Complete initial state QED corrections to off-shell gauge boson pair production in e+e− annihilation

NUCLEAR PHYSICS B ELSEVIER

Nuclear Physics B 477 (1996) 27-58

Complete initial state QED corrections to off-shell gauge boson pair production in e +e- annihilation Dima Bardin 1, Dietrich Lehner 2, Tord Riemann Deutsches Elektronen-Synchrotron DESY, lnstitut far Hochenergiephysik Iflt Zeuthen, Platanenallee 6, D-15738 Zeuthen, Germany

Received 1 March 1996; revised 27 June 1996; accepted 1 July 1996

Abstract We study Standard Model four-fermion production in e+e - annihilation at LEP2 energies and above using a semi-analytical approach. We derive the complete QED initial state corrections (ISR) to the reactions e+e - ~ ( Z ° Z °) --* f l f l f 2 f 2 and e+e - --* ( W + W - ) ~ f ~ f a f ~ f a 2 4- ¢-) with f l :# f2 and f i --/: e , re. As compared to the well-known universal s-channel ISR, additional complexity arises due to non-universal, process-dependent ISR contributions from tand u-channel fermion exchanges. The formulae needed to perform numerical calculations are given together with samples of numerical results. PACS: 12.15.Lk; 13.40.Ks; 14.70.-3 Keywords: Heavy gauge-boson production; Electromagnetic corrections; e+e - annihilation; Four-fermion

production

1. Introduction The LEP2 e+e - accelerator will finally operate at energies between 176 and 205 GeV [ 1 ] and thus pass the production thresholds for W a: and Z pairs. At LEP2, a typical process will be four-fermion production which is much more complex than fermion pair production as known from L E P I . This complexity shows up at tree level already, because o f the many Feynman diagrams involved in four-fermion production. l On leave of absence from Bogoliubov Theor. Lab., JINR, ul. Joliot-Curie 6, RU-141980 Dubna, Moscow Region, Russia. 2Now at Fakult~it f'dr Physik, Albert-Ludwigs-Universitiit, Hermann-Herder-Str. 3, D-79104 Freiburg, Germany. 0550-3213/96/$15.00 Copyright (~) 1996 Elsevier Science B.V. All rights reserved PH S0550-3213 (96)00354-9

D. Bardin et aL/Nuclear Physics B 477 (1996) 27-58

28

e+(k2) ~ f ~ ( P 3 )

e+(k2)

f~(p4)

J'(p.)

~-( kl)

~f~(p4)W+

/ t-(k1)

~"'f~(P3)

''f~(p,)

Fig. 1. The tree level Feynman diagrams for off-shell W pair production (the CC3 process). Left: t-channel diagram. Right: s-channel diagram. The particle momenta are given by the ki and pj.

As tree-level amplitudes are not sufficient to describe experimental data, radiative corrections to four-fermion production are needed. It would be desirable to derive complete O ( a ) electroweak and QCD corrections to four-fermion production, but this has not yet been achieved, although many partial results were reported in [2-5] and in references quoted therein. In view of the anticipated experimental precision of LEP2 and future e+e-colliders, it would be desirable to have theoretical predictions for cross sections and distributions accurate at the level of half a percent. In recent years, three major approaches to four-fermion production in e+e - annihilation have been developed, namely Monte Carlo approaches [3], the semi-analytical approach [6-13], and the "deterministic approach" [ 14]. Monte Carlo and deterministic techniques use numerical integration for all phase space variables. Typically, the semi-analytical method performs analytical integrations over the five (seven, if ISR is included) angular degrees of freedom and uses high precision numerical integration for the remaining two (three, if ISR is included) squared invariant masses. It represents an approach to the high-dimensional, highly singular phase space integration problem inherent in this type of physical problem, which is elegant, fast, and numerically stable. Thus, it may serve as an ideal source of benchmarks for the other two approaches. From LEP1 we know that semi-analytical calculations are also relevant to experimentalists. In this article we will present semi-analytical results for the gauge boson pair production reactions CC3 :

e+e -

( W + W - ) _ _ ~ f -u l f l d f ~, f- ~d ( Y ) ,

NC2 :

e+e - ~

( Z OZ°) --~ f l f l f 2 f 2 ( Y )

NC8 :

e+e - --~ ( Z ° Z ° , Z ° T ,

,

TT) --~ f l f l f 2 f 2 ( Y )

f i # e + , ~Ue' f l 4= f2, f i 4= e +,'-~ue , fl 4= f2, fi 4= e +, (-~Pe (1.1)

with complete initial state QED corrections. The three Feynman diagrams for the charged current CC3 process at tree level are given in Fig. 1. The two or eight diagrams for the neutral current NC2 and NC8 processes are depicted in Fig. 2.3 Classifications of four-fermion processes may be found in Refs. [8,15]. Initial state QED corrections (ISR) represent a dominant correction in e+e - annihilation. Complete ISR to four-fermion production separates into a universal, factorizing, process-independent contribution and a non-universal, non-factorizing, process3 Strictly speaking, the NC2 process is well defined (i.e. observable) only as on-shell reaction.

29

D. Bardin et al./Nuclear Physics B 477 (1996) 27-58

e+(k2)

e+(k2)

A(p,)

el

fl(pl) e--(~1)

~,~.2...~

~'~(p2)

f,(p3)

e-- (/dl)

Fig. 2. The tree level Feynman diagrams for off-shell neutral gauge boson pair production (NC8 process). Left: t-channel. Right: u-channel. The NC2 process is obtained by neglecting diagrams with exchange photons.

dependent part. With the index J labeling the CC3, NC2, and NC8 processes, the ISR corrected cross sections can be generically written as dO'J, QED( S ) f -d s ' [ G-(-s , / s ) ~ j , o ( S , , S , , S 2 dsl ds2 s

) + O-ZQEO non-univ (s, s t, Sl, s2) ]

(1.2)

with invariant boson masses sl and s2, reduced center of mass energy squared s', and tree level four-fermion production cross section o-z0. The factor G(s'/s) contains all mass singularities ln(s/m2e) and incorporates the process-independent ISR radiators as known from s-channel e+e - annihilation [ 16]. In addition, there is a non-universal non-univ . It appears together with t-channel and u-channel amplitudes and contribution O-ZQEO is mass singularity free. For pure s-channel contributions it is absent. Complete initial state QED corrections were shortly communicated for the CC3 process in [7] and for the NC2 and NC8 processes in [ 11]. While a definition of initial state radiation is straightforward in the neutral-current process, there is an arbitrariness for W pair production in its definition. In [7], we restored the U(1) invariance of the initial state photon emission by adding an auxiliary current. This arbitrariness is characteristic of charged current processes. Of course, it is the sum of the corrections that will be finally observable. In this paper the complete analytical formulae for the non-universal corrections, supplemented by a study of their numerical importance, will be presented for the first time. For on-shell production,

e+e - ~ W + W - ( T ) , e+e - ~ Z ° Z ° ( y ) ,

(1.3)

the generic cross section may be obtained as follows: O.j,QED(S) = j

) o'Zo(s , ,My,2 My) 2 +O-zQED -non--univ (S, s~,M~,M2)] . (1.4) -dS'[G(sl/s S

4M2v

Here, My represents the W or Z boson mass. The relation between grz0 and O-zo on the - non--univ non--univ one hand and Orj, QED and Orj,QED on the other will be discussed in Section 2.

30

D. Bardin et al./Nuclear Physics B 477 (1996) 27-58

For Z pair production, there are no additional QED corrections for the on-shell case, while for W pair production there are final state corrections and initial-final interferences. The non-universal initial state QED corrections were not known before as explicit analytical expressions. However, they have been determined as a part of the complete electroweak corrections to the processes (1.3) with numerical integrations in [17]. The outline of this paper is as follows. Section 2 presents general features of our approach to the complete initial state QED corrections. In Section 3 we give a detailed presentation of the non-universal cross section contributions. Section 4 contains numerical results and Section 5 concluding remarks. In a series of appendices, we give technical details of the performed computations. Some notations are introduced in Appendix A. Our phase space parametrization and all relevant relations between particle four-momenta and phase space variables are presented in Appendix B. In Appendix C, the tree level, the real ISR, and the virtual ISR matrix elements are given. These matrix elements represent the starting point for the calculations in this paper. The analytical integrals needed for the integration of the angular phase space variables and, in the case of virtual corrections, loop momenta are collected in Appendix D of [ 18].

2. General structure of initial state QED corrections

To include initial state QED corrections (ISR) to (1.1), the five-particle phase space is required to take into account four final state fermions and a bremsstrahlung photon with momentum p. We make use of the following parametrization: l

~/I~(s, st,O) ~/I~(S',SI,S2)

d F 5 - (27r)14

8s

k/A(sl,m~,m

8s t

× ds' dSl ds2 dcos0 dS2R dS21 d,O2.

8sl

2)

~/A(s2, m3,2m4)2 8s2 (2.1)

In Eq. (2.1) the azimuth angle around the beam has already been integrated over. We have adopted the usual definition of the ,~ function, el(a, b,c)

= a 2 + be + c 2 - 2ab - 2ac - 2bc,

A - A(s, Sl, s2).

(2.2)

We use kl and k2 as the initial electron and positron four-momenta, while pl, P2, P3, and P4 label the final state fermion momenta as indicated in Figs. 1 and 2. The relevant invariant masses are 4 s -- - ( k l

-Jr- k2) 2 = - ( P l

st=--(Pl

+P2

Sl = - - ( P l

-k-P2) 2,

-/- P2 q- P3 q- P4 q- p ) 2 ,

-l-p3 + p 4 ) 2,

4 In our metric, space-like four-vectors k have positive k2. Thus k2 = - m 2 for on-shell particles of mass m.

D. Bardin et al./Nuclear Physics B 477 (1996) 27-58 e+

31

J.'fl

f3

Fig. 3. Generic two-boson production and decay Feynman diagram with real photon initial state radiation.

(2.3)

$2 = --(/93 -I- p4) 2.

The photon scattering angle 0 is defined as the angle between p and k2 in the center of mass system. The solid angle d/2n = dcos0n d~bR

(2.4)

represents the production solid angle of the boson three vector vl -- Pl + P2 in the four-fermion rest frame Pl +P2 +P3 "{'-P4 = 0. J?l ['(22] is the solid angle o f p l [P3] in the two-fermion rest frame Pl +/72 = 0 [P3 + P4 = 0]. In this frame, the three-vectors Pl and P2 [P3 and P4] are back to back. The polar and azimuthal decay angles in the above two-particle rest frames are defined via d~2i = d c o s 0 i dq~i,

i = 1,2.

(2.5)

Further details of the kinematics and the five-particle phase space may be found in Appendix B. The processes (1.1) have the generic structure shown in Fig. 3. Their ISR-corrected cross sections get two types of contributions, namely universal and non-universal ones. The first type is universal in the sense that it arises from photonic insertions to any of the basic diagrams and is independent of the details of the subsequent interactions. These universal corrections are, of course, related to the collinear divergences of the (radiating) initial state electrons and positrons and appear to be known, including higherorder terms, from, e.g., the study of ISR corrections to the (single) Z line shape. They are exactly those of the s-channel ISR contributions. For diagrams with t- and u-channel exchanges and interferences of these diagrams among themselves and with s-channel diagrams, there are additional corrections which are not logarithmically enhanced and which depend on the details of the interfering amplitudes. These additional corrections are thus non-universal. ISR-corrected cross sections for the processes (1.1) with soft photon exponentiation may be described by the ansatz

st; s l, $2) offR(s)= f dsl f ds2j[ dss ' zk d3~(k)(S, ds, ds2ds' with the threefold differential cross section

(2.6)

32

D. Bardin et al./Nuclear Physics B 477 (1996) 27-58

d 3 ~ k~(s, s'; s~, s2)

~,~k~ (s',

dsl ds2 ds ~

s1 , $2) [[~e U[~e-ls(k)J ÷ 7"/(k)] ,

(2.7)

where

St 19=1 -- - - . S

(2.8)

In Eq. (2.7) we have used several additional notations which will now be explained. The subscript J labels different processes, J C {CC3,NC2, NC8} and the superscript index k stands for cross section contributions which stem from squared amplitudes or interferences with distinct Feynman topologies and coupling structures. Using O'(jk'O) ( s; sI, S2 ) ~ - - ~

./7.S2

~jk) ( S; Sl

S2),

(2.9)

the soft + virtual contributions Sj~k) and the hard contributions 7-/~k) take the form _(k) t sJ k~(s, s'; s~, s2) = [1 + ~(s)] ,~,0~(s,; Sl, s2) + o~.~ ~'; ~, s2),

7-[~jk)(S,S';Sl,S2)

=/:/(S, s')

O-~k'O)(s';sl,s2) univer~al part

(k) +O'flj(S,S

t

;Sl,SZ)

(2.10)

non-universal part

with the well-known soft + virtual radiator S and the hard radiator/7/, 2

/Z/(s, s') = - ~

1+

÷

/~e + ( Q ( a 2 ) ,

/3e ÷ O(c~2).

(2.11)

Higher-order terms [16] may be implemented exactly as described in [2,3,13]. We draw the attention to the definition of the (differential) effective tree level cross section

do-j.o(s';sl,s2) _ ,/a(s',sl,s2) y'~C~(S,,Sl,S2)G~k~( ,;Sl,S2)

dsl ds2

7"1"St2

(2.12)

k

which is inherent in the universal ISR correction part. Coupling constants and boson propagators are collected in C~k~, while the Gj~k) represent kinematical functions obtained from the analytical angular integration. Explicit expressions for Cj~k) and e,j e ~ ) may be found in Appendix A. If the index k is associated with s-channel e+e - annihilation only, non-universal ISR contributions are absent. The s-channel ISR diagrams are generically shown in Figs. 4 and 5. They contribute to W pair production. The non-universal cross section contributions originate from the angular dependence of initial state t- and uchannel propagators and contribute both to W- and Z-pair production.

D. Bardin et al./Nuclear Physics B 477 (1996) 27-58

33

t,-

Fig. 4. The amputated s-channel virtual initial state QED Feynman diagram.

e

e

Fig. 5. The amputated s-channel initial state bremsstrahlung Feynman diagrams.

To close this section, we comment on the on-shell cross section (1.4). With the aid of Eq. (A.14) it is obtained by the following replacements: @Z0(s, M2v, M 2) = rv-~olimJ [ d S l / d s 2 0 " j , o ( s ,

Sl,S2)

=f d,, f ds:r~lim--,o[ ~~/Zc(Jk2)(SS'S'l2)G(k)(SS'S'l2)~S

(2.13)

and -non-uniw t .-2 . - 2 Orj,QED (S, S , IV1V , IVIv) = lim [ d s , / d s 2

l'v----~OJ

dSl

on-univ O~jIQE D (S,

SI,SI,S2)

ds2 Z rv~olimc(k)(st, k

+~ro.j(s, s'; sl, s2) .

s1,s2)[Pe

~jiS;SI,62) (2.14)

3. Non-universal initial state corrections

In this section we present the final analytical results for the ISR corrected threefold differential cross sections for the processes ( 1.1 ). The starting point of our calculations are matrix elements of CC3 NC2 and NC8 processes, which are given in Appendix C. With the help of the computer algebra packages SCHOONSCHIP, FORM, and Mathematica [ 19], matrix elements were squared, spin summation was performed, the scalar products were expressed in terms of the phase space variables, and algebraic manipulations of the resulting expressions were carried out. All analytical integrations were

34

D. Bardin et al./Nuclear Physics B 477 (1996) 27-58

Fig. 6. The amputated t- and u-channel virtual initial state QED Feynman diagrams. obtained from hand-made tables of canonical integrals. The kinematical relations needed for the treatment of real bremsstrahlung may be found in Appendix B. For the virtual corrections, tree level kinematics may be used as was explained in Appendix B of [ 13]. In the course of performing the various steps of analytical integrations we proceeded as follows. The virtual photonic corrections have been treated as a net sum of all contributing diagrams. The infrared singularity was isolated and, as a part of the universal corrections, subtracted from the net correction. Thus, the remaining, non-universal virtual corrections are by construction free of infrared problems. 5 After tensor integration over the final state angular variables, the loop momentum integrations and the final integration over the vector boson production angle ~ in the center of mass system were performed with the aid of Appendix D of [ 18]. Also the real photonic corrections were first integrated over the final state angular variables. Then we integrated over the production angles ~bn and On of the vector boson in the two-boson rest frame and finally over the photon production angle 0 in the center of mass system. Again, the corresponding tables of canonical integrals are found in Appendix D of [ 18]. We have used the ultrarelativistic approximation for final state fermions and initial state electrons, i.e. the masses of these particles are neglected wherever possible. The various tables of canonical integrals were checked by Fortran programs with a precision of typically 10 -8 . At the end of our calculations, algebraic manipulations were carried out by hand to yield compactification of our final results 6 given in Eqs. (3.3), (3.5), (3.8), (3.18), (3.20), and in Eq. (3.10) of [18]. In the following two subsections, the non-universal corrections introduced in (2.10) will be explicitly given and commented.

3.1. The neutral-current case In the neutral-current case, initial state QED corrections are represented by the Feynman diagrams of Figs. 6 and 7. The non-universal NC8 corrections are 5 The same was done with the real photonic corrections. Thus, non-universal virtual and real corrections may be treated completely separately. The interested reader may find details in [ 12]. 6 The latter was cross-checked against FORM outputs with the aid of auxiliary FORM codes.

D. Bardin

et al./Nuclear Physics

B 477 (1996) 27-58

35

e

Fig. 7. The amputated t- and u-channel initial state bremsstrahlung Feynman diagrams. SlS2

on-univ

00"~'NC8(S; Sl' $2) -- --Tr8q7"s2 00nv~c8 00f/,NC8 (S, St; S1, $2)

.

(S, Sl, $2),

non-univ = ~Ol S1S2 7rs 00R,NC8 (s,s';sl,s2),

(3.1)

where the subindex V stands for virtual and the subindex R for real photonic corrections. The virtual contribution is a sum of contributions due to the different interferences of loop diagrams with tree level graphs: non-univ . non-univ O'VNC8 ( S , S l , S 2 ) = O'V,t -[-

on-univ 2~,tu +

on-univ

~v,°

.

(3.2)

The squared t- and u-channel contributions are equal: non-univ

00v, t

. non--univ (s,s~,sz) =00v,~ (s,. sl,s2)

31Z2 [sl+ +t~l_

2

+L;I

S(S

O') /12q]

[(s-2o')(s-o')ll2q+--~ss 308 l- s-400 ~l++s-00 ]

+£0 -

/~+002+

112q-~l--40"l++80"-9s

+2 v/~ [4 (2 - /+) + (as -o') 1,2q]. Here, the following notations have been used: O'=S1 -'[- $2, t ~ = S l -- $2,

l+ =In ~ + I n S

L;o = In

s2, S

l_=in-"

S

$2

S1

S

$2

-ln--=ln~

s - 00 + v/-A S _ 00 _ V/'~ '

£1 - s (s - 00) [£0 - - A s--Fj

L

£2=s(s-00) A2

7'

2v~

3 [ £o

[

S--O"

3

~

'

(3.3)

D.

36

ll2q = " ~

Bardinet al./Nuclear Physics B 477 (1996) 27-58

D + --

l_/2_

,

£ - = £ 1 2 -- £34,

s+~+v~

1212 = In

/234 = In

s + , ~ - v/-a '

(_tmax~ _

D+ =Li2

\

,(

tmin=~

sl /

Li2 ( _ tmin~ q- Li2 ( t m a x )

)

s-o'-v/-A

s-6+v~ s-~-v~'

\

sl /

,

1(

tmax=~

\

_ Li2

s2 /

s-o'+~/'A

)

(_tmin), \

s2 /

•

(3.4)

T h e virtual corrections due to the tu-interferences are

non-univ

O'v,tu

.

(s, sl, s2)

_3E2 [s l+ + 8 l_ - s ( s - o')I12q] 2 ( 6)9(sl++~l-)]-~ + £ 1 [(s-~r) l+5s112q+~sl_ +120

G--2/++

dl1

+o'(l+-2l,~)+2

2 l+ - 6 l~ -

+s

+

- ~/202 - 16Dd + 4 D d + -- 4

s-o"

~-

-

s l12q

.G- - ~'~ +

+SV/~ (21,2q+hl-l-)

1 d2

3

1

s --

h2+ -

( s - sl )2 S --

dl-

s1 $2

1 ~ - -

Sl

S2

s -- $2

+ 2 (3s+o') D~.

,

s1 (s - s2) 2 '

-

- ,l+ D~ + ll+De_ 2

In Eq. ( 3 . 5 ) , the f o l l o w i n g additional notations have been used: hl+ = - -

l~

(3.5)

37

D. Bardin et al./Nuclear Physics B 477 (1996) 27-58

S -- S1

d2= - - , $2

S--O r la

= In - - ,

S

Id- = In dl - In d2, ( tmax ~

( tmin ~

D~ =Li2 k s _ o-/ - Li2 k s _ o - / , + Li2 (d2)] ,

Dd=Re[Li2(d,)

Dd+=Li2 ( t m a x ) + L i 2 (

train ) + L i 2 (

\ S -- S 2 /I

\ S -- S2 /I

tmax

Dd- =Li2

-Li2 k s - s2 /

ks-

sl /

\S--Slit

tmin

\ s - Sl / '

(tr~n~

/max

.T,r =Li3

)+Li2 (tmin),

tmax ~ + L i 2 (

(tmin)-Lis(

S -- S 2 J

tmax \S--SIll

- Li3 k s _ o-/ ,

s-o'/

- L i 3 ( tmin ) + Li3 ( tmax ~ - Li3 ( t m i n .Td-= Li3 (tmax , \S--$2/ \ S -- S1/I \ S - - SIlt ks - s2/ f't- = Li3

(

tmax ) _Li3 ( dl train

+Li3(

train ) dl tmax

t_max ~ detmin/ -Li3 (

train ) d2tmax "

(3.6)

The non-universal real bremsstrahlung contribution to the NC8 process is given by non-univ non-univ non-univ non-univ O'R,NC8 (S, St; SI, $2) = OrR,t -~ O'R,tu -~- O'R,u •

Again, the squared t- and u-channel contributions are equal: non-univ ( S , S , t.S 1

O'R, t

'

non-univ $2) -~ O-R,u (S, St; S1, $2)

o- (3s - s ~_ - o-) = V~

St_

+7-

"~ 4S-~ -

1+

+ 1 2 s sl s2 tr

_

~2

1[

o-s1+ SS !

+ 24

1 -- st 2 / q-

S SI $2 O-]

s - o-

s"

2str

( Lcl - Lc2)

2tr

,.2 + T (s'+ -o-)

~-----7--j ( Lcl + Lcs)

(3.7)

38

D. Bardin et al./Nuclear

s:+~+~ 2A

(

+JAI

Physics B 477 (1996) 27-58

)

(L3

4d2

+

L4)

+

U&g_ 2ss’2Lc5

p--s+s~+; (s(sL+g)-4sls2)

--$

+12SS,S2a(s’+v)

Df2

;i2 1

(3.8) In the above equation ’ _ s*=sks’,

A’=

x = A(s’, The additional

( s’(s’-

=ln

=ln

s’_(s’+

VV)

+s’C7+s6

s!_(s’-

fi)

+s’a+s6

1+ (

+s’c--sss +s’Cr-s6

1+

(3.9)

are

JxQ

( Lc4

h(s’,s,,s2),

s’_(s’+fl)

LC2= In

the symbols

-s1, --s.z) = s’_* + 2s’_(r + s2. notations

L,l = In

Lc3

we have introduced

) ’

s’_(s - 6) s’s2

>’

s’(s+6) s’s,

) ’ (3.10)

D. Bardin et al./Nuclear Physics B 477 (1996) 27-58

(3.12)

D~=D~(Sl ~ s2), D]2=Re[-Li2(4

c++a--e+ d

)+Li2 (+ c_+a__e+~

+Li2 (

c+_a__e+ ~} _ Li2 ( c__a__e+]d] d

-Li2

c++a_+e+] +Li2 d

]

+Li2 ( + C + - d - + e + ) - Li2 -Li2 ( + C + - d + + e - ) + Li2 +Lie ( -Li2

39

_ c_+a_+e+

(÷c%+e+) ( q c__a++e_ d ]

c + + a + + e - ) _ L i 2 _c_+a++e_ d c+_a+_e_ ] d + Li2 ( c--a-2+-e-]d c++a+_e_

+Li2 q

d

-

Li2 4 c_+a+_e_

(3.13)

a++ = s + v/--~ ± V/~, c++ = t~ + s'_ i V/~, e + = s' - ~r i v ~ , d = S s s l s2.

(3.14)

The most cumbersome contribution is due to the tu-interference of non-universal real bremsstrahlung, O-R,tu non-univ(S, S'; Sl, S2). Since we will not refer to the explicit expression in the discussions, we will not reproduce it here but refer to Eq. (3.10) of [ 18].

3.2. The charged-current case It is in order to mention that, for the CC3 process, the separation of initial and final state radiation is not unique. Since, in the t-channel contribution, there is electric charge flow from the initial state to the final state, electric current conservation is violated and one faces the problem of finding a gauge-invariant definition of initial state radiation. As a solution we proposed in [7] what we call the current splitting technique (CST). In brief, the CST splits the electrically neutral t-channel neutrino flow into two oppositely flowing electric charges + 1 and - 1 . Charge - 1 is assigned to the initial state, charge + 1 to the final state. This enables a gauge-invariant definition of ISR with photon emission and absorption from the t-channel exchange particle. An auxiliary current is added to the naive charged current for this purpose in Appendix C. Thus, the CC3 t-channel receives the same ISR corrections as the neutral-current t-channel. Of course, when performing a complete calculation, the final state corrections have to take into account opposite auxiliary terms so that the net auxiliary effect will vanish. The non-universal corrections are due to the interferences of the s- and t-channel diagrams of Figs. 4 and 6 with the corresponding Born diagrams and of the diagrams

40

D. Bardin et al./Nuclear Physics B 477 (1996) 27-58

of Figs. 5 and 7 among themselves. These interferences show less symmetry than the interferences of the t- and u-channel diagrams of the neutral-current case. However, as mentioned earlier, the pure s-channel corrections have no non-universal parts: s

s

=0.

fS,CC3 = O'/1,CC3

(3.15)

Further, for the pure t-channel non-universal contributions to the CC3 process one finds, of course, expressions equal to those for the pure t-channel non-universal contributions to the NC8 process, namely t . Of S1 $2 non--univ . O'~,CC 3 ( S , S 1 , $ 2 ) ----7/.87/.$2 fV, t (S, S1,S2) t

°'H,cc3

(S, St; S1, S2) -- Ol S1S 2

non-univ

7"r rrs °'R't

t.

(s,s ,Sl,S2)

(3.16)

non-univ from Eq. (3.8). from Eq. (3.3) and fR,t with fv,non-univ t What remains to be calculated in addition are the virtual and real st-interference contributions. The virtual correction is given by st

-- Ol S1S 2 non-univ . ~7.877. S""2 0"V,st (S, S1,S2)

(S;SI,S2)

f,{,CC3

(3.17)

with on-univ ( s ., s , , s z ) = - v ~ [ 7 s + 3 f + ~ l - + ( 4 s f + 2 a - 6 2 ) l 1 2 q ] o~,st

+£o[5sf+3A

3°2 2

762 ~-+

( 2 s o ' + a - ~ ) l + + 6 ( s f+ ~)

] l_ . (3.18)

The real bremsstrahlung contribution reads st

t.

non--univ

Ol S1S2

f~,cc3(S, s ,sl,s2) - ~r-ff-~ fI~,s,

t.

(3.19)

( s , s ,sl,s2),

with non--univ

fiR,st

+ ~+

f

s

s-

+Zs, s'

t~S t S~_ (ors~_

(s,sp;sl,sx) -

s2 s t

f

1

s +3

S1 S2

+--s

(1)] 2

~-

(Lci+Lc2)

,,ss,(fs'+-8

[o'2s~_2 + - -

+sLs2s,2

ss, ss s+ l ] \

sZs7

2~ (D~- Dr2)+ 2 (s[szs'+

-T

-2\

/

k2ssP + 1/ (Lcl - Lcx)

\ sU

( s! s2 s'+ f SSt2

+~7--1

)

+4

f

+ s'

(Dzlt2+Dz2tl).

Lc5

~ ) (D[ + D~) (3.20)

D. Bardin et al./Nuclear Physics B 477 (1996) 27-58

41

3.3. Discussion

To close the section we will now discuss some formal features of our results. Nonuniversal cross section contributions are analytically rather involved and contain many di- and trilogarithms. Although expected, it is noteworthy that interferences are more involved than matrix element squared contributions, which is especially true for the tu-interference. Investigating the analytical expressions, which are collected above and in Appendix D of [ 18], one realizes that some of them are very complicated (e.g. the function D~2 for the real t-channel correction and some of the integrals for the real tu-interference, which are tabulated as 36), 38), 40); see Ref. [18]). However, the final answers for t-, u-, and st-contributions, both virtual and real, are remarkably compact. The virtual contributions contain only one true dilogarithm, which arises from the three-point scalar integral ll2q (see Eq. (3.4) and integral 15) in Appendix D.2 of [18]). Both these virtual and real contributions do not contain true trilogarithms. In contrast, the tu-interferences are rather cumbersome. This may be traced back to the angular integrations involving products of different space-like fermionic propagators. Virtual tu-interferences contain several true trilogarithms and real bremsstrahlung tuinterference contributions exhibit complicated complex-valued dilogarithms. The latter, however, could indicate that some simplifications were overlooked. We mention that the four non-universal cross section contributions to the CC3 process became much more compact after inclusion of the auxiliary terms than without them. The attentive reader will have noticed that the virtual+soft non-universal contributions are evaluated at s t rather than at s. This is to avoid an unphysical, t~-distribution-like concentration of non-universal virtual+soft ISR corrections at zero radiative energy loss. In Eqs. (3.1), (3.16), (3.17) and (3.19), both reasonable threshold and high-energy properties may be observed. For x/~ or ~ approaching zero, non-universal corrections vanish, which may be verified by inspection of the explicit expressions. Thus also the ISR corrected cross section vanishes for the kinematical zero of the tree level or the universally corrected cross section. At high energies, the screening property may be verified to hold. By screening we mean the assurance that cross sections fall sufficiently fast with rising s. For the CC3 case, this is ensured in Born approximation due to an interplay between the s- and t-channel matrix elements and relies a certain relation between the gauge couplings (see e.g. Ref. [20] ). In the NC2 and NC8 cases, the same is achieved by an interplay between the t- and u-channel amplitudes leading to a cross section proportional to the screening f a c t o r s l s e / s 2 (the couplings of the two channels are equal) [8]. From the universal QED corrections, no additional problems arise, because the interplay between the various contributing tree level matrix elements is not disturbed. This is completely different for the non-universal corrections. Here, the various interferences get different corrections and the interplay between them is destroyed. Fortunately, one may see from the explicit expressions that the screening factor observed in the net NC2 and NC8 tree level cross sections is rediscovered not only for full cross sections, but even for individual non-universal contributions. So, unitarity is not violated. We were not able to find the screening property for non-universal CC3

42

D. Bardin et a l . / N u c l e a r Physics B 477 (1996) 2 7 - 5 8

~. 1.2

~

--

..........::,..:.:.:....:.S2".:.~ ...:.:#...:.::.:~.....

1.1

1i

N

. .:'¢:" /

_

0.9

i

i !

0°8

-

0.8 0.6

-to im

0.7

-

Y 200 300 400 500 600 700 800 900

0.4

0.2

0

/J 200

off-shell Born ....... off-shell univ. I S ~ ..............off-shell complete ISR 300

400

500

600

700

800

900

(G V) Fig. 8. The inclusive total off-shell Z pair production cross section O-to z zt (s). In the inset, the relative deviations of the universally and completely ISR corrected cross sections from the tree level cross section are given.

corrections without including the auxiliary corrections. Also in this respect, auxiliary corrections seem to be a quite natural, if not necessary ingredient of the calculation.

4. Numerical results To illustrate the analytical results of Sections 2 and 3, we present numerical results [21]. The effect of ISR on the boson pair production processes NC2 and CC3 is seen from Figs. 8 and 9 where inclusive cross sections are presented. From both figures one realizes that, as discussed in Section 2, the dominant part of the initial state QED corrections originates from universal ISR. Universal ISR is typically of (_9(10%). Non-universal ISR on the contrary is suppressed and rises from a few parts per thousand in the LEP2 energy region to 69(1%) at 1 TeV. For the NC2 process at 1 TeV the nonuniversal contribution to the cross section is 2.5%, whereas for CC3 at 1 TeV it is 1.4%. For reference we have given cross section values for both Z and W pair production in Table 1. The numerical precision is better than 10 -4. As numerical input for Table 1 we have used the standard LEP2 input (Table 5 in Ref. [3] ) which is reproduced in Table 2. The weak mixing angle is determined by

D. Bardinet al./NuclearPhysicsB 477 (1996)27-58 .D Q..

~

43

20

.,.~:.~:.;.:~.:~.:~?~:'::'~':~':~" 17.5 15 ! 1 !

0.9

12.5

t

0.8

10

0.7

200 300 400 500 600 700 800 900

7.5 5

- off-shell ....... o f f - s h e l l ..............o f f - s h e l l

2.5 0

200

300

400

Born u n i v . ISR complete

500

600

700

ISR

800

900

(G V) Fig. 9, The inclusive total off-shell W pair production cross section o-tower(s). In the inset, the relative deviations of the universally and completely ISR corrected cross sections from the tree level cross section are given. Table 1 Inclusive total off-shell cross section values for the processes NC2 and CC3 with and without initial state QED corrections

V/~

161 175 183 192 205 500 800

ffNC2 [pb] tree

univ

0.0154 0.0648 0.3811 0.9783 1.1833 0.4097 0.2124

0.0127 0.0506 0.2733 0.7788 1.0274 0.4424 0.2388

acc3 [pb] univ

0.0127 0.0508 0.2743 0.7817 1.0313 0.4479 0.2435

4.8087 15.9168 17.6819 18.4479 18.5078 7.3731 3.9971

3.4521 13.2903 15.4036 16.5958 17.1879 7.9168 4.4501

3.4648 t3.3408 15.4637 16.6625 17.2600 7,9810 4.5026

sin 2 0 w = 7 r a ( 2 M w )

x/~GFM2

and w e u s e t h e r e l a t i o n

(4.1)

Gu/x/2 = gE/(8M~v).

In Fig. 10, w e p r e s e n t , as an e x a m p l e for the N C 8 p r o c e s s , the c r o s s section for e + e - - ~ ( Z ° Z °,

Z°y, yy) -~ Ix+tz-bb (y).

(4.2)

44

D. Bardin et aL /Nuclear Physics B 477 (1996) 27-58

Table 2 Input parameters for Table 1 Quantity

Value

Quantity

Value

Mz

91.1888 GeV 2.4974 GeV 80.23 GeV

a(0)

ee(2Mw)

1/137.0359895 1/ 128.07 1.16639 × 10 - 5 GeV - 2 1

Fz

Mw FW

Gu VCKM

3GgM3 / ( ~/'8~r)

140 i~

102Cross ~ sectionports(univ.ISR)

b~120 Li~

10

100 80

~

ZI only "~,....

1 60

40

""~"s.

i ;

................__.YY..°nly ....................

%.

~%,,

15£200250300350400450500550 4

I'¢'~0:'oI ~l.d.i,

20 0 150

Bo~o,.o,, 200 250 300 350 400 450 500 550 600

Fig. 10. The total off-shell cross section for process 4.2. The inset shows how Zand photon pair production contribute to the NC8 cross section. For the particle masses we used numerical values from Ref. [22].

Due to the negative slope of the cross section curve ISR corrections are always positive. Universal ISR corrections vary between approximately 12% at v/'J = 130 GeV and 21% at 600 GeV. Non-universal corrections rise from 0.9% at 130 GeV to 4.1% at 600 GeV. Thus, compared to the NC2 case, the NC8 non-universal corrections are considerably enhanced. Finally it is worth mentioning that non-universal ISR tends to be harder than universal ISR, a fact which emerges from a numerical analysis of cross sections with lower cuts on s r [ 12]. This may be understood as follows: non-universal corrections are infrared finite, whereas the infrared divergent universal corrections are resummed by means of the soft photon exponentiation. Thus the universal corrections contain important soft resummed parts, and the non-universal corrections do not.

O(a)

D. Bardin et al./Nuclear Physics B 477 (1996) 27-58

45

5. Concluding remarks In this paper we have presented the analytical results of the first complete gaugeinvariant calculations of initial state QED corrections to off-shell vector boson pair production in e+e - annihilation. We have decomposed the corrections into a dominant universal contribution and a numerically smaller non-universal contribution. Nonuniversal corrections were found to be smoothly behaving around the two boson production threshold and to be explicitly screened from unitarity violations at high energies. As a by-product, the angular distribution of radiated photons is available from intermediate steps of the calculation but has not been presented here. By construction, we have not taken into account genuine weak corrections [23] or QED corrections related to the final state [ 17]. Compared to ISR, both are known to be smaller but may reach several percent nevertheless. Therefore they are comparable in size to the non-universal corrections calculated here. (See also Refs. [2,24].) Special suppressions of the order O ( F v / M v ) are estimated for interference effects between initial and final state radiation in resonant boson pair production and inter-bosonic final state interferences [25]. The complete photonic corrections to four-fermion production are not available although important steps towards their calculation have been achieved [ 26]. The presented calculation should be considered in connection with the program of semi-analytical treatments of complete processes at tree level [8]. It would be quite interesting to compute complete ISR not only for the off-shell production of vector boson pairs, but also for true four-fermion processes. These computations are simplest for the so-called NC24 [9] and CC11 [ 13] processes, but also the combination of NC24 with Higgs production [ 10] should be straightforward. Because in these reactions all background Feynman diagrams, i.e. those diagrams that must be added to CC3 [ NC8 ] to get the sets for CC11 [NC24], are s-channel diagrams, there is nothing but universal ISR for the pure background. So the only missing pieces are the non-universal ISR contributions to the interferences between the boson pair signal and the background. The results may be of numerical relevance in the NC24 case below the Z Z production threshold. For CC11 and for the inclusion of Higgs production they will be of mainly theoretical interest.

Acknowledgments We would like to thank E Jegerlehner, A. Leike, A. Olchevski, and U. Mtiller for helpful discussions. The contribution of M. Bilenky to the calculation of the virtual corrections to the CC3 process in 1992 is gratefully acknowledged. We are indebted to J. Bltimlein for carefully reading the manuscript.

D. Bardin et al./Nuclear Physics B 477 (1996) 27-58

46

Appendix A. Couplings and factorizing kinematical functions

A.1. Couplings The electroweak couplings and neutral boson propagators enter the Born and QED corrected cross sections in certain combinations. The NC8 process is described by one coupling function [9]: 2 CNC8 -- (6~2)---------5

Z

Re

I •

v.vj,vk,vl---r,z Dv~(Sl)Dvj(s2)Dvk (Sl)D~(s2)

x [L(e, Vi)L(e, Vk)L(e, Vj) L(e, Vt) +R(e, Vi)R(e, Vk)R(e, Vj) R(e, Vz) ] × [L(fl, Vi)L(fl, Vk) + R(fl, Vi)R(fl, Vk)] Nc(fl )Up(Vi, Vk, ml, sl) × [L(f2, Vj)L(f2, Vt) +R(fz, Vj)R(f2, Vt)] Uc(f2)Up(Vj, Vt,m2,s2). (A.1) We have used the left- and right-handed fermion-boson couplings

L(f,y) = eQf 2 '

L ( f , Z ) = g' (I f - s 2 Q y ) , 2cw

R ( f , y ) = eQf2 '

R(f,Z) -

g s2,~ 2cw w~f,

(A.2)

Where Q f is the fermion charge in units of the positron charge e, I f is the fermion's weak isospin third component, and Sw denotes the sine of the weak mixing angle. Qe = - 1 and the fine structure constant is given by e = gsw = ~ . In addition, Eq. (A.1) contains boson propagators

Dr(s) = s - M 2 + isFv/Mv.

(A.3)

The photon has zero mass and width, M z, = F~, = 0. The color factor leptons and three for quarks. The phase space factors

{i

(

Np(Vm, Vn,m,s) =

)

4m 2 1s

f°rVm 4:y °r Vn 4:y

2m 2 1+

Nc(f) is unity for

(A.4) forVm=Vn=y

s

in Eq. (A.1) correctly take into account that fermion masses may be neglected if the corresponding fermion pair couples to a Z resonance but may not if it only couples to a photon. The cross section for the NC2 process is obtained by omitting exchange photons in Eq. ( A . I ) . The CC3 process is described by three coupling functions C(c~)3, one for the t-channel, one for the s-channel, and one for the st-interference contribution [6,13]:

D. Bardin et al./Nuclear Physics B 477 (1996) 27-58

2

47

1

Ccc3 = (6rr2) 2 IDw(sl)l 2 [Dw(s2)[2 xL4(e, W)L2(fl, W) Nc(fl )L2(f2, W) Nc(f2),

2

(A.5)

1

CCC3 = (67r2)2

Z Re IOw(sl)l 2 lOw(s2) 12 Ov,(s)O%(s) v,,vj~,z

x [L(e, Vi)L(e, Vj)+ R(e, Vi)R(e, Vj)] xg3(Vi)g3(Vj)L2(fl, W)Nc(fl)L2(f2, W) Nc(f2), 2 1 C~C3- (67/.2)2 Z Re L2(e, W)

(A.6)

×L(e, Vii)g3(VDL2(fl, W) Nc(fl )L2(f> W) Nc(f2).

(A.7)

v~=~,,z

IDw(sl)lZ lDw(s2)lZ Dv,(s)

The left-handed couplings L(f, W) of a weak isodoublet with the fermion f to a W boson are given by g L(f, W) = 2v~" (A.8) We neglect Cabibbo-Kobayashi-Maskawa mixing effects. The right-handed couplings R(f, W) vanish. The couplings g3 (Vi) originate from the three-boson vertices and are g3(Y) = g sw, g3(Z) =gcw.

(A.9)

Using Gu/v/-2 = f / ( 8 M 2) in the C functions, one easily rewrites the cross sections for heavy boson pair production, namely the processes CC3 and NC2, in terms of Breit-Wigner densities 1 siFv/Mv pv(si) = ¢r Isi - M2v + isiFv/Mv[ z × BR(i),

V = Z, W±

(A.10)

with branching fractions BR(i) into the appropriate fermion pairs. For the process NC2 one gets CNC'2= pZ(Sl)pz(s2) (G/~M2) 2 [ ( 1 - 2s2)4 + (2s2) 4]

sis2

(A. 11)

4

For W+ pair production, the situation is slightly more involved. Defining Ccc3 - (GlzM2w)2pw(sl)PW(S2), SI $2

one obtains by inspection of Eqs. (A.5)-(A.7) CCC3 = Ccc3cvv,

c & , = 74C cc, ( ¢ . +

+

(A.12)

48

D. Bardin et al./Nuclear Physics B 477 (1996) 27-58

C~tc3 = 1Ccc3 s

(c~r + C~z)

(A.13)

with factors Cab as defined in [8] (see also Ref. [6]). For on-shell heavy boson pair production, Fv ---, O, Breit-Wigner densities are replaced by 8-distributions, Fv'--*O

pv(s) ~

8 ( s - M2v) x BR(i).

(A.14)

A.2. Kinematical functions for the factorizing QED corrections The kinematical function for the NC case is [27] ~NC2(S;SI'S2) =~NC8(S;SI'S2)

[S2 + (Sl + S2)2 £-- 2J .

= S1S2 L

;----~1--$2

(1.15)

For the CC case, one finds [6] '

GCC3(S, Sl,S2) =

A+ 12s (sl + s 2 ) + 24 ( s - sl - s2)

G~c3(S, S l , S 2 ) = ~

-- 48SLS2

sis2 £]

(A.16)

,~+12(ssl+ssz+sls2)

G~tc3(S;Sl,S2) = ~ { ( S - S l - S 2 ) [ A - 24

,

÷ 12(SSl ÷

(A.17)

ss2 ÷ sis2)]

(ssl + ss2 + sis2) sis2 £ }

(A.18)

for the kinematical functions. The logarithm 12 contained in integrated t- or u-channel contributions is defined by 1

£ = £(s;sl,s2) = ~ln

s-

sl - s2 ÷ v/~

S--SI

A p p e n d i x B. The 2 ~

.

(A.19)

- - $ 2 --

5 particle phase space

When considering initial state photon bremsstrahlung to a 2 --* 4 process, a photon with momentum p appears as fifth particle in the final state. The intermediate vector bosons (two-fermion systems) have momenta Vl = Pl +P2, v2 = P3 ÷P4. The five-particle phase space has eleven kinematical degrees of freedom. It is convenient to parametrize the five-particle Lorentz-invariant phase space by a decomposition into subsequently decaying particles: drs---

1 v/A(s, s', 0) (27r) 14 8s

~/a(s',sl, s2) V/a(sl,m2,rn~) ~/A(s2, m3,2 m4)2 8s I

8si

8sz

× ds / dsl ds2 d c o s 0 d~bR dcOS0R d~bl dcos01 d~b2 dcos0z,

(B.1)

D. Bardin et al./Nuclear Physics B 477 (1996) 27-58

49

where mi is the mass of the final state fermion with momentum Pi and q~: Azimuth angle around the beam direction. 0: Polar (scattering) angle of the photon with respect to the e+direction k2 in the center of mass frame. ~bR: Azimuth angle around the photon direction p. OR: Polar angle of vl in the (Vl + v2) rest frame with z axis along p. ~bl: Azimuth angle around the vz direction. 01: Decay polar angle of Pl in the vt rest frame with axis along vl. q~2: Azimuth angle around the vz direction. 02: Decay polar angle of P3 in the v2 rest frame with axis along v2. Sl: Invariant mass of the final state fermion pair fir2 : si = -v~. s2: Invariant mass of the final state fermion pair f3f4 : s2 = -v~. s~: Reduced center of mass energy squared. This is the invariant mass of the final state four-fermion system: s ~ = -(Vl + u2) 2 = - ( p l + P2 + P3 d-p4) 2. The ranges of the kinematical variables are

(ml

+ m2 -}- m3 + m4 ) 2 ~ s t <~ s

(ml+m2)2<~Sl <~ ( ~ s t - - m 3 - - m 4 ) 2 (m3 -q- m4)2 ~ s2 <~ (v/-sT- v ~ ) 2 - 1 ~
(B.2)

or, alternatively, (ml + m2) 2

sl

m3 - m4) 2

(m3 + m4) 2

s2

eT;) 2

- 1 ~
0 ~ ~b{R,1,2) ~ 27r - 1 ~
(B.3)

An illustration of the 2 ~ 5 particle phase space is given in Fig. B.1. For the processes under consideration, the integrations over the final state fermion decay angles q~l, 01 and ~b2, 02 are separated from the other integrations and are carried out analytically with the help of invariant tensor integration (see Appendix D. 1 of [ 18 ] ). The remaining integrations are non-trivial. They are collected in Appendices D.4 and D.5 of [ 18]. We will now explain, how the scalar products appearing in the squared matrix elements can be expressed in the center of mass frame with a Cartesian coordinate system as drawn in Fig. B.1 using the phase space variables introduced in Eq. (B.1).

D. Bardin et al./Nuclear Physics B 477 (1996) 27-58

50

e'(kl)

:~.::

jiiji

f ~Cp,)j¢

f~0,~)

f, cp,)

Fig. B.I. Graphical representation of a 2 --* 5 particle reaction. In general, the vectors vl, v2,Pl,P2,P3, and P4 do not lie in the plane of the picture. In the figure, the angle 0 is called 0z, to emphasize that it belongs to the direction of the radiated photon.

The initial state vectors kl and k2 are given by k 1 = (k°, k sin 0, 0 , - k cos 0) ,

(B.4)

k2 = (k °, - k sin 0, 0, k cos 0) with k°

=

x/~ -~-,

k = v/~

-~-~,

~ -

~/

1 -

4m2

--'s

(B.5)

where me is the electron mass. For the momentum of the photon radiated along the z axis one finds in the center of mass system p = ( p O , 0, 0, pO)

with pO_ s2- ~s ~ "

(B.6)

Thus, the invariant products kl k2 and kip may be expressed easily. The virtual boson pair recoils from the photon in the center of mass system:

( s + s'

s-

(v, + v2) = \ 2v G , 0, 0,

2v/~ j .

(B.7)

The vectors vl and v2 in the two-boson rest frame (the R-system with vl.R + vz,R = 0) are easily derived:

I

~/a(s', st, s2)

S t ÷ SI - - $2

=

7--e7

, o, o,

2V7

51

D. Bardin et al./Nuclear Physics B 477 (1996) 27-58

U2,R --

P~2, O, O, --79R =

, O, O,

.

Both Vl,R and u2,R do not depend on angular variables. We need vi in the center of mass system. The boost between the center of mass frame and the two-boson rest frame R with zero component y(0) and modulus of the spatial components y is ~/(o) _

s q- s t

2V/~ 7 '

s -- s'

Y = 2x/~"

(B.9)

Taking into account that the production angles of the virtual vector bosons are not fixed, Eq. (B.8) yields the following center of mass system four-vectors:

U1 = ~(T(0) pR,O 12 -- y79RCOSOR' 7~RsinORCOSdpR, --79RSinORSin~bR,

T(O)79R COS OR -- 7plR~0) , v2 = (y(0) p~0 + y79R cos OR,

--79R sin ORcos ~bR, ~R sin OR sin OR,

~(0) ~RCOSOR -- YP~2) •

(B.10)

The sum (V1,R+ V2,R) in the R-system is simply ( x/~, 0, 0, 0). The boost into the center of mass system yields (vL + v2) = (y(°)x/sT,0,0,-yx/sT), which is exactly (B.7). The above formulae allow to express the products kivj and pvj. The product PiP2 [and analogously P3p4] is most easily calculated in the center of mass system R1 [R2] defined by Pl + P 2 = 0 [P3 +/94 = 0]. This is explained in equations (B.5)-(B.11) of [ 13] for the 2--*4 process. The discussion of Ref. [ 13] remains valid also with an additional photon in the initial state. For the remaining scalar products PPi, kjPi, vjPi, and PiPj we used explicit expressions for the vectors Pi in the center of mass system. The vectors Pi are obtained from the vectors Pi,R in the R-system by rotating and applying the Lorentz boost (B.9):

I Pi=

T(O) pOR(s') -- y [pZR(s') COS0R --pXR(S() sin 0R] [pZR(s') sin OR + pxR (S') COSOR] COS~bR + P]'R(S') sin ~bR z i sin x i • v i -- [Pi,R(S ) OR + Pi,R(S ) COS 0R] sin q~R -k-pi,R(S ) COSq~R

(B.11)

y~o) [pZR( S, ) COSOR --p;R(S') sin 0R] -- y pOR( s') The four-momenta Pi,R(S t) are exactly the final state momenta Pi shown in Eq. (B.21) of [ 13], when these are evaluated at s ~ instead of s. The frame R there has to be understood as frame Rl or Rz introduced here. Now, the derivation of all momenta in the center of mass system in terms of the integration variables is completed for the 2 ~ 5 process and all scalar products appearing

D. Bardin et al./Nuclear Physics B 477 (1996) 27-58

52

in the initial state bremsstrahlung calculation can be expressed in terms of phase space Variables. For the calculation of virtual and soft photon corrections, the 2 ---, 4 phase space is needed together with the corresponding momentum representations. It may be taken over completely from Appendix B of [ 13].

Appendix C. Matrix elements In this appendix, we will present the matrix elements for the CC3 and NC2 processes. The matrix elements for the NC8 process are easily obtained from the NC2 matrix elements by adding amplitudes where Z couplings (propagators) have been replaced by photon couplings (propagators).

C. 1. Tree level amplitudes For the calculation of soft and virtual corrections, the tree level matrix elements for the CC3 and NC2 processes are given in terms of s-channel, t-channel, and u-channel amplitudes corresponding to the Feynman diagrams of Figs. 1 and 2: = Ms"

+

+

M, ,z +

(C.l)

(c.2)

with

.A4sBZ = Dz(s) gy~,, Dw(Sl) g f l ~ , Dw(s2) gac~, Bs'z ~; M12'wM34'wg3(Z) ~' ,~, T ~z, ,

(C.3)

M 8s,y = Dr(s grr' ) Dw(sl) g3~' Dw(s2 g,~,~' )B~'r M12,wM34,w 3' ,~' g3(Y) Tafl~',

(C.4)

B g~fl, g,~,~, 1 ,~fl " " 12,W ,,134,W, "A4t'w= DW(Sl) DW(S2) ~t Bt'wlt'¢l~' alto'

(C.5)

.A4t~,z = DZ(Sl) g/~, Dz(s2) g~, q2 1 Bt,z ,fl M12,zM34, ~, ~, z'

(C.6)

M Rz = ga,~' g~¢~, 1 aft ~a' ~/t#' Dz(sl) Dz(s2) q2Bu'z ' " 1 2 , Z ' " 3 4 , Z "

(C.7)

We have used the vector boson propagator denominators Dv defined in Eq. (A.3), the trilinear coupling g3(V 0) given in Eq. (A.9), and T '~y = (vl - t)2)Yg'~t~ - 2v~'g ~y + 2v~g '~y.

(C.8)

y' In the massless approximation, the annihilation matrix elements Bs.vo, Bt,a Bv, B.,a,av, and the decay matrix elements M~, v are given by

B~.;o:f(-k2)yZ'[L(e,V°)(l+ys)+R(e,V°)(l-ys)]U(kl),

(C.9)

53

D. Bardin et al./Nuclear Physics B 477 (1996) 27-58

[L2(e,V) (1 +)'5)

a# = 2 f i ( - k 2 ) ) " ~ Bt,v

+

RZ(e,V) ( 1 - )'5)] (l,)'~u(k,), (c.10)

'~

[LZ(e,Z)

Bu,z = 2 u ( - k 2 ) ) " '~ -

( +1 ) ' 5 )

+

R2(e,Z)

)'5)] ~tu)"eu(k,) ,

(1 -

(c.11) Mij, v - ~ ( P i ) ) ' "

[ L ( f i , V) (1 +)'5) + R ( f i ,

V) (1 -)'5)] u ( - p j )

(C.12)

with left- and right-handed couplings as defined in Eqs. (A.2) and (A.8). We have further used

l

(C.13)

qt = kl - Ul = - k 2 + u2 = ~ ( kl - k2 - vl + V2) ,

1

q, =kl - u2 = - k 2 + Ul = ~ (kl - k2 + vl - v2).

(C.14)

C.2. Virtual initial state corrections

The amplitudes for virtual initial state corrections are easily obtained from the matrix elements in Eqs. (C.3) to (C.7) by substituting the initial state currents. The t-channel NC2 virtual QED matrix element .h4 vt,Z is represented by the Feynman diagrams of Fig. 6 and given by A4tVz =

g/3~,

Dz(SI)

with

Vt'~ = e2

{

g'~'

v~/3 ~,d3'

DZ(S2) vt,z"ll2,Z

C'~'zZ

--16~-2 -- i ]./(4--.)

(C.15)

~,!~' ~"34,z

i-.(°. ~

Vt,vertl + Vt,vert2 + Vt,self _[_ Vt,box

..

°.

..)

[.£=me

}

(C.161 where dimensional regularization is applied with p being the loop photon momentum, and where Vt,avflert i ---/~ (-k2)

(-2k~ - yU/~) y'~ (~t -- ~) )',u ~t ,),/3 [p 2 - i e ]

×2 [ L 2 ( e , Z )

°:,

[(p-qt)2-ie]

(I+T.5)+R2(e,Z)

[ p 2 + 2 k 2 p _ i e ] qtz (1-ys)]

)"~ ~,)',~ (~, -.~) )'~ (2k,..

-

u(kl)

,

:)'.)

~,..t.--o~ ~., ? i.] [~. q,:-i. 1 [~, .~: ~.] q7 ×2 [ L 2 ( e , Z )

Vt,asflelf=/~ (-k2)

(I+Ts)+R2(e,Z)

(1-)'5)]u(kl)

)" ~,)" (~, - ~) )'. ~, )'~

[.. i.] ['. q,~' i.] [q,l~

,

D. Bardin et al./Nuclear Physics B 477 (1996) 27-58

54

×2 [ L 2 ( e , Z ) (1 + Ys) + R2(e,Z) (1 - Ys)] u(kl) ,

( - 2 k ~ - y~'~) y'~ (¢~, - ~) 7,# (2kl.u - ~3yu)

x

x 2 [LZ(e, Z ) (1 + Ys) + R2(e, Z) (1 -"1'5)] u(kl) .

(C.17)

In the on-shell renormalization scheme, the t-channel counterterm part in Eq. (C.16) originates from the counterterms of two vertex and one electron self-energy loops. It reads

Bt'-----~Z( 2 P

Cta,ff=

+4Pro-

4)

(C.18)

qt2 with Bta,Bvvfrom Eq. (C.10) and the ultraviolet and infrared poles

P -=

1

n - 4 1

pt~ _= _ _ n-4

1 me + -~Ye + In - - - In (2v'~)

#

+

1

y e + In me _ In (2v/-~) 2 /x

u=m, u ....

n = 4 - e,

(C.19)

n = 4 + e.

(C.20)

Here, Ye is Euler's constant. The u-channel virtual QED matrix element A//,V,z is obtained from Jt4Vt,z by the interchanges vl ~-~ v2,

Ad u12,Z +-4 3/t ~z 34,Z"

(C.21)

The symmetry (C.21) implies that, after the angular integrations, AduV,z is not explicitly needed any more, because the terms corresponding to the interference of A/g,V,z with the tree level matrix element is obtained from the interference of .A4vt,Z with the tree level matrix element by interchanging Sl and s2. The virtual initial state QED matrix element for the CC3 t-channel is -- O w ( S l )

(C.22)

Dw(s2) vt'w'"12'w'"34"w

with Vt,~w ~ = e2

Vt~x - i/1. (4-")

~

,,box g ....

(C.23)

where Vt,au x'~/~ = 16~2Ct~w- i ]./,(4-n) /

.B I + Vt,vert ,.8 2 + Vt,self "~) ~ d n P ( gt,vert

~=m. .

(C.24)

The virtual and counter term contributions are those of (C.17) to (C.20) with corresponding Z couplings replaced by W couplings. The electron mass which, in principle,

D. Bardin et al./Nuclear Physics B 477 (1996) 27-58

55

occurs in the NC2 t-channel propagators is absent in the charged current case where a neutrino is exchanged in the t-channel instead. Further, the u-channel diagrams are absent. The virtual initial state QED matrix element for the s-channel annihilation represented by the amputated Feynman diagram in Fig. 4 is given by

v g~' g,8,8' gaa' ~; #' a' Ta#~, Ads'V° - Dvo(S) Dw(sl) Dw(S2) V'v° MI2'wM34'wg3(V°) with

Vf,'vo ~ e

2

{'

Csv° 16~r2

f

driP

~" [

/

i t/, (4-n) ] (-~-~)nVs,vert /Z=me

(C.25)

(C.26)

and

~

(-2k~ - yU,/~)3/~" (2k'~ -/~7 u')

X

[L(e,V°) (I+3/5)+R(e, V°) (I-3/5)]u(kl).

(C.27)

The s-channel counterterm part is a mere vertex counterterm given by 'r' = Bs,vo ~" (2P + 4P IR - 4) Cs,vO

(C.28)

with Bs.vo from Eq. (C.9).

C.3. Initial state bremsstrahlung amplitudes As for the case of virtual initial state QED corrections, initial state bremsstrahlung matrix elements are obtained by altering the amplitudes in Eqs. (C.3) to (C.7). From the Feynman diagrams of Fig. 7, one deduces the t-channel Bremsstrahlung matrix element for the NC2 case:

.A.4R,a = gt~' g,~,~' ,a' o/ a~u a t,Z D z ( s l ) Dz(s2) M12'z M34'zeRt'z eu(P)

(C.29)

with

R~aUt,z = u(-k2){y~ ¢2t2-¢2y#2k~z,-~yu +y~{2 t2--~2 y/~ ~1 tl- ¢1 y/~ + y/z~ z-z-2k~y~l tl-¢l 3/# } x2 [LZ(e, Z) ( 1 + 3/5) + R2(e, Z) ( l - 3/5)] u(kl) .

(C.30)

An analogous result is obtained for the NC2 u-channel initial state bremsstrahlung matrix element,

D. Bardin et al./Nuclear Physics B 477 (1996) 27-58

56

R,a gaa' g##' a' ' z eu(P) "A'~u'Z= D z ( S l ) Dz(s2) M12"z M#34'zeRu, ~au _

_

_

u,Z

U2

Zl

,i

×2 [L2(e,Z) (1 + ys) + RZ(e,Z) (1 - Y s ) ] u(kl).

(C.31)

For the s-channel bremsstrahlung matrix element of the CC3 process one has the Feynman diagrams of Fig. 5 and obtains

get' gl~#' gaa' ,a' a' T,~#r eRr'U a (p) s,VO= Dvo( S) Dw( sI ) Dw( s2) M12'wM34'wg3( V°) s'V° eu

./~R,.~

(c.32) with

R~'# - ~ ( - k 2 ) {Y ~'2k~-t~y~ zl + × [ L ( e , V °) (1 + y s )

z2--

Y~'I

+R(e,V °) (1 - Y s ) ] u(kl).

(C.33)

All the initial state currents introduced so far fulfill the condition of U( 1 ) current conservation, y,u puRs,vo = p~Rt ~

--

puR'~~ = 0.

(C.34)

There is no CC3 u-channel diagram. For the CC3 t-channel diagram, the situation is slightly more complicated. Naively, only the two left diagrams in Fig. 7 contribute to the CC3 t-channel ISR corrections. For these two, current conservation is violated and may be established by adding as an auxiliary current the additional Feynman diagram with radiation from the t-channel propagator [7] (cf. the discussion in Section 3.2). Then, the charged current t-channel matrix element is just given by (C.29) with Z couplings replaced by W couplings. In the above matrix elements the radiatively changed Mandelstam variables were introduced. In terms of the five-particle phase space parametrization from Appendix B they are given by t l = (kl

--

Ul)2

÷

m 2 = a~ + b l cos

OR ÷ Csin ORcos ~bR,

t2 = (re - k2) 2 + m 2 = a~ + b2 cos OR + c sin OR cos qSR, U 1 = (k 1 -

u2) 2 +

m 2 = a~ - bl cos0R - c s i n 0 R c o s ~ R ,

u2 = (ui -- k2) 2 ÷ m~ = a~ - b2 cos 0R - csin 0R COSq~R. Here m,, =

(C.35)

me for NC2 and m,. = m~ = 0 for CC3. In (C.35), the kinematical parameters

D. Bardin et a l . / N u c l e a r Physics B 477 (1996) 2 7 - 5 8

2 2 s'+ a~ = m v - m e -- Sl + ~

a[=m~-m

8

( s + - s'_

s' - & -4-s ' (s'+ + s '

2e - s 2 q -

S t -- 6

a~ = m~ - m 2e - s2 +

2

a~ = m v - m

2e - -

S 1

-~

-4- s'

57

flcos0), ~cosO),

~cos0),

(s'+-s'

s'4 +s------76- (s~ + s'_ ~ cos0)

,

bl = 4s----7 (s~/~ cos 0 - s'_)

vT b2 = ~ (s~/~ cos 0 + s'_), v~7 ~ , / ~

C~'~"- - T

sin0

(C.36)

are derived from the representations (B.4) and (B.10) in Appendix B. The variables 6, s~_, s~_, ~ are introduced in Sections 2 and 3. The two invariants related to radiation from external legs are St

St__

Tel,

Zl = - 2 k i p

= ~ - (1 + / 3 c o s 0 ) -

z2 = - 2 k 2 p

= - ~ (1 - / 3 c o s 0) _-- 2 Z = .

St

St

(C.37)

Eqs. (C.36) and (C.37) are exact. In (C.37) one has to keep the electron mass, since it gives rise to mass singularities. The electron mass in (C.36) may be neglected, but it was retained in some integrals for the sake of numerical stability. References [ 1 ] G. Altarelli, T. Sj6strand and E Zwirner, eds., Report of the Workshop on Physics at LEP2, CERN 96-01 (1996). [2] W. Beenakker et al., Report of the Working Group on WW Cross-sections and Distributions, in [ 1]. [3] D. Bardin et al., Report of the Working Group on Event Generators for WW Physics, in [1]. [4] E Boudjema et al., Report of the Working Group on Standard Model Processes, in [1]. [ 51 M.L. Mangano et al., Report of the Working Group on Event Generators for Discovery Physics, in [ 1]. [6] T. Muta, R. Najima and S. Wakaizumi, Mod. Phys. Lett. A 1 (1986) 203. [7] D. Bardin, M. Bilenky, A. Olchevski and T. Riemann, Phys. Lett. B 308 (1993) 403 [Erratum: B 357 (1995) 7251. [ 8 ] D. Bardin, M. Bilenky, D. Lehner, A. Olchevski and T. Riemann, Semi-analytical approach to fourfermion production in e+e - annihilation, in Proc. of the Zeuthen Workshop on Elementary Particle Theory "Physics at LEP200 and Beyond", ed. T. Riemann and J. Bliimlein, Nucl. Phys. B (Proc. Suppl.) 37B (1994) 148. [91 D. Bardin, A. Leike and T. Riemann, Phys. Lett. B 344 (1995) 383. [10] D. Bardin, A. Leike and T. Riemann, Phys. Lett. B 353 (1995) 513. [ 11 ] D. Bardin, D. Lehner and I". Riemann, Complete initial state radiation to off-shell Z pair production in e+e-annihilation, in Proceedings of the IX International Workshop on High Energy Physics and Quantum Field Theory, ed. B.B. Levtchenko (Moscow Univ. Publ. House, 1995) p. 221.

58

D. Bardin et al./Nuclear Physics B 477 (1996) 27-58

[12] D. Lehner, Ph.D. thesis, Humboldt-Universit~it zu Berlin, Germany (1995), Internal Report DESYZeuthen 95-07 [ hep-ph/9512301 ], http://www.ifh.de/lehner/pub_1995.html and references therein. [ 13] D. Bardin and T. Riemann, Nucl. Phys. B 462 (1996) 3. [ 14] G. Montagna, O. Nicrosini, G. Passarino and F. Piccinini, Phys. Lett. B 348 (1995) 178; G. Passarino, Comput. Phys. Commun. 97 (1996) 261. [15] F.A. Berends, R. Pittau and R. Kleiss, Nucl. Phys. B 424 (1994) 308. [16] F.A. Berends, G.J.H. Burgers and W.L. van Neerven, Nucl. Phys. B 297 (1988) 429 [Erratum: B 304 (1988) 921]; B.A. Kniehl, M. Krawczyk, J.H. Kiihn and R. Stuart, Phys. Lett. B 209 (1988) 337. [17] W. Beenakker, K. Kolodziej and T. Sack, Phys. Lett. B 258 (1991) 469; W. Beenakker, EA. Berends and T. Sack, Nucl. Phys. B 367 (1991) 287; J. Fleischer, K. Kolodziej and F. Jegerlehner, Phys. Rev. D 47 (1993) 830; J. Fleischer, E Jegedehner, K. Kolodziej and G.J. van Oldenborgh, Comput. Phys. Comm. 85 (1995) 29. [18] D. Bardin, D. Lehner and T. Riemann, preprint DESY 96-028 (1996) [hep-ph/9602409]; see also http: / / www.ifh.de/Tehner. [ 19] M. Veltman, "SCHOONSCH1P - A Program for Symbol Handling" ( 1989); H. Strubbe, Comput. Phys. Comm. 8 (1974) 1; J. Vermaseren, Symbolic Manipulations with FORM (Computer Algebra Nederland, Amsterdam, 1991 ) ; M.J. Abell and J.P. Braselton, The Mathematica Handbook (Academic Press, San Diego, CA, 1992); N. Blachman, Mathematica: A practical approach (Prentice Hall, Englewood Cliffs, NJ, 1992). [20] W. Alles, Ch. Boyer and A. Buras, Nucl. Phys. B 119 (1977) 125. [21] D. Bardin, M. Bilenky, D. Lehner, A. Leike, A. Olchevski and T. Riemann, Fortran programs gentle_4fan.f and gentle_nc_qed.f. The codes are available from the authors upon e-mail request or via WWW: http://www.ifh.de/bardin/gentle-4fan.uu and http:// www.ifh.de/lehner/gentle-nc_qed.uu. A short description may be found in I3], p. 68. [22] Particle Data Group (L. Montanet et al.), "Review of Particle Properties", Phys. Rev. D 50 (1994). [23] M. BOhm, A. Denner, T. Sack, W. Beenakker, EA. Berends and H. Kuijf, Nucl. Phys. B 304 (1988) 463; J. Fleischer, E Jegerlehner and M. Zralek, Z. Phys. C 42 (1989) 409. [24] W. Beenakker, Status of Standard-Model Corrections to On-Shell W Pair Production, in Proc. of the Zeuthen Workshop on Elementary Particle Theory "Physics at LEP200 and Beyond", ed. T. Riemann and J. Bliimlein, Nucl. Phys. B (proc. Suppl.) 37B (1994) 59. [25] V.S. Fadin, V.A. Khoze and A.D. Martin, Phys. Lett. B 311 (1993) 311; B 320 (1994) 141; Phys. Rev. D 49 (1994) 2247. [26] A. Aeppli and D. Wyler, Phys. Lett. B 262 (1991) 125; G.J. van Oldenborgh, P.J. Franzini and A. Borrelli, Comput. Phys. Comm. 83 (1994) 14; J. Fujimoto et al., Non-resonant diagrams in radiative four-fermion processes in Proc. of the Zeuthen Workshop on Elementary Particle Theory "Physics at LEP200 and Beyond", ed. T. Riemann and J. B1Omlein, Nucl. Phys. B (Prec. Suppl.) 37B (1994) 169; G.J. van Oldenborgh, Nucl. Phys. B 470 (1996) 71; K. Melnikov and O. Yakovlev, Nucl. Phys. B 471 (1996) 90. [271 V. Baier, V. Fadin and V. Khoze, Sov. J. Phys. JETP 23 (1966) 104. •