On a hidden structure in the MHV Lagrangian Piotr Kotko IFJ PAN - PowerPoint PPT Presentation

THE HENRYK NIEWODNICZAŃSKI INSTITUTE OF NUCLEAR PHYSICS POLISH ACADEMY OF SCIENCES On a hidden structure in the MHV Lagrangian Piotr Kotko IFJ PAN supported by: DEC-2013/10/E/ST2/00656 based on: P .K., A. Stasto, JHEP 1709 (2017) 047 REF2017, Madrid

MOTIVATION Amplitudes in QCD • crucial ingredients of factorization approaches (collinear and High Energy (or k T ) Factorization) • a lot of progress has been made, especially at tree-level • Feynman diagrams (not efficient in general) • recurrence methods • geometry (eg. amplituhedron in supersymmetric theory) • still a tip of the iceberg 1

MOTIVATION Amplitudes in QCD • crucial ingredients of factorization approaches (collinear and High Energy (or k T ) Factorization) • a lot of progress has been made, especially at tree-level • Feynman diagrams (not efficient in general) • recurrence methods • geometry (eg. amplituhedron in supersymmetric theory) • still a tip of the iceberg In this talk 1 We will forget about ordinary Feynman rules (in favor of much more effective Cachazo-Svrcek-Witten (CSW) rules). 2 I’ll discuss what’s truly hidden in the Lagrangian generating the CSW rules (the MHV Lagrangian – equivalent to standard Yang-Mills), and I’ll mention some consequences. 1

On-shell Amplitudes (1) Collinear Factorization � dx A dx B � f a / A ( x A , µ ) f b / B ( x B , µ ) d σ ab → n ( x A , x B ) d σ AB → n + X ∼ x A x B a , b d σ ab → n ∼ |M| 2 dPS M – on-shell amplitude (on-shell limit of the amputated momentum space Green’s function) Gluon amplitudes: a 2 , ε λ 2 2 . . k 2 . M a 1 ... a n � � ε λ 1 1 , . . . , ε λ n = n k 1 a 1 , ε λ 1 1 a i – color index, ε λ i i – polarization vector of gluon with momentum k i and helicity λ i . Ward identities: M a 1 ... a n � � ε λ 1 1 , . . . , k i , . . . , ε λ n = 0 n 2

On-shell Amplitudes (2) Selected methods of calculating on-shell amplitudes Tree-level techniques (some extended to NLO): • Berends-Giele recursion relations 1 • Britto-Cachazo-Feng-Witten (BCFW) recursion relations 2 • Cachazo-Svrcek-Witten (CSW) method 3 Specific for loop corrections: • integrand reduction method 4 • generalized unitarity 5 Automatization: • efficient computer codes for any tree-level amplitude • most amplitudes can be calculated automatically at NLO 1 F.A. Berends, W.T. Giele, Nucl.Phys. B306 (1988) 759-808 2 R. Britto, F. Cachazo, B. Feng, E. Witten, Phys.Rev.Lett. 94 (2005) 181602 3 F. Cachazo, P . Svrcek, E. Witten, JHEP 0409 (2004) 006 4 G. Ossola, C.G. Papadopoulos, R. Pittau, Nucl.Phys. B763 (2007) 147-169 5 Z. Bern, L.J. Dixon, D.C. Dunbar, D.A. Kosower, Nucl.Phys. B425 (1994) 217; Nucl.Phys. B435 (1995) 59 3

Off-shell Amplitudes (1) High Energy Factorization (HEF) � dx A dx B d 2 k TA d 2 k TB F g / A ( x A , k TA ) F g / B ( x B , k TB ) d σ g ∗ g ∗ → n ( x A , x B , k TA , k TB ) d σ AB → n + X ∼ x A x B k 2 k 2 A � 0 , B � 0 p A k A · · · . + . . . + . . . . . connected k A = x A p A + k TA k B · · · k B = x B p B + k TB p B 2 dPS � � � ˜ d σ g ∗ g ∗ → n ∼ � M � � ˜ M – off-shell gauge invariant amplitude. 4

Off-shell Amplitudes (2) Methods of calculating off-shell amplitudes Tree-level techniques: • Lipatov’s effective action 1 • Analog of the Berends-Giele recursion relation (one off-shell leg) 2 • the “eikonalization” method 3 • Matrix Elements of straight infinite Wilson lines 4 • generalization of the BCFW recursion relations 5 Specific for loop corrections: • generalization of the “eikonalization” method to one loop 6 Automatization: • efficient computer code for any tree-level amplitude 7 1 E. Antonov, L. Lipatov, E. Kuraev, I. Cherednikov, Nucl.Phys. B721 (2005) 111-135 2 A. van Hameren, P .K., K. Kutak, JHEP 1212 (2012) 029 3 A. van Hameren, PK, K. Kutak, JHEP 1301 (2013) 078 4 P .K., JHEP 1407 (2014) 128 5 A. van Hameren, JHEP 1407 (2014) 138, A. van Hameren, M. Serino, JHEP 1507 (2015) 010, K. Kutak, A. van Hameren, M. Serino, JHEP 1702 (2017) 009 6 A. van Hameren, arXiv:1710.07609 7 A. van Hameren, arXiv:1611.00680 5

MHV amplitudes (1) The helicity amplitudes with growing complexity • Vanishing amplitudes: ( ± , + , . . . , +) • Maximally Helicity Violating (MHV) amplitudes: ( − , − , + . . . , +) • Next-to-MHV (NMHV) amplitudes ( − , − , − , + . . . , +) • Next-to-next-to-MHV (NNMHV) amplitudes, and so on... 6

MHV amplitudes (1) The helicity amplitudes with growing complexity • Vanishing amplitudes: ( ± , + , . . . , +) • Maximally Helicity Violating (MHV) amplitudes: ( − , − , + . . . , +) • Next-to-MHV (NMHV) amplitudes ( − , − , − , + . . . , +) • Next-to-next-to-MHV (NNMHV) amplitudes, and so on... Color decomposition 1 M a 1 ... a n � � � � 1 λ 1 , . . . , n λ n � ε λ 1 1 , . . . , ε λ n Tr ( t a 1 . . . t a n ) M = n noncyclic permutations For example for four-point MHV amplitude M a 1 a 2 a 3 a 4 � � 2 , ε + 3 , ε + ε − 1 , ε − 4 + − − k 2 k 3 + k 3 k 2 M (1 − , 2 − , 3 + , 4 + ) = M (1 − , 3 + , 2 − , 4 + ) = and so on . + + k 1 k 4 k 1 k 4 − − 1 M.L. Mangano, S.J. Parke, Phys.Rep. 200 301-367 6

MHV amplitudes (3) Spinor algebra • Spinor products ˙ i λ β β ˜ i ˜ β � ij � = u − ( k i ) u + ( k j ) ≡ ǫ αβ λ α λ ˙ α [ ij ] = u + ( k i ) u − ( k j ) ≡ ǫ ˙ j , λ α ˙ j i ≡ u + ( k i ) , ˜ where u ± ( k i ) = 1 λ ˙ 2 ( 1 ± γ 5 ) u ( k i ) and λ α α i ≡ u − ( k i ) . i ˜ λ ˙ Momenta k i are light-like. One can express ( k i ) α ˙ α = λ α α i . 7

MHV amplitudes (3) Spinor algebra • Spinor products ˙ i λ β β ˜ i ˜ β � ij � = u − ( k i ) u + ( k j ) ≡ ǫ αβ λ α λ ˙ α [ ij ] = u + ( k i ) u − ( k j ) ≡ ǫ ˙ j , λ α ˙ j i ≡ u + ( k i ) , ˜ where u ± ( k i ) = 1 λ ˙ 2 ( 1 ± γ 5 ) u ( k i ) and λ α α i ≡ u − ( k i ) . i ˜ λ ˙ Momenta k i are light-like. One can express ( k i ) α ˙ α = λ α α i . • Polarization vectors q ˜ ˜ λ α λ ˙ α λ ˙ q λ α α √ √ � � i i ε + ( ε − α = i ) α ˙ α = 2 � qi � , 2 i [ iq ] α ˙ where q is a null reference momentum. 7

MHV amplitudes (3) Spinor algebra • Spinor products ˙ i λ β β ˜ i ˜ β � ij � = u − ( k i ) u + ( k j ) ≡ ǫ αβ λ α λ ˙ α [ ij ] = u + ( k i ) u − ( k j ) ≡ ǫ ˙ j , λ α ˙ j i ≡ u + ( k i ) , ˜ where u ± ( k i ) = 1 λ ˙ 2 ( 1 ± γ 5 ) u ( k i ) and λ α α i ≡ u − ( k i ) . i ˜ λ ˙ Momenta k i are light-like. One can express ( k i ) α ˙ α = λ α α i . • Polarization vectors q ˜ ˜ λ α λ ˙ α λ ˙ α q λ α √ √ � � i i ε + ( ε − α = i ) α ˙ α = 2 � qi � , 2 i [ iq ] α ˙ where q is a null reference momentum. Parke-Taylor amplitudes 1 � 12 � 4 � 1 − , 2 − , 3 + , . . . , n + � M = � 12 � � 23 � . . . � n 1 � 1 S.J. Parke, T.R. Taylor, Phys.Rev.Lett. 56, 2459 (1986) 7

Off-shell gauge invariant MHV amplitudes Suprising result One off-shell leg pocess g ∗ g → g . . . g 1 M − 1 M − 2 N − m − 1 + + . . . + M N − m m m N − m − k k m = 1 m = 1 k = 1 . . . . . . . . . . . . . . . . . . � 1 ∗ 2 � 4 � 1 ∗ , 2 − , 3 + , . . . , n + � M g ∗ g → g ... g ∼ � 1 ∗ 2 �� 23 �� 34 � . . . � n − 1 n �� n 1 ∗ � Spinor products for off-shell states involve only longitudinal component of the off-shell momentum � 1 ∗ i � = � p 1 i � , where k 1 = p 1 + k T 1 , k 2 1 � 0, p 2 1 = 0. Similar formula holds for g ∗ g ∗ → g . . . g 2 . Note: it is essential that the amplitude is gauge invariant, otherwise we get a mess! 1 A. van Hameren, P .K., K. Kutak, JHEP 1212 (2012) 029 2 A. van Hameren, JHEP 1407 (2014) 138 8

Cachazo-Svrcek-Witten (CSW) Method (1) General idea Glue any amplitude from the MHV amplitudes continued off-shell. 9

Cachazo-Svrcek-Witten (CSW) Method (1) General idea Glue any amplitude from the MHV amplitudes continued off-shell. Off-shell continuation of spinors If k is light-like, we have α = λ k α ˜ α ˜ λ ˙ α k α ˙ λ k ˙ = ⇒ λ k α = k α ˙ q / [ kq ] α where q is auxiliary light-like momentum. If k is off-shell we define the off-shell continuation of spinor in the same way: λ ( ∗ ) α ˜ λ ˙ α k α = k α ˙ q 9

Cachazo-Svrcek-Witten (CSW) Method (1) General idea Glue any amplitude from the MHV amplitudes continued off-shell. Off-shell continuation of spinors If k is light-like, we have α = λ k α ˜ α ˜ λ ˙ α k α ˙ λ k ˙ = ⇒ λ k α = k α ˙ q / [ kq ] α where q is auxiliary light-like momentum. If k is off-shell we define the off-shell continuation of spinor in the same way: λ ( ∗ ) α ˜ λ ˙ α k α = k α ˙ q MHV vertices + i − . . . + � ij � 4 . . . . . � 12 � � 23 � . . . � n 1 � ≡ . + . . . j − + The spinor products are made from off-shell spinors � ij � = ǫ αβ λ ( ∗ ) α λ ( ∗ ) β . i j 9

Cachazo-Svrcek-Witten (CSW) Method (2) Example: NMHV amplitude M ( 1 − , 2 − , 3 − , 4 + , 5 + ) 2 − 2 − 2 − 1 − 1 − 3 − 3 − + + = 3 − − 1 − − 5 + 4 + 4 + 5 + 4 + 5 + 2 − 2 − 1 − 3 − 1 − 3 − − + + − 4 + 4 + 5 + 5 + 10