Modified Mandelstam Representation for Heavy ParticlesJamal N. Islam Citation: Journal of Mathematical Physics 3, 1098 (1962); doi: 10.1063/1.1703852 View online: http://dx.doi.org/10.1063/1.1703852 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/3/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Mandelstam Representation in Potential Scattering J. Math. Phys. 11, 2013 (1970); 10.1063/1.1665360 Remarks on the Polynomial Boundedness in the Mandelstam Representation J. Math. Phys. 5, 1406 (1964); 10.1063/1.1704076 Acnodes and Cusps and the Mandelstam Representation J. Math. Phys. 4, 872 (1963); 10.1063/1.1704012 Mandelstam Representation for Potential Scattering J. Math. Phys. 1, 41 (1960); 10.1063/1.1703635 Fast Detector of Heavy Particles Rev. Sci. Instrum. 27, 1049 (1956); 10.1063/1.1715451

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1098

where

and

I. BIALYNICKI-BIRULA

equations for alA

L = Z{~I'I A, 7J = 0,71 = OJ (A3) o'(o.a..(z) - ojJa.(z»

+ (0; - ill' o')~,(z) - ie Tr 'YI'G(z, z) = 0,

(A4) al'al'(z) + A(z) = o. (AS)

(A9)

Equation (17) differentiated with respect to 11 gives It is convenient to introduce also the expectation the equation for G in the form value of A in analogy to the expectation value of [ -i iJ + m + e-yllall(x)]G(x, y) All' It will be denoted by K and defined as

(~ L-11 oL K,Z, = i OA(z) . (A5)

All functionals of :ijl and A can be expressed in terms of Aji and K. The derivatives with respect to ~jI and A are related to those with respect to ajl and K

through the formulas

(A6)

= o(x - y) - e-yl' ~ _0 - G(x y). (AI0) '/. O~II(X) ,

To obtain from (AS) to (AlO) the Eqs. (30), (31), and (33), we have to differentiate (AS) and (A9) with respect to ~" make use of the formulas

r,,(x,y,z) = ~ oa~(z) G-1(x,y),

o o/C(z) G(x, y) = 0,

(AU)

o ° OA(z) = -o~ oax(z)' (A7) and finally put all = 0 = /c. The generalized Ward identity (32) directly follows from Eq. (22), differ­

Equations (16), (22), and (19) lead to the following entiated with respect to 7J and 71, and from (A7).

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 3, NUMBER 6 NOVEMBER-DECEMBER 1962

Modified Mandelstam Representation for Heavy Particles

JAMAL N. ISLAM Department of Applied Mathematic8 and Theoretical Physics, University of Cambridge,

Cambridge, England (Received May 18, 1962)

The fourth-order Feynman amplitude ceases to satisfy the Mandelstam representation WRen the external masses are sufficiently large. A representation which replaces the Mandelstam one is found in the cases where the four mass invariants are equal in pairs. The physical interpretation is briefly dis­cussed.

1. INTRODUCTION

WE shall be concerned with the partial Feynman amplitude associated with the fourth-order

diagram shown in Fig. 1. It is known1•2 that this

amplitude ceases to satisfy the Mandelstam repre­sentation for a certain set of values of the external masses. If we set

1 S. Mandelstam, Phys. Rev. 115, 1741 (1959). J J. Tarski, J. Math. Phys. 1, 149 (1960).

with P13 = P12 + P23 and P24 = P23 + P34, the stability conditions on the external and internal masses are given by

-1 < Yl2, Y23, Y34, Y41 < + 1

so that for these V's we may put Yii = cos 8H •

o < 8H < 7r. The condition for the validity of the Mandelstam representation is then given by

012 + 023 + 034 + 841 ~ 27r. When the above inequality ceases to be valid, the

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MANDELSTAM REPRESENTATION FOR HEAVY PARTICLES 1099

amplitude A(YI3, Y24) has singularities for complex values of Y13, Y24 in the physical sheets for these variables and this precludes the possibility of a representation such as Mandelstam's. Our objective is to find a representation for A(YI3, Y2.) when it has complex singularities, in the special case when the four invariants Yl2, Y23, Y3., Y.I are set equal in pairs. We find this representation with the aid of the Bergman-W eil formula.

2. THE BERGMAN-WElL FORMULA'

This formula is a generalization of Cauchy's formula and gives the values of a function of several complex variables in its domain of analyticity in terms of its values on a certain subset of the bound­ary of this domain, when the boundary consists of pieces of analytic hypersurfaces. An analytic hypersurface is a surface of the form F(zk, r) = 0, where F is analytic in the complex variables Zk and r is a real parameter varying over a certain interval. For the case of two complex variables, which is the relevant one here, we have, explicitly

A(zl, Z2) = II dtl dt2A(tl, t2) Iq(zl, Z2, tl, t2)1·

Here the integration is over the distinguished bound­ary of the domain of analyticity, which is defined as the two dimensional intersections of the analytic hypersurfaces that form the boundary of this domain taken two at a time. The q factor in the integrand is a 2 X 2 determinant of certain functions qkl defined as follows: We associate two functions qil and qi2 functions of ZI, Z2, tl, and t2 with the kth analytic hypersurface such that these functions are analytic for ZI, Z2 lying inside the domain of ana­lyticity and for til t2 lying in the kth analytic hypersurface which forms a Dart of the boundary

FIG. 1. Fourth­order Feynman diagram for scat­tering process.

a See, for example, A. S. Wightman, Lea Houches Lecture Notes: Dispersion Relations and Elementary Particles, edited by C. R. De Witt and C. R. Omnes (John Wiley &: Sons, Inc., New York, 1959). Also G. Kall~n and J. Toll, Helv. Phys. Acta 33, 753 (1960).

':2' + I r I I I

1

_________ ~~~~----~r_----------+I

-I

FIG. 2. The real plane when the mass invariants are equal.

and such that the following identity is satisfied

(tl - ZI)qkl + (t2 - Z2)qk2 = l. The q factor for the intersection of the kth and lth analytic hypersurfaces is then the 2 X 2 determinant of the q's associated with these two surfaces. Lastly, A(tl, t2) is the value of the function as th t2 ap­proach the distinguished boundary in some suitable manner.

3. SINGULARITIES OF AWu, yu)

The analytic properties of the amplitude associated with the above diagram have been worked out in detail by Tarski2 for general values of the internal and external masses.

From continuity one can readily deduce these properties for degenerate cases. We shall first con­sider the case in which the four mass invariants are equal. We set

812 = 823 = Oa. = On = </I

and choose the masses such that cp > 11"/2, so that the amplitude has complex singularities. The ampli­tude in this case has anomalous threshold branch points at Y24 = cos 2cp, Y13 = cos 2cp in addition to the normal threshold ones at Y24 = -1, Yl3 = -l. It is also singular on the surface

(Zl + I)(z2 + 1) = 4 cos2 </I.

(We set Y24. = Zil YI3 = Z2)' This is essentially the complex surface joining the two branches of the hyperbola (Fig. 2).

(Xl + I)(x2 + 1) = 4 cos2 </I.

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1100 JAMAL N. ISLAM

Zo I

y,

FIG. 3. The Zl plane for a fixed complex value of Z2.

Thus A (z.. Z2) has singularities in the physical sheets for Zl and Z2 which in this case are defined by

-11" < arg (Zl - cos 24» < 11"

-11" < arg (Z2 - cos 24» < 11".

We can now enumerate the analytic hypersurfaces that form the boundary of the domain of ana­lyticity of A(ZI' Z2)' These are just the Zl cut, the Z2 cut, and the cut

..:l(ZI, Z2) == (Zl + 1)(z2 + 1) - 4 cos2 tjJ = p, p ~ O.

(We call this the ..:l cut.) It is necessary to introduce this cut as the singularity associated with the sur­face . ..:l = 0 is known to be of the square root type. This cut can of course be introduced arbitrarily but as we shall see the above choice is a convenient one. We note that since this cut introduces a two­sheeted surface in the Zl plane for a particular value of Z2, the definition of the physical sheet is ambig­uous, but the other singularities in the physical sheet are unambiguously determined. We merely choose a particular one of these sheets.

4. THE BERGMAN-WElL INTEGRAL FOR A(ZI, Z2).

We first determine the q's for the three cuts. For the Zl and Z2 cuts we take the sets [1/(!1 - Zl), 0] and [0, 1/(!2 - Z2)], respectively, as in Cauchy's formula.

For the ..:l cut we see that ..:l(z .. Z2) ¢ p, p ~ 0 as long as z .. Z2 lie inside the domain of analyticity. We may write

1 == {..:l(ZI, Z2) - p} - {..:l(!l, !2) - p} ..:l(ZI, Z2) - p

_ ..:l(Zl! Z2) - ..:l(!I, !2) = ..:l(ZI, Z2) - P

since ..:l(! .. !2) = p for !I, !2 lying in the ..:l cut. We now choose Q .. Q2 such that

(!l - ZI)QI + (!2 - Z2)Q2 == ..:l(ZI' Z2) - ..:l(rl, !2)'

Then the q's for the ..:l cut are given by

qt.l = Ql/[..:l(ZI, Z2) - p], qt.2 = Q2/[..:l(Zl, Z2) - p],

since then

(rl - Zl)qt.l + (!2 - Z2)qt.2 = 1.

We find

QI = -l(r2 + Z2 + z) Q2 = -!(!l + Zl + 2).

The p may be replaced by ..:l(! .. !2)' We thus have finally

2(ZlZ2 + Zl + Z2 - rl!2 - !I - !2) ,

(!l + Zl + 2)

We may remark that for a general analytic hypersurface the q's are not unique. For the ..:l cut the above q's seem to be the simplest ones.

Next we determine the distinguished boundary of our domain. It consists of the intersections of the three cuts taken two at a time. There are thus three contributions to the Bergman-W eil integral, coming from Zl cut (\ Z2 cut, Zl cut (\ ..:l cut, and Z2 cut (\ ..:l cut, respectively. The first of these con­tributions corresponds to the Mandelstam represen­tation. The second intersection one would expect to be the region in the real plane between the branches of the hyperbola and left of the line Xl = cos 24>. But this region has a two-dimensional intersection with the Z2 cut, viz. the portion bounded by the lines Xl = cos 24>, X2 = cos 2tjJ and the lower branch of the hyperbola. We thus get a two-di­mensional region common to the three cuts whereas in general three analytic hypersurfaces (in two complex variables) should have a one-dimensional intersection. This discrepancy is got over by opening out the Zl and Z2 cuts slightly.

Y,

.,

8'

(a)

----------. •• ..--"'+ Cos2i

Xa

(b)

FIG. 4. Plot of the t. cut on the Zl plane as Z2 varies along the Z2 cut.

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MANDELSTAM REPRESENTATION FOR HEAVY PARTICLES 1101

We consider in detail the intersection between the Z2 cut and the Ll cut. This is done conveniently by plotting the intersection in the ZI plane. We note first that for a particular value of Z2, say z~, the ZI plane is as shown in Fig. 3. The Ll cut starts from z~ given by

(z~ + l)(z~ + 1) = 4 cos2 cp

and goes through the ZI cut at ZI = -1 so that only the portion L in the physical sheet is relevant. We now plot Z2 cut (\ Ll cut in the ZI plane. As Z2

varies from - ex) to -1 - 0 Co small > 0) along Y2 = E, CE small> 0), i.e., from ~ to B (Fig. ~) the relevant portion of the intersection plotted III the z plane varies from A'X to B'X. The intersection i~ the surface swept out by the portion A'X going to B'X and gives merely the region Do of the real XIX2 plane as E -+ 0, approaching the ZI rea.! .axis from below. As Z2 goes round the little semIcIrcle from B to C (we take the semicircle for convenience), the intersection on the ZI plane swerves round from B'X to C'X, and as 22 goes from C to D we get the surface swept out by C/X going to D'X, which in the limit E -+ 0 is just the region D2 of the real plane, approaching the 21 real axis from below. For Z2 going round the branch point back to -: ex)

slightly below the cut, we get the same reglons except that these approach the real ZI axis from above. This gives the complete intersection between the 22 cut and the Ll cut. In an analogous manner we obtain the third intersection. We can now write down the various contributions to the Bergman­Wei! integral. Taking account of the way t 1 and t 2

approach the various parts of the distinguished boundary, one gets the following representation for A(zlJ Z2):

fro o 2. dtl dS2P(SIS2) A(ZI, Z2) = l1-a> (SI - ZI)(t2 - 22)

where

+ ff {0'1(StS2)(SI + ZI + 2) JJ D. 2F(z, S)(SI - ZI)

+ 0'2(tIS2)CS2 + Z2 + 2)} dtl dt2 2F(z, t)(t2 - 22)

+ l"r U(Slt2)(SI + ZI + 2) dS1 dS2 l D• 2F(2, t)(SI - ZI)

+ rr U2(SIS2)(S2 + Z2 + 2) dS1

dS2 , lJn , 2F(2, SHS2 - Z2)

P(tIS2) = lim {A(s~, s~) - A(s~, r;) - ACr~, s~) + A(s~, s"2)},

-I plane for 11 ==

f

YI

FIG. 5. The %2

- -----::::::=-+---t----..,!!o. flO + iE, where ~ I ?Xa flO < - 1 and E

---- small > O. y' r.

UI(rs) = lim {A(s~, s;-) - A(s~, s;+)

- A(s~, s;-) + A(s~, s;+)},

and F(z, S) = (Z122 + ZI + Z2 - S1S2 - SI - S2)' The limit IA(s~, s;-) - A(s~, s;+)} represents the discontinuity of A(sl, S2) across the Ll cut, the first superscripts implying that SI is to approach the SI cut from above, in which case the Ll cut in the S2 plane lies below the S2 cut. The other limits are defined similarly. U2 is obtained from 0'1 by interchanging SI and S2' p(tl! S2) corresponds to the Mandelstam spectral function. The first inte­gral may, of course, be over a sma1ler region.

S. DISPERSION RELATION FOR A(Zlo z.)

For a particular value of 22 = z~, we obtain the following Cauchy representation for A(zl, Z2) (see Fig. 3):

A( 0) = {jOOI2. + f } ACs:, z~) - ACr~, z~) ds . ZI, Z2J -"" JL CSI _ 2

1) 1

Thus for a particular value of 22 , we should be able to deduce the above relation from the Berman­Weil representation for A (211 Z2)' We proceed to obtain this relation.

Let SI = s~ = r~ + iE where - co < s~ < -1. The A cut in the 22 plane then lies just below the Z2 cut as shown in Fig. 5. starting from t~ given by

(s: + 1)(t~ + 1) = 4 cos2 cp.

From Cauchy we get, using the same notation for limits as above:

ACl-+ 0) _ f oo02• {A(s~! s~) - A(s~, s;) I ds ~ , Z - _a> (S2 _ z~) 2

f f.- A(r~! r;-) - A(r+ I r-+) + -I (S2 - z~) dS2 '

We get similar relations for -1 < S~ < cos 2¢ and for s~ = s~ - iE. We now write the first term of the Bergman-Weil representation in the form

f_0",,012. drl [fOOl 2. d!;2 A(s~! S~) - ~(S~, S;)

erl - ZI) -a> (S2 - Z2)

_10002• A(s~! S~) - ~(s~, s;) dS2

] -"" (S2 - Z2)

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1102 JAMAL N. ISLAM

Rec.i

FIG. 6. The real plane of the trans-

Re '1. formed variables w ---"""'-0""'". ",",",,"':ki i~oo:r-+, ~-, *, ...,Ol ....... ~-+ and r,.

and substitute for the two terms in the square bracket from the above relations. We get finally for the first term of the Bergman-Weil integral the expression

~ 1, Z2 ~ 1, Z2 ds 1 f OOO 2<1> AC"+ 0) _ AC"- 0)

-CD C\I - ZI)

+ {lrr + for} UIa'I, rz) dSI drg . J D. J D. (ri - Zl)(r2 - Z2)

Substituting this into the Bergman-Weil integral we get, since

1 + (S 1 + ZI + 2) t\l - Zl)(S2 - zz) 2F(z, mSl - Zl)

(S2 + Z2 + 2) 2F(z, mS2 - Z2) ,

A(ZlZ~) = JC002

<1> d51 A(5~,zg) - A(5~,zg) -'" (51 - ZI)

+ lor I U2(51, 52) - CTI(SI, 52) H52 + zg + 2) J D. 2F(z, m52 - z~)

_ for d51 d52 CTI(SI, S2)(S2 + zg t 2) JD• 2F(z, mS2 - Z2)

+ rr d51 dS2

CT2(SI, 52)(52 + Z~ t 2). JiD• 2F(z, r)(r2 - Z2)

For the last three terms, we change the integration variables (SII s,) to (w, S2) where

W = (SI + 1)(S2 + 1) - 4 col cpo

A( 0) = JC082

</> dr A(r~, z~) - A(r~, zg) Zl, Z2 ~ 1 (r _ )

-CD ~l ~

The regions of integration in the real (w, S2) plane are as shown in Fig. 6. Then, noting that the Jacobian ofthe transformation is given by 1/ n-2 + 1) we get for the last three integrals, with F(z, r) w' - w, w' = (Zl + 1)(z2 + 1) - 4 cos2 cp,

{fL. {CTz(rt(W), 52) - UI(SI(W), r2)}

- fL. CTl(Mw), S2) + fL. CT2(~\(W), S2)}

X (S2 + zg + 2) dw dr2 2(w' - w)(52 + 1)(52 - z~)

Jo dw [{J- l

= ( I _ ) (ulrl(w), r2) -400811tIJ W W -co

- UICtlCW), t2» + f'" (-Ul(tl(W), t2» ",/(2 COli' t/J) +1

JCOO 2<1> } (5 + ° + 2) ] + -1 (12(51(W), 52) 2(51

2+ 1;<S2 _ Z~) d52

Now the expressions (1(tl(W) , 52) - O{51 (w), 52), etc., contain terms in a symmetric way for which the limits w + and w - are to be taken. For a fixed value of w + = WO + ie, -4 cos2 cp < WO < 0, the singularities of A(Sl(W), r2) in the r2 plane are as shown in Fig. 7 where the two extra cuts are in fact parts of the r I cut for this fixed value of w. Thus the expression in the square brackets above is the Cauchy integral for A(SI(W), 52) in the S2 variable for a fixed value of W with of course the kernel (52 + z~ + 2)/(\2 + 1). The integration over the 52 cut is given by the first part of the first inte­gral and the third integral in the square brackets above, whereas the integration over Sand T arise from the second part of the first integral and the second integral respectively. We may thus carry out the integration in the square brackets-for this we merely replace 52 by z~. We note that the kernel G'2+z~+2)/2(r2 + 1) cancels out. We obtain finally

, A(W + + 4 cos2

cp _ 1 0) _ A(w -+ 4 C082

cp _ 1 0) + JO dw Z~ + 1 ' Z2 / Z~ + 1 ' ,Z2

-4C08' q, W - W

We put (w + 4 Cos2 cp)/(z~ + 1) - 1 = rl' Then dw/(w - w') = d51/(rl - ZI) and the limits are -1 and 4 cos2 cp/(z~ + 1) - 1 = z~, Thus

A( 0) _ f co•

2</> d" A(r~, z~) - A(s~, z~)

ZI, Z2 - _CD H (rl - Zl)

+ J". A(5~, z~) - A(5~, zg) drl -I (rl - Zl)

This expression is the same as the one we obtained previously for A(zl, z~).

It seems a bit surprising at first that the Bergman­Weil integral for A(Zl, zz) should contain integrations over the real plane only. This becomes somewhat clear if one considers analytic continuation in the external masses. to Let us consider the dispersion

• See S. Mandelstam, Phys. Rev. Letters 4, 84 (1960).

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MAN DEL S TAM REP RES E N TAT ION FOR HE A V Y PAR TIC L E S 1103

relation for A(z17 Z2) for Z2 = z~ (complex) in the situation where there are no complex singularities. It is given by

A( 0) - I-I A,(r" zg) d" Z" Z2 - ,. ~ ,. -a> ~,- Zl

We continue this relation analytically in y, where Y = Yl2 = Y23 = Yu = Y4" For this we include a small imaginary part in y. When Y > 0, the singu­larities of A(Z" zg) are in the unphysical sheet and as Y varies from a value k, 0 < k < 1 to - k, the anomalous threshold and leading singularities move into the physical sheet as shown in Fig. 8, deforming the contour of the above integral with it so that finally we get the dispersion relation we obtained previously. In an analogous manner one can con­tinue the Mandelstam representation, deforming the two dimensional hypercontour as the complex singu­larities appear in the physical sheets for the two variables. In the first part of the Mandelstam representation, viz.,

If ds ds A(tlt2, y) _a> ' 2 Cr, - ZI)(r2 - Z2) ,

we see that it involves the values of A near the real SIS2 plane so that the corresponding leading singularities are on the unphysical sheet near the real plane. As y goes from k to - k, with a small imaginary part, these singularities" graze" along the real plane, go round -1 and back to their original position, deforming the hypercontour in the process, so that the deformed part lies also near the real plane. A detailed consideration shows that we get precisely the regions Do, D, and D2 for the deformed part. It is not straightforward, however, to get the exact representation by this process.

6. UNEQUAL MASS INVARIANTS

We now consider the cases in which the mass invariants are equal in pairs. The case 812 = 834 = q, and 823 = 841 = 1/1 is entirely analogous to the previous one, since here the Landau curve reduces to the surfaces

r T if*e! -:-=--~

b hi ~+I

2Cos2(1

FIG. 7. The z. plane for a fixed value of w.

and

FIG. 8. Path of the singularities in the Zl plane as the mass invariants in­crease.

(ZI + 1)(Z2 + 1) = (cos q, + cos 1/1)\

(z, - 1)(Z2 - 1) = (cos q, - cos 1/1)2,

of which only the former is singular when ¢ + 1/1 > 7r. We consider next the case 812 = 823 = q, and 834 = 841 = 1/1. We describe briefly how in this case the singUlarities arise in the situation we are interested in. This is contained implicitly in Tarski's paper.

Let q" 1/1 < 7r/2 and 1/1 > q,. There are then no anomalous thresholds or complex singularities in the physical sheet. The Landau curve is as shown in Fig. 9, being given by the line X 2 = 1 and the curve

(Z2 + 1)(z~ - 1) - 4z, cos q, cos 1/1

+ 2 cos2 q, + 2 cos2 1/1 = O.

Only the branch r, is singular in the limit (x, ± ie, k2 ± iE), so that the attached complex surface is not singular. The tangents to r 2 and ra are cos 2q, and cos 21/1, respectively. We now increase q, and 1/1 such that 1/1 > q,. As 1/1 exceeds 7r /2, the line Z2 = cos 21/1 goes to -1 and recedes, thereby becoming singular and becoming tangent to r, instead of r 3 •

Next we have q, + 1/1 = 7r. Here the lines Z2 = cos 2q"

Z2 = cos 21/1, and Zl = -1, Zl = cos (q, + 1/1) coincide, while r, and r 2 together form a branch of a hyper­bola and the line Zl = -1. Beyond this point the line Zl = cos (q, + 1/1) becomes singular, the tangents to r, and r 2 are exchanged and we have complex singularities in the physical sheet-being essentially the complex surface joining the portion of r 1 with negative slope to the similar portion of r 2 • As q, crosses 7r /2 we have again the situation depicted in the figure. Here in addition Z2 = cos 2q, is singular and so is the complex surface joining r 1 to the portion of ra with negative slope. We now find a representation for A(Zh Z2) in this situation-we can compare it readily with the previous case by setting q, = 1/1.

H we solve for z, the equation to the Landau curve we get

2 cos q, cos 1/1 ± [(Z2 - cos 2q,)(Z2 - cos 21/1)]112 Zl = (Z2 + 1) •

For any Z2 only one of these values of z, gives a

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1104 JAMAL N. ISLAM

+1

-------f~~--~----~~--------+t

Cos 2'1' _ __ _

FIG. 9. The Landau curve when 812 = 823, 834 = 8.1•

singularity. It can be verified that the sign which gives - (Z2 - cos 2¢) in the limit ¢ ---+ I/; is the one yielding a singularity. Thus the relevant portion of the surface may be taken as

ZI(Z2 + 1) + {(Z2 - cos U)(Z2 - cos 21/;) )1/2

- 2 coscp cos I/; == A'(zl, Z2) = o. We note that in the above expression for ZI if we set cp = I/; we get the singular surface as

ZI = 4 cos2 ¢/(Z2 + 1) - 1,

and the nonsingular one as Zl = 1. This was so in the case considered first.

The q's may then be taken as

q"'i = Q;/[A'(zl,z2) - A'(tl, t2)], i = 1,2.

We note that these q's reduce to those of the first case when cp = 1/;.

Next we determine the distinguished boundary. We note first that the part of the real plane for which A'(zl, Z2) is negative consists of the region between r l and parts of r 2 and ra with negative slope, excluding the strip Z2 = cos 2cp, cos 21/;, where A/(Zl' Z2) is imaginary. We thus expect the con­tribution to the distinguished boundary from the A' cut to be contained in this region, with perhaps additional complex surfaces. We determine first the intersection between the portion of the Z2 cut below Z2 == cos 2cp and the Af cut. For Z2 = x~ + iE, X~ < cos 2cp, the Af cut lies slightly below or above the real Zl axis according as E > 0 or < O. This is to be expected as 1m Zl and 1m Z2 must have opposite signs for a singularity. As before only the portion of the Af cut in the physical sheet is relevant. It can be verified that for any Z2 = x~ + iE, X~ < cos 21/;, the A' cut passes through the Zl cut, though not through Zl = -1 as in the previous case. The exact form of this part of the distinguished boundary is obtained by determining the point, for any given x~, at which the Af cut crosses the Zl cut in the limit E ---+ O. This can be done readily. For Z2 = X2 + it, the A' cut is given by

Zl(X2 + iE + 1) + [(X2 + iE - cos U)

X (X2 + iE - cos 21/;)]112 - 2 cos cp cos I/; = p.

Expanding the square root term in E, setting 1m ZI = 0 and subsequently equating the imaginary parts we obtain We now determine the q's for the new surface­

taking the usual sets for the ZI and Z2 cuts. As before we introduce the cut 2xl [(X2 - cos 2cp)(X2 - cos 21/;)]1/2

To determine the q's we have merely to find Ql and Q2 such that

A'(Zl, Z2) - A'Crl, r2) == Crl - Zl)QI + Cr2 - Z2)Q2 •

Here (tit t2) lies in the cut and (ZI! Z2) does not. We find QI = -l(t2 + Z2 + z) and,

Q __ ! { + + 2(t2 + Z2 - cos 2cp - cos 21/;)} 2 - 2 tl ZI N '

where

N = [(Z2 - cos 2cp)(Z2 - cos 21/;)r /2

+ [(t2 - cos U)(t2 - cos 21/;)]1/2•

+ 2X2 - cos U - cos 21/; = O.

Thus this part of the distinguished boundary con­sists of the region in the real plane bounded by a part of the above curve (the dotted curve in Fig. 9), the relevant portion of the Landau curve and the lines Z2 = cos 2cp, -1, i.e., the regions D~ and D: in Fig. 9. We note that for cp = I/; the above curve reduces to XI = -1 so that D~ and D: reduce to Do and Dz of the previous case. The intersection

Cos III Cos'fi

\ \. "'- .­--'

FIG. 10. Plot of the Af cut in the

x I 21 plane for cos2q. ~ 22 ~ cos2",.

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MAN DEL S TAM REP RES E N TAT ION FOR HE A V Y PAR TIC L E S 1105

d

N

between the fl.' cut and the portion cos 2¢ :::; Z2 :::;

cos 21/1 of the Z2 cut does not lie in the real plane as we get complex values of Zl for real Z2 Csee Fig. 10). As Z2 goes from cos 2¢ to cos 21/1 slightly above the cut, Zl goes from cos I/I/cos tP to cos tP/cos 1/1 along the lower semicircle and for Z2 going back below the cut Zl traces the upper semicircle. The fl.' cut is here parallel to the Zl cut in the limit E -+ 0, starting from a ZIon the circle and going towards - co. The distinguished boundary is thus the complex surface swept out by this cut as Z2

goes from cos 2¢ to cos 21/1 and back, i.e., as Zl goes round the circle. We call this surface 2:.

For the intersection between the Zl cut and the fl.' cut we note that for a given Zl the fl.' cut in the Z2 plane passes through Z2 = -1. A detailed con­sideration shows that this part of the distinguished boundary lies in the real plane consisting of the region bounded by r 2 , Zl = -1, cos (tP + 1/1) and Z2 = -1, excluding the rectangular bit Z2 = cos 2¢, cos 21/1 (the region D~ in Fig. 9) and the region bounded by r l, Zl = -1, Z2 = -1, Le., a region similar to Do of the first case. These regions reduce to DI and Do of the previous case when tP = 1/1.

We can now write down the Bergman-Weil inte­gral for A (Zl, Z2):

iIe08 21/- 008 <.+1/-) p'C"" ) = li, )2 dI'l dI'2

A(Zl, Z2) _00 (I'l - ZI)(I'2 - Z2)

+ rr O"CI'l! I'2)qll.'2 dI'1 dI'2

JJD"+D. (I'l - Zl)

+ f'r T'CI'l, I'2)qll.'l dI'1 dI'2.

JDs '+ D" U (I'a - Z2)

The third portion of the last integral is over the complex surface 2:, p', 0", T' are the various dis­continuity functions obtained by considering the manner in which I'l, I' 2 approach the various parts of the distinguished boundary.

A similar consideration to the above one may be applied to the situation where tP < 7r /2 and tP + 1/1 > 7r. The case 812 = 814 and 823 = 834 is analogous to the above one with Zl and Z2 interchanged.

n n

FIG. 11. Diagrams sa.tisfying the modified Mandelstam representation.

7. PHYSICAL INTERPRETATION

In Fig. 11 are shown three of the diagrams which do not satisfy the Mandelstam representation. The first two belong to the first case and the third one belongs to the second case considered above (neglect­ing spins, etc.). One may ask for possible physical interpretation of the additional terms appearing in the representation. The physical significance of anomalous thresholds is well known. In the case of the third-order vertex function, for example, the presence of an anomalous threshold gives an extra term in the dispersion relation for this func­tion which can be interpreted, in the nonrelativistic limit, as "long-range" contributions due to bound structure effects.6 The terms other than the Mandel­starn one appearing in the above representation could thus be interpreted as a manifestation of the compound structure of the particles involved in the scattering. It is not clear if and how this connection can be made precise. It seems to be of some interest to see what the dispersion relation for a fixed physical value of one of the variables looks like. In the case of equal external and equal internal

4COS _-3 )(a +,

/

-I

FIG. 12. The physica.! region in the equa.! mass case.

Ii R. Oehme, Nuovo cimento 13, 778 (1959).

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1106 JAMAL N. ISLAM

masses, the physical regions are as shown in Fig. 12 (the shaded regions) so that for a· fixed value of the momentum transfer in the physical region we have the following dispersion relation:

A(zlt) = f- I AI(tlt) dtl

-a> (tl - ZI)

+ fOOS 2<1> dr l

A2(t l t) fg(1) A 3(t l t)

-I ~ (tl - ZI) + -I dtl (tl - ZI)'

where get) = 4 cos2 rjJ/(t + 1) - 1 so that -1 < get) < cos 2rjJ for t > 1. Here AI! A 2 , and Aa are the relevant discontinuity functions. We thus get two extra contributions, rather similar to the case of the vertex function, except that the latter has no analogue of the third integral. One gets a similar dispersion relation in the case 812 = 823 , 834 = 8411

except that the lower limit of the third integral is also a function of t and lies below - 1. We also write down a dispersion relation for a fixed value of the energy of the crossed channel, which in the notation of Sec. 1 is given by u = (P12 + P34)2. For the second diagram in Fig. 11 this would be the energy of the channel representing deuteron­deuteron scattering. The condition u = constant is equivalent to ZI + Z2 = constant, the physical values of u being given by ZI + Z2 2:: 2. For a fixed value ~ of (ZI + Z2) in this region we have the follow­ing dispersion relation in ZI:

1<+1 fa 1<+1} d r + A~ + Aa + A~ ~ I ,

.- cos 2<1> -I b (tl - ZI)

where a, b are the roots of

x2 - x~ - ~ - 1 + 4 cos2 rjJ = 0

and are in fact the points at which the ~ cut starts in the physical sheet. We have -1 < a, b < + 1. The first, second, and third pairs of terms come from the normal and anomalous thresholds and the ~ cut, respectively, the A's being the relevant dis­continuities. For rjJ -? 7r (i.e. for deuteron-deuteron scattering) and ~ -? 2 (onset of the physical region) we have a -? 1- and b -? 1 + so that the integrals extend over the entire real axis.

Note added in proof. While this paper was being typed we received a paper by Fronsdal, Mahant­happa, and Norton 6 with work very similar to the first five sections above.

ACKNOWLEDGMENTS

I am grateful to Dr. J. C. Taylor for suggesting the problem and for help and encouragement. I would like to thank Cambridge University for a research maintenance grant.

S C. Fronsdal, K. Mahanthappa, and R. Norton, Phys. Rev. 127, 1847 (1962).

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