VOLUME 45, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 21 OCTOBER 1980

Testing the Spin of the Gluon in Large-Transverse-Momentum Lepton-Pair Production P o r t e r W. Johnson and Wu-ki Tung ( a )

Department of Physics, Illinois Institute of Technology, Chicago, (Received 5 June 1980)

Illinois 60616

A structure function relation sensitive to the spin of the gluon is derived for la rge-transverse-momentum lepton-pair production. A direct test of this relation is devised in terms of a specific angular-correlation coefficient. Calculations of this correlation function, and other angular coefficients are presented.

PACS numbers: 13.40.-f, 13.40.Hq, 13.85.Kf, 14.80.Kx

Many qualitative features of high-energy hadron physics seem to be consistent with perturbative quantum chromodynamics (QCD). Recent obser­vation of three-jet events in high-energy e*e~ an­nihilation has, in particular, given encourage­ment to the QCD picture of quarks and gluons as the underlying constituents of hadron physics. Whereas the spin and "flavor quantum numbers" of quarks are probed in various lepton-hadron processes through their weak-electromagnetic coupling, the properties of the gluon are much more elusive for experimental study. They do not couple directly to the lepton probes. It is the purpose of this paper to show that important in­formation on the spin of the gluon can be extract­ed from the angular distribution of lepton pairs produced at large transverse momentum in had­ron collisions.

Within the conventional perturbative QCD frame­work, the dominant mechanism for producing such lepton pairs (at least for uN and AW reac- .

tions) is quark-antiquark annihilation with the emission of a hard gluon.1*2 This mechanism is shown in Fig. 1. As the large transverse momen­tum of the pair is balanced mainly by that of the recoil gluon, it is apparent that the spin of the latter should have an effect on the angular distr i­bution of the detected leptons. It is then natural to ask whether this effect can be clearly identi­fied and subjected to direct experimental tests. We show that the answer to both questions is yes.

The angular distribution of lepton pairs pro­duced in hadron collisions is determined by struc ture functions which represent independent com­ponents of the virtual-photon polarization tensor (or current correlation function),

= sjd4z e {«* <J}J>2 \JJLz)Jv(Q) \pyp2). Wt \iv (1)

The momentum labels p19p2 refer to the initial-state hadrons, q to the virtual photon (hence the final-state lepton pair), and s is the total center-of-mass energy squared, s = -{pl+p2)

2. We shall use invariant structure functions defined by

W ^ , / ^ ) ^ ^ ^ ^ (2)

where g^v =guv -q^qv/q2 is the gauge-invariant projection tensor, and pafi=gliV pav/s

l/2 {a = 1, 2). The general formalism for analyzing the angular distribution has been developed by Lam and Tung,3 to whose work we refer the reader for details.

We are concerned only with large-transverse-momentum lepton-pair production in TJN or NN scatter­ing. Hence we shall focus our attention on the dominant mechanism1 '2 represented by Fig. 1. The cur­rent correlation function can be written4

W^(PuP2,q)=^ZQ? J ^ ? £ HUt- xJttz- x2)- x,2)

x [u1U1)«<(|a) + M,(|1)«J(|2)K , '(l1/>i, i2P2,q). (3)

Here the sum is over quark flavors. We use xx 2 = {q°±qz)/^ls and x± = qL/^s, where components of the virtual-photon momentum q refer to the initial hadron c m . frame. The parton (antiparton) distribution functions u^^) [u{(^a)] depend on the flavor label rand the longitudinal-momentum fraction £a, where 0 = 1, 2 labels the incoming hadrons. The hard-parton annihilation amplitude w^v is given by

<^yUii>i,U>2,tf) = 4 W ^ ( W (4)

Here T=X1X2~X±2, and as i s t n e effective strong-coupling constant. Substituting Eq. (4) in Eq. (3) and

comparing with Eq. (2), one can derive specific formulas for all the invariant structure functions. We

1382 © 1980 The American Physical Society

VOLUME 45, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 27 OCTOBER 1980

FIG. 1. Dominant mechanism for producing large-transverse-momentum lepton pairs in TCN and NN scattering: hard-gluon emission.

FIG. 2. Kinematics in the center-of-mass frame of the lepton pair.

shall concentrate, however, on two particularly significant relations among the structure func­tions :

2Wl

and

W9 = W+

(5)

(6)

Eq. (6) follows simply from the absence of the (p/p2

v+p2^p1v) term in Eq. (4). Equation (5) is

less obvious but true. It has been pointed out5 that the relation W^

= 2W19 Eq. (5), is intimately tied to the spin-f nature of the quark. One can show that changing the spin of the gluon in Fig. 1 does not affect this relation. Moreover, it is valid regardless of whether the quark lines coupled to the virtual pho­ton are on shell and collinear with the incoming hadrons (i.e., in the Drell-Yan limit) or are off i

^v(^Pl,^P2,cf) = 4:iras{g^(l- T/^Q2/2 -2r[p

in contrast to Eq. (4) for the vector-gluon case. It is not hard to verify that Eq. (7) also leads to

the relation W/ =2W19 Eq. (5), which is only sen­sitive to the quark spin. On the other hand, the presence of the {p^p2

v + p/Pxv) t e r m i n EQ- (?) clearly signals the breakdown of Eq. (6) if the gluon is not a vector particle as in QCD. The conclusion holds true for any parton distribution functions [wt(|) and u{(^)]. The magnitude of this breakdown, as well as other differences implied by Eqs. (4) and (7), will be studied numerically later on in this paper. Firs t , we would like to pose the important qualitative question: Can one devise a clear-cut experimental test of the QCD relation W2-W4 = 0, Eq. (6)?

The answer to the above question appears, at first sight, to be negative, since none of the co-

shell and noncollinear (as in Fig. 1 with high q ±). Therefore, checking the invariance of Eq. (6) over a wide range of q ± (in which the apparent lepton angular distribution is expected to vary substantially) should provide a critical test of the spin of the quark. (The classic Callan-Gross r e ­lation, by comparison, only holds for on-shell quark lines coupled to the virtual photon.)

The main point of the present paper is to gen­eralize the above idea to get a handle on the spin of the much more elusive gluon. (This is one of the few places where effects due to a single gluon can be subjected to detailed study.) For this pur­pose, we take the quark to be a conventional spin-| particle. To see the effect of the gluon spin, we contrast the QCD predictions of the last sec­tion with corresponding formulas resulting from emission of a hard scalar gluon. A straightfor­ward calculation yields, for the latter case,

rpiv^+h»hvk? + {h*p2v+h*i>iv)t1Q}/t£2>ci2, (7)

1 efficients in the angular distribution of the lep-tons in their c m . frame with the traditional choic­es of axes (Gottfried-Jackson, s-channel helicity, Collins-Soper, etc.) relate simply6 to the quantity of interest, W2-W4. However, this first r eac ­tion is misleading because the real physics is ob­scured by any arbitrary choice of axes.

Let us go back to the basic formula daozLllvWilV9

where W^v is defined by Eq. (1) and L^v is the familiar lepton tensor (square of the lepton-vir-tual-photon vertex) L»v= g^-q»qv/q2-k»kv/k2

(k is the relative momentum of the two leptons). Observe that L^v is the projection operator which, when contracted with any four-vector, projects the latter onto the plane perpendicular to both q^ and k^. In the lepton c m . frame, this plane is

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the two-plane perpendicular to the lepton momenta (see Fig. 2). Contracting Lliv with the general form of Wpv given by Eq. (2), we obtain

doa:2W1 + p12(W2-W3+W4)+p2

2(W2+W3+W4)+2pl-p2(W2-W4), (8)

where fix and />2 are the projections of the initial hadron momenta onto the two-plane just described (cf. Fig. 2).

It is important to notice that (i) only physical variables appear in Eq. (8) (i.e., it is independent of the choice of coordinate axes); (ii) the last term involving correlation between px andj>2

measures exactly the combination of structure functions (W2 - W4) that we are interested in. Hence we can make the simple statement: If QCD is correct, the lepton angular distribution should be described by Eq. (8) with the last term (proportional to px* $2) absent. On the other hand, if the gluon is not a vector particle (as exempli­fied by the scalar case), this term is in general of the same order of magnitude as the others (see below).

To go into a little detail, we note px2=x2

2sin2^/ 4T , fi2

2=x12sin262/4r, and p1' p2=x1x2sin61sin92

xcos^ 1 2 /4r where the angles Qx, 62, and <p12are defined in Fig. 2. The last term in Eq. (8) is the only one depending explicitly on the angle cp12. However, for fixed hadron variables (say s,M, #F>#X)> the three angles 919 62, and cp12 are not independent, since cosO = c o s ^ cos#2 - s i n ^ sin02

x cos<£>12 is fixed by them. This fact makes the ex­traction of the terms of interest somewhat more involved than it might at first appear. With high-statistics data, a fit to the angular distribution by Eq. (8) can yield the various combinations of structure functions, including W2 - W4. Alterna­tively, if available data are not sufficient to allow for the model-independent determination of these functions, one can use available information on the parton distributions to calculate the structure functions, compute various measurable angular coefficients for the vector- and scalar-gluon cas­es , and compare with experiment. Some results of this type are presented in the next section.

To get some feeling as to the order of magni­tude of the effects which we are concerned with, we calculated the structure functions and the an­gular coefficients for the case of nN scattering at 225 GeV/c. The parton distribution functions needed in these calculations are already deter­mined by existing experiments at the same ener­gy with use of data dominated by low-q± events. In Fig. 3 we present typical results obtained for the (normalized) QCD-violating angular coeffi­

cient

K=X1X2(W2-W4)/$TW/. (9)

Dependence of this parameter on the variables M, xF, and g± can be readily read off from the fig­ure. K vanishes identically in lowest-order QCD; it is of the order 0.1-0.4 if the gluon has spin 0. For this test case, K is larger for higher values of M, and for lower values of xF and q±.

In addition to the clear-cut K = 0 test , other a s ­pects of the lepton angular distribution can also reflect the influence of the spin of the recoil glu­on. As an example, we calculated the angular co­efficient a in the traditional (1+a eos20*) distr i­bution.7 We found, in the same kinematic region as indicated in Fig. 3, (i) that for the vector-gluon case,4 a >0 and it is a decreasing function of both variables q± and M; and (ii) that for the scalar-gluon case, 0>a > - 0.3 and it only varies slightly with q ± and M (over most regions of the phase space). Details of these calculations will be published elsewhere.

If perturbative QCD is relevant for l a rge- t rans-

K •

1.0

0.8

0.6

0.4 h

0.2

0.0

s = 4 5 0 GeV

SCALAR GLUON

VECTOR GLUON K-0

(FOR ALL M, q ± ) J I I L

q ± (GeV/c)

FIG. 3. The angular coefficient K as a function of q± for var ious values of M and xF if the gluon has spin 0. (QCD p red ic t s K = 0.)

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VOLUME 45, NUMBER 17 PHYSICAL REVIEW LETTERS 27 OCTOBER 1980

verse-momentum lepton-pair production, the an­gular distribution of the leptons can provide valu­able information on the spin of the gluon. This in­formation is not easily obtained elsewhere. (For instance, the tests of gluon spin by three-jet events \xie*e~ interaction are based on similar ideas, but suffer from severe ambiguities on the identity of the gluon jet, as well as limited stat is­tics.) The vanishing of the structure-function combination W2 - W4 seems to be the only known analytic result closely tied to the spin of the glu­on. It appears as one of the angular coefficients in the cross-section formula, Eq. (8), and hence can be related directly to measurement. Back­ground effects due to higher-order diagrams should be small if perturbative QCD makes sense, but clearly they need to be calculated. Ad hoc "intrinsic transverse momentum" of the partons should also introduce breaking of the relation W2

= W4, but that effect may be submerged by the higher-order corrections which are beginning to yield to precise treatment.8 These problems are being pursued vigorously.

We thank C. S. Lam for helpful discussions. This work was supported in part by the National Science Foundation under Grant No. PHY-78-11576.

(a^Also at Fermi National Laboratory, Batavia, 111. 60510.

lrThe literature on numerical calculation of these effects is very extensive. See, for instance, F. Halzen and D. Scott, Phys. Rev. D 18, 3378 (1978), and 19, 216 (1979).

2Although there is the alternative mechanism of gluon-quark "Compton scattering, " calculations (Ref. 1) show that it only matters in NN scattering.

3C. S. Lam and Wu-ki Tung, Phys. Rev. D _18, 2447 (1978). Earlier references can be found in this paper. A compact summary of the formalism as well as many parton-model and QCD results are given in C. S. Lam and Wu-ki Tung, in Proceedings of the Guangzhou Con­ference on Theoretical Particle Physics, 1980 (to be published).

4C. S. Lam and Wu-ki Tung, Phys. Lett. 80B, 228 (1979); J. C. Collins, Phys. Rev. Lett. 42, 291 (1979); R. Thews, Phys. Rev. Lett. 43, 987 (1979).

5Lam and Tung, Ref. 4; C. So Lam and Wu-ki Tung, Phys. Rev. D 27f 2712 (1980), and second paper of Ref. 3.

6The relations between Wt_4 and the various angular coefficients are systematically presented in the first paper of Lam and Tung, Ref. 3.

7K. J. Anderson^al . 9 Phys. Rev. Lett. 42, 944, 948, 951 (1979); J. Badier etal., Phys. Lett. 89B, 145 (1979).

8D. E. Soper, Nucl. Phys. B163, 83 (1980), and Uni­versity of Oregon Institute of Theoretical Science Re­ports No. OITS-128 and No. OITS-134, 1980 (to be published); J. C. Collins, to be published.

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