Electroweak Phase Transition and Sphaleron Eibun Senaha (Natl Taiwan - PowerPoint PPT Presentation

Electroweak Phase Transition and Sphaleron Eibun Senaha (Natl Taiwan U) April 7, 2017 ACFI Workshop: Making the Electroweak Phase Transition (Theoretically) Strong @Umass-Amherst

Outline • part1: EWPT and sphaleron in a complex-extended SM (cxSM) CW Chiang, M. Ramsey-Musolf, E.S., in progress • part2: Band structure effect on sphaleron rate at high-T. K. Funakubo, K. Fuyuto, E.S., arXiv:1612.05431

Introduction - Standard perturbative treatment of EWPT is gauge-dependent. T=T C T C : T at which V eff has degenerate minima v C : minimum of V eff at T C - Gauge-independent methods: v C (1) v C and T C are determined by (2) Patel-Ramsey-Musolf (PRM) scheme [JHEP07(2011)029] v C and T C are determined separately. We analyze EWPT and sphaleron in the cxSM using 2 methods.

SM with a complex scalar (cxSM) H: SU(2)-doublet Higgs, S: SU(2)-singlet Higgs We assume all parameters are real. m 2 , λ , δ 2 , b 2 , d 2 , a 1 , b 1 v 0 , v S 0 , m H 1 , m H 2 , α , m A , a 1

SM with a complex scalar (cxSM) H: SU(2)-doublet Higgs, S: SU(2)-singlet Higgs We assume all parameters are real. m 2 , λ , δ 2 , b 2 , d 2 , a 1 , b 1 v 0 , v S 0 , m H 1 , m H 2 , α , m A , a 1 246GeV

SM with a complex scalar (cxSM) H: SU(2)-doublet Higgs, S: SU(2)-singlet Higgs We assume all parameters are real. m 2 , λ , δ 2 , b 2 , d 2 , a 1 , b 1 v 0 , v S 0 , m H 1 , m H 2 , α , m A , a 1 246GeV 125GeV

SM with a complex scalar (cxSM) H: SU(2)-doublet Higgs, S: SU(2)-singlet Higgs We assume all parameters are real. m 2 , λ , δ 2 , b 2 , d 2 , a 1 , b 1 v 0 , v S 0 , m H 1 , m H 2 , α , m A , a 1 246GeV 125GeV

Patterns of PT Because of 2 fields, there are many patterns of phase transitions. (b) (a) (d) (c)

Patterns of PT Because of 2 fields, there are many patterns of phase transitions. (b) (a) (d) (c) We will focus on type (a) PT.

Leading order analysis where (a) Approximate formulas: - large positive δ 2 (negative α ) gives larger v C /T C . - However, too large positive δ 2 (negative α ) leads to unstable vacuum.

Leading order analysis where (a) Approximate formulas: >0 - large positive δ 2 (negative α ) gives larger v C /T C . - However, too large positive δ 2 (negative α ) leads to unstable vacuum.

Leading order analysis An example: 250 m H 2 = 230 GeV, v S 0 = 40 GeV, T LO a 1 = − (110 GeV) 3 C v LO 200 ¯ C - v C and T C are determined 150 numerically. [GeV] - smaller α (large δ 2 ) gives 100 larger v C /T C . 50 - EW vacuum becomes metastable for a small alpha. 0 -> upper bound on v C /T C -22 -21 -20 -19 -18 -17 -16 -15 α [ � ] - Stronger upper bound on v C /T C comes from bubble nucleation (see later.)

Leading order analysis An example: 250 m H 2 = 230 GeV, v S 0 = 40 GeV, T LO a 1 = − (110 GeV) 3 C v LO 200 ¯ C - v C and T C are determined 150 numerically. [GeV] - smaller α (large δ 2 ) gives 100 larger v C /T C . 50 - EW vacuum becomes EW vacuum is metastable metastable for a small alpha. 0 -> upper bound on v C /T C -22 -21 -20 -19 -18 -17 -16 -15 α [ � ] - Stronger upper bound on v C /T C comes from bubble nucleation (see later.)

NLO analysis - PRM scheme - O(hbar) T C v C V e ff ( v (1) 0 ; T C ) − V e ff ( v (2) 0 ; T C ) = 0 , v C = minimum of high-T potential at T C e.g. 250 0 v E EW E S -5x10 7 200 v C -1x10 8 Vacuum Energies [GeV 4 ] 150 -1.5x10 8 VEVs [GeV] -2x10 8 100 -2.5x10 8 50 -3x10 8 -3.5x10 8 0 T C 0 20 40 60 80 100 120 140 T C 0 20 40 60 80 100 120 140 T [GeV] T [GeV]

μ dependence PRM scheme is gauge independent but scale dependent. origin: 200 180 160 140 120 [GeV] Different scales give different 100 orders of phase transition: 80 60 40 2 nd order for μ ≲ 160 GeV 20 1 st order for μ ≳ 160 GeV 0 50 100 150 200 250 300 [GeV]

Improved-RPM scheme idea: μ dependence is reduced by renormalization group eq. our scheme:

Improved-RPM scheme 200 180 160 140 120 [GeV] 100 80 60 40 T C (RGI) v C (RGI) ¯ 20 T C ¯ v C 0 50 100 150 200 250 300 µ [GeV] μ dependence is significantly reduced by the RG improvement. In this example, phase transition is 1 st order.

LO vs. NLO 250 T C v C ¯ T LO C v LO ¯ 200 C 150 [GeV] 100 50 0 -22 -21 -20 -19 -18 -17 -16 -15 α [ ◦ ]

LO vs. NLO T C < T LO 250 T C C v C ¯ T LO C v LO ¯ 200 C 150 [GeV] 100 50 0 -22 -21 -20 -19 -18 -17 -16 -15 α [ ◦ ]

LO vs. NLO T C < T LO 250 T C C v C ¯ T LO C v LO ¯ 200 C 150 [GeV] 100 50 0 -22 -21 -20 -19 -18 -17 -16 -15 α [ ◦ ]

LO vs. NLO T C < T LO 250 T C C v C ¯ T LO C 1dim. analogy v LO ¯ 200 C 150 [GeV] 100 T LO C 50 T C 0 246 GeV -22 -21 -20 -19 -18 -17 -16 -15 α [ ◦ ]

LO vs. NLO T C < T LO 250 T C C v C ¯ T LO C 1dim. analogy v LO ¯ 200 C 150 [GeV] 100 T LO C 50 T C 0 246 GeV -22 -21 -20 -19 -18 -17 -16 -15 α [ ◦ ]

LO vs. NLO T C < T LO 250 T C C v C ¯ T LO C 1dim. analogy v LO ¯ 200 C 150 [GeV] 100 T LO C 50 T C 0 246 GeV -22 -21 -20 -19 -18 -17 -16 -15 α [ ◦ ]

LO vs. NLO α =-20.5 o T C < T LO 250 T C C v C ¯ T LO C 1dim. analogy v LO ¯ 200 C 150 [GeV] 100 T LO C 50 T C 0 246 GeV -22 -21 -20 -19 -18 -17 -16 -15 α [ ◦ ] In the following, α =-20.5 o is taken.

LO vs. NLO benchmark point: minima of V high − T ( ϕ i ; T ) 250 m H 2 = 230 GeV, v S 0 = 40 GeV, ¯ v ( T ) v S ( T ) ¯ α = − 20 . 5 � , a 1 = − (110 GeV) 3 v S ( T ) ˜ 200 NLO Leading Order: LO 150 [GeV] 100 Next-to-Leading Order: 50 0 0 50 100 150 200 T [GeV]

LO vs. NLO benchmark point: minima of V high − T ( ϕ i ; T ) 250 m H 2 = 230 GeV, v S 0 = 40 GeV, ¯ v ( T ) v S ( T ) ¯ α = − 20 . 5 � , a 1 = − (110 GeV) 3 v S ( T ) ˜ 200 NLO Leading Order: LO 150 [GeV] 100 Next-to-Leading Order: 50 0 0 50 100 150 200 T [GeV] How about nucleation temperature?

Onset of PT - T C is not onset of the PT. - Nucleation starts somewhat V eff below T C . “Not all bubbles can grow” T = Tc 0 expand? or shrink? T = T N 0 50 100 150 200 250 300 volume energy vs. surface energy v [GeV] ∝ (radius) 3 ∝ (radius) 2 There is a critical value of radius -> critical bubble

Nucleation temperature - Nucleation rate per unit time per unit volume [A.D. Linde, NPB216 (’82) 421] - D efinition of nucleation temperature ( T N ) Γ N ( T N ) H ( T N ) − 3 = H ( T N ) .

S 3 (T)/T - LO case - benchmark point: 500 m H 2 = 230 GeV, v S 0 = 40 GeV, α = − 20 . 5 � , a 1 = − (110 GeV) 3 400 T N = 84 . 9 GeV 300 S 3 ( T )/ T S 3 ( T N ) = 152 . 01 GeV 200 T N 152.01 GeV 100 T LO � T N C ' 6 . 1% 84.9 GeV T LO 0 C 80 82 84 86 88 90 cf., MSSM: O(0.1)% T [GeV] T C = 83.1 GeV; T N (LO) = 84.9 GeV , T C (LO) = 90.4 GeV

No nucleation case 10 6 - α =-22.0 o 10 5 - V C /T C = (209.1GeV)/(65.52GeV) = 3.2 10 4 - Too strong 1 st -order EWPT may not be consistent!! 10 3 - No nucleation for α <-21.4 o 10 2 0 10 20 30 40 50 60

解を求める 上の 次元 系 Sphaleron σφαλερο s (sphaleros) “ready to fall” [F .R.Klinkhamer and N.S.Manton, PRD30, 2212 (1984)] Energy vacuum sphaleron N CS =1 configuration space vacuum N CS =0

解を求める 上の 次元 系 Sphaleron in SU(2) gauge-Higgs system How do we find a saddle point configuration? -> use of a noncontractible loop. sphaleron Energy vacuum N CS =1 configuration space vacuum N CS =0

Manton’ s ansatz [N.S. Manton, PRD28 (’83) 2019] Energy functional

Manton’ s ansatz [N.S. Manton, PRD28 (’83) 2019] Energy functional

Manton’ s ansatz [N.S. Manton, PRD28 (’83) 2019] Energy functional input:

Sphaleron energy Equations of motion for the sphaleron with the boundary conditions:

E sph (T)/T in cxSM 10 5 10 4 E sph ( T )/ T 10 3 10 2 10 1 0 20 40 60 80 100 T [GeV] E sph ( T LO E sph ( T C ) E sph ( T N ) C ) = 78 . 00 , = 74 . 23 , = 61 . 31 , T LO T C T N C

T-dependence of E sph (T) E sph ( T ) = 4 π ¯ v ( T ) E ( T ) g 2 If T-dependence comes from v(T) only, one has E sph ( T ) = E sph (0) ¯ v ( T ) v 0 Is this scaling law valid?

T-dependence of E sph (T) 2 E sph ( T ) = 4 π ¯ v ( T ) E ( T ) g 2 1.95 If T-dependence comes from 1.9 v(T) only, one has E ( T ) 1.85 E sph ( T ) = E sph (0) ¯ v ( T ) 1.8 v 0 Is this scaling law valid? 1.75 0 20 40 60 80 100 T [GeV]

T-dependence of E sph (T) 2 E sph ( T ) = 4 π ¯ v ( T ) E ( T ) g 2 1.95 If T-dependence comes from 1.9 v(T) only, one has E ( T ) 1.85 E sph ( T ) = E sph (0) ¯ v ( T ) 1.8 v 0 Is this scaling law valid? 1.75 0 20 40 60 80 100 T [GeV] No, it breaks down especially when T approaches T C .

T-dependence of E sph (T) 2 E sph ( T ) = 4 π ¯ v ( T ) E ( T ) g 2 1.95 If T-dependence comes from 1.9 v(T) only, one has E ( T ) 1.85 E sph ( T ) = E sph (0) ¯ v ( T ) 1.8 v 0 Is this scaling law valid? 1.75 0 20 40 60 80 100 T [GeV] No, it breaks down especially when T approaches T C . ∵ presence of v S (T).