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On spectral properties of parton correlation functions and multiparton wave functions
Volume 131B, number 1,2,3
PHYSICS LETTERS
10 November 1983
ON SPECTRAL PROPERTIES OF P A R T O N C O R R E L A T I O N FUNCTIONS AND M U L T I P A R T O N W A V E F U N C T I O N S
A.V. R A D Y U S H K I N Laboratory of Theoretical Physics, JINR, Dubna, USSR Received 3 May 1983 Using a parametric representation for Feynman integrals it is demonstrated that the functions F(x~ . . . . . Xk), ~b(xl,..., xk), the generalized momenta of which are proportional to the reduced matrix elements of the k-body composite operators, have the spectral properties necessary for their parton interpretation.
1. Parton correlation functions. To give a parton-model interpretation of power corrections (higher twists in deep inelastic leptoproduction processes, one should introduce kparticle parton correlation functions F(X 1. . . . . Xk) [1-3] which are (for k/> 3) generalizations of the usual parton distribution functions related to the lowest twist (2-particle) composite operators. T h e (generalized) m o m e n t a of the parton correlation functions are proportional to the matrix elements of the composite operators containing more than 2 fundamental fields: l
f F(Xl . . . . . Xk)X~(1... X f k d x t . . , dXk =(Pl[(nO/nP)Nlq~l]. . . [(nO/np)Nk~k]lP) ,
(1)
where n~, is an arbitrary lightlike vector introduced to pick out the symmetric-traceless part of the composite o p e r a t o r and [P) is a hadronic state with m o m e n t u m P. The relevant fields are denoted schematically by tpi. Depending on the sign of the xi-parameter one should attribute the corresponding parton either to the initial (xi > 0) or to the final (x~ < 0) states. As a result, the function F ( x b . . . , Xk) is d e c o m p o s e d into a sum of the functions F(t'k-l)(Xil . . . . . Xi~; Xit+l. . . . . Xjk) describing a set
of l partons in the initial state and (k - l) ones in the final state. Such a parton interpretation is self-consistent only if k
~'~ x, = 0
(2)
i=1
( e n e r g y - m o m e n t u m conservation) and moreover 1
m~lXi" ~<1,
(3)
for any set (il . . . . . il). Eq. (3) means that the total longitudinal m o m e n t u m carried by partons in the infinite m o m e n t u m frame, should not exceed that of the hadron. Most surprisingly, a general proof of eq. (3) based only on the definition (3) is not a trivial problem. For instance, analysing the 3-parton correlation functions the authors of ref. [3] succeeded in proving only the much weaker inequality [xll, IX2[, IXl q- X2[ ~ 2.
2. Multiparton wave functions. Studying higher twist effects in hard exclusive processes one should introduce multiparton wave functions absorbing information about the long-distance dynamics. These functions are related to
0 031-9163/83/0000--0000/$03.00 © 1983 North-Holland
179
Volume
131B, n u m b e r
1,2,3
PHYSICS
the (01. . . . . ]P) matrix elements of the corresponding local operators
LETI'ERS
10 N o v e m b e r
1983
and bjl(a)/> 0, D(a)~> 0 are some functions of a-parameters (to be specified below) satisfying the relation
I
k
f q~(Xl. . . . . Xk)X~'... Xf k d X l . . , dXk
D(a) = ~ ~ bi, + C(a),
o
(9)
j = l l
=(O[[(nO/nP)N',;,]. . . [(nO/nP)Nk~l]P) .
(4)
Physically ~(xl . . . . . Xk) should be interpreted as a probability amplitude to find (in the infinite m o m e n t u m frame) the initial hadron in the state where the partons ~p~,. . . , ~Pk carry the fractions Xl. . . . . Xk of its longitudinal momentum. Such an interpretation makes sense only if 0 < x~ < 1 for all x1 and moreover k
x, : 1 .
(5)
i=1
3. Alpha-representation. To prove the validity of eqs. (3), (5) we shall incorporate the well-known parametric representation of the Feynman amplitudes (see, e.g. refs. [4-9]) based on the simple formula for the propagator
where C ( a ) / > O. Using eqs. (1) and (7) one can derive the a-representation for F(Xl . . . . . Xk):
F(x~ . . . . . Xk) = ~ f VIda,~ diagr. d 0
o-
k
iBm(a)-
(10)
Now, incorporating eqs. (8)-(10) one can easily obtain eqs. (2), (3). The multiparton wave function ¢(Xl . . . . . Xk) has a similar a-representation
~(x~ . . . . . Xk) = ~'~ (1-Ida,, diagr. .t 0
or
× ( l ~ 8(xj- Bj(a)/D(a))) q'(a),
(11)
"j= 1
__(/)2 __ m 2) 1 = f d a exp[a(p 2 - m2)]
(6)
0
(Wick's rotation is implied to be performed and, hence, p 2 < 0). After taking the gaussian integrals over the virtual momenta one can write the contribution of any Feynman diagram as an integral over the a-parameters of all lines ~r belonging to the diagram considered. In particular, the matrix element (1) has the following a-representation
(PI[(nO/Pn)N'~,]. . . [(On/Pn)N~ok]lP) : ~' f [ - I d a , , [ F I d B T ( a ) - B T ( a ) ' ~ Ni] diagr,
0
o-
Li=I \"
~--~")
]
j (I)(a) '
(7)
where
B~(a) = ~ bj,(a),
(8a)
Icy
8 ; ( 2 ) = ~'~ b,j(a) I#]
180
(85)
where Bj(a) >~0, D(a) >10 are functions obeying the equality k
D(~) = ~ BAa),
(12)
j-1
from which one can trivially derive eq. (5).
4. Alpha-representation functions and topology of the diagram. The functions B-*(a), B ( a ) , D ( a ) can be connected in a simple way with the topological properties of the corresponding Feynman graph. Recall that a k-tree of the graph G is a subgraph of G which (a) contains all its vertices; (b) has k connected components, and (c) each component has a tree structure (i.e., has no loops). Any k-tree Gt k) is determined in a unique way by the set of lines which one should remove from the initial graph G to get Gt k). The product of a~-parameters related to these lines o- will be referred to as an ( a ) - k - t r e e . The function D(a) is the sum of all ( a ) - l - t r e e s (or, simply, (a)-trees) of the graph G. Next, denote by B(il . . . . . im ] 11. . . . . j,) the
liJ
a}
i dl ~
d)
bl
al
Fig. 1. (a) General structure of graphs contributing to matrix elements of a composite operator. (b) Auxiliary graph with the line corresponding to the ~i-field separated from the other lines entering into the O-vertex.
sum of all its (a)-2-trees possessing the property that the vertices i l , . . . , im belong to one component, jl . . . . . j,, to the other, while the vertices not indicated explicitly may belong to any component. In the a-representation each derivative (nO)~0i related to the O-vertex (fig. la) results in a factor R~(a)*i. T o construct this factor, it is convenient to consider the auxiliary diagram (fig. lb) in which the O-vertex is splitted into two ones (O ~ O1, O2) so as to separate the line corresponding to the q~-field. Then [10,ll] 1
R,(a)- D(a)
10 November 1983
PHYSICS LETFERS
Volume 131B, number 1,2,3
B(Oti),jl O~O)(npj)
(13)
i=l
where j is the vertex into which the external momentum pj (j = 1. . . . . N ) enters. A trivial but crucial observation is that the (a)-2-trees of the graph lb present in B(O~ 0, j ] O ~ °) may be treated also as the ( a ) - l - t r e e s of the graph la. We are interested only in the simplest cases N=I andN=2. (a) N = 1, pl = P (fig. 2a). In this case
(np)-lRi(a) = D ( a ) - I B ( O ~ °, 11 o~)) =--Bi(a)/D(a).
Fig. 2. General structure of graphs contributing to a multiparton wave function.
Consider now an arbitrary tree contributing to D(a). According to the defnition of a tree it should contain a continuous chain of lines joining the vertices 0 and 1. Furthermore such a chain is unique (otherwise we would have a loop graph rather than a tree). If the line of this chain adjacent to the O-vertex corresponds to the ~i-field, then the relevant (a)-tree is a part of Bi(a). Hence, each (a)-tree present in D(a) has its unique counterpart in some of Bi(a)'s. The reversed statement is also true: joining (by Oti)O~°-~ O) the components of a 2-tree contributing to some Bi(a) one obtains a tree contributing to D(a). This gives eq. (12). (b) N = 2, pl = P, p2 = - P (fig. 3a). In such a configuration
( n p ) - l R i ( a ) = V(ol)-l[B(O~i), 1 ] 0 ~ )) -B(Ot'),
2[ O~))].
,1 Strictly speaking, this is true only in a scalar theory while for a QCD diagram there appear additional terms due to numerators of quark propagators and derivatives in the 3-gluon vertices. However, any non-scalar Feynman integral over the virtual momenta p can be represented as a sum of scalar integrals. To do this one should calculate the traces, expand the resulting numerator factors over the denominator ones and omit the terms that are odd in the relevant integration momentum. This procedure is in fact the first step of the most effective modern recipes [12,13] of calculating the QCD Feynman integrals.
(15)
Note that according to the definition of B ( . . . . ] . . . ) one can write B ( O t o, 1 1 0 ~ ))= B(O~ °, 1[O~ °,2) +B(O~i), l, 2 [ O~0) =- B~(a) + Cg(a),
B(O~O, 210~ )) = +B(O~ o, l,
(14)
b}
B ( O ~ °,
(16a)
21 Of ), 1)
2 10~)) =--B?(a) + Ci(a).
(16b)
Hence the 2-trees for which the vertices 1, 2
0OO ¢ ~I P
~
P a/
bl
Fig. 3. General structure of graphs contributing to a multiparton correlation function.
181
Volume 131B, number 1,2,3
PHYSICS LETI'ERS
b e l o n g to t h e s a m e c o m p o n e n t give t h e z e r o t o t a l c o n t r i b u t i o n into R i ( a ) . H o w e v e r , t h e y give a n o n - z e r o c o n t r i b u t i o n C ( a ) = Z C / ( a ) into D ( a ) . T o p r o c e e d f u r t h e r , c o n s i d e r a p a r ticular 2-tree of t h e g r a p h 3b c o n t r i b u t i n g into B+(a). D u e to a b s e n c e of l o o p s t h e r e exists only a single line ( r e l a t e d , say to the ~0t-field) going o u t of the O~)-vertex t h a t starts a cont i n u o u s chain j o i n i n g O~i) with t h e v e r t e x 2. D e f i n e n o w bit(a) t o b e t h e s u m of all such ( a ) - 2 - t r e e s of t h e g r a p h 3b. F u r t h e r m o r e , it is e a s y to r e a l i z e t h a t bit(a) m a y b e t r e a t e d also as t h e s u m of all ( a ) - t r e e s of t h e g r a p h 3a for which t h e c o n t i n u o u s chain j o i n i n g t h e vertices 1 a n d 2 l o o k s like 1 ~ q~i ~ O ~ ~01~ 2. T h e trivial c o n s e q u e n c e s of this o b s e r v a t i o n a r e eqs. (8b) a n d (9).
5. Conclusions. Thus, i n c o r p o r a t i n g t h e a representation we have established that F(xl . . . . . Xk) a n d ~p(Xl. . . . . Xk) possess t h e s p e c t r a l p r o p e r t i e s n e c e s s a r y for t h e i r p a r t o n i n t e r p r e t a t i o n . T h e p r o o f is n o t trivial, a n d o n e m a y p r o p o s e t h a t a s i m p l e r w a y is to use t h e m o r e f a m i l i a r m o m e n t u m r e p r e s e n t a t i o n , e.g., ~(Xl .....
Xk)~-fd411...d41k(~4(£1j-P) -j= 1
x ( l ~ I 6(xi-(lin)/(Pn)))T(ll . . . . .
lk;P).
(17)
"i=1
Indeed, the energy-momentum conservation c o n s t r a i n t s (2), (5) trivially follow f r o m eq. (17). H o w e v e r , to p r o v e t h a t 0 ~< xi ~< 1 for ~o(xl . . . . . Xk) [or eq. (3) for F(xl . . . . . Xk)] o n e
182
10 November 1983
s h o u l d k n o w t h e a n a l y t i c i t y p r o p e r t i e s of the T(ll . . . . . lk, P ) - a m p l i t u d e (cf. ref. [3]), a n d this is in fact a m o r e c o m p l i c a t e d p r o b l e m t h a n t h e original o n e (recall also that o n e of t h e m o s t effective tools to s t u d y t h e s e p r o p e r t i e s is b a s e d on a p a r a m e t r i c r e p r e s e n t a t i o n [6]). I a m grateful to R . K . Ellis for a s t i m u l a t i n g discussion a n d to A . V . E f r e m o v for helpful comments.
References [1] A.V. Efremov and A.V. Radyushkin, JINR E2-80-521 (Dubna, 1980). [2] H.D. Politzer, Nucl. Phys. 172B (1980) 349. [3] R.K. Ellis, W. Furmanski and R. Petronzio, CERNTH-3301 (Geneva, 1982). [4] R. Chisholm, Proc. Cambr. Phys. Soc. 48 (1952) 300, 518. [5] N.N. Bogoliubov and D.V. Shirkov, Introduction to the theory of quantized fields (Interscience, New York, 1959). [6] R.I. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, The analytic S-matrix (Cambridge U. P., London, 1966). [7] N. Nakanishi, Graph theory and Feynman integrals, in: Mathematics and its applications, Vol. 11 (Gordon and Breach, New York, 1971). [8] I.T. Todorov, Analytic properties of Feynman diagrams (Sofia, 1966). [9] O.I. Zavialov, Renormalized Feynman diagrams (Nauka, Moscow, 1979). [10] A.V. Efremov, JINR E-2125 (Dubna, 1965). [11] C.S. Lam and J.P. Lebrun, Nuovo Cimento 59 (1969) 358, 422. [12] A.A. Vladimirov, Teor. Mat. Fiz. 43 (1980) 210. [13] K.G. Chetyrkin and F.V. Tkachov, Nucl. Phys. B192 (1981) 159.
PHYSICS LETTERS
10 November 1983
ON SPECTRAL PROPERTIES OF P A R T O N C O R R E L A T I O N FUNCTIONS AND M U L T I P A R T O N W A V E F U N C T I O N S
A.V. R A D Y U S H K I N Laboratory of Theoretical Physics, JINR, Dubna, USSR Received 3 May 1983 Using a parametric representation for Feynman integrals it is demonstrated that the functions F(x~ . . . . . Xk), ~b(xl,..., xk), the generalized momenta of which are proportional to the reduced matrix elements of the k-body composite operators, have the spectral properties necessary for their parton interpretation.
1. Parton correlation functions. To give a parton-model interpretation of power corrections (higher twists in deep inelastic leptoproduction processes, one should introduce kparticle parton correlation functions F(X 1. . . . . Xk) [1-3] which are (for k/> 3) generalizations of the usual parton distribution functions related to the lowest twist (2-particle) composite operators. T h e (generalized) m o m e n t a of the parton correlation functions are proportional to the matrix elements of the composite operators containing more than 2 fundamental fields: l
f F(Xl . . . . . Xk)X~(1... X f k d x t . . , dXk =(Pl[(nO/nP)Nlq~l]. . . [(nO/np)Nk~k]lP) ,
(1)
where n~, is an arbitrary lightlike vector introduced to pick out the symmetric-traceless part of the composite o p e r a t o r and [P) is a hadronic state with m o m e n t u m P. The relevant fields are denoted schematically by tpi. Depending on the sign of the xi-parameter one should attribute the corresponding parton either to the initial (xi > 0) or to the final (x~ < 0) states. As a result, the function F ( x b . . . , Xk) is d e c o m p o s e d into a sum of the functions F(t'k-l)(Xil . . . . . Xi~; Xit+l. . . . . Xjk) describing a set
of l partons in the initial state and (k - l) ones in the final state. Such a parton interpretation is self-consistent only if k
~'~ x, = 0
(2)
i=1
( e n e r g y - m o m e n t u m conservation) and moreover 1
m~lXi" ~<1,
(3)
for any set (il . . . . . il). Eq. (3) means that the total longitudinal m o m e n t u m carried by partons in the infinite m o m e n t u m frame, should not exceed that of the hadron. Most surprisingly, a general proof of eq. (3) based only on the definition (3) is not a trivial problem. For instance, analysing the 3-parton correlation functions the authors of ref. [3] succeeded in proving only the much weaker inequality [xll, IX2[, IXl q- X2[ ~ 2.
2. Multiparton wave functions. Studying higher twist effects in hard exclusive processes one should introduce multiparton wave functions absorbing information about the long-distance dynamics. These functions are related to
0 031-9163/83/0000--0000/$03.00 © 1983 North-Holland
179
Volume
131B, n u m b e r
1,2,3
PHYSICS
the (01. . . . . ]P) matrix elements of the corresponding local operators
LETI'ERS
10 N o v e m b e r
1983
and bjl(a)/> 0, D(a)~> 0 are some functions of a-parameters (to be specified below) satisfying the relation
I
k
f q~(Xl. . . . . Xk)X~'... Xf k d X l . . , dXk
D(a) = ~ ~ bi, + C(a),
o
(9)
j = l l
=(O[[(nO/nP)N',;,]. . . [(nO/nP)Nk~l]P) .
(4)
Physically ~(xl . . . . . Xk) should be interpreted as a probability amplitude to find (in the infinite m o m e n t u m frame) the initial hadron in the state where the partons ~p~,. . . , ~Pk carry the fractions Xl. . . . . Xk of its longitudinal momentum. Such an interpretation makes sense only if 0 < x~ < 1 for all x1 and moreover k
x, : 1 .
(5)
i=1
3. Alpha-representation. To prove the validity of eqs. (3), (5) we shall incorporate the well-known parametric representation of the Feynman amplitudes (see, e.g. refs. [4-9]) based on the simple formula for the propagator
where C ( a ) / > O. Using eqs. (1) and (7) one can derive the a-representation for F(Xl . . . . . Xk):
F(x~ . . . . . Xk) = ~ f VIda,~ diagr. d 0
o-
k
iBm(a)-
(10)
Now, incorporating eqs. (8)-(10) one can easily obtain eqs. (2), (3). The multiparton wave function ¢(Xl . . . . . Xk) has a similar a-representation
~(x~ . . . . . Xk) = ~'~ (1-Ida,, diagr. .t 0
or
× ( l ~ 8(xj- Bj(a)/D(a))) q'(a),
(11)
"j= 1
__(/)2 __ m 2) 1 = f d a exp[a(p 2 - m2)]
(6)
0
(Wick's rotation is implied to be performed and, hence, p 2 < 0). After taking the gaussian integrals over the virtual momenta one can write the contribution of any Feynman diagram as an integral over the a-parameters of all lines ~r belonging to the diagram considered. In particular, the matrix element (1) has the following a-representation
(PI[(nO/Pn)N'~,]. . . [(On/Pn)N~ok]lP) : ~' f [ - I d a , , [ F I d B T ( a ) - B T ( a ) ' ~ Ni] diagr,
0
o-
Li=I \"
~--~")
]
j (I)(a) '
(7)
where
B~(a) = ~ bj,(a),
(8a)
Icy
8 ; ( 2 ) = ~'~ b,j(a) I#]
180
(85)
where Bj(a) >~0, D(a) >10 are functions obeying the equality k
D(~) = ~ BAa),
(12)
j-1
from which one can trivially derive eq. (5).
4. Alpha-representation functions and topology of the diagram. The functions B-*(a), B ( a ) , D ( a ) can be connected in a simple way with the topological properties of the corresponding Feynman graph. Recall that a k-tree of the graph G is a subgraph of G which (a) contains all its vertices; (b) has k connected components, and (c) each component has a tree structure (i.e., has no loops). Any k-tree Gt k) is determined in a unique way by the set of lines which one should remove from the initial graph G to get Gt k). The product of a~-parameters related to these lines o- will be referred to as an ( a ) - k - t r e e . The function D(a) is the sum of all ( a ) - l - t r e e s (or, simply, (a)-trees) of the graph G. Next, denote by B(il . . . . . im ] 11. . . . . j,) the
liJ
a}
i dl ~
d)
bl
al
Fig. 1. (a) General structure of graphs contributing to matrix elements of a composite operator. (b) Auxiliary graph with the line corresponding to the ~i-field separated from the other lines entering into the O-vertex.
sum of all its (a)-2-trees possessing the property that the vertices i l , . . . , im belong to one component, jl . . . . . j,, to the other, while the vertices not indicated explicitly may belong to any component. In the a-representation each derivative (nO)~0i related to the O-vertex (fig. la) results in a factor R~(a)*i. T o construct this factor, it is convenient to consider the auxiliary diagram (fig. lb) in which the O-vertex is splitted into two ones (O ~ O1, O2) so as to separate the line corresponding to the q~-field. Then [10,ll] 1
R,(a)- D(a)
10 November 1983
PHYSICS LETFERS
Volume 131B, number 1,2,3
B(Oti),jl O~O)(npj)
(13)
i=l
where j is the vertex into which the external momentum pj (j = 1. . . . . N ) enters. A trivial but crucial observation is that the (a)-2-trees of the graph lb present in B(O~ 0, j ] O ~ °) may be treated also as the ( a ) - l - t r e e s of the graph la. We are interested only in the simplest cases N=I andN=2. (a) N = 1, pl = P (fig. 2a). In this case
(np)-lRi(a) = D ( a ) - I B ( O ~ °, 11 o~)) =--Bi(a)/D(a).
Fig. 2. General structure of graphs contributing to a multiparton wave function.
Consider now an arbitrary tree contributing to D(a). According to the defnition of a tree it should contain a continuous chain of lines joining the vertices 0 and 1. Furthermore such a chain is unique (otherwise we would have a loop graph rather than a tree). If the line of this chain adjacent to the O-vertex corresponds to the ~i-field, then the relevant (a)-tree is a part of Bi(a). Hence, each (a)-tree present in D(a) has its unique counterpart in some of Bi(a)'s. The reversed statement is also true: joining (by Oti)O~°-~ O) the components of a 2-tree contributing to some Bi(a) one obtains a tree contributing to D(a). This gives eq. (12). (b) N = 2, pl = P, p2 = - P (fig. 3a). In such a configuration
( n p ) - l R i ( a ) = V(ol)-l[B(O~i), 1 ] 0 ~ )) -B(Ot'),
2[ O~))].
,1 Strictly speaking, this is true only in a scalar theory while for a QCD diagram there appear additional terms due to numerators of quark propagators and derivatives in the 3-gluon vertices. However, any non-scalar Feynman integral over the virtual momenta p can be represented as a sum of scalar integrals. To do this one should calculate the traces, expand the resulting numerator factors over the denominator ones and omit the terms that are odd in the relevant integration momentum. This procedure is in fact the first step of the most effective modern recipes [12,13] of calculating the QCD Feynman integrals.
(15)
Note that according to the definition of B ( . . . . ] . . . ) one can write B ( O t o, 1 1 0 ~ ))= B(O~ °, 1[O~ °,2) +B(O~i), l, 2 [ O~0) =- B~(a) + Cg(a),
B(O~O, 210~ )) = +B(O~ o, l,
(14)
b}
B ( O ~ °,
(16a)
21 Of ), 1)
2 10~)) =--B?(a) + Ci(a).
(16b)
Hence the 2-trees for which the vertices 1, 2
0OO ¢ ~I P
~
P a/
bl
Fig. 3. General structure of graphs contributing to a multiparton correlation function.
181
Volume 131B, number 1,2,3
PHYSICS LETI'ERS
b e l o n g to t h e s a m e c o m p o n e n t give t h e z e r o t o t a l c o n t r i b u t i o n into R i ( a ) . H o w e v e r , t h e y give a n o n - z e r o c o n t r i b u t i o n C ( a ) = Z C / ( a ) into D ( a ) . T o p r o c e e d f u r t h e r , c o n s i d e r a p a r ticular 2-tree of t h e g r a p h 3b c o n t r i b u t i n g into B+(a). D u e to a b s e n c e of l o o p s t h e r e exists only a single line ( r e l a t e d , say to the ~0t-field) going o u t of the O~)-vertex t h a t starts a cont i n u o u s chain j o i n i n g O~i) with t h e v e r t e x 2. D e f i n e n o w bit(a) t o b e t h e s u m of all such ( a ) - 2 - t r e e s of t h e g r a p h 3b. F u r t h e r m o r e , it is e a s y to r e a l i z e t h a t bit(a) m a y b e t r e a t e d also as t h e s u m of all ( a ) - t r e e s of t h e g r a p h 3a for which t h e c o n t i n u o u s chain j o i n i n g t h e vertices 1 a n d 2 l o o k s like 1 ~ q~i ~ O ~ ~01~ 2. T h e trivial c o n s e q u e n c e s of this o b s e r v a t i o n a r e eqs. (8b) a n d (9).
5. Conclusions. Thus, i n c o r p o r a t i n g t h e a representation we have established that F(xl . . . . . Xk) a n d ~p(Xl. . . . . Xk) possess t h e s p e c t r a l p r o p e r t i e s n e c e s s a r y for t h e i r p a r t o n i n t e r p r e t a t i o n . T h e p r o o f is n o t trivial, a n d o n e m a y p r o p o s e t h a t a s i m p l e r w a y is to use t h e m o r e f a m i l i a r m o m e n t u m r e p r e s e n t a t i o n , e.g., ~(Xl .....
Xk)~-fd411...d41k(~4(£1j-P) -j= 1
x ( l ~ I 6(xi-(lin)/(Pn)))T(ll . . . . .
lk;P).
(17)
"i=1
Indeed, the energy-momentum conservation c o n s t r a i n t s (2), (5) trivially follow f r o m eq. (17). H o w e v e r , to p r o v e t h a t 0 ~< xi ~< 1 for ~o(xl . . . . . Xk) [or eq. (3) for F(xl . . . . . Xk)] o n e
182
10 November 1983
s h o u l d k n o w t h e a n a l y t i c i t y p r o p e r t i e s of the T(ll . . . . . lk, P ) - a m p l i t u d e (cf. ref. [3]), a n d this is in fact a m o r e c o m p l i c a t e d p r o b l e m t h a n t h e original o n e (recall also that o n e of t h e m o s t effective tools to s t u d y t h e s e p r o p e r t i e s is b a s e d on a p a r a m e t r i c r e p r e s e n t a t i o n [6]). I a m grateful to R . K . Ellis for a s t i m u l a t i n g discussion a n d to A . V . E f r e m o v for helpful comments.
References [1] A.V. Efremov and A.V. Radyushkin, JINR E2-80-521 (Dubna, 1980). [2] H.D. Politzer, Nucl. Phys. 172B (1980) 349. [3] R.K. Ellis, W. Furmanski and R. Petronzio, CERNTH-3301 (Geneva, 1982). [4] R. Chisholm, Proc. Cambr. Phys. Soc. 48 (1952) 300, 518. [5] N.N. Bogoliubov and D.V. Shirkov, Introduction to the theory of quantized fields (Interscience, New York, 1959). [6] R.I. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, The analytic S-matrix (Cambridge U. P., London, 1966). [7] N. Nakanishi, Graph theory and Feynman integrals, in: Mathematics and its applications, Vol. 11 (Gordon and Breach, New York, 1971). [8] I.T. Todorov, Analytic properties of Feynman diagrams (Sofia, 1966). [9] O.I. Zavialov, Renormalized Feynman diagrams (Nauka, Moscow, 1979). [10] A.V. Efremov, JINR E-2125 (Dubna, 1965). [11] C.S. Lam and J.P. Lebrun, Nuovo Cimento 59 (1969) 358, 422. [12] A.A. Vladimirov, Teor. Mat. Fiz. 43 (1980) 210. [13] K.G. Chetyrkin and F.V. Tkachov, Nucl. Phys. B192 (1981) 159.