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The helicity amplitude approach to low energy theorems
ANNALS
OF PHYSICS:
78, 535-552 (1973)
The Helicity Amplitude Approach to Low Energy Theorems NORMAN
DOMBEY
University of Sussex, School of Mathematical and Physical Sciences, Brighton BNl 9QH, Sussex, England AND
N.
C. MCKENZIE
Research Institute for Fundamental Physics, Kyoto University, Kyoto, Japan Received May 23, 1972
It is shown that “low energy theorems” for charged vector and axial vector currents have a natural derivation in terms of helicity amplitudes. This is done by reconsidering the kinematic constraints as a function of the virtual photon mass, A. The crucial point is that in general a certain limit of the longitudinal amplitudes is required as an input to the theorem. Two new YV sum rules are also obtained.
1. INTRODUCTION Fourteen years after the original derivation [l] a new derivation of the low energy theorem for Compton scattering was given by Goldberger and Abarbanel [2]. They observed that there are kinematic zeros at threshold in the helicity amplitudes for this process which suppress the unknown many particle contribution there since the dynamic cut singularities start well away from threshold. The resulting amplitudes are thus given to first order in a entirely by the Born terms at threshold. This new derivation has a certain elegance and may be considered preferable to alternative derivations, mainly descended from that of Low [l], in as much as it makes the structure of the theorem particularly clear. Naively one might expect that on account of these zeros suppressing the continuum contribution, low energy theorems (hereafter, LETS) for isovector photons or massless virtual weak bosons would again be given simply by the Born terms at threshold. However this is not correct and for example the LETS first derived by Beg [3] for nucleons leading to sum rules such as the Cabbibo-Radicati [4] sum rule cannot be obtained simply from the Born term. It thus seems that the zeros 535 Copyright Al1 rights
0 1973 by Academic Press, Inc. of reproduction in any form reserved.
536
DOMBEY
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MCKENZIE
found by Goldberger and Abarbanel are not present in these cases. Related to this problem is another. Zeros in helicity amplitudes which are manifestly present when the spin one particle has exactly zero mass would seem not to be present for any nonzero mass however small. There is, in short, an apparent discontinuity which we do not expect either on physical grounds (experiments indicate that virtual photon cross sections measured in ep scattering tend smoothly to real photon cross sections) or on theoretical grounds if the spin one particle or virtual particle couples to a conserved current [5]. In this paper we reexamine these problems by considering the constraints imposed by the kinematics as a function of the mass h of a spin one boson undergoing Compton-type scattering with a pion (e.g., yr ---f yn or Wn --f WIT). We restrict our attention to the case of a spinless target because of the greatly simplified analysis, but our conclusions should apply for general spin. The original motivation for studying this problem was to obtain new sum rules for neutrino-nucleon cross sections following the model of the Drell-HearnGerasimov [6] or Beg [3] sum rules. These were to be obtained by first deriving the LET for a Compton-type process with the electromagnetic current replaced by a weak current. The easiest and most elegant way to obtain these LETS seemed to be to follow Goldberger and Abarbanel and simply write down the appropriate Born terms. As we have already remarked such an easy solution was doomed to failure but nonetheless our investigations with helicity amplitudes led us to a rather direct way of obtaining the desired nucleon LETS which are presented together with the sum rules in a separate paper [7]. In this paper we derive the equivalent LETS and sum rules for a pion target but the main purpose of the paper is not the sum rules, which are perhaps more of a theoretical curiosity than an experimental possibility, but rather to study the kinematics of the zero mass limit for cases corresponding to a nonconserved or charged current. We are able to show that such theorems can indeed be proved using helicity amplitudes. They again depend on the presence of kinematic zeros suppressing unknown continuum contributions and these are a result of the massless or nearmassless spin-one nature of the incident object. However, we show that in general the LETS for transverse amplitudes receive a contribution from the longitudinal amplitudes and the limiting behavior of these amplitudes as well as the Born singularities are a necessary input to the theorem. In a previous paper [5] one of us showed that the requirement of current conservation in processes involving a photon could be replaced in S-matrix theory by the statement that longitudinal modes of the photon decoupled for zero photon mass. This work generalizes this result: where the current is conserved the requirement that longitudinal modes tend to zero with photon mass gives rise to conventional LETS, but if the current is nonconserved then longitudinal modes are present at h = 0 and have to be included in the theorems.
THE HELICITY AMPLITUDE
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537
2. THE STRUCTURE OF A Low ENERGY THEOREM We employ the original method of Wang [8] to find the kinematic structure of helicity amplitudes. Wang assumed that the reduced helicity amplitudes f8, defined by 12.1)
1 + 2 - 3 + 4 and fhsA;Ah is the s-channel helicity amplitude with 9, the center-of-mass scattering &gie: have no kinematic singularities in the channel momentum transfer variable t or U. The reduced t- and u-channel amplitudes are likewise assumed to have no kinematical singularities in their channel momentum transfer variables. The only singularities in t or u which can occur infs come from the nonconvergence of the Jacobi series 01 = A, - A, ) p = A, - A, ) where in the s-channel
3” =
1
(2.I+
1) F;&;l,A,(S) P’(!&$‘a+B’)(cos
0,)
J
and these are assumed to be of dynamical origin. This assumption acquires potency when used in conjunction with the crossing matrix first obtained by Trueman and Wick [9]. The crossing matrix relates Schannel and t- or u-channel helicity amplitudes, analytically continued to the same point in the (s, t) or (s, U) plane. From the crossing matrix we obtain a reduced crossing matrix which is defined as the matrix connecting reduced helicity amplitudes. Thus suppose we relate in this way reduced s-channel amplitudes to reduced t-channel amplitudes. The reduced s-channel amplitudes are by assumption already analytic in t and the reduced t-channel amplitudes are analytic in S. The reduced crossing matrix therefore contains all the kinematic singularities and zeros of the s-channel amplitudes in the variable S. Likewise it contains all the t singularities of the t-channel amplitudes. In this sense the reduced crossing matrix contains all the kinematic structure. Other, perhaps more conceptually direct, methods exist for deducing the kinematic structure [IO]. These relate helicity amplitudes to amplitudes already known to have only dynamic singularities (for example the Joos amplitudes [I 11). The kinematic structure is then contained in the coefficients which relate these amplitudes to the helicity amplitudes. All methods agree in any given mass configuration but these other methods tend to be technically involved and whenever two or more masses become equal or a mass becomes zero a fresh approach has to be made. Principally on account of this we favor Wang’s approach. We illustrate the approach by rederiving the result for pion Compton scattering following Abarbanel and Goldberger [2]. Pion Compton scattering is described by two independent s-channel helicity amplitudes f&1, and fi&-lo (in the s-channel
538
DOMBEY
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MCKENZIE
y + n + y + 7r). The related t-channel process is described by f:W1;0o and f&, . Similarly in the u-channel we have f?l,,+.10 and f;1o:--1,,. The crossing matrices are trivial for this case.
f"10:10
=flLoo
=
fJ10;-10
fiso?lO= f&o0 = G:-IO
(2.2)
so between reduced amplitudes
f;“,;,,=
cosm2 40, sinM2 $9&~,;,,
= cos-’ #?, cosm2 &3u3!!10~-10
or
J;“,;,,=
((s - m2)2/t(t - 4m2))3:-1:~ = ((s - m2)2/(u-
m2)2)3~~o;-~~ (2.3)
and
3&,-1o = sin-’ 48,3,“,,,, = sinb2 $0, sin2 ~0,3&-lo or
3;o;-lo= ((s - m2)2/-st)3;~:, = (u(s where we use the usual notation assumption that ps is kinematically and t, we see that
3&o=
6 - m2)2Eo;lo 3:-rGoo = t(t - 4m2)Jit_1;oo 3:~o;-10 = (24- m2)2f&lo
m2)2/s(u -
m2)2)3&40(2.4)
for kinematic quantities. Remembering our regular in t and U, 3” in s and U, and p in s
and and
and
f;80;-10 =
Ns - m2)2/Cf&-10 3,1;00 = ?Koo
fLo
(2.5)
= ((u - m2)2/u>.f&-lo
where the barred amplitudes have no kinematic singularities but only dynamic structure. So since (s - m”) N 1q 1, (U - m”) N 1 q I, t N 1q I2 near threshold, with q the photon momentum, each set of amplitudes has a kinematic zero of order I q I2 at threshold, (s, t) = ( m2, 0). We have performed the calculation symmetrically for s, t, and u because we want to emphasize that the reduced crossing matrix between any two channels contains all the kinematic information. In our later analysis we will take advantage of the greater symmetry to be obtained from s-u crossing. The barred amplitudes have no kinematic singularities but only the usual dynamic structure. The point to note is that the unknown many particle cut starts away from threshold so that its contribution is suppressed by the kinematic zeros. To complete the derivation of a LET we need, aside from kinematic zeros and continuum cuts separated from threshold, to be able to write down the single particle pole terms or Born terms. This is straightforward since the location of the pole is known and the pole residue is in terms of the coupling constants charge, magnetic
THE HELICITY AMPLITUDE
APPROACH TO LOW ENERGY THEOREMS
539
moment, etc. So in the case of pion Compton scattering we can easily write down the pion pole terms in the S- and u-channels. As is well known, however, the LET is not given by these two terms alone. A seagull or contact term must be included as well. This marks an ambiguity in the original S-matrix derivation [2]. In the field theoretic approach the seagull term is forced by the requirement of a gauge invariant Born term. Such a principle lies outside the spirit of the S-matrix approach and if this method is to be self-consistent it should be possible to proceed without recourse to gauge invariant perturbation theory. The ambiguity can be resolved by the requirement that no longitudinal photons are present [5], and this in turn leads to a consideration of (virtual) photon mass h in the problem.
3. REAPPRAISAL
OF THE ZERO MASS LIMIT
We now attempt to generalize the LETS in order to deal with charged or nonconserved currents. Thus we are prompted to reconsider the kinematic constraints as a function of the virtual photon (or W meson) mass h. As the zero mass limit is approached we look for the necessary condition which must be satisfied in order that Goldberger and Abarbanel’s zeros are present. We shall then relax it in the next section and so obtain the generalization of the LET. This is our primary objective. At the same time we find ourselves very well placed to comment on the apparent discontinuity found by a previous author [12] between the identically zero mass case and the nonzero mass case. In this section we will show that no such discontinuities are implied by the kinematics, where the kinematics are defined as the structure introduced by the reduced crossing matrix. We will consider the explicit crossing matrix between the S- and u-channels [ 131. s-channel
1+2+3+4 w+r+ w+?r
u-channel
3+2d+4 w+7T+
The crossing relation has the form (summing
w+rr. repeated indices) (3.1)
and #r = z,/J~= #because of the mass equality. We assume time-reversal invariance for simplicity although this is by no means necessary to the following discussion. Further simplification is obtained by writing the crossing matrix between parity conserving pairs, e.g., hso;~lo f ~-A~~;-A~~. Then noting the following expressions
540
DOMBEY
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MCKENZIE
for the angular quantities cos 40, , sin 40, , cos $0, , cos 4, and sin $ we obtain the reduced crossing matrix tabulated in Table I. cos +e,=
p1q9A
sin
cos ge, = p112/~A
;e, = (-st)1/2/SA
sin +e, = (-24t)112/UA
(3.2)
sin I/J = 2h( -pt)1/2/SAUA
cos # = (p - (m” - X2) t)/SJJ*
where
&2 = [s- (m+ A)2][s - (m- X)2], u,2= [u - (m+ A)21[u - (m- X)2] 5” = SAWA2 + 4Ppt,
(3.3)
p = (m” - h2)2 - su.
Let us then examine the constraints imposed by the reduced crossing matrix on a typical amplitude, 3 11+= f$,;1o + 3!,,;-,, as a function of A. We take as input assumptions the following. (i) Following Wang: that&+@, U, A) is an analytic function of u for all values of s and A, and similarly f;“l+(s, U, A), &+(s, U, A), A..,& U, A) and h2f&, U, A) are analytic functions of s for all u and A. [Of course3S will not necessarrly be an analytic function of u in the presence of dynamic singularities. However the separation into dynamic and kinematic structure is made quite unambiguous by defining the latter to be singularities which come from the reduced crossing matrix under assumption (i). The dynamic structure can then be introduced independently after the kinematic structure has been separated out.] (ii) We assume that A&- and h2& go continuously to zero with h and that the limiting functions &s, U, 0) = lim,,,f’(s, U, A) exist for purely transverse amplitudes. [The first part of this assumption turns out to be too restrictive in the general case and new results come when it is relaxed.] Then the following theorem is a trivial consequence. THEOREM
(a) &+(s, u, A) is an analytic function of s and u for all X (apart from dynamic singularities). (b) the limit lim,+Of:l+(s, u, A) exists and has the form
3A+(s, u, 0)
=
0
-
m2~2.f~l+(s9
4
with
jll+(s, u) analytic in s and u.
542
DOMBEY
Proof.
AND
MCKENZIE
From the crossing matrix
fh+ = [(S,2UA2 + 2pth2)ffi+
+ 2uthy$!,+
+ 2(2uy2
t.g&-
+ 47J,2th2f,“,]/U;, (3.4)
so that (a) by assumption (i)3:1+ is analytic in u. Since all the jU are analytic in s we have by inspection &+ is also analytic in s, and (b) since by assumption (ii)
we have !$J
3A+cG %4
= KS - m”)“/(u - m”)“)
3;+
(s - nz2)2fl,,+(s, u),
withfl,,+(s, u) again analytic in u by assumption (i). Thus there are no discontinuities in the kinematics coming from the reduced crossing matrix and the limit h --t 0 is continuous. Provided assumption (ii) holds the double zero found by Goldberger and Abarbanel[2] at threshold does indeed exist and for nonzero X the zeros move continuously away to new positions whose exact location is given by the dynamics. The significance for the LET is that even for nonzero though sufficiently small X, A, the continuum is still suppressed by zeros near threshold. With strengthened assumptions the crossing matrix yields additional information on the extent of this suppression. For example if lim,,,30; and lim,,,falexist then we can easily show at threshold lims+(rra+A)2f;“l+(s, u, h) = X2j (u, h) where f (u, X) is analytic in u and lim,,,J(u, X) also exists (i.e., the continuum amplitude is of order h2 at threshold). Hence if 3&+ is analytic in h it can be written
3:+
(s - m2)2j$ + X(s - n-22)jifii + h2fiifiii
with each f analytic in s, u, and h. The discussion for the amplitude &-,+ is identical with that for &+ . The case of 3f1- is slightly different in that (l/S,) f;“,- is the amplitude analytic in s and u which has a zero (s - m”) at X = 0. In summary we have shown by a consistency proof (since we assume continuity in X in the u-channel to demonstrate it in the s-channel) that there is no discontinuity in the kinematic structure of the transverse amplitudes as X -+ 0. Provided hfO, and h2f,, vanish with h the zeros found by Goldberger and Abarbanel at h = 0 will indeed be present and as h moves away from zero they will move continuously away from threshold. If in additionf,, andf,, are nonsingular in h then the continuum amplitude will be of order h2 at threshold. Daboul [14] has reached similar conclusions concerning the persistence of kinematic zeros for the case of a single spin one particle scattering off spinless particles.
THE
HELICITY
AMPLITUDE
APPROACH
4. THE GENERALIZED
TO
Low
LOW
ENERGY
THEOREMS
543
ENERGY THEOREM
The crucial point of the last section was the condition imposed on the longitudinal amplitudes in assumption (ii). We now go on to show that if this condition is relaxed the LETS for charged vector currents and axial currents can still be proved in this framework. The longitudinal amplitudes will be seen to make an explicit contribution to the LET and their limiting behavior as a function of h is a necessary input to the theorem. So instead of assuming the limits X(s, 24)= py
A3$-
and
Y(s, u) = pn$ h2&
are zero we assume only that they exist. With this weaker assumption we can still prove that3f1+(s, U, A) is analytic in s and u for all h and that the lim,,,&+(S, U, A) exists. We have however lost the zero (s - ~3)~ in the limit h = 0. Even so the LET may be preserved by calculating the explicit contribution of X and Y to the s-channel amplitudes. In our calculations we take advantage of the continuity in X to work at h = 0 where the equations are simple. We continue to work only with the prototype 311+ and we shall look at Compton scattering off pions by virtual W’s of almost zero mass. G-parity arguments imply that parity-violating amplitudes corresponding to V-A processes vanish identically and there are no pion pole terms in the A-A amplitude. AA Scattering
We consider this case first because the absence of a dynamic singularity, the pion pole, in the threshold region simplifies the discussion. We use the formula obtained from the crossing matrix f;“l+(s, 4 0) = (6 - m”)“l(u - m2)2)flu1+ + Mu - m”>“> z
(4.1)
where Z(s, 24)= (2(2zp(s
- m”)/(u - m”,) Y + 4x
(4.2)
and (4.3)
From 4.1 we have up
- sp
= tZ(S, U),
where S = s - m2,
U = u - m2,
s+u+t=o,
(4.4)
544
DOMBEY
AND
MCKENZIE
and we have dropped the subscripts from J;“,, for the moment. Z has no singularities in Ssince& and& are regular in S. By symmetry it also has no singularities in U. We also need the isotopic spin decomposition Zji = S.jiZO + [Tj 9 Ti] Z- + {Tj 9 Ti} Z+,
(4.5)
where the Ti are the Z-spin matrices of the isovector current. Under s t--f u crossing Z” and Z+ are odd functions, and Z- is even. We can now prove the generalized LETS. (a) Z(S, U) = -Z(U,
S). Write 77 = s - u = S z = $qs,
U. Then
U),
(4.6)
where Z is an even function. From (4.4)
t7$z)/s” = z + U2,j~- Z)/S”
3u = p cannot have any singularity
(Pp
-
(4.7)
at S = 0, hence
35= qs, U)+ O(P) (b)
(4.8)
Z(S, U) = Z(U, S). Write
.w, u>= zo+ SZ,(S)+ uzm + SUZ,(S, U) where Z, is constant, and Z, is an even function. singularity at S = 0. Then from (4.4) p = (up
(4.9
Z,(S) and Z,(S, U) have no
+ (U + S) Z)/S2
+ [Zo+ UZW)+ UZ,(S)+ + wo + u2c38+ Z,(U))1/S2.
= Z,(S) +
uz2
U2Z21/S
(4.10)
As 3” is regular at S = 0 we must have z, = 0
(4.11)
and
3-7= -Z,(U) + Sfz( U)+ O(S2) Z,(U) + Z,(S) + UZ,(S,
U)+ Us(U)= O(S).
(4.12a) (4.12b)
From (4.12b) we have setting S = U = 0, Z,(O) = 0
(4.13)
THE
HELICITY
AMPLITUDE
APPROACH
TO LOW
ENERGY
THEOREMS
545
and setting S = 0 we obtain a(U) = -Z,(O,
U) - Z,(U)/U.
(4.14)
This gives the generalized LET from (4.12a):
tz#J>/u
p =
- Z,(O, U)S + O(P).
Equation (4.15) takes a specially simple form for
t
(4.15)
= 0. Then
3-9= -Z,(O, -S)S + O(P) =
0) s + O(P)
-Z,(O,
= -s(a2zjasau)(o,o)
+ o(9)
(4.16)
In terms oft and 7 this becomes
38= -S(Z,* - Zve)+ O(F)
(4.17)
where ZtB = (a2Z/at2)(t = 0, 77 = 0) etc. For fixed t # 0 and Z = Z(t), (4.15) reduces to
3” _ 4” (1 - Wt) + O(S2).
(4.18)
The theorems for 3:-r+ are found to be the same as for J’$+ . So in the equations above j8 denotes either 3i1+ or &-,+ . V-V Scattering This case is more interesting since there are now pole terms. However the discussion for the continuum part of the amplitudes (i.e., amplitudes with the pole contribution subtracted out) is the same as in the A-A case since these have no dynamic singularities near threshold. So we must calcultate Zc, the continuum part of Z, and then apply the analysis above to Zc. We obtain for the pole terms
3;+ =
W/(u - m”)> x
w2NcG3 Til - [rj 9Til)
(4.19)
and &+ is given by crossing. Using (4.4) we have ZVf
=
-27
.po
zp- = 2t where p denotes pole. Notice that (4
Z,” = 2 # 0;
ZE 0
(4.20)
546
DOMBEY AND MCKENZIE
whereas from (4.13) the continuum contribution Zc- has no linear terms in S and u and hence Z,C- = 0. We shall shortly discuss the significance of Z- for V-V scattering. p+
(ii)
= -2 7.
Consider pion Compton scattering with real photons. In this case Z+ = 0 as we know that there are no longitudinal photons present. Therefore zc+ = -ZPf
= 2rl.
The LET is from (4.8) and (4.19) 3;;
= (2t/(u - mZ)) + 2 + O(S2).
(4.21)
Had we in the first place chosen the Born term as -2(s - m”)/(u - m2), which amount to including the seagull graph as well as the pole terms, then we would have found ZBOrn = 0 and hence Zc = 0 yielding the same LET as derived by Goldberger and Abarbanel [2] by this method. They, however included the seagull in their Born term by appealing to gauge invariance. We have thus shown that the condition Z = 0 for real photons will give equivalent results. So we could say that the limiting behavior of the longitudinal modes plays the role of gauge invariance in a purely S-matrix proof [5]. The requirement that there be no contribution from longitudinal photons in Compton scattering naturally forces the presence of a term corresponding to the seagull even though we start only with the single pion pole terms. In general the limiting behavior of the longitudinal amplitudes, as embodied in Z, is required as well as the dynamic singularities at threshold as an input to the LET. The theorem then follows from the massless or near massless nature of the virtual W. Previously we showed that provided r”,“,- and 34 are nonsingular in h the continuum amplitude will be order O(hz) at threshold. The equivalent result is obtained in the general case provided the generalized Z(s, U, X) is known up to order X2.
5. LONGITUDINAL
AMPLITUDES
AND THE GENERALIZED
LETS
It is acceptable in S-matrix theory to specify as a boundary condition that no longitudinal modes in the scattering amplitude are present in real photon processes; i.e., that Z = 0. We have seen that this condition can be used to replace the field theoretic motions of current conservation or gauge invariance. For virtual W processes, however, Z # 0 and we need to be able to specify how to calculate it in
THE
HELICITY
AMPLITUDE
APPROACH
TO LOW
ENERGY
THEOREMS
547
order to derive LETS. For this we need current algebra and a field theoretic description of the process. As the concept of a virtual particle is indeed field theoretic in origin it is not really surprising that we are led back to field theory. In this framework we describe the scattering process by a scattering tensor Muy = -i(2r)3
1 ein’Yp
where R denotes the retarded commutator polarization vectors of the form E,+ = (l/&2)(0,
-1,
-i,
0)
E,O = wv(q,
1R(J,‘(x)
J,(O)) 1p) dx
(5.1)
of the currents, together with initial E,- = (l/d/z)(O,
1, -i,
0)
(5.2)
0, 0, 40)
when qu = (q. , 0, 0, q) and q2 = h2. E,Ocan be rewritten [5] c,” = (l/4 CL + (M4 + %)ELO,
0, 1).
We can write similar expressions for the final polarization and q’. So as M,,” is nonsingular in h
vectors in terms of h
(The tensors MUy have the dynamic structure given by purely hadronic intermediate states; this is independent of h.) Now
q,‘M,, = CW3 1 ei*Y + 0’ I [Jo’(x), .Wl %x0) I P>) dx (5.4) and in our examples J and J’ are either both vector currents or axial currents so that the commutator is [Joj(x), J,i(O)] 6(x0) = kiia V,“(x) S4(x).
(5.5)
We thus see that qu’Mu,u is given by the matrix element of a single current plus the matrix element of the retarded product of the divergence of a current times a current. The latter is zero by CVC for YV scattering but represents using PCAC the amplitude for pion photoproduction (or electroproduction) for VA scattering, and weak pion production for VA + AA scattering. In fact one directly obtains LETS for such pion processes [15] by replacing the matrix element of qu’Mu, by its
548
DOMBEY AND MCKENZIE
pole terms; such procedures are good to order 4’ N s - m2 for AA scattering and to order qq’ .- (s - m”)” for VA scattering (as here the vector current is conserved). In the case of V-V scattering of pions we can now calculate Z from Eqs. (4.2) and (5.3) zo.+ = 0 z- = 2tF*(t), (5.6) where F,(t) is the pion form factor. There was also a pole contribution
Zp- = 2t which we have already noted. So
zc- = 2t[F,(t) - l]
(5.7)
and this satisfies Zi- = 0 as expected. We can now write the LETS. For t = 0, we can use Eq. (4.17):
3&
= -4(s - m2)(dFJdt)ltzo
+ O((s - m”)“).
(5.8)
On the other hand, for fixed t we can obtain unphysical LETS. From Eqs. (4.18) and (4.19) f&(s, t) = -(2t/(u
- m”)) + 2(F,(t) - l)(l - 2(s - m”)/t) + O((s - m”)“).
(5.9)
For A-A scattering of pions, we write U, = (27r)3 / ei*‘s(p’ I R(W(x) = &P” +
g,q,’
+
g3q”
J”(O) I p) dx (5.10)
with isospin indices understood. The LETS for weak pion production off pionsl are given by g,(s = m2, t = 0) = g,(v = 0, t = 0) = g,(O, 0) and in general g,(O, 0) is given by Eq. (5.4) replacing qP’Muy by its pole terms, but there are no pole terms here; Z = 0 at threshold. The LETS for the transverse AA amplitudes are given by the derivatives of Z; these cannot be predicted by arguments based on PCAC and current algebra and represent first order correction terms to the LETS for weak pion production. The calculation of the LETS in this case is straightforward. We simply need to calculate Z using the expression for q,‘M,, of Eq. (5.4). Remembering that the minus sign in h& implies that we need to consider the polarisation vector E,+ - E,1 The LET’s for pion photoproduction or weak pion production off a pion target should not be. taken too seriously as they involve an extrapolation of the amplitude from q’2 = 0 to ma. For nucleon targets this extrapolation should not be important, but for pion targets of mass m it clearly is important.
THE
HELICITY
AMPLITUDE
APPROACH
TO
and using the crossing matrix and appropriate
LOW
ENERGY
kinematic
THEOREMS
549
factors, we obtain from
(4.3) zji = 2([Ti
For the even amplitudes,
(5.11)
7Til FAt) + 81)t - 21782 -
Z:*” = 0. Hence gi”‘O’(o, 0) = 0
For the odd amplitudes,
(5.12)
Zt- = Z,- = 0. Hence
g;-)(O, 0) = 0
and
g;-‘(0, 0) = -1.
(5.13)
Equation (5.12) and (5.13) give the LETS for weak pion production for A-A scattering we have from Eqs. (4.8) and (4.17) for t = 0
_- -2g$O’(O, f “d+.0) 11+ f$’
= -4(s
0)
+
O(s
- m2)[dF,/dt
-
off pions. Now
(5.14)
m")
It-0 + @g,-ptp-$
0) + (ag,/a$(o,
Having been forced into a field theoretic framework anticipate a more direct calculation of the LET entirely framework. Such a procedure does indeed exist as we looking for is a method of calculating the contribution direction we make the separation
O)] + O((s - m”)“)
to calculate within the show next. of Zc. As
(5.15) Z one might field theoretic What we are a step in this
Muy = M;,?le + M,dj” + R,,
with M$*’ a nonsingular
tensor satisfying q,‘M,d,“’ = q,‘(M,” M:;ffqy = (M,,
-
Mf;le)
- M,gOle) qv .
Then by definition
Now if we derive a set of helicity amplitudes from R,, the construction ensures that the longitudinal amplitudes are nonsingular as h -+ 0 so that the conditions of the theorem obtain. Transverse amplitudes derived from R,, are therefore suppressed by the kinematic zeros and the LET is given entirely by MEFle + Mziff to that order. These manipulations are equivalent to calculating the contribution of Z to the transverse amplitudes and M,dJ” corresponds to Z”. The V-V and A-A theorems
550
DOMBEY AND MCKENZIE
can be derived in this way. In the case of A-A the most general form for &Z$!” should be written down and then its divergence compared with the form for the divergence of IV,,” in terms of the commutator and the amplitude U, . Complete agreement between the various methods ensures [16].
6. SUM RULES We can easily convert the LETS into sum rules. For example, for V-V scattering, we have at t = 0 from Eq. (4.21) -2 mImK(~t=O)dS=2, n- s0 remembering that for t = 0, the integral over the left-hand cut is equal to that over the right-hand cut. Making the conventional normalization so that the Thomson limit is obtained as the LET for neutral isovector photons scattered off 7~+,and using the isospin relations
f” -#2f- -2f+ =fo, f” 4-f- + 3f+ =fi f” -f- +f+ =f2, where f. ,fi ,fi are the amplitudes obtain the sum rule
(6.2)
for the Z = 0, 1, 2 states of 7”~ ---f yv~, we
2rr2a/m = jam[~yo,,o(v)-
~,,~,+(v)ldv
(6.3)
v is the photon laboratory energy, y” is a neutral isovector photon, and u++(v) is the total absorptive cross section. For isoscalar photons ys,
so we could replace the cross sections in (6.3) by the real photon cross sections a,,,~ and a,,+ to obtain the Pagels-Harari sum rule [17]. The integral of the sum rule is expected to converge as f ^s+ only receives contributions from Z = 2 exchange in the t-channel. Similarly the sum rule for the amplitudejic+ follows from Eq. (5.8). We have Irn f&-($
’ = ‘1 ds = -4 $1
. t-o
(6.4)
THE
HELICITY
AMPLITUDE
APPROACH
This becomes the Cabibbo-Radicati
TO LOW
ENERGY
sum rule [4] on normalizing
THEOREMS
551
and using (6.2)
xr,2>= & jm$ [u(yO + 7ro-+z = 0) 0
+cr(yO+7T++z=1)-~u(y0+nO+z=2)]
(6.5)
where the charge radius of the pion r, is given by &(rv2> = (dF,/dt) I,=, . The integral is again expected to converge as fS- has contributions from Z = 1 exchange and there is now an extra power of energy in the denominator. It is now possible to write down sum rules for forward neutrino scattering of pions. The VP’ + AA sum rules are from (4.21) and (5.14) 2
-=
mIm3:A(S, t = s0
S
0)
dS = 2 - 2g,+(O, 0),
(6.6)
and from (5.8) and (5.19,
=--4
[
2% dt It=o+
%(O,O)
+ +o,o)].
The left-hand side of (6.6) and (6.7) now involves the quantities c+( W+~T) where ur = 4(oL + uR). These are total absorptive cross sections2 [18] initiated by a virtual W+ satisfying X2 = 0. T, L and R refer to transverse unpolarised, left and right polarized spin states of the W+ respectively. Then we have m [q(w+?r+)
+ ur( w+?r-)
- 2UT( W’7rO)] = &
(1 - g,+(o, 0))
i 0
(6.8)
and
a 0).
(6.9)
* They are not actually virtual cross sections but are proportional to the cross sections. Conventionally, these virtual weak cross sections do not include the weak coupling constant G; a factor G2 appears in the overall neutrino cross section.
552
DOMBEY AND MCKENZIE
If G-parity conservation is assumed, then
Sum rules could also have been written down for t # 0 from the LETS for t # 0. But these could not be written in terms of total absorption cross sections; also as the helicity amplitude fs is not crossing symmetric under s +-+ ZJexcept at t = 0, it is difficult to obtain a sum rule in terms of physically measurable quantities. The LETS of (4.21) and (5.9) for I’-V scattering could in principle be used to generalize the Cabibbo-Radicati and Pagels-Harari sum rules to fixed t # 0.
ACKNOWLEDGMENTS We would like to thank D. Bailin, G. Barton, and D. Broadhurst for their help and advice; one author (N.C.M.) would like to thank the Royal Society for a Fellowship and Professor H. Yukawa and Professor Z. Maki for their kind hospitality at the Research Institute for Fundamental Physics. REFERENCES 1. F. E. Low, Phys. Rev. 96 (1954), 1248; M. GELL-MANN
AND M. L. GOLDBERGER, 96 (1954), 1433. 2. H. D. I. ABARBANEL AND M. L. GOLDBERGER, Phys. Rev. 165 (1968), 1594. 3. M. BEG, Phys. Rev. Letters, 17 (1966), 333. 4. N. CABIBBO AND L. RADICATI, Phys. Rev. Letters 19 (1966), 697. 5. N. DOMBEY, Nuovo Cimento 32 (1964), 1696; see also, N. DOMBEY, in “Hadronic
6. 7. 8. 9. 10.
11. 12. 13. 14. 15. 16. 17. 18.
Phys. Rev.
Interactions of Electrons and Photons” (J. Cumming and H. Gsbom, Eds.), Academic Press, New York, 1971. S. D. DRELL AND A. C. HEARN, Phys. Rev. Letters, 16 (1966), 908; S. B. GERASIMOV, Soviet Physics J., 2 (1966), 430. N. C. MCKENZIE, Progr. Theoret. Phys. 47 (1972), 246. L-L. C. WANG, Phys. Rev. 142 (1966), 1187. T. L. TRUEMAN AND G. C. WICK, Ann. Physics 26 (1964), 322. G. COHEN-TANNOLJDJI, A. MOREL, AND H. NAVELET, Ann. Physics 46 (1968), 239 (for massive particles); J. P. ADER, M. CAPDEVILLE, AND H. NAVELET, Nuovo Cimento A 56 (1968), 315 (for processes including zero mass particles). H. Joos, Fortschr. Physik. 10 (1962), 65. R. L. OMNES, Phys. Rev. 168 (1968), 1893. Y. HARA, Progr. Theoret. Phys. 45 (1971), 584. J. DABOUL, Phys. Reu. 177 (1969), 2375. For example on a nucleon target, see S. FUBINI AND G. FURLAN, Ann. Physics 48 (1968), 322 (for pion electroproduction); N. PAVER, C. VERZEGNASSI AND E. E. RADESCU, Nuovo Cimento 66 A (1970), 261 (for weak pion production). N. C. MCKENZIE, D. Phil. Thesis, University of Sussex, 1970 (unpublished(. H. PAGELS, Phys. Rev. Letters 18 (1967), 316; H. HARARI, Phys. Rev. Letters 18 (1967), 319. N. DOMBEY, Rev. Mod. Phys. 41 (1969), 236.
OF PHYSICS:
78, 535-552 (1973)
The Helicity Amplitude Approach to Low Energy Theorems NORMAN
DOMBEY
University of Sussex, School of Mathematical and Physical Sciences, Brighton BNl 9QH, Sussex, England AND
N.
C. MCKENZIE
Research Institute for Fundamental Physics, Kyoto University, Kyoto, Japan Received May 23, 1972
It is shown that “low energy theorems” for charged vector and axial vector currents have a natural derivation in terms of helicity amplitudes. This is done by reconsidering the kinematic constraints as a function of the virtual photon mass, A. The crucial point is that in general a certain limit of the longitudinal amplitudes is required as an input to the theorem. Two new YV sum rules are also obtained.
1. INTRODUCTION Fourteen years after the original derivation [l] a new derivation of the low energy theorem for Compton scattering was given by Goldberger and Abarbanel [2]. They observed that there are kinematic zeros at threshold in the helicity amplitudes for this process which suppress the unknown many particle contribution there since the dynamic cut singularities start well away from threshold. The resulting amplitudes are thus given to first order in a entirely by the Born terms at threshold. This new derivation has a certain elegance and may be considered preferable to alternative derivations, mainly descended from that of Low [l], in as much as it makes the structure of the theorem particularly clear. Naively one might expect that on account of these zeros suppressing the continuum contribution, low energy theorems (hereafter, LETS) for isovector photons or massless virtual weak bosons would again be given simply by the Born terms at threshold. However this is not correct and for example the LETS first derived by Beg [3] for nucleons leading to sum rules such as the Cabbibo-Radicati [4] sum rule cannot be obtained simply from the Born term. It thus seems that the zeros 535 Copyright Al1 rights
0 1973 by Academic Press, Inc. of reproduction in any form reserved.
536
DOMBEY
AND
MCKENZIE
found by Goldberger and Abarbanel are not present in these cases. Related to this problem is another. Zeros in helicity amplitudes which are manifestly present when the spin one particle has exactly zero mass would seem not to be present for any nonzero mass however small. There is, in short, an apparent discontinuity which we do not expect either on physical grounds (experiments indicate that virtual photon cross sections measured in ep scattering tend smoothly to real photon cross sections) or on theoretical grounds if the spin one particle or virtual particle couples to a conserved current [5]. In this paper we reexamine these problems by considering the constraints imposed by the kinematics as a function of the mass h of a spin one boson undergoing Compton-type scattering with a pion (e.g., yr ---f yn or Wn --f WIT). We restrict our attention to the case of a spinless target because of the greatly simplified analysis, but our conclusions should apply for general spin. The original motivation for studying this problem was to obtain new sum rules for neutrino-nucleon cross sections following the model of the Drell-HearnGerasimov [6] or Beg [3] sum rules. These were to be obtained by first deriving the LET for a Compton-type process with the electromagnetic current replaced by a weak current. The easiest and most elegant way to obtain these LETS seemed to be to follow Goldberger and Abarbanel and simply write down the appropriate Born terms. As we have already remarked such an easy solution was doomed to failure but nonetheless our investigations with helicity amplitudes led us to a rather direct way of obtaining the desired nucleon LETS which are presented together with the sum rules in a separate paper [7]. In this paper we derive the equivalent LETS and sum rules for a pion target but the main purpose of the paper is not the sum rules, which are perhaps more of a theoretical curiosity than an experimental possibility, but rather to study the kinematics of the zero mass limit for cases corresponding to a nonconserved or charged current. We are able to show that such theorems can indeed be proved using helicity amplitudes. They again depend on the presence of kinematic zeros suppressing unknown continuum contributions and these are a result of the massless or nearmassless spin-one nature of the incident object. However, we show that in general the LETS for transverse amplitudes receive a contribution from the longitudinal amplitudes and the limiting behavior of these amplitudes as well as the Born singularities are a necessary input to the theorem. In a previous paper [5] one of us showed that the requirement of current conservation in processes involving a photon could be replaced in S-matrix theory by the statement that longitudinal modes of the photon decoupled for zero photon mass. This work generalizes this result: where the current is conserved the requirement that longitudinal modes tend to zero with photon mass gives rise to conventional LETS, but if the current is nonconserved then longitudinal modes are present at h = 0 and have to be included in the theorems.
THE HELICITY AMPLITUDE
APPROACH TO LOW ENERGY THEOREMS
537
2. THE STRUCTURE OF A Low ENERGY THEOREM We employ the original method of Wang [8] to find the kinematic structure of helicity amplitudes. Wang assumed that the reduced helicity amplitudes f8, defined by 12.1)
1 + 2 - 3 + 4 and fhsA;Ah is the s-channel helicity amplitude with 9, the center-of-mass scattering &gie: have no kinematic singularities in the channel momentum transfer variable t or U. The reduced t- and u-channel amplitudes are likewise assumed to have no kinematical singularities in their channel momentum transfer variables. The only singularities in t or u which can occur infs come from the nonconvergence of the Jacobi series 01 = A, - A, ) p = A, - A, ) where in the s-channel
3” =
1
(2.I+
1) F;&;l,A,(S) P’(!&$‘a+B’)(cos
0,)
J
and these are assumed to be of dynamical origin. This assumption acquires potency when used in conjunction with the crossing matrix first obtained by Trueman and Wick [9]. The crossing matrix relates Schannel and t- or u-channel helicity amplitudes, analytically continued to the same point in the (s, t) or (s, U) plane. From the crossing matrix we obtain a reduced crossing matrix which is defined as the matrix connecting reduced helicity amplitudes. Thus suppose we relate in this way reduced s-channel amplitudes to reduced t-channel amplitudes. The reduced s-channel amplitudes are by assumption already analytic in t and the reduced t-channel amplitudes are analytic in S. The reduced crossing matrix therefore contains all the kinematic singularities and zeros of the s-channel amplitudes in the variable S. Likewise it contains all the t singularities of the t-channel amplitudes. In this sense the reduced crossing matrix contains all the kinematic structure. Other, perhaps more conceptually direct, methods exist for deducing the kinematic structure [IO]. These relate helicity amplitudes to amplitudes already known to have only dynamic singularities (for example the Joos amplitudes [I 11). The kinematic structure is then contained in the coefficients which relate these amplitudes to the helicity amplitudes. All methods agree in any given mass configuration but these other methods tend to be technically involved and whenever two or more masses become equal or a mass becomes zero a fresh approach has to be made. Principally on account of this we favor Wang’s approach. We illustrate the approach by rederiving the result for pion Compton scattering following Abarbanel and Goldberger [2]. Pion Compton scattering is described by two independent s-channel helicity amplitudes f&1, and fi&-lo (in the s-channel
538
DOMBEY
AND
MCKENZIE
y + n + y + 7r). The related t-channel process is described by f:W1;0o and f&, . Similarly in the u-channel we have f?l,,+.10 and f;1o:--1,,. The crossing matrices are trivial for this case.
f"10:10
=flLoo
=
fJ10;-10
fiso?lO= f&o0 = G:-IO
(2.2)
so between reduced amplitudes
f;“,;,,=
cosm2 40, sinM2 $9&~,;,,
= cos-’ #?, cosm2 &3u3!!10~-10
or
J;“,;,,=
((s - m2)2/t(t - 4m2))3:-1:~ = ((s - m2)2/(u-
m2)2)3~~o;-~~ (2.3)
and
3&,-1o = sin-’ 48,3,“,,,, = sinb2 $0, sin2 ~0,3&-lo or
3;o;-lo= ((s - m2)2/-st)3;~:, = (u(s where we use the usual notation assumption that ps is kinematically and t, we see that
3&o=
6 - m2)2Eo;lo 3:-rGoo = t(t - 4m2)Jit_1;oo 3:~o;-10 = (24- m2)2f&lo
m2)2/s(u -
m2)2)3&40(2.4)
for kinematic quantities. Remembering our regular in t and U, 3” in s and U, and p in s
and and
and
f;80;-10 =
Ns - m2)2/Cf&-10 3,1;00 = ?Koo
fLo
(2.5)
= ((u - m2)2/u>.f&-lo
where the barred amplitudes have no kinematic singularities but only dynamic structure. So since (s - m”) N 1q 1, (U - m”) N 1 q I, t N 1q I2 near threshold, with q the photon momentum, each set of amplitudes has a kinematic zero of order I q I2 at threshold, (s, t) = ( m2, 0). We have performed the calculation symmetrically for s, t, and u because we want to emphasize that the reduced crossing matrix between any two channels contains all the kinematic information. In our later analysis we will take advantage of the greater symmetry to be obtained from s-u crossing. The barred amplitudes have no kinematic singularities but only the usual dynamic structure. The point to note is that the unknown many particle cut starts away from threshold so that its contribution is suppressed by the kinematic zeros. To complete the derivation of a LET we need, aside from kinematic zeros and continuum cuts separated from threshold, to be able to write down the single particle pole terms or Born terms. This is straightforward since the location of the pole is known and the pole residue is in terms of the coupling constants charge, magnetic
THE HELICITY AMPLITUDE
APPROACH TO LOW ENERGY THEOREMS
539
moment, etc. So in the case of pion Compton scattering we can easily write down the pion pole terms in the S- and u-channels. As is well known, however, the LET is not given by these two terms alone. A seagull or contact term must be included as well. This marks an ambiguity in the original S-matrix derivation [2]. In the field theoretic approach the seagull term is forced by the requirement of a gauge invariant Born term. Such a principle lies outside the spirit of the S-matrix approach and if this method is to be self-consistent it should be possible to proceed without recourse to gauge invariant perturbation theory. The ambiguity can be resolved by the requirement that no longitudinal photons are present [5], and this in turn leads to a consideration of (virtual) photon mass h in the problem.
3. REAPPRAISAL
OF THE ZERO MASS LIMIT
We now attempt to generalize the LETS in order to deal with charged or nonconserved currents. Thus we are prompted to reconsider the kinematic constraints as a function of the virtual photon (or W meson) mass h. As the zero mass limit is approached we look for the necessary condition which must be satisfied in order that Goldberger and Abarbanel’s zeros are present. We shall then relax it in the next section and so obtain the generalization of the LET. This is our primary objective. At the same time we find ourselves very well placed to comment on the apparent discontinuity found by a previous author [12] between the identically zero mass case and the nonzero mass case. In this section we will show that no such discontinuities are implied by the kinematics, where the kinematics are defined as the structure introduced by the reduced crossing matrix. We will consider the explicit crossing matrix between the S- and u-channels [ 131. s-channel
1+2+3+4 w+r+ w+?r
u-channel
3+2d+4 w+7T+
The crossing relation has the form (summing
w+rr. repeated indices) (3.1)
and #r = z,/J~= #because of the mass equality. We assume time-reversal invariance for simplicity although this is by no means necessary to the following discussion. Further simplification is obtained by writing the crossing matrix between parity conserving pairs, e.g., hso;~lo f ~-A~~;-A~~. Then noting the following expressions
540
DOMBEY
AND
MCKENZIE
for the angular quantities cos 40, , sin 40, , cos $0, , cos 4, and sin $ we obtain the reduced crossing matrix tabulated in Table I. cos +e,=
p1q9A
sin
cos ge, = p112/~A
;e, = (-st)1/2/SA
sin +e, = (-24t)112/UA
(3.2)
sin I/J = 2h( -pt)1/2/SAUA
cos # = (p - (m” - X2) t)/SJJ*
where
&2 = [s- (m+ A)2][s - (m- X)2], u,2= [u - (m+ A)21[u - (m- X)2] 5” = SAWA2 + 4Ppt,
(3.3)
p = (m” - h2)2 - su.
Let us then examine the constraints imposed by the reduced crossing matrix on a typical amplitude, 3 11+= f$,;1o + 3!,,;-,, as a function of A. We take as input assumptions the following. (i) Following Wang: that&+@, U, A) is an analytic function of u for all values of s and A, and similarly f;“l+(s, U, A), &+(s, U, A), A..,& U, A) and h2f&, U, A) are analytic functions of s for all u and A. [Of course3S will not necessarrly be an analytic function of u in the presence of dynamic singularities. However the separation into dynamic and kinematic structure is made quite unambiguous by defining the latter to be singularities which come from the reduced crossing matrix under assumption (i). The dynamic structure can then be introduced independently after the kinematic structure has been separated out.] (ii) We assume that A&- and h2& go continuously to zero with h and that the limiting functions &s, U, 0) = lim,,,f’(s, U, A) exist for purely transverse amplitudes. [The first part of this assumption turns out to be too restrictive in the general case and new results come when it is relaxed.] Then the following theorem is a trivial consequence. THEOREM
(a) &+(s, u, A) is an analytic function of s and u for all X (apart from dynamic singularities). (b) the limit lim,+Of:l+(s, u, A) exists and has the form
3A+(s, u, 0)
=
0
-
m2~2.f~l+(s9
4
with
jll+(s, u) analytic in s and u.
542
DOMBEY
Proof.
AND
MCKENZIE
From the crossing matrix
fh+ = [(S,2UA2 + 2pth2)ffi+
+ 2uthy$!,+
+ 2(2uy2
t.g&-
+ 47J,2th2f,“,]/U;, (3.4)
so that (a) by assumption (i)3:1+ is analytic in u. Since all the jU are analytic in s we have by inspection &+ is also analytic in s, and (b) since by assumption (ii)
we have !$J
3A+cG %4
= KS - m”)“/(u - m”)“)
3;+
(s - nz2)2fl,,+(s, u),
withfl,,+(s, u) again analytic in u by assumption (i). Thus there are no discontinuities in the kinematics coming from the reduced crossing matrix and the limit h --t 0 is continuous. Provided assumption (ii) holds the double zero found by Goldberger and Abarbanel[2] at threshold does indeed exist and for nonzero X the zeros move continuously away to new positions whose exact location is given by the dynamics. The significance for the LET is that even for nonzero though sufficiently small X, A, the continuum is still suppressed by zeros near threshold. With strengthened assumptions the crossing matrix yields additional information on the extent of this suppression. For example if lim,,,30; and lim,,,falexist then we can easily show at threshold lims+(rra+A)2f;“l+(s, u, h) = X2j (u, h) where f (u, X) is analytic in u and lim,,,J(u, X) also exists (i.e., the continuum amplitude is of order h2 at threshold). Hence if 3&+ is analytic in h it can be written
3:+
(s - m2)2j$ + X(s - n-22)jifii + h2fiifiii
with each f analytic in s, u, and h. The discussion for the amplitude &-,+ is identical with that for &+ . The case of 3f1- is slightly different in that (l/S,) f;“,- is the amplitude analytic in s and u which has a zero (s - m”) at X = 0. In summary we have shown by a consistency proof (since we assume continuity in X in the u-channel to demonstrate it in the s-channel) that there is no discontinuity in the kinematic structure of the transverse amplitudes as X -+ 0. Provided hfO, and h2f,, vanish with h the zeros found by Goldberger and Abarbanel at h = 0 will indeed be present and as h moves away from zero they will move continuously away from threshold. If in additionf,, andf,, are nonsingular in h then the continuum amplitude will be of order h2 at threshold. Daboul [14] has reached similar conclusions concerning the persistence of kinematic zeros for the case of a single spin one particle scattering off spinless particles.
THE
HELICITY
AMPLITUDE
APPROACH
4. THE GENERALIZED
TO
Low
LOW
ENERGY
THEOREMS
543
ENERGY THEOREM
The crucial point of the last section was the condition imposed on the longitudinal amplitudes in assumption (ii). We now go on to show that if this condition is relaxed the LETS for charged vector currents and axial currents can still be proved in this framework. The longitudinal amplitudes will be seen to make an explicit contribution to the LET and their limiting behavior as a function of h is a necessary input to the theorem. So instead of assuming the limits X(s, 24)= py
A3$-
and
Y(s, u) = pn$ h2&
are zero we assume only that they exist. With this weaker assumption we can still prove that3f1+(s, U, A) is analytic in s and u for all h and that the lim,,,&+(S, U, A) exists. We have however lost the zero (s - ~3)~ in the limit h = 0. Even so the LET may be preserved by calculating the explicit contribution of X and Y to the s-channel amplitudes. In our calculations we take advantage of the continuity in X to work at h = 0 where the equations are simple. We continue to work only with the prototype 311+ and we shall look at Compton scattering off pions by virtual W’s of almost zero mass. G-parity arguments imply that parity-violating amplitudes corresponding to V-A processes vanish identically and there are no pion pole terms in the A-A amplitude. AA Scattering
We consider this case first because the absence of a dynamic singularity, the pion pole, in the threshold region simplifies the discussion. We use the formula obtained from the crossing matrix f;“l+(s, 4 0) = (6 - m”)“l(u - m2)2)flu1+ + Mu - m”>“> z
(4.1)
where Z(s, 24)= (2(2zp(s
- m”)/(u - m”,) Y + 4x
(4.2)
and (4.3)
From 4.1 we have up
- sp
= tZ(S, U),
where S = s - m2,
U = u - m2,
s+u+t=o,
(4.4)
544
DOMBEY
AND
MCKENZIE
and we have dropped the subscripts from J;“,, for the moment. Z has no singularities in Ssince& and& are regular in S. By symmetry it also has no singularities in U. We also need the isotopic spin decomposition Zji = S.jiZO + [Tj 9 Ti] Z- + {Tj 9 Ti} Z+,
(4.5)
where the Ti are the Z-spin matrices of the isovector current. Under s t--f u crossing Z” and Z+ are odd functions, and Z- is even. We can now prove the generalized LETS. (a) Z(S, U) = -Z(U,
S). Write 77 = s - u = S z = $qs,
U. Then
U),
(4.6)
where Z is an even function. From (4.4)
t7$z)/s” = z + U2,j~- Z)/S”
3u = p cannot have any singularity
(Pp
-
(4.7)
at S = 0, hence
35= qs, U)+ O(P) (b)
(4.8)
Z(S, U) = Z(U, S). Write
.w, u>= zo+ SZ,(S)+ uzm + SUZ,(S, U) where Z, is constant, and Z, is an even function. singularity at S = 0. Then from (4.4) p = (up
(4.9
Z,(S) and Z,(S, U) have no
+ (U + S) Z)/S2
+ [Zo+ UZW)+ UZ,(S)+ + wo + u2c38+ Z,(U))1/S2.
= Z,(S) +
uz2
U2Z21/S
(4.10)
As 3” is regular at S = 0 we must have z, = 0
(4.11)
and
3-7= -Z,(U) + Sfz( U)+ O(S2) Z,(U) + Z,(S) + UZ,(S,
U)+ Us(U)= O(S).
(4.12a) (4.12b)
From (4.12b) we have setting S = U = 0, Z,(O) = 0
(4.13)
THE
HELICITY
AMPLITUDE
APPROACH
TO LOW
ENERGY
THEOREMS
545
and setting S = 0 we obtain a(U) = -Z,(O,
U) - Z,(U)/U.
(4.14)
This gives the generalized LET from (4.12a):
tz#J>/u
p =
- Z,(O, U)S + O(P).
Equation (4.15) takes a specially simple form for
t
(4.15)
= 0. Then
3-9= -Z,(O, -S)S + O(P) =
0) s + O(P)
-Z,(O,
= -s(a2zjasau)(o,o)
+ o(9)
(4.16)
In terms oft and 7 this becomes
38= -S(Z,* - Zve)+ O(F)
(4.17)
where ZtB = (a2Z/at2)(t = 0, 77 = 0) etc. For fixed t # 0 and Z = Z(t), (4.15) reduces to
3” _ 4” (1 - Wt) + O(S2).
(4.18)
The theorems for 3:-r+ are found to be the same as for J’$+ . So in the equations above j8 denotes either 3i1+ or &-,+ . V-V Scattering This case is more interesting since there are now pole terms. However the discussion for the continuum part of the amplitudes (i.e., amplitudes with the pole contribution subtracted out) is the same as in the A-A case since these have no dynamic singularities near threshold. So we must calcultate Zc, the continuum part of Z, and then apply the analysis above to Zc. We obtain for the pole terms
3;+ =
W/(u - m”)> x
w2NcG3 Til - [rj 9Til)
(4.19)
and &+ is given by crossing. Using (4.4) we have ZVf
=
-27
.po
zp- = 2t where p denotes pole. Notice that (4
Z,” = 2 # 0;
ZE 0
(4.20)
546
DOMBEY AND MCKENZIE
whereas from (4.13) the continuum contribution Zc- has no linear terms in S and u and hence Z,C- = 0. We shall shortly discuss the significance of Z- for V-V scattering. p+
(ii)
= -2 7.
Consider pion Compton scattering with real photons. In this case Z+ = 0 as we know that there are no longitudinal photons present. Therefore zc+ = -ZPf
= 2rl.
The LET is from (4.8) and (4.19) 3;;
= (2t/(u - mZ)) + 2 + O(S2).
(4.21)
Had we in the first place chosen the Born term as -2(s - m”)/(u - m2), which amount to including the seagull graph as well as the pole terms, then we would have found ZBOrn = 0 and hence Zc = 0 yielding the same LET as derived by Goldberger and Abarbanel [2] by this method. They, however included the seagull in their Born term by appealing to gauge invariance. We have thus shown that the condition Z = 0 for real photons will give equivalent results. So we could say that the limiting behavior of the longitudinal modes plays the role of gauge invariance in a purely S-matrix proof [5]. The requirement that there be no contribution from longitudinal photons in Compton scattering naturally forces the presence of a term corresponding to the seagull even though we start only with the single pion pole terms. In general the limiting behavior of the longitudinal amplitudes, as embodied in Z, is required as well as the dynamic singularities at threshold as an input to the LET. The theorem then follows from the massless or near massless nature of the virtual W. Previously we showed that provided r”,“,- and 34 are nonsingular in h the continuum amplitude will be order O(hz) at threshold. The equivalent result is obtained in the general case provided the generalized Z(s, U, X) is known up to order X2.
5. LONGITUDINAL
AMPLITUDES
AND THE GENERALIZED
LETS
It is acceptable in S-matrix theory to specify as a boundary condition that no longitudinal modes in the scattering amplitude are present in real photon processes; i.e., that Z = 0. We have seen that this condition can be used to replace the field theoretic motions of current conservation or gauge invariance. For virtual W processes, however, Z # 0 and we need to be able to specify how to calculate it in
THE
HELICITY
AMPLITUDE
APPROACH
TO LOW
ENERGY
THEOREMS
547
order to derive LETS. For this we need current algebra and a field theoretic description of the process. As the concept of a virtual particle is indeed field theoretic in origin it is not really surprising that we are led back to field theory. In this framework we describe the scattering process by a scattering tensor Muy = -i(2r)3
1 ein’Yp
where R denotes the retarded commutator polarization vectors of the form E,+ = (l/&2)(0,
-1,
-i,
0)
E,O = wv(q,
1R(J,‘(x)
J,(O)) 1p) dx
(5.1)
of the currents, together with initial E,- = (l/d/z)(O,
1, -i,
0)
(5.2)
0, 0, 40)
when qu = (q. , 0, 0, q) and q2 = h2. E,Ocan be rewritten [5] c,” = (l/4 CL + (M4 + %)ELO,
0, 1).
We can write similar expressions for the final polarization and q’. So as M,,” is nonsingular in h
vectors in terms of h
(The tensors MUy have the dynamic structure given by purely hadronic intermediate states; this is independent of h.) Now
q,‘M,, = CW3 1 ei*Y
(5.5)
We thus see that qu’Mu,u is given by the matrix element of a single current plus the matrix element of the retarded product of the divergence of a current times a current. The latter is zero by CVC for YV scattering but represents using PCAC the amplitude for pion photoproduction (or electroproduction) for VA scattering, and weak pion production for VA + AA scattering. In fact one directly obtains LETS for such pion processes [15] by replacing the matrix element of qu’Mu, by its
548
DOMBEY AND MCKENZIE
pole terms; such procedures are good to order 4’ N s - m2 for AA scattering and to order qq’ .- (s - m”)” for VA scattering (as here the vector current is conserved). In the case of V-V scattering of pions we can now calculate Z from Eqs. (4.2) and (5.3) zo.+ = 0 z- = 2tF*(t), (5.6) where F,(t) is the pion form factor. There was also a pole contribution
Zp- = 2t which we have already noted. So
zc- = 2t[F,(t) - l]
(5.7)
and this satisfies Zi- = 0 as expected. We can now write the LETS. For t = 0, we can use Eq. (4.17):
3&
= -4(s - m2)(dFJdt)ltzo
+ O((s - m”)“).
(5.8)
On the other hand, for fixed t we can obtain unphysical LETS. From Eqs. (4.18) and (4.19) f&(s, t) = -(2t/(u
- m”)) + 2(F,(t) - l)(l - 2(s - m”)/t) + O((s - m”)“).
(5.9)
For A-A scattering of pions, we write U, = (27r)3 / ei*‘s(p’ I R(W(x) = &P” +
g,q,’
+
g3q”
J”(O) I p) dx (5.10)
with isospin indices understood. The LETS for weak pion production off pionsl are given by g,(s = m2, t = 0) = g,(v = 0, t = 0) = g,(O, 0) and in general g,(O, 0) is given by Eq. (5.4) replacing qP’Muy by its pole terms, but there are no pole terms here; Z = 0 at threshold. The LETS for the transverse AA amplitudes are given by the derivatives of Z; these cannot be predicted by arguments based on PCAC and current algebra and represent first order correction terms to the LETS for weak pion production. The calculation of the LETS in this case is straightforward. We simply need to calculate Z using the expression for q,‘M,, of Eq. (5.4). Remembering that the minus sign in h& implies that we need to consider the polarisation vector E,+ - E,1 The LET’s for pion photoproduction or weak pion production off a pion target should not be. taken too seriously as they involve an extrapolation of the amplitude from q’2 = 0 to ma. For nucleon targets this extrapolation should not be important, but for pion targets of mass m it clearly is important.
THE
HELICITY
AMPLITUDE
APPROACH
TO
and using the crossing matrix and appropriate
LOW
ENERGY
kinematic
THEOREMS
549
factors, we obtain from
(4.3) zji = 2([Ti
For the even amplitudes,
(5.11)
7Til FAt) + 81)t - 21782 -
Z:*” = 0. Hence gi”‘O’(o, 0) = 0
For the odd amplitudes,
(5.12)
Zt- = Z,- = 0. Hence
g;-)(O, 0) = 0
and
g;-‘(0, 0) = -1.
(5.13)
Equation (5.12) and (5.13) give the LETS for weak pion production for A-A scattering we have from Eqs. (4.8) and (4.17) for t = 0
_- -2g$O’(O, f “d+.0) 11+ f$’
= -4(s
0)
+
O(s
- m2)[dF,/dt
-
off pions. Now
(5.14)
m")
It-0 + @g,-ptp-$
0) + (ag,/a$(o,
Having been forced into a field theoretic framework anticipate a more direct calculation of the LET entirely framework. Such a procedure does indeed exist as we looking for is a method of calculating the contribution direction we make the separation
O)] + O((s - m”)“)
to calculate within the show next. of Zc. As
(5.15) Z one might field theoretic What we are a step in this
Muy = M;,?le + M,dj” + R,,
with M$*’ a nonsingular
tensor satisfying q,‘M,d,“’ = q,‘(M,” M:;ffqy = (M,,
-
Mf;le)
- M,gOle) qv .
Then by definition
Now if we derive a set of helicity amplitudes from R,, the construction ensures that the longitudinal amplitudes are nonsingular as h -+ 0 so that the conditions of the theorem obtain. Transverse amplitudes derived from R,, are therefore suppressed by the kinematic zeros and the LET is given entirely by MEFle + Mziff to that order. These manipulations are equivalent to calculating the contribution of Z to the transverse amplitudes and M,dJ” corresponds to Z”. The V-V and A-A theorems
550
DOMBEY AND MCKENZIE
can be derived in this way. In the case of A-A the most general form for &Z$!” should be written down and then its divergence compared with the form for the divergence of IV,,” in terms of the commutator and the amplitude U, . Complete agreement between the various methods ensures [16].
6. SUM RULES We can easily convert the LETS into sum rules. For example, for V-V scattering, we have at t = 0 from Eq. (4.21) -2 mImK(~t=O)dS=2, n- s0 remembering that for t = 0, the integral over the left-hand cut is equal to that over the right-hand cut. Making the conventional normalization so that the Thomson limit is obtained as the LET for neutral isovector photons scattered off 7~+,and using the isospin relations
f” -#2f- -2f+ =fo, f” 4-f- + 3f+ =fi f” -f- +f+ =f2, where f. ,fi ,fi are the amplitudes obtain the sum rule
(6.2)
for the Z = 0, 1, 2 states of 7”~ ---f yv~, we
2rr2a/m = jam[~yo,,o(v)-
~,,~,+(v)ldv
(6.3)
v is the photon laboratory energy, y” is a neutral isovector photon, and u++(v) is the total absorptive cross section. For isoscalar photons ys,
so we could replace the cross sections in (6.3) by the real photon cross sections a,,,~ and a,,+ to obtain the Pagels-Harari sum rule [17]. The integral of the sum rule is expected to converge as f ^s+ only receives contributions from Z = 2 exchange in the t-channel. Similarly the sum rule for the amplitudejic+ follows from Eq. (5.8). We have Irn f&-($
’ = ‘1 ds = -4 $1
. t-o
(6.4)
THE
HELICITY
AMPLITUDE
APPROACH
This becomes the Cabibbo-Radicati
TO LOW
ENERGY
sum rule [4] on normalizing
THEOREMS
551
and using (6.2)
xr,2>= & jm$ [u(yO + 7ro-+z = 0) 0
+cr(yO+7T++z=1)-~u(y0+nO+z=2)]
(6.5)
where the charge radius of the pion r, is given by &(rv2> = (dF,/dt) I,=, . The integral is again expected to converge as fS- has contributions from Z = 1 exchange and there is now an extra power of energy in the denominator. It is now possible to write down sum rules for forward neutrino scattering of pions. The VP’ + AA sum rules are from (4.21) and (5.14) 2
-=
mIm3:A(S, t = s0
S
0)
dS = 2 - 2g,+(O, 0),
(6.6)
and from (5.8) and (5.19,
=--4
[
2% dt It=o+
%(O,O)
+ +o,o)].
The left-hand side of (6.6) and (6.7) now involves the quantities c+( W+~T) where ur = 4(oL + uR). These are total absorptive cross sections2 [18] initiated by a virtual W+ satisfying X2 = 0. T, L and R refer to transverse unpolarised, left and right polarized spin states of the W+ respectively. Then we have m [q(w+?r+)
+ ur( w+?r-)
- 2UT( W’7rO)] = &
(1 - g,+(o, 0))
i 0
(6.8)
and
a 0).
(6.9)
* They are not actually virtual cross sections but are proportional to the cross sections. Conventionally, these virtual weak cross sections do not include the weak coupling constant G; a factor G2 appears in the overall neutrino cross section.
552
DOMBEY AND MCKENZIE
If G-parity conservation is assumed, then
Sum rules could also have been written down for t # 0 from the LETS for t # 0. But these could not be written in terms of total absorption cross sections; also as the helicity amplitude fs is not crossing symmetric under s +-+ ZJexcept at t = 0, it is difficult to obtain a sum rule in terms of physically measurable quantities. The LETS of (4.21) and (5.9) for I’-V scattering could in principle be used to generalize the Cabibbo-Radicati and Pagels-Harari sum rules to fixed t # 0.
ACKNOWLEDGMENTS We would like to thank D. Bailin, G. Barton, and D. Broadhurst for their help and advice; one author (N.C.M.) would like to thank the Royal Society for a Fellowship and Professor H. Yukawa and Professor Z. Maki for their kind hospitality at the Research Institute for Fundamental Physics. REFERENCES 1. F. E. Low, Phys. Rev. 96 (1954), 1248; M. GELL-MANN
AND M. L. GOLDBERGER, 96 (1954), 1433. 2. H. D. I. ABARBANEL AND M. L. GOLDBERGER, Phys. Rev. 165 (1968), 1594. 3. M. BEG, Phys. Rev. Letters, 17 (1966), 333. 4. N. CABIBBO AND L. RADICATI, Phys. Rev. Letters 19 (1966), 697. 5. N. DOMBEY, Nuovo Cimento 32 (1964), 1696; see also, N. DOMBEY, in “Hadronic
6. 7. 8. 9. 10.
11. 12. 13. 14. 15. 16. 17. 18.
Phys. Rev.
Interactions of Electrons and Photons” (J. Cumming and H. Gsbom, Eds.), Academic Press, New York, 1971. S. D. DRELL AND A. C. HEARN, Phys. Rev. Letters, 16 (1966), 908; S. B. GERASIMOV, Soviet Physics J., 2 (1966), 430. N. C. MCKENZIE, Progr. Theoret. Phys. 47 (1972), 246. L-L. C. WANG, Phys. Rev. 142 (1966), 1187. T. L. TRUEMAN AND G. C. WICK, Ann. Physics 26 (1964), 322. G. COHEN-TANNOLJDJI, A. MOREL, AND H. NAVELET, Ann. Physics 46 (1968), 239 (for massive particles); J. P. ADER, M. CAPDEVILLE, AND H. NAVELET, Nuovo Cimento A 56 (1968), 315 (for processes including zero mass particles). H. Joos, Fortschr. Physik. 10 (1962), 65. R. L. OMNES, Phys. Rev. 168 (1968), 1893. Y. HARA, Progr. Theoret. Phys. 45 (1971), 584. J. DABOUL, Phys. Reu. 177 (1969), 2375. For example on a nucleon target, see S. FUBINI AND G. FURLAN, Ann. Physics 48 (1968), 322 (for pion electroproduction); N. PAVER, C. VERZEGNASSI AND E. E. RADESCU, Nuovo Cimento 66 A (1970), 261 (for weak pion production). N. C. MCKENZIE, D. Phil. Thesis, University of Sussex, 1970 (unpublished(. H. PAGELS, Phys. Rev. Letters 18 (1967), 316; H. HARARI, Phys. Rev. Letters 18 (1967), 319. N. DOMBEY, Rev. Mod. Phys. 41 (1969), 236.