advancements and challenges in multiloop scattering ... · advancements and challenges in multiloop...

Advancements and Challenges in MultiloopScattering Amplitudes for the LHC

Amplitudes – Edinburgh,11July2017

IncollaborationwithA.Primo,A.vonManteuffel,E.Remiddi,J.Lindert,K.Melnikov,C.Wever

LorenzoTancrediTTPKITKarlsruhe

What are we interested in? (Or what are we looking for?)

TheLHC hasdiscoveredtheHiggsBosonandhasopenedawindowontheElectroWeakSymmetryBreakingmechanism,aboutwhichweadmittedlydon’tknowmuch

Beyondanydoubts,thereisstillalotwedon’tunderstandaboutfundamentalparticlephysics andourbestchanceistheLHC

• Higgswidth• Higgscouplingstofermions• Higgsself-couplings• TheHiggspotential• … (notonlyHiggs!)

• Higgs𝒑𝑻 distributioncanbeusedtoconstraincouplingstolightquarks

• 𝝈𝒈𝒈→𝒁𝒁 athighenergytoconstraintheHiggswidth

• Newobservableswillbecrucial

The fascination of precision calculationsPrecisioniscoolbecauseitallowsustopinpointevensmallhintstonewphysics,but(formanyofus)thisisnottheonlyreason!

Astheorists,precisionrequireshigherordercalculationsanddeepunderstandingofthephysicsthatwearesearchingfor

• GoinghigherinperturbationtheoryallowsustoexposefascinatingmathematicalstructuresofQFT

• Anditcanpointustothelimitationsofourcurrentapproach

0 20 40 60 80 100

0.8

1.0

1.2

1.4

pT,h [GeV]

(1/

d/d

pT

,h)/(1/

d/d

pT

,h) S

M

c = -10

c = -5

c = 0

c = 5

[Bishara,Haisch,Monni,Re’16]

We still need to compute Feynman Diagrams!(is a “revolution” finally due?)

Modulorevolutions,westillneedtoputtogetherourphysicalquantitiesstartingfromnotverywellbehavingbuildingblocks…

LO

NLO

NNLO

…

+

+

+

LHCenergiesgiveustheopportunitytostudyprocessesatveryhighenergyandtransversemomentum

Intheseregimes,weprobehighmassresonances andinparticularthetop-quark,whoseinteractionsarecrucialtostudytheHiggsmechanism

Forrealisticdescription,wecannotneglectheavyvirtualparticlesintheloops!

Weneedawaytohandle(multi-)loopscatteringamplitudeswhichdependonmanyscalesand,crucially,allowmassiveinternalstates!

How do we proceed (and can we do better?)

AnyscatteringamplitudeisacollectionofscalarFeynmanIntegrals

Cuts and Feynman Integrals beyond multiple polylogarithms

Z lY

j=1

d

dkj

(2⇡)dS

�1

1

... S�ss

D

↵1

1

...D↵nn

, Sr = ki · pj

+

Z lY

j=1

d

dkj

(2⇡)d

@

@kµj

vµS

�1

1

... S�ss

D

↵1

1

...D↵nn

!= 0 v

µ = k

µj , p

µk

+

@@ xk

Ii (d ; xk) =NX

j=1

cij(d ; xk) Ij(d ; xk)

1 / 1

How do we proceed (and can we do better?)

AnyscatteringamplitudeisacollectionofscalarFeynmanIntegrals


Z lY

j=1

d

dkj

(2⇡)dS

�1

1

... S�ss

D

↵1

1

...D↵nn

, Sr = ki · pj

+

Z lY

j=1

d

dkj

(2⇡)d

@

@kµj

vµS

�1

1

... S�ss

D

↵1

1

...D↵nn

!= 0 v

µ = k

µj , p

µk

+

@@ xk

Ii (d ; xk) =NX

j=1


1 / 1


Z lY

j=1

d

dkj

(2⇡)dS

�1

1

... S�ss

D

↵1

1

...D↵nn

, Sr = ki · pj

+

Z lY

j=1

d

dkj

(2⇡)d

@

@kµj

vµS

�1

1

... S�ss

D

↵1

1

...D↵nn

!= 0 v

µ = k

µj , p

µk

+

@@ xk

Ii (d ; xk) =NX

j=1


1 / 1

Integralsarenotallindependent– thereareIntegration-by-partsidentities(IBPs)


NX

j=1

Cj(d ; xk) Ij(d ; xk)

2 / 1

NMasterIntegrals

[Chetyrkin,Tkachov‘81]

MIsareallknown!Spaceoffunctionsisunderstoodateveryorderin𝜀 weonlyneedMPLs!


A revolution in multi-loop calculations has started when physicists have

re-discovered the so-called multiple polylogarithms

[E.Remiddi, J.Vermaseren ’99; T. Gehrmann, E.Remiddi ’00; ....]

G(0; x) = ln (x) , G(a; x) = ln

⇣1� x

a

⌘for a 6= 0

G(0, ..., 0| {z }n

; x) =

1

n!

ln

n

(x) , G(a, ~w ; x) =

Zx

0

dy

y � a

G(

~w ; y) .

+

Multiple polylogarithms are special because they satisfy

first order di↵erential equations with rational coe�cients

@@ x

G(a, ~w ; x) =

1

x � a

G(

~w ; x) ! purely non-homogeneous equation!

3 / 2

Forfinitepiecein𝑑 = 4only𝑳𝒊𝟐 functions!

@ 1 loop everything is clear now

Reductionisunderstood,weareactuallyabletowriteamplitudesascombinationof4MasterIntegrals intermsof on-shellquantities

one-loop N-point amplitude:

+R

most complicated functions are dilogarithms

“master integrals”: boxes, triangles, bubbles, tadpoles

Cin can be obtained by numerical reduction at integrand level

“rational part”

very different at two loops (and beyond)master integrals/function basis not a priori known

=X

i

Ci4 +

X

i

Ci3 +

X

i

Ci2 +

X

i

Ci1

(and pentagons in D-dim.)

@ 2 loops and beyondit is an entirely different story

Weneedrealisticprocesseswithmasses andmanyscales

Classicexamplesofprocessesweneed

• H+jet productionwithatopquark

• VVproductionabovetopthreshold

4scales,3ratios

5 scales,4ratios

NoideaingeneralwhatMis are,andwhatisthespaceoffunctionsneeded,weonlyknowthatMPLsaresurelynotenough

We made a lot of progress in the last years

Mostimportantdevelopmentisprobablythedifferentialequationsmethod


Z lY

j=1

d

dkj

(2⇡)dS

�1

1

... S�ss

D

↵1

1

...D↵nn

, Sr = ki · pj

+

Z lY

j=1

d

dkj

(2⇡)d

@

@kµj

vµS

�1

1

... S�ss

D

↵1

1

...D↵nn

!= 0 v

µ = k

µj , p

µk

+

@@ xk

Ii (d ; xk) =NX

j=1


1 / 1

FromtheIBPs

[Kotikov ’91;Remiddi ’97;Gehrmann,Remiddi ’00]

WewanttocomputetheMIsasLaurentseries in𝜀 = (4 − 𝑑)/2

We made a lot of progress in the last years

Mostimportantdevelopmentisprobablythedifferentialequationsmethod


Z lY

j=1

d

dkj

(2⇡)dS

�1

1

... S�ss

D

↵1

1

...D↵nn

, Sr = ki · pj

+

Z lY

j=1

d

dkj

(2⇡)d

@

@kµj

vµS

�1

1

... S�ss

D

↵1

1

...D↵nn

!= 0 v

µ = k

µj , p

µk

+

@@ xk

Ii (d ; xk) =NX

j=1


1 / 1

FromtheIBPs

[Kotikov ’91;Remiddi ’97;Gehrmann,Remiddi ’00]

WewanttocomputetheMIsasLaurentseries in𝜀 = (4 − 𝑑)/2

ThereasonwhyDEQsaresousefulisbecauseverytheyoftenbecometriangularInthelimit𝑑 → 4 (bychoosingwisely basisofMIs)

CoefficientsofLaurentserieseffectivelysatisfyFirstorderlinearDEQs!

TheycanexpressedaswellintermsofMPLs


A revolution in multi-loop calculations has started when physicists have

re-discovered the so-called multiple polylogarithms

[E.Remiddi, J.Vermaseren ’99; T. Gehrmann, E.Remiddi ’00; ....]

G(0; x) = ln (x) , G(a; x) = ln

⇣1� x

a

⌘for a 6= 0

G(0, ..., 0| {z }n

; x) =

1

n!

ln

n

(x) , G(a, ~w ; x) =

Zx

0

dy

y � a

G(

~w ; y) .

+

Multiple polylogarithms are special because they satisfy

first order di↵erential equations with rational coe�cients

@@ x

G(a, ~w ; x) =

1

x � a

G(

~w ; x) ! purely non-homogeneous equation!

3 / 2

We discovered canonical bases[Henn’13]

Beautifulsystematizationoftheproceedureabove,i.e.integralsexpressedasMPLsoriteratedintegralsoverdlogschooseMIswithunitleadingsingularities

Theconceptofunitleadingsingularity isunderstood(?)forintegralsthatfulfilthisrequirement [Arkani-Hamedetal’10]


NX

j=1

C

j

(d ; x

k

) I

j

(d ; x

k

)

The equations become

d

~I (d ; x) = (d � 4) A(x)

~I (d ; x)

2 / 3

Differentialequationstakeverysimpleform

A(x)isind-logform

ResultsaretriviallyMultiplePolylogarithms

Thesymbol canbereadoffdirectlyfromA(x)!

Ifsuchabasisexists,itmustbepossibletofinditbytransformationsonDEQsonly [Lee’14]

Wealsohavealmostfullcontrolonanalytical andalgebraic propertiesofMultiplePolylogarithms

Wehavenumericalroutines toevaluateMultiplePolylogarithmswitharbitraryprecison

Two other developments are at least as important…

[Goncharov,Spradlin,Volovich‘10][Duhr,Gangle,Rodes’11][Duhr‘12]

[Vollinga,Weinzierl‘05]

Aslongasweconsidercasesinthissubset,wecandoalot!

Wecancomputescatteringamplitudesefficiently,andgettheminaformthatisusefultodorealphysicswiththem!

Wealsohavealmostfullcontrolonanalytical andalgebraic propertiesofMultiplePolylogarithms

Wehavenumericalroutines toevaluateMultiplePolylogarithmswitharbitraryprecison

Two other developments are at least as important…

[Goncharov,Spradlin,Volovich‘10][Duhr,Gangle,Rodes’11][Duhr‘12]

[Vollinga,Weinzierl‘05]

Aslongasweconsidercasesinthissubset,wecandoalot!

Wecancomputescatteringamplitudesefficiently,andgettheminaformthatisusefultodorealphysicswiththem!

BeautifulExample(abitbiased):

𝑞𝑞 → 𝑉6𝑉7 @2loopinQCD

withouttop-masseffectsbutfullVoff-shellnesseffects

[Caola,Henn,Melnikov,Smirnov,Smirnov‘14,‘15][Gehrmann,vonManteuffel,Tancredi‘15]

Usealltechniquesabovetoputamplitudeinausableformforpheno!

All this works so nicely without masses in the loops…

Whatchangesotherwise?MultiplePolylogsareNOTenoughtospanallspaceoffunctionsneeded@2loops!EllipticFunctionsandpossiblymore…

Intermsof DEQs,theycannotbedecoupledanymorein𝑑 = 4LaurentcoefficientsofMIsfulfillirreducible higherorderdifferentialequations


Let’s look more in detail - we should recall that equations are in block form

I

j

(d ; x

k

) = (m

j

(d ; x

k

) , sub

j

(d ; x

k

))

+

@@ x

k

m

i

(d ; x

k

) =

NX

j=1

h

ij

(d ; x

k

)m

j

(d ; x

k

) +

MX

j=1

nh

ij

(d ; x

k

) sub

j

(d ; x

k

) .

4 / 3






I

j

(d ; x

k

) = (m

j

(d ; x

k

) , sub

j

(d ; x

k

))

+

@@ x

k

m

i

(d ; x

k

) =

NX

j=1

h

ij

(d ; x

k

)m

j

(d ; x

k

) +

MX

j=1

nh

ij

(d ; x

k

) sub

j

(d ; x

k

) .

4 / 3

HomogeneouspartoftheequationremainscoupledConceptofunitleadingsingularityunclear






I

j

(d ; x

k

) = (m

j

(d ; x

k

) , sub

j

(d ; x

k

))

+

@@ x

k

m

i

(d ; x

k

) =

NX

j=1

h

ij

(d ; x

k

)m

j

(d ; x

k

) +

MX

j=1

nh

ij

(d ; x

k

) sub

j

(d ; x

k

) .

4 / 3

HomogeneouspartoftheequationremainscoupledConceptofunitleadingsingularityunclear


We know a more and more examples now

m

-

&%'$

p

��

@@

-

-

-p

p

1

p

2

��

AA

AA

What all these examples have in common is a bulk 2⇥ 2 (or 3⇥ 3)

irreducible system of di↵erential equations

5 / 4


We know a more and more examples now

m

-

&%'$

p

��

@@

-

-

-p

p

1

p

2

��

AA

AA

What all these examples have in common is a bulk 2⇥ 2 (or 3⇥ 3)

irreducible system of di↵erential equations

5 / 4

DEQs for 2-loop sunrise as an example

Hastwomasterintegrals𝑆6 and𝑆7 plusonesubtopology.𝑢 = 𝑝7/𝑚7


Let’s see how this works for the sunrise graph (with u = p

2/m2

)

p

m

m

m

= S(d ; u) =

ZD d

k

1

D d

k

2

[k

2

1

� m

2

][k

2

2

� m

2

][(k

1

� k

2

� p)

2 � m

2

]

,

For ✏ = (2 � d)/2 the sunrise fulfils a 2 system of di↵. equations

d

du

✓S

1

(u)

S

2

(u)

◆= B(u)

✓S

1

(u)

S

2

(u)

◆+ ✏ D(u)

✓S

1

(u)

S

2

(u)

◆+

✓N

1

(u)

N

2

(u)

◆

We need to find a matrix of 2 ⇥ 2 independent homogeneous solutions!

7 / 6


Let’s see how this works for the sunrise graph (with u = p

2/m2

)

p

m

m

m

= S(d ; u) =

ZD d

k

1

D d

k

2

[k

2

1

� m

2

][k

2

2

� m

2

][(k

1

� k

2

� p)

2 � m

2

]

,

For ✏ = (2 � d)/2 the sunrise fulfils a 2 system of di↵. equations

d

du

✓S

1

(u)

S

2

(u)

◆= B(u)

✓S

1

(u)

S

2

(u)

◆+ ✏ D(u)

✓S

1

(u)

S

2

(u)

◆+

✓N

1

(u)

N

2

(u)

◆

We need to find a matrix of 2 ⇥ 2 independent homogeneous solutions!

7 / 6

Dispersion relations and di↵erential equations for Feynman Integrals

The di↵erential equations can be put in the form above

d

du

✓h1

h2

◆= B(u)

✓h1

h2

◆+ (d � 4)D(u)

✓h1

h2

◆+

✓01

◆.

where the two matrices B(u),D(u) are defined as

B(u) =1

6 u(u � 1)(u � 9)

✓3(3 + 14u � u

2) �9(u + 3)(3 + 75u � 15u2 + u

3) �3(3 + 14u � u

2)

◆

D(u) =1

6 u(u � 9)(u � 1)

✓6 u(u � 1) 0

(u + 3)(9 + 63u � 9u2 + u

3) 3(u + 1)(u � 9)

◆

Four regular singular points: u = 0, 1, 9,±1

13 / 31

Analytic solutions reloaded

Ingeneral,givenacoupledsystemofequations,nogeneralmethodtodetermineallhomogeneoussolutions.BUT wecanuseadditionalinformationfromMaximalCut

Acompletesolutioninseriesexpansionin𝜀 requiressolvinghomogeneousequationsin 𝜀 = 0,i.e.findingamatrix2x2suchthat


Other two solutions by di↵erentiation [or cutting the second master integral]

J1(u) /I

C1

dbpR4(b, u)

I1(u) /I

C2

dbp�R4(b, u)

J2(u) /I

C1

db b

2

pR4(b, u)

I2(u) /I

C2

db b

2

p�R4(b, u)

And by construction we find

d

du

✓I1(u) J1(u)I2(u) J2(u)

◆= B(u)

✓I1(u) J1(u)I2(u) J2(u)

◆

15 / 15


Matrix of solutions can be therefore written as the matrix of the maximal cuts

G(u) =

✓I1(u) J1(u)I2(u) J2(u)

◆=

✓Cut

C1 (S1(u)) CutC2 (S1(u))

CutC1 (S2(u)) Cut

C2 (S2(u))

◆

and recall that

G

�1(u) =1

W (u)

✓J2(u) �J1(u)�I2(u) I1(u)

◆! W (u) = det (G(u)) = I1(u)J2(u)�I2(u)J1(u)

where W (u) is the Wronskian of the solutions!

8 / 15



I

j

(d ; x

k

) = (m

j

(d ; x

k

) , sub

j

(d ; x

k

))

+

@@ x

k

m

i

(d ; x

k

) =

NX

j=1

h

ij

(d ; x

k

)m

j

(d ; x

k

) +

MX

j=1

nh

ij

(d ; x

k

) sub

j

(d ; x

k

) .

4 / 3

Analytic solutions reloadedAcompletesolutioninseriesexpansionin𝜀 requiressolvinghomogeneousequationsin 𝜀 = 0,i.e.findingamatrix2x2suchthat



J1(u) /I

C1

dbpR4(b, u)

I1(u) /I

C2

dbp�R4(b, u)

J2(u) /I

C1

db b

2

pR4(b, u)

I2(u) /I

C2

db b

2

p�R4(b, u)


d

du

✓I1(u) J1(u)I2(u) J2(u)

◆= B(u)

✓I1(u) J1(u)I2(u) J2(u)

◆

15 / 15



G(u) =

✓I1(u) J1(u)I2(u) J2(u)

◆=

✓Cut

C1 (S1(u)) CutC2 (S1(u))

CutC1 (S2(u)) Cut

C2 (S2(u))

◆

and recall that

G

�1(u) =1

W (u)

✓J2(u) �J1(u)�I2(u) I1(u)



8 / 15


The Maximal Cut provides us with ONE solution of the homogeneous system![A. Primo, L. T. ’16]

+

@@ x

k

Cut (mi

(d ; xk

)) =NX

j=1

h

ij

(d ; xk

)Cut (mj

(d ; xk

))

11 / 15

MaximalCutprovidessolutionofhomogeneousequation [A.Primo,L.Tancredi‘16]

Ingeneral,givenacoupledsystemofequations,nogeneralmethodtodetermineallhomogeneoussolutions.BUT wecanuseadditionalinformationfromMaximalCut

ComputedefficientlyusingBaikovrepresentation[C.Papadopoulos,H.Frellesvig‘17]

How to get all independent solutions


But cutting the graph maximally we find

m

p=

I

C

dbq±b (b � 4)

�b � (

pu � 1)2

� �b � (

pu + 1)2

�

=

I

C

dbp±R4(b, u)

We want to get independent homogeneous solutions

by integrating along di↵erent contours

The problem of how many independent contours exist is acohomology problem, recent renewed interest from physics community

[... ; Abreu, Britto, Duhr, Gardi ’17 ]

13 / 15

Computemaxcutalongallindependentcontours

[Bosma,Sogaard,Zhang‘17][A.Primo,L.Tancredi‘17][Harley,Moriello,Schabinger‘17]

Im(b)

Re(b)

0 0

Im(b)

Re(b)



J1(u) /I

C1

dbpR4(b, u)

I1(u) /I

C2

dbp�R4(b, u)

J2(u) /I

C1

db b

2

pR4(b, u)

I2(u) /I

C2

db b

2

p�R4(b, u)


d

du

✓I1(u) J1(u)I2(u) J2(u)

◆= B(u)

✓I1(u) J1(u)I2(u) J2(u)

◆

15 / 15



J1(u) /I

C1

dbpR4(b, u)

I1(u) /I

C2

dbp�R4(b, u)

J2(u) /I

C1

db b

2

pR4(b, u)

I2(u) /I

C2

db b

2

p�R4(b, u)


d

du

✓I1(u) J1(u)I2(u) J2(u)

◆= B(u)

✓I1(u) J1(u)I2(u) J2(u)

◆

15 / 15

We obtain at once all solutions from the two master integrals



J1(u) /I

C1

dbpR4(b, u)

I1(u) /I

C2

dbp�R4(b, u)

J2(u) /I

C1

db b

2

pR4(b, u)

I2(u) /I

C2

db b

2

p�R4(b, u)


d

du

✓I1(u) J1(u)I2(u) J2(u)

◆= B(u)

✓I1(u) J1(u)I2(u) J2(u)

◆

15 / 15

CuttingthesecondMI



G(u) =

✓I1(u) J1(u)I2(u) J2(u)

◆=

✓Cut

C1 (S1(u)) CutC2 (S1(u))

CutC1 (S2(u)) Cut

C2 (S2(u))

◆

and recall that

G

�1(u) =1

W (u)

✓J2(u) �J1(u)�I2(u) I1(u)



8 / 15

ItisthematrixofMaximalCuts!

Basis of unit leading singularity?


Let’s rotate the system to a more convenient form

G(u) =

✓I1(u) J1(u)I2(u) J2(u)

◆!

✓S1(u)S2(u)

◆= G(u)

✓m1(u)m2(u)

◆

Such that

d

du

✓m1(u)m2(u)

◆= ✏ G

�1(u)D(u)G(u)| {z }

✓m1(u)m2(u)

◆+ G

�1(u)

✓N1(u)N2(u)

◆

+

Iterated integrals over products of two elliptic integrals and rational functions!

9 / 15

RotateoriginalbasisUsingmatrixG(u)




G(u) =

✓I1(u) J1(u)I2(u) J2(u)

◆!

✓S1(u)S2(u)

◆= G(u)

✓m1(u)m2(u)

◆

Such that

d

du

✓m1(u)m2(u)

◆= ✏ G

�1(u)D(u)G(u)| {z }

✓m1(u)m2(u)

◆+ G

�1(u)

✓N1(u)N2(u)

◆

+


9 / 15


You see what is happening here...

Remember, given a system of di↵erential equations, the matrix of the maximalcuts is the matrix of the homogeneous solutions!

What about our new basis? For ✏ = 0 it’s homogeneous equations is

d

du

✓m1(u)m2(u)

◆= 0

And indeed

✓Cut

C1 (m1(u)) CutC2 (m1(u))

CutC1 (m2(u)) Cut

C2 (m2(u))

◆=

✓1 00 1

◆! The identity !!!!

Unit leading singularity!!! Generalization (?) of [Henn ’13]

17 / 15


Physical intepretation of this rotation

0

@m1(u)

m2(u)

1

A = G

�1(u)

0

@S1(u)

S2(u)

1

A =1

W (u)

0

@J2(u)S1(u) � J1(u)S2(u)

�I2(u)S1(u) + I1(u)S2(u)

1

A

Let maximal-cut it along the two independent contours that we found earlier

CutC1

2

4

0

@m1(u)

m2(u)

1

A

3

5 =1

W (u)

0

@J2(u)I1(u) � J1(u)I2(u)

�I2(u)I1(u) + I1(u)I2(u)

1

A =

0

@1

0

1

A

CutC2

2

4

0

@m1(u)

m2(u)

1

A

3

5 =1

W (u)

0

@J2(u)J1(u) � J1(u)J2(u)

�I2(u)J1(u) + J1(u)I2(u)

1

A =

0

@0

1

1

A

16 / 15

Newbasis’maxcutsalongtwocontoursgiveidentitymatrix



ForSunrise,entriesofmatrixG(u)areproportionaltocomplete ellipticintegrals



G(u) =

✓I1(u) J1(u)I2(u) J2(u)

◆!

✓S1(u)S2(u)

◆= G(u)

✓m1(u)m2(u)

◆

Such that

d

du

✓m1(u)m2(u)

◆= ✏ G

�1(u)D(u)G(u)| {z }

✓m1(u)m2(u)

◆+ G

�1(u)

✓N1(u)N2(u)

◆

+


9 / 15


You see what is happening here...

Remember, given a system of di↵erential equations, the matrix of the maximalcuts is the matrix of the homogeneous solutions!

What about our new basis? For ✏ = 0 it’s homogeneous equations is

d

du

✓m1(u)m2(u)

◆= 0

And indeed

✓Cut

C1 (m1(u)) CutC2 (m1(u))

CutC1 (m2(u)) Cut

C2 (m2(u))

◆=

✓1 00 1

◆! The identity !!!!

Unit leading singularity!!! Generalization (?) of [Henn ’13]

17 / 15


Physical intepretation of this rotation

0

@m1(u)

m2(u)

1

A = G

�1(u)

0

@S1(u)

S2(u)

1

A =1

W (u)

0

@J2(u)S1(u) � J1(u)S2(u)

�I2(u)S1(u) + I1(u)S2(u)

1

A

Let maximal-cut it along the two independent contours that we found earlier

CutC1

2

4

0

@m1(u)

m2(u)

1

A

3

5 =1

W (u)

0

@J2(u)I1(u) � J1(u)I2(u)

�I2(u)I1(u) + I1(u)I2(u)

1

A =

0

@1

0

1

A

CutC2

2

4

0

@m1(u)

m2(u)

1

A

3

5 =1

W (u)

0

@J2(u)J1(u) � J1(u)J2(u)

�I2(u)J1(u) + J1(u)I2(u)

1

A =

0

@0

1

1

A

16 / 15

Newbasis’maxcutsalongtwocontoursgiveidentitymatrix


Iterated integrals over new kernels



G(u) =

✓I1(u) J1(u)I2(u) J2(u)

◆!

✓S1(u)S2(u)

◆= G(u)

✓m1(u)m2(u)

◆

Such that

d

du

✓m1(u)m2(u)

◆= ✏ G

�1(u)D(u)G(u)| {z }

✓m1(u)m2(u)

◆+ G

�1(u)

✓N1(u)N2(u)

◆

+


9 / 15

Thenewbasisfulfilssimpledifferentialequations– homogeneouspartfactorizedin𝜀

Whatarethesefunctions(intheSunrisecase)?

§ EllipticPolylogarithms[Bloch,Vanhove‘13]§ ELifunctions[Adams,Bogner,Weinzierl‘14,’15]§ Iteratedintegralsovermodularforms[Adams,Weinzierl‘17]§ …

Unclearhowtoextendthemtoothertopologieswithmorecomplicatedkinematics

The method is much more general!Weknowafewexamplesat2loops(thenumberisincreasing…)

Differentkinematics(2,3,4-pointfunctions),allreducedto2x2coupledsystemwhosesolutionsgivenbycompleteellipticintegrals



First3x3caseknownhappensat3loops

Bananagraph– 3MIs,3coupledDEQs p

m

m

m

m


d

dx

0

@I1(✏; x)I2(✏; x)I3(✏; x)

1

A =B(x)

0

@I1(✏; x)I2(✏; x)I3(✏; x)

1

A + ✏D(x)

0

@I1(✏; x)I2(✏; x)I3(✏; x)

1

A +

0

@00

� 12(4x�1)

1

A

where B(x) and D(x) are 3 ⇥ 3 matrices, with x = 4m2/p2

B(x) =

0

@1x

4x

0� 1

4(x�1)1x

� 2x�1

3x

� 3x�1

18(x�1)

� 18(4x�1)

1x�1

� 32(4x�1)

1x

� 64x�1

+ 32(x�1)

1

A

D(x) =

0

@3x

12x

0� 1

x�12x

� 6x�1

6x

� 6x�1

12(x�1)

� 12(4x�1)

3x�1

� 92(4x�1)

1x

� 124x�1

+ 3x�1

1

A

19 / 17

[A.Primo,L.Tancredi‘17]



First3x3caseknownhappensat3loops

Bananagraph– 3MIs,3coupledDEQs p

m

m

m

m


d

dx

0

@I1(✏; x)I2(✏; x)I3(✏; x)

1

A =B(x)

0

@I1(✏; x)I2(✏; x)I3(✏; x)

1

A + ✏D(x)

0

@I1(✏; x)I2(✏; x)I3(✏; x)

1

A +

0

@00

� 12(4x�1)

1

A

where B(x) and D(x) are 3 ⇥ 3 matrices, with x = 4m2/p2

B(x) =

0

@1x

4x

0� 1

4(x�1)1x

� 2x�1

3x

� 3x�1

18(x�1)

� 18(4x�1)

1x�1

� 32(4x�1)

1x

� 64x�1

+ 32(x�1)

1

A

D(x) =

0

@3x

12x

0� 1

x�12x

� 6x�1

6x

� 6x�1

12(x�1)

� 12(4x�1)

3x�1

� 92(4x�1)

1x

� 124x�1

+ 3x�1

1

A

19 / 17

3x3coupledhomogeneoussystemNeedamatrixof3x3independentsolutions!

[A.Primo,L.Tancredi‘17]

Same idea applied hereWestudythemaxcutofthethreeloopbananagraphalongallindependentcontoursboundedbybranchcutsandfindallindependentsolutions!


We choose as three independent functions

H1(x) =x K⇣k2

+

⌘K⇣k2�

⌘,

J1(x) =x K⇣k2

+

⌘K⇣1 � k2

�

⌘,

I1(x) =x K⇣1 � k2

+

⌘K⇣k2�

⌘,

where the remaining rows of the matrix G(x) can be obtained bydi↵erentiation. With this choice we have

W (x) = � ⇡3x3

512p

(1 � 4x)3(1 � x)

24 / 17


Interestingly enough, with some e↵ort, and following:[Bailey, Borwein, Broadhurst ’08]

f V1 (x) = 2x K(k2

�) K(k2+)

f V2 (x) = 4x

⇣K(k2

�) K(1 � k2+) + K(k2

+) K(1 � k2�)

⌘,

k± =

p(� + ↵)2 � �2 ±p

(� � ↵)2 � �2

2�with k� =

✓↵�

◆1k+

=2↵k+

↵ =

px +

px(1 � x)

2, � =

px �p

x(1 � x)

2, � =

12

Result expected from studies of Joyce ’73 on cubic lattice Green functions!Elliptic Tri-Log by [Bloch, Kerr, Vanhove ’14]

23 / 17

Generalizationofcompleteellipticintegralsin2x2case

Homogeneoussolutionsareproductsofcompleteellipticintegrals!

Besselmoments[Bailey,Borwein,Broadhurst‘08]

AlreadyJ.Joycecouldsolvethisequationin1973 incontextofcubiclatticeGreenfunctions

MorerecentlyBlochandVanhovewrotethegraphind=2asanEllipticTrilog!

Is this the only approach worth trying?

Verypromisingdevelopments:wehavenowawaytotacklecomplicatedMIsbysolvingdifferentialequationseveniftheyarecoupled!

Wearemakingfastprogressontheclassificationsofthespecialfunctionsinvolved!

Still,amajorissueremains.Calculationswithmanyscalesandinternalmassesgeneratetypicallyhugealgebraiccomplexity.

Complexityofamplitudesforrealisticprocesses,evenwhenwrittenintermsofindependentstructures,increasesfactorially.

Theybecomeaproblemalreadyfor𝟐 → 𝟐 evenforlargestcomputers. Allourmachinerybreaksdown!

Is this the only approach worth trying?

Verypromisingdevelopments:wehavenowawaytotacklecomplicatedMIsbysolvingdifferentialequationseveniftheyarecoupled!

Wearemakingfastprogressontheclassificationsofthespecialfunctionsinvolved!

Still,amajorissueremains.Calculationswithmanyscalesandinternalmassesgeneratetypicallyhugealgebraiccomplexity.

Complexityofamplitudesforrealisticprocesses,evenwhenwrittenintermsofindependentstructures,increasesfactorially.

Theybecomeaproblemalreadyfor𝟐 → 𝟐 evenforlargestcomputers. Allourmachinerybreaksdown!

Verypromisingnumericalapproaches:• ttbar@NNLO[Czakonetal.‘13]• HH@NLO[Borowkaetal.‘16]

Maybeweneedtorethinkentirelywhatwearedoing?

We should remember that we are physicists (mainly!) and that most of our calculations are performed in some sort of approximation…

TakeforexampleH+jetproduction withamassivebottomquark

Importantsourceofuncertainty :

Interferencetop-bottom@NLOinQCDForlarge𝑝= ofHiggs,largelog

ABCD

, log AFAB

We should remember that we are physicists (mainly!) and that most of our calculations are performed in some sort of approximation…

TakeforexampleH+jetproduction withamassivebottomquark

Importantsourceofuncertainty :

Interferencetop-bottom@NLOinQCDForlarge𝑝= ofHiggs,largelog

ABCD

, log AFAB

ProgresstowardsanalyticcomputationofplanarMIs[Bonciani,DelDuca,Frellesvig,Henn,Moriello,Smirnov‘16]

Impressivecalculation,butresultsarenotreallyinaniceshape,noteventheMPLspart!

• Uptotwo-foldintegralrepresentations• Noanalyticcontinuation• ~200MBlargefilesandNPLintegralsarestillmissing

MI with DE method for small 𝑚𝑏 (1/2)DE method

8

• System of partial differential equations (DE) in 𝒎𝒃, 𝒔, 𝒕,𝒎𝒉

𝟐

with IBP relations

• Solve 𝑚𝑏 DE with following ansatz

• Plug into 𝑚𝑏 DE and get constraints on coefficients 𝑐𝑖𝑗𝑘𝑛

• 𝑐𝑖000 is 𝑚𝑏 = 0 solution (hard region) and has been computed before

Step 1: solve DE in 𝒎𝒃

• Interested in 𝑚𝑏 expansion of Master integrals 𝐼𝑀𝐼

expand homogeneous matrix 𝑀𝑘 in small 𝑚𝑏

[Gehrmann & Remiddi ’00]

Inordertocapturethiseffect,noneedofcomputingexactmassdependence!WecanexpandallMIs(andthewholeamplitude)forsmallbottommass!

DeriveDEQsforcoefficientsOnelessscale,nomass,muchsimpler!

[J.Lindert,K.Melnikov,L.Tancredi,C.Wever‘16,’17]

MI with DE method for small 𝑚𝑏 (1/2)DE method

8

• System of partial differential equations (DE) in 𝒎𝒃, 𝒔, 𝒕,𝒎𝒉

𝟐

with IBP relations

• Solve 𝑚𝑏 DE with following ansatz

• Plug into 𝑚𝑏 DE and get constraints on coefficients 𝑐𝑖𝑗𝑘𝑛

• 𝑐𝑖000 is 𝑚𝑏 = 0 solution (hard region) and has been computed before

Step 1: solve DE in 𝒎𝒃

• Interested in 𝑚𝑏 expansion of Master integrals 𝐼𝑀𝐼

expand homogeneous matrix 𝑀𝑘 in small 𝑚𝑏

[Gehrmann & Remiddi ’00]

Inordertocapturethiseffect,noneedofcomputingexactmassdependence!WecanexpandallMIs(andthewholeamplitude)forsmallbottommass!

DeriveDEQsforcoefficientsOnelessscale,nomass,muchsimpler!

Messageis:

Sometimesdoingeverythinganalyticallycanbeoverkill

Ifwefindtherightapproximation,DEQsareverypowerfulalsotogetapproximateresults

Usedalreadyinsimilarcontext

[J.Lindert,K.Melnikov,L.Tancredi,C.Wever‘16,’17]

[R.Mueller,G.Öztürk‘16]

Conclusions§ Scatteringamplitudesneededforrealisticphysicalprocesses@LHCare

(appeartobe?)immenselycomplicated

§ Thankstoprogressintheoreticalunderstanding,alimitedsubsetoftheseamplitudesisnowundermuchbettercontrol(MPLs!)

§ Alsoinmoregeneralcases,westarthavinganideaofhowtoproceedtotackletheproblem(Maxcut,EPLs andModularForms)

§ Still,remainsproblemofenormousalgebraiccomplexity(it’sjustsimplecombinatorics!)

§ Giventhiscomplexity,itisunclearwhetherapurelyanalyticalapproachwillbefeasibleinthenearfuture.

§ HybridNumerical/Analytical(seriesexpansions?)mightbethewaytogo…?

THANKS!

advancements and challenges in multiloop scattering ... · advancements and challenges in multiloop...

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