Scattering Amplitudes of Massive N = 2 Gauge Theories in Three Dimensions Donovan Young NORDITA Abhishek Agarwal, Arthur Lipstein, DY, arXiv:1302.5288 Gauge/Gravity Duality 2013, M¨ unchen, July 30, 2013
Outline I. Invitation to amplitudeology • Twistors • BDS • BCFW recursion relations II. Mass-deformed N = 2 amplitudes in d = 3 • Mass-deformed Chern-Simons theory (CSM) • Yang-Mills-Chern-Simons theory (YMCS) • Massive spinor-helicity in d = 3 • Trouble with YMCS external gauge fields • On-shell SUSY algebras • Four-point amplitudes: superamplitude for CSM • Massive BCFW in d = 3 III. Conclusions and looking forward Gauge/Gravity Duality 2013, M¨ unchen, July 30, 2013 1
Part I: Amplitudeology Gauge/Gravity Duality 2013, M¨ unchen, July 30, 2013 2
Amplitudeology • Parke-Taylor formula: massive simplification of amplitudes in “spinor-helicity” variables; Witten’s twistor string theory • BCFW recursion: n -point amplitudes from n − 1 -point amplitudes • Unitarity methods to construct loop-level amplitudes • BCJ relations: duality between colour and kinematics • KLT relations: gauge-theory amplitudes 2 = gravity amplitudes • Grassmannian formulation BDS formula N =4 , ABJM Dual superconformal symmetry, Yangian, null polygonal Wilson loops N =4 , ABJM Connections to spectral problem integrability N =4 , ABJM Gauge/Gravity Duality 2013, M¨ unchen, July 30, 2013 3
Penrose’s twistors • Penrose’s concept of twistors turns out to be an immensely powerful technique for describing massless amplitudes. • Idea is to coordinatize space by the bundle of light-rays passing through a given point: i.e. by the local celestial sphere. • Imagine two observers at different places in the galaxy. Knowledge of their celestial spheres is enough to determine their locations: Observer A Observer B Gauge/Gravity Duality 2013, M¨ unchen, July 30, 2013 4
Twistors Homogeneous coordinates of CP 3 : Z I = ( Z 1 , Z 2 , Z 3 , Z 4 ) , Z I ∼ λZ I , λ ∈ C . ⇒ null condition) For a given twistor Z I , the incidence relation ( = � � � � Z 1 Z 3 ⇒ Im ( Z 1 Z ∗ 3 + Z 2 Z ∗ = σ µ x µ = 4 ) = 0 , Z 2 Z 4 fixes x µ = (0 , � x 0 ) + k µ τ with k 2 = 0, i.e. specifies a single light ray, going through a specific point in space. Two (or more) twistors Z I and Z ′ I incident to the same point � x 0 specify two (or x 0 )+ k ′ µ τ . x 0 )+ k µ τ and (0 , � more) different light rays through that point, i.e. (0 , � x 0 , the incidence relation takes CP 3 → CP 1 ∼ S 2 which is nothing For fixed � but the celestial sphere at � x 0 . Gauge/Gravity Duality 2013, M¨ unchen, July 30, 2013 5
Spinor-helicity variables On-shell massless particle representations a = p µ ( σ µ ) a ˙ a = λ a ¯ ˙ b ¯ i ¯ p a ˙ λ ˙ a , � ij � = ǫ ab λ a i λ b λ ˙ a b j , [ ij ] = ǫ ˙ λ j , a ˙ with which the Parke-Taylor formula for MHV tree-level gluon scattering am- plitudes is expressed: � � n � ij � 4 � � � ∝ δ 4 λ a i ¯ λ ˙ a − , . . . , − , + , − , . . . , − , + , − , . . . , − � 12 �� 23 � . . . � n 1 � . j ���� ���� ���� ���� i =1 1 i j n Notice that expression is “holomorphic” i.e. does not depend on ¯ λ . Fourier transform w.r.t. ¯ λ [Witten, 2003] � � � � � � � d 2 ¯ λ i � � d 4 x a ¯ λ ˙ a λ a i ¯ λ ˙ a f ( { λ } ) (2 π ) 2 exp i µ i ˙ exp ix a ˙ a i i i i i � � δ 2 ( µ i ˙ d 4 x a λ a f ( { λ } ) → define twistor Z I = ( λ a , µ ˙ = a + x a ˙ i ) a ) . i � �� � INCIDENCE RELATION The particles (light rays) interact at a common point in space-time. Gauge/Gravity Duality 2013, M¨ unchen, July 30, 2013 6
Colour ordering, BDS formula In large-N gauge theories we have fields φ = φ a T a , where T a is (for example) a SU ( N ) generator. Colour ordering refers to (e.g. for 4-particle scattering) � � φ a 1 † ( p 1 ) φ a 2 † ( p 2 ) φ a 3 † ( p 3 ) φ a 4 † ( p 4 ) = M ( p 1 , p 2 , p 3 , p 4 ) Tr[ T a 1 T a 2 T a 3 T a 4 ] + . . . this restricts to the ( p 1 + p 2 ) 2 and ( p 1 + p 4 ) 2 , i.e. adjacent, channels. In N = 4, d = 4 SYM, the MHV amplitudes have a conjectured all-orders form [Bern, Dixon, Smirnov, 2005] � �� n � � � λµ 2 ǫ λµ 2 ǫ log M MHV 1 + 1 + f ( λ ) R � 8 ǫ 2 f ( − 2) 4 ǫg ( − 1) IR IR = − 4 + finite . M tree ( − s i,i +1 ) ǫ ( − s i,i +1 ) ǫ MHV i =1 where f ( − n ) ( λ ) in the n -th logarithmic integral of the cusp anomalous dimension f ( λ ). IR divergences have been regulated by going above four dimensions, i.e. d = 4 − 2 ǫ with ǫ < 0. Gauge/Gravity Duality 2013, M¨ unchen, July 30, 2013 7
Dual superconformal symmetry and Wilson loops Alday & Maldacena taught us that at strong coupling, the dual of the amplitude is the dual of a null-polygonal Wilson loop: i.e. a string worldsheet: √ � 1 � � � � M MHV λ x µ A µ − = N Tr P exp dτ i ˙ = exp 2 π (Area of Min. Surf.) M tree C MHV Moreover, the duality holds also at weak coupling [Brandhuber, Heslop, Travaglini, 2007]. Reason: under T-duality p i ↔ x i +1 − x i , and AdS is mapped to itself . Amplitude is dual to high energy scattering on an IR brane ` a la Gross & Mende, T-duality maps it to the null-polygon in the UV, i.e. on the boundary. The picture which has emerged is that there is a full dual PSU (2 , 2 | 4) symmetry and a Yangian symmetry relating the two [Drummond, Henn, Plefka, 2009]. Gauge/Gravity Duality 2013, M¨ unchen, July 30, 2013 8
Recursion relations p i → ˜ p i = p i + z q , p j → ˜ p j = p j − z q , A ( z ) = p 2 p 2 ˜ i = 0 = ˜ j , q 2 = q · p i = q · p j = 0 . [Britto, Cachazo, Feng, Witten, 2005] • A ( z ) = F ( z ) G ( z ) i.e. amplitude is rational. • Poles in z are simple. • lim z →∞ A ( z ) = 0. ⇒ A ( z ) = � p Res[ A ( z ) , z p ] / ( z − z p ) = Gauge/Gravity Duality 2013, M¨ unchen, July 30, 2013 9
Recursion relations cont’d P = � L p = − � R p = no z dep. P = � L p = − � R p = depends on z A L ( z p ) A R ( z p ) A L ( z p ) A R ( z p ) � � A ( z ) = ⇒ A = A (0) = = , P 2 ( z ) P 2 splittings splittings � � � � A L ( z p ) = A A R ( z p ) = A . . . , p j ( z p ) , . . . , − P ( z p ) . . . , p i ( z p ) , . . . , P ( z p ) , , P 2 ( z p ) = 0 . Gauge/Gravity Duality 2013, M¨ unchen, July 30, 2013 10
A few slides motivating amplitudes in three-dimensions Gauge/Gravity Duality 2013, M¨ unchen, July 30, 2013 11
Three-dimensional theories: ABJM • Propagating degrees of freedom are scalars and fermions. Results have not been interpreted in terms of “helicity”. • A tree = δ 3 ( P ) δ 6 ( Q ) / � � 12 �� 23 �� 34 �� 41 � , six-partilcle result also known [Agar- 4 wal, Beisert, McLoughlin, 2008], [Bargheer, Loebbert, Meneghelli, 2010]. • BCFW and dual super-conformal invariance [Gang, Huang, Koh, Lee, Lip- stein, 2011]. • Extensions to loop-level performed [Chen, Huang, 2011] [Bianchi, Leoni (2) , Mauri, Penati, Santambrogio, 2011] [Caron-Huot, Huang, 2013]. • Yangian constructed [Bargheer, Loebbert, Meneghelli, 2010]. • Grassmanian proposed [Lee, 2010]. • Light-like Wilson loop seems to match amplitudes [Henn, Plefka, Wiegandt, 2010], [Bianchi, Leoni, Mauri, Penati, Santambrogio, 2011]. Gauge/Gravity Duality 2013, M¨ unchen, July 30, 2013 12
Three-dimensional theories: N = 8 SYM and ABJM • Strong coupling IR fixed point of N = 8 SYM is believed to be ABJM. • Can be seen using M2-to-D2 Higgsing of ABJM. • On-shell supersymmetry algebras of the two theories (and analogues with less SUSY) may be mapped to each other [Agarwal, DY (2012)]. • One-loop MHV vanish, one-loop non-MHV are finite [Lipstein, Mason (2012)]. – ABJM 1-loop amps either vanish or are finite. • N = 8 amps have dual conformal covariance [Lipstein, Mason (2012)], ABJM amps have dual conformal invariance. • 4-pt. 2-loop amplitudes agree in the Regge limit between the two theories [Bianchi, Leoni (2013)]. Gauge/Gravity Duality 2013, M¨ unchen, July 30, 2013 13
Mass-deformed three-dimensional theories: N ≥ 4 Chern-Simons-Matter amplitudes [Agarwal, Beisert, McLoughlin (2008)] • Amplitudes computed at the tree and one-loop level. • Exploited SU (2 | 2) algebra to relate amplitudes to one another – same con- traints at play in N = 4 SYM spin chains! Now we will look at massive Chern-Simons-Matter theory with N = 2, and also at another way of introducing mass: Yang-Mills-Chern-Simons theory. Gauge/Gravity Duality 2013, M¨ unchen, July 30, 2013 14
Part II: Mass-deformed N = 2 amplitudes in d = 3 Gauge/Gravity Duality 2013, M¨ unchen, July 30, 2013 15
N = 2 massive Chern-Simons-matter theory � ǫ µνρ Tr( A µ ∂ ν A ρ + 2 i S CSM = κ 3 A µ A ν A ρ ) � � Tr | D µ Φ | 2 + 2 i Tr ¯ Ψ( D µ γ µ Ψ + m Ψ) − 2 � � − 2 � | [Φ , [Φ † , Φ]] + e 2 Φ | 2 � + 2 i Tr([Φ † , Φ][ ¯ Ψ , Ψ] + 2[ ¯ Ψ , Φ][Φ † , Ψ]) Tr κ 2 κ • Gauge field is non-dynamical: external states are Φ’s and Ψ’s. • Mass is set by e : this quantity does not run, m = e 2 /κ . • κ = k/ (4 π ), k is CS level. • Couplings in potential include Φ 6 , Φ 4 , and Φ 2 Ψ 2 . Gauge/Gravity Duality 2013, M¨ unchen, July 30, 2013 16
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