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Polarization of tau leptons in semileptonic B decays
ELSEVIER
Nuclear Physics B 525 (1998) 350-368
Polarization of tau leptons in semileptonic B decays * Marek Jezabek a, Piotr Urban
b
a Institute of Nuclear Physics, Kawiory 26 a, PL-30055 Cracow, Poland b Department of Field Theory and Particle Physics, University of Silesia, Uniwersytecka 4, PL-40007 Katowice, Poland
Received 5 January 1998; revised 13 February 1998; accepted 26 February 1998
Abstract Analytic formulae for the ors order QCD corrections to the differential width of the semileptonic b decay are given with the T polarization taken into account. Thence the polarization of T is expressed by its energy and the invariant mass of the r + P system. The non-perturbative corrections by Falk et al. are incorporated in the calculation. © 1998 Elsevier Science B.V. PACS: 12.38.Bx; 13.20.He Keywords: SemileptonicB decays; Perturbative QCD; Polarization
1. I n t r o d u c t i o n The semileptonic decays of B mesons are now well described with the aid of the heavy quark effective theory ( H Q E T ) [ 1-6] and the QCD perturbative corrections. On the other hand, the non-leptonic processes still suffer from a lack of satisfactory theoretical description. L With this in mind, the former can be successfully employed in the determination of the parameters of the Standard Model ( S M ) . Indeed they have been used to this end, recently yielding the C a b i b b o - K o b a y a s h i - M a s k a w a matrix element * Work supported in part by KBN grants 2P03B08414 and PB659/P03/95/08. l The accuracy of HQET for inclusive processes is still under debate, as the assumptionof quark-hadron duality in the final state might introduce 1/m¢ corrections not seen in the operator product expansion. Phenomenological analyses suggest that these dangerous terms are absent or small for two-quark processes like hadronic decays of r leptons and semileptonicdecays of heavy quarks, see e.g. Ref. [7] and references therein. 0550-3213/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved. PH S0550-32 13 (98)00189-8
M. Je£abek, P. Urban~Nuclear Physics B 525 (1998) 350-368
351
[Vcb] [8]. Moments of lepton spectra can be used in the determination of as, mb and mc [9-11]. The particular processes involving the r lepton make this field yet more
interesting as they are affected by the mass of the charged lepton, now comparable with the masses of the involved quarks mb and mc. The first-order QCD correction to the differential decay width of the b quark has been found analytically [ 12] with the 7- polarization summed over. Now the polarization itself, which is no more suppressed by heavy quark masses, provides information on the SM parameters in its own right. It does not depend on the ]VcKM[ elements, for instance, but can still yield the quark masses. This is all the more important due to the fact that the polarization turns out to be only weakly dependent on the coupling constant as. It should be borne in mind that the first-order QCD correction to the decay width itself is important, amounting to as much as 20% of the Born approximation. Thus the results of this paper render the 7- polarization especially applicable for use in the evaluation of the quark masses. What we calculate here is the longitudinal polarization of the charged lepton. It may be added that the longitudinal polarization is transferred to the lepton system via the intermediating W - boson whose longitudinal polarization in turn takes root in the Higgs mechanism, so that the process may shed light on physics beyond SM. We give the formulae for the decay of b quark into the charmed quark, 7- lepton and 7--antineutrino in terms of the charged lepton energy and the squared four-momentum of the intermediating W- boson, or, equivalently, the invariant mass of the r + system. These formulae, combined with the ones for the unpolarized lepton case, readily give the polarization of the 7- lepton. This expression is then integrated to give the 7energy distribution. Tile correction to the polarization is shown together with the Bornapproximated result. Then the moments of the energy distribution are evaluated for the polarized case. The structure of the paper is as follows. Section 2 is devoted to the kinematical variables. In Section 3 the ideas behind the present calculation of the polarization are discussed. Then in Section 4 QCD corrections are briefly described. The final analytic result is given in Section 5 and in Section 6 the moments of the energy distribution are given.
2. Kinematics 2.1. K i n e m a t i c a l varh~bles
In this section we define the kinematical variables used throughout the article as well as the constraints on those in both cases of a three- and four-body decay. The calculation is performed in the rest frame of the decaying b quark. In order to include the first-order QCD corrections to the decay, one must take into account both the three-body final state with a produced quark c, lepton 7 and an antineutrino ~T and the four-body state with an additional real gluon. The four-momenta of the particles are denoted as follows: Q
352
M. Je~abek, P. Urban~Nuclear Physics B 525 (1998) 350-368
for the b quark, q for the c quark, 7" for the charged lepton, v for the corresponding antineutrino and G for the real gluon. All the particles are assumed to be on-shell, their squared four-momenta equal their masses: Q2 = rob, 2
q2 -_ m c, 2
T2 = m T, 2
p2 = G 2 = 0.
(1)
The four-vectors P = q ÷ G and W = 7- ÷ v characterize the quark-gluon system and the virtual intermediating W bosom respectively. The employed variables are scaled in the units of the decaying quark mass mb, m c2
2Er
m.2r
P - m 2'
~/
m~
x=
W2 t---
p2 (2)
-
Henceforth we scale all quantities so that m~ --- Q2 = 1. The charged lepton is described by the light-cone variables: 7-+ = ½(x :5 V / ~ - 4r/).
(3)
Thus, the system of the c quark and the real gluon is described by the following quantities: P0= ½(1 - t ÷ z ) ,
(4)
P3 = ~/rP~- - z = ½[1 + t 2 + z
2 -2(t+z
+tz)l~,
P + ( z ) = P o ( z ) + P3(z), I In P + ( z )
YP = ~
(6)
P+(z) , _ In
P-(z)
(5)
(7)
v~
where P0 ( z ) and P3 ( z ) are the energy and the length of the momentum vector of the system in the b quark rest frame, 3;p ( z ) is the corresponding rapidity. Similarly for the virtual boson W, 1
W o ( z ) = 7(1 + t -
z),
(8)
W 3 ( z ) = ~ o 2 - t = ½[1 + t 2 + z 2 - 2 ( t + z + t z ) ] 1/2, W + ( z ) = W o ( z ) ± W3(z),
(10)
Y , ( z ) = ½ In W + ( z ) = In W + ( z ) W_(z)
(9)
v'~
(11)
Kinematically, the three-body decay is a special case of the four-body one, with the four-momentum of the gluon set to zero, thus resulting in simply replacing z = p. The
following variables are then useful: p3=P3(p)=~o-p,
(12)
p+=P+(p) =po +p3,
w+ = W~:(p) = 1 - P T : ,
(13)
Yp = y p ( p ) = ½1n p+-, p_
Yw= y w ( p ) = ~ In w+ . w_
(14)
po=Po(p)=l(1--t+p),
M. Je~abek, P Urban~Nuclear Physics B 525 (1998) 350-368
353
We also express the scalar products in terms of the variables used above, so in the units of the b quark mass one gets r.~':g l(t-
Q.P:½(I+z-t),
Q. p= l(1-z
--x-t-t),
Q . 7 - : ½x,
~, . q :
r/),
7-.P=½(x-t-~]), (15)
½( l - x - z + rl).
2.2. K i n e m a t i c a l b o u n d a r i e s
The phase space is divided into two regions. The first of them, henceforth called A, is available for both three- and four-body decay, while the remaining part B corresponds to pure four-body decay. Region A is defined as follows: (16)
2 x / ~ <<. x <~ 1 + ~7 - p = Xm,
tl=r-
1
1--7-_
1-~+
The additional region B of the phase space, where only four-body decay is allowed, has the following boundaries: 2v/~ ~ x ~< x,,,
r/ ~< t ~< tl.
(18)
Conversely, if x should vary at a fixed value of t, the boundaries read r/~
2,
w_ +
~7 <. x <~ w + + W--
~---~-,
(19)
W+
for region A, and V <~ t <~ v / ~
(
1
,
2v~
<~ x ~ w _ + 9-~W_
(20)
for region B. The upper limit of the mass squared of the c-quark-gluon system is in both regions given by Zmax = ( 1 - 7-+) ( 1 - t/7-+ ),
(21 )
whereas the lower limit depends on the region, p
Zmin =
(1 - T _ ) ( I - t / 7 - _ )
in Region A inRegionB
(22)
3. Polarization at tree level In order to calculate the longitudinal polarization we must find the differential decay width for a given polarized final state of the ~- lepton. According to the definition
M. Je2abek, P. Urban~Nuclear Physics B 525 (1998) 350-368
354
F + - F-
P - F + + F-
F-
= 1 - 2-~--,
(23)
where F = F + + F - . Once we know the width with the polarization summed over, we only need to find, for instance, the width for the negatively polarized r lepton. Before going to discuss the QCD corrections to the longitudinal polarization of the T lepton, we stop for a while to look at the tree-level situation where the calculation is easy to follow. Once we choose the b-quark rest frame and decide to look for the longitudinal polarization, we can express the lepton polarization four-vector s in terms of the four-momenta Q and ~- of the b quark and the r lepton, respectively, (24)
s = .AT + 13Q.
This is due to the fact that now only the temporal component of Q does not vanish, whereas the spatial parts of s and w are parallel. The coefficients el, B appearing in the formula above can be evaluated using the conditions defining the polarization fourvector s: s2 = - 1 ,
(25)
s. w=0.
(26)
Upon this one arrives at the following expressions: el+ = + - - 1
x
V/-~T+
,
(27)
-- T _
/~+=T 2 ~ , T+
(28)
-- T _
where the superscripts at el, B denote the polarization of the lepton. This observation combines with another one to make the whole calculation simpler. The total decay width d F o at the tree level for the unpolarized case reads 2 5 un d F o = G FMo[VCKM [2 .Ad0,3dT"~3 (Q," q,
z, z')/7"r5 ,
(29)
where the matrix element amounts to .h4g,n3( r ) = q . 7"0 • ~.
(30)
With this kind of linear dependence on the four-momentum r, it is worth noting that the matrix element with the 7- polarization taken into account is .,A~PO 1
1 A A u n / --
0,3 = ~J"'0,3~ ~ = ~" - m s ) = ½ ( q . K ) ( Q .
~),
(31)
where m stands for the lepton's mass and we have introduced the four-vector K, K = w - ms.
(32)
Applying now the representation (24) of the polarization s we readily obtain the following useful formula for the matrix element with the lepton polarized:
M. Je~abek, P. Urban~Nuclear Physics B 525 (1998) 350-368
j~47"3: 0,3 = q z - - T+
-- T--
.A/[un 0,3(7") -t-
r/ T+
-- 7"_
./~un o,3(Q)-
355 (33)
The first term on the right-hand side of (33) can be calculated immediately once we know the result for the unpolarized case. Thus the problem reduces to performing this calculation again with the only difference amounting to replacing the four-momentum 7- of the lepton with that of the decaying quark, Q. The Born-approximated distribution can be written explicitly as
dF + dx = 12F°f°~ ( x ) ,
(34)
where 2 5 F0-
G F m b IVCKMIz.
(35)
1927r3
The Born level function fo(x) reads, for the unpolarized case, f o ( X ) = ~1 2 7-3 {£°[ x2 -- 3x(1 + r/) + 8r/] + (3x - 6r/) (2 - x ) } ,
(36)
while the polarized cases are obtained using the function Afo:
Afo(x) = ~7-2~r2{((3 -- x -- r/) + 3 ( x -- 2)}
(37)
in the following way:
fo!(X) = ½fo(x) 4- Afo(x ).
(38)
In the formulae above, 7"3 = x / ~ - 4 r / ,
(= 1
P 1 -x+r/
.
(39)
At the tree level, one can express the polarization integrated over the energy of the charged lepton as well: P = 1 - ~ {-24,12 + (X3m- 8r/3/2) (3 + r / - 3p) +12r/(XZm) - 3x~,12 + 3(1 - r / ) 3 ( p +3(-
_
p3/$4 )
1 + r / ) p ( 1 - p/s 2) [3( 1 - 7/) 2 + p ( 3 + 5r/) ] + 3T3S/2
+3(Xm - 2@q) ( - 1 2 r / -
4r/2 + 12r/p - 3p 2 + 3r/p 2 + p 3 )
_ 12r/p3 ln(s2/p) _ 18(2r/2 _ p2 _ r/2p2) In [2V/-~/(xm + r3)] + 1 8 ( 1 -- r/Z)p 2 In ( 2 s 2 v ~ / [ ( 1 -- r/)(1 - 7 / - T3) - p -
r/p])}
/ {T3S/12 + (2r/z - pZ _ r/2p2) ln[ (Xm + T 3 ) / ( 2 v / ~ ) ] -(1
where
- r/2)p2 In
{2v/-~p/
[(1 - 7/)(1 - 7 / + T3) - p - r/p] } } ,
(40)
M. Jegabek, P. Urban~Nuclear Physics B 525 (1998) 350-368
356 s= 1 - v/-~,
7"3 = ~ , 2 _
4r/,
(41)
S= 1 - 7[(1 + r/) ( r / + p 2 ) + p ( 1 +7/2)] + @ + p 3 + 12r/p.
(42)
Anticipating the subsequent discussion of the QCD corrections, let us already note that the specific linear dependence as featured in (30), (31) goes back to the tensorial structure of the matrix element .A40,3,
dun 7.-//zv, .A//~,n3(r) =-**~,0
(43)
where £ and 5ff stand for the leptonic and hadronic tensors, respectively. It is of course the leptonic tensor £ where the linearity derives from, rm
•/zzkv = ~ T+
--
T_
.£,,.(Q). un
7/
un /~/,.(r) 4T+
--
(44)
T_
It is not surprising then that the corrections to the hadronic tensor 7-/will not affect this property.
4. Calculation of QCD corrections The QCD-corrected differential rate for the b --~ c + r - + P reads d F + : d / ' 0 i + d_V~3 + dF~4,
(45)
where dFo• = GFmb[VCKM 2 5 ± [2./g40,3d'R.3 (O,. q, r, v ) / r r 5
(46)
is the Born approximation, while dFli.3, = ~oLsGFmb[ 2 2 5 VCKMI2 A/Iil,3d,-R.3(Q;
q, r, 1.')/77.6
(47)
comes from the interference between the virtual gluon and Born amplitudes. Then,
dFl~,4 = goesGFmb] 2 2 5 VCKM]2./~ li,4dTPo4(Q; q, G, r, v)/q7-7
(48)
is due to the real gluon emission, G denoting the gluon four-momentum. VCKM is the Cabibbo-Kobayashi-Maskawa matrix element corresponding to the b to c or u quark weak transition. The Lorentz invariant n-body phase space is defined as dT"C.,,(P;pl . . . . . Pn) = ~(4)
( p _ Zp
i )1--[ i d3pi2Ei
(49)
The superscript + refers to the fact that now the polarization of the charged lepton is taken into account. In order to evaluate the appropriate rates it is convenient to take advantage of the decomposition (44) which led to the formula for the Bornapproximated matrix element (33). As the QCD corrections influence only the hadronic tensor, the leptonic tensor and thus its linear dependence on r is left intact. This allows
M. Je£abek, P Urban~Nuclear Physics B 525 (1998) 350-368
357
us to represent all the involved matrix elements in an analogous way and, in fact, the whole decay width is of the very same form, 7-+
d F + = q: 7-+
71
(50)
d F u n ( r ) -4- - - d F U n ( Q ) .
-- 7 - _
7-+
-- 7 - _
In Born approximation the contribution to the decay rate into the three-body final state is proportional to the expression jt4~,3= aF - ( x , t ) = ~ql . KQ . v
=
(1 - p - x + t ) -47-r~ -- r_--)
[7-+(x-t-rl)-rl(l+p-t)]
.
(51)
The three-body phase space is parametrized by Dalitz variables, 77-2
dTg3 (Q; q, r, v) = ~ - d x dt.
(52)
The evaluation of the virtual gluon exchange matrix element yields M~n~(7-) = -
[Hoq" 7-0" v + H+pQ . vO . r + H _ q . vq . r
+½P( H+ + H_ ) v . r + l p(H+ - H_ + H L ) [ r . (Q - q - v ) ] ( Q , v) -½Ht[r.
where H o = 4 ( l - Yppo/P3) lnAG + (2po/P3) [Li2(1
-Li2 ( 1 -
(53)
(Q - q - v)] (q. v)] ,
p_w_) p+w+
w----~)-Yp(Yp + 1) + 2 ( l n x / - f i + Y p ) ( Y w + Y p ) ]
+[2p3Yp + (1 - p - 2 t ) l n v / - f i ] / t 1
+ 4,
1
(54)
H+ = ~ [ 1 :t: ( 1 - p ) / t ] Yp/P3 4- t In v/-fi,
(55
1 ~ HL = t ( 1 -- lnVIp) +
(56)
2 pYp In v/P + 72YPP3 + t p 3
and then the polarized case requires A//~,3 _
r+ T+
jM~n3(r)
-- T _
r/ T+
Add,n3(Q).
(57)
-- T _
After renormalization, the virtual correction .M + 1,3 is ultraviolet convergent. However, the infrared divergences are left. They are regularized by a small mass of gluons denoted as Aa. In accordance with the Kinoshita-Lee-Nauenberg theorem, this divergence cancels out when the real emission is taken into account. The rate from real gluon emission is evaluated by integrating the expression
1& (7-) A'//~'n4(r) - ( O . (;)2
132(r)
133(r)
Q . G P . G + -(-P- .- - ~G)'
(58)
M. Je~abek, P Urban~Nuclear Physics B 525 (1998) 350-368
358
where /31(r) = q . " r f Q . ~,(Q. G -
1) + G . p - Q . vQ. G],
/32(r) = q . 7-[G. ~, - q . uQ . G + Q . p ( q . G + Q . v ( Q . 7-q . G -
/33(7") = Q . p ( G . r q . G -
Q . G-
(59) 2 q . Q)]
(60)
G . 7"q . Q ),
(61)
pT-. P ) .
Taking account of polarization amounts to substituting K for 7- in the coefficients B1,2,3: .h4 ~,4 _
7-+ T+
.M~,n4(7-)
-- 7"_
r/
.M~n4(Q)"
(62)
7-+ -- 7-_
The four-body phase space is decomposed as follows: dTZ4(Q; q, G, 7-, u) = d z dTZ3 (Q; P, 7-, z,)dTZ2(P; q, G).
(63)
After employing the Dalitz parametrization of the three-body phase space 7Z3 and integration we arrive at an infrared-divergent expression. The method used in these calculations is the same as the one employed in the previous ones [ 12-14]. The infrared-divergent part is regularized by a small gluon mass l o which enters into the expressions as l n ( l o ) . When the three- and four-body contributions are added the divergent terms cancel out and then the limit l c ~ 0 is performed. This procedure yields well-defined double-differential distributions of the lepton spectra as described below.
5. Analytical results
The following formula gives the differential rate of the decay b ~ z P X , X standing for a c quark or a pair of c and a gluon, once the lepton is taken to be negatively polarized:
,,A(X,t)I
1 2 F o [ F o ( x , t ) - - ~ - - - ~2a, F -
dF-
-
for(x,t) inA
2ors 12Fo-~--~=Fi-8(x, t)
(64) for (x, t) in B
F 1- differs according to which region (A or B) it belongs. Region A is available for the three- and four-body decay, while Region B with a gluon only. The following formulae are given for the negative polarization of the lepton, that is, we take 1
A- =
k/'~ T +
/3--
x
2v'~ T+
-- T _
(65)
--T_'
(66) '
The factor f'0 is defined in Eq. (35), while
M. 3e£abek, P. Urban~Nuclear Physics B 525 (1998) 350-368
Fo (x,t)
=
( 1 - p - x + t) [ 7 - + ( x - t - r / ) - r / ( l + p - t ) ] T+
359
(67)
-- 7-_
and 5
F ~ , a ( x , t ) = F o @ o + Z D na @, + D'~,
(68)
n=l 5
V~,8(x, t) = Fo ~#O+ Z D~qt" + Dg.
(69)
n=l
The factor 12 in the formula (64) is introduced to meet the widely used [10,15,11] convention for Fo(x) and F0. The symbols present in (68) are defined as follows:
+Li2(w_ ) - Li2(w+) + 4Yp In v/-fi/ PoEPJ ~ ln(Zmax - p) - 4 In Zmax, +4 1 - P3
(70)
@l = Li2(w_) + Li2(w+) - Li2(7-+) - Li2(t/7-+) _ Yp @2 - --, P3 @3 = J In v ~ ,
(71)
@ 4 = 7 t In( 1 - 7-+),
(74)
@5=71 ln(l - t/7-+),
(75)
(72) (73)
~° = 4 )(P--P-° ' YP, l, ) I nP (Zmax-~--~-P 3Zmni p + 4 In (Zmax'] k,Z~in / +2P°[Li2( I p 3 -Li?(l
l~+)+Li2( '--:+)p+ - L i 2 ( l
+Li2(,,
+Li2,(
1
lpt_/7-+)
1--t/7-+)p+
J
~1 = Li2 (r+) + Li2 ( t/r+ ) - Li2 (7--) - Li2 ( t/7-_ ), g~2 = ½In( 1 - 7-_ ),
(77) (78)
~'3 = ½ln( 1 - t/7-_),
(79)
360
M. Je~abek, P Urban~Nuclear Physics B 525 (1998) 350-368
(80)
g'4 = ½ l n ( 1 - 7"+), gts= i ln(1 -
t/7"+).
(81)
We introduce C1 . . . C5 to simplify the formulae for D,A and Dff:
1
CI-
[p(-rlx
+ 4tit - fir+ - 471 + x r + - tT.+) - 2p2rl + r l ( x t + x
+ 4 t - 2t 2 + 6 + 2~7) + 7 . + ( - r / x + r/t + 5 r / - 2 x t + x + t + t2)] ,
(82)
C2 =POT - t - 27.+) + ( x t 2 - 2t27.+)/(2r/) + r / ( - x - 2t + 27+ + 3 0 ) / 2 +t(1 + t1
C3 - ~
x) + r+(-2x
+ 2t + 6 ) ,
(83)
[p(-10~Tx + 18rlt - 7r/r+ + x t + 7x7"+ - l l t r + + 4 r + )
+ p 2 ( - 9 r l + 7"+) + (2xt2r+ - x 2 t 2 / 2 - t 2r+)/~7 2 + r12(-7.+ + 10) + x t ( x - t - 1 ) r / ( 4 x t - x7.+ - 2 x - x 2 / 2 + t7.+ + 4t
-4t2+
C4-
1 -xt
117"+ _7"2+ _ 12) + + 7 " + ( - 6 x t - x + 2 x
2 - 3t+5t2-
5)] ,
(84)
[p(2rlxT"+/t - 8rlx + 8~77"+/t + 18r/t - 11r/7"+ - 4r/27"+/t 2
÷ 57"+x - 7t7"+) ÷ p2(2rlT"+/t - 9 7 / - r/27"+/t 2) + x t ( 1 + t - x )
+ ( - 2 x t 2 7 " + + x 2 t 2 / 2 + t27"2+)/rl + r12(57"+/t 2 - 67"+/t - 1 6 / t + 6) + r l ( 2 x T " + / t + 2 x t - 3x7"+ - 4 x + x 2 / 2 - 107.+/t + 3t7.+ + 12t
- 4 t 2 + 37.+ + 7.2 + 4) + 7 . + ( - 4 x t C5 = [ p / ( x
- t -rl/t)]
3x + 2x 2 + l l t + 2t2)] ,
(85)
( - p r l 2 r + / t 2 + 717"+/t - r1 - 712r+/t 2 + 712/t + pT17"+/t
- p r I + prl2 / t ) / 2 + { p / ( 1 - x + r/)] ( - r / t - r/7"+ + 7/z + tT"+)/2 + p ( 3 r l T " + / t + 9 r / - 9t - 37"+)/2 + ( x t 2 / 4 - t27"+)/~7 + r / ( 3 x - 107"+/t
- 1 2 t - 27"+ - 24 + 2 r / ) / 4 + (3t + 5)7"+/2 - x t + 6t + 5 t 2 / 2 ,
(86)
DlA = C , ,
(87)
1 D~ - 4 x 2 x / T L ~4 _~ { p r / ( 3 4 - 6xT"+/t z - 4 x / t 2 + 3 x / t + 5 x t - 2xT+ + 2x + 3 x 2 / t 2 + x 2 + 87.+/t 2 4- 187.+/t - 2 I t + 2t7.+ + 38t - 14t 2 + 207.+)
+ p r / 2 ( 1 6 - 2 x / t 2 - 57"+/t 2 - 27"+/t 4- 1 6 I t - 7"+) 4- pT"+ ( - 4 x t
- 4 x 4- 6t
4-7t 2 4- 11) + p2 [ r / ( - 1 0 + 6xT"+/t 2 4- 6 x / t 2 4- 4 x r + / t - 3 x - 3 x 2 / t 2 -2xZ/t
- 127"+/t 2 - 18T+/t 4- 6 / t 4- 18t - 6 r + ) + r l Z ( x / t z + 7.+/t 2
- ' r + / t - 2 / t ) + 7.+ (2x - 5t - 7) + + p r / ( - 10 - 2xT.+/fl - 4 x / t 2 - x / t
4 - x 2 / t 2 4- 87"+/t 2 4- 67"+/t - 6 / t ) + prlzT"+/t 2 q- pT"+ 4- p Z r l ( x / t 2 - 27"+/t z +2/t)] + r+(-4xt
+ 2 x t 2 + 2 x + 7t +
t 2 --
3t 3 - 5) + r / ( - 14 + 2xT"+/t 2
M. Jegabek, P Urban~Nuclear Physics B 525 (1998) 350-368
361
+ x / t 2 - 4 x T + / t - 2 x / t + 4 x t - 2 x t 2 + 2x7+ - x - x 2 / t 2 + 2 x 2 / t - x 2
- 2 ~ - + / t 2 - 67'+/t - 10tT"+ + 32t - 22t 2 + 4t 3 + 18~-+) + 72(28 + x / t 2 + x + 3T+/t 2 - 5 T + / t - 1 4 / t + t'r+ - 14t + 7"+)} ,
-2x/t
DA3 - ~
1
{p [ 7 ( - 6 -
4x'c+/t 2 - 3x/t 2 - 2xT+/t + x/t-
+ x Z / t + 6"r+/'t: + 1 2 r + / t - 2 I t + 26t - 10T+) + r l 2 ( - x / t : +2/t)
(88) 13x + 2 x 2 / t 2 - 2r+/t e
+ T + ( 6 X + 6 -- 12t)] + p 3 r l ( - - X / t 2 + 2 T + / / 2 -- 2 I t )
+p2 [ 7 ( - 1 0 - 4 - 2 x ~ ' + / t 2 + 3 x / t 2 - x 2 / t 2 - 6 7 + / t 2 - 4~'+/t + 4 / t ) - ~ T Z T + / t 2 - T+] + 72( 14 + x / t 2 - x / t + 3 T + / t 2 - 2~'+/t - 1 4 I t - ~'+)
+7(-
14 + 2 x T + / t 2 + x / t z - 2xT"+/t - x / t + 2 x t - 2 x - x Z / t 2 + x : / t
-2T+/t 2-
8 ; % / t + 1 8 t - 4t 2 + 10T+) + ~ ' + ( - 2 x t + 2x + 2t + 3t 2 - 5)} ,
(89) D4A = - C 3 - C2,
(90)
DA
(91)
-Ca+C, l
1
c5 +
{tp/(l
- x + 7)1
+, +
72(1 + , +
+73 - t~-+)] + [ p / i x - t - 7 1 t ) 1 ( 1 + p ) [n( t - ~-+) + 7 2 ( - 1 + ~ + l t 2 +'r+/t - l/t)
+ 7 3 ( 1 / t 2 - ~'+/t3)] + p 7 ( 4 3 - x~'+/t - 4 x / t - 5 x + 2 x 2 / t
+ 9 r + / t + 13t + 5~'+) + p ' q 2 ( - 5 - 7"+/t ~ + 27"+/t + 1 / t ) + 7"+(9xt + 5x +4t-
6t 2) + 1 2 x t + 5 x t 2 - 2x2t + p ( - 9 x t
+p27(-10
+ 2 x / t - 3T+/t) + p Z r l z ( - T + / t 2
- 3x~'+ + 5t7+) + I / t ) + ( - 6 x t 2 7 " + + x2t 2
+2t2T2+)/(27) + r / ( - 3 5 - xT"+/t + 2 x / t - 2 x t + 4 x r + + 26x -2xZ/t
- 3 x 2 / 2 - 4 r + / t - 2t~'+ - 34t - t2 - 22~-+ - 3~-~+)
+72(-2
+ 2x/t -x
+ 6~'+/t + 2t)} ,
(92)
D,"= C1 ,
(93)
D ~ = 6"2 - C3,
(94)
D 38 = - C 2 - C 4 ,
(95)
D f = C2 + C3,
(96)
D r = - C 2 + C4,
(97)
D ~ = C5.
(98)
One can perform the limit p ---, 0, which corresponds to the decay of the bottom quark to an up quark and leptons. The formulae, which are much simpler in this case, are presented in the same manner as the full results:
M. Je£abek,P Urban~NuclearPhysicsB 525 (1998) 350-368
362
a?- - , dx
I
dt
20[S 1 2 I ' ° [ F ° ( x ' t ) - 3 - - ~ FI-'A(x't)]
for (x, t) in A (99)
20[S
for (x, t) in B
12Fo--~ F1,-n (x, t)
with
Fo (x,t) - ( 1 - x - t ) T+
[r+(x-t-r/)-r/(1-t)]
(lOO)
-- T _
and 5
(101)
FI?A( X, t ) = ~O~O .q_E ~DnA-CI)n_~ ~)A, n=l 5
(102)
F~-,B(X't)=F°~° + E 79~n + 79g, n=l
where t~ 0 =
2 I L i 2 / r + - tx~
k, l _ t
k
l/r+ - 1
f + L i 2 ( ~/t---1 ) + L i z ( t ) ] + ~
1 2
7r
+ln2(1 - ~-+)z lne(1 - t) + ln2(1 - t/T+) - 21n(l - t) In Zmax,
(lO3)
77-2
- ~ + Liz(t) - Liz(r+) - Liz(t/7+), tP203 =
2In(1 - t) 1-t '
(lO4) (1o5) (1o6)
qb4, @5, and ~o__ 2 [Li2 ( r + - t'~ {1/r___--l'~ \-i--~J +Li2k, 1 / t - l J -Li2
Li2(Zl+~--tt )
(1/~-+ - 1 ) ] (Zmax') 1--~--- 1 + In( 1 - t) In k Zmin/
- I n k, 1 - ~ ' - J In [(1 - ~-+) (1 - ~-_)] -In
(I--t~,+) i
~,,=qt, cA _
~
ln[(1-t/r+)(1-t/r_)],
(n=l...5). 1
107)
lO8)
It/(6 + xt -- xr+ + x + tr+ + 4t -- 2t 2 + 57"+) + 2r/2
+ T+(x-- 2xt + t + tZ)] , 109) CA = (xt 2 - 2t2~-+)/(2r/) + r/(30 - x - 2t + 2 r + ) / 2 - xt - 2x~-+ + 2tr+
363
M. Je~abek, P. Urban~Nuclear Physics B 525 (1998) 350-368
+t + tz + 6r+, c 3m -
(110)
[ ( 4 x t 2 r + -- X2t 2 -- 2t272+)/(2"q) + +7/2( 10 - r + )
1
77(-24 + 8xt-
2xr+ - 4x-
x 2 + 2tr+ + 8t-
8t 2 + 2 2 r + - 2r2+)/2
- 6 x t r + -- xt - x t 2 - xr+ + x2t + 2xZr+ - 3 t r + + 5tZ'r+ - 57"+] ,
C4a _
1
[(-4xt2r+
( 11 l )
+ x2t 2 + 2t2r2+)/(2rl) + r / ( 4 + 2 x r + / t + 2 x t
- - 3 x r + - 4 x + x 2 / 2 - l O r + / t + 3tr+ + 12t - 4t 2 q- 3r+ + r2+ )
+ r / 2 ( 6 + 5 r + / t 2 -- 6 r + / t - 1 6 / t ) - 4xtr+ + xt + xt 2 -- 3xr+ - x 2 t q- 2x2r+ + 1 l t r + + 2t2r+] ,
(112)
cA = ( x t 2 -- 4tZr+ ) / ( 4rl) + r1(--24 + 3x -- l O r + / t -- 12t -- 2 r + ) / 4
+ r / 2 / 2 -- x t q- 3 t r + / 2 + 6t + 5 t 2 / 2 q- 57-+/2,
(113)
7:)A = c j , DA03 - 4
(I 14) ~
1
[7/(--14 + 2 x r + / t 2 + x / t 2 -- 4 x r + / t - 2 x / t + 4 x t -- 2xt e
+ 2 x r + -- x - - x 2 / t 2 + 2 x 2 / t -
x 2 -- 2 r + / t z - 6 r + / t -
10tr+ + 32t
- 2 2 t 2 + 4t 3 + 18r+) + r/2(28 + x / t 2 - 2 x / t + x + 3 r + / t 2 - 5 r + / t - 1 4 / t + tr+ - 1 4 t + r + ) + ( - - 4 x t r + + 2xt2r+ + 2 x r + + 7tr~ + t2r+
-3t3r+
--5r+)]
,
(115)
7:)A = - C 2 - C3,
(116)
"DA-=C 2 -- C4,
( l 17)
79~
1 = -~C5 +
4 ~1
[ ( _ 6 x t 2 r + + x2t2 + 2t2r2+)/(2r/)
+ 7 7 ( - 3 5 - x r + / t + 2 x / t - 2 x t + 4x7-+ + 26x - 2 x 2 / t - 3 x 2 / 2 - 4 r ~ / t -2t7-+ - 34t - t 2 - 227"+ - 3r2+) + r / 2 ( - 2 + 2 x / t - x + 67-+/t + 2t) +(9xtr+
+ 12xt + 5 x t 2 + 5 x r + - 2x2t + 4t7-+ - 6 t 2 r + ) ] ,
(118)
On integration over t, one obtains 7- energy distributions according to the formula
1
dF-
12Fo d x
2c~ - f o ( x ) - -~£' f~- ( x ) ,
(119)
where, if we drop the superscript ' - ' , we obtain the c o r r e s p o n d i n g formula for the u n p o l a r i z e d case. The results are presented in Fig. 1, where the polarization is plotted versus the r lepton energy. Both the Born and first-order a p p r o x i m a t i o n s are shown. In order to m a k e the correction more explicit, we also present another diagram, where the function R ( x ) is drawn, defined as follows:
M. Je~abek, 1£. Urban~Nuclear Physics B 525 (1998) 350-368
364
P o l o r i z c t i o n of Lhe "I lepton 0
corrected
-0.2
m
-0.4
-0.6
-0.8
0[8
- 1017
i
0.9
i
1.1 x=2E,/r%
Fig. 1. Polarization o f 7 lepton in the Born approximation (dashed line) and including the first-order Q C D correction (solid line) as functions of the scaled ~- energy x. The mass of the b quark is taken at 4.75 GeV, the mass o f the c quark is 1.35 G e V and the coupling constant a s = 0.2.
2as R (x)]] 1+-~-
1 - P ( x ) = [1-P0(x)] which gives for
(120)
R(x)
R(x) - fl(x) fo(x)
f~(x) fo(X)
(121)
Thus its meaning is how the radiative corrections differ with respect to the state of polarization they act on. In (120) P0 denotes the zeroth-order approximation to the polarization. The function f~ ( x ) / f o ( x ) has been presented in Fig. 2, while Fig. 3 shows the function R(x), so that one can immediately see how small the correction is in comparison with the correction to the energy distribution. Integrating over the charged lepton energy one can obtain its total polarization as well as the corrections to which it is subject. If we take mb = 4.75 GeV as the central value for the decaying quark mass and mb = 4.4 and mb = 5.2 for the limits, we arrive at the following (the mass difference mb - mc = 3.4 GeV everywhere):
I-P=(1-Po)
2as R A~ R1 A2 R2 "~ 1+ 3~ s+-~b-,, p+m-~b..npj
with R s = - ( ~ (~ltq-0'023
-- - - N 7 ~ R R -0.0105 POv ..... +0.0109 '
g l p _ • A~1+0.027
-- ~ . ~ . ~ x -0.025 ,
v'~v+O.Ol7
2
= - - 9 9R -0"22 Rnp . . . . +0.16 "
'
(122)
M. Je~abek, P. Urban~Nuclear Physics B 525 (1998) 350-368
365
QCD correctqon for p o l a 4 z e d "r
3.4
/
2.8
?.~U
2.2
1,6
10:!
019
018
1 1.1 x-2ET/r%
Fig. 2. The ratio /]-(x)/f o (x) representing the radiative correction for the negatively polarized state as dependent on the scaled ~"lepton energy x. Polorizot:on correction R(x) 0.04 0.02
/:
//
.',1_~11//// " /i/
0 -0.02
5.2 CeV
0.04
"-1 /
/
/
0.06 /
_ _
/
4.75 CeV 4.4 OeV
\/ --0.08
0.7
0.8
0.9
1
1.I
1.2
x=PF,/-%
Fig. 3. The QCD-correction function R(x) for the pole mass values of the b quark taken to be 4.4 GeV (dashed), 4.75 GeV (solid) and 5.2 GeV (dash-dotted) as dependent on the scaled ~- lepton energy x.
6. M o m e n t s
o f t h e ~" e n e r g y d i s t r i b u t i o n
T h e m o m e n t s o f the r e n e r g y distribution, w h i c h are useful sources o f information on the p h y s i c a l parameters regarding the d i s c u s s e d decay, can be evaluated a c c o r d i n g to the f o r m u l a
366
M. Je£abek, P Urban~Nuclear Physics B 525 (1998) 350-368 Emax
M~ = f
n dF+ E~--d--~dE,,
(123)
Emin
+ = M~
rn
(124)
Mo• '
where Emin and Emax are the lower and upper limits for the r energy and Mn include both perturbative and non-perturbative QCD corrections to the r energy spectrum. The superscripts denote the polarization states. Since one obviously has M,, = M,+ + M,]-,
(125)
where M n stands for the unpolarized momenta, we only give the values of the momenta for the negative polarization case. The unpolarized distributions were given in [ 12]. The non-perturbative corrections to the charged lepton spectrum from semileptonic B decay have been derived in the HQET framework up to order of 1/m~ [15-17]. The corrected heavy lepton energy spectrum can be written in the following way: A1 F(I) (X] .~2 f(2) , - + -~bbJ,p ( x ) ,
1 dF 2as 12r0 d x - f o ( x ) - ~ - ~ f l ( x )
+ Z--~OPmo
(126)
where -~l and ,~2 are the HQET parameters corresponding to the b quark kinetic energy and the energy of interaction of the b quark magnetic moment with the chromomagnetic field produced by the light quark in the meson B. The functions :(1'2) can be easily ,: np extracted from Formula (2.11) in [ 15]. Formula (126) looks identical if one considers a definite polarization state of the final ~- lepton. The appropriate calculation within the HQET scheme has also been performed [15], see formula (2.12) therein. Following Ref. [ 11 ] we expand the ratios r,, r n = r},°)-
2as ( )__ AI t~(l)_ A2 (2)--) 1 - -~--~6,,p + m--~b_" + --~bbfn _ ,
(127)
where r~°) is the lowest approximation of r,,
r:l0)-
:mb~nf;~-pfO(X)X"dX. \ 2j
y:j-
(128)
fo (z)dz
Each of the ~ i ) - is expressed by integrals of the corresponding correction function f(i) (X) and the tree-level term fo(x),
fl+r/-p
fl+~--p f ( i ) - a2~/~
f(i)- (x)xndx
6 ( 0 - = 32x/~
f2l+n-p x/~ f o ( x)xndx
(x)dx
-
,
(129)
fl+,~-p f o (x)dx a2.,/~
where the index i denotes any of the three kinds of corrections discussed above. The coefficients t3(i) depend only on the two ratios of the charged lepton and the c quark to the mass mb. Following Ref. [ 12] we employ the functional dependence of the form
M. Je£abek, P. Urban~Nuclear Physics B 525 (1998) 350-368 t~/)- ( mb, rtlc , mr ) = t~:i)- ( mb , rrtc ) \m~ ~b
'
367 (130)
The quark masses are not known precisely so we have calculated the coefficients in a reasonable range of the parameters, viz. 4.4 GeV~< mb <~ 5.2 GeV and 0.25 ~< m,./mt, <~ 0.35 and then fitted to them functions of the following form: ~ ( p , q ) = a + b(p - Po) + c ( q - qo) + d ( p - po) 2
= ÷ e ( p - P o ) ( q - qo) + f ( q -
qo) 2,
(131)
where p = mb/m~.,po = 4.75 GeV/I.777 GeV= 2.6730, q = mc/mb, qo = 0.28 and the polynomial coefficients can be fitted for each of the t~(i) separately with a relative error of less than 2%. Our choice of the central values reflects the realistic masses of quarks: mb = 4.75 GeV and mc = 1.35 GeV, for which ~i) = a(ni). To bring out the difference in the extent to which the corrections affect the two different polarization states it is useful to compare these coefficients with the ones obtained with the polarization summed over. This can be done along the lines suggested by the treatment of the polarization itself, see Eq. (120). The corresponding expansion takes the form
r'Tr,, - r~(°) J r })°
1 ---~-2as(s(,Pl--6}TP))+ mi~l(t~(l)-
~}11)) ÷ m---fb'~(~2)_ 2
t~}72)) ] . (132)
With these, one can readily find the actual relative correction of each kind, assuming reasonable values of as,/~1 and ,~2. Here we take a~ = 0.2, 0.15 GeV 2 ~< -3,1 ~< 0,60 GeV 2, ,~2 --- 0.12. GeV 2, keep the b quark mass fixed at 4.75 GeV and the mass of the c quark equal to 1.35 GeV. The corrections then read for n = 1 (n = 5)
Type
Correction to r; /,'~,°)-
Perturbative Kinetic energy Chromomagnetic
-0.00084(-0.0048) 0.008±0.005(0.06 ± 0.04) -0.0097(-0.053)
r, / rn~°)
-0.0009(-0.0052) 0.008 ± 0.005(0.06 i 0.04) -0.0092(-0.0511)
7. S u m m a r y The first-order perturbative QCD correction to the polarization of the charged lepton in semileptonic B decays has been found analytically. It is expressed by the charged lepton energy and the invariant mass of the lepton system. The polarization correction has turned out to be very small, as it does not exceed 1% taking common values for the parameters occurring in the formulae. This makes the polarization a very useful quantity in determining the quark masses. The moments of the ~- lepton energy distribution have
368
M. Je£abek, P Urban/Nuclear Physics B 525 (1998) 350-368
been evaluated for the case o f a p o l a r i z e d lepton and the correction has again been found to be little different f r o m the one in the case o f an unpolarized lepton. The n o n - p e r t u r b a t i v e H Q E T corrections to the m o m e n t s have also been calculated using the f o r m u l a e f r o m [ 15 ].
References [1] [2] 13] [4] [51 [6] 171 [8] [91 [10l [111 [12] [13] 1141 115] [ 16] 117]
M. Voloshin and M. Shifman, Sov. J. Phys. 45 (1987) 292; 47 (1988) 511. H.D. Politzer and M.B. Wise, Phys. Lett. B 206 (1988) 681; B 208 (1988) 504. N. lsgur and M.B. Wise, Phys. Lett. B 232 (1989) 113; B 237 (1990) 527. E. Eichten and B. Hill, Phys. Lett. B 234 (1990) 253. B. Grinstein, Nucl. Phys. Lett. B 339 (1990) 253. H. Georgi, Phys. Lett. B 240 (1990) 447. M. Neubert, Nucl. Phys. B (Proc. Suppl.) 59 (1997) 101. Review of Particle Physics, R.M. Barnett et al., Phys. Rev. D 54 (1996) 1, and references therein. A. Czarnecki, M.JeZabek, J.G. KiJrner and J.H. KOhn, Phys. Rev. Lett. 73 (1994) 317. A. Czarnecki, M. JeZabek and J.H. Kiihn, Phys. Len. B 346 (1995) 335. M.B. Voloshin, Phys. Rev. D 51 (1995) 4934. M. Je2abek and L. Motyka, Acta Phys. Polon. B 27 (1996) 3603; Nucl. Phys. B 501 (1997) 207. M. JeZabek and J.H. Ktihn, Nucl. Phys. B 320 (1989) 20. A. Czarnecki and M. JeZabek, Nucl. Phys. B 427 (1994) 3. A.E Falk, Z. Ligeti, M. Neubert and Y. Nir, Phys. Lett. B 326 (1994) 145. L. Koyrakh, Phys. Rev. D 49 (1994) 3379; PhD Thesis (unpublished) hep/ph9607443. S. Balk, J.G. K6rner, D. Pirjol and K. Schilcher, Z. Phys. C 64 (1994) 37.
Nuclear Physics B 525 (1998) 350-368
Polarization of tau leptons in semileptonic B decays * Marek Jezabek a, Piotr Urban
b
a Institute of Nuclear Physics, Kawiory 26 a, PL-30055 Cracow, Poland b Department of Field Theory and Particle Physics, University of Silesia, Uniwersytecka 4, PL-40007 Katowice, Poland
Received 5 January 1998; revised 13 February 1998; accepted 26 February 1998
Abstract Analytic formulae for the ors order QCD corrections to the differential width of the semileptonic b decay are given with the T polarization taken into account. Thence the polarization of T is expressed by its energy and the invariant mass of the r + P system. The non-perturbative corrections by Falk et al. are incorporated in the calculation. © 1998 Elsevier Science B.V. PACS: 12.38.Bx; 13.20.He Keywords: SemileptonicB decays; Perturbative QCD; Polarization
1. I n t r o d u c t i o n The semileptonic decays of B mesons are now well described with the aid of the heavy quark effective theory ( H Q E T ) [ 1-6] and the QCD perturbative corrections. On the other hand, the non-leptonic processes still suffer from a lack of satisfactory theoretical description. L With this in mind, the former can be successfully employed in the determination of the parameters of the Standard Model ( S M ) . Indeed they have been used to this end, recently yielding the C a b i b b o - K o b a y a s h i - M a s k a w a matrix element * Work supported in part by KBN grants 2P03B08414 and PB659/P03/95/08. l The accuracy of HQET for inclusive processes is still under debate, as the assumptionof quark-hadron duality in the final state might introduce 1/m¢ corrections not seen in the operator product expansion. Phenomenological analyses suggest that these dangerous terms are absent or small for two-quark processes like hadronic decays of r leptons and semileptonicdecays of heavy quarks, see e.g. Ref. [7] and references therein. 0550-3213/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved. PH S0550-32 13 (98)00189-8
M. Je£abek, P. Urban~Nuclear Physics B 525 (1998) 350-368
351
[Vcb] [8]. Moments of lepton spectra can be used in the determination of as, mb and mc [9-11]. The particular processes involving the r lepton make this field yet more
interesting as they are affected by the mass of the charged lepton, now comparable with the masses of the involved quarks mb and mc. The first-order QCD correction to the differential decay width of the b quark has been found analytically [ 12] with the 7- polarization summed over. Now the polarization itself, which is no more suppressed by heavy quark masses, provides information on the SM parameters in its own right. It does not depend on the ]VcKM[ elements, for instance, but can still yield the quark masses. This is all the more important due to the fact that the polarization turns out to be only weakly dependent on the coupling constant as. It should be borne in mind that the first-order QCD correction to the decay width itself is important, amounting to as much as 20% of the Born approximation. Thus the results of this paper render the 7- polarization especially applicable for use in the evaluation of the quark masses. What we calculate here is the longitudinal polarization of the charged lepton. It may be added that the longitudinal polarization is transferred to the lepton system via the intermediating W - boson whose longitudinal polarization in turn takes root in the Higgs mechanism, so that the process may shed light on physics beyond SM. We give the formulae for the decay of b quark into the charmed quark, 7- lepton and 7--antineutrino in terms of the charged lepton energy and the squared four-momentum of the intermediating W- boson, or, equivalently, the invariant mass of the r + system. These formulae, combined with the ones for the unpolarized lepton case, readily give the polarization of the 7- lepton. This expression is then integrated to give the 7energy distribution. Tile correction to the polarization is shown together with the Bornapproximated result. Then the moments of the energy distribution are evaluated for the polarized case. The structure of the paper is as follows. Section 2 is devoted to the kinematical variables. In Section 3 the ideas behind the present calculation of the polarization are discussed. Then in Section 4 QCD corrections are briefly described. The final analytic result is given in Section 5 and in Section 6 the moments of the energy distribution are given.
2. Kinematics 2.1. K i n e m a t i c a l varh~bles
In this section we define the kinematical variables used throughout the article as well as the constraints on those in both cases of a three- and four-body decay. The calculation is performed in the rest frame of the decaying b quark. In order to include the first-order QCD corrections to the decay, one must take into account both the three-body final state with a produced quark c, lepton 7 and an antineutrino ~T and the four-body state with an additional real gluon. The four-momenta of the particles are denoted as follows: Q
352
M. Je~abek, P. Urban~Nuclear Physics B 525 (1998) 350-368
for the b quark, q for the c quark, 7" for the charged lepton, v for the corresponding antineutrino and G for the real gluon. All the particles are assumed to be on-shell, their squared four-momenta equal their masses: Q2 = rob, 2
q2 -_ m c, 2
T2 = m T, 2
p2 = G 2 = 0.
(1)
The four-vectors P = q ÷ G and W = 7- ÷ v characterize the quark-gluon system and the virtual intermediating W bosom respectively. The employed variables are scaled in the units of the decaying quark mass mb, m c2
2Er
m.2r
P - m 2'
~/
m~
x=
W2 t---
p2 (2)
-
Henceforth we scale all quantities so that m~ --- Q2 = 1. The charged lepton is described by the light-cone variables: 7-+ = ½(x :5 V / ~ - 4r/).
(3)
Thus, the system of the c quark and the real gluon is described by the following quantities: P0= ½(1 - t ÷ z ) ,
(4)
P3 = ~/rP~- - z = ½[1 + t 2 + z
2 -2(t+z
+tz)l~,
P + ( z ) = P o ( z ) + P3(z), I In P + ( z )
YP = ~
(6)
P+(z) , _ In
P-(z)
(5)
(7)
v~
where P0 ( z ) and P3 ( z ) are the energy and the length of the momentum vector of the system in the b quark rest frame, 3;p ( z ) is the corresponding rapidity. Similarly for the virtual boson W, 1
W o ( z ) = 7(1 + t -
z),
(8)
W 3 ( z ) = ~ o 2 - t = ½[1 + t 2 + z 2 - 2 ( t + z + t z ) ] 1/2, W + ( z ) = W o ( z ) ± W3(z),
(10)
Y , ( z ) = ½ In W + ( z ) = In W + ( z ) W_(z)
(9)
v'~
(11)
Kinematically, the three-body decay is a special case of the four-body one, with the four-momentum of the gluon set to zero, thus resulting in simply replacing z = p. The
following variables are then useful: p3=P3(p)=~o-p,
(12)
p+=P+(p) =po +p3,
w+ = W~:(p) = 1 - P T : ,
(13)
Yp = y p ( p ) = ½1n p+-, p_
Yw= y w ( p ) = ~ In w+ . w_
(14)
po=Po(p)=l(1--t+p),
M. Je~abek, P Urban~Nuclear Physics B 525 (1998) 350-368
353
We also express the scalar products in terms of the variables used above, so in the units of the b quark mass one gets r.~':g l(t-
Q.P:½(I+z-t),
Q. p= l(1-z
--x-t-t),
Q . 7 - : ½x,
~, . q :
r/),
7-.P=½(x-t-~]), (15)
½( l - x - z + rl).
2.2. K i n e m a t i c a l b o u n d a r i e s
The phase space is divided into two regions. The first of them, henceforth called A, is available for both three- and four-body decay, while the remaining part B corresponds to pure four-body decay. Region A is defined as follows: (16)
2 x / ~ <<. x <~ 1 + ~7 - p = Xm,
tl=r-
1
1--7-_
1-~+
The additional region B of the phase space, where only four-body decay is allowed, has the following boundaries: 2v/~ ~ x ~< x,,,
r/ ~< t ~< tl.
(18)
Conversely, if x should vary at a fixed value of t, the boundaries read r/~
2,
w_ +
~7 <. x <~ w + + W--
~---~-,
(19)
W+
for region A, and V <~ t <~ v / ~
(
1
,
2v~
<~ x ~ w _ + 9-~W_
(20)
for region B. The upper limit of the mass squared of the c-quark-gluon system is in both regions given by Zmax = ( 1 - 7-+) ( 1 - t/7-+ ),
(21 )
whereas the lower limit depends on the region, p
Zmin =
(1 - T _ ) ( I - t / 7 - _ )
in Region A inRegionB
(22)
3. Polarization at tree level In order to calculate the longitudinal polarization we must find the differential decay width for a given polarized final state of the ~- lepton. According to the definition
M. Je2abek, P. Urban~Nuclear Physics B 525 (1998) 350-368
354
F + - F-
P - F + + F-
F-
= 1 - 2-~--,
(23)
where F = F + + F - . Once we know the width with the polarization summed over, we only need to find, for instance, the width for the negatively polarized r lepton. Before going to discuss the QCD corrections to the longitudinal polarization of the T lepton, we stop for a while to look at the tree-level situation where the calculation is easy to follow. Once we choose the b-quark rest frame and decide to look for the longitudinal polarization, we can express the lepton polarization four-vector s in terms of the four-momenta Q and ~- of the b quark and the r lepton, respectively, (24)
s = .AT + 13Q.
This is due to the fact that now only the temporal component of Q does not vanish, whereas the spatial parts of s and w are parallel. The coefficients el, B appearing in the formula above can be evaluated using the conditions defining the polarization fourvector s: s2 = - 1 ,
(25)
s. w=0.
(26)
Upon this one arrives at the following expressions: el+ = + - - 1
x
V/-~T+
,
(27)
-- T _
/~+=T 2 ~ , T+
(28)
-- T _
where the superscripts at el, B denote the polarization of the lepton. This observation combines with another one to make the whole calculation simpler. The total decay width d F o at the tree level for the unpolarized case reads 2 5 un d F o = G FMo[VCKM [2 .Ad0,3dT"~3 (Q," q,
z, z')/7"r5 ,
(29)
where the matrix element amounts to .h4g,n3( r ) = q . 7"0 • ~.
(30)
With this kind of linear dependence on the four-momentum r, it is worth noting that the matrix element with the 7- polarization taken into account is .,A~PO 1
1 A A u n / --
0,3 = ~J"'0,3~ ~ = ~" - m s ) = ½ ( q . K ) ( Q .
~),
(31)
where m stands for the lepton's mass and we have introduced the four-vector K, K = w - ms.
(32)
Applying now the representation (24) of the polarization s we readily obtain the following useful formula for the matrix element with the lepton polarized:
M. Je~abek, P. Urban~Nuclear Physics B 525 (1998) 350-368
j~47"3: 0,3 = q z - - T+
-- T--
.A/[un 0,3(7") -t-
r/ T+
-- 7"_
./~un o,3(Q)-
355 (33)
The first term on the right-hand side of (33) can be calculated immediately once we know the result for the unpolarized case. Thus the problem reduces to performing this calculation again with the only difference amounting to replacing the four-momentum 7- of the lepton with that of the decaying quark, Q. The Born-approximated distribution can be written explicitly as
dF + dx = 12F°f°~ ( x ) ,
(34)
where 2 5 F0-
G F m b IVCKMIz.
(35)
1927r3
The Born level function fo(x) reads, for the unpolarized case, f o ( X ) = ~1 2 7-3 {£°[ x2 -- 3x(1 + r/) + 8r/] + (3x - 6r/) (2 - x ) } ,
(36)
while the polarized cases are obtained using the function Afo:
Afo(x) = ~7-2~r2{((3 -- x -- r/) + 3 ( x -- 2)}
(37)
in the following way:
fo!(X) = ½fo(x) 4- Afo(x ).
(38)
In the formulae above, 7"3 = x / ~ - 4 r / ,
(= 1
P 1 -x+r/
.
(39)
At the tree level, one can express the polarization integrated over the energy of the charged lepton as well: P = 1 - ~ {-24,12 + (X3m- 8r/3/2) (3 + r / - 3p) +12r/(XZm) - 3x~,12 + 3(1 - r / ) 3 ( p +3(-
_
p3/$4 )
1 + r / ) p ( 1 - p/s 2) [3( 1 - 7/) 2 + p ( 3 + 5r/) ] + 3T3S/2
+3(Xm - 2@q) ( - 1 2 r / -
4r/2 + 12r/p - 3p 2 + 3r/p 2 + p 3 )
_ 12r/p3 ln(s2/p) _ 18(2r/2 _ p2 _ r/2p2) In [2V/-~/(xm + r3)] + 1 8 ( 1 -- r/Z)p 2 In ( 2 s 2 v ~ / [ ( 1 -- r/)(1 - 7 / - T3) - p -
r/p])}
/ {T3S/12 + (2r/z - pZ _ r/2p2) ln[ (Xm + T 3 ) / ( 2 v / ~ ) ] -(1
where
- r/2)p2 In
{2v/-~p/
[(1 - 7/)(1 - 7 / + T3) - p - r/p] } } ,
(40)
M. Jegabek, P. Urban~Nuclear Physics B 525 (1998) 350-368
356 s= 1 - v/-~,
7"3 = ~ , 2 _
4r/,
(41)
S= 1 - 7[(1 + r/) ( r / + p 2 ) + p ( 1 +7/2)] + @ + p 3 + 12r/p.
(42)
Anticipating the subsequent discussion of the QCD corrections, let us already note that the specific linear dependence as featured in (30), (31) goes back to the tensorial structure of the matrix element .A40,3,
dun 7.-//zv, .A//~,n3(r) =-**~,0
(43)
where £ and 5ff stand for the leptonic and hadronic tensors, respectively. It is of course the leptonic tensor £ where the linearity derives from, rm
•/zzkv = ~ T+
--
T_
.£,,.(Q). un
7/
un /~/,.(r) 4T+
--
(44)
T_
It is not surprising then that the corrections to the hadronic tensor 7-/will not affect this property.
4. Calculation of QCD corrections The QCD-corrected differential rate for the b --~ c + r - + P reads d F + : d / ' 0 i + d_V~3 + dF~4,
(45)
where dFo• = GFmb[VCKM 2 5 ± [2./g40,3d'R.3 (O,. q, r, v ) / r r 5
(46)
is the Born approximation, while dFli.3, = ~oLsGFmb[ 2 2 5 VCKMI2 A/Iil,3d,-R.3(Q;
q, r, 1.')/77.6
(47)
comes from the interference between the virtual gluon and Born amplitudes. Then,
dFl~,4 = goesGFmb] 2 2 5 VCKM]2./~ li,4dTPo4(Q; q, G, r, v)/q7-7
(48)
is due to the real gluon emission, G denoting the gluon four-momentum. VCKM is the Cabibbo-Kobayashi-Maskawa matrix element corresponding to the b to c or u quark weak transition. The Lorentz invariant n-body phase space is defined as dT"C.,,(P;pl . . . . . Pn) = ~(4)
( p _ Zp
i )1--[ i d3pi2Ei
(49)
The superscript + refers to the fact that now the polarization of the charged lepton is taken into account. In order to evaluate the appropriate rates it is convenient to take advantage of the decomposition (44) which led to the formula for the Bornapproximated matrix element (33). As the QCD corrections influence only the hadronic tensor, the leptonic tensor and thus its linear dependence on r is left intact. This allows
M. Je£abek, P Urban~Nuclear Physics B 525 (1998) 350-368
357
us to represent all the involved matrix elements in an analogous way and, in fact, the whole decay width is of the very same form, 7-+
d F + = q: 7-+
71
(50)
d F u n ( r ) -4- - - d F U n ( Q ) .
-- 7 - _
7-+
-- 7 - _
In Born approximation the contribution to the decay rate into the three-body final state is proportional to the expression jt4~,3= aF - ( x , t ) = ~ql . KQ . v
=
(1 - p - x + t ) -47-r~ -- r_--)
[7-+(x-t-rl)-rl(l+p-t)]
.
(51)
The three-body phase space is parametrized by Dalitz variables, 77-2
dTg3 (Q; q, r, v) = ~ - d x dt.
(52)
The evaluation of the virtual gluon exchange matrix element yields M~n~(7-) = -
[Hoq" 7-0" v + H+pQ . vO . r + H _ q . vq . r
+½P( H+ + H_ ) v . r + l p(H+ - H_ + H L ) [ r . (Q - q - v ) ] ( Q , v) -½Ht[r.
where H o = 4 ( l - Yppo/P3) lnAG + (2po/P3) [Li2(1
-Li2 ( 1 -
(53)
(Q - q - v)] (q. v)] ,
p_w_) p+w+
w----~)-Yp(Yp + 1) + 2 ( l n x / - f i + Y p ) ( Y w + Y p ) ]
+[2p3Yp + (1 - p - 2 t ) l n v / - f i ] / t 1
+ 4,
1
(54)
H+ = ~ [ 1 :t: ( 1 - p ) / t ] Yp/P3 4- t In v/-fi,
(55
1 ~ HL = t ( 1 -- lnVIp) +
(56)
2 pYp In v/P + 72YPP3 + t p 3
and then the polarized case requires A//~,3 _
r+ T+
jM~n3(r)
-- T _
r/ T+
Add,n3(Q).
(57)
-- T _
After renormalization, the virtual correction .M + 1,3 is ultraviolet convergent. However, the infrared divergences are left. They are regularized by a small mass of gluons denoted as Aa. In accordance with the Kinoshita-Lee-Nauenberg theorem, this divergence cancels out when the real emission is taken into account. The rate from real gluon emission is evaluated by integrating the expression
1& (7-) A'//~'n4(r) - ( O . (;)2
132(r)
133(r)
Q . G P . G + -(-P- .- - ~G)'
(58)
M. Je~abek, P Urban~Nuclear Physics B 525 (1998) 350-368
358
where /31(r) = q . " r f Q . ~,(Q. G -
1) + G . p - Q . vQ. G],
/32(r) = q . 7-[G. ~, - q . uQ . G + Q . p ( q . G + Q . v ( Q . 7-q . G -
/33(7") = Q . p ( G . r q . G -
Q . G-
(59) 2 q . Q)]
(60)
G . 7"q . Q ),
(61)
pT-. P ) .
Taking account of polarization amounts to substituting K for 7- in the coefficients B1,2,3: .h4 ~,4 _
7-+ T+
.M~,n4(7-)
-- 7"_
r/
.M~n4(Q)"
(62)
7-+ -- 7-_
The four-body phase space is decomposed as follows: dTZ4(Q; q, G, 7-, u) = d z dTZ3 (Q; P, 7-, z,)dTZ2(P; q, G).
(63)
After employing the Dalitz parametrization of the three-body phase space 7Z3 and integration we arrive at an infrared-divergent expression. The method used in these calculations is the same as the one employed in the previous ones [ 12-14]. The infrared-divergent part is regularized by a small gluon mass l o which enters into the expressions as l n ( l o ) . When the three- and four-body contributions are added the divergent terms cancel out and then the limit l c ~ 0 is performed. This procedure yields well-defined double-differential distributions of the lepton spectra as described below.
5. Analytical results
The following formula gives the differential rate of the decay b ~ z P X , X standing for a c quark or a pair of c and a gluon, once the lepton is taken to be negatively polarized:
,,A(X,t)I
1 2 F o [ F o ( x , t ) - - ~ - - - ~2a, F -
dF-
-
for(x,t) inA
2ors 12Fo-~--~=Fi-8(x, t)
(64) for (x, t) in B
F 1- differs according to which region (A or B) it belongs. Region A is available for the three- and four-body decay, while Region B with a gluon only. The following formulae are given for the negative polarization of the lepton, that is, we take 1
A- =
k/'~ T +
/3--
x
2v'~ T+
-- T _
(65)
--T_'
(66) '
The factor f'0 is defined in Eq. (35), while
M. 3e£abek, P. Urban~Nuclear Physics B 525 (1998) 350-368
Fo (x,t)
=
( 1 - p - x + t) [ 7 - + ( x - t - r / ) - r / ( l + p - t ) ] T+
359
(67)
-- 7-_
and 5
F ~ , a ( x , t ) = F o @ o + Z D na @, + D'~,
(68)
n=l 5
V~,8(x, t) = Fo ~#O+ Z D~qt" + Dg.
(69)
n=l
The factor 12 in the formula (64) is introduced to meet the widely used [10,15,11] convention for Fo(x) and F0. The symbols present in (68) are defined as follows:
+Li2(w_ ) - Li2(w+) + 4Yp In v/-fi/ PoEPJ ~ ln(Zmax - p) - 4 In Zmax, +4 1 - P3
(70)
@l = Li2(w_) + Li2(w+) - Li2(7-+) - Li2(t/7-+) _ Yp @2 - --, P3 @3 = J In v ~ ,
(71)
@ 4 = 7 t In( 1 - 7-+),
(74)
@5=71 ln(l - t/7-+),
(75)
(72) (73)
~° = 4 )(P--P-° ' YP, l, ) I nP (Zmax-~--~-P 3Zmni p + 4 In (Zmax'] k,Z~in / +2P°[Li2( I p 3 -Li?(l
l~+)+Li2( '--:+)p+ - L i 2 ( l
+Li2(,,
+Li2,(
1
lpt_/7-+)
1--t/7-+)p+
J
~1 = Li2 (r+) + Li2 ( t/r+ ) - Li2 (7--) - Li2 ( t/7-_ ), g~2 = ½In( 1 - 7-_ ),
(77) (78)
~'3 = ½ln( 1 - t/7-_),
(79)
360
M. Je~abek, P Urban~Nuclear Physics B 525 (1998) 350-368
(80)
g'4 = ½ l n ( 1 - 7"+), gts= i ln(1 -
t/7"+).
(81)
We introduce C1 . . . C5 to simplify the formulae for D,A and Dff:
1
CI-
[p(-rlx
+ 4tit - fir+ - 471 + x r + - tT.+) - 2p2rl + r l ( x t + x
+ 4 t - 2t 2 + 6 + 2~7) + 7 . + ( - r / x + r/t + 5 r / - 2 x t + x + t + t2)] ,
(82)
C2 =POT - t - 27.+) + ( x t 2 - 2t27.+)/(2r/) + r / ( - x - 2t + 27+ + 3 0 ) / 2 +t(1 + t1
C3 - ~
x) + r+(-2x
+ 2t + 6 ) ,
(83)
[p(-10~Tx + 18rlt - 7r/r+ + x t + 7x7"+ - l l t r + + 4 r + )
+ p 2 ( - 9 r l + 7"+) + (2xt2r+ - x 2 t 2 / 2 - t 2r+)/~7 2 + r12(-7.+ + 10) + x t ( x - t - 1 ) r / ( 4 x t - x7.+ - 2 x - x 2 / 2 + t7.+ + 4t
-4t2+
C4-
1 -xt
117"+ _7"2+ _ 12) + + 7 " + ( - 6 x t - x + 2 x
2 - 3t+5t2-
5)] ,
(84)
[p(2rlxT"+/t - 8rlx + 8~77"+/t + 18r/t - 11r/7"+ - 4r/27"+/t 2
÷ 57"+x - 7t7"+) ÷ p2(2rlT"+/t - 9 7 / - r/27"+/t 2) + x t ( 1 + t - x )
+ ( - 2 x t 2 7 " + + x 2 t 2 / 2 + t27"2+)/rl + r12(57"+/t 2 - 67"+/t - 1 6 / t + 6) + r l ( 2 x T " + / t + 2 x t - 3x7"+ - 4 x + x 2 / 2 - 107.+/t + 3t7.+ + 12t
- 4 t 2 + 37.+ + 7.2 + 4) + 7 . + ( - 4 x t C5 = [ p / ( x
- t -rl/t)]
3x + 2x 2 + l l t + 2t2)] ,
(85)
( - p r l 2 r + / t 2 + 717"+/t - r1 - 712r+/t 2 + 712/t + pT17"+/t
- p r I + prl2 / t ) / 2 + { p / ( 1 - x + r/)] ( - r / t - r/7"+ + 7/z + tT"+)/2 + p ( 3 r l T " + / t + 9 r / - 9t - 37"+)/2 + ( x t 2 / 4 - t27"+)/~7 + r / ( 3 x - 107"+/t
- 1 2 t - 27"+ - 24 + 2 r / ) / 4 + (3t + 5)7"+/2 - x t + 6t + 5 t 2 / 2 ,
(86)
DlA = C , ,
(87)
1 D~ - 4 x 2 x / T L ~4 _~ { p r / ( 3 4 - 6xT"+/t z - 4 x / t 2 + 3 x / t + 5 x t - 2xT+ + 2x + 3 x 2 / t 2 + x 2 + 87.+/t 2 4- 187.+/t - 2 I t + 2t7.+ + 38t - 14t 2 + 207.+)
+ p r / 2 ( 1 6 - 2 x / t 2 - 57"+/t 2 - 27"+/t 4- 1 6 I t - 7"+) 4- pT"+ ( - 4 x t
- 4 x 4- 6t
4-7t 2 4- 11) + p2 [ r / ( - 1 0 + 6xT"+/t 2 4- 6 x / t 2 4- 4 x r + / t - 3 x - 3 x 2 / t 2 -2xZ/t
- 127"+/t 2 - 18T+/t 4- 6 / t 4- 18t - 6 r + ) + r l Z ( x / t z + 7.+/t 2
- ' r + / t - 2 / t ) + 7.+ (2x - 5t - 7) + + p r / ( - 10 - 2xT.+/fl - 4 x / t 2 - x / t
4 - x 2 / t 2 4- 87"+/t 2 4- 67"+/t - 6 / t ) + prlzT"+/t 2 q- pT"+ 4- p Z r l ( x / t 2 - 27"+/t z +2/t)] + r+(-4xt
+ 2 x t 2 + 2 x + 7t +
t 2 --
3t 3 - 5) + r / ( - 14 + 2xT"+/t 2
M. Jegabek, P Urban~Nuclear Physics B 525 (1998) 350-368
361
+ x / t 2 - 4 x T + / t - 2 x / t + 4 x t - 2 x t 2 + 2x7+ - x - x 2 / t 2 + 2 x 2 / t - x 2
- 2 ~ - + / t 2 - 67'+/t - 10tT"+ + 32t - 22t 2 + 4t 3 + 18~-+) + 72(28 + x / t 2 + x + 3T+/t 2 - 5 T + / t - 1 4 / t + t'r+ - 14t + 7"+)} ,
-2x/t
DA3 - ~
1
{p [ 7 ( - 6 -
4x'c+/t 2 - 3x/t 2 - 2xT+/t + x/t-
+ x Z / t + 6"r+/'t: + 1 2 r + / t - 2 I t + 26t - 10T+) + r l 2 ( - x / t : +2/t)
(88) 13x + 2 x 2 / t 2 - 2r+/t e
+ T + ( 6 X + 6 -- 12t)] + p 3 r l ( - - X / t 2 + 2 T + / / 2 -- 2 I t )
+p2 [ 7 ( - 1 0 - 4 - 2 x ~ ' + / t 2 + 3 x / t 2 - x 2 / t 2 - 6 7 + / t 2 - 4~'+/t + 4 / t ) - ~ T Z T + / t 2 - T+] + 72( 14 + x / t 2 - x / t + 3 T + / t 2 - 2~'+/t - 1 4 I t - ~'+)
+7(-
14 + 2 x T + / t 2 + x / t z - 2xT"+/t - x / t + 2 x t - 2 x - x Z / t 2 + x : / t
-2T+/t 2-
8 ; % / t + 1 8 t - 4t 2 + 10T+) + ~ ' + ( - 2 x t + 2x + 2t + 3t 2 - 5)} ,
(89) D4A = - C 3 - C2,
(90)
DA
(91)
-Ca+C, l
1
c5 +
{tp/(l
- x + 7)1
+, +
72(1 + , +
+73 - t~-+)] + [ p / i x - t - 7 1 t ) 1 ( 1 + p ) [n( t - ~-+) + 7 2 ( - 1 + ~ + l t 2 +'r+/t - l/t)
+ 7 3 ( 1 / t 2 - ~'+/t3)] + p 7 ( 4 3 - x~'+/t - 4 x / t - 5 x + 2 x 2 / t
+ 9 r + / t + 13t + 5~'+) + p ' q 2 ( - 5 - 7"+/t ~ + 27"+/t + 1 / t ) + 7"+(9xt + 5x +4t-
6t 2) + 1 2 x t + 5 x t 2 - 2x2t + p ( - 9 x t
+p27(-10
+ 2 x / t - 3T+/t) + p Z r l z ( - T + / t 2
- 3x~'+ + 5t7+) + I / t ) + ( - 6 x t 2 7 " + + x2t 2
+2t2T2+)/(27) + r / ( - 3 5 - xT"+/t + 2 x / t - 2 x t + 4 x r + + 26x -2xZ/t
- 3 x 2 / 2 - 4 r + / t - 2t~'+ - 34t - t2 - 22~-+ - 3~-~+)
+72(-2
+ 2x/t -x
+ 6~'+/t + 2t)} ,
(92)
D,"= C1 ,
(93)
D ~ = 6"2 - C3,
(94)
D 38 = - C 2 - C 4 ,
(95)
D f = C2 + C3,
(96)
D r = - C 2 + C4,
(97)
D ~ = C5.
(98)
One can perform the limit p ---, 0, which corresponds to the decay of the bottom quark to an up quark and leptons. The formulae, which are much simpler in this case, are presented in the same manner as the full results:
M. Je£abek,P Urban~NuclearPhysicsB 525 (1998) 350-368
362
a?- - , dx
I
dt
20[S 1 2 I ' ° [ F ° ( x ' t ) - 3 - - ~ FI-'A(x't)]
for (x, t) in A (99)
20[S
for (x, t) in B
12Fo--~ F1,-n (x, t)
with
Fo (x,t) - ( 1 - x - t ) T+
[r+(x-t-r/)-r/(1-t)]
(lOO)
-- T _
and 5
(101)
FI?A( X, t ) = ~O~O .q_E ~DnA-CI)n_~ ~)A, n=l 5
(102)
F~-,B(X't)=F°~° + E 79~n + 79g, n=l
where t~ 0 =
2 I L i 2 / r + - tx~
k, l _ t
k
l/r+ - 1
f + L i 2 ( ~/t---1 ) + L i z ( t ) ] + ~
1 2
7r
+ln2(1 - ~-+)z lne(1 - t) + ln2(1 - t/T+) - 21n(l - t) In Zmax,
(lO3)
77-2
- ~ + Liz(t) - Liz(r+) - Liz(t/7+), tP203 =
2In(1 - t) 1-t '
(lO4) (1o5) (1o6)
qb4, @5, and ~o__ 2 [Li2 ( r + - t'~ {1/r___--l'~ \-i--~J +Li2k, 1 / t - l J -Li2
Li2(Zl+~--tt )
(1/~-+ - 1 ) ] (Zmax') 1--~--- 1 + In( 1 - t) In k Zmin/
- I n k, 1 - ~ ' - J In [(1 - ~-+) (1 - ~-_)] -In
(I--t~,+) i
~,,=qt, cA _
~
ln[(1-t/r+)(1-t/r_)],
(n=l...5). 1
107)
lO8)
It/(6 + xt -- xr+ + x + tr+ + 4t -- 2t 2 + 57"+) + 2r/2
+ T+(x-- 2xt + t + tZ)] , 109) CA = (xt 2 - 2t2~-+)/(2r/) + r/(30 - x - 2t + 2 r + ) / 2 - xt - 2x~-+ + 2tr+
363
M. Je~abek, P. Urban~Nuclear Physics B 525 (1998) 350-368
+t + tz + 6r+, c 3m -
(110)
[ ( 4 x t 2 r + -- X2t 2 -- 2t272+)/(2"q) + +7/2( 10 - r + )
1
77(-24 + 8xt-
2xr+ - 4x-
x 2 + 2tr+ + 8t-
8t 2 + 2 2 r + - 2r2+)/2
- 6 x t r + -- xt - x t 2 - xr+ + x2t + 2xZr+ - 3 t r + + 5tZ'r+ - 57"+] ,
C4a _
1
[(-4xt2r+
( 11 l )
+ x2t 2 + 2t2r2+)/(2rl) + r / ( 4 + 2 x r + / t + 2 x t
- - 3 x r + - 4 x + x 2 / 2 - l O r + / t + 3tr+ + 12t - 4t 2 q- 3r+ + r2+ )
+ r / 2 ( 6 + 5 r + / t 2 -- 6 r + / t - 1 6 / t ) - 4xtr+ + xt + xt 2 -- 3xr+ - x 2 t q- 2x2r+ + 1 l t r + + 2t2r+] ,
(112)
cA = ( x t 2 -- 4tZr+ ) / ( 4rl) + r1(--24 + 3x -- l O r + / t -- 12t -- 2 r + ) / 4
+ r / 2 / 2 -- x t q- 3 t r + / 2 + 6t + 5 t 2 / 2 q- 57-+/2,
(113)
7:)A = c j , DA03 - 4
(I 14) ~
1
[7/(--14 + 2 x r + / t 2 + x / t 2 -- 4 x r + / t - 2 x / t + 4 x t -- 2xt e
+ 2 x r + -- x - - x 2 / t 2 + 2 x 2 / t -
x 2 -- 2 r + / t z - 6 r + / t -
10tr+ + 32t
- 2 2 t 2 + 4t 3 + 18r+) + r/2(28 + x / t 2 - 2 x / t + x + 3 r + / t 2 - 5 r + / t - 1 4 / t + tr+ - 1 4 t + r + ) + ( - - 4 x t r + + 2xt2r+ + 2 x r + + 7tr~ + t2r+
-3t3r+
--5r+)]
,
(115)
7:)A = - C 2 - C3,
(116)
"DA-=C 2 -- C4,
( l 17)
79~
1 = -~C5 +
4 ~1
[ ( _ 6 x t 2 r + + x2t2 + 2t2r2+)/(2r/)
+ 7 7 ( - 3 5 - x r + / t + 2 x / t - 2 x t + 4x7-+ + 26x - 2 x 2 / t - 3 x 2 / 2 - 4 r ~ / t -2t7-+ - 34t - t 2 - 227"+ - 3r2+) + r / 2 ( - 2 + 2 x / t - x + 67-+/t + 2t) +(9xtr+
+ 12xt + 5 x t 2 + 5 x r + - 2x2t + 4t7-+ - 6 t 2 r + ) ] ,
(118)
On integration over t, one obtains 7- energy distributions according to the formula
1
dF-
12Fo d x
2c~ - f o ( x ) - -~£' f~- ( x ) ,
(119)
where, if we drop the superscript ' - ' , we obtain the c o r r e s p o n d i n g formula for the u n p o l a r i z e d case. The results are presented in Fig. 1, where the polarization is plotted versus the r lepton energy. Both the Born and first-order a p p r o x i m a t i o n s are shown. In order to m a k e the correction more explicit, we also present another diagram, where the function R ( x ) is drawn, defined as follows:
M. Je~abek, 1£. Urban~Nuclear Physics B 525 (1998) 350-368
364
P o l o r i z c t i o n of Lhe "I lepton 0
corrected
-0.2
m
-0.4
-0.6
-0.8
0[8
- 1017
i
0.9
i
1.1 x=2E,/r%
Fig. 1. Polarization o f 7 lepton in the Born approximation (dashed line) and including the first-order Q C D correction (solid line) as functions of the scaled ~- energy x. The mass of the b quark is taken at 4.75 GeV, the mass o f the c quark is 1.35 G e V and the coupling constant a s = 0.2.
2as R (x)]] 1+-~-
1 - P ( x ) = [1-P0(x)] which gives for
(120)
R(x)
R(x) - fl(x) fo(x)
f~(x) fo(X)
(121)
Thus its meaning is how the radiative corrections differ with respect to the state of polarization they act on. In (120) P0 denotes the zeroth-order approximation to the polarization. The function f~ ( x ) / f o ( x ) has been presented in Fig. 2, while Fig. 3 shows the function R(x), so that one can immediately see how small the correction is in comparison with the correction to the energy distribution. Integrating over the charged lepton energy one can obtain its total polarization as well as the corrections to which it is subject. If we take mb = 4.75 GeV as the central value for the decaying quark mass and mb = 4.4 and mb = 5.2 for the limits, we arrive at the following (the mass difference mb - mc = 3.4 GeV everywhere):
I-P=(1-Po)
2as R A~ R1 A2 R2 "~ 1+ 3~ s+-~b-,, p+m-~b..npj
with R s = - ( ~ (~ltq-0'023
-- - - N 7 ~ R R -0.0105 POv ..... +0.0109 '
g l p _ • A~1+0.027
-- ~ . ~ . ~ x -0.025 ,
v'~v+O.Ol7
2
= - - 9 9R -0"22 Rnp . . . . +0.16 "
'
(122)
M. Je~abek, P. Urban~Nuclear Physics B 525 (1998) 350-368
365
QCD correctqon for p o l a 4 z e d "r
3.4
/
2.8
?.~U
2.2
1,6
10:!
019
018
1 1.1 x-2ET/r%
Fig. 2. The ratio /]-(x)/f o (x) representing the radiative correction for the negatively polarized state as dependent on the scaled ~"lepton energy x. Polorizot:on correction R(x) 0.04 0.02
/:
//
.',1_~11//// " /i/
0 -0.02
5.2 CeV
0.04
"-1 /
/
/
0.06 /
_ _
/
4.75 CeV 4.4 OeV
\/ --0.08
0.7
0.8
0.9
1
1.I
1.2
x=PF,/-%
Fig. 3. The QCD-correction function R(x) for the pole mass values of the b quark taken to be 4.4 GeV (dashed), 4.75 GeV (solid) and 5.2 GeV (dash-dotted) as dependent on the scaled ~- lepton energy x.
6. M o m e n t s
o f t h e ~" e n e r g y d i s t r i b u t i o n
T h e m o m e n t s o f the r e n e r g y distribution, w h i c h are useful sources o f information on the p h y s i c a l parameters regarding the d i s c u s s e d decay, can be evaluated a c c o r d i n g to the f o r m u l a
366
M. Je£abek, P Urban~Nuclear Physics B 525 (1998) 350-368 Emax
M~ = f
n dF+ E~--d--~dE,,
(123)
Emin
+ = M~
rn
(124)
Mo• '
where Emin and Emax are the lower and upper limits for the r energy and Mn include both perturbative and non-perturbative QCD corrections to the r energy spectrum. The superscripts denote the polarization states. Since one obviously has M,, = M,+ + M,]-,
(125)
where M n stands for the unpolarized momenta, we only give the values of the momenta for the negative polarization case. The unpolarized distributions were given in [ 12]. The non-perturbative corrections to the charged lepton spectrum from semileptonic B decay have been derived in the HQET framework up to order of 1/m~ [15-17]. The corrected heavy lepton energy spectrum can be written in the following way: A1 F(I) (X] .~2 f(2) , - + -~bbJ,p ( x ) ,
1 dF 2as 12r0 d x - f o ( x ) - ~ - ~ f l ( x )
+ Z--~OPmo
(126)
where -~l and ,~2 are the HQET parameters corresponding to the b quark kinetic energy and the energy of interaction of the b quark magnetic moment with the chromomagnetic field produced by the light quark in the meson B. The functions :(1'2) can be easily ,: np extracted from Formula (2.11) in [ 15]. Formula (126) looks identical if one considers a definite polarization state of the final ~- lepton. The appropriate calculation within the HQET scheme has also been performed [15], see formula (2.12) therein. Following Ref. [ 11 ] we expand the ratios r,, r n = r},°)-
2as ( )__ AI t~(l)_ A2 (2)--) 1 - -~--~6,,p + m--~b_" + --~bbfn _ ,
(127)
where r~°) is the lowest approximation of r,,
r:l0)-
:mb~nf;~-pfO(X)X"dX. \ 2j
y:j-
(128)
fo (z)dz
Each of the ~ i ) - is expressed by integrals of the corresponding correction function f(i) (X) and the tree-level term fo(x),
fl+r/-p
fl+~--p f ( i ) - a2~/~
f(i)- (x)xndx
6 ( 0 - = 32x/~
f2l+n-p x/~ f o ( x)xndx
(x)dx
-
,
(129)
fl+,~-p f o (x)dx a2.,/~
where the index i denotes any of the three kinds of corrections discussed above. The coefficients t3(i) depend only on the two ratios of the charged lepton and the c quark to the mass mb. Following Ref. [ 12] we employ the functional dependence of the form
M. Je£abek, P. Urban~Nuclear Physics B 525 (1998) 350-368 t~/)- ( mb, rtlc , mr ) = t~:i)- ( mb , rrtc ) \m~ ~b
'
367 (130)
The quark masses are not known precisely so we have calculated the coefficients in a reasonable range of the parameters, viz. 4.4 GeV~< mb <~ 5.2 GeV and 0.25 ~< m,./mt, <~ 0.35 and then fitted to them functions of the following form: ~ ( p , q ) = a + b(p - Po) + c ( q - qo) + d ( p - po) 2
= ÷ e ( p - P o ) ( q - qo) + f ( q -
qo) 2,
(131)
where p = mb/m~.,po = 4.75 GeV/I.777 GeV= 2.6730, q = mc/mb, qo = 0.28 and the polynomial coefficients can be fitted for each of the t~(i) separately with a relative error of less than 2%. Our choice of the central values reflects the realistic masses of quarks: mb = 4.75 GeV and mc = 1.35 GeV, for which ~i) = a(ni). To bring out the difference in the extent to which the corrections affect the two different polarization states it is useful to compare these coefficients with the ones obtained with the polarization summed over. This can be done along the lines suggested by the treatment of the polarization itself, see Eq. (120). The corresponding expansion takes the form
r'Tr,, - r~(°) J r })°
1 ---~-2as(s(,Pl--6}TP))+ mi~l(t~(l)-
~}11)) ÷ m---fb'~(~2)_ 2
t~}72)) ] . (132)
With these, one can readily find the actual relative correction of each kind, assuming reasonable values of as,/~1 and ,~2. Here we take a~ = 0.2, 0.15 GeV 2 ~< -3,1 ~< 0,60 GeV 2, ,~2 --- 0.12. GeV 2, keep the b quark mass fixed at 4.75 GeV and the mass of the c quark equal to 1.35 GeV. The corrections then read for n = 1 (n = 5)
Type
Correction to r; /,'~,°)-
Perturbative Kinetic energy Chromomagnetic
-0.00084(-0.0048) 0.008±0.005(0.06 ± 0.04) -0.0097(-0.053)
r, / rn~°)
-0.0009(-0.0052) 0.008 ± 0.005(0.06 i 0.04) -0.0092(-0.0511)
7. S u m m a r y The first-order perturbative QCD correction to the polarization of the charged lepton in semileptonic B decays has been found analytically. It is expressed by the charged lepton energy and the invariant mass of the lepton system. The polarization correction has turned out to be very small, as it does not exceed 1% taking common values for the parameters occurring in the formulae. This makes the polarization a very useful quantity in determining the quark masses. The moments of the ~- lepton energy distribution have
368
M. Je£abek, P Urban/Nuclear Physics B 525 (1998) 350-368
been evaluated for the case o f a p o l a r i z e d lepton and the correction has again been found to be little different f r o m the one in the case o f an unpolarized lepton. The n o n - p e r t u r b a t i v e H Q E T corrections to the m o m e n t s have also been calculated using the f o r m u l a e f r o m [ 15 ].
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