Scattering Amplitudes LECTURE 2 Song He Institute of

Scattering Amplitudes LECTURE 2 Song He Institute of Theoretical Physics, Chinese Academy of Sciences , Beijing

Tree level amplitudes • Tree-level amplitude is a rational function of kinematics momenta polarization vectors Feynman propagators • Only poles, no branch cuts (why tree level special!)

Analytic properties of tree amplitudes •

Analytic properties of tree amplitudes • Feynman diagrams with this propagators recombine on both sides into amplitudes; sum over all possible internal states

Factorization on the pole •

On-shell constructibility • Question: Is knowing all residues on all poles enough to specify the amplitude? • Naively, this is false for Yang-Mill theory gg → gg four point amplitude Contact term Imposing gauge invariance fixes it

On-shell constructibility • Only function which factorizes properly on all poles is the amplitude.

Soft and collinear singularities • The locality requirement does not exclude poles of the form or one can always write There arise from special limits of external momenta: the so-called soft and collinear singularities.

Soft limit

Soft theorems For a soft photon in QED For a soft graviton in GR Soft theorems are very important! More so in recent years: (1) some subleading corrections also universal (2) they are shown to be related to asymptotic symmetries, e. g. BMS (Strominger et al)

Soft theorem: gluons Using color-decompositions, one gets soft theorems for color-ordered amplitudes: with the soft factor depends only on adjacent particles!

Soft theorems: spinor-helicity • The gluon soft factor: • The graviton soft factor: Independent of the helicity or any quantum number of hard particles: the soft emission is longwavelength (IR) and intrinsically classical. Gluon soft factor: not even depend on magnitude of hard momenta, only angular directions (classically conformal invariance: loop level corrections needed) Graviton soft factor: gauge invariance by momentum-conservation (universal to all loops)

Soft theorem: examples

Collinear limits

Collinear limits The amplitude also factorizes as two external momenta become parallel, or collinear! It is singular since the intermediate momentum becomes on-shell: Important: splitting functions for the collinear singularities (here @ tree level)

Collinear limits: examples

Collinear limits: examples

Collinear limits: splitting functions gluon-gluon-quark parity

Spinor magic revisited •

Spinor magic revisited •

Recursion relations

Momentum shift • Let us shift two external momenta • Momentum is conserved, stays on-shell This corresponds to shifting

Shifted amplitude • On-shell tree-level amplitude with shifted kinematics • Analytic structure • Location of poles:

Shifted amplitude • On the pole if • Shifted amplitude: location of poles

Residue theorem • Shifted amplitude • Let us consider the contour integral • Original amplitude • Residue theorem: Residue at z = 0

Residue theorem • Residues on the pole • Unitarity of shifted tree-level amplitude

Residue theorem • Residues on the pole • Unitarity of shifted tree-level amplitude

Residue theorem Final formula

BCFW recursion relations (Britto, Cachazo, Feng, Witten, 2005) Chosen such that internal line is on-shell Sum over all distributions of legs keeping 1, 2 on different sides

BCFW recursion relations (Britto, Cachazo, Feng, Witten, 2005) For ordered amplitudes A(123. . . n) Also sum over helicities of internal particle Sum over all distributions of legs keeping 1, 2 on different sides

Large z behavior

Large z behavior • each FD~1/z, additional z-dependence from polarizations of legs 1 and 2 • naively (the worst behavior):

Large z behavior

Example 1: 4 pt amplitude • Only one term contributes z takes the value when P is on-shell momentum

Example 1: 4 pt amplitude • Use of momentum conservation

Example 1: 4 pt amplitude •

Example 1: 4 pt amplitude •

Example 1: 4 pt amplitude • One gauge invariant object equivalent to three Feynman diagrams

only the first diagram:

Example 3: 6 pt amplitude • vs 220 Feynman diagrams

Example 3: 6 pt amplitude • Spurious pole

Example 3: 6 pt amplitude Comparing with results from off-shell recursion

Remarks on BCFW • Extremely efficient (3 vs 220 for 6 pt, 20 vs 34300 for 8 pt) • Terms in BCFW recursion relations • Gauge invariant • Spurious poles • General idea (later generalized to loops): uses on-shell building blocks only amplitude = sum of them dictated by unitarity • Only a subset of factorization channels are present: highly non-trivial that it gives correct factorizations for other channels (when 1, 2 are on the same side)

Supersymmetry and N=4 SYM

Supersymmetry • (even if it does not exist in nature) deep and powerful ideas for QFT & strings • extremely useful tool for study amplitudes in gauge theories and gravity • in particular, the cousin of QCD -> N=4 SYM: “harmonic oscillator in 21 st century” • Basic idea: consider N=1 chiral model SUSY transformation

Supersymmetry Free-field expansion SUSY transformation generated by anti-commuting Majorana-spinor supercharges

Supersymmetry Ward identities

N= 1 supersymmetry: gauge theory N=1 Super-Yang-Mills covariant derivative

Supersymmetry Ward identities gives Similarly for the gluon amplitude with one negative helicity Tree-level: no difference for YM and SYM gluon amplitudes (similar results for gravity)

Supersymmetry Ward identities In N=1 SYM, supersymmetry Ward identities gives Relate to MHV gluon amplitude Similar but more complicated SWI’s for N^k MHV amplitudes (exercise: 6 pt NMHV) Extended SUSY: N copies of SUSY generators, e. g. N=2 SYM

N= 4 SYM

N=4 SUSY Ward identity N=4 supermultiplet is CPT self-conjegate; the action of supercharges on the operators: SUSY Wald identities (also for YM trees!)

N= 4 superspace •

Superamplitudes • All usual (“component”) amplitudes are contained in the expansion, e. g. • They can be projected out using corresponding Grassmann operators, e. g. • Tree-level: YM gluon amps= SYM gluon amps (more complicated for QCD amps) All tree-level gluon helicity amplitudes are unified into a single object!

Superamplitudes • SUSY Ward identities <=> supercharges annihilate the superamplitudes • Implies momentum conservation: recall that amps actually contain delta functions • Similarly means amps contain Grassmann delta functions, e. g. • Exception: 3 pt anti-MHV (k=-1, degree 4):