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William Javier Torres Bobadilla
Scattering Amplitudes in Gauge Theories: from tree- to multi-loop-level calculations
IFIC Seminar26th April, 2017. Valencia-Spain
Based on the collaborations withP. Mastrolia, A. Primo
• Standard Model (SM) of Particle Physics —> best Quantum Field Theory
• Large Hadron Collider (LHC) confirmed the validity of the SM with the discovery of the Higgs bosonby ATLAS and CMS experiments
• SM does not provide all the information of the subatomic world —> Neutrino oscillations
• SM leaves to much physics without descriptions —> Physics Beyond Standard Model (BSM)
• LHC results demand a refinement of our understanding of the SM physics High precision predictions in background processes —> New physics at the TeV scale
• Relevant observables —> computation of Quantum Chromodynamics (QCD) Scattering Amplitudes
William J. Torres Bobadilla
Introduction
2
LO NLO NNLO NNNLO
0
10
20
30
40
50
σ/pb
LHCpp→h+X gluon fusionMSTW08 68clμ=μR=μF ∈ [mH /4,mH ]Central scale: μ = mH /2
2 4 6 8 10 12 14-0.2
-0.1
0.0
0.1
0.2
0.3
S /TeV
[Anastasiou, Duhr, Dulat, Herzog, Mistlberger (2015)]
• Scattering Amplitudes
• Tree-level (LO) predictions are qualitative due to the poor convergence of the truncated expansion at strong coupling.
• K factors
• Feynman diagrams describe collisions and scattering of elementary particle physics
William J. Torres Bobadilla
Introduction
3
— Practical application in particle physics — Mathematical elegance — Gauge invariant objects
• Perturbative expansion
…tree one-loop two-loop three-loop
LO NLO NNLO NNNLO
0
10
20
30
40
50
σ/pb
LHCpp→h+X gluon fusionMSTW08 68clμ=μR=μF ∈ [mH /4,mH ]Central scale: μ = mH /2
2 4 6 8 10 12 14-0.2
-0.1
0.0
0.1
0.2
0.3
S /TeV
[Anastasiou, Duhr, Dulat, Herzog, Mistlberger (2015)]
• Scattering Amplitudes
• Tree-level (LO) predictions are qualitative due to the poor convergence of the truncated expansion at strong coupling.
• K factors
• Feynman diagrams describe collisions and scattering of elementary particle physics
William J. Torres Bobadilla
Introduction
3
— Practical application in particle physics — Mathematical elegance — Gauge invariant objects
• Feynman diagrams, based on the Lagrangian, are not optimised for these processes.
Simplify the calculations in High-Energy Physics. Discover hidden properties of Quantum Field Theories Towards NNLO is the Next Frontier.
Motivation
OutlineTree-level amplitudes— State-of-the-art— Colour decomposition — Kleiss-Kuijf and Bern-Carrasco-Johansson relations— IR Behaviour— Momentum twistor variables
Analytic calculation of one-loop amplitudes— d dimensional generalised unitarity— Integrand reduction methods— Four dimensional formulation of dimensional regularisation — Applications to gluon and Higgs amplitudes— Automation — UV renormalisation
Extension toward multi-loop scattering amplitudes— Integrand decomposition @ two-loop — Adaptive integrand decomposition — Applications to gluon amplitudes
Conclusions & Outlook William J. Torres Bobadilla
4
William J. Torres Bobadilla5
Hidden Symmetries
Analytic One-loop
Scattering Amplitudes
Multi-lo
op
William J. Torres Bobadilla5
Hidden Symmetries
Analytic One-loop
Scattering Amplitudes
Multi-lo
op
William J. Torres Bobadilla
Simple example: gg —> ggg @ tree-level
6
• Feynman diagram —> gauge dependent quantities
• A factorial growth in the number of terms…
William J. Torres Bobadilla
Simple example: gg —> ggg @ tree-level
6
• Feynman diagram —> gauge dependent quantities
• A factorial growth in the number of terms…
Result of a brute force calculation (just small part of it)
William J. Torres Bobadilla
Simple example: gg —> ggg @ tree-level
6
• Feynman diagram —> gauge dependent quantities
• A factorial growth in the number of terms…
Result of a brute force calculation (just small part of it)
,
Maximally Helicity Violating (MHV) amplitudes
William J. Torres Bobadilla
Simple example: gg —> ggg @ tree-level
7
Reconstruct amplitudes from analytic properties
• Poles
Britto-Cachazo-Feng-Witten (BCFW) recursive relation
Atree
n (1, 2, . . . , n) =X
Partitions r
X
States h
Atree
L (ar, . . . , j, . . . , br, Parbr )i
P 2
ab
Atree
R (�Parbr , br + 1, . . . , l, . . . , br � 1)
[Britto, Cachazo, Feng (2004)] [Britto, Cachazo, Feng, Witten (2005)]
— On-shell amplitudes — Complex momenta — Input —> 3-point amplitudes
Due to the polology theorem[Weinberg]
William J. Torres Bobadilla
Simple example: five gluons @ one-loop
8
Things rapidly become a lot more complicated
…
Calculation using Feynman diagrams is not efficient anymore
Solution —> Chop problem into smaller pieces
Divide the amplitude in terms of gauge invariant pieces —> Colour-decomposition
Take into account the formal properties —> Universal factorisation in IR limit of Scattering amplitudes —> Colour-Kinematics duality
Use simple variables for their calculation —> Spinor-helicity formalism —> Momentum twistor variables
Make use of the fact that the theories we are working in are unitary (probabilities always sum to unity) —> Unitarity based methods
For calculations of multi-loop amplitudes —> work at integrand instead of integral level
William J. Torres Bobadilla
Status
9
[Salam (SILAFAE 2012)]
William J. Torres Bobadilla10
Colour decomposition
In QCD any amplitude can be decomposed as
Momenta Color Helicities
Primitive amplitudes depends on Lorentz variables only
Can contain traces or products of generators T
For the n-gluon tree-level amplitude, the colour decomposition is
+ all non-cyclic permutations
At tree-level
Properties between amplitudes Reflection invariance Cyclic invariance (n-1)! Independent amplitudes
William J. Torres Bobadilla11
Colour decomposition
In QCD any amplitude can be decomposed as
Momenta Color Helicities
Primitive amplitudes depends on Lorentz variables only
Can contain traces or products of generators T
An alternative representation
At tree-level
+ all non-cyclic permutations
[Del Duca, Frizzo and Maltoni (1999)][Del Duca, Dixon and Maltoni (1999)]
Properties between amplitudes Kleiss-Kuijf relations
(n-2)! Independent amplitudes[Kleiss and Kuijf (1989)]
William J. Torres Bobadilla12
Colour-kinematics duality
Jacobi Relation (colour)
[Zhu (1980)] [Bern, Carrasco, Johansson (2008),(2010)]
Write QCD amplitudes in terms of cubic graphs
Satisfy automatically for 4-point tree amplitudes For high multiplicity, is not trivially satisfied
[Bern, Dennen, Huang, Kiermaier (2010)], [Boels, Isermann (2012)] …
[Mastrolia, Primo, Schubert, W.J.T. (2015)] Bern-Carrasco-Johansson relations
(n-3)! Independent amplitudes
IR behaviour of Scattering Amplitudes
• Scattering amplitudes display universal factorisation when a single photon (gluon) or graviton becomes soft: Parametrise soft momentum as δq and take δ→0
• At tree-level
[Low (1958)][Weinberg (1964)]
Soft theorems
13William J. Torres Bobadilla
[Cachazo and Strominger (2014)]Gravity
Yang Mills [Casali (2014)][Bern, Davies, Di Vecchia and Nohle (2014)]
[White (2014)][Luo, Mastrolia, W.J.T. (2014)]
J is the total angular momentum of the emitter
gives arise to the subleading-soft behaviour of the amplitude
IR behaviour of Scattering AmplitudesSubleading Soft theorems
• Universality & factorisation extends to subleading order
14William J. Torres Bobadilla
William J. Torres Bobadilla
Momentum twistor variables
15
Momentum conservation from a different point of view
6-pt kinematics
— Contour defined by edges and cusps — Contraint: — Momenta in terms of cusps
With the Dirac equation
define a new objectFour component spinor variables
Momentum twistor variables
obey certain properties and symmetries— Momentum conservation — On-shell conditions,
— Poincaré symmetries — U(1) symmetry
For an n-point amplitude —> 4n twistor component Because of the symmetries, 4n —> 4n - n - 10 = 3n-10 free components
[Hodges (2009)] [Arkani-Hamed, etal (2010)(2012)]
[Bourjaily (2010)] [Bourjaily, Caron-Huot, Trnka (2010)]
[Elvang, Huang (2015)] [Weinzierl (2016)]
William J. Torres Bobadilla16
Hidden Symmetries
Analytic One-loop
Scattering Amplitudes
Multi-lo
op
William J. Torres Bobadilla16
Hidden Symmetries
Analytic One-loop
Scattering Amplitudes
Multi-lo
op
William J. Torres Bobadilla
Analytic one-loop scattering amplitudes
17
Deal with with integrals of the form
Ii1···ik⇥N (l, pi)
⇤=
Zdd l
Ni1···ik(l, pi)
Di1 · · ·Dik
l · pi, l · "i
Tensor reduction
[Passarino - Veltman (1979)]
Scalar Master Integrals: Made of polylogarithmic functions
- If an amplitude is determined by its branch cuts, it is said to be cut-constructible. - All one-loop amplitudes are cut-constructible in dimensional regularisation. - Master integrals are known
Numerator and denominators are polynomials in the integration variable
William J. Torres Bobadilla
Analytic one-loop scattering amplitudes
18
Standard Unitarity in 4D
Cutting loop = sewing trees
cutting 2x i
p2 �m2! 2⇡�(+)(p2 �m2)
Optical theorem
William J. Torres Bobadilla
Analytic one-loop scattering amplitudes
18
Standard Unitarity in 4D
Cutting loop = sewing trees
cutting 2x i
p2 �m2! 2⇡�(+)(p2 �m2)
Optical theorem
[Bern, Dixon, Dunbar, Kosower (1994)]
Cut internal propagators at amplitude level
Loop reconstruction cut by cut
[Bern, Morgan (1995)]
Generalised Unitarity in 4D[Bern, Dixon, Kosower (1998)]
William J. Torres Bobadilla
Analytic one-loop scattering amplitudes
19
cut-4 :: Britto Cachazo Fengcut-3 :: Forde Bjerrum-Bohr, Dunbar, Ita, Perkins Mastrolia
Quadruple-cut: read our single box coefficient
The cut expansion collapses to a single term
cut-2 :: Bern, Dixon, Dunbar, Kosower. Britto, Buchbinder, Cachazo, Feng. Britto, Feng, Mastrolia.
[Britto, Cachazo, Feng (2004)]
Any one-loop amplitude becomes
In D=4-2ε we can do the decomposition
The on-shell condition
Mass term
William J. Torres Bobadilla
[Ossola,Papadopoulos,Pittau (2006)] [Giele,Kunszt,Melnikov (2008)]
[Badger (2008)] [Mastrolia, Mirabella, Peraro (2012)]
D=4 D=-2ε
Analytic one-loop scattering amplitudes
20
At integral level Z
dd l
(2⇡)d⌘
Zd4l
(2⇡)4d�2✏µ
(2⇡)�2✏
Any one-loop amplitude becomes
In D=4-2ε we can do the decomposition
The on-shell condition
Mass term
William J. Torres Bobadilla
[Ossola,Papadopoulos,Pittau (2006)] [Giele,Kunszt,Melnikov (2008)]
[Badger (2008)] [Mastrolia, Mirabella, Peraro (2012)]
D=4 D=-2ε
Analytic one-loop scattering amplitudes
[Bern, Morgan (1995)]
20
At integral level Z
dd l
(2⇡)d⌘
Zd4l
(2⇡)4d�2✏µ
(2⇡)�2✏
Any one-loop amplitude becomes
In D=4-2ε we can do the decomposition
The on-shell condition
Mass term
William J. Torres Bobadilla
[Ossola,Papadopoulos,Pittau (2006)] [Giele,Kunszt,Melnikov (2008)]
[Badger (2008)] [Mastrolia, Mirabella, Peraro (2012)]
D=4 D=-2ε
Analytic one-loop scattering amplitudes
[Bern, Morgan (1995)]
20
At integral level Z
dd l
(2⇡)d⌘
Zd4l
(2⇡)4d�2✏µ
(2⇡)�2✏
William J. Torres Bobadilla
Analytic one-loop scattering amplitudes
21
Integrand reduction MethodsRecall
Find an identity between integrands. Moreover,
Suppose we find a multipole decomposition
[Ossola, Papadopoulos, Pittau (2006)] [Ellis, Giele, Kunszt, Melnikov (2007)]
[Mastrolia, Ossola, Papadopoulos, Pittau (2008)]
Residues Δ are made of Irreducible Scalar Products Can we find parametric expressions for Δ’s in 4- or d-dimensions?
Yes. General way —> Use multivariate polynomial division
Parametric expressions coefficients are fixed by sampling the numerators on the cut
William J. Torres Bobadilla
Analytic one-loop scattering amplitudes
22
Integrand reduction MethodsLoop parametrisation
Multivariate polynomial division
Consider the ideal generated by the set of denominators
Choose a monomial order and build a Groebner basis
Perform the multivariate polynomial division of module
Express back the elements of in terms of denominators
[Mastrolia, Ossola (2011)] [Badger, Frellesvig, Zhang (2012)]
[Zhang (2012)] [Mastrolia, Mirabella, Ossola, Peraro (2012)]
William J. Torres Bobadilla
Analytic one-loop scattering amplitudes
23
Integrand reduction MethodsLoop parametrisation
Multivariate polynomial division [Mastrolia, Ossola (2011)] [Badger, Frellesvig, Zhang (2012)]
[Zhang (2012)] [Mastrolia, Mirabella, Ossola, Peraro (2012)]Write the numerator in terms
of Irreducible polynomials I ⌘ ND0 · · ·Dn�1
=5X
k=1
X
{i1,··· ,ik}
�i1···ikDi1 · · ·Dik
sum of integrands with five or less denominators
�i1···ikMade of Irreducible Scalar Products Cannot be expressed in terms of denominators
�i1i2i3i4i5 = c0 µ2,
�i1i2i3i4 = c0 + c1 x4,v + µ
2�c2 + c3 x4,v + µ
2c4
�,
�i1i2i3 = c0 + c1 x4 + c2 x24 + c3 x
34 + c4 x3 + c5 x
23 + c6 x
33 ++µ
2 (c7 + c8 x4 + c9 x3) ,
�i1i2 = c0 + c1 x1 + c2 x21 + c3 x4 + c4 x
24 + c5 x3 + c6 x
23 + c7 x1x4 + c8 x1x3 + c9 µ
2,
�i1 = c0 + c1 x1 + c2 x2 + c3 x3 + c4 x4,
Generic structure of the residue
A: Separated computation of cut-constructible and rational terms
A1: Computing the rational term separately (using non gauge invariant terms)
B: D-dimensional unitarity offers the determination of all pieces together
B2: Gamma algebra in extended dimension [Ellis,Giele,Kunszt,Melnikov (2008)]
— R1 and R2 separation [Ossola, Papadopoulos, Pittau(2008); Pittau, Draggiotis, Garzelli (2009)] — Supersymmetric decomposition [Bern, Dixon, Kosower]
— New rules for spinor products — No automatic generator exists
— The explicit representation of the polarisation states is avoid — Gamma algebra has to be extended everywhere. — Automatic generator has to be modified
B3: Don't leave 4 dimensions! [Fazio, Mastrolia, Mirabella, W.J.T (2014)]
B1: 6-dimensional spinor-helicity formalism [Cheung and O’Connell(2009); Davies (2012)]
William J. Torres Bobadilla
How to compute those coefficients?
24
A: Separated computation of cut-constructible and rational terms
A1: Computing the rational term separately (using non gauge invariant terms)
B: D-dimensional unitarity offers the determination of all pieces together
B2: Gamma algebra in extended dimension [Ellis,Giele,Kunszt,Melnikov (2008)]
— R1 and R2 separation [Ossola, Papadopoulos, Pittau(2008); Pittau, Draggiotis, Garzelli (2009)] — Supersymmetric decomposition [Bern, Dixon, Kosower]
— New rules for spinor products — No automatic generator exists
— The explicit representation of the polarisation states is avoid — Gamma algebra has to be extended everywhere. — Automatic generator has to be modified
B3: Don't leave 4 dimensions! [Fazio, Mastrolia, Mirabella, W.J.T (2014)]
B1: 6-dimensional spinor-helicity formalism [Cheung and O’Connell(2009); Davies (2012)]
William J. Torres Bobadilla
How to compute those coefficients?
24
— Explicit 4D representation of polarisation and. spinors — 4D representation of D-reg loop propagators — 4D Feynman rules + (-2ε)-Selection Rules — Easy to implement in existing generators
Four Dimensional Formulation of Dimensional Regularisation
(FDF)
B: D-dimensional unitarity offers the determination of all pieces together
B3: Don't leave 4 dimensions! [Fazio, Mastrolia, Mirabella, W.J.T (2014)]
William J. Torres Bobadilla
How to compute those coefficients?
24
Can we implement it with 4D-object only?
Projections of the vectors and .
The d-dimensional metric tensor can be split as
d-dimensional4-dimensional
-2ε-dimensionalWhere
As well for the gamma matrices
In 4-dimensions, one can infer:
And the Clifford algebra
Excludes any four-dimensional representation of the —2ε-subspace
—2ε-subspace —2ε-Selection Rules (—2ε)-SRs
[Fazio, Mastrolia, Mirabella, W.J.T. (2014)]
FDH: 4D helicity scheme
William J. Torres Bobadilla
[Bern and Kosower (1992)]
Projections of the vectors and .
The d-dimensional metric tensor can be split as
d-dimensional4-dimensional
-2ε-dimensionalWhere
As well for the gamma matrices
[Fazio, Mastrolia, Mirabella, W.J.T. (2014)]
FDH: 4D helicity scheme
William J. Torres Bobadilla
[Bern and Kosower (1992)]
The Clifford algebra conditions are satisfied by imposing
A,B := —2ε-dimensional vectorial indices traded for (—2ε)-SRs
-2ε-Selection Rules
The helicity sum of the transverse polarisation vector is
d = 4 d = -2εmassive gluon
Gluon propagator
William J. Torres Bobadilla
Completeness relations within FDF
Fermion propagator
Allows to generalise the Dirac Equation
[Fazio, Mastrolia, Mirabella, W.J.T. (2014)]
26
Feynman Rules in FDF
William J. Torres Bobadilla
[Fazio, Mastrolia, Mirabella, W.J.T. (2014)]
27
William J. Torres Bobadilla
[Fazio, Mastrolia, Mirabella, W.J.T. (2014)]
28
The gggg amplitude @1-loop
William J. Torres Bobadilla
[Fazio, Mastrolia, Mirabella, W.J.T. (2014)]
28
The gggg amplitude @1-loop
(Gluon loop)
(-2ε-Scalar loop)
Contributions come only from the coefficients:
which amounts
Being the full one-loop amplitude
[Bern & Kosower (1992)]William J. Torres Bobadilla
The gggg amplitude @1-loop[Fazio, Mastrolia, Mirabella, W.J.T. (2014)]
29
William J. Torres Bobadilla30
NLO QCD corrections to Higgs to partons
[Adler, Collins and Duncan (1977)]
• For 2 gluons Higgs, we use an effective operator with
Feynman Rules in FDF
[Fazio, Mastrolia, Mirabella, W.J.T. (2014)]
Draw all possible topologies
Build tree amplitudes
IntegrandGeneration
Perform integrand reduction methods
Analytical coefficients of each Master Integral
Automation
FeynArts[Hahn (2001)]
[Mertig et al (1991) (2016)]FeynCalc
[Maitre & Mastrolia (2008)]S@M
T@M[W.J.T.]
Performchecks
Gauge invariance
Ward Identity
(-2ε)—SRs
William J. Torres Bobadilla31
Draw all possible topologies
Build tree amplitudes
IntegrandGeneration
Perform integrand reduction methods
Analytical coefficients of each Master Integral
Automation
FeynArts[Hahn (2001)]
[Mertig et al (1991) (2016)]FeynCalc
[Maitre & Mastrolia (2008)]S@M
T@M[W.J.T.]
Performchecks
Gauge invariance
Ward Identity
(-2ε)—SRs
William J. Torres Bobadilla31
Draw all possible topologies
Build tree amplitudes
IntegrandGeneration
Perform integrand reduction methods
Analytical coefficients of each Master Integral
Automation
FeynArts[Hahn (2001)]
[Mertig et al (1991) (2016)]FeynCalc
[Maitre & Mastrolia (2008)]S@M
T@M[W.J.T.]
Performchecks
Gauge invariance
Ward Identity
(-2ε)—SRs
William J. Torres Bobadilla31
William J. Torres Bobadilla32
Applications to gluon and Higgs amplitudes
[Bern and Kosower (1992)]
[NJET]
[NJET]
[Schmidt (1997)]
[Badger, Glover, Mastrolia, Williams (2009)]
[Kunszt, Signer and Trocsanyi (1993)]
To be checked with GoSamPreliminary results
William J. Torres Bobadilla33
UV renormalisation at one-loop
Simple exercise —> renormalised QED Lagrangian
Renormalised fields
Within FDF
William J. Torres Bobadilla33
UV renormalisation at one-loop
Simple exercise —> renormalised QED Lagrangian
Renormalised fields
Within FDF With the use of integration-by-parts identities
Renormalisation in the on-shell scheme
William J. Torres Bobadilla
Results
34
1�
2+ 3+
3+
1�
2+
4+
5+
4+
6+1�
2+ 3+
4+
5+
[Badger, Frellesvig, Zhang (2013)] [Badger (2016)]
Parametrisation of momentum twistor variables
William J. Torres Bobadilla35
Hidden Symmetries
Analytic One-loop
Scattering Amplitudes
Multi-lo
op
William J. Torres Bobadilla35
Hidden Symmetries
Analytic One-loop
Scattering Amplitudes
Multi-lo
op
William J. Torres Bobadilla
What about multi-loop level?
[Bern, Dixon, Kosower (2000)]
loop momenta decomposed as before l↵i = l↵i + l↵i
li · pj , li · "j , li · lj , µij
New subtlety µij = �li · lj
gg —> gg @2-loop
Follow a diagrammatic approach
A(2)4 ({pi}) =
Zdd l1(2⇡)d
dd l2(2⇡)d
N({li}, {pi})Q7k=1 Dk({li}, {pi})
Traditional approach Perform poly div module Groebner basis Fit residue at the cut
36
William J. Torres Bobadilla
Adaptive integrand decomposition[Mastrolia, Peraro, Primo (2016)]
• Split d=4-2ε into parallel and orthogonal directions
• Nice properties for less than 5 external legs
• Loop momenta become
• Numerator and denominators depend on different variables
Polynomial division is reduced to a substitution rule (of reduce variables and physics ISP)
• Straightforward identification of spurious terms 37
William J. Torres Bobadilla
Adaptive integrand decomposition[Mastrolia, Peraro, Primo, W.J.T. (2016)]
gg —> ggg @2-loop
38[Badger, Frellesvig, Zhang (2013)]
William J. Torres Bobadilla
Adaptive integrand decomposition[Mastrolia, Peraro, Primo, W.J.T. (2016)]
gg —> ggg @2-loop
Consider Loop momenta parametrisation
Parallel components expressed in terms of orthogonal ones + denominators
38[Badger, Frellesvig, Zhang (2013)]
William J. Torres Bobadilla
Adaptive integrand decomposition[Mastrolia, Peraro, Primo, W.J.T. (2016)]
gg —> ggg @2-loop
Consider Loop momenta parametrisation
Parallel components expressed in terms of orthogonal ones + denominators
38[Badger, Frellesvig, Zhang (2013)]
William J. Torres Bobadilla
Adaptive integrand decomposition[Mastrolia, Peraro, Primo, W.J.T. (2016)]
gg —> ggg @2-loop
Consider Loop momenta parametrisation
Parallel components expressed in terms of orthogonal ones + denominators
38[Badger, Frellesvig, Zhang (2013)]
Outlook
William J. Torres Bobadilla
• NN...LO — many legs — massive particles in the loops (full mt dependence)
• Extend our Four-dimensional-Formulation at two-loops — Workout explicit loop component — Should selection rules be redefined?
• Integrand reduction methods at multi-loop level— Provide automation of the Adaptive Integrand Decomposition
39
• Computational techniques of scattering amplitudes.— Efficient techniques for tree- and multi-loop-level amplitudes
• We have proposed a new regularisation scheme (Four-dimensional-Formulation)— Computes d-dimensionally scattering amplitudes using four-dimensional ingredientes — Nobel calculations of relevant for QCD processes: gg —> ng & gg —> ng H — Colour-Kinematics duality for dimensionally regulated amplitudes — UV renormalisation at one-loop
• Intensive study of integrand reduction methods — Calculation of one-loop scattering amplitudes —> Automated— Extension towards multi-loop scattering amplitudes — Warming up exercises using traditional integrand reduction methods and adaptive ones
Summary
gg ! t t, gg ! t t ggg ! ngH
Outlook
William J. Torres Bobadilla
• NN...LO — many legs — massive particles in the loops (full mt dependence)
• Extend our Four-dimensional-Formulation at two-loops — Workout explicit loop component — Should selection rules be redefined?
• Integrand reduction methods at multi-loop level— Provide automation of the Adaptive Integrand Decomposition Thanks
• Computational techniques of scattering amplitudes.— Efficient techniques for tree- and multi-loop-level amplitudes
• We have proposed a new regularisation scheme (Four-dimensional-Formulation)— Computes d-dimensionally scattering amplitudes using four-dimensional ingredientes — Nobel calculations of relevant for QCD processes: gg —> ng & gg —> ng H — Colour-Kinematics duality for dimensionally regulated amplitudes — UV renormalisation at one-loop
• Intensive study of integrand reduction methods — Calculation of one-loop scattering amplitudes —> Automated— Extension towards multi-loop scattering amplitudes — Warming up exercises using traditional integrand reduction methods and adaptive ones
Summary
gg ! t t, gg ! t t ggg ! ngH
loops