Thursday, September 29, 2011

Neutrinos on the brain-2

My previous post demonstrated that the proton curve was of the form:

y = c * exp( - exp( -a * x + b ))

Specifically, my eyeball fit was

y=exp(3.58)* exp( - exp( -0.0061*x - 2.9201))

Let me use t (for time) instead of x, and p(t) for the proton pulse as a function of t,
and rearrange it a bit:

p(t) = exp( c + d * ( 1 - exp( -a * ( t - t0 ) ) )   { for t ≥ t0 }

The inner (1-exp()) is the current in a L-R circuit turning on, starting at time t0, creating a magnetic field.
The outer exp() is the proton spill increasing exponentially as the magnetic field turns on.

Suppose I shift the time t0 to t0+m.

p(t) = exp( c + d * ( 1 - exp( -a * (t - t0 - m ) ) )
≡   exp( c' + d' * ( 1 - exp ( -a * (t - t0) ) )

d' = d * exp( a * m )
c' = c + d * ( 1 - exp ( a * m ))

Notice the problem this presents for us.  Given samples of a pulse of this form, I want to do a curve fit  to find the starting point of the pulse, t0.   But the starting point t0 cannot be determined from the fit because I can redefine the constants and move t0 around.

For the proton curve, I have a physical start of time, when the current turns on the kicker magnet.   For the neutrino pulse however, this is fatal, I do not have a good physical start - it is the physical start that I want to determine (and that yields this strange faster-than-light puzzle).

PS: If I know exactly when the neutrino curve lifts up from zero, I can fix t0. But this is unfortunately where the statistics are the worst. There is definitely a greater than 60ns ambiguity in where the neutrino curve lifts up from zero. And if I knew t0, then I umambiguously know whether neutrinos are superluminal or not.

PPS: Or fit the neutrino curve howsoever, draw it and see where it kisses 0, and that tells you the timing, reparametrize -- you need good statistics at the upper portions of the curve, and I bet you still have more than 60ns ambiguity of the origin of time.

PPPS: Of course, I have yet to show that three other curves follow this very model. But I'm not after certainty just yet; my obsession will clear if I think I know why OPERA measured what it did. If I have 50+% of the correct required reasoning, I'll be more than happy.

PPPPS: On a ten nano-second clock, the neutrino curve will look something like this (entirely made up,
except for the sum across 5 clock ticks matches the binning result)
0,0,1,1,0,   (2)
1,0,1,2,1,   (5)
1,0,0,1,2,   (4)
1,2,4,2,2,   (11)
0,1,3,2,2,   (7), etc., you get the idea.
The rest of the series is:
PPPPS: Next steps:
1. Digitize the three other edges, check whether they follow this model. Assuming yes, then:

2. Find an industrial-strength curve fitter and fit the protons and neutrinos to this model.

3. Estimate the error of estimate of the neutrinos - see if it swallows the 60ns that OPERA found.

4. To be determined.