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Spin symmetry in e+e− annihilation into heavy mesons
Volume 247, number 2,3
PHYSICS LETTERS B
13 September 1990
Spin s y m m e t r y in e + e - annihilation into heavy m e s o n s ¢r T h o m a s M a n n e l La n d Zbigniew R y z a k 2 Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA Received 25 June 1990
We employ the spin symmetry of the heavy quark effective theory to discuss the exclusive total cross sections for the production of heavy mesons in electron-positron annihilation. We verify and extend the results derived previously on the basis of spin counting arguments by De Rtljula, Georgi and Glashow.
1. Introduction
The question of theoretical predictions for the production rates of heavy mesons (in the following generally denoted H ) has been studied ever since the discovery of the charm quark. In particular the ratios of the total cross sections of e+e---,HIZI, e + e - - , H * I r t and e + e - ~ H * I : I * have been a subject of some controversy over a decade ago. Several approaches have been pursued to derive predictions for these ratios. Using only spin counting arguments De Rtijula, Georgi and Glashow [ 1 ] suggested that near threshold the ratios should be O'e+e- ~HI7t : O'e+e- o H * l q + O'e+e- ~ H f i * : O'e+e- oHITI* =
1 : 4: 7.
( 1)
Their result was later analyzed by Close [2 ] who used the helicity formalism and the multipole expansion to discuss the cross sections in detail. He concludes that the result of ref. [ 1 ] corresponds to special assumptions for the ratio of the intrinsic longitudinal and transverse cross section. Using a linear potential model Eichten and Lane [ 3 ] give another prediction for the ratio of the cross sections: ae+e-~Hfl:ae+e-~H.f~+ae+e-~Hn.:ee+e-~Hn.=l:4:
[ ( ~ 2(PH*~4~ 7+6
PH* \mu./
+
-, \mH*/ _1
(2)
where PH* and m H*are the absolute value of the three-momentum and the mass of the heavy meson respectively. In this paper we will readdress the issue of the exclusive e+e - production rates ( 1 ) using the recently formulated effective theory for heavy quarks [ 4-10 ]. In this approach the heavy quarks are simply static color sources and do not feel the recoil of the QCD interactions with the light quarks. In this non-recoil limit there are two additional symmetries. One is an SU(Nf) flavor symmetry, where Nf denotes the number of heavy flavors. This symmetry arises because the mass dependence may be scaled out so that no reference to the heavy quark masses is left in the effective theory. The second new symmetry is due to the fact that in the non-recoil limit the spin degrees of freedom of the heavy quarks decouple. An SU(2 ) rotation of the spin of the heavy quark, at any fixed velocity, thus generates an additional symmetry. Both of these additional symmetries are broken by terms of the order AQco/m, where AOCD is the QCD scale Work supported by the National Science Foundation under Grant #PHY-8714654. On leave from Technische Hochschule Darmstadt, D-6100 Darmstadt, FRG. 2 Supported by the Department of Energy under Grant #DE-AC02-76ER03064.
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Volume 247, number 2,3
PHYSICSLETTERSB
13 September 1990
and m is the mass of the heavy quark involved in the particular process, and by higher order QCD corrections of the order aQCo (m). In what follows we use the ideas of the heavy quark effective theory to discuss the production of heavy mesons in e+e - collisions. Using spin symmetry alone we rederive the result of De Rfijula, Georgi and Glashow; in addition we make predictions on how the ratios of the total cross sections will behave above threshold as a function of the CMS energy. We restrict ourselves to the lowest order in AQCD/mand aQCD(m); however, the approach using the effective heavy quark theory allows to include some of those effects in a systematic way [ll]. The paper is organized as follows. In order to make the present note self-contained we give in section 2 a brief introduction into the formalism of spin symmetry. In section 3 we give the cross sections and their ratios; in section 4 we discuss our results and conclude. Finally, let us note that the issue of the heavy quark effective theory predictions for the ratio ( 1 ) have been independently discussed in ref. [ 11 ].
2. Formalism of spin symmetry In this paper we employ the formalism appropriate for the heavy quark (spin symmetry) limit: m ~ and AocD=Const. [ 5,7,4]. The formalism uses the so-called "velocity" superselection rule and was first presented by Georgi. As he notes, in the spin symmetry limit all heavy quark states are described by two four-vectors: the velocity vector vu (v~vU= l, Vo> 0) and the residual momentum vector k u. The total four-momentum pu equals
Pl'=mvU+k u,
(3)
where rnv u corresponds to the "infinite" part of PU and k u is the "finite" part of pu (i.e. [ku[ "~AQcD). In the spin symmetry limit QCD interactions neither can connect states of different velocity nor can create quarkantiquark pairs out of the vacuum. This is true up to corrections of the order 1/m and corrections due to hard gluon exchange suppressed by aQcD(m). Thus it follows that the heavy quark part of the QCD lagrangian in the heavy quark limit can be written as [ 9 ]
~eavy= ~ d3v (i[[+ ~h+ +ih-i-
(4)
where ~ is the covariant derivative. The field h + (h-~+ ) creates (annihilates) only quark states of a given velocity v and the field h-i- (h i- ) creates (annihilates) only antiquark states of the velocity v. In each velocity sector of the lagrangian (4), there are two separate SU (2) spin symmetries: one for the quark states and one for the antiquark states [ 9 ]. The generators of these symmetries can be associated with three unit vectors eu ( i = 1, 2, 3) which are orthogonal to vu: ei'v=0, e2 = _ 1. Let us introduce a triplet of quark spin operators S ÷ ( v, c ) ( e = e1, e2, ~3), and a triplet of antiquark operators S - ( v, E) ( ~= ~~, ~2, e3). The commutation relations for the operators S are [S+(v, ~), h + ] =75~b~h+,
[ S - ( v , ~), hi-] = - y s ~ * h i - ,
[S~±)(v, ~), h~ v)] =0.
(5,6,7)
Now consider a heavy meson built of the heavy quark and a light antiquark. The mass M of the meson in the spin symmetry limit tends to the quark mass M = m [ 1 + O (AQcD/m) ]. Such a meson is moving with an "infinite" momentum p u = My ~'and obviously the only heavy quark states present in the meson wave-function come from the v u sector of the theory. In this sector we have the symmetry generated by the operators S + (v, E) and all hadron states in the sector must occur in multiplets of SU (2)~pin. On the other hand, the operators S + (v, e) do not commute with total angular momentum, which means that they connect states of different spins. In particular, consider pseudoscalar and vector meson states IP ( v ) ) and IV(v, e) ), with momentum PU=MvU. Spin symmetry tells us that [ 7 ] 413
Volume 247, number 2,3
PHYSICS LETTERS B
13 September 1990
S + (v, E)IP(v) > = q l V ( v , e) >,
(8)
where t/is a phase factor which may be chosen to be q = 1. We can use the machinery introduced above in order to derive relations between the matrix elements of the QCD vector current [[Tuh between the vacuum and heavy meson-antimeson states. First let us note that for the two pseudoscalars the matrix element may be parametrized by one form-factor and we shall denote it as
( H( vl )ft( v2) l [[TuhlO ) =m~( vl "v2) ( Vl - V2)/,,
(9)
where { is the analogue of the dimensionless Isgur-Wise function for timelike momentum transfers and the meson states are normalized relativistically. We may use eqs. (8) and ( 5 ) to write ( H * ( v , , e)ft(v2) Ih-yah I0 ) = ( H ( v , ) I : I (Va)[ h-yuT,~b1~h [0).
(10)
In order to use (10) let us note the following gamma matrix identities: 7uTs ~ = ½7u(1 + 7s)75tb~+ ½7"(1 -- 7s)75tb~= ½BU"(v, ~)7.(1 +75) - ½AU"(v, e)7.(1 - T s )
= ½[BU"(v, e) -Aa"(v, e) ]y. + ½[BUd(v, e) +AU"(v, e) ]Y.Ts,
(1 1 )
where we have defined
AU~(v, e) --vUe"-v"eU-ieu"abVa%,
BU"(v, e) - v~e~-v"e**+ieu"aOVa%.
(12,13)
Now we can plug the formula ( 1 1 ) into eq. (10). Because the matrix element of the pseudovector operator h-Tu75h between the vacuum and the HFI state is zero one derives the relation ( H * ( V l , e)I:IIh-Tuhl0) = -im~(v,
•
V2)(.laabcV2Vl~.. a
b
C
(14)
Similarly, a relation for the matrix element of the vector meson-antimeson pair can be obtained: (H*(v~, el)H*(v2, e2)[ hyuh I0 ) -- ( H ( v ~ ) ,
IXI(/)2)[ h-Y5~2~2YU75~l ~ l h [ 0 )
= ½[A"U(Vl, ~l)B(xfl(v2, ez)+A~,~(v2, ~2)BaU(Vl, e l ) ]
H(vl)IZI(v2)Ih-Tahl0)
=m~(vl "v2) [ - e f e 2 'vl +e~el "v2 + (v, -v2)Uel .e2].
(15)
Note that in the spin symmetry limit all the current matrix elements (9), (10), ( 15 ) are given in terms of one universal coefficient function, the Isgur-Wise function {(v~.v2). However, the ratios for the exclusive production of the HIZI, H*FI (HIZI*) and H*I:I* meson pairs in the e + e - annihilation experiments will be independent of this unknown coefficient function.
3. Cross sections and ratios
In this section we make use of the relations between current matrix elements which are implied by spin symmetry. Since the matrix elements of the currents are given in terms of one function, the Isgur-Wise function ((v~'v2), we can make predictions on the ratios of the cross sections for all combinations of 0 - and 1- final states of the heavy mesons. The cross section for e+e - -,HI:I, where H is a heavy meson, is given by ~e+e-~Hn(S)= ~e4 S3 I dcI)Lu, W ~ ,
(16)
where s = 4EZm~. In (16) we introduced the leptonic tensor L,,, + -L~,,=PuP,, +PUP,+ - ( P
414
+.p
-)gu,',
(17)
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PHYSICS LETTERSB
13 September1990
where p - (p + ) corresponds to the momentum of the incoming e - (e + ) and dqb= J-dP~ dP2 (27~)4~4(p+ + p - - - P 1 - P 2 ) is the phase space volume element of the two outgoing heavy particles. The hadronic tensor Wu. is given in terms of (9) Wuv= ( H (vl)I=I (v2)I h-yuhI0) (OlEy~,hlH(v~)fl(v2)) --m2 ]((vl "V2)12Vuv ; .
(18)
Here and in the following we use the notation v+=vl +-vz,
(19)
which is convenient, since vu+ L
#v
/~P
=L
v~+ =0,
(20)
due to current conservation. Putting everything together we find for the cross section in the case of two pseudoscalar final states ae+e-_Hn(S)= ~ - S I~(vt'vz)12
1-
~
(1-cos20),
(21)
where O is the CMS scattering angle of the heavy mesons and S - 2 --1. Vl "v2 = -2m
Integrating this result over the phase space yields for the total cross section a e + ~ - ~ H n ( s ) = ~ l ( ( V l ' V 2 ) l 2 1--
(22)
Along the same lines the cross section for the decay into one vector and one pseudoscalar heavy meson may be calculated. Starting from the current matrix element (10) derived in their last section we have the hadronic tensor of the form 3
W~=
~ (H*(Vl, e(i))f-I(vz)l[iyuhlO)(Ol[i~uhlH*(v~, ~(i))IZI(v2)) i=1
=mZl((vl'vz)lz{[1--(vl"vz)Z]guv--½(v+v+ +VuV-;)+~(vl"v2)(v+ vv+ --v u- v~-)},
(23)
where we have summed over the final state polarizations. As noted above all terms proportional to v u+ or v + will not contribute due to the conservation of the lepton current and we find for the cross section in this case 72)3/2 0"e+e-~H*I--l(S)= ~not 2 I((Vl "V2)I 2 ( 41--
f dO 7 4) not 2 I((v~.vz)l 2 ( 1-( l + c o s 2 0 ) = g-~imZ
z 3/2"
(24)
Let us also note that in this order of the effective theory the cross section a,+e-~Hn* equals a¢+¢-on*n and therefore is given by eq. (24). Finally the cross section for the final state with two vector mesons may be calculated from the current matrix element ( 15 ). In this case the hadronic tensor is given by
415
Volume 247, number 2,3 3
PHYSICS LETTERS B
3
W'u*= ~ ~
(H*(v,,
i=lj=l
13 September 1990
¢~(i))f-I*(v2, ¢2(J))l&,hlO) (OIf[7~hlH*(v~,E(i))I=I*(v2, ~2(J)))
:m2l((Ul "/32)12{211 -- (Vl 'V2)2]guy + [ 2 - (vl 'u2)]/)~
VU- }'JFU +
terms,
(25)
where we have not displayed the terms involving v + since they will not contribute. The cross section in this case is na 2
a~+e-~n.n.(S)=~-~-S IC(Vl'V2)I2 - ha2 3s I~(vl'v2)12
(
1-4
(
4 1-
2)3/2fd~2[s
j~L~+
7
2 3/2 3 + ~ 5
(
s ) ] 3 - ~ 5 m 2 (1-cosZO)
•
(26) (27)
Taking the ratios of our results for the cross section the unknown coefficient function I~(v~ 'v2) 12 drops out and we get for the ratios using (22), (24) and (26): s
0 " e + e - ~ H H ( S ) :0"e+ e ~H*I=I(S) "3ff0"e+e-~HI=I*(S) : 0"e+e-~H*i=i*(S ) = 1 :
s
~-~ : 3 + rn 2 .
(28)
Note that near the threshold for the meson production we have for the ratios the result 1 : 4: 7 which is the result obtained in ref. [ 1 ] by means of spin counting.
4. D i s c u s s i o n and c o n c l u s i o n s
In this paper we have shown that in the limit where one may neglect effects of the order of
AQCD/mand
aQCD( m ) the ratio (28) of the exclusive cross section goes like 1 :s/m 2:3 + s/m 2. This verifies and extends the results of ref. [ 1 ] where the ratio 1 : 4: 7 appropriate for the limit Ecru ~ 2m was derived. Naturally our prediction has a finite region of validity. At the lower end around the threshold energy E c m ~ 2m one must expect important resonance effects. It has been shown both experimentally [ 12 ] and theoretically [ 13 ] that the effects of the resonant intermediate states of the W family enhance the production of D* vector mesons way over the ratio predicted in ref. [ 1 ]. The same should be true for all heavy mesons. We conclude that only in the energy region a couple of GeV higher than Ecru = 2m one can reasonably expect the prediction (28) to work. The upper limit of validity of (28) is slightly less clear. We expect different effects to correct (28): QCD radiative corrections as well as effects of higher order in AQCD/m.The former have been discussed in detail in ref. [ 11 ] using the heavy quark effective theory. The latter are difficult to discuss, since they involve nonperturbative QCD contributions. One can give an example of such an effect assuming that at high energies the form factor ( may be parametrized by an exponential ....
((vl'v2)
oc
exp
-
.
If we denote the mass difference between H and H* by Am then for energies Ecru/> taxi--re~Amthere will be large exponential corrections to (28). On the other hand the question of the high energy behavior of the ratios (28) may be devoid of any practical consequences. The cross sections for exclusive processes like e + e - production of heavy mesons fall rapidly at high energies and at some point there will be no experimental data to verify any theoretical prediction. Of course, to turn this into a quantitative discussion one would have to know the form of ( for Ecm >> 2m, but this is a highly nontrivial problem beyond the scope of the present note. 416
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Acknowledgement We t h a n k A. F a l k a n d B. G r i n s t e i n for c o m m u n i c a t i n g the results o f t h e i r w o r k (ref. [ 11 ] ) p r i o r to publication. We are also i n d e b t e d to H. G e o r g i a n d W. R o b e r t s for helpful discussions.
References [ 1 ] A. De Rtljula, H. Georgi and S. Glashow, Phys. Rev. Lett. 37 (1976) 398. [2] F. Close, Phys. Lett. B 65 (1976) 55. [ 3 ] K. Lane and E. Eichten, Phys. Rev. Lett. 37 (1976) 477. [4] M. Voloshin and M. Shifman, Sov. J. Nucl. Phys. 45 (1987) 292; 47(1988) 511. [5] E. Eichten and B. Hill, Phys. Lett. B 234 (1990) 511. [ 6 ] H. Politzer and M. Wise, Phys. Lett. B 206 ( 1988 ) 681; B 208 ( 1988 ) 504. [7] N. Isgur and M. Wise, Phys. Lett. B 232 (1989) 113; B 237 (1990) 527. [8 ] B. Grinstein, Harvard preprint HUTP-90/A002 (1990). [9] H. Georgi, Phys. Lett. B 240 (1990) 447. [ 10] A. Falk, H. Georgi, B. Grinstein and M. Wise, Harvard preprint HUTP-90/A011 (1990). [ 11 ] A. Falk and B. Grinstein, Harvard preprint HUTP-90/A042 (1990). [ 12] G. Goldhaber et al., Phys. Lett. B 69 (1977) 503. [ 13 ] A. Le Yaouanac, L. Olivier, O. Pene and J.C. Raynal, Phys. Lett. B 71 ( 1977 ) 397.
417
PHYSICS LETTERS B
13 September 1990
Spin s y m m e t r y in e + e - annihilation into heavy m e s o n s ¢r T h o m a s M a n n e l La n d Zbigniew R y z a k 2 Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA Received 25 June 1990
We employ the spin symmetry of the heavy quark effective theory to discuss the exclusive total cross sections for the production of heavy mesons in electron-positron annihilation. We verify and extend the results derived previously on the basis of spin counting arguments by De Rtljula, Georgi and Glashow.
1. Introduction
The question of theoretical predictions for the production rates of heavy mesons (in the following generally denoted H ) has been studied ever since the discovery of the charm quark. In particular the ratios of the total cross sections of e+e---,HIZI, e + e - - , H * I r t and e + e - ~ H * I : I * have been a subject of some controversy over a decade ago. Several approaches have been pursued to derive predictions for these ratios. Using only spin counting arguments De Rtijula, Georgi and Glashow [ 1 ] suggested that near threshold the ratios should be O'e+e- ~HI7t : O'e+e- o H * l q + O'e+e- ~ H f i * : O'e+e- oHITI* =
1 : 4: 7.
( 1)
Their result was later analyzed by Close [2 ] who used the helicity formalism and the multipole expansion to discuss the cross sections in detail. He concludes that the result of ref. [ 1 ] corresponds to special assumptions for the ratio of the intrinsic longitudinal and transverse cross section. Using a linear potential model Eichten and Lane [ 3 ] give another prediction for the ratio of the cross sections: ae+e-~Hfl:ae+e-~H.f~+ae+e-~Hn.:ee+e-~Hn.=l:4:
[ ( ~ 2(PH*~4~ 7+6
PH* \mu./
+
-, \mH*/ _1
(2)
where PH* and m H*are the absolute value of the three-momentum and the mass of the heavy meson respectively. In this paper we will readdress the issue of the exclusive e+e - production rates ( 1 ) using the recently formulated effective theory for heavy quarks [ 4-10 ]. In this approach the heavy quarks are simply static color sources and do not feel the recoil of the QCD interactions with the light quarks. In this non-recoil limit there are two additional symmetries. One is an SU(Nf) flavor symmetry, where Nf denotes the number of heavy flavors. This symmetry arises because the mass dependence may be scaled out so that no reference to the heavy quark masses is left in the effective theory. The second new symmetry is due to the fact that in the non-recoil limit the spin degrees of freedom of the heavy quarks decouple. An SU(2 ) rotation of the spin of the heavy quark, at any fixed velocity, thus generates an additional symmetry. Both of these additional symmetries are broken by terms of the order AQco/m, where AOCD is the QCD scale Work supported by the National Science Foundation under Grant #PHY-8714654. On leave from Technische Hochschule Darmstadt, D-6100 Darmstadt, FRG. 2 Supported by the Department of Energy under Grant #DE-AC02-76ER03064.
412
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Volume 247, number 2,3
PHYSICSLETTERSB
13 September 1990
and m is the mass of the heavy quark involved in the particular process, and by higher order QCD corrections of the order aQCo (m). In what follows we use the ideas of the heavy quark effective theory to discuss the production of heavy mesons in e+e - collisions. Using spin symmetry alone we rederive the result of De Rfijula, Georgi and Glashow; in addition we make predictions on how the ratios of the total cross sections will behave above threshold as a function of the CMS energy. We restrict ourselves to the lowest order in AQCD/mand aQCD(m); however, the approach using the effective heavy quark theory allows to include some of those effects in a systematic way [ll]. The paper is organized as follows. In order to make the present note self-contained we give in section 2 a brief introduction into the formalism of spin symmetry. In section 3 we give the cross sections and their ratios; in section 4 we discuss our results and conclude. Finally, let us note that the issue of the heavy quark effective theory predictions for the ratio ( 1 ) have been independently discussed in ref. [ 11 ].
2. Formalism of spin symmetry In this paper we employ the formalism appropriate for the heavy quark (spin symmetry) limit: m ~ and AocD=Const. [ 5,7,4]. The formalism uses the so-called "velocity" superselection rule and was first presented by Georgi. As he notes, in the spin symmetry limit all heavy quark states are described by two four-vectors: the velocity vector vu (v~vU= l, Vo> 0) and the residual momentum vector k u. The total four-momentum pu equals
Pl'=mvU+k u,
(3)
where rnv u corresponds to the "infinite" part of PU and k u is the "finite" part of pu (i.e. [ku[ "~AQcD). In the spin symmetry limit QCD interactions neither can connect states of different velocity nor can create quarkantiquark pairs out of the vacuum. This is true up to corrections of the order 1/m and corrections due to hard gluon exchange suppressed by aQcD(m). Thus it follows that the heavy quark part of the QCD lagrangian in the heavy quark limit can be written as [ 9 ]
~eavy= ~ d3v (i[[+ ~h+ +ih-i-
(4)
where ~ is the covariant derivative. The field h + (h-~+ ) creates (annihilates) only quark states of a given velocity v and the field h-i- (h i- ) creates (annihilates) only antiquark states of the velocity v. In each velocity sector of the lagrangian (4), there are two separate SU (2) spin symmetries: one for the quark states and one for the antiquark states [ 9 ]. The generators of these symmetries can be associated with three unit vectors eu ( i = 1, 2, 3) which are orthogonal to vu: ei'v=0, e2 = _ 1. Let us introduce a triplet of quark spin operators S ÷ ( v, c ) ( e = e1, e2, ~3), and a triplet of antiquark operators S - ( v, E) ( ~= ~~, ~2, e3). The commutation relations for the operators S are [S+(v, ~), h + ] =75~b~h+,
[ S - ( v , ~), hi-] = - y s ~ * h i - ,
[S~±)(v, ~), h~ v)] =0.
(5,6,7)
Now consider a heavy meson built of the heavy quark and a light antiquark. The mass M of the meson in the spin symmetry limit tends to the quark mass M = m [ 1 + O (AQcD/m) ]. Such a meson is moving with an "infinite" momentum p u = My ~'and obviously the only heavy quark states present in the meson wave-function come from the v u sector of the theory. In this sector we have the symmetry generated by the operators S + (v, E) and all hadron states in the sector must occur in multiplets of SU (2)~pin. On the other hand, the operators S + (v, e) do not commute with total angular momentum, which means that they connect states of different spins. In particular, consider pseudoscalar and vector meson states IP ( v ) ) and IV(v, e) ), with momentum PU=MvU. Spin symmetry tells us that [ 7 ] 413
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13 September 1990
S + (v, E)IP(v) > = q l V ( v , e) >,
(8)
where t/is a phase factor which may be chosen to be q = 1. We can use the machinery introduced above in order to derive relations between the matrix elements of the QCD vector current [[Tuh between the vacuum and heavy meson-antimeson states. First let us note that for the two pseudoscalars the matrix element may be parametrized by one form-factor and we shall denote it as
( H( vl )ft( v2) l [[TuhlO ) =m~( vl "v2) ( Vl - V2)/,,
(9)
where { is the analogue of the dimensionless Isgur-Wise function for timelike momentum transfers and the meson states are normalized relativistically. We may use eqs. (8) and ( 5 ) to write ( H * ( v , , e)ft(v2) Ih-yah I0 ) = ( H ( v , ) I : I (Va)[ h-yuT,~b1~h [0).
(10)
In order to use (10) let us note the following gamma matrix identities: 7uTs ~ = ½7u(1 + 7s)75tb~+ ½7"(1 -- 7s)75tb~= ½BU"(v, ~)7.(1 +75) - ½AU"(v, e)7.(1 - T s )
= ½[BU"(v, e) -Aa"(v, e) ]y. + ½[BUd(v, e) +AU"(v, e) ]Y.Ts,
(1 1 )
where we have defined
AU~(v, e) --vUe"-v"eU-ieu"abVa%,
BU"(v, e) - v~e~-v"e**+ieu"aOVa%.
(12,13)
Now we can plug the formula ( 1 1 ) into eq. (10). Because the matrix element of the pseudovector operator h-Tu75h between the vacuum and the HFI state is zero one derives the relation ( H * ( V l , e)I:IIh-Tuhl0) = -im~(v,
•
V2)(.laabcV2Vl~.. a
b
C
(14)
Similarly, a relation for the matrix element of the vector meson-antimeson pair can be obtained: (H*(v~, el)H*(v2, e2)[ hyuh I0 ) -- ( H ( v ~ ) ,
IXI(/)2)[ h-Y5~2~2YU75~l ~ l h [ 0 )
= ½[A"U(Vl, ~l)B(xfl(v2, ez)+A~,~(v2, ~2)BaU(Vl, e l ) ]
H(vl)IZI(v2)Ih-Tahl0)
=m~(vl "v2) [ - e f e 2 'vl +e~el "v2 + (v, -v2)Uel .e2].
(15)
Note that in the spin symmetry limit all the current matrix elements (9), (10), ( 15 ) are given in terms of one universal coefficient function, the Isgur-Wise function {(v~.v2). However, the ratios for the exclusive production of the HIZI, H*FI (HIZI*) and H*I:I* meson pairs in the e + e - annihilation experiments will be independent of this unknown coefficient function.
3. Cross sections and ratios
In this section we make use of the relations between current matrix elements which are implied by spin symmetry. Since the matrix elements of the currents are given in terms of one function, the Isgur-Wise function ((v~'v2), we can make predictions on the ratios of the cross sections for all combinations of 0 - and 1- final states of the heavy mesons. The cross section for e+e - -,HI:I, where H is a heavy meson, is given by ~e+e-~Hn(S)= ~e4 S3 I dcI)Lu, W ~ ,
(16)
where s = 4EZm~. In (16) we introduced the leptonic tensor L,,, + -L~,,=PuP,, +PUP,+ - ( P
414
+.p
-)gu,',
(17)
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where p - (p + ) corresponds to the momentum of the incoming e - (e + ) and dqb= J-dP~ dP2 (27~)4~4(p+ + p - - - P 1 - P 2 ) is the phase space volume element of the two outgoing heavy particles. The hadronic tensor Wu. is given in terms of (9) Wuv= ( H (vl)I=I (v2)I h-yuhI0) (OlEy~,hlH(v~)fl(v2)) --m2 ]((vl "V2)12Vuv ; .
(18)
Here and in the following we use the notation v+=vl +-vz,
(19)
which is convenient, since vu+ L
#v
/~P
=L
v~+ =0,
(20)
due to current conservation. Putting everything together we find for the cross section in the case of two pseudoscalar final states ae+e-_Hn(S)= ~ - S I~(vt'vz)12
1-
~
(1-cos20),
(21)
where O is the CMS scattering angle of the heavy mesons and S - 2 --1. Vl "v2 = -2m
Integrating this result over the phase space yields for the total cross section a e + ~ - ~ H n ( s ) = ~ l ( ( V l ' V 2 ) l 2 1--
(22)
Along the same lines the cross section for the decay into one vector and one pseudoscalar heavy meson may be calculated. Starting from the current matrix element (10) derived in their last section we have the hadronic tensor of the form 3
W~=
~ (H*(Vl, e(i))f-I(vz)l[iyuhlO)(Ol[i~uhlH*(v~, ~(i))IZI(v2)) i=1
=mZl((vl'vz)lz{[1--(vl"vz)Z]guv--½(v+v+ +VuV-;)+~(vl"v2)(v+ vv+ --v u- v~-)},
(23)
where we have summed over the final state polarizations. As noted above all terms proportional to v u+ or v + will not contribute due to the conservation of the lepton current and we find for the cross section in this case 72)3/2 0"e+e-~H*I--l(S)= ~not 2 I((Vl "V2)I 2 ( 41--
f dO 7 4) not 2 I((v~.vz)l 2 ( 1-( l + c o s 2 0 ) = g-~imZ
z 3/2"
(24)
Let us also note that in this order of the effective theory the cross section a,+e-~Hn* equals a¢+¢-on*n and therefore is given by eq. (24). Finally the cross section for the final state with two vector mesons may be calculated from the current matrix element ( 15 ). In this case the hadronic tensor is given by
415
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3
W'u*= ~ ~
(H*(v,,
i=lj=l
13 September 1990
¢~(i))f-I*(v2, ¢2(J))l&,hlO) (OIf[7~hlH*(v~,E(i))I=I*(v2, ~2(J)))
:m2l((Ul "/32)12{211 -- (Vl 'V2)2]guy + [ 2 - (vl 'u2)]/)~
VU- }'JFU +
terms,
(25)
where we have not displayed the terms involving v + since they will not contribute. The cross section in this case is na 2
a~+e-~n.n.(S)=~-~-S IC(Vl'V2)I2 - ha2 3s I~(vl'v2)12
(
1-4
(
4 1-
2)3/2fd~2[s
j~L~+
7
2 3/2 3 + ~ 5
(
s ) ] 3 - ~ 5 m 2 (1-cosZO)
•
(26) (27)
Taking the ratios of our results for the cross section the unknown coefficient function I~(v~ 'v2) 12 drops out and we get for the ratios using (22), (24) and (26): s
0 " e + e - ~ H H ( S ) :0"e+ e ~H*I=I(S) "3ff0"e+e-~HI=I*(S) : 0"e+e-~H*i=i*(S ) = 1 :
s
~-~ : 3 + rn 2 .
(28)
Note that near the threshold for the meson production we have for the ratios the result 1 : 4: 7 which is the result obtained in ref. [ 1 ] by means of spin counting.
4. D i s c u s s i o n and c o n c l u s i o n s
In this paper we have shown that in the limit where one may neglect effects of the order of
AQCD/mand
aQCD( m ) the ratio (28) of the exclusive cross section goes like 1 :s/m 2:3 + s/m 2. This verifies and extends the results of ref. [ 1 ] where the ratio 1 : 4: 7 appropriate for the limit Ecru ~ 2m was derived. Naturally our prediction has a finite region of validity. At the lower end around the threshold energy E c m ~ 2m one must expect important resonance effects. It has been shown both experimentally [ 12 ] and theoretically [ 13 ] that the effects of the resonant intermediate states of the W family enhance the production of D* vector mesons way over the ratio predicted in ref. [ 1 ]. The same should be true for all heavy mesons. We conclude that only in the energy region a couple of GeV higher than Ecru = 2m one can reasonably expect the prediction (28) to work. The upper limit of validity of (28) is slightly less clear. We expect different effects to correct (28): QCD radiative corrections as well as effects of higher order in AQCD/m.The former have been discussed in detail in ref. [ 11 ] using the heavy quark effective theory. The latter are difficult to discuss, since they involve nonperturbative QCD contributions. One can give an example of such an effect assuming that at high energies the form factor ( may be parametrized by an exponential ....
((vl'v2)
oc
exp
-
.
If we denote the mass difference between H and H* by Am then for energies Ecru/> taxi--re~Amthere will be large exponential corrections to (28). On the other hand the question of the high energy behavior of the ratios (28) may be devoid of any practical consequences. The cross sections for exclusive processes like e + e - production of heavy mesons fall rapidly at high energies and at some point there will be no experimental data to verify any theoretical prediction. Of course, to turn this into a quantitative discussion one would have to know the form of ( for Ecm >> 2m, but this is a highly nontrivial problem beyond the scope of the present note. 416
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Acknowledgement We t h a n k A. F a l k a n d B. G r i n s t e i n for c o m m u n i c a t i n g the results o f t h e i r w o r k (ref. [ 11 ] ) p r i o r to publication. We are also i n d e b t e d to H. G e o r g i a n d W. R o b e r t s for helpful discussions.
References [ 1 ] A. De Rtljula, H. Georgi and S. Glashow, Phys. Rev. Lett. 37 (1976) 398. [2] F. Close, Phys. Lett. B 65 (1976) 55. [ 3 ] K. Lane and E. Eichten, Phys. Rev. Lett. 37 (1976) 477. [4] M. Voloshin and M. Shifman, Sov. J. Nucl. Phys. 45 (1987) 292; 47(1988) 511. [5] E. Eichten and B. Hill, Phys. Lett. B 234 (1990) 511. [ 6 ] H. Politzer and M. Wise, Phys. Lett. B 206 ( 1988 ) 681; B 208 ( 1988 ) 504. [7] N. Isgur and M. Wise, Phys. Lett. B 232 (1989) 113; B 237 (1990) 527. [8 ] B. Grinstein, Harvard preprint HUTP-90/A002 (1990). [9] H. Georgi, Phys. Lett. B 240 (1990) 447. [ 10] A. Falk, H. Georgi, B. Grinstein and M. Wise, Harvard preprint HUTP-90/A011 (1990). [ 11 ] A. Falk and B. Grinstein, Harvard preprint HUTP-90/A042 (1990). [ 12] G. Goldhaber et al., Phys. Lett. B 69 (1977) 503. [ 13 ] A. Le Yaouanac, L. Olivier, O. Pene and J.C. Raynal, Phys. Lett. B 71 ( 1977 ) 397.
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