Adventures in Loop Integration Jacob Bourjaily Niels Bohr - - PowerPoint PPT Presentation

Jacob Bourjaily

Niels Bohr International Academy

Adventures in Loop Integration

Amplitudes in the LHC Era

based on work in collaboration with Caron-Huot, Herrmann, Trnka; Dixon, Dulat, Panzer; 
 He, McLeod, Spradlin, von Hippel, Wilhelm; Duhr, Dulat, Penante,…

✦Loop Integrands (prescriptive representations of) 
 ✦Loop Integration (better technology for)

• dual-conformal sufficiency
• momentum twistor reducibility 


✦Loop Integrals (general structure of)

• beyond multiple polylogarithms
• beyond elliptic polylogarithms
• a bestiary of irreducible


loop-integral geometries

Roadmap (past Elliptic Polylogs)

!2

[JB, Dixon, Dulat, Panzer (to appear)] [JB, McLeod, von Hippel, Wilhelm (2018)] [JB, He, McLeod, von Hippel, Wilhelm (2018)] [JB, McLeod, Spradlin, von Hippel, Wilhelm (2017)] [JB, Herrmann, Trnka (2016)] [JB, Herrmann, McLeod, Trnka (in prep)] [JB, McLeod, von Hippel, Wilhelm (2018)] [JB, McLeod, von Hippel, Wilhelm (in prep.)]

Constructing Integrands for Loop Amplitudes
 (constructively)

Novel Representations of Integrands

!4

✦Powerful new tools now exist for understanding

and computing integrands in perturbation theory

Q-cuts and Forward Limits Recursion Relations Prescriptive Unitarity Correlator Bootstraps

Generalized/Prescriptive Unitarity

✦Integrands are rational functions—so may be

expanded into an arbitrary (but complete) basis:

!5

with coefficients determined by cuts

✦A representation is called prescriptive if all the

coefficients are individual field-theory residues

[Bern, Dixon, Kosower; Dunbar; …]

AL=2

n

=

X

L

fL

L

∈ 8 > < > : , , 9 > = > ; fL ∈ 8 > > > > < > > > > : , , 9 > > > > = > > > > ;

[JB, Herrmann, Trnka]

Generalized/Prescriptive Unitarity

✦Integrands are rational functions—so may be

expanded into an arbitrary (but complete) basis:

!5

with coefficients determined by cuts

✦A representation is called prescriptive if all the

coefficients are individual field-theory residues

[Bern, Dixon, Kosower; Dunbar; …]

AL=3

n

=

X

W

fW

+ X

L

fL where the integrals span all contact-term topologies of those drawn above,

[JB, Herrmann, Trnka]

Building Bases of Loop Integrands

✦In order to define a (finite-dimensional) basis of

loop integrands, two things must be specified:

!6

• A fixed spacetime dimension (or )
• A bound on the (loop-dependent) polynomial degrees— 


the “power-counting” of the theory

A Notational Triviality: WLOG 
 use inverse propagators for everything!

✦Consider one-loop integrands in d dimensions


A classic result (of P-V) is that all integrands with 
 propagators or more are reducible:

Basics of Basis Reduction

!7

[Passarino, Veltman; van Neerven, Vermaseren] ✦Moreover, the only independent integrands with 


propagators can be chosen to be parity-odd

Power-Counting & Constructibility

✦An integrand has ‘ -gon power-counting’ if:

!8

(much less* ambiguous for integrand bases than amplitudes)

✦Let denote a complete basis of integrands with

p-gon power-counting. Because ,

✦An amplitude is ‘p-gon constructible’ if:

(nota bene: this may be loop-order (L) dependent!)

One Loop Unitarity Redux (4d)

✦Re-considering one loop unitarity in 4 dimensions

!9

[Ossola, Papadopoulos, Pittau; Forde, Kosower]

One Loop Unitarity Redux (4d)

✦Re-considering one loop unitarity in 4 dimensions

!9

1 6 1 6 20 1 6 20 50 1 6 20 50 105 1 6 20 50 105 196

[Ossola, Papadopoulos, Pittau; Forde, Kosower]

One Loop Unitarity Redux (4d)

✦Re-considering one loop unitarity in 4 dimensions

!9

1+0 1+5 1+0 2+4 0+20 1+0 3+3 2+18 0+50 1+0 4+2 5+15 2+48 0+105 1+0 5+1 9+11 7+43 2+103 0+196

[Ossola, Papadopoulos, Pittau; Forde, Kosower]

One Loop Unitarity Redux (4d)

✦Re-considering one loop unitarity in 4 dimensions

!9

1+0 0+5 1+0 1+4 0+14 1+0 2+3 0+14 0+30 1+0 3+2 2+12 0+30 0+55 1+0 4+1 5+9 2+28 0+55 0+91

[Ossola, Papadopoulos, Pittau; Forde, Kosower]

Triangularity & Diagonalization

✦Consider a theory that is bubble constructible

(such as SYM)

!10

Triangularity & Diagonalization

✦Consider a theory that is bubble constructible

(such as SYM)

!10

✦Rather than starting from an arbitrary basis of loop

integrands, tailor each to manifestly match one cut

✦one loop:

Using Prescriptive Unitarity

!11

[JB, Caron-Huot, Trnka (2013)] ✦two loop: [JB, Trnka (2015)]

Extending the Prescriptive Reach

✦This procedure continues to work at three loops: ✦Generalizing this to non-planar 


theories is quite easy, provided less-
 than-best power counting is considered

!12

[JB, Herrmann, Trnka (2017)] [JB, Herrmann, McLeod, Trnka (in prep)]

Beyond Planar Prescriptivity

✦Abandoning box power-counting, however,

immediately allows for prescriptive bases

!13

I1 8+1604 (8+221) [`1]3[`2]⊕[`1][`2]3 ⊕[`1]2[`2]2[`1+`2] I2 4+1301 (4+160) [`1]4⊕[`1]2[`2]2 ⊕[`1]3[`2][`1+`2] I3 4+996 (4+116) [`1]3[`2]2 I4 4+396 (4+32) [`1]2[`2]2 I5 22+530 (10+45) [`1]3⊕[`1][`2]2 ⊕[`1]2[`2][`1+`2] I6 16+384 (8+28) [`1]2[`2]2 I7 6+294 (2+18) [`1]3[`2] I8 6+114 (2+4) [`1]3[`2] I9 32+149 (6+6) [`1]2⊕[`2]2⊕ [`1][`2][`1+`2] I10 17+103 (3+3) [`1]2[`2] I11 9+27 (1+0) [`1][`2] I12 15+21 (1+0) [`1][`2] I13 2+103 [`1]4 Ia

14

7+43 [`1]3 Ib

14

2+48 [`1]3 I15 2+18 [`1]2 Ia

16

9+11 [`1]2 Ib

16

5+15 [`1]2 I17 3+3 [`1] Ia

18

5+1 [`1] Ib

18

4+2 [`1] I19 1+0 1 I20 1+0 1

[Feng, Huang] [JB, Herrmann, McLeod, Trnka (in prep.)]

Recent Advances in Loop Integration Technology

When it’s been Integrated

!15

but what this really means depends on who’s talking (& why)

Obviously, “loop integrands should be integrated” This is so even when the integral is “just” a number

When it’s been Integrated

!15

but what this really means depends on who’s talking (& why)

Obviously, “loop integrands should be integrated” This is so even when the integral is “just” a number

When it’s been Integrated

!15

but what this really means depends on who’s talking (& why)

Obviously, “loop integrands should be integrated” This is so even when the integral is “just” a number

When it’s been Integrated

!15

but what this really means depends on who’s talking (& why)

Obviously, “loop integrands should be integrated” This is so even when the integral is “just” a number Are these “numbers” MZVs?

implications for BES…

[O. Schnetz (private corr.)]

YES!

[JB, Heslop, Tran (2015)]

When the result is a function, this is more subtle— depending on various (often valid) criteria

When’s it been Integrated?

!16

AL=2,MHV

n

=

X

a<b<c<d<a

≡

≡ (`1,N1)(`2,N2) (`1,a)(`1,a+1)(`1,b)(`1,b+1)(`1,`2)(`2,c)(`2,c+1)(`2,d)(`2,d+1)

When the result is a function, this is more subtle— depending on various (often valid) criteria

When’s it been Integrated?

!16

AL=2,MHV

n

=

X

a<b<c<d<a

≡

≡ (`1,N1)(`2,N2) (`1,a)(`1,a+1)(`1,b)(`1,b+1)(`1,`2)(`2,c)(`2,c+1)(`2,d)(`2,d+1)

Z

d4`1d4`2

When the result is a function, this is more subtle— depending on various (often valid) criteria

When’s it been Integrated?

!16

[Hodges (1977)] [Symanzik (1972)]

When the result is a function, this is more subtle— depending on various (often valid) criteria

When’s it been Integrated?

!16

✦ built of functions known to

• undergrads (Euler/Abel/…)
• Goncharov/Brown/Bloch…
• Mathematica/GiNaC…

[Hodges (1977)] [Symanzik (1972)]

• numerically fast (and reliable)
• manifest “(transcendental) weight”
• minimal cancellation among terms
• manifest physical symmetries 


(non-redundantly)

When the result is a function, this is more subtle— depending on various (often valid) criteria

When’s it been Integrated?

!16

✦ built of functions known to

• undergrads (Euler/Abel/…)
• Goncharov/Brown/Bloch…
• Mathematica/GiNaC…

[Hodges (1977)] [Symanzik (1972)]

When the result is a function, this is more subtle— depending on various (often valid) criteria

When’s it been Integrated?

!16

✦Certifiability

• against some reference (symbology, fibration bases,…)
• by checking physical limits/branch cuts/…

⇒ Z d8~ ` (a0, b0)(a1, b1)(a2, b2) (`1, a0)(`1, a1)(`1, b1)(`1, `2)(`2, a2)(`2, b2)(`2, b0) =

∞

Z d6~ ↵ U F 3 =

i∞

Z

−i∞

• d7~

z h Γ(−z1)2 · · · ih uz1

1 · · · uz7 7

i =

∞

Z d4~ ↵ 1 f1f2 g2 = Z ds p 4s3−g2s −g3 H3(s)

[JB, McLeod, Spradlin, von Hippel, Wilhelm (2017)]

When the result is a function, this is more subtle— depending on various (often valid) criteria

When’s it been Integrated?

!16

✦Certifiability

• against some reference (symbology, fibration bases,…)
• by checking physical limits/branch cuts/…

⇒ Z d12~ ` (a0, b0)(a1, b1)(a2, b2)(a3, b3) (`1, a0)(`1, a1)(`1, b1)(`1, `2) · · · (`3, b0) =

∞

Z d8~ ↵ U 2 F 4 =

i∞

Z

−i∞

• d16~

z h Γ(−z1)2 · · · ih uz1

1 · · · uz16 16

i =

∞

Z d6~ ↵ 1 f1f2f3 g3 = Z ds dz p 4s3−g2(z)s −g3(z) H4(s, z)

[JB, He, McLeod, von Hippel, Wilhelm (2018)]

When the result is a function, this is more subtle— depending on various (often valid) criteria

When’s it been Integrated?

!16

✦Certifiability

• against some reference (symbology, fibration bases,…)
• by checking physical limits/branch cuts/…

[JB, He, McLeod, von Hippel, Wilhelm (2018)]

When the result is a function, this is more subtle— depending on various (often valid) criteria

When’s it been Integrated?

!16

✦Certifiability

• against some reference (symbology, fibration bases,…)
• by checking physical limits/branch cuts/…

✦Partial fraction to death (e.g. use HyperInt)

Rationalizing Loop Integration

✦Feynman parameterize in 4d, one loop at a time ✦Maintain manifest dual conformal invariance:

• regulate IR divergences with `DCI masses’
• rescale Feynman parameters to trivialize DCI

!17

✦Parameterize kinematic variables using: 


momentum twistors 
 chosen non-redundantly A surprisingly large class of planar UV finite multi- loop integrals can be directly integrated provided the right kind of naïveté (and mild cleverness):

[JB, McLeod, von Hippel, Wilhelm (2018)] [JB, Dixon, Dulat, Panzer (to appear)] [JB, Caron-Huot, Trnka (2013)] [Panzer (2014)]

Planarity & Dual-Conformality

✦We may parameterize momenta of planar loop

(Feynman) integrals by their dual-graphs

!18

(a, b)=(b, a)≡(xb−xa)2 =(pa+ . . . +pb−1)2 ≡sa···b−1 and (`, a)≡(x`−xa)2

pa ≡ (xa+1−xa)

⌘

2 ⌘

(ab;cd) ⌘ (a, b)(c, d) (a, c)(b, d)

[Drummond, Henn, Smirnov, Sokatchev; 


Drummond, Korchemsky, Henn; …]

=

✦Dual-Conformal Invariance: conformality in x’s ≡ Z d4`1d4`2 (a, c)(b, e)(d, f) (`1, a)(`1, b)(`1, c)(`1, `2)(`2, d)(`2, e)(`2, f)

The Dual-Conformal Regulator

✦The basic idea of the dual-conformal regulator is to

give legs masses, but controlled by a parameter that is dimensionless & has no conformal weight

!19

‘δ’

p2

a 7! p2 a + δ(pa−1+pa)2(pa+pa+1)2

(pa−1+pa+pa+1)2

(a, a +1) 7! (b a, d a +1) = (a, a +1) + δ(a1, a +1)(a, a +2) (a1, a +2) I = Z

R3,1 L

Y

i=1

d4`i I 7! Iδ ⌘ Z

R3,1 L

Y

i=1

" d4`i Y

a

(`i, a) (`i,b a) !# I

[JB, Caron-Huot, Trnka (2013)]

xa 7! xb

a ⌘ xa + δ(xa+1xa) (a2, a)

(a2, a +1)

Persevering Dual-Conformality

✦ Using the dual-conformal regularization scheme, 



 
 
 all(?) UV-finite planar loop integrals take the form:

!20

[JB, Dixon, Dulat, Panzer (to appear)] ✦ Coefficients of each divergence can be obtained as

strictly finite (Feynman-) parametric integrals— which can always be rendered manifestly DCI

Restoring Conformality

✦Feynman parameterization is naïvely at odds with

maintaining (dual) conformal invariance

!21

⇒ Z d4` 1 (`, 1)(`, 2)(`, 3)(`, 4)

=

∞

Z ⇥ d3~ ↵ ⇤ 1 F 2

F ≡ α1α2(1, 2)+α2α3(2, 3)+α1α3(1, 3)

+α1α4(1, 4)+α2α4(2, 4)+α3α4(3, 4)

At least when integrating one loop (at a time), conformality is always(?) restorable by rescaling Feynman parameters:

α1 7!α1(2, 3) α2 7!α2(1, 3) α3 7!α3(1, 2) α4 7!α4 (1, 2)(2, 3) (2, 4)

F 7! (1, 2)(2, 3)(1, 3) ⇣ α1α2+α2α3+α1α3+α4(α1v+α2+α3u) ⌘ | {z }

(f1+α4f2)

Restoring Conformality

✦Feynman parameterization is naïvely at odds with

maintaining (dual) conformal invariance

!21

⇒ Z d4` 1 (`, 1)(`, 2)(`, 3)(`, 4)

=

∞

Z ⇥ d3~ ↵ ⇤ 1 F 2

At least when integrating one loop (at a time), conformality is always(?) restorable by rescaling Feynman parameters:

α1 7!α1(2, 3) α2 7!α2(1, 3) α3 7!α3(1, 2) α4 7!α4 (1, 2)(2, 3) (2, 4)

F 7! (1, 2)(2, 3)(1, 3) ⇣ α1α2+α2α3+α1α3+α4(α1v+α2+α3u) ⌘ | {z }

(f1+α4f2)

∞

Z ⇥ d3~ ↵ ⇤ 1 F 2 ∝

∞

Z ⇥ d2~ ↵ ⇤

∞

Z d↵4 1 (f1+↵4f2)2 =

∞

Z ⇥ d2~ ↵ ⇤ 1 f1f2

Symanzik Polynomial Obstructions

✦It is easy to see that the trick just used breaks down

at higher loops if one uses the Symanzik (graph) polynomial formalism. For example, consider:

!22

✦Nevertheless, it appears that this obstruction is

always avoidable simply by parameterizing one loop at a time (true through at least three loops)

F = (a, c)αaαc

• βd + βe + βf + γ
• + (d, f)βdβf
• αa + αb + αc + γ
• + . . .

Iell

db ∝ ∞

Z ⇥ d6~ ↵ ⇤ U F 3

∞

Z [d~ ↵]U n−2(L+1) F n−2L

Symanzik Polynomial Obstructions

✦It is easy to see that the trick just used breaks down

at higher loops if one uses the Symanzik (graph) polynomial formalism. For example, consider:

!22

✦Nevertheless, it appears that this obstruction is

always avoidable simply by parameterizing one loop at a time (true through at least three loops)

[JB, McLeod, Spradlin, von Hippel, Wilhelm (2017)] Iell

db ∝ ∞

Z ⇥ d4~ ↵ ⇤ 1 f1 f2 f3

f1 ⌘ ↵(1+1)+1, f2 ⌘ 1+u1↵+v11+u22+v23, f3 ⌘ (1+u3↵)2+(1+u41)3+23+u3u4u5f1,

u1 ⌘(ab;ce), u2 ⌘(bd;ef), u3 ⌘(ab;cf), v1 ⌘(ea;bc), v2 ⌘(fb;de), u4 ⌘(bc;da), u5 ⌘(ac;d f).

Dual-Conformal Sufficiency

✦We may now (regulate &) represent all of the

following integrals in the space of finite, manifestly conformal (Feynman-)parametric integrals

!23

where u’s are parity-even cross-ratios:

⌘

2 ⌘

(ab;cd) ⌘ (a, b)(c, d) (a, c)(b, d)

Ik ∈ span ( ∞ Z d~ ↵ N(~ ↵) F(~ ↵, ~ u) )

[JB, Dixon, Dulat, Panzer (to appear)]

Dual-Conformal Sufficiency

✦We may now (regulate &) represent all of the

following integrals in the space of finite, manifestly conformal (Feynman-)parametric integrals

!23

where u’s are parity-even cross-ratios:

⌘

2 ⌘

(ab;cd) ⌘ (a, b)(c, d) (a, c)(b, d)

Ik ∈ span ( ∞ Z d~ ↵ N(~ ↵) F(~ ↵, ~ u) )

[JB, McLeod, von Hippel, Wilhelm (2018)]

Dual-Conformal Sufficiency

✦We may now (regulate &) represent all of the

following integrals in the space of finite, manifestly conformal (Feynman-)parametric integrals

!23

where u’s are parity-even cross-ratios:

⌘

2 ⌘

(ab;cd) ⌘ (a, b)(c, d) (a, c)(b, d)

I(L)

8,B ⌘ ∞

Z d2L~ ↵ [d] u1 (f1 · · · fL−1) g1 g2 g3 ✓↵L

2 (1n1 1+2n1 2)

g1

+ 1n2

1+2n2 2

g2

1

◆ fk ⌘

• ↵1

1+. . . +↵k 1

• 2u2 +

↵1

2+. . . +↵k 2

• 1u3 + 12u2u3u4 +

k

X

i,j=1

↵i

1↵j 2 ;

g1 ⌘ fL−1+ ↵1

2+. . . +↵L−1 2

• ↵L

1 +↵L 2

• +↵L

1 2 u2+↵L 2

• 1+2
• ;

g2 ⌘

• ↵1

1+. . . +↵L 1

• +↵L

2 u5+1+2u1;

g3 ⌘

• ↵1

1+. . . +↵L 1

• +↵L

2 +1u3,

[JB, McLeod, von Hippel, Wilhelm (2018)]

Conformal Complications

✦Although a good start, we haven’t yet eliminated

all conformal redundancies—just the rescalings

!24

—which is to say that parity-even cross-ratios are:

• too great in number
• the “wrong” variables…

n(n−5)/2

# rescaling-independent cross ratios:

3n−15

# actually independent cross ratios:

Conformal Complications

✦Although a good start, we haven’t yet eliminated

all conformal redundancies—just the rescalings

!24

—which is to say that parity-even cross-ratios are:

• too great in number
• the “wrong” variables…

n(n−5)/2

# rescaling-independent cross ratios:

3n−15

# actually independent cross ratios:

Conformal Complications

✦Although a good start, we haven’t yet eliminated

all conformal redundancies—just the rescalings

!24

—which is to say that parity-even cross-ratios are:

• too great in number
• the “wrong” variables…

n(n−5)/2

# rescaling-independent cross ratios:

3n−15

# actually independent cross ratios:

Conformal Complications

✦Although a good start, we haven’t yet eliminated

all conformal redundancies—just the rescalings

!24

—which is to say that parity-even cross-ratios are:

• too great in number
• the “wrong” variables…

n(n−5)/2

# rescaling-independent cross ratios:

3n−15

# actually independent cross ratios:

Conformal Complications

✦Although a good start, we haven’t yet eliminated

all conformal redundancies—just the rescalings

!24

—which is to say that parity-even cross-ratios are:

• too great in number
• the “wrong” variables…

n(n−5)/2

# rescaling-independent cross ratios:

3n−15

# actually independent cross ratios:

• over-count the degrees of freedom
• insensitive to the rank of the Gramian
• do not rationalize Gramian dets
• satisfy (complex) algebraic relations

p (1−u−v−w)2−4uvw

Momentum-Twistor Magic

✦Unsurprisingly (to most of us), momentum twistors

are (closer to) the right kind of conformal variables

!25

[Golden, Paulos, Spradlin,Volovich; Harrington; McLeod, …] [Hodges (2009)]

• manifest the rank of the Gramian
• no constrained extra degrees of freedom
• rationalize all 6x6 Gram determinants
• positive domain Euclidean domain
• positive domain is a cluster variety
• cluster coordinates given by plabic graphs

⊂

Momentum-Twistor Magic

✦Unsurprisingly (to most of us), momentum twistors

are (closer to) the right kind of conformal variables

!25

[Golden, Paulos, Spradlin,Volovich; Harrington; McLeod, …] [Hodges (2009)]

• manifest the rank of the Gramian
• no constrained extra degrees of freedom
• rationalize all 6x6 Gram determinants
• positive domain Euclidean domain
• positive domain is a cluster variety
• cluster coordinates given by plabic graphs
• easy to expose/probe kinematic boundaries
• easy to eliminate redundant parameters

⊂

[JB, McLeod, von Hippel, Wilhelm (2018)]

<b<c <d<a

Loop Integral Zoology

general complexity beyond polylogs
 (& beyond elliptic polylogs)

The Two-Loop ‘Master’ Integrals

!27

✦How hard are the general “masters” at two loops? [JB, Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, (in progress)]

The Two-Loop ‘Master’ Integrals

!27

✦How hard are the general “masters” at two loops? [JB, Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, (in progress)]

AL=2

n

=

X

L

fL

The Two-Loop ‘Master’ Integrals

!27

✦How hard are the general “masters” at two loops? [JB, Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, (in progress)]

AL=2

n

=

X

L

fL

L

∈ 8 > < > : , , 9 > = > ; fL ∈ 8 > > > > < > > > > : , , 9 > > > > = > > > > ;

The Two-Loop ‘Master’ Integrals

!27

✦How hard are the general “masters” at two loops? [JB, Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, (in progress)]

AL=2

n

=

X

L

fL

The Two-Loop ‘Master’ Integrals

!27

✦How hard are the general “masters” at two loops? [JB, Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, (in progress)]

#d.o.f. # cross
 ratios # Kinematic Square Roots 4x4(+6x6)+cuts/coeffs # elliptic
 curves

The Two-Loop ‘Master’ Integrals

!27

✦How hard are the general “masters” at two loops? [JB, Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, (in progress)]

#d.o.f. # cross
 ratios # Kinematic Square Roots 4x4(+6x6)+cuts/coeffs # elliptic
 curves

17 20 70(+56)+10 16

The Two-Loop ‘Master’ Integrals

!27

✦How hard are the general “masters” at two loops? [JB, Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, (in progress)]

#d.o.f. # cross
 ratios # Kinematic Square Roots 4x4(+6x6)+cuts/coeffs # elliptic
 curves

17 20 70(+56)+10 16 13 14 35(+7)+1 4

The Two-Loop ‘Master’ Integrals

!27

✦How hard are the general “masters” at two loops? [JB, Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, (in progress)]

#d.o.f. # cross
 ratios # Kinematic Square Roots 4x4(+6x6)+cuts/coeffs # elliptic
 curves

17 20 70(+56)+10 16 13 14 35(+7)+1 4 9 9 15(+1)+0 1

The Two-Loop ‘Master’ Integrals

!28

✦How hard are the general “masters” at two loops? [JB, Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, (in progress)]

#d.o.f. # cross
 ratios # Kinematic Square Roots 4x4(+6x6)+cuts/coeffs+curves # elliptic
 curves

17 20 70(+56)+10+16 16 13 14 35(+7)+1+4 4 9 9 15(+1)+0+1 1

Elliptic Curves at Two Loops

!29

[JB, Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, (in progress)]

#d.o.f. # cross
 ratios # Kinematic Square Roots 4x4(+6x6)+cuts/coeffs+curves # elliptic 
 curves

17 20 70(+56)+10+16 16 13 14 35(+7)+1+4 4 9 9 15(+1)+0+1 1

⊃

⊃

✦How hard are the general “masters” at two loops?

Elliptic Curves at Two Loops

!30

[JB, Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, (in progress)]

⊃

⊃

✦How do the parents see their elliptic duaghters?

Elliptic Curves at Two Loops

!30

[JB, Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, (in progress)]

⊃

⊃

✦How do the parents see their elliptic duaghters?

Elliptic Curves at Two Loops

!30

[JB, Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, (in progress)]

⊃

⊃

✦How do the parents see their elliptic duaghters?

⊃

(

)

Elliptic Curves at Two Loops

!30

[JB, Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, (in progress)]

⊃

⊃

✦How do the parents see their elliptic duaghters?

⊃

(

)

Elliptic Curves at Two Loops

!30

[JB, Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, (in progress)]

⊃

⊃

✦How do the parents see their elliptic duaghters?

⊃

(

)

(

)

⊃

Elliptic Curves at Two Loops

!30

[JB, Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, (in progress)]

⊃

⊃

✦How do the parents see their elliptic duaghters?

⊃

(

)

(

)

⊃

Elliptic Curves at Two Loops

!30

[JB, Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, (in progress)]

⊃

⊃

✦How do the parents see their elliptic duaghters?

⊃

(

)

(

)

⊃

(

)

⊃

Traintracks Past Polylogarithms

✦Despite their ubiquity at low multiplicity and low

loop orders, iterated polylogarithms are far from the only class of integrals that are needed in QFT

!31

[Bloch, Kerr, Vanhove; Broadhurst;…]

T(L) ≡

[JB, He, McLeod, von Hippel, Wilhelm (2018)]

Traintracks Past Polylogarithms

✦Despite their ubiquity at low multiplicity and low

loop orders, iterated polylogarithms are far from the only class of integrals that are needed in QFT

!31

[Bloch, Kerr, Vanhove; Broadhurst;…]

T(L) ≡

[JB, He, McLeod, von Hippel, Wilhelm (2018)]

pression: T(L) =

1

Z ⇥ dL~ ↵ ⇤ dL~

• 1
• f1 · · · fL
• gL

,

fk ⌘(a0ak1;akbk1)(ak1bk;bk1a0)(akbk;ak1bk1)fk1

+↵0(↵k+k)+↵kk +

k1

X

j=1

h ↵j↵k(bja0;ajak)

+↵jk(bja0;ajbk)+↵kj(a0aj;akbj)+jk(a0aj;bkbj)

i , gL ⌘ ↵0 +

L

X

j=1

h ↵j(bja0;ajb0)+j(a0aj;b0bj) i .

Traintracks Past Polylogarithms

✦Despite their ubiquity at low multiplicity and low

loop orders, iterated polylogarithms are far from the only class of integrals that are needed in QFT

!31

[Bloch, Kerr, Vanhove; Broadhurst;…]

T(L) ≡

[JB, He, McLeod, von Hippel, Wilhelm (2018)]

pression: T(L) =

1

Z ⇥ dL~ ↵ ⇤ dL~

• 1
• f1 · · · fL
• gL

,

fk ⌘(a0ak1;akbk1)(ak1bk;bk1a0)(akbk;ak1bk1)fk1

+↵0(↵k+k)+↵kk +

k1

X

j=1

h ↵j↵k(bja0;ajak)

+↵jk(bja0;ajbk)+↵kj(a0aj;akbj)+jk(a0aj;bkbj)

i , gL ⌘ ↵0 +

L

X

j=1

h ↵j(bja0;ajb0)+j(a0aj;b0bj) i .

) =

Z dx dL2~ z p 4x3g2(~ z)xg3(~ z) G0(x, ~ z),

Traintracks Past Polylogarithms

✦Despite their ubiquity at low multiplicity and low

loop orders, iterated polylogarithms are far from the only class of integrals that are needed in QFT

!31

[Bloch, Kerr, Vanhove; Broadhurst;…]

T(L) ≡ A

• L+1

z }| { '12, . . ., '12, '13, '13,

L+1

z }| { '34, . . ., '34, '24, '24 [JB, He, McLeod, von Hippel, Wilhelm (2018)] pression: T(L) =

1

Z ⇥ dL~ ↵ ⇤ dL~

• 1
• f1 · · · fL
• gL

,

fk ⌘(a0ak1;akbk1)(ak1bk;bk1a0)(akbk;ak1bk1)fk1

+↵0(↵k+k)+↵kk +

k1

X

j=1

h ↵j↵k(bja0;ajak)

+↵jk(bja0;ajbk)+↵kj(a0aj;akbj)+jk(a0aj;bkbj)

i , gL ⌘ ↵0 +

L

X

j=1

h ↵j(bja0;ajb0)+j(a0aj;b0bj) i .

) =

Z dx dL2~ z p 4x3g2(~ z)xg3(~ z) G0(x, ~ z),

Traintracks Past Polylogarithms

✦Despite their ubiquity at low multiplicity and low

loop orders, iterated polylogarithms are far from the only class of integrals that are needed in QFT

!31

[Bloch, Kerr, Vanhove; Broadhurst;…]

T(L) ≡ A

• L+1

z }| { '12, . . ., '12, '13, '13,

L+1

z }| { '34, . . ., '34, '24, '24 [JB, He, McLeod, von Hippel, Wilhelm (2018)] pression: T(L) =

1

Z ⇥ dL~ ↵ ⇤ dL~

• 1
• f1 · · · fL
• gL

,

fk ⌘(a0ak1;akbk1)(ak1bk;bk1a0)(akbk;ak1bk1)fk1

+↵0(↵k+k)+↵kk +

k1

X

j=1

h ↵j↵k(bja0;ajak)

+↵jk(bja0;ajbk)+↵kj(a0aj;akbj)+jk(a0aj;bkbj)

i , gL ⌘ ↵0 +

L

X

j=1

h ↵j(bja0;ajb0)+j(a0aj;b0bj) i .

) =

Z dx dL2~ z p 4x3g2(~ z)xg3(~ z) G0(x, ~ z),

A Three-Loop Calabi-Yau 3-Fold

!32

✦Consider the simplest finite 3-loop wheel integral: [JB, McLeod, von Hippel, Wilhelm (in prep.)]

A Three-Loop Calabi-Yau 3-Fold

!32

✦Consider the simplest finite 3-loop wheel integral: [JB, McLeod, von Hippel, Wilhelm (in prep.)]

A Three-Loop Calabi-Yau 3-Fold

!32

✦Consider the simplest finite 3-loop wheel integral: [JB, McLeod, von Hippel, Wilhelm (in prep.)]

A Three-Loop Calabi-Yau 3-Fold

!32

Z d4x

Ad4x Bd4xC

(a1,a2)(b1,b2)(c1,c2) (A,a1)(A,a2)(A,B)(B,b1)(B,b2)(B,C)(C,c1)(C,c2)(C,A)

✦Consider the simplest finite 3-loop wheel integral: [JB, McLeod, von Hippel, Wilhelm (in prep.)]

A Three-Loop Calabi-Yau 3-Fold

!32

|YA) := |a1)α1+|a2)α2+|C)α3+|B)η1 =: |QA)+|B)η1

Z d4x

Ad4x Bd4xC

(a1,a2)(b1,b2)(c1,c2) (A,a1)(A,a2)(A,B)(B,b1)(B,b2)(B,C)(C,c1)(C,c2)(C,A)

✦Consider the simplest finite 3-loop wheel integral: [JB, McLeod, von Hippel, Wilhelm (in prep.)]

A Three-Loop Calabi-Yau 3-Fold

!32

|YA) := |a1)α1+|a2)α2+|C)α3+|B)η1 =: |QA)+|B)η1

|YB) := |b1)β1+|b2)β2+|QA)β3+|C)η2 =: |QB)+|C)η2

Z d4x

Ad4x Bd4xC

(a1,a2)(b1,b2)(c1,c2) (A,a1)(A,a2)(A,B)(B,b1)(B,b2)(B,C)(C,c1)(C,c2)(C,A)

✦Consider the simplest finite 3-loop wheel integral: [JB, McLeod, von Hippel, Wilhelm (in prep.)]

A Three-Loop Calabi-Yau 3-Fold

!32

|YA) := |a1)α1+|a2)α2+|C)α3+|B)η1 =: |QA)+|B)η1

|YB) := |b1)β1+|b2)β2+|QA)β3+|C)η2 =: |QB)+|C)η2

Z d4x

Ad4x Bd4xC

(a1,a2)(b1,b2)(c1,c2) (A,a1)(A,a2)(A,B)(B,b1)(B,b2)(B,C)(C,c1)(C,c2)(C,A)

∞

Z h d2~ ↵ i h d2~

• i

Z d4xC (a1,a2)(b1,b2)(c1,c2) (QA,QA)(QB,QB)(QB,C)(C,c1)(C,c2)

=

✦Consider the simplest finite 3-loop wheel integral: [JB, McLeod, von Hippel, Wilhelm (in prep.)]

A Three-Loop Calabi-Yau 3-Fold

!32

|YA) := |a1)α1+|a2)α2+|C)α3+|B)η1 =: |QA)+|B)η1

|YB) := |b1)β1+|b2)β2+|QA)β3+|C)η2 =: |QB)+|C)η2

Z d4x

Ad4x Bd4xC

(a1,a2)(b1,b2)(c1,c2) (A,a1)(A,a2)(A,B)(B,b1)(B,b2)(B,C)(C,c1)(C,c2)(C,A)

α1 7!α1(C,a2) α2 7!α2(C,a1) α3 7!(a1,a2) β1 7!β1 (C,a1)(a1,a2) (a1,b1) β2 7!β2 (C,a1)(a1,a2) (a1,b2) β3 7!1

∞

Z h d2~ ↵ i h d2~

• i

Z d4xC (a1,a2)(b1,b2)(c1,c2) (QA,QA)(QB,QB)(QB,C)(C,c1)(C,c2)

=

✦Consider the simplest finite 3-loop wheel integral: [JB, McLeod, von Hippel, Wilhelm (in prep.)]

A Three-Loop Calabi-Yau 3-Fold

!32

|YA) := |a1)α1+|a2)α2+|C)α3+|B)η1 =: |QA)+|B)η1

|YB) := |b1)β1+|b2)β2+|QA)β3+|C)η2 =: |QB)+|C)η2

Z d4x

Ad4x Bd4xC

(a1,a2)(b1,b2)(c1,c2) (A,a1)(A,a2)(A,B)(B,b1)(B,b2)(B,C)(C,c1)(C,c2)(C,A)

α1 7!α1(C,a2) α2 7!α2(C,a1) α3 7!(a1,a2) β1 7!β1 (C,a1)(a1,a2) (a1,b1) β2 7!β2 (C,a1)(a1,a2) (a1,b2) β3 7!1

∞

Z h d2~ ↵ i h d2~

• i

Z d4xC (a1,a2)(b1,b2)(c1,c2) (QA,QA)(QB,QB)(QB,C)(C,c1)(C,c2)

=

∞

Z d2~ ↵ d2~

• Z d4xC (a1,a2)2(b1,b2)(c1,c2)/(a1,b1)

(↵1+↵2+↵1↵2)(C,R)(C,S)(C,c1)(C,c2)

=

✦Consider the simplest finite 3-loop wheel integral: [JB, McLeod, von Hippel, Wilhelm (in prep.)]

A Three-Loop Calabi-Yau 3-Fold

!32

|YA) := |a1)α1+|a2)α2+|C)α3+|B)η1 =: |QA)+|B)η1

|YB) := |b1)β1+|b2)β2+|QA)β3+|C)η2 =: |QB)+|C)η2

Z d4x

Ad4x Bd4xC

(a1,a2)(b1,b2)(c1,c2) (A,a1)(A,a2)(A,B)(B,b1)(B,b2)(B,C)(C,c1)(C,c2)(C,A)

α1 7!α1(C,a2) α2 7!α2(C,a1) α3 7!(a1,a2) β1 7!β1 (C,a1)(a1,a2) (a1,b1) β2 7!β2 (C,a1)(a1,a2) (a1,b2) β3 7!1

|YC) := |c1)γ1+|R)γ2+|S)γ3+|c2)η3 =: |Q)+|c2)η3

∞

Z h d2~ ↵ i h d2~

• i

Z d4xC (a1,a2)(b1,b2)(c1,c2) (QA,QA)(QB,QB)(QB,C)(C,c1)(C,c2)

=

∞

Z d2~ ↵ d2~

• Z d4xC (a1,a2)2(b1,b2)(c1,c2)/(a1,b1)

(↵1+↵2+↵1↵2)(C,R)(C,S)(C,c1)(C,c2)

=

✦Consider the simplest finite 3-loop wheel integral: [JB, McLeod, von Hippel, Wilhelm (in prep.)]

A Three-Loop Calabi-Yau 3-Fold

!32

|YA) := |a1)α1+|a2)α2+|C)α3+|B)η1 =: |QA)+|B)η1

|YB) := |b1)β1+|b2)β2+|QA)β3+|C)η2 =: |QB)+|C)η2

Z d4x

Ad4x Bd4xC

(a1,a2)(b1,b2)(c1,c2) (A,a1)(A,a2)(A,B)(B,b1)(B,b2)(B,C)(C,c1)(C,c2)(C,A)

α1 7!α1(C,a2) α2 7!α2(C,a1) α3 7!(a1,a2) β1 7!β1 (C,a1)(a1,a2) (a1,b1) β2 7!β2 (C,a1)(a1,a2) (a1,b2) β3 7!1

|YC) := |c1)γ1+|R)γ2+|S)γ3+|c2)η3 =: |Q)+|c2)η3

∞

Z h d2~ ↵ i h d2~

• i

Z d4xC (a1,a2)(b1,b2)(c1,c2) (QA,QA)(QB,QB)(QB,C)(C,c1)(C,c2)

=

∞

Z d2~ ↵ d2~ d2 ~ (a1,a2)2(b1,b2)(c1,c2)/(a1,b1) (↵1+↵2+↵1↵2)(Q,c1)(Q,Q)

=

∞

Z d2~ ↵ d2~

• Z d4xC (a1,a2)2(b1,b2)(c1,c2)/(a1,b1)

(↵1+↵2+↵1↵2)(C,R)(C,S)(C,c1)(C,c2)

=

✦Consider the simplest finite 3-loop wheel integral: [JB, McLeod, von Hippel, Wilhelm (in prep.)]

A Three-Loop Calabi-Yau 3-Fold

!33

Z d4x

Ad4x Bd4xC

(a1,a2)(b1,b2)(c1,c2) (A,a1)(A,a2)(A,B)(B,b1)(B,b2)(B,C)(C,c1)(C,c2)(C,A)

✦Consider the simplest finite 3-loop wheel integral: [JB, McLeod, von Hippel, Wilhelm (in prep.)]

|YC) := |c1)γ1+|R)γ2+|S)γ3+|c2)η3 =: |Q)+|c2)η3

∞

Z d2~ ↵ d2~ d2 ~ (a1,a2)2(b1,b2)(c1,c2)/(a1,b1) (↵1+↵2+↵1↵2)(Q,c1)(Q,Q)

=

∞

Z d2~ ↵ d2~

• Z d4xC (a1,a2)2(b1,b2)(c1,c2)/(a1,b1)

(↵1+↵2+↵1↵2)(C,R)(C,S)(C,c1)(C,c2)

=

A Three-Loop Calabi-Yau 3-Fold

!33

Z d4x

Ad4x Bd4xC

(a1,a2)(b1,b2)(c1,c2) (A,a1)(A,a2)(A,B)(B,b1)(B,b2)(B,C)(C,c1)(C,c2)(C,A)

✦Consider the simplest finite 3-loop wheel integral:

∞

Z d2~ ↵ d2~ d2 ~

• n0

f1 f2 f3

=

[JB, McLeod, von Hippel, Wilhelm (in prep.)]

|YC) := |c1)γ1+|R)γ2+|S)γ3+|c2)η3 =: |Q)+|c2)η3

∞

Z d2~ ↵ d2~ d2 ~ (a1,a2)2(b1,b2)(c1,c2)/(a1,b1) (↵1+↵2+↵1↵2)(Q,c1)(Q,Q)

=

∞

Z d2~ ↵ d2~

• Z d4xC (a1,a2)2(b1,b2)(c1,c2)/(a1,b1)

(↵1+↵2+↵1↵2)(C,R)(C,S)(C,c1)(C,c2)

=

A Three-Loop Calabi-Yau 3-Fold

!33

Z d4x

Ad4x Bd4xC

(a1,a2)(b1,b2)(c1,c2) (A,a1)(A,a2)(A,B)(B,b1)(B,b2)(B,C)(C,c1)(C,c2)(C,A)

✦Consider the simplest finite 3-loop wheel integral:

∞

Z d2~ ↵ d2~ d2 ~

• n0

f1 f2 f3

=

x1:= (c1a1;a2b2) x2:= (a1b1;b2c2) x3:= (b1c1;c2a2) y1:= (a1a2;b1c2) y2:= (b1b2;c1a2) y3:= (c1c2;a1b2) z1:= (b2c1;c2b1) z2:= (c2a1;a2c1) z3:= (a2b1;b2a1)

[JB, McLeod, von Hippel, Wilhelm (in prep.)]

|YC) := |c1)γ1+|R)γ2+|S)γ3+|c2)η3 =: |Q)+|c2)η3

∞

Z d2~ ↵ d2~ d2 ~ (a1,a2)2(b1,b2)(c1,c2)/(a1,b1) (↵1+↵2+↵1↵2)(Q,c1)(Q,Q)

=

∞

Z d2~ ↵ d2~

• Z d4xC (a1,a2)2(b1,b2)(c1,c2)/(a1,b1)

(↵1+↵2+↵1↵2)(C,R)(C,S)(C,c1)(C,c2)

=

A Three-Loop Calabi-Yau 3-Fold

!33

Z d4x

Ad4x Bd4xC

(a1,a2)(b1,b2)(c1,c2) (A,a1)(A,a2)(A,B)(B,b1)(B,b2)(B,C)(C,c1)(C,c2)(C,A)

✦Consider the simplest finite 3-loop wheel integral:

∞

Z d2~ ↵ d2~ d2 ~

• n0

f1 f2 f3

=

n0 := y1(x1x2x3y1y2y3) f1 := α1+α2+α1α2 ; f2 := α1(1+α2+β1+β2+γ2)+α2(1+x1z2(z3β1+β2)+γ2)

+β1y1(1+x1x3y2z2β2+γ2)+x2y1(x1y3γ1+β2(1+γ2)) ;

f3 := α1(1+α2+β1+β2+γ2) h γ1+β2(1+α2+x3y1y2β1+γ2)

+z3β1(1+α2+γ2)

i

+γ1

h α2(1+x1(z3β1+β2)+γ2)

+x3y1(x2z1β2(1+γ2)+β1(1+x1y2β2+γ2)

i

+(1+γ2)(1+α2+β1+β2+γ2)(z3α2β1+(α2+x3y1y2β1)β2)

x1:= (c1a1;a2b2) x2:= (a1b1;b2c2) x3:= (b1c1;c2a2) y1:= (a1a2;b1c2) y2:= (b1b2;c1a2) y3:= (c1c2;a1b2) z1:= (b2c1;c2b1) z2:= (c2a1;a2c1) z3:= (a2b1;b2a1)

[JB, McLeod, von Hippel, Wilhelm (in prep.)]

A Three-Loop Calabi-Yau 3-Fold

!33

Z d4x

Ad4x Bd4xC

(a1,a2)(b1,b2)(c1,c2) (A,a1)(A,a2)(A,B)(B,b1)(B,b2)(B,C)(C,c1)(C,c2)(C,A)

✦Consider the simplest finite 3-loop wheel integral:

∞

Z d2~ ↵ d2~ d2 ~

• n0

f1 f2 f3

=

x1:= (c1a1;a2b2) x2:= (a1b1;b2c2) x3:= (b1c1;c2a2) y1:= (a1a2;b1c2) y2:= (b1b2;c1a2) y3:= (c1c2;a1b2) z1:= (b2c1;c2b1) z2:= (c2a1;a2c1) z3:= (a2b1;b2a1)

[JB, McLeod, von Hippel, Wilhelm (in prep.)]

= Z d3~ q p Q(~ q) H3(~ q)

A Three-Loop Calabi-Yau 3-Fold

!33

Z d4x

Ad4x Bd4xC

(a1,a2)(b1,b2)(c1,c2) (A,a1)(A,a2)(A,B)(B,b1)(B,b2)(B,C)(C,c1)(C,c2)(C,A)

✦Consider the simplest finite 3-loop wheel integral:

∞

Z d2~ ↵ d2~ d2 ~

• n0

f1 f2 f3

=

[JB, McLeod, von Hippel, Wilhelm (in prep.)]

x1:= (c1a1;a2b2) x2:= (a1b1;b2c2) x3:= (b1c1;c2a2) y1:= (a1a2;b1c2) y2:= (b1b2;c1a2) y3:= (c1c2;a1b2) z1:= 0 z2:= 0 z3:= 0

= Z d3~ q p Q(~ q) H3(~ q)

A Three-Loop Calabi-Yau 3-Fold

!33

Z d4x

Ad4x Bd4xC

(a1,a2)(b1,b2)(c1,c2) (A,a1)(A,a2)(A,B)(B,b1)(B,b2)(B,C)(C,c1)(C,c2)(C,A)

✦Consider the simplest finite 3-loop wheel integral:

∞

Z d2~ ↵ d2~ d2 ~

• n0

f1 f2 f3

=

[JB, McLeod, von Hippel, Wilhelm (in prep.)]

= Z d3~ q p Q(~ q) H3(~ q)

x1:= (c1a1;a2b2) x2:= (a1b1;b2c2) x3:= 1 y1:= 1 y2:= 1 y3:= (c1c2;a1b2) z1:= 0 z2:= 0 z3:= 0

A Three-Loop Calabi-Yau 3-Fold

!33

Z d4x

Ad4x Bd4xC

(a1,a2)(b1,b2)(c1,c2) (A,a1)(A,a2)(A,B)(B,b1)(B,b2)(B,C)(C,c1)(C,c2)(C,A)

✦Consider the simplest finite 3-loop wheel integral:

∞

Z d2~ ↵ d2~ d2 ~

• n0

f1 f2 f3

=

[JB, McLeod, von Hippel, Wilhelm (in prep.)]

x1:= (c1a1;a2b2) x2:= (a1b1;b2c2) x3:= 1 y1:= 1 y2:= 1 y3:= (c1c2;a1b2) z1:= 0 z2:= 0 z3:= 0

= Z d3~ q G(~ q)H3(~ q)

A Three-Loop Calabi-Yau 3-Fold

!33

Z d4x

Ad4x Bd4xC

(a1,a2)(b1,b2)(c1,c2) (A,a1)(A,a2)(A,B)(B,b1)(B,b2)(B,C)(C,c1)(C,c2)(C,A)

✦Consider the simplest finite 3-loop wheel integral:

∞

Z d2~ ↵ d2~ d2 ~

• n0

f1 f2 f3

=

[JB, McLeod, von Hippel, Wilhelm (in prep.)]

= Z d3~ q G(~ q)H3(~ q)

x1:= 1 x2:= 1 x3:= 1 y1:= 1 y2:= 1 y3:= 1 z1:= 0 z2:= 0 z3:= 0

A Three-Loop Calabi-Yau 3-Fold

!33

Z d4x

Ad4x Bd4xC

(a1,a2)(b1,b2)(c1,c2) (A,a1)(A,a2)(A,B)(B,b1)(B,b2)(B,C)(C,c1)(C,c2)(C,A)

✦Consider the simplest finite 3-loop wheel integral:

∞

Z d2~ ↵ d2~ d2 ~

• n0

f1 f2 f3

=

[JB, McLeod, von Hippel, Wilhelm (in prep.)]

= Z d3~ q G(~ q)H3(~ q)

x1:= 1 x2:= 1 x3:= 1 y1:= 1 y2:= 1 y3:= 1 z1:= 0 z2:= 0 z3:= 0

→ 20ζ5

A Three-Loop Calabi-Yau 3-Fold

!33

Z d4x

Ad4x Bd4xC

(a1,a2)(b1,b2)(c1,c2) (A,a1)(A,a2)(A,B)(B,b1)(B,b2)(B,C)(C,c1)(C,c2)(C,A)

✦Consider the simplest finite 3-loop wheel integral:

∞

Z d2~ ↵ d2~ d2 ~

• n0

f1 f2 f3

=

[JB, McLeod, von Hippel, Wilhelm (in prep.)]

= Z d3~ q G(~ q)H3(~ q)

x1:= (c1a1;a2b2) x2:= 1 x3:= 1 y1:= 1 y2:= 1 y3:= 1 z1:= 0 z2:= 0 z3:= 0

A Three-Loop Calabi-Yau 3-Fold

!33

Z d4x

Ad4x Bd4xC

(a1,a2)(b1,b2)(c1,c2) (A,a1)(A,a2)(A,B)(B,b1)(B,b2)(B,C)(C,c1)(C,c2)(C,A)

✦Consider the simplest finite 3-loop wheel integral:

∞

Z d2~ ↵ d2~ d2 ~

• n0

f1 f2 f3

=

[JB, McLeod, von Hippel, Wilhelm (in prep.)]

= Z d3~ q G(~ q)H3(~ q)

x1:= (c1a1;a2b2) x2:= 1 x3:= 1 y1:= 1 y2:= 1 y3:= 1 z1:= 0 z2:= 0 z3:= 0

x1 1−x1 " G({0, 0, 0, 1, 0, 0}, x1)−G({0, 0, 1, 0, 0, 0}, x1)+G({0, 1, 1, 0, 0, 0}, x1)−G({0, 0, 0, 1, 1, 0}, x1)

+G({0, 0, 1, 0, 1, 0}, x1)−G({0, 1, 0, 1, 0, 0}, x1)+G({0, 1, 0, 1, 1, 0}, x1)−G({0, 1, 1, 0, 1, 0}, x1) + ζ2

h G({0, 0, 0, 1}, x1)−G({0, 0, 1, 0}, x1)+G({0, 1, 1, 0}, x1)−G({0, 1, 0, 1}, x1) i

+2ζ3

h G({0, 1, 0}, x1)−G({0, 0, 1}, x1)+G({0, 1, 1}, x1)−G({0, 0, 0}, x1) i

+6ζ4

h G({0, 0}, x1)−G({0, 1}, x1) i

−2(5ζ5+ ζ2 ζ3)G({0}, x1)+4(ζ4

2− ζ2 3)+3ζ6

#

=

A Three-Loop Calabi-Yau 3-Fold

!33

Z d4x

Ad4x Bd4xC

(a1,a2)(b1,b2)(c1,c2) (A,a1)(A,a2)(A,B)(B,b1)(B,b2)(B,C)(C,c1)(C,c2)(C,A)

✦Consider the simplest finite 3-loop wheel integral:

∞

Z d2~ ↵ d2~ d2 ~

• n0

f1 f2 f3

=

[JB, McLeod, von Hippel, Wilhelm (in prep.)]

= Z d3~ q G(~ q)H3(~ q)

x1:= (c1a1;a2b2) x2:= 1 x3:= 1 y1:= 1 y2:= 1 y3:= 1 z1:= 0 z2:= 0 z3:= 0

x1 1−x1 " G({0, 0, 0, 1, 0, 0}, x1)−G({0, 0, 1, 0, 0, 0}, x1)+G({0, 1, 1, 0, 0, 0}, x1)−G({0, 0, 0, 1, 1, 0}, x1)

+G({0, 0, 1, 0, 1, 0}, x1)−G({0, 1, 0, 1, 0, 0}, x1)+G({0, 1, 0, 1, 1, 0}, x1)−G({0, 1, 1, 0, 1, 0}, x1) + ζ2

h G({0, 0, 0, 1}, x1)−G({0, 0, 1, 0}, x1)+G({0, 1, 1, 0}, x1)−G({0, 1, 0, 1}, x1) i

+2ζ3

h G({0, 1, 0}, x1)−G({0, 0, 1}, x1)+G({0, 1, 1}, x1)−G({0, 0, 0}, x1) i

+6ζ4

h G({0, 0}, x1)−G({0, 1}, x1) i

−2(5ζ5+ ζ2 ζ3)G({0}, x1)+4(ζ4

2− ζ2 3)+3ζ6

#

=

→ 20ζ5

Bestiary of Loop Integral Geometry

!34

CY1

CY2

CY3

(?)

CYL−1

(?)

CYL−1

[Bloch, Kerr, Vanhove; Broadhurst;…] [JB, He, McLeod, von Hippel, Wilhelm (2018)] [JB, McLeod, Spradlin, von Hippel, Wilhelm (2018)] ✦The bad news is that even elliptic polylogarithms


are far from sufficient for loop integrals in QFT

Bestiary of Loop Integral Geometry

!34

CY1

CY2

CY3

CY3

(?)

CYL−1

(?)

CYL−1

[Bloch, Kerr, Vanhove; Broadhurst;…] [JB, He, McLeod, von Hippel, Wilhelm (2018)] [JB, McLeod, Spradlin, von Hippel, Wilhelm (2018)] ✦The bad news is that even elliptic polylogarithms


are far from sufficient for loop integrals in QFT

Bestiary of Loop Integral Geometry

!34

CY1

CY2

CY3

CY3

(?)

CYL−1

(?)

CY4

(¿?)

CYL−1

[Bloch, Kerr, Vanhove; Broadhurst;…] [JB, He, McLeod, von Hippel, Wilhelm (2018)] [JB, McLeod, Spradlin, von Hippel, Wilhelm (2018)] ✦The bad news is that even elliptic polylogarithms


are far from sufficient for loop integrals in QFT

Bestiary of Loop Integral Geometry

!34

CY1

CY2

CY3

CY3

(?)

CYL−1

(?)

CY4

(¿?)

CYL−1

[Bloch, Kerr, Vanhove; Broadhurst;…] [JB, He, McLeod, von Hippel, Wilhelm (2018)] [JB, McLeod, Spradlin, von Hippel, Wilhelm (2018)] ✦Exactly how rigid—aka far from polylogarithmic—

can Feynman integrals be?

Bestiary of Loop Integral Geometry

!34

CY1

CY2

CY3

CY3

(?)

CY2

(¿?)

CYL−1

(?)

CY4

(¿?)

CYL−1

[Bloch, Kerr, Vanhove; Broadhurst;…] [JB, He, McLeod, von Hippel, Wilhelm (2018)] [JB, McLeod, Spradlin, von Hippel, Wilhelm (2018)] ✦Exactly how rigid—aka far from polylogarithmic—

can Feynman integrals be?

Bestiary of Loop Integral Geometry

!34

CY1

CY2

CY3

CY3

(?)

CY4

(¿?)

CY2

(¿?)

CYL−1

(?)

CY4

(¿?)

CYL−1

[Bloch, Kerr, Vanhove; Broadhurst;…] [JB, He, McLeod, von Hippel, Wilhelm (2018)] [JB, McLeod, Spradlin, von Hippel, Wilhelm (2018)] ✦Exactly how rigid—aka far from polylogarithmic—

can Feynman integrals be?

✦Exactly how rigid—aka far from polylogarithmic—

can Feynman integrals be?

Bestiary of Loop Integral Geometry

!35

CY1

CY2

CY3

CY3

(?)

CY4

(¿?)

CY2

CYL−1

(?)

CY4

(¿?)

[JB, He, McLeod, von Hippel, Wilhelm (2018)] [JB, McLeod, Spradlin, von Hippel, Wilhelm (2018)]

(¿?)

CYL−1

[Bloch, Kerr, Vanhove; Broadhurst;…]

✦Exactly how rigid—aka far from polylogarithmic—

can Feynman integrals be?

Bestiary of Loop Integral Geometry

!35

CY1

CY2

CY3

CY3

(?)

CY4

(¿?)

CY2

CYL−1

(?)

CY4

(¿?)

[JB, He, McLeod, von Hippel, Wilhelm (2018)] [JB, McLeod, Spradlin, von Hippel, Wilhelm (2018)] [JB, McLeod, von Hippel, Wilhelm (2018)]

(¿?)

maximally!

CYL−1

[Bloch, Kerr, Vanhove; Broadhurst;…]

✦Exactly how rigid—aka far from polylogarithmic—

can Feynman integrals be?

Bestiary of Loop Integral Geometry

!35

CY1

CY2

CY3

CY3

(?)

CY4

(¿?)

CYL−1

(?)

CY4

(¿?)

[JB, He, McLeod, von Hippel, Wilhelm (2018)] [JB, McLeod, Spradlin, von Hippel, Wilhelm (2018)] [JB, McLeod, von Hippel, Wilhelm (2018)]

maximally!

CYL−1

[Bloch, Kerr, Vanhove; Broadhurst;…]

CY2

✦Exactly how rigid—aka far from polylogarithmic—

can Feynman integrals be?

Bestiary of Loop Integral Geometry

!35

CY1

CY2

CY3

CY3

(?)

CYL−1

(?)

CY4

(¿?)

[JB, He, McLeod, von Hippel, Wilhelm (2018)] [JB, McLeod, Spradlin, von Hippel, Wilhelm (2018)] [JB, McLeod, von Hippel, Wilhelm (2018)]

CY4

maximally!

CYL−1

[Bloch, Kerr, Vanhove; Broadhurst;…]

CY2

✦Exactly how rigid—aka far from polylogarithmic—

can Feynman integrals be?

Bestiary of Loop Integral Geometry

!35

CY1

CY2

CY3

CY3

(?)

CYL−1

(?)

CY4

(¿?)

[JB, He, McLeod, von Hippel, Wilhelm (2018)] [JB, McLeod, Spradlin, von Hippel, Wilhelm (2018)] [JB, McLeod, von Hippel, Wilhelm (2018)]

CY4

maximally!

CY6

CY2

✦Exactly how rigid—aka far from polylogarithmic—

can Feynman integrals be?

Bestiary of Loop Integral Geometry

!35

CY1

CY2

CY3

CY3

(?)

CYL−1

(?)

CY4

(¿?)

[JB, He, McLeod, von Hippel, Wilhelm (2018)] [JB, McLeod, Spradlin, von Hippel, Wilhelm (2018)] [JB, McLeod, von Hippel, Wilhelm (2018)]

CY4

maximally!

CY6

CY2

CY8

✦Exactly how rigid—aka far from polylogarithmic—

can Feynman integrals be?

Bestiary of Loop Integral Geometry

!35

CY1

CY2

CY3

CY3

(?)

CYL−1

(?)

CY4

(¿?)

[JB, He, McLeod, von Hippel, Wilhelm (2018)] [JB, McLeod, Spradlin, von Hippel, Wilhelm (2018)] [JB, McLeod, von Hippel, Wilhelm (2018)]

CY4

maximally!

CY2

CY2L−2

✦Exactly how rigid—aka far from polylogarithmic—

can Feynman integrals be?

Bestiary of Loop Integral Geometry

!35

✦The good news is that the relevant geometries are

extremely special (and small in number!)

CY1

CY2

CY3

CY3

(?)

CYL−1

(?)

CY4

(¿?)

[JB, He, McLeod, von Hippel, Wilhelm (2018)] [JB, McLeod, Spradlin, von Hippel, Wilhelm (2018)] [JB, McLeod, von Hippel, Wilhelm (2018)]

CY4

maximally!

CY2

CY2L−2

Questions?