An invitation to Scattering Amplitudes

Sangmin LeeSeoul National University

29 January 2015Joint Winter Conference on Particle Physics, String and Cosmology

References

N. Arkani-Hamed, talks and lectures

F. Cachazo, talks and lectures

M. Spradlin, “An Introduction to Amplitudes,”Scattering Amplitudes in Hong Kong, Nov. 2014

J. Bourjaily, overview talk, Grassmannian Geometry of Scattering Amplitudes, Dec. 2014

H. Elvang, Y.-t. Huang, “Scattering Amplitudes in Gauge Theory and Gravity”Cambridge Univ. Press, Mar. 2015

Progress in scattering amplitudes of gauge theory and gravity

Simple results

Efficient algorithms

Deeper understanding

Old and new ideas are being unified!

Field theory vs S-matrix theory

a f = Âi

S f iai , SS† = 1

[Jacob Bourjaily, talk 2014]

The final result was summarized in 8 pages.

220 Feynman diagrams, thousands of terms.

=hiji4

h12ih23i · · · hn1i d4(P)

Simplification and generalization

+

++

+++

+

++

i �

� j

n-point MHV amplitude

[Parke-Taylor ’86][Giele-Berends ’88]

[Marcus Spradlin, talk 2014]

1. Simplifications do not happen by accident.

2. This is an experimental science.

“The Philosophy of Amplitudeology”

N=4 SYM

One-loop

6-gluon

amplitude

(schematically)

in modern notation.

= [12345]Li2(u)+ [12356]Li2(v)+ [13456]Li2(w)

[Zvi Bern, KITP colloquium] [Marcus Spradlin, talk 2014]

Color decomposition

Full amplitude (permutation symmetry)

“Color-ordered” amplitude(cyclic symmetry only)

Color ordered amplitude

Color ordered Feynman rules (Less diagrams, simpler Feynman rules)

Ma1,a2,...,ann (p1, p2, . . . , pn) = Â

s�Sn/Zn

Tr(Ts(1) · · · Ts(n))An(s(1), · · · , s(n))

Spinor helicity notation

h = +1 h = �1

[Berends, Kleiss, Troost, Wu, Xu... 80’s]

Massless momentum in bi-spinor notation: paa = lala

Lorentz invariants: hiji ⌘ eab(li)a(lj)b , [ij] ⌘ eab(li)a(lj)b

Helicity weights:

(l, l) ! (tl, t�1l) , e+ ! t�2e+ , e� ! t+2e�

=) An(tili, t�1i li, hi) =

n

’i=1

t�2hii

!An(li, li, hi)

3-gluon amplitude

p1 + p2 + p3 = 0

1-2-

3+

An(tili, t�1i li; hi) =

’i

t�2hii

!An(li, li; hi)

An(sli, sli; hi) = s�2(n�4)An(li, li; hi)

A3 =h12i3

h23ih31i .

A3 is the unique Lorentz invariant satisfying the following scaling properties:

Caution: momentum complexified

Twistor

Twistors “linearize” conformal symmetry

Twistor transformation

Paa = lala ! la∂

∂µa

Kaa =∂

∂la

∂

∂la! µa

∂

∂la

A(⇥, ⇥) ! A(⇥, µ) =Z

d2⇥ eiµ� ⇥� A(⇥, ⇥)

ZA ⌘ (⇥�, µ�)

MAB = ZA ∂

∂ZB � (trace) 2 SU(2, 2)

[Penrose 67, Witten 03]

Half-way Fourier transform

e

ip·x !Z

e

ix

aalalae

�iµalad

2l µ d(µa � x

aala)

BCFW recursion [Britto-Cachazo-Feng(-Witten) 04(05)]

Higher point amplitude from lower ones

Idea: on-shell deformation of momenta

lj ! lj � zll , ll ! ll + zlj

(p2j = 0, p2

l = 0, p1 + · · ·+ pn = 0 unaffected)

An = An(z = 0) =Z dz

2piAn(z)

z

(deformed contour, fall-off at infinity, residue theorem)

j lJ Lj l. . .

12 n

. ..

n-1

....

..

1PJ(0)2

.. .... .

.

. .. .

�h +h

pj(zJ)

ÂJ[L=All

pl(zJ)

BCFW and gauge symmetry

No need for 4-point vertex!

Build up higher point amplitude from lower ones

An = ÂJ,±

A±J (zJ)

1PJ(0)2 A⌥

L (zJ)

j lJ Lj l. . .

12 n

. ..

n-1

....

..

1PJ(0)2

.. .... .

.

. .. .

�h +h

pj(zJ)

ÂJ[L=All

pl(zJ)

Grassmannian

Momentum conservation in spinor-helicity

BCFW deformation is a special case of

li ! Sijl

j , li ! lj(S�1)ji , S 2 GL(n, C)

[Britto, Cachazo, Feng, Witten 04-05]

[Arkani-Hamed,Cachazo,Cheung,Kaplan 09]

P-conservation at each vertex: Grassmannian

Cmi 2 Gr(k, n)

l (2-plane)

l (2-plane)C (k-plane)

n

Âi=1

(pi)aa =n

Âi=1

lialia =

�l1a · · · lna

�0

B@l1

a...

lna

1

CA

Grassmannian Integral [Arkani-Hamed,Cachazo,Cheung,Kaplan 09]

C = k

(i + k � 1)(i)

Mi = em1···mk Cm1(i)Cm2(i+1) · · ·Cmk(i+k�1)

(C · W)m = CmiWi

n

GL(k)⇥ GL(n)

“Contour integral over Grassmannian”

An,k(W) =Z dk⇥nC

vol [GL(k)]d4k|4k(C · W)

M1

M2

· · · Mn�1

Mn

W = (µ, ⇥; �)

Dual Super-conformal

Dual super-conformal symmetry

yi+1 � yi = pi

y2

y3

yn+1 ⌘ y1

yn

p1

p2

First noticed in perturbation theory

Physical explanation from AdS/CFT, T-duality & Wilson loops.[Alday-Maldacena 07; Berkovits-Maldacena 08, Beisert-Ricci-Tseytlin-Wolf 08]

Original + dual super-conformal >> Yangian [Drummond-Henn-Plefka 09]

... mutually non-local and non-linear (spin-chain analogy)

[Drummond-Henn-Korchemsky-Sokatchev 07]

Yangian and Integrability

Heisenberg spin chain

Original + dual super-conformal Yangian of SL(4|4)

H = �J Âi

~Si · ~Si+1

integrable = infinite number of conserved charges

Yangian of SU(2)

(Near) integrability of string theory in AdS5

On-shell diagrams and permutations[Arkani-Hamed,Bourjaily, Cachazo,Goncharov,Postnikov,Trnka 12]

All tree amplitudes & loop integrands from 3-point amplitudes via “BCFW bridging”

Permutation, positive Grassmannian

On-shell diagrams

TREE 4-point MHV =

??? =

Amplituhedron

Amplitude as “volume”of some polytope.

Hidden symmetries become manifest.

Unitarity and locality “emerge” from positivity of the polytope.

[Arkani-Hamed,Trnka 13]

Scattering equation [Cachazo,He,Yuan 13]

Âb 6=a

pa · pbza � zb

= 0 (za 2 CP1)

Valid for massless spin 0, 1, 2 particles in all dimensions

Generalizes Witten’s twistor string theory

Same equation appears in the high energy string scattering

[Gross,Mende 87]

KK/BCJ/KLT

Kleiss-Kuijf identities

Bern-Carrasco-Johansson identities

#(independent amplitudes) : (n � 1)! ! (n � 2)!

#(independent amplitudes) : (n � 2)! ! (n � 3)!

[Bern,Carrasco,Johansson 08]

f abc = Tr(TaTbTc)� Tr(TbTaTc) , f abe fecd + f bce fe

ad + f cae febd = 0

An(pi, hi, ai) = Âs2Sn/Zn

Tr(Tas(1) · · · Tas(n) )An(s(1h1), · · · , s(nhn))

A4 =nscs

s+

ntctt

+nucu

ucs + ct + cu = 0

ns + nt + nu = 0

color (Jacobi)

kinematics An(gauge) = Â

I

cInIDI

=) Mn(gravity) = ÂI

nI nIDI

!

BCJ doubling

[Kleiss-Kuijf 89]

Loop integration, dLog, …

Polylogarithms

Symbols

Cluster coordinates

Lik+1(z) =Z z

0Lik(t)

dtt

Beyond D=4, N=4 SYM

Stringy corrections

Other dimensions (e.g. D=3, N=6 ABJM)

Non-SUSY theories

Massive particles

…

Outlook

We will see a lot more within our lifetime!