On the theory of dispersion relations for virtual processes

-~

Nuclear Physics 10 (1959) 71--81; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or mic~rofilmwithout written permission from the pt~hllRher

ON THE THEORY OF DISPERSION RELATIONS FOR VIRTUAL PROCESSES A. A. L O G U N O V

Laboratory ot Theoretical Physics, Joint Institute at Nudear Research, Dulma, USSR Received 4 M a y 1958 Abstract:

A d e d u c t i o n of t h e dispersion relations for virtual p h o t o p r o d u e t i o n is presented.

1. I n t r o d u c t i o n

Dispersion relations for virtual processes have been considered in ref.X). These relations permit one to investigate inelastic processes in which electrons,/~ mesons, etc., participate along with strongly interacting particles (nucleons, n mesons, etc.). Examples of such processes are e + p --> e + n + n , n + p --> e + + e - + p ,

7 + P -+ e + + e - + P e + p -+ e + 7 + p.

Processes of production of n-mesons in electron-nucleon collisions and their relation to virtual photoproduction were examined in ref.2), and the dispersion relations for virtual photoproduction were investigated. It should be especially noted that the invariant functions FlP, n(k~), FzP, n(k2), which characterize the electromagnetic structure of the nucleon enter the dispersion relations for this process. A consequence of this is that independent information on the electromagnetic structure of the nucleon can be derived from experimental data relating to this process. In the present paper (which is a direct continuation of ref. ~) and uses the same notations) we give a proof of the dispersion relations for virtual processes for the particular case of virtual photoproduction, the method of N. N. Bogolubov s) being used for this purpose. 2. A n a l y t i c P r o p e r t i e s of the V i r t u a l P h o t o p r o d u c t i o n A m p l i t u d e in the F i c t i t i o u s R e g i o n

For the investigation of the analytic properties of the virtual photoproduction amplitude we assume that the squares of the 4-dimensional vectors k and q have the following values: k ~ = ~(r),

~(~) = ~ - ~ - m ~ ,

qZ = 3,

r < --pZ(1--e) 2. 71

(~.1)

72

A . A . LOGUNOV

We shall assume as usual that k, q, p and p' are related b y the equation

k+/~ = q+~'. Then in the coordinate system in which p + p ' = k = 2e--(l+e)p, q = 2e+(1--e)p,

(2.2) 0 we get

4ep.2 = / z 2 + m r ~ 2 = V~E~--(1--e)'p~--z.

(2.3)

It is obvious that As > E 2 since z < --p2(1--e)2. This is a very important circumstance as in this case Im E > l i m a / E * - - (1--e)*p2--~[,

Im E : # 0.

(2.4)

The retarded and advanced amplitudes of the process can be written in the form

~L(k+---~) ---- --fe"C~+')'F~,L (z)~ F ~t,,,,,(z--y)

=

e-~'(P"P)(a~+') ( # ' ,

_

adv

F~,.(z--y) =

_

s'l o~p~(v)(x)I#, s>,

(2.5)

~i(/~-i-q)z adv

e_ti(p, p)(~_,} ( p , ,

~/'p (x)

s'l~

I#,s).

(2.6)

These amplitudes are proportional to the quantities T and T+ introduced in ref.2); thus

T~t = _ (p0p,0l~n~)-~ T and similarly for T aav and T +. It m a y be recalled that according to the causality principle /~(z)=O adv F,,,,,(z)=O

if if

x~O, z_> O.

(2.7)

In the reference system in w h i c h . p + p ' = 0 the retarded and advanced amplitudes can be written in the form ret~,(E, lr) = Ta,

~ , o ( E , T) = - -

d.

fe *(E®°-e'ffi~/~t'(x-')'pt-r)+*cp'= Fret (z)dz,

-- J

f

--~gp 09

e *(E'0"-e'z~/Et--(l-s)'pt'-;)+~O'z

F a,advo(z)dz"

(2.8)

(2.9)

Since in (2.8) the integration is performed in the region

Zo > Ixl,

Zo > o,

one has

< --pZ(1--8) ~, Im E > [ImV'E ~ - (1--,)~p~--TI;

(2.10)

ON

THE* T H E O R Y

OF DISPERSION

RELATIONS

FOR. V I R T U A L

PROCESSES

73

and hence the function S± Tret(E, 3) will be analytic in the upper half plane of the complex variable E. It should be recalled t h a t S . means symmetrization or antisymmetrization with respect to the vector e. In an analogous way it can be proved that S.TVav(E, 3) is an analytic function in the lower half plane of the complex variable E. It can be shown that under certain conditions S T ret and S T ~v are identical on a segment of the real axis of E. To prove this the behaviour of the difference of ST(E, 3) for real values of E should be investigated:

ST~,o(k+q) = S T ~` -- --Sfe

__

ST*d* ~(~-ff)~o

(2.11)

{F.,rctt.(x)--F=,a d v.(x)}dx.

As shown in ref.*) and *) this can be written in the form

ST~,,~,(E,3) = 2rri ~ S (p', s'l/'p(0)tn, Ae--ep)(n, ~e--ePli~(O)]#, s) n

• 6 ( E + M~-~-+p * - ~ / M . a + a 2 + r 2 p~) -- 2~ri ~ S (p', s'[i~ (0)In,--)te+rp)(n, --he +ePlip (0)IP, s>

(2.12)

n

• 8 ( E - - ~/M~-+ p * + ~ / M . * + A * + , 2 p2).

Separating the term referring to the intermediate single-nucleon state and performing elementary transformations of the 8 functions we write (2.12) in the form

STy,, o,(E, 3) = S[~,,~,(E, 3) + 2rdSA (p, 3)8(E-~-Ep) -- 2rdSB(p, 3)8('E--Et,),

(2.13)

where

S[~,,o(E, 3) = 2rd,,>t~. l 1 + ~ o

S(p"s'[iP(O)[n'Ae--eP)

• (n, Xe-epliv(0)l p, I Ep.

s)8(E--Ep.(3))

(2.14)

-2~i , ,X> x [ 1 + -~o S(p',s I~,(0)In,"- a e + e p ) • (n, --~te+eplip (0)IP,

t

s)~(E+ Ep~(.)),

SA (p, 3) ---- ] 1 -- E p [ • S' Po I ,-

• (s", Ae--ep]i~(0)]p, s),

• (s",

-,~e+~Plip (o)IP, s ) ,

(2.15)

74

A. A. LOGUNOV

(2.16)

Vm2+p~Ep = p2+2a~--¼(m,'+p~ ).

If the usual assumption is made that between M and M + p there are no bound states in the meson-nucleon system, i.e., that

M,, ~ M + p

(n > 1),

one finds for the scatterer momenta

p~ < ½Mp+~#2+4~m,9--~L¢.

(2.17)

The function T,,.o,(E , ,) has the energy spectrum represented on fig. 1.

Ee=

,V/(pa+MS ) Fig. 1

Summing up we can say that in this section it has been proved that if

- - V < r < --p2(l--e)~'

(2.18)

STret(E, ~) will be an analytic function in the upper half plane of E and ST ~v in the lower half plane of E. On the other hand if the condition p2 <

(2.19)

is valid there will always be a segment (--Eq, Eq) on the real E axis for which STret(E) and STaY(E) are identical:

STret(E, ~) ~ STSav(E, ~), Im E = 0, [El < Eq. (2.20) Thus the set of functions STret(E, ~) and S~ a~v(E, ~) can be represented b y the function

S~(E,

= l sTret (g, x)

Im E > 0

| STSaV(E, ~)

Im E < 0

T)

(2.21)

which is analytic over the whole complex plane of the variable E, excluding the cuts along the real axis --oo
e,

E e
oo,

ImE=0,

(2.22)

and which has simple poles at the points E = + Ep and does not increase at infinity more rapidly than a polynomial of the n tu degree.

ON

THE

THEORY

OF

DISPERSION

RELATIONS

FOR

VIRTUAL

75

PROCESSES

Thus the Cauchy theorem, with the contour indicated on fig. 2, can be applied to the function

S~(E, ~)

g ( E , "~) =

(2.23)

(E--Eo)"+a

where "

"v'~r-Y--I-P21Eol < M#--p2--F}p2-.l.-¼(rn~,2-Fp2)--2-~(Eova Ep).

Fig. 2

When the radius of the large circle approaches infinity and the radii of the semi-circles approach zero we obtain

S~(E, T) -- (E--E°)"+' f 2=i

J

S/(E', ~)dE'

IE'I> ~e

(E'--Eo)"+~(E'--E)

SA(p,v)(Eo--E'~"+I

SB(lb,'~)(E--Eo'~"+'

E+Ep \Eo+Ep/ "[- E C,(~, P, Eo)E'.

(2.24)

Ep--E \Ep--Eo]

O
The dispersion relations (2.24) were derived only for the fictitious range of values of z. 3. A n a l y t i c

Continuation

with

Respect

to Variable

T

In order to change from the fictitious values of • to the value v ----/,3 of interest to us we must perform an analytic continuation. To do this we make use of the following result obtained in ref. 5, ~). If the variables E, x, R are real, - - V ~ • ~ (1--[-p)p2

(3.1)

and 2 0 < p2 < Pm,,,

P*m~.-- M+pM p2+m~4 2 ~-~(2M+p)p V § M4/z2-~-mr2 (2M--#) 1+

:+

4/z2+m~, 2

M

(M+p) 2

2M+p

13 ' if

m~,2 < 3p 2,

76

A. A. LOGUNOV"

then the function/(E,

w) in expression (2.13) can be represented in the form

S/(E, z) = Fx(~+2EV/Kr'-f-p2, v)-f-F2(~--2Ev/-M-T+p ', w)

(3.2)

where the functions F~(y, ~) have the following properties: a) F d g , z) are generalized functions of variable y, b) F~(y, ~) are analytic functions of variable z and are regular in the region

ptt~, c) Fdy, w) = 0 for y < 2M#--2p~+#~+{(mv~+#" ). --V ~ Re w < ( l + p ) p ~,

[Im w] <

(3.3)

We shall now demonstrate that from the representation (3.2) follows the validity of the dispersion relatioh (2.31) for the value = ~* (3.4) of interest to us. Let us take some negative ~, which satisfies (2.25). Then for the given the dispersi6n relations (2.31) as well as the representation (3.2) will be satisfied simultaneously. Inserting (3.2) into (2.31) we obtain

(Eo-E~"+~

SA(p,~)

S~(E,w) = ~b(E, w)

\Eo+E p/ SB(p,r) [ E--Eo\"+I

(E+Ep(z))

(3.5)

(E~(~)--E) ~E---p-~--Eo) + o<,<.2c,(w, p, ~o)E" where

~(E,w) =

(E--E°)"+1 f 2~i

(E--Eo)"+t f + 2~ti J

dE" F~(2E'%/M~+p', w) E"--E--

(e"-E-

,I[~,,-Eo-

~

2V'M2-4-p']~ 2~/~1 dE" F2(2E"V'M2+p 2, ~)

Z_L___~[E"-Eo-

~

V +~

(3.6)

V+'

The function ¢(E, w) will be an analytic function of E in the region in which neither of the denominators

vanishes; that is, providing that .g

It is obvious that the condition (3.8) will always be satisfied if lira w[ < 2v~-~--k-p'l I m El.

(3.9)

ON

THE

THEORY

OF

DISPERSION

RELATIONS

FOR

VIRTUAL

77

PROCESSES

If the inequality (3.9) is satisfied, dp(E, T) will be an analytic function with respect to the variable E. Consider the analytic properties of the function ~b(E, ~) with respect to the variable ,. Since the F~ (y, v) are analytic functions of r in the region --V < R e , < (1-}-p)/z2,

Jim t[ < p/zz,

(3.10)

the function ¢(E, T) will also be analytic in this region if neither of the denominators gt_

Eo -

2

T

~n+l

(__E, Eo+ "

'

2

7:

'~-+1

(3.11)

vanishes in the region (3.10). If a real number is ch.osen for Eo, neither of the denominators (3.11) will vanish for any complex values of r from region (3.10) and the function ~(E, ~) will be analytic for all complex values of in the range (3.10). For real values of r in the region (3.10) the denominators (3.11) will also not vanish if E o is chosen so that IE0[ <

2 M p + p 2 - 2p~--~+ ½ (mr~+p ~)

(3.12)

However, since T < (1-~-p)~u~,

p~ < pZ

,

the inequality (3.12) will be satisfied if [Ed <

2Mp-{-Pa--2P2-~-- (1-t-P)Pz-{-½(m~'z-bPz)

(3.13)

2~/M~+p ~ Thus, if E o is chosen in a suitable manner, the function ~b(E, T) will also be analytic for real values of z from the region (3.10). Summing up to results obtained above, we m a y say that if E o is restricted by the inequality (3.13), the integrals (3.6) will define a function ~(E, z) which is an analytic function m the region --V < Re r < ( l + p ) p 2, IIm r[ < p,uL lira r[ < 2 V ' ~ - + p ~ l I m El.

(3.14)

However, it has been previously established by us that the function S ~ (E, ~) is regular in the region I m E > lira ~/E2--(1--e)2p~--~I, I m E < - - [ I m x/E2--(1--e)*p2--rl.

(3.15)

{S~(E, r)--~b (E, v)} (E~-- Ep2 (T))

(3.16),

Thus the difference

78

A. A. LOGUNOW

should be an analytic function of the variables E in the common part of regions (3.14) and (3.15), i.e., --V < Re 3 < (l+p)p 2, IIm 31 < p/~, tim 31 < 2 ~ / ~ + p ~ l l m El, IIm El > [Im~/E~--(1--*)~P~--3].

(3.17)

However, for real 3 < -- max [(1--e)2p 2, 2pl], this difference, according to the dispersion relation (2.31), is a polynomial in E,

--SA(t,,3)(F.--ED) { E°--E F+I \Eo+Ep !

+(E'--Ep'(3))

X

(

"+1

+SB(p, 3)(E+Ep) \Ep--Eol

(3.18)

Cr(3, P, Eo) Er.

0
In virtue of the uniqueness of the analytic continuation it will be a polynomial in E over the complete region (3.18). The functions A (p, 3), B(p, 3) and C,(3, p, Eo) can therefore be analytically continued to a certain region of complex values of 3. It can be shown that this region contains at least the whole region --V < R e 3 < ( l + p ) p 2,

iim3 [ < ppS.

(3.19)

Indeed, let us take an arbitrary ~ lying in the region (3.19). For a given 3+* = %+i~/, ~/=/= 0, we choose E+* ---- E , + i E ~ so that the pair 3+*, E+* lies in the region (3.17). This can be done if E is chosen so that the following conditions be satisfied: lira El > 1 Im~/E'--(1--e)~P~--3], IIm 31 < 2~/M2+PS] Im El.

(3.20)

It is easy to verify that the conditions (3.20) will hold if E, and E~ are chosen as follows: E, = ~l~q/Er,

E r < M,

Er,--~Lq2/Er,--3r--(1--e)2p" > 0.

(3.21)

Thus for any point 3~* from the region (3.19), Im 3 , * ~ 0, it is always possible to find such a E** that the pairs 3~* and E** will lie in the region (3.17). A consequence of this is that the functions A(p, 3), B(p, 3) and Cr(3, p, E0) will be analytic with respect to 3 over the whole region (3.19) and possibly possess a cut along the real axis of 3. It can be shown that as a matter of fact this cut does not exist. Indeed, choose the real ~, so that --V < 3r < ( l + # ) p ~ and assume 3 -= 3, = 3r4-i~, E = E, = E,+in/2E,,

n > 0;

~/is a sufficiently small number. The pairs 3+, E+ and 3_, E_ will be in the

O1q THE THEORY OF DISPERSION RELATIONS FOR VIRTUAL PROCESSES

79

region (3.17) only if, in accord with (3.21), the following inequalities are fulfilled: g,~--{gq'/g,a--~,--(1--8)'p 2 >0, E, < M. However, if the points T, E are in the region (3.17) t h e y will also be in the broader region (3.14) in which the function ~(E, ~) is analytic. However, such values of z will be in the region (3.14) for which Im ~ = 0. Therefore, b y continuity we have lira ~(E~, T.) = lim qb(E,-4-ie, ~,). t/--~0

(3.22)

6--~0

Using (3.6), the following important relation can be derived: lim {4,(E+, ~+)--4,(E_, ~_)} = lim {~(E,+ie, ~,)--~(E,--ie, ~,)} ~--~0

*-~0

•>o

(3.23)

=

Taking into account (2.19) we get lira [~b(E+, ~+)--~(E_, ~_)](E,'--Ep'(~)) = ST(E,, ~r)(Er'--Ep'(x)). (3.24) On the other hand, since E+, ~+ and E_, T_ enter the region (3.17) one can employ expressions (2.10) and (2.11) for such pairs of points; in the given ease these expressions are meaningful and for ~/--~ 0 tend to the functions

sIret(E,, ~,), Sitar(E,, ~,). We thus obtain the following limiting relation: lim {S~(E+, ~+)--S~(E_, r_)}

,~o

= {Slret(E,, T,)--ST~v(E,, ~,)} = ST(E,, ~,). (3.25) Multiplying (3.25)by (E,~--Ep~(,,)) and combining with (3.24)we obtain (E,'--Ep'(T,)) lira {S~(E+, z+)--~(E+, T+)} ,t~o

= (E,~--Ep2(T,)) lira {S~(E_, ~_)--~(E_, T_)}.

(3.26)

~--+0

Inasmuch as the pairs of points (E+, ~+) and (E_, ~_) lie in region (3.17), the functions on both sides in eq. (3.26) will be identical with the polynomial in E. Denoting this polynomial by P,(E) we can rewrite (3.26) as follows: lim P,+(E+) = lim P,_(E_) ~---~0

(3.27)

~/---~0

or by continuity

lim P,+(E,) = lim P,_(E,). ~/--~0

(3.28)

~--+0

But this signifies that the function P,(E) is continuous when ~ passes across the real axis. Since P,(E) is an analytic function of ~ above and below the real axis it will also be an analytic function for points on the real

80

A.A.

LOGUNOV

axis, that is, no cuts will exist. Functions A (T, ~), B(z, p) and C(T, p, E0) will therefore be analytic in the region lIm TI <

--V < Re ~ < (l+p)# 2,

p,u2.

But this means that the last three terms in the dispersion relations (2.24) are analytic functions of ~ and E in the region

Re T < (l+p)p 2,

Jim ~[ < p/P,

Im E ~ 0.

(3.29)

Since ~(E, T) is analytic in region (3.14) the whole fight hand side of the dispersion relations (2.24) will also he an analytic function in the region (3.14). We can therefore extend the region in which the function S~(E, T) is defined so that it coincides with the right hand part throughout the entire region (3.14). However, since the point v = / , 2 which is of interest to us is in the region (3.14) the dispersion relations (2.24) will also be valid for

ST(E, p~)

(E_Eo) ~ooFz (2E,,~/~~-+p=..l_/z~,p~)-i-F,,(--2E"~/M~-+PLI-p', p') d E ' 2=i

J-~

(E --E)(E --Eo) "+1

(Eo--Ey~+zSA(p,p 2) -

+

(E--Eo~"+zSB(p, p2) sp-E;

(3.30)

Ep-s

:E c,(#,, ~, ~o)E'.

O~T
It should be noted that the expression

F~ (2E ~/M-i-+pa+# ~, p')+ F, (-- 2 E ~ / ~ - + p ~ + # ', p') which replaces

s/(E, p2)

is identical with the latter in the region

E~Et (E t is the threshold energy) and is its analytic continuation to the region Ec ~_ E ~ Et in which a direct definition of function s/(E, p2) in terms of the integral (2.13) would be meaningless. In order to go over to real values of E we put E = E,+i, and let e approach zero. Taking into account that for the extended function S ~ (E, T) the following limiting relations hold:

lim S~(E +ie, z) g--~0

we obtain

ret

=

STUdy(E,T),

(3.31)

ON THE THEORY OF DISPERSION RELATIONS FOR VIRTUAL PROCESSES

~t STId,(E, p2) = q _ { S T ( E 2, p2)+ (E--E0)"+1 p 2rd

--\Eo+E.] +

Y

81

s/(E',s t, 2~)

fOO

dE'

-~o ( E ' - - E ) ( E ' - - E o ) ~+1

Ep+E+\Ep--Eo/ E--Ep

(3.32)

C,(p 2,p, Eo)E'.

O
The Hermitian and anti-Hermitian parts of the process amplitude are defined by the following relations: D~,,,~(E) = ltTr*t 2~ ~,~tE~+T,,,o,(E)), J ~d~ 1 t7~, t E ) _ Z , , .adv o(E)). A~'; ° (E) = 2i ' "'"" '

(3.33)

Dispersion relations connecting the Hermitian and anti-Hermitian parts of the process amplitude can easily be obtained with aid of (3.32) and (3.33): ~oo SA~, o(E')dE' SD.o(E) -- (E-E°)"+I-~z P.I -o0 ( E ' - - - - - - - E ) - ( ~ "+a

(Eo--E~"+ISA(I~z,#) -

+

(E--Eo]n+ISB(I-~',t ~) +

Z

e

(3.34)

-Eo/

C,(#', p, Eo)E'.

O
If should be noted that SA,,~(E) = 0, IEI < E°. The negative energy region in the dispersion relations (3.34) can be excluded if the following symmetry property of A is employed: A,,.,o(E) = --P,,A~,,o,(--E); P,e is the nucleon spin state interchange operator. It should be mentioned that the analysis of the terms SA (p, 3) and S B ( p , 3) is similar to the analysis of the single-nucleon terms in the Compton effect dispersion relations 3.e). In conclusion I take the opportunity to express my deep appreciation to Academician N. N. Bogolubov for valuable discussions. References 1) A. A. Logunov, Doklady Akad. Nauk SSSR 117 (1957) 792 2) A. A. Logunov and L. D. Solovyov, Nuclear Physics 10 (1959) 60 3) N . N . Bogolubov, Report a t International Conference on Theoretical Physics, Seattle (1956); N. N. Bogolubov and D. V. Shirkov, I n t r o d u c t i o n to the Theory of Quantized Fields (Gostekhizdat, 1957); N. N. Bogolubov, B. V. Medvedev a n d M. K. Polivanov, Problems Encountered in t h e Theory of Dispersion Relations (Gostekhizdat, 1958) 4) A. A. Logunov, L. D. Solovyov and A. N. Tavkhelidze, Nuclear Physics 4 (1957) 427 5) A. A. Logunov, Scientific Reports of Higher School (in print) 6) A. A. Logunov and P. S. Isayev, Nuovo Cimento (in print) 7) V. S. Vladimirov a n d A. A. Logunov, report 260 of J I N R (Joint .Institute for Nuclear Research) (1958)