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One-particle exchange theory of pion photoproduction
One-Particle
Exchange
Theory
I. General A. Instifut
jiir
Theoretische
r\ I.
of Pion
Calculation*
HARUX-AR-RAGHID
h’emphysik
Photoproduction
der
i
Technische
Hochschule,
Karlsruhe,
Germany
AND
J.
MICHAEL Lawrence
Radiation
Laboratory,
MORAVCSIK
c7niversify
of California,
Livermore,
California
A calculation of single pion photoprodllction from single nucleons is given using the one-particle exchange model involving the exchange of one pion, one p, one W, one $, one nucleon, one L1’*, and one ‘ V**. Crossed diagrams, previously neglected, have also been included in the calculation. An imaginary part of the amplitudes is included using a Breit-Wigner type formula and experimentally observed widths. The t,heory should be a good representation of pion photoproduction up to about 800 MeV incident laboratory photon energy. A numerical evalnat,ion of the theory will be given in a subsequent paper. I. INTRODUCTION
Ever since the beginnings of pion physics in the late 1940’s, photoproduction of pions from nucleons, together with elastic pion-nucleon scattering, has played a central role in our knowledge of pi mesons. Accordingly, this process has received a large amount of theoretical attention. At the beginning, first order perturbation calculations (1) were carried out, using, as intermediate particles, only nucleons and pions. These calculations were in violent disagreement with the experimental results, and were soon abandoned. About the same time, particularly in connection with the ratios of the various pion charged states, some very simple-minded classical models (2) were also used, which gave qualitatively the right results. In the early 1950% the so-called static theory (3) was developed, which, from the point of view of the present paper, can be described as a first * Work performed mission. 7 On leave from Dacca, Pakistan.
partly the
under
Pakistan
the auspices Atomic
Energy 331
of the United Commission’s
States
Atomic Atomic
Energy Energy
ComCentre,
332
HARUN-AR-RASHID
AND
MORAVCSIK
order perturbation theory using pions and nucleons as intermediate particles, plus semiphenomenological terms describing the resonance in the J = 35, T = ,35 state. To be sure, the static theory was at the time presented in a different spirit, and its justification was also argued on a different basis. Nevertheless, its numerical success was based on the fact that, at low energies, the above mentioned resonance is the dominant feature of the pion-nucleon interaction. A somewhat refined version of the static theory was the so-called Chew-Low theory (4), with almost identical results but a somewhat more erudite justification. The next step in the theory of pion phenomena came with the introduction of one-dimensional dispersion theories. This, applied to the pion photoproduction process (5), yielded the CGLN theory. Again, from our present point of view, this can be described as a first order perturbation theory using only nucleons and pions as intermediate particles, plus a resonance term in the $5, N state, in which, in addition to certain phenomenological elements, restrictions of the dispersion relations themselves were also made use of. It was thought that this theory should give a good description of pion photoproduction up to, say, 400 R!IeV photon energy in the lab system. The CGLN theory in its original form used various approximations in the detailed evaluation of its formulae. Several alternate approximations (6) have been explored since, and it appears clear that in view of the uncertainties in these approximations the CGLN t.ype of theory even in the low energy region gives a prediction for the experimental observables which carries an error not smaller than 10%. As pion-pion interaction was “discovered” through the study of the nucleon electromagnetic form factors, attempts were made to incorporate this interaction into the pion-photoproduction calculations. The results (7) gave correction terms to the CGLN predictions which in general were small, but for some quantities (such as the negative to positive pion ratio near threshold) could be quite significant In the meantime, however, the experimental techniques have also been extended and it became possible to measure photoproduction of pions up to about 1.2 BeV. The first interpretation of such experiments was done in terms of phenomenology and in this way the so-called second, third, and fourth resonances were discovered. A similar phenomenological analysis, for pion-nucleon scattering, has reached a quite sophisticated stage by now (8), and led to the discovery of further structure in terms of angular momentum states. At the same time, however, the need also arose to calculate the physical observables in this new energy region in some more fundamental way, and to include in such a calculation the newly discovered heavy mesons of strangeness zero. Since even today the only calculational technique which can be employed without the necessity of
THEORY
OF
PIOh-
PHOTOPRODUCTION
333
crude approximations and without the lengthy use of high speed computers is first order perturbation theory (now rechristened “pole contribution”), the logical scheme to be used for calculating phot’oproduction turned out to be the so-called one-particle exchange theory, OPET, in which the pole contributions of all those particles are calculat’ed which are experiment’ally krlowrl and whose quantum numbers are appropriate for the particular diagram. Such OPET can of course also be used for other processes, and calculations have been carried out also for pion-nucleon scattering (9), nucleon-nucleon scattering (IO), pion production in pion-nucleon scattering (II), etc. In all cases the first results have been highly encouraging. In suc*h calculations the coupling constants other than that bet’ween the pion and the nucleon are used as adjustable parameters, but one would expect to get the same value for a given coupling constant from various react’ions, so that there is room for a check. It might be mentioned t#hat from the point of view of OPET the success of the static theory and t,he CGLS t)heory is easily understandable, since they are basically special cases of OPET. The OPET theory in it,s simplest form, in which the contributions are simply calculated from the first order pert’urbation diagram, can of course not be exactly correct. The specific objections one can raise are as follows. 1. The pole contributions to t’he amplitudes are always real. Hence unitarity is not satisfied and many experiment,al observables (such as polarization) are identically zero. It is possible, however, in several ways, to modify the amplitudes so as to make t,hem unitary (12). For example, when one includes as intermediat,e states particles which have a finite width, their propagator can be modified to include an imaginary part’, and this makes the corresponding amplitude also complex. Alt’hough the widths must be taken from experimental data, at least formally one can preserve unitarity this way. 2. One dimensional dispersion relations are not satisfied (12). This is a different problem from that, considered under 1, since the amplitudes could easily be unitary and still not satisfy a given form of dispersion relations. At the same time, it is also a less serious shortcoming, since in most cases the dispersion relations which are asked to be satisfied are not proven and have no empirical verification either. As things stand now, if one does demand that some dispersion relations be satisfied, one must insist on the presence of cert’ain contributions to the amplitudes which at present, cannot be calculated at all and whose precise physical origin is unltiown. 3. The OPET theory is obviously quite incomplete, since it must use experimental values for the masses and widths of the int’ermediate particles and contains coupling constants which are determined by fit to experimental data. This is readily admitted to be true, and simply means that once the validity of OPET
334
HARUN-AR-RASHID
AND
MORAVCSIK
in a form modified to eliminate objections 1 and perhaps 2 is established empirically, some additional dynamical principle (perhaps bootstrap, perhaps something else) will have to be introduced to explain why continuum contributions are negligible and why only those masses, widths, and coupling constants occur in nature which are observed. When presenting the results of OPET calculations, it has become customary in the literature to deduce the OPET as an approximation to something more highbrow, usually the Mandelstam representation. We would like to deplore this habit, since we believe that the road leading, for instance, from the full Mandelstam representation to an OPET scheme is so much loaded with drastic approximations and ad hoc assumptions as to make the OPET a completely incredible method of calculation. Furthermore, such a procedure might also give the completely erroneous impression that the OPET in some way preassumes the validity of the Mandelstam representation or any other part of the mathematical framework of X matrix or field theory. We believe that at this point it is much preferable to use the OPET as an assumption in itself, whose validity appears to be justified by its practical success. In the present paper we give an OPET calculation of pion photoproduction. This is not the first time that this has been done (13). Our justification for presenting such results is manifold. First’, we included in our calculations certain crossed diagrams which were omitted before on the basis that their contribution is likely to be much smaller than the contribution of those diagrams which were kept. We are not only unsure about the veracity of such statements in general, but in addition believe that because of the large cancellations often occurring in complicated processes involving many diagrams, it is unwise to neglect any terms a priori if they can be calculated wit,h relat,ively little additional effort. Our numerical evaluation of the results, to be presented in a later paper, will show if in this case the omission of the crossed diagrams in previous calculations lead to appreciable errors or not. Secondly, since the last calculations, certain new resonances were found which should also be included in the calculation. Thirdly, we plan to carry out a more thorough numerical investigation of OPET as applied to photoproduction, including data on polarization, studying the possibility of several solutions for the coupling constants, and giving predictions for those observables which have not been measured but whose measurement might now be feasible. Recently several authors (14) have developed modifications of the OPET in which corrections are included to take into account the presence of inelastic channels. Since our calculations do not intend to cover energies above 800 MeV and up to this energy inelastic channels do not play a dominant role, we believe that these absorption corrections will not be important in the first approximation. A more detailed investigation of this point is in progress.
THEORY
II.
SUMMAltY
OF
OF
PION
335
PHOTOPRODUCTION
PR,OCESSES
AND
INTERACTIONS
The diagrams included in the present calculation are shown in Fig. 1. The last three graphs are processes which were not included in previous calculations (1s). The vertex functions used in the calculations are listed in Table I. It was assumed that the quantum numbers of the various particles are those given in Table II. The notation used in the vertex functions, t,oget*her with all notation used in this paper, is given in the Appendix. The width of an intermediate particle is taken into account by formally substituting in the expressions for the mass of the particle 1~ + iI’, where I? is the width, taken from experiments. This will yield an imaginary part of the amplitudes which otherwise would be pure real. The precise form is given in the next section. The propagators used in the caIculat,ion are given in Table III. The spin 1 propagator agrees with that of ref. 15, and t,he propagator for spin $5 is the same as in ref. 13. It might be worth emphasizing that throughout this work we assume,like all others who have carried out similar calculations, that the coupling constants fol the int’eraction between two kinds of particles do not depend on the charge stat’e of the particles involved. This assumption is not, only questionable but in fact
\ 1 N
(5)
(6)
(IO) FIG. 1. Processes used in the present calculation. Wavy dashed lines pions, heavy dashed lines heavy zero-strangeness nucleons, heavy solid lines nucleon isobars.
lines denote photons, mesons, light solid
light, lines
336
HARUN-AR-RASHID
AND
TABLE FORMS
OF INTERACTIONS
MORAVCSIK
I
USED
IN
THIS
Particles
CALCULATION Interaction
T-N-N r--N-N* r - N -
N**
y-?r---w
y-N-N y-N-N* -,, -
N -
N**
p-N-N o-N-N 4-N-N
TABLE QUANTUM
NUMBERS Approximate mass (MeV)
N N* N** K P
w *
938 1238 1512 139
750 780 1020
OF PARTICLES
II USED
IN
THE
Spin
Parity
34 N 35 0 1 1 1
i+ -
PRESENT
CALCULATION Isotopic spin
% 5% % 1 1 0 0
G Parity
+ -
clearly untrue, sincefor instance the magnetic moments of the proton and neutron are distinctly different. The extension of the calculations to the more general case in which this assumption is not made is of course trivial, consisting only in carrying a superscript Eon the coupling constants. We did this in fact but only for the magnetic coupling constant of the y - N - N vertex. In previous numerical
THEORY
OF
PION
TABLE
337
PHOTOPRODUCTION
III
USED IN THE PRESENT CALCULATIONS
PROPAGATORS Spin
1
0
a The
letter
m2 -
q denotes
the
momentum
q2
of the intermediate
particle
with
mass
m.
work (as well as in our projected numerical analysis) the extra freedom of the charge dependence of all coupling constants is not taken advantage of, which might be worth keeping in mind in case “irreconcilable” discrepancies with experiments arise. Using the interactions given in Table I, the contributions of diagrams (l)-(5) to the 5a amplitudes of CGLN are well-known while those of diagrams (6) and (7)-also calculated by Gourdin and Salin (IS)-are given as follows:
& = --
NM” cos e ,m*z - s w,*
3 2 = -AN,*+ 3
N,* m*2-s
33 = ___ m*2 - s
34 = 0
3 -;N;*+N; l - 3 m**2 - s
Nn”; 32 = m**z - s cos e
(2.1)
and (2.2)
33 = 0 The calculations for the crossed diagram (9) are much simplified by neglecting the Pauli interaction term and the contributions to the F, amplitudes of CGLN are given by
R(W) =i!$ ; [w- m:+em + W* wy: 1; 2nz] m~2 l- 2L Fz(W)
= --R(-W)
Fd(W)
= Fd -W>
(2.3)
338
HARUN-AR-RASHID
AND
MORAVCSIK
while those of diagram (10) can be simply obtained by the transformation w* = -m** In Table I, we have used the gauge-noninvariant vertex ‘u
=
M-0
G6iY6QJ
coupling
u(p)
+&mh:,cr
(
but as pointed variant form
out by Gourdin
for the -yNN*-
>
and Salin one can write
the following
gauge-in-
‘UI = - ~‘,(P’)[G&w~~xQ - bx>lu(p> and can easily deduce that in that case G6
=
(m
+
m*>
Fa
(26’
Using the coupling 2)1 the contributions
-
2m
=
of diagram
G6’
(6) are given by
5 = w + m/3 CN 2 W + m ?)z*~- s 53=--
(2.5)
2w CN W + m ms2--s
54
=
0
where CN = ‘$
pk Z/(El
+ m) (E2 + m)
It is easy to see that Eqs. (2.1) and (2.5) satisfy the relations (2.4). The great advantage of using the gauge-invariant form is that with this we can maintain crossing symmetry and the contributions of diagram (9) can be simply obtained. The results are better given in terms of the invariant amplitudes Ai of CGLN:
AI = m?25[
-q-k
-;
+ &*
fh
+ WL*>(2 p’.q - p.k)
- iP’.q
m(m
+ m*)
- m1
+ pz*k (2.6)
Aa = m.f[A, = ,f
:’
-i(m+m*) U
where P’eq = (m*2 - m” + 1)/2.
-&{i-2Pr*q}-m*P’eq]
THEORY
OF
PION
For the second resonance, Gourdin
and Salin have suggested the coupling
G’PJ=x(p’)u(p) and using this the contributions
339
PHOTOPRODUCTION
(kc~ - ICXE~)
(2.7)
of diagram (7) are given by
where CN’ = (gTGT’/4a)kW v’(E, + m)(Ez +m)($/(Ez + m)) while those of diagram (10) which need not be quoted here are again obtained simply from crossing symmetry. III.
RESULTS
FOR
THE
AMPLITUDES
Invariant amplitudes for pion photoproduction from nucleons have been defined in a number of ways. Among these, there are the %i functions of ref. 5, the Bi functions of ref. 16, and the A, B, C, and D functions of ref. 17. In this paper we will disgust the reader by defining another set of amplikrdes, which however are quite closely related either to the A, B, C, D or to the Bi’s. In particular we have al = A(sin 0)-’ = B4 ct2 = B = B1 + BB cos 0 a3 = C = Bz a4 = Dsin0
= -B1cos6
(3.1) + BSsin2B
The advantage of these amplitudes lies in the fact that while they maintain the features of simplicity which characterize the A, B, C, D amplitudes, they do not become infinite at 0 = 0” or HO”, but at the same time leave the formulae for the observables in a quite symmetric form. The connection (18) between the amplitudes ai and the amplitudes pi is given as follows: s = -alcOse + a3 32 = --a1 53
=
a1
+
54
=
a,
-
cos sin2
B a3
+
1 -;---a3-7
he cos
sin2 0
a4
(3.2)
e
sin2 0
a4
and conversely 61 = -32 a2
All these invariant
=
31
+
54
+
53 cos
cf2cos
e
a4
5h cos
e +
=
amplitudes
52
-
are functions
e
(3.3)
a3 = cfl -
53 sin2 e
of W and cos 8.
340
HARUN-AR-RASHID
AND
MORAVCSIK
The experimental observables are given in terms of the amplitudes C?,;by the formulas in Table IV. The constant K in front of all observables is given by K = 389.35 barns q MT2
(3.4)
i%
The observables are given for linearly polarized photons, with Q = 0 denoting photons polarized parallel to the production plane, and C$= 90” denotes photons polarized in the k X q direction. For unpolarized photons one simply makes the substitutions cos24 = sin” I$ = >s
sin I$ cos @I= 0
(3.5)
Each of the amplitudes can be written as a sum of contributions from the various processes (for the notation on the subscripts and superscripts, see the Appendix). Thus we have a?( w, cos e) = c af- “( w, cos e>
(3.6)
A
There are three c&% for each i, corresponding to the three values of the isospin index, and the total amplitude for a given process (indexed by j) is given by (j’& = c (j)PJ’@e= (313+ai+ + W3- ai- + (A30a;
(3.7)
E
TABLE IV EXPERIMENTAL
OBSERVABLES
OF PION
Ai INVARIANT
PHOTOPRODUCTION
IN TERMS
OF THE
AMPLITUDES*
= K([I Al 12sin2B+ 1A8 Ia] sin24 + [I AZ 1%sin2 0 + / A4 I21cos241 ZOPl = K(2 Im (Aa*A4 - A,*Az sin2 0) sin 4 cos 41 ZOP, = K( -2 Im A1*A3 sin o sin2 4 - 2 Im At*Ad sin 0 cos24) zoprl = K{2 Im (A2*Aa - AI*A4) sin 0 sin 4 cos 41 IO
= K( -2 Im = K( -2 Im = K(-2 Im
IOTZI
=
Id1
IoTm = IOT, = IoTm = IoTmr IoTz
= =
IoTrsz = ITo mn = IOTl&, 9 The
(A3*A4 + A1*A2 sin2 0) sin 4 cos 4) A1*A3 sin e sin2 4 + 2 Im A2*A4 sin e cos2 41 (A2*Aa + A1*A4) sin 0 sin 4 cos 4) K([I Al 1%sin* e - 1 As 121sin2 4 + (I A:! I* sin20 - 1A4 I21COS* 41 K( [I A’ 12 sin2 e + 1 Al 121 sin2 4 - (I AS I2 sin2e + I Ad I21cos24t K( [I A1 12sin2 e - I Al I21sin24 + (I A4 I2 - I A2 I2sin28) COS* 41 K(2 Re (A1*A4 + A2*A,) sin 0 sin 4 cos 4) K(2 Re (AZ*& - AI*A4) sin 0 sin 4 cos 41 K( -2 Re A,*A, sin 0 sin2 4 + 2 Re Az*Ah sin B cos24) K(2 Re A,*Aa sin e sin2 4 + 2 Re A2*A4 sin 0 COG41 K{2 Re (A1*A2 sin2 e + Aa*Ad) sin 4 cos 41 K(2 Re (Aa*A4 - A1*A, sine 0) sin 4 cos 4)
Id, IllA,
=
constant
K is given
by 389.35
qM;%-1
barns.
OF
THEORY
where the assignment
PION
341
PHOTOPRODUCTION
of the j’s is as follows j=l
r+P
j=a
y+p+n+r+
j=3
y+n-+n+3r”
j=4
Y+n--tp+7r-
-+p+rO (3.8)
and the (i)~t’~ are con&ants given by Table V. One should note that forj = 1, 3 in the kinematic relations M,o is to be used, while forj = 2, 4 we must use M,+. Thus a given W will correspond to a different E, forj = 1, 3 than forj = 2,4. Each of the ai* “s can be written in terms of real and imaginary parts @f*“(W, cos8) = Re@f”(W,
cos8) + iIm@,!‘X(W,
cos8)
(3.9)
and we further write Re @f*“(W, cos 0) =
&“(W>
+ p(w) cos 8 + cfq w> co2 e cP(W) + e”(W) cos 0
TABLE CHARGE
STATE
(3.10)
V COEFFICIENTS+ j
5 1
+
2
01 1
0 9 See Eq.
Y
10
$5 a
-1
-z/5 6
(3.7). TABLE
RELATIONSHIP
3
BETWEEN
THE
Two
VI
SETS OF NUMERATOR
FUNCTIONS
IN SECTION
III
up (WI = -I$“, (W)
a!” (W) = f S:?(W)
bf,’ (W) = -j$
cp (W) = -j$$ (W)
biX (WI = f f*: (WI - fii:; (WI c5” (W) = fS4 (WI - rf:; on
up (W) = ri:: CM’)+ f 6:: (WI bisxW = f f:: (W) + f 6,; (WI cp (W) = If:: (W) + f is (WI
UP (W) = f i$ (WI + f s:“3 (W) btx (W) = f f:; (W) + f f:; (W) - fi:: (WI CP (W) = -jf:: (W) - f&i (WI
(W)
A=
29 Z" Xl{ ___ W+m
28k 0 0 0
-2wk
2q2-cl X1r (Ez + m) (W -
-
1
AMPLITUDES
m)
,+2 -2qk 0 0 0
V,8
m)VdI
WSm
wk
W-m
wk
2
+ (W -
+ (W -
1 + 2wk
2
X2E &I-
X& q [- V.8
-W-m+-
W-m-----
PHOTOPRODUCTION
Ez+m
-&?a
FOR PION
m)VdI
(1 -
2&V&‘)
2wk)VM
1 1-(1 -
VII
THE VARIOUS
TABLE FROM
p} q +
m
vg”
-
m-&I
EXCHANGE
p,2 -2qk 0 0 0
X.4 &
1 + 2wk
[- vsw
+
-
(1 -
(w - m)VM”l
+ (W - m)V.+f”l
3
PROCESSES~
{[-w-~~+~I-(‘-“‘)“u.
&
X$q(-VVEw
.
-X35
X&{VBw[W
ONE-PARTICLE
~klVrw}
0
X=
2qk
kW+m El+m
m + m)
~
W+m*
-
28
2q dW-mW+m -
El + m
+ m
R + m (W + m)”
&+mW-m m* + 2kE1
&tNe*
2qk 0 0 0
-f
f &tNE*
El + (W + m)(Ez
W-m*-
m**
m**+m*-1-mm*9
m**
9
2ok
m’“-mfl 2m*
W+m k
w - m* - W - m+
2wk -+ W-m
qE1+mm**+m2-1-mm*W+m*
; X&NE* __
X&NE*
X&NE*
m** -
e W+m
Fd
)I
El + m ~ q(W + m)
VII-Continued
-f ; --
; X&NE*
-2qk 0 0 0
Ez+m
___
- 2h&
-x*rz,Q--
2qk
- X&Z, -e W-m -
F&
W-m-
8
TABLE
W-m
W-m W$m*-2mm+2-m2fl
W+m*-2m
2m*
1
1
q2
2 W (ES -t
q
2qk 0 0 0
In**2 -
-l&CL-
xfo-q---
1
m2 + 2kEx
qE2fmW2-mZ
vi’ + wt
2k2
2W(Et
m**l
q
___-
10
+ m) iv;
m**z m**z
-
IV - nz
1
2m,**
- 2m m**
tn**
w - m - m2 + 1 IV -
m2 + 1 W -
2m**
-
.%A --w - m
-
2WCE2 + m) )\,;* __- k IV + m P2
w + w1** - $$&
1 + mm**
m) N;*
we - WI,* m,*** + IQ -
p W + m EQ + m
k q -Xf&---
1 lc qW+m
XEO -
where Xhc is a matrix as follows
A=
-
2m
1
Liz Q,
& = Y!2 &
1
1
4rm,
vr*
=
VMW = __ 4m,
v~P=--Gz-
4mi,
1 v,p = __
km
G,F” 2m &
Ga ~3 em
:L
W+m
G2 yz JfL + m) (Ez f
m)11i2
[(E,
[(E,
m)ll’*
+ m) (Ez -I- m)l’”
+ m) (Ez f
[(El + m) (E2 + m)P
[(El
ICEI + m) (E2 + ndl’”
[(El + ml (E2 + m)l’”
4-l
1 0 0
0
0
p*
N’ M
NM*=--
9
6 7 8
x
” G6 4rE1+m
10
1
1
2
-1 -1
m)P*
2W(E1
+ m) (Ez + m)
k2 q8
[(E,
0
1
0
0 1 +1
+ m) 6% + nW2
gr Gr + m) qe L% + m) t.& + m)P2
m) (E2 +
1
1
-1
+ m) (EP + m)P2
+1
1
2
+
5
between the graphs and values of X, see Fig. 1. For all
4~ 2m
= --1 YTFT
= 4,2W(&
f
qk [(El
’ ysFe qk [(El 4~ 2m
NE*=-- l
v&f@ =
VII-Continued
8 For the definition of the functions, see Section III. For the relationship graphs ff,~ = ff,4 = fh = .f,‘,, = f-h = fh4 = 0.
and
1 1 +1
1 1
0
$1
2 0
0
-
1
+
E
2 3 4 5
x
TABLE
THEORY
OF
PION
347
PHOTOPRODUCTION
Note that dX( W) and e’(W) do not depend on i. The constants do not depend either on 5 or on i. Finally we will write the numerator functions appearing in Eq. of fun&ions j’k$ (W), (T = 0, 1, 2). The incentive for this is the .j$;t ( W) functions would be the ones appearing in the numerator 5, amplitudes in a form similar to that of Eq. (3.10), i.e.
C?‘, D ‘, and r A (3.10) in terms fact that these if we wrote the
3i = fO,i + f1.j cos 8 + fi,< co2 e
(3.11)
and these j::: are already available in the literature for the few processes t’hat have been calculated previously. The relationships between the two kinds of numerator functions are given in Table VI. As we will see, in the present calculations only the fi,i’s, jf,, and j’f.2 are nonzero. In terms of the other set of numerator functions given by Eq. (3.10) this represents less of a simplification, since we have only Cl
I, A
=
E.A c2
=
0
(3.12)
There are of course other relations between these functions, 6EeX(W) = ci,“(W’J
and
biqh(W)
such as
= -cE,“(w)
(3.13)
The final results are given in Table VII. In as much as they have been calculat,ed before they agree with previous references. A detailed numerical evaluation of the present theory will be given in a forthcoming paper. We plan to determine the best values of the parameters by comparison with experimental data up to about 800 MeV laboratory phot,on energy (which is the approximate range of validity of the present calculation), and then make predictions for those physical observables which have not. been measured so far but which might be feasible with the latest experimental techniques. ACKNOWLEDGMENT
One of us (H. R.) is grateful to Professor while work on this paper was in progress. APPENDIX.
A a,
aft”(W) B bf ’ A( W)
R,
NOTATION
G. Hiihler AND
for the hospitality
extended
to him
KINEMATICS
one of the four coefficients of the invariant amplitudes as defined in ref. 17 (i = 1, 2, 3, 4) the coefficients of the four invariant amplitudes defined by Eq. (3.1) one of the numerator functions for af’ ‘( W, 0) (see Eq. (3.10) ) one of the four coefficients of the invariant amplitudes as defined in ref. 1’7 one of the numerator functions for @,f’“( W, 0) (see Eq. (3.10) ) (i = 1, 2, 3, 4) t,he four coefficients of the invariant amplitudes as defined in ref. 16
348
HARUN-AR-RASHID
c p(w) eA D diX( W) DA E, El E2 ei”(W>
GA
k
M,+, M,o 1Fb m* ** i’?,*,
N;*, Pl P2 9
NM* NY
AND
MORAVCSIK
one of the four coefficients of the invariant amplitudes as defined in ref. 17 one of the numerator functions for a!’ “( W, 0) (see Eq. (3.10) ) constant in the numerator of the imaginary parts of the af’ A (see Eq. (3.10)) one of the four coefficients of the invariant amplitudes as defined in ref. 17 one of the denominator functions for a!* “( W, ~9) (see Eq. (3.10) ) constant in the denominator of the imaginary parts of the af* “s (see Eq. (3.10)) photon energy in the laboratory system in MeV incoming nucleon total energy in ems, in units of pion mass outgoing nucleon total energy in ems, in units of pion mass one of the denominator functions for a!‘“( W, 0) (see Eq. (3.10) ) the electromagnetic coupling constant s ( 1/13i’)“2 (i = 1, 2, 3,4) the coefficients of the four invariant amplitudes as defined in ref. 5 coefficient functions in a cos 6 expansion of the ~i( W, e)‘s (see Table VI) “magnetic moment” coupling constant for the vertex in diagram h “charge” coupling constant for the vertex involving a photon in diagram X “charge” coupling constant for the vertex involving no photon but a boson in diagram X index denoting one of the four invariant amplitudes isospin coefficients (see Eq. (3.7) and Table V) index denoting one of the four photoproduction reaction, (see Eq. (3.8)) photon momentum in ems, in units of pion mass positive and neutral pion mass in MeV nucleon mass in units of pion mass N*( 1220) mass in units of pion mass N**( 1510) mass in units of pion mass combinations of coupling constants and kinematic factors (see Table VII for definition) incoming nucleon momentum in ems, in units of pion mass outgoing nucleon momentum in ems, in units of pion mass pion momentum in ems, in units of pion mass index on the numerator functions f?,;(W)
THEORY
ZN
OF
PION
PHOTOPRODUCTIOK
349
spinor for a spin 55 particle spinor for a spin j%$particle combination of coupling constants and kinematic factors (see Table VII for definition) rNN* vertices (Eq. 2.4) total energy in ems in units of pion mass a constant matrix giving t’he dependence of the contribution from the ten diagrams on the charge state index E combination of coupling constants and kinematic factors (see Table VII for definition) width of the particle forming the intermediat.e state in diagram X photon polarizat’ion four-vector = 0 if two indices are equal; = (?:} for an (‘o’;“d”)perturbation of the indices vector meson polarization four-vector production angle in ems index denoting one of the ten diagrams in Fig. 1 omega meson rest mass in units of pion mass rho meson rest mass in units of pion mass phi meson rest mass in units of pion mass index denot,ing one of the three c,harge state amplitudes (t = +, -, 0) = ?,i‘( YjlYY - Y.YU) angle of polarization of the photon wit,h respect to the production plane pion energy in ems in unit,s of pion mass four-vector three-vector
w a a x2 = x2 - Q2 a.b = a-b - a&,, pl? = p; = -& QL -1 k2 = 0 All kinematic quantities
will be calculated
from E, as follows:
q = (u" - 1y2
El = (m” + k2p2
w=k+-“.r,
El + k
Ez = (vb2 + q2)Ii2
w = k + (rn” + ii2)1/1
350
HARUN-AR-RASHID
AND
MORAVCSIK
In these formulas the pion massdifference is taken into account, but the nucleon massdifference is not. RECEIVED,
August 9, 1965 REFERENCES
1. K. A. BRUECKNER, Phys. Rev. 79, 641 (1950); M. F. KAPLON, Pkys. Rev. 83, 712 (1951). 2. K. A. BRUECKNER AND M. L. GOLDBERGER, Phys. Rev. 76, 1725 (1949). S. G. F. CHEW,
Phys.
Rev.
89, 591 (1953);
ibid.
94, 1749
(1954);
4. G. F. CHEW AND F. Low, Phys. Rev. 97, 1392 (1954). 5. G. CHEW, M. GOLDBERGER, F. Low, AND Y. NAMBU, Phys. 6. P. FINKLER, thesis, Purdue University, 1964 (unpublished);
ibid.
96, 285, 1669
(1954).
Rev. 106, 1345 (1957). J. M. MCKINLEY, Univ. G. HBHLER, AND K. DIETZ,
of Illinois Tech. Report No. 38 (1962) (unpublished); Z. Physik 160, 453 (1960) and private communication. 7. E. g., J. S. BALL, Phys. Rev. 134, 2014 (1961). 8. L. D. ROPER, Phys. Rev. Letters 12, 340 (1964); L. D. ROPER, R. M. WRIGHT, AND B. T. FELD, Phys. Rev., in press. 9. E. G. M. KIKUGaWA, Progr. Theoret. Phys. (k’yoto) 31, 656 (1964). 10. See, e. g., BRYAN, DISMUKES, AND RAMSEY,NTEZ. Phys. 46, 353 (1963). 11. F. SELLERI, Nuovo Cimento, in press. quoted therein. 12. M. J. MORAVCSIK, Ann. Phys. 30, 10 (1964) and references 13. M. GOURDIN .~ND PH. SALIN, Nuovo Cimento 27,193 and 310 (1963); PH. SALIN, Nuooo Cimento 28, 1294 (1963) and referewes quoted therein. 14. E.g. K. GOTTFRIED AND J. D. JACKSON, Nuovo Cimento 34, 735 (1964). 16. M. J. MORL4VCSIK, Phys. Rev. 126, 734 (1962). 16. B. ANDERSSON,NUC~. Phys., in press. 17. M. J. MORAVCSIK, Phys. Rev. 126, 1088 (1962). 18. H. A. RASHID, Nuovo Cimento 33, 965 (19G4).
Exchange
Theory
I. General A. Instifut
jiir
Theoretische
r\ I.
of Pion
Calculation*
HARUX-AR-RAGHID
h’emphysik
Photoproduction
der
i
Technische
Hochschule,
Karlsruhe,
Germany
AND
J.
MICHAEL Lawrence
Radiation
Laboratory,
MORAVCSIK
c7niversify
of California,
Livermore,
California
A calculation of single pion photoprodllction from single nucleons is given using the one-particle exchange model involving the exchange of one pion, one p, one W, one $, one nucleon, one L1’*, and one ‘ V**. Crossed diagrams, previously neglected, have also been included in the calculation. An imaginary part of the amplitudes is included using a Breit-Wigner type formula and experimentally observed widths. The t,heory should be a good representation of pion photoproduction up to about 800 MeV incident laboratory photon energy. A numerical evalnat,ion of the theory will be given in a subsequent paper. I. INTRODUCTION
Ever since the beginnings of pion physics in the late 1940’s, photoproduction of pions from nucleons, together with elastic pion-nucleon scattering, has played a central role in our knowledge of pi mesons. Accordingly, this process has received a large amount of theoretical attention. At the beginning, first order perturbation calculations (1) were carried out, using, as intermediate particles, only nucleons and pions. These calculations were in violent disagreement with the experimental results, and were soon abandoned. About the same time, particularly in connection with the ratios of the various pion charged states, some very simple-minded classical models (2) were also used, which gave qualitatively the right results. In the early 1950% the so-called static theory (3) was developed, which, from the point of view of the present paper, can be described as a first * Work performed mission. 7 On leave from Dacca, Pakistan.
partly the
under
Pakistan
the auspices Atomic
Energy 331
of the United Commission’s
States
Atomic Atomic
Energy Energy
ComCentre,
332
HARUN-AR-RASHID
AND
MORAVCSIK
order perturbation theory using pions and nucleons as intermediate particles, plus semiphenomenological terms describing the resonance in the J = 35, T = ,35 state. To be sure, the static theory was at the time presented in a different spirit, and its justification was also argued on a different basis. Nevertheless, its numerical success was based on the fact that, at low energies, the above mentioned resonance is the dominant feature of the pion-nucleon interaction. A somewhat refined version of the static theory was the so-called Chew-Low theory (4), with almost identical results but a somewhat more erudite justification. The next step in the theory of pion phenomena came with the introduction of one-dimensional dispersion theories. This, applied to the pion photoproduction process (5), yielded the CGLN theory. Again, from our present point of view, this can be described as a first order perturbation theory using only nucleons and pions as intermediate particles, plus a resonance term in the $5, N state, in which, in addition to certain phenomenological elements, restrictions of the dispersion relations themselves were also made use of. It was thought that this theory should give a good description of pion photoproduction up to, say, 400 R!IeV photon energy in the lab system. The CGLN theory in its original form used various approximations in the detailed evaluation of its formulae. Several alternate approximations (6) have been explored since, and it appears clear that in view of the uncertainties in these approximations the CGLN t.ype of theory even in the low energy region gives a prediction for the experimental observables which carries an error not smaller than 10%. As pion-pion interaction was “discovered” through the study of the nucleon electromagnetic form factors, attempts were made to incorporate this interaction into the pion-photoproduction calculations. The results (7) gave correction terms to the CGLN predictions which in general were small, but for some quantities (such as the negative to positive pion ratio near threshold) could be quite significant In the meantime, however, the experimental techniques have also been extended and it became possible to measure photoproduction of pions up to about 1.2 BeV. The first interpretation of such experiments was done in terms of phenomenology and in this way the so-called second, third, and fourth resonances were discovered. A similar phenomenological analysis, for pion-nucleon scattering, has reached a quite sophisticated stage by now (8), and led to the discovery of further structure in terms of angular momentum states. At the same time, however, the need also arose to calculate the physical observables in this new energy region in some more fundamental way, and to include in such a calculation the newly discovered heavy mesons of strangeness zero. Since even today the only calculational technique which can be employed without the necessity of
THEORY
OF
PIOh-
PHOTOPRODUCTION
333
crude approximations and without the lengthy use of high speed computers is first order perturbation theory (now rechristened “pole contribution”), the logical scheme to be used for calculating phot’oproduction turned out to be the so-called one-particle exchange theory, OPET, in which the pole contributions of all those particles are calculat’ed which are experiment’ally krlowrl and whose quantum numbers are appropriate for the particular diagram. Such OPET can of course also be used for other processes, and calculations have been carried out also for pion-nucleon scattering (9), nucleon-nucleon scattering (IO), pion production in pion-nucleon scattering (II), etc. In all cases the first results have been highly encouraging. In suc*h calculations the coupling constants other than that bet’ween the pion and the nucleon are used as adjustable parameters, but one would expect to get the same value for a given coupling constant from various react’ions, so that there is room for a check. It might be mentioned t#hat from the point of view of OPET the success of the static theory and t,he CGLS t)heory is easily understandable, since they are basically special cases of OPET. The OPET theory in it,s simplest form, in which the contributions are simply calculated from the first order pert’urbation diagram, can of course not be exactly correct. The specific objections one can raise are as follows. 1. The pole contributions to t’he amplitudes are always real. Hence unitarity is not satisfied and many experiment,al observables (such as polarization) are identically zero. It is possible, however, in several ways, to modify the amplitudes so as to make t,hem unitary (12). For example, when one includes as intermediat,e states particles which have a finite width, their propagator can be modified to include an imaginary part’, and this makes the corresponding amplitude also complex. Alt’hough the widths must be taken from experimental data, at least formally one can preserve unitarity this way. 2. One dimensional dispersion relations are not satisfied (12). This is a different problem from that, considered under 1, since the amplitudes could easily be unitary and still not satisfy a given form of dispersion relations. At the same time, it is also a less serious shortcoming, since in most cases the dispersion relations which are asked to be satisfied are not proven and have no empirical verification either. As things stand now, if one does demand that some dispersion relations be satisfied, one must insist on the presence of cert’ain contributions to the amplitudes which at present, cannot be calculated at all and whose precise physical origin is unltiown. 3. The OPET theory is obviously quite incomplete, since it must use experimental values for the masses and widths of the int’ermediate particles and contains coupling constants which are determined by fit to experimental data. This is readily admitted to be true, and simply means that once the validity of OPET
334
HARUN-AR-RASHID
AND
MORAVCSIK
in a form modified to eliminate objections 1 and perhaps 2 is established empirically, some additional dynamical principle (perhaps bootstrap, perhaps something else) will have to be introduced to explain why continuum contributions are negligible and why only those masses, widths, and coupling constants occur in nature which are observed. When presenting the results of OPET calculations, it has become customary in the literature to deduce the OPET as an approximation to something more highbrow, usually the Mandelstam representation. We would like to deplore this habit, since we believe that the road leading, for instance, from the full Mandelstam representation to an OPET scheme is so much loaded with drastic approximations and ad hoc assumptions as to make the OPET a completely incredible method of calculation. Furthermore, such a procedure might also give the completely erroneous impression that the OPET in some way preassumes the validity of the Mandelstam representation or any other part of the mathematical framework of X matrix or field theory. We believe that at this point it is much preferable to use the OPET as an assumption in itself, whose validity appears to be justified by its practical success. In the present paper we give an OPET calculation of pion photoproduction. This is not the first time that this has been done (13). Our justification for presenting such results is manifold. First’, we included in our calculations certain crossed diagrams which were omitted before on the basis that their contribution is likely to be much smaller than the contribution of those diagrams which were kept. We are not only unsure about the veracity of such statements in general, but in addition believe that because of the large cancellations often occurring in complicated processes involving many diagrams, it is unwise to neglect any terms a priori if they can be calculated wit,h relat,ively little additional effort. Our numerical evaluation of the results, to be presented in a later paper, will show if in this case the omission of the crossed diagrams in previous calculations lead to appreciable errors or not. Secondly, since the last calculations, certain new resonances were found which should also be included in the calculation. Thirdly, we plan to carry out a more thorough numerical investigation of OPET as applied to photoproduction, including data on polarization, studying the possibility of several solutions for the coupling constants, and giving predictions for those observables which have not been measured but whose measurement might now be feasible. Recently several authors (14) have developed modifications of the OPET in which corrections are included to take into account the presence of inelastic channels. Since our calculations do not intend to cover energies above 800 MeV and up to this energy inelastic channels do not play a dominant role, we believe that these absorption corrections will not be important in the first approximation. A more detailed investigation of this point is in progress.
THEORY
II.
SUMMAltY
OF
OF
PION
335
PHOTOPRODUCTION
PR,OCESSES
AND
INTERACTIONS
The diagrams included in the present calculation are shown in Fig. 1. The last three graphs are processes which were not included in previous calculations (1s). The vertex functions used in the calculations are listed in Table I. It was assumed that the quantum numbers of the various particles are those given in Table II. The notation used in the vertex functions, t,oget*her with all notation used in this paper, is given in the Appendix. The width of an intermediate particle is taken into account by formally substituting in the expressions for the mass of the particle 1~ + iI’, where I? is the width, taken from experiments. This will yield an imaginary part of the amplitudes which otherwise would be pure real. The precise form is given in the next section. The propagators used in the caIculat,ion are given in Table III. The spin 1 propagator agrees with that of ref. 15, and t,he propagator for spin $5 is the same as in ref. 13. It might be worth emphasizing that throughout this work we assume,like all others who have carried out similar calculations, that the coupling constants fol the int’eraction between two kinds of particles do not depend on the charge stat’e of the particles involved. This assumption is not, only questionable but in fact
\ 1 N
(5)
(6)
(IO) FIG. 1. Processes used in the present calculation. Wavy dashed lines pions, heavy dashed lines heavy zero-strangeness nucleons, heavy solid lines nucleon isobars.
lines denote photons, mesons, light solid
light, lines
336
HARUN-AR-RASHID
AND
TABLE FORMS
OF INTERACTIONS
MORAVCSIK
I
USED
IN
THIS
Particles
CALCULATION Interaction
T-N-N r--N-N* r - N -
N**
y-?r---w
y-N-N y-N-N* -,, -
N -
N**
p-N-N o-N-N 4-N-N
TABLE QUANTUM
NUMBERS Approximate mass (MeV)
N N* N** K P
w *
938 1238 1512 139
750 780 1020
OF PARTICLES
II USED
IN
THE
Spin
Parity
34 N 35 0 1 1 1
i+ -
PRESENT
CALCULATION Isotopic spin
% 5% % 1 1 0 0
G Parity
+ -
clearly untrue, sincefor instance the magnetic moments of the proton and neutron are distinctly different. The extension of the calculations to the more general case in which this assumption is not made is of course trivial, consisting only in carrying a superscript Eon the coupling constants. We did this in fact but only for the magnetic coupling constant of the y - N - N vertex. In previous numerical
THEORY
OF
PION
TABLE
337
PHOTOPRODUCTION
III
USED IN THE PRESENT CALCULATIONS
PROPAGATORS Spin
1
0
a The
letter
m2 -
q denotes
the
momentum
q2
of the intermediate
particle
with
mass
m.
work (as well as in our projected numerical analysis) the extra freedom of the charge dependence of all coupling constants is not taken advantage of, which might be worth keeping in mind in case “irreconcilable” discrepancies with experiments arise. Using the interactions given in Table I, the contributions of diagrams (l)-(5) to the 5a amplitudes of CGLN are well-known while those of diagrams (6) and (7)-also calculated by Gourdin and Salin (IS)-are given as follows:
& = --
NM” cos e ,m*z - s w,*
3 2 = -AN,*+ 3
N,* m*2-s
33 = ___ m*2 - s
34 = 0
3 -;N;*+N; l - 3 m**2 - s
Nn”; 32 = m**z - s cos e
(2.1)
and (2.2)
33 = 0 The calculations for the crossed diagram (9) are much simplified by neglecting the Pauli interaction term and the contributions to the F, amplitudes of CGLN are given by
R(W) =i!$ ; [w- m:+em + W* wy: 1; 2nz] m~2 l- 2L Fz(W)
= --R(-W)
Fd(W)
= Fd -W>
(2.3)
338
HARUN-AR-RASHID
AND
MORAVCSIK
while those of diagram (10) can be simply obtained by the transformation w* = -m** In Table I, we have used the gauge-noninvariant vertex ‘u
=
M-0
G6iY6QJ
coupling
u(p)
+&mh:,cr
(
but as pointed variant form
out by Gourdin
for the -yNN*-
>
and Salin one can write
the following
gauge-in-
‘UI = - ~‘,(P’)[G&w~~xQ - bx>lu(p> and can easily deduce that in that case G6
=
(m
+
m*>
Fa
(26’
Using the coupling 2)1 the contributions
-
2m
=
of diagram
G6’
(6) are given by
5 = w + m/3 CN 2 W + m ?)z*~- s 53=--
(2.5)
2w CN W + m ms2--s
54
=
0
where CN = ‘$
pk Z/(El
+ m) (E2 + m)
It is easy to see that Eqs. (2.1) and (2.5) satisfy the relations (2.4). The great advantage of using the gauge-invariant form is that with this we can maintain crossing symmetry and the contributions of diagram (9) can be simply obtained. The results are better given in terms of the invariant amplitudes Ai of CGLN:
AI = m?25[
-q-k
-;
+ &*
fh
+ WL*>(2 p’.q - p.k)
- iP’.q
m(m
+ m*)
- m1
+ pz*k (2.6)
Aa = m.f[A, = ,f
:’
-i(m+m*) U
where P’eq = (m*2 - m” + 1)/2.
-&{i-2Pr*q}-m*P’eq]
THEORY
OF
PION
For the second resonance, Gourdin
and Salin have suggested the coupling
G’PJ=x(p’)u(p) and using this the contributions
339
PHOTOPRODUCTION
(kc~ - ICXE~)
(2.7)
of diagram (7) are given by
where CN’ = (gTGT’/4a)kW v’(E, + m)(Ez +m)($/(Ez + m)) while those of diagram (10) which need not be quoted here are again obtained simply from crossing symmetry. III.
RESULTS
FOR
THE
AMPLITUDES
Invariant amplitudes for pion photoproduction from nucleons have been defined in a number of ways. Among these, there are the %i functions of ref. 5, the Bi functions of ref. 16, and the A, B, C, and D functions of ref. 17. In this paper we will disgust the reader by defining another set of amplikrdes, which however are quite closely related either to the A, B, C, D or to the Bi’s. In particular we have al = A(sin 0)-’ = B4 ct2 = B = B1 + BB cos 0 a3 = C = Bz a4 = Dsin0
= -B1cos6
(3.1) + BSsin2B
The advantage of these amplitudes lies in the fact that while they maintain the features of simplicity which characterize the A, B, C, D amplitudes, they do not become infinite at 0 = 0” or HO”, but at the same time leave the formulae for the observables in a quite symmetric form. The connection (18) between the amplitudes ai and the amplitudes pi is given as follows: s = -alcOse + a3 32 = --a1 53
=
a1
+
54
=
a,
-
cos sin2
B a3
+
1 -;---a3-7
he cos
sin2 0
a4
(3.2)
e
sin2 0
a4
and conversely 61 = -32 a2
All these invariant
=
31
+
54
+
53 cos
cf2cos
e
a4
5h cos
e +
=
amplitudes
52
-
are functions
e
(3.3)
a3 = cfl -
53 sin2 e
of W and cos 8.
340
HARUN-AR-RASHID
AND
MORAVCSIK
The experimental observables are given in terms of the amplitudes C?,;by the formulas in Table IV. The constant K in front of all observables is given by K = 389.35 barns q MT2
(3.4)
i%
The observables are given for linearly polarized photons, with Q = 0 denoting photons polarized parallel to the production plane, and C$= 90” denotes photons polarized in the k X q direction. For unpolarized photons one simply makes the substitutions cos24 = sin” I$ = >s
sin I$ cos @I= 0
(3.5)
Each of the amplitudes can be written as a sum of contributions from the various processes (for the notation on the subscripts and superscripts, see the Appendix). Thus we have a?( w, cos e) = c af- “( w, cos e>
(3.6)
A
There are three c&% for each i, corresponding to the three values of the isospin index, and the total amplitude for a given process (indexed by j) is given by (j’& = c (j)PJ’@e= (313+ai+ + W3- ai- + (A30a;
(3.7)
E
TABLE IV EXPERIMENTAL
OBSERVABLES
OF PION
Ai INVARIANT
PHOTOPRODUCTION
IN TERMS
OF THE
AMPLITUDES*
= K([I Al 12sin2B+ 1A8 Ia] sin24 + [I AZ 1%sin2 0 + / A4 I21cos241 ZOPl = K(2 Im (Aa*A4 - A,*Az sin2 0) sin 4 cos 41 ZOP, = K( -2 Im A1*A3 sin o sin2 4 - 2 Im At*Ad sin 0 cos24) zoprl = K{2 Im (A2*Aa - AI*A4) sin 0 sin 4 cos 41 IO
= K( -2 Im = K( -2 Im = K(-2 Im
IOTZI
=
Id1
IoTm = IOT, = IoTm = IoTmr IoTz
= =
IoTrsz = ITo mn = IOTl&, 9 The
(A3*A4 + A1*A2 sin2 0) sin 4 cos 4) A1*A3 sin e sin2 4 + 2 Im A2*A4 sin e cos2 41 (A2*Aa + A1*A4) sin 0 sin 4 cos 4) K([I Al 1%sin* e - 1 As 121sin2 4 + (I A:! I* sin20 - 1A4 I21COS* 41 K( [I A’ 12 sin2 e + 1 Al 121 sin2 4 - (I AS I2 sin2e + I Ad I21cos24t K( [I A1 12sin2 e - I Al I21sin24 + (I A4 I2 - I A2 I2sin28) COS* 41 K(2 Re (A1*A4 + A2*A,) sin 0 sin 4 cos 4) K(2 Re (AZ*& - AI*A4) sin 0 sin 4 cos 41 K( -2 Re A,*A, sin 0 sin2 4 + 2 Re Az*Ah sin B cos24) K(2 Re A,*Aa sin e sin2 4 + 2 Re A2*A4 sin 0 COG41 K{2 Re (A1*A2 sin2 e + Aa*Ad) sin 4 cos 41 K(2 Re (Aa*A4 - A1*A, sine 0) sin 4 cos 4)
Id, IllA,
=
constant
K is given
by 389.35
qM;%-1
barns.
OF
THEORY
where the assignment
PION
341
PHOTOPRODUCTION
of the j’s is as follows j=l
r+P
j=a
y+p+n+r+
j=3
y+n-+n+3r”
j=4
Y+n--tp+7r-
-+p+rO (3.8)
and the (i)~t’~ are con&ants given by Table V. One should note that forj = 1, 3 in the kinematic relations M,o is to be used, while forj = 2, 4 we must use M,+. Thus a given W will correspond to a different E, forj = 1, 3 than forj = 2,4. Each of the ai* “s can be written in terms of real and imaginary parts @f*“(W, cos8) = Re@f”(W,
cos8) + iIm@,!‘X(W,
cos8)
(3.9)
and we further write Re @f*“(W, cos 0) =
&“(W>
+ p(w) cos 8 + cfq w> co2 e cP(W) + e”(W) cos 0
TABLE CHARGE
STATE
(3.10)
V COEFFICIENTS+ j
5 1
+
2
01 1
0 9 See Eq.
Y
10
$5 a
-1
-z/5 6
(3.7). TABLE
RELATIONSHIP
3
BETWEEN
THE
Two
VI
SETS OF NUMERATOR
FUNCTIONS
IN SECTION
III
up (WI = -I$“, (W)
a!” (W) = f S:?(W)
bf,’ (W) = -j$
cp (W) = -j$$ (W)
biX (WI = f f*: (WI - fii:; (WI c5” (W) = fS4 (WI - rf:; on
up (W) = ri:: CM’)+ f 6:: (WI bisxW = f f:: (W) + f 6,; (WI cp (W) = If:: (W) + f is (WI
UP (W) = f i$ (WI + f s:“3 (W) btx (W) = f f:; (W) + f f:; (W) - fi:: (WI CP (W) = -jf:: (W) - f&i (WI
(W)
A=
29 Z" Xl{ ___ W+m
28k 0 0 0
-2wk
2q2-cl X1r (Ez + m) (W -
-
1
AMPLITUDES
m)
,+2 -2qk 0 0 0
V,8
m)VdI
WSm
wk
W-m
wk
2
+ (W -
+ (W -
1 + 2wk
2
X2E &I-
X& q [- V.8
-W-m+-
W-m-----
PHOTOPRODUCTION
Ez+m
-&?a
FOR PION
m)VdI
(1 -
2&V&‘)
2wk)VM
1 1-(1 -
VII
THE VARIOUS
TABLE FROM
p} q +
m
vg”
-
m-&I
EXCHANGE
p,2 -2qk 0 0 0
X.4 &
1 + 2wk
[- vsw
+
-
(1 -
(w - m)VM”l
+ (W - m)V.+f”l
3
PROCESSES~
{[-w-~~+~I-(‘-“‘)“u.
&
X$q(-VVEw
.
-X35
X&{VBw[W
ONE-PARTICLE
~klVrw}
0
X=
2qk
kW+m El+m
m + m)
~
W+m*
-
28
2q dW-mW+m -
El + m
+ m
R + m (W + m)”
&+mW-m m* + 2kE1
&tNe*
2qk 0 0 0
-f
f &tNE*
El + (W + m)(Ez
W-m*-
m**
m**+m*-1-mm*9
m**
9
2ok
m’“-mfl 2m*
W+m k
w - m* - W - m+
2wk -+ W-m
qE1+mm**+m2-1-mm*W+m*
; X&NE* __
X&NE*
X&NE*
m** -
e W+m
Fd
)I
El + m ~ q(W + m)
VII-Continued
-f ; --
; X&NE*
-2qk 0 0 0
Ez+m
___
- 2h&
-x*rz,Q--
2qk
- X&Z, -e W-m -
F&
W-m-
8
TABLE
W-m
W-m W$m*-2mm+2-m2fl
W+m*-2m
2m*
1
1
q2
2 W (ES -t
q
2qk 0 0 0
In**2 -
-l&CL-
xfo-q---
1
m2 + 2kEx
qE2fmW2-mZ
vi’ + wt
2k2
2W(Et
m**l
q
___-
10
+ m) iv;
m**z m**z
-
IV - nz
1
2m,**
- 2m m**
tn**
w - m - m2 + 1 IV -
m2 + 1 W -
2m**
-
.%A --w - m
-
2WCE2 + m) )\,;* __- k IV + m P2
w + w1** - $$&
1 + mm**
m) N;*
we - WI,* m,*** + IQ -
p W + m EQ + m
k q -Xf&---
1 lc qW+m
XEO -
where Xhc is a matrix as follows
A=
-
2m
1
Liz Q,
& = Y!2 &
1
1
4rm,
vr*
=
VMW = __ 4m,
v~P=--Gz-
4mi,
1 v,p = __
km
G,F” 2m &
Ga ~3 em
:L
W+m
G2 yz JfL + m) (Ez f
m)11i2
[(E,
[(E,
m)ll’*
+ m) (Ez -I- m)l’”
+ m) (Ez f
[(El + m) (E2 + m)P
[(El
ICEI + m) (E2 + ndl’”
[(El + ml (E2 + m)l’”
4-l
1 0 0
0
0
p*
N’ M
NM*=--
9
6 7 8
x
” G6 4rE1+m
10
1
1
2
-1 -1
m)P*
2W(E1
+ m) (Ez + m)
k2 q8
[(E,
0
1
0
0 1 +1
+ m) 6% + nW2
gr Gr + m) qe L% + m) t.& + m)P2
m) (E2 +
1
1
-1
+ m) (EP + m)P2
+1
1
2
+
5
between the graphs and values of X, see Fig. 1. For all
4~ 2m
= --1 YTFT
= 4,2W(&
f
qk [(El
’ ysFe qk [(El 4~ 2m
NE*=-- l
v&f@ =
VII-Continued
8 For the definition of the functions, see Section III. For the relationship graphs ff,~ = ff,4 = fh = .f,‘,, = f-h = fh4 = 0.
and
1 1 +1
1 1
0
$1
2 0
0
-
1
+
E
2 3 4 5
x
TABLE
THEORY
OF
PION
347
PHOTOPRODUCTION
Note that dX( W) and e’(W) do not depend on i. The constants do not depend either on 5 or on i. Finally we will write the numerator functions appearing in Eq. of fun&ions j’k$ (W), (T = 0, 1, 2). The incentive for this is the .j$;t ( W) functions would be the ones appearing in the numerator 5, amplitudes in a form similar to that of Eq. (3.10), i.e.
C?‘, D ‘, and r A (3.10) in terms fact that these if we wrote the
3i = fO,i + f1.j cos 8 + fi,< co2 e
(3.11)
and these j::: are already available in the literature for the few processes t’hat have been calculated previously. The relationships between the two kinds of numerator functions are given in Table VI. As we will see, in the present calculations only the fi,i’s, jf,, and j’f.2 are nonzero. In terms of the other set of numerator functions given by Eq. (3.10) this represents less of a simplification, since we have only Cl
I, A
=
E.A c2
=
0
(3.12)
There are of course other relations between these functions, 6EeX(W) = ci,“(W’J
and
biqh(W)
such as
= -cE,“(w)
(3.13)
The final results are given in Table VII. In as much as they have been calculat,ed before they agree with previous references. A detailed numerical evaluation of the present theory will be given in a forthcoming paper. We plan to determine the best values of the parameters by comparison with experimental data up to about 800 MeV laboratory phot,on energy (which is the approximate range of validity of the present calculation), and then make predictions for those physical observables which have not. been measured so far but which might be feasible with the latest experimental techniques. ACKNOWLEDGMENT
One of us (H. R.) is grateful to Professor while work on this paper was in progress. APPENDIX.
A a,
aft”(W) B bf ’ A( W)
R,
NOTATION
G. Hiihler AND
for the hospitality
extended
to him
KINEMATICS
one of the four coefficients of the invariant amplitudes as defined in ref. 17 (i = 1, 2, 3, 4) the coefficients of the four invariant amplitudes defined by Eq. (3.1) one of the numerator functions for af’ ‘( W, 0) (see Eq. (3.10) ) one of the four coefficients of the invariant amplitudes as defined in ref. 1’7 one of the numerator functions for @,f’“( W, 0) (see Eq. (3.10) ) (i = 1, 2, 3, 4) t,he four coefficients of the invariant amplitudes as defined in ref. 16
348
HARUN-AR-RASHID
c p(w) eA D diX( W) DA E, El E2 ei”(W>
GA
k
M,+, M,o 1Fb m* ** i’?,*,
N;*, Pl P2 9
NM* NY
AND
MORAVCSIK
one of the four coefficients of the invariant amplitudes as defined in ref. 17 one of the numerator functions for a!’ “( W, 0) (see Eq. (3.10) ) constant in the numerator of the imaginary parts of the af’ A (see Eq. (3.10)) one of the four coefficients of the invariant amplitudes as defined in ref. 17 one of the denominator functions for a!* “( W, ~9) (see Eq. (3.10) ) constant in the denominator of the imaginary parts of the af* “s (see Eq. (3.10)) photon energy in the laboratory system in MeV incoming nucleon total energy in ems, in units of pion mass outgoing nucleon total energy in ems, in units of pion mass one of the denominator functions for a!‘“( W, 0) (see Eq. (3.10) ) the electromagnetic coupling constant s ( 1/13i’)“2 (i = 1, 2, 3,4) the coefficients of the four invariant amplitudes as defined in ref. 5 coefficient functions in a cos 6 expansion of the ~i( W, e)‘s (see Table VI) “magnetic moment” coupling constant for the vertex in diagram h “charge” coupling constant for the vertex involving a photon in diagram X “charge” coupling constant for the vertex involving no photon but a boson in diagram X index denoting one of the four invariant amplitudes isospin coefficients (see Eq. (3.7) and Table V) index denoting one of the four photoproduction reaction, (see Eq. (3.8)) photon momentum in ems, in units of pion mass positive and neutral pion mass in MeV nucleon mass in units of pion mass N*( 1220) mass in units of pion mass N**( 1510) mass in units of pion mass combinations of coupling constants and kinematic factors (see Table VII for definition) incoming nucleon momentum in ems, in units of pion mass outgoing nucleon momentum in ems, in units of pion mass pion momentum in ems, in units of pion mass index on the numerator functions f?,;(W)
THEORY
ZN
OF
PION
PHOTOPRODUCTIOK
349
spinor for a spin 55 particle spinor for a spin j%$particle combination of coupling constants and kinematic factors (see Table VII for definition) rNN* vertices (Eq. 2.4) total energy in ems in units of pion mass a constant matrix giving t’he dependence of the contribution from the ten diagrams on the charge state index E combination of coupling constants and kinematic factors (see Table VII for definition) width of the particle forming the intermediat.e state in diagram X photon polarizat’ion four-vector = 0 if two indices are equal; = (?:} for an (‘o’;“d”)perturbation of the indices vector meson polarization four-vector production angle in ems index denoting one of the ten diagrams in Fig. 1 omega meson rest mass in units of pion mass rho meson rest mass in units of pion mass phi meson rest mass in units of pion mass index denot,ing one of the three c,harge state amplitudes (t = +, -, 0) = ?,i‘( YjlYY - Y.YU) angle of polarization of the photon wit,h respect to the production plane pion energy in ems in unit,s of pion mass four-vector three-vector
w a a x2 = x2 - Q2 a.b = a-b - a&,, pl? = p; = -& QL -1 k2 = 0 All kinematic quantities
will be calculated
from E, as follows:
q = (u" - 1y2
El = (m” + k2p2
w=k+-“.r,
El + k
Ez = (vb2 + q2)Ii2
w = k + (rn” + ii2)1/1
350
HARUN-AR-RASHID
AND
MORAVCSIK
In these formulas the pion massdifference is taken into account, but the nucleon massdifference is not. RECEIVED,
August 9, 1965 REFERENCES
1. K. A. BRUECKNER, Phys. Rev. 79, 641 (1950); M. F. KAPLON, Pkys. Rev. 83, 712 (1951). 2. K. A. BRUECKNER AND M. L. GOLDBERGER, Phys. Rev. 76, 1725 (1949). S. G. F. CHEW,
Phys.
Rev.
89, 591 (1953);
ibid.
94, 1749
(1954);
4. G. F. CHEW AND F. Low, Phys. Rev. 97, 1392 (1954). 5. G. CHEW, M. GOLDBERGER, F. Low, AND Y. NAMBU, Phys. 6. P. FINKLER, thesis, Purdue University, 1964 (unpublished);
ibid.
96, 285, 1669
(1954).
Rev. 106, 1345 (1957). J. M. MCKINLEY, Univ. G. HBHLER, AND K. DIETZ,
of Illinois Tech. Report No. 38 (1962) (unpublished); Z. Physik 160, 453 (1960) and private communication. 7. E. g., J. S. BALL, Phys. Rev. 134, 2014 (1961). 8. L. D. ROPER, Phys. Rev. Letters 12, 340 (1964); L. D. ROPER, R. M. WRIGHT, AND B. T. FELD, Phys. Rev., in press. 9. E. G. M. KIKUGaWA, Progr. Theoret. Phys. (k’yoto) 31, 656 (1964). 10. See, e. g., BRYAN, DISMUKES, AND RAMSEY,NTEZ. Phys. 46, 353 (1963). 11. F. SELLERI, Nuovo Cimento, in press. quoted therein. 12. M. J. MORAVCSIK, Ann. Phys. 30, 10 (1964) and references 13. M. GOURDIN .~ND PH. SALIN, Nuovo Cimento 27,193 and 310 (1963); PH. SALIN, Nuooo Cimento 28, 1294 (1963) and referewes quoted therein. 14. E.g. K. GOTTFRIED AND J. D. JACKSON, Nuovo Cimento 34, 735 (1964). 16. M. J. MORL4VCSIK, Phys. Rev. 126, 734 (1962). 16. B. ANDERSSON,NUC~. Phys., in press. 17. M. J. MORAVCSIK, Phys. Rev. 126, 1088 (1962). 18. H. A. RASHID, Nuovo Cimento 33, 965 (19G4).