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Dispersion K-matrix approach to nuclear reactions and its application to Nα scattering
) 6.B:2.L 1
Naclear
Not to
Physics
A224
be reproduced
(1974)
by photoprint
61-
76; @
DISPERSION K-MATRIX APPROACH AND ITS APPLICATION A. G. BARYSHNICKOV,
North-Holland
or microfilm
without
written
Publishirrg Co., Amsterdam permission
TO NUCLEAR
from the publisher
REACTIONS
TO Na SCATTERING
L. D. BLOKHINTSEV, A. N. SAFRONOV and V. V. TUROVTSEV of Nuclear Physics, Moscow State Unicersity
Institute
Received 10 December 1973 Abstract: A description of nuclear reactions is suggested based on the assumption that the main contribution to the reaction K-matrix elements, regarded as analytic functions of the variable z = cos 6 (8 being the scattering angle), comes from the singularities of these functions which are nearest to the physical region. Analytic properties of K-matrix elements with respect to .z are discussed. As an illustration, elastic Ncz scattering is examined and vertex constants values are found for the vertices cc 2 t+p and G(2 7+1x
1. Introduction Recently, various dispersion and graph methods [e.g., refs. ‘-“)I have found a wide application in the theory of nuclear reactions. These methods are based on the assumption that the reaction amplitude T is an analytic function of kinematic variables (scattering angle, energy) and that the main contribution to T comes from singularities of this function in the above variables nearest to the physical region, which are singularities of the simplest Feynman graph amplitudes. In the case of binary reactions (of the type A+x -+ B + y), it may be rigorously shown “) that for sufficiently large values of orbital angular momentum I the partial amplitudes of T are dominated by the nearest singularities of the amplitude T with respect to the variable z = cos 9, where 9 is the scattering angle in the c.m. system. However, for small I the assumption about the dominant role of the nearest singularities is usually invalid, which may be manifested, in particular, in a serious failure of the unitarity relation for partial amplitudes with small I when attempting to approximate them by the contribution from the nearest singularities. The unitarity condition fails even in such favourable cases as deuteron stripping (or pick-up) reactions for which the pole singularity, corresponding to the one-nucleon transfer mechanism, is usually appreciably closer to the physical region than are the remaining singularities. It seems attractive to have a method which would allow the combination of the nearest singularities principle and the unitarity condition. Such a method employing a formalism of the reaction K-matrix is suggested in the present work. In this method, K-matrix elements are considered to be analytic functions of z and the main contribution to matrix elements is assumed to come from these function singularities closest to the physical region. 61
A. G. BARYSHNKKOV
62
et ai.
In sect. 2, various formulae are given which relate elements of K- and T-matrices. In sect. 3, the main points of this approach are elucidated and the analytic properties of K-matrix elements with respect to z are discussed. Finally, in sect. 4 the approach suggested is applied to describe elastic Na scattering, thus enabling us to obtain the values of vertex constants for the virtuai processes K 7rt t +p and c1Z? r + n. 2. Some general relations for the K-matrix The relations between the K- and T-matrices are given byt s = (1 -#C)(l
+&Y)-1,
T = K(1++X)-“,
s = 1 -iz-.
(1)
From this definition it follows that given any Hermitian K-matrix, the S-matrix automatically turns out to be unitary. To be specific, let us consider the interaction of a non-relativistic spinless particle at energies at which only two-particle channels are open. Then from eq, (1) one readily obtains the equations relating partial amplitudes of T- and K-matrix elements: r;‘” = Kfk- i 5 pj Kyl;:‘“, i, k = 1,2,
. . ., n,
j=l
where
Here 12is the nttmber of open channels; Pj = ~j~j~2~; Fj 3pr and Ej are the reduced mass, the relative moments and the relative kinetic energy in the j-channel, reQik is the c.m. system scattering angle for the process i -+ k; spectively; Z@= COS 9& ; Pi(z) are the Legendre polynomials, h = c = 1. The normalization used above curresponds to the following expression for the differential cross section for the reaction i -+ k:
Formulae (Z)-(4) and all subsequent arguments can be trivially generalized for the case of particles with spin. For instance, in terms of the total angular momentum J, orbital angular momentum I and channel spin S, eqs. (2) take the form
From eqs. (2), it follows that the amplitudes Tik can be expressed in terms of t While calculating the operator product in eq. (I), the summation over the intermediate involves only the physical states of the system with on-mass-shell particles.
states
DISPERSION K-MATRIX
Kik by solving a set of linear algebraic tion has the form
For two open channels, 8’”
the solution
Given
one open channel
this solu-
reads
= ((1 + ip, K:“)K:’
TF2 = ((l+ q12 = K&?/D,
equations.
63
ip, K:l)K:z-
D = (1 f ip, K:l)(l+
- ip#C;“)“f/D, ~~~(~~*)2~~~, ip, K:2)+p,
(7) p2(#2)‘.
We note that Tik = Tii and Kik = Kfi owing to time reversal invariance; hence it follows that Im Kfk = 0 since the K-matrix is Hermitian. In some cases, a somewhat different procedure may be useful in which the amplitudes of inelastic processes, Tik(i # k), are expressed in terms of the K-matrix non-diagonal elements, K{‘“(j # nz), and the elastic scattering amplitudes, T”;j, using eqs. (2). Then, however, eqs. (2) turn out to be non-linear with respect to the new unknowns, Tfk(i # k) and KY and the analytic solution can be found only in the twochannel case (see also ref. lo), taking the form Ii=
= fl-[I-4p,p2g~“‘g:2’(iY:2)2]~f/2p,~~K:2. gj”’ = 1 - ip, X/” = *(l +e
2i81ck)
(8) ),
where 6ik)(Ek) is the elastic scattering phase shift in the k-channel for orbital angular momentum t. In the multichannel case (n > 2), relations of the type (8) cannot be obtained and eqs. (2) should be solved numerically. Approximate formulae may also be obtained which express Tik(i # k) in terms of K{‘“(j # m) and T/j. To do so, we solve eq. (2) at i = k with respect to Ki’: $i
= (T,“+ i z] pj @qji)j#. j+i
W
Substituting eq. (9) into eq. (2) with i # k this equation lowing form: T;‘k = giOKfkg:“)_
ig:i)
c pj Ki’l-yk+pi z pj KyTfT,‘k. j*i,k
Solving eq. (IO) by iteration,
may be reduced
we obtain
j#i
to the fol-
(10)
in the first order in Kim
T;‘k = gji)#kg(k> 2 .
(11)
A formula of this type has been derived in refs. “) in order to take account of the interaction in the initial and final states of nuclear reactions. We observe that in the two-channel case, the expression (11) follows from formula (8) if the following condition 14P,
P2 s1”‘d”‘(~~‘>‘l
c
1
(12)
A. G. BARYSHNICKOV
64
ef al.
is fulfilled. Substituting eq. (11) into the right-hand side of eq. (IO), we obtain Tik to second order in KY, etc. The expression for Tfk to third order in K/“’ is as follows:
(13)
f
pi pk g;i’g;k’(K;k)3}$J~k)
+
O[(Ki”)“].
Making use of formulae (8), (11) and (13), amplitudes of inelastic processes may be calculated if the matrix elements Kf”(i # k) and the elastic scattering amplitudes Tp are known. The quantities Ti’j may be taken either from experimental data on elastic scattering or from a model (e.g. the optical model). Such an approach is meaningful when the cross sections for inelastic processes are small in comparison with those for elastic processes. It should be noted that, in contrast to the approach based on the use of formulae of the type (7), this approach does not ascertain the automatic ful~lment of the unitarity condition. 3. Some analytic properties of the K-matrix
The K-matrix is usually defined in the physical region of the variables E and z. Nevertheless, we can analyticaIly continue the K-matrix beyond the physical region, by taking advantage of the analyticity of S-matrix and of formulae such as (I), (2) and (5) which relate the K-matrix to the S- (or T-) matrix. The analytic properties of the K-matrix with respect to energy have been discussed in a number of works (e.g., refs. ‘I “)). Therefore, we shall not dwell upon them at length. It will be only noted that with the usual definition of the K-matrix when the number N of independent matrix elements Kik is determined by the number IZof channels open at a given energy (N = @z(n+ I)), the values Kik vary non-analytically as the energy passes through the value corresponding to a threshold of a channel. This non-analytic behaviour can be avoided by introducing the matrix elements Kik for closed channels. We are, however, primarily concerned with direct nuclear reactions which are characterized by a strong dependence of the differential cross sections on the scattering angle; the analytic properties of Kik with respect to zik are, therefore, more important to us. We proceed from the assumption that the main contribution to the K-matrix elements, Kik, regarded as analytic functions of the variables zik, comes from the singularities closest to the physical region. In this connection, the problem is to investigate the singularities of K-matrix elements with respect to zjk. Consider, for example, the one-channel case. Omitting for brevity the channel index, we rewrite eq. (6) in the form
(14)
DISPERSION It is known
‘) that the T-matrix
K-MATRIX
singularities
65
on the physical
sheet can be found by
investigating singularities of the Feynman graphs. In so doing, the leading singularities+ of the simplest graphs (pole, triangle, etc) correspond to the nearest singularities of T-matrix. If I>> 1, the partial amplitudes can be written as a sum of contributions from the various singularities in the following manner “):
T,(E)
73
C A,(& ~i)e~z’nsit-fri+e),
(15)
where pi =r Ti $ ([f - l)*; the quantities Ai(E, li) are I-independent; ri = 213-$(Q-I- 1) = +(3ni_4Ui+3); ni, Ii and Of are, respectively, the number of internal lines, that of inde~ndent closed contours and that of vertices in the graph whose leading singularity is the point z = ii_ From eq. (15) it is seen that at I >> 1 the partial amplitudes fall off exponentially; the greater 1~~1, i.e., the farther the singularity ii from the physical region - 1 5 z 5 + 1, the faster is the fall-off. Therefore, in the case of sufficiently large I, T,(E) is dependent solely on the contribution from the nearest singularity. In view of
P
(a)
n
P
a fb)
P
P
P
(cf
Fig. I. Simplest Feynman graphs for the pr scattering.
decrease in X,(E) with /, from eq. (8) it follows that at 1 > 1 the approximate equality becoming more exact with increasing 1. Taking into account the results obtained in ref. 4), one may deduce from this equaiity that at least the nearest to the physical region singularity with respect to z which determines the asymptotic behaviour of T,(E) and K,(E) at I -+ w is common for K and T. As regards the remaining singularities, the situation is more complex. It may be stated that the function K(E, z) must have the singularities of T(E, z) which are the leading sing~arities of irreducible Feynman graphs and that it must not have the leading singularities of reducible Feynman graphs. By reducible (irreducible) graphs we mean the graphs which can (cannot) bz represented as two (or more) blocks connected by internal lines such that each of the blocks describes a process allowed at a given energy. Indeed, it can be readily shown that in eq. (14) the contribution from any irreducible block F to the term Kim) I> _- cancelkd by the contribution to E;{” = Ti that comes from the Izading singularity of reducible ladder graph consisting of 1~ blocks E’; the contribution from the block F to K{” cannot bz compensated for by the exponential &(E)
M I;(E),
+ The Feynman graph amplitude, apart from its leading singularities characterizing the graph in question, also has singularities of “contracted” graphs, which can be obtained from the initial graph by leaving out one or several lines and merging of vertices connected by these lines ‘).
A. G. BARYSHNICKOV
66
et al.
the remaining terms. For example, in the case of pa scattering, the contribution from the leading singularity of a square graph (fig. lb) to the term K,‘l’ = T,is cancelled by the contribution from a closer-lying pole singularity (fig. la) to the term Ki2’ = OPT, sothat as a whole K(E, z) has a pole singularity and has no leading singularity of the graph of fig. lb. It should be noted that the reducible graphs, aside from their leading singularities, also have the lower-order singularities of the “contracted” graphs (see preceeding footnote) which may provide a contribution to the K-matrix. To calculate this contribution, one should, after performing the integration over the virtual particle energies in the expression for the amplitude of a reducible graph, leave out the vanishing imaginary terms in the propagators and consider the resultant singular integrals over threedimensional momenta in the sense of their principal value. Such a recipe agrees with the fact that in Lippmann-Schwinger formalism the integral equations for the K-matrix differ from the corresponding equations for the T-matrix only by the presence of the principal value sign before the integral term ‘* “). The considerations outlined above are also applicable to the multichannel case. In particular, it follows from them that if the T-matrix element Tik(Ek, zik) has a pole at zili = I, the corresponding K-matrix element J?(E,, Zik) also has a pole at the same point and with the same residue. Similarly, if Tik(Ek, Zik) has a logarithmic singularity in Zik corresponding to a triangle graph, then Kik(&, Zik) has the same singularity, and the discontinuity of Tikand Kikon the cuts due to this singularity are the same. Thus the contribution from the closest singularities in Zik to the matrix elements Kikcan be taken into account by taking the sum of amplitudes of the simplest FeynThen the amplitudes Tikare found by solving man graphs (pole, triangle, etc.) as Kik. the set of equations (2)+.
4. An analysis of the elastic Consider, as an example, of the reaction p+ 01+ d+ At such energies, the nearest corresponding to the triton
Na scattering
the pu scattering at proton energies below the threshold 3He which is equal to 22.9 MeV (in the lab system). to the physical region singularity in z is the pole z = - 5 transfer mechanism (fig. la)++:
(16) t This procedure is also applicable if we take account of many-particle channels; in this case, however, eqs. (2) change from algebraic into integral ones. tt It will be noted that the one-pion exchange graph does not contribute to the Ncr scattering, since the vertex of CL--f cr+n becomes zero because of the laws of conservation of angular momentum and parity (also isospin); however, for other processes (e.g., the Nt scattering) the pole corresponding to this graph may turn out to be the nearest singularity.
DISPERSION
K-MATRIX
67
where mi is the mass of particle i, and p is the proton momentum in the c.m. system. Choosing the amplitude of the graph shown in fig. la, which we designate as K(l) (E, z), being a matrix element of K(E, z) for a single open channel, we obtain+ K(E, z) = K’l’(E, z) = - m,G2/2p2([ -i-z), K,(E;) = K&E)
= (- I)‘+ ‘iv, G2Ql(C)/2p’,
(17)
where Q,(i) is a Legendre function of the second kind, and G = Gcrtpis the vertex constant ‘*lz) for the CI-+ t +p vertex. The elastic pa scattering differential cross section has been calculated for various energies and values of the constant G. As an illustration, fig. 2 shows the theoretical curves and experimental points for the elastic pcl scattering differentiai cross section at large angles and at the incident proton energy 20.62 MeV; for this energy, 5 = 4.38.
60
40 -5 D‘ E % ;: 73
20
60
120
160
0
Gn.
Fig. 2. Comparison between theory and experiment for the elastic large-angle pa scattering ED = 20.62 MeV. Experimental points have been taken from ref. 13).
at
+ The presence of spin of the proton leads to the appearance of a trivial factor &,,,uj in eq. (171, which we omit QJ,*(~4,) is the proton spin projection in the initial (finai) state].
A. G. BARYSHNICKOV
65
Fig. 3. A graphic
representation
of the elastic pa scattering for the K-matrix.
et al.
amplitude
in the
pole approximation
Curve 1 in fig. 2 has been obtained by formulae (3), (4), (6) and (18) at G2 = 7 fm. As can be seen, this curve agrees with the experimental cross section only qualitatively; a similar situation takes place also at other values of G’. The cause of such a discrepancy apparently lies in the fact that at small Z,especially at I = 0, singularities, more distant than the pole one and corresponding to other mechanisms, may contribute appreciably to the partial amplitudes K,(E). This contribution may be approximately estimated by adding a real constant to the partial pole amplitude K,,(E) and by fitting it by the x2 method. Curve 2 in fig. 3 calculated in this way furnishes a good description of the experimental data at 9 2 100”. It will be noted that the results of the calculations are sensitive to the value of G2, the best agreement being reached at G2 = 7 fm. This fact is illustrated by curve 3 in fig. 2, which has been calculated in the same manner as curve 2, but with G2 = 4 fm; it is seen that the curve deviates perceptibly from the experimental points. However, with the same value G2 = 7 fm and at other proton energies, theory and experiment agree satisfactorily. For comparison, fig. 2 shows also the pa scattering differential cross section at 20.62 MeV calculated within the framework of the peripheral model with a pole mechanism 14*‘) (curve 4). It can be seen that the approach suggested within the framework of the same pole mechanism allows one to improve the agreement between theory and experiment, as compared to the peripheral model. It is of interest to note that in the case of the peripheral model the best fit to the experimental data is also attained at G2 = 7 fm. Expanding the right-hand side of eq. (6) into a series in Kz it can be readily seen that taking the pole graph of fig. la as K(E, z), we obtain the amplitude T(& z) as a series of ladder graphs (fig. 3). Crosses on the lines signify that the corresponding virtual particle is on the mass shell, i.e., that its non-relativistic propagator (E-k2/2m+iy)’ is replaced by - inG(E- k2/2nz). It is interesting to try to estimate corrections for the pure pole approximation for the K-matrix. Within the framework of the same mechanism for triton exchange the first such correction K’2)(E, z) is due to the allowance for the double triton exchange in the K-matrix. According to the sect. 3, as Kc2)(E, z), one should take the expression which differs from the amplitude of the square graph of fig. lb in that the integral over the virtual momenta is calculated in the sense of its principal value; in the case under consideration, such a recipe results in Kc2)(E, z) coinciding with the real part of the amplitude Mc2)(E, z) of the graph shown in fig. lb. Proceeding from the
DISPERSION
Feynman integral partial amplitude:
for M(‘)(E,
@‘(E)
z) it is easy to obtain
= (C/$)P s C =
P being the principal We observe that
K-MATRIX
G4m,
mdk(QJS(p,
0
69
the following
k)l)‘/(p’
~1, n1~/47r2(1~z, + ni,).
expression
for the
-k’>,
(18)
value sign. M?(E)
= K!*‘(E) - ip[K{“(E)]‘,
(19)
where K,(‘)(E) is defined by formula (17). The effect of the correction Kc2) on the pa scattering differential cross section at 20.62 MeV is shown in fig. 4, where the experimental points are the same as in fig. 2. The solid curve 1 has been obtained for K(E, .z) = K(“(E, z) and differs from the curve 2 in fig. 2 in being corrected for the Coulomb interaction which is essential at small scattering angles. The pa scattering partial amplitude, which includes both the nuclear and Coulomb interactions, was taken in the following form T* = T/C) + e2ibzT(N)1 1 >
C-20)
where T{” is the partial amplitude of pure Coulomb scattering, c1 is the Coulomb scattering phase shift andT/N’ is the nuclear scattering amplitude as defined by eq. (6). The dashed line 2 in fig. 4 has been obtained by formulae (20) and (6) at K(E, z) = K(“(E, z)+Kc2)(E, 2). Finally, the dash-and-dot line 3 corresponds to the inclusion of only the terms with an odd number of triton lines in the right-hand side of the graphic equality shown in fig. 3 which leads to the relation
T/N) = @‘/[l
+(pK:“)*].
(21)
In constructing all the three curves we took G* = 7 fm; the partial amplitudes with I = 0 were fitted in the same way as in the case of the curves 2 and 3 in fig. 2. We see that at large 3 all three curves are practically coincident and describe the experiment well. The curve 2 is close to the curve 1 at all values of 3, which suggests that the role of the K(*) correction is insignificant. At 3 5 90” all the theoretical curves lie below the experimental points and the omission of terms corresponding to an even number of triton exchanges whose amplitudes are maximum at small 3, increases the discrepancy between theory and experiment (curve 3). The unsatisfactory description of experiment at 3 < 90” by the curves shown in fig. 4 is evidently due to the fact that in our calculations the main contribution to the partial amplitudes TiN’ comes from the pole graph of fig. la whose amplitude
A. G. BARYSHNICKOV
70
0
60
et al.
120
180
Gl.
Fig. 4. The effect of the contribution from the double triton exchange into the K-matrix on the elastic pr scattering differential cross section at E,, = 20.62 MeV. Experimental points are from ref. 13).
K”‘(E, z) is maximum at 9 = 180” and decreases monotonically with the decreasing 3; therefore, at small 9 the contribution from other mechanisms may be essential. The simplest of such mechanisms corresponds to the triangle graph of fig. lc which describes the scattering of an incident proton from one of nucleons of an a-particle. At the proton energy 20.62 MeV, the amplitude of this graph has the leading logarithmic singularity at z = z, w 5 (more exactly at zd = 5.00 for scattering from a proton and at z, = 5.16 for scattering from a neutron). Thus, the point z, is the nearest singularity relative to the point z = 1 corresponding to the forward scattering. In the physical region, the amplitude K““(E, z) of the graph depicted in fig. lc is
DISPERSION
K-MATRIX
71
maximum at 9 = 0. To estimate roughly the contribution from this graph, we assume that the vertex functions of the graph are constant. The very crude assumption about the constancy of the pN scattering vertex is to a certain extent justified in our case because the nearest singularity of this vertex corresponding to the one-pion exchange in the t-channel does not contribute, according to the aforesaid, to the amplitude K”‘(E, z). Further, we neglect, in conformity with the isobaric invariance, the mass difference between 3H and 3He and between the proton and the neutron and take Gzrm = G& = G2. Then, taking advantage of the expression for the nonrelativistic triangle graph amplitude given in ref. “), we can obtain for the partial
Fig. 5. Elastic pee scattering differential cross section at Ep = 20.62 MeV with the pole and triangle mechanisms included in the K-matrix. Experimental points are from ref. 13).
et al.
A. G. BARYSHNICKOV
72
3ood
x 3
IOO-
- 300
I
50-
$ E G
- 50
5 -0 IO-
5-
Fig. 6. Comparison between theory and experiment for the elastic nr* scattering at the neutron energies 17.6 MeV (curve 1, full circles, left-hand scale) and 20.9 MeV (curve 2, open circles, right-hand scale). The curves have been obtained in the pole approximation for the K-matrix. Experimental points are from ref. 16).
amplitude Kid’(E) Kid’ = - (9/4$)(m,/p)G2fJ,, J,
=
s
1 l arctgcat1-‘>“>
2 -1
P,(z)dz
(1 -z)f
=
sm
Q,(l + s’)dx,
a-1
(22)
a = (f$)pl(m, d&g+, f = fig +_fr$% where constants ~$4 have the meaning of the NN scattering amplitudes in the state with spin s, which are normalized by the condition da$$dQ = /f$l'. As before, the spin factor a,,, is omitted in eq. (22).
DISPERSION
K-MATRIX
73
The pa scattering differential cross section at 20.62 MeV calculated by formulae (20) and (6) for K, = K[“+K,‘d’ is presented in fig. 5. In our calculations, we took G2 = 7 fm andf = 14 fm. It can be seen that the inclusion of the triangle graph into the K-matrix allows a satisfactory description of experiment in the entire range of scattering angles. The valuef = 14 fm is close to 18.3 fm which is obtained if we put f$2 = -t&d, where ~$2 are the NN scattering lengths. The calculated differential cross section shows a Iow sensitivity to the variations of the values offin the range 10 to 20 fm. Hitherto we have considered the pa scattering; the rm scattering may be treated in a similar manner. Fig. 6 shows a comparison between theory and experiment 1“) for the nm scattering differential cross section at the neutron energies 17.6 MeV (curve 1, full circles, left-hand scale) and 20.9 MeV (curve 2, open circles, right-hand scale). Curves 1 and 2 have been obtained analogously to curve 1 in fig. 4, i.e., in the pole approximation for K(E’, 2). The pole graph for the na scattering can be obtained from the graph of fig. la by the following substitution: p -+ n, t + z. The partial amplitude J&,(E) and the constant G2 = G& were fitted by the x2 method. The value G2 = 8.5 fm corresponds to curve 1, and G2 = 8.0 fm corresponds to curve 2. These values are close to the value G&, = 7.0 fm, which has been obtained from the pa scattering, as might be expected since G&, = G& under isobaric invariance. At 9 2 90”, the theoretical curves show a close correspondence to the experimental points. As in the case of the pa scattering, better agreement between theory and experiment at 9 < 90” is attained when the triangle graph is taken into account. In the examples considered above the energy of incident nucleons was chosen away from resonances in the Ncl system; however, if necessary, the resonance phenomena may be included in the scheme under consideration. 5. Conclusion As the example of Na scattering indicates, the suggested approach, which ensures the S-matrix unitarity, enables one to achieve a reasonable agreement with experiment and to extract the values of vertex constants G. The values G& = 7.0 fm and G&,, = 8.0 or 8.5 fm we have obtained are consistent with the results of peripheral model analysis of elastic pet scattering at the proton energy 48.8 MeV [ref. “)I (G&, = 7.0 fm), of the 3He(d, p)4He reaction [ref. ‘“)I (G.‘,, = 7.1 fm) and of the 3H(p, y)4He process at the proton energy 156 MeV [ref. “>I (G&, = 7.7 fm); our values are somewhat lower than G&, = il.3 fm which has been obtained in ref. 2“) by using the dispersion relations for the forward elastic na scattering amplitude. The vertex constant GABc, corresponding to the virtual decay (synthesis) A z B+C, coincides with the on-shell matrix element for this process to within kinematic factors. In contrast to the conventionally used reduced widths, the vertex constants are model-independent and similar to the coupling constants in the theory
74
A. G. BARYSHNICKOV
et a/.
of elementary particles. The value GABCis proportional to the coefficient in the asymptotic behaviour of the wave function for the channel A + B+C and can be expressed in terms of the Fourier component of the overlap integral for the nuclei A, B and C {refs. ‘, ““)I. The vertex constant for the triton break-up (t -+ d+n) has been calculated in ref * *’ )Yb solving the Faddeev equation for two different NN potentials. For the Malfliet-Tjon potential 2”) G& = 1.9 fm, for the Darewich-Green potential 23), G:,, = 0.1 fm; from an analysis of experimental data on various reaction it follows that G&, M 1 fm, which agrees we11with the value obtained in ref. ““) using the Reid NN potential “‘). From these examples it follows that the vertex constants can be highly sensitive to the form of the NN interaction. It would be, therefore, of interest to compare the phenomenological values of constants GoltPand G,,,, which have been found in this work and elsewhere, with the theoretical values calculated by methods of nuclear structure theory. Unfortunately, such calculations have not so far been made. A reliable determination of constants G& and GsLiadwould yield a highly valuable information; at present, only the order of magnitude of these constants is known 17,26). Some data on the vertex constants for the lightest nuclei are given in ref. 27). W e note that our vertex constant GABCis related to the dimensionles quantity &.., of ref. 27) by
Gh, = (- 1)“2n*(hlpc)(g’~~;i)‘,
(23)
p and 1 being the reduced mass and the relative orbital angular momentum of fragments B and C. For most reactions of the type A+x -+ B + y, the pole corresponding to the mechanism of nucleon or cluster transfer is the nearest to the physical region singularity in z. In accordance with what has been stated in sect. 3 (eq. (15) and below), in such a case, the knowledge of the vertex constants of the corresponding pole graph fully determines the partial amplitudes of the above reaction at sufficiently large E. Taking this into consideration may be of great help in an anaIysis of experimental data 28). A similar situation arises in the NN scattering, where the partial amplitudes for large 2 are determined by the one-pion exchange graph and expressed in terms of the pionnucleon coupling constant 2“). In the present work, only examples of pure elastic scattering have been considered. The next step should be the application of the suggested approach to the multichannel processes, primarily to the reactions involving light nuclei at low energies, when the number of open channels is small. It would be interesting to examine the polarization phenomena which would require to allow more accurately for the spin dependences in vertices of the appropriate graphs. It should be mentioned that even the pure pole approximation for the K-matrix leads to non-zero polarization of reaction products with non-polarized initial particles if the orbital momenta in vertices of the pole graph are non-zero. In the case of the pole approximation for the T-matrix, polarization is absent.
DISPERSION
K-MATRIX
75
Needless to say, the approach advanced in the present work is not the sole way to guarantee the unitarity of the S-matrix within the framework of dispersion methods. This can be done, for example, by using the Pad& approximant 30) or the multichannel N/D method 31). As compared to the Pad& approximant, the K-matrix approach is characterized by having a clearer physical meaning. Moreover, the tentative calculations of pcl scattering at the proton energy 20.62 MeV, which have been performed by us using the Pad& approximant, yield the results very similar to those obtained in sect. 4 of this paper. As regards the N/D method, it seems to us that in this case the practical calculations are more involved than those in the K-matrix approach. It should be noted that recently there have appeared works devoted to the applications of the K-matrix (or similar) methods to the three-body problem (see, e.g. refs. 32-39)). Some preliminary results of the pa scattering analysis within the framework of the approach suggested have been published in ref. 40). References 1) I. S. Shapiro, Teoriya pryamykh yadernykh 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27)
reaktsii (Theory of direct nuclear reactions) (Gosatomizdat, Moscow, 1963); Usp. Fiz. Nauk 92 (1967) 549 E. I. Dolinsky, P. 0. Dzhamalov and A. M. Mukhamedzhanov., Nucl Phys. A202 (1973) 97 M. P. Lecher, Nucl. Phys. B23 (1970) 116 V. S. Popov, ZhETF (USSR) 47 (1964) 2229 R. H. Dal&z and S. F. Tuan, Ann. of Phys. 3 (1960) 307 K. L. Kowalsky, Phys. Rev. DS (1972) 395 R. J. Eden, P. V. Landshoff, D. I. Olive and J. C. Polkinghorne, The analytic S-matrix (Cambridge Univ. Press, 1966) M. L. Goldberger and K. M. Watson, Collision theory (Wiley, New York, 1964) ch. 5 W. Tobocman and M. A. Nagarajan, Phys. Rev. 163 (1967) 1011 E. 0. Ah, P. Grassberger and W. Sandhas, Nucl. Phys. Al39 (1969) 209 M. H. Ross and E. L. Shaw, Phys. Rev. Lett. 12 (1964) 627; G. Cramer and K. Schilling, Z. Phys. 191 (1966) 51 M. M. Al-Beidovi, L, D. Blokhintsev, E. I. Dolinsky and V. V. Turovtsev, Vestnik MGU, ser. fiz.-as&. 6 (1967) 3 P. W. Allison and R. Smythe, Nucl. Phys. A121 (1968) 97 I. Borbei, E. I. Dolinsky and V. V. Turovtsev, Yad. Fiz. 8 (1968) 492 L. D. Blokhintsev, E. I. Dohnsky and V. S. Popov, Nucl. Phys. 40 (1963) 117 A. Niiller, M. Drosg, J.C. Hopkins, J. D. Seagrave and E. C. Kerr, Phys. Rev. C4 (1971) 36 A. G. Baryshnickov and L. D. Blokhintsev, Phys. Lett. 36B (1971) 205 E. I. Dolinsky, Lzv. Akad. Nauk SSSR (ser. fiz.) 34 (1970) 165 A. G. Baryshnickov, L. D. Blokhintsev, A. M. Mukhamedzhanov and V. V. Turovtsev, Phys. Lett. 45B (1973) 1 M. P. Lecher, Nucl. Phys. B36 (1972) 634 Yu. V. Orlov and V. B. Belyaev, ZhETF Pibma 17 (1973) 385 R. A. Mahhet and I. A. Tjon, Nucl. Phys. Al27 (1969) 161 G. Darewich and A. E. S. Green, Phys. Rev. 164 (1967) 1324 Y. E. Kim and A. Tubis, Phys. Rev. Lett. 29 (1972) 1017 R. V. Reid, Ann. of Phys. 50 (1968) 411 G. V. Avakov, E. I. Dolinsky and V. V. Turovtsev, Nucl. Phys. AI96 (1972) 529 A. S. Rinat (Reiner). L. P. Kok and M. Sting], Nucl. Phys. A190 (1972) 328
76
A. G. BARYSHNICKOV
et al.
28) P. E. Shanley, Plays. Rev. Lett. 24 (1970) 18 29) P. Cziffra, M. H. MacGregor, M. J. Moravcsik and II. P. Stapp, Phys. Rev. 114 (1959) 880 30) G. A. Baker and J. L. Gammel, ed., The Pad6 approximant in theoretical physics (Academic Press, NY, 1970) 31) A. S. Rinat (Reiner) and M. Sting& Ann. of Phys. 65 (1971) 141 32) I. H. Sloan, Phys. Rev. 165 (1968) 1587 33) R. W. Finkel and L. Rosenberg, Phys. Rev. 168 (1968) 1841 34) K. L. Kowalsky, Phys. Rev. D5 (1972) 395; D6 (1972) 3705 35) T. Sasakawa, Nucl. Phys. A186 (1972) 417; A203 (1973) 496 36) R. T. Cahill, Nucl. Phys. Al94 (1972) 599 37) P. C. Tandy, R. T. Cahill and I. E. McCarthy, Phys. Lett. 41B (1972) 241 38) M. G. Fuda, Nuovo Cim. 1lA (1972) 701 39) I. Manning, Phys. Rev. D5 (1972) 1472 40) A. G. Baryshnickov, L. D. Blokhintsev, A, N. Safronov and V. V. Turovtsev, ZhETF PiSma 16 (1972) 414
Naclear
Not to
Physics
A224
be reproduced
(1974)
by photoprint
61-
76; @
DISPERSION K-MATRIX APPROACH AND ITS APPLICATION A. G. BARYSHNICKOV,
North-Holland
or microfilm
without
written
Publishirrg Co., Amsterdam permission
TO NUCLEAR
from the publisher
REACTIONS
TO Na SCATTERING
L. D. BLOKHINTSEV, A. N. SAFRONOV and V. V. TUROVTSEV of Nuclear Physics, Moscow State Unicersity
Institute
Received 10 December 1973 Abstract: A description of nuclear reactions is suggested based on the assumption that the main contribution to the reaction K-matrix elements, regarded as analytic functions of the variable z = cos 6 (8 being the scattering angle), comes from the singularities of these functions which are nearest to the physical region. Analytic properties of K-matrix elements with respect to .z are discussed. As an illustration, elastic Ncz scattering is examined and vertex constants values are found for the vertices cc 2 t+p and G(2 7+1x
1. Introduction Recently, various dispersion and graph methods [e.g., refs. ‘-“)I have found a wide application in the theory of nuclear reactions. These methods are based on the assumption that the reaction amplitude T is an analytic function of kinematic variables (scattering angle, energy) and that the main contribution to T comes from singularities of this function in the above variables nearest to the physical region, which are singularities of the simplest Feynman graph amplitudes. In the case of binary reactions (of the type A+x -+ B + y), it may be rigorously shown “) that for sufficiently large values of orbital angular momentum I the partial amplitudes of T are dominated by the nearest singularities of the amplitude T with respect to the variable z = cos 9, where 9 is the scattering angle in the c.m. system. However, for small I the assumption about the dominant role of the nearest singularities is usually invalid, which may be manifested, in particular, in a serious failure of the unitarity relation for partial amplitudes with small I when attempting to approximate them by the contribution from the nearest singularities. The unitarity condition fails even in such favourable cases as deuteron stripping (or pick-up) reactions for which the pole singularity, corresponding to the one-nucleon transfer mechanism, is usually appreciably closer to the physical region than are the remaining singularities. It seems attractive to have a method which would allow the combination of the nearest singularities principle and the unitarity condition. Such a method employing a formalism of the reaction K-matrix is suggested in the present work. In this method, K-matrix elements are considered to be analytic functions of z and the main contribution to matrix elements is assumed to come from these function singularities closest to the physical region. 61
A. G. BARYSHNKKOV
62
et ai.
In sect. 2, various formulae are given which relate elements of K- and T-matrices. In sect. 3, the main points of this approach are elucidated and the analytic properties of K-matrix elements with respect to z are discussed. Finally, in sect. 4 the approach suggested is applied to describe elastic Na scattering, thus enabling us to obtain the values of vertex constants for the virtuai processes K 7rt t +p and c1Z? r + n. 2. Some general relations for the K-matrix The relations between the K- and T-matrices are given byt s = (1 -#C)(l
+&Y)-1,
T = K(1++X)-“,
s = 1 -iz-.
(1)
From this definition it follows that given any Hermitian K-matrix, the S-matrix automatically turns out to be unitary. To be specific, let us consider the interaction of a non-relativistic spinless particle at energies at which only two-particle channels are open. Then from eq, (1) one readily obtains the equations relating partial amplitudes of T- and K-matrix elements: r;‘” = Kfk- i 5 pj Kyl;:‘“, i, k = 1,2,
. . ., n,
j=l
where
Here 12is the nttmber of open channels; Pj = ~j~j~2~; Fj 3pr and Ej are the reduced mass, the relative moments and the relative kinetic energy in the j-channel, reQik is the c.m. system scattering angle for the process i -+ k; spectively; Z@= COS 9& ; Pi(z) are the Legendre polynomials, h = c = 1. The normalization used above curresponds to the following expression for the differential cross section for the reaction i -+ k:
Formulae (Z)-(4) and all subsequent arguments can be trivially generalized for the case of particles with spin. For instance, in terms of the total angular momentum J, orbital angular momentum I and channel spin S, eqs. (2) take the form
From eqs. (2), it follows that the amplitudes Tik can be expressed in terms of t While calculating the operator product in eq. (I), the summation over the intermediate involves only the physical states of the system with on-mass-shell particles.
states
DISPERSION K-MATRIX
Kik by solving a set of linear algebraic tion has the form
For two open channels, 8’”
the solution
Given
one open channel
this solu-
reads
= ((1 + ip, K:“)K:’
TF2 = ((l+ q12 = K&?/D,
equations.
63
ip, K:l)K:z-
D = (1 f ip, K:l)(l+
- ip#C;“)“f/D, ~~~(~~*)2~~~, ip, K:2)+p,
(7) p2(#2)‘.
We note that Tik = Tii and Kik = Kfi owing to time reversal invariance; hence it follows that Im Kfk = 0 since the K-matrix is Hermitian. In some cases, a somewhat different procedure may be useful in which the amplitudes of inelastic processes, Tik(i # k), are expressed in terms of the K-matrix non-diagonal elements, K{‘“(j # nz), and the elastic scattering amplitudes, T”;j, using eqs. (2). Then, however, eqs. (2) turn out to be non-linear with respect to the new unknowns, Tfk(i # k) and KY and the analytic solution can be found only in the twochannel case (see also ref. lo), taking the form Ii=
= fl-[I-4p,p2g~“‘g:2’(iY:2)2]~f/2p,~~K:2. gj”’ = 1 - ip, X/” = *(l +e
2i81ck)
(8) ),
where 6ik)(Ek) is the elastic scattering phase shift in the k-channel for orbital angular momentum t. In the multichannel case (n > 2), relations of the type (8) cannot be obtained and eqs. (2) should be solved numerically. Approximate formulae may also be obtained which express Tik(i # k) in terms of K{‘“(j # m) and T/j. To do so, we solve eq. (2) at i = k with respect to Ki’: $i
= (T,“+ i z] pj @qji)j#. j+i
W
Substituting eq. (9) into eq. (2) with i # k this equation lowing form: T;‘k = giOKfkg:“)_
ig:i)
c pj Ki’l-yk+pi z pj KyTfT,‘k. j*i,k
Solving eq. (IO) by iteration,
may be reduced
we obtain
j#i
to the fol-
(10)
in the first order in Kim
T;‘k = gji)#kg(k> 2 .
(11)
A formula of this type has been derived in refs. “) in order to take account of the interaction in the initial and final states of nuclear reactions. We observe that in the two-channel case, the expression (11) follows from formula (8) if the following condition 14P,
P2 s1”‘d”‘(~~‘>‘l
c
1
(12)
A. G. BARYSHNICKOV
64
ef al.
is fulfilled. Substituting eq. (11) into the right-hand side of eq. (IO), we obtain Tik to second order in KY, etc. The expression for Tfk to third order in K/“’ is as follows:
(13)
f
pi pk g;i’g;k’(K;k)3}$J~k)
+
O[(Ki”)“].
Making use of formulae (8), (11) and (13), amplitudes of inelastic processes may be calculated if the matrix elements Kf”(i # k) and the elastic scattering amplitudes Tp are known. The quantities Ti’j may be taken either from experimental data on elastic scattering or from a model (e.g. the optical model). Such an approach is meaningful when the cross sections for inelastic processes are small in comparison with those for elastic processes. It should be noted that, in contrast to the approach based on the use of formulae of the type (7), this approach does not ascertain the automatic ful~lment of the unitarity condition. 3. Some analytic properties of the K-matrix
The K-matrix is usually defined in the physical region of the variables E and z. Nevertheless, we can analyticaIly continue the K-matrix beyond the physical region, by taking advantage of the analyticity of S-matrix and of formulae such as (I), (2) and (5) which relate the K-matrix to the S- (or T-) matrix. The analytic properties of the K-matrix with respect to energy have been discussed in a number of works (e.g., refs. ‘I “)). Therefore, we shall not dwell upon them at length. It will be only noted that with the usual definition of the K-matrix when the number N of independent matrix elements Kik is determined by the number IZof channels open at a given energy (N = @z(n+ I)), the values Kik vary non-analytically as the energy passes through the value corresponding to a threshold of a channel. This non-analytic behaviour can be avoided by introducing the matrix elements Kik for closed channels. We are, however, primarily concerned with direct nuclear reactions which are characterized by a strong dependence of the differential cross sections on the scattering angle; the analytic properties of Kik with respect to zik are, therefore, more important to us. We proceed from the assumption that the main contribution to the K-matrix elements, Kik, regarded as analytic functions of the variables zik, comes from the singularities closest to the physical region. In this connection, the problem is to investigate the singularities of K-matrix elements with respect to zjk. Consider, for example, the one-channel case. Omitting for brevity the channel index, we rewrite eq. (6) in the form
(14)
DISPERSION It is known
‘) that the T-matrix
K-MATRIX
singularities
65
on the physical
sheet can be found by
investigating singularities of the Feynman graphs. In so doing, the leading singularities+ of the simplest graphs (pole, triangle, etc) correspond to the nearest singularities of T-matrix. If I>> 1, the partial amplitudes can be written as a sum of contributions from the various singularities in the following manner “):
T,(E)
73
C A,(& ~i)e~z’nsit-fri+e),
(15)
where pi =r Ti $ ([f - l)*; the quantities Ai(E, li) are I-independent; ri = 213-$(Q-I- 1) = +(3ni_4Ui+3); ni, Ii and Of are, respectively, the number of internal lines, that of inde~ndent closed contours and that of vertices in the graph whose leading singularity is the point z = ii_ From eq. (15) it is seen that at I >> 1 the partial amplitudes fall off exponentially; the greater 1~~1, i.e., the farther the singularity ii from the physical region - 1 5 z 5 + 1, the faster is the fall-off. Therefore, in the case of sufficiently large I, T,(E) is dependent solely on the contribution from the nearest singularity. In view of
P
(a)
n
P
a fb)
P
P
P
(cf
Fig. I. Simplest Feynman graphs for the pr scattering.
decrease in X,(E) with /, from eq. (8) it follows that at 1 > 1 the approximate equality becoming more exact with increasing 1. Taking into account the results obtained in ref. 4), one may deduce from this equaiity that at least the nearest to the physical region singularity with respect to z which determines the asymptotic behaviour of T,(E) and K,(E) at I -+ w is common for K and T. As regards the remaining singularities, the situation is more complex. It may be stated that the function K(E, z) must have the singularities of T(E, z) which are the leading sing~arities of irreducible Feynman graphs and that it must not have the leading singularities of reducible Feynman graphs. By reducible (irreducible) graphs we mean the graphs which can (cannot) bz represented as two (or more) blocks connected by internal lines such that each of the blocks describes a process allowed at a given energy. Indeed, it can be readily shown that in eq. (14) the contribution from any irreducible block F to the term Kim) I> _- cancelkd by the contribution to E;{” = Ti that comes from the Izading singularity of reducible ladder graph consisting of 1~ blocks E’; the contribution from the block F to K{” cannot bz compensated for by the exponential &(E)
M I;(E),
+ The Feynman graph amplitude, apart from its leading singularities characterizing the graph in question, also has singularities of “contracted” graphs, which can be obtained from the initial graph by leaving out one or several lines and merging of vertices connected by these lines ‘).
A. G. BARYSHNICKOV
66
et al.
the remaining terms. For example, in the case of pa scattering, the contribution from the leading singularity of a square graph (fig. lb) to the term K,‘l’ = T,is cancelled by the contribution from a closer-lying pole singularity (fig. la) to the term Ki2’ = OPT, sothat as a whole K(E, z) has a pole singularity and has no leading singularity of the graph of fig. lb. It should be noted that the reducible graphs, aside from their leading singularities, also have the lower-order singularities of the “contracted” graphs (see preceeding footnote) which may provide a contribution to the K-matrix. To calculate this contribution, one should, after performing the integration over the virtual particle energies in the expression for the amplitude of a reducible graph, leave out the vanishing imaginary terms in the propagators and consider the resultant singular integrals over threedimensional momenta in the sense of their principal value. Such a recipe agrees with the fact that in Lippmann-Schwinger formalism the integral equations for the K-matrix differ from the corresponding equations for the T-matrix only by the presence of the principal value sign before the integral term ‘* “). The considerations outlined above are also applicable to the multichannel case. In particular, it follows from them that if the T-matrix element Tik(Ek, zik) has a pole at zili = I, the corresponding K-matrix element J?(E,, Zik) also has a pole at the same point and with the same residue. Similarly, if Tik(Ek, Zik) has a logarithmic singularity in Zik corresponding to a triangle graph, then Kik(&, Zik) has the same singularity, and the discontinuity of Tikand Kikon the cuts due to this singularity are the same. Thus the contribution from the closest singularities in Zik to the matrix elements Kikcan be taken into account by taking the sum of amplitudes of the simplest FeynThen the amplitudes Tikare found by solving man graphs (pole, triangle, etc.) as Kik. the set of equations (2)+.
4. An analysis of the elastic Consider, as an example, of the reaction p+ 01+ d+ At such energies, the nearest corresponding to the triton
Na scattering
the pu scattering at proton energies below the threshold 3He which is equal to 22.9 MeV (in the lab system). to the physical region singularity in z is the pole z = - 5 transfer mechanism (fig. la)++:
(16) t This procedure is also applicable if we take account of many-particle channels; in this case, however, eqs. (2) change from algebraic into integral ones. tt It will be noted that the one-pion exchange graph does not contribute to the Ncr scattering, since the vertex of CL--f cr+n becomes zero because of the laws of conservation of angular momentum and parity (also isospin); however, for other processes (e.g., the Nt scattering) the pole corresponding to this graph may turn out to be the nearest singularity.
DISPERSION
K-MATRIX
67
where mi is the mass of particle i, and p is the proton momentum in the c.m. system. Choosing the amplitude of the graph shown in fig. la, which we designate as K(l) (E, z), being a matrix element of K(E, z) for a single open channel, we obtain+ K(E, z) = K’l’(E, z) = - m,G2/2p2([ -i-z), K,(E;) = K&E)
= (- I)‘+ ‘iv, G2Ql(C)/2p’,
(17)
where Q,(i) is a Legendre function of the second kind, and G = Gcrtpis the vertex constant ‘*lz) for the CI-+ t +p vertex. The elastic pa scattering differential cross section has been calculated for various energies and values of the constant G. As an illustration, fig. 2 shows the theoretical curves and experimental points for the elastic pcl scattering differentiai cross section at large angles and at the incident proton energy 20.62 MeV; for this energy, 5 = 4.38.
60
40 -5 D‘ E % ;: 73
20
60
120
160
0
Gn.
Fig. 2. Comparison between theory and experiment for the elastic large-angle pa scattering ED = 20.62 MeV. Experimental points have been taken from ref. 13).
at
+ The presence of spin of the proton leads to the appearance of a trivial factor &,,,uj in eq. (171, which we omit QJ,*(~4,) is the proton spin projection in the initial (finai) state].
A. G. BARYSHNICKOV
65
Fig. 3. A graphic
representation
of the elastic pa scattering for the K-matrix.
et al.
amplitude
in the
pole approximation
Curve 1 in fig. 2 has been obtained by formulae (3), (4), (6) and (18) at G2 = 7 fm. As can be seen, this curve agrees with the experimental cross section only qualitatively; a similar situation takes place also at other values of G’. The cause of such a discrepancy apparently lies in the fact that at small Z,especially at I = 0, singularities, more distant than the pole one and corresponding to other mechanisms, may contribute appreciably to the partial amplitudes K,(E). This contribution may be approximately estimated by adding a real constant to the partial pole amplitude K,,(E) and by fitting it by the x2 method. Curve 2 in fig. 3 calculated in this way furnishes a good description of the experimental data at 9 2 100”. It will be noted that the results of the calculations are sensitive to the value of G2, the best agreement being reached at G2 = 7 fm. This fact is illustrated by curve 3 in fig. 2, which has been calculated in the same manner as curve 2, but with G2 = 4 fm; it is seen that the curve deviates perceptibly from the experimental points. However, with the same value G2 = 7 fm and at other proton energies, theory and experiment agree satisfactorily. For comparison, fig. 2 shows also the pa scattering differential cross section at 20.62 MeV calculated within the framework of the peripheral model with a pole mechanism 14*‘) (curve 4). It can be seen that the approach suggested within the framework of the same pole mechanism allows one to improve the agreement between theory and experiment, as compared to the peripheral model. It is of interest to note that in the case of the peripheral model the best fit to the experimental data is also attained at G2 = 7 fm. Expanding the right-hand side of eq. (6) into a series in Kz it can be readily seen that taking the pole graph of fig. la as K(E, z), we obtain the amplitude T(& z) as a series of ladder graphs (fig. 3). Crosses on the lines signify that the corresponding virtual particle is on the mass shell, i.e., that its non-relativistic propagator (E-k2/2m+iy)’ is replaced by - inG(E- k2/2nz). It is interesting to try to estimate corrections for the pure pole approximation for the K-matrix. Within the framework of the same mechanism for triton exchange the first such correction K’2)(E, z) is due to the allowance for the double triton exchange in the K-matrix. According to the sect. 3, as Kc2)(E, z), one should take the expression which differs from the amplitude of the square graph of fig. lb in that the integral over the virtual momenta is calculated in the sense of its principal value; in the case under consideration, such a recipe results in Kc2)(E, z) coinciding with the real part of the amplitude Mc2)(E, z) of the graph shown in fig. lb. Proceeding from the
DISPERSION
Feynman integral partial amplitude:
for M(‘)(E,
@‘(E)
z) it is easy to obtain
= (C/$)P s C =
P being the principal We observe that
K-MATRIX
G4m,
mdk(QJS(p,
0
69
the following
k)l)‘/(p’
~1, n1~/47r2(1~z, + ni,).
expression
for the
-k’>,
(18)
value sign. M?(E)
= K!*‘(E) - ip[K{“(E)]‘,
(19)
where K,(‘)(E) is defined by formula (17). The effect of the correction Kc2) on the pa scattering differential cross section at 20.62 MeV is shown in fig. 4, where the experimental points are the same as in fig. 2. The solid curve 1 has been obtained for K(E, .z) = K(“(E, z) and differs from the curve 2 in fig. 2 in being corrected for the Coulomb interaction which is essential at small scattering angles. The pa scattering partial amplitude, which includes both the nuclear and Coulomb interactions, was taken in the following form T* = T/C) + e2ibzT(N)1 1 >
C-20)
where T{” is the partial amplitude of pure Coulomb scattering, c1 is the Coulomb scattering phase shift andT/N’ is the nuclear scattering amplitude as defined by eq. (6). The dashed line 2 in fig. 4 has been obtained by formulae (20) and (6) at K(E, z) = K(“(E, z)+Kc2)(E, 2). Finally, the dash-and-dot line 3 corresponds to the inclusion of only the terms with an odd number of triton lines in the right-hand side of the graphic equality shown in fig. 3 which leads to the relation
T/N) = @‘/[l
+(pK:“)*].
(21)
In constructing all the three curves we took G* = 7 fm; the partial amplitudes with I = 0 were fitted in the same way as in the case of the curves 2 and 3 in fig. 2. We see that at large 3 all three curves are practically coincident and describe the experiment well. The curve 2 is close to the curve 1 at all values of 3, which suggests that the role of the K(*) correction is insignificant. At 3 5 90” all the theoretical curves lie below the experimental points and the omission of terms corresponding to an even number of triton exchanges whose amplitudes are maximum at small 3, increases the discrepancy between theory and experiment (curve 3). The unsatisfactory description of experiment at 3 < 90” by the curves shown in fig. 4 is evidently due to the fact that in our calculations the main contribution to the partial amplitudes TiN’ comes from the pole graph of fig. la whose amplitude
A. G. BARYSHNICKOV
70
0
60
et al.
120
180
Gl.
Fig. 4. The effect of the contribution from the double triton exchange into the K-matrix on the elastic pr scattering differential cross section at E,, = 20.62 MeV. Experimental points are from ref. 13).
K”‘(E, z) is maximum at 9 = 180” and decreases monotonically with the decreasing 3; therefore, at small 9 the contribution from other mechanisms may be essential. The simplest of such mechanisms corresponds to the triangle graph of fig. lc which describes the scattering of an incident proton from one of nucleons of an a-particle. At the proton energy 20.62 MeV, the amplitude of this graph has the leading logarithmic singularity at z = z, w 5 (more exactly at zd = 5.00 for scattering from a proton and at z, = 5.16 for scattering from a neutron). Thus, the point z, is the nearest singularity relative to the point z = 1 corresponding to the forward scattering. In the physical region, the amplitude K““(E, z) of the graph depicted in fig. lc is
DISPERSION
K-MATRIX
71
maximum at 9 = 0. To estimate roughly the contribution from this graph, we assume that the vertex functions of the graph are constant. The very crude assumption about the constancy of the pN scattering vertex is to a certain extent justified in our case because the nearest singularity of this vertex corresponding to the one-pion exchange in the t-channel does not contribute, according to the aforesaid, to the amplitude K”‘(E, z). Further, we neglect, in conformity with the isobaric invariance, the mass difference between 3H and 3He and between the proton and the neutron and take Gzrm = G& = G2. Then, taking advantage of the expression for the nonrelativistic triangle graph amplitude given in ref. “), we can obtain for the partial
Fig. 5. Elastic pee scattering differential cross section at Ep = 20.62 MeV with the pole and triangle mechanisms included in the K-matrix. Experimental points are from ref. 13).
et al.
A. G. BARYSHNICKOV
72
3ood
x 3
IOO-
- 300
I
50-
$ E G
- 50
5 -0 IO-
5-
Fig. 6. Comparison between theory and experiment for the elastic nr* scattering at the neutron energies 17.6 MeV (curve 1, full circles, left-hand scale) and 20.9 MeV (curve 2, open circles, right-hand scale). The curves have been obtained in the pole approximation for the K-matrix. Experimental points are from ref. 16).
amplitude Kid’(E) Kid’ = - (9/4$)(m,/p)G2fJ,, J,
=
s
1 l arctgcat1-‘>“>
2 -1
P,(z)dz
(1 -z)f
=
sm
Q,(l + s’)dx,
a-1
(22)
a = (f$)pl(m, d&g+, f = fig +_fr$% where constants ~$4 have the meaning of the NN scattering amplitudes in the state with spin s, which are normalized by the condition da$$dQ = /f$l'. As before, the spin factor a,,, is omitted in eq. (22).
DISPERSION
K-MATRIX
73
The pa scattering differential cross section at 20.62 MeV calculated by formulae (20) and (6) for K, = K[“+K,‘d’ is presented in fig. 5. In our calculations, we took G2 = 7 fm andf = 14 fm. It can be seen that the inclusion of the triangle graph into the K-matrix allows a satisfactory description of experiment in the entire range of scattering angles. The valuef = 14 fm is close to 18.3 fm which is obtained if we put f$2 = -t&d, where ~$2 are the NN scattering lengths. The calculated differential cross section shows a Iow sensitivity to the variations of the values offin the range 10 to 20 fm. Hitherto we have considered the pa scattering; the rm scattering may be treated in a similar manner. Fig. 6 shows a comparison between theory and experiment 1“) for the nm scattering differential cross section at the neutron energies 17.6 MeV (curve 1, full circles, left-hand scale) and 20.9 MeV (curve 2, open circles, right-hand scale). Curves 1 and 2 have been obtained analogously to curve 1 in fig. 4, i.e., in the pole approximation for K(E’, 2). The pole graph for the na scattering can be obtained from the graph of fig. la by the following substitution: p -+ n, t + z. The partial amplitude J&,(E) and the constant G2 = G& were fitted by the x2 method. The value G2 = 8.5 fm corresponds to curve 1, and G2 = 8.0 fm corresponds to curve 2. These values are close to the value G&, = 7.0 fm, which has been obtained from the pa scattering, as might be expected since G&, = G& under isobaric invariance. At 9 2 90”, the theoretical curves show a close correspondence to the experimental points. As in the case of the pa scattering, better agreement between theory and experiment at 9 < 90” is attained when the triangle graph is taken into account. In the examples considered above the energy of incident nucleons was chosen away from resonances in the Ncl system; however, if necessary, the resonance phenomena may be included in the scheme under consideration. 5. Conclusion As the example of Na scattering indicates, the suggested approach, which ensures the S-matrix unitarity, enables one to achieve a reasonable agreement with experiment and to extract the values of vertex constants G. The values G& = 7.0 fm and G&,, = 8.0 or 8.5 fm we have obtained are consistent with the results of peripheral model analysis of elastic pet scattering at the proton energy 48.8 MeV [ref. “)I (G&, = 7.0 fm), of the 3He(d, p)4He reaction [ref. ‘“)I (G.‘,, = 7.1 fm) and of the 3H(p, y)4He process at the proton energy 156 MeV [ref. “>I (G&, = 7.7 fm); our values are somewhat lower than G&, = il.3 fm which has been obtained in ref. 2“) by using the dispersion relations for the forward elastic na scattering amplitude. The vertex constant GABc, corresponding to the virtual decay (synthesis) A z B+C, coincides with the on-shell matrix element for this process to within kinematic factors. In contrast to the conventionally used reduced widths, the vertex constants are model-independent and similar to the coupling constants in the theory
74
A. G. BARYSHNICKOV
et a/.
of elementary particles. The value GABCis proportional to the coefficient in the asymptotic behaviour of the wave function for the channel A + B+C and can be expressed in terms of the Fourier component of the overlap integral for the nuclei A, B and C {refs. ‘, ““)I. The vertex constant for the triton break-up (t -+ d+n) has been calculated in ref * *’ )Yb solving the Faddeev equation for two different NN potentials. For the Malfliet-Tjon potential 2”) G& = 1.9 fm, for the Darewich-Green potential 23), G:,, = 0.1 fm; from an analysis of experimental data on various reaction it follows that G&, M 1 fm, which agrees we11with the value obtained in ref. ““) using the Reid NN potential “‘). From these examples it follows that the vertex constants can be highly sensitive to the form of the NN interaction. It would be, therefore, of interest to compare the phenomenological values of constants GoltPand G,,,, which have been found in this work and elsewhere, with the theoretical values calculated by methods of nuclear structure theory. Unfortunately, such calculations have not so far been made. A reliable determination of constants G& and GsLiadwould yield a highly valuable information; at present, only the order of magnitude of these constants is known 17,26). Some data on the vertex constants for the lightest nuclei are given in ref. 27). W e note that our vertex constant GABCis related to the dimensionles quantity &.., of ref. 27) by
Gh, = (- 1)“2n*(hlpc)(g’~~;i)‘,
(23)
p and 1 being the reduced mass and the relative orbital angular momentum of fragments B and C. For most reactions of the type A+x -+ B + y, the pole corresponding to the mechanism of nucleon or cluster transfer is the nearest to the physical region singularity in z. In accordance with what has been stated in sect. 3 (eq. (15) and below), in such a case, the knowledge of the vertex constants of the corresponding pole graph fully determines the partial amplitudes of the above reaction at sufficiently large E. Taking this into consideration may be of great help in an anaIysis of experimental data 28). A similar situation arises in the NN scattering, where the partial amplitudes for large 2 are determined by the one-pion exchange graph and expressed in terms of the pionnucleon coupling constant 2“). In the present work, only examples of pure elastic scattering have been considered. The next step should be the application of the suggested approach to the multichannel processes, primarily to the reactions involving light nuclei at low energies, when the number of open channels is small. It would be interesting to examine the polarization phenomena which would require to allow more accurately for the spin dependences in vertices of the appropriate graphs. It should be mentioned that even the pure pole approximation for the K-matrix leads to non-zero polarization of reaction products with non-polarized initial particles if the orbital momenta in vertices of the pole graph are non-zero. In the case of the pole approximation for the T-matrix, polarization is absent.
DISPERSION
K-MATRIX
75
Needless to say, the approach advanced in the present work is not the sole way to guarantee the unitarity of the S-matrix within the framework of dispersion methods. This can be done, for example, by using the Pad& approximant 30) or the multichannel N/D method 31). As compared to the Pad& approximant, the K-matrix approach is characterized by having a clearer physical meaning. Moreover, the tentative calculations of pcl scattering at the proton energy 20.62 MeV, which have been performed by us using the Pad& approximant, yield the results very similar to those obtained in sect. 4 of this paper. As regards the N/D method, it seems to us that in this case the practical calculations are more involved than those in the K-matrix approach. It should be noted that recently there have appeared works devoted to the applications of the K-matrix (or similar) methods to the three-body problem (see, e.g. refs. 32-39)). Some preliminary results of the pa scattering analysis within the framework of the approach suggested have been published in ref. 40). References 1) I. S. Shapiro, Teoriya pryamykh yadernykh 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27)
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