Scattering Amplitudes from the Amplituhedron NMHV Volume Forms

Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry

Scattering amplitudes from the amplituhedron NMHV volume forms

Andrea Orta

Ludwig-Maximilians-Universit¨atM¨unchen

V Postgraduate Meeting on Theoretical Physics Oviedo, 17th November 2016

Based on 1512.04954 with Livia Ferro, TomaszLukowski, Matteo Parisi Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry Table of contents

1 Scattering amplitudes in planar N = 4 super Yang-Mills Heavy computations for simple amplitudes N = 4 SYM

2 Introduction to the tree-level Amplituhedron Positive geometry Amplitudes as volumes

3 NMHV volume forms from symmetry Capelli diﬀerential equations The k = 1 solution Examples of NMHV volume forms

4 Conclusions Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry Scattering amplitudes . . .

Scattering amplitudes are central objects in QFT. Interesting as an intermediate step to compute observables; as a means to gain insight into the formal structure of a speciﬁc model.

How are they traditionally computed? 1 Stare at Lagrangian and extract the Feynman rules; 2 draw every possible Feynman diagram contributing to the process of interest; 3 evaluate each one of those and add up the results. Straightforward enough. What could possibly go wrong? Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry . . . are complicated?

Consider tree-level gluon amplitudes in QCD. 2g → 2g : 4 diagrams Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry . . . are complicated?

Consider tree-level gluon amplitudes in QCD. 2g → 3g : 25 diagrams

[slide by Z. Bern] Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry . . . are simpler than expected!

Consider tree-level gluon amplitudes in QCD. 2g → 3g : 10 colour-ordered diagrams

tree ± ± ± ± ± A5 (1 , 2 , 3 , 4 , 5 ) = 0 tree ∓ ± ± ± ± A5 (1 , 2 , 3 , 4 , 5 ) = 0

h12i4 Atree(1−, 2−, 3+, 4+, 5+) = 5 h12ih23ih34ih45ih51i

+ − 3 1 1 2 α β λi λj hiji = αβλi λj = 2 2 + λi λj 4 1− 5+ Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry . . . are simpler than expected!

Consider tree-level gluon amplitudes in QCD. 2g → 3g : 10 colour-ordered diagrams

tree ± ± ± ± ± A5 (1 , 2 , 3 , 4 , 5 ) = 0 tree ∓ ± ± ± ± A5 (1 , 2 , 3 , 4 , 5 ) = 0

h12i4 Atree(1−, 2−, 3+, 4+, 5+) = 5 h12ih23ih34ih45ih51i

hiji4 AMHV(1+,..., i −,..., j−,..., n+) = n h12ih23i · · · hn1i

MHV = maximally helicity-violating [Parke, Taylor] (Ordinary + Dual) superconformal symmetries give rise to an inﬁnite-dimensional Yangian algebra Y psu(2, 2|4) [Drummond, Henn, Plefka] At weak coupling : more constrained, easier to compute At strong coupling : amenable to AdS/CFT techniques

Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry

The simplest quantum ﬁeld theory [Arkani-Hamed, Cachazo, Kaplan]

Most symmetric theory in 4D is planar N = 4 super Yang-Mills. Maximal susy: spectrum is organized in a single supermultiplet with 2 gluons (g ±), 8 gluinos (ψ±), 6 scalars (ϕ), all massless.

1 Ω = g + + ηAψ + ηAηBϕ + A 2 AB 1 1 + ηAηBηC ψ¯ D + ηAηBηCηD g − 3! ABCD 4! ABCD Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry

The simplest quantum ﬁeld theory [Arkani-Hamed, Cachazo, Kaplan]

Most symmetric theory in 4D is planar N = 4 super Yang-Mills. Maximal susy: spectrum is organized in a single supermultiplet with 2 gluons (g ±), 8 gluinos (ψ±), 6 scalars (ϕ), all massless. (Ordinary + Dual) superconformal symmetries give rise to an inﬁnite-dimensional Yangian algebra Y psu(2, 2|4) [Drummond, Henn, Plefka] At weak coupling : more constrained, easier to compute At strong coupling : amenable to AdS/CFT techniques

planar N =4 SYM

N =4 SYM SYM massless massive [picture by L. Dixon] Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry

The simplest quantum ﬁeld theory [Arkani-Hamed, Cachazo, Kaplan]

QCD N =4 N =1 scalar An = An − 4An + An Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry The power of momentum twistors

“Masslessness” of the spectrum + conformal symmetry −→ introduce momentum supertwistors for describing the kinematics. Instead of four-momenta pµ µ = 0, 1, 2, 3 and Grassmann-odd ηA A = 1, 2, 3, 4 α, α˙ = 0, 1, 0˙, 1˙ use mom. supertwistors ZA A = A = 1, 2, 3, 4

3|4 The geometry of momentum twistor superspace CP ensures masslessness of momenta and momentum conservation. Generating function for every tree-level N = 4 SYM amplitude Z 1 dk×nc L = αi δ4|4(C ·Z) n,k GL(k) (1 ··· k)(2 ··· k + 1) ··· (n ··· k − 1)

[(Arkani-Hamed, Cachazo, Cheung, Kaplan) (Mason, Skinner)] Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry Towards the amplituhedron

Two remarkable results inspired the amplituhedron:

1 Z dk×nc L = αi δ4|4(C ·Z) n,k GL(k) (1 ··· k)(2 ··· k + 1) ··· (n ··· k − 1)

One-to-one correspondence between residues of Ln,k and cells of the positive Grassmannian G+(k, n).

[Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, Trnka]

NMHV tree-level amplitudes can be thought of as volumes of polytopes in twistor space.

[Hodges] Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry

1 Scattering amplitudes in planar N = 4 super Yang-Mills Heavy computations for simple amplitudes N = 4 SYM

2 Introduction to the tree-level Amplituhedron Positive geometry Amplitudes as volumes

3 NMHV volume forms from symmetry Capelli diﬀerential equations The k = 1 solution Examples of NMHV volume forms

4 Conclusions Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry Positive means inside

Y 2 Z3 Triangle in RP

Interior of a triangle

A A A A Y = c1Z1 + c2Z2 + c3Z3 , c1, c2, c3 > 0

Points inside are described by the positive triple

(c1 c2 c3)/GL(1) Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry Positive means inside

Y n−1 Simplex in RP

Interior of a simplex

A X A Y = ci Zi , ci > 0 i

Points inside are described by the positive n-tuple

(c1 c2 ... cn)/GL(1) , a point in G+(1, n) . Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry Positive also means convex

Z4 Z5

Y Polygon in m Zn RP Z3

Z1 Z2

Interior of a n-gon with vertices Z1,..., Zn is only canonically deﬁned if

 1 1 1  Z1 Z2 ... Zn  . . .  Z =  . . .  ∈ M+(1 + m, n) 1+m 1+m 1+m Z1 Z2 ... Zn Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry Tree-level amplituhedron

m Interior of an n-polyhedron in RP

( ) tree A X A C = (c1 ... cn) ∈ G+(1, n) A [Z] = Y = ci Z , n,1;m i Z = (Z ... Z ) ∈ M (1 + m, n) i 1 n + Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry Tree-level amplituhedron

m Interior of an n-polyhedron in RP

( ) tree A X A C = (c1 ... cn) ∈ G+(1, n) A [Z] = Y = ci Z , n,1;m i Z = (Z ... Z ) ∈ M (1 + m, n) i 1 n +

Generalize this picture to account for Nk MHV amplitudes

Tree-level amplituhedron

tree C ∈ G+(k, n) An,k;m[Z] = Y ∈ G(k, k+m): Y = C·Z , Z ∈ M+(k + m, n)

[Arkani-Hamed,Trnka] Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry The volume form

Volume form ˜ (m) tree Top-dimensional diﬀerential form Ωn,k deﬁned on An,k;m with only logarithmic singularities on its boundaries.

˜ (m) top-dimensional: Y ∈ G(k, k + m) −→ Ωn,k is an mk-form dα log-singularity : approaching any boundary, Ω˜ (m) ∼ n,k α Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry The volume form

Volume form ˜ (m) tree Top-dimensional diﬀerential form Ωn,k deﬁned on An,k;m with only logarithmic singularities on its boundaries.

˜ (m) top-dimensional: Y ∈ G(k, k + m) −→ Ωn,k is an mk-form dα log-singularity : approaching any boundary, Ω˜ (m) ∼ n,k α

Z2 If Y = α1Z1 + α2Z2 + Z3 , Y Z 2 2 3 ˜ (2) dα1 dα2 1 h123i hY d Y i Ω3,1 = ∧ = α1 α2 2 hY 12ihY 23ihY 31i Z1 Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry The volume form

Volume form ˜ (m) tree Top-dimensional diﬀerential form Ωn,k deﬁned on An,k;m with only logarithmic singularities on its boundaries.

˜ (m) top-dimensional: Y ∈ G(k, k + m) −→ Ωn,k is an mk-form dα log-singularity : approaching any boundary, Ω˜ (m) ∼ n,k α

Z2 If Y = α1Z1 + α2Z2 + Z3 , Y 2 Z3 (2) 1 h123i Ω3,1 = ≡ [1 2 3] Z 2 hY 12ihY 23ihY 31i 1 | {z } Area of (dual) triangle Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry Get to the amplitude

Morally . . . Z An,k ˜ (m) = Ωn,k (Y , Z) An,0 tree An,k;m

“Scattering amplitudes are volumes of (dual) amplituhedra”

The physics, i.e. the kinematics of scattering particles, is encoded in Z A variables, bosonized version of momentum supertwistors ZA:

 α  λi  µ˜α˙  (λα, µ˜α˙ ) are bosonic d.o.f. of ZA  i  i i A φ1 · χi  A A Zi =   , with χi are fermionic d.o.f. of Z  .  A  .  φα are auxiliary fermionic d.o.f. φk · χi Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry Get to the amplitude

Precisely . . . Z An,k m·k (m) ∗ = d φ Ωn,k (Y , Z) An,0

“Scattering amplitudes are volumes of (dual) amplituhedra”

The physics, i.e. the kinematics of scattering particles, is encoded in Z A variables, bosonized version of momentum supertwistors ZA:

1 Scattering amplitudes in planar N = 4 super Yang-Mills Heavy computations for simple amplitudes N = 4 SYM

2 Introduction to the tree-level Amplituhedron Positive geometry Amplitudes as volumes

3 NMHV volume forms from symmetry Capelli diﬀerential equations The k = 1 solution Examples of NMHV volume forms

4 Conclusions Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry

Covariance and scaling properties [Ferro,Lukowski, AO, Parisi]

Integral representation of the volume

Z k×n k (m) d cαi Y Ω (Y , Z) = δk+m(Y A − c Z A) n,k (1 ··· k) ··· (n ··· k − 1) α αi i γ α=1

Look for symmetry properties: obvious ones are ? GL(k + m)-covariance

1 Ω(m)(Y · g, Z · g) = Ω(m)(Y , Z) n,k (det g)k n,k

? GL+(k) ⊗ GL+(1) ⊗ · · · ⊗ GL+(1)-scaling 1 Ω(m)(h · Y , λ · Z) = Ω(m)(Y , Z) n,k (det h)k+m n,k Example: m = 2, k = 1, n = 4

Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry

The Capelli diﬀerential equations [Ferro,Lukowski, AO, Parisi] New observation: Capelli equations ! ∂ det Ω(m)(Y , Z) = 0 , W A = (Y A, Z A) Aν n,k a α i ∂Waµ 1≤ν≤k+1 1≤µ≤k+1 Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry

The Capelli diﬀerential equations [Ferro,Lukowski, AO, Parisi] New observation: Capelli equations ! ∂ det Ω(m)(Y , Z) = 0 , W A = (Y A, Z A) Aν n,k a α i ∂Waµ 1≤ν≤k+1 1≤µ≤k+1

Example: m = 2, k = 1, n = 4

  ∂Y 1 ∂Z 1 ∂Z 1 ∂Z 1 ∂Z 1 1 2 3 4 (2) det 2×2 ∂Y 2 ∂Z 2 ∂Z 2 ∂Z 2 ∂Z 2  Ω = 0  1 2 3 4  4,1 ∂ 3 ∂ 3 ∂ 3 ∂ 3 ∂ 3 Y Z1 Z2 Z3 Z4 Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry

The Capelli diﬀerential equations [Ferro,Lukowski, AO, Parisi] New observation: Capelli equations ! ∂ det Ω(m)(Y , Z) = 0 , W A = (Y A, Z A) Aν n,k a α i ∂Waµ 1≤ν≤k+1 1≤µ≤k+1

Example: m = 2, k = 1, n = 4

∂Y A ∂Z A (2) ∂Z A ∂Z A (2) i i j Ω4,1(Y , Z) = 0 , Ω4,1(Y , Z) = 0 ∂ B ∂ B ∂ B ∂ B Y Zi Zi Zj for all values of A, B = 1, 2, 3 Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry

All k = 1 volume forms [following Gel’fand, Graev, Retakh]

Master formula

Z +∞ 1+m ! n (m) Y m! Y Ω (Y , Z) = ds θ(s · Z ) n,1 A (s · Y )1+m i 0 A=2 i=m+2 Z (m) dc1 ... dcn 1+m A A to be compared with Ωn,1 (Y , Z) = δ (Y − ci Zi ) γ c1 ··· cn Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry

All k = 1 volume forms [following Gel’fand, Graev, Retakh]

Master formula

Z +∞ 1+m ! n (m) Y m! Y Ω (Y , Z) = ds θ(s · Z ) n,1 A (s · Y )1+m i 0 A=2 i=m+2 Z (m) dc1 ... dcn 1+m A A to be compared with Ωn,1 (Y , Z) = δ (Y − ci Zi ) γ c1 ··· cn

New integral lives in the dual Grassmannian G(1, 1 + m). Fixed number of integration variables s2,..., s1+m. (m) integration domain Dn is shaped by θ-functions. Integrand( s · Y )−(1+m) is free of singularities. Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry

All k = 1 volume forms [following Gel’fand, Graev, Retakh]

Master formula

Z +∞ 1+m ! n (m) Y m! Y Ω (Y , Z) = ds θ(s · Z ) n,1 A (s · Y )1+m i 0 A=2 i=m+2 Z (m) dc1 ... dcn 1+m A A to be compared with Ωn,1 (Y , Z) = δ (Y − ci Zi ) γ c1 ··· cn Known results are

(2) X (4) X Ωn,1 = [1 i i + 1] , Ωn,1 = [1 i i + 1 j j + 1] i i

h1 i i + 1i2 h1 i i + 1 j j + 1i4 [1 i i + 1] = , [1 i i + 1 j j + 1] = hY 1 iihY i i + 1ihY i + 1 1i hY 1 i i + 1 ji · · · hY j + 1 1 i i + 1i Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry Two-dimensional examples

Toy model: m = 2, useful for visualization purposes.

Three-point volume form Z +∞ (2) ds2 ds3 Ω3,1 = 2 3 = [1 2 3] 0 (s · Y )

(2) D3

s2 Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry Two-dimensional examples

Toy model: m = 2, useful for visualization purposes.

Four-point volume form Z +∞ (2) θ(s · Z4) Ω4,1 = 2 ds2 ds3 3 = [1 2 3] + [1 3 4] 0 (s · Y )

`Z4

(2) D4

s2 Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry Two-dimensional examples

Toy model: m = 2, useful for visualization purposes.

n-point volume form Z +∞ Qn n−1 (2) θ(s · Zi ) X Ω = 2 ds ds i=4 = [1 i i + 1] n,1 2 3 (s · Y )3 0 i=2

s3 `Zn ` W Zn−1 (2) Dn `Z5

`Z4

s2 Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry Physical examples

Realistic case: m = 4, hard to visualize.

Five-point volume form Z +∞ (4) ds2 ... ds5 Ω5,1 = 4! 5 = [1 2 3 4 5] 0 (s · Y )

(4) D5

s2 Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry Physical examples

Realistic case: m = 4, hard to visualize.

Six-point volume form Z +∞ (4) θ(s · Z6) Ω6,1 = 4! ds2 ... ds5 5 0 (s · Y )

`Z6

(4) D6 s2 Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry Physical examples

Realistic case: m = 4, hard to visualize.

Six-point volume form

Z +∞ Z a Z a−s2 (4) −5 Ω6,1 = 4! ds3 ds5 ds2 ds4 (s · Y ) 0 0 0

a = 1 + s3 + s5 a

(4) D6 a s2 Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry Physical examples

Realistic case: m = 4, hard to visualize.

Six-point volume form

(4) Ω6,1 = [1 2 3 4 5] + [1 2 3 5 6] + [1 3 4 5 6]

s4 s4 s4 s4

= (4) − + D5 (4) D6 s2 s2 s2 s2 What needs to be done? ? Understand whether the Capelli equations hint at a realization of Yangian symmetry in the amplituhedron framework. ? Use this knowledge to move beyond NMHV volume forms.

Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry Conclusions and outlook

Summarizing, The amplituhedron construction allows to think of scattering amplitudes in planar N = 4 SYM as volumes of “polytopes”. Volume forms corresponding to tree-level NMHV amplitudes are fully constrained by symmetry −→ Capelli equations. Our master formula explicitely computes the “volume” of a region in a dual Grassmannian. Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry Conclusions and outlook

What needs to be done? ? Understand whether the Capelli equations hint at a realization of Yangian symmetry in the amplituhedron framework. ? Use this knowledge to move beyond NMHV volume forms. Amplitudes in planar N = 4 SYM Introduction to the tree-level Amplituhedron NMHV volume forms from symmetry

Thank you!

[picture by A. Gilmore]