On the large QCD corrections to the multijet diffential cross sections in e+e− annihilation

Nuclear Physics B188 (1981) 1 0 9 - 1 1 7 © North-Holland Publishing C o m p a n y

ON THE LARGE QCD CORRECTIONS TO THE MULTIJET DIFFERENTIAL

C R O S S S E C T I O N S I N e+e - A N N I H I L A T I O N D.A. ROSS

Department of Physics, University of Southampton, Southampton S09 5NH, UK Received 23 March 1981 All the large s u m m a b l e corrections to the differential cross section with respect to a particular quantity, C, which is a m e a s u r e of "multijetness", for e÷e - annihilation are extracted and it is suggested how they m a y be s u m m e d to all orders in perturbation theory. The remaining corrections are small. The theoretical predictions are compared with s o m e of the existing data. T h e double distributions in the quantities C and D, which is a m e a s u r e of acoplanarity, are also discussed.

In a recent p a p e r [1] by the present author, Ellis, and Terrano we presented the differential cross sections dtr/dC and do,/dD for the process e + e - ~ a n y t h i n g , calculated up to second order in the Q C D strong coupling constant, as. The quantities ½C and 27~D are defined to be the sum of the principal minors and the determinant respectively* of the matrix i i

o"•" ~Papa/lPal =

°

,

(1)

where the sum runs over all the particles in the final state. C is a measure of multijetness; it has the value zero for two-jet events, 0 < C <¼ for planar events and 0 < C < 1 for acoplanar events. D is a measure of acoplanarity, being zero for all planar events. T h e quantity do'/dD(D # 0), calculated in Q C D neglecting all non-perturbative effects due to the hadronization of the final-state partons, begins in order a~ since at least four particles are needed in the final state to obtain an acoplanar event, whereas the quantity do'/dC(C # 0) begins in order as, so for the latter quantity we have the first subleading corrections, which is required if a comparison of the theoretical predictions with the data is to be used to determine as. Unfortunately the corrections are very large even for the small values of &s(s) expected at P E T R A energies. This can be seen from fig. 1 in which we have plotted the quantity (C/o') der/dC against C at s = 900 G e V 2 and for A ~ = 0.3 GeV. The solid line is the histogram obtained by using a M o n t e Carlo m e t h o d to integrate the order a~ differential cross section over all phase space in a particular bin for C and the dotted line is the differential cross section to order as. • C and D are trivially related to the W o l f r a m - F o x [2] parameters/-/2 and ~r3. 109

110

D.A. Ross / Multijet differential cross sections

0"6

0"5

~

~ =30 GeM AM---~.=.3GeV ~''.- L.O.

.

0-4

0"3

%~.%%

0"2

0.1

1__ C Fig. 1. Plot of (C/o,) d ~ / d C against C to order a 2 (solid line) and to order as (broken line).

In ref. [1] it was suggested that this large correction may be due to the fact that the o n e - l o o p diagrams with three final-state particles contained a double pole term

where T is the matrix element for the lower order process in n = 4 - 2e dimensions, and that this double pole is cancelled by a double pole term in the differential cross section with four final-state particles,

2 (s)~T,

(3)

and that the difference 0ts'W

(Cv+ ½CA)T

T

(4)

accounted for nearly all of the higher order corrections. Since it is believed that the leading infrared divergences exponentiate in Q C D as in Q E D , these large

D.A. Ross / Multi/et differential cross sections

111

corrections m a y be summable to all orders, leaving a converging perturbation expansion. However, the a b o v e - m e n t i o n e d terms are not the only source of large corrections and not the only corrections that can be analytically subtracted and s u m m e d to all orders. The fact that the total order a 2 correction was so well represented by eq. (4) is a coincidence of Q C D with five flavours. That this is so m a y be seen from fig. 2 in which we have plotted on separate graphs the coefficients of C 2 (as/2~r) 2, CFCA(Ots/2~) 2, and CFTR(cts/2~) 2 (solid histograms) and the broken lines are the corresponding coefficients of eq. (4). W e still have a large positive correction for the CvCA(Ots/2~r) 2 coefficient and large negative corrections for the other two. This situation is greatly improved by the inclusion of two further summable large corrections: (i) The higher order corrections to the three particle cross sections contains a term

e, (11CA-- 4TR) In (slas23)r, 6

-2~r

80

(5)

1 ootq

"'-...

40

120

"CFCA(~)

0 80 "...

.opI

- 8 0 ~ "2

"4

.............

400~'"'".

C

"6

"8

.2

20~

-4

C

-6

'8

Fig. 2

0 ~.8 I

I

I

I

I

I

I r ~

-20 -40

I

-60 #

r~s2

XCFTa(2"-~)

-80 I

I

•2

I

I

.4

I

C

I

-6

I

I

I

-8

Fig. 2. Comparison of the coefficientsof C 2, C F C A and C F T R for (C/~r) dtr/dC with Cv + ~CA 1 ~a~r 1 lowest order cross section.

x

112

D.A. Ross / Multijet differential cross sections

where sij = (Pi + pj)2, pi and Pj are the m o m e n t a of particles i and ]. This is indicative of the fact that whereas the total energy, s, is the correct argument of ffs for the total cross section, for three jets the correct argument should be s13s23/s. N o w over most of the range of C, in the case of three final-state particles C-6

S13S23

s

2

,

(6)

so the correct argument of c~s should be ~sC instead of s, and this enables us to sum to all orders the correction term as ( l l C A - - 4 T R ) In 27r 6

(lC)T.

(7)

(ii) It has been shown [3] that the thrust (t) distribution as t-> 1 behaves like - e x p (--a/zr)CF In 2 (1 -- t). N o w the limit t --> 1 is equivalent to the limit C --> 0, so

120 2O0

80 160

as 2

4O ']20 0

' 8O 40 I

-2

.4

.6

!

.8

0

I

I

•2

I

I

I

.4

I I

.6

8

C

C

20

0

.8

-20 -40 -60

-80 •2

.4

.6

.8

C

Fig. 3. Comparison of the coefficients of C~, CFC A and CFTR for (C/o') dtr/dC with all the large summable corrections.

D.A. Ross / Multijet differential cross sections

113

we have the behaviour as C -~ 0 do" dC

O -a c - , o ' ~ exp ~ CFln 2 (½C)A.

(8)

This enables us to sum to all orders the correction term -or 2CF In 2 (½C)T. 2~"

(9)

In fig. 3 the broken curve is the sum of the three large summable terms, eqs. (4), (7) and (9). We see now that this gives a very good qualitative reproduction of the exact corrections for each of the coefficients, leaving only small terms. These small corrections are well reproduced by the formula

[ 3CF In 5C _ ~CA In ( I OC~ +~TRln ( ~3C -~)] 3 /

~t~s T+0(C-0.75)

X [--2CA ln3 ( 4 0 C - ~ 30) +. ~135"''2 - t c F In 2 (4C - 3) + (1 - C)(6 In C + 1))

\ ~--~/

(10)

.

This can be seen from fig. 4 in which the broken line is the sum of eqs. (4), (7), (9) and (10). We emphasize that this is a very good fitting interpolating formula and has not been derived analytically. There are, in fact, other large terms in the correction to do"/dC, but all these large terms appear to cancel in each coefficient. Without performing the calculation to order ot3 we cannot be sure that this will happen in the next order so it may still be the case that the perturbation expansion is not converging. However, the qualitative fit of fig. 3 leads us to be optimistic about having isolated and summed all the large higher order corrections, leaving a well behaved perturbation expansion. In fig. 5 we show a curve in which we have exponentiated the correction factors of eqs. (4) and (9) and incorporated the corrections (7) into the argument of ffs. This increases the theoretical prediction for the differential cross section (cf. the histogram of fig. 1) except for very low values of C. We have also plotted the data from the Mark J detector [4] on the same plot*. The fit is not impressive and if the Q C D prediction is at all believable there must be considerable smearing of the pure Q C D curve due to non-perturbative effects of final-state hadronization. We know that these hadronization effects will produce significant changes to the curve at low values of C since the curve presented here requires a substantial negative contribution at C = 0 in order to satisfy the requirement 1 Io1 __do" ~-dC

=

1.

(11)

* Unfortunately this is the only group that has presented data in the form of C (or/-/2) distributions.

114

D.A. Ross / Multijet differential cross sections

120 200 80 160

es 2

40 120 0 80 -40 40 -80 .2

.4

"6

0

"8

C

I

I

I

-2

I

I

I

I

.4

C

.6

~~,_

.8

20 4~

0

-2C

.....

. . . . . . .

:

/

.

8

-40

-60

~ ~ ~(~s 2 "CFT~ 1~)

f

-80 I

I

•2

I

I

.4

I

I

.6

I

i

.8

I

C Fig. 4. Comparison of the coefficients of C 2, CFCACFTR for (C/a) do'/dC with all the large summable corrections plus the interpolating formula eq. (10) for the remaining correction.

A n example of the effects of such smearing is shown by the b r o k e n line of fig. 5 in which the solid line [together with a deita function at C = 0 with negative coefficient in o r d e r to satisfy (11)] has b e e n folded with a transverse m o m e n t u m distribution. S o m e i m p r o v e m e n t in the fit of this curve with the data can be seen particularly away f r o m the two-jet r e g i o n C = 0. H o w e v e r , in o r d e r to obtain such an i m p r o v e m e n t an average transverse m o m e n t u m of 2 G e V was required. A l t h o u g h this is very m u c h larger than the 300 M e V usually taken for average transverse m o m e n t u m , this m a y not be entirely u n r e a s o n a b l e if all effects which decrease as a p o w e r of s are to be simulated by a transverse m o m e n t u m distribution since some of these effects m a y be controlled by the masses of b a r y o n s p r o d u c e d in the jets. It appears f r o m fig. 5 that the value of a , using A~-g = 0.3 G e V is not obviously wrong, but a detailed analysis of the effects of hadronization is required if as at 30 G e V is to be d e t e r m i n e d f r o m e+e - scattering to better than 3 5 % .

D.A. Ross / Multijet differential cross sections

115

C do o

dC versus C

0-7

0-6

0"5 ,°°.

0.4

[

0.3

]

0.2

i F t

0-1

0.2

0-4

0"6

()8

C Fig. 5. Solid curve: plot of (C/g) dtr/dC up to order a 2 with the summable corrections summed to all orders. The data points are from the Mark J group. Broken curve: the theoretical curve folded with a transverse momentum distributions with average transverse momentum taken to be 2 GeV. A n o t h e r useful q u a n t i t y is the d o u b l e differential cross section d 2 t r / d C dD. I n fig. 6 we p r e s e n t a scatter plot s h o w i n g the d o u b l e d i s t r i b u t i o n in C a n d D. T h e plot is b a s e d o n a total of 500 e v e n t s at s = 9 0 0 G e V 2 with A~-g t a k e n to b e 0.3 G e V . T h e differential cross section actually diverges as D ~ 0 a n d t h e r e is a n e g a t i v e delta f u n c t i o n at D = 0 which cancels the divergence. A g a i n h a d r o n i z a t i o n effects w o u l d s m e a r these singularities out, a n d in fig. 6 the delta f u n c t i o n has b e e n s m e a r e d e v e n l y over the r a n g e 0 < D < 0.1 or 0 < D < Dmax(C), for values of C w h e r e the m a x i m u m p e r m i t t e d v a l u e of D, Dm~,,(C) is less t h a n 0.1. A n o t h e r m e a s u r e of the n u m b e r of t h r e e - j e t e v e n t s is the q u a n t i t y F = Io~ d D I01 d C -d2tr .5 dC dD '

(12)

w h e r e ~/ is s o m e low cutoff for the q u a n t i t y D. This restricts us to n e a r l y c o p l a n a r events. T h e q u a n t i t y F has a n o n - z e r o v a l u e in o r d e r as, so it can also b e used to m e a s u r e a~. W e find that for ~ = 0.1

F = l'4 as L [1-6"9 ~]

(13)

D.A. Ross / Multi jet differential cross sections

116

C 0-2

0.4

!

i

0-6

0-8

0"8

/

0-6 D 0-4

0-2

Fig. 6. Scatter plot in the variables C and D based on a total of 500 events at s = 900 G e V 2 with A~-g = 0.3 GeV.

whereas for ~7 = 0.2 F=l'4~[

l+l'8~s]'zrj'

(14)

both of these are well within the limits of acceptance for a perturbation expansion, and in the first case the correction is negative. The reason for this is the abovementioned (double) infrared divergence of the differential cross section as D--> 0. Because of this F is logarithmically dependent on rl in order a 2 and contains a term - a In z r / + b In r/. For sufficiently small values of ~ this can cancel or even overwhelm the large positive correction discussed above*. The coefficient of the In 2 71 is connected with the double pole and may therefore also be exponentiated along with the ¢r2. Eqs. (13) and (14) should m o r e properly be written

F=l.4~exp[(CF+½CA)('rr2-21n2rl)~--~][l+f(rl)~],

(15)

where f(rl) is a reasonably small, slowly varying function of r/ which is well approximated by f(~7) = - 2 . 7 + 1.37 In 7?

(16)

for 0.05 < 7? < 0.25. * In this region only the ~r2 term is important. The effect of exponentiating the CF(as/~r) In (~C) or incorporating/3o(as/2zr) In (~C) into the argument of 6s are very small.

D.A. Ross / Multijet differential cross sections

117

0.4

c 2 31°30.2o.1

, 0.2

0.4

0.6

0.8

S--~ (,~,2÷.~,3) Fig. 7. Scatter plot in the variables sphericity (S) and acoplanarity (A) based on a total of 500 events

at s = 900 GeV 2 with A~-~= 0.3 GeV. A2 and '~3 a r e the small eigenvalues of the matrix Oii. Finally in fig. 7 we present a scatter plot in terms of the quantities sphericity and acoplanarity defined by the Tasso [5] group. These are linear c o m b i n a t i o n s of the eigenvalues of the matrix 0 ij [eq. (1)]. T h e quantities are not quite the same as in ref. [5], since they have defined their 0 ii without dividing by particle m o m e n t u m for each particle. Their quantities are not infrared c o n v e r g e n t in Q C D which m e a n s that their distribution is logarithmically sensitive to cutoff's (e.g. the pion mass) p r o v i d e d by the process of hadronization. Nevertheless, a qualitative c o m p a r i s o n of fig. 7 with their distribution suggests that m a n y m o r e of their multijet events are also acoplanar, whereas in fig. 7 m o s t of the events are c o n c e n t r a t e d near the line representing c o p l a n a r events. This suggests that a substantial n u m b e r of the multijet events r e p o r t e d in ref. [5] are due t o smearing f r o m the h a d r o n i z a t i o n of events which are treated in Q C D as two jet events, and not to gluon bremsstrahlung. This adds further c r e d e n c e to o u r previous s t a t e m e n t that the multijet events cannot be used as a direct test of Q C D , w i t h o u t a detailed u n d e r s t a n d i n g of the m e c h a n i s m of hadronization. T h e a u t h o r is grateful to Keith Ellis and D a v i d Politzer for useful conversations,

References [1] R.K. Ellis, D.A. Ross and A.E. Terrano, Nucl. Phys. B178 (1981) 421 [2] G.C. Fox and S. Wolfram, Phys. Rev. Lett. 41 (1978) 1581; Nucl. Phys. B149 (1979) 413; Phys. Lett. 82B (1979) 134 [3] P. Binetruy, Phys. Lett. 91B (1980) 245 [4] D.P. Barber et al., Phys. Rev. Lett. 43 (1979) 830 [5] R. Brandelik et al., Phys. Lett. 86B (1979) 243.