Thermal phase transition at the 2-loop level beyond temperature - PowerPoint PPT Presentation

Thermal phase transition at the 2-loop level beyond temperature expansions Eibun Senaha (Nagoya U, E-ken) Feb. 14, 2014 @U of Toyama in collaboration with Koichi Funakubo (Saga U) Ref. PRD87, 054003 (2013)

Outline Motivation A tractable calculation scheme for 2-loop effective potential at finite T. Application to thermal phase transition Summary

Motivation ❒ 1 st -order electroweak phase transition (EWPT) is necessary for successful electroweak baryogenesis (EWBG). scalar e.g. ❒ Sunset diagram can change strength of 1 st - vector order EWPT. (e.g. stop-stop-gluon in the MSSM) Complication scalar ∼ T 2 ❒ Thermal resummation: dominant at high T. - such thermal corrections have to be resummed. − 1 − 1 ❒ Noncovariant: D µ ν ( p ) = L ( T ) L µ ν ( p ) + T ( T ) T µ ν ( p ) , p 2 − m 2 p 2 − m 2 L μν , T μν are projections: T ij = g ij − p i p j (in thermal rest frame) T 00 = T 0 i = T i 0 = 0 , − p 2 , P µ ν = g µ ν − p µ p ν (4dim. transverse) p µ T µ ν = p µ L µ ν = 0 L µ ν = P µ ν − T µ ν , p 2 ,

❒ standard calculation methods: - Effective theory (Hard Thermal Loop, etc) - High-temperature expansion (HTE) ❒ HTE has been exclusively used in the EWBG calculation. 40 30 HTE formula: 20 K HTE ( a ) = − π 2 10 3 (ln a 2 + 3 . 48871) a=m/T 0 [R.R.Parwani, PRD45, 4695 (1992)] -10 -20 ❒ Approximation breaks down around m/T=O(1) -30 0.001 0.01 0.1 1 10 ❒ This raise a question about the reliability of a the EWPT analysis in the MSSM. Our aim: to devise a tractable calculation scheme beyond HTE.

❒ standard calculation methods: - Effective theory (Hard Thermal Loop, etc) - High-temperature expansion (HTE) region relevant to EWBG ❒ HTE has been exclusively used in the EWBG calculation. 40 30 HTE formula: 20 K HTE ( a ) = − π 2 10 3 (ln a 2 + 3 . 48871) a=m/T 0 [R.R.Parwani, PRD45, 4695 (1992)] -10 -20 ❒ Approximation breaks down around m/T=O(1) -30 0.001 0.01 0.1 1 10 ❒ This raise a question about the reliability of a the EWPT analysis in the MSSM. Our aim: to devise a tractable calculation scheme beyond HTE.

Abelian-Higgs model ❒ As an illustration, we consider Abelian-Higgs model. L = − 1 4 F µ ν F µ ν + | D µ Φ | 2 − V 0 ( | Φ | 2 ) where F µ ν = ∂ µ A ν − ∂ ν A µ , D µ Φ = ( ∂ µ − ieA µ ) Φ scalar potential: V 0 ( | Φ | 2 ) = − ν 2 | Φ | 2 + λ 4 | Φ | 4 scalar fields: 1 � � Φ ( x ) = v + h ( x ) + ia ( x ) √ 2 field-dependent masses: h = − ν 2 + 3 λ a = − ν 2 + λ m 2 4 v 2 , m 2 4 v 2 , m 2 A = e 2 v 2 .

Resummed perturbation theory ❒ Dominant thermal corrections are taken into account. ❒ We adopt a resummation method in which thermal masses are added and subtracted in the Lagrangian. L B ( R ) : bare (renormalized) Lagrangian, [Buchmuller et al, NPB407 (’93) 387 .] L B = L R + L CT L CT : counterterms Φ Φ † Φ + 1 2 A µ � � → L R − ∆ m 2 ∆ m 2 L L µ ν ( i ∂ ) + ∆ m 2 T T µ ν ( i ∂ ) A ν Φ Φ † Φ − 1 2 A µ � � + L CT + ∆ m 2 ∆ m 2 L L µ ν ( i ∂ ) + ∆ m 2 T T µ ν ( i ∂ ) A ν Explicitly, Φ = 3 e 2 + λ L = e 2 ∆ m 2 T 2 , ∆ m 2 ∆ m 2 3 T 2 T = 0 , 12

Resummed perturbation theory ❒ Dominant thermal corrections are taken into account. ❒ We adopt a resummation method in which thermal masses are added and subtracted in the Lagrangian. L B ( R ) : bare (renormalized) Lagrangian, [Buchmuller et al, NPB407 (’93) 387 .] L B = L R + L CT L CT : counterterms Φ Φ † Φ + 1 2 A µ � � → L R − ∆ m 2 ∆ m 2 L L µ ν ( i ∂ ) + ∆ m 2 T T µ ν ( i ∂ ) A ν Φ Φ † Φ − 1 2 A µ � � + L CT + ∆ m 2 ∆ m 2 L L µ ν ( i ∂ ) + ∆ m 2 T T µ ν ( i ∂ ) A ν Explicitly, Φ = 3 e 2 + λ L = e 2 ∆ m 2 T 2 , ∆ m 2 ∆ m 2 3 T 2 T = 0 , 12

Resummed perturbation theory ❒ Dominant thermal corrections are taken into account. ❒ We adopt a resummation method in which thermal masses are added and subtracted in the Lagrangian. L B ( R ) : bare (renormalized) Lagrangian, [Buchmuller et al, NPB407 (’93) 387 .] L B = L R + L CT L CT : counterterms Φ Φ † Φ + 1 2 A µ � � → L R − ∆ m 2 ∆ m 2 L L µ ν ( i ∂ ) + ∆ m 2 T T µ ν ( i ∂ ) A ν Φ Φ † Φ − 1 2 A µ � � + L CT + ∆ m 2 ∆ m 2 L L µ ν ( i ∂ ) + ∆ m 2 T T µ ν ( i ∂ ) A ν new counterterm Explicitly, Φ = 3 e 2 + λ L = e 2 ∆ m 2 T 2 , ∆ m 2 ∆ m 2 3 T 2 T = 0 , 12

Resummed propagators 1 1 ∆ h ( p ) = ∆ a ( p ) = h ( T ) , a ( T ) , p 2 − m 2 p 2 − m 2 − 1 − 1 D µ ν ( p ) = L ( T ) L µ ν ( p ) + T ( T ) T µ ν ( p ) , p 2 − m 2 p 2 − m 2 m 2 h,a ( T ) = m 2 h,a + ∆ m 2 m 2 L,T ( T ) = m 2 A + ∆ m 2 L,T Φ With these, we compute 1- and 2-loop effective potentials. 1-loop resummed potential R ( v ; T ) = 1 + 1 � � (2 π nT ) 2 + k 2 + ¯ (2 π nT ) 2 + k 2 + ¯ V (1) m 2 m 2 � � � � � � ln h ( T ) ln a ( T ) 2 2 k k + 1 + ( D � 2)1 � � (2 π nT ) 2 + k 2 + ¯ (2 π nT ) 2 + k 2 + ¯ m 2 m 2 � � � � � � ln L ( T ) ln T ( T ) 2 2 k k � ¯ � ¯ � ¯ � ¯ � T 4 m 2 m 2 m 2 m 2 � � � � �� h ( T ) a ( T ) L ( T ) T ( T ) + I B + I B + 2 I B I B 2 π 2 T 2 T 2 T 2 T 2 At 1-loop level, there is no complication to obtain the resummed effective potential.

2-loop resummed sunset diagrams Resummed SSV and SVV type diagrams ∆ h D µ ν � � D R k µ k ν D µ ν ( q ) ∆ 1 ( k ) ∆ 2 ( k + q ) SSV ( m 1 , m 2 ; m L , m T ) = − 4 � � k q ∆ a D µ ν ∆ h � � D µ ν ( k ; m 1 ) D µ ν ( q ; m 2 ) ∆ ( k + q ) D R SV V ( m ; m 1 L , m 1 T ; m 2 L , 2 T ) = − 4 � � k q noncovariant due to T µ ν ( k ) , L µ ν ( k ) It is not easy to evaluate the divergence and finite parts.

Devised gauge boson propagator Resummed gauge boson propagator is devised as follows. Using P µ ν ( p ) = g µ ν − p µ p ν = T µ ν ( p ) + L µ ν ( p ), L µ ν ( p ) or T µ ν ( p ) can be elimi- p 2 nated. − 1 � − 1 − 1 � T µ ν ( p ) ≡ D r =0 D µ ν ( p ) = P µ ν ( p ) + µ ν ( p ) , − p 2 − m 2 p 2 − m 2 p 2 − m 2 L T L − 1 � − 1 − 1 � L µ ν ( p ) ≡ D r =1 D µ ν ( p ) = P µ ν ( p ) + µ ν ( p ) . − p 2 − m 2 p 2 − m 2 p 2 − m 2 T L T In general, ( r ∈ R ) D µ ν ( p ) = (1 − r ) D r =0 µ ν ( p ) + r D r =1 µ ν ( p ) � − (1 − r ) � � � � − 1 − 1 − r � = + P µ ν ( p ) + T µ ν ( p ) − rP µ ν ( p ) . − p 2 − m 2 p 2 − m 2 p 2 − m 2 p 2 − m 2 L T T L covariant noncovariant

Devised gauge boson propagator The loop calculation is simplified if r is determined in the following way. r = d − 2 g µ ν � � T µ ν ( p ) − rP µ ν ( p ) = 0 d − 1 D µ ν ( p ) = D cov µ ν ( p ) + δ D µ ν ( p ) , � P µ ν ( p ) � − 1 + − ( d − 2) D cov µ ν ( p ) = d − 1 , p 2 − m 2 p 2 − m 2 L T � − 1 − 1 � � T µ ν ( p ) − d − 2 � δ D µ ν ( p ) = d − 1 P µ ν ( p ) . − p 2 − m 2 p 2 − m 2 T L µ ν ( p ) = p µ δ D µ ν ( p ) = 0 and g µ ν δ D µ ν ( p ) = 0 p µ D cov

Decomposition Diagrams involving the gauge boson can be decomposed in the following way. covariant part D cov D µ ν µ ν = resummed diagrams -> renormalization is possible +1 � � µ ν � ˆ Π ( q ) − ∆ m 2 L L ( q ) − ∆ m 2 � T T ( q ) δ D µ ν ( q ) 2 q noncovariant sector

Decomposition Diagrams involving the gauge boson can be decomposed in the following way. covariant part D cov D µ ν µ ν = resummed diagrams -> renormalization is possible +1 � � µ ν � ˆ Π ( q ) − ∆ m 2 L L ( q ) − ∆ m 2 � T T ( q ) δ D µ ν ( q ) 2 q from thermal CT noncovariant sector

In general, self-energy can be written as + Π S ( q ) q µ u T ν + q ν u T Π µ ν ( q ) = Π L ( q ) L µ ν ( q ) + Π T ( q ) T µ ν ( q ) + Π G ( q ) q µ q ν µ q 2 � q 2 µ = u µ − q µ ( u · q ) /q 2 with u µ = (1 , 0 ). where u T � � g µ ν � q µ q ν T=0: Π T =0 � Π 0 ( q ) Π T =0 µ ν ( q ) δ D µ ν ( q ) = 0 µ ν q 2 � � ( q 0 = 0 , q → 0) L,T + � 2 � ( � ) T ≠ 0: Π T � =0 ∆ m 2 µ � ( q ) → L,T ( T ) L µ � , T µ � , Finally, the noncovariant part is reduced to � D µ � ( q ) → m 2 L − m 2 1 � � µ � � � � � ( � ) T ( T ) − � ( � ) ˆ Π ( q ) − ∆ m 2 L L ( q ) − ∆ m 2 T � T T ( q ) L ( T ) + O ( � ) 48 � 2 2 q finite! To leading order no v-dependence. -> no effect on phase transition.

Application

Toy model ❒ We study the thermal phase transition in the MSSM-like toy model. stop and gluon-like particles L = L Abelian − Higgs + ∆ L L Abelian − Higgs = − 1 4 F µ ν F µ ν + ( D µ Φ ) ∗ D µ Φ − V ( | Φ | 2 ) , 1 V ( | Φ | 2 ) = − ν 2 | Φ | 2 + λ 4 | Φ | 4 , D µ Φ = ( ∂ µ − ieA µ ) Φ , Φ = ( v + h + ia ) . √ 2 ˜ ∆ L = 1 λ 4( ∂ µ G ν − ∂ ν G µ ) 2 + | ( ∂ µ − ig 3 G µ )˜ 0 | ˜ 4 | ˜ t | Φ | 2 | ˜ t | 2 − m 2 t | 2 − t | 4 − y 2 t | 2 . Note: gluon-like particle is a U(1) gauge boson ❒ Landau gauge ξ =0 is taken. v-dependent masses: 0 + y 2 µ � h = − ν 2 + 3 λ µ � a = − ν 2 + λ µ � m 2 v 2 , m 2 4 v 2 , m 2 A = e 2 µ � v 2 , m 2 t = m 2 v 2 , m 2 ¯ ¯ ¯ ¯ G = 0 . ˜ 4 2