# Projects

## Neutrino Oscillations

Neutrino oscillations means that the neutrinos, that can be produced and detected in three species - flavors, oscillate between the flavors as they propagate from source to detector.

Neutrino oscillations can occur if neutrinos have masses that are nonzero and different. In the Standard Model of elementary particles this cannot take place. Evidence for neutrino oscillations is therefore an indication of New Physics beyond the Standard Model.

In the early summer of '98 the Super-Kamiokande collaboration reported evidence for neutrino oscillations related to the atmospheric neutrino problem. This has rekindled an interest to interpret all deviations from expected results in neutrino physics in terms of neutrino oscillations.

We are working on several problems in the realm of neutrino oscillations. The results so far are described in more detail below. For an introduction to Neutrino Oscillations in Swedish you can consult Neutrinooscillationer on my homepage. For an introduction in English you can read this lecture.

### Neutrino Oscillations with three flavors in matter: Applications to neutrinos traversing the Earth.

T. Ohlsson, H. Snellman

Analytic formulas are presented for three flavor neutrino oscillations in matter in the plane wave approximation. We calculate in particular the time evolution operator in both mass and flavor bases. We also find the transition probabilities expressed as functions of the vacuum mass squared differences, the vacuum mixing angles, and the matter density parameter. The application of this to neutrino oscillations in a mantle-core-mantle step function model of the Earth's matter density profile is discussed.[15]

### Three flavor neutrino oscillations in matter

T. Ohlsson, H. Snellman

We derive analytic expressions for three flavor neutrino oscillations in matter in the plane wave approximation using the Cayley-Hamilton formalism. Especially, we calculate the time evolution operator in both flavor and mass spaces. Furthermore, we find the transition probabilities, matter mass squared differences, and matter mixing angles all expressed in terms of the vacuum mass squared differences, the vacuum mixing angles, and the matter density. The conditions for resonance are also studied by some examples. [14]

### Domain of Mixing Angles in Three Flavor Neutrino Oscillations

J. Lundell and H. Snellman

We clarify the domain needed for the mixing angles in three flavor neutrino oscillations and show that it is necessary and sufficient to let all mixing angles have $\left[ 0, \pi/2 \right]$ as their domain. This holds irrespectively of any assumptions on the neutrino mass squared differences. [13]

### Neutrino oscillations and mixings with three flavors

T. Ohlsson, H. Snellman

Global fits to all data of candidates for neutrino oscillations are presented in the framework of a three-flavor model. The analysis excludes mass regions where the MSW effect is important for the solar neutrino problem. The best fit gives $\theta _{1} \approx 28.9^\circ$, $\theta_{2} \approx 4.2^\circ$, $\theta_{3} \approx 45.0^\circ$, $m_{2}^{2}-m_{1}^{2} \approx 2.87 \cdot 10^{-4} \; {\rm eV}^2$, and $m_{3}^{2}-m_{2}^{2} \approx 1.11 \; {\rm eV}^2$ indicating essentially maximal mixing between the two lightest neutrino mass eigenstates. [11]

The strong interaction theory of today is Quantum ChromoDynamics, QCD. This theory is a quantum gauge-field theory of quarks and gluons exhibiting chiral symmetry. QCD can only be solved perturbatively in the regime of high energy physics. At low energies the theory undergoes spontaneous breaking of the chiral symmetry. Various methods have been devised to study low energy hadron physics in models incorporating chiral symmetry breaking.

Our research has been focussed on the low energy properties of baryons, for which there are good measurements using weak and electromagnetic interactions, that can be used to investigate hadron structure. We have studied various forms of effective Lagrangian techniques, including Skyrmions and bound state models of quark-antiquark states. At present our interest is focused on the Chiral Quark Model. In this model the active constituents of the hadrons are quarks, pseudoscalar Goldstone bosons and gluons. The Goldstone bosons, appearing due to the chiral symmetry breaking, polarize the vacuum in the hadrons in a non-trivial way, that seems consistent with experimental findings.

We have studied magnetic moments of the octet and decuplet baryons, their quark spin polarizations and for the octet baryons also their weak form factors. A part of these investigations concerns the role of flavor symmetry breaking in the analysis of the data.

Hadron physics includes many interesting and challenging mathematical methods and unsolved physical problems, such as relativistic quantum field theory, group theory, bound state problems, quantization of non-linear field theories, etc.

Below you find abstracts of recent works.

### Chiral Quark Model Analysis of Nucleon Quark Sea Isospin Asymmetry and Spin Polarization

T. Ohlsson, H. Snellman

We analyze recent measurements of the nucleon quark sea isospin asymmetry in terms of the chiral quark model. The new measurements indicate that the SU(3) model with modest symmetry breaking and no $\eta'$ Goldstone boson gives a satisfactory description of data. We also discuss the matching parameter for the axial-vector current. Finally, we analyze the nucleon quark spin polarization measurements directly in the chiral quark model without using any SU(3) symmetry assumption on the hyperon axial-vector form factors. The new data indicate that the chiral quark model gives a remarkably good and consistent description of all low energy baryon measurements. [10]

### Weak Form Factors for Semileptonic Octet Baryon Decays in the Chiral Quark Model

T. Ohlsson, H. Snellman

We calculate weak vector and axial-vector form factors of first- and second-class currents for the semileptonic octet baryon decays in the chiral quark model. Our results for the chiral quark model are in good agreement with existing experimental data and are compared to other model calculations. [9]

We have also discussed some features of the chiral quark model prediction for the weak magnetism and compared to the corresponding result for the chiral quark soliton model. [12]

### Decuplet Baryon Magnetic Moments in the Chiral Quark Model

J. Linde, T. Ohlsson, H. Snellman

We present calculations of the decuplet baryon magnetic moments in the chiral quark model. As input we use parameters obtained in qualitatively accurate fits to the octet baryon magnetic moments studied earlier. The values found for the magnetic moments of Delta++ and Omega- are in good agreement with experiments. We finally calculate the total quark spin polarizations of the decuplet baryons and find that they are considerably smaller than what is expected from the non-relativistic quark model.[8]

### Octet Baryon Magnetic Moments in the Chiral Quark Model with Configuration Mixing

J. Linde, T. Ohlsson, H. Snellman

The Coleman-Glashow sum-rule for magnetic moments is always fulfilled in the chiral quark model, independently of SU(3) symmetry breaking. This is due to the structure of the wave functions, coming from the non-relativistic quark model. Experimentally, the Coleman-Glashow sum-rule is violated by about ten standard deviations. To overcome this problem, two models of wave functions with configuration mixing are studied. One of these models violates the Coleman-Glashow sum-rule to the right degree and also reproduces the octet baryon magnetic moments rather accurately.[7]

### Is there a mass dependence in the spin structure of baryons?

J. Linde and H Snellman

We analyze the axial-vector form factors of the nucleon hyperon system in a model with SU(3)flavor breaking due to mass dependent spin polarizations. This mass dependence is deduced from an earlier analysis [2,4] of the magnetic moment data and implies that the spin contributions from the quarks to a baryon decrease with the mass of the baryon. When applied to the axial-vector form-factors, these mass dependent spin polarizations bring the various sum-rules from the model into better agreement with experimental data. Our analysis leads to a reduced value for the total spin polarization of the proton.[5]

### Magnetic moments of the 3/2 resonances and their quark spin structure

J. Linde and H. Snellman

We discuss magnetic moments of the J=3/2 baryons based on an earlier model for the baryon magnetic moments, allowing for flavor symmetry breaking in the quark magnetic moments as well as a general quark spin structure. From our earlier analysis of the nucleon-hyperon magnetic moments and the measured values of the magnetic moments of the $\Delta^{++}$ and the $\Omega^{-}$ we predict the other magnetic moments and the spin structure of the resonance particles. We find from experiment that the total spin polarization of the decuplet baryons, $\Delta\Sigma(3/2)$, is considerably smaller than the non-relativistic quark model value 3, although the data is still not good enough to give a precise determination.[4]

### Magnetic moments of quarks and baryons and the proton spin structure

J. Linde, H. Snellman

We discuss the magnetic moments of the baryons allowing for flavor symmetry breaking in the quark magnetic moments. We show that there is a correlation between isospin symmetry breaking and data for the nucleon spin structure obtained from deep inelastic scattering. For small values of the isospin symmetry breaking, of the order of 5 %, the magnetic moments and weak axial-vector form factors alone indicate a value of the spin polarization of about 0.20. Larger values of the spin polarization are compatible only with large isospin symmetry breaking. We also calculate weak axial-vector form factors, which are independent of the symmetry breakings, from magnetic moment data and find good agreement with experiment [2].

### Charmonium in the instantaneous approximation

J. Linde, H. Snellman

The charmonium system is studied in a Salpeter model with a vector plus scalar potential. We use a kinematical formalism based on the one developed by Suttorp, and present general eigenvalue equations and expressions for decay observables in an onium system for such a potential both in the Feynman and Coulomb gauges. Special attention is paid to the problem with renormalization of the lepton pair decays, and we argue that they must be defined relative to one of the experimental decay widths because renormalization of the vertex function is not possible. The parameters of the model are determined by a fit to the mass spectrum and the lepton pair decay rates. Two gamma decays and E1 and M1 transitions are then calculated and found to be well accounted for. No significant differences in the results in Feynman or Coulomb gauge are found. A comparison is made, regarding the electromagnetic transitions, between the full and reduced Salpeter equation. A large difference is found showing the importance of using the full Salpeter equation[6].

### Quantization of the chiral sigma model

C. Cronström, H. Snellman

We study the field theoretic quantization of the chiral sigma model by using lagrangian multipliers to implement the geometry in phase space.

### Production of bound states in relativistic high-energy scattering

J. Blom, H. Snellman

Production of bound states in high-energy collisions is studied in a relativistic frame-work. The generic case is production of hydrogen atoms in high-energy inelastic electron-proton scattering. The same formalism is with minor modifications applicable to several other systems like quark-quark scattering with meson production etc. [3]

1997-09-25