Dynamical Generation of Fermion Mixing Luca Smaldone in - - PowerPoint PPT Presentation

Dynamical Generation of Fermion Mixing Luca Smaldone

in collaboration with Massimo Blasone and Petr Jizba

Department of Physics ”E.R.Caianiello” and INFN Sezione di Napoli, Gruppo Collegato di Salerno

29th Indian Summer School Prague, 5th September 2017

1

Contents

• 1. Neutrino Mixing in QFT
• 2. Patterns of Dynamical Symmetry Breaking
• 3. Conclusions and Perspectives

2

Motivations

• Flavor mixing is a central ingredient in the Standard Model;
• Mixing transformations for fields have been shown to be

non–trivial since they induce a condensate structure in the vacuum1;

• This suggests the idea of dynamical generation of mixing in a

similar way as it happens for the masses.

1M.Blasone, G.Vitiello (1995).

3

Neutrino Mixing in QFT

Massive neutrinos

Mixing transformations defines fields with definite mass ν1, ν2: νe(x) = cos θ ν1(x) + sin θ ν2(x) νµ(x) = − sin θ ν1(x) + cos θ ν2(x) with tan 2θ = 2meµ me − mµ These relations define neutrinos with definite mass.

4

Mass eigenstates

Free fields with definite masses can be expanded as: νi(x) = 1 √ V

• k,r
• ur

k,i(t)αr k,i + vr −k,i(t)βr† −k,i

• eik·x ,

i = 1, 2 A mass-eigenstate neutrino is defined as: |νr

k,i = αr† k,i|01 2 ,

i = 1, 2 where the mass vacuum satisfies: αr

k,i|01 2 = βr k,i|01 2 ,

i = 1, 2

5

Flavor Charges

Flavor charges2: Qνe(t) =

• d3x ν†

e(x)νe(x) ,

Qνµ(t) =

• d3x ν†

µ(x)νµ(x)

Total flavor charge: Q = Qνe(t) + Qνµ(t) Because of mixing, only this last is conserved.

2M.Blasone, P.Jizba, G.Vitiello (2001)

6

Standard Flavor eigenstates

Flavor eigenstates, are usually taken as a simple combination of mass eigenstates: |νr

k,eP

= cos θ |νr

k,1 + sin θ |νr k,2

|νr

k,µP

= − sin θ |νr

k,1 + cos θ |νr k,2

ACHTUNG! These are NOT eigenstates of the flavor charges:

P νr k,e| : Qνe(0) : |νr k,eP

= cos4 θ + sin4 θ + 2|Uk| sin2 θ cos2 θ < 1

P νr k,e| : Qνµ(0) : |νr k,eP

= 2 (1 − |Uk|) sin2 θ cos2 θ > 0 with Uk = ur†

2,kur 1,k. 7

Mixing generator

In a finite volume, mixing relations are rewritten να

e (x)

= G−1

θ (t)να

1 (x) Gθ(t)

να

µ(x)

= G−1

θ (t) να

2 (x)Gθ(t)

where mixing generator has been introduced Gθ(t) = exp

• θ
• d3x
• ν†

1(x)ν2(x) − ν† 2(x)ν1(x)

• 8

Decomposition of the mixing generator

The mixing generator can be decomposed as3: Gθ = B(Θ1, Θ2) R(θ) B−1(Θ1, Θ2) where

R(θ) ≡ exp   θ

• k,r

αr†

k,1αr k,2 + βr† −k,1βr −k,2

• eiψk −
• αr†

k,2αr k,1 + βr† −k,2βr −k,1

• e−iψk
• 

  Bi(Θi) ≡ exp

k,r

Θk,i ǫr αr

k,iβr −k,ie−iφk

, i − βr†

−k,iαr† k,ieiφk,i

• , i = 1, 2

and B(ϑ1, ϑ2) ≡ B1(ϑ1) B2(ϑ2).

3M.Blasone, M.V.Gargiulo, G.Vitiello (2015)

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Bi(Θk,i), i = 1, 2 are Bogoliubov transformations which induces a mass shift and R(θ) is a rotation. Their action on the mass vacuum is: | 01,2 ≡ B−1(Θ1, Θ2)|01,2 =

• k,r
• cos Θk,i + ǫr sin Θk,iαr†

k,iβr† −k,i

• |01,2

R−1(θ)|01,2 = |01,2 A rotation of fields is not a rotation at the level of creation and annihilation operators.

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Flavor Vacuum

The flavor vacuum is defined by4: |0(t)e,µ ≡ G−1

θ (t) |01,2

In the infinite volume limit: lim

V →∞ 1,20|0(t)e,µ = lim V →∞ e V

• d3k

(2π)3 ln(1−sin2 θ |Vk|2) 2

= 0 where |Vk|2 ≡

• r,s

| vr†

−k,1us k,2 |2 = 0

for m1 = m2

4M.Blasone, G.Vitiello (1995).

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Vacuum condensate

• Condensation density: e µ0|αr†

k,iαk,i|0e µ = sin2 θ |Vk|2, with

i = 1, 2. Same result for antiparticles.

• |Vk|2 ≃ (m2−m1)2

4k2

for k ≫ √m1m2.

12

Flavor eigenstates

A neutrino flavor-state can be defined: |νr

k,σ(t) = αr† σ,k|0(t)e µ ,

σ = e, µ At any time these are eigenstates of the flavor charges: Qσ(t)|νr

k,σ(t) = |νr k,σ(t)

Note that |νr

k,σ(t) = |νr k,σP because

• B(m1, m2) , R−1(θ)
• = 0

13

Bogoliubov vs Pontecorvo

Bogoliubov and Pontecorvo do not commute! As a result, flavor vacuum gets a non-trivial term: |0e,µ ≡ G−1

θ |01,2 = |01,2 +

• B(m1, m2) , R−1(θ)
• |

01,2

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Flavor Vacuum and Condensate Structure

The flavor vacuum is characterized by a condensate structure:

|0e,µ =

• k
• r
• (1 − sin2 θ |Vk|2) − ǫr sin θ cos θ |Vk|(αr†

k,1βr† −k,2 + αr† k,2βr† −k,1)

+ǫr sin2 θ |Vk||Uk|(αr†

k,1βr† −k,1 − αr† k,2βr† −k,2) + sin2 θ |Vk|2αr† k,1βr† −k,2αr† k,2βr† −k,1

• |01,2
• SU(2) (Perelomov) coherent state.
• Condensate structure on vacuum as in systems with SSB (e.g.

superfluids, superconductors).

• Exotic condensates: mixed pairs due to a non-diagonal

Bogoliubov transformation.

• Note that |0e µ = |a1 ⊗ |b2 ⇒ entanglement.

15

Why is vacuum structure so important?

• Dark Energy contribution of the neutrino mixing5:

Λ = 128π3G K dk k2(ωk,1 + ωk,2)|Vk|2 If the cut-off is chosen of the order K ∼ √m1m2, with m1 = 7 × 10−3eV and m2 = 5 × 10−2eV one gets: Λ ∼ 10−56cm−2 which is compatible with modern experimental upper bounds.

5M.Blasone, A.Capolupo,S.Capozziello, S.Carloni, G.Vitiello, (2005)

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Patterns of Dynamical Symmetry Breaking

Chiral symmetry

Consider the Lagrangian L = i ¯ ψγµ∂µψ + U(ψψ) where ψ =

• ψI

ψII

• .

U

• ψψ
• is assumed to be invariant under chiral transformations

U(2)V × U(2)A: g = (g, g5) , g = eiωα σα

2 ,

g5 = eiωα

σαγ5 2

, α = 0, 1, 2, 3 so the entire Lagrangian is invariant as well.

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Chiral Symmetry Breaking

The vector and axial Noether charges are: Jα

µ

= ψγµ σα 2 ψ Jα

5µ

= ψγµγ5 σα 2 ψ If we add a diagonal mass term LM = −mψψ the conservation law of axial currents is explicitly broken: ∂µJα

µ

= ∂µJα

5µ

= iψ γ5m ψ

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Isospin Symmetry Breaking

Adding a mass-shift term L∆m = −ψ

• ∆m

−∆m

• ψ

the Isospin symmetry is broken to U(1)0

V × U(1)3 V (the subscript

index indicates the generator) ∂µJ0

µ

= ∂µJ1

µ

= i 2∆mψ σ1 ψ ∂µJ2

µ

= 1 2ψ ∆mσ1 ψ ∂µJ3

µ

=

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Total Flavor Charge Conservation

Finally we add to the Lagrangian, an off-diagonal term Lh = −ψ

• h

h

• ψ

The current evolution are now ∂µJ0

µ

= ∂µJ1

µ

= i 2(mI − mII)ψ σ1 ψ ∂µJ2

µ

= −1 2ψ [2hσ3 − (mI − mII)σ1] ψ ∂µJ3

µ

= −ihψ σ1 ψ Conservation of the total flavor charge Q0.

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Dynamical Generation of Fermion Mixing

Dynamical generation of mixing occurs if6 U(2)V × U(2)A − → U(1)0

V ,

at the ground state level. SSB is characterized by the existence of some (quasi)-local operators Φi so that Ω| [Qα(0), Φi(0)] |Ω = Ω|ϕα

i |Ω = 0 ,

• n some dressed vacuum. ϕα

i are called order parameters. We look at

• rder parameters of the form ψiψj ± ψkψl with i, j, k, l = I, II.
• 6M. Blasone, P. Jizba, L. S., in preparation (2017).

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Patterns of SSB

Symmetry Group Order Parameter Broken Charges U(2)V × U(2)A

• U(2)V

ψIψI + ψIIψII = 0 Qα

5

U(1)0

V × U(1)3 V

ψIψI ± ψIIψII = 0 Qα

5 ; Q1; Q2

U(1)0

V

ψIψI ± ψIIψII = 0 ψIψII + ψIIψI = 0 Qα

5 ; Q1; Q2; Q3 22

Vacuum degeneracy (1)

All charges are time independent: [Qα, H] = [Qα

5 , H]

= 0 , α = 0, 1, 2, 3 Degenerate vacua: | θ, θ5 = eiθαQα+iθ5 ,αQα

5 |Ω

|Ω is a fiducial vacuum.

23

Vacuum degenracy (2)

Consider a fiducial vacuum |Ω(m), where only v =

• j=I,II

Ω(m)|ψj(x)ψj(x)|Ω(m) = 0 We find

• j=I,II

θ5

0|ψj(x)γ5ψj(x)|θ5 0 = i sin θ5 0 v

θ5

3|ψII(x)γ5ψII(x) − ψI(x)γ5ψI(x)|θ5 3 = i sin θ5 3 v

θ5

1|ψI(x)γ5ψII(x) + ψII(x)γ5ψI(x)|θ5 1 = i sin θ5 1 v

θ5

2|ψI(x)γ5ψII(x) − ψII(x)γ5ψI(x)|θ5 2 = sin θ5 2 v

in contrast with our hypothesis.

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Vacuum degenracy (3)

Consider |Ω(m1, m2), leaved invariant Q0, Q3. We introduce Q+ = Q1 + iQ2 , Q− = Q1 − iQ2 The degenerate vacua are Perelomov SU(2) coherent states: |θ ≡ exp [θ ( Q+ − Q−)] |Ω(m1, m2) One gets θ|Q3|θ = sin 2θ Ω(m1, m2)|Q1|Ω(m1, m2) = 0 The residual symmetry, for every θ = 0, is U(1)0

V : Dynamical origin

• f mixing.

25

Two Flavor NJL model

Two-flavor NJL model7 is described by the Lagrangian: L = iψγµ∂µψ − ψLMψR − ψRM †ψL − 1 2Gtr

• MM †

M is an auxiliary boson field: Mab = −2Gψ

b Rψa L

M †

ab = −2Gψ b Lψa R.

(1) In the mean-field approximation, M is substituted by its vev: Mc =

• me

me µ me µ mµ

• .

(2)

7Y.Nambu, G.Jona Lasinio (1961).

26

Gap equations and dynamical generation of mixing

We derive, thanks to 1-loop Effective Action8 me = i G 4π4

• d4p me p2 − mµ d

p4 − p2t + d2 mµ = i G 4π4

• d4p mµ p2 − me d

p4 − p2t + d2 me µ = i G 4π4

• d4p

p2 + d p4 − p2t + d2 where d = det Mc , t = trM 2

c

These equations present all solutions previously discussed.

8M.Blasone, P.Jizba, L.S. (2015).

27

Conclusions and Perspectives

Conclusions

Conclusions:

• The non-trivial condensate structure of the flavor vacuum

suggests a dynamical origin of mixing. The same structure has been proved to be a mark of this phenomenon.

• For models with chiral U(2)V × U(2)A, we have shown how to

derive general information on various patterns of symmetry breaking and about dynamical generation of mixing. These are confirmed in specific examples.

28

Perspectives

Perspectives:

• The application of FI methods has suggested the study of

appearance of inequivalent representations in this framework9.

• More general situations, eventually including Lorentz-Poincar´

e symmetry breaking were only slightly touched10. The use of FI’s in studying these situations requires a generalization of the QM and QFT partition function11.

• 9M. Blasone, P. Jizba, L. S. (2017).

10 M.Blasone, P. Jizba, G. Lambiase, N. Mavromatos (2014). 11

• M. Blasone, P. Jizba, L. S. (2017).

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Thank you for the attention!

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