the rise of on-shell smeft massive amplitudes€¦ · crash course on massive kinematics massive...

The Rise of On-Shell SMEFT Massive AmplitudesRafael Aoude and C. S. Machado

hep-ph/1905.11433

Rafael Aoude - JGU Mainz

EFTs are an useful way to parametrize new physics, while…

What can we learn building EFTs from Massive on-shell language?

On shell methods and Spinor Helicity formalism are efficient tools to calculate scattering amplitudes.

SMEFT three-point amplitudes in massive on-shell language

Recursion relations to built n-point amplitudes

!2

Putting EFTs On-Shell

Rafael Aoude - JGU Mainz !3

Putting EFTs On-Shell

The ideal path is … everything built from on-shell Amplitudes info

• Recursion Relations with EFTs • Symmetry breaking … • Is it constructible? • Soft and low-energy theorems..

More realistic first step:

feynamn rules on-shell massive define dim-6 basis with three-point

!

Recursive Relations and consistent fact. ! Build four-point amps


some (few) directions…

Exploring soft behavior massless EFTs and its RRs

how to build massless operators [Henning et al 2019] SMEFT massless amplitudes [Ma et al. 1029]

Exploring massive EFTs

also exploring massless SMEFT

subtracted recursion relations, soft bootstrap, modified soft theorems, RR for EFTs, etc. [Elvang et al. 2018, Cheung et al. 2015,2018, etc..]

Massive higgs + gluons amplitudes [Shadmi et. al 2019] Black holes S-matrix [Chung et. al 2018] Massive quarks [Ochirov et. al 2018]

… and a lot more


Crash course on Massless Kinematics

Spinor-helicity building blocks

Rank 1 matrix !det(p) = 0 ! p↵↵̇ = �↵�̃↵̇

p↵↵̇ = pµ�µ↵↵̇



Under each particle LG U(1)Spinor-helicity building blocks

polarization vectors are represented by the same obj.


p↵↵̇ = pµ�µ↵↵̇



Under each particle LG U(1)Spinor-helicity building blocks

polarization vectors are represented by the same obj.

Special kinematics and little group uniquely ….

|h| = 1� [g] For a dimension 6 coupling, the total helicity is ±3


p↵↵̇ = pµ�µ↵↵̇


Crash course on Massive Kinematics

[Nima et. al 2017]

Massive LG is SU(2) !det(p) = m2 ! p↵�̇ = �1↵�̃

1�̇+ �2

↵�̃2�̇= �I

↵✏IJ �̃J�̇

Rank 2 matrix !



Massive Polarization vector

hi jiIJ [i j]IJhi|p|j]IJmassive building blocks

[Nima et. al 2017]


1�̇+ �2

↵�̃2�̇= �I

↵✏IJ �̃J�̇

Rank 2 matrix !





[Nima et. al 2017]

Special kinematics and little group do not uniquely define

… and the three-point amplitude depends on the masses configuration

i.e, how many massive legs and degenerate masses


1�̇+ �2

↵�̃2�̇= �I

↵✏IJ �̃J�̇

Rank 2 matrix !





[Nima et. al 2017]

Special kinematics and little group do not uniquely define …

… and the three-point amplitude depends on the masses configuration

e.g: 1 scalar + 2 vectors, all massive

M(1h2I1,I2V 3J1,J2

V̄) = g0 h23i[23] + g1 h23i2 + g2 [23]

2

* assumed SU(2) indices symm.


1�̇+ �2

↵�̃2�̇= �I

↵✏IJ �̃J�̇

Rank 2 matrix !

i.e, how many massive legs and degenerate masses


Holomorphic SMEFT

We start with the purely bosonic SMEFT in the holomorphic form

+ Holom (H) - Anti-Holom (AH)

Non-Holom. (NH)

[Alonso et. al 2014]


Holomorphic SMEFT

…. here we want to fix the kinematic structure, map the feynman rules into the dimensionless functions

which encodes the symmetries and UV properties of the theory

We start with the purely bosonic SMEFT in the holomorphic form

+ Holom (H) - Anti-Holom (AH)

Non-Holom. (NH)


Some Higgs Couplings

SM interactions where

Some Gauge-Gauge couplings


Three-point Map

H/AH

NH

SM couplings Dim-6 Wilson Coefficients


Three-point Map

H/AH

NH


* depends on input scheme

• The holom. of the operators maps to holom. of the massive coeffs.

• Non holom. operators only enters in SM-like structures

• Inherit SU(2) structures from SMEFT lagrangian in our coeffs.


Three-point Map

H/AH

NH


To define a basis, choose 11 coeffs…


Three-point Map

H/AH

NH


* depends on input scheme


… and the others are related by

+ …


Three-point Map

H/AH

NH



… and the others are related by

… some SU(2) relations in on-shell coeffs.

Ideal will be get this from soft limits/unitarity, but we already know how it looks..


Shift a subset of the external momenta pi ! p̂i(z) = pi + z ⌘i

… such that the shifted p’s still on shell and mom. conserv.

What about 2 2 scattering amplitudes?!





M = �X

K

M̂L(zK)M̂R(zK)

p2K �m2+B1

M̂L M̂Rand are lower point amplitudes

Using Cauchy …

Only works if and usually this is the problem for EFTs.

B1 ! 0

B1 residue at infinity




Asking for consistent factorization helps to calculate the boundary term… in the massive case, the HE limit helps SM with the non-factorizable


M = �X

K

M̂L(zK)M̂R(zK)

p2K �m2+B1 Only works if and

usually this is the problem for EFTs.B1 ! 0

B1

M̂L M̂Rand are lower point amplitudes

Using Cauchy …

residue at infinity


Two factorization channels t and u:

WW̄��+ � SM

Let’s do it with the two massless legs

+

lower point amps


Two factorization channels t and u:

final amplitude

symm under 3,4 ; good naive large-z behavior; consistent factorization

WW̄��+ � SM

Let’s do it with the two massless legs

+


WW̄hh� SM

naive bad large-z behavior … “lack of massless polarization vectors”

… reconstruct with poles residues (gluing) and well behaved UV

4-pt contact int.

+ + +


WW̄hh� SM



4-pt contact int.

+ + +


WW̄hh� SM



4-pt contact int.

In the HE limit we have

fixing the contact

+ + +


What changes for BSM?

dim-6 operator insertion in both ‘Left’ and ‘Right’ amplitudes and in both t and u channels

Can appear new particle poles induced by the BSM operator

BCFW + consistent factorization still works in some cases but not a general procedure

+

Unitarity HE limit cannot be used for fixing contact terms


Conclusions and future directions

… kinematic structures fixed by Lorentz, LG and Bose but symmetries unknown

BCFW application for SM massive and discussion on SMEFT

Three point amplitude map between purely boson SMEFT and on-shell coefficients

+


Conclusions and future directions

n-point massive (mandarin) tree

• General method to obtain n-point massive amplitudes with higher dim ops

… kinematic structures fixed by Lorentz, LG and Bose but symmetries unknown

BCFW application for SM massive and discussion on SMEFT

Future?

• Everything purely from on-shell: 4-point test, unitarity, soft/collinear limits

• On-shell map for HEFT Lagrangian and its difference for SMEFT looking to some process with multi boson legs.

• Symmetry breaking understanding via on-shell methods.

Three point amplitude map between purely boson SMEFT and on-shell coefficients

+


Thank you

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