A low-energy S-matrix theory of neutron-deuteron scattering

ANNALS

OF PHYSICS:

73, 372416

(1972)

A Low-Energy S-Matrix Theory of Neutron-Deuteron Scattering* RICHARD H. J. BOWER Department of Physics, Princeton University, Princeton, New Jersey 08540

We examinethe feasibilityof a pure S-matrix descriptionfor low-energyneutrondeuteronscattering.Dispersionrelationsare assumed to existand the leadingleft hand contributionsare determinedincludingthosearisingfrom severalnormalsingularities and from two anomalous singularities. The distantleft handsingularities werehandled parametrically.Sumrules,which usedonly the unitarity and analytic&ypropertiesof partial wavesand very importantly were independentof break-upchannels,were constructedto examineseveralimportantpropertiesof partial waveswith thesesingularities.By adjustingonly the distantleft handsingularitystructure,we wereableto producedoubletS waveswith a triton bound state,the triton couplingconstantand low-energyphaseshiftsin goodagreement with the experimentalphaseshifts.For the quartet S wave, we obtainedlow-energyphaseshiftswhich agreedwell with experimentalresultsandwhichwereinsensitiveto the distantleft handsingularitystructure. Thesesolutionsalsocontaineda ghostwhosepositionwassensitiveto the left hand singularitystructure.

I. INTRODUCTION The three-nucleon problem has been a subject of study for many years now. A detailed discussionof the history of the problem can be found in any number of review articles including the article by Delves and Phillips [l] and the article by Amado [2] and so I will confine myself to mentioning a few of the highlights. Variational Calculations

The early calculations of three-body parameters (e.g., triton binding energy, neutron-deuteron doublet and quartet scattering lengths) used variational methods and local two-body potentials. In 1935, Thomas [3] showed that a zero-range potential would predict an infinite triton binding energy, and thus indicated the sensitivity of the triton binding energy to short-range nucleon-nucleon forces. The two-body local potentials used in such calculations have since become progressively more sophisticated. In 1969, Blatt et al. [4] using the Hamada-Johnston potential * This work constituteda partialfulfillmentof the Ph.D. requirementat PrincetonUniversity.

372 Copyright All rights

6 1972 by Academic Press, Inc. of reproduction in any form reserved.

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[5] obtained a triton binding energy of 6.7 f 1.0 MeV and a neutron-deuteron doublet. scattering length of 1.2 & 1.0 F. Experimentally the triton binding energy is 8.48 MeV, the neutron-deutron doublet scattering length is 0.11 f 0.07 F [6] or 0.7 & 0.3 F [7] and the neutron-deutron quartet scattering length is 6.14 f .06 F [6] or 6.38 f .06 F [7]. Separable Potentials

Studying the three-nucleon problem for complicated local two-nucleon potentials is a formidable task. The partial-wave solution in a Fadeev formulation [8] involves solving coupled integral equations in at least two continuous variables [9]. For nonlocal separable potentials the problem is reduced to solving coupled integral equations continuous in one variable [9]. Amado, using a separable potential, obtained a triton binding energy of 11.01 MeV, a doublet scattering length of -1.04 F a quartet scattering length of 6.32 F [IO]. By introducing a parameter “Z” which Weinberg interpreted as the probability of finding the deuteron in a bare elementary particle state [I 11, Amado was able to produce a triton binding energy of 8.53 MeV, a doublet scattering length of 0.7 F and a quartet scattering length of 6.20 F for Z = 0.0496. These results agree very well with the earlier experimental results of Hurst and Alcock, but not the later results of van Oers and Seagraves. Overbinding of the triton was generally characteristic of separable potential calculations. Sitenko and Kharchenko [12] obtained a triton binding energy of 12.5 MeV. Mitra et al. [13] overbound the triton even when tensor forces are included, finding values which range from 8.85 to 12.2 MeV. In 1967, Phillips [14] showed that his separable potential calculation cannot simultaneously give the correct triton binding energy and the van Oers-Seagrave value for the doublet scattering length. (Because his calculation is representative, the results can be extended to other separable calculations.) Phillips obtained a triton binding energy of 9.5 MeV for a doublet scattering length of 0.11 F and a triton binding energy of 8.5 MeV for a doublet scattering length of 0.7 F. Recently, Brayshaw and Buck [15] have calculated phase shifts which are in good agreement with van Oers and Seagraves’ experimental results and they have also obtained a reasonable value of the triton binding energy of 8.73 MeV using a separable potential approximation for a square-well two-nucleon interaction. Brayshaw and Buck concluded that because it is possible to correlate the important features of the two-and three-nucleon low-energy data with their “nonrealistic potentials,” that one must either have reservations about a program whose laurels rest on such an approach, or believe that the “real” potential (should one exist) has the same off-shell behavior as their potentia1.l 1 off-shell in the nonrelativistic limit means off-the-energy-shell, i.e., where E # E’. In the relativistic limit off-shell means off-the-mass-shell, i.e., P2 = Pop0 - P * P # ma.

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Ambiguities

of the Potential Method

The Thomas investigation demonstrated the sensitivity of the triton binding energy to the shape of the two-body nucleon-nucleon potential. This sensitivity is attributed to the off-shell behavior of the potential. In a review article, Noyes and Fiedeldey [16] illustrated the triton binding energy dependence on the shape of the two-body potentials. The results of Noyes and Osborn [17] for the bound state energy of three equivalent bosons using several two-body potentials, each of which fits the low-energy two-nucleon parameters (scattering lengths and effective ranges), are quoted. Noyes and Osborn obtained a bound-state energy of 12.76 MeV for Yukawa potentials, 10.5 MeV for exponential potentials and 11.24 MeV for Yamaguchi potentials. Noyes and Fiedeldey concluded that until reliable models for the off-shell as well as the on-shell potentials are established, calculations of the triton binding energy which use potentials will not be very convincing. S-Matrix

Approach

During the 1950’s a new approach to high-energy physics evolved, the disperison theoretic approach, a method dealing only with on-shell quantities. In 1959, Blankenbeckler et al. [18] applied this technique to study low-energy elastic neutron-deuteron scattering. Assuming the existence of forward dispersion relations, they calculated low-energy forward scattering including only the proton exchange diagram as a contribution to the left hand cut. Using the then available experimental data, they claimed to obtain results compatible with the quasiexperimental phase shifts of Christian and Gammel [19]. The dispersion theoretic or S-matrix approach was then largely ignored in threenucleon investigations. In 1969, Barton and Phillips [20] used dispersion techniques to study the low-energy neutron-deuteron scattering. As input for the left hand cut they too employed only the simplest contribution, the proton exchange diagram. Using a Pagels [21] type of N/D solution they obtained a quartet scattering length of 6.6 F and low-energy phase shifts which are in moderate agreement with the experimental results. Poor results were achieved for the low-energy doublet S-wave phase shifts, the doublet scattering length and triton binding energy. By subtracting the N equation at threshold (introducing a C.D.D. pole [22] and thereby fixing the doublet scattering length, they found that the general shape of the low-energy doublet phase shift could be reproduced (i.e.. for D2 < 0.05 F-2 where D = three momentum in the center-of-mass frame of neutron-deutron system) and a triton binding energy of 6 MeV was obtained. Their calculation assumes elastic scattering and makes no attempt to account for the nearby deuteron break-up channel. A more ambitious calculation was attempted by Avishai, Ebenhoh, and Reiner [23] who build a model to account for the “two-nucleon exchange” as well as the

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proton exchange. Their approach is not purely an S-matrix method because their model for the two-nucleon exchange requires the off-energy shell behavior of the one-nucleon exchange. An S-matrix calculation is strictly an on-shell calculation and the input for such a calculation consists of coupling constants, and on-shell scattering amplitudes. Also, an S-matrix calculation involves an analysis of the singularity structure as a whole, in order to assess the importance of the left hand singularities, namely, their proximity to the right hand cut. Motivation

and Outline of Present Work

An S-matrix description is based only on the analytic structure of the scattering amplitude, a structure determined by the experimentally known mass spectrum, Lorentz invariance, and general conservation laws. Technically it involves only on-shell quantities; consequently, this approach, when feasible is independent of the vagaries that a potential scheme introduces. In this paper, the feasibility and adequacy of such a description for low-energy neuton-deuteron scattering are investigated. Rubin, Sugar and Tiktopoulos [24] provided the basis for this work. They proved in potential theory using two-body Yukawa interactions that the neutron-deuteron scattering amplitude satisfied dispersion relations in D2 for fixed scattering angles. Thus, our first task was to determine the important left hand singularities and the appropriateness of an S-matrix description. The two leading left hand singularities are amenable to an S-matrix calculation. The nearest singularity is due to proton exchange. The calculation of this contribution requires only the deuteron-proton-neutron coupling constant which is obtainable from low-energy neutron-proton data. The second singularity, known as the “anomalous triangle” singularity and subsequently denoted d is due to a three nucleon pinch, (see Fig. 1). This contribution requires the on-shell nucleon-nucleon amplitude. This nucleon-nucleon amplitude is studied in detail and a good (in our view) representation of it in the appropriate energy region is obtained. Others [23] employ a “two-nucleon” contribution which would not give a good fit to the nucleon-nucleon amplitude. Other more distant left hand singularities are also included. Immediately to the left of the d singularity, is the d’ singularity due to a two-nucleon-pion pinch and its contribution requires the OTT -+ PN on-shell amplitude. In this work a pole approximation is used. Graphs of the type shown in Fig. 2 include this singularity. The two-pion exchange in the t-channel is the next singularity to the left of the d’ singularity. (One pion exchange is forbidden by isospin.) This contribution is obtained by approximating the two pions by a u meson resonance. Other singularities are regarded as distant and consequently neglected.

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FIG.

1. This graph contains the A singularity.

Because we do not have accurate knowledge of the distant left hand singularities, all the discontinuities are cut off for large negative values of D2. A standard procedure in dispersion calculations has been to determine the left hand singularity contributions to various partial waves and then to construct a partial wave amplitude using N/D techniques. This construction is both tedious and expensive. Fortunately it is possible to determine many important properties of the partial wave without this machinery. Using the analyticity and unitarity properties, sum rules can be constructed to test whether the left hand discontinuities are compatible with partial wave solutions which have desirable properties.2 Such sum rule results can be independent of break-up channel uncertainties. When we examine the effects of break-up, as we do not want to restrict ourselves to a particular model, the inelasticity is parameterized in a simple manner. As our calculation is exploratory in nature, we make no claims of mathematical precision, and will be satisfied, for example, to obtain a triton binding energy of 8.5 A .3 MeV. Outline

In Section II, the locations of the important left hand singularities of the neutrondeuteron system are discussed. Section III contains technical information: a definition of the invariant amplitudes, a discussion of the partial wave analysis, and a description of the interaction hamiltonians. Section IV describes the nucleon2 The existence of a partial wave solution is not being tested. Rubin, Sugar and Tiktopoulos have established that partial wave solutions do exist. Our approximation for the left hand discontinuity is being examined.

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nucleon amplitude in the energy region where it is needed to calculate the discontinuity of the A singularity for the neutron-deuteron amplitude. Section V shows our calculations of the neutrondeuteron discontinuities. Section VI contains a discussion of the sum rules for testing the properties of partial waves. Section VII describes our N/D approach. In Section VIII the results of our calculation are presented, and in Section IX, conclusions are drawn.

II. SINGULARITY LOCATIONS Before establishing the locations of the leading left hand singularities of the neutron-deuteron amplitude, some definitions are useful to introduce, namely: 3 = deuteron binding energy, or = triton binding energy, z z scattering angle in the center of momentum (i.e. in Fig. 2. z = D . D’/D”).

system

First normal singularities and then anomalous singularities are discussed. The proton pole is at u = m2 [or D2 = -2mB/(4.5 + 4z)]. The triton pole is

/’ // PION

FIG. 2. The A’ singularity.

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at s = mt2 [or D2 = --$rn (or - B)]. The 2n exchange in the t-channel has its leading contribution at t = (2mJ2 (or D2 = -2mr2/(1 - z)) where m, = pion mass. If the 27~exchange is approximated by a u meson exchange, the resulting pole singularity is at t = mo2, where m, = CTmeson mass. A nucleon and pion exchange in the u-channel gives rise to a singularity with its leading contribution at u = (m + m,)2 [or D2 = -(2mm, + m2 + 2mB)/(2.5 + 2z)]. ANALYTIC STRUCTURE OF N-D AMPLITUDE IN THE i3’

PLANE

NT CUT

SiT

’

“4v

-40 - .76

.L I

I a2

IS IN UNITS OFmB;

mB-Xl90

MeV2

FIGURE 3

The leading anomalous singularity 3 d is located at t = 16mB (or D2 = -8mB/(l - z)). The anomalous singularity d’ is located at u = (2mv2 + m2) [D2 = (mm + ~%@~/(1.25 + z)]. The locations of the poles and cuts which these singularities contribute to the S-wave amplitudes are shown in Fig. 3. The triton pole is, of course, present only in the doublet S-wave amplitude. A c meson mass of 3.42 pion masses is used [25]. a For discussions of the Landau singularities of Feynman graphs, one has many choices. Some discussions which I found useful include: R. J. EDEN, “Complex Variable Theory and Elementary Particle Physics,” 1961 Brandeis Summer Institute Lectures in Theoretical Physics, Benjamin, New York, 1962, or J. D. BJORKEN AND S. D. DRELL, “Relativistic Quantum Fields” McGrawHill, New York, 1965. For a discussion of the Landau singularities of nonrelativistic graphs, the treatment by M. RUBIN, R. SUGAR, AND G. TJKTOPOUL~S, Phys. Rev. 146 (1966), 1130; 162 (1967), 1555, is recommended.

S-MATRIX THEORY 0~ N-D SCATTERING

III.

379

TECHNICALDETAILS

Our S-matrix and invariant amplitudes are defined as follows:4 S,‘i = (II out 1i in) = ana- i(2r)4 S4(P, - Pi) NiN,M,+i , Pitn) E four momentum

of initial (final) state,

k :iB 3 energy of iB’-th Boson in state i, k tSiF = energy of iF’-th Fermion in state i, mieiF = mass of iF’-th Fermion in state i,

N,, is analogously defined to Ni for state n. For two-particle elastic scattering in the center-of-mass frame Mni = (k’s,m,fs,m, 1M ( ks,mlis,m,i),

with k = the initial 3 momentum, k’ 2 the final 3 momentum, si = the spin of particle i, i = 1, 2, and mjk’s j = 1, 2 k = i, f the spin projections. To obtain partial-wave unitarity equations for a time reversal, parity and rotationally invariant interaction, one proceeds in a well-prescribed manner, as described, for example, in Goldberger and Watson [26]. Primarily to introduce notation, two steps of this procedure are mentioned below. The partial-wave expansion of Mni leads to the expression (ir’s,mlfs,m,f 1M / irs,mlis,m,i) = C (slmlfs2m2f I Sv’) (i? j I’m’) (Sv’rm’ x (S’I’J’M’

/ J’M’)

1M / SIJM) (JM / Svlm) (Im I k) (Sv I slm,Ggnm,i),

4 See, for example, J. D. BJORKEN AND S. D. DRELL, “Relativistic Hill, New York, 1965.

Quantum Fields,” McGraw-

380

BOWER

where the symbols represent the usual quantities and the sum is over the repeated symbols. The rotational invariance of our theory is the statement (S’I’J’M’

1M / SIJM)

= 6,,6WM,M,,s,~~s(k2).

In this study the partial-wave unitarity equations for the nucleon-nucleon singlet S-wave amplitude, M,&,, , for the nucleon-nucleon triplet S-wave amplitude, M&, , for the neutrondeuteron doublet S-wave amplitude, M,& , and for the neutron-deuteron quartet S-wave amplitude M,&+ are used extensively and thus we explicitly display these equations:

J=

0, 1,

As the terms M,&, are quadratic in the three momentum (k or k’) and thus of the order k2/m2 times M,&, , such terms are neglected in the analysis. Since much information is expressed in terms of the S-wave amplitudes, fj , which have the nonrelativistic forms: fj = e@j sin S,/l k 1, Sj real in the elastic region, j = 1 (3) denoting nucleon-nucleon singlet (triplet) waves, j = 2 (4) denoting neutron-deuteron doublet (quartet) waves, we relate the above partial-wave amplitudes to these, namely: M ,JfoJ = - 16rr2jJm,

J = j = 1, 3,

M,,f,,

J = j = 2, 4.

To describe identical-particle of the fj’s.

= -48~~5,

scattering requires another factor 2 in the coefficient

Interaction Hamiltonians and Feynman Rules An important intermediate step is the calculation of Feynman graphs. To calculate such graphs requires interaction Hamiltonians. Where we deal with interactions whose Hamiltonians are well known5, we use these Hamiltonians. 5 See,for example, BJORKEN AND DRELL,

“Relativistic

Quantum Fields,” Appendix B.

S-MATRIX

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381

SCATTERING

Where the interactions are not wellknown, we determine an appropriate description, obtaining, in the notation of Bjorken-Drell, the following Hamiltonian to describe the nonrelativistic deuteron-neutron-proton interaction [ 181:

fW = g, : 6d-4 y“C$,(x) 4uW: + h.c.3 where &.&x)

= neutron (proton) field 4, = deuteron field,

h.c. SE hermitian

conjugate of preceding expression,

c = iy2yo, g, = deuteron-neutron-proton and the following Hamiltonian deuteron interaction:

coupling constant,

to describe the nonrelativistic

triton-neutron-

ff(x) = g, : &: + h-c., where z&(x) E triton field, g, z triton-neutron-deuteron

coupling constant.

The calculation of Feynman graphs for a given interaction Hamiltonian straightforward exercise and the reader is referred to any standard text.

IV. THE NUCLEON-NUCLEON

is a

AMPLITUDE

The anomalous triangle singularity A is very near the triton pole for backscattering angles. Consequently the contribution of this singularity to the triton binding energy and other low-energy parameters can be very important in a dispersion theoretic calculation. The discontinuity is obtained by placing the three intermediate nucleons in Fig. 2 on the mass shells (i.e., Qi2 = m2, i = 1,2, 3) and thus requires a good description of the nucleon-nucleon amplitude. Defining the quantities sOlnand t,, as follows: t cm = W - Nj2,

s,, = (Q2 + W2,

the condition for the three intermediate nucleons determines the domain where the nucleon--nucleon amplitude is desired, namely: t,, > 16mB,

Re(s,,) < 4m2 - 16mB.

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Using the nucleon-nucleon amplitude for the four-point blob generates a contribution to the d singularity, the d’ singularity, the 277 singularity and many other discontinuities which are further to the left. Sometimes the discontinuities of several singularities are calculated jointly. In such cases, identifying the discontinuity for each singularity is a delicate matter. For example, examine what happens when the pion-exchange graph of Fig. 4 is inserted into the blob. The graph of

v

FIG. 4. This nucleon-nucleon graph represents pion exchange in the u channel. It is referred to in the appendix as an exchange graph.

NUCLEON NUCLEON ANALYTIC STRUCTURE IN

HEAVY

THE

i;l’ PLANE

0’ CUTS

UNITAAITY lTCUT

-2

-2.34

i2

IS

IN

UNITS

OFmB;

mB=

CUl

DK.U -1.0

2090

Mev’

FIG. 5. These diagrams are the source of the left hand singularities deuteron calculations.

used in the neutron-

Fig. 6ii is the result, a graph which contains the A singularity and the d’ singularity. (In our analysis, such a separation was not necessary as we wanted both contributions, although it turns out that such a separation was possible.)

S-MATRIX

THEORY

OF

N-D

ii

i

-‘r/-A-..

y

111

iv

V

vi FIQURJI6

595i73b6

383

SCATTERING

384

BOWER

vii

where

denote6

nut km

the lght

nu~eon

hand

cut contrlbUtion

of the

apiitude.

... Vlll

FIGURE 6 Important graphs contributing to A and A’ singularity.

S-MATRIX

385

THEORY OF N-D SCATTERING

For the nucleon-nucleon amplitudes, we need a knowledge of the important exchange forces and direct poles. A deuteron-pole, pion exchanges, sigmaexchanges (scalar, isoscalar particles to represent the two-pion exchange contribution) and phenomenological heavy sigma exchanges (heavy scalar, isoscalar particle to represent short-range forces or hard core effects) were chosen. S-wave dispersion relations were then written for the neutron-neutron singlet channel using experimental phase shift data for the right hand integrand and the abovementioned exchange forces and direct poles for the left hand integrand. The parameters of the heavy sigmas are determined by the requirement that the dispersion amplitude reproduce the experimentally known scattering lengths and effective range. The S-wave Dispersion Relations

The analytic structure of the neutron-proton triplet S-wave is shown in Fig. 5. The singlet (also neutron-neutron) S-wave has this structure minus the deuteron pole. The dispersion relations are: Im f(w’) dw’ h(w)= j, .H.c. :"wFy;)y + & f,, "y'$+ + jR.H*c. (wl _I_w)-g > w = square of three momentum

in the center of the mass.

The integral over the left hand cut (L.H.C.) includes the contribution from rrexchanges, a-exchanges and heavy sigma exchanges. The $ indicates that we do the integral around the deuteron pole at zD , and & denotes that the deuteron pole contributes only to the triplet S-wave. The requirement that our amplitudes reproduce the proper scattering lengths gives us the following equations: h(O) = fj”*R***(o)

j = 1, 3

where f y.R.A. (w) is effective range amplitude. The stipulation that our amplitudes reproduce the experimental effective range yields the following equations: (a/aw)y;(w) - fj”*“-*‘(W)]lo

= 0,

j=

1,3.

B Other dispersion representations of the nucleon-nucleon amplitude exist ; for example, BALL, A. SCOTTI AND D. WONG, Phys. Reo. 142 (1965), 1000. ‘Their model is not well suited for this calculation as its low-energy behavior is not accurate; in fact, it has a deuteron with a binding energy of 12 MeV.

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The Right Cut Comparison with experimental results’ shows that a reasonable representation forf,(w), w > 0, is the effective range approximation

f2E.R.A.(~) = C,/(l/iF

- i &Z&z/W

- i d\/z,).

The integral over the right cut gives C,/(dW

+ i dZ,)(d/w

+ i dZ).

The Deuteron Pole The deuteron pole contributes to the triplet S-wave amplitude, f2D(~), where fsD(W) = -mg,2/8r(w + mB).

an amount

Matching the residue of this amplitude with the residue of the effective range amplitude at the position of the deuteron pole determines the value of the deteronneutron-proton coupling constant g, , namely, gD2 = 32rr/3mr, = 4.05, rt 3 effective range of triplet S wave. Pion Exchange Pion exchange contributes equally to the neutron-proton proton singlet S-wave and to the neutron-neutron amount [27]:

triplet S-wave, neutronS-wave amplitude an

wherefm2 = 0.08 [28]. Sigma Exchanges Sigma exchange, like pion exchange, contributes equally to the neutron-proton ’ R. WILSON, “The Nucleon-Nucleon Interaction,” Interscience, New York, 1963. For more recent information on the nucleon-nucleon low-energy parameters, see the review article by H. P. NOYES ANLI H. FIEDELDEY in ‘Three Particle Scattering in Quantum Mechanics,” Benjamin, New York, 1968. The changes in these experimental values are very small and do not affect our results. * The sigma meson is to be regarded as a representation of the two pion contribution. Values go2 = 3.42 and tn, = 461 MeV, as given by H. P. NOYES, D. S. BAILEY, R. A. ARNDT, AND M. H. MACGREGOR, Phys. Rev. B 139 (1963, 380, are used.

S-MATRIX

triplet S-wave, neutron-proton amplitude, the amount being:

387

THEORY OF N-D SCATTERING

singlet S-wave and to the neutron-neutron

S-wave

f”(w) = (g,2/16nw) ln[(4w + WZ,~)/WZ,~]. The analysis for the heavy sigmas is precisely that for the u-exchange with m, and go2 replaced by m, and gz2, respectively. We now proceed to determine the phenomenological sigma masses and couplings. For convenience, the following notation is introduced: dji

E

xji&2,

m,ji s Njimo ,

pij E XjJNj, 9

i = 1, 2 to account for the two heavy sigmas, j = 3, 1, 1 N denoting neutron-proton triplet, neutron-proton singlet and neutron-neutron

singlet.

The requirement of reproducing the scattering lengths and effective ranges gives numerical results for Kj , where Kj is Ki = i pji ,

j = 3, 1, In.

i=l

In determining these heavy 0 parameters we have the freedom to choose Nj2 > Nj, > 1 and then determine values for xji . Because the calculation of the d discontinuity is nonrelativistic, a precise knowledge of Nji is unnecessary (the region where the mass dependence becomes interesting is relativistic). Effectively, our dispersion relations are Im h(w’) dw’

h:(w) = j L.f$.C. (w’ - WI r +s R.H.C. lc;r;‘;jd;’

+ &

for

w<
where the integral over the left hand cut (L.R.C.) includes only the r-cut and the a-cut contributions, and wL is a large value -m2. One might be tempted to start with the above equation and remove the restriction w < wL . Proceeding thus would be inconsistent with unitarity constants for this would predict that f;,(w) + Ki as w + co. Combining this with the unitarity equation. yields Imf(w) Such behavior of Imf(w)

= 2m[w/(w + 4m)]l/” 1f I2+ 2mKj2. would give a divergent representation

for the integral

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over the right hand cut (R.H.C.). Moreover, the scattering length and effective range cannot be reproduced with only one parameter, Kj . If we had parameterized our nucleon-nucleon amplitude by choosing different phenomenological particle exchanges (e.g., p mesons and heavy p mesons), the same constants Kj would be obtained and, hence, our calculation did not depend on the model.

V. CALCULATING THE LEFT HAND DISCONTINUITIES FOR N-D

SCATTERING

The important left hand singularities of the neutron-deuteron amplitude have been previously located. We now calculate the discontinuities of these singularities employing, as usual, Bjorken and Drell notation. Proton Singularitiy

The invariant amplitude for the proton exchange diagram is i-&p = [g&(N)

9 * y(P, - p41 - m) $I* . yu(N)]/[(D

- iv)2 - my.

Forming the nonrelativistic limit, calculating the discontinuity across the D2 axis, for a given scattering angle and performing the partial-wave projections yield a contribution to disc& , namely, disc f E2= s disc f

z4=

-2

O(4mB + D2) d(9D2 + 4mB),

* disc f z2. A and A’ Singularity

General Discussion

To calculate the discontinuity associated with the A singularity, we require a representation of the on-shell nucleon-nucleon representation (for the four-point blob of Fig. 2). We define, for convenience, an “unprojected nucleon-nucleon amplitude” as the sum of particle graphs and another term (as described in Section IV) where the S-wave projection of the sum is the S-wave dispersion theoretic amplitude. It is this “unprojected nucleon-nucleon amplitude” which we insert for the four-point blob. The obvious expansion is made yielding graphs i-vii of Fig. 6, and the contribution of each of these graphs to the discontinuity is separately obtained. For each graph which corresponds to particle exchange in the nucleon-nucleon amplitude (graphs i-vi), the Feynman amplitude is calculated, and then the contri-

S-MATRIX

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389

SCATTERING

bution to disc& is determined. When the Feynman integral representation for the discontinuity is infinite, (i.e., graphs i, iii, v, and vi), the integral (both real and imaginary parts) is regularized and the leading nonrelativistic term is used to calculate the contribution to disc.fi (i.e., we are cutting off the discontinuity). The graph in Fig. 6 with the most interesting singularity structure is graph ii which contains two nonrelativistic anomalous singularities whereas the others contain at most one. It also manifests a normal singularity at t = 4m2 which vanishes in the nonrelativistic limit. We describe the analysis for this graph in some detail. The analysis appropriate to the other graphs will then be obvious. The Feynman integral representation of graph ii is:

Nm(Q, D 'a!

=

s #4(Q2

-

*2)[@

-

Q)2

-

, D',

N, N',

S)

m2][(D’ - Q)z - m2][(D - N’ - Q)2 - mr2]

S = all spins and polarizations and, as usual, m and m, also have a negative imaginary part. We first do the Q” integration by the well-known method of residues. Before doing the Q integration, we expand the integrands on the assumption that the square of any three momenta is small with respect to the square of the nucleon mass (the leading terms retained.) The result of this integration defines our nonrelativistic amplitude. For N,, = 1, the above prescription yields NR L! - Lz =--

st

(2 ) “I

[mB+

(q-Q)2][mB+

(zQJ2]

[m,2+(D+D’-Q)2]’

Consider now a simpler integral-the graph in Fig. 1 with the four-point nucleon-nucleon blob replaced by a constant. Its Feynman integral representation is IA = s &

NAQ, D, D, N, N, S> (Q2 - m2)[(D - Q)2 - m2][(D’ - Q)2 - m2] .

Set Nd = 1. (Note that Id = limm,a,m - m,,21es .) In the nonrelativistic Id - IT”:

1 (2T)3 mB f

1

(Q - q)”

mB +

(Q

_ $)”

This integral is rewritten using Feynman’s integral identity [29]:

IdNR = 2

J’o (x12t -

4

x,t + 4mB)li2 ’

*

limit

390

BOWER

which is easily integrated yielding: -i -ln 7T25m4 t

NR IA-

4dmB+4 ( 42/mB-d1/T’

1

Thus, disc I,““(D, z) = -e(t - 16mB)/16mt1/2.9 The calculation of disc IzR is similar, but more tedious, yielding: -4rri disc I”(D2, z) = c1

2 1 ___ PI2 r2 - r 1 w+ (rI t1/2 & w+&2* 2 1

w -I- mB - m,2 - 0.25r2t (r, - s1)1/2 (r2 - s2)1/2 mB - mr2 - 0.25r,t - s#/~ (rl - s2)l12 mB + mm2 - 2r,(w - 0.125t) (r2 - s&a (r2 - sq)lj2 w - mB + m,2 - 2r,(w - 0.125t) (rl - s3)lj2 (rl - s,)~/~ 1’

(-

+ 2

where

w a, bl cl

5 = = =

-(1.25 + z) D2, (w - mB + m,“)” - 4m,2w, (t/2)(w - mB + mT2), -tw + 0.0625t2,

r1t2) = k-b, + (-1 tb2 - 4~~)1’21/2~~, s1t2) = + + (-) $[l - (16mB/t)11i2, s2t4) = [w - mB + m,,2 - (+) [(w + m,J2 - mB]li2 [(w - mJ2 - mB2]1/2]/2w. Deuteron Box Diagram The invariant amplitude

for graph i of Fig. 6 is

d4Q C(N’) y““[m - (@’ - @)I cj’*

. rtm + t?) 4 * rb - (b - @,)IY”WM I @-)a

(Q2

-

m2)[(D

-

Q)”

-

m2]

* [(D’ - Q)” - m”][(D + N - Q)2 - mD2] I

* (&,u, -

(D + N -

QL, tD + N - Q>,, mD2

)*

9 This result agrees, as one would expect, up to a phase, with that obtained by a naive application of the Cutkosky rules. See R. E. C~OSKY, J. Math. Phys. 1 (1960), 429.

S-MATRIX

The nonrelativistic

THEORY

discontinuities

s

= -24rrigDm2

W[(21D2z

-1

disc j&

N-D

391

SCATTERING

are

1

discj,,

OF

O(t - 16mB) dz + 12mB)2 - 72D2(t - 16mJ3)(1 + z)]li2 ’

= 4 . disc fo2 .

Pion Exchange

The pion exchange graphs iii in Fig. 6 do not contribute to the neutron-deuteron amplitude because isospin is conserved. The sum of pion graphs ii of Fig. 6 yields the invariant amplitude h4, = .-3igD2g02 .-

d4Q E(N’) y5(E3 - $2 + m) $ - d-52 + m) $‘*r@’ - $2 + m> y54N) s (27~)~ [(D - Q)” - m2][(D’ - Q)2 - m”][(D’ - N - Q)2 - m,2][Q2 - m2] .

The analysis required is that appropriate to I,, with IV,, # 1. The numerator must also be expanded, and the quadratic powers are retained. The analysis is considerably more difficult technically, but we will content ourselves with quoting the results, namely:

fn,==-%J2g02 24 Jl dz {1°(2.5m,2)

+ 115(-w

+ mB - m,,2)

-1

+ f11[5w + 0.5(D + D’)2] + 112[5w - 0.625t + 0.625(D + D’)“]}

L, ==

--gD2g02 l dz {iom,2 + f12(- w + mB - mr2) 24 J + @[2w + &67(D

+ D’)2] + 112[2w - 0.25t + 0.208(D + D’)“]},

where pi E

p E

I

dx, dx, dx, S(l - C:=‘=, xi) xixi

’

(b2 - 0.25a2)3/2 dx, dx, dx, S(1 - C:=, xi) xi

s

(b2 - 0.25a2)3/2

’

i,j=

1,2,

i = 0, 1, 2;

i = 0 indicates that there is no xi in the numerator.

a = -[x,D

+ x,D’ + 2x,(D + D’)],

b2 = (xl + x,)(mB

+ 0.25D2) + x~((D + D’)2 + mw2).

The calculations of the discontinuity p’ij and IC involve similar techniques to those employed in the calculation of the discontinuity of Jo.

392

BOWER

Sigma Meson Exchange The nonrelativistic

discontinutities

for diagram v in Fig. 6 are:

disc fy2 = disc fy4 _

-i3Lyy2 + 8(-4D2

The nonrelativistic

- m,2) ln \ ( [(-4D2)W - 16mB) In i ( [(-4D2)lj2 je(-4D2

(r

amplitude

(16mB)1’2 - m, (16mB)li2 + m, - mo][(16mB)1/2 + m,][(16mB)1’2

) ! f m,] - m,]

*

for diagram iv of Fig. 6 is

M (I = m2g,2g02xN+o . 40 . +*xi~ 21 PY ’ 1 [mB + (OSD - Q)“][mB

+ (O.fi’-

Q)2][m,2 + (D + D’ - Q)“]

and an integral of this type, namely, I,, , has been previously analyzed. The discontinuity f y2 has a contribution due to the d singularity and a contribution due to the singularity which arises from a sigma pole in the t-channel. The calculation thus implicitly invokes a model for calculating the deuteron-sigma coupling constant. The latter contribution accounts for the first term in the equation for disc f z2 . The discontinuity of MU” has a contribution from the d singularity and from the A, singularity. (d,’ is analogous to d’ with m, replacing m, .) Heavy Sigma Meson Exchanges The invariant amplitude for diagram vii in Fig. 6 for which an intermediate proton scatters off the heavy sigma, gives in the limiting cases of m$, --+ 00, i = 1,2 and j = 3, 1, 1 N, the discontinuities disc f y2(D2) = 1.5(3K, + KS) - i3i-gD2g,2m2 [(-4D2)l/” D2mU2 -1 s 1.5(3Kl + K3) * disc f %, disc

- 4(mB)‘12] 8(-4D2

-

16mB)!

f QD”) = 6K, * disc f r.

The invariant amplitudes for diagrams vi and vii in Fig. 6 for which an intermediate neutron scatters off the heavy sigma yield in the same limits: disc f gN(D2) = 12KIN * disc f 2‘, disc f zN(D2) = -SKIN

. disc f 2.

S-MATRIX

THEORY

OF

N-D

393

SCATTERING

Right Hand Cut Contribution of the Nucleon-Nucleon Amplitude The right hand cut part of the “unprojected nucleon-nucleon amplitude” contributes to the d singularity of the neutron-deuteron amplitude. Symbolically, this contribution is shown in diagram vii of Fig. 6. Only the S-wave contribuions are explicitly considered because the contributions of higher partial waves are much smaller in the nonrelativistic limit. A heuristic argument suggests that the P-wave ~4% of the S-wave at the leading edge of the d singularity (where Re w PQ -16mB). Because the right hand cut contribution is less than a 10 % effect, the inclusion of higher order partial waves is not necessary. The right hand cut contributions of the neutron-proton singlet, neutron-proton triplet, and nucleon-nucleon singlet S-wave amplitudes are handled separately, but similarly, and so we examine only the n - p triplet case. Here the spin dependence is identical to that of the deuteron pole graph. In the deuteron box diagram, the deuteron pole contribution can be isolated. By comparing the contributions in the n - p triplet S-wave scattering, we determine the necessary replacement. Defining Ii,

I9=2s1(1$2)1,2 [fj(-A

- icy) +fj(-A

where f j is the modified effective range amplitude A =_ mB - D2(3 - 0.752) 4

9

+ icy)],

(both poles on second sheet) and

C = 3[(-D2)(t

-

16mB)(.5)(1 + z)]~/~,

and where j = NPT, NPS, and NNS refer to the neutron-proton triplet, neutronproton singlet, and neutron-neutron singlet waves, respectively, we can state the discontinuity formulas: disc fr&‘h’,zr(D”) = -i6rgD2mCt

I

disc fFii(D’)

= 4 disc frL”(D2),

disc f fb:@“)

= -i18rrgD2mC,,,

disc fL:f(D’)

= 0,

disc f fh$(D2)

= -i36rrgD2mC,,,

disc fz$i(D’)

= 0,

1, $

O(t - 16mB) I”“,

l -cc l!?(t s -1 PI2

16mB) I”‘“,

l * e(t i -1 PI2

16mB) I”““,

BOWER

394

where rt = triplet n-p effective range, rs = singlet n-p effective range, c wan _= __2

rsnn

3

r Snll c singlet n-n effective range.

VI. ANALYTICI~Y-UNITARITY

BOUNDS

Neutron-deuteron scattering is a three-body problem because the nearest break-up channel at D2 = 1.33mB is very near the elastic threshold at D2 = 0, and the triton pole at D2 = -3.76 mB. Any nonrelativistic calculation must account for this channel. The formal inclusion of break-up in N/D calculations is straightforward, but a good description of break-up is lacking. Moreover, the N/D solutions may include C.D.D. poles which have emerged through the inelastic cut. We now calculate sum rules which require information about only the elastic channel [30]. If the nearby singularities are the only important ones and the elastic S-waves of the neutron-deuteron system satisfy unitarity constraints, our results are compelling. As usual, the first step to obtain sum rules is to write Cauchy integrals:

$+(Y).~(Y)

4 = 0

Y = D2

j = 294,

where 4(y) is a test function analytic in the cut y plane. Our knowledge of h(y) determines the optimal form for 4(y). The discontinuities offj(y) for y < 0 have been calculated. Information about fj( y) for y > 0 is available. Therefore 4(y) may have cuts only on the line y > 0. If 4(y) were to have cuts elsewhere, on these cutsfj( y) would have to be known and this would require information not available to us. As our calculation is nonrelativistic, d(y) should emphasize the nonrelativistic region and not the relativistic region. For a 4(y) with the optimal structure the sum rule becomes

4(y) dy- ~&#~YT) +jmImMb9 $(v)l du =0. s Imhb9 L.H.C.

0

(At the triton pole, f2( y) = &/( y - yT).) Examples analytic properties are d(y) = l/(d-(y

- b) - a)”

a > 0

of +(y)‘s with optimal

b > 0

N = integer

S-MATRIX

THEORY

0~

N-D

395

SCATTERING

Where no experimental information exists, relativistic unitarity obtain a bound for the integrand on the right hand cut, namely,

I Im[f#WhWll

1 < 3

z/v

+ 3122+ z/u + m2

=f”(u) I &Y>l

3m

can be used to

I 4(Y)l

j = 2, 4.

Assuming the availability of experimental information for f,(y) 0 < C, < y < C, , the sum rule can be written as the inequality

in the region

I
#Y)l f”(Y) dY,

which can be rewritten as an inequality for gT2(RT = 3gT2m/4n). If for all 4(y) we can show that 0 < A < gT2 < B in the doublet S-wave channel, thenf,( y) exists with the properties: (i) it has a triton bound state at the experimentally coupling constant gT2, A < gT2 < B. (ii) it has no ghosts or extraneous bound states. (iii) (iv)

known value, and a

it has the left hand discontinuities used in the calculation. it satisfies unitarity on the right hand cut.

If, however, a contradiction occurs (for one #(w) A, < gT2 < Bl , and for another $/o~~t{~ < gr2 < B, , B2 < A, or Bl < A,) then one or more of the conditions is If for all 4(y) we can show --I A I < gT2 < 1B 1in the quartet S-wave channel, then f&v) exists with the properties: (i) (ii) (iii)

it has no ghosts or extraneous bound states. it has the left hand discontinuities used in the calculations. it satisfies unitarity on the right hand cut.

The occurrence of a contradiction

indicates that a condition has been violated.

396

BOWER

VII. N/D SOLUTIONS After determining the left hand discountinuities for a partial wave, historically one has usually proceeded to do “N over D”. In Section VI, we described a technique to determine whether the left hand discontinuities are compatible with partial waves having desirable properties. Because no N/D solution with desirable properties can exist if the sum rule is violated, the sum rule can be used to restrict the number of N/D constructions. Satisfying the sum rule guarantees the existence of N/D solutions with desirable properties; however, the solution may require many C.D.D. poles. After testing the forces using the sum rules, partial wave solutions are constructed using N/D techniques. For details about this well-known construction, the interested reader can consult any number of sources.1o Thus, only a few highlights will be noted here. The N and D functions are defined by the equalities:

A single-channel approximation to the coupled channel unitarity equation is employed, (the coupled channel term being neglected as it is relatively small for y < m2), namely,

= -2ipxjj*.

A simple parameterization of break-up a Froissart [31] is employed to demonstrate the general nature of the role break-up plays. Operationally pe is replaced by pe(e(y) + RB( y - 1.33 mB)), with R a constant. For a nonrelativistic approximation to pe and a substraction point at co, the Dj equation becomes Q(Y) = 1 - f

m dz” Im[h(-z”)] --rL l+R

-1 04

Dj(-z”)

97

-+JE l/lyl

1 Tr

lyl-zz”

for

y < 0,

lo A discussion and a list of references encompassing the single-channel problem, the manychannel problem, and various parameterizations of inelasticities in contained in E. G. HOHLER, of Springer Tracts in Modem Physics, Vol. 45, Chap. 6, Springer-Verlag, Berlin, 1968.

397

S-M.~TRIX THEORY OF N-D SCATTERING Re Q(y)

= 1 - 1 m dz Im[h(-z”)] -YL. = (1 + R)dz” z# - y

D,(--2”)

I 2R 1 n Z”fY

for

y > 0,

yL designates the nearest point of the left hand cut. The logarithmic divergence in Dj at y = 1.33 mB is artificial, being a result of our treatment of the three-body threshold dependence, which should be of the form (z - QmB)” (phase space). Phase Shifts

The doublet and quartet S-wave amplitudes can be written f; = l/(1 D 1cot aj - i/ D I). Experimental information for / D 1cot Si is available, and this can be expressed in terms of Ni and Dj as follows: Re Dj = Nj Cot 6j , where Sj becomes complex when break-up occurs.

VIII.

RESULTS

The sum rule results and N/D solutions are now presented. Sum rules results are for test functions c$(y) of the form 4(Y) = W-Y

- 4”.

The N/1) solutions are obtained by approximating

the linear integral equation for

D by a matrix equation.

Results are shown for two cases: (1) proton pole is the only left hand singularity; (2) proton pole, d singularity, A’, sigma pole, and A,’ singularity are the left hand singularities. For case two, two values of the cut-off parameter and several parametric values for the sigma deuteron coupling constant are employed.

398

BOWER

DOUBLET

SUM RULE

RESULTS FOR gT2

(PROTON EXCHANGE ONLY)

--I-

IO-.

FIG. 7. a is in units of (mB))'l"; mB = 2090 MeVZ; A unitarity bound is used for the entire right hand cut; A doublet S-wave solutions is possible.

QUARTET

SUM RULE RESULTS FOR Cj 2

(PROTON EXCHANGE ONLY) I.2 IO

.’ .

8

.’

6

..

4

‘.

2

”

FIG. 8. a is in units of (mB)>"*; mB = 2090 MeV2 ; A unitarity bound is used for the entire right hand cut; No quartet S-wave solution without ghosts is possible.

S-MATRIX

THEORY

0~

N-D

399

SCATTERING

Proton Exchange Alone

The S-wave doublet and quartet sum rule results for a nonrelativistic unitarity bound on the right cut are shown in Figs. 7 and 8, respectively. In Fig. 7, the upper and lower bounds of gT2 (triton coupling constant) are plotted against the test parameter “a”. In Fig. 8, the upper and lower bounds of g2, a coupling constant associated with a bound state having spin 312, even parity and the experimental triton binding energy are plotted. As such a particle does not exist, a value of 0 for g2 must emerge if a quartet partial wave with desirable properties can exist. These graphs show that a solution with desirable properties is possible for the doublet but not the quartet. The elastic S-wave N/D solutions (for a 10 mesh point approximation) are consistant with the sum rule predictions. The doublet solution contains a bound state with energy 2.3 MeV, a doublet scattering length of approximately 100 F, &/‘b

QUARTET (PROTON

PHASE SHlflS act-kws~

VS. ENERGY CNL~)

.

-sol

.ol .or i .04 ’ c& - .OB ’ - 5’ (i-2) --i

FIG. 9. * van Oers and Seagrave experimental values; R = 0.10; . . . . . . R = 0.20. 595/73b7

lb

I2

R = 0; -.-.-.

R = 0.05; - - - - - -

400

BOWER

and a triton coupling constant gT2 = 0.14 f o0.14 07. The sum rule results predict 1 < gT2 < 3.5, but only for a solution with a bound state having an energy of 8.5 MeV. The introduction of inelasticity into the N/D equations by means of the “R” parameter is not sufficient to simultaneously reproduce the triton binding energy, the triton coupling constant and the doublet scattering length. The quartet solution, as predicted by the sum rule, contains a ghost with energy 18 MeV. The value for the quartet scattering length of 6.5 P1 is in reasonable agreement with the experimental values (6.14 f. 0.06) F [6] or (6.38 i .06) F [7]. In Fig. 9, the quartet N/D phase shifts, 1D I cot 6, are plotted against the energy and compared with the experimental phase shifts of van Oers and Seagrave [7]. Several curves, each corresponding to different inelasticity values as characterized by R, are shown. The bump at three-body break-up, D = 0.071 F-2, in the curves with nonzero inelasticity is artificial, a consequence of our cavalier description of the break-up threshold behavior. The elastic solution and solutions with R values less than 0.2 fit the data very well. Fixing the doublet scattering length by subtracting the N equation at threshold (and introducing a C.D.D. pole), we obtain a binding energy of 5.7 MeV. and a triton coupling constant of 1.25 (20 mesh points)12. For R = 1, a binding energy of 4.8 MeV and gT2 of 0.8 are obtained. Incorporating experimental information (i.e., using the formula in the van Oers and Seagraves paper [7]) yields more restrictive sum rules. Figure 10 shows the sum rule results for the following situations: (1) A unitarity bound for the entire right hand cut; (2) Experimental results for the regions: (i) 0.01 F-2 < D2 < 0.3 F-2 (ii) 0.001 F-2 < D2 < 0.3 F-2 (iii) 0 < D2 < 0.3 F-2 and a unitarity bound for the rest of the right hand cut. The results using the formula for the region 0.1 F-2 < D2 < 0.3 F-2 would be indistinguishable in Fig. 10 from those for case 2i. If the formula is valid for LL < D < 0.3 F-2, with LL < 0.001 F-2, the sum rule predicts that a doublet S-wave with the desirable properties can not exist. As there are two experimental I1 To test the appropriateness of the approximation, the number of mesh points is varied from 10 to 25. No discernible change in D occurred. If, in addition, a linear interpolation scheme for D is employed to calculate the scattering lengths and phase shifts, a quartet scattering length of (6.2 + 0.05)F is obtained which is 0.2 F less than the results quoted above for a lO-mesh approximation where no interpolation scheme is used. The latter results are henceforth labeled as “crude approximation” results. I2 G. B. BARTON AND A. C. PHILLIPS, Nucl. Phys. A 132 (1969), 97 subtract N at threshold and obtain a triton binding energy of 6 MeV.

S-MATRIX

SUM

THEORY

RULE

OF

RESULTS

INFORMATION (PROTON

N-D

401

SCATTERING

FOR

C&’

WHEN

IS INCLUDED EXCHANGE

ONLY)

FIG. 10. a is in units of mlW2 ; mB = 2O!JO MeV2 ; unitarity bound over entire right hand cut; -a-.-. experimental information for 0.01 F-’ Q D2 < 0.3 F-* ; . * . * . . experimental information for 0.001 F-2 Q Da < 0.3 Fe2 ; - - - - - - experimental information for 0 < D2 < 0.3 F-2.

values for the doublet scattering length (0.7 f 0.3) F [8] and (0.11 & 0.07) F [7], the van Oers formula is suspect for D2 < 0.01 F-2. This being the case, the sum rule results are not significantly more restrictive than those which incorporate unitarity for the entire right hand cut. Including all the Singularities

Since our calculations are nonrelativistic, a cut-off is used for the left hand discontinuities, and nonrelativistic unitarity (as opposed to relativistic) is employed. For the sum rules, the larger “a” becomes, the more the relativistic as opposed to nonrelativistic behavior, is emphasized. With no cut-offs, the contribution of the left hand cut to the sum rule dominates the nonrelativistic unitarity bound contribution as a + 00. The relativistic unitarity bound contribution, however, dominates the left hand cut and reinforces our argument for using nonrelativistic unitarity and a cut-off procedure. In Fig. 11, the doublet S-wave rule results for a cut-off at D2 = -50 mB are shown. (The curve labeled SNDG = 1.) This curve reveals that no doublet S-wave solution with all the desirable properties can exist. The situation can be remedied

402

BOWER

DOUBLET

SUM

RULE

RESULTS

.-.-

, ‘;‘“y :,i

7 ‘\

9

,\ I1

,,(,. 13’+5,17

IS \

‘\

‘\ ‘\

21 23 --a-t

_ _ ‘I,., 25 Z’r

FOR

. . i+

. . . 31 ‘*.

..A

‘\ ‘..

FIG.

11.

-SNDG=l;C.O.=--5OmB;...*

..SNDG=O;C.O.=--5OmB;------

= -1; C.O. = -50 mB; No doublet solution for SNDG possible for SNDG = 0.

SNDG

= 1, -1;

Doublet

solution

by changing the contribution of the o-exchange in the t-channel. Figure 11 shows the results for the cases in which the sigma discontinuity is 0 times the original value (the curve labeled SNDG = 0)and -1 times the original value (the curve labeled SNDG = -l), and suggests that acceptable solutions can exist for SNDG values satisfying -a, < SNDG < a2 , where 0 < ai < 1 (i = 1, 2). The sum rule results for a cut-off at D2 = -25mB are compatible with the existence of desirable doublet S-wave solutions. This cut-off eliminates the contribution of a-exhange in the t-channel. The N/D results shown are for an 18 mesh point approximation, 9 mesh points in the proton cut region and 9 points in the “additional cuts” regiorP. In Fig. 12, I* To test the approximation, the number of mesh points in the proton cut region is increased from 9 to 25. If, in addition, a linear interpolation scheme for D is employed, scattering lengths 0.3 F lower than the doublet scattering lengths for a 9-mesh-point approximation and no linear interpolation scheme (“crude approximation”) are obtained. These results are stable. The number of mesh points in the “additional cuts” region are not varied for lack of time and money. Uncertainties of the order of 0.3 F are anticipated from this region based on the above results. Our best estimates of the scattering length values henceforth appear in quotation marks. Scattering lengths with no quotation marks are for the “crude approximation.”

S-MI.~TRIX

THEORY

N/a DOWIT

0~

ALL

SCATTERING

PARAMETERS THE

(FOR

N-D

SINWLARITIES

403

VS.

INELAsTiCiTY

FACTOR

AND A CUT-OFF, %--r%ni?)

6 .' 5 '. 4 .. 3 -. 2 .t

_.-

,, ,,

_..'.

'\ I .. 0 'lx, i '\ A \ -I -2 -. .3 '. -4 '.

‘\

__. . .. ...'

d '\

A

ID

-R--b

'\ '\

'\

'\

-5 .. -6 .. -7 -8 . .

FIG. 12. &(MeV); - - - - - “‘a2 ” (F); * . . . . gr2. The error bars indicate the uncertainty in four mathematical approximation. Experimental values: or = 8.5 MeV a2 = (0.11 * 0.07)F [6] or (0.7 f 0.3) F [7].

the N/D doublet parameters using a cut-off at D2 = -25mB are plotted against the inelasticity parameter R. Solutions which reproduce the experimental values have R values near 0.2. For R = 0.2 the theoretical values are cT = (7.9 & 0.2) MeV, a2 = 0 . 1 F 7 "a2" = (0.4 f. 0.4) F, gT2 = 2.6 f 0.2 14. I4 Other estimates of gr2 are given by L. P. KOK AND A. S. RINAT (Reiner), Nucl. Phys. A 156 (70) 593 AND M. P. LOCHER, Nucl. Phys. B 23 (70) 116. The latter obtains a value of gfb = 2.8 k 0.3, a value in good agreement with our results.

404

BOWER

bjb DOUBLET S-WAVE PHASE SHIFTS VS. ENERGY

-I6 .. -1.7 .. .04

A8

.I2

.I

20

.24

.28

52(f-2) FIG. 13. * van Oers and Seagraves’ experimental results; -p 0-5OmB;***. .p = -1.0; R = 0.35; C.O. = -50 mB; -----R

= -0.35; R = 0; C.O. = = 0.2; C.O. = -25 mB.

The g$ value is compatible with sum rule predictions that suggest gT2 satisfied the inequality 2.5 < gT2 < 5.3. In Fig. 13, the doublet phase shifts are plotted (the curve labeled R = 0.2 and C.0 = -25mB). Recall that the bump at D2 = 0.071 F-2 is artificial and has its origins in our approximate treatment of the break-up threshold behavior. In Fig. 14, the N/D doublet parameters for a cut-off at D2 = -50mB are plotted against p, a parameter proportional to SNDG5. I6 Our sum rules are more flexible with regard to varying the strength of the sigma contribution than our N/D solutions (questions of time and money are involved). We can, however, obtain an estimate of the effect of varying the strength in the N/D by keeping the same mesh points and varying the contribution of the sigma exchange for these mesh points. If SNDG characterizes the factor multiplying the original strength in the N/D solution (at the fixed mesh points), p and SNDG satisfy either of the inequalities P
S-MATRIX

THEORY

OF

N-D

SCATTERING

405

b@ DOUBLET PARAMETERS vs. DIFFERENT SIGMA cONTRIEJJTI~NS 22 20 I8

t

16 14 I2 IO 8 6 4 2

-2 -4 -6 -8 -10

FIG. 14. -&.feV); sigma exchange contribution. (0.7 f 0.3)F 171.

-----“$“F; Experimental

. . . . *gr2; p characterizes the strength of the values: l r = 8.5 MeV; d = (0.11 f 0.07)F [6] or

For p values near -0.39, the low energy three-body parameters are reproduced. Folklore, and our experience with proton exchange, suggest that increasing the inelasticity should increase the binding energy (also decrease the energy of a ghost). In Fig. I5 the doublet parameters for p = -0.35 are plotted against the inelasticity parameter and the preceding suggestion is demonstrated. The graph also shows that the scattering length decreases with increasing inelasticity. In the solution which reproduces the experimental values, p = -0.35 and R = 0, the theoretical results being: E* = (8.65 f .2) MeV a2 = (0.3 f 0.3) F, “d" = (0.6 i 0.4) F, g,2 = 3.4 & 0.2. In Fig. 13, the doublet phase shifts are plotted (the appropriate curve being labeled

406

BOWER

N/D DOUBLET PARAMETERS VS. THE INELASTICITY FACTOR

FIG. 15. a2 = (0.11

f 0.07)F

sr(MeV);

- - - _ - “,2”

[6] or (0.7

f 0.3)F

F; . . . . . gra;

Experimental values: Ed= 8.5 (MeV);

[7].

in an obvious manner). The theoretical calculations are in good agreement with experimental values. To determine (independently of the above approach) values for g,2, a pole dominance method is used to estimate the triton photodisintegration cross section. Our approach is based on analogous calculations which we did for the photodisintegration of the deuteron. A pole dominance calculation for this process gives a reasonable fit to the low energy El cross section data and underestimates the Ml cross section near threshold. (For the latter process, rescattering effects are important as there is no centrifugal barrier. Our calculation neglects this rescattering effect.) Experimental data [32] show that El cross section dominates the Ml cross section except at extremely low energies and so the pole fit gives a reasonable

S-MATRIX

THEORY OF N-D SCATTERING

407

fit within a factor of 2. As the triton and deuteron are both diffuse states (weakly bound), a similar behavior is expected for the photodisintegration of the triton; in particular, we expect that the El cross section of the triton dominates the Ml cross section near threshold. With this assumption, the photodisintegration data of Fetisov et al. [33] are fitted with pole diagrams and the value gr2 = 3 is obtained. If this value of gT2 is used to calculate the cross sections, the Ml cross section is much smaller than the El cross section (except for extremely low energies), as expected. The quartet S-wave sum rule results for a cut-off at D2 = -5OmB predict independently of the sigma exchange contribution that there are no quartet S-wave solutions with all the desirable properties. The N/D solutions as predicted contain ghosts. In Fig. 16, the ghost values, ghost couplings and quartet scattering lengths are N/D QUARTET PARAMETERS DIFFERENT ALL

SINGULARITIES

vs

SIGMA CONTRIBUTIONS AND

A CUTOFF

AT

.. 22 .. 20 .. 18‘24

16.. \ I2. IO.. *.-* -.. .. : *. 8 .. ,**. : _______ __-_--_----:L 6 -. -------;L----:. : : 4 I’ :,,+ 2.. : -. . -0 -6 $’ -2 0 I 14

FIG.

16.

-

<&,feV);

exist; b’ = (6.14 h 0.06)F

[6]

- - _ _ _ “a@ F; . . . . . or (6.38 f 0.06)F 171.

-g2;

Experimental

values: EG does not

408

BOWER

plotted against p. The quartet scattering length is very insensitive to change in p, but the ghost moves to the left as p -+ -co. For p = -10, the N/D solution contains in addition to this ghost, another zero of D at an energy of 92 MeV. For p = -1000, the ghost moves to 90 MeV. The low energy phase shifts are as insensitive to variations in p as the scattering length and indistinguishable from those for which proton exchange is the only source of the left-hand cut discontinuity. Thus the graph of Fig. 9 is appropriate for any of the following cases: (1) A cut-off at D2 = -5OmB and -1000 (2) A cut-off at D2 = -25mB.

< p < 1;

IX. CONCLUSION Our work demonstrates that a purely S-matrix approach provides a good basis for understanding the low-energy behavior of the three-nucleon system. Good fits to the low-energy neutron-deuteron quartet and doublet S-wave phase shifts, the triton binding energy and the triton coupling constant are obtained. To arrive at these results approximations for the distant singularities and inelasticity have been made. The low-energy behavior of the quartet S-wave does not depend on a detailed knowledge of the distant singularities. With proton exchange as the only source of the left hand discontinuity, a good fit to the low-energy quartet data is obtained. In confirmation of our sum rule, our solution contains a ghost. Barton and Phillips [20] who made essentially the same calculation (proton exchange only, no inelasticity) obtained results similar to ours and we conclude that their solution must also contain a ghost.l6 When we include other singularities our good fit to the lowenergy quartet phase shifts is maintained. Also, as before, a ghost is found at approximately the same position. Varying the strength of the a-exchange term did not change our good fit to the low energy quartet data but did move the ghost to the left. The point of view that the appearance of a ghost indicates the neglect of a short-range force is adopted, a somewhat analogous position to that taken by Noyes and Wong [27]. Its presence seems essential, but the low-energy quartet behavior is insensitive to its position. A good fit to the quartet data is also seen in other recent calculations which use variational approaches [34] and separable potentials [lo]. In our calculation the disagreement between the theoretical and experimental quartet phase shifts for energies greater than the break-up threshold energy is reduced by the introduction of inelasticity. I6 Because Barton and Phillips have normalized their D equation (the D of N/D) at the origin instead of at to, it is easy to overlook this ghost.

S-MATRIX THEORY OF N-D SCATTERING

409

In the doublet S-wave channel, the low-energy phase shifts and the triton binding energy are somewhat sensitive to the three parameters at our disposal which are: (i) (ii) (iii)

the cut-off position of the left hand discontinuities, the strength of u-exchange, inelasticity.

The cut-off and u-exchange represent distant singularities, and are the dispersion equivalents of short-range repulsive forces. The inelasticity contribution vanishes quadratically with energy at the break-up threshold and, consequently, represents a distant singularity, a singularity which acts like a short-range attractive force. These three parameters thus describe the distant singularities of the amplitude. Hence, adjusting only the distant singularity structure produces the entire lowenergy doublet S-wave amplitude: the triton binding energy, the triton coupling constant and the low-energy phase shifts. In closing, the following research suggestions are made: (1) Use a more realistic representation of break-up. (Break-up is especially easy to incorporate in this approach.) (2) Determine a better value for the A discontinuity. This requires a better description of the amplitude Dn -+ PN than the pole approximation. (3) Study the distant singularities (short-range forces) in more detail.

APPENDIX In the appendix, the spin projections are described. The spin dependence of a nonrelativistic graph is denoted by S, to which are attached descriptive appendages (e.g., sD represents the spin factor of the nonrelativistic deuteron box diagram). The following notation for the spin projections is introduced: j j j j

= = = =

1 -singlet spin amplitude, 2-doublet spin amplitude, 3-triplet spin amplitude, 4-quartet spin amplitude.

The Condon-Shortly convention for calculating Clebsch-Gordon coefficients is observed and the notation (j,m,j,m, I JM) is employed for the coefficient. The graphs for nucleon-nucleon scattering are discussed first and then the graphs for neutron-deuteron scattering.

410

BOWER

Deuteron Pole

- i) a:a(fiu3

= a(--iu3

Since (+ 4 4 - 4 / 00) = -($

+ i) a = 2.

- $ 4 4 100), (1 1SD ) 1) = 0.

Pion Exchange s,= EE x&u . (N’ - N) xNlx$m

* (N’ - N) x+-.

Introduce the notation N- = N’ - N, (3~S,-~3)=(11I~~~3)ola.N-olaro.N-ol(~~Q~Ill) = G2, where 3 denotes the third component of the vector (1 I Sdn I 1) = (00 / 4 + + - Q) o1~. N-o@ * N-/3 <+ + + - 4 ) 00) + (00 1 4)8 ; - $) au . N-@u . N-a (4 - 4 4 $100) + (00 14 - s + 4) /3u . N-/&xi . N-a (4 4 + - $100) u . N-mu . N-/3 ($ - + 4 Q100) +s = i(-NF2

- (Nl-

- NF2 - (Nl= -N-2, S;= = x&s

+ iN,-)(N,+ iN,-)(N,-

- (N’ + N) xNa . x&s

- iN,-) - iN,-)) - (N’ + N) xN1 .

Introduce the notation N+ = N + N’: (3~S,“,/3)=(11~~~

*g)olo.N+ololG*N+CL(+&

(llS&;,l)=(OO~~~

$*Ill>=N$q

+-+)~.N+oc/b.N+~(fr+ $--hlOO> . N+@o - Nfor (+ - + + + 100) +(ooI+g ~-~)m3 + (00 1 4 - 4 + 4) ,&s . N+$b. N’a <+ + 3 - 2 100) +(001&--Q ~~)/3u-N+~cm~N+a(fr-~ +&lOO) zzz*{NC2 + (N,+ + iN,+)(N,+ - iN,+)

+ Ni2 + (N,+ + iN,‘)(N,+ = Nf2.

- iN,‘>}

S-MATRIX

THEORY

OF N-D

SCATTERING

411

u Meson Exchange

r-

Substituting we obtain

1.

the appropriate

Clebsch-Gordon

Coefficient for the quartet projection,

(4 I s,u I 4) = 2.

412

BOWER

Deuteron Box Diagram

By the appropriate

substitution

of Clebsch-Gordon

Coefficients, we obtain

(4 ( SD 14) = 4. Pion Exchange

ST1 = x;m . (D + D’) a . &~&*o.

(D + D’) xN.

Defining D+ = D + D’, we proceed with the projections.

.D+(u1+iu2)(u1-iu2)0.D+~(~~ 42 1/2 - (+$I

II g-&>flcr.D+u”

&-&I++) (u1 ;2iu2’

u . D+cx (10 4 4 1 4 Q)

+ (Q $1 10 4 3) a(~ . D+u3u% . D+cu (10 8 B I 4 4) - (+ 4 / 10 g 4) OIQ. D+ = $0;”

+ $0;”

(u’ + ia2)

z/z

+ D+2 + $0,‘” = 20;’

u3a.D+/3(ll

Q-

$I*$>

+ iD+2.

In our calculation we are interested in the S wave and make use of the following equalities: j- dsZ 0;”

= s dQ 0;”

= j- df2 0;”

= 6 j- dQ o+~.

S-MATRIX

Make the appropriate

Making the appropriate

THEORY

OF

N-D

413

SCATTE~G

changes in the Clebsch-Gordon

changes in the Clebsch-Gordon

coefficients to obtain

coefficients, we obtain

(4 IS,1 / 4) = 4 - g + 2 - Q = 2 u Exchange

Substituting we obtain

the appropriate Clebsch-Gordon (4IS,u1/4)

coefficient for the quarter projection = 1.

This latter answer is verified by inspection. For the spin projections of the other cr term s& = x&s . 40 . 4’*xN, see the discussion in the proton exchange section.

414

BOWER

Heavy 0 Exchange

The only new spin term which occurs in the section describing the heavy sigma exchange is the following:

Replacing the Clebsch-Gordon

coefficients appropriately

gives

(4 1S,l 14) = Q - 3 + Q - Q = 0. The other spin terms which occur in the heavy sigma exchange have been discussed above. Right Hand Cut Contribution

of the Nucleon-Nucleon

Amplitude

All the spin terms which occur in the section dealing with the right hand cut contributions of the nucleon-nucleon amplitude have been previously discussed. Triton Pole Diagram

S-MATRIX

THEORY

OF

415

N-D SCATTERING

ACKNOWLEDGMENT I wish to thank my thesis advisor, Professor George Tiktopoulos, for his valuable contribution to this paper. His critical comments and words of encouragement were much appreciated.

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