Two-particle twist-3 distribution amplitudes of the pion and kaon in the light-front quark model

1.4 MB·

by  Ho-Meoyng Choi

Two-particle twist-3 distribution amplitudes of the pion and kaon in the light-front
quark model
Ho-Meoyng Choi
Department of Physics, Teachers College, Kyungpook National University, Daegu, Korea 41566
Chueng-Ryong Ji
Department of Physics, North Carolina State University, Raleigh, NC 27695-8202
Weinvestigatethetwo-particletwist-3distributionamplitudes(DAs)ofthepseudoscalarmesons,
inparticularpseudoscalar(φP (x))andpseudotensor(φσ (x))DAsofpionandkaon,inthelight-
3;M 3;M
front quark model based on the variational principle. We find that the behavior of the conformal
symmetry in each meson distribution amplitude depends on the chiral limit characteristics of the
7 light-fronttrialwavefunctiontakeninthevariationalprinciple. Wespecificallytakethetwodifferent
1 light-fronttrialwavefunctions,Gaussianvs. power-lawtype,anddiscusstheircharacteristicsofthe
0
conformal symmetryin thechiral symmetry limit as well as their resulting degree of theconformal
2 symmetry breaking in φP (x) and φσ (x) depending on the trial wave function taken in the
3;M 3;M
b computation. We present numerical results of transverse moments, Gegenbauer-moments and ξ-
e moments and compare them with other available model estimates. The SU(3) flavor-symmetry
F breakingeffect is also quantified with thenumerical computation.

6
2
I. INTRODUCTION

]
h
Hadronic distribution amplitudes (DAs) are the longitudinal projection of the hadronic wave functions obtained
p
- by integrating the transverse momenta of the fundamental constituents [1–3]. These nonperturbative quantities are
p defined as vacuum-to-hadronmatrix elements of particular nonlocal quark or quark-gluonoperators and thus encode
e
important information on bound states in QCD. Especially, the electromagnetic and transition form factors at high
h
Q2 aswellastheB-physicsphenomenologyinthe contextofSU(3)flavorsymmetrybreakingeffectrequireadetailed
[
information of meson DAs. Meson DAs are also indispensable for the analysis of hard exclusive electroproduction

3 based on the QCD factorization [4]. In particular, the shape of the pion DA has been extensively discussed due
v
to the nature of the pion as the massless Nambu-Goldstone boson [5, 6]. Finding the fundamental nonperturbative
2
information of QCD motivated many theoretical studies to calculate meson DAs using nonperturbative methods
0
4 such as the QCD sum rule [3, 7–15], the chiral-quark model from the instanton vacuum [16–18], the Nambu-Jona-
2 Lasinio (NJL) model [19, 20], the Dyson-Schwinger equation (DSE) approach [21, 22], and the light-front quark
0 model (LFQM) [23, 24]. Among them, the LFQM appears to be one of the most effective and efficient tools in
. studying hadron physics as it takes advantage of the distinguished features of the light-front dynamics (LFD) [25].
1
0 WorkinginMinkowskispace,theLFDallowsthestudyofphysicalobservablesbothinspacelikeandtimelikekinematic
7 regions. The rational energy-momentum dispersion relation of LFD, namely p− = (p2 +m2)/p+, yields the sign
⊥
1 correlation between the light-front (LF) energy p−(= p0 p3) and the LF longitudinal momentum p+(= p0 +p3)
: and leads to the suppression of vacuum fluctuations in LF−D. It facilitates the partonic interpretation of the hadronic
v
i amplitudes. TheLFDalsocarriesthemaximumnumber (seven)ofthekinetic(orinteractionindependent)generators
X
and thus the less effort in dynamics is necessary in order to get the QCD solutions that reflect the full Poincare´
r symmetries. Based on the advantage of LFD, the LFQM has been quite successful in describing various static and
a
non-static properties of hadrons [26–42] such as meson mass spectra [26, 27], the decay constants (i.e. the lowest
momentsoflight-coneDAs)[23,28],electromagneticandweaktransitionformfactors[29–40]andgeneralizedparton
distributions(GPDs)[41,42]. TheLFQManalysisofthepionformfactor[38,39]hasalsoprovidedcompatibleresults
both in spacelike and timelike regions with the holographic approach to LF QCD [43] based on the 5-dimensional
anti-de Sitter (AdS) spacetime and the conformal symmetry which has given insight into the nature of the effective
confinement potential and the resulting LF wave functions for both light and heavy mesons [44].
Throughtherecentanalysisofthetwist-2andtwist-3DAsofpseudoscalarandvectormesons[45–48],wediscussed
also the link between the chiral symmetry of QCD and the LFQM. In Ref. [24], we have analyzed the two-particle
twist-2 DAs of pseudoscalar (φA (x)) and vector (φ|| (x)) mesons using our LFQM [26]. We then extended our
2;M 2;V
LFQMto analyze two-particletwist-3 pseudoscalar(φP (x)) DAs of pseudoscalarmesons [45, 46] andchirality-even
3;M
twist-3(φ⊥ (x))DAsofvectormesons[47]todiscussthelinkbetweenthechiralsymmetryofQCDandthenumerical
3;V
results of the LFQM. In particular, through the analysis of twist-3 DAs of π and ρ mesons, we observed that the
LFQMwitheffectivedegreesoffreedomrepresentedbytheconstituentquarkandantiquarkcouldprovidetheviewof
effectivezero-modecloudaroundthequarkandantiquarkinsidethemeson. Ournumericalresultsappearedconsistent
with this view and effectively indicated that the constituent quark and antiquark in the LFQM could be considered
2
as the dressed constituents including the zero-mode quantum fluctuations from the vacuum.
To discuss the wave function dependence of the LF zero-mode [49–58] contributions to φP (x) and φ⊥ (x), we
3;M 3;V
analyzedboththe exactlysolvablemanifestly covariantBethe-Salpeter(BS) modelandthe morephenomenologically
accessible realistic LFQM[24, 26] in the standardLF approach. The purpose of taking the exactly solvable covariant
BS model was to check the existence (or absence) of the zero mode in each channel, e.g. φP (x) or φ⊥ (x), without
3;M 3;V
any ambiguity. For example, performing the LF calculation in the covariant BS model with the multipole type
qq¯ bound state vertex function, we not only showed that the twist-3 φP (x) and φ⊥ (x) receive both the zero-
3;M 3;V
mode and the instantaneous contributionsbut alsoidentified the zero-modeoperatorcorrespondingto the zero-mode
contribution. As discussed in Refs. [45, 47], we also found the universal mapping [see e.g. Eq. (35) in [45]] between
the covariantBSmodelandthe standardLFQMforany two-pointandthree-pointfunctions. With this mapping, we
were able to boost the exactly solvable covariantBS model computation into the more phenomenologicallyaccessible
LFQM computation. In practice, the LF vertex function obtained in the covariant BS model was mapped into the
phenomenological, typically Gaussian, LF trial wave function which has been scrutinized by the standard LFQM
analysisofmesonmassspectroscopybasedonthevariationalprinciple andothermesonphenomenology[26,27]. The
remarkable finding from this practice was that the zero-mode contribution as well as the instantaneous contribution
revealed in the covariant BS model became absent in the LFQM with the LF on-mass-shell constituent quark and
antiquark degrees of freedom. Without involving the zero-mode and instantaneous contributions, our LFQM with
the Gaussian trial wave function provided the result of twist-3 DAs φP (x) and φ⊥ (x) which not only satisfied
3;M 3;V
the fundamental constraint (i.e., symmetric form with respect to x) anticipated from the isospin symmetry but also
provided the consistency both with the chiral symmetry and the conformal symmetry (e.g., the correct asymptotic
form in the m 0 limit) expected from the QCD. Our LFQM predictions with the Gaussian wave function such
q
→
as φ|| (x) 6x(1 x) and φ⊥ (x) (3/4)[1+(2x 1)2] for ρ and φP (x) 1 for π in the chiral symmetry limit
2;ρ → − 3;ρ → − 3;π →
reproducetheexactfunctionalformsanticipatedfromQCD’sconformallimit[7,59]. ThisexemplifiesthatourLFQM
predictionwiththeGaussianwavefunctionsatisfiesboththechiralsymmetryandtheconformalsymmetryconsistent
with the QCD if one correctly implements the zero-mode link to the QCD vacuum.
It is important, however,to realize that satisfying both the chiral symmetry and the conformal symmetry depends
on the choice of the LF trial wave function. The key in the Gaussian LF wave function is the factorization of the
transversemomentum dependence fromthe dependence ofscale independent parameterssuchas mass. It allowsthat
the m 0 limit satisfies both the chiral symmetry and the conformal symmetry simultaneously. If the LF trial
q
→
wave function is not taken as Gaussian but for example taken as power-law (PL) type, then the factorization of the
transverse momentum dependence from the scale independent parameter dependence cannot be fulfilled and thus
the m 0 limit may not satisfy the conformal symmetry although it may still satisfy the chiral symmetry. This
q
→
dependence on the LF trial wave function indicates that some particular meson DAs may not satisfy the conformal
symmetry while they still satisfy the chiral symmetry consistent with QCD. Similarly, the DSE approach in [22]
provided the asymptotic form of the pion φP (x) with a broad downward concave shape in the central region of x
3;π
rather than φP (x) 1 anticipated from QCD’s conformal limit [7]. There are two independent two-particle twist-3
3;π →
DAs of a pseudoscalar meson, namely, pseudoscalarDA φP and pseudotensor DA φσ [7–11, 17, 20]. The authors
3;M 3;M
in [22] also analyzed the pseudotensor DA φσ (x), and found that the asymptotic form of the pion φσ (x) coincide
3;M 3;π
with the anticipated expression of QCD’s conformal limit, 6x(1 x).
−
These developments motivate our present work for the more-in-depth analysis of the two-particle twist-3 pion and
kaonDAs inLFQMwithdifferentformsofLFtrialwavefunctions. We firstextendourpreviouswork[45]toanalyze
the twist-3 pseudotensor DA φσ (x) of a pseudoscalar meson within the LFQM. We also discuss the discrepancy
3;M
of the asymptotic forms of φP (x) between DSE approach [22] and QCD’s conformal limit expression [7] from the
3;π
perspective of dependence of DA on the formof LF trial wavefunctions such as Gaussian wavefunction vs. PL wave
function. Althoughthe two-particletwist-3pionDAswerebrieflydiscussedinLC2016[48], weelaboratemoreinthis
work on the dependence of DA on the form of LF trial wavefunctions as well as the SU(3) flavor-symmetrybreaking
effect through the complete analysis of two-particle twist-3 DAs of pseudoscalar meson. In order to compute the
twist-3 pseudotensor DA φσ (x), we again utilize the same manifestly covariant BS model used in [45–47] to check
3;M
the existence (or absence) of the LF zero-mode contribution. We then apply the previously found universalmapping
[see e.g. Eq. (35) in [45]] between the covariant BS model and the standard LFQM to map the vertex function
obtained in the exactly solvable covariant BS model into the more phenomenologically accessible Gaussian and PL
radial wave functions provided from our LFQM variational principle computation.
Thepaperisorganizedasfollows. InSec.II,wecomputethetwist-3pseudotensorDAφσ (x)inanexactlysolvable
3;M
model based on the covariant BS model of (3+1)-dimensional fermion field theory. We then link the covariant BS
model to the standard LFQM with the previously found universal mapping between the two as discussed above and
presentthe resultingformofφσ (x) aswellasφP (x) inourLFQM.InSec.III,we presentournumericalresultsof
3;M 3;M
3
p=P k
−
P
Γ Γ
P
ΓP =γ5 Γ=γµγ5,γ5,σµνγ5
k
FIG. 1: Feynmandiagram for theone-quark-loop evaluation of themeson decay amplitude in themomentum space.
φσ (x) and φP (x) for the pion and kaon and discuss the results in the chiral vs. conformal symmetry limit. The
3;M 3;M
SU(3) flavor symmetry breaking effects on the twist-3 DAs for the kaon are also discussed. Summary and discussion
follow in Sec. IV. In the Appendix, the derivation of twist-3 DAs of pseudoscalar meson is presented.
II. MODEL DESCRIPTION
A. Manifestly Covariant BS Model
The φP and φσ are defined in terms of the following matrix elements of gauge invariant nonlocal operators in
3;M 3;M
the light-front gauge [7–9]:
1
0q¯(z)iγ q( z)M(P) =f µ dxeiζP·zφP (x), (1)
h | 5 − | i M MZ 3;M
0
and
i 1
0q¯(z)σ γ q( z)M(P) = f µ (P z P z ) dxeiζP·zφσ (x), (2)
h | αβ 5 − | i −3 M M α β − β α Z 3;M
0
where z2 = 0 and P is the four-momentum of the meson (P2 = m2 ) and the integration variable x corresponds
M
to the longitudinal momentum fraction carried by the quark and ζ = 2x 1 for the short-hand notation. The
normalization parameter µ = m2 /(m +m ) results from quark condensa−te. For the pion, µ = 2 q¯q /f2 from
M M q q¯ π − h i π
the Gell-Mann-Oakes-Renner relation [60]. We also note from the argument in [21, 22] that the pseudoscalar DA
of the pion, φP (x), i.e. pseudoscalar projection of the pion’s LF wave function, might be understood as describing
3;π
the probability distribution of the chiral condensate within the pion [61]. The normalization of the two twist-3 DAs
Φ= φP ,φσ is given by
{ 3;M 3;M}
1
dxΦ(x)=1. (3)
Z
0
Defining zµ =τηµ using the lightlike vector η =(1,0,0, 1),one can rewrite Eqs. (1) and (2) as [see Appendix for
−
the explicit derivation of Eqs. (4) and (5)]
2(P η) ∞ dτ
φP (x)= · e−iζτ(P·η) 0q¯(τη)iγ q( τη)M(P) , (4)
3;M f µ Z 2π h | 5 − | i
M M −∞
and
φσ (x)= 12 ∞ dτ xdx′e−iζ′τ(P·η) 0q¯(τη)i(P//η P η)γ q( τη)M(P) , (5)
3;M −f µ Z 2π Z h | − · 5 − | i
M M −∞ 0
respectively. The nonlocal matrix elements 0q¯(τη)iΓ q( τη)M(P) for pseudoscalar (Γ = γ ) and pseu-
α α α 5
M ≡ h | − | i
dotensor (Γ = (P//η P η)γ ) channels are given by the following momentum integral in two-point function of the
α 5
− ·
manifestly covariantBS model (see Fig. 1)
d4k H
=N e−iτk·ηe−iτ(k−P)·η 0 S , (6)
Mα cZ (2π)4 NpNk α
4
where N denotes the number of colors and S = Tr[iγ (/p+m )γ ( /k+m )] for pseudoscalar channel and
c α 5 q 5 q¯
Tr[i(P//η P η)γ (/p+m )γ ( /k+m )] for pseudotensor channel. The−denominators N (= p2 m2 + iε) and
− · 5 q 5 − q¯ p − q
N (=k2 m2+iε)comefromthequarkpropagatorsofmassm andm carryingtheinternalfour-momentap=P k
k − q¯ q q¯ −
and k, respectively. In order to regularize the covariant loop, we use the usual multipole ansatz [47, 52, 62, 63] for
the qq¯bound-state vertex function H =H (p2,k2) of a meson: H (p2,k2)=g/Nn, where N =p2 Λ2+iε, and g
0 0 0 Λ Λ −
and Λ are constant parameters. We note that the power n for the multipole ansatz should be n 2 to regularize the
≥
loop integral and our essential results in terms of the zero-mode issue do not depend on the value of n.
For the LF calculation, we use the metric convention a b = 1(a+b− +a−b+) a b and separate the trace
· 2 − ⊥ · ⊥
term S into the on-mass-shell propagating part [S ] and the off-mass-shell instantaneous part [S ] , i.e. S =
α α on α inst α
[S ] +[S ] via/q =/q +1γ+(q− q−). Inthe referenceframewhereP =0,i.e., P =(P+,M2/P+,0),the LF
α on α inst on 2 − on ⊥
energiesoftheon-mass-shellquarkandantiquarkaregivenbyp− =(k2 +m2)/xP+ andk− =(k2 +m2)/(1 x)P+,
on ⊥ q on ⊥ q¯ −
respectively, where x=p+/P+ is the LF longitudinal momentum fraction of the quark.
After a little manipulation, we can rewrite Eq. (4) for the pseudoscalar channel as
N d4k k η H
φP (x) = c δ 1 x · 0 S
3;M f µ Z (2π)4 (cid:18) − − P η(cid:19)N N P
M M p k
·
N d2k χ(x,k )
c ⊥ ⊥
= [S ] , (7)
f µ Z 16π3 (1 x) P full
M M
−
where
g
χ(x,k )= , (8)
⊥ [x(m2 M2)][x(m2 M2)]n
M − 0 M − Λ
and
k2 +m2(Λ2) k2 +m2
M2 = ⊥ q + ⊥ q¯. (9)
0(Λ) x 1 x
−
The full resultofthe traceterm[S ] has beenobtainedin [45]andit receivesnot only[S ] and[S ] but also
P full P on P inst
the zero-modecontribution[S ] in this manifestly covariantBSmodel, i.e. [S ] =[S ] +[S ] +[S ] ,
P Z.M. P full P on P inst P Z.M.
where [S ] = 4(p k + m m ) = 2[M2 (m m )2], [S ] = 2k+(p− p−) = 2(1 x)(m2 M2),
and [S ]P on = 2[xon(m· 2on M2q)+q¯m2 m20+−(1 q2−x)mq¯2 ], resPpeincsttively. The d−etaiolend proced−ure to oMbt−ain t0he
P Z.M. − M − 0 q − q¯ − M
zero-mode calculation is given in [45]. However, as we have explained in great detail in [45], the full result of trace
term [S ] in the more realistic LFQM using the Gaussian or PL type wave functions gives the same result for
P full
the decay amplitude only with the on-mass-shell contribution involving neither the zero-mode contribution nor the
instantaneous contribution. Effectively, it indicates that the on-mass-shell constituent quark and antiquark in the
LFQMcanbeconsideredasthedressedconstituentsincludingthezero-modeandinstantaneousquantumfluctuations
from the vacuum. The same observation has been made for the calculation of the twist-2 and-3 DAs of the vector
meson [47] as well as the pion electromagnetic form factor [45].
Similarly, Eq. (5) for the pseudotensor twist-3 φσ (x) can be rewritten as
3;M
6 N d4k x k η H
φσ (x) = c dx′δ 1 x′ · 0 S ,
3;M −f µ (P η)Z (2π)4 Z (cid:18) − − P η(cid:19)N N σ
M M · 0 · p k
6 N d2k x χ(x′,k )
= c ⊥ dx′ ⊥ [S ] , (10)
−f µ P+ Z 16π3 Z (1 x′) σ full
M M 0 −
where χ(x′,k ) = χ(x x′,k ). We should note for this pseudotensor channel that, due to the nature of the
⊥ ⊥
→
second rank tensor operator contracting meson momentum, the r.h.s. of Eq. (A.6) is not the DA itself but the
derivative of DA so that the x′-integration appears in Eq. (10) with the integration range from 0 to x. One may
find without any difficulty that the manifestly covariant calculation of the trace term S would give zero result for
σ
the decay amplitude if the x′-integration is done from zero to 1 since DA at the end point x = 1 must be zero. As
the x′ integration range from 0 to x, the decay amplitude is in general not zero unless x = 1 or x = 0. In the LF
calculation,the same observationcanbe made if we include allthree contributions,i.e. on-mass-shell,instantaneous,
and zero-mode contributions, in the full result of the trace term [S ] =[S ] +[S ] +[S ] , where [S ] =
σ full σ on σ inst σ Z.M. σ on
4[(P k )p+ (P p )k+]=2P+[(2x′ 1)M′2+m2 m2],[S ] = 2k+P+(p− p−)= 2P+(1 x′)(m2 M′2),
· on − · on − 0 q¯− q σ inst − − on − − M− 0
and[S ] =2P+[x′(m2 M′2)+m2 m2+(1 2x′)m2 ]withM′ =M (x x′),respectively. Thisindicatesthat
σ Z.M. M− 0 q− q¯ − M 0 0 →
not only the on-mass-shellcontributionbut also both the instantaneouscontributionand the zero-mode contribution
in principle exist in the LF calculationto coincide with the manifestly covariantBS result. However,it is remarkable
5
to observe that the full result of trace term [S ] in the more realistic LFQM using the Gaussian or PL type wave
σ full
functionswhichwediscussinthenextsubsection,Sec.IIB,isidenticaltotheresultwhen[S ] isreplacedby[S ] as
σ full σ on
discussedinthecaseofpseudoscalarchannel. Itassuresthattheon-mass-shellconstituentquarkandantiquarkinthe
LFQM can be regardedas the dressed constituents including the zero-mode and instantaneous quantum fluctuations
from the vacuum.
B. Application to Standard Light-Front Quark Model
InthestandardLFQM[26–42],thewavefunctionofagroundstatepseudoscalarmeson(JPC =0−+)asaqq¯bound
state is given by
Ψλλ¯(x,k⊥)=ΦR(x,k⊥)Rλλ¯(x,k⊥), (11)
whereΦR istheradialwavefunctionandthespin-orbitwavefunctionRλλ¯ withthehelicityλ(λ¯)ofaquark(antiquark)
thatisobtainedbytheinteraction-independentMeloshtransformation[64]fromtheordinaryspin-orbitwavefunction
assigned by the quantum numbers JPC. The covariant form of the spin-orbit wave function Rλλ¯ is given by
Rλλ¯ = √2[Mu¯2λ(pq()mγ5vλ¯(mpq¯))2]1/2, (12)
0 − q− q¯
and it satisfies λλ¯R†λλ¯Rλλ¯ =1. The normalization of our wave function is then given by
P
dxd2k dxd2k
Z 16π3⊥|Ψλλ¯(x,k⊥)|2 =Z 16π3⊥|ΦR(x,k⊥)|2. (13)
Xλλ¯
For the radial wave function Φ , we try both the Gaussian or harmonic oscillator (HO) wave function Φ and
R HO
the power-law (PL) type wave function Φ [31] as follows
PL
4π3/4 ∂k
Φ (x,k )= zexp( ~k2/2β2), (14)
HO ⊥ β3/2 r ∂x −
and
128π ∂k 1
Φ (x,k )= z , (15)
PL ⊥ r β3 r ∂x (1+~k2/β2)2
where ~k2 = k2 + k2 and β is the variational parameter fixed by the analysis of meson mass spectra [26]. The
⊥ z
longitudinal component k is defined by k = (x 1/2)M + (m2 m2)/2M , and the Jacobian of the variable
z z − 0 q¯− q 0
transformation x,k ~k =(k ,k ) is given by
⊥ ⊥ z
{ }→
∂k M m2 m2 2
z = 0 1 q− q¯ . (16)
∂x 4x(1 x)(cid:26) −(cid:20) M2 (cid:21) (cid:27)
− 0
As discussed in the previous section, Sec. I, the transverse momentum k dependence factorizes as exp( ~k2/2β2)=
⊥
−
exp( k2/2β2)exp( k2/2β2) in Φ while such factorization of k dependence of 1/(1+~k2/β2)2 is not feasible in
− ⊥ − z HO ⊥
Φ . Thus,thescale(orconformal)invarianceofthetransversemomentumk aswellasthelongitudinalmomentum
PL ⊥
fractionxisachievedinthemassless(chiral)limitforΦ whiletheconformalinvarianceofthetransversemomentum
HO
k doesn’t hold in the chirallimit for Φ . This distinguishes the behavior of the chirallimit between Φ and Φ
⊥ PL HO PL
and leads to the difference in the chiral limit for φP (x) depending on which LF model wave function is applied for
3;M
the computation. We present more details of the chiral limit behaviors for each case of the LF trial wave functions
discussed in this work.
Inourpreviousanalysesoftwist-2andpseudoscalartwist-3DAsofapseudoscalarmeson[45]andthechirality-even
twist-2 and twist-3 DAs of a vector meson [47], we have shown that the results in the standard LFQM is obtained
by the mapping of the LF vertex function χ in BS model into our LFQM wave function Φ as follows (see Eq. (35)
R
in [45] or Eq. (49) in [47])
χ(x,k ) Φ (x,k )
⊥ R ⊥
2N , m M , (17)
c 1 x → k2 +A2 M → 0
p − ⊥
p
6
TABLE I: Model parameters for the Gaussian wave function with the linear and HO confining potentials [24, 26, 28] and for
thepower-law wave function [31]. q=u and d.
Model mq (GeV) ms (GeV) βqq¯ (GeV) βqs¯ (GeV)
Linear 0.22 0.45 0.3659 0.3886
HO 0.25 0.48 0.3194 0.3419
Power-Law 0.25 0.37 0.335 0.41
where A = (1 x)m +xm and m M implies that the physical mass m included in the integrand of BS
q q¯ M 0 M
− →
amplitude has to be replaced with the invariant mass M since the results in the standard LFQM are obtained from
0
the requirementofallconstituents being ontheir respectivemassshell. The correspondenceinEq.(17) isvalidagain
in this analysis of a pseudotensor twist-3 DA φσ (x).
3;M
We now apply the same mapping to both φP (x) in Eq.(7) andφσ (x) in Eq. (10) to obtainthem in our LFQM
3;M 3;M
as follows:
√2N d2k Φ (x,k )
φP (x) = c ⊥ R ⊥ [M2 (m m )2], (18)
3;M fMµM Z 16π3 k2⊥+A2 0 − q− q¯
p
and
6√2N d2k x Φ (x′,k )
φσ (x) = c ⊥ dx′ R ⊥ [(1 2x′)M′2+m2 m2], (19)
3;M fMµM Z 16π3 Z0 k2 +A′2 − 0 q− q¯
⊥
q
respectively, where A′ = A(x x′). It is remarkable to observe that both the zero-mode contribution and the
→
instantaneous contribution are absorbed into the LF on-mass-shell constituent quark and antiquark contribution as
shown in Eqs.(18) and (19).
For the point of view of QCD, one should note that the quark-antiquark DAs of a hadron depend on the scale µ
that may separate nonperturbative and perturbative regimes. In our LFQM, we can associate µ with the transverse
integration cutoff via k µ. The dependence on the scale µ is then consistently given by the QCD evolution
⊥
| | ≤
equation [1], while the DAs at a certain low scale can be obtained by the necessary nonperturbative input from
LFQM. As the cutoff dependence becomes marginal beyond a certain nonperturbative cutoff scale, the Gaussian (or
HO) and PL wave functions given by Eqs. (14) and (15) are allowed to perform the integral up to infinity without
any appreciable loss of accuracy.
III. NUMERICAL RESULTS
In the numerical computations, we use the linear and HO confining potential model parameters for the Gaussian
wavefunction givenin Table I, whichwere obtained fromthe calculationof mesonmass spectra using the variational
principleinourLFQM[24,26,28]. Forthesensitivityanalysisdependingontheformofthemodelwavefunctions,we
also use the PL wave function with the model parametersadopted from Ref. [31]. Since our numerical results for the
twist-2 φA (x) and twist-3 φP (x) of π an K mesons were presented in our previous works [24, 45], we shall focus
2;M 3;M
on the calculation of the twist-3 φσ (x) of π and K mesons together with some new results for φP (x) including
3;M 3;M
the PL wave function in this work.
Defining the LF wave function ψP(σ)(x,k ) for the twist-3 pseudoscalar (pseudotensor) channel as
3;M ⊥
∞
φP(σ)(x)= d2k ψP(σ)(x,k ), (20)
3;M Z ⊥ 3;M ⊥
0
the n-th transverse moment is obtained by
∞ 1
kn P(σ) = d2k dxψP(σ)(x,k )kn. (21)
h ⊥iM Z ⊥Z 3;M ⊥ ⊥
0 0
For the pion case, our results of the second transverse moments for ψP (x,k ) and ψσ (x,k ) obtained from the
3;π ⊥ 3;π ⊥
linear [HO] parameters are k2 P = (553 MeV)2[(480 MeV)2] and k2 σ = (481 MeV)2[(394 MeV)2], respectively.
For the kaon case, we obtahin⊥ikπ2 P = (582 MeV)2[(510 MeV)2] ahnd⊥iπk2 σ = (481 MeV)2[(428 MeV)2] for the
h ⊥iK h ⊥iK
7
FIG. 2: 3D plots for ψ3P;π(x,k⊥) (upper panel) and ψ3σ;π(x,k⊥) (lower panel) obtained from the HO (left panel) and the PL
(right panel) wave functions, respectively.
linear [HO] parameters, respectively. Since the PL wave function given by Eq. (15) is not enough power suppressed
to give finite transverse moments unless the transverse integration cutoff is performed, we do not estimate them for
the PL wave function case.
Fig. 2 shows the 3D plots for the twist-3 pion LF wave functions ψP (x,k ) (upper panel) and ψσ (x,k ) (lower
3;π ⊥ 3;π ⊥
panel)obtainedfromthe Gaussianwavefunctions with HO modelparameters(left panel) andthe PLwavefunctions
(right panel), respectively. For the case of pseudoscalar ψP (x,k ), it shows the concave shape for low k2 for both
3;π ⊥ ⊥
GaussianandPLwavefunctions butits DAφP (x) afterthe k -integrationupto infinityshowsratherconvexshape
3;π ⊥
in the central region of x as we show in Fig. 3. On the other hand, for the case of pseudotensor ψσ (x,k ), it
3;π ⊥
shows the convex shape for any value of k2 regardless the choice of the wave functions. For both pseudoscalar and
⊥
pseudotensorchannels,thePLwavefunctionshavemorehighmomentumtailsthanthecorrespondingGaussianwave
functions for k 1 GeV. Thus, the PL wave functions are rather sensitive to the transverse momentum cutoff
⊥
| | ≥
values. We also should note that ψP (x,k ) is much more sensitive to the choice of the LF wave functions than
3;π ⊥
ψσ (x,k ). This may lead to different asymptotic behaviors for different LF wave functions in the chiral symmetry
3;π ⊥
limit.
WeshowinFig.3thecorrespondingtwo-particletwist-3pionφP (x)(leftpanel)andφσ (x)(rightpanel)obtained
3;π 3;π
fromthenonzeroconstituentquarkmassesusingGaussianwavefunctionswithHO(solidlines)modelparametersand
PLwavefunctions(dashedlines). Wealsoplotourresultsinthechiralsymmetry(m 0)limitforbothGaussian
u(d)
→
(dotted lines) and PL (dot-dashed lines) wave functions and compare them with the chiral-limit prediction of DSE
8
1.5 2
1.5
1
Pφ(x)3;π σφ(x)3;π 1
HO HO
0.5 HO (m=0 limit) HO (m=0 limit)
q q
PL PL
PL (m=0 limit) 0.5 PL (m=0 limit)
q q
DSE 6x(1-x)
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
x x
FIG. 3: Thetwist-3 DAsφP (x)(left panel) and φσ (x) (right panel) of pion.
3;π 3;π
approachemployingthedynamicalchiralsymmetrybreaking(DCSB)improved(BD)kernels[22](double-dot-dashed
line) as well as the asymptotic result 6x(1 x) for the case φσ (x). Our results for both φP (x) and φσ (x) are
− 3;π 3;π 3;π
normalized without the momentum cutoff (i.e. k ).
⊥
For the φP (x) case in Fig. 3, our results wi|th n|o→nz∞ero constituent quark masses show rather convex shapes for
3;π
both Gaussian and PL wave functions but they show quite different end point behaviors, i.e. the end points are
more enhanced for the PL wave function than the Gaussian wave function. The difference between the two wave
functions are more drastic in the chiral symmetry limit, where the result of Gaussian wave function reproduces the
result φP (x) 1 anticipated from the QCD’s conformal limit [7] but the result of PL wave function shows the
3;π →
concave shape similar to the result of DSE approach [22], in which the following asymptotic form was obtained:
φP (x) 1+(1/2)C(1/2)(2x 1). This rebuts the remark made in Ref.[22] that our LFQM has curvature of the
3;π → 2 −
oppositesignonalmosttheentiredomainofsupportinconflictwithamodel-independentpredictionofQCD.Wehave
shownin ourprevious works[45, 47] that ourLFQM is indeed consistentwith the nature of chiralsymmetry in QCD.
While the authors in [22] explained that the difference, i.e. (1/2)C(1/2)(2x 1) term in chiral symmetry limit, may
2 −
come fromthe mixing effect between the two-andthree-particle twist-3 amplitudes, we observethe similardifference
taking the power-law type LF wave function in which the transverse momentum dependence cannot be factorized
from the scale independent parameter dependence. Especially, we find that the end point behaviors of φP (x) also
3;π
affect the asymptotic form in the chiral symmetry limit. The cutoff dependent behaviors of φP (x) obtained from
3;π
both Gaussianand PL wavefunctions are also presented in Ref. [46], where the concaveshape for the Gaussianwave
function can also be seen with the cutoff scale µ = 1 GeV or less being taken but the cutoff dependence was shown
to be more sensitive for the PL wave function than the Gaussian one.
For the φσ (x) case in Fig. 3, our results with nonzero constituent quark masses for both Gaussian (solid line)
3;π
and PL (dashed line) show again different end point behaviors, i.e. the end points are more enhanced for the PL
wave function than the Gaussian wave function. However, in the chiral symmetry limit, Gaussian (dotted line) and
PL (dot-dashed line) wave functions show very similar shapes each other. Thus, the degree of conformal symmetry
breaking depends on the channel of DAs, φP (x) vs. φσ (x). As expected, the result from Gaussian wave function
3;π 3;π
reproduces exactly the asymptotic form 6x(1 x). The same chiral-limit behavior was also obtained from the DSE
approach[22]. AsonecanseefromFig.3,the−twist-3pseudoscalarφP (x)ismoresensitivetotheshapeofthemodel
3;π
wave functions (Gaussian vs. PL) than the twist-3 pseudotensor φσ (x). It is quite interesting to note in the chiral
3;π
symmetry limit that while φP (x) is sensitive to the shapes of model wave functions, φσ (x) is insensitive to them.
3;π 3;π
Fig.4showsthe 3Dplotsforthe twist-3kaonLFwavefunctions ψP (x,k )(upper panel)andψσ (x,k )(lower
3;K ⊥ 3;K ⊥
panel)obtainedfromthe Gaussianwavefunctions with HO modelparameters(left panel) andthe PLwavefunctions
(right panel), respectively. For the kaon case, we assign the momentum fractions x for s-quark and (1 x) for the
−
lightu(d)-quark. Due to the SU(3) flavor-symmetrybreakingeffect, the twist-3 kaonLF wavefunctions are distorted
infavorofthe heaviers-quark. Otherthanthe SU(3) flavor-symmetrybreakingeffect, the generalbehavioris similar
9
FIG. 4: The 3D plots for ψ3P;K(x,k⊥) (upperpanel) and ψ3σ;K(x,k⊥) (lower panel) obtained from theHO (left panel) and the
PL (right panel) wave functions.
to the pion case.
We show in Fig. 5 the corresponding two-particle twist-3 kaon φP (x) (left panel) and φσ (x) (right panel)
3;K 3;K
obtainedfromGaussianwavefunctionswithHO(solidlines)andLinear(dottedlines)modelparametersandPLwave
functions (dashed lines). We also compare our results with the prediction of DSE approachemploying the dynamical
chiralsymmetrybreaking-improved(BD)kernels[22](dot-dashedline). OurresultsforbothφP (x)andφσ (x)are
3;K 3;K
normalized without the transverse momentum cutoff. In both pseudoscalar and pseudotensor twist-3 kaon DAs, the
difference between the HO and Linear model parameters using the same Gaussian wave functions is less significant
than the difference between the Gaussian and PL wave functions. On the other hand, the SU(3) flavor-symmetry
breaking effect is more pronouncedin the Gaussianwavefunction than the PL wavefunction. As in the case of pion,
while some disagreements between our LFQM prediction and DSE prediction are seen in φP (x), some agreements
3;K
betweenthemcanalsobe seeninφσ (x). Especially,forthe pseudotensorDAφσ (x), ourpredictionfromPLwave
3;K 3;K
function is in goodagreementto the resultfrom DSE approachincluding the end points behaviors. As was discussed
in [22], the SU(3) flavor-symmetrybreaking effect of two-particle twist-3 kaon DAs may be quantified by considering
a ratio, viz.
1/2 P(σ)
dx¯φ (1 x¯)
δφP3;(Kσ) = R0 1/2dx3φ;KP(σ)(−x) , (22)
0 3;K
R
10
2 2
1.5 1.5
Pφ(x)3;K 1 σφ(x)3;K 1
HO
Linear
PL
0.5 HO 0.5 DSE
Linear
PL
DSE
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
x x
FIG. 5: The twist-3 DAsφP (x)(left panel) and φσ (x) (right panel) of pion.
3;K 3;K
where x¯=1−x. We obtain δφP3;K =(1.28,1.38,1.06)and δφσ3;K =(1.33,1.43,1.05)for (Linear, HO, PL) parameters,
respectively. The same formula as in Eq. (22) should hold for twist-2 DA (φP ) [24, 45], and we obtain the ratio as
2;K
δφP2;K =(1.15,1.28,1.16)for(Linear,HO,PL)parameters. OurresultsshouldbecomparedwiththeDSEapproach[22]
results using two different procedures, i.e. rainbow-ladder (RL) truncation and the DCSB-improved (DB) kernels:
δφP3;K =δφσ3;K =(1.28,1.12) for (RL, DB) and δφP2;K =1.14 for DB, respectively. As one can see from our results, the
SU(3) flavor-symmetry breaking effect is larger for Gaussian wave function than for PL wave function. Overall our
resultsfromthePLwavefunctionagreequantitativelywiththeDSEresultsfromDCSB-improvedkernels. Regarding
ontheflavorsymmetrybreakingeffect,ourLFQMresults[24,28]ofleptonicdecayconstantratiosf /f =1.24[1.18]
K π
and f /f =1.24[1.32]obtained from Gaussian wave functions with Linear [HO] parameters can also be compared
Bs B
with the experimental data f /f = 1.22 [65] and the recent unquenched lattice-QCD f /f = 1.22(8) [66],
K π Bs B
respectively.
The twist-3 pseudoscalar DA φP (x) and pseudotensor DA φσ (x) are usually expanded in terms of the Gegen-
3;M 3;M
bauer polynomials C1/2 and C3/2, respectively, as follows [17]:
n n
∞
φP = aP C1/2(2x 1),
3;M n,M n −
nX=0
∞
φσ = 6x(1 x) aσ C3/2(2x 1). (23)
3;M − n,M n −
nX=0
The coefficients aP(σ) are called the Gegenbauer moments and can be obtained by
n,M
1
aP (x) = (2n+1) dxC1/2(2x 1)φP (x),
n,M Z n − 3;M
0
4n+6 1
aσ (x) = dxC3/2(2x 1)φσ (x), (24)
n,M 3n2+9n+6Z n − 3;M
0
using the orthogonal condition for the Gegenbauer polynomials
1 π21−4lΓ(2l+n)
dx[x(1 x)]l−1/2Cl (2x 1)Cl(2x 1)= δ . (25)
Z − m − n − n!(n+l)Γ2(l) mn
0
TheGegenbauermomentswithn>0describehowmuchtheDAsdeviatefromtheasymptoticone. Inadditiontothe
Gegenbauermoments, one canalso define the expectation value of the longitudinal momentum, so-calledξ-moments,