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Rotational covariance and light-front current matrix elements

arXiv:hep-ph/9303264v1 16 Mar 1993Rotational covariance and light-front current matrix elementsB. D. KeisterDepartment of Physics, Carnegie Mellon University, Pittsburgh, PA 15213Light-front current matrix elements for elastic scattering from hadrons with spin 1or greater must satisfy a nontrivial constraint associated with the requirement of ro-tational covariance for the current operator.

Using a model ρ meson as a prototypefor hadronic quark models, this constraint and its implications are studied at bothlow and high momentum transfers. In the kinematic region appropriate for asymp-totic QCD, helicity rules, together with the rotational covariance condition, yield anadditional relation between the light-front current matrix elements.Typeset Using REVTEX1

I. INTRODUCTIONLight-front dynamics has found frequent application in particle and nuclear physics. Firstintroduced by Dirac [3], it has the advantage that, of the ten generators of transformationsfor the Poincar´e group, only three of them depend upon interaction dynamics.

In particular,certain forms of Lorentz boosts are interaction independent. This is in contrast to the moretraditionally used instant-form dynamics, which has four interacting generators, includingall boosts.However, as with all forms of relativistic dynamics, more generators than just the light-front “Hamiltonian” P −= P 0 −P · n, where n is a spatial unit vector, must dependnontrivially on the interaction, and these generators correspond to rotations about axesperpendicular to n. This is especially important when computing matrix elements of elec-tromagnetic and weak currents in the front form.

Because the current operator must havethe transformation properties of a four-vector, and some of these transformations are in-teraction dependent, it must also depend upon the strong interaction. In particular, thecurrent operator cannot satisfy all of the covariance requirements associated with transverserotations without containing interaction dependent components.

This is sometimes calledthe “angular condition” [1,2].Rotational covariance has been studied in a variety of contexts. For hadrons composedof quarks, Terent’ev and others have examined the extent to which the current operatoris uniquely determined without knowing its two-body components.For elastic electron-deuteron scattering, the lack of uniqueness modulo two-body currents has been explored byemploying a variety of “schemes” to satisfy the rotational covariance requirement [5,7].In this paper, we examine a specific rotational constraint for light-front current matrixelements of hadrons composed of quarks.

The current operator must satisfy certain non-trivial commutation relations with the interacting generators of the Poincar´e group. Therequirement of current covariance for rotations about an axis perpendicular to n gives riseto constraints on current matrix elements for elastic scattering from particles of spin 1 or2

greater [5,7]. While form factors of the pion [10] and the nucleon [11] have frequently beencalculated, the particular test discussed here is not applicable because j < 1.

We thereforetake a model of a ρ meson as a prototype hadron to which to apply the rotational covariancetest.II. ROTATIONAL COVARIANCEIn light-front dynamics, full rotational covariance implies a nontrivial set of conditionswhich any hadronic model must in principle satisfy.First, the state vector for the hadron must be an eigenstate of the total spin operator.This condition is satisfied in light-front models which use a quantum mechanical Hilbertspace with a fixed number of particles [10,11].

Models using a field theory, in particularthose motivated by QCD, may not necessarily use rotationally covariant state vectors. Athigh Q2, this deficiency may be irrelevant if the corrections to rotational covariance fall as apower of Q2.

At moderate and low Q2, the issue may be important. In any event, we considerhere only models whose state vectors have the proper rotational covariance properties.Second, the current operator Iµ(x) must satisfy the conditions of Lorentz covariance.

IfΛµν is the matrix for a homogeneous Lorentz transformation and aµ is a spacetime transla-tion, thenU(Λ, a)Iµ(x)U(Λ, a)† = (Λ−1)µνIν(Λx + a). (1)It must be conserved with respect to the four-momentum.

If P µ is the generator of spacetimetranslations, thengµν[P µ, Iν(0)] = 0. (2)These constraints have two implications.

First, it is possible to express the physical con-tent of current matrix elements between any two states in terms of a limited number ofLorentz invariant functions of the masses of the states and the square of the momentum3

transfer. Second, the operator Iµ(0) must have in general a complicated structure, since itobeys nontrivial commutation relations with at least some generators which are interactiondependent.To illustrate these two points, consider current matrix elements with spacelike momentumtransfer.

It has been shown [7,9] that all spin matrix elements of the current four-vector canbe computed from the set of matrix elements of I+(0) in a frame in which q+ = q0+q·n = 0.Alternatively, it means that all invariant form factors can be computed from matrix elementsof I+(0). However, the covariance requirements imply that the matrix elements of I+(0) mustsatisfy a set of constraints.

This is particularly relevant if a one-body operator has been usedto compute the matrix elements. If a constraint involves transformations which use Poincar´egenerators which are non-interacting, it will in general be satisfied for matrix elementscomputed with one-body operators.However, constraints which involve transformationsusing interacting generators will in general not be satisfied with one-body current matrixelements.An interaction-dependent constraint can be derived by requiring that Breit-frame matrixelements of the transverse current in a helicity basis vanish if the magnitude of the helicityflip is 2 or greater.

For light-front current matrix elements ⟨˜p′µ′|I+(0)|˜pµ⟩correspondingto elastic electron scattering from a target of mass M and spin j, the condition is expressedas follows:Xλ′λD(j)†µ′λ′(R′ch)⟨˜p′α′|I+(0)|˜pα⟩D(j)λµ(Rch) = 0,|µ′ −µ| ≥2. (3)In Eq.

(3), ˜p ≡(p⊥, p+) is a light-front momentum, and the matrix element is evaluated ina frame where q+ = p′+ −p+ = 0, and the perpendicular component of ˜p′ and ˜p lies alongthe x axis. The rotationRch = Rcf(˜p, M)Ry( π2);R′ch = Rcf(˜p′, M)Ry( π2),(4)where Rcf is a Melosh rotation which, together with the rotation Ry( π2), transforms thestate vectors from light-front spin to helicity.

For inelastic excitation of a state with mass4

M′ and spin j′, Eq. (3) is modified only by the use of M′ and j′ in the rotation matrices forthe final state.

For elastic scattering, Eq. (3) is applicable only to states with j ≥1.

For aspin-1 particle, Eq. (3) can be expressed explicitly in terms of individual light-front matrixelements as follows [5]:∆(Q2) ≡(1 + 2η)I1,1 + I1,−1 −q8ηI1,0 −I0,0 = 0,(5)where Iµ′,µ ≡⟨˜p′µ′|I+(0)|˜pµ⟩is the matrix element of I+(0):p′⊥= −p⊥= 12q;p′+ = p+ =qM2 + 14q2,(6)Q2 = −q2 is the square of the four-momentum transfer, η ≡Q2/4M2.Earlier studies of the deuteron form factors using models with one-body currents androtationally covariant state vectors indicate that ∆(Q2) can be relatively small for low andmoderate Q2, though, not surprisingly, ∆(Q2) is an increasing function of Q2 [5,7].

Animportant feature of the deuteron is that the characteristic nucleon momentum is very smallcompared to a nucleon mass.For most quark models of hadrons, the characteristic constituent momentum is not smallwith respect to the quark mass, and questions of rotational covariance therefore require aseparate investigation. There have been previous studies of the pion [10] and the nucleon [11]form factors using models with one-body currents and rotationally covariant state vectors.Equation (5) has no counterpart for spin zero and spin 12, so the dynamical aspect of ro-tational covariance was not addressed in those works.

Some other conclusions from thosestudies which are relevant to the discussion below include1. For small quark masses (10 MeV), relativistic effects such as those of Melosh rotationscan be substantial, even at low Q2.2.

In the limit mq →0, it can be shown [12] that Q2Fπ(Q2) →const as Q2 →∞.In what follows, we examine a model for a ρ meson similar to those used for the pionpublished earlier, in light of the rotational covariance condition (5) at both low and highQ2.5

III. THE MODELThe model ρ meson is composed of a valence quark and an antiquark.

Since it must becolor-antisymmetric and flavor-symmetric (isospin 1), the space-spin wave function must besymmetric. For the ground state, we take the coupled states 3S1 −3D1.

This is the samespace-spin combination as that of a deuteron composed of two nucleons. Since the detailsof such a deuteron form factor calculation are discussed extensively elsewhere [7], only theunique features of the quark model will be presented here.

The mass operator isM2 = M20 + 2mU + U0,(7)where M20 = 4(m2 + k2) is the non-interacting mass operator, m is the quark mass and k isthe relative three-momentum. The potential U is a harmonic-oscillator potential:U = 12κr2,r ≡i∇k,(8)and U0 is a constant The S- and D-state wave functions are Gaussians, just as for a non-relativistic harmonic-oscillator problem:φ0(k)= NSe−k2/2α2;φ2(k)= NDk2e−k2/2α2.

(9)The interacting mass eigenvalue isM2 = 4(m2 + 3α2) + U0. (10)In an extensive study of mesons using a nonrelativistic quark model, Godfrey and Isgurobtained D-state admixtures with amplitude 0.04 for an excited ρ-meson state, with noreported admixture for the ρ (750) [13].

To test sensitivity, we use a D-state admixtureamplitude coefficient of 0.04; this is an extreme choice, but in the end, the results differ littlefrom those obtained by ignoring the D-state admixture entirely. An oscillator parametervalue α = 0.45 GeV/c has been extracted from the Godfrey-Isgur results [14].6

IV. FORM FACTORSMatrix elements of matrix elements of I+(0) can be written as [7]⟨˜p′µ′|I+(0)|˜pµ⟩= ⟨p+112q⊥µ′1|I+quark(0)|p+1 −12q⊥µ1⟩Zdk(2π)3∂(˜pk)∂(˜p1˜p2)12 ∂(˜p′1˜p2)∂(˜p′k′)12×D( 12)¯µ1µ1 [Rcf(k′)] D( 12)¯µ2µ2 [Rcf(−k′)]×⟨12 ¯µ′112 ¯µ′2|1µ′S⟩⟨lµ′l1µ′S|1µ′⟩Yl′µ′l(ˆk′)φl′(k′)×D( 12)†µ1 ¯µ1 [Rcf(k)] D( 12 )†µ2 ¯µ2 [Rcf(k)]×⟨12 ¯µ112 ¯µ1|1µS⟩⟨lµl1µS|1µ⟩Ylµl(ˆk)φl(k).

(11)The internal kinematics in the integral arep′1+= p+1 =qm2 + 14q2⊥;p′2+ = p+2 ;p′1⊥= p1⊥+ q⊥;p′2⊥= p2⊥k⊥= (1 −ξ)p1⊥−ξp2⊥;k′⊥= k⊥+ (1 −ξ)q⊥;ξ ≡p+1 /(p+1 + p+2 )k3= (ξ −12)vuutm2 + k2⊥ξ(1 −ξ) ;k′3 = (ξ −12)vuutm2 + k′⊥2ξ(1 −ξ) ;k= (k⊥, k3);k′ = (k′⊥, k′3). (12)The three elastic form factors G1, G2 and G3 can be expressed in terms of the matrixelements Iµ′,µ as follows [7]:G0(Q2)=12(1 + η)(1 −23η)(I1,1 + I0,0) + 53q8ηI1,0 −13(1 −4η)I1,−1G1(Q2)=1(1 + η)"I1,1 + I0,0 −I1,−1 −(1 −η)s2ηI1,0#G2(Q2)=√83(1 + η)−η2(I1,1 + I0,0) +q2ηI1,0 −(1 + 12η)I1,−1.

(13)The right-hand sides in Eq. (13) are not unique.

One can always replace one of the Iµ′,µ,or linear combinations of them, with a combination which satisfies the rotational covariancecondition (5). A common procedure has been to choose a particular combination of Iµ′,µas calculated from one-body current operators, and eliminating the remaining terms from7

Eq. (13) via the rotational covariance condition.By implication, the eliminated termsdepend upon two-body current operators.

Thus, for different choices of one-body matrixelements, each form factors Gi(Q2) will differ by a multiple of ∆(Q2)one−body, which is neverzero.V. LOW Q2The requirement of rotational covariance can be studied at low momentum transfers byexamining the behavior of the magnetic and quadrupole moments µ and ¯Q and the chargeradius.

They are related to the form factors Gi(Q2) as follows:µ≡limQ2→0 G1(Q2)¯Q≡limQ2→0 3√2G2(Q2)Q2⟨r2⟩≡limQ2→06Q2[1 −G0(Q2)]. (14)Extracted values of ⟨r2⟩12 are shown in Table I for quark masses of 10, 300 and 1000 MeV.Also shown is the effect of including or leaving out the Melosh rotations, which gives ameasure of the size of relativistic effects.

In addition, the quantityδ ≡limQ2→0∆(Q2)1 −G0(Q2)(15)gives a measure of the sensitivity to rotational covariance uncertainty. Already one can seefrom this table corresponding to very low Q2 that simply raising the value of the quarkmass is not the same as the nonrelativistic limit.

That limit depends upon the quark mass,the value of Q2, the momentum scale α, and the composite mass. For comparison, we alsoshow results using the same parameter, except that the ρ meson is given a fictitious valueof 2 GeV.

In this last case, especially for a quark mass of 1 GeV, one can see that both theMelosh rotations (the measure of relativistic effects) and the rotational covariance parameterδ are small.8

VI. MODERATE Q2In Figs.

1, 2 and 3 we show the calculated results for G0(Q2), as obtained using Eq. (13).The relativistic effects, as characterized by turning the internal Melosh rotations on and off,are largest for the smallest quark mass.

For all three choices of quark mass, the rotationalcovariance uncertainty function ∆(Q2) becomes comparable to G0, and hence the currentmatrix elements themselves, in the region 1–2 GeV/c2.In Figs. 4, 5 and 6 we show calculated results for G0(Q2) using a fictitious ρ-meson massof 2 GeV.

In all three cases, the relativistic effects are smaller than the corresponding caseswhere Mρ = 750 MeV, but the nonrelativistic limit is still not really reached until the quarkmasse is considerably larger than the momentum scale α. For this choice of meson mass,and for all three choices of quark mass, the covariance function ∆(Q2) is much smaller forthe same range of Q2 than in the previous three figures.From the results shown here, along with those of other parameter sets, it becomes clearthat the rotational covariance uncertainty function ∆(Q2) becomes comparable to G0, andhence the current matrix elements themselves, when η = Q2/4M2 is of order unity.

Thus,for a ρ meson with physical mass 750 MeV, the breakdown of rotational covariance occursin the region 1–2 GeV/c2. This can be understood from the fact that the dimensionlessargument of the Melosh rotations in Eq.

(3) is Q/2M, which manifests itself in terms ofthe η factors in Eq. (5).

The dynamical nature of the rotational covariance condition iscontained in the presence of the interacting mass M.Note also that, for elastic scattering, the current matrix elements Iµ′,µ depend upon thequark mass m and the momentum transfer, but they do not depend upon the composite massM. The internal Melosh rotations, which give a measure of size of relativistic effects, dependupon the quark mass, but the composite mass enters only at the point of computing formfactors and evaluating the rotational covariance condition.9

VII. ASYMPTOTIC BEHAVIORAs noted above, it has been shown that, for the pion form factor, models such as the oneused here have the property that Q2Fπ(Q2) →const as Q2 →∞if mq = 0.

For a model ρmeson, the differing feature is the presence of an overall spin index and some momentum-independent Clebsch-Gordan coefficients. We therefore expect that, as for the case of thepion form factor [12], Q2Iµ′,µ →const (perhaps dependent upon µ′, µ) as Q2 →∞formq = 0 and any µ′, µ.

On the other hand, power counting rules of perturbative QCD [16]predict that the matrix element I0,0 dominates as Q2 →∞, and that I1,0 is suppressed by onpower of Q and I1,−1 by two powers of Q. Thus, a constituent quark model with one-bodycurrents only cannot reproduce the asymptotic limit.While simple models may fail to describe the asymptotic limit appropriate for pertur-bative QCD, we note that the rotational covariance condition takes a simple form at veryhigh Q2.

In the Q2 →∞limit of Eq. (5), I1,−1 drops out due to power suppression.

Theremaining terms give2ηI1,1 −q8ηI1,0 −I0,0 = 0,(16)The Breit frame (1, 1) matrix element of the transverse current in a helicity basis is identicallyzero [16]. The light-front matrix element I1,1 is not identically zero, but is suppressed bytwo powers of Q relative to a specific combination of I0,0 and QI1,0.Note also that, in Eq.

(16), the factors of η which correspond to the dynamical nature ofthe rotational covariance condition are now very large. At the same time, the gluon-exchangeterms used to derive the power-counting helicity rules in perturbative QCD correspond totwo-body currents in a constituent model such as that presented here.

At high Q2, therefore,the η factors, the dynamics of perturbative QCD and rotational covariance are linked in away which cannot be described in a constituent model with one-body currents.10

VIII. CONCLUSIONSThe requirement of rotational covariance for matrix elements of electromagnetic currentsis nontrivial to satisfy for for elastic scattering from systems with spin j ≥1.

For the caseof the deuteron, the fact that its structure is essentially nonrelativistic (all masses are largecompared to the momentum scale of the system) suggests that the rotational covariancerequirement can be satisfied in a satisfactory quantitative way using only one-body currentmatrix elements.For hadrons composed of quarks, typical quark masses are not smallcompared to typical momentum scales, and issue of rotational covariance therefore mustbe studied separately. In this paper, we have investigated the behavior of current matrixelements in a simple model of a ρ meson.Our results using variable input parametersindicate a breakdown of rotational covariance of current matrix elements when η = Q2/4M2is of order unity.

The dynamical nature of the rotational covariance constraint is reflectedin the presence of the interacting mass eigenvalue of the composite particle. In addition,rotational covariance implies a specific power-law relation among the spin matrix elementsat high Q2.

That relation is consistent with the power-counting rules of perturbative QCD,which in turn are derived from gluon-exchange contributions that correspond to two-bodycurrents in a constituent-quark framework. The quark model used in this paper does notcontain such two-body currents, and also does not satisfy the high-Q2 power-law relation.IX.

ACKNOWLEDGEMENTSThis work was supported in part by the U.S. National Science Foundation un-der Grant PHY-9023586.The author wishes to thank Professor C. E. Carlson,Dr. P.-L. Chung and Professor N. Isgur for helpful conversations and correspon-dence.11

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TABLESTABLE I. Calculated r.m.s.

charge radius, together with the dimensionless rotational covari-ance parameter δ, for various model parameters.Mρ (MeV)mq (MeV)Melosh⟨r2⟩12 (fm)δ75010ON1.17.03OFF0.76.16300ON0.61.17OFF0.54.361000ON0.51.36OFF0.50.42200010ON1.11.11OFF0.66.05300ON0.45.19OF0.37.111000ON0.33.04OFF0.32.1513

FIGURESFIG. 1.

Composite form factor G0(Q2) computed using a quark mass of 10 MeV and a ρ-mesonmass of 750 MeV. The solid curve denotes the full calculation and the dashed curve correspondsto the calculation where the Melosh rotations are turned off.

The dot-dashed curve describes thecovariance function ∆(Q2) defined in Eq. (5).FIG.

2. Composite form factor G0(Q2) computed using a quark mass of 300 MeV and a ρ-mesonmass of 750 MeV.

The legend is the same as that of Fig. 1.FIG.

3. Composite form factor G0(Q2) computed using a quark mass of 1 GeV and a ρ-mesonmass of 750 MeV.

The legend is the same as that of Fig. 1.FIG.

4. Composite form factor G0(Q2) computed using a quark mass of 10 MeV and a ρ-mesonmass of 2 GeV.

The legend is the same as that of Fig. 1.FIG.

5. Composite form factor G0(Q2) computed using a quark mass of 300 MeV and a ρ-mesonmass of 2 GeV.

The legend is the same as that of Fig. 1.FIG.

6. Composite form factor G0(Q2) computed using a quark mass of 1 GeV and a ρ-mesonmass of 2 GeV.

The legend is the same as that of Fig. 1.14

01234Q2 (GeV2)10-22510-125100mq = 10 MeV; M = 750 MeV

01234Q2 (GeV2)10-22510-125100mq = 300 MeV; M = 750 MeV

01234Q2 (GeV2)10-22510-125100mq = 1000 MeV; M = 750 MeV

01234Q2 (GeV2)10-22510-125100mq = 10 MeV; M = 2000 MeV

01234Q2 (GeV2)10-22510-125100mq = 300 MeV; M = 2000 MeV

01234Q2 (GeV2)10-22510-125100mq = 1000 MeV; M = 2000 MeV

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