Deep electroproduction and deep electron-positron annihilation

ANNALS

OF PHYSICS:

Deep

79,

1-33 (1973)

Electroproduction

and Deep

Electron-Positron

Annihilation

R. GATTO Istituto

Istituto di Fisica dell’llniversitd, Nazionale di Fisica Nucleare,

Roma, Sezione

Italy and di Roma, Italy

P. MENOTTI lstituto

Scuol~ Normale Superiore, Nazionale di Fisica Nucleare,

Pisa, ltaly and Sezione di Pisa,

Italy

AND 1. VENDRAMIN Istituto

di Fisica

dell’lJniversitd,

Padot?a,

Italy

Received July 19, 1972

The problem of continuation of the scaling functions for deep inelastic electron scattering into deep inelastic annihilation is discussed. Dynamical models are produced for which analytic continuation connects the two scaling functions and the general conditions are discussed. The dynamical characterization of those situations for which the analytic continuation rule is invalid is also given. The study also illustrates the structure of the singularities of the scaling functions in the w-plane. Models studied include ladder diagrams and particular diagrams with unstable particles propagated. In the latter case cuts generally appear in the scaling functions in the kinematical region for annihilation.

1. INTRODUCTION

The experimental evidence for a scaling behavior of the structure functions in the deep inelastic electron scattering has led to the suggestion that a similar scaling behavior occurs in the electron-positron annihilation channel [I]. The problem arises whether a connection exists between the scaling functions for the two channels. It has been suggested in particular by Drell et al. [2] that the two scaling functions are related by analytic continuation. The argument is based on formal manipulations of the expression for the inelastic scattering structure functions in terms of 1 Copyright 0 1973 byAcademic Press, Inc. All rightsof reproduction in anyformreserved.

2

GATTO, MENOTTI,

AND VENDRAMIN

the electromagnetic current. The authors [2] show that for a particular class of graphs the scaling functions for the two processes are given by the same integrals, with the only difference that the scaling variable w belongs in the two cases to different intervals; this leads to the result that the two scaling functions are related by analytic continuation. On the other hand, Landshoff [3] has constructed a dual model which exhibits scaling for electroproduction but not for annihilation. The problem of the possible relation between the scaling functions of the two crossed channels through analytic continuation is essentially of a dynamical nature. We recall in fact that the connected forward matrix element of the current commutator (q. > 0 in the lab.)

W,,(q,P>= Im 7’,,(q,P>= t 1 @YP I [J,(x), Jv(0)lI P> d4x = ii c (W4 %I + P - PA

(n I J,(O) IP> n =-

(g!A”-

9LL4” 2 W,(q2, P * 4) 4 )

+ (P, - qu Y)(P”

- q” y,

W2(q29P . a>

U-1)

can be decomposed as the sum of three pieces, the connected part C, the semidisconnected part D, and the “pair” part P, according to the nature of the intermediate states in (1. I), see Fig. 1. From the stability of the hadron target we easily derive that C can contribute only for -q2/2p * q < 1, D only for -q2/2p * q < 0 and P for -q2/2q - p < -1. C is the contribution relevant to the inelastic scattering and as we have just seen it coincides in the physical region of this channel, i.e., 0 < -(q2/2q * p) < 1 with the full matrix element of the current commutator [4]. The scaling functions F,(w) and F2(w) for the scattering channel are defined by

where in the scattering channel we define w = -q2/2v with v = p * q.

FIG. 1. Decomposition of the forward matrix element of the connected commutator connected, semidisconnected, and pair contributions.

into

DEEP

ELECTROPRODUCTION

On the other hand the correlation

AND

ANNIHILATION

function wti, for the annihilation

Wuy(q,p) = &(27~)~ S(q -p -p&O

channel,

j J,(O)! n. pXn,p / J,(O)] O>,

(1.3)

is given only by the pair contributions to the commutator in (1 .l). The scaling function in the annihilation channel are analogously defined by F,(w) = hil

F&p,

v),

F,(w) = lip-v) 1

iv2(q’, v),

(1.4)

where now w = -q2/2v, with v = -q ‘p, p being the four momentum of the detected outgoing hadron; the physical region for the annihilation channel is for l
F,(w) = &F2(w),

(1.5)

where the upper and lower sign refer to Fermi or Bose statistics of the target hadron. Our aim is to analyze whether the relations (1.5) hold (of course, for the analytic continuations); in other words, whether the knowledge of F<(w) in the physical region for scattering 0 < w ==c1 enables us to obtain through analytic continuation F,(o) in the physical region for annihilation. We know that only superrenormalizable theories exhibit scaling properties in perturbation theory; in simply renormalizable theories the scaling behavior does not hold in perturbation theory. We shall examine in detail within a superrenormalizable theory the properties of the scaling functions Fi and F, . Our analysis will consist of two parts; first, we shall determine in various specific exampIes the position of the singularities of F(w) in the w-plane to find out in which cases an analytic continuation is possible for w > 1; then we shall examine whether such analytic continuation coincides with F(w). Particular attention will be devoted to the occurrence of real cuts on the physical sheet for w > 1, which spoil the analyticity of F(w) along the positive real axis and invalidate the analytic continuation relation (1.5). It is well known that the scaling function of the box graph (we refer for simplicity to the case of a scalar photon) with suitable stability conditions for the propagated particles (see also Section 4) has a very simple analytic structure: it

4

GATTO,

MENOTTI,

AND VENDRAMIN

allows for an analytic continuation for w > 1 and this continuation gives the annihilation scaling function [5]. The scaling function for the double box graph has a much more complex structure. After setting down the general formalism in Sections 2 and 3, we shall give in Section 4 a complete analysis of the singularities of the scaling function for the double box graph. This analysis is carried through by reducing the scaling function to a double integral over Feynman parameters and giving a complete Eden-Hadamard type analysis of the singularities [6]. Both Landau and nonLandau singularities are examined. As long as the propagated particles are stable and no zero-mass particles appear in the theory the continuation of F(o) above w = 1 is possible; the calculation of Section 5, showing the coincidence of the analytically continued F(o) with -P(w), gives a direct check of the recently proposed argument by Landshoff and Polkinghorne and of the result of Drell et al. on the validity of the relation F(w) = ---F(U) for ladder graphs in which the propagated particles are stable [S]. In Section 6 we write down the scaling function for a general double box diagram in terms of the DGS spectral function; the analyticity along the positive real axis is easily evidenced as long as the stability conditions are satisfied. As the propagated particles become unstable, cuts arise on the physical sheet with real branch points for w > 1. The occurrence of real branch points for w > 1 poses the problem whether there is any simple connection between the two scaling functions F(o) and F(w) in the general case. To this propose we examine in Section 7 a model (a box diagram with dressed propagators) which possesses such real branch points for w > 1 and which is simple enough to allow for an explicit evaluation of the two scaling functions. No simple connection emerges in this case [S, 91. The breakdown of the analytic continuation rule for the scaling function is easily related to the nonvanishing double discontinuity in the parton mass and in the parton-proton energy of the parton-proton amplitude. In Section 8 we summarize the conclusions and we give a general outlook on the relations between Bjorken scaling in the scattering and the annihilation channel.

2. GENERAL EXPRESSION FOR THE SCALING FUNCTION The dominant contribution in the deep inelastic region is expected to arise from diagrams with contiguous electromagnetic vertices [ll, 121, whose general structure is shown in Fig. 2. The amplitude A describes virtual scattering on the target as illustrated in Fig. 3. We consider the forward (t = 0), virtual Compton scattering of scalar photons, interacting with scalar particles [13]. The scaling function F(U) is given by the

DEEP ELECTROPRODUCTION

AND ANNIHILATION

FIG. 2. Forward virtual Compton scattering for contiguouseleckomagnetic are indicated in parenthesis.

vertices. Masses

(fi) FIG.

3. Forward

virtual scattering amplitude.

Fourier transform of the weight function of the E(XJ 6(x2) singularity on the light cone [ 141, so that, with reference to Figs. 2 and 3 we can write F(w)

=

-

q

j eiMWS9(77x0)

dx, .

(2.1)

In Eq. (2.1) p2 = M” is the target mass, u = --(q2/2v),

b’=p.cl,

and we have introduced a four-vector n = (1; 1, 0,O).

G.2)

6

GA-I-TO,

MENOTTI,

AND

VENDRAMIN

In Eq. (2.3), A(k2, k * p) is the virtual meson-meson amplitude of Fig. 3, m, is the mass of the boson field propagated in Fig. 2. It is convenient to introduce, besides n s (1; I, 0, 0), also 5 = (- 1; 1, 0,O) and to decompose the virtual particle momentum k as K=pn+oE+&,

(2.4)

where the two-dimensional vector & gives the transverse component of k. From Eqs. (2.1), (2.3), and (2.4) exchanging the order of integration we obtain

F(f.0)= -M-g$J

dp da d2& (-4pa

-

ET2 - mo2 + ie)-’

0

* A(-4pa

&.2;

-

M(~

-

a))

. j- dxo

,9(2~+~~h

(2.5) dR (-R * A(-

R + 2Mwp;

@&J

+ 2Mwp

- mo2 + ic)-l

+ Mp),

where we have put RT2 = R. The meson-meson amplitude A(k2, k * p) can be expressed by a Nambu representation. For a graph with N internal lines and I independent loops, the Nambu representation [6] is CN-21-2

A(k2,k.p)=G~~‘~duiS(~vi--)

[Flk2 - 2F2k . p - F3 4 icC]N-21 ‘(2e6)

l=i

where C, and Fj (j = 1,2,3) are homogeneous functions of the Feynman parameters vi (i = 1 *** N) of degree 2 and I + 1, respectively, and G is a constant. The representation in Eq. (2.6), for k2 = -R + 2Mcup and k * p = &%12~ + Mp, can be inserted into Eq. (2.5), and the two denominators parametrized a la Feynman using 1 da an-l =-----*1 (2.7) llX”Y o [ax + (1 - 4 yP+l

s

We obtain, after exchange of the order of integration, F(w) = (- l)N +T~MG &

Jo1le dui S ( :

vi

-

l=i

- 1) + 11 + -~e~dp~owdR{N4&

1)

CN-2’~2

* 1’

(N

-

24

da

UN-21-1

0

2Mp[a(F2 - 04

+ a> -

+ a(M2wF2 + F3 - mo2) + mo2 - ie(aC + 1 - a)}-N+2z+1.

~1 (2.8)

DEEP

ELECTROPRODUCTION

AND

7

ANNIHILATION

After integration over R (note that the coefficient multiplying and over p, by the use of the identity +a, I -*

dp (xp + I’ -

= $gy

k)-”

R is nonnegative)

6(x),

12.9)

we find F(w) =

(-1)N 57% a ~ 1’ fi dt,i 6 (ii pi - I) CN-2’ -? iv - 21 - I amo2 o l=i da ‘+22-l

8 a-

- 1F, + ~(1 - FJ-l

F2 + w’;; - FJ 1 I) -- 11-l

[a(F, -

(2.10)

- [a(M2wF, + F3 - moz) + m02]-N+l’-‘.

Finally we perform the integration

on the variable a and obtain the result

F(w)= (-1)NT3iG ~jlfidci$+ N - 21 - 1 iimo2 o l=i ’ (F2 2 ,?Fl)

)

I)? l=i

sgW2 + 4

2

’

iEA-t

0’;; - Fl) 1

- FJI N-21-1

9

w(M2wF2 + FJ”+

(2.1 I)

mo2(F2- wF,) I

The support properties of F(w) are implicit in the &functions in Eq. (2.11). We have for

F, + ~(1 - Fl) > 0,

0 < w :< (F,/F,),

for

F2 + ~(1 - FJ < 0,

(F,/F,) < w < 0.

(2.12)

From the structure of the functions Fl and F2 in the Nambu representation, as can be deduced for instance from the set of graphical rules by Eden et al. [6], we have 1F, j < F1 , and, thus, -1 ,cw < 1.

(2.13)

In particular for “direct” graphs, i.e., for A given by any graph which can be drawn in a plane without either internal or external lines crossing, the external lines being attached round the diagram in the order of Fig. 3, we have F2 > 0 and correspondingly 0 < w < (F,I&)

< 1.

(2.14)

8

GATTO, MENOTTI, AND VENDRAMIN

3. SPECIALIZATION TO LADDER GRAPHS We shall specialize the general results obtained in the previous section to a ladder graph. We take the contribution to A(/?, k . p) from the ladder in Fig. 4 with n + 1 horizontal lines.

FIG.

4. Forward virtual scattering amplitude for ladder graph.

FIG. 5. Graph for the contracted amplitude A’ and its Feynman parameters.

DEEP

ELECTROPRODUCTION

AND

9

ANNIHILATION

We follow the technique described in the previous section taking advantage of the relation A(k2, k . p) = (fi. &)

A’W,

(3.1)

k . I’);

At(k2, k . p) is the contracted Feynman graph given in Fig. 5, where the Feynman parametrization is also illustrated. In this way we obtain for the scaling function F(w) the contribution

f’(%4 = C-1)”Gn(l@-)&a .

F'"' 2

_

Jb’ da, fi dcxi d/c?+6 ( a0 T- i

(JL~ + PJ -

I)

1 ==i

l=i

,.&Z'

'

C(n,F(")

8(w) d(Fvjn' - wFin))

2

- [o(M2wF,'"' + F;"') + mo2(F,(n' - cd+')]-'.

(3.2)

CY, Fp', Fp', and Fp' are given through the formulas [ 151 C'"'(~o

7 % =

I...,

011 ...

%a a,

+

; p1 (010

,.*., +

13,) pl,

C'-ycd,

...

2,

; p,

...

/In)

+ LYJ~~C(~-~)(~Y~ ,..., a, ; fl, ,..., /2in) - 1.1 i- x1 ‘.. cd,~,/3,

(3.3a)

and cya:, F%o

) a, 7 8, = a0 + ml + p1 ,

, '111,..., 2, ; p, ,...' Pn) = no@'(ao = 0: '2' . .. . . fY, : /3, I.' pn>,

(3.3b) (3.3c)

(3.3e)

(3.3f)

GATTO, MENOTTI, AND VENDRAMIN

10

The expression for G:“’ in Eq. (3.3e) is obtained from Fp) substituting according t0 0l.j f--) CY,-j (j = O,.**, n) and pj t) Pn+l-j (j = I ,..., n). The support property for direct graphs described at the end of Section 2 is evident since Fp’ > 0 and (Fp)/Fp)) = (q, m-ecx,jo1,,..* CL, + .**) ,< 1. 4. SINGULARITIES OF THE SCALING FUNCTION IN THE U-PLANE 4.1. Simple Box Diagram

Before examining the analytic properties of the double box graph we shall briefly recall the singularity structure of the scaling function for the simple box, as these are connected to the more complicated situation of the double box. The complete scaling function for the simple box diagram as given in Fig. 6 (with addition of the crossed diagram) can be calculated explicity [5] F(w)

=

&

+

j@)

fql

+

@l

0)

4-w)

-

w)

u2

_

w(2

5;,~2jj

+

1

1+w

02 + w(2 - (p2/M2))

+ 1 I*

(4.1)

The function F(w) in Eq. (4.1) is finite and continuous for -1 < w < 1 and vanishes linearly at both extremes of such interval. The analytic function F(U), for complex W, (4.2) S(w)= j&5 oJ2 - w(2 1-W - (p2/M2)) + 1 coincides with F(u) for 0 < w < 1.

FIG.

6. Simple box diagram for virtual Compton scattering.

DEEP

Its singularities

ELECTROPRODUCTION

AND

11

ANNIHILATlON

are two simple poles at

cc(i) = 9{2 - (p2/M2) & i(l*/M)(4

- (p2/M2))‘,‘2:.

(4.3)

For p2 = 0 the two singularities collide at w = 1. As a preliminary example [5] of more general situations the propagator for the exchanged boson (of mass CL)in Fig. 6 can be replaced by a dressed propagator obtaining for ,F(w) p(p2) d$ -- . w* - ~(2 - (p2/M2)) -!- 1

(4.4)

The cut structure of 9(w) as given by Eq. (4.4) is shown in Fig. 7. The end points C and C’ are determined by the lowest mass in the spectrum of the exchanged particle. Only when such lowest mass vanishes, C and C’ both end at w = j-1. Excluding such a possibility F(w) can be continued from the “scattering” interval 06 P w < 1, to the region w > 1. By direct calculation we verify [5] that such analytic continuation gives the scaling function F(w) for annihilation, defined for w > 1, through the relation F(w) = --F(u). This feature is related to the stability of the target hadron and of the particles appearing in the vertical propagators.

FIG. 7. Singularities of the scaling function boson replaced by a dressed propagator.

F(W) for the simple box with the exchanged

4.2. Double Box Diagram

We shall now make use of the formalism developed in the preceding sections to study the analyticity properties in w of the scaling function for the double box diagram. We refer to the diagram in Fig. 4 for the virtual scattering amplitude A(k2, li . p) taken for n = 1. To simplify the discussion at this stage we put nz = m, = 1, and at the end we shall also specialize to nz, = m. From Eqs. (3.2) and (3.3) we obtain for such a diagram

12

GATTO, MENOTTI,

After integrating

AND VENDRAMIN

over /I1 we can extract the analytic function S(w) in the form

Red,

Redo

Im dt

FIG. 8. Undistorted

region for the integral in Eq. (4.6) giving the scaling function.

The undistorted integration region is shown in Fig. 8. To study the singularities of S(w) as given in Eq. (4.6) we shall employ the usual Hadamard-Eden procedure and look for the end-point singularities and pinch-singularities. We put m, = 1 and write down the expressions for the singularity surfaces S’l’ E a1 = 0, S2’ = a&l

(4.7a)

- a>2 + w[(aO + 01~)~- (01~+ 013(2 - p2) + l] = 0, (4.7b)

and the equations specifying the boundaries of the integration

region

,z:‘l’ Es a0 = 0,

2’2’ = 010+ a1 - 1 = 0, #z(3) = wolo + q - w = 0. The singularities

(4.8a) (4.8b) (4.8c)

are given by the solutions of the set of equations &S(*) = 0 (i = 1, 2), j&‘p’ = 0 (j = 122, 3), (4.9)

DEEP

ELECTROPRODUCTION

AND

ANNIHILATIOK

13

The following conclusions are obtained from the discussion of Eqs. (4.9). The study of end-point singularities shows that w=o

(4.10)

and cG(+)= 1 - ($/2)

& +/2)(4

(4.11)

- $)ll’L

are branch-points [16]. The singularity at w = 0 also obtains in other ways (e.g. as a pinch between S(r) and P)). Obviously, such singularities lie on the physical sheet of the analytic function F(w). From pinches between SQ’ and Dz) we obtain the branch points (4.12)

3(&) = I - 2p2 * 2ip( 1 - $)1;?,

for 01~= UI~ = 4, lying on the boundary of the undistorted integration region. Such singularities are, thus, again on the physical sheet. Finally for Sc2) becoming locally cone-like we have the branch points U(i) = -1

+ (2 -

2

P2Y

! 2 -

2

P2 /+$

_ 4)1,2

(4.13)

for iy,, = CQ= (2 - p”)-l, that is for values external to the undistorted contour implying that WC*) do not belong to the physical sheet. In any case for 0 < p c 2 (defining the stability region for the decay p --f 2~2, with tn = 1) w(+) lie on the unit circle / w(~)I” = 1; whereas, for p > 2 they lie on the real axis and satisfy 0 -C ~3(-) < 1 and WC+)> 1. To study the singularities occuring for infinite values of the Feynman parameters we substitute in Eq. (4.6) y = %;I

p = yal

(4.14)

obtaining

rP - W(Y - l)l[y - (1 -r ml PCBCl -~ WI2 + o[(l

c p)” - y(1 + /I)(1 + m12 - $) -t y2”r,2];.2-. (4.15)

From pinch between SsPCl

and 22 1~ 4~

-w)2+w[(1

$8)2-y(1

+-)(I

+m,2-$)~+71,2]

- 1) - P we find, for y = 0, the singularities w L- 1

(4.16)

14

GATTO,

MENOTTI,

AND

VENDRAMIN

and w = 1 --2,

(4.17)

obviously on an unphysical sheet. The solution w = 1 - p2 is, however, fictitious as can be seen by explicitly performing one of the two integrations in the expression for S(w). In Eq. (4.6) we substitute x = (111- w(1 - CXo)

y=l-CXcu,--ol,

(4.18)

obtaining X

s(w) = -G1& s’-” dy(OJy + x)(f(x;w)y + g(x* 0 dxs’“-” , (0))’ 0 where f and g are defined later. Integration

over y gives

S(W) = -G1&/lPYdx&ln

[**$$-$-I.

10

(4.19)

>

(4.20) 3

In Eqs. (4.19) and (4.20) f(X;

w)

=

-X(1

+

w)

--0(--w

+

ml2 - p2),

(4.21b)

g(x; w) = x(1 - 0 - x) + /A, h(X;

(4.21a)

w) = X2 + wx(1 - ml2 + p2) + p2c0,

I(X; co) = (1 - 0 - x)(m12 - w - x - p2) + ~2.

(4.21~) (4.21d)

The Hadamard-Eden method can be applied to Eq. (4.20) before or after performing the derivation on m12, or to Eq. (4.19) and confirms the results already found, but shows the absence of the singularity at 1 - p2. The form of the numerator in Eq. (4.15) is responsible for such cancellation [18]. The cuts attached to the physical sheet branch points can be drawn by discussing the motion with 01~of the singularities in 01~of Eq. (4.6). From the end-point singularities in CQ= 1 - 01~we have 2 kd~o)

=

1 -

2aoclpM

ao)

4~

i

2010(lP- ao) (4a,(l - a01 - kw.

(4.22)

We have &h)(4)

4,) --+ 0-3

G(-) --t -co,

= &)

for

3

01~-+ 0+ (or a0 -+ l-).

(4.23)

DEEP

ELECTROPRODUCTION

AND

ANNIHILATION

I5

We note that, for 0 < p < 1, (G(+))* = ~5~) and j &(,)I” =: 1. For p > 1, ~3~.are real and satisfy -1 < G(+) < 0 and 63-j < -1. For 01~varying inside (0, 1) the cuts for t.~ < 1, run along the negative real axis and the fraction of the unit circle for which Re w < 1 - 2p2, while for TV> 1 they run along the real intervals (-co; +1 - 2~~ - 2p(p2 - l)‘/“) and (1 - 2~~ + 2p(pL2 - 1)lj2; 0). Finally from the end-point singularities at CY~= w( I -- a,,) we have &)(a,J with 0 < cuts -1 and

= 1 - (~72)

i +/2)((4/1

- 010)- PV.

(4.24)

&j(O) = &(+) ; for 0 < LX,< 1 they provide the cuts ending at &(+). For p < 2, i.e., in the stability region Lj(+) = &F-_) and / &q1)12 = 1. Their relative can be drawn as vertical lines, Re w = 1 - .$$. For p > 2, &) are real and 1 n < w(+) < 0, w(-) < -1. The cuts run along the real interval between &c_, d(-) and along the line Re o = 1 - 4~~.

\ I

II r--f-------) 41 I’

Rew

a FIG. 9a. The cuts on the w-planefor the function F(w), on the physical sheet for 0 < p < 1 (CL= exchangedparticlemass). .595/79/r-2

1<)1<2

I

b FIG. 9b.

The cuts on the w-plane for the function F(O) on the physical sheet for 1 < p < 2.

FIG. 9c.

The cuts on the w-plane for the function F(W) on the physical sheet for p > 2.

DEEP

ELECTROPRODUCTION

AND

ANNIHILATION

17

The situation is described for each case in Figs. 9a, 9b, and 9c. The relative positions of &) and ij(,) is as shown in the figures as G(+) is obtained from L;)(!) by substituting p with 2~. We note that, because of the location of the singularities on the unphysical sheets, it is in any case possibIe to deform the physical-sheet cuts such as to have them lying along the unit circle with branch points at & , and along the negative real axis. This is the same structure of singularities as shown in Fig. 7. 4.3. General Ladder Diagrams

The complete analysis of all singularities of the general ladder diagram looks like a rather hard task. We shall extract here a class of singularities which have a simple interpretation. These are the branch points analogous to the c&) = 1 - $(2p)2 $ ;(2p)((2p)2 --- 4)‘/2. which are obtained by replacing in cG(*) , p with 2~. Let us take for the amplitude A a ladder with n + 1 rungs, and consider the reduced graph obtained by contracting all vertical propagators in Fig. 5, i.e., by setting all pi equal to zero. We have (4.25) and also

(4.26)

Then in order to obtain a pinch between the singularity

surface (4.27)

and the contour (4.25), we have to impose that all derivatives

a IL _aa,IO orj)Iw2 -

w (2 - p2 ogj ajj + I]\

(k = CA..., n)

(4.28)

O=j

be equal. A solution of these equations is clearly given by “0 = ... = cd:,= (I/n + 1)

(4.29)

18

GATTO, MENOTTI, AND VENDRAMIN

and a,(;\ = 1 - &((n + 1) /A)2 & ((n + 1) /L/2)(((n + 1) /J)2 - 4)l’2.

(4.30)

Clearly these are branch points on the physical sheet. We see that, for increasing n, these branch points move along the circle of radius 1 toward the left and afterwards along the negative real axis with i;l_“’ -+ -co and 6:’ + O- as y1-+ co (see Fig. IO). M3) +a) wcty

,R------N

'\

: ---*--

I

: \

-4 w&l

w(+)

-$

----

41) q w(+) ' 40) t w (+I -$--------

zt77

\ \

\

i \

,'

-&-w t-1

I& $ A (1 w(- 1

/'C(2) -7231

w f-1

WC-1

FIG. 10. The branch points C& In’, for A given by the ladderwith n + 1 rungs,

5. THE SCALING FUNCTIONIN THE ANNIHILATION CHANNEL In the previous section we have shown that the deep inelastic scaling function F(w), corresponding to the double-box diagram admits an analytic continuation all along the positive real axis. In this section we shall outline the calculation which explicitly proves that for w > 1 F(w) = --F(w), with F(w) analytically continued above w = 1; F(w) is the scaling function for the annihilation diagram shown in Fig. 11 [20]. The result can be understood as follows: the expression given by Eqs. (2.1) and (2.3) (with A given by the box graph) coincides (as it can be explictly proved) with the Bjorken limit of the structure function given by the graph shown in Fig. 12, as calculated from the squared matrix element integrated over the phase-space of the outgoing particles. We go over from the scattering to the annihilation region by changing p = (M, 6) into (-A4, a). The possibility of having a relation of analyticcontinuation between the two regions is intimately connected to the fact that whenever the propagated particles are stable (the target particle is always assumed

DEEP

ELECTROPRODUCTION

AND

ANNIHILATION

19

stable) with respect to decay modes implicit in the graph under consideration, the propagators never become singular, either in the scattering or in the annihilation region. We shall in fact show in Section 6 that as soon as the stability conditions on the propagated particles are relaxed, cuts appear for w > 1, and the relation between the two scaling functions is modified.

9 ‘i --

q+k -----

k

FIG.

11. Annihilation diagram corresponding to the double box in the scattering channel.

FIG.

12. Inelastic scattering graph corresponding to the double box.

20

GATTO,

MENOTTI,

AND

VENDRAMIN

The structure function w( p, q) in the annihilation

. S(+‘((q + k)2 -

channel is given by

6”) S’+‘((l - k)2 - ,u”) S’+‘((~II - 1)~ (k2 - PZ,,~)~(1” - ml”)

@) . (5 1)

The ie can be omitted from the propagators which never become singular because of the stability of the propagated particles. We go over to light-cone and transverse-momentum variables putting

4 = an + t% k=pn+aii+&,

(5.2)

1 = pin + a@ + IT ,

p = -(M/2)(n

- it),

so that -q2 = 4&l, v =p.q

= -M(a-/I),

and for LX+ +co 6J = -(q2/2v) -+ -(2/3/M).

(5.3)

In this limit we have ~m+F(o)

8MG1 a = - 7 w s dp dp, da, d2i& d21r 1

7 S (-4(pl * 8 (h

-

- p) (a1 -

P -

+)

-

(27 -

ET)2 -

p2)

a1 + +)

* 6 (4 (7 + pl)(F

- al) - L2 - pa) 4---M

- PI + al>

1 * (-2Mwp

- ii r2 - mo2)” (--4p,a,

First we perform the pl- and p-integration transverse momentum I& .

- IT2 -

ml”)

-

and then integrate explicitly

(5.4)

over the

DEEP ELECTROPRODUCTION

21

AND ANNIHILATION

We have then CM/t&o

F(w) = -t?(w - 1) X2M $- &

jM 1

12

du, 4o,(4a,

1

I

2M IT2 + 2Ma, + 404Tp;M 4a, - 2M WW

L,4011’;M

+ Mb

- 2M)

- ml21

- .a2 + 2Mwp2

(4ul

_t

2M

+

2h4w’ -4a,

= B(w - 1) =q.o),

))

(5.5)

which shows explicitly the correct support property of the scaling function in the annihilation channel. Integrating over diT2, performing the substitution u r = (M/2)(x + w), and putting, as in Section 4, M = m, = 1, we obtain c

.F(w) = G, &

.Y

l--w

j

l2 0

dx

(

+J------

w

1(x; a)

w + x

(5.6)

g(x: w) 1'

where g, h, and I are functions of x and w given by Eqs. (4.2 1) (now we have w > 1). This shows directly that F(w) is given by the analytic continuation of F(w) (Eq. 4.20) for w > 1 along the real axis.

6. OCCURRENCE OF REAL BRANCH POINTS FOR u > 1

In this section we shall express the scaling function in term of the DGS [7] spectral function for the amplitude A. After specifying a box diagram for A we shall study the analyticity of P(w) along the whole positive real axis. We shall always assume the stability of the target hadron. We shall see that the analyticity for w > 1 is related to the stability of the vertically propagated particles. The equivalence of the present treatment with the one obtained from the Nambu representation is shown in Appendix A. The DGS representation for the amplitude A (see Fig. 2) is A(k2, P * W = j-l1 dP jlI;,, dh -k2 + 2;;A.;[)_

x _ iE .

(6.1)

For greater generality we take M2 = p2, in general different from mo2= m2. For F(w) we obtain (see Eq. (2.5))

F(w) = id j-y 4 4) I,,,

(1 - (w/B)) &0/B) &l - (w/S)) H(X; B) mo2@ -

w) + w(M2q8

+ A)

’

(6’2)

22

GATTO,

MENOTTI,

AND

VENDRAMIN

and for direct graphs (see Section 2) we have

= e(w) t?(l - W) 9 w), which defines the analytic function S(W). The simple box requires H(X, /?) for the Born term, which gives H(h, j?) = -(igz/(2r)3

8(/l - 1) 8(h - (p2 -fW))

(6.4)

l--o - w(m02 + A42 - p2) + m,2 *

(6.5)

and

SqlJ) =-e167r M2u2

Stability of the target requires M < m, + p. From Eq. (5.5) we find that (i) For /.L > M + m, the poles in o lie on the real negative axis. (ii) For m, > M + p (unstable parton) they lie on the real interval from +1 to too. (iii) Otherwise, i.e., m, - M < p < M + m, , the poles are complex conjugate and lie on the circle of radius (ma/M). This shows that for sufficiently high m, we expect singularities for w > 1; the problem of singularities for w > 1 will be considered in Section 7.

FIG. 13. Double box diagram.

DEEP

ELECTROPRODUCTION

AND

23

ANNIHILATION

We shall examine the double box shown in Fig. 13. For this double-box we need H((x, p) for the lower box in Fig. 13, which is given by

graph

where

The support of H(h, p) is easily understood, for stable target, by looking at the domain for A, X = +(01, p), as 01varies from 0 to 1. In particular it is immediately found that the lower integration bound A@) is given by the larger solution of the equation 4A

P, = (A- m2 + po2 + /3(m2 - p12 - M2))2 - 4p02(A + P2M2) = 0.

(6.8)

This has the form 4P)

= -M2P2

(6.9)

+ (PO + ( f@, M2, moZ,p12))1/2)2,

where (6.10)

f CA M2, mo2, p12) = M2/3* - p(m2 + M2 - p12) + m2, which has already been discussed previously. H(h, 8) can be explicitly evaluated by performing we obtain

the integral in Eq. (6.6) and

MA, p> = e(p) w - P)(l - p)ca/m

(6.11)

fl@, B>,

with

00 - 4P))P(k R(h~

I@

=

jGl

(d(h,

p))l/Z

[p2442

_

p(m2

P) - &Jo21 +

M2

-

p12)

+

m2]'

(6.12)

where B(X, /I) = h - m2 + P,,~ 5 p(m2 4 M* - p12). We notice that as M -c ,ul + m, p”p” - &m” + M2 - p12) + m2 > 0

for

O
If, in addition to M < p1 + m, also m < M + p1 holds, the inequality satisfied for 0 < /3 < co.

(6.13)

(6.13) is

24

GATTO,

MENO’ITI,

AND

VENDRAMIN

We see that i!Z shows the typical square-root (integrable) divergence on the boundary as h ---f (1@) + 0. The scaling function gets now the form

iw) = i?T3eb)e(i - w) w J: dp (1 - p) (1 - $-) (6.14)

= ecu) e(l - W)SC04 We shall next show how we can simply derive the analyticity of S(o) along the real axis for o > 0 for stable target and stable propagated particle of mass m. The proof makes use of the following facts: 1. II@) is an analytic function of j3. 2. R(h, p) is an analytic function of /3 for A > (1(p) and the integral over X in Eq. (6.14) converges. 3. For w > 0 the denominator in the integral in Eq. (6.14) never vanishes during the integration. Points 1 and 2 have already been examined. As for point 3 we have that (/I - w) m2 + w(M2c@

+ A)

3 (6 - W) m2 + w(M2wfi

+ A(@)) > (S - a) m2 + 4M2wS

= (p - W) m2 + w(M%@

- P(m2 + M2 - PI’) + m2),

+4(P))

(6.15)

where L’&@) is the value of L’I@) for p0 = 0. The last expression in Eq. (6.15) is positive for 0 < o < /3 < 1 as M < ~1~+ m; in fact, it is linear in p and positive for p = w and /3 = 1. On the other hand, if in addition we have m < p1 + M (stability of the particle of mass m) the same argument leads to the positivity of (6.15) also for 1 < ,8 < w. We point out that, as soon as the stability condition m < p1 + M is relaxed, branch points appear on the real axis for w > 1. In fact in this case, two branch points for o > 1 appear as end-point singularities (/3 = w) for the denominator /PM2 + m2 - p(m2 + Ma - p12) in a(& j?). Moreover if m > p,, + pl + M (unstability of the upper vertical propagator) two additional branch points appear for w > 1 as end-point singularities (/3 = 1, h = /1(l) = (pO + c”.1)2- M2 in the denominator (p - w) m2 + (M2wp + h) of the integrand in Eq. (6.14)). We can recover from the representation in Eq. (6.14), the position of the branch points of the double-box scaling function examined in Section 4 (M = m and

DEEP ELECTROPRODUCTION

25

AND ANNIHILATION

p0 = p1 = p). The branch points 8(*) are given by the end point p = w in the denominator rn2(p2 - 1) + p2j3; also they are given by the cone-like structure in A of A(A, /3) at the end-point /3 = w. The branch points 3(*) are given by the endpoint h = A(l), /3 = 1 in the denominator (/3 - w) W? -C w(um2/l + A). The endpoint h = A(p), /I = w in the same denominator gives the solutions w = 1 and w = 1 - p* on the second sheet, which were discussed in Section 4. The branch points W+ are given by the pinch between h = A(p) and m2(/3 - co) + w(m%$

The model we have just been considering, CL,,= 0 limit. For pu = 0, in fact, we have

-+ A) = 0.

admits also of a finite, nontrivial

(6.16) This gives for the scaling function

43 (1 - rB)U- (w/P))

(6.17)

which can be written in terms of elementary functions (logs). The scaling function takes a particularly simple form in the mass configuration considered by Nakaniski [21], i.e., M = m, p. = 0, p1 = 2M which simplifies R to the form W(X, /3) = 6,

and gives for

w

- M2(1 + 28)) M2U + p,’

(6.18)

F(W)

rr3G1 -F(w)

=

M4(1

+

w)2

[(~CIJ + 1) In (*)

+ 2(w - I )I.

(6.19)

Such a scaling function has only singularities along the negative real axis and no complex singularity. This feature is due to the large mass which is exchanged in the lowest rung.

7. CONTINUATION

OF THE SCALING

FUNCTION

IN PRESENCE OF REAL

CUTS

In this section we shall examine the problem of the analytic continuation of the scaling function to the annihilation region in presence of cuts for w > 1. In this case the scaling function in the annihilation channel cannot simply be given by the analytic continuation of F(U) for w > 1, because of the ambiguity introduced by

26

GATTO, MENOTTI,

AND VENDRAMIN

the cut. As we have already mentioned in discussing the double box in the previous section, such cuts are expected to be present whenever some kind of instability occurs in the vertical propagators. In this model we can evaluate explicitly both the scaling function F(w) in the annihilation region and the analytic continuation of F(w) above w = 1. We can, thus, compare the two results and find that there is no simple connection between the two scaling functions [22].

FIG.

14. Box diagram with dressed propagators.

The model we shall examine is the diagram shown in Fig. 14, where the vertical propagators are dressed propagators given by the Lehmann representation

17(k2)= j In order to understand evaluate the Bjorken limit exchanged on the vertical function on the light-cone

-3

p(m”) dm2 + m2 - ic ’

(7.1)

the properties of the graph in Fig. 14, we first have to of a box with unequal masses (to be called m and m’) lines. By evaluating the Fourier transform of the wave we get for such limit

iMg2 iMOXQ @4 ma,m’2) = - 2(2n)4 s dx, e d4k

e-ikwz,

(-k2 + m2 - ie)(-k2 + ml2 - i#-(p

- k)2 + ,LL,,~ - ie)

= O(OJ)O(l - 0~) S(W, m2, m’2),

(7.2)

with 2

S(w, m2, mt2) = -Ig6a

1 m2

_

m’2

In

M2u2 - w(M2 + ma - po2) + m2 M2u2 - u(M2 + ml2 - ~~2) + ml2 * (7.3)

The singularities of this function are given by the solutions of the equation f (w, m”) 3 M2u2 - u(M2+m2-p,,2)+m2=0

(7.4)

DEEP

ELECTROPRODUCTION

AND

27

ANNIHILATION

and of the equation obtained by replacing m with IN’ in (7.4). This is the same equation which characterizes the position of the singularities of the simple box. which have been discussed in Section 6. In particular we recall that for m > M + f.~~(unstable parton), w+ both lie on the positive real axis and CO* > 1. Moreover it is easily seen that for m = A4 I- p,, . WA ==-w 1 and for increasing m3, w+ and CJLmove in opposite directions (Bw-/8m2) < 0,

(~w+/ihP) > 0).

Form2++rx,onehasw_-+1+andw+++co. Integrating over the weight function appearing in Eq. (7.1), we set for the scaling function of the graph in Fig. 14 .F(co) = j dm2 dml2@T’~) p(nP) T(w, UP, nP).

(7.6)

The weight function p(m2) will be assumedto be of the form p(n22)= const 6(m2 - 3) + 6(m2 - (M t p,,Y)f(m2)3

(7.7)

and the stability of the target implies ~3 > A4 - p,) . The scaling function 9(w), as given by Eq. (7.6), is analytic for real positive w except for a cut from 1 to + co. Moreover, ,9(w) has two complex branch points at

G(i) == (1/2M”)[fi2 -

/co2

+

A42

&

i(4FPIt42

-

(‘442

+ E? -

I-Loz)2)1/2].

(7.8)

These lie at finite distance from the real axis, thus, allowing for the analytic continuation of F(w) along the real axis above and below the cut 1 < w < CO. We now calculate the scaling function in the annihilation channel as due to the processof Fig. 1.5.

Frc.

15.

Diagramfor photonannihilationcorresponding to the scatteringdiagramof Fig. 14.

28

GATTO,

MENOTTI,

AND

VENDRAMIN

We have clearly (P = --p * q), v V(q, p) + F(w)

=

dm2 dmf2 p(m”) p(mf2) F(w, m2, m12),

I

(7.9)

where F(w, m2, m’2) is calculated from the Bjorken limit of S W(q, p, m2, mf2)

= $

j & St-G + d2 + W K$, + qo) SC-(p (-k2

+ m2 -

k)2 + Po2) O(p, + m’2 + ie)

ie)(-k2

k,,)

(7.10)

Notice the opposite signs of it- in the denominators, as the final result Eq. (7.9) corresponds to a sum of squares of matrix elements. Following for instance Ref. (5) we obtain F(w, m2, mf2) = $$I

dp do d2RT

* S(a -

@hJ)

q--M2 q--M

+ 4pa + /CT2 -

p -

2M(p

-

u) + po2)

u)

(Er2 + 4pu + m2 - ic)(RT2 + 4pu + ml2 + ic) = 8(6~ - 1) ~(co, m2, m12),

(7.11)

where g(o,

m2, m’2) = Af16~

1 ml2 -

m2 + 2i.z

. {ln[f(w, m”) + ic(w - l)] - lnLf(w, m’2) - ie(u -

l)]}, (7.12)

andf(w, m”) was defined in Eq. (7.4). We now compare 9(w, m2, m’2) to the analytic continuation above w = 1. We have S(t.0

of S(w, m2, m’2)

- k, m2, m’2)

= ( g2/16?r)(l/m2 - m12)(ln[f(w, m2) + ic(M2 + m2 - po2 - 2M2w)] (7.13) - ln[f(w, mf2) + ie(M2 + m” - po2 - 2M2w)]). From Eqs (7.12) and (7.13) we immediately

get

Re ~(oJ, m2, mr2) = -Re F(w, m2, m’2) + (rg2/8)

6(m2 - m12) O(-j(w,

m”))

(7.14) and Im S(w, m2, m’2) = -1m F(w, mf2, m”).

(7.15)

DEEP ELECTROPRODUCTION

AND ANNIHILATION

29

lntegrating over the weight functions p to obtain F(w) and taking into account Eqs. (7.14) and (7.15), we find F(w)

= j dtrP drn12 p(m”) p(m12) F(w, == J’ dm2 dttz’2 p(m2) p(m’“) == -Re

m2, m’2)

Re .F(w, m2, tti2)

1 dm’ dm’2 p(m”) p(m’2) F(w

-

ic, tn2, m”)

+ $

jzuj

p2(m2)dtn”

CL

= -ReF(w

-

it)+%

I L(w)

(7.16)

p2(m2) dm2,

where the lower limit L(w) is given by L(w) = W(W + (w/w - l)(/Lo2/M2)).

(7.17)

This shows that, where cuts are present for u > 1, the relation between F(w) and F(w) is not one of simple analytic continuation. The cut, whenever present, has its lower extreme below the point w = 1 + &,/M) and the integral on the right hand side of Eq. (7.16) contributes as soon as w lies on the cut. If p(m2), as expected. stretches up to co, the cut starts from w = 1.

8. GENERAL

CONCLUSIONS

AND

OUTLOOK

In the present paper we have considered the question of the validity of the analytic continuation rules (see Eq. (1.5)) connecting the scaling functions in the annihilation channel to those of the scattering channel. Our analysis has been carried out through explicit evaluation of various examples, showing different features, in order to reach a general understanding of the problem. We have concentrated on the model of a superrenormalizable field-theory, since this is the only example of field theory which shows scaling in perturbation theory without cut-off; the photon can be taken to be scalar without any qualitative change. The first point we have examined is the analyticity of the scaling function F(w); in other words, the singularity structure of F(w) in the complex w-plane and the consequent possibility of giving an analytic continuation of F(w) for w > I (annihilation region). As far as the propagated particles are stable we find that (for a simple box, with or without dressed propagator or for a double box) the cut structure can be given the form shown in Fig. 7. The presence of a finite gap around w = 1 in the circular cut is related to the absence of zero mass particles in the theory, and it allows the continuation of F(w) for w > 1. An important result is obtained by varying the masses appearing in the internal lines of the double box.

30

GATTO,

MENOTTI,

AND

VENDRAMIN

When the unstability limit is reached for a vertically propagated particle, cuts appear for w > 1 along the real axis on the physical sheet, thus showing (as the analytic continuation of F(U) is complex) that the analytic continuation rules (1.5) are invalidated. The second point examined (which is also related to the occurrence of cuts for u > 1) is that of the coincidence, apart from a possible sign depending on the statistics of the target, of the analytic continuation of F(o) with the scaling function in the annihilation region, F(w). In the previously mentioned examples with stable vertically propagated particles we explicitly prove the relation F(w) = --F(w) for w > 1; for the double box we, thus, have an explicit check of the recently proposed argument by Landshoff and Polkinghorne [8] on the coincidence of --F(w) with the analytic continuation of F(w) for ladder graphs provided the propagated particles are all stable. The essential ingredient in their proof is the expression of the scaling function in the annihilation channel in terms of the $-discontinuity (s’ = (p - /c)~) of the parton-proton amplitude A(s’, k12 + ic, Ic,~ - ie)

of Fig. 3, where one virtual parton mass is taken above and the other below the relative /?-cuts. The coincidence of F(w) with the analytic continuation of F(w) for w > 1 is due to the vanishing of the s’-discontinuity of the k,2-discontinuity of A. In other words the presence of double discontinuity graphs is responsible for the failure of the analytic continuation rules, a fact which has also been pointed out by Suri [2]. As we have seen above (see expecially Section 6) the presence of unstability is responsible for the appearance of cuts for w > 1 on the physical sheet. We have, thus, been led to an explicit investigation of a simple model showing unstability and, consequently, cuts for w > 1. The model is a box graph with dressed vertical propagators; the scaling functions F(w) and F(w) can be evaluated in terms of the spectral function p(m2) of the dressed propagators obtaining the relation (8.1)

where L(w) is defined in Eq. (7.17). Equation (8.1) has to be contrasted to the naive analytic continuation rule F(w) = --F(w), the failure of which is easily connected with the nonvanishing of the double discontinuity of the considered graph. In fact, the double discontinuity in our case is given by S(S

-

p2)

p ( ““:

+_ if”

+

M2w)

p(m2) dm2

J “2

T

fT2

+

~2~

_

m2 -

(8.2) ie ’

DEEP

ELECTROPRODUCTION

AND

31

ANNIHILATlON

which for m > M + ,u and w > 1 does not vanish. A nonvanishing double discontinuity is also present in graphs having the box structure with “dressed” lower vertices. This is the case for, e.g., the Drell and Lee [23] model where the lower vertex is given by a composite particle vertex function. The k2-discontinuity of the vertex function is responsible for the failure of the exact analytic continuation rule; however, as the discontinuity is vanishing more rapidly than the vertex function, the analytic continuation rule holds for w around I at least at the leading order in (w - 1) [23,24]. A general analysis of the continuation problem based on the light-cone formalism has recently been given [25]. Both channels are expressed in terms of the analytically continued nonforward Compton amplitude, and the conditions for validity of the analytic continuation are determined. Furthermore, one shows that nontrivial Bjorken scaling exists in the annihilation channel provided the asymptotic total annihilation cross-section scales canonically. Summarizing the analysis performed in the present work we believe that a complete clarification has emerged of the principal dynamical circumstances which are relevant to the question whether by analytically continuing the scattering scaling functions we can determine the annihilation scaling functions. Furthermore, the structure of the singularities of the scaling function has been studied in different models, such that we can now have some definite and general ideas about them. The crucial role in the continuation problem is played by the “double discontinuity” discussed previously, and situations for which such double discontinuity vanishes or does not vanish can be concretely examined. When experiment will be able to tell us whether the idea of analytic continuation applies or not, we shall be able to infer important conclusions on the underlying dynamical structure of the theory and of its light-cone limit.

APPENDIX

A

Relation Between the Expressions for the Scaling Function Obtained from the DGS Representation and from the Nambu Representation In the following we shall employ a technique due to Nakanishi validity of all the relevant hypotheses. We insert into Eq. (3.2) the identity

[7], assuming the

(Al) where the limits in p are evident from Eqs. (3.3~ and 3.3d), and (1(/I) and +x 595/79/I-3

are

32

GATTO, MENOTTI,

AND VENDRAMIN

in fact the boundaries for Fp)/Fin’ when the Feynman parameters are varied subject to the constraints

l=j

O=i

With the assumptions required by the Nakanishi’s

*Iomdolo 3 daidPi %010

+

C1"=i

(%

method we arrive at

+

pi)

-

1)

c(n'FP'

(A3)

It is then enough to verify that the function H(n)@; A) = Srn duo fi dai dpi 0

14

. 6(ao + CFzi (ai + pi) -

1) S(p - (Fp’/Fp’)) C(n)I;(W 1

6(X - (F,(n)/Fp))) (A4)

is the DGS. spectral function, in order to have the analytic function F(w) in the form

as provided from the DGS representation. REFERENCES D. J. LEVY, AND T. M. YAN, Phys. Rev. 187 (1969), 2159; see also J. PESTIEAU Phys. Lett. 30B (1969), 483. D. J. LEVY, AND T. M. YAN, Phys. Rev. 187 (1969), 2159 and P&s. Rev. Dl (1970), 1617; a general analysis is given by P. V. LANDSHOFF, J. C. POLKINGHORNE, AND R. D. SHORT, Nucl. Phys. B28 (1971), 225; A. SURI, Phys. Rev. D4 (1971), 570; G. PREPARATA, Frascati Meeting on Electromagnetic Interactions, Frascati, Italy, 1972. 3. P. V. LANDSHOFF, Phys. Lett. 32B (1970), 57. 4. R. JACKIW, R. P. VAN ROYEN, AND G. B. WEST, Phys. Rev. D2 (1970), 2473; H. LEUTWYLER AND J. STERN, Nucl. Phys. 20B (1970), 77. For general light-cone analysis see K. G. WILSON,

1. S. D. DRELL, AND P. ROY, 2. S. D. DRELL,

DEEP

5. 6. 7. 8. 9. 10. 11. 12.

13. 14. 15.

16.

17.

18.

19. 20.

21. 22. 23. 24. 25.

ELECTROPRODUCTION

AND

ANNIHILATION

33

P&S. &a. 179 (1969), 1499; R. BRANDT AND G. PREPARATA, Nucl. P&T. 27B (1971), 541; Y. FRISHMAN, Phys. Rev. Lat. 25 (1970), 966; H. FRITXH AND M. GELL-MANN, “Scale invariance and light-cone,” Gordon and Breach (1971); J. M. CORNWALL AND R. JACKIW, Phys. Rev, D4 (1971), 367; D. A. Drcus, R. JACKIW, AND V. L. TEPLITZ, Phys. Rev. D4 (1971), 1733. See e. g. R. GATTO AND P. MENOTTI, Nuovo Cimento 7A (1972), 118. See R. J. EDEN, P. V. LANDSHOFF, D. I. OLIVE, AND J. C. POLKINGHORNE, “The analytic S-matrix,” Cambridge University Press, London/New York, 1966. S, DESER, W. GILBERT, AND E. C. G. SUDARSHAN, Phy~ Rec. 115 (1959), 731; M. IDA, Pvogr. Theor. Phys. Kyoto 23 (1960), 1151; N. NAKANISHI, Progr. Theor. Phys. Suppl. 18 (1961), 1. P. V. LANDSHOFF AND J. C. POLKINGHORNE, Preprint DAMTP 72116. R. GATTO, P. MENOT-Q AND I. VENDRAMIN, Lett. Nuoro Cimrnto 4 (1972), 79. See S. D. DRELL et al. Ref. [2]; A. SURI, Ref. [2]. J. D. BJORKEN AND E. A. PASCHOS, Whys. Rev. 185 (1969), 1975; see P. V. LANDSHOFF, J. C. POLKINGHORNE, AND R. D. SHORT, Ref. [2]. This is true provided the interaction is sufficiently regular (superrenormalizable interaction) such as to give a wave function finite on the light-cone in coordinate space. See e.g. R. GATTO AND P. MENOTTI, NUODO Cimento A 2 (1971), 881. The extension to vector photons is trivial; the scaling function Ez differs from the one discussed here by a factor w*. R. GATTO AND P. MENOTTI, Ref. [5]. I. G. HALLIDAY, Nuovo Cimento 51A (1967), 971; G. ALTARELLI AND H. R. RUEINSTEIN, Phys. Rec. 187 (1969), 2111; T. R. GAISSER AND J. C. POLKINGHORNE, Nuouo Cimemo A 1 (1971), 501: S. I. CHANC; AND P. M. FISHBANE, Phys. Rec. D2 (1970). 1084. The two branch points &;(+) (which coincide with the poles of the simple box) collide for p + 0 in w = 1. At the same time, however, the function F(w) blows up for all values of w due to an infrared divergence. We consider here pL2 > 0. The change of variables given by Eq. (4.14) analyses the situation at ca in the Feynman parameters with 01~ growing no more quickly than 01~ . A different change of variables explointing the opposite situation does not show any further singularity. Such “singularity killing” phenomenon is not considered in the current treatment of Feynman graph singularities. The role played by the numerator is evidenced by modifying in Eq. (4.15) the numerator into an expression non vanishing at ,5’ = OJ (y - 1): the integrated form, corresponding to Eq. (4.20) then shows an actual singularity at w ; 1 -- cc?. S. D. DRELL et al., Ref. [l]. For the general ladder graph in Yukawa theory with cut off see S. D. DRELI., D. J. LEVY AND T. M. YAN, Ref. [2]. For a general discussion of ladder graphs see P. V. LANDSHOFF AND J. C. POLK~NGKORNE, Ref. [22]. N. NAKANISHI, Phys. Rev. 135B (1964), 1431. See also P. V. LANDSHOFF AND J. C. POLKINGHORNE, DAMTP, preprint and to be published; R. GATTO, P. MENOTTI, AND I. VENDRAMIN, Lett. Nuoco Cimento 4 (1972), 79. S. D. DRELL AND T. D. LEE, Whys. Rev. D 5 (1972), 1738. P. MENOTTI in Proceedings of the Informal Meeting on Electromagnetic Interactions, Frascati, Italy May 1972. R. GATTO AND G. PREPARATA, Nwl. Phys. B 47 (1972). 313.