Phase Shifts in Colliding-Beam Experiments

P H Y S I C A L R E V I E W D V O L U M E 6 , N U M B E R I 1 J U L Y 1 9 7 2

Measuring T-n Phase Shifts in Colliding-Beam Experiments*

C a r l E . Car l son Enrico Fevlni Institute, C?ziversity o f Chicago, Chicago, Illinois 60637

and

Wu-Ki Tung Enrico F e r m i Institute and the Department oj- Phys ics , C?ziz,ersity of Chicago, Chicago, Illinois 60637

(Received 7 February 1972 )

The possibility of exploiting the unique features of the two-photon process ee - eenn in colliding-beam experiments to measure the even partial-wave a-n phase shifts i s examined in detail. The commonly used Weizsgcker-Williams approximation i s shown to be inadequate. An exact procedure of phenomenological analysis for extracting the relevant phase shifts i s formulated. Dynamical (model-dependent) calculations a r e performed separately to study the expected effects due to the final-state pion-pion interaction. The sensitivity of the measured cross sections to various prominent features of the s-wave n-n phase shifts (scattering length, existence of the scalar-meson resonance) i s investigated.

I. INTRODUCTION

It has been generally recognized by now that in the forthcoming generation of e 'e - and e-e- colliding-beam experiments a t energ ies of 3 GeV o r higher, the two-photon processes will play a domi- nant role . Calculations based on quantum e lec t ro- dynamics indicate that the c r o s s sect ion f o r this new type of p r o c e s s becomes l a r g e r than that of the single-photon annihilation processes (which has dominated colliding-beam physics up to now) a t about 1.5-2.0 GeV energy p e r beam.'-3 Motivated by these developments, we studied, in a previous paper4 (hereafter r e f e r r e d to a s I), the genera l fea tures of two-photon p r o c e s s e s in colliding-beam experiments fo r a r b i t r a r y hadronic final s ta tes5 and investigated the new possibilities offered by this type of p r o c e s s in studying hadron physics. We pointed out the possibility of probing the con- stituents of a near-mass-shel l photon by the other one in the deep-inelastic region. This was a l so done by other groups and, in part icular , in some detail by Brodsky et ~ 1 . ~ We a lso emphasized the unique possibilities fo r studying the s-wave n-n interaction in the kinematically mos t favored ex- clusive channel, the two-pion production ~ h a n n e l , ~ i.e., ee - eenn. It is toward a detailed study of this la t ter problem that this paper is devoted.

The uniqueness of this react ion (essentially y y - nn) l i es in two facts : (i) There is no other final- s ta te hadron except the two pions (in contrast to a l l s tudies of the nn sys tem in pure hadronic react ions where a baryon is always presen t in the final s ta te) ; (ii) because the initial s ta te has positive charge conjugation, the two pions mus t have

J ~ = O + , 2 + , . . . and I' = 0 + , 2+

(in con t ras t to the corresponding one-photon annihilation p r o c e s s where J' = I - , 1' = 1'). The ad- vantages a r e obvious: Because of (i), no theoret i - cally ambiguous final-state correct ion effects (e.g., between the outgoing nucleon and pions) have to be taken into account in o r d e r to ex t rac t information on the n-n interaction; because of (ii), the domi- nant p-wave interaction ( p meson) is absent, allow- ing a n ideal chance to study the s-wave n-n interaction cleanly over a l a rge energy range. In our previous work4 (I), we outlined a phenomenological procedure to ex t rac t the s-wave n-n phase shif ts 6: and proposed a model to calculate the y y - nn amplitude when 6: a r e given. Some related works8 have a l so appeared in the l a s t year . In a l l these works (including I) one a s s u m e s the s tandard equivalent-photon o r WeizsHcker -Williams (WW) approximation formula which reduces the ee - eenn p r o c e s s to that of y y - nn for on-shell photons. It is not hard to s e e , however, that although the WW approximation is appropriate fo r order-of-magnitude calculations (which w e r e what the or iginal in- vestigations intended), i t can induce e r r o r s of up to 5w0 in the differential c r o s s section. This is comparable to, o r , in cer tain regions, bigger than the effects due to n-n interactions which one t r i e s t o isolate in the present case. In o r d e r to extract useful information f r o m this p r o c e s s one clear ly needs to improve the approximation scheme.

In this paper we do th ree things: ( i ) We investigate the range of validity of the WW approximation and propose subst i tutes fo r it which a r e appropr i - a t e fo r the purpose of extracting information on n-n interactions; (ii) we propose phenomenological

C . E . C A R L S O N A N D W . - K . T U N G

FIG.

-- -c- q I . -. .

4; 1. The two-pion production process via two-pho-

ton exchange in e-e collision. The dashed lines a r e electrons, wavy lines a r e virtual photons, and solid lines a r e pions.

procedures for extracting n-n phase shifts which a r e a s model-independent a s possible and discuss built-in checks; (iii) we present model calculations which illustrate the effects of the 8-8 interaction on the measured c ros s sections and test the sensitivity of the lat ter to certain qualitative features of the s-wave n-a phase shifts, for instance the value of the scattering length and the existence of the a resonance.

An attempt i s made to make this paper reason-

FIG. 2 . (a) Kinematics of process (1) defined in the laboratory frame. The particle momenta correspond to those of Fig. 1. The quantities in parentheses after the momentum labels a r e the associated energy variables in the lab. (b) Kinematics of process (1) defined in the c.m. frame of the two outgoing pions. The pion momenta p 1 and p2 a re in the x-z plane and a re not shown in the figure to avoid confusion. x and X' a re the respective azimuthal angles of the planes of scattering for the two pai rs of leptons. The variables 4 and $' cannot be given a simple geometrical interpretation in this figure (see Ref. 9).

ably self-contained. We try to make the procedure scattering angle of a third particle, say a pion,

and reasoning clear and the results explicit with- which also defines the x-z plane [Fig. 2(a)]. For

out giving too many details which either can be calculational purposes, i t is more convenient to

found in I o r a r e relegated to the Appendixes. choose a s independent variables k2 and q2 (the two

11. EXACT FORMULATION OF THE SCATTERING AMPLITUDES AND CROSS SECTION

Consider the process

via the two-photon-exchange mechanism. This i s depicted in Fig. 1, which also specifies our momentum variables. The kinematics of this process i s fully specified by 8 independent variables. In the laboratory (the c.m. system of the incoming leptons) these can be chosen a s E, the incoming energy; E , 6, 4, E' , B ' , $', the energy and angles of two outgoing particles, say the two leptons; and 8" the

virtual-photon masses); the "hadronic variables" s (the total c.m. energy squared) and 8, (the sca t - tering angle) for the process yy - nn in i t s c.m. frame; and "lepton variables" ($, X ) and ($', x') which specify, respectively, the configuration of each of the two pa i rs of lepton momenta in the " res t frame" of the corresponding virtual photon [i.e., the frame in which the photon 4-momentum has only one nonvanishing component, Fig. 2(b)]. These two se ts of variables a r e defined in Fig. 2 and in I; the relations between them a r e given in Appendix A of I. We shall write out these relations explicitly when needed in subsequent discussions.

The exact amplitude for the above process corresponding to Fig. 1 can be written a s

Decomposing the virtual photons into their helicity components, each of the two vector products can be written as

where ~ ( ' ~ ) ( k ) , m = 1 ,0 , -1, a r e the helicity polarization vectors associated with the virtual photon k. Substi- tuting into (21, we obtain the following physically transparent expression for the scattering amplitude:

M E A S U R I N G n - n P H A S E S H I F T S I N C O L L I D I N G - B E A M . . .

where

i s clearly the helicity amplitude for the "hadronic process":

~ ( k , m) +y(q, a)- s(Pl) +n(P2) .

The cross section for e e - eenn i s

where the same factorized form a s in Eq. (4) i s explicitly displayed. We have

=invariant phase-space element for the outgoing leptons,

d o , =solid angle element in c.m. system of nn . In Eq. (7), the leptonic matrix elements which a r e known functionsQ of the leptonic variables a r e completely factored from the hadron amplitudes which a r e unknown functions of k2, q2, and the other hadron variables. This formula gives the exact connection between the measured ee- eenn cross section and the "density matrix" w,,,, for the yy- nn process. The diagonal elements of this matrix UTmn,,, a r e clearly just proportional to the c ros s section for yy - nn, Eq. (6).

In principle, by measuring the distributions in the pure lepton variables (+, X, #', X ' o r their laboratory equivalents) one can isolate a l l the elements WIJ,, , and f rom them determine the scattering amplitudes T,, for W - nn. We emphasized in I that because the kinematics favor the low-energy region and because of the exclusion of the strong p wave by charge conjugation, the s partial wave plays the dominating role for this process. Since in the elastic unitary region the phases of the partial-wave amplitudes a r e just those of the n-s scattering,'' one should be able to get very interesting information on the s-wave n-n phase shifts.

The exact relation Eq. (7) contains 20 distinct te rms a s given in (A9) of I. In practice, one needs to simplify the expression in order to make the problem more tractable, both experimentally and theoretically. One way is to average over the azimuthal angles x and x ' . [These a r e the azimuthal angles of the scattering planes of the two pai rs of leptons with respect to the scattering plane of the strong process yy - nn in the c.m. frame of the latter, Fig. 2(b).] We get

where

d p ' = d p / d x d x ' ,

C O S ~ ~ = -q.(k, tk , ) / [ (k . q)' -J22q2]112, (10)

C O S ~ $ ' = -k . (q, + q2)/[(k q)' - k2q2]'12 . Equation (9), which i s still an exact expression

for our process [Eq. (1) and Fig. 11, serves a s the basis for our subsequent discussions.

So far , we have discussed only one diagram (i.e.,

I

Fig. 1) contributing to process (1). In principle there a r e other diagrams of the same order in a which could also contribute to this process, e.g., Figs. 3(a) and 3(b). It i s clear, however, that diagrams of the type Fig. 3(a) a r e a factor cu down from the single-photon annihilation processes which themselves a r e insignificant at high energies. Diagrams of the type Fig. 3(b) have been studied by several and shown to be negligible a s compared to the main diagram, Fig. 1,

C . E . C A R L S O N AND W . - K . T U N G

FIG. 3. Additional diagrams of the same order in o! a s Fig. 1 which may contribute to process (1).

a s long a s one does not restr ict oneself to the large k2, q2 region. None of these should create a s e r i - ous background problem for process (1) for ener - gies of the order E 2 3 GeV.

111. THE W APPROXIMATION AND ITS RANGE OF VALIDITY

It i s customary to approximate Eq. (9) by noting that the small virtual-photon mass region k2- 0, q2-0 is most important. Setting k2=0, q2=0 in the square bracket in Eq. (9), two significant s im- plifications occur: (i) The hadron amplitudes involving longitudinal photons vanish, thus only the f i r s t te rm survives; (ii) the kinematics simplify, thus the lepton lab scattering angles 8 = 8' = 0, and one can show

After integrating over al l the lepton variables, one gets the standard f o r m ~ l a ' - ~ . ' ~ :

where d~(~y) /d (cos8 , ) is just the on-shell yy- nn differential c ross section. The great majority of recent works on process (1) a r e based on the use of (12), which allows one to concentrate on the simpler process yy - ns with on-shell photons.

What i s the range of validity of Eq. (12)? To answer this question, we have studied in some detail the case of pointlike pions, for which one knows the exact form of the matrix elements T,, (and thus W, ,,,) . I3 One can then calculate the cross section from the exact formula, Eq. (9). The relevant Feynman diagrams a r e given in Fig. 4. The calculations a r e quite involved. We relegate the details to Appendix A and only discuss the r e - sults here.

Making use of some previous results of Baier and Fadin,14 our detailed calculations yield

i a ) ( b ) ( c )

FIG. 4. Gauge-invariant "Born diagramsJ' for pointlike pions.

where

p =pion mass . Equation (13) i s accurate up to te rms which a r e quadratic in the large logarithmic factors L and L,. Terms linear in these a s well a s te rms of the form ( S / ~ E ' ) L ~ o r (s/4E2)LL, which a r e small in the energy range we consider a r e neglected. The standard WW formula, Eq. (12), corresponds to keeping only the te rm L2(Ls -$)A on the right-hand side of Eq. (13) and approximating L2 by

L2=41n2(E/m).

To get some feeling about Eq. (13), let u s substitute some typical values of the variables. Thus, for E = 3 GeV, ~ = 6 = 2 . 2 5 ~ we get

The coefficient of the A term in (13) comes out to be -880 while the corresponding number obtained from the WW formula, Eq. (12), i s -1300. The difference i s -50% of the true answer. The main correction effect comes from the -$LL, term. The second term i s the square brackets and the B term make relatively small corrections.

Previously, Brodsky et a1 .3 investigated the val-

6 - M E A S U R I N G r - n P H A S E S H I F T S I N C O L L I D I N G - B E A M . . .

idity of the WW approximation for the simpler case IV. EXTRACTION OF r-r PHASE SHIFTS-

of single no ~roduct ion . They found the exact r e - PHENOMENOLOGICAL ANALYSIS

sult t o be about 25O/0 higher than that given by Eq. (12) and the effect i s reduced by inserting a form factor for the no vertex. The fact that the corrections to Eq. (12) in these two cases occur in opposite directions can be readily understood. This i s explained in a footnote.15 We shall show later on that strong-interaction effects a r e not likely to r e - duce the size of the corrections to Eq. (12) for our case.

Clearly, if one hopes to extract useful information from the two-photon process ee - eenn, 5% inaccuracy can not be tolerated and Eq. (12), a s i t stands, i s useless. One must go back to the exact formula, Eq. (9).

This i s not necessarily a setback a s it might ap- pear to be. A moment's reflection will reveal that the simple formula Eq. (12) cannot be used even if it were accurate. The reason i s that to make certain that one i s indeed measuring the ee - eenn process, one has to tag one of the outgoing elec- t rons in addition to detecting the pions. Only in this way can one exclude the one-photon annihilation background a s well a s eliminate the possibility of producing additional neutral particles. The WW approximation formula, Eq. (12), having integrated over a l l lepton variables, does not describe this situation. One will have to go back to the complete differential formula Eq. (7), o r the partially integrated formula Eq. (9). We believe the latter i s a more practical choice.

Two additional points a r e worth mentioning: ( i) The deviation from the WW approximation

mainly comes from the large k2 and/or q2 region. Therefore, the equivalent photon approximation can be used with good success if one only integrates over the small k2, q2 region. A very interesting study in this respect i s given by Kessler et a l , in the last paper of Ref. 2.

(ii) Equation (9), which will be the basis of much of the subsequent discussions, i s obtained by ave r - aging over the azimuthal angles x and X ' which a r e defined in the c.m. frame of the two outgoing pions. These variables a r e not to be confused with the lab azimuthal angles $, $' which most authors use. (The latter cannot be varied independently from s for fixed E, E'. Thus averaging over @, 4 ' for fixed s and other variables a s i s implicitly done in most papers is not always a meaningful operation.) Our averaging procedure i s done after the transformation to the n-n c.m. frame i s car r ied out and for each fixed value of s. We note that for this colliding process, the lab and pion c.m. f rames a r e not very much different, the transformation from the former to the lat ter does not significantly distort the kinematics of the individual events.

We begin our phenomenological analysis by a s - suming the ideal situation where measurements of the process can be done with great accuracy and outline the model-independent procedure to ex- t rac t the n-n phase shifts. Later on, we introduce the usual plausible dynamical assumptions which simplify the analysis with l e s s than perfect data.

Consider the helicity amplitudes T,,,(k2, q2, s , cos 0,) introduced ear l ie r [Eq. (5)] for the process yy - nn [Eq. (6)]. We shall take the variables k2, q2, and s to be fixed and only explicitly display the 6, dependence. Each of the helicity amplitudes has the partial-wave expansion

T,,(e,) = c (21+ 1)~:; d:2nno(e,), I even

where d(')(6,) a r e the usual rotational matrices. The important features to notice are : (i) Because the charge-conjugation quantum number for the y y

state i s +, the two pions can only be in states with even isospin and even angular momentum; (ii) to lowest order in a , the unitarity condition demands that the phase of each partial-wave amplitude a;; (for given isospin I =0, 2) is the same a s the corresponding partial-wave amplitude for n -n sca t - tering, a t least when s i s below the inelastic threshold, l.e.,

a;; )~ a ( i ) f l e i 6 i Inn (16)

where 6: is the n-n phase shift for isospin I and angular momentum 1.

Substituting (15) and (16) into Eq. ( 7 ) o r (9) we get the connection between the measured c ros s section and the n-n phase shifts. To be more explicit, let us consider Eq. (9). The W',,,, which enter into this formula can be written out (we omit the isospin index)

Wll,,, = I a!~'\" 101 a ~ ~ ) l I a!;'\ cos(bo - 62)diqA(B,)

+ 2 5 / a~~ ' j 2 [d~~ .$0 , ) ]2 +. . , W, -1,1 = 25 1 a!:)1 I2[d$:d(0,)l2 +. . , (17)

Wol,ol=251~6~~12~~12~,0(~n)12,

with similar expressions for Wl0,,, and If one can separate the various t e rms in the above formulas (by measuring the distributions in + and 6, for fixed k2, q2, and s ) one can eliminate the unknowns 1 a i d 1 and solve for cos(6, - 6,,), 1, 1' =even integers. A very important feature of this type of phase-shift analysis i s the fact that the factors cos(6, - 6 , , ) enter linearly in the c ros s - section formula. Consequently, the well-known ambiguities associated with phase-shift analysis in hadron-hadron scattering do not a r i s e a t a l l in this

152 C . E . C A R L S O N A N D W . - K . T U N G

case. To simplify the parametrization, one can invoke

the argument of angular momentum ba r r i e r s for higher part ial waves and assume that the final- s tate interaction is negligible except in the lowest angular momentum states. Further, since higher angular momentum corresponds to la rger impact parameter where the interaction is dominated by the exchange of the least massive particles, one expects the higher partial waves to be dominated by the single-pion-exchange Born diagram, Fig. 4(a). Gauge invariance demands that the crossed Born diagram and seagull diagram, Figs. 4(b) and 4(c) be also included. One can, therefore, se t in Eq. (17)

6 , = 0 , for l > l o a;; = ( ' ) ( E l

(18) a m ,

where aL;)f8) a r e the partial-wave amplitudes calculated from the Born diagrams of Fig. 4 and 2, is some small even integer.

More directly, one can write

and substitute into Eq. (9) o r Eq. (7). The phase shifts 6, can then be extracted from data by sys - tematically eliminating the unknowns 1 a&) 1 , 1=0, . . . , I,. For energies not too far from the threshold, one can simply take I , = 0 (we remind the reader again that the p wave is absent) and the procedure outlined above then reduces to that of I. The validity of the assumptions which go into this phenomenological procedure can, of course, be directly checked by experiment. As the quality of the data improves, one can increase the value of l o and see if the approximation (18) is valid o r not. A similar procedure is known to be very useful in N-N phase-shift analysis" (where the situation i s more complex due to nonlinear unitarity).

Except for Eq. (16), we have not included the isospin in our discussion. In principle, the only model-independent way of disentangling the I = 0 and I = 2 amplitudes is to measure both the n'a- and non0 final states. However, there a r e both theoretical reasons and experimental evidence17*18 that the I = 2 final-state interaction i s not important in the low-energy region. Therefore, a s a f i r s t approximation, one can again use the gauge- invariant Born diagram result for the I = 2 part. As the range and quality of data improve, this approximation can be removed.

In an actual experiment, one may not be able to

completely average over the two variables x and X ' a s proposed, In that case, Eq. (7) still applies. With incomplete integration over x and x', some t e rms in addition to those in Eq. (9) will survive. They can be treated in the same manner a s outlined in the above. An important point to bear in mind is that most additional amplitudes involve one o r more longitudinal photons (helicity index 0) and give small contributions to the c ros s section.

In addition to the phenomenological analysis, d is - cussed above, one can make dynamical assumptions which allow one to "calculate" the unknowns ( a:,) / and 6 , in the formulas given. One can then "predict" the measured ee- eenn c ros s sections through Eq. (7) o r (9). In particular, dispersion relations (which connect the r ea l and imaginary par t s of an analytic function) can be used to relate ( a s ) , ) to 6, provided the latter i s known for a l l energies. On the other hand, some predictions on 6, itself can be made if one imposes some combina- tions of constraints due to crossing symmetry, analyticity, unitarity, partial conservation of axial- vector current (PCAC), and current algebra - to the best of one's ability. The dynamical theories involved, however, a r e f a r f rom perfected. A di- rec t comparison of experimental data with such "predictions" which a r e obtained after a long chain of model-dependent calculations probably will not be particularly revealing. Since theoretically the nn phase shifts 6: a r e much more interesting than the absolute values of the partial-wave amplitudes 1 a:,,),',)(, i t is clearly desirable to extract 6, from the data in a s model-independent a way a s possible and then compare the results with theoretical models for a-a scattering. This is the approach advocated here.

V. MODEL CALCULATIONS -SENSITIVITY OF MEASURED CROSS SECTION TO QUALITATIVE FEATURES OF THE s-WAVE n-n PHASE SHIFT

In spite of our caution against making theoretical predictions, if one wants to get some feeling of the effectiveness of extracting the n-n phase shifts from e + e - e + e + a + n, it is useful to estimate the order of magnitude of correction effects due to n-n interaction on the measured c r o s s section. Such studies could be useful in providing a guide for the planning of relevant experiments. The pre- c i se results may be somewhat model-dependent, but the qualitative features should be instructive if the dynamical assumptions made a r e reasonable.

Our model consists of using dispersion relations to calculate I a',", 1 from assumed phenomenological test phase shifts 6,. We then test the sensitivity of the measured ee - eena c ros s section to varia- tions of some important qualitative features of the

6 - M E A S U R I N G n - n P H A S E S H I F T S IN C O L L I D I N G - B E A M . . . 153

s-wave n-n phase shifts, such a s the magnitude of the scattering length and the existence of the sca- la r meson around 700 MeV.

As mentioned before, in an actual experiment i t i s probably necessary to tag one o r both of the outgoing electrons, thus specifying the full kinematics of ee - eenn. However, in order to under- stand the qualitative features, it i s more transparent to present results for the case where the lepton variables a r e integrated. This we shall do in what follows. Our basic formula i s therefore Eq. (9) with fixed s and cos0, but a l l remaining lepton variables integrated over. We have discussed in some detail the results of such a calculation for the case of pointlike pions (Born approximation) in Sec. Ill. and Appendix A. For the pres-

I

ent case, we perform the same calculation with only one modification, namely, the s-wave Born amplitudes a r e replaced by the unitarity-corrected /a:,! 1 eei60. [See Eq. (19) with 1, =0.]

The essential features of our model were de- scribed in our previous paper.* We shall outline them here.

(1) From our experience in Sec. I11 and Appendix A, we know that Woo,oo, being proportional to kZq2, does not contribute significantly to the c ros s section. [This i s because the factor k2q2 cancels the denominator (l/k2q2) from the virtual-photon propagator, so the integrated c ros s section then loses two powers of L = ln(4~~/m:) which i s the dominating factor.] We can write therefore

where dp"=dp/dXdXfds and coshq' and cash.$' a r e given by (10).

(2) Now, we apply Eq. (19), which reads in this case

Tmn = TE) for ( m , n) = (1, -I), (0, I) , and (1,O).

The amplitudes T Z ) for charged pions a r e given in Appendix A. The only s-wave partial-wave amplitude [for (m, n) = (1, I ) ] i s

(22)

where

The corresponding amplitudes for n0n0 production a r e obviously zero.

The amplitudes with definite isospin a r e related to the charged (c) and neutral (n) amplitudes by

(3) The unitarized s-wave amplitude a(s ) in Eq. (21) i s calculated from the dispersion relation for the partial-wave amplitude. Assuming elastic unitarity, this dispersion relation becomes a standard Omn6s-type integral equation. This equation can be solved in a standard way for any given se t of n-n phase shifts. The form of this integral equation and explicit expressions for the solution a r e given in Appendix B. We only make two relevant remarks here:

(i) Strictly speaking, the solution to the OmnZs equation i s not unique; possible polynomial t e rms could be added. The arb i t ra r iness can be fixed by additional physical requirements on the asymptotic behavior of the amplitude. In obtaining our solution, we have assumed that the s-wave part ial wave does not r i s e Linearly in s a s s -m.

(ii) In this model, the n0n0 production proceeds entirely through the s-wave partial amplitude and i s isotropic in the n-n c.m. frame. The c ros s section for ee - eenOnO can be obtained from Eq. (20) by setting

Now, we have everything expressed in t e rms of 6,(s). We can therefore perform the calculation by assuming various possible se ts of phenomenological phase shifts. Since we a r e interested only in the qualitative features of our model calculations, we choose simple analytic forms for the phase shifts which possess the desired properties. In choosing these phase shifts we obviously also bear

C . E . C A R L S O N A N D W . - K . T U N G

I

P H A S E S H I F T S I

FIG. 5. Four se ts of phase shifts used in the calculation of the magnitude of the s-wave amplitude for yy- sr.

in mind the available experimental information a s well a s theoretical models for 6,(s). There a r e three features of the phase shift which determine i ts qualitative behavior over the elastic scattering region. These a r e (i) the scattering length; (ii) possible resonances (6 = lr/2); (iii) asymptotic behavior (6 increases to nn o r decreases to zero). We have chosen many se ts of phase shifts which exhibit different types of behavior under these three categories and tested the effects on the measured c ros s section.

We present here results obtained from four se ts of such phase shifts which illustrate the qualitative features of our model calculations. One of these i s chosen to agree qualitatively with the phase shifts calculated by Kang and eel^ o r Brown and Goble18 using current algebra, unitarity, dispersion the- ory, and crossing symmetry. The features of this phase shift a r e i ts small scattering length, resonant behavior near &= 700 MeV, and asymptotic return to zero. Each of the other three phase shifts differs from the f i r s t qualitatively in one r e - spect, that is, one has no o resonance, one has a large scattering length, and one approaches the value n a s s - m. These phase shifts, a s functions of the pion c.m. momentum a r e depicted in Fig. 5. Analytic forms for these test phase shifts and the corresponding Omnes functions a r e given in Ap- pendix B.

Some typical results a r e presented in Figs. 6 and 7. In Fig. 6, the integrated c ros s section do/ ds i s plotted a s a function of the pion momentum. The incident electron energy i s taken to be 3 GeV.

- B O R N 1 . 2 1 .........

8 I

FIG. 6. Integrated cross section for process (I), d a / d s , a s a function of p , the magnitude of pion momentum in the dipion c.m. frame. The solid line i s the c ros s section for pointlike pions. The other lines a r e unitarity- corrected using the 4 sets of phase shifts given in Fig. 5. The incident electron energy is E = 3 GeV.

The solid curve is the c ros s section due to the Born diagrams without a -a interaction. The other curves a r e the unitarity-corrected c ros s sections with the 4 different se ts of phase shifts mentioned previously. Note that (i) the 8-lr interaction can produce a very significant effect on the measured c ros s section; (ii) the c ros s section i s concentrat- ed in the near-threshold region, it becomes quite small beyond 200-MeV/c pion cam. momentum.

In Fig. 7 we plot the differential c ros s section do/dsd(cos9,) a s a function of the pion c.m. scattering angle 9, for four different values of s. The various lines have the same significance a s in Fig. 6.

From these curves we can see that the measured c ros s section for ee - eenlr should be quite sensi- tive to the n-n interaction effect. In particular, the se t of phase shift with scattering length of o r - der unity gives r i s e to c ros s sections which a r e significantly different from the others. The other sets , all taken to have zero scattering lengths, tend to behave in qualitatively similar ways with 10-20% differences in the actual c ross sections. Other phase shifts with small but nonzero scattering length give similar results . Stated explicitly, in the region near the threshold where our model should be fairly reliable (and where the bulk of the c ros s section l ies) we see grea t sensitivity to the scattering length. We might point out that the absolute normalization for this process can be mea-

M E A S U R I N G n - n P H A S E S H I F T S IN C O L L I D I N G - B E A M . . . 7- - --r- 7 r T - 1 1

f i . 3 1 2 M e V I I

.5 - f i = 296 MeV 7

I

6 : 395 M e V

I

I' ' ' T - ' --7 f i - 5 0 0 M e V

FIG. 7. Differential c ros s sections do/dsd(cos8,) f o r fixed s plotted against the pion c.m. scattering angle 8,. The various lines have the same meaning a s in Fig. 6.

sured by comparing with the pure quantum-electro- dynamic (QED) process ee - eepp .

In contrast, the model calculation yields small c ross section and relative insensitivity to the various phase shifts in the 700-MeV region where the suspected scalar meson o lies. In fact, the one phase shift which does not r i s e above 48" through- out yields cross sections almost indistinguishable from those from the other se ts which do go through 90" near 700 MeV. In the context of our model, this happens because the unitarity -corrected s -

wave amplitude becomes small a s compared to the remaining Born amplitudes and the distinction between various phase shifts becomes insignificant. This fact i s in turn related to the assumed asymptotic behaviors of the phase shifts and the s-wave amplitude. The predictions in this region can be easily changed by modifying these two factors. We see, therefore, in our model, the cross section has little sensitivity to the existence o r nonexistence of the o meson. Its effect is mixed up with the asymptotic behavior of the phase shift wherein lies the

156 C . E . C A R L S O N AND W . - K . T U N G

most ambiguous aspect of this model. The failure of the model to demonstrate a clear

signal for the o meson does not mean, however, that this very interesting subject can not be studied in this reaction. The model i s simply not reliable in this respect. This circumstance only underlines the need of pursuing a model -independent proce - dure of extracting the n-n phase shifts a s outlined in Sec. ZV. There exist in the literature other mod- e l calculations on this process either using a procedure similar to ours o r involving some form of resonance parametrization for the relevant amplitudes. So far a s we can tell, a l l suffer from ambiguities in this respect not l e s s severe than those we explicitly spelled out.

After completing this work we received a revised version of the work by Goble and ~ o s n e r , ' a s well a s the Cornell conference p r ~ c e e d i n g s , ~ in which similar results obtained previously by the same authors were discussed. Since these results have considerable overlap with the present investigation, and since the conclusions with regard to the a meson seem, on the surface, to differ, some specific comments a r e called for . The overlap is in their section on yy - nn and our Sec. V on the model calculations. The two models a r e essentially equivalent. Given the same 11-11 interaction a s input, the solutions a r e also the same. The differences lie in emphasis and interpretation. Spe- cifically:

(i) In Ref. 8, the input information on 71-71 scattering amplitudes i s taken from model calculations based on current algebra while attention is focused on the effects of the a meson in the process yy- nn. In the present investigation, the 71-11 phase shifts a r e treated a s phenomenological quantities to be extracted from the experiment; the current-algebra solution to the n-n phase shift i s selected a s one of several representative test phase shifts. It seems desirable that the results from this type of experiments a r e used both to extract the n-71 scattering length, a s an important test of current algebra ideas, and to investigate the o meson problem.

(ii) Because of this difference in emphasis, the curves of Ref. 9 a r e plotted against the variable p, the pion c.m. velocity, which emphasizes the high- momentum region. The most striking feature is the appearance of a wiggle o r shoulder around p - 0.8-0.9 in &/dp for ee - eenr which i s attributed to the existence of the o meson. We have per - formed similar calculations using the variable p in our model. The results a r e almost the same a s given in Ref. 8, if the phase shift corresponding to the current-algebra solution (6, in Fig. 5) is used. However, the nonresonating phase shift (for instance, 6, of Fig. 5) also gives qualitatively simi-

l a r results except the wiggle is l e s s pronounced. In view of this fact and the inherent ambiguities of these model calculations in this energy region (stressed both in Ref. 8 and the present work), i t should be clear that the interpretation of such a structure, if it were indeed found experimentally, must be treated with great care. We also note that the same curve when plotted versus the variable p (as i s done in our Fig. 6) does not show such a structure a t all.

ACKNOWLEDGMENTS

We wish to thank Professor Jack Sandweiss for several helpful comments. We a r e grateful to Professor Jonathan Rosner for discussions and for sending us copies of his work with Dr. R. Goble prior to publication. A correspondence with Dr. Tom Walsh was responsible for the addition of the comments a t the end of Sec. 111 to the preprint version of this paper. We express our thanks to him also.

APPENDIX A

Here we outline the calculation of the differential c ros s section &!~~ee,+n-/dsd(cosO,) for the case of pointlike pions. The result i s used in the text both in Sec. 111 where the validity of the WW approximation is discussed and in Sec. V where the model calculation including unitarity corrections i s reduced to a form similar to the present case.

For pointlike pions, the scattering amplitudes a r e given by the gauge-invariant Feynman diagrams, Figs. 4(a)-4(c). By neglecting t e rms proportional to (k2q2) (justification of this approximation is given later in this appendix), we get

where p = (1 - 4p2/s)'12 is the velocity of the pions in their c.m. frame. In the same approximation the remaining lepton variables 3, $' in Eq. (9) a r e given by

Substituting (Al) and (A2) into Eq. (9) and inte-

M E A S U R I N G a - a P H A S E S H I F T S I N C O L L I D I N G - B E A M . . .

grating over a l l variables except s and case, we get

where Z3(s, case,) = 1 ,

and

The integrals involved in evaluating zi(s), Eqs. (A5), a r e threefold integrals over, say, k2, q 2 , and E (or E ' ) . Although very complicated, these integrals can be in principle done exactly. For dimen- sional reasons, the result can only involve quanti- t ies like P and

1

E integration). Thus we a r e justified in neglecting terms proportional to k2q2 in our calculation. Using similar arguments, one can easily isolate the coefficients of the leading terms proportional to L2Ls, Ls3, LLs2, . . . , etc.

Similar calculations for the cross section integrated over cose,, i.e., du:fJee, ,/ds, have been done by Baier and Fadin14 up to te rms linear in the logarithmic factors. The integrals z i (s ) can also be inferred from their results by integrating our Eq. (A3) over cos8, and comparing the results. One then obtains

For a l l practical purposes, it suffices to obtain r e - sults which a r e a t least quadratic in the large logarithmic factors L, L, and these only with constant coefficients (i.e., one can neglect te rms like xnL, xnLLs, . . . , n = l , 2 , . . .). The task then becomes considerably simpler. For instance, one immediately recognizes that the L factor can only come from the lower limit of the k2 (or q2) integral with the photon propagator factor l /k2 in the integrand. Thus, if the propagator factor l/kZq2 i s canceled by other factors in the integrand, then the result can, a t most, be of order Ls (from the

If one only retains te rms quadratic in L and L,, then z,, z3, and z4 become proportional. The corresponding terms in (A3) can be combined and we get the result quoted in Sec. 111, Eq. (13). In our direct calculations of the integrals zi(s) , Eqs. (A5), we have checked a l l the coefficients except those of LL, and L,.

APPENDIX B

In this appendix we show some of the formulas relevant to our dispersion (0mn6s equation) treatment of the s-wave strong-interaction modified amplitude, including a list of the test phase shifts that we employed.

In writing down the dispersion relation for d ( s ) (we shall omit the angular momentum and helicity labels 1 , m, n ) , we assume that the right-hand cut i s dominated by elastic unitarity and that the only important contribution to the left-hand cut comes from the Born term. The relation is

158 C . E . C A R L S O N AND W . - K . T U N G

This standard Omnts equation has the solution

with

For the test phase shifts we use, the ambiguity associated with the 0mn& solution i s fixed by requiring the s-wave amplitude not to r i se linearly in energy a s s -m .

The Born amplitude over the elastic unitarity region can be well approximated by one o r two poles on the left-hand cut. For example, for the charged amplitude

is accurate to 4% over the range 4p2 C s < 16p2, and

i s accurate to 1% for 4 p 2 c s < 100p2. If the pole approximation i s made, the Omn6s solution can be integrated analytically.

Thus, writing

= % for I = 0 and (2 n / 3 ) for I = 2 ,

we get

where

In obtaining (B7) we have again neglected t e rms of higher order in k2 and q2 for reasons already mentioned. The most important feature of Eq. (B7) i s that ar(s , k2, $) is written in the same form a s the Born amplitude, with only the substitutions ~ ; ( ~ ) ( s ) - ai(s) and a:(B) - ai(s). Since the integration involved in evaluating Eq. (20) i s done for fixed s , there a r e no changes in any of the integrals that one needs to evaluate. Thus one needs only to isolate the s-wave part of the Born calculation discussed in Appendix A, make the appropriate substitutions of ar(s, k2, Y2) for ar(')(s, k2, $), and then one can write down the integrated answer directly. The substitutions affect only the Z i of Appendix A, and we shall give just the modified forms of these Z i here:

6 - M E A S U R I N G n - n P H A S E S H I F T S I N C O L L I D I N G - B E A M . . . 159

with D = 1 - p2cos201:. We would like to call the reader 's attention to the fact that the large corrections to the WW approximation exhibited in Eq. (13) remain also in this case. This i s important because these corrections a r e in the opposite direction to the unitarity corrections and of comparable magnitude in a wide region of the variables.

For completeness, we give the analytic forms for the test phase shifts and the corresponding 0mn6s functions. The isospin-zero phase shifts a r e

Three of these phase shifts, 6,, 6,, and 6,, pass through 90" at s = 2 4 . The Omnss functions a r e

For the isospin-2 corrections, which a r e small, we always took the I = 2 phase shift to be -hl(s).

*Supported in p a r t by the U. S. Atomic Energy Commis- s ion under Contract No. AT(l1-1)-264.

'v. E. Balakin, V. M. Budnev, and I. F. Ginsburg, ZhETF Pis . Red. 11, 559 (1970) [ J E T P Let te rs 11, 388 (1970)).

2 ~ . Arteaga-Romero, A. Jaccar in i , and P . K e s s l e r , Phys. Rev. D 3, 1569 (1971), and e a r l i e r re fe rences cited therein; J . P a r i s i , N. Arteaga-Romero, A. Jaccar in i , and P . K e s s l e r , ibid. 4, 2927 (1971).

J . Brodsky, T. Kinoshita, and H. Terazawa, Phys. Rev. Le t te rs 5, 972 (1970); Phys. Rev. D 4 , 1532 (1971).

'c. E. Carlson and Wu-Ki Tung, Phys. Rev. D 4 , 2873 (1971).

5 ~ . Brown and I. Muzinich, Phys. Rev. D 4, 1496 (1971). 6 ~ . J. Brodsky, T. Kinoshita, and H. Terazawa, Phys.

Rev. Le t te rs 27, 280 (1971); s e e also T. F . Walsh, DESY repor t (unpublished).

'P. C. DeCelles and J. F . Goehl, Jr., Phys. Rev. 184, 1617 (1969).

8 ~ . H. Lyth, Nucl. Phys. m, 195 (1971);=, 640(E) (1971); H. Schierholz and K. Sundermeyer, Nucl. Phys. B40, 125 (1972); R. Goble and J. Rosner , Phys. Rev. D - 5, 2345 (1972): s e e a l so S. Brodsky, in Proceedings of the International Synzposium on Electron and Photon Interactions at High Energies, edited by N. B. Mistry (Cornell Univ. P r e s s , Ithaca, N. Y., 1972).

' ~ x p l i c i t l y ,

where

F o r detai ls s e e Ref. 4 and T. P. Cheng and Wu-Ki Tung, Phys. Rev. D 3, 733 (1971). 'k. Watson, Phys. Rev. 88, 1163 (1952). "K. Fujikawa, University of Chicago Report No. E F I

71-51 (unpublished). ore complete versions of the equivalent photon ap-

proximation can be found in R. B. Cur t i s , Phys. Rev. 104, 211 (1956); R. H. Dalitz and D. R. Yennie, ibid. 105, 1598 (1957). 1 3 ~ h e question of the validity of the WW approximation

was also ra i sed by H. Cheng and T. T. Wu, Nucl. Phys. B32, 461 (1971). -v. N. Baier and V. S. Fadin, ZhETF Pis . Red. 13, 293 (1971) [ J E T P Let te rs 13, 208 (1971)l.

l S ~ h e logari thmic factor L can only come f r o m the lower l imi t s of the k 2 and q2 integration. The L, factor can come both f r o m the upper l imi t s of the k 2 and q2 integration and f rom the E (or E ' ) integration. In the c a s e of yy- na, the mat r ix elements T,, a r e at most constant in the virtual-photon m a s s e s k 2 and g 2 , there a r e enough damping fac tors f r o m the other fac tors in Eq. (9), how- e v e r , to cut off the k 2 and q2 integration a t about s (in- s tead of extending to the upper l imit 4 ~ ' ) . No L, factor can come f r o m this source. One consequence is that the coefficient of the LL, t e r m i s zero. On the other hand,

160 C . E . C A R L S O N A N D W . - K . T U N G - 6

for yy - 8' the matrix element is proportional to [ ( s + k * + q2)' - 4 ~ ~ ~ ~ 1 ~ ' ~ which cancels part of the damping factors in the k 2 and q2 integrations mentioned above and it is easy to demonstrate that the LL, term does occur with a positive contribution. This then becomes the main correction effect. When a form factor is put in, however, the upper ranges of the k Z , q2 integrations are again damped, thus reducing the correction effect f o r the sin-

gle ro production. '"ee for instance, G. Breit and R. D. Haracz, in High

Energy Physics, edited by E . Burhop (Academic, New York, 1 9 6 7 ) , Vol. 1 .

' I D . Morgan and G. Shaw, Phys. Rev. D 2, 5 2 1 (1970) . '*E. Tryon, Phys. Rev. Letters 9, 769 (1968) ; L. S.

Brown and R. L. Goble, Phys. Rev. D 4, 723 (1971) ; J . S . Kang and B. W. Lee, ibid. 3, 2814 (1971) .

PHYSICAL REVIEW D VOLUME 6 , NUMBER 1 1 J U L Y 1 9 7 2

- pp Backward Scattering and s-Channel Resonances: Upper Limit to the

p p Coupling of the S(1929) Meson

R. Bizzar r i , P. Guidoni, and F. Marzano Istituto di Fisica dell'liniuersitz -Roma, Istituto Nazionale di Fisica Nucleare -Sezione di Roma, Italy

and

E. Castelli , P. Poropat, and M. Sessa Istituto di Fisica dell'liniversitk - Tries te , Istituto Nazionale di Fisica Nucleare - Sezione di Tr ies te , Italy

(Received 1 3 September 1971)

From the energy dependence of the Fp and pn inelastic cross sections we deduce an upper limit to the resonant contribution in Fp backward scattering for c.m. energies between 1915 and 1950 MeV. This limit is smaller than the expected contribution from diffraction scattering. The energy dependence of the 180" Fp elastic cross section in this energy range cannot therefore be directly related to the formation of s-channel resonances.

I. INTRODUCTION

The observat ion with the CERN missing-mass spec t rometer of a very narrow resonance (rs 35 MeV) a t a m a s s of 1929 MeV1 has stimulated the s e a r c h for i t s possible coupling to the Fp system. However, the low-energy Pp e las t ic scat ter ing c r o s s sect ion is dominated by a conspicuous f o r - ward peak apparently due to diffraction, and consequently the detection of a highly inelast ic r e s o - nance f r o m the energy dependence of the total e last ic c r o s s sect ion appears very difficult. It h a s s ince been pointed out by Cline et aL2 that a be t te r sensitivity to a possible resonant fip in te r - action could be achieved by looking a t the energy dependence of the backward scat ter ing c r o s s s e c - tion, away f r o m the forward diffraction peak.

I t is the purpose of this paper to show that pub- lished data on the pp c r o s s section, together with new da ta on the fin annihilation c r o s s sect ion i n the momentum range 350-580 MeV/c, allow one to place a significant upper l imit on the possible contribution of a single resonant interaction to the

backward fip elast ic c r o s s sect ion if the resonance h a s a m a s s between 1915 and 1945 MeV and a width between 10 and 40 MeV. At the s a m e t ime we will point out that a t these low momenta, even in the backward direction, diffraction scat ter ing is capable of masking possible effects of s-channel resonances.

11. DATA

The values of the f i and $n inelastic c r o s s s e c - tions in th i s energy interval a r e shown in Fig. 1. The Fp da ta a r e those of Loken and Derr ick3 and Amaldi et ul.* The fin data a r e new and we shal l briefly descr ibe how they have been obtained.

The experiment is very s i m i l a r to that of Ref. 4. The deuterium-filled 81-cm Saclay bubble cham- b e r h a s been exposed to a separated antiproton beam f r o m the CERN proton synchrotron. Three exposures a t beam momenta of 467, 549 , and 612 M e ~ / c (at the entrance of the illuminated region of the chamber ) have been made. All p interactions have been measured if they occur in a r e -

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Phase Shifts in Colliding-Beam Experiments