Scattering Amplitudes LECTURE 1 Jaroslav Trnka Center for Quantum - PowerPoint PPT Presentation

Scattering Amplitudes LECTURE 1 Jaroslav Trnka Center for Quantum Mathematics and Physics (QMAP), UC Davis ICTP Summer School, June 2017

Particle experiments: our probe to fundamental laws of Nature

Theorist’s perspective: scattering amplitude Initial Final states states

Phenomenology Tool how to learn Initial Final about the dynamics: states states interactions, theories, symmetries

What does the blob really represent?

What does the blob really represent? but there is more than that…..

It can be for example a sum of different pictures _ 2 _ _ 1 2 _ _ _ _ 2 3 1 3 3 _ 6 + + + _ _ _ + + + 1 4 6 + 4 4 5 + + + + 6 5 5 (c)

And in a special case even something more surprising 2 1 3 6 7 5 4

Overview of lectures ✤ Lecture 1: Review of scattering amplitudes Motivation On-shell amplitudes Kinematics of massless particles ✤ Lecture 2: New methods for amplitudes Recursion relations for tree-level amplitudes Unitarity methods for loop amplitudes On-shell diagrams ✤ Lecture 3: Geometric formulation Toy model: N=4 SYM theory Positive Grassmannian Amplituhedron

Motivation

Quantum Field Theory (QFT) ✤ Our theoretical framework to describe Nature ✤ Compatible with two principles Special relativity Quantum mechanics

Perturbative QFT (Dirac, Heisenberg, Pauli; Feynman, Dyson, Schwinger) ✤ Fields, Lagrangian, Path integral Z D A D ψ D ψ e iS ( A, ψ , ψ ,J ) 4 F µ ν F µ ν + i ψ 6 D ψ � m ψψ L = � 1 ✤ Feynman diagrams: pictures of particle interactions Perturbative expansion: trees, loops

Great success of QFT ✤ QFT has passed countless tests in last 70 years ✤ Example: Magnetic dipole moment of electron g e = 2 Theory: 1928 g e ∼ 2 Experiment:

Great success of QFT ✤ QFT has passed countless tests in last 70 years ✤ Example: Magnetic dipole moment of electron g e = 2 . 00232 Theory: 1947 g e = 2 . 0023 Experiment:

Great success of QFT ✤ QFT has passed countless tests in last 70 years ✤ Example: Magnetic dipole moment of electron g e = 2 . 0023193 1957 Theory: g e = 2 . 00231931 1972 Experiment:

Great success of QFT ✤ QFT has passed countless tests in last 70 years ✤ Example: Magnetic dipole moment of electron g e = 2 . 0023193044 Theory: 1990 g e = 2 . 00231930438 Experiment:

Dualities ✤ At strong coupling: perturbative expansion breaks ✤ Surprises: dual to weakly coupled theory Gauge-gauge dualities (Montonen-Olive 1977, Seiberg-Witten 1994) Gauge-gravity duality (Maldacena 1997)

Incomplete picture ✤ Our picture of QFT is incomplete ✤ Also, tension with gravity and cosmology If there is a new way of thinking about QFT, it must be seen even at weak coupling ✤ Explicit evidence: scattering amplitudes

Colliders at high energies ✤ Proton scattering at high energies LHC - gluonic factory ✤ Needed: amplitudes of gluons for higher multiplicities gg → gg . . . g

Early 80s ✤ Status of the art: gg → ggg Brute force calculation 24 pages of result ( k 1 · k 4 )( ✏ 2 · k 1 )( ✏ 1 · ✏ 3 )( ✏ 4 · ✏ 5 )

New collider ✤ 1983: Superconducting Super Collider approved ✤ Energy 40 TeV: many gluons! ✤ Demand for calculations, next on the list: gg → gggg

Parke-Taylor formula (Parke, Taylor 1985) ✤ Process gg → gggg ✤ 220 Feynman diagrams, 100 pages of calculations ∼ ✤ 1985: Paper with 14 pages of result

Parke-Taylor formula ✤ Process gg → gggg ✤ 220 Feynman diagrams, 100 pages of calculations ∼

Parke-Taylor formula

Parke-Taylor formula ✤ Within a year they realized Spinor-helicity variables p µ = σ µ a λ a ˜ h 12 i 3 λ ˙ M 6 = a ˙ a h 12 i = ✏ ab � (1) a � (2) h 23 ih 34 ih 45 ih 56 ih 61 i b � (1) � (2) b ˜ a ˜ [12] = ✏ ˙ a ˙ ˙ ˙ b (Mangano, Parke, Xu 1987)

Parke-Taylor formula m Fermi National Accelerator Laboratory FERMILAB-Pub-86/42-T March, 1986 ✤ Within a year they realized AN AMPLITUDE FOR n GLUON SCATTERING h 12 i 3 M n = h 23 ih 34 ih 45 i ... h n 1 i STEPHEN 3. PARKE and T. R. TAYLOR Fermi National Accelerator Laboratory P.O. Box 500, Batavia, IL 60510. Abstract A non-trivial, squared helicity amplitude is given for the scattering of an arbitrary number of gluons to lowest order in the coupling constant and to leading order in the number of colors. *rated by Unlversitles Research Association Inc. under contract with the United States Department 01 Energy

Problems with Feynman diagrams ✤ Particles on internal lines are not real Individual diagrams not gauge invariant ✤ Obscure simplicity of the final answer Most of the terms in each diagram cancels ✤ Lesson: work with gauge invariant quantities with fixed spin structure

Birth of amplitudes ✤ New field in theoretical particle physics New methods and Uncovering new effective calculations structures in QFT “Road map” Explicit New structure New method calculation discovered which exploits it

What are scattering amplitudes

Scattering process ✤ Interaction of elementary particles ✤ Initial state and final state | i i | f i ✤ Scattering amplitude M if = h i | f i e + e − → e + e − e + e − → γγ ✤ Example: or etc. Z |M| 2 d Ω ✤ Cross section: probability σ =

Scattering amplitude in QFT ✤ Scattering amplitude depends on types of particles and their momenta M if = F ( p i , s i ) ✤ Theoretical framework: calculated in some QFT ✤ Specified by Lagrangian: interactions and couplings L = L ( O j , g k ) ✤ Example: QED L int = e ψγ µ ψ A µ

Perturbation theory ✤ Weakly coupled theory M = M 0 + g M 1 + g 2 M 2 + g 3 M 3 + . . . ✤ Representation in terms of Feynman diagrams ✤ Perturbative expansion = loop expansion M = M tree + M 1 − loop + M 2 − loop + . . .

Divergencies ✤ Loop diagrams are generally UV divergent Z ∞ d 4 ` ( ` 2 + m 2 )[( ` + p ) 2 + m 2 ] ∼ log Λ ∼ −∞ ✤ IR divergencies: physical effects, cancel in cross section ✤ Dimensional regularization: calculate integrals in 4 + ✏ dimensions Divergencies ∼ 1 ✏ k

Renormalizable theories ✤ Absorb UV divergencies: counter terms Finite number of them: renormalizable theory Infinite number: non-renormalizable theory ✤ Mostly only renormalizable theories are interesting ✤ Exceptions: effective field theories Example: Chiral perturbation theory - derivative expansion L = L 2 + L 4 + L 6 + L 8 + . . . Different loop orders are mixed

Analytic structure of amplitudes ✤ Tree-level: rational functions g 2 Only poles ∼ ( p 1 + p 2 ) 2 ✤ Loops: polylogarithms and more complicated functions Branch cuts ∼ log 2 ( s/t )

Kinematics of massless particles

Massless particles ✤ Parameters of elementary particles of spin S Spin s = ( − S, S ) On-shell (physical) particle Mass m p 2 = m 2 Momentum p µ p 2 = 0 ✤ Massless particle: m = 0 spin = helicity: only two extreme values h = { − S, S } h = (+ , − ) Example: photon s = 0 missing

Spin functions ✤ At high energies particles are massless Fundamental laws reveal there ✤ Spin degrees of freedom: spin function s=0: Scalar - no degrees of freedom s=1/2: Fermion - spinor u s=1: Vector - polarization vector ✏ µ s=2: Tensor - polarization tensor h µ ν

Spin functions ✤ At high energies particles are massless Fundamental laws reveal there ✤ Spin degrees of freedom: spin function s=0: Scalar - no degrees of freedom s=1/2: Fermion - spinor u s=1: Vector - polarization vector ✏ µ s=2: Tensor - polarization tensor h µ ν

Polarization vectors ✤ Spin 1 particle is described by vector ✏ µ 2 degrees of freedom 4 degrees of freedom ✏ · ✏ ∗ = 0 ✤ Null condition: 3 degrees of freedom left ✤ We further impose: ✏ · p = 0 Identification ✏ µ ∼ ✏ µ + ↵ p µ Feynman diagrams depend on α gauge dependence

Spinor helicity variables ✤ Standard SO(3,1) notation for momentum p µ = ( p 0 , p 1 , p 2 , p 3 ) p j ∈ R p 2 = p 2 0 + p 2 1 + p 2 2 − p 2 ✤ We use SL(2,C) representation 3 ✓ p 0 + ip 1 ◆ p 2 + p 3 p ab = σ µ ab p µ = p 2 − p 3 p 0 − ip 1 p 2 = det( p ab ) = 0 On-shell: Rank ( p ab ) = 1

Spinor helicity variables ✤ We can then write p ab = λ a κ b b = λ a e ✤ SL(2,C): dotted notation λ ˙ p a ˙ b e λ is complex conjugate of λ ✤ Little group transformation λ → t λ leaves momentum p → p λ → 1 e e unchanged λ t 3 degrees of freedom

Spinor helicity variables ( p 1 + p 2 ) 2 = ( p 1 · p 2 ) ✤ Momentum invariant p µ 2 = σ µ b λ 2 b e a λ 1 a e p µ 1 = σ µ λ 2˙ λ 1˙ b ˙ b a ˙ a ✤ Plugging for momenta ( p 1 · p 2 ) = ( σ µ b ) ( λ 1 a λ 2 b )( e a e b ) a σ µ b ˙ λ 1˙ λ 2˙ a ˙ b e a e = ( ✏ ab � 1 a � 2 b )( ✏ ˙ b ) � 1˙ � 2˙ a ˙ ✏ ab ✏ ˙ a ˙ b b e a e h 12 i ⌘ ✏ ab � 1 a � 2 b Define: [12] ≡ ✏ ˙ � 1˙ � 2˙ a ˙ b