Angular distributions of two jets in pp and pp[over ] collisions

PHYSICAL RE VIE% 0

VOLUME

Angular distributions

28, %UMBER

of Physics,

1981

of two jets in pp and pp colbsions

K. O. Mikaelian, * K. Mita, Department

1 MAY

and M. A. Samuel

Oklahoma State University, Stillwater, Oklahoma (Received 19 December 1980j

74078

Assuming that quark, antiquark, and gluon jets can be identified in pp and pp collisions with two high-PT jets in the final state, we calculate the angular distribution for each pair of jets. Our purpose is to suggest experiments which come as close as possible to measuring the various parton-parton scattering cross sections ,d. Our predictions for the, angular distributions are based on quantum chromodynamics, but we also explore how our results would be

g,

modified if the three-gluon

and four-gluon couplings were absent.

I. INTRODUCTION

The impressive list of successful applications of quantum chromodynamics' (QCD) suggests that QCD is indeed the correct theory of strong interactions. Most of the remaining problems (confinement of quarks and gluons, prescription dependence of &zcz„etc. ) are theoretical in nature and a phenomenological approach to bypass these problems (temporarily) has allowed the successful comparison of recent experimental data with the predictions of QCD. One of the earlier applications of QCD was to high-P~ hadronic processes which, according to the parton model, were described by the hard scattering of constituent quarks and gluons and the subsequent decay of one of the scattered partons into a physical particle, e. g. , m, A, etc. , having a large momentum perpendicular to the initial colliding hadronic beams. While the distribution and fragmentation functions are taken f rom other experiments serniphenornenologically (in particular, deep-inelastic e-P scattering), the hard scattering is described by QCD. Calculated to lowest order in the running coupling constant &„all 2-2 processes involving quarks and gluons are given by Combridge, Kripfganz, and Ranft and by Cutler and Sivers. That recent experiments can be understood within such an approach is, of course, one of the successes of QCD. It is clear, however, that the theoretical prediction in question is a sum over a, very large number of subprocesses that can contribute, for example, to pp-7tX, and one can legitimately question how rigorous a test of QCD this is. Ideally one would like to measure the basic processes qq -qq, gq-gq, etc. , as predicted by QCD (in analogy with Moiler and Compton scattering in QED). But of course, it is impossible to prepare initial states of pure quarks or antiquarks or gluons: even protons carry a substantial amount of gluons in addition to the three quarks. However, we may fare better in

"

st~dying final states in that we may be able to distinguish among the three types of jets: quark, antiquark, or gluon jets. This optimism is based on the full-scale effort which has been underway for some time to study jets produced in e'e collisions, and we expect that the considerable experience gained from e'e physics will be applied to the separation and identification of jets produced in high-energy hadronic collisions. With this assumption we can come close to measuring the basic parton-parton scattering cross sections, though one still has to sum over the constituents of the initial hadrons. In our case, these will be protons or antiprotons. The assumption is that from a large sample of two high-P~

jet events,

p+p (or p)-jet+jet+X, one can select and classify final states by the nature of each jet:

(i) f)P (or P) -qqK

(ii) pp (or p)-qqX,

(iii) PP (or 0)-qqX (iv) PP (or (v)

(vi)

P)-qgX,

ff (»W-qg» PI (or f) —ggX.

The number of subprocesses contributing to each of the above reactions is greatly reduced by the selection procedure, bringing us closer to a, measurement of the basic parton-parton cross sections, which we denote collectively by o,~«, where a, b, c, and d stand for a quark, an antiquark, or a gluon. One would like to measure o, ~«essentially as a function of two parameters': c. m. energy vs and c. m. angle 8*. The dependence on Ps is reflected by the power n in d'o/d'P ~1/Pr" and has been studied extensively in ea, rlier work. ' Except for the relatively mild scale breaking in1960

1981 The American Physical Society

DISTRIBUTIONS OF T%0 JETS IN pp

ANGULAR

troduced by (2, (Q'), a(x 1/s for all parton-parton cross sections. In this paper we focus on the angular' dependence of two-jet cross sections which is different in each subprocess and which, in our opinion, reveals more about @CD. Following the notation of Ref. 4, we write do'g() (:g

s

dc()))

d cose*

2

dt

~

&&g (Q

)

AND pp

~

.

1961

~

100

~

The Z, ~,„are (dimensionless) functions of 8* only and are plotted in Figs. 1 and 2. In Sec. II we discuss the angular distribution of jets in reactions (i)-(vi). After summing over flavors and integrating over momenta, we find that the angular distributions in the laboratory (pp or pp center-of-mass) frame are similar, but not identical to », even after making energy cuts on each jet. In an attempt to measure Z, ~,» more directly we calculate in Sec. III several angular distributions in the parton-parton center-of-mass f rame. These distributions turn out to be highly distorted by experimental cuts on energies and angles. In Sec. IV we discuss three processes sensitive to the three-gluon and four-

Z„,

qs

qg

I

l

I

I

j

I

—. 4 —. 2 8 —. 6 —.

l

l

I

0

i

.2

I

j

.4

T

.6

.8

cos 8* gg

FIG. 2. The angular distributions & for the reactions gg, qg qg, and qq gg. The dashed curves are

without the three-gluon coupling in qg the four-gtuon coupling in gg gg.

and in

gluon couplings,

clusions

Sec.

qg and without

V we present our con-

. II. ANGULAR DISTRIBUTIONS IN THE LABORATORY FRAME

In this paper we take the laboratory. frame to be the center-of-mass frame of the colliding protonproton or proton-antiproton beams, having in mind ISABELLE, the CERN p accumulator, or the Fermilab pp project. For ea.ch pair of jets, specifically qq, qq, gq, andgg, we calculatethe angular distribution of one of the jets, do(/tB- cdX)/d cos 8, where g is the angle between c and &. g is the proton beam and 8 is a proton or antiproton beam. With the usual definition of x, (x, ) as the fraction of momentum carried by parton a (b) (B), the angular distribution is given by

$0

in'

do

-

0 cos8 (AB cdX)

I

I

l

I

l

I

—. 8 —. 4 —. 2 6 —.

I

I

I

0

I

.2

I

I

A

I

I

.6

P2

I

.8

(1 —P cos8)

cos |I*

'(2

2(q2)

2s

qq

FIG. 1. The angular distributions & for the reactions —qq, qq —qq, and gg qq. The subscripts 1 and 2 refer to flavor. The dashed curve is without the three-

As discussed in the Introduction, we have to sum over g and 5, but we assume that c and d can be

gluon coupling.

identified.

K.

I962

0.

K. MITA,

MIKAKLIAN,

M. A. SAMUEL

AND

In the above equation the P's are probability functions (we used the same parametrization as in Ref. 11), and P is defined as xtl

xg

Xt2

XQ

variables of the subprocess s, t, s=xzx&S, where S = (&g+&s)', and t = -(s/2)(1 —cos8*) As. indicated by our notation in Eq. (2), we neglect all scale-breaking effects in the probability functions, ' then all such effects are found only in the strong coupling "constant" o', (Q ), The Mandelstam and u satisfy

2)

12K

27 in(Q'/A')

=2stu/(s

CO

0V

pp-qgX

'U ,

10'

'

where we have assumed only three flavors u, d, and s. For Q we have taken the symmetric combination Q

10-'

s+t+u=0,

+t +u ),

and set A=0. 5 GeV. Except for the appearance of S in the logarithm of o.', (Q ), the cross section depends on S only through do/d cos8 ~ 1/S. For illustration we have chosen v S= 30 GeV, equivalent to a 450-GeV beam on a fixed target, and found that the 1/S behavior can be used to extend our results to other energies. 8* As mentioned earlier, ~ are functions of only, which is related to 6) via

10, —. —. 8 —. 6 —. 4 2 I

I

I

I

I

l

I

I

I

I

I

I

I

.4

.2

0

I

I

.8

.6

cos 8

FIG. 3. The angular distributions

do/d cos0 for pp qgX, pp ggX, and pp qqX in the center of mass of the two protons. The center-of-mass energy v S =30 GeV, and an energy cut of 6 GeV has been imposed on each final jet. The effect of dropping the gluon self-couplings is shown in Fig. 6.

—qqX,

pp

Z„,

cos8* = (cos8 —p)/(1 —P cos 8) . The limits of integration over x, and x~ cover the full range 0 to 1. However, we impose lower limits on the energies E, and E~ of the two jets, and perform the integrations subject to the conditions E, ~ 6 GeV and E„~ 6 GeV (for v S= 30 GeV). There are two reasons for doing this: first, we do not expect that low-energy jets can be identified as such; second, it ensures that the event was due to a hard-scattering process. Throughout the integration we monitor Q and check that it is large enough so that n, never exceeds 0. 35. The jet energies are given by

vs

E, =

+x,)(1 —P 4 (x,

)/(1-Pcos8)

(7a)

pp

.01— l

I

i

I

I

I

—(x, + x,)(1+ Ws

~

4

I

I

.6

I

.8

cos 0 p

—2p cos8)/(1 —p cos8)

.

(Vb)

FIG. 4. The angular distributions

—ggX,

—qqX,

The results of our calculations are given in 3 and 4 for pp and pp collisions, respec-

do/d cose for pp

—qgX in the

pp center-of-mass is relative to the proton direction. The dashed lines are without the gluon self-couplings. ~S =30 GeV and E~« —6 GeV. pp

frame. The angle

Figs.

I

I

—. 4 —. 2 6 —. 8 —.

and

E~ =

—qgX

and pp 0

ANGULAR

DISTRIBUTIONS OF T%0 JETS IN pp

In our probability functions we have included sea quarks and antiquarks, but found that their contribution was very small compared to valence quarks and antiquarks and gluons. For this reason we have not plotted the angular distributions for processes requiring at least one sea component, e. g. , pp —qqX or qqX, etc. %e have assumed that the flavor of the final q or q jet cannot be determined, and hence sum over three final flavors, I, d, and s. Production of heavier flavors, c, 5, . . . , would require a more detailed calculation including quark masses (see, e. g. , Ref. 12). Even after specifying the nature of the final two jets as we have done in this paper, we find that the summation indicated in Eg. (2) includes a large number of terms. Had we dropped the sea contributions, the number of terms would have For example, there are decreased substantially. seven terms in each reaction pp-ggX or pp -ggX (gg ggq Bs gg~ dd ggq ss ggy QB ggq dd g'g~ and ss-gg). Without the sea, pp-ggX has only one term and pp-ggX has three. Obviously, the most complicated reaction is pp -qqX which has 33 terms (11 without the sea). To get a feeling for the relative importance of each component (valence quarks, gluons, and the q-q sea) in the proton, we ran our programs with For and without one or two of these components. pP-qqX the contribution of the sea ranges from less than 8% in the backward hemisphere to less For gluons, than 1% in the forward hemisphere. the corresponding numbers are 8/0 and 2%. In the case of t)p-ggX, however, the sea contributes less than 0. 5% at all angles, while the gluons contribute the lion's share, about 76% at all angles. Note that in this case the small content of gluons relative to valence quarks inside a proton is compensated by the large gg-gg cross section see Fig. 2. Experimentally, the gluon contribution may be isolated by comparing pp-ggX with pp

tively.

AND

pp. . .

1963

This involves the triply differential cross section d o/dx, dx~dcos8*, for example, where x„x~, and 8~ may be determined by measuring E„E~, and 8 see Eqs. (6) and (7). In this section we briefly discuss do/d cos8*. Our original hope was that by integrating over x, and x, for fixed 8~ one could extract Z, ~,~(8*) more easily. We found, to our surprise, that these two quantities bear little resemblance to each other, and that the method fails essentially because of the singularities at small x, and x» and the need to make cuts on E, and E~. do/d cos8* is given by an expression identical to Eg. (2) but without the Jacobian factor (1 —P )/ (1 —P cos8} . Note that now we can pull out Z, ~,~ from under the integral signs. The integrals are done subject to the same cuts E, ~ 6 GeV and E~ ~ 6 GeV, plus a cut of 30' on 8, and 8~ in the forThe experiward and backward hemispheres. mental situation we have in mind is the following: in the laboratory, i. e. , pp or pp c. m. frame, two jets of energy larger than 6 GeV each are detected in an angular region between 30' and 150 with respect to each beam. The results are shown in Fig. 5. The central bump of each curve is certainly not due to Z, ~ « (compare with Figs. '1 and 2}, but is a product of the cuts and the peaking of the functions P(x) for small x. The details are discussed in the Appendix', suffice it here to say that only around 0*= 90' small values of both x, and xb are allowed, and is subthe value of the integral multiplying stantially increased by the inclusion of this small

—

Z„«

—

~ ggX.

CC)

III. ANGULAR DISTRIBUTIONS TWO-JET c.m. FRAME

CO

IN THE

Our aim was to find a way to measure Z, ~« for each a, 5, c, and d. We have already mentioned one difficulty: the necessity to sum over a and b inside each colliding particle. Another difficulty is that the quantity do/dcos8 treated in the previous section [see Eg. (2)] is still rather crude because it involves an average over each Z, ~,~— by integrating over x, and x~ for fixed 8 one effectively averages over 8* appearing in Z, ~«. Clearly one needs to measure all the variables x„x» and 8 to reconstruct Z, ~« from the data.

Q 'Q

.01 —. 4 —. 2 8 —. 6 —. I

I

I

I

I

I

I

I

0

cos t)*

FIG. 5. The angular distributions do./d cose* in the center of mass of the two final jets. The cuts are discussed in the text. v S =30 GeV.

K. O.

I964

M

IKAE LIA1V, K. MITA,

AÃ D M.

A. SAMUEL

and xb region in the phase space. The lesson to be learned is that the experimental cuts greatly distort the pure @CD cross sections, and therefore the angular distributions of the two jets in the laboratory frame, discussed in Sec. We have II, are more faithful representations. checked this sensitivity of do/dcos8 to „by calculating with different Z, ~,~, keeping the same probability functions. In fact, we can make use of this sensitivity to try and determine certain quantities, like the triple-gluon and the quartic-gluon couplings, which are difficult to measure in other experi-

x,

Z„,

These tests are discussed next.

ments.

IV. THE TRIPLE- AND QUARTIC-GLUON COUPLINGS

pv-qgx

In this section we analyze how some of the we have considered are modified by removing the triple- or quartic-gluon couplings. Of course, these self-couplings are a consequence of the non-Abelian nature of the theory, and there have been some suggestions on how to measure

processes

them.

"

.01

We concentrate on three reactions: gq-gq, gg-qq, and gg-gg. In the first two we drop the three-gluon coupling, and in the third we drop the

four-gluon

The results are as follows:

coupling.

", = -4(u

+ s')/9us + (u' + s

Z'",

+without 3-g Se-4'e

4(u2

Z',"",

+without2-&

(

rs-ee

z'"'„'

=

2

2

(sb)

—3(u'+ t')/Ss'

(9a)

+ t2)/6 t

(3 —ut/s

—us/t

(9b)

—st/u

),

I

I

I

I

I

I

0

~

8

FIG. 6. Comparison without the three-gluon

of angular distribution with and and four-gluon couplings in pp

(sa)

+ S2)/9uS

, = (u'+ t')/sut

'I

. cose

collisions.

)/t,

I

—. 8 —. 2 6 —. 4 —.

(loa)

The curves without the multigluon couplings are shown as dashed lines in Figs. 1 and 2 along with the full curves. Clearly gq -gq is most sensitive

to the three-gluon coupling. Proton-proton collisions are best suited to search for the three- and four-gluon couplings. For example, if we neglect sea antiquarks in the proton, then pp-qqXis due only to gg-qq, and PP -ggX comes only from gg-gg. As to testing

gq-gq, either pp-gqX, or pp-gqX will do. Fig. 6 we compare several of these processes

In

with and without the triple- or quartic-gluon couplings. The absence of the forward peak when we drop the three-gluon coupling in pp -gqX is particularly striking, an observation also made re-

The peak goes away because it cently by Reya. came from the t-channel gluon-exchange diagram involving the three-gluon coupling. It is interesting to note the sign of the contribution coming from the three-gluon coupling: it is positive in gq -gq and negative in gg- qq. The four-gluon coupling also is negative in gg-gg. Consequently, both pp-qqX and pp ggX increase, while pp -gqX decreases when we drop the multigluon couplings. V. REMARKS AND CONCLUSIONS

If our basic postulate that the nature of high-P~ jets can be identified turns out to be correct (and there is some positive evidence "), then the ex-

periments we have suggested come as close as possible to measuring the basic parton-parton cross sections. A p-p colliding machine like ISABELLE is particularly suited for such experiments, because we have found the antiquark (sea) content to be rather unimportant, and, consequently, the proton can be regarded as almost a pure state of (valence) quarks and gluons. Practically all parton-parton cross sections can be measured, in particular qq -qq, gq-gq,

"

ANGULAR

2-

1.0

DISTRIBUTIONS OF

1+ ~cose*~ ~

1

TWO

JETS

IN pp AND

pp. . .

1965

simply adds all the contributions given in Figs. The individual components 3 and 4, respectively. on which we have concentrated in this paper, however, reveal substantially more about @CD and we hope that future experimental efforts will dis- . tinguish among jets and try to measure each distribution separately.

—~cose*~

25

1-cose* X.

ACKNOW( LEDGMENTS

1+ jcose*i

c

2-

25

~

1

—Icose*l

1+ cose* 26

26

1.0

We wish to thank R. W. Brown for a helpful discussion on this subject. This work was supported by the U. S. Department of Energy under Contract No. EY-76-S-05-5074.

1-cose*

1+ cose*

APPENDIX

X,

FIG. 7. The phase-space cuts on x, and

xb

as discus-

sed in the Appendix. The energy and angular cuts remove the shaded area from the region of integration over x~ and xb.

The angular distribution in the center-of-mass frame of the two jets is given by 8~ d cos0*

- cd&) =g —Z„„(8*) (AB

1T

gg-gg,

and

gg-qq,

by choosing the proper final

state in pp -2 jets+X. Of course, all our calculations are based on @CD. However, we discussed in Sec. IV how certain cross sections are changed if we modify the theory, specifically if we drop the tripleFrom gluon and the quartic-gluon couplings. Figs. 1, 2, 4, and 6 we conclude that pp-gqX and pp -ggX are good candidates for such tests while pp -qqX is only slightly affected by dropping the three- gluon coupling. We chose a specific c. m. energy (v S = 30 GeV) and specific energy cuts on the two jets (E„,~ 6 GeV) in our numerical work. One can extend our results to other energies by using the fact that do/d cos8 almost scales like 1/S; the scaling would be exact were it not for some mild logarithmic dependence of n, (Q ) on S. It is very important, however, to remember to scale the enFor example, all our ergy cuts proportionally. curves need simply be multiplied by an almost constant factor of 1.44 to go over to v S= 25 GeV, provided the cuts are also changed to 5 GeV. We explain in the Appendix why our results for the two-jet c. m. distributions are so sensitive to the cuts. We point out here that the bumps in Fig. 5 would disappear were we to directly impose constant cuts on x, and xb. Although one can impose such cuts, we find that the remaining data include events of too low energy and/or too close to the colliding beams, unless the cuts on x, and x, are severe. We have tried to remain faithful to experimental cuts which, of course, involve energies and angles. To find the inclusive pp -2 jets+X rates one

a, b

X

A1

Xb

where 0* is the angle between c and a in that frame. Since o'. ,(a ) is a rather mild (logarithmic) of Q, one would expect do/d cos8* to be approximately proportional to Z, ~,~(8*). The results, however, are quite different when we compare Fig. 5 with Figs. 1 and 2, and here we explain that difference which essentially has to do with the cuts on x, and xb. These cuts depend nontrivially on 8*. In the laboratory frame, i. e. , c. m. of A. and B, we require that each detected jet have a minimum energy E, ~ 5v S/2 and angle such that cos8, In our numerical examples 5 =0. 4 and

,

&

,

~

~

=0. 13. Since

vS

[x + xg+ (x

xp)

cos8

],

in the x, -x„plane only the region above the lines given by 25 —x, (l + cos8*) 1 + cos~

(A3)

is allowed (see Fig. I). The angles 8, , satisfy cos8 cy

tg

P+ cos0~ 1 p P cosg4

and, therefore,

lines given by

(A4)

the allowed region is above the

K.

0.

1+I cos8~ In

Fig.

7 we show

space are forbidden From Eqs. (A3) and region, unshaded in 8* moves away from

M

IKAK I. IA 5, K. MITA,

I

cosg what regions of the phase because of the above cuts. (A5), we see that the allowed Fig. 6, becomes smaller as 90'. In other words, max-

*Present address: Lawrence Livermore National Laboratory, Livermore, California. C. H. Llewellyn Smith, in High Energy Physics 1980,

—

proceedings of the XX International Conference, Madison, Wisconsin, edited by L. Durand and L. G. Pondrom (AIP, New York, 1981); E. Beya, DESYBeport No. 79/88, DO- TH 79/20 (unpublished). W. Celmaster and D. Sivers, Report No. ANL-HEP-PR80-28, 1980 (unpublished). A. Buras, Rev. Mod. Phys. 52, 199 (1980); B. P. Feynman, B. D. Field, and G. C. Fox, Phys. Bev. D 18, 3320 {1978). B. L. Combridge, J. Kripfganz, and J. Banft, Phys Lett. 70B, 234 {1977). B. Cutler and D. Sivers, Phys. Rev. D 17, 196 (1978). 6See, e.g. , C. Bromberg et a/. , Phys. Bev. Lett. 45, 769 (1980) and references therein. Unless glueballs exist and one can make glueball beams. H. P. Nilles and K. H. Streng, Phys. Bev. D 23, 1944 (1981); E. G. Floratos et al. , Phys. Lett. 90B, 297 (1980); S. J. Brodsky and F. Gunion, Phys. Bev.

J.

~

J.

A Ã D

M. A. SAM

V

I I.

imum phase space is allowed when 8*= 90', and, therefore, the bumps in Fig. 5 are interpreted as due to the increase in the coefficient of Z, ~,~ in Eq. (Al). Note, in particular, that the integrand I',"(x,)I', (x,) n, '(Q')/x, x, diverges as x, -0 and/or x, -0, but the small x, and x~ region is eliminated by these cuts.

Lett. 37, 402 (1976).

J. Ellis

and I. Karliner, Nuel. Phys. B148, 141 {1979); K. Koller, T. F. Walsh, and P. M. Zerwas, Phys. Lett. 82B, 263 (1979). ' The other variables controlling 0,&. & are the spin polarizations of the partons which are very hard to measure. In this paper we average/sum over all helicities. "B.W. Brown, D. Sahdev, and K. O. Mikaelian, Phys. Rev. Lett. 43, 1069 (1979). J. Babcock, D. Sivers, and S. Wolfram, Phys. Rev. D 18, 162 (1978). E. Beya, in High Energy Physics 1980 (Bef. 1); R. W. Brown, E. M. Haacke, and J. D. Stroughair, Phys. Bev. Lett. 45, 1060 (1980); K. Fabricius et al. , ibid. 45, 867 (1980). E. Beya, Dortmund Report No. DO-80-0765, 1980 (unpublished); M. Gluck and E. Reya, Dortmund Report No. DO-TH 80/11, 1980 (unpublished). ' J. P. Berge et al. Report No. FEBMILAB-PUB-80/62, EXP, 1980 (unpublished). 6We have used P&(x) =3(1 —x) /x.

—