Renormalisation flow in a Yukawa model with quadratic symmetry - PowerPoint PPT Presentation

Renormalisation flow in a Yukawa model with quadratic symmetry breaking Istv´ an Kaposv´ ari, Antal Jakov´ ac and Andr´ as Patk´ os Institute of Physics, E¨ otv¨ os University, Budapest Outline: • The model, its symmetries and spectra (Reminder of the Cakovec-2016 talk) • Spectra in Local Potential Approximation ( LPA ) interpretation through RG-flow structure • Effect of the wavefunction renormalisation ( LPA ′ ): adiabatic deformation of the LPA -flow • Conclusions

The model Complex scalar and single Dirac-fermion with chiral Yukawa-interaction Φ = 1 ψ R/L = 1 √ 2(Φ 1 ( x ) + i Φ 2 ( x )) , 2(1 ± γ 5 ) ψ. The symmetric part of the scale dependent action k < Λ : � ψ R ψ L Φ ∗ + ¯ Γ ( k ) Z ψ ¯ ∂ψ + Z φ ∂ m Φ ∗ ∂ m Φ + U k (Φ ∗ Φ) + h k ( ¯ d 4 x � � SY M = ψ / ψ L ψ R Φ) . Symmetries: ψ → e − iα ¯ ¯ ψ → e iα ψ, ψ Φ → Φ , U(1) , ψ → ¯ ¯ ψ → e iγ 5 Θ ψ, ψe iγ 5 Θ , Φ → e − 2 i Θ Φ , U A (1) . U A (1) : fermion mass-term can be generated only via non-zero Φ -condensate.

Quadratic explicit breaking of U A (1) � Γ ( k ) [Φ( − p )Φ( p ) + Φ ∗ ( − p )Φ ∗ ( p )] . ESB = Π k p Symmetry: U (1) × U A (1) → U (1) × Z (2) Π k : � ΦΦ + Φ ∗ Φ ∗ � k � = 0 Superposition of oppositely charged condensates Spontaneous breaking: � 2 Φ ∗ Φ − v 2 6 (Φ ∗ Φ) 2 → U ( k ) � U ( k ) k Φ ∗ Φ + λ k SSB = λ k INV = M 2 k 6 2 0 = u k Field expectation: Φ 0 = Φ ∗ 2 → equations of the condensate: √ � � k − 2 | Π k | + λ k u k M 2 6 u 2 2 = 0 , ← INV √ k � � λ k u k 6 ( u 2 k − v 2 k ) − 2 | Π k | 2 = 0 , ← SB √

Infrared spectrum of the scalar sector ( k = 0 ) Denominator of the bosonic propagators: INV ( u 0 = 0 , ψ = 0 ) B ) = ( q 2 + M 2 0 ) 2 − 4 | Π 0 | 2 ∆( Γ (2) m 2 1 = M 2 m 2 2 = M 2 → 0 − 2 | Π 0 | , 0 + 2 | Π 0 | SB ( u 0 � = 0 , ψ = 0 ) � � q 2 + λ 0 ( q 2 + 4 | Π 0 | ) ∆( Γ (2) 3 v 2 B ) = 0 + 4 | Π 0 | 1 = λ 0 0 + 4 | Π 0 | = λ 0 → m 2 3 v 2 3 u 2 m 2 0 , 2 = 4 | Π 0 | . Separation of the symmetric and broken symmetry regions (critical surface): M 2 → m 2 1 = 0 , m 2 0 = 2 | Π 0 | 2 = 4 | Π 0 | The masses m 2 1 and m 2 2 go continuously through the critical surface. The second mass corresponds in the SB-phase to the pseudo-Goldstone field from the U A (1) breakdown. Fermion mass: u 0 m ψ = h k =0 √ 2

Strategy for solving RGE Toy model of top-Higgs system with enlarged scalar sector: Is it possible to identify the PGB with the Higgs ( m G = m 2 = m Higgs )? How large can be made m hb /m G ? How does influence Π Λ the m hb /m G ratio? Strategy: Tune h Λ , M 2 Λ with fixed Π Λ /M 2 Λ and λ Λ to get � 2 m 2 √ 2 m top = 173 8Π 0 � 125 Higgs h 0 = 246 , = = . h 2 0 u 2 m 2 u SM 173 top SM Read out m 2 m 2 = 3 h 2 Higgs 0 G m 2 m 2 2 λ 0 top hb

Wetterich RGE’s for the effective action I. symmetric phase Z φ q 2 R + M 2 1 + 2 λ � � ∂ t M 2 = − 2 h 2 ˆ ˆ ∂ t ∂ t , Z 2 ψ q 2 ∆( q 2 3 R ) q q R R + M 2 ) 2 + 16 | Π | 2 − λ 2 5( Z φ q 2 � 1 � ˆ ˆ ∂ t λ = 6 h 4 ∂ t ∂ t , Z 4 ψ q 4 ∆ 2 ( q 2 3 R ) q q R 1 � ˆ ∂ t h = h 3 Π ∂ t R ) , Z 2 ψ q 2 R ∆( q 2 q ∂ t Π = − λ Π � 1 ˆ ∂ t R ) . ∆( q 2 3 q Note: Evolution of the Yukawa coupling is very slow with diminishing Π Λ /M 2 r Λ

Wetterich RGE’s for the effective action II. broken symmetry phase 4 Z φ q 2 R + m 2 hb + m 2 3 � 1 + λ � G = − 2 h 2 ˆ ˆ 4 ∂ t m 2 G ∂ t ∂ t , Z 2 ψ q 2 R + m 2 ∆( q 2 6 R ) q q ψ − Z 2 ψ q 2 R + m 2 � hb = 2 h 2 ˆ ψ ∂ t m 2 ∂ t ( Z 2 ψ q 2 R + m 2 ψ ) 2 q G − m 2 + λ 1 � � 2 � � ˆ 4 Z φ q 2 R + 7 m 2 hb + 3 m 2 4 Z φ q 2 R + m 2 hb + m 2 hb � ∂ t , G ∆( q 2 ∆( q 2 6 R ) R ) q − Z 2 ψ q 2 R + m 2 ∂ t h = − h 3 � ψ G )ˆ 2 ( m 2 hb − m 2 ∂ t ( Z 2 ψ q 2 R + m 2 ψ ) 2 ∆( q 2 R ) q + h 3 m 2 1 � 3 1 � � ˆ hb ∂ t hb ) 2 − . Z 2 ψ q 2 R + m 2 ( Z φ q 2 R + m 2 ( Z φ q 2 R + m 2 G ) 2 4 q ψ

Wetterich RGE’s for the effective action III. broken symmetry phase ∆ nm = δ n + m ∆ ∂ t λ = ∂ F t λ + ∂ B t λ, δ Φ n δ Φ ∗ m 2 � t λ = 4 h 4 ˆ 3 ∂ F G 2 ψ [1 − 4 m 2 ψ G ψ + m 4 ψ G 2 ∂ t ψ ] , p � 2 t λ = 1 ∆ 22 ∆ − 1 � ˆ 3 ∂ B 2∆ 12 ∆ 10 + 2∆ 21 ∆ 01 + 2∆ 2 � � ∂ t 11 + ∆ 20 ∆ 02 ∆ 2 2 p � − 6∆ 2 10 ∆ 2 + 2 ∆ 20 ∆ 2 01 + ∆ 02 ∆ 2 01 � � 10 + 4∆ 11 ∆ 10 ∆ 01 . ∆ 3 ∆ 4 1 1 1 G G ( q 2 G hb ( q 2 R ) = , R ) = , G ψ = , Z Φ q 2 R + m 2 Z Φ q 2 R + m 2 Z 2 ψ q 2 R + m 2 G hb ψ ∆ = G G G hb

Phase structure and spectra in LPA, ( η ψ = η φ = 0) 0.0006 0.0005 m hb2 0.0004 Λ 2 m G 2 0.0003 Λ 2 m ψ 2 0.0002 Λ 2 0.0001 t - 10 - 8 - 6 - 4 - 2 | Π Λ | /M 2 Λ = 0 . 01 , h ≡ h Λ

Scalar mass ratio vs. Π Λ /M 2 Λ m G 2 m G 2 m hb2 m hb2 0.560 ● 0.7 ● 0.555 ● ● λ = 0.5, slope: 0.007184 ● ● λ = 0.5 0.6 0.550 ▲ λ = 0.8, slope: 0.0010221 ▲ λ = 2 ● ● ■ λ = 0.94002 ■ λ = 5 0.545 ● 0.5 ◆ λ = 1, slope: - 0.0002739 ◆ λ = 8 ▲ ▲ ▲ ▲ 0.540 ★ λ = 1.4, slope: - 0.000963 ★ λ = 13 ■ ■ ■ ◆ ◆ ◆ 0.4 0.535 ★ ■ ★ ★ ★ ◆ ● ▲ ■ ▲ ● ★ ◆ ■ ★ 0.530 Π Λ Π Λ M Λ 2 M Λ 2 10 - 15 10 - 10 10 - 5 10 - 8 10 - 6 10 - 4 10 - 2 h Λ ≈ 0 . 7 h Λ ≈ 3 Important observation: Scalar mass ratio apparently approaches a value insensitive to Π Λ /M 2 Λ Conjecture: λ ( k = 0) for | Π Λ | /M 2 Λ → 0 is a unique function of h Λ What is behind this systematics observed from the numerical solution of the RGE’s?

The RG-flow in the symmetric phase RGE’s of dimensionless couplings 1 + M 2 ∂ t Π r + 2Π r = 4 λ r Π r v d r r ) 2 , r ) 2 − 4Π 2 ((1 + M 2 3 r ) 2 + 4Π 2 (1 + M 2 r v d − 4 λ r v d r ∂ t M 2 r + 2 M 2 r = 4 h 2 r ) 2 , r ) 2 − 4Π 2 ((1 + M 2 3 4 λ 2 r v d (1 + M 2 r ) r ) 2 + 52Π 2 ∂ t λ r + (4 − d ) λ r = − 24 h 4 5(1 + M 2 � � r v d + , r ) 2 − 4Π 2 r 3((1 + M 2 r ) 3 2Π r h 3 2(1 + M 2 � � ∂ t h r + 1 r v d r ) 2(4 − d ) h r = − + 1 . r ) 2 − 4Π 2 r ) 2 − 4Π 2 (1 + M 2 (1 + M 2 r r Fixed point solution in d = 4 : � � M ∗ 2 � λ ∗ 2 ≈ 18 2 r,UV UV v 4 = (32 π 2 ) − 1 . Π ∗ 5 , ≈ 2 v 4 1 − , r,UV = 0 , h 4 h 2 5 UV UV With h Λ = 173 / 246 it suggests λ ∗ UV ≈ 0 . 94 . Compare with λ Λ belonging to the limiting m 2 G /m 2 hb ratio!

The RG-flow in the symmetric phase 1.4 1.2 1.0 ◆ 0.8 λ 0.6 0.4 0.2 0.0 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 M r2 Two classes of trajectories: i) running into Landau-singularity, ii) running into instability separated by the renormalized trajectory ending in the fixed point. ∆ t needed for the evolution in the broken phase is independent of Π Λ . The smaller is Π Λ the longer is the evolution along a flow-line in the symmetric phase.

The RG-flow in the symmetric phase 1.5 1.0 λ 0.5 0.0 10 20 30 40 50 60 log  M 2 Π  Fixed λ Λ , M 2 Λ / Π Λ → ∞ selects the renormalized trajectory Predicts(!) m 2 G /m 2 � 5 / 18 × m 2 G /m 2 hb ≈ 1 . 5 ψ (= 0 . 42 compare to figure!!) Is this picture robust enough when one improves solutions of the RGE’s?

Effect of wavefunction renormalisation ( LPA ′ ) 1 + M 2 ∂ t Π r + (2 − η φ )Π r = 4 λ r Π r v 4 1 − η φ � � r , r ) 2 − 4Π 2 ((1 + M 2 r ) 2 3 6 r ) 2 + 4Π 2 (1 + M 2 1 − η ψ − 4 λ r v 4 1 − η φ � � � � ∂ t M 2 r + (2 − η φ ) M 2 r = 4 h 2 r r v 4 , r ) 2 − 4Π 2 ((1 + M 2 r ) 2 5 3 6 4 λ 2 r v d (1 + M 2 1 − η ψ r ) � � ∂ t λ r − 2 η φ λ r = − 24 h 4 r v 4 + r ) 3 × r ) 2 − 4Π 2 3((1 + M 2 5 1 − η φ r ) 2 + 52Π 2 � � � 5(1 + M 2 � × , r 6 4Π r h 4 2(1 + M 2 r v 4 � r ) 1 − η φ + 1 − η ψ � � � ∂ t h 2 r − ( η φ +2 η ψ ) h 2 r = − . r ) 2 − 4Π 2 r ) 2 − 4Π 2 (1 + M 2 (1 + M 2 6 5 r r Algebraic equations of the anomalous dimensions: r ) 2 + 4Π 2 (1 + M 2 1 − η φ � � η ψ = h 2 r η φ = h 2 r v 4 , r v 4 (4 − η ψ ) . r ) 2 − 4Π 2 ((1 + M 2 r ) 2 5

Slow (logarithmic) variation of the Yukawa-coupling h h 2 � � r ( t 0 ) 2 3 h 2 r ( t ) = r ( t 0 ) C ( t − t 0 ) , C ≈ v 4 4 + < 1 + M ∗ 2 1 − 2 h 2 16 π 2 rUV 3 2.0 2 1.5 1 1.0 λ λ 0 0.5 - 1 0.0 - 0.002 0.000 0.002 0.004 0.006 0.008 0.000 0.001 0.002 0.003 0.004 0.005 h r ( t = −∞ ) = 0 . 7 2 2 M r M r Initial data ( h r Λ , Π r Λ ) labels the points in the plane. ”Neutral” flow-line separates regions of RG-trajectories ending with instability and RG-trajectories ending with Landau-singularity LPA ′ -flow → adiabatic overlay of LPA flow patterns belonging to slowly varying h r ( t )