T HE S PINOR H ELICITY F ORMALISM IN SUGRA PHENOMENOLOGY Bryan - PowerPoint PPT Presentation

Index Motivation The spinor helicity formalism Monophoton signal in LSP gravitino production Conclusions T HE S PINOR H ELICITY F ORMALISM IN SUGRA PHENOMENOLOGY Bryan Larios J. Lorenzo Diaz Cruz ısico M´ Facultad de Ciencias F´ atematico BUAP XXX Reuni´ on Anual de la Divisi´ on de Part´ ıculas y Campos de la SMF Bryan Larios 3 body Stop decay with LSP gravitino/goldstino in the f.s.

Index Motivation The spinor helicity formalism Monophoton signal in LSP gravitino production Conclusions Outline Motivation, Spinor Helicity Formalism (SHF), SHF to the rescue in SUGRA phenomenology, Conclusions. Bryan Larios 3 body Stop decay with LSP gravitino/goldstino in the f.s.

Index Motivation The spinor helicity formalism Monophoton signal in LSP gravitino production Conclusions Motivation We have been working with some process and reactions in supersymmetric models where the gravitino is very light (LSP) and then a good dark matter candidate. Recently we computed t 1 → b + W + ˜ the 3-body stop decay ( ˜ Ψ µ ) where we use the MATHEMATICA power to handle the huge traces that appear in the scattering amplitudes. Bryan Larios 3 body Stop decay with LSP gravitino/goldstino in the f.s.

Index Motivation The spinor helicity formalism Monophoton signal in LSP gravitino production Conclusions t 1 → b + W + ˜ ˜ Ψ µ Lorenzo Diaz-Cruz & Bryan Larios-Lopez, EPJC (2016) 76 :157. Bryan Larios 3 body Stop decay with LSP gravitino/goldstino in the f.s.

Index Motivation The spinor helicity formalism Monophoton signal in LSP gravitino production Conclusions Considering just one channel (chargino), we obtained for the chargino ∣ M 1 χ ∣ 2 . Bryan Larios 3 body Stop decay with LSP gravitino/goldstino in the f.s.

Index Motivation The spinor helicity formalism Monophoton signal in LSP gravitino production Conclusions Bryan Larios 3 body Stop decay with LSP gravitino/goldstino in the f.s.

Index Motivation The spinor helicity formalism Monophoton signal in LSP gravitino production Conclusions Motivation There are several promising search channels (in SUSY models) to find new physics at both lepton and hadron colliders but, we t 1 → b + W + ˜ learned with the last process ( ˜ Ψ µ ) that it is neces- sary to implement new calculations methods. In order to learn how to compute scattering amplitudes efficiently and then ap- ply the new methods to some reaction with spin-3/2 particle, we revisited the monophoton plus mising energy ( e + e − → γ ˜ G ˜ G ). Bryan Larios 3 body Stop decay with LSP gravitino/goldstino in the f.s.

Index Motivation The spinor helicity formalism The spinor helicity formalism (SHF) (a pragmatic point of view) Monophoton signal in LSP gravitino production Conclusions The spinor helicity formalism (a pragmatic point of view) Bryan Larios 3 body Stop decay with LSP gravitino/goldstino in the f.s.

Index Motivation The spinor helicity formalism The spinor helicity formalism (SHF) (a pragmatic point of view) Monophoton signal in LSP gravitino production Conclusions The SHF is based in the following observation Fields with spin-1 transform in the ( 1 2 ) representation of the 2 , 1 Lorentz group. So we are able to express 4-moments of any particle as a biespinor: p µ → p a ˙ a a = p µ σ µ (1) p a ˙ a ˙ a γ µ = ( 0 0 ) σ µ σ µ ¯ a = ( I, ⃗ σ ) , ¯ aa = ( I, −⃗ σ ) σ µ σ µ ˙ a ˙ Bryan Larios 3 body Stop decay with LSP gravitino/goldstino in the f.s.

Index Motivation The spinor helicity formalism The spinor helicity formalism (SHF) (a pragmatic point of view) Monophoton signal in LSP gravitino production Conclusions We also know that: u s (⃗ p ) ¯ u s (⃗ p ) = 1 2 ( 1 + sγ 5 )(−/ p ) with s = ± , para s = − , tenemos: − p a ˙ u − (⃗ p ) ¯ u − (⃗ p ) = 1 2 ( 1 − γ 5 )(−/ p ) = ( 0 ) a 0 0 where u − (⃗ p ) = ( φ a 0 ) and φ a is a two component spinor that solves the Weyl equation. a explicit formula for this spinor is the following: φ a = ( − sin ( θ 2 ) e − iφ ) cos ( θ 2 ) u − (⃗ p ) = ( 0 ,φ ∗ a ) , and a = − φ a φ ∗ also ¯ p a ˙ a ˙ ˙ Bryan Larios 3 body Stop decay with LSP gravitino/goldstino in the f.s.

Index Motivation The spinor helicity formalism The spinor helicity formalism (SHF) (a pragmatic point of view) Monophoton signal in LSP gravitino production Conclusions The key of the SHF is to considerate φ a as the fundamental object and express the 4-moments of the particles in terms of φ a . Notation If p and k are two 4-momentos and φ a , κ a their corresponding spinors, we can define the following products of spinors: k ) = − [ kp ] p ) u − (⃗ [ pk ] = φ a κ a = ¯ u + (⃗ (2) similarly, we also have k ) = − ⟨ kp ⟩ p ) u + (⃗ a κ ∗ ˙ a = ¯ ⟨ pk ⟩ = φ ∗ u − (⃗ (3) ˙ Bryan Larios 3 body Stop decay with LSP gravitino/goldstino in the f.s.

Index Motivation The spinor helicity formalism The spinor helicity formalism (SHF) (a pragmatic point of view) Monophoton signal in LSP gravitino production Conclusions Polarizations 4-vectores The contraction with γ matrix is as follows: : E + ( k,q ) = / √ ⟨ q ∣ γ µ ∣ k ] γ µ = 2 √ (∣ k ]⟨ q ∣ + ∣ q ⟩[ k ∣) 1 2 ⟨ qk ⟩ 2 √ ⟨ qk ⟩(∣ k ]⟨ q ∣ + ∣ q ⟩[ k ∣) , 2 (4) = E − ( k,q ) = 1 1 / √ [ q ∣ γ µ ∣ k ⟩ γ µ = √ ⟨ k ∣ γ µ ∣ q ] γ µ 2 [ qk ] 2 [ qk ] √ [ qk ](∣ k ]⟨ q ∣ + ∣ q ⟩[ k ∣) . 2 (5) = Bryan Larios 3 body Stop decay with LSP gravitino/goldstino in the f.s.

Index Motivation The spinor helicity formalism The spinor helicity formalism (SHF) (a pragmatic point of view) Monophoton signal in LSP gravitino production Conclusions Before to start with our computation, I will show all the formulas that are needed to compute the scattering amplitudes: [ ij ] = − [ ji ] , ⟨ i ∣ γ µ ∣ j ] = [ j ∣ γ µ ∣ i ⟩ , ⟨ i ∣ γ µ ∣ j ]⟨ k ∣ γ µ ∣ l ] = 2 ⟨ ik ⟩[ lj ] ⟨ ij ⟩ = [ ji ] ∗ , ⟨ ab ⟩⟨ cd ⟩ = ⟨ ac ⟩⟨ bd ⟩ + ⟨ ad ⟩⟨ cb ⟩ , ⟨ ij ⟩[ ji ] = ⟨ ij ⟩⟨ ij ⟩ ∗ = ∣⟨ ij ⟩∣ 2 , n ⟨ ij ⟩[ ji ] = − 2 k i ⋅ k j = s ij , ∑ ⟨ ik ⟩[ kj ] = 0 , k = 1 Bryan Larios 3 body Stop decay with LSP gravitino/goldstino in the f.s.

Index Motivation The spinor helicity formalism Computing the amplitudes with SPH Monophoton signal in LSP gravitino production Conclusions e + e − → γ ˜ G ˜ G Using the SUSY QED model constructed by Mawatari and Oexl (arXiv:1402.3223v2), and applying the SHF we will compute the scattering amplitude for the e + e − → γ ˜ G ˜ G reaction. It is important to mention that the cross sections for this reaction has been computed numerically, so it is interesting have an analytical result. Bryan Larios 3 body Stop decay with LSP gravitino/goldstino in the f.s.

Index Motivation The spinor helicity formalism Computing the amplitudes with SPH Monophoton signal in LSP gravitino production Conclusions Feynman Diagrams Figure: Feynman Diagrams for e + e − → γ ˜ G ˜ G Bryan Larios 3 body Stop decay with LSP gravitino/goldstino in the f.s.

Index Motivation The spinor helicity formalism Computing the amplitudes with SPH Monophoton signal in LSP gravitino production Conclusions e + e − → γ ˜ G ˜ G We have 5 Feynman diagrams each on them with 5 external (massless) particles, each particle have two helicity states ( ± ), in principle we need to compute 2 5 helicity amplitudes for each diagram. Bryan Larios 3 body Stop decay with LSP gravitino/goldstino in the f.s.

Index Motivation The spinor helicity formalism Computing the amplitudes with SPH Monophoton signal in LSP gravitino production Conclusions Feynman Diagrams Figure: All the helicity amplitudes Bryan Larios 3 body Stop decay with LSP gravitino/goldstino in the f.s.

Index Motivation The spinor helicity formalism Computing the amplitudes with SPH Monophoton signal in LSP gravitino production Conclusions e + e − → γ ˜ G ˜ G One of the several and marvelous advantages of the helicity amplitudes is that it is easy to identify the symmetries as well as the null helicity amplitudes. We already know that the terms ⟨ xy ] and [ xy ⟩ are zero, a small program could help us to find which helicity amplitude is zero. A a λ 1 λ 1 λ 2 λ 3 λ 4 λ 5 ≈ ( ¯ v λ 2 ( 2 )/ ǫ λ 3 ( 3 )/ qu λ 1 ( 1 ) ¯ u λ 5 ( 5 ) v λ 4 ( 4 )) , (6) A a −+−−+ ≈ ( ¯ v + ( 2 )/ ǫ − ( 3 )/ qu − ( 1 ) ¯ u + ( 5 ) v − ( 4 )) , (7) A a −+−−+ ≈ [ 54 ⟩ = 0 . (8) Bryan Larios 3 body Stop decay with LSP gravitino/goldstino in the f.s.

Index Motivation The spinor helicity formalism Computing the amplitudes with SPH Monophoton signal in LSP gravitino production Conclusions Counting helicity amplitudes Figure: Counting helicity amplitudes Bryan Larios 3 body Stop decay with LSP gravitino/goldstino in the f.s.