Advancements and Challenges in Multiloop Scattering Amplitudes for the LHC Amplitudes – Edinburgh, 11 July 2017 In collaboration with A. Primo, A. von Manteuffel, E. Remiddi, J. Lindert, K. Melnikov, C. Wever Lorenzo Tancredi TTP KIT Karlsruhe
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
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
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
Integralsarenotallindependent– thereareIntegration-by-partsidentities(IBPs)
Cuts and Feynman Integrals beyond multiple polylogarithms
NX
j=1
Cj(d ; xk) Ij(d ; xk)
2 / 1
NMasterIntegrals
[Chetyrkin,Tkachov‘81]
MIsareallknown!Spaceoffunctionsisunderstoodateveryorderin𝜀 weonlyneedMPLs!
Cuts and Feynman Integrals beyond multiple polylogarithms
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
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
FromtheIBPs
[Kotikov ’91;Remiddi ’97;Gehrmann,Remiddi ’00]
WewanttocomputetheMIsasLaurentseries in𝜀 = (4 − 𝑑)/2
We made a lot of progress in the last years
Mostimportantdevelopmentisprobablythedifferentialequationsmethod
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
FromtheIBPs
[Kotikov ’91;Remiddi ’97;Gehrmann,Remiddi ’00]
WewanttocomputetheMIsasLaurentseries in𝜀 = (4 − 𝑑)/2
ThereasonwhyDEQsaresousefulisbecauseverytheyoftenbecometriangularInthelimit𝑑 → 4 (bychoosingwisely basisofMIs)
CoefficientsofLaurentserieseffectivelysatisfyFirstorderlinearDEQs!
TheycanexpressedaswellintermsofMPLs
Cuts and Feynman Integrals beyond multiple polylogarithms
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]
Cuts and Feynman Integrals beyond multiple polylogarithms
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
Cuts and Feynman Integrals beyond multiple polylogarithms
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
All this works so nicely without masses in the loops…
Whatchangesotherwise?MultiplePolylogsareNOTenoughtospanallspaceoffunctionsneeded@2loops!EllipticFunctionsandpossiblymore…
Intermsof DEQs,theycannotbedecoupledanymorein𝑑 = 4LaurentcoefficientsofMIsfulfillirreducible higherorderdifferentialequations
Cuts and Feynman Integrals beyond multiple polylogarithms
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
HomogeneouspartoftheequationremainscoupledConceptofunitleadingsingularityunclear
All this works so nicely without masses in the loops…
Whatchangesotherwise?MultiplePolylogsareNOTenoughtospanallspaceoffunctionsneeded@2loops!EllipticFunctionsandpossiblymore…
Intermsof DEQs,theycannotbedecoupledanymorein𝑑 = 4LaurentcoefficientsofMIsfulfillirreducible higherorderdifferentialequations
Cuts and Feynman Integrals beyond multiple polylogarithms
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
HomogeneouspartoftheequationremainscoupledConceptofunitleadingsingularityunclear
Cuts and Feynman Integrals beyond multiple polylogarithms
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
Cuts and Feynman Integrals beyond multiple polylogarithms
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
Cuts and Feynman Integrals beyond multiple polylogarithms
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
Cuts and Feynman Integrals beyond multiple polylogarithms
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
Cuts and Feynman Integrals beyond multiple polylogarithms
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
Cuts and Feynman Integrals beyond multiple polylogarithms
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
Cuts and Feynman Integrals beyond multiple polylogarithms
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
Analytic solutions reloadedAcompletesolutioninseriesexpansionin𝜀 requiressolvinghomogeneousequationsin 𝜀 = 0,i.e.findingamatrix2x2suchthat
Cuts and Feynman Integrals beyond multiple polylogarithms
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
Cuts and Feynman Integrals beyond multiple polylogarithms
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
Cuts and Feynman Integrals beyond multiple polylogarithms
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
Cuts and Feynman Integrals beyond multiple polylogarithms
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)
Cuts and Feynman Integrals beyond multiple polylogarithms
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
Cuts and Feynman Integrals beyond multiple polylogarithms
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
We obtain at once all solutions from the two master integrals
Cuts and Feynman Integrals beyond multiple polylogarithms
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
CuttingthesecondMI
Cuts and Feynman Integrals beyond multiple polylogarithms
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
ItisthematrixofMaximalCuts!
Basis of unit leading singularity?
Cuts and Feynman Integrals beyond multiple polylogarithms
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)
Basis of unit leading singularity?
Cuts and Feynman Integrals beyond multiple polylogarithms
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
Cuts and Feynman Integrals beyond multiple polylogarithms
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
Cuts and Feynman Integrals beyond multiple polylogarithms
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
RotateoriginalbasisUsingmatrixG(u)
Basis of unit leading singularity?
ForSunrise,entriesofmatrixG(u)areproportionaltocomplete ellipticintegrals
Cuts and Feynman Integrals beyond multiple polylogarithms
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
Cuts and Feynman Integrals beyond multiple polylogarithms
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
Cuts and Feynman Integrals beyond multiple polylogarithms
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
RotateoriginalbasisUsingmatrixG(u)
Iterated integrals over new kernels
Cuts and Feynman Integrals beyond multiple polylogarithms
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
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
The method is much more general!Weknowafewexamplesat2loops(thenumberisincreasing…)
Differentkinematics(2,3,4-pointfunctions),allreducedto2x2coupledsystemwhosesolutionsgivenbycompleteellipticintegrals
First3x3caseknownhappensat3loops
Bananagraph– 3MIs,3coupledDEQs p
m
m
m
m
Cuts and Feynman Integrals beyond multiple polylogarithms
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]
The method is much more general!Weknowafewexamplesat2loops(thenumberisincreasing…)
Differentkinematics(2,3,4-pointfunctions),allreducedto2x2coupledsystemwhosesolutionsgivenbycompleteellipticintegrals
First3x3caseknownhappensat3loops
Bananagraph– 3MIs,3coupledDEQs p
m
m
m
m
Cuts and Feynman Integrals beyond multiple polylogarithms
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!
Cuts and Feynman Integrals beyond multiple polylogarithms
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
Cuts and Feynman Integrals beyond multiple polylogarithms
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!