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Coherent states and structure functions in QED
Volume 228, number 1
PHYSICS LETTERS B
7 September 1989
C O H E R E N T STATES AND STRUCTURE FUNCTIONS IN QED F. AVERSA and M. GRECO 1NFN, Laboratori Nazionali di Frascati, C.P. 13, 1-00044 Frascati, Italy Received 26 May 1989
The methods of coherent states and of structure functions in QED are considered in detail for a precise evaluation of the radiative effects at LEP/SLC. They are explicitly shown to give identical results for the exponentiated infrared factors and the o ( a ) terms corresponding to the initial and final state radiation. Then the combined use of both techniques further improves the theoretical predictions to an accuracy better than 1% - including interference effects - and provides simple analytical formulae of immediate phenomenological application.
The role played by QED radiative corrections in e+e - reactions at LEP/SLC energies for the precise determination of electroweak parameters is well known ~. The resummation of the leading double and single logarithms of soft origin [ 3 ], together with the exact determination of the left-over o ( a ) terms [ 4, 5 ], is essential to reach a degree of accuracy of o (1%) in the theoretical predictions. The calculation of all logarithmic o (o?) corrections, as recently done [6,7] for the reaction e+e - ~ t + ~ t -, is the further necessary tool to get absolute control of the QED corrections, as required for precision tests of the theory. To this aim the method of the structure functions [8], extended to both initial and final states [6 ] in the reaction e +e- --, ~t+~t-, has been shown to be quite powerful, suggesting a systematic approach to other processes as, for example, Bhabha scattering. However, only numerical solutions have been obtained so far in the case of a resonant cross section, leaving unclear the connection to other analytical approaches to the problem. The method of coherent states [ 9 ], for example, has been proved to be very successful in describing the main features of multiphoton emission with good accuracy, giving explicit analytical results for resonant processes and interference effects with pure QED background [ 3,4 ]. The determination of the left-over o ( a 2) logarithms is needed only for the evaluation of the radiative effects beyond the 1% level. A close connection between the two above methods of resummation has been explicitly shown [ 10 ] to exist in the non-singlet case for QED non-resonant processes, as well as for QCD. The aim of the present paper is to further explore this relation for e+e - reactions when they proceed via Z exchange also. More in detail we will explicitly show that the method of structure functions applied to initial and final state radiation coincides with the coherent state approach with an accuracy of o (1%), giving explicitly in addition the desired o ( a 2) left-over corrections. The basic formula which describes the reaction e+e---,~t+~t - in the approach of structure functions is the following [ 6 ]:
a(s) = i dx ao( ( 1 -x)s)He(x, s)F.(E--x, ( 1 --x)s),
(1)
o
where e = AE/E and the initial and final state radiation kernels are expressed in terms of the electron and muon structure functions as ~' See ref. [ 1 ] for recent reviews of LEP/SLC physics. See ref. [2] for a review of radiative corrections at LEP/SLC.
134
0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 228, number 1
PHYSICS LETTERS B
Hc(X,S)= fl d--Zzhe(z,s)De(~-~,s),
FF,(X,S)=
1--~.
7 September 1989
f dyno(y,(1-y)s),
(2,3)
0
and Ho(x, s) is defined as in eq. (2). By taking into account the effect of the soft radiation to all orders and of the hard one up to o ( a 2) one obtains ~2 /
1 1 2 { (2-x)[31n(1-x)-41nx]-4-H~(x, S) ~.ZJe(S)]~eXfle-I __~fl~(2--x)+~fle
ln(1-X)x + x - 6 ) ,
(4)
k
with fie= (2c~/~z) ( L e - 1 ), Le=ln(s/m2e ) and O/
Je(s) = 1 + -- {3Le + 2 [ ( ( 2 ) 7~
1 l}
2
+(~)
{[~-2¢'(2)]L~ + [3~(3)+~((2)-~]Le+
[-6(2(2)-9~(3)-6~(2)
In 2 + 3 ~ ( 2 ) + l~]}
2
- - l + - - aAA e~ ' )( +2( ~)) ' z c
(5)
By insertion of (4) in eqs. (2) and ( 1 ) one easily obtains a(s)=
i0 d x a o ( S ( 1 - x ) ) [ A ~ ( s ) J u ( s ) f l e X P ° - ~ "( e - x ) P " + R ( x ,
...)],
(6)
where the first term on the RHS of eq. (6) is proportional to the leading soft contribution, while R (x .... ) gives further correction terms of order (f12) and fie in the final cross section. We will assume the fractional energy resolution ~ = A E / E of order 10- i_ 1 0 - 2. By splitting the Born cross section ao ( s ) as ao =aQo ED a--ao~NV----a0RES, with ~3 a0QED(s) =A --Is'
a~NT(s) = B Re
s_MS-+iFM
,
a~ES(s)=C(s_M2)2+F2M2,
(7)
the corresponding radiatively corrected cross sections are obtained as follows: o'QED (S)
o'INT (S)
xRe
= A_ (Ae(s)Au(s)eao+/~, F ( I +fie)F(1 +flu) 2F1 (1 fie; 1 +fie +fl~; e) + . . . ) s F(l+fl~+fl,) ' ' = B ~ A ~ ( s ) A . ( s ) e p°+a" F( 1 +fie)F( 1 + flu) ( F ( l +/L + # . ) 1
( l - - z ) - P e 2F,
(
z)l }
fie + ft,, fie ; 1 + fie + ft,; Z-'-~-i -
+...
,
(8)
(9)
~2 Eq. (4) agrees with the complete second order result of ref. [7] up to hard terms of o(a2Le). In the resonant region this is not of numerical importance. The generalization of our results using the more complete but longer expression of ref. [ 7 ] is straightforward. ~3 The modifications of the Born cross sections (7) due to electroweak corrections, with the usual replacement in M and F, can be correspondingly introduced in our final expressions. 135
Volume 228, number 1 (TRES(s) = - - C
xIm
PHYSICS LETTERS B
7 September 1989
S~A~(s)a.(s)¢,~o+~t F(I+ flo)r(l + fl.) r ( l + f l ~ +fl,)
1
(l-z)-Po~f,
flo+fl.,fl~;l+flo+fl~;z_--_-- i
+ ....
(10)
where M~ = M R- iFM, z = e s / ( s - M~ ) and the dots indicate next to leading terms corresponding to R (x, ... ) in eq. (6). By expanding in fie the hypergeometric functions one then finds aQED(s)
=GQoED(s)/le(S)A.(S)(.fle+fl~a"~ ....
(11)
a,NT(s)=a,oNT(S)Ae(S)A.(s)ep F(l+fle)F(l+fl~) F( 1 + fie +, 6 . )
1
-cos- dR Re
[
exp(idR)
(
1 + (es/MF) ~n 8R exp(idR)
)1 (12)
"J~°oo,
a~S(s ) = a ~ r S ( s ) A e ( S ) A . ( s ) e ~ ~ F( 1 + flDF( 1 +fl~) F(1 +fie + f l , )
1 + (es/MF) sin
dR
exp(idR) (13)
X (cos f i e 0 - cot dR sin flee) +..., where (MZ_s)_,=
sin dR exp(idR)
Mr
'
tandR=
MF (M2__S)
[ es+ M 2 - s'~
0=arctan ~
~
( M 2 - s~
) - a r c t a n \---M--fi-].
We now wish to c o m m e n t briefly on the various factors appearing in the RHS of eqs. ( 11 ) - ( 13 ) and compare them with the analogous expressions obtained in the framework of coherent states, where, to leading order ~4, the differential cross section is given by [ 2,4 ]
da
S "da~°') (C~i~ra--~Cbi)),
dg2 - , ~ . - ~
(i=QED,
(14)
RES, INT),
with - (~fie+ ft. + 2flint c infra ( Q E---D )
( 15 fie
,-~(,NT)_ _COSdR _ Re •-~inr~ f, RES)__efl.
nr~. --
1
exp(iSR)
1+ (es/MF) s'[nSR exp(i~R)"
e
+ (es/MF) sin dR exp(idR)
l~e+(MF/s)
)
flint
e+ (MF/s) e x p ( - - i d R ) / s i n S R J"~ ]3 '
e
e x p ( - - i d a ) / s i n fiR
O 2,m.( l + f l_~s_M2
) ,
(16) (17)
and flint = ( 4 a / ~ ) l n tan ½0. Furthermore the finite terms Cb i) can be written as C~,) = a { 3 ( L e + L , ) + 4 [ ~ ( 2 ) zc
- 1]}+C~,) '
(18)
with C~i) containing other o ( a ) finite terms, coming from bremsstrahlung and box diagrams, odd in the exchange 0 ~ r t - 0, etc. Then, after putting fli,t = 0 in eqs. (15 ) - ( 1 7 ) , the exponentiated factors coincide with those in eqs. (11 ) ( 13 ). Furthermore the finite terms C bi) contain the o (fie, fl,) expansion in Ae (s)A, (s) in eqs. ( 11 ) - ( 13 ). The ~4 This corresponds to keep C~g) to o(a) only. 136
Volume 228, number 1
PHYSICS LETTERS B
7 September 1989
remaining non factorizable real and virtual terms can also be included in the approach o f structure functions [ 6 ]. On the other hand, the extra terms contained in eqs. ( 1 1 ) - ( 13 ) are o f o (,82) _ in particular the factor ,8202/2 in eq. ( 13 ) arising from the expansion o f cos Ofle - and o(Efl), and are not written explicitly. From the above discussion it follows that the two formalisms give identical answers for both the exponentiated and o (fl) finite terms relative to initial and final state radiation. On the other hand they add complementary informations for the tim, dependence and o (f12, eft) terms, respectively. Then, from the combined informations o f eqs. ( 1 1 ) - (13) and ( 15 ) - ( 17 ) one can transform eq. (14) into the following form: da dO -
do'(dO oQED) [ C}n~rEaD) ( 1 "q- C ~ Q E D ) )
"~ c ~ Q E D ) ] "~
do'~lNT)d-------~ tF]f~(INT) infra ~~, g "1 -"a- *'-'F Im(INT) " /~ "~ C ~ INT)
dao(REs) [-C~(RES)COS(fle~)--cot dR sin(fl~0) (1 + C~RES))+ C~ REs) ] + - -dg2 L ,.fra 1 +fie[ ( s - M 2 ) / M F ] O
,
(19)
with the definitions ( 15 ) - ( 17 ) of C}~'~ra and C(FQED )
/ k2 O~ (ZJ(,) +Z~ ( I ) ) __ j~gl, + (___~ (/~ (2) 3,_/~ (2) 3,_/] ( l ) A ( l ) , 1 2R/"~ IR2 l-fie -~e --~t --~e ~/.t j - - ~ 7 ~ /JeH~t-- ~p.e ~ , 7C
~(FINT) ~ __OL( z j ( l ) + Z j 7E
(I)).
/~.\ 2 j ~ t , + (~.~_~ (/]e(2) + A ~2) +Zje(l )m ~l ) ) _ ~ 1 2 ] ~ e ~ t
_fleCOSO(fle+l)+tan6RsinO(fl~+l) cos Ofl~ + tan 6R sin 0fl~ --
(2O)
e
--s)
1 + eS/(M2R
l 2 cos0+tan6Rsin~ E --s) ~-/~¢ nile cos qfl~ + t a n 6R sin 0fl~ 1 + es/(-M 2 '
(21)
2
~ (FRES)
(OL~(A(2).4_A(2).4_/((I)A(I), \rtj
Og (z~e( I ) .~_Zj ( 1 ) 7t
12
--2flecosO(fl~+l)--cot~RsinO(fl~+l) cos 0fl~ - cot 6R sin Ofl~ 1 2
COS 0 - - c o t 6R sin ¢
-- 4fie COS OJ~e - - c o t ~R sin
Ofle
1
e + Es/ (-M 2 - s)
e 1
+ e s / ( M 2 - s ) ~-~°"
(22)
The above equations represent our final result, which describes the radiative correction factors to an accuracy better than (1%). The effect o f the new terms of o (f12, eft) in eqs. ( 18 ) is shown in figs. 1, 2, where we plot the ratios R (~ = d a (~ [ 1 + o ( a ) + o (fie) + o ( a 2) ]/da (i) [ 1 +o(c~) ], for i = Q E D , INT, RES, for e=0.01 and e =0.05. Notice that the factors C~ra of eqs. ( 15 ), (17) do not appear in the ratios R (~). We have taken the scattering angle 0 = ½z~in the factors C~ ~) to automatically cancel the box diagram and other non factorizable contributions. As is clear from figs. 1, 2 the Q E D and I N T corrections are practically constant in the resonant region to a value o f about - 0 . 0 1 , while the RES correction is modulated essentially by the term 1 - f l 2 0 2 / 2 , with an extra factor of o (0.01-0.02 ). The extension of our results to the case o f the Z line shape is straightforward. It simply corresponds to take the limit f t , = 0, d , = 1 and e = 1 - 4 l f l / s in eq. (10). Then one simply obtains for s - M 2
a(s) ~--ao(S)
MZ-s M~
~sin(1--fl~)~R ~fl~ sin 6R sin gfl~ de(S),
(23)
which agrees with refs. [ 3,4,11 ] up to constant factors of o (f12). 137
Volume 228, number 1
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'
'
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PHYSICS LETTERS B
. . . .
I
. . . .
I
7 September 1989
'
'
R(i)
R(q
O. 9 9 5
B.995
Q.99~
O. 9 9 f i
Q.985
O. 9 8 5
'
'
'
'
'
'
'
'
'
'
I
'
'
/
\ / \ / \
/ \
0.988
/ \
0. 9 8 8
/ N
o.975
.
98
.
.
.
I
92
.
.
.
.
I
94
.
.
.
.
J
96
....
Vq
98
Fig. 1. Ratio R(i)=da(i)[l+o(a)+o(fle)+o(a2)]/ dcr(~) [ l + o ( a ) ] for e+e---,~+~t -, with i=QED (solid line), INT (dashed line), RES (dot-dashed line) for e=0.01. The values of Mand Fare taken to be 92 GeV and 2.6 GeV, respectively.
0. 975
i
'
90
92
i
i
I 94
J
J
9s
l
A
J
98
Fig. 2. Same as in fig. 1 for ~=0.05.
To c o n c l u d e , we h a v e explicitly s h o w n that t h e a p p r o a c h o f t h e c o h e r e n t states a n d that o f the s t r u c t u r e functions offer two c o m p l e m e n t a r y m e t h o d s in Q E D to i m p r o v e the t h e o r e t i c a l a c c u r a c y r e q u i r e d for p r e c i s i o n m e a s u r e m e n t s at L E P / S L C energies. T h e y give i d e n t i c a l results for the e x p o n e n t i a t e d a n d finite o ( a ) factors r e l a t i v e to the initial a n d final state r a d i a t i o n . M o r e o v e r , the r e m a i n i n g l o g a r i t h m i c o ( a 2 ) corrections, o b t a i n e d w i t h the m e t h o d o f s t r u c t u r e f u n c t i o n s c o m p l e t e the i n f o r m a t i o n on i n t e r f e r e n c e effects p r o v i d e d by the app r o a c h o f c o h e r e n t states. T h e o v e r a l l p i c t u r e p r o v i d e s s i m p l e analytical f o r m u l a e w h i c h can be easily e x t e n d e d to e + e - r e a c t i o n s o t h e r t h a n e + e - - - , ~ t + p - . We are grateful to G. Altarelli, O. N i c r o s i n i a n d L. T r e n t a d u e for discussions.
R e f e r e n c e s
[ 1 ] E.g.J. Ellis and R. Peccei, eds., Physics at LEP, CERN report CERN 86-02 ( 1986); G. Alexander et al., eds., Polarization at LEP, CERN report CERN 88-06 (1988). [2] E.g., M. Greco, Riv. Nuovo Cimento 11, no. 5 (1988). [ 31 M. Greco, G. Pancheri and Y. Srivastava, Nucl. Phys. B 101 ( 1975 ) 234. [4] M. Greco, G. Pancheri and Y. Srivastava, Nucl. Phys. B 171 (1980) 118. [5] 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. [6] O. Nicrosini and L. Trentadue, Z. Phys. C 39 (1988) 479. [7 ] F.A. Berends, W.L. van Neerven and G.J.H. Burgers, Nucl. Phys. B 297 (1988) 429. [8] E.A. Kuraev and V.S. Fadin, Sov. J. Nucl. Phys. 41 (1985) 466; G. Altarelli and G. Martinelli, in: Physics at LEP, eds. J. Ellis and R. Peccei, CERN report CERN 86-02 ( 1986); O. Nicrosini and L. Trantadue, Phys. Lett. B 196 (1987) 551. [9] V. Chung, Phys. Rev. B 140 (1965) 1110; M. Greco and G. Rossi, Nuovo Cimento 50 (1967) 168. [ 10] G. Curci and M. Greco, Phys. Lett. B 79 (1978) 406. [ 11 ] R.N. Cahn, Phys. Rev. D 36 ( 1987 ) 2666.
138
PHYSICS LETTERS B
7 September 1989
C O H E R E N T STATES AND STRUCTURE FUNCTIONS IN QED F. AVERSA and M. GRECO 1NFN, Laboratori Nazionali di Frascati, C.P. 13, 1-00044 Frascati, Italy Received 26 May 1989
The methods of coherent states and of structure functions in QED are considered in detail for a precise evaluation of the radiative effects at LEP/SLC. They are explicitly shown to give identical results for the exponentiated infrared factors and the o ( a ) terms corresponding to the initial and final state radiation. Then the combined use of both techniques further improves the theoretical predictions to an accuracy better than 1% - including interference effects - and provides simple analytical formulae of immediate phenomenological application.
The role played by QED radiative corrections in e+e - reactions at LEP/SLC energies for the precise determination of electroweak parameters is well known ~. The resummation of the leading double and single logarithms of soft origin [ 3 ], together with the exact determination of the left-over o ( a ) terms [ 4, 5 ], is essential to reach a degree of accuracy of o (1%) in the theoretical predictions. The calculation of all logarithmic o (o?) corrections, as recently done [6,7] for the reaction e+e - ~ t + ~ t -, is the further necessary tool to get absolute control of the QED corrections, as required for precision tests of the theory. To this aim the method of the structure functions [8], extended to both initial and final states [6 ] in the reaction e +e- --, ~t+~t-, has been shown to be quite powerful, suggesting a systematic approach to other processes as, for example, Bhabha scattering. However, only numerical solutions have been obtained so far in the case of a resonant cross section, leaving unclear the connection to other analytical approaches to the problem. The method of coherent states [ 9 ], for example, has been proved to be very successful in describing the main features of multiphoton emission with good accuracy, giving explicit analytical results for resonant processes and interference effects with pure QED background [ 3,4 ]. The determination of the left-over o ( a 2) logarithms is needed only for the evaluation of the radiative effects beyond the 1% level. A close connection between the two above methods of resummation has been explicitly shown [ 10 ] to exist in the non-singlet case for QED non-resonant processes, as well as for QCD. The aim of the present paper is to further explore this relation for e+e - reactions when they proceed via Z exchange also. More in detail we will explicitly show that the method of structure functions applied to initial and final state radiation coincides with the coherent state approach with an accuracy of o (1%), giving explicitly in addition the desired o ( a 2) left-over corrections. The basic formula which describes the reaction e+e---,~t+~t - in the approach of structure functions is the following [ 6 ]:
a(s) = i dx ao( ( 1 -x)s)He(x, s)F.(E--x, ( 1 --x)s),
(1)
o
where e = AE/E and the initial and final state radiation kernels are expressed in terms of the electron and muon structure functions as ~' See ref. [ 1 ] for recent reviews of LEP/SLC physics. See ref. [2] for a review of radiative corrections at LEP/SLC.
134
0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 228, number 1
PHYSICS LETTERS B
Hc(X,S)= fl d--Zzhe(z,s)De(~-~,s),
FF,(X,S)=
1--~.
7 September 1989
f dyno(y,(1-y)s),
(2,3)
0
and Ho(x, s) is defined as in eq. (2). By taking into account the effect of the soft radiation to all orders and of the hard one up to o ( a 2) one obtains ~2 /
1 1 2 { (2-x)[31n(1-x)-41nx]-4-H~(x, S) ~.ZJe(S)]~eXfle-I __~fl~(2--x)+~fle
ln(1-X)x + x - 6 ) ,
(4)
k
with fie= (2c~/~z) ( L e - 1 ), Le=ln(s/m2e ) and O/
Je(s) = 1 + -- {3Le + 2 [ ( ( 2 ) 7~
1 l}
2
+(~)
{[~-2¢'(2)]L~ + [3~(3)+~((2)-~]Le+
[-6(2(2)-9~(3)-6~(2)
In 2 + 3 ~ ( 2 ) + l~]}
2
- - l + - - aAA e~ ' )( +2( ~)) ' z c
(5)
By insertion of (4) in eqs. (2) and ( 1 ) one easily obtains a(s)=
i0 d x a o ( S ( 1 - x ) ) [ A ~ ( s ) J u ( s ) f l e X P ° - ~ "( e - x ) P " + R ( x ,
...)],
(6)
where the first term on the RHS of eq. (6) is proportional to the leading soft contribution, while R (x .... ) gives further correction terms of order (f12) and fie in the final cross section. We will assume the fractional energy resolution ~ = A E / E of order 10- i_ 1 0 - 2. By splitting the Born cross section ao ( s ) as ao =aQo ED a--ao~NV----a0RES, with ~3 a0QED(s) =A --Is'
a~NT(s) = B Re
s_MS-+iFM
,
a~ES(s)=C(s_M2)2+F2M2,
(7)
the corresponding radiatively corrected cross sections are obtained as follows: o'QED (S)
o'INT (S)
xRe
= A_ (Ae(s)Au(s)eao+/~, F ( I +fie)F(1 +flu) 2F1 (1 fie; 1 +fie +fl~; e) + . . . ) s F(l+fl~+fl,) ' ' = B ~ A ~ ( s ) A . ( s ) e p°+a" F( 1 +fie)F( 1 + flu) ( F ( l +/L + # . ) 1
( l - - z ) - P e 2F,
(
z)l }
fie + ft,, fie ; 1 + fie + ft,; Z-'-~-i -
+...
,
(8)
(9)
~2 Eq. (4) agrees with the complete second order result of ref. [7] up to hard terms of o(a2Le). In the resonant region this is not of numerical importance. The generalization of our results using the more complete but longer expression of ref. [ 7 ] is straightforward. ~3 The modifications of the Born cross sections (7) due to electroweak corrections, with the usual replacement in M and F, can be correspondingly introduced in our final expressions. 135
Volume 228, number 1 (TRES(s) = - - C
xIm
PHYSICS LETTERS B
7 September 1989
S~A~(s)a.(s)¢,~o+~t F(I+ flo)r(l + fl.) r ( l + f l ~ +fl,)
1
(l-z)-Po~f,
flo+fl.,fl~;l+flo+fl~;z_--_-- i
+ ....
(10)
where M~ = M R- iFM, z = e s / ( s - M~ ) and the dots indicate next to leading terms corresponding to R (x, ... ) in eq. (6). By expanding in fie the hypergeometric functions one then finds aQED(s)
=GQoED(s)/le(S)A.(S)(.fle+fl~a"~ ....
(11)
a,NT(s)=a,oNT(S)Ae(S)A.(s)ep F(l+fle)F(l+fl~) F( 1 + fie +, 6 . )
1
-cos- dR Re
[
exp(idR)
(
1 + (es/MF) ~n 8R exp(idR)
)1 (12)
"J~°oo,
a~S(s ) = a ~ r S ( s ) A e ( S ) A . ( s ) e ~ ~ F( 1 + flDF( 1 +fl~) F(1 +fie + f l , )
1 + (es/MF) sin
dR
exp(idR) (13)
X (cos f i e 0 - cot dR sin flee) +..., where (MZ_s)_,=
sin dR exp(idR)
Mr
'
tandR=
MF (M2__S)
[ es+ M 2 - s'~
0=arctan ~
~
( M 2 - s~
) - a r c t a n \---M--fi-].
We now wish to c o m m e n t briefly on the various factors appearing in the RHS of eqs. ( 11 ) - ( 13 ) and compare them with the analogous expressions obtained in the framework of coherent states, where, to leading order ~4, the differential cross section is given by [ 2,4 ]
da
S "da~°') (C~i~ra--~Cbi)),
dg2 - , ~ . - ~
(i=QED,
(14)
RES, INT),
with - (~fie+ ft. + 2flint c infra ( Q E---D )
( 15 fie
,-~(,NT)_ _COSdR _ Re •-~inr~ f, RES)__efl.
nr~. --
1
exp(iSR)
1+ (es/MF) s'[nSR exp(i~R)"
e
+ (es/MF) sin dR exp(idR)
l~e+(MF/s)
)
flint
e+ (MF/s) e x p ( - - i d R ) / s i n S R J"~ ]3 '
e
e x p ( - - i d a ) / s i n fiR
O 2,m.( l + f l_~s_M2
) ,
(16) (17)
and flint = ( 4 a / ~ ) l n tan ½0. Furthermore the finite terms Cb i) can be written as C~,) = a { 3 ( L e + L , ) + 4 [ ~ ( 2 ) zc
- 1]}+C~,) '
(18)
with C~i) containing other o ( a ) finite terms, coming from bremsstrahlung and box diagrams, odd in the exchange 0 ~ r t - 0, etc. Then, after putting fli,t = 0 in eqs. (15 ) - ( 1 7 ) , the exponentiated factors coincide with those in eqs. (11 ) ( 13 ). Furthermore the finite terms C bi) contain the o (fie, fl,) expansion in Ae (s)A, (s) in eqs. ( 11 ) - ( 13 ). The ~4 This corresponds to keep C~g) to o(a) only. 136
Volume 228, number 1
PHYSICS LETTERS B
7 September 1989
remaining non factorizable real and virtual terms can also be included in the approach o f structure functions [ 6 ]. On the other hand, the extra terms contained in eqs. ( 1 1 ) - ( 13 ) are o f o (,82) _ in particular the factor ,8202/2 in eq. ( 13 ) arising from the expansion o f cos Ofle - and o(Efl), and are not written explicitly. From the above discussion it follows that the two formalisms give identical answers for both the exponentiated and o (fl) finite terms relative to initial and final state radiation. On the other hand they add complementary informations for the tim, dependence and o (f12, eft) terms, respectively. Then, from the combined informations o f eqs. ( 1 1 ) - (13) and ( 15 ) - ( 17 ) one can transform eq. (14) into the following form: da dO -
do'(dO oQED) [ C}n~rEaD) ( 1 "q- C ~ Q E D ) )
"~ c ~ Q E D ) ] "~
do'~lNT)d-------~ tF]f~(INT) infra ~~, g "1 -"a- *'-'F Im(INT) " /~ "~ C ~ INT)
dao(REs) [-C~(RES)COS(fle~)--cot dR sin(fl~0) (1 + C~RES))+ C~ REs) ] + - -dg2 L ,.fra 1 +fie[ ( s - M 2 ) / M F ] O
,
(19)
with the definitions ( 15 ) - ( 17 ) of C}~'~ra and C(FQED )
/ k2 O~ (ZJ(,) +Z~ ( I ) ) __ j~gl, + (___~ (/~ (2) 3,_/~ (2) 3,_/] ( l ) A ( l ) , 1 2R/"~ IR2 l-fie -~e --~t --~e ~/.t j - - ~ 7 ~ /JeH~t-- ~p.e ~ , 7C
~(FINT) ~ __OL( z j ( l ) + Z j 7E
(I)).
/~.\ 2 j ~ t , + (~.~_~ (/]e(2) + A ~2) +Zje(l )m ~l ) ) _ ~ 1 2 ] ~ e ~ t
_fleCOSO(fle+l)+tan6RsinO(fl~+l) cos Ofl~ + tan 6R sin 0fl~ --
(2O)
e
--s)
1 + eS/(M2R
l 2 cos0+tan6Rsin~ E --s) ~-/~¢ nile cos qfl~ + t a n 6R sin 0fl~ 1 + es/(-M 2 '
(21)
2
~ (FRES)
(OL~(A(2).4_A(2).4_/((I)A(I), \rtj
Og (z~e( I ) .~_Zj ( 1 ) 7t
12
--2flecosO(fl~+l)--cot~RsinO(fl~+l) cos 0fl~ - cot 6R sin Ofl~ 1 2
COS 0 - - c o t 6R sin ¢
-- 4fie COS OJ~e - - c o t ~R sin
Ofle
1
e + Es/ (-M 2 - s)
e 1
+ e s / ( M 2 - s ) ~-~°"
(22)
The above equations represent our final result, which describes the radiative correction factors to an accuracy better than (1%). The effect o f the new terms of o (f12, eft) in eqs. ( 18 ) is shown in figs. 1, 2, where we plot the ratios R (~ = d a (~ [ 1 + o ( a ) + o (fie) + o ( a 2) ]/da (i) [ 1 +o(c~) ], for i = Q E D , INT, RES, for e=0.01 and e =0.05. Notice that the factors C~ra of eqs. ( 15 ), (17) do not appear in the ratios R (~). We have taken the scattering angle 0 = ½z~in the factors C~ ~) to automatically cancel the box diagram and other non factorizable contributions. As is clear from figs. 1, 2 the Q E D and I N T corrections are practically constant in the resonant region to a value o f about - 0 . 0 1 , while the RES correction is modulated essentially by the term 1 - f l 2 0 2 / 2 , with an extra factor of o (0.01-0.02 ). The extension of our results to the case o f the Z line shape is straightforward. It simply corresponds to take the limit f t , = 0, d , = 1 and e = 1 - 4 l f l / s in eq. (10). Then one simply obtains for s - M 2
a(s) ~--ao(S)
MZ-s M~
~sin(1--fl~)~R ~fl~ sin 6R sin gfl~ de(S),
(23)
which agrees with refs. [ 3,4,11 ] up to constant factors of o (f12). 137
Volume 228, number 1
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Fig. 1. Ratio R(i)=da(i)[l+o(a)+o(fle)+o(a2)]/ dcr(~) [ l + o ( a ) ] for e+e---,~+~t -, with i=QED (solid line), INT (dashed line), RES (dot-dashed line) for e=0.01. The values of Mand Fare taken to be 92 GeV and 2.6 GeV, respectively.
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Fig. 2. Same as in fig. 1 for ~=0.05.
To c o n c l u d e , we h a v e explicitly s h o w n that t h e a p p r o a c h o f t h e c o h e r e n t states a n d that o f the s t r u c t u r e functions offer two c o m p l e m e n t a r y m e t h o d s in Q E D to i m p r o v e the t h e o r e t i c a l a c c u r a c y r e q u i r e d for p r e c i s i o n m e a s u r e m e n t s at L E P / S L C energies. T h e y give i d e n t i c a l results for the e x p o n e n t i a t e d a n d finite o ( a ) factors r e l a t i v e to the initial a n d final state r a d i a t i o n . M o r e o v e r , the r e m a i n i n g l o g a r i t h m i c o ( a 2 ) corrections, o b t a i n e d w i t h the m e t h o d o f s t r u c t u r e f u n c t i o n s c o m p l e t e the i n f o r m a t i o n on i n t e r f e r e n c e effects p r o v i d e d by the app r o a c h o f c o h e r e n t states. T h e o v e r a l l p i c t u r e p r o v i d e s s i m p l e analytical f o r m u l a e w h i c h can be easily e x t e n d e d to e + e - r e a c t i o n s o t h e r t h a n e + e - - - , ~ t + p - . We are grateful to G. Altarelli, O. N i c r o s i n i a n d L. T r e n t a d u e for discussions.
R e f e r e n c e s
[ 1 ] E.g.J. Ellis and R. Peccei, eds., Physics at LEP, CERN report CERN 86-02 ( 1986); G. Alexander et al., eds., Polarization at LEP, CERN report CERN 88-06 (1988). [2] E.g., M. Greco, Riv. Nuovo Cimento 11, no. 5 (1988). [ 31 M. Greco, G. Pancheri and Y. Srivastava, Nucl. Phys. B 101 ( 1975 ) 234. [4] M. Greco, G. Pancheri and Y. Srivastava, Nucl. Phys. B 171 (1980) 118. [5] 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. [6] O. Nicrosini and L. Trentadue, Z. Phys. C 39 (1988) 479. [7 ] F.A. Berends, W.L. van Neerven and G.J.H. Burgers, Nucl. Phys. B 297 (1988) 429. [8] E.A. Kuraev and V.S. Fadin, Sov. J. Nucl. Phys. 41 (1985) 466; G. Altarelli and G. Martinelli, in: Physics at LEP, eds. J. Ellis and R. Peccei, CERN report CERN 86-02 ( 1986); O. Nicrosini and L. Trantadue, Phys. Lett. B 196 (1987) 551. [9] V. Chung, Phys. Rev. B 140 (1965) 1110; M. Greco and G. Rossi, Nuovo Cimento 50 (1967) 168. [ 10] G. Curci and M. Greco, Phys. Lett. B 79 (1978) 406. [ 11 ] R.N. Cahn, Phys. Rev. D 36 ( 1987 ) 2666.
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