Analytic approach to the complete set of QED corrections to fermion pair production in e+e−-annihilation

Nuclear Physics B351 (1991) 1-48 North-Holland

D. BAIL IN, M. BILENKY. A. CHIZHOV and A. SAZONOV Wa h"w 0 %&ar Research . ubiza, Jqead Post SU-101 Moscow. USSR

ffice P.0. Do. 79, 1

0. FEDORENKO P, ,trosarodsk State ON" SUIM18 %rasaa tMk, W USS, T. RIEMANN and M. SACHWITZ histifiii für Hoclieiier,-iepkvsik-. Plafanenallée 6. 0-1615 Zeitilieii / Brairdeizbiem, Génnaîaç° Received 17 May 1990 (Revised 17 September 1990) We present the convolution integral for fermion pair production in the electroweak standard thee, to order 0(a) including also soft photon exponential ior.- The result is complete in the sense that it includes initial- and final-state radiation and their interference. From the basic result - analytic formulae for the differential cross section - we also derive the corresponding expressions for the total cross section a-T and the integrated for-ward-back-ward asymmetry A The numerical importance of different contributions for the analysis of experiments at LEP/SLC energies is discussed. F113

.

1. Introduction Stimulated by the possibility of very precise experiments at e + e --storage rings, the QED corrections for the reaction e + (k2) +e - (k1) ---)'(Yl Z) ---)' f+(P 2 ) +f - (P,) +"y(p) have been studied in great detail for a large energy region including the Z-resonance. A comprehensive collection and comparison of available results within the electroweak standard theory [1] may be found in ref. [2] and in references quoted therein . Analytic and semi-analytic formulae proved to be of great value both for a deeper understanding of the process (1) and for ensuring a high reliability of Monte Cario codes [3]. Recently, the analytic approach to QED has also been used 0550-3213/91/$03 .50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)

D. Bardill et al. /

2

e ' e --annihilation

for the analysis of Z-line shape data at LEP/SLC [4]. The line shape is the integrated total cross section O'T(S) as a function of s = 4E 2 , where E is the beam energy. This most inclusive observable has been studied theoretically since many years and is well described to order O(a) [5-101 with soft photon exponentiation [10, 11] and also with several massive intermediate bosons [12], higher-order leading logarithmic corrections [1114] and the complete QED initial-state radiation corrections to order O(a-' ) [15]. A compilation of many important contributions is contained in ref. [16]. my recently, similar analytic results have been published for the integrated forward-baclard asymmetry AFB . The complete O(a) convolution integral with soft photon exponentiation for AFB was obtained in ref. [17] (see also ref. [8]) and the leading logarithmic approximation for initial-state radiation including O(a2) terms in ref. [ 18]. A survey on results about A Fs is contained in ref. [ 19] . Compared to ~ T and A FB, analytic results for the differential cross section dor/dcosO are scarce. Earlier attempts are contained in refs. [20,21]. Other distributions are treated in ref. [22] . In pure QED, the first compact analytic expressions for hard bremsstrahlung corrections to the differential cross section have been derived in ref. [23]. A formalism for leading logarithmic approximations may be found in ref. [18], though without application to dcr/dcosO. Recently, semi-analytic formulae allowing for quite realistic cuts have been applied in ref. [24] . This paper contains the systematic presentation of an analytic calculation of E corrections to the angular distribution for the reaction (1). Though computer codes relying on the present results have already been released and used [25] for applications in LEP physics, cry few of the material has been presented so far [26]; see also refs. [3, 16, 19]. In sect. 2 we introduce the notation and describe in short two derivations of the analytic expressions for the angular distribution . Sects. 3-5 contain the basic result which consists of compact, explicit expressions for the soft and hard photon radiator functions in the convolution integral due to initial(sect. 3) and final- (sect. 5) state radiation and their interference (sect. 4) to order (a). They also include higher-order corrections and some formulae for o--T and .4 FB' Sect. 6 contains numerical results and conclusions .

2. Notations an

hase space para etrisatiun

The explicit analytic formulae for the differential cross section proved to be too cumbersome to be described in detail here . For that reason, we choose a semi-analytic presentation with the following notation : du dc

E

inn = 0,1

E

E

A=T,FBa=e,i,f

Re[pl"(s,s ;ni,n)RI(cyn,n)p 4 A

(2)

D. Bardin et al. / e t e --annihilation

3

The scattering angle c = cos O is defined between the produced fermion fT (f = j,, v, q) and the positron beam. The functions tTÂ .O(s, s; in, n) are reduced cross sections defined below and the corrections Ra(c; in, n) can be expressed by semi-analytic formulae for soft and hard parts of Q bre sstra lung radiator functions SAa(c,,E ; in, n) and HAM, , 0, 0' ,o(s,s'

;~n,n)

R~(c'm,n) -,~ de° a o [s(OS~(c,E ; m,n) +O (L°-E)H~( t ' , c) ra. (s, s; in, n ) In eq. (3), we allow for a possible cutoff 4 (0 < A < 1 - 4m'/s) on the energy v of the emitted photon in units of the beam energy. The effective energy s' of the created fermion pair is s' _ (1 - Os. The sums over in, n in eq. (2) include photon (fn = 0) and Z-boson (fn = 1) exchange. A generalisation to the case of additional vector bosons Z, ft > 1, is straightforward. Indices a = e, f, i are used for initialand final-state radiation and their interference. The angular dependence of the Born cross section has been formally included into the initial-state contribution a = e. The P-even and P-odd cross section parts carry index A = T and A = Uidcr CP-invariance, they are also C-even (C-odd), correspondingly. After symmetric (anti-symmetric) integration over cos O the RT-functions (R -functions) yield the total cross section ~T (the integrated forward-backward asymmetry A Fd) fl

UT = f dc AFB =

UFB T

=

u dc 1 T

(4) ~~

i

du du 0 dc dc - f , dc dc

(5)

Within our terminology in,n=0, I a=e,i,f

Re[dA~~,o(s~s ;m,n)Ra(m,n)],

i RÂ(m,n)=d,~ ' f dc R~(c ;m,n), u

A=T,FB,

(6) (7)

d FB = 1 . The additional numerical factors in eqs. (6) and (7) ensure the usual total cross section normalisations for eT and AFB .

D. Bardhn et al.

4

/ e + e - -annihilation

The dependence of da/dc on weak neutral couplings and electric charges of the fermions and on possible polarisations and weak loop corrections together with the typical Born-like resonance behaviour have been collected in the explicit Born factors a~'~°(s, s; in, n). As a consequence, the functions Ra(c; iï1, i1) depend only on the Z-boson mass ML = , and width FZ = F, [ 16,271 and on kinematic variables s, cos 0, and the energy cutoff 1 . ®f course, fermion masses which are assumed here to be small compared to s, MZ, Fz show up in certain logarithmic mass singularities. The reduced Born cross sections s'; in, n) have different energy dependence for a = e, i, f, Q4 .O (S,S' ;in,n)

= U4(l(S',S' ;in,n),

a. 1 ,11 (S ' S' ;in .11)

=U-, '(S,S' ;in,11),

S,S' ;ii1,i1) = 0~''(s,s ;in,11) .

Here,

4- =

1®) (11)

FB or T if A = T or FB, and

0~4,(S,S' ;n1,i1)

_

«2s

7r,

i CJin,n ;A,,A,,h,,h,,)ct ;[Xnr (S ) Xri (S) +Xm(S)Xri (S I )l

(12)

X

rT ( S )

=

g rr

4 rra

S , S - 111~ 11

j,=M -î1V1 T (s),

(13) (14)

(15) In eq. (12), the fermions have a color factor c f = 3 in case of quarks and c f = 1 for leptons. Their vector and axial vector couplings vf(n) and a f(n) to gauge boson Z,, are contained in CA , CT( n1,n ;A,,À,,hl,h,) = i Al

i , *(n)

+a,(iï1)a* (il)] +si~[1',(tn)a*(i1) +l'* (n)ae(m)]1

X(h,[L'f(m)t'y*(11)

+af(in)a'~(n)]

+h , [t'f(n1)af (i1) +t'f (n)af(ni) ]) , (16)

CF13(n1 ,i 1 ;A i ,Àl,il l ,%1l)

(17)

D. Burclin et al. / e + e --annihilation

5

We allow for longitudinal polarisations A-, A + of both the electron and positron beams and for final states with helicities h -, h +. Due to the C?invariance of the problem and the (r ; a) structure of the interactions, the following combinations of polarisations are possible: (18) h, = ~(1 -h + h_),

h,= -I(h + - h -) .

(19)

In case of the standard theory [1], we use the conventional couplings of photon and Z-boson, go = e,

cf( 0 )

=

Qf ,

a, (0) = 0, vf(1)

af( g, = ( ~ Gt, zI =I'"(f) - 2Qfsin -

(20 O`v,

1)

=1;(f) , ( 21 )

where 1L(f) is the third component of the weak isospin of fermion f; 21~(e) = Qe = -1 . The inclusion of non-QED weak loop effects (see refs. [2,28-30] and references quoted therein) is trivial if one uses the form factor approach for their description 29]. Coupling constants 9,,g,, and the vector couplings i,,0), 1-f0) become complex and s-dependent . Further details are explained in refs. [19,301 . For an exact treatment, an additional form factor "ef(1) replaces the product ve(1)t-f(1) in eq. (16) whenever it appears there. In principle, energy-dependent form factors should be understood to be part of the integrand in eqs. (2) and (3). We have checked by explicit comparisons that, due to the minor s-dependence of them, the numerical error which is implied by the approximation is practically negligible [3,311 . For notational convenience, all angular dependences have been included into the radiator functions. The functions RÄ(c; tn, n) of eq. (3) are the result of an incoherent sum of real and virtual photonic corrections. They are obtained by straightforward but lengthy Feynman diagram calculations with extensive use of the analytic manipulation programs SCHOONSCHIP [32] and REDUCE [33]. The RÄ(c; m, n) are gauge invariant and, if integrated without cuts, also L.orentz invariant . Determinations of the vertex and box corrections [34], soft photon and hard photon contributions deserve different techniques, correspondingly. We would like to give short comments on the two methods which have been used for the O(«)-bremsstrahlung phase space integrations . In one approach, we used the so-called R y-system, the rest system of (f- y), where the three-momentum

D. Bardin et al. / e + e - -annihilation

relation p(f- ) + p(y) = 0 is fulfilled. The phase space parametrisation chosen is T

, 4s~ 1 dT =

f~

n,

c ,

dcos O S dx ()

s -A 1 s - x +'ti -'- 4 7r

2T d

1 dcos OR y ~0 1

R

~Py '

(22)

where (OR, cpR ) are photon angles in the R .,-system and x the momentum fraction of f } in the c.m.s. in units of the beam energy, x= -2p(f+) [p(e +) +p(e )] = 26E(f }) Some details of the calculation have been described in ref. [23], including the 't-dimensional treatment of soft photon contributions which follows the methods developed in ref. [35]. The hard photon integration has been done with SCHOONSCHIP using tables of integrals [36]. The total cross section UT and the integrated forward-backward asymmetry without cut [8] have been determined with this technique as well as the angular distribution which remained unpublished so far with exclusion of the pure QED case [23]. Another set of integration variables allows for a cut on the photon energy,

f dT =

IT

4s

f

~ m

d cos

c

m

~ m

Of)

dc f

l

~ . ma .

1" , min

dt, ,

27T

f

dcpy ,

(24)

()

_7

= 1 - 4m f/s .

t m(s)

In eq. (24), cpy

iF

(26)

the azimuthal angle of the photon in the c .m.s. and L' 2

=

-2P

+ = 1 - 2E(f -) / 6s_ . s (f )p(y)

(27)

We should also comment on the integration boundaries for cos O, c n, = 1 - 2m2/s .

(28)

The radiators for the angular distribution contain terms behaving in the hard photon corner like ln(1 + c), (1 ± c) -2, etc., which become singular and for initial-state radiation even non-integrable at c = ± 1 . In fact these quantities are

D. Bardin et at. / e +e - -annihilation

related to t = - 2p(e +)p(f+) and x, t

x

_

1

_

= 2 - 2s_

S2

-

4inéS

VX 2

-41112S

(29)

COS ® ,

From eq. (28), it becomes evident that the phase space integrals (22) and (24) are properly regulated . In order to get a convolution representation for da/d COs one has to perform in eq. (24) two integrations. This has been done with SCHOONSCHIP and REDUCE, based on tables of integrals for hard bremsstrahlung. The soft photon part is the same as determined earlier : see e.g. refs. [7-10]. While all the hard photon parts of the radiators for the angular distribution remain independent of the type of exchanged gauge boson, this is not the case for the box diagram contributions to the soft photon part of the QED interference corrections as has been indicated in the notation in eq. (3). After summing up the vertex or box contributions with those due to soft photons, the resulting SÂ(c, E ; in, n) are infrared finite but yet dependent on the infinitesimal parameter E which discriminates between soft and hard photons. The integral over the photon energy in eq . (3) as a whole is independent of E as may be seen from the formulae of sect. 3. Restricting oneself to a semi-analytic (i .e. containing one numerical integration) result for du/dc, one can take into account even cuts on the a-collinearity and on the fermion energies as has been demonstrated recently in ref. [24] where eq. (24) was also used as the starting point.

3. Initial-state radiation In this section, we give a systematic presentation of initial-state corrections .

3.1 . THE RADIATOR FUNCTIONS FOR THE ANGULAR DISTRIBUTION

There are two radiator functions for the QED initial-state corrections, both of them containing well-known soft photon parts and hard photon parts which will deserve some comments. The soft photon parts including vertex corrections are S4(c,E ;in,n) = DA (c)[1 +S(E,ß e ) ], DT(c) = 1 +c2 ,

DFB(c) =2c,

A =T,F13,

(31) (32),(33)

D. Barclin et al. / e + e --annihilation

where we include in eq. (31) also the Born cross section. Further, + 3) + a Q2 7r 77' 3 - -i j I

2«

(34)

s

(35) Le =1n , . QC(Le Pe = - 1), Ili e The hard photon parts are symmetrised for the C-even part (A = T) and anti-symmetrised for the C-odd part (A = FB), e.FB(

C) = T Qt hT.FB(t' , c) ± hT .FB(i') - 01 °

(36)

As already stated, the hard radiator parts depend on only two variables v, c, h e-( t', c) -

Le z3'-, y-

2z

2z

Y

Y`

-I' +

,

+

2 - -r4+'0°2-4z + 3z 3yz (1°2''3)

z z (31°4 + $zl°,_ + z-') + _ (gj' 3 + -3-z,'1 ) (37) ( r, + 4z) 1, Y' hFB(c) _

Le

2z

Y-

Y

2 4 2z +-r,- ~rf(r , +z) + 3 (3r,+2z) Y Y,,

(38)

The following abbreviations are used : L c = In Y + L e , z

(39)

z = 1 - c. , = 1 + Z", 1-" c + ='-,(1 ±c),

(40)

Y=c + +zc_ .

(41)

As one should expect, the distributions are singular for v approaching 1 (hard photon emission) and c approaching ± 1 (collinear radiation) . In the soft photon corner (c - ---> E), the hard radiators become singular too, lim A(t , ,c) _ e Djc) +®(1), -~ e t'

t

A =T,FB .

(42)

Frorn eq. (42) the necessary compensation of the dependence on the soft photon

D. Bardin et al. / e + e --annihilation

9

parameter E in eq. (34) is evident. Performing the integration over the photon energy variable c leads to analytic expressions for the initial-state corrections RÂ(c; m, n) to du/dc as defined in eqs. (2) and (3). These integrations are cumbersome but not too complicated. The structure of the integrands Ore . c'(s, s'; m, n)HÂ(t , , c) leads to integrals containing rational functions, logarithms and Euler dilogarithms . The interested reader may envisage the Fortran program MUCUTCOS in the package ZBIZON [38], where the corresponding analytic expressions for the angular distribution are coded. We only would like to comment on the origin of the radiative tail within the present formalism. For initial-state radiation, it is useful to linearise in s' the resonating function (I2), (13): mi

- mP' . - m~

mn X

ma

xS'-iila S' S' S' -

(43)

if a.t least one of the interfering gauge bosons is massive . For n =p, there arises the factor s

)n 2*  - m ;,

i M s

2

r,

(44)

2 Mrr

setting the scale of tail effects, M/T,. The threshold is defined by the onset of influence of the imaginary quantity (44) onto the real type cross section. So, it needs another imaginary quantity which is found from (s' - m,-,) - ' . After integration over the photon energy, functions of the following type arise:

f

dc,

1

1 -R -E -In - Rn 1 R -4 I1

_ - LR,r( , ) ,

(4j)

If the cutoff ® is not as infinitesimal as E (i .e . if hard photons are radiated), a substantial imaginary part is developed if s > M,?. The photonic case, n = p = 0, must be treated with care. In order to get the exact hard bremsstrahlung correction for pure photon exchange one must use a reduced Born crass section (12) which includes an additional phase space factor under the integral, OÂ(S',S' ;®,0) =1 - 4fnf/S'(1

+2n1flS') oA'(S',S' ; ®, ®) .

(47)

The additional threshold factor in (47) influences only a minor part of the calculation and it is not too difficult (though lengthy) to take it into account in an analytic calculation. The pure photonic corrections to the angular distribution may

D. ~ardi~a et a!.

10

/ e + e - -raaa'ailailrati®'a

e found in ref. X23 . The function F~~ defined there ire eqs. (S) and (10) corresponds to T(c ; 0, 0) introduced here in e . (3). Interesting enough, in the existing literature there is no explicit mentioning of the correction Fa(c; , 0) or of its integral which contributes to Fa (see below). Since QED corrections due to photon exchange do not contain axial-type couplings, there does not arise any contribution of ~a(cj 0, 0) to the cross section - if not at least one beam polarisation and one final-particle helicity are non~ero ; see eqs. ( lh)-(19).

3 .2. SO~T P

®TaDN EXP+DNENTiATIC~N

s may be seen from eq. (~?), the hard parts of the radiator functions become singular in the soft photon corner of the phase space. Their divergent part can e combined `vith the soft photon radiator (~ 1), (34) in order to get the lowest order of an infinite sung over soft photon contributions . The treatment can follow exactly the arguments given in refs . 1 , ISO for the total cross section. The result is the following modification of the initial-state radiative corrections for the angular

lniti®l state rodi®tion

d~ dcos ~ ïnb®rn)

~r

r -i

i

a

e

a

v

!

a

t

a

fs = 30 GeV

.5

- 0.5

-1 .0

- 0.5

0 .0

0.5

cos 0

1 .0

F'ig. l . The C-even and C-odd contributions to dQ/d cos O due to Z exchange (solid line), y exchange (dashed line) and yZ interference (dotted line) in nbarn as functions of the scattering angle at >~ = 30 GeV (a), 91.1 GeV (b) and 200 GeV (c) . The vülues for the parameters are Mz = 91 .1 GeV, l'z = 2.5 GeV, sin g ~ N, = 0.23 and .~ = 1 .

A Bardin et ad. / e ' e - -annihilation

Initial state radiation

Initial state radiation

-1 .0

-0 .5

0.0

Fig. 1 (continued).

0.5

cos 0

. Bardà~t c~t cal. / e + e -aaaiaàltilutàoaa

D I?

-

distribution : ~.~)(Sr,S''i31,t1)

~,d_

.( -t C ;IPI,Id) _

~ i~ .-t

~~

S

~

,-t

i71, fI

(C)

1 +S(

e

)

,L'ß~-t -~Ij~(d',C) },

c

(48) ( 49) _ _ ~( Ne ~ - ~

i®

e

t

J

n fig. 1 the contributions of the initial-state radiation corrections to the differential cross section are shown as fu~~ctions of the scattering angle. e choose as typical energies ~ _ ~0 C~eV ("I'1ZISPI'A ); 91 .1 C~eV (the Z-boson n~ass ®~alue; L. , SL. ) and 200 C~eV (tail region ; CEP 200). °I' e relative i portance of the cross section contributions depends on three different components ; the coupling combinations ~.~( 3 ;~, ;~ ) which are in addition channel dependent, the factors (1. ,~, J,x. J ~ ) which have been split off for reasons explained above, and then the Born plus EI~ correction factors. Shown are the contributions for soft photon exponentiation. ~~ lo`ver energies, pure p otonic corrections naturally dominate ir~lt~~l st®te radiation 1

0

,

ä-

dG dcos ~ (nbl

Sol-ra 0(oL~+soft ahotcn exponentiation (spel saee ~ _ ® .25

______

i

-1 .0

~

i

i

i

lJidjvd +

v

v

~

I

-0.5

i

i

i

~

~

~

v

~

i

I

0.0

i

0.5

cos0

9.0

Fig. 2. The differential cross section with Born and initial-state radiation contributions in nbarn as functions of the scattering angle and of the photon energy cutoff ~] . For the other parameters see fig. 1 .

- D. Buab4 et ul /e c -ama ihlamion

13

Initial state radiation . . .'`,.`,I ,I ,-, `

d6

---- Born ---O(w) . . ........ O( a)~soft photon exponen t'atio n (spe) --- O(of- )+ spe, u= &25 -- -- Q (oG )+s pe' E _- YG eV

dcos8 (n b)

f

J- =g1.1 GeV

V., Y

,6

-1.0

-Q5

dcosQ
0.5

oos8

Initial state radiation

00 du

0.0

----_Born _- --0 W

'0(a)-s o ft

photon e xpome nt ^z t / on ( spm)

------ 0 (ot) ~spe, & =0.25 -- -- Q(cW ° s P e ' E =I Gev y

moi

E =2OOGev

O OU1

- lD

-05

O0 Fig . 2(cmndnue d ) .

05

cos8

1.0

D. Bardin et al. / e + e --annihilation

14

for QT while they are zero for O'FB (if there are no polarisation phenomena) . At the Z peak and beyond in the tail region, the Z exchange dominates. For I cos O1 approaching 1 (beam directions), the cross section rises extremely due to the collinear hard photon emission . In fig. 2 the net Born plus initial-state radiation cross section is shown as a function of the scattering angle with the photon energy cutoff ® as parameter. In the lower-energy range the angular distribution is nearly symmetric due to the dominance of pure QED. The same is true at resonance, but here it is due to the smallness of the coupling combination accompanying the nonsymmetric pure Z-exchange contribution (in case of muon production). In the tail region, a pronounced non-syrnineiric angular behavior occurs. Hard bremsstrahlung leads to a rising of the cross section with exclusion of the resonance region where due to initial-state radiation the reduced effective energy falls below the peaking value. A cut on the maximal photon energy reduces the cross section while soft photon expo-nentiation compensates at least partly the negative virtual plus soft photon terms. 3.3. INTEGRATED CROSS SECTION AND FORWARD-BACKWARD ASYMMETRY

Integrating over the angular dependence of the initial-state radiation, eqs. (31) and (36), one gets the radiator functions for QT [5] and A FB [171. These are definitely different from each other already in the O(a) leading logarithmic approximation . To some extent, this remained unclear in the comment on that point in ref. [18]. Nevertheless, at the Z peak the differences are small [42]. For more details we refer to ref. [171 and references therein. The one remaining integral over the photon momentum can also be calculated . For that purpose, one has to take care of the energy dependence of the width function in the Z-boson propagator (13). Sufficiently far away from fermion thresholds, the approximation (15) is excellent . This fact allows to apply the Z-boson transformation [9], thus ensuring that the s-dependence of the :J width does not complicate the analytic integrations, (51) XI(s) =X1(s),

r gl 1 +l Z 4-rra MZ 2

XI(s) =

1

Mz =MZ

®Z

+

r

-I

S

2

M2 Z

FZ,

-MZ+AZ,

= - 21 M - -37 MeV, Z

r2

r

s -12 MZ + rMZrZ

,

52 (53) (54) (55)

D. Bardifr et al. / e +e --annihilation

15

where we used MZ = 91 .1 GeV and TZ = 2 .6 GeV [4]. As a result, one obtains for the total cross section (4) the QED correction functions Re (in, n) to order O(a) and Re (m, n) with soft photon exponentiation, Re(fn,n) = 1 +S(E,,S e ) +H2(m,n) +He (m,n), (56) RT(fn,n) = [1 +S(13e)]H;(m,n) +H;(in,n) . (57) The soft photon corrections SCE, ße) and S(ße ) have been introduced above. The hard photon parts consist of the residual hard part He(m, n) and another one, H2 3 (ßn, r), which depends on the treatment of the soft photon corner of the hard photon phase space integral, H,e(m,n)

= g[Kj(R» . ) - Ki(R

)] ,

i = 1,3 .

(5s)

Here, g is a resonating kinematic factor, g=

( 1 -R»>)(1-R, )

(59)

The threshold behaviour and further details of the dynamics are contained in the K-functions, K,( R)

= -

1ßeR[ û + (1 + R)1(1 - R) ] ,

K2 (R)

1 = 1 - R l6e [RI(1 -R) -1(0)1,

(61)

R K3 (R)= -R 4 ß-.J 1 ( 1-R )' I(Z)

f

J( a)

= 2FI(1 «)ße'

E

dt,

L' -Z

-In

(60)

(62) (63)

E-Z

1 +ße' a) = Pe

f dt' 0

1 -ai,

(64)

The above definitions are valid if at least one of the R,, R are non-zero. Although the limit of both masses M,, M vanishing formally exists, it differs from the correct form of Re (0, 0) by a constant . This is due to the occurrence of the double pole (1 - 0- ' from QT(s', s'; 0, 0) under the integral. Such a double pole behaviour cannot be calculated as continuous limit of the two single poles

fl. BRrdln et aL / e + e -annihilation

16

arising in the massive case,

.~(~) o,0) +s(,ae ) + s;'+S [1 1 ~ 1 1

+_

2« Qé(L 11 e - 1)( +'-! )ln Tr 1 -® (65)

whefe the upper, barred (lower, unbarred) case is with (without) soft photon exponentiation. Further, if the cutoff ® is removed, an additional finite error would occur due to lacking phase space factors (47). We quote here the corresponding correction without [8,201 and with soft photon exponentiation,

0,0) -

~

j(p) 1

1

+s(ß~)

+sz -~-5 + ]

2

+ ` 1 11 « Q2(L C -1) (+ -,)ln ( -2 7r m ,t

'

S

p= 1 -41nf/s . The logarithmic final-state mass dependence regularises the massless photon propagator for emission of very hard photons and the corresponding singularity of ln(1 - A) in the limit A - 1 . For applications, we add to the soft photon correction s(13,) the next-to-leading order logarithmic correction 5' s exactly as obtained in refs. [15,16] . For ,(1, 1) which is the dominating hard photon contribution at LEP energies, we also add terms arising from an explicit integration with the corresponding hard photon terms in 511 as quoted in refs. [15,161 . This ensures the necessary high accuracy of the order OK I %) but is not described here . Due to the occurrence of logarithms in the hard radiator function for O-FB (see ref. [171 for details), the corresponding integrated expressions have a slightly more complicated structure than those for O'T . For the photon exchange contribution, RFB(u,o) = RT(0,0)

+2 2-

L, +4dL,+l4 +l 3 .

(66)

-

D. Bardin et al. / e + e -annihilation

17

The following abbreviations are used: 1 d (67) , 2 2 -' L, =1n(1 -A), (68) L, = In(1 - '-,®),

(69)

1 ; = Li 2 W) - 2(70) 1.~=Li,( -,,) -Li, 2-à

Li,(z) _ -

I dx

Jo x

The initial-state correction to

O"FB

2a 77'

Q2

-In 2L,-+ - --,L;,

ln(1 - xz) .

(71)

(72)

due to the yZ interference is

,113.)~J[Al(l -R)l ~[j 1 1

+

)

+ 900 + 52V +S ]

( 0 )(L e - 1)[In-4+LR(")] 1 1-1 +2(Le - 1)[dr-'-,LR(" )] +4dr+2r 2-~ L,

+

(Le

1)

1+R

(1 +3R) +4d rL,

+(I -2R)1 4 +13 +2R

1 +R2

, D3

(73)

D. Bardin et at. / e t e --annihilation

18

The resonance logarithm 1 .,(_) as defined in eq. (45) and the additional abbreviations depend on the complex Z-boson mass parameter, 2D., - D, + 1-1 D, = Li,

+

(Le - 1 - 21n2)L R(v) ,

- Li,

1 +R

1 +

(74)

+ ln 21n( -r) - (L, + In 2)In 1 - 1 + R (75)

1 D, = Li, - Li ., R R 1.

=

(76)

(77)

1 +R *

The third correction to

Re-1 3,)F FB( 1 1) = 1

-L, ln 1 -

-1 13,) ( 1

+

2« 11

AFB

is due to Z-e:change,

1m R(1 -R~)J (

0~

®

« ' ) IM(R) 1 -R ./

[1 +S(ß~) +ô~ +s1

(L~ - 1) [In .1 + t1. R('I)] - t(Le - 1)LR(,) 2

+211 . 1 d(L~ + 1) + 2R

1 + R-

, tD~ (1+R)-

1-J R'-1 1-6R +21 r 1' L, +4djr1 -L,+(Le - 1) L,_ 8 4 +4 , +5 2- .1 I1 +RI 11 + RI_ +(-1 +211 -R12)14+13 (78) ), t=R 2(1 -Rx) . R - R*

(79)

The C-odd initial-state corrections behave very similar to those for O .T . They have also the logarithmic electron mass singularity L e . Both Z-exchange corrections R',(1,1) develop a radiative tail beyond the resonance . This is due to their

' D. ~rarcdiaa et cal. / e e - -tree®rüailr~~i®,r

structure as already discussed for the angular distribution (see eq. (4~ ). )~L~(-~ (L

e-

ecause of t eir complexi , we quote ere t e ctio s ~(1 Q ) a = 1 taken at R = 1, i.e. at re~ona ce (we leave ut ere t e cou li g c t e e inition of X, eq. ( 3)),

P = rz~

la~- =(Le - 1)(2-31nP) .

( ~)

laFa=~ ---',1n2+41n~2-~f-i~(1)-~1nP

rrom these ®(cr) expressions it is easy to derive the corresponding ones for soft photon exponentiation . The most interesting feature is connected with hard p otons - or better. with their absence at resonance. The initial-state e fission of a hard photon for s =1Viz leads to a largely reduced effective energy s' and thus a non-resonant behaviour, i.e. a much reduced cross section . As a consequence of the resulting soft photon dominance, C-even and C-odd observables behave similarly and consequently the leading-order coefficients h;, b are equal far them at resonance. This has been observed m~merically first in ref. [42] and explained in ref. [17]. The yZ interference corrections at the peak position are exclusively due to the imaginary parts of RÂ(1, 0) since X becomes imaginary for R = 1, 1 a Re[XRq(~ . ®)] ^' - ~,4 .T + ,iQe~T(Le- 1) +Pg:~+P,g :~J P gT=2(L e - 1)(1nP -'-,~, gFe=2(Le-

gT= -'-, ;;(L e - 1),

(87 (1~~)

1)(lnp- +1n2)+ [ -2+21n2-41n-2+L.i-,(1) ] .

" D. Baa-riira c't nl. j ~ e --raaaaaiia~lratâe~ra

?()

The orn contribution to Qr8 vanishes at the peak. 1\ievert eless, the leading corrections to ~ r and ~Fe are equal again. Integrated cross section and asy merry as functions of energy and the photon energy cutoff .~ have been st ie in detail in refs. [~, 39]. e `vill come back to that point in sects. ~ and 6.

itial- ~ ai-~ a e i e e e ce

i i

The initial-final-state interference corrections have an interesting property : only those `which are diagonal in the arguments indicating the exchanged vector bosons are independent observables . The other interference contributions may be detern'ined by the following simple relation: B.-~( c ; ia~ , i~ )

_ ;

°~(

c ; iü, iïi )

+

~(

c ; il , id ) ~

( 91)

n nrdPr to get (91), it is essential to separate the reduced Born factor, eqs. (1 ) and (12), from the i~BD contents . The validity of eq. (91) may be seen immediately from eqs. (3), (12) and (13). An explicit proof has been given in ref. [37J. ~ .1 . THE RADIATOR FUNCTIONS FOR THE ANGULAR DISTRIBUTIONS

For interference corrections, the soft photon part has to include besides the soft photon emission terms S 4(c, E, Jl ) also the contributions B.4(c, A ; ii1, is ) originating from photon-Z-boson and photon-photon box diagrams . Their sum is infrared finite (and independent of the infrared cutoff ~ ) while the infinitesimal soft photon cutoff parameter E disappears only after integrating over the photon energy variable u, a

S,'~(C,E ;ii1,i1) _ -~c~t~[SA(c,E,A) +8,4(C~~~~1~i1)~ , 7T E

C_

S,q (c,E,A) =2D~(cl 21n ~ ln c

92

+Li,(c + ) -Li,(c_) -'-'-,(In-'c + -ln'c_) . (93)

ere, IAA (c) is defined in eqs. (32) and (33) with A = T(FB) if A = FB(T), and c + in eq. (41). The box diagram contributions [34J are dependent on the type of the gauge bosons exchanged. We write them in (anti-) symmetrised form, Br,r-~a(c,A~in,n) =b(c,A ;m,n) ±b(-c,~l ;m,n) .

(94)

-

D. Bardiii et

The two different b(c, A

e -annihilation

x functions are

0) _-2c-+ 1

C[ -

14 In

2E - 2-,, i A

2c + + c(In c - + 2-,. ifl

- 2-,. ic -

Mc,A ;nai) = -4c+(I #1 - 2c-)[I(I) -

2c 2,

1(c

M

2E C21n - + 4L,g + In(c +c-) In - + 21(c-) - 21(c A 1 c-

The following abbreviations are used: /(a) = L'2 (1 - aR,-, 1 ),

L,, = In(] - R,-,') ~ LR~ 1 ) *

(97) .(98)

and R,, is defined in eq. (46), L R( ) in eq. (45). The hard radiator parts are independent of the gauge boson exchanged,

0

H'T . FB(

Jh T. FB( F'C) + A FB(

QCQ

While the box terms for A = FB and A = T could be expressed by one and the same function, this is not the case here. h'T (v, c) = 2c,

C~[4 --3-z(2+z) In-LI

4

+2

t,

1

+4+z(2+lnz) +

+C 2 )In cB( L") 0 = 2(1

MFF

+4c + +

2 'y

2

C+

4 2

L, 2

2

(

11

(j

4 2 y

t.

+

Z)2

In

c _+ zc + C s +m-

2 -z(l +z)

_ Z _Z2

5 4( 2 +4+2z -z(l-z)lnz + - 2 +--3-z 11 V I y 6 Li

-4-2z- z2 )

+ 2( Z 2 - 1)c, In c +c-

2)] ln y . + 2[(1 _ Z + Z2) +c(1 -z 2 ) + c'- (1 + z + z

(101)

ail / e ` e D. Borifin ci .

It is easy to check that in the soft photon limit 0 , -- 0) the functions h A6 0% 0 behave such that they compensate after integration over the photon momentum the dependence of the box contributions on line. In fig . 3, the interference contributions are shown as functions of the scattering angle for three different energies . Compared to initial-state radiation, the interference corrections are small . They do not contain fermion mass singularities though kinematic singularities at cos 0 = ± I occur. In contrast to initial-state radiation, these are integrable so that formally the full angular range in eqs. (22) and (24) may be used . Of course, the interference contributions depend on the hard photon cutoff -1 . For more severe cuts, they grow up. This is due to the fact that for the totally inclusive problem (-I = 1-ni - 1) there exists a fine-tuned cancellation of box and bremsstrahlung contributions (for more details see subsect. 4.2) which becomes more and more disbalanced if the phase space of the bremsstrahlung integral becomes more and more restricted . For very tight cut values, -1 << 1 . the cross s- ectior, to order. 0(~T) starts to diverge and may become even negative . This can be seen from the leading soft photon contribution R.'_,""(c-, in, n). After integration over the photon energ:y , the dependence of S,-,(c,,E, A) in eq. (92) on E drops out . It is replaced by the following terms arising from the soft photon corner

Initiol - final Interference

Born 0 (Cx)

= op 5 Ed= l GeV

__ 0 ( a) , a

- -MoO

Fig. 3. QED contribution to d(r/d cos (7) due to initial-final interference as a function of the scattering angle and A . For the parameters see figs. 1 and 2.

6n6~A~t -~a~~~

~~tt~~~~~~~c~

_____o

fl

-1 .®

C

C

~,  ~ ® ~ . ~. _

a. C ~ g, ~

®- F

-®.5

0.4

~f09~6~~-~fi~E~f~

0~

eQs ~

6t+~~~tCE'Eft~~

0 .003

0.002

0.0 01

0.

-1,0

- 0,5

0,0 Fig . 3 (continued) .

0.5

c®s F3

1 .0

D. Bardin et cil. / e - e _-annihilation

24

of the phase space integral: i . soft (C -. 171, 11) .4

(c),S,

In -1,

(102) (103)

The complete interference contributions to the differential cross section may be found in the code MUCUTCOS [38]. In analogy to initial-state radiation, the logarithmic dependence on the cutoff -1 in eq. (102) has to be cured by an adequate soft photon resummation which should lead to a replacement of %Qj In E + by some function of the following type :

In order to get really a smooth, well-defined behaviour of (105), the exponent therein has to be larger than -1, i.e. 8j should be positive definite. This is not the case in (105) as may be seen from eq. (103). A possible way out is the exponentiation of interference radiation together with initial- and final-state radiation as is dictatcd by soft photon theorems (see e .g, ref. [10]). A collection of the relevant terms leads to the following order OW expressions after integration (one should have in mind that s' = s for the soft photon case): da' dc

aAO (s, s ; in, n) DA(c)

A, in, it e

2a 1

7

Q2 -

S

In in 2 + Qf2 In

1~

S

1 -c + 2QeQf In 1+c

Wf - I

In A

(106)

The final-state expressions have been taken from sect. 5. Again, the correction is not positive definite due to the expression in square brackets which is a possible candidate to replace the exponent in (105). This is a consequence of too crude approximations . The soft photon contribution as a whole must be positive for any

mdiiae î aL

ara

/e -e -`atiimiliil otioia

eter combination since it is a comUolete module m auare

fact.

(s, s ; ni, n)

The f

--~

-- 2p

After integration

/(z~k

~

.~--_-__ ~-- k2 -~ ~-- kpp)( - p / p)

x~mrs»n==

A.m ni .

pi

photon momenthe corrections which are relevant for soft

the soft photon corner of the n -dimensional

phase space [351, one gets photon exponentiation -~ ~' --

I

k~ -` z 4ni f '

1" ± - 1c --2p24 z k`= /

tum

.

I

for

z S 2m/ - ` /s --~mn . . .- . ./ x~~K s, s ; mm,~Ö ~~ .K~~(D'~--~~~~-------i' ~ ' -- = c ' ~~ U mr ~ ~ \

\

-'

'~ a. -- IT-=

z f (Tf 2a s- 2m r -- ~ + ---Q z -------~ I m-----~ S T --m f S «r m "f ~

+ ç~+ -~~~ + '~~~2a c ~~ c ~~ r ~~+ ---~~ In -~------ -- '~' In -----~~' ='"~ ! u,+ -R ~ R+ vr c -~~ L ~~ // z z m~(n == /2 -4 mnc(M,s~~ , ~~ i = S z + c qq , m~

~~ ±= ±~(a,

± z -z z zlVz . ~~n~cmm~~ y

( DX 2) ( ll3

'

(}14)

D. Barclin et al. / e + e --annihilation

26

In the ultra-relativistic limit of eq. (111), one gets eq. (106). While eq. (111) as a whole is positive definite, this property is lost in eq. (106) as a result of the approximations . Since we do not intend in our applications to apply such stringent cuts as to make inevitable the adequate exponentiation of initial-final interference contributions we would like to remain at this point of clarification. In a recent series of papers, there have been presented quite interesting results on soft photon exponentiation [43] including the interference [44]. Tracing back also to earlier work on that subject [101, the problem of exponentiation of soft photon interference radiation has been solved there in the ultra-relativistic limit. From the above discussion we conclude that this is adequate for values of cos O not too close to 1, i.e. assuming some realistic acceptance cut for the experimental set-up. Further, the authors make the implicit assumption that one may use the radiators for vT also for the differential cross section. This is not the case in general as shown in this paper and in ref. [ 17]. Nevertheless, the radiators for QT can be used as an approximation. Further, we see no problem in combining the hard radiator parts presented here for the differential cross section with the result of ref. [44] for the refined treatment of soft photon exponentiation . -I.2.

INTEGRATED CROSS SECTION AND FORWARD-BACKWARD ASYMMETRY

The integrated interference contributions are composed of box corrections B.4(A, n, n) and of bremsstrahlung contributions bA(A, n, n), a RA(n,n) = -QeQf[B4(A ;n,n) +bA(A ;n,n)] . u

(115)

For the total cross section one gets BT(A ;0,0)

2E = 61n - - y ,

bT (A ;0,0)

= -61n

BT (A ;

bT (A

2 EA

-

2E

1 1) =61n -

:l ) 1) _ -61n

+ 3(2+A),

---9+3R[1 +(1 +R)L R(1)] +3L z ,

2 E®

Lz = In (SIMz2 ),

+6+3[®(1 -R) -R(1 +R)L R( ®)] ,

(117)

(118)

(119) (120)

where L Ft(®) is defined in eq . (45) and M' in eq . (53). Without cut (® = l), the

D Barda®a

~~ a!

/ e {e -drjaaaa~aa~c~eralaa

resulting corrections to ~-r are quite small [8, 1], a

RT(®~ ®) - -

12

~Yf ? a

The interference QED corrections ue to p t cross section part aga are

~(1 2

ec

2 .

+26 :1-5,')L,+x(15-2 .~-E5:1=)L~

+~ Li,(1) +~In-2-4Li,(~) +BLi,

2

- ~ Li~(d - 1) . (12 )

Their sum is a simple constant for ~ = 1 [23]. 125) The Z-exchange corrections are much more involved. For ~ = 1, they may found in ref. [8]; with d < l, BFB( Jl ;

l , l ) _ (1 + 8 In

2E 2) In ~ - + R

- ( 9 - 4R - 4R2 ) In 2

- 2 ln- 2

+'-,(5-4R}LZ-f,[ 4~-9R+3R-+2(-5+3R-6 :.-')In2]LR(1)

- (1

-

- 3R + 6R~ - 8R~) Li 1

4R; Li 2(1) - Li e f -

1

R

1,

2R~ -L~2 ( 1

RI1 (126}

D. Bardin et aL / e 'e --annihilation

28 -(1 MFB (A ; 11

2 E-11

+8 In 2)ln- + j (2 A

- M + 5RA)

+ 14 (4 - 16RA + SRS-' - 10RI + 26A - 5® -' )ln2 + 41n -' 2 + 2L20) 4 (3 - R + 2R-,l - R .12 -2R 2 + 2R2A - 4A + à2 )LI -5 + 3R - 4R--,î + 3R A2 +5R 2- 25 RI -

6

+ 13a - 5A2 L 2

6 A 5 + - + R L R(-I) -4 Li 2(1) +8L'2

(2) -

-1 (5 + 3R + 3R 2 + 5R 3 )I),, Wth

(127)

K, defined in eqs. (75) and (76). Again, the Z-boson parameters have to be

understood in primed quantities . A generalization of the initial-final interference radians to the case of several massive gauge bosons [12] is trivial due to relation (91). At the Z peak, a fine-tuned cancellation of box and bremsstrahlung terms occurs. Therefore, the resulting Z-exchange interference contributions T 1,1 ) and Ran{ , 1) become small there. This has been noticed from numerical results in refs. 17, & 42J and is known also from earlier investigations of J14i physics 1451 For i-{ 1,1) it is evident from eq. (122). A Taylor expansion around S=M27 }Melds 16 In

R - 1 +0 3

+19(R- I)ln( R - I +a) +®(R- 1)

(128) FB(",

a

-Qe Qf 7

1+81n(' - .1) - 41n(1 - Afl In

R

I +A

A

-3(In 2)(R - I)ln(R - 1) + 3'i [1 + 8 ln(2 - A) - 6 In( I - A) I ( R - 1) In( R - I + A) +4(R - I)ln(I -,A) + O(R - 1)

)-

(129)

For -i approaching 1 and using in the peak region In R = R - I + OR R - 1)2], one

D. Bardin et al.

/ e 'e - -annihilation

29

gets

a RT (l, I ; A= 1) = -Qe Qf I -3(R- 1) +0[(R- 1)21), 7r

a R'F1301 1 ; A = 1) = -Q.Qf(-3(R- 1)(In2)ln(R- 1) 7T

+ -41 [7 - 101n 2 - 61n 2 2 - 6 Li,(1)] (R - 1) +

1)211 .

(131) The real parts of R'(1, 1) contribute to oA in a product together with IXI.2 A IR - I I _'. As a consequence of their proportionality to (R - 1) whose real vanishes at the peak, the Z-exchange interference corrections become extrem small. For A * 1, there are at the peak larger contributions due to ln(R - I + 1) In -4 in eqs. (128) and (129). A similar discussion applies to the yZ-interference corrections. Due to eq. (91), half the R'(1, 1) are combined there with the resonating function X which A becomes imaginary at R = 1 . So, the imaginary parts of RA", 1) are relevant . Again, the QED corrections are suppressed compared to the Z-exchange rill cross section behaving like I I - RI -2 So, all the interference contributions are small since the photonic corrections RA(01 0) are also non-resonanting at the peak. In case of cuts to the photon ener&zy, -4 < 1, the 0(a) interference bremsstrahlung has to be taken into account properly. For very small A, logarithms of the type In A may become even dominating and then one should try to apply some soft photon summation procedure. 5. Final-state ra iation 5. L THE RADIATOR FUNCTIONS FOR THE ANGULAR DISTRIBUTION

The final-state radiation contributions to the differential cross section have a simple angular dependence compared to the expressions discussed in the preceding sections. The soft photon parts of the final-state radiators SAf(c, C; m, n) may be obtained from S(E,fl e) as defined in eq. (34) by replacing Me by the final-state mass in f, Sf(c,,E ; m, n) = DA(c)S(c,8f ),

A = T,1713,

(132)

where DAW is introduced in eqs . (32) and (33). The hard photon radiators are = -Q a 2 H'(v, c) 1D )[Hf (v) - 3v] + 4vl , (133) T 7r f T(C a Qf2 (134) D HFFB( f L3, 0 FB(c)hf(v),, where the radiators Hf(v), h f(0 have been derived earlier for the integrated cross

D. Bardia et a~ / e + e - -annihilation

3p

sections

T and a°Fg [ 17),

~

integration of the symmetric radiator function C ~4 = ~T) over the photon energy gi`~es a contribution to the differential cross section, he corresponding anti-sy~nrnetric radiator integral vanishes, As as ee~1 pointed out e.g. in ref® [ 19], the soft photon corrections (132) to even an c~dd cross section parts are equal and will, consequently, cancel for the for~~9ar ~bacl`- ward as merry. After integration over the full photon phase space, due to the correction in the eqs. C 137) and ( 3 ), there is minor influence on deno instar . Ne~~ertheless, for s all cut o values ~ e~ 1, s+nft photon exponentiation is recornn~ended in order to get reliable numerical results. A naive addition of initial- and final-state radiation corrections as has been perfo ed in ref. i17] c sages this dependence and leads to an overestimation of final-state radiation ue to an imbalance of corrections. n order to maintain the smallness of isolated final-state corrections after combination with the large contributions from the initial state, one 8~as to proceed more carefully. A treatment based on the intuitive picture of subsequent radiation of photons from initial and final sta es is proposed in su sect. ~ .3. An isolated soft photon exponentiation for finâl-state radiation is analogue to the pio~:~ûure described in ref. [ 15] and consists of the replacements Fn

Numerical results due to final state radiation are shown in fig. 4. >.2. INTEGRATED CROSS SECTION AND FORWARD-BACKWARD ASYMMETRY TO ORDER Oia)

T'he integrated final state O(«) QED corrections are simple,

a 1~~(m,n) _ ~Qf~'-,(Lf- 1)(41n® +3-4~ +û2 ) -2Li2(4) +2Li~(1) +fÂ(~) ], (141) r-fr( ®) = z( 1 -®)(3 - ®)L, i + (6® - 2

_,2)

~

(142)

D. Bardin et al. / e+e -annihilation

31

-

Final state radiation d6

Born 0 (00 ........ 0 lo`) + soft photon exponentiation (spe( 0 (ot) +spe, A=0.25 - - 0 (od + spe, E I = 1 GeV

dcos 0 ( nb)

rrs = 30 GeV 0.05

d'/

W

fl

W

fl

-1.0 0

a

fl

S

fl

l~ L fl

B

9

fl

W

W

A

W

fl

I

A

fl

W

!

W

A

A

A

W

fl

W

W

05 fl

T--v-

A

P

®

4

P.~

W

P

Cos

Final state radiation

i

d6

A

0.0

-0.5

W

ti

,

-b6

--

,

Born

dcos

®(a1 . .. .. . . . 0(oG)+soft photon exponentiation (spei

(n b)

______ 0 (a)+spe, ß=0 .25 - - 0 (od +spe, Ey =1 GeV

,(s = 91 .1 GeV

0.7 e ~

w L

-1 .0

W

W

fl

l-W~WW

I

-0 .5

.

W

~

.

~

~

~

~

I

0.0

~

.

~

o

.

.

.

(

0.5

W

W

s

A

~

~

W

cos0

e

e

1.0

Fig . 4. Born plus final-state correction contributions to da/d cos 19 in nbarn as a function of the scattering angle and .1 . For the parameters see figs. 1 and 2.

D. Bardin et aL / e + e --annihilation

32

1

I

Fingt state radiation ..I I I , I I . . i I

. I

I

.

I , I-

Born __- -0(oG) . . . . . . ...0 (cc.)+soft photon exponentiation (spe) ------ 0 (c4) +spe, o = 0.25 -- 0(oc ) +spe, E 9 = 1 GeV

dG dcos 0 (n b) 0.002

/

vls = 200 GeV /

0.001

-1.0

-0.5

0.0

0.5

cos 0

1.0

Fig . 4 (continued) .

dT

0.16

(n b)

4s = 30 GeV 0.12

0.08

C

0.04

0.00

Born Born +Initial (spe) Born +Initial (spe)+Final ------- Born + Initial (spe)+Fingt (spe)

0 .0

Fig. 5. The total cross section

0.2 UT

0.4

. . . . . . 0.6

(a) :

~~II~I~II~~~~~I .II=

0.8

A

1 .0

as a function of the photon energy cutoff 4. For the parameters see figs. 1 and 2.

D. Bardin et al. / e + e --annihilation

6r (nb )

= 9I .1 GeV

0.5

Bo rn Born + Initial (spe) Born + Initial (spe) + Final Born + Initial (spe) + Final (spe)

Initial (spe) Initial (spe) + Initial (spe) +

Fig . 5 (continued).

f. Bardin et al. / e + e --annihilation

34 -4.0

e

A FB (%)

fs= 30

.

.

.

.

.

.

.

.

rrn

I''

GeV

-5.0 Born - - - Born + Initial (spe) Born + Initial (spe) + Final ------ Born + initial (spe ) + Final (spe)

1~ I l

-6.0

-7.0 teee

0.0 2.0

m

elveeeeee

Q2

.eleeeeeeeeelW

0.4

0.6

eeeee

(a)

m

Ieeeee .m

A

0.8

W:

1.0

us = 91 .1 GeV

Born -- - Born + Initial (spe) Born + Initial (spe) +Final ------ Born + Initial (spe) + Final (spe )

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

0.0

0.0

0 .2

0.4

.-,- _. _ _ -.-_ -:,-:. r

0.6

0.8

0

1.0

Fig. 6. The integrated forward-backward asymmetry A Ï:I3 as a function of the photon energy cutoff A. For the parameters see figs. I and 2.

D. Bardin et al. / e } e - -annihilation

35

Born ____ Born + Initial (spe) . . . . . . . .. Born + Initial (spe) + Final Born + Initial (spe) + Final (spe)

Fig. 6 (continued).

If the photon energy cut is removed, one gets 1Z-fr(rn,n) _ -

Qc

RFB(in,n) = 0 .

(144),tl45)

Figs. 5 and 6 illustrate the above discussion of the corrections to the angular distribution due to final-state radiation. A comparison of (138) with (136) and (140) shows that the contribution of final-state radiation to 0"FB vanishes if neither a cut on the photon energy is applied [8] nor finite-mass effects are taken into account [46] . Then, the only (and minor) influence of final-state radiation on A FB is due to the C-even correction in eq. (5). If a tight cut is applied (A << 1), the C-even and C-odd final-state corrections (141)-(143) approach each other because from soft photons there is no influence on the angular behaviour of the emitted fermions . Due to the logarithmic cut dependence, one again should exponentiate the soft photon contributions in that case but we leave out the details here . The ansatz is defined in (139) and (140). 5.3. SOFT PHOTON EXPONENTIATION

As may be seen from eqs. (133)-(136), the final-state radiation contributions to the differential cross section behave as a function of the normalised photon energy c , quite similar to the integrated quantities QA , A = T, FB. Based on their simple

D. Ba rciin et al. / e + e - -annihilation

36

dependence on the scattering angle, we now derive a common treatment of soft photon exponentiation for initial- and final-state corrections. We start from the definition (2), (3): du (e+ f ) dc

A=T.F .B

Re

~fl dt'Q~)(S',S' ;II1,I1) (,

xlc3(i')[1 +(T (S,S ;In,il)D-1(C)

-1

f(),

+S(E,ße)]D,4(c) +0(i'-E)HÂ(t,,c)}

dit [ 5(u)S(E,ß f )

+®(tl - E)HÂ(ii,c)j

~

,

(146)

4(it -C)

=

(147)

a(c)HA(11"c),

where RFa is independent of the scattering angle and T has an almost negligible dependence . Further, eq. (146) can be rewritten as follows: do,( ` + f) dC

tn , ta = (), 1 A = T, FB

x 3(v)[1

Re

f -1 di'?( tT, S', S' ;

in, n )

9

+S(E,ße ) +I'a(S' , c,,)]DA(c) +0(1,-E)HA(t,,c) }, (148)

iA(S , C,

) =

()-

du [ 3(11)S(E,ß f ) +0("

-E)H~(lt,C)] .

(149)

The dependence of râ on c is very weak for A = T and absent for A = FB. Now the following ansatz seems to include a quite reasonable approximation of higherorder soft photon corrections: d&(e +f1 dc

ttt, ti = 0, 1 A = T, FB

Ref, dt'Q()(s',s' ;m,n)FÂ(L',c)FÂ(L',c,s'), (150)

FA(t' , C) =& (L')DA(C)[1 +S(E,ß e )] +®(t'-E)HÂ(L ,~~C)~ (151) FÂ (t, , c, s') = 1 + )Â(s ' , c, ®) . (152)

D. Bardin et al. / e ' e --annihilation

37

The exponentiation of soft photon contributions leads to the following replacements: FAeC. ( L' C ) --3>FACe ("' C )

=

I

FAf( v, c, s')

--3-FAf ( A

v, c, s')

DA( C )ieel "oe-l [l +g(i6e)] +iÎAe(l* .jC) ?

=

(153)

.11

f f0 du(pf'u",1- 1 [ I + 9(m)]

154)

jLff(U,C,P ;) f

+ "A

2a Q, sr P'= - -f In -, - 1 f 7r mi (

(155)

Here again the residual hard radiator parts H are

P HAa'3 ( L"P C, à6) = HAa( 1,~ C~ à6) - -DA(C),

(156)

The integrations over the variable u can be performed in eq. (154), f ~;fA ( 1 C, SI =

L

GA (v, c, s') = 4 f

G4( t', c, s

(157)

a 4) _ _Q2[2 Li,(A") +gA(-'-",C)l e 77 f

9T('A ' -g 0 = - 1) ( I -X) (3 - A') In( I - A') 6)

- 13 -

( 158) 4

I+C (159)

( ,jl,, C )

9FB

! 2

A o,2

In the above formulae, we introduced the variable A' which is connected with the reduced s' in the same way as A with s, Ar = 1 -

I

Smin

S

F

1)

A'

1 - 1,

9

1 ? 11

if one wants to obtain from the above expressions for the angular distribution those for aT and A FB9 it is helpful to start with the observation that the F4f are independent of the scattering angle with exclusion of 9T("") C). Since that depen= 0.1 % Q2, one can neglect it. Then the dence is minor, i .e. less than Ral7r)Q2 2 f f final-state radiation factor is completely independent of the scattering angle and

D. Bardin et aL / e ' e - -annihilation

38

c) only, the integration over c has to be done for Mv, .4 ~

5:(e+f) A

FÀ

ni, il = 0, 1

Re

f(~'

'Idt,o,('(s',s ~4 ;m,n)Fe(t,,)FAf(t~,s'), 4 A

(163) (164)

'f ' dcFe(L 4 , , c) . 4 0

This integration yields the normal convolution representation for the total cross section 5~4 , but now modified by the additional factor FAf. So, by a direct Feynman diagram calculation we have shown that eq. (163) is an approximation of sufficient accuracy though some well-defined terms of the order OW have been neglected. A similar formula in the case of O`T has been obtained by other methods in ref. [ 141. The dependence of ffT and A FB on the cutoff parameter A is shown in figs. 5 and 6 for different handlings of initial- and final-state corrections. For smaller A the cross section becomes smaller since the positive hard photon part is restricted then . In the tail region (figs. 5c and 60, one observes a steeply falling cross section at values of A which cut away the radiative tail. This happens if the effective energy s' cannot reach the resonance energy, i.e. A < I - Mz2Is. Even for infinitesimal A the net negative bremsstrahlung corrections remain finite due to soft photon exponentiation as discussed in the text . In the figures, we applied the simplified but sufficiently accurate procedure explained in subsect. 4.2. Since the corresponding soft photon corrections in units of the Born cross sections Cr T and (TFB are equal, the net final-state correction to A FB vanishes for vanishing A . For sufficiently large cutoff values, the final-state contributions also remain nearly negligible. In the intermediate region, they must be taken into account at least for precision measurements .

6.

iscussion

In the foregoing sections, basic formulae for and characteristic features of the QED corrections to three different observables in fermion pair production from e + e - -annihilation have been presented separately. Numerical results have been produced with the codes MUCUT and MUCUTCOS of the package ZBIZON [381. In fig. 7, the net QED corrections to the differential cross section are shown in combination with the weak loop corrections . The latter have been determined with the code DIZET which is also part of ZBIZON . For Mz = 91 .1, mt = 100, M11 = 100 (all masses in GeV), and a, = 0.12 we get in accordance with refs. [16,271 sin 20 W =0.2314, F7 =2 .477 GeV. The corrected inuon decay constant, running QED coupling, and weak neutral vector and axial vector couplings to be

D. Bardin et aL / e ' e --annihilation d6 dcos E) (n b)

39

Born WOO

t,

. . . . 0(od+ soft photon exponentiation (spe) ------ 0 (od + s Pe, A= 0.2 5 0 (cc) + spe, E y = 1 GeV

.05

-1.0 d6 dcos 0

-0.5

0.0

0.5 cos 0

1.0

Born 0(0(,) . . . . .. .. .. 0(od+soft photon exponentiation (spe) ------ 0 (o~) + spe, A= 0.25

(n b)

1 GeV

0(o(,)+spe, E

Fs

91.1 GeV

.5

-1.0

-0.5

0.0

0.5

Cos 0

1.0

Fig. 7. The differential cross section with complete electroweak corrections as a function of the scattering angle and 1. The values for the parameters are MZ = 91 1, 1711 = 100, M H :- 100 (all masses in GeV) and a, = 0. 12.

40

D. Bardin et al. / e + e --annihilation

Fig . 7 (continued) .

Fig. 8. The total cross section trT as a function of an acceptance cut on the scattering angle, cos N I-< cos (")m :,X . For the parameters see fig . 7.

D. Bardin et al. / e + e --annihilation

0.008

fs

GT In bl

= 200 GeV

0.0ûô

0.004

0.002

0.000

0.0

0.2

0.4

0.6

Fig. 8 (continued).

0 .8 Cmax

1.0

D. Bardift et al / e + e --aru :iltilatiott

42

0 .0

0.2

0.4

0.6

0.8 C max

1.0

Fig . 9. The integrated forward-backward asymmetry A M as a function of an acceptance cut on the scattering angle . For the parameters see fig. 7.

D. Bardirt et al. / e'e - -annihilation

43

Fig. 9 (continued).

50

100

150

200

f

250

300

SIGeV

Fig. 10 . Cross section contributions due to initial-final state interference (a) and final-state radiation (b) in percent as functions of and .]. For the parameters see figs. 1 and 2.

C

I3. Bardu et al / e + e --annihilation

4-s ,GeV

r

GeV

Fig . 11 . Asymmetry contributions due to initial-final state interference (a) and final-state radiation (b) . For the parameters see fig . 10.

D. Bardiii et al. / e - e --annihilation

45

120 A FS

M)

A = .01

70

20

-30

initial state initial - f inat interference final state ------ all

-80

-130 30

M I

- I

80

130

I

180

-

I

2 301

4-s , 6eV,

28 10J

Fig. 11 (continued).

used in eqs. (16), (17), (20) and (21) are [291 aFA (Mz2 ) = a(l .063 - MAN),

(165)

Gt,pz( M2) z = Gl,(1 .000 - iO .005),

(166)

ae L,

2

e(MZ)

11

2

(167)

-2 [1 - 4sin 2 OW (1 .01 I + iO.013)],

(168)

1

ee(IVZ2 ) = '41[1-

8sin 2 O W (I .01 I + iO.013) + 16 sin4 0w ( 1 .023 + iO.027)] (169)

If, for the sake of comparison, the initial-final interference terms are excluded, the fully corrected total cross section o-T as obtained here :n +0.2% with that obtained from the code ZSHAPE [15,161 wb ch is ex,- -' tc, order O(a 2). For A FB and duld cos 0, we do not know of a progri- which .Vould allow for a similar comparison. While the weak loop corrections remain small [16,191, the QED corrections amount typically to several percents or more, essentially due to initial-state ;

-

T-,,

46

D. Bardiù et aL / e ' e - -annihilation

radiation . Even in the tail region, the angular dependence of the corrected differential cross section follows closely the Born cross section behaviour with exclusion of scattering angles near I cos i9l = 1 . This is reflected also in figs. 8 and 9 where aT and A FB are shown as functions of an acceptance cut, 1191 emax' In general. the asymmetry is not a monotonic, rising with 19. a, function though there are regions with an almost linear behaviour. For loose cut values, the tail effect is pronounced at s > M2. z The total asymmetry is small at resonance with corrections nearly of the order of the Born contribution. The relatively large asymmetry value for a tight cut. E,, < I GeV, may be understood from fig. 2b where a systematic distortion of the differential cross section is present in that case. In some recent determinations of aT [41 and A FB [251 at LEP, the QED corrections [11, 1-5, 381 have been applied to data corrected to correspond to a detector with a full anatular acceptance (4-,. geometry). From figs. 8 and 9 one can see that this may be justified although the present formulae allow a refined analytic appro, ach to the data. This seems to be recommended at least for the interpretation of precision measurements. In view of the relative smallness of interference and final-state corrections, it is interesting to know where they can be neglected completely. Of course, this depends on the energy region and photon energy cutoff chosen. In figs. 10 and I I corresponding regions of relevance are shown. To summarise. the analytic approach to QED corrections has been proven quite for consistency checks of Monte Carlo programs [31 and for the interpretation of experimental data. This article contains a first, systematic presentation of our analytic formulae for the complete set of QED corrections to the differential cross section da/dcosO, total cross section O'T and integrated forward-back-ward asymmetry AFB' We would like to thank F.A. Berends, M. Greco, W. Hollik, R. Kleiss and Yu. Sedykh for enlightening discussions and fruitful cooperation.

[II

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