Analysis of structures in p-p spin correlation measurements for 90°c.m. geometry at intermediate energies

Nuclear Physics A370 (1981) 46878 © North-Holland Publishing Company

ANALYSIS OF STRUCTURES IN p-p SPIN CORRELATION MEASUREMENTS FOR 90~. GEOMETRY AT INTERMEDIATE ENERGIES A . ~VARC and rL. BAJZER "RudjerBoskovié" Institute, Zagreb, Yugoslavia and M . FURIE Prirodoslovreo-matematicki fakulte~ Marulicev ng 19, University of Zagreb, Zagreb, Yugoslavia Received 26 March 1981 (Revised 25 May 1981) Abstract: Simplifications in the relationships between spin-dependent observables and components of helicity amplitudes are exhibited at 90~m. . Moduli for three components of definite spin are determined numerically from existing experimental data . Additional measurements of cross sections are suggested to clarify the origin of structures observed in the data.

1. Introduction The study of excitation functions for spin-dependent observables in protonproton systems has revealed unexpected behaviour . Prominent structures in dam,,, d~T and A,~,(90~,m.) in the region of incident momenta from 1 to 3 GeV/c were observed by the Argonne t .2) and Rice University 3) groups . A phase-shift analysis of an extended set of data performed by Hoshizaki indicated the existence of tD2 and 3F3 partial-wave resonances 4). Alternative interpretations of structures exhibiting no resonant behaviour were also given 3-'). General agreement upon the origin of the phenomena was not reached. Recent Rice University data added another, yet unexplained, sharp enhancement a) in the excitation functions for A (90~.~.)~ In this article we make a model-free analysis of spin correlation observables in 90~.m. geometry. We decompose the helicity amplitudes at 90~.~, into three parts. The moduli of these three parts are shown to be uniquely specified with only three observables at 90~,m.: d~/dt, AL~(90~ .m.) and A,,v(90~.,~,). This procedure extracts exact information on the three parameters from the available incomplete set of experiments. Besides, another advantage of the suggested amplitude analysis is that the structure in the observables can be attributed entirely to amplitudes of well-defined spin without applying the cumbersome and delicate overall fitting procedure. Our approach offers direct insight into the mechanism of how the 468

A. ~`uarr et al . / pp spin correlation

469

structures could be created as sheer interference phenomena, without invoking the assumption of resonance. Available experimental data are used to perform an explicit numerical analysis . Although the surprising behaviour of A,,(90~.m,) could be interpreted as an interference phenomenon, it is Hoshizaki's recent phase-shift parametrization, with a resonant 'DZ partial wave, which comes fairly close to reproducing such behaviow . We find that the existing data on d~/dt at 90~,m. [refs. 9-11 )] are neither entirely consistent nor available in sufficiently fine steps in the domain of interest . We therefore suggest additional unpolarized experiments at 90~,m, to resolve the issue of the observed structures . 2. Formalism The analysis of spin-dependent observables in p-p elastic scattering can be performed in terms of the s-channel helicity amplitudes (~tiAi~M~Ai.12) . Their iz) : expansions over the total angular momentum J are shown to be the following

_ ~ {(2J+1)MJ +JMJ_l,J +(J+1)MJ+i,J -2 J(~NJ}PJ(x), J even

_ ~ {- (2J+1)MJ +JMJ_l,J +(J+1)MJ +I,J-2 J(~NJ}PJ(x), ! even

J even

{(J+ 1)MJ_ 1.J +JMJ+i,J +2 J(~NJ }dii (x) + ~ (ZJ+1)MJJdii(x), J odd

J even

(1)

{(J+1)MJ_l .r+JMJ+i,r+2 J(~NJ}d;_1(x) -

where x = oos Be.m ., momentum .

t = -2g2(1-cos

6~.~n.),

~ (2J+1)MJJdi-i(x)~ J oaa

s = 4(mP +q 2) and q is the c.m .

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A. Srvarc et al. / pp spin correlation

The partial-wave projections of the M-matrix, MJ, MJtt.J, MJJ and NJ, as well as the d-functions are given in detail in ref. tz)+. For Bc .,n. = 90° , the helicity amplitudes can be simplified as follows: fit

=f+8 .

fi2 = -f+g~

fi3 = ~ B(J)KJ + ~ C(J)MJJ, J even

7 odd

fi4 =

-fi3 r

fis=0,

where

B(J)=(J+1)MJ _ t,J +JMJ+t ..r +2 J(~NJ , t KJ = for even J,

(iJI

_i 2 C(J)=(J+1)(2(J

_i

for odd J.

1)1 + \z(J+1)1

The five helicity amplitudes are not mutually independent; they are reduced to three non-trivial terms, fit, fit and fi3 or equivalently f, g and fi3. The f-function contains only the spin singlet part of the partial-wave projections of the M-matrix, while g and fi3 contain the contributions of the triplet part . Using the existing expressions for experimental variables in terms of helicity amplitudes t2) together with the set of equations (2), we obtain formulae for spin-correlation parameters and the cross section at 90~.m.: d_~_ ~ I, dt 4q 4

I=It12 +IsI 2 +Ifi3I 2 ,

IA,v, IA~,ç

and

'

= Is12 +Ifi31

2

-If1 2 ,

= ISIZ-IfI 2 -Ifi3I

2

(3)

,

IA~L=Ifi3I 2 -IfI 2 -IgI 2 ;

IK~ - 2 Re (ffi3 ) cos e~gebco ; ; Note the missing parenthesis in the expression for di g (x).

.

(4)

A. Swarc et al. / pp spin correlation

471

As follows directly from the set of equations (3), the observable ANN at 90~.m. can be expressed in the form ANN =1- II11 2 . The structures in ANN at 90~.m, can therefore be understood as an interplay of the square of the absolute value of the singlet part of helicity amplitudes and the differential cross section, d~/dt, at 90~ .m . . Using the set of equations (3), one can express the singlet and both triplet parts of the differential cross section in terms of the observables ANN, ALL and d~/dt at 9°.m. = 90°. This enables one to perform the analysis in a representation where the singlet and triplet contributions are separated explicitly :

~Î~2

~~hs~

=

2=

29

4

~(1 -ANN)

,

4

24 ~(1+A LL )

A very indicative relation for the sum of the triplet parts of the differential cross section can be obtained directly from (6): 2

a

IBIZ +I~3IZ= ~tT(1+ANN)

"

(7)

Several physically significant characteristics of the 90~.m, geometry follow from the preceding considerations . (i) The number of independent amplitudes is reduced to threes because of the symmetry properties and identity of the particles:

(ü) Moduli of helicity-amplitude components are uniquely determined by a subset of observables d~/dt, Ac.c.(90~.m.) and ANN(90~.m .), eq . (6). Their relative phases are contained in DNN and Kss [eq. (4)J . (iii) The singlet part of the cross section is proportional to Q(1-ANN ), while the triplet part is proportional to ~(1+ANN) . (iv) Expression (s) indicates that a peak in ANN can appear not only because of the enhancement in the triplet part, but also because of the minimtun in the singlet part . ' Similar observations, although in different formalisms, were made by other authors ") .

472

A . Swarc et al. / pp spin conelation

so~~ 90°

40

x

o

T

0

U

K.ABE et al . D.T.WILLIAMSetal . M.G .ALBROW etal .

-->K--o~ _ Q_ 1,2

1,3

1,6 P1ab~

1, 5

GdV/c )

1,6

Fig . i . Momentum dependence of the cross section ~(90~.m.) . Solid circles are from ref. 9), crosses ere from ref . i~ and open circles are from ref. 11 ) . The broken line represents such behaviour of v(90~, m.) which results in a smooth momentum dependence of the ringlet . The dash~otted line is as analogue with smooth triplet behaviour .

For the sake of completeness, we remind the reader that only the non-central part of the triplet interaction contributes at 90~.m.. Large values of A,vnr(90~.m .), i.e. the large triplet/ringlet ratio, reflect the important role of spin in nucleon-nucleon dynamics . 0,9 Oa z0,7 z

a

O,B

OS

0 ~

~

1,1

1,Z

1,3

LG

P 1ab

L5

(G~Vk l

1,B

1,7

1,8

Fig . 2 . A n  data points are from ref . s). The curve was obtained assuming a gauasian distribution with the smooth background .

A. Ssvarc et al. / pp spin correlation

47 3

0,3-

Fig. 3. ALL data points are from ref. Z). The curve was obtained assuming a gaussian distribution with the smooth background .

Fig. 4. Momentum dependence of the singlet amplitude ~I~ Z . Points are obtained directly from the data. Solid, broken and dash-dotted lines were obtained assuming a smooth momentum dependence for 0-(90~,m,), ~fjZ and ~g~2+~~3~Z, respectively. The dotted line was calculated from phase shifts of ref. °).

47 4

A. Ssvarc et al. / pp spin correlation

3. Numerical analysis of available data Figs . 1-3 summarize the experimental data on rr(90~,m,), Annr(90~,m,) and A,s(90~,,n,) . The data on the unpolarized cross section (fig . 1) contain two sets 9'to), which match smoothly while the third set does not 11 ). Experiments at ~(90~,m,) were not performed in sufficiently fine momentum steps as compared to the a structures observed in ALA and A,vr,. Fig. 2 shows values of Anne [ref, )] . Since ref. s) provides a recent systematic wide-range A,r, experiment, we have chosen to fit these data . The fit is shown in fig. 2; it is used as a representative of A(90~,~,) behaviour. Fig. 3 shows values for Ac,c,(90~,m,) . Using eqs. (6) and (7), we calculated values for ~}~ 2 , ~g~ Z and ~~3~ 2 from the above data points and under various assumptions on the behaviour of unpolarized cross sections . Figs . 4-7 show the calculated values . Our first approach was to assume a smooth trend for rr(90~,m,) and use the structure observed in A,m"(90~,m,) . This led to a dip in the singlet part and an enhancement in the triplet part (solid lines in figs. 4-7) . Such simultaneous effects in different spin channels are somewhat surprising as spin singlet and spin triplet channels are completely decoupled due to parity conservation in the strong interaction . We therefore examined two other

~i 9d°

1,i

1,5

1;8

p~~(GeV/C 1

Fig. 5. Momentum dependence of the wmponent ~g~2. The symbols âre the same as in fig. 4.

A. Swarc ti al. / pp spin correlation

475

m c_

e y d

_w

m d N

.

C

v d

ä

C

d s

ô

b b a v

e0

[Z(

~/

11aJ)gwlZl~ I

476

A. Swarc et al. / pp spin correlation

O N

T m N

G ~ouC d 8 e d E 0 ew

'w

N

O

A. Swarc et al. l pp spin correlation

4~~

extreme possibilities. We calculated the dashed lines in figs . 4-7 using the same Atr, values as for the solid lines, but assuming such behaviour of tr(90~.m .) as to obtain a smooth singlet part . The dash-dotted lines in these figures reflect the possibility that the triplet part may contain no structure. The required values of tr(90~,m.) are shown in fig. 1 with corresponding lines. We observe that the assumed values of ~(90~.m.) differ essentially for the various alternatives . The structure in A2,m,(90~.m.) can be a consequence of the singlet part of the cross section having a minimum and/or a triplet part of the cross section having a maximum. Our analysis emphasizes the need for additional systematic measurements of ar(90~.~ .) in finer momentum steps to resolve between essentially different possibilities. A minimum in the singlet part might be due tô an interference phenomenon . A maximum in the triplet part might be consistent with resonant behaviour. We have also calculated values for ~f~ 2, ~g~2 and ~fi3~Z from Hoshizaki's phase shifts . In figs . 4-7 we compare Hoshizaki's predictions (dotted lines) with the values obtained in this work . The curves bear resemblance in shape. Our computer calculations indicate that small variations of the 'D Z parameters (SZ changed from 1 .7° to 0.8 ° and elasticity from 0.707 to 0.67) at p,ab = 1 .34 GeV/c can either produce or destroy the dip in ~f~ 2. This fact also enlightens the delicate nature of the phase-shift approach. The structure in Ac.c.(90~.m .) observed by the Argonne group z) does not substantially affect the momentum dependence of either the singlet or the triplet part . In fact, the structure in AL~(90~,m,) is responsible for the structures in ~g~2 and ~~s~ Z studied separately ; however, its impact on the combined contribution of these two components vanishes .

4. Conclusions Specific properties of the 90~.,°. geometry have been used to connect a subset of spin correlation parameters and absolute values of helicity-amplitude components with defined spin . The data on A,mv(90~.m .), AI,Z,~90~.,°.) and ~(90~.m.) have been employed to calculate the excitation functions for the moduli of these components ; in this way a simplification in the data analysis has been achieved. The results obtained suggest a possible dip in the singlet part of the cross section (interference phenomenon) and an enhancement in the triplet part (consistent with resonance) at p~,°= 1.34 GeV/c. Additional fine-step measurements of unpolarized cross sections at 90~.m. are required to determine whether the structures observed reflect a resonânce or an interference phenomenon .

Note added in proof: During the submission procedure we learned of similar efforts made by Goldstein and Moravcsik 1° ) to extract exact information from incomplete sets of experiments.

478

A. S~oarc et al . l pp spin correlation References

1) LP. Auer, E. Colton, H. Halpern, D. Hill, H. Spinka, G. Theodosiou, D. Underwood, Y. Watanabe and A. Yokosawa, Phys . Rev. Lett 41 (1978) 354 2) LP. Auer, A. Beretvas, E. Colton, H. Halpern, D. Hill, K. Nield, B. Sandler, H. Spinka, G. Theodosiou, D. Underwood, Y. Watanabe and Y. Yokosawa, Phys . Rev. Lett. 41 (1978) 1436 3) Ed .K . Biegert, J.A. Buchanan, J.M . Clement, W.H . Dragoset, R.D . Felder, J.H . Hoftiezer, K.R. Hogstrom, J. Hudomelj-Gabitz4ch, J.D . Lesikar, W.P. Madigan, G.S . Mutchler, G.C . Phillips, J,B . Roberts, T .M. Williams, K. Abe, R.C . Fernow, T.A . Mules, S. Bart, B.W . Mayes and L. Pinsky, Phys . Lett, 73B (1978) 235 4) N. Hoshizaki, Prog. Theor. Phys . 61 (1979) 129 5) S. Minami, Phys . Rev. D18 (1978) 3273 6) C.L. Hollas, Phys . Rev. Lett . 44 (1980) 1186 7) D.W . Joynaon, J. of Phys. G2 (1976) L65 8) D.A. Bell, J.A . Buchanan, M.M. Calkin, J.M. Clement, W.H . Dragoset, M. Furié, K.A. Johns, J.D . Lesikar, H.E . Miettinen, T.A . Mules, G.S . Mutchler, G.C . Phillips, J.B . Roberts and S,E. Turpin, Phys. Lett . 94B (1980) 310 9) K. Abe, B.A. Barnett, J.H . Golduran, A.T. Laasanen, P.H. Steinberg, G.J. Mariner, D.R. Moffet and E.F. Farker, Phys . Rev. D12 (1975) 1 10) D.T. Willisins, LJ . Bloodworth, E. Eisenhandler, W.R. Gibson, P.I . Kalmus, L.C.Y . Lee Chi Kwong, G.T.J. Arnison, A. Astbury, S. Gjesdal, E. Lillethun, B. Stave, O. Ullaland and LL. Walkirre, Nuovo Ciur . 8A (1972) 447 11) M.G. Albrow, S. Anderson/Ahnehed, B. Bo~njakovib, C. Dauen, F .C . Erné, J.P . Lagnaux, J .C. Sens and F. Udo, Nucl . Phys . B23 (1970) 445 12) M. Arik and P.G . Williams, Nucl . Phys. B136 (1978) 425 13) M.H . MacGregor, MJ . Moravcsik and H.P . Stapp, Ann. Rev. Nucl . Sci. 10 (1960) 291 14) G.R. Goldstein and M.J . Moravcsik, Phys. Lett. B, to be published