Extracting |Vbc|, mc and mb from inclusive D and B decays

Physics Letters B 321 (1994) 88-94 North-Holland PHYSICS LETTERS B

Extracting [ 1, mc and mb from inclusive D and B decays

Michael Luke a,b and Martin J. Savage b,c a Department of Physics, University of Toronto, Toronto, Canada MSS 1A 7 b Department of Physics, University of California at San Diego, La Jolla, CA 92037, USA c Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA

Received 30 August 1993 Editor: H. Georgi

Using recent results for nonperturbative contributions to the B and D meson inclusive semileptonic widths, a model indepen- dent extraction of I Vb~l, mc and mb is made from the experimentally measured B and D lifetimes and semileptonic branching ratios. Constraining the parameters of the HQET at (9 (1/m ~ ) by the D semileptonic width, I Vb~l is found to lie in the range 0.040 < I V~I < 0.057. The c and b quark masses are not well constrained due to uncertainty in the relevant scale of as. These results assume the validity of perturbative QCD at the low scales relevant to semileptonic charm decay. Without making this assumption, somewhat less stringent bounds on V~ from B decay alone may be obtained.

There has been much recent interest in the appli- cat ion of the techniques o f the heavy quark effective theory ( H Q E T ) [1 -5 ] to inclusive decays o f had- tons containing a single c or b quark [ 6-12 ]. As first made explicit in [6] , the differential decay rate for an inclusive factorisable process such as B - , X c e ue or B--,X,7 may be written, using an opera tor product expansion, in terms of the matr ix elements of local op- erators between B mesons. Using the techniques of HQET, it was shown in [6] that the leading term of the expansion in 1 ~rob reproduces the par ton model, and that the (9(1~rob) correct ions vanish by the equations of mot ion for the heavy quark. More re- cently, the (9 ( 1 /m 2 ) correct ions to the par ton model have been calculated for semileptonic decays [ 7-11 ], as well as for the rare decays B--,Xsy [8,12] and B-~Xse+e - [12]. A crucial observat ion which emerges from the opera tor product expansion is that the semileptonic width of a heavy meson is propor- t ional to the fifth power of the heavy quark mass, not the meson mass. The heavy quark mass mq is a well- defined quant i ty in HQET, and is defined such that the residual mass term in the effective heavy quark Lagrangian vanishes (for a detai led discussion of the defini t ion of mq, see [ 13 ] ).

In addi t ion to these nonper turba t ive corrections, there are calculable (9(cts(mq)) corrections to the free

quark decay picture coming from real and vir tual gluon emission [14]. In this letter, the complete expression to (9 ( as (mq), 1 / m 2 ) for the semileptonic width of a heavy meson is used to extract the c quark mass from the measured D + lifetime and semileptonic branching ratio. F rom this, the b quark mass is de te rmined and the B lifetime predic ted in terms of the K M matr ix element I Vbc I. Compar ing this with the exper imental ly measured l ifetime allows I Vbc I to be determined.

Throughout this letter the mass of a heavy meson will be denoted by MQ and the mass of the corresponding quark by mq. The meson and quark masses are related by [ 15 ]

M Q = m q + A "~" 1 + 3"~'2 O ( 1 ) 2 m ~ + ~ . (1)

The paramete r X in (1) may be interpreted as the energy of the light degrees of f reedom of the meson. It corresponds to the const i tuent mass of the light quark in the meson, and so is expected to be of the order of a few hundred MeV. F rom considerat ion of QCD mass inequalities analogous to those used in the light quark sector to prove that rap> rn~, Guralnik and Manohar [ 16 ] have shown the existence of a rigorous lower bound

88 0370-2693/94/$ 07.00 © 1994 Elsevier Science B.V. All fights reserved. SSDI 0370-2693 (93)E11 52-N

Volume 321, number 1,2 PHYSICS LETTERS B 20 January 1994

l /> 237 GeV. (2)

We will also take A< 800 MeV in this work when quoting limits on rn~, mb and Vbc. This bound is consistent with the QCD sum rules estimate [ 17,18 ] (see also [ 19 ] )

~s.r. = 570+ 70 MeV (3)

and is also consistent with the quark model in which .d is expected to be a few hundred MeV.

21 and 22 are defined in terms of the expectation values in the heavy quark effective theory,

(HQ.I[ i ( iD)2hlHe) = 2Me2 , ,

( H e I h-( ½i)aU~Gu,hlHe ) = 6MQ,~ 2 (#) , (4)

where H e is the pseudoscalar heavy meson (B or D) and h is the heavy quark field in the effective theory. 22 parameterises the effects of the chromomagnetic moment operator, and may be extracted from the B * -

B mass splitting

)-2(mb) = ~mb(MB* - M n ) ~- ¼ (MB*2 --MB)2

=0.12 GeV 2 (5)

(corresponding to 22(m~)=0.10 GeV 2 [20]) . 21 parameterises the kinetic energy of the b quark inside the hadron. Since it contributes equally to mn and mn., 21 cannot be extracted from the meson masses.

At (~ ( 1/m 2 ), the nonperturbative corrections to the parton model are also parameterised in terms of 21 and 22. Combining the results from [7-11 ] with the as corrections to the free quark model [ 14 ] gives the complete expression for the semileptonic D width to Cg( l lmZ, oq(#<) )

2 5 F(D__,e+v~Xq)= Gvmc

1927r 3

"~1 )<(IVc~I2{[ 1 2°q(#~)g(m~)+2--~2]f~(m~)3n

922

+ i V~dl2( 1 2as(#¢) ~ -- 92:'~'~ 3 ~ g (O)+ 2m 2 ,],]

+ O ~33,°t~(U~) 2, m2 ,]. (6)

The functions fl,2 arise from the finite mass of the quark in the final state,

fl (x) = 1 - 8x 2 + 8x 6 - x 8 - 24x 4 log x ,

f2(x)=l--8xE--8X4+8x6+ SxS+8X410gx, (7)

while the function g(x ) arising from one-gluon graphs is tabulated in ref. [ 14] (we note that g (0 )=3 .62 , g (0 .2 /1 .6)=3 .15 and g(1 .6 /4 .8 )=2 .40) . We also take ms= 200 + 80 MeV and md ~ -- O.

The choice of the appropriate scale/tc in (6) is a potentially troublesome one. The scale is set roughly by mc; however, since the final state hadron typically carries off only a fraction of the available energy, the appropriate scale may be somewhat lower. Formally this choice of scale is a higher order effect; however, in practice there is a considerable difference between, for example, ~xs (mc) and ors ( me~ 3 ). In partic- ular, at scales much less than mc, QCD perturbation theory cannot be trusted. Since we have not done the two-loop calculation, we cannot resolve this scale ambiguity. In this letter we will take/t~--- me, and the sensitivity of the results to higher order terms in as will be estimated by formally varying/~¢ between m~ and mc/3. Taking A~4~D = 250 MeV and A~3~D = 300 MeV, and using the two-loop expression for as(#) gives ors (me) = 0.09~r and ors (mc/3 ) = 0.4rr. We stress that we are not claiming that QCD perturbation theory is valid at/~---m~/3, but rather that if perturbation theory is meaningful for D decays, we expect this approach to give a reasonable estimate of the uncertainty due to higher order QCD corrections ~1. For- tunately, the determination of Vbc is relatively insen- sitive to the uncertainty in m~ extracted from the D width.

From the observed semileptonic branching ratio, lifetime and mass of the D -+ [ 21 ]

Br(D -+ --.e -+ + anything) = ( 17.2 + 1.9)%,

zo± = 10.66+0.23)< 1 0 - 1 3 S ,

MD_+ = 1869.3+0.3 MeV, (8)

it is a simple matter to extract mc and 21 from eqs. (6) and (1) for any value of.4. (We have checked

#~ Note that the energies encountered in this problem are similar to those encountered in hadronic ~ decays, for which the corrections to (9(cq(m~) 4) have been calculated, and for which the perturbation series appears to be converging [21 ].

89


that the D o gives results consistent with the D -+; the D ± meson is used, as the uncertainty in its semileptonic branching fraction is somewhat smaller than for the DO.) The results are shown in figs. l a and lb. In each figure the narrow band corresponds t o / ~ = me,

while the broader band corresponds to rn~/3 < #¢ < me.

The uncertainty corresponding to 1 a errors on all exper imenta l input parameters is included in the width o f each band. F rom fig. I, the G u r a l n i k - M a n o h a r bound (2) on At rans la tes to the l imit

1.5

)~1 1 ( G e V ) 2

0.5

-0.5

-1

2000

300 400 500 600 700 800

A ( M e V )

1900

m c 1800

( M e V )

1700

1600

1500

1400 300 400 500 600 700 800

X ( M e V )

Fig. I. ,t~, m,, and mb as functions of,'/. In each graph, the vertical line corresponds to the Guralnik-Manohar bound d> 237 GeV. The shaded regions correspond to the experimental error in the input parameters and, for the broader band, the variation in #~,

90

Volume 321, number 1,2

5300

5200

m b 5100

(MeV) 5000

4900

4800

(c)

PHYSICS LETTERS B 20 January 1994

• , . . . . , . . . . , . . . . , . . . . , . . . .

tc=mc

~ mc/3<Pc<m c

4700

4600 300 400 500 600 700 800

X (MeV)

Fig. 1. Continued.

21 > - 0 . 5 G e V ( l a ) . (9)

Because of the uncertainty in fic, the c quark mass is not well constrained from the data, although taking A< 800 MeV suggests mc>~ 1460 MeV. Since the upper limit of the error bar in fig. 1 a is both very sensitive to the value of as at low scales and corresponds to me> mn, a useful upper bound on mc cannot be extracted.

For B mesons, formulas analogous to ( 1 ) and (6) hold, with the replacements MD-~MD, mc-*mb, Vcs-~ Vbc, Vcd~ Vbu and m,-~ me. Since 2 ~ is now determined from D decays for each value of A, the b quark mass mb may be solved for as a function only of,4. Using M s = 5.278 _+ 0.002 GeV gives the results shown in fig. lc. Since Ms is less sensitive to 21 than Mn, it is not surprising that mb is roughly a linear function of A, with smaller error bars than inc. mb is found to lie in the range 5140 >mb > 4600 MeV.

Given mb and 21, the semileptonic width of the B may now be predicted, allowing the KM matrix element I Vbcl to be determined as a function of A. The inclusive branching ratio of the B to electrons is measured to be [22]

Br (B-~ e+ anything) = ( 10.7 _+ 0.5 )% (10)

(where the charge of the B is not determined). Av- eraging the recent measurements of the B lifetime by the DELPHI Collaboration [23] and by the CDF Collaboration [ 24 ] gives a B lifetime of

zB= 1.32_+0.09)< 1 0 - 1 2 S (11)

(statistical and systematic errors added in quadra- ture). This gives I Vbcl as a function of A, as shown in fig. 2. The two bands in the figure correspond to the two choices of scale tib-~mb, t ic=mc and me~3 <fic< me, mb/3 < tib < mb (with tic~fib fixed). Over the range of values for A shown in fig. 2, I Vbcl is found to lie in the range 0.040< ] Vbc] <0.057. The lower bound is set by the rigorous lower bound on ,4, while the upper corresponds to .,t= 800 MeV. This extraction is consistent with the current value of [Vbc[ (1.32 pS/ZB)i/2=O.051 +0.008 extracted from the exclusive decay B~D*ef 'e using HQET [ 18,19 ]. We note that choosing a lower scale for tie and tib tends to raise the preferred values of I Vb~[.

For a given value of A the effects of higher order terms in 1/mq and as on this extraction of [ Vbc[ may

91


0.06

PHYSICS LETTERS B 20January1994

0.055

0.05

IVbcl

0.045

0.04

0.035

0.03 300 400 500 600 700 800

S ( M e V )

Fig. 2. The weak mixing angle [ V~[ as a function of A for rB = 1.32 + 0.09 ps. The vertical line corresponds to the constraint imposed by the bound on A. The error bar on the left indicates the current value of I Vb~l extracted from the exclusive decay B~D*eY~ and its associated uncertainty. The shaded region corresponds to the experimental error in the input parameters and, for the broader band, the variations in/t~ and/tb.

be est imated. These terms may be parameter ized by 8bx and eb,c, defined by

MD=m~( 1+ "~ ~'1+3~'2 ) mc 2 m ~ + ~ '

Ms =mb( 1+ d 2'+322 ) mb 2m'-""~b + fib (12)

and

F(D--'Xqe + ~'e) oc m~( 1 + E~) ,

F(B~Xqe + ~'e) ocm~( 1 +eb) • (13)

They correspond to a change in I Vb~ I of

A, Vbcl l -~- ) IVb~I 2 ,,,b (Ec+5~C)-(~b+5ab)

-~ ~ [ e ~ - 3eb + 5 ( ~ - 3~b) ] • (14)

The leading contr ibut ions to ~c are o f order (X/rnc) 3 and [oq(m~)/rr] ).2/m~ ,~2, which we est imate to be

~2 The operator h-(iD)2h receives no strong interaction corrections to its coefficient, by reparameterisation invariance [ 25 ].

at most a few percent, while ~b is expected to be at least an order of magni tude smaller. We have at- t empted to take the leading contr ibut ions to Ec, b arising from higher order QCD corrections into account by varying the scales ~tc, b between mb,c and mbx/3. From (14) , we can unders tand why our results were relat ively insensit ive to the choice of scale: an uncertainty in the charm width only gives ~ ~ this uncerta inty in [ Vbc ]. Fur thermore , the uncer ta inty in scale tends to cancel between E~ and ~b.

Since the extract ion of 21 as a function of A de- pends on the val idi ty of per turbat ion theory for QCD at a low (/z<m~) scale, it is instructive to compare this extract ion of ] Vb~ [ with that obta ined by ignor- ing the D decay data altogether and s imply varying 21 between + 1 GeV 2 for each given value of ,4. This amounts to working at (9 ( 1/mq) in the heavy quark expansion with a reliable est imate of the uncer ta inty from 1 /m 2 terms. As before, the scale/tb is also var- ied between flb=mb/3 and /lb= mb. The results are shown in fig. 3. The shaded area on each graph corresponds to both the exper imental errors on the input parameters and the variat ions in 21 and/zb. As ex-

92


0.06

PHYSICS LETTERS B 20 January 1994

0.055

0.05

IVbcl

0.045

0.04

0.035

0.03 300 400 500 600 700 800

~, ( M e V )

Fig. 3. The weak mixing angle I Vb~l as a function of/ /for zB = 1.32+0.09 ps, without imposing constraints on 21 from D decay. The vertical line corresponds to the lower bound on A, and the shaded regions correspond to the experimental error in the input parameters, varying 2 ~ from - 1 GeV to 1 GeV and #b from mb to rnb/3. The error bar on the left indicates the current value of I V~ I extracted from the exclusive decay B--*D*eG and its associated uncertainty.

pected, the uncer ta inty in I Vbcl is somewhat larger than when the D decay da ta are included, par t icular ly at large values of A; however, I Vbc[ is still well con- s trained since the predic t ion for the B width is rela- t ively insensi t ive to 2 i. For 237 < / 1 < 800 MeV, I Vbc I is found to lie in the range 0 .04< I Vb~l <0.06.

It is encouraging that the extract ion of I Vbc I f rom inclusive decays is consistent with that obta ined from exclusive decays over the full al lowed range of A. Fur thermore , the extract ion of I Vbc I f rom inclusive decays has some theoret ical and exper imenta l advan- tages over exclusive decays. No exclusive hadronic final state needs to be identif ied, and the extrapola- t ion of form factors to zero recoil is not required. The method is l imi ted by the uncerta int ies in X and the relevant hadronic scales/~b and /~ . Addi t iona l input, such as a lat t ice measurement of A or 2~, or a two- loop calculation of the semileptonic decay rate, would further constrain I Vb~l. For example, from fig. 1 a, the QCD sum rules est imate (3 ) o f / 1 = 5 7 0 + 7 0 MeV corresponds to 0.1<21~<1.5 GeV 2 and 0 .044< I Vbcl < 0.054 (although the upper bound on 2 t should not be taken too seriously, since both the 1/mc ex-

pansion and QCD per turba t ion theory are unrel iable at this po in t ) . On the other hand, the nonrelat ivis t ic quark model suggests that A ~ 350 MeV, the mass of the light const i tuent quark, and that 2~~ - (350 M e V ) 2 ~ - 0 . 1 GeV 2 (note that 2~ < 0 corresponds to a posi t ive mass shift from the kinetic energy of the heavy quark) . F rom fig. la , these values of 21 and X are consistent with one another, and are consistent with somewhat lower values of I Vbc I.

We thank M.B. Wise for useful discussions, part ic- ularly concerning the relevant scales/to and/~b and the possibil i t ies of extracting Vbc without using the D decay data. We are also grateful to A. Falk, B. Holdom, A. Manoha r and L. Wolfenstein for useful comments and criticisms. M.J.S. acknowledges the support o f a Superconduct ing Supercoll ider Nat ional Fel lowship from the Texas Nat ional Research Labora tory Com- mission under grant FCFY9219. This research was supported in part by TNRLC grant RGFY93-206 and by the Depar tmen t of Energy under contracts DE- FG03-90ER40546 (UC San Diego) and DE-FG02- 91ER40682 ( C M U ) .

93


References

[ 1 ] M.B. Voloshin and M.A. Shifman, Yad. Phys. 45 (1987) 463 [Sov. J. Nucl. Phys. 45 (1987) 292]; 47 (1988) 801 [ 4 7 ( 1 9 8 8 ) 5 1 1 ] .

[2] N. Isgur and M.B. Wise, Phys. Lett. B 232 (1989) 113; B 237 (1990) 527.

[3] H. Georgi, Phys. Lett. B 240 (1990) 247. [4] B. Grinstein, Nucl. Phys. B 339 (1990) 253. [ 5 ] E. Eichten and B. Hill, Phys. Lett. B 243 (1990) 427. [6] J. Chay, H. Georgi and B. Grinstein, Phys. Lett. B 247

(1990) 399. [7] I.I. Bigi, N.G. Uraltsev and A.I. Vainshtein, Phys. Lett. B

293 (1992) 430. [811.1. Bigi, B. Blok, M. Shifman, N.G. Uraltsev and A.

Vainshtein, TPI-MINN-92/67-T (1992); Phys. Rev. Lett. 71 (1993) 496.

[9] B. Blok, L. Koyrakh, M. Shifman and A. Vainshtein, NSF- ITP-93-68, hep-ph/9307247.

[ 10] A.V. Manohar and M.B. Wise, UCSD-PTH 93-14, hep-ph/ 9308246.

[ 11 ] T. Mannel, IKDA 93/16, hep-ph/9308262.

[ 12] A.F. Falk, M. Luke and M.J. Savage, UCSD-PTH 93-23. [13]A.F. Falk, M. Luke and M. Neubert, Nucl. Phys. B 388

(1992) 363. [ 14 ] N. Cabibbo and L. Maiani, Phys. Lett. B 79 ( 1978 ) 109. [ 15 ] A.F. Falk and M. Neubert, Phys. Rev. D 47 ( 1993 ) 2965. [ 16] Z. Guralnik and A.V. Manohar, Phys. Lett. B 302 (1993)

103. [ 17] M. Neubert, Phys. Rev. D 46 (1992) 1076. [ 18] M. Neubert, Phys. Lett. B 264 ( 1991 ) 455. [19]M. Neubert, SLAC-PUB-6263 (1993), Phys. Rep., to

appear. [20] A.F. Falk, B. Grinstein and M. Luke, Nucl. Phys. B 357

(1991) 185; E. Eichten and B. Hill, Phys. Lett. B 243 (1990) 427.

[21] E. Braaten, S. Narison and A. Pich, Nucl. Phys. B 373 (1992) 581, and references therein.

[22] Particle Data Group, K. Hikasa et al; Review of particle properties, Phys. Rev. D 45 (1993) 1.

[23 ] DELPHI Collab., P. Abreu et al., CERN-PPE/93-80 (1993). [24] M.L. Mangano (CDF Collab.), FERMILAB-Pub-93/139-

E (1993). [ 25 ] M. Luke and A.V. Manohar, Phys. Lett. B 286 ( 1992 ) 348.

94

Documents

Extracting |Vbc|, mc and mb from inclusive D and B decays