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Hard bremsstrahlung processes at LEP II
Volume 262, number 1
PHYSICS LETTERS B
13 June 1991
Hard bremsstrahlung processes at LEP II A. Aeppli a,b,1 and D. Wyler a a InstitutJ~rtheoretischePhysik, UniversitdtZiirich, CH-8001 Ziirich, Switzerland b Physics Department 2 Brookhaven National Laboratory, Upton, N Y 11973, USA
Received 25 March 1991
The matrix element for single hard photon bremsstrahlung in the dominant LEP II process e+e - ~W +W---,4f with massless fermions is given. An appropriate Monte Carlo event generator is described and the results for total cross sections and the photon spectrum are presented. The narrow width approximation for the W and the background are discussed.
1. Introduction LEP II, the electron-positron storage ring with a CM energy x/~ ~ 2Mw promises to directly probe the three-gauge boson vertex and yield a precise determination of the W mass, Mw. This measurement allows a very stringent test o f the standard model: the three quantities e, GF and Mz fix the three unknown gauge parameters of the model uniquely, making Mw a true prediction which only depends on m t and, more weakly, on the mass of the Higgs boson [ 1-3 ]. Consequently, several methodes o f accurately fixing Mw at LEP II have been devised [4]. At LEP II, a W + W - pair is produced. Each W decay into a fermion pair; the cross section for the process e+e ---, W W ~ 4 fermions (as function of V~) has a characteristic shape and magnitude which depends on Mw [ 5,6 ]; we also expect the threshold behaviour to reflect the kinematical value o f Mw [ 6 ]. In view of the importance of this measurement, radiative corrections must be included. In this paper we present the (hard) photonic bremsstrahlung, which can easily be measured by observing an energetic photon in addition to the fermions. Previously, complete one-loop calculations including soft photon corrections have been presented for on-shell production of the W pair [ 7,8 ]. Combining these results with the hard photon bremsstrahlung Partially supported by SchweizerischerNationalfonds. 2 Addressafter 1 February 1991.
cross section, the full O (or) calculation has been given in ref. [ 9 ]. Since the observed process is really a fourfermion production, and there could be important threshold effects, a calculation o f this process is needed. The hard bremsstrahlung evaluated in this work constitutes the first step; the additional corrections will be presented elsewhere. As we expected the on-shell (peaking) approximation to be quite reliable for x/~> 2Mw, we present for comparison the results for this simplification. We also study the background processes and show them to be small at the energies considered.
2. The calculation 2. I. The m a t r i x element [lO]
Since m t >~Mw, all the fermions involved in the process e+e - - - , W + W - ~ 4 f can be taken to be massless for x / ~ 2Mw (the rate e + e - ~6tf1I'2 is presumably very small if m t is appreciably larger than Mw). Then, the massless helicity formalism is appropriate [ 1 1-14 ]. The collinear singularities occurring in this situation can be removed by introducing a finite mass as discussed below [ 15 ]. Since we are interested in fermions produced by a W + W - pair, we will take the outgoing fermions to be lefthanded. Then, the amplitudes depend on the helicities z and 2 of the incoming electron and the photon, respectively. The graphs that contribute are shown in fig. 1. The
0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
125
Volume 262, number l
PHYSICS LETTERS B
(1.1)
-6
(t.2)
(3.1)
13 June 1991
(6.1)
(3.2)
(6.2)
(7.3)
(7.4)
>4
(7.6)
(7.5)
(3.3)
(5.1)
(5.2)
(9.3)
(9.4)
(9.5)
(9.6)
Fig. 1. Diagrams contributing to the radiative process e+e---,4f+7 in unitary gauge.
a m p l i t u d e corresponding to the graph i is written as
cos 0w = M w / M z .
Mr(z, 2) =eA,(x, t)Jl~(z, 2 ) ,
D e p e n d i n g on whether the fine structure constant a = eE/4~z or the F e r m i constant G u is chosen as input parameter, we set g = e / s i n 0w or g = 4 M 2 x / ~ G u. The kinematics is defined in fig. 2. We write
( 1)
where e is the electric charge a n d A i ( z ) is a factor corresponding to the graph without photon. The ~ ( z , 2) are discussed below. The relevant A~(z) are
ig2 ( e 2 G cos O. g 2 As(x, t) =--~- \ x - x - M 2 + i M z e z ] -ig 4 1
A,(x, ~ ) =
4
(6)
po=pi+pj, Pijk=Pi+pj+pk, '
(7)
(2) f3, P 3
(3)
x'
f4 ,P4
with c_ -
1 COS 0 w
l - - -
C+ -- COS 0 w
126
fs,P5 (sin20w - ½) ,
(4) "{6 ,P6
sin20w,
(5)
e/
'
"Lq),., p 7
Fig. 2. The kinematics and momentum conventions.
Volume 262, number 1 s=(pI+p2)
2,
13 June 1991
where r is a suitably chosen reference momentum
t = ( p 2 - - P 5 6 ) 2,
S'= (Pl +P2 --P7) 2,
so =p~,
PHYSICS LETTERS B
r•k. t ' = ( --Pl -FP34) 2 ,
2 • S(]k= P/jk
(7cont'd)
The ~ ( z , ;t) are presented in terms of spinor products
(i+lj_)=_=a+(p~)u_(py) ,
(8)
where u(pi) are free spinors and
u+ (p) = ~ ( 1 +_~5)u(P)
(9)
It is useful and easy to split the diagrams into gauge invariant subsets. They are indicated in table 1. Within gauge invariant sets of diagrams, r must be fixed. An appropriate choice of r reduces the number of terms considerably. In order to include leptonic and hadronic (quark) decays of the W+W - pair, it is convenient to introduce a charge Qo such that
are the helicity projections. The corresponding states are denoted by LP+- >. For antiparticles
Q3 ---Qo,
v+ = ½(l-T- ~'5)v.
For leptonic decays, only Qo = 0 contributes; for the hadronic decays of the W's also Q0 = 2 must be added. The helicity amplitudes are listed in table 1. For each gauge invariant subset a convenient reference m o m e n t u m r, as indicated, has been selected; only non-vanishing contributions are given. Sets 1-8 contribute to all possible W+W - decays ( Q o = 0 ) . If the W + decays hadronically, sets 9, 11 must be added, if the W - decays into quarks, sets 10, 12. The various factors in the table are defined as follows: "radiation" factors:
(10)
An explicit form is (Po, qo> 0)
~
p~ -ip2
q,-iq2
(11)
The momenta are to be defined with respect to any axis, such that ( 11 ) can be used. A convenient choice for the z-axis is the beam direction. In order to keep (11 ) meaningful, we exclude a very small cone around the beam direction for the particle momenta. For completeness, we list some of the properties of the spinor products:
=O ,
04 = Qo - 1 ,
Q5 = Q o - 1,
~/~
r o = . < j _ 17+ >,
,
=*
(po, q o > 0 ) ,
I 12=2(p'q) •
(17)
Dok=D(Pok),
1 D(p2) = pZ-M~v + iMwFw " (12)
(16)
"propagator-functions":
Dij=D(po),
=-
Q6--Qo •
(18)
The functions F1, F2 represent the triple gauge couplings and are given by
Spinor-strings are reduced by using the Fierz identity FI ( 1, 2, 3, 4, 5, 6, 7) = [ (P34 -P56)" e+ ] (a.b)
lY#
Iq2 - >
=2
+ 2(ps6-a) (e+" b) - 2(ps4"b) (e+ .a) , (13)
together with the relation
= .
-2[ (P34+P7)'bl (e+ .a) - 2 (P~6 .e+ ) (a.b), (14)
The notation #=pu),u is employed. The polarization vector Eu (k, 2 ) of the photon with momentum k is
2 ¢ ( k , Z , r ) = x~~ < r - A I k 2 > '
F2( 1, 2, 3, 4, 5, 6, 7) = [ (2p56 +PT)'a] (e+ .b)
(15)
(19)
where
eU~=
aU=<3-1yul4-),
bu=<5-l?Ul6-) .
(20)
The explicit form is 127
Volume 262, number 1
PHYSICS LETTERS B
13 June 1991
Table 1 Helicity amplitudes for the diagrams in fig. 1. The arguments o f M o (z, 2, r) are the (incoming) electron and photon helicities z, 2 and the chosen reference momentum (see text). The table refers to Qo=0 (leptonic decays) and to Qo = ~ [see eq. (16) ]. Qo = 0
subset 1
JCl i ( +, +, 2)
~2(-,
+, 1 )
, ~ 1 2 ( + , - - , 1)
subset 2
~#H ( - , - , 2 ) ~//v4 ( +, +, 5 )
~V,a ( + , +, 5) ~ v ( + , +, 5) ~G3 (+, +, 5) ~ v ( _ , +, 5) ~V,a(--, +, 5) .av2. ( - , +, 5) ~63 ( - , +, 5)
subset 3
.//vs ( +, - , 4) j/v (+,_,4) .A/v2. ( +, - , 4) ~/33 ( + , - , 4 ) ~/v5 ( - , - , 4) ,¢tv~ ( - , - , 4) Jr'v2. ( - , - , 4) •~33 ( - - , - - , 4) ~Vlb ( +, +, 3)
~v(_,
subset 4
+, 3)
-¢/~',b ( +, - , 4 ) agVtb ( - , - - , 4 ) ~'~2b ( +, +, 5 ) ¢f/V2b(-- , + , 5 )
subset 5
J/vzb ( +, --, 6 ) J/~'Zb ( --, --, 6) J/Sl ( - , - , 4) d/v,. ( - , - , 4)
.#v ( _ , +, l) ~ / V l a ( - - , + , 1)
subset 6
JtV~b ( --, --, 4 ) d/v~b( - , +, 3)
subset 7
Jr'52 ( - , +, 5 )
subset 8
~t'v5 ( - , - , 2 ) ~/~'~. ( - , - , 2) ~/V2b( --, --, 6 )
~¢v2~( - , +, 5)
~t6~ ( - , +, 5) Qo = ]
subset 9
J/v3 ( +, - , 4) ~//v3 ( - , - , 4) ,¢/v4 ( + , +, 3)
~,v (_, +, 3) subset 10
,¢¢v ( +, - , 6 ) ~/v(_, -,6)
~ (+, +, 5) ~ v ( + , - , 5) subset 11
.//v3 ( - , - , 4)
~ v ( _ , +, 3) subset 12
~tv5 ( - , - , 6)
~¢v (_, +, 5) 128
D34D56r~2 [ ( 2 - I 1 + )FI (1, 2, 3, 4, 5, 6, 7 ) - (1,--,7) ] D34D56r12 [ ( 2 - I 1 + )F1 (2, 1, 3, 4, 5, 6, 7 ) - ( 2 ~ 7 ) ] D34D56rT2 [ ( 1 + 12 - )Ft ( 1, 2, 3, 4, 5, 6, 7) - (2*-*7) ] D34Ds6rT2 [ ( 1 + 12 - ) F~ ( 2, 1, 3, 4, 5, 6, 7 ) - ( 1.-.7 ) ] D347D56r15 [ ( 4 - 15+ )F2( 1, 2, 3, 4, 5, 6, 7) + (4,--,7) ] D34D347D56r45 ( 4 - 17 + ) ( 7 + 1-634I 5 + ) F 2 ( 1, 2, 3, 4, 5, 6, 7 ) D34D56D567r45 ( 4 - 17 + ) ( 7 + 1.6615+ ) Fz ( 1, 2, 5, 6, 3, 4, 7 ) D34D56r45 ( 7 - 1 4 + ) [ ( 7 + 10÷15+ ) ( a . b ) - ½ ( 7 + 1~15+ ) (e+.b) ] D347D56rl5 [ ( 4 - 15+ )F2(2, 1, 3, 4, 5, 6, 7) + (4,--,7) ] D34D347D56r45 ( 4 - 17+ ) ( 7 + 1-63415+ )F2(2, 1, 3, 4, 5, 6, 7) D34D56D567r45(4- 17+ ) ( 7 + 1#615+ )F2(2, 1, 5, 6, 3, 4, 7) D34D56r45(7 - 14+ ) [ ( 7 + 10_ 15+ ) ( a . b ) - ½(7+ I~15+ ) (e_.b) ] D34D567r~5 [ ( 4 + 1 5 - )F2 (1, 2, 5, 6, 5, 6, 7 ) + (5.--}7)] D34D347D56r~5 ( 7 + 15 - ) ( 7 - I/)314- )F2( 1, 2, 3, 4, 5, 6, 7 ) D34D56D567r~5 ( 7 + 1 5 - ) ( 7 -
1-65614- )F2( 1, 2, 5, 6, 3, 4, 7)
D34D56r~5 ( 5 + 1 7 - ) [ ( 7 - 1 0 + 1 4 - ) (a'b)-½ ( 7 - 1 6 1 4 - ) (e+.a) ] D34D567r~5 [ ( 4 + ] 5 - ) F2 (2, 1, 5, 6, 5, 6, 7) + (5,--,7) ] D34D347D56r~5
( 7 + [ 5 -- ) ( 7 -- 11)314-- ) Fz (2, 1, 3, 4, 5, 6, 7 )
D34D56D567r~5 ( 7 + ] 5 - ) ( 7 - [-656 ]4-- )/72 (2, 1, 5, 6, 3, 4, 7 ) D34D56r~5 ( 5 + 1 7 - ) [ ( 7 - 1 0 _ 1 4 - ) ( a . b ) - ~ ( 7 - 1 6 1 4 - ) ( e _ . a ) ] D34D347D56~2 ( 7 + 1 4 - ) Fz ( 1, 2, 3, 7, 5, 6, 4) D340347D56~2 ( 7 + 1 4 - )F2(2, 1, 3, 7, 5, 6, 4) D34D347056~r~__( 3 -- ]7 + ) F2 ( 1, 2, 7, 4, 5, 6, 3 ) D34D347D56f~/~ ( 3 - ]7 + ) F2 (2, 1, 7, 4, 5, 6, 3) D34D56D567~__ ( 7 + [6 - ) F2 ( 1, 2, 5, 7, 3, 4, 6) D34D56D567f~_ ( 7 + 1 6 - )F2(2, 1, 5, 7, 3, 4, 6) D34056D567~/2 ( 5 - - 17+ )F2( 1, 2, 7, 6, 3, 4, 5) D34056D567f~/2 ( 5 -- ] 7 + ) F2 (2, 1, 7, 6, 3, 4, 5 ) O34D56r~4 [ ( 1 + 14-- )GI( 1, 3, 4 ) - (1"-'7)] D34D347D56r~4( 7 + I 1 - ) ( 7 - 1-6314- ) G l ( 1, 3, 4) D347D56rl4 [ ( 4 - I 1 + ) Gl ( 1, 3, 4) -- (4"-'7) ] D349347D56rI4( 4 - [ 7 + ) ( 7 + 1J03411 + ) Gl ( 1, 3, 4) D34D347D56N/~_ ( 3 - 17 + ) G l ( 1, 7, 4) D340347D56~/2 ( 7 + 1 4 - ) G l ( 1 , 3, 7) D34D56r25 [ ( 2 - [5+ )G2(2, 5, 6) - (2*-.7) ] D34D56D567r25 ( 7 - 12 + ) ( 7 + 1.66] 5 + ) G2 (2, 5, 6 ) D34D567r'~5 [ ( 5 4 12 - ) G2 (2, 5, 6) + ( 5,--,7 ) ] D34D56D567r~5 ( 5 + 1 7 - ) ( 7 - 1-65612- )G2(2, 5, 6) -- O349560567~2 ( 5 -- ] 7 + ) G2 (2, 7, 6 ) -D34056D567%/2 ( 7 + 1 6 - ) G2(2, 5, 7) -- ~D347056r34 [ ( 3 + 1 4 - ) F2 ( 1, 2, 3, 4, 5, 6, 7 ) + ( 3 ~ 7 ) ] --20347D56r34 [ ( 3 + 1 4 - ) F 2 ( 2 , 1, 3, 4, 5, 6, 7 ) + (3"*7)] --23D347D56r'~4 [ ( 4 - - 13+ )F2(1, 2, 3, 4, 5, 6, 7)+(4*-*7)] -]D347D56r'~4 [ ( 4 - 13+ )F2(2, 1, 3, 4, 5, 6, 7 ) + (4*-*7)] --]D34D567r56 [ ( 6 + 1 5 - )/72( 1, 2, 5, 6, 3, 4, 7 ) + (5'--'7)] -- {D34D567r56 [ ( 6 + 15-- )/72(2, 1, 5, 6, 3, 4, 7) + ( 5 ~ 7 ) ] r ~ r*56 [ ( 5 - 1 6 + ) F 2 ( 1 , 2 , 5 , 6 , 3 , 4 , 7 ) + ( 6 " - - ' 7 ) ] --720 34,--567 2 * [ ( 5 - - 16+ )F2(2, 1, 5, 6, 3, 4, 7)+(6*--'7)] -- ~O340567r56 2 * [ ( 3 + 1 4 - ) G ~ ( 1 , 3, 4) + ( 3 ~ 7 ) ] --~D347D56r34 - ]D347056r34 [ ( 4 - [3 + ) G l ( 1, 3, 4 ) + (4"-'7) ] 2 * [ ( 5 + 1 6 - ) G 2 ( 2 , 5, 6 ) + ( 5 ' - - ' 7 ) ] -- 30349567r56 2 -- 3D349567r56 [(6--15+)G2(2, 5, 6)+(6*-*7)]
Volume 262, number 1
PHYSICS LETTERS B
F I ( 1 , 2, 3, 4, 5, 6, 7) =+2(
1+ I / ) 3 4 - / ) 5 6 1 2 + ) ( 3 - 1 5 + ) ( 6 +
+4(3-
[/)5614-)(1+ 16-)(5-
-4(5-
1/)3416-) ( 1 + 1 4 - ) ( 3 - 1 2 + ) ,
14-)
12+) (21a)
/72(1,2,3,4,5,6,7) =-4(1+1/)5612+)(3-15+)(6+14-) +2(3-12/)56+/)714-)(1+
1 6 - ) ( 5 - 12+ )
-4(5-1/)34+/)716-)(1+14-)(3-12+). (21b) The functions GI, G2 arise in neutrino exchange diagrams. They are given by
Gl(i,j,k) =4(i-
Ij + ) ( k + 1(/)2 --/)56)1 5 + ) ( 6 + 12-- ),
Gz(i,j, k) = 4 ( 1 - 13+ ) ( 4 + I ( -/)1 +/)34) lJ + ) ( k + l i - ) • (22)
13 June 1991
2.2. Singularities(peaks) of the matrix element and IR cuts The singularities of M occur when the various propagators get close to zero. They are important for the numerical evaluation and must be carefully analyzed. We have the following singularities: (i) (7, Z) peaks: if the initial electrons radiate a very hard photon, the s-channel diagrams (see fig. 1 ) peaks when s ' = ( s - 2 E r x / ~ ) becomes close to 0 or M2z. We can eliminate these singularities by requiring
Ev < ]Mz
(25)
for x/~ ~ 2Mz. For larger E~, the Z peak can occur; however we expect it to be unobservably small [ 16 ]. (ii) Forward peaks: arise for small t, t'. Since the kinematics does not favor these values (see below), they are not very strong. (iii) Breit-Wigner peaks: large contributions due to nearly on-shell W bosons, characterized by a resonance factor 1
Finally, the factors A are
(p 2 - M w2)
As(S', z)
for subset 1 ,
As(s, z)
for subsets 2, 3, 4, 9, 10,
At(t,z)
for subsets 5, 6, 11, 12,
At(t', z).
forsubsets 7, 8 .
for a W boson with momentum/~1. (iv) Infrared singularities: usual poles, when the photon energy approaches zero. We avoid them by a cutoff o9~ (see table 2 for values of tot). (v) Collinear singularities: familiar peaks for collinear photon radiation from external massless particles. They are eliminated by a suitable cut ev on the
(23)
The total amplitudes are then M(2, z ) = ~ M~(2, z ) ,
(24)
i
where i ranges over the appropriate graphs.
2
+ M w2 F w2
(26)
~ In order to investigate the effects of a momentum dependent width, we have replaced MwFwby p2Fw/Mw,where p is the W momentum. The deviations from a constant width are negligible.
Table 2 Total cross sections for e+e---, 4f+ photon in units of 10-~ pb. The (lower) energy and angle cuts for the photon are given; the upper cut on the photon energy is 60 GeV. ev (deg)
1 5 10 15
Purely leptonic case
Semileptonic case
toy= 1GeV
oo~=5 GeV
toy= 10 GeV
to~=15 GeV
toy= 1GeV
toy=5 GeV
toy= 10 GeV
to~=15 GeV
0.5367 0.3272 0.2347 0.1792
0.2765 0.1695 0.1234 0.0960
0.1697 0.1043 0.0760 0.0591
0.1121 0.0691 0.0504 0.0391
1.4184 0.8655 0.6201 0.4713
0.7164 0.4407 0.3183 0.2430
0.4350 0.2678 0.1933 0.1470
0.2814 0.1793 0.1256 0.0956 129
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Table 3 trwwv in peaking approximation including finite mass effects. The results are given for one decay channel. The energy cut to~ as well as the angle cut discussed in section 2.4 depend on the beam energy Eb. For xr (see footnote 3 ), we take 10- 3. In order to compare the corrections introduced applying (38), we chhose the two values yr= 1, 10. Eb (GeV)
82 85 90 95 100 105 110 115 120 125
aww (pb)
0.102 0.173 0.217 0.229 0.231 0.226 0.219 0.212 0.204 0.196
O'wwv y~= 10
y~= 1
0.0213 0.0596 0.0912 0.106 0.113 0.115 0.116 0.114 0.112 0.110
0.0247 0.0695 0.107 0.125 0.132 0.135 0.136 0.134 0.131 0.128
13 June 1991
A
f dx c(x)g(x), o+~ where c(x) is flat and g(x) ~ 1/x is transformed into an unproblematic intergral over u if du = g ( x ) d x . For our calculation, the integration routine VEGAS [20,21 ] was used, where importance sampling is numerical. Since it works best for "flat" functions, we transform the variables, in which the rates peak "away", according to the outlined prescription. We say that x is "sampled with g ( x ) " . Consider now a function f ( x ) with several peaks x~. We write
a= f f(x) dx -
f f(x)
g,(x)
(29)
angle (see table 3). Both (iv) and (v) arise from the radiation factors ( 17 ).
where gi(x) peaks at x i and falls off sufficiently fast for x away from x~. Then it is clear that f ( x ) / Z g ~ (x) is well behaved as x~x~. The change of variable
2.3. Cross sections and Monte Carlo integration
dui =gi(x) d x ,
(30)
with The e+e ---,f3i~4f5I'6~/cross section is given by 1 1
da= 2s4~
IM(2'z)12dC)ll"
a= ~ ~ f(x(u,) ) dui, (27)
Here d01 ~ is the I 1-dimensional phase space of the five partial particles d4pi6(p2)(27~) 4¢~4
d0,1 i=3
+P2
Pi i
(28) Since the various 3//,. peak at different positions in phase space (see previous discussion), we must choose suitable variables to obtain acceptable numerical results. This is done by a multichannel Monte Carlo integration (MC) [ 17,18 ] which we briefly review. The most important method to reduce the error in a MC integration is importance sampling [ 18,19 ]. Roughly, one samples an integrandf(x) in such a way that there are many points in regions where f ( x ) is large. This can be achieved by a suitable change in the integration variables. For instance, the integral 130
• Y~gi(x(ui))
(31)
yields the desired result. The various choices ui(x) are called the "channels". For multidimensional integrals, with strong peaks in several variables, such a procedure is essential for obtaining stable results. The differential cross section exhibits several peaks. We found it sufficient to choose four channels for purely leptonic and six for semileptonic processes; in some cases their number can be reduced by symmetry arguments. The integration variables must be chosen such that this behaviour can be recognized and tranformed away.
2. 3.1. Initial radiation channels (diagrams I, 5 of
fig. l) The graphs 1, 5 show all of the above peaks. [Where not explicitly stated, the s' peak at M2z is avoided by a cut. Since it requires Ev~ ~Mz, the cross section is small in that region anyway. ] In order to find the most sensible "channels", it is
Volume 262, number 1
PHYSICS LETTERS B
useful to view the process as e + e - --,T + X, and choose the photon energy E r and its angle, f2r= (cpr, 0r) as variables. The E~ -1 and ( 1 - c o s 2 0r) -1 peaks are eliminated by the transformation (30) with g = x - i, ( l - x 2 ) -~, respectively. The ranges are [0+co r, ~Mz] for Ev, and [ ( - 1 +¢r), ( 1 - E r ) ] for cos 0r, respectively, where cor, Er are the cutoffs. To parameterize the "decay" of X, we use the angle of W +, £2w= (~w, 0w) and the invariant masses s34, s56. Due to the Breit-Wigner poles, s34 and s56 are sampled with (26) symmetrically around M 2. The angle 0w is properly accumulated by the t or t' peaks, respectively. This then requires two channels. Because of the symmetry between t and t' diagrams it is sufficient to evaluate one channel and double the result. The final variables are the angles ~r~34and f256 of the decay products relative to their "parent" W boson. The phase space element is now given by dO1, ( s ) = ~ o ~ -zr ~1/2(S~,S34, S56 ) X dE r d~2r d.Qw ds34 ds56 dg-234d~56,
( 32 )
with 2(a, b, c ) = a 2 + b 2 + c z - 2 a b - 2 b c - 2 a c .
(33)
Finally, events with a photon m o m e n t u m generated parallel to one of the final state (charged) fermion's m o m e n t u m have to be rejected (see discussion on mass effects).
2.3.2. Final radiation channels (diagrams 3, 6, 7, 9 offig. 1) This time, it is natural to view the total process as e+e - ~ W + W - , followed by W + --' ~fT and W - ~T'f or vice versa. [ When both W decay leptonically or hadronically, the integration variables corresponding to these two cases appear symmetrically. This symmetry allows a check of the matrix element and reduces the integration work. ] To parameterize the first step, it is obvious to use the variables $347, $56 and f2w, where $347, $56 are sampled according to the Breit-Wigner distribution and Ow as described in section 2.3.1. While W---,]'f is trivially described by f256, the process W + ~I'f7 needs some comments. The emission of the photon from the charged lepton or down quark with m o m e n t u m
13 June 1991
P4 results in a radiation factor x/~/(P4 "P7); for had-
ronic decays an additional factor r~4=2/(p4"P7) × (P3"P7) arises (both quarks are charged). The threeparticle phase space can be described by d¢5 = ~EaEb d~2a d~ab,
(34)
where Ea, Eb, d~ab, d~'2, are the energies of two of the particles, their relative azimuthal angle and the solid angle of the first one, respectively. We choose particle b to be the photon. To treat the first factor, we note that (p4.P7)=2Es(EB-E3), where EB= s~347/2. Thus we take Ea = E3 as a variable and sample it with 1/EB (EB--E3). A second channel takes into account the radiation factor r24 which can be written as [ 2 E 2 ( E B - E 3 ) ( E 7 + E 3 - E B ) ] -1 in the P347 rest frame. We use again E3 as variable, sampled now accordingly to r24 . The range of the variables is over the allowed region, except for the cust. [These are defined in the laboratory system and cannot be applied "directly" at the time of generating events. However given a cut in the laboratory system, we can calculate a minimal value for the cut in the rest frame and therefore reduce the number of subsequently rejected events. ] The remaining variables, ~24 and (047pose no problem. Thus, we have the following channels for W+-~tfT: 1 21/Z(s', $347, $56 ) doll ( S ) - 29 S × d~"2wds347 ds56 dEr dE3 d Q3 d~o37d~'~56 ,
(35)
with two different variable samplings as discussed. In conclusion, purely leptonic events require the two channels (32) and one of the channels (35) (along with an analogous one for W - ). For semileptonic decays both channels (35) (and the corresponding one for W - ) must be taken.
2. 4. Finite mass effects Neglecting the fermion masses results of course in collinear divergencies which require the introduction of an angle cutoff ~r. However in order to combine the hard photon cross section with the virtual O ( a ) corrections we have to include finite mass effects in the case that the photon makes a small angle of O (mr/ Ef) with one of the fermions. The procedure is well 131
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PHYSICS LETTERS B
known [ 15 ]. Essentially there are two effects to consider: Firstly, the inclusion of nonleading terms of O ( m 2 ) requires multiplying IMI 2 with a mass correction factor,
~
q
t
m 2
wf=l-
s'(s-s') (pf.k) s 2 + s '2 '
13 June 1991
(36)
for, e.g., an initial state fermion. Similar expressions apply to final state fermions [ 15,17 ]. Secondly, we have to take care of the right peaking behaviour of the cross section in case that the fermions are generated with massive kinematics. Evaluating ( 11 ) for a massive momentum p and the photon momentum k yields I(P+ Ik-)12=2(p'k)+m
2 ko q-k3
(37)
Po -t-P3
which causes indeed an unwanted peaking behaviour of the cross section in a not strictly collinear case. This can be removed by multiplying IMI 2 with a compensation factor, w~=
I (pflk)12 2(pf'k)
(38)
We have checked that this procedure allows to remove the original angle cut ¢v. However there are finite corrections surviving for angles around ~ (mr/ Er) which are of the order of several per cent. These are "numerator" effects and we can eliminate them by appropriate cuts.
3. Background The resonance characteristics of the WW channel for the 4 fermion production should eliminate any background, especially if appropriate cuts on the fermions are applied. However, if it is impossible to apply such cuts, the background should be reexamined. In order to simplify the analysis, we consider here only final states which have no electrons in them (e.g., Ix+x-v~9~; ~t+V,~lq). Then, only the diagrams of fig. 3 contribute. The corresponding matrix elements can be found, e.g., in ref. [ 22 ]. (For a more complete list, see ref. [ 23 ]. ) Whereas these effects are appreciable at lower energies [ 16 ], they turn out to be negligible here. A more complete study will be given elsewhere. 132
Fig. 3. Diagrams other than W+W- production contributing to the 4 fermion final state without electrons. Depending on the nature of the fermions, there are several of these.
4. Results The numerical integration was carried out using VEGAS [20,21] with a sufficiently fine grid to reduce the statistical error to less than 1%. For all numerical results we have taken G~ as input parameter (compare section 2). The values for Mw used are 80.9 and 82.0 GeV; the corresponding widths are (QCD corrected) 2.14 and 2.23 GeV. Throughout the paper we are using M z = 91.16 GeV. While the results strongly depend on the lower energy cut ogv of the photon, the cross section falls so strongly with increasing energy that the value of the upper cut [see eq. (25) ] is irrelevant. We will use 60 GeV. The Z pole yields only an unmeasurably small peak. The lower cut toy is kept fixed except for the results of table 3. In order to identify the W pair most clearly, we consider purely leptonic decays (both W decay into a given lepton-neutrino pair) and semileptonic decays, where one of the W decays hadronically into one pair of flavors [4]. In table 2 we give the total cross sections for the processes e+e---~lVl~2927 and ~lv2Od 7 (or vice versa) for a beam energy of 100 GeV, as a function of the cutoffs for energy and angle between the photon and any charged particle. [ See also the above discussion on the effects of a finite electrom mass. ] Fig. 4 shows the photon spectrum for a beam energy of 100 GeV as a function of the angle cutoff. Fig. 5 gives the excitation curves for e + e - ~ 4 f and 4 f + photon. Whereas the leptonic and the (properly scaled) semileptonic cases are the same without a photon, they differ when the photon is included. The cross sections for low cutoffs are large as expected especially well above threshold. It is interesting to compare the exact calculation
Volume 262, number 1 0.24
PHYSICS LETTERS B
i
i
i
13 June 1991 (39)
dtTapprox = do*oy+ daww~,
i
,1(~a
0.20 3
where the peaking approximation is used on the fighth a n d side and daov is given in curve ( b ) . This curve, labelled ( d ) , is again different from the exact one (c). Finally, the exact h a r d p h o t o n result can be split in the following way:
0.t6 o.t2
b
~o 0.08
dal~,=daww~,+daw(~,)w+daww(v) + A a ,
0.04 0.00
I0
40
20 30 Ey (GeV)
50
Fig. 4. Photon spectrum for three values o f the angle cut ev. The
W mass is 80.9 GeV, the angle cuts are e~= 1°, 5 ° and 10 °.
0.3
,
,
,
,
,
incl. photon
~, 0.2 tn
o
/
(40)
where dawwv has been given above, a n d daw(v)w is the cross section for two on-shell W, one o f this decays into a lepton pair a n d a photon. In fact, there is no " d o u b l e counting" o f diagrams with the p h o t o n e m i t t e d from the W-line. Neglecting A a a n d adding davy to curve ( a ) yields curve (e). A comparison with (c) shows that A a is small. Finally, we list in table 3 the results for awwv as discussed in (39) including finite mass effects for the initial electrons. The r e m a i n i n g angle cut discussed above is taken to be mE/gb times some n u m b e r Yr a p p r o p r i a t e l y choosen to keep the corrections small. F o r c o m p a r i s i o n we also give the results for y r = 1. In this calculation, the energy cut tgv is i n t r o d u c e d as a
0.t 0.3
0.0
'
80
'
'
t00
'
I
I
I
I
I
I
'
120
0.2
E beam [GeV] Fig. 5. Radiative and nonradiative excitation curves as a function of the beam energy for Mw=80.9 GeV (solid line) and
M w = 82.0 GeV (dashed line). The upper curves represent the sum ao~+ alv. The dotted curve is the semileptonic contribution for M w = 80.9 GeV divided by three (color!). The applied cuts
0
"- 0.t
are a~v=2.0 GeV and Ev= 5 °. with finite W - w i d t h to the a p p r o x i m a t e t r e a t m e n t where the W are on-shell (peaking a p p r o x i m a tion ~2). F o r the tree-level calculation without photon the exact curve ( a ) and the a p p r o x i m a t i v e curve ( b ) o f fig. 6 show a sizeable difference even above threshold. Including the h a r d photon, we write
#2 In the peaking approximation the Breit-Wigner propagator 1/ (paw -Maw + i3/wFw) is replaced by (n/MwFw)~(paw-Maw).
0.0
J i
80
I
t00 E beam [GeV]
I
120
Fig. 6. Exact and approximate excitation curves for the purely leptonic case (Mw=80.9 GeV). Curves (a), (c) exact results, curve (b) peaking approximation of the nonradiative processes, curve (d) aww+aww~ where both W are treated in peaking approximations, curve ( e ) approximation in eq. (40). The applied cuts are o~v= 1 GeV, e~= 1°. 133
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PHYSICS LETTERS B
f u n c t i o n Of Eb as discussed, e.g., in ref. [ 8 ] t,3 In conclusion, we h a v e s h o w n that in the case where no k i n e m a t i c a l cuts are i m p o s e d , the finite w i d t h effects d u e to off-shell W ' s result in a m e a s u r a b l e c o n t r i b u t i o n to the cross sections. T h e p h o t o n s p e c t r u m as well as the total cross section are g i v e n for different masses M w a n d different (regularizing) cuts. O u r results are d e r i v e d w i t h i n the s t a n d a r d m o d e l . H o w ever, the e v e n t g e n e r a t o r is general a n d can be u s e d for any e x t e n s i o n o f the m o d e l . In the case w h e r e there are no electrons in the final state, we h a v e s h o w n that the b a c k g r o u n d is negligible.
Acknowledgement T h e helicity a m p l i t u d e o f table 1 was o b t a i n e d by H. Bijnens, Z. K u n s z t a n d D. W y l e r [ 10,4 ]. We w o u l d like to t h a n k H. Bijnens a n d Z. K u n s z t for the early c o l l a b o r a t i o n a n d for m a n y helpful discussions. W e also t h a n k J. V e r m a s e r e n for p r o v i d i n g us w i t h his M C routines. #3 In narrow width approximation, an expression for the photon energy cutoff in 09.t=x/~Xr/{ 1+Xr+ [ ( 1 --Xr) 2 - 4M2/s] 1/2} with xr the part of the W energy shifted to the soft photons.
References [ 1 ] J. Ellis and R. Peccei, eds., Physics with LEP, CERN yellow report CERN 86-02 (CERN, Geneva, 1986). [2] A. B6hm and W. Hoogland, eds., ECFA Workshop on LEP 200, CERN report CERN/ECFA 87-08 ( 1987 ).
134
t 3 June 1991
[3] W.J. Marciano, Nucl. Phys. Proc. Suppl. B 11 (1989) 5-34. [4] P. Roudeau et al., in: ECFA Workshop on LEP 200, CERN report CERN/ECFA 87-08, eds. A. BShm and W. Hoogland (1987). [5] T. Muta, R. Najima and S. Wakaizumi, Mod. Phys. Lett. A 1 (1986) 203. [6] Z. Hioki, Nucl. Phys. B 316 (1989) 1. [7] M. Lemoine and M. Veltman, Nucl. Phys. B 164 (1980) 445; R. Philippe, Phys. Rev. D 26 (1982) 1588; M. B6hm, A. Denner, T. Sack, W. Beenacker, F. Berends and H. Kuijf, Nucl. Phys. B 304 (1988) 463. [8 ] J. Fleischer, F. Jegerlehner and M. Zralek, Z. Phys. C 42 ( 1989 ) 409. [9] H. Tanaka, T. Kaneko and Y. Shimizu, KEK preprint 9029. [ 10] H. Bijnens, Z. Kunszt and D. Wyler, unpublished. [ 11 ] P. De Causmaecker, R. Gastmans, W. Troost and T.T. Wu, Phys. Lett. B 105 (1981) 215; Nucl. Phys. B 206 (1982) 53; F.A. Berends et al., Nucl. Phys. B 206 (1982) 61; B 239 (1984) 382. [ 12] R. Kleiss and W.J. Stirling, Nucl. Phys. B 262 ( 1985 ) 235. [13]Z. Xu, D.-H. Zhang and L. Chang, Tsinghua preprints TUTP-84/3. [ 14] J.F. Gunion and Z. Kunszt, Phys. Lett. B 161 (1985) 333. [ 15] R. Kleiss, Z. Phys. C 33 (1987) 433. [ 16 ] W. Marciano and D. Wyler, Z. Phys. C 3 (1979) 181. [ 17 ] F.A. Berends, W. Hollik and R. Kleiss, Nucl. Phys. B 304 (1988) 712. [ 18 ] R. Kleiss, in: Z Physics at LEP I, Vol 3, CERN yellow report (CERN 89-08). [19] E. Byckling and K. Kajantie, Particle kinematics (Wiley, New York). [20] G.P. Lepage, J. Comput. Phys. 27 (1978) 192. [21 ] S. de Jong, R. Laterveer and J.A.M. Vermaseren, AXO user manual, report NIKHEF-H, Amsterdam ( 1987 ). [22 ] V. Barger and T. Han, Phys. Lett. B 241 (1990) 127. [23] J.C. Roma~o and P. Nogueira, IFM-preprint 8/88.
PHYSICS LETTERS B
13 June 1991
Hard bremsstrahlung processes at LEP II A. Aeppli a,b,1 and D. Wyler a a InstitutJ~rtheoretischePhysik, UniversitdtZiirich, CH-8001 Ziirich, Switzerland b Physics Department 2 Brookhaven National Laboratory, Upton, N Y 11973, USA
Received 25 March 1991
The matrix element for single hard photon bremsstrahlung in the dominant LEP II process e+e - ~W +W---,4f with massless fermions is given. An appropriate Monte Carlo event generator is described and the results for total cross sections and the photon spectrum are presented. The narrow width approximation for the W and the background are discussed.
1. Introduction LEP II, the electron-positron storage ring with a CM energy x/~ ~ 2Mw promises to directly probe the three-gauge boson vertex and yield a precise determination of the W mass, Mw. This measurement allows a very stringent test o f the standard model: the three quantities e, GF and Mz fix the three unknown gauge parameters of the model uniquely, making Mw a true prediction which only depends on m t and, more weakly, on the mass of the Higgs boson [ 1-3 ]. Consequently, several methodes o f accurately fixing Mw at LEP II have been devised [4]. At LEP II, a W + W - pair is produced. Each W decay into a fermion pair; the cross section for the process e+e ---, W W ~ 4 fermions (as function of V~) has a characteristic shape and magnitude which depends on Mw [ 5,6 ]; we also expect the threshold behaviour to reflect the kinematical value o f Mw [ 6 ]. In view of the importance of this measurement, radiative corrections must be included. In this paper we present the (hard) photonic bremsstrahlung, which can easily be measured by observing an energetic photon in addition to the fermions. Previously, complete one-loop calculations including soft photon corrections have been presented for on-shell production of the W pair [ 7,8 ]. Combining these results with the hard photon bremsstrahlung Partially supported by SchweizerischerNationalfonds. 2 Addressafter 1 February 1991.
cross section, the full O (or) calculation has been given in ref. [ 9 ]. Since the observed process is really a fourfermion production, and there could be important threshold effects, a calculation o f this process is needed. The hard bremsstrahlung evaluated in this work constitutes the first step; the additional corrections will be presented elsewhere. As we expected the on-shell (peaking) approximation to be quite reliable for x/~> 2Mw, we present for comparison the results for this simplification. We also study the background processes and show them to be small at the energies considered.
2. The calculation 2. I. The m a t r i x element [lO]
Since m t >~Mw, all the fermions involved in the process e+e - - - , W + W - ~ 4 f can be taken to be massless for x / ~ 2Mw (the rate e + e - ~6tf1I'2 is presumably very small if m t is appreciably larger than Mw). Then, the massless helicity formalism is appropriate [ 1 1-14 ]. The collinear singularities occurring in this situation can be removed by introducing a finite mass as discussed below [ 15 ]. Since we are interested in fermions produced by a W + W - pair, we will take the outgoing fermions to be lefthanded. Then, the amplitudes depend on the helicities z and 2 of the incoming electron and the photon, respectively. The graphs that contribute are shown in fig. 1. The
0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
125
Volume 262, number l
PHYSICS LETTERS B
(1.1)
-6
(t.2)
(3.1)
13 June 1991
(6.1)
(3.2)
(6.2)
(7.3)
(7.4)
>4
(7.6)
(7.5)
(3.3)
(5.1)
(5.2)
(9.3)
(9.4)
(9.5)
(9.6)
Fig. 1. Diagrams contributing to the radiative process e+e---,4f+7 in unitary gauge.
a m p l i t u d e corresponding to the graph i is written as
cos 0w = M w / M z .
Mr(z, 2) =eA,(x, t)Jl~(z, 2 ) ,
D e p e n d i n g on whether the fine structure constant a = eE/4~z or the F e r m i constant G u is chosen as input parameter, we set g = e / s i n 0w or g = 4 M 2 x / ~ G u. The kinematics is defined in fig. 2. We write
( 1)
where e is the electric charge a n d A i ( z ) is a factor corresponding to the graph without photon. The ~ ( z , 2) are discussed below. The relevant A~(z) are
ig2 ( e 2 G cos O. g 2 As(x, t) =--~- \ x - x - M 2 + i M z e z ] -ig 4 1
A,(x, ~ ) =
4
(6)
po=pi+pj, Pijk=Pi+pj+pk, '
(7)
(2) f3, P 3
(3)
x'
f4 ,P4
with c_ -
1 COS 0 w
l - - -
C+ -- COS 0 w
126
fs,P5 (sin20w - ½) ,
(4) "{6 ,P6
sin20w,
(5)
e/
'
"Lq),., p 7
Fig. 2. The kinematics and momentum conventions.
Volume 262, number 1 s=(pI+p2)
2,
13 June 1991
where r is a suitably chosen reference momentum
t = ( p 2 - - P 5 6 ) 2,
S'= (Pl +P2 --P7) 2,
so =p~,
PHYSICS LETTERS B
r•k. t ' = ( --Pl -FP34) 2 ,
2 • S(]k= P/jk
(7cont'd)
The ~ ( z , ;t) are presented in terms of spinor products
(i+lj_)=_
(8)
where u(pi) are free spinors and
u+ (p) = ~ ( 1 +_~5)u(P)
(9)
It is useful and easy to split the diagrams into gauge invariant subsets. They are indicated in table 1. Within gauge invariant sets of diagrams, r must be fixed. An appropriate choice of r reduces the number of terms considerably. In order to include leptonic and hadronic (quark) decays of the W+W - pair, it is convenient to introduce a charge Qo such that
are the helicity projections. The corresponding states are denoted by LP+- >. For antiparticles
Q3 ---Qo,
v+ = ½(l-T- ~'5)v.
For leptonic decays, only Qo = 0 contributes; for the hadronic decays of the W's also Q0 = 2 must be added. The helicity amplitudes are listed in table 1. For each gauge invariant subset a convenient reference m o m e n t u m r, as indicated, has been selected; only non-vanishing contributions are given. Sets 1-8 contribute to all possible W+W - decays ( Q o = 0 ) . If the W + decays hadronically, sets 9, 11 must be added, if the W - decays into quarks, sets 10, 12. The various factors in the table are defined as follows: "radiation" factors:
(10)
An explicit form is (Po, qo> 0)
~
p~ -ip2
q,-iq2
(11)
The momenta are to be defined with respect to any axis, such that ( 11 ) can be used. A convenient choice for the z-axis is the beam direction. In order to keep (11 ) meaningful, we exclude a very small cone around the beam direction for the particle momenta. For completeness, we list some of the properties of the spinor products:
04 = Qo - 1 ,
Q5 = Q o - 1,
~/~
r o =
,
(po, q o > 0 ) ,
I
(17)
Dok=D(Pok),
1 D(p2) = pZ-M~v + iMwFw " (12)
(16)
"propagator-functions":
Dij=D(po),
Q6--Qo •
(18)
The functions F1, F2 represent the triple gauge couplings and are given by
Spinor-strings are reduced by using the Fierz identity FI ( 1, 2, 3, 4, 5, 6, 7) = [ (P34 -P56)" e+ ] (a.b)
lY#
Iq2 - >
=2
+ 2(ps6-a) (e+" b) - 2(ps4"b) (e+ .a) , (13)
together with the relation
-2[ (P34+P7)'bl (e+ .a) - 2 (P~6 .e+ ) (a.b), (14)
The notation #=pu),u is employed. The polarization vector Eu (k, 2 ) of the photon with momentum k is
2
F2( 1, 2, 3, 4, 5, 6, 7) = [ (2p56 +PT)'a] (e+ .b)
(15)
(19)
where
eU~=
aU=<3-1yul4-),
bu=<5-l?Ul6-) .
(20)
The explicit form is 127
Volume 262, number 1
PHYSICS LETTERS B
13 June 1991
Table 1 Helicity amplitudes for the diagrams in fig. 1. The arguments o f M o (z, 2, r) are the (incoming) electron and photon helicities z, 2 and the chosen reference momentum (see text). The table refers to Qo=0 (leptonic decays) and to Qo = ~ [see eq. (16) ]. Qo = 0
subset 1
JCl i ( +, +, 2)
~2(-,
+, 1 )
, ~ 1 2 ( + , - - , 1)
subset 2
~#H ( - , - , 2 ) ~//v4 ( +, +, 5 )
~V,a ( + , +, 5) ~ v ( + , +, 5) ~G3 (+, +, 5) ~ v ( _ , +, 5) ~V,a(--, +, 5) .av2. ( - , +, 5) ~63 ( - , +, 5)
subset 3
.//vs ( +, - , 4) j/v (+,_,4) .A/v2. ( +, - , 4) ~/33 ( + , - , 4 ) ~/v5 ( - , - , 4) ,¢tv~ ( - , - , 4) Jr'v2. ( - , - , 4) •~33 ( - - , - - , 4) ~Vlb ( +, +, 3)
~v(_,
subset 4
+, 3)
-¢/~',b ( +, - , 4 ) agVtb ( - , - - , 4 ) ~'~2b ( +, +, 5 ) ¢f/V2b(-- , + , 5 )
subset 5
J/vzb ( +, --, 6 ) J/~'Zb ( --, --, 6) J/Sl ( - , - , 4) d/v,. ( - , - , 4)
.#v ( _ , +, l) ~ / V l a ( - - , + , 1)
subset 6
JtV~b ( --, --, 4 ) d/v~b( - , +, 3)
subset 7
Jr'52 ( - , +, 5 )
subset 8
~t'v5 ( - , - , 2 ) ~/~'~. ( - , - , 2) ~/V2b( --, --, 6 )
~¢v2~( - , +, 5)
~t6~ ( - , +, 5) Qo = ]
subset 9
J/v3 ( +, - , 4) ~//v3 ( - , - , 4) ,¢/v4 ( + , +, 3)
~,v (_, +, 3) subset 10
,¢¢v ( +, - , 6 ) ~/v(_, -,6)
~ (+, +, 5) ~ v ( + , - , 5) subset 11
.//v3 ( - , - , 4)
~ v ( _ , +, 3) subset 12
~tv5 ( - , - , 6)
~¢v (_, +, 5) 128
D34D56r~2 [ ( 2 - I 1 + )FI (1, 2, 3, 4, 5, 6, 7 ) - (1,--,7) ] D34D56r12 [ ( 2 - I 1 + )F1 (2, 1, 3, 4, 5, 6, 7 ) - ( 2 ~ 7 ) ] D34D56rT2 [ ( 1 + 12 - )Ft ( 1, 2, 3, 4, 5, 6, 7) - (2*-*7) ] D34Ds6rT2 [ ( 1 + 12 - ) F~ ( 2, 1, 3, 4, 5, 6, 7 ) - ( 1.-.7 ) ] D347D56r15 [ ( 4 - 15+ )F2( 1, 2, 3, 4, 5, 6, 7) + (4,--,7) ] D34D347D56r45 ( 4 - 17 + ) ( 7 + 1-634I 5 + ) F 2 ( 1, 2, 3, 4, 5, 6, 7 ) D34D56D567r45 ( 4 - 17 + ) ( 7 + 1.6615+ ) Fz ( 1, 2, 5, 6, 3, 4, 7 ) D34D56r45 ( 7 - 1 4 + ) [ ( 7 + 10÷15+ ) ( a . b ) - ½ ( 7 + 1~15+ ) (e+.b) ] D347D56rl5 [ ( 4 - 15+ )F2(2, 1, 3, 4, 5, 6, 7) + (4,--,7) ] D34D347D56r45 ( 4 - 17+ ) ( 7 + 1-63415+ )F2(2, 1, 3, 4, 5, 6, 7) D34D56D567r45(4- 17+ ) ( 7 + 1#615+ )F2(2, 1, 5, 6, 3, 4, 7) D34D56r45(7 - 14+ ) [ ( 7 + 10_ 15+ ) ( a . b ) - ½(7+ I~15+ ) (e_.b) ] D34D567r~5 [ ( 4 + 1 5 - )F2 (1, 2, 5, 6, 5, 6, 7 ) + (5.--}7)] D34D347D56r~5 ( 7 + 15 - ) ( 7 - I/)314- )F2( 1, 2, 3, 4, 5, 6, 7 ) D34D56D567r~5 ( 7 + 1 5 - ) ( 7 -
1-65614- )F2( 1, 2, 5, 6, 3, 4, 7)
D34D56r~5 ( 5 + 1 7 - ) [ ( 7 - 1 0 + 1 4 - ) (a'b)-½ ( 7 - 1 6 1 4 - ) (e+.a) ] D34D567r~5 [ ( 4 + ] 5 - ) F2 (2, 1, 5, 6, 5, 6, 7) + (5,--,7) ] D34D347D56r~5
( 7 + [ 5 -- ) ( 7 -- 11)314-- ) Fz (2, 1, 3, 4, 5, 6, 7 )
D34D56D567r~5 ( 7 + ] 5 - ) ( 7 - [-656 ]4-- )/72 (2, 1, 5, 6, 3, 4, 7 ) D34D56r~5 ( 5 + 1 7 - ) [ ( 7 - 1 0 _ 1 4 - ) ( a . b ) - ~ ( 7 - 1 6 1 4 - ) ( e _ . a ) ] D34D347D56~2 ( 7 + 1 4 - ) Fz ( 1, 2, 3, 7, 5, 6, 4) D340347D56~2 ( 7 + 1 4 - )F2(2, 1, 3, 7, 5, 6, 4) D34D347056~r~__( 3 -- ]7 + ) F2 ( 1, 2, 7, 4, 5, 6, 3 ) D34D347D56f~/~ ( 3 - ]7 + ) F2 (2, 1, 7, 4, 5, 6, 3) D34D56D567~__ ( 7 + [6 - ) F2 ( 1, 2, 5, 7, 3, 4, 6) D34D56D567f~_ ( 7 + 1 6 - )F2(2, 1, 5, 7, 3, 4, 6) D34056D567~/2 ( 5 - - 17+ )F2( 1, 2, 7, 6, 3, 4, 5) D34056D567f~/2 ( 5 -- ] 7 + ) F2 (2, 1, 7, 6, 3, 4, 5 ) O34D56r~4 [ ( 1 + 14-- )GI( 1, 3, 4 ) - (1"-'7)] D34D347D56r~4( 7 + I 1 - ) ( 7 - 1-6314- ) G l ( 1, 3, 4) D347D56rl4 [ ( 4 - I 1 + ) Gl ( 1, 3, 4) -- (4"-'7) ] D349347D56rI4( 4 - [ 7 + ) ( 7 + 1J03411 + ) Gl ( 1, 3, 4) D34D347D56N/~_ ( 3 - 17 + ) G l ( 1, 7, 4) D340347D56~/2 ( 7 + 1 4 - ) G l ( 1 , 3, 7) D34D56r25 [ ( 2 - [5+ )G2(2, 5, 6) - (2*-.7) ] D34D56D567r25 ( 7 - 12 + ) ( 7 + 1.66] 5 + ) G2 (2, 5, 6 ) D34D567r'~5 [ ( 5 4 12 - ) G2 (2, 5, 6) + ( 5,--,7 ) ] D34D56D567r~5 ( 5 + 1 7 - ) ( 7 - 1-65612- )G2(2, 5, 6) -- O349560567~2 ( 5 -- ] 7 + ) G2 (2, 7, 6 ) -D34056D567%/2 ( 7 + 1 6 - ) G2(2, 5, 7) -- ~D347056r34 [ ( 3 + 1 4 - ) F2 ( 1, 2, 3, 4, 5, 6, 7 ) + ( 3 ~ 7 ) ] --20347D56r34 [ ( 3 + 1 4 - ) F 2 ( 2 , 1, 3, 4, 5, 6, 7 ) + (3"*7)] --23D347D56r'~4 [ ( 4 - - 13+ )F2(1, 2, 3, 4, 5, 6, 7)+(4*-*7)] -]D347D56r'~4 [ ( 4 - 13+ )F2(2, 1, 3, 4, 5, 6, 7 ) + (4*-*7)] --]D34D567r56 [ ( 6 + 1 5 - )/72( 1, 2, 5, 6, 3, 4, 7 ) + (5'--'7)] -- {D34D567r56 [ ( 6 + 15-- )/72(2, 1, 5, 6, 3, 4, 7) + ( 5 ~ 7 ) ] r ~ r*56 [ ( 5 - 1 6 + ) F 2 ( 1 , 2 , 5 , 6 , 3 , 4 , 7 ) + ( 6 " - - ' 7 ) ] --720 34,--567 2 * [ ( 5 - - 16+ )F2(2, 1, 5, 6, 3, 4, 7)+(6*--'7)] -- ~O340567r56 2 * [ ( 3 + 1 4 - ) G ~ ( 1 , 3, 4) + ( 3 ~ 7 ) ] --~D347D56r34 - ]D347056r34 [ ( 4 - [3 + ) G l ( 1, 3, 4 ) + (4"-'7) ] 2 * [ ( 5 + 1 6 - ) G 2 ( 2 , 5, 6 ) + ( 5 ' - - ' 7 ) ] -- 30349567r56 2 -- 3D349567r56 [(6--15+)G2(2, 5, 6)+(6*-*7)]
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F I ( 1 , 2, 3, 4, 5, 6, 7) =+2(
1+ I / ) 3 4 - / ) 5 6 1 2 + ) ( 3 - 1 5 + ) ( 6 +
+4(3-
[/)5614-)(1+ 16-)(5-
-4(5-
1/)3416-) ( 1 + 1 4 - ) ( 3 - 1 2 + ) ,
14-)
12+) (21a)
/72(1,2,3,4,5,6,7) =-4(1+1/)5612+)(3-15+)(6+14-) +2(3-12/)56+/)714-)(1+
1 6 - ) ( 5 - 12+ )
-4(5-1/)34+/)716-)(1+14-)(3-12+). (21b) The functions GI, G2 arise in neutrino exchange diagrams. They are given by
Gl(i,j,k) =4(i-
Ij + ) ( k + 1(/)2 --/)56)1 5 + ) ( 6 + 12-- ),
Gz(i,j, k) = 4 ( 1 - 13+ ) ( 4 + I ( -/)1 +/)34) lJ + ) ( k + l i - ) • (22)
13 June 1991
2.2. Singularities(peaks) of the matrix element and IR cuts The singularities of M occur when the various propagators get close to zero. They are important for the numerical evaluation and must be carefully analyzed. We have the following singularities: (i) (7, Z) peaks: if the initial electrons radiate a very hard photon, the s-channel diagrams (see fig. 1 ) peaks when s ' = ( s - 2 E r x / ~ ) becomes close to 0 or M2z. We can eliminate these singularities by requiring
Ev < ]Mz
(25)
for x/~ ~ 2Mz. For larger E~, the Z peak can occur; however we expect it to be unobservably small [ 16 ]. (ii) Forward peaks: arise for small t, t'. Since the kinematics does not favor these values (see below), they are not very strong. (iii) Breit-Wigner peaks: large contributions due to nearly on-shell W bosons, characterized by a resonance factor 1
Finally, the factors A are
(p 2 - M w2)
As(S', z)
for subset 1 ,
As(s, z)
for subsets 2, 3, 4, 9, 10,
At(t,z)
for subsets 5, 6, 11, 12,
At(t', z).
forsubsets 7, 8 .
for a W boson with momentum/~1. (iv) Infrared singularities: usual poles, when the photon energy approaches zero. We avoid them by a cutoff o9~ (see table 2 for values of tot). (v) Collinear singularities: familiar peaks for collinear photon radiation from external massless particles. They are eliminated by a suitable cut ev on the
(23)
The total amplitudes are then M(2, z ) = ~ M~(2, z ) ,
(24)
i
where i ranges over the appropriate graphs.
2
+ M w2 F w2
(26)
~ In order to investigate the effects of a momentum dependent width, we have replaced MwFwby p2Fw/Mw,where p is the W momentum. The deviations from a constant width are negligible.
Table 2 Total cross sections for e+e---, 4f+ photon in units of 10-~ pb. The (lower) energy and angle cuts for the photon are given; the upper cut on the photon energy is 60 GeV. ev (deg)
1 5 10 15
Purely leptonic case
Semileptonic case
toy= 1GeV
oo~=5 GeV
toy= 10 GeV
to~=15 GeV
toy= 1GeV
toy=5 GeV
toy= 10 GeV
to~=15 GeV
0.5367 0.3272 0.2347 0.1792
0.2765 0.1695 0.1234 0.0960
0.1697 0.1043 0.0760 0.0591
0.1121 0.0691 0.0504 0.0391
1.4184 0.8655 0.6201 0.4713
0.7164 0.4407 0.3183 0.2430
0.4350 0.2678 0.1933 0.1470
0.2814 0.1793 0.1256 0.0956 129
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Table 3 trwwv in peaking approximation including finite mass effects. The results are given for one decay channel. The energy cut to~ as well as the angle cut discussed in section 2.4 depend on the beam energy Eb. For xr (see footnote 3 ), we take 10- 3. In order to compare the corrections introduced applying (38), we chhose the two values yr= 1, 10. Eb (GeV)
82 85 90 95 100 105 110 115 120 125
aww (pb)
0.102 0.173 0.217 0.229 0.231 0.226 0.219 0.212 0.204 0.196
O'wwv y~= 10
y~= 1
0.0213 0.0596 0.0912 0.106 0.113 0.115 0.116 0.114 0.112 0.110
0.0247 0.0695 0.107 0.125 0.132 0.135 0.136 0.134 0.131 0.128
13 June 1991
A
f dx c(x)g(x), o+~ where c(x) is flat and g(x) ~ 1/x is transformed into an unproblematic intergral over u if du = g ( x ) d x . For our calculation, the integration routine VEGAS [20,21 ] was used, where importance sampling is numerical. Since it works best for "flat" functions, we transform the variables, in which the rates peak "away", according to the outlined prescription. We say that x is "sampled with g ( x ) " . Consider now a function f ( x ) with several peaks x~. We write
a= f f(x) dx -
f f(x)
g,(x)
(29)
angle (see table 3). Both (iv) and (v) arise from the radiation factors ( 17 ).
where gi(x) peaks at x i and falls off sufficiently fast for x away from x~. Then it is clear that f ( x ) / Z g ~ (x) is well behaved as x~x~. The change of variable
2.3. Cross sections and Monte Carlo integration
dui =gi(x) d x ,
(30)
with The e+e ---,f3i~4f5I'6~/cross section is given by 1 1
da= 2s4~
IM(2'z)12dC)ll"
a= ~ ~ f(x(u,) ) dui, (27)
Here d01 ~ is the I 1-dimensional phase space of the five partial particles d4pi6(p2)(27~) 4¢~4
d0,1 i=3
+P2
Pi i
(28) Since the various 3//,. peak at different positions in phase space (see previous discussion), we must choose suitable variables to obtain acceptable numerical results. This is done by a multichannel Monte Carlo integration (MC) [ 17,18 ] which we briefly review. The most important method to reduce the error in a MC integration is importance sampling [ 18,19 ]. Roughly, one samples an integrandf(x) in such a way that there are many points in regions where f ( x ) is large. This can be achieved by a suitable change in the integration variables. For instance, the integral 130
• Y~gi(x(ui))
(31)
yields the desired result. The various choices ui(x) are called the "channels". For multidimensional integrals, with strong peaks in several variables, such a procedure is essential for obtaining stable results. The differential cross section exhibits several peaks. We found it sufficient to choose four channels for purely leptonic and six for semileptonic processes; in some cases their number can be reduced by symmetry arguments. The integration variables must be chosen such that this behaviour can be recognized and tranformed away.
2. 3.1. Initial radiation channels (diagrams I, 5 of
fig. l) The graphs 1, 5 show all of the above peaks. [Where not explicitly stated, the s' peak at M2z is avoided by a cut. Since it requires Ev~ ~Mz, the cross section is small in that region anyway. ] In order to find the most sensible "channels", it is
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useful to view the process as e + e - --,T + X, and choose the photon energy E r and its angle, f2r= (cpr, 0r) as variables. The E~ -1 and ( 1 - c o s 2 0r) -1 peaks are eliminated by the transformation (30) with g = x - i, ( l - x 2 ) -~, respectively. The ranges are [0+co r, ~Mz] for Ev, and [ ( - 1 +¢r), ( 1 - E r ) ] for cos 0r, respectively, where cor, Er are the cutoffs. To parameterize the "decay" of X, we use the angle of W +, £2w= (~w, 0w) and the invariant masses s34, s56. Due to the Breit-Wigner poles, s34 and s56 are sampled with (26) symmetrically around M 2. The angle 0w is properly accumulated by the t or t' peaks, respectively. This then requires two channels. Because of the symmetry between t and t' diagrams it is sufficient to evaluate one channel and double the result. The final variables are the angles ~r~34and f256 of the decay products relative to their "parent" W boson. The phase space element is now given by dO1, ( s ) = ~ o ~ -zr ~1/2(S~,S34, S56 ) X dE r d~2r d.Qw ds34 ds56 dg-234d~56,
( 32 )
with 2(a, b, c ) = a 2 + b 2 + c z - 2 a b - 2 b c - 2 a c .
(33)
Finally, events with a photon m o m e n t u m generated parallel to one of the final state (charged) fermion's m o m e n t u m have to be rejected (see discussion on mass effects).
2.3.2. Final radiation channels (diagrams 3, 6, 7, 9 offig. 1) This time, it is natural to view the total process as e+e - ~ W + W - , followed by W + --' ~fT and W - ~T'f or vice versa. [ When both W decay leptonically or hadronically, the integration variables corresponding to these two cases appear symmetrically. This symmetry allows a check of the matrix element and reduces the integration work. ] To parameterize the first step, it is obvious to use the variables $347, $56 and f2w, where $347, $56 are sampled according to the Breit-Wigner distribution and Ow as described in section 2.3.1. While W---,]'f is trivially described by f256, the process W + ~I'f7 needs some comments. The emission of the photon from the charged lepton or down quark with m o m e n t u m
13 June 1991
P4 results in a radiation factor x/~/(P4 "P7); for had-
ronic decays an additional factor r~4=2/(p4"P7) × (P3"P7) arises (both quarks are charged). The threeparticle phase space can be described by d¢5 = ~EaEb d~2a d~ab,
(34)
where Ea, Eb, d~ab, d~'2, are the energies of two of the particles, their relative azimuthal angle and the solid angle of the first one, respectively. We choose particle b to be the photon. To treat the first factor, we note that (p4.P7)=2Es(EB-E3), where EB= s~347/2. Thus we take Ea = E3 as a variable and sample it with 1/EB (EB--E3). A second channel takes into account the radiation factor r24 which can be written as [ 2 E 2 ( E B - E 3 ) ( E 7 + E 3 - E B ) ] -1 in the P347 rest frame. We use again E3 as variable, sampled now accordingly to r24 . The range of the variables is over the allowed region, except for the cust. [These are defined in the laboratory system and cannot be applied "directly" at the time of generating events. However given a cut in the laboratory system, we can calculate a minimal value for the cut in the rest frame and therefore reduce the number of subsequently rejected events. ] The remaining variables, ~24 and (047pose no problem. Thus, we have the following channels for W+-~tfT: 1 21/Z(s', $347, $56 ) doll ( S ) - 29 S × d~"2wds347 ds56 dEr dE3 d Q3 d~o37d~'~56 ,
(35)
with two different variable samplings as discussed. In conclusion, purely leptonic events require the two channels (32) and one of the channels (35) (along with an analogous one for W - ). For semileptonic decays both channels (35) (and the corresponding one for W - ) must be taken.
2. 4. Finite mass effects Neglecting the fermion masses results of course in collinear divergencies which require the introduction of an angle cutoff ~r. However in order to combine the hard photon cross section with the virtual O ( a ) corrections we have to include finite mass effects in the case that the photon makes a small angle of O (mr/ Ef) with one of the fermions. The procedure is well 131
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known [ 15 ]. Essentially there are two effects to consider: Firstly, the inclusion of nonleading terms of O ( m 2 ) requires multiplying IMI 2 with a mass correction factor,
~
q
t
m 2
wf=l-
s'(s-s') (pf.k) s 2 + s '2 '
13 June 1991
(36)
for, e.g., an initial state fermion. Similar expressions apply to final state fermions [ 15,17 ]. Secondly, we have to take care of the right peaking behaviour of the cross section in case that the fermions are generated with massive kinematics. Evaluating ( 11 ) for a massive momentum p and the photon momentum k yields I(P+ Ik-)12=2(p'k)+m
2 ko q-k3
(37)
Po -t-P3
which causes indeed an unwanted peaking behaviour of the cross section in a not strictly collinear case. This can be removed by multiplying IMI 2 with a compensation factor, w~=
I (pflk)12 2(pf'k)
(38)
We have checked that this procedure allows to remove the original angle cut ¢v. However there are finite corrections surviving for angles around ~ (mr/ Er) which are of the order of several per cent. These are "numerator" effects and we can eliminate them by appropriate cuts.
3. Background The resonance characteristics of the WW channel for the 4 fermion production should eliminate any background, especially if appropriate cuts on the fermions are applied. However, if it is impossible to apply such cuts, the background should be reexamined. In order to simplify the analysis, we consider here only final states which have no electrons in them (e.g., Ix+x-v~9~; ~t+V,~lq). Then, only the diagrams of fig. 3 contribute. The corresponding matrix elements can be found, e.g., in ref. [ 22 ]. (For a more complete list, see ref. [ 23 ]. ) Whereas these effects are appreciable at lower energies [ 16 ], they turn out to be negligible here. A more complete study will be given elsewhere. 132
Fig. 3. Diagrams other than W+W- production contributing to the 4 fermion final state without electrons. Depending on the nature of the fermions, there are several of these.
4. Results The numerical integration was carried out using VEGAS [20,21] with a sufficiently fine grid to reduce the statistical error to less than 1%. For all numerical results we have taken G~ as input parameter (compare section 2). The values for Mw used are 80.9 and 82.0 GeV; the corresponding widths are (QCD corrected) 2.14 and 2.23 GeV. Throughout the paper we are using M z = 91.16 GeV. While the results strongly depend on the lower energy cut ogv of the photon, the cross section falls so strongly with increasing energy that the value of the upper cut [see eq. (25) ] is irrelevant. We will use 60 GeV. The Z pole yields only an unmeasurably small peak. The lower cut toy is kept fixed except for the results of table 3. In order to identify the W pair most clearly, we consider purely leptonic decays (both W decay into a given lepton-neutrino pair) and semileptonic decays, where one of the W decays hadronically into one pair of flavors [4]. In table 2 we give the total cross sections for the processes e+e---~lVl~2927 and ~lv2Od 7 (or vice versa) for a beam energy of 100 GeV, as a function of the cutoffs for energy and angle between the photon and any charged particle. [ See also the above discussion on the effects of a finite electrom mass. ] Fig. 4 shows the photon spectrum for a beam energy of 100 GeV as a function of the angle cutoff. Fig. 5 gives the excitation curves for e + e - ~ 4 f and 4 f + photon. Whereas the leptonic and the (properly scaled) semileptonic cases are the same without a photon, they differ when the photon is included. The cross sections for low cutoffs are large as expected especially well above threshold. It is interesting to compare the exact calculation
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PHYSICS LETTERS B
i
i
i
13 June 1991 (39)
dtTapprox = do*oy+ daww~,
i
,1(~a
0.20 3
where the peaking approximation is used on the fighth a n d side and daov is given in curve ( b ) . This curve, labelled ( d ) , is again different from the exact one (c). Finally, the exact h a r d p h o t o n result can be split in the following way:
0.t6 o.t2
b
~o 0.08
dal~,=daww~,+daw(~,)w+daww(v) + A a ,
0.04 0.00
I0
40
20 30 Ey (GeV)
50
Fig. 4. Photon spectrum for three values o f the angle cut ev. The
W mass is 80.9 GeV, the angle cuts are e~= 1°, 5 ° and 10 °.
0.3
,
,
,
,
,
incl. photon
~, 0.2 tn
o
/
(40)
where dawwv has been given above, a n d daw(v)w is the cross section for two on-shell W, one o f this decays into a lepton pair a n d a photon. In fact, there is no " d o u b l e counting" o f diagrams with the p h o t o n e m i t t e d from the W-line. Neglecting A a a n d adding davy to curve ( a ) yields curve (e). A comparison with (c) shows that A a is small. Finally, we list in table 3 the results for awwv as discussed in (39) including finite mass effects for the initial electrons. The r e m a i n i n g angle cut discussed above is taken to be mE/gb times some n u m b e r Yr a p p r o p r i a t e l y choosen to keep the corrections small. F o r c o m p a r i s i o n we also give the results for y r = 1. In this calculation, the energy cut tgv is i n t r o d u c e d as a
0.t 0.3
0.0
'
80
'
'
t00
'
I
I
I
I
I
I
'
120
0.2
E beam [GeV] Fig. 5. Radiative and nonradiative excitation curves as a function of the beam energy for Mw=80.9 GeV (solid line) and
M w = 82.0 GeV (dashed line). The upper curves represent the sum ao~+ alv. The dotted curve is the semileptonic contribution for M w = 80.9 GeV divided by three (color!). The applied cuts
0
"- 0.t
are a~v=2.0 GeV and Ev= 5 °. with finite W - w i d t h to the a p p r o x i m a t e t r e a t m e n t where the W are on-shell (peaking a p p r o x i m a tion ~2). F o r the tree-level calculation without photon the exact curve ( a ) and the a p p r o x i m a t i v e curve ( b ) o f fig. 6 show a sizeable difference even above threshold. Including the h a r d photon, we write
#2 In the peaking approximation the Breit-Wigner propagator 1/ (paw -Maw + i3/wFw) is replaced by (n/MwFw)~(paw-Maw).
0.0
J i
80
I
t00 E beam [GeV]
I
120
Fig. 6. Exact and approximate excitation curves for the purely leptonic case (Mw=80.9 GeV). Curves (a), (c) exact results, curve (b) peaking approximation of the nonradiative processes, curve (d) aww+aww~ where both W are treated in peaking approximations, curve ( e ) approximation in eq. (40). The applied cuts are o~v= 1 GeV, e~= 1°. 133
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f u n c t i o n Of Eb as discussed, e.g., in ref. [ 8 ] t,3 In conclusion, we h a v e s h o w n that in the case where no k i n e m a t i c a l cuts are i m p o s e d , the finite w i d t h effects d u e to off-shell W ' s result in a m e a s u r a b l e c o n t r i b u t i o n to the cross sections. T h e p h o t o n s p e c t r u m as well as the total cross section are g i v e n for different masses M w a n d different (regularizing) cuts. O u r results are d e r i v e d w i t h i n the s t a n d a r d m o d e l . H o w ever, the e v e n t g e n e r a t o r is general a n d can be u s e d for any e x t e n s i o n o f the m o d e l . In the case w h e r e there are no electrons in the final state, we h a v e s h o w n that the b a c k g r o u n d is negligible.
Acknowledgement T h e helicity a m p l i t u d e o f table 1 was o b t a i n e d by H. Bijnens, Z. K u n s z t a n d D. W y l e r [ 10,4 ]. We w o u l d like to t h a n k H. Bijnens a n d Z. K u n s z t for the early c o l l a b o r a t i o n a n d for m a n y helpful discussions. W e also t h a n k J. V e r m a s e r e n for p r o v i d i n g us w i t h his M C routines. #3 In narrow width approximation, an expression for the photon energy cutoff in 09.t=x/~Xr/{ 1+Xr+ [ ( 1 --Xr) 2 - 4M2/s] 1/2} with xr the part of the W energy shifted to the soft photons.
References [ 1 ] J. Ellis and R. Peccei, eds., Physics with LEP, CERN yellow report CERN 86-02 (CERN, Geneva, 1986). [2] A. B6hm and W. Hoogland, eds., ECFA Workshop on LEP 200, CERN report CERN/ECFA 87-08 ( 1987 ).
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t 3 June 1991
[3] W.J. Marciano, Nucl. Phys. Proc. Suppl. B 11 (1989) 5-34. [4] P. Roudeau et al., in: ECFA Workshop on LEP 200, CERN report CERN/ECFA 87-08, eds. A. BShm and W. Hoogland (1987). [5] T. Muta, R. Najima and S. Wakaizumi, Mod. Phys. Lett. A 1 (1986) 203. [6] Z. Hioki, Nucl. Phys. B 316 (1989) 1. [7] M. Lemoine and M. Veltman, Nucl. Phys. B 164 (1980) 445; R. Philippe, Phys. Rev. D 26 (1982) 1588; M. B6hm, A. Denner, T. Sack, W. Beenacker, F. Berends and H. Kuijf, Nucl. Phys. B 304 (1988) 463. [8 ] J. Fleischer, F. Jegerlehner and M. Zralek, Z. Phys. C 42 ( 1989 ) 409. [9] H. Tanaka, T. Kaneko and Y. Shimizu, KEK preprint 9029. [ 10] H. Bijnens, Z. Kunszt and D. Wyler, unpublished. [ 11 ] P. De Causmaecker, R. Gastmans, W. Troost and T.T. Wu, Phys. Lett. B 105 (1981) 215; Nucl. Phys. B 206 (1982) 53; F.A. Berends et al., Nucl. Phys. B 206 (1982) 61; B 239 (1984) 382. [ 12] R. Kleiss and W.J. Stirling, Nucl. Phys. B 262 ( 1985 ) 235. [13]Z. Xu, D.-H. Zhang and L. Chang, Tsinghua preprints TUTP-84/3. [ 14] J.F. Gunion and Z. Kunszt, Phys. Lett. B 161 (1985) 333. [ 15] R. Kleiss, Z. Phys. C 33 (1987) 433. [ 16 ] W. Marciano and D. Wyler, Z. Phys. C 3 (1979) 181. [ 17 ] F.A. Berends, W. Hollik and R. Kleiss, Nucl. Phys. B 304 (1988) 712. [ 18 ] R. Kleiss, in: Z Physics at LEP I, Vol 3, CERN yellow report (CERN 89-08). [19] E. Byckling and K. Kajantie, Particle kinematics (Wiley, New York). [20] G.P. Lepage, J. Comput. Phys. 27 (1978) 192. [21 ] S. de Jong, R. Laterveer and J.A.M. Vermaseren, AXO user manual, report NIKHEF-H, Amsterdam ( 1987 ). [22 ] V. Barger and T. Han, Phys. Lett. B 241 (1990) 127. [23] J.C. Roma~o and P. Nogueira, IFM-preprint 8/88.