# Kinematics for eSTAR

Here are some kinematic plots for the EIC, with various regions in eta shown.  For the "jet", these are the properties of the struck quark.  Hadronization will smear the angle and decrease the momentum relative to the quark.

Just as a pointer about the effect of the kinematic peak, and resolution requirements.  At x = Ee/Ep, e.g. 0.05 for 5x100, the x for electrons is independent of pseudorapidity, the change in x for a given change in electron energy (i.e. dx/dE) is very large, and this, combined with the smooth underlying 1/xQ^4 dependence of the cross section, leads to a peak in the final-state electron energy about the incoming electron energy.  I illustrate this below, with lines of constant electron energy of 4.8,4.85,4.9,4.95,5.0,5.05,5.1,5.15 GeV.  A classic "Atari" shape shows up, and you can see that at, say, Q^2 of 2 GeV, a 1% change from 5.0 to 4.95 GeV, or from 5.0 to 5.05 GeV, leads to a factor of 2 change in x.  This makes for very strict resolution and absolute calibration requirements if you want reasonable measurements in this region.

In terms of tracking resolution, it is much better to think in terms of pt than in E, since the solenoidal field measures only in the transverse plane.  Additionally, one really measures the sagitta (maximal deviation of the chord from straight over a length L).  One can directly translate position accuracy into sagitta resolution.  For the STAR magnetic field (assuming uniform solenoid), the sagitta is sagitta [cm] = 1.9e-4 [GeV/cm]* L[cm]*L[cm]/pT [GeV], where L is the transverse length in cm.  If we take primaries, i.e. assume we can get the vertex with high precision, then a simplified picture of the STAR magnetic field has L as L = (tan(theta)<1) ? 200*tan(theta) : 200.  To give a scale: at L=200 cm (mid-rapidity primary) the linear term in our d(1/pt)/(1/pt) resolution is 0.5%/GeV.  Take a fixed sagitta resolution ds, 1/pt = sagitta/(1.9e-4*L*L), so d(1/pt) = ds /(1.9e-4*L*L)

0.5% [GeV^-1] * 1.9e-4 [GeV/cm] * 200 [cm]*200[cm] = 400 microns, which is reasonably close to the fundamental resolution of the TPC.  A 10 GeV track with 0.8 cm sagitta has as linear term a 5% resolution.

Through the hole in the poletip (2.5 to 4 in eta), this underestimates the information we can get, since we can also use bending in the radial field through the hole in the poletip and to outside.  In any case, I show below the kinematics with lines constant in pt = E sin(theta) and in sagitta, for both hadrons and electrons.

One can see that the region from -1 to -2 in eta has reasonable but somewhat small sagittas (0.5 - 2 cm) for electrons and sagittas on the order of 1-6 cm for hadrons for 5x100.  At 0.5 cm, if you want dpt/pt~5% measurements you probably need something like 250 micron measurement resolution out at the maximal radius at the endcap.  Resolution scales linearly down with increasing magnetic field and decreasing position resolution.

Measurements with this simplified field aren't as easy in the 2.5-4 region.  For 30x100, sagittas are on the order of 200 microns to 2 mm for hadrons and maybe half that for electrons.   But, again, there is extra bending power by using the radial fields in the hole in the poletip; just needs a full simulation.  For the electrons, it's probably better to measure E with calorimetry in this region.

Thinking about gas volumes for dE/dx or other uses: below is the total pathlength between different radial positions, as a function of eta."Outer" is basically the radius covered by the outer padrows, "inner" the radius covered by the inner padrows of the tpc, and "tiny" the region of air inside the TPC inner field cage, available for instrumentation beyond the TPC.  Since the dE/dx is dominated by the outer sector currently, since only 20% of the inner sector is currently instrumented, 60 cm or so gives good dE/dx.  Reinstrumenting the inner sector to cover the full range of gas available would push the dE/dx coverage out to eta of 1.5; to go to eta of 2 likely needs further instrumentation (either inside the TPC, or extending the x range beyond the endcaps of the TPC).  For electrons from 5x100, eta of -1.5 is Q^2 of ~5 GeV^2 for x>0.1 or so (about halfway in the red).  To get full coverage down to Q^2 of 1 GeV^2, you need to cover to eta of -2.3.

One can see that there is more than enough path length if you fill the entire space inside the TPC with gas ("tiny").  One can start to play with what is the minimum amount of gas you need; cut the radius, cut the length in z, offset the volume in z, etc. so that you have space for other detector systems.  Below are two options, and I attach the macro (adding an extension .txt so it can actually be uploaded) so others can play.  In the "FGT-like"(blue), I go in as far as possible in the current cone structure (the cones end at 60 cm, and the FGT starts at 70 cm, so this pushes in a bit relative to the FGT).  For this geometry, one would need to combine the reinstrumented inner sector with this new detector in order to get enough path length.  In the "Ring" (purple), I started at somewhat higher radius, but also pushed the gas volume closer to the vertex.  In this geometry, one has sufficient coverage with either the inner sectors (to eta of ~1.5) or the ring (eta ~1.5 to 2.8), and there is space for another type of detector at lower z and also at lower radius.  One could think about shortening its z length a little on the other end, if drifting of 160 cm is an issue (as it might be if you also want accurate spacepoints).

To first order, it looks like we could leave the region from 0-40 cm in z free for a silicon system of some sort, and so have both electron and heavy flavor capability, if we stay careful about material.  One option might be to stuff the support material for the silicon inside the ring, so it only really impacts the very forward eta.  Of course, that would be an optimization for the 5 GeV electron energy that would be bad for a higher electron energy.

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