The photoproduction of vector mesons (,,) is of renewed interest because intense high energy ... density matrix of the vector meson and hence its vector and.
PHYSICAL REVIEW C
VOLUME 53, NUMBER 2
FEBRUARY 1996
Polarization observables in vector meson photoproduction Michael Pichowsky, C¸etin S¸avkli, and Frank Tabakin Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260 ~Received 11 September 1995! The photoproduction of vector mesons ( r , v , f ) is of renewed interest because intense high energy beams of polarized electrons and photons are under development. These beams and also polarized targets make it possible to explore the dynamics of basic baryon structure. As a step toward that goal, an analysis of all possible polarization observables for the case of vector meson photoproduction from a nucleon target is presented. The question of which observables are needed to determine completely the basic photoproduction amplitudes and the relationships between spin observables are addressed. Such theorems are most readily demonstrated by representing all observables as bilinear products of helicity amplitudes and using known properties of Dirac gamma and spin-1 matrices. The general angular dependence of spin observables, especially near thresholds and resonances, is examined for the vector meson case. The criteria for a complete set of observables and the relationships between observables are then presented. PACS number~s!: 24.70.1s, 25.20.Lj, 13.60.Le, 13.88.1e
I. INTRODUCTION
The photoproduction and electroproduction of mesons is of renewed interest now that CEBAF has arrived. The thresholds for r , v , and f mesons will be readily attained and exceeded. The new high-intensity continuous beams, and the development of polarized targets and beams, will allow measurement of spin observables with unprecedented precision. In addition, the measurement of recoil baryons, including the L ~which due to its weak decay L→p1 p 2 is spin selfanalyzing!, will provide even more spin observables. Vector meson decays ( r → pp ; v → ppp ; f →K 1 K 2 ,K¯ ,K) provide a measure of their density matrices, which is equivalent to determining the intensity, polarization, and tensor polarization of these vector mesons. Such measurements were made in the 1960’s @1#. With CEBAF, data of higher precision and completeness should be achievable @2#. The polar¯ ,K, and of the r ization of both the f , using its decay to K using ( r → pp will be measured at the CLAS detector at CEBAF @3#. From measurement of the angular distribution of the decay mesons, it should be possible to obtain the full density matrix of the vector meson and hence its vector and tensor polarization. It is therefore timely and important to study the photoand electroproduction of vector mesons. The associated form factors and intermediate isobar states should test quark models. The f meson is of particular interest because of its simple (s¯ s) 3 S 1 2 quark content. It is the positronium of strange quarks and, in parallel to charmonium (c¯ c) states, should provide insights into basic QCD dynamics in the nonperturbative region. Perhaps an intimate connection between production of s¯ s pairs and baryon polarization could provide detailed tests of quark descriptions. For example, the family of reactions g p→K 1 L, g p→K 1 S 0 , p¯ p→ ff , and g p→ f p all involve s¯ s strangeness production. In addition, they could also involve preexisting s¯ s content of the initial protons and antiprotons. That possibility has been explored in a series of papers dedicated to finding direct evidence for an admixture of s¯ s pairs in baryons @4 – 6#. Spin observables 0556-2813/96/53~2!/593~18!/$06.00
53
are probably most sensitive to such a configuration mixture. In view of the above motivation, in this paper we discuss some general features of spin observables for the photoproduction of vector mesons. Our approach is similar to that used in studies of pseudoscalar meson production and of the ¯ L reaction near threshold @7–9#. However, for a vecp¯ p→L tor meson the spin 1 complicates the dynamics and a more general approach is required to keep track of all spin observables and to demonstrate their general features, e.g., their nodal structure and their normal energy evolution. For example, the question of which spin observables vanish at 0° and 180°, which have nodes, where these nodes occur and how they are expected to evolve with energy ~based on simple centrifugal barrier and/or resonance constraints! are addressed here. In addition, we examine the question of what constitutes a complete set of measurements, e.g., which measurements are needed to determine the magnitude and phases of the basic amplitudes. Also, we wish to know which measurements are redundant, based on assumed symmetries. Such questions have been answered for the pseudoscalar meson case @10–12#; those discussions are extended here to the vector meson case. Indeed, generalized discussions already exist in the literature @13,14#, which are however quite different from our treatment. Our description uses a space of bilinear products of helicity amplitudes, along with well-known properties of 434 and 333 matrix bases, to reveal the general features of spin observables. We also use the idea of transversity amplitudes @15#. For clarity, we also include the pseudoscalar case using this bilinear form description and show how it generalizes to the vector meson case and indeed to many other reactions.
II. THE BASIC AMPLITUDE AND SPIN OBSERVABLES
W →VW 1N W 8 , our For vector meson photoproduction, gW 1N basic amplitude is of the form F [ ^ qW l V l 2 u T u kW ll 1 & , 593
~2.1!
© 1996 The American Physical Society
594
S
PICHOWSKY, S¸AVKLI, AND TABAKIN
F5
H 21
H 11
H 321
2H 421
H 41
H 31
2H 121
H 221
H 20
H 10
2H 30
H 40
H 40
H 30
H 10
2H 20
H 221
H 121
H 31
2H 41
H 421
H 321
2H 11
H 21
D
53
, ~2.3!
where the JW parity relation,
^ qW l V l 2 u T u pW ll 1 & 5 ~ 21 ! L f 2L i ^ qW 2l V 2l 2 u T u pW 2l2l 1 & ,
~2.4!
has been used, where L i [l2l 1 and L f [l V 2l 2 . The rows of F are labeled by the final state helicities (l V ,l 2 ) and the columns by the initial photon and proton helicities (l,l 1 ). 1 The helicity amplitudes in F are defined by the following:
FIG. 1. The coordinate system and kinematical variables for vector meson photoproduction. Here V denotes the vector meson and l V its helicity.
where the variables and the coordinate system conventions are presented in Fig. 1. The incident and final relative momenta are kW and qW , respectively. Jacob-Wick ~JW! @16# phase conventions are used throughout this paper. We use l561 to describe the two transverse helicity states of the initial photon, l 1 561/2 to describe the target (N) proton helicity, l V 561,0 for the final vector meson helicity and l 2 for the final (N 8 ) baryon helicity. For a real photon, the above amplitude represents 3323232524 complex numbers. However, by virtue of parity invariance there are 12 relations between these amplitudes and consequently we have 12 independent complex helicity amplitudes or 24 real numbers, at each energy and angle. We denote these 12 helicity amplitudes by
^ qW l V l 2 u T u pW ll 1 & →H al V ~ u ! ,
~2.2!
where a51, . . . ,4 and l V 561,0, and the particular matrix element assignments are given below. The pseudoscalar case is recovered by simply taking l V →0 and then we have 132323258 complex numbers and, after considering parity symmetries, we obtain the usual four independent (a51, . . . ,4) helicity amplitudes H a ( u ) @10#. The vector meson amplitude can be displayed as a 634 matrix in helicity space
H 1l V [ ^ l V ,l 2 511/2u T u l51,
l 1 521/2& ,
H 2l V [ ^ l V ,l 2 511/2u T u l51,
l 1 511/2& ,
H 3l V [ ^ l V ,l 2 521/2u T u l51,
l 1 521/2& ,
H 4l V [ ^ l V ,l 2 521/2u T u l51,
l 1 511/2& ,
~2.5!
along with the JW parity rules. The pseudoscalar meson case is described by the l V 50(234) part of the above matrix. General spin observables, V, for vector meson photoproduction can be expressed in the following trace form: V5
Tr@ F ~ A g A N ! F † ~ B V B N 8 !# , Tr@ F F † #
~2.6!
where the trace is over spin-space helicity quantum numbers l,l 1 ,l V ,l 2 . This basic expression for observables is an average over a classical ensemble of particles in the beam. Interference occurs only at the quantum level for the basic two-body reaction. The matrices have the dimensions F (634), A g A N (434), F † (436), and B V B N 8 (636). Here A g denotes the usual 232 Hermitian spin matrices (1,sW g ), which describe the photon’s two spin degrees of freedom. The 232 matrix A N is similarly the (1,sW N ) spin matrices, as is B N 8 (1,sW N 8 ), which describes the recoil baryon’s spin state. The vector meson matrix B V is a 333 matrix and a complete set of these is provided by the 333 unit operator (1), the usual spin-1 operators SW , plus five independent rank-2 operators t 262 , t 261 , and t 20 . These operators are given in terms of the spin-1 spherical tensor operators S 161 57
S x 6iS y
A2
57
S6
A2
,
S 10 5S z ,
We often use l 1 561 to designate the nucleon’s helicity, which really has the values l 1 561/2. 1
53
595
POLARIZATION OBSERVABLES IN VECTOR MESON . . .
by the tensor operator coupling t 2 m 5 @ S 1 3S 1 # 2 m . Associated with the above ensemble average, Eq. ~2.6!, there are density matrices describing the polarization state of each particle. For example, the incident photon and the target proton are described by Ig W g• P W S!, ~1 1s 2 g
~2.7!
IN W N• P W N!. ~1 1s 2 N
~2.8!
r g5 and
r N5
W S above is the Stokes vector, which is used to describe The P the photon’s polarization as discussed in Ref. @8#. The vector W N describes the target polarization; its component in the yˆ P W N •yˆ . The final state density madirection is denoted by T[ P trices are, for the final baryon, r N85
I N8 W N •P W ~1 1s 8 N8!, 2 N8
~2.9!
and, for the final vector meson,
r V5
IV W V 13 t •T V ! . ~ 1 1SW • P 3 V
~2.10!
s~ u !5r0 where
r f 5F r i F † ,
1 s ~ u ! 5 r 0 Tr@ F F † # [ r 0 I ~ u ! . 4
for which each component is normalized to be 3/2 J>1/2 J>3/2 J>3/2
0 1 1 0
0 0 0 0
H 1,0 ( u ) H 2,0 ( u ) H 3,0 ( u ) H 4,0 ( u )
H J1,0 H J2,0 H J3,0 H J4,0
J d 21/2,3/2 (u) J d 21/2,1/2 (u) J d 1/2,3/2 (u) J d 1/2,1/2 (u)
J>3/2 J>1/2 J>3/2 J>1/2
0 0 0 1
0 1 0 0
H J1,21 H J2,21 H J3,21 H J4,21
J d 23/2,3/2 (u) J d 23/2,1/2 (u) J d 21/2,3/2 (u) J d 21/2,1/2 (u)
J>3/2 J>3/2 J>3/2 J>1/2
0 0 0 0
1 0 0 1
with 1 J J* b J , ab J L,L 8 @ ab;l V ,l V8 # [6 Ga v H 1 H 2 2 J 1 ,J 2 ab l V l V8 al V bl V8
(
3 3
S S
J1
J2
J
l V2 z a
l V8 2 z b
L
J1
J2
J
ja
jb
L8
D
.
D
~4.4!
Note that L depends on the helicity labels a,b and l V ,l V8 , while L 8 depends only on the helicity labels a,b. To understand the consequence of the above result, select a particular choice of spin observable by designating the associated values of a , and b ; for example, take 10,1 for the target polarization, see Eq. ~2.15!. Now consider the full family of G3 v matrices with the same matrix ‘‘shape’’ for both G and v . That family is called a ‘‘Legendre class.’’ For the target polarization case, the matrix products G 10,11,13,143 v 1,4 are all of the same shape and thus the associated spin observables form a ‘‘Legendre class.’’ ~The members of this target polarization class will be discussed later.! Since all members of a ‘‘Legendre class’’ vanish for the same (a,b;l V l V8 ) values, they are all formed from the same J set of Wigner d functions, d L,L 8 ( u ) of Eqs. ~4.3! and ~4.4!. If, for example, every member of that set of Wigner d functions vanishes at 0° and 180°, then every member of that ‘‘Legendre class’’ of spin observables will also vanish at 0° and 180°. Similarly, if every member of that set of Wigner d functions has a zero or a node at 90°, then every member of that ‘‘Legendre class’’ of spin observables will also have a zero or a node at 90°. These observations follow from the fact that these families of spin observables are all expressed by various combinations ~see J, above! of the same set of Wigner d functions. The mixture coefficients, J, do depend on the partial wave helicity amplitudes, which is how dynamics of the reaction affects the detailed angular dependences. If one truncates the partial wave expansion due to either threshold or resonance considerations, then the blend of Wigner d’s is strongly restricted and one can demonstrate explicit associated angle dependences of the spin observable profiles. For example, if only one J 1 ,J 2 set of partial wave helicity amplitudes are nonzero, then using the triangle rule J 5J 1 1J 2 •••J 1 2J 2 , only a limited number of J values appear, which severely restricts the nodal structure.
H 1,21 ( u ) H 2,21 ( u ) H 3,21 ( u ) H 4,21 ( u )
That allows one to test and extract specific dynamical information from the nodal structure and energy evolution of spin observables ~see Ref. @9#!. Thus the angular dependences of spin observables can be grouped into classes @8,9# with the same potential nodal structure ~hence the nomenclature Legendre class!. To examine the general role of resonances on the nodal structure of spin observables, it is most convenient to introduce explicit orbital angular momentum quantum numbers. Therefore, in a separate paper, Ref. @17#, the electric and magnetic multipole amplitudes for vector meson photoproduction are studied in detail. Another way of examining the angular dependence of a Legendre class of spin observables is to return to the partial wave expansion, Eq. ~4.1!. For a given Legendre class, only selected values of a and l V appear. Thus, one can pick the corresponding helicity amplitudes from the first column of Table I, then proceed to the partial wave helicity amplitudes and the associated Wigner d functions in the second and third columns. The fourth column gives the range of J needed for that amplitude to contribute; while the last two columns indicate the value of the Wigner d function at the 0° and 180° end points. Using this table, one can deduce which observables are zero at the end points. For example, consider the Legendre class of observables for which both G a and v b are diagonal, e.g., the class: G 1,2,9,153 v 1,4. These observables depend on linear combinations of ( al V 6 u H a,l V u 2 . From Table I it follows that the associated observables do not necessarily vanish at the end points. In this diagonal Legendre class the single and double observables are I , C zz 8 , C zz 88 , C z 88z 8 , NV NN gN C zz , C zz 8 , and C zz 8 8 . This information is extracted from gV
gN
N V
53
601
POLARIZATION OBSERVABLES IN VECTOR MESON . . .
TABLE II. Spin observables in BHP form as products of G a and v matrices. Here the a 51, . . . 16, b 5129, A,B,C range is displayed. Single-, double-, and triple-, but not quadruple-, spin observables are shown. The entries of ‘‘phase class’’ G˜ PL 3 v˜ P do not appear—they are all quadruple-spin observables.
TABLE III. Spin observables in BHP form as products of G a and v b matrices. Here the a 51, . . . 16, b 5126 range is displayed.
a\b
1
2
1
1
I
2
2C zz 88
b
gN
g NN 2C zxy 8 8
4
NN 2C y y 88
6 7
8 9 10
g NN 2C zzy 8 8 NN 2C xx 88 g NN C zyx 8 8 NN C zx 8 8 gN C zz
2T
11
gN C zx
12
P N8
13 14
4
C zx 8
PV
C zz 8
2C z 88x 8
C zz 88y 8
2C z 88z 8
gV
3
5
3
C xz 8 8
gN V
N V
NN V 2C xy 88x 8 8V C NN xz5 gV C y6
NV C gxx6 NN V C zy 8 8x 8
gN V C xz 886 NN V C yx 88x 8 NV 2C zx 8
gN V C y y 886 g NV 2C zyx 8 NV 2C xx 8 gN V C zy 88x 8 NV 2C gy y6
g NV
g NN
gV
C xy 8
NN V 2C y y 88y 8 g NV 2C yxy 8
NN V C xx 88y 8 gN V 2C yz 88y 8 NN V 2C zx 8 8y 8 g NV 2C zzy 8 C Ny y8 V NV 2C y y 8 gN V C xy 88y 8
g NV
2C zxy 8 g NV
2C xy y 8 C y 88y 8 N V
C xz 8 8y 8
16
C zz 8 8
2C xx 886
C yx 88y 8
gN V
C yz 8 8x 8
NN
2C zx 88 gN
gV C z4
A 23 T 21
2C z 884
N V
C gx4V
NV 2C gxx4
gV
C yz 8 g NV C xxz 8
NN V C zy 8 8z 8
NN V C zy 8 84
gN V 2C xz 884 NN V C yx 88z 8
gN V C xz 88z 8 NN V C yx 884
NV 2C zz 8
2C NV z4
C zz 885
gN V g NV
gN V 2C y y 884 g NV 2C zyz 8 NV 2C xz 8 gN V C zy 88z 8 NV C gy y4
gN V
gN V C y y 88z 8
2C NV x4 C zy 884
gN V g NV
2C y yz 8
C xx 884
gN V
C yz 8 8z 8
2C xx 88z 8
gN V
C yz 8 84
gN V
N V
NN V 2C zx 8 58 g NV 2C zz5
gN V C xy 885 g NV 2C zy6 g NV 2C zx5 g NV 2C xy5
C y 885
C xz 8 58
C yx 88y 8
2 3 4 5
9
A
gV C z6
T 22
T 20
T y8y8
2C z 886
N V
2C xzx 88
C zz 887
C zz 888
2C y y 887
2C y y 888
C xx 887
C xx 887
gN V
gN V
NN V
gV
2C yx 8 C zy 8 86
NN V
NN V
NN V
B
C
C gyzN
NV 2C gyz7
NV 2C gyz8
S
C gx7V
C gx8V
C gyxN
NV C gyx7
NV C gyx8
2C xy y 8 8 2C y yx 8 8 g NN
g NV
2C xxx 8
NN V
6 7
2C xz 88x 8
gN V
C yx 886
NN V
2C xzz 8 8 g NN
2C yz 88 2C yz 887 gN
gN V
2C yz 888
gN V
2C zz 8 58
2C zx 8 78
2C zx 8 88
C Ny78 V
C Ny88 V
NN V
g NV 2C zz7
9
2C NV z6
10
2C y y 88x 8
11
2C NV x6
g NV 2C zx7
12
C zy 886
C y 887
gN V
gN V g NV
C y yx 8 14
2C NV y7
C xx 88x 8 gN V
C yz 8 86
NN V
g NV g NN 2C zz8 2C yxy 8 8
2C NV y8
C xy 88 gN
g NV g NN 2C zx8 2C yzy 8 8
C y 888
gN 2C xy
2C xz 8 78
2C xz 8 88
C xzx 8 8
2C zz 8 78
2C zz 8 88
2C zx 887
2C zx 888
N V
NN V
13
2C xxz 8 8 g NN
8
N V
NN V
C xy 887
C xy 888
g NV 2C xy7
g NV 2C xy8
C yx 887
C yx 888
gN V
gN V
g NN
C yx 88 gN
gN V
gN V
NN V NN V
15
gN V
16
2C x 886
N V
gN V
NN V
C xxx 8 8 g NN
gN V
C y yz 8 8 g NN
NN V
2C x 88z 8 N V
2C x 884
N V
2C zx 885
gN V
class are S, C yz 8 , C yz 88 , C gyzN , C xy 884 , C xzz 88 , C yx 884 , gN V g NN C xz 88z 8 , and C xzz 8 8 . gN
8
g NN
N V
Tables II and III, where the relation between the a and b choices of the matrices are related to explicit spin observables. Another example of how to use Tables I–III to deduce the end-point behavior of a given Legendre class of observables is seen by examining the Legendre class of observables for which both G a and v b are antidiagonal, e.g., the class: G 3,4,6,7 3 v 5,A . Only the helicity amplitude products with a,b5(1,4),(2,3) and l V8 52l V appear for these observables. Now using Table I with those products, it follows that this Legendre class involved bilinear helicity products that vanish at both 0° and 180°. From Tables II and III, we learn that the single-, double-, and triple-spin observables of this gV
7
NN V
NN V C xx 885 gN V 2C yz 885
NN V
NN V
2C x 88x 8 2C zx 88y 8
NN V 2C y y 885 NV C gyx5
NN V
NN V
2C zz 8 y8 8
C gx5V
C Ny58 V 2C NV y5
NN V
NN V
15
N V
6
2C yzy 8 2C NN 8 V 2C NN 8 V 2C xy y 8 xy 8 z 8 xy 8 4 NV NN 8 V NN 8 V 2C gyz5 2C xz4 C xzz 8
NN
C zyz 8 8
gV
5
a\b
NN V
NN V
NN V
The procedure consists of using Tables I–III along with the Legendre class information to determine the end-point rules. Alternately, one can use the general results Eqs. ~4.3! and ~4.4!. V. TRANSVERSITY, PHASE CLASS, AND COMPLETE EXPERIMENTS A. Transversity 1. Pseudoscalar meson transversity
Let us deal with the question of which experiments are needed to determine the magnitude and phase of the 12 helicity amplitudes for vector meson production. Clearly, 23 experiments are needed at each energy and angle. It is well known that for pseudoscalar meson photoproduction, it is possible to determine the magnitude of the four transversity amplitudes by measuring all four single-spin observables. The remaining three phases ~one overall phase is arbitrary! can be determined by selecting three additional double-spin
602
PICHOWSKY, S¸AVKLI, AND TABAKIN
observables, following the Barker-Donnachie-Storrow ~BDS! @11,12# rules. The pseudoscalar meson case can be recovered from Eqs. ~3.7!–~3.10!, by setting v A →1 and omitting all observables which involve a final meson spin of 1. In that limit, the BDS rules can be understood by performing a unitary transformation on the G matrices along with a unitary transformation on the helicity amplitudes to generate ˜ &: the transversity amplitudes u H ˜ &[ uH a
(b U~a,b4 ! u H b & ,
~5.1!
and a new set of G matrices G˜ a [U~ 4 ! G a U† ~ 4 ! .
~5.2!
The 16 spin observables are invariant under such a unitary transformation in ‘‘helicity space:’’ ˜ u G˜ a u H ˜ &. ˇ a} ^ H u G au H & [ ^ H V The physically meaningful unitary operator is the transversity choice,
åU~ 4 ! 5
1 2
S
1
2i
i
1
1
i
2i
1
1
i
i
21
1
2i
2i
21
D
~5.3!
,
which involves rotating the helicity quantization axis (zˆ and zˆ 8 ) to the direction normal to the scattering plane yˆ 5yˆ 8 , see Fig. 1. With the above 434 unitary transversity transformation, the following matrices now are diagonal:
˜ uH a,l V & [
8 V ,l V
u H b,l 8 & . V
a,l V
v b → v˜ b . In this new representation the diagonal terms are indicated by the boxed matrices in Eqs. ~3.7!– ~3.22!. The shapes of the tranversity-transformed matrices are presented in Appendices C and D. B. Phase class
As discussed in Appendices C and D the shape of the G˜ , v˜ allows us to group these matrices into ‘‘phase classes.’’ As in the ‘‘Legendre classes’’ of the original matrices G, v , the shapes are defined by where nonzero entries appear in the matrix. The classification into diagonal, antidiagonal, left parallelogram, right parallelogram, crossed, and polygon shapes is of significance in that these shapes select the contributing bilinear helicity products. For example, if in the product G˜ a v˜ b both matrices are diagonal, then that observable depends on linear combinations of the product 2 ˜ ˜ a ˜ b has uH a,l V u . If on the other hand the matrix product G v an entry at the location a,b in the G˜ space and at the location
˜ ˜ ˜ ˜* H H b,l 8 5 u H a,l V uu H b,l 8 u expi ~ f b,l 8 2 f a,l V ! , a,l V
G˜ 2 G˜ 7 G˜ 14G˜ 16. The remaining eight matrices are either of one class ~called PL for left parallelogram form!,
or of another class ~called PR for right parallelogram form!, G˜ 3 G˜ 5 G˜ 9 G˜ 11. 2. Vector meson transversity
The above procedure can now be extended to the vector meson case by introducing an additional unitary transversity transformation in the 333 space: 21
A2i
1
2 A2i
0
2 A2i
1
A2i
21
D
V
V
V
˜ ˜ where H a,l V 5 u H a,l V u expi(fa,lV), etc. Thus, the shapes of the matrices in the tranversity description tell us which phases b,a f b,l 8 2 f a,l V [ f l 8 l are needed to determine the associated V
G˜ 6 G˜ 8 G˜ 13G˜ 15,
S
b,l V 8
~3!
4! U~a,b Ul
For the vector meson part of the transversity v˜ space, the following matrices now have diagonal (D) form, v˜ 1 v˜ 3 v˜ A ; antidiagonal (AD) form, v˜ 6 ; crossed (X) form, v˜ 8 v˜ 9 v˜ B v˜ C ; and diamond or polygon ( P) shaped form, v˜ 2 v˜ 4 v˜ 5 v˜ 7 . We have extended the definition of transversity amplitudes to the case of a vector meson. The original exˇ ab are of the pressions for the spin observable profiles V same form as given in Eqs. ~3.7!–~3.22!, except that the helicity amplitudes H al V are replaced by the transversity am˜ , and the matrices are replaced by G a →G˜ a and plitudes H
whereas, the following are now antidiagonal:
1 2
(
l V ,l V8 in the v˜ space, then that observable depends on the following product:
G˜ 1 G˜ 4 G˜ 10G˜ 12,
U~ 3 ! 5
53
V V
spin observable. In Fig. 2, the basic problem of determining the 12 amplitudes is illustrated, where the lengths of the vectors correspond to the magnitude of the transversity amplitudes, and the phases of these complex amplitudes are also shown. To fix this diagram, we need to determine 12 magnitudes and then 11 phases; one overall phase and the overall orientation of the diagram of Fig. 2 is arbitrary. This situation is a generalization of the pseudoscalar case, which is described in Appendix E and by Fig. 3. C. Complete experiments
,
~5.4!
which makes the yˆ axis the quantization axis for the spin-1 meson. Correspondingly, there is now a transformation in ˜ u G˜ a G˜ b u H ˜ & , with ‘‘helicity space:’’ ^ H u G a v b u H & [ ^ H
The phase classification of the transversity matrices are a guide to the task of picking a complete set of experiments. Thus the procedure is to first select experiments which give information about the magnitudes of the 12 transversity amplitudes and then to pick experiments which yield nonredundant phase information. In the pseudoscalar case, the singlespin observables yield the magnitudes of all four transversity
53
603
POLARIZATION OBSERVABLES IN VECTOR MESON . . .
FIG. 2. This diagram displays the magnitudes and selected relative phase angles of the 12 transversity amplitudes ˜ ˜ H 1l V , . . . ,H 4l V ~with l V 50,61) for photoproduction of vector mesons. The 12 complex amplitudes are determined within an overall phase if the 12 magnitudes and 11 angles are fixed by 23 appropriate measurements. The magnitudes are determined by measuring six single-spin observables plus six double-spin observables, see the text. Then independent phase angle measurements are made by selecting spin observables from different ‘‘phase class’’ observables. st The phase angles are labeled by the convention f l l 8 . V V
amplitudes. Then three double-spin observables, selected using the BDS @11,12# rules, yield three phases. The task is similar for the vector case, but the result is more complicated. To extract the magnitudes of the transversity amplitudes, we need to examine all observables for which both G˜ a and
v˜ b are of the diagonal phase class. These observables are produced by all products: G˜ 1,4,10,123 v˜ 1,3,A , see Appendices C and D. In the vector meson case there are only eight independent single-spin observables, so we learn that this case is not as favorable as the pseudoscalar meson case, where the singlespin observables sufficed to determine all four amplitude magnitudes. Indeed, the situation is that only six of the vector meson photoproduction single-spin observables are of diagonal form; namely, I ,Tˇ , Pˇ N 8 ,Sˇ , Pˇ V , and the tensor polarization Tˇ y y . Thus we need to turn to the double-spin observables for the remaining six diagonal phase class matrices reside. Therefore, for vector meson photoproduction it is not possible to determine the magnitudes of the 12 transversity amplitudes by only measuring six single-spin observables. VI. CONCLUSIONS
Several conclusions can be drawn from describing spin observables for vector meson photoproduction in bilinear helicity product form. Here one essentially extends the BDS @11,12# rules to include the vector meson degree of freedom. The diagonal matrices in the transversity basis are indicated by boxes in Eqs. ~3.7!–~3.22!. For the corresponding observables, the diagonal nature of the product G˜ a v˜ b means that those observables depend on linear combinations of the squared magnitude of the transversity amplitudes, e.g., they 2 ˜ depend on combinations of 6 u H i,l V u . Those experiments which depend only on the magnitude of the transversity amplitude are the following six singlespin observables: I ,Tˇ , Pˇ N 8 ,Sˇ , Pˇ V ,Tˇ y y , plus the following six double-spin observables: ˇ g N ,Cˇ g N 8 ,Cˇ NN 8 ,Cˇ g V ,Cˇ NV ,Cˇ N 8 V . C xy xy yy xy yy yy Note the above x component for the photon beam corresponds to a photon linearly polarized perpendicular to the scattering plane @8#. In contrast to the pseudoscalar meson production case, the magnitudes of the transversity amplitudes for the vector meson case cannot be determined by just single-spin observable measurements, one needs to also perform six double-spin measurements. Only six of the eight independent single-spin observables are of diagonal form; the remaining two single-spin observables provide transversity amplitude phase information. In addition, of those six double-spin observables three involve measuring the spin state of the final vector meson ~via its decay!
FIG. 3. This diagram displays the magnitudes and selected rela˜ , . . . ,H ˜ tive phase angles of the four transversity amplitudes H 1 4 for photoproduction of pseudoscalar mesons. The four complex amplitudes are determined within an overall phase if the four magnitudes and three angles are fixed by seven appropriate measurements. The magnitudes are determined by measuring the four single-spin observables, then independent phase angle measurements are made by selecting three double-spin observables from at least two different ‘‘phase class’’ observables. The phase angles are labeled by the convention f st .
ˇ g V ,Cˇ NV ,Cˇ N 8 V , C xy yy yy two require a polarized photon beam Cˇ gxyN ,Cˇ gxyN 8 , and one requires a polarized target and measurement of the spin state of the recoil final baryon 8 Cˇ NN yy .
604
PICHOWSKY, S¸AVKLI, AND TABAKIN
That sets the task for determining the magnitudes of the transversity amplitudes, which provide the most convenient situation. Note that near threshold, the two spin observables ˇ g N vanish. ˇ g V and C C xy xy Observables for which both G˜ a and v˜ b are not diagonal depend on transversity amplitude phase differences. As an extension of the BDS rules for the pseudoscalar meson case, these transversity phase-dependent observables form phase classes in which some experiments provide redundant phase information. A graphical procedure for analyzing the redundancy and phase class of experiments for vector meson production is illustrated in Fig. 2. The procedure consists of picking a set of phases between transversity amplitudes that correspond to a given phase class for the G and v matrices. Then list all the G3 v matrices of that same phase class. It is best to start with the two single-spin observables that are not diagonal in the transversity basis; for example T 20 ,T 21 . The corresponding experiments can be selected from Tables II and III, where only the experiments corresponding to the number of unknown phases need to be performed. In that process, some experiments prove to be feasible, some are difficult if not impossible; indeed, in some cases one needs to go to triple-spin observables. In any event, this selection procedure can be used to answer the question of the experiments needed for a full determination of all phases and also which ones will be extremely difficult to determine because of realistic experimental conditions. In addition to these features of using the bilinear helicity product form, one can deduce many other aspects of spin observables based on general knowledge of the properties of the G a v b matrices. For example, when a ‘‘Lorentz’’ (S g m S 21 5 l mn g n ) transformation or a parity, chirality, or time reversal operation is performed on the G a and v b matrices in the ‘‘helicity amplitude space,’’ they reveal linear relations between spin observables. If Fierz transformations are made on the G a and v b matrices, again in helicity space, then quadratic relationships between spin observables are readily deduced. An important part of our analysis is the use of transversity amplitudes. Many other types of amplitudes can be defined. For example, one can use the unitary transformations from the Dirac g matrices to the chiral, Majorana, or other g matrices sets to define new amplitudes; corresponding unitary transformations in the vector meson helicity three-space can also be invoked. Thus the amplitude basis is hardly unique and one can deduce other sets and therefore deduce other spin observables as the ones to measure to determine the magnitude and/or phases of these alternately defined amplitudes. However, there is something very special about the transversity amplitudes, which is closely related to using the normal to the scattering plan as the spin-quantization axis. The special property is that the transversity amplitudes make the optimum number of single-spin observables independent of amplitude phases. Correspondingly, phase information is relegated to the more complicated spin observables. This is most dramatic for the four single-spin observables for pseudoscalar meson photoproduction in that single-spin observables provide the magnitude of all four transversity ampli-
53
tudes. That simplification is also of benefit for vector mesons, which suggests that there are compelling reasons to use transversity amplitudes. We hope that the procedure described here will be useful in ascertaining the information content and the nodal behavior of vector meson spin observables and in planning experiments. Measurement of all spin observables needed to fully determine the 12 amplitudes is probably not feasible, although it is good to know what is needed for that full task. Even without a full experimental determination of the photoproduction amplitudes, it is possible to extract useful dynamical information, which is not an uncommon situation in strong interaction physics. Note added in proof. It was called to our attention by Professor R. Workman that quadrant ambiguities for the phases are discussed in a paper by N. W. Dean and Ping Lee, Phys. Rev. D 5, 2741 ~1972!. This is under study using our BHP method. ACKNOWLEDGMENTS
We wish to thank Dr. S. N. Yang, Dr. B. Saghai, and Dr. C. Fasano for their helpful comments. The research on this paper by F.T. was supported in part by the NSF. APPENDIX A: BASIS MATRICES 1. Four 3 four matrices
The basic properties of the 16 Hermitian G matrices are presented here. These are well-known matrices, except that we have made them Hermitian in order to generate real observables when used in the four-dimensional part of our helicity space amplitudes. The basic properties are Tr@ G a G b # 54 d ab , G a † 5G a . These permit one to expand any 434 matrix X in terms of the G’s and to extract the expansion coefficients C b using: X5 ( b C b G b and C b 5(1/4)Tr@ G b X # . The definition of the G matrices in terms of g m , g 5 , s m n , . . . is presented in Eq. ~3.3!. 2. Three 3 three matrices
The basic properties of the nine Hermitian v matrices are presented here. These are Hermitian in order to generate real observables when used in the three-dimensional part of our helicity space amplitudes. The basic properties are Tr @ v a v b # 53 d ab , v a † 5 v a . As in the 434 case, these properties allow one to expand a general 333 matrix and to extract the associated expansion coefficients. The nine v matrices are defined in the text as a unit matrix, plus three vector spin-one matrices, plus a rank-2 tensor. As given earlier the Cartesian form of the rank-2, symmetric Hermitian tensor is ti j5
S i S j 1S j S i 2 2 dij . 2 3
The associated spherical tensor form is
53
605
POLARIZATION OBSERVABLES IN VECTOR MESON . . .
t 205
At 3 2
zz 52
1
A6
t 261 57 ~ t xz 6it yz ! 57
S
The third Legendre class is of left parallelogram shape ( PL) and has G 10,11,13,14 as its members:
~ v 81 v 9 !,
1
A6
~ v 6i v ! , 6
7
G PL 5
D S
D
t xx 2t y y 1 v 82 v 9 i 6it xy 5 t 262 5 1 6 v 5, 2 2 A3 A6 where these are also expressed in terms of the Cartesian rank-2 tensor or the v 529 matrices. The following diagonal terms are of particular interest: t 22 1t 222 t 20 2 2 t y y 52 5S 2y 2 , 2 3 A6 t 22 1t 222 t 20 2 t xx 51 2 5S 2x 2 . 2 3 A6
APPENDIX B: ORIGINAL BASIS MATRICES
a
0
0
b
0
0
0
0
0
0
c
0
0
d
0
F G 0
0
a
0
0
0
0
b
c
0
0
0
0
d
0
0
G D5
F G a
0
0
0
0
b
0
0
0
0
c
0
0
0
0
d
,
a
b
c
d
G1
11
11
11
11
G2
11
11
21
21 .
G9
11
21
11
21
G 15 21
11
11
21
The second Legendre class is of antidiagonal shape (AD) and has G 3,4,6,7 as its members:
G AD 5
F G 0
0
0
a
0
0
b
0
0
c
0
0
d
0
0
0
,
a
b
c
d
G3
1i
1i
2i
2i
G4
11
21
21
11 .
G6
21
21
21
21
G7
1i
2i
1i
2i
b
c
d
1i
2i
1i
2i
G 11 21
21
21
21 .
G 13 21
21
11
11
1i
2i
2i
1i
G 10 ,
G 14
a
b
c
d
G5
1i
2i
2i
1i
G8
21
11
21
11 .
G 12
2i
2i
1i
1i
G 16 11
11
11
11
,
These shapes are important for the determination of the angular dependence of spin observables, e.g., their nodal structure. Later, the associated shapes for the transversity transformed matrices G˜ will be presented, which are useful for the analysis of a complete set of experiments.
1. Original four 3 four
The 16 G matrices can be grouped into four classes with four members in each class according to their ‘‘shape.’’ By shape, we mean the location of nonzero entries. For the original ~Dirac! Hermitian matrices the shapes are diagonal (D), antidiagonal (AD), left parallelogram ( PL), and right parallelogram ( PR). The first Legendre class is of diagonal shape (D) and has G 1,2,9,15 as its members:
a
The third Legendre class is of right parallelogram shape ( PR) and has G 10,11,13,14 as its members:
G PR 5
Here the matrix t i j is mapped to the rank-2 spherical tensor operator t 2 m ; it is used in the BHP form for spin observables. The same rules apply to the operator t 2 m , which appears in the trace form for spin observables.
F G 0
2. Original three 3 three
The nine v matrices can also be grouped into four classes according to their ‘‘shape.’’ By shape, we mean the location of nonzero entries. For the original Hermitian matrices, the shapes are diagonal (D), antidiagonal (AD), diamond or polygon ( P), and crossed (X). In this case there are two D, one AD, four P, and two X matrices, which accounts for the nine v b matrices. In addition, we classify the three matrices v A,B,C , which are particular linear combinations of v 1,8,9, see Eqs. ~3.17! and ~3.16!. The matrix v A is antidiagonal; whereas v B and v C are of crossed X shape. The explicit forms are
F G a
0
0
v D5 0 0
b
0 , c
0
F G 0
0
a
v AD 5 0 c
b
0 , 0
0
v1 v4
a
b
c
11
11
11
A3
0
2
a
v5 vA
2i
2
b
A6 2
21
0 11
A3
,
2
c 1i
A6 2
21
,
606
PICHOWSKY, S¸AVKLI, AND TABAKIN
F G 0
v P5 b 0
a
0
0
c , 0
d
G˜ D 5
a
v2
A3
1
v3
A3
2 2
1i
2
A3 A3
1
2
2i
2
1i
2
A3
1
v7
A3
c
A3
1
2
2i
v6
b
A3 2
1 2i 2 1i
d
A3
1
2
A3
1i
2
A3
2
2
A3
2i
2
A3 2
0
b
v X5 0 d
c
0 , e
0
v8 v9
b
1 2 2
vB
2
vC
1
A3 2
A3 2
0
0
0
0
b
0
0
0
0
c
0
0
0
0
d
,
a
b
c
d
G˜ 1 G˜
11
11
11
11
11
11
21
21 .
G˜ 10 G˜
21
11
11
21
21
11
21
11
4
12
2 ,
A3 2 G˜ AD 5
A3 2
F G 0
0
0
a
0
0
b
0
0
c
0
0
d
0
0
0
,
a
b
c
d
11
11
11
11
2i
2i
1i
1i .
G˜ 14 21 G˜ 16 1i
11
11
21
2i
1i
2i
G˜ 2 G˜
7
The third phase class is of left parallelogram shape ( PL) and has G˜ 6,8,13,15 as its members:
a 1 2 2
F G a
The second phase class is of antidiagonal shape (AD) and has G˜ 2,7,14,16 as its members:
A3
F G a
53
1 2
c
A3 2
A3 2
11 11
d 1 2
e
A3 2
A3 2
1 2 2 1 2 2
1
1 2
11
1 2
2
1
1 2
11
1 2
1
A3
G˜ PL 5
F G 0
a
0
0
b
0
0
0
0
0
0
c
0
0
d
0
,
a
b
c
d
G˜ 6 G˜
21
21
11
11
2i
1i
2i
1i .
G˜ 13 G˜
1i
2i
2i
1i
21
21
21
21
8
15
.
2
The fourth phase class is of right parallelogram shape ( PR) and has G˜ 3,5,9,11 as its members:
A3 2
The classification of these matrices is helpful in the analysis of the angular dependence of spin observables, e.g., of the ‘‘Legendre class’’ and the associated nodal structure. The associated shapes for the transversity transformed matrices v˜ , which are useful for the analysis of a complete set of experiments, will be presented later. APPENDIX C: TRANSVERSITY BASIS MATRICES
Introducing the transversity amplitudes involves a unitary transformation of the basis matrices, see Eqs. ~5.1!–~5.4!. These transformed matrices are presented here for the 434, G˜ and 333, v˜ cases. 1. Transversity four 3 four
After the transversity transformation, the 16 G˜ matrices still form four classes, with four members in each class. Since these matrices are part of the analysis of which experiments are needed to determine the magnitude and phase of the transversity amplitudes, we refer to these as the ‘‘phase class.’’ The first phase class is of diagonal shape (D) and has G˜ 1,4,10,12 as its members:
G˜ PR 5
F G 0
0
a
0
0
0
0
b
c
0
0
0
0
d
0
0
,
G˜ 3 G˜
5
G˜ 9 G˜
11
a
b
c
d
2i
2i
1i
1i
11
21
11
21 .
11
11
11
11
1i
2i
2i
1i
APPENDIX D: TRANSVERSITY THREE 3 THREE
After the transversity transformation, the nine v˜ matrices still form four classes. For these transversity Hermitian matrices, the shapes are still diagonal (D), antidiagonal (AD), diamond or polygon ( P), and crossed (X). In this case there are two D, one AD, four P, and two X matrices, which accounts for the nine v˜ b matrices. In addition, we classify the three matrices v˜ A,B,C , which are particular linear combinations of v˜ 1,8,9, see Eqs. ~3.17! and ~3.16!. The matrix v˜ A is now diagonal; whereas v˜ B and v˜ C are of crossed X shape. The first phase class is of diagonal shape (D) and has v˜ 1,3,A as its members:
F G a
0
0
v˜ 1
v˜ D 5 0 0
b
0 , c
v˜ 3
0
v˜ A
a
b
c
11
11
11
1
A6 2
11
0 21
2
A6 2
11
.
53
607
POLARIZATION OBSERVABLES IN VECTOR MESON . . .
The second phase class is of antidiagonal shape (AD) and has v˜ 6 as its sole member:
F G 0
0
a
v˜ AD 5 0 c
b
0 , 0
0
a
v˜ 6
1i
b
A6
c
2
A6
2i
0
1. Single-spin observables
There are four single-spin observables for pseudoscalar meson photoproduction, where we include the cross section:
.
cross section:
2
F G a
0
v˜ P 5 b 0
0
c , 0
d
b
A3
2
v˜ 4
2i
v˜ 5
2
v˜ 7
2i
2
A3 2
A3 2
A3 2
c
b
v˜ X 5 0 d
c
0 , e
0
A3
2
2
2
1i
A3 2
2i
A3
2
1
2
1i
a
b
v˜ 8
12 A3 4
31 A3 4
2
v˜ 9
11 A3 4
v˜ B v˜ C
n
~E3!
A3 2
1i
n
A3
2
2
A3 2
1i
A3
1
2
A3 2
2i
A3 2
A3 2
.
A3
c
Note that all of these single-spin observables are diagonal in the transversity amplitude case, which is the meaning of the boxed G matrices. Therefore, measurement of the four spin observables yields the magnitudes of all four transversity ˜ , . . . ,H ˜ . To determine the amplitude phases, amplitudes H 1 4 one needs to measure double-spin observables.
A3
2. Double-spin observables
2
There are four transversity amplitudes and hence four amplitude phases; however, one overall phase is arbitrary. Therefore, one needs to perform three measurements to fix these three phases, see Fig. 3:
d
e
12 A3 2
31 A3 4
12 A3 4
32 A3 4
11 A3 2 2
32 A3 4
11 A3 4 .
12 A3 4
31 A3 4
1
12 A3 2
31 A3 4
12 A3 4
11 A3
32 A3
11 A3
32 A3
11 A3
4
4
2
4
4
1
~E4!
2
F G 0
~E2!
1 beam: Sˇ 5 ^ H u G 4 u H & . 2
d
The fourth phase class is of crossed shape (X) and has v˜ 8,9,B,C as its four members: a
n
1 recoil: Pˇ N 8 5 ^ H u G 12 u H & , 2
a
v˜ 2
~E1!
1 target: Tˇ 52 ^ H u G 10 u H & , 2
The third phase class is of polygon shape ( P) and has v˜ 2,4,5,7 as its four members: 0
n
1 I5 ^ H u G 1 u H & , 2
C
gN
We can return to the case of pseudoscalar meson photoproduction by omitting all vector meson spin observables and by replacing all v matrices by zero, except for the v 1,A →1 case. In addition, the 12 amplitudes reduce to 4: H a,l V →H a . This limit is equivalent to looking at the l V →0 terms only. For the pseudoscalar meson case, we next present the BHP spin observable profiles.
S
0 G G
n
2 G 12
0
0
G
3
0
G
9
5
11
C
gN8
5
C NN 8 5
S
S
DS 5
0
2 P N8
0
H
0
G
F
0
E
D
1 check C gi jN 8 5 ^ H u C gi jN 8 u H & , 2
beam-recoil:
These phase classes are useful in specifying the experiments needed to determine the magnitude and phases of the 12 complex transversity amplitudes. APPENDIX E: PSEUDOSCALAR MESONS
5
1 beam-target: Cˇ gi jN 5 ^ H u C gi jN u H & , 2
0
n G 10
0
G 14
0
2G 7
2G 16
0
2G 2
DS 5
0
2T
0
O x8
0
O z8
C x8
0
C z8
,
D
1 ˇ NN 8 5 ^ H u C NN 8 u H & , target-recoil: C ij ij 2 2G 6 0 G
8
0
2G 13
2 G4
0
n 0
G
15
DS 5
T x8
0
0
2S
L x8
0
T z8 0 L z8
,
D
.
The three boxed G matrices in the above double-spin observables already appeared in the single-spin observables. Thus there are only four, instead of five, members in each of the above double-spin categories. In particular, the following double-spin observables are equal to single-spin observables:
608
PICHOWSKY, S¸AVKLI, AND TABAKIN
C gxyN 52 P N 8 ,
C xy 88 52T, gN
53
C y y 88 52S.
1 ˇ g NN 8 5 ^ H u C g NN 8 u H & , C zi j zi j 2
NN
These equalities are direct consequences of parity conservation. For the vector meson case, the appearance of the v A matrix in the corresponding double-spin observables yields linear relations between observables, not the above restriction; hence, in that case there are five double-spin observables in these categories. In addition, the G a matrices for each double-spin observable are of the same ‘‘phase class.’’ For C g N , a 53,5,9,11 appear, which are of phase class PR. Based on this shape category, these observables depend on the following relative phases: f 1,3 , f 2,4 . For C g N 8 , a 52,7,14,16 appear, which are of phase class AD. Based on this shape category, these observables depend on the following relative phases: f 1,4 , f 2,3 . For C NN 8 , a 56,8,13.15 appear, which are of phase class PL. Based on this shape category, these observables depend on the following relative phases: f 1,2 , f 3,4 . Here, we label the phases using f a,b for the phase difference ˜ and f a,b 5 f a 2 f b between the transversity amplitudes H a ˜ H b , see Fig. 3. Making three measurements of the same phase class observables would be redundant; only two are needed for the two phases. The third measurement should be taken from another phase class. Thus one needs three double-spin measurements, but not more than two from a given phase class. Here we have presented a derivation of the BDS @11,12# rules, based on the shape of the G˜ matrices and the geometric picture of the transversity amplitudes shown in Fig. 3. The advantage of this rendition of the BDS theorem is that it can be generalized to the case of vector meson photoproduction, see Fig. 2.
8 C gxiNN j 5
8 C gyiNN j 5
1 ˇ g NN 8 5 ^ H u C g NN 8 u H & , C yi j yi j 2
~E6!
6 3 13 8V C NN xi j 5 ~ G 2G G !
7 8V C NN yi j 5 ~ G 2
n G G 4
14
!
S
S
n
2 G
S
13
2G
2G 9
2G
2
2G
0
G
11
2G 3
0 7
0
0 G
G
0
DS DS
16
0
0 5
6
0 G
0
DS 5
0
0
0
S
1
14
2L z 8
0
0
2I
T z8
0
0
5 C z8 0 0
5 2O z 8 0
L x8 0 2T x 8
2E 0 F G 0 H
D
,
D
0
2C x 8 , 0 0
D
O x8 . 0
Here all three particles with spin are involved. Of the 339 possible triple-spin observables, 15 are nonzero. Three of these are equal to spin spin observables and the remaining 12 are equal to double-spin observables. Therefore, there is no new information in triple-spin observables for pseudoscalar meson photoproduction, and it is fortunately not necessary to consider such complicated measurements. This is not the case for vector meson photoproduction. APPENDIX F: VECTOR MESONS
The single- and double-spin observables were presented in the text. Using the BHP approach and MAPLE, it is possible to derive explicit expressions for the triple- and quadruple-spin observables for vector meson photoproduction. Triple-spin observables involve the spin of three particles, including the vector and tensor polarization of the vector meson.
In the pseudoscalar meson case, the following triple-spin observables can be derived in the BHP form. Again, the triple-spin observables are displayed in a Cartesian format with ~E5!
G
2G 8
0
0
8 C gziNN j 5
3. Triple-spin observables
1 ˇ g NN 8 5 ^ H u C g NN 8 u H & , C xi j ij 2
S
G 15
~E7!
1. Triple-spin observables
There are four types of triple-spin observables. The first three types, N,N 8 ,V, g ,N 8 ,V, and g ,N,V involve the vector meson. The fourth type involves the g ,N,N 8 particles and does not include the vector meson:
0
n v 3
0
0
v6
0
v8
v9
v2
0
v4
v5
0
v7
0
0
3
0
0
v6
0
v8
v9
v4
v5
0
v7
0
0
0
v
v9
v7
0
0
n v
v2
0
0
3
n v
0
0
v
v2
0
v4
v5
0
6
8
0
D
D
,
,
53
POLARIZATION OBSERVABLES IN VECTOR MESON . . .
8 V 52 ~ G 8 2G 5 G 15! C NN zi j
n
C gxiNj 8 V 5 ~ G 14 G 10 2G 7 !
S
n
C gyiNj 8 V 5 ~ G 142 G 10 2G 7 !
S
n
n
S
n
5 12 G3! C gyiNV j 52 ~ G 2 G
n
8 C gxiNN j 5
S
G 15
0
0
2G
G
13
0
2G 8 1
0 2G
6
D
n v , A
0
0
v6
0
v8
v9
v2
0
v4
v5
0
v7
0
0
3
0
0
v6
0
v8
v9
0
2v5
v4
0
2v2
0
0
3
0
0
v6
0
vB
vC
0
2v
4
0
2v
0
0
n v
7
5
v
2
3
0
0
v6
0
vB
vC
v7
0
2v5
v4
0
2v2
0
0
0
v
0
v
v
S
n v
0
6
B
0
n v 3
0
0
v6
0
v8
v9
v2
0
v4
v5
0
v7
0
0
0
v
0
v
v
n v 3
0
0
2v5
0
n v
7
3
0
0
2v
v4
0
0
v
v
5
6
0
9
D
2v2
0
0
0
v
vC
6
4
8
2v
2
B
0
0
C
n v 3
0
0
v6
0
vB
vC
v7
0
2v5
v4
0
2v2
0
0
3
0
0
v6
0
vB
vC
n v
0 0
n v 3
0
0
v6
0
v8
v9
v2
0
v4
v5
0
v7
0
0
0
v
0
v
v
0
8 C gyiNN j 5
S
n v 3
2G 9
0 2G
0
2
0
G
0 2G
0
11
16
0
6
D
n v ,
8
9
8 C gziNN j 5
A
D
S
,
D
,
~F2!
,
D
0
S
D
n v 3
~F1!
,
0
0
v
n v
0
v7
S
11 10 G G9! C gziNV j 52 ~ G
3
0
C gziNj 8 V 52 ~ G 162 G 12 G 2 !
5 12 G3! C gxiNV j 52 ~ G G
n v
0
S
D
0
v7 v
609
,
D
,
~F3!
,
0
2G 3
G
0
7
G
0
5
0
D
n
G 14 v 1 . ~F4! 0
2. Quadruple-spin observables
Quadruple-spin observables involve the spin of all four particles, including the vector and tensor polarization of the vector meson. Of course all of these involve the vector meson, including its vector and tensor polarization; hence, the 338 Cartesian display appears again. The question arises: Are all of the quadruple-spin observables redundant? In the pseudoscalar meson case the full case of triple-spin observables was redundant in that it was were all determined by single- and/or double-spin observable measurements. In the vector meson case, we have
C gxxiNNj 8 V 5 ~ G 152G 9 2G 8 !
n
NN 8 V 52 ~ G 2 G 1 G 16! C gxyi j
S S
0
n v 3
0
0
v6
0
vB
vC
v7
0
2v5
v4
0
2v2
0
0
0
0
v
0
v
v
0
n v
v7
0
2v5
v4
0
2v2
0
0
3
6
B
C
0
n v 3
0
0
v6
0
vB
vC
v7
0
2v5
v4
0
2v2
0
0
D
D
,
,
610
PICHOWSKY, S¸AVKLI, AND TABAKIN
C gxziNNj 8 V 5 ~ G 132G 112G 6 !
NN 8 V 5 ~ 2G 15G 9 G 8 ! C gyxi j
n
S
S
NN 8 V 5 ~ 2G 2 G 1 G 16! C gy yi j
0
n v 3
0
0
v6
0
vB
vC
v7
0
2v5
v4
0
2v2
0
0
3
0
0
v6
0
vB
vC
2v5
v4
0
2v2
0
0
0
v
0
v
vC
n v
0
v7
0
n v 3
0
7
0
2v
0
n v 3
v7
0
0
v
S
S S
n v 3
0
v7
C gyziNNj 8 V 5 ~ 2G 132G 11G 6 !
n
0
0
0
v6
2v5
v4
0
0
0
v
2v5
v4
0
n v
0
0
v
v7
0
2v5
v4
0
v2
2
B
0
0
vB
vC
2v2
0
0
0
v
v
B
0
0
v
vC
2v2
0
B
0
0
v4
v5
0
v7
0
0
n v 3
0
0
v6
0
v8
v9
v2
0
v4
v5
0
v7
0
0
D
0
n v 3
0
0
v6
0
v8
v9
v2
0
v4
v5
0
v7
0
0
0
v
v
0
3
n v
0
0
v
v2
0
v4
v5
0
v7
0
0
3
0
0
v6
0
v8
v9
0
v
v
0
v
0
0
0
v
n v
2
@1# J. T. Dakin, G. J. Feldman, W. L. Lakin, F. Martin, Martin L. Perl, E. W. Petraske, and W. T. Toner, Phys. Rev. Lett. 30, 142 ~1973!. @2# J. M. Laget and R. Mendez-Galain, Nucl. Phys. A581, 397 ~1995!. @3# P. L. Cole et al., M. Anghinolfi et al., A. R. Dzierba et al., and V. Burkert et al., CEBAF approved experiments. @4# E. M. Henley and A. G. Williams, Phys. Lett. B 281, 178 ~1992!. @5# E. M. Henley, G. Krein, J. Pollock, and A. G. Williams, Phys. Lett. B269, 31 ~1991!. @6# A. I. Titov, Yongseok Oh, and Shin Nan Yang, Chin. J Phys. 32, 1351 ~1994!. @7# Frank Tabakin, Robert A. Eisenstein, and Yang Lu, Phys. Rev. C 44, 1749 ~1991!. @8# C. G. Fasano, Frank Tabakin, and Bijan Saghai, Phys. Rev. C
4
5
7
8
9
,
,
D
0
0
S
C
2v2
6
6
D
0
6
0
S
C gzziNNj 8 V 5 ~ 2G 8 G 5 2G 15!
2v
4
3
C gzxiNNj 8 V 5 ~ G 6 2G 3 G 13!
C gzyiNNj 8 V 5 ~ G 7 2 G 4 G 14!
0
v
5
6
D
53
D
,
,
,
D D
,
.
~F5!
46, 2430 ~1992!. @9# Bijan Saghai and Frank Tabakin, Phys. Rev. C ~to be published!. @10# R. L. Walker, Phys. Rev. 182, 1729 ~1969!. @11# I. S. Barker, A. Donnachie, and J. K. Storrow, Nucl. Phys. B79, 431 ~1974!. @12# I. S. Barker, A. Donnachie, and J. K. Storrow, Nucl. Phys. B95, 347 ~1975!. @13# M. Simonius, Phys. Rev. Lett. 19, 279 ~1967!. @14# H. E. Conzett, Rep. Prog. Phys. 57, 1 ~1994!. @15# G. R. Goldstein, J. F. Owens, III, J. P. Rutherfoord, and M. J. Moravcsik, Nucl. Phys. B80, 164 ~1974!. @16# M. Jacob and G. C. Wick, Ann. Phys. 7, 404 ~1959!. @17# C¸etin S¸avkli, F. Tabakin, and S. N. Yang, Phys. Rev. C ~to be published!.
