Gauge invariant regularization for perturbative chiral gauge theory Yu Hamada Kyoto Univ. 2018/7/30 @Strings and Fields 2018 Based on YH, H. Kawai, arXiv:1705.01317, YH, H. Kawai, K. Sakai, arXiv:1806.00349 YH, H. Kawai, K. Sakai, to appear 1 / 20
Motivation Standard Model is a chiral gauge theory (CGT) SU (3) C × SU (2) L × U (1) Y (EW sector) However, regularization of CGT is difficult! • No lattice regulator [cf. Grabowska-Kaplan, 2015] • No manifestly gauge-invariant perturbative regulator because fermion mass term is forbidden by chiral gauge symmetry... 2 / 20
Regularization problem for CGT Eg. Dimensional Regularization � / � / L reg. = ¯ D (4) + / � � ψ D ( ǫ ) P L ψ, D ( ǫ ) , γ 5 = 0 • ǫ -dimensional kinetic term behaves as “mass term” • Gauge sym. is broken even when anomaly-free theory • Need extra local counter terms to restore gauge sym: Γ[ A ] + ∆Γ[ A ] s.t. δ ω (Γ[ A ] + ∆Γ[ A ]) = 0 • However, the procedure is rather complicated... Is there a gauge-invariant regularization for CGT? 3 / 20
Our answer ✓ ✏ 5D Domain-Wall fermion with PV regulators + (4 + ǫ ) -D gauge field ✒ ✑ This regularization is quite useful! 4 / 20
One-loop diagrams 5 / 20
One-loop diagrams 5 / 20
Domain-wall fermion [Kaplan, 1992] � ∞ � d 4 x ¯ � / � S DW = ds ψ ( x, s ) ∂ (4) + γ 5 ∂ s − Mǫ ( s ) ψ ( x, s ) −∞ • ǫ ( s ) is the sign function ( M > 0) → induce LH massless mode ∝ e − M | s | M� ( s ) • s -direction’s size is infinitely large + M → RH massless mode is decoupled s − M • Massive modes form a continuous spectrum and give rise to IR divergence s = 0 → cancel by bosonic field φ ( x, s ) with constant mass ( − M ) 6 / 20
Action ✓ ✏ � / � � d 4 x ¯ � S = ds ψ ( x, s ) D (4) + γ 5 ∂ s − Mǫ ( s ) ψ ( x, s ) � / � � d 4 x ds ¯ � + φ ( x, s ) D (4) + γ 5 ∂ s + M φ ( x, s ) 1 � d 4 x tr( F µν ( x ) F µν ( x )) + 4 g 2 ✒ ✑ • boson φ will cancel IR div. from ψ • Gauge field is 4-dimensional one A µ ( x ) : s -indep. & A 5 = 0 s • Dirac fermion (boson) are expected to be regularized in a gauge-invariant way 7 / 20
Action in 4D form [Narayanan-Neuberger, 1993] ✓ ✏ � / � � d 4 x ¯ � S = ds ψ ( x, s ) D (4) + γ 5 ∂ s − Mǫ ( s ) ψ ( x, s ) � / � � ds ¯ d 4 x � + φ ( x, s ) D (4) + γ 5 ∂ s + M φ ( x, s ) 1 � d 4 x tr( F µν ( x ) F µν ( x )) + 4 g 2 ✒ ✑ � ˆ M ψ ≡ − ∂ s − Mǫ ( s ) ⇓ “mass operators” ˆ M φ ≡ − ∂ s + M ✓ ✏ � � � � d 4 x ¯ D (4) + ˆ M ψ P L + ˆ M † S = ds ψ ( x, s ) / ψ P R ψ ( x, s ) � � � � d 4 x ds ¯ D (4) + ˆ M φ P L + ˆ M † / + φ ( x, s ) φ P R φ ( x, s ) 1 � d 4 x tr( F µν ( x ) F µν ( x )) , + 4 g 2 ✒ ✑ s -space looks like an internal space of 4D spinors ψ, φ 8 / 20
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sha1_base64="2NElDx1lUoUQGeDbuy189XNog=">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</latexit> Vacuum polarization diagram • Propagator � � � � M † p + ˆ 1 p + ˆ 1 G ψ ( p ) = − i/ M ψ M ψ P R + − i/ P L ψ p 2 + ˆ M † ψ ˆ p 2 + ˆ M ψ ˆ M † ψ p + ˆ M φ P R + ˆ M † − i/ φ P L G φ ( p ) = p 2 + ˆ M † φ ˆ M φ • Vaccum polarization diagram p k k − Π µν ( k ) = p 0 = p − k d 4 p � G ψ ( p ) γ µ G ψ ( p ′ ) γ ν G φ ( p ) γ µ G φ ( p ′ ) γ ν � � � � �� = Tr − Tr (2 π ) 4 • Tr includes the trace over s -space where ˆ M ψ and ˆ M φ act. • We will regulate the UV divergence later. 9 / 20
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