IC/73/U3

REFERENCE

INTERNATIONAL CENTRE FOR

THEORETICAL PHYSICS

SOME TOPOLOGICAL AND GRAPHICAL ASPECTS

OF PHASE CONTOURS

A. Shafee

INTERNATIONALATOMIC ENERGY

AGENCY

UNTTED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION 1973 MIR AM ARE-TRIESTE

• f t

1C/73A3

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

SOME TOPOLOGICAL AND GRAPHICAL ASPECTS

OF PHASE CONTOURS *

A. Shafee

International Centre for Theoretical Physics* Trieste, Italy.

ABSTRACT

Using the topological properties of phase contours, the phase contour

diagram of a function in the complex plane is reduced to the simpler form of

a l>ichromatic multigraph,which contains all the combinatorial characteristics

regarding links between zeros and singularities of the function. It is

found that the saddle points of the phase play an essential role in any such

qualitative global description of the function, as can be anticipated from

MorBe theory. In the case of the scattering amplitude, constraints from

symmetries, periodicity or unitarity are seen to yield simplifications with

interesting results concerning factorization and asymptotic behaviour.

MIRAMARE - TRIESTE

May 1973

To be submitted for publication.

I. IMTRODUCTION

Qualitative methods of analysis have teen known to provide useful

insight into the nature of the solutions of complicated dynamical problems,

•where a quantitative solution turns out to be impossible or extremely

difficult to obtain. They are valuable in establishing existence and

consistency conditions and in indicating peculiarities of the solution, so

that the subsequent application of quantitative methods for an exact or

approximate solution may be facilitated. It is of course also possible that

the particular problem of interest has no quantitative content, so that

qualitative methods alone can suffice.

The richness of the singularity spectrum of the strong interaction

amplitude, vhich makes an analytical solution difficult, provides a non-

trivial and interesting field for the application of certain qualitative

methods, especially those of a topological nature. Of all the different

approaches to the Btudy of symmetry and consistency conditions for scattering

amplitudes, that which most successfully exploits the global topological2)-Q)

properties is probably the phase contour method ,where patterns of

singularities and zeros of amplitude functions related by crossing or some

other symmetry have been investigated using specific dynamical models, like

Regge asymptotic behaviour, or more general physical principles, like

unitarity or hermiticity, but without resorting to numerics for most results.

On the other hand, by analysing the data with dispersion relations, one can

obtain ' the phenomenological picture of such patterns, showing intimate

relations between specific zeros and poles of the amplitude. Such relations

probably, have more topological than geometric significance, because the

complications from a complex background or kinematic peculiarities usually

destroy the usefulness of metric-dependent geometrical concepts like linearity

or convexity, while topological invariants are less likely to be affected

by errors in the data or by perturbative correction terms in a theoretical

study.

In previous applications of the phase contour method more emphasis

was laid on the physical content than on the topological side of the approach.

This left certain mathematically undesirable features in the treatment, e.g.

ambiguity in the labelling of phase and arbitrariness in the spacing of

phase contours. Besides, the crucial dependence on visual presentation made

any generalization of such techniques to complex spaces of multiple dimensions

obscure and impracticable. The usefulness of topological methods, especially• 12) lU) l)

of the critical point theory of Morse, " in the global analysis of

-2-

complex functions has been of great interest to mathematicians in recent

years. In this paper we intend to introduce the concept of topological

invariants in the study of phase contour diagrams and show the relevance

of critical points and of Morse theory in this context. In the case of a

single complex variable we shall show how phase contour diagrams {PCDs) can "be

reduced to graphs containing all the combinatorial inforraation relating poles,

zeros and critical points. We Shall follow a rather heuristic line so that

the formalism may remain sufficiently transparent to show its relation to the

previous works with the phase contour method.

In Sec.II the phase contour diagram is re-defined on a more abstract

footing than previously. We use the concept'of bundles of phase contours

connecting sources of opposite polarity. In Sec.Ill we show hov homotopical

equivalence of contours allovs us to derive dichromatic multigraphs ** from

the PCDs. We indicate the importance of the critical points of the function

in a comprehensive topological description of these graphs and hence also of

the original PCDs. In Sec.TV. we look into the consequences of physical

constraints, like hermiticity, positivity over the cut from unitarity,

asymptotic "behaviour and periodicity in the patterns of zeros and poles.

II. PHASE CONTOUR DIAGRAMS

Let F be a complex function defined in an n-dimensional complex

space C with points z = (z ,z ,...,z ) . The phase of the function is

defined,as usual,as the real-valued function

<f>(z) = Im log F(z) . (2.1)

We define phase contours P. in the space Cn as the connected sets of

points P.CCn , such that 4>{z) = K. (a constant) for all ze.?± . Since

the n-dimensional complex space corresponds to a 2n-dimensional real space,

the P. are, because of the single constraint (2.l), (2n~l)-dimensional

real subspaces in this real space. If we consider functions of a single

complex variable z = z , the phase contours are simply the familiar" 2)-ll)

contour curves in the complex z plane. Usually a PCD is defined

as the collection of P. drawn at regular intervals A with '.K̂ = K̂ + iA

(i = 0,1,2,...), on the whole z space or a section of this space. However,

the arbitrariness of K and A introduces an unsatisfactory discrete

description of a generally continuous function <j) . In this paper we shall

define a phase contour diagram D as the collection of all P. corresponding

to every.value K in the range of <ji> . According to this definition, any

point where the phase <}> is defined must lie on a phase contour P. .

In a similar way one can define a modulus contour diagram vith the

modulus y of F given by

p(z) - Re log F(z) . (2.2)

for all z£ C where log F is defined. These again are (2n-j)-dimensional

real subspaces, or one-dimensional curves if n = 1 . In the latter case,phase

and modulus contours form two orthogonal families of curves, by Cauchy-Riemann

conditions, spanning all points of the complex space where log F is defined.

The logarithm has branch points at the origin and at infinity. Hence ((> and

y are defined everywhere, except where F. has a singularity or a zero and

along cuts joining the zeros and the singular points. The behaviour of phase

contours in the neighbourhood of a singular point or a zero depends on its

type and order. In the neighbourhood of an essential singularity or an

exponential zero, phase or modulus contours are not well defined. We shall

consider only isolated singular points and zeros of finite order. Let F

be factorizable into the form

F(z) = U-z.) * f(z) , (2.3)

where a, is a positive or negative integer, f(z) being regular and non-zero

at a. . The phase <J> is the sum of the phases of the two factorsI

<J>(z) = a± Im log(z-z1) + Im log f(z) . (2.10

Since the second term is regular near z. » by choosing a disc small enough

around z. its variation can be made as small as desired, so that we can

effectively replace it by a constant within the disc and on its boundary.• a

At the boundary of the disc z = z. + r e we have

c|>(6) — a.e + a constant. . (2.5)

On a complete rotation around z.,'<J> changes by 2a^i\ , necessitating a branch

cut. However, the different sheets of this logarithmic singularity differ

only by multiples of this constant 2OUTT , changing the label <fi o f every

phase contour Pi , leaving the P. themselves end,hence,the PCD invariant.

It is, therefore, sufficient to consider Just one sheet.

If a i > 0 , i , e . z is a zero, <J> increases for an anticycllc

rotation around z. . We shall call z. a source of strength a. . Similarly,

if OL < 0 , i.e. if zi is a singularity pf F{z) , § decreases for an

anticyclic rotation around z. , which vill be called a sink of strength

a. , or a source of strength -ot̂ . These terms are borrowed from the obvious

electrostatic and hydrodynamic analogies. We shall also use the term "source"

generally to indicate either a source or a sink.

Lemma 2.1

P. are continuous open sets with sources and sinks as their limit

points. Proof: The continuity of E. at regular points follows from <J>

"being a harmonic function and satisfying Laplace's equation. Closed loops

of P. are not possible because then the orthogonal modulus contours

entering the region bounded by P. must either converge to one or more

points, which is impossible even at singularities of F , or pass out of the

region through another point of P. . In the latter case,modulus contours

must exist whose two points of intersection with P. approach coincidence,

i.e. the modulus contour becomes tangential to P. . This too is forbidden

"by the orthogonality condition. Hence all P. must be open and have

singularities of log- F as limit points.

Let B be the set of all P. such that the subspace X C C covered

by the set B = {P.} is arcwise connected. We call B a phase bundle, in

loose analogy with fibre bundles, as the connected and continuous set B may

be considered to fibrate the space X , though the definition of the base space

beoomes complicated because different ends of different P.€B may have

different singularities as limit points. If all P.€B have the same two

limit points, we shall call it a closed bundle. Henceforth only closed bundles

will be considered and for brevity these will be called simply bundles. We

shall call the difference between the extremal phases in B its thickness-

Lemma 2.2

The limit points of a phase bundle of non-zero thickness are sources

of opposite polarity. Proof: If DA is a small disc around the limit point

A , with the boundary 3D. , we have seen in Eq.(2.5) that the Pi form a

monotonic sequence on an arc of 3D, . Since different P.̂ cannot intersect,

on a continuous deformation and translation of the arc from dD^ to 3DB ,

the boundary of a small disc around the other limit point B , the sequence

-5-

will remain unaltered with respect to A , but the orientation with respect to

B will be opposite to that at A . Hence, again from Eq. (2.5), the a at B

must have a sign opposite to that of A .

Lemma 2.3

If B a b and B b are two different bundles sharing the limit point

b , "but with the other end points a and c different, and if B,/1i 4 <b ,— — ~~~ SD DC

then this intersection contains a critical point of the function. Proof: If

the intersection is non-zero, it must be the whole or part of the adjacent

boundary contour. It cannot be the whole contour because,by assumption,the

other limit points a and c_ are different. -Hence it can only be part of

the boundary contour which must at some point split into two branches to join

the different limit points a/ and c_ .;; At. the branching,point, a tangent to

the contour must become indeterminate. From the definition of the phase,

Eq.(2.l), and from the Cauchy-Riemann conditions,it can be seen that at the

branching pointdUn F) = ^ = 0 . (2.6)

However F cannot be infinite at an analytic point of the function, so that

we must have dF - 0 , which makes the branching point of the contour a

critical point of the functions $ and y . If it is a non-degenerate

(x, - x , x2 = y) does notcritical point, i.e. if the Hessian

12) " * -vanish, then it would have two eigenvalues of opposite sign and the

critical point would be a saddle point. We shall call the contour attached

to the critical point, the critical Value contour. Obviously the thickness

of a bundle is equal to the difference between the phases on the critical

value contours bounding it.

If a. is a non-integral real number, F has an,.algebraic branch point

at z. with a finite (if a. is a rational fraction) or an infinite (if a^

is irrational) number of Riemannian sheets. If F has only algebraic

singularities, it can be seen from Eqs.{2.3) to (2.5) that the PCDs on all

sheets of F are Identical, except for a constant shift in the labelling of

all contours on each sheet. If the branch cuts are taken along the phase

contours, no contours move from one sheet to another, so that each sheet

becomes self-contained. Similarly the logarithmic cut associated with poles

and zeros of integral order are also made harmless by choosing it along a P̂^

in a bundle B . Then B becomes effectively the union of two bundles

B = BXUB2 (2.7)

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separated by the cut. However, if explicit labelling of the phase is avoided

(e.g. if ve consider only properties of the differential d$ , which is veil defined

and has no logarithmic cut at regular points of F , and is sufficient to

determine the singularities and zeros of F as well as the critical points of

<}>), then B can be considered to be a single effective bundle with a thickness

t B given by

*B = \ + \ '

which is invariant with respect to the choice of the P. chosen for the cut,

i.e. with respect to the particular partition of B into B and B

By mapping a cut sheet stereographically in the usual way onto the unit

sphere, the "point at infinity" aan be treated as any other point. From

Eq.(2.3) ve can get a source-sink duality on this sphere; every source of

strength a (positive or negative) at a finite point must be accompanied by

a source of strength -a at the point at infinity.

Theorem 2.1

For a function with only algebraic singularities (including non-rational

orders) the algebraic sum of the thicknesses of the phase bundles leaking into

adjoining sheets through the cuts is zero. Proof: Since F is by assumption

completely factorizable into the form of Eq,.(2.3)9 the point at infinity is a

source of strength equalf but opposite in sign,to the algebraic sum of the

strengths of sources at finite points. By Laplace's equation,no other sources

exist on a particular sheet; hence,bundles crossing into another sheet must

return to join a source of opposite strength on the same sheet, or be cancelled

by a bundle of opposite thickness emerging from the tmcompensated source

passing into another sheet, or the bundle may have both end points on other

sheets. In each case we can see that the net thickness of flux bundles passing

out of the branch cuts is zero.

Corollary 2.1.1.

The algebraic sum of the thicknesses of phase bundles passing into each

algebraic branch cut is zero. This follows from the fact that all the sheets

of an algebraic branch point are interconnected through the same branch cut,

and the net flow of phase bundles into each sheet is zero (by Theorem 2.l) ,

Therefore, all leakages into other sheets may be regarded as ineffective2) 3)

crossings of the branch cut which were mentioned by Eden et al. * However,

this will not be true in general when different sheets have different singularity

structures, as in the case of the physical scattering amplitude or a properly

unitarized model of it. 7

III. GRAPHICAL REPRESENTATIONS

Given a bundle B , all P. £. B (except the extremals, "which,

being critical value contours, have branches and hence a more complicated

topology) can be continuously transformed into one another, i.e. there exist

mappings homotopic relative to the end points giving all the P. e B - 3B(SB) .

In other words, all P. £ B are derivable from mappings belonging to a unique

homotopic equivalence class. This would also extend to the case of the

compound bundle (Eq.(2.?)) of an algebraic function,if contours on other

sheets are identified with their images on the original sheet. So far as

the combinatorial problem of the linkages between zeros and poles is concerned,

for many purposes it is sufficient to consider only the equivalence classes of

contours, as individual contours within a bundle are of little topological

interest.

We shall call the set of singular points of the phase and the homo-

topically equivalent classes of P, {or the open sets B ) represented by arcs

connecting the relevant limit points, a graphical representation of the PCD ,

or a phase contour graph (PCG). A PCG has the same homotopy type as the

PCD from which it is derived.

In general a PCG would be a multigraph, i.e. multiple arcs between the

same vertices would exist, provided they are homotopically inequivalent. We

can give an arc a weight equal to the thickness of the bundle it represents.

Obviously the algebraic sum of the weights of the arcs of W. meeting at a

vertex i indicates the strength and type of its singularity:

(3.1)

where a. is the index used in Eq.(2.3) to show the nature of the phase

singularity at z. . We shall use the sign convention that an anticyclic

rotation of increasing phase in the PCD corresponds to an outgoing positive

weight for the arc representing the bundle, in conformity with our use of the

terms "source" and "sink" in the previous section.

Since B. ^ B"2 = 0 for any two bundles B1 and B2 in a PCD, the

arcs representing 1L and IL cannot intersect either." Hence a PCG is a

planar graph. If only arcs of finite weight are considered, we can see from

Lemma 2.2 ,which states that bundles of finite thickness must connect

sources of opposite sign, that the PCG must be a bichromatic graph, i.e. a

graph where the vertices belong to two distinct classes and the arcs join

vertices of only different types. The PCDs of an algebraic function being

identical on all sheets, so will be the PCGs.

- 8 -

Lemma 3.1

The faces of a PCG are homotopic to the critical value contours and the

associated critical points. Proof: Let D be a PCD with sinks and sources1,

S- and phase bundles B. . connecting S. with S. . Let G "be the PCG

of D with vertices v. and arcs a., connecting v. with v. . Let h

be a set of homotopic mappings h: B^, >a.. . But D = Us, U Bj , U 3B. .H 11 k , t lj lJ I ij IJ

and G = vi a n £ » w^ere f. are the faces. The mappings h being

homotopic relative to the set {S.} , v. can be identified with S. .

Hence the rest of D is mapped onto the rest of G, i.e. U 9B* >^f0 •

Matching individual connected components we get a one-to-one correspondence

between the critical value contours of D and the faces of G .

Theorem 3.1

The PCG is maximally connected in the absence of degenerate critical

points. Proof: Let D be the PCD of the function and G its graphical

representation. If G is not maximally connected, there exists a source-

sink pair(Vj;v,)€ G , which can be connected by an arc a., homotopic ally

distinct from the arcs already present. Let h be the set of homotopic

mappings h: D -»• G . Now, (h~ a, .) d B = 0 (for B with m = i ,ij m,n ^mn ^ mn

n = j does not exist). Hence h~̂ - a, . C 8B,, U 9B.p , for some k * J snd

some • H ^ i . Let 3D. be the boundary of a small disc around i . A point

on 3D. on the other side of B., must belong to another bundle, B., , .

Similarly, there is another bundle B.p, separated, in part, by 3B .,

So a., is the image of part of a critical value contour which has at least

six end-points: i, j,k , I, k1 and V . But by Morse theory a non-

degenerate critical value contour in a two-dimensional manifold can only have

four branches. Hence, if there is no degeneracy, Bi- and its graphical

image a.. must already exist, making G a maximally connected graph.

Corollary 3.1.1

The faces of a PCG are quadrilaterals. The proof is similar to the case

for maximal bichromatic graphs without multiple arcs. First, it is obvious

that a face of a bichromatic graph can have only an even number of vertices

and sides. As a face cannot be bounded by two homotop'ically distinct arcs,

the minimum number of sides is four. If there are more than two vertices

of either colour, then, in addition to its two arcs connecting a vertex

with adjacent vertices of opposite colour, a diagonal can be drawn from it to n

third vertex of opposite colour on the periphery of the face. If the graph

is maximal, such an arc must already exist, but the face would then split

_ o _

Into tvo faces. Hence each face must contain exactly four vertices (tvoof each colour) on its boundary.

From the PCG, two graphs of interest can be derived - the dual graph and

the medial graph. The dual graph is constructed by Joining the midpoint (in

the topological sense, meaning any interior point) of every face with the

midpoints of its boundary arcs. We shall also consider the domain exterior

to the graph as a face, though unlike the other faces this will not be a

quadrilateral.

Lemma 3-,2

The outermost circuit of a maximal bichromatic multigraph consists of

only two arcs. Proof; If there is a second vertex of either colour on

this circuit, it can be joined to the adjacent vertex of opposite colour with

an arc surrounding the entire graph except the original linkage between the

two vertices (Fig. 1 ) that forms a new periphery. But this is impossible

because the graph is already maximally connected.

Theorem 3.2

The dual graph of the PCG is the graphical equivalent of the modulus

contour diagram. By a graphical equivalent we mean that non-intersecting

cycles of this dual graph represent all the different homological equivalence

classes of modulus contour cycles of the modulus contour diagram. Proof: The

modulus contours, being orthogonal to the phase contours, should encircle the

sources and sinks. The homological equivalence class of a modulus contour

depends only on the particular pole or zero it encloses. If the modulus

increases away from the region bounded by the contour we shall give the contour a

cyclic sense. In this case we have a pole inside,and with opposite orientation

and growth rate we can associate a zero. Since each face represents a critical

point, joining adjacent critical points by lines bisecting the arcs of the

original PCG, we essentially generate loops around the sources and sinks, and

hence the homology classes of the modulus contours. The proper orientation

of the cycles follows by giving each segment between successive critical points

a direction agreeing with the Cauchy-Riemann conditions.

It is easy to verify, by simple counting, that the topology of the PCG,

even when accompanied by the specification of the arc weights, cannot determine

the function completely. For example, the function

. (z-zWz-Zo)F(z) * C T — 7 S_ (3.2)

( H )

- 10 -

will have a graph of four vertices, two of each colour, and in general five

topologically-distinct arcs (Fig. 2 ). There are only two independent weights

corresponding to the loops, "but the function F has ten real parameters from

the five complex parameters C, z^, z 2, z and z^ . This indicates tbe

existence of a class of transformations under which the topology and the weights

of the arcs remain invariant.

A change in the topology occurs when one or more of the arc weights go

to zero, which can happen (as we shall see in the next section) whenever the

sinks and sources move into a pattern with a reflection symmetry. The weights

remain unchanged if the critical values remain invariant under the transformation

which may change the positions of the sources and the sinks and also of the

critical points.

If z = z(z.) is a multivalued function giving the critical points z

in terms of the sources and sinks z. , it is obtained hy solving the condition

for the critical point: (df/dz) = 0 , with f = logF. If we now consider

f(z,z.) with z = z(z.) , we get

df =3[f_ Sz_92 9z.

9fdz, dzi , (3.3)

because (9f/3z) =• G as z is a critical point. Hence for stationary

critical values the infinitesimal transformations of the poles and zero3 dz

must satisfy the condition

dz± = 0

where z is obtained from the solution, in z , of

= 0

i

(3.!+)

(3.5)

If F(z,z.) is factorizable into its sources and sinks, as in the case of an

algebraic function

F(z,zi) = ||F i(z,z i) ,

then the critical point is given by (with F^ = (dF^dz))

(3.6)

(3.7)

- 11 -

and the condition for veight-invariant transformations "becomes:

V r - r - T to. = 0 . (3.8)

All l inear transformations

Z i —> a z i + t , (3.9)

where a and b are complex constants indicating translation, rotation or

dilatation of the PCD on the complex plane, keep the weights invariant. This

reduces the number of parameters of the function from 2V + 2 (where V is

the total number of vertices) to 2V - 2 , indicating thai many other solutions

to Eqs.(3.7) and (3.8) remain outside the class given by the transformations

of (3.9).

The only non-trivial Betti number of a graph is the number of faces F

in it. As we have seen before, this is equal to the number of saddle points

of the phase, in the absence of any degeneracy resulting from symmetries or

other special relations between the strengths and positions of the zeros and

singularities of the function, and preventing the graph from being maximally

connected.

Theorem 3.3

For a maximally connected bichromatic multigraph the number of faces

F - V - 2 , where V is the total number of vertices of either colour. The

proof is by induction. We have seen in Lemma 3.2 that the circumference of

a maximally-connected bichromatic multigraph contains one vertex of each

colour. If a new vertex appears outside this graph, it can be connected to

the vertex of opposite colour by two homotopically distinct arcs, which,together

with the two arcs of the previous circumference, form the four edges of a new

quadrilateral face. With V = k we get F = 2 , as can be verified by

explicit construction. Hence, constructing graphs with V > H by adding

new vertices and consequently faces from one with V • k , we get the general

relation F » V - 2 . .

- 12 -

IV. PHYSICAL CONSTRAINTS

Our discussion of the PCGs in the previous section was mostly concerned

with algebraic functions with similar Riemannian sheets. Models for the

scattering amplitudes with only meromorphic functions exist, which satisfy

important constraints like asymptotic Regge behaviour, crossing symmetry,

direct and crossed channel poles (though on the real axis of the single sheet).

Unitarity involves the introduction of branch points of non-algebraic nature -

except the elastic branch point . However, treating the cuts as effective

sinks and sources according to the amount of phase leaking into or out of

them, it is possible to consider only the physical Bheet or a submanifold in

it instead of the complete Riemannian surface and apply some of our results

to the subgraph belonging to this region.

A. Symmetries and factorization

We make the following observations:

a) Hermitian analyticity — A(S) * A*(S*) — makes the PCD on the

physical sheet, and hence the PCG on this sheet, symmetric on reflection by

the real axis. Only the labels of the P. change.

b) Other symmetries may exist, giving left-right symmetry of the

graph', e.g. the crossing symmetry of the A' or B~ amplitudes of TTN

scattering or the IT "if scattering amplitude. Crossing antisymmetry as

in A'~ and B irN-» amplitudes would also produce symmetry on either side

of the imaginary axis.

Theorem U.I

Every reflection symmetry of the function leads to an increase in the

number of components of the graph and the appearance of a degenerate critical

value contour. Proof: Let S be the line of symmetry dividing the graph

G into identical subgraphs A and B . We prove that A and B are

disconnected. If any arc connects A and B , it must either be symmetrical

itself under reflection on S , or have a symmetrical partner.

Since an arc (which represents a phase bundle of finite thickness) can connect

only a source with a sink - by Lemma 2.2 - reflection symmetry rules out a

symmetrical arc. Similarly, a symmetric partner is made impossible by

planarity. Hence A and B must be disconnected components of the graph

of the function. Since no phase bundles cross S , S itself must be a

phase contour, say with phase <f> . Even if A (or B) is maximally connected,

- 13 -

its periphery must contain a source and a sink (from Lemma 3.3). The phase

contours with phase <t> from these singularities must "be connected with S ,

because (by Lemma 2.1) all phase contours must have sources and sinks as

their limit points. Hence S must be part of a contour with at least six

components (Fig. 3 ). By Morse theory, a non-degenerate critical value

contour can have only four components. So S must be a part of a degenerate

critical value contour. The critical points can be either distinct or co-

incident. By the same arguments, it can be proved,in the case with

vertices on the symmetry line,that though the graph does not split into

disconnected components, it ceases to be maximally connected.

The existence of symmetries indicates the possibility of factorization

of the amplitude into functionally similar components. Let F(z) be a

crossing-symmetric amplitude, i.e. F(z) = F(-z) . We can write the17)

phase representation

F(z) = exp

o

IfTF I

* 1

6(z')dz'

z - z1ih.l)

where ± z. are the zeros, ± z, the poles and 6(z') the phase of the

contours leaking out of the physical sheet through the cuts. In Eq., (U.l)

we can see that F(z) factorizes into f(z) and f(-z) , with the two

symmetric branch cuts and the symmetric poles and zeros separating into the

two factors in any complementary combination.

However, factorization of the amplitude does not in general lead to

factorization of the graph into subgraphs corresponding to the factors.

Because of the requirement of planarity, the graph of a function must be

different in general from the superposition or any simple interconnection

of the graphs of its factorial components. Connectivity, being a global

property, depends not only on the individual strengths of the relevant

sources and sinks, but also on the nature and position of all other sources.

We mentioned towards the end of Sec.Ill how the weights of the arcs of the

graph of an algebraic function are controlled by the positions and strengths

of all the sources on any sheet. For a more complicated function like the

scattering amplitude, where each sheet is expected to have a different pole-

zero structure, the poles and zeros on other sheets influence the weights

of the arcs on the sheet of interest in a complicated way. On the other

hand, we know from the phase representation that the amplitude is fully known

on the physical sheet if its poles and zeros on the same sheet and its phase

along the branch cuts of this sheet are known. Since the full content of

- lU -

analyticity is contained in the branch cuts, the weaker topological

constraints on the PCGs can also be obtained by replacing the cuts (and

therefore the connected sheets) by an effective combination of sources and

sinks with strength related to the phase leak into the cut. If there are

no oscillations in the phase along the cut, then obviously a single source

or sink would suffice for the whole cut.' In the presence of oscillations

the cut may act as a source or a sink locally, but,if the scale of such

oscillations is small compared with the distance between the zeros and poles,

we may take semilocal averages and reduce it to a single vertex of an effective

strength.

B. Periodicity and infinite graphs

Periodic pole and zero structures in amplitudes induce periodicity

in the PCGs with simplifications resulting in interesting predictions. Zero

width dual models provide a non-trivial test case* Howevert the linearity of

the graph destroys all faces and removes the possibility of obtaining some

knowledge of the non-degenerate critical points, which undoubtedly exist in

the physical amplitude and would appear in properly unitarized models,

though at the price of breaking exact linearity and periodicity.

Let us take a function F ,vith poles along the real axis at

x. = xQ + ia [i = 0, 1, 2„ '••» °° ] , and zeros also along the real axis

and with the period b , x| = x' + ib [i = 0, 1, 2, • • • , 0 0 ] . If a = b ,

we shall have an infinite linear graph (Fig. k ) with alternate poles and

zeros, except for a possible sequence of only poles or zeros at the finite

end. At sufficiently high z values we can expect the end effects to be

minimal and the weights of the arcs connecting adjacent poles and zeros to

settle down to constant values depending on the separation between adjacent

poles and zeros. That there are no arcs going to the point at infinity

from the vertices far from the finite end can be visualized as follows. If

we take a large circle with its whole circumference far from the finite end

and with equal numbers of poles and zeros inside, then because of the

constancy of the arc weights (̂ and w 2 (Fig k ) the net phase leaking

to adjacent vertices will be zero. Consequently, because of the equality

of poles and zeros within the circle, the amount, of phase going to infinity

will also be zero. Hence we must have u. + (Op = 2TT . The power behaviour

at infinity is essentially determined by the uncompensated zeros or poles at

the finite end as well as some phase leakage to infinity from all vertices

near the origin. If we sum the total weight a from all the poles and

zeros, except the point at infinity, with aQ equal to the number of un-

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44

paired zeros or poles

a = aQ + 1 - 1 + 1 - 1 + ••• (U.2)

The sum of the series is indeterminate but bounded between ou + 1 and ou

If we use the factorized form for the function:

= C

\a a

with C1 = Ca , k being the integral part of (xo-XQ)/a . This has an

asymptotic power "behaviour off the real axis

F(a) "* o o>(~) *°~x° a , (k.k)

where the index is actually the number of unmatched zeros at the finite end

together with a fraction indicating the leakage of phase bundles from the

nearby matched pairs as an end effect. That the analytic method is more

powerful and gives the exact index, compared with the partial indeterminacy

of the graphical method, can be expected from the fact that the latter does

not use the separation between the zeros and the poles, but only the number

of the excess. This uncertainty could possibly be remedied if the weights

could be be calculated without, of course, using full analyticity properties.

However, as we have already seen, the exact weights of arcs cannot be

determined by only graphical and topological means, because different

functions may have the same graphs but different,or even the same^weights.

The constraints from periodic symmetry would work only for vertices away

from the finite end.

C. Unitarity and asymptotic behaviour

We have just seen how even a purely meromorphic function could have

a non-integral asymptotic power behaviour due to an infinite number of poles

and zeros. An algebraic function also, in general, must have non-integral

power behaviour, because the source or sink at the point at infinity must be

equal and opposite in sign to the algebraic sum of the strengths of all the

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sources and sinks in the finite plane, on each sheet as the sheets are

identical.

For a scattering amplitude the nature of the branch points are

generally unknown. From the unitarity equation it can be shown that

the elastic branch point is of the square root type. The other branch

points are expected tc be infinite-sheeted, with no symmetry to make the

PCDs on different sheets identical> as in the case of the purely algebraic

function.

However, we can Btill think of the total phase flux moving out of

the physical sheet into the branch points or into the singularities of the

unphysical sheets to be absorbed by an effective sink. For an amplitude

without any Born poles and with identical left- and right-hand cuts, e.g.

the symmetric pion-pion amplitude A(ir t •+ IT ir ) , we get a very simple

FCGr (Fig. 5 ) with two equal effective sources representing the two cuts,

and one at infinity that balances them and gives the asymptotic behaviour.

For UDL. > t £ 0 , the imaginary part of the amplitude must remain positive

throughout the cuts. Hence phase oan vary, at mostj from 0 to .ir over

each side of a cut, giving each cut a maximum possible strength of a = +1 .

So the sink at infinity has,at most,strength a = -2 . Or,

A(s,t) Z s"2 , (k.5)

as |s| -*0O •' O S t < UMJ . This result is identical to Jin. and Martin fs 1 '

classic lower bound on the scattering amplitude. However, we have ignored

the possibility of logarithmic terms, either at infinity or over the cuts.

It is only to be expected that such a simple qualitative approach, which does19)not employ dispersion relations or the properties of Herglotz functions ,

cannot be as powerful as Jin and Martin's more detailed analysis. Never-

theless, a graphical picture does, give us a more natural insight into the

result which can be hidden in the details of analytical calculations.

V. CONCLUSIONS

We have tried in this work to indicate the topological aspects of the

phase contour method for the analysis of scattering amplitudes, with some

help from the results of Morse theory. Although the difference in the

topological nature of ordinary contours and critical value contours is well

known from Morse theory, we have been able to show also that this difference

in the homotopy types in the simple case of a one-dimensional complex

manifold leads to the different components of a unique graph representing all

topologically equivalent phase contour diagrams. We believe the phase contour

graph is easier to handle mathematically and should provide as much qualitative,

particularly combinatorial, information regarding links between zeros and

singularities of the amplitude as the full PCD. Observing that such a graph

must in general be a maximally connected bichromatic multigraph, ve found

numerical relations between the number of poles and zeros and the number of

critical points of an algebraic function on any sheet. We also saw that

periodicity can give us simple predictions about asymptotic behaviour, and,in

the case of the physical scattering amplitude;unitarity can constrain the

strengths of the vertices representing the cuts and hence the asymptotic

power behaviour - a result previously obtained by Jin and Martin using

dispersion relations. We have observed how the number of components of the

graph can indicate the presence of symmetries of the function in the complex

plane.

Although the use of PCGs is intuitive and heuristic and the results

obtainable from it are only qualitative, this method probably concerns the

more basic aspects of the scattering function and avoids the manifold problems

of detailed quantitative analysis. However, the full power of Morse

theory and the topological aspects of PCDs can probably be felt in the

multidimensional case, e.g. for production amplitudes with zeros and poles in

many complex variables, with higher dimensional simplicial complexes

generalizing the concept of phase contour graphs.

ACKNOWLEDGMENTS

The author is indebted to Professor Abdus Salam as well as

the International Atomic Energy Agency and UNESCO for the kind

hospitality at the International Centre for Theoretical Physics, Trieste.

He would also like to thank Professor A.O. Barut for reading the manuscript.

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REFERENCES

1) L.E. Elsgolc, Qualitative Methods iri Mathematical Analysis

(American Mathematical Society, Providence, R.I.,196*0.

2) C.B. Chiu, R.J. Eden and C-I. Tan, Fhys. Rev. Letters 20_, ko6 (1968).

3) C.B. Chiu, R.J. Eden and C-I. Tan, Phys. Rev. 170., 1U90 (1968).

h) R.J. Eden and C-I. Tan, Phys. Rev. 1JO_, 15l6 (1968).

5) R.J. Eden and C-I. Tan, Phys. Rev. 1J_2, 1583 (1968).

6) C.B. Chiu, R.J. Eden, M.B. Green and F. Guerin, Phys. Rev. 182.,

1669 (1969).

7) C.B. Chiu, R.J. Eden and M.B. Green, Phys. Rev. _l85, 1731* (1969).

8) G.D.. Kaiser, Phys. Rev. ljJJ, 2372 (1969).

9) G.D. Kaiser, Phye. Rev. l8_3_, 1U99 (1969).

10) S. Jorna and J.A. McClure, Nucl. Phys. B13_, 68 (1969).

11) A. Shafee, Nucl. Phys. B3J., 556 (1971).

12) J. Milnor, Morse Theory (Princeton Univ. Press, Princeton, N.JV1969)-

13) M. Morse, Topologioal Methods in the Theory of Functions of a Complex

Variable (Princeton Univ. Press, Princeton, N.J.,191*7)-

lit) M. Morse and S.S. Cairns, Critical Point Theory in Global Analysis

and Differential Topology (Academic PreBS, New York I969).

15) 0. Ore, Four-Colour Problem (Academic Press, New York 1967).

16) R.J. Eden, P.V. Landshoff, D. Olive and J.C, Polkinghome, The

Analytic S-Matrix (Cambridge Univ. Press, Cambridge, UK,1966).

17) M. Sugawara and A. Tubis, Phys. Rev. 130, 2127 (1963).

18) Y.S. Jin and A. Martin, Phys. Rev. 135B, 1369 and 1375 (196M,

19) J.A. Shohat and J.D. Tamarkin, The Problem of Moments

(American Mathematical Society, Providence, R.I.;1963).

H.g.1 A maxinm.l'bi chromatic mult igraph with

of the graph.

vertex V1 would result in a new are C* . A is the Inner part

Fig.2 A maximal "bichromatic multigraph with four vertices.r r • •

- 20 -

1

sto oo tOoo

Fig.3 Reflection symmetry: the subgraphs A and B are reflection-

symmetric vith respect to the line S .

to oo

Q), Q),

to oo

Periodic linear graph with alternate poles and zeros.

ootHCO -* ORHC

Fig. 5 PCG with two cuts and point at infinity.

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