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Multiple polylogarithms and Feynman integrals

Christian Bogner (HU Berlin)

joint work with Francis Brown

arXiv:1302.6215 with M. Lüders, arXiv:1302.7004 and 1405.5640 with L. Adams and S. Weinzierl,

Many thanks to Erik Panzer!

Les Houches, June 2014

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Outline:

The computational problem: Integration over Feynman parameters

Multiple polylogarithms in several variables and the program MPL

Applications: Feynman integrals, hypergeometric functions

Outlook: Beyond multiple polylogarithms

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The computational problem: Integration over Feynman parameters

*

Feynman rules

Feynman integrals ∝ 1 + R

... + R

... ...

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Scalar Feynman integrals

For a generic Feynman graphG withNedges and loop-numberL(first Betti number) we consider the scalar Feynman integral

I(Λ) = Z YL

i=1

dDkiD/2

YN j=1

1

−qj2+m2jνj, N,L, νj∈Z,D∈C,

Λ :external parameters, i.e. kinematical invariants and massesmj;qj:momenta Using the “Feynman trick” we can re-write this as

I(Λ) =Γ (ν−LD/2) QN

j=1Γ(νj) Z

0

...

Z 0

YN i=1

dxixiνi−1

!

δ 1−

XN i=1

xi

! Uν−(L+1)D/2

(F(Λ))ν−LD/2,

whereν=PN

j=1νj,ǫ= (4−D)/2.

U andF are the first and the second Symanzik polynomial.

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Symanzik polynomialsfor a graphG with Feynman parametersx1, ...,xN:

U = X

spanning treesTofG

Y

edges∈T/

xi

F = − X

spanning 2-forests(T1,T2)

 Y

edges∈(T/ 1,T2)

xi

 X

edges∈(T/ 1,T2)

qi

2

+U XN

i=1

xim2i

Example:

x4 x3

x1 x2

x5

U=x3x4+x2x4+x1x2+x1x3+x5(x1+x2+x3+x4)

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Symanzik polynomialsfor a graphG with Feynman parametersx1, ...,xN:

U = X

spanning treesTofG

Y

edges∈T/

xi

F = − X

spanning 2-forests(T1,T2)

 Y

edges∈(T/ 1,T2)

xi

 X

edges∈(T/ 1,T2)

qi

2

+U XN

i=1

xim2i

Example:

x4 x3

x1 x2

x5

U=x3x4+x2x4+x1x2+x1x3+x5(x1+x2+x3+x4)

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Symanzik polynomialsfor a graphG with Feynman parametersx1, ...,xN:

U = X

spanning treesTofG

Y

edges∈T/

xi

F = − X

spanning 2-forests(T1,T2)

 Y

edges∈(T/ 1,T2)

xi

 X

edges∈(T/ 1,T2)

qi

2

+U XN

i=1

xim2i

Example:

x4 x3

x1 x2

x5

U=x3x4+x2x4+x1x2+x1x3+x5(x1+x2+x3+x4)

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Assume singularities are taken care of, i.e. ...

... the Feynman integral is finite.

... by renormalization under the integral(Brown, Kreimer 2011)

... by some approach to separate UV and IR singularities, e.g.Panzer (2014).

Computational problem:

Compute a finite integral over Feynman parameters with an integrand of the type:

(Q

Qi)(multiple) polylogarithms of{Pi} QPi

where thePi andQi are polynomials in the Feynman parameters. Usually: Symanzik polynomials

Concept:Try to integrate out all Feynman parameters:

choose a Feynman parameterxjin which allPi arelinear,

integrate overxjby use of an appropriate class of functions, given by iterated integrals

Recent success of this concept in work byE. Panzer,C. Duhr et al,L. Dixon et al, F.

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Multiple polylogarithms in several variables

Iterated integrals

I(t) = Zt

0

fw(t(w))dt(w)

| {z }

ωw

...

Z t′′′

0

f2(t′′)dt′′

| {z }

ω2

Z t′′

0

f1(t)dt

| {z }

ω1

≡ [fw(t)dt|...|f2(t)dt|f1(t)dt] (short-hand notation) We use the termiterated integralfor linear combinations of such integrals.

The differential one-formsfi(t)dtbelong to a chosen setΩ.

Examples:

Polylogs=n

dt t, 1−tdt o

classical polylogarithms:Liw(t) = dt

t|...|dx t | dx

1−t

| {z }

wtimes

,

multiple polylogarithms in one variable:

Li (t) = [...|dt

|...|dt

| dt

|dt

|...|dt

| dt ]

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Examples:

HPL=n

dt

t, 1−tdt ,1+tdt o

Harmonic Polylogarithms(Remiddi, Vermaseren 1999), (implementationMaitre ’05, ’07)

Two-dimensional Harmonic Polylogarithms(Gehrmann, Remiddi ’01):

xvariable and one additional fixed parameter ΩCyclotomic=n

dt t, φtldt

k(t)|k∈N+,0≤l≤ϕ(k), φk(t) : cyclotomic polyn.o Cyclotomic Harmonic Polylogarithms(Ablinger, Blümlein, Schneider ’11), (implementationAblinger)

Hypn = (

dtn tn, tdtn

n−1, ...,

Q

a≤i≤n−1ti dtn Q

a≤i≤nti−1 ,1≤a≤n )

:Hyperlogarithms

Poincare, Kummer 1840, Lappo-Danilevsky 1911also seeGoncharov ’01,applications and implementationby Panzer ’13, ’14

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LetΩnbe the set of differential 1-forms dff withf ∈n

t1, ...,tn,Q

a≤i≤bti−1o

,for 1≤a≤b≤n:

n=

 dt1

t1 , ..., dtn

tn , dQ

a≤i≤bti

Q

a≤i≤bti−1 where1≤a≤b≤n

Examples:

1=n

dt1 t1,t1dt−11 o

(→multiple polylogs in one variable)

2=n

dt1 t1,dtt2

2, tdt1

1−1, tdt2

2−1,t1dtt2+t2dt1

1t2−1

o

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FromΩnwe constructhomotopy invariantiterated integrals.

Viewed as integrals along pathsγ,this means Z

γ1

ωk...ω1= Z

γ2

ωk...ω1for homotopic pathsγ12.

Problem:Not every sequence ofωi ∈Ωnwill provide a homotopy invariant integral.

Theorem(Chen ’77)⇒The integral is homotopy invariant if the sequence (tensor-product)[ω1|...|ωm]satisfies

Xm i=1

1|...|ωi−1|dωii+1|...ωm] +

m−1X

i=1

1|...|ωi−1i∧ωi+1|...|ωm] =0.

There is anexplicit symbol mapψfor constructing such homotopy invariant iterated integrals, seeCB, Brown ’12(closely related to the “symbol” inDuhr, Gangl, Rhodes

’11,Goncharov et al ’10).

⇒Construction provides themultiple polylogarithms in several variablesB(Ωn).

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Some details on the implementation:

n= ΩFibern ∪ΩBasen where allfi ∈ΩFibern depend on the last variabletnand all bi ∈ΩBasen ≡Ωn−1do not.

The bijectiveliftingmapλ: ΩHypn →ΩFibern is defined by λ

Q

a≤i≤n−1ti dtn Q

a≤i≤nti−1 =d

Q a≤i≤nti Q

a≤i≤nti−1.

For each pairfi,fj∈ΩFibern we have an explicit relation (due to Arnol’d) fi∧fj=X

k

ckbk∧αkwithck∈Q,bk∈ΩBasen , αk∈ΩFibern . W.r.t these relations we defineρi[f1|...|fm] =P

kckbk⊗[f1|...|fi−1−1αk|fi+2|...|fm] where the pairfi,fi+1is replaced by the r.h.s. of their Arnol’d relation.

Let[a1|...|am]be a hyperlogarithm with allai ∈ΩHypn .Thesymbol mapis recursively computed byρ:

ψ[a1|...|am] =λa1⊔ψ[a2|...|am]− X

1≤i<m

⊔(id⊗ψ)ρi[a1|...|am].

The procedure of takingprimitivesinvolves similar steps.

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PropertiesofB(Ωn)(Brown ’05):

They are well-defined functions ofnvariables, corresponding to end-points of paths.

On these functions, functional relations are algebraic identities.

They can be decomposed to an explicit basis.

B(Ωn)is closed under taking primitives.

LetZbe theQ-vector space of multiple zeta values. The limits at 0 and 1 of functions inB(Ωn)areZ-linear combinations of elements inB(Ωn−1).

Consequence:We can integrate over these functions from 0 to 1.

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Integration strategy for a Feynman parameterxj: In Feynman parameters:R

0 dxm...R

0 dxj(QQQi)I({Pi})

Pi with allPi linear inxj; In cubical coordinates:R

0 dx1...R1 0dtnP

jfjβj, β∈ B(Ωn), fjhaving denominators inn

t1, ...,tn,Q

a≤i≤bti−1o , integrate hereovertn(i.e. over thexjdependence)

In Feynman parameters:R

0 dxm...R

0 dxj+1(QQi)I({Pi})

QPi

We can continue if there is anext Feynman parameterxj+1in which all polynomials of thenew set{Pi}arelinear. When is this the case?

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Which are the new polynomialsPi? Example:

Start with the set of polynomials{P1,P2}:P1=A1xj+B1,P2=A2xj+B2, R

0 1

P1P2dxj=R

0 1

(A1xj+B1)(A2xj+B2)dxj

=R

0

A1

(A1B2−B1A2)(A1xj+B1)dxj−R

0

A2

(A1B2−B1A2)(A2xj+B2)dxj

= lnA1−lnA1BA22−B−ln1BA12+lnB2

New set:{A1,B1,A2,B2,A1B2−B1A2}

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Linear reducibility

Linear reduction algorithm (Brown ’08)

If the polynomialsS={P1, ...,Pm}are linear in a Feynman parameterxr1, consider:

Pi=Aixr1+Bi,Ai=∂Pi

xr1

,hi =Bi|xr1=0

S(r1)=irreducible factors of{Ai}1≤i≤n,{Bi}1≤i≤n,

BiAj−AiBj 1≤i<j≤n

iterate for a sequence(xr1,xr2, ...,xrn)⇒S(r1),S(r1,r2), ...,S(r1, ...,rn) take intersections like:S[r1,r2]=S(r1,r2)∩S(r2,r1),...

xr1,xr2, ...,xrn ⇒ S(r1),S[r1,r2], ...,S[r1, ...,rn]

S={P1, ...,Pm}islinearly reducibleiffor all1≤k≤nevery polynomial in S[r1, ...,rk]is linear inxrk+1.,

IfS={UG,FG}is linearly reducible we call theFeynman graphG linearly reducible.

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Some linearly reducible (massless) Feynman graphs :

all vacuum graphs with vertex width 3⇒corresponding propagator-type graphs

(Brown ’09)

all two-loop graphs with four on-shell legs (and many with three- and four loops)(CB, Lueders, ’13)

all minors of linearly reducible graphs(Brown ’09, CB, Lueders, ’13)

all propagator-type graphs with≤4 loops(Panzer ’13)

all graphs with three off-shell legs and≤3 loops(Panzer ’14)

all graphs with vertex width 3 with three off-shell legs(Panzer PhD thesis)

all ladder-shaped graphs with four off-shell legs(Panzer PhD thesis)

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Applications

1) Parametric Feynman integrals:

As an example consider theone-loop hexagonintegral inD=6 dimensions with on-shell conditionsp12=m2,pi2=0,i=2, ...,6 to the external momenta:

I= Z

xi≥0

Y6 i=1

dxiδ(1−x6) 2 F3,

X 2 Xj−1

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Del-Duca, Duhr and Smirnov (2011)computed the integral, after a simplification to

I= 1

s142s252s362 Z

xi≥0

Q3 i=1dxi

(u2+x1+x2)(u3x1+u1x3+x2)(u4x1x2+x2+x1x3+x3) using cross-ratios

u1=s262s352

s252s362 ,u2=s132 s462

s362 s142 ,u3=s152 s224

s142 s225,u4= s212s362 s213s226. We introduce new variablesu,v,x,yby

u1=1+y1 ,u2=1+v−u1+v ,u3= (1+y)(−1+u−v)(1−u)(−y−x) ,u4=1+v1+v−x.

With this choice, the limit of eachui at a tangential base-point corresponding to the ordering(x2,x3,x1,u,v,x,y)is 1.

⇒We can integrate outx2,x3,x1.

Our result agrees with the program byPanzer (2014).

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2) Expansion of generalized hypergeometric functions Gaussian hypergeometric function:

2F1(a,b;c;z) =P

m≥0(a)m(b)m (c)m

zm m!

for|z|<1 or|z|=1 andRe(cab)>0; with Pochhammer-symbol(x)y=Γ(x+y)Γ(x)

Generalized hypergeometric functions:

pFq(a1, ...,ap;b1, ...,bq;z) =P

m≥0 Qp

i=1(ai)m Qq

j=1(bj)m zm m!

forqporq=p1 and (|z|<1 or|z|=1 andRe Pp−1

i=1 biPp i=1ai

>0

Appell functions:In

2F1(a,b;c;x)·2F1(a,b;c;y) =P m≥0

P n≥0

(a)m(a)n(b)m(b)n (c)m(c)n

xm y n m!n!

replace terms like(a)m(a)nby(a)m+nto obtain

F1(a;b,b;c;x,y) =P m≥0

P n≥0

(a)m+n(b)m(b)n (c)m+n

xm y n

m!n!, |x|,|y|<1, F2(a;b,b;c,c;x,y) =P

m≥0 P

n≥0

(a)m+n(b)m(b)n (c)m(c)n

xm y n

m!n!, |x|+|y|<1, F3(a,a;b,b;c;x,y) =P

m≥0P

n≥0

(a)m(a)n(b)m(b)n (c)m+n

xm y n

m!n!,|x|,|y|<1, F4(a;b;c,c;x,y) =P

m≥0 P

n≥0

(a)m+n(b)m+n (c)m(c)n

xm y n m!n!,|x|

1 2,|y|

1 2<1,

Horn functions, Lauricella functions, Kampé de Feriét functions, ...

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Hypergeometric-functions-approach:

Step 1:Express the Feynman integral by hypergeometric functions, e.g. using the Mellin-Barnes approach.

⇒The hypergeometric functions depend on the regularization parameterǫ.

E.g. inpFq(a1, ...,ap,b1, ...,bq;z)theai andbi are of the form λj+ǫρj

massless case: allλj are integers massive case: someλj are half-integers

Step 2:Use differential properties to reduce hyp. fct. by loweringai andbi by integers (e.g usingHYPERDIREbyBytev, Kalmykov, Kniehl, Moch).

Step 3:Expansion of the hypergeometric functions atǫ=0.

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Solutions to the expansion problem:

Moch, Uwer, Weinzierl ’02: Use of nested sums as

Z(n;m1, ...,mk;x1, ...,xk) = X

n≥i1>...>ik>0

x1i1 i1m1...xkik

ikmk

withZ(∞;m1, ...,mk;x1, ...,xk) =Limk, ...,m1(xk, ...,x1)

for the expansion of four types of sums called A, B, C, D.(programs:xsummer (Moch, Uwer ’05),nestedsums(Weinzierl ’02))

Examples: The generalized hypergeometric functionspFp−1are covered by type

A:Pn i=1 xi

(i+c)mΓ(i+a1+b1ǫ)

Γ(i+c1+d1ǫ)...Γ(i+Γ(i+ak+bkǫ)

ck+dkǫ)Z(i+o1,m1, ...,ml,x1, ...,xl),aj,cj,oZ;cN

The Appell functionF2requires the combination of all four algorithms A, B, C, D.

Huber, Maitre ’05: Combination of nested sums with anintegral-approachfor

2F1. (programs:HypExp,HypExp2)

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Expansion by use of integral representations:

2F1(a,b;c;z) = Γ(b)Γ(c−b)Γ(c) R1

0 tb−1(1−t)c−b−1(1−tz)−adt

forRe(c)>Re(b)>0 and|arg(1z)|π

Example:

2F1(1,1+ǫ;3+ǫ;z) = Γ(3+ǫ)Γ(ǫ+1)R1 0

tǫ(1−t) 1−tz dt

=R1 0

2(t−1) tz−1 dt+ǫR1

0

(3+2 lnt)(t−1)

tz−1 dt+ǫ2R1 0

(1+3 lnt+ln2t)(t−1)

tz−1 dt

3R1 0

(9 lnt+2 ln2t+6)(t−1)lnt

6(tz−1) dt+O(ǫ4)

= z12(2z+2(1−z)ln(1−z)) +ǫz12(z+3(1−z)ln(1−z) +2(1−z)Li2(z))

2 1z2(1−z) (ln(1−z) +3Li2(z)−2Li3(z))

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Integral representations of generalized hypergeometric functions:

pFq(a1, ...;b1, ...;z)=

Γ(bq) Γ(ap)Γ(bq−ap)

R1

0tap−1(1−t)bq−ap−1p−1Fq−1(a1, ...;b1, ...;zt)dt

forRe(bq)>Re(ap)>0 and (pqorp=q+1 and|arg(1z)|< π)

Appell functions:

F1(a;b,b;c;x,y) = Γ(a)Γ(c−a)Γ(c) R1 0

ua−1(1−u)c−a−1

(1−ux)b(1−uy)bdu,Re(c)>Re(a)>0, F2(a;b,b;c,c;x,y)

= Γ(b)Γ(bΓ(c)Γ(c)Γ(c−b)Γ(c) −b)

R1 0

R1

0ub−1vb−1(1−u)c−b′−1(1−ux(1−v)−vy)ac′−b′ −1du dv,

F3(a,a;b,b;c;x,y)

= Γ(b)Γ(bΓ(c)Γ(c−b−b) )

R R

u,v≥0,u+v≤1

ub−1vb−1(1−u−v)c−b−b

(1−ux)a(1−vy)adudv,

F4(a;b;c,c;x(1−y),y(1−x)) =Γ(a)Γ(b)Γ(c−a)Γ(cΓ(c)Γ(c) −b)

×R1 0

R1

0 ua−1vb−1(1−u)(1−ux)c−a−1b(1−v)(1−vy)c′−b−1a

1−(1−ux)(1−vy)uvxy

c+c−a−b−1

du dv

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Example forp+1Fp:

3F2(2,1+ǫ,1+ǫ;3+ǫ,2+ǫ;z) =Γ(3+ǫ)Γ(2+ǫ) Γ(1+ǫ)2

R1 0

R1 0

x(1−x)ǫyǫ (1−xyz)1+ǫdx dy

=−R1 0

R1 0 2x

2xy−1dx dy+ǫR1 0

R1 0

x(2 ln(1−xyz)−2 ln(1−x)−2 lny−5

xyz−1 dx dy

−ǫ2R1 0

R1 0

x

xyz−1 ln2(1−xyz) +ln2(1−x) +ln2(y) +4+5 ln(y) +2 ln(1−x)ln(y) +5 ln(1−x)−2 ln(1−xyz)ln(1−x)−2 ln(1−xyz)ln(y)−5 ln(1−xyz))dx dy

+O(ǫ3)

= z22(z+ (1−z)ln(1−z)) +ǫz12(5z+7(1−z)ln(1−z) +2Li2(z)−4zLi2(z))

2 1z2(4z+9(1−z)ln(1−z) + (7−12z)Li2(z)−(2−6z)Li3(z)) +O(ǫ3)

(integrated withMPL, checked withHypExp)

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Example for Appell F1:

F1(a;b1,b2;c;x,y) = Γ (c) Γ(a)Γ(c−a)

Z 1 0

ta−1(1−t)c−a−1(1−tx)−b1(1−ty)−b2dt

= Γ (c)

Γ(a)Γ(c−a) Z 1

0

ta−13 (1−t3)c−a−1(1−t1t2t3)−b1(1−t2t3)−b2

is in the appropriate form after introducing the variablest1=x/y,t2=y,t3=t.

As an example we compute

F1(a;b1,b2;c;x,y) = Γ(2+ǫ) Γ(1+ǫ)

Z1 0

(1−z3)ǫ

(1−z1z2z3)(1−z2z3)dz3

= 1

x−y(ln(1−y)−ln(1−x)) + ǫ

x−y

ln(1−y)−ln(1−x) +1

2ln(1−y)2−1

2ln(1−x)2 Li2(x) +Li2(y)) +O ǫ2

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Outlook: Beyond multiple polylogarithms

A forbidden minor:K4with four on-shell legs(CB, Lüders ’13)

J. Henn, A. Smirnov, V. Smirnov 2013, using the differential equations approach: Evaluation of theK4up to functions of weight six in theǫ-expansion in terms of harmonic polylogarithms,

E. Panzer 2014: a change of variables linearizing the polynomials at the critical step⇒integration over Feynman parameters⇒evaluation in terms of

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The massive two-loop sunrise integral (finite inD=2 dimensions) m1

p m2

m3 S p2,m1,m2,m3

=R

σ

x1dx2∧dx3+x2dx3∧dx1+x3dx1∧dx2 F

withσ={[x1:x2:x3]∈P2|xi≥0,i=1,2,3}

and the Second Symanzik polynomial:

F=−x1x2x3p2+ (x1x2+x2x3+x1x3)(x1m21+x2m22+x3m32) linearly irreducible, defining an elliptic curve

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Casem1=m2=m3:

Broadhurst, Fleischer, Tarasov (1993):second order differential equation Groote, Pivovarov (2000), Laporta, Remiddi (2004):elliptic integrals Bloch, Vanhove (2013):elliptic dilogarithm

Case of arbitrary masses:

Berends, Buza, Böhm, Scharf (1994):Lauricella functions Müller-Stach, Weinzierl, Zayadeh (2012):second order differential equation

Adams, CB, Weinzierl (2013):elliptic integrals Adams, CB, Weinzierl (2014):elliptic dilogarithm

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Adams, CB, Weinzierl (2014):

With functions

ELin;m(x;y;q) = X j=1

X k=1

xj jn

yk kmqjk,

E2;0(x;y;q) =1 i

1

2Li2(x)−1

2Li2(x−1) +ELi2;0(x;y;q)−ELi2;0(x−1;y;q)

the result for arbitrary masses takes the form

S p2,m1,m2,m3

=ψ(q) π

X3 i=1

ELi2;0(wi(q);−1;−q)

while for equal masses S p2,m

=3ψ(q)

π ELi2;0(r3;−1;−q),withr3=e2πiq .

Hereψ(q)solves the homogeneous differential equation (complete elliptic integral), andwi are functions ofq,m1,m2,m3determined by transformations on intersection points of the elliptic curve withσ.

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Conclusions:

Multiple polylogarithms in several variables are homotopy invariant iterated integrals with particularly good properties. They are useful for the computation of Feynman integrals by integrating over Feynman parameters.

The expansion of (generalized) hypergeometric functions is a further application of the integration program for multiple polylogarithms. The approach of expansion via integral represantions may extend the existing approaches.

The two-loop sunrise integral is an example for a case beyond multiple polylogarithms. For arbitrary particle masses, the integral can be expressed by integrals over elliptic integrals or - more interestin

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Basic definitions:

Riemann zeta function:ζ(s) =P n=1 1

ns,

Multiple zeta values:ζ(s1, ...,sk)P

n1>n2>...>nk≥1 1

ns11...nskk fors2, ...,sk>0;s1≥2 Expansion of the logarithm:−ln(1+z) =P

n=1 (−z)n

n

Multiple polylogarithms:

Li(s1, ...,sk)(z1, ...,zk) =P

n1>n2>...>nk≥1 z1n1...zknk ns11...nskk n1

,si≥1,|zi|<1 Euler’s Gamma-function:Γ(x) =R

0 tx−1e−tdtforRe(x)>0

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Multiple polylogarithms in several variables

Let

k be a field (eitherRorC), Ma smooth manifold overk, γ: [0,1]→Ma smooth path onM, ω1, ..., ωnsmooth differential 1-forms onM, γi) =fi(t)dt the pull-back ofωi to[0,1]

Def.:Theiterated integralofω1, ..., ωnalongγis Z

γ

ωn...ω1= Z

0≤t1≤...≤tn≤1

fn(tn)dtn...f1(t1)dt1.

We use the termiterated integralfork-linear combinations of such integrals.

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From thisΩnwe want to construct iterated integrals which arehomotopy invariant,i.e.

Z

γ1

ωn...ω1= Z

γ2

ωn...ω1 for homotopic pathsγ12.

Consider tensor productsω1⊗...⊗ωm≡[ω1|...|ωm]overQ.

Define an operatorDby

D([ω1|...|ωm]) = Xm

i=1

1|...|ωi−1|dωii+1|...ωm] +

m−1X

i=1

1|...|ωi−1i∧ωi+1|...|ωm].

Def.:AQ−linear combination of tensor products

ξ= Xm l=0

X

i1, ...,il

ci1, ...,ili1|...|ωil],ci1, ...,il∈Q

is calledintegrable wordif

D(ξ) =0.

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Consider the integration map Xm

l=0

X

i1, ...,il

ci1, ...,ili1|...|ωil]7→

Xm l=0

X

i1, ...,il

ci1, ...,il

Z

γ

ωi1...ωil

Theorem(Chen ’77): Under certain conditions onΩthis map is an isomorphism from integrable wordstohomotopy invariant iterated integrals.

Our class of homotopy invariant functions:

Construct the integrable words of 1-forms inΩn.

(for an explicit construction seeCB, Brown ’12 and cf.Duhr, Gangl, Rhodes ’11,Goncharov et al ’10)

By the integration map obtain the set ofmultiple polylogarithms in several variablesB(Ωn).

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Integration procedure for a Feynman parameterxj:

Given: Integrand P{Q}·I({P}){P} withQ,Ppolynomials in Feynman parameters, allPlinear inxj andI({P})iterated integrals with differential forms dPP Let{P}={A(xj)} ∪ {B}where allA(xj)depend onxjand allB do not. By a reverse shuffle we factorI({P}) =I({B})·I′′ {A(xj)}

. Factor out “trailing zeroes”:I′′ {A(xj)}

=P

lnk(xj)·I′′′ {A(xj)}

such that noI′′′ {A(xj)}

begins with dxxj

j

Fornpolynomials in{A(xj)}introducencubical coordinatest1, ...,tnas rational functions in thexi such that:

each form is replaced byω∈ΩHypn and forms independent ofxj

each point where all 0≤ti ≤1 corresponds to a point where allxi≥0.

IntegrationR1

0 dtn...: a) Primitives by concatenation and“symbol map”⇒ iterated integrals inB(Ω). b)Limits attn=0 andtn=1.

Back to Feynman parameters, introducing integration constants due to different vanishing conditions of thex−andt−integrals.

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A well knownfunctional equationis the five-term-relation:

−Li2 1−y 1−1x

!

−Li2 1−x 1−y1

!

+Li2(xy)−Li2(x)−Li2(y) =1

2ln2(1−x)+1

2ln2(1−y) Writing each function as iterated integral on the total space (usingψ), the relation becomes obvious:

Li2

1−y 1−1x

!

= dx

x + dx 1−x − dy

1−y|xdy+ydx 1−xy

− dx

1−x| dy 1−y

− dx

x + dx 1−x| dx

1−x

Li2 1−x 1−1y

!

= dy

y + dy 1−y − dx

1−x|xdy+ydx 1−xy

+

dx 1−x| dy

1−y

− dy

y + dy 1−y| dy

1−y

Li2(xy) = dx

x +dy

y |xdy+ydx 1−xy

,Li2(x) = dx

x | dx 1−x

,Li2(y) = dy

y | dy 1−y

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Example 1: Vacuum graphs withν=2LandD=4: IG=

Z

xj≥0

YN i=1

dxixiνi−1

!

δ 1−

XN i=1

xi

! 1 UG2

Example 2: Sunrise graph withν=L+1 andD=2: IGG) =

Z

xj≥0

YN i=1

dxixiνi−1

!

δ 1−

XN i=1

xi

! 1 FGG)

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From thisΩnwe want to construct iterated integrals which arehomotopy invariant.

Def.:Smooth pathsγ1, γ2onM arehomotopicif their end-points coincide (i.e.γ1(0) =γ2(0),γ1(1) =γ2(1)) andγ1 can be continuously transformed intoγ2.

Def.:An iterated integral is calledhomotopy invariantif it satisfies Z

γ1

ωn...ω1= Z

γ2

ωn...ω1

for homotopic pathsγ1, γ2.

By such integrals we obtain function of variables given only by the end-points of paths.

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Forean edge ofG consider thedeletion(G\e) andcontraction(G//e) ofe

The deletion and contraction of different edges iscommutative.

⇒IfC,Dare disjoint sets of edges ofG thenG\D//C is a unique graph.

Any such graph is calledminorofG.

Def.:A setGof graphs is calledminor-closedif for eachG∈ G all minors belong to Gas well.

Example:The set of all planar graphs is minor-closed.

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B is a minor of A

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B is a minor of A

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Theorem(Robertson and Seymour): Any minor-closed set of graphs can be defined by a finite set of graphs which arenotin the set (so-calledforbidden minors).

Example:

The set of planar graphs is the set of all graphs which have neitherK5norK3,3as a minor. (Wagner’s theorem)

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Theorem (Brown ’09, CB and Lüders ’13)

The set of linearly reducible Feynman graphs is minor-closed.

⇒Search for the forbidden minors!

Case study by M. Lüders:

LetΛbe the set of massless Feynman graphs with four on-shell legs. (On-shell condition:pi2=0,i=1, ...,4)

At two loops all graphs are linearly reducible.

First forbidden minors at three loops.

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