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Mesons and the structure of nucleons
ANNALS
OF PHYSICS:
4, 189-232 (19%)
Mesons
and
the Structure
Part II. The Nucleon
Isobar
BERN.~RD
Physics Department
of Nucleons
and Pion Dynamics*
T. FELD
and Laboratory for Nuclear Science, Massachusetts of Technology, Cambridge, Massachusetts
Institute
An “atomic” model of the physical nucleons, previously applied to the nucleon ground states, is extended to describe the excited isobar nucleon state, of angular momentum and isotopic spin 3/2. The model is used to compute the cross sections for resonant photoproduction of pions and Compton scattering of photons on protons. Other, nonresonant processes are ta$en into account in a phenomenological fashion, and the computed cross sections for the aforementioned processes are compared with available experimental data. The model is also used as a guide for the phenomenological interpretation of other high-energy processes-in particular, photodisintegration of the deuteron in the region of the photomeson threshold, and the resonant K - P interaction at ~1 Bev. INTRODUCTION
In an earlier communication (1)’ a model was developed in which the physical nucleons (proton and neutron) were depicted as bound systems of a nucleon-like core and a single pion. The main features of the physical nucleons were seen to be determined by the condition bhat the core (spin and isotopic spin l/2) and the pion (spin 0, isotopic spin 1) are bound in a pllz state of total isotopic spin T = l/2. In considering, in I, the field-theoretic basis for this model, it was observed that the pion-nucleon forces in Dhe p3/2 state with T = 3/2 are also expected to be attractive, although less strongly than in the ground state. The existence of such an attraction is manifest in the well-known resonance phenomena associated with p-wave pion scattering and production in the T = J = 3/2 state. Indeed, these resonance phenomena are so pronounced as to suggest the existence of a quasi-stationary or virtual state of bhe nucleon, the so-called2 (3,3) or ‘5sobar” state. * This work was supported in part by the joint program of the Office of Naval Research and the U. S. Atomic Energy Commission. I, 1 Henceforth referred to as I. . /,I‘ 2 It is customary, in this field, to characterize the various states of a single nucleon and 189
190
FELD
The isobar hypothesis was first suggested by Brueckner (2) on the basis of fragmentary evidence on pion-nucleon scattering. Subsequently, careful measurements by Anderson, Fermi, and co-workers and by other groups3 on the cross sections and angular distributions of the various experimentally accessible pionnucleon scattering processes have lent weight to Brueckner’s suggestion through relatively accurate determinations of the parameters (energy dependence of the 3,3 scattering phase-shift, (~33, and the value of the resonance energy) which define the properties of the scattering in the resonant state. These experiments have also shown that the phase-shifts corresponding to scattering in the other three p-states ((~$1, (~13, (~11)are small, at least for pion energies below the resonant energy, NlSO-Mev pion kinetic energy (in the laboratory system); the s-wave scattering phase shifts are also relatively small, although non-negligible, in both isotopic spin states, while the d-wave phase shifts are too small to detect,, with the presently available experimental accuracy, in the region of and below the resonance. An important consequence of the pion-nucleon scattering measurements has been the demonstration of charge independence (isotopic spin conservation) in the pion-nucleon interaction. The concept of the isobar resonance, together with charge independence, was first applied to the interpretation of the other “fundamental” pion-nucleon reactions-photoproduction and production in nucleon-nucleon collisions-by Brueckner and Watson (3). These ideas have proved exceedingly fruitful for the interpretation and analysis of these reactions (4). In the case of the photoproduction of neutral pions from nucleons t’he (3,:~) resonance completely dominates. Two types of phenomenological approach have been applied in the interpretation of photopion phenomena involving the resonance: Brueckner and Watson (3) describe the resonance by a Breit-Wigner formula, adopting the same formalism to the description of this resonance as is conventionally used in low-energy nuclear physics (6). Fermi (6) and Gell-Mann and Watson (7) assume that the first step in the reaction, the absorption of the photon by the nucleon, can be deany number of pions (real and virtual) by the symbol (2T, 24, where T is the total isotopic spin and J the total angular momentum of the system. Whenever there is any possible ambiguity concerning the parity of the state concerned, this is indicated by a + or - superscript on 2.J. Thus the state of a proton and a single s-nave positive pion is (3,1-) while for a p,,~ pion, the state would be (3,1+) ; i.e., owing to the negative intrinsic parity of the (pseudoscalar) pion, the parity of a state with one nucleon and a single pion is (-l)C+l. 3 No attempt will be made, in this paper, at a complete documentation of the volumenous experimental results upon which the information concerning the isobar state is based. Instead, the reader is referred to excellent summaries presented at the 19.56 CERN symposium on pion physics, as well as to the papers representing the most recent results (9). In particular, attention is called to the reviews of pion-nucleon scattering by L. C. L. Yuan, of photopion production on nucleons by G. Bernardini, and of the production of mesons in nucleonnucleon collisions by A. W. Merrison, not only for excellent reviews of the present status of these fields but also for the complete sets of references to previous work contained therein.
MESONS
AND
NUCLEONS
191
scribed by the conventional field-theoretic equations in a perturbation approximation (8), but that the reaction in the (3,3) state is enhanced by the strong pion-nucleon interaction in the final state; the “enhancement factor” for the final state wave function is then computed by straightforward quantum mechanical techniques, utilizing the properties of the pion-nucleon system in the (3,3) state as observed in the scattering process. In the photoproduction of charged pions from nucleons another process, in addition to the excitation of the resonance, is important. This is the photoelectric effect4 in which, owing to the pseudoscalar nature of the pion, the pion emerges mainly in an s-state for photon energies not far above threshold. This is a process for which the field-theoretic perturbation calculations (8) appear to offer an adequate description. A number of phenomenological analyses, based on the concepts and methods outlined above, have been applied to the data on photomeson production, with fair success. Among the most recent and complete is that of Watson et al. (9). However, some questions remain unsettled, both from the experimental and theoretical points of view. One is the nature of photomeson production in states other than those accounted for by the major effects mentioned above. Another concerns the electromagnetic processes involved in the excitation of the isobar resonance: Conservation of angular momentum and parity limits the possible processes responsible for photomeson production in the (3,3) state to magnetic dipole and electric quadrupole photon absorption (10). The experimental evidence indicates that both processes may occur, although magnetic dipole absorption dominates, at least at energies well below the resonance (11). We shall return to these questions in a following section. With regard to other reactions involving nucleons and mesons, we have already alluded to the considerations of Brueckner and Watson (3) on the effects of the isobar on meson production in nucleon-nucleon collisions. They proposed that the interaction proceeds mainly through the excitation of one of the nucleons into the isobar state. The consequences of this mechanism are most easily analyzed for the reaction in which one of the products is a deuteron Application
of conservation
of isotopic spin, angular momentum,
and parity,
4 Qualitatively, at least, it is clear that the photoelectric effect is much more important in charged than neutral pion production; since the nucleon mass is so much greater than that of the pion, the electric dipole moment in the initial (physical nucleon) state will only be appreciable for those configurations of nucleon core and (bound) pion cloud in which the pion carries charge. However, such arguments neglect the finiteness of the nucleon mass and the possibility that a charged pion may, before emerging from the nucleon field, suffer a “charge-exchange” scattering and emerge as a neutral pion. Such effects lead to an observable, albeit small, photoelectric production of neutral mesons which must be considered in the analysis of &production phenomena.
192
FELD
together with the assumption that the two nucleons (one excited) in the intermediate state have zero orbital angular momentum, defines uniquely the twonucleon state as ID2 (3, 6). This model, when applied to reaction (1) and its inverse, accounts for the main features of pion production and absorption in twonucleon systems at energies up to -600 Mev (I 2, 4. TO the extent that the phenomena discussed above can be described in terms of models in which the nucleon isobar is treated as a resonant state, a phenomenological approach is suggested for treating other high energy reactions involving the isobar (IS, IQ). Thus, the same resonance may be regarded as responsible for photon scattering (nucleon Compton effect), for the photodisintegration of the deuteron, and, at least in part, for nucleon-nucleon scattering in the appropriate energy range. The parameters which determine the resonant features of these reactions (the appropriate partial widths) can be derived from the observed properties of the pion scattering, photoproduction, and nucleonnucleon production cross sections. Also at higher energies, both for pion-nucleon (15) and nucleon-nucleon (16) interactions, the isobar resonance appears to play a major role. The phenomenological isobar models, discussed above, derive their justification not only from their success in describing the observations but also on the basis of meson field-theoretic considerations. The major contributions in this direction have resulted from the demonstration by Chew (17) that the (3,3) resonance in pion-nucleon scattering follows from a nonrelativistic, “fixed source” approximation to conventional meson theory, upon application of the (renormalization) techniques developed in the quantum held-theoretic treatment of electrodynamics.5 Of primary interest, in this connection, is the demonstration by Chew and Low that the (3,3) scattering phase shift may be described in terms of an “effective-range” formula (19) and that the photomeson production cross section near threshold is given in terms of the same parameters as determined the properties of the resonant pion scattering (ZO).6 A development of major importance in the interpretation of meson scattering and production is the recent application of “dispersion relations” to these phenomena. Briefly, it has been shown that the requirement of microscopic causality (i.e., the limitation of the speed of propagation of wave phenomena to speeds not exceeding that of light) leads to connections between the real part of the scattering (or production) amplitude and certain integrals involving observable cross sections, analogous to the well-known Kramers-Kronig dispersion relations for the propagation of electromagnetic waves. The application of these relations to pion scattering and photoproduction (21) has recently been summarized by 5 An excellent summary of meson field theory is given by Wick (18). 6 Similar connections can also be shown to hold for Compton scattering private communication).
(F. E. Low,
193
MESONS AND NUCLEONS
Chew, Goldberger, Low, and Nambu. These dispersion relations yield results which do not differ markedly from the field-theoretic expressions at low energies (19, 20) and, consequently, tend to fortify the conclusions of the nonrelativistic, fixed-source approximation to the meson field-theoretic treatment of isobar resonance phenomena. The work reported below represents an extension of the atomic model developed in I to dynamic phenomena involving the nucleon isobar state. This approach lies between the field-theoretic and the purely phenomenological treatments. Thus, although the existence of the virtual state is postulated, its wave function and, in particular, the description of its decay properties are assumed to have the forms dictated by meson field theory: The wave function describing t’he isobar is derived from considerations such as those used in I to describe the nucleon ground states; the decay properties, specifically the decay width, are expressed in terms of the Chew-Low effective range relation for the (3,3) scattering phase shifts, with the parameters chosen to fit the scattering observations. Such considerations serve to specify the properties of the nucleon states between which transitions may be induced by various means. Thus, it is possible, on the basis of this model, to compute directly the matrix elements corresponding to electromagnetic excitation of the isobar. Decay of the isobar by meson emission corresponds to photoproduction. Decay by the emission of electromagnetic radiation (related to photon absorption by argument(s of detailed balancing) corresponds to Compton scattering. One feature of this model is that it provides a mechanism for computing, in an unambiguous fashion, the effects of the isobar at energies considerably removed from the resonance energy. In the case of photon Compton scattering, this perm its us to extend the calculations to energiesbelow the meson-production threshold into the range most extensively covered by the available experimental data. Such considerations also provide a guide for extending the phenomenological computations on the isobar effects in the deuteron photo-disintegration into the energy region below meson-production threshold. DESCRIPTION
WAVE
OF THE
ISOBAR
STATE
FUXCTION
We assume the existence of a quasi-stationary state, in the conventional sense,7of spin 3/2 and isotopic spin 3/2. There are (2J + 1) (2T + 1) = 16 sub-states, whose wave functions we write as %2T8
. 2J3
*
=
%T,
) 2J3
exp ( - B/2@,
(2)
in which the a’s represent the wave functions of the (stationary) isobar sub’ See Blatt
and Weisskopf
(6), pps. 412-417.
194
FELD
states and I’ is the “decay width”, r = fi/ (isobar mean lifetime). Following the procedure of I, 3 9253 = X~T~%J,G(T), in which the x’s represent the four possible charge states of the isobar, @2T
x3
= pa+
Xl
= (2/3)“‘p?r’ + (1/3)“‘7za+
x-1 = (2/3)“2n~o + (1/3)1’2p7rx-3
(3)
(4)
= mr-,
the \E’s the angular momentum states
(5) and G(r) is the appropriate radial wave function of the isobar. The exact form of G(r) is of no great concern in the following considerations, and we make no attempt at deriving it. Its form, for r 5 the range of the pion-nucleon forces, is presumably similar to the radial wave function R(r) for the nucleon ground state, Eq. (7b) of I; the important consideration is that its spatial extension is relatively small, of magnitude a few times the nucleon Compton wavelength. Thus, in treating the absorption of photons, it is not too unreasonable to utilize an approximation in which the photon wavelength is treated as large compared to the nucleon size, even though photons in the resonance region have wavelengths’ x y w fi/2mc = (l/2)?. In computing the matrix elements for electromagnetic transitions between the ground and excited nucleon states, we will be concerned with the off-diagonal matrix elements of the magnetic moment operator and the electric quadrupole operator & =
T,3T2Ygm.
(7) The diagonal elements of these operators in the state JZ = J correspond to the static moments. The results of such a computation of the magnetic moments of 8 The pion Comptonwavelength, K- 1 = h/me S 1.4 X lo-‘3 cm, is used throughout this paper as the characteristic length for pion-nucleon interactions. We also adopt, for the ratio of nucleon to pion mass, M/m = 6.75.
MESONS
AND
195
NUCLEONS
the ground nucleon states have already been given in I (the ground state, with spin l/2, has, of course, no electric quadrupole moment). We may, similarly, compute the magnetic moments of the isobar states, Taking gs = 2(wT/3 + b),
(64
gz = (Mlm)T.a3 ,
(6b)
=t&ia
(6~)
we obtain /.93 =
=k b +
p&l = zt(>4u
f
Jfl4~0,
3b + M/m),,
.
W)
The values of a and b depend on the assumption concerning the intrinsic moments of the nucleon cores: for “Dirac” cores, a = 1, b = l/2; this assumption leads to roughly equal neutron and proton moments in the ground state. It is possible, as shown in I, to obtain the observed nucleon moments by appropriate choice of a and b. However, in view of the approximate nature of such computations, as well as other possible effects (e.g., K-meson states), we shall not attempt such refinements. Assuming the “Dir&c” values of a and b, the static moments of the isobar states are shown in Table I. For the static electric quadrupole moments, defined as the diagonal elements of (16 7r,‘5)1’2& of Eq. (7),
=
=
00 G2(dr4 dr- (XZT~ ,
s0
-%(r2>3*(X2T3,
T,3x2T3
>. $ *‘3*(3
cos” 8 -
l)$
an
(74
7S3XZT,).
Thus, referring to Eqs. (4), we obtain the values shown in Table I. TABLE
I
STATIC MOMENTS OF THE ISOBAR STATES 2T3
r/Pa"
3
7.75
1 -1
2.92 -1.92
-3
-6.75
QIQC
1 -; -1
a PO = eh/2Mc is the nucleon Bohr magneton. b &O = -%(~~)a S -2.5 X IO+ cm2, assuming the same rms radii for the isobar and ground nucleon states, (T~)~E (0.8 X lo-l3 cm)’ GZ (r2)8 .
196 DECAY
FELD
PROPERTIES
We assume that the resonance scattering is described by an effective range formula of the form given by Chew and Low (19) 4.f2 --- q3 cot 3 hc w
cY33 =
1 -
W/WY )
in which r], the pion (c.m.) momentum, w, the total (c.m.) energy minus the nucleon rest-energy, and wr , the resonance energy, are all defined in Appendix I. The experimental values of the parameters in Eq. (8) are9 f”/lic = 0.082 f 0.007, wr E 2.12-2.15, corresponding to a laboratory pion kinetic energy at resonance of -190-200 Mev (3). The connection with the resonance width,” I’ of Eq. (a), is made by way of the usua1 resonance scattering formula (5) tan
~3~
= r/2(w,
- w),
(9)
giving
(10) in units of me”. PHOTOPRODUCTION MAGNETIC
DIPOLE
EXCITATION
OF THE
OF PIONS ISOBAR
Rather than compute directly the photoproduction process, we take advantage of the known isobar scattering properties and compute the ratio
01) 9 U. Haber-Schaim (private communication). 10 For p-wave scattering, resonance theory l- = 2r,(qR)“/[l
See, also, Orear (6) gives
+ (qRY1,
(20).
@a)
and J?Othe “reduced” where g is the pion wave number, R the “range ” of the interaction, width. Anderson and Metropolis (221) have analyzed the pion scattering data to obtain a “best fit” in the form of Eq. (9). They report (8b) kinetic energy), which corresponds to E, = (r,. - 1)mc 2 = 133 Mev (199.Mev pion laboratory R = 0.89 ~-1, ~CJ= 0.34 mc2 = 47 Mev. This value of r. corresponds to a lifetime for the 70 = h/m? = 5 X 10bz4 set, which virtual state of only -3 times the “natural lifetime”, is uncomfortably short for a virtual state. Nevertheless, the pion may be thought of as making a few oscillations, at least, before being re-emitted from the “compound” system. It will be noted that we make no distinction between the total width and the scattering width. In view of the large value of I’, this is clearly a valid approximation.
MESONS
AND
197
NUCLEONS
in which J?ir is the width for the decay from the intermediate state (j) to the final state (f). The sum over initial states (i) is simplified by the fact that a spin-0 incident pion can cause transitions only to states with the same magnetic quantum number, if the z-axis is chosen to lie along the direction of the incident pion. Thus, the sum over initial states contains just two terms, corresponding to the two possible polarizations of the target nucleons and, if the target nucleons are initially unpolarized, these two terms are equal. It suffices, then, to compute just two transition widths, I’ljz, 1,Zand l?1,2,--1/Z, for each mode of de-excitation of the isobar. (We have indicated in the subscripts only the magnetic quantum numbers of the isobar and final nucleon states; the other relevent quantum numbers are understood and, in any event, obvious for any given reaction.) For pion emission, the I’jfcT’are simply the pion width I’, Eq. (lo), multiplied by the appropriate Clebsch-Gordan coefficients connecting the isobar state and the final pion-nucleon state (the nucleon goes from T = J = 3,‘2 to T = J = l/2) (rf) b, 1j2 r1/2, 112(*O)= pp, (124 = ?Q, (Tt) (+I) = ggr rl12, -l/2 b2, -ll2 Wb) = w, (To)~ r(r’) = ? @ ‘= 2r(R*). WC) wjf For magnetic dipole photon emission, we have
rjf(Y) -- g3 I Jfif I21 Y
03)
in which the iUj, are the appropriate matrix elements of the magnetic moment operator p of Eqs. (6), Mjf = F / F=
s0
Xl.r-l,f*
k!x3,i*3.i
dQ,
m R(r)G(r)r2 dr.
(14) (15)
We may writ,e p
=
P+E-
+
/A-E+
+
(16)
P3C0,
in which /&t = wv2(Pz
f G&J,
Ps
(17)
EO= zo
(18)
P3 =
and q = (1/2)l’“(xo f iyo),
198
FELD
are combinations of the unit vectors along the three coordinate axes. The three terms in Eq. (16) lead to the emission of left-circularIy polarized radiation, with rnf = mi + 1, the emission of right-circularly polarized radiation, with mi = mj - 1, and unpolarized radiation, with rn/ = mj , respectively. In general n/l,,
=
CjfPeff
,
(19)
in which the Cjf are the appropriate combinations of CIebsch-Gordan coefficients and pcff may be regarded as an effective magnetic moment governing the transition. Straightforward computation yields I c*1/2.
w3, I c*1/2,11/2 I = d2/3,
I =
*1/z
and hff
=
(l/3)
(2~
+
M/m)Fpo.
(20)
Accordingly
with
4 r-l
=
3n,3
(22)
kkrr2;
and the ratio of cross sections, Eq. (II), is simply I’,/I’ for a’ = a0 and 2I’,/r for 7r’ = ?r+. Now, we are really interested in the cross section for the inverse reaction uy+* , rather than un+y . These are connected by the detailed-balancing condition (6) (23) (The factor 2 arises because of the two possible incident photon polarizations.) Accordingly
KY2r, u’y+r __ = -_. 2~2 r ua-vr, Actually,
1 for
7r’ = nP
{ 2 for 7r’ = a* i .
there are four possible photoproduction
y+P*P+?rO y+N--tN+n'
(24)
reactions to consider:
(25a)
y+P+N+n+ r+N+P+r-.
Wb)
MESONS
AND
199
NUCLEONS
These are all covered by letting r’ = a’, with P = a0 for the first two, and ?r = n” for the second group, It is to be noted that the us-a1are the cross sections for resonant pion scattering, 167r XT2sin’ a33 9
u~~,,o = -
,
lyaO*,O= 2u,++,a.
(27)
Hence uy-mo= 2Uy*&
(28)
with
Substituting Eqs. (10) and (22) for I? and I’? , we obtain
(29) In Eq. (29), (e2/j2) is th e ratio of coupling constants for the electromagnetic and meson fields,
The factor (W/W,)is a slowly increasing function of the incident photon energy, going from z1/2 at threshold to 1 at the resonance. The prediction of the atomic model, Eq. (29), differs from the results of Chew and Low (.B) and of the dispersion approach (W) in only a rat,her minor way; the results of these field-theoretic computations may be obtained from Eq. (29) by the substitution of the difference between the static nucleon moments, pP - kC(N = 4.70 po, for the ‘(effective dynamic moment”, perf’= peff (w/w,)~‘~, of the atomic model. Numerically, this can make an appreciable difference. Thus, assuming the “Dirac” moments for the core nucleons, Eq. (20) yields p,ff = 2.92 Fpo . However, since, as shown in I, the core moments must be assumed considerably different from the Dirac moments in order to account for the observed static moments of the physical nucleons, we may expect the value of Meffto differ considerably from this (rather small) value. The assumption of anomalous core moments increases the value of peff , by approximately a factor 2. Accordingly, we would obtain pefr’M pp - pC(N at resonance for a (not unreasonable) value of the overlap integral, F x 0.8.
200
FELD
This is in accord with the observed value of the peak cross section (9),
ELECTRIC
QUADRUPOLE
a,,,o(max)
% 220-230Nb.l’
EXCITATION
OF THE ISOB,ZR
The excited isobar state, which can decay by pion or magnetic dipole photon emission, as considered in the previous section, has available also t.he emission of electric quadrupole radiation as a possible, competing decay mode. Since the same detailed balancing relationship, Eq. (23), holds for electric quadrupole (J?2) processes as for magnetic dipole (Ml), we may compute the cross section for E2 pion production t,hrough the relationship
for which the denominator has already been computed, Eqs. (21) and (22). The required partial widths are given by the expression12
where Qu is the appropriate the previous case, we write
matrix
element of the E2 operator, Eq. (7). As in (32)
Qij = CuQeii , and obtain, in this case, / C*I,~, *I/Z 1’ = J/15, / C&I/Z, +,2 1’= 6/15,
Qeff= (&)'I2
Combining
(
R(r)G(r)r* dr = (&)1’2
(r2),ff ,
(33)
this result with that of the previous section,
11From our point of view, the important result is the demonstration that the atomic model reproduces, in a simple and straightforward fashion, the essential results of the much more complicated field-theoretic computations. We do not pretend to be able to derive an accurate value of the effective magnetic moment. Nor can we insist on the kinematical factor, w/w~, especially in view of the inability of the atomic model to take into account that feature of the Yukawa interaction which permits the creation and destruction of pions. Accordingly we shall, in subsequent numberical computations, arbitrarily set ~~rf’ = pp - /4Jg= 4.70 /Jo. 12Blatt and Weisskopf (6)) Chapter XII.
MESONS AND NUCLEONS r (EZ) (E2) Y (‘?a = ry(M1) = ___ uy+.r u-f+r
201 (36)
in which k/~ = E,/mc’ (see Appendix I). According to Eq. (36) the relative E2 to Ml photoproduction increases with the square of the photon energy. To estimate the ratio we assume (~~r~),n $Z (K~T’)~F, with (K~?), z g, which was the value required in I to account for the electron scattering experiments. Taking pEl,ff= 4.70 PO z 6 poF, as suggested in the previous section, 2 (W fJY-wr = 0.0019~. l 0.075 ‘ c (36a) ay+*(M1) s tj
( > K
is expected Thus, even in the resonance region (Icr !X 2~) E2 photoproduction to have a negligible effect on the observed cross sections; however, its effects on the angular distributions in neutral photopion production, as will be seen in a following section, can by no means be neglected. PHOTOELECTRIC
PION PRODUCTION
Although, strictly speaking, our atomic model is not competent, for reasons to be discussed below, to deal in a reliable fashion with the other main reaction in photomeson production, the photoelectric effect, it is nevertheless interesting as well as illuminating to consider briefly this process from the point of view of the atomic model. The cross section for the “photodisintegration” of the physical nucleon by electric dipole radiation’* is,
where
&I,~ = e / r YI*\~~*T,~*~ dr3
(38)
is the electric dipole matrix element connecting the initial (nucleon) state and the final state, in which there are a nucleon and a free pion in an s-state. Aside from numerical factors, the magnitude of the matrix element near threshold is I QL
m I =
e (&)(&>“;:
(384
since the initial nucleon wave function extends over a range 4i/mc and the final pion s-wave function, (sin qr)/qr, is essentially constant over the region of the nucleon. Equation (38a) gives, for the electric dipole cross section near
202
FELD
threshold,
(39) Equation (39) contains the required dependence of the cross section on the momentum of the outgoing pion as well as the main dimensional and dimensionless constants which determine its magnitude. To probe into the other features of the process (aside from numerical constants) it is necessary to specify the wave functions to be used in Eq. (38). For the target nucleons, the atomic model, as developed in I, specifies #i as the wave function of the physical proton or neutron. For the final state, if we accept the model on its face value, we would have 9r = xm
~>cphPs(~),
(40)
where x specifies the final (nucleon and pion) charge state, p the polarization of the final nucleon, and 9, the s-wave function of the outgoing free pion. Taking q8(n) = (sin p-)/p leads immediately, in Eq. (38), to the following selection rules : (1) Nonvanishing matrix elements occur only for states in which the final nucleon polarization is opposite to that of the target nucleon; i.e., El photoproduction is a pure “spin-flip” process. (2) Only charged pions are produced--a+ from protons, 6 from neutrons. Furthermore, in this approximation the process is charge-symmetric, leading to equal cross sections for xi and r- production on protons and neutrons, respectively. It is relatively easy to take into account, at least to first order, the deviations from charge symmetry and the To-electric dipole production. In the first place, the electric dipole operator, in Eq. (38), assumes infinite nucleon mass. For finite nucleon mass the dipole moments of the states (pr-), (n?r+), (pro), (MO) are in the ratio (1 + m/M) : 1: (m/M) :O; their squares, which determine the corresponding cross sections, are in the ratio 1.32: 1.0: 0.022:O. Furthermore the possibility of “charge exchange” ?r”-production can be taken into account, in our approximation, by noting that the outgoing s-wave pion wave function, \E,(s) in Eq. (40), should strictly speaking be an eigenfunction of the isotopic spin of the final state
Since (~1 + CQwe note that, despite the choice of a dipole operator which normally leads to pure charged pion production, the actual final state mixture
MESONS
AND
NUCLEONS
203
contains slightly different amounts of the two isotopic spin states and, accordingly, leads to a small ?y”electric dipole photoproduction. The above-mentioned effects will be included in the final expressions for the electric dipole photoproduction amplitudes given below. They, are, of course, well known and have been extensively discussed in the literature (6). However, the major difficultyI with the application of the atomic model lies in the description of that part of the final state pertaining to the nucleon charge. X(X, r) in Eq. (40). In all of the computations described above, the final state consists, perforce, of a nucleon core and a free pion, since the process computed is one in which the physical nucelon is stripped of its pion cloud. However, the final nucleon observed is not bare, but a fully “clothed” physical nucleon. The process of converting a core to a physical nucleon is one which involves meson creation and, accordingly, the quantum nature of the meson field, an aspect which our atomic model chooses to ignore. Accordingly, we cannot expect a cross section computed on the basis of the atomic model to be correct in all of its details (which is the reason why we have not bothered about such minor matters as numerical constants). Nevertheless, as a crude first approximation, we might consider multiplying the cross section of the atomic model, Eq. (39), by the factor (f’/ti), the “renormalized” coupling constant for the meson field of a nucleon, as representing, in some sense, the probability for a bare nucleon to create the meson cloud required to convert it to a physical nucleon. Such a crude approximation cannot provide the correct numerical constants, nor does it give the other, presumably slowly varying, energy dependent factors which arise from the dynamic nature of the meson field equations. Nevertheless, the atomic model, as so modified, contains essentially all of the major features of the meson field-theoretic results as described below. Actually, the computation of the photoelectric pion production, by the use of meson field theory in a perturbation approximation, is well known (8) ; it is one of the few cases where the application of perturbation techniques in the most straightforward fashion yields reliable results. Starting with the Yukawa equation for a pseudoscalar pion14
q 2q? - K2cQ = - 4?rfT(d.v)p(r) 3 K
(42)
the electromagnetic field is introduced by the substitution V -+ V f (ie,&)A, and the pion wave function is written G Q= 90 + 91 where 90 is the static solution 13There are other difficulties inherent in the use of such a model, such as those involving the proper choice of the dipole operator, as discussed by Fermi (6). We believe, however, that these have their source in the same problem as is considered below. 14See I and Appendix I for details of notation and definition.
204
FELD
of Eq. (42) and ~1 has the form of an outgoing wave {F’ exp [i(‘r With these substitutions, Eq. (42) becomes
q 2~l*-&*=
-hc47rief ~~(d.V)p(r) K
=F 2
The first term on the right leads to the aforementioned section,
-
(A.Th*. El production
rt)]].
Wk.) cross
which, aside from the inverted dependence on photon energy, is as computed by the crude model discussed above. The second term in Eq. (42a) takes into account the absorption of higher electric moments and leads to what is known in the atomic photoelectric effect as the retardation term. In the summary which follows, the perturbation fieldtheoretic results will be assumed for the photoelectric cross sections with, however, the slight modifications required to take into account the finite nucleon mass and the charge-exchange scattering, as discussed above. SUMMARY:
CROSS SECTIONS FOR PHOTOPION PRODUCTIOX
In this section, we give expressions for the differential and integral cross sections for the four fundamental photopion production reactions, Eqs. (25). In these expressions, the photoelectric process is described by the results of the perturbation computation, discussed in the preceding section; to the extent of the validity of this approximation, this includes all electric multipole orders of the incident photon beam. For all other processes, on the other hand, we confine ourselves to those terms leading to p-wave pion production.15 In particular these include magnetic dipole and electric quadrupole resonance absorption. It is most convenient to describe the photoproduction process in terms of the elements of the “scattering matrix” S,f, in which the subscripts i and j refer, respectively, to the initial and final nucleon spin states and the superscripts + and - distinguish between the two possible states of polarization of the incident photon.16 Then, for an initially unpolarized photon beam falling on initially unpolarized nucleons, the differential cross section in the c.m. system for photopion production is
au - = f 2 / sji* 12, cm
2
(44)
15Only electric dipole absorption leads to s-wave pions, and this is included in the photoelectric terms. 16One advantage of this description in terms of the S’s is that it enables rapid computation of the effect of initial polarizations, or of the final nucleon polarization (84).
MESONS
AND
205
NUCLEONS
where the summation is to be taken over all values of initial and final photon and nucleon polarizations. With the restrictions stated above, St*-
= -Se+++ = Y sin 0 eTip,
EL;-;-
= -S&+
s-g-
= simi+ = -El
= X sin 0 eTiq,
(45)
+ K cos 0,
Xi+-*- = S-rx+ 11 = E,’ sin2 0 eF2+2
in which El = E[l - g2sin2 O /(K”$ 1k - q j2)],
(464
EI’ = E$/(K* + j k - q I”>,
(46b)
K
= -Ml+Ms-E3,
X
= 2Ma - s((Ma - I&) + Eq(k - q cos o)/(K”
Y
= Ml + x(M,
(47) +
j
k - q I’),
- I%) - Eq(k - q cos 0)/(/f” + j k - q I”).
(48) (49)
E, Ml , M3 , and E3 are the amplitudes, respectively, for electric dipole, magnetic dipole with j = l/2, magnetic dipole with j = 3/2, and electric quadrupole with j = 3/2 absorption. In the factors which multiply E, representing the “retardation” effects, k and q are the photon and pion wave-number vectors respectively, The amplitudes are, of course, not the same for all four of the photoproduction reactions, Eqs. (25). However, there are well-known connections (?‘) according to which the four values of a given amplitude may be expressed in terms of three real parameters, with the phases determined by the (known) pion scattering phase-shifts in the corresponding states. Aj = -aalei”‘i
+ fla3ezn3i,
(Y + P + P + TO)
= dsal’eia’j
+ aageia3i,
(Y + N -+ N + TO)
= flaleia’i
+ fla3eiasi,
(Y + P -+ N + ?r+)
= --al’eia”
+ *aaeiu3j,
(50)
(y + N -+ P + r-).
Thus, to describe the four processeswe need, in Eqs. (45), 4 X 3 = 12 real parameters. For practically all purposes, this number may be reduced considerably, as follows: (1) O f the processesleading to p-wave pions, the predominant effects arise from the isobar excitation, which gives rise to an enhancement of the T = 3/2 terms (us) in Ma and Es . The rest of the amplitudes for p-wave pions are relatively small and their effects are essentially negligible, except possibly on TO-
206
FELD
production near threshold (25). The usual procedure is to compute these small amplitudes from meson theory in the Born approximation (22, 23). We shall, in the following, neglect the nonresonant p-waves and, accordingly, take M, = 0, M3 = tm3ae2a33,
(51)
ES = Ee33ei”“3,
in which the factor [ = 1 or dl/a for ?y”or r* production, respectively, and a38is the resonant scattering phase shift, Eq. (8). The value of mu is obtained from Eqs. (26-29) with by equating ~~~~~~~~~ ~-t-t*0Of’) = &- m3t
(52)
from Eqs. (44-49). Correspondingly, from Eqs. (36) and (W _ 8n Uy-nO - ;;- e332,
we obtain p = e33/m33 = 0.075
(54)
AJ/K.
(2) For the photoelectric amplitudes, E, it is necessary to determine all three parameters, el , c<, and e3. We evaluate these by considering the threshold limit (p -+ 0), at which cyl = LY~-+ 0 and E(g -+ 0) = EO is real. In this limit we have E,,(y + P -+ N + ir+) = t~o~‘~
= (Q(3?yz
z (25 x 2.J
(55a)
[see Eq. (43)]; E,,(r + N --+ N + TO) = 0
(55b)
from the vanishing of the dipole moment in the final state; Eo(-y + N + P + 6)
= (1 + m/M)ao”2
(55c)
from the ratio of the final state dipole moments. These three conditions determine the threshold values ~~(0) = (1 + m/M)(ao/3)“‘, ~(0) = (1 - m/M)(2ao/3)“‘, d(0) = -(l
+ m/M)(2ao/3)1’2,
(56)
MESONS
AND
207
NUCLEONS
from which follows
E&y + P + P + 7r0)= (m/M)(ao/2)'12.
(56’)
Taking m/M = 0.15, the above gives for the ratios of the threshold (s-wave) photopion cross sections,
u&r-):&+):ao(Pn"):ao(Nao)
= 1.32:1:0.011:0.
To obtain the amplitudes as a function of photon energy, it is necessary to know the variation of the e’s with energy. We shall, in the following, assume that the energy dependenceis all in the factor” (q/k) of a0 . Then, for finit)e q, we substitute our e(O)%,Eqs. (56), into Eqs. (50), using Orear’s fits to the experimentally determined scattering phase shifts (5), (Y~= 0.16 r], a3 = -0.11 7. The results, obtained by straightforward algebraic manipulation, are shown in TABLE
II
ELECTRIC DIPOLE AMPLITUDES FOR PWOTOPION PRODUCTION, PIOCESS -y+P-+P+#
r+N-tN+r”
r+P-+N+r+ rfN+P+?r-
rr
0.105 0 1.00 1.15
E = [u~'I~ i-i
-0.127~ -O.l46?j 0.0571 o.om
Table II, in which we give the real and imaginary parts of the coefficients, I = tr + 4-i , in
E = [ull’2,
(57)
for the four reactions. The values in the table are the lowest order terms in an expansion in powers of q = q/K. However, the higher order corrections, even at photon energies well above the resonance, are only a few percent or less. We are now in possession of the necessary factors for the determination, within the limits of our approximations, of the required cross sections through Eqs. (4449). Owing to the complexity of the retardation factors, the general expressions for the cross sections are still quite complicated. However, they may be further simplified by neglecting, in the retardation factors, the nucleon recoil energy (more specifically, the difference between the incident photon and I7 Actually, Chew and Low (%9) have indicated how to take into account, to first order, the nucleon recoil effects. This may be achieved by, first, modifying u&2 by multiplication with the factor (1 - mk/2Mx)/(l + mk/MK) and, second, sqbstituting (mk/Mtc)for (m/M) in Eqs. (53). These modifications, beside introducing a slight alteration in the total cross sections for charged pion production, have very little effect on the predictions up to E7 = 400 Mev. Accordingly, we have not included them.
208
FELD
outgoing pion energies in the cm); in this approximation lc2z q2 + Kz, q/k E (u/c), = p,
(58)
~~+ 1k - q 1’c 2kZ(1 - 0 cos e), and the amplitudes become (all in units of ao”‘) El = [[l - /3’sin’ e/2(1 - 6 cos 19)], E1’ = @/2(1 - p cos 0)) K
= fR(1 - /+P’,
X Y
= ?4{5RI2 + (1 + p)lei”33+ = ?+Cj{{R(l - p)eiort3- c/3},
(59) 031,
with”
Finally, we write out the expression for (1) the differential cross sections 1 clcr z z = ( { j2 [l - @(l - @) sin2 e/2(1 - /3 cos 0)“] + ~2R2{(1 - p)” + [2 + %(l + P)~ - (1 - p)“] sin2 Bj + @(l + p)P(C, cos a33
030 +
Pi
sin
(~33)
sin2
- 2ER(l - p) (lr cos a33+ li sin (YS) 1 -
tJ
8; 1cos
p2 sin’ e 2(1 - p cos e>
(2) the differential cross sections at 90” in the cm. i $ (go”) = 1r 1’[I - ; ~“(1 - D’)] + 2t2R2 [ 1 + ; (1 + d2]
(62)
+ .$(I + P)P(~~ cm (~33+ ti sin 4; (3) The total cross section k.
= I !: 1’[l - (1 - P”>(tanh-’ P - P>/Pl + %tR2{ (1 - ~1”+ 411 +
X/4(1+ ~1~11 + %tRO + P>
+P(lr cos 01~3 -I- bi sin ad + 2gR (1 - P>kr cos a33
tank-‘/j I* Note previous footnotes:
(17) for kinematical
corrections
to (ro ; (11) for doff’.
(63)
MESONS
AND
209
NUCLEONS
24
Y+P--,N+
?r+
Ey( lab) = 265 Mev
01 0
30’
600
ecm.
900
120°
150”
I600
FIG. 1. Angular distribution of R+ photoproduction at E,(lab) = 265 Mev. The solid curve is from Eq. (61). The broken curve is Eq. (61) without the retardation terms. The data are from: (A)-Univ. of California (27); (X)-M.I.T., L. S. Osborne and B. Richter (unpublished); (O)-Cornell (28); (C/)-Cal. Tech (.%‘S). Data from Illinois (SO) at 225 Mev show a similar flat behavior at small angles.
The’ comparison of the above expressions with the experimental observations has been discussed by a number of authors (4). The agreement between theory and observation is excellent in general, especially if due attention is paid to the details of the theory (%?, 23). In particular, the presence of the retardation terms renders the interpretation of angular distribution data rather more complicated than it might otherwise be (266). Unfortunately, the effects of the retardation terms are not easy to observe, as is illustrated in Fig. 1, in which the experimental observations (27-30) on rf-production are compared with Eq. (61) with (solid curve) and without (broken curve) the retardation terms, for 265Mev photons. Although there is some experimental indication of the expected minimum at small angles, the experiments are clearly not, yet good enough to establish conclusively this characteristic effect of the retardation terms. The effect is expected to become more pronounced at higher photon energies, but data at these energies and small angles are not, yet available. Another example of the importance of the retardation terms is in the analysis of the low-energy charged pion photoproduction from neutrons and protons to derive the threshold value of the ?r/?r+ production ratio (31). Here the situation is still not completely clear, despite an excellent series of measurements, owing to the possible importance of other small terms neglected in the above analysis (32). In the photoproduction of neutral pions, the analysis given above indicates two effects which are of considerable interest. One is the small s-wave production, especially in y + P --+ P + 7r”, arising from both the dipole moment in the final state and “charge-exchange” scattering of photoproduced ?r+. Since 1 { 1’ is negligible for this reaction-see Table II-this effect manifests itself mainly
210
FELD
0.2-
g -Q5-
I 200
-
I 250 Ey(lab)
= A+BcosB*Ccas2
I 300 in Mev
B I 350
400
FIG. 2. Comparison of observed asymmetry coefficient B/A in y + P --) P + POwith the predictions of Eq. (61). The data are from: (X)-M.I.T. (33); (O)-Cal. Tech. andcornell (34). The solid curve takes into account retardation effects by a method outlined in Appendix II. The broken curve neglects the retardation correction (i.e., is a plot of Bo/Ao). A decrease in the magnitude of & , Table II, shifts the theoretical curves to the right.
as a small asymmetric
term, in the usual description
of the angular distributions
da(7r0)= A + B cos 0 + C cos20, 0%
arising from s-p interference. Figure 2 shows a comparison between the observed asymmetry (53, 5’4) and the predictions of Eq. (Bl), as a function of the photon energy. Such a comparison is rendered somewhat ambiguous by the fact that, the asymmetry coefficient, B in Eq. (64), contains an angular dependent factor arising from the retardation effects in the last term in Eq. (61). We have included a correction for this effect in Fig. 2. The details of comparison of Eq. (61) with experimental data are discussed in Appendix II. The main point is that the theory clearly reproduces the essential features of the data. In particular, we obtain the change of sign of B/A in the region E, M 250-300 Mev. This effect results from the fact that {,. and pi (Table II) are approximately equal and opposite for the reaction under consideration. Hence, the interference between the electric dipole and the resonant photoproduction vanishes for as3 - 45O. The precise value of photon energy at which B = 0 depends on the exact values of !z-,. and l, ; these may be somewhat different from the values in Table II. The effect, however, is clearly present in the experiments. The other phenomenon, on which the preceding analysis can shed some light, is the variation with energy of the ratio C/A in Eq. (64). For pure Ma production,
MESONS
AND
211
NUCLEONS
I
I
I
I
250
300
350
4co
Ey(lab)
I
in Msv
3. Comparison of observed anisotropy coefficient C/A in y + P -+ P + ?yOwith the predictions of Eq. (61) and Appendix II (heavy solid curve). The dashed curve is Eq. (61) without the electric dipole terms (< = 0). The dot-dashed curve is Eq. (61) without electric quadrupole resonance absorption (p = 0). The value C/A = -0.6 (light line) is for magnetic dipole resonance absorption only (r = p = 0). The data are from: (X)-M.I.T. (SS), averaged; (O)-Cal. Tech. and Cornell (34). FIG.
CIA has the well-known value -0.6. However, the introduction of even small amounts of E3 leads to appreciable changes in the value of C/A.l’ The effects of the retardation, through the third term in Eq. (61), are not very important. Figure 3 shows a comparison of the predicted and observed values of C/A; also shown (broken curves) are the predictions in the absence of the El terms and of the electric quadrupole resonanceabsorption. It would appear from this comparison that we may have overestimated the magnitude of the electric quadrupole contribution. However, the data are not yet sufficiently accurate to draw any firm conclusions. COMPTON
SCATTERING
BY NUCLEONS
A system which undergoes resonant photon absorption must also exhibit a photon scattering resonance. Thus, following the early observations of the resonant nature of the photoproduction of neutral pions from protons, it was recognized (14) that the Compton scattering of photons by protons should be anomalously large in the region of the photoproduction resonance. Effects of this resonant scattering were first established in the observations of Pugh, Gomez, Frisch, and Janes (35), although these observations were confined to an energy region, below the photomeson production threshold, where the anomalous scattering cross section is so small as to be observable only through the effects of its interference with the normal (Thomson) scattering on the anguiar distribution of the scattered photons. Subsequently, Yamagata et al. (36), exI9 In the absence of the s-wave terms, -C/A 1.00forp=l.ForpureE~(p-+m),C/A=+1.00.
increases with increasing p, to the value
212
FELD
tending the observations to ~280 Mev photon energies, have provided a conclusive demonstration of the resonant character of the scattering. On a purely phenomenological basis, the resonant Compton scattering cross section can be deduced from the ratio of the measured photomeson production and meson scattering cross sections. Such a computation has been presented by Austern (13), with results in excellent agreement with the observations of the Illinois group (36). The discussion of the resonant scattering to be presented below does not differ in principle from that of Austern. However, by employing a specific model, we are able to derive an expression for the photon scattering amplitude which can be extrapolated directly into the energy region below the photomeson threshold (~150 Mev) and, thereby, to compute effects of the resonance in this region. Another technique for the computation of photon scattering is by use of dispersion relations (37, 38). This approach relates the photon scattering amplitude to the total photon cross section which, in the energy region under consideration, is dominated by the isobar resonance; accordingly, it yields relations which do not differ in their essential features from those of the phenomenological approach. As in the foregoing, we concentrate our attention on the effects of the isobar resonance although we shall consider, albeit briefly, the other main contributions to the Compton scattering of photons by nucleons at energies up to the isobar resonance energy. CROSS SECTION FOR RESONANT SCATTERING OF PHOTOSS In order to compute the photon resonance scattering cross section, it suffices to compute the relative probability of photon to pion decay from the isobar state, UYY -=--y UyrO
r yI‘,
uu+y fJa+ro
(65)
since the initial stage (excitation of the isobar) is the same in both reactions. The widths, F* and I?7 , have been derived in the previous sections, Eqs. (10) and (22 and 35), respectively; the cross section cr.,,+ is given by Eqs. (29) and (36) for magnetic dipole and electric quadrupole photoproduction, respectively. We consider first the resonant scattering of magnetic dipole photons. Combining the appropriate expressions, all given in the previous sections. gyy
(Ml)
=
xy2 ~
rr2 -
2X,2 r2
u‘,o+,o
)
@a)
MESONS
AND
213
NUCLEONS
It is convenient to express uyy in terms of the classical Thomson cross section for the scattering of photons by protons,
Equation (66a) may be evaluated directly (13) ; alternatively, using the constants derived from the pion scattering and photoproduction data, ii? cd sin
4*1-----.
K2 Wr
a!33
q3
2 1
(68a)
Equation (68a) contains, aside from kinematic factors, the resonant scattering phase shift, (y33. At resonance(cy33 = go”, rl, = 1.63, I&/K = 1.88), uyycM1)/uT Z 11, testifying to the importance of the resonant scattering. A significant feature of Eqs. (68 and 68a), however, is that t,hey lead to a finite resonant scattering cross section even in the region of the photopion production threshold (9 ---f 0), since2’CY.~~ a q3for 71<< 1. Thus, in the region of E, m 150 Mev (k/K m 1, w/w, z l/2, sin G%3/g3 z I/4) we expect uyy’M1)/uT ti I/& While the magnitude of this resonant contribution to Compton scattering is relatively small near the photopion threshold, its effects on the angular distribution of the scattered photons are, as will be shown in the following section, even more striking. These effects will persist to even lower energies.Accordingly, it is important to be able to extrapolate Eq. (68a) below the meson photoproduction threshold. We are able to do this by using the “effective range” expression for the (3,3) phase-shift, Eq. (8), which we rewrite in the form hc w sin cy33 0 Tf w,---=v3 {(]
4 me2 --
- ;>’ :p$)3)y7
(6g)
and which, when substit’uted with the appropriate constants into Eq. (68), gives LMl) u-fy
UT
0.045(k/K)*
I(1 - w/w)~ + (0.11173mc2/~)2) ’
(68b)
The effective range formula, as derived from meson theory (19), is valid in the “unphysical region” of total energy of the meson less than its rest-energy, mc’ (’i.e., below photomeson threshold). Equation (68b) extrapolates smoothly 2oSee footnote 10,
Eq. (Sb)
214
FELD
awl,
’ Kn
200
300 Ertlab)
4co in Mev
500
FIG. 4. Ratio of magnetic resonant Compton scattering to Thomson scattering of pho tons by protons, Eqs. (68). The kinematical factors are computed as outlined in Appendix I. Also plotted is the ratio of the amplitudes, [u~~(M~)/u~]~/~.VT = (8r/3) (e2/Mc2)e = 0.20
microbarn. through the photomeson threshold, the second term in the resonance denominator becoming negative for E, < E, (threshold); see Appendix I. We have evaluated Eq. (68b) as a function of photon energy; the results are shown in Fig. 4. Before proceeding to the angular distributions, we consider and dispose of the scattering resulting from electric quadrupole excitation of the resonance. By the same arguments as used in the preceding, we may write (E2) r (82) u7-t =L* @a) uyro(B2) r7r Hence, using Eqs. (65), (36), (52-54) (@-kc)
Thus, even at resonance, (r-,r(E2)/~rr(Y1)E (0.007)2, and the electric quadrupole
215
MESONS AND NUCLEONS
resonance scattering is neglegible, even in so far as the effects of its amplitude on the angular distribution are concerned. Accordingly, we neglect these effects in all of the following considerations. ANGULAR
DISTRIBUTION
OF SCATTERED PHOTONS
A. Thomson plus Resonant Scattering Only We consider a plane unpolarized electromagnetic wave incident along the positive z-axis on a target of unpolarized protons. We characterize the scattering by two amplitudes: (1) aT , the Thomson scattering amplitude, which describes an electric dipole, spin-independent scattering process; (2) aM , the amplitude for scattering through the excitation of an intermediate state of j = 3/2 by magnetic dipole radiation. Then (taking due account of the relative phases of the electric and magnetic dipole components in the incident plane wave and making use of the appropriate products of Clebsch-Gordan coefficients for the relative weights of the various possible transitions) the differential scattering cross section may be described in terms of the elements of the scattering matrix, Sf P, where the =t refer to the two possible states of polarization of the incident photon and i and j refer to the initial and final proton spin orientations
EL*-+- = -iaTX~,-lE
M + aMXl,-l ,
S-f.++ = iuTXl,lE + xaMXl,lM, S+;- = -iaTX1,-lE
+ j/SaMX1,-lM,
(70)
X 18Al, St-++ = se ??- = ?!I? a ‘+I 3
S*-t+= s+-*- = 0. The X1,,BOrM are the vector spherical harmonics of order one, as defined in Blatt and Weisskopf (5) X I,fl E = Fi
+
Xl,?
=
i J-
X l,ztlM =
e+@(cos 8 80
f
i(e0),
u
d-
3 iG
sin 0 60 , (71)
$- efi’(80 f i cos 0 (eo>, u J-
Xl,oM = i
J
$- sin 0 90 . n-
216
FELD
The photon scattering cross section is
-
3 {(I UT I2 + 1 I UM 12) (1 + cos2 e) 167r
(7%
-!- i 1 UM 1’sin2 0 + i Re aToM* cos 8 , in which the summation is taken over all states of the photon and proton polarization. We have purposely, in Eq. (72), kept separate the sin2 0 term which represents the contribution of proton “spin-flip” scattering; this term would be absent from the coherent elastic scattering by nucleons in complex nuclei (35). The amplitudes UT and oM may be related to the total Thomson and magnetic resonance scattering cross sections, as given in the previous section, by integrating Eq. (72). u YY =
+ uM = 1 aT I2 + 35 ( aM 12.
(724 Furthermore, recognizing that both aT and aM are real and that &T is inherently negative (37) (aT = -.T”‘)) UT
16n du --~=(1+;~)+(1+~~)cos28-R(~)li2cost’, in which the ratio ~~/a~ is given by Eqs. (68) and Fig. 4. Most of the available experimental measurements are of the differential section at B = 90”. In Fig. 5 the observations are compared with
(72b)
cross
@g&i?
(90°) = 1 + ; %” (72~) T from Eq. (72b). The agreement is, on the whole, excellent, although these data seem to suggest a slightly smaller value of the resonance energy than the E,, &Z 340 Mev assumed in these computations, or else an additional contribution to the cross section at intermediate energies. A more interesting and sensitive comparison with experiment would involve the interference term in Eq. (72b). An important effect of this term is to reduce the cross section in the forward direction, an effect already predicted by the dispersion considerations (37). Thus, Eqs. (72) predict for the ratio of forward (e = 0) to 90” scattering
(724
MESONS
AND
NUCLEON.5
217
FIG. 5. The differential photon scattering cross section at 90” compared with experiment. . of Illinois ($6); (A)-Univ. of Illinois, The data are from: (o)-M.I.T. (85); (0)-U mv. unpublished. We are grateful to A. 0. Hanson for communication of the latest Illinois results.
This ratio” has a minimum at uM/uT = 1.25; the value of the ratio at the minimum is 0.176. From Fig. 4, this corresponds to a photon energy E, g 217 Mev. The ratio of forward to 90” scattering, Eq. (72d), is plotted as a function of photon energy in Fig. 6. Unfortunately, for obvious experimental reasons, there are no data available on the forward scattering of photons. We may, however, compare the (rather scanty) available experimental evidence on the angular distributions of the scattered photons with the predictions of Eq. (72b). This comparison is shown in Fig. 7 at two photon energies, 120 and 240 Mev. The agreement is quite reasonable, especially when we consider that the theory outlined above neglects a number of effects which, although relatively unimportant for the total cross section, may be expected to have considerable influence on the angular distributions. B. Other Processes A$ecting the Scattering In the preceding, the non-resonant scattering is assumed to be given by the classical Thomson formula. This should, of course, be replaced by the KleinNishina expression for the scattering of photons by a particle obeying the Dirac equation (39), with further corrections, as derived by Powell and by Gomez and Walecka (40), for the effects of the anomalous nucleon moments. These corrections are quite important for photons in the region of the meson threshold and below and are extensively discussed in the literature (35, 41). 21The differential cross section itself has a minimum value (16*/3u~) (du,,/&) for CM/UT = 0.96. According to Fig. 4, this corresponds to Ey g 209 Mev.
(0”) = 0.4
218
FELD
0.6 -
Ey(Mev) FIG. 6. Predicted ratio of 0” to 90” photon scattering us photon energy. The broken curve
indicates the type of effect to be expected from a simple modification factor”, as discussed in the text.
10.0-
(b)
of or through a “form
240 Mev
0.0 -
6.0 -
0
30
60
90
120
150
100
ecm cdegree?.)
FIG. 7. Angular distributions in Compton scattering of photons by protons compared with theory. (a) E., = 120 Mev: the data are from the M.I.T. group (85); (b) E7 = 240 Mev: the data are from the Illinois group (as), reduced by 20 percent. We have arbitrarily reduced these experimental values in order to compare the shapes, since our (incomplete) theory appears to underestimate the cross sections in this energy region.
MESONS
AND
NUCLEONS
219
In the scattering of x-rays by atomic systems the classical Thomson amplitude is modified by a “form factor”, F(qz), where 2 = (~~)l’~is the root-mean-square radius of the charge distribution and q is proportional to the photon momentum transfer q = 2k sin e/2.
(73)
In adopting an atomic model to describe the nucleons, such form factor effects would be expected in the scattering of high-energy photons or electrons. In the case of electron scattering, these effects have been observed and measured (42) and the atomic model has been shown (1) to be consistent with these observations.22For photon scattering, however, the form factor effects are expected to be more complicated. In particular, they are not, at least not in any simple or obvious fashion, directly derivable from the electron scattering observations.23 Nevertheless, entirely for the purposes of illustration, we consider briefly the possible effects of a simple atomic form factor on the Thomson scattering. Assuming x = 0.80 X lo-l3 cm, and the “exponential” charge distribution used by Hofstadter to fit the electron scattering observations (@), we have aTr = aJ%d
(74)
with F(qs) = (1 + q222/12)-2
(744
qx = 1.11 (AT/K) sin e/2.
(74b)
and The broken curves in Figs. 6 and 7 illustrate the effects of the substitution of aT’for aT in Eqs. (72). However, the most important mesonic effects (even aside from the resonant scattering) are not describable in terms of a simple form factor. Some of the possible mesonic modifications of the Thomson scattering amplitude have been discussed, from a general field-theoretic point of view, by Klein (44). A more specific computation of such effects, using the Chew-Low-Wick approximations to meson field theory, has been reported by Watson, Zachariasen, and Karmas (45), but the details of this computation have not yet been published.24In any 22The scattering of high energy electrons from nucleons differs in a number of crucial respects from the scattering of photons. In particular, effects of the isobar resonance are not significant for electron scattering up to electron energies exceeding 500 Mev (4). 23 These difficulties in describing electric dipole photon scattering are associated with the large electric dipole photoproduction, the real process whose virtual counterpart leads to an enhanced electric dipole photon scattering, and the failures of the atomic model in predieting such processes. We are indebted to M. L. Goldberger for a number of explanations concerning this question. 24 A qualitative indication of the nature of the effects of virtual s-wave pion excitation
220
FELD
event, as may be seen in Figs. 5 and 7, the major features of the nucleon Compton scattering process for photon energies up to the resonance are quite adequately described by the simple theory of the previous section. The elucidation of the importance of other mesonic effects, such as those mentioned in this section, awaits further improvements in the experiments. ISOBAR PROCESSES INVOLVING
EFFECTS
Two
ON OTHER
REACTIONS
NUCLEONS
Other high-energy nuclear reactions, in which the energies available are sufficient to excite nucleons to the isobar state, should exhibit effects of the (3,3) resonance. In particular, reactions involving two nucleons, such as nucleon-nucleon scattering, pion production in two nucleon collisions, pion- and photodisintegration of the deuteron, all show such effects in the appropriate energy ranges. Since these reactions have been extensively discussed in the literature (3, 4, 6, 12, 13, 14, 18), we confine ourselves to a few remarks concerning the behavior of such reactions near the meson-production threshold. In particular, nucleon-nucleon scattering and the photodisintegration of the deuteron involve no real mesons and, in a fashion similar to Compton scattering, should exhibit resonance effects in the region below the meson production threshold. Resonance effects in nucleon-nucleon scattering can be shown (13, 14) to be very small in the energy range under consideration (Enuoleon5 5-600 Mev). In may be obtained from the following considerations: We assume that the anomalous electric dipole scattering involves an intermediate state of j = l/2- (nucleon plus one s-wave pion). Let the amplitude for scattering in this state be es . Then the additional terms which must be added to the S-matrix, Eqs. (70), are S-*-$‘+ = S**l-
=
-i
2
3’3aE&", ?'$~EX1.-lE,
The cross section
contains contributions, both in its magnitude and in the angular distribution, proportional to the amplitude of a3 . For real, positive us we expect a decrease in the cross section (and its value at 99”) if oE < $5 1 er j ; for as > g 1 ar ( the cross section at 90” becomes greater than its value in the absence of as . Such effects are obtained in the meson-theoretic calcula.tions (46).
MESONS
AND
221
NUCLEONS
the reactions r+D
eN+P,
(75a)
?r’+D+N+P,
(75b)
on the other hand, the isobar plays a major role. Assuming t,hat both reactions proceed through the same intermediate state, in which one of the nucleons is excited, we have a(y + D + N + P) = 4r + p + p + To> a(?ro+ P + P + dj * a(?r”+ D + N + P)
(76)
The right-hand side of Eq. (76) has been evaluated in the preceding, Eq. (29) ; to obtain the cross section for reaction (75a), however, we require an expression for the cross section for pion-disintegration of the deut)eron.Here, for real pion momenta, we may use the experimental values (4), as has been done by Austern W). The evaluation of Eq. (76) at photon energies below meson threshold (5150 Mev) requires a means of extrapolating the cross section for reaction (75b) into the nonphysical region; unfortunately, we do not have a meson-theoretic expression in this case. Nevertheless, we may note (12, 14) that for small pion momenta u(n” + D-+N
+ P) sA7,
(77)
which,25on substitution into Eq. (76), gives an expressionfor o(y + D ---f N + P) which is independent of q, This expression should be valid for photon energies both above and below the threshold, provided the condition ( v2 1 < 1 is satisfied.26In the region of the photomeson threshold, substitution of the experimental values into Eqs. (76), (29), and (77) yields a(y + D -+ N + P) N” 5 fib. Since this is only of the order of 10 percent of the observed cross section at F‘Y - 150 Mev, it is necessary to consider other sources of enhancement of the reaction in the threshold region. One such possibility could arise from the photoproduction of an s-wave pion on one nucleon and its reabsorption, before escape, in the two-nucleon system 25The experimentsgive (S), in the threshold region, u(N + P -+ D + ~0) s 0.41$ millibarns for the resonant part of the cross section. From detailed balancing (12), u(+’ + D + N + P) = (4p2/3q2)u(N + P --f D + T”) = %W~c)2(1/~2)u(N
+ P * D + 4,
(77’)
where p and q are, respectively, the proton and pion c.m. momenta for the same (total) cm. energies. Thus, A G 0.55 (p/me)’ millibarns in the threshold region. z6 Below the photoineson threshold, the (imaginary) pion momentum is defined in Appendix I.
222
FELD
(14,46-48). Such s-wave pions are produced mainly by electric dipole absorption. Accordingly, the intermediate-state meson will be charged and the nucleons in the intermediate state are identical. Near threshold, the available energy is small, and the nucleons will be in a relative ‘So state. Reabsorption of the s-wave pion results in a 3Pofinal state.” Austern (49) has demonstrated how a process of this nature leads to cross sections and angular distributions which are in reasonable accord with the observations in the region of the photomeson threshold. THE
SECOND RESONANCE
IN THE PION-NUCLEON
CROSS SECTION
As a final example of a reaction in which the isobar state may play a prominent role we consider the interaction of negative pions with protons in the region of -1 Bev pion laboratory energies. A peak in the total cross section has been interpreted (50) as a resonance in the isotopic spin T = l/2 state, since it is absent in the a+-P cross section. Figure 8 is a reproduction of the analysis of Cool, Piccioni, and Clark (50) of the observations in terms of the cross sections for interaction in the two possible isotopic spin states. It is a general property of reaction theory (5) that, for a resonance in a state of angular momentum j, the peak cross section is ures= 47mj + %hAlutot .
(78)
It is seen from Fig. 8 that the observed peak cross section in u1/2requires, if it is to be interpreted as due to a single resonance involving a single j-value, j 2 6 for U,I/U tot N l/2 (the value observed at ~1 Bev) and, at best, j 2 3 for u,~/ut~~II 1. A suggestion, for accounting for the observed cross section by means of a pion-pion scattering resonance, has been advanced by Piccioni et al. (50), by Dyson and by Takeda (51). This model has the attractive feature that the c.m. wave-length X, appropriate to a pion-pion collision, is sufficiently large to account for the observed peak cross section even for an s-wave pion-pion resonance. Owing to the large momentum spread of the target pions, a consequence of their confinement in the small volume associated with the physical nucleons, a large “doppler” broadening of the resonance is expected. Roughly speaking, for a pion cloud of dimensions x(l/2)fi/mc (49)) the uncertainty principle predicts a momentum spread of wf2mc or an energy spread of .~600 Mev in the c.m. system of the two pions; the contribution to the resonance width in the laboratory system would be - twice this. 27The final two-nucleon state for the resonant photodisintegration, resulting from isobar excitation, is ‘Dz . Since there is no interference between electric and magnetic dipole photodisintegration (they lead to final triplet and singlet states, respectively), another reason must be sought for the observed asymmetry in the angular distribution. This probably arises out of a retardation denominator in the electric disintegration amplitude (14,@9).
MESONS
0
I
0.4
I
I
0.6
I
AND
I
0.8 El(lab)
223
NUCLEONS
I
I
1.0
I
I
1.2
I
I
I.4
I
,
in 8ev
FIG. 8. The total cross sections u(,+ + P) = 4312, u(?r- + P) = $&I/Z + >~us/z, and cI,2 vs pion lab energy, as deducted by Cool, Piccioni, and Clark (60) from the experimental data.
Another possibility, has been suggested by the author and by Lindenbaum and Sternheimer (15). According to this suggestion, the process responsible for the resonance involves the inelastic excitation of the nucleon isobar by the incident pion. Lindenbaum and Sternheimer have derived the threshold crosssection behavior of such an inelastic isobar excitation process, and are able to reproduce the rapid rise on the low-energy side of the resonance. In addition, they have shown that the observations on the charged pion energy spectra at 1.0 and 1.37 Bev are consistent with the reaction. ?r+P-+7r+xn*--tN+?r-+?r+.
(79)
The following considerations are concerned with a crude, qualitative attempt to fit the shape of the observed cross-section by a model which assumesthat a T = l/2 resonanceis exhibited by those processesin which the isobar (excited by the incident pion) and the inelastically scattered pion are in an orbital p-state. There are three such states, corresponding to the possible combinations of the isobar spin, j = 3/2+, and the orbital angular momentum of the scattered pion, C’ = 1; i.e., j = l/2+, 3/2+, and 5/2+. The first two can be excited by incident pions of orbital angular momentum C = 1 only; the last requires 4 = 3. If all three states exhibit a resonance at roughly the same energy, the sum Z(j + l/2) = 6 is sufficient to account for the peak cross section, Eq. (7S), even with uJutot II l/2. We apply resonance theory, in its simplest form (5), to each of these reso-
224
FELD
nances. The total cross sect.ion is
I‘,r 2 > 4(-Y - -YrY + IQ'
utot (') = 47&.2 j + I
(
(80)
in which 71is the pion cm. energy, I?, is the width corresponding to the emission (absorptioin) of the incident pion (elastic scattering), and 17 is the total width, r = r, t r*e ) where I‘,! is the width for emission of a lower energy pion, leaving the nucleon in the (3,3) state (inelastic scattering). We make the further ,simplifying assumption that inelastic scattering always excites the isobar to the (same) energy corresponding to the peak of the isobar resonance (approximately two pion masses above the nucleon ground state). For the pion widths we adopt the simple forms, appropriate to low-energy resonance theory (5) r*
= rowc(d,
r+ = r,+h(d),
@Ia)
@lb)
with I’0 an adjustable constant and vc(7), ~~~(7’) the angular momentum barrier penetration factors for the elastically and inelastically scattered pions, respectively,28 211(x) = 22/(l
uz(z) = &(225
+ &
+ 45x2 + 6x4 + z").
(824
(82b)
(For all three resonances, I’ = 1; forj = l/2 and3/2,4 = 1; forj = 5/2, P = 3.) We are left with two parameters, Y,. and I?o , which we may attempt to adjust to fit the observations. Figure 9 illustrates one such attempt, in which we have assumed E, = 0.95 Bev (pion kinetic energy in the laboratory system) and To = 0.12 Bev. Also shown in Fig. 9 is the computed ratio
For each j-value
(83) The observed ratio at ~1 Bev is -0.5 (50). the fit to the experimental data Considering the crudeness of the “theory”, may be regarded as quite reasonable. Actually, there is considerable leeway in the interpretation of the experiments since it is not necessary to assign all of the cross section, CQ,~in Fig. 8, to the resonance. Thus, an alternative analysis by Cool et al. (50), reproduced in Fig. 10, attributes an appreciable fraction of the scattering to nonresonant interactions. In this case, both the width and the height of the T = l/2 resonance are reduced considerably, and the demands on 28These expressions assume that the range of the interactions is ~-1 = h/me.
MESONS
225
NUCLEONS
AND
1.0
‘\ ‘\ 80-
“ ‘1 .‘-..
-.-_
, ---
.-.__
$1;
__ ---.-.-.-.-.-
60 c
4~‘-~~,~,-------,
-----------
I
I
0.6
I
I
0.8
I
I
1.0
I
I
I
1.4
1.2
FIG. 9. Comparison of experimental ~~12with the predictions of the simple resonance model described in the text. The separate curves labeled u(j) are the contributions to the total cross section of resonances with total angular momentum j = l/Z, 3/2, and 5/2. The curve u is the sum, c = &u(j). The monotonically decreasing, dot-dashed curve shows the ratio u,r/q,,t = ~o,r(j)/&~t(j) predicted on the basis of the simple theory; the scale for the ratio is on the right side of the figure.
0.4 Edob)
0.8 in 8ev
1.2
1.6
FIG. 10. Alternative analysis of the pion-proton total cross section data (60) in which a nonresonant contribution is subtracted from the observed total cross section curves. The resulting resonances are narrower, more symmetrical, and of smaller peak cross section. They present less difficulty in “fitting” than does the curve labeled mr,2 in Figs. 8 and 9.
226
FELD
TABLE III RELATIVE FINAL STATE PROBABILITIES IN THE T = 35 RESONANT INTERACTION OF T- AND PROTONS PKOWSS
Final
Elastic scattering
Inelastic
scatteringo
All (50% each)
state
Relative
probability
NT0
1
Pn-
2
NTOTO NKfTN*-?r”
4 (1) (0) :, 10 (2) (2) )
P&aP*-*O
;
N
P 7r+ 79 7r-
i
4 (0) (1) 5 4 1 1.8
2.6
(1The order of pions is as in Eq. (84b).
the theory are relaxed to a large extent. Furthermore, since there is no experimental evidence avaiIabIe concerning the relative contributions of the resonant and nonresonant interactions to the inelastic scattering, we are free to choose the intrinsic widths, I’,, in Eqs. (81), different for the elastic and inelastic scattering. Under these circumstances, it is even possible to assign the resonance to one angular momentum, j = 5/2, or to a combination of any two. Clearly, the clarification of this situation awaits further experimental information. In addition to the energy distribution of the pions resulting from reaction (79), discussed by Lindenbaum and Sternheimer (lb’), the interpretation given above may be utilized to yield predictions concerning the charge distribution in the various possible final states resulting from the resonant a- + P interaction, since the isotopic spins are specified in all stages of the interaction.2s These predictions, which follow from a straightforward application of the principle of charge independence (isotopic spin conservation) to the reactions
are summarized
T-+P+%+n,
(844
lr- + P ---f 3x* + lrl -+ 3-z + 7rn + d,
@4b)
in Table III,
in which we list separately the results for elastic
29We disregard, in this analysis, possible charge-exchange processes in the intermediate stages. The validity of such an analysis requires, especially, a negligible pion-pion interaction.
MESONS
AND
NUCLEONS
227
scattering, inelastic scattering, and a total based on the assumption of equal elastic and inelastic contributions to the resonance. Also shown in the last column, in parentheses, are the predictions of the pion-pion scattering resonance theories of Dyson and Takeda, respectively (51). It is also possible to compute the angular distributions corresponding to the various contributions to the resonance, by straightforward application of reaction theory. However, owing to the complications arising from the mixture of states assumed in the above interpretation, as well as from the motion of the isobar in the c.m. system, the results are not expressible in any simple form. In view of the tentative nature of the interpretation and the lack of experimental data, such an analysis does not seem particularly fruitful at this stage. DISCUSSION
The examples discussed in the preceding serve to demonstrate the importance of the nucleon isobar resonance for high energy interactions involving mesons (both real and virtual) and nucleons. We have, we believe, shown how such resonant processes may be treated in terms of a relatively simple model describing the structure of physical nucleons. Such a treatment is not a substitute for a proper field theoretical approach to these problems; rather, it should be regarded as a means of distilling, from the field theory, those features which, as a consequence of general invariance and conservation principles, must result from any reasonable approach to the meson-nucleon interactions. In addition, it appears to us that the approach adopted in this paper is a useful tool for elucidating the physical nature of the major processes involved in the interactions of pions and nucleons. Another important use of the more-or-less phenomenological approach adopted in the foregoing is that it lays the groundwork for educated speculations concerning the nature and form of processes for whose description the field-theoretic treatment may be prohibitively complicated. Two such examples have been discussed in the last section3’ These discussions have, however, not exhausted the possible applications of this approach. Thus, for example, it would be of 3* With respect to the last example, the 1’ = l/2 pion scattering resonance, it is possible that our analysis, if it should prove tenable, is of somewhat broader interest than the immediate problem to which it is applied. We have in mind the question of the parity of the K-meson. The reaction, ?r + P + A0 + K”, whose threshold falls inside this resonance, appears to have a cross section which, following an initial steep rise from the threshold, assumes the shape of the ?r- + P cross section (61). Consequently, a determination of the energy dependence near threshold and of the angular distribution associated with the resonant part of the reaction will yield the relative intrinsic parity of the ~0 - KO system, provided that the parity of the resonant ?r- - P state is known. Our model for this resonance assumes positive parity for the resonant states. Thus, the A” - KO intrinsic parity, on this assumption, is (-l)‘s, where es is the orbital angular momentum associated with the resonant ho - KO production process.
228
FELD
interest to consider some aspects of the nucleon-nucleon interaction, for which the fieId-theoretic treatment appears to yield a static potential in reasonable agreement with observations on low-energy nucleon nucleon interactions (5~). Using the atomic model, the nucleon-nucleon interaction might be treated in a fashion similar to the Heitler-London treatment of the interactions between two hydrogen atoms. In particular, it would be of great interest to apply such techniques to the interaction between moving nucleons, in an attempt to provide a somewhat more quantitative basis for the speculations of Frisch (54) concerning the possible effects of virtual isobar excitation in giving rise to the spin-orbit force between nucleons. Such computations are clearly of a higher order of difficulty than those undertaken in this paper, but they may nevertheless prove to be more tractable than the corresponding field-theoretic calculations. APPENDIX KINEMATICS
I
OF PHOTOMESON PRODUCTION
The following conventions are used throughout this paper: All energies are in units of mc” = 137 Mev; all momenta in units of mc; all lengths in units of K-I 3 &/mc = 1.44 X lo-l3 cm; all masses in units of the pion mass, m. Unless otherwise specified, all expressions apply to the baryocentric system, c.m. The symbols have the following meanings: E,,, = photon or pion kinetic energy, k, 4 = photon or pion wave-number (momentum/@, = pion momentum, rl = total pion energy (-y’ = 1 + q2; E, = y - l), Y = UT/C = q/r, B w = total c.m. energy minus the nucleon rest-energy E y + q2/2M in a nonrelativistic approximation for the recoil nucleon. We review briefly the relativistic kinematics of photomeson production: Let E = the incident phot,on (laboratory) energy, P = E = the incident photon momentum, U = E + M = the total laboratory energy (M is the mass of the target). Then, the Lorenta transformation to the c.m. system is uo = (1 - po”)-1’2(u - POE),
(Al)
PO = (1 - po2)-1’2(E - p&Q = 0,
W)
yielding PO = E/U, U. = (U” - E2)1’2 = (M2 + 2ME)“‘.
b43) (A4)
We define w=Uo-M
(A5)
229
MESONS AND NUCLEONS
and obtain w/M
= (1 + 2E/M)l”
-
Lw
1.
Now from y = (1 + ?#‘2 = w -
v2/2M,
(A7)
in which the recoil nucleon has been treated nonrelativistically, y/M Finally,
= [(l + 2w/M + M-2)1’2 -
k = the photon c.m. momentum k = [(l - Po),‘(l
+
we obtain
I].
W9
in units of mc,
/30)]“2E = E(1 + 2E/M)--l”.
(A%
We note, from Eq. (A8), that below threshold (w < I), y < 1 and, (A7), $ < 0; i.e., q becomes imaginary, q6 becomes negative [see Eq. We assume, for the resonance energy, E,, (lab) s 190 Mev. For c.m. energy, this corresponds to E,, (lab) s 340 Mev. Application of expressions yields
from Eq. (68b)]. the same the above
k, = 1.88, wr = 2.11, -yr = 1.92, q,. = 1.63. APPENDIX
INTERPRETATION
OF no-PHOTOPRODUCTION
II
EXPERIMENTS
For ?y* photoproduction, Table II indicates that we may neglect the 1 l (* term in Eq. (61), and write
8, (AlO) ,I +cocos2
--1 du uo dO
p2 sin” 0 2(1 - pcose)
where AO
= R2P +
Bo = -2R(l
(1
+
,0)~/21+
R(1
+
P)I~C~+~cos
- P>(Sr COS a33 + li sin a33),
Co = -A.,, + R%
~33+
bi sin
ad,
(All)
- p)“.
However, most of the experiments are confined to a limited angular range, around 90”, and are of insufficient accuracy to detect effects of the retardation factor which multiplies Bo in Eq. (AlO). Accordingly, such experiments can be
230
FELD
and have been interpreted
by fitting
da
s
to an expression
= A + B cos 19+ C cos”8.
L412)
In order to compare the experimental values of the coefficients with the predictions of Eqs. (AlO-11), we make the arbitrary assumption that an average experiment consists of measurements of du/dQ at three angles: 45”, 90°, 135’. In terms of these measurements, the coefficients become _B = da(45’) A 4 C -= A
- du(135’) du(90”) ’
(Al3)
da(45’) + do(135“) - 2 du(90’) da&W)
(A14)
Using Eqs. (A13-14) to define the experimental expression, Eq. (AlO), gives
B-=A
coefficients,
the theoretical
B” { 1 - P2/4(l - @?a I,
Ao
C Co -=--A Ao
Bo 3 P /4(1 - p”l2).
Ao
It is these ratios which have been plotted under various assumptions in Figs. 2 and 3. Also shown, in Fig. 2, is the uncorrected asymmetry ratio Bo/Ao ; it is seen that the corrections to B/A, due to the retardation effects, become appreciable only for photon energies appreciably greater than ~300 Mev. The same is true for the anisotropy coefficient, C/A. RECEIVED:
March 4, 1958 REFERENCES
1. B. T. FELD, Annals of Physics 1, 58 (1957). 2. K. A. BRUECKNER, Phys. Rev. 66, 106 (1952). 3. K. A. BRUECKNER AND K. M. WATSON, Phys. Rev. 86,923 (1952). 4. “Pion Physics,” A. Citron, G. von Dardel, B. d’ Espagnat, Y. Goldschmidt-Clermont, C. Peyrou, eds., CERN Symposium, 1956, Vol. 2. CERN, Geneva, 1956. 6. .J. M. BLATT AND V. F. WEISSKOPF, “Theoretical Nuclear Physics,” Chapter VIII. Wiley, New York, 1952. 6. E. FERMI, Nuovo cimento 2, Suppl. 1, 17 (1955). 7. M. GELL-MANN AND K. M. WATSON, in “Annual Review of Nuclear Science,” Vol. 4. Annual Reviews, Stanford, 1954. 8. G. WENTZEL, “The Quantum Theory of Fields.” Interscience, New York, 1949. 9. K.M. WATSON, J.C. KECK, A.V. TOLLESTRUP,AND R. L. WALKER, Phys.Rev. 101,1159 (1956).
MESONS
AND
NUCLEONS
231
B. T. FELD, Phys. Rev. 89, 330 (1953). B. T. FELD, Nuovo cimento 2, Suppl. 1, 139 (1955). A. H. ROSENFELD, Phys. Rev. 98, 139 (1954). N. AUSTERN, Phys. Rev. 87,208(A) (1952); 100, 1522 (1955). B. T. FELD, Phys. Rev. 99, 342(A) (1953); 91, 454(A) (1953); Nuovo cimento 2, Suppl. 1, 145 (1955). 16. B. T. FELD, in “Proceedings of the Sixth Annual Rochester Conference on High-Energy Physics,” Chapter IV, p. 11. Interscience, New York, 1956; S. J. LINDENBAUM AND R. M. STERNHEIMER, Phys. Rev. 166, 1107 (1957); 199, 1723 (1958). 16. D. C. PEASLEE, Phys. Rev. 94, 1085 (1954); 96, 1580 (1954); S. J. LINDENBAUM AND R. M. STERNHEIMER, Phys. Rev. 106, 1874 (1957). 17. G. F. CHEW, Phys. Rev. 89, 591 (1953); 96, 1669 (1954).
i0. 11. 18. lb. 1.4.
18. G. C. WICK, Revs. Modern Phys. 27, 339 (1955). 19. G. F. CHEW AND F. E. Low, Phys. Rev. 101, 1570 (1956). 20. J. OREAR, Nuovo cimento 4, 856, (1956). 21. H. L. ANDERSON, in “Proceedings of the Sixth Annual Rochester Conference on High
Energy Physics.” Interscience, New York, 1956. 2%‘.G. F. CHEW AND F. E. Low, Phys. Rev. 101, 1579 (1956). 23. G. F. CHEW AND M. L. GOLDBERGER, F. E. Low, AND Y. NAMBU, Phys. Rev. X%,1337, 1345 (1957). 94. B. T. FELD, Nuovo cimento 12, 425 (1954). $5. L. J. KOESTER AND F. E. MILLS, Phys. Rev. 106.1900 (1957). 26. M. J. MORAVCSIK, Phys. Rev. 104, 1451 (1956); 107,600 (1957). $7. N. JARMIE, G. W. REPP, AND R. S. WHITE, Phys. Rev. 91, 1023 (1953); E. KNAPP, W. IMHOF, R. W. KENNEY, AND V. PEREZ-MENDEZ, Phys. Rev. 107,323 (1957). 28. T. L. JENKINS, D. LUCKEY, T. R. PALFREY, AND R. R. WILSON, Phys. Rev. 96,179 (1954). 29. R. L. WALKER, J. G. TEASDALE, V. Z. PETERSON, AND J. I. VETTE, Phys. Rev. 99, 210 (1955). 30. J. H. MALMBERGAND C. S. ROBINSON, Phys. Rev. 109,158 (1958). 31. M. J. MORAVCSIK, Phys. Rev. 106, 267 (1957). z@. M. BENEVENTANO, G. BERNARDINI, D. CARLSON-LEE, G. STOPPINI, AND L. TAU, Nuovo cimento 4, 323 (1956). $7. Y. GOLDSCHMIDT-CLERMONT, L. S. OSBORNE, AND M. B. SCOTT, Phys. Rev. 97,188 (1955). 34. W. S. MCDONALD, V. Z. PETERSON, AND D. R. CORSON, Phys. Rev. 107,577 (1957). 36. G. E. PUGH, R. GOMEZ, D. H. FRISCH, AND G. S. JANES, Phys. Rev. 106,982 (1957). 36. T. YAMAGATA, L. B. AUERBACW, G. BERNARDINI, I. FILOSOFO, A. 0. HANSON, AND A. C. ODIAN, Bull. Am. Phys. Sot. Ser. ZZ, 1, 350 (1956). 87. M. GELL-MANN, M. L. GOLDBERGER, AND W. THIRRING, Phys. Rev. 96, 1612 (1954). 58. R. H. CAPPS, Phys. Rev. 168, 1032 (1957). 39. See W. HEITLER, “The Quantum Theory of Radiation.” Oxford Univ. Press, London and New York, 1936. 40. J. L. POWELL, Phys. Rev. 76,32 (1949); R. G O M E Z AND D. WALECKA, Phys. Rev. 104.1479 (1956). 41. C. L. OXLEY AND V. L. TELEGDI, Phys. Rev. 100,435 (1955); also in “Proceedings of the Seventh Annual Rochester Conference on High-Energy Physics.” Interscience, New York, 1957; also, C. L. OXLEY, Phys. Rev., in press (1958). 42. R. HOFSTADTER, Revs. Modern Phys. 28, 214 (1956). 4.3. S. D. DRELL AND M. A. RUDERMAN, Phys. Rev. 106, 561 (1957). 44. A. KLEIN, Phys. Rev. 99, 998 (1955).
232
FELD
46. W. K. R. WATSON, F. ZACHARIASEN, AND W. J. KARZAS, Bull. Am. Phys. Sot. Ser. II, 1, 383 (1956). .@. F. ZACHARIASEN, Phys. Rev. 101,371 (1956). Q. E. A. WHALIN, B. D. SCHRIEVER, AND A. 0. HANSON, Phys. Rev. 101, 377 (1956). 48. R. R. WILSON, Phys. Rev. 104, 218 (1956). .@. N. AUSTERN, Phys. Rev. 108, 973 (1957). 60. R. COOL, 0. PICCIONI, AND D. CLARK, Phys. Rev. 103, 1082 (1956). 51. F. J. DYSON, Phys. Rev. 99, 1037 (1955); G. TAKADA, Phys. Rev. 100,440 (1955). 6%‘. F. S. CRAWFORD, JR., M. CRESTI, M. L. GOOD, K. GOTTSTEIN, E. M. LYMAN, F. T. SOLMITZ, M. L. STEVENSON, AND H. K. TICHO, Bull. Am. Phys. Sot. Ser. ZZ, 3, 25 (1958). 55. S. GARTENHAUS, Phys. Rev. 100, 900 (1955). 64. D. H. FRISCH, Phys. Rev. 101, 1833 (1956).
OF PHYSICS:
4, 189-232 (19%)
Mesons
and
the Structure
Part II. The Nucleon
Isobar
BERN.~RD
Physics Department
of Nucleons
and Pion Dynamics*
T. FELD
and Laboratory for Nuclear Science, Massachusetts of Technology, Cambridge, Massachusetts
Institute
An “atomic” model of the physical nucleons, previously applied to the nucleon ground states, is extended to describe the excited isobar nucleon state, of angular momentum and isotopic spin 3/2. The model is used to compute the cross sections for resonant photoproduction of pions and Compton scattering of photons on protons. Other, nonresonant processes are ta$en into account in a phenomenological fashion, and the computed cross sections for the aforementioned processes are compared with available experimental data. The model is also used as a guide for the phenomenological interpretation of other high-energy processes-in particular, photodisintegration of the deuteron in the region of the photomeson threshold, and the resonant K - P interaction at ~1 Bev. INTRODUCTION
In an earlier communication (1)’ a model was developed in which the physical nucleons (proton and neutron) were depicted as bound systems of a nucleon-like core and a single pion. The main features of the physical nucleons were seen to be determined by the condition bhat the core (spin and isotopic spin l/2) and the pion (spin 0, isotopic spin 1) are bound in a pllz state of total isotopic spin T = l/2. In considering, in I, the field-theoretic basis for this model, it was observed that the pion-nucleon forces in Dhe p3/2 state with T = 3/2 are also expected to be attractive, although less strongly than in the ground state. The existence of such an attraction is manifest in the well-known resonance phenomena associated with p-wave pion scattering and production in the T = J = 3/2 state. Indeed, these resonance phenomena are so pronounced as to suggest the existence of a quasi-stationary or virtual state of bhe nucleon, the so-called2 (3,3) or ‘5sobar” state. * This work was supported in part by the joint program of the Office of Naval Research and the U. S. Atomic Energy Commission. I, 1 Henceforth referred to as I. . /,I‘ 2 It is customary, in this field, to characterize the various states of a single nucleon and 189
190
FELD
The isobar hypothesis was first suggested by Brueckner (2) on the basis of fragmentary evidence on pion-nucleon scattering. Subsequently, careful measurements by Anderson, Fermi, and co-workers and by other groups3 on the cross sections and angular distributions of the various experimentally accessible pionnucleon scattering processes have lent weight to Brueckner’s suggestion through relatively accurate determinations of the parameters (energy dependence of the 3,3 scattering phase-shift, (~33, and the value of the resonance energy) which define the properties of the scattering in the resonant state. These experiments have also shown that the phase-shifts corresponding to scattering in the other three p-states ((~$1, (~13, (~11)are small, at least for pion energies below the resonant energy, NlSO-Mev pion kinetic energy (in the laboratory system); the s-wave scattering phase shifts are also relatively small, although non-negligible, in both isotopic spin states, while the d-wave phase shifts are too small to detect,, with the presently available experimental accuracy, in the region of and below the resonance. An important consequence of the pion-nucleon scattering measurements has been the demonstration of charge independence (isotopic spin conservation) in the pion-nucleon interaction. The concept of the isobar resonance, together with charge independence, was first applied to the interpretation of the other “fundamental” pion-nucleon reactions-photoproduction and production in nucleon-nucleon collisions-by Brueckner and Watson (3). These ideas have proved exceedingly fruitful for the interpretation and analysis of these reactions (4). In the case of the photoproduction of neutral pions from nucleons t’he (3,:~) resonance completely dominates. Two types of phenomenological approach have been applied in the interpretation of photopion phenomena involving the resonance: Brueckner and Watson (3) describe the resonance by a Breit-Wigner formula, adopting the same formalism to the description of this resonance as is conventionally used in low-energy nuclear physics (6). Fermi (6) and Gell-Mann and Watson (7) assume that the first step in the reaction, the absorption of the photon by the nucleon, can be deany number of pions (real and virtual) by the symbol (2T, 24, where T is the total isotopic spin and J the total angular momentum of the system. Whenever there is any possible ambiguity concerning the parity of the state concerned, this is indicated by a + or - superscript on 2.J. Thus the state of a proton and a single s-nave positive pion is (3,1-) while for a p,,~ pion, the state would be (3,1+) ; i.e., owing to the negative intrinsic parity of the (pseudoscalar) pion, the parity of a state with one nucleon and a single pion is (-l)C+l. 3 No attempt will be made, in this paper, at a complete documentation of the volumenous experimental results upon which the information concerning the isobar state is based. Instead, the reader is referred to excellent summaries presented at the 19.56 CERN symposium on pion physics, as well as to the papers representing the most recent results (9). In particular, attention is called to the reviews of pion-nucleon scattering by L. C. L. Yuan, of photopion production on nucleons by G. Bernardini, and of the production of mesons in nucleonnucleon collisions by A. W. Merrison, not only for excellent reviews of the present status of these fields but also for the complete sets of references to previous work contained therein.
MESONS
AND
NUCLEONS
191
scribed by the conventional field-theoretic equations in a perturbation approximation (8), but that the reaction in the (3,3) state is enhanced by the strong pion-nucleon interaction in the final state; the “enhancement factor” for the final state wave function is then computed by straightforward quantum mechanical techniques, utilizing the properties of the pion-nucleon system in the (3,3) state as observed in the scattering process. In the photoproduction of charged pions from nucleons another process, in addition to the excitation of the resonance, is important. This is the photoelectric effect4 in which, owing to the pseudoscalar nature of the pion, the pion emerges mainly in an s-state for photon energies not far above threshold. This is a process for which the field-theoretic perturbation calculations (8) appear to offer an adequate description. A number of phenomenological analyses, based on the concepts and methods outlined above, have been applied to the data on photomeson production, with fair success. Among the most recent and complete is that of Watson et al. (9). However, some questions remain unsettled, both from the experimental and theoretical points of view. One is the nature of photomeson production in states other than those accounted for by the major effects mentioned above. Another concerns the electromagnetic processes involved in the excitation of the isobar resonance: Conservation of angular momentum and parity limits the possible processes responsible for photomeson production in the (3,3) state to magnetic dipole and electric quadrupole photon absorption (10). The experimental evidence indicates that both processes may occur, although magnetic dipole absorption dominates, at least at energies well below the resonance (11). We shall return to these questions in a following section. With regard to other reactions involving nucleons and mesons, we have already alluded to the considerations of Brueckner and Watson (3) on the effects of the isobar on meson production in nucleon-nucleon collisions. They proposed that the interaction proceeds mainly through the excitation of one of the nucleons into the isobar state. The consequences of this mechanism are most easily analyzed for the reaction in which one of the products is a deuteron Application
of conservation
of isotopic spin, angular momentum,
and parity,
4 Qualitatively, at least, it is clear that the photoelectric effect is much more important in charged than neutral pion production; since the nucleon mass is so much greater than that of the pion, the electric dipole moment in the initial (physical nucleon) state will only be appreciable for those configurations of nucleon core and (bound) pion cloud in which the pion carries charge. However, such arguments neglect the finiteness of the nucleon mass and the possibility that a charged pion may, before emerging from the nucleon field, suffer a “charge-exchange” scattering and emerge as a neutral pion. Such effects lead to an observable, albeit small, photoelectric production of neutral mesons which must be considered in the analysis of &production phenomena.
192
FELD
together with the assumption that the two nucleons (one excited) in the intermediate state have zero orbital angular momentum, defines uniquely the twonucleon state as ID2 (3, 6). This model, when applied to reaction (1) and its inverse, accounts for the main features of pion production and absorption in twonucleon systems at energies up to -600 Mev (I 2, 4. TO the extent that the phenomena discussed above can be described in terms of models in which the nucleon isobar is treated as a resonant state, a phenomenological approach is suggested for treating other high energy reactions involving the isobar (IS, IQ). Thus, the same resonance may be regarded as responsible for photon scattering (nucleon Compton effect), for the photodisintegration of the deuteron, and, at least in part, for nucleon-nucleon scattering in the appropriate energy range. The parameters which determine the resonant features of these reactions (the appropriate partial widths) can be derived from the observed properties of the pion scattering, photoproduction, and nucleonnucleon production cross sections. Also at higher energies, both for pion-nucleon (15) and nucleon-nucleon (16) interactions, the isobar resonance appears to play a major role. The phenomenological isobar models, discussed above, derive their justification not only from their success in describing the observations but also on the basis of meson field-theoretic considerations. The major contributions in this direction have resulted from the demonstration by Chew (17) that the (3,3) resonance in pion-nucleon scattering follows from a nonrelativistic, “fixed source” approximation to conventional meson theory, upon application of the (renormalization) techniques developed in the quantum held-theoretic treatment of electrodynamics.5 Of primary interest, in this connection, is the demonstration by Chew and Low that the (3,3) scattering phase shift may be described in terms of an “effective-range” formula (19) and that the photomeson production cross section near threshold is given in terms of the same parameters as determined the properties of the resonant pion scattering (ZO).6 A development of major importance in the interpretation of meson scattering and production is the recent application of “dispersion relations” to these phenomena. Briefly, it has been shown that the requirement of microscopic causality (i.e., the limitation of the speed of propagation of wave phenomena to speeds not exceeding that of light) leads to connections between the real part of the scattering (or production) amplitude and certain integrals involving observable cross sections, analogous to the well-known Kramers-Kronig dispersion relations for the propagation of electromagnetic waves. The application of these relations to pion scattering and photoproduction (21) has recently been summarized by 5 An excellent summary of meson field theory is given by Wick (18). 6 Similar connections can also be shown to hold for Compton scattering private communication).
(F. E. Low,
193
MESONS AND NUCLEONS
Chew, Goldberger, Low, and Nambu. These dispersion relations yield results which do not differ markedly from the field-theoretic expressions at low energies (19, 20) and, consequently, tend to fortify the conclusions of the nonrelativistic, fixed-source approximation to the meson field-theoretic treatment of isobar resonance phenomena. The work reported below represents an extension of the atomic model developed in I to dynamic phenomena involving the nucleon isobar state. This approach lies between the field-theoretic and the purely phenomenological treatments. Thus, although the existence of the virtual state is postulated, its wave function and, in particular, the description of its decay properties are assumed to have the forms dictated by meson field theory: The wave function describing t’he isobar is derived from considerations such as those used in I to describe the nucleon ground states; the decay properties, specifically the decay width, are expressed in terms of the Chew-Low effective range relation for the (3,3) scattering phase shifts, with the parameters chosen to fit the scattering observations. Such considerations serve to specify the properties of the nucleon states between which transitions may be induced by various means. Thus, it is possible, on the basis of this model, to compute directly the matrix elements corresponding to electromagnetic excitation of the isobar. Decay of the isobar by meson emission corresponds to photoproduction. Decay by the emission of electromagnetic radiation (related to photon absorption by argument(s of detailed balancing) corresponds to Compton scattering. One feature of this model is that it provides a mechanism for computing, in an unambiguous fashion, the effects of the isobar at energies considerably removed from the resonance energy. In the case of photon Compton scattering, this perm its us to extend the calculations to energiesbelow the meson-production threshold into the range most extensively covered by the available experimental data. Such considerations also provide a guide for extending the phenomenological computations on the isobar effects in the deuteron photo-disintegration into the energy region below meson-production threshold. DESCRIPTION
WAVE
OF THE
ISOBAR
STATE
FUXCTION
We assume the existence of a quasi-stationary state, in the conventional sense,7of spin 3/2 and isotopic spin 3/2. There are (2J + 1) (2T + 1) = 16 sub-states, whose wave functions we write as %2T8
. 2J3
*
=
%T,
) 2J3
exp ( - B/2@,
(2)
in which the a’s represent the wave functions of the (stationary) isobar sub’ See Blatt
and Weisskopf
(6), pps. 412-417.
194
FELD
states and I’ is the “decay width”, r = fi/ (isobar mean lifetime). Following the procedure of I, 3 9253 = X~T~%J,G(T), in which the x’s represent the four possible charge states of the isobar, @2T
x3
= pa+
Xl
= (2/3)“‘p?r’ + (1/3)“‘7za+
x-1 = (2/3)“2n~o + (1/3)1’2p7rx-3
(3)
(4)
= mr-,
the \E’s the angular momentum states
(5) and G(r) is the appropriate radial wave function of the isobar. The exact form of G(r) is of no great concern in the following considerations, and we make no attempt at deriving it. Its form, for r 5 the range of the pion-nucleon forces, is presumably similar to the radial wave function R(r) for the nucleon ground state, Eq. (7b) of I; the important consideration is that its spatial extension is relatively small, of magnitude a few times the nucleon Compton wavelength. Thus, in treating the absorption of photons, it is not too unreasonable to utilize an approximation in which the photon wavelength is treated as large compared to the nucleon size, even though photons in the resonance region have wavelengths’ x y w fi/2mc = (l/2)?. In computing the matrix elements for electromagnetic transitions between the ground and excited nucleon states, we will be concerned with the off-diagonal matrix elements of the magnetic moment operator and the electric quadrupole operator & =
T,3T2Ygm.
(7) The diagonal elements of these operators in the state JZ = J correspond to the static moments. The results of such a computation of the magnetic moments of 8 The pion Comptonwavelength, K- 1 = h/me S 1.4 X lo-‘3 cm, is used throughout this paper as the characteristic length for pion-nucleon interactions. We also adopt, for the ratio of nucleon to pion mass, M/m = 6.75.
MESONS
AND
195
NUCLEONS
the ground nucleon states have already been given in I (the ground state, with spin l/2, has, of course, no electric quadrupole moment). We may, similarly, compute the magnetic moments of the isobar states, Taking gs = 2(wT/3 + b),
(64
gz = (Mlm)T.a3 ,
(6b)
=t&ia
(6~)
we obtain /.93 =
=k b +
p&l = zt(>4u
f
Jfl4~0,
3b + M/m),,
.
W)
The values of a and b depend on the assumption concerning the intrinsic moments of the nucleon cores: for “Dirac” cores, a = 1, b = l/2; this assumption leads to roughly equal neutron and proton moments in the ground state. It is possible, as shown in I, to obtain the observed nucleon moments by appropriate choice of a and b. However, in view of the approximate nature of such computations, as well as other possible effects (e.g., K-meson states), we shall not attempt such refinements. Assuming the “Dir&c” values of a and b, the static moments of the isobar states are shown in Table I. For the static electric quadrupole moments, defined as the diagonal elements of (16 7r,‘5)1’2& of Eq. (7),
=
=
00 G2(dr4 dr- (XZT~ ,
s0
-%(r2>3*(X2T3,
T,3x2T3
>. $ *‘3*(3
cos” 8 -
l)$
an
(74
7S3XZT,).
Thus, referring to Eqs. (4), we obtain the values shown in Table I. TABLE
I
STATIC MOMENTS OF THE ISOBAR STATES 2T3
r/Pa"
3
7.75
1 -1
2.92 -1.92
-3
-6.75
QIQC
1 -; -1
a PO = eh/2Mc is the nucleon Bohr magneton. b &O = -%(~~)a S -2.5 X IO+ cm2, assuming the same rms radii for the isobar and ground nucleon states, (T~)~E (0.8 X lo-l3 cm)’ GZ (r2)8 .
196 DECAY
FELD
PROPERTIES
We assume that the resonance scattering is described by an effective range formula of the form given by Chew and Low (19) 4.f2 --- q3 cot 3 hc w
cY33 =
1 -
W/WY )
in which r], the pion (c.m.) momentum, w, the total (c.m.) energy minus the nucleon rest-energy, and wr , the resonance energy, are all defined in Appendix I. The experimental values of the parameters in Eq. (8) are9 f”/lic = 0.082 f 0.007, wr E 2.12-2.15, corresponding to a laboratory pion kinetic energy at resonance of -190-200 Mev (3). The connection with the resonance width,” I’ of Eq. (a), is made by way of the usua1 resonance scattering formula (5) tan
~3~
= r/2(w,
- w),
(9)
giving
(10) in units of me”. PHOTOPRODUCTION MAGNETIC
DIPOLE
EXCITATION
OF THE
OF PIONS ISOBAR
Rather than compute directly the photoproduction process, we take advantage of the known isobar scattering properties and compute the ratio
01) 9 U. Haber-Schaim (private communication). 10 For p-wave scattering, resonance theory l- = 2r,(qR)“/[l
See, also, Orear (6) gives
+ (qRY1,
(20).
@a)
and J?Othe “reduced” where g is the pion wave number, R the “range ” of the interaction, width. Anderson and Metropolis (221) have analyzed the pion scattering data to obtain a “best fit” in the form of Eq. (9). They report (8b) kinetic energy), which corresponds to E, = (r,. - 1)mc 2 = 133 Mev (199.Mev pion laboratory R = 0.89 ~-1, ~CJ= 0.34 mc2 = 47 Mev. This value of r. corresponds to a lifetime for the 70 = h/m? = 5 X 10bz4 set, which virtual state of only -3 times the “natural lifetime”, is uncomfortably short for a virtual state. Nevertheless, the pion may be thought of as making a few oscillations, at least, before being re-emitted from the “compound” system. It will be noted that we make no distinction between the total width and the scattering width. In view of the large value of I’, this is clearly a valid approximation.
MESONS
AND
197
NUCLEONS
in which J?ir is the width for the decay from the intermediate state (j) to the final state (f). The sum over initial states (i) is simplified by the fact that a spin-0 incident pion can cause transitions only to states with the same magnetic quantum number, if the z-axis is chosen to lie along the direction of the incident pion. Thus, the sum over initial states contains just two terms, corresponding to the two possible polarizations of the target nucleons and, if the target nucleons are initially unpolarized, these two terms are equal. It suffices, then, to compute just two transition widths, I’ljz, 1,Zand l?1,2,--1/Z, for each mode of de-excitation of the isobar. (We have indicated in the subscripts only the magnetic quantum numbers of the isobar and final nucleon states; the other relevent quantum numbers are understood and, in any event, obvious for any given reaction.) For pion emission, the I’jfcT’are simply the pion width I’, Eq. (lo), multiplied by the appropriate Clebsch-Gordan coefficients connecting the isobar state and the final pion-nucleon state (the nucleon goes from T = J = 3,‘2 to T = J = l/2) (rf) b, 1j2 r1/2, 112(*O)= pp, (124 = ?Q, (Tt) (+I) = ggr rl12, -l/2 b2, -ll2 Wb) = w, (To)~ r(r’) = ? @ ‘= 2r(R*). WC) wjf For magnetic dipole photon emission, we have
rjf(Y) -- g3 I Jfif I21 Y
03)
in which the iUj, are the appropriate matrix elements of the magnetic moment operator p of Eqs. (6), Mjf = F / F=
s0
Xl.r-l,f*
k!x3,i*3.i
dQ,
m R(r)G(r)r2 dr.
(14) (15)
We may writ,e p
=
P+E-
+
/A-E+
+
(16)
P3C0,
in which /&t = wv2(Pz
f G&J,
Ps
(17)
EO= zo
(18)
P3 =
and q = (1/2)l’“(xo f iyo),
198
FELD
are combinations of the unit vectors along the three coordinate axes. The three terms in Eq. (16) lead to the emission of left-circularIy polarized radiation, with rnf = mi + 1, the emission of right-circularly polarized radiation, with mi = mj - 1, and unpolarized radiation, with rn/ = mj , respectively. In general n/l,,
=
CjfPeff
,
(19)
in which the Cjf are the appropriate combinations of CIebsch-Gordan coefficients and pcff may be regarded as an effective magnetic moment governing the transition. Straightforward computation yields I c*1/2.
w3, I c*1/2,11/2 I = d2/3,
I =
*1/z
and hff
=
(l/3)
(2~
+
M/m)Fpo.
(20)
Accordingly
with
4 r-l
=
3n,3
(22)
kkrr2;
and the ratio of cross sections, Eq. (II), is simply I’,/I’ for a’ = a0 and 2I’,/r for 7r’ = ?r+. Now, we are really interested in the cross section for the inverse reaction uy+* , rather than un+y . These are connected by the detailed-balancing condition (6) (23) (The factor 2 arises because of the two possible incident photon polarizations.) Accordingly
KY2r, u’y+r __ = -_. 2~2 r ua-vr, Actually,
1 for
7r’ = nP
{ 2 for 7r’ = a* i .
there are four possible photoproduction
y+P*P+?rO y+N--tN+n'
(24)
reactions to consider:
(25a)
y+P+N+n+ r+N+P+r-.
Wb)
MESONS
AND
199
NUCLEONS
These are all covered by letting r’ = a’, with P = a0 for the first two, and ?r = n” for the second group, It is to be noted that the us-a1are the cross sections for resonant pion scattering, 167r XT2sin’ a33 9
u~~,,o = -
,
lyaO*,O= 2u,++,a.
(27)
Hence uy-mo= 2Uy*&
(28)
with
Substituting Eqs. (10) and (22) for I? and I’? , we obtain
(29) In Eq. (29), (e2/j2) is th e ratio of coupling constants for the electromagnetic and meson fields,
The factor (W/W,)is a slowly increasing function of the incident photon energy, going from z1/2 at threshold to 1 at the resonance. The prediction of the atomic model, Eq. (29), differs from the results of Chew and Low (.B) and of the dispersion approach (W) in only a rat,her minor way; the results of these field-theoretic computations may be obtained from Eq. (29) by the substitution of the difference between the static nucleon moments, pP - kC(N = 4.70 po, for the ‘(effective dynamic moment”, perf’= peff (w/w,)~‘~, of the atomic model. Numerically, this can make an appreciable difference. Thus, assuming the “Dirac” moments for the core nucleons, Eq. (20) yields p,ff = 2.92 Fpo . However, since, as shown in I, the core moments must be assumed considerably different from the Dirac moments in order to account for the observed static moments of the physical nucleons, we may expect the value of Meffto differ considerably from this (rather small) value. The assumption of anomalous core moments increases the value of peff , by approximately a factor 2. Accordingly, we would obtain pefr’M pp - pC(N at resonance for a (not unreasonable) value of the overlap integral, F x 0.8.
200
FELD
This is in accord with the observed value of the peak cross section (9),
ELECTRIC
QUADRUPOLE
a,,,o(max)
% 220-230Nb.l’
EXCITATION
OF THE ISOB,ZR
The excited isobar state, which can decay by pion or magnetic dipole photon emission, as considered in the previous section, has available also t.he emission of electric quadrupole radiation as a possible, competing decay mode. Since the same detailed balancing relationship, Eq. (23), holds for electric quadrupole (J?2) processes as for magnetic dipole (Ml), we may compute the cross section for E2 pion production t,hrough the relationship
for which the denominator has already been computed, Eqs. (21) and (22). The required partial widths are given by the expression12
where Qu is the appropriate the previous case, we write
matrix
element of the E2 operator, Eq. (7). As in (32)
Qij = CuQeii , and obtain, in this case, / C*I,~, *I/Z 1’ = J/15, / C&I/Z, +,2 1’= 6/15,
Qeff= (&)'I2
Combining
(
R(r)G(r)r* dr = (&)1’2
(r2),ff ,
(33)
this result with that of the previous section,
11From our point of view, the important result is the demonstration that the atomic model reproduces, in a simple and straightforward fashion, the essential results of the much more complicated field-theoretic computations. We do not pretend to be able to derive an accurate value of the effective magnetic moment. Nor can we insist on the kinematical factor, w/w~, especially in view of the inability of the atomic model to take into account that feature of the Yukawa interaction which permits the creation and destruction of pions. Accordingly we shall, in subsequent numberical computations, arbitrarily set ~~rf’ = pp - /4Jg= 4.70 /Jo. 12Blatt and Weisskopf (6)) Chapter XII.
MESONS AND NUCLEONS r (EZ) (E2) Y (‘?a = ry(M1) = ___ uy+.r u-f+r
201 (36)
in which k/~ = E,/mc’ (see Appendix I). According to Eq. (36) the relative E2 to Ml photoproduction increases with the square of the photon energy. To estimate the ratio we assume (~~r~),n $Z (K~T’)~F, with (K~?), z g, which was the value required in I to account for the electron scattering experiments. Taking pEl,ff= 4.70 PO z 6 poF, as suggested in the previous section, 2 (W fJY-wr = 0.0019~. l 0.075 ‘ c (36a) ay+*(M1) s tj
( > K
is expected Thus, even in the resonance region (Icr !X 2~) E2 photoproduction to have a negligible effect on the observed cross sections; however, its effects on the angular distributions in neutral photopion production, as will be seen in a following section, can by no means be neglected. PHOTOELECTRIC
PION PRODUCTION
Although, strictly speaking, our atomic model is not competent, for reasons to be discussed below, to deal in a reliable fashion with the other main reaction in photomeson production, the photoelectric effect, it is nevertheless interesting as well as illuminating to consider briefly this process from the point of view of the atomic model. The cross section for the “photodisintegration” of the physical nucleon by electric dipole radiation’* is,
where
&I,~ = e / r YI*\~~*T,~*~ dr3
(38)
is the electric dipole matrix element connecting the initial (nucleon) state and the final state, in which there are a nucleon and a free pion in an s-state. Aside from numerical factors, the magnitude of the matrix element near threshold is I QL
m I =
e (&)(&>“;:
(384
since the initial nucleon wave function extends over a range 4i/mc and the final pion s-wave function, (sin qr)/qr, is essentially constant over the region of the nucleon. Equation (38a) gives, for the electric dipole cross section near
202
FELD
threshold,
(39) Equation (39) contains the required dependence of the cross section on the momentum of the outgoing pion as well as the main dimensional and dimensionless constants which determine its magnitude. To probe into the other features of the process (aside from numerical constants) it is necessary to specify the wave functions to be used in Eq. (38). For the target nucleons, the atomic model, as developed in I, specifies #i as the wave function of the physical proton or neutron. For the final state, if we accept the model on its face value, we would have 9r = xm
~>cphPs(~),
(40)
where x specifies the final (nucleon and pion) charge state, p the polarization of the final nucleon, and 9, the s-wave function of the outgoing free pion. Taking q8(n) = (sin p-)/p leads immediately, in Eq. (38), to the following selection rules : (1) Nonvanishing matrix elements occur only for states in which the final nucleon polarization is opposite to that of the target nucleon; i.e., El photoproduction is a pure “spin-flip” process. (2) Only charged pions are produced--a+ from protons, 6 from neutrons. Furthermore, in this approximation the process is charge-symmetric, leading to equal cross sections for xi and r- production on protons and neutrons, respectively. It is relatively easy to take into account, at least to first order, the deviations from charge symmetry and the To-electric dipole production. In the first place, the electric dipole operator, in Eq. (38), assumes infinite nucleon mass. For finite nucleon mass the dipole moments of the states (pr-), (n?r+), (pro), (MO) are in the ratio (1 + m/M) : 1: (m/M) :O; their squares, which determine the corresponding cross sections, are in the ratio 1.32: 1.0: 0.022:O. Furthermore the possibility of “charge exchange” ?r”-production can be taken into account, in our approximation, by noting that the outgoing s-wave pion wave function, \E,(s) in Eq. (40), should strictly speaking be an eigenfunction of the isotopic spin of the final state
Since (~1 + CQwe note that, despite the choice of a dipole operator which normally leads to pure charged pion production, the actual final state mixture
MESONS
AND
NUCLEONS
203
contains slightly different amounts of the two isotopic spin states and, accordingly, leads to a small ?y”electric dipole photoproduction. The above-mentioned effects will be included in the final expressions for the electric dipole photoproduction amplitudes given below. They, are, of course, well known and have been extensively discussed in the literature (6). However, the major difficultyI with the application of the atomic model lies in the description of that part of the final state pertaining to the nucleon charge. X(X, r) in Eq. (40). In all of the computations described above, the final state consists, perforce, of a nucleon core and a free pion, since the process computed is one in which the physical nucelon is stripped of its pion cloud. However, the final nucleon observed is not bare, but a fully “clothed” physical nucleon. The process of converting a core to a physical nucleon is one which involves meson creation and, accordingly, the quantum nature of the meson field, an aspect which our atomic model chooses to ignore. Accordingly, we cannot expect a cross section computed on the basis of the atomic model to be correct in all of its details (which is the reason why we have not bothered about such minor matters as numerical constants). Nevertheless, as a crude first approximation, we might consider multiplying the cross section of the atomic model, Eq. (39), by the factor (f’/ti), the “renormalized” coupling constant for the meson field of a nucleon, as representing, in some sense, the probability for a bare nucleon to create the meson cloud required to convert it to a physical nucleon. Such a crude approximation cannot provide the correct numerical constants, nor does it give the other, presumably slowly varying, energy dependent factors which arise from the dynamic nature of the meson field equations. Nevertheless, the atomic model, as so modified, contains essentially all of the major features of the meson field-theoretic results as described below. Actually, the computation of the photoelectric pion production, by the use of meson field theory in a perturbation approximation, is well known (8) ; it is one of the few cases where the application of perturbation techniques in the most straightforward fashion yields reliable results. Starting with the Yukawa equation for a pseudoscalar pion14
q 2q? - K2cQ = - 4?rfT(d.v)p(r) 3 K
(42)
the electromagnetic field is introduced by the substitution V -+ V f (ie,&)A, and the pion wave function is written G Q= 90 + 91 where 90 is the static solution 13There are other difficulties inherent in the use of such a model, such as those involving the proper choice of the dipole operator, as discussed by Fermi (6). We believe, however, that these have their source in the same problem as is considered below. 14See I and Appendix I for details of notation and definition.
204
FELD
of Eq. (42) and ~1 has the form of an outgoing wave {F’ exp [i(‘r With these substitutions, Eq. (42) becomes
q 2~l*-&*=
-hc47rief ~~(d.V)p(r) K
=F 2
The first term on the right leads to the aforementioned section,
-
(A.Th*. El production
rt)]].
Wk.) cross
which, aside from the inverted dependence on photon energy, is as computed by the crude model discussed above. The second term in Eq. (42a) takes into account the absorption of higher electric moments and leads to what is known in the atomic photoelectric effect as the retardation term. In the summary which follows, the perturbation fieldtheoretic results will be assumed for the photoelectric cross sections with, however, the slight modifications required to take into account the finite nucleon mass and the charge-exchange scattering, as discussed above. SUMMARY:
CROSS SECTIONS FOR PHOTOPION PRODUCTIOX
In this section, we give expressions for the differential and integral cross sections for the four fundamental photopion production reactions, Eqs. (25). In these expressions, the photoelectric process is described by the results of the perturbation computation, discussed in the preceding section; to the extent of the validity of this approximation, this includes all electric multipole orders of the incident photon beam. For all other processes, on the other hand, we confine ourselves to those terms leading to p-wave pion production.15 In particular these include magnetic dipole and electric quadrupole resonance absorption. It is most convenient to describe the photoproduction process in terms of the elements of the “scattering matrix” S,f, in which the subscripts i and j refer, respectively, to the initial and final nucleon spin states and the superscripts + and - distinguish between the two possible states of polarization of the incident photon.16 Then, for an initially unpolarized photon beam falling on initially unpolarized nucleons, the differential cross section in the c.m. system for photopion production is
au - = f 2 / sji* 12, cm
2
(44)
15Only electric dipole absorption leads to s-wave pions, and this is included in the photoelectric terms. 16One advantage of this description in terms of the S’s is that it enables rapid computation of the effect of initial polarizations, or of the final nucleon polarization (84).
MESONS
AND
205
NUCLEONS
where the summation is to be taken over all values of initial and final photon and nucleon polarizations. With the restrictions stated above, St*-
= -Se+++ = Y sin 0 eTip,
EL;-;-
= -S&+
s-g-
= simi+ = -El
= X sin 0 eTiq,
(45)
+ K cos 0,
Xi+-*- = S-rx+ 11 = E,’ sin2 0 eF2+2
in which El = E[l - g2sin2 O /(K”$ 1k - q j2)],
(464
EI’ = E$/(K* + j k - q I”>,
(46b)
K
= -Ml+Ms-E3,
X
= 2Ma - s((Ma - I&) + Eq(k - q cos o)/(K”
Y
= Ml + x(M,
(47) +
j
k - q I’),
- I%) - Eq(k - q cos 0)/(/f” + j k - q I”).
(48) (49)
E, Ml , M3 , and E3 are the amplitudes, respectively, for electric dipole, magnetic dipole with j = l/2, magnetic dipole with j = 3/2, and electric quadrupole with j = 3/2 absorption. In the factors which multiply E, representing the “retardation” effects, k and q are the photon and pion wave-number vectors respectively, The amplitudes are, of course, not the same for all four of the photoproduction reactions, Eqs. (25). However, there are well-known connections (?‘) according to which the four values of a given amplitude may be expressed in terms of three real parameters, with the phases determined by the (known) pion scattering phase-shifts in the corresponding states. Aj = -aalei”‘i
+ fla3ezn3i,
(Y + P + P + TO)
= dsal’eia’j
+ aageia3i,
(Y + N -+ N + TO)
= flaleia’i
+ fla3eiasi,
(Y + P -+ N + ?r+)
= --al’eia”
+ *aaeiu3j,
(50)
(y + N -+ P + r-).
Thus, to describe the four processeswe need, in Eqs. (45), 4 X 3 = 12 real parameters. For practically all purposes, this number may be reduced considerably, as follows: (1) O f the processesleading to p-wave pions, the predominant effects arise from the isobar excitation, which gives rise to an enhancement of the T = 3/2 terms (us) in Ma and Es . The rest of the amplitudes for p-wave pions are relatively small and their effects are essentially negligible, except possibly on TO-
206
FELD
production near threshold (25). The usual procedure is to compute these small amplitudes from meson theory in the Born approximation (22, 23). We shall, in the following, neglect the nonresonant p-waves and, accordingly, take M, = 0, M3 = tm3ae2a33,
(51)
ES = Ee33ei”“3,
in which the factor [ = 1 or dl/a for ?y”or r* production, respectively, and a38is the resonant scattering phase shift, Eq. (8). The value of mu is obtained from Eqs. (26-29) with by equating ~~~~~~~~~ ~-t-t*0Of’) = &- m3t
(52)
from Eqs. (44-49). Correspondingly, from Eqs. (36) and (W _ 8n Uy-nO - ;;- e332,
we obtain p = e33/m33 = 0.075
(54)
AJ/K.
(2) For the photoelectric amplitudes, E, it is necessary to determine all three parameters, el , c<, and e3. We evaluate these by considering the threshold limit (p -+ 0), at which cyl = LY~-+ 0 and E(g -+ 0) = EO is real. In this limit we have E,,(y + P -+ N + ir+) = t~o~‘~
= (Q(3?yz
z (25 x 2.J
(55a)
[see Eq. (43)]; E,,(r + N --+ N + TO) = 0
(55b)
from the vanishing of the dipole moment in the final state; Eo(-y + N + P + 6)
= (1 + m/M)ao”2
(55c)
from the ratio of the final state dipole moments. These three conditions determine the threshold values ~~(0) = (1 + m/M)(ao/3)“‘, ~(0) = (1 - m/M)(2ao/3)“‘, d(0) = -(l
+ m/M)(2ao/3)1’2,
(56)
MESONS
AND
207
NUCLEONS
from which follows
E&y + P + P + 7r0)= (m/M)(ao/2)'12.
(56’)
Taking m/M = 0.15, the above gives for the ratios of the threshold (s-wave) photopion cross sections,
u&r-):&+):ao(Pn"):ao(Nao)
= 1.32:1:0.011:0.
To obtain the amplitudes as a function of photon energy, it is necessary to know the variation of the e’s with energy. We shall, in the following, assume that the energy dependenceis all in the factor” (q/k) of a0 . Then, for finit)e q, we substitute our e(O)%,Eqs. (56), into Eqs. (50), using Orear’s fits to the experimentally determined scattering phase shifts (5), (Y~= 0.16 r], a3 = -0.11 7. The results, obtained by straightforward algebraic manipulation, are shown in TABLE
II
ELECTRIC DIPOLE AMPLITUDES FOR PWOTOPION PRODUCTION, PIOCESS -y+P-+P+#
r+N-tN+r”
r+P-+N+r+ rfN+P+?r-
rr
0.105 0 1.00 1.15
E = [u~'I~ i-i
-0.127~ -O.l46?j 0.0571 o.om
Table II, in which we give the real and imaginary parts of the coefficients, I = tr + 4-i , in
E = [ull’2,
(57)
for the four reactions. The values in the table are the lowest order terms in an expansion in powers of q = q/K. However, the higher order corrections, even at photon energies well above the resonance, are only a few percent or less. We are now in possession of the necessary factors for the determination, within the limits of our approximations, of the required cross sections through Eqs. (4449). Owing to the complexity of the retardation factors, the general expressions for the cross sections are still quite complicated. However, they may be further simplified by neglecting, in the retardation factors, the nucleon recoil energy (more specifically, the difference between the incident photon and I7 Actually, Chew and Low (%9) have indicated how to take into account, to first order, the nucleon recoil effects. This may be achieved by, first, modifying u&2 by multiplication with the factor (1 - mk/2Mx)/(l + mk/MK) and, second, sqbstituting (mk/Mtc)for (m/M) in Eqs. (53). These modifications, beside introducing a slight alteration in the total cross sections for charged pion production, have very little effect on the predictions up to E7 = 400 Mev. Accordingly, we have not included them.
208
FELD
outgoing pion energies in the cm); in this approximation lc2z q2 + Kz, q/k E (u/c), = p,
(58)
~~+ 1k - q 1’c 2kZ(1 - 0 cos e), and the amplitudes become (all in units of ao”‘) El = [[l - /3’sin’ e/2(1 - 6 cos 19)], E1’ = @/2(1 - p cos 0)) K
= fR(1 - /+P’,
X Y
= ?4{5RI2 + (1 + p)lei”33+ = ?+Cj{{R(l - p)eiort3- c/3},
(59) 031,
with”
Finally, we write out the expression for (1) the differential cross sections 1 clcr z z = ( { j2 [l - @(l - @) sin2 e/2(1 - /3 cos 0)“] + ~2R2{(1 - p)” + [2 + %(l + P)~ - (1 - p)“] sin2 Bj + @(l + p)P(C, cos a33
030 +
Pi
sin
(~33)
sin2
- 2ER(l - p) (lr cos a33+ li sin (YS) 1 -
tJ
8; 1cos
p2 sin’ e 2(1 - p cos e>
(2) the differential cross sections at 90” in the cm. i $ (go”) = 1r 1’[I - ; ~“(1 - D’)] + 2t2R2 [ 1 + ; (1 + d2]
(62)
+ .$(I + P)P(~~ cm (~33+ ti sin 4; (3) The total cross section k.
= I !: 1’[l - (1 - P”>(tanh-’ P - P>/Pl + %tR2{ (1 - ~1”+ 411 +
X/4(1+ ~1~11 + %tRO + P>
+P(lr cos 01~3 -I- bi sin ad + 2gR (1 - P>kr cos a33
tank-‘/j I* Note previous footnotes:
(17) for kinematical
corrections
to (ro ; (11) for doff’.
(63)
MESONS
AND
209
NUCLEONS
24
Y+P--,N+
?r+
Ey( lab) = 265 Mev
01 0
30’
600
ecm.
900
120°
150”
I600
FIG. 1. Angular distribution of R+ photoproduction at E,(lab) = 265 Mev. The solid curve is from Eq. (61). The broken curve is Eq. (61) without the retardation terms. The data are from: (A)-Univ. of California (27); (X)-M.I.T., L. S. Osborne and B. Richter (unpublished); (O)-Cornell (28); (C/)-Cal. Tech (.%‘S). Data from Illinois (SO) at 225 Mev show a similar flat behavior at small angles.
The’ comparison of the above expressions with the experimental observations has been discussed by a number of authors (4). The agreement between theory and observation is excellent in general, especially if due attention is paid to the details of the theory (%?, 23). In particular, the presence of the retardation terms renders the interpretation of angular distribution data rather more complicated than it might otherwise be (266). Unfortunately, the effects of the retardation terms are not easy to observe, as is illustrated in Fig. 1, in which the experimental observations (27-30) on rf-production are compared with Eq. (61) with (solid curve) and without (broken curve) the retardation terms, for 265Mev photons. Although there is some experimental indication of the expected minimum at small angles, the experiments are clearly not, yet good enough to establish conclusively this characteristic effect of the retardation terms. The effect is expected to become more pronounced at higher photon energies, but data at these energies and small angles are not, yet available. Another example of the importance of the retardation terms is in the analysis of the low-energy charged pion photoproduction from neutrons and protons to derive the threshold value of the ?r/?r+ production ratio (31). Here the situation is still not completely clear, despite an excellent series of measurements, owing to the possible importance of other small terms neglected in the above analysis (32). In the photoproduction of neutral pions, the analysis given above indicates two effects which are of considerable interest. One is the small s-wave production, especially in y + P --+ P + 7r”, arising from both the dipole moment in the final state and “charge-exchange” scattering of photoproduced ?r+. Since 1 { 1’ is negligible for this reaction-see Table II-this effect manifests itself mainly
210
FELD
0.2-
g -Q5-
I 200
-
I 250 Ey(lab)
= A+BcosB*Ccas2
I 300 in Mev
B I 350
400
FIG. 2. Comparison of observed asymmetry coefficient B/A in y + P --) P + POwith the predictions of Eq. (61). The data are from: (X)-M.I.T. (33); (O)-Cal. Tech. andcornell (34). The solid curve takes into account retardation effects by a method outlined in Appendix II. The broken curve neglects the retardation correction (i.e., is a plot of Bo/Ao). A decrease in the magnitude of & , Table II, shifts the theoretical curves to the right.
as a small asymmetric
term, in the usual description
of the angular distributions
da(7r0)= A + B cos 0 + C cos20, 0%
arising from s-p interference. Figure 2 shows a comparison between the observed asymmetry (53, 5’4) and the predictions of Eq. (Bl), as a function of the photon energy. Such a comparison is rendered somewhat ambiguous by the fact that, the asymmetry coefficient, B in Eq. (64), contains an angular dependent factor arising from the retardation effects in the last term in Eq. (61). We have included a correction for this effect in Fig. 2. The details of comparison of Eq. (61) with experimental data are discussed in Appendix II. The main point is that the theory clearly reproduces the essential features of the data. In particular, we obtain the change of sign of B/A in the region E, M 250-300 Mev. This effect results from the fact that {,. and pi (Table II) are approximately equal and opposite for the reaction under consideration. Hence, the interference between the electric dipole and the resonant photoproduction vanishes for as3 - 45O. The precise value of photon energy at which B = 0 depends on the exact values of !z-,. and l, ; these may be somewhat different from the values in Table II. The effect, however, is clearly present in the experiments. The other phenomenon, on which the preceding analysis can shed some light, is the variation with energy of the ratio C/A in Eq. (64). For pure Ma production,
MESONS
AND
211
NUCLEONS
I
I
I
I
250
300
350
4co
Ey(lab)
I
in Msv
3. Comparison of observed anisotropy coefficient C/A in y + P -+ P + ?yOwith the predictions of Eq. (61) and Appendix II (heavy solid curve). The dashed curve is Eq. (61) without the electric dipole terms (< = 0). The dot-dashed curve is Eq. (61) without electric quadrupole resonance absorption (p = 0). The value C/A = -0.6 (light line) is for magnetic dipole resonance absorption only (r = p = 0). The data are from: (X)-M.I.T. (SS), averaged; (O)-Cal. Tech. and Cornell (34). FIG.
CIA has the well-known value -0.6. However, the introduction of even small amounts of E3 leads to appreciable changes in the value of C/A.l’ The effects of the retardation, through the third term in Eq. (61), are not very important. Figure 3 shows a comparison of the predicted and observed values of C/A; also shown (broken curves) are the predictions in the absence of the El terms and of the electric quadrupole resonanceabsorption. It would appear from this comparison that we may have overestimated the magnitude of the electric quadrupole contribution. However, the data are not yet sufficiently accurate to draw any firm conclusions. COMPTON
SCATTERING
BY NUCLEONS
A system which undergoes resonant photon absorption must also exhibit a photon scattering resonance. Thus, following the early observations of the resonant nature of the photoproduction of neutral pions from protons, it was recognized (14) that the Compton scattering of photons by protons should be anomalously large in the region of the photoproduction resonance. Effects of this resonant scattering were first established in the observations of Pugh, Gomez, Frisch, and Janes (35), although these observations were confined to an energy region, below the photomeson production threshold, where the anomalous scattering cross section is so small as to be observable only through the effects of its interference with the normal (Thomson) scattering on the anguiar distribution of the scattered photons. Subsequently, Yamagata et al. (36), exI9 In the absence of the s-wave terms, -C/A 1.00forp=l.ForpureE~(p-+m),C/A=+1.00.
increases with increasing p, to the value
212
FELD
tending the observations to ~280 Mev photon energies, have provided a conclusive demonstration of the resonant character of the scattering. On a purely phenomenological basis, the resonant Compton scattering cross section can be deduced from the ratio of the measured photomeson production and meson scattering cross sections. Such a computation has been presented by Austern (13), with results in excellent agreement with the observations of the Illinois group (36). The discussion of the resonant scattering to be presented below does not differ in principle from that of Austern. However, by employing a specific model, we are able to derive an expression for the photon scattering amplitude which can be extrapolated directly into the energy region below the photomeson threshold (~150 Mev) and, thereby, to compute effects of the resonance in this region. Another technique for the computation of photon scattering is by use of dispersion relations (37, 38). This approach relates the photon scattering amplitude to the total photon cross section which, in the energy region under consideration, is dominated by the isobar resonance; accordingly, it yields relations which do not differ in their essential features from those of the phenomenological approach. As in the foregoing, we concentrate our attention on the effects of the isobar resonance although we shall consider, albeit briefly, the other main contributions to the Compton scattering of photons by nucleons at energies up to the isobar resonance energy. CROSS SECTION FOR RESONANT SCATTERING OF PHOTOSS In order to compute the photon resonance scattering cross section, it suffices to compute the relative probability of photon to pion decay from the isobar state, UYY -=--y UyrO
r yI‘,
uu+y fJa+ro
(65)
since the initial stage (excitation of the isobar) is the same in both reactions. The widths, F* and I?7 , have been derived in the previous sections, Eqs. (10) and (22 and 35), respectively; the cross section cr.,,+ is given by Eqs. (29) and (36) for magnetic dipole and electric quadrupole photoproduction, respectively. We consider first the resonant scattering of magnetic dipole photons. Combining the appropriate expressions, all given in the previous sections. gyy
(Ml)
=
xy2 ~
rr2 -
2X,2 r2
u‘,o+,o
)
@a)
MESONS
AND
213
NUCLEONS
It is convenient to express uyy in terms of the classical Thomson cross section for the scattering of photons by protons,
Equation (66a) may be evaluated directly (13) ; alternatively, using the constants derived from the pion scattering and photoproduction data, ii? cd sin
4*1-----.
K2 Wr
a!33
q3
2 1
(68a)
Equation (68a) contains, aside from kinematic factors, the resonant scattering phase shift, (y33. At resonance(cy33 = go”, rl, = 1.63, I&/K = 1.88), uyycM1)/uT Z 11, testifying to the importance of the resonant scattering. A significant feature of Eqs. (68 and 68a), however, is that t,hey lead to a finite resonant scattering cross section even in the region of the photopion production threshold (9 ---f 0), since2’CY.~~ a q3for 71<< 1. Thus, in the region of E, m 150 Mev (k/K m 1, w/w, z l/2, sin G%3/g3 z I/4) we expect uyy’M1)/uT ti I/& While the magnitude of this resonant contribution to Compton scattering is relatively small near the photopion threshold, its effects on the angular distribution of the scattered photons are, as will be shown in the following section, even more striking. These effects will persist to even lower energies.Accordingly, it is important to be able to extrapolate Eq. (68a) below the meson photoproduction threshold. We are able to do this by using the “effective range” expression for the (3,3) phase-shift, Eq. (8), which we rewrite in the form hc w sin cy33 0 Tf w,---=v3 {(]
4 me2 --
- ;>’ :p$)3)y7
(6g)
and which, when substit’uted with the appropriate constants into Eq. (68), gives LMl) u-fy
UT
0.045(k/K)*
I(1 - w/w)~ + (0.11173mc2/~)2) ’
(68b)
The effective range formula, as derived from meson theory (19), is valid in the “unphysical region” of total energy of the meson less than its rest-energy, mc’ (’i.e., below photomeson threshold). Equation (68b) extrapolates smoothly 2oSee footnote 10,
Eq. (Sb)
214
FELD
awl,
’ Kn
200
300 Ertlab)
4co in Mev
500
FIG. 4. Ratio of magnetic resonant Compton scattering to Thomson scattering of pho tons by protons, Eqs. (68). The kinematical factors are computed as outlined in Appendix I. Also plotted is the ratio of the amplitudes, [u~~(M~)/u~]~/~.VT = (8r/3) (e2/Mc2)e = 0.20
microbarn. through the photomeson threshold, the second term in the resonance denominator becoming negative for E, < E, (threshold); see Appendix I. We have evaluated Eq. (68b) as a function of photon energy; the results are shown in Fig. 4. Before proceeding to the angular distributions, we consider and dispose of the scattering resulting from electric quadrupole excitation of the resonance. By the same arguments as used in the preceding, we may write (E2) r (82) u7-t =L* @a) uyro(B2) r7r Hence, using Eqs. (65), (36), (52-54) (@-kc)
Thus, even at resonance, (r-,r(E2)/~rr(Y1)E (0.007)2, and the electric quadrupole
215
MESONS AND NUCLEONS
resonance scattering is neglegible, even in so far as the effects of its amplitude on the angular distribution are concerned. Accordingly, we neglect these effects in all of the following considerations. ANGULAR
DISTRIBUTION
OF SCATTERED PHOTONS
A. Thomson plus Resonant Scattering Only We consider a plane unpolarized electromagnetic wave incident along the positive z-axis on a target of unpolarized protons. We characterize the scattering by two amplitudes: (1) aT , the Thomson scattering amplitude, which describes an electric dipole, spin-independent scattering process; (2) aM , the amplitude for scattering through the excitation of an intermediate state of j = 3/2 by magnetic dipole radiation. Then (taking due account of the relative phases of the electric and magnetic dipole components in the incident plane wave and making use of the appropriate products of Clebsch-Gordan coefficients for the relative weights of the various possible transitions) the differential scattering cross section may be described in terms of the elements of the scattering matrix, Sf P, where the =t refer to the two possible states of polarization of the incident photon and i and j refer to the initial and final proton spin orientations
EL*-+- = -iaTX~,-lE
M + aMXl,-l ,
S-f.++ = iuTXl,lE + xaMXl,lM, S+;- = -iaTX1,-lE
+ j/SaMX1,-lM,
(70)
X 18Al, St-++ = se ??- = ?!I? a ‘+I 3
S*-t+= s+-*- = 0. The X1,,BOrM are the vector spherical harmonics of order one, as defined in Blatt and Weisskopf (5) X I,fl E = Fi
+
Xl,?
=
i J-
X l,ztlM =
e+@(cos 8 80
f
i(e0),
u
d-
3 iG
sin 0 60 , (71)
$- efi’(80 f i cos 0 (eo>, u J-
Xl,oM = i
J
$- sin 0 90 . n-
216
FELD
The photon scattering cross section is
-
3 {(I UT I2 + 1 I UM 12) (1 + cos2 e) 167r
(7%
-!- i 1 UM 1’sin2 0 + i Re aToM* cos 8 , in which the summation is taken over all states of the photon and proton polarization. We have purposely, in Eq. (72), kept separate the sin2 0 term which represents the contribution of proton “spin-flip” scattering; this term would be absent from the coherent elastic scattering by nucleons in complex nuclei (35). The amplitudes UT and oM may be related to the total Thomson and magnetic resonance scattering cross sections, as given in the previous section, by integrating Eq. (72). u YY =
+ uM = 1 aT I2 + 35 ( aM 12.
(724 Furthermore, recognizing that both aT and aM are real and that &T is inherently negative (37) (aT = -.T”‘)) UT
16n du --~=(1+;~)+(1+~~)cos28-R(~)li2cost’, in which the ratio ~~/a~ is given by Eqs. (68) and Fig. 4. Most of the available experimental measurements are of the differential section at B = 90”. In Fig. 5 the observations are compared with
(72b)
cross
@g&i?
(90°) = 1 + ; %” (72~) T from Eq. (72b). The agreement is, on the whole, excellent, although these data seem to suggest a slightly smaller value of the resonance energy than the E,, &Z 340 Mev assumed in these computations, or else an additional contribution to the cross section at intermediate energies. A more interesting and sensitive comparison with experiment would involve the interference term in Eq. (72b). An important effect of this term is to reduce the cross section in the forward direction, an effect already predicted by the dispersion considerations (37). Thus, Eqs. (72) predict for the ratio of forward (e = 0) to 90” scattering
(724
MESONS
AND
NUCLEON.5
217
FIG. 5. The differential photon scattering cross section at 90” compared with experiment. . of Illinois ($6); (A)-Univ. of Illinois, The data are from: (o)-M.I.T. (85); (0)-U mv. unpublished. We are grateful to A. 0. Hanson for communication of the latest Illinois results.
This ratio” has a minimum at uM/uT = 1.25; the value of the ratio at the minimum is 0.176. From Fig. 4, this corresponds to a photon energy E, g 217 Mev. The ratio of forward to 90” scattering, Eq. (72d), is plotted as a function of photon energy in Fig. 6. Unfortunately, for obvious experimental reasons, there are no data available on the forward scattering of photons. We may, however, compare the (rather scanty) available experimental evidence on the angular distributions of the scattered photons with the predictions of Eq. (72b). This comparison is shown in Fig. 7 at two photon energies, 120 and 240 Mev. The agreement is quite reasonable, especially when we consider that the theory outlined above neglects a number of effects which, although relatively unimportant for the total cross section, may be expected to have considerable influence on the angular distributions. B. Other Processes A$ecting the Scattering In the preceding, the non-resonant scattering is assumed to be given by the classical Thomson formula. This should, of course, be replaced by the KleinNishina expression for the scattering of photons by a particle obeying the Dirac equation (39), with further corrections, as derived by Powell and by Gomez and Walecka (40), for the effects of the anomalous nucleon moments. These corrections are quite important for photons in the region of the meson threshold and below and are extensively discussed in the literature (35, 41). 21The differential cross section itself has a minimum value (16*/3u~) (du,,/&) for CM/UT = 0.96. According to Fig. 4, this corresponds to Ey g 209 Mev.
(0”) = 0.4
218
FELD
0.6 -
Ey(Mev) FIG. 6. Predicted ratio of 0” to 90” photon scattering us photon energy. The broken curve
indicates the type of effect to be expected from a simple modification factor”, as discussed in the text.
10.0-
(b)
of or through a “form
240 Mev
0.0 -
6.0 -
0
30
60
90
120
150
100
ecm cdegree?.)
FIG. 7. Angular distributions in Compton scattering of photons by protons compared with theory. (a) E., = 120 Mev: the data are from the M.I.T. group (85); (b) E7 = 240 Mev: the data are from the Illinois group (as), reduced by 20 percent. We have arbitrarily reduced these experimental values in order to compare the shapes, since our (incomplete) theory appears to underestimate the cross sections in this energy region.
MESONS
AND
NUCLEONS
219
In the scattering of x-rays by atomic systems the classical Thomson amplitude is modified by a “form factor”, F(qz), where 2 = (~~)l’~is the root-mean-square radius of the charge distribution and q is proportional to the photon momentum transfer q = 2k sin e/2.
(73)
In adopting an atomic model to describe the nucleons, such form factor effects would be expected in the scattering of high-energy photons or electrons. In the case of electron scattering, these effects have been observed and measured (42) and the atomic model has been shown (1) to be consistent with these observations.22For photon scattering, however, the form factor effects are expected to be more complicated. In particular, they are not, at least not in any simple or obvious fashion, directly derivable from the electron scattering observations.23 Nevertheless, entirely for the purposes of illustration, we consider briefly the possible effects of a simple atomic form factor on the Thomson scattering. Assuming x = 0.80 X lo-l3 cm, and the “exponential” charge distribution used by Hofstadter to fit the electron scattering observations (@), we have aTr = aJ%d
(74)
with F(qs) = (1 + q222/12)-2
(744
qx = 1.11 (AT/K) sin e/2.
(74b)
and The broken curves in Figs. 6 and 7 illustrate the effects of the substitution of aT’for aT in Eqs. (72). However, the most important mesonic effects (even aside from the resonant scattering) are not describable in terms of a simple form factor. Some of the possible mesonic modifications of the Thomson scattering amplitude have been discussed, from a general field-theoretic point of view, by Klein (44). A more specific computation of such effects, using the Chew-Low-Wick approximations to meson field theory, has been reported by Watson, Zachariasen, and Karmas (45), but the details of this computation have not yet been published.24In any 22The scattering of high energy electrons from nucleons differs in a number of crucial respects from the scattering of photons. In particular, effects of the isobar resonance are not significant for electron scattering up to electron energies exceeding 500 Mev (4). 23 These difficulties in describing electric dipole photon scattering are associated with the large electric dipole photoproduction, the real process whose virtual counterpart leads to an enhanced electric dipole photon scattering, and the failures of the atomic model in predieting such processes. We are indebted to M. L. Goldberger for a number of explanations concerning this question. 24 A qualitative indication of the nature of the effects of virtual s-wave pion excitation
220
FELD
event, as may be seen in Figs. 5 and 7, the major features of the nucleon Compton scattering process for photon energies up to the resonance are quite adequately described by the simple theory of the previous section. The elucidation of the importance of other mesonic effects, such as those mentioned in this section, awaits further improvements in the experiments. ISOBAR PROCESSES INVOLVING
EFFECTS
Two
ON OTHER
REACTIONS
NUCLEONS
Other high-energy nuclear reactions, in which the energies available are sufficient to excite nucleons to the isobar state, should exhibit effects of the (3,3) resonance. In particular, reactions involving two nucleons, such as nucleon-nucleon scattering, pion production in two nucleon collisions, pion- and photodisintegration of the deuteron, all show such effects in the appropriate energy ranges. Since these reactions have been extensively discussed in the literature (3, 4, 6, 12, 13, 14, 18), we confine ourselves to a few remarks concerning the behavior of such reactions near the meson-production threshold. In particular, nucleon-nucleon scattering and the photodisintegration of the deuteron involve no real mesons and, in a fashion similar to Compton scattering, should exhibit resonance effects in the region below the meson production threshold. Resonance effects in nucleon-nucleon scattering can be shown (13, 14) to be very small in the energy range under consideration (Enuoleon5 5-600 Mev). In may be obtained from the following considerations: We assume that the anomalous electric dipole scattering involves an intermediate state of j = l/2- (nucleon plus one s-wave pion). Let the amplitude for scattering in this state be es . Then the additional terms which must be added to the S-matrix, Eqs. (70), are S-*-$‘+ = S**l-
=
-i
2
3’3aE&", ?'$~EX1.-lE,
The cross section
contains contributions, both in its magnitude and in the angular distribution, proportional to the amplitude of a3 . For real, positive us we expect a decrease in the cross section (and its value at 99”) if oE < $5 1 er j ; for as > g 1 ar ( the cross section at 90” becomes greater than its value in the absence of as . Such effects are obtained in the meson-theoretic calcula.tions (46).
MESONS
AND
221
NUCLEONS
the reactions r+D
eN+P,
(75a)
?r’+D+N+P,
(75b)
on the other hand, the isobar plays a major role. Assuming t,hat both reactions proceed through the same intermediate state, in which one of the nucleons is excited, we have a(y + D + N + P) = 4r + p + p + To> a(?ro+ P + P + dj * a(?r”+ D + N + P)
(76)
The right-hand side of Eq. (76) has been evaluated in the preceding, Eq. (29) ; to obtain the cross section for reaction (75a), however, we require an expression for the cross section for pion-disintegration of the deut)eron.Here, for real pion momenta, we may use the experimental values (4), as has been done by Austern W). The evaluation of Eq. (76) at photon energies below meson threshold (5150 Mev) requires a means of extrapolating the cross section for reaction (75b) into the nonphysical region; unfortunately, we do not have a meson-theoretic expression in this case. Nevertheless, we may note (12, 14) that for small pion momenta u(n” + D-+N
+ P) sA7,
(77)
which,25on substitution into Eq. (76), gives an expressionfor o(y + D ---f N + P) which is independent of q, This expression should be valid for photon energies both above and below the threshold, provided the condition ( v2 1 < 1 is satisfied.26In the region of the photomeson threshold, substitution of the experimental values into Eqs. (76), (29), and (77) yields a(y + D -+ N + P) N” 5 fib. Since this is only of the order of 10 percent of the observed cross section at F‘Y - 150 Mev, it is necessary to consider other sources of enhancement of the reaction in the threshold region. One such possibility could arise from the photoproduction of an s-wave pion on one nucleon and its reabsorption, before escape, in the two-nucleon system 25The experimentsgive (S), in the threshold region, u(N + P -+ D + ~0) s 0.41$ millibarns for the resonant part of the cross section. From detailed balancing (12), u(+’ + D + N + P) = (4p2/3q2)u(N + P --f D + T”) = %W~c)2(1/~2)u(N
+ P * D + 4,
(77’)
where p and q are, respectively, the proton and pion c.m. momenta for the same (total) cm. energies. Thus, A G 0.55 (p/me)’ millibarns in the threshold region. z6 Below the photoineson threshold, the (imaginary) pion momentum is defined in Appendix I.
222
FELD
(14,46-48). Such s-wave pions are produced mainly by electric dipole absorption. Accordingly, the intermediate-state meson will be charged and the nucleons in the intermediate state are identical. Near threshold, the available energy is small, and the nucleons will be in a relative ‘So state. Reabsorption of the s-wave pion results in a 3Pofinal state.” Austern (49) has demonstrated how a process of this nature leads to cross sections and angular distributions which are in reasonable accord with the observations in the region of the photomeson threshold. THE
SECOND RESONANCE
IN THE PION-NUCLEON
CROSS SECTION
As a final example of a reaction in which the isobar state may play a prominent role we consider the interaction of negative pions with protons in the region of -1 Bev pion laboratory energies. A peak in the total cross section has been interpreted (50) as a resonance in the isotopic spin T = l/2 state, since it is absent in the a+-P cross section. Figure 8 is a reproduction of the analysis of Cool, Piccioni, and Clark (50) of the observations in terms of the cross sections for interaction in the two possible isotopic spin states. It is a general property of reaction theory (5) that, for a resonance in a state of angular momentum j, the peak cross section is ures= 47mj + %hAlutot .
(78)
It is seen from Fig. 8 that the observed peak cross section in u1/2requires, if it is to be interpreted as due to a single resonance involving a single j-value, j 2 6 for U,I/U tot N l/2 (the value observed at ~1 Bev) and, at best, j 2 3 for u,~/ut~~II 1. A suggestion, for accounting for the observed cross section by means of a pion-pion scattering resonance, has been advanced by Piccioni et al. (50), by Dyson and by Takeda (51). This model has the attractive feature that the c.m. wave-length X, appropriate to a pion-pion collision, is sufficiently large to account for the observed peak cross section even for an s-wave pion-pion resonance. Owing to the large momentum spread of the target pions, a consequence of their confinement in the small volume associated with the physical nucleons, a large “doppler” broadening of the resonance is expected. Roughly speaking, for a pion cloud of dimensions x(l/2)fi/mc (49)) the uncertainty principle predicts a momentum spread of wf2mc or an energy spread of .~600 Mev in the c.m. system of the two pions; the contribution to the resonance width in the laboratory system would be - twice this. 27The final two-nucleon state for the resonant photodisintegration, resulting from isobar excitation, is ‘Dz . Since there is no interference between electric and magnetic dipole photodisintegration (they lead to final triplet and singlet states, respectively), another reason must be sought for the observed asymmetry in the angular distribution. This probably arises out of a retardation denominator in the electric disintegration amplitude (14,@9).
MESONS
0
I
0.4
I
I
0.6
I
AND
I
0.8 El(lab)
223
NUCLEONS
I
I
1.0
I
I
1.2
I
I
I.4
I
,
in 8ev
FIG. 8. The total cross sections u(,+ + P) = 4312, u(?r- + P) = $&I/Z + >~us/z, and cI,2 vs pion lab energy, as deducted by Cool, Piccioni, and Clark (60) from the experimental data.
Another possibility, has been suggested by the author and by Lindenbaum and Sternheimer (15). According to this suggestion, the process responsible for the resonance involves the inelastic excitation of the nucleon isobar by the incident pion. Lindenbaum and Sternheimer have derived the threshold crosssection behavior of such an inelastic isobar excitation process, and are able to reproduce the rapid rise on the low-energy side of the resonance. In addition, they have shown that the observations on the charged pion energy spectra at 1.0 and 1.37 Bev are consistent with the reaction. ?r+P-+7r+xn*--tN+?r-+?r+.
(79)
The following considerations are concerned with a crude, qualitative attempt to fit the shape of the observed cross-section by a model which assumesthat a T = l/2 resonanceis exhibited by those processesin which the isobar (excited by the incident pion) and the inelastically scattered pion are in an orbital p-state. There are three such states, corresponding to the possible combinations of the isobar spin, j = 3/2+, and the orbital angular momentum of the scattered pion, C’ = 1; i.e., j = l/2+, 3/2+, and 5/2+. The first two can be excited by incident pions of orbital angular momentum C = 1 only; the last requires 4 = 3. If all three states exhibit a resonance at roughly the same energy, the sum Z(j + l/2) = 6 is sufficient to account for the peak cross section, Eq. (7S), even with uJutot II l/2. We apply resonance theory, in its simplest form (5), to each of these reso-
224
FELD
nances. The total cross sect.ion is
I‘,r 2 > 4(-Y - -YrY + IQ'
utot (') = 47&.2 j + I
(
(80)
in which 71is the pion cm. energy, I?, is the width corresponding to the emission (absorptioin) of the incident pion (elastic scattering), and 17 is the total width, r = r, t r*e ) where I‘,! is the width for emission of a lower energy pion, leaving the nucleon in the (3,3) state (inelastic scattering). We make the further ,simplifying assumption that inelastic scattering always excites the isobar to the (same) energy corresponding to the peak of the isobar resonance (approximately two pion masses above the nucleon ground state). For the pion widths we adopt the simple forms, appropriate to low-energy resonance theory (5) r*
= rowc(d,
r+ = r,+h(d),
@Ia)
@lb)
with I’0 an adjustable constant and vc(7), ~~~(7’) the angular momentum barrier penetration factors for the elastically and inelastically scattered pions, respectively,28 211(x) = 22/(l
uz(z) = &(225
+ &
+ 45x2 + 6x4 + z").
(824
(82b)
(For all three resonances, I’ = 1; forj = l/2 and3/2,4 = 1; forj = 5/2, P = 3.) We are left with two parameters, Y,. and I?o , which we may attempt to adjust to fit the observations. Figure 9 illustrates one such attempt, in which we have assumed E, = 0.95 Bev (pion kinetic energy in the laboratory system) and To = 0.12 Bev. Also shown in Fig. 9 is the computed ratio
For each j-value
(83) The observed ratio at ~1 Bev is -0.5 (50). the fit to the experimental data Considering the crudeness of the “theory”, may be regarded as quite reasonable. Actually, there is considerable leeway in the interpretation of the experiments since it is not necessary to assign all of the cross section, CQ,~in Fig. 8, to the resonance. Thus, an alternative analysis by Cool et al. (50), reproduced in Fig. 10, attributes an appreciable fraction of the scattering to nonresonant interactions. In this case, both the width and the height of the T = l/2 resonance are reduced considerably, and the demands on 28These expressions assume that the range of the interactions is ~-1 = h/me.
MESONS
225
NUCLEONS
AND
1.0
‘\ ‘\ 80-
“ ‘1 .‘-..
-.-_
, ---
.-.__
$1;
__ ---.-.-.-.-.-
60 c
4~‘-~~,~,-------,
-----------
I
I
0.6
I
I
0.8
I
I
1.0
I
I
I
1.4
1.2
FIG. 9. Comparison of experimental ~~12with the predictions of the simple resonance model described in the text. The separate curves labeled u(j) are the contributions to the total cross section of resonances with total angular momentum j = l/Z, 3/2, and 5/2. The curve u is the sum, c = &u(j). The monotonically decreasing, dot-dashed curve shows the ratio u,r/q,,t = ~o,r(j)/&~t(j) predicted on the basis of the simple theory; the scale for the ratio is on the right side of the figure.
0.4 Edob)
0.8 in 8ev
1.2
1.6
FIG. 10. Alternative analysis of the pion-proton total cross section data (60) in which a nonresonant contribution is subtracted from the observed total cross section curves. The resulting resonances are narrower, more symmetrical, and of smaller peak cross section. They present less difficulty in “fitting” than does the curve labeled mr,2 in Figs. 8 and 9.
226
FELD
TABLE III RELATIVE FINAL STATE PROBABILITIES IN THE T = 35 RESONANT INTERACTION OF T- AND PROTONS PKOWSS
Final
Elastic scattering
Inelastic
scatteringo
All (50% each)
state
Relative
probability
NT0
1
Pn-
2
NTOTO NKfTN*-?r”
4 (1) (0) :, 10 (2) (2) )
P&aP*-*O
;
N
P 7r+ 79 7r-
i
4 (0) (1) 5 4 1 1.8
2.6
(1The order of pions is as in Eq. (84b).
the theory are relaxed to a large extent. Furthermore, since there is no experimental evidence avaiIabIe concerning the relative contributions of the resonant and nonresonant interactions to the inelastic scattering, we are free to choose the intrinsic widths, I’,, in Eqs. (81), different for the elastic and inelastic scattering. Under these circumstances, it is even possible to assign the resonance to one angular momentum, j = 5/2, or to a combination of any two. Clearly, the clarification of this situation awaits further experimental information. In addition to the energy distribution of the pions resulting from reaction (79), discussed by Lindenbaum and Sternheimer (lb’), the interpretation given above may be utilized to yield predictions concerning the charge distribution in the various possible final states resulting from the resonant a- + P interaction, since the isotopic spins are specified in all stages of the interaction.2s These predictions, which follow from a straightforward application of the principle of charge independence (isotopic spin conservation) to the reactions
are summarized
T-+P+%+n,
(844
lr- + P ---f 3x* + lrl -+ 3-z + 7rn + d,
@4b)
in Table III,
in which we list separately the results for elastic
29We disregard, in this analysis, possible charge-exchange processes in the intermediate stages. The validity of such an analysis requires, especially, a negligible pion-pion interaction.
MESONS
AND
NUCLEONS
227
scattering, inelastic scattering, and a total based on the assumption of equal elastic and inelastic contributions to the resonance. Also shown in the last column, in parentheses, are the predictions of the pion-pion scattering resonance theories of Dyson and Takeda, respectively (51). It is also possible to compute the angular distributions corresponding to the various contributions to the resonance, by straightforward application of reaction theory. However, owing to the complications arising from the mixture of states assumed in the above interpretation, as well as from the motion of the isobar in the c.m. system, the results are not expressible in any simple form. In view of the tentative nature of the interpretation and the lack of experimental data, such an analysis does not seem particularly fruitful at this stage. DISCUSSION
The examples discussed in the preceding serve to demonstrate the importance of the nucleon isobar resonance for high energy interactions involving mesons (both real and virtual) and nucleons. We have, we believe, shown how such resonant processes may be treated in terms of a relatively simple model describing the structure of physical nucleons. Such a treatment is not a substitute for a proper field theoretical approach to these problems; rather, it should be regarded as a means of distilling, from the field theory, those features which, as a consequence of general invariance and conservation principles, must result from any reasonable approach to the meson-nucleon interactions. In addition, it appears to us that the approach adopted in this paper is a useful tool for elucidating the physical nature of the major processes involved in the interactions of pions and nucleons. Another important use of the more-or-less phenomenological approach adopted in the foregoing is that it lays the groundwork for educated speculations concerning the nature and form of processes for whose description the field-theoretic treatment may be prohibitively complicated. Two such examples have been discussed in the last section3’ These discussions have, however, not exhausted the possible applications of this approach. Thus, for example, it would be of 3* With respect to the last example, the 1’ = l/2 pion scattering resonance, it is possible that our analysis, if it should prove tenable, is of somewhat broader interest than the immediate problem to which it is applied. We have in mind the question of the parity of the K-meson. The reaction, ?r + P + A0 + K”, whose threshold falls inside this resonance, appears to have a cross section which, following an initial steep rise from the threshold, assumes the shape of the ?r- + P cross section (61). Consequently, a determination of the energy dependence near threshold and of the angular distribution associated with the resonant part of the reaction will yield the relative intrinsic parity of the ~0 - KO system, provided that the parity of the resonant ?r- - P state is known. Our model for this resonance assumes positive parity for the resonant states. Thus, the A” - KO intrinsic parity, on this assumption, is (-l)‘s, where es is the orbital angular momentum associated with the resonant ho - KO production process.
228
FELD
interest to consider some aspects of the nucleon-nucleon interaction, for which the fieId-theoretic treatment appears to yield a static potential in reasonable agreement with observations on low-energy nucleon nucleon interactions (5~). Using the atomic model, the nucleon-nucleon interaction might be treated in a fashion similar to the Heitler-London treatment of the interactions between two hydrogen atoms. In particular, it would be of great interest to apply such techniques to the interaction between moving nucleons, in an attempt to provide a somewhat more quantitative basis for the speculations of Frisch (54) concerning the possible effects of virtual isobar excitation in giving rise to the spin-orbit force between nucleons. Such computations are clearly of a higher order of difficulty than those undertaken in this paper, but they may nevertheless prove to be more tractable than the corresponding field-theoretic calculations. APPENDIX KINEMATICS
I
OF PHOTOMESON PRODUCTION
The following conventions are used throughout this paper: All energies are in units of mc” = 137 Mev; all momenta in units of mc; all lengths in units of K-I 3 &/mc = 1.44 X lo-l3 cm; all masses in units of the pion mass, m. Unless otherwise specified, all expressions apply to the baryocentric system, c.m. The symbols have the following meanings: E,,, = photon or pion kinetic energy, k, 4 = photon or pion wave-number (momentum/@, = pion momentum, rl = total pion energy (-y’ = 1 + q2; E, = y - l), Y = UT/C = q/r, B w = total c.m. energy minus the nucleon rest-energy E y + q2/2M in a nonrelativistic approximation for the recoil nucleon. We review briefly the relativistic kinematics of photomeson production: Let E = the incident phot,on (laboratory) energy, P = E = the incident photon momentum, U = E + M = the total laboratory energy (M is the mass of the target). Then, the Lorenta transformation to the c.m. system is uo = (1 - po”)-1’2(u - POE),
(Al)
PO = (1 - po2)-1’2(E - p&Q = 0,
W)
yielding PO = E/U, U. = (U” - E2)1’2 = (M2 + 2ME)“‘.
b43) (A4)
We define w=Uo-M
(A5)
229
MESONS AND NUCLEONS
and obtain w/M
= (1 + 2E/M)l”
-
Lw
1.
Now from y = (1 + ?#‘2 = w -
v2/2M,
(A7)
in which the recoil nucleon has been treated nonrelativistically, y/M Finally,
= [(l + 2w/M + M-2)1’2 -
k = the photon c.m. momentum k = [(l - Po),‘(l
+
we obtain
I].
W9
in units of mc,
/30)]“2E = E(1 + 2E/M)--l”.
(A%
We note, from Eq. (A8), that below threshold (w < I), y < 1 and, (A7), $ < 0; i.e., q becomes imaginary, q6 becomes negative [see Eq. We assume, for the resonance energy, E,, (lab) s 190 Mev. For c.m. energy, this corresponds to E,, (lab) s 340 Mev. Application of expressions yields
from Eq. (68b)]. the same the above
k, = 1.88, wr = 2.11, -yr = 1.92, q,. = 1.63. APPENDIX
INTERPRETATION
OF no-PHOTOPRODUCTION
II
EXPERIMENTS
For ?y* photoproduction, Table II indicates that we may neglect the 1 l (* term in Eq. (61), and write
8, (AlO) ,I +cocos2
--1 du uo dO
p2 sin” 0 2(1 - pcose)
where AO
= R2P +
Bo = -2R(l
(1
+
,0)~/21+
R(1
+
P)I~C~+~cos
- P>(Sr COS a33 + li sin a33),
Co = -A.,, + R%
~33+
bi sin
ad,
(All)
- p)“.
However, most of the experiments are confined to a limited angular range, around 90”, and are of insufficient accuracy to detect effects of the retardation factor which multiplies Bo in Eq. (AlO). Accordingly, such experiments can be
230
FELD
and have been interpreted
by fitting
da
s
to an expression
= A + B cos 19+ C cos”8.
L412)
In order to compare the experimental values of the coefficients with the predictions of Eqs. (AlO-11), we make the arbitrary assumption that an average experiment consists of measurements of du/dQ at three angles: 45”, 90°, 135’. In terms of these measurements, the coefficients become _B = da(45’) A 4 C -= A
- du(135’) du(90”) ’
(Al3)
da(45’) + do(135“) - 2 du(90’) da&W)
(A14)
Using Eqs. (A13-14) to define the experimental expression, Eq. (AlO), gives
B-=A
coefficients,
the theoretical
B” { 1 - P2/4(l - @?a I,
Ao
C Co -=--A Ao
Bo 3 P /4(1 - p”l2).
Ao
It is these ratios which have been plotted under various assumptions in Figs. 2 and 3. Also shown, in Fig. 2, is the uncorrected asymmetry ratio Bo/Ao ; it is seen that the corrections to B/A, due to the retardation effects, become appreciable only for photon energies appreciably greater than ~300 Mev. The same is true for the anisotropy coefficient, C/A. RECEIVED:
March 4, 1958 REFERENCES
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