EFTs from the soft limits of scattering amplitudes Jaroslav Trnka - PowerPoint PPT Presentation

EFTs from the soft limits of scattering amplitudes Jaroslav Trnka Center for Quantum Mathematics and Physics (QMAP) University of California, Davis with Clifford Cheung, Karol Kampf, Jiri Novotny, Chia-Hsien Shen, Congkao Wen Amplitudes in the LHC era, GGI, October 31, 2018

Motivation ✤ Tree-level amplitudes of massless particles in EFTs ✤ Normally not considered: bad powercounting, problems with loops ✤ Standard procedure: Lagrangian Symmetry Properties of amplitudes

Motivation ✤ In this talk: opposite approach Start with generic Lagrangian with free couplings = free parameters in the amplitude Impose kinematical constraints: fix all parameters Find corresponding theory Construct recursion relations to calculate amplitudes ✤ Classify interesting EFTs, perhaps find some new ones ✤ It is easier to impose kinematical constraints on amplitudes than to search in space of all symmetries

T ypical example ✤ Single scalar L = 1 2( ∂φ ) 2 + c 4 ( ∂φ ) 4 + c 6 ( ∂φ ) 6 + c 8 ( ∂φ ) 8 + . . . Is there a symmetry which fixes relates these couplings? ✤ 6pt amplitude ( . . . ) X c 2 A 6 = + c 6 ( . . . ) 4 s 123 A 6 = Impose kinematical condition on A 6

EFT setup

Three point interactions ✤ Consider scalar field theory given by L = 1 2( ∂φ ) 2 + L int ( φ , ∂φ , . . . ) ✤ Simplest interaction is 3pt but there are no 3pt L int = λφ 3 amplitudes except for ✤ Any derivatively coupled term can be written as L int = ( ⇤ φ )( . . . ) and removed by EOM

Fundamental interaction ✤ Let us start with a 4pt interaction term L int = λ 4 ( ∂ m φ 4 ) many terms ✤ Four point amplitude: special kinematics ✤ Six point amplitude: presence of contact terms Powercounting ∂ m ∂ m = ∂ 2 m − 2 L 6 = ∂ 2 m − 2 φ 6 ∂ 2 ✤ For no contact terms possible L int = λ 4 φ 4

EFT setup ✤ We consider the infinite tower of terms L = 1 2( ∂φ ) 2 + λ 4 ( ∂ m φ 4 ) + λ 6 ( ∂ 2 m − 4 φ 6 ) + . . . ✤ Even if we start with the 4pt term we can do field redefinitions and generate infinite tower ✤ We get a generic amplitude A n ( λ 4 , λ 6 , . . . ) ✤ Find constraints which uniquely specifies all couplings

On-shell constructibility ✤ On the pole the amplitude must factorize ✤ Contact terms vanish on all poles: not detectable ✤ Therefore, EFT amplitudes are not specified only by factorization - unfixed kinematical terms ∼ ( s 12 + s 123 ) s 56 s 12 s 56 on the pole s 123 s 123

On-shell constructibility ✤ Naively, this problem arises also in YM theory ✤ In fact, the contact terms there is completely fixed Contact term Imposing gauge invariance fixes it ✤ In our case, contact terms are unfixed with free parameters, there is no gauge invariance

Extra constraints ✤ If we want to fix the amplitude completely we have to impose additional constraints! ✤ It must link the contact terms to factorization terms None of them individually satisfy condition X ✤ Natural condition for EFTs at low energies Soft limit p → 0

Simplest case

Free theory ✤ Single scalar field φ ✤ Minimal derivative coupling L = 1 2( ∂φ ) 2 + c 4 φ 2 ( ∂φ ) 2 + c 6 φ 4 ( ∂φ ) 2 + . . . ✤ Looks like interesting interacting theory but it is not free theory with L = 1 all amplitudes X s ij = 0 2( ∂φ ) 2 φ → F ( φ ) are zero ij

Non-trivial example ✤ Multiple scalars φ = φ a T a ✤ Write the same Lagrangian: now it is not just free L = 1 2( ∂φ ) 2 + c 4 φ 2 ( ∂φ ) 2 + c 6 φ 4 ( ∂φ ) 2 + . . . traces, more couplings ✤ We can do “color”- ordering (Kampf, Novotny, Trnka, 2013) Tr( T a 1 T a 2 . . . T a n ) A (123 . . . n ) X A n = σ

Non-trivial example ✤ Example: six point amplitude 4 3 3 4 X A 6 = important: 5 5 2 2 same power-counting cycl 1 6 1 6 p 2 × p 2 A 6 ∼ c 2 + c 6 p 2 4 p 2 A 6 → 0 ✤ Impose: vanishing in soft limit for p → 0 c 6 ∼ c 2 fixes 4

Non-linear sigma model (Weinberg 1966) A n → 0 ✤ Continue to higher points: for fixes all coefficients and gives a unique theory (up to a gauge group) p → 0 (Susskind, Frye 1970) L = F 2 F φ a T a i 2 h ( ∂ µ U )( ∂ µ U ) i where U = e SU(N) non-linear sigma model ✤ Symmetry explanation: shift symmetry Low energy QCD φ → φ + a

Uniqueness in minimality ✤ When renormalizing the SU(N) non-linear sigma model we need higher derivative terms L χ P T = L 2 + L 4 + L 6 + . . . ( ∂ µ U )( ∂ µ U ) ( ∂ µ ∂ ν U )( ∂ µ ∂ ν U ) etc [( ∂ µ U )( ∂ ν U )] 2 ✤ They all have just a soft-limit vanishing ✤ Only the minimal coupling (NLSM) is uniquely fixed

Exceptional theories (Cheung, Kampf, Novotny, JT 2014)

Single scalar ✤ The first non-trivial is the original example L = 1 2( ∂φ ) 2 + c 4 ( ∂φ ) 4 + c 6 ( ∂φ ) 6 + c 8 ( ∂φ ) 8 + . . . ( ∂φ ) 2 n = [( ∂ µ φ )( ∂ µ φ )] n ✤ Calculate 6pt amplitude 3 4 3 4 X A 6 = 2 5 2 5 σ 1 6 1 6 ( s 12 s 23 + s 23 s 13 + s 12 s 13 )( s 45 s 56 + s 45 s 46 + s 46 s 56 ) X c 2 = + c 6 s 12 s 34 s 56 = 4 4 s 123 σ Lagrangian trivially invariant trivial soft-limit vanishing p i → 0 ↔ φ → φ + a

Single scalar ✤ The first non-trivial is the original example L = 1 2( ∂φ ) 2 + c 4 ( ∂φ ) 4 + c 6 ( ∂φ ) 6 + c 8 ( ∂φ ) 8 + . . . ( ∂φ ) 2 n = [( ∂ µ φ )( ∂ µ φ )] n ✤ Calculate 6pt amplitude 3 4 3 4 X A 6 = 2 5 2 5 σ 1 6 1 6 ( s 12 s 23 + s 23 s 13 + s 12 s 13 )( s 45 s 56 + s 45 s 46 + s 46 s 56 ) X c 2 = + c 6 s 12 s 34 s 56 = 4 4 s 123 σ p i → tp i Impose quadratic A 6 → O ( t 2 ) t → 0 vanishing

Single scalar ✤ The first non-trivial is the original example L = 1 2( ∂φ ) 2 + c 4 ( ∂φ ) 4 + c 6 ( ∂φ ) 6 + c 8 ( ∂φ ) 8 + . . . ( ∂φ ) 2 n = [( ∂ µ φ )( ∂ µ φ )] n ✤ Calculate 6pt amplitude 3 4 3 4 X A 6 = 2 5 2 5 σ 1 6 1 6 ( s 12 s 23 + s 23 s 13 + s 12 s 13 )( s 45 s 56 + s 45 s 46 + s 46 s 56 ) X c 2 = + c 6 s 12 s 34 s 56 = 4 4 s 123 σ c 6 = 4 c 2 There is a single solution and it fixes: 4

Single scalar ✤ The first non-trivial is the original example L = 1 2( ∂φ ) 2 + c 4 ( ∂φ ) 4 + c 6 ( ∂φ ) 6 + c 8 ( ∂φ ) 8 + . . . ( ∂φ ) 2 n = [( ∂ µ φ )( ∂ µ φ )] n ✤ Apply to higher point amplitudes p i → tp i A n = O ( t 2 ) for t → 0 ✤ Cancelations between diagrams required, a unique c 2 n ∼ c # solutions exists and relates 4

Single scalar ✤ The Lagrangian becomes L = 1 2( ∂φ ) 2 + c 4 ( ∂φ ) 4 + 4 c 2 4 ( ∂φ ) 6 + 20 c 3 4 ( ∂φ ) 8 + . . . ( ∂φ ) 2 n = [( ∂ µ φ )( ∂ µ φ )] n ✤ Apply to higher point amplitudes p i → tp i A n = O ( t 2 ) for t → 0 ✤ Cancelations between diagrams required, a unique c 2 n ∼ c # solutions exists and relates 4

Single scalar ✤ The Lagrangian becomes L = − 1 p 1 − g ( ∂φ ) 2 where g = 8 c 4 g ✤ Apply to higher point amplitudes p i → tp i A n = O ( t 2 ) for t → 0 ✤ Cancelations between diagrams required, a unique c 2 n ∼ c # solutions exists and relates 4

Result: DBI action (Dirac, Born, Infeld 1934) ✤ The Lagrangian becomes L = − 1 p 1 − g ( ∂φ ) 2 where g = 8 c 4 g ✤ It describes the fluctuation of D-dimensional brane in (D+1) dimensions φ ✤ What is the symmetry principle behind this?

Result: DBI action (Dirac, Born, Infeld 1934) ✤ Symmetry of the action: (D+1) Lorentz symmetry φ → φ + ( b · x ) + ( b · φ∂φ ) ✤ It can be shown that this implies the soft limit behavior ✤ But we can also derive the action based on the soft limit p 2 L 0 ( X ) /g = 2 X L 0 ( X ) − L ( X ) L ( X ) ∼ 1 − gX → X = ( ∂φ ) 2 where

Galileon ✤ Let us consider the next Lagrangian 2 ( ∂φ ) 2 + λ 4 ( ∂ 6 φ 4 ) + λ 6 ( ∂ 10 φ 6 ) + . . . L 2 = 1 ✤ Calculate amplitudes: impose again A n = O ( t 2 ) Galileons Fully specifies a family of solutions Relevant for φ → φ + a + ( b · x ) Galilean symmetry cosmological models ✤ There are (d-2) Lagrangians: L n = φ det[ ∂ µ j ∂ ν k φ ] n n ≤ d j,k =1

Special Galileon ✤ Not enough for us: not minimal, not unique ✤ We impose even stronger condition p i → tp i A n = O ( t 3 ) for t → 0 ✤ And there exists an unique solution, linear combination of Galileon Lagrangians: we called it special Galileon ✤ No symmetry explanation at that time

Special Galileon ✤ Not enough for us: not minimal, not unique ✤ We impose even stronger condition p i → tp i A n = O ( t 3 ) for t → 0 ✤ And there exists an unique solution, linear combination of Galileon Lagrangians: we called it special Galileon φ → s µ ν x µ x ν + λ 4 12 s µ ν ( ∂ µ φ )( ∂ ν φ )

Classification ✤ Use soft-limit as classification tool (Cheung, Kampf, Novotny, Shen, JT 2016) (Elvang, Hadjiantonis, Jones, Paranjape 2018) ✤ No more interesting theories with 4pt vertices O ( t 4 ) no theory with non-trivial behavior ✤ Starting with 5pt vertices: WZW model but nothing more at higher points ✤ There are also analogues of DBI and Galileon for multiple scalars but nothing more

Recursion relations