Vacuum Stability in Left-Right Symmetric Model Garv Chauhan - PowerPoint PPT Presentation

Vacuum Stability in Left-Right Symmetric Model Garv Chauhan Washington University in St. Louis, USA PHENO 2019 University of Pittsburgh May 7, 2019 ArXiV: 1905.XXXXX

1 • Hypercharge generator arises in a more natural sense 2 L B T 3 R T 3 L Q from (B-L) see-saw mechanism. • The Standard Model(SM) has been highly successful but it thus explains small masses of left-handed neutrinos via • It features heavy right-handed Majorana neutrinos, and Mohapatra, Pati (PRD ’75) Senjanovic, Mohapatra (PRD ’75) Pati, Salam (PRD ’74) footing at high-energies. • Left-Right model treats left & right chiralities on equal needs extension as the scalar potential for SM becomes Left-Right Symmetric Model unbounded at ∼ 10 10 GeV.

1 • Hypercharge generator arises in a more natural sense 2 L B T 3 R T 3 L Q from (B-L) see-saw mechanism. • The Standard Model(SM) has been highly successful but it thus explains small masses of left-handed neutrinos via • It features heavy right-handed Majorana neutrinos, and Mohapatra, Pati (PRD ’75) Senjanovic, Mohapatra (PRD ’75) Pati, Salam (PRD ’74) footing at high-energies. • Left-Right model treats left & right chiralities on equal needs extension as the scalar potential for SM becomes Left-Right Symmetric Model unbounded at ∼ 10 10 GeV.

1 • Hypercharge generator arises in a more natural sense 2 L B T 3 R T 3 L Q from (B-L) see-saw mechanism. • The Standard Model(SM) has been highly successful but it thus explains small masses of left-handed neutrinos via • It features heavy right-handed Majorana neutrinos, and Mohapatra, Pati (PRD ’75) Senjanovic, Mohapatra (PRD ’75) Pati, Salam (PRD ’74) footing at high-energies. • Left-Right model treats left & right chiralities on equal needs extension as the scalar potential for SM becomes Left-Right Symmetric Model unbounded at ∼ 10 10 GeV.

• The Standard Model(SM) has been highly successful but it needs extension as the scalar potential for SM becomes • Left-Right model treats left & right chiralities on equal footing at high-energies. Pati, Salam (PRD ’74) Mohapatra, Pati (PRD ’75) Senjanovic, Mohapatra (PRD ’75) • It features heavy right-handed Majorana neutrinos, and thus explains small masses of left-handed neutrinos via see-saw mechanism. • Hypercharge generator arises in a more natural sense from (B-L) 2 1 Left-Right Symmetric Model unbounded at ∼ 10 10 GeV. Q = T 3 L + T 3 R + B − L

2 L N e R 1 1 2 0 L L 3 L 2 R R R R 2 e L 2 1 d R u L d L 1 3 u R Particle Content of LRSM SU ( 2 ) L SU ( 2 ) R U ( 1 ) B − L ( ) 2 1 Q L ≡ ( ) 1 2 Q R ≡ ( ) ν L 2 1 ψ L ≡ − 1 ( ) 1 2 ψ R ≡ − 1 ( ) φ + φ 0 2 2 Φ = φ − φ 0 ( 1 ) 2 ∆ + ∆ ++ √ 3 1 ∆ L = 2 ∆ + ∆ 0 − 1 √ ( 1 ) 2 ∆ + ∆ ++ √ 1 3 ∆ R = 2 ∆ + ∆ 0 − 1 √

Maiezza, Senjanovic, Vasquez (PRD ’17) Deshpande, Gunion, Kayser, Olness (PRD ’91) 3 Scalar Potential

• For viability of the model to be extension for SM, the potential for the theory should be stable. • Given a random set of quartic couplings, it does not ensure the stability of the vacuum. • Using the concepts of boundedness, copositivity and gauge orbit spaces, conditions for stability of the potential can be derived. Dev, Mohapatra, Rodejohann, Xu (JHEP ’19) Kim (JMP ’84) Kannike (EPJC ’12) 4 Vacuum Stability

• For viability of the model to be extension for SM, the potential for the theory should be stable. • Given a random set of quartic couplings, it does not ensure the stability of the vacuum. • Using the concepts of boundedness, copositivity and gauge orbit spaces, conditions for stability of the potential can be derived. Dev, Mohapatra, Rodejohann, Xu (JHEP ’19) Kim (JMP ’84) Kannike (EPJC ’12) 4 Vacuum Stability

• For viability of the model to be extension for SM, the potential for the theory should be stable. • Given a random set of quartic couplings, it does not ensure the stability of the vacuum. • Using the concepts of boundedness, copositivity and gauge orbit spaces, conditions for stability of the potential can be derived. Dev, Mohapatra, Rodejohann, Xu (JHEP ’19) Kim (JMP ’84) Kannike (EPJC ’12) 4 Vacuum Stability

5 2 V Scalar Potential: λ terms ( ) Tr [˜ φφ † ] + Tr [˜ 1 Tr [ φ † φ ] − µ 2 φ † φ ] ⊃ − µ 2 + λ 1 Tr [ φ † φ ] 2 + λ 2 ( φφ † ] 2 + Tr [˜ φ † φ ] 2 ) Tr [˜ + λ 3 Tr [˜ φφ † ] Tr [˜ φ † φ ] ( ) Tr [˜ φφ † ] + Tr [˜ + λ 4 Tr [ φ † φ ] φ † φ ]

6 4 (4) (3) 4 (2) (1) Stability : λ terms • By analyzing the boundedness of λ sector of the potential, λ 1 > 0 λ 2 λ 1 − > 0 2 λ 2 + λ 3 λ 1 + λ 3 − 2 λ 2 − λ 2 > 0 4 λ 2 λ 1 + λ 3 + 2 ( λ 2 − | λ 4 | ) > 0

7 Numerical case 1: λ ’s Values of set λ ’s are: 8 λ 2 = 0 λ 4 = 1 6 λ 1 4 2 0 - 5 0 5 λ 3

7 Numerical case 1: λ ’s Values of set λ ’s are: 8 λ 2 = 0 λ 4 = 1 6 λ 1 4 λ 1 > 0 λ 1 − 1 λ 3 > 0 2 λ 1 + λ 3 > 2 0 - 5 0 5 λ 3

7 Numerical case 1: λ ’s Values of set λ ’s are: 8 λ 2 = 0 λ 4 = 1 6 λ 1 4 λ 1 > 0 λ 1 − 1 λ 3 > 0 2 λ 1 + λ 3 > 2 0 - 5 0 5 λ 3

8 Numerical case 2: λ ’s Values of set λ ’s are: 8 λ 2 = 1 λ 4 = − 2 6 λ 1 4 2 0 - 5 0 5 λ 3

8 4 Numerical case 2: λ ’s Values of set λ ’s are: 8 λ 2 = 1 λ 4 = − 2 6 λ 1 4 λ 1 > 0 λ 1 − 2 + λ 3 > 0 2 λ 1 + λ 3 > 3 λ 1 + λ 3 > 2 0 - 5 0 5 λ 3

8 4 Numerical case 2: λ ’s Values of set λ ’s are: 8 λ 2 = 1 λ 4 = − 2 6 λ 1 4 λ 1 > 0 λ 1 − 2 + λ 3 > 0 2 λ 1 + λ 3 > 3 λ 1 + λ 3 > 2 0 - 5 0 5 λ 3

9 3 V Scalar Potential : ρ terms ( ) Tr [∆ L ∆ † L ] + Tr [∆ R ∆ † ⊃ − µ 2 R ] ( L ] 2 + Tr [∆ R ∆ † R ] 2 ) Tr [∆ L ∆ † + ρ 1 ( ) Tr [∆ L ∆ L ] Tr [∆ † L ∆ † L ] + Tr [∆ R ∆ R ] Tr [∆ † R ∆ † + ρ 2 R ] + ρ 3 Tr [∆ L ∆ † L ] Tr [∆ R ∆ † R ] ( ) Tr [∆ L ∆ L ] Tr [∆ † R ∆ † R ] + Tr [∆ † L ∆ † + ρ 4 L ] Tr [∆ R ∆ R ]

(5) (6) (7) 10 Stability : ρ terms • For boundedness of ρ sector of the potential, ρ 1 > 0 ρ 1 + ρ 2 > 0 ρ 3 − 2 | ρ 4 | + 2 ( ρ 1 + ρ 2 ) > 0

11 Numerical case 1: ρ ’s Values of set ρ ’s are: 8 ρ 2 = 1 ρ 4 = − 1 6 ρ 1 4 2 0 - 5 0 5 ρ 3

11 Numerical case 1: ρ ’s Values of set ρ ’s are: 8 ρ 2 = 1 ρ 4 = − 1 6 ρ 1 4 ρ 1 > 0 ρ 1 + 1 > 0 2 ρ 3 + 2 ρ 2 > 0 0 - 5 0 5 ρ 3

12 Numerical case 2: ρ ’s Values of set ρ ’s are: 8 ρ 2 = − 3 ρ 4 = − 1 6 ρ 1 4 2 0 - 5 0 5 ρ 3

12 Numerical case 2: ρ ’s Values of set ρ ’s are: 8 ρ 2 = − 3 ρ 4 = − 1 6 ρ 1 4 ρ 1 > 0 ρ 1 − 3 > 0 2 ρ 3 + 2 ρ 2 − 8 > 0 0 - 5 0 5 ρ 3

12 Numerical case 2: ρ ’s Values of set ρ ’s are: 8 ρ 2 = − 3 ρ 4 = − 1 6 ρ 1 4 ρ 1 > 0 ρ 1 − 3 > 0 2 ρ 3 + 2 ρ 2 − 8 > 0 0 - 5 0 5 ρ 3

13 4 4 4 • For boundedness of the potential, 4 Dreaded case: α 1 ̸ = 0 √( ) λ 2 α 1 + 2 λ 1 − ( ρ 1 + ρ 2 ) > 0 2 λ 2 + λ 3 √( ) λ 2 α 1 + λ 1 − ( ρ 3 − 2 | ρ 4 | + 2 ( ρ 1 + ρ 2 )) > 0 2 λ 2 + λ 3 √( ) λ 1 + λ 3 − 2 λ 2 − λ 2 α 1 + 2 ( ρ 1 + ρ 2 ) > 0 4 λ 2 √( ) λ 1 + λ 3 − 2 λ 2 − λ 2 α 1 + ( ρ 3 − 2 | ρ 4 | + 2 ( ρ 1 + ρ 2 )) > 0 4 λ 2 √ α 1 + 2 ( λ 1 + λ 3 + 2 ( λ 2 − | λ 4 | )) ( ρ 1 + ρ 2 ) > 0 √ α 1 + λ 1 + λ 3 + 2 ( λ 2 − | λ 4 | )) ( ρ 3 − 2 | ρ 4 | + 2 ( ρ 1 + ρ 2 )) > 0

14 • Values of couplings are: Numerical case 1: α 1 ̸ = 0 8 6 λ 2 = λ 4 = ρ 2 = 0 λ 1 4 λ 3 = ρ 1 = ρ 3 = 1 2 ρ 4 = − 1 • Condition 2: α 1 + √ λ 1 > 0 0 - 5 0 5 α 1

15 • Values of quartic • Condition 2: couplings are: Numerical case 2: α 1 ̸ = 0 8 6 λ 1 = λ 3 = ρ 3 = 1 ρ 1 4 λ 2 = λ 4 = ρ 2 = 0 2 ρ 4 = − 1 0 α 1 + √ 2 ρ 1 − 1 > 0 - 5 0 5 α 1

breaking scale. GC, Dev, Mohapatra, Zhang (JHEP ’19) • We obtained necessary and sufficient conditions for the stability of the potential. • Renormalization group analysis of the potential requires these conditions and can put important constraints on • For correct symmetry breaking, conditions for presented here are required. • Work presented here can be generalized. Conditions on quartic couplings of a higgs potential for a theory can be obtained completely. 16 Conclusions

• We obtained necessary and sufficient conditions for the stability of the potential. • Renormalization group analysis of the potential requires these conditions and can put important constraints on • For correct symmetry breaking, conditions for presented here are required. • Work presented here can be generalized. Conditions on quartic couplings of a higgs potential for a theory can be obtained completely. 16 Conclusions breaking scale. GC, Dev, Mohapatra, Zhang (JHEP ’19)