# DIS Kinematics

This page collects the results of simulations concerning the kinematic coverage of the EIC machine and detector.
Plots are shown for representative permutations of beam energies that span the full range of the BNL proposal, as discussed in the EIC White Paper, varying between:

• A 5 GeV electron beam on a 50 GeV proton beam, the lowest center-of-mass energy during Phase-I eRHIC.
• A 30 GeV electron beam on a 250 GeV proton beam, the highest center-of-mass energy reachable at eRHIC Phase-II.

Unless stated otherwise plots are produced from PYTHIA events using ROOT.

## Kinematic Variables for DIS

The kinematics of the event, defined via the variables x, Q2 and y, can be reconstructed in a number of ways.

#### From the scattered lepton

 Kinematic variables, calculated solely from the scattered lepton Resolution functions for leptonic kinematics

The figure on the left shows that measurement of the energy and angle of the scattered lepton, coupled with knowledge of the incident beam energies, completely defines the parton-level kinematics of the collision. From the equations on the right hand side it is obvious that the x and y resolutions diverge for low y. To avoid this, the kinematics are commonly reconstructed from the other methods, utilising the hadronic final state, where the resolution δy/y ~ constant. The resolution for Q2 worsens as the momentum and/or energy resolution of the scattered lepton worsens.

#### From the hadronic final state

The Jacquet-Blondel method reconstructs event kinematics solely from the hadronic final state.

 Kinematic variables based solely on hadronic final state

This method generally gives good resolution in y. Resolution in Q2 degrades the more pT of the event is missed by the detector (i.e. as particles escape down the beam pipe). Hence it generally gives poor resolution in Q2 at low Q2, but better resolution as Q2 increases and more particles are scattered into the detector acceptance. As x is calculated from y and Q2, x resolution generally follows that of Q2. Note that for charged-current events this is the only available method by which to calculate event kinematics.

#### Other methods

Further methods of kinematic reconstruction utilise a mixture of both electron and hadronic information, in order to yield the benefits of both. For more details please see arXiv:1208.6087

## Q2 and Bjorken x coverage

Different beam energies provide access to different ranges in Q2 and x, via the (approximate) relation Q2 = sxy. The figures below show the Q2 vs. x planes accessible with different beam energy permutation, for Q2 greater than 1 GeV2. The limits imposed by selecting y greater than 0.01 and less than 0.95 are shown by the solid and dashed lines respectively.

 5 x 50 GeV collisions 10 x 100 GeV collisions 20 x 250 GeV collisions

## Beam Direction Convention

All plots obey the following convention regarding beam direction, following what was done at HERA:

• The hadron beam goes to +z and such to positive rapidities, very often referred to as "forward".
• The lepton beam goes to -z and such to negative rapidities, very often referred to as "backward".

## Inclusive DIS: scattered lepton kinematics

The Q2 of a DIS interaction is related the polar angle of the scattered lepton via Q2 = 2EeEe(1+cosθe).
The measurement of low Q2 therefore relies on detecting electrons at forward angles, especially at high incident electron energy.
The figure below shows the Q2 of the interaction as a function of the rapidity of the scattered lepton, for Q2 greater than 0.1 GeV2 and different lepton-hadron beam energy combinations.

 Q2.rap.png

The figure below shows the relation between scattered lepton momentum and its rapidity, Y.

 Lep.mom.png

The higher the center-of-mass energy, the smaller the scattering angle of the lepton is. The small scattering angles are also correlated with the highest scattered lepton energies and low Q2 values. The high Q2 events enter in the barrel. This can been also seen in the following picture, which shows the momentum of the scattered lepton for three different beam energy combinations in bins of the scattered lepton rapidity for Q2 > 0.1 and 0.01 < y < 0.95.

 Lepton.sqrt(s).rap.png

For semi-inclusive DIS it is critical to detect the hadronic final state in addition to the scattered lepton. The plot below show the relation between hadron (π) momentum and rapidity, Y. The following cuts have been applied:

• Q2 > 1 GeV2
• 0.01 < y < 0.95
• z > 0.1

z is defined the fraction of the virtual photon energy, ν, carried by the hadron in the rest frame of the proton beam.

 Pion.kinematics.png

The smaller the center-of-mass energy, the more hadrons are produced in the original hadron beam direction (i.e. at forward/positive rapidities). The maximum hadron momentum in a specific rapidity bin increases with the hadron beam energy. With increasing center-of-mass energy the hadrons are boosted into the central detector, and later into the original lepton beam direction (i.e. backward/negative rapidities) at the highest lepton beam energies.

 For semi-inclusive physics full coverage in the kinematic variables z, pT and φ (see figure) is critical for the physics. pT is not measured with respect to the beam line, as it is in p + p collisions, but with respect to the virtual photon. For TMDs φ is the angle between the lepton scattering plane and the hadron production plane. Parton-level SIDS kinematics

The plots below show the correlation of the pion z and pT as function of rapidity with the following cuts applied:

• Q2 > 1 GeV2
• 0.01 < y < 0.95
• p > 1 GeV

The plots show clearly that in rapidity range of -3 < Y < 3 all z and pT values are covered.

 Pion.z.rapidity.png Pion.pt.rapidity.png

## Exclusive Reactions

For exclusive DIS it is critical to reconstruct the final state of the particle produced in the interaction (for example, a photon for DVCS, a lepton pair for j/ψ → e+e- etc.) in addition to the scattered lepton. For the studies shown below, we consider the photon produced in a DVCS event. The following cuts have been applied:

• Q2 > 1 GeV2
• 0.01 < y < 0.95
• Eγ > 1 GeV
• -5 < rapidity < 5

The plot below shows the energy vs. rapidity in the laboratory frame for photons from DVCS for different center-of-mass energies.

It is clear that at the low lepton beam energies the DVCS photon goes into both the barrel and the backward direction, whereas at larger lepton energies the backward direction alone dominates.

The following plot shows the correlation between the scattering angle of the DVCS photon and the scattered lepton for three different center-of-mass energies:

 ThetaVsTheta.DVCS.png

At large lepton beam energies the produced photon and the scattered lepton, both go into the backward direction and tend to be very close to each other. A suitable detector must be able to discriminate them in order to reconstruct the full final state.

To reconstruct the Mandelstam variable t, which represents the momentum transfer to the proton in exclusive reactions, it is critical to detect the forward going scattered proton. t is essential in exclusive reactions as it can be fourier transformed to the impact parameter b, which gives the transverse spatial distribution of partons in the proton. The figure below shows the scattering angle θ for three different center-of-mass energies as a function of the scattered proton momentum.

 Exclusive.p.angle.png These protons cannot be detected in the main detector. The standard detectors used to detect the scattered proton are roman pots placed at large distances ~20+ m from the IR.

## Diffractive Reactions

For exclusive diffraction it is important, as in exclusive reactions, to reconstruct the complete final state. This includes the scattered lepton, the final state (jet, J/ψ, etc.) and the scattered nucleus or proton, which is at a small angle to the hadron beam direction. It is not possible to detect the nucleus in Roman Pots as one does the protons from DVCS, as the beam angular divergence limits the smallest outgoing angle, θmin, that can be measured. The pT kicks that are needed to bring a nucleus out of the beam with θmin = 0.1 mrad (1.58 GeV/c for Au, 1.9 GeV/c for U) are much larger than the energy to break up the nucleus (8 MeV). Therefore the technique to ensure the nucleus did not break up has to be different. If a nucleus breaks up the first thing that happens is that it emits neutrons. Therefore the idea is to veto for elastic diffractive reactions by observing neutrons in the Zero-Degree-Calorimeters.

The figure below shows the scattering angle distribution for breakup neutrons from a gold nucleaus for different excitation energies of the nucleus.

 Neutron.10MeV.png Neutron.50MeV.png Neutron.500MeV.png

If the outgoing hadron beam line has an angular acceptance of +/- 5mrad, then the decay neutron of the heavy ion beam can be transported to the zero degree calorimeter. More details about this simulation can be found here

Plots of the kinematics of diffractive jets and J/ψ decay products will follow soon.