散乱振幅で理論的に探る電弱対称性の破れ The electroweak effective field theory from on-shell amplitudes 北原 鉄平 名古屋大学 素粒子宇宙起源研究所 (KMI) / 高等研究院 基研研究会 素粒子物理学の進展 2020 2020 年 9 月 4 日 , オンライン
「中間子の精密測定におけるア こちらをクリック 興味のある方は一緒に共同研究しましょう 本題に入る前に … フレーバーは出てきません グラフや実験結果は出てきません 最新のフレーバーのレビュートークは ノマリーの現状と新物理の識別」 於 物理学会第 75 回年次大会 ( 招待講演 ) , 京都大学セミナー hep-ph と hep-th の境界領域の研究です , Minkowski metric ( + , − , − , − ) D = 4 The electroweak effec@ve field theory from on-shell amplitudes 2 / 24 Teppei Kitahara : Nagoya University, PPP2020, September 4, 2020, online talk
Based on [1709.04891] Nima Arkani-Hamed, Tzu-Chen Huang, Yu-tin Huang Novel formalism [1809.09644] Yael Shadmi, Yaniv Weiss [1909.10551] Technion, scattering Gauthier Durieux, TK , Yael Shadmi, Yaniv Weiss amplitudes group [2008.09652] Gauthier Durieux, TK , Camila S. Machado, Yael Shadmi, Yaniv Weiss
Introduction (1/2) Effective field theory (EFT) can be generally constructed by assuming field contents and Lorentz, global and gauge symmetries, e.g. , SMEFT, HEFT, HQET, SCET, … EFT is bottom-up and natural approach (when one does not discover any new resonance) General problems of (effective) Lagrangian treatment: Find nice operator basis: operator redundancy via field redefinitions and EOMs e.g. , Warsaw basis (dimension-six SMEFT) [Grzadkowski, Iskrzynski, Misiak, Rosiek '10] Gauge redundancy (=gauge-fixing dependence), which is canceled out at amplitude level (after the complicated calculations) The electroweak effec@ve field theory from on-shell amplitudes 4 / 24 Teppei Kitahara : Nagoya University, PPP2020, September 4, 2020, online talk
J6+er1m7fd3rtLmRUCkxnOWCYWEZKEU 5mi pGFrkgKI0YmUfX06o+vyFC0oz/VtucrFKUcBpTjJRGYfc2SJG6woiVP3bhaOCEyfr kVtnT+cvR36dR1X+bJ1ACpNQrj0YxALhMmCIJ4xAx4WBqOXa3+3RBj4A12sAfC e3yb+qNU0chqwC7t9e2jXAR8LpxF90MRF2P0TbDJcpIQrzJCUS8fO1apEQlGsz7OCQpIc4WuUkKW HKVErsra1B38pMkGxpnQiytY0/2OEqVSbtNI76wslO1aBf9XWxYqPl6VlOeFIhzfXxQXDKoMVhOCGyoIVmyrBcKC6rdCfIW03UrP0dImO 0vPxaX7tDxhu4v z85a+zogEPwEQyA 8ZgAr6DCzAD2OgZY2NinJofzG/m1Dy/32oaTc978E+YP+8AivrEsQ= </latexit> ClHicbZFda9swFIZlr9sy7yvZYDe9EQuDjHXBXyG9WCFtKOxmo4OlCcSJkRXZFZVlI8mFYPKL+m92t39T2XUhc3dAh5fn6Oj PVHOqFS2/dcwnxw8f a8 <latexit sha1_base64="Ujo+T7SfZOEH+Ycv rAPdoMlK4M=">A Introduction (2/2) Scattering amplitude (on-shell amplitude, modern amplitude method, or spinor-helicity formalism) is an alternative way to EFTs (will explain at on after next slide) Scattering amplitudes can be bootstrapped from Lorentz symmetry, locality and unitarity Advantages: No operator and gauge redundancies. Gauge invariance is manifest Bypassing Lagrangian, operators, and Feynman rules/diagrams Drastically simple results compared to Feynman methods h 12 i 4 e .g. , gg → ggg corresponds to sum of 25 diagrams. g , 3 + g , 4 + g , 5 + g ) = ig 3 M 5 (1 − g , 2 − s h 12 ih 23 ih 34 ih 45 ih 51 i n is impossible by the Feynman methods g [Mangano, Parke '91] The electroweak effec@ve field theory from on-shell amplitudes 5 / 24 Teppei Kitahara : Nagoya University, PPP2020, September 4, 2020, online talk
On-shell approach to the SMEFT Derive anomalous dimension matrix (one- and two-loop levels) [Cheung, Shen ’15; Bern, Parra-Martinez, Sawyer ’19, ’20; Elias Miro, Ingoldby, Riembau ’20; Jiang, Ma, Shu ’20] Derive non-interference theorem for the new physics operators [Azatov, Contino, Machado, Riva ’16; Craig, Jiang, Li, Sutherland ’20, Jiang, Shu, Xiao, Zheng ’20; Gu, Wang ‘20] Enumeration of independent massless operators (consistent with Hilbert series approach) [Shadmi, Weiss ’18; Ma, Shu, Xiao ’19; Falkowski ’19; Durieux, Machado ’19; Durieux, TK, Machado, Shadmi, Weiss ’20] Hilbert series [Henning, Lu, Melia, Murayama ’15, '17] Investigate the electroweak symmetry (relations from SU(2) L ×U(1) Y SSB) using massive scattering amplitudes [Christensen, Field ’18; Aoude, Machado ‘19; Christensen, Field, Moore, Pinto ’19; Durieux, TK, Shadmi, Weiss ’19; This talk Bachu, Yelleshpur ‘19] The electroweak effec@ve field theory from on-shell amplitudes 6 / 24 Teppei Kitahara : Nagoya University, PPP2020, September 4, 2020, online talk
Spinor-helicity formalism (massless scattering amplitudes) (1/2) reviews e.g ., [Elvang, Huang ’13, Dixon ’13; Schwartz ‘14] Massless particle is an irreducible representations of the Poincaré group; particle i = | p i , h i ⟩ is particle’s helicity h = ± 1/2, ± 1 2 , …, p h n M n ( p h 1 1 , p h 2 Massless n -pt amplitudes are given by (all particles are incoming) n ) Little-group (LG) is subgroup of the Lorentz group, which leaves invariant; p i → p i p i In , SO(2) U(1) LG for massless particle ≃ D = 4 Massless amplitudes are scaled by their helicities { } under U(1) LG transformation h 1 , h 2 , … M n ( p h 1 1 , …, p h n n ) → e 2i ξ ∑ h i M n ( p h 1 1 , …, p h n Little group scaling; n ) The electroweak effec@ve field theory from on-shell amplitudes 7 / 24 Teppei Kitahara : Nagoya University, PPP2020, September 4, 2020, online talk
̂ Spinor-helicity formalism (massless scattering amplitudes) (2/2) Lorentz group irreducible representation symbol (A, B) . spinor-helicity formalism B = 1 A , ̂ 2 ( ̂ J ± i ̂ K ) undotted spinor 2 : (1/2, 0) | i ⟩ α → e − i ξ | i ⟩ α (under LG) λ i , α = u − ( p i ), ¯ v − ( p i ) ⟨ ij ⟩ = − ⟨ ji ⟩ · α → e + i ξ | i ] · · α (under LG) ˜ dotted spinor α 2 *: (0, 1/2) ⟨ ii ⟩ = [ ii ] = 0 λ i = u + ( p i ), ¯ v + ( p i ) | i ] α = p μ α = | i ⟩ α [ i | · p i , α · i σ μ , α · 4-vector p μ 2 × 2 *: (1/2, 1/2) α = p 2 det p i , α · i = 0 α i constrained | ζ ⟩ α [ i | · α α = ε μ ,+ ε + σ μ , α · α = 2 4-vector i , α · i ⟨ i ζ ⟩ ε μ ,± polarization vector auxiliary spinor ζ i p i ⋅ ε ± | i ⟩ α [ ζ | · i = 0, ε ± i ⋅ ( ε ± i )* = − 1 α α = ε μ , − ε − σ μ , α · α = 2 i , α · ∑ ε μ , λ ( ε ν , λ i )* = − η μν i [ i ζ ] i λ =± … … … The electroweak effec@ve field theory from on-shell amplitudes 8 / 24 Teppei Kitahara : Nagoya University, PPP2020, September 4, 2020, online talk
massless → massive [Kleiss, Stirling ’85; Dittmaier ’98; Cohen, Elvang, Kiermaier ‘10] formalize/generalize for any mass and spin particles [1709.04891] Arkani-Hamed, Huang, Huang
Massive-spinor formalism (1/4) [Arkani-Hamed, Huang, Huang ‘17] p i , α · α p 0 i + p 3 p 1 i − ip 2 i ) 2 − ( p 1 i ) 2 − ( p 2 i ) 2 − ( p 3 i i = ( p 0 i ) 2 α = det p i ⋅ σ = det p i , α · p 1 i + ip 2 p 0 i − p 3 i i = m 2 > 0 = p 2 i = 0 : rank 1 → product of two vectors rank 2 → sum of two products of two vectors p i , α · α α ≡ ∑ α = | i 1 ⟩ α [ i 1 | · α + | i 2 ⟩ α [ i 2 | · | i I ⟩ α [ i I | · α = | i ⟩ α [ i | · p i , α · p i , α · α α I =1,2 In , SO(3) SU(2) LG for massive particles; leaves invariant; ≃ α → p i , α · D = 4 p i , α · p i , α · α α Amplitudes are transformed by SU(2) LGs (for massive external particles) | i I ⟩ , | i I ] Bold spinors carry the SU(2) LG index I = 1,2 The electroweak effec@ve field theory from on-shell amplitudes 10 / 24 Teppei Kitahara : Nagoya University, PPP2020, September 4, 2020, online talk
CfHicbZFNSwMxEIaz63f9qnr0YLAKSrHsVlEvgujFYwWrQreU2XTahmazS5IV69pf4T/z5k/xIqa1SG0dCLy8 0ySmQkTwbXxvA/HnZmdm19YXMotr6yurec3Nu91nCqGVRaLWD2GoF wiVXDjcDHRCFEocCHsHs9yD8 odI8lneml2A9grbkLc7AWKuRfwtCbHOZJREYxZ/7EAQ0DFA2f53cBQU6QfmW8v5SR pOUp6l/Om7XouN7KUfKJBtgXRY+Ho0bjXyBa/kDYNOC38kCmQUlUb+PWjGLI1QGiZA65rvJa egTKcCeznglRjAqwLbaxZKSFCXc+Gw+vTfes0aStW9khDh+54RQaR1r0otKTtoqMncwPzv1wtNa3zesZlkhqU7OehViqoielgE7TJFTIjelYAU9z+lbIOKGDG7itnh+BPtjwt7s l/7hUvj0pXF6NxrFItskuOSA+OSOX5IZUSJUw8unsOAfOofPl7rlF9+gHdZ1RzRb5E+7pNy awaU=</latexit> ⃗ <latexit sha1_base64="tOFX2CrFq96zx5VmQv1rQDt9d1A=">A Massive-spinor formalism (2/4) One can use the SU(2) LG rotation for the spin-quantization axis Convenient choice (for any spin particles): Arbitrary spin polarization can be spin axis I = 1 given by two opposite spin states I = 2 ✓ ◆ ✓ ◆ ✓ ◆ a 1 0 p = a | + z i + b | � z i = a + b b 0 1 In this choice, in high energy limit, spinor corresponds to positive (negative) I = 1 ( I = 2) helicities Any choice of spin-quantization axis is possible in general (“SU(2) LG covariant”) The electroweak effec@ve field theory from on-shell amplitudes 11 / 24 Teppei Kitahara : Nagoya University, PPP2020, September 4, 2020, online talk
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