Neutrino Mass Sensitivity Calculation

Scripts used in sensitivity calculations

The mermithid sensitivity calculations are written in python and a python script can be used to configure the calculation and make sensitivity plots. Mermithid has several processors for this purpose, in mermithid/mermithid/processors/Sensitivity/. For cavity sensitivity calculations, use the CavitySensitivityCurveProcessor.

Mermithid processors are all designed to be used in the same way:

  1. Define a dictionary with the processor’s configurable parameters;

  2. Instantiate a processor and pass the configuration dictionary to its Configure() method;

  3. Call the processor’s Run() method to make it perform its task.

In mermithid/tests, the Sensitivity_test.py script contains several examples of how to perform sensitivity calculations using mermithid processors. In test_SensitivityCurveProcessor, the CavitySensitivityCurveProcessor is used as described above to calculate and plot sensitivity to the neutrino mass as a function of gas density. Other working examples for creating sensivitiy plots vs. frequency or exposure can be found in mermithid/test_analyses/Cavity_Sensitivity_analysis.py.

The dictionary to configure the processor (see #1, above) can read in a separate configuration file (.cfg) with sensitivity-specific parameters. The documentation page sensitivity_configurable_parameters.rst describes all parameters in such a .cfg.

Analytic sensitivity formula

Test

Systematic uncertainty contributions

The following contributions to energy broadening of the beta spectrum are included:

  1. sigma_trans: Translational Doppler broadening due to thermal motion of tritium atoms or molecules.

  2. sigma_f: Energy broadening due to mis-reconstruction of the start frequencies of events from the range of electron pitch angles (after axial frequency corrections), and from the process of fitting chirp tracks that have finite length and SNR to find where the tracks start.

  3. sigma_B: Energy broadening due to radial, azimuthal, and temporal varation of the magnetic field. This is the remaining broadening after any event-by-event corrections for the field have been performed.

  4. sigma_Miss: Energy broadening due to the detection of events after one or more tracks have been missed.

  5. sigma_Plasma: Energy broadening due to plasma effects of the charged particles in the source volume of an atomic tritium experiment.

In the list above, each variable that starts with sigma is a standard deviation of a distribution of energies. The measured spectrum is the convolution of the underlying beta spectrum with such distributions of energies. Asymmetries in these distributions are not accounted for in this approximate model.

Each sigma has an associated delta, which is the uncertainty on sigma from calibration, theory, and/or simulation (see the previous section). For example, sigma_trans has an associated variable delta_sigma_trans.

In the script mermithid/mermithid/misc/SensitivityCavityFormulas.py, lists of these energy broadening standard deviations (sigmas) and uncertainties on them (deltas) are returned by the get_systematics method of the CavitySensitivity class.

Contributions 4 and 5 are simply inputted in the sensitivity configuration file; we do not yet have a way to calculate these in mermithid. Contributions 1, 2, and 3 are calculated in mermithid, as described below.

Translational Doppler broadening (sigma_trans)

The thermal translational motion of tritium atoms causes a Doppler broadening of the \({\beta}\) energy spectrum. sigma_trans is the standard deviation of this broadening distribution. There are two options for how to include translational Doppler broadening in your sensitivity calculations, in mermithid:

1. Manually input values for sigma_trans and its uncertainty delta_trans, calculated outside of mermithid. This is done in the DopplerBroadening section of the configuration file, by setting UseFixedValue to True and providing values for Default_Systematic_Smearing and Default_Systematic_Uncertainty.

  1. Have mermithid calculate sigma_trans, for you (default option).

  • For an atomic tritium experiment, mermithid will also calculate delta_trans, conservatively assuming that the gas temperature uncertainty will be determined simply by our knowledge that the gas is not so hot that it escapes the atom trap. These calculations for an atomic experiment are described further, below.

  • For a molecular tritium experiment, you need to input a number fraction_uncertainty_on_doppler_broadening (which equals delta_trans/sigma_trans) in the DopplerBroadening section of the configuration file.

Calculation of sigma_trans for option 2: For a thermalized source gas, the translational Doppler broadening is described by a Gaussian with standard deviation \({\sigma_{\text{trans}} = \sqrt{\frac{p_{\text{rec}}^2}{2m_T}2 k_B T}}\), where \({m_T}\) is the mass of tritium and \({T}\) is the gas temperature.

Calculation of delta_trans for option 2, with atomic T:

Track start frequency determination and pitch angle correction (sigma_f)

Radial, azimuthal, and temporal field broadening (sigma_B)