Relativistic corrections to the vector meson light front wave function

T. Lappi, H. Mantysaari, and J. PenttalaDepartment of Physics, University of Jyvaskyla P.O. Box 35, 40014 University of Jyvaskyla, Finland and

Helsinki Institute of Physics, P.O. Box 64, 00014 University of Helsinki, Finland

We compute a light front wave function for heavy vector mesons based on long distance matrixelements constrained by decay width analyses in the Non Relativistic QCD framework. Our ap-proach provides a systematic expansion of the wave function in quark velocity. The first relativisticcorrection included in our calculation is found to be significant, and crucial for a good descriptionof the HERA exclusive J/ψ production data. When looking at cross section ratios between nuclearand proton targets, the wave function dependence does not cancel out exactly. In particular the fullynon-relativistic limit is found not to be a reliable approximation even in this ratio. The importantrole of the Melosh rotation to express the rest frame wave function on the light front is illustrated.

I. INTRODUCTION

At large densities or small Bjorken-x, non-linear QCDdynamics is expected to manifest itself in nuclear struc-ture. To describe the QCD matter in this non-linearregime, an effective field theory known as the Color GlassCondensate (CGC) has been developed, see e.g. [1, 2].Diffractive scattering processes at high energies are es-pecially powerful probes of this region of phase space.The advantage in diffractive, with respect to inclusive,scattering is that since no color charge transfer is al-lowed, even at leading order in perturbative QCD at leasttwo gluons have to be exchanged with the target. Con-sequently, the cross sections approximatively probe thesquare of the gluon density [3], and can be expected tobe highly sensitive to non-linear dynamics.

An especially interesting diffractive process is exclusivevector meson production in collisions of real or virtualphotons with the target, where only one meson with thesame quantum numbers as the photon is produced. Inthese processes only vacuum quantum numbers are ex-changed between the target and the diffractive system.Thus the target can remain intact, and the transversemomentum transfer can be used to probe the spatialstructure of the target. This momentum transfer is bydefinition the Fourier conjugate to the impact parameter.As such, it becomes possible to study the target struc-ture differentially in the transverse plane. A particularlyimportant channel is the production of J/ψ mesons. Thecharm quark is heavy enough to enable a weak couplingdescription of its elementary interactions. Neverthelessthe quark mass is not large enough to make the processinsensitive to saturation effects. Also experimentally theJ/ψ is relatively easily identifiable and produced withlarge enough cross sections to be seen.

Exclusive J/ψ production in electron-proton deep in-elastic scattering has been studied in detail at HERAby the H1 and ZEUS experiments [4–9]. Additionally,lighter ρ and φ [10–12] and heavier Υ states [13, 14] havebeen measured. Recently, it has also become possibleto measure exclusive vector meson production at RHICand at the LHC in ultra peripheral collisions [15, 16]where the impact parameter between the two hadrons

is large enough such that the scattering is mediated byquasi-real photons, see Refs. [17–26] for recent measure-ments. These developments have also enabled vector me-son photoproduction studies with nuclear targets, whichare more sensitive to gluon saturation. Indeed signa-tures of strong nuclear effects (e.g. saturation, or gluonshadowing) are seen in J/ψ photoproduction (see e.g.Refs. [27–29]). The effects seen in these exclusive pro-cesses are consistent with inclusive measurements such asparticle spectra in proton-nucleus collisions (see e.g. [30–35]). However, in exclusive scattering the non-linear ef-fects are larger, since inclusive cross sections at leadingorder are only sensitive to the first power of the gluondensity.

One major source of model uncertainties in the theoret-ical description of vector meson production follows fromthe nonperturbative vector meson wave function. Forthe J/ψ , a natural first approximation is to treat it as afully nonrelativistic bound charm-anticharm state, whichis the limit taken in the seminal work in Ref. [3]. Thecalculation of Ref. [36] recovers the same nonrelativisticresult in the dipole picture (see also Ref. [37]). Alreadyearly on, it has been argued that this nonrelativistic ap-proximation obtains important corrections from the mo-tion of the charm quark pair in the bound state [38, 39].More recently, much of the phenomenological literatureon J/ψ photoproduction has used phenomenological lightcone wave functions to describe the meson bound state.This has the advantage that the light cone wave functionis invariant under boosts in the longitudinal direction,and is thus naturally more suited to high energy collisionphenomena. A disadvantage of some recent phenomeno-logical parametrizations has been that they do not fullyuse the information on the nonperturbative bound statephysics, most importantly decay widths, of quarkoniumstates that are usually analyzed in terms of nonrelativis-tic wavefunctions.

In recent literature, the applied different phenomeno-logical wave functions result in e.g. J/ψ production crosssections that differ up to ∼ 30% from each other [27, 40,41]. This is a large model uncertainty, compared to theprecise data that is already available from HERA andthe LHC, and especially given that the Electron Ion Col-

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lider (EIC) [42, 43] is in the horizon (and similar plansexist at CERN [44] and in China [45]). The EIC will per-form vast amounts of precise DIS measurements over awide kinematical range, which calls for robust theoreticalpredictions.

To reduce the model uncertainty related to the vec-tor meson wave function, we propose in this work anew method to constrain the wave function for heavymesons based on input from the Non Relativistic QCD(NRQCD) matrix elements. These matrix elements cap-ture non-perturbative long-distance physics, and can beobtained by computing the vector meson decay widthsin different channels as a systematic expansion in boththe coupling constant αs and the quark velocity v. Aswe will demonstrate, these matrix elements can be usedto determine the value and the derivative of the vectormeson wave function at the origin. As such, this ap-proach provides more constraints than the phenomeno-logical parametrizations widely used in the literature. Inparticular, starting from manifestly rotationally invariantrest frame wave functions, one by construction obtainsconsistent parametrizations of longitudinally and tran-versally polarized vector mesons simultaneously, whichis not obvious in many light cone approaches.

This manuscript is organized as follows. First, inSec. II we review how vector meson production is com-puted in the dipole picture within the Color Glass Con-densate framework, and how the cross section depends onthe vector meson light front wave function. In Sec. III wefirst present how to obtain the rest frame wave functionin terms of the NRQCD matrix elements, and then showhow this is transformed to the light cone by applyingthe Melosh rotation [46, 47]. We compare the obtainedNRQCD based wave function to other widely used wavefunctions that are reviewed in Sec. IV. The numericalanalysis including vector meson-photon overlaps and J/ψproduction cross sections is presented in Sec. V.

II. VECTOR MESON PRODUCTION IN THEDIPOLE PICTURE

A. Exclusive scattering

At high energies exclusive vector meson production invirtual photon-proton (or nucleus) scattering can be de-scribed in a factorized form. The necessary ingredientsare the virtual photon wave function Ψλ

γ describing theγ∗ → qq splitting, the dipole-target scattering amplitudeN and the vector meson wave function ΨV describing thetransition qq → V . The scattering amplitude reads [48](note that the correct phase factor coupling the dipolesize r to the transverse momentum transfer ∆ is deter-mined in Ref. [49])

Aλ = 2i

∫d2bd2r

dz

4πe−i(b+( 1

2−z)r)·∆

×Ψλ∗γ (r, Q2, z)ΨV (r, z)N(r,b, xP). (1)

Here Q2 is the photon virtuality, r the transverse size ofthe dipole, b the impact parameter and z the fractionof the photon light cone plus momentum carried by thequark. The photon polarization is λ, with λ = ±1 re-ferring to the transverse polarization and λ = 0 to thelongitudinal one.

In this work will study coherent vector meson V pro-duction. The coherent cross section refers to the scatter-ing process where the target proton (or nucleus) remainsintact. In this case, the cross section as a function ofsquared momentum transfer t ≈ −∆2 can be written as

dσγ∗p→V p

dt= R2

g

(1 + β2

) 1

16π|AT,L|2. (2)

The dipole amplitude N depends on the longitudinal mo-mentum fraction xP the target loses in the scattering pro-cess, which reads

xP =M2V +Q2 − t

W 2 +Q2 −m2N

. (3)

HereMV is the mass of the vector meson V andmN is theproton mass. The scattering amplitude AT,L is obtainedfrom Eq. (1) by summing over the quark helicities, andin the case of transverse (T) polarization, averaging overthe photon polarization states λ = ±1.

In Eq. (2) two phenomenological corrections are in-cluded following Ref. [48]. First, β = tan

(πδ2

)is the

ratio between the real and imaginary parts of the scat-tering amplitude. It can be obtained from an analyticityargument as

δ =∂ lnAT,L∂ ln(1/xP)

. (4)

The so called skewedness correction is included interms of the factor Rg, which reads,

Rg =22δ+3

√π

Γ(δ + 5/2)

Γ(δ + 4). (5)

This correction can be derived by considering the vectormeson production in the two-gluon exchange limit, as-suming that the two gluons carry very different fractionsof the target longitudinal momentum [50]. In this case,the cross section can be related to the collinearly factor-ized parton distribution functions scaled by the factorRg.In the dipole picture applied here, where the two quarksare color rotated in the target color field and undergomultiple scattering, this limit is not reached. In this workwe include both of the real part and skewedness correc-tions widely used in the previous literature, but empha-size that these numerically large corrections should beused with caution when predicting absolute normaliza-tions for the cross sections.

In addition to coherent scattering, one can study inco-herent diffraction where the target breaks up, but thereis still no exchange of color charge between the producedvector meson and the target remnants. These processes

3

are recently studied extensively in the literature as theyprobe, in addition to saturation effects [41], also theevent-by-event fluctuations of the scattering amplituderesulting from the target structure fluctuations, see e.g.Refs. [51–54] or Ref. [55] for a review. As the focus inthis work is on the vector meson wave function which en-ters in calculations of both incoherent and coherent crosssections similarly, from now on we only consider coherentscattering here.

B. Virtual photon wave function

The virtual photon splitting to a cc dipole is a simpleQED process, and the photon wave function Ψγ can becomputed directly by applying the light cone perturba-tion theory (see e.g. [56, 57]). Using the diagrammaticrules of light front perturbation theory and the conven-tions used in Refs. [58, 59], the wave function can bewritten as

Ψλγ,hh(k) =

efe√Nc

q− − k− − k′−uh(k)/ε

λ(q)vh(k′)

4√πk+k′+q+

. (6)

Here ef is the fractional charge of the quark (in this workwe consider only charm quarks with ef = 2/3), k, k′ andq are the quark, antiquark and photon momenta respec-tively, e =

√4παem and h and h refer to the quark and

antiquark light front helicities [60]. The factor√Nc is in-

cluded to obtain a squared wave function proportional tothe number of colors Nc. The spinors which are the eigen-states of light front helicity read, in the Lepage-Brodskyconvention

uh(k) =1

21/4√p+

(√2p+ + γ0m+ αT · k

)χh (7)

vh(k) =1

21/4√p+

(√2p+ − γ0m+ αT · k

)χ−h, (8)

where the four-component helicity spinors read χh=+1 =1√2(1, 0, 1, 0)T and χh=−1 = 1√

2(0, 1, 0,−1)T , and αT =

(γ0γ1, γ0γ2). We use the light cone variables defined asp± = 1√

2(p0 ± p3). The spinor normalization convention

is uhuh = −vhvh = 2mδhh, where m is the quark mass.In the light cone gauge, in which ε+ = 0, the photon

polarization vectors read

ελ=0(q) =

(0, 0, 0,

Q

q+

)(9)

ελ=±1 =

(0, ελT ,

q · ελTq+

), (10)

where

ελ=±1T = (∓1,−i)/

√2, (11)

and Q2 = −q2.The wave function can be evaluated by substituting

the polarization vectors and explicit expressions for the

spinors in Eq. (6) and setting the photon transverse mo-mentum q to zero. It is convenient here to define a wavefunction in terms of the momentum fraction z and pullout a factor 4π. This should be done so that probabilityis conserved,

∫dk+|Ψλ

γ,hh(k+,k)|2 =

∫dz

4π|Ψλγ,hh(z,k)|2, (12)

so that we can write Ψλγ,hh

(z,k) =√

4πq+Ψλγ,hh

(k+,k).

In momentum space, the wave functions read

Ψλ=0γ,hh(z,k) = −efe

√Nc

2Qz(1− z)(k2 + ε2)

δh,−h (13)

Ψλ=+1γ,hh

(z,k) = −efe√

2Nc(k2 + ε2)

[keiθk(zδh+δh−

−(1− z)δh−δh+) +mδh+δh+

](14)

Ψλ=−1γ,hh

(z,k) = −efe√

2Nc(k2 + ε2)

[ke−iθk((1− z)δh+δh−

−zδh−δh+) +mδh−δh−]

(15)

where ε2 = Q2z(1 − z) + m2 and keiθk = kx + iky. Thewave function in the mixed transverse coordinate, longi-tudinal momentum fraction space entering in the vectormeson production cross section (1) is then obtained byperforming a Fourier transform

Ψλγ,hh(z, r) =

∫d2k

(2π)2eik·rΨλ

γ,hh(z,k). (16)

The mixed space wave function for the longitudinal po-larization is

Ψλ=0γ,hh(z, r) = −efe

√Ncδh,−h2Qz(1− z)K0(εr)

2π. (17)

Similarly, for the transverse photon with λ = ±1 thewave function reads

Ψλ=+1γ,hh

(z, r) = −efe√

2Nc

[ieiθr

εK1(εr)

2π(zδh+δh−

−(1− z)δh−δh+) +mK0(εr)

2πδh+δh+

]

Ψλ=−1γ,hh

(z, r) = −efe√

2Nc

[ie−iθr

εK1(εr)

2π((1− z)δh+δh−

−zδh−δh+) +mK0(εr)

2πδh−δh−

]. (18)

We note that the these wave functions agree with thosederived in Ref. [59] using the same convention, exceptfor the overall sign in case of transverse polarizationswhich does not affect any of our results. On the otherhand, when compared to the widely used wave functionsreported in Ref. [48], the relative sign between the massterm and z terms in the λ = +1 case is different.

4

We emphasize that the quark light cone helicity struc-ture above does not exactly correspond to the spin struc-ture in the rest frame of the meson (there is no rest framefor the spacelike photon). In particular, when trans-formed to the meson rest frame, there are both S andD wave contributions in both longitudinally and trans-versely polarized photons. The transformation betweenthe light front wave function expressed in terms of thequark light front helicities, and the rest frame wave func-tion in terms of the quark spins is discussed in Sec. III B.We will discuss the decomposition of light cone wavefunctions, including the virtual photon one, into the Sand D wave components in more detail in Appendix A.

C. Dipole-target scattering

The dipole-target scattering amplitude N in Eq. (1) isa correlator of Wilson lines, corresponding to the eikonalpropagation of the quarks in the target color field. Inprinciple, it satisfies perturbative evolution equations de-scribing the dependence on momentum fraction xP, theso called JIMWLK equation [61–67], or the BK equa-tion [68, 69] that is obtained in the large-Nc limit. Theseperturbative evolution equations, combined with a non-perturbative input obtained by fitting some experimentaldata, can in principle be used to evaluate the dipole am-plitude at any (small) xP. This has been a successfulapproach when considering structure functions in DIS orinclusive particle production in hadronic collisions, seee.g. Refs. [31–35, 70, 71].

In diffractive scattering considered here one explicitlymeasures the transverse momentum transfer ∆, whichis the Fourier conjugate to the impact parameter. Con-sequently, the dependence on the transverse geometryneeds to be included accurately in the calculation. How-ever, perturbative evolution equations generate long dis-tance Coulomb tails that should be regulated by somenon-perturbative physics in order to avoid unphysicalgrowth of the cross section [72]. There have been at-tempts to include effective confinement scale contribu-tions in the BK and JIMWLK evolutions and use theobtained dipole amplitudes in phenomenological calcu-lations of e.g. vector meson production [73–76] (seealso [77]). As the main focus of this work is in vectormeson wave functions, we apply a simpler approach anduse the so called IPsat parametrization to describe thedipole-proton scattering amplitude.

The IPsat parametrization [78] consist of an eikon-alized DGLAP-evolved [79–82] gluon distribution, com-bined with an impact parameter b dependent transversedensity profile. The advantage of this parametrizationis that it matches perturbative QCD result in the dilute(small dipole size |r|) limit, and respects unitary in thesaturation regime. The dipole amplitude in the IPSat

parametrization reads

N(r,b, x) = 1− exp

(− π2

2Ncr2αs(µ

2)xg(x, µ2)Tp(b)

),

(19)where the proton transverse density profile is assumed tobe Gaussian:

Tp(b) =1

2πBpe−b

2/(2Bp) (20)

with B = 4 GeV−2. The initial condition for the DGLAPevolution is obtained by fitting the HERA structure func-tion data [83–86], and the fit results in an excellent de-scription of the total reduced cross section and the charmcontribution [87]. The scale choice is µ2 = C/r2 + µ2

0,with the parameters C and µ0, among with the DGLAPinitial condition, are determined in the fit performed inRef. [87] (see also [88]).

Following Ref. [78] (see also [87]), the dipole-protonscattering amplitude can be generalized to coherent scat-tering in the dipole-nucleus case as

NA(r,b, x) = 1−exp

(− π2

2Ncr2αs(µ

2)xg(x, µ2)ATA(b)

).

(21)This estimate is valid in case of large nuclei, assumingthat the dipole size |r| is not very large, which is the casein heavy vector meson production. Here TA(b) is theWoods Saxon distribution integrated over the longitudi-nal coordinate, with the normalization

∫d2bTA(b) = 1.

The nuclear radius used here is RA = (1.13A1/3 −0.86A−1/3) fm.

In order to calculate vector meson production, it is stillnecessary to determine the vector meson wave function.It can not be computed perturbatively, and consequentlythere are many phenomenological parametrizations usedin the literature. The main goal of this paper is to obtainthe meson wave function in a systematic expansion inquark velocities given by the NRQCD approach. We willalso discuss, for comparison, some other wave functionparametrizations in Sec. IV.

III. LIGHT CONE WAVE FUNCTION FROMNRQCD

NRQCD is an effective field theory describing QCD inthe limit where quark masses are large, or v = p/m issmall, where p is e.g. quark momentum and m is thequark mass. In this approach, it becomes possible tofactorize cross sections into universal long distance ma-trix elements and perturbatively calculated process de-pendent hard factors.

A. Vector meson wave function in the rest frame

The J/ψ decay width in the NRQCD approach is writ-ten as an expansion in the quark velocity v [89]. At lowest

5

order in v, the decay width is only sensitive to the longdistance matrix element 〈O1〉J/ψ, which itself is deter-mined by the value of the (renormalized) wave functionat the origin. At next order, one finds a contributionproportional to the long distance matrix element 〈~q 2〉J/ψwhich is suppressed by a relative v2. This matrix ele-ment is sensitive to the derivative of the wave functionat the origin (see also Ref. [90, 91] for a discussion ofthe velocity suppressed contributions to the distributionamplitude).

In this work we follow Ref. [92], where these matrix ele-ments are determined. There, a subset of higher order (inv) contributions to the decay width including higher pow-ers of ∇2 are resummed to all orders following Ref. [93].As a result, the J/ψ decay width in the leptonic channelcan be written as

Γ(J/ψ → e−e+) =8πe2

qα2em

3M2V

[1− f

(〈~q 2〉J/ψm2c,NR

)

− 2CFαs

π

]2

〈O1〉J/ψ (22)

with

f(x) =x

3(1 + x+√

1 + x). (23)

Here, eq = 2/3 is the fractional charge of the charm quarkand MV is the J/ψ mass. At this order in v, the J/ψ is apure S wave state, and its wave function can be factorizedinto a spin part and a scalar part. We will discuss the spinand angular momentum structure in more detail later.

The extraction of the matrix elements that we use [92]has been done in a calculation that includes both ve-locity and αs corrections, such as in (22). Here, on theother hand, we will be using the light cone wave func-tions in a leading order calculation of cross sections, in-cluding only velocity corrections to the wavefunction. Ina strict NRQCD power counting sense in αs, the αs cor-rections could be considered more important. Althoughsteps have been taken to take them into account in thedipole picture exclusive cross section calculations [94] (seealso recent work in a different formalism [95]), fully in-cluding them in the cross section is not yet possible atthis point since the full photon to heavy quark pair wavefunction is not known to one loop accuracy. Thus wewill leave a computation that includes also the pertur-bative αs calculations to future work, and continue withour focus on the velocity corrections to the wave functionhere.

Since our cross section calculation does not includepure αs corrections, taking the wave function to be givenby just the operator 〈O1〉J/ψ in (22) would lead to aninconsistent treatment of the αs corrections between thedecay width and the cross section. Even in a more generalsense, the αs contributions that appear as corrections tothe decay widths or cross sections expressed in terms of

nonrelativistic wavefunctions should, in light cone per-turbation theory, be thought of as perturbative correc-tions to the light cone wave function itself [39, 94]. Thiscan be understood in the sense that the degrees of free-dom in the nonrelativistic wavefunction are constituentquarks as opposed to bare quarks in the light cone wavefunction, see the discussion in [39]. To obtain a consis-tent picture here, we will absorb the αs correction to thescalar part of the wavefunction φ(r), which is then trans-formed to the light cone wave function. We thus relatethe value and derivative at the origin of φ(r) to the longdistance matrix elements as[

1− 2CFαs

π

]2

〈O1〉J/ψ = 2Nc|φ(0)|2 +O(v4), (24)

〈~q 2〉J/ψ = −∇2φ(0)

φ(0)+O(v2). (25)

The non-perturbative long distance matrix elementshave been determined in Ref. [92] by considering simul-taneously the J/ψ → e+e− and ηc → 2γ decays. As aresult of this analysis, the matrix elements for J/ψ read

〈O1〉J/ψ = 0.440+0.067−0.055 GeV3, (26)

〈~q 2〉J/ψ = 0.441+0.140−0.140 GeV2. (27)

The analysis in Ref. [92] is done by using the charm quarkmass mc,NR = 1.4 GeV. In general, the charm quarkmass in NRQCD can differ from the charm quark massused in the IPsat fits discussed in Sec. II. In our numericalanalysis, we will use the NRQCD value for the charmquark mass in both the meson and photon wave functionswhen using the NRQCD results. Everywhere else in thiswork we use the charm mass mc = 1.3528 GeV obtainedin the IPsat fit to the HERA structure function data.

The uncertainties quoted above for the long distancematrix elements are not independent, and the correlationmatrix is also provided in Ref. [92]. To implement thesecorrelated uncertainties, we use a Monte Carlo methodand sample parameter values from the Gaussian distribu-tion taking into account the full covariance matrix. Theuncertainty is then obtained by calculating the one stan-dard deviation band with respect to the result obtainedby using the best fit values.

To construct the meson wave function, we start fromthe meson rest frame where we can use the NRQCD ma-trix elements to constrain the wave function as discussedabove. In the rest frame, we require that the quark spinsare coupled into a triplet state, and the total spin andangular momentum to a J = 1 vector state. Thus we canin general write the spin-structure of the wave functionin the following form:

ψλss(~r) =∑

L,mL,mS

〈L mL 1 mS |1 λ〉

×⟨

1

2s

1

2s

∣∣∣∣∣1 mS

⟩Y mL

L (θ, φ)ψL(r). (28)

6

Here Y mL

L are the spherical harmonics, ψL is the radialwave function corresponding to the orbital angular mo-mentum L and 〈j1mj1j2mj2 |JmJ〉 are Clebsch-Gordancoefficients. In general, the conservation of spin-paritytells us that for J/ψ the orbital angular momentum canonly take values L = S,D. Since J/ψ should be domi-nated by the S wave contribution, we will from now onconsider the case where only the S wave component isnon-zero. We note that in principle in the NRQCD ap-proach one finds the D wave contribution to the vectormeson wave function to be suppressed by v2 compared tothe S wave, and this is of the same order as the first rela-tivistic correction included in terms of the wave functionderivative above. However, the D wave contribution tothe decay width is suppressed by an additional v2 and assuch the D-wave contribution is not constrained by thedecay widths at this order. Thus it is most consistent toset it to zero. In this case the wave function simplifiesto:

ψλss(~r) =

⟨1

2s

1

2s

∣∣∣∣∣1 λ⟩φ(r) (29)

where φ(r) is the scalar part of the wave function andrelated to the long distance matrix elements as shown inEqs. (24) and (25). Using the 3-dimensional polarizationvectors in Eq. (11) we can also write this as

ψλss(~r) = Uλssφ(r) (30)

where

Uλss =1√2ξ†s~ελ · ~σξs (31)

in the case of transverse polarization and

Uλ=0ss =

1√2ξ†sσ3ξs (32)

when the vector meson is longitudinally polarized. Hereξ+ = (1, 0) and ξ− = (0, 1) are the two component

spinors describing spin-up and spin-down states and ξs =iσ2ξ

∗s is the antiquark spinor.

The behavior of the quarkonium wavefunction at longdistances is determined by non perturbative physics.This long distance physics affects short distances throughthe requirement of the normalization of the wave func-tion. The NRQCD approach broadly speaking consists ofparametrizing the nonperturbative long distance physicsby measurable coefficients that serve as coefficients inthe short distance expansion, which is used to calculatea physical process happening at short distance scales. Inpractice this amounts to expressing the wave function asa Taylor expansion around the origin:

φ(~r) = A+B~r2. (33)

The linear term does not appear to ensure that the Lapla-cian of the wave function is finite at the origin. The coeffi-cients can also be written as A = φ(0) and B = 1

6∇2φ(0),

and using equations Eqs. (24) and (25) we get the values

A =

[1− 2CF

αs

π

]√1

2Nc〈O1〉J/ψ = 0.213 GeV3/2,

(34)

B = −1

6A〈~q 2〉J/ψ = −0.0157 GeV7/2. (35)

The uncertainties in the long distance matrix elementsare correlated as discussed above, and in our numericalcalculations this correlated uncertainty is propagated tothe coefficients A and B.

We then want to write our wave function ansatz (33)in light cone coordinates (k, z). We do this by first goingto momentum space:

ψλss(~k) =

∫d3~re−i

~k·~rψλss(~r) = Uλssφ(k)

= Uλss(2π)3(Aδ3(~k)−B∇2

kδ3(~k)

)(36)

where ~k = (k, k3). We then want to change the longitu-dinal momentum variable from k3 to the plus momentumfraction carried by the quark: z. Unfortunately there isno unique way to do this, due to the different nature ofinstant form and light cone quantization. In principlewe would want to define z as the ratio of the quark k+

to the meson P+ = MV /√

2, working in the rest frameof the meson. However, a quark inside a bound state

described as a superposition of different ~k modes is notexactly on shell, its energy being affected by the bindingpotential. Thus we do not precisely know the k0 requiredto calculate k+ from k3. The rest frame wave functionalso includes values of k3 that are very large, leadingto values of k+ that are larger than MV /

√2. This is

perfectly possible in instant form quantization with thetime variable t. However, in light cone quantization k+

is a conserved momentum variable, and has to satisfy0 < k+ < P+. The procedure that we adopt here is(similarly to e.g. [96]) to define the momentum fractionin practice as z = k+

q /(k+q + k+

q ) where kq and kq are the

quark and antiquark momenta, with k+ calculated as-

suming k0 =√m2c,NR + ~k2. In other words, we normal-

ize by the total plus momentum of the quark-antiquarkpair, instead of the meson plus momentum, and assumean on-shell dispersion relation. This choice has the ad-vantage that it leads to 0 < z < 1 by construction. Thisleads us to the expression for the longitudinal momentumin the meson rest frame k3 as

k3 = M

(z − 1

2

)(37)

where

M =

√k2 +m2

c,NR

z(1− z) (38)

7

is the invariant mass of the quark-antiquark pair. Weemphasize that since this choice is not unique, we mightexpect corrections or ambiguities proportional to powersof the difference between the meson mass and the quark-antiquark pair invariant mass M2

V −M2 to appear. Suchcorrections are, however, higher order corrections in thenonrelativistic limit and also numerically very small forJ/ψ for the values of mc,NR and 〈~q 2〉 used here. Wecould also hope that since the invariant mass is a rota-tionally invariant quantity, these ambiguities would notlead to serious violations of rotational invariance (whichexpresses itself here as the equality of physical proper-ties such as decay widths of transverse and longitudinalpolarization states). We will see an example of such acorrection explicitly in Appendix B.

To change the variables in our wave function, one needsto be careful with the delta functions and their deriva-tives. We therefore make the change by requiring thatthe overlap

∫d3~k

(2π)3ψλss(

~k)ϕ(~k) =

∫d2k

(2π)2

dz

4πψλss(k, z)ϕ(k, z),

(39)where ϕ is an arbitrary wave function, does not changeunder the change of variables. This requirement tells us

that the scalar part φ(~k) changes to

φ(k, z) = (2π)3

√2∂z

∂k3

(Aδ

(z − 1

2

)δ2(k)

−B(∂z

[∂z

∂k3∂z

[∂z

∂k3δ

(z − 1

2

)]]δ2(k)

+δ

(z − 1

2

)∇2

kδ2(k)

))(40)

where

∂z

∂k3=

4z(1− z)M

. (41)

Equation (40) is the scalar part of the NRQCD basedvector meson wave function in the meson rest frame, ex-pressed in momentum space. We note that this wavefunction is not normalizable due to the presence of thedelta functions. However, as the NRQCD approach canonly be used to constrain the coordinate space wave func-tion and its derivative at the origin, we are forced to usethe expansion of Eq. (33) which can not result in a nor-malizable wave function. However, for the purposes ofthis work this is not a problem, as the vector meson pro-duction is sensitive to the vector meson wave functionoverlap with the virtual photon wave function, and thephoton wave function is heavily suppressed at long dis-tances where the expansion (33) is not reliable.

B. Wave function on the light front

The NRQCD wave function obtained in the previoussection is written in the vector meson rest frame in terms

of the quark and antiquark spin states s, s. In order tocalculate overlaps with the virtual photon wave function(17) and (18), we need to express it in terms of the lightcone helicities h, h. The transformation between thesetwo bases, usually expressed in terms of the 2-spinors, isknown as the “Melosh rotation” [46, 47].

The Dirac spinors that are used to factorize the nonrel-ativistic wavefunction into a a spin- and scalar part, areeigenstates of the spin-z operator in the zero transversemomentum limit. In terms of the two component spinvectors ξ defined above in Eqs. (31) and (32) they read

us(p) =1√N

(ξs

~σ·~pEp+mξs

)(42)

vs(p) =1√N

(~σ·~p

Ep+m ξs

ξs

). (43)

The normalization factor N is determined from the con-dition usus = −vsvs = 2mδs,s.

Both the Dirac spinors in terms of the spin-z compo-nent us and the helicity spinors uh (see Eqs. (7), (8))are solutions to the Dirac equation, and as such can beobtained as linear combinations of each other. This map-ping is the Melosh rotation Rsh. It can be computed fromthe spinor inner products (see also Ref. [97]) as

Rsh(k, z) =1

2mus(k, z)uh(k, z) (44)

where k+ = zq+ and q+ is the meson plus momentum,and s and h refer to the spin and light front helicity,respectively.

The helicity spinors uh and vh can also be written in asimilar form as the spinors in the spin basis, Eqs. (42) and(43), by introducing the two component helicity spinorsχh. To do this we write the helicity spinors (7), (8) inthe form

uh(p) =1√N

(χh

~σ·~pEp+mχh

)(45)

vh(p) =1√N

( ~σ·~pEp+m χhχh

)(46)

where N is again determined by the normalization re-quirement and χh = iσ2χ

∗h. Using this form one can

check that the Melosh rotation also connects the twocomponent spin and helicity spinors as

Rsh(k, z) = ξ†sχh (47)

The coefficients Rsh can also be expressed as a 2×2matrix rotating the 2-spinors

R(k, z) =mc,NR + zM − i(~σ × ~n) · (k, k3)√

(mc,NR + zM)2 + k2. (48)

Here M is the invariant mass of the qq system fromEq. (38) and n = (0, 0, 1) is the unit vector in the longi-tudinal direction. In terms of this matrix the 2-spinors

8

ξs and χh are related by

χ± = R(k, z)ξ± (49)

Using Eq. (47) we can now express the NRQCD wavefunction in the light front helicity basis. We write

Ψλhh(k, z) = Uλh,hφ(k, z), (50)

where the scalar part is given in Eq. (40). The helicitystructure Uλ

hhis obtained by applying the transform (47)

in Eq. (31) and (32), i.e.,

Uλhh =∑

ss

R∗sh(k, z)R∗sh(−k, 1− z)Uλss. (51)

After the Melosh rotation, we compute the Fouriertransform to obtain the light front wave function in themixed transverse coordinate – longitudinal momentumfraction space as

Ψλhh(r, z) =

∫d2k

(2π)2eik·rΨλ

hh(k, z)

=

∫d2k

(2π)2eik·rUλh,h(k, z)φ(k, z). (52)

The different helicity components of the final light frontwave function resulting from this procedure are

Ψλ=0+− (r, z) = Ψλ=0

−+ (r, z) =π√

2√mc,NR

[Aδ(z − 1/2) +

B

m2c,NR

((5

2+ r2m2

c,NR

)δ(z − 1/2)− 1

4∂2zδ(z − 1/2)

)]

Ψλ=1++ (r, z) = Ψλ=−1

−− (r, z) =2π

√mc,NR

[Aδ(z − 1/2) +

B

m2c,NR

((7

2+ r2m2

c,NR

)δ(z − 1/2)− 1

4∂2zδ(z − 1/2)

)]

Ψλ=1+− (r, z) = −Ψλ=1

−+ (r, z) =(Ψλ=−1−+ (r, z)

)∗=(−Ψλ=−1

+− (r, z))∗

= − 2πi

m3/2c,NR

Bδ(z − 1/2)(r1 + ir2)

Ψλ=1−− (r, z) = Ψλ=−1

++ (r, z) = Ψλ=0++ (r, z) = Ψλ=0

−− (r, z) = 0 (53)

The first relativistic correction to the wave function,proportional to B or the wave function derivative, mixesthe helicity and spin states. In particular, in thecase of transverse polarization the h, h = ±∓ termsare non-vanishing when the relativistic correction is in-cluded. These terms also bring a non-zero contributionto photon-vector meson overlaps. In general, we expectthat if higher order corrections in v were included in thewave function parametrization, we would also find othercomponents to be non-vanishing.

The Melosh rotation is crucial here, as it generateshelicity structures that are not visible in the spin ba-sis. This is in contrast to some early attempts to trans-form the wave functions obtained by solving the poten-tial models to the light front as done e.g. in Ref. [78].The role of the Melosh rotation in the context of vectormeson light front wave functions and exclusive scatter-ing was first emphasized in Ref. [47]. More recently itwas applied to J/ψ production in the dipole picture inRef. [98], and in [99] different quark-antiquark potentialswere studied in this context. In the case of excited statessuch as ψ(2S) the role of the Melosh rotation is expectedto be even more significant [100].

Let us in passing briefly compare our approach to theone in the recent work of Krelina et al in Ref. [98]. In ourapproach, we take the NRQCD wave function which onlyincludes the S wave contribution (D wave part is sup-pressed by v2). The quark spin dependence is now triv-

ial, as the total angular momentum must be provided bythe quark spins which gives us the structure of Eq. (30).In Ref. [98], the authors assume, unlike we do here, thatthe spin structure of the vector meson wave function inthe rest frame has the same form as the light cone he-licity structure of the photon light cone wave function,Eq. (17) and (18). This structure is then supplementedby a wave function obtained from the potential model,and a Melosh rotation to the light front is applied at theend. Such a procedure leads to a large D-wave contri-bution in the wavefunction, which we do not have. Wediscuss the structure of the wavefunctions in terms of S-and D-waves in more detail in Appendix A.

To determine the role of the relativistic corrections inthe vector meson wave function, we will also study forcomparison the fully non-relativistic wave function whereour starting point for the scalar part is

φ(~r) = A′. (54)

Following the previous procedure, the final result for thelight cone wave function can be read from Eq. (53) withthe substitutions A = A′, B = 0. One notices that thiscan now be written as

ΨλJ/ψ,hh(r, z) =

π√

2√mc,NR

UλhhA′δ

(z − 1

2

). (55)

In this extreme non-relativistic limit (k = 0, z = 1/2) theMelosh rotation simply corresponds to an identity matrix

9

so that the spin and helicity bases are interchangeablehere. The normalization A′ is obtained from the vanRoyen-Weisskopf equation for the leptonic width [101],which is also obtained from Eq. (22) by neglecting the rel-ativistic correction proportional to 〈q2〉J/ψ/m2

c,NR, and

the higher order QCD correction ∼ αs (note that para-metrically αs ∼ v):

Γ(J/ψ → e−e+) =16πe2

fαem

M2J/ψ

|φ(0)|2 (56)

By using the experimental value for leptonic width [102],we can calculate the coefficient A′ to be

A′ = φ(0) = 0.211 GeV3/2. (57)

C. Overlap with photon

Using the obtained J/ψ wave function on the lightfront, Eq. (53), we can directly compute overlaps with thevirtual photon, Eqs. (17) and (18). In these overlaps, wealso include the phase factor exp

(i(z − 1

2

)r ·∆

)present

in the vector meson production amplitude in Eq. (1).We also assume that the dipole amplitude does not de-pend on the orientation θr of r as is the case in the IP-sat parametrization, and integrate over θr. The overlapssummed over the quark helicities read

r∑

hh

∫ 2π

0

dθr

∫ 1

0

dz

4π(ΨL

J/ψ)∗ΨLγ e

i(z−1/2)r·∆ =reefQ

2

√Nc

2mc,NR

[AK0(rε)

+B

m2c,NR

(9

2K0(rε) +m2

c,NRr2K0(rε)− Q2r

4εK1(rε) +

1

4∆2r2K0(rε)

)](58)

and

r∑

hh

∫ 2π

0

dθr

∫ 1

0

dz

4π(ΨT

J/ψ)∗ΨTγ e

i(z−1/2)r·∆ = reef

√Ncmc,NR

2

[AK0(rε)

+B

m2c,NR

(7

2K0(rε) +m2

c,NRr2K0(rε)− r

2ε(Q2 + 2m2

c,NR)K1(rε) +1

4∆2r2K0(rε)

)], (59)

where ε2 = Q2/4 + m2c,NR and ∆ = |∆| and r = |r|. In

the case of transverse polarization, the result is identicalin cases with λ = +1 and λ = −1. We will study theseoverlaps numerically in Sec. V A. We note that thanksto the delta function structure in z in our wave func-tion (53), many phenomenological applications becomenumerically more straightforward as the z integral canbe performed analytically.

IV. PHENOMENOLOGICAL WAVEFUNCTIONS

To provide a quantitative point of comparison for theeffect of the relativistic corrections, we want to comparethe light cone wave functions obtained in Sec. III to otherparametrizations used in the literature. For this purpose,let us now discuss two specific alternative approachesused for phenomenological applications in the literature.

A. Boosted Gaussian

A commonly used phenomenological approach to con-struct the vector meson wave function is to assume thatit has the same polarization and helicity structure as thevirtual photon. This can be done by replacing the scalarpart of the photon wave functions (17) and (18) by anunknown function as [48]

efez(1− z)K0(εr)

2π→ φT,L(r, z), (60)

with the explicit factor Q in the longitudinal wave func-tion replaced by the meson mass as 2Q → MV . Thescalar function φ(r, z) is then parametrized, and the pa-rameters can be determined by requiring that the result-ing wave function is normalized to unity and reproducesthe experimental leptonic decay width. As we will dis-cuss in more detail in Appendix B, this procedure doesnot correspond to the most general possible helicity struc-ture. Nevertheless, our result at this order in the nonrel-ativistic expansion can in fact also be written in terms ofthe “scalar part of light cone wave functions.” However,

10

at higher orders in v different a different structure couldappear.

In the Boosted Gaussian parametrization, the qq in-variant mass distribution is assumed to be Gaussian, withthe width of the distribution R and the normalizationfactors NT,L being free parameters. In mixed space, theparametrization reads

φT,L(r, z) = NT,Lz(1− z) exp

(− m2

cR2

8z(1− z)

−2z(1− z)r2

R2+m2cR2

2

). (61)

In this work we use the parameters constrained inRef. [87] by using the same charm quark mass mc =1.3528 GeV as is used when fitting the IPsat dipole am-plitude to the HERA data. The parameters are deter-mined by requiring that the longitudinal polarization canbe used to reprodcue the experimental decay width. Theobtained parameters are R = 1.507 GeV−1, NT = 0.589and NL = 0.586 with MV = 3.097 GeV.

The specific functional form and helicity structure ofthe Boosted Gaussian parametrization imply that in thevector meson rest frame there are both S and D wavecontributions. This is demonstrated explicitly in Ap-pendix A by performing a Melosh rotation from thelight front back to the J/ψ rest frame. This feature ishard to describe in potential model calculations, and ourNRQCD based wave function in particular has only the Swave component in the rest frame. The D-wave contribu-tion in the Boosted Gaussian wavefunction is, however,quite small.

B. Basis Light-Front Quantization (BLFQ)

The second wave function we study here for compar-isons is based on explicit calculations on the light front.In this approach, one constructs a light front HamiltonianHeff, which consists of a one gluon exchange interaction,and a non-perturbative confining potential inspired bylight-front holography. The formalism is developed inRefs. [103–109].

The quarkonium states are obtained by solving theeigenvalue problem

Heff

∣∣∣ψJPC

mJ

⟩= M2

V

∣∣∣ψJPC

mJ

⟩. (62)

As a solution, one obtains the invariant mass M2V spec-

trum and the light front wave functions in momentumspace

ψJPC

mJ(k, z, h, h) =

⟨k, z, h, h

∣∣∣ψJPC

mJ

⟩(63)

Here J, P,C and mJ are the total angular momentum,parity, C-parity and the magnetic quantum number ofthe state, respectively. The free paramters, value of the

coupling constant, strength of the confining potential,quark mass and the effective gluon mass, can be con-strained by the charmonium and bottonium mass spec-tra [110, 111]. In this work, we use the most up-to-dateparametrizations from Ref. [111].

The obtained BLFQ wave functions have been ap-plied in studies of the J/ψ production in the dipolepicture at HERA [112], and in the context of exclusiveJ/ψ production in ultra peripheral heavy ion collisions atthe LHC in Ref. [113]. Following the prescription usedin Refs. [112, 113], we consider the fitted quark massmBLFQc = 1.603 GeV in the “BLFQ wave function” to

be an effective mass of the quarks in the confining po-tential, including some non-perturbative contributions.Consequently, when calculating the overlaps we use, asin [112, 113], mc = 1.3528 GeV for the charm mass inthe photon wave function, as constrained by the charmstructure function data in the IPsat fit [87].

V. VECTOR MESON PRODUCTION

A. Photon overlap

The exclusive vector meson production cross sectiondepends on the overlap between the cc component of thevirtual photon wave function with the vector meson wavefunction, see Eq. (1). In Fig. 1 these overlaps for ∆ = 0are shown as a function of the transverse size r = |r|of the intermediate dipole, using four vector meson wavefunctions:

1. NRQCD expansion, which is constructed byparametrizing the wave function and its deriva-tive at the origin based on NRQCD matrix ele-ments including corrections ∼ v2, and performingthe Melosh rotation to the light front. This is ourresult from Sec. III.

2. Delta, which is the fully non-relativistic limit(Eq. (55)) of the above wave function, without anyinformation about the wave function derivative.

3. Boosted Gaussian, the phenomenologicalparametrization discussed in Sec. IV A

4. BLFQ wave function based on Basis Light-FrontQuantization, discussed in Sec. IV B.

In Fig. 2 we show the same overlaps plotted as ratios tothe fully nonrelativistic limit, i.e. the Delta parametriza-tion. For the NRQCD expansion based wave function, wealso show the model uncertainty related to the NRQCDmatrix elements that control the value of the wave func-tion and its derivative at the origin. The uncertaintyband is in this case computed as discussed in Sec. III.

The effect of the first relativistic correction can be de-termined by comparing the Delta and NRQCD expansionwave functions. At large dipoles the negative velocity

11

10−3 10−2 10−1 1000.0000

0.0005

0.0010

0.0015

0.0020Q2 = 0.05 GeV2

Longitudinal

NRQCD expansion

Delta

Boosted gaussian

BLFQ

10−3 10−2 10−1 1000.0000

0.0025

0.0050

0.0075

0.0100

0.0125 Q2 = 3.2 GeV2

10−3 10−2 10−1 100

r [fm]

0.000

0.005

0.010

0.015

0.020 Q2 = 22.4 GeV2

10−3 10−2 10−1 1000.000

0.005

0.010

0.015

0.020

0.025Transverse

10−3 10−2 10−1 1000.000

0.005

0.010

0.015

0.020

10−3 10−2 10−1 100

r [fm]

0.0000

0.0025

0.0050

0.0075

0.0100

0.0125

Overlap r/2∫

dz(Ψ∗γΨJ/ψ) [GeV]

FIG. 1: Forward (∆ = 0) virtual photon-J/ψ wave function overlaps computed using the different vector meson wave functionsas a function of the dipole size r at different photon virtualities.

suppressed ∼ r2 contribution suppresses the vector me-son wave function1 compared to the fully non-relativisticform. This is especially visible at small Q2. At largerphoton virtualities, the exponential suppression in thephoton wave function becomes dominant before the rel-ativistic −r2 correction becomes numerically important.Thus, while the effect of the relativistic correction is dra-matic in the ratio in Fig. 2, at large Q2 it is insignificantfor the actual overlap, as is seen in Fig. 1.

For small dipoles the wavefunctions are most strictlyconstrained by the quarkonium decay widths. TheNRQCD parametrization does not, however, reduce ex-actly to the fully nonrelativistic Delta parametrizationin the small r limit. This can be traced back to the factthat the gradient correction also affects the decay width,

1 The wave function would change sign at r0 = 0.73 fm. As thereshould be no node in the J/ψ wave function, we set the wavefunction to zero at r > r0. We have checked that this cutoff hasa negligible effect on our numerical results.

as seen in Eq. (22) (and from the fact that the constantsA in (34) and A′ in (57) are different). A part of the3-dimensional gradient correction becomes a correctionto the functional form in z even at r = 0. This leadsto the overlaps at small r being slightly different, eventhough the same decay width data is used to obtain theparameters of the rest frame wavefunctions.

Both the Boosted Gaussian and BLFQ wave func-tions are even more suppressed at large dipole sizesthan the NRQCD parametrization. This is most clearlyseen on the ratio plot, Fig. 2. This is a straightfor-ward consequence of the fact that in these parametriza-tions the wavefunction normalization imposes an addi-tional suppression at large r. For the Boosted Gaussianparametrization this additional suppression happens atsuch a large r that the overlap is already very small,and thus has a negligible effect on the overall overlap inFig. 1. The Boosted Gaussian parametrization is veryclose to our NRQCD also for small dipoles. The BLFQparametrization yields a a somewhat larger wave functionoverlap at small r than our NRQCD one, or the Boosted

12

10−2 10−1 1000.00

0.25

0.50

0.75

1.00

1.25 Q2 = 0.05 GeV2

Longitudinal

NRQCD expansion

Boosted gaussian

BLFQ

10−2 10−1 1000.00

0.25

0.50

0.75

1.00

1.25 Q2 = 3.2 GeV2

10−2 10−1 100

r [fm]

0.00

0.25

0.50

0.75

1.00

1.25 Q2 = 22.4 GeV2

10−2 10−1 1000.00

0.25

0.50

0.75

1.00

1.25

Transverse

10−2 10−1 1000.00

0.25

0.50

0.75

1.00

1.25

10−2 10−1 100

r [fm]

0.00

0.25

0.50

0.75

1.00

1.25

Overlap normalized by delta-overlap

FIG. 2: Ratios of the forward (∆ = 0) virtual photon-J/ψ wave function overlaps computed using the different vector mesonwave functions to the fully nonrelativistic Delta parametrization as a function of the dipole size r at different photon virtualities.

Gaussian.2

The suppression with respect to the nonrelativisticlimit is larger for the longitudinal polarization state thanfor the transverse one. This can be understood as fol-lows. The longitudinal virtual photon wave function de-pends on the quark momentum fraction as ∼ z(1 − z)(see Eq. (17)), and as such is peaked at z = 1/2. Thez-structure of the fully non-relativistic wave function isδ(z − 1/2), and when the first relativistic corrections areincluded, the z = 1/2 region still dominates the overlap.On the other hand, the transverse photon wave functionis not peaked at z = 1/2, see Eq. (18). Thus, the sup-pression from the ∂2

zδ(z − 1/2) term in the relativisticcorrection is smaller for the transverse polarization.

2 The parameters in the BLFQ wave function are constrained bythe charmonium mass spectrum, and not the decay widths thatprobe the wave function at r = 0. Consequently the BLFQwave function is not required to result in exactly the same decaywidth as the other wave functions, which explains the differenceat small r.

B. J/ψ production

The total exclusive DIS J/ψ production cross sectionfor a proton target at W = 90GeV is shown in Fig. 3,compared with the H1 [4] and ZEUS [8] data. The overallnormalization of the cross section has a relatively largetheoretical uncertainty. We note that the two correc-tions discussed in Sec. II, the real part and especially theskewedness correction are numerically significant, up to∼ 50% (see e.g. Ref. [40]). As discussed in Sec. II, espe-cially the skewedness correction is not very robust and itsapplicability in the dipole picture used here is not clear.In addition to the possibly problematic skewedness cor-rections, the fact that our NRQCD based wave functionsare not normalized affects the absolute normalization ofthe vector meson production cross sections. Thus our fo-cus here is rather on the relative effects of different mesonwave functions and the dependence on Q2.

The vector meson cross section is dominated by dipole

13

101 102

Q2 +M2J/ψ [GeV2]

10−1

100

101

102

σL

+σT

[nb]

γ∗p→ J/ψp, W = 90 GeV

NRQCD expansion

Delta

Boosted gaussian

BLFQ

ZEUS

H1

FIG. 3: Total J/ψ production cross section as a function ofvirtuality computed using different vector meson wave func-tions compared with H1 [4] and ZEUS [8] data.

sizes of the order of 1/(Q2 + M2V ) as can be seen3 from

Fig. 1. Consequently, it is more instructive to look at thedependence of the J/ψ cross section on Q2 than the over-all normalization. From Fig. 3 one sees that the fully non-relativistic wave function results in a too steep Q2 depen-dence compared to the HERA data. The first relativisticcorrection slows down the Q2 evolution close to the pho-toproduction region and leads to a better agreement withthe experimental data. This is a consequence of the basicbehavior of the relativistic correction as a ∼ −r2 modi-fication that suppresses the vector meson wave functionstrongly at large dipoles. Thus the reduction from therelativistic correction is larger for smaller Q2. At largeQ2 the exponential suppression from the photon wavefunction starts to dominate at smaller dipole sizes, andthe relativistic −r2 correction becomes negligible. How-ever, the relativistic contribution to the momentum frac-tion z structure is present at all Q2, and suppresses thelongitudinal cross section more than the transverse one.

A similar trend in the Q2 dependence is also visiblewith both the Boosted Gaussian and BLFQ wave func-tions. For the Boosted Gaussian case, the agreementwith HERA data has been established numerous timesin the previous literature, e.g. in Ref. [48]. The Q2 de-pendence of the cross section is slightly weaker when theBLFQ wave function is used, but the difference is compa-rable to the experimental uncertainties. We note that inRef. [112] the BLFQ wave function is found to result in across section underestimating the HERA data in the pho-toproduction region. In this work, compared to the setup

3 Note, however, that as the dipole amplitude scales as N ∼ r2

at small r, the dominant dipole size scale for the cross section islarger than the maximum of the overlap peaks in Fig. 1.

0 5 10 15 20 25

Q2 [GeV2]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

R=σL/σ

T

γ∗p→ J/ψp, W = 90 GeV

NRQCD expansion

Delta

Boosted gaussian

BLFQ

ZEUS

H1

FIG. 4: Longitudinal J/ψ production cross section divided bythe transverse cross section as a function of photon virtuality.Results obtained with different wave functions are comparedwith the H1 [4] and ZEUS [8] data.

used in Ref. [112], we use an updated BLFQ parametriza-tion from Ref. [111] which was shown in Ref. [113] to re-sult in a good description of the J/ψ production in ultraperipheral proton-proton collisions at the LHC, which inpractice probe vector meson photoproduction [15, 16].

To cancel normalization uncertainties, we next studycross section ratios. In Fig. 4 the longitudinal-to-transverse ratio of the J/ψ production cross section isshown as a function of the photon virtuality. The re-sults are compared with the H1 and ZEUS data fromRefs. [4, 8]. The first relativistic correction reduces thelongitudinal cross section more than the transverse one.As discussed above, this is due to the fact that a part ofthe correction shifts the meson wave function away fromthe δ(z − 1/2), which is the structure preferred by lon-gitudinal photons but not by transverse photons. Thisshows up as a decrease in the longitudinal to transverseratio as a function of Q2. The effect is even stronger withthe Boosted Gaussian and BLFQ wavefunctions.

Finally, we study vector meson production in the fu-ture Electron Ion Collider. As the diffractive cross sectionat leading order in perturbative QCD is approximativelyproportional to the squared gluon density, exclusive vec-tor meson production is a promising observable to lookfor saturation effects at the future Electron Ion Collider(see e.g. [114]).

To quantify the non-linear effects, we compute the nu-clear suppression factor

σγ∗A→J/ψA

cA4/3σγ∗p→J/ψp. (64)

The denominator corresponds to the so called impulseapproximation, which is used to transform the photon-proton cross section to the photon-nucleus case in theabsence of nuclear effects, but taking into account the

14

10−1 100 101 102

Q2 [GeV2]

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00σγ∗ A→

J/ψA/(

1 2A

4/3σγ∗ p→

J/ψp)

Nuclear suppression, W = 90 GeV

NRQCD expansion

Delta

Boosted gaussian

BLFQ

FIG. 5: Nuclear suppression factor for total coherent J/ψproduction as a function of Q2 computed using the differentvector meson wave functions.

different form factors (transverse density profiles Fouriertransformed to the momentum space). The A4/3 scal-ing can be understood to originate from the fact thatthe coherent cross section at t = 0 scales as ∼ A2, andthe width of the coherent spectra (location of the firstdiffractive minimum) is proportional to 1/R2

A ∼ A−2/3.The numerical factor c depends on the proton and nuclearform factors, and is found to be very close to c = 1

2 inRef. [87]. In the absence of non-linear effects (or shadow-ing effects in the gluon distribution), with dipole ampli-tudes (19) and (21) that depend linearly on r2xg(x, µ2),this ratio is exactly 1.

The obtained nuclear suppression factor is shown inFig. 5 in the Q2 range accessible at the Electron IonCollider. We emphasize that all the nuclear modifica-tions in this figure are calculated with exactly the samedipole cross sections, corresponding to the same nuclearshadowing (as measured e.g. by the nuclear suppressionin FL or F2). Thus the difference between the curvesresults purely from vector meson wave function effects.When using the NRQCD wave function with the rela-tivistic corretion, the Boosted Gaussian wave function orthe BLFQ wave function, the obtained nuclear suppres-sion factors are practically identical. Even though largemass of the vector meson renders the scale in the processlarge, a moderate suppression ∼ 0.75 is found at smalland moderate Q2. In the small Q2 region the uncer-tainty obtained by varying the NRQCD matrix elementsis large.

The fully non relativistic wave function results in amuch stronger suppression at small Q2. This can be un-derstood, as it was already seen in Fig. 1 that this wavefunction gives more weigh on larger dipoles compared tothe other studied wave functions. As the larger dipolesare more sensitive to non-linear effects, a larger nuclearsuppression in this case is anticipated. The first rela-

tivistic correction ∼ −r2 suppresses the overlap at largedipole sizes, and consequently the nuclear suppression.At higher Q2 the photon wave function again cuts out thelarge dipole part of the overlap in all cases, and as suchthe results obtained by applying the fully non-relativisticwave function do not differ from other wave functionsany more. At asymptotically large Q2 only small dipolescontribute and the dipole amplitudes can be linearized.Consequently, the suppression factor approaches unity atlarge Q2 independently of the applied wave function.

The fact that the fully non-relativistic wave functionresults in a very different nuclear suppression demon-strates that the dependence on the meson wave functiondoes not completely cancel in the nucleus-to-proton crosssection ratios. Consequently, a realistic (and relativistic)description of the vector meson wave function is neces-sary for interpreting the measured nuclear suppressionfactors. This indicates that there is a large theoreticaluncertaintly in using the fully nonrelativistic formula ofRyskin [3], not only for extracting absolute gluon distri-butions, but even for extracting nuclear modifications tothe dipole cross section (or the gluon density) from crosssection ratios.

VI. CONCLUSIONS

In this work we proposed a new parametrization forthe heavy vector meson wave function based on NRQCDlong-distance matrix elements. These matrix elementscan be used to simultaneously constrain both the valueand the derivative of the vector meson wave function atthe origin using quarkonium decay data. This approachprovides a systematic method to compute the vector me-son wave function as an expansion in the strong couplingconstant αs and the quark velocity v.

Compared to many phenomenological approaches usedin the literature, our approach uses two independentconstraints (the wave function value and its deriva-tive). The obtained wave function is rotationally sym-metric in the rest frame and contains only the S wavecomponent. Consequently, we simultaneously obtain aconsistent parametrization for both polarization states.This is unlike in some widely used phenomenologicalparametrizations where the virtual photon like helicitystructure is assumed on the light front. Relating lightcone wavefunctions to rest frame ones also provides aconsistent way to discuss the effect of a potential D-wavecontribution to the meson wavefunction. We do not seeindications, neither theoretically nor phenomenologically,that a significant D-wave contribution would be requiredor favored for the J/ψ .

The first relativistic correction to the wave function,controlled by the wave function derivative at the origin,is found to have a sizeable effect on the cross section.The negative ∼ −r2 relativistic contribution in terms ofthe transverse size r suppresses the obtained wave func-tion at larger dipole sizes. The momentum fraction part

15

of the correction partially compensates for this effect forthe transverse photon by shifting the wave function awayfrom the fully non relativistic configuration where bothquarks carry the same fraction of the longitudinal mo-mentum, a configuration which is not preferred by thetransverse photon.

A disadvantage in our approach is that it is not pos-sible to obtain a wave function which is normalized tounity. In the NRQCD framework the value of the wave-function at long distances is parametrized by a nonper-turbative matrix element, whose effect is felt in the valueof the wavefunction near the origin. This can lead toan overestimation of the cross section at Q2 = 0, whereone is most sensitive to the long distance behavior of thewave function. In practice, however, we obtain cross sec-tions that are quite similar to what is given by e.g. theBoosted Gaussian parametrization. The wave functionoverlap with the photon is also smaller than with theBLFQ approach. Thus the lack of normalization in thewavefunction does not seem to be an important effect forJ/ψ . The situation would be different for lighter vectormesons.

The structure of the wave function can be probed bystudying cross sections (and cross section ratios) at differ-ent photon virtualities where the dipole sizes contribut-ing to the cross section vary. The first relativistic correc-tion is found to weaken the Q2 dependence of the totalJ/ψ production cross section and the longitudinal-to-transverse ratio. These effects are broadly similar to pre-dictions obtained by the Boosted Gaussian parametriza-tion, or by the BLFQ wave function that is based on anexplicit calculation on the light front including confine-ment effects.

When comparing vector meson production off protonsto heavy nuclei, we find that the wave function doesnot completely cancel in the nuclear suppression factor,which compares the γ∗A cross section to the γ∗p in theimpulse approximation. This demonstrates that a realis-tic vector meson wave function is necessary to properlyinterpret the nuclear suppression results, and in partic-ular a fully non-relativistic approach can not be reliablyused to extract the non-linear effects on the nuclear struc-ture.

In addition to the corrections in velocity, it wouldbe important to include perturbative corrections in thestrong coupling αs in the calculation of exclusive vectormeson production. Indeed some recent advances [94, 115]are gradually making it possible to do so in the dipolepicture. However, a study of the phenomenological im-plications of these αs corrections remains to be done. Interms of understanding current and future experimen-tal collider data, it would also be important to explorewhether this approach can be extended to excited statessuch as the ψ(nS).

Acknowledgements

We thank M. Escobedo and M. Li for discussions.This work was supported by the Academy of Finland,projects 314764 (H.M), 321840 (T.L and J.P) and 314162(J.P). T.L is supported by the European Research Coun-cil (ERC) under the European Union’s Horizon 2020 re-search and innovation programme (grant agreement NoERC-2015-CoG-681707). The content of this article doesnot reflect the official opinion of the European Union andresponsibility for the information and views expressedtherein lies entirely with the authors.

Appendix A: Orbital decomposition

In Sec. III we highlighted how it is crucial to properlytransform the NRQCD based vector meson wave functionto the light front by performing the Melosh rotation. Inparticular, we demonstrated that this rotation gives riseto the helicity structures absent in the rest frame spinstructure (e.g. non-zero Ψλ=±1

h=±1,h=∓1).

In this section, we illustrate the role of the Melosh ro-tation by considering both the J/ψ and virtual photon(in the case of charm quarks) wave functions, and deter-mining the contributions from the S and D wave com-ponents. The NRQCD based wave function obtained inSec. III contains only the S wave structure. For the J/ψwave function, we study here the commonly used BoostedGaussian parametrization (see Sec. IV A).

The S and D waves are properly defined in the restframe. Consequently, we take the vector meson or thevirtual photon wave functions on the light front writtenin momentum space and perform the Melosh rotation totransform them to the meson rest frame. In the restframe we then remove either the S or D wave contribu-tion, and transform the final wave function back to thelight front and Fourier transform to transverse coordinatespace.

The Boosted Gaussian parametrization of the J/ψwave function is decomposed to S and D componentsin Fig. 6. In principle the angular momentum structureof the parametrization could turn out to correspond to alarge D-wave component in the rest frame. Indeed it ismostly constrained by the choice of having helicity struc-ture on the light front exactly the same as that of thephoton, which has a large D-wave component as we willsee. However, in practice the S-wave only result is a goodapproximation of the full result. This is due to the smallquark velocities contributing to the wave function, as inthe momentum space the Boosted Gaussian wave func-tion is exponentially suppressed at large invariant mass

M2 =k2+m2

c

z(1−z) . Thus, large transverse momentum |k| or

large longitudinal momentum (z → 0 or z → 1) contri-butions are heavily suppressed, and do not generate asignificant D-wave component.

A similar discussion can be carried out for the BLFQ

16

10−3 10−2 10−1 100

r [fm]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Longitudinal

S+D

S-wave

D-wave

10−3 10−2 10−1 100

r [fm]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Transverse

r/2∫

dz∑

hh |ΨBG,S/D

hh,λ|2 [GeV]

FIG. 6: Vector meson wave function from the Boosted Gaus-sian parametrization decomposed into S and D wave com-ponents in the J/ψ rest frame as a function of the quark-antiquark transverse separation. Left panel shows the longi-tudinal polarization and right panel transverse polarization.

wave function described in Sec. IV A. As shown in [116],in the rest frame, the squared J/ψ BLFQ wave functionis dominated by the S wave component, the D wave con-tributing only a small fraction of the order of 0.1 . . . 4%(depending on the polarization). In heavier mesons, thiscontribution is even smaller. This is comparable to theBoosted Gaussian case discussed above.

Overall, based on neither the Boosted Gaussian nor theBLFQ parametrizations, we do not see any confirmationfor the result of Ref. [98], where the D wave part of theJ/ψ wave function was found to result in tens of percentcontribution on the vector meson production cross sec-tion. Part of this discrepancy might be merely a questionof terminology. In our discussion here, we have insistedthat the terms S-wave and D-wave refer to the angularmomentum components of the 3-dimensional wave func-tion in the meson rest frame. Thus the mere presence,in the light cone wave function, of terms proportional totransverse momenta originating from the Melosh rotationcannot be taken as an indication of a D-wave componentin the meson.

Let us now move to the case of a virtual photon. Sincea spacelike virtual photon does not have a rest frameand is not a bound state, it is not customarily thoughtof in terms of an S – D wave decomposition. Now, how-ever, we have an explicit light cone wave function forthe photon just like for the meson, and we can use thesame procedure to determine its S- and D-wave compo-nents in the meson rest frame. The resulting squaredlight front wave functions summed over quark helicitiesare shown in Fig. 7. The full photon wave function, writ-ten in Eqs. (17) and (18), is denoted by S + D, as itcan be written as a sum of these two components. When

10−3 10−2 10−1 1000.0

0.2

0.4

0.6

0.8

1.0×10−5

Q2 = 0.05 GeV2

Longitudinal

10−3 10−2 10−1 100

r [fm]

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025Q2 = 22.4 GeV2

10−3 10−2 10−1 1000.00

0.02

0.04

0.06

0.08

0.10Transverse

S+D

S-wave

D-wave

10−3 10−2 10−1 100

r [fm]

0.00

0.02

0.04

0.06

0.08

0.10

r/2∫

dz∑

hh |Ψγ,S/D

hh,λ|2 [GeV]

FIG. 7: Virtual photon wave function integrated over the lon-gitudinal momentum fraction z decomposed to S and D wavecomponents as a function of the quark-antiquark transverseseparation.

compared to the full result, the squared D wave onlycontribution is found to be strongly suppressed. There isalso a contribution originating from the overlap betweenthe S and D wave contributions. This term would van-ish if we integrated over all the angles. Here, we onlyintegrate over the azimuthal direction of r. Integrationover the momentum fraction z corresponds to the eval-uation of the coordinate space wave function at x3 = 0,and consequently one angular integral is not performedand the overlap does not vanish. The relative importanceof different contributions is found to be approximativelyindependent of Q2.

The S −D overlap contribution is numerically signif-icant, which is reflected by the large difference betweenthe full result and the S wave only contribution. Thissuggests that even though the D wave contribution issuppressed by the quark velocity, the charm quark massis not large enough to render this contribution negligi-ble. This is due to the fact that the photon wave func-tion in the momentum space behaves as ∼ 1/k2 wherek is the quark transverse momentum, and this power-like tail brings numerically large contributions from rela-tively large momenta. Additionally, the integration overthe longitudinal momentum fraction z includes high mo-mentum contributions, as the photon wave function hasa support over a large range of z.

The contribution from the S −D overlap changes signat large transverse separations in case of the longitudi-nal polarization. There is no node in the radial part ofthe wave function, but the spherical harmonic functiondescribing the angular part of the D wave componentchanges sign, which explains the sign flip. In the S −Doverlap mostly the helicity-+− and −+ components ofthe D wave contribute by coupling to the S wave. On

17

the other hand, in the D wave squared wave function,one sums all helicity components. As the D wave compo-nent itself is a relativistic correction, none of the helicitystructures dominates unlike in the S wave part. More-over, only the +− and −+ helicity components changesign at large distances, and as the ++ and −− compo-nents do not vanish in this region, no node appears in thesquared D wave result. In the case of transverse polar-ization with λ = ±1, the helicity component ±± in theD wave also changes the sign at large distances, but thiseffect is not easily visible in Fig. 7 as the other helicitycomponents that do not change sign dominate.

Appendix B: Photon-like parametrizations of lightcone wave functions

As discussed in Sec. IV, an often used approach toparametrize vector meson wave functions is to start fromthe helicity structure of the virtual photon light conewave functions (17), (18). One then replaces the Besselfunction K0 in the photon wave functions (17) and (18)by an unknown function as [48]

efez(1− z)K0(εr)

2π→ φT,L(r, z), (B1)

with the explicit factor Q in the longitudinal wave func-tion replaced by the meson mass as 2Q → MV . Thisleads, with our sign conventions, to the wave functionbeing written as

ψλ=0hh (r, z) =

√Ncδh,−h

×[MV +

m2c −∇2

r

MV z(1− z)

]φL(r, z) (B2)

ψλ=±1hh

(r, z) =√

2Nc1

z(1− z)(mcδh,±δh,±

∓ie±iθr(zδh,±δh,∓ − (1− z)δh,∓δh,±

)∂r)φT (r, z).

(B3)

The scalar functions φT,L(r, z) are then parametrized,and the parameters can be determined by requiring thatthe resulting wave function is normalized to unity andreproduces the experimental leptonic decay width. Interms of Lorentz-invariant form factors this means thatthe meson is assumed to have a nonzero Dirac form fac-tor but a vanishing Pauli form factor, since this is thestructure dictated by the gauge-boson-fermion vertex atleading order perturbation theory. The procedure there-fore does not generate the most general possible helicitystructure.

This photon-like parametrization approach starts froma spacelike photon, where the photon momentum breaksrotational symmetry that is manifested here as a sym-metry between longitudinal and transverse meson polar-ization states. The common approach is to separately

parametrize the longitudinal and transverse functionsφT,L(r, z). The helicity structure obtained by generaliza-tion from the photon wave function is of course consistentwith rotational symmetry, since the decay of a timelikevirtual photon is rotationally symmetric. Thus one couldderive a constraint relating φL(r, z) and φT (r, z) by re-quiring the meson rest frame wavefunctions to be thesame. To our knowledge this approach has not, however,been used in the literature. Using separate parametriza-tions for φT,L(r, z) should be contrasted with the ap-proach in this paper. Here, we maintain rotational in-variance in the meson rest frame, in particular startingfrom the same decay constants calculated from the restframe wave functions. Our procedure for going from therest frame to the light cone wave function therefore si-multaneously determines the wavefunction for both lon-gitudinal and transverse polarization states.

One can take the parametrization (B2), (B3) in mo-mentum space, perform the inverse Melosh rotation andseparate the S- and D- wave components to get a restframe 3-dimensional wavefunction. Assuming that theFourier transforms of the scalar functions are rotation-ally invariant, i.e. φT,L(k, z) depend only on k = |~k| =√

(M/2)2 −m2c , the result of this exercise in momentum

space is:

ψλ=0S = φL(k)

(MV +

4E2

MV

)

×√Ncπ

2

(E +mc)(2E)3/2

(1

3k2 + (E +m)2

)(B4)

ψλ=+1,−1S = φT (k) · 4E

×√Ncπ

2

(E +mc)(2E)3/2

(1

3k2 + (E +m)2

)(B5)

ψλ=0D = φL(k)

(MV +

4E2

MV

)

×√Ncπ

2

(E +mc)(2E)3/2

4

3√

2k2 (B6)

ψλ=+1,−1D = φT (k) · 4E

×√Ncπ

2

(E +mc)(2E)3/2

4

3√

2k2. (B7)

These expressions are written in terms of the energy of

the quark in the meson rest frame E =

√~k2 +m2

c =

M/2, and k = |~k|. Let us point out a few aspects of theseexpressions. Firstly, as discussed above, in the photon-like parametrization there is always a D-wave compo-nent in the meson wave function. It is, as expected,explicitly a relativistic correction, i.e. proportional tothe squared 3-momentum of the quark. Secondly, the

18

rest frame wave functions of the transverse and longitu-dinal polarizations are not the same, but differ by a factor(MV + 4E2

MV

)/(4E) ≈ 1 + O

((M −MV )2

), where M is

the invariant mass of the quark pair and MV the massof the meson. As discussed earlier in Sec. III, the coor-dinate transformation from k3 to z inevitably introducesambiguities that are proportional to this difference, sothis should not come as a surprise. In an NRQCD powercounting, this difference is of the order of the bindingenergy of the meson, which is higher order than we areconsidering here.

In spite of this discussion, the wavefunction that weobtained in Sec. IV can in fact be written in a photon-likeform in terms of scalar parts of light cone wave functions.In the notation of [48] these read

φL(r, z) =π√

Nc(2mc,NR)3/2

4MVmc,NR

4m2c,NR +M2

V

·[Aδ(z − 1/2)

+B

m2c,NR

((34m2

c,NR + 52M

2V

4m2c,NR +M2

V

+m2c,NRr2

)δ(z − 1/2)

− 1

4∂2zδ(z − 1/2)

)](B8)

φT (r, z) =π√

Nc(2mc,NR)3/2

[Aδ(z − 1/2)

+B

m2c,NR

((11

2+m2

c,NRr2

)δ(z − 1/2)

− 1

4∂2zδ(z − 1/2)

)](B9)

We emphasize that we do not expect that writing downsuch a paramterization in terms of two scalar parts ofa light cone wave function having the helicity structureof a photon would be possible at higher orders in thenonrelativistic expansion.

As a side remark, we discussed above that the photon-like structure generically implies a non-zero D wave com-

ponent, see Eqs. (B6) and (B7). On the other hand, ourNRQCD based wave function by construction has no Dcomponent. However, the D-wave component resultingfrom inserting the scalar parts (B8) and (B9) into the for-mulae for the D-wave contribution, Eqs. (B6) and (B7),

behaves as ∼ k2∇2~kδ(3)(~k). Such a function actually

yields zero when convoluted with any test function f(~k),since the angular integral picks out the ` = 2 componentof f , which must vanish at k = 0. Thus the D-wave con-tribution corresponding to (B8) and (B9) is in fact zeroin a distribution sense.

It is interesting to note, that when one calculates fromthese expressions the decay constants for the different po-larization states using the light cone perturbation theoryexpressions (26) and (27) in Ref. [48], one obtains:

fL =

√2Nc

mc,NRef

(A+

5

2

B

m2c,NR

)(B10)

fT =

√2Nc

mc,NRef

2mc,NR

MV

(A− 1

2

B

m2c,NR

)(B11)

The results are not exactly equal. However, as discussedabove, if one approximates the meson mass by the quarkpair invariant mass, as we did in transforming to themomentum fraction z, they do reduce to the same re-sult. This can be seen explicitly by replacing MV in

(B11) by 〈M〉 ≈ 2√m2c,NR + 〈q2〉 and using Eq. (35) to

write, at lowest nontrivial order in the quark velocity,2mc,NR/MV ≈ 1 + 3B/(m2

c,NRA) (note that B < 0). In

this approximation Eqs. (B10) and (B11) also give backthe same decay width expression that we are using to de-termine the rest frame wavefunction. We reiterate that adifference such as this can be expected in our procedure.We are constructing our wavefunctions by requiring thedecay widths calculated from the rest frame wave func-tions to have the correct value, and to be the same forthe different polarization states. The coordinate trans-formation to light cone wave functions does not conservethese properties exactly, but only up to a given order inthe nonrelativistic expansion.

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