Pole masses of Neutrinos in the GrimusNeufeld model Vytautas D ud - - PowerPoint PPT Presentation

Pole masses of Neutrinos in the Grimus–Neufeld model

Vytautas D¯ ud˙ enas

Vilniaus Universitetas In collaboration with Thomas Gajdosik, Darius Jurčiukonis

September 6, 2019

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Outline

1

Model for masses

2

Pole mass calculation

3

Comparing two approximations

4

Conclusions

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Motivation

BSM physics already:

neutrinos mix and have mass...

but what is the exact mechanism?

Unknown BSM physics: More scalars?

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Motivation

BSM physics already:

neutrinos mix and have mass...

but what is the exact mechanism?

Unknown BSM physics: More scalars? Being general but minimal:

2HDM + 1 Seesaw neutrino → Grimus–Neufeld model [GN ’89].

Incorporates masses and mixings at one loop.

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Yukawa couplings

Singlet Weyl spinor neutrino N: LM = −1 2M(NN +h.c.) Take 2HDM in the Higgs basis (H1 =

1 √ 2v, H2 = 0) ,

Leptons are in the Flavour basis (Y 1

ℓi = √ 2mi v

, i = e,µ,τ, ) Neutrino Yukawa couplings in GN Y 1

ν and Y 2 ν are 2 general

complex 3-vectors: LYuk = −Y 1

νiℓi ˜

H1N −Y 2

νiℓi ˜

H2N +H.c.,i = e,µ,τ ; ˜ HI ≡ εijHIj , ε12 = −ε12 = 1, By a unitary transformation U we can pick a basis: Y 1

ν U = (0,0,y) , Y 2 ν U =

• 0,d,d′

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Neutrino mass generation

Y 1

ν U = (0,0,y) , Y 2 ν U =

• 0,d,d′

. . ν3 N ν3 y y H1 H1 . . ν2 N H2 ν2 d d . . ν3 N H2 ν3 d′ d′

Figure: Seesaw and radiative mass generation in Weyl spinor notation. The arrow shows chirality propagation.

Approximate mass eigenstate basis ⇒ U ≈ VPMNS

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Neutrino mass generation

Y 1

ν VPMNS = (0,0,y) , Y 2 ν VPMNS =

• 0,d,d′

. . ν3 N ν3 y y H1 H1 . . ν2 N H2 ν2 d d . . ν3 N H2 ν3 d′ d′

Task: take PMNS, ∆m2

12, and ∆m2 13 from experiment and

relate them to d, d′,y at one loop.

Also relates the scalar sector to Yukawa. Implement in FlexibleSUSY (FS) spectrum generator.

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Pole masses

Effective two point functions for Majorana neutrinos:

.

νj νi

iΓij =

.

νj νi

iσpΣij =

We get poles: det(µ2 −ΓΣ−1ΓΣ−1) = 0 ⇒ det(µ ±ΓΣ−1) = 0 We solve it perturbatively: µ = ∑

n

εnµ(n) , Σ = ∑

n

εnΣ(n) , Γ = ∑

n

εnΓ(n) ε – perturbation series ordering parameter

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Pole mass 1 loop

When ε is a loop order parameter, then GN model has: Γ[0] = −     m3 m4     , Γ[1] =     ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗     , m3 ≈ y2v2 2M , m4 ≈ M Solving det(µ +ΓΣ−1) = 0 for ε = 1: µ1 = 0 µ3 = m3 −Γ[1]

33 −m3Σ[1] 33

µ2 = −Γ[1]

22

µ4 = m4 −Γ[1]

44 −m4Σ[1] 44

Off diagonal 2pt functions, e.g. Γ23 enter only at ε = 2.

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Pole mass 1 loop

µ1 = 0 µ3 = m3 −Γ[1]

33 −m3Σ[1] 33

µ2 = −Γ[1]

22

µ4 = m4 −Γ[1]

44 −m4Σ[1] 44

µ2 is finite without any UV subtractions. µ3 and µ4 needs a subtraction scheme. Interested only in µ2 and µ3

Grimus and Lavoura (GL) appoximation [GL ’02]: they are finite, gauge invariant, no UV subtraction. How?

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Getting µ3 without subtraction scheme ?

We need all three masses, µ1, µ2 and µ3, to be zero at O

• ε0

:

There is no multiplicative counterterm possible for O

• ε1

Then O

• ε1

result is finite and gauge invariant for µ2 and µ3 Counterterms should appear only at O

• ε2

.

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Getting µ3 without subtraction scheme ?

We need all three masses, µ1, µ2 and µ3, to be zero at O

• ε0

:

There is no multiplicative counterterm possible for O

• ε1

Then O

• ε1

result is finite and gauge invariant for µ2 and µ3 Counterterms should appear only at O

• ε2

.

Ordering series in terms of couplings?

. . ν3 N ν3 y y H1 H1 . . ν2 N H2 ν2 d d . . ν3 N H2 ν3 d′ d′

Figure: These diagrams have the same number of Yukawa couplings

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Grimus –Lavoura approximation

(my interpretation)

Assign ordering parameter to all Yukawa couplings, i.e. Y → ε

1 2 Y , then:

ΓGL[0] = −     m4     , ΓGL[1] =     ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗     , m3 ≈ y2v2 2M = O

• ε1

, m4 ≈ M Solving det(µ +ΓΣ−1) = 0 for ε = 1: µ2µ3 = ΓGL[1]

22

ΓGL[1]

33

−

• ΓGL[1]

23

2 , µ2 + µ3 = −ΓGL[1]

22

−ΓGL[1]

33

Checked: we get the same ΓGL[1]

ij

as in [GL ’02]

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Determining Yukawas in GL

In GL we have: (µ2,µ3)GL

i

= fi

• mS,US,UPMNS,µ4,Z3,d,d′

, m3 ≡ Z3µ3 ≈ y2v2 2µ4 , mS,US -masses and mixings of scalars Assuming normal hierarchy for neutrinos, we solve for d and d′: µGL

3

=

• ∆m2

31 , µGL 2

=

• ∆m2

21

⇒

• dGL,d′

GL

• i = ˜

fi (mS,US,UPMNS,µ4,Z3)

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GL vs MSbar vs FlexibleSUSY poles

Put solutions for dGL and d′

GL in calculation X (MSbar or

FlexibleSUSY): (µ2,µ3)X

i = Fi

• mS,US,UPMNS,µ4,Z3,dGL,d′

GL

• Define an ”error” function:

∆µi µi (mS,US,µ4,Z3,UPMNS) ≡ µX

i −

• ∆m2

i1

• ∆m2

i1

, i = 2,3 We take a benchmark 2HDM point: mH = 300, mA = 411, mH± = 442, GeV, sinθH−A ≈ 0.07 Look how it depends on Z3 = m3

µ3 and µ4

Set charged lepton Yukawa couplings to the H2 to zero for simplicity.

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GL vs MSbar vs FlexibleSUSY poles

0.5 1.0 1.5 2.0 2.5 3.0 m3 μ3

• 0.5

0.5 1.0 1.5 2.0 Δμ2 μ2 0.5 1.0 1.5 2.0 2.5 3.0 m3 μ3 0.1 0.2 0.3 0.4 0.5 0.6 Δμ3 μ3

Figure: µ4 = 105, mH = 300, mA = 411, mH± = 442 GeV; sinθH−A ≈ 0.07. MSbar - blue, FS - orange

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GL vs MSbar vs FlexibleSUSY poles

2 4 6 8 Log10(μ4)

• 1.0
• 0.5

0.5 1.0 Δμ2 μ2

2 4 6 8 Log10(μ4) 0.1 0.2 0.3 0.4 0.5 Δμ3 μ3

Figure: µ3

m3 = 1.5, mH = 300, mA = 411, mH± = 442 GeV;

sinθH−A ≈ 0.07. MSbar - blue, FS - orange

Numerical effects start at µ4 = 106 and kills the calculation at 108 (MSbar with LoopTools). Radiative mass is more off for FS.

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Conclusions

GL approximation – coupling ordering instead of loop (they differ for a seesaw)

first order in GL is close to 1 loop approximation But how close, depends on parameters Which perturbation ordering is more accurate?

Can we formulate some condition and/or estimate error?

it is possible to use FS to reproduce neutrino data

but up to M < 106GeV (Maybe 108 in some cases) and not too big loop corrections for m3

µ3 .

The dependence on scalar potential is not yet fully researched...

Further step: relate scalar potential with neutrino Yukawa couplings and make it work with FS.

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