Conclusion

Chapter V: The future of UCN 𝜏

5.8 Conclusion

This section discussed upgrades to the UCN𝜏 experiment that could enable the measurement of the free neutron lifetime to a total uncertainty of less than 0.10 s.

Section 5.2 discussed how increasing the number of UCN produced by the source would decrease the statistical uncertainty of an extracted lifetime, and Sections 5.3 and 5.4 discussed how to make better use of however many UCN may be available.

Section 5.5 discussed the need to upgrade the primary detector to accommodate an increase in the number of UCN loaded into the trap. Sections 5.6 and 5.7 discussed how to decrease some of the largest systematic uncertainties of this experiment. If most of the potential upgrades presented in this section can be realized, then a future iteration of the UCN𝜏experiment will be able to achieve a measurement of the free neutron lifetime to a total uncertainty of less than 0.10 s.

A p p e n d i x A

OPTIMIZING THE HOLDING LENGTH

Consider a simplified version of a UCN𝜏with the following assumptions:

1. The number of UCN loaded into the trap is a random variable 𝑛∼ Integer[Gaussian(𝑁 , 𝑓 𝑁)],

where 𝑓 ≥ 1 is fixed and known and𝑁 is fixed but unknown

2. The number of background events during the unloading process is 𝑏 ∼ Poisson(𝐵),where𝐵is fixed and known

3. Other than background, there are no systematic effects

4. All UCN are counted exactly 70 s after the beginning of the unloading process, and the counting of UCN is perfectly efficient

For a holding length𝑡 ,the proportion of UCN that are expected not to decay between the end of the filling process and when the UCN are counted is 𝑝=𝑒^{−(120+𝑡)/𝜏}.The 120 s in the exponent comes from two sources: 50 s of cleaning, as well as 70 s of delay between the beginning of the unloading process and when the UCN are counted.

As explained in Section 3.2.1, there is benefit to having multiple short holding lengths. Anoctetis a set of eight runs: four short runs with holding lengths of 20 s, 50 s, 100 s, and 200s; and four long runs with a holding length of𝑡_long.Starting from Equation 3.14 and making the simplifying assumption that𝑠=1 and𝜎_𝑆=𝜎_𝐶 =0, the likelihood function simplifies to

L (𝜏, 𝑁|𝑢_𝑖, 𝑡_𝑖) =−Õ

𝑖

ln Gaussian 𝑢_𝑖

𝑁 𝑝_𝑖+𝐵, 𝑓 𝑁 𝑝²

𝑖 +𝑁 𝑝_𝑖(1−𝑝_𝑖) +𝐵

, (A.1) where𝑖sums over all runs in the analysis.

The maximum likelihood of𝑁 and𝜏,as well as the uncertainties of those estimates (𝜎_𝑁, 𝜎_𝜏), was found using the methodology described in Section 3.9. 𝜎_𝜏is inversely proportional to the square of the amount of data gathered, so maximizing the data gathered per unit of time (Δ𝑡) is equivalent to maximizing𝜎⁻²

𝜏 /Δ𝑡 .

Figure A.1: Relative amounts of data gathered per hour as a function of𝑡_𝐿. Each run takes a total time𝑡 +𝛿 to complete, where𝑡 is the holding length of this run and𝛿is some constant amount of overhead time per run. An octet takes a total time of 370 s+4𝑡_𝐿 + 8𝛿 to complete, where 𝑡_𝐿 is the holding length of the long runs and 370 s= (20+50+100+200)s. Monte Carlo simulations were modelled after the simplified version of the UCN𝜏experiment described at the beginning of this appendix. These simulation were repeated with different choices for the long holding length𝑡_𝐿. Figure A.1 shows the rate of data gathered per hour as a function of long holding length for reasonable choices of𝑁 =10,000, 𝐵=50,and𝛿=630 s, and for two options for 𝑓 . Larger values for 𝑓 (larger scales of the uncertainty of the normalization estimate𝜎_𝑁 relative to the ideal case of𝜎_𝑁 =√

𝑁) led to a larger value of the optimum holding length and decreased the statistical precision of a measurement made from a fixed number of runs.

A p p e n d i x B

UCN EVENT RECONSTRUCTION ALGORITHMS

In Algorithm 1:

• 𝑡_𝑖is the time that the𝑖^thphotoelectron was observed

• ch𝑖 ∈ {1,2} is which photomultiplier tube of the primary detector the 𝑖^th photoelectron was observed in

• 𝑇_𝐶 is the coincidence window

• 𝑇_𝑇 is the tail window

• 𝑁_PEis the photoelectron threshold

Algorithm 1UCN reconstruction without a prompt window INPUTS:{(𝑡_𝑖,ch𝑖)}_𝑖^𝐿₌₁where𝑡_𝑖 ≤ 𝑡_𝑖+1

PARAMETERS:𝑇_𝐶, 𝑇_𝑇, 𝑁_PE 𝑖← 1

while𝑖 ≤ 𝐿do

𝑗 , 𝑛,state←𝑖+1,1,0 while 𝑗 ≤ 𝐿do

ifstate=0then if𝑡_𝑗 −𝑡_𝑖 > 𝑇_𝐶 then

breakloop over 𝑗 else

𝑛← 𝑛+1 if ch𝑖 ≠ch𝑗 then

state← 1 end if end if else

if𝑡_𝑗 −𝑡_𝑗−1> 𝑇_𝑇 then breakloop over 𝑗 else

𝑛← 𝑛+1 end if end if

𝑗 ← 𝑗+1 end while

if state=1 and𝑛 ≥ 𝑁_PEthen

push backreconstructed UCN event with start time𝑡_𝑖, end time𝑡_𝑗−1+𝑇_𝑇, and𝑛photoelectron

𝑖← 𝑗 else

𝑖←𝑖+1 end if end while

return reconstructed UCN events

In Algorithm 2:

• 𝑡_𝑖is the time that the𝑖^thphotoelectron was observed

• ch𝑖 ∈ {1,2} is which photomultiplier tube of the primary detector the 𝑖^th photoelectron was observed in

• 𝑇_𝐶 is the coincidence window

• 𝑇_𝑃is the prompt window

• 𝑇_𝑇 is the tail window

• 𝑁_PEis the photoelectron threshold

Algorithm 2UCN reconstruction with a prompt window INPUTS:{(𝑡_𝑖,ch𝑖)}_𝑖^𝐿₌₁where𝑡_𝑖 ≤ 𝑡_𝑖+1

PARAMETERS:𝑇_𝐶, 𝑇_𝑃, 𝑇_𝑇, 𝑁_PE 𝑖← 1

while𝑖 ≤ 𝐿do

𝑛_𝑃, 𝑛_𝑇,state, 𝑗 ←1,1,0, 𝑖+1 while 𝑗 ≤ 𝐿do

ifstate=0then if𝑡_𝑗 −𝑡_𝑖 > 𝑇_𝐶 then

breakloop over 𝑗 else

𝑛_𝑃, 𝑛_𝑇 ←𝑛_𝑃+1, 𝑛_𝑇 +1 if ch𝑖 ≠ch𝑗 then

state← 1 end if end if

else if state=1then if𝑡_𝑗 −𝑡_𝑖 > 𝑇_𝑃 then

if 𝑛_𝑃 ≥ 𝑁_PEthen state← 2 else

breakloop over 𝑗 end if

else

𝑛_𝑃, 𝑛_𝑇 ←𝑛_𝑃+1, 𝑛_𝑇 +1 end if

end if

ifstate=2then

if𝑡_𝑗 −𝑡_𝑗−1> 𝑇_𝑇 then breakloop over 𝑗 else

𝑛_𝑇 ← 𝑛_𝑇 +1 end if

end if 𝑗 ← 𝑗+1 end while if state=2then

push backreconstructed UCN event with start time𝑡_𝑖, end time𝑡_𝑗−1+𝑇_𝑇, 𝑛_𝑃prompt photoelectrons, and𝑛_𝑇 total photoelectrons

𝑖← 𝑗 else

𝑖←𝑖+1 end if end while

return reconstructed UCN events

In Algorithm 3:

• 𝑡_𝑖is the start time of the reconstructed UCN event

• 𝛿_𝑖is the length of each reconstructed UCN event

• 𝑁^𝑃

𝑖 is the number of photoelectrons found in the prompt window of each reconstructed UCN event

• 𝑁^𝑇

𝑖 is the total number of photoelectron found in each reconstructed UCN event

• 𝑇_𝑃is the prompt window from the reconstruction algorithm

• 𝑇_𝑇 is the tail window from the reconstruction algorithm

• 𝑁_PEis the photoelectron threshold from the reconstruction algorithm

• cdf is a cumulative distribution function of the measured time of detection of photoelectron in a reconstructed UCN event, relative to the time of the first photoelectron in a reconstructed UCN event

Algorithm 3UCN-event-tail correction INPUTS:

𝑡_𝑖, 𝛿_𝑖, 𝑁^𝑃

𝑖 , 𝑁^𝑇

𝑖

^𝐿

𝑖=1where𝑡_𝑖 ≤ 𝑡_𝑖+1, 𝛿_𝑖 ≥ 𝑇_𝑇,and𝑁_PE ≤ 𝑁^𝑃

𝑖 ≤ 𝑁^𝑇

𝑖

PARAMETERS:𝑇_𝑃, 𝑇_𝑇, 𝑁_PE,cdf {𝐹_𝑖}^𝐿_𝑖=1,{𝑊_𝑖}_𝑖^𝐿₌₁←0,1

for𝑖 ←1 to 𝐿do 𝐴_𝑖← 𝑁^𝑇

𝑖 /cdf(𝛿_𝑖) for 𝑗 ←𝑖+1 to 𝐿do

𝐹_𝑗 ← 𝐹_𝑗+ 𝐴_𝑖

cdf(𝑡_𝑗 −𝑡_𝑖+𝑇_𝑃) −cdf(𝑡_𝑗 −𝑡_𝑖) 𝑁^𝑇

𝑗 ← 𝑁^𝑇

𝑗 − 𝐴_𝑖

cdf(𝑡_𝑗 −𝑡_𝑖+𝛿_𝑗) −cdf(𝑡_𝑗−𝑡_𝑖) end for

for 𝑗 ← 𝑁^𝑃

𝑖 −𝑁_PE+1 to𝑁^𝑃

𝑖 do 𝑊_𝑖 ←𝑊_𝑖−Poisson(𝑗|𝐹_𝑖) end for

end for

return {𝑊_𝑖}_𝑖^𝐿₌₁

A p p e n d i x C

THE THINNED POISSON DISTRIBUTION

Denote the probability of observing a random value𝑥 drawn from a Poisson dis- tribution with mean 𝜇 as 𝑃_𝑃(𝑥|𝜇). Denote the probability of observing a random value 𝑦from a Binomial distribution with𝑁 trials, each of which has a probability of success 𝑝,as𝑃_𝐵(𝑦|𝑁 , 𝑝).

Let𝑛 ∼Poisson(𝑁), and then let𝑥 ∼Binomial(𝑛, 𝑝).Then 𝑃(𝑥|𝑁 , 𝑝) =Õ

𝑌

𝑃_𝐵(𝑥|𝑌 , 𝑝) ·𝑃_𝑃(𝑌|𝑁)

=

∞

Õ

𝑌=𝑥

𝑌!

𝑥!(𝑌 −𝑥)!𝑝^𝑥(1− 𝑝)^𝑌−𝑥· 𝑁^𝑌𝑒^−𝑁 𝑌!

= 𝑝^𝑥𝑒^−𝑁 𝑥!

∞

Õ

𝑌=𝑥

𝑁^𝑌(1−𝑝)^𝑌−𝑥 (𝑌 −𝑥)!

= (𝑁 𝑝)^𝑥𝑒^−𝑁 𝑥!

∞

Õ

𝑌=𝑥

[𝑁(1− 𝑝)]^𝑌−𝑥 (𝑌 −𝑥)!

= (𝑁 𝑝)^𝑥𝑒^{−𝑁 𝑝} 𝑥!

∞

Õ

𝑌=𝑥

[𝑁(1− 𝑝)]^𝑌−𝑥𝑒^{−𝑁(1−𝑝)} (𝑌 −𝑥)!

= (𝑁 𝑝)^𝑥𝑒^{−𝑁 𝑝} 𝑥!

∞

Õ

𝑌=0

[𝑁(1− 𝑝)]^𝑌𝑒^{−𝑁(1−𝑝)} 𝑌!

= (𝑁 𝑝)^𝑥𝑒^{−𝑁 𝑝} 𝑥! ·1

= (𝑁 𝑝)^𝑥𝑒^{−𝑁 𝑝} 𝑥!

=𝑃_𝑃(𝑥|𝑁 𝑝).

Therefore𝑥 ∼ Poisson(𝑁 𝑝).