Three neutrino oscillations in matter

Physics Letters B 782 (2018) 641–645

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Physics Letters B www.elsevier.com/locate/physletb

Three neutrino oscillations in matter Ara Ioannisian a,b,∗ , Stefan Pokorski c a b c

Yerevan Physics Institute, Alikhanian Br. 2, 375036 Yerevan, Armenia Institute for Theoretical Physics and Modeling, 375036 Yerevan, Armenia Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL-02-093 Warsaw, Poland

a r t i c l e

i n f o

Article history: Received 15 April 2018 Received in revised form 22 May 2018 Accepted 1 June 2018 Available online 4 June 2018 Editor: G.F. Giudice

a b s t r a c t Following similar approaches in the past, the Schrödinger equation for three neutrino propagation in matter of constant density is solved analytically by two successive diagonalizations of 2 × 2 matrices. The final result for the oscillation probabilities is obtained directly in the conventional parametric form as in the vacuum but with explicit simple modification of two mixing angles (θ12 and θ13 ) and mass eigenvalues. In this form, the analytical results provide excellent approximation to numerical calculations and allow for simple qualitative understanding of the matter effects. © 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

The MSW effect [1] for the neutrino propagation in matter attracts a lot of experimental and theoretical attention. Most recently, the discussion is focused on the DUNE experiment [2]. On the theoretical side, a large number of numerical simulations of the MSW effect in matter with a constant or varying density has been performed. Although, in principle, sufficient for comparing the theory predictions with experimental data, they do not provide a transparent physical interpretation of the experimental results. Therefore, several authors have also published analytical or semi-analytical solutions to the Schrödinger equation for three neutrino propagation in matter of constant density, in various perturbative expansions [3–5]. The complexity of the calculation, the transparency of the final result and the range of its applicability depend on the chosen expansion parameter. In this short note we solve the Schrödinger equation in matter with constant density, using the approximate see-saw structure of the full Hamiltonian in the electroweak basis. This way one can diagonalize the 3 × 3 matrix by two successive diagonalizations of 2 × 2 matrices (similar approaches have been used in the past, in particular in Refs. [4] and [5]). We specifically have in mind the parameters of the DUNE experiment but our method is applicable for their much wider range. The final result for the oscillation probabilities is obtained directly in the conventional parametric form as in the vacuum but with modified two mixing angles and mass eigenvalues,1 similarly to the well known results for the

*

Corresponding author. E-mail address: [email protected] (A. Ioannisian). 1 The results of this paper have been presented as private communication by one of us (A.I.) to the members of the T2HKK collaboration in December 2017.

two-neutrino propagation in matter. The three neutrino oscillation probabilities in matter have been presented in the same form as here in the recent Ref. [6], where the earlier results obtained in Ref. [5] are rewritten in this form. The form of our final results can also be obtained after some simplifications from Ref. [4]. Our approach can be easily generalized to non-constant matter density by dividing the path of the neutrino trajectory in the matter to layers and assuming constant density in each layer. The starting point is the Schrödinger equation

i

d dx

ν = Hν

(1)

where H is the Hamiltonian in matter. In the electroweak basis it reads

⎛

0

⎜ H=U⎜ ⎝0

0 m2 2E

0

0

⎞

⎞ ⎛ V (x) 0 0 ⎟ † ⎝ 0 0 0⎠ 0 ⎟ ⎠U + 0

ma2

0

0

(2)

0

2E

The matrix U is the neutrino mixing matrix in the vacuum. The mass squared differences are defined as m2 ≡ m22 − m21 (≈ 7.5 × 10−5 eV2 ) and ma2 ≡ m23 − m21 (≈ ±2.5 × 10−3 eV2 , positive sign is for normal mass ordering and negative sign for inverted one). Here V (x) is the neutrino weak interaction potential energy V = √ 2G F N e (N e is electron number density) and we take it in this section to be x-independent. The neutrino oscillation probabilities are determined by the S-matrix elements

S αβ = T e

−i

 xf x0

H(x)dx

(3)

https://doi.org/10.1016/j.physletb.2018.06.001 0370-2693/© 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

642

A. Ioannisian, S. Pokorski / Physics Letters B 782 (2018) 641–645

For a constant V and in order to obtain our results in the same form as for the oscillation probabilities in the vacuum, it is convenient to rewrite the S-matrix elements as follows:

S αβ = e

†

−iU m Hm U m (x f −x0 )

= Ume

−i Hm L

† Um

Using eqs. (2), (8) we obtain T T H = O13 U δ∗ O23 H O23 U δ O13 ⎛ 2 V c 13

⎜ 2 ⎜ = ⎜s12 c 12 m 2E ⎝

(4)

The matrix Hm is the Hamiltonian in matter in the mass eigenstate basis:

⎛

0

0 0 ⎠

(5)

H3

0

⎛

⎡

1 0 S α β = ⎣U m ⎝ 0 e −i φ21 0 0

0 0

⎞

e −i φ31

⎤

† ⎦ ⎠ Um

(6)

Here we neglect irrelevant overall phase, e −i H1 L . The neutrino transition probabilities do not depend on the overall phase of the S matrix. The remaining task is to find the eigenvalues of H and the mixing matrix U m : †

H = U m Hm U m

(7)

It is convenient to do it in two steps, first calculating the hamiltonian in a certain auxiliary basis. This way, to an excellent approximation, we can diagonalize the 3 × 3 matrix by two successive diagonalizations of the 2 × 2 matrices. The auxiliary basis [7,8] is defined by the following equation 

H =U

aux†

HU

aux

and S = U

aux (−i H L )

e

U

aux†

(8)

⎛

δ∗

O12

c 13 c 12 = ⎝−s12 c 23 − c 12 s23 s13 e i δ s12 s23 − c 12 c 23 s13 e i δ

c 13 s12 c 12 c 23 − s12 s23 s13 e i δ −c 12 s23 − s12 c 23 s13 e i δ

⎛ mT T  O12 O13 H

⎛

0



m O13 O12

0

⎜ ≡⎜ ⎝0 0

⎞

0

2E

m231

0

⎛

2E

⎞

1 0 0 U =⎝0 1 0 ⎠ 0 0 eiδ

(11)

(c 12 ≡ cos θ12 , s12 ≡ sin θ12 etc.). The matrices Oi j are orthogonal matrices. It is more convenient to rewrite the matrix U in another form

U → U˜ = U · U δ = O23 U δ O13 O12 c 13 c 12

(13)

+ V s213 (14)

0

0

0

⎞

0 ⎠

H2

(15)

H3

0

⎛

⎞ 1 0 0 ⎟ ⎟ + H1 ⎝ 0 1 0 ⎠ ⎠ 0

0

(16)

1

2E

After the first rotation we have T   O13 H O13 ⎛  sin2 θ13

2 mee

2E

 )V + cos2 (θ13 + θ13

⎜ = ⎝cos θ13 s12 c12 2Em

 s c cos θ13 12 12 2 (c 12 − s212 )

⎞

m2

0

2E

m2

 s c sin θ13 12 12

2E

 s c sin θ13 12 12

m2

 cos2 θ13

2E

2 mee

2E

⎟ ⎠

m2 2E

 )V + sin2 (θ13 + θ13

(17) where  sin 2θ13 =

a sin 2θ13

(18)

,

(cos 2θ13 − a )2 + sin2 2θ13

and

a =

2E V

(19)

2 mee

We can safely neglect the (23), (32) elements which are generated after the first rotation (see Appendix A) and diagonalize the remaining 2 × 2 sub-matrix with the second rotation

δ

⎛

H1

=⎝ 0

0

m221

⎞

s13 e −i δ c 13 s23 ⎠ c 13 c 23 (10)

where

0 2 mee

(9)

and the rotations Oi j are defined by the decomposition of the mixing matrix U in the vacuum (see eq. (2)) as follows:

U = O23 U O13 U

⎟ ⎟ ⎟, ⎠

2

The term s212 2E has been subtracted from the diagonal elements; it gives an overall phase to the S-matrix and according to the comments after eq. (6) is irrelevant. 2 The definition of mee coincides with one of the definitions of the effective mass squared differences measured at reactor experiments [9,10]. This matrix has a see-saw structure, with the (13), (31) elements much smaller than the (33) element and can be put in an almost diagonal form by two rotations

0

δ

2E

m s212 ) 2E

2



U aux = O23 U δ O13

s13 c 13 V

0

2

where

−

⎞

m2

2 2 mee = c 12 ma2 + s212 (ma2 − m2 )

m

and the U m is the neutrino mixing matrix in matter. Defining φ21 = (H2 − H1 ) L and φ31 = (H3 − H1 ) L, we can write

2 (c 12

s13 c 13 V

⎞

H1 0 ⎝ 0 H2

s12 c 12

c 13 s12

s13

⎞

m sin 2θ12 =

where

 sin 2θ cos θ13 12  sin2 2θ (cos 2θ12 −  )2 + cos2 θ13 12

 =

2E V

m2

 (cos2 (θ13 + θ13 )+

 sin2 θ13

a

,

).

(20)

The eigenvalues of H are

= ⎝−s12 c 23 − c 12 s23 s13 e i δ c 12 c 23 − s12 s23 s13 e i δ c 13 s23 e i δ ⎠ s12 s23 − c 12 c 23 s13 e i δ −c 12 s23 − s12 c 23 s13 e i δ c 13 c 23 e i δ

H2 − H1 ≡

(12)

=

m221 2E

m2 2E

 sin2 2θ , (cos 2θ12 −  )2 + cos2 θ13 12

(21)

A. Ioannisian, S. Pokorski / Physics Letters B 782 (2018) 641–645

H3 − H1 ≡



m231

m 2 P νμ →νe = sin2 2θ13 s23

2E

 = cos2 θ13

2 mee

2E

1

2 − [(c 12 −

2

+

 + sin2 (θ13 + θ13 )V 2

2 m  mee s212 ) + sin2 θ13

2E

2E

1 m221

1 4

2E

+ V]+

1 4E

(m221 − m2 cos 2θ12 ) (23)

 m  m m m U˜ m = U aux O13 O12 = O23 U δ O13 O13 O12 = O23 U δ O13 O12 .

(24)

For the matrix U defined in eq. (10) we get

⎞⎛

m cos θ13 0 1 0 0 U m = O23 ⎝ 0 1 0 ⎠ ⎝ 0 1 m 0 0 eiδ − sin θ13 0

⎞⎛

⎛

m cos θ12 1 0 0 ⎜ m × ⎝ 0 1 0 ⎠ ⎝ − sin θ12 − iδ 0 0 e 0

m sin θ13

⎞

0⎠ m cos θ13

⎞

m sin θ12

0

m cos θ12

0⎠

⎟

(25)

0 1

m  and with θ13 = θ13 + θ13

m sin 2θ13 =

m cos 2θ13

sin 2θ13

,

(cos 2θ13 − a )2 + sin2 2θ13

=

(26)

cos 2θ13 − a

(cos 2θ13 − a )2 + sin2 2θ13

m cos 2θ12

 sin 2θ cos θ13 12  sin2 2θ (cos 2θ12 −  )2 + cos2 θ13 12

=

φi j =

m2i j 2E

2

φ21

φ31 + φ32

,

cos 2θ12 − 

(28)

2

(27)

 sin2 2θ (cos 2θ12 −  )2 + cos2 θ13 12

In summary the mixing matrix in matter, U m , is given by the following change of the parameters from the vacuum solution: m θ12 → θ12 (eq. (27)) m (eq. (26)) θ13 → θ13 m ≡ θ23 θ23

δm ≡ δ. The mass eigenvalues are given by eqs. (21), (22), (23). The oscillation probabilities P να →νβ (α , β = e , μ, τ ) have the same forms as for the vacuum oscillations with mass eigenstates m m as above and with replacements θ12 → θ12 and θ13 → θ13 . For the νμ → νe transition we have

L

i , j = 1, 2, 3

( L = 1285 km for DUNE) .

(29)

This approximate solution is valid for all energies. Numerically our result is identical to the approximation of two angles rotation in [4] and the 0th order result of [5]. For anti-neutrino oscillations P ν¯ α →ν¯ β , V → − V and δ → −δ . For normal mass hierarchy ma2 is positive and for inverted mass hierarchy it is negative. Our solutions are illustrated in Figs. 1 and 2 for νμ → νe oscillation at DUNE distance for several values of δC P and compared with the oscillation probabilities in the vacuum, shown by the dotted curves. They are the reference point of our discussion. The matter effects and their dependence on δC P observed in those plots have easy explanation in terms of our analytic formulas. For the sake of definiteness, we focus on the region of the first maximum (E = (1–6) GeV), accessible in the DUNE experiment. First of all we notice that the modification in matter of the solar sector parameters, angle θ12 and m221 , have very small effect when the phase φ21 1 (as it is for DUNE distance and energies). In the first term of the rhs eq. (28) the dependance on the φ21 is sub-leading. In the 2nd and 3rd terms we have combinations φ m sin 2θ12 sin 221 and that for small phases can be rewritten as m sin 2θ12 sin

and m sin 2θ12 =



where

Finally, for the mixing matrix in matter we obtain

⎛

φ32

m m cm 13 sin 2θ13 sin 2θ12 sin 2θ23 cos δ

4

4 2E

+ [

2

2 + sm2 12 sin

 1 φ21 m m + cm sin 2 θ sin 4 θ sin 2 θ cos δ sin2 23 13 13 12

(cos 2θ13 − a )2 + sin2 2θ13 2 1 mee − (cos 2θ13 − a )2 + sin2 2θ13 2E

2 mee

φ31

φ21 φ31 φ32 × sin sin sin 2 2 2  m2 2 m 2 + c13 sin 2θ12 (c 23 − s223 sm2 13 )

(22)

2 2E

2 mee

2

2 cm2 12 sin

sin 2 2 m m −cm 13 sin 2θ13 sin 2θ12 sin 2θ23 sin δ

+ cos (θ13 + θ13 ) V ]

=

1

× sin



2

+

643

m221 4E

m L sin 2θ12

m221 4E

 sin 2θ12 L = cos θ13

m2 4E

L,

 1 at E = which is almost independent on matter (cos θ13 1–6 GeV). Therefore the dependence on matter due to change of θ12 − m221 is very suppressed at DUNE. The most important effect is the dependence of the oscillation probability on the angle θ13 which has larger (smaller) values in matter than in the vacuum for normal (inverted) neutrino mass hierarchies (and opposite for antineutrinos). Thus the oscillation probabilities have larger (lower) oscillation amplitudes for normal (inverted) neutrino mass hierarchies (and opposite for antineutrinos). In oder words the matter of the Earth is amplifying the effect of the mass ordering on neutrino oscillations. The dependence on the angle θ13 enters multiplicatively in the first three terms of eq. (28), whereas the fourth term is small in the region of the first maximum. Therefore the matter effects relative to the oscillations in the vacuum do not depend on the value of δC P , as it is seen in Figs. 1 and 2. Moving to the next resonances (lower energies) the difference between oscillations in matter and in the vacuum remain qualitatively similar, although some small differences can be seen due to the fact that the change in the angle θ13 is smaller. Finally, in Fig. 3 we show the accuracy of the analytical solutions comparing them with numerical/exact results.

644

A. Ioannisian, S. Pokorski / Physics Letters B 782 (2018) 641–645

Fig. 1. νμ → νe oscillation probability at DUNE for normal mass hierarchy, δcp = 0 (red), δcp = π2 (green), δcp = π (black), δcp = − π2 (blue). Thickness of the plots are from varying constant/uniform matter density 2.5–3 g/cm3 . Dotted plots are for vacuum oscillations. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

Fig. 2. νμ → νe oscillation probability at DUNE for inverted mass hierarchy, δcp = 0 (red), δcp = π2 (green), δcp = π (black), δcp = − π2 (blue). Thickness of the plots are from varying constant/uniform matter density 2.5–3 g/cm3 . Dotted plots are for vacuum oscillations.

Fig. 3.

| P | P

≡

anl | P νnum μ →νe − P νμ →νe |

P νnum μ →νe

. The relative error of our analytic result to the exact (numeric) νμ → νe oscillation probability for normal mass hierarchy, δcp = 0 (red), δcp = π2

(green), δcp = π (black), δcp = − π2 (blue). Matter density 2.6 g/cm3 .

A. Ioannisian, S. Pokorski / Physics Letters B 782 (2018) 641–645

Acknowledgements

m  sin θ13 A = sin θ12

645

sin 2θ12 m2 2

m231

(1 − e −i φ31 )

One of us (A.I.) is grateful to the CERN Theory group for its hospitality and to the DUNE members for useful discussions at Fermilab in summer 2017. A.I. is especially grateful to Pilar Coloma and Maury Goodman for underlining the importance of presenting the oscillation parameters in a simple form. S.P. is partially supported by the National Science Centre, Poland, under research grants DEC-2015/19/B/ST2/02848, DEC-2015/18/M/ST2/00054 and DEC-2014/15/B/ST2/02157.

S 0 is our solution for the neutrino transition matrix elements and the S 1 is its first order corrections. |A|, |B | are at least smaller than 0.5% for all energies, therefore our 0th order solution, S 0 , is working excellently.

Appendix A

References

In ordinary perturbation expansion in the basis of our solutions we estimate the size of effect of the neglected elements (23), (32) in eq. (17). We divide the hamiltonian into two parts

S = e −i H L = e −i H0 L −i H L m221

(30) m231

†

Here H0 ≡ U m diag (0, 2E L , 2E L ) U m with our solutions for U m (eq. (25)) and mass square differences (eqs. (21)–(23)). H(23) = m2

 sin 2θ  H(32) = sin θ13 and all other elements in H are 12 4E zeros. We treat H as a perturbation. 1 −a t b ea t By making use well known identity ea+b = ea T e 0 dt e

we get

S = S0 + S1 + . . .

⎛

1 0 S 0 = U m ⎝ 0 e −i φ21 0 0

⎛

0 S1 = Um ⎝ 0

0 0

A B

where

0 0

e −i φ31

⎞ A † B ⎠ Um 0

⎞ † ⎠ Um ,

(31) (32)

(33)

m  B = cos θ12 sin θ13

sin 2θ12

m2

2

m231 − m221

(e −i φ31 − e −i φ21 )

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