Valencia November 2006
Neutrino Oscillations in Split Supersymmetry
Marco Aurelio Díaz Pontificia Universidad Católica de Chile
. – p.1/31 Neutrino Oscillations The neutrino mass matrix is diagonalized by the rotation matrix VPMNS, such that the mass eigenstates evolve in time as
iE jt i j ψi = e− VPMNS ψ j Xj Calculating transition probabilities in the ultra-relativistic limit, where m2 E ~p + i i ≈ | | 2 ~p | | leads to results like (in the two neutrino approximation) 2 2 2 ∆m L Pν ν = sin (2θ) sin 1.27 i→ j E where θ is the mixing angle.
. – p.2/31 Three Neutrinos A general 3 3 neutrino mass matrix is diagonalized by a Pontecorvo-Maki- × Nakagawa-Sakata matrix of the type
1 0 0 c13 0 s13 c12 s12 0 V = 0 c s 0 1 0 s c 0 PMNS 23 23 − 12 12 0 s23 c23 s13 0 c13 0 0 1 − − θ • 23: atmospheric angle θ • 13: reactor angle θ • 12: solar angle ∆ 2 • m23: atmospheric mass squared difference ∆ 2 • m12: solar mass squared difference
. – p.3/31 Experimental Constraints Experimental results are consistent with the following values of the mixing angles and masses: 2 0.52 < tan θ23 < 2.1 2 tan θ13 < 0.049 2 0.30 < tan θ12 < 0.61 3 2 3 2 1.4 10− < ∆m < 3.3 10− eV × 23 × 5 2 5 2 7.2 10− < ∆m < 9.1 10− eV × 12 × mee < 0.84 eV
There is no direct measurement of the scale of neutrino masses. From Table 1 (3 σ values) hep-ph/0405172, M. Maltoni, T. Schwetz, M. Tortola, J. W. F. Valle
. – p.4/31 Bilinear R-Parity Violation R-Parity and Lepton Number are violated by bilinear terms in the superpotential.
The three parameters 1,2,3 have units of mass:
W = WMSSM + i Li Hu
These terms induce sneutrino vacuum expectation bvaluesb < ν˜i >= vi, which contribute to the gauge boson masses (although in a negligible amount):
v2 + v2 + v2 + v2 + v3 = v2 (246 GeV)2 u d 1 2 3 ∼ In the soft supersymmetry breaking potential the following terms are added:
so f t so f t V = VMSSM + Bii Li Hu
e
. – p.5/31 Neutralinos and Neutrinos 0 T 3 1 2 In basis the (ψ ) = ( iλ , iλ , H , H , ν , νµ, ντ ) the neutralino/neutrino mass − 0 − 1 2 e matrix is e e 1 1 1 1 1 M1 0 2 g0vd 2 g0vu 2 g0v1 2 g0v2 2 g0v3 −1 1 −1 −1 −1 0 M2 2 gvd 2 gvu 2 gv1 2 gv2 2 gv3 1 1 − g0vd gvd 0 µ 0 0 0 − 2 2 − M = 1 g v 1 gv µ 0 N 2 0 u − 2 u − 1 2 3 1 1 2 g0v1 2 gv1 0 1 0 0 0 − 1 1 g0v2 gv2 0 2 0 0 0 − 2 2 1 g v 1 gv 0 0 0 0 2 0 3 2 3 3 − and a neutrino 3 3 mass matrix is induced. ×
. – p.6/31 Low Energy See-Saw Low energy see-saw mechanism with three neutrinos
T χ0 m 1 T MN = M = me f f = m χ−0 m " m 0 # ⇒ − · M · defining Λi = µvi + ivd, the effective mass matrix is Λ2 Λ Λ Λ Λ M g2+M g 2 1 1 2 1 3 1 2 0 2 (0)Λ Λ me f f = Λ1Λ2 Λ2 Λ2Λ3 A i j 4 det( χ0 ) 2 ∼ M Λ1Λ3 Λ2Λ3 Λ 3 and only one neutrino acquire mass at tree level
2 2 M1g + M2g0 ~ 2 mν3 = Tr(me f f ) = Λ 4 det( 0 ) | | Mχ . – p.7/31 Neutrino Angles at Tree Level T The diagonalization Vν me f f Vν = diag(0, 0, mν) is performed with two rotations:
1 0 0 cosθ 0 sinθ 13 − 13 Vν = 0 cosθ sinθ 0 1 0 23 − 23 × 0 sinθ23 cosθ23 sinθ13 0 cosθ13 The atmospheric angle θ23 and the reactor angle θ13 are simple functions of the Λi
Λ2 Λ1 tanθ23 = tanθ13 = −Λ3 − 2 2 Λ2 + Λ3 q and the solar angle is undefined at tree level.
. – p.8/31 Effect of loops The neutrino mass matrix at tree level has the form,
ν(0) (0) Mi j = A ΛiΛ j and at one loop,
ν Mi j = AΛiΛ j + B(Λi j + Λ ji) + Ci j
For each individual loop we have,
Z, W, t˜: AΛ Λ • ∼ i j χ+, χ0, b˜: AΛ Λ + B(Λ + Λ ) + C • ∼ i j i j j i i j The first three loops renormalize the atmospheric mass. The second three loops break the symmetry, inducing a solar mass.
. – p.9/31 Neutralino Decays
1.8 © ¨ ¦ § ¦
In the presence of BRpV a neutralino
¤ 1.6 LSP is not stable: £ ¢ ¡
τ µ 1.4
, , e
0 © ¨ χ ¦ §
1 ¦
q ¥ 1.2 W ¤ £
q0 ¢ ¡ 1
Ratios of BR are closely related to the 0.8
Λi parameters. 0.6
-1.5 -1 -0.5 0 0.5 1 1.5 Collider Physics Neutrino Physics ⇔ . – p.10/31 Split Supersymmetry
Based on work by, M.A. Díaz, Pavel Fileviez-Perez, and Clemencia Mora hep-ph/0605285
. – p.11/31 Split Susy, Rp Conserved All scalars except for one neutral Higgs doublet H are heavy, with a mass of the order of m. The higgsino mass and the Higgs-higgsino-gaugino vertices in the supersymmetric lagrangian valid above m are, e
T Hu† a ae Hd† a a susy = µHu iσ2 Hd gσ W + g0B Hu gσ W + g0B Hd L − √2 − √2 The fine-tunedelight Higgse doublet H =e c iσeH e+ s H , and theeequivalente e − β 2 d∗ β u couplings in the effective lagrangian below m are given by,
T H† a a e H iσ2 a a split = g˜uσ W + g˜u0 B Hu g˜dσ W + g˜0 B Hd L −√2 − √2 − d with the following matching econditionse atethe scale m: e e e
g˜u(m) = g(m) sin β(m) g˜d(m) =e g(m) cos β(m)
g˜u0 (m) = g0(m) sin β(m) g˜d0 (m) = g0(m) cos β(m) e e e e e e . – p.12/31 e e e e e e Running
The couplings g˜u, g˜d, g˜u0 , and g˜d0 , run with their own RGE [Giudice, Romanino, Nucl.Phys.B699, 65 (2004)]. In particular we have,
g˜u cos (2β) 2 2 m˜ (m ) tan β(m˜ ) 1 + (7g 3g0 ) ln tan β g˜ W ≈ 64π2 − m ≡ d ( m˜ W )
g˜u0 cos (2β) 2 2 m˜ (m ) tan β(m˜ ) 1 (9g + 3g0 ) ln tan0 β g˜ W ≈ − 64π2 m ≡ d0 ( m˜ W )
2 2 2 2 2 2 Definition: g˜ g˜ (m ) + g˜ (m ) , g˜0 g˜0 (m ) + g˜0 (m ) . ≡ u W d W ≡ u W d W In this case, the neutralino mass matrix loops like
M 0 1 g˜ c v 1 g˜ s v 1 − 2 0 β0 2 0 β0 1 ˜ 1 ˜ 0 M2 2 gcβv 2 gsβv Mχ0 = 1 1 − 2 g˜0cβ0 v 2 g˜cβv 0 µ −1 1 − 2 g˜0sβ0 v 2 g˜sβv µ 0 . – p.13/31 − − Split Susy, Rp Violated The higgsino-lepton mixing and the Lepton-slepton-gaugino vertex in the supersymmetric lagrangian valid above m are,
T Li† a a susy = i Hu iσ2Li gσ W + g0B Li L −e√2 Since the sleptons and the Higgse bosons mix, this timee the fine-tunede e light Higgs is equal to
H = cβiσ H∗ + sβ H s iσ L∗ − 2 d u − i 2 i and the equivalent couplings in the effective lagrangian beloe w m are given by,
ai T a a e split = H iσ2 g˜dσ W + g˜0 B Li L −√2 − d with the following matching conditions at the scale em: e
ai(m)g˜d(m) = g(m)si(m) ai(m)g˜ed0 (m) = g0(m)si(m) where the si are the Higgs-slepton mixing angles. e e e e e e e e . – p.14/31 S.S. Low Energy See-Saw In Split Susy the low energy see-saw mechanism is similar to the MSSM: T χ0 m 1 T MN = M = me f f = m χ−0 m " m 0 # ⇒ − · M · but the mixing looks like, 1 g˜ c a v 1 g˜c a v 0 − 2 0 β0 1 2 β 1 1 m = 1 g˜ c a v 1 g˜c a v 0 − 2 0 β0 2 2 β 2 2 1 g˜ c a v 1 g˜c a v 0 − 2 0 β0 3 2 β 3 3 If we define λi = aiµ + i, which are related to the MSSM-BRpV parameters by
Λi = λivd, the effective neutrino mass matrix is given by 2 2 2 2 2 λ λ1λ2 λ1λ3 M1g˜ c +M2g˜0 c0 1 β β 2 2 (0)λ λ me f f = v λ1λ2 λ2 λ2λ3 A i j 4 det( 0 ) ∼ Mχ λ λ λ λ λ2 1 3 2 3 3 and again only one neutrino acquire mass at tree level. . – p.15/31 S.S. Tree Level Results T The diagonalization Vν me f f Vν = diag(0, 0, mν) is performed with two rotations:
1 0 0 cosθ 0 sinθ 13 − 13 Vν = 0 cos θ sinθ 0 1 0 23 − 23 × 0 sinθ cosθ sinθ 0 cosθ 23 23 13 13 with the atmospheric angle θ23 and the reactor angle θ13 satisfying
λ2 λ1 tanθ23 = tanθ13 = − λ3 − 2 2 λ2 + λ3 and the solar angle is undefined at tree level. The treeqlevel mass for the only massive neutrino is:
M g˜2c2 +M g˜ 2c 2 1 β 2 0 β0 2 ~ 2 mν3 = Tr(me f f ) = v λ 4 det( 0 ) | | Mχ
. – p.16/31 Higgs boson loop The Higgs boson contribution to the mass matrix is, i j h 1 h 2 2 ∆Mν = ∆Π (0) = G m B (0; m , m ) i j − 16π2 i jk k 0 k h Xk The loop can be representede by the diagram χ0 k
ν j ν h i ∆Πi j(0) =
e h This diagram has the form,
h ∆Πi j(0) = AΛiΛ j + B(Λi j + Λ ji) + Ci j generating a solar neutrino mass. To this diagram we need to add the Goldstone e boson contribution, which tends to cancel the Higgs contribution but not enough to spoil the solution [See Davidson & Losada, Haber & Grossman]. . – p.17/31 Higgs boson loop in detail The Higgs boson loop in detail is, 1 ∆Πh (0) = (E λ + F )(E λ + F ) m B (0; m2, m2) i j − 64π2 k i k i k j k j k 0 k h Xk and parteof it can be represented by the diagram χ0 Nk2 k Nk2 W W ν j ν h Hd Hd i ∆Πi j(0) = g˜cβ g˜cβ e e e e e λ ξ3 /µ h λ ξ /µ j − j i 3 − i with,
E = (g˜sβ N g˜0s0 N )ξ N (g˜sβ ξ g˜0s0 ξ ) k − k2 − β k1 4 − k4 2 − β 1 +(g˜cβ N g˜0c0 N )ξ +N (g˜cβ ξ g˜0c0 ξ ) k2− β k1 3 k3 2 − β 1 1 Fk = (g˜cβ Nk2 g˜0cβ0 Nk1) −µ − . – p.18/31 A Split Susy Solution The chosen Split Supersymmetry benchmark is:
M1 = 50 GeV , µ = 200 GeV ,
M2 = 300 GeV , mh = 120 GeV , tan β = 50 , and the BRpV solution is characterized by,
= 0.107 GeV , λ = 0.0001 GeV , 1 − 1 = 0.001 GeV , λ = 0.031 GeV , 2 − 2 = 0.217 GeV , λ = 0.036 GeV . 3 3 − The values of the neutrino observables in this scenario are,
2 3 2 2 ∆m = 2.3 10− eV , tan θ = 1.22 , atm × atm 2 5 2 2 ∆m = 8.4 10− eV , tan θ = 0.56 , sol × sol 2 mee = 0.0037 eV , tan θ13 = 0.0083 ,
within experimental bounds. . – p.19/31 Gaugino-Higgsino mass plane
BRpV and Split Susy parameters are fixed to the benchmark values, except M2 and µ, which are varied.
Perturbative expressions for the neutrino masses are,
~λ (~ ~λ) 2 mν = C | × × | 2 ~λ 4 | | ~ 2 2 (~ λ) mν = A ~λ +2B(~ ~λ) + C · 3 | | · ~λ 2 | | with A = 620 GeV, B = 0.95 GeV, C = 0.25 GeV, and − − 2 0 0 0 0 B1λ2 B1λ3 C1 0 C13 e f f 2 Mν =0 Aλ2 Aλ2λ3+ B1λ2 0 B3λ2 + 0 0 0 0 Aλ λ Aλ2 B λ B λ 2B λ C 0 C2 2 3 3 1 3 3 2 3 3 1 3 3 . – p.20/31 Gaugino-Higgsino mass plane
The eigenvectors at tree level are,
~ ~λ ~v1 = × ~ ~λ | × | ~λ (~ ~λ) ~v2 = × × ~λ (~ ~λ) | × × | ~λ ~v3 = ~λ | | The rotation matrix is,
v11 v21 v31 c13c12 c13s12 s13 V = v v v = s s s s s s + c c s c PMNS 12 22 32 − 23 13 12 − 23 13 12 23 12 23 13 v v v c s c c s s s c c c 13 23 33 − 23 13 12 − 23 13 12 − 23 12 23 13 from where the atmospheric, solar, and reactor angles can be obtained. . – p.21/31 Conclusions
• Supersymmetry with Bilinear R-Parity Violation provides a framework for neutrino masses and mixing angles compatible with experiments.
• In Split Supersymmetry with BRpV the Higgs boson forms the only and crucial loop, and trilinear RpV couplings are essentially irrelevant.
• Neutrino parameters can be extracted from collider physics, specially from neutralino decays.
. – p.22/31 Effect of the Decoupling For simplicity, we assume all sneutrinos very heavy and neglect the running from m. The CP-even and CP-odd Higgs loops depend on the couplings: 0 0 e Fi Fi H A
0 0 Fj Fj
nnH nn nn nnA nn nn = i O = i (Q cα S sα) = O γ = (Q s S c )γ i j − i j − i j i j 5 i j β − i j β 5 0 0 Fi Fi h G
0 0 Fj Fj
nnh nn nn nnG nn nn = i Oi j = i (Qi j sα + Si j cα) = Oi j γ5 = (Qi j cβ + Si j sβ)γ5 − . – p.23/31 Effect of the Decoupling The loops involving the kth neutralino contribute with
m 0 χ νχ νχ νχ νχ ∆Π0 = k O HO H B0kH + O hO hB0kh i j − 16π 2 ik jk 0 ik jk 0 h νχ νχ νχ νχ O AO AB0kA O GO G B0kG − ik jk 0 − ik jk 0 i and in the limit of degenerate scalars and pseudoscalars: νχ νχ νχ νχ νχ νχ νχ νχ O HO H + O hO h O AO A O GO G = 0 ik jk ik jk − ik jk − ik jk obtaining a cancellation between scalars and pseudoscalars. In Split Susy H and A are decoupled, with sα = c and cα = s . The contribution to the neutrino mass − β − β matrix is, m 0 χ νχ νχ νχ νχ ∆Π0 = k O hO hB0kh O GO G B0kG i j − 16π 2 ik jk 0 − ik jk 0 h i and in the limit of degenerate scalars and pseudoscalars: νχ νχ νχ νχ νχ νχ νχ νχ O hO h O GO G = 2s c (Q S + Q S ) ik jk − ik jk − β β ik jk jk ik and the cancellation is incomplete. . – p.24/31 Sugra: Three Neutrinos
Based on work by, M.A. Díaz, Clemencia Mora, and Alfonso Zerwekh Eur.Phys.J.C44,277(2005)
. – p.25/31 Sugra Parameters at the GUT Scale Sugra is characterized by the following parameters, all defined at the GUT scale, except for tan β which is defined at the SUSY scale:
• m0: Universal scalar mass.
• M1/2: Universal gaugino mass.
• tan β: Ratio between vev’s.
• A0: Common trilinear coupling.
• sign(µ): Sign of higgsino mass.
In BRpV we add i and Λi as input at the SUSY scale.
. – p.26/31 Neutrinos in Supergravity Solutions to neutrino physics in a Sugra model with universal soft terms at the GUT scale, except for i and B ( Λ ), which are free at the weak scale. i ⇒ i Input: = 0.0004 GeV 1 − 2= 0.052 GeV 3= 0.051 GeV
2 Λ1= 0.022 GeV 2 Λ2= 0.0003 GeV 2 Λ3= 0.039 GeV . – p.27/31 Sugra: scan on neutrino parameters
For a fixed sugra point in parameter space, i and Λi are randomly varied, accepting solutions with good masses and mixing angles. Spectrum:
0 m(χ1)=99 GeV
m(χ1 )=175 GeV m(t˜1)=376 GeV
m(b˜1)=492 GeV m(h)=111 GeV m(H)=408 GeV
. – p.28/31 Sugra scenario predictions Sugra benchmark predicts,
2 3 2 2 ∆matm = 2.7 10− eV , tan θatm = 0.72 , 2 × 5 2 2 ∆msol = 8.1 10− eV , tan θsol = 0.55 , × 2 mee = 0.0036 eV , tan θ13 = 0.0058 , within experimental bounds.
. – p.29/31 Atmospheric Mass The atmospheric mass can be approximated as ∆m2 3 √5(AΛ2 + C2)C2 32 ≈ 2 3 3 2 explaining the quadratic depen- 2 dence of ∆m32 on 2, 3, and Λ3, and the mild dependence on Λ1. A 8 eV/GeV4 ≈ B 1 eV/GeV3 ≈ − C 9 eV/GeV2 ≈ . – p.30/31 Atmospheric Angle The atmospheric angle can be approximated as 2C23 (0) 2Λ2Λ3 tan 2θ23 = tan 2θ = ≈ AΛ2 + C(2 2) 6 23 Λ2 Λ2 3 3 − 2 3 − 2
If 2 0 then 2 → tan θ23 0, as seen in→ frame (a).
If 3 0 then 2 → tan θ23 π/2, as seen→in frame (b).
. – p.31/31