Explicit solutions of the multi{loop integrl recurrence reltions nd its ppliction? P. A. BAKOV ; nstitute of ucler Physics, Moscow Stte University, Moscow 9899, Russi The pproch to the constructing explicit solutions of the recurrence reltions for multi{loop integrls re suggested. The resulting formuls demonstrte high eciency, t lest for 3{loop vcuum integrls cse. They lso produce new type of recurrence reltions over the spce{time dimension. Vcuum cse Recently [,] new pproch to implement recurrence reltions [3] for the Feynmn integrls ws proposed. n this work we extend the generl formuls for the solutions of the recurrence reltions to the multi{loop cse. Let us consider rst vcuum L-loop integrls with = L(L + )= denomintors (so tht one cn express through them ny sclr product of loop moment) of rbitrry degrees: B(n;D)=m n i D = X ij LD d D p :::d D p L D n :::D n A (ij) p i p j m ; p k p l = X = ; () (A ) (kl)(d + m ): () The recurrence reltions tht result from integrtion by prts, by letting (@=@p i ) p k ct on the integrnd [3], re:? Tlk presented t the AHEP'96 (EPFL-UL Lusnne, Sept. -6 996) Supported in prt by the RBRF (grnt 96{0{00654), TAS (grnt 93-80- ext); e-mil: bikov@theory.npi.msu.su Preprint submitted to Elsevier Preprint 9 ovember 996
D i k B(n;D)= X LX ;d= l= A (il) d n d d+ (A ) (kl)( + )B(n;D); where c B(:::;n c;:::) = B(:::;n c ;:::), in prticulr c n = n c. Using the reltions [n d d+ ; ]= d ; X = A (il) (A ) (kj) = (i (k l) j) ; they cn be represented s D L i k B(n;D)= X LX ;d= l= (A ) (kl)( + )A (il) d n d d+ B(n;D): (3) The common wy of using these reltions is step{by{step reexpression of the integrl () with some vlues of n i through set of integrls with shifted vlues of n i, with the nl gol to reduce this set to liner combintion of severl "mster" integrls k (D) with some "coecient functions" F k (n;d):. B(n;D)= X k F k (n;d) k (D) evertheless, to nd proper combintions of these reltions nd proper sequence of its use is the mtter of rt even for the tree{loop integrls with one mss [4]. Then, even in cses when such procedures were constructed, they led to very time nd memory consuming clcultion becuse of lrge reproduction rte t every recursion step. nsted, let us construct the F k (n;d) directly s solutions of the given recurrence reltions. ote, tht if we nd ny set of the solutions, we could construct F k (n;d) s their liner combintions. Let us try the solution of (3) in the following form: f k (n) = ({) dx dx x n xn g(x ) (4) where integrl symbols denote subsequent complex integrtions with contours which will be described lter. Acting by some opertor O i ( ;n + ) (ll decresing opertors should be plced to the left) on (4) nd performing the integrtion by prts one gets (s.t. re surfce terms):
O i ( ;n + )f k (n)= ({) dx dx x n xn O i (x ;@ )g(x ) + (s. t.): So, if we choose the g(x ) s the solution of O i (x ;@ )g(x ) = 0 nd cncel the surfce terms by proper choosing of integrtion contours (for exmple, closed or ended in the zero points) we nd tht (4) is solution of reltions O i ( ;n + )f k (n) = 0, nd dierent choices of contours correspond to dierent solutions. The dierentil equtions for (3) hve the solution g(x )=P(x + ) (D L )=, where P (x ) = det( X = (A ) (kl) x ) is the polynomil in x of degree L, so we get the desirble solutions of (3): f k (n;d)= ({) dx dx x n xn det((a ) (kl)(x + )) D L : (5) Finlly, let us derive from (3) the recurrence reltions with D-shifts. ote tht if f k (n i ;D) is solution of (3), then by direct substitution to (3) one cn check tht P ( + )f k (n i ;D ) lso is solution. Hence, if f k (n i ;D)is complete set of solutions, then f k (n i ;D)= X n S k n(d)p( + i )f n (n i ;D ); where the coecients of mixing mtrix S is numbers, tht is do not ct on n i. For the solutions (5) the S is the unit mtrix (the incresing of D by leds to ppering of fctor P (x ) in the integrnd of (5)), but the desire to come to some specic set of mster integrls my led to nontrivil mixing. These reltions look dierent from recently proposed in [5], lthough further investigtions cn give some connections with them. To check the eciency of this pproch we evluted (using REDUCE) the rst 5 moments in the smll q expnsion of the 3-loop QED photon vcuum polriztion. The 3-loop contribution to the moments re expressed through bout 0 5 three{loop sclr vcuum integrls with four mssive nd two mssless lines. The integrl (5) in this cse cn be solved to nite sums of the Pochhmmer's symbols (see []). Moreover, it is not necessry to evlute these integrls seprtely. nsted, we evluted few integrls of (5) type, 3
but with P D= producted by long polynomil in x i (the results see in [,6], they re in greement with QCD clcultions [7] mde by FORM). The comprison with the recursive pproch shows resonble progress: the common wy used in [6] demnds severl CPU hours on DEC-Alph to clculte full D dependence of the rst moment, nd further clcultions becme possible only fter trunction in (D= ). n the present pproch the full D clcultion for the rst moment demnds few minutes on PC. on{vcuum cse Suppose tht integrls () depend on R externl moment p i (L<iL+R). The number of the denomentors re now = L(L +)=+LR, nd the number of dditionl ("externl") invrints re = R(R +)=. Let us expnd the integrls in forml seri over "denomintor{like" objects D of () type with = +; ::; +, depending on externl moment only: B(n l;(l=;:::; );p k;(k=l+;:::;l+r) )= d D p :::d D p L D n :::Dn = = X n i (i> ) m n i+ +LD b(n i ; (i=;:::; + ) ) Y+ i= + D n i i : (6) We dene such generl expnsion in order to write the recurrence reltions in compct form, in prctice the coecients A (ij) nd my bevery simple. The expnsion with negtive n i corresponds to the lrge moment expnsion, with positive ones to the expnsion ner points m. The n i cn lso be noninteger, but with unit shifts. Acting by (@=@p i ) p k, (i = ;:::;L;k = ;:::;L + R) on the integrnd we get recurrence reltions. The dditionl reltions we get cting by p k (@=@p i ), (i; k = L +;:::;L+R) on both sides of (6). These new reltions look like the old ones with only exception tht they hve no terms proportionl to spce{time dimention D. The complete set of recurrence reltions is now ((D L R ) i k (D R ) ^ i k) b(n;d)= X+ = ;d= X L+R l= (A ) (kl)( + )A (il) d n d d+ b(n;d); 4
where ^ i k=( i k if i; k > L, else 0). The corresponding dierentil equtions hve the solution g(x )=g 0 (x + ), where g 0 (x) = det (A ) (kl) x D L R det 0 (A ) (kl) x D R ; (7) nd det 0 denotes the minor with k; l > L. So, one cn use the representtion (4), but the problem of resolving it to explicit formuls demnds futher investigtions. Finlly note tht one cn formlly obtin the formuls (5, 7) by "chnge of integrtion vribles" from loop moment to "denomentor{like objects" D. The weight function for this chnge is d D p d D p L Y i (D i =m x i ) / det((a ) (kl)(x + )) D L : References [] P. A. Bikov, Phys. Lett. B385 (996) 404, hep-ph/960367. [] P. A. Bikov, hep-ph/960454. [3] K. G. Chetyrkin, F. V. Tkchov, ucl. Phys. B9 (98) 59; F. V. Tkchov, Phys. Lett. B00 (98) 65. [4] D. J. Brodhurst,. Phys. C54 (99) 599; L. V. Avdeev, Comp. Phys. Commun. 98 (996) 5. [5] O. V. Trsov, Phys. Rev.D54 (996) 6479. [6] P. A. Bikov, D. J. Brodhurst Preprint OUT-40-54, hep-ph/9504398, P- 95-3/377, Proc. of 4th nt. Workshop on Softwre Engineering nd Articil ntelligence for High Energy nd ucler Physics, Pis, tly, 995, eds. B. Denby, D. Perret{Gllix, 67. [7] K. G. Chetyrkin, J. H. Kuhn, M. Steinhuser, TTP-96-3 (996), hep-ph/960630. 5