Scattering Amplitudes in Gauge Theories: from tree- to ... · • Scattering Amplitudes •...

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


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


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


Simple example: gg —> ggg @ tree-level

6

• Feynman diagram —> gauge dependent quantities

• A factorial growth in the number of terms…



6



Result of a brute force calculation (just small part of it)



6



Result of a brute force calculation (just small part of it)

,

Maximally Helicity Violating (MHV) amplitudes



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]


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


Status

9

[Salam (SILAFAE 2012)]


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


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)]


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


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)]


Hidden Symmetries

Analytic One-loop


Multi-lo

op


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



18

Standard Unitarity in 4D

Cutting loop = sewing trees

cutting 2x i

p2 �m2! 2⇡�(+)(p2 �m2)

Optical theorem



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)]



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


[Ossola,Papadopoulos,Pittau (2006)] [Giele,Kunszt,Melnikov (2008)]

[Badger (2008)] [Mastrolia, Mirabella, Peraro (2012)]

D=4 D=-2ε


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


[Ossola,Papadopoulos,Pittau (2006)] [Giele,Kunszt,Melnikov (2008)]

[Badger (2008)] [Mastrolia, Mirabella, Peraro (2012)]

D=4 D=-2ε


[Bern, Morgan (1995)]

20

At integral level Z

dd l

(2⇡)d⌘

Zd4l

(2⇡)4d�2✏µ

(2⇡)�2✏



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



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)]



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)]


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)]


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


[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


FDH: 4D helicity scheme



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


Completeness relations within FDF

Fermion propagator

Allows to generalise the Dirac Equation


26

Feynman Rules in FDF



27



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


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


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



Applications to gluon and Higgs amplitudes


[NJET]

[NJET]

[Schmidt (1997)]

[Badger, Glover, Mastrolia, Williams (2009)]

[Kunszt, Signer and Trocsanyi (1993)]

To be checked with GoSamPreliminary results


UV renormalisation at one-loop

Simple exercise —> renormalised QED Lagrangian

Renormalised fields

Within FDF


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


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


Hidden Symmetries

Analytic One-loop


Multi-lo

op


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


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


Adaptive integrand decomposition[Mastrolia, Peraro, Primo, W.J.T. (2016)]

gg —> ggg @2-loop

38[Badger, Frellesvig, Zhang (2013)]


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


• 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


• 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

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