QCD and final-state jet measures in leptoproduction

Nuclear Physics B156 (1979) 43-52 © North-Holland Publishing Company

QCD AND FINAL-STATE JET MEASURES IN LEPTOPRODUCTION P.M. STEVENSON Physics Department, Blackett Laboratory, Imperial College, London SW7 2BZ, UK Received 17 April 1979

Various infrared-safe measures of jet-like-hess in leptoproduction are studied in QCD perturbation theory using a general formalism. This quantifies the effects predicted in a previous paper. The nucleon remnants jet is found to be distinctly narrower than the quark jet.

1. Introduction In many ways this is an addendum to a previous paper on this subject [1]. There I studied the 2-jet fraction in leptoproduction, defined by analogy with Sterman and Weinberg [2], but argued that the qualitative features of the results would carry over to other, experimentally more convenient, measures of jet-like-ness. Here I present (in sect. 2) a method for calculating the first-order QCD value of a general infrared-finite jet-measure in leptoproduction: this reduces the computation involved to the evaluation of one-dimensional integrals over parton distribution functions. This formalism is applied (in sect. 3) to give quantitative results for various jet-measures related to spherocity [3] and thrust [4]. Such variables have of course been studied before in the context of leptoproduction [5, 6], but the present approach is somewhat different. The usual procedure is to study the distributions de/dS, d~r/dT etc., to look for the "perturbative tail" (at large S, low T) where the QCD result emerges from beneath an estimate of the non-perturbative, fragmentation effects. Refs. [5, 6] thus follow the example of De Rfijula et al. [7], in the e*e- case. This is a perfectly valid approach, but here I shall follow the earlier example of refs. [3, 4] and study the average values of these quantities, as a way of quantifying the jet-like-hess of events. This might at first seem to be a retrograde step but it does have the advantages of directness and simplicity. Moreover, it is interesting to consider the averaged quantities as functions of the three independent variables needed to specify the "initial state" (in the sense of leptoproduction as a virtual-boson nucleon collision). 43

44

P.M. Stevenson / Leptoproduction

I shall work in the hadronic c.m. frame and use IV, XBi and y as the three independent initial-state variables. 0 2 is regarded as a dependent variable given by Q2

XBi

W2

(1)

(1 -xm) (neglecting the nucleon mass). Although 0 2 plays a major role in the dynamics it is not so convenient for describing the kinematics, The importance of W is becoming increasingly recognized: in the c.m. frame its significance is natural and obvious. For instance in the parton model the partonic "final state" (before fragmentation into hadrons) has its kinematics fixed uniquely by IV. As argued in [1] this implies that the non-perturbative fragmentation jets have widths which are independent of x m and y at fixed W. This suggests that by studying the variation with XBi and y of various measures of jet-like-ness one can test the QCD predictions without having to estimate the background from fragmentation effects. This is the view adopted here. Unfortunately the effects in question are small, so there is no possibility of a dramatic verification of QCD. However, if it were clearly established that jetqike-ness does vary with XBi and/or y (at fixed W), it would indicate that the 2-stage parton-model picture (W +- + N ~ quark + nucleon remnants ~ quark jet + nucleon remnants jet) is not wholly tenable. Furthermore if the xBj and y variations were in the right direction and of the expected size it would be useful encouragement for QCD. I regard the y-dependence as particularly important since its direction is predicted by a very simple argument [1], from fundamental considerations.

2. General formalism The effects discussed above could be looked for with various jet-measures. Fortunately the QCD calculation of a general infrared-safe jet-measure, U, can be expressed in a general form. The result, to O(~s(O2)) in QCD, for the average value of U as a function of W, XBi, Y, is

(U)= 4 c~,(02)[L(y)CL + R(y)cR + S(Y)cs] 3 ~ [ ~ ~ ] '

(2)

where CL = qL[F] + qR[F] + gT[G], Ca = qk[f] + qR[F] + gT[G], CS = qs[F] + qs[F] + gs[G].

(3)

P.MI Stet,enson / Leptoproduction

45

The notation is adapted from [1]. The factors L(y), R(y), S(y) represent the proportions of left, right and scalar polarizations of the virtual boson:

L:R:S= 1 :(1-y)2:2(1-y),

v scattering,

(l-y)2:l :2(l-y),

~ scattering,

~(I + ( 1 - y)2):~(l + (1 - 1')2): 2(1 - y),

e(#) scattering.

The functionals qL[F], etc., are generalizations of the corresponding quantities in [1], and are now given by drt

qs[F] = XBi

I

1

~F(r/)J, 7/

rl ~ , g.r[G] =3 f d~G(rl)i(rl-XBi)-+xBi)(J'-J), rt s i f d--5 r l G(r/)(rl - XBi)J, gs[O] = ~XB

(4)

where the integrations are for xm < rt < 1. F(rt), ~'(r/) and G(~t) are the quark, antiquark and gluon distribution functions (the first two being averaged over active flavours, weighted by the squares of weak or electromagnetic charges as appropriate). The above result depends on the specific choice of jet-measure only through the quantities J and J' (see table 1). This simple generalization of the calculation in [1] comes about by regarding that calculation as computing the cross section from the O(oL~) diagrams, but weighted by a "slab function" which is 1 in the 3-jet region and 0 in the 2-jet region. To calculate the average value of spherocity, thrust, etc., the only change is to replace this slab function by the spherocity function, thrust function or whatever. The first step is therefore to express the required jet-measure (defined in terms of the final-state kinematics) as a weight function ~t'(v, z), using the relations

l+cosOl=2z/(z+v(1 [kll=~W(z+v(1

z)),

z)), (5)

and the same expressions, with z--,(1 - z ) , for cos 02 and Ik21. (kj, k2 are the 3-momenta of the final-state partons, and 01, 02 are the angles between them and the current direction.)

46

P.M. Stevenson / Leptoproduction

Table 1 The quantities

1-T

Z J' (see eqs. (4), (7), (8)) for various jet measures

(L Ip;l'~

1-max \ ~ ]

J=

v(l + v-2v ~) (l+v)2

J'=ll-v)Iog(l S

/4~ 2 . [2ilp~l~ 2 ~g) m m ~ , )

,

+v)

v log t,

64 [(1--9V--38V2--58V3--63V4--25V s) 6(1+ v)3(l + v2 )

"=Tv[

J'

_2(1+; ),og((l+c2)]] (1',(1 + v ) / J 32 v(1 v ) ( l + 2 v + v 2 - 2 v 3) 2 rr (1 +v)(1 + v 2)

1 r

1 _ ( 2 , p'[~

J=

Y

( 4 5 2 ( Y p~rl) 2 \TrJ \V, lp'l]

32 J = ~ 3~r" v

(4)2(

64 v , 3v2_v3 ) J - 3 7 r 2 ( l + v ) ~(2+6v

£q

Z~

\E, lp']/

(4)2(

'~'

2

IV) 2 [,o4-]/

2'V']p}I/

IV) 2

v (t+v)

J'=(l+v)log(l+v)-vlogv 32 J ' = - -7r2r'

64 v3 v) J - 3 r r 2 ( l + v ) 3(3+

J'

j,=~2v(2-v/ (l+v)

64 v 2 -rr 2 (l+c)

Except for S and T, the jet axis is taken to be the virtual boson direction. The sum V i runs over all the particles in the final state (but for EQ and Er the sum is restricted to the appropriate hemisphere).

2 i 2 For example, for the quantity £ = (4/rr) (5~, Ipx]/W) (where the transverse momenta, p~, are defined with respect to the current direction) one finds ~/'~(v, z)= (64/rrZ)vz(1- z). Usually, however, ~V(v, z) has three distinct pieces according to whether z < v/(1 + v), v/(1 + v) < z < 1/(1 + v), or z > 1/(1 + v) (i.e., kl in the remnants half; kl, kz both in the quark-jet half; k2 in the remnants half). It is necessary, but not sufficient, that ~/'(v, z)--, 0 as z ~ 0, z + 1, and v -+ 0 in order for the result to be infrared safe. Following the analogy with the calculation in [1] the quantities qp, gp (P = L, R, S) are now given by 1

qp[F]=½

1

drlF(rl)f

f

dz[MPI 2°H/'(Gz),

rl

XBj

11

l

1

g p [ G ] = , 3 f drlG(~/)f XBi

(}

dzIMf,]2~'(v,z)

(6)

P.M. Stevenson / Leptoproduction

47

where IMpI2, IMP, 12 are the matrix elements (tabulated in [1]) and v = t,(r/)--- ~ -- xBj--2.

(7)

--XBj)

Because of the symmetry of ~(L,, z) under z ~ (1 - z ) , arising from the invariance of the jet-measure under kl ~ k 2 , it is possible to evaluate the z integrals in (6) in terms of 1

J=J(n)=-f

~/'(V, 2) dz,

0 1

J'=J'(n)=-f 7C(v,z)dz-,.,

(8)

0

leading to eqs. (4) above. (J and J ' for various jet-measures, are given in table 1.) The final step is to evaluate the integrals of eqs. (4) using one's favourite distribution functions, and substitute the results into eqs. (2), (3). Some results obtained by this procedure are reported in sect. 3.

3. Numerical results for various jet-measures In leptoproduction it is most natural to use the virtual-boson direction as the jet axis. Unfortunately it seems that this axis is not easy to determine accurately in present experiments. Consequently, I will show only the results for spherocity (S) and thrust (T), defined as in ref. [6], where the jet axis is fitted so as to maximize T (see figs. 1 and 2.) The analogous quantities Y_,and r, defined with the boson direction as the jet axis [5]), have also been investigated. These show much the same behaviour. The main features of the results are fairly insensitive to changes in the distribution functions, particularly the gluon distribution. An exception to this rule is the detailed x m dependence (i.e., the actual of the curves in figs. 1, 2) which, not unnaturally, depends on the precise shape of the distributions. The 0 2 dependent distributions used here are taken from ref. [8], sect. IIA. I assume 4 active flavours and take A to be 0.5 GeV. The perturbative results are not, of course, valid near xaj = 0, because O 2~ XBiW 2 must be large. The scale of the variations of S and T with xBi is unfortunately very small, particularly at large W where the results are most reliable. One reason for this seems to be the Q2 dependence of the distribution functions: computations using 0 2 independent functions show a more distinct fall of S and (1 - T) with xB~. Using 0 2 dependent functions lowers S and (1 - T) considerably and flattens out the

(cf.,

shape

P.M. Stevenson

48

Leptoproduction

01

S .05

.- ~ I

100

L

h

l

l

,

l

i

~

0.5

Xaj

Fig. 1. T h e a v e r a g e s p h e r o c i t y as a function of x~3j for W - 5, 10, 30, l l t 0 G e V , in t,-isoscalar interactions. T h e d a s h e d and solid curves are for y - 0 and y - 1 respectively. T h e m a j o r

n o n - p e r t u r b a t i v e effect is to raise these curves by an a m o u n t i n d e p e n d e n t of x m and y. T h e results for e(tx) scattering are very similar.

O"

1-T

.0~

J

0

i

h

i

I

0"5

i

i

i

XBj

Fig. 2. A s fig. 1, but for thrust.

curves. Thus it seems that the QCD effects conspire to mimic the naive partonmodel expectation remarkably well, which is somewhat disappointing. The y-dependence, although not large, is consistently in one direction in e,/x and u scattering. Thus even if the experimental statistics are not sufficient to detect a significant y dependence in any particular W, x m bin, the results could be combined to enhance the visibility of the effect. (See appendix A). In figs. 1, 2 I have shown only the curves for y = 0 and y = 1 ; the interpolation (at fixed W and xm) is approximately linear in y. The y-dependence in ~ scattering is very interesting since at large y and reasonably large xm (-0.5) events are significantly less

P.M. Stet~enson / Leptoproduction

49

0.1 \x

S .05

0 0

05

X~

Fig. 3. As fig. 1, but for :-isoscalar interactions. The dashed and solid curves are for y= 0 and y= 0.9 respectively. 2-jet-like than normal (see fig. 3). The reason is that the parton-model cross section is very small in this region, so the less 2-jet-like O(a~(O2)) effects are relatively more important. Another application of the present method is to study the differences between the quark jet and the nucleon remnants jet. This can be done as follows: divide the final state of each event into two halves, in the c.m. frame. Evaluate ~2~]P!r[ in each half separately, and form '~q,r = (4/7r)2(2 }~, Jp!r [ / w ) ~ for each half (with the boson direction as the jet axis). (These are essentially the spherocities of the "fake" events formed by doubling each half (@, [1]). Notice the factor of 2 to allow for this.) Remarkably, the theoretical result from first-order QCD is that the quark jet is very much wider than the nucleon remnants jet, in the sense that Yq> !0Er. (But, of course, if one constructed thrust-like variables rq, rr, these would be identically the same!) This difference between the two jets arises partly because of the 3-body nature of the final state in first-order QCD. If the gluon is radiated into the remnants half then the balancing px appears in the other half; whereas if the gluon is radiated into the quark half then the balancing pT also appears in this half. Thus the sum of IpTl's in the quark half is much larger on average than that in the remnants half, then the balancing p r appears in the other half; whereas if the gluon is radiated into the quark half, then the balancing p r also appears in this half. Thus (It is interesting to note that the same qualitative difference between the two types of jet has been found by Konishi, Ukawa and Veneziano [9] using leading logarithm techniques. There the difference is due to the strong ordering of the transverse momenta and its relation to the energy ordering in the ladder diagrams. It is important to recognize that the leading log calculations and the present infrared-safe perturbation theory approach are addressing rather different questions: the former is concerned with the internal structure of the jets, the latter with

P.M. Stevenson / Leptoproduction

50

x

\

0.2

d 0.1

"9, / 9 5 - . - - - ~ a00

0

. . . .

[ l i l t

0

O5

x~

Fig. 4. The difference, d = (Y-;q- Z,.), calculated ignoring hadronization, as a function of W, XBi, y. (Conventions as in fig. 1.1

the global energy/momentum structure of the hadronic final state. Nevertheless it is encouraging that they agree on common qualitative features.) The perturbative result for d ~ (Zq- Z~) is shown in fig. 4. It would be unrealistic, however, to expect such a large effect at present energies, since the nonperturbative effects seem to be dominant. As a very rough guide to the experimental expectation one might consider the following procedure (somewhat in the spirit of refs. [7, 6]). Assume that the pT spread due to non-perturbative effects is roughly the same for each type of jet so that, in the absence of gluon bremsstrahlung, ZNe q = -,gNe r = ZNe. Divide the first-order QCD partonic final states into "2-jet" and "3-jet" configurations according to whether their X, value is less than, or greater than, yNe. For the 2-jet configurations the observed ~xobs yobs will both be about q ~m r equal to ,~NP: for the 3-jet configurations the observed _qV °bS, mrS'°bswill be roughly the perturbative values Xq, Yr. Hence A°bs = ~'°bs--

X robs ~ ' ~ f-r ( ~ ' q -- ~ ' r ) ,

l.e.,a. o b s = f 7) d , •

(9)

where f' is the fraction of events with X > Z NP, as calculated in perturbative QCD. This fraction is easily calculated by applying the formalism of sect. 2: the required weight function ~P(v, z) is once again a slab function which is 1 in the 3-jet region ,Z>X ~e, i.e., when 4 v z ( 1 - z ) > C, where C = Y,Ye(¼cr)2. This immediately gives Jp=(l-C/v)l/20(v-C), Jr'

(1+(1 - C / v ) 1/z' , _ =log \ 1 _ ( 1 C/c : )l/2)O(t,-C

.~,

l.

(10)

P.M. Stevenson / Leptoproduction

51

Given that one has an estimate of ,~NP, and hence C, f ' can now be computed from eqs. (2)-(4). At low energies this fraction is small so the observable difference between the two jets, d TM, is tiny compared to the purely perturbative result, d. But as the energy increases the non-perturbative effects die out and the observable difference becomes a substantial fraction of the naive prediction. To illustrate this I have shown in fig. 5 an estimate of d °bs, = f'd, made by using E Np = (S)Np as given by the second paper of ref. [6]. (This figure is for illustrative purposes only. It should not be taken too seriously because it depends strongly on the guessed value of the nonperturbative contribution.) Note that, since yNP falls rapidly with W, f ' - - t h e fraction of events in which gluon bremsstrahlung effects determine the PT spread-grows from - 0 . 0 4 at W = 5 GeV to - 0 . 5 at W = 100 GeV. This rise completely outweighs the fall of d with energy (fig. 4) and results in the difference between the two jets becoming increasingly clear at higher energies (fig. 5).

4. Conclusions

I have described a general method for obtaining the first-order QCD results for various infrared-safe jet-measures. At present energies the values of S, (1 - T), etc., obtained in this way are likely to be considerable underestimates, because of nonperturbative effects. However I have stressed that the predicted y-dependence, and possibly the xm dependence, (at fixed W) could be observable in current experiments. Also there is a clear qualitative prediction that the quark jet and the nucleon remnants jet have different PT spreads; an effect that becomes clearer at higher energies.

- - - - - _ 0.04

100 - -

d 002

t

o

i

I

0.5

i

i

t

t

Xa j

Fig. 5. An estimate of the observable difference, d °bs =(Eqobs - E r obs), taking into account the estimated non-perturbative effects. (Conventions as in fig. 1.)

P.M. Steeens'on / Leptoproduction

52

It is a pleasure to thank Dr. H.F. Jones for many discussions and for much helpful advice. I am also grateful to Dr. K. Konishi for several interesting discussions. I would also like to thank the S.R.C. for financial support.

Appendix The purpose of this appendix is to suggest a possible procedure (which experimentalists are invited to improve upon) for observing the ),-dependence predicted by QCD in e, ~ or ~, scattering. Imagine that the events have been binned in terms of I~: xBi and y, and that the value of some jet-measure, U (= S, (1 - T), etc.), has been measured for each event~ For each W, x~i bin find the best-fit straight line through the plot of U against y. Use this to evaluate, by extrapolation, AU(W, XBi):-(U(y=O)

U ( y = 1))[ w.~,~,.

This is expected to be zero (up to statistical fluctuations) in the naive parton picture, but should be positive according to QCD (see figs. 1, 2). The significance of these results is likely to be small because of poor statistics in each bin. However one could examine the sum over several, or all, bins, i.e., form A U , ot =

v

(AU(W,

XB0),

~V, v ~,.i b i n s

to improve on this. The OCD expectation for this quantity can be inferred from the figures presented here (but will depend on the data sample and the bin widths, etc.). An idea of the statistical significance of a positive result could be obtained by repeating the same procedure after randomly permuting the U values of the events in each W, XBi bin.

References [1] [2] [3] [4] [5] [6] [7] [8] [9]

P.M. Stevenson, Nucl. Phys. B150 (19791 357. G. Sterman and S. Weinberg, Phys. Rev. kett. 39 (19771 1436. H. Georgi and M. Machacek, Phys. Rev. Lett. 39 (19771 1237. E. Farhi, Phys. Rev. Lett. 39 (19771 1587. H. Georgi and J. Sheiman, Harvard preprint HUTP-78/A034 (19781: A. M6ndez and T. Weiler, Phys. Lett. 838 (19791 221. J. Ranft and G. Ranft, Phys. Lett. 82B (19791 129; P. Bin6truy and G. Girardi, Nucl. Phys, B155 (19791 150. A. De Rfijula, J. Ellis, E.G. Floratos and M.K. Gaillard, Nucl. Phys. B138 (19781 387. J.F. Owens and E. Reya, Phys. Rev. D17 (1978) 3003. K. Konishi, A. Ukawa and G. Veneziano, Rutherford Lab. preprint RL-79-026, Nucl. Phys. B, lo be published.