Scattering amplitudes
from soft theorems
Yohei Ema
University of Tokyo

PPP2017 @ YITP 2017.08.02

On-going work with S. Chigusa, H. Shimizu, Y. Tachikawa and T. Yamaura

Introduction

Scattering amplitude theory•

Scattering amplitude is of central importance in particle physics.

•It sometimes shows a surprising simplicity that is not obvious from thestandard Feynman diagrammatic method.
ex. 6pt Maximally Helicity Violated (MHV) amplitude of pure YM:
4h12iA6 [1 2 3+ 4+ 5+ 6+ ] =h1 2ih2 3ih3 4ih4 5ih5 6ih6 1iafter summing over 220 (!) diagrams.
•Scattering amplitude program tries to construct amplitudes fromanalytical properties, not relying (heavily) on Feynman diagrams.

Goal of this talk• Show that tree-level YM/gravity amplitudes are recursively

constructible, with leading soft theorem being an input.

• We also review basic ingredients of modern scattering

amplitude theory.

Outline
1. Introduction
2. Review A: spinor helicity formalism
3. Review B: on-shell recursion
4. Review C: soft theorems
5. Idea (and explicit computation)
6. Summary

Outline1. Introduction
2. Review A: spinor helicity formalism
3. Review B: on-shell recursion
4. Review C: soft theorems
5. Idea (and explicit computation)
6. Summary

Spinor helicity formalism••Consider 4-dim theory with only massless particles.
The momentum pȧbdet p =•µ ȧb⌘ pµ (¯ ) satisfiesµpµ p = 0pȧb=ȧb|pi [p| .

The momentum product in this language is
2p · q = hp qi[p q] where hp qi ⌘ ✏ȧḃ |piȧ |qiḃ and [p q] ⌘ ✏ab [p|a [q|b .

Amplitudes are constructed from these products.
•Little group keeps momentum intact.
1|pi!
t|pi,|p]!
t|p].
In terms of angle/square brackets:•Amplitude transforms due to the external lines as:
1An ..., {ti |ii, ti |i], hi }, ... = ti•

2hi

An (..., {|ii, |i], hi }, ...) .

Three point amplitude is determined solely from little group scaling.

Outline
1. Introduction
2. Review A: spinor helicity formalism
3. Review B: on-shell recursion
4. Review C: soft theorems
5. Idea (and explicit computation)
6. SummaryComplex momentum shift[Britto, Cachazo, Feng, 04; Britto, Cachazo, Feng, Witten, 05]• Consider the following complex momentum shift:p̂i (z) ⌘ pi + zqi , where pi · qi = qi2 = 0 : on-shell condition of p̂i (z)X
qi = 0 : momentum conservation of p̂i (z) .
andi

Shifted amplitude is a function of z : Ân (z)(original amplitude is An = Ân (0)).
• Poles: associated with on-shell intermediate particle (Locality).
Oltshenamp\• Amplitude factorizes near the poles as IÂ(z) ! ÂL (zI )1P̂I2 (z).µACHAu,.

2,3Ar

¥-1-;

d→yFi/

fdiacpnnaitpnns ?

n'.

NLÂR (zI ), P̂I2 (z) / (z.

zI ).

I.

On-shell recursion[Britto, Cachazo, Feng, 04; Britto, Cachazo, Feng, Witten, 05]• From the standard complex analysis, we obtain

1Â(0) =2⇡i

Idz
Â(z) =|z|=0 z

IX
1dz
1ÂL (zI ) 2 ÂR (zI ) + B1 .
2⇡iP̂I (z)|z zI |=0 z
I(b)

(a)

• (a) : products of lower point on-shell amplitudes.
• (b) : contribution from |z| = 1, which vanishes whenlim Â(z) = 0.

|z|!1

B1 = 0 , on-shell constructibility of the theory• Two (or more) ways to achieve on-shell constructibility:(1) Invent a good momentum shift (such as BCFW shift)
Â(z)
Â(z)
!
.
(2) Modify the integrand aszzf (z)[Cheung, Kampf, Novotny, Shen, Trnka, 15](We should know how the amplitude behaves as f (z) ! 0. )On-shell recursion[Britto, Cachazo, Feng, 04; Britto, Cachazo, Feng, Witten, 05]• From the standard complex analysis, we obtain

1Â(0) =2⇡iIdz
Â(z) =|z|=0 z

IX
1dz
1ÂL (zI ) 2 ÂR (zI ) + B1 .
2⇡iP̂I (z)|z zI |=0 z
I(b)

(a)

• (a) : products of lower point on-shell amplitudes.
• (b) : contribution from |z| = 1, which vanishes whenlim Â(z) = 0.

|z|!1

B1 = 0 , on-shell constructibility of the theory• Two (or more) ways to achieve on-shell constructibility:(1) Invent a good momentum shift (such as BCFW shift)Â(z)Â(z)!
.
(2) Modify the integrand aszzf (z)[Cheung, Kampf, Novotny, Shen, Trnka, 15](We should know how the amplitude behaves as f (z) ! 0. )(Anti-)holomorphic shift[Cohen, Elvang, Kiermaier, 10]• We will stick to the following complex momentum shifts.

Holomorphic shift: |îi = |ii

ai z|Xi, |î] = |i] with

Anti-holomorphic shift: |îi = |ii, |î] = |i]* qi =Xai |i] = 0.
Xai |ii = 0.
ai z|X] withai |Xi[i| ! qi2 = qi · pi = 0 for holomorphic shift.

** Similar relation holds for anti-holomorphic shift.

• We will take |Xi = |1i or |X] = |1] in the following.

z = 1/a1 corresponds to the soft limit of the particle 1.

Large z behavior P

h...ian [...]sn• If coupling dimension is unique, amplitude is An = g P.
asddh...i [...]* ai , si : common due to little group scaling and mass dimension.

Let us define a ⌘ anad , s ⌘ s nsd .

• Dimensional analysis:a+s=4n[g] where [g] : mass dimension of coupling g.

• Little group scaling:aXs=hi where hi : helicity of i-th particle.
[Cohen, Elvang, Kiermaier, 10]i

Hol shift: lim Ân (z) ! O(z a ) with 2a = 4
z!1n* YM: [g] = 0, Einstein gravity: [g] =hi .

iAnti-hol shift: lim Ân (z) ! O(z s ) with 2s = 4
z!1[g]Xnn + 2.

[g] +Xihi .

Outline
1. Introduction
2. Review A: spinor helicity formalism
3. Review B: on-shell recursion
4. Review C: soft theorems
5. Idea (and explicit computation)
6. Summary

Leading soft theorems[Low 58; Weinberg 65; …]• Leading soft theorem:

pp
1 (0)
An { ✏|1i, ✏|1], h1 }, ... = S An 1 ({|2i, |2], h2 }, ...) + O(✏0 )✏(0)where S =nX[1 k]hx kihy ki

k=2and S(0)h1 kihx 1ihy 1i

hx 2i=
hx 1ih1 2i

for positive helicity graviton

hx 4i
for positive helicity gluon (color-ordered).
hx 1ih1 4i• From little group scaling, it behaves under hol/anti-hol soft limit asAn ({✏|1i, |1], h1 }, ...) = ✏ 1 h1 S (0) An 1 ({|2i, |2], h2 }, ...) + O(✏ h1 )andAn ({|1i, ✏|1], h1 }, ...) = ✏1+h1S(0)h1An 1 ({|2i, |2], h2 }, ...) + O(✏ ).

Outline
1. Introduction
2. Review A: spinor helicity formalism
3. Review B: on-shell recursion
4. Review C: soft theorems
5. Idea (and explicit computation)
6. Summary

What we learned so far• Integrand should fall off at large z for on-shell

constructibility.
• Under holomorphic or anti-holomorphic shift:+ +lim Â4 [1 2 3 4 ] ! const for pure YM theory

z!1and lim M̂n (z) ! z at worst for Einstein gravity.
z!1• Under holomorphic/anti-holomorphic soft limit:

An ({✏|1i, |1], h1 }, ...) = ✏ 1 h1 S (0) An 1 + O(✏ h1 )1+h1 (0)h1A({|1i,✏|1],h},...)=✏SA+O(✏).
and n1n 1

Idea• Main idea: use soft theorem to take better integrand.
•Consider the worst caseXhi = 0 and assume h1 > 0.

Under anti-holomorphic soft shift,large z behavior is lim Â4 (z) ! z 0 for YM and lim M̂n (z) ! z 1 for gravity.
z!1z!1Soft limit is Â4 (z) = Ŝ (0) Â3 |z=1/a1 + O(✏1 ) for YMand M̂n (z) = ✏Ŝ (0) M̂n 1 |z=1/a1 + O(✏2 ) for gravity with ✏ ⌘ 1

a1 z.

Take integrand asIIdz M̂n (z)dz Â4 (z)z (1 a1 z)2 for gravity!
z 1 a1 z for YM andIntegrand falls off rapidly enough at large z.
Residue at z = 1/a1 is nothing but the leading soft term.

Computation: gluon 4pt• Consider 4pt (color-ordered) YM amplitude A4 [1+ 2+ 34 ].

• Under anti-holomorphic soft shift, pole is only at z = 1/a1 .
2* (p̂i (z) + p̂j (z)) / (12a1 z) (pi + pj ) .

We need to consider only the soft factor (soft limit is ``exact’’):3pt from little group
A4 [1+ 2+ 3 4 ] = Ŝ (0) Â3 [2+ 3 4 ]|z=1/a1
✓◆
hx 2i
hx 4i
h3 4i4=hx 1ih1 2i hx 1ih1 4i h2 3ih3 4ih4 2i
h3 4i4=,h1 2ih2 3ih3 4ih4 1iwhere |1ih2 4i + |2ih4 1i + |4ih1 2i = 0 : Schouten identity is used.

It correctly reproduces the Parke-Taylor MHV amplitude.
[Parke and Taylor 86]

Computation: graviton 4pt• Consider 4pt gravity amplitude M4 (1+ 2+ 3•4 ).

Again we only need to consider the soft factor:
M4 (1+ 2+ 3 4 ) = Ŝ (0) M̂3 (2+ 3 4 )|z=1/a13pt from little group
018X [1 k]hx kihy ki
h34i
A=@.
222h1 kihx 1ihy 1i h2 3i h3 4i h4 2i
k=2,3,4(*)[1 4]h2 4ih3 4i• (*) is simplified after using Schouten identity as (⇤) =.
h1 2ih1 3ih1 4i

We finally obtain
M4 (1+ 2+ 3 4 ) = (p1 + p4 )2 A4 [1+ 2+ 3 4 ]A4 [1+ 3 2+ 4 ].

It reproduces the KLT relation: (gravity) = (gauge)2[Kawai, Lewellen, Tye, 86]Comparison
There are of course other recursion methods.
For instance,[Britto, Cachazo, Feng, 04; Britto, Cachazo, Feng, Witten 05]• BCFW shift: |1̂i = |1iz|2i, |2̂] = |2] + z|1]

Recursion relation is simple, especially for MHV amplitudes.
Large z behavior is non-trivial (analyzed by Feynman diagrams).
No need for soft theorem.
• [1, m[Cheung, Shen, Trnka, 15]

1i-line shift: |îi = |ii + zci |1i (i 6= 1), |1̂] = |1]• Recursion relation is not so simple compared to BCFW.
• For m = n, large z behavior analysis can be simple.

No need for soft theorem.

Our recursion relation is new, but otherwise…

zX
i6=1ci |i]Outline
1. Introduction
2. Review A: spinor helicity formalism
3. Review B: on-shell recursion
4. Review C: soft theorems
5. Idea (and explicit computation)
6. Summary

Summary• It is important to control large z behavior to achieve

on-shell constructibility of a given theory.
• We demonstrate (tree-level) on-shell constructibility ofYM/Einstein gravity with soft theorem being an input.
• Recursion with soft theorems can be extended to othertheories.