Algebraic Geometry and Feynman Amplitudes

Fortschritte der Physik 19, 613-627 (1971)

Algebraic Geometry and Feynman Amplitudes

JOHN CUNNINGHAM

Department of Applied Mathematics, University College of North Wales, Bangor Wales, U. K .

Contents

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613 2. The Landau Criteria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 3. Multiple Point Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 4. Curve Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 5. Differential Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620 AppendixC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627

Summary

The representation method of obtaining first and second type Landau curves from the Landau criteria is described. The Plucker characteristic analysis of singular curves is outlined and its relationship with the theory of automorphic functions indicated. A differential approach to singularity classification is discussed and a possible extension of the method to obtain asymptotic behaviour is conjectured. A preliminary form of some of the material here presented was the basis of a lecture given by the author at the University of Wales Colloquium in Applied Mathematics, Gregynog, 1969.

1. Introduction

While many theorists tend to look upon the study of the analytic structure of collision amplitudes via perturbative field theory as at best an isoteric mathematical exercise and at worst old fashioned, it should be emphasised that the perburbation series provides the only really managable theoretical model for elementary particle processes which both formally satisfies the usual axioms of field theory and also exhibits, in some sense, the desirable properties of unitarity (analyticity), crossing, and Regge asymptotics. Most modern dynamical theories

614 JOE“ CUNKMUHAM

are based on postulates arising more or less directly out of intuition developed in perturbative field theory. The modern theory of analytic properties of perturbative collision ampltudes dates back to a paper by EDEN [ I ] in 1952 which soon became the heuristic basis for dispersion relations and the Mandelstam conjecture. The analyses of LANDAU [2], POLKINGHORNE and SCREATON [3] and the elegant discussion by TARSKI [a] of the “box” diagram for elastic scattering led in the late 1950’s and early 1960’s to a number of proofs of dispersion relations and the Mandelstam conjecture which, although incomplete, represented a very substantial contribution to the developmenta in high energy physics of the ensuing decade. The theory up to this point has been outlined by the present author [5] and the text of EDEN, LANDSROFF, POLKINGHORNE and OLIVE [S] presents an up to date and definitive description of the subject against the broad canvas of elementary particle physics. While much of the recent development of the subject as represented by the literature concerns high energy behaviour (see, e.g., RISK [7]) we shall concentrate in this paper on those aspects of the proofs of dispersion relations (other than convergence problems and subtractions) which were incompletely dealt with in the early 1960’s and which remain a partially unsolved problem. The details described in Ref. [5] will be assumed in the following discussions.

2. The Landau Criteria

The Landau curves are loci of points in the. multi dimensional complex space of the invariants describing a collision process whereat, on some sheet, the quanta1 collision amplitudes have non-analytic behaviour and herein lies their importance for dispersion theory. Some details are recapitulated in Appendix A. One incomplete aspect of the early perturbative proofs of dispersion relations was that the criteria of LANDAU [2] and those of POLKINCHORNE and SCREATON [3) were not strictly equivalent. Singular points, so called second type singularities (CUTKOSKY [8 ] , FAIRLIE, LANDSHOFF, NUTTAL and POLKINCHORNE [g]), could arise from Jacobian factors introduced in Ref. [3] by performing the k-integrations, and these were initially overlooked. Studies of such singularities have been slight and our ignorance of this area of the subject is mathematically unsatis- factory although a belief, based on a somewhat limited experience of particular examples, that such singularities do not manifest themselves in the physical sheets of amplitudes has produced no serious conflict with empirical reality. In any full mathematical theory properties on unphysical sheets may be necessary if the mathematics requires continuations off the physical sheet. To see simply how second type singularities arise as solutions of the Landau equations it is instructive to use the representation method of Landau curve determination [ lo , 111. Consider a “triangle” vertex diagram

( 1 ) PI = q 3 - 42 P2 = Q1 - 4 3

P3 = Qz - Q1. Because of the energy-momentum conservations and the Landau loop equation

Algebraic Geometry and Feynman Amplitudes 615

the vectors describing the lines of the diagram span a two dimensional space and may be represented as

The solutions for which the vectors pi do indeed span a two dimensionaI space lead in straightforward fashion to the usual first type Landau curve. Explicitly, choosing the rest frame of the particle 1, a, = ml, the conservation equations yield directly

(6)

(7)

(8)

or, on elimination, the usual cubic surface in the variables zl, z 2 , z3 . We remark that the general vertex problem can be attacked using two dimensional vectors and the method always yields directly expressions for the invariants zl, z,, z3 in terms of parameters. There may be additional constraint equations for the parameters ai arising from the consistency conditions which must be imposed when there are more than sufficient linear scalar equations from which to determine the a-values. I n practicc we have found it a most convenient scheme for numerical comput,ation [ lo , 11, 12, 13, 141. However, when some of the qi's fail to span two dimensions we obtain not only first type Landau curves corresponding to Yelf energy loops (contracted diagrams associated with normal thresholds) but also other solutions of second type. For suppose

z, = m2a + ma% - 2a2a3 f i- i(a32 - m32)

z2 = m32 + m,2 - 2m,a3

z3 = mI2 + m22 - 2m,a2

then

where

Solutions with Ai = &mi/mj correspond to first type curves whereas those with aj infinite are of second type. Explicitly, if q2 and q3 are linearly dependent either

m2 q 3 q 2 = f-

m, and we obtain the normal thresholds

or (13)

(14)

the second type curve of vertex type. The matematical basis of the Landau criteria has been carefully reexamined and the reader is referred for rigorous discussion of the singularity structure of functions defined, as are the perturbative amplitudes, by multiple integrals to the works of BLOXHAM, OLIVE and POLKINGHORNE [IS], AKS, GILBERT and HOWARD [I61 and to the text of HWA and TEPLITZ [17].

616 JOHN CUNNIN~JUM

3. Multiple Point Structure

On the physical sheet the Landau curves represent all possible points of singularity of the amplitude studied. Whenever two branch points of an amplitude coincide paths of analytic continuation which might have threaded around these points had they been distinct are inadmissable and at such coincidences the analytic structure may change nature. The simple mechanism whereby the tangency of the two Landau curves permits a pinch sinplarity to become a harmless coincidence by slipping over an end point is well known. The more complicated mechanism whereby at a multiple point on a Landau curvc a pinch sin-oularity dissolves into a harmless coincidence due to a repairing of zeros of the Feynman denominator function (at least three zeros must be involved) was discovered by EDEN, LANDSFIOFF, POLKINGRORNE and TAYLOR [18] and, bafore this discovery, had been overlooked in perturbative studies. This, along with the neglect of second type singularities, waa the main deficiency in the early perturbative proofs of dispersion relations. The physical implication of the appearance of multiple points on Landau curves is extremely serious because they provide a mechanism whereby the Mandelstam conjecture may fail (through the appearance of complex singularities) without the occurrence of anomalous thresholds. Examples have been given by OLIVE and TAYLOR [I91 and ISLAM [20].

4. Curve Classification

The Landau curvc for which no Feynman parameter a vanishes is called the leading curve for the given diagram while curves with one or several vanishing parameters are called lower order curves. This labelling has been the basis of several attempts to classify Landau curves. Such labelling is more than just convenient because, in general, the non vanishing of a particular a corrcsponds to the pinch mechanism (P) in this integration variable while the vanishing of an (Y corresponds to the end point mechanism (E). Unfortunately this is not a strict correspondance: CUNNINGHAM and RAFIQUE [ lo] have given an example in which the Feynman parameter pinch persists a t (Y = 0 a t every point on a curve. The strict correspondance is merely

a = O * E (15)

An interesting point may be made here. Taski [a] has proved that, for single loop diagrams, curves of adjacent orders touch one another whenever they meet so establishing an elegant geometrical hierarchy of curvcs. Such a property has besn conjuctured for curves corresponding to arbitrary diagrams but considerable care must be exercised because, in its simple form, the tangency theorem fails to hold. For example the leading curve for the “wigwam” vertex diagram has non tangen- tial intersections with “ice-cream cone” type curves of second highest order [lo], the contracted diagram behaving as a pseudo “triangle” vertex diagram which effectively lowers its order by one. Further problems in using the equations of curves to classify singularities have been spotlighted: examples have been given [l l] in which curves of different


orders and also of different types have the same equation. That a strict hierarchy based on Landau curve equation did not exist was first pointed out by LANDS- HOFF, OLIVE and POLKINOHORNE [ZZ]. I n this paper we are particularly interested in the multiple point structure of Landau curves and for this purpose it is convenient to use the methods of algebraic geometry first applied in this connection by REOOE and BARUCCHI [22]. Plucker ’s characteristics for an irreducible algebraic curve are defined as follows: n is the degree of the curve, the number of intersections with a straight line; rn is the class of the curve, the number of tangents which can be drawn from a given point; b is the number of nodes; x is the number of cusps; i is the number of inflections; t is the number of bitangents; d is the deficiency or genus. These parameters are related by the following equations.

m = n(n - 1) - 26 - 3%

n = m(m - 1) - 2 t - 3i

k = 3m(m - 2) - 6 t - 8i

i = 3n(n - 2) - 66 - 8%

d = 1/2(n - l)(n - 2J - 6 - 5~

(17)

(18)

(19)

(20)

(21)

(22) = 1/2(m - l ) (m - 2) - t - i. Wc shall illustrate their use by means of a simple example discussed in detail by EVANS [23]. She considered the “box” diagram for the elastic scattering of a pion (p) on a nucleon (M) imposing the constraint of nucleon number conservation on the lines of the diagram all of which have mass ,u or M obtaining the equation

t ( -s2+2s(&!a+p2)- ( M 2 - p 2 ) 2 ) + + 4 ( M Z 9 - s(2M4 + p4) + H” + 2p” -

-33p4MS) = 0 (23)

where s and t are the usual Mandelstam variables. This curve is a cubic and must because of (17)-(22) conform to one of three types

m d x t i d

6 0 0 0 9 1

4 1 0 0 3 0

3 0 1 0 1 0

For curves of low de,pee tables such as the above are to be found in standard text books [24] but for curves of higher degree tables can be produced trivially by enumeration by computer. In order to select the appropriate set of characteristics Evans considered the number of tangents which can be drawn to touch the curve and which pass through the origin. There are four such tangents and a line (s = 0) passing through a node a t infinity so one concludes that this cubic has class m = 4. From this analysis, were it not already well known from previous analyses, we could conclude that there are no features such as acnodes or cusps

618 JOHN C U N ~ - G H ~

which could lead to a failure of the Mandelstam conjecture and that we need concern ourselves only with the possible appearance of anomalous thresholds. In a more complex problem, of course, the analysis would indicate the number of multiple points requiring investigation. Several diagrams have been investigated [25,26] wherein the key to the success of the analysis lies in the simplicity of the parametrisation possible for the Landau curves; for example, if s and t can be expressed as rational polynomials of some parameter then the deficiency of the curve is zero; as each characteristic is detemined so the complexity of the residual calculation diminishes. When the Landau curve is specified in terms of serveral parameters (as in the representation method [ l o ] ) information may be deduced from relations among parameters using Zeuthen’s rule. The example of the “Mercedes” vertex diagram dealt with by several authors [12,13,14, 271 has proved too awkward to handle analytically. Instead computer drawings of Landau curves obtained by the representation method [ l o ] were scrutinised and it was shown empirically that a proper parametrisation in terms of simple automorphic functions is not possible (a theorem of SALMON [28] relates the genus, and hence the type of automorphic function required, to the number of independent circuits possible on a curve; some details are given in Appendix B). While clearly then the general problem remains open REGGE and BARUCCHI [22] have succeeded in classifying the infinite set of “ladder” diagrams using the properties of Cayley determinants to show that for a ladder of N rungs

(24)

m = Z N - l ( 3 N - 2 ) (25 ) 6 = 2 2 ~ - 3 ~ 2 - 2 ~ - 1 ( 2 ~ - 1) (26)

x = o (26) d = 2N-2(N - 2 ) + 1. (28)

n = 2*-1N

While the results of [ l a ] suggest that the property x = 0 may be untypical this analysis is very satisfying because the infinite family of “ladder” diagrams provides a perturbative model in which both multiple point structure and high energy behaviour are known; the sun1 of all “ladder” diagrams has been shown using Mellin transforms to be Regge behaved (see, e.g., Ref. [ S ] ) . The author believes that a worthwhile unsolved problem consists in performing Plucker characteristic analysis for other infinite classes of diagrams, in particular for those classes whose asymptotic properties are already well known.

5. Differential Properties

A new approach to the analytic properties of perturbative amplitudes has been proposed by DE ALFARO, JACXSIC and REGGE [29] and applied to single loop diagrams by GIFFON [30]. While it is certainly based on the Feynman parameter representation of amplitudes it does not appeal directly to the analytic properties of functions defined by multiple integrals. Instead the Feynman integral (whose denominator function may be written down in principle using the SYMANZIK [31]. rules) is used to derive, by differentiation under the sign of integration with respect to the invariant variables, sets of coupled differential equations for the


amplitude. Some details are given in Appendix C. The Landau singularities may then be read off from the differential equations without necessarily performing the integrations. Also the author [32] has pointed out that the high energy behaviour can be obtained by integrating the differential equations in asymptotic form. Explicitly in the case of the “box” diagram (all masses equal) the amplitude F satisfies asymptotic equations of the form

where

implying that

a - (d F) - d-le In s at w m

- a (d F) - d-’G(t) a s a-m - s2t (t - 4m2)

8-00

Non leacling terms in the expansion may also be obtained by retaining additional terms in the various equations. At the present time this attractive method suffers from a serious defect in that the derivation of the necessary differential equations for large classes of interesting diagrams has not been accomplished, and until this is done the method remains somewhat academic. SYMANZIK [33] has used a differential approach to determine the asymptotic forms of vertex functions in quantum electrodynamics.

Appendix A

The Feynman parameter representation for a term in the perturbation series for a given collision amplitude has the form

where

and qi is the four momentum of the particle of mass mi corresponding to the i-th internal line of the diagram; the qi depend via energy-momentum conservation at the vertices of the diagram on a set of I independent internal four vectors hi and on the external four momenta pi. The symbol 2 summarises the

620 JOEN CUNNINGRAM

R

) independent invariant variables essentially pi - p i , C p i = 0 which describe

a diagram with n external lines (when n 2 4 it is usual to fix the n quantities pi2 at the appropriate mass squared values and treat the analytic properties in the remaining variables only). The Landau equations [2] determining possible singularities of F (2) are

( i = l

L

i = l C a i = 1

(a 7)

(-4 8)

q.2 , - m ; 2 , - i = l , 2 ,..., L

2’ aiq; = 0

together with the V - 1 independent conservation equations where V is the total number of vertices. There are 1 equations of type (A 8) where sums are taken round independent closed loops of the diagram. If /3 be the largest dirncnsion of space spanned by the vectors of the problem (p = 4 if n 2 5, p = n - 1 for n < 5) then to be found are L parameters a, /3L components of the q’s, and LV invariant variables. To find the a’s there are P I + 1 scalar linear equations so that the a’s may be found in terms of other quantities i f y 5 0 where

y = L - / 9 ( L - V + I ) - l . (-4 9)

If y 5 0 we may use the first L equations involving a’s to find them in tcrrns of other quantities leaving 1 + pL scalar equations from which to find ,8L q- components and N invariants. So N - 1 quantitics remain undetermined, or, in othcr words, one relationship (the Landau curve) exist,s connecting the 2‘s. In the POLKINGHORNE and SCREATON [3] approach thc k integrations are per- formed and singularity locations are found by considering criteria for zeros of d to coincide (possibly trapping the integration hypercontour) or move into a boundary point of the integration region. The situation y 5 0 corresponds to the coincidence mechanism while y > 0 corresponds to the boundary point mechanism (some of the a‘s have to be set equal to zero to solve the Landau equations).

Appendix B

We present here some details of the analysis of Ref. [13, I d ] of the “Mercedes” vertex diagram (see Fig. 1). I n the symmetrical case m, = m2, m, = m5 we may cast the Landau equations (which in the representation method does not involve the u equations except when the a‘s are explicitly sought) into the form depending on three parameters a, b, c,

z1 = p I 2 = F (b, C)

z2 = pZ2 = F (a, -c)

z3 = p32 = F, (a, b )

0 = F,(u, b, C) .

(R 1)

(B 3

(B 3)

(B 4)


Manipulative simplifications result when m, = 2m, and since the geometrical problems are not reduced it is this case which has been studied in detail. The curves

@(% 22) = 0

arising when z3 and the internal masses are fixed at various values have been plotted by computer techniques and are exemplified in Figs. 2-5 (the normal threshold lines zl, z p = (ml & m3 f m5)2 are indicated in Fig. 2): in each figure

I 7-

I I

I

Fig. 2

622

Fig. 4

Algebraic Geometry and Feynmrtn Amplitudes 623

Fig. 5

nz3 = 1.4, m, = 0.8 , z3 = 4 while m, successively assumes the values 1.0, 1.6, 1.8, 2.1. If the degree of the curve is known (in the case illustrated it is believed to be twelve) a table of all Plucker characteristics possible is easily compiled and empirical observations of numbers of nodes and cusps, tangents and bitangents, etc. dramatically reduce the large number of possibilities. Other more sttbtle points may also be taken into consideration. For example in Fig. 4 there are, on the axis of symmetry, four ordinary nodes, one acnode, and two other real points (twelve points counting multiplicities) exhausting all possibilities if the degree is twelve. Thus by symmetry and reality of the Landau curve the remaining finite nodes must occur in real of complex con- jugate pairs: so the number must be odd. Features such as the isolated point (acnode) in Fig. 4 cannot, of course, be found by computer! Points with a = b, c = 0 clearly lie on the axis of symmetry (and there are others) and were investigated analytically. Indeed, as shown in the work of RISK [25], one of the vital first steps in any such analysis is to investigate multiple points on special lines such as axes of symmetry and the line at infinity. A further example is the application of Salmon’s theorem which states that the number of independent circuits possible on a Plucker curve, in general, exceeds by unity the genus of the curve. The illustrated curves are certainly not unicursal (indeed we believe there to be three independent circuits) and so cannot have genus zero. I n turn this means that a proper psrametrisation in terms of rational functions of a single parameter is not possible. The general “box” diagram has as leading Landau curve a quartic curve of genus one (two independent circuits are possible) and so elliptic functions (meromorphic doubly periodic functions in the

4s Zeitschrift ..Fortschritte dcr Physil;“, Heft 9

624 JOHN C n m a ~ 4 ~

complex plane) are required to effect a proper parametrisation; the higher the genus the more unfamiliar are the automorphic functions which are required. However, this unfamiliarity on the part of high energy theoretical physicists may well be at an end. In the theory of dual resonance models LOVELACE [34] and ALESSANDRTNI [35] have shown that the harmonic problem on Riemann surfaces is directly relevant to the problem of constructing multiloop Venezimo amplitudes ; in its context this work could well be as significant as the first Writing down of the Feynman rules in perturbative field theory! The basic mathematics involves Abelian differentials defined on spheres with several handles and the corresponding automorphic functions associated with various groups of transformations. In the case of elliptic functions single valued functions on a torus (a sphere with one handle) become doubly periodic functions in the complex plane. A pay-off in our problem could be a curve classification with a group theoretic basis through automorphic functions. Since a single variable parametrisation has not been found an analytic approach based on the locus (B 3) in the (a, b) plane and on other loci obtained by elimination between (B3) and (B4) could yield information using Zeuthen’s rule. Although this has been tried [Z7] it has not, to the author’s knowledge, been carried to a successful final stage of positively identifying @. Incidentally, although th is problem is not treated in the present paper, the com- putation of actual numerical values of the a‘s for any set of parameters a, b, c is straight forward because the equations are linear in the u‘s. The importance of this advantage of the representation method lies in the fact that, in many cases, the sign or size of t,he a‘s can be used to decide whether or not a given point of a Landau curve is actually singular.

Appendix C

Consider the c c b ~ ~ ” scattering diagram for which the amplitude is, in the notation of Appendix A with 1 = 1,

The denominator d may be written in form

where Pi.5 = Pi

the four-momentum of the i - th internal line and

siS5 = mia (C 4)

the square of the corresponding mass. If now we define a new parameter dr5 by


then, performing the four dimensional loop integration, we obtain essentially

where D is a homogenous quadratic form given by 6

dcj i=l

D = 2 L X ; S ~ , ~ U ~ .

We may recas t (7) into the form used by GIFBON [3U] namely

We have ignored for simplicity the factors independent of the si,i arising from the Jacobian introduced in performing the loop integration. We shall deal with the case s1,3 = s, = t with all other non zero si,i = m2. For the present however we shall treat the ten S ~ J , i < j (s~,~, s2,4 the usual Mandelstam variables, the the internal masses, and the si,j, j $. 5 the external masses) on an equivalent footing. We introduce the 6 x 6 Cayley matrix

and denote by

the usual signed minors of the determinant A 2 of (C 10) obtained by ths indicated row and column deletions. Differentiating with respect to s ; , ~ one obtains

where

and

The result

I,m=1

has been used. Giffon now rewrites (C 12) Using

48*

JOHN CTNXIXGHAN 626

and

to obtain

4

F,' = 2 8:' Fm,,, n=1

m+n

m*n

Next Fm,n is attacked in a similar fashion and a set of differential equations is obtained [30]. In our case

a at

Fm,n a - (AP) = as

where A 2 is the Cayley determinant

0 1 1 1 1 1

1 0 i n 2 t m2 m2

1 in2 0 m2 s m2

1 t m2 0 m2 m2 A2(s , t ) =

1 m? s m2 0 m2

1 m2 ?n2 ma rn2 0

The Fi,i = s and t only respectively. For example

are all constants excepting F183 and F2,4 which are functions of

It is not too hard to read off the Landau singularities, e.g., A = 0 is the usual leading curve and various lower curves arise from the square root and logarithmic factors. We remark that, since we have not taken second type singularities into account and the analysis requires some (trivial) emmendations if a serious discussion (of say absorptive parts) is intended. A feasible programme extending the above analysis would be to treat the general ladder diagram, hopefully obtaining the exact differential equations for the amplitude or, a t least, their asymptot,ic form (see text).


References

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[7] C. RISE, High Energy Bchaviour of Feynman Diagrams, Michigan lectures 1969. [8] R. CUTKOSKY, 3. Math. Phys. 1,429 (1960). [9] D. B. FAERLIE, P. V. LANDSHOW, J. NUTTAL and J. C. POLKINOHORNE, J. Math. Phys.

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[I61 S. AKS, R. P. GILBERT and H. C. HOWARD, J. Math. Phys. 6, lG2G (1965). [I71 R. C. Hwa and V. L. TEPLITZ, Homology and Feynman Integrals, Benjamin 1966. [I81 R. J. EDEN, P. V. LA~DSHOFF, J. C. POLKIX~HORNE and J. C. TAYLOR. J. Math. Phys.

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[30] M. GIFFON, Nuovo Cimento 61 A, 663 (l9G9). [3I] K. SYUXZIK, Pros. Theor. Phys. 20, 690 (1958). [32] J. C~NINGHAM: A Differential Approach to High Energy Behaviour of Feynmm

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Algebraic Geometry and Feynman Amplitudes

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