TPC spatial distortions effects on pT
Updated on Tue, 2015-06-09 09:57. Originally created by genevb on 2015-03-27 16:40.
dpT represents the error in reconstructed transverse momentum
d(1/pT) = -q*B
or
dpT = q*B*pT2
dpT[h-] = - B*pT2
dpT[h+] = + B*pT2
True physical functions are functions of pT: h-(pT), h+(pT)
Measured data functions are recorded at the wrong pT: h-'(pT), h+(pT), such that
h-'(pT) = h-(pT+dpT[h-])
h+'(pT) = h+(pT+dpT[h+])
Here I explore three scanarios:
-Gene
The TPC measures sagitta, and determines pT from it:
pT = RCurvature * CDCurvature * BField
RCurvature = (s/2) + (L2/(8*s)) <= radius of curvature as a function of sagitta s and track length L
CDCurvature = 0.000299792458 [GeV/c/kGauss/cm]
BField = ±4.984778 kGauss <= reversed and forward full field
We absorb the direction of the sagitta into a coefficient of ±1 which represents q (or equivalently 1/q since 1/-1 = -1), the charge of the track. For small s (large pT), this can be approximated as:
pT/q = CDCurvature * BField * L2 / (8*s)
And for long tracks of approximately the same length, the dependence is simple:
pT/q ∝ 1/s
...or...
q/pT ∝ s
...or...
1/pT ∝ q*s
Errors in TPC measurements are errors in s, and can thus be treated as errors in q/pT. In the following text, we assume an error in s which, after including the aforementioned coefficients, results in an error on 1/pT of -q*B.
pT = RCurvature * CDCurvature * BField
RCurvature = (s/2) + (L2/(8*s)) <= radius of curvature as a function of sagitta s and track length L
CDCurvature = 0.000299792458 [GeV/c/kGauss/cm]
BField = ±4.984778 kGauss <= reversed and forward full field
We absorb the direction of the sagitta into a coefficient of ±1 which represents q (or equivalently 1/q since 1/-1 = -1), the charge of the track. For small s (large pT), this can be approximated as:
pT/q = CDCurvature * BField * L2 / (8*s)
And for long tracks of approximately the same length, the dependence is simple:
pT/q ∝ 1/s
...or...
q/pT ∝ s
...or...
1/pT ∝ q*s
Errors in TPC measurements are errors in s, and can thus be treated as errors in q/pT. In the following text, we assume an error in s which, after including the aforementioned coefficients, results in an error on 1/pT of -q*B.
dpT represents the error in reconstructed transverse momentum
d(1/pT) = -q*B
or
dpT = q*B*pT2
dpT[h-] = - B*pT2
dpT[h+] = + B*pT2
True physical functions are functions of pT: h-(pT), h+(pT)
Measured data functions are recorded at the wrong pT: h-'(pT), h+(pT), such that
h-'(pT) = h-(pT+dpT[h-])
h+'(pT) = h+(pT+dpT[h+])
Here I explore three scanarios:
- If h- or h+ is flat in pT, then NO EFFECT is discernable in them!
- If exponential in mT = √(m2 + pT2)...
h-(pT) ~ C*e-F*mT
h+(pT) ~ D*e-G*mT
dmT = (pT/mT)*dpT
h-'(pT) ~ h-(pT+dpT[h-]) ~ C*e-F*(mT+(pT/mT)*dpT[h-])
h+'(pT) ~ h+(pT+dpT[h+]) ~ D*e-G*(mT+(pT/mT)*dpT[h+])
dpT has a different sign depending on track charge sign:
h-'(pT) ~ C*e-F*(mT-B*(pT3/mT)) = h-(pT)*e+F*B*(pT3/mT)
h+'(pT) ~ D*e-G*(mT+B*(pT3/mT)) = h+(pT)*e-G*B*(pT3/mT) - If h- and h+ scale as ~ 1/pTn (power law region)...
h-(pT) ~ C/pTn
h+(pT) ~ D/pTn
h-'(pT) ~ h-(pT+dpT[h-]) ~ C/(pT+dpT[h-])n
h+'(pT) ~ h+(pT+dpT[h+]) ~ D/(pT+dpT[h+])n
dpT has a different sign depending on track charge sign:
h-'(pT) ~ C/(pT-B*pT2)n
h+'(pT) ~ D/(pT+B*pT2)n
h-(pT)/h+(pT) ~ C/D
h-'(pT)/h+'(pT) ~ (C/D) * (pT-B*pT2)n/(pT+B*pT2)n
h-'(pT)/h+'(pT) ~ (h-(pT)/h+(pT)) * (pT-B*pT2)n/(pT+B*pT2)n
h-'(pT)/h+'(pT) ~ (h-(pT)/h+(pT)) * (1-B*pT)n/(1+B*pT)n
...and for |dpT/pT| ≪ 1...which means |B|*pT ≪ 1
h-'(pT)/h+'(pT) ~ (h-(pT)/h+(pT)) * (1-B*pT)2*n
h-'(pT)/h+'(pT) ~ (h-(pT)/h+(pT)) * (1-2*n*B*pT)
[ h-'(pT)/h+'(pT) ] / [ (h-(pT)/h+(pT)) ] ~ (1-2*n*B*pT)
-Gene
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