- Email: [email protected]
Distributions of leptons in decays of polarised heavy quarks
NH NUCLEAR PHYSICS B ELSEVIER
Nuclear Physics B 427 (1994) 3—21
Distributions of leptons in decays of polarised heavy quarks * A. Czarnecki a M. Jezabek b,c Instituf fir Physik, Johannes Gutenberg-Universita:, D-55099 Mainz, Germany 1 b Institute of Nuclear Physics, Kawiory 26a, PL-30055 Cracow, Poland C Institut für Theoretische Teilchenphysi/c, Universitdt Karisruhe, D-76128 Karlsruhe, Germany a
Received 23 February 1994; revised 15 March 1994; accepted 26 May 1994
Absfract Analytic formulae are given for QCD corrections to the lepton spectra in decays of polarised up and down type heavy quarks. These formulae are much simpler than the published ones for the corrections to the spectra of charged leptons originating from the decays of unpolarised quarks and polarised up type quarks. Distributions of leptons in semileptonic A~and Ab decays can be used as spin analysers for the corresponding heavy quarks. Thus our results can be applied to the decays of polarised charm and bottom quarks. For the charged leptons the corrections to the asymmetries are found to be small in charm decays whereas for bottom decays they exhibit a non-trivial dependence on the energy of the charged lepton. Short lifetime enables polarisation studies for the top quark. Our results are directly applicable for processes involving polarised top quarks.
1. Introduction Inclusive semileptonic decays of heavy flavours play an important role in present day particle physics. In the near future with increasing statistics at LEP and good prospects for B-factories quantitative description of these processes may offer the most interesting tests of the standard quantum theory of particles. At the high energy frontier semileptonic decays of the top quark will be instrumental in establishing its properties [1,21. *
Work supported in part by BMFT under contract 056KA93P, by KBN under grant 2P30225206 and by
DFG. 1
Pennanent address.
0550-3213/94/$07.00 ® 1994 Elsevier Science By. All rights reserved SSDIO55O-321 3(94)00244-9
4
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
Recent theoretical developments, discovery of heavy quark symmetries [3,4] and effective theory [5] as well as subsequent applications of an expansion in inverse powers of the heavy quark mass mQ [6], have led to a consistent treatment of semileptonic decays of charmed and beautiful hadrons. In the limit mQ oo the inclusive decays of heavy flavour hadrons are described by the decays of the corresponding heavy quarks. For finite quark masses there arise non-perturbative corrections due to the heavy quark motion inside the heavy hadrons [71. A systematic approach to these corrections based on 1 /mQ expansion is described in Ref. [6] and other articles cited in Ref. [41. In the present article we consider only perturbative QCD corrections to the decays of heavy quarks. Non-perturbative effects have to be treated according to the recipes provided by the simple model of Ref. [7] or by the more formal approach of Ref. [6]. On the other hand inclusion of perturbative corrections is necessary for a quantitative description of semileptonic decays because these corrections are of the order of 20 percent. Interesting new opportunities are provided by decays of polarised heavy quarks. It is well known [1,2] that the heavy quarks produced in Z0 decays are polarised. According to the Standard Model the degree of longitudinal polarisation is fairly large, amounting to (Pb) = —0.94 for b and (PC) = —0.68 for c quarks [1]. The polarisations depend weakly on the production angle. QCD corrections to the production cross section have been calculated recently [8]. Unfortunately, this original high polarisation is to large extent washed out during hadronisation into mesons. At the time being only charmed and beautiful A baryons seem to offer a practical method to measure the polarisation of the corresponding heavy quarks. This method proposed by Bjorken [9] has been successfully applied to both semileptonic [101 and hadronic [11] decays of Lambda baryons. The polarisation transfer from a heavy quark Q to the corresponding AQ baryon is 100% [12], at least in the limit mQ oc. In view of the growing sample of A~and ~1b baryons produced at LEP there arises an opportunity to measure the polarisation of the c and b quarks originating from the decays of the Z bosons. The angular distributions of charged leptons [13,14] and neutrinos [15] from semileptonic decays of A~and Ab can be used as spin analysers for the decaying heavy quarks. Polarisation studies are much more direct for the top quark which is a short lived object and decays before hadronisation takes place. Angular distributions of leptons from semileptonic decays of the top quark can provide a very interesting information on the space—time structure of the corresponding electroweak current [16]. First order QCD corrections to the energy spectra of charged leptons in the decays of c and b quarks have been calculated analytically in Ref. [17] and Ref. [18]. The results of a later analytical calculation presented in Ref. [19] disagree with those of Ref. [17] and Ref. [18] and agree with a Monte Carlo for charm and bottom [20] as well as with numerical results for top [211. In Ref. [211 the formulae from Ref. [20] were used and the accuracy achieved was good enough to observe discrepancies with Ref. [17] and Ref. [18]. The formulae for the virtual corrections used in Ref. [20] and Ref. [21] were taken from the classic articles on muon decays [22]. The formulae given in the present article agree with the ones given in Ref. [19] but are much simpler. They agree also with those given in Ref. [231 for the joint angular and energy charged lepton distribution in top quark decays. In the latter case simplification is even more striking. Some numerical results for polarised charm and bottom quarks —*
—*
.4. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
5
have already been presented in Ref. [15]. In Section 2 the formulae are given for the QCD corrected distributions of leptons originating from weak decays of polarised heavy quarks: Subsection 2.1 contains our main result the triple differential angular and energy distributions of leptons, and in Subsections 2.2 and 2.3 the double differential distributions are described. The corresponding formulae for a massless quark in the final state are presented in Section 3. In Section 4 QCD corrections to the angular distributions in charm and bottom decays are presented. For the sake of completeness we summarize our calculations based on Ref. [19] and Ref. [23] in Appendix A. —
2. Angular and energy distributions In this Section we give the formulae2 for the distributions of leptons from the weak decays of a polarised heavy quark Q of mass m 1 and the weak isospin 13 = ±1/2.The mass of the quark q onginating from the decay is denoted by m2, and = m2/mI. These formulae are written in the Q rest frame. However, the two variables which we use, namely the scaled energy x = 2t°/mi = 2Q £/m~,where £ is the four-momentum of the 2/m~f charged lepton, and the scaled effective mass squared of the leptons y = (t + v) are Lorentz invariants. For the massless lepton case considered in the present article also the third variable cos 0, can be related to the scalar product £. s, where ~15= (0, s) is the spin four-vector of the decaying quark and B denotes the angle between s and the direction of the charged lepton. S an Is! = 1 corresponds to fully polarised, S = 0 to unpolarised decaying quarks. The distribution of the neutrino for 13 = ±1/2 is given by the formulae describing the distribution of the charged lepton from the decay of Q with the weak isospin 13 = ~ 1/2. .
2.1. Triple differential distributions The QCD corrected triple differential distribution for the semileptonic decay of the polarised quark with 13 = ±1/2 can be written in the followingway: dF~ dxdydcosO
G~m~ IVciu~sI2 32ir3 (1—y/y)2+y2 x
F~(x,y)+ScosOJ~(x,y)
—~[F~(x,y)+ScosOJ~(x,y)]~. 3ir j
(1)
Vcyj,~denotes the element of the Cabibbo—Kobayashi—Maskawa quark mixing matrix corresponding to the decay channel under consideration. The effects of W propagator are included as a factor 1/[(1 —y/9)2+y2], where 9 =m~/m~ and y= r~/m~ cf. 2The Fortran77 version of the formulae given in this article is available upon request from [email protected].
6
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
Ref. [19]. These effects are importantfor the top quark [19,24]. For charm and bottom quarks m1 is much smaller than the mass of W boson and the four-fermion limit can be employed in which the factor mentioned above is replaced by 1. The Dalitz variables x and y fulfil kinematic constraints: 0
~ X
0 ~
1
Xm
=
Ym
= X(Xm
~
—
—
x)/(1
—
x),
(2)
or 2, 0~y~(1—) w_~
(3)
The functions w±and other useful kinematic functions are defined as follows: p~=~(1 —y+e2) ~
—
P±~VØ ±~3= 1 w~, }~,=~ln(p~/p_) =ln(p~/), —
Y~=~ln(w+/w....) =ln(w+/~/~), (4)
Zm(1X)(1y/X).
The functions F~(x,y) and J~(x, y) corresponding to Born approximation read (5)
F~(X,Y)X(XmX),
J 1~(x,y)=F~(x,y),
(6)
F~(X,y)(Xy)(XmX+y),
(7)
J~(X,y)(Xy)(XmX+y2y/X).
(8)
The first order QCD corrections and the corresponding functions F~(x, y) have been calculated in Ref. [19] whereas ijf(x, y) has been given in Ref. [23]. In the course of the present work we rederived these old results and from them we obtained the much simpler expressions given in this3article. The result for Jj (x, y) is new. The formulae for F~(x, y) and J~(x, y) read 3Throughout this article polylogarithms are defined as real functions. In particular:
Li 2(x)
=
—
JdYlnll
—
y~/y,
Li3(x)
=
Jd~Li2(~/Y.
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
7
F~(x,y) =F~(x,y)~o+
+ A~,
(9)
~
+B~,
(10)
where [Li2(1~_~-~)+Li2(1_
~
—Li2 (i
—
l_Y/x) _Li2(1_~_L~)
1 ;Y/x) +Li2(w) —Li;(w÷)+4Y~ln]
+4(1_~Yp)lfl(Zm_2)_4lflZm, ~Pt =Li2(w..)
=
+Li2(w~)—Li2(x) —Li2(y/x),
ln ,
cP4=~ln(1—x) ~5=~ln(1—y/x), At
=
(11)
x+
2)—2e2y, A~=—p~(3+2x+y+e At=—(1 —y)(3+2x+y) +22(1 +5x) +~, A~=5—5x—2(4+5x) A~= —2xy + 9x
—
4x2
_~‘l,
—
—
y2
—
72x,
A~=~(—4xy+4y —y2 + y2/x) + ~2[x +y Bt
=
—
x/(1
—
y/x)],
(12)
—x + E2x,
B~=p~[(5—2x—y—2y/x—2/x)+2(—1+2/x)] +e2(—xy—x+2y—y/x—Y2/x)+e4(x+y/x), B~=(1—y)(5--2x—y—2y/x—2/x)+2e2(—3+7x+2/x) +~(1 —2/x) B~=—3+x+2/x+2(4—9x—4/x)+4(—1+2/x), B~=
—2xy
+ 3x
—
4x2 + 6y
—
y2
—
2Y2/x
—
7e2x,
B~=~(2—2x2+2Y—3Y2—2y/x+3y2/x) +~e2[—4+x+y+2y/x—x/(1—y/x)]+e4,
(13)
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A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
Aj=—2xy+x+y+y2+2(x—y), A~=p~(—5+2x—3y+2)—22y, A~=(l—y)(—5+2x—3y)+6e2(l+x—2y)—4, A~=5 +4xy
—
5x+ 3y
—
5y2
—
2y2/x+
~2(4
5x+ lly)
—‘~,
A~=6xy+9x—4x2 lly—2y2+2y2/x+72(—x+Y), —
A~=~(3xy+2y—3y2—2Y2/x)+~2[3Y—y/(1—x)], Bj
=
—2xy
—
x
—
5y + Y2
—
2y2/x + e2(x
—
(14)
y),
B~=pft(3+10x+y+10y/x—2/x)+2(1+2/x)] +e2xy+x—2y+y/x+Y2/x—4x+y/x), B~=(1—y)(3+10x+y+l0Y/x—2/x) +2e2(—l+5x—4y—2y/x+2/x)—e4(1+2/x), ~
=—3 + l2xy+x—7y—y2
—
12y/x+8y2/x+2/x
+e2(4—9x+7y—4/x)+4(—l +2/x), B
2—y—2y2+10y2/x+7e2(—x+y), 5=6xy—9x—4x B~=~(2—5xy—2x2+2y+7y2 —2Y/x—2Y2/x) +~2[—4—9y+2y/x+Y/l—xfl+e4. 2.2. Double differential x
—
(15)
B distributions
For the top quark in the narrow W width limit y 0 the mass squared of the leptons is fixed at value y = (m~/m~)2 Thus, in this limit the triple differential distributions of subsection 2.1 reduce to the double differential x B distributions. In the four-fermion limit (m~ oc) the double differential x 0 distributions can be calculated from the following formulae: —#
—
—+
—
dF~ 1d dI’~ dxdcosef ~dxdydcos9 —
G~rn! vc~2(f~(x)+Scosof~(x)
—
2aS[f±(
) +Scos9J~(x)])~
(16)
where f~(X)X2(XmX)2/(1X),
(17)
j~(x)=f~(x,y),
(18)
A.
Czarnecki,
M. Jezabek/Nuclear Physics B 427 (1994) 3—21
9
f~(x)=~x2(xm—x)2[3—2X+2+22/(1—x)]/(1—X)2
(19) (20)
and f~(x)=fdYFi~(x~Y)~
(21)
j~(x)=fdYJt~(x~Y).
(22)
The integrals in Eqs. (21)—(22) can be calculated numerically. In the massless limit 0 analytic formulae for f~(x) and j~(x) have been derived. These formulae are listed in Subsection 3.2. The results for chann and bottom quarks have already been published in Ref. [15]. For the polarisation independent parts of the distributions they are in perfect agreement with the results of Ref. [19]. Thus, they are in conflict with the results of Ref. [17] and Ref. [18]; see Section 5 in Ref. [19] where a detailed comparison is given and cross checks described. A convenient way to present the results is to express the double differential distributions as products of the energy distributions and the asymmetry functions, cf. Eqs. (5)— (7) in Ref. [15], —~
~
dxdcosB where
=—~[1±a~(x)cos9] dx
(23)
,
a~(x) ±S[fo±(x)~~Ji±(x)]/[fo±(x)
~ft±(x)]
.
(24)
The functions a±(x) for charm and bottom quarks have been given in Ref. [15]. 2.3. Double differential Y
y
—
B distributions
We follow the normalisation convention of Ref. [25] and write the double differential B distributions in the following way:
—
w+
dI’~ dydcosB
=
j
If dx
—
dI’~ dxdydcosO 2rn5 (1 ~~j2
~-~[F 1(y) 3ir
where
+ScosOJ~(y)
~(~0(Y)
+ Scos9J~(y)]’L j
(25)
10
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
Fo(y)=4p3[(1
2], ...~2)2+y(l +2
—2Y
,J~(y)=Fo(y) +48y(yY~~
(26) (27) (28)
and Fi(y) =12fdxF~(x~Y)~
(29)
Ji~(Y)=12fdxJi~(x~Y).
(30)
Eq. (29) provides a very non-trivial cross check of this calculation. It has been verified both analytically and numerically that for F~(x, y) given by Eq. (9) the integrals in (29) are equal. Moreover, .F~(y) can be calculated in an easier way, see formula (2.6) in Ref. [25], if the phase space integrals over the momenta of leptons are performed first. This cross check is also fulfilled. It has been shown, see Section 5 of Ref. [19], that when F~(y) is analytically continued it agrees with the analytic results of Ref. [26] for QCD corrections to the width of W boson. All these tests indicate that our formulae for F~(x, y) are correct. Since all the calculations for both F~(x, y) and J~(x, y) were performed in parallel using the same algebraic computer programs we believe that the same is true also for J~(x,y). An analytic expression for F~(y) has first been given in Ref. [25]. For the sake of completeness we rewrite this expression using the notation of the present article and some identities for dilogarithms. Our formulae for F 1 (Y) and J~(y) read F1(y)=Fo(y)~Po+At~Pi+A2Y~+A3Y~+A4p3ln+A5p3,
(31)
Y)J~(Y)!t’o+8t!Pl +I3~Yw+l3tYp+l3~p3lnE+I3~p3 +13~Y~Y,,+B~Y~lne, 1~o+BI~1~ Y)~(Y)! +B~YwYp +13fl’wlfle+48Y2 (13j +B~p0/p3),
(32) (33)
where !ko=2[4Li 2(2p3/p+) —4Y~,ln(2p3/p÷) —lnp.lnw~+lnp+lnw]po/p3 21~Y, (34) +8ln(2p3) !Pi =4Li 2(w) —4Li2(w~) (35) and A1 =Fo(y)po/p3, 2)[1+y—4y2—2(2—y)+41 A2—8(1 —
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
11
A3=—2[3+6y--21y2+12y3—e2(1+12y+5y2)+e4(11+2y)—e6], A 2—e2(4—y)+34] 4=—12[1+3y—4y A5=—2[5+9y—6Y2—2(22—9y) +5~]
(36)
Bt=5—9y2+4Y3—e2(8—8y+6y2)+4+2e6, 5~=—8[1 +4y Bt=—2[3(1
—
—
+ 3y
~2(3
—
4y2) + ~4(3 y) —
—
~6]
—y)2(1+4y) +e211 —5y2) —4(13—2y)
~
B~=—12[(1 —y)(l+4y) —2(4—y)+34] B~=2[15—y+2y2—2(12+7y)—3e4], f3~=—24(1+y—e2)(2p3+y2/p3), (37) =
5
—
42y + 45y2 + 4y3
—
2e2(4
+ 8y + 3y2) + ~ + 26,
8~=8[—1+8y+l0y2+2(3—9y—4y2)—e4(3—y)+6], B~=2[—3(1—y)(3—23y—4y2)+e2(1+24y+5y2)+4(7—2Y)+e6], B~~=—12[3—23y—4y2—e2(6—y)+34], L3~=—2[—15+37y—2y2+e2(l2+7y)+34], B~=—24[2p3(l —5y—2)—2(1+y—2)y/p3], 8~=—24[(1
—
y)(l
—
SY)
2e2(1
—
—
y) +
[Li
—
=
2(w~_) _Li2(P_/P÷)] w+p+ Y~[Li2(w_) + Li2(w~)+ 8ln(2p3)
4 [Li3(1) + Li3 (~i~)
—
—
Li3 (p_/p+)
2ln)’ —41n(w÷)], —
}
Li3 (w /w÷ )
+2Y~[Li2(w+)—Li2(w_) —4Li2(2p3/p+) +4Y~1n(2p3/p+) +2Y~lnp+—4Y~lnw+] + 41~,[Li2
(~‘~) w+p+
—
Li2(P.../P+)] .
(38)
3. Massless case 3.1. Triple differential distributions For m2 = 0 the functions in Eq. (1) simplify considerably. The limit easily obtained for F~(x,y) and J~(x,y). In this limit
—*
0 can be
12
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
F~(x,y)=x(l—x),
(39)
J~(x,y)=F~(x,Y),
(40)
F~(x,y)=(x—y)(l—x+y),
(41)
J~(x,y)=(x—y)(l—x+y—2Y/x)
(42)
and Fj~(x,y)=F~~Po+x~i—(3+2x+y)P203+5(1—x)P4 22YY2)~5+ ~y(4—4x—Y+Y/x), +(2XY+9X4X F~(x,y)=F~cP 2)~i+ (—5 +2x 3y)~2o3 0+ (—2xy +x+ Y +y +(5+4xy—5x+3y—5y2 —2y2/x)~4
(43)
—
+ (6xy + 9x
—
4x2
—
lly
2y2 +2y2/x)P5
—
+~y(2+3x—3y—2y/x),
(44)
Jt(x,y)=J~o—x~I+(5—2x—y—2y/x—2/x)~ 2G3 2+6y—y2—2y2/x)~ +(—3+x+2/x)P4+(—2xy+3x—4x 2+2y—3y2—2y/x+3y2/x), +~(2—2x Jj(x,y)=J 2 —2y2/x)P 1~Po+(—2xY—x—5y+y 1 +(3+lOx+y+ l0y/x—2/x)~2,,3 2 12y/x + 8y2/x + 2/x)P + (—3 + l2xy + x y 4 2 2y~+ 10y2/x)P + (6xy 9x 4x 5 2+2y+7y2 —2y/x—2y2/x), + ~(2—5xy—2x
5 (45)
—
—
—
—
—
—
—
(46)
where
=
+ 2Li
2 (11 ~/x) 2(x) + 2Li2(Y/x) + 1n
= ~1T2
+ Li2(Y)
—
Li 2(x)
—
Li2(y/x)
,
~P2~3=~(1 —y)ln(1 —y) cP4=~ln(l—x) ~5=.~ln(l—y/x).
(47)
Eqs. (43)—(45) agree with those given in the literature; cf. Eqs. (3.9) and (4.9) in Ref. [19] and Eq. (3.3) in Ref. [23].
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
3.2. Massless limitfor x
13
0 distributions
—
For e = 0 explicit analytic formulae can be derived for the double differential x B distributions. The functions f~(x),j~(x),f~(x) and j~(x)which appear in Eq. (16) are given by the following expressions, cf. Eqs. (3.10) and (4.10) in Ref. [19]: —
f~(x)=x2(l —x), f 1~(x)= ~x2(3 2x),
(48) (49)
—
(50) (51)
2(1—2x), j~(x)=~x f~(x)=f~(x)~o(x)+(1—x)[~(5+8x+8x2)ln(1—x) +j~x(l0—19x)1,
(52)
f~(x)=f~(x)~o(x)+~(41—36x+42x2—l6x3)ln(1—x) + ~x(82
—
153x + 86x2),
(53)
jRx) =j~(x)çbo(x) + (1— x)[~(—3+ 12x + 8x2 +4/x) ln(1 + ~(8
—
2x
—
~O(X)
For
—*
—
—
(54) 36x + 14x2
—
l6x3
—
4/x)ln(l
—
x)
l03x2 + 78x3)
(55)
2(1
(56)
=2Li
2(x) + 3.3. Massless limit for y
x)
l5x2)],
jj(x) =j~(x)çbo(x) + ~(11 + ~1(—8+ 18x
—
—
x).
~ir2+ln —
0 distributions
0 the functions in Eq. (25) read, cf. Eqs. (3.l)—(3.2) in Ref. [25],
.Fo(y)2(l—y)2(1+2y),
(57)
3~(y)=Fo(y),
(58)
J~7y)=2(1—y)(l—1ly—2y2)—24y2lny,
(59)
Fl(y)=Fo[~1r2+4Li
2)
2(y)+2lnYln(1—Y)]--(1—y)(5+9y—6Y +4y(l—y—2y2)lny+2(1—y)2(5+4y)ln(1—y),
(60)
J~(y)=—~ir2(1—y)(1+y+4y2)+(l—y)(15—y+2y2) +4(1 —y)(5+5y—4y2)Li2(y) +8(1—y)(2+2y—y2)ln(1—y)lny +8y(2+y—y2)lny+2(1—y)2(5+4y)ln(1—y),
(61)
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
14
y)=_~ir2(1+6y+9y2_4y3)+(1_y)(15_37y+2y2) + 240y2 [Li (1) 3 Li3 (Y) I 2+4y3 + l8y2lny]Li +4[5 —42y+45y 2(y) 2)ln(1 —y)lny +8(1 —y)(2— 13y—y +4y[4—(19+ir2)y—2y2] lny —2(1 —y)(l + 19y+4y2)ln(1 —y). —
(62)
4. Angular distributions Angular distributions are related to the double differential y —0 distributions discussed in Subsection 2.3, (1_E)2
dF~ dcos0
=
J 0
dy
dr~ dydcos0
(63)
,
with dI’~/dy d cos B given in Eq. (25). For the top quark and the narrow W width the double differential distributions are proportional to Dirac delta function t5(y’ m~/m?) and the integral in (63) is trivial. For the charm and for the bottom quarks the fourfermion approximation can be used. In order to illustrate the size of the radiative corrections discussed in the present article we employ the four-fermion limit and write the resulting angular distributions in the following way: —
dr± =FoR.()[1+a~() cosB]
d cos B
,
(64)
where r 2 2/384~r~. In Born approximation for completely polarised 0 = 1)G 5.m~I Vc~j~ I quarks (S= R.o()=l—82+86—8—244ln,
(65) (66)
a~()=~[1—122—364+44e6+38—244(3+22)lne]/7~~(). (67) QCD corrections decrease the total decay rate. In Fig. la the ratios R.(e)/lZo(e) are shown as functions of for a 5 = 0.1, 0.2, 0.3 and 0.4 as the solid, dashed, dotted and dash-dotted lines, respectively. The corrections to Eq. (66) are of the order of one percent or smaller. In Fig. lb the function a() is shown for a5 = 0, 0.2 and 0.4 as the solid, dashed and dotted lines, respectively.
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
15
0.5
1 0.95
~
&(c)
~
0.65 0.6
0
0.2
.1.,.
0.4
I
0.6
0.8
1
-01
0
0.2
0.4
I
0.6
0.8
1
Fig. 1. (a) QCD reduction of the total decay rate: the ratios R~(e)/7~o(e)as functions of e for a
5 = 0.1, 0.2, 0.3 and 0.4 — solid, dashed, dotted and dash-dotted lines; (b) QCD comacted angular asymmetry a (e) for a5 = 0, 0.2 and 0.4 — solid, dashed and dotted lines, respectively.
Acknowledgement
The authors gratefully acknowledge the fruitful collaboration with Hans KUhn on the subject presented in this article. M.J. thanks Kacper Zalewski for useful comments. Note added We have been urged to give more information on the pending conflict between the results of Ref. [19] and Refs. [17,18,7] in addition to what was stated in the Introduction. For the lifetime of the t quark the calculation of Ref. [19] leads to the result of Ref. [25], which has been confirmed by many other groups [28]. After this paper had been submitted for publication a preprint [29] appeared confinning the results of Ref. [19] on the e+ spectrum in charm decays. For bottom decays the controversy was partly clarified earlier, see footnote in Ref. [19]on page 27. Adding this to the crosschecks described in Ref. [19]and in the present article we consider the controversy as solved in favour of Ref. [19] and the related subsequent articles.
Appendix A. Derivations and formulae In this Appendix we sketch our derivation of Eq. (1). The method has been described in our earlier papers Ref. [19] and Ref. [23]. Many formulae presented here have already been given in these articles. We rewrite them using the notation adopted in the present article. Some formulae are simplified and misprints corrected. In the following Q, q, £, z.’ and G denote the scaled four-momenta of the decaying quark and the decay products: quark, andThus, gluon. common scaling factor at is 2 = 1,charged q2 = 2,lepton, £2 = neutrino = G2 = 0. the The gluon is massless. However, 1/mi, and Q
16
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
intermediate steps we introduce the scaled gluon mass A0 in order to regularize those 2, s the expressions which WeRintroduce P formulae = q + G, are z = given P spin four-vector of are theinfrared decayingdivergent. quark and = Q s. also Some only for the weak isospin 13 = 1/2 of the decaying quark. This implies that the analogous formulae for 13 = —1/2 can be obtained by replacing £ with ii and vice versa. —
—
A.1. Differential decay rate The QCD corrected differential rate is given by the following formula:
dr~= dr~+ dI’~ 3+ dF~4,
(68)
where
2M~ dr~=G~m~IVcrj,,i~
5
(69)
3dR.3(Q;q,e,i’)/ir
in Born approximation,
dr~= ~
(70)
describes the virtual gluon contribution and dV~ 4=~
(71)
comes from real gluon emission. Lorentz invariant n-body phase space is defined as dR.~(P;pi,p2,...,p~)=8(4)(P_~pj)JJ.~_~~!-,
(72)
A.2. Three-body contributions In Born approximation the rates for the decays into three-body final states are proportional to the expressions 2+y2], M~3=q~R.~/[(1 —y/9) M~ 2+y2]. (73) 3=q.eR.v/[(1—y/7) The formulae for F~(x, y) and J~(x, y) can be easily derived from (69) and (73) when Dalitz parametrization of the three-body phase space dR.
1~dxdydad(cos0)df3 (74) 3(Q;q,f,v) = 1 is employed. Integration over Euler angle fi is trivial, and non-trivial integrals over a read, see Appendix B in Ref. [23],
J
das .q2ir5cos9[~(1 +y —q2)
fdas.v=21rScos0[y/x_~(1+y_q2_x)].
(75)
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
17
As a final step the scalar products in M~3are expressed in terms of the variables x, y and cos 0, Q.s0,
£.s=—~xScosB 2—x+y)
(76)
,
Q.~~(l—q
v.q(1—q2—x)
and Eqs. (5) —(8) follow immediately. The same steps are performed for the three-body radiative corrections which are given by Eq. 70), where
Mt
2Q
2 +y2]~[q~vR~£H
3 = —[(1— y/9) + q~v(q £ + R £Q q .
+
£ + Q. yR. £
‘2(i.’
—
—
Q
.
pR~
0+ £R. q)H
.
R.
. £)(H÷+ H4],
(77)
—poY~/p 3)lnA0+(2po/p3)[Li2(i —Li2 (1_
—
____
p+w+
Y~(Y~ + 1) +2(lne+ ~)(Y~+ ~) 2—2y)1n]/y+4,
—
\
W.1~j
(78)
+[2p3Y~+(l—e H±=~[1±(l2)/y]Yp/p 3±uln.
(79)
A.3. Four-body contributions Neglecting those terms whose contributions to the decay rates are O( AG ln AG) or smaller (i.e. vanishing in the limit AG 0) one can cast the contribution of real radiation into the following expression: —~
1 ~+ 2+y2 L’Q.G2
1
i,4
(l—y/7)
________
I
1
—
5+ Q.GP.G 2
+
5+ (P.G)2
~
(80)
where 5~q. v[R.~(Q .G— 1) +G.I!RQ — Q.fR~G+G~tR~G], p [G . £R. q — q £R. G+ R . £(q. G — Q . G — 2q. Q)]
+ R £(Q. p q~G .
—
G. p q~Q),
5~=R.e(G.vq.G—q2P.p).
(81)
The four-body phase space is decomposed as follows:
dR. 4(Q;q,G,f,p)=dzdR.3(Q;P,t,~)dR.2(P;q,G).
(82)
18
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
G integration of M~4over dR.2 ( P; q, G) is performed. Due to Lorentz invariance all the integrals which appear in this calculation can be reduced to the scalar integrals After q is substituted by P
—
1nJdR.2(P;q~G) (Q.G)~.
(83)
Explicit formulae are listed in Appendix C of Ref. [231 for the reduction of tensor integralsfdR.2(P;q,G)(Q.G)~IGa,fdR.2(P;q,G)(Q.Gy1GaG$andfdR.2(P;q,G)(Q. G)nGaGflGY to the scalar ones. In the next step Dalitz parametrization is employed for the three-body phase space dR.3 (Q; P, £, v) in the same way as described in the2 previous subsection, = ~2 is replaced withbut P2 now = z. for q replaced with P in Eqs. (74)—(76). In particular q After integrations over Euler angles a and ~8the contribution of real radiation is split into two pieces: the infrared divergent part of the form const. [F~(x,y)
+ ScosBJ~(x,y)]I~~,
where 1djV1_
(84)
2)1_I/(PG)+21O/(PG), 2(1Y+E
and the rest which is infrared finite. For AG << 1 the infrared divergent part can be integrated over z and the result is proportional to
f
-~-
dz
‘div =
4 (i
—
~ ~)ln
(zrn—2) —2 lfl(Zm/2)
2
(e+Aa)2
_Li 2(1_ 1)7k)
_Li2(1_~)
_Y~(Y~+1)] .
(85)
When this result is added to the contribution of virtual corrections the infrared divergent terms ln AG cancel out. The sum is simplified using the following identity: ‘~
Li2
\.
1
—
____
—
p~w~j
Li2
\
1
—
w.~j
—
Li2
=Li2(w....) —Li2(w÷)—2Y~lnp+.
\1
—
P+ (86)
For the infrared finite part the limit AG 0 is performed. The formulae for the scalar integrals I,~simplify considerably in this limit and read —~
‘~2
~/(P. G),
I—i =irY~(z)/~’~,
A. Czarnecki, M. Jezabek/ Nuclear Physics B 427 (1994) 3—21
19
~
2(n+1)(P2)~~1’/~A
‘I
(n~0),
(87)
where4 .~i=Q2P2
—
(Q
.
P)2
=
A(z)=z2—2(l+y)z+(1—y)2, ~ p(Z)~ —‘1 ~ P.G=~(z—2).
(88)
Integration of the infrared finite part over z is tedious. It would be difficult to accomplish this task without FORM [27]. All the integrals which appear can be reduced to integrals dz z’1/Am(z) and j’dz Z~ZY~(Z)/A/(Z). Thus, they can be derived from the recursion relations given in Appendix B of Ref. [19].
f
A.4. FormulaeforF~(x,y)and J~(x,y) After some algebra mentioned in preceding subsections one derives the following formulae for F~(x,y): F~(x,y)=~(x,y)+fl~(x,y,e2)—fl~(x,y,zm),
(89)
where the first term in r.h.s. is the sum of the three-body virtual corrections and the infrared divergent four-body ones. The two other terms originate from the infrared finite four-body piece. For J~(x, y) one has J~(x,y)=lC~(x,y)+JC~(x,y,e2)—K~(x,y,zm).
(90)
The results of Ref. [19] and Ref. [23] are presented in this way; see Eqs. (3.2)—(3.6) and (4.5)—(4.8) in Ref. [19], and (3.1), (A.1)—(A.l5) in Ref. [23]. Both fl~(x, y, z) and X~(x, y, z) are given by lenghty expressions: 7i~(x,y,z) =(a~+za~)Y~(z)/A312(z) + (a~+za~)Yp(z)/~/~3 +a~Yp(z)v5+(a~+za~)/A(z)+a~Z +a~z1 +a~lnZ+a~[Li 2(W+(z))+Li2(W..(z))],
(91)
~ (b~+ zb~)Yp(z)/’/Z~3~+ ~ 2(z)+ (b~+ zb~)/A(z)+ b~z+ b~z~ + (b~+ zb~)/A +b~lnz+b~[Li 2(W~(z))+Li2(W_(z))], (92) calculation only n ( 1 is needed. Thus, instead of Eq. (87) one can use I,, = for n = 0, 1. +
41n the present G)’+’(Q. p)fl/(p2)fl+l
20
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
where a~and b~are rational functions of x, y and
~2, and
W~(z)=~[1+y—z±~/~3].
(93)
These complicated expressions simplify dramatically for z special points
=
2
and z
= Zm.
At these
=
W±(2)=W±, A(Zm)
(x
=
—
YP(Zm)/\/5=
y/x)2,
1/)lfl(’l~)~ 2(
Li 2(W+(Zm)) +Li2(W_(Zm)) =Li2(x) +Li2(y/x)
,
(94)
and after some algebra Eqs. (9) and (10) are derived. References [1] J.H. KUhn and P.M. Zerwas, in Heavy flavours, ed. AJ. Buras and M. Lindner (World Scientific, Singapore, 1992), p. 434. [2] J.H. Kuhn et al., DESY Orange Report 92-123A (1992), Vol. I, p. 255; J.H. Kuhn, Top quark at a linear collider, in Physics and experiments with linear e~e colliders, ed. F.A. Harris et al. (World Scientific, Singapore, 1993), p. 72. [31N. Isgur and MB. Wise, Phys. Lett. B 232 (1989) 113; B 237 (1990) 527. [4] For excellent introduction and reviews see: J.D. Bjorken, in Proc. 4th Rencontres de Physique de la Vallee d’Aoste, La Thuile, Italy, 1990, ed. M. Greco (Editions Frontières, Gif-sur-Yvette, France, 1990); N. Isgur and MB. Wise, Heavy quark symmetry, in B decays, ed. S. Stone (World Scientific, Singapore, 1992), p. 158; M. Neubert, preprint SLAC-PUB-6263 (1993), to appear in Phys. Rep.; K. Zalewski, Heavy flavours (Theory), plenary talk at the EPS HEP93 Conference, Marseille, 22—28 July 1993, preprint CERN-TH.698l/93; MB. Wise, Status of Heavy Quark Theory, plenary talk at XVI lot. Symp. on Lepton-.photon interactions, Cornell University, Ithaca, New York, 10—15 August 1993. [5] E. Eichten and B. Hill, Phys. Lett. B 234 (1990) 511; H. Georgi, Phys. Lett. B 240 (1990) 447; B. Grinstein, NucI. Phys. B 339 (1990) 253. [6] J. Chay, H. Georgi and B. Grinstein, Phys. Lett. B 247 (1990) 399; I. Bigi, M. Shifman, N. Uraltsev and A. Vainshtein, Phys. Rev. Lett. 71(1993) 496; I. Bigi, M. Shifman, N. Uraltsev and A. Vainshtein, mt. J. Mod. Phys. A 9 (1994) 2467; A.V. Manohar and MB. Wise, Phys. Rev. D 49 (1994) 1310; A. Falk, E. Jenkins, A.V. Manohar and MB. Wise, Phys. Rev. D 49 (1994) 4553; R.L. Jaffe and L. Randall, Nucl. Phys. B 412 (1994) 79; M. Neubert, Phys. Rev. D 49 (1994) 4623. [7]G. Altarelli et aL, Nucl. Phys. B 208 (1982) 365. [8] J.G. Kdrner, A. Pilaftsis and M.M. Tung, preprint MZ-Th/93-30, Z. Phys. C, in print. [9] J.D. Bjorken, Phys. Rev. D 40 (1989) 1513.
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21 [10] See e.g. P. Bialas, J.G. KUmer, M.
Kramer and K.
Zalewski, Z. Phys. C 57 (1993) 115, and
therein. [11] See e.g. M. Jezabek, K. Rybicki and R. Rylko, Phys. Lett. B 286 (1992) 175. [12] FE. Close, J.G. KOmer, R.J.N. Phillips and D.J. Summers, J. Phys. G 18 (1992) 1716. [131 G. Kopp, L.M. Sehgal and P.M. Zerwas, NucI. Phys. B 123 (1977) 77. [14] B. Mele andG. Altarelli, Phys. Lett. B 299 (1993) 345. [15] A. Czarnecki, M. Je±abek,J.G. Kdrner and J.H. Kuhn, Phys. Rev. I_eU. 73 (1994) 317. [16]M. Je~abekand J.H. KUhn, Phys. Left. B 329 (1994) 317. [17] N. Cabibbo, G. Corbo and L. Maiani, Nucl. Phys. B 155 (1979) 93. [18] G. Corbo, Phys. Lett. B 116 (1982) 298; NucI. Phys. B 212 (1983) 99. [19] M. Jezabek and J.H. KUhn, Nucl. Phys. B 320 (1989) 20. [20] A. Ali and E. Pietarinen, NucI. Phys. B 154 (1979) 519. [21] M. Je~abekand J.H. Kuhn, Phys. Lett. B 207 (1988) 91. [22] R.J. Finkelstein, RE. Behrends and A. Sirlin, Phys. Rev. 101 (1956) 866; S. Berman, Phys. Rev. 112 (1958) 267; T. Kinoshita and A. Sirlin, Phys. Rev. 113 (1959) 1652. [23] A. Czarnecki, M. Je~abekand J.H. Kuhn, Nucl. Phys. B 351 (1991) 70. [24] M. Jezabek and J.H. KUhn, Phys. Rev. D 48 (1993) Rl910. [25]M. Je~abekand J.H. KUhn, Nucl. Phys. B 314 (1989) 1. [26] T.H. Chang, K.J.F Gaemers and W.L. van Neerven, Nucl. Phys. B 202 (1982) 407. [27] J.A.M. Vermaseren, Symbolic Manipulation with FORM, CAN, Amsterdam 1991. [28] CS. Li, R.J. Oakes and T.C. Yuan, Phys. Rev. D 43 (1991) 3759; A. Czarnecki, Phys. Left. B 252 (1990) 467; J. Liu and Y.-P. Yao, Int. J. Mod. Phys. A 6 (1991) 4925. [29] C. Greub, D. Wyler and W. Fetscher, Phys. Left. B 324 (1994) 109.
21
references
Nuclear Physics B 427 (1994) 3—21
Distributions of leptons in decays of polarised heavy quarks * A. Czarnecki a M. Jezabek b,c Instituf fir Physik, Johannes Gutenberg-Universita:, D-55099 Mainz, Germany 1 b Institute of Nuclear Physics, Kawiory 26a, PL-30055 Cracow, Poland C Institut für Theoretische Teilchenphysi/c, Universitdt Karisruhe, D-76128 Karlsruhe, Germany a
Received 23 February 1994; revised 15 March 1994; accepted 26 May 1994
Absfract Analytic formulae are given for QCD corrections to the lepton spectra in decays of polarised up and down type heavy quarks. These formulae are much simpler than the published ones for the corrections to the spectra of charged leptons originating from the decays of unpolarised quarks and polarised up type quarks. Distributions of leptons in semileptonic A~and Ab decays can be used as spin analysers for the corresponding heavy quarks. Thus our results can be applied to the decays of polarised charm and bottom quarks. For the charged leptons the corrections to the asymmetries are found to be small in charm decays whereas for bottom decays they exhibit a non-trivial dependence on the energy of the charged lepton. Short lifetime enables polarisation studies for the top quark. Our results are directly applicable for processes involving polarised top quarks.
1. Introduction Inclusive semileptonic decays of heavy flavours play an important role in present day particle physics. In the near future with increasing statistics at LEP and good prospects for B-factories quantitative description of these processes may offer the most interesting tests of the standard quantum theory of particles. At the high energy frontier semileptonic decays of the top quark will be instrumental in establishing its properties [1,21. *
Work supported in part by BMFT under contract 056KA93P, by KBN under grant 2P30225206 and by
DFG. 1
Pennanent address.
0550-3213/94/$07.00 ® 1994 Elsevier Science By. All rights reserved SSDIO55O-321 3(94)00244-9
4
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
Recent theoretical developments, discovery of heavy quark symmetries [3,4] and effective theory [5] as well as subsequent applications of an expansion in inverse powers of the heavy quark mass mQ [6], have led to a consistent treatment of semileptonic decays of charmed and beautiful hadrons. In the limit mQ oo the inclusive decays of heavy flavour hadrons are described by the decays of the corresponding heavy quarks. For finite quark masses there arise non-perturbative corrections due to the heavy quark motion inside the heavy hadrons [71. A systematic approach to these corrections based on 1 /mQ expansion is described in Ref. [6] and other articles cited in Ref. [41. In the present article we consider only perturbative QCD corrections to the decays of heavy quarks. Non-perturbative effects have to be treated according to the recipes provided by the simple model of Ref. [7] or by the more formal approach of Ref. [6]. On the other hand inclusion of perturbative corrections is necessary for a quantitative description of semileptonic decays because these corrections are of the order of 20 percent. Interesting new opportunities are provided by decays of polarised heavy quarks. It is well known [1,2] that the heavy quarks produced in Z0 decays are polarised. According to the Standard Model the degree of longitudinal polarisation is fairly large, amounting to (Pb) = —0.94 for b and (PC) = —0.68 for c quarks [1]. The polarisations depend weakly on the production angle. QCD corrections to the production cross section have been calculated recently [8]. Unfortunately, this original high polarisation is to large extent washed out during hadronisation into mesons. At the time being only charmed and beautiful A baryons seem to offer a practical method to measure the polarisation of the corresponding heavy quarks. This method proposed by Bjorken [9] has been successfully applied to both semileptonic [101 and hadronic [11] decays of Lambda baryons. The polarisation transfer from a heavy quark Q to the corresponding AQ baryon is 100% [12], at least in the limit mQ oc. In view of the growing sample of A~and ~1b baryons produced at LEP there arises an opportunity to measure the polarisation of the c and b quarks originating from the decays of the Z bosons. The angular distributions of charged leptons [13,14] and neutrinos [15] from semileptonic decays of A~and Ab can be used as spin analysers for the decaying heavy quarks. Polarisation studies are much more direct for the top quark which is a short lived object and decays before hadronisation takes place. Angular distributions of leptons from semileptonic decays of the top quark can provide a very interesting information on the space—time structure of the corresponding electroweak current [16]. First order QCD corrections to the energy spectra of charged leptons in the decays of c and b quarks have been calculated analytically in Ref. [17] and Ref. [18]. The results of a later analytical calculation presented in Ref. [19] disagree with those of Ref. [17] and Ref. [18] and agree with a Monte Carlo for charm and bottom [20] as well as with numerical results for top [211. In Ref. [211 the formulae from Ref. [20] were used and the accuracy achieved was good enough to observe discrepancies with Ref. [17] and Ref. [18]. The formulae for the virtual corrections used in Ref. [20] and Ref. [21] were taken from the classic articles on muon decays [22]. The formulae given in the present article agree with the ones given in Ref. [19] but are much simpler. They agree also with those given in Ref. [231 for the joint angular and energy charged lepton distribution in top quark decays. In the latter case simplification is even more striking. Some numerical results for polarised charm and bottom quarks —*
—*
.4. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
5
have already been presented in Ref. [15]. In Section 2 the formulae are given for the QCD corrected distributions of leptons originating from weak decays of polarised heavy quarks: Subsection 2.1 contains our main result the triple differential angular and energy distributions of leptons, and in Subsections 2.2 and 2.3 the double differential distributions are described. The corresponding formulae for a massless quark in the final state are presented in Section 3. In Section 4 QCD corrections to the angular distributions in charm and bottom decays are presented. For the sake of completeness we summarize our calculations based on Ref. [19] and Ref. [23] in Appendix A. —
2. Angular and energy distributions In this Section we give the formulae2 for the distributions of leptons from the weak decays of a polarised heavy quark Q of mass m 1 and the weak isospin 13 = ±1/2.The mass of the quark q onginating from the decay is denoted by m2, and = m2/mI. These formulae are written in the Q rest frame. However, the two variables which we use, namely the scaled energy x = 2t°/mi = 2Q £/m~,where £ is the four-momentum of the 2/m~f charged lepton, and the scaled effective mass squared of the leptons y = (t + v) are Lorentz invariants. For the massless lepton case considered in the present article also the third variable cos 0, can be related to the scalar product £. s, where ~15= (0, s) is the spin four-vector of the decaying quark and B denotes the angle between s and the direction of the charged lepton. S an Is! = 1 corresponds to fully polarised, S = 0 to unpolarised decaying quarks. The distribution of the neutrino for 13 = ±1/2 is given by the formulae describing the distribution of the charged lepton from the decay of Q with the weak isospin 13 = ~ 1/2. .
2.1. Triple differential distributions The QCD corrected triple differential distribution for the semileptonic decay of the polarised quark with 13 = ±1/2 can be written in the followingway: dF~ dxdydcosO
G~m~ IVciu~sI2 32ir3 (1—y/y)2+y2 x
F~(x,y)+ScosOJ~(x,y)
—~[F~(x,y)+ScosOJ~(x,y)]~. 3ir j
(1)
Vcyj,~denotes the element of the Cabibbo—Kobayashi—Maskawa quark mixing matrix corresponding to the decay channel under consideration. The effects of W propagator are included as a factor 1/[(1 —y/9)2+y2], where 9 =m~/m~ and y= r~/m~ cf. 2The Fortran77 version of the formulae given in this article is available upon request from [email protected].
6
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
Ref. [19]. These effects are importantfor the top quark [19,24]. For charm and bottom quarks m1 is much smaller than the mass of W boson and the four-fermion limit can be employed in which the factor mentioned above is replaced by 1. The Dalitz variables x and y fulfil kinematic constraints: 0
~ X
0 ~
1
Xm
=
Ym
= X(Xm
~
—
—
x)/(1
—
x),
(2)
or 2, 0~y~(1—) w_~
(3)
The functions w±and other useful kinematic functions are defined as follows: p~=~(1 —y+e2) ~
—
P±~VØ ±~3= 1 w~, }~,=~ln(p~/p_) =ln(p~/), —
Y~=~ln(w+/w....) =ln(w+/~/~), (4)
Zm(1X)(1y/X).
The functions F~(x,y) and J~(x, y) corresponding to Born approximation read (5)
F~(X,Y)X(XmX),
J 1~(x,y)=F~(x,y),
(6)
F~(X,y)(Xy)(XmX+y),
(7)
J~(X,y)(Xy)(XmX+y2y/X).
(8)
The first order QCD corrections and the corresponding functions F~(x, y) have been calculated in Ref. [19] whereas ijf(x, y) has been given in Ref. [23]. In the course of the present work we rederived these old results and from them we obtained the much simpler expressions given in this3article. The result for Jj (x, y) is new. The formulae for F~(x, y) and J~(x, y) read 3Throughout this article polylogarithms are defined as real functions. In particular:
Li 2(x)
=
—
JdYlnll
—
y~/y,
Li3(x)
=
Jd~Li2(~/Y.
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
7
F~(x,y) =F~(x,y)~o+
+ A~,
(9)
~
+B~,
(10)
where [Li2(1~_~-~)+Li2(1_
~
—Li2 (i
—
l_Y/x) _Li2(1_~_L~)
1 ;Y/x) +Li2(w) —Li;(w÷)+4Y~ln]
+4(1_~Yp)lfl(Zm_2)_4lflZm, ~Pt =Li2(w..)
=
+Li2(w~)—Li2(x) —Li2(y/x),
ln ,
cP4=~ln(1—x) ~5=~ln(1—y/x), At
=
(11)
x+
2)—2e2y, A~=—p~(3+2x+y+e At=—(1 —y)(3+2x+y) +22(1 +5x) +~, A~=5—5x—2(4+5x) A~= —2xy + 9x
—
4x2
_~‘l,
—
—
y2
—
72x,
A~=~(—4xy+4y —y2 + y2/x) + ~2[x +y Bt
=
—
x/(1
—
y/x)],
(12)
—x + E2x,
B~=p~[(5—2x—y—2y/x—2/x)+2(—1+2/x)] +e2(—xy—x+2y—y/x—Y2/x)+e4(x+y/x), B~=(1—y)(5--2x—y—2y/x—2/x)+2e2(—3+7x+2/x) +~(1 —2/x) B~=—3+x+2/x+2(4—9x—4/x)+4(—1+2/x), B~=
—2xy
+ 3x
—
4x2 + 6y
—
y2
—
2Y2/x
—
7e2x,
B~=~(2—2x2+2Y—3Y2—2y/x+3y2/x) +~e2[—4+x+y+2y/x—x/(1—y/x)]+e4,
(13)
8
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
Aj=—2xy+x+y+y2+2(x—y), A~=p~(—5+2x—3y+2)—22y, A~=(l—y)(—5+2x—3y)+6e2(l+x—2y)—4, A~=5 +4xy
—
5x+ 3y
—
5y2
—
2y2/x+
~2(4
5x+ lly)
—‘~,
A~=6xy+9x—4x2 lly—2y2+2y2/x+72(—x+Y), —
A~=~(3xy+2y—3y2—2Y2/x)+~2[3Y—y/(1—x)], Bj
=
—2xy
—
x
—
5y + Y2
—
2y2/x + e2(x
—
(14)
y),
B~=pft(3+10x+y+10y/x—2/x)+2(1+2/x)] +e2xy+x—2y+y/x+Y2/x—4x+y/x), B~=(1—y)(3+10x+y+l0Y/x—2/x) +2e2(—l+5x—4y—2y/x+2/x)—e4(1+2/x), ~
=—3 + l2xy+x—7y—y2
—
12y/x+8y2/x+2/x
+e2(4—9x+7y—4/x)+4(—l +2/x), B
2—y—2y2+10y2/x+7e2(—x+y), 5=6xy—9x—4x B~=~(2—5xy—2x2+2y+7y2 —2Y/x—2Y2/x) +~2[—4—9y+2y/x+Y/l—xfl+e4. 2.2. Double differential x
—
(15)
B distributions
For the top quark in the narrow W width limit y 0 the mass squared of the leptons is fixed at value y = (m~/m~)2 Thus, in this limit the triple differential distributions of subsection 2.1 reduce to the double differential x B distributions. In the four-fermion limit (m~ oc) the double differential x 0 distributions can be calculated from the following formulae: —#
—
—+
—
dF~ 1d dI’~ dxdcosef ~dxdydcos9 —
G~rn! vc~2(f~(x)+Scosof~(x)
—
2aS[f±(
) +Scos9J~(x)])~
(16)
where f~(X)X2(XmX)2/(1X),
(17)
j~(x)=f~(x,y),
(18)
A.
Czarnecki,
M. Jezabek/Nuclear Physics B 427 (1994) 3—21
9
f~(x)=~x2(xm—x)2[3—2X+2+22/(1—x)]/(1—X)2
(19) (20)
and f~(x)=fdYFi~(x~Y)~
(21)
j~(x)=fdYJt~(x~Y).
(22)
The integrals in Eqs. (21)—(22) can be calculated numerically. In the massless limit 0 analytic formulae for f~(x) and j~(x) have been derived. These formulae are listed in Subsection 3.2. The results for chann and bottom quarks have already been published in Ref. [15]. For the polarisation independent parts of the distributions they are in perfect agreement with the results of Ref. [19]. Thus, they are in conflict with the results of Ref. [17] and Ref. [18]; see Section 5 in Ref. [19] where a detailed comparison is given and cross checks described. A convenient way to present the results is to express the double differential distributions as products of the energy distributions and the asymmetry functions, cf. Eqs. (5)— (7) in Ref. [15], —~
~
dxdcosB where
=—~[1±a~(x)cos9] dx
(23)
,
a~(x) ±S[fo±(x)~~Ji±(x)]/[fo±(x)
~ft±(x)]
.
(24)
The functions a±(x) for charm and bottom quarks have been given in Ref. [15]. 2.3. Double differential Y
y
—
B distributions
We follow the normalisation convention of Ref. [25] and write the double differential B distributions in the following way:
—
w+
dI’~ dydcosB
=
j
If dx
—
dI’~ dxdydcosO 2rn5 (1 ~~j2
~-~[F 1(y) 3ir
where
+ScosOJ~(y)
~(~0(Y)
+ Scos9J~(y)]’L j
(25)
10
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
Fo(y)=4p3[(1
2], ...~2)2+y(l +2
—2Y
,J~(y)=Fo(y) +48y(yY~~
(26) (27) (28)
and Fi(y) =12fdxF~(x~Y)~
(29)
Ji~(Y)=12fdxJi~(x~Y).
(30)
Eq. (29) provides a very non-trivial cross check of this calculation. It has been verified both analytically and numerically that for F~(x, y) given by Eq. (9) the integrals in (29) are equal. Moreover, .F~(y) can be calculated in an easier way, see formula (2.6) in Ref. [25], if the phase space integrals over the momenta of leptons are performed first. This cross check is also fulfilled. It has been shown, see Section 5 of Ref. [19], that when F~(y) is analytically continued it agrees with the analytic results of Ref. [26] for QCD corrections to the width of W boson. All these tests indicate that our formulae for F~(x, y) are correct. Since all the calculations for both F~(x, y) and J~(x, y) were performed in parallel using the same algebraic computer programs we believe that the same is true also for J~(x,y). An analytic expression for F~(y) has first been given in Ref. [25]. For the sake of completeness we rewrite this expression using the notation of the present article and some identities for dilogarithms. Our formulae for F 1 (Y) and J~(y) read F1(y)=Fo(y)~Po+At~Pi+A2Y~+A3Y~+A4p3ln+A5p3,
(31)
Y)J~(Y)!t’o+8t!Pl +I3~Yw+l3tYp+l3~p3lnE+I3~p3 +13~Y~Y,,+B~Y~lne, 1~o+BI~1~ Y)~(Y)! +B~YwYp +13fl’wlfle+48Y2 (13j +B~p0/p3),
(32) (33)
where !ko=2[4Li 2(2p3/p+) —4Y~,ln(2p3/p÷) —lnp.lnw~+lnp+lnw]po/p3 21~Y, (34) +8ln(2p3) !Pi =4Li 2(w) —4Li2(w~) (35) and A1 =Fo(y)po/p3, 2)[1+y—4y2—2(2—y)+41 A2—8(1 —
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
11
A3=—2[3+6y--21y2+12y3—e2(1+12y+5y2)+e4(11+2y)—e6], A 2—e2(4—y)+34] 4=—12[1+3y—4y A5=—2[5+9y—6Y2—2(22—9y) +5~]
(36)
Bt=5—9y2+4Y3—e2(8—8y+6y2)+4+2e6, 5~=—8[1 +4y Bt=—2[3(1
—
—
+ 3y
~2(3
—
4y2) + ~4(3 y) —
—
~6]
—y)2(1+4y) +e211 —5y2) —4(13—2y)
~
B~=—12[(1 —y)(l+4y) —2(4—y)+34] B~=2[15—y+2y2—2(12+7y)—3e4], f3~=—24(1+y—e2)(2p3+y2/p3), (37) =
5
—
42y + 45y2 + 4y3
—
2e2(4
+ 8y + 3y2) + ~ + 26,
8~=8[—1+8y+l0y2+2(3—9y—4y2)—e4(3—y)+6], B~=2[—3(1—y)(3—23y—4y2)+e2(1+24y+5y2)+4(7—2Y)+e6], B~~=—12[3—23y—4y2—e2(6—y)+34], L3~=—2[—15+37y—2y2+e2(l2+7y)+34], B~=—24[2p3(l —5y—2)—2(1+y—2)y/p3], 8~=—24[(1
—
y)(l
—
SY)
2e2(1
—
—
y) +
[Li
—
=
2(w~_) _Li2(P_/P÷)] w+p+ Y~[Li2(w_) + Li2(w~)+ 8ln(2p3)
4 [Li3(1) + Li3 (~i~)
—
—
Li3 (p_/p+)
2ln)’ —41n(w÷)], —
}
Li3 (w /w÷ )
+2Y~[Li2(w+)—Li2(w_) —4Li2(2p3/p+) +4Y~1n(2p3/p+) +2Y~lnp+—4Y~lnw+] + 41~,[Li2
(~‘~) w+p+
—
Li2(P.../P+)] .
(38)
3. Massless case 3.1. Triple differential distributions For m2 = 0 the functions in Eq. (1) simplify considerably. The limit easily obtained for F~(x,y) and J~(x,y). In this limit
—*
0 can be
12
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
F~(x,y)=x(l—x),
(39)
J~(x,y)=F~(x,Y),
(40)
F~(x,y)=(x—y)(l—x+y),
(41)
J~(x,y)=(x—y)(l—x+y—2Y/x)
(42)
and Fj~(x,y)=F~~Po+x~i—(3+2x+y)P203+5(1—x)P4 22YY2)~5+ ~y(4—4x—Y+Y/x), +(2XY+9X4X F~(x,y)=F~cP 2)~i+ (—5 +2x 3y)~2o3 0+ (—2xy +x+ Y +y +(5+4xy—5x+3y—5y2 —2y2/x)~4
(43)
—
+ (6xy + 9x
—
4x2
—
lly
2y2 +2y2/x)P5
—
+~y(2+3x—3y—2y/x),
(44)
Jt(x,y)=J~o—x~I+(5—2x—y—2y/x—2/x)~ 2G3 2+6y—y2—2y2/x)~ +(—3+x+2/x)P4+(—2xy+3x—4x 2+2y—3y2—2y/x+3y2/x), +~(2—2x Jj(x,y)=J 2 —2y2/x)P 1~Po+(—2xY—x—5y+y 1 +(3+lOx+y+ l0y/x—2/x)~2,,3 2 12y/x + 8y2/x + 2/x)P + (—3 + l2xy + x y 4 2 2y~+ 10y2/x)P + (6xy 9x 4x 5 2+2y+7y2 —2y/x—2y2/x), + ~(2—5xy—2x
5 (45)
—
—
—
—
—
—
—
(46)
where
=
+ 2Li
2 (11 ~/x) 2(x) + 2Li2(Y/x) + 1n
= ~1T2
+ Li2(Y)
—
Li 2(x)
—
Li2(y/x)
,
~P2~3=~(1 —y)ln(1 —y) cP4=~ln(l—x) ~5=.~ln(l—y/x).
(47)
Eqs. (43)—(45) agree with those given in the literature; cf. Eqs. (3.9) and (4.9) in Ref. [19] and Eq. (3.3) in Ref. [23].
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
3.2. Massless limitfor x
13
0 distributions
—
For e = 0 explicit analytic formulae can be derived for the double differential x B distributions. The functions f~(x),j~(x),f~(x) and j~(x)which appear in Eq. (16) are given by the following expressions, cf. Eqs. (3.10) and (4.10) in Ref. [19]: —
f~(x)=x2(l —x), f 1~(x)= ~x2(3 2x),
(48) (49)
—
(50) (51)
2(1—2x), j~(x)=~x f~(x)=f~(x)~o(x)+(1—x)[~(5+8x+8x2)ln(1—x) +j~x(l0—19x)1,
(52)
f~(x)=f~(x)~o(x)+~(41—36x+42x2—l6x3)ln(1—x) + ~x(82
—
153x + 86x2),
(53)
jRx) =j~(x)çbo(x) + (1— x)[~(—3+ 12x + 8x2 +4/x) ln(1 + ~(8
—
2x
—
~O(X)
For
—*
—
—
(54) 36x + 14x2
—
l6x3
—
4/x)ln(l
—
x)
l03x2 + 78x3)
(55)
2(1
(56)
=2Li
2(x) + 3.3. Massless limit for y
x)
l5x2)],
jj(x) =j~(x)çbo(x) + ~(11 + ~1(—8+ 18x
—
—
x).
~ir2+ln —
0 distributions
0 the functions in Eq. (25) read, cf. Eqs. (3.l)—(3.2) in Ref. [25],
.Fo(y)2(l—y)2(1+2y),
(57)
3~(y)=Fo(y),
(58)
J~7y)=2(1—y)(l—1ly—2y2)—24y2lny,
(59)
Fl(y)=Fo[~1r2+4Li
2)
2(y)+2lnYln(1—Y)]--(1—y)(5+9y—6Y +4y(l—y—2y2)lny+2(1—y)2(5+4y)ln(1—y),
(60)
J~(y)=—~ir2(1—y)(1+y+4y2)+(l—y)(15—y+2y2) +4(1 —y)(5+5y—4y2)Li2(y) +8(1—y)(2+2y—y2)ln(1—y)lny +8y(2+y—y2)lny+2(1—y)2(5+4y)ln(1—y),
(61)
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
14
y)=_~ir2(1+6y+9y2_4y3)+(1_y)(15_37y+2y2) + 240y2 [Li (1) 3 Li3 (Y) I 2+4y3 + l8y2lny]Li +4[5 —42y+45y 2(y) 2)ln(1 —y)lny +8(1 —y)(2— 13y—y +4y[4—(19+ir2)y—2y2] lny —2(1 —y)(l + 19y+4y2)ln(1 —y). —
(62)
4. Angular distributions Angular distributions are related to the double differential y —0 distributions discussed in Subsection 2.3, (1_E)2
dF~ dcos0
=
J 0
dy
dr~ dydcos0
(63)
,
with dI’~/dy d cos B given in Eq. (25). For the top quark and the narrow W width the double differential distributions are proportional to Dirac delta function t5(y’ m~/m?) and the integral in (63) is trivial. For the charm and for the bottom quarks the fourfermion approximation can be used. In order to illustrate the size of the radiative corrections discussed in the present article we employ the four-fermion limit and write the resulting angular distributions in the following way: —
dr± =FoR.()[1+a~() cosB]
d cos B
,
(64)
where r 2 2/384~r~. In Born approximation for completely polarised 0 = 1)G 5.m~I Vc~j~ I quarks (S= R.o()=l—82+86—8—244ln,
(65) (66)
a~()=~[1—122—364+44e6+38—244(3+22)lne]/7~~(). (67) QCD corrections decrease the total decay rate. In Fig. la the ratios R.(e)/lZo(e) are shown as functions of for a 5 = 0.1, 0.2, 0.3 and 0.4 as the solid, dashed, dotted and dash-dotted lines, respectively. The corrections to Eq. (66) are of the order of one percent or smaller. In Fig. lb the function a() is shown for a5 = 0, 0.2 and 0.4 as the solid, dashed and dotted lines, respectively.
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
15
0.5
1 0.95
~
&(c)
~
0.65 0.6
0
0.2
.1.,.
0.4
I
0.6
0.8
1
-01
0
0.2
0.4
I
0.6
0.8
1
Fig. 1. (a) QCD reduction of the total decay rate: the ratios R~(e)/7~o(e)as functions of e for a
5 = 0.1, 0.2, 0.3 and 0.4 — solid, dashed, dotted and dash-dotted lines; (b) QCD comacted angular asymmetry a (e) for a5 = 0, 0.2 and 0.4 — solid, dashed and dotted lines, respectively.
Acknowledgement
The authors gratefully acknowledge the fruitful collaboration with Hans KUhn on the subject presented in this article. M.J. thanks Kacper Zalewski for useful comments. Note added We have been urged to give more information on the pending conflict between the results of Ref. [19] and Refs. [17,18,7] in addition to what was stated in the Introduction. For the lifetime of the t quark the calculation of Ref. [19] leads to the result of Ref. [25], which has been confirmed by many other groups [28]. After this paper had been submitted for publication a preprint [29] appeared confinning the results of Ref. [19] on the e+ spectrum in charm decays. For bottom decays the controversy was partly clarified earlier, see footnote in Ref. [19]on page 27. Adding this to the crosschecks described in Ref. [19]and in the present article we consider the controversy as solved in favour of Ref. [19] and the related subsequent articles.
Appendix A. Derivations and formulae In this Appendix we sketch our derivation of Eq. (1). The method has been described in our earlier papers Ref. [19] and Ref. [23]. Many formulae presented here have already been given in these articles. We rewrite them using the notation adopted in the present article. Some formulae are simplified and misprints corrected. In the following Q, q, £, z.’ and G denote the scaled four-momenta of the decaying quark and the decay products: quark, andThus, gluon. common scaling factor at is 2 = 1,charged q2 = 2,lepton, £2 = neutrino = G2 = 0. the The gluon is massless. However, 1/mi, and Q
16
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
intermediate steps we introduce the scaled gluon mass A0 in order to regularize those 2, s the expressions which WeRintroduce P formulae = q + G, are z = given P spin four-vector of are theinfrared decayingdivergent. quark and = Q s. also Some only for the weak isospin 13 = 1/2 of the decaying quark. This implies that the analogous formulae for 13 = —1/2 can be obtained by replacing £ with ii and vice versa. —
—
A.1. Differential decay rate The QCD corrected differential rate is given by the following formula:
dr~= dr~+ dI’~ 3+ dF~4,
(68)
where
2M~ dr~=G~m~IVcrj,,i~
5
(69)
3dR.3(Q;q,e,i’)/ir
in Born approximation,
dr~= ~
(70)
describes the virtual gluon contribution and dV~ 4=~
(71)
comes from real gluon emission. Lorentz invariant n-body phase space is defined as dR.~(P;pi,p2,...,p~)=8(4)(P_~pj)JJ.~_~~!-,
(72)
A.2. Three-body contributions In Born approximation the rates for the decays into three-body final states are proportional to the expressions 2+y2], M~3=q~R.~/[(1 —y/9) M~ 2+y2]. (73) 3=q.eR.v/[(1—y/7) The formulae for F~(x, y) and J~(x, y) can be easily derived from (69) and (73) when Dalitz parametrization of the three-body phase space dR.
1~dxdydad(cos0)df3 (74) 3(Q;q,f,v) = 1 is employed. Integration over Euler angle fi is trivial, and non-trivial integrals over a read, see Appendix B in Ref. [23],
J
das .q2ir5cos9[~(1 +y —q2)
fdas.v=21rScos0[y/x_~(1+y_q2_x)].
(75)
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
17
As a final step the scalar products in M~3are expressed in terms of the variables x, y and cos 0, Q.s0,
£.s=—~xScosB 2—x+y)
(76)
,
Q.~~(l—q
v.q(1—q2—x)
and Eqs. (5) —(8) follow immediately. The same steps are performed for the three-body radiative corrections which are given by Eq. 70), where
Mt
2Q
2 +y2]~[q~vR~£H
3 = —[(1— y/9) + q~v(q £ + R £Q q .
+
£ + Q. yR. £
‘2(i.’
—
—
Q
.
pR~
0+ £R. q)H
.
R.
. £)(H÷+ H4],
(77)
—poY~/p 3)lnA0+(2po/p3)[Li2(i —Li2 (1_
—
____
p+w+
Y~(Y~ + 1) +2(lne+ ~)(Y~+ ~) 2—2y)1n]/y+4,
—
\
W.1~j
(78)
+[2p3Y~+(l—e H±=~[1±(l2)/y]Yp/p 3±uln.
(79)
A.3. Four-body contributions Neglecting those terms whose contributions to the decay rates are O( AG ln AG) or smaller (i.e. vanishing in the limit AG 0) one can cast the contribution of real radiation into the following expression: —~
1 ~+ 2+y2 L’Q.G2
1
i,4
(l—y/7)
________
I
1
—
5+ Q.GP.G 2
+
5+ (P.G)2
~
(80)
where 5~q. v[R.~(Q .G— 1) +G.I!RQ — Q.fR~G+G~tR~G], p [G . £R. q — q £R. G+ R . £(q. G — Q . G — 2q. Q)]
+ R £(Q. p q~G .
—
G. p q~Q),
5~=R.e(G.vq.G—q2P.p).
(81)
The four-body phase space is decomposed as follows:
dR. 4(Q;q,G,f,p)=dzdR.3(Q;P,t,~)dR.2(P;q,G).
(82)
18
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
G integration of M~4over dR.2 ( P; q, G) is performed. Due to Lorentz invariance all the integrals which appear in this calculation can be reduced to the scalar integrals After q is substituted by P
—
1nJdR.2(P;q~G) (Q.G)~.
(83)
Explicit formulae are listed in Appendix C of Ref. [231 for the reduction of tensor integralsfdR.2(P;q,G)(Q.G)~IGa,fdR.2(P;q,G)(Q.Gy1GaG$andfdR.2(P;q,G)(Q. G)nGaGflGY to the scalar ones. In the next step Dalitz parametrization is employed for the three-body phase space dR.3 (Q; P, £, v) in the same way as described in the2 previous subsection, = ~2 is replaced withbut P2 now = z. for q replaced with P in Eqs. (74)—(76). In particular q After integrations over Euler angles a and ~8the contribution of real radiation is split into two pieces: the infrared divergent part of the form const. [F~(x,y)
+ ScosBJ~(x,y)]I~~,
where 1djV1_
(84)
2)1_I/(PG)+21O/(PG), 2(1Y+E
and the rest which is infrared finite. For AG << 1 the infrared divergent part can be integrated over z and the result is proportional to
f
-~-
dz
‘div =
4 (i
—
~ ~)ln
(zrn—2) —2 lfl(Zm/2)
2
(e+Aa)2
_Li 2(1_ 1)7k)
_Li2(1_~)
_Y~(Y~+1)] .
(85)
When this result is added to the contribution of virtual corrections the infrared divergent terms ln AG cancel out. The sum is simplified using the following identity: ‘~
Li2
\.
1
—
____
—
p~w~j
Li2
\
1
—
w.~j
—
Li2
=Li2(w....) —Li2(w÷)—2Y~lnp+.
\1
—
P+ (86)
For the infrared finite part the limit AG 0 is performed. The formulae for the scalar integrals I,~simplify considerably in this limit and read —~
‘~2
~/(P. G),
I—i =irY~(z)/~’~,
A. Czarnecki, M. Jezabek/ Nuclear Physics B 427 (1994) 3—21
19
~
2(n+1)(P2)~~1’/~A
‘I
(n~0),
(87)
where4 .~i=Q2P2
—
(Q
.
P)2
=
A(z)=z2—2(l+y)z+(1—y)2, ~ p(Z)~ —‘1 ~ P.G=~(z—2).
(88)
Integration of the infrared finite part over z is tedious. It would be difficult to accomplish this task without FORM [27]. All the integrals which appear can be reduced to integrals dz z’1/Am(z) and j’dz Z~ZY~(Z)/A/(Z). Thus, they can be derived from the recursion relations given in Appendix B of Ref. [19].
f
A.4. FormulaeforF~(x,y)and J~(x,y) After some algebra mentioned in preceding subsections one derives the following formulae for F~(x,y): F~(x,y)=~(x,y)+fl~(x,y,e2)—fl~(x,y,zm),
(89)
where the first term in r.h.s. is the sum of the three-body virtual corrections and the infrared divergent four-body ones. The two other terms originate from the infrared finite four-body piece. For J~(x, y) one has J~(x,y)=lC~(x,y)+JC~(x,y,e2)—K~(x,y,zm).
(90)
The results of Ref. [19] and Ref. [23] are presented in this way; see Eqs. (3.2)—(3.6) and (4.5)—(4.8) in Ref. [19], and (3.1), (A.1)—(A.l5) in Ref. [23]. Both fl~(x, y, z) and X~(x, y, z) are given by lenghty expressions: 7i~(x,y,z) =(a~+za~)Y~(z)/A312(z) + (a~+za~)Yp(z)/~/~3 +a~Yp(z)v5+(a~+za~)/A(z)+a~Z +a~z1 +a~lnZ+a~[Li 2(W+(z))+Li2(W..(z))],
(91)
~ (b~+ zb~)Yp(z)/’/Z~3~+ ~ 2(z)+ (b~+ zb~)/A(z)+ b~z+ b~z~ + (b~+ zb~)/A +b~lnz+b~[Li 2(W~(z))+Li2(W_(z))], (92) calculation only n ( 1 is needed. Thus, instead of Eq. (87) one can use I,, = for n = 0, 1. +
41n the present G)’+’(Q. p)fl/(p2)fl+l
20
A. Czarnecki, M. Jezabek/Nuclear Physics B 427 (1994) 3—21
where a~and b~are rational functions of x, y and
~2, and
W~(z)=~[1+y—z±~/~3].
(93)
These complicated expressions simplify dramatically for z special points
=
2
and z
= Zm.
At these
=
W±(2)=W±, A(Zm)
(x
=
—
YP(Zm)/\/5=
y/x)2,
1/)lfl(’l~)~ 2(
Li 2(W+(Zm)) +Li2(W_(Zm)) =Li2(x) +Li2(y/x)
,
(94)
and after some algebra Eqs. (9) and (10) are derived. References [1] J.H. KUhn and P.M. Zerwas, in Heavy flavours, ed. AJ. Buras and M. Lindner (World Scientific, Singapore, 1992), p. 434. [2] J.H. Kuhn et al., DESY Orange Report 92-123A (1992), Vol. I, p. 255; J.H. Kuhn, Top quark at a linear collider, in Physics and experiments with linear e~e colliders, ed. F.A. Harris et al. (World Scientific, Singapore, 1993), p. 72. [31N. Isgur and MB. Wise, Phys. Lett. B 232 (1989) 113; B 237 (1990) 527. [4] For excellent introduction and reviews see: J.D. Bjorken, in Proc. 4th Rencontres de Physique de la Vallee d’Aoste, La Thuile, Italy, 1990, ed. M. Greco (Editions Frontières, Gif-sur-Yvette, France, 1990); N. Isgur and MB. Wise, Heavy quark symmetry, in B decays, ed. S. Stone (World Scientific, Singapore, 1992), p. 158; M. Neubert, preprint SLAC-PUB-6263 (1993), to appear in Phys. Rep.; K. Zalewski, Heavy flavours (Theory), plenary talk at the EPS HEP93 Conference, Marseille, 22—28 July 1993, preprint CERN-TH.698l/93; MB. Wise, Status of Heavy Quark Theory, plenary talk at XVI lot. Symp. on Lepton-.photon interactions, Cornell University, Ithaca, New York, 10—15 August 1993. [5] E. Eichten and B. Hill, Phys. Lett. B 234 (1990) 511; H. Georgi, Phys. Lett. B 240 (1990) 447; B. Grinstein, NucI. Phys. B 339 (1990) 253. [6] J. Chay, H. Georgi and B. Grinstein, Phys. Lett. B 247 (1990) 399; I. Bigi, M. Shifman, N. Uraltsev and A. Vainshtein, Phys. Rev. Lett. 71(1993) 496; I. Bigi, M. Shifman, N. Uraltsev and A. Vainshtein, mt. J. Mod. Phys. A 9 (1994) 2467; A.V. Manohar and MB. Wise, Phys. Rev. D 49 (1994) 1310; A. Falk, E. Jenkins, A.V. Manohar and MB. Wise, Phys. Rev. D 49 (1994) 4553; R.L. Jaffe and L. Randall, Nucl. Phys. B 412 (1994) 79; M. Neubert, Phys. Rev. D 49 (1994) 4623. [7]G. Altarelli et aL, Nucl. Phys. B 208 (1982) 365. [8] J.G. Kdrner, A. Pilaftsis and M.M. Tung, preprint MZ-Th/93-30, Z. Phys. C, in print. [9] J.D. Bjorken, Phys. Rev. D 40 (1989) 1513.
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21
references