Application of Dispersion Relations to Proton Compton Scattering Citation Mathews, Jon (1957) Application of Dispersion Relations to Proton Compton Scattering. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/GFXM-F011. https://resolver.caltech.edu/CaltechETD:etd-07142004-160311 Abstract Dispersion theory has been applied to proton Compton scattering at all angles. Unitarity of the S matrix determines the imaginary part of the scattering amplitude; the only absorptive process considered is pion photoproduction. The problem of analytic continuation of amplitudes has been handled in a natural way, which should be valid for scattering energies up to several hundred Mev. Numerical results calculated in a static approximation are in essential agreement with present experimental data. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Physics and Mathematics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Minor Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Gell-Mann, Murray Thesis Committee: Unknown, Unknown Defense Date: 1 January 1957 Record Number: CaltechETD:etd-07142004-160311 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-07142004-160311 DOI: 10.7907/GFXM-F011 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 2883 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 16 Jul 2004 Last Modified: 11 Oct 2023 23:05 Thesis Files Preview PDF (Mathews_j_1957.pdf) - Final Version See Usage Policy. 2MB Repository Staff Only: item control page

APPLICATION OF DISPERSION RELATIONS TO PROTON COMPTON SCATTERING Thesis by Jon Mathews In Partial Fulfillment of the Requirements For the Degree of Dector of Philosophy California Institute of Technology 1957 ACKNOWLEDGEMENTS The author is greatly indebted to Professor Murray Gell-Mann for sugg¢gsting this problem and for giving considerable advice during the course of the ensuing research. The author also wishes to thank Dr. Ricardo Gomez for many discussions of the M.I.T. experimental results and of Compton scattering in general, Dr. Ge Bernardini for forwarding to the author the results of the Illinois group, and Drs. V. 2 Peterson and W. Se McDonald for communicating their analysis of neutral pion photoproduction in advance of publication. Finally, the author is very grateful to the National Science Foundation for the predoctoral fellowship which was awarded him for his graduate studys ABSTRACT Dispersion theory has been applied to proton Compton scattering at all angles. Unitarity of the S matrix determines the imaginary part of the scattering amplitude; the only absorptive process considered is pion photoproduction. The problem of analytic continuation of amplitudes has been handled in a natural way, which should be valid for scattering energies up to several hundred Mev. Numerical results calculated in a static approximation are in essential agreement with present experimental data. TABLE OF CONTENTS PART TITLE Acknowledgements Abstract Introduction T Unitarity Relations IT Dispersion Formulas III Transformation of Amplitudes IV Numerical Applications and Discussion References Appendices Figures PAGE 10 26 31 42 63 Introduction Even if one accepts present-day field theory - including pions - as valid, it is very difficult to calculate any quantity involving pions, since the size of the pion-nucleon coupling constant renders perturbation methods of dubious value. Hence it is very desirable to seek out properties of field theory not requiring a perturbation expansion. An example is the derivation of so-called dispersion relations for scattering amplitudes. This originated with the work of Kramers (1) and Heisenberg, which relied on the correspondence principle and closely paralleled the classical theory of dispersion. In recent years the concept of "microscopic causality," the commuting of field operators at points separated by a space-like interval, has been applied by Gell-Mann, Goldberger and others (2)~(4) to derive dispersion relations within the framework of quantum field theory. Another important property of field theory is the basic idea of conservation of probability, or, more formally, the unitarity of the (5) S matrix. The most familiar theorem related to unitarity is the well-known relation between the imaginary part of a forward seattering amplitude and the total cross-section. The program of this thesis is as follows: Dispersion theory is to be used to predict the elastic gamma-ray scattering from free protons. Since dispersion relations are conveniently written as relations between the real and imaginary parts of scattering amplitudes, we use the unitarity of the 5 matrix to deduce the imaginary Compton amplitude, and then get the real part from dispersion theory. The unitarity principle gives the imaginary part of the scattering amplitude as a bilinear form in amplitudes for all processes resulting from gamma-rays incident on protons. The only such process we shall take into consideration is pion photoproduction. There are at least three significant processes which occur when gamma-rays with energies of several hundred Mev strike photons. The predominant one is electron pair preduction. The dispersion scattering resulting from this process is known as Delbruck scattering.* Not much is known quantitatively about non-forward Delbruck scattering, but the cross-section is certainly negligible except for extreme forward angles (ane k/m, where k is the photon energy and m is the electron mass)» At such angles experimental difficulties make it impossible to measure Compton scattering anyway. The second absorptive process is pion photoproduction, which has been studied quite thoroughly for photon energies up to 500 Mev. Since the charged and neutral cross-sections both show a resonance at about 300 Mev, we should be able to obtain reliable low enorgy results without knowing the photoproduction cross-sections at higher energies; the relevant integrals converge rapidly and contributions from beyond 500 Mev are apparently only small corrections which can be estimated by extrapolating the experimental curves. Incidentally, one can estimate the effect of this resonance by the Breit-Wigner single~level formila for total cross~section. If the pion and gamma-ray widths are determined from scattering and photoproduction * A brief discussion and some references are given in reference (6) data, ‘”) one finds that the total gamma-ray cross-section contributed by this resonance at 300 Mev should be about 3.5 times the zero-energy cross-section. The more precise calculations of this thesis, and the few experimental points in this region, suggest that this considerably underestimates the true effect, which is some five or more times the zero-—enorgy cross-section. Finally, the third process is elastic gamma-ray scattering, which is the process we are trying to predict. Fortunately, the cross~section is several orders of magnitude smaller than photomeson cross-sections, so that it is a reasonable approximation to neglect the absorptive effects of Compton scattering. This procedure is clearly equivalent to neglecting all terms of order e* and higher in the Compton scattering amplitudes The situation is quite different for pion scattering; here the dispersion relations become integral equations for the scattering amplitudes, rather than integrals which give the amplitudes directly. In Section I of this thesis we derive the unitarity relations which give us the imaginary parts of the Compton amplitudes in terms of photo- production amplitudes. In Section II we discuss the dispersion formalism in general, and its application to Compton scattering amplitudes. The unitarity relations are expressed most naturally in the center-of- momentum (hereafter CM) system, whereas a different coordinate system is used for the dispersion relations; the transformation of amplitudes between these two systems is discussed in Section TJI. Finally, in Section IV, approximations are made and some numerical computations are given. Several appendices contain formulas and calculations which would be inconvenient interruptions in the main body of the thesis» I - Unitarity Relations We begin by mentioning some conventions to be used in this thesis. Units are chosen so that fi =c=1. In numerical work M, the proton mass, is also set equal to unity. The unit of cross-section is then (n/Me)* = 442 x 107-8 om” = 442 microbarns -e denotes the rationalized charge of the electron: 2 e“/4ntic = 1/137 Three-vectors are written @; if @-@-=1 we often write a instead. In general by 4 we mean a /\A. A four-vector is written a or a3 by a-b we mean ab. = a>, - a,b) - anb, - as We shall begin with the formal relation - 4 Seq = Opg ~ (2m) Slop = Py) Rog relating the R and S matrices for scattering from a state i, with total four-momentum Py to a state f with total four-momentum Pee When we calculate a scattering amplitude, even in the forward direction, we get R, not S, since we are calculating lime,; S, so that we do not pick up the Os term. Calculation with the well-known Feynman rules gives -iR. In this thesis we shall refer to R as the "scattering amplitude." The cross-section for a process i-f is - 4 2 where the summation extends over all final states observed. For example, if we have a two-particle final state, 3 3 = ( ap. (: Ph (2n)* §(B.4D,-2,) $ (5,481, 189, |? Va) (an)?! (2m)? 3 a(% Pe (an) $(2 48,8, ) |r, |? Vs" (on)? 2 = an erlReal (1) i which is the usual rule. vA We now assume that the S matrix is unitary, which is equivalent to conservation of probability. (>) From the relation s*s=1 we deduce immediately Re, - Re = -(21)* i= 8(p -p) RAR. (2) i if n n nf “ni (p, = Dp = p) We shall now specialize: State i is a photon key @ and a proton X, Be State f is a photon k', e! and a proton “kt, Bue State n is a meson Ge ont plus a nucleon “a, Eee The notation is fairly conventional, k denoting photon momenta; e, polarizations; q, meson momenta; W, meson energies, and EB, proton energies. us = q” + mn” ne = Kk" + we It is clear we must work in the CM system. If, for example, the initial proton were at rest, the final one would be recoiling at some angle, so that Rar would refer to photoproduction in some odd system, neither lab nor CM nor anything else in particular. The states in (2) are related by their total four-momenta p3 for the CM system p = O, whereas no such characterization in terms of total momentum defines the lab systems With these substitutions, (2) becomes Rit fi if ch Ra. = w(2n)* 4 | 1 dle +E ok) R# Rs (an)? (25)? al 4, e E 2 { aX, REAR (3) T E+ +B =k t ( “a q A) Note that we must sum over the possible charges (+ and 0) of the pion. Also we have omitted reference to the nucleon spin states; the O matrices will take care of themaclves if we interpret * as Hermitian conjugation. We write our photoproduction amplitude as ok + £, (yx) e-qia-g a A A ~m A A A Ria = fy (kyx) e-kxg + £,, (kx) id-e + £ (keyx) e-q i (4) OQ > — (x =k. uw Since R must be of the form $A, with A a pseudovector, the four amplitudes in (4) are the only ones possible. We shall always assume transverse gauge in this thesis; k-e = Os In order to substitute (4) into (3) and integrate over q, we must do some integrals, such as 1, =|aQ £, (xt) £,(x) (xskeg y x'ke 4) 47 d 4 f, x f, x x=k-q » x's". q Choosing k as axis of spherical polar coordinates, and changing the integration variables to x and x', we obtain I= af dx dx! f, (xt) f(x) (y = kk!) where K"(xyx",y) =lex. (xt)? - yx” + 2xxty and integration is to be performed only over regions for which K is real. Proceeding in similar fashion we do the other required integrals; the results are in Appendix A. We now write the Compton scattering amplitude in the form Ros = By (kyy) 6-6! + go(kyy) e-k! eu k + ga (kyy) ig-e xt PA By ov 85 eV £3 95 + g, (key) @-o! 1d-k xk! 4 8 (kyy) ig (e-k! ot xk = olek @ x kt) nA ~ + ap (ky) 46-(O-kt Ot x kt - Otek 3 x k) (5) We have omitted two amplitudes, obtainable by replacing the minus signs by plus signs in the fifth and sixth terms. In general, these amplitudes will be expected to occur, but only in order a’; we are neglecting such terms, The vanishing of these amplitudes to order e” results from two effects: There are no such terms in the Born approximation (i.e., the Klein-Nishina amplitude), and the appearance of such terms in the dispersion contributions is forbidden by the theorem!” equating the phase of a meson photoproduction matrix element to the corresponding meson scattering phase shift. This theorem breaks down for terms of order e? in the photoproduction amplitude, and so terms of order et will appear in the Compton amplitude. If the amplitude (5) is squared and averaged over spins and polarizations, and if the appropriate density of states is used, we find for the Compton cross-section in the OM system: 2 2 2 2 4, (a 5/AQ) * (kE, /2nE) | ale (1+cos~6) + le, sin’6 + (Re eye. *Re ge, tORe etec-2Re g3e,-2Re B78) sin“Gcos® - 2Re 878. sin“® cos’6 + \e.|* sin“® (142c0s"@) (6) + 2fe,|* (3-cos“6) + 2|e,\* sin’6 (1+cos“0) +(3 fa 2. Ake e3e,) sin*e | We shall write this in the form (ag/aQ) 7 (e7/4nm)* (A + B cos® + C cos*® + D cos*® + E cos“8) (7) If we now do the integration in (3) by means of the formulas in Appendix A, we obtain the imaginary parts of the amplitudes Ss as bilinear integral forms in the quantities fi and f, » These integrals are given in Appendix B. The formulas appear to be too cumbersome for direct application. In the forward direction (y=1) the kernels of Appendix B become singular. For example 1 = 1 S(x = x!) K(x,x"y1) Working out the limits of the remaining kernels as y1, we obtain the formulas of Appendix C for Im g, (ky1). We have a check available for the first and third of these, since the former must be (-£/2E, ) times the total photomeson cross-section for unpolarized light, and the latter must be (-E/4E, ) times the total cross-section for circular polarization parallel to proton spin minus the total cross-section for polarization opposite to proton spine Hoth of these checks, which are derivable from (2), are easily verified. It should be noted that the correct velocity v, to use in formula (1) is the sum of photon and proton velocities, 1 + (k/E,,) » if we are in the CM system. (See Appendix F) Since we need the imaginary parts of Bs at all angles, and the formulas in Appendix B are very complicated, we shall take advantage of the fact that photomeson production at energies up to 500 Mev appears to be capable of description in terms of a very small number of multipole amplitudes. We shall adopt the notation of Gell-Mann and Watson”) and write Ea fe aA nN 6-(k °) ; Rad = fr iS-e ~ fn xeqg=i60(k xe) xq “ a 4 ay 4 m a > Ann a nA -f [2k x 6-4 +io(k xe) xq} +] fy, 16(k eg + © k-q) (8) M3 9 2 Note that we use M3 to mean magnetic dipole in the J=3/2 state, not magnetic octupole. In the notation of (4), fy = fy + 2 fg *3 =~ fm * tug * 3S te fo = aoa +X fn - X fu3 +i x fro f) = 0 2 We may now insert (8) directly into the unitarity relation (3), and obtain formulas for Im Bs which are much more tractable than those of Appendix B: Im g = ~ (qu 8/208) * (lee 24 lf 4 2y fh ° + legal” ) 10 Im gy = ~(qu,i,/2nB) z (=|fal* = altel? + 2 ltl *.) Im gy * ~(qu,B,/2n8) = ( -|z,.| 2 y lta 24 y \fy3 2 L. y\fuo| 2 +y fitsfpo ) (9) Im B, > ~(qa EB /enk) x ( ~ [fy + \s,,] - 1 \tep ~ Re 3" E2 ) ~ (qu 5 /2nE) a ( ~ (tal? + \eyal? + L lt sa\* ) Im Bs Fr IJ_-— Dispersion Formulas Before discussing our specific applications, we shall briefly review the dispersion formalism. It should be emphasized that we are not concerned with the rigorous foundations of dispersion theory, a subject which has received considerable attention recently. (8) The derivations of this section will be heuristic in nature, with little or no mention of necessary and sufficient conditions, etc. We begin with Dyson's formula for the S matrix (in interaction representation) 5 = 2%) (os)? P|, (t,) H(t) = Fye9 Gi) dty + + dt, PIH,(t,) « . . H(t, Hy being the interaction Hamiltonian and P the time-ordering operator. The S matrix element for scattering a photon with four~momentum q and polarization i to a photon q', f , while a proton p goes into a proton p', is Seg 2 =" (iy fat oe edt, (Poe , ap(G")P|Hy(t,) ase H,(t,)] a, @)e >) ll & denotes a bare proton; a and ae are photon destruction and + creation operators. By commuting the a and a_ through the Pebracket, (?) we obtain (assuming initial and final photon are not identical) “Va { 4 44 miqex qty (5 J = t Sos = (49,93) d"x d'y e ) (Vo » P 5, (@)3p(y) ¥) YW denotes a& physical nucleon, and we are in Heisenberg representation. Making a change of variable and using translational invariance, this becomes - 71/2 r\(a4, .ix-(qtq!)/2 . : \ Sp, = ~(4a,a5) SS o+q-0'-< ifa x ox: (ata!)/ cm sf, @i_-a) We now follow Goldberger 2) and consider instead of Ra new quantity M, defined by gg = ~ilba, gi)? f aby eimlarat/2 cy , [s:@, al P) n(x) (0) where nen = {3 if x, 0 0 if x,< 0 R =M for physical scattering energies, i.e., positive energies such that all momenta are real. We shall adopt the coordinate system of Figure 1: S 3-$ NX ‘N Figure 1 > The initial and final protons have three-momenta -A /2 and +h /2, respectively; the initial and final photon momenta are pth and p~A : The photon energy is 2. With these substitutions, (10) becomes ates = ifahe MEE H,, [j 0/2), Coa] oo aD 2 2 We now consider M to be defined by (11) for complex values of 2, with A being held fixed. Since P = p(t}, with p(t) = 4? a a? 4 there are branch points at £ = + A/2 3 the precise way we cut the {plane is not important provided the cuts are confined to the lower half plane. A convenient arrangement of cuts is given in reference (4). It then follows from the assumption of microscopic causality [ 5,(x/2), 5, (-x/2) = 0 for x 0 that 2{M has no Singularities above the real axis. This is the key step in establishing dispersion relations. The conclusion may be rendered plausible by the following reasoning: 24M is of the form ~ t pene fat fax gitt - ix Ve"-(0°/4) F(x, +t) 6 7t This is # function of 4, If we take its Fourier transform by multiplying by eist eo Cael t 2 7,2 fot Jar Jax eit(stt) - ix VE-(A"/4) wry 4) ~o 98 = and integrating over L from = to +0, we obtain If the real part of s is positive, and if we can interchange the order of integration so as to integrate over L first, we can complete 13 the 2 contour by a large semicircle in the upper half of the L-plane, and it is easily seen that the integral vanishes. Therefore 2M is a function whose Fourier transform vanishes in the upper half plane. With certain mathematical assumptions about boundedness . at infinity, it then follows that 2iM is analytic in the upper half of the d-plane. The necessary and sufficient conditions are discussed in great detail by to11. 8) From this fact we can deduce alternative dispersion relations, depending on the behavior of M for large le al, For example, if 27u(2) is bounded for | Re t|>-, an elementary contour integration gives at (2,47) = (ifn) | 2ut M(Lt a") att (12) where P indicates that the principal value is to be taken at deok , If M does not decrease rapidly enough at high energies for (12) to be valid, we may obtain an alternative relation by formally subtracting from (12) the corresponding relation for 4=0 (or any other constant value). Thus, for example, at (2,47) - [pe m(s47)} 0 = (4t/n) of as mt a att (13) This "subtraction" technique is very important in obtaining dispersion relations. Note that (13) could alternatively have been derived by applying the relation (12) to the function M(2)_ - M(0) The actual relations which we shall use are obtained by taking UY the real part of both sides of (12) or (13); thus we obtain the real part of M as an integral involving the imaginary part. We now write M in a form analogous to (5): A av & —_ —> A a Mog (tyA°) = hy(b,d7) Get + hg (2,7) od ot. A + hy (0°) ioe x et 2 2 a > > —> + h, (2°) eel iS-pxAa ° (14) 1A > ' A aA + hg (2,07) io: (e-4 et xh -el.Ae x A) 2 2 2 2 > 7 aA a +h, (2,07) is- (e-4 et x p + eta ex p) 2 2 Again two amplitudes have been omitted, for reasons discussed after (5). Since the only important requirement for the derivation of (12) and (13) was analyticity for Im 4>0 , we conjecture without proof that the quantities h, (2,47) obey independent dispersion relations. A fact which makes this assumption very plausible is the following: If we attempt to derive the Klein-Nishina formula fron dispersion relations, we can formally obtain an imaginary, or absorptive, part of M from the process represented by this diagram: The process is forbidden by momentum conservation, but it has a non-zero matrix element. We obtain the Klein-Nishina formula exactly if we "factor out" the amplitudes h, and apply dispersion relations separately to each. See Appendix D for details. 15 Thus we have the basic relation _ 42 2b Re h,(2,*) = (1/n) » {22 abst si) at (15) In order to obtain Im h, for £'< 0, we consider the symmetries of Mp,e If we write the n (x) of (11) as wi \" if x <0 n(x) =Le(x) +1 > where £(x) 2 2 +1 if Xo? 0 Mes becomes the sum of two terms, called its dispersive and absorptive parts, respectively: Mey = Deg * toy Correspondingly, each hy can be written hn, = no) 4 ap (2? i i i It then follows from (11), by reasoning practically identical to that of Capps and Takeda, “) that the functions ng?) and p{2) are real, that for i=1,2 , ned) ig an even function of 2 and nia) is an odd function, and that for i=3-6, pid) is odd while ni?) is even. Therefore (15) becomes ao 12 ‘i 2h Re h(t) = (2/n) p (2 Tm by (O') age (4=1,2) (16) © Bre _ 22 2b Re h,(t) = (2t/n) p {2 Im by (l") ayy (4-3-6) (17), Our formalism has all been derived for fixed A’. We shall expand relations (16) and (17) in powers of A’, applying dispersion 16 relationa separately to each power, without worrying about convergence of the series for large A*. We are not particularly interested in large a* for two reasons: In our numerical applications we will be forced to make approximations which limit us to low energies, and the dispersion formalism used here appears to require serious modification for large A”, At the end of this Section we shall discuss this matter in somewhat more detail. After expanding into a power series in A, we shall neglect the terms in a4, a®, e « » This is quite reasonable in view of our emphasis on low energies. Jf the difficulty mentioned in the preceding paragraph can be resolved, and if it appeared that significantly greater precision could be obtained, terms in a’, e eo could be included in the framework of this thesis. With the assumptions we shall make about photoproduction (Section IV), it is most reasonable to consider only the first two terms. We noted before that a relation such as (15) is valid only if h, (4) approaches zoro rapidly enough for large 4, Otherwise we must make a subtraction, as in equation (13). For example, in the forward direction (A720) it is well-known that limp, h,(t) = e/2tM . But the right side of (16) for i=l, A*=0, 420 is certainly negative, since In h, (A*=0) is just one-half the negative of the total cross- section for wmpolarized gamma-rays on protons. Therefore (16) cannot be correct, and for A*=0 we write instead fo, 8) ob ne y(t) = (e%/M) + (at? /n) {241 Im BE") ays (18) : ° bth? . 2%) This relation, with a similar one for hg» has been used by 17 (10) Gell-Mann and Goldberger in a discussion of Compton scattering for near-forward angles, and by Gell-Mann, Goldberger and Thirring =) in an extension to all angles by considering the multipoles expected (12.) to dominate low-energy Compton scattering. Capps has recently made a similar calculation with the more refined photoproduction data now available. In principle the method of this thesis is much more general, since we apply dispersion relations at all angles, and are not forced to make assumptions ahout the relative strengths of various multipole amplitudes. In practice, however, we make assumptions about photoproduction which are essentially equivalent to these previous approximations, with the result that our low-energy calculations should closely resemble those of Cappse It is important to observe that (18) mst be valid; we cannot be required to make another subtraction. The reason is that as L>o, the total cross-section should not increase indefinitely. It follows that 2t |p, (2) must remain finite as 14>. Therefore 2t|n, Yr? is square~integrable at infinity, and satisfies a dispersion relation. We shall use similar reasoning to limit the number of subtractions we must make in other dispersion formulas. The CM differential cross-section may be written ax = (1/4n*) (te,/B)* lm\2 a0 2 This formula is derived in Appendix F, formula (F-~2). Now consider the limit 4+, with A remaining finite (<M). Then we are considering near-forward scattering, into a solid angle dO~K*, The kinematical relations we shall need are 18 1/2 ~ E~ge £ pt Bs /2 M and in order that the total cross-section in this forward cone shall not diverge, we must therefore require \ml < las t+, To derive restrictions on the high-energy behavior of the hy we apply this condition to formula (F-3). Considering each term in (F~3) separately, we deduce the following restrictions as Lr cos lnel< a 4=Ly 25355 (19) ln, 4 g7t i=4,6 There is one possible source of error, however; the h, may be larger at high energies than (19) suggests, with the dominant terms in the cross-section cancelling each other. It can be seen from (F=3) that this is in fact the case for h, and hy. Ifh,~4 and h, = ah, + 0(1), then (ul? is still of order unity. If one examines M for different spins and polarizations, we find that this cancellation oceurs in every case. The possibility haw h~ 4 can therefore not be excluded by these arguments. We believe it quite unlikely that this relation between h, and hg at high energies holds (it certainly is not true in Born approximation), but even if it were true our dispersion relations and all succeeding arguments would be unaltered. We must also consider restrictions in the other direction, requiring us to make subtractions, We pointed out that equation (16) would be incorrect for hy in the forward direction, since the low-energy amplitude has the wrong signe A stronger argument results if we "switch off" all meson effects. Then the dispersion integral vanishes to order e”, since there are now no absorptive effects of this order. But we should still get the Thomson term 19 eo” Ma Therefore our unsubtracted formula is wrong. The same reasoning applies to all e terms in the Born approximation. We may go farther and represent the proton anomalous moment He by a Pauli term in the Hamiltonian. Terms in ep, and Ke now occur in the scattering amplitude. The following theorem has been (12) and by Gell-Mann and Goldberger 2?) ; If we set proved by Low, 4~A~ some low energy o<M, then it follows from field theory, to all orders in a perturbation expansion, that the Born approximation calculation is correct for terms in M of order w ana 1. We shall hereafter refer to this theorem as "the low~energy theorem." We shall certainly subtract out such terms, since we are thereby assured of getting the low-energy terms exactly right, and the new integrals will converge faster at high energies. The status of the remai ning terms in eh and Be is somewhat dubious; we have no assurance that the validity of a Pauli-type moment extends to such terms. If we write Im h,(4,0°) = Im h, (4,0) + A” Tm hi(L,0) +6 6 4, and make use of (19), the most general form for our dispersion relations is 2b Re h, (44°) =A, + (207 /w) P at! [ats Tm b, (4,0) a ft (£177) + a" A, + Ag + (2870? Ar) f att x 22 © £1 (£18 _p?) 20 2t Re hy (2,47) = BL +B, + (207 /r) of att 2h) Im ng (2*40)| 2 o 21(L1*_p2) + A? B, + Be + ») + (2877 /m) r( at! {oe Im nycts,o) Toa Vad Shr (£1 p?) at Re h, (4°) = ct + (283 /p) at att ja Im 25 (2,0) 6 £17 (£124?) 4 (8 4 | + (20707 /n) >{ at! Bx im ny (A',0)] 2t Re h, (bya*) = 2 + (2bfr) of at! [2c Im h, (2",0)] 6 py2_p2 dred (20) + (22 e + (2ba? fr) >| al! Le Tm 13 (¢4,0) ey 6 £r2_pe 2t Re h,(2,a°) = 4 Bt + (20? fn) ef att | an Im ng(t',o) t J ar2 (e122) + a” (3 + + tl + (203 2 Jn) Pf att |22 Im ng(ts,o)| Te J 2 (4174) 2t Re h, (2,07) =P, 4 (2t/1) P{ at! Bx Im ng(t",0)| tT J 2272 oo +a (2 4 | + (2ba?/n) Pf alt |e Tm ng(tt,o)| 2? } gr2_p2 21 We must now attempt to determine the constants appearing in these dispersion relations. Our first step will be to use the low-energy theorem. This fixes the following values: 2 = = = zz A, = e“/M A, = 0 B, = 0 B, = 0 C, = -2h C. =p D. = 4p? D, = 0 B= 4p e+ p E.= 0 2-4 1 h 2 1 M a 3 Fy = Abe Fy = 0 (eas +4) This fixes 12 of our 22 constants. The remaining 10 constants can not be fixed precisely at present. Three alternatives suggest themselves for each: (a) The constant may be equated to its value in Born approximation. (b) The constant may be evaluated by eliminating a subtraction. For example, we may suppose that co A, = (2/n) P| age 24" Im ht (41,0) in which case the first dispersion equation becomes ob Re h, (244°) =A, + (207/n) Pf att Ba Tm n, (tt,0)] oO Lt (212_p2) + ab Ae + (20°/r) Pf dtabe lat: Im nt (24,0)| , gre _p2 (ce) Finally, we may suppose that neither (a) nor (b) is correct, and leave the constant unknown, ultimately to be fitted to experimental data. 22 In this thesis we have decided to adopt alternative (a), but we shall briefly discuss to what extent (b) may be applicable. If the Born approximation for a constant contains an e* term, then (b) cannot be correct, as we mentioned earlier. This applies to Ags Ds and E,, whose Born approximations we give here: oe vw The remaining seven constants admit all three alternatives. We can Ag = ans D3 “ue Be ~(1/4M") [2 + 2 + ou® not decide unambiguously among these, but we can compare the values assigned to each constant by (a) and by (b). We list these values below; the dispersion integrals have been evaluated numerically with the static approximations of Section IV. Constant fa) (o) B, tale + ' = 2620 coms) {ads [a Im ng (t,o) = +320 ~2y = { - By “ale + ra) = =.310 (2/n)) teats [at Im ns(tt,o)| = 0 B “Ha fe + = 6155 (2/n) | att BLY Im ni (2t,0)| = 0 5 3 (2 ') /n fan 20 oe C, va € + ra] = 039 cr fat lat Im by 1,0) = 610 Me M re) yre BE, “3 (§ + ) = 2155 a as late Im h ,(t,0)| = 244 23 Constant, fa) fe 16m \M oat -o8 By “ “2,(8 + 3) -.078 cay | ats Im ng (t,o =0 ove ° The integrals are over experimental photomeson cross=sections up to 500 Mev and are unlikely to be in error by more than 20%. No small differences of large quantities are involved. The zeros under (b) result from neglecting meson angular momenta greater than 1 in the photoproduction process. It appears that (a) and (b) are quite distinct alternatives. In connection with our choosing (a), it is interesting to note that in every case the Born approximation supplies exactly the desired number of constants; all terms involving higher powers of d vanish. Finally, there is the possibility that Cy and Fys two of the constants guaranteed by the low-energy theorem, may in fact be given by dispersion relations, as in alternative (b) ahove.e The two possible sum rules are oo Cy: ~2ye = (ofa) abso Im hy (2,0) oO A or —olh = e240 + 9222 Fh Aup, = (2/n) } ati [oes Im ng(ttso)| 0288 13 or 4,56 24, The integral in Cy reduces to the difference of two nearly equal quantities, which are given separately. The agreement is not satis- factory in either case, Before leaving this Section, we shall briefly discuss the manner in which the simple dispersion formalism discussed here becomes Aineorrect above a certain critical photon energy s* We remarked after equation (10) that the Goldberger amplitude M equals the conventional scattering amplitude R for physical scattering energies. However, the integrals in (20) extend from 2=0 to ¢=09 for any A’; clearly the range o<t<a/2 is an unphysical energy range, Thus we must examine the behavior of the absorptive part of M for such energies. It follows from (11) by inserting a complete set of intermediate states V with energies EO and momenta By that 2b hey =o (an) 2 (Hs OKIE, , 50¥ 2) $68) 8 at) 2 2 (21) (ys 3,009 En , 5,0) 3) 88) 6 at) 2 2 2 This expression differs from the absorptive part of the true scattering amplitude R by the sign between the two terms within the square brackets, Therefore as long as only the first term is non-zero we are all right, but when the second term begins to contribute, our results cease to be rigorously correct. The first term exists at the following energies: Los a = “0° [LE for V = 1 nucleon ® A good discussion for the case of meson-nucleon scattering is given in reference 4. 25 t7h, = Mm + (m”/2) ~ (7/4) for V = 1 nucleon Eafe plus 1 meson, etces where m is the pion rest mass. Similarly the spectrum of the second term consists of the single point 4s 4, and a continuous spectrum Lew a We do not need to consider the contributions at Lz + 4, 3; these just give us the Born approximation, as we have seen earlier in this section. Therefore the criterion for validity of our dispersion relations is t> 0 3; the spectra of the two terms in (21) are then isolated as shown here: wt, 0 vA a a and in integrating from 420 to t= © we only get contributions from the first term, which has the correct sign. If t< O, however, the situation is different. The two spectra now overlap, as shown below: When we integrate from t=0 to t= co we now make an error, since in the interval o<t< it. we should integrate the absorptive part of M, not of Re In order to remedy this defect, a method needs to be found to relate the absorptive part of M directly to an observed process, as we have done for R in Section I. Solving the equation Mm + (m”/2) ~ (a*/2) = 0 26 we find for 4/2 a value of about 370 Mev. Thus for momentum transfers in the CM system of less than 370 Mev the two absorptive spectra do not overlap. Since the static approximations of Section IV would be quite unreliable at such a high energy, we shall confine ourselves to lower energies and the limitation discussed above is not of practical importances Finally, one more point should be mentioned. For a given value of A’, we must integrate over all 4 from 0 to ©. But Im h, (2,4”) vanishes except in the regions L>d. ’ Lt » so that the lower limit of the integral is t if t> OQ. However, the lowest 4 which is physically meaningful is 4=A/2. When b/2>t, » which occurs for 4/2>130 Mev, we have the problem of analytically continuing our scattering amplitude to the unphysical range 1 < L<a/2. This can be done by expanding in powers of A*/1h? and then analytically continuing the amplitude, This amounts to the same thing as the expansion into Legendre polynomials discussed by Capps and Takeda, “4 and is apparently valid until the limit 420 is reached; beyond this point such a continuation clearly fails. Ill — Transformation of Amplitudes We have derived in Section I the imaginary parts of our amplitudes in the CM system, and in Section II we have seen that the dispersion formalism is conveniently done in a different system, shown in Figure le We must now consider how to transform our amplitudes from one coordinate system to the other. There are two effects which complicate the transformation. 27 The first we shall call "spin precession"; this effect, closely related to the well-known Thomas precession of atomic electrons, simply reflects the fact that an electron's spin direction depends on the Lorentz frame in which it is viewed. In other words, "spin up" in one system is not necessarily “spin up" in another. The second effect has to do with our choice of transverse gauge for the electromagnetic field. The condition ork =O is not relativ- istically invariant, and so if it is satisfied in the CM system it may not be satisfied in some other system. Therefore the gauge used in the two systems will not be identical; there will be a gauge transformation from one to the other. We consider the spin precession first. We start with the well-known transformation properties of the Dirac equation. If a Lorentz transform- ation (x,t)>(x',t') is described by = + v/ By (y- 1) 34] yw (22) t' = y(t ~ ¥-x) 2,-1/2 where Y = (1-v‘) / » then the Dirac spinor u becomes u'=Su, where S=ytl- yo-F (23) 2(¥ +1) Ye This use of the symbol ‘if should not be confused with the Dirac —> matrices or e yor By Bquation (23) is obtained from the familiar condition SY S=a,y, , : t= where aay is defined by x, ay yxy e It then follows that the spinor for a moving proton is 28 ~~ > 7 U> = arp + EO tM Uy = D u, (24) 2M(E + M)\/ P where Uy is the spinor for a proton at reste The normalization is tu = constant. Note that (24) defines what we mean by "spin up," or any other direction of spin; we transform the proton to rest and then consider its spin directions Now denote quantities in two Lorentz systems by the subscripts 1 and 2. Consider a matrix element Up M u, (25) We evaluate this matrix element in practice by setting Up =D Ue, » U, =D,u,) » and omitting the u,'8e That is, in system 1 we calculate Dei 6M, =I, However, if Uy SU, then 5S "8 = My « From this and the relation gm, = pS it follows that mM 17 D - SBM, SD il In system 2 on the other hand, we calculate My = PesiM2D; > Therefore the matrices Il, and n, are connected by the relation M, = 1 N,3; (26) where T = Dy 1, “, « Straightforward calculation gives 29 . = (z, 4M) ( ¥ +1) ~ {Py Vv + iO-¥ x Py (27) 1/2 (2c y 2) 99 49) We now consider system 1 to be the CM system, and system 2 to be the one shown in Figure 1. The kinematical relations between the two coordinate systems are summarized here: k= &+ (&p/o) B & =k - x R= 3+ Gyp/e) 3 Bay = (M+ 1 x (1-y)| ¥? B, = (By jo/p) (LB, jp) B = (6/28, 70) (tok) y= PES fy - "a" /t _ (28 70 + 4 “y E =p £ = (k/2E 1/2) | 2x, + K(14y)| 9 is defined by * = Ei fp + UE, py 4 gf The velocity of system 2 relative to system 1 is > > Vv >> =e p = = kt+k! tE a /2 2k, With these substitutions, (27) becomes T= (M+E, 7) +E alae +46+px w/e (28) The replacement B= 8 converts qT; into Tpe Direct use of (28) in (26) is tedious; the results can be expressed 30 > , >> more simply. re M, =A + 4io-B » then Mm, = C+ ic-D , where C A cos 2h - BS sin 2h _s D ~> ad sin 2} +B=29B sinh 3 22 > > the z-axis being chosen along the direction of pxA. \ is defined by 2 cos 2h = EA /2 * MEY. By /2 We now turn to the problem of the gauge transformation between the two coordinate systems. It is well-known that the polarization four-vector e is always undefined in that the replacement e—> ethk, \ being an arbitrary number, is without physical meaning. Consider a photon with momentum k=(ie,k ) and transverse polarization e=(e,0); @-k=06 Performing a Lorentz transformation (22) we obtain new four-vectors kt=(k",k!) > et=(e',e!) 3 et is no longer zero. However, an equivalent polarization is e" =e! — (et/c!) k' , in which the time component now vanishes, Carrying this operation through, we find that the polarizations in our two systems are related as follows: e, = 8, + Brey 5 (3 + 5/2" 3) Py pot a/2'? The formula for final polarizations is obtained by making the substi- “a ~? > 3785 » A2e Ae We are now in a position to relate the amplitudes g, of (5) . Aa Ay a tutions ee] » 8 to the h, of (14). Since we are considering invariant matrix elements of the form (25), the quantities to which we must apply the above transformations are the scalars kE,R and CE, joe The resulting 31 formulas, which are rather lengthy, are given in Appendix E. A reasonable computational procedure might be to expand these formulas in powers of a’, since this is the way we have handled the dispersion relations. This, however, leads to very cumbersome formulas, and the numerical work would be of an order of magnitude to be handled by an electronic computer. Furthermore, these transformation relations are kinematical in nature and vary slowly and smoothly with 4 and a’. This is to be contrasted with the dispersion integrals in (20), which fluctuate quite considerably over the energy range we are considering. In order to get a reasonable estimate of the effects of the absorptive photomeson process, in particular the well-known resonance, we shall therefore make a static approximation in these transformation formulas; LM and A? fv" are neglected. In this static limit, we have ath, = 2k, ath, = -(4/27) ke, 2th, = 2ke, 2th, = (2/27) 2k, ath, = -(2/27) kes + (2/27) akg, (29) ang = =(2/27) akg, - (2/27) Zhe IV — Numerical Applications and Discussion Since the unitarity relations, which we shall take in the form (9), involve photoproduction amplitudes, we must deduce these amolitudes from available cross-section data. In this we shall be guided by the successful phenomenological approach of Brueckner, Watson and others, 32 who show that the data strongly suggest, and can be satisfactorily explained in terms of, a resonance in the magnetic dipole amplitude to the state T=J=3/2, plus smaller amplitudes in the other S and P statese The usual approach is to analyze differential cross-sections in the form (as /aQ) =A +Bcos @ + C cos” © If we know the three quantities A, B and C as functions of energy, we have enough information to determine any three multipole amplitudes, since the phases of these amplitudes are very closely equal to the (7) corresponding meson scattering phase shifts. As a preliminary, we arbitrarily set f,,,=0 and solved for f, ML £1’ fn and £33 the notation is that of (4). We used the experimental (14) n° data of Corson, Peterson and McDonald. The results are that fro is negligible, fu has the expected resonance behavior, and fr is quite small, of the order of its Born approximation with a reasonable coupling constant (2? /4n ~ 10). It therefore seems valid to consider the photoproduction amplitudes as Rorn approximation plus an enhanced M3(T=3/2) matrix elements This was checked by computing the charged amplitudes in this way - the Born approximation matrix elements are now quite large —- and comparing with experiment. The agreement is quite satisfactory except that at extreme forward and backward angles the computed cross-section is low; in other words the computed C is too lows This discrepancy is still present if the exact Born approximation is used, rather than its 5- and P-wave parts alone. 33 In view of all this, it seemed very reasonable to neglect all amplitudes except the charged El and M3(T=3/2) amplitudes. This is a very good approximation for neutral production, and for charged production these two amplitudes are certainly the dominant ones. The work of Chew and Low (25) also suggests that these two amplitudes should be predominant in low-energy photoproduction. To be sure, we will be doing integrals of photoproduction amplitudes over all energies, but the integrals converge rapidly beyond the resonance and contributions from above 350 Mev are quite small. Thus as a first approximation we simplify (9) to the following: it +\2 eo {2 Im y ~(quB/2nE) (let, + 3y| £°| ) Im gy * ~(q0,8,/2n8) ( -3 [294]? ) Im gy = ~(qu,E,/2n8) ( -|22,]? + 3 y| £05)? ) (30) 2 Im g, = -(qu,E,/2nk) ( 3th ) 2) (e] In g, = ~(q0,8,/2nE) ( 2 | = 0 5 x With only these photoproduction amplitudes present, the total photoproduction cross-sections are 0, = (qo EB /nB) ( a|e0,|? ) 34 6, = (qu BB yn?) | len m4 lee, ) so that (30) may be rewritten in terms of total cross-sections: Im g, = ~(E/2E,) (o, = 19, + 209) Im g, = -(E/2E,) ( 4°,) Im g, = -(8/2H,) (~0, + Lo, * 209) on Im B, = ~(£/2E, ) ( 2°.) Im g, = ~(B/2E,,) ( 20,) 0 In 6 The fact that we have been able to express our amplitudes in terms of the two experimental quantities oC, and 0, is a consequence of our limiting ourselves to the two independent amplitudes fy and El° We should now convert these Im 8s into Im hy» as discussed in Section III. For purposes of numerical calculation, however, we shall make the static approximation discussed at the end of that Section. The result is that we are essentially applying dispersion relations to the Ese We will be guided very significantly, of course, by our discussion of the correct dispersion relations, as regards subtractions and Born approximation terms. The most direct procedure would be to substitute (25) into the dispersion relations (20), replacing Am by 2k" (1ey) and 4 by k, and then integrate over the experimental photomeson cross-sections. 35 This is very nearly what we shall do, but two points need some discussione In the first place, we should try to get the low~energy amplitudes correct, and in the second place we should try to understand which energy, lab or GM, to use in various places in the dispersion relations in order to make the least error in our static epproximation. To get the low-energy amplitudes correct, we shall use the Born approximations for the B59 rather than trying to transform the Born approximations for the hye We shall neglect terms of order (Kf)? and smaller. In order to decide the second question, we refer to the exact forwurd~direction dispersion relations in the laboratory aystem, (10s21) The procedure which seems most reasonable is then to set the factor E/E, of (31) equal to unity, and interpret all energies appearing in or multiplying dispersion integrals as laboratory energies. We equate the free variable in the dispersion integrals to the lab energy corresponding to the CM energy k which is the argument of the rest of the equations We thus take for approximate dispersion relations: 2k Ro g, = (eA) » (ain) { a. \o, Gt) + 6 ait) ° fader + (dfn) ny) fo) oat (kt?) 2k Re gy = (efi?) + (axe /n) for Cut) 2 19271412 kt (kt =k; ) 36 2k Re g, = 2k (yu “i, 2) Qkp*y - cay {aH | 50, (kt) -o 3 xt (ict? yt4 + alfa on) | a, (K.] 2k Re g, = 2p? = (342 /am) i dky, or (kt) Pid 2k Re gs = =2ky* — (3%7/2m) f__ SSL o (kt é eis -k *) 2k Re gg = epk/M We define the following integrals: (ky) Oo,+ T34(e) = (x2 /n of xf -4? dkt Jo 4 (ey) = (x2 /n ) [- (kt) Toy+ xt (ict wk 2) ul (i/n) (- ot) K ok) r ie} (ict K?) tE (uP /n) f__ Oo et) L (Ky) ékt? (x17?) Then (32) becomes (with M=1): eo” Fa | 2k Re gy ~ 2(I, + I,) + 3(1+y)K, i 2k Re gy = nek + 3K, cs) (32) 37 _ 2 2 ‘ty 2 _ 2k Re gy = 2k(w* = pl) — 2kn’y - (5/2)3, + 29, + Zu, (1 y) kg (33) 2k Re &, = =2kp” 25 2 2 2k Re B= ~Akp 2 2k Re By > epk The arguments of the dispersion integrals in (33) are understood to be kre The integrals I, J, K, L were computed numerically, using experimental photomeson data up to 500 Mev. The results are graphed in Figures 2 and 3. In our caleulations we have made small corrections for extrapolated tails beyond 500 Mev; it was assumed that o~ KS? in this region. The real parts of the amplitudes g, were computed from (33) and are given in Table le Table 1 k, (Mev) k/M 2k Reg, 2k Reg, 2k Reg, 2k Reg, 2k Reg, 2k Regy 2 2 2 2 2 2 e e e e e e 0 ) 1 0) ) ) 9) f°) 50 20507 -6959™,00ly #6050 .117-.197y 39-199 199 e071 100 40968 68 25".013y 0084 Be 79y 0396 eG 136 40 61309 604-0061y 2070 B720525y e571 #0 STI, 0183 190 61709 o2e298y 127 70=0775y 0885 e885 239 230 2008) o235—0674y oo A73. «299100297 10234 10234 281 270 2293. oA Qme9lly 82. +0 208-16199y 12297-10297 0321 In Table 2 we list the imaginary parts of the E59 obtained from (31) without further approximation: k, (Mew) 190 230 270 38 Table 2 2k Img 2k Img 2k Img 2k Img 1 Se ee 3 L 2 2 2 2 e (= 8 e ~366—,078y 0078 0366—.0397 e039 —.716-.A4ly eAAL oT l6m_221y —e221 =o 748-1. 7199 1.719 o hS= »860y —.860 2k Ing, 2 e e039 =e221 860 We can now use (6) to calculate the coefficients A-E of (7). In Table 3 we list the results, and in Table 4 we give the differential cross-section at representative angles, in units of the Thomson 32 2 cross~section: (e?/4nm)* = 2.36 x 10 cm « k, (Mev) 50 100 140 190 230 270 Table 3 A B C D 059 0 050 0 048 200 o41 meO5 250 -10 oak 07 056 wo31 205 205 1.17 73 -020 Ol 2055 052 =~o51 08 481 065 206 003 200 000 201 201 200 201 39 Table 4 - (aa /aQ) + (e*/4net)® ik, (Mev) 0° 45° 90° 135° 180° 0 1.00 075 250 275 1.00 “50 084, 067 48 70 094 100 056 251 249 271 290 140 226 235 256 283 298 190 226 356 1.17 1.59 1.70 230 1.60 | 1.96 2.55 2.63 | 2.48 270 5.56 5.31 481 4037 4020 The most significant comparison of these predictions with experimental data would appear to be that of Figure 4, where the 90° excitation curve from Table 4 is plotted against the existing experimental data (16) from MIT and Tllinois. '47) In Figure 5 we show the predicted angular distributions at various energies; the experimental data at angles other than 90° is so fragmentary that no real comparison is possible. In Figure 6 we give the predicted 135° excitation curve, With the existing experimental points. We have performed an alternative calculation along the same lines as the one described in this thesis, except for the subtraction procedure (and some minor kinematical differences). In Section II we discussed the alternatives available to us, and chose to make the maximum number of subtractions allowed if our total forward cross~ section is to remain finite at high energies. In our previous calculation we made fewer subtractions than would be allowed by our criteria of Section II; subtractions were made only when the subtracted 40 term was guaranteed correct by the low-energy theorem. In several instances, for example the A” part of the first equation of (20), this meant that we were omitting a” terms which must be present even in the absence of mesons. In Figure 7 we give the predictions of this medel. It is interesting to note that the excitation curves at 90° for the two models are practically identical up to the highest energies considered. The angular distributions at high energies are markedly different, howevere In the calculation discussed in this thesis, forward scattering exceeds backward scattering by 30% at 270 Mev; in our previous calculation the 180° cross-section is nearly twice the 0° cross-section at this energy. Further experimental data at high energies would help distinguish among the alternative subtraction possibilities. Capps 22) has recently considered this same problem, by a somewhat different approach. He has only used the two forward=direction dispersion relations in the lab system, and has extended his work to other angles by considering the multipole amplitudes expected to dominate the scattering. We have included two angular distributions predicted by Capps in Figure 5. He states that he expects his curves to overestimate the backward scattering by a factor of about 1e5. Watson, Zachariasen and Karzas are attacking this problem in the (15), (18) framework which Chew and Low (19) have applied to meson scattering and photoproduction. Numerical results are not available at the present time, but are expected shortly. Finally, we should mention that Compton scattering may be analyzed in terms of multipole amplitudes, although we have not chosen to Al do so in our calculations. Details of the general analysis are given in Appendix G. 1. Re 36 he 5 Te Be 90 10. ll. 12. 13. 156 16. REFERENCES He Ao Kramers, Nature 113, 673 (1924) H. A. Kramers and W. Heisenberg, 2. 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Weisskopf, Theoretical Nuclear Physics, Wiley, 1952 Appendix A_— Useful Integrals dH wD OQ > = | iu ht > hy > = A A We use the notation x=k-q, xt aA & ay by c are comstant vectorse K(x,x',y) = l- x “ xt? ~ y~ + 2xxly All double integrals range over those values of x and x' for which K is reale 45 fda, fle) fledged pope -aff dade id con y) ALA ~ AA ne A +acbhe bc an) + x x xy R(AbEE + + Qa: R') - sys [Geng Org) + Bay targa RY Rb ER DRE EG 2K RRbR) + ay s[ ay) (xorg) Gary rey) KJ » 46 Appendix B - General Unitarity Relations Again we set K* =zi1- x - xt? - y + 2xxfy and integrate only over regions where K is reale The common Factor - as has T been omitted, tow yolgls aff de fT gan’ gE] po) fol HOKE K? Ig K* 4K fhee)$.e) + a Re f(x!) $,() Elo sla) 2 43(x) ) $4(x) or Re 43(x) Im guly)= aff deb (Pale) Weg ffl) fb) ry? (1 4 Xx (xxt-y) 2 x oe ake 2x) f,(2) pe, ak | 4 (x) 4,6) XX ay KE" wes ¢{ X + [ Sh + GR ee ) falx) + ap leo yienag) yk] 2m HOT HO [st BEL] 47 Ol 47 4 ot Lm 93 yl ap eers. £'(x) f. Ge) +L y- xx! t wate FOI A) + [yarn - oh | t (x') f(x) + [6- xy] Pict xy/ + K*(3xy-2 ‘y *_axt) x 28e fC Ald) }+ (axy -2r'y' «J E} G4 [4,1 - pp [ory Pct Lxuj + “{- = me are p00) fi} dx dy! yo */ ry (-y*) + ge 2 ke Mey) fala (- yo Im ge ly) if 4 oy - 2Re fle) fbx) | op 2 */-4 + xLxy . [ pny) . eg TE |. Zhe f, (x/ 4) any oy alg) x & Ke LG) fi (x) - yl rae ~ aan {, (x') f, (x) (- (~y Sry “+ X- ax'y@4x'y) } * + ‘ xb xe duy yt) +E. sayy Long ny) Orsary eg) Er "sey? [ + (x-n'y) Crt ry/ + E'(Sxy*+x _axty?-3x'yl > 2 Re fs (x) fy | | _ dx dx! ~xy! 2 * tm fe (4) : aff Ee {| Cy" - yl 2Re f, (x') 4, fx) + ~xXy! Kt +7, ; ; Ca 2 [ Taye ~ itor {, (x) ¢, G1 ~ ey? [ Grip Vebag) Graxtyeny) ~ £ (3xt + 3xty*- ~ 2ke L*C') Fy x) +] foe _E]. E G Sixty boxy. )] ake f*Cx') f (x. Ly Fil « 2 ke ts’ C«’) f(x) eg? [ 20rxg) "be xy) 1 ESky Sey rey?-v) «2% £Rxy ful} 49 Appendix C - Unitarity Relations in Forward Direction m fous wy & ry, + (t-x* *O) f be) $ me) by zm gi =~ E85 fod fl £240 + F2Od etext)» Re Akl) fle) + 4 Co f5'l0) f@) six (1-92) » 2 Re {*(y) 4, (x) +4 (rt $.°G) f, Ow) Iw gf) - = $8 [def lor) £762) £0) wht! MU + (x4 ~2 Re Of) ~ 2 (a0 PL Gd 4.0) Lyte 2 Re fo (x) f£'G) + £ (tex) fe ‘(x) f(x) +4 (I-x ea (iery 2k Le) L'a) # bx(i-x}s 2ke ff Al fh) wale feb fel} Sm gol ~ AL [def Eliw othe f°) f 60) YrE ~« + ix(i-y?) * 2Re °C) Al ~ £,°0) AG) ~ 4 (fa-xt)~2 ke FQ) fF, J} 50 Ba gy id = ~ $45 f ef 4 (ort 2 Re $6) $00) ~F (1-9 } x 2 Re {"t) flr) -4 (1-29) £20) f,00) <a (alte 2 Re $80) fr Got Lum 9s =. pepht [def -4 lox »2ke f* IL lx) Lert ake (0) 4,6) tb lan fs 2Re fi OE Ga) + i (t-x'l"s re (4°04 ‘Od - 45° "bd f ‘M) | Sb (ten 2& fA) FO) + fv) x 2ke fr bd 4,'b) th (xd APG) G+ olives 2 ke FO) fad) + uP (tx 2 Ke f, fC 51 Fn Gul= FAG Pe] mip lea ae $0) Ha) r 7 li-x') *, 2 Re f*(x) L(x) vt ert sake (4G) 0) ~ 6°01 46) ° ié (I-n'I"s 2 Re fe ‘O) ty (x) ~ ; (i-x°} Od, (0) -% (i-x*]'« 2 Re fb) ty lx} - rr (1°) 28 fy (a) Lo) 52 Appendix D ~ "Derivation" of Klein-Nishina Amplitude by Dispersion Theory We start with equation (21): 2b Aes = =(1/2)(2n*) E (Yes IE ICY, » 1,00 Vg) § BB) § G-H,-4) 2 2 2 > (We, OVI, . sp) g) 8G,8) §@,-0,12) 2 2 2 The states V must either contain one fermion or two fermions and one antifermione If the first term is to contribute, we must have =" Ph u dy ° F BE =t + Ey 2 The only possibility is a one-fermion state, with L = I (07/48) 5 BL = W/E 2 2 Similarly the only contribution from the second term occurs for a one-fermion estate, with Le+ (*/ue,) , B= M/E 2 The matrix elements of j are easily worked out: rt A (Pe. iV) = (Py. vee, vVa) =e up BS-e, vs >A ~a > 2] ; , oS. - e[(E +™) Coke e, + (E, +™) o,f Pp (45, AE, 40] Ve 53 The last expression is a matrix in the initial and final spins; we have omitted the spinors. We therefore find for the contribution of the first term in (21): 2 2.92 ,2 ~ne” §( ade: be) og yee ys 2. MOAT et +E, O- et Op x Bas 2 = LM Ey 2 2 7? QRA 7~aat > «| pre-MoesA = 2 2 Using the identity A ADAM vAw a AANA aA ann a 4-A A aad A 4a a AAA a CO af-baoedd =a-be-d=a-cbd +tadbc Ho-(a-be xd +ec-da xb ~acbxd-=bdaxe+adbxc + b-ca xd) we find the following contributions to the amplitudes defined in (14): 2 s Im h, ? MA~/4, Im h, + Ey 2 Tm hy * Ma*/2, Imby M45 noi> We have omitted the common factor ~(ne*/anE,) § Varta? ate, ) 2 4 2 To get the contribution from the second term we (1) change the overall sign (2) change the Signs of 4 and Pp (3) exchange @ and e!. If we integrate over positive energies 4, the second term is the only one whose delta-function is picked up. 5d, The Klein-Nishina amplitudes in this coordinate system are 2k Re hy = 6” 2b Re h, = e L on ah 075? at 3 1 3 16 2b Re hy = eo MAZR 2b Re hy = Pe (Fat + ") L Pe gh By ve at 7 3 16 2 3 16 It is now easy to verify that relations (16) and (17) are satisfied; (16), with one subtraction, for h,, and (17) for the other three. 1 The important fact is that the amplitudes h, themselves obey the dispersion relations. It would be very difficult to apply the 2772 ~—CU« > relations to a quantity such as h, (t) iG.p x 4/2, since p would vary with L, and perhaps the polarizations should also vary in order to remain transverse. Appendix E - Transformation of Amplitudes Alp ly, +42 +E y, (M+p}* 4p Oey, +e (Hep) (Len, 4°) LE, hy, = * (63) + _ (kee gy } = 2 a” (y Ly, +2h by, +2] 55 4( 7 +2lEy, }(p Ey +e rey, ) [« E,. a) LE, hs - L " (4p) (LEy, + ey" _ 8L* Ewe *(p Ey, tM +H Ey, ) (e&9.) + dL Eon lee (stop }? (tl + S$)" oop Clee 4) W) - ¥¢ Co, p. ole + 26F na) (xé aa) (rt1+p]* (06, + # q (arp) (Ey, + als Ful pin" fEy, + =a “ey - Bp tO Eu (« &, } (mep)” (fey, + aa)" “ te, hb. - ee ($+ 26E 4) ql Ew tp * © vey) (Ey, + 4°)” (eeg)- “Gap (te = u (Rage) ‘ep {Ey # a, fe +p ate lye balla . Z tg (m+p)* (KE, 29) , 1p Eon -" a (20Ey), + ¢) (p ay. + ME hey, | (LEx 94) (+p /* (CE y, + aye «p4 -Y RE, 2 r we — a | Ea En H](Cr Exp] oars 24 Eas ft ME) a2 LE wy, + 77 (HE g | 4 RF EanllrEg A) 52 te . , (aH #( reg] a0 (Eng) (4+ OE I] (ke & 46) 56 4p Ey, (p Eo, + Me +ME,, ) DE aby = pos keg) - ; LE, hs _ 2H (4+ 20Ey, ) (ee 3 / - 40°Ey, ptt : "te +p |” (Stl) (M+p)” (LEq, + 4°)" (Ex Js] f + ft 2LEy, (L+ Ey, ~ p) E (ee o/s* eyo 3)" [1° (m+ p)° (M+ Exy,) | (« “gs Hy Eon, p20 y+ FUP Ent MONE) Ey,) (xy) - tpley, (ap )* (Ea, + 4 a2)" (M+p)*(LEq,+ 4] 1 [lee Npta,ritones,) + mo UEnt! | zptty (PEyyrttotén Ic, 4, | Ey, +t LEa,+ 4% 7 » tpt Fn [¢ +E, HM ply, t1l+ Mey | - Ex {frEy,-f) | (eee go] Ging ye (bE, +2 |’ Eo,+K ly, L, ; ap (+ 2CEy) (cE, 9,) $ oT Ey ee (« Ex ga] “Gp? (lege? (Mep)* (CE n,2 =)" + Ulin, (Eye ti) , (KE) - 4p bon, ¥ (20, Fee +N MEG) ep }* (lb, + 4 © “ ba (E+ e)" (KE, 9.) 4e4 Ean en pe, 25 Ena lp Et Mle ME) “ a) (ep) (lly +5 ko il Pe een, LE ay, + & J 4 L Ey, aba) | a Limp ote Me) Ha] 6) 57 Appendix F - Cross-Sections in Arbitrary Coordinate Systems This discussion is based on part of an article by Moller ‘2° on the scattering matrix. The problem is to derive a cross~section from a matrix element in an arbitrary coordinate system. We shall consider the problem of a particle 1 colliding with a particle 2 to give a particle 3 and a particle 4. The generalization to arbitrary numbers of particles is immediate. In the lab systen, with particle 2 at rest, ; 3 2f3 { 3 a 9 7 9 xO(E,. + 318 E oy) (F-1) We must write this in a covariant way, so that the subscripts L can 3 be removed. In the first place, integration over 3 variables dp is not invariant, so we multiply and divide by E55, and Baw giving ap 3L Ein x $(z,_4 -E,,) 3L aie Ay 2L Secondly, we recall that lpi? is not an invariant, since it contains the reciprocal energy of each particle. Therefore we write . 3 [43 d= (1/dm") (1/ p45 Bo) (LR, L Ey, Bop Bay a) PaL { OP x “an 6 Ban 36.2.2 -_— (54,4 2-Py1-Poy,) & (B 3rey Ea Bop) 58 Finally, we must find an invariant which equals Py Foy, in the lab systeme Moller suggests the scalar V-C/2(%,Poy ~ PayPoy) - Vos Ho Bf > ©, * P,)° = EXE, (@, ~¥Q)" = (x %)° Therefore an invariant expression for cross~section is as = (a/en) |v) - Gr) 7p, 2o, lel? S0p,-2,-2,-9,) x x § (2,48, -E, -E,) (¥-2) This is the correct generalization of (F-1) te an arbitrary coordinate systems > > v.)* => The "effective velocity" ¥ (v. Vyr-V5)° = (vy xv xv,)* has the following values in the three coordinate systems with which we are concerned: Lab: vy = 1 => CM: lv. | FV, + Vy F E/E, System of Figure 1 : (tz, + a*)/tz, — 4 pad 2 2 The density of states is obtained when we integrate over the energy delta~function of (F+1) or (F-2). Various expressions for the density of states in the CM and lab systems are well known. For Compton scattering in our special system the integration over the delta-functions gives fap, fap, 8 (pyap-Py-Pp) §(x,- +E, =By-E,) = ant a(at/2) It is more convenient to have an element of CM solid angle in our 59 expression. Therefore we obtain a cross-section from M by writing R= (LE, /&E, ) M 2 and substituting into the cross-section formula in the CM systeme The result is ag = (27/4 B*) Iml? ag 2 Finally, for reference we give the summed and averaged \ul 2 of (14): = av |u|? = thi rr (p+ 2°) + (Re boh,- te Ki hal #e ¢t (47-25) +(ns te beh) so + iby! (48s 298) -2Re heh, fe (t= 24] 60 Appendix G - Multipole Analysis of Compton Seattering In reference (2) a phase-shift analysis of the scattering of electromagnetic radiation by a particle of spin 1/2 is given. However, the possibility of mixed electric-magnetic scattering (for example, Ml-E2 with J=3/2) is omitted. To include this, we merely note that unitarity and time-reversal invariance require the S matrix for a channel of given total angular momentum and parity to have the form 216 4 4 sind oil bs4 6.) cosé e (G~1) 4(8.+8,) 2i § i sin® e 1 ~2 cos6 e 2 where the two rows (or columns) refer to electric and magnetic radiation. The quantities ®, 8,; s., are real. It then follows in a quite straightforward fashion that the scattering amplitude for circularly polarized light on spin-up particles is s(2)= h rid wir ~ sa): oJ -(z)* PP ( “TW 1) 2« F “or “S, 1,4 Cos Os 5. -y, © l es sve pat (rey Fn re = a) “Sj ahd (toy e -1) Lx soot! €,J- T+; oo, (hy he > si sn Garey @ EEE oS) [aa PAS al) Nw 7,I+Z Cc Fey A Oras per WT Tyas Ware st! £742 2135, * THe 44 ~ ae) et ( Cos OF 444 e ? _ i) Zs 344,44! (¢-2) or j | awd ob 28 Mid-% * ( ir )* Sain ( tas O7,5+5 e “Te JEsnu, hae 4 Lz . SUPE Yat pat g an At t4 : = Te)" ey #13m6, 5, a4 © 7,743,441 =( 40) Seat Yy,1-4 a a 2 61 For J=1/2, 6570 since the only multipoles are El and Ml and no mixing occurs. We have defined: aad Lt _ ” Zour * 2 Ca e-s.s ZLeyn-s *s 2 4 Xv is the normalized angular distribution of A for (EL), radiation; a Yom Similarly characterizes (ML) y radiation. In terms of vector . »__ (21) spherical harmonics, mn a f= gn Xun 3 et. Yow in ei ae a" fara | Zon = Yo. 14/2 defines the spin of the scatterer in the final state. 7 If J=3/2, the six terms in (G-2) give El, M2, El-M2, E2, M1, and E2-Ml scattering, respectively. We do not feel that a phase-shift analysis of Compton scattering is very meaningful, however, because of the absorptive effects of photomeson production. When this channel is taken into account, the form of (G-1) becomes A + da VAB + ip (G-3) YAR + ig B+ iy where A and B are squares of photoproduction amplitudes, and Ay Bay By ¥ are real. Thus the scattering is now characterized by five real parameters, not the three of (G-1), and we see no natural phase shift description of this scattering. One might try making 8, and 8, in (Gel) complex, but this form is not equivalent to (G-3). We shall therefore analyze our amplitude (5) into E1(1/2), M1-E2(3/2), etc. without using phase shifts. We expand R in normalized multipole amplitudes: 62 -_—3 «A -t(o.@xe Then if we define ass 8 4 by , i Ne) nw an “ st n N rf) a — on i rl bo a Pe ct a + a eee Qo > A Le) a “ er 2 moO rt cy a bs) + i) ri 8 @ f rc i ° bet py or * ‘3 =I iow (2, + 20, - 38g ~ 3%y +6«,) LL zo hr + ue ( 6.-%«) Me éd M(-E2 * -~63—~ e SO lore Lie 200 250 kK. (MEV) FIGURE 2 — DISPERSION INTEGRALS —64- Ko 4 Tk, O | | | l | 0 50 100 ISO 200 250 kK, (MEV) FIGURE 3—DISPERSION INTEGRALS (do /dQ) — (e2/40M)* —65— | i 1 | THEORETICAL | qT x MILT RESULTS o ILLINOIS RESULTS = 9-4 XX x as x YX l l l | 50 \OO I50 200 250 kK, (MEV) FIGURE 4 — _ EXCITATION CURVE FOR 6@,,=90° (do /dQ) ~ (e2/40M)* -~66- MN —— THESIS ---- CAPPS 20 MEV I85 MEV 140 MEV | — 45 90 135 G. (DEGREES) FIGURE 5 —- ANGULAR DISTRIBUTION -67- (do /dQ),,, > (e?/4aM/P | | | THEORETICAL eo ILLINOIS RESULTS xX M.IT RESULTS > 4 P ¢ | | | l 50 I00 150 200 P50 kK, (MEV) FIGURE 6 — EXCITATION CURVE FOR 6,,=135° ~68— 247 MEV er I6| MEV | O MEV | | | 0 a5 930 135 180 9.,, (DEGREES) FIGURE 7- ANGULAR DISTRIBUTIONS OF OLD MODEL