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Bhabha scattering near the Z0
Volume
177, number
BHABHA
PHYSICS
1
SCATTERING
LETTERS
B
4 September
1986
NEAR THE Z,
M. GRECO INFN,
Laboratori
Received
Nazionali di Frascati, I-00044 Frascati, Iialy
16 May 1986
A complete analysis of EM radiative corrections to Bhabha scattering near the Z, is presented Compact analytic formulae are given which include exact one-loop results and soft-and-collinear photon effects resummed to all orders. Detection of back-to-back e+e- pairs can be therefore used as a high precision monitor of luminosity at LEP/SLC energies.
e’e- annihilation at LEP/SLC energies will provide precision tests for the standard electroweak model only if QED radiative corrections are under control at the level of 5 1%. Indeed first-order corrections [l-3], for example, reduce the Z, peak cross section by more than 50%, or shift the zero in the forward-backward asymmetry by about (+300) MeV, for an energy resolution of (10-‘~10~2). It is therefore important that higher-order corrections are properly taken into account if the electroweak parameters have to be measured to the required accuracy *‘. Previous studies of these effects have been presented over the past few years, in particular for the process e + e- + ~_l+p~ [l-4]. The reaction e+e- -+ e-e-, on the other hand, is particularly interesting for its large cross section and could provide a high precision monitor of the beam luminosity. Detailed studies of electro-weak radiative corrections to this process, which have been performed earlier [5-g], are all incomplete in some respects. Indeed the calculation of electro-weak first order corrections, performed in ref. [5], does not extend to the energy range around the Z,, because of the lack of finite width effects. Those were included in ref. [6], together with the complete treatment of soft photon effects, resummed to all orders. The analytical expressions for the box diagrams in the s and t channels however, were only given in the limit s - M2, the left-over terms being of order ((Y/T). An attempt to improve these results has been made in ref. [7]. Finally a treatment of collinear hard photon effects, quite relevant in calorimetric-type experiments, has been given in ref. [8]. The aim of the present paper is to give a final and complete description of QED radiative effects for Bhabha scattering, including exact analytical expressions for all one-loop diagrams, and soft and collinear hard photon effects resummed to all orders. Therefore our results include all double logarithmic terms of the form (a/n) ln(s/m’) ln(A, T/M), (a/n) In S2 In A, simple logs as ((Y/T) ln( S/m2), (a/a) ln(A, T/M, S’), resummed to all orders, and all finite terms of orders (Y/T. In the above M and r are the mass and width of the Z, boson, and A and 6 the energy and angular resolutions of the experiment, better defined below. Hard photon effects of order (a/r)(A, S) have been neglected. Our considerations apply to a typical experiment in which the following requirements are satisfied: (i) The electron-positron pair should be detected back-to-back within a certain acollinearity angle J of a few degrees (J 5 5” ). The energy resolution Aw depends upon J (see eq. (16)). (ii) An electromagnetic calorimeter of finite and small angular resolution S is centred along the electron and positron directions. In principle it does not discriminate between a charged particle and the accompanying collinear photons. *’ For a recent review see ref. [4].
0370-2693/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
97
V o l u m e 177, n u m b e r 1
PHYSICS LETTERS B
(a)
(b)
K,
4 S e p t e m b e r 1986
/d)
(c)
T /
q,
MQED(s) o
MRES(s) o
--....
MW(tl
MQED(t) o
o
M~ED(s)XOv(S) MRoES(s)X6v(S) M~ED(L)x6v(t) Mwo(t)×fiv(t)
Z
M~ED(t)×d~(t)
tl'oQED s)xd~(s)
+cK
+CF,
MQED(s) pox
MRES(s) box
M~ED(t)
M~ox(t)
DO×
Fig. 1. Virtual g r a p h s in the s and t chalmel.
Then, using (i) and (ii), one would be sure that all but a fraction ._4-= A~o/E of the beam energy (~- = 2E) is taken by the electrons and the accompanying hard photons. For small A and 8, fully analytic expressions can be used, neglecting hard-photon effects of order [(a/~r)A (a/w)6)]. We give a brief account of the derivation of our formulae. Our notations are as in ref. [6]. unless stated explicitly. The relevant virtual graphs are shown in fig. 1. We have
M(s,
t) = M,QH)(s)[1 + 6QH)(s)] -- M,QED(t)[1 + 6QED(t)] + MRES(s)[1 + 26.,(S)] -M,W(t)[l+26.(t)]
AAcQED/ +MhoQED X (s)--l..bo~ , t ) + M , ~ S ( s ) - M h o x w (t).
(1)
where
M~VD(s) = (e2/s)4(s)4'(s), MoR~+S(s)=
M,~H)(t) = (e2/t)4(t)4(t),
[e2/(s - M~)][fv4(S) + fAA.(s)][fv4(s) + fAA'.(s)].
M<~'(,) = [ e 2 / ( , - MZ)] [ / v 4 ( , ) + : A A . ( t ) ] [ f v 4 ' ( , ) + / A A , ' . ( , ) ] . , , b,,~ ,s)=(2a~-/s){4~(s)J~(s)[V~V(s)+ ZwiV2V(s)] +A~.(s)A~(s)[A{(s)+ 2~iA~(s)]}, M Q b,,. ED (, 't , i= ( 2 a 2 / t ) { J , ( t ) J ; ( t )~[ V l V ( t ) + 2wiV2v(t)] +A,,(t)A.(t)[A'{(t)+ 2wiA~(t)]}. Mb~,~S(s) = (a/2w){ M(pES(s)[V,Z(s) + 2wiVf(s)] + MfES(s)[AZ(s) + 2wiAZ(s)] }. M~,{w~(,) = (,~/2~,){ M2(t)[ V#(,) + 2~iEz(¢)] + M2(¢)[A~(,) + 2~iAZ(,)] }, MQED/
,
(2)
with
4.(s)=9(k2)y,u(k,), Jff(s)=fi(q,)y.v(q:). A.(s)=9(k2)y~ysu(k,), A;~(s)=fi(ql)y, ysv(q2). J,.(t)=fa(q,)Y, utk,), J~'(t)=V(k2)y.t,(q2), A,(t)=fi(q,)yuVsu(k]), A'~(t)=?(k2)T.Vsv(q2). 98
(3)
Volume 177, number 1
PHYSICS LETTERS B
4 September 1986
and f v = (4 sin20w - 1 ) / 4 sin 0w cos 0w,
)CA= -- ¼sin 0w COS 0W,
0,, being the weak mixing angle. Moreover the following notations are used:
s = ( k l+k2) 2=4E 2, z=cos0,
t=(k,-q,)2=-s½(1-cos0),
a=sin½0,
b=cos
!O 2 ,
fie=
u=(k 1-q2) 2=-s½(l+cos0),
(2a/~r)[ln(s/m 2) --
1],
flint = ( 4 a / ~ r )
ln(a/b).
The matrix elements MsnES(s) and MW(t) are defined as M5REs'w = MoRES'W(7 --+ ~, 75 7 75 ~ 7 ) 2 # 2 g ' P" # " The weak boson is taken as a resonance of mass M and width F, with M R = M - iMF and phase shift 8R(S ) where tg 8R(s)= MF/( M 2- s). The radiative factors 8 in eq. (1) are defined as follows (see fig. 1):
8eED(x)=ZS,(X)+8~(X)
( X = S , t),
(4)
with the vertex and vacuum polarization parts given by (X is the p h o t o n mass) 8v(S ) -
6if(s) + iSvI(S) =
(a/2~r)[½1nZ(s/m 2) - ln(s/m2)] + ~fl~ + (a/vr)(½cr 2- ~)} + i ( a / ~ r ) [ r r l n ( Z E / ) t ) - ~ v ] ,
{ - ½/~c l n ( Z E / A ) +
8v(t ) = -½fl~ l n ( 2 E / ? t ) - ( 2 a / ~ r ) I n a ln(ZE/~.) +
(5)
(a/Z~r)[½1nZ(s/m 2) -ln(s/rnZ)]
3 + gt3e + ( a / w ) ( ~ l n a - ln2a) + ( a / f r ) ( 112~r2 - }),
and 8 , ( s ) --- 8 ~ ( s ) + i6~(s) + ~
i=Ed'Q2[ln(s/m2) ,q
- 2] + i - -5 i=¢,,q
(6)
O/
(~,~(t)=-5~~
~2 i=
QZ[ln(-t/m~)-})],
¢',q
with Q2 = 1 ,
Q~ = 4j ( u p ) ,
Q2 = ~(down).
The y Z box diagrams contributions Mbo RES W ) can be casted in a very compact formula, ~ (s) and Muox(t which holds generally for every s and t, and in the limit M 2 --+ )t2 reduces to the known results for the "fy QED QED box diagrams Mbox (s) and Mbo× (t). One has [9]
MRES, box tS) =--2a2{ [f(s, t,
U) - f(s, U, t)][fv4(S)+fAA,(s)][fvJ/,(s)+fAA'~(s)]
+ [f(s, ,, .)+f(s,., t)]
(7)
where
f(s, t, u ) = ( M ~ . - s ) + [sp(1 + +[(u-t-
'{[ln[(ut)'/:/A 2] + I n ( I - s / M 2 ) 2] In(u/t) u/MZ)-sp(1 + t/M2)]} MR2 ) / u 2] {ln(1 --
+u-'{(M~/s-
1) ln(1 -
s/M 2)
ln(-t/s)
+ sp(1
s/M 2) + ln(-t/M~)},
+
t/M~) -
sp(1 -
s/M~)} (8) 99
Volume 177, number 1
PHYSICS LETTERS B
4 September 1986
with sp(x)
= - f0
ln(l
- t),
a n d s =- s + i~. T h e n for MR2 ~ X2 one recovers the 7"/ box d i a g r a m s results [10]
f ( s , t, u ) - f ( s , f ( s , t, u ) + f ( s , with
u, t ) ~ - s a[I~qV(s)+ 2~iVf(s)], u, t ) ~ - s - a [ A ~ ( s ) + 2 7 r i A ~ ( s ) ] ,
(9)
,2
VlV(S)=-81n(a/b)ln(2E/?t)-z(b
-4 l n 2 a + a
4 ln2b)+b-2
In a - a
-2 In b
=- - 8 In(a/b) l n ( 2 E / ) ~ ) + Vl7(S), Vf(s ) = 2 ln( a/b ) - ½z( b -4 In a + a -4 In b) - z/(1 - z2), A~(s)=-z(b
-4 l n 2 a - a
A~(s)=-½z(b
41n2b)+b-2
-4In a - a
In a + a
2 In
(10)
b,
4in b)+l/(1-z2).
O n the other hand, with the definition of m b R~s o x ( S ) given in eq. (2), one o b t a i n s v l Z ( s ) + 2~riv2Z(s) =
( M~-s)[f(s,
t, u) - f ( s ,
u, t)]
=-4 ln(b/a)In[( M~-s)Z/X2s] + v,Z(s) + 2~riv2Z(s),
A((s) + 2~riAZ(s)=( M ~ - s ) [ f ( s , t, u) + f(s, u,
(11)
t)].
The expressions for the t-channel box d i a g r a m "*box A/IQEDtt) a n d Mbox(t w t ) can be easily o b t a i n e d by a p p l y i n g the crossing relation s ~ t in eq. (7), or equivalently in eqs. (9) a n d (11), with M~ ~ M 2 a n d t --, t + ie, when necessary. One recovers then the 3'3' results [6] ,3
V~V(t) = 8 In b l n ( 2 E / X ) + 8 In a In b + ~7r2(1 - b 4) 21_ [(1 - b a ) / b 4] ln2a + (1 -- b 4 ) In 2 ( a / b )
~- 8 l n b l n ( 2 E / ~ ) +
A~(t) =
- k~(a
a 2 In(
a/b )
V(f(t),
- b 4) + [(1 - b 4 ) / b 4] ln~a - (1 - b 4) l n ~ ( a / b )
Vf(t) = 2 ln(2E/~)
A~(t) = - ½{[(1
+ (a 2/b2 ) In a +
-
+ ½{4 In a - [(1 6 4 ) / 6 4 ] In
+ ( a ~ / b 2) In a - a ~ l n ( a / b ) ,
(12)
b4)/b 4] In a + a2/2b 2 } - 2 l n ( 2 E / ~ ) + Vff(t),
a + a2/2b2},
and, similarly to eq. (1l),
vjZ(t) + Z ~ r i V f ( t ) = ( M 2 - t ) [ f ( t , s, u ) - f ( t , u, s)] =- 4 ln(~/~/?~)(2 In b + iTr) +
AZl (t) + ZTriAZz(t)=( M 2 - t ) [ f ( t ,
vlZ(t) + 2~riV2z ( t )
(13)
s, u) + f ( t , u, s ) ] .
:~2 The functions ~V(s) and A~(s) differ from the definition adopted in ref. [6] by a factor 2~. ,3 As for V~'(s) and AV2(s),notice the different definition of Vf(t) and A~(t), with respect of ref. [6], by a factor 2~v. Notice also a sign misprint in ref. [6], in the expressions of V~(t) and A'~(t). 100
Volume 177, number 1
PHYSICS LETTERSB
4 September 1986
Let us briefly comment the above formulae. The s-channel ,/Z box diagrams have been first calculated [1] in the approximation s - M 2, where
(2a/~r)M~ES(s){ln(b/a)ln[(M 2 - s)2/M~X 2] + VZ(s), AZ(s), AZ(s) are of order (s - M2).
MbRo~S(s) =
½sp(a 2 ) - ½sp(b 2 ) - l n 2 a + ln2b},
(14)
and Similarly the ,{Z box diagrams in the t channel have been previously computed [6] in the same soft limit, namely not including those contributions which vanish in the limit of k---, 0, and are of the same order of magnitude of the left-over weak corrections. This amounts to approximate as
MWx(t)
W Mbox(t )=
MW(t)(a/Tr)(ln[(-t + M2)Z/MZ?t2](i~r+ 2 In b) + 2 l n 2 b - s p ( b 2) - 4 In a In b + ~Tr2 - sp((s + M2)/M2)}.
(15)
It is clear that the exact expression for the ~,Z box diagrams, given in the simple forms of eqs. (8), (11), (13), allows for the complete evaluation of all EM contributions to the process under consideration. A similar attempt has been made in ref. [7]. Although the box contributions have been put in closed form in terms of two functions, analogous to the non-infrared parts of ~ and A,, their analytical expressions are rather cumbersome and include many simple and double logarithms of ( s - M2). This concludes the discussion of the virtual contributions to one-loop corrections. The analysis of the photon emission contributions follows closely that of ref. [6], as far as soft effects are concerned. Let us first define operatively the energy resolution A¢0, in an experiment where the electron-positron pair is detected almost back-to-back. For a given acollinearity angle J, the maximum energy kma x taken by undetected soft photons, which defines the energy resolution Ao~, is given by
(a/~r)
kma×-&o=[V~/(l+cosJ)](-(1-cosJ)+2[(1-cosJ)½-(m2/s)(l+cosJ)]l/2}. Then for J = 1°, 3 ° and 5 °, one obtains A -= Ao~/E= (1.7)%, (5.1)% and (8.3)%, respectively.
(16)
The first order bremsstrahlung contributions, in the soft-photon approximation, can be grouped, following ref. [1] in three different classes: "QED-like" terms, pure resonant terms and interference with the resonating amplitude. To this aim it is useful to define the various lowest-order cross section as follows: d o 0 [ y ( s ), V(s)] = (c~2/4s)(1 + z 2) - d o 0 ( a ) ,
(17a)
do0 [ v ( s ) , V(t)] = - (~2/4s)2(1 + z)2/(1 - z) --- do(2),
(17b)
d%iT(t), 7(t)]
=(a2/4s)[2/(1-z)2][(1 +z)2+41-do0(3 ),
(17c)
(a2/4s)2R'(t)(1 + z ) 2 ( f 2 + f 2 ) _ doo(4), =(a2/4s)[2/(1-z)]2R'(t)[(fd+f~)(l+z)2+4(fv-f~)]-
(17d)
doo[~,(s), Z ( t ) ] = doo[V(t), Z ( t ) ] d o 0 [ Z ( t ), Z ( t ) ]
=(a2/4s)2R'2(t){(1 + z ) 2 [ ( f v2 +f2)2+4f2v/2 ] + 4 [ ( y d +y~)2--4f2vy2A]}=--d%(6),
doo[Z(s), v(s)l = (~2/4s)2R'(s)[fd(1 + z:) +f~2z] do0 [ Z ( s ) , y(t)] = - (,~2/4,)2R'(s)[(a
d%[Z(s),
Z(t)] =
~- d % ( 7 ) ,
+ z)~/(1 - z)] ( f v~ + f ~ ) --- do0(8),
-(a2/4s)R'(s)2R'(t)(1 + z ) 2 [ ( / v2 +/AZ)2+4fZv/2]-- d % ( 9 ) ,
do0(5),
(17e)
(17 0 (17g)
(17h) (17i) 101
Volume 177, number 1
doo[Z(s),
Z(s)l
4 September 1986
PHYSICS LETTERS B
= (~2/4s)[ R'2(s) + l'2(s)l( f2v + f2)2{l + [4f2f2A/(f2 + f2 ) 2 ] 2 Z } -
Rt(t)=ls/(m2-l),
+z
2
d%(10),
(17j)
R'(s)+iI'(s)=s/(s-m2).
(17k)
Then in terms of these elementary cross sections the bremsstrahlung terms read as 6
9
do(l~,) = 6QED(Iy) E d % ( i ) + 3im(l~,) Y', d % ( i ) + 3RES(ly) do0(10 ), i=1
(18)
i=7
with
(2~/v)B(m z) + (2a/~)F(a, b), Re({exp[i3R(S)]/cos 3R(S)}
3OED(lv) = (2/~e + 2flim ) ln(2E/X) + (2/~e + 2/3im) In A 3im(1Y) = (2Be + 2Bim) ln(2E/X) + (Bim + Be) in A + X ( ~ e -1- /~int)
ln{ a[1 +
(as/MF)
exp[i3R(,)] sin 3R(S)]
(19a)
})
- ( 2 ~ / ~ ) B ( m 2) + (2~/,,)f(a, b), 3R~S(lv) = (2fl¢ + 2/~int) ln(2E/?,) +/3~ In A -
(19b)
fl¢3(s, Aoo) cot
+ (fl~ + 2fl~m) In la{1 +
(as~Mr) exp[i3R(S)] -(2~/v)B(m 2) + (2a/v)F(a, b),
3R(S )
sin 3R(S ) -~1 (19c)
and B ( m 2)
3(s,
=
jq71 2 __
l n ( s / m 2)
+
lln2(s/m2),
F(a,
b)
=
A~0) = arctg qba + arctg q~b, Oa = (2a~0~S- + M 2 - -
2 ln2a + sp(b 2) - 2 ln2b - sp(a2), s)/MF,
(Pb = ( S --
M2)/MF.
(20)
In experiments where the electrons are observed as a single particle track, one obtains the final corrected cross section by simply adding the virtual and real corrections from eqs. (1) and (18) respectively, exponentiating the soft part as usual [1,11]. When however collinear hard radiation (k >__k~0) from the final particles is also detected, as in calorimetric-type experiments, one has to include further corrections, as explicitly indicated in ref. [8]. We will first consider the former case. Then, as in ref. [6] we obtain ,4 10
do, ot(e+e --* e+e ) = ~[~ doo(i)(Cg~ra + C~F"),
(21)
i=1
where [1]
C(i) = infra
(A)(2fl¢+2fl,m)
(i =
1 , " ..,6),
(22a)
C~.{?ra= ( A )(Be+&")[1/COS aR(S)] Re(exp[i3 R(s)]{ a { 1 + ( A s / M F ) exp[iaR(S )] sin 3R(S ) } - 1 }
× { A { k + ( M r / s ) exp[-i6R(S)]/sin3R(s)}-'} l~") c~,O, infra = a <
] k{1 q-
(as/MF)
exp[i3R(s)] sin 3R(S)}-ll/~"
(i=7,8,9),
la{a +
(22b)
(Mr/,)
× e x p [ - i a R ( s ) ] / s i n aR(S)} -'12"m' [1 -- flea(S, a~) cot aR(S)], ,4 Notice the rearrangement of the factors with respect to the analogous equation of ref. [6].
102
fl,
(22C)
Volume
177, number
PHYSICS
1
and the finite factors C!+!) *’ also include CF’ = :p, + (2a/a)( + j?z/(l
LETTERS
now the contributions
f7r2 - :> + (2cu/+o,
b) + 2%%)
&77’ - :) + (2a/a)(
(2a/71)(ir2
+ :)
+(+q{wf>
+ (09{%(J)
:ln
a - In* u) + (2a/7r)F(
+
a, b)
G(f)+&(t)]>
(4a/57)($ln
(23b)
a - ln2u) + (2a/77)F(a,
b) + 26,(t)
+ [(b4-l)/(b4+1)lA:(~)},
cp) = Gp, + (2a/r)(&7r2
-
+(a/277)[G(s) cc)=
neglected
(234
+6,R(S)+tS,(t)+((Y/2~)[T/IYf(S)+tA:(S)+ :p,--
of order (Y/T previously
1986
+ z’)] A:(s)},
cg) = ;p, + (2u/77)(
@=
4 September
B
i)
+
+K(s)
+p, - (2u/77)(kr2
(2u/7r)(+ln + fG(t)
(23c) a - ln2u) + (2a/77)F(a,
b) + Q(s)
+~:(~)I~
+ +) + (4u/7r)(+ln
(23d)
a - ln2u) + (2u/m)F(u,
6) + a7(t)
+ (u/271)
+(cu/2~){[(f:+f,Z)b4-(f:-fA2)]/[(f:+fAZ)h4+(f:-fA2)1} (23e) x [W) +80)17
x[v,:(t)+
v:(t)]
CF) = :p, - (2a/rr)(
&r’ + ;> + (4a/m)(
+4f:C]
+(ol/a)Af(t)(h4[(f:+~~)2 +(fv’
:1 n a - ln2u) + (2a/77)F(u,
-(f:-fA2)2)/(b4[(~~+fAZ)2+4f:~~] (23f)
-fAzJ2)>
CF) = s/3, + (2+7)(
$72 - :> + (2+r)F(u,
b) + 6%)
+ v:(s) + 277[(Z’(4/R’(41 pi+) cp=
b) + (~~/rr)V:(t)
+
[mP’b)l
- K_w]} + W277){
W)
+ b/274{ Y::(s)
[fA2(1 + 2’)
+f:2z]/[f:(l+z~)+~~2z]}{A:(s)+A~(s)+2~[z’(s~/~’~s~l[~~~.F~-~:~~~]}~ b) +8,(t) :p, + (2a/a)(&2 - :> + (2a / 77)( + 1n a - 1n2u) + (2cu/a)F(u, +(a/277)[5W
+ fG(s)
t-&(r)
ltcu[Z’(s)/R’(s)][V~(t)-
Cb9) = $/3, + (2c*/n)(&s”’ +(u/277)[
G(s)
+( r; +fAZ)*2+[
+&)I
Vz”(S)+AY,(f)-A:(S)+:],
- $) + (2a/a)( + G(t)
+~[z’(s)/R’(s)][~~(t) c$tO’ = :p, + (2a/?r)(f7r2
:ln
(23h)
a - ln*u) + (2a/n)F(
+X(s)
+M)]
- V?(J)
+A;([)
- :) + (2a/m)F(u, (f:
(2%)
-A:.(s)
(23i)
+ :I,
b) + (+)(vI:(s)
+fA2J2(1 + z’) + vxz]
a, b)
+ [4.Mx+
z’)
A:(s)).
(23j)
We consider now the case of calorimetric-type measurements, where collinear hard radiation (k > Aw) from the final particles is detected within a small cone of half opening angle S (8 -K 1). Then one has to add the following correction factor [8,12-141 to each term in the RHS of eq. (21), taken to first order in cy: P”(i)
= da,( i)(4cr/n)[
(ln( E/do)
*5 The vacuum polarization corrections e*(s) = e*/[lQ(s)] in MoQED(s).
- a) ln( E6/m) due
to S:(s)
can
- $ln( E/do) be resummed
by
+ :( z - $r2)]. introducing
[4] the
(24) running
coupling
constant
103
Volume 177, number 1
PHYSICS LETTERS B
4 September 1986
T h e n in agreement with the K i n o s h i t a - L e e N a u e n b e r g theorem on the mas singularities [15], the m - d e p e n d e n c e c o m i n g from the final e l e c t r o n - p o s i t r o n pair disappears after adding eq. (24) to eq. (21) and the overall correction factor to the Born cross sections can be simply o b t a i n e d from eq. (21), to first order in a, by the substitution /3e(ln A + 3) ~ ( 2 ~ / ~ r ) [ l n ( 4 / 8 2 ) ( l n
,3 + 3) + (3 _ ~ r 2 ) ] .
(25)
F r o m the k n o w n results on the e x p o n e n t i a t i o n of soft and collinear divergences ,6, one then obtains the final result, 10
dOtot(e+e
~ e+e
) = y" d o 0 ( i ) [ ( ~ r ~ + C~v"],
(26)
i-1
where infra
"~ infra
C(vi) = C F ' + 3 ( ! 3 8 - / ? e ) + ( 2 a / ~ r ) ( 3
~r12
),
with 13, = ( 4 a / v ) l n ( 2 / 8 ) . So far large-angle hard b r e m s s t r a h l u n g effects have not been considered. As long as the e l e c t r o n - p o s i t r o n pair is detected back-to-back with good collinearity, the accuracy of the formulae given above is of order ( a / ~ ' ) ( J , 8). H a r d p h o t o n effects have to be taken into account otherwise [16]. Finally, weak interactions have been only considered to renormalize the mass a n d the width of the vector boson. To conclude we have presented a complete analysis of EM radiative corrections to Bhabha scattering near the Z 0. Our results, in fully analytic form, include the exact c o n t r i b u t i o n s of one-loop diagrams and the whole series of double a n d simple logarithms from soft a n d collinear divergences in exponentiated form. The process of e+e scattering, with the electron-positron pair detected almost back-to-back, can be therefore used as a high precision m o n i t o r of l u m i n o s i t y at L E P / S L C energies. I a m grateful to M. Consoli and Y. Srivastava for discussions. ,6 See for example ref. [13] and references therein.
References [1] [2] [3] [4]
M. Greco, G. Pancheri and Y. Srivastava, Nucl. Phys. B 171 (1980) 118; B 197 (1982) 543(E). F.A. Berends, R. Kleiss and S. Jadach, Nucl. Phys. B 202 (1982) 63. M. BOhm and W: Hollik, Nucl. Phys. B 204 (1982) 45. G. Altarelli et al., Precision tests of the electroweak theory at the Z °, Physics at LEP, eds. J. Ellis and R. Peccei. CERN 86-02 (1986). [5] M. Consoli, Nucl. Phys. B 160 (1979) 208. [6] M. Consoli, S. Lo Presti and M. Greco, Phys. Lett. B 113 (1982) 415. ]7] R. Sommer, M. B0hm and W. Hollik, Wiirzburg preprint (1983). [8] M. Caffo, R. Gatto and E. Remiddi, Nucl. Phys. B 252 (1985) 378. [9] R.W. Brown, R. Decker and E.A. Paschos, Phys. Rev. Lett. 52 (1984) 1192; see also M. Consoli and A. Sirlin, Precision tests of the electroweak theory at the Z °, Physics at LEP, eds. J. Ellis and R. PecceL C E R N 86-02 (1986). [10] I.B. Khriplovich, Yad. Fis. 17 (1973) 298 [Sov. J. Nucl. Phys. 17 (1973) 576]; R.W. Brown, V.K. Cung, K.O. Mikaelian and E.A. Paschos, Phys. Lett. 43 B (1973) 403. [11] M. Greco, G. Pancheri and Y. Srivastava, Nucl. Phys. B 101 (1975) 234. [12] G. Sterman and S. Weinberg, Phys. Rev. Lett. 39 (1977) 1436. 104
Volume
177, number
1
PHYSICS
LETTERS
B
4 September
1986
[13] G. Curci and M. Greco, Phys. Lett. B 79 (1978) 406. [14] M. Greco, Precisioa tests of the electroweak theory at the Z ‘, Physics at LEP, eds. J. Ellis and R. Peccei, CERN 86-02 (1986). [15] T. Kinoshita, J. Math. Phys. 3 (1962) 650; T.D. Lee and M. Nauenberg, Phys. Rev. 133 (1964) 1549. [16] See for example, R. Kleiss, Precision tests of the electroweak theory at the Z’. Physics at LEP, eds. J. Ellis and R. Peccei, CERN 86-02 (1986).
105
177, number
BHABHA
PHYSICS
1
SCATTERING
LETTERS
B
4 September
1986
NEAR THE Z,
M. GRECO INFN,
Laboratori
Received
Nazionali di Frascati, I-00044 Frascati, Iialy
16 May 1986
A complete analysis of EM radiative corrections to Bhabha scattering near the Z, is presented Compact analytic formulae are given which include exact one-loop results and soft-and-collinear photon effects resummed to all orders. Detection of back-to-back e+e- pairs can be therefore used as a high precision monitor of luminosity at LEP/SLC energies.
e’e- annihilation at LEP/SLC energies will provide precision tests for the standard electroweak model only if QED radiative corrections are under control at the level of 5 1%. Indeed first-order corrections [l-3], for example, reduce the Z, peak cross section by more than 50%, or shift the zero in the forward-backward asymmetry by about (+300) MeV, for an energy resolution of (10-‘~10~2). It is therefore important that higher-order corrections are properly taken into account if the electroweak parameters have to be measured to the required accuracy *‘. Previous studies of these effects have been presented over the past few years, in particular for the process e + e- + ~_l+p~ [l-4]. The reaction e+e- -+ e-e-, on the other hand, is particularly interesting for its large cross section and could provide a high precision monitor of the beam luminosity. Detailed studies of electro-weak radiative corrections to this process, which have been performed earlier [5-g], are all incomplete in some respects. Indeed the calculation of electro-weak first order corrections, performed in ref. [5], does not extend to the energy range around the Z,, because of the lack of finite width effects. Those were included in ref. [6], together with the complete treatment of soft photon effects, resummed to all orders. The analytical expressions for the box diagrams in the s and t channels however, were only given in the limit s - M2, the left-over terms being of order ((Y/T). An attempt to improve these results has been made in ref. [7]. Finally a treatment of collinear hard photon effects, quite relevant in calorimetric-type experiments, has been given in ref. [8]. The aim of the present paper is to give a final and complete description of QED radiative effects for Bhabha scattering, including exact analytical expressions for all one-loop diagrams, and soft and collinear hard photon effects resummed to all orders. Therefore our results include all double logarithmic terms of the form (a/n) ln(s/m’) ln(A, T/M), (a/n) In S2 In A, simple logs as ((Y/T) ln( S/m2), (a/a) ln(A, T/M, S’), resummed to all orders, and all finite terms of orders (Y/T. In the above M and r are the mass and width of the Z, boson, and A and 6 the energy and angular resolutions of the experiment, better defined below. Hard photon effects of order (a/r)(A, S) have been neglected. Our considerations apply to a typical experiment in which the following requirements are satisfied: (i) The electron-positron pair should be detected back-to-back within a certain acollinearity angle J of a few degrees (J 5 5” ). The energy resolution Aw depends upon J (see eq. (16)). (ii) An electromagnetic calorimeter of finite and small angular resolution S is centred along the electron and positron directions. In principle it does not discriminate between a charged particle and the accompanying collinear photons. *’ For a recent review see ref. [4].
0370-2693/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
97
V o l u m e 177, n u m b e r 1
PHYSICS LETTERS B
(a)
(b)
K,
4 S e p t e m b e r 1986
/d)
(c)
T /
q,
MQED(s) o
MRES(s) o
--....
MW(tl
MQED(t) o
o
M~ED(s)XOv(S) MRoES(s)X6v(S) M~ED(L)x6v(t) Mwo(t)×fiv(t)
Z
M~ED(t)×d~(t)
tl'oQED s)xd~(s)
+cK
+CF,
MQED(s) pox
MRES(s) box
M~ED(t)
M~ox(t)
DO×
Fig. 1. Virtual g r a p h s in the s and t chalmel.
Then, using (i) and (ii), one would be sure that all but a fraction ._4-= A~o/E of the beam energy (~- = 2E) is taken by the electrons and the accompanying hard photons. For small A and 8, fully analytic expressions can be used, neglecting hard-photon effects of order [(a/~r)A (a/w)6)]. We give a brief account of the derivation of our formulae. Our notations are as in ref. [6]. unless stated explicitly. The relevant virtual graphs are shown in fig. 1. We have
M(s,
t) = M,QH)(s)[1 + 6QH)(s)] -- M,QED(t)[1 + 6QED(t)] + MRES(s)[1 + 26.,(S)] -M,W(t)[l+26.(t)]
AAcQED/ +MhoQED X (s)--l..bo~ , t ) + M , ~ S ( s ) - M h o x w (t).
(1)
where
M~VD(s) = (e2/s)4(s)4'(s), MoR~+S(s)=
M,~H)(t) = (e2/t)4(t)4(t),
[e2/(s - M~)][fv4(S) + fAA.(s)][fv4(s) + fAA'.(s)].
M<~'(,) = [ e 2 / ( , - MZ)] [ / v 4 ( , ) + : A A . ( t ) ] [ f v 4 ' ( , ) + / A A , ' . ( , ) ] . , , b,,~ ,s)=(2a~-/s){4~(s)J~(s)[V~V(s)+ ZwiV2V(s)] +A~.(s)A~(s)[A{(s)+ 2~iA~(s)]}, M Q b,,. ED (, 't , i= ( 2 a 2 / t ) { J , ( t ) J ; ( t )~[ V l V ( t ) + 2wiV2v(t)] +A,,(t)A.(t)[A'{(t)+ 2wiA~(t)]}. Mb~,~S(s) = (a/2w){ M(pES(s)[V,Z(s) + 2wiVf(s)] + MfES(s)[AZ(s) + 2wiAZ(s)] }. M~,{w~(,) = (,~/2~,){ M2(t)[ V#(,) + 2~iEz(¢)] + M2(¢)[A~(,) + 2~iAZ(,)] }, MQED/
,
(2)
with
4.(s)=9(k2)y,u(k,), Jff(s)=fi(q,)y.v(q:). A.(s)=9(k2)y~ysu(k,), A;~(s)=fi(ql)y, ysv(q2). J,.(t)=fa(q,)Y, utk,), J~'(t)=V(k2)y.t,(q2), A,(t)=fi(q,)yuVsu(k]), A'~(t)=?(k2)T.Vsv(q2). 98
(3)
Volume 177, number 1
PHYSICS LETTERS B
4 September 1986
and f v = (4 sin20w - 1 ) / 4 sin 0w cos 0w,
)CA= -- ¼sin 0w COS 0W,
0,, being the weak mixing angle. Moreover the following notations are used:
s = ( k l+k2) 2=4E 2, z=cos0,
t=(k,-q,)2=-s½(1-cos0),
a=sin½0,
b=cos
!O 2 ,
fie=
u=(k 1-q2) 2=-s½(l+cos0),
(2a/~r)[ln(s/m 2) --
1],
flint = ( 4 a / ~ r )
ln(a/b).
The matrix elements MsnES(s) and MW(t) are defined as M5REs'w = MoRES'W(7 --+ ~, 75 7 75 ~ 7 ) 2 # 2 g ' P" # " The weak boson is taken as a resonance of mass M and width F, with M R = M - iMF and phase shift 8R(S ) where tg 8R(s)= MF/( M 2- s). The radiative factors 8 in eq. (1) are defined as follows (see fig. 1):
8eED(x)=ZS,(X)+8~(X)
( X = S , t),
(4)
with the vertex and vacuum polarization parts given by (X is the p h o t o n mass) 8v(S ) -
6if(s) + iSvI(S) =
(a/2~r)[½1nZ(s/m 2) - ln(s/m2)] + ~fl~ + (a/vr)(½cr 2- ~)} + i ( a / ~ r ) [ r r l n ( Z E / ) t ) - ~ v ] ,
{ - ½/~c l n ( Z E / A ) +
8v(t ) = -½fl~ l n ( 2 E / ? t ) - ( 2 a / ~ r ) I n a ln(ZE/~.) +
(5)
(a/Z~r)[½1nZ(s/m 2) -ln(s/rnZ)]
3 + gt3e + ( a / w ) ( ~ l n a - ln2a) + ( a / f r ) ( 112~r2 - }),
and 8 , ( s ) --- 8 ~ ( s ) + i6~(s) + ~
i=Ed'Q2[ln(s/m2) ,q
- 2] + i - -5 i=¢,,q
(6)
O/
(~,~(t)=-5~~
~2 i=
QZ[ln(-t/m~)-})],
¢',q
with Q2 = 1 ,
Q~ = 4j ( u p ) ,
Q2 = ~(down).
The y Z box diagrams contributions Mbo RES W ) can be casted in a very compact formula, ~ (s) and Muox(t which holds generally for every s and t, and in the limit M 2 --+ )t2 reduces to the known results for the "fy QED QED box diagrams Mbox (s) and Mbo× (t). One has [9]
MRES, box tS) =--2a2{ [f(s, t,
U) - f(s, U, t)][fv4(S)+fAA,(s)][fvJ/,(s)+fAA'~(s)]
+ [f(s, ,, .)+f(s,., t)]
(7)
where
f(s, t, u ) = ( M ~ . - s ) + [sp(1 + +[(u-t-
'{[ln[(ut)'/:/A 2] + I n ( I - s / M 2 ) 2] In(u/t) u/MZ)-sp(1 + t/M2)]} MR2 ) / u 2] {ln(1 --
+u-'{(M~/s-
1) ln(1 -
s/M 2)
ln(-t/s)
+ sp(1
s/M 2) + ln(-t/M~)},
+
t/M~) -
sp(1 -
s/M~)} (8) 99
Volume 177, number 1
PHYSICS LETTERS B
4 September 1986
with sp(x)
= - f0
ln(l
- t),
a n d s =- s + i~. T h e n for MR2 ~ X2 one recovers the 7"/ box d i a g r a m s results [10]
f ( s , t, u ) - f ( s , f ( s , t, u ) + f ( s , with
u, t ) ~ - s a[I~qV(s)+ 2~iVf(s)], u, t ) ~ - s - a [ A ~ ( s ) + 2 7 r i A ~ ( s ) ] ,
(9)
,2
VlV(S)=-81n(a/b)ln(2E/?t)-z(b
-4 l n 2 a + a
4 ln2b)+b-2
In a - a
-2 In b
=- - 8 In(a/b) l n ( 2 E / ) ~ ) + Vl7(S), Vf(s ) = 2 ln( a/b ) - ½z( b -4 In a + a -4 In b) - z/(1 - z2), A~(s)=-z(b
-4 l n 2 a - a
A~(s)=-½z(b
41n2b)+b-2
-4In a - a
In a + a
2 In
(10)
b,
4in b)+l/(1-z2).
O n the other hand, with the definition of m b R~s o x ( S ) given in eq. (2), one o b t a i n s v l Z ( s ) + 2~riv2Z(s) =
( M~-s)[f(s,
t, u) - f ( s ,
u, t)]
=-4 ln(b/a)In[( M~-s)Z/X2s] + v,Z(s) + 2~riv2Z(s),
A((s) + 2~riAZ(s)=( M ~ - s ) [ f ( s , t, u) + f(s, u,
(11)
t)].
The expressions for the t-channel box d i a g r a m "*box A/IQEDtt) a n d Mbox(t w t ) can be easily o b t a i n e d by a p p l y i n g the crossing relation s ~ t in eq. (7), or equivalently in eqs. (9) a n d (11), with M~ ~ M 2 a n d t --, t + ie, when necessary. One recovers then the 3'3' results [6] ,3
V~V(t) = 8 In b l n ( 2 E / X ) + 8 In a In b + ~7r2(1 - b 4) 21_ [(1 - b a ) / b 4] ln2a + (1 -- b 4 ) In 2 ( a / b )
~- 8 l n b l n ( 2 E / ~ ) +
A~(t) =
- k~(a
a 2 In(
a/b )
V(f(t),
- b 4) + [(1 - b 4 ) / b 4] ln~a - (1 - b 4) l n ~ ( a / b )
Vf(t) = 2 ln(2E/~)
A~(t) = - ½{[(1
+ (a 2/b2 ) In a +
-
+ ½{4 In a - [(1 6 4 ) / 6 4 ] In
+ ( a ~ / b 2) In a - a ~ l n ( a / b ) ,
(12)
b4)/b 4] In a + a2/2b 2 } - 2 l n ( 2 E / ~ ) + Vff(t),
a + a2/2b2},
and, similarly to eq. (1l),
vjZ(t) + Z ~ r i V f ( t ) = ( M 2 - t ) [ f ( t , s, u ) - f ( t , u, s)] =- 4 ln(~/~/?~)(2 In b + iTr) +
AZl (t) + ZTriAZz(t)=( M 2 - t ) [ f ( t ,
vlZ(t) + 2~riV2z ( t )
(13)
s, u) + f ( t , u, s ) ] .
:~2 The functions ~V(s) and A~(s) differ from the definition adopted in ref. [6] by a factor 2~. ,3 As for V~'(s) and AV2(s),notice the different definition of Vf(t) and A~(t), with respect of ref. [6], by a factor 2~v. Notice also a sign misprint in ref. [6], in the expressions of V~(t) and A'~(t). 100
Volume 177, number 1
PHYSICS LETTERSB
4 September 1986
Let us briefly comment the above formulae. The s-channel ,/Z box diagrams have been first calculated [1] in the approximation s - M 2, where
(2a/~r)M~ES(s){ln(b/a)ln[(M 2 - s)2/M~X 2] + VZ(s), AZ(s), AZ(s) are of order (s - M2).
MbRo~S(s) =
½sp(a 2 ) - ½sp(b 2 ) - l n 2 a + ln2b},
(14)
and Similarly the ,{Z box diagrams in the t channel have been previously computed [6] in the same soft limit, namely not including those contributions which vanish in the limit of k---, 0, and are of the same order of magnitude of the left-over weak corrections. This amounts to approximate as
MWx(t)
W Mbox(t )=
MW(t)(a/Tr)(ln[(-t + M2)Z/MZ?t2](i~r+ 2 In b) + 2 l n 2 b - s p ( b 2) - 4 In a In b + ~Tr2 - sp((s + M2)/M2)}.
(15)
It is clear that the exact expression for the ~,Z box diagrams, given in the simple forms of eqs. (8), (11), (13), allows for the complete evaluation of all EM contributions to the process under consideration. A similar attempt has been made in ref. [7]. Although the box contributions have been put in closed form in terms of two functions, analogous to the non-infrared parts of ~ and A,, their analytical expressions are rather cumbersome and include many simple and double logarithms of ( s - M2). This concludes the discussion of the virtual contributions to one-loop corrections. The analysis of the photon emission contributions follows closely that of ref. [6], as far as soft effects are concerned. Let us first define operatively the energy resolution A¢0, in an experiment where the electron-positron pair is detected almost back-to-back. For a given acollinearity angle J, the maximum energy kma x taken by undetected soft photons, which defines the energy resolution Ao~, is given by
(a/~r)
kma×-&o=[V~/(l+cosJ)](-(1-cosJ)+2[(1-cosJ)½-(m2/s)(l+cosJ)]l/2}. Then for J = 1°, 3 ° and 5 °, one obtains A -= Ao~/E= (1.7)%, (5.1)% and (8.3)%, respectively.
(16)
The first order bremsstrahlung contributions, in the soft-photon approximation, can be grouped, following ref. [1] in three different classes: "QED-like" terms, pure resonant terms and interference with the resonating amplitude. To this aim it is useful to define the various lowest-order cross section as follows: d o 0 [ y ( s ), V(s)] = (c~2/4s)(1 + z 2) - d o 0 ( a ) ,
(17a)
do0 [ v ( s ) , V(t)] = - (~2/4s)2(1 + z)2/(1 - z) --- do(2),
(17b)
d%iT(t), 7(t)]
=(a2/4s)[2/(1-z)2][(1 +z)2+41-do0(3 ),
(17c)
(a2/4s)2R'(t)(1 + z ) 2 ( f 2 + f 2 ) _ doo(4), =(a2/4s)[2/(1-z)]2R'(t)[(fd+f~)(l+z)2+4(fv-f~)]-
(17d)
doo[~,(s), Z ( t ) ] = doo[V(t), Z ( t ) ] d o 0 [ Z ( t ), Z ( t ) ]
=(a2/4s)2R'2(t){(1 + z ) 2 [ ( f v2 +f2)2+4f2v/2 ] + 4 [ ( y d +y~)2--4f2vy2A]}=--d%(6),
doo[Z(s), v(s)l = (~2/4s)2R'(s)[fd(1 + z:) +f~2z] do0 [ Z ( s ) , y(t)] = - (,~2/4,)2R'(s)[(a
d%[Z(s),
Z(t)] =
~- d % ( 7 ) ,
+ z)~/(1 - z)] ( f v~ + f ~ ) --- do0(8),
-(a2/4s)R'(s)2R'(t)(1 + z ) 2 [ ( / v2 +/AZ)2+4fZv/2]-- d % ( 9 ) ,
do0(5),
(17e)
(17 0 (17g)
(17h) (17i) 101
Volume 177, number 1
doo[Z(s),
Z(s)l
4 September 1986
PHYSICS LETTERS B
= (~2/4s)[ R'2(s) + l'2(s)l( f2v + f2)2{l + [4f2f2A/(f2 + f2 ) 2 ] 2 Z } -
Rt(t)=ls/(m2-l),
+z
2
d%(10),
(17j)
R'(s)+iI'(s)=s/(s-m2).
(17k)
Then in terms of these elementary cross sections the bremsstrahlung terms read as 6
9
do(l~,) = 6QED(Iy) E d % ( i ) + 3im(l~,) Y', d % ( i ) + 3RES(ly) do0(10 ), i=1
(18)
i=7
with
(2~/v)B(m z) + (2a/~)F(a, b), Re({exp[i3R(S)]/cos 3R(S)}
3OED(lv) = (2/~e + 2flim ) ln(2E/X) + (2/~e + 2/3im) In A 3im(1Y) = (2Be + 2Bim) ln(2E/X) + (Bim + Be) in A + X ( ~ e -1- /~int)
ln{ a[1 +
(as/MF)
exp[i3R(,)] sin 3R(S)]
(19a)
})
- ( 2 ~ / ~ ) B ( m 2) + (2~/,,)f(a, b), 3R~S(lv) = (2fl¢ + 2/~int) ln(2E/?,) +/3~ In A -
(19b)
fl¢3(s, Aoo) cot
+ (fl~ + 2fl~m) In la{1 +
(as~Mr) exp[i3R(S)] -(2~/v)B(m 2) + (2a/v)F(a, b),
3R(S )
sin 3R(S ) -~1 (19c)
and B ( m 2)
3(s,
=
jq71 2 __
l n ( s / m 2)
+
lln2(s/m2),
F(a,
b)
=
A~0) = arctg qba + arctg q~b, Oa = (2a~0~S- + M 2 - -
2 ln2a + sp(b 2) - 2 ln2b - sp(a2), s)/MF,
(Pb = ( S --
M2)/MF.
(20)
In experiments where the electrons are observed as a single particle track, one obtains the final corrected cross section by simply adding the virtual and real corrections from eqs. (1) and (18) respectively, exponentiating the soft part as usual [1,11]. When however collinear hard radiation (k >__k~0) from the final particles is also detected, as in calorimetric-type experiments, one has to include further corrections, as explicitly indicated in ref. [8]. We will first consider the former case. Then, as in ref. [6] we obtain ,4 10
do, ot(e+e --* e+e ) = ~[~ doo(i)(Cg~ra + C~F"),
(21)
i=1
where [1]
C(i) = infra
(A)(2fl¢+2fl,m)
(i =
1 , " ..,6),
(22a)
C~.{?ra= ( A )(Be+&")[1/COS aR(S)] Re(exp[i3 R(s)]{ a { 1 + ( A s / M F ) exp[iaR(S )] sin 3R(S ) } - 1 }
× { A { k + ( M r / s ) exp[-i6R(S)]/sin3R(s)}-'} l~") c~,O, infra = a <
] k{1 q-
(as/MF)
exp[i3R(s)] sin 3R(S)}-ll/~"
(i=7,8,9),
la{a +
(22b)
(Mr/,)
× e x p [ - i a R ( s ) ] / s i n aR(S)} -'12"m' [1 -- flea(S, a~) cot aR(S)], ,4 Notice the rearrangement of the factors with respect to the analogous equation of ref. [6].
102
fl,
(22C)
Volume
177, number
PHYSICS
1
and the finite factors C!+!) *’ also include CF’ = :p, + (2a/a)( + j?z/(l
LETTERS
now the contributions
f7r2 - :> + (2cu/+o,
b) + 2%%)
&77’ - :) + (2a/a)(
(2a/71)(ir2
+ :)
+(+q{wf>
+ (09{%(J)
:ln
a - In* u) + (2a/7r)F(
+
a, b)
G(f)+&(t)]>
(4a/57)($ln
(23b)
a - ln2u) + (2a/77)F(a,
b) + 26,(t)
+ [(b4-l)/(b4+1)lA:(~)},
cp) = Gp, + (2a/r)(&7r2
-
+(a/277)[G(s) cc)=
neglected
(234
+6,R(S)+tS,(t)+((Y/2~)[T/IYf(S)+tA:(S)+ :p,--
of order (Y/T previously
1986
+ z’)] A:(s)},
cg) = ;p, + (2u/77)(
@=
4 September
B
i)
+
+K(s)
+p, - (2u/77)(kr2
(2u/7r)(+ln + fG(t)
(23c) a - ln2u) + (2a/77)F(a,
b) + Q(s)
+~:(~)I~
+ +) + (4u/7r)(+ln
(23d)
a - ln2u) + (2u/m)F(u,
6) + a7(t)
+ (u/271)
+(cu/2~){[(f:+f,Z)b4-(f:-fA2)]/[(f:+fAZ)h4+(f:-fA2)1} (23e) x [W) +80)17
x[v,:(t)+
v:(t)]
CF) = :p, - (2a/rr)(
&r’ + ;> + (4a/m)(
+4f:C]
+(ol/a)Af(t)(h4[(f:+~~)2 +(fv’
:1 n a - ln2u) + (2a/77)F(u,
-(f:-fA2)2)/(b4[(~~+fAZ)2+4f:~~] (23f)
-fAzJ2)>
CF) = s/3, + (2+7)(
$72 - :> + (2+r)F(u,
b) + 6%)
+ v:(s) + 277[(Z’(4/R’(41 pi+) cp=
b) + (~~/rr)V:(t)
+
[mP’b)l
- K_w]} + W277){
W)
+ b/274{ Y::(s)
[fA2(1 + 2’)
+f:2z]/[f:(l+z~)+~~2z]}{A:(s)+A~(s)+2~[z’(s~/~’~s~l[~~~.F~-~:~~~]}~ b) +8,(t) :p, + (2a/a)(&2 - :> + (2a / 77)( + 1n a - 1n2u) + (2cu/a)F(u, +(a/277)[5W
+ fG(s)
t-&(r)
ltcu[Z’(s)/R’(s)][V~(t)-
Cb9) = $/3, + (2c*/n)(&s”’ +(u/277)[
G(s)
+( r; +fAZ)*2+[
+&)I
Vz”(S)+AY,(f)-A:(S)+:],
- $) + (2a/a)( + G(t)
+~[z’(s)/R’(s)][~~(t) c$tO’ = :p, + (2a/?r)(f7r2
:ln
(23h)
a - ln*u) + (2a/n)F(
+X(s)
+M)]
- V?(J)
+A;([)
- :) + (2a/m)F(u, (f:
(2%)
-A:.(s)
(23i)
+ :I,
b) + (+)(vI:(s)
+fA2J2(1 + z’) + vxz]
a, b)
+ [4.Mx+
z’)
A:(s)).
(23j)
We consider now the case of calorimetric-type measurements, where collinear hard radiation (k > Aw) from the final particles is detected within a small cone of half opening angle S (8 -K 1). Then one has to add the following correction factor [8,12-141 to each term in the RHS of eq. (21), taken to first order in cy: P”(i)
= da,( i)(4cr/n)[
(ln( E/do)
*5 The vacuum polarization corrections e*(s) = e*/[lQ(s)] in MoQED(s).
- a) ln( E6/m) due
to S:(s)
can
- $ln( E/do) be resummed
by
+ :( z - $r2)]. introducing
[4] the
(24) running
coupling
constant
103
Volume 177, number 1
PHYSICS LETTERS B
4 September 1986
T h e n in agreement with the K i n o s h i t a - L e e N a u e n b e r g theorem on the mas singularities [15], the m - d e p e n d e n c e c o m i n g from the final e l e c t r o n - p o s i t r o n pair disappears after adding eq. (24) to eq. (21) and the overall correction factor to the Born cross sections can be simply o b t a i n e d from eq. (21), to first order in a, by the substitution /3e(ln A + 3) ~ ( 2 ~ / ~ r ) [ l n ( 4 / 8 2 ) ( l n
,3 + 3) + (3 _ ~ r 2 ) ] .
(25)
F r o m the k n o w n results on the e x p o n e n t i a t i o n of soft and collinear divergences ,6, one then obtains the final result, 10
dOtot(e+e
~ e+e
) = y" d o 0 ( i ) [ ( ~ r ~ + C~v"],
(26)
i-1
where infra
"~ infra
C(vi) = C F ' + 3 ( ! 3 8 - / ? e ) + ( 2 a / ~ r ) ( 3
~r12
),
with 13, = ( 4 a / v ) l n ( 2 / 8 ) . So far large-angle hard b r e m s s t r a h l u n g effects have not been considered. As long as the e l e c t r o n - p o s i t r o n pair is detected back-to-back with good collinearity, the accuracy of the formulae given above is of order ( a / ~ ' ) ( J , 8). H a r d p h o t o n effects have to be taken into account otherwise [16]. Finally, weak interactions have been only considered to renormalize the mass a n d the width of the vector boson. To conclude we have presented a complete analysis of EM radiative corrections to Bhabha scattering near the Z 0. Our results, in fully analytic form, include the exact c o n t r i b u t i o n s of one-loop diagrams and the whole series of double a n d simple logarithms from soft a n d collinear divergences in exponentiated form. The process of e+e scattering, with the electron-positron pair detected almost back-to-back, can be therefore used as a high precision m o n i t o r of l u m i n o s i t y at L E P / S L C energies. I a m grateful to M. Consoli and Y. Srivastava for discussions. ,6 See for example ref. [13] and references therein.
References [1] [2] [3] [4]
M. Greco, G. Pancheri and Y. Srivastava, Nucl. Phys. B 171 (1980) 118; B 197 (1982) 543(E). F.A. Berends, R. Kleiss and S. Jadach, Nucl. Phys. B 202 (1982) 63. M. BOhm and W: Hollik, Nucl. Phys. B 204 (1982) 45. G. Altarelli et al., Precision tests of the electroweak theory at the Z °, Physics at LEP, eds. J. Ellis and R. Peccei. CERN 86-02 (1986). [5] M. Consoli, Nucl. Phys. B 160 (1979) 208. [6] M. Consoli, S. Lo Presti and M. Greco, Phys. Lett. B 113 (1982) 415. ]7] R. Sommer, M. B0hm and W. Hollik, Wiirzburg preprint (1983). [8] M. Caffo, R. Gatto and E. Remiddi, Nucl. Phys. B 252 (1985) 378. [9] R.W. Brown, R. Decker and E.A. Paschos, Phys. Rev. Lett. 52 (1984) 1192; see also M. Consoli and A. Sirlin, Precision tests of the electroweak theory at the Z °, Physics at LEP, eds. J. Ellis and R. PecceL C E R N 86-02 (1986). [10] I.B. Khriplovich, Yad. Fis. 17 (1973) 298 [Sov. J. Nucl. Phys. 17 (1973) 576]; R.W. Brown, V.K. Cung, K.O. Mikaelian and E.A. Paschos, Phys. Lett. 43 B (1973) 403. [11] M. Greco, G. Pancheri and Y. Srivastava, Nucl. Phys. B 101 (1975) 234. [12] G. Sterman and S. Weinberg, Phys. Rev. Lett. 39 (1977) 1436. 104
Volume
177, number
1
PHYSICS
LETTERS
B
4 September
1986
[13] G. Curci and M. Greco, Phys. Lett. B 79 (1978) 406. [14] M. Greco, Precisioa tests of the electroweak theory at the Z ‘, Physics at LEP, eds. J. Ellis and R. Peccei, CERN 86-02 (1986). [15] T. Kinoshita, J. Math. Phys. 3 (1962) 650; T.D. Lee and M. Nauenberg, Phys. Rev. 133 (1964) 1549. [16] See for example, R. Kleiss, Precision tests of the electroweak theory at the Z’. Physics at LEP, eds. J. Ellis and R. Peccei, CERN 86-02 (1986).
105