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A simple formula for reducing Feynman diagrams to scalar integrals
Volume 263, number 1
PHYSICS LETTERS B
4 July 1991
A simple formula for reducing Feynman diagrams to scalar integrals A.I. D a v y d y c h e v Institute for Nuclear Physics, Moscow State University, SU- 119 899 Moscow, USSR Received 23 March 1991
An explicit general formula is obtained which makes it possible to reduce tensor Feynman integrals (corresponding to arbitrary one-loop N-point diagrams) to scalar integrals.
1. It is very important to develop methods and algorithms which enable one to evaluate effectively various types o f F e y n m a n diagrams. This is connected with the necessity of performing a large number of calculations in various gauge theories (QCD, electroweak model, etc. ). Since in realistic calculations we are often confronted with spinor and vector particles, we are compelled to deal with integrals with tensor structures in the numerator. A standard approach to evaluating such tensor integrals has been developed in refs. [ 1-4 ] (a modified procedure has been proposed in ref. [ 5 ] ). It involves the following steps: (i) the tensor integral is represented as a sum o f independent tensor structures (formed from the external m o m e n t a and the metric tensor) multiplied by scalar quantities; (ii) by considering various contractions, a system of linear equations is obtained from which we define these scalar quantities as integrals with scalar numerators; (iii) scalar numerators are represented in terms of the denominators and we obtain a representation in terms of initial scalar integrals. Although this approach makes it possible to solve the problem in principle, nevertheless, the expressions obtained turn out to be very cumbersome when the n u m b e r of independent external m o m e n t a increases (see, e.g. ref. [ 3 ] ). In addition, this approach has some complications connected with the appearance of "kinematic" determinants in the expressions obtained. In the present paper we propose another approach to this problem. It is based on some relations connecting integrals in different space-time dimensions (see below). This approach enables us to derive a simple general formula for one-loop N-point tensor integrals. It should be noted that a particular case o f such a formula for a class o f three-point integrals has been examined earlier in ref. [ 6 ]. 2. The one-loop N-point tensor integrals we are interested in are of the following form: in'v,,
j(N) ,!~l"'FtM \ " ~
"'"
U N ) ~ f qu~'''qt'Mdnq PN D ,Pl ...D N
(1)
where D j - (pj+ q ) 2 _ m 2 + i 0 are the massive denominators (some o f the masses can be taken equal to zero). We will consider the powers o f the denominators us and the space-time dimension n as arbitrary parameters. This enables us to deal with analytically- and dimensionally-regularized integrals. Sometimes we will use the notation { vj} -= (Vl, ..., vN). The corresponding scalar integral ( M = 0 ) is o f the form JN(n; v,,
"'"
/.IN) =
f
,,dn q ,,u . D l ...D N
(2)
The integrals (1) and (2) correspond to the N-point Feynman diagram o f fig. 1. It should be noted that the 0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
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J
Fig. 1. The one-loop N-point diagram.
integration m o m e n t u m q is usually chosen so that one of the external m o m e n t a pj vanishes (e.g., p N = 0 and m 2N+ i 0). For the purpose of obtaining symmetric (with respect to the indices 1.... , N) results we will keep all the m o m e n t a pj arbitrary (if necessary one can put PN= 0 in the final expressions). Let us consider first the vector integral j~N) corresponding to the case when M = 1 (see ( 1 ) ). This integral can be obtained from (2) by differentiation with respect to the external m o m e n t u m (e.g., p~ )
DN= q 2 _
J~uN)(n;{Vj)) = --ptuJ(N)(n; ( Vj}) -
1 - - -- 1) 2(Vl
0
- - u J(N)(n; Op~
(lPj--~jt
}) ,
(3)
where 8ik is the Kronecker symbol. Below we will need some relations connecting the different scalar integrals j(N). TO obtain these formulae, it is convenient to use the a-representation for the integral (2). Representing all the denominators D; -~' in the form of the integrals with respect to the a r p a r a m e t e r s , integrating over the m o m e n t u m q, performing a standard variables substitution ( Z a ~ = A , ai=Afl~, Zfl~= 1 ) and integrating over A, we obtain
xi...iI-Iflyi-'dfl,~(~fli-1)(~<~t 0
flflt(pj-p,)2-~.,fl~m2) n/2-2'' .
(4)
0
Here and henceforth Y and E denote the sum and the product from 1 to N (if these limits are not written explicitly). By using the representation (4), one can easily prove two useful formulae: 0
-J~m(n; uOpl
{vj}) = 2 v ,
u
k=2E(P~ --Pk)uvkn-~J(N)(n+2;
{ vJ +gs* +gjk}) ,
(5)
N
J~U)(n; { v j } ) = - n - ' ~ v#~N~(n+2;{Vj+~jk}) •
(6)
k=l
Note that the scalar integrals on the RHS of ( 5 ) and (6) have the space-time dimension (n + 2 ) rather than n. 108
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The n - l_factors appear from n" / 2 in ( 4 ). If we consider, for example, n - n/2j (N) instead of J (N) such factors will not appear. Finally, combining (5) and (6) we get 0
--J(U)(n;{Pj--~j,})=--2(P,--1)pLuj(U)(n;{vj})--2(9,--1) 0PI~
N
~ p.~uPk~-'J(U)(n+Z;{gj+~j.~}). k=l
(7)
Inserting the formula (7) into (3) yields N
J},N)(n; V, ..... UN)= ~ PkuVkn--~J(N)(n+2; {Vs+dSk}).
(8)
k=l
We see that the vector integral J~uN) is expressed through the sum of the external vectors Pku multiplied by the scalar integrals in (n + 2) dimensions, each coefficient at Ps,u being expressed through one scalar integral only. For example, when N = 2 and Pe = 0 (p~ = p ) we have J},2) (n; vl, v2)=pl, vl n - l J ( 2 ) ( n + 2 ; vl + 1, 9 2 )
.
Let us examine now the case M = 2 of formula (1). By analogy with (4), the tensor integral Ivu,u2 (N) can be expressed through J(uU) : [ ( N ) (H" { g j } )
~u,~,z,-,
1 (N) . =-P~u2Ju, (n, { g s } ) - 2 ( u , - 1 )
0 - - J ( u , U ) ( n ; {gj--dj,}) Op,u:
(9)
Inserting the expression (8) for J(u,u) and using (7) we obtain
J(N) tn'91
N
9N)=--½gu,u27~--lj(N)(tl+2;{gj})
+ ~ Pku, P k u y k ( g k + l ) r g - - 2 J ( N ) ( n + 4 ; { P j + 2 ~ j k } ) k=l
+ ZZ
k
(Pku, Pk'u: +Pk'u'Pku2)Pkl')k'TC--2J(N)( n+4; {gJdl-l~Jk"l-(~Jk" } ) "
(I0)
The scalar factor at gu,u~ is expressed through the integral in (n + 2) dimensions, while other integrals on the RHS of (10) have the dimension n + 4. As in the case of (8), each scalar factor is expressed through one scalar integral only. 3. Examination of the expressions (8) and (10) (as well as the way of obtaining these results) enables us to construct a general formula for the integrals ( 1 ) (for arbitrary values of M): j(U) , .ul....UM(n; Pl ..... PN)=
E
(--½)a{[g]'~[Pl]K''"[PN]XU}U,...UM
22+ ZKi=M
× (Vl),,,...(VN),~AAr'I-MJ(N)(n+2(M--2); Vl +Xl .... , Vu+tCU) ,
( 11)
where ( v ) , ~ = F ( v + x ) / F ( v ) is the Pochhammer symbol. The structure { [g]a[pl ]'~...[pu]"~'}u,...uu is the symmetric (with respect to/zl, ..., PM) tensor combination, each term of which is constructed from 2 metric tensors g, x~ momenta pl, ..., XN momenta PN. For example, {gPl }u,u2u3 =g.utu2Plj,3
-t-g~lu3Pl~2 + gu2u3Plu~ .
In the formula ( 11 ) the sum extends over all possible non-negative values of 2, ~1..... XN, with the restriction that the rank of the tensor structures, (22+ ZKi), should be equal to M. Therefore, max x i = M a n d max 2 = [ M / 2 ] (integer part of M / 2 ) . We will prove the formula ( 11 ) by induction. Let this formula be true for some value of M. Then we have, by analogy with (9), that
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j(N) In" 91,'", 9N) = --PluM+tJ,ul...UM( (N) - I(N) t,," 9~-- 1, ..., 9N) ,Ul....uMU~+I".--, n,. 91, "", 9N) -- 2- ( U- l - 1 ) OPluM+I --U'...UM'--'
(12)
Inserting the expression (11) for I(N) ~,ul.../~M into (12), using the formula (7) and taking into account that ( 9 1 - 1 )~1 = (91 - 1 ) (91)~1 we obtain (N) J,u,...UM+I (17, 9 1 , . . . , 9 N )
0 =
Z
J]-,tel ,...,t
{[gl'~[Plltel...[PNltel~}ul...UM( ~=l (9,)te,_a,,)r&-MjtN)(n+ 2(M--J.); {9j+ICj--t~jl })
0Pl/~M+ I
22 + Z tei = M N
+E k= 1
( _ ½)z{ [gla [Pl ] tel... [PN] te'v}ul...uuPku,~t+l
E
2,Xl,...,teN 2~.+ 5"/¢i = M
X (i=I~ (gi)te,+a,k)~za-M-Ij(N)(n+2(M--,~+l);{Uj+~Cj+t~jk}).
(13)
Substituting (x~ °td) = x t new) + 1 ) a n d ( 2 ( ° 1 d ) = 2 ("ew)- 1 ) in the first term of the R H S of (13), and (x~ °ld) = x~ "ew) - 1 ) in the sums of the second term, we have
J(uU.!uM(n; 91 ..... VN) =
Y, 2,tel ,---,ten 22+ Zm =M+
(-½)
2
0
{[g] 2- [Pl
]Kl+l
[P2lte2...[PN]"}u,...u,~ +
[Pi] tei-aik
[g]a = 1
i= 1
Pk,Uu+l .//I-.-./~M
1
X (1)l)~<1 "'" ( 9N)x,v 7~2-M- 1j(N) (n+ 2 ( M - 2 +
1 ); 91 + xl, ..., 9N + XN).
( 14)
In formula (14) it is understood that [ g i g - l = 0 at 2 = 0 and [Pk]te~-1= 0 at Xk= 0. One can easily see that the first term in the large parentheses on the R H S of (14) produces all the tensor structures with gu,uM+, (i = 1.... , M ) , while the second term gives all the structures with PkuM+, ( k = 1.... , N). AS a result, the terms in the large parentheses correspond to { [g] a [Pl ] tel... [PN] XN}Ul...UM+ 1' and we obtain the formula ( 11 ) with M substituted by ( M + 1 ). Thus, we proved the general formula ( 11 ). Let us illustrate the general formula ( 1 1 ) by a simple example of massless two-point integrals with P2 = 0
(Pl-P), qul""quMd"q m---U. . .i n. " 91' 92)----~ [(p+q)2]pl(q2)t,2"
(15)
j(2)
Using the well-known formula for scalar massless integrals, J(2) (n; 91, 92) =Ttn/Zil--n(P 2) n/2--vl--v2 F ( n / 2 - - 91 ) F ( n / 2 -
92)/"(91 + 1/2 - n / 2 ) F(91 )F(gE)F( n - 91 - 92)
(16)
we obtain from ( 1 1 ) that j(2) , 9 1 , 9 2 ) -_ n , / E i l - n ( p 2 ) ~/2-~,-,:( _ l ) M [ F ( 9 ~ ) F ( 9 2 ) F ( n - t,l - 92 - M ) ]-1 m...uM~n; IM/2]
x
~
(½p2)Z{[g]a[p]~-2a}~l...~MF(n/2-- 91 + 2 ) F ( n / 2 -
92+M-2)F(9,
+ 92-n/2-2).
(17)
2=0
In particular, the term with ;t = 0 coincides with the result of ref. [ 7 ]. 4. Thus, in the present paper we obtained the simple general formula ( 1 1 ) for reducing one-loop N-point 110
Volume 263, number I
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Feynman diagrams to scalar integrals. It should be noted that if we choose the integration m o m e n t u m in ( 1 ) so that PN= 0 then only the terms with x s = 0 survive in the formula ( 11 ). From our point of view, a closed general form o f the formula (11 ) has some advantages over the approach [ 1-4]. In particular, we have one scalar integral at each o f the independent tensor structures only, and "kinematic" determinants do not appear. In addition, the coefficients o f ( 11 ) do not depend on the masses rnj. These facts are rather useful for the algorithmization of calculations. On the other hand, the application o f this formula to realistic calculations requires expressions for the corresponding scalar integrals (2) with various values of the space-time dimension n and the powers of denominators p~. For some simple cases such expressions are well known (see, e.g., (16) ). We also note that recently in refs. [ 8-10 ] some new general results for one-loop integrals corresponding to diagrams with various numbers of external lines have been presented. In particular, in refs. [ 9,10 ] expressions for scalar N-point massive integrals (2) (with arbitrary values of n and uj) have been obtained in the form of multiple hypergeometric functions. Note that these explicit expressions satisfy the relations (5), (6) (this fact confirms that the results are self-consistent). The formula ( 11 ) enables us to apply those results to tensor integrals and to obtain expressions in the form o f functions o f the same type. The author is grateful to E.E. Boos and V.A. Ilyin for useful discussions. I wish to thank T. Yano for sending me a copy o f the extended variant of ref. [ 4 ].
References [ 1] L.M. Brown and R.P. Feynman, Phys. Rev. 85 (1952) 231. [ 2] G. Passarino and M. Veltman, Nucl. Phys. B 160 ( 1979) 151. [3] R.G. Stuart, Comput. Phys. Commun. 48 (1988) 367; R.G. Stuart and A. Gongora-T., Comput. Phys. Commun. 56 (1990) 337. [4] T. Yano et al., Mem. Fac. Eng. Ehime Univ. XI(4) (1989) 1. [5] G.J. van Oldenborgh and J.A.M. Vermaseren, Z. Phys. C 46 (1990) 425. [6] B.A. Arbuzov, E.E. Boos, S.S. Kurennoy and K.Sh. Turashvili, Yad. Fiz. 40 (1984) 836. [7] D.W. Duke, J.D. Kimel and G.A. SoweU,Phys. Rev. D 25 (1982) 71. [8] E.E. Boos and A.I. Davydychev, INP MSU preprint 90-11/157 (Moscow, 1990). [9] A.I. Davydychev, J. Math. Phys. 32 ( 1991), to be published. [ 10] A.I. Davydychev, INP MSU preprint 90-56/202 (Moscow, 1990).
111
PHYSICS LETTERS B
4 July 1991
A simple formula for reducing Feynman diagrams to scalar integrals A.I. D a v y d y c h e v Institute for Nuclear Physics, Moscow State University, SU- 119 899 Moscow, USSR Received 23 March 1991
An explicit general formula is obtained which makes it possible to reduce tensor Feynman integrals (corresponding to arbitrary one-loop N-point diagrams) to scalar integrals.
1. It is very important to develop methods and algorithms which enable one to evaluate effectively various types o f F e y n m a n diagrams. This is connected with the necessity of performing a large number of calculations in various gauge theories (QCD, electroweak model, etc. ). Since in realistic calculations we are often confronted with spinor and vector particles, we are compelled to deal with integrals with tensor structures in the numerator. A standard approach to evaluating such tensor integrals has been developed in refs. [ 1-4 ] (a modified procedure has been proposed in ref. [ 5 ] ). It involves the following steps: (i) the tensor integral is represented as a sum o f independent tensor structures (formed from the external m o m e n t a and the metric tensor) multiplied by scalar quantities; (ii) by considering various contractions, a system of linear equations is obtained from which we define these scalar quantities as integrals with scalar numerators; (iii) scalar numerators are represented in terms of the denominators and we obtain a representation in terms of initial scalar integrals. Although this approach makes it possible to solve the problem in principle, nevertheless, the expressions obtained turn out to be very cumbersome when the n u m b e r of independent external m o m e n t a increases (see, e.g. ref. [ 3 ] ). In addition, this approach has some complications connected with the appearance of "kinematic" determinants in the expressions obtained. In the present paper we propose another approach to this problem. It is based on some relations connecting integrals in different space-time dimensions (see below). This approach enables us to derive a simple general formula for one-loop N-point tensor integrals. It should be noted that a particular case o f such a formula for a class o f three-point integrals has been examined earlier in ref. [ 6 ]. 2. The one-loop N-point tensor integrals we are interested in are of the following form: in'v,,
j(N) ,!~l"'FtM \ " ~
"'"
U N ) ~ f qu~'''qt'Mdnq PN D ,Pl ...D N
(1)
where D j - (pj+ q ) 2 _ m 2 + i 0 are the massive denominators (some o f the masses can be taken equal to zero). We will consider the powers o f the denominators us and the space-time dimension n as arbitrary parameters. This enables us to deal with analytically- and dimensionally-regularized integrals. Sometimes we will use the notation { vj} -= (Vl, ..., vN). The corresponding scalar integral ( M = 0 ) is o f the form JN(n; v,,
"'"
/.IN) =
f
,,dn q ,,u . D l ...D N
(2)
The integrals (1) and (2) correspond to the N-point Feynman diagram o f fig. 1. It should be noted that the 0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
107
Volume 263, n u m b e r 1
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PHYSICS LETTERS B
\
4 July 1991
J
Fig. 1. The one-loop N-point diagram.
integration m o m e n t u m q is usually chosen so that one of the external m o m e n t a pj vanishes (e.g., p N = 0 and m 2N+ i 0). For the purpose of obtaining symmetric (with respect to the indices 1.... , N) results we will keep all the m o m e n t a pj arbitrary (if necessary one can put PN= 0 in the final expressions). Let us consider first the vector integral j~N) corresponding to the case when M = 1 (see ( 1 ) ). This integral can be obtained from (2) by differentiation with respect to the external m o m e n t u m (e.g., p~ )
DN= q 2 _
J~uN)(n;{Vj)) = --ptuJ(N)(n; ( Vj}) -
1 - - -- 1) 2(Vl
0
- - u J(N)(n; Op~
(lPj--~jt
}) ,
(3)
where 8ik is the Kronecker symbol. Below we will need some relations connecting the different scalar integrals j(N). TO obtain these formulae, it is convenient to use the a-representation for the integral (2). Representing all the denominators D; -~' in the form of the integrals with respect to the a r p a r a m e t e r s , integrating over the m o m e n t u m q, performing a standard variables substitution ( Z a ~ = A , ai=Afl~, Zfl~= 1 ) and integrating over A, we obtain
xi...iI-Iflyi-'dfl,~(~fli-1)(~<~t 0
flflt(pj-p,)2-~.,fl~m2) n/2-2'' .
(4)
0
Here and henceforth Y and E denote the sum and the product from 1 to N (if these limits are not written explicitly). By using the representation (4), one can easily prove two useful formulae: 0
-J~m(n; uOpl
{vj}) = 2 v ,
u
k=2E(P~ --Pk)uvkn-~J(N)(n+2;
{ vJ +gs* +gjk}) ,
(5)
N
J~U)(n; { v j } ) = - n - ' ~ v#~N~(n+2;{Vj+~jk}) •
(6)
k=l
Note that the scalar integrals on the RHS of ( 5 ) and (6) have the space-time dimension (n + 2 ) rather than n. 108
Volume 263, number 1
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The n - l_factors appear from n" / 2 in ( 4 ). If we consider, for example, n - n/2j (N) instead of J (N) such factors will not appear. Finally, combining (5) and (6) we get 0
--J(U)(n;{Pj--~j,})=--2(P,--1)pLuj(U)(n;{vj})--2(9,--1) 0PI~
N
~ p.~uPk~-'J(U)(n+Z;{gj+~j.~}). k=l
(7)
Inserting the formula (7) into (3) yields N
J},N)(n; V, ..... UN)= ~ PkuVkn--~J(N)(n+2; {Vs+dSk}).
(8)
k=l
We see that the vector integral J~uN) is expressed through the sum of the external vectors Pku multiplied by the scalar integrals in (n + 2) dimensions, each coefficient at Ps,u being expressed through one scalar integral only. For example, when N = 2 and Pe = 0 (p~ = p ) we have J},2) (n; vl, v2)=pl, vl n - l J ( 2 ) ( n + 2 ; vl + 1, 9 2 )
.
Let us examine now the case M = 2 of formula (1). By analogy with (4), the tensor integral Ivu,u2 (N) can be expressed through J(uU) : [ ( N ) (H" { g j } )
~u,~,z,-,
1 (N) . =-P~u2Ju, (n, { g s } ) - 2 ( u , - 1 )
0 - - J ( u , U ) ( n ; {gj--dj,}) Op,u:
(9)
Inserting the expression (8) for J(u,u) and using (7) we obtain
J(N) tn'91
N
9N)=--½gu,u27~--lj(N)(tl+2;{gj})
+ ~ Pku, P k u y k ( g k + l ) r g - - 2 J ( N ) ( n + 4 ; { P j + 2 ~ j k } ) k=l
+ ZZ
k
(Pku, Pk'u: +Pk'u'Pku2)Pkl')k'TC--2J(N)( n+4; {gJdl-l~Jk"l-(~Jk" } ) "
(I0)
The scalar factor at gu,u~ is expressed through the integral in (n + 2) dimensions, while other integrals on the RHS of (10) have the dimension n + 4. As in the case of (8), each scalar factor is expressed through one scalar integral only. 3. Examination of the expressions (8) and (10) (as well as the way of obtaining these results) enables us to construct a general formula for the integrals ( 1 ) (for arbitrary values of M): j(U) , .ul....UM(n; Pl ..... PN)=
E
(--½)a{[g]'~[Pl]K''"[PN]XU}U,...UM
22+ ZKi=M
× (Vl),,,...(VN),~AAr'I-MJ(N)(n+2(M--2); Vl +Xl .... , Vu+tCU) ,
( 11)
where ( v ) , ~ = F ( v + x ) / F ( v ) is the Pochhammer symbol. The structure { [g]a[pl ]'~...[pu]"~'}u,...uu is the symmetric (with respect to/zl, ..., PM) tensor combination, each term of which is constructed from 2 metric tensors g, x~ momenta pl, ..., XN momenta PN. For example, {gPl }u,u2u3 =g.utu2Plj,3
-t-g~lu3Pl~2 + gu2u3Plu~ .
In the formula ( 11 ) the sum extends over all possible non-negative values of 2, ~1..... XN, with the restriction that the rank of the tensor structures, (22+ ZKi), should be equal to M. Therefore, max x i = M a n d max 2 = [ M / 2 ] (integer part of M / 2 ) . We will prove the formula ( 11 ) by induction. Let this formula be true for some value of M. Then we have, by analogy with (9), that
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j(N) In" 91,'", 9N) = --PluM+tJ,ul...UM( (N) - I(N) t,," 9~-- 1, ..., 9N) ,Ul....uMU~+I".--, n,. 91, "", 9N) -- 2- ( U- l - 1 ) OPluM+I --U'...UM'--'
(12)
Inserting the expression (11) for I(N) ~,ul.../~M into (12), using the formula (7) and taking into account that ( 9 1 - 1 )~1 = (91 - 1 ) (91)~1 we obtain (N) J,u,...UM+I (17, 9 1 , . . . , 9 N )
0 =
Z
J]-,tel ,...,t
{[gl'~[Plltel...[PNltel~}ul...UM( ~=l (9,)te,_a,,)r&-MjtN)(n+ 2(M--J.); {9j+ICj--t~jl })
0Pl/~M+ I
22 + Z tei = M N
+E k= 1
( _ ½)z{ [gla [Pl ] tel... [PN] te'v}ul...uuPku,~t+l
E
2,Xl,...,teN 2~.+ 5"/¢i = M
X (i=I~ (gi)te,+a,k)~za-M-Ij(N)(n+2(M--,~+l);{Uj+~Cj+t~jk}).
(13)
Substituting (x~ °td) = x t new) + 1 ) a n d ( 2 ( ° 1 d ) = 2 ("ew)- 1 ) in the first term of the R H S of (13), and (x~ °ld) = x~ "ew) - 1 ) in the sums of the second term, we have
J(uU.!uM(n; 91 ..... VN) =
Y, 2,tel ,---,ten 22+ Zm =M+
(-½)
2
0
{[g] 2- [Pl
]Kl+l
[P2lte2...[PN]"}u,...u,~ +
[Pi] tei-aik
[g]a = 1
i= 1
Pk,Uu+l .//I-.-./~M
1
X (1)l)~<1 "'" ( 9N)x,v 7~2-M- 1j(N) (n+ 2 ( M - 2 +
1 ); 91 + xl, ..., 9N + XN).
( 14)
In formula (14) it is understood that [ g i g - l = 0 at 2 = 0 and [Pk]te~-1= 0 at Xk= 0. One can easily see that the first term in the large parentheses on the R H S of (14) produces all the tensor structures with gu,uM+, (i = 1.... , M ) , while the second term gives all the structures with PkuM+, ( k = 1.... , N). AS a result, the terms in the large parentheses correspond to { [g] a [Pl ] tel... [PN] XN}Ul...UM+ 1' and we obtain the formula ( 11 ) with M substituted by ( M + 1 ). Thus, we proved the general formula ( 11 ). Let us illustrate the general formula ( 1 1 ) by a simple example of massless two-point integrals with P2 = 0
(Pl-P), qul""quMd"q m---U. . .i n. " 91' 92)----~ [(p+q)2]pl(q2)t,2"
(15)
j(2)
Using the well-known formula for scalar massless integrals, J(2) (n; 91, 92) =Ttn/Zil--n(P 2) n/2--vl--v2 F ( n / 2 - - 91 ) F ( n / 2 -
92)/"(91 + 1/2 - n / 2 ) F(91 )F(gE)F( n - 91 - 92)
(16)
we obtain from ( 1 1 ) that j(2) , 9 1 , 9 2 ) -_ n , / E i l - n ( p 2 ) ~/2-~,-,:( _ l ) M [ F ( 9 ~ ) F ( 9 2 ) F ( n - t,l - 92 - M ) ]-1 m...uM~n; IM/2]
x
~
(½p2)Z{[g]a[p]~-2a}~l...~MF(n/2-- 91 + 2 ) F ( n / 2 -
92+M-2)F(9,
+ 92-n/2-2).
(17)
2=0
In particular, the term with ;t = 0 coincides with the result of ref. [ 7 ]. 4. Thus, in the present paper we obtained the simple general formula ( 1 1 ) for reducing one-loop N-point 110
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4 July 1991
Feynman diagrams to scalar integrals. It should be noted that if we choose the integration m o m e n t u m in ( 1 ) so that PN= 0 then only the terms with x s = 0 survive in the formula ( 11 ). From our point of view, a closed general form o f the formula (11 ) has some advantages over the approach [ 1-4]. In particular, we have one scalar integral at each o f the independent tensor structures only, and "kinematic" determinants do not appear. In addition, the coefficients o f ( 11 ) do not depend on the masses rnj. These facts are rather useful for the algorithmization of calculations. On the other hand, the application o f this formula to realistic calculations requires expressions for the corresponding scalar integrals (2) with various values of the space-time dimension n and the powers of denominators p~. For some simple cases such expressions are well known (see, e.g., (16) ). We also note that recently in refs. [ 8-10 ] some new general results for one-loop integrals corresponding to diagrams with various numbers of external lines have been presented. In particular, in refs. [ 9,10 ] expressions for scalar N-point massive integrals (2) (with arbitrary values of n and uj) have been obtained in the form of multiple hypergeometric functions. Note that these explicit expressions satisfy the relations (5), (6) (this fact confirms that the results are self-consistent). The formula ( 11 ) enables us to apply those results to tensor integrals and to obtain expressions in the form o f functions o f the same type. The author is grateful to E.E. Boos and V.A. Ilyin for useful discussions. I wish to thank T. Yano for sending me a copy o f the extended variant of ref. [ 4 ].
References [ 1] L.M. Brown and R.P. Feynman, Phys. Rev. 85 (1952) 231. [ 2] G. Passarino and M. Veltman, Nucl. Phys. B 160 ( 1979) 151. [3] R.G. Stuart, Comput. Phys. Commun. 48 (1988) 367; R.G. Stuart and A. Gongora-T., Comput. Phys. Commun. 56 (1990) 337. [4] T. Yano et al., Mem. Fac. Eng. Ehime Univ. XI(4) (1989) 1. [5] G.J. van Oldenborgh and J.A.M. Vermaseren, Z. Phys. C 46 (1990) 425. [6] B.A. Arbuzov, E.E. Boos, S.S. Kurennoy and K.Sh. Turashvili, Yad. Fiz. 40 (1984) 836. [7] D.W. Duke, J.D. Kimel and G.A. SoweU,Phys. Rev. D 25 (1982) 71. [8] E.E. Boos and A.I. Davydychev, INP MSU preprint 90-11/157 (Moscow, 1990). [9] A.I. Davydychev, J. Math. Phys. 32 ( 1991), to be published. [ 10] A.I. Davydychev, INP MSU preprint 90-56/202 (Moscow, 1990).
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