Distribution amplitudes and decay constants for $(π,K,ρ,K^*)$ mesons in light-front quark model

0.28 MB·English

by  Ho-Meoyng Choi

Distribution amplitudes and decay constants for (π,K,ρ,K∗) mesons in light-front
quark model
Ho-Meoyng Choia and Chueng-Ryong Jib
a Department of Physics, Teachers College, Kyungpook National University, Daegu, Korea 702-701
b Department of Physics, North Carolina State University, Raleigh, NC 27695-8202
We present a calculation of the quark distribution amplitudes(DAs), the Gegenbauer moments,
∗
anddecayconstantsfor π,ρ,K andK mesonsusingthelight-front quarkmodel. Whilethequark
DA for π is somewhat broader than the asymptotic one, that for ρ meson is very close to the
∗
asymptotic one. The quark DAs for K and K show asymmetric form due to the flavor SU(3)-
symmetrybreakingeffect. Thedecayconstantsforthetransversely polarized ρ andK∗ mesons(fT
ρ
and fKT∗) as well as thelongitudinally polarized ones(fρ and fK∗) are also obtained. Ouraveraged
values for fVT/fV, i.e. (fρT/fρ)av = 0.78 and (fKT∗/fK∗)av = 0.84, are found to be consistent with
7
other model predictions. Especially, our results for the decay constants are in a good agreement
00 with theSU(6) symmetry relation, fρT(K∗) =(fπ(K)+fρ(K∗))/2.
2

n I. INTRODUCTION the hard contribution) [24, 25, 26, 27], our results indi-
a catedthatthe suppressionofthe endpointregionfor the
J
quark DA corresponds to the suppression(enhancement)
2 Hadronic distribution amplitudes(DAs) are important
of the soft(hard) contribution.
2 ingredients in applying QCD to hard exclusive processes
Another important area that requires detailed study
1 via the factorization theorem [1, 2, 3]. They provide of meson DAs is the B-physics phenomenology under in-
essentialinformationonthenonperturbativestructureof
v tense experimental investigation at BaBar and Belle ex-
hadrondescribing the distributionofpartons interms of
7 periments. The K,ρ and K∗ DAs have attracted at-
the longitudinal momentum fractions inside the hadron.
7 tention rather recently [28, 29, 30, 31] due to the deep
1 Both the electromagnetic form factors at high Q2 and relevance to the exclusive B-meson decays to (K,ρ,K∗)
1 the B-physics phenomenology are highly relevant to the
mesons. Inparticular,theSU(3)flavorsymmetrybreak-
0 detailed computation of hadronic DAs.
ingeffectinthemesonquarkDAincludingstrangequark
7
0 During the past few decades, there have been many isimportantforthepredictionsofexclusiveBu,d,s-decays
/ theoretical efforts to calculate the pion DA using non- to light pseudoscalar and vector mesons in the context
h
perturbativemethods suchasthe QCDsumrule[3,4,5, of CP-violationand Cabibbo-Kobayashi-Maskawaquark
p
6,7,8],latticecalculation[9,10,11,12,13],chiralquark mixingmatrixstudies. TheSU(3)breakingeffectisreal-
-
p model from the instanton vacuum [14, 15, 16], Nambu- izedinthe difference betweenthe longitunalmomentaof
e Jona-Lasinio(NJL) model [17, 18], and light-front quark thestrangeandnonstrangequark, x x =0,inthe
h model(LFQM) [19, 20]. It is well known that the shape two particle Fock components of thhesm−esuo(nd.)iT6he similar
:
v ofthepionquarkDAisveryimportantinthepredictions effect was also found in our PQCD analysis [32] for the
Xi ofpionelectromagneticformfactorinbothnonperturba- exclusiveheavymesonpairproductionine+e− annihila-
tive and perturbative momentum transfer regimes. The tionsat√s=10.6GeV.Notonlytheshapeoftheheavy
r
a QCD sum-rule based analysis [6, 7] of the π γ tran- meson quark DAs matters in the prediction of the cross
sition form factor Fπγγ∗(Q2) measured by th−e CLEO sectionfortheheavymesonpairproductions,butalsothe
experiment [21] has shown that neither double-humped cross section ratios for σ(e+e− D+D−)/σ(e+e−
DA for the pion predicted by Chernyak and Zhitnit- D+D−) and σ(e+e− B0B¯0)/→σ(e+es− s B+B−) d→e-
sky [3] nor the asymptotic one are favored at the 2σ viate from 1 apprecia→bly dsues to the SU→(3) symmetry
level of accuracy. It is also interesting to note that breaking.
the recent anti-de Sitter space geometry/conformal field A particularly convenient and intuitive framework
theory(AdS/CFT) prediction [22] for the meson DA is in applying PQCD to exclusive processes is based
φAdS/CFT(x) x(1 x),whichwouldapproachtothe upon the light-front(LF) Fock-state decomposition of
∝ −
asymptoticformpx(1 x)onlyinthelimitoflnQ2 . hadronic state. In the LF framework, the valence quark
− →∞
The shape of φ (x) increases the usual perturba- DA is computed from the valence LF wave function
AdS/CFT
tive QCD (PQCD) predictions for the pion form factor Ψ (x ,k ) of the hadron at equal LF time τ = t +
n i ⊥i
andπ γ transitionformfactorby16/9and4/3,respec- z/c which is the probability amplitude to find n con-
−
tively. In our recent LFQM application to the PQCD stituents(quarks,antiquarks, and gluons) with LF mo-
analysis of the pion form factor [23], we further found menta k = (x ,k ) in a hadron. Here, x and k are
i i ⊥i i ⊥i
a correlation between the shape of quark DA and the the LF momentum fraction and the transversemomenta
amount of soft and hard contributions to the pion form oftheithconstituentinthen-particleFock-state,respec-
factor. SimilartothepreviousfindingsfromtheSudakov tively. IfthefactorizationtheoreminPQCDisapplicable
suppression of the soft contribution (or enhancement of to exclusive processes, then the invariant amplitude
M
2
forexclusiveprocessfactorizesintotheconvolutionofthe DAs are found to be zero due to isospin symmetry, the
process-independent valence quark DA φ(x,µ) with the oddmoments for K andK∗ mesonDAs arenonzerodue
process-dependent hard scattering amplitude T [1], i.e. to the flavor SU(3)-symmetry breaking effect. We com-
H
pute the decay constants for the transversely polarized
ρ and K∗ mesons(fT and fT ) as well as the longitudi-
= [dxi] [dyi]φ(xi,µ)TH(xi,yi,µ)φ(yi,µ), (1) ρ K∗
M Z Z nally polarized ones(fρ and fK∗) and compare with the
light-conesumrule(LCSR)calculations,inwhichthe ra-
wofhqeurear[dkxsii]n=thδe(1v−alΣennk=ce1xFko)cΠknks=t1adtxe.kHanedren, µisdtheneontuems btheer ptiroedoficftViTonasndoffVthiesBanimpρoarntadntBingreKdi∗enttrafonrsitthioenLfCoSrmR
separation scale between perturbative and nonperturba- → →
factors. We also confirm that our results for the decay
tive regime. Since the collinear divergences are summed
constants follow an old SU(6) symmetry relation [39],
sinystφe(mxia,tµic)a,lltyhecohmaprudtesdcaatstearinpgertaumrbpalittiuvdeeexTpHancsaionnbine fρT(K∗) =(fπ(K)+fρ(K∗))/2.
The paper is organizedas follows: In Sec.II, we briefly
α (µ). To implement the factorization theorem given by
s describe the formulation of our LFQM [35, 36] and the
Eq. (1) at high momentum transfer, the hadronic wave
procedureoffixingthemodelparametersusingthevaria-
function plays an important role linking between long
tionalprinciplefortheQCD-motivatedeffectiveHamilto-
distancenonperturbativeQCDencodedinDAandshort
nian. The shape ofthe quarkDAis then uniquely deter-
distance PQCD encoded in T .
H mined in our model calculation. In Sec.III, the formulae
ThequarkDAofameson,φ(x,µ),istheprobabilityof
for the quark DAs and decay constants of pseudoscalar
findingcollinearquarksuptothescaleµintheL =0(s-
z and vector mesons are given in our LFQM. The Gegen-
wave) projection of the meson wave function defined by
bauer and ξ(= x x ) moments are also given in this
1 2
−
|k⊥|<µ section. In Sec. IV, we present the numerical results for
φ(x ,µ)= [d2k ]Ψ(x ,k ), (2) thedecayconstants,thequarkDAs,theGegenbauerand
i ⊥i i ⊥i
Z ξ moments for (π,K,ρ,K∗) mesons and compare with
other theoretical model predictions. Summary and con-
where
clusions follow in Sec.V. The relations between ξ and
n d2k Gegenbauer moments are presented in Appendix A.
[d2k ]=2(2π)3δ k Πn ⊥i . (3)
⊥i (cid:20) ⊥j(cid:21) i=12(2π)3
Xj=1
II. MODEL DESCRIPTION
Simple relativistic quark-model based on the LF frame-
workhasbeen studied forvariousmesons[19, 20,33, 34].
InourLFQM[35,36],themesonwavefunctionisgiven
Although the proof of duality between the LFQM and
by
the first principle QCD is not yet available, we have at-
tempted to fill the gapbetween the model wavefunction ΨJJz(x,k ,λλ¯) = φ (x,k ) JJz(x,k ), (4)
and the QCD-motivated effective Hamiltonian[35, 36]. M ⊥ R ⊥ Rλλ¯ ⊥
TheessentialfeatureofourLFQM[35,36]istotreatthe where φ (x,k ) is the radial wave function and
R ⊥
gaussian radial wave function as a trial function for the JJz(x,k ) is the spin-orbit wave function obtained by
variationalprincipletotheQCD-motivatedHamiltonian. Rλλ¯ ⊥
the interaction-independent Melosh transformation [40]
We saturate the Fock state expansionby the constituent
fromtheordinaryequal-timestaticspin-orbitwavefunc-
quark and antiquark, i.e. Hqq¯ = H0 +Vint, where the tionassignedbythe quantumnumbersJPC. The meson
interaction potential V consists of confining and hy-
int wave function in Eq. (4) is represented by the Lorentz-
perfine interaction terms. From the variational princi- invariant variables x = p+/P+, k = p x P and
ple minimizing the central Hamiltonian with respect to i i ⊥i ⊥i− i ⊥
λ , where P,p and λ are the meson momentum, the
i i i
the gaussianparameter,we can find the optimum values
momentaandthe helicities oftheconstituentquarks,re-
of our model parameters and predict the mass spectra
spectively.
for the low-lying ground state pseudoscalar and vector
The radial wave function φ (x,k ) of a ground state
R ⊥
mesons [35, 36]. We applied our LFQM for various ex- pseudoscalar meson(JPC =0−+) is given by
clusiveprocessessuchastheelectromagneticformfactors
of π, K and ρ [35, 37] mesons and semileptonic and rare 1 1/2
B decays to π and K [36, 38]. Our results for the above φR(x,k⊥) = (cid:18)π3/2β3(cid:19) exp(−~k2/2β2), (5)
exclusive processes were in a good agreement with the
available data as well as other theoretical model predic- where ~k2 = k2 + k2 and the gaussian parameter β is
⊥ z
tions. relatedwiththesizeofthemeson. Here,thelongitudinal
The purpose of this work is to calculate the quark component k of the three momentum is given by k =
z z
DAs, the Gegenbauer moments, and decay constants for (x 1)M +(m2 m2)/2M with the invariant mass
π,ρ,K and K∗ mesons using our LFQM and compare 1− 2 0 2− 1 0
with other theoretical model predictions. As expected, k2 +m2 k2 +m2
M2 = ⊥ 1 + ⊥ 2, (6)
while the odd Gegenbauer moments for π and ρ meson 0 x x
1 2
3
where x =x and x =1 x. The covariantform of the
1 2
spin-orbit wave functions−for pseudoscalar(JPC = 0−+) TABLEI:Theconstituentquarkmassesmq(inGeV)andthe
and vector(1−−) mesons are given by gaussianparametersβqq¯(inGeV)forthelinear[HO]potential
obtained from thevariational principle. q=u and d.
00 = u¯(p1,λ)γ5v(p2,λ¯) , mq ms βqq¯ βqs¯
Rλλ¯ −√2[M2 (m m )2]1/2 0.22[0.25] 0.45[0.48] 0.3659[0.3194] 0.3886[0.3419]
0 − 1− 2
u¯(p ,λ) ǫ(J ) ǫ·(p1−p2) v(p ,λ¯)
1J3 = 1 (cid:20)6 z − M0+m1+m2(cid:21) 2 . (7) where Vconf = br(br2) for the linear(HO) potential and
Rλλ¯ − √2[M2 (m m )2]1/2 Sq Sq¯ =1/4( 3/4)forthevector(pseudoscalar)meson.
0 − 1− 2 h W·e tihen take−φ (x,k ) as our trial function to mini-
R ⊥
The polarization vectors ǫµ = (ǫ+,ǫ−,ǫ ) used in this mize the central Hamiltonian via
⊥
analysis are given by ∂ Ψ[H +V ]Ψ
0 0
h | | i =0. (13)
∂β
2
ǫµ( 1) = 0, ǫ ( ) P ,ǫ ( 1) ,
± (cid:20) P+ ⊥ ± · ⊥ ⊥ ± (cid:21) From the above constraint, only 4 parameters are inde-
pendent among the light-quarkmasses andthe potential
(1, i)
ǫ ( 1) = ± , parameters (m ,β ,a,b,κ)(q = u,d). In order to de-
⊥ q qq¯
± ∓ √2
termine these four parameters from the two experimen-
1 P2 M2 tal values of ρ and π masses, we take the string tension
ǫµ(0) = P+, ⊥− 0,P . (8)
M (cid:20) P+ ⊥(cid:21) b=0.18GeV2 andtheconstituentuanddquarkmasses
0
m = m = 0.22(0.25) GeV for the linear (HO) poten-
u d
Note that RJJz†RJJz = 1. The normalization of tial,whichareratherwellknownfromotherquarkmodel
λλ¯ λλ¯ λλ¯
our wave fuPnction is given by analysescommensurate with Regge phenomenology[42].
More detailed procedure of determining the model pa-
d3k ΨJJz(x,k ,λλ¯)2 rameters of light quark sector(u(d) and s) can be found
Z | M ⊥ | in[35,36]. Ourmodelparametersforthelightquarksec-
Xλλ¯
torobtainedbythe variationalprinciple aresummarized
1 ∂k
= dx d2k z φ (x,k )2 =1, (9) in Table I.
⊥ R ⊥
Z Z (cid:18) ∂x (cid:19)| |
0
where the Jacobian of the variable transformation
III. QUARK DISTRIBUTION AMPLITUDES
x,k⊥ ~k =(k⊥,kz) is given by AND DECAY CONSTANTS
{ }→
∂k M (m m )2 2
z = 0 1 1− 2 . (10) The quark DA of a hadron in our LFQM can be
∂x 4x1x2(cid:26) −(cid:20) M02 (cid:21) (cid:27) obtained from the hadronic wave function by integrat-
ing out the transverse momenta of the quarks in the
The effect of the Jacobi factor has been analyzed in
hadron(see Eq. (2)),
Ref. [41].
The key idea in our LFQM [35, 36] for mesons is to |k⊥|<µ d2k ∂k
treat the radial wave function φR(x,k⊥) as a trial func- φ(x,µ) = Z √16π⊥3r ∂xzΨ(x,k⊥,λλ¯).(14)
tion for the variational principle to the QCD-motivated
Hamiltonian saturating the Fock state expansion by the For K and K∗ meson cases, we assign the momentum
constituent quark and antiquark. The QCD-motivated fractions x for s-quark and (1 x) for the light u(d)-
−
effectiveHamiltonianforadescriptionofthemesonmass quark. The quark DA describes probability amplitudes
spectra is given by [42] to find the hadron in a state with minimum number of
Fock constituents and small tranverse-momentum sepa-
ration defined by an ultraviolet(UV) cutoff µ > 1GeV.
H =H +V = m2+~k2+ m2+~k2+V . (11)
qq¯ 0 qq¯ q q q q¯ qq¯ The dependence on the scale µ is then given∼by the
QCD evolution equation[1] and can be calculated per-
In our LFQM [35, 36], we use the two interaction po-
turbatively. However,the DAs ata certainlow scale can
tential V for the pseudoscalar and vector mesons: (1)
qq¯ beobtainedbythenecessarynonperturbativeinputfrom
Coulombplusharmonicoscillator(HO),and(2)Coulomb
LFQM.Moreover,the presenceofthedampingGaussian
plus linear confining potentials. In addition, the hyper-
factor in ourLFQM allowsus to performthe integralup
fine interaction, which is essential to distinguish vector
to infinity without loss of accuracy. The quark DAs for
frompseudoscalarmesons,isincludedforbothcases,viz.,
pseudoscalar(P) and vector(V) mesons are constrained
4κ 2S S by
V =V +V =a+ + q· q¯ 2V ,
qq¯ 0 hyp conf Coul.
V − 3r 3mqmq¯∇ 1 fP(V)
φ (x,µ)dx= , (15)
(12) Z P(V) 2√6
0
4
where the decay constant is defined as where Φ (x)=6x(1 x) is the asymptotic DA and the
as
−
coefficients a (µ) are Gegenbauer moments [1, 17, 43].
0q¯γµγ q P =if Pµ, (16) n
h | 5 | i P TheGegenbauermomentswithn>0describehowmuch
for a pseudoscalar meson and the DAs deviate from the asymptotic one. The zeroth
Gegenbauer moment is fixed by the decay constant [1],
0q¯γµq V(P,λ) = fVMVǫµ(λ), e.g. for the pion:
h | | i
0q¯σµνq V(P,λ) = ifT[ǫµ(λ)P ǫν(λ)P ], (17)
h | | i V ν − µ 1 3
for a vector meson with longitudinal(λ = 0) and a0 = 6 [dx]φπ(xi,µ)= fπ, (24)
Z √6
transverse(λ= 1) polarizations, respectively. The con- 0
±
straint in Eq. (15) must be independent of cut-off µ up where f 131 MeV. In addition to the Gegenbauer
π
to corrections of order Λ2/µ2, where Λ is some typical moments,w≃ecanalsodefine theexpectationvalue ofthe
hadronic scale(< 1 GeV) [1]. For the non-perturbative longitudinal momentum, so-called ξ-moments:
valence wave fu∼nction given by Eq. (5), we take µ 1
∼ 1 1
GeV as an optimal scale for our LFQM.
ξn = dξξnΦˆ(ξ)= dxξnΦ(x), (25)
The explicit form of a pseudoscalar decay constant is h i Z Z
−1 0
given by
where Φ(x) = 2Φˆ(2x 1) normalized by ξ0 = 1. In
fP =Z01dxZ [d2k⊥] A2A+k2⊥φR(x,k⊥), (18) AexpppliecnitdliyxgAiv,enthuepretolatn−io=ns6.between hξni ahndian(µ) are
p
where =(1 x)m +xm . Thedecayconstants,f and
1 2 V
A −
fT, for longitudinally and transversely polarized vector
V IV. NUMERICAL RESULTS
mesons, respectively, are given by
1 φ (x,k ) 2k2 In our numerical calculations, we use two sets of the
f = dx [d2k ] R ⊥ + ⊥ ,
V Z0 Z ⊥ A2+k2⊥(cid:20)A M0+m1+m2(cid:21) mtoor(dHelOp)acroanmfientienrgspfoortenthtieallsingeiavrenainndThaabrlme oI.nic oscilla-
p (19)
We showinTable IIourpredictionsforthe decaycon-
fT = 1dx [d2k ] φR(x,k⊥) + k2⊥ . stants of (π,K,ρ,K∗) mesons and compare with other
V Z0 Z ⊥ pA2+k2⊥(cid:20)A M0+m1+m(22(cid:21)0) tAhseoorneeticcaalnmsoedeelfrpormediTcatibolnesI[I9,, o3u0]rarseswuelltlsafsodratthae[4d5e]-.
The pion decay constant fexp. 131 MeV is meaured cay constants of (π,K,ρ,K∗) obtained from both lin-
fromπ µνandtheρmesoπndec≃ayconstantfexp. 215 ear and HO potential models( especially, those from HO
MeV is→measured from ρ e+e− with the loρngitu≃dinal potential) are compatable with the data [45]. Our val-
→ ues for the ratio of fT and f , i.e. fT/f = 0.76[0.80]
polarization. While the constant fV are known from ex- V V ρ ρ
periment, the constant fVT are not that easily accessible and fKT∗/fK∗ = 0.82[0.86] obtained from the linear[HO]
in experiment and hence can be estimated only theoret- potential, are quite comparable with the recent QCD
ically. sum rule results, fρT/fρ = (0.78±0.08) and fKT∗/fK∗ =
Theaveragevalueofthetransversemomentumisgiven (0.78 0.07) [30]. We also find that our results for the
±
by decayconstantsagreesurprisinglywellwithanoldSU(6)
symmetry relation [39], fρT(K∗) = (fπ(K)+fρ(K∗))/2 via
hk2⊥iQQ¯ =Z d3k|k2⊥||φR(x,k⊥)|2. (21) thesumruleφTρ(K∗) =(φπ(K)+φLρ(K∗))/2,whereφL(T)(x)
isthelongitudinally(transversely)polarizedvectormeson
Numerically, we have confirmed that hk2⊥i1Q/Q2¯ = βQQ¯. DA.
We show in Fig. 1 the normalized quark DAs Φ (x)
This is a nonperturbative measure of the transverse size P
for π and K mesons obtained from linear(solid line) and
in the mesonic valence state.
HO(dashed line) potentials. For the pion DA, we also
We may also redefine the quark DA as Φ (x) =
P(V)
compareourresults with the asymptotic resultΦ (x)=
(2√6/f )φ(x) so that as
P(V) 6x(1 x)(dotted line) as well as the AdS/CFT pre-
−
1 diction [22] Φ (x) = π x(1 x)/8(double-dot-
AdS/CFT
Φ (x)dx=1. (22) −
Z P(V) dashedline). For the pion case,pour quark DAs obtained
0
frombothmodelparametersaresomewhatbroaderthan
The quark DA Φ(x) evolved in the leading order(LO) the asymptotic one. We also note from the normalized
of αs(µ) is usually expanded in Gegenbauer polynomials pion DAs that the suppression of the end point(x 0
C3/2 as and 1) region has the following order in DAs, Φ (x→)>
n HO
Φ (x) > Φ (x) > Φ (x). As discussed in
∞ as Linear AdS/CFT
Φ(x,µ) = Φ (x) 1+ a (µ)C3/2(2x 1) ,(23) Ref. [23], there exists correlation between the shape of
as (cid:20) nX=1 n n − (cid:21) nonperturbative quark DA and the amount of low/high
5
2
TABLE II: Decay constants(in MeV) for the linear[HO] po- Linear potential(m=0.22 GeV,β=0.3659 GeV)
tential models compared with othermodels and data. HO potential(m=0.25 GeV,β=0.3194 GeV)
φ (x)~x(1-x)
as
fM Linear[HO] SR[30] Lattice[9] Exp.[45] φAdS/CFT(x)~[x(1-x)]1/2
1.5
fπ 130[131] - 126.6(6.4) 130.70(10)(36)
fK 161[155] - 152.0(6.1) 159.80(1.4)(44)
fρ 246[215] 205(9) 239.4(7.3) 220(2)(a), 209(4)(b)
ffKρT∗ 128586[[127233]] 126107((150)) 255.5-(6.5) 217(-5)(c) Φ(x)π 1
fKT∗ 210[191] 170(10) - -
(a) Exp. value for Γ(ρ0 →e+e−).
(b) Exp. valuefor Γ(τ →ρντ). 0.5
(c) Exp. value for Γ(τ →K∗ντ).
Q2 contributions to the pion form factor. As the end- 00 0.2 0.4 0.6 0.8 1
x
point region for the quark DA is more suppressed, the
soft(hard) contribution to the pion form factor gets sup-
pressed(enhanced). This finding is rather similar to the
previous findings from the Sudakov suppression of the 2
soft contribution[24, 25, 26, 27]. For the kaon case, the
Linear potential
quarkDAis asymmetricdue tothe flavorSU(3) symme- HO potential
try breaking effect. The peak points of quark DAs for
1.5
two potential models are moved slightly to the right of
x = 0.5 point indicating that s-quark carries more lon-
gitudinal momentum fraction than the light u(d)-quark.
Φ(x)K 1
In the LO QCD [1], the information of the leading-
twist pion DA can be extracted from the pion-photon
transition form factor F (Q2) as follows:
πγ
0.5
Q2FLO(Q2) 1 Φ (x,Q)
πγ π
= dx . (26)
√2f (cid:12) Z 6x(1 x)
π (cid:12)twist−2 0 −
(cid:12)
The experimental (cid:12)value obtained in CLEO [21] is 00 0.2 0.4 0.6 0.8 1
x
Q2F (Q2) = (16.7 2.5 0.4) 10−2 GeV at Q2 = 8
πγ
± ± ×
GeV2, which goes to √2f 0.185 GeV in the asymp- FIG. 1: Normalized DAs Φ(x) for π and K mesons obtained
π
≃
totic Q2 limit. With our leading twist pion DA from linear(solid line) and HO(dashed line) potential models
shown in→Fig∞. 1, we obtain Q2FLO(Q2) = 0.202[0.181] compared with asymptotic result(dotted line) as well as the
πγ
AdS/CFT prediction [22](double-dot-dashedline).
GeV for the linear[HO] potential.
For comparison between the leading twist and next-
to-leading twistcontributions to Q2F (Q2), we show in
πγ
Fig.2ourpreviousLFQM[35]predictionforQ2Fπγ(Q2) from the asymptotic Φas(x)(dot-dashed line) and the
compared with the data [21, 44]. The thick solid and AdS/CFT ΦAdS/CFT(x)(double-dot-dashed line). One
thick dashed lines represent our linear and HO potential shouldnotethattheAdS/CFTprediction( x(1 x))
∼ −
model predictions including the higher twist effects(i.e. increases the usual PQCD prediction( x(p1 x)) by
k⊥ and the constituent mass m = mu = md) obtained 4/3. Our higher twist results for both∼potent−ial mod-
from els are not only very similar to each other but also
in good agreement with the experimental data up to
√N 1 ∂k
FNLO(Q2) = (e2 e2) c dx d2k z Q2 10 GeV2 region. At large Q2 region, our higher
πγ u− d π3/2 Z0 Z ⊥r ∂x twis∼t prediction for the linear[HO] potential approaches
φ (k2) (1 x)m Q2FNLO(Q2) = 0.194[0.180] compared to the leading
R − , (27) πγ
× m2+k2 k′2 +m2 twist result 0.202[0.181]. While the higher twist ef-
⊥ ⊥ fect on Q2F (Q2) is large for the low and interme-
p πγ
where N is the color factor and k′ = k + (1 diate Q2(< 10 GeV2) region, its effect becomes very
c ⊥ ⊥
x)q . The thin solid and thin dashed lines repr−e- small for l∼arge Q2 region compared to the leading twist
⊥
sent our leading twist contribution(see Eq. (26)) from contribution. Incidentally, it has been found that the
Φ (x) and Φ (x), respectively. We also com- leading Fock-state contribution to F (Q2) fail to re-
Linear HO πγ
pare our results with the leading twist contributions produce the Q2 = 0 value corresponding to the axial
6
CLEO-data analysis [5, 6, 7]. For the kaon case, the
first moment is proportional to the difference between
0.25 the longitudinal momenta of the strange and nonstrange
quarkinthetwo-particleFockcomponentofthekaon,i.e.
aK = (5/3) x x (x = x,x = 1 x). The knowl-
0.2 ed1ge of aKhissi−mpou¯ritanst for preu¯dictin−g SU(3)-violation
V] 1
Ge effectswithinanyQCDapproachthatemploysthequark
2Q)[ 0.15 DAs of mesons. In our model calculation, we obtain
2QF(πγ FπγNLO with ΨHO(x,kT) a positive value of aK1 = 0.09[0.13] for the linear[HO]
0.1 FFππγγNLOL Ow witiht hφ ΨHOL(ixne)a r(x,kT) tphootseentoiablt.aiOneudr frreosmultostfhoerraeK1stiamraeteqsuistuecchonassisttheentprweivtih-
FπγLO with φLinear (x) ous LFQM [20](aK = 0.08), the chiral-quark model [15]
0.05 FFCππLγγLLEOOO wwiitthh φφAads(Sx/C)F T(x) (0a.0K15 =0.00.20)9.6)Thaen1dpotshiteivQeCsiDgnsoufma-Krucleasn[b2e8,un29d]e(rasK1too=d
CELLO ± 1
0 intuitively since the heavier strange quark(antiquark)
0 5 10 15 20 25 30
Q2[GeV2] carriesalargerlongitudinalmomentumfractionthanthe
lighter nonstrange antiquark(quark). It is interesting to
FIG. 2: The leading twist FπLγO(Q2) and the next-to-leading note that aK for the HO potential model is greater than
twistFNLO(Q2)contributionstoπ−γtransitionformfactor. 1
πγ thatforthelinearonealthoughtheconstituentmassdif-
Data are taken from Refs. [21, 44].
ferencem m =0.23GeVisthesameforbothmodels.
s u
−
This difference is attributed to the fact that the strange
quark mass for HO potential model(m = 0.48 GeV) is
s
anomaly[47,48],i.e. itgivesonlyahalfofwhatisneeded larger than for the linear one(ms =0.45 GeV) and leads
to get the correct π0 γγ rate [49]. However,as shown to more asymmetric shape for the HO potential. For
in Refs. [50, 51, 52], t→he leading Fock-state contribution the second Gegenbauer moment aK2 , however, the linear
to F (Q2) has been enhanced by replacing the leading potential model gives positive value(0.03), while the HO
πγ
Fock-state wave function to an ‘effective’ valence quark one gives negative value(-0.03). We note that the QCD
wave function that is normalized to one. By taking the sum-rules[28,29]andlatticecalculation[11]givepositive
‘effective’ pion wave function with the asymptotic-like values while the chiral quark model [15] gives a negative
DAs,theauthorsin[50,51,52]foundanagreementwith value.
the experimental data. Our LFQM prediction [35] also We show in Fig. 3 the normalized quark DAs for
uses the same approach as Refs. [50, 51, 52], i.e. the π(upper panel) and K(lower panel) mesons obtained
leading Fock-state ‘effective’ wave function that is nor- fromthelinearpotentialmodel(exactsolution)andcom-
malized to one. The reason why our model is so suc- pare with those from the truncated Gegenbauer polyno-
cesful for Fπγ transition form factor is because the Q2 mials up to n = 6(approximate solution). For the pion
dependence( 1/Q2) is due to the off-shell quark prop- case,thetruncationupton=4(dashedline)seemsmore
∼
agator in the one-loop diagram and there is no angular closetoourexactsolution(solidline)thanthetruncation
condition [53] associated with the pseudoscalar meson. up to n=6(dot-dashed line) although the end-point be-
In Tables III and IV, we list the calculated ξn and havior of n = 6 case is more close to the exact solution
Gegenbauer moments a (µ) for the pion(TablehIIIi) and than n 4 case. Since the result up to n=2 truncation
n ≤
thekaon(TableIV)DAsobtainedfromthelinear[HO]po- is not much different from that up to n = 4 truncation,
tential models at the scale µ 1 GeV. We also present wedonotshowtheresultforn=2caseinthefigure. For
the comparison with other mo∼del estimates at the scale thekaoncase,whilebothtruncationsupton=4(dashed
of 1 µ 3 GeV. While the odd Gegenbauer mo- line) and n = 6(dot-dashed line) show good agreement
ments≤of th≤e π meson DA become zero due to isospin with the exact solution(solid line), the truncation up to
symmetry, the odd moments for the kaon are nonzero n=2(dottedline)deviatesalotfromthe exactsolution.
due to a flavor-SU(3) violation effect of O(m m ). Thus, it seems not sufficient to truncate the Gegenbauer
s u(d)
For the pion case, our result for the second Ge−genbauer polynomials only up to n=2 for the kaon case. In both
moment, aπ = 0.12[0.05] obtained from linear[HO] po- cases of π and K mesons, our model calculation shows
2
tential is quite comparable with other theoretical model thatthetruncationoftheGegenbauerpolynomialsupto
predictions given in Table III. A fair average is, how- n = 4 seems to give a reasonable approximation to the
ever, aπ = 0.17 0.15 with still large errors [11]. The exact solution.
2 ±
LCSR based CLEO-data analysis [5, 6, 7] on the tran- We show in Fig. 4 the normalized quark DAs Φ (x)
ρ
sition form factor Fπγ suggests a negative value for aπ4, forthelongitudinally(solidline)andtransversely(dashed
which is consistent with the result aπ(1GeV2) > 0.07 line) polarized ρ meson obtained from the linear(upper
obtained in Ref. [4]. Our result aπ4 = 0.003[−0.03] panel) and HO(lower panel) potential models and com-
4 − −
obtained from the linear[HO] potential also prefers a pare with the asymptotic result(dotted line). For both
negative value consistent with the recent LCSR based potential models, the quark DA ΦL with longitudinal
ρ
7
2 metry. Our ξ moments obtained from both linear and
Φ (x)[1+ up to n=4] HO potential models are similar to the asymptotic re-
as
Φas(x)[1 + up to n=6] sults and quite consistent with the previous LFQM [20]
Φ (x)(Linear potential)
LFQM
and QCD sum-rules [30]. However, slight differences of
1.5
ξ moments among different model predictions turn out
to be quite sensitive in terms of Gegenbauer moments.
Forinstance,thesecondGegenbauermomentaρ forthe
2L
Φ(x)π 1 longitudinally polarized ρ meson obtained from the lin-
ear potential model gives positive value(0.02), while the
samefromtheHOpotentialmodelgivesnegativevalue(-
0.02). This may be still comparable with the previous
0.5 LFQM [20] giving negative value(-0.03) and the QCD
sum-rules [30] giving positive value(0.09+0.10). Our re-
−0.07
sults for the higher Gegenbauer moments aρ(n 4) ob-
n ≥
tained from both linear and HO potential models show
0
0 0.2 0.4 0.6 0.8 1
x negative values regardless of polarization states. These
negative values for the higher Gegenbauer moments are
relatedwiththeconcaveshapesofquarkDAs,ΦL(x)and
ρ
ΦT(x), at the end-point region.
2 ρ
Φas[1 + up to n=2] We show in Fig. 5 the normalized quark DAs ΦK∗(x)
Φas[1 + up to n=4] forlongitudinally(solidline)andtranversely(dashedline)
Φ [1 + up to n=6]
Φas (x)(Linear potential) polarized K∗ meson obtained from the linear(upper
LFQM
1.5 panel) and HO(lower panel) potential models. As in the
caseoftheρmeson,theshapeofΦT forthetransversely
K∗
polarized K∗ meson near the central(x = 1/2) region is
somewhat broader than that of ΦL for the longitudi-
Φ(x)K 1 nallypolarizedK∗ mesoninbothliKne∗arandHOmodels.
Also, the peak points for both ΦL and ΦT are shifted
K∗ K∗
totherightofx=1/2pointduetotheSU(3)flavorsym-
0.5 metry breaking as in the case of K meson. Our quark
DAs for the K∗ meson satisfy also the SU(6) symmetry
relation [39], ΦT =(Φ +ΦL )/2.
K∗ K K∗
In Tables VI and VII, we list the calculated ξn
0 h i
0 0.2 0.4 0.6 0.8 1 and Gegenbauer moments an(µ) for the longitudi-
x
nally(Table VI) and transversely(Table VII) polarized
FIG. 3: Normalized DAs for π(upper panel) and K(lower K∗ meson DAs obtained from the linear[HO] potential
panel)mesonsobtainedfromlinearpotentialmodelcompared models at µ 1 GeV and compare with the available
∼
with those obtained from the truncated Gegenbauer polyno- QCD sum-rule results [30, 31]. While the odd Gegen-
mials up to n=6. bauer moments of the ρ meson DAs become zero due to
the isospin symmetry, the odd moments for the K∗ me-
sonDAs arenonzerobecause the SU(3)flavorsymmetry
polarization is not much different from the asymptotic is broken. Our values of the first Gegenbauer moments,
resultandthequarkDAΦT withtransversepolarization aK∗ =0.11[0.14]and aK∗ =0.10[0.14]for the linear[HO]
ρ 1L 1T
is somewhat broader than both ΦL and Φ . Overall, potential model are in a good agreement with the QCD
ttohteicqureasrukltDtAhsanforthtohseeρfomretshoenpairoenρccloasseer. tAoalstthheouagshymthpe- sµu=m-1ruGleeVre.suNlotste[3t0h]aatK1tL∗he=paoK1sTi∗tiv=e0a.K10∗±re0f.e0r7sattotKhe∗sccoanle-
1
overall shapes of our ΦL and ΦT are not much different taining an s quark but the sign will change for K¯∗ with
from the asymptotic reρsult, theρ end-point behaviors of an s¯quark. Our predictions for the even powers of aK∗
2nL
our model calculation exhibiting the concave shape are andaK∗ (n 1)givenegativevalueswhiletheLCSRre-
2nT ≥
differentfromthe asymptoticresultofthe convexshape. sults [30] give positive values. However, the recent QCD
WealsoshouldnotethatourDAsfortheρmesonsatisfy sum-rule result aK∗ =0.02 0.02 [31] at the scale µ=1
the SU(6) symmetry relation [39], ΦTρ =(Φπ+ΦLρ)/2. GeV is consisten3tTwith our±value aK3T∗ = 0.04[0.03] ob-
In Table V, we list the calculated ξn and Gegen- tained from the linear[HO] potential model. Also, the
bauer moments a (µ) for the ρ mesohn DiAs obtained ξ2n moments are not much different between the two
n
h i
from the linear[HO] potential models at µ 1 GeV and models. We show in Fig. 6 the normalized quark DAs
compare with other model estimates. The∼odd Gegen- for transversely polarized ρ(upper panel) and K∗(lower
bauer moments of both longitudinally and transversely panel) mesons obtained from the linear potential model
polarized ρ meson DAs become zero due to isospin sym- and compare with those from the truncated Gegenbauer
8
2 2
Linear potential model Linear potential model
ΦLρ(x) ΦLK*(x)
ΦTρ(x) ΦTK*(x)
Φ (x)
1.5 as 1.5
Φ(x)ρ 1 Φ(x)*K 1
0.5 0.5
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
x x
2 2
HO potential model HO potential model
ΦLρ(x) ΦLK*(x)
ΦTρ(x) ΦTK*(x)
Φ (x)
1.5 as 1.5
Φ(x)ρ 1 Φ(x)*K 1
0.5 0.5
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.4: NormalizedDAsforthelongitudinally(solidline)and FIG. 5: Normalized DAs for longitudinally(solid line) and
∗
transversely(dashedline)polarizedρmesonobtainedfromlin- tranversely(dashed line) polarized K meson obtained from
ear(upperpanel)andHO(lowerpanel)potentialmodelscom- linear(upper panel) and HO(lower panel) potential models,
pared with asymptotic result(dotted line). respectively.
V. SUMMARY AND DISCUSSION
polynomials up to n = 6. For the ρ meson case(upper In this work, we investigated the quark DAs, the
panel), since the truncation up to n = 2 does not much Gegenbauer moments, and decay constants for π,ρ,K
differ from that up to n = 4(dashed line), we do not andK∗ mesonsusingtheLFQMconstrainedbythevari-
showtheresultforn=2caseinthefigure. Bothtrunca- ationalprinciple forthe QCD-motivatedeffectiveHamil-
tions up to n=2 and n=4 are quite close to our exact tonian. Our model parameters obtained from the varia-
solution(solid line). The truncation up to n = 6(dot- tionalprincipleuniquelydeterminetheabovenonpertur-
dashed line) having a deep at x = 1/2 point shows a bative quantities.
slightdeviationfrom the exact solution. For the K∗ me- Ourpredictions for the quarkDAs for π and ρ mesons
son case(lower panel), both truncations up to n=4 and showsomewhatbroadershapesthantheasymptoticone.
n = 6 show good agreement with the exact solution, The odd Gegenbauer moments for π and ρ meson DAs
while the truncation up to n = 2(dotted line) deviates become zero due to isospin symmetry. Our predic-
a lot from the exact solution. For both ρ and K∗ meson tions for aπ and aπ are consistent with the recent light-
2 4
cases, the truncation of the Gegenbauer polynomials up cone sum-rule based CLEO data analysis for the pion-
to n=4 seems to give an overall reasonable approxima- photontransition formfactor. Interestingly,we also find
tion to the exact solution. that our leading twist result for the pion-photon tran-
9
2 for the transversely polarized ρ and K∗ mesons(fT and
ρ
ΦΦas[[11 ++ uupp ttoo nn==46]] fKT∗) using our LFQM. Our predicted values of fVT/fV
Φas (x)(Linear potential) averaged between the linear and HO potential cases
LFQM
1.5 are (fρT/fρ)av = 0.78 and (fKT∗/fK∗)av = 0.84. They
are consistent with the recent QCD sum rule results,
fρT/fρ =(0.78±0.08)and fKT∗/fK∗ =(0.78±0.07)[30].
Moreover, our results for the decay constants are in a
TΦ(x)ρ 1 good agreement with the SU(6) symmetry relation [39],
fρT(K∗) = (fπ(K) +fρ(K∗))/2 via the sum rule φTρ(K∗) =
(φπ(K)+φρ(K∗))/2. Further investigations to utilize our
LFQM are underway.
0.5
Acknowledgments
0
0 0.2 0.4 0.6 0.8 1
x
This work was supported by a grant from the U.S.
Department of Energy under Contract No. DE-FG02-
03ER41260. H.-M.ChoiwassupportedinpartbyKorea
2 ResearchFoundation under the contract KRF-2005-070-
Φas(x)[1 + up to n=2] C00039. ThisresearchalsousedresourcesoftheNational
Φ (x)[1 + up to n=4]
as
Φ (x)[1 + up to n=6] Energy Research Scientific Computing Center, which is
as
Φ (x)(Linear potential)
LFQM supportedbytheOfficeofScienceoftheU.S.Department
1.5 of Energy under Contract No. DE-AC02-05CH11231.
TΦ(x)*K 1 APPENDIGXEAG:ENRBEALUATEIROMNOBMETEWNTESEN ξ AND
Theξ-moments ξn definedbyEq.(25)canbe related
h i
0.5 to the Gegenbauer moments a (µ) in Eq.(23). The rela-
n
tions up to n=6 are given by
3
ξ1 = a ,
1
0 h i 5
0 0.2 0.4 0.6 0.8 1
x 12 1
ξ2 = a + ,
2
h i 35 5
FIG. 6: Normalized DAs for the transversely polarized
ρ(upper panel) and K∗(lower panel) mesons obtained from ξ3 = 9 a + 4 a ,
1 3
linear potential model compared with those obtained from h i 35 21
Gegenbauer polynomials up to n=6. ξ4 = 3 + 8 a + 8 a ,
2 4
h i 35 35 77
1 40 8
ξ5 = a + a + a ,
sition form factor, Q2FLO(Q2) = 0.202[0.181] GeV ob- h i 7 1 231 3 143 5
πγ
tained from the linear[HO] potential model, is reduced ξ6 = 1 + 12a + 120 a + 64 a . (A1)
to Q2FNLO(Q2) = 0.194[0.180] if we include the higher h i 21 77 2 1001 4 2145 6
πγ
twist effects such as the transverse momentum and the
Also,thefirstsixGegenbauerpolynomialsinEq.(23)are
constituent mass. Our result is quite compatible with
as follows:
the CLEO data, Q2F (Q2)=(16.7 2.5 0.4) 10−2
GeV at Q2 = 8 GeπVγ2 [21]. The±quark±DAs×for K C13/2(ξ) = 3ξ,
and K∗ show asymmetric forms due to the flavorSU(3)- 3
C3/2(ξ) = (5ξ2 1),
symmetry breakingeffect. This leadsto the nonzeroval- 2 2 −
ues of the odd Gegenbauer moments. In our model cal- 5
culations of the quark DAs for (π,K,ρ,K∗) mesons, the C33/2(ξ) = 2ξ(7ξ2−3),
truncation of the Gegenbauer polynomials up to n = 4 15
seems to give a reasonable approximation to the exact C43/2(ξ) = 8 (21ξ4−14ξ2+1),
solution. 21
C3/2(ξ) = ξ(33ξ4 30ξ2+5),
Ourpredictionsforthedecayconstantsforπ,K,longi- 5 8 −
tudinallypolarizedρandK∗ mesonsareinagoodagree- 1
ment with the data. We also obtain the decay constants C63/2(ξ) = 16(3003ξ6−3465ξ4+945ξ2−35).(A2)
10
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