Scattering Amplitudes in gauge theories - IFIC Indico Server (Indico)

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William J. Torres Bobadilla

Institut de Física Corpuscular  Universitat de València - CSIC

Scattering Amplitudes in gauge theories

XI CPAN DAYS

22.10.2190,Eurostars Palacio de Cristal

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Amplitudes?

What are they? Where do they appear?

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William J. Torres Bobadilla 2

Amplitudes?

What are they?

Intensity of light (wave)

Optics

Electromagnetism

Electric and magnetic field

Quantum Mechanics

h

^out

|

ⁱⁿ

i

Quantum Field theory

h

^out

| S |

ⁱⁿ

i

Where do they appear?

(4)

Amplitudes?

What are they?

Intensity of light (wave)

Optics

Electromagnetism

Quantum Mechanics

h

^out

|

ⁱⁿ

i

Quantum Field theory

h

^out

| S |

ⁱⁿ

i

Pr ob ab ilit y( x ) = | am pli tu de ( x ) | 2

(5)

Amplitudes?

What are they?

Intensity of light (wave)

Optics

Electromagnetism

Quantum Mechanics

h

^out

|

ⁱⁿ

i

Quantum Field theory

h

^out

| S |

ⁱⁿ

i

Pr ob ab ilit y( x ) = | am pli tu de ( x ) | 2

(6)

Scattering Amplitudes

Particle interactions

1 + 2 ! 3 + 4

Quantum probability

The simplest process

2

⇠

2->2 scattering

Amplitudes ~ Feynman diagrams

Tree-level one-loop two-loop

…

Perturbation expansion

(7)

William J. Torres Bobadilla

Scattering Amplitudes

1 + 2 ! 3 + 4

Quantum probability

2

⇠

2->2 scattering

_pre^cisio^{n d}

epends on the

number of couplings =

Tree-level one-loop two-loop

…

(8)

Scattering Amplitudes

1 + 2 ! 3 + 4

Quantum probability

2

⇠

2->2 scattering

_pre^cisio^{n d}

epends on the

number of couplings =

Tree-level one-loop two-loop

…

e^ix =

✓ 1 + x²

2!

+ x⁴

4!

+ x⁶

6!

+· · ·

◆ +i

✓ x+ x³

3!

+ x⁵

5!

+ x⁷

7!

+· · ·

◆

=cosx+i sinx

(9)

Scattering Amplitudes

(10)

[Salam (SILAFAE 2012)]

(11)

Status @ N…NLO

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Introduction

Standard Model (SM) of Particle Physics —> best Quantum Field Theory

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

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Introduction

Scattering Amplitudes

— Practical applications in particle physics

— Mathematical elegance

— Gauge invariant objects

Perturbative expansion

…

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Introduction

Scattering Amplitudes

— Practical applications in particle physics

— Mathematical elegance

— Gauge invariant objects

Simplify the calculations in High-Energy Physics.

Discover hidden properties of Quantum Field Theories Towards NNLO is the Next Frontier.

Motivation

LO NLO NNLO NNNLO

0 10 20 30 40 50

σ/pb

LHC pp→h+X gluon fusion MSTW08 68cl μ=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

(15)

Tree-level Scattering amplitudes

(16)

Simple example: gg —> ggg @ tree-level

Feynman diagram —> gauge dependent quantities

A factorial growth in the number of terms …

(17)

Simple example: gg —> ggg @ tree-level

9

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

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Simple example: gg —> ggg @ tree-level

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

,

Maximally Helicity Violating (MHV) amplitudes

(19)

Reconstruct amplitudes from analytic properties

Poles

Britto-Cachazo-Feng-Witten (BCFW) recursive relation

A^tree_n (1, 2, . . . , n) = X

Partitions r

X

States h

A^tree_L (a_r, . . . ,ˆj, . . . , b_r,Pˆ_a_r_b_r) i

P_ab² A^tree_R ( P_a_r_b_r, b_r + 1, . . . ,ˆl, . . . , b_r 1) [Bri%o, Cachazo, Feng (2004)]

[Bri%o, Cachazo, Feng, Wi%en (2005)]

— On-shell amplitudes 

— Complex momenta

— Input —> 3-point amplitudes

Due to the polology theorem

[Weinberg]

Simple example: gg —> ggg @ tree-level

(20)

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

Simple example: gg —> ggg @ one-loop

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Status

12

[Salam (SILAFAE 2012)]

(22)

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

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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

+ 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)]

At tree-level

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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

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Relations between kinematic numerators Provides symmetries among amplitudes

Strong Connection between gravity and Yang-Mills amplitudes Construction of gravity from knowledge of Yang-Mills amplitudes

[Bern, Carrasco, Johansson (2008),(2010)]

[Johansson, Ochirov (2014),(2015)]

[de la Cruz, Kniss, Weinzierl (2015),(2016)]

…

Construct an off-shell current [Llanes, Rodrigo, W.J.T. (2017)]

Colour-kinematics duality

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Multi-loop Scattering amplitudes

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Dimensional regularisation schemes

Before computing multi-loop amplitudes…

I₀ =

Z ₁

0

dx x Consider

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I₀ =

Z ₁

0

dx x Consider

does not exist

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Dimensional regularisation schemes

I₀ =

Z ₁

0

dx x Consider

does not exist

I_✏ =

Z ₁

0

dx

x^1+✏ =

Z 1 0

dx

x^1+✏ +

Z ₁

1

dx x^1+✏

Tweak the integrand

(with ✏ 2 C )

well defined

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I₀ =

Z ₁

0

dx x Consider

does not exist

I_✏ =

Z ₁

0

dx

x^1+✏ =

Z 1 0

dx

x^1+✏ +

Z ₁

1

dx x^1+✏

Tweak the integrand

(with ✏ 2 C )

1

✏

<(✏) < 0

+1

✏

<(✏) > 0

well defined

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Dimensional regularisation schemes

I₀ =

Z ₁

0

dx x Consider

does not exist

I_✏ =

Z ₁

0

dx

x^1+✏ =

Z 1 0

dx

x^1+✏ +

Z ₁

1

dx x^1+✏

Tweak the integrand

(with ✏ 2 C )

1

✏

<(✏) < 0

+1

✏

<(✏) > 0

we are physicists well defined

I_✏ = 0 , 8✏ 2 C

I

₀

= 0

(analytical continuation)

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I₀ =

Z ₁

0

dx x Consider

does not exist

I_✏ =

Z ₁

0

dx

x^1+✏ =

Z 1 0

dx

x^1+✏ +

Z ₁

1

dx x^1+✏

Tweak the integrand

(with ✏ 2 C )

1

✏

<(✏) < 0

+1

✏

<(✏) > 0

we are physicists well defined

I_✏ = 0 , 8✏ 2 C

I

₀

= 0

(analytical continuation)

This was dimensional regularisation

— All computations made in d=4-2ε

— Singularities manifest as poles in ε

— physical observables don’t depend on ε

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Dimensional regularisation schemes

For all dimensional schemes

Z d

⁴

l

(2⇡ )

⁴

! µ

⁴_DS^d

Z d

^d

¯ l (2⇡)

^d

[Gnendiger (RADCOR 2017)]

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Z d

⁴

l

(2⇡ )

⁴

! µ

⁴_DS^d

Z d

^d

¯ l (2⇡)

^d

[Gnendiger, et al (2017)]

(35)

Z d

⁴

l

(2⇡ )

⁴

! µ

⁴_DS^d

Z d

^d

¯ l (2⇡)

^d

[Gnendiger, et al (2017)]

[Fazio, Mastrolia, Mirabella, W.J.T. (2014)]

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Z d

⁴

l

(2⇡ )

⁴

! µ

⁴_DS^d

Z d

^d

¯ l (2⇡)

^d

with the unified framework

The most used DS can be understood as

n_✏ = d_s d CDR/HV : n_✏ = 0 FDH/DRED : n_✏ = 2✏

[Gnendiger, et al (Fazio, W.J.T.) (2017)]

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One-loop scattering amplitudes

Deal with with integrals of the form

I_i₁_···_i_k ⇥

N (¯l, p_i)⇤

= Z

d^d¯l Nⁱ1···i_k(¯l, p_i) D_i₁ · · · D_i_k

¯l², ¯l · p_i, ¯l · "_i

Numerator and denominators are

polynomials in the integration variable

Tensor reduction

[Passarino - Veltman (1979)]

Cut-constructible amplitude -> determined by its branch cuts

All one-loop amplitudes are cut-constructible in dimensional regularisation.

Master integrals are known

d=4 d=-2ε

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One-loop scattering amplitudes

Deal with with integrals of the form

I_i₁_···_i_k ⇥

N (¯l, p_i)⇤

= Z

d^d¯l Nⁱ1···i_k(¯l, p_i) D_i₁ · · · D_i_k

¯l², ¯l · p_i, ¯l · "_i

Numerator and denominators are

polynomials in the integration variable

Tensor reduction

[Passarino - Veltman (1979)]

Unitarity based methods

cut-4 :: Britto Cachazo Feng cut-3 :: Forde

Bjerrum-Bohr, Dunbar, Ita, Perkins Mastrolia

cut-2 :: Bern, Dixon, Dunbar, Kosower.

Britto, Buchbinder, Cachazo, Feng.

Britto, Feng, Mastrolia.

Isolate the leading discontinuity!

i

q²_i m² i✏ ! 2⇡ ⁽⁺⁾ q_i² m²_i

d=4 d=-2ε

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William J. Torres Bobadilla 22 A⁽¹⁾ ({p_n}) =

Z

` N (`; {p_n}) ⌦ G_D (↵)

G_D (↵) = X

i2↵

˜ (q_i)Y

j6=i

G_D (q_i, q_j)

The loop-tree duality theorem @ 1L

[Catani, Gleisberg, Krauss, Rodrigo, Winter (2008)]

One-loop integrals decompose as a linear combination of N single-cut phase-space integrals

˜ (q_i) = ı2⇡ q_i² m²_i ✓ (q_i,0)

G_D (q_i, q_j) = 1

q_j² m²_j ı0⌘ · k_ji

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A⁽¹⁾ ({p_n}) =

Z

` N (`; {p_n}) ⌦ G_D (↵)

G_D (↵) = X

i2↵

˜ (q_i)Y

j6=i

G_D (q_i, q_j)

All internal

momenta sets on shell the internal propagator with momentum

and selects its positive energy mode q_i = ` + k_i

dual prop linear in

the loop momentum Differs from Feynman prop only by the imaginary part prescription

⌘ ! future-like, ⌘^µ = (1, 0) k_ji = q_j q_i

The loop-tree duality theorem @ 1L

˜ (q_i) = ı2⇡ q_i² m²_i ✓ (q_i,0)

G_D (q_i, q_j) = 1

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William J. Torres Bobadilla 22 A⁽¹⁾ ({p_n}) =

Z

` N (`; {p_n}) ⌦ G_D (↵)

G_D (↵) = X

i2↵

˜ (q_i)Y

j6=i

G_D (q_i, q_j)

The loop-tree duality theorem @ 1L

˜ (q_i) = ı2⇡ q_i² m²_i ✓ (q_i,0)

Modify +i0 prescription of the Feynman props. 

It compensates for the absence of multiple-cut contributions that  appear in the Feynman Tree Theorem

Lorentz-covariant dual prescription —> η a future-like vector Number of single cuts = the number of legs.

Singularities of the loop diagram —> singularities of the dual integrals.

Loop-Tree Duality works only on propagators.  

Same procedure for Tensor loop integrals and scattering amplitudes.

G_D (q_i, q_j) = 1

[Driencourt-Mangin, Rodrigo, Sborlini, W.J.T. (To appear)]

[Buchta, Chachamis, Draggioas, Malamos, Rodrigo (2014)]

[Aguilera et al (W.J.T.) (2019)]

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One-loop integrand decomposition

Recall

Find an identity between integrands. Moreover,

Suppose a multipole decomposition

[Ossola, Papadopoulos, Pi%au (2006)]

[Ellis, Giele, Kunszt, Melnikov (2007)]

[Mastrolia, Ossola, Papadopoulos, Pi%au (2008)]

Residues Δ are made of Irreducible Scalar Products

Can we find parametric expressions for Δ’s in 4- or d-dimensions?

Parametric expressions Yes. General way —> Use multivariate polynomial division coefficients are fixed by sampling the numerators on the cut

(43)

One-loop integrand decomposition

Loop parametrisation

Multivariate polynomial division

[Mastrolia, Ossola (2011)]

[Badger, Frellesvig, Zhang (2012)]

[Zhang (2012)]

[Mastrolia, Mirabella, Ossola, Peraro (2012)]

[Mastrolia, Peraro, Primo (2016)]

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

Write the numerator in terms of

Irreducible polynomials I ⌘ N

D₀ · · · D_n ₁ =

X5

k=1

X

{i_1,···,i_k}

i₁···i_k

D_i₁ · · · D_i_k

sum of integrands with five or less denominators i1···i_k Made of Irreducible Scalar Products  

Cannot be expressed in terms of denominators

i1i2i3i4i5 = c₀ µ²,

i₁i₂i₃i₄ = c₀ + c₁ x_4,v + µ² c₂ + c₃ x_4,v + µ² c₄ ,

i₁i₂i₃ = c₀ + c₁ x₄ + c₂ x²₄ + c₃ x³₄ + c₄ x₃ + c₅ x²₃ + c₆ x³₃ + +µ² (c₇ + c₈x₄ + c₉x₃),

i1i2 = c₀ + c₁ x₁ + c₂ x²₁ + c₃ x₄ + c₄ x²₄ + c₅ x₃ + c₆ x²₃ + c₇ x₁x₄ + c₈ x₁x₃ + c₉µ²,

i₁ = c₀ + c₁ x₁ + c₂ x₂ + c₃ x₃ + c₄ x₄,

Generic structure of the residue

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Adaptive integrand decomposition (AID)

Einstein-Yang-Mills Amplitudes

g

g g

h

g(p₁) + g(p₂) ! g( p₃) + h( p₄)

=

h^µ⌫(p_i) ! "^µ

i(p_i)"^⌫_i(p_i) 4-point process depending on 2 scales + d

s = (p₁ + p₂)² t = (p₂ + p₃)²

Warming up exercise

More gravitons

{I₄[µ₁₁], I₄[µ²₁₁], I₄[µ³₁₁], I₄[µ⁴₁₁], I₃[µ₁₁], I₃[µ²₁₁], I₂[µ₁₁], I₂[µ²₁₁]} L^EYM = 2

²

p gR 1 4

p gg^µ⇢g^⌫ F_µ⌫^a F_⇢^a + L^gf

R_µ

⌫ = @_µ _⇢

⇢⌫ @_⇢ ⇢

µ⌫ + ⇢

µ ⇢⌫ ⇢

⇢ µ⌫ ,

µ⇢⌫ = 1 2 g^⇢

@_µg_⌫

+@_⌫g

µ @ g_µ

⌫ , g_µ

⌫ = ⌘_µ_⌫

+h_µ

⌫ .

[W.J.T. (2018)]

(45)

Initialisation

Identify parent topologies from Feynman graphs

AIDA for EYM amplitudes

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(46)

Initialisation

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(47)

Initialisation

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