We revisit neutrino oscillations in constant matter density for a number of different scenarios: three flavors with the standard Wolfenstein matter potential, four flavors with standard matter potential and three flavors with non-standard matter potentials. To calculate the oscillation probabilities for these scenarios one must determine the eigenvalues and eigenvectors of the Hamiltonians. We use a method for calculating the eigenvalues that is well known, determination of the zeros of determinant of matrix  $$(\lambda I -H)$$  <math> <mrow> <mo>(</mo> <mi>λ</mi> <mi>I</mi> <mo>-</mo> <mi>H</mi> <mo>)</mo> </mrow> </math>  , where H is the Hamiltonian, I the identity matrix and  $$\lambda $$  <math> <mi>λ</mi> </math>   is a scalar. To calculate the associated eigenvectors we use a method that is little known in the particle physics community, the calculation of the adjugate (transpose of the cofactor matrix) of the same matrix,  $$(\lambda I -H)$$  <math> <mrow> <mo>(</mo> <mi>λ</mi> <mi>I</mi> <mo>-</mo> <mi>H</mi> <mo>)</mo> </mrow> </math>  . This method can be applied to any Hamiltonian, but provides a very simple way to determine the eigenvectors for neutrino oscillation in matter, independent of the complexity of the matter potential. This method can be trivially automated using the Faddeev–LeVerrier algorithm for numerical calculations. For the above scenarios we derive a number of quantities that are invariant of the matter potential, many are new such as the generalization of the Naumov–Harrison–Scott identity for four or more flavors of neutrinos. We also show how these matter potential independent quantities become matter potential dependent when off-diagonal non-standard matter effects are included.

United States

Asli Abdullahi

Stephen Parke

Neutrino oscillations in matter using the adjugate of the Hamiltonian

Henry Tye

Remembering Stephen Weinberg

Light sterile neutrinos can be probed in a number of ways, including electroweak decays, cosmology and neutrino oscillation experiments. At long-baseline experiments, the neutral-current data is directly sensitive to the presence of light sterile neutrinos: once the active neutrinos have oscillated into a sterile state, a depletion in the neutral-current data sample is expected since they do not interact with the Z boson. This channel offers a direct avenue to probe the mixing between a sterile neutrino and the tau neutrino, which is currently only weakly constrained by current data from SuperK, IceCube and NOvA, however, these constrains will continue to improve as more data is collected by these experiments. In this work, we study the potential of the DUNE experiment to constrain the mixing angle which parametrizes this mixing, θ 34, through the observation of neutral-current events at the far detector. We find that DUNE will be able to improve significantly over current constraints thanks to its large statistics and excellent discrimination between neutral- and charged-current events.

Pilar Coloma

Brazil

David Forero

DUNE sensitivities to the mixing between sterile and tau neutrinos

In this Addendum we rewrite the neutrino mixing angles and mass squared differences in matter given in our original paper, [1], in a notation that is more conventiona 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 &amp; T2HK, NOvA, T2HKK and DUNE are also given for neutrinos and anti-neutrinos, disappearance and appearance channels, and for both normal and inverted orderings.

Denmark

Peter Denton

Addendum to “Compact perturbative expressions for neutrino oscillations in matter”

In this work we have studied the Kleiss-Kuijf relations for the recently introduced Parke-Taylor factors at one-loop in the CHY approach, that reproduce quadratic Feynman propagators. By doing this, we were able to identify the non-planar one-loop Parke-Taylor factors. In order to check that, in fact, these new factors can describe non-planar amplitudes, we applied them to the bi-adjoint Φ3 theory. As a byproduct, we found a new type of graphs that we called the non-planar CHY-graphs. These graphs encode all the information for the subleading order at one-loop, and there is not an equivalent of these in the Feynman formalism.

South Korea

Naser Ahmadiniaz

Colombia

Humberto Gomez

Cristhiam Lopez-Arcos

Non-planar one-loop Parke-Taylor factors in the CHY approach for quadratic propagators

In this paper we reconsider the Cachazo-He-Yuan construction (CHY) of the so called scattering amplitudes at one-loop, in order to obtain quadratic propagators. In theories with colour ordering the key ingredient is the redefinition of the Parke-Taylor factors. After classifying all the possible one-loop CHY-integrands we conjecture a new one-loop amplitude for the massless Bi-adjoint Φ 3 theory. The prescription directly reproduces the quadratic propagators of the traditional Feynman approach.

Spain

Pedro Talavera

One-loop Parke-Taylor factors for quadratic propagators from massless scattering equations

We further develop and extend a recent perturbative framework for neutrino oscillations in uniform matter density so that the resulting oscillation probabilities are accurate for the complete matter potential versus baseline divided by neutrino energy plane. This extension also gives the exact oscillation probabilities in vacuum for all values of baseline divided by neutrino energy. The expansion parameter used is related to the ratio of the solar to the atmospheric ∆ m 2 scales but with a unique choice of the atmospheric ∆ m 2 such that certain first-order effects are taken into account in the zeroth-order Hamiltonian. Using a mixing matrix formulation, this framework has the exceptional feature that the neutrino oscillation probability in matter has the same structure as in vacuum, to all orders in the expansion parameter. It also contains all orders in the matter potential and sin θ 13 . It facilitates immediate physical interpretation of the analytic results, and makes the expressions for the neutrino oscillation probabilities extremely compact and very accurate even at zeroth order in our perturbative expansion. The first and second order results are also given which improve the precision by approximately two or more orders of magnitude per perturbative order.

Ecuador

Hisakazu Minakata

Compact perturbative expressions for neutrino oscillations in matter

We reformulate perturbation theory for neutrino oscillations in matter with an expansion parameter related to the ratio of the solar to the atmospheric Δ m 2 scales. Unlike previous works, we use a renormalized basis in which certain first-order effects are taken into account in the zeroth-order Hamiltonian. We show that the new framework has an exceptional feature that leads to the neutrino oscillation probability in matter with the same structure as in vacuum to first order in the expansion parameter. It facilitates immediate physical interpretation of the formulas, and makes the expressions for the neutrino oscillation probabilities extremely simple and compact. We find, for example, that the ν e disappearance probability at this order is of a simple two-flavor form with an appropriately identified mixing angle and Δ m 2 . More generally, all the oscillation probabilities can be written in the universal form with the channel-discrimination coefficient of 0 , ± 1 or simple functions of θ 23 . Despite their simple forms they include all order effects of θ 13 and all order effects of the matter potential, to first order in our expansion parameter.