Advancements and Challenges in Multiloop Scattering ... · Advancements and Challenges in Multiloop Scattering Amplitudes for the LHC Amplitudes–Edinburgh, 11 July 2017 In collaboration
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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!
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