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Scattering Amplitudes LECTURE 1

Jaroslav Trnka Walter Burke Institute for Theoretical Physics, Caltech Center for Quantum Mathematics and Physics (QMAP), UC Davis

Asian winter school, Okinawa, January 2016 The first instance of this phenomenon is extremely simple and trivial. Consider an analog of the “factorization channel” diagram (2.22), but connecting two black vertices. Because these vertices require that all the ’s be parallel, it makes no physical di↵erence how they are connected. And so, on-shell diagrams related by, e

(2.28)

represent the same on-shell form. Thus, we can collapse and re-expand any chain of connected black vertices in anyway we like; the same is obviously true for vertices. Because of this, for some purposes it may be useful to define composite black and white vertices with any number of legs. By grouping black and white vertices together in this way, on-shell diagrams can always be made bipartite—with (internal) edges only connecting white with black vertices. We will, however, preferentially draw trivalent diagrams because of the fundamental role played by the three- amplitudes. There is also a more interesting equivalence between on-shell diagrams that will play an important role in our story. We can see this already in the BCFW represen- tation of the four-particle amplitude given above, (2.20). The picture is obviously not cyclically invariant—as a rotation would exchange its black and white vertices. But the four-particle amplitude of course is cyclically invariant; and so there is another generator of equivalences among on-shell diagrams, the “square move”, [80]:

(2.29)

The merger and square moves can be used to show the physical equivalence of many seemingly di↵erentScattering on-shell diagrams. For amplitudes instance, the following two diagrams generate physically equivalent on-shell forms: 2

1 3 6 (2.30)

7 5

4

–16– Overview of lectures

✤ Review of scattering amplitudes Motivation On-shell amplitudes, kinematics Recursion relations, unitarity methods

✤ Geometric formulation Toy model: N=4 Super Yang Mills theory On-shell diagrams Amplituhedron Motivation (QFT)

✤ Our theoretical framework to describe

✤ Compatible with two principles

Special relativity Quantum mechanics Perturbative QFT (Dirac, Heisenberg, Pauli; Feynman, Dyson, Schwinger)

✤ Fields, Lagrangian, Path integral

iS(A, , ,J) = 1 F F µ⌫ + i D m A e 4 µ⌫ D D D L 6 Z ✤ 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

Theory: ge =2 1928 Experiment: ge 2 ⇠ Great success of QFT

✤ QFT has passed countless tests in last 70 years

✤ Example: Magnetic dipole moment of electron

Theory: ge =2.00232 1947 Experiment: ge =2.0023 Great success of QFT

✤ QFT has passed countless tests in last 70 years

✤ Example: Magnetic dipole moment of electron

1957 Theory: ge =2.0023193

1972 Experiment: ge =2.00231931 Great success of QFT

✤ QFT has passed countless tests in last 70 years

✤ Example: Magnetic dipole moment of electron

Theory: ge =2.0023193044 1990 Experiment: ge =2.00231930438 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 k )(✏ k )(✏ ✏ )(✏ ✏ ) 1 · 4 2 · 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 3 µ µ ˜ 12 p = aa˙ aa˙ 6 = h i (1) (2) 23 34 45 56 61 12 = ✏ a M h ih ih ih ih i h i ab b [12] = ✏ ˜(1)˜(2) a˙ b˙ 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

12 3 = h i Mn 23 34 45 ... n1 h ih ih i h 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 .

*rated by Unlversitles Research Association Inc. under contract with the United States Department 01 Energy Birth of amplitudes

✤ New field in theoretical

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

✤ Initial state i and final state f | i | i

✤ Scattering amplitude if = i f M h | i

+ + + ✤ Example: e e e e or e e etc. ! ! 2 ✤ : = d ⌦ probability |M| Z Scattering amplitude in QFT

✤ Scattering amplitude depends on types of particles and their momenta = F (p ,s ) Mif i i ✤ Theoretical framework: calculated in some QFT

✤ Specified by Lagrangian: interactions and couplings = ( ,g ) L L Oj k ✤ Example: QED = e Aµ Lint µ Perturbation theory

✤ Weakly coupled theory = + g + g2 + g3 + ... M M0 M1 M2 M3 ✤ Representation in terms of Feynman diagrams

✤ Perturbative expansion = loop expansion tree 1 loop 2 loop = + + + ... M M M M Divergencies

✤ Loop diagrams are generally UV divergent 4 1 d ` log ⇤ ⇠ (`2 + m2)[(` + p)2 + m2] ⇠ Z1

✤ IR divergencies: physical effects, cancel in cross section

✤ Dimensional regularization: calculate integrals in 4+✏ 1 dimensions Divergencies ⇠ ✏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 L2 L4 L6 L8 Different loop orders are mixed Analytic structure of amplitudes

✤ Tree-level: rational functions

g2 2 Only poles ⇠ (p1 + p2)

✤ Loops: polylogarithms and more complicated

log2(s/t) Branch cuts ⇠ Kinematics of massless particles Massless particles

✤ Parameters of elementary particles of spin S Spin s =( S, S) On-shell (physical) particle Mass m p2 = m2 Momentum pµ ✤ Massless particle: m =0 p 2 =0 spin = helicity: only two extreme values h = S, S { } Example: h =(+, ) 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 - vector ✏µ s=2: Tensor - polarization tensor hµ⌫ 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 hµ⌫ Polarization vectors

✤ Spin 1 particle is described by vector ✏µ

2 degrees of freedom 4 degrees of freedom

✤ Null condition: ✏ ✏ ⇤ =0 3 degrees of freedom left ·

✤ We further impose: ✏ p =0 Identification · ✏ ✏ + ↵p Feynman diagrams depend on ↵ µ ⇠ µ µ gauge dependence Spinor helicity variables

✤ Standard SO(3,1) notation for momentum µ p =(p0,p1,p2,p3) pj R 2 ✤ We use SL(2,C) representation p + ip p + p p = µ p = 0 1 2 3 ab ab µ p p p ip ✓ 2 3 0 1 ◆ 2 On-shell: p =det(pab)=0

Rank (pab)=1 Spinor helicity variables

✤ We can then write pab = ab

✤ SL(2,C): dotted notation pab˙ = ab˙

is complex conjugate of e ✤ Little groupe transformation t leaves momentum ! 1 p p unchanged ! ! t e e 3 degrees of freedom Spinor helicity variables

✤ Momentum invariant (p + p )2 =(p p ) 1 2 1 · 2 µ µ pµ = µ p1 = aa˙ 1a1˙a 2 bb˙ 2b 2b˙

✤ Plugging for momentae e

µ (p p )=( ˙ )( )( ˙ ) 1 · 2 aa˙ µbb 1a 2b 1˙a 2b

=(✏ )(✏ ˙ ˙ ) ✏ab ✏a˙ b˙ ab e1a e2b a˙ b 1˙a 2b

Define: 12 ✏ab1a2b [12] e✏ ˙ e1˙a ˙ h i⌘ ⌘ a˙ b 2b e e Invariant products

✤ Momentum invariant

s =(p + p )2 = ij [ij] ij i j h i Square brackets Angle brackets ij = ✏ [ij]=✏ h i ab ia jb a˙ b˙ ia˙ jb˙ ✤ Antisymmetry e e 21 = 12 [21] = [12] h i h i ✤ More momenta (p + p + p )2 = 12 [12] + 23 [23] + 13 [13] 1 2 3 h i h i h i Invariant products

✤ Shouten identity 13 24 = 12 34 + 14 23 h ih i h ih i h ih i ✤ Mixed brackets 1 2+34] 12 [24] + 13 [34] h | | ⌘h i h i ✤ Momentum conservation n Non-trivial conditions: iaia˙ =0 i=1 Quadratic relation X e between components Polarization vectors

✤ Two polarization vectors ⌘ µ µ ⌘aa˙ ✏µ = µ a a˙ ✏+ = aa˙ aa˙ ⌘ [⌘ ] Note that h ei e where are auxiliary spinors (✏+ ✏ )=1 ee · ✤ Freedom in choice of ⌘ , ⌘ corresponds to ✏µ ✏µ + ↵ pµ ⇠ e ✤ Gauge redundancy of Feynman diagrams Scaling of amplitudes

✤ Consider some amplitude A( + + ) ···

A =(✏1✏2 ...✏n) Q · depends only on momenta ✤ Little group scaling

p p 2 t ! A(i) t A(i) ! 1 ! · ✏+ ✏+ 1 ! t2 · 1 A(i+) A(i+) ! t ✏ t2 ✏ ! t2 · ! · e e Back to Parke-Taylor formula

+ + + + ✤ Let us consider A(123 4 5 6 )

✤ 1 2 Scaling A ti, i = t A(i, i) for particles 1,2 t · ✓ ◆ 1 1 A t , e = A( ,e ) for particles 3,4,5,6 i t i t2 · i i ✓ ◆ ✤ Check for explicit eexpression e If only ij allowed 12 3 h i A = h i the form is unique 6 23 34 45 56 61 h ih ih ih ih i Helicity amplitudes

✤ In Yang-Mills theory we have + or - “gluons” + + + + A6(123 4 5 6 )

✤ We denote k: number of - helicity gluons

✤ Some amplitudes are zero

An(+ + + +) = 0 First non-trivial: k=2 ··· An( ++ +) = 0 ··· An( + +) An( )=0 ··· A (+···)=0 Parke-Taylor formula for tree level n ··· Three point amplitudes Three point kinematics

✤ Gauge invariant building blocks: on-shell amplitudes 2 2 2 p1 = p2 = p3 =0 p1 + p2 + p3 =0 ✤ Plugging second equation into the first (p + p )2 =(p p )=0 1 2 1 · 2 ✤ Similarly we get for other pairs (p p )=(p p )=(p p )=0 1 · 2 1 · 3 2 · 3 These momenta are very constrained! Three point kinematics

✤ Use spinor helicity variables trivializes on-shell condition

p1 = 11,p2 = 22,p3 = 33

✤ The mutual conditionse then translatee to e (p p )= 12 [12] = 0 1 · 2 h i ✤ And similarly for other two pairs (p p )= 13 [13] = 0 (p p )= 23 [23] = 0 1 · 3 h i 2 · 3 h i Two s o l u t i o n s

✤ We want to solve conditions 12 [12] = 13 [13] = 23 [23] = 0 h i h i h i

1: 12 =0 which implies 2 = ↵1 h i Then we also have 23 = ↵ 13 h i h i And we set 13 =0 by demanding = h i 3 1 1 ⇠ 2 ⇠ 3 Two s o l u t i o n s

✤ Solution 2: [12] = [23] = [13] = 0

1 ⇠ 2 ⇠ 3

✤ Let us take this solutione e e p = ,p= ↵ ,p=( ↵ ) 1 1 1 2 2 1 3 1 2 1

e complexe momenta e No solution for real momenta Three point amplitudes

✤ Gauge theory: scattering of three gluons (not real)

✤ Building blocks: 12 , 23 , 13 , [12], [23], [13] h i h i h i ✤ Mass dimension: each term m ⇠ 3 ✤ Three point amplitude A3 ✏ p p m ⇠ ⇠ ⇠ Three point amplitudes

✤ Two options (1) (2) A = 12 a1 13 a2 23 a3 A = [12]b1 [13]b2 [23]b3 3 h i h i h i 3 + ✤ Apply to A3(1, 2, 3 )

1 a1+a2 A3(t1,t 1)=t A3 a1 + a2 =2 a1 =3 1 a +a · A (t ,t )=t 1 3 A a1 + a3 =2 a2 = 1 3 2 2 · 3 1e a2+a3 A3(t3,t 3)=t A3 a2 + a3 = 2 a3 = 1 e · e Three point amplitudes

+ + ✤ Similarly for A3(1 , 2 , 3)

✤ Two fundamental amplitudes

3 3 + 12 + + [12] A (1, 2, 3 )= h i A (1 , 2 , 3)= 3 13 23 3 [13][23] h ih i This is true to all orders: just kinematics Three point amplitudes

✤ Collect all ( +) amplitudes 3 + 12 A3(1, 2, 3 )= h i 13 23 4 h ih i ab 3 h i + 13 A (1, 2 , 3)= h i 12 23 31 3 12 23 h ih ih i h ih 3 i + 23 ) where a,b are - helicity gluons A (1 , 2, 3)= h i 3 12 13 h ih i [ab]4 ✤ Similarly for (+ + ) amplitudes [12][23][31]

where a,b are + helicity gluons Three point amplitudes

✤ Using similar analysis we find for gravity

ab 8 [ab]8 h i 12 2 23 2 31 2 [12]2[23]2[31]2 h i h i h i where a,b are - helicity gravitons where a,b are + helicity gravitons

✤ Note that there is no three-point scattering They exist only for complex momenta

✤ Important input into on-shell methods Tree-level amplitudes Feynman diagrams

✤ Yang-Mills Lagrangian 1 = F F µ⌫ (@A)2 + [email protected] + A4 L 4 µ⌫ ⇠

f abc g p f abef cdeg g ⇠ µ⌫ ↵ ⇠ µ⌫ ↵ ✤ Draw diagrams Feynman rules Sum everything Locality and unitarity

✤ Only poles: Feynman 1 where P = pk Locality 2 P k X2P ✤ On the pole Feynman diagrams recombine on both Unitarity sides into amplitudes 1 L R M !P 2=0 M P 2 M Factorization on the pole

2 1 On P =0 Res = L R M M P 2 M

✤ For P 2 =0 the internal leg: on-shell physical particle

✤ Both sub-amplitudes are on-shell, gauge invariant

✤ On-shell data: statement about on-shell quantities 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 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

✤ In gravity we have infinity tower of terms

pgR h2 + h3 + h4 + ... L ⇠ ⇠ ✤ Only h 3 terms important, others fixed by diffeomorphism symmetry

✤ On-shell constructibility of Yang-Mills, GR Only function which factorizes properly on all poles is the amplitude. decomposition

✤ Sum of Feynman diagrams in Yang-Mills = (Color) (Kinematics) M ⇥ FD X Polarization vectors Gauge dependent ✤ Color factors Tr(T a1 T a2 T a3 ...Tan ) ✤ Decomposition

= Tr(T 1 T 2 T 3 ...Tn ) A(123 ...n) M X Color decomposition

✤ Sum of Feynman diagrams in Yang-Mills = (Color) (Kinematics) M ⇥ FD X Polarization vectors Gauge dependent ✤ Color factors Tr(T a1 T a2 T a3 ...Tan ) ✤ Decomposition

= Tr(T 1 T 2 T 3 ...Tn ) A(123 ...n) M X Color ordered amplitude

A(123 ...n)

Particles are ordered, other orderings: permutations Gauge invariant ✤ This is a key object of our interest

✤ Consider: All particles massless and on-shell All momenta incoming Helicities fixed Thank you for attention!