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Five jet production with heavy quarks at e+e− colliders
Physics Letters B 323 (1994)53-64 North-Holland
PHYSICS LETTERS B
Five jet production with heavy quarks at e + e- colliders* Alessandro Ballestrero 1 and Ezio Maina 2 Dtparttrnento dl Ftstca Teortca, Untverstth dt Tonno, Turm, Italy and INFN, Seztone dt Tonno, v Gturza 1, 10125 Turm, Italy
Recexved 24 November 1993 Editor R Gatto
Heavy quark production m five jet events at e+e - colhders is studied at tree level using hellclty amphtudes Total production rates for 2b3j and 4bj are gtven and compared with the corresponding results for massless quarks The process e+e - ~ q(tgg? which is the dominant contribution to 4J7 produchon is briefly dascussed
1. Introduction The great n u m b e r o f hadronlc decays of the Z ° observed at LEP provides the opportunity to test our understanding o f strong lnteracttons an unprecedented detml Large samples of multi-jet events have been accumulated and analyzed [ 1,2 ]. In particular the A L E P H Collaboration and the O P A L Collaboration have studied the jet-frachon distributions as a function of ycut for up to five jets. Recent advances in b-tagging techniques based on the introduction o f vertex detectors and on a refinement of the selection procedures, with their large efficlencles and the resulting high purmes, have p a v e d the way to the study o f heavy-quark production in association with light-quark and gluon jets. The flavour Independence o f the strong couphng constant has been verified by several groups [3] using both jet-rates and shape variables It has been lately pointed out [4,5 ] that the effects of the b-quark mass are substantial and increase with the n u m b e r of jets. In this paper we plan to study heavy-quark p r o d u c h o n in five-jet events at e + e - colhders. Two different mechamsms contribute to five-jet production for hght quarks, the 'polnt-hke' annihilation o f e + e - to Z °, 7 [6], which dominates at LEP I energy, and W + W - production [ 7 ], followed by the emission of a gluon from the decay products o f the W ' s , whxch dominates above the W W treshold This latter contribution ~s severely suppressed for b-quarks by the smallness o f the Fbc element o f the C K M matrix and by the large mass o f the top and will not be considered here. W e have studied the anmhflatlon m e c h a m s m for five-jet production at tree level taking full account ofT, Z interference and o f quark masses Therefore our matrix elements can be used at all e + e - colhders. As a b y p r o d u c t o f the m a i n calculation we have also examined the d o m i n a n t contribution, e+e - ~ q(tggT, to the p r o d u c h o n o f a smgle hard p h o t o n in assoc~atlon w~th four jets. Both initial state and final state radlat~on have been included at tree level, without ISR resummatlon We have computed all matrix elements both in the formahsm of [8,9] and 111 that o f [10] using the former for the actual calculation. The relevant formulae are summarized m the following section Both methods can * Work supported in part by Mlmstero delrUnlversltfi e della Rlcerca Sclentlfica 1 E-mad baUestrero@to lnfn it. 2 E-mall [email protected] it 0370-2693/94/$ 07.00 @ 1994-Elsevier Science B V All rights reserved SSDI 0 3 7 0 - 2 6 9 3 ( 9 4 ) 0 0 0 2 8 - 6
53
Volume 323, number 1
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3 March 1994
6
4 5
6
3 4
3
5 7
(1)
4 5
7
(2)
7
(3)
4
(41
3
4
4 5
5
7
(51
6
7
7
(61
6
7
6
7
(r)
7
(8) 6
4
(9) 6
6
3
(10)
4
(11)
7
7 (12)
Fig 1 Representative diagrams contributing to e + e - ~ q ~ I g g g All others diagrams can be obtained through permutations of the gluon labels The contrlbuUon of the two diagrams involving a four-gluon vertex, which are not shown, are mluded in the analytmal expression given in the text for the chagrams with two connected three-gluon vertices
be easily i m p l e m e n t e d in a small set o f nested subroutines. This however results in a computer program which is far too slow for the present calculation. We have mstead used the symbohc package Mathemattca [ 11] to write down the Fortran expression for each hehclty amphtude. The individual Z-functions are computed first and then c o m b m e d in larger structures that describe chunks of increasing size o f the F e y n m a n diagrams shown 111 figs. 1-2. Every sub-diagram is saved a n d then used several times. With this procedure we have produced a rather large piece o f code, which however runs quite fast, and therefeore can be used in high statistics Monte Carlo runs. As an example the program for q~ggg production is about 24,000 hnes long, but reqmres only about 54
Volume 323, number 1
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6
7
6
7
(13)
7
(14) 6
(151 6
4
4
6
5
5
3 3
(16)
7
(17)
6
7
(18)
6
7
6
4 3
(19)
7
5
(2O)
6
3
(21)
6
7
6
4
7 (2~)
7
7 (~)
7 (~4)
Fig 2 Representative diagrams contributing to e + e - ~ qclqq The remmmng diagrams can be obtained through simultaneous substitutions q(Ps,25) ~ q(P7,~,7) and q(p4,),4) ~ q(p6,26) The addttlonal ones needed for identical flavour can be obtained exchanging q(Ps, 25 ) with ~ (PT, 27 ) including their color quantum numbers and changing the overall sign
5 × 10 -2 s e c o n d s t o e v a l u a t e o n a V a x s t a t l o n 4 0 0 0 / 9 0 . T h e a m p h t u d e s h a v e b e e n c h e c k e d f o r gauge m v a n a n c e . W e h a v e u s e d Mz = 91.1 G e V , F z = 2 5 G e V , s m 2 ( 0 w ) = .23, m b = 5 G e V , aem = 1 / 1 2 8 a n d as = .115 i n t h e n u m e r i c a l p a r t o f o u r work. 55
Volume 323, number 1
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Table 1 The Z-functions for all independent hehclty combmatmns m terms of the functions s, t, tl and # defined in the text The remaining Z-functions can be obatlned changing sign to the hellcRles and exchanging + with - , s with t and R with L 21222324
! Z(Pt,~.l,P2,'2-2,P3,'~3,P4,~'4, CR, C L, C R, CLt x1
++++ +++++-+ +-++ ++-+-++--+
- 2 [s(3, 1 ) t(4, 2)dRCR - #I#2Yl3Y]4CtRCL -- ~]ffl2fl3fl4CtLCR] -2q2c R [s (4, 1 )/z3c~ - s (3, 1 )/t4c~] -2t/lCR It (2, 3)~4c ~ - t(2, 4)#3c~] -2rl4C ~ [s (3, 1 )It2cR -- s (3, 2)#ICL] --2 [S( 1, 4) t(2, 3)C~CR -- #ll~2q3q4CtLCL -- t/lr/2,u 3/Z4C~C/~] 0 - 2 [#1 # 4112q 3¢tLCL "l- # 2f13q 1q 4CtRCR -- # 2fl 41]1173CtLCR -- fl Ill 3712?142 tl~ 1-] -- 2r/3C~ [S (2, 4)ltlCL -- S ( l, 4)IL2CR ]
..1-
--
____
Table 2 The X-functions for the two independent behclty combinations m terms of the functions s, t, r/ and/~ defined in the text The remalmng X-functions can be obatlned changing sign to the hehcltms and exchanging + with - , s with t and R with L 2123
X (Pl,21,P2,P3,,~.3)
+ + +-
(/tit/2 + / t 2 q l ) (#3q2 + #2r/3) + s(1, 2) t(2, 3) (#1t/2 +/z2r/1)s(2, 3) + (/z2t/3 + #3q2)s(1,2)
2. Calculation W e present the h e h c l t y a m p h t u d e s in the f o r m a h s m o f [8,9] writing the arnphtudes are.
T h e two basic functions which are n e e d e d m
Z (p,, 2,, Ps, 2j, Pk, ).k, Pl, 2l, CR, CL, C'R, C~) = [ft ( p . 2z )-FUu (p j, 2j ) ] [ft (Pk, J.k)F;U (Pt, 2t) ],
( 1)
where
F (')u = yU(c~')p R + C~')pL)
(2)
and
X (p,,2.pj;pk,2k
) = [ft(pz,2,)l) j U ( P k , 2 k ) ]
(3)
Thear expressions in t e r m s o f splnor p r o d u c t s s (l, j ) = ~ ( p . + )u ( P s , - ) = t* (j, z ), tl a n d / z functions can be f o u n d m tables 1 a n d 2. A c o n v e n i e n t , t h o u g h n o n - u n i q u e , choice for the splnor p r o d u c t s and for the funcUons t / a n d / z is:
o
x)
s O , j ) = (p', + ~p:) { P _ ; - P _ 2 \ pO, - px, r/(z) = ( 2 ( p ° - p ~ ) )
V2 ,
(p
1/2 -
_p,)1/2
o x (P; + ~P: ) \ p , o - p,~
,
I~(,) = ± m,
~(~)
(5) '
where the sign + ( - ) refers to a particle ( a n t l p a m c l e ) o f mass m,. 56
(4)
Volume 323, number 1
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PHYSICS LETTERS B
Table 3 The color factors, multiplied by 9, for e+e - ---, q~tggg
345 453 534 543 354 435
345
453
534
543
354
435
64 1 1 10 --8 --8
1 64 1 --8 10 --8
1 1 64 --8 --8 10
10 --8 --8 64 1 1
--8 10 --8 1 64 1
--8 --8 10 1 1 64
The polarization vectors o f a gluon o f m o m e n t u m p, can be written as (6)
8# (p~,2) = Nz [~t(p,,2)TUU(pa(z),2) ], where Pa(z) lS an auxiliary massless m o m e n t u m not parallel to p, and the normalization factor p , ) ] - 1 / 2 All a m p h t u d e s are independent o f the Pa(t)'s There are fifty F e y n m a n diagrams contributing to e-(pb2)
+ e+ ( p 2 , - 2 ) ~ q(p6,26) -F q(p7,27) -b g(P3,23) -b g(P4,24) -b g(Ps,25)
lS N l =
[4(pa(,)
(7)
Some o f them are shown in fig. 1. All others can be obtained through permutations o f the gluon labels. The two diagrams with four-gluon vertices are not shown. Their contribution is included in the expressions gwen below for the diagrams wzth two connected three-gluon vertices The amplitude squared for this process is. 2 1 l g 3 e Z N 3 N 4 N 5 × ~ ~ ~l/t(,t}~1,¢().).(. , IMIqoggg = 3 /--, z-, ' {;Q n,n' -
-
(8)
where ~r is a permutation o f the indexes (3, 4, 5), {2} indicates the hehcltles o f all external parhcles, M . is the coefficient o f the color matrix Tl~ = (Tn(3)Tn(4)TrC(5))ts,
(9)
in the full a m p l i t u d e and C ~ , is the appropriate color factor, Cn~, = T r ( T ~ ( T ~' )t ). The color factors are given in table 3. To fix our notation, diagrams 1-4 have color structure 345, diagrams 5-7 have color structure [3,4] 5 where [3,4] is the usual commutator, [3,4] = 3 4 - 4 3 , diagrams 8-10 have color structure 5 [ 3,4 ] and diagrams 11-12 have color structure [ [ 3,4 ], 5 ] The production o f four quarks and one gluon e- (pl,2) + e+(p2,-2)
--~ g(P3,23) + q(P4,24) + q(Ps,25) -I- q(P6,26) -b q(P7,27),
(10)
IS described in lowest order by twenty-four F e y n m a n diagrams if the two quarks have different flavour (case I) and by forty-eight if their flavour is the same (case II). Twelve o f the diagrams for case I are shown in fig. 2, all others can be obtained with the simultaneous substitutions ~ (Ps, 25 ) ~ q (PT, 27 ) and q (P4, 24) ~ q (P6, 26 ). The additional ones needed for identical flavour can be obtained exchanging ~ (Ps, 25 ) with ~ (P7, 27 ) including their color q u a n t u m numbers and changing the overall sign 57
Volume 323, number 1
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Table 4 The color factors, multiplied by 9, for e + e - ~ qlqlq2q2
A1 A2 Aa A4 A5
A1
A2
A3
A4
As
24 -3 -6 21 -27
-3 24 21 -6 27
-6 21 24 -3 27
21 -6 -3 24 -27
-27 27 27 -27 54
Table 5 The addmonal color factors, multiplied by 9, needed for quarks of identical flavour e + e - ~ qglq(l.
B1 B2 Ba B4 B5
A1
A2
A3
-8
l 10 1 -8 9
10 1 -8 1 -9
1 10 1 9
A4
A5
1
9 9 -9 -9 0
-8 1 10 -9
T h e a m p l i t u d e s q u a r e d for t h e s e p r o c e s s e s is: 24 1 32~ [~ 2 [qoq,o,g= ~g;e iv3 ~--~ ~-~M,
(11)
(2] t , j = l 48
IMlqo,o~ -
4 (21) 2g3e2N3 ~
Z
C:j,
(12)
{,t} ~,j= 1 In case I a n d case II respectively. C,j a n d C[j are t h e a p p r o p r i a t e t a b l e 4 a n d 5. We define the following colour structures
Am llJl "~ = (TaT e) llJ 1 (T b) t2J2
A a , 2 J 2 = (TbTa),,j 1
A a t232 = 4 llJ 1
A a t',J2 = 5 llJ 1
(T b)
(Tbra] llJ I x .'12j 2 ~
~2 llj 1
(Tb)
llJ1
b
2
c (T)/2J2 Cabc
c o l o u r factors w h i c h c a n b e e x t r a c t e d f r o m
A a ,zJ2
=
b
(raTb),zj z (13)
and a t2j 2
a 12J1
Bn qJt = An llJ2"
(14)
T h e c o l o r f a c t o r is g i v e n by:
C.m
1212A t am 22z2 j~,,.
= A .~ ' l J , ~
(15)
i n case I a n d b y a n a l o g o u s f o r m u l a e i n case II, p o s s i b l y w i t h An,m --+ Bn,m W e a d o p t t h e f o l l o w i n g s h o r t h a n d n o t a t i o n , w h e r e i n t h e left h a n d side t h e i n d e x t s t a n d s b o t h f o r t h e m o m e n t u m p, a n d f o r t h e c o r r e s p o n d i n g p o l a r i z a t i o n 2~, a n d [t] s t a n d s for t h e set o f f o u r l n d m e s a s s o c m t e d w i t h a g l u o n m o m e n t u m a n d to h i s a u x l h a r y m o m e n t u m
X(t,j,k) 58
= X(Pz,,~,Pj;Pk,~k),
(16)
Volume 323, number 1
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PHYSICS LETTERS B
X ( [ t ] , y ) = X(p,,2z,pj,pa(,),2z),
(17)
Z(t,j;k,l)
(18)
= Z(P,,2,,Pj,2j,Pk,Xk,Pt,2I, 1, 1; 1, 1),
Z ( [ t ] , J , k ) = Z(P,,2,;Pa(,),2z;Pj,2j,Plc, 2k, 1, 1, 1, 1),
(19)
Z ( [t]; [ J ] ) = Z(p,,X~,pao),~,;pj,2j;pao),J,j; 1, 1, 1, 1),
(20)
Ze(l,J)
=
Z (p~,,~z,pj,~j,pz,2,pb2, 1, 1; 1, 1) x dy + Z (I,X~,J,~j, 2,),, 1,2;CR, CL, C'R,C'L) X dzo ,
(21)
where dr, dzo are the p h o t o n and Z ° propagator functions. The attachment o f two external gluons to a quark hne through a three-gluon vertex is conveniently described with the help of:
Z2([t,j],k,l) = Z([t]; [j]) × (X(k;j,l)-X(k,t;l))
+2(X([j];t)×Z([t];k,I)-X([z];j)
xZ([j];k,l)). (22)
The function that descrxbes the insertion o f two connected three-gluon vertices, and of the term with the same color structure in the associated four-gluon vertex, on a fermlon hne is: Z3([l,J], [k];l,m)
= 4Z([t];l,m) x X([j];z) x (X([k],j) +X([k];t)) + X([t],j) x (X([j];t)
+ 2X([j],k)))
× (Z([t]; [j]) × X([k],j)-
+2Z([k],l,m)
× ( Z ( [ l ] , [ j ] ) x p , Pk
+ (X(l;k,m) - X(l,t,m) - X(l,j,m))
2Z([k]; [j]) × X([1],j))
+ 2X(l;j;m) × Z([t]; [j]) × (X([k],j)
+ X ( [ k ] ; t ) ) - 2 Z ( [ t ] , l , m ) x Z ( [ k ] , [ j ] ) x Pz Pj - (t +-~ j ) .
(23)
The d e n o m i n a t o r associated with this function is (p~ + pj )2 (p, + p~ + Pk)2 In order to slmphfy our formulae we make the convention that all repeated indices are s u m m e d over, and that the sum extends to all possible values, 2 = t l , o f all internal hehcltles. The last line m eq 25-48 gives the set o f values taken independently by each m o m e n t u m index. Defining the sign factors e (z) as:
e(t) =
-1 +1
t = 1,2 t > 2 '
(24)
the s p m o r functions necessary for q(tggg production, which correspond to the diagrams shown in fig. 1 are" M1 = - Z ( [ 3 ] ; 6 ,
t) x Z ( [ 4 ] , t , j ) x Z ( [ 5 ] , j , k ) x Ze(k, 7) x e ( k ) ,
Pl ----"{P6,P3}, Pj = {P6,P3,P4}, Pk = {Pl,P2,PT}, M2 = - Z ( [ 3 ] , 6 , / )
x Z([4];t,J)
(25)
x Ze(J,k) x Z([5],k, 7),
p, = {p6,p3}, pj = {P6,p3,p4}, pk = {ps,p7},
(26)
M3 = Z ( [ 3 ] , 6 , t) x Ze(z,J) x Z ( [ 4 ] , j , k ) x Z ( [ 5 ] ; k , 7 ) , P, = {P6,P3}, Pj = {P4,Ps,P7}, Pk = {Ps,P7},
(27) 59
Volume 323, number 1 M4 = Ze(6, z) x Z ( [ 3 ] , t , J )
PHYSICS LETTERS B
× Z([4],J,k)
3 March 1994
x Z ( [ 5 ] , k , 7) × e O ) ,
Pl -" {P6,Pl,P2}, Pj = {P4,P5,PT}, Pk = {Ps,P7},
(28)
3"/5 = - Z 2 ( [ 3 , 4], 6, t) x Z ( [5], l , J ) x Ze(J, 7) x e ( J ) ,
Pl = {P6,P3,P4}, Pj = {Pt,P2,PT},
(29)
M6 = - Z 2 ( [ 3 , 4 ] , 6 , t) x Ze(Z,J) X Z ( [ 5 ] ; J , 7 ) ,
(30)
Pt = {P6,P3,P4}, Pj = {P5,P7}, M7 = - Z e ( 6 , t) x Z 2 ( [ 3 , 4 ] ; t , j )
x Z ( [ 5 ] , j , 7) x e ( t ) ,
P, = {Pl,P2,P6}, Pj = {Ps,P7}, Ms = - Z ( [ 5 ] , 6 ,
t) × Z z ( [ 3 , 4 ] , z , . / ) × Ze(J, 7) x e ( J ) ,
P, = {Ps,P5}, Pj = {PbPz,P7}, M9 = - Z ( [ 5 ] , 6 ,
(31)
(32)
t) x Z e ( t , J ) X Z 2 ( [ 3 , 4 ] ; J , 7 ) ,
P~ = {P6,Ps}, Pj = {P3,P4,P7},
(33)
MlO = - Z e ( 6 , l) × Z ( [51, l , j ) × Z2( [3, 4 ] ; j , 7) x e ( t ) ,
Pl =- {Pl,P2,p6}, Pj -=- {P3,P4,p7},
(34)
Mll = - Z 3 ( [ 3 , 4 1 , [ 5 ] , 6 , t) × Z e ( l , 7 ) x e ( t ) , (35)
P, -'= {Pl,P2,P7}, M12 = Ze(6, t) x Z 3 ( [ 3 , 4 ] , [ 5 1 , / , 7 ) × e ( l ) ,
(36)
P, = {Pl,P2,P6}, The s p m o r functions for the diagrams m fig. 2, which describe q(tq(tg p r o d u c t i o n are M13 = Z ( [ 3 ] , t , 5) × Z ( 4 , t , 6 , j ) x Ze(J, 7) × e ( j ) , Pz = {P3,Ps}, P.1 ---- {Pl,P2,P7}, M14 = - Z ( [ 3 ] ; 4 ,
Z) x Z ( t , 5,6, J) × Ze(J, 7) × e ( J ) ,
P~ = {P3,P4}, Pj = {Pl,PZ,PT}, M15 = - Z ( [ 3 ] ,
(38)
l, 5) × Z ( 4 , t , j , 7) × Ze (6, J ) × e ( J ) ,
P, ---- {P3,PS}, Pj = {Pl,P2,P6},
6O
(37)
(39)
PHYSICS LETTERS B
Volume 323, number 1 M16
=
3 March 1994
Z ( [ 3 ] , 4 , z) × Z O , 5 ; j , 7) x Z ~ ( 6 , j ) × e ( J ) , (40)
P~ = {P3,P4}, P; = {Pl,P2,P6}, M17 = - Z ( 6 , / , 4 , 5 )
x Z([3];z,3)
x Z e ( J , 7) x e ( J ) ,
(41)
P, = {P4,Ps,P6}, Pj = { P l , P z , P 7 } , M18 = - Z ( 6 ,
1,4, 5) × Z e ( l , J ) × Z ( [ 3 ] , J ,
7),
(42)
P, = {P4,Ps,P6}, Pj = {P3,PT}, m19
=
- Z e ( 6 , l) × Z ( t , J , 4 , 5 ) × Z ( [ 3 ] , J , 7) x e ( l ) , (43)
P, = {PbP2,P6}, Pj = {P3,P7}, M2o = - Z ( [ 3 ] ; 6 , t) × Z ( t , J , 4 , 5 ) x Ze(J, 7) x e ( J ) ,
(44)
P, = {P3,P6}, Pj = {Pl,P2,P7}, t) × Z e ( l , J ) × Z ( J , 7 , 4 , 5 ) ,
M21 = - Z ( [ 3 ] ; 6 ,
(45)
Pz = {P3,P6}, Pj = {P4,Ps,P7}, M22 = --Ze(6, l) × Z ( [ 3 ] , t , J ) × Z ( J , 7 , 4 , 5 ) x e ( t ) ,
(46)
Pl = {Pl,P2,P6}, Pj = {P4,Ps,P7},
M23 = Ze(6, t) × ( 2 ( X ( [ 3 ] , 4 ) + X ( [ 3 ] , 5 ) ) + X ( t , 5;7) - X ( t ; 3 , 7 ) )
× Z([3];4,5))
× Z(z,7,4,5) - 2X(4,3;5) × Z([3];t,7) - (X(t,4,7) × e(t),
Pz = { P b P 2 , P 6 } ,
M24 = - Z e ( l , 7 )
(47)
× (2 ( X ( [ 3 ] , 4 ) + X ( [ 3 ] , 5 ) )
+ X(6,5;t) - X(6,3, t)) × Z([3];4,5))
P, = { P l , P z , P 7 } .
× Z(6, t;4,5)-
2 X ( 4 ; 3 , 5) × Z ( [ 3 ] ; 6 , t ) -
(Y(6,4;t)
× e(t), (48)
F r o m the examples above the one can to construct the full amphtudes for 5j and 4j y production with little effort.
3. Results Our results are presented in figs. 3 and 4. In fig. 3 we show the total cross section for five-jet production as a function o f Ycut in the J A D E scheme [ 12 ] and in the D U R H A M scheme [13 ] We have neglected the mass o f the quarks o f the first two generations, and have s u m m e d over all their contributions. The cross section for four b-quark plus one gluon-jet production is given by the dotted hnes The dashed lines represent the cross 61
Volume 323, number 1
PHYSICS LETTERS B
3 March 1994
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0 04
Yeut
Fig. 3. Total cross section as a function of Ycut in the JADE and in the DURHAM scheme for four b-quarks plus one gluon-jet production (dotted line), for the production of two b-quarks plus any comblnatlon of three light-patton jets (dashed line), and for the sum of all contributions to five-jet producUon including only light partons (continuous line) section for the production o f two b-quarks plus any combination o f three hght-parton jets. Finally the sum o f all contributions to five-jet production including only hght, strongly interacting, particles is shown by the continuous hne. It is to be noticed that, at least in the J A D E scheme in which y, neglecting the mass o f the hght hadrons which are effectively detected, represents an actual lnvarlant mass, there is a natural lower limit to the allowed range in Ycut for b-quark production, which corresponds to the lnvarlant mass o f the b, y~n ,~ (mb/Mz ° )2 ~ 3 × 10 -3. At lower Ycut one begins to resolve the decay products o f the heavy quark In fig. 3 we also give, in the J A D E scheme, the cross section for e+e - ~ q(tggT, summing over five flavours In this case we have taken for the c-quark mc = 1 7 GeV. Additional cuts are necessary in order to screen our results from the colhnear smgularltles due to initial state radiation We have required ptr > 5 GeV and I cos 0 r I< .72; these cuts are similar to the selection criteria adopted by the LEP collaborations. When our results for five-jet production are compared with the data presented by A L E P H [ 1 ] and O P A L [2] it is clear that the absolute n o r m a h z a t l o n is about a factor o f five too small. The simplest explanation for this discrepancy is our choice for as In fact we have used as = 115 which corresponds t o Q 2 = Mz20with A~-~ = 200 MeV wRh five active flavours in the standard formula
1 [ bllog(log(Q2/A2))] o~s(Q) - bolog(Q2/A2 ) 1 bElog(Q2/A2 )
.
(49)
The analysis o f shape variables and jet rates to 69 (as2) has shown that, in order to get agreement between the data and the theoretical predictions, the scale o f the strong coupling constant has to be chosen to be Q = xuMzo, with xu ~ 0.1 [2]. It has later been shown that when the relevant logarithms are properly resummed [14] 62
Volume 323, number 1
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0 95 0 90 0 85
z I"'~
0 80 O 75 0 70 0 65 0 60
/ 0 55
/
DURHAM
0 50
055
0 005
O OlO
0 015
0 02:0
0 95 0 9O 0 85
J
0 8O 0
0 75
-
0 70
-
2b3g/2d3g
/ 0 65
........
2u2bg/2u2dg
/
0 60
2d2bg/2d2sg
0 55
JADE
0 50 I
0 O0
I
r
4
I
0 01
,
,
o o2
,
,
I 003
004
Ycut
Fig 4 Ratio of massive to massless cross sections for tr (2b3g)/a (2d3g) (continuous hne), tr (2u2bg)/tr (2u2dg) (dashed hne) and tr(2d2bg)/a(2d2sg) (dotted hne) in the JADE and DURHAM recombination schemes as a function of Ycut.
agreement 1s obtained for much larger values of the scale, x u ~ 1. [ 15 ]. It therefore not surprismg that our tree level expressions require a relatively small scale in order to describe the data. A related Issue, for processes with massive quarks, is the choice of the scale of the r u n n i n g mass Within our tree-level approach we have mgnored this problem, using m b = 5 GeV. The cross section for 2b3j is quite large, of the order of several hundreds pb at ycut= 001 an the D U R H A M scheme and ycut = .005 in the JADE scheme, and it remains sizable out to ycut ~ 01 and to y~ut ~ .02 respectively. Over this limited range a (Your) decreases by more than one order of magnitude This behavlour should be clearly observable with an efficiency of order .3 for detecting each b-quark It will be more difficult to detect the production of 4bj, whose cross section is only a few pb at the lowest values ofy~.t considered in this paper In fig 4 we present the cross section ratios a ( 2 6 3 g ) / a ( 2 d 3 g ) (continuous line), a (2u2bg)/a (2u2dg) (dashed hne) and tr(2d2bg)/tr(2d2sg) (dotted line) in the two recombination schemes as a function of Y~t These curves confirm our previous conclusions that mass effects increase with the n u m b e r of final state light partons. The ratio for the d o m i n a n t 2q3g production process is equal to 58 at Ycut= 001 in the D U R H A M scheme and to 67 at Ycm = .005 in the JADE scheme. It is even smaller for the processes with four quark jets in the final state. This corresponds to a 6 - 8 % decrease in the predictions for the total five-jet cross section An estimate of the effects due to the charm mass can be obtained from the ratio a (2c2g7)/tr (2u2gT) which is equal to .93 at Yc~t = 005 in the JADE scheme. The corresponding value for a(2b2gT)/tr(2d2gT) is .63, slightly smaller than in the case of 2q3g production. Five-jet production will be visible also at higher energies At ~ = 200 GeV we obtain, with Yc,t = .005 m the JADE scheme, a ( e + e - ~ dd3g) = 9.8 × 10 - 2 p b and a ( e + e - ~ b[~3g) = 8 8 × 10 - 2 p b , showing that the mass of the b decreases the cross section by 10% even at the highest energy envisaged for LEP II. 63
Volume 323, number 1
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4. Conclusions The cross section for the production of b-quarks in five-jet events has been computed using hehclty amphtudes and the Z - f u n c t i o n formahsm The differences with the corresponding massless rates and the total cross sections for 2b3j and 4bj production have been studied at the Z ° peak. Mass effects result in a 30T40% reduction on individual cross sections and in a 6+8% reduction of the total five jet production. The produchon of a hard photon in association with four jets has been investigated
Acknowledgement We gratefully acknowledge the collaboration of S. Morettl in checkang the amplitudes for purely hadronlc final states and the collaboration of M. Ughettl for the calculation of the processes with a hard photon
References [1] L3 Collab, B Adeva et al, Z Phys C 55 (1992) 39, ALEPH Collab, D Buskuhc et al, Z Phys C 55 (1992) 209 [2] DELPHI Collab., P Abreu et al, Z Phys C 54 (1992) 55, OPAL Collab, P.D Acton et al, Z Phys C 55 (1992) 1 [3] L3 Collab, B. Adeva et al, Phys Lett B 271 (1991) 461, DELPHI Collab, P. Abreu et al, Phys. Lett B 307 (1993) 221, OPAL Collab, R. Akers et al, prepnnt CERN-PPE/93-118, July 1993 [4] A. Ballestrero, E Mama and S Morettl, Phys Lett B 294 (1992) 425 [5] A Ballestrero, E Mama and S Morettl, Torlno preprlnt DFTT 53-92, October 1992, to appear m Nucl Phys B [ 6 ] N K Falck, D. Graudenz and G Kramer, Nucl Phys B 328 (1989) 317 [7]N Brown, Z Phys C 51 (1991) 107 [8] R Klelss and W J Stlrllng, Nucl Phys B 262 (1985) 235 [9] C Mana and M Martlnez, Nucl Phys B 287 (1987) 601 [10] K Haglwara and D Zeppenfeld, Nucl Phys B 274 (1986) 1 [ 11 ] Mathematlca is a registered trademark of Wolfram Research, Inc S Wolfram, Mathematlca (Addison-Wesley, Redwood City, CA, 1991) [12] JADE Collab, W Bartel et al, Z Phys C 33 (1986) 23, JADE Collab, S. Bethke et al., Phys Lett B 213 (1988) 235 [13]N Brown and W J Stlrllng, Phys Lett B 252 (1990) 657, Z Phys C 53 (1992) 629 [14] S Catanl, L Trentadue, G Turnock and B R Webber, Phys Lett B 263 (1991) 491, S Catanl, L Trentadue, G Turnock and B R. Webber, preprlnt CERN-TH 6640/92, S Catanl, G Turnock and B R Webber, Phys Lett B 272 (1991)368, B 295 (1992)269, G. Turnock, Cambridge preprlnt Cavendlsh-HEP-92/3, Ph D. Thesis, U of Cambridge (1992), J C Colhns and D E Soper, Nucl Phys B 193 (1981) 381, B 197 (1982) 446, B 213 (1983) 545 (E), B 284 (1987) 253, J Kodalra and L Trentadue, Phys Lett B 112 (1982)66, R. Flore, A Quartarolo and L. Trentadue, Phys Lett B 294 (1992) 431, S Catam, YuL Dokshltzer, M Olsson, G Turnock and B.R. Webber, Phys Lett. B 269 (1991) 432, S Catanl, Yu L Dokshltzer, F Floram and B.R Webber, Nucl Phys B 377 (1992) 445 [15] OPAL Collab, P D Acton et al, preprlnt CERN-PPE/93-38, March 1993, DELPHI Collab, P Abreu et al., preprlnt CERN-PPE/93-43, March 1993
64
PHYSICS LETTERS B
Five jet production with heavy quarks at e + e- colliders* Alessandro Ballestrero 1 and Ezio Maina 2 Dtparttrnento dl Ftstca Teortca, Untverstth dt Tonno, Turm, Italy and INFN, Seztone dt Tonno, v Gturza 1, 10125 Turm, Italy
Recexved 24 November 1993 Editor R Gatto
Heavy quark production m five jet events at e+e - colhders is studied at tree level using hellclty amphtudes Total production rates for 2b3j and 4bj are gtven and compared with the corresponding results for massless quarks The process e+e - ~ q(tgg? which is the dominant contribution to 4J7 produchon is briefly dascussed
1. Introduction The great n u m b e r o f hadronlc decays of the Z ° observed at LEP provides the opportunity to test our understanding o f strong lnteracttons an unprecedented detml Large samples of multi-jet events have been accumulated and analyzed [ 1,2 ]. In particular the A L E P H Collaboration and the O P A L Collaboration have studied the jet-frachon distributions as a function of ycut for up to five jets. Recent advances in b-tagging techniques based on the introduction o f vertex detectors and on a refinement of the selection procedures, with their large efficlencles and the resulting high purmes, have p a v e d the way to the study o f heavy-quark production in association with light-quark and gluon jets. The flavour Independence o f the strong couphng constant has been verified by several groups [3] using both jet-rates and shape variables It has been lately pointed out [4,5 ] that the effects of the b-quark mass are substantial and increase with the n u m b e r of jets. In this paper we plan to study heavy-quark p r o d u c h o n in five-jet events at e + e - colhders. Two different mechamsms contribute to five-jet production for hght quarks, the 'polnt-hke' annihilation o f e + e - to Z °, 7 [6], which dominates at LEP I energy, and W + W - production [ 7 ], followed by the emission of a gluon from the decay products o f the W ' s , whxch dominates above the W W treshold This latter contribution ~s severely suppressed for b-quarks by the smallness o f the Fbc element o f the C K M matrix and by the large mass o f the top and will not be considered here. W e have studied the anmhflatlon m e c h a m s m for five-jet production at tree level taking full account ofT, Z interference and o f quark masses Therefore our matrix elements can be used at all e + e - colhders. As a b y p r o d u c t o f the m a i n calculation we have also examined the d o m i n a n t contribution, e+e - ~ q(tggT, to the p r o d u c h o n o f a smgle hard p h o t o n in assoc~atlon w~th four jets. Both initial state and final state radlat~on have been included at tree level, without ISR resummatlon We have computed all matrix elements both in the formahsm of [8,9] and 111 that o f [10] using the former for the actual calculation. The relevant formulae are summarized m the following section Both methods can * Work supported in part by Mlmstero delrUnlversltfi e della Rlcerca Sclentlfica 1 E-mad baUestrero@to lnfn it. 2 E-mall [email protected] it 0370-2693/94/$ 07.00 @ 1994-Elsevier Science B V All rights reserved SSDI 0 3 7 0 - 2 6 9 3 ( 9 4 ) 0 0 0 2 8 - 6
53
Volume 323, number 1
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3 March 1994
6
4 5
6
3 4
3
5 7
(1)
4 5
7
(2)
7
(3)
4
(41
3
4
4 5
5
7
(51
6
7
7
(61
6
7
6
7
(r)
7
(8) 6
4
(9) 6
6
3
(10)
4
(11)
7
7 (12)
Fig 1 Representative diagrams contributing to e + e - ~ q ~ I g g g All others diagrams can be obtained through permutations of the gluon labels The contrlbuUon of the two diagrams involving a four-gluon vertex, which are not shown, are mluded in the analytmal expression given in the text for the chagrams with two connected three-gluon vertices
be easily i m p l e m e n t e d in a small set o f nested subroutines. This however results in a computer program which is far too slow for the present calculation. We have mstead used the symbohc package Mathemattca [ 11] to write down the Fortran expression for each hehclty amphtude. The individual Z-functions are computed first and then c o m b m e d in larger structures that describe chunks of increasing size o f the F e y n m a n diagrams shown 111 figs. 1-2. Every sub-diagram is saved a n d then used several times. With this procedure we have produced a rather large piece o f code, which however runs quite fast, and therefeore can be used in high statistics Monte Carlo runs. As an example the program for q~ggg production is about 24,000 hnes long, but reqmres only about 54
Volume 323, number 1
PHYSICS LETTERS B 6
3 March 1994
6
7
6
7
(13)
7
(14) 6
(151 6
4
4
6
5
5
3 3
(16)
7
(17)
6
7
(18)
6
7
6
4 3
(19)
7
5
(2O)
6
3
(21)
6
7
6
4
7 (2~)
7
7 (~)
7 (~4)
Fig 2 Representative diagrams contributing to e + e - ~ qclqq The remmmng diagrams can be obtained through simultaneous substitutions q(Ps,25) ~ q(P7,~,7) and q(p4,),4) ~ q(p6,26) The addttlonal ones needed for identical flavour can be obtained exchanging q(Ps, 25 ) with ~ (PT, 27 ) including their color quantum numbers and changing the overall sign
5 × 10 -2 s e c o n d s t o e v a l u a t e o n a V a x s t a t l o n 4 0 0 0 / 9 0 . T h e a m p h t u d e s h a v e b e e n c h e c k e d f o r gauge m v a n a n c e . W e h a v e u s e d Mz = 91.1 G e V , F z = 2 5 G e V , s m 2 ( 0 w ) = .23, m b = 5 G e V , aem = 1 / 1 2 8 a n d as = .115 i n t h e n u m e r i c a l p a r t o f o u r work. 55
Volume 323, number 1
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Table 1 The Z-functions for all independent hehclty combmatmns m terms of the functions s, t, tl and # defined in the text The remaining Z-functions can be obatlned changing sign to the hellcRles and exchanging + with - , s with t and R with L 21222324
! Z(Pt,~.l,P2,'2-2,P3,'~3,P4,~'4, CR, C L, C R, CLt x1
++++ +++++-+ +-++ ++-+-++--+
- 2 [s(3, 1 ) t(4, 2)dRCR - #I#2Yl3Y]4CtRCL -- ~]ffl2fl3fl4CtLCR] -2q2c R [s (4, 1 )/z3c~ - s (3, 1 )/t4c~] -2t/lCR It (2, 3)~4c ~ - t(2, 4)#3c~] -2rl4C ~ [s (3, 1 )It2cR -- s (3, 2)#ICL] --2 [S( 1, 4) t(2, 3)C~CR -- #ll~2q3q4CtLCL -- t/lr/2,u 3/Z4C~C/~] 0 - 2 [#1 # 4112q 3¢tLCL "l- # 2f13q 1q 4CtRCR -- # 2fl 41]1173CtLCR -- fl Ill 3712?142 tl~ 1-] -- 2r/3C~ [S (2, 4)ltlCL -- S ( l, 4)IL2CR ]
..1-
--
____
Table 2 The X-functions for the two independent behclty combinations m terms of the functions s, t, r/ and/~ defined in the text The remalmng X-functions can be obatlned changing sign to the hehcltms and exchanging + with - , s with t and R with L 2123
X (Pl,21,P2,P3,,~.3)
+ + +-
(/tit/2 + / t 2 q l ) (#3q2 + #2r/3) + s(1, 2) t(2, 3) (#1t/2 +/z2r/1)s(2, 3) + (/z2t/3 + #3q2)s(1,2)
2. Calculation W e present the h e h c l t y a m p h t u d e s in the f o r m a h s m o f [8,9] writing the arnphtudes are.
T h e two basic functions which are n e e d e d m
Z (p,, 2,, Ps, 2j, Pk, ).k, Pl, 2l, CR, CL, C'R, C~) = [ft ( p . 2z )-FUu (p j, 2j ) ] [ft (Pk, J.k)F;U (Pt, 2t) ],
( 1)
where
F (')u = yU(c~')p R + C~')pL)
(2)
and
X (p,,2.pj;pk,2k
) = [ft(pz,2,)l) j U ( P k , 2 k ) ]
(3)
Thear expressions in t e r m s o f splnor p r o d u c t s s (l, j ) = ~ ( p . + )u ( P s , - ) = t* (j, z ), tl a n d / z functions can be f o u n d m tables 1 a n d 2. A c o n v e n i e n t , t h o u g h n o n - u n i q u e , choice for the splnor p r o d u c t s and for the funcUons t / a n d / z is:
o
x)
s O , j ) = (p', + ~p:) { P _ ; - P _ 2 \ pO, - px, r/(z) = ( 2 ( p ° - p ~ ) )
V2 ,
(p
1/2 -
_p,)1/2
o x (P; + ~P: ) \ p , o - p,~
,
I~(,) = ± m,
~(~)
(5) '
where the sign + ( - ) refers to a particle ( a n t l p a m c l e ) o f mass m,. 56
(4)
Volume 323, number 1
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PHYSICS LETTERS B
Table 3 The color factors, multiplied by 9, for e+e - ---, q~tggg
345 453 534 543 354 435
345
453
534
543
354
435
64 1 1 10 --8 --8
1 64 1 --8 10 --8
1 1 64 --8 --8 10
10 --8 --8 64 1 1
--8 10 --8 1 64 1
--8 --8 10 1 1 64
The polarization vectors o f a gluon o f m o m e n t u m p, can be written as (6)
8# (p~,2) = Nz [~t(p,,2)TUU(pa(z),2) ], where Pa(z) lS an auxiliary massless m o m e n t u m not parallel to p, and the normalization factor p , ) ] - 1 / 2 All a m p h t u d e s are independent o f the Pa(t)'s There are fifty F e y n m a n diagrams contributing to e-(pb2)
+ e+ ( p 2 , - 2 ) ~ q(p6,26) -F q(p7,27) -b g(P3,23) -b g(P4,24) -b g(Ps,25)
lS N l =
[4(pa(,)
(7)
Some o f them are shown in fig. 1. All others can be obtained through permutations o f the gluon labels. The two diagrams with four-gluon vertices are not shown. Their contribution is included in the expressions gwen below for the diagrams wzth two connected three-gluon vertices The amplitude squared for this process is. 2 1 l g 3 e Z N 3 N 4 N 5 × ~ ~ ~l/t(,t}~1,¢().).(. , IMIqoggg = 3 /--, z-, ' {;Q n,n' -
-
(8)
where ~r is a permutation o f the indexes (3, 4, 5), {2} indicates the hehcltles o f all external parhcles, M . is the coefficient o f the color matrix Tl~ = (Tn(3)Tn(4)TrC(5))ts,
(9)
in the full a m p l i t u d e and C ~ , is the appropriate color factor, Cn~, = T r ( T ~ ( T ~' )t ). The color factors are given in table 3. To fix our notation, diagrams 1-4 have color structure 345, diagrams 5-7 have color structure [3,4] 5 where [3,4] is the usual commutator, [3,4] = 3 4 - 4 3 , diagrams 8-10 have color structure 5 [ 3,4 ] and diagrams 11-12 have color structure [ [ 3,4 ], 5 ] The production o f four quarks and one gluon e- (pl,2) + e+(p2,-2)
--~ g(P3,23) + q(P4,24) + q(Ps,25) -I- q(P6,26) -b q(P7,27),
(10)
IS described in lowest order by twenty-four F e y n m a n diagrams if the two quarks have different flavour (case I) and by forty-eight if their flavour is the same (case II). Twelve o f the diagrams for case I are shown in fig. 2, all others can be obtained with the simultaneous substitutions ~ (Ps, 25 ) ~ q (PT, 27 ) and q (P4, 24) ~ q (P6, 26 ). The additional ones needed for identical flavour can be obtained exchanging ~ (Ps, 25 ) with ~ (P7, 27 ) including their color q u a n t u m numbers and changing the overall sign 57
Volume 323, number 1
PHYSICS LETTERS B
3 M a r c h 1994
Table 4 The color factors, multiplied by 9, for e + e - ~ qlqlq2q2
A1 A2 Aa A4 A5
A1
A2
A3
A4
As
24 -3 -6 21 -27
-3 24 21 -6 27
-6 21 24 -3 27
21 -6 -3 24 -27
-27 27 27 -27 54
Table 5 The addmonal color factors, multiplied by 9, needed for quarks of identical flavour e + e - ~ qglq(l.
B1 B2 Ba B4 B5
A1
A2
A3
-8
l 10 1 -8 9
10 1 -8 1 -9
1 10 1 9
A4
A5
1
9 9 -9 -9 0
-8 1 10 -9
T h e a m p l i t u d e s q u a r e d for t h e s e p r o c e s s e s is: 24 1 32~ [~ 2 [qoq,o,g= ~g;e iv3 ~--~ ~-~M,
(11)
(2] t , j = l 48
IMlqo,o~ -
4 (21) 2g3e2N3 ~
Z
C:j,
(12)
{,t} ~,j= 1 In case I a n d case II respectively. C,j a n d C[j are t h e a p p r o p r i a t e t a b l e 4 a n d 5. We define the following colour structures
Am llJl "~ = (TaT e) llJ 1 (T b) t2J2
A a , 2 J 2 = (TbTa),,j 1
A a t232 = 4 llJ 1
A a t',J2 = 5 llJ 1
(T b)
(Tbra] llJ I x .'12j 2 ~
~2 llj 1
(Tb)
llJ1
b
2
c (T)/2J2 Cabc
c o l o u r factors w h i c h c a n b e e x t r a c t e d f r o m
A a ,zJ2
=
b
(raTb),zj z (13)
and a t2j 2
a 12J1
Bn qJt = An llJ2"
(14)
T h e c o l o r f a c t o r is g i v e n by:
C.m
1212A t am 22z2 j~,,.
= A .~ ' l J , ~
(15)
i n case I a n d b y a n a l o g o u s f o r m u l a e i n case II, p o s s i b l y w i t h An,m --+ Bn,m W e a d o p t t h e f o l l o w i n g s h o r t h a n d n o t a t i o n , w h e r e i n t h e left h a n d side t h e i n d e x t s t a n d s b o t h f o r t h e m o m e n t u m p, a n d f o r t h e c o r r e s p o n d i n g p o l a r i z a t i o n 2~, a n d [t] s t a n d s for t h e set o f f o u r l n d m e s a s s o c m t e d w i t h a g l u o n m o m e n t u m a n d to h i s a u x l h a r y m o m e n t u m
X(t,j,k) 58
= X(Pz,,~,Pj;Pk,~k),
(16)
Volume 323, number 1
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PHYSICS LETTERS B
X ( [ t ] , y ) = X(p,,2z,pj,pa(,),2z),
(17)
Z(t,j;k,l)
(18)
= Z(P,,2,,Pj,2j,Pk,Xk,Pt,2I, 1, 1; 1, 1),
Z ( [ t ] , J , k ) = Z(P,,2,;Pa(,),2z;Pj,2j,Plc, 2k, 1, 1, 1, 1),
(19)
Z ( [t]; [ J ] ) = Z(p,,X~,pao),~,;pj,2j;pao),J,j; 1, 1, 1, 1),
(20)
Ze(l,J)
=
Z (p~,,~z,pj,~j,pz,2,pb2, 1, 1; 1, 1) x dy + Z (I,X~,J,~j, 2,),, 1,2;CR, CL, C'R,C'L) X dzo ,
(21)
where dr, dzo are the p h o t o n and Z ° propagator functions. The attachment o f two external gluons to a quark hne through a three-gluon vertex is conveniently described with the help of:
Z2([t,j],k,l) = Z([t]; [j]) × (X(k;j,l)-X(k,t;l))
+2(X([j];t)×Z([t];k,I)-X([z];j)
xZ([j];k,l)). (22)
The function that descrxbes the insertion o f two connected three-gluon vertices, and of the term with the same color structure in the associated four-gluon vertex, on a fermlon hne is: Z3([l,J], [k];l,m)
= 4Z([t];l,m) x X([j];z) x (X([k],j) +X([k];t)) + X([t],j) x (X([j];t)
+ 2X([j],k)))
× (Z([t]; [j]) × X([k],j)-
+2Z([k],l,m)
× ( Z ( [ l ] , [ j ] ) x p , Pk
+ (X(l;k,m) - X(l,t,m) - X(l,j,m))
2Z([k]; [j]) × X([1],j))
+ 2X(l;j;m) × Z([t]; [j]) × (X([k],j)
+ X ( [ k ] ; t ) ) - 2 Z ( [ t ] , l , m ) x Z ( [ k ] , [ j ] ) x Pz Pj - (t +-~ j ) .
(23)
The d e n o m i n a t o r associated with this function is (p~ + pj )2 (p, + p~ + Pk)2 In order to slmphfy our formulae we make the convention that all repeated indices are s u m m e d over, and that the sum extends to all possible values, 2 = t l , o f all internal hehcltles. The last line m eq 25-48 gives the set o f values taken independently by each m o m e n t u m index. Defining the sign factors e (z) as:
e(t) =
-1 +1
t = 1,2 t > 2 '
(24)
the s p m o r functions necessary for q(tggg production, which correspond to the diagrams shown in fig. 1 are" M1 = - Z ( [ 3 ] ; 6 ,
t) x Z ( [ 4 ] , t , j ) x Z ( [ 5 ] , j , k ) x Ze(k, 7) x e ( k ) ,
Pl ----"{P6,P3}, Pj = {P6,P3,P4}, Pk = {Pl,P2,PT}, M2 = - Z ( [ 3 ] , 6 , / )
x Z([4];t,J)
(25)
x Ze(J,k) x Z([5],k, 7),
p, = {p6,p3}, pj = {P6,p3,p4}, pk = {ps,p7},
(26)
M3 = Z ( [ 3 ] , 6 , t) x Ze(z,J) x Z ( [ 4 ] , j , k ) x Z ( [ 5 ] ; k , 7 ) , P, = {P6,P3}, Pj = {P4,Ps,P7}, Pk = {Ps,P7},
(27) 59
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x Z ( [ 5 ] , k , 7) × e O ) ,
Pl -" {P6,Pl,P2}, Pj = {P4,P5,PT}, Pk = {Ps,P7},
(28)
3"/5 = - Z 2 ( [ 3 , 4], 6, t) x Z ( [5], l , J ) x Ze(J, 7) x e ( J ) ,
Pl = {P6,P3,P4}, Pj = {Pt,P2,PT},
(29)
M6 = - Z 2 ( [ 3 , 4 ] , 6 , t) x Ze(Z,J) X Z ( [ 5 ] ; J , 7 ) ,
(30)
Pt = {P6,P3,P4}, Pj = {P5,P7}, M7 = - Z e ( 6 , t) x Z 2 ( [ 3 , 4 ] ; t , j )
x Z ( [ 5 ] , j , 7) x e ( t ) ,
P, = {Pl,P2,P6}, Pj = {Ps,P7}, Ms = - Z ( [ 5 ] , 6 ,
t) × Z z ( [ 3 , 4 ] , z , . / ) × Ze(J, 7) x e ( J ) ,
P, = {Ps,P5}, Pj = {PbPz,P7}, M9 = - Z ( [ 5 ] , 6 ,
(31)
(32)
t) x Z e ( t , J ) X Z 2 ( [ 3 , 4 ] ; J , 7 ) ,
P~ = {P6,Ps}, Pj = {P3,P4,P7},
(33)
MlO = - Z e ( 6 , l) × Z ( [51, l , j ) × Z2( [3, 4 ] ; j , 7) x e ( t ) ,
Pl =- {Pl,P2,p6}, Pj -=- {P3,P4,p7},
(34)
Mll = - Z 3 ( [ 3 , 4 1 , [ 5 ] , 6 , t) × Z e ( l , 7 ) x e ( t ) , (35)
P, -'= {Pl,P2,P7}, M12 = Ze(6, t) x Z 3 ( [ 3 , 4 ] , [ 5 1 , / , 7 ) × e ( l ) ,
(36)
P, = {Pl,P2,P6}, The s p m o r functions for the diagrams m fig. 2, which describe q(tq(tg p r o d u c t i o n are M13 = Z ( [ 3 ] , t , 5) × Z ( 4 , t , 6 , j ) x Ze(J, 7) × e ( j ) , Pz = {P3,Ps}, P.1 ---- {Pl,P2,P7}, M14 = - Z ( [ 3 ] ; 4 ,
Z) x Z ( t , 5,6, J) × Ze(J, 7) × e ( J ) ,
P~ = {P3,P4}, Pj = {Pl,PZ,PT}, M15 = - Z ( [ 3 ] ,
(38)
l, 5) × Z ( 4 , t , j , 7) × Ze (6, J ) × e ( J ) ,
P, ---- {P3,PS}, Pj = {Pl,P2,P6},
6O
(37)
(39)
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=
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Z ( [ 3 ] , 4 , z) × Z O , 5 ; j , 7) x Z ~ ( 6 , j ) × e ( J ) , (40)
P~ = {P3,P4}, P; = {Pl,P2,P6}, M17 = - Z ( 6 , / , 4 , 5 )
x Z([3];z,3)
x Z e ( J , 7) x e ( J ) ,
(41)
P, = {P4,Ps,P6}, Pj = { P l , P z , P 7 } , M18 = - Z ( 6 ,
1,4, 5) × Z e ( l , J ) × Z ( [ 3 ] , J ,
7),
(42)
P, = {P4,Ps,P6}, Pj = {P3,PT}, m19
=
- Z e ( 6 , l) × Z ( t , J , 4 , 5 ) × Z ( [ 3 ] , J , 7) x e ( l ) , (43)
P, = {PbP2,P6}, Pj = {P3,P7}, M2o = - Z ( [ 3 ] ; 6 , t) × Z ( t , J , 4 , 5 ) x Ze(J, 7) x e ( J ) ,
(44)
P, = {P3,P6}, Pj = {Pl,P2,P7}, t) × Z e ( l , J ) × Z ( J , 7 , 4 , 5 ) ,
M21 = - Z ( [ 3 ] ; 6 ,
(45)
Pz = {P3,P6}, Pj = {P4,Ps,P7}, M22 = --Ze(6, l) × Z ( [ 3 ] , t , J ) × Z ( J , 7 , 4 , 5 ) x e ( t ) ,
(46)
Pl = {Pl,P2,P6}, Pj = {P4,Ps,P7},
M23 = Ze(6, t) × ( 2 ( X ( [ 3 ] , 4 ) + X ( [ 3 ] , 5 ) ) + X ( t , 5;7) - X ( t ; 3 , 7 ) )
× Z([3];4,5))
× Z(z,7,4,5) - 2X(4,3;5) × Z([3];t,7) - (X(t,4,7) × e(t),
Pz = { P b P 2 , P 6 } ,
M24 = - Z e ( l , 7 )
(47)
× (2 ( X ( [ 3 ] , 4 ) + X ( [ 3 ] , 5 ) )
+ X(6,5;t) - X(6,3, t)) × Z([3];4,5))
P, = { P l , P z , P 7 } .
× Z(6, t;4,5)-
2 X ( 4 ; 3 , 5) × Z ( [ 3 ] ; 6 , t ) -
(Y(6,4;t)
× e(t), (48)
F r o m the examples above the one can to construct the full amphtudes for 5j and 4j y production with little effort.
3. Results Our results are presented in figs. 3 and 4. In fig. 3 we show the total cross section for five-jet production as a function o f Ycut in the J A D E scheme [ 12 ] and in the D U R H A M scheme [13 ] We have neglected the mass o f the quarks o f the first two generations, and have s u m m e d over all their contributions. The cross section for four b-quark plus one gluon-jet production is given by the dotted hnes The dashed lines represent the cross 61
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lel ........
10'
2b 4b
-~1oo 10-1 DURHAM 1 0 -z
, 0
,
i
,
000
.
i
r
,
,
,
0 005
.
.
.
i
i
i
,
i
,
0 010
.
.
.
.
i
i
i
,
i
,
0 015
.
.
.
.
E
I
,
0 020
.
.
.
.
i
.
I
i
Ob -..
........
2b
ld
~w b
10-'
10-~
JADE I
O0
I
,
,
I
I
,
I
,
0 O!
~
,
,
0 02
I
I
I
0 03
I
I
I
I
0 04
Yeut
Fig. 3. Total cross section as a function of Ycut in the JADE and in the DURHAM scheme for four b-quarks plus one gluon-jet production (dotted line), for the production of two b-quarks plus any comblnatlon of three light-patton jets (dashed line), and for the sum of all contributions to five-jet producUon including only light partons (continuous line) section for the production o f two b-quarks plus any combination o f three hght-parton jets. Finally the sum o f all contributions to five-jet production including only hght, strongly interacting, particles is shown by the continuous hne. It is to be noticed that, at least in the J A D E scheme in which y, neglecting the mass o f the hght hadrons which are effectively detected, represents an actual lnvarlant mass, there is a natural lower limit to the allowed range in Ycut for b-quark production, which corresponds to the lnvarlant mass o f the b, y~n ,~ (mb/Mz ° )2 ~ 3 × 10 -3. At lower Ycut one begins to resolve the decay products o f the heavy quark In fig. 3 we also give, in the J A D E scheme, the cross section for e+e - ~ q(tggT, summing over five flavours In this case we have taken for the c-quark mc = 1 7 GeV. Additional cuts are necessary in order to screen our results from the colhnear smgularltles due to initial state radiation We have required ptr > 5 GeV and I cos 0 r I< .72; these cuts are similar to the selection criteria adopted by the LEP collaborations. When our results for five-jet production are compared with the data presented by A L E P H [ 1 ] and O P A L [2] it is clear that the absolute n o r m a h z a t l o n is about a factor o f five too small. The simplest explanation for this discrepancy is our choice for as In fact we have used as = 115 which corresponds t o Q 2 = Mz20with A~-~ = 200 MeV wRh five active flavours in the standard formula
1 [ bllog(log(Q2/A2))] o~s(Q) - bolog(Q2/A2 ) 1 bElog(Q2/A2 )
.
(49)
The analysis o f shape variables and jet rates to 69 (as2) has shown that, in order to get agreement between the data and the theoretical predictions, the scale o f the strong coupling constant has to be chosen to be Q = xuMzo, with xu ~ 0.1 [2]. It has later been shown that when the relevant logarithms are properly resummed [14] 62
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0 95 0 90 0 85
z I"'~
0 80 O 75 0 70 0 65 0 60
/ 0 55
/
DURHAM
0 50
055
0 005
O OlO
0 015
0 02:0
0 95 0 9O 0 85
J
0 8O 0
0 75
-
0 70
-
2b3g/2d3g
/ 0 65
........
2u2bg/2u2dg
/
0 60
2d2bg/2d2sg
0 55
JADE
0 50 I
0 O0
I
r
4
I
0 01
,
,
o o2
,
,
I 003
004
Ycut
Fig 4 Ratio of massive to massless cross sections for tr (2b3g)/a (2d3g) (continuous hne), tr (2u2bg)/tr (2u2dg) (dashed hne) and tr(2d2bg)/a(2d2sg) (dotted hne) in the JADE and DURHAM recombination schemes as a function of Ycut.
agreement 1s obtained for much larger values of the scale, x u ~ 1. [ 15 ]. It therefore not surprismg that our tree level expressions require a relatively small scale in order to describe the data. A related Issue, for processes with massive quarks, is the choice of the scale of the r u n n i n g mass Within our tree-level approach we have mgnored this problem, using m b = 5 GeV. The cross section for 2b3j is quite large, of the order of several hundreds pb at ycut= 001 an the D U R H A M scheme and ycut = .005 in the JADE scheme, and it remains sizable out to ycut ~ 01 and to y~ut ~ .02 respectively. Over this limited range a (Your) decreases by more than one order of magnitude This behavlour should be clearly observable with an efficiency of order .3 for detecting each b-quark It will be more difficult to detect the production of 4bj, whose cross section is only a few pb at the lowest values ofy~.t considered in this paper In fig 4 we present the cross section ratios a ( 2 6 3 g ) / a ( 2 d 3 g ) (continuous line), a (2u2bg)/a (2u2dg) (dashed hne) and tr(2d2bg)/tr(2d2sg) (dotted line) in the two recombination schemes as a function of Y~t These curves confirm our previous conclusions that mass effects increase with the n u m b e r of final state light partons. The ratio for the d o m i n a n t 2q3g production process is equal to 58 at Ycut= 001 in the D U R H A M scheme and to 67 at Ycm = .005 in the JADE scheme. It is even smaller for the processes with four quark jets in the final state. This corresponds to a 6 - 8 % decrease in the predictions for the total five-jet cross section An estimate of the effects due to the charm mass can be obtained from the ratio a (2c2g7)/tr (2u2gT) which is equal to .93 at Yc~t = 005 in the JADE scheme. The corresponding value for a(2b2gT)/tr(2d2gT) is .63, slightly smaller than in the case of 2q3g production. Five-jet production will be visible also at higher energies At ~ = 200 GeV we obtain, with Yc,t = .005 m the JADE scheme, a ( e + e - ~ dd3g) = 9.8 × 10 - 2 p b and a ( e + e - ~ b[~3g) = 8 8 × 10 - 2 p b , showing that the mass of the b decreases the cross section by 10% even at the highest energy envisaged for LEP II. 63
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4. Conclusions The cross section for the production of b-quarks in five-jet events has been computed using hehclty amphtudes and the Z - f u n c t i o n formahsm The differences with the corresponding massless rates and the total cross sections for 2b3j and 4bj production have been studied at the Z ° peak. Mass effects result in a 30T40% reduction on individual cross sections and in a 6+8% reduction of the total five jet production. The produchon of a hard photon in association with four jets has been investigated
Acknowledgement We gratefully acknowledge the collaboration of S. Morettl in checkang the amplitudes for purely hadronlc final states and the collaboration of M. Ughettl for the calculation of the processes with a hard photon
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