[PPT] - Modern Aspects of Perturbative QFT and Gravity Bryan Larios in PowerPoint Presentation

SLIDE 1

Modern Aspects of Perturbative QFT and Gravity

Bryan Larios

in collaboration with

J. Lorenzo Diaz-Cruz

XXXI Annual Meeting DPyC-SMF May 24, 2017

SLIDE 2

Overview

Introduction / Motivation.
Basics of the Spinor Helicity Formalism (SHF).
Frontier in Scattering Amplitudes (SA).
Our contributions.
Final comments.

2

SLIDE 3

Motivation

Particle Physics: The key
bservable measured in particle

scattering experiments is the scattering cross section.

Mathematics: It has been

realized in recent years that amplitudes themselves have a very interesting mathematical structure.

dσ dΩ ∝ |A|2

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SLIDE 4

Motivation

The calculation with Feynman diagrams are cumbersome, however final results often strikingly simple.

Brice S. DeWitt, 1967

It is well known that the number of Feynman diagrams tends to grow very quickly with the number of particles involved, e.g. for gluon scattering at tree level in QCD one have

Mangano & Parke, 1991

gg → gg, 4 diagrams gg → ggg, 25 diagrams gg → gggg, 220 diagrams gg → gggggg, more than 1 million of diagrams

4

SLIDE 5

If one compute , part of the result is gg → ggg

From Z. Bern talk, ICTP-SAIFR, 15

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SLIDE 6

“When the number of external particles grow, the mathematical expression for each diagram becomes significantly more complicated”.

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SLIDE 7

grad student Who can defend me now?

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“When the number of external particles grow, the mathematical expression for each diagram becomes significantly more complicated”.

SLIDE 8

SHF me!!!!!, The Spinor Helicity Formalism grad student Who can defend me now?

“When the number of external particles grow, the mathematical expression for each diagram becomes significantly more complicated”.

SLIDE 9

The Spinor Helicity Formalism

The key of the SHF is to express the 4-momentum of each external particle in terms of 2-component numerical spinor , and consider these spinors as the fundamental blocks of the amplitude. ¯ u+(~ p)u−(~ k) = aa = [pk] = −[kp] ¯ u−(~ p)u+(~ k) = ∗

˙ a∗˙ a = hpki = hkpi

¯ u+(~ p)u+(~ k) = [pki = 0 ¯ u−(~ p)u−(~ k) = hpk] = 0

Massless case

pa˙

a = −φaφ∗ ˙ a

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SLIDE 10

The SHF is implemented to massive particles as well. With these ingredients it is possible in principle to compute processes and reactions in the SM, we only need to know the rules of these 2-component spinors.

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SLIDE 11

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SLIDE 12

Returning to the 5 gluons amplitude in QCD and using the SHF, it is possible to find the following helicity amplitude

Parke & Taylor, 1986

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SLIDE 13

Just to remind you…….

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SLIDE 14

Frontier in Scattering Amplitudes

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SLIDE 15

on-shell recursion relations

During the last decade, a lot of progress has been done to understand the mathematical structure of the scattering amplitudes, I would like to point out some of the most important

SLIDE 16

An = X

diagramas I

ˆ AL(zI) 1 ˆ P 2

I

ˆ AR(zI)

on-shell recursion relations

BCFW

Britto, Cachazo, Feng and Witten, 2005

16

During the last decade, a lot of progress has been done to understand the mathematical structure of the scattering amplitudes, I would like to point out some of the most important

SLIDE 17

During the last decade, a lot of progress has been done to understand the mathematical structure of the scattering amplitudes, I would like to point out some of the most important

on-shell recursion relations

KLT This amazing result that came from String Theory relates SA of gravity with SA of Yang-Mills theory (tree level)

A4(Gravity) = A4(YM)2

Kaway, Lewellen, Tye, 1986

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SLIDE 18

on-shell recursion relations
Geometrization of SA

During the last decade, a lot of progress has been done to understand the mathematical structure of the scattering amplitudes, I would like to point out some of the most important

SLIDE 19

During the last decade, a lot of progress has been done to understand the mathematical structure of the scattering amplitudes, I would like to point out some of the most important

on-shell recursion relations
Geometrization of SA The Amplituhedron

Applying BCFW and complex analysis in several variables it is possible to express (in some toy model theories) the SA as the volume of a polyhedron.

N.Arkani-Hamed, et.al.

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SLIDE 20

on-shell recursion relations
Geometrization of SA
SA from first principles

During the last decade, a lot of progress has been done to understand the mathematical structure of the scattering amplitudes, I would like to point out some of the most important

SLIDE 21

on-shell recursion relations
Geometrization of SA
SA from first principles

Just applying momentum conservation and gauge invariance it is possible to express some tree level amplitudes for gluons and gravitons.

R. Medina, et.al.
N. Arkani-Hamed, et.al.

21

During the last decade, a lot of progress has been done to understand the mathematical structure of the scattering amplitudes, I would like to point out some of the most important

SLIDE 22

Gravitino Phenomenology with SHF

In SUSY theories with gravity, the spin-3/2 gravitino is the superpartner of the graviton and it is considerate one candidate for DM, when this is the LSP. In order to investigate the nature of the gravitino and the NLSP (Cosmology and Collider Physics), it is necessary to compute scattering amplitudes that involve gravitinos in the final state. Using the traditional Feynman approach (Trace technology) result extremely laborious to compute observables.

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Some progress has been done in order to express the 4 gravitino states in terms of spinor variables With these states at hand it has been possible to evaluate several processes and reactions considering the full (massive) gravitino and also with the goldstino approximation.

L. Diaz-Cruz, BL, 2017

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SLIDE 24

e+e− → ˜ Ψµ ˜ χ0

Associated production of
4-Body stop decay ˜

t → ˜ Ψµ b l+ νl

Some calculations with gravitino

L.Diaz-Cruz, BL, 2017 L.Diaz-Cruz, BL, 2017

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SLIDE 25

Final Comments

It will be interesting if in the future the scattering

amplitudes involved in realistic theories are computed from first principles or with a geometric approach.

Our next step (wish) is to implement recursion

relations to SA with gravitinos in the final state.

Apply our results to Cosmology.

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SLIDE 26

Modern Aspects of Perturbative QFT and Gravity

Bryan Larios

Overview

Motivation

dσ dΩ ∝ |A|2

Motivation

The calculation with Feynman diagrams are cumbersome, however final results often strikingly simple.

It is well known that the number of Feynman diagrams tends to grow very quickly with the number of particles involved, e.g. for gluon scattering at tree level in QCD one have

gg → gg, 4 diagrams gg → ggg, 25 diagrams gg → gggg, 220 diagrams gg → gggggg, more than 1 million of diagrams

If one compute , part of the result is gg → ggg

“When the number of external particles grow, the mathematical expression for each diagram becomes significantly more complicated”.

“When the number of external particles grow, the mathematical expression for each diagram becomes significantly more complicated”.

“When the number of external particles grow, the mathematical expression for each diagram becomes significantly more complicated”.

The Spinor Helicity Formalism

The key of the SHF is to express the 4-momentum of each external particle in terms of 2-component numerical spinor , and consider these spinors as the fundamental blocks of the amplitude. ¯ u+(~ p)u−(~ k) = aa = [pk] = −[kp] ¯ u−(~ p)u+(~ k) = ∗

¯ u+(~ p)u+(~ k) = [pki = 0 ¯ u−(~ p)u−(~ k) = hpk] = 0

pa˙

The SHF is implemented to massive particles as well. With these ingredients it is possible in principle to compute processes and reactions in the SM, we only need to know the rules of these 2-component spinors.

Returning to the 5 gluons amplitude in QCD and using the SHF, it is possible to find the following helicity amplitude

Just to remind you…….

Frontier in Scattering Amplitudes

During the last decade, a lot of progress has been done to understand the mathematical structure of the scattering amplitudes, I would like to point out some of the most important

BCFW

During the last decade, a lot of progress has been done to understand the mathematical structure of the scattering amplitudes, I would like to point out some of the most important

During the last decade, a lot of progress has been done to understand the mathematical structure of the scattering amplitudes, I would like to point out some of the most important

KLT This amazing result that came from String Theory relates SA of gravity with SA of Yang-Mills theory (tree level)

A4(Gravity) = A4(YM)2

During the last decade, a lot of progress has been done to understand the mathematical structure of the scattering amplitudes, I would like to point out some of the most important

During the last decade, a lot of progress has been done to understand the mathematical structure of the scattering amplitudes, I would like to point out some of the most important

During the last decade, a lot of progress has been done to understand the mathematical structure of the scattering amplitudes, I would like to point out some of the most important

During the last decade, a lot of progress has been done to understand the mathematical structure of the scattering amplitudes, I would like to point out some of the most important

Gravitino Phenomenology with SHF

Some progress has been done in order to express the 4 gravitino states in terms of spinor variables With these states at hand it has been possible to evaluate several processes and reactions considering the full (massive) gravitino and also with the goldstino approximation.

e+e− → ˜ Ψµ ˜ χ0

t → ˜ Ψµ b l+ νl

Some calculations with gravitino

Final Comments

amplitudes involved in realistic theories are computed from first principles or with a geometric approach.

relations to SA with gravitinos in the final state.

Thank you