Understanding Kinematics Challenges in High-Energy Physics Experiments

Delve into the complex world of kinematics in high-energy physics research, exploring challenges faced in collision simulations and event analysis. Discover how concepts like light-cone variables and covariant definitions play a crucial role in interpreting scattered electron data. Explore practical applications in event generation using tools like Pythia to reconstruct particle properties accurately in detector frames.

• Kinematics Challenges
• High-Energy Physics
• Electron Scattering
• Particle Collisions
• Event Generation

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1. UNIVERSAL KINEMATICS CHARLES HYDE OLD DOMINION UNIVERSITY NORFOLK VA

2. THE KINEMATIC CHALLENGE Most Generators assume head-on collisions Many Cross Section formulae for DIS, SIDIS, DVES are not-fully Lorentz invariant: 2?2 ???2 Electron azimuthal variable ?edefined in head-on frame Some ion polarization conventions refer to an event-by-event frame Virtual photon and target ion momenta are anti-collinear. EIC designs (and HL-LHC): non-zero collision angle Awkward to boost variables back and forth. 1 ? (??)2?2 ?? ?2+ ??2?1 DIS: ???????= ?2 C. HYDE, LIGHT-CONE KINEMATICS 22-26 July 2019 2

3. LIGHT-CONE VARIABLES From electron, ion 4-vectors k , P , define light-cone vectors (me2=0): ? ??1 2 ? ??= ?? ??= ? ?? 2 (2? ?) (? ?) n2= 0 = 2 In convention with k |k| and P |P|cos?Crossing |P| Target rest-frame ( ?=1): Define Lorentz Covariant Transverse Unit Vectors: X , Y X2 = 1 = Y2, X Y = 0 ,X n= X = 0 = Y n= Y Start with = (0,0,1,0) = up ?? ? ? ?? ( ? ?)?? 1+2 ? ? ( ? ?) n = 1 n [1,0,0, 1]/21/2, [1,0,0,1]/21/2 ?? ?? ????? ??????, 0123= 1 , C. HYDE, LIGHT-CONE KINEMATICS 22-26 July 2019 3

4. RIGHT-HANDED COORDINATE SYSTEM ??= ??+ ??/ 2 ?????????????= 1 =??????? ?????? + convention: ??= ?+ ?+ ? ? ? ? ? Incident Electron (me=0) and Incident Proton k+=0 ? = P+/?= M/ 2 ??= ?? ??/ 2 ?= ? ? ?? ? ? ?? ? 2 ? ? /? k =0 P =0 P ?= M/ ( 2) C. HYDE, LIGHT-CONE KINEMATICS 22-26 July 2019 4

5. COVARIANT DEFINITION OF SCATTERED ELECTRON Compute ? ? from k , P , xB, y, ?e: ? ?= ? ? ??+ ? ? ??+ |k | ???????+ ??????? ? += (? ?) ? = ? ? ?2= ? ? 2= 2? ? + 2? ? = 2? ? ? = 2? ?+ 2? ?++ 2? +? Invert: ? += ?2/(2? ) |k |2 = 2k +k 2? ?+= 2? ? 1 ? + 2? +? ?2 ? += 2? ?+?+= ????+ ? = 1 ? ? + ???? C. HYDE, LIGHT-CONE KINEMATICS 22-26 July 2019 5

6. APPLICATION: DIS EVENT GENERATOR E.G. PYTHIA Compute coordinate vectors ??, ??, ??,?? separately in Reference Frames of generator (with correct s = (k+P)2) and Collider Detector. Run Generator For all output four-vectors pa , a = 1,2, Compute invariants(p n), (p ), (p X), (p Y)using basis vectors in generator frame Reconstruct p in true detector frame, using vectors ??, ??, ??,?? defined to detector frame: ?? ?? No Lorentz boosts, rotations necessary. All is accomplished by defining ??, ??, ??,?? separately in two frames ? ???+ ?? ? ?? ?? ?? ???= ?? ? ??? ?? ? ??? ?? ? ??? C. HYDE, LIGHT-CONE KINEMATICS 22-26 July 2019 6

7. CUSTOM GENERATOR WITH CROSS SECTION FORMULA DEFINED ONLY IN PROTON-REST OR HEAD-ON COLLISION FRAME Define k , P in Collider (non-zero crossing angle) Frame = Lab frame. Compute reference vectors ??, ??, ??,?? in Collider Frame Choice of ? value is arbitrary, ? ?=1 puts reference system in target rest frame ? = (? ?2)/ ? 2 , ?+= ? = 2 = ??+ ? ????????????/? puts system in a head-on collider frame with ? =?0 ?? 2 = n.b. Proton energy EpLab [ (P++P )/ 2 ]Head-on frame Generate DIS event by (Q2, xB, ?e) or (y, xB, ?e) ?2 2?+ 2?0, ?+= ??+ ? ????????????/ 2, ? = C. HYDE, LIGHT-CONE KINEMATICS 22-26 July 2019 7

8. NUCLEI M MA xB xA=Q2/(2q PA) y y = (q PA)/(k PA) C. HYDE, LIGHT-CONE KINEMATICS 22-26 July 2019 8

9. SPECTATOR TAGGING D(e,e ps)X d F2,n/t 2 t = Mn2 (PD pp)2 0.004 GeV2 Uncertainty in Beam emittances are major systematics in defining t Proton recoil resolution comparable to beam emittance High resolution neutron tagging 35%/ En Measure F2p of bound proton in D C. HYDE, LIGHT-CONE KINEMATICS 22-26 July 2019 9

10. FINAL STATE RECONSTRUCTION Current Jet and Target fragmentation jet reconstruction requires proper kinematics in detector frame C. HYDE, LIGHT-CONE KINEMATICS 22-26 July 2019 10

11. CONCLUSIONS Many legacy codes and published cross section codes are not fully covariant Light-cone (& Sudakov) variables ??, ??, ??,?? defined in both the detector frame and a head-on collider frame provide a natural adaptation, without the need for repeated rotations and boosts of all variables. C. HYDE, LIGHT-CONE KINEMATICS 22-26 July 2019 11