Loop integrals, integration-by-parts and MATAD

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1 Loop integrals, integration-by-parts and MATAD Matthias Steinhauser TTP, University of Karlsruhe CAPP 009, March/April 009, DESY, Zeuthen Matthias Steinhauser p.

2 Outline I. Integration-by-parts II. MATAD III. Examples Matthias Steinhauser p.

3 I. Integration-by- Parts (IBP) [Many thanks to Robert Harlander for the possibility to recycle some of his transparencies.] Matthias Steinhauser p.3

4 A. Notation F(n,...,n N ) = d d k d d k l ( p + m )n ( p N + m N )n N N number of propagators n i indices l number of loops p i linear combinations of loop momenta k i and external momenta q i Dimensional Regularization d = 4 ǫ scaleless integrals = 0 Matthias Steinhauser p.4

5 B. Example: -loop self energy k + q q = F(n,n ) = d d k ( p + m )n ( p + m )n k p = k, p = k + q special case: m = m = 0: [n = a, n = b] I q (a,b) = iπ d/ ( q ) d/ a b Î(a,b) Î(a,b) = Γ(a + b d/) Γ(d/ a) Γ(d/ b) Γ(a) Γ(b) Γ(d a b) Matthias Steinhauser p.5

6 Example: -loop self energy Î(a,b) = Γ(a + b d/) Γ(d/ a) Γ(d/ b) Γ(a) Γ(b) Γ(d a b) remarks: valid for (almost) all a, b, d expansion in ǫ through zγ(z) = Γ( + z) and Γ( + ǫ) = ǫγ E + ǫ ) E π (γ + 6 (γ E is cancelled by our normalization) +... Matthias Steinhauser p.6

7 Example: -loop self energy k + q q = F(n,n ) = d d k ( p + m )n ( p + m )n k p = k, p = k + q special case: n = 0 [equivalently: q = 0, m = m ] k = F(n)= d d k ( k + m ) n = iπ d/ (m ) d/ nγ(n d/) Γ(n) Matthias Steinhauser p.7

8 Example: -loop self energy k = F(n)= d d k ( k + m ) n = iπ d/ (m ) d/ nγ(n d/) Γ(n) note: massless tadpoles = 0 (scaleless integrals) e.g. = 0 in massless case Matthias Steinhauser p.8

9 C. Products and convolutions of -loop factorization = a=n +n 3 +n +n 5 d/ d d l (l ) n +n 3[(l q) ] n 4 Î(n, n 5 ) d d k (k ) n 5 [(k + l) ] n }{{} (l ) d/ n 5 n Î(n,n 5 ) d d l (l ) a [(l q) ] n 4 (q ) d P ni Î(n + n 3 + n + n 5 d/, n 4 ) Î(n, n 5 ) [Euclidian momenta] Matthias Steinhauser p.9

10 Products and convolutions of -loop () analogously: (for m = 0) convolutions of -loop self-energy diagrams however: genuine -loop: T topology Matthias Steinhauser p.0

11 Products and convolutions of -loop () analogously: (for m = 0) convolutions of -loop self-energy diagrams however: genuine -loop: T topology but: shrinking any line convolution of -loop! Matthias Steinhauser p.0

12 D. IBP relations Dimensional regularization: Integrals are finite Surface terms vanish d d k k µ p µ f(k,p,...) = 0 p µ : loop or external momentum Example: F(n) = d d k ( k +m ) n Consider: 0 = d d k k µ k µ ( k + m ) n Matthias Steinhauser p.

13 D. IBP relations Example: F(n) = d d k ( k +m ) n 0 = d d k k µ k µ ( k + m ) n = d d d k ( k + m ) n + k µ( n)( k µ ) ( k + m ) (n+) = = Matthias Steinhauser p.

14 D. IBP relations Example: F(n) = d d k ( k +m ) n 0 = d d k k µ k µ ( k + m ) n = d d d k ( k + m ) n + k µ( n)( k µ ) ( k + m ) (n+) = d d d n k ( k + m ) n + nm ( k + m ) (n+) = Matthias Steinhauser p.

15 D. IBP relations Example: F(n) = d d k ( k +m ) n 0 = d d k k µ k µ ( k + m ) n = d d d k ( k + m ) n + k µ( n)( k µ ) ( k + m ) (n+) = d d d n k ( k + m ) n + nm ( k + m ) (n+) = (d n)f(n) + nm F(n + ) F(n) = d n + (n )mf(n ) Any F(n),n > can be expressed by F() F(n) = 0 if n 0 F() = master integral Matthias Steinhauser p.

16 D. IBP relations Example: F(n) = d d k ( k +m ) n F(n) = d n + (n )mf(n ) Raising and lowering indices: Define operators ±, ±,...: I ± F(n,...,n I,...) = F(n,...,n I ±,...) F(n) = d n + (n )m F(n) Matthias Steinhauser p.

17 E. Topology T (massless) == act on integrand with d d l d d k [l ] n [k ] n [(k q) ] n 3 [(l q) ] n 4 [(l k) ] n 5 l µ l µ, l µ k µ, l µ q µ, k µ k µ, k µ l µ, k µ q µ, example: l µ l µ = d + l µ l µ l µ l µ [l ] n = n [l ] n Matthias Steinhauser p.

18 Topology T (massless) == d d l d d k [l ] n [k ] n [(k q) ] n 3 [(l q) ] n 4 [(l k) ] n 5 l µ l µ l µ l µ [l ] n = n [l ] n l µ [(l q) ] n = n 4 4 l µ [(l q) ] n = n 4 4 { } [(l q) ] n l l q 4+ [(l q) ] n 4+ { l + [ (l q) l q ] } Matthias Steinhauser p.3

19 Topology T (massless) == d d l d d k [l ] n [k ] n [(k q) ] n 3 [(l q) ] n 4 [(l k) ] n 5 l µ l µ l µ l µ [l ] n = n [l ] n l µ [(l q) ] n = n 4 4 l µ [(l q) ] n = n 4 4 { [(l q) ] n l + 4+ [(l q) ] n 4+ { l + [ (l q) l q ] } [ (l q) l q ] } Matthias Steinhauser p.3

20 Topology T (massless) == d d l d d k [l ] n [k ] n [(k q) ] n 3 [(l q) ] n 4 [(l k) ] n 5 l µ l µ l µ [l ] n = n [l ] n l µ [(l q) ] n = n 4 4 { [(l q) ] n 4+ }{{} 4 + l }{{} + [ (l q) }{{} ]} }{{} l q 4 [ d n n 4 n 5 + n 4 (q )4 + + n 5 ( )5 +] T = 0 Matthias Steinhauser p.4

21 IBP identities for T [ ] d n n 4 n }{{} 5 + n 4(q }{{} )4 + }{{} +n 5( )5 + T = 0 task: combine all IBP identities such that T (n,n,n 3,n 4,n 5 ) simpler integrals, i.e. convolutions of -loop low values of n i Matthias Steinhauser p.5

22 IBP identities for T [ ] d n n 4 n }{{} 5 + n 4(q }{{} )4 + }{{} +n 5( )5 + T = 0 task: combine all IBP identities such that T (n,n,n 3,n 4,n 5 ) simpler integrals, i.e. convolutions of -loop low values of n i Exercise: compute IBP with l µ k µ Matthias Steinhauser p.5

23 IBP identities for T d n + n 4 ( + ( q ) 4 + ) + n 5 ( + ( + )5 + ) = 0 n ( + ( q + 4 ) + ) + n 4 ( + (q )4 + ) + ( ) n = 0 n ( + ( + 5 ) + ) + ( q ) n n 5 ( + ( + )5 + ) = 0 n ( + ( + 5 ) + ) + (q ) n n 5 ( + ( )5 + ) = 0 n ( + ( q + 3 ) + ) + n 3 ( + (q )3 + ) + ( ) n = 0 d n + n 3 ( + (q )3 + ) + n 5 ( + ( )5 + ) = 0 () (3): [ d n 5 n n 4 }{{} n (5 ) + n 4 (5 3 )4 + }{{} ] T = 0 Matthias Steinhauser p.6

24 F. Triangle rule T(n,n,n 3,n 4,n 5 ) = d n 5 n n 4 ] [n (5 ) + + n 4 (5 3 )4 + T (n,n,n 3,n 4,n 5 ) recurrence relation example: n = n = n 3 = n 4 = n 5 = T (,,,, ) = [ T (,,,, 0) T (, 0,,, ) d 4 ] + T (,,,, 0) T (,, 0,, ) = ǫ [ ] 6ζ(3) Matthias Steinhauser p.7

25 Triangle rule indices > apply rec.-rel. repeatedly T (,3,,,) T (,3,,,0) T (,,,,) T (3,,,,0) T (,3,,,0) T (3,,,,) T (4,,,,0) T (,3, 0,,) T (,,,,0) T (4,0,,,) generates many terms needs computer algebra! T (,,0,,) T (3,,,,0) T (3,, 0,,) [n (5 ) + + n 4 (5 3 )4 + ] Matthias Steinhauser p.8

26 Triangle rule observation: sub-loop! e.g. 3-loop: this rec.-relation applies to any triangular d n 5 n n 4 = n (5 ) + + n 4 (5 3 )4 + = ǫ [ ] = ǫ [ ] Matthias Steinhauser p.9

27 Triangle rule exercise Which convolutions of - and -loop integrals does the following integral reduce to? Matthias Steinhauser p.0

28 G. Reduction to MIs modern : Laporta-, Gröbner-, Baikov-,... algorithm old-fashioned (but much more effective): look ( by hand ) for combinations of IBP relations which systematically reduce complicated integrals to simpler ones MINCER: massless -point function;,, 3 loops Grinder: HQET integrals;,, 3 loops MATAD: vaccum bubbles;,, 3 loops [Larin,Tkachov,Vermaseren] [Grozin] [Steinhauser] Matthias Steinhauser p.

29 II. MATAD (MAssive TADpoles) [MATAD3 for FORM3] Matthias Steinhauser p.

30 MATAD: MAssive TADpoles written in FORM (MATAD3 FORM3) vaccum bubbles up to 3 loops propagators: mass M or massless zero external momentum final result up to (incl.) O(ǫ 3 l ) Matthias Steinhauser p.3

31 MATAD: applications moments of Π γ (q ) m c,m b ρ parameter: 3-loop corrections decoupling of α s and m q matching coefficient of eff. Lagrangian (gg H, B X s γ) Γ(H γγ)... Matthias Steinhauser p.4

32 A. Flowchart read FD translate notation traces expansion Wick rotation... reduction MIs simple integrals expansion in ǫ print and save result FD projectors FRs... Input files:.dia file: FORM fold/diagram GLOBAL: overall settings, for interaction, projectors,... TREAT -files declare file for user-defined variables Matthias Steinhauser p.5

33 A. Flowchart read FD translate notation traces *--#[ expansion GLOBAL : Wick rotation... *--#] GLOBAL : *--#[ reduction TREAT : MIs simple integrals.sort *--#] TREAT : expansion in ǫ print and save result FD projectors FRs #define LOWLIMM "-"... #define GAUGE "xi" multiply, deno(3,-)*( Input -d files: (mu,mu) + q(mu)*q(mu)/q.q.dia file: ); FORM fold/diagram GLOBAL: overall settings, for interaction, projectors,... TREAT -files declare file for user-defined variables Matthias Steinhauser p.5

34 B. Notation/Topologies Æ Å Æ Æ½ ƾ Æ Å Å½ ž Matthias Steinhauser p.6

35 masses: M,M,... ( ) em = µ ǫ, M... B. Notation loop momenta: p,p,... external momenta: q,q,... massless scalar denominators: /p.p,/p.p,... massive scalar denominators: sm,sm,...; sm M p internal indices: nu,nu,... external indices: mu,mu,... deno(a,b) = a+bǫ scalar denominator: Den(L,#, momenta,masses,exp, momenta,masses ),#,L) Matthias Steinhauser p.7

36 B. Notation scalar denominator: Den(L,#, momenta,masses,exp, momenta,masses ),#,L) example: Den(L,8,+p5,pM,8,L) p 5 Den(L,8,-p7,pM,M,pM,8,L) M p 7 Den(L,3,-p6,pM,M,pM,exp,+q,pQ,3,L) M ( p 6+q expand in q ) pm,pq: powercount variables; needed for expansion spinor lines: FT, FT,... FT(L,3,+p5,pM,M,pM,3,L) M γ ν p ν 5 FT(L,6,-p3,pM,M,pM,exp,+q,pQ,6,L) M γ ν ( p 3 +q ) ν FT(mu) γ µ Matthias Steinhauser p.7

37 B. Notation gluon propagator: Dg(nu3,nu4,L,6,+p,pM,6,L) gν 3 ν 4 GAUGE p ν 3 p ν 4 p Vgh: ghost-gluon vertex V3g: 3-gluon vertex Vggs, Dsig: implementation of 4-gluon vertex other vertices: use TREAT0 Matthias Steinhauser p.7

38 B. Notation colour diagram: fqcd... fundamental, external: i,i,... fundamental, internal: j,j,... adjoint, external: a(),a(),... adjoint, internal: b(),b(),... d (i,i) δ i,i GM(a(),i,i) λ a i,i prop(a(),a())) δ a,a Matthias Steinhauser p.7

39 More notation/convention imaginary unit i: M p W.R. M +p all vertices: i (...) all propagators: i (...) don t care about i of internal lines note: i ( i d d ) l=,,3 W.R. k j i ( d d ) l=,,3 k j,e overall i (independent of l) is omitted γ E multiply with (e γ Eǫ ) l γ E disappears from final result normalization of the integrals: dk k e.g.: iπ d/ = e γeǫ em (M ( n) Γ(n d/) ) Γ(n) Matthias Steinhauser p.8

40 ms/software.html. download matad3.tar.gz. mkdir matad3; cd matad3 3. gunzip matad3.tar.gz 4. tar -xvf matad3.tar 5. adapt path to form3 in startform3 Matthias Steinhauser p.9

41 C. Examples -loop tadpole *--#[ dl : sm ; #define INT "tadl" #define MASS "M" *--#] dl : *--#[ expdl : multiply, em*pmˆ4; #call denoexp{m} #include matad.info # time *--#] expdl : Matthias Steinhauser p.30

42 C. Examples -loop tadpole > cd calc3 > startform3 -S form.set -d CLASS=c misc -d PROBLEM=test -d LOOPS= -d DIAFILE=test.dia -d RESDIR=results -d DIAGRAM=dl generic/maindia.frm Matthias Steinhauser p.30

43 C. Examples -loop tadpole dl = + epˆ- * ( - Mˆ*eM ) + ep * ( - /*Mˆ*eM*z - Mˆ*eM ) > cd calc3 + epˆ * ( - /*Mˆ*eM*z + /3*Mˆ*eM*z3 - Mˆ*eM ) > startform3 -S form.set -d CLASS=c misc -d PROBLEM=test - Mˆ*eM; -d LOOPS= -d DIAFILE=test.dia -d RESDIR=results -d DIAGRAM=dl generic/maindia.frm Matthias Steinhauser p.30

44 C. Examples () -loop expanded in external momentum *--#[ dl : Den(L,,p,pM,M,pM,,L) * Den(L,,p,pM,M,pM,exp,q,pQ,,L) ; q #define INT "tadl" #define MASS "M" *--#] dl : Matthias Steinhauser p.3

45 C. Examples () -loop expanded in external momentum q > cd calc3 > startform3 -S form.set -d CLASS=c misc -d PROBLEM=test -d LOOPS= -d DIAFILE=test.dia -d RESDIR=results -d DIAGRAM=dl generic/maindia.frm Matthias Steinhauser p.3

46 C. Examples () -loop expanded in external momentum q Q: Euclid. momentum dl > = cd calc3 + epˆ- * ( em ) > startform3 -S form.set -d CLASS=c misc -d PROBLEM=test + ep -d * ( LOOPS= /*em*z -d+ /60*Q.Qˆ*Mˆ-4*eM DIAFILE=test.dia ) -d RESDIR=results -d DIAGRAM=dl generic/maindia.frm + epˆ * ( - /3*eM*z3 - /*Q.Q*Mˆ-*eM*z + /0*Q.Qˆ* Mˆ-4*eM*z ) - /6*Q.Q*Mˆ-*eM + /60*Q.Qˆ*Mˆ-4*eM; Matthias Steinhauser p.3

47 D. (MIs) Symbols in MATAD D6 = 6ζ 3 7ζ 4 4ζ ln + 3 ln 4 «+ 6Li 4 4» π «Cl 3 D5 = 6ζ ζ » π «X Cl 6 3 m>n>0 D4 = 6ζ 3 77» π «ζ 4 6 Cl 3 D3 = 6ζ 3 5» π «ζ 4 6 Cl 4 3 DM = 6ζ 3» π «ζ 4 4 Cl 3 DN = 6ζ 3 4ζ ln + ln 4 «ζ 4 + 6Li 4 3 B4 = 4ζ ln + ln 4 3 «ζ 4 + 6Li 4 3 E3 = 39 S = 4 3 OepS = 763 Tep = Cl π 3 3 π 3 ln 3 «8 9π 3 ln 3 6 π 3 ln 3 8 7π3 3 7 ( ) m cos(πn/3) m 3 n ζ + ζ π 3Cl «6 3Im 4Li 3 e iπ/ π ζ 5 ζ ζ π «Cl 7 3Im e iπ/6 3 A π ζ + ζ π 3Cl 3 «6 3Im 4Li 3 e iπ/6 3 3 A5 3 A5 Matthias Steinhauser p.3

48 Literature M. Steinhauser, MATAD: A program package for the computation of massive tadpoles, Comput. Phys. Commun. 34 (00) 335 [arxiv:hep-ph/000909]. ms/software.html V. A. Smirnov, Feynman integral calculus, Berlin, Germany: Springer (006) 83 p S. A. Larin, F. V. Tkachov and J. A. M. Vermaseren, The Form Version Of Mincer, S. G. Gorishnii, S. A. Larin, L. R. Surguladze and F. V. Tkachov, MINCER: Program for Multiloop Calculations in Quantum Field Theory for the Schoonschip System, Comput. Phys. Commun. 55 (989) 38. K. G. Chetyrkin and F. V. Tkachov, Integration By Parts: The Algorithm To Calculate Beta Functions In 4 Loops, Nucl. Phys. B 9 (98) 59. D. J. Broadhurst, Three Loop On-Shell Charge Renormalization Without Integration: Lambda-Ms (QED) To Four Loops, Z. Phys. C 54, 599 (99). L. V. Avdeev, Recurrence Relations for Three-Loop Prototypes of Bubble Diagrams with a Mass, Comput. Phys. Commun. 98 (996) 5 [arxiv:hep-ph/9544]. Matthias Steinhauser p.33

49 III. Examples Matthias Steinhauser p.34

50 A. Scalar 3-loop integral see: problems/c misc/test Matthias Steinhauser p.35

51 B. Photon polarization fucntion ( q ) g µν + q µ q ν Π(q ) = i dxe iqx 0 Tj µ (x)j ν(0) 0 = Π (0) (q ) + α s 4π Π() (q ) + = +... ( αs ) Π () (q ) 4π Projector: q Π(q ) = d ( g µν + q µq ν q ) Π µν (q ) see: problems/c pi/pi Matthias Steinhauser p.36

52 Exercises I. Compute the ǫ expansion for the one-loop tadpole with index n =,,3, 4,... using (a) the explicit analytic result, (b) the recurrence relation.. Derive the triangle rule for the two-loop integral of type T where each line has a different mass. 3. (a) Derive the recurrence relations for the massless two-loop integral of type T. (b) Program the triangle rule in order to compute the ǫ expansion of the integral T for arbitrary indices up to (including) terms of order ǫ. 4. (a) Derive the recurrence relations for the two-loop tadpole with massive and massless lines. (b) Compare with explicit analytic result. Matthias Steinhauser p.37

53 Exercises II. Compute (a) one-loop tadpole, (b) two-loop tadpole with massive and massless line (c) three-loop tadpole with 4 lines where each of them connects vertex with vertex ; massive and massless lines, with MATAD for various choices of the indices. Compare with the analytic results.. Compute three-loop vacuum integrals (with six lines, tad3l ) with various massive/massless combinations. Which constant appears in the final result? For which choice of indices is the result finite? 3. Check that the sum of the (two) singlet diagrams contributing to Π(q ) at three loops is zero. Why? 4. Compute Π(q ) for arbitrary gauge parameter ξ and check that ξ drops out in the sum of the bare diagrams. 5. Renormalize Π(q ) and compare with the result in the literature. 6. Compute the axial-vector, scalar and pseudo-scalar correlator. Matthias Steinhauser p.38