Introduction FRG Sine-Gordon models C-function Results Summary I SG Higgs potential Summary II Sine-Gordon models: from Conformal Field Theory to Higgs physics István Nándori , (V. Bacsó, I. G. Márián, N. Defenu, A. Trombettoni) MTA-DE Particle Physics Research Group, University of Debrecen ERG2016, Trieste
Introduction FRG Sine-Gordon models C-function Results Summary I SG Higgs potential Summary II Motivation Why Sine-Gordon (SG) models? � 1 � � 2 ( ∂ µ ϕ ) 2 + u cos ( βϕ ) d d x S SG [ ϕ ] = BKT (Berezinski-Kosterlitz-Thouless) phase transition ⇒ e.g. to study the effect of amplitude fluctuations Higgs physics – sine-Gordon type Higgs potential ⇒ e.g. to solve stability problem? CFT (Conformal Field Theory) – Zamolodchikov c-function ⇒ e.g. to determine the c-function in the RG flow ⇒ Functional Renormalization Group (FRG)
Introduction FRG Sine-Gordon models C-function Results Summary I SG Higgs potential Summary II Wetterich RG equation Functional Renormalization Group Wetterich RG equation � � k ∂ k Γ k = 1 k ∂ k R k 2 Tr , Γ ( 2 ) + R k k R k ( p ) ≡ p 2 r ( y ) , y = p 2 k 2 regulator: R k → 0 ( p ) = 0 , R k → Λ ( p ) = ∞ , R k ( p → 0 ) > 0 Approximations � V k ( ϕ ) + Z k ( ϕ ) 1 � � 2 ( ∂ µ ϕ ) 2 + ... Γ k [ ϕ ] = x N cut g n ( k ) � ( 2 n )! ϕ 2 n V k = n = 1
Introduction FRG Sine-Gordon models C-function Results Summary I SG Higgs potential Summary II Massive and Layered Sine-Gordon → topological β 2 Sine-Gordon, Z 2 + periodic − c = 8 π � 1 � � 2 ( ∂ µ ϕ ) 2 + u cos ( βϕ ) d 2 x S SG [ ϕ ] = ⇒ Coulomb-gas, 2D XY spin model, 2D superfluid, bosonised Thirring model → Ising-like ( β 2 Massive Sine-Gordon, Z 2 − c = ∞ ) � 1 � � 2 ( ∂ µ ϕ ) 2 + 1 2 M 2 ϕ 2 + u cos ( βϕ ) d 2 x S MSG [ ϕ ] = ⇒ Yukawa-gas, 2D XY spin model + external field, 2D charged superfluid, bosonised QED 2 N → β 2 Layered Sine-Gordon, Z 2 + (periodic) − c ( N ) = 8 π N − 1 � N � 2 N N � 2 ( ∂ µ ϕ n ) 2 + 1 1 � � � d 2 x 2 M 2 S LSG [ ϕ ] = ϕ n + u n cos ( βϕ n ) n = 1 n = 1 n = 1 ⇒ Layered vortex-gas, layered 2D XY spin model, layered superconductor, bosonised multiflavour QED 2
Introduction FRG Sine-Gordon models C-function Results Summary I SG Higgs potential Summary II Sinh- and Shine-Gordon → topological β 2 Sine-Gordon, Z 2 + periodic − c = 8 π � 1 � � 2 ( ∂ µ ϕ ) 2 + u cos ( βϕ ) d 2 x S SG [ ϕ ] = = ⇒ Conformal Field Theory, characterised by the central charge C = 1 Sinh-Gordon, Z 2 − → phase-structure? � � 1 � 2 ( ∂ µ ϕ ) 2 + u cos ( i βϕ ) d 2 x S ShG [ ϕ ] = � � 1 � 2 ( ∂ µ ϕ ) 2 + u cosh ( βϕ ) d 2 x = = ⇒ Conformal Field Theory, characterised by the central charge C = 1 Ising: C = 1 / 2 Shine-Gordon, Z 2 + (periodic β 2 = 0) − → phase-structure? � 1 � � 2 ( ∂ µ ϕ ) 2 + u Re cos [( β 1 + i β 2 ) φ ] d 2 x S Shine [ ϕ ] = � 1 � � 2 ( ∂ µ ϕ ) 2 + u cos ( β 1 φ ) cosh ( β 2 φ ) d 2 x = = ⇒ Conformal Field Theory
Introduction FRG Sine-Gordon models C-function Results Summary I SG Higgs potential Summary II Conformal invariance and fixed points of the FRG method Global dilatation symmetry ( a → λ a ) ⇒ symmetry under the dilatation of the length scale 1.0 ⇒ scale invariance ⇒ phase transition point 0.8 ⇒ fixed point of the FRG 0.6 < - u < For example: SG model ⇒ > > > > 0.4 < > > 0.2 < < < < > 0.0 0 5 10 15 20 25 30 35 40 ~ 1/ z Local dilatation symmetry ( a → λ ( x ) a ) ⇒ conformal transformations: relative angles unchanged ⇒ conformal group: finite dimensional for d � = 2 d = 2 dimensions: ⇒ conformal group: infinite dimensional ⇒ central charge (C) of the Virasoro algebra conformal invariance ⇔ scale invariance fixed points of FRG ⇒ central charges (C)
Introduction FRG Sine-Gordon models C-function Results Summary I SG Higgs potential Summary II Conformal invariance and fixed points of the FRG method Global dilatation symmetry ( a → λ a ) ⇒ symmetry under the dilatation of the length scale ⇒ scale invariance c=0 1.0 III ⇒ phase transition point 0.8 ⇒ fixed point of the FRG 0.6 < - u < For example: SG model ⇒ > > > > 0.4 < > > 0.2 < < < > < I II 0.0 5 10 15 20 25 30 35 40 2 = 1/ z c=1 c=1 Local dilatation symmetry ( a → λ ( x ) a ) ⇒ conformal transformations: relative angles unchanged ⇒ conformal group: finite dimensional for d � = 2 d = 2 dimensions: ⇒ conformal group: infinite dimensional ⇒ central charge (C) of the Virasoro algebra conformal invariance ⇔ scale invariance fixed points of FRG ⇒ central charges (C)
Introduction FRG Sine-Gordon models C-function Results Summary I SG Higgs potential Summary II C-function Out of the fixed points? → C-theorem in d = 2! C-theorem → c-function A. B. Zamolodchikov, JETP Lett. 43 , 730 (1986) ⇒ function of the couplings: c(g) c=0 1.0 III ⇒ decreasing from UV to IR 0.8 ⇒ at fixed points: c ( g ⋆ ) = C 0.6 < - u < > > > > 0.4 < > > 0.2 < < < < I > II 0.0 5 10 15 20 25 30 35 40 2 = 1/ z c=1 c=1 C-function in FRG A. Codello, G. D’Odorico, and C. Pagani, JHEP 1407 , 040 (2014) ⇒ expression in LPA k ∂ k c k = [ k ∂ k ˜ k ( ϕ 0 )] 2 V ′′ k ( ϕ 0 )] 3 , [ 1 + ˜ V ′′ Questions: ⇒ c-function for the SG? ⇒ phase diagram, c-function for the ShG? ⇒ (interpolating models?)
Introduction FRG Sine-Gordon models C-function Results Summary I SG Higgs potential Summary II C-function for the SG model c=0 sine-Gordon, LPA + z 1.0 III 0.8 Rescaling z = 1 /β 2 , ˜ u = uk − 2 0.6 < � 1 - u < � � > > > > 0.4 d 2 x z ( ∂ µ ϕ ) 2 − (˜ uk 2 ) cos ( ϕ ) S SG [ ϕ ] = < 2 > > 0.2 < < < I > II < 0.0 5 10 15 20 25 30 35 40 2 = 1/ z c=1 c=1 FRG equations (power-law regulator b = 1) u 2 u k ) 2 ˜ ( k ∂ k ˜ 1 � � 1 � u 2 k ( 2 + k ∂ k )˜ 1 − ˜ u k = 1 − , k ∂ k z k = − , k ∂ k c k = k 3 2 π z k ˜ ( 1 + ˜ u k ) 3 u k 24 π [ 1 − ˜ u 2 k ] 2 c-function results (power-law regulator b = 2) 1.0 1.0 z + LPA, region III z + LPA, region I 0.2 1.0 0.8 0.8 0.19 0.8 c k=0 c k=0 0.18 0.6 0.17 0.4 0.6 0.6 0.16 0.2 c k c k 0.15 0.0 0.4 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 5 10 15 20 25 u k= 2 2 k= k= = 5.6 2 2 0.2 k= = 3.4 0.2 u k= = 0.9, k= = 40 2 2 u k= = 0.8, k= = 40 k= = 1.0 2 2 k= = 0.1 u k= = 0.7, k= = 40 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 k/ k/ V. Bacso, N. Defenu, A. Trombettoni, I. Nandori, Nucl. Phys. B 901 (2015) 444.
Introduction FRG Sine-Gordon models C-function Results Summary I SG Higgs potential Summary II C-function for the ShG model I sinh-Gordon, LPA Linearised FRG equation, LPA V k ( ϕ ) = − 1 ( 2 + k ∂ k ) ˜ ˜ k ( ϕ ) + O ( ˜ V ′′ 2 V ′′ k ) 4 π SG model ˜ V SG ( ϕ ) = ˜ u k cos ( βϕ ) � − 2 + 1 � 4 π β 2 β 2 k ∂ k ˜ u k = ˜ u k → c = 8 π ⇒ topological phase transition ShG model ˜ V ShG ( ϕ ) = ˜ u k cos ( i βϕ ) � − 2 − 1 � 4 π β 2 No β 2 k ∂ k ˜ u k = ˜ u k → c ⇒ No topological phase transition N. Defenu, V. Bacso, I. G. Márián, I. Nandori, A. Trombettoni, in preparation
Introduction FRG Sine-Gordon models C-function Results Summary I SG Higgs potential Summary II C-function for the ShG model II sinh-Gordon, LPA + z Taylor expansion ( Z 2 symmetry) � 1 + 1 2 β 2 ϕ 2 + 1 � 4 ! β 4 ϕ 4 + ... ˜ u k cos ( i βϕ ) ∼ V ShG ( ϕ ) = ˜ = ˜ u k initial values: symmetric phase of the Ising ⇒ single phase FRG equations (power-law regulator b = 1) β → i β 1.2 � � u k = − β 2 � ShG model, mass-cutoff ( 2 + k ∂ k )˜ 1 − ˜ u 2 1 − 1.0 2 π ˜ u k k 0.8 β 4 u 2 k ˜ k ∂ k β 2 1 k = − k 0.6 - u 24 π 3 [ 1 − ˜ u 2 k ] 2 > > > > > > > > > > > > > > 0.4 0.2 0.0 0 5 10 15 20 25 30 35 40 2 ⇒ single phase! N. Defenu, V. Bacso, I. G. Márián, I. Nandori, A. Trombettoni, in preparation
Introduction FRG Sine-Gordon models C-function Results Summary I SG Higgs potential Summary II Summary I C-function for the SG model obtained in FRG No topological-type phase transition for the ShG model No Ising-type phase transition for the ShG model Phase structure of the ShG model ( β → i β ) C-function for the ShG model ( β → 0) Outlook Interpolating models → Poster (No. 2)
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