High energy fixed angle scattering in a field theory model

Nuclear Physics B83 (1974) 189-217. North-Holland Publishing Company

HIGH ENERGY

FIXED ANGLE

IN A FIELD THEORY

SCATTERING MODEL

I.G. HALLIDAY, J. HUSKINS and C.T. SACHRAJDA

Physics Department, lmperial College, London S. W. 7 Received 26 June 1973

Abstract: We treat massive electrodynamics as a model for fixed angle scattering. The dominant momentum flow through Feynman diagrams at high energies is examined. In spinor electrodynamics this leads to the "eikonal" answer. In scalar electrodynamics we must also consider other flows which cancel between different diagrams. The analysis is carried out in Feynman parameter space. As well as the d-line contributions we also analyse the pinch effects. Although for a given diagram these may be larger they again cancel betweeff diagrams. To leading powers of log s we thus prove the "eikonal" answer for spinor and scalar electrodynamics.

1. Introduction Over the past few years there has been a growing interest in high-energy twob o d y scattering at fixed angle [1 ]. The major theoretical problem in this field is to avoid having the amplitude dominated b y the Born term which gives typically A(s, cos 0) _~ 1Is. This dominance holds in two-body potential scattering and in an analysis o f g ~ 3 perturbation theory diagrams [2]. The experimental curves have a much steeper fall with s. The most popular approach [ 3 - 5 ] in avoiding the Born term has been to define it away b y saying that the incoming particles are really bound states o f "partons". In diagramatic language this means that we must have four vertices for splitting say mesons into partons and rejoining them. These partons then may or m a y not scatter on each other depending on the choice of model. This structure is clearly not a Born term. In fig. 1 we draw various graphs which have been put forward as the dominant contributions in 7rlr scattering. These models clearly have many points in common with scattering two bound states in potential theory, but insisting that the final state contains the same two states. This is, of course, hard to achieve when we scatter through large angles at high energies. In this paper we wish to investigate the relativistic analogue o f another potential theory device for obtaining fixed angle amplitudes smaller than the Born term. Tiktopoulos [6] pointed out that an absorptive energy-dependent potential with a

190

I.G. Halliday et aL, Fixed angle scattering

(a)

(b]

(c)

Fig. 1. Bound state models of fixed angle scattering. singularity worse than I/r 2 as r-~ o gives rise to an exponential fall off for A(s, cos 0) in x/s. In field theory, this r = 0 singularity corresponds to the existence of powers of momenta in the numerators of Pr9Pagators. We therefore study the massive electrodynamics of a single vector particle. Clearly we would prefer to study a Yang-Mills theory with isospin triplet pmesons. However the structure of the diagrams is so much more complicated that we have restricted ourselves to the single massive vector case. We shall show that on summing the leading terms in each order of perturbation theory

A(s, cos 0) ~ -1$ eB(O)ln2(S/So)

(1.1)

This form, with different B, s o, holds for scalar-scalar, scalar-spinor and spinorspinor scattering. It has previously been used to fit the pp data [7]. The ln2s exponential leads to a very natural explanation of the curvature found in the log-log plot of do/dt versus s. The major part of this paper consists in seeking out the large momentum flows which give the dominant contributions in ~3 field theory. This knowledge then leads to an easy evaluation of the spinorial numerator factors. We find that, in technical language, only end-point singularities [8] contribute to our calculation. Pinch singularities are lower order due to some strange cancellations. The above result was previously obtained by Cardy [9] for spinor-spinor scattering. He guessed the correct momentum flows for the spinor-spinor case. We prove this result. Fortuitously he did not attempt scalar-scalar scattering where his "eikonal" ansatz is in fact incorrect in general. Our result (1.1) still holds due to some unlikely cancellations caused by Gauge invariance. Due to an error in [2] Cardy however claimed that pinch contributions eventually turn e x p ( - B ln2s) into exp(-x/s). This is incorrect and at least in leading order our calculations are unchanged by pinch contributions. Many other authors [10] have employed eikonal techniques in studying fixed angle limits. These have usually employed an arbitrary cut-off in the four-momentum squared carried by any line. This we believe to be incorrect. They also do not prove their kinematic assumptions. Typically they also have problems with scalar particles.

I.. G. Halliday et aL, Fixed angle scattering

191

It is clear from the fact that our result is much smaller than the Born term that our calculation may be extremely unstable against lower order terms that are not cancelled. This should be contrasted with say the Regge limit of ladder diagrams where the sum is typically much greater than individual terms [8]. We have no "real defence against this possibility. Any such contribution may be dressed up with multiple "photon" exchange between the external lines which will eventually give an 2 . . extra e-bIn s factor. However summation of such uncancelled terms may cancel this factor. In the context of this model a form similar to (1.1) has been proven for the nucleon form factors [11 ]. The comparison of this form with experiment is shown in fig. 2. Due to the enormous technical complexity of our problem we have organised this paper in a slightly unusual manner. In the next section we outline our results in some detail without proof. We hope this will be useful. The following sections then contain the detailed proofs. Sect. 3 contains the computation of the dominant momentum flows in ladders in ¢3 theory. This is done using a-space techniques for end-point singularities. These results are used to calculate the spinor electrodynamics contributions in sect. 4 and scalar electrodynamics in sect. 5. In sect. 6 we discuss the effects of pinch contributions for low order diagrams.

2. Summary of results In the conventional technique for calculating perturbation theory diagrams the object of interest is the quadratic form

lO-1~ GM(O) 10-2.

10-3.

- -

lO-~

0.15 exp{-0.2/, In21q2]) q2

I

I

I

Fig.' 2. Massive e l e c t r o d y n a m i c s fit to n u c l e o n e l e c t r o m a g n e t i c f o r m factor.

1.G. Halliday et al., Fixed angle scattering

192 r

=~ ~(k 2- m2). i=I

(2.1)

Here k i is the momentum in the ith line written in terms of external momenta p / a n d loop momenta kl. This form is diagonalised by means of a translation and rotation

t~ = 7¢TAlc - 2kTBp + p T p p _ ~ t~irn2 = ~ X i K 2 + D / C ,

(2.2)

where we have introduced an obvious matrix notation. The Xi are the eigenvalues of the matrix A and the Ki are the diagonal integration momenta. D and C are the usual Feynman parametric determinants. We show that in many asymptotic situations, including our own, the dominant asymptotic contribution comes entirely from the region IKi[~ X/s for all components. Moreover we prove that

k i = (aRn)i + Y.,

(2.3)

where Q, R are matrix functions of the cti only and Yi is a function due to Nakanishi [13]. The Yi are linear functions of the Pi and simple graphical rules exist for their calculation in terms o f the a i. Thus from (2.3) since [Kil ~ x/s for each component we see that the only way k i can pick up terms ~ ~/s is through the Yi" For the planar ladder diagrams we now show that the dominant terms A ,(s, cos 0 ) ~ (log s) 2u f(O)

(2.4)

come from regions shown in fig. 3. We stress that it is the components of the momenta which ~ x/s. The propagator may still be near mass-shell. Thus the lines in a contain only Pl and therefore their square is finite. However the lines in/3 have momenta proportional to (p 1 + P2) and hence their mass squared "~ s. For non-planar diagrams such as fig. 4 we have similar contributions to the above.

P

~

~

P

~

,~

Ca)

(b)

Fig. 3. Dominant momentum flows for planar ladder diagrams. Only the ~ lines contain momentum components "-, x/s.

p,

I-./5.L

p~

Fig. 4. A non-planar diagram with short circuit.

1. G. Halliday et aL, Fixed angle scattering

193

However the wiggly line is a d-line of length 3 and ,4 " -

1

s3

(2.5)

f(O)

instead of the expected 1/s 5. However all the unscaled lines depend only on P3 and so are almost collinear. These scalings are called short circuits. In spinor electrodynamics the m o m e n t u m flows of fig. 3a just give the eikonal result assumed by Cardy [9]. In the dominant terms the spinor propagator numerators for the lines in o~, 6 give T'Pl and T'P2 respectively. These eventually always give Pl'P2 "" ½s. Our other contributions must go away. The contributions of fig. 3b go away because the spinor propagator numerators of the lines in T are not ~ (T'~/s) in this case. Thus we lose factors o f x / s relative to fig. 3a. Moreover in our short circuit example we claim that all the non-scaled lines only depend on P3, thus all these unscaled propagators ~ (T'P3) and on obtaining the clot-product give rn 2, not s. This is exactly as happened in the fixed-t eikonal of Tiktopoulos and Treiman [ 12]. In scalar electrodynamics we have the complication that there is also a 4~2A2 coupling and hence we have the diagrams of fig. 5. In this case the m o m e n t u m factors come from the vertices and not the spinor propagators. Thus fig. 3b is only negligible compared to fig. 3a iffi contains more than one line. Otherwise fig. 3b gives a contribution which, along with the crossed diagram, exactly cancels the ~b2A2 diagram obtained by removing/3. This cancellation always happens. Thus we are again left with fig. 3a. Once again short circuits are negligible and for the same reason. Finally, as is well known [2], pinches may also occur in the fixed angle limit. We have not carried out an exhaustive survey of these contributions but have restricted ourselves to the lowest order cases. In individual diagrams these contributions may be much larger than the d-line terms. However for our electrodynamic couplings these pinch terms always cancel between diagrams in a manner which does not occur for the d-line terms. Thus, at least in the lowest order of perturbation theory where pinches occur, they do not contribute to the eikonal sum. This should be contrasted with the situation in parton models where these cancellations apparently do not occur and these terms upset many of the nice results of the model [14,15]. Whether on calculating the pinch terms to higher order and summing them we change the eikonal form (2.4) we do not know. 3.~b 3 m o m e n t u m flow

First of all we shall prove that, after we have introduced the diagonal integration to replace the usual loop momenta, then the dominant region of the in-

v a r i a b l e s Ki

JJA Fig. 5. Further scalar diagrams needed in scalar electrodynarnics.

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I.G. Halliday et al., Fixed angle scattering

tegral as s -+ oo at fixed 0 has all components of all Iril < e%/s for arbitrarily small e. We write our original momentum in line i

ki = (ak)i + (EP)i'

(3.1)

where Q, E are matrices with entries -+ 1,0 only. Q is an r x p matrix and E is an r × 4 matrix where r is the number of internal lines and p is the number of loops or independent integration four-momenta. In the usual way

= IoTA k - 2kTBp + (pTr'p- ~ aim2 )

(3.2)

is rearranged by writing

t~' = k - A - I B p

(3.3)

and then rotating •' by the matrix R. Then

k i = (QRK)i + (QA-1Bp +Ep) i = (QRK)i + Y . ,

(3.4)

la

OZ= ~-* XiK2i + D/C.

(3.5)

1

The crucial point is that as proven in appendix A we have simple graphical rules for calculating Yi just as we do for D, C Thus (3.4) tells us the momentum flow in line L If we can prove that only small Ki matter then the second term will give us our dominant momentum. To prove our original hypothesis we study the Feynman integral where we integrate the subset of t~i, ~ (v = 1,2, 3, 4 Lorentz label) (i, v)ex over the range IKi,v[ > X / ~ . This should be negligible compared to our original Feynman integral. Thus we must perform

.

r

5(~o~--1)

Ax =f~l d4Kif~ da/ (¢+ie)N

,

(3.6)

using (3.5) but over our limited range of d4gi:

f Ax ~x

dc~j6

c~j-1 Cu -1 ~+1 '

(3.7)

where on letting e ~ 0 we obtain the usual result. For simplicity let us now restrict ourselves to the planar ladders where the scalings leading to the dominant contributions have previously been studied [2], fig. 6.

I.G.

Halliday et aL, Fixed angle scattering A,

.~,

, , :.m[~pIk-pl t.l

195

#n

~, .

.

.

.

.

p: ,p*p,

Fig. 6. The Feynman parameters for the ladder. In the canonical manner [2] the region of c~-space giving the dominant contribution to (3.7) as s ~ t -+ ,,o is obtained by studying the zeros of ( ~ Xi), f, g. In the case where the Xi were not present [2] the sets of lines which when scaled give zeros off, g are shown in fig. 7. There are also disconnected scalings possible (see appendix B). At most 2/~ such scalings may be performed in sequence. We now need to know which scalings give zeros of the X's. The ki are the eigenvalues of the matrix A, and clearly C = 1-I ~ki. The k's are thus non-polynomial functions of the c~-parameters. For the/~-l'oop diagram they are solutions of a/ath order algebraic equation. Fortunately it is rather easy to study the zeros of the X's without solving the equation explicitly. In appendix C it is shown that we can label the eigenvalues so that ki vanishes like (Of), where {si} is the set of all scalings that close the ith loop. (In the usual way-v~e have scaled our set of (x-parameters by p/.) Therefore, when one or more of the Xi are present all V-set scalings have to include the loops associated with thege X's (which implies that d-line scalings are no longer allowed). But, it is not possible to perform 2/1 scalings with this restriction. When the final V-set has been scaled, if no X's were present, we could scale at least 2 more d-lines using only those lines in the V-set, and still scale all subsequent L-sets, thereby obtaining two more scalings, (see fig. 8 for example). Therefore, terms with X's in the denominator obtained by integrating some of the tc's only over IKi,ul > e~/s are at least two powers of log s down on the full amplitude. We have therefore established that only the region It
[]jE£si}

d-lines

2]_LLU V- sets

IL_II

lILt21 L-sets

Fig. 7. The ~3 dominant scalings.

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L G. Halliday et al., Fixed angle scattering

I Iv-., d-line

d-lil~.

Fig. 8. Two d-lines that can always be scaled after the V-set. Let us now study the Yi in (3.4) which we now know carry the large momenta if any flow through the line i. One rule for calculating Yi for any graph is (a) Cut the lines at. , a / E ~27 such that the graph has 2 separate, singly connected parts. If external m o m e n t u m P'r flows into the two halves then

Yi=~

I-I (xjP3,1C, je~'r

(3.11)

where 7 labels all possible ways of doing this. Thus the study of the m o m e n t u m flow in a given diagram corresponding to a given set of scalings reduces to the study of the zeros of the above functions. Rather than attempt an exhaustive list of such scalings we only give one example and leave the dedicated reader to check our resuits in an arbitrary case. Thus consider the 2 sets of scalings of the 3 rung ladder, shown in fig. 9a, b. In fig. 9a the second scaling leads by (3.11) to the result that the lines marked 3 only carry P3" The coefficient of (Pl +P2), for example, vanishes when the d-line shown is scaled. Similarly for the later scalings. Clearly, lines labelled both 3 and 1 have no large m o m e n t u m flow. Thus the large momenta flow as in fig. 10a. For case b the first 2 scalings clearly fix the m o m e n t u m flow as in fig. 10b. The formula (3.11) clearly satisfies four-momentum conservation. It is incidentally easy to check that these scalings do not give zeros of any eigenvalues of A. 1

3

3

],

2

3

3

1

3

Lb 1

3

,[_l_J

l._l [

'L_[3

i'll

,e,',

,(b)

Fig. 9. 2 scalings and their momentum flow.

L G. Halliday et aL, Fixed angle scattering

P,

~__~

P~ P,+P2P+ P~.

197

. ~ _ e,_ _ 5 _P,

P~

(a)

P:<

P~. P+ (b)

Fig. 10. The large m o m e n t u m flow for our two simple examples.

Our claim is that the above structure always holds. Thus the external momenta never split in the dominant contributions and always flow entirely down one line. This is not true for the non-leading terms or the pinch terms which we will treat at the end. Thus for the ladders we have only the flows shown in fig. 3. Similar results hold for crossed ladders.

4. Spinor electrodynamics Again let us start by considering the planar ladder diagrams where the sides of the ladders are now spinor propagators and the rungs are photon exchanges, in the Feynman gauge, for definiteness. From our arguments in sect. 2 we expect now to obtain A ~ (log s) 2u f(O) X (spinors).

(4.1)

s

We could study this problem by the techniques of the last section. However we prefer to use an a-parameter representation of the spinor Feynman diagram. This may be found in Nakanishi [13]. Suppose we have a Feynman diagram with some propagators corresponding to the lines l 1 ..., lk having terms in the numerator linear in the momentum in that line with Lorentz indices/a 1 .../~k- Then the Feynman parametric form is ~k

m

AoeS ll'dclj ~--o=(~ i=|HXl'l"'it lG~mEl) C2(V-ie) v-m 2m ( l - v ) (2-v)...(m-v)' v =N-

2/~,

(4.2) r

#t

where ~ div sums all divisions o f l 1 ... lk into m pairs (l'1l'~), (l'2l'~)... (lm, lm) and the remainder Jm" The function V, is defined by V = D/C the ratio of the two usual determinants and the two index Lorentz tensor XI, r is computed for two lines l, l' as follows. We cut the graph into two simply connected halves with l, I' cut and the lines li, iES also cut. Then

I.G. Halliday et al., Fixed angle scattering

198

X~,l~, =

~(+-)

(i~sOqi)guJC,

(4.3)

where ~ is summation over all ways of doing this. For the sign we refer readers to Nakanishi. Thus remembering that Y/is a vector, this expression must be dotted into the appropriate 7u's etc.. To obtain (4.1), after all scalings by Pi have been performed, then for any term in ~ u, _ ~ - ' d i v there must remain an O(1) term in each YI and all Xl,l' must ~(1--[ ~ " p ; l). This is crucial since as we saw in sect. 2 we need to obtain a ~/s factor from each 7"P. (Notice that, in our gauge, the photon lines contain no momenta in the numerator.) In our Feynman parameter language this means that each Yl must contain at least one external momentum and each Xl, l,, which is momentum independent, must reduce the effective scaling length by 1. The above constrain the allowed sets of scalings to an enormous extent. Thus the Xl, l, condition implies that each scaled set must include a circuit containing l, l', i.e., in (4.3) Cvanishes faster than the numerator by one power of the scaling parameter as the parameter tends to zero. Then we see that all d-lines must have the same photon, F, and it is included in all V-sets. If this does not happen then there is a fermion line l which gives either YI ~ P or XI, 1, ~ 1 depending on where it occurs. This is shown for a simple case in fig. 11 where the Y1 of the line shown vanishes on performing the two scalings indicated. These two conditions (a) "Xl, l, implies/, l' in a scaled circuit" and (b) "all V-sets containing the same photon" prevent us obtaining 2~ scalings for all terms in (4.2) containing an Xl,1, , so that these terms may be dropped. Now consider the line l in fig. 12. Then [13] we use the representation r

r

V=D/C=-j~=I ~m2+j~=laiY2

(4.4)

to study the momentum flow. Notice the sums in (4.4) run over all lines of the diagram. The function D = fs + gt + d is linear in s and t and all our d-line and V-set

'F , Fig. 11. The line I drops a x/s factor,

P~

t:..- t; rt;.,-.-

~L,

P*

Fig. 12. The F scaling m o m e n t u m flow.

L G. Halliday et al., Fixed angle scattering

199

scalings making f, g vanish linearly relative to d. Thus in (4.4), on carrying out any scaling making f, g vanish, for each YI must be proportional to at most one external momentum or a l ~ O. Thus we have obtained our result if we can prove that in any 2/a scalings there is always a d-line or V-set scaling which does not include a l. This follows because clearly lr+l-., lu+l °~ P3 whereas l~+1..-l'u+l °: P4 and the Y! conserve momentum. Because of conservation clearly Yl = P3 and we have the eikonal flow, i.e., the large momenta flow along the fermion lines except for the distinguished photon, and the loop momenta are smaller. Again we have only been able to show this scaling result by explicit enumeration. (We have examined all possible scaling sequences for the box diagram and three rung ladder graphs, together with a large number of sequences for higher order graphs.) For non-planar graphs we have a similar result but extra unusual scalings are dlines, e.g., as shown in fig. 13. The real problem is to show that the "short circuit" contributions which in 4~3 give much larger contributions may be ignored in electrodynamics. For t ~ 0 this has been discussed by Tiktopoulos and Treiman [12]. Consider any (no-loop) d-line with n d elements, a l (l~ d) and scale by p. Then for ale d

YI = O(1)Pi + O(p)(p's)

(4.5)

for some i = i(l, d). Now consider the original graph minus the lines in d. We remove the vertices into which at least one a I (IE d) runs. This gives a set of disconnected graphs G i. We now use another representation of the YI, Xl, t' due to Nakanishi [13]

YI = ~b" Wl(alb)pb (for any vertex a in the graph).

(4.6)

Here we sum over the 4 vertices where the external momenta Pb are attached and

W(alb) = ~ (+ Up). P

(4.7)

In this last formula we sum over all paths between a and b containing/, and Up is the determinant of the graph with the path P contracted to a point. The vertex a is arbitrary, we take it to lie on the d-line. Any path from a through an unscaled line, l, to vertex b which has a common vertex with the d-line between b and l (or = b) contributes O(0) to Y/as it completes a loop and so Up vanishes. This follows since contraction of P and the d-line contracts a loop to a point. Thus from the usual topological formula Up ~ p. Then we define allowed paths as those which do not complete loops with d. Equation

V

v

--

Fig. 13. An unusual scaling set.

200

1.G. Halliday et aL, Fixed angle scattering

(4.5) then says that all allowed paths through a given line I go to the same external vertex. Since the allowed paths close no loops with the d-line the disconnected graphs G i have at most 4 sections which may be labelled Pl, P2, P3 and P4" They then give Yl = (O(1)Pi + O(p)P l) where PI is a linear combination of external momenta. Within all other sections we find Yl = O(P)PI" Now we turn to Xl, l,. This also has a representation in terms of circuits [13]

Xl, i'' = l ~ c i ( + Uci) g~v.

(4.8)

In this formula C is the usual determinant and we sum over circuits Ci including/, l'. Uci is the determinant of the graph with C i contracted out. The signs are defined in [13]. It is then clear that if l, l' are within different sections Xl, t, ~ p since when any circuit containing l, l' is contracted out, the resulting graph has a circuit consisting solely of lines in the d-line. Now let us compute the maximum allowed energy dependence using eq. (4.2). Look at each fermion path in a section bounded by contact with a spinor or the dline. Now commute the ~'s towards the spinor. Many of these terms are O(1) as ~i ~i = m2 and ~iu(Pi) = tIlu(Pi ). Thus not all O(1)Pi terms in YI give X/s. Let us fist the contributions. The O(1)Pi terms in YI can eventually (i) contract into a Pl' ; (ii) contract into the d-line through an unscaled photon line; (iii) contract into the d-line through an XI, l,; (iv) contract into another Pi giving m2; (v) contract into an O(1)pj term in a different section through an Xl, I, ; (vi) encounter a spinor giving m; (vii) in a continuous set of fermion lines bounded at both ends by contact with the d-line there may remain a single Pi. For l E d-line we find YI = O(1)P/where PI is a linear combination of external momenta. We now set an upper bound on the powers of s by counting the powers of s available when all Pi are assumed to give x/s. This may not necessarily happen. In other words the coefficient of our term may be zero. Again we stress that we are considering one term out of the sum in (4.2). Thus the Pi must occur in (i), (ii), Ctii), (v), (vii) above. Then finally we allow (i) ~/s for each ~ in the d-line; (ii) x/s for eachXl, l, l e d , l ' ~ d ; (iii) ~/s for each photon between a vertex in the d-line and one not in d; (iv) ps for each ~ in an unscaled fermion line; (v) ps for each XU,, lCsection i, l ' E s e c t i o n j ; (vi) x/s for each contiguous fermion path bounded at both ends by contact with the d-line. Thus (iv) and (v) may be ignored, the extra p from the scaling killing the extra s.

L G. Halliday et aL, Fixed angle scattering

201

1

2-1(nd+l-2nd'y-2nTT) X s~(nd'y-1) 1 A "~" s•-~(nd-nd'r-2nxdd) X S ,

(4.9)

where nv.r is the number o f unscaled photons coupling to the d-line at both eflds, nd~. is the number of photons in d and nxd d the number of Xl, l,'s with l, f @d. The above counting is left to the reader to check. Thus finally we get, including a p rid-1 from the scaling, -- (nd,r +nxd d + n r~ ) = --1.

(4.10)

Thus the only relevant d-lines have one scaled photon with n.y,r = nxd d = 0, i.e., they are of the same topological structure as the planar ladder scalings as in fig. 7. However the unscaled photons may be crossed over as in fig. 14. It is straightforward to extend this analysis to include V-sets. Let the V-set contain n d elements and n v loops. Scale all the elements by p. The V-set divides the graph into sections as before. Observe that Xll, now has the additional property that if l and l' are both contained in a circuit included in the V-set, then X l l , ~ p-1. The factor of C -2 in the Feynman integrand ~p-2nv. In setting an upper bound on the power of s we must now add to our list of six "allowances" for d-lines: (vii) p -2nxc, where nxc = the number o f Xll,'S with l and I ' E V-set circuit; (viii) O-2nvRecounting (i) to (viii): 1

1

1

A ~ f i (nd-nd'y-2nxdd) X fi (nd+l-2nd'y-2n'y'r-nv) X s~(nd-l-nv) X p n d - l - n x c - 2 n v . (4.11) Clearly the scaling is only important if no,r - n v = l

and

n rv=(nxd d-nxc)=0,

(4.12)

so that all of the photon lines must be contained in scaled loops, and can not be used as short circuits. We may now study the momentum flow as for the planar diagrams with the same conclusions. Thus for the generalized ladder diagrams we believe that the only important momentum flow are those corresponding to the eikonal model. The eikonal model may then be used to calculate our answer. In the above we have neglected vertex corrections. Appelquist and Primack [11 ] have studied the spinor electromagnetic form factor in this model and have found similar momentum flow to dominate. Thus we have not studied these insertions in detail.

Fig. 14. The d-lines for crossed ladders.

202

L G. Halliday et al., Fixed angle scattering

(a) (b) Fig. 15. Two graphs with connected photon exchanges. Graphs containing fermion loops have been discussed by Cardy [9]. Only those graphs with a photon connecting the two incident fermion lines are important, e.g., fig. 15a. Graphs such as fig. 15b do not have sufficiently short scalings. The exchanged photon (or one of them, if there are several in the graph) carries all of the momentum transfer, and the eikonal model again applies. In a weak coupling model with g2 ~ (ln 2 s)-l, graphs with fermion loops are asymptotically negligible. As the ladders and crossed ladders cancel to leading order, the asymptotic behaviour is given simply by the vertex corrections to the Born term in such a model.

5. Scalar electrodynamics In this section we examine whether the eikonal ideas of the previous section are also valid in scalar quantum electrodynamics at fixed angle. The interaction Lagrangian is £I = -ie: c~*(~ 2 -~u)(pAU: + e2:A2~b*¢: , which gives the two vertices in fig. 16 [16]. The vertex of fig. 16a has associated with it a factor o f - i e ( p +P')u and the vertex of fig. 16b a factor of 2ie2guv. We begin by considering the generalised ladder graphs containing only threeparticle vertices. From our analysis of 9 3 and spinor QED, we expect the eikonal momentum flow, i.e.: scattering through a single hard vector meson, to be important, and possibly also scattering through hard scalar propagators (i.e.: the flows in figs. 3a and b in the planar graphs). At the end of this section we show that short circuit scalings are again unimportant, so that the momentum flows associated with these scalings may be ignored.

(a)

(b)

Fig. 16. The vertices (a) -ie(p +P')u and (2) 2ie2g~v.

L G. Halliday

et al., Fixed angle scattering

1

203

3

Fig. 17. A region of eikonal momentum flow. Consider the diagram of fig. 17 in the region of momentum space indicated. The spin factors associated with propagator (1) is ~ 2Pl'2P2 = 2s, (2) is 2s, (3) is - 2 u and with propagator (4) is (Pl +P3)'(P2 +P4) = (s - u). The contribution to the amplitude is therefore - 8 s 2 u ( s - u)A¢~ where Ae is the equivalent amplitude in ~3 . . . . = log~'s. It is straightforward to theory. The behaviour is therefore -4 s (1/s4)logbs see that for the generalised ladder graphs with/a loops the contribution from eikonal momentum flow is [ln2Us × (a function of the scattering angle)]. To see what happens in the regions of momentum space where one or more of the scalar propagators goes off shell like s let us first look at the box diagram, fig. 18a. The large s, fixed angle behaviour of this graph is easily calculated and is found to be e4n 2

s

T(s, t) ~ ( ~ ) 4 l°g2s(5 t + 2 ) .

(5.1)

This amplitude consists of two dominant contributions, T(s, t) = Ts(S, t) +

Tv(s, t), where T(s, t) = n2e----~4Slog2s (2n)4 t

(5.2)

is the contribution from the two regions of m o m e n t u m space where one or other of the scalar propagators has mass 2 ~ s, and rv(s, t) -- 2~r2e4 (1 + 2 t ) l o g 2 s

(5.3)

is the contribution from where one or other of the vector propagators has mass2~ t. The existence of the T s term appears to imply that we can not use the eikonal model for scalar QED, for the model would miss out this term. However we have also to consider the diagrams of fig. 18b and 18c. For the crossed box, fig. 18b, we find the amplitude

TX(s, t) = TsX(s, t) + TX(s, t ) , where

(o)

-U (b)

(c)

-V-

Fig. 18. Fourth order graphs in scalar QED.

L G. Halliday et aL, Fixed angle scattering

204 7r2e4 TX = ~ t l°g2s'

(5.4)

TX=2zr2e4 (l + 2t)log2s

(5.5)

s

(21r)4

v

(2~r)4

and the subscripts s and v have the same meaning as in (5.2) and (5,3). The two diagrams of fig. 18c contribute _

:r

7r2e4 (2~) 4 log2s,

(5.6)

We therefore find that T s + T x + Tzx = 0 to leading order, so that the contributions that were not present in spinor QED fixed angle scattering have cancelled each other to leading order. (Notice also that T v + T x = 0 to leading order, though this is also the case in spinor QED. There still remain fourth order graphs of order log2s from the vertex corrections, as in spinor QED.) The remaining fourth order graphs are listed in fig. 19. The only one which is of order log2s is fig. 19a. The graphs of fig. 19b and 19c ~ constant, and that o f fig. 19d ~ log s. Therefore to fourth order, scalar QED behaves in a similar way to spinor QED. Let us now generalise these arguments to high order graphs, and as an example consider the diagram of fig. 20a. We are interested in the contributions from regions such as k 2 ~ s and all other propagators soft, which are comparable to the eikonal contributions in ~b3 theory but not in spinor QED. Consider the region of momentum space indicated in fig. 20b. We write this contribution to the amplitude for the diagram as s 2 (" d4q T(s, t) ~- const -s aq2q2q2 s3jf d4 ql d4 q2d4 q 51(ql ,q2,q 5) = const ~llr 1. 2 log2(s)ts3

fd4qld4q2d4qsi(ql,q2,qs).

(5.7)

From this we see that this contribution is cancelled in an analogous way to the cancellations in the fourth order graphs, i.e.: by the contributions of fig. 21 a and 2 lb in the regions of the momentum space indicated. Moreover, contributions such as those in fig. 22 are not of leading order since the spin factor associated with the marked propagator is ~ es due to the fact that the momentum flowing through the

(a)

(b)

[c}

(d)

Fig. 19. The remaining fourth order graphs in scalar QED.

205

L G. Halliday et aL, Fixed angle scattering k

t',

'~,°e,. ('I, el,

q

p.

Ca)

(b)

Fig. 20. Non-eikonal momentum flows in higher order graphs in scalar QED.

(a)

(b)

Fig. 21. Graphs cancelling the contribution from fig. 20. $

S

S

Fig. 22. Non-leading contributions in scalar QED. bottom vertex is small compared to x/s. It should be noted that in each diagram with at least one triangle insertion of the type of fig. 23a, the spin factors imply that if this diagram contributes to the leading behaviour then the (momentum transfer) 2 through the triangle is at least s'r, where 3' > 0. Therefore this diagram will be cancelled by replacing the triangle by the insertions of fig. 23b. Since we know from arguments given above and in sect. 3 that we cannot get the leading behaviour from contributions with two or more non-consecutive hard scalar propagators, we conclude that diagrams with more than one triangle insertion, such as fig. 24, may be ignored. (This can be checked by looking at the sets of scalings.) Diagrams of the type of fig. 25 are powers of logarithms down on the leading behaviour since they log s if there is an even number of loops and ~ log2s if there is an odd number of loops [2]. S

(a)

U

(b)

Fig. 23. A set of insertions which cancels to leading order.

Fig. 24. A graph with two seagull vertices which does not contribute to leading in order in scalar QED.

206

L G. Halliday et al., Fixed angle scattering

Fig. 25. Bjorken-Wu diagrams. rl--.x

Fig. 26. A graph containing a vertex correction. Let us now consider the vertex corrections. We have already stated that to fourth order only the diagrams of fig. 19a contribute to the leading behaviour. In higher order diagrams such as that in fig. 26, we find that in order to get the maximum number of logarithms from the loop integrations together with the required power of s from the spin factors, the vector meson propagator in the vertex correction must begin and end on opposite sides of a vector propagator whose mass 2 ~ t. The region of phase space where one or more of the scalar propagators in the top line ~ 1/s gives a contribution down by at least a power o f log s on the leading behaviour. Hence these diagrams behave just as they do in spinor QED. Diagrams with insertions such as those in fig. 27 can be easily, though tediously checked to be of lower order, since either there are not enough scalings of the required length to give the leading behaviour, or the spin factors are not sufficient to do so. Thus the behaviour of vertex corrections is similar to that in spinor QED. Finally, we consider diagrams with connected vector meson amplitudes such as those in fig. 15. As in spinor QED a d-line analysis shows that the diagram of fig. 15b is down on fig. 15a by a power of s and in fig. 15a the region where the right hand propagator ~ 1/t dominates the region where t goes through the tower. Also, by a generalisation of the arguments presented for the box diagram, we have the cancellations indicated in fig. 28. The computation of connected vector meson amplitudes is therefore the same as in spinor QED, and the result of summing the iterations of such amplitudes "~ e x p ( - a log2s). We therefore argue that the end-point contributions in scalar QED behave the same as those in spinor QED. They are calculated correctly in the eikonal model by ignoring the seagull vertices. The unimportance of the non-eikonal momentum flows is emphasised by using the Landau gauge where the numerator of the vector meson propagator is ( g u y - kuku/k2) instead ofguu. Consider the box diagram in fig. 18a. In the Landau gauge, the spin factor asso-

Fig. 27. Insertion which may be ignored.

1.G, Halliday et aL, Fixed angle scattering 5

207

U

Fig. 28. Cancelling contributions in graphs containing scalar meson loops. ciated with the left vector meson propagator in the region where the top scalar propagator ~-- 1/s is P~

(g P2"2°I ,u

p2

](2p~+P~)~0"

(5.8)

Such a zero also occtars in higher order graphs in the momentum region where,a scalar propagator --~ 1/s. Therefore, these momentum flows are now of lower order in each diagram separately. For the same reason, diagrams with triangle insertions are also of lower order independently. The gauge transformation mixes the contributions among the graphs, so that the non-eikonal contributions obtained in the Feynman gauge, and which cancelled, now do not appear at all. This property is also found in the pinch contribution discussed in sect, 6. Where explicit calculations have to be performed we have used the Feynman gauge, as the Landau gauge is much less convenient. We complete our discussion of end-point contributions in scalar QED by showing that short circuit scalings give asymptotically negligible contributions, just as in spinor QED. We start by considering d-lines in diagrams which have no seagull vertices, for example fig. 29. The d-line is marked in heavy type. There is now a momentum factor at each vertex. The a-parameter form is again given by eq. (4.2), but there is a term corresponding to rrj = (N-2/~) of the form ta+l

C -2 ~

N Xl~l~,(log A - log V),

div i=1

where A is a mass out-off parameter.

Fig. 29. An example of a d-line (,,x~) in a non-planar graph.

(5.9)

I.G. Halliday et al., Fixed angle scattering

208

Scale some d-line with n d elements by P- The Xll,'s are all of order I or smaller. Consider the factors of s that can be obtained from the Yl'S. If there are enough Y f s in the Jm we can obtain: (i) A factor o f s for each vertex in the d-line with an unscaled p h o t o n attached. Let there be n u of these. (ii) A factor o f s for each p h o t o n in the d-line, and let there be n&r of these. (iii) All other possible factors o f s come in the combination ps and thus may be ignored as in spinor QED. Including the p nd-1 from the scaling, the maximum asymptotic behaviour is

S-nd+nu+nd~/X (logarithms).

(5.10)

But b y inspection we see that n d = n u + 2rid,r -

1,

(5.11)

so that the power behaviour for the d-line is -nd3,+l s

(5.12)

i.e.: the scaling can only be important if there is one photon in the d-line. Let us now consider what happens if we introduce seagull vertices. If such a vertex is introduced into the d-line by deleting one o f the scalar lines in the d-line, then the new d-line is shorter b y one than the old one, but the maximum power of s is also down b y one since there is no m o m e n t u m factor associated with seagull vertices. Similar arguments can be applied to seagull vertices not on the d-line, so that even with such vertices, short circuit d-line scalings do not lead to the dominant behaviour. We now go back to eq. (5.9) and consider what happens if we scale by p a V-set containing l loops and of length n d. As before let us start b y considering the case o f a diagram with no seagull vertices. Then we get a factor p nd-1 from the Jacobian of the scaling, and a factor of p-21 from the C - 2 factor. Let us first consider the rn = 0 term in eq. (4.2). There are then no X factors and the power o f s obtained from the Yt factors is the same as for the d-line. The maximum asymptotic behaviour from the V-set (modulo logarithms) is therefore

s-nd+21+nu+nd'y = s -(nd3'-l)+l ,

(5.13)

so that short circuits (i.e.: na~ > l+ 1) are again unimportant. The other terms in (4.2) with m < k have factors ofXll,. As in spinor QED, if l and l' lie in a scaled circuit of the V-set then Xlt, ~ 1/p, thus reducing the effective length o f the V-set, and compensating for the loss of two factors of Y. The behaviour is then the same as for the m = 0 term. The In V term in (5.9) does not contribute to the leading behaviour as there exist no scalings that can enhance the power of the logarithm. The introduction of seagull vertices presents no problem, and does not change the above conclusions.

209

I. G. Halliday et al., Fixed angle scattering

In scalar quantum electrodynamics, divergences appear in all of the graphs with one or more loops. The log A term in eq. (5.9) is a primitive divergence. Divergences o f subgraphs appear when subsets o f the a-parameters are integrated near zero. The theory is renormalizable, so that all of these divergences are cancelled by counter terms. The contributions to the amplitude with the highest power of log s ((log s)2u) are all finite, so that we can ignore the divergent terms. All of the arguments that we have applied to scalar-scalar and spinor-spinor scattering can also be applied to scalar-spinor scattering. As an illustration let us again consider lowest order diagrams in the Feynman gauge, fig. 30. We find that in the box and crossed box there are three dominant regions, where q2 _~ t, q2 ~ t and k 2 ~ s. The contribution from the region where k 2 -- s is down on the leading behaviour since in this region the components of k2 ~ ex/s so that the spin factor containing ~2 cannot give a factor of s when it is contracted into another of the momenta. The contribution from k 2 ~ s in the box and crossed box cancel. The diagram of fig. 30c is found to be down on the leading behaviour, and the only relevant vertex corrections are those shown in fig. 30d, just as in spinor and scalar QED. All the arguments of sect. 4 for spinor QED and of this section for scalar QED can now be carried through and we find that where in scalar QED we need a cancellation of the type shown in fig. 31 a, we now have that shown in fig. 3 lb. The leading behaviour of diagrams with n vector mesons is ~ (Pl +P3)Uu(P4)?Uu(P2)(1/s) log 2(n- 1)s, and the combinatorics works just as in scalar-scalar scattering or spinor-spinor scattering to give d____O(s, dt 0 ) = ~2 A (0) e - a l og2s+O(1og s).

(5.14)

- X .....I .... .... (a)

......... (¢)

(b)

(d)

....... l" ...... ,.,_.._/

....

-..... - spinorSCalar

Fig. 30. Fourth order graphs for scalar-spinor scattering. ((3)

-..

r'm--- • --'5~--- ÷ ---57S-

=0

(b)

Fig. 31. Contributions which cancel, (a) in scalar QED, and (b) in scalar-spinor QED.

210

I.G. Halliday et aL, Fixed angle scattering

Fig. 32. A diagram with pinch contributions.

6. Pinch contributions In the above we have only studied the contributions coming from the edges of a-parameter space. The possibility of pinches for non-planar diagrams has previously been studied in ¢3 theory. In that theory they were shown to be irrelevant but, in a given order, gave contributions greater than the non-pinch terms. This raises the alarming possibility that, when combined with the spinor factors of electrodynamics, they will dominate the d-line contributions in a given order of perturbation theory. This does not happen due to some miraculous cancellations between diagrams. There are two different types of pinch contributions which have been found. The original is the dominant term in the q~3 ladder o f fig. 32. After a pinch in a-space I r t ! corresponding to x = alt~ 3 - a2a4, and y = C~la3 - a 2 a 4 being zero the smallest dr t lines are obtained by scaling a 4, a4, 72 or a 2 a 3 71a2 a3; these are o f the same effective length. The a ' s are scaled b y X/P and the 7's b y p. Notice that because of the pinch conditions which introduce f i ( x ) 8 0 ' ) factors a 1 ~ a~ ~ x/p for the first scaling. t F If we contract out alala4a4"), 2 we obtain the contracted diagram shown in fig. 32. From our rules for the YI, the m o m e n t u m flow in the a 2 line is ~ (x/P)Pi. Similarly the a 3 momentum is ~ @ Pi. Thus the dominant m o m e n t u m flow is as indicated by the wiggly lines. The other scaling corresponds to (Pl +P2) flowing through 71. The sophisticated reader of Gelfand and Shilov [17] may wonder at the implied use of the approximation 5 ( a l a 3 - a2a4) "" ~ ( a l a 3 ) for a2, a 4 -+ 0. Thus on p. 327 we have the statement 5(xy - c) "" - 2 In cS(x) 60"), c ~ O. This is only true for integration over x y > 0 while we necessarily integrate over x > 0, y > 0. We have checked in simple integrals that the naive answer is correct for our case. Thus the dominant contribution for fig. 32 is 1/s 3 *. The numerator factors however have lost at least s 3/2 compared with the d-line contribution. Hence these terms may be ignored in our approximation. The same results hold trivially for scalar electrodynamics. The second type of pinch has been discovered b y Landshoff [14] in the context o f the parton model for fixed angle scattering. This is not the dominant contribution in ~3 theory but it is for the mixed q~3,q~4 diagram of fig. 33a. Here it gives 1 / x / ~ . The topology of this diagram occurs in scalar electrodynamics. The numerator factors give the same powers of s as in the d-line contributions since the impor* In ref. [2] there is an algebraic error in computing the scaling Jacobian and the answers there should be multiplied by 1/x/s.

I.G. Hallidayet aL, Fixedanglescattering p

~, .k~ P3 p, k,;.

k~=R~ p k,.

p

P, P

P~

(o)

p~ (b)

211

k,. p P

(c)

Fig. 33. Diagrams in scalar QED which have pinch contributions which cancel within the set. tant region of phase space is where k i ~ xPi in fig. 33 *. Thus large momenta flow in all lines and we apparently have an amplitude ~ x/s. This would be a disaster. Fortunately this contribution is cancelled between the diagrams shown in figs. 33a, b and c in scalar electrodynamics. A similar cancellation occurs in spinor electrodynamics. The behaviour for scalar particles of fig. 33a is 1 1

f dxf(x),

(6.1)

0 where the integral is convergent. The numerator spin factors are c(Pl(l +x)P3(1 +x)) (P2(2 - x ) P 4 ( 2 - x ) ) ( - 4 ) = - c ( 1 +x)2(2 - x ) 2 t 2 .

(6.2)

where c contains the factors common to all diagrams and ( - 4 ) comes from the two seagull 2i factors. Thus we obtain Aa

1

t2 - c ~

f d x f ( x ) (1 +x)2(2 - x ) 2 . 0

(6.3)

For fig. 33b we obtain 1

c JO r dxf(x) Ab ~v'S-~ (xzs) (p 1(1 +x)P3(1 +x)) (2XPl(2--x)P2)2xP3(2 --x)P4) X 2i(-i) 2 i, where the factor (2i) (-4) 2 i takes into account the seagull vertex, the two new three-particle vertices and the extra propagator:

A b "" - - ~

fdxf(x)(1 +X)2(2--x) 2 .

(6.4)

Similarly,

* This momentum flow corresponds to a pinch in c~space on the surface a2/al ~/o;1 = ~;&~,.

= &3/a4

=

212

L G. Halliday et al., l~xed angle scattering

cut A c ~---~J

"dx J"t x ") ( 1 + x ) 2 ( 2 - x )

2 .

(6.5)

Thus A a + A b + A c ~-- const.

(6.6)

This latter result is only obvious once we have carried out a detailed study of the non-leading scalings and the corrections to the m o m e n t u m flow. We have investigated this but do not give the details here. There are six other such contributions which also cancel as shown (fig. 34). In spinor electrodynamics we obtain the cancellation between the non-seagull graphs of fig. 35. The details are deferred to appendix D. The problem appears in parton models where the Landshoff pinch has caused trouble with the scaling laws [ 14]. Here the particles propagating are mainly fertalons from ~r -~ qq and there is apparently no cancellation.

7. Conclusions Having obtained the appropriate m o m e n t u m flows, the evaluation of the sum of dominant terms is exactly equivalent to Cardy's work [9]. We therefore do not repeat it here. The resulting amplitude for spinor QED is A spinor ~ a u(p 3 ) "/~zu(p 1) "u(P4 ) 3,t~ u(P2) e -Bln2s-c Ins

(7.1)

The coefficients a and c are both angle dependent functions. With the inclusion of graphs containing fermion loops, B may also acquire an angle dependence. In scalar QED the amplitude has a similar functional form A scalar "~ ~"e - ~ ln2s-c-lns "

(7.2)

The fundamental unsolved problem is, of course, whether our answer is stable

Fig. 34. Further graph having pinch contribution which cancel within the set.

Fig. 35. Graphs in spmor QED which have cancelling pinch contributions.

L G. Halliday et al., Fixed angle scattering

213

under summing lower order terms. Our only defence is that if any term arises which is larger than our answer (1.1) then there also exist terms where this term is surrounded by photon exchanges. This will eikonalise the original term. However, it is not clear that this process may be continued for ever without double-counting. Two of us (J.H. and C.T.S.) would like to thank the S.C.R. for financial support.

Appendix A We show that the elements of the vector Y defined in eq. (3.4) are the functions

Y! that appear in Nakanishi [13]. We then have simple graphical rules for their evaluation. We go back to eq. (3.1) and arbitrarily separate the internal momenta into two parts,

kl = fcl + ql '

(A.1)

where ~cI is an "integration piece" and the ql's are fixed. The split is done so that the kl's and qfs separately satisfy four-momentum conservation at each vertex, the external momenta being included in the q's. For a graph with # loops,/a of the kt's, (kT), can be chosen as independent variables, and the others expressed in terms of them through the matrix Q:

i, l = (Qk*)l.

(A.2)

Then eq. (3.2) reads:

qs - ~ al(k 2 - m z) = ~ ( k * t Qt)lal(Qk*)l l

l

- 2 ~ ~l(Qk*)lq[ + ~l a t ( q 2 - m2)"

(A.3)

This expression is conveniently rewritten by defining the matrix q/ and vector q* by ell i] = ~, (Qt)il °tl Ql] ' l

(A.4a)

qi* = ~ l (Qt)i l°~lql '

(A.4b)

so that

t~ = k*tQl, k * - 2q*tk * + ~

1

al(q2- m2).

(A.5)

1.G. Halliday et al., Fixed angle scattering

214

For our/1 independent momenta, the split (A.1) may be determined by imposing the ~t conditions

qi = 0

(i = 1.... ,/a)

(A.6)

on the ql's. Nakanishi tells us that the solution to the set of eqs. (A.1) and (A.6) is

ql = Yl"

(A.7)

But, by setting each qi to zero, the "diagonal" momenta are now obtained simply by a rotation of k* (the translation being proportional to q*) k* = R K .

(A.8)

Eq. (A.1) then reads

kl = (AR~)I + Yl

(A.9)

and the Y1 can be evaluated using the graphical representations in [13].

Appendix B The possible scaling sets for the planar ladder graphs in q53 have been discussed in [2]. There are, in addition to the connected sets discussed there, disconnected sets of lines. Consider fig. 36a. If the wiggly lines are scaled by p before performing any other scaling, then the coefficient of s in the denominator function does not vanish, as there is a term proportional to a~7/C, which is O(1). However, if we first perform the scaling (by Pl) indicated in fig. 36b, the coefficients o f s and t are both O(Pl), and the term a~7/Cis O(p2). Therefore, if we now scale 36a by P2, 1 >> 192 >> Pl, the coefficient functions are both O(plP2), and the scaling is allowed. At any point in a sequence of scalings, the allowed disconnected scalings are always strongly restricted by those scalings already performed. The existence of these scalings does not appear to change any of our conclusions about the important m o m e n t u m paths. The possible existence of important disconnected short circuit scalings is complex to consider in generality, but we can find no examples where a disconnection can sufficiently enhance a short circuit scaling to make it important in QED.

t

(o)

t

(b)

Fig. 36. An example of a disconnected scaling.

I.G. Halliday et aL, Fixed angle scattering

215

Appendix C We derive the properties of the eigenvalues of the A m a t r i x required in sect. 3. Consider the ladder graph with (¢t + 1) rungs and F e y n m a n parameters as shown in fig. 6. M o m e n t a m a y be chosen so that the matrix A is: • C1

a2

a2

C2

o~3

a3

C3

q-1

@

/a

o~# ,

c /,

(c.1)

where C i = %. + o~i+1 +/3i + ")'i- The eigenvalue equation then gives the properties o f the Xi, without solution, if we r e m e m b e r that for an algebraic equation: ?`u _ d l X U - 1 + d2?`u-2 ... + du = 0 ,

(C.2)

then #

du = H

?'i '

1

d.-1 = ,~. ?'1 "" Xi "" ?'u'

~t

d l = ~1 ?`i •

(C.3)

F o r the m a t r i x A, the d's are explicitly .u--m

dm = ~

{hi)

C(1, n 1 - 1) C(n 1 + 1, n 2 - 1) ... C(nu_ m + 1,/l),

(C.4)

where ~iuff..{n is the sum over all ways o f choosing (/a - m) out of/1 numbers such ~. zJ . . . . . . . that n i < ni+l, and CO, l) is the C function of the graph consisting o f the hnes {ai "" a]+l, [3i "" ~j, 7i "" "tj} only, so that C(i, t3 - q , and C(i, i - 1 ) -= 1.

216

I.G. Halliday et al., Fixed angle scattering

When a sequence of n scalings has been performed with parameters/~1 "'" Pn, each C i ~ rli, where Hi is the product of the O's for those scalings which include the ith loop. For the scalings which are relevant to us Y C(i, 1) ~ 1-[ ~ k "

(C.5)

Let the order of the ~'s be rim I> ~P2 ~> "'" ~>r/p,

(C.6)

for some permutation P, and label the eigenvalues so that XI~>x 2>1...1>x .

(C.7)

Comparing the expressions for each d m given by eqs. (C.3) and (C.4) we see that it is necessary that: Xi - n~- •

(C.8)

In other words, we can label the X's so that 3.] vanishes like lli~ {sfl (Pi) where {Sj} is the set of all scalings that close the ]th loop.

Appendix D In spinor electrodynamics the graphs of fig. 35 possess the Landshoff pinch. In X~3 this pinch is ~/s down on the leading term (the first type of pinch), but in the spinor theory the numerator factors enhance the behaviour to a potential amplitude of ~/s, which would again dominate over the eikonal contribution. However, there is again a cancellation of this pinch among the four graphs. The numerator factors for fig. 35a are -ff(3)7o(XlJ3)"fo(xlJ1 +X~2) 7v(x~ 1)'yu u(1)

X u(4) 7o(1 - x ) ]b47o(1 - x ) ~1+~2)~, (1 - x ) Jb23, u(2).

(D.1)

By anticommuting the 7-matrices, the leading term in this expression is found to be 4 s2x3(1 - x ) 3"u(3)~ 4 u(l)'u(4)~b 3 u(2).

(D.2)

The denominator is x(1 - x ) s 2 X (factors common to all four graphs).

(D.3)

The numerator factors of fig. 35b reduce to - 4 su(1 -x)3x 3"u(3)/b4u(1)'u(4)~b3u(2 )

(D.4)

I.G. Halliday et al., Fixed angle scattering

217

and the denominator is x(1 - x ) su × (common factors).

(D.5)

therefore, figs. 35a and 35b cancel. Similarly, there is a cancellation between figs. 35c and 35d.

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