Scattering in Susy large N Chern Simons Theory Shiraz Minwalla Department of Theoretical Physics Tata Institute of Fundamental Research, Mumbai. Cambridge, March 2015 Shiraz Minwalla References Based on K. Inbasekar, S. Jain, S. Mazumdar, S.M. V. Umesh, S. Yokoyama: to appear soon Related earlier work S. Jain, M. Manglik, S.M. , S. Wadia, T. Takimi, S. Yokoyama : ArXiv 1404.6373 Y. Dandekar, M. Manglik, S.M. ArXiv 1407.1322 Shiraz Minwalla Introduction Last three years; fairly intensive study of U(N) Chern Simons theories with fundamental matter. Initial motivations: AdS/CFT. Limit of ABJ theories. Vasiliev Duality. Field theories turn out to be solvable in the large N limit. Effectively vector like (Chern Simons has matrices but with no degrees of freedom). Detailed study has thrown up surprises. First strong weak (Bose-Fermi) duality even without susy. Second, many unusual aspects of dynamics. Shiraz Minwalla Introduction In particular: recent intensive study of 2 2 scattering in purely bosonic and purely fermionic theories.× Results in perfect agreement with duality. However conflict with unitarity unless we assume two modified structural features of S matrices. First: delta function term at forward scattering. Second, modifed rules of crossing symmetry. This talk: present explicit computations for the in S matrix of the most general renormalizable = 1 SUSY Chern Simons theory at all orders in the t’N Hooft coupling. Will find results in perfect agreement with duality. Unitarity requires the same delta function piece at forward scattering and modification of crossing symmetry conjectured for the bosonic and fermionic theories. Strong evidence for universality. Generalization to higher susy (ABJ) underway. Expect same features. Along the way make precise expression of the Bose Fermi nature of Giveon Kutasov dualities. Shiraz Minwalla Theories Z S = d 3xd 2θ − κ 1 β α 1 α β 1 α β Tr DαΓ DβΓ D Γ Γα; Γβ Γ ; Γ Γα; Γβ 2π −4 − 6 f g − 24f gf g 1 α α πw 2 (D Φ¯ + iΦΓ¯ )(DαΦ iΓαΦ) + m ΦΦ¯ + (ΦΦ) ; (1) − 2 − 0 κ Gauge multiplet Γα. Single fundamental matter multiplet φ. Parameters: N; κ (integers).w (dimensionless). m0 (dimensionful). New effectively continuous dimensionless N parameter λ = κ in the large N limit. Shiraz Minwalla Conjectured Duality Motivated by computations of the thermal partition function on S2 S1, about two years ago Jain, Yokoyama and myself× conjectured that this class of theories is self dual. Conjectured dualiy map as follows. Define 3 w 2m λ0 = λ Sgn(λ) ; w 0 = − m0 = − 0 : (2) − 1 + w 0 1 + w N0 = κ N + 1; κ0 = κ j j − − Generalization of level rank duality. Turns out pole mass (see below) 2m m = 0 : (3) 2 + ( 1 + w)λ Sgn(m) − Under duality m0 = m: − Shiraz Minwalla Scattering amplitudes and the duality map Would like to understand duality in more detail. Ideally to construct φD in terms of φ. Question imprecise as φD and φ are not gauge invariant. While arbitrary insertions of φ and φD are not meaningful, insertions that are taken to infinity along lines of equations of motion - i.e. S matrices - are gauge invariant and well defined. For this reason the map between S matrices is a ‘poor mans bosonization’ map between these two theories. This talk: We indpependently compute the S matrix for φ and φD and examine their interrelationship under duality. Also uncover interesting structural features along the way. Shiraz Minwalla S matrices: Susy Solution Z 2 d p 2 α ip:x Φ(x; θ) = p a(p)(1 + mθ ) + θ uα(p)α(p) e 2p0(2π)2 cy 2 α cy −ip:x + a (p)(1 + mθ ) + θ vα(p)α (p) e Susy c ∗ c iQα = uα(p )(a@α + a @αc ) + u (p )( α@a + α @ac ) − i α i − y y c y ∗ y y c y + vα(pi ) a @α + (a ) @(αc )y + vα(pi ) α @a + (α ) @(ac )y (4) Shiraz Minwalla On Shell Superfield Ai (p) = ai (p) + αi (p)θi y y y (5) Ai (p) = ai (p) + θi αi (p) 1 [Qα; Ai (pi ; θi )] = QαAi (pi ; θi ) (6) y 2 y [Qα; Ai (pi ; θi )] = QαAi (pi ; θi ) ! 1 −!@ Q = i uβ(p) vβ(p)θ β − @θ − ! (7) 2 −!@ Q = i vβ(p) uβ(p)θ β @θ − Shiraz Minwalla Dual Supersymmetry Conjectured duality transformation: bose-fermi interchange. a α $ Action of dual susy on A; Ay ! 1 −!@ Q = i vβ(p; m)θ uβ(p; m) β − − − − @θ ! (8) 2 −!@ Q = i uβ(p; m)θ vβ(p; m) β − − − @θ Using u(m; p) = v( m; p); v(m; p) = u(m; p) − − − we see that the dual and original susy generators are proportional to each other. So an S matrix invariant under original susy is automatically invariant under dual susy. Susy compatable with Bose-Fermi interchange. Shiraz Minwalla S matrix in onshell superspace Consider the S matrix S(p ; θ ; p ; θ ; p ; θ ; p ; θ )(2π)3δ(p1 + p2 p3 p4) = 1 1 2 2 3 3 4 4 − − 0 A (p ; θ )A (p ; θ )UAy (p ; θ )Ay (p ; θ ) 0 h j 4 4 4 3 3 3 2 2 2 1 1 1 j i (9) Invariance under susy implies 1 1 −!Q α(p1; θ1) + −!Q α(p2; θ2) 2 2 + −!Q α(p3; θ3) + −!Q α(p4; θ4) S(p1; θ1; p2; θ2; p3; θ3; p4; θ4) = 0 (10) Shiraz Minwalla Solution to susy equation We have explicitly solved (10); the solution is given by S(p1; θ1; p2; θ2; p3; θ3; p4; θ4) = f1 + F2 θ1θ2θ3θ4+ 1 1 C f C∗ F θ θ 2 12 1 − 2 34 2 1 2 1 1 1 1 + C f C∗ F θ θ + C f + C∗ F θ θ 2 13 1 − 2 24 2 1 3 2 14 1 2 23 2 1 4 1 1 1 1 + C f + C∗ F θ θ + C f C∗ F θ θ 2 23 1 2 14 2 2 3 2 24 1 − 2 13 2 2 4 1 1 + C f C∗ F θ θ 2 34 1 − 2 12 2 3 4 (11) Shiraz Minwalla Solution to susy equation Where 1 1 1 1 C = v ∗(p )v ∗(p ) C = v ∗(p )u∗(p ) 2 12 −4m 1 2 2 23 −4m 2 3 1 1 1 1 C = v ∗(p )u∗(p ) C = v ∗(p )u∗(p ) 2 13 −4m 1 3 2 24 −4m 2 4 1 1 1 1 C = v ∗(p )u∗(p ) C = u∗(p )u∗(p ) 2 14 −4m 1 4 2 34 −4m 3 4 (12) Note that susy determines 6 of the 8 scattering amplitudes in terms of the other two. Indpendent data = four boson and four fermion scattering. Shiraz Minwalla Scattering Amplitudes: Colour Kinematics We refer to a quantum (bosonic or ferionic) that transforms in the fundamental of U(N) as a particle and a quantum that tranforms in the antifundamental of U(N) as an antiparticle. We study the most general 2 2 scattering process. There are three such processes × P + Pm P + Pn i ! j j n P + A Pn + A (13) i ! Aj + An Ai + Aj ! All these S matrices are contained in appropriate on shell limits of the single correlator ¯j ¯n < φi φmφ φ > Shiraz Minwalla Color Kinematics It follows from U(N) invariance that ¯j ¯n j n n j < φi φmφ φ >= aδi δm + bδi δm Consequently, in each channel we need to compute 2 distinct scattering amplitudes. For particle-particle scattering it is convenient to work in the following basis for U(N) singlets j n j n n j j n n j < φ φmφ¯ φ¯ >= Tsym δ δ + δ δ + Tas δ δ δ δ i i m i m i m − i m a+b TSym = 2 is the S matrix in the channel of symmetric a−b exchange, while Tas = 2 is the S matrix in the channel of antisymmetric exchange. Shiraz Minwalla 1 Colour Kinematics, N On the other hand for particle - antiparticle exchange the following basis is most useful j n j n ! j n δi δm n j δi δm < φ φmφ¯ φ¯ >= T + T δ δ i Sing N Adj i m − N TSing = Na + b is the scattering matrix in the singlet channel, while TAdj = b is the scattering matrix in the adjoint channel. It is not difficult to convince oneself that, at leading order in 1 the N expansion 1 a b Tsym Tas T ∼ ∼ ∼ ∼ Adj ∼ N On the other hand S (1) Sing ∼ O Shiraz Minwalla 1 Unitarity and N We have chosen the colour structures multiplying TSing etc to be simply projectors onto the exchange representations. As projectors square to unity it follows from unitarity that y y i(Tc T ) = TcT − c c Tc is any one of the four T matrices described above. We have used the fact that e.g. 2 4 production contributes ! 1 to unitarity only at subleading order in N . The unitarity equation is rather trivial for Tsym, Tas, Tadj , as 1 the RHS is subleading in N . However it imposes a highly nontrivial nonlinear constraint on Tsing, the most nontrivial of the four scattering matrices. Shiraz Minwalla Scattering: non relativistic Limit The S matrix for non relativistic particles interacting via Chern Simons exchange was worked out in the early 90s, most notably by Bak, Jackiw and collaborators. The main result is strikingly simple. Consider the scattering of two bosonic particles in representations R1 and R2, in exchange channel R.