Five-loop massive tadpoles

Transcription

1 Five-loop massive tadpoles York Schröder (Univ del Bío-Bío, Chillán, Chile) recent work with Thomas Luthe and earlier work with: J. Möller, C. Studerus Radcor, UCLA, Jun 0

2 Motivation pressure of hot QCD phenomenology needs physical NLO [Braaten/Nieto ; KLRS 0] hardest building block: -loop tadpoles(m,0) at d = ɛ QCD beta function and anomalous dimensions moments known since years at -loop accuracy [vanritbergen/vermaseren/larin ] -loop needed e.g. for improved ρ-parameter, decoupling equation,... [loop: 0] partial results appear since years mainly from Karlsruhe (β QED, γ m,...) [Baikov/Chetyrkin/Kühn/Rittinger 0-] many problems allow for asymptotic expansions mapping on tadpoles, often for price of many dots in this talk: focus on master integrals basic building block for -loop problems interested in methods that allow to choose d in the end main progress via refinement of Laporta approach (problem-specific) reduction not covered here [ see talk of T.Luthe] [ see other talks] York Schröder, UBB Chillán /

3 Classification consider fully massive -loop tadpoles Euclidean space-time same mass in all propagators /(q i + ) the -loop integral family needs propagators / lines q i {k, k, k, k, k, k, k, k, k, k, k, k, k, k, k } where k a...bc = k a k b k c trivalent graphs have lines: classification: label sectors by binary rep identify unique graphs find all isometries and corresponding momentum shifts choose largest representative from each class normalization: divide out [-loop tadpole] #loops recall that in d, [-loop tadpole] /ɛ York Schröder, UBB Chillán /

4 Classification: non-trivial shift relations suppose you were given a -loop massive tadpole [ID of our list above] (d) k.. k + (k k ) + (k k ) + (k k ) + (k k ) + (k k ) + (k +k k ) + shift k µ i k µ i = M ijk µ j with M = and det M = then gives = d (d) k.. k + k + (k k ) + k + (k k ) + k + k + = d [ID of our list above] does this (evil) momentum-labelling occur in practice? York Schröder, UBB Chillán /

5 Classification: -loop 0 0 some numerology for specific momentum list combinatorics wins! L(L+)/ + LE = scalar products = possible sectors + do not correspond to a Feynman graph zero-sectors shifts + unique sectors ( with -loop factors not shown) York Schröder, UBB Chillán /

6 Classification: -loop sc prod =.M sectors.m no graph K zeros K shifts unique (show -conn. cubic graphs) 0 York Schröder, UBB Chillán /

7 -loop Sectors arrive at unique -loop sectors (+ factorized ones not shown) t = : t = : t = : t = : t = 0 : t = : t = : recall that at ///-loop there were ///0 unique sectors (plus 0/// fact) York Schröder, UBB Chillán /

8 -loop Masters a (small) IBP reduction reveals that some sectors contain multiple master integrals need in addition (+ factorized ones) masters with dots. some examples: recall that at ///-loop there were 0/0/0/ masters with dots how to evaluate these + (++) zero-scale master integrals? various methods, e.g. explicit integration in x-space differential eqs (in mass ratio); solve iteratively with HPLs explicit solution of low-order difference equations: P F P etc. numerical solution of difference equations via factorial series [Laporta 00] Mathematical structure interested in the coefficients of an ɛ expansion in many cases, these are from a generic class of functions/numbers e.g. harmonic polylogarithms HPL(x) [Remiddi/Vermaseren 00] e.g. harmonic sums S(N) [Vermaseren ] relation: H m () S m ( ) if solution numerical: use some PSLQ York Schröder, UBB Chillán /

9 Evaluation: differential equations perform IBP reduction with two masses: M, m get differential eqn in z = M/m use boundary values at z = 0 (z = ) use symmetry relations like z /z typically, want the integral at z = (z = 0) simple -loop example (basketball type) B (z) (d) J p.. p + z p + z p + (p + p + p ) + B (z) = x d d Γ( d B (/z), B (0) = )Γ( d )Γ(d ) Γ( d )Γ( d ) satisfies { z( z ) z ( z } )(d ) z z(d )(d ) B (z) = (d ) z d ( z d ) solution standard, via variation of constants, in terms of HPL(z); set z = and use algebra of HPL() resp. S( ) B = ep^ * ( / - /*z ) ep^ * ( 0/ + /*pi^ + /*pi^ - 0*s -...) +... York Schröder, UBB Chillán /

10 Evaluation: difference equations perform IBP reduction with symbolic power x on one line derive difference equation for generalized master I(x) D x D...D N R p j (x)i(x + j) = F (x) j=0 typically, want I(); solve the difference equation explicitly (if st order) numerically (very general setup) [Laporta 00] [see talk by T.Luthe] solve via factorial series I(x) = I 0 (x) + R j= I j(x), where I j (x) = µ x j Γ(x + ) a j (s) Γ(x + + s K j ) s=0 need boundary condition for fixing, say, a j (0): use decoupling at large x I(x) = k g(k )/(k + )x I(x) () x x d/ g(0) York Schröder, UBB Chillán /

11 Choice of basis of transcendentals to absorb single powers of π as well as powers of ln, def h n k=0 Γ(k + /) Γ(k + )Γ(/) (/) k (k + ) n [ ] H n h n + h Coefficient ɛ/ Γ( ɛ) Γ ( ɛ/) + O(ɛn ), ɛ, n H = h = π, H = h h ln, etc I.. = +( ) ɛ0 + ( ) ɛ + (H ) ɛ + (H H ) ɛ + (H H + H ) ɛ +... to absorb powers of ln, def as elements of the MZV basis A n Li n ( ) + ln n ( ( )n n! n(n ) ζ ln ) I.. = +(0) ɛ 0 + (0) ɛ + ( ζ ) ɛ + ( A + H + ζ ) ɛ +... York Schröder, UBB Chillán 0/

12 Sample results (d) I.. = +( ) ɛ 0 + ( ) ɛ + ( ) ɛ + ( 0 ) ɛ + ( ) ɛ ɛ +... I.. = +(0) ɛ 0 + ( ) ɛ + ( ) ɛ + ( 0 ) ɛ + ( ) ɛ +.00 ɛ +... I 0.. = +( ) ɛ0 + ( 0 ) ɛ + ( ζ 0 ) ɛ + ( ζ + ζ + ) ɛ + ( ζ ζ + ζ + ) ɛ ɛ +... I 0.. = +( ) ɛ0 + ( 0 ) ɛ + ( 0 ) ɛ + ( ζ ) ɛ + ( ζ + ζ ) ɛ ɛ +... I 0.. = +(0) ɛ 0 + (0) ɛ + ( ζ ) ɛ + ( ζ + ζ + ζ ) ɛ + ( H ζ + ζ.0 ɛ +... I 0.. = +(0) ɛ 0 + (0) ɛ + (0) ɛ + (0) ɛ + ( ζ ) ɛ ɛ ζ ζ + ζ ζ ) ɛ York Schröder, UBB Chillán /

13 Invitation to UTFSM Valparaíso, Chile - Jan 0: th Chilean HEP (Summer!) School [ - Jan 0: th ACAT workshop [ York Schröder, UBB Chillán /