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Song He Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing
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Motivations: QFT and Amplitudes
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Quantum Field Theory (QFT) Our theoretical framework to describe Nature Essentially the consequence of two major principles
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Perturbative QFT Feynman diagrams: pictures of particle interactions Perturbative expansion: trees, loops
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Great success of QFT QFT has passed countless tests in last 70 years Example: Magnetic dipole moment of electron
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Incomplete picture Our understanding of QFT is incomplete! Also, tension with gravity and cosmology Explicit evidence: scattering amplitudes If there is a new way of thinking about QFT, it must be seen even at weak coupling
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Colliders at high energies Proton scattering at high energies Needed: amplitudes of gluons for higher multiplicities LHC - gluonic factory gg → gg … g
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Early 80s Status of the art: gg → ggg Brute force calculation 24 pages of result
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New collider 1983: Superconducting Super Collider approved Energy 40 TeV: many gluons! Demand for calculations, next on the list: gg→gggg
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Parke-Taylor formula Process : gg→gggg 220 Feynman diagrams, ~ 100 pages of calculations 1985: Paper with 14 pages of result (Parke, Taylor 1985)
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Parke-Taylor formula Within a year they realized
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Parke-Taylor formula Within a year they realized
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Birth of amplitudes
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Spinor-Helicity Formalism
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Lorentz group representations Spinors Infinitesimally with These are 6 generators of the Lorentz group: We see that the SO(1, 3) may be mapped to two commuting copies of SU(2)
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Spinors Weyl spinors Chiral solutions of the massless Dirac equation Basis e.g.
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Lower spin representations of 4-D Lorentz group Spinors
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Spinor-helicity formalism for massless particles For a massless particle
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An explicit realization Requiring the four-momentum to be real with Lorentz-signature translates into the relation The sign in this relation follows the sign of the energy of the associated four-momentum Spinor-helicity formalism for massless particles
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Spinor invariants By definition, Lorentz invariant objects= functions of spinor invariants (“angle” and “square” brackets) Mandelstam invariants For real momenta, we have invariants of the two SU(2)’s
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anti-symmetry Momentum conversation Schouten Identity Spinor invariants
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Rescaling freedom (little group) for real momenta for complex momenta Little group
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Under little group scaling, the amplitude transforms homogeneously. e.g. Little group
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U(1) generator of helicity Helicity Lorentz invariant quantity for massless particles
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Three point amplitudes which is a singular kinematic point for real momenta! Momentum conversation requires has two chirally conjugate solutions
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solution 1: for solution 2 Up to an overall constant Three point amplitudes The 3-pt amplitude is non-singular, and completely fixed by Poincare invariance! Little group scaling fixes
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Consider the special case of 3pt amps with identical spin (s) particles e.g. --+ or but the second case is obviously non-local! Poincare + locality → MHV three point amplitudes Three point amplitudes
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Gluon polarizations Basic properties (null, transverse, etc.) are automatically satisfied
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transforms like helicity +1 Gluon polarizations
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pick Gluon polarizations: 3 pt
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Graviton polarization ~ 200 terms
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The on-shell amplitude is remarkable simple (the square of gluon 3pt amplitudes!) Graviton polarization: 3 pt
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Unimaginable to calculate 4 or more gravitons amplitudes by brute force thousands of terms Graviton polarization: 4 pt
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Gauge-theory Amps: Helicity, Color etc.
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Helicity amplitudes Helicities of gluons (gravitons) +1 or -1 (+2 or -2) helicity amplitudes e.g. A(+++---), A(++----), A(--++++), … Huge simplifications, with different properties and structures! Number of negative-helicity gluons (gravitons): k k=0: A(+, +, +, …,+)=0 k=1: A(-, -, -, …, -)=0 k=n-1: A(-, +, +, …, +)=0 k=n: A(+, -, -, …, -)=0 both at tree level in general and For all loops with supersymmetries
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Vanishing tree amplitudes
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Similarly, the gluon tree-amplitude with one flipped helicity state vanishes which follows from the choice
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Simplest amplitudes: MHV
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MHV classification
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Non-Abelian Gauge Theories invariant under the local gauge transformation
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Feynman rules for non-abelian gauge theory
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Color decompositions organize color d.o.f.→ colors x kinematics → pieces with simpler analytic properties
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the SU(N c ) Fierz identity for simplifying the resulting traces photonphoton
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Reducing the color factor to a single color trace
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Different orderings related by permutations Gauge invariant This is a key object of our interest Color-summed cross section is indeed made up of color-ordered amplitudes Color-ordered amplitudes simpler pole structure
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Color ordered Feynman rules
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QCD amplitudes: e.g. similar decomposition for those with one fermion line
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Properties of color-ordered amplitudes Cyclicity Reflection U(1) decoupling
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Kleiss-Kuijf relations (fancy version of U(1) decoupling) Bern-Carrasco-Johansson relations
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Some Examples
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A simple four-point example little group check
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exchanges labels 1 and 2 Other helicity structures two other terms can be obtained by parity
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A five-point amplitude
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four gluon amplitude
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A nice choice of reference left only 2 diagrams vanish
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