Classifying local four gluon S-matrices Subham Dutta Chowdhury November 20, 2020 YITP Strings and Fields 2020 Subham Dutta Chowdhury 1/24 Classifying local four gluon S-matrices
References � 1910.14392 with Abhijit Gadde, Tushar Gopalka, Indranil Halder, Lavneet Janagal, Shiraz Minwalla � 2006.12458 with Abhijit Gadde. Subham Dutta Chowdhury 2/24 Classifying local four gluon S-matrices
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. s →∞ A ( s, t ) ≤ s 2 lim (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 3/24 Classifying local four gluon S-matrices
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. A Y M = f abc ( ǫ 1 .ǫ 2 ( k 1 − k 2 ) .ǫ 3 + ǫ 1 .ǫ 3 ( k 3 − k 1 ) .ǫ 2 + ( k 2 − k 3 ) .ǫ 1 ǫ 2 .ǫ 3 ) , 2- derivative (2) A F 3 = Tr ( T α 1 T α 2 T α 3 ) F 1 ab F 2 bc F 3 ca + perm. , 3- derivative (3) where, F 1 µν = ( k 1 ,µ ǫ 1 ,ν − k 1 ,ν ǫ 1 ,µ ) � The most general S-matrix then becomes aA Y M + bA F 3 . Subham Dutta Chowdhury 4/24 Classifying local four gluon S-matrices
Analytic S-matrices � Polynomial in s, t and u (the mandelstam variables), ǫ a 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. � ∞ n ( m ) x m . Z S-matrix ( x ) = (4) m =0 � Furthermore, we require Lorentz invariance, gauge invariance, S 4 permutation invariance and G -invariance. . Subham Dutta Chowdhury 5/24 Classifying local four gluon S-matrices
Building blocks I: Scattering data and gauge invariance � Momenta and mandelstam variables. s := − ( p 1 + p 2 ) 2 = − ( p 3 + p 4 ) 2 = − 2 p 1 .p 2 = − 2 p 3 .p 4 t := − ( p 1 + p 3 ) 2 = − ( p 2 + p 4 ) 2 = − 2 p 1 .p 3 = − 2 p 2 .p 4 (5) u := − ( p 1 + p 4 ) 2 = − ( p 2 + p 3 ) 2 = − 2 p 1 .p 4 = − 2 p 2 .p 3 . � Polarisations and Gauge invariance: gluons ǫ ( i ) ,a → R a b ǫ ( i ) ,b ǫ ( i ) ,a → ǫ ( i ) ,a + p ( i ) µ ζ ( i ) ,a . , (6) µ µ µ µ 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 ǫ a µ = ǫ µ ⊗ τ a . τ ( i ) ,a → R a b τ ( i ) ,b , ǫ ( i ) µ → ǫ ( i ) µ + p ( i ) µ ζ ( i ) . (7) Subham Dutta Chowdhury 6/24 Classifying local four gluon S-matrices
� It is convenient to think of the gluon S-matrix as the sum of products, S ( ǫ ( i ) ,a , p ( i ) µ ) = S photon ( ǫ ( i ) µ , p ( i ) µ ) S scalar ( τ ( i ) ,a ) + . . . (8) µ � Summary: � Gluon S-matrix: S photon ( ǫ ( i ) µ , p ( i ) µ ) S scalar ( τ ( i ) ,a ) � S photon ( ǫ ( i ) µ , p ( i ) µ ) : Gauge Invariant, Lorentz invariant � S scalar ( τ ( i ) ,a ) : G invariant. � S 4 invariant. � Apply these in steps. Subham Dutta Chowdhury 7/24 Classifying local four gluon S-matrices
S 4 permutations: Quasi invariant and non-quasi-invariant S-matrices � The total S-matrix must be S 4 invariant. Z 2 ⊗ Z 2 subgroup leaves the mandelstam variables s, t and u invariant. I, P 12 P 34 , P 13 P 24 , P 14 P 23 � Since S 4 is the semi-direct product S 3 ⋉ ( Z 2 × Z 2 ) , we denote the irreducible representations of ( Z 2 × Z 2 ) by charges under ( P 12 P 34 , P 13 P 24 , P 14 P 23 ) . � The state with (+ , + , +) charge is Z 2 ⊗ Z 2 invariant polynomial of ( ǫ i , p i , τ a i ) . We term as ”quasi-invariant” S-matrix. � The state with (+ , − , − ) , ( − , + , − ) and ( − , − , +) charge are the Z 2 ⊗ Z 2 non-invariant polynomials of ( ǫ i , p i , τ a i ) . We term as ”non quasi-invariant” S-matrix. Subham Dutta Chowdhury 8/24 Classifying local four gluon S-matrices
� Finally the full subset of Quasi-invariant gluon S-matrices. M inv ⊕ M non − inv , M gluon = M photon ⊗ V scalar , M inv ≡ M photon , non − inv ⊗ V scalar , non − inv M non − inv ≡ (9) � The S-matrix must also be invariant under the remaining S 3 . S 4 ( Z 2 × Z 2 ) = S 3 . (10) � S 3 has three irreducible representations. 1 S , 1 A , 2 M The fundamental representation is 3 = 1 s + 2 M . The left action of S 3 onto itself is 6 = 1 S + 1 A + 2 . 2 M . We also define 3 A = 1 A + 2 M . Subham Dutta Chowdhury 9/24 Classifying local four gluon S-matrices
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 S 3 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 S 3 with a polynomial of Mandelstam invariants transforming in the same irrep. Analogous to contracted derivatives acting on the local Lagrangian Subham Dutta Chowdhury 10/24 Classifying local four gluon S-matrices
Colour Module � Let us denote the colour representation corresponding to the gauge group G be given by ρ � The Z 2 × Z 2 invariant singlet corresponding to tensor product of four representations ρ of the gauge group G . = ρ ⊗ 4 − 3( S 2 ρ ⊗ ∧ 2 ρ ) ρ ⊗ 4 | Z 2 × Z 2 (11) = n 1 S + n 2 M + n 1 A (12) � The Z 2 × Z 2 non-invariant singlet from the tensor product of four representations of the gauge group G . S 3 acts non trivially on states with these charges. − S 4 ρ + S 3 ρ ⊗ ρ, ρ 3 = (13) S 4 ρ − S 3 ρ ⊗ ρ + ∧ 2 ρ ⊗ S 2 ρ. ρ 3 A = Subham Dutta Chowdhury 11/24 Classifying local four gluon S-matrices
Table: The counting of S 3 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 ) n S n M n A SU ( N ) n S n M n A N ≥ 9 2 2 0 N ≥ 4 2 2 0 N = 8 3 2 0 N = 3 1 2 0 N = 4 3 3 0 N = 2 1 1 0 Table: The counting of S 3 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 ) n 3 n 3 A SU ( N ) n 3 n 3 A N ≥ 7 0 0 N ≥ 3 0 1 N = 6 0 1 N = 2 0 0 N = 4 1 0 Subham Dutta Chowdhury 12/24 Classifying local four gluon S-matrices
Explicit example of construction of V scalar for SO ( N ) and SU ( N ) � For SO ( N ) ( N ≥ 9 ), there are two quasi-invariant color structures χ 3 , 1 and χ 3 , 2 both transforming under 3 . χ (1) = Tr(Φ 1 Φ 2 )Tr(Φ 3 Φ 4 ) 3 , 1 χ (1) = Tr(Φ 1 Φ 2 Φ 3 Φ 4 ) . (14) 3 , 2 Both the structures are automatically symmetric under Z 2 × Z 2 . � For SU ( N ) ( N ≥ 4 ), there are two quasi-invariant color structures ξ 3 , 1 and ξ 3 , 2 both transforming under 3 . ξ (1) = Tr(Φ 1 Φ 2 )Tr(Φ 3 Φ 4 ) 3 , 1 ξ (1) = Tr(Φ 1 Φ 2 Φ 3 Φ 4 ) | Z 2 × Z 2 . (15) 3 , 2 Here the first structure is automatically symmetric under Z 2 × Z 2 while the second one requires explicit symmetrization. � We have a systematic classification for all lower N . Subham Dutta Chowdhury 13/24 Classifying local four gluon S-matrices
Explicit example of construction of V scalar , non − inv for SO ( N ) and SU ( N ) � Quasi non-invariant modules for SO (6) and SO (4) χ SO (6) , (1) = ε ijklmn Φ ij 1 φ kl 3 Φ mα Φ nα ε ijklmn Φ ij 1 Φ kl 3 Φ mα Φ nα � 4 | (+ −− ) = 2 2 4 + ε ijklmn Φ ij 2 Φ kl 4 Φ mα Φ nα 3 . 1 (16) This transforms in a 3 A of S 3 . χ SO (4) , (1) � = Φ 1 ∧ Φ 2 Tr(Φ 3 Φ 4 ) | (+ −− ) = Φ 1 ∧ Φ 2 Tr(Φ 3 Φ 4 ) + Φ 2 ∧ Φ 1 Tr(Φ 4 Φ 3 ) − Φ 3 ∧ Φ 4 Tr(Φ 1 Φ 2 ) − Φ 4 ∧ Φ 3 Tr(Φ 2 Φ 1 ) . (17) This structure is symmetric under P 12 hence transforms as 3 . Subham Dutta Chowdhury 14/24 Classifying local four gluon S-matrices
� For SU ( N ) ( N ≥ 3 ) we have one quasi non-invariant structure, ξ (1) � 3 A = Tr(Φ 1 Φ 3 Φ 2 Φ 4 ) | (+ −− ) = Tr(Φ 1 Φ 3 Φ 2 Φ 4 ) + Tr(Φ 2 Φ 4 Φ 1 Φ 3 ) − Tr(Φ 3 Φ 1 Φ 4 Φ 2 ) − Tr(Φ 4 Φ 2 Φ 3 Φ 1 ) . 1 = 2 (Tr( { Φ 1 , Φ 3 } [Φ 2 , Φ 4 ]) + Tr( { Φ 2 , Φ 4 } [Φ 1 , Φ 3 ])) This state is anti-symmetric under P 12 hence transforms as 3 A . Subham Dutta Chowdhury 15/24 Classifying local four gluon S-matrices
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