perturbation theory of neutrino oscillation with and without...
TRANSCRIPT
Perturbation Theory of Neutrino Oscillation with and without NSI �
Hisakazu Minakata Tokyo Metropolitan
University
Part I: Comments on Kamioka-Korea 2 detector setting �
• Search for CP violation and ν mass hierarchy; discussion supplementary to Suzuki-san’s and Volpe-san’s talks �
September 20, 2009 Erice School 09: Neutrinos
September 20, 2009 Erice School 09: Neutrinos
Kamioka-Korea 2 detector setting
Why don’t you bring one of the 2 tanks to Korea? (@EPP2010)
September 20, 2009 Erice School 09: Neutrinos
Original idea: sensitive because dynamism in 2nd oscillation maximum
September 20, 2009 Erice School 09: Neutrinos
Two detector method is powerful
September 20, 2009 Erice School 09: Neutrinos
Spectral information solves intrinsic degeneracy
from 1000 page Ishitsuka file
SK momentum resolution ~30 MeV at 1 GeV
T2K T2KK
2 detector method powerful!
Ishitsuka-Kajita-HM-Nunokawa 05 �
September 20, 2009 Erice School 09: Neutrinos
Two-detector setting is powerful
• With the same input parameter and Korean detector of 0.54 Mt the sign-Δm2 degeneracy is NOT completely resolved
T2KK Korea only
September 20, 2009 Erice School 09: Neutrinos
10 -2
10 -1
0 1 2 3 4 5 6
normal
10 -2
10 -1
0 1 2 3 4 5 6
inverted
s
in 2
2
13
10 -2
10 -1
0 1 2 3 4 5 6
normal
Kamioka 0.54 Mton
Kamioka 0.27 Mton + Korea 0.27 Mton
10 -2
10 -1
0 1 2 3 4 5 6
inverted
s
in 2
2
13
Total mass of the detectors = 0.54 Mton fid. mass 4 years neutrino beam + 4 years anti-neutrino beam
3 σ (thick) 2 σ (thin)
Mass hierarchy CP violation (sinδ≠0)
hep-ph/0504026
T2K�
T2KK�
September 20, 2009 Erice School 09: Neutrinos
Relative cross section error does matter
• Identical 2 detector setting robust to larger systematic error • It gives conservative lower bounds on sensitivity estimate
of mass hierarchy and CP
Barger et al. 07
T2K II
T2KK
September 20, 2009 Erice School 09: Neutrinos
T2KK can solve θ23 degeneracy in situ T2K-II + phase II reactor T2KK
δ=0 assumed si
n2 2θ 1
3
sin2 θ23
sin2
2θ 1
3
> 3σ 2~3σ
T2KK 2σ (rough)
T2KK has better sensitivity at sin2 2θ13 < 0.06~0.07 .
hep-ph/0601258
Conclusion: part I �
• Kamioka-Korea 2 detector setting = reasonable thing to think about for sensitivities both CPV and mass hierarchy
• Sensitivity to CPV comparable between in T2K and T2KK staged approach possible
September 20, 2009 Erice School 09: Neutrinos
September 20, 2009 Erice School 09: Neutrinos
Part II: This talk is originally meant to
be…
Perturbation Theory of Neutrino Oscillation with and without NSI �
• This is meant to be a pedagogical talk for students
• Nothing terribly exciting, but serves for better understanding of ν oscillations, a useful tool for understanding lepton flavor mixing �
September 20, 2009 Erice School 09: Neutrinos
General feature (standard interaction only) �
September 20, 2009 Erice School 09: Neutrinos
General feature (continued) �
September 20, 2009 Erice School 09: Neutrinos
Our goal �
Because Hamiltonian has this form, it is natural to define the tilde basis (next page) �
Tilde basis �
September 20, 2009 Erice School 09: Neutrinos
Akhmedov et al. 04�
September 20, 2009 Erice School 09: Neutrinos
Perturbation theory
of ν oscillation
Perturbation theory of ν oscillation �
• No general perturbative scheme for ν oscillation many dimensionless parameters: Δm2
solar/Δm2atm, Δm2L/E, a/Δm2
• Only known small parameter: Δm2solar/Δm2
atm ~ 0.03
• Need to choose the right framework suitable for your problem (case by case)
• I explain you only a particular example
September 20, 2009 Erice School 09: Neutrinos I work mostly for LBL experiments�
ε perturbation theory �
• I take the assumption
• I assume a/Δm2atm~O(1), Δm2
atm E/L~O(1) • most natural perturbation theory of ν
oscillation Peµ consists only of order ε2 terms widely used formula (Cervera et al. 00)
September 20, 2009 Erice School 09: Neutrinos
Are you happy with such small θ13? �
Perturbation theory in tilde basis �
September 20, 2009 Erice School 09: Neutrinos
``Interaction representation’’ �
zeroth-order �1st-order �
We follow standard path �
September 20, 2009 Erice School 09: Neutrinos
Seµ and Peµ �
September 20, 2009 Erice School 09: Neutrinos
Cervera et al. 00�
Constant matter density approximation �
Matter hesitation �
September 20, 2009 Erice School 09: Neutrinos
Matter effect enters into Pαβ at second order in ε => no first order a-dependent term �
Highly nontrivial, in particular for Pee, but in fact very easy to prove !�
“matter hesitation” indicates that detecting the matter effect is not easy in LBL experiments �
order ~ ε2 �Kikuchi-HM-Uchinami JHEP09 �
Magic baseline= solar amplitude zero �
September 20, 2009 Erice School 09: Neutrinos
At magic baseline, = 0, the solar amplitude vanishes �
which occurs at L ~ 7200 km �
Then there is no δ dependence in Peµ �
September 20, 2009 Erice School 09: Neutrinos
Analytic treatment of degeneracy can be done with 2nd order formula
• You can draw two ellipses from a point in P-Pbar space
• Intrinsic degeneracy
• doubled by the unknown sign of Δm2
• 4-fold degeneracy
September 20, 2009 Erice School 09: Neutrinos
ν oscillation with NSI
New physics at TeV scale? �
• Waiting for LHC for new physics at TeV scale
• <φ> = 250 GeV • => E~TeV as a
new physics scale • • Extra dimensions • Supersymmetry • …. September 20, 2009 Erice School 09: Neutrinos
September 20, 2009 Erice School 09: Neutrinos
ν's Non-standard interactions (NSI) • If there exists New Physics at TeV scale there
might be NSI of ν’s expressed by higher dimensional operators
• But coefficient ε may be small (MNP = 1 TeV)
• if dimension 6, ε~ (MW/MNP)2 = 0.01 • if dimension 8, ε~ (MW/MNP)4 = 0.0001
Wolfenstein, Grossman, Berezhiani-Rossi, Davidson et al. … many people
September 20, 2009 Erice School 09: Neutrinos
Non Standard Interaction (NSI)
Grossmann, Ota-Sato-Yamashita, Huber et al. …
I this talk, I focus on effects of NSI in ν propagation in matter governed by evolution eq. => vector type interactions
• It comes in into 3 places, production, propagation, & detection
ν oscillation with NSI �• Given the structure
• the system is highly nontrivial ! • all εαβ may be related, but likely to be
present with similar magnitudes • At the moment nobody knows how these
elements are related with each other • No stringent constraints on particular εαβ
September 20, 2009 Erice School 09: Neutrinos include all of them �
A general theorem of phase reduction �
September 20, 2009 Erice School 09: Neutrinos
For Δm221=0, or at the magic baseline, number
of CP violating phases decreases by 1 �
natural because of effective 2 generations�
Kikuchi-HM-Uchinami JHEP09 �
September 20, 2009 Erice School 09: Neutrinos
Perturbation theory with
NSI
ε perturbation theory with NSI �• We include NSI by assuming
• and expand to ε2; NSI 2nd order formula • If dimension 8, 1st order formula would be
enough but, 2nd order formula is simpler !
September 20, 2009 Erice School 09: Neutrinos
NSI 2nd order formula for Peµ �
SI formula -> NSI 2nd order formula �
c12s12 (Δm221 / a)
s13 e-iδ (Δm231/a)
c12s12 (Δm221 / a)
c12s12 (Δm221 / a) s13 e-iδ (Δm2
31/a)
s13 e-iδ (Δm231/a)
Cervera et al , hep-ph/0002108 September 20, 2009 Erice School 09: Neutrinos
SI formula -> NSI 2nd order formula �
September 20, 2009 Erice School 09: Neutrinos T. Kikuchi, H.M., S. Uchinami, arXiv:0809.3312 => JHEP �
Generalized atmospheric and solar variables �
“θ13” “θ12”
SI-NSI confusion ! �
September 20, 2009 Erice School 09: Neutrinos
Secret is in tilde hamiltonian !�
Invariance of tilde hamiltonian �
September 20, 2009 Erice School 09: Neutrinos
Seτ = Seµ (extended transf.) �
tilde-H is invariant under the extended transf. �
Undoing … �
How big is the contribution of εαβ? Bird-eye view
• decoupling of εµτ and εµµ to P(νe -> νµ) to ε2
• Determine εeµ and εeτ from P(νe -> νµ) • εµτ and εµµ dependent term is common to all P(νµ -> ντ), P(νµ -> νµ) and P(ντ -> ντ)
September 20, 2009 Erice School 09: Neutrinos
If θ23 is maximal Matter hesitation Direct transition by NSI
Characteristics �
spectrum analysis required �
NSI perturbation theory; temporary summary �
• Structure of ν oscillation with NSI is in fact very simple within ε perturbation theory
• But in real life, it is complicated even with single εαβ because the system is enriched with 2 phases lepton KM phase + NSI phase requires hard work �
September 20, 2009 Erice School 09: Neutrinos
ν oscillation only with
single type of NSI εαβ is already a
complicated phenomenon;
(example with εeµ) �
September 20, 2009
0 1 2 3 4 5 6 7 8
P(!e
!!"#"$x10
-4]
0
1
2
3
4
5
6
7
8
P(!
e!!"#"$x
10
-4]
sin22"
%&"'"%(
#&
E = 20 GeV, |$e!)"'"(*((+
% : varied continuasly
&e!
: varied discontinuously
Erice School 09: Neutrinos
2 phase confusion with εeτ; synergy �
September 20, 2009 Erice School 09: Neutrinos
Gago-HM-Nunokawa-Uchinami-Zukanovich Funchal, arXiv:0904.3360 �
Degeneracy; εeτ�
September 20, 2009 Erice School 09: Neutrinos
September 20, 2009 Erice School 09: Neutrinos
Conclusion • General relations between S matrix exist
because of the special structure of matter effect
(both with and without NSI) • Qualitative understanding of neutrino
oscillation can be achieved by perturbative formulation
• SI only: neat Peµ and Peτ 2nd-order formula, matter hesitation,
• With NSI: generalized solar and atmospheric variables, εαβ dependence of Peµ, Peτ, Pµτ, etc.
diag(a, 0, 0)�