This was emphasized in ref. [11], because the approximation α/A 1 was used when the
formula was derived. We will reformulate the derivation of the formula in section 2 to show
the problem more explicitly but here we take the solar mixing angle θ
12
as a good example
to show the problem. The effective sin 2θ
12
in matter, denoted as sin 2θ
m
12
, expanded in α
to first order, is [11]
sin 2θ
m
12
∼ α/A. (1.4)
The solar resonance is at A = α cos 2θ
12
≈ 0.4α so near the solar resonance sin 2θ
m
12
is quite
likely to be larger than 1. As will be shown in section 2, sin 2θ
m
12
> 1 does appear in the
expansion when the energy is lower than 0.34GeV, which makes the calculation invalid.
Originally, sin 2θ
m
12
in the calculation was expected not only less than 1 but also small, i.e.
sin 2θ
m
12
1, otherwise the unitarity of the effective mixing matrix will be badly violated,
thereby invalidating the calculation.
Despite the claimed bound (1.3) in [11], in practice this formula works well below the
bound (see figures 6, 7 presented later in this paper). For example T2K has used this
formula in their recent publication [12] because eq. (1.1) exhibits excellent accuracy near
the solar resonance.
1
So (1.3) is most likely not the true bound of validity. We would like to know to what
extent the formula is accurate or valid. The main goal of this paper, is to mathematically
demonstrate that there is no lower bound of A for the domain of validity. We will provide
explicit errors of the formula, among which the main error related to the matter effect is
only O(s
2
13
αA∆
2
). This implies that the formula is still accurate when A is close to α and
one may apply (1.1) below the bound.
Note that a higher order calculation in the original perturbative approach will not work
since the series in α/A can not converge at the resonance if the branch cut singularity is
not treated carefully. Actually a higher order correction to the formula (1.1) is computed
in ref. [13] but the correction blows up when taking the vacuum limit A → 0. Thus it can
not give a correct estimation when A is small. This is due to a lack of careful treatment of
the branch cut singularity related to the solar resonance.
Branch cuts in the oscillation system with the matter effect are essentially related
to level crossings [14, 15], but less noticed before. Note that the three eigenvalues of the
oscillation system come from the same cubic equation but they are different. The difference
originates from the different branches in the square roots and cubic roots in the general
solutions of a cubic equation. At a level crossing two of the eigenvalues are very close
to each other which makes the problem quite non-perturbative and this just corresponds
to the starting point of the branch cuts, which are called branch cut singularities. The
branch cut singularities are essentially origins of all non-perturbativities in the oscillation
system. In this paper, we will remove the singularity corresponding to the solar resonance
in our analytic calculation by transformation of the eigenvalues to some singularity-free
variables and compute the S-matrix using the Cayley-Hamilton theorem. In this way the
1
Note that for T2K, the energy range is 0.1-1.2 GeV and the spectrum peaks at 0.6 GeV [5]. A part of
the current measured range 0.1-0.34 GeV is below the bound (1.3) which would lead to sin 2θ
m
12
> 1 in the
expansion.
– 2 –
