- Email: [email protected]
Use of spin-correlations to study low energy ΛΛ and ΛΛ space symmetries and resonances
__ __ BB
s
8 June 1995
PHYSICS
c
ELSEVIER
LETTERS
B
Physics Letters B 352 (1995) 162-168
Use of spin-correlations to study low energy RR and AR space symmetries and resonances Gideon Alexander a, Harry J. Lipkin a,b a School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel ’ Department of Nuclear Physics, Weizmann Institute of Science, Rehovot 76100, Israel
March 1995 Editor: L. Montanet
Received 27
Abstract A general method to investigate spin correlations between two weakly decaying spin l/2 hyperons is presented and applied to measurement of the relative amounts of the SAA= 1 or SAA= 0 states as a function of the CM energy of any state of a AA or Ax system with nonrelativistic relative motion and arbitrary orbital angular momentum. The large statistics experimental data existing in the multi-hadron decay of the Z” and in pp annihilations at low energies can be used to search for nontrivial spin correlations expected if there are low-mass or subthreshold resonances in either the AA pair, like the proposed H dibaryon, or the AA system when emerging from a resonance decay. In the absence of such structures, the relative S = 0 part should show a low energy enhancement due to the Pauli exclusion principle for the AA system and angular momentum barrier considerations similar to those present in the Bose-Einstein correlation of identical bosons and seen in the GG system when emerging from 93 pairs. This enhancement may be used as an estimate of the two-hyperon emitter dimension in the Z” hadronic decays.
1. Introduction In reactions between particles which lead to multihadronic final states the constructive interference between two identical bosons, the so called BoseEinstein Correlation (BEC), is well known. These correlations lead to an enhancement of the number of identical bosons over that of non-identical bosons when the two particles are close to each other in phase space. Experimentally this effect, also known as the GGLP effect, was for the first time observed in particle physics by Goldhaber et al. [ 11, in like-sign charged pions produced in pj? annihilations at fi = 2.1 GeV. In addition to the quantum mechanical aspect of the BEC, these correlations are also used
Elsevier Science B.V. SSDI 0370-2693(95)00492-O
to estimate the dimension of the emitting source of the identical bosons [ 21. Some recent reviews which summarize the underlying BEC theoretical aspects and the experimental results are given, for example, in Refs. [ 3-51. It has further been shown, that under certain conditions, a Bose-Einstein like enhancement can also be expected in a boson-antiboson system like the Kc?? pair [ 6,7] if the “Chaoticity” A parameter is larger than zero. In fact, in a sample of spinless boson-antiboson system, which is a mixture of C = + 1 and C = - 1 states, the C = $1 part behaves like identical bosons, that is, it produces a BEC like low mass enhancement whereas the C = -1 part decreases to zero as EK,z+ 0.This has been experimentally demonstrated for the
G. Alexander, HJ. Lipkin /Physics
first time by the OPAL Collaboration [ 81 in their study of the ge system and was later confirmed by the DELPHI [ 91 and ALEPH [ lo] groups at LEP This GGLP low mass enhancement is the result of the spatial symmetric state which is always the case for identical Bosons and is also true for the C = +l part of the Boson-Antiboson system. It has been pointed out [6] that a pair of identical fermions, ff, having a total spin of S = 1, will have an Ef,f dependence near threshold similar to that of a Boson-Antiboson pair with the eigenvalue of C = - 1. This follows from the Pauli exclusion principle and the expectation that in the absence of low mass pwave resonances the relative contribution of the e = 0 state increases as Ef,f decreases due to the angular momentum barrier. The measurement of this behavior will allow to estimate for the first time the dimension Ra of the identical fermions emitter in a similar way as it has been done in the past for identical bosons emitter. In view of the fact that now high statistics Z” 4 hadrons data exist which contain pairs of AA and An [ IO], it became experimentally feasible to study their correlations. In this note we describe a method which allows experimentally to measure in a given sample of Ah or Ax pairs the fractions of events with a total spin 0 and a total spin 1. This method, which is based on the angular correlation between the decay products of the AA (An) pairs, is also applicable to any system of two spin l/2 hyperons or hyperon - antihyperon which decay weakly. Using this method we explore the possibility to study the AA (or Ah) system as a function of the variable Q = Jm,
used often
in BEC studies.
2. Selection of S = 0 and S = 1 states The AA (or Ah) pairs produced at low Q in inclusive Z” + hadrons decays are expected to be unpolarized, in contrast to single A’s and to pairs produced in exclusive strong interaction processes. A’s produced in strong interactions tend to be polarized normal to the reaction plane; this is observed both in inclusive single A production and in exclusive pair production; e.g. the reaction pp -+ Ai [ 111. The kinematics in Z decay is completely different. There is no reaction plane. The
Letters B 352 (1995) 162-l 68
163
interaction is a weak interaction which tends to produce longitudinally polarized strange quark-antiquark pairs that can hadronize into longitudinally polarized fast A particles [ 121. But the A and A in a low Q pair have essentially the same momentum in the laboratory and cannot be produced from a polarized SS pair emitted directly from the Z with equal and opposite lab momenta. Either one or both will be produced by a strong fragmentation process which does not favor a given helicity. Furthermore any polarization relative to the initial beam or initial jet direction is probably completely washed out in multiparticle fragmentation. The new ingredient here is the relative polarization of the two A’s; i.e. whether their spins are parallel or antiparallel. This can be measured by decay angular correlations and give information on the presence of resonances and on space-symmetry effects analogous discussed above for the C = f 1 parts of the bosonantiboson system [ 6,7]. In contrast to other analyses which assume a particular production mechanism for A polarization [ 11,121, our treatment holds for any production mechanism, does not assume any hyperon polarization in the laboratory system and gives results completely independent of any such polarization. Let us consider a AA pair in its center of mass system. For large Q values, the maximum radial angular momentum emax is larger than zero and several J = L + S are present, where S = Sr + S2 is spin sum of the two hyperons. However for a sufficiently small Q and near zero, where a Bose-Einstein enhancement occurs for identical bosons, the s-wave (l = 0) prevails ’ . In fact, if &, = 0 then J = S. Whereas S = 0 is allowed for two identical fermions, the state S = 1 is forbidden by the Pauli exclusion principle which should be experimentally observed through the Q behavior of the AA system. In order to avoid the Pauli exclusion principle let us first consider a Al? system at, or very near, its threshold E,,i = rnA + ml\, being in a pure f? = 0 state. We will further select the simple measurable case where the A decays into p1 +T- and the h to p2 +rr+. Here pi stands for the decay proton or anti-proton both of which we further will refer to as protons. Let us denote by y the cosine of the angle (in space) between p1 and p2 in the AR center of mass system and (y) its average. ’ From angular momentum barrier consideration.
164
G. Alexander, H.J. Lipkin /Physics
In the decay of a single polarized A, the proton angular distribution in the A center of mass system, is given by [ 131 dw/dcos8
c( 1 -LX,, cost9
(1)
where the parameter (Y~, arising from the parity violation, was measured experimentally [ 141 to be LX,,= , ;s;;t$fs,‘,Z
=0.642
+ 0.013
(2)
Here fs and fp are the s-wave and p-wave decay amplitudes. Since CZ~= -(YA one has ffi = -0.642
* 0.013
Setting x = cos 0 then the average 277
1 (xi\> = 27rN
xm
J J d+
0
(XA) is given by
Letters B 352 (1995) 162-168
For the case where angles are also measured with respect to another external axis; e.g. the normal to a production plane, such an angular distribution can have a quadrupole component or tensor polarization when the two spins of l/2 are coupled to spin 1 and are aligned with respect to this axis. Such tensor polarizations are observed, for example in Ai production by low-energy pp annihilation [ 111. Since we average over all orientations with respect to any fixed axis, only the scalar product of the two dipoles survives. To calculate the relation between (y) and the proton decay cosine averages, (XA) and (xi) of each of the hyperons 2 , we utilize the relation for the matrix elements of any vector operator V between states of the same total angular momentum J following from the Wigner-Eckart theorem [ 151,
(V . J)(J?WIJ, M’)
(.I, MIVIJ, M’) = (1 - cuAx)xdx
J(J+
(3)
I>
Let Pi and Pz be the unit vectors in the direction of flight of the pi and p2 protons respectively. Then, (XI\), averaged over the whole angular range, is equal to
Xrnill
where Xm8.X
N=
J
(1 -a,,x)dx
(4)
(6)
Xi”,” In particular the average over the whole x range i.e., XG” = -1 and .rrnax = +l, yields
and (xi) is given by (7)
(xA) = -0.214
l 0.004
and
(x;;;} = +0.214 & 0.004
We now note that general arguments do not allow dN/dy distribution of a two spin-112 hyperon system at threshold to have a y dependence higher than its first power. It is essentially the same argument that prevents a spin-l /2 particle from having a quadrupole moment. The distribution of the angle between the proton momenta from the two hyperons can depend only upon the dipole moments of the individual angular distributions. It must be also be a scalar under rotations in the laboratory, since there is no external reference frame here. Thus the only allowed nontrivial distribution is the scalar product of the two dipoles which is proportional to y. This means that dN/dy has the form dN/dy=A.[l+Bs.y] The factor Bs can then be determined of (YL
from the value
From Eqs. (6) and (7) follows that the average (y) of the cosine angle between the two decay protons: (Y) =
vl *pz)=
(4
. Sl) (Sl * S2)
&(SI
+
l)Sz(S2
(9
. S2) +
(8)
1)
From this one finds that (Y) R = (X*)(Xx)
=2S(S$l)
(fi . pz) = (P,,,)(P2,z)
- 3
= 4.
lS1 . s2)
(9)
The total spin S can have one of the two possible values, S = 0 and S = 1 so that -3 for S = 0 (Y> RQ=” = (x,,)(xA) = { -TV for s = 1 2 Note that at Q = 0 the individual of the two hyperons system.
hyperon CMS is equal to that
G. Alexander, HJ. Lipkin /Physics
Due to the Pauli exclusion principle, at e = 0 only S = 0 is allowed for the AA system and therefore one expects
R(Q)
‘so
RQ4
= -
(Y)
w
=
(Y) = 0.0458
-
3
165
Letters B 352 (1995) 162-l 68 2,
1.8
L
(10) x1.2
This limit can be experimentally verified by studying the ratio R(Q) as function of Q. Note that in the measurement of this limit there is no need for a reference sample as is the case in the study of the n-r and KK Bose-Einstein correlations. We thus find that for S = 0 one has for AA system
9 =;l f CO.8
0.6
0.4
B. = -9(x,,)*
L
= -0.4122 0.2 1
and for S = 1 01
Bi = +3(x*)2
As a consequence, one has for the AA system at, or very near, its threshold dN/dyI,,
0: 1 - 0.4122.
For comparison Pauli principle)
y
(11)
the C = 0, S = 1 state (forbidden by the should have the angular distribution
dN/dyl,=, 0: 1 + 0.1374.
y
(12)
These two distinctly different distributions, which are shown in Fig. 1, are derived at the limit of Q = 0 but as will shown in the next section are also valid for larger Q values. Finally for the Ah system one has 0: 1 + 0.4122.
y
dN/dyI,=, 0; 1 - 0.1374.
y
dN/dyl,,
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
= +0.1374
and
Fig. 1. The distribution of dN/dy as a function of v of the Ah system with a total spin of S = 0 and S = 1. These distributions are compared to a flat one shown by the dotted line.
The AA and AA wave functions in the center-ofmass system can be expressed in an “L - S-coupling” basis where the spins of the two baryons are coupled to a total spin S, and that this spin is coupled with some orbital angular momentum L to a total angular momentum J. For such a wave function we can immediately apply the Wigner-Eckart theorem Eq. (5) for any vector operator V and generalize Eq. (8) to apply to any two vector operators depending only upon the spin variables St and Sz: t 14)
(13)
3. Q dependence 3.1. Extension of results to finite Q values Experimentally one does not expect to have any significant number of AA events very near to EnA = 2nz~ unless there exists a AA resonance just above threshold. However, we shall now show that Eq. (8) can be applied directly to events with finite Q values in the region where the relative momentum of the two A’s is nonrelativistic.
This result is completely independent of the orbital angular momentum L and the total angular momentum J. In order to use Eq. (8) for this case, we note that the quantities (XA) and (xX) are defined in the rest frame of the decaying A and similarly for the proton momenta, Pr and Pa. However, in Eq. (14) all vectors are defined in the center-of-mass system. We shall now show that as long as the relative motion of the two baryons is nonrelativistic we can ignore these differences between frames, determine the experimental value of (y) s (PI .4) from the values for each proton momentum taken in the rest frame of its own A,
166
G. Alexander, H.J. Lipkin/Physics
and use this value in Eq. (9) and the known values of (XI\) and (XI\) with no boost corrections. We first note that Eqs. (6) and (7) hold also when the definition of Pr and Pz is generalized to be any unit vectors in the directions of some linear combination of the pion and proton momenta in the rest frame of the individual A and/or A respectively. When the baryon is moving, the pion and proton momenta are no longer back-to back, but one suitable combination of the pion and proton momenta can be found to define (x1\) and (xii) for moving baryons and obtain the same value as in the rest frame. For totally nonrelativistic motion, the correct vector is the pion-proton relative velocity for each A decay, which would be the same in the center-of-mass system of the two A’s as in the rest system of each A. A nonrelativistic boost changes the velocities of the decay proton and decay pion, but not their relative velocity. The momentum of the decay pion is definitely relativistic, even though the momenta of the A and decay proton may not be. However, one can still define a relative velocity vector which gives the correct value for (xi\) and (xX). When a A and its decay proton and pion are boosted by a velocity /3, the momenta of the decay proton and decay pion in the new frame are given to first order in p by the Lorentz transformations P,, (P> = P,,(O) - P&(O)
+ Q(P2)
(15)
P,(P)
+ CYP2)
t 16)
= P,(O)
- P&(O)
The “relative velocity” linear terms in p,
can be defined to cancel the
P,(P) PAP) ~,l(P) = - &CO) &r(O) =--pp (0) EP,(O) + W2) E,>(o) n =
&l(O) +
W)
Letters B 352 (1995) 162-168
We can now use the relative velocities for the decays of the two A’s as the vectors VI and V2 in Eq. (14) which holds in the center-of-mass system of the two A’s. But we now note that relative velocities as defined by Eq. (17) do not change to first order in p under a Lorentz boost to the rest frame of each A. We can evaluate the scalar products (5 . I$), (VI . Sl) and (Vz . 82) using the rest frame of each A for its corresponding relative velocity. But in the rest frame of each A a unit vector in the direction of the relative velocity is the same as a unit vector in the direction of the proton momentum. Thus we again recover Eq. (8) now for the general case of finite Q value and all values of orbital angular momentum L and total angular momentum J. Eq. (8) can be used directly for states with definite spin and finite Q-values, with the value of (y) s (PI . Pz) determined from the values for each proton momentum taken in the rest frame of its own A. The results for (y) and R depend only on the total spin S of the two A’s in their center of mass system and are completely independent of their relative motion in space and their orbital angular momentum. 3.2. Measurement of the spin correlation We now have a clear way to differentiate between the S = 0 and S = 1 contribution to the AA (Ah) for all values of Q where the relative motion is nonrelativistic. A straightforward way to achieve it is by fitting the sum of the expected dN/dy distributions for S = 0 and S = 1, given in Eqs. (11) and (121, to the experimental distribution at a given Q range. In this way the relative contributions Fo and FI from the S = 0 and S = I respectively will be determined. Next one can define the two ratios: co =
2Fo Fo + 6 /3
(171
The relative velocity defined in this way is exactly equal to the usual relative velocity in the nonrelativistic limit where Ep(0) and E,(O) are just the proton and pion masses. In the nonrelativisitic limit the relative proton-pion velocity is independent of the velocity of their center-of-mass. Our relativistic generalization (17) has the same property of being invariant under Lorentz boosts to first order in the velocity 0.
and
C = 2Fll3 ’ Fo+F,/3
where F, is divided by 3 to offset the 2S+ 1 factor. At high Q values, where C,,, is large, one expects CI g C2 = 1. At Q x 0, due to the exclusion principle, Co = 2 whereas Ct = 0. Thus CO behavior as a function of Q is similar to the @@ state originating from a l@Ko system whereas Ct behaves like the @K”L pair. The dependence of CO and Ct on Q, given in arbitrary units, are shown schematically in Fig. 2. The behavior of Ce( Q) may then be fitted to an expression,
G. Alexander, H.J. Lipkin / Physics Letters B 352 (1995) 162-168
Fig. 2. A schematic view of the correlation functions Co and Cl as a function of Q for the S = 0 part and S = 1 part respectively for the AA system. The dotted line represents the expectation for a Ax system in the absence of resonances.
used previously in the study of two identical correlations, of the type Co(Q) = (1 +
/KQzR+
bosom
(18)
where Rs may be interpreted as an estimate of the two hyperons emitter dimension and the parameter A, the strength of the effect, should be equal to 1.
4. Low energy dibaryon spectroscopy The possibility of bound states or low-mass resonances in both AA and AA systems have been suggested theoretically and investigated experimentally, so far with inconclusive results. The study of the invariant mass spectra and spin correlations of pairs produced in Z” decay can give additional information. In the framework of QCD derived models, the existence of dibaryon states (apart from the deuteron) has been discussed. A AA state, referred to as the dibaryon H, has been proposed as early as 1977 [ 161 and examined in various analyses (see e.g. Refs. [ 1620] ) . It should have a mass not far below or above its threshold of 2mA. So far no conclusive experimental evidence has been found for its existence. However,
167
it is only recently that large samples of events exists which do contain a AA or Ai pairs in their final state. If this H state is above the AA threshold, it may be detected directly by looking at the invariant mass distribution of the AA system. At the same time, if it is as expected an !! = 0, S = 0 state, it will not affect the behavior of (y) as a function of Q and its limit as Q --t 0. The AA system has been investigated recently in low-energy pp annihilation experiments, with the result that Ah production is dominated by the triplet spin (S = 1) state [ 111. It is not clear whether this spin correlation arises from the dynamics of the AA interaction or from the reaction mechanism; e.g. kaon exchange. This ambiguity can be resolved by examining the spin correlation for pairs produced in Z” decay where the reaction mechanism is completely different. If the same S = 1 dominance also occurs here it would support the suggestions of a low-lying resonance just above threshold.
5. Summary The method described here allows a measurement of the spin content of two spin 1I2 hyperons or a pair of hyperon and antihyperon samples irrespective of the orbital angular momentum. This can be utilized in the study of the exclusive reactions like pp + Ax and p jj -+ Z+p in order to verify the presence of s-channel resonances and to investigate the reaction mechanism. In the search and study of high mass resonances, like the .I/ti -+Ax, the final spin state of the hyperon-antihyperon is determined by the spin-parity values of the resonance as long as its decay does conserve parity. Thus the measurement of the spin state of the hyperon-antihyperon can serve as an additional tool in establishing the identity of the resonance and its properties. In multi-hadron final states at high energies, like the Z0 decay into multi-hadrons, the question of fragmentation and hadronization are handled in different models and compared to data via Monte Carlo calculations. An important check as to the validity of these models can by obtained from measurements of the typical hadronic emitter dimension. Such an information was extracted from the Bose-Einstein Correlation studies of identical bosons, mainly the r*&
168
G. Alexander, H.J. Lipkin /Physics
systems. Already the large existing statistics of Z” decays accumulated at the LEP experiments presents a sizable sample of multi-hadrons final states which contain pairs of identical AA and n hyperons. Utilizing the method outline here, they can be analyzed in terms of their total spin content as a function of the variable Q. The corresponding correlation functions Co(Q) and Cl (Q) will allow an estimate of the typical emitter dimension of two hyperons which in turn can then be confronted with the various fragmentation and hadronization models.
References III G. Goldhaber et al., Phys. Rev. Lett. 3 (1959) 181; Phys. Rev. 120 (1960) 300. [21 See e.g., M.G. Bowler, Particle World (Gordon and Breach Science Publishers) 2 ( 1991) 1. 131 M. Gyulassy et al., Phys. Rev. C 20 (1979) 2267; W. Hofmann, A Fresh Look at Bose-Einstein Correlations, LBL 23108 (1987). [41 W.A. Zajc, in Hadronic Multiparticle Production, P Carruthers, ed., World Scientific ( 1988) 235; M.I. Podgoretskii, Sov. J. Part. Nucl. 20 (1989) 266. I51 For a recent concise summary of BEC results of like-sign charged pions see e.g., S. Marcellini, Proc. of the Joint Int. Lepton-Photon Symp. and EPS Conf. on HEP, Geneva, 25th July-1st August 1991 (eds. S. Hegarty et al.), Vol. I, page 750. 161 G. Alexander, Bose-Einstein Effect in the K”? BosonAntiboson System, presented at the Workshop DAQNE and Other topics in Particle Physics, 18-20 November 1992, Frascati, Italy, LNF Preprint LNF-93/001 (p); G. Alexander, Bose-Einstein Correlation Effect in a BosonAntiboson System, Proc. of the Int. Conf. on Bose and the 20th Century Physics, 30 December-5 January 1994, Calcutta, India (in print).
Letters B 352 (1995) 162-168
[7] A detailed handling of the Ku? probability amplitudes in terms of symmetric and anti-symmetric states is given by H.J. Lipkin, Phys. Rev. Len. 69 (1992) 3700; see also H.J. Lipkin, Phys. Lett. B 219 (1989) 474; Argonne report ANL-HEP-PR-88-66. (81 OPAL Collaboration, PD. Acton et al., Phys. Lett. B 298 ( 1993) 456. [9] DELPHI Collaboration, P Abreu et al., Phys. Lett. B 323 ( 1994) 242. [ 101 ALEPH Collaboration, D. Buskulic et al., Z. Physik C 64 (1994) 373. [ 1I] See e.g., PD. Barnes et al., Nucl. Phys. A 558 (1993) 277~; N.H. Hamann. Nucl. Phys. A 558 (1993) 287~. [ 121 G. Gustafson and J. Hakkinen, Phys. Lett. B 303 (1993) 350. [ 131 See e.g., E. Segr?, Nuclei and Particles, 2nd Edition (Benjamin/Cummings Publishing Co.) p. 876. [ 141 Particle Data Group, Phys. Rev. D 50 (1994) 1173. [ 151 E.P. Wigner, Z. Physik 43 (1927) 624; C. Eckart, Rev. Mod. Phys. 2 (1930) 305. [ 161 R.L. Jaffe, Phys. Rev. Lett. 38 (1977) 195. [ 171 S.A. Yost and C.R. Nappi, Phys. Rev. D 32 (1985) 816. [ 181 M.A. Moinester, C.B. Dover and H.J. Lipkin, Phys. Rev. C 46 (1992) 1082. [ 191 D. Pal and J.A. McGovern, J. Phys. G: Nucl. Part. 18 ( 1992) 593. [20] J.L. Ping et al., Proc. of the XIII Int. Conf. On Particles and Nuclei, 28 June-2 July 1993, Perugia, Italy, p. 526; in the same proceedings see also: S. Paul, p. 646.
s
8 June 1995
PHYSICS
c
ELSEVIER
LETTERS
B
Physics Letters B 352 (1995) 162-168
Use of spin-correlations to study low energy RR and AR space symmetries and resonances Gideon Alexander a, Harry J. Lipkin a,b a School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel ’ Department of Nuclear Physics, Weizmann Institute of Science, Rehovot 76100, Israel
March 1995 Editor: L. Montanet
Received 27
Abstract A general method to investigate spin correlations between two weakly decaying spin l/2 hyperons is presented and applied to measurement of the relative amounts of the SAA= 1 or SAA= 0 states as a function of the CM energy of any state of a AA or Ax system with nonrelativistic relative motion and arbitrary orbital angular momentum. The large statistics experimental data existing in the multi-hadron decay of the Z” and in pp annihilations at low energies can be used to search for nontrivial spin correlations expected if there are low-mass or subthreshold resonances in either the AA pair, like the proposed H dibaryon, or the AA system when emerging from a resonance decay. In the absence of such structures, the relative S = 0 part should show a low energy enhancement due to the Pauli exclusion principle for the AA system and angular momentum barrier considerations similar to those present in the Bose-Einstein correlation of identical bosons and seen in the GG system when emerging from 93 pairs. This enhancement may be used as an estimate of the two-hyperon emitter dimension in the Z” hadronic decays.
1. Introduction In reactions between particles which lead to multihadronic final states the constructive interference between two identical bosons, the so called BoseEinstein Correlation (BEC), is well known. These correlations lead to an enhancement of the number of identical bosons over that of non-identical bosons when the two particles are close to each other in phase space. Experimentally this effect, also known as the GGLP effect, was for the first time observed in particle physics by Goldhaber et al. [ 11, in like-sign charged pions produced in pj? annihilations at fi = 2.1 GeV. In addition to the quantum mechanical aspect of the BEC, these correlations are also used
Elsevier Science B.V. SSDI 0370-2693(95)00492-O
to estimate the dimension of the emitting source of the identical bosons [ 21. Some recent reviews which summarize the underlying BEC theoretical aspects and the experimental results are given, for example, in Refs. [ 3-51. It has further been shown, that under certain conditions, a Bose-Einstein like enhancement can also be expected in a boson-antiboson system like the Kc?? pair [ 6,7] if the “Chaoticity” A parameter is larger than zero. In fact, in a sample of spinless boson-antiboson system, which is a mixture of C = + 1 and C = - 1 states, the C = $1 part behaves like identical bosons, that is, it produces a BEC like low mass enhancement whereas the C = -1 part decreases to zero as EK,z+ 0.This has been experimentally demonstrated for the
G. Alexander, HJ. Lipkin /Physics
first time by the OPAL Collaboration [ 81 in their study of the ge system and was later confirmed by the DELPHI [ 91 and ALEPH [ lo] groups at LEP This GGLP low mass enhancement is the result of the spatial symmetric state which is always the case for identical Bosons and is also true for the C = +l part of the Boson-Antiboson system. It has been pointed out [6] that a pair of identical fermions, ff, having a total spin of S = 1, will have an Ef,f dependence near threshold similar to that of a Boson-Antiboson pair with the eigenvalue of C = - 1. This follows from the Pauli exclusion principle and the expectation that in the absence of low mass pwave resonances the relative contribution of the e = 0 state increases as Ef,f decreases due to the angular momentum barrier. The measurement of this behavior will allow to estimate for the first time the dimension Ra of the identical fermions emitter in a similar way as it has been done in the past for identical bosons emitter. In view of the fact that now high statistics Z” 4 hadrons data exist which contain pairs of AA and An [ IO], it became experimentally feasible to study their correlations. In this note we describe a method which allows experimentally to measure in a given sample of Ah or Ax pairs the fractions of events with a total spin 0 and a total spin 1. This method, which is based on the angular correlation between the decay products of the AA (An) pairs, is also applicable to any system of two spin l/2 hyperons or hyperon - antihyperon which decay weakly. Using this method we explore the possibility to study the AA (or Ah) system as a function of the variable Q = Jm,
used often
in BEC studies.
2. Selection of S = 0 and S = 1 states The AA (or Ah) pairs produced at low Q in inclusive Z” + hadrons decays are expected to be unpolarized, in contrast to single A’s and to pairs produced in exclusive strong interaction processes. A’s produced in strong interactions tend to be polarized normal to the reaction plane; this is observed both in inclusive single A production and in exclusive pair production; e.g. the reaction pp -+ Ai [ 111. The kinematics in Z decay is completely different. There is no reaction plane. The
Letters B 352 (1995) 162-l 68
163
interaction is a weak interaction which tends to produce longitudinally polarized strange quark-antiquark pairs that can hadronize into longitudinally polarized fast A particles [ 121. But the A and A in a low Q pair have essentially the same momentum in the laboratory and cannot be produced from a polarized SS pair emitted directly from the Z with equal and opposite lab momenta. Either one or both will be produced by a strong fragmentation process which does not favor a given helicity. Furthermore any polarization relative to the initial beam or initial jet direction is probably completely washed out in multiparticle fragmentation. The new ingredient here is the relative polarization of the two A’s; i.e. whether their spins are parallel or antiparallel. This can be measured by decay angular correlations and give information on the presence of resonances and on space-symmetry effects analogous discussed above for the C = f 1 parts of the bosonantiboson system [ 6,7]. In contrast to other analyses which assume a particular production mechanism for A polarization [ 11,121, our treatment holds for any production mechanism, does not assume any hyperon polarization in the laboratory system and gives results completely independent of any such polarization. Let us consider a AA pair in its center of mass system. For large Q values, the maximum radial angular momentum emax is larger than zero and several J = L + S are present, where S = Sr + S2 is spin sum of the two hyperons. However for a sufficiently small Q and near zero, where a Bose-Einstein enhancement occurs for identical bosons, the s-wave (l = 0) prevails ’ . In fact, if &, = 0 then J = S. Whereas S = 0 is allowed for two identical fermions, the state S = 1 is forbidden by the Pauli exclusion principle which should be experimentally observed through the Q behavior of the AA system. In order to avoid the Pauli exclusion principle let us first consider a Al? system at, or very near, its threshold E,,i = rnA + ml\, being in a pure f? = 0 state. We will further select the simple measurable case where the A decays into p1 +T- and the h to p2 +rr+. Here pi stands for the decay proton or anti-proton both of which we further will refer to as protons. Let us denote by y the cosine of the angle (in space) between p1 and p2 in the AR center of mass system and (y) its average. ’ From angular momentum barrier consideration.
164
G. Alexander, H.J. Lipkin /Physics
In the decay of a single polarized A, the proton angular distribution in the A center of mass system, is given by [ 131 dw/dcos8
c( 1 -LX,, cost9
(1)
where the parameter (Y~, arising from the parity violation, was measured experimentally [ 141 to be LX,,= , ;s;;t$fs,‘,Z
=0.642
+ 0.013
(2)
Here fs and fp are the s-wave and p-wave decay amplitudes. Since CZ~= -(YA one has ffi = -0.642
* 0.013
Setting x = cos 0 then the average 277
1 (xi\> = 27rN
xm
J J d+
0
(XA) is given by
Letters B 352 (1995) 162-168
For the case where angles are also measured with respect to another external axis; e.g. the normal to a production plane, such an angular distribution can have a quadrupole component or tensor polarization when the two spins of l/2 are coupled to spin 1 and are aligned with respect to this axis. Such tensor polarizations are observed, for example in Ai production by low-energy pp annihilation [ 111. Since we average over all orientations with respect to any fixed axis, only the scalar product of the two dipoles survives. To calculate the relation between (y) and the proton decay cosine averages, (XA) and (xi) of each of the hyperons 2 , we utilize the relation for the matrix elements of any vector operator V between states of the same total angular momentum J following from the Wigner-Eckart theorem [ 151,
(V . J)(J?WIJ, M’)
(.I, MIVIJ, M’) = (1 - cuAx)xdx
J(J+
(3)
I>
Let Pi and Pz be the unit vectors in the direction of flight of the pi and p2 protons respectively. Then, (XI\), averaged over the whole angular range, is equal to
Xrnill
where Xm8.X
N=
J
(1 -a,,x)dx
(4)
(6)
Xi”,” In particular the average over the whole x range i.e., XG” = -1 and .rrnax = +l, yields
and (xi) is given by (7)
(xA) = -0.214
l 0.004
and
(x;;;} = +0.214 & 0.004
We now note that general arguments do not allow dN/dy distribution of a two spin-112 hyperon system at threshold to have a y dependence higher than its first power. It is essentially the same argument that prevents a spin-l /2 particle from having a quadrupole moment. The distribution of the angle between the proton momenta from the two hyperons can depend only upon the dipole moments of the individual angular distributions. It must be also be a scalar under rotations in the laboratory, since there is no external reference frame here. Thus the only allowed nontrivial distribution is the scalar product of the two dipoles which is proportional to y. This means that dN/dy has the form dN/dy=A.[l+Bs.y] The factor Bs can then be determined of (YL
from the value
From Eqs. (6) and (7) follows that the average (y) of the cosine angle between the two decay protons: (Y) =
vl *pz)=
(4
. Sl) (Sl * S2)
&(SI
+
l)Sz(S2
(9
. S2) +
(8)
1)
From this one finds that (Y) R = (X*)(Xx)
=2S(S$l)
(fi . pz) = (P,,,)(P2,z)
- 3
= 4.
lS1 . s2)
(9)
The total spin S can have one of the two possible values, S = 0 and S = 1 so that -3 for S = 0 (Y> RQ=” = (x,,)(xA) = { -TV for s = 1 2 Note that at Q = 0 the individual of the two hyperons system.
hyperon CMS is equal to that
G. Alexander, HJ. Lipkin /Physics
Due to the Pauli exclusion principle, at e = 0 only S = 0 is allowed for the AA system and therefore one expects
R(Q)
‘so
RQ4
= -
(Y)
w
=
(Y) = 0.0458
-
3
165
Letters B 352 (1995) 162-l 68 2,
1.8
L
(10) x1.2
This limit can be experimentally verified by studying the ratio R(Q) as function of Q. Note that in the measurement of this limit there is no need for a reference sample as is the case in the study of the n-r and KK Bose-Einstein correlations. We thus find that for S = 0 one has for AA system
9 =;l f CO.8
0.6
0.4
B. = -9(x,,)*
L
= -0.4122 0.2 1
and for S = 1 01
Bi = +3(x*)2
As a consequence, one has for the AA system at, or very near, its threshold dN/dyI,,
0: 1 - 0.4122.
For comparison Pauli principle)
y
(11)
the C = 0, S = 1 state (forbidden by the should have the angular distribution
dN/dyl,=, 0: 1 + 0.1374.
y
(12)
These two distinctly different distributions, which are shown in Fig. 1, are derived at the limit of Q = 0 but as will shown in the next section are also valid for larger Q values. Finally for the Ah system one has 0: 1 + 0.4122.
y
dN/dyI,=, 0; 1 - 0.1374.
y
dN/dyl,,
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
= +0.1374
and
Fig. 1. The distribution of dN/dy as a function of v of the Ah system with a total spin of S = 0 and S = 1. These distributions are compared to a flat one shown by the dotted line.
The AA and AA wave functions in the center-ofmass system can be expressed in an “L - S-coupling” basis where the spins of the two baryons are coupled to a total spin S, and that this spin is coupled with some orbital angular momentum L to a total angular momentum J. For such a wave function we can immediately apply the Wigner-Eckart theorem Eq. (5) for any vector operator V and generalize Eq. (8) to apply to any two vector operators depending only upon the spin variables St and Sz: t 14)
(13)
3. Q dependence 3.1. Extension of results to finite Q values Experimentally one does not expect to have any significant number of AA events very near to EnA = 2nz~ unless there exists a AA resonance just above threshold. However, we shall now show that Eq. (8) can be applied directly to events with finite Q values in the region where the relative momentum of the two A’s is nonrelativistic.
This result is completely independent of the orbital angular momentum L and the total angular momentum J. In order to use Eq. (8) for this case, we note that the quantities (XA) and (xX) are defined in the rest frame of the decaying A and similarly for the proton momenta, Pr and Pa. However, in Eq. (14) all vectors are defined in the center-of-mass system. We shall now show that as long as the relative motion of the two baryons is nonrelativistic we can ignore these differences between frames, determine the experimental value of (y) s (PI .4) from the values for each proton momentum taken in the rest frame of its own A,
166
G. Alexander, H.J. Lipkin/Physics
and use this value in Eq. (9) and the known values of (XI\) and (XI\) with no boost corrections. We first note that Eqs. (6) and (7) hold also when the definition of Pr and Pz is generalized to be any unit vectors in the directions of some linear combination of the pion and proton momenta in the rest frame of the individual A and/or A respectively. When the baryon is moving, the pion and proton momenta are no longer back-to back, but one suitable combination of the pion and proton momenta can be found to define (x1\) and (xii) for moving baryons and obtain the same value as in the rest frame. For totally nonrelativistic motion, the correct vector is the pion-proton relative velocity for each A decay, which would be the same in the center-of-mass system of the two A’s as in the rest system of each A. A nonrelativistic boost changes the velocities of the decay proton and decay pion, but not their relative velocity. The momentum of the decay pion is definitely relativistic, even though the momenta of the A and decay proton may not be. However, one can still define a relative velocity vector which gives the correct value for (xi\) and (xX). When a A and its decay proton and pion are boosted by a velocity /3, the momenta of the decay proton and decay pion in the new frame are given to first order in p by the Lorentz transformations P,, (P> = P,,(O) - P&(O)
+ Q(P2)
(15)
P,(P)
+ CYP2)
t 16)
= P,(O)
- P&(O)
The “relative velocity” linear terms in p,
can be defined to cancel the
P,(P) PAP) ~,l(P) = - &CO) &r(O) =--pp (0) EP,(O) + W2) E,>(o) n =
&l(O) +
W)
Letters B 352 (1995) 162-168
We can now use the relative velocities for the decays of the two A’s as the vectors VI and V2 in Eq. (14) which holds in the center-of-mass system of the two A’s. But we now note that relative velocities as defined by Eq. (17) do not change to first order in p under a Lorentz boost to the rest frame of each A. We can evaluate the scalar products (5 . I$), (VI . Sl) and (Vz . 82) using the rest frame of each A for its corresponding relative velocity. But in the rest frame of each A a unit vector in the direction of the relative velocity is the same as a unit vector in the direction of the proton momentum. Thus we again recover Eq. (8) now for the general case of finite Q value and all values of orbital angular momentum L and total angular momentum J. Eq. (8) can be used directly for states with definite spin and finite Q-values, with the value of (y) s (PI . Pz) determined from the values for each proton momentum taken in the rest frame of its own A. The results for (y) and R depend only on the total spin S of the two A’s in their center of mass system and are completely independent of their relative motion in space and their orbital angular momentum. 3.2. Measurement of the spin correlation We now have a clear way to differentiate between the S = 0 and S = 1 contribution to the AA (Ah) for all values of Q where the relative motion is nonrelativistic. A straightforward way to achieve it is by fitting the sum of the expected dN/dy distributions for S = 0 and S = 1, given in Eqs. (11) and (121, to the experimental distribution at a given Q range. In this way the relative contributions Fo and FI from the S = 0 and S = I respectively will be determined. Next one can define the two ratios: co =
2Fo Fo + 6 /3
(171
The relative velocity defined in this way is exactly equal to the usual relative velocity in the nonrelativistic limit where Ep(0) and E,(O) are just the proton and pion masses. In the nonrelativisitic limit the relative proton-pion velocity is independent of the velocity of their center-of-mass. Our relativistic generalization (17) has the same property of being invariant under Lorentz boosts to first order in the velocity 0.
and
C = 2Fll3 ’ Fo+F,/3
where F, is divided by 3 to offset the 2S+ 1 factor. At high Q values, where C,,, is large, one expects CI g C2 = 1. At Q x 0, due to the exclusion principle, Co = 2 whereas Ct = 0. Thus CO behavior as a function of Q is similar to the @@ state originating from a l@Ko system whereas Ct behaves like the @K”L pair. The dependence of CO and Ct on Q, given in arbitrary units, are shown schematically in Fig. 2. The behavior of Ce( Q) may then be fitted to an expression,
G. Alexander, H.J. Lipkin / Physics Letters B 352 (1995) 162-168
Fig. 2. A schematic view of the correlation functions Co and Cl as a function of Q for the S = 0 part and S = 1 part respectively for the AA system. The dotted line represents the expectation for a Ax system in the absence of resonances.
used previously in the study of two identical correlations, of the type Co(Q) = (1 +
/KQzR+
bosom
(18)
where Rs may be interpreted as an estimate of the two hyperons emitter dimension and the parameter A, the strength of the effect, should be equal to 1.
4. Low energy dibaryon spectroscopy The possibility of bound states or low-mass resonances in both AA and AA systems have been suggested theoretically and investigated experimentally, so far with inconclusive results. The study of the invariant mass spectra and spin correlations of pairs produced in Z” decay can give additional information. In the framework of QCD derived models, the existence of dibaryon states (apart from the deuteron) has been discussed. A AA state, referred to as the dibaryon H, has been proposed as early as 1977 [ 161 and examined in various analyses (see e.g. Refs. [ 1620] ) . It should have a mass not far below or above its threshold of 2mA. So far no conclusive experimental evidence has been found for its existence. However,
167
it is only recently that large samples of events exists which do contain a AA or Ai pairs in their final state. If this H state is above the AA threshold, it may be detected directly by looking at the invariant mass distribution of the AA system. At the same time, if it is as expected an !! = 0, S = 0 state, it will not affect the behavior of (y) as a function of Q and its limit as Q --t 0. The AA system has been investigated recently in low-energy pp annihilation experiments, with the result that Ah production is dominated by the triplet spin (S = 1) state [ 111. It is not clear whether this spin correlation arises from the dynamics of the AA interaction or from the reaction mechanism; e.g. kaon exchange. This ambiguity can be resolved by examining the spin correlation for pairs produced in Z” decay where the reaction mechanism is completely different. If the same S = 1 dominance also occurs here it would support the suggestions of a low-lying resonance just above threshold.
5. Summary The method described here allows a measurement of the spin content of two spin 1I2 hyperons or a pair of hyperon and antihyperon samples irrespective of the orbital angular momentum. This can be utilized in the study of the exclusive reactions like pp + Ax and p jj -+ Z+p in order to verify the presence of s-channel resonances and to investigate the reaction mechanism. In the search and study of high mass resonances, like the .I/ti -+Ax, the final spin state of the hyperon-antihyperon is determined by the spin-parity values of the resonance as long as its decay does conserve parity. Thus the measurement of the spin state of the hyperon-antihyperon can serve as an additional tool in establishing the identity of the resonance and its properties. In multi-hadron final states at high energies, like the Z0 decay into multi-hadrons, the question of fragmentation and hadronization are handled in different models and compared to data via Monte Carlo calculations. An important check as to the validity of these models can by obtained from measurements of the typical hadronic emitter dimension. Such an information was extracted from the Bose-Einstein Correlation studies of identical bosons, mainly the r*&
168
G. Alexander, H.J. Lipkin /Physics
systems. Already the large existing statistics of Z” decays accumulated at the LEP experiments presents a sizable sample of multi-hadrons final states which contain pairs of identical AA and n hyperons. Utilizing the method outline here, they can be analyzed in terms of their total spin content as a function of the variable Q. The corresponding correlation functions Co(Q) and Cl (Q) will allow an estimate of the typical emitter dimension of two hyperons which in turn can then be confronted with the various fragmentation and hadronization models.
References III G. Goldhaber et al., Phys. Rev. Lett. 3 (1959) 181; Phys. Rev. 120 (1960) 300. [21 See e.g., M.G. Bowler, Particle World (Gordon and Breach Science Publishers) 2 ( 1991) 1. 131 M. Gyulassy et al., Phys. Rev. C 20 (1979) 2267; W. Hofmann, A Fresh Look at Bose-Einstein Correlations, LBL 23108 (1987). [41 W.A. Zajc, in Hadronic Multiparticle Production, P Carruthers, ed., World Scientific ( 1988) 235; M.I. Podgoretskii, Sov. J. Part. Nucl. 20 (1989) 266. I51 For a recent concise summary of BEC results of like-sign charged pions see e.g., S. Marcellini, Proc. of the Joint Int. Lepton-Photon Symp. and EPS Conf. on HEP, Geneva, 25th July-1st August 1991 (eds. S. Hegarty et al.), Vol. I, page 750. 161 G. Alexander, Bose-Einstein Effect in the K”? BosonAntiboson System, presented at the Workshop DAQNE and Other topics in Particle Physics, 18-20 November 1992, Frascati, Italy, LNF Preprint LNF-93/001 (p); G. Alexander, Bose-Einstein Correlation Effect in a BosonAntiboson System, Proc. of the Int. Conf. on Bose and the 20th Century Physics, 30 December-5 January 1994, Calcutta, India (in print).
Letters B 352 (1995) 162-168
[7] A detailed handling of the Ku? probability amplitudes in terms of symmetric and anti-symmetric states is given by H.J. Lipkin, Phys. Rev. Len. 69 (1992) 3700; see also H.J. Lipkin, Phys. Lett. B 219 (1989) 474; Argonne report ANL-HEP-PR-88-66. (81 OPAL Collaboration, PD. Acton et al., Phys. Lett. B 298 ( 1993) 456. [9] DELPHI Collaboration, P Abreu et al., Phys. Lett. B 323 ( 1994) 242. [ 101 ALEPH Collaboration, D. Buskulic et al., Z. Physik C 64 (1994) 373. [ 1I] See e.g., PD. Barnes et al., Nucl. Phys. A 558 (1993) 277~; N.H. Hamann. Nucl. Phys. A 558 (1993) 287~. [ 121 G. Gustafson and J. Hakkinen, Phys. Lett. B 303 (1993) 350. [ 131 See e.g., E. Segr?, Nuclei and Particles, 2nd Edition (Benjamin/Cummings Publishing Co.) p. 876. [ 141 Particle Data Group, Phys. Rev. D 50 (1994) 1173. [ 151 E.P. Wigner, Z. Physik 43 (1927) 624; C. Eckart, Rev. Mod. Phys. 2 (1930) 305. [ 161 R.L. Jaffe, Phys. Rev. Lett. 38 (1977) 195. [ 171 S.A. Yost and C.R. Nappi, Phys. Rev. D 32 (1985) 816. [ 181 M.A. Moinester, C.B. Dover and H.J. Lipkin, Phys. Rev. C 46 (1992) 1082. [ 191 D. Pal and J.A. McGovern, J. Phys. G: Nucl. Part. 18 ( 1992) 593. [20] J.L. Ping et al., Proc. of the XIII Int. Conf. On Particles and Nuclei, 28 June-2 July 1993, Perugia, Italy, p. 526; in the same proceedings see also: S. Paul, p. 646.