Higher derivative corrections from mixing color and kinematics - - PowerPoint PPT Presentation

Higher derivative corrections from mixing color and kinematics

Laurentiu Rodina


IPhT, CEA Saclay

with J.J. Carrasco, Z. Yin, S. Zekioglu

December 12, 2019 
 QCD meets Gravity V

Locality +

gauge invariance

Adler zero soft theorems UV scaling color/ kinematics Yang-Mills X X X Gravity X X X Einstein-Yang-Mills X X X Bi-adjoint scalar X X X NLSM X X X X DBI X X special Galileon X X X Born-Infeld X conformal dilaton X

2

Amplitude bootstrap without unitarity

[Arkani-Hamed, LR, Trnka ’16] [LR ’16, ’18] [Carrasco, LR ’19]

Proposed that soft particles carry black hole information 


[Hawking, Perry, Strominger ’16]
 


How much BH information can soft particles carry anyway?


3

How much amplitude information can soft particles carry?

What can black holes teach us about amplitudes?

• Naive answer: IR and UV are disjoint

• In fact, soft theorems completely constrain amplitudes!
• The UV info of one particle is hidden in the IR of a different one

p = zp, z → 0

4

An+1 = 1 z A(−1) + z0A(0) + zA(1) + z2A(2) + … = ( 1 z S0 + z0S1) An +

∞

∑

i=2

ziA(i)

= IR (soft theorem satisfying) + UV (soft theorem avoiding)

e1⋅e2 e3⋅p1 e4⋅p1 p1 . p2

⟶ O(z)

zp1 → 0

⟶ O(1/z)

zp2 → 0

IR vs UV

Adler zero

• NLSM, scalar theory of Nambu-Goldstone bosons


• Adler zero: 


• Adler zero uniquely fixes NLSM amplitudes (DBI, sGal)


[Cheung, Kampf, Novotny, Trnka, ’14][Arkani-Hamed, LR, Trnka, ’16][LR ’16]

• Implies Yang-Mills uniqueness from soft theorems

pi → zpi, z → 0 ⇒ Anlsm → O(z)

5

A6 = p1⋅p3 p4⋅p6 (p1 + p2 + p3)2 + …

A4 = p1⋅p3

AYM

n+1 → (

1 z S0 + z0S1) An + O(z); BYM

n+1 → (

1 z S0 + z0S1) An + O(z)

AYM

n+1 − BYM n+1 → O(z)

“Adler 1/0”

• UV scaling under two particle BCFW shifts fixes YM, GR [LR ’16]
• GR integrands via UV [Edison, Herrmann, Parra-Martinez, Trnka ’19]
• Scalar theories? 


• Fixes NLSM, sGal, bi-adjoint scalar [Carrasco, LR ’19]

pi → z pi, z → ∞

Possibility: IR/UV duality

• IR or UV behavior ⇒ unique S-matrix solution
• Conformal symmetry? [Loebbert, Mojaza, Plefka ’18]

7

• Color-dual representation:

satisfy antisymmetry and Jacobi

• 1. Double copy


• 2. Amplitude relations


c, n c → n ⇒ ℳ = ∑ nini di

Color-kinematic duality

𝒝 = ∑ cini di

k12A(2134...n) + k13A(2314...n) + . . . + k1,n−1A(234...1n) = 0

• Test subject: NLSM
• Immensely simpler: BCJ relations + cyclic inv.

• BCJ relations + Locality uniquely fix NLSM amplitudes


[Carrasco, LR ’19]

• Add Unitarity => Higher derivative NLSM
• But (h.d.) NLSM = abelian Z-theory…

Bootstrapping via color/kinematics

9

k12A(2134...n) + k13A(2314...n) + . . . + k1,n−1A(234...1n) = 0

Higher derivatives

• Open superstring? (Z-theory)

(SYM) 


[Carrasco, Mafra, Schlotterer ’17][Mafra, Schlotterer ’17]

• Open bosonic string? (Z-theory)

(YM+ ) 


[Huang, Schlotterer, Wen ’16][Azevedo, Chiodaroli, Johansson, Schlotterer ’18]

• NLSM?

; 


[LR ’18][Low, Yin ’19]

• Natural question: local building blocks that reach all orders in mass

dim?

⊗ ⊗

(DF)2

O(p4) A4 = u2 + 2 s t A4 = u2 − s t

𝒝 = ∑ cini di ?

Yes! Very few building blocks are needed

Mixing color with kinematics

• Traditionally color and kinematics separated
• Allow mixing:
• New solutions possible!
• Constructive approach?

c = a1s12Tr(T1T2T3T4) + a2s23Tr(T1T4T3T2) + … 𝒝 = ∑ cini di

Building blocks and a composition rule

• Kinematic permutation invariants spanned by


• Composition rule (New Jacobi from Old Jacobi)
• Simplest kinematic numerator (linear):

=> YMS

• Next simplest (quadratic):

=> NLSM

• Next:

…

• In general scalar kin:

σ2 = s2 + t2 + u2 σ3 = s3 + t3 + u3 = stu Js(k, j) = kt jt − kuju nss

s = t − u

nnl

s = Js(nss, nss) = s(t − u)

Js(nnl, nss) = nss

s σ2

n = nss f(σ2, σ3) + nnl g(σ2, σ3)

Modifying color factors

• Color permutation invariant:
• 


̂ cs ≡ cs σY

2 σX 3

̂ css

s = Js(nss, c) σX 2 σY 3 = (nss t ct − nss u cu) σY 2 σX 3

dabcd = ∑ Tr(TaTbTcTd) ̂ cnl,d

s

= dabcdnnl

s σX 2 σY 3

̂ C = ∑

X,Y

( ̂ c + ̂ css + ̂ cnl,d) ⟶ 𝒝 = ̂ Csns s + ̂ Ctnt t + ̂ Cunu u

• String field theory limit can rearranged to manifest building blocks


(see our paper!)


Corrections to supergravity amplitudes

• 

• Replace color with kinematics

• Diff. inv. only for

𝒝ym = ̂ Cs nym

s

s + ̂ Ct nym

t

t + ̂ Cu nym

u

u ̂ c → ̂ nym = ̂ c|c→nym ̂ c = c σY

2 σX 3 →

̂ nym = nymσY

2 σX 3

GRhd = GR∑ aX,YσY

2 σX 3

• Non-susy h.d.? Need to modify
• Only 7 gauge invariant tensor structures


[Bern, Edison, Kosower, Parra-Martinez ’17]

nym

Non-susy h.d.

• Only 4 new building blocks (

), , with 


[Broedel, Dixon ’12][Garozzo, Queimada, Schlotterer ’18][Azevedo, Chiodaroli, Johansson, Schlotterer ’18]

• conjectured to be fixed by gauge inv., unitarity, BCJ relations, and

scaling in the limit 


nF3, n(F3)2+F4, nd2F4, nd4F4 σY

2 σX 3

̂ C

(DF)2

α′ → ∞ Bosonic string = (Z theory) ⊗ (YM + (DF)2)

A = ∑ cini di

Summary and Outlook

• Just a few building blocks required to capture the full expansion of
• pen superstring, a few extra to capture bosonic string
• This was only single trace color structure, can be extended to

double trace [Low, Yin ’19]

• Composition rules exist at higher multiplicity
• General building blocks, loop-level building blocks? Stay tuned!