33. Additional limit information    Appendix

33.1. Conversion probability as a function of mass    Appendix

Fig. 1 shows how the axion-photon conversion probability changes as a function of the axion mass. This implements eq. \eqref{eq:theory:axion_interaction:conversion_probability}, reproduced here with slightly changed notation,

\[ P_{a↦γ}(z) = \left( \frac{g_{aγ} B L}{2} \right)² \left(\frac{\sin\left(\frac{q L}{2}\right)}{\frac{q L}{2}}\right)², \]

with \(q = \frac{m²_γ - m²_a}{2 E_a}\). \(E_a\) is the energy of the axion (or in the context of a limit calculation the energy of a candidate). In the vacuum setup \(m_γ = 0\). The figure shows this conversion probability for different axion energies and based on the CAST magnet. We see that the conversion probability starts falling off roughly around \(m_a \approx \SI{0.01}{eV}\), with the exact value depending on energy (and personal \(ΔP\) cutoff).

axion_conversion_probability_vs_mass.svg
Figure 1: Figure 194: Axion-photon conversion probability as a function of axion mass. Using \(g_{aγ} = \SI{1e-12}{GeV⁻¹}, B = \SI{8.8}{T}, L = \SI{9.26}{m}\). Different axion energies indicated by color.

\clearpage

33.1.1. Generate the plot of the conversion probability    extended

Taken and adapted from ./../org/Code/CAST/babyIaxoAxionMassRange/axionMass.html. 

import ggplotnim, unchained, sequtils
proc momentumTransfer(m_γ, m_a: eV, E_a = 4.2.keV): eV =
  ## calculates the momentum transfer for a given effective photon
  ## mass `m_gamma` and axion mass `m_a` at an axion energy of 
  ## 4.2 keV `E_a` (by default).
  result = abs((m_γ * m_γ - m_a * m_a) / (2 * E_a))

proc vacuumConversionProb(E_a: keV, m_a: eV, B: Tesla, L: Meter): float =
  ## calculates the conversion probability in BabyIAXO for the given axion 
  ## mass `m_a`
  # both `g_agamma` and `B` only scale the absolute value `P`, does not matter
  const g_aγ = 1e-12.GeV⁻¹
  # convert length in `m` to `eV`
  let q = momentumTransfer(m_γ = 0.0.eV, m_a = m_a, E_a = E_a)
  let term1 = pow(g_aγ * B.toNaturalUnit() * L.toNaturalUnit() / 2.0, 2)
  let term2 = pow(sin(q * L.toNaturalUnit() / 2.0) / (q * L.toNaturalUnit() / 2.0), 2.0)
  result = term1 * term2

let energies = arange(1, 9, 2)
var df = newDataFrame()
for E_a in energies:
  let masses = logspace(-6, 0, 1000)
  let Ps = masses.mapIt(vacuumConversionProb(E_a.keV, it.eV, 8.8.T, 9.26.m))
  df.add toDf({"masses" : masses, "Ps" : Ps, "E_a [keV]" : E_a})
ggplot(df, aes("masses", "Ps", color = "E_a [keV]")) +
  geom_line() +
  xlab("Axion mass [eV]") + ylab("Conversion probability") +
  themeLatex(fWidth = 0.9, width = 600, baseTheme = singlePlot) +
  scale_x_log10() + scale_y_log10() + 
  ggsave("~/phd/Figs/axions/axion_conversion_probability_vs_mass.pdf")

33.2. Expected limit table with percentiles

The table shown in tab. 31 shows the same table as tab. 26 in the main part of the thesis, but with a focus on the variation of the limits. The focus is on different percentiles of the distribution of sampled toy limits. The \(P_i\) columns correspond to the limit at the \(i^{\text{th}}\) percentile of all toy limits. \(P_{50}\) would be the median and thus expected limit. The table yields insight into the probabilities with which limits are expected for certain setups, given the pure statistical fluctuation possible by the measured candidates.

The veto information has been merged into the 'Type' column. A suffix (L) indicates 'line veto', 'S' the 'septem veto' and 'SL' both vetoes. '-' means no vetoes. FADC and scintillators are implicitly included if any of septem or line vetoes are in use. The units are excluded in the column names to save space. For the axion-electron and axion-photon tables they are all in \(\si{GeV⁻¹}\). For the expected limit (last column) the uncertainty is again a bootstrapped standard deviation.

The same table for the expected axion-photon limit and chameleon limits are tab. 32 and tab. 33, respectively. They only have a single row, because we only computed the expected limit for one veto setup.

\footnotesize

Table 31: Table of the expected limits for different veto setups, comparable to tab. 26, with a focus on the percentiles \(P_i\) of the computed toy limits. For example \(P_{25}\) is the \(25^{\text{th}}\) percentile of the distribution of toy limits. All values in units of \(\si{GeV⁻¹}\).
εeff nmc Type εtotal \(P_5\) \(P_{16}\) \(P_{25}\) \(P_{75}\) \(P_{84}\) \(P_{95}\) Expected
0.98 1000 MLP - 0.98 6.44e-23 6.82e-23 7.09e-23 8.65e-23 9.09e-23 1.03e-22 7.805(37)e-23
0.91 1000 MLP - 0.91 6.59e-23 6.96e-23 7.21e-23 8.75e-23 9.25e-23 1.03e-22 7.856(43)e-23
0.95 1000 MLP - 0.95 6.53e-23 6.87e-23 7.14e-23 8.74e-23 9.18e-23 1.02e-22 7.860(51)e-23
0.95 2500 MLP L 0.8 6.77e-23 7.07e-23 7.26e-23 8.72e-23 9.17e-23 1.03e-22 7.862(29)e-23
0.98 15000 MLP L 0.82 6.7e-23 7e-23 7.2e-23 8.72e-23 9.2e-23 1.02e-22 7.868(11)e-23
0.95 50000 MLP L 0.8 6.75e-23 7.04e-23 7.25e-23 8.72e-23 9.18e-23 1.02e-22 7.8782(65)e-23
0.95 15000 MLP L 0.8 6.75e-23 7.04e-23 7.24e-23 8.72e-23 9.16e-23 1.03e-22 7.879(12)e-23
0.98 2500 MLP L 0.82 6.73e-23 7.01e-23 7.19e-23 8.72e-23 9.22e-23 1.02e-22 7.883(30)e-23
0.86 1000 MLP - 0.86 6.74e-23 7.08e-23 7.31e-23 8.88e-23 9.35e-23 1.03e-22 7.960(51)e-23
0.91 2500 MLP L 0.76 6.91e-23 7.18e-23 7.38e-23 8.9e-23 9.3e-23 1.03e-22 7.99(16)e-23
0.91 15000 MLP L 0.76 6.9e-23 7.18e-23 7.38e-23 8.87e-23 9.34e-23 1.04e-22 8.004(11)e-23
0.98 2500 MLP SL 0.76 6.93e-23 7.2e-23 7.42e-23 8.97e-23 9.47e-23 1.06e-22 8.085(29)e-23
0.95 2500 MLP S 0.78 6.91e-23 7.22e-23 7.43e-23 9.08e-23 9.53e-23 1.07e-22 8.113(36)e-23
0.95 2500 MLP SL 0.73 6.99e-23 7.29e-23 7.49e-23 9e-23 9.46e-23 1.05e-22 8.125(31)e-23
0.98 2500 MLP S 0.8 6.82e-23 7.16e-23 7.42e-23 9.02e-23 9.46e-23 1.06e-22 8.131(32)e-23
0.86 2500 MLP L 0.72 7.03e-23 7.32e-23 7.54e-23 9.09e-23 9.58e-23 1.06e-22 8.156(30)e-23
0.86 15000 MLP L 0.72 7.03e-23 7.32e-23 7.54e-23 9.06e-23 9.51e-23 1.06e-22 8.183(13)e-23
0.91 2500 MLP S 0.74 7.03e-23 7.33e-23 7.54e-23 9.12e-23 9.63e-23 1.07e-22 8.22(19)e-23
0.9 2500 LnL L 0.75 6.96e-23 7.28e-23 7.49e-23 9.13e-23 9.61e-23 1.06e-22 8.217(37)e-23
0.91 2500 MLP SL 0.7 7.1e-23 7.42e-23 7.62e-23 9.17e-23 9.64e-23 1.08e-22 8.287(33)e-23
0.86 2500 MLP S 0.7 7.19e-23 7.5e-23 7.72e-23 9.27e-23 9.71e-23 1.08e-22 8.401(29)e-23
0.9 2500 LnL SL 0.69 7.21e-23 7.52e-23 7.74e-23 9.38e-23 9.89e-23 1.11e-22 8.427(34)e-23
0.86 2500 MLP SL 0.66 7.32e-23 7.6e-23 7.79e-23 9.38e-23 9.76e-23 1.08e-22 8.459(35)e-23
0.8 2500 LnL L 0.67 7.3e-23 7.6e-23 7.83e-23 9.4e-23 9.91e-23 1.09e-22 8.499(32)e-23
0.9 2500 LnL - 0.9 6.91e-23 7.43e-23 7.73e-23 9.57e-23 1.01e-22 1.12e-22 8.579(37)e-23
0.8 2500 LnL - 0.8 7.13e-23 7.59e-23 7.88e-23 9.79e-23 1.03e-22 1.15e-22 8.738(39)e-23
0.8 2500 LnL SL 0.62 7.52e-23 7.82e-23 8.03e-23 9.68e-23 1.02e-22 1.13e-22 8.747(41)e-23
0.7 2500 LnL L 0.59 7.72e-23 8.02e-23 8.21e-23 9.86e-23 1.04e-22 1.16e-22 8.930(40)e-23
0.7 2500 LnL - 0.7 7.4e-23 7.87e-23 8.23e-23 1.01e-22 1.07e-22 1.19e-22 9.086(33)e-23
0.7 2500 LnL SL 0.54 8.01e-23 8.28e-23 8.51e-23 1.02e-22 1.08e-22 1.2e-22 9.257(35)e-23

\normalsize

\footnotesize

Table 32: Table of the different percentiles for the single axion-photon expected limit. All values in units of \(\si{GeV⁻¹}\).
εeff nmc Type εtotal P5 P16 P25 P75 P84 P95 Expected
0.95 10000 MLP L 0.8 8.24e-11 8.5e-11 8.66e-11 9.56e-11 9.83e-11 1.04e-10 9.0650(75)e-11

\normalsize

\footnotesize

Table 33: Table of the different percentiles for the single chameleon expected limit.
εeff nmc Type εtotal P5 P16 P25 P75 P84 P95 Expected
0.95 10000 MLP L 0.8 3.22e+10 3.35e+10 3.43e+10 3.82e+10 3.93e+10 4.16e+10 3.6060(39)e+10

\normalsize

33.2.1. Generate the expected limit table with percentiles    extended

Following sec. 13.13.3, 

./generateExpectedLimitsTable --path ~/org/resources/lhood_limits_21_11_23/ --prefix "mc_limit_lkMCMC" --precision 2
33.2.1.1. Axion-photon:
./generateExpectedLimitsTable --path ~/org/resources/lhood_limits_axion_photon_11_01_24// --prefix "mc_limit_lkMCMC" --precision 2 --coupling ck_g_aγ⁴
33.2.1.2. Chameleon
./generateExpectedLimitsTable --path ~/org/resources/lhood_limits_chameleon_12_01_24/ --prefix "mc_limit_lkMCMC" --precision 2 --coupling ck_β⁴

33.3. Observed limit - axion photon \(g_{aγ}\)

Fig. 2 shows the sampled coupling constants in \(g⁴_{aγ}\) of the calculation for the observed limit, i.e. the marginal posterior likelihood function for the real candidates for the axion-photon coupling.

mcmc_real_limit_likelihood_ck_g_aγ⁴.svg
Figure 2: Figure 195: Marginal posterior likelihood function of the real candidates for the axion-photon coupling constant in \(g⁴_{aγ}\) space. The yellow line is a numerical integration of the likelihood function using Romberg's method &romberg_integration. Limit at \(g⁴ \approx \SI{6.56e-41}{GeV⁻⁴} ⇒ g \approx \SI{9e-11}{GeV⁻¹}\).

33.4. Observed limit - chameleon \(β_γ\)

Fig. 3 shows the sampled coupling constants in \(β⁴_γ\) of the calculation for the observed limit, i.e. the marginal posterior likelihood function for the real candidates for the chameleon coupling.

mcmc_real_limit_likelihood_ck_β⁴.svg
Figure 3: Figure 196: Marginal posterior likelihood function of the real candidates for the chameleon coupling constant in \(β⁴_γ\) space. The yellow line is a numerical integration of the likelihood function using Romberg's method &romberg_integration. Limit at \(β⁴ \approx \num{9.2e41} ⇒ β \approx \num{3.1e10}\).
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