- Email: [email protected]
Angular correlations at some multiparticle annihilation of slow antiprotons
Volume 6, number 3
PHYSICS
LETTERS
ing energy. Errors quoted for each run are uncertainties derived from fittin the observed X-ray line to a Gaussian shape 5p’). These errors reflect the effects of drifts which reduced the quality of some of the runs more than in others. Accordingly, each measurement was assigned a weight taken to be the reciprocal of the sums of the squares of the errors quoted in columns 2 and 3 of table 2. The weighted average (including a ‘770correction for the isotopic abundances in the two samples) is K,(Ca44) - Ko(Ca40) = 4.8 + 2.0 keV, where the error quoted is the weighted rms deviation of the weighted mean. The annihilation radiation observed in the X-ray spectra provided an opportunity for additional checks on the detecting system. When the difference between the positions of this line in the two samples was analysed in the same way as for the K, line, the weighted mean difference obtained was 0.7 rt 3.0 keV. This null result adds to our confidence in the difference observed for the K, lines. A systematic error would be introduced if gamma rays due to nuclear interactions with the muons should fall in the KoI energy region. However, when the muon gate was widened and a suitable delay was inserted in order to search for such capture gamma
15 September 1963
rays in natural calcium, no lines intense enough to cause significant error were observed in the region around 790 keV. No such search was made for Ca44, but examination of the level structures of the nuclei that would be formed suggests that no error should arise from this source. Our result makes it seem unlikely that the charge radius of Ca44 is greater than that of Ca40. We are extremely grateful to R. Gabriel and W. Stanula for their invaluable help with the electronics during the experiment. We appreciate the assistance of J. Peregrin and Carol Hagaman in preparing the calibration sources and analysing the data.
1) H. L.Anderson, C S. Johnson and E. P. Hincks, Phys. Rev. 130 (1963) 2468. This paper contains references to previous work. 2) C.C.Trail and S.Rahoy, Rev. Sci. Instr. 30 (1959) 425. 3) E.H. Moltehn Jr., to be published. 4) D.R.Alburger, Phys. Rev. 92 (1953) 1257. 5) J. E.Monahan, S.Raboy and C. C. Trail, Phys. Rev. 123 (1961) 1373. 6) R. T. Julke et al., Argonne National Laboratory Report ANL-6499, unpublished.
*****
ANGULAR
CORRELATIONS OF
AT SOME MULTIPARTICLE SLOW ANTIPROTONS
J. WERLE, Institute for Nuclear
ANNIHILATION
St. DYMUS and T. ZIELINSKI Research
and University
of Warsaw,
Poland
Received 26 August 1963
Several highly interesting features of strongly interacting particle systems have been disclosed in the last few years by studying various multiparticle processes l). Many new resonances are already well established and still more of them suspected. Nevertheless, all this is only the very beginning of more systematic and fuller studies of multiparticle processes. In fact only a few simplest theoretical arguments were used up till now at the analysis of the experimental data. They consist mainly in comparing the invariant mass distributions with the corresponding phase space expectations and in the use of various selection rules. This type of reasoning allows frequently to establish not only the existence but also some basic properties of isolated resonances treated like unstable particles. However, several important problems concerning the multi264
particle processes themselves or the conditions under which the resonances are created together with other particles cannot be investigated with the help of the mentioned methods alone. It is obvious that the studies of angular correlations and polarisations can supply further valuable information about the mechanism of multiparticle processes. Angular correlations should be also rather easy to measure. They are, of course, much more involved than in the case of binary reactions and one must know how to look for them and how to interprete them in terms of some definite quantum numbers. It is quite impossible to give a universal and useful prescription for all feasible types of multiparticle processes. One can only propose several approaches which may be useful
Volume 6, number 3
PHYSICS
when applied to some restricted classes of multiparticle processes. Thus in ref. 2, relativistic partial wave expansions for processes of the type a+b -I l+...
+N= (l+...
+n) + (l’+...
+n’)
(1)
are obtained on the basis of a formal division of the final N particle system into two subsystems A and A ’ comprising n and n’ particles respectively (72+ n ’ = N 2 3). For the sake of brevity we shall write down the corresponding partial wave expansion only for the special case: n = 3, n’ = 2. The result can be then easily extended to other numbers of particles in the subsystems. We shall use three different coordinate systems: 1. XYZ system attached to the overall centre of mass frame, 2. xyz attached to the c.m. frame of the subsystem A, 3. x ‘y ‘Z ’ attached to the c. m. frame of the subsystem A’. The systems of axes xyz and xly % ’ are obtained from XYZ by performing the Lorentz transformations R (C 6 - @L,(V) and R(c 8 - @)L,(u’) respectively. Here R(@B -@) denotes a rotation through the Euler angles C, 6 which are the polar angles of the resulting momentum p of the subsystem A. The symbols L,(v) or L_Z(v’) denote the special Lorentz transformations paralellel or antiparallel to t$e 2 axis with velocities ZI =p(p2+,2)-z or v’ =p@2+2~~2)-Z respectively. Thus the systems xyz and x’y ‘z ’ move with respect to XYZ in opposite directions with all three pairs of axes correspondingly parallel. The linear momenta and polarizations of the final particles will be described by the following set of quantum numbers (A)
Eo, r=(a,p,Y),
X=(X1+2),
h=(Xl,X2,i3),
(A’ )
w’ 3 w=((p,@, Xf=(Xi,Xb),
(A+A*) p=o,
w=(p2+2$)~+(p2+~72)~,
(2) (3)
rr=(cp,e).
(4)
The symbols w , w ’ and W denote the invariant masses of the subsystem A,A’ and the total system (A +A’). The quantum numbers (2), (3) and (4) are referred to the xyz , x'y ‘2 ’ and XYZ systems of axes respectively. Thus Y are the Euler angles which specify the orientation in xyz of the plane of the final momenta kl, k2, kg of the particles A with respect to some standard orientation coinciding in our case with the xy plane 2). Similarly 50, f3are the polar angles of the momentum k’l(= -k’2) of the particle 1’ in the xyz system. The quantum numbers x specify the energies and relative angles between the momenta ki. The Xi or X’j denote the individual particle helicities, again referred to the xyz or x’y’z’ systems respectively. Now we expand our three- and two-particle momentum helicity eigenvectors for the subsystems A and A’ in their respective c. m. frames into the
LETTERS
15 September 1963
angular momentum s , p or s ’ , cc’ eigenvectors according to the procedures given in refs. 374). Next we perform the same Lorentz transformations on these substates as before at the construction of the xyz and x’y’z’ coordinate systems. The resulting products of two quasi-particle states can be once more expanded into the total angular momentum J,M eigenstates again using the Jacob and Wick formula 4). Thus we obtain the following expansion for the individual linear momentum helicity eigenvectors labelled by the quantum numbers (2-4)
2
1
JMSS
pv/A’
= 2n
)P=oIdw;rwx~;ww’~‘)
NJNsNs,
x Di&4I(Q) D;&-) D$lv4w) X (P=OMJ~‘;vswxX;s’w’X’)
(5)
with Nk2 = (2k + l)/ 8n 2. According to our conventions the quantum numbers ~1and g’ have the meaning of helicities of the subsystems A and A’ respectively. The component of the spin vector s in the direction normal to the $ plane is denoted by the symbol v . We define all the helicities ~1, M’; ~1, X and X’ in a Lorentz invariant manner so that they coincide with the usual helicities if referred respectively to the xyz, x’y’z’ or XYZ frames (for details see ref. 2)). After separating out the energy momentum conservation 6 we obtain the following partial wave expansion for the S matrix elements which describe the process (1) (0; rwxh; WW’X I S(w) 100X&) = (27i)i
2 Jss’w’v
$
N,
Ns
L$&
x &A’; vswx)c; S’W’X 1Q(w) 1A&)
S* D,,(y)
,
Qv’ s’*
(w 1
(6)
v’ = xl’- x2’. It is to where c = X, - $,, x = p-p’, be noted that the spins of individual particles are not coupled in our procedure so no Clebsch-Gordan coefficients occur in the expansion (6). Further more, due to the use of Lorentz invariant helicities we have avoided cumbersome spin transformation functions. Although the expansion (6) is quite rigorous it is most useful if the formal division.into two subsystems reflects some essential differences between them, e.g., if one of the subsystems is a resonance. We have applied expansions of this type for obtaining angular correlations in several m ltiparticle annihilations of slow antiprotons 2 95 73. Here we state briefly some simple results for the following three processes P+p’(K+m
+(~++n-+no),
fs+d-(A+Ko)+(,++.-+.o),
(7) (8) 265
PHYSICS
Volume 6, number 3 p + d - p + [7r- + (n+ + rr- + no)] .
+ lSll\ 2
+ (IS1012 - IS1112~~~&
9
(10)
where SJl L11 are the S matrix elements corresponding to the ‘indicated values of J and 1/A1 and some fixed values of the continuous parameters W,w ,ZU’ and x. For the processes (8) and (9) the mentioned approximations reduce the angular correlations to the following simple form c + d sin28 .
(11)
However, the expressions for c and d in terms of the S matrix elements are now more involved. We shall quote here only the simpler result for the process (9) for which 6) *****
266
15 September 1963
c = IsPI2 + 4132
(9)
The way of division into two subsystems is indicated in each case by parentheses. After having taken account of the conventional selection rules due to the space parity, charge parity and isospin conservation laws, we make several simplifying assumptions. First we take the low energy limit, i. e. , we assume the annihilation to take place nearly at rest. Then we assume that the three pions in small parentheses For the reaction are in a l-- state (w resonance). (7) this seems to be well justified by the experimental data 7). Next we keep tentatively only a few lowest permissible values of the orbital momentum of the pair which forms the subsystem A’ in (7) and (8). Thus we take 1 ’ = S’ = 0.1 for the m pair in (‘7) and I ’ = 0 for the h K pair in (8). The calculat$ns with 1’ = 1 have been actually performed in ref. but the results are rather involved and will not be given in this note. Somewhat different division for the process (9) corresponds to the situation in which the final proton is also nearly at rest and thus may be expected to play the role of a “spectator” in the simpler but experimentally inaccessible reaction lj + n -) 7r- + w. According to this assumption we take the lowest value of the relative orbital momentum L’ = 0 of the proton with respect to the nwsubsystem. Only the lowest permitted value l = 1 of the relative orbital momentum of the nwpair has been kept. We state here the results obtained under these restrictions for un polarized initial beams. For the process (7) one gets the following angular distribution ISOl\ 2 sin28 sin28 sin2(0!*)
LETTERS 2
2
2
2
a + IQ2 - Isfla - Is?1 2 7
(12)
d = ,Si,2
where the index j in S,? indicates the value of the resulting angular momentum of the VW system. It is interesting to note that the angular correlation for the basic process p + n _ IT-+U will be obtained from (11) only after neglecting the elements SL’ and S $ 2 which are not allowed for this procesg. Ther; is, however, no convincing reason for neglecting these elements in the process (9) in which the correlation between the spins of the nucleons in the deuteron prevents the proton from being a pure spectator. In order to increase statistics one can perform integration over some suitable ranges of the variables zu,TV’, W and X which are the arguments of the correlation coefficients. The general form of the angular distribution will obviously not be changed by such an integration. If our formulae do not fit the experimental data it can only mean that our restrictions concerning the quantum numbers of the final state were too severe. One may try then to improve the agreement by adding the next permissible values of the relative momenta of the subsystem A’. Another possible source of disagreement may lie in some admixture of other states of the three pion system. Comparing the experimental data with the theoretical expectations one can estimate the degree of such an admixture. The angular correlations (10) and (11) are comparatively simple. However, we should like to stress once more that in order to see them one must perform a few Lorentz transformations as the angles Y, wand h have to be measured in different inertial systems. More details concerning the angular correlations in the processes (7-9) including some obtained for polarized targets will be given in refs. 2, 5 26). Proc. of the 1962 Int. Conf. on High Energy Physics at CERN. 2) J. Werle, Angular correlations in some type of multiparticle processes, Nuclear Phys., to be published. 3) J.Werle, Physics Letters 4 (1963) 12’7; Nuclear Phys. 1) E.g.,
4) 5) 6) 7)
44 (1963) 579, 637. M. Jacob and G. C.Wick, Ann. Phys. 6 (1959) 404. St.Dymus, to be published. T. Zielinski, to be published. R.Armenteros et al., ref. l), p. 90.
PHYSICS
LETTERS
ing energy. Errors quoted for each run are uncertainties derived from fittin the observed X-ray line to a Gaussian shape 5p’). These errors reflect the effects of drifts which reduced the quality of some of the runs more than in others. Accordingly, each measurement was assigned a weight taken to be the reciprocal of the sums of the squares of the errors quoted in columns 2 and 3 of table 2. The weighted average (including a ‘770correction for the isotopic abundances in the two samples) is K,(Ca44) - Ko(Ca40) = 4.8 + 2.0 keV, where the error quoted is the weighted rms deviation of the weighted mean. The annihilation radiation observed in the X-ray spectra provided an opportunity for additional checks on the detecting system. When the difference between the positions of this line in the two samples was analysed in the same way as for the K, line, the weighted mean difference obtained was 0.7 rt 3.0 keV. This null result adds to our confidence in the difference observed for the K, lines. A systematic error would be introduced if gamma rays due to nuclear interactions with the muons should fall in the KoI energy region. However, when the muon gate was widened and a suitable delay was inserted in order to search for such capture gamma
15 September 1963
rays in natural calcium, no lines intense enough to cause significant error were observed in the region around 790 keV. No such search was made for Ca44, but examination of the level structures of the nuclei that would be formed suggests that no error should arise from this source. Our result makes it seem unlikely that the charge radius of Ca44 is greater than that of Ca40. We are extremely grateful to R. Gabriel and W. Stanula for their invaluable help with the electronics during the experiment. We appreciate the assistance of J. Peregrin and Carol Hagaman in preparing the calibration sources and analysing the data.
1) H. L.Anderson, C S. Johnson and E. P. Hincks, Phys. Rev. 130 (1963) 2468. This paper contains references to previous work. 2) C.C.Trail and S.Rahoy, Rev. Sci. Instr. 30 (1959) 425. 3) E.H. Moltehn Jr., to be published. 4) D.R.Alburger, Phys. Rev. 92 (1953) 1257. 5) J. E.Monahan, S.Raboy and C. C. Trail, Phys. Rev. 123 (1961) 1373. 6) R. T. Julke et al., Argonne National Laboratory Report ANL-6499, unpublished.
*****
ANGULAR
CORRELATIONS OF
AT SOME MULTIPARTICLE SLOW ANTIPROTONS
J. WERLE, Institute for Nuclear
ANNIHILATION
St. DYMUS and T. ZIELINSKI Research
and University
of Warsaw,
Poland
Received 26 August 1963
Several highly interesting features of strongly interacting particle systems have been disclosed in the last few years by studying various multiparticle processes l). Many new resonances are already well established and still more of them suspected. Nevertheless, all this is only the very beginning of more systematic and fuller studies of multiparticle processes. In fact only a few simplest theoretical arguments were used up till now at the analysis of the experimental data. They consist mainly in comparing the invariant mass distributions with the corresponding phase space expectations and in the use of various selection rules. This type of reasoning allows frequently to establish not only the existence but also some basic properties of isolated resonances treated like unstable particles. However, several important problems concerning the multi264
particle processes themselves or the conditions under which the resonances are created together with other particles cannot be investigated with the help of the mentioned methods alone. It is obvious that the studies of angular correlations and polarisations can supply further valuable information about the mechanism of multiparticle processes. Angular correlations should be also rather easy to measure. They are, of course, much more involved than in the case of binary reactions and one must know how to look for them and how to interprete them in terms of some definite quantum numbers. It is quite impossible to give a universal and useful prescription for all feasible types of multiparticle processes. One can only propose several approaches which may be useful
Volume 6, number 3
PHYSICS
when applied to some restricted classes of multiparticle processes. Thus in ref. 2, relativistic partial wave expansions for processes of the type a+b -I l+...
+N= (l+...
+n) + (l’+...
+n’)
(1)
are obtained on the basis of a formal division of the final N particle system into two subsystems A and A ’ comprising n and n’ particles respectively (72+ n ’ = N 2 3). For the sake of brevity we shall write down the corresponding partial wave expansion only for the special case: n = 3, n’ = 2. The result can be then easily extended to other numbers of particles in the subsystems. We shall use three different coordinate systems: 1. XYZ system attached to the overall centre of mass frame, 2. xyz attached to the c.m. frame of the subsystem A, 3. x ‘y ‘Z ’ attached to the c. m. frame of the subsystem A’. The systems of axes xyz and xly % ’ are obtained from XYZ by performing the Lorentz transformations R (C 6 - @L,(V) and R(c 8 - @)L,(u’) respectively. Here R(@B -@) denotes a rotation through the Euler angles C, 6 which are the polar angles of the resulting momentum p of the subsystem A. The symbols L,(v) or L_Z(v’) denote the special Lorentz transformations paralellel or antiparallel to t$e 2 axis with velocities ZI =p(p2+,2)-z or v’ =p@2+2~~2)-Z respectively. Thus the systems xyz and x’y ‘z ’ move with respect to XYZ in opposite directions with all three pairs of axes correspondingly parallel. The linear momenta and polarizations of the final particles will be described by the following set of quantum numbers (A)
Eo, r=(a,p,Y),
X=(X1+2),
h=(Xl,X2,i3),
(A’ )
w’ 3 w=((p,@, Xf=(Xi,Xb),
(A+A*) p=o,
w=(p2+2$)~+(p2+~72)~,
(2) (3)
rr=(cp,e).
(4)
The symbols w , w ’ and W denote the invariant masses of the subsystem A,A’ and the total system (A +A’). The quantum numbers (2), (3) and (4) are referred to the xyz , x'y ‘2 ’ and XYZ systems of axes respectively. Thus Y are the Euler angles which specify the orientation in xyz of the plane of the final momenta kl, k2, kg of the particles A with respect to some standard orientation coinciding in our case with the xy plane 2). Similarly 50, f3are the polar angles of the momentum k’l(= -k’2) of the particle 1’ in the xyz system. The quantum numbers x specify the energies and relative angles between the momenta ki. The Xi or X’j denote the individual particle helicities, again referred to the xyz or x’y’z’ systems respectively. Now we expand our three- and two-particle momentum helicity eigenvectors for the subsystems A and A’ in their respective c. m. frames into the
LETTERS
15 September 1963
angular momentum s , p or s ’ , cc’ eigenvectors according to the procedures given in refs. 374). Next we perform the same Lorentz transformations on these substates as before at the construction of the xyz and x’y’z’ coordinate systems. The resulting products of two quasi-particle states can be once more expanded into the total angular momentum J,M eigenstates again using the Jacob and Wick formula 4). Thus we obtain the following expansion for the individual linear momentum helicity eigenvectors labelled by the quantum numbers (2-4)
2
1
JMSS
pv/A’
= 2n
)P=oIdw;rwx~;ww’~‘)
NJNsNs,
x Di&4I(Q) D;&-) D$lv4w) X (P=OMJ~‘;vswxX;s’w’X’)
(5)
with Nk2 = (2k + l)/ 8n 2. According to our conventions the quantum numbers ~1and g’ have the meaning of helicities of the subsystems A and A’ respectively. The component of the spin vector s in the direction normal to the $ plane is denoted by the symbol v . We define all the helicities ~1, M’; ~1, X and X’ in a Lorentz invariant manner so that they coincide with the usual helicities if referred respectively to the xyz, x’y’z’ or XYZ frames (for details see ref. 2)). After separating out the energy momentum conservation 6 we obtain the following partial wave expansion for the S matrix elements which describe the process (1) (0; rwxh; WW’X I S(w) 100X&) = (27i)i
2 Jss’w’v
$
N,
Ns
L$&
x &A’; vswx)c; S’W’X 1Q(w) 1A&)
S* D,,(y)
,
Qv’ s’*
(w 1
(6)
v’ = xl’- x2’. It is to where c = X, - $,, x = p-p’, be noted that the spins of individual particles are not coupled in our procedure so no Clebsch-Gordan coefficients occur in the expansion (6). Further more, due to the use of Lorentz invariant helicities we have avoided cumbersome spin transformation functions. Although the expansion (6) is quite rigorous it is most useful if the formal division.into two subsystems reflects some essential differences between them, e.g., if one of the subsystems is a resonance. We have applied expansions of this type for obtaining angular correlations in several m ltiparticle annihilations of slow antiprotons 2 95 73. Here we state briefly some simple results for the following three processes P+p’(K+m
+(~++n-+no),
fs+d-(A+Ko)+(,++.-+.o),
(7) (8) 265
PHYSICS
Volume 6, number 3 p + d - p + [7r- + (n+ + rr- + no)] .
+ lSll\ 2
+ (IS1012 - IS1112~~~&
9
(10)
where SJl L11 are the S matrix elements corresponding to the ‘indicated values of J and 1/A1 and some fixed values of the continuous parameters W,w ,ZU’ and x. For the processes (8) and (9) the mentioned approximations reduce the angular correlations to the following simple form c + d sin28 .
(11)
However, the expressions for c and d in terms of the S matrix elements are now more involved. We shall quote here only the simpler result for the process (9) for which 6) *****
266
15 September 1963
c = IsPI2 + 4132
(9)
The way of division into two subsystems is indicated in each case by parentheses. After having taken account of the conventional selection rules due to the space parity, charge parity and isospin conservation laws, we make several simplifying assumptions. First we take the low energy limit, i. e. , we assume the annihilation to take place nearly at rest. Then we assume that the three pions in small parentheses For the reaction are in a l-- state (w resonance). (7) this seems to be well justified by the experimental data 7). Next we keep tentatively only a few lowest permissible values of the orbital momentum of the pair which forms the subsystem A’ in (7) and (8). Thus we take 1 ’ = S’ = 0.1 for the m pair in (‘7) and I ’ = 0 for the h K pair in (8). The calculat$ns with 1’ = 1 have been actually performed in ref. but the results are rather involved and will not be given in this note. Somewhat different division for the process (9) corresponds to the situation in which the final proton is also nearly at rest and thus may be expected to play the role of a “spectator” in the simpler but experimentally inaccessible reaction lj + n -) 7r- + w. According to this assumption we take the lowest value of the relative orbital momentum L’ = 0 of the proton with respect to the nwsubsystem. Only the lowest permitted value l = 1 of the relative orbital momentum of the nwpair has been kept. We state here the results obtained under these restrictions for un polarized initial beams. For the process (7) one gets the following angular distribution ISOl\ 2 sin28 sin28 sin2(0!*)
LETTERS 2
2
2
2
a + IQ2 - Isfla - Is?1 2 7
(12)
d = ,Si,2
where the index j in S,? indicates the value of the resulting angular momentum of the VW system. It is interesting to note that the angular correlation for the basic process p + n _ IT-+U will be obtained from (11) only after neglecting the elements SL’ and S $ 2 which are not allowed for this procesg. Ther; is, however, no convincing reason for neglecting these elements in the process (9) in which the correlation between the spins of the nucleons in the deuteron prevents the proton from being a pure spectator. In order to increase statistics one can perform integration over some suitable ranges of the variables zu,TV’, W and X which are the arguments of the correlation coefficients. The general form of the angular distribution will obviously not be changed by such an integration. If our formulae do not fit the experimental data it can only mean that our restrictions concerning the quantum numbers of the final state were too severe. One may try then to improve the agreement by adding the next permissible values of the relative momenta of the subsystem A’. Another possible source of disagreement may lie in some admixture of other states of the three pion system. Comparing the experimental data with the theoretical expectations one can estimate the degree of such an admixture. The angular correlations (10) and (11) are comparatively simple. However, we should like to stress once more that in order to see them one must perform a few Lorentz transformations as the angles Y, wand h have to be measured in different inertial systems. More details concerning the angular correlations in the processes (7-9) including some obtained for polarized targets will be given in refs. 2, 5 26). Proc. of the 1962 Int. Conf. on High Energy Physics at CERN. 2) J. Werle, Angular correlations in some type of multiparticle processes, Nuclear Phys., to be published. 3) J.Werle, Physics Letters 4 (1963) 12’7; Nuclear Phys. 1) E.g.,
4) 5) 6) 7)
44 (1963) 579, 637. M. Jacob and G. C.Wick, Ann. Phys. 6 (1959) 404. St.Dymus, to be published. T. Zielinski, to be published. R.Armenteros et al., ref. l), p. 90.