Exact results for three- and four-point ladder diagrams with an arbitrary number of rungs

Physics Letters B 305 (1993) 136-143 North-Holland

PHYSICS LETTERS B

Exact results for three- and four-point ladder diagrams with an arbitrary number of rungs N.I. Ussyukina a,1 and A.I. Davydychev b,2,3 B Institute for Nuclear Physics, Moscow State University, I 19899, Moscow, Russian Federation b Instituut Lorentz, Universiteit Leiden, P.O.Box 9506, NL-2300 RA Leiden, The Netherlands

Received 23 February 1993

Exact results for L-loop ladder graphs with three and four external lines (in the case of massless internal particles and arbitrary off-shell external momenta) are obtained in terms of polylogarithms.

1. The problem of evaluating ladder graphs in q u a n t u m field theory has been examined for a long time (see, e.g., refs. [ 1,2 ] and references therein). Some of such diagrams are important for the calculation of radiative corrections (especially in multi-jet processes and Bhabha scattering). Note that we are often confronted with the cases when internal particles are massless (photons, gluons), or their masses can be neglected in high-energy processes. While the high-energy asymptotic behaviour of ladder diagrams has been studied in many publications [ 3 - 7 ] (see also ref. [ 1 ] and references therein), exact results have been known only for some special cases. In the present paper we show that the approach presented in ref. [ 8 ] (where the one- and two-loop diagrams have been considered) is rather powerful, and it can be extended to the case of an arbitrary number of loops (rungs). We shall examine three- and four-point ladder diagrams with massless internal particles and arbitrary off-shell external m o m e n t a (formally, such diagrams correspond to massless ~3 theory). The remainder of the paper is organized as follows. Section 2 contains necessary information about one-loop triangle and box diagrams. In section 3 we examine the L-loop three-point ladder diagram, while in section 4 the four-point case is considered. Section 5 (conclusion) discusses the main results. 2. In this section we shall briefly review some useful properties of one-loop triangle and box diagrams shown in figs. 1a, 1b. We shall use these formulae below, when evaluating L-loop diagrams. More detailed information can be found e.g. in ref. [ 8 ]. Definition o f the three-point F e y n m a n integral (see fig. la): E-mail address: ussyuk@compnet, msu. su. 2 Permanent address: Institute for Nuclear Physics, Moscow State University, 119899 Moscow, Russian Federation. 3 E-mailaddress: davyd@compnet, msu. su. /J3

P2

(a) 136

kl

171

(b)

k4

Fig. 1. Elsevier Science Publishers B.V.

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PHYSICSLETTERSB

J3(n;vl,/)2, valpE,pE,p2) = _

6 May 1993

dn r [ ( p 2 _ r ) 2 ] v , [ ( p l +r)2]~2(r2)V3,

(1)

where n is the space-time dimension. Here and below the usual "causal" prescription in pseudo-euclidean momentum space is understood, (p2) -v,._~(p2q_ i0)- u. In some cases we shall omit the momentum arguments p2, 2 2 in J3. P2,P3 Double Mellin-Barnes contour integral representation [ 9, l 0 ]: ioo io~ 7~n/2il--n(p2)n/2--Ym l f f J3(n; 1/1, rE,/)3) = F ( n - ~/)i)HF(/)~) (2hi) 2 du dv x U y V F ( - u ) F ( - v) - ioo - io~

1/3-u)F

- 1/3-v)r(1/3 + u+ v)r(Z /),- In+u+v)

(2 )

where dimensionless variables are defined by X - ppA ~,

y - - -pJ p].

(3)

In formula (2) the integration contours should separate the "right" and "left" series of poles of gamma functions in the integrand (see, e.g., ref. [ 11 ] ). The "uniqueness" conditions (see, e.g., refs. [12-15, 8] ): J3(n;/)1, 1/5,/)3) [ 2u,=n =Trn/Zil--" ~=t fi

F ( ½ n - 1/1) F(1/i) (p2),,-n/2,

(4)

[/)tJ3(n; 1/1 + 1, 1/2, /)3) "~ 1/2J3(n;/)l, 1/5+ 1, /)3) "~ 1/3J3(n; /)1, /)2, /)3 -4- 1 ) ] I ~p,=.-2 f l r ( ½ n - 1/i- 1 ) r(1/i) (p2)U,-n/2+l

i=1

(5)

Parametric representation for n = 4, 1/1= 1 - ~, /)2 = 1 + ~, /)3= I: 1 J3(4; l - J , 1+~, 1 ) = p~ ~d° d ~ y ~ 2 + ( l _ x _ y ) ~ + x ,

(6)

where x and y are defined by ( 3 ). The denominator of (6) can be written as p2 [y~2+ (1 - x - y ) ~ + x ]

= (Pt +~P2) 2.

(7)

Definition of the four-point Feynman integral (see fig. 1b): J4(n;/)l,/)2,/)3,1/4lk2, k2, k~,k2, s,t) = -

f

dn r [(k4_r)2]~[(k2+k3+r)2]~e[(k3+r)2]V3(r2)~,,

(8)

where the independent Mandelstam variables are s --- (k~ + k2 ) 2, t = ( k2 + k3 ) 2. "Pairing" of the arguments in the case n = 4, 1/~+ 1/2+ 1/3= 3,/)4 = 1: 2 k3,2 k,,2 s, t)=s"+~(k~)-U(k2)-~J3(4; l - v , l - u , llk2k2, k2kE, st). J4(4; l - v , l + u + v , l - u , like,2 k2,

(9)

In the four-point case the dimensionless variables are given by X - --k2k2 st '

Y - --k22k2 st

(10)

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The main steps of the proof of formula (9) have been presented in ref. [8 ]. They involve (i) Feynman parametric representation, (ii) Mellin-Barnes representation and (iii) the "uniqueness" condition (4). 3. In this section we shall deal with the L-loop case of the three-point ladder diagram shown in fig. 2 (where the arrangement of momenta is also indicated). The corresponding Feynman integral is

c,L,( P2l , P22 ,2P 3f) -

...

f (rx--r2)2(r2--r3)2...(rL_l--rL)2r~i=l d4r~...d4rL fi

1 (Pl +ri)2(p2--ri) 2"

(11)

Note thatC
2

L

2 /ire "X c(L)(pZ,p~,p3)=~-~3 ) ~(L)(x,y),

with x and y defined by in ref. [8]:

(12)

( 3). For the one- and two-loop cases, simple integral representations have been derived

l t~ (1) ( X ' Y ) ----"- -

f y ~ 2 + ( 1 - xd,- y ) ~ + x

(ln xy_ _t_2l n , )

(13)

o 1

d, l n , ( l n y +ln ( ) ( I n y +2 In ~) qb{2)(X,.V)_- -- ½f y~2+ (1 --X--y)~+X

(14)

0

By decomposition of the denominator,

1

1(1

y~2+(1--x--y)~+x -- 2

1 x

)

~+(py)-~

(15)

'

where

2(x,y)~x/(1--x--y)Z-4xy,

p(x,y)~

1- x - y + 2

(16)

'

these integrals can be evaluated in terms of polylogarithms (see, e.g., ref. [ 16] ), 1

(--1) N ! "'lnN--l~ Liu(Z) -- (AT5i~ ~¢¢__Z_1 .

(17)

The results are [ 8 ]

~

p~

pl+r2 J P3 ~

J

r2--r3

P2 P2 -- r2 r2

. . .

~

~

I rL-I - - r L

J

P2

138

FL

rL

I)2

Fig. 2.

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'(

)

q~(1)(x, y ) = ~ 2 [ L i 2 ( - p x ) + L i 2 ( - p y ) ] +lnY In 1 +py ~(2)(x,y)= ~ +½1n2~

x

6[Li4(-px)+Li4(-py)]+31n

6 May 1993

+ln(px)ln(py)+~n z

1+px

(18)

[Li3(-px)-Li3(-py)]

[LiE(-px)+LiE(-PY)]+~ln2(px)lnE(pY)+½rc21n(px)ln(pY)+hnElnE~x + ~rc4) "

(19)

Note that for negative values of x and y we should take into account the ( + i0 )-prescription for the denominators of (11 ). This requires the following substitutions in the logarithmic terms of (18) and (19) (with a--sgn p~): l n ( p x ) ~ l n ( - p x ) +irca,

ln(py)--+ln(-py) + i ~ a .

(20)

When evaluating the two-loop diagram in ref. [ 8 ], we used a special analytic regularization and the uniqueness conditions (4) and ( 5 ). In such a way, the result was reduced to the functions (6). If we continue to apply this procedure (with a suitable analytic regularization, the same as in ref. [ 14 ] ), we can generalize the results ( 13 ) and (14) to the L-loop case: 1

~(L)(x'Y)=

1 ! L!(L--1)!

L--1

d,

y¢2+(l~-xx--y)¢+X

lnL_,~(

y ¢) ln-+lnx

(

In

y

+2In

~)

.

(21)

Note that the formula (21 ) can also be written as

,)--

1

L

1 { y¢2+(1-x-y)~+xd¢L £rln 'ln_Y +in ¢) ]. "\ x

(22)

(L!) 2

In the present paper we shall prove the representation (21 ) by induction. Suppose, the formula (21 ) holds for some value of L (for L = 1 and L = 2, it coincides with the results ( 13 ) and (14), respectively). Transition from L loops to (L + 1 ) yields an additional integration (see fig. 2 ):

(I)(L+I)(x,y)=

p2f rE(pi +

in--5

(I)(L)('(P~r)2 (P2 ~ r ) 2 )

d4r

r)Z(pE-r) 2

'

P3

(23)

1

If we insert the representation (21) for • (L) into (23), we get a fourth denominator, [p~ +¢PE + ( 1 - ~ ) r ] z. The logarithms In [ ( P 2 - r)2/(Pl + r) 2] occurring in the numerator can be transformed into derivatives with respect to the powers of the denominators by

1

(Pl +r)Z(Pz-r) 2

lnk((P2--r)2 ~ (Ok \(-P-~+ - ~ ] =

1 [(P~ + 0 2] ~+a[ ( P z - r ) 2] ~-s

) t~=0

'

(24)

where 0~- 0/0~. The remaining box integral over the momentum r is J4(4;l_6,1,1+O, ll(l_~)-Ep2,~z(1

2 2 2 2, ( 1 - ~ ) - 2 ( p i + ¢ p 2 ) 2 ) . -¢) -2 p3,pl,p2,p

(25)

By use of the "pairing" property (9), it can be reduced to the three-point function ( 1 ), (p2)-~( 1 -~)2~-2aj3(4; 1 -~, 1 +6, l lp~, ~2p2, (Pl +~P2) 2) •

(26)

Using the representation (6) and substituting the variables (to restore a usual form of the denominator (7) ), we find 139

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PHYSICS LETTERSB 1

~L+')(x'Y)=

L!(L-1)!

6 May 1993

1

y~2+(1-x-y)~+x

--~

× [lnl--'rl(O,~+lnq)L+lnLq(O,~+lntl)L-t](½q-a[¢-'5--(~y/x)~])a=o.

(27)

Then, after using the obvious "commutation" rule, (Oa+ln q)Lq-6f(O) =~l-6 O~f(~),

(28)

we can put r/-6= 1 and integrate over q. Finally, evaluating the derivatives with respect to ~, we arrive at the same result as (21) withL substituted by ( L + I ); q.e.d. The integral (21 ) can be also evaluated in terms of polylogarithms (17). Using the decomposition of denominator (15), we find

1 2L (_l)Jfiln2L-j(y/X)[Lij( 1)_Lij(_py)] ~(L)(x'Y)=-- ~.2j~L (j--L)!(2L--j)! ---~

(29)

with 2 and p defined by (16). Note that the highest order of polylogarithms is 2L (for L-loop diagram). The expression (29), however, is not explicitly symmetric with respect to x and y. To restore this "hidden" symmetry, we can transform Lij ( - 1/px) to the inverse variable (see ref. [ 16 ] ), and we get

1[ 1 2L j!lnZL-j(y/x) [ L i j ( - - p x ) + ( - - 1 ) J L i j ( - - p y ) ] O-L)~ ( 2 L - j ) ! + 2 k ~= o ~t=o (k!l!(L-k)!(L-l)! k+l)!(1--2'-k-¢)

¢(k+I)lnL-k(px)lnL-I(py)

'

(30)

k + l even

where the coefficients of the second sum on the RHS are expressed in terms of the Riemann ¢ function of even arguments (for example, ¢(0) = - ½, ¢(2) = ~zt=, ((4) = ~og4, ( ( 6 ) = ~-~5~6, etc. ). In general, rational factors at the powers of ~ can be related to Bernoulli numbers (see, e.g., re£ [ 16] ). Note that the leading asymptotic behaviour as x--+0, y--+0 is given by the term with k = 1= 0 of the double sum in large parentheses on the RHS of (30), (m (x, y) Ix~o,y~O~

(L!) -2 lnL(px)lnL(py) ,

(31 )

while polylogarithms in the first sum of (30) vanish in this limit (for L>~ 1 ). It is easy to check that for L = 1 and L = 2 the expression (30) coincides with the results (18) and (19), respectively (we should remember that Li ~(z) = - In ( 1 - z) ). Moreover, if we consider the case L = 0 (remembering that Lio (z) = z~ ( 1 - z ) ), we obtain, from (30), a correct "tree" vertex ~(o)= 1. Another way to check the result (21 ) is to apply it to the calculation of the L-loop propagator-type ladder diagram, B(L)(k2) =

f

d4r

r=(k+r) 2

C(L-I)(r

2,

(k+r) 2, k 2) .

(32)

Inserting the expression (21 ) for the three-point function and using the formulae (24), (6) and (28), we arrive at the well-known result (see refs. [ 14,17 ] )

B(L)(k2) = (irt2) L (2L)!

(k2)L_ ~ (L!) 2 ¢ ( 2 L - 1 ) .

140

(33)

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6 May 1993

4. Let us consider the L-loop four-point ladder diagram ("multiple box") shown in fig. 3 (the number of rungs is L + 1 ). The corresponding Feynman integral is

D(L)(k 2, k 2, k 2, k 2 , s, t) f ... f

d4r'"'d4rL I-It1 (k2 +k3 +rl)2(rl-r2)2(r2-r3)2...(rt,_l--rt)2r 2 i=l (k3 + r i ) 2 ( k , - r i ) 2'

(34)

where, as usual, s= (kl +k2) 2 and t= (k2 +k3) 2. In ref. [ 8 ] it was shown that, for L = 1 and L = 2, the functions (34) can be reduced to the corresponding three-point functions ( 1 1 ) by

DO)(k21,k22, k2,k2, s,t ) = C o) ( k 2l k 32, k 22k ,2, s t ) ,

(35)

Dt2)(kE, kE, kE, kE, s , t ) = t C ( 2 ) ( k l k 3 , k2k4, st)

(36)

2

2

2

2

(note that the formula ( 35 ) follows from (9) if we put u = v = 0). Moreover, in the "zero-loop" (tree) case we get D co)= 1 / t = t - 1CtO). Looking at these formulae, we can assume that an analogous "pairing" property is also valid for the L-loop case, namely

D(L)(k~,k 2,k 2,k ] , s , t ) = t L - ' C (L) (k,k3, 2 2 k2k4, z 2 st) .

(37)

We shall prove the formula (37) by induction. Let us suppose that (37) holds for some value of L (we know that it is true for L = 0, 1, 2). It is convenient to introduce the Mellin-Barnes representation, 2

17[

C~L)(p2'p2'p3)= ~3

1

dudv

(2hi) 2

k,p2 ] J U L ) ( u , v ) .

(38)

--ioo--ioo

In fact, d/(~-) (u, v) is the Mellin-transformed image of the function ~tt-) (x, y) ( 12 ). For example, for L = 1 we find, from (2) (see also ref. [ 18 ] ), that dtt~)(u, v) =F2( - u ) F 2 ( - v ) F 2 ( 1 + u + v ) .

(39)

Considering the (L + 1 )-loop function (23) and using (2), it is easy to obtain that io~

1

Jt(L+l)(u,v)=F(-u)F(-v)F(l+u+v)(--~

f

ioo

f du'dv'

--iov--i~

X F(u'-u)F(v'-v)F(1 +u+v-u'-v') F ( 1 - u ' ) F ( 1 - v ' ) F ( l + u' + v')

..CL)- , ~u, v').

(40)

Let us consider then the ( L + 1 )-loop four-point function,

DtL+l)(k 2,k 2,k 2,k ] , s , t ) =

d4r r2(k3+r)2(k4_r)2 D(L)(k 2, k 2, (k3+ r) 2, (k4-r) 2, s, (k2+k3+r)2).

/ Cq-r3 1

p3+r2

k3+rL

(41)

k~

[k2+k3+rl -~4--rl

~

k4

Fig. 3. 141

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By assumption, in the L-loop case D (L) can be reduced to C (r) ( 3 7 ) . Using the Mellin-Barnes representation for C (L) (38), we find that L

/5 (L+I)g/r2~r~l, t~2,1~ 2

ioo ioo

k2k],s,t)=(~) (2zri) el f~.

×J4(4;1-v,l+u+v,l-u,

dudvJg(L)(u,u)

ll kt, 2 k2, 2 k3, 2 k4, 2 s, t) .

(42)

The box integral J4 has exactly the same powers of denominators as needed to apply the "pairing" property (9). Using (9) and (2), we arrive at the representation o f D {L+t) that coincides with that of C ¢L+1) (see ( 3 8 ) (40)) if we substitutep2 - - , k l2k 32, p2-.-,k2k4, 2 2 2 p2-+st and multiply the result by t t. Thus, we proved that the formula (37) is valid also for the ( L + 1 )-loop case; q.e.d. So, the L-loop ladder four-point function is expressed in terms of the three-point function ( 12 ), (iTz2)L q){re(X, y) , D
(43)

where X and Y are defined by (10). Thus, the function D
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[2] P.D.B. Collins, An introduction to Regge theory and high energy physics (Cambridge U.P., Cambridge, 1977). [ 3 ] I.F. Ginzburg and V.V. Serebryakov, Yad. Fiz. 3 ( 1966 ) 164. [ 4 ] P. Osland and T.T. Wu, Nucl. Phys. B 288 (1987) 77, 95. [ 5 ] L. Lukaszuk and L. Szymanowski, Z. Phys. C 43 ( 1989 ) 133; M.C. Berg6re and L. Szymanowski, Phys. Lett. B 237 (1990) 503; Nucl. Phys. B 350 ( 1991 ) 82. [6] R. Gastmans and W. Troost, Phys. Lett. B 249 (1990) 523; R. Gastmans, W. Troost and T.T. Wu, Nucl. Phys. B 365 ( 1991 ) 404. [ 7 ] K.S. Bj~rkevoll, G. F~tldt and P. Osland, Nucl. Phys. B 386 ( 1992 ) 280, 303. [8] N.I. Ussyukina and A.I. Davydychev, Phys. Lett, B 298 (1993) 363. [9] E.E. Boos and A.I. Davydychev. Theor. Mat. Fiz. 89 ( 1991 ) 56. [ 10] A.I. Davydychev, J. Phys. A 25 (1992) 5587. [ 11 ] W.N. Bailey, Generalized hypergeometric series (Cambridge U.P., Cambridge, 1935). [ 12] M.D.'Eramo, L. Peliti and G. Parisi, Lett. Nuovo Cimento 2 ( 1971 ) 878; A.N. Vassiliev, Yu.M. Pis'mak and Yu.R. Honkonen, Teor. Mat. Fiz. 47 ( 1981 ) 291. [ 13 ] N.I. Ussyukina, Teor. Mat. Fiz. 54 (1983) 124. [ 14 ] V.V. Belokurov and N.I. Ussyukina, J. Phys. A 16 ( 1983 ) 2811. [ 15 ] D.I. Kazakov, Teor. Mat. Fiz. 58 (1984) 345. [ 16 ] L. Lewin, Polylogarithms and associated functions (North-Holland, Amsterdam, 1981 ). [ 17 ] D.J. Broadhurst, Phys. Lett. B 164 ( 1985 ) 356. [ 18] N.I. Ussyukina, Teor. Mat. Fiz. 22 (1975) 300. [ 19 ] G. 't Hooft and M. Veltman, Nucl. Phys. B 44 ( 1972 ) 189; C.G. Bollini and J.J. Giambiagi, Nuovo Cimento 12 B (1972 ) 20. [20] R.J. Gonsalves, Phys. Rev. D 28 (1983) 1542; W.L. van Neerven, Nucl. Phys. B 268 ( 1986 ) 453; G. Kramer and B. Lampe, J. Math. Phys. 28 (1987) 945.

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