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Radiative corrections to B quark production
16 June 1994
PHYSICS LETTERS B
ELSEVIER
Physics Letters B 329 (1994) 369-373
Radiative corrections to B quark production Jin Woo Jun Department of Physics, lnje University, Kimhae 621-749, South Korea
Changkeun Jue Department of Physics, Kyungpook University, Taegu 702-701, South Korea Received 20 January 1994 Editor: M. Dine
Abstract
The analytic formulae for the partial decay width F(Z~b6) including the electroweak and QCD l-loop corrections and photonic and gluonic bremsstrahlung, are presented for arbitrary masses of t and b quarks.
e +e --annihilation experiments at LEP have led to an accurate knowledge of various Z boson properties such as its mass and width. In particular the possibility of a precise determination of its partial decay rate into hadrons and the prospect of reducing the error to a few per mille in oncoming high statistics measurements have induced a considerable amount of theoretical work to calculate the QCD corrections to this quantity. Of all the observable processes at the Z resonance, e ÷ e - ~ bb is unique in that it receives contributions from l-loop Feynman diagrams that have top quarks attached to external fermion lines. We shall be concerned with the electroweak and QCD l-loop corrections to the partial decay width F ( Z ~ bb). Until recently, most calculations adopted the massless fermion approximation [ 1-3]. In Ref. [4], Chang et al. calculated the QCD corrections to the widths of Z boson taking the quarks to be massive in the renormalization scheme of Antonelli et al. [5]. Recently in Ref. [6] and Ref. [7], similar calculations are carried out in the scheme and in the on-shell scheme of B0hm et al. [ 8 ], respectively. In the on-shell renormalization scheme of Sirlin [2,3], Akhundov et al. [9] and Bardin et al. [ 10] obtained the finite parts taking into account the massive t quark effect only, neglecting the massive b quark effect. In this note we explicitly take into account the mass effects of t and b quarks in the on-shell renormalization scheme of Sirlin [2,3]. The mb dependent bremsstrahlung will be obtained from the mass dependent bremsstrahlung integrals of Denner and Sack [ 11 ]. Since the IR pole terms, i.e. the coefficients of 1 / ( n - 4) are found without any approximation in Refs. [ 2,3 ], they are already mb dependent. Therefore, the mb dependent IR part of the bremsstrahlung integrals of Ref. [ 11 ] can be automatically added to vanish. We work in the unitary gauge and use the Euclidean metric. We use all the notations of Refs. [ 2,3,9]. Let us start with the electroweak one-loop corrected matrix element for the decay Z ~ bb. ..g~w=
g
E,~a[%,(l+.ys).Tr[b_4S2 lQbl%,3r~]u,
4Cw 0370-2693/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI0370-2693 ( 9 4 ) 0 0 4 7 6 - N
(1)
370
J. W. Jun, C. Jue / PhysicsLettersB 329 (1994)369-373
where S 2 = 1 - C~v = sin20w, g = 41ra/S~v. The one-loop form factors ~r ~b and ~r ~ are 5r[b=~-~,+ ~
g2 {
~
R
[W'(-1)-Z'(-1)]-M'(-1)
+ ~--~ - ~f---ff- I Qb [ + 2 -----ff--+ [ -R+ ~-ib=l+
½- (1 - R ) IQb I1 IVY( --ME, ME) + ~] +R[V~( -M~, M2w,ME) + ~1 + VCt'), ~
(1--R)Qb2 [ 4 - 2 ( - M ~ + 2 m ~ ) S ( - M ~ , m
2,mb2)lPm
- l n m~w - 3 In R - 4 + M E z K ( - M ~ , mE, mE)
(
+ I z e ' ( - 1 ) - ~ ( 1 - R ) 1+2 Tr Q~ Ln
m)
- ( 1 ) l - ~ lR [ W ' ( - 1 ) - Z ' ( - 1 ) ] + M ' ( - 1 ) +
( 1 - R ) a Q ba [ V I, ( - - M z2, ME) +~21] ,
(2)
where R = cosZ0w and the functions j ( q 2 , m 2, mE) and ~5~(q2, m~, m 2) are defined by formulae (2.16) and (3.4) of Ref. [2]. The primed quantities reflect the finite mass effects of m, and rob. Thus, W'( - 1) = W( - 1) + Wt,b( -- 1),
Z'( - 1) =Z( - 1) +Z,,b( -- 1),
M'(-1)=M(-1)+Mt.b(-1),
zF'(--1)=zF(--1)+Zffb(--1),
V~( -M2z, M~v) = I V,b 12[V~( -M2z, M2w)+ V~( -M~, M~v, m~)] , V~( - M z ,2 Mz) 2 = Vl( - M 2, M2z) + V~( - M 2, ME, m2) , V~( --Mz,2 M 2w, M2w) = [Vtb 12[ V2( --M2z, M2w,ME) + V~( - M 2, M~v, M2w, rn~)] ,
~(5r-11)+3 1-rrln
vct'=-±(l+2R)3
r)
]Vtb[2,
(3)
where Vtb is the Kobayashi-Maskawa matrix element and r = m t2/ M w2. The finite functions W ( - 1), Z ( - 1), M(-1),zF(-1) Vl( - M z ,2 Mw), 2 V:( --ME, M~v, MEw) can be found in the appendix A of Ref. [3] and the functions V~ ( -MZz, M~v, m 2) and V~( -Mz,2 Mw,2 M 2, mE) are defined by formulae (A.9) and (A. 10) of Ref. [9]. The other mass dependent parts are ,
( m --MEw, ~ )mr,2 W,.b( -- 1) = IV,hi 2 --6[/3( mE ~ + 3 "~2 ll ( -- M 2w, m,z , mb2) +3 ,va W
~"
-/3( - M 2 , 0, 0)] I,(-M2w, mbz, mt2)
)
W
1 Zt.b(--1) = -- ~ [ 3 - 8(1 --R) +-~( 1 - R ) z] [13(--ME, m,2, m 2) --13(-M 2, 0, 0)]
J.W.Jun, C. Jue/PhysicsLettersB 329 (1994)369-373
371
1 - ~ [ 3 - 4 ( 1 - R ) +8(1 - R ) 21 [13( - M 2, m 2, m 2) -I3( -M2, 0, 0)1 (~)
+
2
U'mt--~wIo( - U l , m,,2 mZt) +
(3) m 2 - M2
m 2, m 2) ,
Io( --ME,
Mt.b(-- 1) = [ - - 4 + 3 ~ ( 1 - R ) ] [ 1 3 ( - M 2, mr,: m 2 ) - I 3 ( - M 2 , 0 , 0 ) ] + [-2+8(1-R)] r Zt,b(--1)=-×
(~)
[ / 3 ( - M 2, m 2, m 2 ) - 1 3 ( - M 2, 0, 0 ) 1 ,
2 _m, _ [ 1 - m t 2 J ( - - M 2,mr,2 m 2 ) ] +
M~v
mt mt + ( -- '~1 + 2M~: [12 In j-~-T ,vlz mEt + M--~z:
3) M---~w m~ [1-m~5(-M2z'
[ 1 ln~.2m~
× ~ M~
+
m~
- -
Mi
(
+
-
[3_8(1_R) +~(l_R)21
Mm'~4'M~I:~vJ(--ME, mr,2 mE) ] W
-
1
Z I
1
m~, m~,)] + ~-~ [ 3 - 4 ( 1 - R )
1
-~
+
m~
°4 '"~ ~ M2 ~ , z
z
2M~, MwMJ
w~"
+8(l-R)
t
2]
_ . , 2 m~.,,,~)
]
,.. z,
•
(4)
Therefore, FEW-- FBom 1 + 2 Re(A') -~3Sw2 Re(B')
=/'Born + FEWH~p,
(5)
where /"born =
Mza ~ _ 16C2S 2
4m~, M--T G'o ,
['~[4mZ~'~2m~ A'=Sr ]b--1, B ' : S r ~--Sr ,b, G'o:[l+ "-~z)(1-4S2)Z+[l-
--~z ).
(6)
Let us turn into the QCD virtual corrections which can be obtained from the previous results by keeping only the pure QED terms, setting Qb = 1, replacing a by the strong coupling constant a~ and applying an overall factor CF = ~. Thus, QCD Fl-loop =/'Born' 2 Re(A') ,
(7)
where A'---~
'lb,c
-- 1
,
~ r ~ b ' ~ - - l + 4~s---~w
m:
(l-R)
- I n M---~w- 3 In R - 4 + M ~ (
[4-2(-M~+2m~):(-M~,mLm~,)IPI~
)
-M~, mE, m~) .
(8)
Since the converting back and forth between the Euclidean metric and the Minkowskian metric is straightforward in practice and the results are independent of metric when the physical quantities are expressed in terms of the squared masses, we will use the Minkowskian metric for convience in the bremsstrahlung integrals. Thus, the electroweak bremsstrahlung is
372
J. W. Jun, C. Jue / Physics Letters B 329 (1994) 369-373
EW /"l~rem.
a 1 f d3p d3q d31 4 (2~r)gMz 2po 2qo ~ o 3 ( p + q + l - k )
~_, I"/~ brem. Ew ]2,
(9)
spin
where EW "~bmm.--
2q~+ly~], , --eQo n (l)eU(k)ff(p)[.[2p ~-~+ ~'d - Y u Y u -~2 ] t a T s ) v (q), 4CwSw
(lO)
4 2 and NI = 2p. l, N2 = 2q. l, a = 1 - gSw. Therefore,
EW
Fbrem" =
4.._a~(Qb) 2 7r
1
{
v/l_4m2/M2z FBom_(ME-2m~)I12--mb2(lll
1
+•22)--11--12
m2
2M~_2mz[(l+Mz'~z)(I°+I°)+2I])
(11)
•
The integrals I~2, 11I, 122, 11, 12, I °, I °, I can be obtained from the corresponding integrals of Ref. [ 11 ] by the substitutions Mz~Mw, ms ~ ml, mb ~ m2. Following the previous-mentioned recipe, the QCD bremsstrahlung, FbQCD . . . . can be read easily. The IR divergences of the bremsstrahlung integrals are regularized by a finite photon mass A and the IR divergencs of the virtual corrections are dimensionally regularized. Using the correspondence 2/ (4 - n) - y + In 47r ~ In Az, Pm can be replaced by In (~'/A) with an arbitrary parameter ~"with dimension of mass. Then, it can be easily checked that the partial width F( = Fao~ + --lr'EWqoop+/'~r~m. +/'i-Q~l~p-+ Fb~°~:,D.) is IR finite. In on-shell renormalization scheme of Sirlin, Mw should be determined by the equation from the/z-decay process:
M2w( 1 - M2w~= ora
MU
1
(12)
v~c~ 1- Ar'
where Ar = ~ a
Re{X (Mw, Mz, MH, my) }
--~Re
( -~ (l+2Zra~LnM~ 2
"v
+
R
(I_R)~[W'(-~)-Z'(-I)]
+ ~I-R [ W ' ( O ) - W ' ( - 1 ) - ~ R ( R + I ) + - - f + ( 1|~ J /911_~RR lnR ] )
(13)
As before, the function W'(0) consists of two parts:
w'(o) = w(o) + w,,e(o),
(14)
where W(0) can be found in Eq. (A.2) of Ref. [3], and m----~t2 W,.o(O)=lV,~l 3M2Ii(O, m2, m 2) + 3 - -
)
11(0, m 2, m 2) •
Putting all together, the l-loop corrected partial width F(Z--* bb) can be summarized as follows:
(15)
373
J. W. Jun, C. Jue / Physics Letters B 329 (1994) 369-373
GF F(Z~bb)- M 81/~¢rG6 1- 4m ME I.f 1-Ar+2Re(A'+A')-~3S~vRe(B') 4 1 [a(Qb)Zq-~3as][(M~--2m~)l~z--m~(ll~ + ~/1-4m~/MZz ¢r
1
+122) --11 --I2
1 ((m~tlO+IO)+21)]) 2M~- 2m~ 1+ MU' '
(16)
In conclusion, we obtain the analytical formulae for the partial width F(Z ~ bb) including the electroweak and QCD l-loop radiative corrections and photonic and gluonic bremsstrahlung for arbitrary masses of t and b quarks. If we physically determine the arbitrary mass parameter ~', we can also obtain the numerical results by use of the input parameters GF, Mz, Mn and my. In the near future, we wish to use Eq. (16) in the numerical analysis of the results of LEP precision experiments to find the improved bounds on mt and MH. This work was supported in part by the '93 program of the Inje Research and Scholarship Foundation.
References [ 1 ] W. Beenakker and W. Hollik, Z. Phys. C 40 (1988) 141; F.A. Berends, in: Z Physics at LEP, eds. G. Altarelli, R. Kleiss and C. Verzegnassi CERN report CERN 89-08, Vol. 1 (CERN, Geneva, 1989) p, 89; D. Albert, W.J. Marciano, D. Wyler and Z. Parsa, Nuci. Phys. B 166 (1980) 460. [2] D.Y. Bardin, P.C. Christova and O,M. Fedorenko, Nucl. Phys. B 175 (1980) 435. [3] D.Y. Bardin, P.C. Christova and O.M. Fedorenko, Nucl. Phys. B 197 (1982) 1. [4] T.H. Chang, K.J.F. Gaemers and W.L.V. Neerven, Nucl. Phys. B 202 (1982) 407. [5] F. Antonelli, M. Consoli and G. Corbo, Phys. Lett. B 91 (1980) 90. [6] K.G. Chetyrkin, J.H. Kiihn and A. Kwiatkowsk, Phys. Lett. B 282 (1992) 221; S. Narison, Phys. Lett. B 260 (1991) 411; K.G. Chetyrkin and J.H. Kiihn, Phys. Lett. B 248 (1990) 359. [7] J. Bernaeu, A. Pich and A. Santamaria, Phys. Lett. B 200 (1988) 569. [8] M. BiShm,W. Hollik and H. Spiesberger, Fortschr. Phys. 34 (1986) 687. [9l A.A. Akhundov, D.Y. Bardin and T. Riemann, Nucl. Phys. B 276 (1986) 1. [10] D.Y. Bardin and V.A. Dokuchaeva, Nucl. Phys. B 246 (1984) 221. [11] A. Denner and T. Sack, Z. Phys. C 40 (1990) 653.
PHYSICS LETTERS B
ELSEVIER
Physics Letters B 329 (1994) 369-373
Radiative corrections to B quark production Jin Woo Jun Department of Physics, lnje University, Kimhae 621-749, South Korea
Changkeun Jue Department of Physics, Kyungpook University, Taegu 702-701, South Korea Received 20 January 1994 Editor: M. Dine
Abstract
The analytic formulae for the partial decay width F(Z~b6) including the electroweak and QCD l-loop corrections and photonic and gluonic bremsstrahlung, are presented for arbitrary masses of t and b quarks.
e +e --annihilation experiments at LEP have led to an accurate knowledge of various Z boson properties such as its mass and width. In particular the possibility of a precise determination of its partial decay rate into hadrons and the prospect of reducing the error to a few per mille in oncoming high statistics measurements have induced a considerable amount of theoretical work to calculate the QCD corrections to this quantity. Of all the observable processes at the Z resonance, e ÷ e - ~ bb is unique in that it receives contributions from l-loop Feynman diagrams that have top quarks attached to external fermion lines. We shall be concerned with the electroweak and QCD l-loop corrections to the partial decay width F ( Z ~ bb). Until recently, most calculations adopted the massless fermion approximation [ 1-3]. In Ref. [4], Chang et al. calculated the QCD corrections to the widths of Z boson taking the quarks to be massive in the renormalization scheme of Antonelli et al. [5]. Recently in Ref. [6] and Ref. [7], similar calculations are carried out in the scheme and in the on-shell scheme of B0hm et al. [ 8 ], respectively. In the on-shell renormalization scheme of Sirlin [2,3], Akhundov et al. [9] and Bardin et al. [ 10] obtained the finite parts taking into account the massive t quark effect only, neglecting the massive b quark effect. In this note we explicitly take into account the mass effects of t and b quarks in the on-shell renormalization scheme of Sirlin [2,3]. The mb dependent bremsstrahlung will be obtained from the mass dependent bremsstrahlung integrals of Denner and Sack [ 11 ]. Since the IR pole terms, i.e. the coefficients of 1 / ( n - 4) are found without any approximation in Refs. [ 2,3 ], they are already mb dependent. Therefore, the mb dependent IR part of the bremsstrahlung integrals of Ref. [ 11 ] can be automatically added to vanish. We work in the unitary gauge and use the Euclidean metric. We use all the notations of Refs. [ 2,3,9]. Let us start with the electroweak one-loop corrected matrix element for the decay Z ~ bb. ..g~w=
g
E,~a[%,(l+.ys).Tr[b_4S2 lQbl%,3r~]u,
4Cw 0370-2693/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI0370-2693 ( 9 4 ) 0 0 4 7 6 - N
(1)
370
J. W. Jun, C. Jue / PhysicsLettersB 329 (1994)369-373
where S 2 = 1 - C~v = sin20w, g = 41ra/S~v. The one-loop form factors ~r ~b and ~r ~ are 5r[b=~-~,+ ~
g2 {
~
R
[W'(-1)-Z'(-1)]-M'(-1)
+ ~--~ - ~f---ff- I Qb [ + 2 -----ff--+ [ -R+ ~-ib=l+
½- (1 - R ) IQb I1 IVY( --ME, ME) + ~] +R[V~( -M~, M2w,ME) + ~1 + VCt'), ~
(1--R)Qb2 [ 4 - 2 ( - M ~ + 2 m ~ ) S ( - M ~ , m
2,mb2)lPm
- l n m~w - 3 In R - 4 + M E z K ( - M ~ , mE, mE)
(
+ I z e ' ( - 1 ) - ~ ( 1 - R ) 1+2 Tr Q~ Ln
m)
- ( 1 ) l - ~ lR [ W ' ( - 1 ) - Z ' ( - 1 ) ] + M ' ( - 1 ) +
( 1 - R ) a Q ba [ V I, ( - - M z2, ME) +~21] ,
(2)
where R = cosZ0w and the functions j ( q 2 , m 2, mE) and ~5~(q2, m~, m 2) are defined by formulae (2.16) and (3.4) of Ref. [2]. The primed quantities reflect the finite mass effects of m, and rob. Thus, W'( - 1) = W( - 1) + Wt,b( -- 1),
Z'( - 1) =Z( - 1) +Z,,b( -- 1),
M'(-1)=M(-1)+Mt.b(-1),
zF'(--1)=zF(--1)+Zffb(--1),
V~( -M2z, M~v) = I V,b 12[V~( -M2z, M2w)+ V~( -M~, M~v, m~)] , V~( - M z ,2 Mz) 2 = Vl( - M 2, M2z) + V~( - M 2, ME, m2) , V~( --Mz,2 M 2w, M2w) = [Vtb 12[ V2( --M2z, M2w,ME) + V~( - M 2, M~v, M2w, rn~)] ,
~(5r-11)+3 1-rrln
vct'=-±(l+2R)3
r)
]Vtb[2,
(3)
where Vtb is the Kobayashi-Maskawa matrix element and r = m t2/ M w2. The finite functions W ( - 1), Z ( - 1), M(-1),zF(-1) Vl( - M z ,2 Mw), 2 V:( --ME, M~v, MEw) can be found in the appendix A of Ref. [3] and the functions V~ ( -MZz, M~v, m 2) and V~( -Mz,2 Mw,2 M 2, mE) are defined by formulae (A.9) and (A. 10) of Ref. [9]. The other mass dependent parts are ,
( m --MEw, ~ )mr,2 W,.b( -- 1) = IV,hi 2 --6[/3( mE ~ + 3 "~2 ll ( -- M 2w, m,z , mb2) +3 ,va W
~"
-/3( - M 2 , 0, 0)] I,(-M2w, mbz, mt2)
)
W
1 Zt.b(--1) = -- ~ [ 3 - 8(1 --R) +-~( 1 - R ) z] [13(--ME, m,2, m 2) --13(-M 2, 0, 0)]
J.W.Jun, C. Jue/PhysicsLettersB 329 (1994)369-373
371
1 - ~ [ 3 - 4 ( 1 - R ) +8(1 - R ) 21 [13( - M 2, m 2, m 2) -I3( -M2, 0, 0)1 (~)
+
2
U'mt--~wIo( - U l , m,,2 mZt) +
(3) m 2 - M2
m 2, m 2) ,
Io( --ME,
Mt.b(-- 1) = [ - - 4 + 3 ~ ( 1 - R ) ] [ 1 3 ( - M 2, mr,: m 2 ) - I 3 ( - M 2 , 0 , 0 ) ] + [-2+8(1-R)] r Zt,b(--1)=-×
(~)
[ / 3 ( - M 2, m 2, m 2 ) - 1 3 ( - M 2, 0, 0 ) 1 ,
2 _m, _ [ 1 - m t 2 J ( - - M 2,mr,2 m 2 ) ] +
M~v
mt mt + ( -- '~1 + 2M~: [12 In j-~-T ,vlz mEt + M--~z:
3) M---~w m~ [1-m~5(-M2z'
[ 1 ln~.2m~
× ~ M~
+
m~
- -
Mi
(
+
-
[3_8(1_R) +~(l_R)21
Mm'~4'M~I:~vJ(--ME, mr,2 mE) ] W
-
1
Z I
1
m~, m~,)] + ~-~ [ 3 - 4 ( 1 - R )
1
-~
+
m~
°4 '"~ ~ M2 ~ , z
z
2M~, MwMJ
w~"
+8(l-R)
t
2]
_ . , 2 m~.,,,~)
]
,.. z,
•
(4)
Therefore, FEW-- FBom 1 + 2 Re(A') -~3Sw2 Re(B')
=/'Born + FEWH~p,
(5)
where /"born =
Mza ~ _ 16C2S 2
4m~, M--T G'o ,
['~[4mZ~'~2m~ A'=Sr ]b--1, B ' : S r ~--Sr ,b, G'o:[l+ "-~z)(1-4S2)Z+[l-
--~z ).
(6)
Let us turn into the QCD virtual corrections which can be obtained from the previous results by keeping only the pure QED terms, setting Qb = 1, replacing a by the strong coupling constant a~ and applying an overall factor CF = ~. Thus, QCD Fl-loop =/'Born' 2 Re(A') ,
(7)
where A'---~
'lb,c
-- 1
,
~ r ~ b ' ~ - - l + 4~s---~w
m:
(l-R)
- I n M---~w- 3 In R - 4 + M ~ (
[4-2(-M~+2m~):(-M~,mLm~,)IPI~
)
-M~, mE, m~) .
(8)
Since the converting back and forth between the Euclidean metric and the Minkowskian metric is straightforward in practice and the results are independent of metric when the physical quantities are expressed in terms of the squared masses, we will use the Minkowskian metric for convience in the bremsstrahlung integrals. Thus, the electroweak bremsstrahlung is
372
J. W. Jun, C. Jue / Physics Letters B 329 (1994) 369-373
EW /"l~rem.
a 1 f d3p d3q d31 4 (2~r)gMz 2po 2qo ~ o 3 ( p + q + l - k )
~_, I"/~ brem. Ew ]2,
(9)
spin
where EW "~bmm.--
2q~+ly~], , --eQo n (l)eU(k)ff(p)[.[2p ~-~+ ~'d - Y u Y u -~2 ] t a T s ) v (q), 4CwSw
(lO)
4 2 and NI = 2p. l, N2 = 2q. l, a = 1 - gSw. Therefore,
EW
Fbrem" =
4.._a~(Qb) 2 7r
1
{
v/l_4m2/M2z FBom_(ME-2m~)I12--mb2(lll
1
+•22)--11--12
m2
2M~_2mz[(l+Mz'~z)(I°+I°)+2I])
(11)
•
The integrals I~2, 11I, 122, 11, 12, I °, I °, I can be obtained from the corresponding integrals of Ref. [ 11 ] by the substitutions Mz~Mw, ms ~ ml, mb ~ m2. Following the previous-mentioned recipe, the QCD bremsstrahlung, FbQCD . . . . can be read easily. The IR divergences of the bremsstrahlung integrals are regularized by a finite photon mass A and the IR divergencs of the virtual corrections are dimensionally regularized. Using the correspondence 2/ (4 - n) - y + In 47r ~ In Az, Pm can be replaced by In (~'/A) with an arbitrary parameter ~"with dimension of mass. Then, it can be easily checked that the partial width F( = Fao~ + --lr'EWqoop+/'~r~m. +/'i-Q~l~p-+ Fb~°~:,D.) is IR finite. In on-shell renormalization scheme of Sirlin, Mw should be determined by the equation from the/z-decay process:
M2w( 1 - M2w~= ora
MU
1
(12)
v~c~ 1- Ar'
where Ar = ~ a
Re{X (Mw, Mz, MH, my) }
--~Re
( -~ (l+2Zra~LnM~ 2
"v
+
R
(I_R)~[W'(-~)-Z'(-I)]
+ ~I-R [ W ' ( O ) - W ' ( - 1 ) - ~ R ( R + I ) + - - f + ( 1|~ J /911_~RR lnR ] )
(13)
As before, the function W'(0) consists of two parts:
w'(o) = w(o) + w,,e(o),
(14)
where W(0) can be found in Eq. (A.2) of Ref. [3], and m----~t2 W,.o(O)=lV,~l 3M2Ii(O, m2, m 2) + 3 - -
)
11(0, m 2, m 2) •
Putting all together, the l-loop corrected partial width F(Z--* bb) can be summarized as follows:
(15)
373
J. W. Jun, C. Jue / Physics Letters B 329 (1994) 369-373
GF F(Z~bb)- M 81/~¢rG6 1- 4m ME I.f 1-Ar+2Re(A'+A')-~3S~vRe(B') 4 1 [a(Qb)Zq-~3as][(M~--2m~)l~z--m~(ll~ + ~/1-4m~/MZz ¢r
1
+122) --11 --I2
1 ((m~tlO+IO)+21)]) 2M~- 2m~ 1+ MU' '
(16)
In conclusion, we obtain the analytical formulae for the partial width F(Z ~ bb) including the electroweak and QCD l-loop radiative corrections and photonic and gluonic bremsstrahlung for arbitrary masses of t and b quarks. If we physically determine the arbitrary mass parameter ~', we can also obtain the numerical results by use of the input parameters GF, Mz, Mn and my. In the near future, we wish to use Eq. (16) in the numerical analysis of the results of LEP precision experiments to find the improved bounds on mt and MH. This work was supported in part by the '93 program of the Inje Research and Scholarship Foundation.
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