Multiple polylogarithms and Feynman integrals

Transcription

1 Multiple polylogarithms and Feynman integrals Erik Panzer Institute des Hautes E tudes Scientifiques Amplitudes 215 July 7th ETH/University of Zu rich

2 Topics 1 hyperlogarithms & iterated integrals 2 multiple polylogarithms 3 functional equations, symbols 4 parameter integrals 5 other stuff 6 coaction (Brown, Goncharov) 1 / 32

3 Some FI/amplitudes are expressible via multiple polylogarithms (MPL) k z 1 1 Li n1,...,n d (z 1,..., z d ) = zk d d k n 1 <k 1 < <k d 1, z kn d i z d < 1 d and their special values, e.g. multiple zeta values (MZV) ζ n1,...,n d = Li n1,...,n d (1,..., 1). Example (p1 2 = 1, p2 2 = z 2, p3 2 = 1 z 2 ) ( ) Φ = 4i Im [Li 2(z) + log(1 z) log z ] z z ( ) Φ = 252ζ 3 ζ ζ 3, ζ4 2 expressible means each term of the ε-expansion is a rational/algebraic linear combination of MPL whose arguments are rational/algebraic functions 2 / 32

4 iterated integrals (Chen 1973) Take a manifold X and differential forms ω 1,..., ω n Ω 1 (X). Integrating these along a path γ C 1 ([, 1], X), we can construct functions (on γ): 1 tn t2 ω n ω 1 := γ (ω n )(t n ) γ (ω 1 )(t 1 ) γ 1 If ω = df is exact, γ ω = f (γ(1)) f (γ()) is boring. 2 Not all iterated integrals are homotopy invariant. Example Take ω = ydx Ω 1 (R 2 ), then γ ω is the area between γ and the x-axis. (x,y) (x,y) γ γ (,) (,) integrability condition (Chen), simplest case: ω homotopy invariant dω = γ 3 / 32

5 Hyperlogarithms (Poincaré 1884, Lappo-Danilevsky 1927) Let X = C \ Σ for a finite set of points Σ. The regular, non-exact forms ω σ = d log(z σ) = dz z σ generate homotopy invariant iterated integrals, called hyperlogarithms. Examples γ ω = log γ(1) γ(), z ω 1 = Li 1 (z), All MPL can be written this way: z z ω ω 1 = Li 2 (z) ω n d 1 ω σd ω n 1 1 ω σ1 = ( 1) d Li n1,...,n d ( σ2 σ 1,, σ d σ d 1, Notation and some special cases: z ω σ n ω σ1 = I(; σ 1,..., σ n ; z) = G(σ n,..., σ 1 ; z) [Goncharov] Σ = { 1,, 1} harmonic polylogarithms (HPL) [Remiddi & Vermaseren] Σ = {, 1, 1 y, y} 2 dimensional HPL [Gehrmann & Remiddi] z σ d ) 4 / 32

6 Path concatenation Let γ η denote the concatenation of γ and η at γ(1) = η() = (γ η)( 1 2 ): γ η To decompose ω 2 ω 1 = γ η split the interval {t 1 t 2 } }{{} γ η ω 2ω 1 γ t 1 ω 2 ω 1 + ω 2 ω 1 + ω 2 ω 1, η γ η = {t 1 t }{{ } {t } t 2} } {{ } γ ω 2ω 1 η ω 2 γ ω 1 t 2 { 1 2 t 1 t 2 } }{{} η ω 2ω 1 More generally, the path concatenation formula reads n ω n ω 1 = ω n ω k+1 ω k ω 1. γ η k= η γ 5 / 32

7 Path concatenation G( σ; z) is analytic in z = γ(1) on C \ Σ, but multivalued. Remember Even though γ is suppressed in the notation G( σ; z) and z w, these functions still depend on the homotopy class of γ. If η is a closed loop with η() = η(1) =, analytic continuation M η gives z M η ω n ω 1 = n z ω n ω k+1 ω k ω 1. k= η Note: This only adds lower weight functions (corresponding to prefixes). Similar: change of basepoint z n z ω n ω 1 = ω n ω k+1 ω k ω 1 b k= b 6 / 32

8 Shuffle product The shuffle product of two words w n+m w n+1 w n w 1 = σ w σ(n+m) w σ(1) is the sum of all their shuffles σ, i.e. permutations which preserve the relative order of letters in both factors: σ 1 (1) < < σ 1 (n) and σ 1 (n + 1) < < σ 1 (n + m). For arbitrary words u and v, we find that ( γ is linearly extended) ( ) ( ) u v = (u v). Example ω 3 γ γ ω 2 ω 1 = γ γ γ (ω 3 ω 2 ω 1 + ω 2 ω 3 ω 1 + ω 2 ω 1 ω 3 ) {t 3 } {t 1 t 2 } = {t 1 t 2 t 3 } {t 1 t 3 t 2 } {t 3 t 1 t 2 } γ 7 / 32

9 Singularities Singularities when {γ(), γ(1)} z τ { } Σ are logarithmic: There exist (uniquely determined) functions f k,w (z), analytic at z = τ, such that γ w = k Definition (logarithmic regularization) log k (z τ)f k,w (z). The regularized limit is Reg z τ γ w := f,w(τ). Example γ ω = log z γ() G(; z) = z ω := Reg ω = log(z) γ() γ These expansions and limits can be computed algorithmically using shuffles and rescalings like G(λ σ, λz) = G( σ; z) (σ 1 ). 8 / 32

10 Differentials By definition, z G(σ n, σ; z) = 1 z σ n G( σ, z). But now consider the σ as variables themselves. Differentiating under the integral sign in one finds G(σ n,..., σ 1 ; z) = dg( σ; z) = n i=1 z dt n t n σ n tn dt n 1 t n 1 σ n 1 t2 dt 1 t 1 σ 1 G(, σ i, ; z) d log σ i σ i 1 σ i σ i+1 σ := z, σ n+1 := Equivalently, in the sum representation of MPL we see d Li n ( z) = k Li n ei ( z) dz k z k and can replace any occurring zeros (resulting from n k = 1 say) by Li n (..., z k z k 1,...) dz k dz k Li n (..., z k z k+1,...) 1 z k z k (1 z k ). 9 / 32

11 Functional equations Remember Proving functional relations between MPL is easy - just differentiate! Question Can we write Li 2 (1 x) as a hyperlogarithm with argument x? 1 Take a derivative: x Li 2 (1 x) = 1 x 1 Li 1(1 x) 2 Express all MPL in the integrand as hyperlogs (recursion): 3 integrate back: Li 1 (1 x) = log(x) = G (; x) Li 2 (1 x) = C G (1, ; x) = C log(1 x) log(x) Li 2 (x) 4 Fix constant C = π2 6 = ζ 2 via some limit, for example x 1 1 / 32

12 Products of hyperlogarithms basis The representation in terms of hyperlogarithms is free of relations: Lemma All hyperlogarithms G( σ; z) are linearly independent functions of z (even with algebraic coefficients). The recursive algorithm (differentiation & integration & limits) solves Problem Given some MPL G( σ( x), z( x)) or Li n ( z( x)) whose arguments ( σ, z or z) are rational functions of variables x 1,..., x n, write it in the basis σ 1,..., σ n G( σ 1 (x 2,..., x n ); x 1 )G( σ 2 (x 3,..., x n ); x 2 ) G( σ n ; x n ). completely algebraic (no numerics), programmed (HyperInt) in general not the shortest or simplest representation dependence on order of the variables x 1,..., x n 11 / 32

13 Higher dimensions: multiple polylogarithms Idea: treat all variables equally; via higher dimensional iterated integrals Example d Li 1,1 (z 1, z 2 ) = dz 1 1 z 1 Li 1 (z 2 ) = dz 1 1 z 1 = z2 (z1,z 2 ) (,) dz 1 z 1 (1 z 1 ) Li 1(z 1 z 2 ) + dz 2 Li 1 (z 1 z 2 ) 1 z 2 ( dz 2 dz2 dz 1 dz ) (z1,z 1 2 ) d log(1 z 1 z 2 ) 1 z 2 1 z 2 z 1 1 z 1 [ ( dz1 dz 2 dz1 + + dz 1 dz ) ] 2 z1 dz 2 + z 2 dz 1 1 z 1 1 z 2 z 1 1 z 1 1 z 2 1 z 1 z 2 Corollary (MPL as iterated integrals) ( Every Li n ( z) of d arguments is an iterated integral on C d \ V the singularities (alphabet) S MPL d = {z 1,..., z d } {1 z i z j : 1 i j d} S MPL d ) for [Brown & Bogner] 12 / 32

14 Higher dimensions & simplification Every subset S Q[z 1,..., z d ] of polynomials determines a manifold X = C d \ V (S) = C d \ {f = } which defines a space of iterated integrals. We consider homotopy invariant (Chen condition on w!) integrals γ w of forms d log(f ), f S. Theorem (Kummer 184) Every iterated integral of at most three rational forms can be expressed as a sum of products of log, Li 2 and Li 3. Conjecture (Duhr, Gangl & Rhodes) To express every MPL up to weight 4, it suffices to add the functions Li 4, Li 2,2. At weight 5 it suffices to add Li 5 and Li 2,3. f S Method: basis Ansatz, fit coefficients via symbol (differentiation) or motivic coaction 13 / 32

15 Basepoints Remember The symbol w determines the function z b w completely if one fixes a choice b for the basepoint. when b is forgotten, then the symbol w only captures the highest weight with b fixed, w z b w is injective the motivic coaction is not necessary different basepoints can make expressions simpler physics can sometimes hint a good basepoint Programs Symbol techniques are standard now, many people have programs which can do at least some of the work in an automatized way. Ask around; public: HyperInt,...?... yours here! 14 / 32

16 Motivic coaction Idea: abstract from the integral/function and define algebraic objects (motivic periods) which have more structure. The construction is complicated. [Brown,Dupont,Goncharov,... ] Hint In recent lectures at the IHES (available online), Brown constructed motivic Feynman periods in very general cases. Thus, a coaction is defined under very mild assumptions (in particular not restricted to polylogarithmic integrals). Upshot: There exist algebraically/geometrically defined objects IG m ( s) for a wide class of Feynman graphs G and kinematics s. These are elements of some algebra H m with a period map per: H m C sending I m G ( s) to the actual Feynman integral (number) I G( s). 15 / 32

17 However, there is more structure: A coaction : H m H dr H m which is very powerful to obtain relations, in particular for multiple polylogarithms (where differentiation does not buy us anything!). Example ζ m 2,3 = 1 ζ m 2,3 + ζ dr 2, ζ dr 3 ζ m 2 + ζ dr 2 ζ m 3 Note ζ2 dr = and that {ζm 5, ζm 2 ζm 3 } is a basis of motivic MZV in weight 5, therefore we conclude ζ2,3 m = 2ζm 2 ζm 3 + cζm 5, with c undetermined. An explicit formula for the coproduct/coaction for MPL is due to Goncharov/Brown. Caution If people speak about a coproduct H dr H dr, they necessarily work with de Rham periods, which have no associated real period (ζ2 dr = ). For multiple polylogarithms, this means that multiples of iπ are dropped. 16 / 32

18 Motivic coaction Caution The similarity of both sides, H dr and H m in the case of MPL is rather misleading. In fact, by construction both sides are very different. In the case of Feynman integrals, the m-side corresponds to Feynman integrals again, whereas the dr-side, at least empirically, relates to cuts of integrals, i.e. residues. [talk by Abreu] 17 / 32

19 Further points elliptic MZV and integrals [talk by Bogner,Schlotterer] numeric evaluation of MPL, e.g. Ginac, but also optimizations [Manteuffel, Tancredi] algebraic functions in the forms (root-valued letters) single-valued MPL cluster polylogarithms special values (number theory): relations between MZV, MPL at roots of unity, / 32

20 Feynman integrals polylogarithms 1 differential equations in normal form [talk by Henn,von Manteuffel] d f (ε, x) = ε d log(s)a s f (ε, x) s S 2 summation: extremely powerful computer algebra tools for sums [Ablinger, Raab, Schneider, Weinzierl... ] 3 bootstrap, Ansätze, recursions [talk by Drummond, von Hippel] 4 integration of parameter integrals / 32

21 Definite parameter integrals They are everywhere [talk by Borowka]: hypergeometric functions p F q ( ; z) ( a, b ) Γ(c) 1 2F 1 z = t b 1 (1 t) c b 1 (1 zt) a dt c Γ(b)Γ(c b) Appell s functions F 1, F 2, F 3, F 4 ( α, α ) γ x Γ(γ) F 3 y β, β = Γ(β)Γ(β )Γ(γ β β ) 1 1 v u β 1 v β 1 (1 u v) γ β β 1 (1 ux) α (1 vy) α dudv Feynman integrals in Schwinger parameters Phase-space integrals [talk by Dulat]... Idea: If the representation is simple enough (linearly reducible), there is an order of integration for the parameters such that each intermediate partial integral is a MPL of rational arguments. 2 / 32

22 Schwinger parameters With the superficial degree of divergence sdd = E(G) D/2 loops(g), Φ(G) = Γ(sdd) ( ) 1 ψ sdd e Γ(a e) (, ) E ψ D/2 δ(1 α N ) αe ae 1 dα e ϕ e Graph polynomials: U = α e T e / T F = q 2 (T 1 ) α e + U e / F e F =T 1 T 2 m 2 eα e 21 / 32

23 Example: massless triangle ( ) Φ = = 1 z z dα 2 dα 3 (1 + α 2 + α 3 )(α 2 α 3 + z zα 3 + (1 z)(1 z)α 2 ) ( dα2 α 2 + z dα ) 2 log (α 2 + 1)(α 2 + z z) α 2 + z (1 z)(1 z)α 2 Singularities of the original integrand: S = {ψ, ϕ}, i.e. at α 3 = σ i for σ 1 = 1 α 2 and σ 2 = α 2(1 z)(1 z) α 2 + z z After integrating α 1 from to, the integrand has singularities S 3 = { } 1 + α }{{ 2, α } 2, 1 z, 1 z, α }{{} 2 + z z, z + α }{{} 2, z + α }{{ 2 } σ 1 = σ 2 = σ 2 = σ 1 =σ 2 (pinch) With the same logic, predict the possible singularities after dα 2 : S 3,2 = {z, z, 1 z, 1 z, z z, z z 1} 22 / 32

24 Polynomial reduction [F. Brown] Definition Let S denote a set of polynomials, then S e are the irreducible factors of { } lead e (f ), f, D αe= e(f ): f S and {[f, g] e : f, g S}. Lemma If the singularities of F are cointained in S, then the singularities of F dα e are contained in S e. Improvements Fubini: intersect over different orders Compatibility graphs 23 / 32

25 HyperInt: massive triangle Graph polynomials: > E:=[[2,3],[1,3],[1,2]]: > M:=[[1,s1],[2,s2],[3,s3]],[m1ˆ2,m2ˆ2,m3ˆ3]: > psi:=graphpolynomial(e): > phi:=secondpolynomial(e,m): Polynomial reduction: > L[{}]:=[{psi,phi}, {{psi,phi}}]: > cgreduction(l, {s1,s2,s3,m1,m2,m3}, 2): > L[{x[1],x[2]}][1]; s i + (m j ± m k ) 2, si 2 s i s j i i j s 1s 2 s 3 si 2 mi 2 + s i (mi 2 mj 2 )(mi 2 mk) 2 + i i i<j s i s j (mi 2 + mj 2 ) 24 / 32

26 Linear reducibility Definition If for some order of variables (edges), all S 1,...,k are linear in α k+1, then S (the Feynman graph G with S = {ψ, ϕ}) is called linearly reducible. Lemma If S is linearly reducible, the integral e dα e f of any rational function f with singularities in S is a MPL with symbol letters in S 1,...,N. Integration of linearly reducible integrands in terms of MPL can be automatized via hyperlogarithms. 25 / 32

27 HyperInt open source Maple program integration of hyperlogarithms transformations of MPL to G( ; z) Note: No further simplification (e.g. rewrite as Li 2,2 ) provided! polynomial reduction graph polynomials symbolic computation of constants (no numerics) Example > read "HyperInt.mpl": > hyperint(polylog(2,-x)*polylog(3,-1/x)/x,x=..infinity): > fibrationbasis(%); 8 7 ζ3 2 computes Li 2 ( x) Li 3 ( 1/x)dx = 8 7 ζ / 32

28 HyperInt: propagator > E := [[2,1],[2,3],[2,5],[5,1],[5,3],[5,4],[4,1],[4,3]]: > psi := graphpolynomial(e): > phi := secondpolynomial(e, [[1,1], [3,1]]): > add((epsilon*log(psiˆ5/phiˆ4))ˆn/n!,n=..2)/psiˆ2: > hyperint(eval(%,x[8]=1), [seq(x[n],n=1..7)]): > collect(fibrationbasis(%), epsilon); ( 254ζ ζ 5 2ζ 2 ζ 5 196ζ ζ ) ε 2 ( + 28ζ ζ ) 7 ζ3 2 ε + 2ζ ζ2 2ζ 3 27 / 32

29 HyperInt: triangle Graph polynomials: > E:=[[1,2],[2,3],[3,1]]: > M:=[[3,1],[1,z*zz],[2,(1-z)*(1-zz)]]: > psi:=graphpolynomial(e): > phi:=secondpolynomial(e,m): Integration: > hyperint(eval(1/psi/phi,x[3]=1),[x[1],x[2]]): > factor(fibrationbasis(%,[z,zz])); (G (z; 1) G (zz; ) G (z; ) G (zz; 1) + G (zz;, 1) G (zz; 1, ) + G (z; 1, ) G (z;, 1))/(z zz) Polynomial reduction: > L[{}]:=[{psi,phi}, {{psi,phi}}]: > cgreduction(l): > L[{x[1],x[2]}][1]; { 1 + z, 1 + zz, zz + z} 28 / 32

30 Linearly reducible families (fixed loop order) 1 all 4 loop massless propagators (Panzer) 2 all 3 loop massless off-shell 3-point (Chavez & Duhr, Panzer) also in position space; give conformal 4-point integrals 3 all 2 loop massless on-shell 4-point (Lüders) 29 / 32

31 Linearly reducible massive graphs (some examples) 3 / 32

32 Linearly reducible families (infinite) 3-constructible graphs (3-point functions) [Brown, Schnetz, Panzer] Theorem (Panzer) All ε-coefficients of these graphs (off-shell) are MPL over the alphabet {z, z, 1 z, 1 z, z z, 1 z z, 1 z z, z z z z}. minors of ladder-boxes (up to 2 legs off-shell) Theorem (Panzer) All ε-coefficients of these graphs are MPL. For the massless case, the alphabet is just {x, 1 + x} for x = s/t. 31 / 32

33 4-point recursions Start with the box or (double box) and repeat, in any order: v 1 v 4 v 2 v 3 v 1 v 4 e v 2 v 3 v 3 or v 1 v 4 v 2 v 3 e v 4 or v 1 v 2 v 4 v 3 e Example v 1 v 4 v 1 v 4 v 1 v 4 v 1 v 4 v 2 v 3 v 2 v 3 v 2 v 3 v 2 v 3 32 / 32