An integral representation for the four-point function analogous to the Bergman-Weil integral

I 7.A.1

[

Nuclear Phystcs 81 (1966) 557--574; (~0 North-Holland Pubhshmg Co., Amsterdam N o t to be reproduced by photoprmt or microfilm without written permismon from the pubhsher

AN I N T E G R A L

REPRESENTATION

FOR THE FOUR-POINT FUNCTION

ANALOGOUS TO THE BERGMAN-WEIL INTEGRAL PETER ARRHI~N Department of Theorencal Physics Unwerstty of Lund, Lund, Sweden Received 8 November 1965 Abstract: The fact that the analyticlty domain -#4 of the four-point function is bounded by non-

analytic hypersurfaces lmphes that the Bergman-Wefl integral is not apphcable to the fourpoint functmn However, as shown by Svensson 5), there exists a special integral representahon for the n-point functmn which, when apphed to the three-point function, can be interpreted as the Bergman-Well integral We use this integral representatmn to obtain a Bergman-Wedhke integral for the four-point function (without local comrnutatlwty) This integral relates the value of the four-point function at any point in d¢/4to the boundary value of the function on a sJx-dlmenslonal subset of the boundary of -/¢4-

1. Introduction

During the last ten years, the mathematical properties o f the v a c u u m expectation values o f field operators have been studied, starting f r o m three basic requirements: (i) Lorentz invariance, (li) a reasonable e n e r g y - m o m e n t u m spectrum and (nl) local commutativlty (only sometimes used). As these assumptions are believed to be valid for any theory, the results should also be generally valid. (For an expos6 of the general basis and results o f field theory, see the article by KSJ16n in ref. 7)). It has been shown by Hall and Wightman 8), using only assumptions (1) and (ii), that the v a c u u m expectation value of the p r o d u c t of n scalar fields ( 0 [ A l ( x j ) . . . A,(x,)hO) can be u n d e r s t o o d as the b o u n d a r y value o f a certain analytic function depending only o f the Lorentz-mvarlant quantities Z,k

=

--

¢, Ck = (X,-- X, + ~ - -

t~,)(Xk--

Xk + ,

-- iqk),

3, = x , - x,+ 1 - iq,,

t/,2 < 0; n o > 0 [q, in the forward light cone; also written t/l e C+], (we use the metric x z = 2 2 - x 2 ) . For brevity we call this function, the n-point function, F.(Z,k ). The analyticlty d r . of the n-point function is the domain over which Z,k varies when ¢~ takes sible values in (1). For the two-point function there is only one variable z, domain o f analytlClty is the whole complex plane except for a cut along the 557

(i)

(2a) (2b)

domain all posand the positive

558

P. ARRn~N

real axis (the "cut plane"). From the work of Kfill6n-Wlghtman 1), K/illdn 2), M611er 3) and Manoharan 9) the domain ~ ' , is exphcltly known for n = 3, 4 and 5. It is also known that in the case n = 2 local commutativity does not extend the domain of analyticlty, but when n > 3 it does. This extended domain, or rather the "envelope of holomorphy" of the extended domain, is so far known completely only for n = 3 1). However, in the last year Wu 10) has perhaps taken a first step in obtainmg the envelope of holomorphy for the case n > 3. Closely related to the problem of analytic~ty is the problem of deriving representation formulas for the n-point function. As the basic formula we have here the Fourier integral. In the case n = 2 the Cauchy integral formula gives another integral representation. The connection between these two representanons is shown m ref. 5). For the three-point function, we have the Bergman-Weal integral which Kfilldn and Toll 4) have used to obtain integral representatlons of the most general function analync m the above mentioned extended domain following from local commutativity. Also, in the case n = 3, Svensson 5) has shown the connection between the Fourxer transform and the Bergman-Weil integral for the three-point function without myoking local commutatiwty. In fact, he uses a function K°)(z. a) which represents the three-point funcnon m the following way: (See further below) F3(z,~) = j d3a ~po)(_ a)KO)(z, a),

K°)(z, a) = O(a~2-a,, a22)f f

dy,dyz

fi(y~"~-al 1)~(Y2+a2z)6(Y,Y2 +a,2)~(y)

4re6

[(y~ - ~)2(y 2 - ~2)z] 2 ZI, k =

(3)

--~t~k"

(4) ' (4a)

Here q~(3)(-a) is a certain weight function and the vector y is a tlme-hke linear combination of yt and Yz. The integration in formula (3) is to be performed over the interval - o o < a,k < oo. In ref. 5), it is shown that the integral formula (3) can, after solving for the integral K(3)(z, a), be interpreted as the Bergman-Wei1 integral for the three-point function (without local commutativity) by appropriate choice of the weight function q~(3)(_ a). The fact that the boundary of ~ ' 4 does not consist of analytical hypersurfaces makes it impossible to write down an integral representation for the four-point functmn making use of the Bergman-Weil formula. This fact, when combined with the above menttoned result of Svensson, makes us interested in studying the generalization of (3) to the case of the four-point function. That is, we want to see if the fourpoint functmn formula corresponding to (3) can be understood as a Bergman-Weillike integral for the four-point function (without local commutativity). 2. The Four-Point Function 2.1. K N O W N

FACTS

Consider the vacuum expectation value of four scalar field operators Al(x~),

559

FOUR-POINT FUNCTION ANALOGOUS

Aa(x2), Aa(xa) and A4(x4):

(Olal(xl)&(x2)a3(x3)a4(x~)Bo) = FA~"2"3a"(xl-x2, x 2 - x 3 , x3-x4).

(5)

From the general properties of the theory 2) ,, it follows that the function F is the boundary value of a certain analytic function depending on the complex variables Zjk defined by X2 --X3, X3--X4) = B.V. Fg(Z11, z22, z33, z12, zx3, Zza),

FaIA~A3A"(Xl--X2,

(6) (6a)

Z~k = -- (Xj -- Xj + 1 -- iqj)(Xk-- Xk + 1 -- iVlk) = -- ~j ~k.

The function F4(z) is regular when the vectors t/above are all real vectors lying in the forward light cone. K/ill6n has proved that the analyticity domain .A' 4 of this functlon is bounded by the following curves 2) zsj cut: Fjk:

zj~ = r,

r > 0,

2Zjk -- ZJJZ~k + r ,

(7a),

r > O,

j # k,

(7b)

k < O,

z ~ j.

(7c)

r

S,j:

2z,j = kz,, + z!J- ,

k (The curves F~k are relevant when the imaginary parts ofzjj and Zkk have the same sign, and the curves S 4 when they have opposxte signs.) Z = ~a'~.

(7d)

Here, Z i s the 3 x 3 matrix Z = I[z~d[ and ~ is a symmetric 3 x 3 matrix with elements ~jk (J ~ k) real and ~, complex, the imaginary parts all having the same sign. The ! t 3 x 3 matrix a' = [la~k[[ has ajk real and positive f o r j ~ k and ajj! = 0. We can express eqs. (7d) in a slightly different way2). Introduce three linearly independent lightlike vectors v~, v2, and v3 satisfying the equations V1, V k 3V

a~k

O,

a,k

>

0.

(7e)

The surface in (7d) can be rewritten in terms of these vectors as ~l

m

0~I1V1--]-0~t2V2--~-(Zt3V3 ,

i = 1,2,3,

(7f)

the c~ being the same as described above. Eqs. (7d) and (7f) are entirely equivalent. Later, we shall need eqs. (7f) in the so-called Kleltman form 2¢,¢k = A,k(k,k¢ 2 + k k , ¢ 2 ) - - ( 1 - - A , k )

F ¢2¢2 + r , k l

,

(7g)

I_ r~k t

A, k =

!

aa av ~kt aRt = Ak,, ! (e,k a;k + e,t a;t)(~k, a'k, + ekl akl

* See also, e . g , the article by Kall6n m ref. 7).

(7h)

560

P. ARRHEN !

r,k = 2~tk[~ik a,k' + ~,l a,,, +Ctktakl ] = rk, , k, k --

1

-

l(kt

(7i)

(7j)

O~kl O~ll

All the parameters k~k , rzk and A,k are real and fulfil the relations (1-A12-Ala-A23)

(7k)

2 = 4A12A23A13,

(71)

k,k = r k l . l'tl

In all the above formulas (7) i # k # I. The inequalities for the quantities ~tk and a~k mentioned above imply the existence of certain inequalities for the coefficxents A,k, r,k and k~k. These we shall, however, not write down. They can be obtained in 2). 2 2. D E R I V A T I O N OF T H E F O R M U L A F 4 ( z ) = ~d~a~4~(--a)KC4~(z, a) F O R T H E F O U R POINT F U N C T I O N

We here closely follow the lines of ref. 5). For the four-point function, we get

(OlAl(xOAz(xz)A3(x3)A4(x4)[O> = B.V. F4(z11, z22, z33, z12, z13, z23), F4(z) _

1

(6)

(dpldpzdp3e,Pi~l+,p2~2+,p3¢3

× G(p2, p2, p2, Pl P2, Pt P3, Pzp3)O(pl)O(pz)O(p3),

(8)

~/j ~ C + ,

(8a)

~j = xj-A~+ l-itlj,

(8b)

z,~ = - L ~ k -

(8c)

N o w we take explicit note of the fact that the function G(pjPk) vanishes when p2 > 0 and, therefore, introduce further 0-functmns in the following way:

F4(z)

_

1

( A , d - A,

(2~)~j~, ~3~

~tpl~l+tp2~2+tp3~3

xr(p~ , P2, P3, P, P2, P, P3, P2 p 3 ) 0 ( - p2)0(_ p2)O( _ ~

2

p3)O(PI)

× O(p2)O(p3)e(p).

(9)

We have here introduced a new function F(p,pk) to indicate that although G and F have to be equal in the physical domain an p-space defined by: p2 < O, PjPk <

_,/p2p2 and det ]P,Pk] < O, they need not be equal for other values The function F is considered to be an extension of the function G to domain. An extra factor e(p) has been inserted. The vector p is a c o m b m a n o n of the vectors p~, P2 and P3- I f all p2 < 0, p is defined e.g., p = P l .

of the variables. the non-physical time-like, linear as one of the p,,

561

FOUR-POINT FUNCTION ANALOGOUS

We now introduce into (9) the integral representation 1 f dy [y232' e 'py O(-p2)O(p) = 2-~

p2 # 0,

(9a)

where the subscript R (read "retarded") means that a vector it/, q e C+, should be added to y. After the integration has been performed, th,s vector should tend to zero t. We then get F4( 0 = J d6a # ' ) ( - a)K(')(z, a).

(10)

Here, the weight function q~(4)(_ a) = t h ( - al 1, - a22, - a33, - a12, - al 3, - a23) is related to F by the formula

,

Y3, YlY2, YlYa, Y2Y3) = '

dptdpzdp3 e'my'+'~r~+'wy~

×r(p

,

,p3,e,p

,e

p3,p2p3)

(p).

(10a)

To assure the existence of a time-hke vector p we define F to vanish unless det IP,Pk[ < 0. The formula (10a) implies that ~b(-a) = 0 unless det la,kl >= 0, and consequently there exists a time-like vector y. The kernel K(4)(z, a) in eq. (19) ,s given by 3

K('~)(z, a) - O(la'kl) ; d ~ d(2d~"3

~ J

¢~

,_-<,=1

[(~ _ ~,)2(~2_~2)2(~3_~3)232

,

(lOb)

where the subscript R in eq. (10b) is now unnecessary because ~ has an imaginary part according to eq. (Sb). The vector ¢" in e(O is a time-like combination of ¢'t, (2, (3 analogous to the vector y in eq. (10a). 2.3. THE CHOICE OF THE FUNCTION ~bl4~(--a) To interprete the formula (10) as a Bergman-WeiMike integral, we must first prescribe the extension of the functmn F(p,pk). We define F(p,pk) = G(p,Pk) inside the domain p2 < O, p,pk + ~/p2p2k < 0 and det IP,Pkl < O. Outs,de this domain we take F(p,pk) = 0. We thus consider G(p,pk) defined for negative values of allp °, pO, and pO by the equation G(p,p~) = G(p, pk), (11) all pO < 0,

pO

_pO.

We obtain

yl y2, yl y3, y2y3) _

e(y) )12j(dpldp2dp3e,Vlrl+,v2v2+,v3y3F(p2,p2,p2

(z

'

PtPz,PlP3 , P2Pa)e(P)

We also here assume that our theory contains no particle at zero mass (,see ref. 5)).

562

e. Aggrl~N ~(Y)

(2~) '~ -

f

I f dpl dp2dp3e'P~Y'+'v2r2+'P3"30(pl)O(p2)O(P3)G(p'Pk)

alP1 dp2dp3e'P'(-Y')+tPE(-Y2)+'Pa(-Y3)O(pl)O(p2)O(P3)G(p, = e(Y)[F2 ( - Y, Yk) -- FZ (-- y, Yk)], F ~ ( - - Y, Yk) = B.V. F4(Z,k), Z,k = -- (Y i-- iq,)(yk-- iqk),

pk) ]

(12) (13a) (13b)

rh ~ C+,

(13c)

F 2 ( - - y, yk) = B.V. F4(Z,k),

(laa)

Z ,k = -- (-- Y ,-- i G ) ( - - Yg-- iqk),

(14b)

t/, ~ C+.

(14c)

F4(z) = f d6a K(4)(z, a)[F~ ( a ) - F~-(a)].

(15)

Formula (107 now reads

(Here it is assumed that e(y) = 1 m formula (127. 7 Note: The above prescribed extension of the function G(p,pk) to the function F ( p , p k ) can also be performed in a quite analogous way for the cases of the two- and the three-point functions. We thus take (for the three-point function) F(p,pk) = G(p,pk) in the domain p~ < 0 p22 < 0 and p l p 2 - q - \ / p l 2p 2 2 ~ 0 and zero otherwise. In the same manner as above we now obtain for the three-point function F3(z) =

fd3a

K(3)(z, a)~(3)(- a),

(16)

K(3)(z, a) being defined in (47 and

(~(3)(__ a)

= F 3 + ( a ) - F 3 ( a ),

(17)

F3 (--Y, Yk) = B.V. F3(z,,~),

(lSa)

Z,k = -- (Y , -- iG)(Yk -- iqk),

(18b)

q, e C + ,

(18c)

F 3 ( - - y , yk) = B.V. Fa(z,k),

(19a)

Z ,k = -- ( -- y , - itl,)( -- Yk -- itlk ),

(19b)

I/, ~ C+.

(19c)

+

In ref. 5), Svensson could determine the function ~b(a)(-a) dlrectly by comparing the mtegrands of the Bergman-Well antegral with the integral (16), with K(3)(z, a) explicitly solved. This means that he also implicltly determined the funchon F(p,pk)

FOUR-POINT FUNCTION ANALOGOUS

563

with the aid of the formula corresponding to (10a). It is easy to see that our formula (17) equals the explicit expression for qS(3)(-a) given in ref. 5) (see formulae 53a-e in ref. 5). Note that 53b should be multiphed by - 1.) This fact is really a reason for choosing F(p,pk) as above for the four-point function. It is clear that the above performed constructmn of the function F can be carried out also for the n-point function, with n > 4. 2.4 F O R M A L C O N N E C T I O N B E T W E E N T H E W E I G H T F U N C T I O N q~141(--a) A N D T H E FOURIER TRANSFORM G

We can write the formula (10a) by introducing the function

A4(--Y,Yk, b) =-i~(dpldp2dp3 (2re) .J

3 e'pwl+~v2r2+'p3y3 ]-Ii~(p~pk+b,k)g(p ). ,=
(20)

Here, the vectors y and p are the same sort ofhnear combinations as in formula (10a). We then get q~,4)(_ a) = ,

jd6bG(- b)A4(b, a).

(21)

According to the orthonormality relation

f A,(b, a)A4(c, b)d6b

-

1

(2=) 12

6(a-c),

(22)

(this formula can be proved in a way similar to the corresponding formula in ref. 5), cE (A2.10)-(A2.14) in ref. 5)). Eq. (21) can be solved for G and gives G ( - b) = - i(2rr) 'z f d6a~b(4)( -

a)A4(a, b).

(23)

The formula (23) gives the connection between the Fourier transform G(-b) and the weight function qS(4)(-a). It is to be compared to the corresponding formula for the three-point function in ref. 5), (formula (55).) 2.5. I N T E R P R E T A T I O N

OF THE FORMULA

(15) F O R T H E F O U R - P O I N T

FUNCTION

F4(z) ~ f d"aKl4l(z, a)[Fa-(a) -- F4+(a)] The formula (15) for the four-point function is evidently a direct generahzataon of the corresponding formula (16) for the three-point function, i.e., the Bergman-Wed integral. The first question that now arises is whether we can express the integral K(4)(2 ', a ) in terms of elementary functions. It turns out that elliptic integrals appear very soon in such an evaluation (cf. appendix A.) As is shown in appendix B, the singularities of the kernel K(4)(,z , a) are given by the

564

P. ARRHEN

following equations cuts:

2kk ~- akk

Zt t Zkk 2Ztk . . . . . . + r r Fzk : atl Ukk

[ 2a~k = - -

Y

+r

relevant when

akk > O,

(24)

relevant when r > 0; relevant part of the boundary when Im z,, Im Zkk > 0

(25a) (25b)

i#k 1

2Ztk S,k :

k z t t + k Zkk

i2a, k = ka,,+ ~1 akk

relevant when k < 0; relevant part of the boundary when [m z,, Im Zkk < 0

(26a) (26b)

i#k S(--Ztk ,

a,k) = 0

defined implicitly by:

2Z,k = A,k(k,kZ,,+kk, Zkk)+(1--A,k)( z''zkk +r,k), \

2a,k = A,k(k,ka,~+kk, akk)+(1--A,k ) (a"-a-t~k +r,k), \

k~k --

Fkl

(1-A12-A13-A23)

(3 eqs.)

(27a)

(3 eqs.)

(27b)

rtk

rtk

2 =

4A12A13Aza,

(27c)

?'tl rtk = rkt ,

(27d)

Ak, = A,~,

(27e)

z-C k--/: l (plus certain conditions of relevance).

"Z'kk-CUtS" and the curves F,k and S,k are generalizations of the singularities of K(a)(z, a). The new manifold with singularities S(z, a) = 0 has Z = ~a'c~ (7d) as an The

envelope. This is shown m appendlx B (except for conditions of relevance). That singularities off the boundary also appear indicates that the integral (15) for the fourpoint function is more complicated than the corresponding integral (16) for the threepoint function. The domain of integration in (15) is clearly a six-dimenslonal subset of the boundary of J/'4. From (24)-(27), it is clear that the domain of integration for the variables

FOUR-POINT FUNCTION ANALOGOUS

565

a,k consists of points satisfying every equation for the boundary curve in (24)-(27) which is relevant. The integral (15) is an integral representation for the four-point function F4(z) analytic in ~g4. It relates the value of F,,(z) at any point m ~g4 to the boundary value of the function on a six-dimensional subset of the boundary of dr' 4. As it is obvious that these boundary points are physical points, a knowledge of F4(z) in the physical regmn provides a knowledge of F4(z) in the whole domain ~//'4. Thus we can look upon the integral (15) as a Bergman-Weil-like integral for the four-point function in the domain .1/*/4, though not bounded by analytic hypersurfaces. The author would hke to express his gratitude to Professor G. K/illdn for most generous help and advice during th~s investigation, which he suggested. He is also indebted to Professor T. Gustafson for his constant encouragement and kind interest.

Appendix A THE INTEGRATION

O V E R ~1 A N D ~2 IN T H E I N T E G R A L

KI41(z, a)

Assuming a33 > 0 we can take ( = (3. Starting with the integration over (1, we get I1((2~1, (3~1)

= f (~(~2ball)t~((l(2 +al2)d((I(3-ba'3)d(l --[((l-~1)2] 2

d -- a 1 2

--a22

- Q23

--a13

--a23

--a33

4

a11--¢ 2

[D(-a,1,-azz,-a33,{~,-a,z,-az3,~i(3,-a13,

(A.a) We have here used the notation D ( ~ I 1, (~22 ~ 0[33 ~ ~44. ~ 0112 ~ (x23 ~ ~34 ~ ~13 ~ °~24 ~ °[14)

- -

~11

~12

~13

~12

~22

~23

~24

~13

~23

~33

~34

[~14

~24

~34

~44

,

~14

(A.la)

To perform the next integration over (z we choose a coordinate system where ~l = 0 and (Y3, (~ = 0. After integration over the space components of (2, we obtain the

566

P. A R R H E N

expression: i2 = f d(2 d(, 6(( 2 -ka,,)6(~-k-az2)~(~1~3-ka,3)6(~2~3-ka23)6(~,~2+a,2) [-(~1 ~'1)2(~2 ~'2)2] 2 -

-

--(¢1 ~2) 0 dC°0(R(¢°)), 4

--¢1ff3

--a23

--a33

IT

(

--C3¢1i

-oo [ O ( - a 2 2 , - a 3 3 , ~f, ¢2z , - - a 2 3 ,

g2)

--~1~3, ~1~2'

, --~2~3, ½(a22--~22))] ~(A.2)

Here

2 ffOea33~2 R((°) - a23-a22a33 + a33 (~1 ~3)2 -1-a33 ~2/ '

(A.2a)

(¢1 ~2)0 = _~l~2nt_~o 0 023(41 if3) a33

(A.3)

Substituting (~1~2)0

= S = - ~ l-;0Z- 0 + -az3(~lff3-) a33

e.g.

~°d~° = - d S ,

(A.4)

we get

Z=(~, ~c~) [-a22

--a23 0 1 /. . -. S.

-aza

-S

--a33

-- ~1 ~'3

- ~ ., ( 3.

j.

.

.

.

ff~

]

1

--a12

--a22

--a23

-a,3

-a23

-a33

163-o~ [ D ( - a a l , --a22, - a 3 3 , ffl, --a12,

--a23, --¢1~3, --a13, --S, ½(all--el))] ~ --S --a23 1

--~1~3 --a33 -2

-¢~

~(.22-~)

X [D(--.=~,

- a3~,

¢~, ~,

- a=3,

- ¢, ~3,

~f --ff3~l dS

¢'¢~

~, ~,

- S, - #S~,

~("~=-

~))]<

(A.5)

This is an elliptic integral. Further, we see that if we try to perform the integratmn over (3 before that over S in K(4) =

I2(~l~3,~2(3)e(g3)o(g?+a33)d~3

[(~3-~3)2] 2 elhptic integrals are again obtained.

FOUR-POINT FUNCTION ANALOGOUS

567

Appendix B T H E A N A L Y T I C P R O P E R T I E S O F T H E I N T E G R A L KC41(z,a)

From the construction of formula (10), it is clear that the analyticity domain of is that of the four-point function. In this appen&x, we show algebraically how the singularities appear in K(4)(z, a).

K(g)(z, a) (a,k real)

B.1. T H E M A I N S I N G U L A R I T Y

The main singularity of the integral can be obtained as a coincidence singularity between all three denominators. Since the integral is depending on only three external vectors 41, ~z, and ~3 it is sufficient to study the integral in three dimensions (x, y, and t). In addition we also require a condition on the derivatives 6). Thus we must satisfy the following set of l0 equations. q51 = ½ [ ~ 2 - 2 ~ l ~ t - a l l ] = 0, 1

~b2

2

(B.la)

~[~2-2~2~2-a22] = 0,

(B.lb)

t~3 = ½ [ ~ 2 - - 2 ~ 3 ~ 3 - - a 3 3 ] = 0,

(B.lc)

])4-

I 2 = 7[(1+a11] = 0,

(8.1d)

q55 = (1(2+a12 = 0,

(B.le)

q56 = (1(3+a13

= 0,

(B.lf)

q~7 = ½[#2+a22] = O,

(B.lg)

0,

(B.lh)

~9 = ½[~2"[-a33] = 0,

(B.li)

q~lo = D = det[ ~qS~ = O.

(B.lj)

~8 = ~2~3q-a23

I

=

Ouk

In the derivative condition D, we have relabelled the variables (1, (2, (3 in the following way:

C1 ~ ¢~ =

Ul~

C2

C3

U2,

(~ = /25,

(~ = U 8 ,

~0 = /'/3,

¢0 = /'/6,

¢0 = //9,

~ /'/4~

~U 1 " ' "

"~-- 117,

(8.2)

~U9]

D =

! GU 1

0U 9

(B.3)

568

P ARRHEN

Explicitly the 9 × 9 determinant D has the following form:

47 4~ -4~ 0 0

o o

o o

o

o

4~ 4~ o o 0 o

(1 ~ -~ (~ ~ - ~ 47 (~ (~ -(~ o o

D=-

0 0 I

:0

o o o

o o o

(~ (; o

(~ (~ o

o

o

o

-4~ o o ~ 0 0

o

o 4~ 0

o -~ 0

o (7 ~

o -~

o ~ (~

o -~ -(~

-(~ o -(~ -(~ o

o

o (~ 4;

(B.4)

In principle, eqs. (B.1) now yield, after elimination of the nine variables ul • • • u9 (or (1 , (2, (3) a function S(42 , 42 , 423 , 4 1 ~ 2 , 4 1 ~ 3 , 4 2 4 3 , a t k ) = O for which the integral might have singularities. (No condlnons on relevance are considered here.) It has turned out to be impossible to perform this elimination directly. However, we can reformulate this algebraic problem m the following way: The first three equations ~bl, q52, and q53 can be interpreted as a necessary condmon that the vectors 4k-- (k be hght-hke. Assuming these vectors to be linearly independent, we can introduce these three hght-like vectors as a base system and write 3

(, = ~, fl,~v k

i = 1, 2, 3,

(B.5a)

i = 1, 2, 3,

(B.5b)

k=l 3

¢i = Z ~,kVk k=l

where the coefficients O%k and fl,k evidently fulfil 7,k = fl,k

for

i ~ k.

(B.6a)

Thus ~ - 4, = ( e , - f t , ) v , ,

(B.6b)

which means that eqs. (B.la-c) are satisfied. We now turn to the equatmn D = 0 (B.lj). The 9 x 9 determinant D can be reduced to the following simpler form D=

(~ (2

(3

¢3 ~1 ~3 -~ ~1 ~1 %3

~ ~2 ~1

~s ~3 ~2

(B.7)

We have here used the notation

I¢, (l [ff2

¢7 ¢I = j x ¢f

_¢o _¢o (B.7a)

FOUR-POINT FUNCTIONANALOGOUS

569

We now substitute the expressions (B.5a) and (B.5b) in D and, after some rearrangements, we obtain: Y1

13

~3

X

fill

ill2

f2t

fl22 fl23

flat

f32 f13

ill3 (flt2f123f131--f121f132f113)"

(B.8)

Assuming the determinant Jfl,k] (or la,k]) to be non-zero, the condition D = 0 now reads fl12f123f131

(B.9)

-~- f 1 2 1 f 1 3 2 f l 1 3 "

We now turn to the rest of the equations in (B.1). After eliminating tilt, fl22, and fl33 with the aid of eqs. (BAd), (B.lg), and (B.h), the remaining three equations (B.le), (B.lf), and (B.lh) read:

2atz = At2(kx2all + k 2 1 a z z ) + ( 1 - A t ~ )

( alia-22 + r 1 2 ) , \

(B.10a)

r12

2a23 = Az3(k23a22+k32a3a)+(1-Az3)( a22a33 +r23) , \

(B.10b)

/"23

2a3t = A3t(k3ta33+k13at1)+(1-A31 ) ( alia33 +rx3 ) . \

(B.10c)

/'13

Here, we have fla a,i flu a~t

= Ak,,

(B.10d)

r,k ---- 2(fl,k ilk, a,k + fi,, ilk, a a + fk, fl,k akl) = /'k,,

( .10e)

B,k =

(fl,k a,k + fl,, a;t)(fk, a~, + ilk, a'kZ) t

t

]( tk - -

t

flkl_ ! ft. kk,

(B.10f)

The quantities a,'k are defined by V, V k - b a ~ k

= O.

(B.10g)

In the above formulae i ¢ k ¢ l. According to eq. (B.9), the (complex) numbers Ajk , kjk and rjk are related by k,k = rkt,

(1-At2-A23-A31)

2 = 4Ax2A23A31.

(B.1 la) (B.11b)

If we use the same elimination technique for the vectors ~ and use eqs. (B.6a) and

570

P. ARRHEN

(B.5b), we get the following impliclt expressions for the manifold S(~,4k,

24,4k = A,k(k,k43+kk,42)--(1--A,k)( ~242 +r,k),

a,k) =

0

(3 eqs.)

(B.12a)

(3 eqs.)

(B.12b)

\ r~k

2a,k

=

A,k(k,ka,,+kk, akk)+(1--A,~ ) (a-"ak~k +r,k), \

kl k __ Ykl rd

r:k

( 1 - A 1 2 - A I 3 - A 2 3 ) 2 = 4A12AI3A23, Gk =

A~k : Aki"

I'kt,

(B.12c) (B.lZd)

B.2. THE ENVELOPE OF THE MAIN S I N G U L A R I T Y OF THE I N T E G R A L

We want to calculate the envelope of the singularities S(~,4k, a,k) = 0 when the real parameters a,k vary in the interval - ~ < a,k < + o9. We use essentially the techmque used by K/ill6n in ref. 2), appendix l, with minor extensions. There it is shown that the envelope of the family of curves defined by the equation F(z, a,) = 0 (a, are real parameters, and z a set of complex variables) is given by the simultaneous equations F(z, a,) = 0, (B.13a) 0F

p,;,

-

(B.13b)

Oal where j is a complex number and the p, are real numbers. We adopt this result with the following generalization. In our case, we have the function F given implicitly by the ten equations ~)j(Ul " ' " 1"19, 41, 42, 43,

j=

a,k) = O,

1...10.

(B.14)

We obtain the condition for the envelope from the following equations ~,= 0

i=

1...10, ...

Oa,k

OUl

(~(fil

~)1

(B.15a)

#4ho

Ou9

A(a,k) = da,k

~Ul

~U 9

~cb9

c3~9

Oa,k

63U 1

e4~9 Ou9

= P,t3"

(B.15b)

We will now see what condxtions these equations impose on our coefficients ~,k i ~ k in (B.5b) or on the numbers A,k, r,k and k,k in (B.12).

571

FOUR-POINT FUNCTION ANALOGOUS

We first write down the quantity OD

c~D

c~D

o

~

~

_~o

0

0

0

0

0 1

~t ~

o o ~ ~

0

_A(a12) =

--

x

,;~

o o ~ o o _~o o - ~ ~o ~, _,.;o o

0D

~D

c~D OD

o G o o ~ o

o o _~o o o ~ o o _,.;o o o ~ _~o

c~D

o o

o o ~ _~o o o o o ~ _~o

0

0

o

o

~

~

o

o

0

0

o

o

~;

~.~ _~o ~

~

o _~o

0

0

o

o

o

o

~

_~o

,gradxO ~1

c~D

~3

'grad20

~1

~2

~3

o

~;

igrad30

~2

~3

(B.16) ~ ~3

Here ~D ~

grad, D

0D

c~D

~

0~ °

Cl _~o

(B. 16a)

Now, using the eqs. (B.6b) and the inverse equations of (B.5a), i.e., 3

v, = ~ flS'(t,

i = 1, 2, 3,

(B.17)

l=l

we get

- A(a,z) =

B;)la,kl(~,.~-fl,,)(=22-flz2)(=33-B33)

× EflX o , + fl;)O2 + fl[2XO3] (B.18)

or

A(a,2) = fll-2'J

(B.19)

with 3

J = -la,~l l-[ (~kk--flkk)[f123'D1-l-fll31D2 -l-fll)D3], k=l

[a,kl =detla,kl = -- ~2,1 ' (3L

(B.19a)

(B.19D)

572

r. ARRHI~N

gradkD Ok

(B.19c}

Vk

By cyclic permutation, we obtain i # k.

A(aa,) = flS~),

For the quantity A ( a ~ )

~D

_~ (7

~

_(o

o

o

o o (x

o o ~

o o -~1o

~ o 0

~ ~

G -~ ~ -~

0 A(al~ ) =

t

0 _½ 0 0 0

=k-

~D

we get

~D

0

¢3D c3D

~D

3D

riD

3D

o

o

o

~

o -~

o

o

o

o 0

o 0

(~

(~ -(~

0

0

0

~7

~

-~

o

o

o

o ~

o ~

o -~

0

o o

o o

o o

G ~;

~ ~

-~ -~

o ~

o ~

o -~

0

o

o

o

o

o

o

G

G

-~

/ ii,gradl° (~

"J-

(B.20}

-~+(1

-

grad2 D

~3

-~i+(1

~2 (2

(2 (3

(2 (~3

G

-]-

rgradl -~+~,

grad3 D

~2

(3 (3

(2 (3

- ~ 1 + ( 1 l] (2 (3

It •

Using eqs. (B.6b) and (B.17), we get 3 A(all) = --[a,k[ H (O~kk--flkk)2flll 1 - ' [fl23 - ' Ol -l-flieD2 +fl[21D3] = ~-flll 1 --1 3. k=l

(B.21)

(B.22).

Again, by cychc permutation, we get A(a,,) = 7ft, 1 - , 3,

i = 1, 2, 3.

(B.23)

Thls, and the eqs. (B.20), gives us the following expression of the envelope conditions (B.15b) A(a,~) = fl,~19, I 4: k (B.24a) A(a,) = 7ft, ' - ' 3,

t = 1, 2, 3.

(B.24b)

This means that the matrix fl-1 and hence also fl is real and, further, from eqs. (B.6a) that the numbers ~,k (i # k) are real. Also, It follows from the reahty of the

FOUR-POINT FUNCTION ANALOGOUS

573

matrix (a,k) that the numbers alk are all real. Thus, the envelope condition can be put in the following form ~ = ~tlVl"[-~,2V2"~O~t3V3

v, vk C~,k,

( i = 1,2,3),

(B.25)

real,

(B.25a)

i ~ k real,

(B.25b)

0~120~230~31 =

(B.25c)

0~210~320~13.

Also, we see that eqs. (B.25) are eqmvalent to the reahty of the coefficients A,k, r,g and k,k in eqs. (B.12). Note 1. The numbers ~,g in eq. (B.5b) are not elements of a symmetric matrix, but are only restricted by the eq. (B.25c). However, if we redefine the vectors v,, mulUplying them by suitable factors and use eq. (B.25c), we can obtain a matrix ~ in (B.5b) which is symmetric. Note 2. From the above, we know nothing about the signs of the quantites alk and a,~ since we have not studied the condition for the singularities to be relevant. B.3. "LOWER" SINGULARITIES We also want to indicate that our integral K(g)(z, a) in eq. (10b) contains the generalizations of the singulariUes of those of the integral K(3)(z, a) for the threepoint function, 1.e. the surfaces (7a), (7b), and (7c). To see this, we perform the integration In K(4)(z, a) over the variables (3 and get I(4)(~t~3,~2~3 ) = (d(36(~'2k-a33)8((3~2-~-a32)6((3(l+a13) J

2

K(4)(z, a) ----- ( d~l d(z 6((2z+ a22)6((" (2 + a i z)6((~ + a, i)e(~) i(,)((,

j

~_(2)~¢ _-(~)~]2

'

(B.26a)

¢3, ¢2¢3). (B.26b)

Among its singularities, this integral must contain the singularities of the integral

K(3~(z, a) = f d(1d(zb(~z+a'I)b(~l~z+a12)b((z+a22)e(~)

(B.27)

[(¢2 -- (~2)2(¢ I -- ~I)2] 2

As this integral is explicitly solved in ref. 5), we can at once write down the singularities of K(3)(z, a)

ZR, = akk,

k ---- 1, 2 relevant when akk > 0,

F12: 2z12 - - 2"112"22 " ~ - ~ "

relevant when r > 0,

(B.28) (B.29)

r

r satisfies the equation:

2a12 - a l l a 2 2

+r,

(B.29a)

574

P ARRHEN

$12:

2z12 =

kz11+ z2~2

relevant when k < 0,

(B.30)

ka11+ a!~.

(B.30a)

k k satisfies the equation: 2a12 =

k

Note. The

" K (3) (z, a) are restricted a,k m

2 -- al 1 a22 ~= 0 , by the condition a12 while we do not have this restriction in K~4)(z, a). Hence, we must check that the 1ntegral K~3)(z, a) has no singularltles w h e n aj2 2 - al 1 a22 < 0. It turns out that K~3)(z, a) 2 =

0

for

,, masses ,,

a x 2 - - a l l a 2 2 < O.

Hence, we have s h o w e d that the "lower" singularities o f K~4~(z, a) consist of the curves ( B . 2 8 ) - ( B . 3 0 ) and the obvious cychc permutation o f these formulas. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)

G. Kall6n and A. S Wlghtman, Mat Fys. Skr. Dan. Vld Selsk. 1, No G. Kall6n, Nuclear Physics 25 (1961) 568 N. H Moller, Nuclear Physics 35 (1962) 434 G. Kall6n and J. Toll, Helv. Phys. Acta 33 (1960) 753 B. E. Y. Svensson, Nuclear Physics 39 (1962) 198 J. C. Polklnghorne, Dispersion relations, Scothsh Univ. Summer School Boyd, 1961) Ecole d'Et6 de Physique Th6orlque Les Houches: Relation de Dispersion et talre (Hermann, Paris (1960)) D. Hall and A. S W~ghtman, Mat Fys Medd. Dan Vld Selsk 31, No. A. C Manoharan, J. Math. Phys. 3 (1962) 853 A. C T Wu, Phys Rev. 135B1 (1964) 2 2 2

6 (1958)

1960 (Oliver and Partlcule El6men5 (1957)