Application of Mössbauer-type sum rules for B meson decays

Physics Letters B 308 (1993) 105-110 North-Holland

PHYSICS LETTERS B

Application of M/Sssbauer-type sum rules for B meson decays H a r r y J. Lipkin Department of Nuclear Physws, Welzmann Instttute of Sctence, Rehovot 76100, Israel and School of Phystcs and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel A vlv Umverstty, Tel Awv, Israel Received 2 February 1993, revised manuscript received 17 April 1993 Editor. H. Georga

Sum rules originally derived for the M6ssbauer effect are apphed to weak semfieptonlc B decays The sum rules follow from assuming that the decay by electroweak boson emission of an unstable nucleus or heavy quark m a bound system is described by a pomthke couphng to a current which acts only on the decaying object, that the Hamdtoman of the bound state depends on the momentum of the decaying object only m the kinetic energy and that the boson has no final state mteractmns The decay rate and the first and second moments of the boson energy spectrum for fixed momentum transfer are shown to be the same as for a nonmteractmg gas of such unstable objects with a momentum dlstnbut~on the same as that of the bound state. B meson semlleptomc decays are shown to be dominated by the lowest-lying states m the charmed meson spectrum.

Simple features of the spectroscopy and weak transitions of hadron states containing heavy quarks have recently been discussed in detail [ 1,2 ]. The contributions to a Bjorken sum rule [3] have been examreed [4,5 ]. The Bjorken sum rule recalls sxmilar sum rules derived from the Mtissbauer effect [ 6-8 ] which can also be v~ewed as the emission of an electroweak boson in a pomtlike vertex by a heavy object bound to a complicated system, where the transition revolves a change m the momentum and the mass of the heavy object without a change in the interactions between the heavy object and the rest of the system. We derive sum rules for the moments of the energy spectrum of the emitted boson for decays of bound heavy quarks, analogous to those previously derived for the M6ssbauer effect [ 9 ]. In addition to the Bjorken sum rule for the decay rate we find two additional sum rules showing that the mean and mean square energies of the emitted boson or lepton pair are independent of the detailed dynamics of the bound state, depend only on the momentum spectrum of the heavy quark m the lmtml state and are thus identical to those for the decay of a free heavy quark in a gas with the same momentum spectrum as m the bound state. The weak decay from an initial state denoted by l Elsevier Science Pubhshers B.V.

to a final state denoted b y f i s described by the Ferma golden rule ofttme dependent perturbation theory Wl~f = I (fl nweak It ) 12p(Ef) ,

(la)

where Hwe~kdenotes the weak Hamiltonian and p (El) is the density of final states. For the case ofa semileptonic decay where a lepton pair is emitted with momentum - q, the matrix element for a transition from an imtlal hadron state at rest IZb) containing a b quark to a final state I f ~ ( q ) L ( - q ) ) o f a hadronf~ with total momentum q containing a c quark and a lepton pair with momentum - q factorizes into a leptonic factor depending only on the lepton variables and a weak vertex describing the heavy quark transition
lb)

where gk(q) is a function 0fthe lepton variables and Jk(q) is the Fourier component of the flavor-changing weak current for momentum transfer q and the index k=0, 1 describes the spin character of the heavy quark transition; e.e. J~ transforms hke a vector under spin rotations and describes spm-fhp transitions while Jo ~s s~mply the identity operator and describes nonflip transitions. 105

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The result ( 1 ) is exact to first order in the electroweak interaction described by the standard model and exact to all orders m strong interactions if the initial and final states considered, I/b) and If~), are exact eigenstates of the strong interaction hamiltonian. We do not need to assume the validity of QCD, perturbative or otherwise, or even the validity of local field theory at this state. All we need is the existence of a hamiltonian for strong interactions and its eigenstates. In heavy quark transitions this description applies to semileptonic decays and also to nonleptonic decays in cases where factorization holds; i.e. there are no final state mteractmns between the hadrons produced by the W and the final bound state. We now derive the sum rules explicitly for the case of a heavy quark transition ( 1 ). The relevant matrix elements of the weak current (f~lJk(q) lib) depend only on the variables of the heavy quark and determine completely the dependence of the transition on the hadronic wave functions. It is convenient to normalize the current and the coefficxent gk ( q )

(2a)

where the factor gk(q) depends upon the strength of the interaction and depends upon the kinematic variables only via the magnitude of the momentum transfer q and not upon the initial momentum of the heavy quark and 27kIS a spin factor acting on the spin of the heavy quark and normalized to satisfy the relation If~)

I (f~127kltb) IZ= 1 .

(2b)

(2c)

This IS just the Bjorken sum rule. To obtain additional sum rules it is convenient to define the reduced matrix element 106

(3a)

which expresses the Bjorken sum rule in the form IJb)

I (fclMk(q) ltb )12= l .

(3b)

The Hamiltonian of the bound state of a heavy quark o f f l a v o r f a n d a light quark "brown muck" can be written as a function of the co-ordinate X, the momentum P and the mass m f o f the heavy quark, and all the degrees of freedom in the brown muck denoted by ~,

Hs=H(P, ms, X, ~ ) ,

(4)

where all the flavor dependence is in the heavy quark mass my The transition operator (3a) contains a momentum displacement operator, exp ( - iq. X ) P exp (iq. X) = P + q,

(5a)

exp ( - iq. X) Hfexp ( xq. X)

=Hf[ (P-t-q), my, X, ~1 .

(5b)

([Ec(q)]n)k - ~ (Ec)nl(£1Mk(q)lib)l 2 IX)

= (tb IZk[exp ( -iq'X)Hcexp(lq'X)]nSk [tb) =(ZblZk{n[(P+q),m~,X,~]}"Xkllb)

,

(6a)

( l E e ( q ) ]n)k = ( Bkl {Hb + H [ (P+q), mc, X, ~,] - H I (P), rob, X, ~ ] } " [ B k ) ,

(6b)

( [ E c ( q ) 1")k = (Bk [{//~ + H [ (P+q), m~, X, ~1 - H [ (P), me, X, ~ , ] } " l B k ) ,

The transition probability or branching ratm is propomonal to the square of this matrix element and multiplied by kinematic; e.g. phase space factors which do not depend on the explicit form & t h e hadron wave functions but only on hadron masses and the momenta of the external particles. This matrix element is seen to sausfy the sum rule

~'. ]gk(q)(fclJk(q)Ilb)12: Igk(q)I 2 If~)

(f~lMk(q) l t b ) = ( f c l Z k e X p ( l q ' X ) [ i b ) ,

We now derive sum rules for the moments of the energy distribuUon Ec of the charmed f n a l state,

gk(q) (f~lJk(q) Itb ) =gk(q) (f~ IXk exp(iq'X)[tb ) ,

24 June 1993

(6C)

were IBk) denotes the state produced by the operator Xk acting on the imtial state tb:

IBk) = S k l l b ) •

(7)

We have thus obtained sum rules for moments of the final state energy distribution at a fixed momentum transfer q in terms of expectation values calculated in the initial state ]lb). The case n = 0 is just the Bjorken sum rule, showing that the sum of the squares of all transxtion matrix elements at fixed momentum transfer q is completely independent of the Hamdtonian H describing

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the dynamics of the bound state and is the same as for a free b quark decaying to charm. The sum rules reduce to a particularly simple form for the first and second moments, n = 1 and n = 2, if one simple additional assumption is made which holds in all conventional models; namely that the state Bk, produced by the operator -rk acting on the initial state tb is an eigenfunction of the bound state Hamlltonian Hb

lib IBk> -~HbSkltb > =M(Bk)IB~ > .

(8)

This assumption holds trivially for the case k = 0 where the spin operator 2Yois just the identity. It is a good approximation for the case k = 1 if spin effects are neglected as well as in all models where flapping the spin of the heavy b quark in the B meson produces the vector B* state. These include both simple constituent quark models as well as more general models which assume the approximation of heavy quark symmetry. It also holds in all simple models for the Ab baryon where the b quark carries the spin of the baryon, all other degrees of freedom are coupled to zero angular momentum and flipping the spin of the b quark simply flips the spin of the whole baryon. For the first and second moments, n = 1 and n = 2, the operator Hs appears in eq. (6b) only as acting either to the right on IBk> or to the left on ~ =,

(9a)

24 June 1993

the heavy quark kinetic energy, which is assumed to contain all the dependence of H on P and the heavy quark mass. The quantity R (q) is just the free recoil energy; i.e. the change in kinetic energy of a free charmed quark when its momentum changes from P to P + q. The quantity Ib~ is just the "isomer" or "isotope" shift, i.e. the change in the binding energy of the bound state when the mass of the heavy quark changes from m b to rnc. The two sum rules are expressed in terms of expectation values of operators depending only on the change in the Hamiltonian when the heavy quark momentum Pis replaced by P + q and the heavy quark mass is changed from mb to mc. They thus depend only on the momentum distribution of the heavy quark in the initial state and are independent of the brown muck in all models where P enters only into the kinetic energy term in the bound state Hamiltonian and the change in hyperfine energy with quark mass is neglected. They are therefore the same as for the decay of free b quarks in a gas with the same momentum distribution as in the bound state. The sum rules can be written in various ways appropriate for different applications. These depend upon which experimental quantities are measured and upon assumptions about values of parameters which are not directly measured like quark masses and hyperfine energy contributions. The sum rules can be expressed in terms of the energy Ew(q) carried offby the Wand observed as the energy of the lepton pair or hadrons produced from the Wwhlch must have momentum q for momentum conservation. From energy conservation

where

5m=mb--mc,

(9b)

R ( q ) = H [ ( P + q ) , mc, X, ~,,l - H [ (P), rn~, X, ~,,] = x / ( P + q)2 + m 2 - v / ~ + rn 2

I.~>

(9c)

Ie~= ~m+ H[e, m~, X, ~] - H [ P , me, X, ~] 6m 2mcmb '

~p2 _ _

Thus

k = ~ Ewl I=

q2 2m~'

(10)

Ew(q) =M(/b) - E c ( q ) •

=M(lb)-- < [Ec(q) ] >k = ~m-- +M(tb) - M ( B k )

(9d)

where the approximate equalities hold for the case where the nonrelativistic approximation is used for

~m--

q2 8m - - -- ( B k Ip21nk> - 2mc 2mcmb

+M(ib)-M(Bk) ,

(lla)

107

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([Ew(q)12>k-=- ~ (Ew)2l(f~lMk(q)[to>[ 2

the state Bk. The LHS of eq. (12) is just the mass

= ([Ew(q)] >2+ (Bkl [R(q)+Ibc]2LBk>

M(Dk) m the heavy quark symmetry hmit and ~s exact to first order xn the difference ( 1~me- 1/mo). The

-- 2

( [Ew(q)] >~+ +

q2 3m~

(Sm) 2 2 ~ ( ( B k l P 4 1 B k > - - ( B k l P 2 I B k > 2) .

4rncmb

(11b)

The mean energy carried by the W is seen to differ from the heavy quark mass difference 8m by three corrections: the free recoil kinetic energy, the "isomer shift" and a spin correction M(Zb)-M(Bk) which is present in the case of spin-flip transitions and is equal to the hyperfine splitting. If we make the nonrelatlvistlc approximation for the heavy quark motion, and neglect the dependence of the hyperfine energy on the heavy quark mass the two sum rules reduce to simple expressions for the mean energy and the dispersion of the energy distribution of the emitted W at fixed momentum q when appropriate kinemanc factors (i.e. phase space) are taken out of the observed distributions. When the momentum transfer q and the mean momentum P of the heavy quark m the bound state are both small m comparison with the heavy quark mass, the dispersion vanishes and the mean lepton energy is Just the quark mass difference mb-- mc. The transition is always to the ground state of the charmed system and the differences m the brown muck wave functions m the two cases is negligible. When the momentum transfer q ~s negligibly small, the correcnon resulting from the finite momentum of the heavy quark m the bound state Is just the difference in the heavy quark kinetic energies m the initial and final ground states. This is just equal to the difference in binding energies to first order m the mass difference. We now derive a sum rule for the mean excitation energy of the charmed final state above the ground state of the charmed system. The exact ground state energy is not known since it depends upon the bindmg energy of the brown muck to the heavy quark. We express this ~gnorance by wrmng

( Bk IH,. IBk > = M ( Dk) + E ,

(12)

where M(Dk) is the mass of the charmed analogue of 108

correcnon is therefore second order and can also be seen to be poslnve, since the perturbation result is also variational and g~ves an upper bound for the mass. The mean excltanon energy AEc of the observed spectrum above the ground state is obtained by combining eqs. (6c), (9c) and ( 12 ) to obtain ( A E c ) k =- ( E c ) k - - M ( D k )

= ~ EcI (f~lMk(q)Itb> 12-M(Dk) IJ~>

= + e .

(13)

In nearly all models generally considered all the dependence of H on the heavy quark momentum P is in the kinetic energy; i.e. all dependence of the forces between the heavy quark and the brown muck upon the velocity of the heavy quark is neglected. In that case R(q) is just the free recoil energy; the kinetic energy gained by a free quark with momentum P as a result of absorbing a momentum transfer q. Thus the mean excitation energy including corrections to the binding energxes which are first order in the difference of 1/m is just the free recoil energy R(q). In contrast to the M6ssbauer effect, where the whole system has a very high mass and the recoil kinetic energy of the whole system is negligible, the contribution of the recoil kinetic energy of the final charmed state to the mean excitation energy (13) is appreciable. We therefore express the sum rule (13) in terms of the invariant mass My of the final state If~> and the excitation energy of the first p-wave resonance in the charmed system denoted by AE~k.

( x/M~c + ~ >k-- x/M( Dk)2 +q 2 zS,Elk

(BklR(q) IBk>-x/M(Dk)2+q2 +M(Dk)+~ Z~lk

(14) The sum rule (14) can be simplified by introducing parameters a=

dM(Ok)2+q 2 M(Dk)

'

(15a)

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PHYSICS LETTERS B

It=-M(Dk)- ( Dk Ix/P2 + m~ I D k ) <~M(Dk) -inc.

(15b)

~

M(Dk)-mc a(a-1)M(Dk) z~kglk (a-1)M(Dk)+mc '

M] -M(D~) ~ M~f+qz+x/~M(Dk)2+q2

>~M f - M ( D k )

(16)

M(Dk) - mc

z~kElk

(Bk I R ( q ) I B k ) -- ( a - - 1 ) M ( D k )

=dot2M(Dk)2--2M(Dk)it+ it 2 -otM(Dk) +it 2(a-

1 )M(Ok)it (17a)

(BklR (q)IBk ) -- ( a - - 1 ) M ( D k ) (17b)

where we have assumed that P.q which averages to zero over the angular d i s t r i b u t i o n o f P can be neglected a n d that the correction ~ to the first order result for M(Dk) IS negligible in c o m p a r i s o n with excitation energies, ZXElk .

(18)

Substituting eqs. ( 15 ) - ( 18 ) into the sum rule ( 1 4 ) gives

( M f ) k - M ( D k ) <~ It a ( a - - 1)M(Dk) l~kElk z~EIk a M ( Dk ) - i t M(Dk)-mc a(a-1)M(Dk) <~ Z~'lk (a-1)M(Dk)+m~ "

aq 2 q2"b ( a + 1 )M(Dk)mc '

(20b)

while for any given excited final state other than IDkl) I£-) ¢ IDk), I (f~ I M p ( q ) l i b ) 1 2

= x / a 2 M ( D k ) 2 - 2M(Dk) It+ It2 + otM(Dk) - It'

( a - 1 )M(Ok)it ~< , aM(Dk) - It

(20a)

Y, I(f-IMk(q) llb)l 2 Ifc)~=]Dk)

OL

a~<<

I(fclMk(q) llb)] 2

[Jc) ,~ [Dk)

q 2 - x/M( Dk) 2+q2 =

b o u n d s for the probability o f B decays into exclted c h a r m e d states.

Z

Then

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(19)

The excitation spectrum o f the final charmed states is seen to have a m e a n excitation energy above the ground state Dk whose scale is set by the p a r a m e t e r It. This p a r a m e t e r is just the difference between the c h a r m e d h a d r o n mass and the energy o f the c h a r m e d quark and is roughly equal to a light constituent quark mass and b o u n d e d by eq ( 1 5 b ) . F o r low values o f q 2 the m e a n e x o t a t i o n energy is even lower by the ratio

q2/2M(Dk) 2. F r o m these inequalities ( 1 9 ) we obtain u p p e r

M ( D k ) - m c ( a ( a - 1)M(Dk) M c - M ( D ~ ) (or- 1 )M(Dk) +mc"

(21)

This result (eqs. ( 2 0 ) ) is an upper b o u n d for the total probability o f decays into states other than the ground state, when the phase space factors are assumed to be equal for all final states. Including phase space factors will give an even stronger upper bound. The m a x i m u m value o f q2 which leaves the energy available just at the threshold for producing the lowest p-wave c h a r m e d state [10], D ( 2 4 2 0 ) is q2~ 1.25M(Dk)2, which gives a = 1.5. This occurs when the lepton m o m e n t a are exactly parallel and they carry the m i n i m u m possible energy for a given value o f q2. This gives an upper b o u n d on the probability o f producing an excited c h a r m e d meson state o f less than 56% for the quoted values [10], M(D) = 1.87 GeV, rn~= 1.35 GeV, A E I k = 0 . 5 5 GeV. An even smaller upper b o u n d is obtainable if some estimate o f the kinetic energy o f the heavy quark can be used in eq. (15b). An upper b o u n d for the probablhty o f producing a given state [fc) with higher mass is given by eq. (21) for the extreme case where no lower excited states are produced. The probability o f producing the ground state is thus ~ 50% even under these extreme conditions and increases rapidly with more realistic lower values o f q2. Thus the excitation spectrum will be d o m i n a t e d by the ground state Dk and the first low-lying D*'s. The algebraic techniques used in deriving M6ssbauer sum rules a p p e a r similar to those used in deriving optical sum rules in atomic physics. However, the 109

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underlying physics is very different. The separation o f the degrees o f f r e e d o m in the MSssbauer sum rules into the internal degrees o f f r e e d o m o f the particle emitting the b o s o n a n d the " s p e c t a t o r " degrees o f freedom describing the b i n d i n g does not exist m a t o m i c physics. There is therefore no obvious simple relation between our sum rules a n d a recently derived " o p t i c a l sum rule" [ 11 ] for dipole vector transitions, d e r i v e d by analogy with the T R K dipole sum rule in a t o m i c physics. The MSssbauer sum rules [7] give the effects o f b i n d i n g on the m o m e n t s o f the energy spectrum for the decay o f an unstable particle a n d relate these moments for the case o f a b o u n d particle to the case o f the decay o f a free nucleus or free heavy quark. These have no analog in a t o m i c physics. The Bjorken sum rule, for example states that the decay rate and lifetime o f a b o u n d heavy quark are essentially the same as for a free heavy quark. In a t o m i c physics the free electron is stable a n d there is no factorizatlon o f the transition matrix element into a factor describing the decay o f a free electron and a factor describing the m o m e n t u m transfer to the b o u n d system. A n o t h e r difference is the absence o f a multipole decomposition in the derivation o f the MSssbauer sum rules [ 7 ], which include contributions from all multipoles. Stimulating

110

and

clarifying

discussions

with

24 June 1993

N a t h a n Isgur and J o n a t h a n L. Rosner are gratefully acknowledged. This research was partially s u p p o r t e d by the Basic Research F o u n d a t i o n a d m i n i s t e r e d by the Israel A c a d e m y o f Sciences a n d H u m a n i t i e s a n d by grant No. 90-00342 from the U n i t e d - S t a t e s - I s r a e l Binational Science F o u n d a t i o n ( B S F ) , Jerusalem, Israel.

References [ I ] N Isgur and M.B. Wise, Phys. Len B 232 (1989) 113; B 237 (1990) 527 [2] N. Isgur and M B Wise, Phys. Rev Lett 66 ( 1991 ) 1130 [3] J.D. Bjorken, in. Proc Rencontre de Physique de la Vallee D'Aoste (La Thufie, Italy), SLAC Report No SLAC-PUB5278 (1990), to be pubhshed. [4] N. Isgur, M.B Wise and M Youssefmlr, Phys. Len. B 254 (1991) 215 [ 5 ] N. Isgur and M B Wise, Phys Lett. D 43 ( 1991 ) 819. [6] H J. Llpkm, Ann Phys. 9 (1960) 332 [7]H.J Llpkm, Ann Phys 18 (1962)182 [8] For a general revxew see H.J. Llpkm, Quantum mechames (North-Holland, Amsterdam, 1973) pp 33-110. [ 9 ] For a general mtroduetxon see H J. Llpkm, Argonne prepnnt ANL-HEP-PR-92-86, submitted to Nucl Phys. A [ 10 ] Particle Data Group, J J. Hernandez et al, Revxewof particle properties, Phys. Lett. B 239 (1990) 1. [ 11 ] M B Voloshm, Phys Rev D 46 (1992) 3062