Deuteron photodisintegration at high energies

Nuclear Physics A364 (1981) 219-252 @ North-Holland Publishing Company

DEUTERON

PHOTODISINTEGRATION

AT HIGH ENERGIES

M. ANASTASIO Physics Department,

Brooklyn

College of CUNY,

Brooklyn,

NY 11210,

USA

and M. CHEMTOB CEN, Saclay, BP 2, 91191

Gif-sur- Yvette, Cedex, France

Received 9 January 1981 Abstract: The reaction y + d+ p+ n is studied at photon energies in the range 0.1 to 0.7 GeV. The approach is based on a non-covariant description of the deuteron (d) (accounting for the pn S- and D-components as well as for the virtual AA component) and a plane-wave description of the final pn state. The scattering amplitude incorporates, in addition to the impulse term, all the relevant one-pion exchange current terms corresponding to the Born and the rescattering terms in the process yN+ N+ rr. We discuss contributions to the latter process induced by the N&(A) resonance as well as by the N*(Sii, Pii, Dis) resonances in the second bump. We consider two treatments of the Nf contribution based 07 the isobaric model and a dispersion theory model. Predictions are presented for differential cross sections and for proton polarizations. We assess the roles of the various terms in the scattering amplitude. Some refinements are considered occasionally, bearing on final-state interaction effects and the p-meson exchange terms. We also examine in detail the sensitivity of predictions to various inputs and approximations, the most relevant being the range parameters for the aNN vertex, the off-shell extrapolation of the amplitude yN + TN and the retardation corrections. Recognizing that the indefiniteness in certain inputs may be the cause of appreciable uncertainties, we are led to seek for plausible prescriptions for these inputs by referring to measurements. We find that instrumental ifems, in order to achieve acceptable fits to differential cross sections, are a range parameter for the PNN vertex around 700-900MeV/c and an energy-dependent N& width. The predictions for the proton polarizations feature a much stronger sensitivity. In particular, we fail to explain the large enhancement observed at around 500 MeV photon energy.

1. Introduction Recent measurements of spin observables in the processes p + p + p + p [ref. ‘)I and y+d+p+n [refs. 2*3)] reveal interesting structures at several invariant masses between 2.1 and 2.5 GeV which are strongly suggestive of dibaryon resonance signals. The theoretical interpretation is at present very preliminary. These resonances could eventually fit in a potential model scheme as tightly bound NN* or N*N* states 4). Quark models suggest, on the other hand, a radically different description as six-quark bound states ‘). While the theoretical discussion promises to be difficult, one feels that a new, promising perspective is thus offered for studies of the two-nucleon system at intermediate energies. 219

220

M. Anastasia,

M. Chemtob J Deuteron photodisintegration

One of the clearer experimental

signals indicative of the existence of one, and perhaps even two, dibaryon resonances has been in the measurements of the polarization of protons in the process y + d -+p + n [refs. 2*3)].For fixed scattering angle, say 90” c.m., and variable photon energy one finds that the polarization rises gradually from rather small values (-- 10% around 300 MeV), reaches a maximum (of nearly -80% around 500 MeV), then falls. The process of deuteron photodisintegration occupies an important position in nuclear physics and these new results make its study all the more exciting. It is of interest to determine whether the observed enhancement is a genuine resonance signal or whether it is instead a threshold effect, reflecting the opening of inelastic NN* channels. Two other motivations can also be given for a study of the process y +d+ p+n. As is well known, photon absorption on a single nucleon is permitted only for an off-massshell, interacting nucleon. The higher the photon energy, the further removed from the mass shell the nucleon must be. As a result of this, the questions of highmomentum components inside the deuteron and the small but finite probability of the AA component admixed in the deuteron, which have been widely debated in the past few years 6*7), are especially relevant to this process. As for the second motivation, one should mention the availability of a large body of precise, good quality data [for differential cross sections **“) as well as for polarizations 11,12)],with the promise of still more to come, owing to several high-energy photon beams soon to come into existence. In this paper we report on a study of the process y +d + p + n in the interval of photon energies 0.1
M. Anastasio, M. Chemtob / Druteron photodisintegration

221

a region where it is nearly energy independent I*). At photon energies above 300 MeV, cross sections are observed to decrease rapidly with increasing photon energy. However, a closer look at the data reveals a slight transition around v 2: 400-500 MeV from a sharp fall-off to a slower one. The above features should in principle have a simple explanation on a qualitative basis. Indeed, the structure at 300 MeV is unmistakably associated with excitation of the N& isobar. Also, the changes in slopes and the strong enhancement in polarizations could be due to the opening of inelastic channels. However, the question of what level of agreement can be achieved on a quantitative basis is not at all clear. When it comes to quantitative predictions several difficulties manifest themselves: in the dependence on wave functions, in the choice of inputs for coupling constants and form factors and in the analytic continuation of these inputs to unphysical regions. There are several theoretical works in the literature dating from about 20 years ago r3-r5) and also later 16-r8).These works were chiefly satisfied in explaining gross features so that one has a poor appreciation from the discussions there of the sensitivity to various inputs. Two recent works, one by Laget r9) using a non-relativistic approach, the other by Ogawa et al. *‘) using a covariant approach, represent serious attempts at a quantitative description. While some discrepancies persist here and there, especially for angular distributions near forward or backward angles, these works definitely achieve substantial improvement in the quality of agreement with data. However, this is so only for photon energies below 500 MeV. Above this energy agreement becomes poorer. Moreover, predicted proton polarizations as a function of photon energy are typically structureless, showing no sign of an enhancement. The point of view adopted in our present work comes very close to that of Laget 19), and we try in our discussion to build on his findings as well as to emphasize complementary aspects. One should also mention in this connection the complementarity which exists between the photon and the pion absorption break-up processes. Useful insight has been gained recently on the latter process regarding the treatment of retardation effects as well as the uncertainties in the meson-nucleon coupling constants and form factors *l). We shall also discuss in this work two mechanisms which have remained largely unexplored so far. These are concerned with the excitation of nucleon resonances higher than the N& and with the exchange currents induced by the deuteron’s intrinsic AA component. The contents of the paper fall into 4 sections. In sect. 2 we discuss the nonrelativistic approach to the process y + d + p + n. We construct the scattering amplitudes for impulse and one-meson exchange graphs, describe inputs for deuteron wave functions and for vertex functions of elementary amplitudes and explain the reduction of the complicated spin-momentum dependence of matrix elements. (In appendix A we discuss phase space considerations and quote formulas for the differential cross section du/dR and for the polarization P. In appendix B we discuss technical aspects of the calculations. In appendix C we collect the final formulas expressing the irreducible tensor decompositions of the various scattering

222

M. Anastasio, M. Chemtob / Deuteron photodisintegration

amplitudes.) In sect. 3 we present predictions for da/d0 and P, discuss what bearing these predictions have on various inputs and compare with measurements. In sect. 4 we summarize our conclusions. 2. Non-relativistic 2.1. SCATTERING

model for y + d + p + n

AMPLITUDES

The process y(q) +d(p) + p(k) + n(l) is discussed in the kinematics described by fig. la. We consider the construction of scattering amplitudes, that is of matrix elements of the e.m. current w:(q) (S is the total intrinsic spin of final state), for the graphs shown in figs. lb-d. These are: the impulse approximation amplitude (I), the standard one-meson exchange amplitude (X), and the one-meson exchange amplitude induced by the virtual dd component of the deuteron (d). Fig, 1 is schematic, as we have only drawn representative graphs in a Feynman diagram representation and in reference to specific mechanisms. Each graph in figs. lb-d has a companion graph obtained by the interchange p -n. The construction of observables from the current matrix elements is discussed in appendix A. We find it convenient to work in the isospin formalism and to define, therefore, the matrix elements %Y(q) = ((Nk)NO)S~li*

Kw(P)~

= 0) 3

(1)

D(P)

(0)

(cl

(b)

(d)

Fig. 1. Deuteron photodisintegration amplitude (a). Nucleon current impulse graph (b). Exchange current one-meson-exchange graph (c), AA exchange current one-meson-exchange graph (d). Heavy lines represent the A-resonance.

M. Anastasio, M. Chemtob / Deuteron photodisintegration

with respect to two-nucleon one has

223

states coupled to good total spin and isospin. Naturally,

~~(4)=~(~~l(q)+ce~O(4))

*

(2)

The following are general formulas for the amplitudes I, X and A associated with graphs b, c and d, respectively, in fig. 1. These formulas are valid irrespective of whether or not the final-state interactions are included: (~~~(4))*=Xs’.ll:[tf;~~~(~q, x)+0%&,

W3clNx= -1&xx

cc (4”))P(Tm(q”>)”

[

+ (~“(q”))Pv~ 4:

Wfw)*

qsI(s

qZ

(4”))”

+p2-ie

= -1 %xX

[

+w(4”Np(f~

q2n+p2-ie

(34

x)1770,

+p2-k

Psd-4

W-h

(3b)

xl] 770I


x)

,

@&.

x,] 77: *

,

x)

(34

In these equations the suffices p, it are labels in the two-nucleon spin and isospin subspaces, m is an isospin index for the exchanged meson, the integration variable S is such that the exchanged meson momentum is qn = $q - 8, and the two terms inside brackets correspond to graphs related by the interchange p c, )t. Before explaining further the notations used in eqs. (3), we discuss briefly the assumptions entering into their derivation. Our approach is not Lorentz invariant, so that the current matrix elements depend on the choice of frame. We shall choose Aere to work in the deuteron rest frame. The discussion of the exchange matrix elements suffers from a further ambiguity which can be traced to the difficulty in accounting simultaneously for the following two aspects: the strong interaction amplitudes and form factors, on the one hand, and the recoil and retardation effects, on the other. These two aspects are associated with two different interpretations of the space-time sequence of interactions, which are realized in the Feynman and time-ordered perturbati0.n theories. Thus, it is clear that the Feynman graph representation leads to inconsistencies in a non-covariant description of the deuteron when one comes to incorporate recoil effects, by way of the nucleon kinetic energy terms. One advantage of the Feynman graph picture is that one can talk consistently of renormalized vertices for 7rNN and yN + TN and consider realistic, model-independent inputs for the relevant amplitudes. One is then only making the implicit assumption that the other spectator nucleon acts effectively as a spectator in not distorting the renormalized vertices by making them medium dependent. There is sense then in talking about off -shell mesons although it should be stressed that the meson mass is not unambiguously defined, if at the same time one talks about

224

M. Anastasio, M. Chemtob / Deuteron photodisintegration

on-mass-shell nucleons. The reason is that the energy conservation constraints at the emission and absorption vertices give relations such as q”, = p? + q” - px and 4: = n: - n’: which are incompatible if it is simultaneously assumed that p? = EPl, . . .. . In the time-ordered approach one can treat recoil effects consistently. However, the meaning of meson form factors is ill-defined or at best restrictive, as such form factors have more nearly the meaning of the cut-off factors familiar from recoilless field theories. Moreover, one is restricted then to isobaric-like models for the process yN+ TN, and it is known that such models are inaccurate in certain details. To be more precise, let us consider the N* isobar excitation graph. In the time-ordered rules, one writes the energy denominators for the three distinct time-orderings as: 1 C-55) 4” ( @?O+md)-&$+,+EQ) ’

1

1

1 [ (4o+md)-(El,+EQ+~~“)+ie+(q,+ma)-(EB+*+Oq.+Eq-l)+iE

1

1 +(40+md)-(E~+,+o,“+E,-k)

(40+md)-(40+EQ+Wq,+Eq-k)

I

*

Here Md is the deuteron mass, Q =pl, E$ = (Q2+ m$)1’2, EQ = (Q2+ mi)1’2 and 0,” = (4; +p2)1’2 with the understanding E& + (Eg -&(Q)). In the Feynman graph prescription, one would retain instead only the first two terms. These would sum up to the Feynman propagator factor (4: + p2 - in) = (--((I:)~ + 4: + p2 - k), as given in eq. (3), only under the rather special assumption of setting 4: = Ea_k - EQ in the first term and & = E&+, - Ek in the second term, which correspond to the energy conservation constraints at the absorption and emission vertices, respectively. The important conclusion at this point is thus that the non-covariant treatment has an inherent ambiguity with respect to the description of retardation and rescattering effects. This ambiguity is related to the time sequence of the interactions. In order to get a useful insight into both of these effects, it appears necessary to discuss both the time-ordered and the Feynman graph pictures. (An important question at this point is whether energy denominators develop poles as one accounts for the retardation effects. The denominators with an N* intermediate state are regulated by the width term and cause no problem. On the other hand, one can convince oneself that the other denominators are always negative and are never vanishing within the range of integration.) In the rest of this subsection we summarize some useful notations in connection with eqs. (3) and also the later discussion: Nucleon mass = mN; A (N”) isobar mass = mA (mN*); meson mass = CL; pion (rho) mass = m, (m,). Relative momentum of final pn state x= k -$q. Spin (isospin) coupled two-nucleon wave function = xs (~1). Spin (isospin) coupled two-delta wave function = ,& (7~;). Solid spherical harmonics: %,Ji&)

= WL(~)

Xx&2,

(44

M. Anastasio,

225

M. Chemtob / Deuteron photodisintegration

wLfcf4

(4b)

xx31ki’*

= [YLNl)

The deuteron wave functions in the position and momentum representations defined as

are

where

dr r[u(r)io(pr), -w(rh(~r)l . The deuteron AA component is generally given by (LSJ basis)

An approximate operator representation of the AA component, by perturbation theory and will be used in calculations, is

which is suggested

~,“(p)=[(Sp.S”)~(~)+(3(SP.~)(Sn~~)-Sp.Sn)~(p)](T~.T”)~~.

(7)

As it turns out, this is a rather accurate representation of the full results of eq. (6). The connection with the full wave function in the LSJ basis is 5

3 C(P) = -UOl(P) 2JzL

9

l+‘(p) = -mu&)

4J35,r

=

15 --21(p),

4JiG

03)

0 = W(P).

The wave functions for the two-nucleon scattering states are given by IL!&(r). In the plane-wave approximation: c~;;Lr(~)+, J$(eix.r- (_)S+r e-ix’r) . The deuteron break-up matrix elements are

@b) (G’(r) is the AA component plane-wave approximation:

wave function

in position representation.)

In the

3

(104

P:z(s, X)-*~[~A(S-X)-(-)S+z~A(S+X)].

(lob)

ry,,(&

X)+.Mwx)-(-)s+z&~+X)l

226

M. Anastasia,

M. Chemtob / Deuteron photodisintegration

The meson absorption vertices T’“(q,) for r and p are given by: rho

pion N+N:

-igrrNNF=NN(4~)Tm(u’4”),

A+N:

-igrrNAFnNA(q~)T+m(SC’4n),

A-+A:

-i&AfiwAA(d

) Tm

-~~NN(dhbxq”~*E N

9

i&NA -~~NA’lq~)~~(S+Xq.)‘E,

(s

’ 4n ) ,

i&AA -,mdF,AAb&ti+

(11)

Trn)(Sx&)‘E.

Here eA = ~~(4,) are the spin-l polarization vectors and we have not considered coupling terms involving the time components. All form factors are normalized to unity at the meson mass q’, = -,u*(mt or mz). The electromagnetic current absorption vertices t*(q) are given by the following formulas where time and space components are written in succession:

+i(G:(q2)+T3G~(q2))~], N

(12) N+A:

0, iG~A($--]

sx9

, A

A+A:

Gt’(q*),

iGiA(q2)z]

. A

The amplitudes for y(4) + N + 7rm(qn) + N in the non-relativistic

limit read as:

~,“(4,)-[VVl”iq~~+V2mi4~~~(q-qn)4n+V3mi(~x4n)X4+V4m(4nX4)1,(13) v,” = ;[ vi73 + v,’ ${r*, 73) + v; $[r,, 7311.

(14)

We have written e.m. vertices in the general virtual photon case of non-vanishing 4’. The couplings and normalizations for pure nucleon transitions are wellGz” = $(Gg * GE), Gs” = $(GL f G&); established: G:=O, G,p, (0) = GL (0)/2.79 = G&(O)/ - 1.91= 1; g&N = g/x& with g = 13.5. For NA and AA transitions we have limited ourselves to the dominant spin couplings. Our conventions for spin and isospin transition operators are: (3lPllb = 2 ,

<~llS’llD = -2 ,


,

(13

with equivalent definitions for T. Note the following useful rules to help in relating with other transitions: Emission and absorption vertices are related by the replacement 4,, e-4” ; the transposition of N and A amounts to the replacements S-S+, To TC. The quantities V”,, V: in eq. (14) are the usual invariant amplitudes such

M. Anastasio, M. Chemtob / Deuteron photodisintegration

that V:* = V2’ (sp,tp,up), where S, = -(pi +q)*, tp = -(q -qn)*, u, = -(pl We defer to subsect. 2.3 a discussion of the choice of other inputs.

2.2. DEUTERON

WAVE

227

-q,)*.

FUNCTIONS

For photon energies 0.1 s v s 1 GeV, one is probing wave functions in situations where the D-state and short-range correlations are especially relevant. This is most directly seen in the plane-wave approximation where matrix elements are either proportional directly to momentum space wave functions (impulse term) or to integrals of these wave functions weighted by the meson propagator and by form factors (exchange terms). For exchange amplitudes, the relevant momenta cover a wide region centered around $Y.The relevant momenta in the impulse amplitude are (k -4) in the forward hemisphere and k in the backward hemisphere. (Note the useful symmetry between these two momenta about 8* = $rr.) For c.m. angles 0 d 0* s &r it is found that Ik - 4 1rises linearly with v, being larger at smaller angles, and thus covering a band centered at 0.5 GeV/c at v = 0.2 GeV and 1.3 GeV/c at v = 1 GeV. The situation is illustrated in fig. 2. We also show in this figure the relative kinetic energy in the final state given by X2/mN. At Y = 0.3 GeV, it is equal to 0.27 GeV which is a region where the elastic pn interaction is relatively weak, presumably restricted to the low partial waves. This can perhaps be taken as an indication that final-state interactions in waves higher than 1= 0 are not important for Y3 0.3 GeV. The present study of the process y + d + p + n tests the deuteron wave functions in an interval of internal momenta d 1 GeV/c. It is now well appreciated that potential

(11

a3

05 v(GeV)

ae

03

Fig. 2. Recoil momentum of spectator nucleon (q -k) in the lab frame versus photon lab energy Y for a set of fixed scat&zing angles B* in the c.m. frame, indicated on each curve. The lowermost curve drawn in full line is for x = (k - $q1at tJ* = 0”. This quantity is practically angle independent.

228

M. Anastasio,

M. Chemtob / Deuteron photodisintegration

model adjustments to scattering data leave a large freedom in the short distance region below, say, 0.7 fm. Conventional potential models suffer also from another limitation which concerns the consideration of inelastic channels associated with the explicit N” degrees of freedom. As is well known, the virtual NN” and N*N* components are described by wave functions which are peaked at large momenta. This is simply on account of the fact that they are made out of states of high intrinsic spin. Thus the discussion of high-momentum short-distance components inside nuclei should go hand in hand with that of short-lived virtual components. Over the past few years useful insight has been gained into the role of N* inelastic channels in descriptions based on frameworks using transition potentials 6*7*22). We shall rely upon the predictions obtained in ref. 23). The two-nucleon system is solved in the momentum representation with an exact treatment of retardation effects, nonetheless subject to the usual limitations, alluded to in part in subsect. 2.1. In fig. 3, we show typical predictions for the deuteron momentum space wave functions quoted in this work. These predictions are obtained in a calculation of box diagram potentials which allow for exchanges of 7r and p and intermediate propagation of the Nz3. Some features, which are more or less standard, are: the increased importance of the normal D-state around momenta ~300 MeV/c, the dominance of the ‘Di and to a lesser extent, of the 3S1 channels in the AA component, and the relative model independence for momenta up to 500 MeV/c. We see indeed, in relation to the last

lo4

0

I

I

a5

1

15

K (G&/c) Fig. 3. Deuteron NN and AA wave functions in the momentum representation (units MeVm3’*). The upper curves give i(K) and -P(K), full lines corresponding to Reid soft-core potential (pd = 6.5%) and dashed lines to the potential MDFPAII (pd=4.5%) [ref. 23)]. The lower curves give -O(K) (long dash-dot) and +@(K) (short dash-dot) for the same potential (pdd = 0.5%).

M. Anastasia,

229

M. Chemtob / Deuteron photodisintegration

point, that deviations with respect to the Reid soft-core potential wave functions start being significant at 500 MeV/c. The D-state from ref. 23) that we shall use has a weaker probability and also weaker high momentum components, both features reflecting a softer tensor force than Reid’s. Two interesting features of the approach used in ref. 23) are to be noted: (i) retardation corrections are effective in modifying the normal NN components and AA components from momenta of about 400 MeV/c on, but they are not so large in the AA component; (ii) the inclusion of the p-meson contribution is to a large extent reducible to a renormalization of the r-meson contribution, achieved in terms of TNA form factors having a lower cut-off mass. 2.3. ELEMENTARY

AMPLITUDES

For the information on coupling constants and form factors we shall draw heavily on the SU(6) symmetry and especially on the explicit quark model of Feynman et al. 24). This is a relativistic four-dimensional quark model with a built-in simple prescription for boosting hadronic states. Thus, this model embodies all standard SU(6) symmetry predictions and gives in addition detailed information on elastic and transition form factors which comes in fact very close to that of non-relativistic quark models. Part of this information has also been found to compare satisfactorily with data. We shall quote the formulas for coupling constants and form factors (in the general case q2 # 0 for the e.m. current) in terms of the notations introduced in subsect. 2.1. No proofs will be supplied here. Electromagnetic vertices GE (q2)/G(q2) = Gh (q2)W(q2)

= Gb (q2)/-2G(q’)

= 1,

G~*:d(q2)/G(q2)= 345/(1+q*/(m~+mA)~), G~(q2)/G(q2)=(1-q2/2m~),

G&*(q2)/G(q2) = 2/(1 +q2/4m:)

(17) .

As is clear in these equations, the function G(q2> is a common form factor normalized as G(0) = 1. While the expression of G(q2) is prescribed in the model, this is somewhat deficient and it is recommended by the authors of ref. 24) that one use for G(q2) the standard dipole fit (1 +q2/0.71 (GeV/c)2)-2. The q2-dependent factors in the above equations are essentially kinematical factors which are very close to unity and in fact exactly unity at q2 = 0. The present prediction for G;*(O) disagrees (it is a factor 3 smaller) with the one quoted in ref. 25). We believe however that we give the correct result, since this is consistent with the standard SU(6) symmetry prediction for magnetic moments in the spin-$ decuplet, ,u(A++) = 2t.~(A+) = -p(fi-) = 2~(p) [ref. 2”>]. We also note incidentally that there is an indirect indication (from a study of r + p + r + p + r> that the true value of *(A++) is quite close to the SU(6) prediction 27).

M. Anastasio,

230

M. Chemtob / Deuteron photodisintegration

The use we made of distinct N* and N masses in eqs. (12) is essentially no more than a mere prescription. This kind of SU(6) symmetry violation is not usually under control in standard quark models. On this point, and in anticipation of future discussions, we would like to call attention to the fact that useful, detailed information can be extracted from quark bag model predictions. As is well known, octetdecuplet mass splittings are satisfactorily explained in this model in terms of the color magnetic gluon exchange interactions *‘). Meson vertices pion: rho

:

The vector-meson couplings are deduced from the e.m. current couplings by invoking VMD (vector-meson dominance). Thus the four-vector amplitude for emission of a vector meson is given schematically as:

where from e-e+ data, ‘yp= 2.84. In eq. (19), Kv = 3.70 is the anomalous nucleon isovector magnetic moment. The present quark model prediction for the ?rNA coupling gives g,,NA= 1.7/m,, which compares favorably with the prediction g&A = 1.61/m, based on a comparison with the N& width as calculated with the covariant TNA coupling. We shall describe the pion absorption form factors for rrNN* by functions of the form FwNN((I:)

=

[(A*-d>/(A*+&l”,

where the power index it = 1 or 2 and the cut-off momentum A is regarded as an adjustable parameter. Empirical information can be made available only indirectly by invoking the strong version of PCAC. It follows from this version that one can identify pion vertices with vertices associated to the non-pion pole part of the axial current. Recent analyses of neutrino-nucleon elastic and inelastic scattering data favor for the latter a dipolar form (1 + ~*/rn~)-’ with mA = 0.7-0.9 GeV. For the description of the invariant amplitudes V?” for YN+ TN, we shall draw heavily on the application of CGLN theory by Adler 29)and the construction method described in ref. 30). We summarize briefly the main ingredients of this method. The starting point is the fixed t one-dimensional dispersion relations for each of the four (six for q2 # 0) amplitudes VJfpo.These have the useful property that the (nucleonpole and pion-pole) Born terms are explicitly separated (in their quality of inhomogeneous terms) from the rescattering, non-Born terms (in their quality of dispersion

231

M. Anastasia, M. Chemtob / Deuteron photodisintegration

terms), We quote incidentally the contribution

of these Born terms:

@a)

s,

(4OVY)~ion=$

where Fy

and FI

are isovector

Wb) n

nucleon and pion form factors normalized

as

F:(O)=F:(O)=l.

The non-Born parts are specified throughout the energy plane simply by the knowledge of absorptive parts in the physical region for the energy variable s’, but outside as well as inside the physical region for the momentum transfer variable t’. Explicitly,

W” (s, f))non-Born =;

ds’ Im V?” Lm mr.a+m,f

(s’, I)(

d-s

’

-iq

*Jzp-&-)

,

(21) where the second term refers to the crossed cut and other notations are standard. The absorptive parts are constructed by means of the multipole moment expansions of the invariant amplitudes truncated to the dominant Mr, moment contributing to the 33 channel. In this construction, we use Adler’s solutions to the relativistic dispersion relations for Mr,. For a fixed value of the photon energy in the overall lab frame, that is for fixed v, the invariant variables S, t, u span a range of values on account of internal motion. These intervals coalesce to one point in the limit of recoilless nucleons, s = rnh + 2vmN - q2, u = rni - 2vmN - q*, t = 0. The predictions we obtain for imaginary and real parts of the Vc in the recoilless case are shown in fig. 4. Note that the pion form factor is not included at this stage. Resonant-like shapes are observed in these results, as expected. The strongest energy dependence occurs at the expected position, v = 0.33 GeV, that is 4s = (2mi + 2mNv)1’2 = 1.23 GeV. It is interesting to compare these predictions with the predictions based on the simpleminded static isobaric model. Performing the calculation of the N3*3pole graph in the static approximation and incorporating a width term, it is found that the only non-vanishing contributions to the invariant amplitudes are given by

v3

=-&&NA( l )[ -2

v:=g;NA 97

A

1 40-(mA_mN)+~ir*“~“+mA_mN

1

1’

We note the absence of a contribution to VI in spite of the fact that this is quite important in the fully relativistic treatment. When one compares the predictions for Vi and V: in fig. 3 with those of the isobaric model using tentatively &,A = 1.61/m,, it is found that the isobaric model underestimates the true results by a

232

M. Anastasia, M. Ckemtob j Deuteron pko~djsinfegratioa

Fig. 4. Non-Born parts of isovector amplitudes V; for yN + trN (in units of pion mass) versus photon energy in the dispersion model for the N&resonance. Real parts are given in (a) and imaginary parts in (b). We have not drawn all amplitudes as the 33 dominance assumption involves relations among amplitudes which hold to an excellent accuracy, Vi = 3 V; = VT, V; - 0.

factor nearly as large as 2. The dependence on photon energy reveals also specific differences. With a constant width parameter r = 120 MeV, the isobaric model amplitudes have, in comparison with the present amplitudes, less pronounced peaks for the imaging parts, slower slopes at the positions where the real parts change sign and shorter high-energy tails. The peak positions are also slightly shifted downwards to Y = 0.28 GeV. An improvement of the isobaric model predictions can be achieved by raising the coupling constant and incorporating a barrier penetration factor in the width, r = roqz3

,

(23)

where q,* stands for the c.m. pion momentum calculated on the basis of an off -shell NT3 mass, (M; +2vmN)“’ . By adjusting to a free width of I’ = 120 MeV, we find that I’, = 1 15/qz3 where qo* is the pion c.m. momentum in the free decay A -, NT.

2.4. REDUCTION

OF MATRIX ELEMENTS

The computational phase of the problem calls, in view of its practical feasibility, for a systematic procedure. This is on account of the large number of terms one needs to carry along and the need to feed the relevant information into a computer program.

233

M. Anastasio, M. Chemtob / Deuteron photodisintegration

The expressions that result when wave functions and spin-momentum decompositions of elementary amplitudes are inserted into the basic formulas for current matrix elements consist of a long list of terms involving products of Pauli spin and isospin operators multiplied in varied combinations by the various three-momentum variables. The reduction of these products is a straightforward although tedious task and the following remarks expose some technical aspects of the procedure. Most of these remarks are pertinent to the plane-wave Born approximation case. First of all, we find it convenient to orient the lab frame in which we work in such a way that the Oz axis lies along q. The transverse current components are then easily picked up by projecting onto the x-y plane. Note in particular that in this frame (VXq)p=-ipVp

(p = *l)

.

(24)

We further specify the orientation of our frame by defining the XOZ plane so that k has a vanishing azimuthal angle. Any product of Pauli spin or isospin operators is of course reducible to forms which are at most linear in u or 7. Recognizing furthermore that these operators will act on an initial deuteron state which carries good total intrinsic spin 1 and isospin 0, the independent irreducible tensor operators turn out to be quite few: spin:

1,

S=~(uP+u”),

tit = [a” x u”f2’,

~=~(d-u”),

(254

isospin:

1,

& = ;(r: - 7;) .

(25b)

It is of course always true that isospin and spin-momentum dependences occur in the factorized form, (9YYj; + S”S,“). Letting S still denote the total intrinsic spin of the final state, it is easy to see that, due to the initial deuteron spin and isospin configuration, the two terms p c, n are simply related, in fact merely by a phase factor (YO”)= (Y4”“)(-)s+‘. One further useful symmetry is &(Q) = 6(-Q) and 6,“(Q) = +&“(-Q) where b(Q) and 6A(Q) are defined by eqs. (5b) and (7). This symmetry which only holds in the plane-wave case is a consequence of the even intrinsic parity of deuteron. Let us now discuss the reduction of current matrix elements for the various amplitudes. Consider first the impulse term, eq. (3a), and use the notation ti = Gp(qZ)9’f. The current matrix element as defined by eq. (2) for the final pn state is then given as

(%(q))r = 4

c xh: W%‘) + e,@'(~*W'W&,

I=O.l

x)

+~GS(qZ)-~~GV(q2)1~I;ry,,(-~q,~)~~~~o = G"(q*)[d+-x

+$q) + (-)s+‘8&

+kq)] .

(26)

234

M. Anastasio,

M. Chemtob / Deuteron photodisintegration

Here the first equation applies to the general case with final-state interactions and the second uses the various simplifications mentioned above, and hence applies only in the plane-wave Born approximation. Let us digress here for a moment to discuss the inclusion of final-state interactions. A very convenient method of doing this is by inserting the scattering state wave function in a subtracted form such that the impulse plane-wave term is only modified by the addition of correction terms. Applying this subtraction procedure to the uncoupled S-waves gives 1 C~~,~,~~s(~2)+~~~~s~~v(~2)l~~

(%(4))r+(U1))r+~;

x

JI

f omdrr[u(r)io(t4r)~oll(~)-

x (ei+dss(x,

w(~)h(~qr)%~(8)1

r) -i&r)) *

(27)

Here S& is the phase shift and u’Oss the radial function normalized to the asymptotic form sin (xr -6’ Oss)/hr), where the lower indices stand for LSJ. We turn now to exchange amplitudes in the Born approximation and start with the one-pion exchange term associated with the normal NN deuteron component. The isospin operator dependence can be readily isolated by referring to eqs. (3b) and (14) so that the relevant current matrix element with respecfto the final pn state is given as

=~~;[&-s,(--~v~-~v:

+ v;)+(-)s+l~(x+s)(-~v::+:v:

-

Vi)]Xl

.

(28) On account of the various symmetries mentioned above, it turns out that the interchange term p t*n is given by an exactly analogous expression. It is thus incorporated by simply multiplying the r.h.s. of eq. (28) by 2. For the one-pion exchange term associated with the AA component, we shall limit ourselves to the direct uncrossed graph only and hence leave out the graph with crossed photon and meson lines. The reason is that, in comparison with the direct term, this crossed st, u term is obviously smaller and, moreover, has a weaker energy dependence, both points resulting from the additional pion energy term in the energy denominator. The current matrix element involves now multiple products of the spin and isospin transition operators. A convenient method of reducing these products consists of converting the entire products into forms involving the nucleon Pauli spin and isospin operators. By isolating the isospin operator dependence and taking matrix elements with respect to the final pn state, it is found that A=4

1 ~sf~:[&&A(~-S)-(-)s+l~A(~+S)) I=O.l

xC(T~)“(~+~~)P(~~)“(~P.~n)lX1770 m =3*s’[~~A(X-S)+(-)S~~A(X+S)]X1,

(2%

M. Anastasio, M. Chemtob / Deuteron photodisintegration

where the term corresponding contribution.

to the interchange

p t*n

235

gives an exactly equal

3. Results and discussion 3.1. FINAL-STATE

INTERACTIONS

The final pn state is generally described in our applications in the Born approximation. In the impulse terms, however, we account for the final-state interactions in the ‘So and 3S1 waves, by using the Tabakin wave functions 31) in the subtraction method discussed in subsect. 2.4. Let us note immediately that the corrections found in this way are small. In the most sensitive low-energy region I/ < 200 MeV, they result in approximately =20%-10% reduction of da/d0 in the forward hemisphere. This reduction effect is expected on the basis of the net repulsive np interactions at these energies. The corrections are significant in the polarization P in so far as this quantity vanishes identically for the impulse terms in Born approximation. In fact, the polarizations contributed by the impulse amplitude are generally negative (positive) for YB 300 MeV (Y d 100 MeV) in the forward hemisphere never exceeding 4% in absolute value. The neglect of final-state interactions in the exchange amplitudes is the principal shortcoming of our present work. It cannot clearly be avoided without reconsidering an entirely different approach. One can envisage, however, a qualitative discussion based on a description of the final state in the eikonal approximation. Assuming furthermore that the np potential can be taken as a uniform, smooth well of depth (V) (a typical expectation is (V)= -50 MeV), then the main physical effect is incorporated in a modified local relative momentum given as x’ = x(1 + mr.JI(v)I/X2)1’2. While our applications are generally carried with x, we shall comment later on the effect of replacing x by x’. 3.2. ONE-PION-EXCHANGE

(BORN TERMS)

There is already convincing evidence that the Born terms in the one-pionexchange amplitude are important at low photon energies, v d 100 MeV [ref. ‘)I. On our results given in fig. 5 we see that these exchange-current terms stand out as a significant correction to the impulse term up to the highest energies. (Since the complete calculation of the pion current term contribution due to the D-state entailed considerable technical work, we have ignored the D-state component in calculating the pion current amplitude. We have convinced ourselves that this simplification is not a dangerous one by recognizing that in the pair current term the D-state contribution to da/d0 is marginal compared to that of the S-state, especially for v b 0.3 GeV.)

236

M. Anastasio, M. Chemtob / Deuteron photodisintegration

lO0

I 300

I 500

nm

v (M&I

Fig. 5. Differential cross section at fixed 8* = 90”. The labels refer to the predictions obtained by adding successively the impulse (a), pair (b) and pion (c) current terms. The predictions with the wave functions based on MDFPAII are labelled as a, b, c (drawn in dashed lines) and those with Reid soft-core wave functions are labelled as a’, b’, c’ (drawn in full lines). Data points lo) are shown as crosses.

We see on fig. 5 that the exchange corrections have individually a specific dependence on the photon energy. When both pion and pair current terms are added together, the correction is akin to an overall uniform enhancement of the impulse cross sections by a factor x2-3. However, the separate effect of the pair term, with respect to the impulse term, gives instead a constructive interference followed by a destructive interference with increasing photon energy. The dependence of the predictions on the wave functions is important to note, keeping in mind however that we are sampling a restricted class of soft-core potentials. That Reid’s wave function results in a larger impulse amplitude is easily explained on the basis of the comparison discussed in subsect. 2.2 (cf. fig. 3). The large discrepancy between impulse terms is however somewhat attenuated by the incorporation of exchange terms. This results from the predominance of the exchange amplitudes and their greater sensitivity to the lower momentum components of wave functions. It is convenient to interpret the added contributions of impulse and exchange terms as a background, continuum amplitude, as opposed to a resonant one. The comparison with data in fig. 5 indeed lends itself naturally to such an interpretation. The important statement that one can make at this point is one of increased confidence in the accuracy and model-independence of our description of the background amplitude.

237

M. Anastasia, M. C~e~tob / Deuteton photodisjntegrat~on 3.3. ONE-PION

EXCHANGE

(NT3 EXCITATION)

The visible structure present in the data for dg/df2 and P around 1/==200300 MeV are evident manifestations of the N?3 excitation in TN+ TN. [Let us remark here on the status of experimental measurements. We rely generally on the latest meas~ements and compilation reported by Dougan et al. lo). We also interpret their cross section vahtes as referring to the lab frame da/da, although this information is not stated explicitly in their work. In the alternative assumption that the quoted values lo>refer to da/da*, our data points in figs. 5,6,8 and 10 must be renormalized by the Jacobian factor d cos 6*/d cos 8. For 8” = 90”, this factor is very close to 1, increasing from 1.04 at Y = 0.1 GeV to 1.24 at it = 0.7 GeV.] These deuteron measurements contain useful, new information bearing on the off-shell description of the process yN + qN. Our theoretical predictions for du/d,lt are given in fig. 6. An important feature to recognize here is their crucial dependence on the meson-exchange contributions, especially in the enhancement region 200300 MeV.

I

3lo

t

v (M&i

I

!m

7m

Fig. 6. Differential cross sections at B*= 90” based on the MDFPAII wave functions. Figs. A and B differ in the background amplitudes, given for A by the impulse term and for B by the impulse and exchange terms. Data points are shown as crosses ‘O).The predictions incorporate the N$ resonance contribution, the labels serving to identify various prescriptions: (a) static limit, A = 850 MeV, F = 120 MeV (full line), (b) recoil corrections (long dashes), (c) recoil corrections with A = 700 MeV, F = 80 MeV (short dashes), (d) A = 800 MeV and energy-dependent width F = r0q*3 (dots) (e) static limit with amplitudes V, in the dispersion model (long dashes-short dashes).

Our principal aim will be to circumscribe the main theoretical uncertainties arising from the following items: the role of retardation corrections, of the parametrizations of the rrNN form factor and of the N& width. We shall use for our starting choices a constant N& width, equal to the free width r = I20 MeV, and the following two forms for the eNN and n-NA form factors F +&qz) and FwNd(q~), in reference to

238

M. Anastasio, M. Chemtob / Deuteron photodisintegration

the time-ordered

and Feynman graph treatments,

respectively,

r(~2-diM~2+d)1

[time-ordered],

[0.28+0.72(A2-mZ)/(A’+qZ)]

[Feynman] .

(30)

[The latter is of similar type to the Ferrari-Selleri 32) form factor, with the added freedom in our case of an adjusted cut-off parameter A.1 It turns out that a sensible procedure for choosing LI consists of adjusting the theoretical prediction for da/d0 right at the height of the maximum. (As one sees in fig. 6 this adjustment is largely independent of other inputs.) We find on this basis the estimates A = 0.850 GeV for the monopolar form factor in the time-ordered case and A = 0.400 GeV for the Ferrari-Selleri form factor in the Feynman case. [This latter value is to be compared with the value 0.334 GeV estimated in ref. 32).] It appears from fig. 6 that the static limit case (recoilless nucleons) is to be preferred, especially in the intermediate energy region Y< 400 MeV. To the extent that this case incorporates the minimal amount of information, it also optimizes the errors incurred in the off-shell extrapolations. The inclusion of retardation corrections has adverse effects on du/dfl with wrong shapes, wider peaks shifted to higher energies and too long tails. A lower cut-off A or an appreciably lowered width cannot prevent these large distortions of dr/dR. However, an energy-dependent width of a standard type r = r0q*3 (with retardation corrections included) is capable of re-establishing a reasonable matching with data. It is difficult to decide whether this is more favorable than in the static limit case, since the high-energy tail is now better accounted for. The important conclusion at this point is that an intrinsic interpretation of the retardation corrections cannot be inferred independently of the parametrization of the NT3 resonance. The comparison between the predictions using the time-ordered and Feynman graph treatments (the latter corresponding to the dispersion theory model calculations of the Vk) reflects from another point of view the difficulty of disentangling rescattering from retardation effects. Thus, already in the static limit situation, accounting for a realistic description of the N T3 resonance shape results in a high-energy tail much longer than in the constant width case. This is a feature that could be anticipated on the basis of the results for the Vk discussed in subsect. 2.3 (cf. fig. 4). While the agreement with data is not significantly improved in the enhancement region, better matching is now achieved in the high-energy tail region. The treatment of retardation effects in the Feynman graph case is made on the basis of an off -shell pion four-momentum qn. We shall only summarize some of the problems raised in this treatment without attempting a numerical calculation. The pion propagator develops now a pole inside the integration interval, which must be handled by evaluating a principal value integral. The definition of q,, depends on whether one prescribes q”, by energy conservation at the emission e.m. vertex or the absorption vertex. This inescapable ambiguity raises important uncertainties in

M. Anastasio, M. Chemtob / Deuteron photodisintegration

239

defining the elementary amplitudes. For example, with 4: fixed by the e.m. vertex, one finds that 4: is increasingly negative with v. Clearly the information on the qi dependence of yN + rrN for hard, time-like pions is very meager. The situation at the absorption vertex TN + N is different, since energy conservation requires instead space-like pions 4; > 0. The origin of this asymmetry is simply a reflection of the inconsistent assignment of on-shell energies to the intermediate nucleons. It is important at this point to remark on the systematic discrepancies featured by all our predictions with respect to the position of the maximum. None of the inputs discussed so far is able to account for the required left shift to lower photon energies of about 50 MeV. These discrepancies are not unique to our non-covariant approach I’). A plausible explanation might be eventually sought in the neglected pn final-state interactions. We have tested this suggestion by considering the simple prescription of a modified relative np momentum x’ in the wave function form factors. The answer is a negative one. We find a small effect amounting roughly to an overall reduction of (da/do) by 20-30%, all other parameters remaining unchanged. No appreciable shift of the peak position is obtained. We also find that the correction can be roughly simulated by raising the cut-off parameter to, say, A = 0.95 GeV. Let us stress, however, that this prescription cannot be relied upon in detail, so one cannot exclude that a realistic account of pn final-state interactions could indeed explain the required shift in the peak position. Another interesting explanation could also be sought in attractive AN rescattering effects of higher order than one-meson exchange, as would be incorporated in a coupled-channel approach. Our results for the proton polarization P are given in fig. 7. One notices here a much stronger sensitivity to inputs. The incorporation of the meson-exchange background terms results in a reversal of the sign of P, which improves greatly the agreement with measurements. This matter of sign has been the subject of some concern to us, since contradictory results for it were found in former studies. For 8* 4 90”, refs. 17S19*20) predict P < 0, while ref. 16) predicts P > 0, as also noted in ref. 12). Our results demonstrate clearly that the incorporation of the pion exchangecurrent terms is the cause of the sign difference. One should perhaps count this nice feature of exchange currents as one further confirmation of their existence. To the extent that our results give indeed small positive polarizations for v s 400 MeV, this can be considered as a reasonably satisfactory situation on a qualitative level. Unfortunately, the situation is not conclusive on a quantitative level. We see indeed that the retardation corrections deteriorate the agreement achieved in the static limit case. The use of an energy-dependent width re-establishes a better matching, but the small values of P found in the Feynman graph treatment’indicate that this agreement is coincidental. It is clear, however, there is no way to substantially improve on the agreements with the measurements for P. Our approach can in principle claim a reasonable qualitative understanding of the data for v =S400 MeV. We conclude this on the basis of the large statistical errors in the data points and the missing item of final-state

240

M. Anastasio, M. Chemtob / Deuteron photodisintegration

-1.0

I

I o

-. e-

41 100

I

1

TOKYO [3]

l

STANFORO[fl]

A

BONN [12]

I

I

I

300

I

-

(A)

I

I

500

I

700

v(tw)

Fig. 7. Polarization of recoil proton at 0* = 90” based on the MDFPAII wave functions. Conventions identical to fig. 6 are used.

interactions, which is generally expected to be important in spin observables. The large enhancement at 500 MeV is left entirely unexplained. In particular, we infer from our findings that an increased da/d0 does not necessarily go together with an increased P, quite to the contrary.

3.4. DEUTERON

AA COMPONENT

The treatment of the AA component entails a definite prescription for the complex energy terms in the energy denominators. We follow here a somewhat conservative procedure which consists in assigning zero-width to the spectator nucleon and an energy-dependent width r = TOADY to the active nucleon, where 4” is the pion c.m. momentum calculated for an effective isobar mass JsT = (m&+ 2~nz~)~‘~. We see on the results for du/dR in fig. 8 that this AA component gives a negligible small correction, showing up mainly in the high-energy tail region. The correction to P, as given in fig. 9, is also seen to result in a marginally small enhancement. Except in the high-energy region, the predictions are generally very insensitive to the AA component. The probability of the AA component amounts here to 0.5%, so it appears likely that a higher probability has visible effects at photon energies 0.6 GeV and beyond. Let us comment briefly here on a very qualitative treatment of the AA component accounting for a possible strong attraction of the two A’s. We simulate this attraction by assigning a net constant width of order of one to several hundreds MeV, keeping the same kinematics appropriate to a continuum state. The results one finds in this way feature now an enhancement in do/d0 around v = 600 MeV, which is clearly

M. Anastasio, M. Chemtob / Deuteron photodisintegration

16'

I

300

lm

0

I

Vb4dJ)

600

241

0

700

Fig. 8. Differential cross sections at 6* = 90” and 127’ based on the MDFPAII wave functions. For B* = 90”, we show predictions for impulse +exchange (Born and NA terms) using A = 0.8 GeV (a) (full line), +AA component (b) (dash)+second bump resonances (c) (long dash-short dash) using for these latter terms constant widths, the Ferrari-Selleri form factor (A = 0.4 GeV) and the + 1 relative phase. The predictions with a -1 phase are shown as (c’) (dot) and with a -i phase as (I.?‘)(double dash-dot). The predictions with identical inputs as (c) but a cut-off A = 0.7 GeV are shown as (d) (long dash-double short dash). For 0* = 127”, we show corresponding predictions to the cases (c) and (d) above, labelled respectively as (C) (long dash-triple dot) and (D) (long dash-double dot).

absent in measurements. On the other hand, the shape of P is not altered qualitatively. One possible conclusion from this study is that the interpretation of an enhancement in P in terms of a dibaryon resonance is constrained by the absence of an analogous enhancement in du/da.

-06 -

Fig. 9. Polarization of recoil proton at f3*= 90”. Specifications identical to fig. 8 are used.

242

M. Anastasio, M. Chemtob / Deuteron photodisintegration

3.5. SECOND-BUMP RESONANCES The

visible second bump in the free process yN + trN around photon energies of 700 MeV is currently associated to the contributions of a compound of three isospin-~resonancesP~~(1450),S~~(1510)andD~~(1530).Fortheparametrizationof these resonances we shall rely on a recent multipole moment analysis by Berends and Donnachie 33).We follow here a systematic procedure which consists in expanding in the static limit the amplitudes Vi in terms of the lowest relevant multipole moments, using VI =+(Eo+-MI_+~(~-x,)M~_+(~-~x,)E~), 141 -(Mu_ v3

v4=,k;l,l -(MI-

+ 6x&f2-)

=

+

3(2x, - l)Mz- + 3&-) ,

hz,

,

(31)

where Ikl and 141are y and rr c.m. momenta and xs = k *q/lk/141.The latter variable is fixed somewhat arbitrarily to xs = 1. One then substitutes in these equations the expressions for the resonating multipoles quoted in ref. 33). It is not possible to infer from this work the full isospin structure, unless one makes the reasonable approximation of negligible isoscalar amplitudes relative to isovector ones. Then, one has V: = Vi and, for isospin-3 final states, V(kl”) = i&V:. From the solutions A in ref. 33) we infer the results (units m, = 1)

U%+h= (E:- ha =

-0.190 E, - 5.63 - i0.33 ’ -0.275 E, - 5.45 - iO.607 ’

(ME )P1l= (H-

)DIS=

-1.04 E,-4X3-i1.56’ -0.240

(32)

E, - 5.45 - iO.607 ’

Nearly all the V, (k = 1,. . . , 4) receive contributions from these resonances. These contributions are found to be of comparable size to those of the Nz3 resonance for photon energies above 400 MeV. We should emphasize at this point the qualitative exploratory character of our present treatment. In particular, we have not been able to sort out with confidence the relative phase between these contributions to the Vk and those of the Nz3, although the most likely convention is +I. Thus in our results given in fig. 9 for P, we consider the choice +l for the relative phase, along with the other two choices - 1 and i. (The latter choice is not necessarily contradictory with T-reversal.) The predictions found here are obtained in the static limit treatment with constant widths for the second bump resonances. One sees that the corrections are generally small in da/da. The two alternative choices of phases have some adverse effects. One notes from fig. 9 that the -1 phase is not favored for P and from fig. 8 that the i phase enhances adversely dcr/dL! out to the lower energies.

M. Anastasia, M. Chemrob / Deuteron photodisintegration

243

I 400

20

30

60

90 ?2ol%lleoO e%*gr**s)

3l

8)

90 1P 150 M e*(Degrres)

Fig. 10. Differential cross section in c.m. frame versus proton c.m. angle for the four indicated incident photon energies. The predictions based on the MDFPAII wave functions incorporate the impulse and all r-exchange (Born + N& + second bump resonances + AA component) terms in the static limit treatment: Data are from Anderson et al. ‘) (crosses), Buon et al. ‘) (circles) and Dougan et al. lo) (black circles). Full (dashed) line curves correspond to the lrNN cut-off parameter A = 0.8 (0.7) GeV.

If one restricts then to the more favorable + 1 relative phase, one sees that while the effect on dc/dR is small, the effect on P results in an interesting broad structure around v = 500-600 MeV. This structure does not however compare in size with measurements. The important conclusion here is that the second bump resonances should be instrumental in describing quantitatively the large enhancement in P with or without dibaryon resonances. To be decisive on this matter will require, however, a more quantitative analysis.

3.6. RHO-MESON

EXCHANGE

We discuss briefly a qualitative treatment of the p-meson contribution. A simple calculation of the direct-term amplitude for yN-*pN, based on the couplings discussed in subsect. 2.3, gives us

x (-+(a” x (4” x E)) + 2(& x E)) x 4.

(33)

There exists a simple relationship between the p- and r-exchange amplitudes which manifests itself after one sums over the p-meson polarizations. By expressing the

M. Anastasio, M. Chemtob / Deuteron photodisintegration

244

direct terms in the N& pole graph for both cases in terms of the quantity (T,“(q,)P(Tm(4n))n(qo-(m~ - mN)+$iI’), one finds: pion:

-kgdA V3

-

283Nia”

x 4”

-

3fimNmA

2qd x qb”

. qd , (34)

rho :

-

2q,)(a” *qn) - i(aP X a”) + 2u”] x q

,

where T = i(~’ +T”). Thus we remark that one piece of the p-meson contribution can be absorbed in the r-meson contribution as a sort of vertex renormalization. ‘A similar relationship holds for the one-meson terms induced by the AA components. Since the general qualitative expectation for the p-meson terms is an approximate overall renormalization of the r-meson terms, our procedure of adjusting to data the cut-off parameter A presumably accounts for the missing p-meson terms. To get further support for this statement we have attempted to incorporate those pieces of the p-meson contributions having the same spin and isospin structure as the r-meson terms. Referring to the discussion above and to appendix C, we find that these contributions entail the following multiplicative factors

(

l-O.,,@)

”

) P

(l-0.42%

n

P)

for direct terms only in the yN+ TN and ?A + TA cases, respectively, where the numerical estimates are based on the coupling constant values quoted in subsect. 2.3. Analyzing the predictions for the case where impulse, exchange and Nz3 terms are included together, using A = 0.850 GeV, we find that the p-meson correction induces typically a reduction of (da/do) at low energies v 6 400 MeV but an enhancement at higher energies. The surprising enhancement correction can be explained on the basis of a partial cancellation between the direct and the spectator terms, noting that different values of the r-momentum are involved in each of these terms. (The reduction is of the order of 30% and the enhancement may be substantially larger, a factor 2 or so.) Naturally, the prescription used is not very reliable and one cannot exclude qualitatively different predictions from a more realistic calculation.

3.7. FURTHER

COMPARISON

WITH

DATA

To complete the comparison with existing measurements, we show in figs. 10 and 11 angular distribution predictions for da/d&?* and P, respectively, at several energies. The shapes are on the whole reasonably well reproduced, especially the flat distributions for (do-/da*) at c.m. angles 30” s 8* s 150”. It would be interesting to see whether data for (da/do*) feature also the rises at forward and backward angles when Y2400 MeV. The magnitudes for (da/do*) are not, however, accounted for. We must conclude here that our previous assignment of the cut-off parameter A, on

245

M. Anastasio, M. Chemtob / Deuteron photodisintegration

30

60

90 120 l3l 8*(Degcres 1

lKl

Fig. 11. Proton polarization versus c.m. proton angle for the four indicated incident photon energies. Identical inputs are used as in fig. 10. Data are from Ikeda etal. 3, (black circles), Liu et al. If) (open circles) and Kose et al. ‘*) (triangles). Full (dashed) line curves correspond to the mNN cut-off parameter A = 0.8 (0.7) GeV.

the basis of an overall fitting to the enhancement region at fixed angle (0* = 90”), is of limited accuracy. One could perhaps argue that fitting A to the angular distributions at Y= 300 or 400 MeV is more appropriate. Indeed, substantial improvement is achieved with a slightly lowered value (A = 0.7 GeV). We see that a lower A affects mainly the overall size of (dv/dR*), with corrections decreasing rapidly with Y.This is particularly apparent on the polarizations which involve ratios of amplitudes. Thus, none of our conclusions reached in earlier subsections are modified with the present lower assignment for A.

4. Conclusions The strongest limitations in our study of the process y +d+ p+ n concern the non-covariant description of the deuteron and the neglect of final-state interactions. Both of these limitations are presumably not too relevant in an intermediate interval of photon energies within our wide interval 0.1-0.7 GeV. We reach this conclusion from the fact that our approach accounts well for the most general qualitative features: the enhancement around 200-300 MeV, the change to a slower decrease of da/d0 at 400 MeV, the negative polarizations of about -20% around 300 MeV. The one-pion-exchange mechanisms constitute the main dynamical effect necessary to achieve an acceptable fit to measurements. A crucial, specific role here is played by the background terms corresponding to the standard meson-exchange currents. The interpretation of retardation effects (nucleon kinetic energies or retarded pion

M. Anastasio,

246

M. Chemtob / Deuteron photodisintegration

propagator) is interconnected to that of the rescattering effects (Feynman versus time-ordered graph treatments). Thus the consideration of an energy-dependent N& width is compelling if one incorporates retardation corrections. Also, the more realistic account of the NT3 resonance shape is characterized by a strong N& contribution surviving out to the high-energy region. The p-meson terms are likely to improve the picture and eventually allow one to narrow more closely the true estimate of the range of the TN interaction. There are two notable qualitative features not accounted for in our approach; the exact location of the enhancement around 300 MeV in da/da and the strong enhancement of P around 500 MeV. The former discrepancy is typical of attractive pn interactions of non-perturbative character. It is most likely to be removed by the incorporation of final-state interactions. As for the latter discrepancy, our study gives a further convincing support for a possible dibaryon resonance signal there. We have found indeed that two potential candidates, the dd component and the second bump resonances, fail to explain the measured polarizations rising to a peak of -80%. The incorporation of the latter effect is necessary for a quantitative description of the measurements.

Appendix A We collect expressions for some observables of the process y + d + p + n. Let the photon lab energy be denoted by Y. For documentation purposes we shall also consider the general case of virtual space-like photons of finite mass squared 4* > 0. Consider the rank-two covariant tensor: 4&,

4) = zz,

(d(p)lj,(O)lp(k)nU))(p(k)n(I)lj,(O)ld(p))

xS’4’(p+q-k-I)dp,

(A4

and its scalar and transverse projections J&(q) = e: (q)J+!Mveiv(q)(i = S, T) where &s(q) denotes the spin-l scalar polarization vector (for a system of four-momentum 4) and e=(q) denotes the two (left and right) spin-l transverse polarization vectors. In order to deduce differential cross sections with respect to the final proton p(k), one must integrate over all final states with a density of states dp = dk df for fixed solid angle &. It is convenient to perform the integration over Ikl in the overall c.m. frame. The result reads:

(A.2) where s = rni -q* +2mdv, and the star suffix is used to mean that the starred momentum or energy variables are evaluated in the c.m. frame. The absence of such a suffix means as always lab frame variables. Here VA(q) is essentially the e.m. current matrix element, the exact connection in the non-covariant lab frame

M. Anastasio,

247

M. Chemtob / Deuteron photodisintegration

description being the one given by eqs. (1) and (2) in the text. The consideration of the c.m. frame is useful as an intermediate step because of the simplicity of the derivation and also on account of the simple connection between direct and inverse processes which is given for spin-averaged cross sections by (A.3)

*(nP+d?‘)=i($)2$(yd+pn). d@ To switch over from c.m. to lab, one needs simply to invoke the jacobian, dL?X dcosO* k/k* -= -= d cos 13 y sin2 t9+ (k*/k) cos 8 cos e* ’ df&c

(A.4)

where y = (1 - 02)-l/2 and u is the c.m. target velocity u = 1~*]/(~*2+m~)1’2. The c.m. momenta are of course independent of angle and given by the useful formulas: k*=[(s-m:

- m;)2-4m~m;]“2/2Js,

4*=[(s+q2-m:)2+4

42m:]“2/2Js. (A.3

The differential cross section and proton polarization are given for y + d + p + n in the real photon case by the equations: da -= da:

2?r2a -.&(42 v

= 0, v) ,

(A.6) (A.7)

where II = (q x k)/lq x kl in conformity with the Madison convention and %‘r= %‘*((q)~=~(q)is understood as an operator in the two-nucleon spin subspace. An average over the two (left and right) polarization states is implicit in eq. (A.6). In anticipation of future electrodisintegration studies we also quote the relevant formulas for differential cross section and proton polarization of the process e + d + e’ + p + n summed (averaged) over lepton polarizations:

Tr [(a* n*) Py =

(A.9) Tr

--&(r&

+ &rG)

Appendii

+ tg2 (ied o,o:]

B

We discuss the reduction of the spin-momentum operator dependence of the current matrix elements into irreducible tensors. This is a long and cumbersome

248

M. Anastasia,

M. Chemtob / Deuteron photodisintegration

calculation, the details of which will be spared here. When the main work is done the current matrix elements are given by long expressions of the form:

03.1) where the discrete index n runs over a finite sum of terms, the coefficients a, are scalar functions of the momentum variables which are at most linear in the projection p = *l, rcAo) and tiAs’ are irreducible tensors with respect to momentum variables and spin operators respectively. Detailed formulas for the various amplitudes are given in appendix C. Having the current matrix elements in the above explicit representation makes the numerical evaluation.of observables easier. The following formulas obtain for the differential cross section and for the proton polarization (defined in conformity with the Madison convention)

“,Lkl a,(Ao”,A;, dd%“, x c r~~J,;f’l* mo X

ho”

(

m0

A,“’ A: ( m0 m-m0

A:‘,P) 1 -m >

AZ m-m0

1 -m > ’

03.2)

pwI,p)(S’l P, = 4i s5, (-)l+” 1 (-) *a+%+%‘+%‘(Sq n.n’

x (Sl py

11) c

JS+ x1 IS)

&(A:, A;,p)a:,(Ao”‘, A;‘, p)

p=*l

x .;*I

,xm,

~-,--d~~‘ry*(

““’ m0

09 0

X

Ao” ( m0

At m-m0

1

Ai

1

-m )( m-m0

r

G’

m-m;

l

)

-m

Ai’ A; m-rnb >I S’

1 1

A:’ 03.3) S I .

We use standard notations for 3j and 6j symbols and for reduced matrix elements. The difficult part of the numerical calculations is in the evaluation of the coefficients a,. These are in general complex functions. For the impulse amplitude they are given in terms of the deuteron momentum space wave functions and for the exchange amplitudes in terms of three-dimensional integrals (over the meson momentum variable 8). The integrals over 6 are performed in polar coordinates in the frame defined in sect. 2. No general symmetry seems to exist which could spare us the considerable but necessary effort of performing all three integrals numerically. One particular property worthy of note is the symmetry (antisymmetry) of real

M. Anastasia, M. Chemtob / Deuteron ~~o~disiategration

249

(imaginary) parts of the integrands under the change of azimuthal angle 4 + @ + w. It should also be noted that the dependences over polar and azimuthal angles are not so smooth. Acceptable accuracy could only be achieved with quadratures using a minimum of 10 mesh points. Appendix C We collect the expressions for the current matrix elements in the reduced form given by eq. (B.l). Let us first introduce a few useful notations, themselves based on notations defined in the main body of the text. k,=$q-x,

s,=s-x,

s,=s+x,

k,=$qq+x

(x=k-kQ),

qn=44-s,

qp=h+s,

q=(qnxW,

We shall denote the impulse amplitudes by the suffix I and the one-pion exchange amplitudes by X and d, whether associated to the normal NN or to the AA component of the deuteron. We shall also use compact notations, deIeting suffices and parentheses in those cases where there is no danger of confusion. For documentary purposes, and in anticipation of electrodisintegration studies, decompositions will be given for the time component of the current as well. This decomposition is not written exphcitly for the amplitude X as the time component here is given explicitly in terms of the space component of the current by invoking the divergenceless relation qnTI; (4”) = 0 satisfied by our model for yN + TN. For the p-exchange contributions (identified by the suffix rho) we shall utilize the connection discussed in subsect. 2.4 with the pion exchange contributions and specify

M. Anastasio,

250

M. Chemtob / Deuteron photodisintegration

the common term in a compact self-evident notation.

x(

1 (2J + 1)“2[~[37(~,, R,) x tij?) I= 1.2.3

-(pen))]

+~[G~(4~)([ri(k,)Y’~‘(I;.)-t~(k,)Y’~’(~~)XS][b’ x ~]~‘+[;(k”)YrO’(L”,)+ liqk”)Yr2’(k”) x x1$

-~&((k,)[YC2’(/C”)

+(p-n;

f~++n)l)x~,

-~~~~[~“‘(1”,a,)x~J?+3S+2P+~~]~!

+~~~[~~2’(b”,~“)xs-22+~~l~‘l-40~%~~~

xc

(2J +

1)1’2(-&oJ

-Jz[Y’.’

x C(2J + 1)“2[&$/?Z’o’ + &‘2J

-

x #];I) - &23’

x 2];‘+

[V2.’x &T&l’) +$r&&;

f[Z”.’ x 3S + 221:’ x s - 281;’ + Jzl[223J x

fplql 1q”12QOg[ YC”‘(ff”) +gYc2’(&) x xl;‘+g5[ + g%iJg($&ap(g”, + W’(6,b”)

6”)+*[-2@“‘(4,

+ Jw2’(&,

YC2’(tf”n) x !zfp

6,)

&)I x sly

- [y&&[O’(~“,

6”) + $/g&V2’(~“*

- 4~@“(q,,

1”) x xl;’

- ~J7?r[J7~~1’(&,

ctl~‘l1

5,) - Js2’(6,

6,) + $J$YL2’(b”)

5,) - 2JzW”‘(~“, 5,) x ~]~‘)I

+~~~~~~n~2[f04P[Y~o’(~,)-JZYC2’(q^”)XS-Xl~1+b~it,P x (-[ YL2’(5”)x s + 28 + &@+ - #w($, - $s,[-

6,) + &@&[“(& &/w”(,-,,

+(pf*n,pf,n)

I

2&r[&5%“‘(&,

6,)

8,) x s + 2z1y

6,) + $&%w2’(~“, 6,) - @w3’(&, XI,

6,) x !9];‘)1

251

M. Anastasio, M, Chemtob / Deuteron photodisintegration

J

(~:,S(4))~=-G~'(q2)g~lrlA

xLt]~“‘)+~ey-7j2J5G

x[&(OP- ~)(lo+~[Y[2'(q") x [ Y[“($)

x g+]“’ - 1 + 12(& . 6,)’ + i24J;;q

[ Y”‘(Gj) x S][“‘)

+bc*n)

X [&( ti - ep)(3s,

I

XI.

+ 108, + 8J2?r[ Yr2’(&) X xjcd’

+$J;;[Y[“‘-JzY[21(~~)

xs]~‘+~[Yc2’(~“)

WI:‘)

+~~‘P(-~~~~~[Y~“‘-~Y~~‘(~)XS-~]~~+2(~”~~”)~~* x .=;,

2 (1+ (-)9(2J

99

+8(& &)‘(S, -2J&#+,-

+ 1)1’2[[Yc1’(&J x P(&)Y’x

+&)-ilO&,(&

s+

Xl’d’

a&,)[ Y”‘(+j)X~++~~

J ;7) ; [-J$(Q”‘(&b, Bn)+SO’ejl, $72 B”,$“)I,

-~[9”(~,~,,~n>+~1~(~,~n,b”)x~lcd’ +JJ[+2J($,iin,I,>+ +2J(9,$.,k) xf8lf’)l+(p++n,pC*n) x1, I

x

J

d6

1

+&GV@(~n)[Yc2’(6n)

(qg(q))y+g&))p[ x

14”12 I (qO+md)-(E&+q+E:n)

(2*)3(q~+n&bfsi

J$“:(

x

~lcoll+(p-4

I

r1oml)

XI .

-(--)2$$]-~(+-)2

”1~;

[$( e-

(4~+m~)((qo+mA)-(Et+,+E$)

W)(S

+ 3rn)

252

M. Anastasia,

M. Chemtob / Deuteron photodisintegration

+~~P(3s+13~+f~[YC”‘-~Y[*‘(~,)XS-7~]~’ - $G[ y[O’+ fl- Y[*‘(b,> x zp +(pHn,pon)

I

- &i[

YC2’(b”)x &‘)]

XI.

References 1) A. Yokosawa, Proc. Int. Meeting on frontier of physics, Singapore 1978, ed. K.K. Phua, C.K. Chew and Y.K. Lim (Singapore Nat. Acad. of Science, 1978) 2) T. Kamae et al., Phys. Rev. Lett. 38 (1977) 468 3) H. Ikeda et al., Phys. Rev. Lett. 42 (1979) 1321 4) T. Kamae and T. Fujita, Phys. Rev. Lett. 38 (1977) 471 5) P.J.G. Mulders, A.Th.M. Aerts and J.J. de Swart, Phys. Rev. Lett. 40 (1978) 1543 6) L.S. Kislinger, Theor. methods in medium energy and heavy-ion collisions, ed. K.W. McVoy and W.A. Friedman (Plenum Press, New York, 1978) 7) H. Arenhiivel, 11th Summer School on nuclear physics, Mikolajki (Poland) 1978 8) J. Buon, V. Gracco, J. Lefrancois, P. Lehmann, B. Merkel and Ph. Roy, Phys. Lett. 26B (1968) 595 9) R.L. Anderson, R. Prepost and B.H. Wiik, Phys. Rev. Lett. 22 (1969) 651 10) P. Dougan, T. Kivikas, K. Lugner, V. Ramsay and W. Stiefler, Z. Phys. A276 (1976) 55 11) F.F. Liu, D.E. Lundquist and B.H. Wiik, Phys. Rev. 165 (1968) 1478 12) R. Kose, W. Paul, K. Steckhorst and K.H. Kessler, Z. Phys. 220 (1969) 305 13) N. Austern, Phys. Rev. 100 (1955) 1522 14) F. Zachariasen, Phys. Rev. 101 (1956) 371 15) L.D. Pearlstein and A. Klein, Phys. Rev. 118 (1960) 193 16) D. Hasselmann, DESY Report no. 68/38 (1968) unpublished 17) D.J. George, Phys. Rev. 167 (1968) 1357 18) W.M. Wynn, Ph.D. thesis, Georgia Inst. of Technology (1971) 19) J.M. Laget, Nucl. Phys. A312 (1978) 265 20) K. Ogawa, T. Kamae and K. Nakamura, Nucl. Phys. A340 (1980) 451 21)\ M. Brack, b.0. Riska and W. Weise, Nucl. Phys. A287 (1977) 525 22) A.M. Green, Few-body systems and electromagnetic interactions, ed. C. Cioffi Degli Atti and E. de Sanctis (Frascati, 1978) (Springer, Berlin, 1978) 23) M.R. Anastasio, A. Faessler, H. Miither, K. Holinde and R. Machleidt, Nucl. Phys. A322 (1979) 369 24) R.P. Feynman, M. Kislinger and F. Ravndal, Phys. Rev. D3 (1971) 2706 25) H.J. Weber and H. Arenhavel, Phys. Reports 36C (1978) 277 26) M.A. Beg, B.W. Lee and A. Pais, Phys. Rev. Lett. 13 (1964) 514 27) B.M.K. Nefkens et al., Phys. Rev. Dl8 (1978) 3911 28) A.J.G. Hey, Int. Conf. on high-energy physics, Geneva, 1979 (CERN, Geneva, 1979) 29) S.L. Adler, Ann. of Phys. 50 (1968) 189 30) M. Chemtob, Mesons in nuclei, ed. M. Rho and D. Wilkinson (North-Holland, Amsterdam, 1979) 31) F. Tabakin, Ann. of Phys. 30 (1964) 51 32) E. Ferrari and F. Selleri, Nuovo Cim. 26 (1963) 1450 33) F.A. Berends and A. Donnachie, Nucl. Phys. B136 (1978) 317