The mass ratio mcmb in semi-leptonic b-decays

Volume 221, number 2

PHYSICS LETTERS B

27 April 1989

THE MASS RATIO mJmb IN S E M I - L E P T O N I C b-DECAYS -AYosef N I R

Stanford LinearAcceleratorCenter, Stanford University, Stanford, CA 94306, USA Received 6 February 1989

The calculation of the s23 mixing angle from semi-leptonic B-decay depends on the mass ratio me~rob.The energy scales at which the running masses mc and mb should be taken are determined once terms of order (cq (mc/mb)) are taken into account. We give an analytic expression for the QCD correction to the decay rate to all orders in the ratio between on-shell masses. We explain how it should be modified when both masses are taken at a single common scale.

1. Introduction The exact determination of the quark sector parameters is most important both as a test o f the three-generation standard model and as a probe of physics beyond this model. With three generations, the quark mixing matrix is parametrized by three real mixing angles and one complex phase. Two of these mixing angles, s23 = I Voul and s~ 3= I Vuu l, are extracted from B-meson decay measurements. The best experimentally measured and theoretically understood decay modes are the inclusive semi-leptonic decays. The quark-level process is b-~q£-'~, where q is either a c-quark or a u-quark and ~ is a charged lepton. The masses o f the e and g leptons and of the u-quark can be safely neglected (we later comment on the neglect of mu). However, the charmed semi-leptonic decay rate strongly depends on the ratio me~rob [ 1,2]. The values of the masses mo and mb that should be used seem ambiguous [ 3,4 ]. Quark masses run with the energy scale and it is not obvious what the relevant energy scales are. In particular, we want to decide whether to use the ratio between on-shell masses or the ratio between the masses taken at a c o m m o n energy scale. We show that the calculations for these two possibilities differ at order (as ( m J rnb) ). Previous calculations o f Q C D corrections to the decay rate (beyond zeroth order in the mass ratio) used numerical integration. We argue that they correspond to the ratio between on-shell masses. To show that, we perform the integration analytically. We then modify the calculation for the use o f the ratio between masses at a single energy scale, finding that in this case there is no term of the form ( m o / m u ) l n ( m J m b ) , as expected on general grounds [ 51. We further remark on the implications o f our calculation on Q C D corrections to charmless B-decays and on Q E D corrections to lepton decays, e.g. x ~ gg~v~. We note that our results cannot be directly applied tO top decays: we assume that the fermionic masses involved are all small compared to the mass of the intermediate boson. This does not hold for mt ~ M w , and modifications are necessary [ 6 ].

2. Semi-leptonic quark decay We take the general case of a heavy quark h of charge - -~ and a lighter quark 1 of charge + 2. The value o f the Work supported by the Department of Energy, contract DE-AC03-76SF00515. 184

0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Volume 221, number 2

PHYSICS LETTERS B

27 April 1989

mixing term I V~hl is extracted from the inclusive semi-leptonic decay rate Mh~Xl~v~, where Mh is a weaklydecaying meson that contains the quark h. At the quark level one uses the spectator quark model, assuming that the partial width is given by the h-quark W-mediated decay: F(Mh-~X~v~) =F(h--*l~9~) = (Gvmh 2 5 I V~h12/192n3)FpsFQcD,

(1)

where Fps is a phase space factor and FQCD is a QCD correction factor. Both Fps and FQCD depend on the mass ratio

p=m~/m 2 .

(2)

The calculation of Fps is well known:

Fps(p) = 1 - 8p+ 8 p 3 - p 4 - 12pZln(p).

(3)

The FQCD parameter is of the form FQcD(p) = 1 -- (20Z~/3n)f(p).

(4)

The crucial point is that with different definitions ofp the functionf(p) is modified beyond zeroth order in p. We now show this explicitly. Suppose we use for the ratio p:

p= [ml(mi) ]Z/[mh(mh) ] 2 ,

(5)

namely, each mass is taken on its mass shell. We should use certain functions Fp~(p) a n d f ( p ) . Then we make the calculation using a different ratio p': p' = [ml(mh) ]2/[mh(mh) ]~',

(6)

namely, both masses are taken on a single energy scale. Now we should use functions F'os( p' ) a n d f ' ( p ' ) . The ratios p and p' are related, to first order in oz~,by

p=p' [1 - (2o~Jn)ln(p') ] .

(7)

As the difference is O (o~) while the phase space factor is, by definition, zeroth order in o~, clearly:

F'v~(x) = F p ~ ( x ) .

(8)

However, modifications at order (¢x~o) are required. To first order in p we have Fp~(p) = 1 - 8p= 1 - 8/9' + ( 16o~Jn)p' ln(p' ) =Fp~(p' ) [1 + ( 16ozJn)p' ln(p' ) ] .

(9)

The O (o~) correction should be absorbed in a new function F'QcD(p' ): F~cD(p' ) = 1 -- (2oz~/3n)f' (p'),

(10)

where f ' (x) = f ( x ) - 24x In (x) + ....

( 11 )

Indeed, the calculations with either p or p' differ at order (¢x~(ml/mh)). Moreover, when we use p, the ratio between on-shell masses, we expect terms of the form oz~p ln(p) to appear in FQCD(p), as the gluon loops on external legs are calculated at different scales. However, when we use p', the ratio between masses at a single common scale; we expect no terms of the form oe~p' ln(p' ) [ 5 ]. Eq. ( 11 ) then tells us that the coefficient of the p In (p)-term in f(p) is 24. In the next section we give the analytic expression for f(p) and show that this is indeed the case. Previous calculations of the QCD corrections [ 1,2,7 ] are a modification of earlier QED calculations [ 8 ] of Ix decay. In the QED calculation, the masses are by definition on-shell masses: lepton masses are experimentally 185

Volume221, number2

PHYSICSLETTERSB

27 April 1989

measurable, and these physical masses identify with the on-shell masses. Consequently, the existing calculations of QCD-corrected quark decays correspond to the mass ratio between on-shell masses.

3. An analytic expression for the mass-dependent corrections In previous calculations [ 1,2,7 ] the differential cross section is given analytically. However, to derive the correction to the decay rate, the integration, being mathematically rather non-trivial, is carried out numerically. As we are interested in the coefficient of the p In(p) term, we carried out an analytic integration to all orders in p. Our starting point was the differential cross section as given in ref. [7 ]. We now give the expression for the function h (p ) = Fp~(p )f (p ): h (p) = - ( 1 _p2 ) ( ~ _ 2_~_~9p+ ~p2 ) +p In (p) (20 + 9 0 p - 4p2 + ~Zp3) +pZln2p(36 +p2) q- ( 1 _p2) ( ~ _ .~p+ ~p2) ln( 1 - p ) - 4 ( 1 q- 30p2 +p 4) In(p) In( 1 - p ) - ( 1 + 16p 2+p4) [6 Li2(p) - 7r2] -32p3/~(1 +p) F [ n 2 - 4 Li2 (x/P) +4 L i 2 ( - v / p ) - 2 In(p) I n / 1 - V ~ 7 \i +x/~,] J "

(12)

The result is finite when p ~ 0 as guaranteed by the Kinoshita theorem [ 9 ]. The dilogarithm function Li2 (x) is defined as in ref. [ 10 ]:

Li2 ( x ) = -

~ ln(1-z) x x2 x3 z dz=]-i+~+-3~+..,

J

forlxl~
(13)

0

Our result agrees with the numerical integration results given for specific cases in refs. [ 1,2 ]. For the cases of interest p ~ 0.1, for which an approximation to O (p 3) works well:

h(p) =Tr 2 - ~ + p [ 6 8 + 2 4 In(p) ] _p3/2 32 7~2-bp2 [ 167r2+ 2 7 3 - 36 In(p) + 36 In2 (p) ] _p5/2 32zc2 +p3 [ 19~___A_2 _t~ ln(p) ] + O ( p a ) .

(14)

To find f(p) one has to divide h (p) by Fos (p) where Fps (p) is given in eq. ( 3 ). To first order in p we get

f(p)=7~z- ~ +p[18 + 8rcZ+ Z41n(p) ] .

(15)

The coefficient of the p ln(p) term is indeed 24. The functionf' (p') that should be used when the masses are taken at a single common scale can be derived from eq. (14) by applying eq. ( 11 ). To first order in p' we get f (p') = z c 2 - ~ +p ' (18+8rt 2) .

(16)

There is no term of the form p' In (p'). Thus we proved our above statement: Previous numerical calculations, being in agreement with eq. (15) but not with eq. (16), correspond to the mass ratio between on-sheU masses. The function h(p) can be used directly, by rewriting eq. ( 1 ) as F(h~l~9~) = F ~°)- (2Ots/3~r)F ~1) , F~°)= ( G2m~ l V~h12/192~r3)Fps(p),

/ ' ~ ) = ( G2m~ l V~hIZ/1921r3)h(p) •

(17)

4. The s23 mixing angle The above discussion is most relevant for the calculation of the s23= [ Vcb [ mixing angle (we use the parametrization of ref. [ 11 ] ) from the charmed semileptonic B-decay [ 12 ]: 186

Volume 221, number 2

PHYSICSLETTERSB

(s23)2= (192~r3/G 2) [BR(b~cJ~9~)/Tb] ( 1/~loFos, c )/mb,5

27 April 1989 (18)

with p< = [rnc(m¢) ]51 [mb(mb) ]5,

F~s =Fps(Pc),

t/3 =FQcD(Pc) = 1 - ( 2 a J 3 n ) f ( & ) .

(19)

Weuse [13]

rnc(mc) = 1.27_+0.05 GeV, mb(mb) =4.25_+0.10 GeV.

(20)

The mass ratio is then (p~) 1/2= 0.30-+ 0.02.

(21)

This gives [ 2 ] f(p~) = 2.51 -+0.06,

F~s =0.52_+0.04.

(22)

To find t/~ one has to give a value to as. However, unlike p, the value of as (or equivalently the scale at which it should be taken) is not determined until we calculate to O((oq)2). The best we can do is try to estimate the scale at which the O ( ( a s ) 2) corrections are smallest. We take 1.5 GeV~<#~<2.5 GeV which, for AQCD= 150 MeV, gives [ 13] as=0.20_+0.02. We get r/~ =0.89_+ 0.01.

(23)

The value of t/~ = 0.87 given in ref. [ 3 ] corresponds to as = 0.24. The r/~-value is not sensitive to the uncertainty in Pc. For the semi-leptonic branching ratio we use the world average [ 14 ] of measurements both in the continuum and on the Y(4S) peak: B R ( b ~ e v e X ) =0.115_+0.004.

(24)

We assume for the present calculation R ---F(b~u£v~)/F(b~c~v~) = 0. This may give an Sz3-value higher by up to 4% than the true value. The world average for the B lifetime is [ 15 ] zb = (1.18_+0.14) X 10 -12 s.

(25)

The largest uncertainty in the extraction of s23 from the semi-leptonic decay width comes from the m~ dependence. A fit to the leptonic spectrum gives [4 ] =4.95_+0.05 GeV.

(26)

This fit is based on the model by Altarelli et al. [ 16]. Theoretically, it is plausible to use mb(/t) at a scale # which corresponds to the average mass of the l~v system [ 4 ]. With 1.5 GeV ~ / l ~ 2.5 GeV and AQcD= 150 MeV, the range is 4.6 GeV ~
(27)

The determination 0fs23 from the B-meson semileptonic decay (eq. ( 18 ) ) is thus subject to both experimental and theoretical uncertainties. Due to the rn ~-dependence, we cannot determine (s23)2 to an accuracy better than 30%. Adding the errors in quadrature we get (s23)2+ (2.1 _+0.7) ×

10 -3 ,

(28)

which gives 187

Volume 221, number 2

PHYSICS LETTERS B

27 April 1989

s23 =0.046 + 0.008.

(29)

Calculations ofs23 within models other than the free quark model usually give somewhat higher s23 values [ 1618].

5. Charmless semi-leptonic b-decay The s~3 = I Vubl mixing angle is given by (s13)2= (192n3/G~) [BR(b~u~9~)/zb] (1/qoFps)/mb. ,, u 5

(30)

with

F~s=F~(pu), , t o = Qco(p~)=l-(2aJ3n)f(m)

Pu ~ . [mu(mu)]Z/[mb(mb)] 2,

Ut

.ll

F

•

(31)

In all existing calculations the u-quark mass is neglected. However, as the u-quark is lighter than AQco, its mass is not well-defined on-shell. What we would really like to use is the ratio mJrnb at a common mass scale, as m~ at scales above AQCDis well defined and known [ 13 ]: m , ( 1 G e V ) = 5.1 _+ 1.5 MeV,

rob(1GeV) = 5.6 +_0.1GeV,

p ~ , ~ (8_4) +6 × 10 --7 .

(32)

The value of mb given here corresponds to A = 150 MeV. From eqs. (16) and (32) we can see that indeed p" can be safely put to 0: F~s=Fo~(0) = 1, f ' (0) = n 2 - ~ 3 . 6 2 .

(33)

Using the same range for a~ as in the calculation of~/~ we get n~ = 0 . 8 5 + 0 . 0 1 .

(34)

Again, the value given in ref. [3], q~ =0.82, corresponds to as=0.24. The ratio q'o/q~ does not depend on our choice of ors. We get

S13/$23 =0.74 [F(B~X~v~)//'(B~XcI~V~) ]1/2

(35)

6. Lepton decays The above calculation of QCD corrections to quark decays can be easily modified to calculate QED corrections to lepton decay, ~,~ ~jgjvi, by the replacement

as~]azra.

(36)

For the z~lx%v~ decay, we have m,=105.659MeV,

m~=1784.2MeV,

p = 3 . 5 × 1 0 -3 .

(37)

We get

Fos(p=3.5× 10 -3) =0.9728, f ( p = 3.5 × 10 -3) = 3 . 4 8 .

(38)

Although p is of order 10 - 3 , f ( p ) is modified from its zeroth order value by as much as 4%. However, as arm is small, FQzD(p) is modified from its zeroth order value by only 2 parts in 104: FQEo(p=0) =0.9957,

FQZD+(p=3.5× 10 -3) =0.9959.

(39)

In ref. [ 19 ] QED corrections to z decay rates are calculated with terms of order (~Era" ( m 2, / m~2 ) ) neglected. 188

Volume 221, number 2 T h e y get

F(x~

PHYSICS LETTERS B

27 April 1989

g v g ( 7 ) )/_r'(,c--,evg(7 ) ) = 0 . 9 7 2 8 . We find a 0.02% correction to this value:

F(z--,Ixv9 ( 7 ) ) / F ( z - - , e v 9 ( 7 ) ) = 0.9730.

(40)

7. Conclusions The need for accuracy in the d e t e r m i n a t i o n o f the quark mixing angles necessitates a refinement o f the ingredients involved in the calculation. We concern ourselves with one such aspect: the quark mass ratio that should be used in the calculation o f semi-leptonic decay widths. We are interested in the difference between calculations using the ratio between the on-shell masses a n d those using the ratio between the masses taken at a c o m m o n energy scale. There are three possible cases: ( a ) The ratio ml/mh is close to 1. In this region the question is u n i m p o r t a n t b o t h in principle, as the two possible mass ratios are very close to each other, a n d in practice, as nature has not p r o v i d e d us yet with such a case. ( b ) The ratio mi/mh is close to 0. Here the question is interesting in principle, as for light quarks there is no well-defined on-shell mass. However, in practice the question is, again, u n i m p o r t a n t because the mass ratio can be a p p r o x i m a t e d to zero, a n d the calculations identify to zeroth order. (c) The ratio mdm~ is non-negligible, but not too close to 1. Here the question is i m p o r t a n t both in principle a n d in practice. We find that previous calculations, which were all numerical, correspond to the ratio between the on-shell masses. We give an analytic expression for the Q C D correction factor to all orders in the ratio between on-shell masses. We also give useful a p p r o x i m a t i o n s to the Q C D correction when either the ratio between on-shell masses or the ratio between the masses at a single scale is used.

Acknowledgement I t h a n k F r e d G i l m a n , H o w a r d H a b e r a n d Michael Peskin for their i m p o r t a n t advice, a n d Isi Dunietz and Russel K a u f f m a n for c o m m e n t s on the manuscript.

References [ 1] N. Cabibbo and L. Maiani, Phys. Lett. B 79 ( 1978 ) 109; G. Corb6, Phys. Lett. B 116 (1982) 298; Nuel. Phys. B 212 (1983) 99. [ 2 ] A. All and E. Pietarinen, Nucl. Phys. B 154 (1979) 519. N. Cabibbo, G. Corbo and L. Maiani, Nucl. Phys. B 155 (1979) 83. [3] A.J. Buras, W. Slominski and H. Steger, Nucl. Phys. B 238 (1984) 529; B 245 (1984) 369. [ 4 ] E.H. Thorndike, in: Proc. 1985 Intern. Symp. on Lepton and photon interactions at high energies, eds. M. Konuma and K. Takahashi (Kyoto, 1985). [ 5 ] M. Peskin, private communication. [ 6 ] M. Je~,abekand J.H. KiJhn, Phys. Lett. B 207 ( 1988 ) 91. [7] Q. Hokim and X.-Y. Pham, Ann. Phys. 155 (1984) 202. [ 8 ] R.E. Behrends, R.J. Finkelstein and A. Sirlin, Phys. Rev. l 01 (1956) 866; S.M. Berman, Phys. Rev. 112 (1958) 267; T. Kinoshita and A. Sirlin, Phys. Rev. 113 (1959) 1652. [9] T. Kinoshita, J. Math. Phys. 3 (1962) 650. [ 10 ] L. Lewin, Dilogarithms and associated functions (Macdonald, London, 1958). [ 11 ] H. Harari and M. Leurer, Phys. Lett. B 181 (1986) 123. 189

Volume 221, number 2

PHYSICS LETTERS B

[ 12] See e.g.J.S. Hagelin, Nucl. Phys. B 193 ( 1981 ) 123; F.J. Gilman and J.S. Hagelin, Phys. Lett. B 133 (1983) 443. [ 13 ] J. Gasser and H. Leutwyler, Phys. Rep. 87 (1982) 77. [ 14 ] HRS Collab., C.R. Ng et al., Argonne preprint ANL-HEP-PR-88-11 ( 1988 ). [15] S.L. Wu, Nucl. Phys. B (Proc. Suppl.) 3 (1988) 39. [ 16] G. Altarelli, N. Cabibbo, G. Corbo, L. Maiani and G. Martinelli, Nucl. Phys. B 208 (1982) 365. [17] M. Wirbel, B. Stech andM. Bauer, Z. Phys. C 29 (1985) 637. [ 18 ] B. Grinstein, M.B. Wise and N. Isgur, Phys. Rev. Lett. 56 ( 1986 ) 298; Toronto preprint UTPT-88-12 (1988). [ 19 ] W.J. Marciano and A. Sirlin, Phys. Rev. Lett. 61 ( 1988 ) 1815.

190

27 April 1989