking Ho-Yeon Won1June-Young Kim1 2yand Hyun-Chul Kim1 3z

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INHA-NTG-11/2022

Gravitational form factors of the baryon octet with ﬂavor SU(3) symmetry breaking

Ho-Yeon Won,1, ∗June-Young Kim,1, 2, †and Hyun-Chul Kim1, 3, ‡

1Department of Physics, Inha University, Incheon 402-751, South Korea

2Theory Center, Jeﬀerson Lab, Newport News, VA 23606, USA

3School of Physics, Korea Institute for Advanced Study (KIAS), Seoul 02455, South Korea

(Dated: October 10, 2022)

We investigate the gravitational form factors of the baryon octet within the framework of the

SU(3) chiral quark-soliton model, considering the eﬀects of ﬂavor SU(3) symmetry breaking, and

the corresponding energy-momentum tensor distributions. We examine the eﬀects of ﬂavor SU(3)

symmetry breaking to the mass, angular momentum, pressure, and shear force distributions of the

baryon octet. We ﬁrst ﬁnd that a heavier baryon is energetically more compact than a lighter

one. For the spin distributions of the baryon octet, they are properly normalized to their spins

and are decomposed into the ﬂavor-singlet axial charge and the orbital angular momentum even

when the ﬂavor SU(3) symmetry is broken. While the eﬀects of the ﬂavor SU(3) symmetry breaking

diﬀerently contribute to the angular momentum distributions for the octet baryons, they are found

to be rather small. The spin and orbital angular momentum almost equally contribute to the

angular momentum distributions for the octet baryons. We also estimate the eﬀects of the ﬂavor

SU(3) symmetry breaking to the pressure and shear force distributions. Interestingly, even if we

include the eﬀects of the SU(3) ﬂavor symmetry breaking, the shear force distributions are kept to

be positive over r. It indicates that the Polyakov & Schweitzer local stability condition is kept to

be intact with the ﬂavor SU(3) symmetry broken. Lastly, we discuss how much the gravitational

form factors vary with the eﬀects of ﬂavor SU(3) symmetry breaking considered.

I. INTRODUCTION

It is of great importance to understand the mechanical structure of a baryon as much as the electromagnetic (EM)

one, since it reveals how the baryon is mechanically shaped by its partons. The gravitational form factors (GFFs) of

a baryon provide information on its mechanical properties such as the mass, spin, pressure, and shear force. At an

early stage, the GFFs were considered as a purely academic subject [1, 2] due to the diﬃculty in having access to

them experimentally. However, the generalized parton distributions (GPDs) have paved way for extracting the GFFs

experimentally, since the EM form factors and GFFs are deﬁned respectively as the ﬁrst and second Mellin moments

of the GPDs that can be measured by the hard exclusive process such as deeply virtual Compton scattering (DVCS)

or hard exclusive meson production. Recently, the ﬁrst measurement of the nucleon D-term form factors from DVCS

was reported [3–5]. The transition GPDs will soon be extracted from the experimental data on the hard exclusive

meson production p→∆++π−at Jeﬀerson Lab (JLab) [6, 7]. This measurement will lead to the N→∆ transition

GFFs [8]. Moreover, the upcoming Electric-Ion Collider (EIC) project will unveil the fractions of the mass and spin

of the nucleon, which are taken up by quarks and gluons inside it. It is well known that the quark content of the

nucleon spin is small (see a recent review [9]) and the strange quark is polarized negatively (∆s∼ −0.10 [10]). This

implies that the gluon spin and the orbital motion of the quarks and gluon should considerably contribute to the

nucleon spin. The future EIC project will provide a clue to the spin structure of the nucleon.

The GFFs for spin-1/2 particles parametrize the matrix element of the energy-momentum tensor (EMT) current [1,

2, 11, 12]. It was recently generalized to higher-spin particles [13] in a systematic way. Based on this parametrization,

the GFFs of the nucleon have been intensively investigated in various approaches [14–42]. The parity ﬂip transition [43–

45] and N→∆ transition [8] matrix elements of the EMT current were also parametrized. For a spin-1 particle,

the model-independent formalism for the GFFs and distributions were studied in Refs. [46–50] and the GFFs were

obtained by many theoretical works [37, 51–53]. The GFFs for a spin-3/2 particle were also examined [37, 54–56].

On the other hand, the GFFs of the baryon octet were much less studied [57]. To compute them, we need to consider

the ﬂavor SU(3) symmetry and its breakdown. Since the eﬀects of the ﬂavor SU(3) symmetry breaking on the GFFs

and related distributions have never been examined, it is worthwhile to investigate them. In particular, it is critical

to check whether the local and global stability conditions are satisﬁed with the ﬂavor SU(3) symmetry broken.

∗E-mail: hoywon@inha.edu

†E-mail: Jun-Young.Kim@ruhr-uni-bochum.de

‡E-mail: hchkim@inha.ac.kr

arXiv:2210.03320v1 [hep-ph] 7 Oct 2022

While the three-dimensional (3D) EMT distributions, which show how partons are spatially distributed inside a

baryon in the Breit frame (BF) [58], were obtained by the 3D Fourier transform of the corresponding GFFs, there

have been serious criticisms of the 3D distributions of the nucleon [59–63]. The 3D distributions depend on the shape

of the wave packet of a baryon, and this wave packet cannot be localized below the Compton wavelength. It brings

about ambiguous relativistic eﬀects of which the contribution is approximately by around 20 % for the nucleon. Thus,

they cannot be neglected anymore. To circumvent these ambiguous relativistic eﬀects, the two-dimensional (2D)

spatial EMT distributions have been considered in the inﬁnite momentum frame (IMF) or on the light-front (LF).

The ambiguous relativistic corrections are kinematically suppressed then [26, 41, 42]. However, we have to pay the

price that we lose information in the longitudinal direction.

There is yet another way of understanding the 3D distributions by deﬁning them using the Wigner phase-space

distribution. While it does not furnish the 3D distributions with the probabilistic meaning, it allows us to treat their

relativistic eﬀects. Moreover, it shows that the 3D BF and 2D IMF distributions can naturally be interpolated in

the Wigner sense. Thus, we can trace down the origin of the relativistic corrections to the 2D IMF distributions.

At the same time, a direct connection between the 3D BF and 2D IMF distributions was found to be the IMF Abel

transform [64, 65]. Note that, very recently, a novel concept of the 3D strict probabilistic distribution was introduced

to remove ambiguous relativistic corrections [66, 67]. In this work, we ﬁrst deﬁne the 3D BF distributions in the

Wigner sense and then map out the 2D IMF ones by using the IMF Abel transform.

In the current work, we will scrutinize the GFFs of the baryon octet and pertinent three-dimensional distributions

within the framework of a pion mean-ﬁeld approach or the chiral quark-soliton model (χQSM) [68–70]. E. Witten

in his seminal paper [71, 72] inspired the idea of the meson mean-ﬁeld approach. In the limit of a large number of

colors (Nc), the quantum ﬂuctuations are of order 1/Nc, so that it can be ignored. Thus, a baryon can be viewed

as Ncvalence quarks bound by a pion mean ﬁeld that arises from a classical solution of the equation of motion.

To put more explicitly, the presence of the Ncvalence quarks polarizes the vacuum, which produces the pion mean

ﬁeld. Then the Ncvalence quarks are also inﬂuenced by the pion mean ﬁeld in a self-consistent way. As a result, a

classical baryon appears as a chiral soliton with a hedgehog symmetry, which is composed of the Ncvalence quarks.

While we ignore the 1/Ncmesonic quantum ﬂuctuations, we have to consider the ﬂuctuations of the pion ﬁeld along

the zero-mode direction. The translational and rotational zero modes are related to the symmetries of the baryon.

Integrating over the zero modes completely, we can restore the correct quantum numbers of the baryon [68, 70]. The

χQSM successfully described various properties of the baryon octet and decuplet such as the EM properties [73–

79], axial-vector structures [80–82], tensor charges [83, 84], GFFs [65, 85–88], and partonic structures [89–97]. It

has also been extended to singly heavy baryons [98–106]. The GFFs of the singly heavy baryons were also studied

within the χQSM [88]. The χQSM can also be associated with quantum chromodyanmics (QCD) via the instanton

vacuum [107, 108]. The low-energy QCD eﬀective partition function can be derived from the instanton vacuum.

The dynamical quark mass, which is obtained from the Fourier transform of the fermionic zero mode, is originally

momentum-dependent. In the present work, we turn oﬀ the momentum dependence and introduce a regularization

scheme to tame the divergence coming from the quark loops.

The present work is organized as follows: In Section II, we deﬁne the GFFs of a spin-1/2 baryon from the matrix

elements of the EMT current. In Section III A, we explain the general formalism for the EMT distributions in both 3D

and 2D cases. In Section IV, we show how the GFFs and the EMT distributions can be computed within a framework

of the SU(3) χQSM, considering the eﬀects of the ﬂavor SU(3) symmetry breaking. In Sec V, the numerical results

for the GFFs and the EMT distributions of the octet baryons are presented and discussed. The last Section devotes

to the summary of the present work and draw conclusions.

II. GRAVITATIONAL FORM FACTORS OF A SPIN-1/2 PARTICLE

The symmetric EMT current in QCD can be derived by varying the QCD action under the Poincar´e transformation

according to N¨other’s theorem with the symmetrization imposed for a particle with nonzero spin [109–111]. More

directly, one can derive the symmetric EMT current by taking a functional derivative of the QCD action [1, 112] with

respect to the metric tensor of a curved background ﬁeld. The symmetric total EMT operator consists of the quark

(q) and gluon (g) parts, which are respectively expressed as

Tµν

q=i

4¯

ψqγµ−→

Dνψq+¯

ψqγν−→

Dµψq−¯

ψqγµ←−

Dνψq−¯

ψqγν←−

Dµψq−gµν ¯

ψqi

2−→

D − i

2←−

D − ˆmqψq,

Tµν

g=Fa,µηFa, ν

η+1

4gµν Fa,κηFa,

κη.(1)

Here, the covariant derivatives are deﬁned as ←→

Dµ=←→

∂µ+igtaAa

µ.tarepresent the SU(3) color group generators that

satisfy the commutation relations [ta, tb] = ifabctcand are normalized to be tr(tatb) = 1

2δab.ψqdenotes the quark

ﬁeld with ﬂavor qand ˆmqdesignates the corresponding current quark mass. Fa,µη stands for the gluon ﬁeld strength

expressed as Fa

µν =∂µAa

ν−∂νAa

µ−gfabcAb

µAc

ν. The total EMT operator is conserved as follows:

∂µˆ

Tµν = 0,ˆ

Tµν =X

Tµν

q+ˆ

Tµν

g,(2)

For the lowest-lying octet baryon, the matrix element of the EMT current can be parametrized in terms of the

three GFFs [1, 2, 113]:

hB, p0, J0

3|ˆ

Tµν (0)|B, p, J3i

= ¯u(p0, J0

3)"AB(t)PµPν

+JB(t)i(Pµσνρ +Pνσµρ)∆ρ

2mB

+DB(t)∆µ∆ν−gµν ∆2

4mB#u(p, J3),(3)

which depends on the spin polarizations J3and J0

3, the average momentum P= (p+p0)/2 of the initial and ﬁnal

states, and the four-momentum transfer ∆ = p0−p. The squared momentum transfer is denoted by t= ∆2. The

on-shell conditions of the ﬁnal and initial four momenta are given by p02=p2=m2

Bwhere mBdenotes the mass

of the octet baryon. In the BF, these GFFs AB(t), JB(t), and DB(t) are traditionally understood as the mass,

angular momentum, and D-term form factors, respectively. Here, one should keep in mind that in the level of the

quark and gluon degrees of freedom we have one additional form factor ¯c, which is constrained to satisfy the relation

Pa=q,g ¯ca(t) = 0. It can be dropped because of the conservation of the total EMT current.

III. ENERGY-MOMENTUM TENSOR DISTRIBUTIONS

In the BF, a 3D distribution is traditionally deﬁned as a Fourier transformation of the corresponding form factor.

Since, however, the baryon cannot be localized below the Compton wavelength, it causes ambiguous relativistic correc-

tions. These corrections are up to 20 % for the nucleon. In the non-relativistic picture, they are often neglected. In the

large Nclimit, the frame dependence of the distribution was carefully examined in Ref. [114]. These 3D distributions

in the BF can be understood quasi-probabilistic distributions in phase space or the Wigner distributions [26, 115–117].

To obtain the quantum-mechanical probabilistic distributions, one should take the IMF or the LF frame such that

the relativistic corrections are kinematically suppressed and the nucleon is described as a transversely localized state.

This yields 2D transverse densities in the IMF or on the LF.

The matrix element of the EMT current for a physical state |ψican be expressed in terms of the Wigner distribution

as [116]

hˆ

Tµν (r)i=Zd3P

(2π)3Zd3RW(R,P)hˆ

Tµν (r)iR,P,(4)

where W(R,P) represents the Wigner distribution given by

W(R,P) = Zd3∆

(2π)3e−i∆·R˜

ψ∗P+∆

2˜

ψP−∆

2

=Zd3ze−iz·Pψ∗R−z

2ψR+z

2.(5)

The average position Rand momentum Pare deﬁned as R= (r0+r)/2 and P= (p0+p)/2, respectively. ∆=p0−p

denotes the three-momentum transfer, which enables us to get access to the internal structure of a particle. The

variable z=r0−rstands for the position separation between the initial and ﬁnal particles. The Wigner distribution

contains information on the wave packet of a particle

ψ(r) = hr|ψi=Zd3p

(2π3)eip·r˜

ψ(p),˜

ψ(p) = 1

p2p0hp|ψi,(6)

where the plane-wave states |piand |riare respectively normalized as hp0|pi= 2p0(2π)3δ(3)(p0−p) and hr0|ri=

δ(3)(r0−r). The position state |rilocalized at rat time t= 0 is deﬁned as a Fourier transform of the momentum

eigenstate |pi

|ri=Zd3p

(2π)3p2p0e−ip·r|pi.(7)

If we integrate over the average position and momentum, then the probabilistic density in either position or momentum

space is recovered to be

Zd3P

(2π)3WN(R,P) = |ψN(R)|2,Zd3RWN(R,P) = |˜

ψN(P)|2.(8)

Given Pand R, the matrix element hˆ

Tµν (r)iR,Pconveys information on the internal structure of the particle

localized around the average position Rand average momentum P. This can be expressed as the 3D Fourier transform

of the matrix element hB, p0, J0

3|ˆ

Tµν (0)|B, p, J3i:

hˆ

Tµν (r)iR,P=hˆ

Tµν (0)i−x,P=Zd3∆

(2π)3e−ix·∆1

p2p0p2p00hp0, J0

3|ˆ

Tµν (0)|p, J3i,(9)

with the shifted position vector x=r−R. Note that, very recently, a novel concept of the 3D strict probabilistic

distribution was introduced to remove ambiguous relativistic corrections [66, 67].

A. Three-dimensional energy-momentum tensor distributions in the Breit frame

Having integrated over Pof Eq. (4), we ﬁnd that the part of the wave packet can be factorized. Thus, the target

in the BF is understood as a localized state around Rfrom the Wigner perspective. In this frame, Eq. (9) is reduced

Tµν

BF,B(r, J0

3, J3) = Zd3∆

(2π)32P0

e−i∆·rhB, p0, J0

3|ˆ

Tµν (0)|B, p, J3i.(10)

From now on we use rinstead of x, i.e., x=r−R→r. In the Wigner sense, the temporal component of the EMT

current yields mass distribution:

T00

BF,B(r, J0

3, J3) = εB(r)δJ0

3J3=mBZd3∆

(2π)3e−i∆·rAB(t)−t

4m2

BAB(t)−2JB(t) + DB(t)δJ0

3J3.(11)

By Integrating T00

BF,B over 3D space, one obviously gets the mass of a baryon in the rest frame

Zd3rT 00

BF,B(r, J0

3, J3) = mBAB(0) = mB,(12)

with the normalization AB(0) = 1. Note that, for a higher-spin particle (J≥1), a quadrupole distribution of the

energy inside the particle appears [46–49, 56, 118, 119]. The size of the mass distribution can be quantiﬁed by the

mass radius. It is given by either integral of the mass distribution or derivative of the mass form factor AB(t) with

respect to the momentum squared,

hr2

εiB=Rd3r r2εB(r)

Rd3r εB(r)=6

AB(0)

dAB(t)

dt t=0

.(13)

The 0k-component of the EMT current is related to the spatial distribution of the spin carried by the partons inside

a baryon:

B(r, J0

3, J3) = ijkrjT0k

BF,B(r, J0

3, J3)

= 2Sj

3J3Zd3∆

(2π)3e−i∆·rJB(t) + 2

3tJB(t)

dt δij +∆i∆j−1

3∆2δij JB(t)

dt .(14)

In principle, both the monopole and quadrupole distributions should be considered when we deal with the spin

distribution. However, we drop the quadrupole contribution for simplicity, which does not aﬀect the normalization of

the spin form factor JB(0). The quadrupole structure of the spin distribution was intensively discussed and related

to the monopole distribution in Refs. [120, 121]. The monopole contribution to the spin distribution, which is the

ﬁrst term in Eq. (14), is deﬁned as

ρB

J(r) := Zd3∆

(2π)3e−i∆·rJB(t) + 2

3tJB(t)

dt .(15)

Integrating Ji

B(r, J0

3, J3) over space gives the spin of the baryon as follows

Zd3rJi

B(r, J0

3, J3) = 2 ˆ

3J3Zd3r ρB

J(r)=2ˆ

3J3JB(0) = ˆ

3J3,(16)

which is just the spin operator of a baryon. The quadrupole contribution, the second term in Eq. (14), obviously

vanishes after the integration over the 3D space.

The spatial components EMT current Tij

BF,B provides information on the mechanical properties of a baryon. It can

be decomposed into isotropic and anisotropic contributions. This anisotropic contribution plays a signiﬁcant role in

the mechanical structure of a baryon [26, 113]. They are respectively referred to as the pressure pB(r) and shear force

sB(r) and expressed as [58, 113]

Tij

BF,B(r, J0

3, J3) = pB(r)δij δJ0

3J3+sB(r)rirj

r2−1

3δij δJ0

3J3,(17)

where the pressure and shear force distributions are respectively deﬁned as

pB(r) = 1

6mB

dr r2d

dr ˜

DB(r), sB(r) = −1

4mB

dr ˜

DB(r),with ˜

DB(r) = Zd3∆

(2π)3e−i∆·rDB(t).(18)

From Eq. (18), it is easy to see that the 3D von Laue stability condition for the pressure is automatically satisﬁed:

Zd3r pB(r)=0.(19)

It indicates that the pressure should have at least one nodal point. In addition, the pressure and shear force distribu-

tions automatically comply with the diﬀerential equation derived from the total EMT conservation:

∂iTij

BF,B(r, J0

3, J3) = rj

r2

∂sB(r)

∂r +2sB(r)

r+∂pB(r)

∂r δJ0

3J3= 0.(20)

It gives a number of the integral relations between pressure and shear force. One of them is the 2D von Laue stability

condition that is derived as

Z∞

dr r −1

3sB(r) + pB(r)= 0.(21)

The combination −1

3sB(r) + pB(r) carries the meaning of the tangential force distribution. It is an eigenvalue of the

stress tensor and it must at least have one nodal point such that it complies with the 2D von Laue condition (21).

Moreover, in Refs. [26, 113, 122], the local stability conditions were conjectured:

3sB(r) + pB(r)>0, sB(r)>0.(22)

The combination 2

3sB(r) + pB(r) bears the meaning of the normal force distribution and is again identiﬁed as an

eigenvalue of the stress tensor. Equation (22) implies that at any distance rthe normal force should be directed

outwards. This Polyakov-Schweitzer local stability condition was examined in various contexts [26, 88, 113, 122]. The

value of the D-term form factor at zero momentum transfer is obtained by integrating the pressure or shear-force

distributions over 3D space as

DB(0) = −4mB

15 Zd3r r2sB(r) = mBZd3r r2pB(r),(23)

and the positive shear force for any value of rimplies the negative D-term. In addition, the positivity of the normal

forces (22) enables us to deﬁne the mechanical radius:

hr2

mechiB=Rd3r r22

3sB(r) + pB(r)

Rd3r2

3sB(r) + pB(r)=6DB(0)

−∞ DB(t)dt.(24)

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INHA-NTG-11/2022GravitationalformfactorsofthebaryonoctetwithavorSU(3)symmetrybreakingHo-YeonWon,1,June-YoungKim,1,2,yandHyun-ChulKim1,3,z1DepartmentofPhysics,InhaUniversity,Incheon402-751,SouthKorea2TheoryCenter,JeersonLab,NewportNews,VA23606,USA3SchoolofPhysics,KoreaInstituteforAdvancedStudy(KIAS...

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INHA-NTG-112022 Gravitational form factors of the baryon octet with avor SU3 symmetry breaking Ho-Yeon Won1June-Young Kim1 2yand Hyun-Chul Kim1 3z

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