Compact Perturbative Expressions for Neutrino Oscillations in Matter: II

In this paper we rewrite the neutrino mixing angles and mass squared differences in matter given, in our original paper, in a notation that is more conventional for the reader. Replacing the usual neutrino mixing angles and mass squared differences in the expressions for the vacuum oscillation probabilities with these matter mixing angles and mass squared differences gives an excellent approximation to the oscillation probabilities in matter. Comparisons for T2K, NOvA, T2HKK and DUNE are also given for neutrinos and anti-neutrinos, disappearance and appearance channels, normal ordering and inverted ordering.


tion

In this
paper we rewrite the neutrino mixing angles and mass squared differences in matter given in our original paper, [1], in a notation that is more conventional for the reader.Replacing the usual neutrino mixing angles and mass squared differences in the expressions for the vacuum oscillation probabilities with these matter mixing angles and mass squared differences gives an excellent approximation to the oscillation probabilities in matter.Higher orders are also easily calculated and provide several orders of magnitude improvement per order.

In Sectio 2, we give the approximation to the mixing angles and mass squared difference in matter and discuss how to use these to calculated the oscillation probabilities in matter both at 0th order and 1st order.We also give expansions of the mixing angles and mass squared differences in matter in powers of (a/∆m 2 ).In Section 3, we make a detailed comparison between the exact and the approximate oscillation probabilities in matter for the T2K, NOvA, T2HKK and DUNE experiments.Section 4 is the Summary.


Mixing Angl

In this sect
on, a simple and accurate way to evaluate oscillation probabilities, recently shown in [1], is given. 1 Details as to the why's and how's of this method are contained in that paper.

The mixing a gles in matter, which we denote by a θ 13 and θ 12 here, can also be calculated in the following way, using ∆m where a ≡ a cos2 θ 13 + ∆m 2 ee sin 2 ( θ 13 − θ 13 ) is the θ 13 -modified matter pot [0, π/2].θ 23 and δ are unchanged in matter for this approximation.

The neutrino mass squared differences in matter, i.e. the ∆m .In Fig. 1 and Fig. 2 the values of a, a θ 13 , sin 2 θ 12 , m 2 j and ∆ m 2 jk as a function of the neutrino energy for a density of 3.0 g.cm −3 .

To calculate the oscillation probabilities, to 0th order, use the above ∆ m 2 jk instead of ∆m which is small and vanishes in vacuum, so t at our perturbation theory reproduces the vacuum oscillation probabilities exactly.

If P να→ν β (∆m 2 31 , ∆m 2 21 , θ 13 , θ 12 , θ 23 , δ) is the oscillation probability in vacuum then P να→ν β (∆ m 2 3 nction but replace the mass squared differences and mixing angles with the matter values given in eq.2.1.1 -2.1.3.The resulting oscillation probabilities are identical to the zeroth order approximation given in Denton, Minakata and Parke, [1].


Higher Orders

If the 0th order is not accurate enough, going to 1st order is simple and gives another two orders of magnitude in accur cy.First the ∆ m 2 jk remain unchanged but the mixing matrix is modified by
U M M N S ⇒ V ≡ U M M N S (1 + W 1 ),(2.2.1)
wher the matrix W 1 is given by
W 1 = sin( θ 13 − θ 13 ) s 12 c 12 ∆m 2 21    0 0 − s 12 e −iδ /∆ m 2 31 0 0 + c 12 e −iδ / ∆ m 2 32 + s 12 e +iδ /∆ m 2 31 − c 12 e +iδ /∆ m 2 32 0    . (2.2.2)
where s 12 = sin θ 12 and c 12 = cos θ 12 etc.The ∆ m 2 jk and the V mixing matrix can be used to calculate the oscillation probabilities and improves the accuracy by two orders of magnitude.We call this the 1st or ering (NO): Top left, the matter potentials, a and a , top right, sine squared of mixing angles in matter, sin 2 θ jk , bottom left, the mass squ ight, the mass squared differences in matter, ∆ m 2 jk .E ν ≥ 0 (E ν ≤ 0) is for neutrinos (anti-neutrinos).E ν = 0 is the vacuum values for both neutrinos and anti-neutrinos.up to O(a/∆m 2 ee ) 2 .The expansion for a can be used to calculate ∆ m 2 31 as follows,


Expansions in
∆ m 2 31 =    ∆m 2 31 + ( a − a ) + 1 2 ∆ m 2 21 − ∆m 2 21 − a , a, a > 0 ∆m 2 31 + ( a − 2a ) + 1 2 ∆ m 2 21 − ∆m 2 21 + a , a, a < 0 (2.3.2)
where the quantities in
[• • • ] is of O(

2 21
) for al
values of E ν .As can be seen from Fig. 1  .This can be used to obtain the asymptotic values for neutrino mass squareds in matter, which agree with the values given in [1].


Oscillation Probabilit t for oscillations: ∆m 2 ee = ± 2 ing (NO) neutrino spectrum and ∆m 2 ee < 0 for inverted ordering (IO).Note we have avoided the special points: θ 23 = π/4 as well as δ = 0, ±π/2, π, so as not to overestimate the precision.

We consider four experimental setups: to be comprehensive, the energy windows are chosen to be wider than that xperiment.

• T2K & T2HK: with baseline, L = 295 km, neutrino energy 0.2 < E ν /GeV < 3.0, and density,

n ∆P changes
sign.

In Table 1, we give the maximum difference and fractional difference of the 0th order approximation to the exact probability.T2K/HK NOvA ∆P/P 10 −3 10 −3 10 −2.5 10 −2

Table 1: The maximum ∆P and ∆P/P at 0th order in the DMP approximation.The largest fraction difference occurs at oscillation maximum for ν µ disappearance channel, where the oscillation probability is a few %.None of the experiments included here, T2K, NOvA, T2HKK and DUNE will be within an order a magnitude of being sensitive to any of these diff rences.


Summary

In summary, the simple 0th order approximation of DMP, [1], is sufficiently accurate for all of the accele ator based neutrino oscillation experiments operating or planned: T2K, NOvA, T2HKK and DUNE.First order, which is also imple to use, improves the accuracy by a further two orders of magnitude for these experiments.In each figure, the top panel i exact oscillation probability in matter , P ex mat , from [3], and in vacuum, P vac .The Middle panel is difference betw en exact oscillation probabilities in matter and vacuum (black), and the difference between exact and 0th (red) and exact and 1st (green) approximations to the matter probabilities using the DMP scheme, [1].Bottom panel is similar to middle panel but plotting the fractional differences, ∆P/P .The density use is 2.3 g.cm −3 .In each figure, the top panel is exact oscillation probability in matter , P ex mat , from [3], and in vacuum, P vac .The Middle panel is difference between exact oscillation probabilities in matter and vacuum (black), and the difference between exact and 0th (red) and exa t and 1st (green) approximations to the matter probabilities using the DMP scheme, [1].Bottom panel is similar to middle panel but plotting the fractional differences, ∆P/P .The density use is .3 g.cm −3 .Right is νµ → νe appearance.In each figure, the top panel is exact oscillation probability in matter , P ex mat , rom [3], and in vacuum, P vac .The Middle panel is difference between exact oscillation probabilities in matter and vacuum (black), and the difference between exact and 0th (red) and exact and 1st (green) approximations to the matter probabilities using the DMP scheme, [1].Bottom panel is similar to middle panel but plotting the fractional differences, ∆P/P .The density use is 3.0 g.cm −3 .Right is νµ → νe appearance.In each figure, the top pane

is exact
oscillation probability in matter , P ex mat , from [3], and in vacuum, P vac .The Middle panel is difference between exact oscillation probabilities in matter and vacuum (black), and the difference between exact and 0th (red) and exact and 1st (green) approximations to the matter probabilities using the DMP scheme, [1].Bo

om panel is simil
r to middle panel but plotting the fractional differences, ∆P/P .The density use is 3.0 g.cm −3 .Right is νµ → νe appearance.In each figure, the top panel is exact oscillation probability in matter , P ex mat , from [3], and in vacuum, P vac .The Middle panel is difference between exact oscillation probabilities in matter and vacuum (black), and the difference between exact and 0th (red) and exact and 1st (green) approximations

nce.In each f
gure, the top panel is exact oscillation probability in matter , P ex mat , from [3], and in vacuum, P vac .The Middle panel is difference between exact oscillation probabilities in matter and vacuum (black), and the difference between exact and 0th (red) and exact and 1st (green) approximations to the matter probabilities using the DMP scheme, [1].Bottom panel is similar to middle panel but plotting the fractional differences, ∆P/P .The density use is 3.0 g.cm −3 .Right is νµ → νe appearance.In each figure, the

, and t
e difference, ∆P , between exact a ional differences, ∆P/P .The density use is 3.0 g.cm −3 .Right is νµ → νe appearance.In each figure, the top panel is exact oscillation probability in matter , P ex mat , from [3], and in vacuum, P vac .The Middle panel is difference between exact oscillation probabilities in matter and vacuum (black), and the difference between exact and 0th (red) and exact and 1st (green) approximations to the matter probabilities using the DMP scheme, [1].Bottom panel is similar to middle panel but plotting the fractional differences, ∆P/P .The d

sity us
is 3.0 g.cm −3 .

Figure 2 :
2
Fi uared eigenvalues in matter, m 2 j , and bottom right, the mass squared differences in matter, ∆ m 2 jk .E ν ≥ 0 (E ν ≤ 0) is for neutrinos (anti-neutrinos).E ν = 0 is the vacuum values for both neutrinos and anti-neutrinos.


Figure 3 :
3
Figure 3: T2K, for normal ordering (NO): Top Left figure is ν µ disappearance, Top Right figure is νµ disappearance, Bottom Left figure is ν µ → ν e appearance, and Bottom Right is νµ → νe appearance.In each figure, the top panel is exact oscillation probability in matter , P ex mat , from[3], and in vacuum, P vac .The Middle panel is difference between exact oscillation probabilities in matter and vacuum (black), and the difference between exact and 0th (red) and exact and 1st (green) approximations to the matter probabilities using the DMP scheme,[1].Bottom panel is similar to middle panel but plotting the fractional differences, ∆P/P .The density use is 2.3 g.cm −3 .


Figure 4 :
4
Figure 4: T2K, for inverted ordering (IO): Top Left figure is ν µ disappearance, Top Right figure is νµ disappearance, Bottom Left figure is ν µ → e appearance, and Bottom Right is νµ → νe appearance.In each figure, the top panel is exact oscillation probability in matter , ∆P/P .The density use is 2.3 g.cm −3 .


Figure 5 :
5
Figure 5: NOvA, for normal ordering (NO): Top Left figure is ν µ disappearance, Top Right figure is νµ disappearance, Bottom Left figure is ν µ → ν e appearance, and BottomRight is νµ → νe appearance.In each figure, the top panel is exact oscillation probability in matter , P ex mat , from[3], and in vacuum, P vac .The Middle panel is difference between exact oscillation probabilities in matter and vacuum (black), and the difference between exact and 0th (red) and exact and 1st (green) approximations to the matte 6 :
6
Figure 6: NOvA, for inverted ordering (IO): Top Left figure is ν µ disappearance, Top Right figure is νµ disappearance, Bottom Left figure is ν µ → ν e appearance, and BottomRight is νµ → νe appearance.In each figure, the top panel is exact oscillation probability in matter , P ex mat , from[3], and in vacuum, P vac .The Middle panel is difference between exact oscillation probabilities in matter and vacuum (black), and the difference between exact and 0th (red) and exact and 1st (green) approximations to the matter probabilities using the DMP scheme,[1].Bottom panel is similar to middle panel but plotting the fractional differences, ∆P/P .The density use is 3.0 g.cm −3 .


Figure 7 :
7
Figure 7: T2HKK, for normal ordering (NO): Top Left figure is ν µ disappearance, Top Right figure is νµ disappearance, Bottom Left figure is ν µ → ν e appearance, and BottomRight is νµ → νe appearance.In each figure, t e top panel is exact oscillation probability in matter , P ex mat , from[3], and in vacuum, P vac .The Middle panel is difference between exact oscillation probabilities in matter and vacuum (black), and the difference between exact and 0th (red) and exact and 1st (green) approximations to the matter probabilities using the DMP scheme,[1].Bottom panel is similar to middle panel but plotting the fractional differences, ∆P/P .The density use is 3.0 g.cm −3 .


Figure 8 :
8
Figure 8: T2HKK, for inverted ordering (IO): Top Left figure i