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Heavy quarks spin correlations in e+e− annihilations
Volume 215, number 4
PHYSICS LETTERS B
29 December 1988
H E A V Y Q U A R K S P I N C O R R E L A T I O N S I N e+e - A N N I H I L A T I O N S
R.H. DALITZ, Gary R. GOLDSTEIN 1 Department of Theoretical Physics, Oxford University, 1 Keble Road, Oxford OX1 3NP, UK
and R. MARSHALL Rutherford Appleton Laboratory, Chilton, Didcot OXI I OQX, UK Received 1 August 1988; revised manuscript received 8 October 1988
It is pointed out that in e+e - annihilation the physical situation is especially favourable for the observation of the strong spin correlation between the D* and 13" in opposing c- and e-jets, predicted by QCD. Some assessment is given of the possibility of observing this in PEP and PETRA data, and with LEP. The possibilities for similar observations for (B*, 13") and (T*, "i'*) are briefly discussed.
Quantum chromodynamics has the property of helicity conservation for quark lines, in the limit of zero mass for the quark involved. As a result, it is commonly believed [ 1 ] that hadronic processes resuiting from the interaction between unpolarized hadrons can generate only small polarizations, of order mq/E~. Here we point out that this helicity conservation implies very strong polarization correlations between particles in the final state. It would be of much interest, as a check on QCD, to confirm these correlations experimentally, and this can be conveniently done using e+e - annihilation data, by measuring correlations between the leading particles in the q and q jets produced. We begin by considering the electromagnetic excitation of a pair of charmed jets, the first step being e+e-~ce.
(1)
We have in mind CM energy about 30 GeV, where the Z ° contribution is not yet of major importance. For unpolarized e ÷ and e - beams, the density matrix for the (ce) system produced at angle O to the incident e - direction is readily calculated. In helicity Permanent address: Physics Department, Tufts University, Medford, MA 02155, USA.
space, and with the approximation mc = 0, it is given by
p(c,-C),~'2 ~x =6(k', -k' )6(L -k) ( 1 + 4 , 0 / c o s 2 0 ) 2( 1 + cos20)
(2)
We note that this agrees with a more general expression given by Lee [2] in another context, but disagrees with the angle-independent expression given by Anselmino et al. [3]. The c-quark density matrix p(c)a,a obtained from (2) by averaging over e-quark helicities is a unit matrix; the individual c and e quarks are unpolarized. The physical situation ( 1 ) is very favourable for observing the correlations implied by (2). This reaction has a rate of about 4 of the total e+e - cross section, so it is rather prominent. The mass mc is sufficiently low, mc/Ee~ 1 , for the (ce) density matrix to be adequately represented by (2), its mc = 0 limit. At the same time, the mass mc is sufficiently large that secondary (c, e) pairs will only rarely be formed in either of the two jets. A D* seen in a jet is then necessarily the leading particle, containing the initial cquark and therefore carrying the spin information about that quark. The empirical evidence is consis-
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tent with the formation of one D* meson in every cquark jet, although the data are also consistent with the ratio D / D * = ~ given by spin counting, as is appropriate for a simple quark model. When a D meson is the leading particle, the c-spin information is lost, since the D meson, being spinless, cannot carry spin information. More generally [ 4 ], D / D * = f / ( 2 - a 0 is adopted in the phenomenological treatments of jet development, where f i s to be determined from the data. For convenience, we shall define a parameter a = 1/( 1 - f ) , for which D / D * = ( a - 1 ) / ( a + 1 ), with a = 2 for the case of spin counting. In these circumstances, the (c, e) helicity correlation will generate a strong helicity correlation between D* and I7)*. D* formation from c involves the process c~--+DA +X~-,
(3)
the c-quark picking up a ~ or a quark from the vacuum, the u or d quark left then being hadronized to the hadron state X, where the suffices A and x denote helicities, so that 2 = A + x. The (D*, [3* ) density matrix RAAA 'm can be calculated using (2) and the matrix e l e m e n t MKA), for (3), and summing over all hadronic states X. A general formula for R(D*, I)*) is inconvenient, so we simply tabulate all of its non-zero elements: ++
R+0
1 +a) 2 ,
D O -+ _+0 = - R
0-_4 =*~
TO
(4a) 0T
+_0----- - R o - +
= (~/2) [sin20/(I +cos20) ]/ (I +~) 2 .
We note that the quark-model value a = 2 makes RA,3 a multiple of the unit matrix. The recent D* decay angular distributions measured by the HRS group [ 5 ] give a = 1.75 _+0.35 for this parameter; they also give R+_ =0.07_+0.04. There are two D* decay modes to be considered, (a) D*--,Dn,
(b) D*--*D7.
(6)
Decay mode (a) is characterized by a p-wave amplitude A3 (0, ¢) = YA (0, (b) and the angular distribution is w ( x , ~) =
w(o, O, O, ~)
1. d ' d = ~'AA,AA, A A , A A R A , A A d ,
tAA,.
(7)
Retaining only the non-zero elements of (4) leads to the general form for W(x, x ): (R + + + R + + )sin20 sin20 + 2 ( R J- ~ +R6- 3 )sin 2 0cos20 + 2 (R o++o+ R o_o ) cosZ0 sin20 + 4R o% ocos20 cos20 +sin 20sin 2 0 { e x p [ i ( ~ - 0 ) ] X ( R % - R -°o_- R°+ +R%)+h.c.}o-
.
(8)
W c ( x , ~ ) = [9/16~2(1 + a ) 2] x {[ ½a+ ( 1 - ½a)cos2O] [ l a + ( 1 - l a ) c o s 2 0 ]
(4b)
We note that: (a) parity conservation in the creation (1) and X'X -A',-X hadronization (3) steps e n s u r e s RA'A =R-A'.--A, (b) charge-conjugation symmetry in steps ( 1 ) and A'A (2) e n s u r e s R /1'/1 A, A = R x, j , (c) hermiticity of the density matrix ensures RZ'~ A'A ~ ~~Io' A~A'' )t. Further, the calculation of R involves summation over a large number of states X and we have assumed the phases of the off-diagonal elements to be real, the contributions to the imaginary part of R cancelling out in the sum. Again, the sum over all (A', A) values leads us to a D* density matrix RAA (D*), whose only non-zero elements are 784
(5)
_=(al2)Roo=(a/2)/(l+a).
Inserting the expression (4), and normalizing the angular distribution, the final result is
R++ = R _+_+ =~R66- = o'R °+ __° = (o'2/2)R °° = (a2/2)/(
R++ = R
29 D e c e m b e r 1988
+ ½a[sin20/( 1+cos20) ]
X c o s ( 0 - 0 ) sin 20sin 20} .
(9)
Decay mode (b) is characterized by a magnetic dipole amplitude B~A where/t denotes the photon helicity, over which there is a summation. The structures of the distributions W(x, y) and W(,{, 7) are quite similar to (8). W(x, 1') can be obtained from (8) by replacing sin20 by ( 1 + cos20)/2, and cos20 by sin20, in the first two lines of (8), leaving the 0terms stand; W0/, y) can be obtained by making this replacement for both 0 and ff It is interesting to note that, for a = 2, the three correlation functions W(x, x ), W(x, y) and W('y, y) have precisely the same form. We may note also that these correlation functions involve spherical harmonics of second rank in both 0 and & The
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decay interactions D * - , D n and D * ~ D 7 conserve parity and are therefore independent of first-rank spin tensors ~. Consider now the practical possibilities for the observation of the correlated angular distribution (9) using the published data [ 7 ] of the JADE group for illustration. There also exists a great deal of data, similar but with higher measurement accuracy, under study by the HRS Collaboration [ 8,9 ]. The JADE data consists of some 24 thousand annihilations to hadronic states, about 4 of which have (ce) as their primary quark-antiquark pair. A further sample of similar size is under analysis by JADE. Each (ce) event leads to two charm particles, which may be pseudoscalar states (D °, D +, D~+ and I) °, IT)-, I3~- ) or vector states (D *°, D *÷, D *+ and I) *°, 13"-, 17)~*- ) or their excited states at higher mass. The latter generally decay to a D or D*, but the fraction of D*'s thus produced is quite small [ 10], so attention will be confined to directly produced D* states. O f these, only the D* -+ states are useful, since the only D *° decays are D°n ° and D°7, which are more difficult to deal with experimentally. The total number of D* -+ mesons may be estimated as 2 × ~ × ~ = ½ per (ce) event, thus about 4400. Since its decay processes D-+n ° and D-+7 are difficult for accurate measurement, attention is confined to the DOn + mode, which has branching ratio 54 + 5% [ 11 ]. The D o meson has a number of decay modes with reasonably ~t The remarks of Donoghue [6] about the empirical determination of the density matrix for pOmesons in jets do not take this into account. Since p°~n+n- conserves parity, its decay angular distribution cannot determine the odd elements of the density matrix. The same comment holds for his remarks on correlations between pOdecays occurring in the q and dljets. Wo~dcannot appear even in correlated pO decays. His terms Wevencorrespond to the product assumption (in our notation) RA'A =PAA'P
.
29 December 1988
high branching ratios which are convenient for identification with quite high efficiency, as shown in table 1. With the figures in table 1, the total number o f events containing one identifiable D* -+ may be expected to be (0.0156) × (0.54) × 4 4 0 0 = 37 + 6. This expectation is to be compared with the figure of 92 Y - 4 6 deduced from the total number of D *÷ and D * - mesons reported by the JADE group [7 ]. The higher resolution in the HRS experiment leads to a rather higher detection efficiency [ 9 ], typically of order 40%, and to the ability to include the K - n + n ° decay mode, which has a higher branching ratio. Whereas B F × DE,,~ 1.6% for the JADE experiment, a value B F × D E = 13.5% might be achieved in the HRS experiment, where this analysis is at present under way [ 9 ]. The discussion suggests that, with the JADE apparatus, the chance of finding one identifiable D *-+ meson in a (ce) two-jet event is (empirically) only 92 1.05%. When a D *÷ meson is identified in one 880 -jet, the chance of identifying a D * - in the opposing jet is only 0.25%, which means about one identifiable D * + D * - event in the whole JADE sample. The situation may be significantly better in the HRS experiment, perhaps by two orders of magnitude, which would permit the evaluation o f the moments o f (9) to better than 10%. We wish to emphasize that the factors contributing to the experiment proposed here are all highly favourable. The production o f (ce) pairs has a high rate, the yield o f (ce) --, ( D*I3* + hadrons ) where D* and I3" are leading particles in the opposing jets could scarcely be higher, the decays D *÷ -~n+D ° and D * - - , n 13° have a high branching ratio, and the (D °, I3°) mesons resulting have clearly identifiable decay modes. The final efficiencies (BF × DE) are admittedly somewhat low but are un-
Table 1 Decay
Branching fraction (BF) a) (%)
Detection efficiency (DE) (%) (JADE)
Product BFxDE (%)
D°-K n + D°-,K n+n° D°-,K-n+n-n -
4.15_+0.5 13.0 _+1.7 8.6 _+3.0
25
1.04_+0.13
6
0.52_+0.18
alRe£ [11]. 785
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likely to be improvable by more than the factor of 5 achieved by HRS. At P E T R A / P E P energies, the role o f Z ° in process (1) is not negligible, but gives both an observed backward/forward asymmetry in the e + e - ~ c e angular distribution and a longitudinal polarization to the c (e) quark, both of order 10-15%. With LEP, and CM energies around the Z ° mass, the Z ° contribution will be outstandingly dominant. Therefore, with this case in mind, the calculations outlined above have been repeated including both ./and Z ° contributions. We give here only the result, using the standard notation [12] which we now briefly summarize. The electric charge is denoted by Qq or Q~, in units of the proton charge, and the weak vector and axial-vector charges are denoted by (vq, aq) and (v~, a~), for quarks and leptons, respectively. Thus, for example, (Q~, v~, ac) are ( + ~, 1 - 8 sin20w/3, + 1 ) and (Q¢; u~, G ) are ( - 1, - 1 + 4 sin20~, - 1 ); we note that G vanishes for the Weinberg angle given by sinZ0w= ~, so that it is essentially zero for these calculations. We denote by Z the Z ° resonance factor
sGz Z= (s_m2)+imzFz ,
(10)
where Gz = 1 / (2 sin 20w)-=0.357_ ~ + 0.004, s=4E 2 and (mz, ['z) are the Z ° mass and width. In the approximation ve_ 0, the expressions for p;,;~;';are p++~ = { ( l + c o s Z O ) [ Q g + a 2 ( v q + a q ) 2 1 z 2 1 ] _+4a¢cos O(vq +aq)QqRe(z)}/d,
( 1 la)
v --
(lib)
+
2 z
2
2-
P+~+= - s i n 2 0 [ Q q ~ a c t v q - a q ) [ Z l Z ] / d , where d is then
d=2(l+cos20)[Qq+a~(v~+aq)rZl2] + 8 cos Oa~aqQq R e ( z ) .
(12)
The (D*, IS)*) density matrix R may be calculated from p as before. As result of the violation of both parity and charge conjugation invariance in the (ce) creation process ( 1 ) now, R has many more non-zero elements. Not all of these non-zero elements enter into the final distribution W(n, n), because the decay processes D*-, Dn and D./conserve parity. We give only the distribution W(n, n) here, and that only with the approximation v~= 0; 786
29 December 1988
Wc(n, n ) = [9/16n2(1 + a ) 2] × { [½a+ ( 1 - ½cr)cos20] [ ½a+ ( 1 - ½~)cos20]
- (a/d)sin 20sin 20 sinZO X
{ [Qq2 + a e2 ( ~ - a c )2
Izl 2]cos(6-~)}}.
In general, W(n, n) will also have a term proportional to sin ( 6 - O) inside the square brackets, but this term is proportional to re, whose value now stands at ( - 0 . 0 8 + 0 . 0 2 ) , and to I m z , which is tq'naller than Izl 2 by a factor of Fz/MzGz~ 12. Observations of the kind discussed here can also be considered for the b and t quarks. (a) B* production is certainly expected to be strong in e+e - annihilations, but ~he (B*, B) mass difference is only about 52 MeV [ 13 ] so that its only observable decay mode is B*-*B./. For this mode, the (B*, 13") spin correlation will lead to the distribution Wb(7, ./), given by (9) after replacing the two square brackets on the second line of (9) by [ ( 2 + a ) / 4 - ( 1 - a / 2 ) cos20] times the same square bracket in t~ Wb (./, ./) will be difficult to measure because (i) it requires identifying a low-energy "/-ray in a strong background of ./-rays from other sources, such as n°--.././decays, and (ii) the B-meson decays which allow unique identification have quite low branching fractions, typically less than 1%, e.g. 0.5+0.2% for B + - , D ° n + and 0.33_+0.15% for BO_,D * ( 2 0 1 0 ) - n + [ 14], and these modes require that the D o meson is also to be identified, for which the present (JADE) efficiency is about 1.6%, as we noted above. (h) The situation is qualitatively different for t quarks. The T* and T mesons have not yet been observed but, from the present limits on the t-quark mass, in,> 50 GeV, it is to be expected that the (T*, T) mass difference will be less than 5 MeV. If so, the dominant T* decay modes will be through weak interactions; the process T*--.~+v~+ (hadrons with b = + 1 )
(14)
will be very prominent decay modes [ 15 ]. Since parity is violated in these decay modes (14), the distribution W,(~, ~; 0, 0, 0, ~) will have more terms than appear in (7), because all possible elements of R j ' , j will come into play. However, if m , > 50 GeV does hold, as appears likely, there will be many other corn-
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p l i c a t i o n s arising, a n d we shall not go f u r t h e r into these q u e s t i o n s here. O n e o f us ( G . R . G . ) gratefully a c k n o w l e d g e s the h o s p i t a l i t y o f the T h e o r e t i c a l Physics D e p a r t m e n t , O x f o r d U n i v e r s i t y . T h e w o r k o f G . R . G . was partially s u p p o r t e d by a grant f r o m the U n i t e d States D e p a r t m e n t o f Energy.
References [ 1] G.R. Farrar, Phys. Rev. Lett. 56 (1986) 1643. [2]S.-C. Lee, Polarization density matrix for heavy QQ production in hadron colliders preprint Fermilab - PUB87/13-T (February 1987). [3] M. Anselmino, P. Kroll and B. Pire, Z. Phys. C 29 (1985) 135. [4] R.P. Feynman and R.D. Field, Phys. Rev. D 15 (1977) 2590.
29 December 1988
[ 5 ] HRS Collab., S. Abachi et al. Phys. Lett. B 199 ( 1987 ) 585. [ 6 ] J.F. Donoghue, Phys. Rev. D 19 ( 1979 ) 2806. [7] JADE Collab., W. Barrel et al., Phys. Lett. B 146 (1984) 121. [8] HRS Collab., P. Baringer et al., Phys. Len. B 206 (1988) 551. [9] HRS Collab., B. Musgrave, private communication (July 1988). [10] ARGUS Collab., H. Albrecht et al., Phys. Rev. Lett. 56 (1986) 549; CLEO Collab., C. Bebek et al. ( 1987 ). [11 ] D. Hitlin, in: Proc. 1987 Intern. Symp. on Lepton and photon interactions at high energy (Hamburg, July 1987 ), eds. W. Bartel and R. Riickl (North-Holland, Amsterdam, 1988~ p. 179. [ 12 ] R. Marshall, Acta Phys. Austr. Suppl. 24 ( 1982 ) 63. [ 13 ] Particle Data Group, M. Aguilar-Benitez et al., Review of particle properties, Phys. Len. B 170 (1986) 1. [14]W. Schmidt-Parzefall, in: Proc. 1987 Intern. Symp. on Lepton and photon interactions at high energy (Hamburg, July 1987), eds. W. Bartel and R. Riickl (North-Holland, Amsterdam, 1988) p. 257. [ 15 ] J.H. Kuhn and K.H. Streng, Nucl. Phys. B 198 ( 1984 ) 71.
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PHYSICS LETTERS B
29 December 1988
H E A V Y Q U A R K S P I N C O R R E L A T I O N S I N e+e - A N N I H I L A T I O N S
R.H. DALITZ, Gary R. GOLDSTEIN 1 Department of Theoretical Physics, Oxford University, 1 Keble Road, Oxford OX1 3NP, UK
and R. MARSHALL Rutherford Appleton Laboratory, Chilton, Didcot OXI I OQX, UK Received 1 August 1988; revised manuscript received 8 October 1988
It is pointed out that in e+e - annihilation the physical situation is especially favourable for the observation of the strong spin correlation between the D* and 13" in opposing c- and e-jets, predicted by QCD. Some assessment is given of the possibility of observing this in PEP and PETRA data, and with LEP. The possibilities for similar observations for (B*, 13") and (T*, "i'*) are briefly discussed.
Quantum chromodynamics has the property of helicity conservation for quark lines, in the limit of zero mass for the quark involved. As a result, it is commonly believed [ 1 ] that hadronic processes resuiting from the interaction between unpolarized hadrons can generate only small polarizations, of order mq/E~. Here we point out that this helicity conservation implies very strong polarization correlations between particles in the final state. It would be of much interest, as a check on QCD, to confirm these correlations experimentally, and this can be conveniently done using e+e - annihilation data, by measuring correlations between the leading particles in the q and q jets produced. We begin by considering the electromagnetic excitation of a pair of charmed jets, the first step being e+e-~ce.
(1)
We have in mind CM energy about 30 GeV, where the Z ° contribution is not yet of major importance. For unpolarized e ÷ and e - beams, the density matrix for the (ce) system produced at angle O to the incident e - direction is readily calculated. In helicity Permanent address: Physics Department, Tufts University, Medford, MA 02155, USA.
space, and with the approximation mc = 0, it is given by
p(c,-C),~'2 ~x =6(k', -k' )6(L -k) ( 1 + 4 , 0 / c o s 2 0 ) 2( 1 + cos20)
(2)
We note that this agrees with a more general expression given by Lee [2] in another context, but disagrees with the angle-independent expression given by Anselmino et al. [3]. The c-quark density matrix p(c)a,a obtained from (2) by averaging over e-quark helicities is a unit matrix; the individual c and e quarks are unpolarized. The physical situation ( 1 ) is very favourable for observing the correlations implied by (2). This reaction has a rate of about 4 of the total e+e - cross section, so it is rather prominent. The mass mc is sufficiently low, mc/Ee~ 1 , for the (ce) density matrix to be adequately represented by (2), its mc = 0 limit. At the same time, the mass mc is sufficiently large that secondary (c, e) pairs will only rarely be formed in either of the two jets. A D* seen in a jet is then necessarily the leading particle, containing the initial cquark and therefore carrying the spin information about that quark. The empirical evidence is consis-
0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
783
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tent with the formation of one D* meson in every cquark jet, although the data are also consistent with the ratio D / D * = ~ given by spin counting, as is appropriate for a simple quark model. When a D meson is the leading particle, the c-spin information is lost, since the D meson, being spinless, cannot carry spin information. More generally [ 4 ], D / D * = f / ( 2 - a 0 is adopted in the phenomenological treatments of jet development, where f i s to be determined from the data. For convenience, we shall define a parameter a = 1/( 1 - f ) , for which D / D * = ( a - 1 ) / ( a + 1 ), with a = 2 for the case of spin counting. In these circumstances, the (c, e) helicity correlation will generate a strong helicity correlation between D* and I7)*. D* formation from c involves the process c~--+DA +X~-,
(3)
the c-quark picking up a ~ or a quark from the vacuum, the u or d quark left then being hadronized to the hadron state X, where the suffices A and x denote helicities, so that 2 = A + x. The (D*, [3* ) density matrix RAAA 'm can be calculated using (2) and the matrix e l e m e n t MKA), for (3), and summing over all hadronic states X. A general formula for R(D*, I)*) is inconvenient, so we simply tabulate all of its non-zero elements: ++
R+0
1 +a) 2 ,
D O -+ _+0 = - R
0-_4 =*~
TO
(4a) 0T
+_0----- - R o - +
= (~/2) [sin20/(I +cos20) ]/ (I +~) 2 .
We note that the quark-model value a = 2 makes RA,3 a multiple of the unit matrix. The recent D* decay angular distributions measured by the HRS group [ 5 ] give a = 1.75 _+0.35 for this parameter; they also give R+_ =0.07_+0.04. There are two D* decay modes to be considered, (a) D*--,Dn,
(b) D*--*D7.
(6)
Decay mode (a) is characterized by a p-wave amplitude A3 (0, ¢) = YA (0, (b) and the angular distribution is w ( x , ~) =
w(o, O, O, ~)
1. d ' d = ~'AA,AA, A A , A A R A , A A d ,
tAA,.
(7)
Retaining only the non-zero elements of (4) leads to the general form for W(x, x ): (R + + + R + + )sin20 sin20 + 2 ( R J- ~ +R6- 3 )sin 2 0cos20 + 2 (R o++o+ R o_o ) cosZ0 sin20 + 4R o% ocos20 cos20 +sin 20sin 2 0 { e x p [ i ( ~ - 0 ) ] X ( R % - R -°o_- R°+ +R%)+h.c.}o-
.
(8)
W c ( x , ~ ) = [9/16~2(1 + a ) 2] x {[ ½a+ ( 1 - ½a)cos2O] [ l a + ( 1 - l a ) c o s 2 0 ]
(4b)
We note that: (a) parity conservation in the creation (1) and X'X -A',-X hadronization (3) steps e n s u r e s RA'A =R-A'.--A, (b) charge-conjugation symmetry in steps ( 1 ) and A'A (2) e n s u r e s R /1'/1 A, A = R x, j , (c) hermiticity of the density matrix ensures RZ'~ A'A ~ ~~Io' A~A'' )t. Further, the calculation of R involves summation over a large number of states X and we have assumed the phases of the off-diagonal elements to be real, the contributions to the imaginary part of R cancelling out in the sum. Again, the sum over all (A', A) values leads us to a D* density matrix RAA (D*), whose only non-zero elements are 784
(5)
_=(al2)Roo=(a/2)/(l+a).
Inserting the expression (4), and normalizing the angular distribution, the final result is
R++ = R _+_+ =~R66- = o'R °+ __° = (o'2/2)R °° = (a2/2)/(
R++ = R
29 D e c e m b e r 1988
+ ½a[sin20/( 1+cos20) ]
X c o s ( 0 - 0 ) sin 20sin 20} .
(9)
Decay mode (b) is characterized by a magnetic dipole amplitude B~A where/t denotes the photon helicity, over which there is a summation. The structures of the distributions W(x, y) and W(,{, 7) are quite similar to (8). W(x, 1') can be obtained from (8) by replacing sin20 by ( 1 + cos20)/2, and cos20 by sin20, in the first two lines of (8), leaving the 0terms stand; W0/, y) can be obtained by making this replacement for both 0 and ff It is interesting to note that, for a = 2, the three correlation functions W(x, x ), W(x, y) and W('y, y) have precisely the same form. We may note also that these correlation functions involve spherical harmonics of second rank in both 0 and & The
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PHYSICS LETTERS B
decay interactions D * - , D n and D * ~ D 7 conserve parity and are therefore independent of first-rank spin tensors ~. Consider now the practical possibilities for the observation of the correlated angular distribution (9) using the published data [ 7 ] of the JADE group for illustration. There also exists a great deal of data, similar but with higher measurement accuracy, under study by the HRS Collaboration [ 8,9 ]. The JADE data consists of some 24 thousand annihilations to hadronic states, about 4 of which have (ce) as their primary quark-antiquark pair. A further sample of similar size is under analysis by JADE. Each (ce) event leads to two charm particles, which may be pseudoscalar states (D °, D +, D~+ and I) °, IT)-, I3~- ) or vector states (D *°, D *÷, D *+ and I) *°, 13"-, 17)~*- ) or their excited states at higher mass. The latter generally decay to a D or D*, but the fraction of D*'s thus produced is quite small [ 10], so attention will be confined to directly produced D* states. O f these, only the D* -+ states are useful, since the only D *° decays are D°n ° and D°7, which are more difficult to deal with experimentally. The total number of D* -+ mesons may be estimated as 2 × ~ × ~ = ½ per (ce) event, thus about 4400. Since its decay processes D-+n ° and D-+7 are difficult for accurate measurement, attention is confined to the DOn + mode, which has branching ratio 54 + 5% [ 11 ]. The D o meson has a number of decay modes with reasonably ~t The remarks of Donoghue [6] about the empirical determination of the density matrix for pOmesons in jets do not take this into account. Since p°~n+n- conserves parity, its decay angular distribution cannot determine the odd elements of the density matrix. The same comment holds for his remarks on correlations between pOdecays occurring in the q and dljets. Wo~dcannot appear even in correlated pO decays. His terms Wevencorrespond to the product assumption (in our notation) RA'A =PAA'P
.
29 December 1988
high branching ratios which are convenient for identification with quite high efficiency, as shown in table 1. With the figures in table 1, the total number o f events containing one identifiable D* -+ may be expected to be (0.0156) × (0.54) × 4 4 0 0 = 37 + 6. This expectation is to be compared with the figure of 92 Y - 4 6 deduced from the total number of D *÷ and D * - mesons reported by the JADE group [7 ]. The higher resolution in the HRS experiment leads to a rather higher detection efficiency [ 9 ], typically of order 40%, and to the ability to include the K - n + n ° decay mode, which has a higher branching ratio. Whereas B F × DE,,~ 1.6% for the JADE experiment, a value B F × D E = 13.5% might be achieved in the HRS experiment, where this analysis is at present under way [ 9 ]. The discussion suggests that, with the JADE apparatus, the chance of finding one identifiable D *-+ meson in a (ce) two-jet event is (empirically) only 92 1.05%. When a D *÷ meson is identified in one 880 -jet, the chance of identifying a D * - in the opposing jet is only 0.25%, which means about one identifiable D * + D * - event in the whole JADE sample. The situation may be significantly better in the HRS experiment, perhaps by two orders of magnitude, which would permit the evaluation o f the moments o f (9) to better than 10%. We wish to emphasize that the factors contributing to the experiment proposed here are all highly favourable. The production o f (ce) pairs has a high rate, the yield o f (ce) --, ( D*I3* + hadrons ) where D* and I3" are leading particles in the opposing jets could scarcely be higher, the decays D *÷ -~n+D ° and D * - - , n 13° have a high branching ratio, and the (D °, I3°) mesons resulting have clearly identifiable decay modes. The final efficiencies (BF × DE) are admittedly somewhat low but are un-
Table 1 Decay
Branching fraction (BF) a) (%)
Detection efficiency (DE) (%) (JADE)
Product BFxDE (%)
D°-K n + D°-,K n+n° D°-,K-n+n-n -
4.15_+0.5 13.0 _+1.7 8.6 _+3.0
25
1.04_+0.13
6
0.52_+0.18
alRe£ [11]. 785
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likely to be improvable by more than the factor of 5 achieved by HRS. At P E T R A / P E P energies, the role o f Z ° in process (1) is not negligible, but gives both an observed backward/forward asymmetry in the e + e - ~ c e angular distribution and a longitudinal polarization to the c (e) quark, both of order 10-15%. With LEP, and CM energies around the Z ° mass, the Z ° contribution will be outstandingly dominant. Therefore, with this case in mind, the calculations outlined above have been repeated including both ./and Z ° contributions. We give here only the result, using the standard notation [12] which we now briefly summarize. The electric charge is denoted by Qq or Q~, in units of the proton charge, and the weak vector and axial-vector charges are denoted by (vq, aq) and (v~, a~), for quarks and leptons, respectively. Thus, for example, (Q~, v~, ac) are ( + ~, 1 - 8 sin20w/3, + 1 ) and (Q¢; u~, G ) are ( - 1, - 1 + 4 sin20~, - 1 ); we note that G vanishes for the Weinberg angle given by sinZ0w= ~, so that it is essentially zero for these calculations. We denote by Z the Z ° resonance factor
sGz Z= (s_m2)+imzFz ,
(10)
where Gz = 1 / (2 sin 20w)-=0.357_ ~ + 0.004, s=4E 2 and (mz, ['z) are the Z ° mass and width. In the approximation ve_ 0, the expressions for p;,;~;';are p++~ = { ( l + c o s Z O ) [ Q g + a 2 ( v q + a q ) 2 1 z 2 1 ] _+4a¢cos O(vq +aq)QqRe(z)}/d,
( 1 la)
v --
(lib)
+
2 z
2
2-
P+~+= - s i n 2 0 [ Q q ~ a c t v q - a q ) [ Z l Z ] / d , where d is then
d=2(l+cos20)[Qq+a~(v~+aq)rZl2] + 8 cos Oa~aqQq R e ( z ) .
(12)
The (D*, IS)*) density matrix R may be calculated from p as before. As result of the violation of both parity and charge conjugation invariance in the (ce) creation process ( 1 ) now, R has many more non-zero elements. Not all of these non-zero elements enter into the final distribution W(n, n), because the decay processes D*-, Dn and D./conserve parity. We give only the distribution W(n, n) here, and that only with the approximation v~= 0; 786
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Wc(n, n ) = [9/16n2(1 + a ) 2] × { [½a+ ( 1 - ½cr)cos20] [ ½a+ ( 1 - ½~)cos20]
- (a/d)sin 20sin 20 sinZO X
{ [Qq2 + a e2 ( ~ - a c )2
Izl 2]cos(6-~)}}.
In general, W(n, n) will also have a term proportional to sin ( 6 - O) inside the square brackets, but this term is proportional to re, whose value now stands at ( - 0 . 0 8 + 0 . 0 2 ) , and to I m z , which is tq'naller than Izl 2 by a factor of Fz/MzGz~ 12. Observations of the kind discussed here can also be considered for the b and t quarks. (a) B* production is certainly expected to be strong in e+e - annihilations, but ~he (B*, B) mass difference is only about 52 MeV [ 13 ] so that its only observable decay mode is B*-*B./. For this mode, the (B*, 13") spin correlation will lead to the distribution Wb(7, ./), given by (9) after replacing the two square brackets on the second line of (9) by [ ( 2 + a ) / 4 - ( 1 - a / 2 ) cos20] times the same square bracket in t~ Wb (./, ./) will be difficult to measure because (i) it requires identifying a low-energy "/-ray in a strong background of ./-rays from other sources, such as n°--.././decays, and (ii) the B-meson decays which allow unique identification have quite low branching fractions, typically less than 1%, e.g. 0.5+0.2% for B + - , D ° n + and 0.33_+0.15% for BO_,D * ( 2 0 1 0 ) - n + [ 14], and these modes require that the D o meson is also to be identified, for which the present (JADE) efficiency is about 1.6%, as we noted above. (h) The situation is qualitatively different for t quarks. The T* and T mesons have not yet been observed but, from the present limits on the t-quark mass, in,> 50 GeV, it is to be expected that the (T*, T) mass difference will be less than 5 MeV. If so, the dominant T* decay modes will be through weak interactions; the process T*--.~+v~+ (hadrons with b = + 1 )
(14)
will be very prominent decay modes [ 15 ]. Since parity is violated in these decay modes (14), the distribution W,(~, ~; 0, 0, 0, ~) will have more terms than appear in (7), because all possible elements of R j ' , j will come into play. However, if m , > 50 GeV does hold, as appears likely, there will be many other corn-
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p l i c a t i o n s arising, a n d we shall not go f u r t h e r into these q u e s t i o n s here. O n e o f us ( G . R . G . ) gratefully a c k n o w l e d g e s the h o s p i t a l i t y o f the T h e o r e t i c a l Physics D e p a r t m e n t , O x f o r d U n i v e r s i t y . T h e w o r k o f G . R . G . was partially s u p p o r t e d by a grant f r o m the U n i t e d States D e p a r t m e n t o f Energy.
References [ 1] G.R. Farrar, Phys. Rev. Lett. 56 (1986) 1643. [2]S.-C. Lee, Polarization density matrix for heavy QQ production in hadron colliders preprint Fermilab - PUB87/13-T (February 1987). [3] M. Anselmino, P. Kroll and B. Pire, Z. Phys. C 29 (1985) 135. [4] R.P. Feynman and R.D. Field, Phys. Rev. D 15 (1977) 2590.
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[ 5 ] HRS Collab., S. Abachi et al. Phys. Lett. B 199 ( 1987 ) 585. [ 6 ] J.F. Donoghue, Phys. Rev. D 19 ( 1979 ) 2806. [7] JADE Collab., W. Barrel et al., Phys. Lett. B 146 (1984) 121. [8] HRS Collab., P. Baringer et al., Phys. Len. B 206 (1988) 551. [9] HRS Collab., B. Musgrave, private communication (July 1988). [10] ARGUS Collab., H. Albrecht et al., Phys. Rev. Lett. 56 (1986) 549; CLEO Collab., C. Bebek et al. ( 1987 ). [11 ] D. Hitlin, in: Proc. 1987 Intern. Symp. on Lepton and photon interactions at high energy (Hamburg, July 1987 ), eds. W. Bartel and R. Riickl (North-Holland, Amsterdam, 1988~ p. 179. [ 12 ] R. Marshall, Acta Phys. Austr. Suppl. 24 ( 1982 ) 63. [ 13 ] Particle Data Group, M. Aguilar-Benitez et al., Review of particle properties, Phys. Len. B 170 (1986) 1. [14]W. Schmidt-Parzefall, in: Proc. 1987 Intern. Symp. on Lepton and photon interactions at high energy (Hamburg, July 1987), eds. W. Bartel and R. Riickl (North-Holland, Amsterdam, 1988) p. 257. [ 15 ] J.H. Kuhn and K.H. Streng, Nucl. Phys. B 198 ( 1984 ) 71.
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