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Production and decay of P wave charmonium states in p̄p collisions
Volume 147B, number 1,2,3
PHYSICS LETTERS
1 November 1984
P R O D U C T I O N AND DECAY OF P WAVE C H A R M O N I U M STATES IN Op C O L L I S I O N S A.D. MARTIN, M.G. OLSSON 1 and W.J. S T I R L I N G 2 Department of Physics, University of Durham, England Received 5 July 1984
We study the angular distribution of the cascade process ~p ~ X j --* de'/~ e+ e ' / a n d find that such data may offer tests of the QCD helicity selection rules and also determine the multipole structure of the Xj --" de'~ radiative decays.
G o o d momentum resolution in ~p collisions allows the formation of charmonium states which are not directly accessible in e +e- annihilation. In particular the four P-wave states, 1P1 and 3pj = Xj with J = 0, 1,2, can be directly formed and studied by their radiative decays [1]: I3p --* X j ---~tpy
L e + e -,
PP ~
1p1 ~
(1)
~cY
L~ 3,Y,
(2)
as well as the S wave states ~k(3S1) and 7/c(1S0) via ~p~k~e+e
,
pp--'7/~77.
We explore the general structure of the angular distribution of the decay products and indicate how physically relevant parameters may be extracted from the data. We find that the processes offer an excellent opportunity to study (i) the validity of the helicity selection rules [2], based on applying QCD perturbation theory at pre-asymptotic energies, and (ii) the multipole character of the P ~ S wave radiative transitions. It has been argued [2] that the helicity and angular dependence of large momentum transfer 1 Permanent address: Department of Physics, University of Wisconsin, USA. 2 Present address: Theory Division, CERN, Geneva, Switzerland.
exclusive processes are a sensitive test of the gluon spin and other basic elements of perturbative QCD. The ~p coupling to charmonium states has been advocated as a possible testing ground. The helicity selection rule [2] is a direct consequence. Q C D requires the annihilating (massless) q and ~1 to have opposite helicities, and this requirement feeds through to the annihilating p and ~. That is, annihilation takes place from states of net helicity = ( ) ~ - )~p) = +1. Annihilation from the )~ = 0 configuration is predicted to be suppressed by O(1/Q2). Thus an immediate prediction is that the J = 0 states X0, *lc, and also, (for the reason given below) the 1Pt state couple only weakly to ~p, and that the other states have the following angular decay distributions ~p~-oe+e-:
do/dcosOcx(l+cos20),
(3)
~p~xt,2~k-/:
do/dcosOcc(1-½cos20),(4)
where 0 is the centre-of-mass scattering angle between the incident and outgoing particle directions. The last result [3] assumes XJ ~ ~kY is a pure electric dipole (El) radiative transition. However there is evidence to suggest that the above picture is oversimplified. First the ~/c coupling to ~p is not suppressed; indeed, to within the experimental errors, the */c ---' PP and ~k~ PP branching ratios are approximately equal [4]. Moreover the angular distribution [5] for the reverse reaction e+e ----, ~k--' PP indicates the presence of a )~ = ~ - h v = 0 component, and
0370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
203
V o l u m e 147B, n u m b e r 1,2,3
PHYSICS L E T T E R S
1 N o v e m b e r 1984
m a y possibly suggest the dominance of the ~ ~ ~p electric form factor [6]. Another problem for the helicity selection rule is that the decay ~b ~ pTr is appreciable [7], whereas a suppression is predicted [2]. To use processes (4) to test the helicity selection rule is informative, but requires care. If, for example, for X 2 we allow for helicity k = 0 and _+ 1 (with amplitudes BIxl) then we obtain
,/ g
d o ( ~ p --, X2 ~ ~bv)/d cos 0
IBl1 (1 -
cos20) + ¢1B012(1 - } cos20), (5)
and to identify a B o component appears to require large statistics. However the Crystal Ball data [8] indicate a sizeable magnetic quadrupole ( M 2 ) contribution to the X2 - ' q~Y transition. We find that the ~ p --, X2 - ' q~Y data should provide a sensitive determination of the E t / M 2 admixture, which in turn should help with the test of the helicity selection rule. The complete angular distribution for the production and decay PP -~ X j ~
~Dy e+e -
is, following the notation of Karl et al. [9], J
W(O;O',dp')=~_,B~xl
~_, u,v'= -J
XA J as
~_, d S ~ ( O ) d ~ , ( O ) # = ++_1
_o'ota, ~,~
i~l,~lJ,,ip
I u ,~1,
(6)
where the ~b helicity a - 1, - # and o ' -= ~,'-/~, and the density matrix for the ~b decay into an unpolarized e + and e - is
Y'~ ~],~(q~:,0',-q¢)~,*(ff',0',-q¢).
Bxlx: = 7 h , ( - 1)JB_xl_x2 BXlX2 = " t i c ( - - 1 ) S B x 2 x l
(from parity), (from C conjugation),
where ~h, and */c are the parity and C parity of the charmonium state. It follows that B + _ = B_+ = 0 for 1P1 production and so it can only be produced with k = k l - ~ 2 = 0. In eq. (6) we have used BIA I ~ nk~x2 tO denote the two independent production amplitudes. Similar parity considerations permit the X j ~ ~by transition to be described by ( J + 1) independent helicity amplitudes A{ with p = 0,1 . . . . . J. These in turn can be expressed [9] in terms of amplitudes of definite multipolarity j = 1 . . . . , J + 1 2 j + 1 )1/2 ~ ( j , 1;1, v - l l J ,
v ).
(8)
J
(7)
~=il
The angles and helicities are indicated in fig. 1; 0', q~' specify the ~b ~ e+e - decay in the ~k rest frame with z axis aligned with the ff direction in the X rest frame. If we denote the ~, p helicities by X1, ~ 2 then the amplitudes Bxlx~ describing ~p ~ Xj produc204
tion satisfy
A~=Y'~aj
¢'°(0',~')
=
Fig. 1. Symbolic sketch o f ~ p ~ XJ "" ~/'Y ~ e ÷ e - y showing the particle helicities; note o = 1 , - ~. The production of PP ~ Xj is described by amplitudes BbxI and the decay X j --* ~Y by AI~I.
For instance, for X2 ---' ~PY the amplitudes a 1, a2, a 3 correspond to E 1, M 2 and E 3 radiative transitions. A single quark transition (IALI = 1, IASI < 1) prohibits [10] an E 3 contribution to X2 ~ ~kY and so we set a 3 = 0 f r o m n o w o n . N o w the crystal ball data [8] determine the M 2 contribution to X2--* ~k7 to be +0.292
a 2 = _ (0.333_0.116),
(9)
whereas X1 --->~k3' is found to be an essentially pure
Volume 147B, number 1,2,3
PHYSICS LETTERS
E 1 transition [
+ 0.020
(10)
a 2 = - ~.0.002 _ 0.0o81.
1 November 1984
of the single angular distributions, obtained by integrating W ( O ; 8% @') over the other angles. If we write the ~p --, Xj --' 6Y distribution in the form
T h e convention is a 1 > 0 and the normalisation is such that
Wp(O) = Wp(½~r)[1 + OtCOS2O+ flCOS40],
la112 + la212 = 1.
then for X1 and X2 we have
(11)
D a t a on ~p P-wave c h a r m o n i u m production can be analysed in terms of the double decay distribution of eq. (6) to determine the fraction of IXl = 1 p r o d u c t i o n
(2 - 3 R ) ( 1 + 6 a , a 2 ) a ( X 1 ) = (2 + R ) + 2 ( 3 R - 2 ) a a a 2'
(13)
f l ( X l ) = 0,
"(x2) 3R - 6 + 4(13R - 6)a22 + 6¢5-(2 - R ) a , a 2
R - 2 B Z / ( B 2 + 2 B 2)
(12)
and, in the case of XL2, the precise E 1 , M 2 admixture. F o r illustration we give the structure
10 - R - 4RaZ9 - 2¢~-(2 - R ) a l a 2
B(x2) =
5 2 40(1 -3R)a 2
10 - R - 4 R a ~ - 2 ~ ( 2 - R ) a , a 2 '
a I
R=O (~=0 prod.)
R= T
R=I 0,=-+1 prod.)
1.0
i
1-0
0'8
0.8
=--
0.8
0-6
0.6
0.6
04,
O.Z,
04,
0.2
0"2
0-2
1"0
i=_1
.
I
1 0. 2 = - [
wp(e}
0 -I
0 -1
0
COS 8
£0S 8
I
0 -1
0
0
COS 0
b 1-0
i
0.8 0-6
wd81
04, C12=-~ 0.2
01 COS 8' Fig. 2. The angular distributions, Wp(0), f o r ~ p --, X2 "" q~Y in the X2 rest frame, and Wd(O' ) for ~p~ e+e various choices of the parameters R and a 2 of eqs. (12) and (8) respectively.
in the ~ rest frame for
205
Volume 147B, number 1,2,3
PHYSICS LETTERS
whereas the X0 distribution is isotropic. On the other h a n d for the ~k ~ e + e - decay distribution integrated over ~' (as well as 9) we obtain
wd(o') =
wd(l
)[1 + o , ' c o s 2 0 ' ] ,
(14)
with a ' for the various Xj states given by - 1 + 6ala 2
a ' ( X 0 ) = 1, =
a'(Xa)-
3- 2ala 2
1 + 4a 2 - 6v~ala 2
, 2 + 2v/~ala2 " 13 - ~a As expected Wd(0') depends only on the Xj ~ ~k7 multipole structure. Finally, for ~p ~ 1P1 ~ 7/c't we obtain eq. (13) with
1 November 1984
helicity zero state and for the importance of the magnetic quadrupole transitions in X2 ---' ~kT- The data from the I S R experiment [1] should be able to determine the multipole and helicity structure of this process. The helicity structure results in a direct test of the Q C D helicity conservation hypothesis, and moreover, the verification of a large magnetic quadrupole transition is of great interest in heavy quark spectroscopy, possibly indicating a large anomalous magnetic m o m e n t of the c h a r m q u a r k [11]. W e thank Dr. A. L u n d b y for several informative discussions concerning the ISR c h a r m o n i u m experiment. W e also thank the British Science and Engineering Research Council for support. References
a(1P1) = - 1 ,
1~(1p1) ~---0.
T h e recent I S R charmonium experiment [1] has collected the most data for the X 2 state and so in fig. 2 we show the spread of possible angular distributions, Wp(0) and Wd(0' ), for ~p ~ X2 ~k7 assuming different E 1 / M 2 admixtures (specified b y the parameter a2) and for X2 p r o d u c t i o n in different helicity configurations (specified b y the parameter R of eq. (12)). We see that the angular distributions should allow a good determination of a 2, and provided the data covers the forward (or backward) directions in 8 should yield a value for R. W e have indicated some of the results which m a y be obtained from direct production of c h a r m o n i u m states by ~p annihilation. There is already some evidence for production through a
206
[1] Annecy-CERN-Geneva-Lyon-Oslo-Rome-TurinCollab., ISR experiment, R704, A. Lundby, private communication. [2] S.J. Brodsky and G.P. Lepage, Phys. Rev. D24 (1981) 2848. [3] A. Andrikopoulou, Z. fur Phys. C 22 (1984) 63. [4] Mark III Collab., K.F. Einsweiler et al, Proc. HEP Conf. (Brighton, 1983) p. 348. [5] M.W. Eaton et al., Phys. Rev., D29 (1984) 804. [6] M. Claudson, S.L. Glashow and M.B. Wise, Phys. Rev. D25 (1981) 1345. [7] Particle Data Group, Rev. Mod. Phys. 56 (1984) $1. [8] M. Oreglia et al., Phys. Rev. D25 (1982) 2259. [9] G. Karl, S. Meshkov and J.L. Rosner, Phys. Rev D13 (1976) 1203. [10] F.J. Gilman and I. Karliner, Phys. Lett. 46B (1973) 426; Phys. Rev. D10 (1974) 2194. [11] G. Karl, S. Meshkov and J.L. Rosner, Phys. Rev. Lett., 45 (1980) 215; R. McClary and N. Byers, Phys. Rev. D28 (1983) 1692.
PHYSICS LETTERS
1 November 1984
P R O D U C T I O N AND DECAY OF P WAVE C H A R M O N I U M STATES IN Op C O L L I S I O N S A.D. MARTIN, M.G. OLSSON 1 and W.J. S T I R L I N G 2 Department of Physics, University of Durham, England Received 5 July 1984
We study the angular distribution of the cascade process ~p ~ X j --* de'/~ e+ e ' / a n d find that such data may offer tests of the QCD helicity selection rules and also determine the multipole structure of the Xj --" de'~ radiative decays.
G o o d momentum resolution in ~p collisions allows the formation of charmonium states which are not directly accessible in e +e- annihilation. In particular the four P-wave states, 1P1 and 3pj = Xj with J = 0, 1,2, can be directly formed and studied by their radiative decays [1]: I3p --* X j ---~tpy
L e + e -,
PP ~
1p1 ~
(1)
~cY
L~ 3,Y,
(2)
as well as the S wave states ~k(3S1) and 7/c(1S0) via ~p~k~e+e
,
pp--'7/~77.
We explore the general structure of the angular distribution of the decay products and indicate how physically relevant parameters may be extracted from the data. We find that the processes offer an excellent opportunity to study (i) the validity of the helicity selection rules [2], based on applying QCD perturbation theory at pre-asymptotic energies, and (ii) the multipole character of the P ~ S wave radiative transitions. It has been argued [2] that the helicity and angular dependence of large momentum transfer 1 Permanent address: Department of Physics, University of Wisconsin, USA. 2 Present address: Theory Division, CERN, Geneva, Switzerland.
exclusive processes are a sensitive test of the gluon spin and other basic elements of perturbative QCD. The ~p coupling to charmonium states has been advocated as a possible testing ground. The helicity selection rule [2] is a direct consequence. Q C D requires the annihilating (massless) q and ~1 to have opposite helicities, and this requirement feeds through to the annihilating p and ~. That is, annihilation takes place from states of net helicity = ( ) ~ - )~p) = +1. Annihilation from the )~ = 0 configuration is predicted to be suppressed by O(1/Q2). Thus an immediate prediction is that the J = 0 states X0, *lc, and also, (for the reason given below) the 1Pt state couple only weakly to ~p, and that the other states have the following angular decay distributions ~p~-oe+e-:
do/dcosOcx(l+cos20),
(3)
~p~xt,2~k-/:
do/dcosOcc(1-½cos20),(4)
where 0 is the centre-of-mass scattering angle between the incident and outgoing particle directions. The last result [3] assumes XJ ~ ~kY is a pure electric dipole (El) radiative transition. However there is evidence to suggest that the above picture is oversimplified. First the ~/c coupling to ~p is not suppressed; indeed, to within the experimental errors, the */c ---' PP and ~k~ PP branching ratios are approximately equal [4]. Moreover the angular distribution [5] for the reverse reaction e+e ----, ~k--' PP indicates the presence of a )~ = ~ - h v = 0 component, and
0370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
203
V o l u m e 147B, n u m b e r 1,2,3
PHYSICS L E T T E R S
1 N o v e m b e r 1984
m a y possibly suggest the dominance of the ~ ~ ~p electric form factor [6]. Another problem for the helicity selection rule is that the decay ~b ~ pTr is appreciable [7], whereas a suppression is predicted [2]. To use processes (4) to test the helicity selection rule is informative, but requires care. If, for example, for X 2 we allow for helicity k = 0 and _+ 1 (with amplitudes BIxl) then we obtain
,/ g
d o ( ~ p --, X2 ~ ~bv)/d cos 0
IBl1 (1 -
cos20) + ¢1B012(1 - } cos20), (5)
and to identify a B o component appears to require large statistics. However the Crystal Ball data [8] indicate a sizeable magnetic quadrupole ( M 2 ) contribution to the X2 - ' q~Y transition. We find that the ~ p --, X2 - ' q~Y data should provide a sensitive determination of the E t / M 2 admixture, which in turn should help with the test of the helicity selection rule. The complete angular distribution for the production and decay PP -~ X j ~
~Dy e+e -
is, following the notation of Karl et al. [9], J
W(O;O',dp')=~_,B~xl
~_, u,v'= -J
XA J as
~_, d S ~ ( O ) d ~ , ( O ) # = ++_1
_o'ota, ~,~
i~l,~lJ,,ip
I u ,~1,
(6)
where the ~b helicity a - 1, - # and o ' -= ~,'-/~, and the density matrix for the ~b decay into an unpolarized e + and e - is
Y'~ ~],~(q~:,0',-q¢)~,*(ff',0',-q¢).
Bxlx: = 7 h , ( - 1)JB_xl_x2 BXlX2 = " t i c ( - - 1 ) S B x 2 x l
(from parity), (from C conjugation),
where ~h, and */c are the parity and C parity of the charmonium state. It follows that B + _ = B_+ = 0 for 1P1 production and so it can only be produced with k = k l - ~ 2 = 0. In eq. (6) we have used BIA I ~ nk~x2 tO denote the two independent production amplitudes. Similar parity considerations permit the X j ~ ~by transition to be described by ( J + 1) independent helicity amplitudes A{ with p = 0,1 . . . . . J. These in turn can be expressed [9] in terms of amplitudes of definite multipolarity j = 1 . . . . , J + 1 2 j + 1 )1/2 ~ ( j , 1;1, v - l l J ,
v ).
(8)
J
(7)
~=il
The angles and helicities are indicated in fig. 1; 0', q~' specify the ~b ~ e+e - decay in the ~k rest frame with z axis aligned with the ff direction in the X rest frame. If we denote the ~, p helicities by X1, ~ 2 then the amplitudes Bxlx~ describing ~p ~ Xj produc204
tion satisfy
A~=Y'~aj
¢'°(0',~')
=
Fig. 1. Symbolic sketch o f ~ p ~ XJ "" ~/'Y ~ e ÷ e - y showing the particle helicities; note o = 1 , - ~. The production of PP ~ Xj is described by amplitudes BbxI and the decay X j --* ~Y by AI~I.
For instance, for X2 ---' ~PY the amplitudes a 1, a2, a 3 correspond to E 1, M 2 and E 3 radiative transitions. A single quark transition (IALI = 1, IASI < 1) prohibits [10] an E 3 contribution to X2 ~ ~kY and so we set a 3 = 0 f r o m n o w o n . N o w the crystal ball data [8] determine the M 2 contribution to X2--* ~k7 to be +0.292
a 2 = _ (0.333_0.116),
(9)
whereas X1 --->~k3' is found to be an essentially pure
Volume 147B, number 1,2,3
PHYSICS LETTERS
E 1 transition [
+ 0.020
(10)
a 2 = - ~.0.002 _ 0.0o81.
1 November 1984
of the single angular distributions, obtained by integrating W ( O ; 8% @') over the other angles. If we write the ~p --, Xj --' 6Y distribution in the form
T h e convention is a 1 > 0 and the normalisation is such that
Wp(O) = Wp(½~r)[1 + OtCOS2O+ flCOS40],
la112 + la212 = 1.
then for X1 and X2 we have
(11)
D a t a on ~p P-wave c h a r m o n i u m production can be analysed in terms of the double decay distribution of eq. (6) to determine the fraction of IXl = 1 p r o d u c t i o n
(2 - 3 R ) ( 1 + 6 a , a 2 ) a ( X 1 ) = (2 + R ) + 2 ( 3 R - 2 ) a a a 2'
(13)
f l ( X l ) = 0,
"(x2) 3R - 6 + 4(13R - 6)a22 + 6¢5-(2 - R ) a , a 2
R - 2 B Z / ( B 2 + 2 B 2)
(12)
and, in the case of XL2, the precise E 1 , M 2 admixture. F o r illustration we give the structure
10 - R - 4RaZ9 - 2¢~-(2 - R ) a l a 2
B(x2) =
5 2 40(1 -3R)a 2
10 - R - 4 R a ~ - 2 ~ ( 2 - R ) a , a 2 '
a I
R=O (~=0 prod.)
R= T
R=I 0,=-+1 prod.)
1.0
i
1-0
0'8
0.8
=--
0.8
0-6
0.6
0.6
04,
O.Z,
04,
0.2
0"2
0-2
1"0
i=_1
.
I
1 0. 2 = - [
wp(e}
0 -I
0 -1
0
COS 8
£0S 8
I
0 -1
0
0
COS 0
b 1-0
i
0.8 0-6
wd81
04, C12=-~ 0.2
01 COS 8' Fig. 2. The angular distributions, Wp(0), f o r ~ p --, X2 "" q~Y in the X2 rest frame, and Wd(O' ) for ~p~ e+e various choices of the parameters R and a 2 of eqs. (12) and (8) respectively.
in the ~ rest frame for
205
Volume 147B, number 1,2,3
PHYSICS LETTERS
whereas the X0 distribution is isotropic. On the other h a n d for the ~k ~ e + e - decay distribution integrated over ~' (as well as 9) we obtain
wd(o') =
wd(l
)[1 + o , ' c o s 2 0 ' ] ,
(14)
with a ' for the various Xj states given by - 1 + 6ala 2
a ' ( X 0 ) = 1, =
a'(Xa)-
3- 2ala 2
1 + 4a 2 - 6v~ala 2
, 2 + 2v/~ala2 " 13 - ~a As expected Wd(0') depends only on the Xj ~ ~k7 multipole structure. Finally, for ~p ~ 1P1 ~ 7/c't we obtain eq. (13) with
1 November 1984
helicity zero state and for the importance of the magnetic quadrupole transitions in X2 ---' ~kT- The data from the I S R experiment [1] should be able to determine the multipole and helicity structure of this process. The helicity structure results in a direct test of the Q C D helicity conservation hypothesis, and moreover, the verification of a large magnetic quadrupole transition is of great interest in heavy quark spectroscopy, possibly indicating a large anomalous magnetic m o m e n t of the c h a r m q u a r k [11]. W e thank Dr. A. L u n d b y for several informative discussions concerning the ISR c h a r m o n i u m experiment. W e also thank the British Science and Engineering Research Council for support. References
a(1P1) = - 1 ,
1~(1p1) ~---0.
T h e recent I S R charmonium experiment [1] has collected the most data for the X 2 state and so in fig. 2 we show the spread of possible angular distributions, Wp(0) and Wd(0' ), for ~p ~ X2 ~k7 assuming different E 1 / M 2 admixtures (specified b y the parameter a2) and for X2 p r o d u c t i o n in different helicity configurations (specified b y the parameter R of eq. (12)). We see that the angular distributions should allow a good determination of a 2, and provided the data covers the forward (or backward) directions in 8 should yield a value for R. W e have indicated some of the results which m a y be obtained from direct production of c h a r m o n i u m states by ~p annihilation. There is already some evidence for production through a
206
[1] Annecy-CERN-Geneva-Lyon-Oslo-Rome-TurinCollab., ISR experiment, R704, A. Lundby, private communication. [2] S.J. Brodsky and G.P. Lepage, Phys. Rev. D24 (1981) 2848. [3] A. Andrikopoulou, Z. fur Phys. C 22 (1984) 63. [4] Mark III Collab., K.F. Einsweiler et al, Proc. HEP Conf. (Brighton, 1983) p. 348. [5] M.W. Eaton et al., Phys. Rev., D29 (1984) 804. [6] M. Claudson, S.L. Glashow and M.B. Wise, Phys. Rev. D25 (1981) 1345. [7] Particle Data Group, Rev. Mod. Phys. 56 (1984) $1. [8] M. Oreglia et al., Phys. Rev. D25 (1982) 2259. [9] G. Karl, S. Meshkov and J.L. Rosner, Phys. Rev D13 (1976) 1203. [10] F.J. Gilman and I. Karliner, Phys. Lett. 46B (1973) 426; Phys. Rev. D10 (1974) 2194. [11] G. Karl, S. Meshkov and J.L. Rosner, Phys. Rev. Lett., 45 (1980) 215; R. McClary and N. Byers, Phys. Rev. D28 (1983) 1692.