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Classifying local four gluon S-matrices

Subham Dutta Chowdhury November 20, 2020 YITP Strings and Fields 2020

Subham Dutta Chowdhury Classifying local four gluon S-matrices 1/24

References

1910.14392 with Abhijit Gadde, Tushar Gopalka, Indranil Halder, Lavneet

Janagal, Shiraz Minwalla

2006.12458 with Abhijit Gadde. Subham Dutta Chowdhury Classifying local four gluon S-matrices 2/24

Introduction and motivation

Takeaway from Shiraz’s talk: Conjectured the uniqueness of graviton scattering Classification of photon and gravitational S-matrices Constraints on the space of such S-matrices For D ≤ 6, Einstein gravity was found to be unique at the level of 4-point

scattering. lim

s→∞ A(s, t) ≤ s2

(1)

While from structural arguments it can be argued that such universality classes

don’t exist for four point gluon S-matrices, nevertheless their classification remains an open and interesting question.

Subham Dutta Chowdhury Classifying local four gluon S-matrices 3/24

3-gluon scattering

Kinematic considerations (no mandelstam variables) force the most general flat

space S-matrix to be a linear combination of Yang-Mills and a cubic field strength term. AY M = f abc (ǫ1.ǫ2 (k1 − k2) .ǫ3 + ǫ1.ǫ3 (k3 − k1) .ǫ2 + (k2 − k3) .ǫ1ǫ2.ǫ3) , 2- derivative (2) AF 3 = Tr(T α1T α2T α3)F 1

abF 2 bcF 3 ca + perm.,

3- derivative (3) where, F 1

µν = (k1,µǫ1,ν − k1,νǫ1,µ)

The most general S-matrix then becomes aAY M + bAF 3. Subham Dutta Chowdhury Classifying local four gluon S-matrices 4/24

Analytic S-matrices

Polynomial in s, t and u (the mandelstam variables), ǫa

i (the adjoint-valued

polarisation tensors)- seemingly infinite

Can be graded by number of derivatives. The number of parameters, n(m),

needed to specify the most general dimension m S-matrix is finite. ZS-matrix(x) =

∞

• m=0

n(m)xm. (4)

Furthermore, we require Lorentz invariance, gauge invariance, S4 permutation

invariance and G-invariance. .

Subham Dutta Chowdhury Classifying local four gluon S-matrices 5/24

Building blocks I: Scattering data and gauge invariance

Momenta and mandelstam variables.

s := −(p1 + p2)2 = −(p3 + p4)2 = −2p1.p2 = −2p3.p4 t := −(p1 + p3)2 = −(p2 + p4)2 = −2p1.p3 = −2p2.p4 u := −(p1 + p4)2 = −(p2 + p3)2 = −2p1.p4 = −2p2.p3. (5)

Polarisations and Gauge invariance: gluons

ǫ(i),a

µ

→ Ra

b ǫ(i),b µ

, ǫ(i),a

µ

→ ǫ(i),a

µ

+ p(i)

µ ζ(i),a.

(6) Here µ and a are the Lorentz and G-adjoint color index respectively.

It is useful to impose this invariance by thinking of the adjoint valued

polarization vector as a product ǫa

µ = ǫµ ⊗ τ a.

τ (i),a → Ra

b τ (i),b,

ǫ(i)

µ → ǫ(i) µ + p(i) µ ζ(i).

(7)

Subham Dutta Chowdhury Classifying local four gluon S-matrices 6/24

It is convenient to think of the gluon S-matrix as the sum of products,

S(ǫ(i),a

µ

, p(i)

µ ) = Sphoton(ǫ(i) µ , p(i) µ ) Sscalar(τ (i),a) + . . .

(8)

Summary: Gluon S-matrix: Sphoton(ǫ(i)

µ , p(i) µ ) Sscalar(τ (i),a)

Sphoton(ǫ(i)

µ , p(i) µ ): Gauge Invariant, Lorentz invariant

Sscalar(τ (i),a): G invariant. S4 invariant. Apply these in steps. Subham Dutta Chowdhury Classifying local four gluon S-matrices 7/24

S4 permutations: Quasi invariant and non-quasi-invariant S-matrices

The total S-matrix must be S4 invariant. Z2 ⊗ Z2 subgroup leaves the

mandelstam variables s, t and u invariant. I, P12P34, P13P24, P14P23

Since S4 is the semi-direct product S3 ⋉ (Z2 × Z2), we denote the irreducible

representations of (Z2 × Z2) by charges under (P12P34, P13P24, P14P23).

The state with (+, +, +) charge is Z2 ⊗ Z2 invariant polynomial of (ǫi, pi, τ a

i ).

We term as ”quasi-invariant” S-matrix.

The state with (+, −, −), (−, +, −) and (−, −, +) charge are the Z2 ⊗ Z2

non-invariant polynomials of (ǫi, pi, τ a

i ). We term as ”non quasi-invariant”

S-matrix.

Subham Dutta Chowdhury Classifying local four gluon S-matrices 8/24

Finally the full subset of Quasi-invariant gluon S-matrices.

Mgluon = Minv ⊕ Mnon−inv, Minv ≡ Mphoton ⊗ Vscalar, Mnon−inv ≡ Mphoton, non−inv ⊗ Vscalar, non−inv (9)

The S-matrix must also be invariant under the remaining S3.

S4 (Z2 × Z2) = S3. (10)

S3 has three irreducible representations.

1S, 1A, 2M The fundamental representation is 3 = 1s + 2M. The left action of S3 onto itself is 6 = 1S + 1A + 2.2M . We also define 3A = 1A + 2M.

Subham Dutta Chowdhury Classifying local four gluon S-matrices 9/24

Building Blocks II

The space of quasi-invariant and non quasi-invariant S-matrices form a

’module’ over the ’ring of polynomials of s, t, u’.

Local modules: Obtained from local Lagrangians. (Always polynomial in

(s, t)).

Project local modules onto S3 singlets → S-matrix One to one map between equivalence classes of local lagrangians and

S-matrices.

Descendants: Scalar product of a Local module transforming in a particular

irreducible rep of S3 with a polynomial of Mandelstam invariants transforming in the same irrep. Analogous to contracted derivatives acting on the local Lagrangian

Subham Dutta Chowdhury Classifying local four gluon S-matrices 10/24

Colour Module

Let us denote the colour representation corresponding to the gauge group G be

given by ρ

The Z2 × Z2 invariant singlet corresponding to tensor product of four

representations ρ of the gauge group G. ρ⊗4|Z2×Z2 = ρ⊗4 − 3(S2ρ ⊗ ∧2ρ) (11) = n1S + n2M + n1A (12)

The Z2 × Z2 non-invariant singlet from the tensor product of four

representations of the gauge group G. S3 acts non trivially on states with these charges. ρ3 = −S4ρ + S3ρ ⊗ ρ, (13) ρ3A = S4ρ − S3ρ ⊗ ρ + ∧2ρ ⊗ S2ρ.

Subham Dutta Chowdhury Classifying local four gluon S-matrices 11/24

Table: The counting of S3 representations of quasi-invariant color structures. The results for SO(N) with N = 7, 6, 5 are the same as those for N ≥ 9.

SO(N) nS nM nA N ≥ 9 2 2 N = 8 3 2 N = 4 3 3 SU(N) nS nM nA N ≥ 4 2 2 N = 3 1 2 N = 2 1 1

Table: The counting of S3 representations of non-quasi-invariant color structures. The results for SO(N) with N = 5, 3 are the same as those for N ≥ 8.

SO(N) n3 n3A N ≥ 7 N = 6 1 N = 4 1 SU(N) n3 n3A N ≥ 3 1 N = 2

Subham Dutta Chowdhury Classifying local four gluon S-matrices 12/24

Explicit example of construction of Vscalar for SO(N) and SU(N)

For SO(N) (N ≥ 9), there are two quasi-invariant color structures χ3,1 and

χ3,2 both transforming under 3. χ(1)

3,1

= Tr(Φ1Φ2)Tr(Φ3Φ4) χ(1)

3,2

= Tr(Φ1Φ2Φ3Φ4). (14) Both the structures are automatically symmetric under Z2 × Z2.

For SU(N) (N ≥ 4), there are two quasi-invariant color structures ξ3,1 and

ξ3,2 both transforming under 3. ξ(1)

3,1

= Tr(Φ1Φ2)Tr(Φ3Φ4) ξ(1)

3,2

= Tr(Φ1Φ2Φ3Φ4)|Z2×Z2. (15) Here the first structure is automatically symmetric under Z2 × Z2 while the second one requires explicit symmetrization.

We have a systematic classification for all lower N. Subham Dutta Chowdhury Classifying local four gluon S-matrices 13/24

Explicit example of construction of Vscalar, non−inv for SO(N) and SU(N)

Quasi non-invariant modules for SO(6) and SO(4)

• χSO(6),(1) = εijklmnΦij

1 φkl 3 Φmα 2

Φnα

4 |(+−−)

= εijklmnΦij

1 Φkl 3 Φmα 2

Φnα

4

+εijklmnΦij

2 Φkl 4 Φmα 1

Φnα

3 .

(16) This transforms in a 3A of S3.

• χSO(4),(1)

= Φ1 ∧ Φ2Tr(Φ3Φ4)|(+−−) = Φ1 ∧ Φ2Tr(Φ3Φ4) + Φ2 ∧ Φ1Tr(Φ4Φ3) −Φ3 ∧ Φ4Tr(Φ1Φ2) − Φ4 ∧ Φ3Tr(Φ2Φ1). (17) This structure is symmetric under P12 hence transforms as 3.

Subham Dutta Chowdhury Classifying local four gluon S-matrices 14/24

For SU(N) (N ≥ 3) we have one quasi non-invariant structure,

• ξ(1)

3A = Tr(Φ1Φ3Φ2Φ4)|(+−−)

= Tr(Φ1Φ3Φ2Φ4) + Tr(Φ2Φ4Φ1Φ3) −Tr(Φ3Φ1Φ4Φ2) − Tr(Φ4Φ2Φ3Φ1). = 1 2 (Tr({Φ1, Φ3}[Φ2, Φ4]) + Tr({Φ2, Φ4}[Φ1, Φ3])) This state is anti-symmetric under P12 hence transforms as 3A.

Subham Dutta Chowdhury Classifying local four gluon S-matrices 15/24

Mphoton

There are two four derivative generators in the 3 and one six derivative

generators in the 1S of S3. The total number of generators are 7 for D ≥ 5.

The most general parity even gauge invariant photon Lagrangian in D ≥ 5 can

be obtained by taking linear combinations of pairs of contracted derivatives on the three ‘generator’ Lagrangians (also called descendants) Tr(F 2)Tr(F 2), Tr(F 4), − F ab∂aF µν∂bF νρF ρµ (18)

The local module generator dual to the first structure is given by,

E(1)

3,1

= 8Tr(F1F2)Tr(F3F4), E(2)

3,1 = 8Tr(F1F3)Tr(F2F4),

E(3)

3,1

= 8Tr(F1F4)Tr(F3F2), (19)

For D ≥ 5, the most general photonic S-matrix is obtained by taking projection

• f the Local module generators dual to (18) onto S3 singlets.

Subham Dutta Chowdhury Classifying local four gluon S-matrices 16/24

Mphoton,non−inv

Due to interplay between derivatives and polarisation vectors, the simple group

theoretic counting outlined for the scalars does not work.

We use the counting techniques outlined in Minwalla et al 2019 and guess the

answer at each dimension D. We check our prediction through plethystic counting.

Example: For large D, there is only one quasi invariant photon module,

• E(1)

≃ −6F ab

1 ∂aF µν 2 ∂bF νρ 3 F ρµ 4 |(+−−)

= 6

• −F ab

1 ∂aF µν 2 ∂bF νρ 3 F ρµ 4

− F ab

2 ∂aF µν 1 ∂bF νρ 4 F ρµ 3

+F ab

3 ∂aF µν 4 ∂bF νρ 1 F ρµ 2

+ F ab

4 ∂aF µν 3 ∂bF νρ 2 F ρµ 1

• .

(20) We observe that this structure is symmetric under P12.

More such quasi non-invariant modules for D = 7, 6, 5 and 4. Subham Dutta Chowdhury Classifying local four gluon S-matrices 17/24

Tensor Product

Minv ≡ Mphoton ⊗ Vscalar

GSO(N) =

• m,n

am,n

m

• b=1

n

• c=1

Tr

• T eT f

Tr

• T gT h

∂µb∂νcF e

ijF f ji

• ∂µbF g

kl∂νcF h lk

• (21)

Mnon−inv ≡ Mphoton, non−inv ⊗ Vscalar, non−inv

HSU(N) =

• m,n

am,n

m

• b=1

n

• c=1

Tr({T e, T f}[T g, T h])

• ∂µb∂νcF e

αβ∂αF f jk∂µb∂βF g kl∂νcF h lj

−∂µb∂νcF f

αβ∂αF e jk∂µb∂βF g kl∂νcF h lj

• .

(22)

Complete listing done for all N and D ≥ 8. Subham Dutta Chowdhury Classifying local four gluon S-matrices 18/24

Plethystic counting

The Local module generators coming from the Lagrangian, are characterised by

their momentum dimension (∆J) and S3 transformation properties.

The number of independent gauge invariant polynomials at a particular order in

derivatives can be enumerated and re-arranged in the form of a counting partition function. When freely generated, the generating function of their S-matrices is given by, ZS-matrix(x) =

• J

x∆J ZRJ(x) (23)

Example: TrF 2TrF 2 → ZS-matrix(x) = x4(1+x2+x4)

(1−x4)(1−x6).

Subham Dutta Chowdhury Classifying local four gluon S-matrices 19/24

As far as four gluon terms are concerned, most of all gauge invariant

Lagrangians are obtained by taking the products of derivatives of four linearised field strengths

Write down a generating function (called letter partition function)

corresponding to a single gluon field using adjoint characters of group G and photon letter partition function.

We, therefore, can obtain the counting alternatively in the form of a matrix

integral that projects the multiparticle partition function over degree four polynomials of derivatives of the field strength into SO(D) as well as G singlets, modulo equations of motion, modulo bianchi identities and modulo total derivatives.

We obtain a perfect match for both Minv and Mnon−inv. Subham Dutta Chowdhury Classifying local four gluon S-matrices 20/24

SO(N) Gluon partition function N ≥ 9 2x4(4 + 7x2 + 7x4 + 3x6)D N = 8 (2x4(5 + 8x2 + 8x4 + 3x6) + x6)D N = 6 (2x4(4 + 7x2 + 7x4 + 3x6) + x6(x2 + x4 + x6))D N = 4 (3x4(4 + 7x2 + 7x4 + 3x6) + x6(1 + x2 + x4))D SU(N) Gluon partition function N ≥ 4 (2x4(4 + 7x2 + 7x4 + 3x6) + x6(x2 + x4 + x6))D N = 3 (x4(6 + 11x2 + 12x4 + 6x6) + x6(x2 + x4 + x6))D N = 2 x4(4 + 7x2 + 7x4 + 3x6)D

Table: Partition function over the space of Lagrangians in D ≥ 8 involving four F a

αβ’s.

Recall D ≡ 1/((1 − x4)(1 − x6)). Contribution of Mnon−inv is underlined. The rest is the contribution of Minv. Partition function for SO(5) and SO(7) is the same as that for SO(N) for N ≥ 9.

Subham Dutta Chowdhury Classifying local four gluon S-matrices 21/24

Use as a basis

Although we have evaluated local modules over ring over polynomials, we can

think of them as basis vectors for the (finite-dimensional) vector space of S-matrices over s, t, u.

In particular the classification of S-matrices we provide serve as a basis for

expressing all 4 gluon scattering amplitudes with not-necessarily-analytic S-matrices.

As a check of our basis for gluon structures, we express the four gluon

amplitude coming from pure Yang-Mills theory in terms of our structures.

Subham Dutta Chowdhury Classifying local four gluon S-matrices 22/24

Conclusions and future directions

We have classified four-point local gluon S-matrices in arbitrary number of

dimensions and for gauge group SO(N) and SU(N).

Our method is general and can be applied in the straightforward way to other

gauge groups as well.

In effect, the classification of four-point S-matrices is equivalent to

classification of equivalence classes of quartic Lagrangians.

We have used the fact that the gluon S-matrix admits a type of “factorization”

into the S-matrix of adjoint scalars and that of photons. More precisely, Mgluon = Minv ⊕ Mnon−inv (24)

Our classification is done by identifying all the individual scalar and photon

components involved in equation (24).

Subham Dutta Chowdhury Classifying local four gluon S-matrices 23/24

We define a partition function over the space of local S-matrices

Z(x) = Tr x∂ (25) where ∂ is the derivative order. We have tabulated the partition functions in all the dimensions and for all SO(N), SU(N) gauge groups.

Local scalar S-matrices give rise to the so called “truncated solutions” i.e.

solutions having support over only finitely many spins of the conformal crossing

• equation. We expect the gluon S-matrices to parametrize the truncated

solutions for the crossing equation of non-abelian currents- test of bulk locality.

[Polchinski et al ’09]

As a zeroth order problem, one can try to extend the scalar case to the

coloured scalar crossing equation: Scalars charged under Adjoint or fundamental representation of SO(N) or SU(N). Fermions?

In our classification the factorized structure plays an important role. It is

tempting to guess that Colour Kinematics dualities may lead to such quartic corrections to Yang-Mills giving rise to gravitational theories different from Einstein gravity.

[Dixon et al ’12, Huang et al ’12, Johansson et al ’17 etc ]

Subham Dutta Chowdhury Classifying local four gluon S-matrices 24/24