Independence of current components polarization vectors and reference frames in the light-front quark model analysis of meson decay constants Ahmad Jafar Ari

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Independence of current components, polarization vectors, and reference frames

in the light-front quark model analysis of meson decay constants

Ahmad Jafar Ariﬁ ,1, ∗Ho-Meoyng Choi ,2, †Chueng-Ryong Ji ,3, ‡and Yongseok Oh 4, 1, §

1Asia Paciﬁc Center for Theoretical Physics, Pohang, Gyeongbuk 37673, Korea

2Department of Physics Education, Teachers College,

Kyungpook National University, Daegu 41566, Korea

3Department of Physics, North Carolina State University, Raleigh, NC 27695-8202, USA

4Department of Physics, Kyungpook National University, Daegu 41566, Korea

(Dated: February 6, 2023)

The issue of resulting in the same physical observables with diﬀerent current components, in

particular from the minus current, has been challenging in the light-front quark model (LFQM)

even for the computation of the two-point functions such as meson decay constants. At the level

of one-body current matrix element computation, we show the uniqueness of pseudoscalar and

vector meson decay constants using all available components including the minus component of

the current in the LFQM consistent with the Bakamjian-Thomas construction. Regardless of the

current components, the polarization vectors, and the reference frames, the meson decay constants

are uniquely determined in the non-interacting constituent quark and antiquark basis while the

interactions of the constituents are added to the meson mass operator in the LFQM.

Keywords:

Introduction.— Light-front dynamics (LFD) [1–3] is a

useful framework for studying hadron structures with its

direct applications in Minkowski space. The distinct fea-

tures of LFD compared to other forms of Hamiltonian dy-

namics include that the rational energy-momentum dis-

persion relation in the LFD induces the suppression of

vacuum ﬂuctuations and that the LFD carries the maxi-

mal number (seven) of the kinematic generators of trans-

formations for the Poincar´e group.

The light-front quark model (LFQM) based on the

LFD has been quite successful in describing the mass

spectra and electroweak properties of mesons by treating

mesons as quark-antiquark bound states [4–17]. Typ-

ically in the LFQM [4–12], the constituent quark (Q)

and antiquark ( ¯

Q) are constrained to be on their respec-

tive mass shells, and the spin-orbit (SO) wave function

is thus obtained by the interaction-independent Melosh

transformation [18] from the ordinary equal-time static

representation. While the hadronic form factors and de-

cay constants are obtained from the matrix elements of a

one-body current directly in the three-dimensional light

front (LF) momentum space eﬀectively with the plus cur-

rent, J+=J0+J3, the calculation with diﬀerent current

components such as the transverse current J⊥and the

minus current, J−=J0−J3, should in principle yield

the same results of the hadronic form factors and decay

constants as the physical observables must be Lorentz in-

variant. However, in practice, the issue of resulting in the

same physical observables with diﬀerent current compo-

nents has been challenging in LFQM and led discussions

∗Electronic address: ahmad.jafar.ariﬁ@apctp.org

†Electronic address: homyoung@knu.ac.kr

‡Electronic address: ji@ncsu.edu

§Electronic address: yohphy@knu.ac.kr

on the Fock space truncation [19], the zero-mode con-

tribution [20,21], etc., in a variety of contexts [22–27].

Thus, clarifying this long-standing issue even in the two-

point function level, such as the computation of decay

constants, is of great importance to construct a reliable

light-front model to study hadron structure.

Focusing on the vector meson decay amplitudes with

the matrix element of one-body current [28], two of us

showed that the decay constants obtained from J+with

longitudinal polarization and J⊥with transverse polar-

ization are numerically the same by imposing the on-

shellness of the constituents consistently throughout the

LFQM analysis. In fact, it was demonstrated that those

two decay constants obtained from using the so-called

“Type II” [28] link between the manifestly covariant

Bethe-Salpeter (BS) model and the standard LFQM are

exactly equal to those obtained directly in the standard

LFQM imposing the on-shellness of the constituents.

This on-mass shell condition is equivalent to impos-

ing the four-momentum conservation P=p1+p2at the

meson-quark vertex, where Pand p1(2) are the meson

and quark (antiquark) momenta, respectively, which im-

plies the self-consistent replacement of the physical me-

son mass Mwith the invariant mass M0of the quark-

antiquark system. The generalization of the results in

Ref. [28] to any possible combination of current compo-

nents and of polarization is the main object of the present

work.

We notice in retrospect that this condition for the one-

body current matrix element computation is consistent

with the Bakamjian-Thomas (BT) construction [29,30]

up to that level of computation, where the meson state

is constructed by the noninteracting Q¯

Qrepresentations

while the interaction is included into the mass operator

M:= M0+VQ¯

Qto satisfy the group structure or commu-

tation relations. The main purpose of the present work is

arXiv:2210.12780v2 [hep-ph] 3 Feb 2023

to demonstrate that the long-standing issue of resulting

in the same physical observables with diﬀerent current

components can be resolved for the two-point physical

observables, explicitly in the analysis of the decay con-

stants for the one-body current matrix element compu-

tation with the aforementioned self-consistent condition

stemmed from the BT construction. We note that the

meson system of the constituent quark and antiquark

presented in this work is immune to the limitation of the

BT construction regarding the cluster separability for the

systems of more than two particles [31].

Within the scope described above, we show for the

ﬁrst time the uniqueness of pseudoscalar and vector me-

son decay constants using all available components of the

current in our LFQM being consistent with the BT con-

struction for the one-body current matrix element com-

putation. We explicitly demonstrate that the same de-

cay constants are resulted not only for all possible current

components but also for the polarization vectors indepen-

dent of the reference frame. Our explicit demonstration

is in fact related to the Lorentz invariant property that

could not be obtained in the relativistic quark models

based on LFD without implementing the aforementioned

self-consistency condition.

Theoretical framework.— While our demonstration

can be applied to the mesons composed of unequal-mass

constituents in general, here we focus on the equal mass

case of the constituents for simplicity. The essential as-

pect of the standard LFQM for the meson state [4–10] is

to saturate the Fock state expansion by the constituent

quark and antiquark and treat the Fock state in a non-

interacting representation. The interactions are then en-

coded in the LF wave function ΨJJz

λ1λ2(p1,p2), which is

the mass eigenfunction. The meson state |M(P, J, Jz)i ≡

|Mi of momentum Pand spin state (J, Jz) can be con-

structed as

|Mi =Zd3p1d3p22(2π)3δ3(P−p1−p2)

×X

λ1,λ2

ΨJJz

λ1λ2(p1,p2)|Q(p1, λ1)¯

Q(p2, λ2)i,(1)

where pµ

iand λiare the momenta and the helicities

of the on-mass shell (p2

i=m2

i) constituent quarks, re-

spectively. For the equal mass case, we set mi=m.

Here, p= (p+,p⊥) and d3pi≡dp+

id2pi⊥/(16π3).

The LF relative momentum variables (x, k⊥) are deﬁned

as xi=p+

i/P +and ki⊥=pi⊥−xiP⊥, which sat-

isfy Pixi= 1 and Piki⊥= 0. By setting x≡x1

and k⊥≡k1⊥, we decompose the LF wave function

as ΨJJz

λ1λ2(x, k⊥) = φ(x, k⊥)RJJz

λ1λ2(x, k⊥), where φ(x, k⊥)

is the radial wave function and RJJz

λ1λ2is the SO wave

function obtained by the interaction-independent Melosh

transformation.

The covariant forms of the SO wave functions are

RJJz

λ1λ2= ¯uλ1(p1)Γvλ2(p2)/(√2M0), where Γ = γ5and

−ˆ

(Jz) + ˆ(Jz)·(p1−p2)/(M0+ 2m) for pseudoscalar

and vector mesons, respectively,1and M2

0=Pi(k2

i⊥+

i)/xi. The polarization vectors ˆµ(Jz) of the vector me-

son are given by ˆµ(±1) = (0,2⊥(±1)·P⊥/P +,⊥(±1))

with ⊥(±1) = ∓(1,±i)/√2 for transverse polarizations

and ˆµ(0) = (P+,(P2

⊥−M2

0)/P +,P⊥)/M0for longitu-

dinal polarization [4,5]. One of the important char-

acteristics of our LFQM in contrast to other covariant

ﬁeld theoretic computations in LFD [13–15] is to use

M0other than the physical mass Min deﬁning RJJz

λ1λ2

and ˆµ(0) as well. Because of this property imposed by

the on-shellness of the constituents, which is consistent

with the BT construction, the SO wave functions satisfy

the unitary condition, Pλ1,λ2RJJz†

λ1λ2RJJz

λ1λ2= 1, indepen-

dent of model parameters. Furthermore, the longitudi-

nal polarization vector satisﬁes P·ˆ(0) = 0 only when

P=p1+p2or equivalently P2=M2

0, which we call the

self-consistency condition. We should note that the LF

energy conservation (P−=p−

1+p−

2) in addition to the LF

three-momentum conservation at the meson-quark vertex

is required for the calculations of the physical observables

using the matrix element with the one-body current to

be consistent with the BT construction [29,30] up to the

level of computation presented in this work as the meson

state is constructed by the noninteracting Q¯

Qrepresen-

tations. The interaction between quark and antiquark

is implemented in the radial wave function through the

mass spectroscopic analysis as discussed below. This con-

dition will be shown to be important in the complete

covariant analysis of the meson decay constants in the

LFQM.

The interactions between quark and antiquark are in-

cluded in the mass operator [29,30] to compute the mass

eigenvalue of the meson state. In our LFQM, we treat

the radial wave function as a trial function for the varia-

tional principle to the QCD-motivated eﬀective Hamilto-

nian HQ¯

Q, i.e., HQ¯

Q|Ψi= (M0+VQ¯

Q)|Ψi=M|Ψi, so

that the mass eigenvalue is obtained from the interaction

potential VQ¯

Qin addition to the relativistic free energies

of quark and antiquark. The detailed mass spectroscopic

analysis can be found in Refs. [11,12]. For the radial

wave function of the 1Sstate meson, we use the Gaus-

sian wave function φ(x, k⊥) = p∂kz/∂x ˆ

φ(k) as a trial

wave function, where ˆ

φ(k) = (4π3/4/β3/2) exp−k2/2β2

and βis the variational parameter ﬁxed by mass spectro-

scopic analysis. It should be mentioned, however, that

our observation and discussion about the independence

of the model predictions with respect to the components

of the current, the polarization vectors, and the refer-

ence frames is completely irrelevant to any speciﬁc form

of the radial wave function as far as the Jacobian factor

p∂kz/∂x, which is crucial for the Lorentz invariance of

1The Lorentz invariant properties with the BT construction dis-

cussed here would in general apply to other types of the wave

function vertices as well, e.g., the axial vector coupling for the

pseudoscalar meson vertex in the analysis of axial anomaly.

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Independenceofcurrentcomponents,polarizationvectors,andreferenceframesinthelight-frontquarkmodelanalysisofmesondecayconstantsAhmadJafarAri,1,Ho-MeoyngChoi,2,yChueng-RyongJi,3,zandYongseokOh4,1,x1AsiaPacicCenterforTheoreticalPhysics,Pohang,Gyeongbuk37673,Korea2DepartmentofPhysicsEducation,Teachers...

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Independence of current components polarization vectors and reference frames in the light-front quark model analysis of meson decay constants Ahmad Jafar Ari

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