TPC spatial distortions effects on pT

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
q/pT ∝ s
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
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)