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 1 / 16

Outline Model for masses 1 Pole mass calculation 2 Comparing two approximations 3 Conclusions 4 2 / 16

Motivation BSM physics already: neutrinos mix and have mass... but what is the exact mechanism? Unknown BSM physics: More scalars? 3 / 16

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. 3 / 16

Yukawa couplings Singlet Weyl spinor neutrino N : L M = − 1 2 M ( NN + h . c . ) 1 Take 2HDM in the Higgs basis ( � H 1 � = √ 2 v , � H 2 � = 0) , √ 2 m i Leptons are in the Flavour basis ( Y 1 ℓ i = , i = e , µ , τ , ) v Neutrino Yukawa couplings in GN Y 1 ν and Y 2 ν are 2 general complex 3-vectors: ν i ℓ i ˜ ν i ℓ i ˜ L Yuk = − Y 1 H 1 N − Y 2 H 2 N + H . c ., i = e , µ , τ ; ˜ H I ≡ ε ij H Ij , ε 12 = − ε 12 = 1 , By a unitary transformation U we can pick a basis: 0 , d , d ′ � Y 1 ν U = ( 0 , 0 , y ) , Y 2 � ν U = 4 / 16

Neutrino mass generation Y 1 ν U = ( 0 , 0 , y ) , Y 2 0 , d , d ′ � � ν U = . � H 1 � � H 1 � . N ν 3 ν 3 y y H 2 H 2 . . . . ν 2 N ν 2 ν 3 N ν 3 d d d ′ d ′ Figure: Seesaw and radiative mass generation in Weyl spinor notation. The arrow shows chirality propagation. Approximate mass eigenstate basis ⇒ U ≈ V PMNS 5 / 16

Neutrino mass generation Y 1 ν V PMNS = ( 0 , 0 , y ) , Y 2 0 , d , d ′ � � ν V PMNS = . � H 1 � � H 1 � . N ν 3 ν 3 y y H 2 H 2 . . . . N N ν 2 ν 2 ν 3 ν 3 d d d ′ d ′ Task : take PMNS, ∆ m 2 12 , and ∆ m 2 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. 6 / 16

Pole masses Effective two point functions for Majorana neutrinos: . . ν j ν i ν j ν i i Γ ij = iσp Σ ij = We get poles: det ( µ 2 − ΓΣ − 1 ΓΣ − 1 ) = 0 ⇒ det ( µ ± ΓΣ − 1 ) = 0 We solve it perturbatively: ε n µ ( n ) , Σ = ∑ ε n Σ ( n ) , Γ = ∑ µ = ∑ ε n Γ ( n ) n n n ε – perturbation series ordering parameter 7 / 16

Pole mass 1 loop When ε is a loop order parameter, then GN model has:  0 0 0 0   0 0 ∗ ∗  0 0 0 0 0 ∗ ∗ ∗ Γ [ 0 ] = −  , Γ [ 1 ] =      ,     0 0 0 ∗ ∗ ∗ ∗ m 3   0 0 0 m 4 ∗ ∗ ∗ ∗ m 3 ≈ y 2 v 2 2 M , m 4 ≈ M Solving det ( µ +ΓΣ − 1 ) = 0 for ε = 1: µ 3 = m 3 − Γ [ 1 ] 33 − m 3 Σ [ 1 ] µ 1 = 0 33 µ 2 = − Γ [ 1 ] µ 4 = m 4 − Γ [ 1 ] 44 − m 4 Σ [ 1 ] 22 44 Off diagonal 2pt functions, e.g. Γ 23 enter only at ε = 2. 8 / 16

Pole mass 1 loop µ 3 = m 3 − Γ [ 1 ] 33 − m 3 Σ [ 1 ] µ 1 = 0 33 µ 2 = − Γ [ 1 ] µ 4 = m 4 − Γ [ 1 ] 44 − m 4 Σ [ 1 ] 22 44 µ 2 is finite without any UV subtractions. µ 3 and µ 4 needs a subtraction scheme. Interested only in µ 2 and µ 3 Grimus and Lavoura (GL) appoximation [GL ’02]: they are finite, gauge invariant, no UV subtraction. How? 9 / 16

Getting µ 3 without subtraction scheme ? We need all three masses, µ 1 , µ 2 and µ 3 , to be zero at � ε 0 � O : � ε 1 � There is no multiplicative counterterm possible for O � ε 1 � Then O result is finite and gauge invariant for µ 2 and µ 3 � ε 2 � Counterterms should appear only at O . 10 / 16

Getting µ 3 without subtraction scheme ? We need all three masses, µ 1 , µ 2 and µ 3 , to be zero at � ε 0 � O : � ε 1 � There is no multiplicative counterterm possible for O � ε 1 � Then O result is finite and gauge invariant for µ 2 and µ 3 � ε 2 � Counterterms should appear only at O . Ordering series in terms of couplings? . � H 1 � � H 1 � . ν 3 N ν 3 y y H 2 H 2 . . . . N N ν 2 ν 2 ν 3 ν 3 d d d ′ d ′ Figure: These diagrams have the same number of Yukawa couplings 10 / 16

Grimus –Lavoura approximation (my interpretation) Assign ordering parameter to all Yukawa couplings, i.e. 1 Y → ε 2 Y , then:  0 0 0 0   0 0 0 ∗  0 0 0 0 0 ∗ ∗ ∗ Γ GL [ 0 ] = −  , Γ GL [ 1 ] =      ,     0 0 0 0 0 ∗ ∗ ∗   0 0 0 m 4 ∗ ∗ ∗ ∗ m 3 ≈ y 2 v 2 ε 1 � � 2 M = O , m 4 ≈ M Solving det ( µ +ΓΣ − 1 ) = 0 for ε = 1: � 2 � µ 2 µ 3 = Γ GL [ 1 ] Γ GL [ 1 ] Γ GL [ 1 ] , µ 2 + µ 3 = − Γ GL [ 1 ] − Γ GL [ 1 ] − 22 33 23 22 33 Checked: we get the same Γ GL [ 1 ] as in [GL ’02] ij 11 / 16

Determining Yukawas in GL In GL we have: ( µ 2 , µ 3 ) GL m S , U S , U PMNS , µ 4 , Z 3 , d , d ′ � � = f i , i m 3 ≡ Z 3 µ 3 ≈ y 2 v 2 , m S , U S -masses and mixings of scalars 2 µ 4 Assuming normal hierarchy for neutrinos, we solve for d and d ′ : � � µ GL 31 , µ GL ∆ m 2 ∆ m 2 = = 3 2 21 d GL , d ′ i = ˜ � � ⇒ f i ( m S , U S , U PMNS , µ 4 , Z 3 ) GL 12 / 16

GL vs MSbar vs FlexibleSUSY poles Put solutions for d GL and d ′ GL in calculation X (MSbar or FlexibleSUSY): ( µ 2 , µ 3 ) X m S , U S , U PMNS , µ 4 , Z 3 , d GL , d ′ � � i = F i GL Define an ”error” function: � µ X ∆ m 2 i − ∆ µ i i 1 ( m S , U S , µ 4 , Z 3 , U PMNS ) ≡ , i = 2 , 3 � µ i ∆ m 2 i 1 We take a benchmark 2HDM point: m H = 300, m A = 411, m H ± = 442, GeV, sin θ H − A ≈ 0 . 07 Look how it depends on Z 3 = m 3 µ 3 and µ 4 Set charged lepton Yukawa couplings to the H 2 to zero for simplicity. 13 / 16

GL vs MSbar vs FlexibleSUSY poles Δμ 3 Δμ 2 μ 3 μ 2 2.0 0.6 0.5 1.5 0.4 1.0 0.3 0.5 0.2 m 3 0.5 1.0 1.5 2.0 2.5 3.0 μ 3 0.1 - 0.5 m 3 0.5 1.0 1.5 2.0 2.5 3.0 μ 3 Figure: µ 4 = 10 5 , m H = 300, m A = 411, m H ± = 442 GeV; sin θ H − A ≈ 0 . 07. MSbar - blue, FS - orange 14 / 16

GL vs MSbar vs FlexibleSUSY poles Δμ 3 Δμ 2 μ 3 μ 2 1.0 0.5 0.4 0.5 0.3 Log 10 ( μ 4 ) 2 4 6 8 0.2 - 0.5 0.1 Log 10 ( μ 4 ) 2 4 6 8 - 1.0 Figure: µ 3 m 3 = 1 . 5, m H = 300, m A = 411, m H ± = 442 GeV; sin θ H − A ≈ 0 . 07. MSbar - blue, FS - orange Numerical effects start at µ 4 = 10 6 and kills the calculation at 10 8 (MSbar with LoopTools). Radiative mass is more off for FS. 15 / 16

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 < 10 6 GeV (Maybe 10 8 in some cases) and not too big loop corrections for m 3 µ 3 . 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. 16 / 16