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

Left-Right Symmetric Model

• The Standard Model(SM) has been highly successful but it

needs extension as the scalar potential for SM becomes unbounded at ∼ 1010 GeV.

• 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) Q T3L T3R B L 2

1

Left-Right Symmetric Model

• The Standard Model(SM) has been highly successful but it

needs extension as the scalar potential for SM becomes unbounded at ∼ 1010 GeV.

• 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) Q T3L T3R B L 2

1

Left-Right Symmetric Model

• The Standard Model(SM) has been highly successful but it

needs extension as the scalar potential for SM becomes unbounded at ∼ 1010 GeV.

• 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) Q T3L T3R B L 2

1

Left-Right Symmetric Model

• The Standard Model(SM) has been highly successful but it

needs extension as the scalar potential for SM becomes unbounded at ∼ 1010 GeV.

• 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) Q = T3L + T3R + B − L 2

1

Particle Content of LRSM

SU(2)L SU(2)R U(1)B−L QL ≡ ( uL dL ) 2 1

1 3

QR ≡ ( uR dR ) 1 2

1 3

ψL ≡ ( νL eL ) 2 1 −1 ψR ≡ ( N eR ) 1 2 −1 Φ = ( φ0

1

φ+

2

φ−

1

φ0

2

) 2 2 ∆L = ( 1

√ 2∆+ L

∆++

L

∆0

L

− 1

√ 2∆+ L

) 3 1 2 ∆R = ( 1

√ 2∆+ R

∆++

R

∆0

R

− 1

√ 2∆+ R

) 1 3 2

2

Scalar Potential

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

3

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

• 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

Scalar Potential: λ terms

V ⊃ −µ2

1Tr[φ†φ] − µ2 2

( Tr[˜ φφ†] + Tr[˜ φ†φ] ) +λ1Tr[φ†φ]2 + λ2 ( Tr[˜ φφ†]2 + Tr[˜ φ†φ]2) + λ3Tr[˜ φφ†]Tr[˜ φ†φ] +λ4Tr[φ†φ] ( Tr[˜ φφ†] + Tr[˜ φ†φ] )

5

Stability : λ terms

• By analyzing the boundedness of λ sector of the potential,

λ1 > 0 (1) λ1 − λ2

4

2λ2 + λ3 > 0 (2) λ1 + λ3 − 2λ2 − λ2

4

4λ2 > 0 (3) λ1 + λ3 + 2(λ2 − |λ4|) > 0 (4)

6

Numerical case 1: λ’s

• 5

5 2 4 6 8 λ3 λ1

Values of set λ’s are: λ2 = 0 λ4 = 1

7

Numerical case 1: λ’s

• 5

5 2 4 6 8 λ3 λ1

Values of set λ’s are: λ2 = 0 λ4 = 1 λ1 > 0 λ1 − 1

λ3 > 0

λ1 + λ3 > 2

7

Numerical case 1: λ’s

• 5

5 2 4 6 8 λ3 λ1

Values of set λ’s are: λ2 = 0 λ4 = 1 λ1 > 0 λ1 − 1

λ3 > 0

λ1 + λ3 > 2

7

Numerical case 2: λ’s

• 5

5 2 4 6 8 λ3 λ1

Values of set λ’s are: λ2 = 1 λ4 = −2

8

Numerical case 2: λ’s

• 5

5 2 4 6 8 λ3 λ1

Values of set λ’s are: λ2 = 1 λ4 = −2 λ1 > 0 λ1 −

4 2+λ3 > 0

λ1 + λ3 > 3 λ1 + λ3 > 2

8

Numerical case 2: λ’s

• 5

5 2 4 6 8 λ3 λ1

Values of set λ’s are: λ2 = 1 λ4 = −2 λ1 > 0 λ1 −

4 2+λ3 > 0

λ1 + λ3 > 3 λ1 + λ3 > 2

8

Scalar Potential : ρ terms

V ⊃ −µ2

3

( Tr[∆L∆†

L] + Tr[∆R∆† R]

) +ρ1 ( Tr[∆L∆†

L]2 + Tr[∆R∆† R]2)

+ρ2 ( Tr[∆L∆L]Tr[∆†

L∆† L] + Tr[∆R∆R]Tr[∆† R∆† R]

) +ρ3 Tr[∆L∆†

L]Tr[∆R∆† R]

+ρ4 ( Tr[∆L∆L]Tr[∆†

R∆† R] + Tr[∆† L∆† L]Tr[∆R∆R]

)

9

Stability : ρ terms

• For boundedness of ρ sector of the potential,

ρ1 > 0 (5) ρ1 + ρ2 > 0 (6) ρ3 − 2|ρ4| + 2(ρ1 + ρ2) > 0 (7)

10

Numerical case 1: ρ’s

• 5

5 2 4 6 8 ρ3 ρ1

Values of set ρ’s are: ρ2 = 1 ρ4 = −1

11

Numerical case 1: ρ’s

• 5

5 2 4 6 8 ρ3 ρ1

Values of set ρ’s are: ρ2 = 1 ρ4 = −1 ρ1 > 0 ρ1 + 1 > 0 ρ3 + 2ρ2 > 0

11

Numerical case 2: ρ’s

• 5

5 2 4 6 8 ρ3 ρ1

Values of set ρ’s are: ρ2 = −3 ρ4 = −1

12

Numerical case 2: ρ’s

• 5

5 2 4 6 8 ρ3 ρ1

Values of set ρ’s are: ρ2 = −3 ρ4 = −1 ρ1 > 0 ρ1 − 3 > 0 ρ3 + 2ρ2 − 8 > 0

12

Numerical case 2: ρ’s

• 5

5 2 4 6 8 ρ3 ρ1

Values of set ρ’s are: ρ2 = −3 ρ4 = −1 ρ1 > 0 ρ1 − 3 > 0 ρ3 + 2ρ2 − 8 > 0

12

Dreaded case: α1 ̸= 0

• For boundedness of the potential,

α1 + 2 √( λ1 −

λ2

4

2λ2+λ3

) (ρ1 + ρ2) > 0 α1 + √( λ1 −

λ2

4

2λ2+λ3

) (ρ3 − 2|ρ4| + 2(ρ1 + ρ2)) > 0 α1 + 2 √( λ1 + λ3 − 2λ2 − λ2

4

4λ2

) (ρ1 + ρ2) > 0 α1 + √( λ1 + λ3 − 2λ2 − λ2

4

4λ2

) (ρ3 − 2|ρ4| + 2(ρ1 + ρ2)) > 0 α1 + 2 √ (λ1 + λ3 + 2(λ2 − |λ4|)) (ρ1 + ρ2) > 0 α1 + √ λ1 + λ3 + 2(λ2 − |λ4|)) (ρ3 − 2|ρ4| + 2(ρ1 + ρ2)) > 0

13

Numerical case 1: α1 ̸= 0

• 5

5 2 4 6 8 α1 λ1

• Values of couplings are:

λ2 = λ4 = ρ2 = 0 λ3 = ρ1 = ρ3 = 1 ρ4 = −1

• Condition 2: α1 + √λ1 > 0

14

Numerical case 2: α1 ̸= 0

• 5

5 2 4 6 8 α1 ρ1

• Values of quartic

couplings are: λ1 = λ3 = ρ3 = 1 λ2 = λ4 = ρ2 = 0 ρ4 = −1

• Condition 2:

α1 + √2ρ1 − 1 > 0

15

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 breaking scale. GC, Dev, Mohapatra, Zhang (JHEP ’19)

• 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

• btained 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 breaking scale. GC, Dev, Mohapatra, Zhang (JHEP ’19)

• 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

• btained 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 breaking scale. GC, Dev, Mohapatra, Zhang (JHEP ’19)

• For correct symmetry breaking, conditions for λ presented

here are required.

• Work presented here can be generalized. Conditions on

quartic couplings of a higgs potential for a theory can be

• btained 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 breaking scale. GC, Dev, Mohapatra, Zhang (JHEP ’19)

• For correct symmetry breaking, conditions for λ presented

here are required.

• Work presented here can be generalized. Conditions on

quartic couplings of a higgs potential for a theory can be

• btained completely.

16

Thank you !!

17