Z. Phys. C 59, 283-293 (1993)

ZEITSCHRIFT FORPHYSIKC 9 Springer-Verlag 1993

Covariant trace formalism for heavy meson s-wave to p-wave transitions S. Balk 1, J.G. K6rner 1'*, G. Thompson 1'**, F. Hussain 2 x Institut ffir Physik, Johannes Gutenberg-Universitfit,Staudinger Weg 7, W-6500 Mainz, Germany 2 International Centre for Theoretical Physics, Trieste, Italy Received 15 June 1992; in revised form 11 November 1992

b--,c transitions are studied in the context of the heavy quark effective theory using covariant meson wave functions. We use the trace formalism to evaluate the weak transitions. As expected from heavy quark symmetry, the eight transitions between s- and p-wave states are described in terms of only two universal form factors which are given in terms of explicit wave function overlap integrals. We present our results in terms of both invariant and helicity amplitudes. Using our helicity amplitude expressions we discuss rate formulae, helicity structure functions and joint angular decay distributions in the decays /~--*D** (~(D, D*)+ n)+ W-(--*l- vl). The heavy quark symmetry predictions for the one pion transitions D**---,(D, D*)+ n are similarly worked out by using trace techniques. Abstract. Heavy meson, s- to p-wave, weak

1 Introduction

In a series of recent papers [1-6], we have presented a covariant formulation of heavy meson and heavy baryon decays in the leading order of the heavy /tuark effective theory (HQET) [4, 7, 8]. This method was based on a Bethe-Salpeter formulation in the limit of the heavy quark mass going to infinity. The starting point of our investigation was the demonstration that the equal velocity assumption, arising from the heavy quark limit, could be formulated in a covariant manner using the spin-parity projectors developed by Delbourgo et al. [9]. Our way of deriving the consequences of heavy quark symmetry provides an alternative approach to the spinalgebra approach of Isgur and Wise [10-13]. It is perhaps closest to the "trace formalism" techniques used in [14] for mesons and in [15, 16] for baryons, and provides an explicit dynamical formulation of current-induced transitions among heavy hadrons in terms of wave func* Supported in part by the BMFT, Germany, under contract 06MZ730 ** Supported by the BMFT, Germany, under contract 06MZ730

tion overlap integrals of Bethe-Salpeter bound state wave functions. In our previous works on heavy meson transitions only the ground state s-wave mesons were considered. Here we extend the Bethe-Salpeter approach in the HQET to present covariant wave functions of p-wave heavy meson resonances containing one heavy quark and a light antiquark and to derive explicit expressions for the current-induced s- to p-wave transitions. As expected from heavy quark symmetry the s- to p-wave transitions are described in terms of only two universal reduced form factor functions r and ~*/2(09) for transitions from the degenerate s-wave multiplet j r = (0-, 1-) to the two degenerate p-wave multiplets Je=(O+, 1~-/2) and J P = (1 3/2, + 2 + ), respectively. We confirm the results of [12] on weak 0 - ~ ( 0 +, 1 +/2) and 0---*(1~-/2,2 +) transitions*. Our results on the 1-~(0+,1i~/2) and 1----,(1 3/2, + 2 + ) transitions are new. Our results are presented in terms of both invariant and helicity amplitudes. Using the helicity amplitudes we write down rate formulae, helicity structure functions and discuss joint angular decay distributions in the two-sided cascade decay B~D**(~(D,D*)+n)+ W-(~l-~Tt)**. The joint angular decay distributions and their heavy quark symmetry structure can be usefully employed in Monte Carlo event generation programs to analyze the importance of charm meson p-wave contributions to semileptonic /3-decays. In an Appendix we derive the heavy quark symmetry predictions for the helicity structure of the heavy meson one-pion transitions D**-,(D, D*)+ n again by using trace techniques. 2 Heavy meson wave functions

We begin by briefly reviewing the Bethe-Salpeter approach to heavy mesons. In the heavy quark mass limit, * Semileptonic B-decaysinto p-wave charm mesons have also been investigated in [171 using QCD sum rule techniques ** We employ a generic notation and use D** for any of the four p-wave states

284

m a i m , in the leading order of the HQET, the heavy quark spin is decoupled from the light degrees of freedom and the projection of the heavy quarks four-momentum pe in the direction of the four-velocity v of the heavy meson is fixed to be the mass of the heavy quark. If we just restrict our attention to the projection of all momenta in the direction of v, the heavy quark itself is also effectively free of coupling to the light degrees of freedom; neither light quarks nor gluons are able to change its four-velocity. Since the heavy quark "label" is free in this approximation, the corresponding heavy index c~ in the heavy meson's (Qgl) Bethe-Salpeter amplitude q~(P, Pa, k) (see, e.g. [4]) satisfies the free Dirac equation, i.e. M)~ q)~,=0. The upper index fi represents the Dirac index of the light degrees of freedom. P and M are the meson momentum and mass and k is the momentum of the light degrees of freedom with P=Pa+k. We ignore flavour and colour for the moment. The Bethe-Salpeter amplitude is defined as usual as the matrix element of the time ordered gauge invariant product of quark and antiquark fields taken between the vacuum and the heavy meson state. We shall sometimes loosely refer to the Bethe-Salpeter amplitude as the Bethe-Salpeter wave function or just the wave function. In general, the BetheSalpeter amplitudes will involve both spin and orbital angular momentum degrees of freedom. Now since in the direction of motion (or four-velocity) the heavy quark propagates as an on-shell quark we have i f - 1)~'~,=0,

(1)

as mentioned above, where v~ = Pu/M, i.e. the BargmannWigner equation on the heavy quark index. Equation (1) implies that, in the leading order of HQET, all the heavy meson wave functions are of the form 1)~~'(b,,(v, 'p k).

(2)

The covariant s- and p-wave heavy meson wave functions can then be derived by constructing interpolating fields with the correct jPc quantum numbers (see e.g. [20]). For the s-wave states one has*

1)7~8aflkfl~P(V' k),

1

1Pl(l + - ) : 2 ~/M(} + 1)?sk.e~p(v, k),

1

So(0-

v/i(

+

k),

3S1(1- - ) : ~1 x / ~ ( ~ + 1)#~b~(v, k),

(3)

and for the p-wave states one has

k):=x/MZ~ ~ ~ ,k),

(5) where j = s, p for s- and p-wave states, respectively. This may be understood as follows: the projecting functions (1 +~)F(1 -~)/4:=)~ are the Bargmann-Wigner wave functions and are appropriate for picking out the required particle states. The reduced spin wave functions F for the cases of interest are written out in Table 1. To get to the Bethe-Salpeter wave function, we first create an appropriate interpolating field in the LSZ framework. Following [20] this then leads us directly to the form of the Bethe-Salpeter wave function which we have introduced above. In general, an appropriate interpolating field has the form (b(e, X ) = ~

1

I ~,(X + nx)z(P, k)O,(X - zx)

+)" - 5

3P 1(1 + +):

1;i

x/-M(} + 1)(r

2/fix/2x ~ ( } +

k),

1)e~Pr~v~e~k~?oOp(v, k),

* Note that the relative phase of our two s-wave states differ from the relative phase chosen in [12]

(6)

where in this equation k is the relative momentum of the heavy quark and the light degrees of freedom 4 and X := (mhx h + mlxt)/(mh+ml) is the center of mass, x:= Xh--Xl is the relative position, r/:= (ml)/(mh + mr), "c:= 1 -- t/and h and I denote heavy and light quarks, respectively. For s-waves as z(P, k)= z(P) it takes the simpler form

eps_ ..... (P, X) = ~h(X ) z( P )tPt(X )e- ~e.x.

4)~(v, k)=

(4)

where we have used the analogue of the LS-coupling scheme to construct the p-wave states, e~ and e~flare spin s = 1 and s = 2 polarization four-vectors and tensors, respectively. In (3) and (4) we have removed the heavy mass scale dependence of the heavy meson wave function by factoring the square root of the heavy mass. r k) and ~bp(v, k) are the s- and p-wave orbital functions which, in the general case, would be bispinor-valued functions. Note that the momentum k appearing in (4) is scaled by the light mass scale, i.e. by a mass factor of O(AQco), which also provides the scale for the orbitial p-wave function ~bprelative to Cs. We have used a generic notation for the orbital p-wave function ~bpwhich in general would be different for different spin configurations due to, for example, spin-orbit coupling effects as described later on. The wave functions are mostly used in the form of (3) and (4) but at a closer look they have the form

9eikxe-ie'Xd4xd4k,

(b~(v, k)=

3p~

1

3p2(2+ +):5 ~ ( r

(7)

Note that the covariant spin wave functions (3) and (4) have first been constructed for heavy (Q(~) quarkonium systems [18, 19] and have been in wide use since then to compute heavy quarkonium decays. However, the concept of charge conjugation parity that is appropriate for 4 From the p-wave projectiors in Eq. (4) one sees that the momentum of the light degreesof freedom can be shiftedby any fraction of the meson momentum P. Thus the momentum of the light degrees of freedom can be replaced by its relative momentum in z(P, k)

285 Table 1. Reduced spin wave functions F for mesonic s- and p-wave states

State

F

ISo(0- +)

Ys

3S1(1- ) 3P o (0 + + )

J~ _ ~/.,/~

3P l {1 + + ) 3p2(2+ +) IPl(1 + )

__i75a,t%=ka/x/2

P1 - P2

k

7~e~l~k~ ?sk.e. ,t5r162

(1 +/z) (1 ~-/2)

2s(3k.e-t/~)/x/g

the (Q(~) states and that arises naturally in the LS-coupling scheme is no longer so useful for (Qq) states. In fact the appropriate degenerate heavy-light states are determined by coupling the orbital angular momentum l with the spin of the light quark system just as in the hydrogen atom. For the s-wave states (l=0) one has the usual degenerate ground state multiplet with JP= (0-, 1-). For the p-wave states with l= 1, on the other hand, one has two degenerate multiplets J P = ( 0 + , I + / = ) a n d JP=(1~/2, 2+). The subscript on the 1 + states labels the total angular momentum of the light quark system which is j = l / 2 and j = 3 / 2 according to the decomposition 1/2 | 1 = 1/2 | 3/2. The JP = 1 + mesonic p-wave states I 1 ~-/2) and I 1 ~/2 ) are linear combinations of the J = 1 p-wave spin states in (4). One has

I17/~>=

~-I1+ >+

I1++>,

I1~/2)=~-~ 11+- ) - ~

11+ + ).

(8)

Note that the two states in (8) are not charge conjugation parity eigenstates. Using standard four-dimensional 7-matrix identities the 1 ~/2 and 1 J/2 wave functions calculated according to the composition (8) can be simplified to read

r

P2 = M 2 V 2

~/=)= ~ x / M

(fJ+l)75(-#(r

(9} The corresponding reduced spin wave functions are listed in Table 1.

P1 =M1 Vl Fig. 1. Feynman amplitude for current-induced mesonic transitions to lowest order of perturbation theory. Vertex functions are related to B.S. amplitudes by multiplication with inverse propagators SOFt

M~=(Mz(vz)IJ v a(z=O)lM,(v,)) d4kl d4k2 Tr {~2(Ve, k2) - 5 (2x)4 (2x)4

"J-u(kl,

(10)

where q~= 704/7o- To lowest order in perturbation theory (if applicable) one would have Y,(k,, k2; Vl, v2; z = 0 ) = - ~ o 1 (k,)(2rt)4a(k,-k=) | 7.(1 -'/5) (see Fig. 1) where ~ov is the bare Feynman propagator, O~OF~(k)= (r One then recovers the Mandelstam-Nishijima formula for current-induced transitions between bound states [21,223. However, in the heavy quark limit, going beyond perturbation theory in the coupling, one finds generally in H Q E T that the interaction kernel factorizes according to

~.(kl,

k2; vl, v2; z = 0 )

= ~ - ( k l , k2; vl, v2; z = 0 ) | 7 . ( 1 - 7 5 ) ,

(1 I)

where J - connects the light quark legs and the weak current vertex connects the heavy quarks. The general case of a current-induced transition between two heavy mesons is depicted in Fig. 2. There are Wilson lines of glue running down from the weak interaction vertex to the light quark propagator and also Wilson lines of glue along the light quark propagator itself, but the heavy quarks propagate freely. To proceed further we rewrite the spin wave functions (3, 4, 9) in a factorized form. In the notation of (5) we write r = x/MzA,

3 Weak transitions

k2; Vl, I.)2; z = 0)1~)1 (/21, k,)},

for s-waves,

eb= x/-Mz~UAp for p - waves

(12)

For the s- to p-wave transitions one then finds Given the wave functions (3-9) it is now an easy matter to compute the heavy quark symmetry structure of currentinduced transitions between heavy mesons. Firstly, the transition has the general form

Mu=Tr{~27,(1--75)zt(Vj(oo)v,=+V)(oo)7=)},

(13)

where j = l / 2 and 3/2 for the degenerate multiplets (0 +, 1 i~/2) and (1 ~/2, 2+), respectively. In (13) we have used

286 (remember that the heavy quark side transition is completely specified). In fact there is one invariant amplitude each for the transitions 1/2---+1/2 + and 1/2--+3/2 + in (15). The easiest way to see this is by writing down the number of "LS"-amplitudes for the transition (1/2-)~(1/2+,3/2+)+(0+-spurion). This gives one "LS'-amplitude each for the 1/2-~1/2+(L= 1, S = 1/2) and 1/2-->3/2 + transitions (L= 1, S=3/2) as stated above. Similarly the ground state to ground state transition 1/2---+1/2- is described by one "LS" amplitude (L=0, S = 1/2) and thus by one reduced form factor function, ~(co) as is well known. In the following we denote the two reduced form factor functions that describe the (0-, 1-)-+(0 +, 1 i~/2) and (0-, 1-)~(1~/2,2 +) transitions by ~*/z(co) and ~*/2(co), respectively. By an inspection of (4, 9, 13) they can be seen to be related to the set Fj(co) and F)(co) introduced earlier in (13, 14) by

P2 =M272

~

kl +P1

Pq =Mq Vq Fig. 2. Current-induced transitions between heavy mesons in the heavy quark effective theory going beyond perturbation theory. There are Wilson lines of glue running down from the weak interaction vertex to the light quark propagator and also Wilson lines of glue along the light quark propagator itself. In the heavy quark limit the heavy quarks are free of glue

the expansion s d4kl d4k2

(2~g)4 (27C)4 Al(kl)J-(kl, k2; Vl,/)2;

= Fj(co) v, ~+ F)(co)?,,

z=-O)Az(kz)k2~

~*/2 (co)-- (co + 1)F1/2(co)- 3F'1/2 (co),

~/2(co)= f 3/2(co).

(16)

Incidentally, the fact that the form factor function F~/2(CO ) does not contribute to ~/2(co) can be easily understood by noting that the vanishing of the contraction of 7, with the s = 3/2 spin wave function is nothing but the well-known Rarita-Schwinger subsidiary condition for spin 3/2 fields. Collecting our previous results we then obtain the following structure for s- to s-wave and s- to p-wave transitions:

(14)

where the overlap integral depends on whether one has a transition into the j = 1/2 or j = 3 / 2 p-wave multiplet. The velocity transfer variable co is defined by co=vl.v2. Note that the most general expansion of the lhs of (14) requires the 12 covariants v l , ( } l + l ) , vl,(}z+l), V2~t(r "t- 1), /)2~t(~2 ~ 1), 1) and Y,(~2 -+ 1). However, when contracted with the spin wave function Zl and Z2,, the expansion (14) is the most general form. Nevertheless the most general expansion in terms of 12 covariants must be retained if one wants to project out Fj(co) and F)(co) from the overlap integral on the lhs of (14). Returning to the effective form (14) we note that the form factor functions Fj(co) and F)(co) are functions of the velocity transfer variable co = vl. v2 only as we have scaled out the heavy mass dependence from the wave functions 4~. The expansion (14) when inserted into the trace (13) shows that there are two independent form factor functions that describe mesonic s- to p-wave transitions. This is in accord with general heavy quark symmetry counting arguments which are best done by counting the number of LS-amplitudes needed to describe the light transition [23] as follows. In the present case we have the light side transitions (JP= 1/2-)~(J e= 1/2 +, 3/2 +) where the change in parity going from the initial to the final state is due to the p-wave (l = 1) nature of the final state. The aim of the game is to determine the number of independent amplitudes for the hadronic transition matrix element

err((

75 ~',~ +1)

7,(~bl_+

M(light) = (1/2 +, 3/2+ I TI 1/2- )

(15)

,ii,

,((1)

Mu=~/~~~Tr

r

(~2+ 1)

2+ )

M"=~:Tr((

-~-~((co+l,r7,'e*':' 2

] ,,9,

Similar covariant trace expressions for the s- to s-wave and s- to p-wave transitions can be written down using purely group theoretical tensor/spinor contraction methods (see, e.g. [30]). The Bethe-Salpeter approach used here to derive the trace expressions has the advantage that it affords the opportunity to look inside the bound-state system and thereby provides a great deal of

287 physics insight of how a heavy quark and a light antiquark interact with one another inside a heavy hadron and how non-Abelian gluon dynamics is at work during the current-induced transition. In addition, our approach establishes a connection to the more traditional way of thinking about bound-state problems as it has developed over the last 40 years.

C. (0-, 1-)-+(13/2, + 2+)

0--~152:M,~=

M,/-~M~ ( - (~o2- 1)~*, - 3~./)1/),,

(27)

4 Covariant amplitudes

*ct'~t

0 - - - 2 + ' MV = - Mx/~IM~ez In this section we shall list the results of explicitly evaluating the trace expressions (17-19). We shall separately write down the vector and axial vector current matrix elements M v and M A, resp., where, in our notation, M a is contributed to from @uT~b. We mention that we use the Bjorken and Drell conventions throughout this paper. One obtains

9

, a *

/)1~lt;.~z'pa/)l/)2~3/2(fD),

M A = x/-]~1M~2 e*" % l , "((~o+ l)g~,u-/)l~,/)2u)~*/2(o)),

(28)

"((co + 1)ie.~a~e~e *~(/)a -/)2)" *

A. (0-, ~-)~(0-, 1-) O - ---+0 - : Mt~g = N / ~ I M 2 ( / ) I 0---+1-

" v :N~l .mu

@V2).~((D),

a ~((-0), M2 lg~ivo,rg2,v /),p V2 9

1 - ~ 0 - :from 0 - + 1 -

with 1 ~ 2 and e*--'Sl

1 -- -+1 -- : MV . - ~

* (~2./)~e~u+v

(21)

(23)

M2((CO-- 1)e*. - ~2./)1/)2..)~1/2(~o), * * ~

9

*v

Ma~= - - ~ / 5 x/M~M21eu~oae2 vqv~2~/2(c~ (24) M1M2

.

a

(30)

~ l e . v p a F , 1" / ) l U 2v~ l p/ 2 (ao ) *) ,

(25)

"43 1 + g~" el (121 --/)2),) ~ / 2 ((D),

(/)1 -/)2) . M .V =0,

(31)

(/)1 + v2)UM~ = 0.

(32)

This follows since the current vertex is sandwiched between the positive energy projectors (}1 + 1) from the left and (~2+ 1) from the right. As can be seen by explicit calculation the covariant amplitudes listed in this section do indeed satisfy the relations (31) and (32). In this context it is interesting to check on the validity of vector current conservation for the corresponding (diagonal) neutral current c-~c (or b ~ b ) transition amplitudes. Taking (31) into account one has v 1 q U M u = ~ ( M 1 - M 2 ) ( / ) I +/)2) u M .v,

9((co - 1)ca, -e1./)2/)1u)~*/2(tn),

1-~7/2"

" g2,,~" /) 1, le/~,'pa g o1(/) 1 "~-/)2)a ~ ~/2 ((D) 9

A nice check of our covariant results is afforded by the observation that the vector and axial vector amplitudes must satisfy the relations

0- ~ 11/2. M . =

\/~

(29)

%I-g./)2

Ma=x/~iM2

0 - ...+0 + " MuA__ - - ~/~ x/M1M2(vl-v2)u~/2(09),

M .A =

1 --+2 + . M.v

(22)

B. (0-, 1 - ) ~ ( 0 +, li~/2)

1 ~0+.Mu = -

(--(co+ 1)e~.el(/)l--/)2).

+(/)l +/)2).~'-/)1~'<.)~*/2(co),

-- 'g~ "gl (1)1 -1-/)2),) ~((D),

/1

a

+ 3e~.vl/)2.el/)l.)~/2(e)),

* 2.~1e2"

N ~ Y l m 2 " (lg.vpo-,glg v 2*O(v~+/)2)'){(co),

p

- ( e ) + 1)v2.elg~u-- 2(09 + 1)e~./)lelu

( - (~ + 1)~*. + ~*. ~/)2.) ~ (~o),

M2 = ~

M .a-

= -

(20)

v

(26)

(33)

where M1 and M2 now refer to masses of mesons with the same heavy flavour quantum numbers. If one considers transitions between members of different degenerate states (Ma r the vector current is no longer conserved as (33) shows. However, the breaking of vector current conservation occurs only at the next-to-leading order as can also be seen from (33) and thus does not affect the vector current conservation at the leading order of the heavy mass expansion.

288 Table 2. Table of LS-couplingamplitudes. Entries are given in terms of (LS) specification. Rightmost column shows the relevant heavy quark symmetry form factor dP ~ j , e '

V.:

1-

0+

0-~0 0-~1-

(11) (11)

(00) -

1---0-

(11)

-

Au:

1+

0-

(00)

(11)

Form factor

(22) (01)

(10)

(21) 1 --,1-

0-~0

+

0--~ 1+/2 1-~0 + 1---*1~/2

(10)

(01)

(11) (21) (12) (32)

(21) (22)

-

(11)

-

(00) (11) (22) (01) (10) (21) (01) (11) (21) (22)

0-~lk2 0- -~2 +

as in 0--,1 i)2 (22) -

1 -~lk2

as in 1----~1~/2 (01) (12) (21) (32) (22) (23) (43)

1---~2+

(01)

~(~)

(11)

Fig. 3. Definition of polar angles O and O* and azimuthal angle Z in the two-sided cascade decay B~D**(--*D+n, D * + n ) + W (--,l- ~) (00)

(11) (11) ~*:2(c0) (10) (01) (11) (21) (12) (32) (11) (22) (33) (11) (22) ~/z(o~) (12) (32) (33) (33)

As Fig. 3 shows, the Woff~ I moves along the negative z-axis. Its helicity components are projected by the polarization four-vectors eu(t, -* _+, 0) where the bar over the polarization vector reminds one of the fact that the z-axis is chosen to lie along the recoiling charm meson. It is convenient to project out the z- (or "0") component (longitudinal part of the spatial spin 1 projection) and the time-component (scalar or spin 0 projection) covariantly by writing / 2 5 . (0) = ~1/775 x/qZgu x / o z - 1 ((M 1 + M2)(/) 1 +/)2)/2/(0) At-]) + ( M I - M2)(v~ - v2)u/(0)- 1)),

(34)

1

xff~gu*(t) = qu = ~ (M1 + M 2 ) ( v l -- v2)u The predictive power of heavy quark symmetry in mesonic b--,c transitions can be nicely assessed by comparing the number of transition form factors allowed by Lorentz invariance to the number of independent form factors that remain in the heavy quark limit. The counting of the number of independent form factors in the general case is readily done by counting the number of LS-coupling amplitudes in the decay: meson (JV)-,meson (J'P') .AftWoff.shell (g/z : 1 -, 0 + ; A u : 1 +, 0-), This has been done in Table 2, where we list the relevant LS coupling amplitudes. The reduction of the number of independent amplitudes is quite substantial: 20--* 1, 2 0 ~ 1, and 30~ 1 for the three cases (0-, 1 )--,(0-, 1-), (0 , 1 - ) ~ ( 0 +, 1;/2) and (0-, 1-)-~(1~-/2,2+), respectively, when the heavy quark spin symmetry is used to constrain the transition amplitudes. This comparison becomes even more impressive when the neutral current type b - ~ b and c-~c transitions are added to the list of heavy meson transitions using also heavy flavour symmetry.

5 Helicity

amplitudes

For the purpose of calculating decay rates, partial rates into polarized current components and angular decay distributions it is much more convenient to work in terms of the helicity amplitudes of the process. For definiteness we shall be using the conventions of [24] (see also Fig. 3) in defining our coordinate systems and phase conventions.

1

+ ~ ( M 1 - M2)(/)1 + V 2 )

(35)

u9

Bearing in mind that the current vertex is sandwiched between positive energy projectors one will have the replacements Vu: x f q S g * ( O ) T U - - * x ~ - 1 (M1 + M2)/(0) -t- 1),

A.: ,j

e*

I(M

1),

For the spin 1 charm meson in the final state the covariant projection onto the z- (or "0") component is given by ~(0) = ~

1

( --/)1~ + 0)/)2~)"

(36)

For the spin 2 charm meson in the final state one always has the contracted form v~'e*,,,(m) with m=0, + 1(m = _+2 does not contribute in the helicity frame Fig. 3). Using the appropriate Clebsch-Gordan coefficients to couple two spin 1 objects to a spin 2 object one finds ,,

/)1

/1

= x/2

/)1'e2~,(0) = i ~ x / ~ -

1 1 e~(0).

(37)

289 The transverse helicity components can also be projected out covariantly, for example by defining an auxiliary four-vector with spatial components along the y-axis in Fig. 3. However, for our purposes, it is more convenient to use the explicit noncovariant representations g*(• ~,(+)=

1

(0, + 1, i, 0),

H+=HV T H A + : L ~ x / - ~ - I

43

9( ~ 1

+_vf~+ 1)~*/:(co),

C. (0-)-+(1~-/2, 2 +)

1 (0,-T-l,i, 0).

(38)

0= -* 1;/2 "x/~HoV = - /~ Mx/MT~IM2(co2- 1)

42 9(M, + M2)~/2(CO),

Finally, for the completely antisymmetric 8-tensor we use the convention 80~z3 = 1 as in Bjorken-Drell Having physics applications in mind we shall only list the helicity amplitudes H ; ~ : for the decays of the lower lying ground state pseudoscalar bottom meson/~ (or B) meson. The/~* decay is dominantly electromagnetic. For the/~-decays one can then drop one helicity index on the helicity amplitudes. Thus we shall label the helicity amplitudes by the 'current' index as in [-24] according to

9(Mr - ME) ~*/z(co),

H+_=HV

1~

9x / ~ - 1(co + 1 ) ( ~ -

H,~w,&=;.w:= H2~.

(43)

x/MIM: 1

The helicity amplitudes read: /7~ A /2 0- --+2+ : x / q ' H o = i f 5 ~ V f O S -

A. (0-)-~(0-, 1-) 0- - , 0 - : x f ~ H ~ = ~ x f f ~

0- - , 1 - : x / ~ H 8 = - ~ ( c o

9( M , - M2) {~'/2 (co),

( i l + ME)~ (co),

x / ~ H , v = , ~ M , M 2 ( c o + 1)(i~-M2)~(co),

1 (co + 1)

/72HA, = ~ //2 x/q~x/~M2(co2-1)

(39)

+ 1)(M~- M2) ~(co),

"(M, + Mz)~*/2(co),

(44)

/

H+_=__+ H v -Ha+ =-- q ~ 4 M I M 2

H+ =H+_--H+ v A __~+_Hr+--HA+

" x / ~ - 1 (co + 11(_+~/co- 1 -~)~(co),

(40)

The transverse contribution H+ in (40) is defined to be H x = H ~v- H ~ , , according to the left-chirality of the Standard Model b-,c current by,(1-ys)c.

+~

The longitudinal and scalar helicity amplitudes can be seen to follow a remarkable heavy quark symmetry pattern for transitions to degenerate pairs of states. One has H~(0--~0-)=-H~(0

B. ( 0 ) - , ( 0 +, 1 ~/z)

-~1-)

H ~ ( 0 - - . 1 +/2)= - Hta(0- -*0+),

1

o--,o+

+ 1)~.*/2(co).

Hot(0 -* 13/:)= -- H~(0- -,2 + ), and

"(Ma -- Mz)~*/2(CO),

H/(0--,0-)= ~Hff=+~3

M~lM2(co-1)

9(M1 + M2) {~'/2(co),

-H~(0--,I-),

Hy(0 -, 1 ~/2)= -- S ~ ( 0 - -,0 +), (41)

HV(0 - -, I ~/2) = -- HS(0- -,2 + ).

(46)

The equalities (45) and (46) bear witness to the fact that the mesonic transitions can be considered to be dual to the free quark decay (FQD) transitions where one has the corresponding equalities

1

"(M1 + M2)~*/2(CO), 1 = -,j

HV I o(FQD)= + Ha+ ~t(FQD) 1

- x / ~ 2 m , , / ~m~ 1 (ml + m2) x / ~ 7 1, 9 (M~ -- M 2 ) ~ 1"/2 (CO),

(451

(42)

(47)

290 H v ~,(FQD) = __H ~ ~o (FQD)

into pairs of degenerate states. One then finds

1

- x/~ ~ ( m i

- m 2 ) x / ~ + 1.

(48)

0---*(0

~o+1

O 1-): Hi=Ki(~o)~-1~(~o)] 2,

(54)

(m~ and m2 denote the initial and final quark masses). For completeness and for later applications we shall also list the transverse F Q D helicity amplitudes

0-+(0+@ 11"/2): fl, = K,(~)~ (co- 1)1~T/2(o~)[2,

H~ ~ + 1(FQD) : - v / 2 ~

x / ~ - - 1,

(49)

0- - q l 3+/2@2 + ):/~i = Kd o~)3 (co - 1)(co + 1)2 [{~/2(co)1z,

HA ~+1 ( F Q D ) = -T-x//2 ~

~

(50)

(55)

1

1.

(56) where the universal factors Kdco) read Ku = 8Ml M2co,

6 Rate functions and joint angular decay distributions

q2K r = 4M ~M2 (co(M~ + m 2)- 2M1 m2), Let us exhibit the full structure of the exclusive / ~ D , D*, D** . . . . s.1. decays by writing down the twicedifferential rate distribution with regard to q2 and the cosine of the lepton's polar angle O in the (l-~) CM system (measured with regard to the direction of the charm meson (see Fig. 3)). One has [24, 25] dE G2 (q2_mZ)p dq2dcos O - ( 2 n ) 31Vb~12 8m~q z Lu~HU~,

(51)

where

Lu~H"~=~(q2-m~)

(1 + c o s 2 0 ) / q v

3 2 ^ 3 + ~ sin OHL+~ cosOHp

+m213 (asln .

2 _ _

^

3

+~1/~ s +3 c o s O H s L ) ) .

2

~

(52)

The lepton's mass is denoted by ml. The I4i(i= U, L, P, S, SL) are the five (helicity) structure functions that are measurable in the decay process. They can be expressed in terms of the helicity amplitudes defined in Sect. 5 via / t u = I H + I 2 + I H - ] 2,

Kp=-SM1M2 ~x//~-l,

Ks = 3KL, q2KsL = 4M1 M2 ~

1= ( ~ - )

I~(m)12 + : (co- 1)! ~*/2(a))l 2

1

+ ~ (~o- 1)(~o + 1)21 ~*/2 (co)12 + . . . . .

IS1e=ln +l z - l n _ [z, /~s = 3 IH, I2, (53)

As (52) shows, the spin 1 structure functions Hu (unpolarized transverse), HL (longitudinal) and /4e (parityodd) are measurable in the mt= 0 limit whereas the measurement of spin 0 and spin 0 - s p i n 1 interference structure functions Hs (scalar) and HsL (scalar-longitudinal interference), resp., requires consideration of rn~r 0 effects. The helicity structure functions can easily be worked out for the cases of interest using the helicity amplitudes calculated in Sect. 5. In order to exhibit the duality structure between the mesonic transitions on the one hand and the F Q D transition on the other it is quite instructive to compute the helicity structure functions for transitions

(57)

One can now see how the duality between the mesonic and partonic transitions is at work. The spin kinematical factors Ki(co) in (57) are identical to the corresponding F Q D structure functions as calculated from (47-50) (see also [24]). Thus the mesonic transition spin structure is perfectly mimicking the F Q D spin structure. This makes physical sense as there is no spin communication between the light side and the heavy side in the mesonic transitions. A similar duality structure holds true for the spin structure functions when the spins of the heavy quarks are not summed. The above observations can be cast into a quantitative statement in the form of the Bjorken's sum rule. When the F Q D rates are equated to the sum of the contributions from the mesonic transitions in the sense of duality the spin kinematic factors Ki(o)) cancel on both sides in the heavy quark limit. Remembering to include the spinstatistics factor 1/2 in the F Q D rate the contributions of the lowest lying s- and p-wave states to the Bjorken's sum rule [14, 26] (see also [12]) are given by

fl~ = I/-/ol ~,

/4sL = Re(Hill ~).

1(M ~-- M~).

(58)

where the dots stand for contributions from higher excitations and the inelastic continuum. From (58) one immediately derives an upper bound for the elastic form factor function ~(m). One finds ~(~o)< x/2/(m + 1). Defining the charge radius p of the elastic transition by the expansion {(co) = 1 - p2 g o - 1) + . . .

(59)

one finds from the sum rule (58) 1

1

,

2

2

p2=~+1214,/2(1)1 +3 I~/dl)12+ . . . .

(60)

where again the dots stand for contributions from higher

291 excitations and the inelastic continuum.* Equation (60) provides a lower bound on the charge radius, i.e. pz > 1/4. The structure of the weak transition amplitudes may be further probed by considering in addition also the hadron-side polar angle dependence of the decay D**~ (D, D*)+n, and the azimuthal correlation of the lepton and hadron decay planes. Accordingly we define an azimuth Z between the two decay planes and a polar angle O* in the D** rest frame, as specified in Fig. 3. One can then write down a three-fold joint angular decay distribution for the two sided cascade decay B~D**(~(D*, D)+ re)+ W-(~l-~l). In the zero lepton mass case ml=0 relevant for s.1. decays involving the e- and, for most purposes, the p-lepton one has the three-fold angular decay distribution [24] W(O,Z , O*)=

~

d l _ l , 2 , ( T z - O ) d l _ l , 2 , , ( r c - O ) e -il;'-'t'')z.

H 2,112~ ~ * . d JD** tO*xnJ~ 2~,,2~t )u2~,,2~tIO*~lh~ J[ 2,1129

(61)

Using the conventions of [34], (61) can be checked to reproduce the corresponding joint angular decay distribution for the cascade decay B--,D*(~D + 7t)+ W-(~l-9~) given in [24, 29]. The correct normalization of the decay distribution (61) can be obtained by comparison with (51) and (52). The generalization to m~r 0 relevant for decays with ~-leptons is straight-forward [24]. A few sign changes translates (61) into the corresponding distribution for B-~ O**(--*(D*, D)+Tr)+ W+ (~ l +v,) [24]. Equation (61) contains the usual Wigner d-functions. The angular decay structure is specified in terms of the helicity amplitudes H;~W (for the weak decay / ~ D * * + W-) and the one-pion transition amplitudes h~o.~. (for the strong decay D**-,(D*, D)+rc) defined in the Appendix. An analysis of the joint angular decay distribution may then be used to unravel the quark dynamics contained in the weak and the pionic transition amplitudes Hx Wand hxo,. Note that the heavy quark symmetry predicts the complete angular decay structure of a given decay chain. Let us exhibit some of the general joint angular structure by discussing the decays into the D**(0 +) and D**(1 +) states. The angular decay distribution in /~---, D**(~Dn)+ W (~I-~7~) lbr D**(0 +) is quite simple. It reads W(O, Z, O*) ~ The decay/~D**(~D*Tr)+ W-(--*l-g~) involving the D**(1 +) has more structure. One finds

W(O, Z, 0") =sin2 0[ Ho[2(cosZ O*lh~lZ + sin2 0* lh~+[2) 1 ((1 + C O S 2 0 ) ( ] H + 12+ IH-12)

1/ 2

~sin O Re (H + - H_ )H* cos Z

+~1 sin20 Re(H+ +H_)Ho* cos Z)

9sin20*(lh~12-lh+[ 2 ) - s 1s i n 2 ORe(H+H*_) "COS 2Z s i n 2 0 * ( [ h g [ 2 - ] h ~ - [ 2 )

(62)

where we have omitted an overall factor of 1/2. The terms proportional to ]h~[ 2 give the three-fold angular decay distribution for the cascade decay B--D*(--D+rc)+ W-(--l-qz) treated in [24,29]. We have dropped the so-called T-odd contributions proportional to Im(HiH*)(i#j) in (62). They are expected to be quite small since one is below particle production threshold in the decay region. After O and g integration the remaining cos O* distribution agrees with the corresponding distributions given in [33]. When one is not summing over the spins of the D*, but follows the decay chain further by measuring also the angular decay distribution in the decay D*-~Drc (with polar anlge 69* and azimuthal angle Z2 in the D* rest system) one correspondingly has a five-fold angular decay distribution which can be obtained from (61) by the replacement dJ,**tt~,~AJD ** ( ~ , ~ l h ~ 12__~ dJD**I,'~*XAJo** IAO*Xt.n L~*A1 {Z2h*lA1 [t2~*]~i(2~--J..,jZ2 2z,2a~(~' Ju2,,2a,~t7 ln2att2'a u2a,0~t~2 ]U2a,,O~lJ2]~

(63) The three-fold angular decay distribution involving the p-wave state D**(2 +) can be worked out from (61) along similar lines using the spin-two Wigner d,,,,,,,(02,) functions (see e.g. [34]) but will not be done here. An analysis of the three-fold and five-fold angular decay distributions (61) and (63) will prove to be quite useful in disentangling the contributions of the charm p-wave states to the s.1./~-decays through their characteristic angular structure. For example, one could use the heavy quark symmetry predictions for the weak helicity amplitudes H;~w, and the one-pion helicity amplitudes h~ to generate Monte Carlo events and compare them with the data. For the normalized elastic form factor function ~(co) one would use a dipole type co-dependence with ~(co)=(1 + 1/2(co- 1))-" and n ~ 2 [35]. Taking into account the power dependence of the contributions of the p-wave states to Bjorken's sum rule (58) one would correspondingly use ~*/2(co)= ~~'/z(1) (1 + 1/2 (co- 1)) - 2 and ~*/2(co)=~/2(1)(1 + 1/2(co- 1)) -3 for the s- to p-wave form factor functions with zero recoil values ~*/2(1) and (~/2(1) as free fit parameters.

+ 2 c o s O ( l H + ] 2 [H_[2)) 9(sin20*[ hgl 2 +(1 +cosZO*)[h~+ 12)

* The zero recoil values ~1"/2(1) and ~/2(1) have been evaluated in the explicit "mock-meson" harmonic oscillator model of [-27] with the result ~*/2(1)=2~*/2(1)=1.07. Present data require higher values of ~ */2( 1) and ~*/2 ( 1) by about a factor of square root of three

E28]

Acknowledgements. While completing this manuscript A.F. Falk drew our attention to a preprint of his on related work 1-30]. Where there is overlap between [30] and this paper our results agree. We would like to thank T. Mannel for discussions and making available to us a preliminary version of a paper in progress 1-31]. Finally we would like to thank K. Reim of the ARGUS collaboration for keeping us honed to the needs of experimentalists when analyzing t h e / 3 ~ D * * s- to p-wave data. J.G.K. acknowledges the hospitality of the DESY theory group and thanks the DESY directorate for support.

292 Appendix

hn++= In this Appendix we determine the heavy quark symmetry structure of the one-pion transitions D * * 4 ( D , D * ) + n . Again we employ trace techniques to calculate the covariant transition amplitudes. From these we obtain the pion transition helicity amplitudes h~o.~, needed in the joint angular decay distribution (61). The covariant amplitudes are obtained by tracing the wave functions with a pseudo-scalar 75-coupling inserted at the appropriate place*. The reduced matrix elements multiplying the traces will be denoted by R'~/2 and R~/2 in analogy to the form factor functions ~*/2(~o) and ~*/2(o~).

ii) 2 + 4 0

( 0+ ).(0-)

\1- + "

Tr (y5 ( ~ ) ( r

+ 1)(~ + 1)(y~r

(64)

i) 0 + 4 0 - + n

M = h ~ = - ~ / i (co+ 1) Mx//M~I M2 R*/2,

(70)

+n

M = ~ 1 M 2 / ) 2 / ) 2 ~lztct,R3/2,

(71)

h~ = - k / i (c~

(72)

iii) 2'+41 - + n M = Mx/~xM~"1~~'ae6~l~,v2,~2fl/)l?/)26R3/2, ~ * * h~ = -T-

A. \ 1 ; . )

(~2--1)x/-M1M2R*/2 ,

72

~ (co - 1 ) ~ R * / 2 .

(73) (74)

We would like to mention that the heavy quark symmetry results (65-67) and (70-74) agree with corresponding predictions made in [13] where spin-algebra techniques were used. The s- and d-wave composition of the 1 + 4 1 - + n transitions can be analyzed by doing a partial wave projection. Denoting the partial wave amplitudes by mt one finds [32] ms=

(h~ +2h~_),

(75)

md =

( - hg + h~ ).

(76)

(65)

ii) 1 [ / 2 4 1 - + n

y/=

(66)

h~= -

h~+ _

=

(e)+ 1) ~ R * / 2 ,

- -

/ ~ ( ~ o + I ) x / M ~ M 2 Nil2, * v3

(67)

Equation (75) and (76) show that the partial wave nature of the decay l i ~ / 2 ~ l - + n is purely s-wave, whereas 1 + / 2 4 1 - + n is purely d-wave just as the transitions of their associated degenerate partners. The partial wave nature of the pion transitions can easily understood by analyzing the partial waves of the light side transitions: 1/2 + 4 1/2 + + n must be s-wave and 1/2 + ~ 3 / 2 - + n must be a d-wave transition. In the same vein one can understand the rate equalities [13] of the one-pion transitions

F2+~l-+n-'[-F2+~o + n = ffl+/2--~l- +n, ffl+~/2~l-+n=Fo+~o +~, as may be verified by squaring the one-pion transition amplitudes h~ (65-67) and (70-74). M = ~ ~

Tr (75 ( ~ ) ( } z + 1)(~a + 1) References

. (~61T5((co-- 1)r + 3v2

\

/)~'gX
"t11))) R~/2,

(68)

i) 1+/24--1- + n

/ 11

,

M= X/6 M,,/-M~M~M~((co+2)e~./)I/)2.~-(,,~-1)~L~)R~/:, (69)

h~ = - 2 . / ~ ((D2 x/u

1)Mx/~M21M2R*/2,

* The c-*c p-wave to s-wave one-pion transition amplitudes have recently also been calculated in the chiral symmetryframework [36]

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