Physics Letters B 294 (1992) 425-430
North-Holland PHYSICS LETTERS B
Heavy quark production at e + e - colliders
in three- and four-jet events
Alessandro Ballestrero a, Ezlo M a l n a b,a a n d Stefano Morettl
aa INFN, Sezlonedt Tormo, 1-10125 Turm, Italy
b Dtparttmento dz Flszca Teortca, Umverstta dt Tortno, 1-10125 Turm, Italy Received 10 August 1992
The contrabutlon of massive quarks to three- and four-jet dxstrlbutlons in e+e - collisions is studied at LEP I energies using exact hehclty amplitudes Total cross sections as a function of Your, in both the JADE and the kT algorithms, are presented and compared with massless results Some dlstnbutxons for variables of interest for QCD stu&es at LEP are examined
H a d r o n l c decays of the Z ° produced in e+e - a n n i - hilations provide a n ideal e n v i r o n m e n t
for
precise tests of Q C D [ 1] All LEP e x p e r i m e n t s have per- formed a large n u m b e r o f m e a s u r e m e n t s I n particular- as has been measured from j e t rates [ 2 ] a n d from
global event shape variables [2,3 ] F l a v o r i n d e p e n d - ence of the coupling has been verified [ 4 ]
- Three- [5 ] a n d four-jet [6 ] distributions have been studied a n d c o m p a r e d with Q C D predictions Addi- tional evidence for the presence of the three-gluon vertex has b e e n f o u n d T h e color factors, which de- t e r m i n e the gauge group which is responsible for strong Interactions, have been m e a s u r e d [ 7 ]
- The differences between gluon a n d quark jets have been investigated [ 8 ]
In some cases the posslbihty of tagging quarks jets using the semlleptonic decays of b a n d c quarks has been used All Q C D results m e n t i o n e d are based o n a careful c o m p a r i s o n of e x p e r i m e n t a l data with M o n t e Carlo s i m u l a t i o n s To our knowledge, all available programs are based o n massless m a t r i x ele- ments a n d kinematics Mass effects are k n o w n to gave corrections at the percent level to q u a n t i t i e s like / ' ( Z ~ b b ) in the Born a p p r o x i m a t i o n , a n d one ex- pects their relevance to increase with the n u m b e r of
Work supported in part by Mlmstero deU'Unlverslta e della Rlcerca Sclentlfica
final partons A close e x a m i n a t i o n of the differences between the results o b t a i n e d with massive a n d mass- less matrix elements would d e t e r m i n e the range of applicability of the latter a n d would gave better con- trol o n this source of systematic errors
T h e partial width of the Z ° i n bl~, though very dif- ficult to measure with great accuracy, is potentially very interesting because of the presence of one-loop vertex corrections, sensitive to the top mass, which do not c o n t r i b u t e for other f e r m l o n species Clearly a n u n d e r s t a n d i n g of mass effects in multijet events is highly desirable
Q C D p r o d u c t i o n of heavy quarks is also relevant for Hlggs searches at LEP since the Hlggs decay to bl~ is by far the d o m i n a n t one for hght Hlggs masses, a n d probably the only channel with rates large enough for detection at e+e - colhders [ 9 ]
In the c o m i n g years, i m p r o v e m e n t s in statistics, i n secondary vertex reconstruction with silicon vertex detectors a n d in particle identification will allow m u c h more detailed studies of heavy quark produc- tion at LEP
I n this paper we present cross sections for three- a n d four-jet events at LEP energies which include all quark mass effects in leading order A m o n g previous calculations [ 1 0 - 1 5 ] , the only one which does not neglect quark masses is that o f r e f [ 15 ], where, how- ever, the Z ° c o n t r i b u t i o n Is not i n c l u d e d The matrix e l e m e n t for all processes has b e e n c o m p u t e d at the
Volume 294, n u m b e r 3,4 PHYSICS LETTERS B 19 N o v e m b e r ! 992
a m p l i t u d e level following b o t h the m e t h o d o f ref [ 1 6 ] a n d t h a t o f ref [ 1 7 ] as a check o f the correct- ness o f o u r results I n our experience it is easier to p r o d u c e fast routines with the f o r m a l i s m o f r e f [ 1 7 ] b u t ~t is clearly m u c h s~mpler to give c o m p a c t , easy to i m p l e m e n t f o r m u l a e using the f o r m a l i s m o f r e f
[ 1 6 ] The higher speed o f the routines b a s e d on r e f [ 17] is certainly related to the smaller n u m b e r o f s u b r o u t i n e calls we are using in this case a n d m a y not reflect an intrinsic s u p e r i o r i t y o f the m e t h o d M o r e details on o u r calculation a n d the e x p h c l t expres- sions o f the a m p h t u d e s for p r o d u c t i o n o f h e a v y quarks a n d / o r l e p t o n s in events with three o r f o u r final p a r t o n s at LEP energies will a p p e a r in a sepa- rate p a p e r [18] T h r e e - p a r t o n q u a n t i t i e s are c o m - p u t e d at O ( a s ) a n d f o u r - p a r t o n q u a n t i t i e s at O ( a 2) Virtual corrections to e + e - - , q q g w o u l d be n e e d e d for a t r e a t m e n t o f three p a r t o n q u a n t i t i e s to O ( a 2 )
The a m p h t u d e s h a v e been c h e c k e d for gauge in- v a r i a n c e a n d m the a p p r o p r i a t e l i m i t r e p r o d u c e the f o r m u l a e m refs [ 14,15 ] We have used a s = 0 l 15, v / ~ = M z = 9 1 1 GeV, F z = 2 5 GeV, s l n 2 ( 0 w ) = 0 23, me = 1 7 G e V a n d mu = 5 0 G e V
T h e e x p e r i m e n t a l d e f i n i t i o n o f a j e t is b a s e d on a clustering p r o c e d u r e T h e m o s t widely used scheme is the J A D E a l g o r i t h m The two ( p s e u d o ) p a r t i c l e s t a n d j for which the q u a n t i t y
y J = 2 W ( 1 - c o s
Or)
Ev,s( 1 )
IS m l m m u m are c o m b i n e d into a single p s e u d o p a r t l -
cle
kofmomentumpk=p,+p~
The process is i t e r a t e duntil all p s e u d o p a r t l c l e pairs satisfy y,j 1> Y~ut T h e ob- jects g e n e r a t e d b y this p r o c e d u r e are called j e t s T h e J A D E algorithm has been recently criticized [ 19 ] and a new clustering v a r m b l e
m l n (E~, E 2)
y,~ = 2 ( 1 - c o s
Or)
( 2 )E21s
has been p r o p o s e d T h e c o r r e s p o n d i n g a l g o r i t h m is k n o w n as the kr-clusterlng In what follows we ne- glect all h a d r o n l z a t m n effects, a n d a p p l y cuts at the p a r t o n l c level
In fig 1 we present the cross sections for e+e - - - , qqg, q = c , b as a function Of Ycut for b o t h d e f i n i t i o n s o f y The range in the two cases has been chosen in such a way that the qqg cross sections for the smallest Ycut values are o f s i m i l a r m a g n i t u d e As o b v i o u s f r o m their d e f i n i t i o n ( 1 ), ( 2 ) , a cut in yT iS m u c h stronger t h a n a c o r r e s p o n d i n g cut in yJ a n d the effect b e c o m e s m o r e p r o m i n e n t for smaller Ycut This can be clearly seen in the different slopes o f the two groups o f curves In fig 2 the ratios o f the cross section for m a s s i v e
6 0 0 0 i , i , f , J f i i I , J , i , i , i , i J i ) I 5 o o o ~ q q g 4 0 0 0 " " " . . . . C 3 0 0 0 b 2 0 0 0 k T " ' - . . . JADE I000 t 0 -~ 10 -2 10 -1 Yeut
Volume 294, number 3,4 PHYSICS LETTERS B 19 November 1992 I 0 0 5 O g O 0 5 5 0 0 0 0 7 5 0 7 0 10 -~ l ' l ' I , i , i J l i I . . . . - k ¸ / ~ / d q q g g ... J A D E , L . t , * , i i t i I , , . i , * , * , i i t , I l f f z 1 0 - * Y c u t
Fig 2
R]"=a(ceg)/a(ufig), R~=tr(b6g)/a(dclg)and R4bd=cr(bbgg)/cr(dagg)asafunetmnofyc.tfory s
(dashed) andy v (contxnuous)quarks to the cross section for massless quarks with xdentxcal charge a n d 13,
R~U=a(c¢g)/cr(ufig)
a n d R bo = a ( b b g ) / t r ( d d g ) , are shown as a f u n c u o n of Ycut F o r small Yc.t the cross section for b quarks ~s almost 20% smaller t h a n for d's As expected the ratxos be- come closer to one for larger Ycut, b u t for YCut as large as 0 2, stdl Rb°~<0 96 I n the same figure, for com- parison, we also report R bd = cr (blagg) / a ( d d g g )In fig 3 we give the cross sections for e + e - ~q~lgg, q = d, c, b The qClgg final state is the d o m i n a n t con- trlbutxon to the four-jet cross section The cross sec- t m n for two u quarks a n d two gluons xs w i t h i n 5% o f the result for c h a r m e d quarks It is interesting that the charm a n d the b o t t o m cross sections are approx- imately equal m this case
In fig 4 the cross sections for e+e-~q~lqC:l, q = d ,
2 0 0 0 qqgg 1 5 0 0 - - b i o o o ' " ~ d b i L , i . i . i . i t i i I i i - , ~ , i 10-~ lff ~ tO-* Ycut
Volume 294, number 3,4 PHYSICS LETTERS B 19 November 1992 2 0 I i , i , i , i , i f r i 1 5 1 0 b 4q 1 ~ 3 1 ~ z 1 0 ' kr . . . c " ' . ~ d " ' . . ~ ""~'- , JADE i Yeut
Fig 4 Cross sections for e ÷e- --,qClCl, q = d (dotted), u (dash-dotted), c (dashed) and b (continuous) as a functmn of );cut for both definitions ofy u, c, b is s h o w n as a f u n c t i o n o f Ycut T h o u g h v e r y s m a l l c o m p a r e d w i t h t h e qdlgg case th~s case c o u l d b e q m t e i n t e r e s t i n g i f h e a v y q u a r k s c a n b e t a g g e d w~th h i g h e f f i c i e n c y F o r 106 Z °, c o r r e s p o n d i n g to an in- t e g r a t e d l u m i n o s i t y o f a b o u t 30 p b - 1, o n e e x p e c t s ap- p r o x i m a t e l y 100 e v e n t s w~th 4b q u a r k s F r o m figs 1, 3 a n d 4 it c a n be s e e n that, w h i l e f o r t h e t h r e e - p a r t o n case t h e cross s e c t i o n s f o r Y Jut = 0 01 are a p p r o x l -
m a t e l y m a t c h e d b y t h e cross s e c t i o n s f o r Ycut = T 0 0 0 1 5 , m t h e f o u r - p a r t o n case th~s h a p p e n s f o r
T
Ycut= 0 003
T h e d i s t r i b u t i o n s m y ~ f o r all j e t pairs m e + e - qClg, q = d, b is s h o w n m fig 5 J e t s are o r d e r e d ac- c o r d i n g to t h e i r energies, t e , E, >t E,+ 1 Since t h e t h i r d j e t is in m o s t cases t h e g l u o n j e t , fig 5 s h o w s t h a t t h e g l u o n j e t is o n a v e r a g e shghtly h a r d e r f o r h g h t q u a r k s . . . . i . . . . i . . . . i . . . . i . . . . i . . . . i . . . . i . . . . i , , , - 1 - 0 1 5 qqg b i o to tzz3 . . . d ::k b 0 0 5 . . . . i . . . . i . . . . i . . . . ~ , , . i . . . . i . . . . i , . , 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 /z (GeV)
Fig 5 DlStrlbuUon for #,j = ~ in e %---,q~lg for q = d (dashed) and b (continuous) Jets are ordered according to their energies Ycut = 0 0015 in the kT algonthm
Volume 294, number 3,4 PHYSICS LETTERS B 19 November 1992 0 1 2 qqgg o l o - - b ,, d "~ E 4 E 1 0 0 8 . --'- .E 3 . . . . E 2 ,/' b 0 0 6 r.fl b 0 0 4 10 2 0 3 0 4 0 5 0 E (GeV)
Fig 6 Energy distribution of all four jets in e+e--,qqgg for q = d (dashed) and b (continuous) Jets are ordered according to their energxes yc.t=0 0015 m the kT algorithm
then for h e a v y quarks This is also c o n f i r m e d by the d i s t r i b u t i o n in x2---2E2/EcM, which we do not show
In fig 6 we present the energy distributions o f all four, energy ordered, jets in e + e - -~ qqgg, q = d, b Be- side the d i s t r i b u t i o n shown we h a v e e x a m i n e d other variables like the m o d i f i e d N a c h t m a n n - R e l t e r angle [20,21 ] and the B e n g t s s o n - Z e r w a s angle [22] T h e results for massive b and c quarks differ very little f r o m the c o r r e s p o n d i n g massless distributions Masses seem to p r i m a r i l y affect total cross sections T h e i r effect d e p e n d s on the n u m b e r o f jets, on the clustering a l g o r i t h m and on Yc.t T h e r e f o r e j e t rates are sensitive to q u a r k masses A precise d e t e r m m a - tlon o f mass effects w o u l d require the calculatton o f two- a n d three-jet rates to O ( a 2) F r o m our prelimi- nary analysis it appears that distributions o f e v e n t shape variables at the parton level are only slightly m o d i f i e d by the Inclusion o f masses O n one hand, this m e a n s that all p r e v i o u s analyses based on these variables m a i n t a i n their validity On the o t h e r hand. this shows that e v e n t shape variables, as studied In this paper, are unable to d i s c r i m i n a t e m a s s i v e q u a r k events f r o m massless q u a r k events
Conclustons
We h a v e c o m p u t e d the exact m a t r i xelements for e + e - ~ q q g , e + e - ~ qqgg and
e + e - - . q q q q with massive quarks Total cross sec- tions h a v e been presented as a f u n c t i o n o f the cut which defines the j e t r e c o n s t r u c t i o n a l g o r i t h m F o r b quarks and a not too large Ycut the mass effects on to- tal cross sections are large S o m e distributions for variables o f interest for Q C D studies at LEP have been e x a m m e d They show that analyses based on e v e n t shape variables are not affected by mass effects but also that they do not p r o v i d e efficient m e a n s for enriching the h e a v y quark sample
R e f e r e n c e s
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