Fortschritte der Physik / Progress of Physics: Band 24, Heft 5 1976 [Reprint 2021 ed.] 9783112520444, 9783112520437

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BSßOTSffiOOtPS BSlSfl@KIMTrBS©BfiIS« ffiEOTOOoOK W S IF. &ħ p 3 + 4. 1 denotes the beam, 2 the target, and we call 3 and 4 respectively the particles which share some physical properties with the beam and the target (for instance ,1 and 3 are mesons, 2 and 4 are baryons) so that the t- and M-channels are well defined. The 4-momenta of the particles are denoted by Pi(i = 1, 2, 3, 4) with + p2 = p3 + p4, their spins by ji and their masses by wi; (m; =f= 0).

M. G. DONCEL, P. MINUAERT and L. MICHEL

266

2.1. Co variant quantization systems To describe the polarization of the initial and final states one must fix a quantization frame for each spinning particle. Several different choices are possible, the most popular are the helicity and transversity frames in the s-, t- and «-channels3). Unfortunately there is not as yet a universal agreement on the definition of these quantization frames; the most usual conventions are the following. i) For each particle and for each channel, the transversity quantization axis Tn and the helicity second axis are along the "Basel normal" n to the reaction plane, defined by = 0

(» = 1,2, 3),

n2=—

1,

det(w, px, p2, p3) > 0

(2.1)

where the last condition is equivalent to n • p1 X p3 > 0 in the laboratory system, or in the center of mass system. ii) For each particle and for each channel the helicity and the transversity frames have the same first axis, Tn{-V> = %('), Note that with these two conventions, the transversity second axis rw')) = /I(Q, Q') Y^Q)

Y%,(Q') dQ dQ'.

(2.20)

Amplitude Reconstruction for Usual Quasi Two Body Reactions

271

2.4.3. Cascade decay angular distribution Consider the cascade decay C A -f B , A A l + -Bi (with spins j ( C ) = j , j ( A ) = 7(^1) = 1/2, j ( B ) = j { B ] ) = 0); the first decay is parity conserving and the second decay is parity violating, (e.g. 21* —> A n , A - » prc). We denote b y 0 and the angles of A with respect to the quantization frame of C , and b y 6 1 and 4>i the angles of A , with respect to the canonical quantization frame for A , deduced from the quantization frame of C b y a pure Lorentz transformation (boost). Then the cascade angular distribution is

E E C(L, L 7(0, 4>-e u M,N,M1

X ^ r ^ . ^ W i , ^ ) .

(2.21a)

Instead of the canonical quantization frame for particle A one m a y uste the helicity frame, deduced from the previous one b y the rotation JR((j>, 6, 0). We denote b y 0XA, 1A the angles of A x with respect to this frame, then the cascade angular distribution is 7(0,

V , W)

=

L=0

E

E C h{L, L l t Mx)Et LM

l / ^ r ^ M,Mi }/

0

d l(,

e, 0 ) $

5^,(0/, V ) . (2.21b)

The coefficients C { L , L u J ) in E q . (2.21a) and C h { L , L u M x ) in E q . (2.21b) depend on the spins and parities of the particle and on the dynamics of the decays, if these involve more than one amplitude. The most usual decay of this type is 3+/2 - > 1+/2 0~, 1+/2 - > 1+/2 0~. F o r this cascade decay the non vanishing coefficients G ( L , L u J ) and C h ( L , L u M J are

4 n C ( 0 , 0, 0) = 1,

4 n C ( 2 , 0, 2) =

4TIC(1, 1, 0) = « 1/5/9,

4JIC(1, 1, 2) = « -|/2/45

-1 (2.22a)

471(7(3, 1, 2) = —oc 1/7/5,

(4,71) C » ( 0 , 0, 0) = 1,

(4ar) C » ( 2 , 0, 0) =

( i n ) C * ( l , 1, 0) = « 1/1/15,

( 4 n ) C ( l , 1, ± 1 ) = «1/4/15

( 4 a ) C " ( 3 , 1, 0) = - o c 1/3/5,

( i n ) G h ( 3 , 1, ± 1 ) = - a ]/2/5

-1 (2.22b)

where oc is the asymmetry parameter of the second decay. With these known values of the coefficients, the tjfc parameters are deduced b y a best fit adjustment of the decay angular distribution or b y a moment analysis (¿2 = (0, (j>), Q 1 = (61, (p^, Q x h = ( 0 / , 1A))

C(L, L C"(L, L

u

u

J ) t M

1

L M

= E (JL1NM N,Mi

) % =

1

IL M ) { Y

JN

(Q) Y^(Qy)),

E ( D L ( 4 > , 6 , ry, rz are the Pauli matrices, and x, y, z are the projections of the vector g on the s-transversity axes n*1*, n(3> respectively. These components can be written £: x — PT sin \p,

y ='PL,

[z = jP r cos

(3.6)

where 4> is the angle between n and I with the sign of n X I • Pi, see fig. 1. By definition PT is the length of the projection of 5 on the (x, z) plane; it is the degree of transverse polarization 0 PT ^ 1. PL is the projection of 5 on the beam; it may be positive or negative and its modulus \PL\ is the degree of longitudinal polarization 0 iS \PL\ iS 1. Note that P r 2 + PL2 = is the degree of polarization of the target. It is important to remark that in general the initial state is not B-symmetric. Indeed the matrices 1 and t z are B-symmetric in transversity quantization but t x and r y are not. Then, except in the case of normal polarization, 5 = the initial state is not invariant by reflection through the reaction plane.

274

M . G . DONCEL, P . MINNAERT a n d L . MICHEL

3.2.2. The density matrix of the final state The density matrix of computed from eq. (3.1) with the initial state (3.5) is linear in the components of It can be written in the form '. 3.3.1. The double differential cross sections From the general form (3.7) of p) d f ,

3^/2 ai = J o{ip) dty,

-)t/2

(3.15 a)

n/2

and we use the notation In = / / « < ) # • o Then the asymmetry can be obtained by

(3.15 b)

'

(3 16b)

P r P t

~

(a(f) 2 cos f ) {am •

(3

-16c)

3.3.2. Production and single decay angular distribution Assume first that only one final particle undergo a two body decay (e.g. -rep -> TCA, A -> 7iN; or 7ip —pN, p 7t7t). Then the final state is characterized by 3 angles ip, d, 4>. Les us call T the reaction transition matrix and M the decay transition matrix. The normalized combined angular distribution is defined by FL, * ) =

.

7

(3.17)

/ tr MTq,TW^ d(cos 0) d dtp From eq. (3.1) this can be written tr M — q,M1 I(*P;0,) = • j tr M — eyJtft d(cos 6) d dip J o-o

(3.18)

Then, by comparison of the multipole expansion of a/a0 Qf (cf. eq. (3.10)) with the expansion (2.8) and from the usual decay angular distribution (2.16) one gets the combined normalized angular distribution -i

+

p

f C 4jt

0 S

U S c { L ) i L {WM L= 1 M=-L

+ PT sin r'tLM + PlK}

YLM(e, 4>)

+

pTooSrFM (3.19)

where the coefficients C(L) are defined in section 2.4 and are, for the most usual decays, well known numerical coefficients (cf. eq. (2.17)). By inspection of this expression one sees that it allows the measure of the quantities PRPT, tjh + Plv^m> Pt'^m, Pt^m >

Amplitude Reconstruction for Usual Quasi Two Body Reactions

277

either by a best fit adjustment or by a moment analysis which yields PRPT = (2 cos 4>) = j I(i> ; Q) 2 cos tp dQ dtp C(L) (t\t + PL%) = -, Q) Yi(Q) dQ di,

C(L) PTHLM = (2 eos rpYf¡(Q)) =J

(3.20a) (3.20b)

I(4>; Q) Y^[Q) 2 cos ip dû dtp

(3.20 c)

C(L) PTxt\ = (2 sin >pY^(Q)) = j I{rp; Q) YLM{Q) 2 sin tp dQ dip

(3.20d)

with Q = (6, ) and dQ = e£(cos 6) d. Note that eq. (3.20a) is a particular case of eq. (3.20c) f o r i = 0 (with the conventions % o = pR, 0(0) = and is equivalent to eq. (3.16c). By different choices of the initial polarization i.e. of PT and PL, one easily deduces from these equations the value of the observables: i) The target is unpolarized, i.e., PL = PT = 0. One must first verify that the^angular distribution is isotropic around the direction of the beam. Then "eq. (3.20b) gives the B-symmetric parameters tjj, and one must verify that the B-antisymmetric moments vanish. (We recall that in transversity quantization B-symmetric parameters have M = even and B-antisymmetric parameters have M = odd). ii) The target is transversally polarized, i.e., PT 4= 0, PL = 0. Eq. (3.20c) gives the B-symmetric polarization transfer parameters and one must verify that the B-antisymmetric moments vanish. Similarly, eq. (3.20d) gives the B-antisymmetric polarization transfer parameters and one must verify that the corresponding B-symmetric moments vanish. Furthermore one may verify that eq. (3.20b) yields the same results as in case i). iii) The target is longitudinally polarized, i.e., PT = 0, PL =|= 0. One must verify that the angular distribution is isotropic around the common direction of the beam and of the polarization vector Then eq. (3.20b) gives the B-symmetric parameters t\t (which should be equal to the parameters obtained in i)) and the B-antisymmetric polarization transfer parameters iv) The target is arbitrarily polarized, i.e., PT 4= 0, PL #= 0. One obtains all the parameters. Eq. (3.20b) gives the B-symmetric parameters tLM and the B-antisymmetric parameters vtML. Eq. (3.20c, d) give the B-antisymmetric parameters H\t and xtl,, and one may verify that their B-symmetric moments vanish. Of course, if the decay is parity conserving, this analysis yields only the L = even parameters (cf. section 2.4.1). The moments with L = odd must be found compatible with zero. 3.3.3. Production and joint decay angular distribution If both final particles (3 and 4) are untable and undergo two body decay (e.g. Trp -> pA p 7T7r, A Nn) the combined production and joint decay angular distribution is (cf. eq. (2.9, 2.19, 3.11)) ; 0, 4>, 0', 4>') = ~ £ X ¿TI L=0 L'= 0

C(L) C'(L') X

+ PT sin rtttu- + W 20*

M=-L

)

M' =

X

-L'

{tK'

+ Pt COS

Ylm(6, ) Y%,(6', ')

(3.21)

278

M . G . DONCEL, P . MINNAERT a n d L . MICHEL

where C = 1, = PR C(0) = C'{0) = 1/]/4TT, 7 0 ° = l/]/4n. The moment analysis of this distribution gives (Q = (0, ), Q' = (0', ')). C(L) C'(L') [TWM. + PL'*WM-) = (N(®) C(L) C'(L') PTHTFM. =

(2 cos * TI{Q)

YLMW))

(3.22)

Y^Q'))

C(L) C'(L') PT*T%M, = Y&Q)

(£>')>.

A similar discussion to that of the preceding section 3.3.2. can be made. We shall not repeat it. We only recall that in the present case, the B-symmetric parameters (in transversity) have M -f M' = even and the B-antisymmetric ones have M + M' = odd. Furthermore, if both decays are parity conserving, one gets only the L = even and L' = even parameters; all the moments with L or L' odd must vanish. If the decay in (D',4>') IS parity violating one gets the multipoleparameters with L — even and U = even or odd. All other moments vanish. 3.3.4. Production and cascade decay angular distribution Assume again that only one final particle decays, but that it undergoes a cascade decay of the type discussed in section 2.4.3. (e.g. Kp -> 7tS*, S * AN, A -> pu). Then one may study the angular distribution of the production and of the cascade decay. It reads (cf. eq. (2.21a) for the canonical quantization frame and (2.21b) for the helicity quantization frame). UR, 0, 4> ; EU fa) = ^

i?

¿71 L=0

H =

E

0 J even

+ PT sin

C(L, LUJ)Z

(.JL, NML I LM)

+ PT cos

M,MltN

+ PTFT^} Yjn{B, ) Y^(6,

FA)

(3.23a) ; 6,;

fa")

=

J?

E

L = 0

X i = 0

C"(L, L„ MJ £

+ PT cos

M.MI

+ PT SIN * ^

+ PLWM)

DI®,

0, O f t

fa").

(3.23b) where the coefficients C{L, LU J) of Eq. (3.23a) and CH{L, LU MJ of (3.23b) are given in Eq. (2.22 a) and (2.22 b). With these known values of the coefficients, the parameters TML and "TML(TX = X, Y, Z) are deduced by a best fit adjustment of the combined angular distributions or by a moment analysis {Q = (0, ), QX = (0,, fa), QX" = (0/, fah)) ' C(L, LU J)

+ PJNFO = E {JLTNMI N,MI

I LM) (YJN(Q)

C(L, LU J) PTHLM = £ (JL1NM1 H.MI

I LM) (2 cos * YJ, (fl)

C(L, LU J) PTHLM = £

| LM) (2 sin , 6, 0 ) ^

G»(L, Llt M,)

, 0, 0 ) ^ 7 ^ * ) ) .

=

279

V)>,

(3-24b)

A discussion identical to that of Section 3.3.2. can be made. Note however that in this case all parameters (L = even and L = odd) can be measured. 3.3.5. More complex combined production and decay angular distributions One may consider more intricate situations. For example, if the two final particles are unstable and one of them undergo a cascade decay (e.g. Kp pE*, p -> rnz, E* -> An, A p-rc) the complete angular distribution involves 7 angles. Still more complex is the case where both final particles undergo cascade decays (e.g. Kp -> A^*, A1 -> prr, p nn, 2 * -> Arc, A pjr). Then the complete angular distribution involves 9 angles. The expressions of such angular distributions are easily written down, however the present day experimentalists are not yet interested in such complex reactions with polarized target. 4. Amplitude Reconstruction in Usual Reactions

In the previous sections we have shown the way of measuring the observables of a reaction with unpolarized target (section 2) or with polarized target (section 3). They are embodied in the final polarization ao/ or in the transfer polarization matrix W, which are quadratic expressions of the transition matrix T or f , namely (c.f. eq. (2.14) and (3.9')). < 7 0 / = - J - T T t ,

Zt

W=^-Tf ¿t

t.

We call amplitudes the elements of these transition matrices. Their reconstruction consists essentially in obtaining an explicit expression for T or T by inverting the quadratic expressions TTt or f f i . Theoretically this can easily be done, and one obtains T or i up to some unknown phases (by the procedure of "conventional amplitude reconstruction" of Appendix 1,2). Practically each concrete case needs a separate study since generally the observable matrices OQJ or W are not completely measured. In this section we present the practical method of reconstructing the amplitudes in usual reactions with unpolarized and/or polarized target. For pedagogical purposes we first recall the method of reconstructing the amplitudes in the simplest reaction type 7rp —> KA, by measurement of the classical Wolfenstein parameters P, R, A. In view of further generalizations to higher final spins, we introduce, already in this simple case, the multipole formalism and some complex spin rotation parameters. This is discussed in section 4.1., and summarized in table 4. In the following sections we present the details of the generalization to reactions of the types 7tp KE*, nA, K*A, pN as simple comments to the tables 5 to 10, in which all the recipes for measurements and amplitude reconstructions have been gathered.

280

M . G . DONCEL, P . MINNAERT a n d L . MICHEL

Table 4 Amplitude reconstruction for reactions of type Tip KA with polarized target (Comparison with the PRA parameters and the spin flip and non flip helicity amplitudes)

a) Combined production and decay angular distribution and measurement of the polarization transfer by the method of moments. Ity, 6) = i- £ C(L) £ [(£ + PT{cos Wit + sin ¿Jt L M

+ Pj^f}

YLM(H)

C(L) (t* + Pjvtfo =