Nuclear Effects in the Proton-Deuteron Drell-Yan Reaction. Peter Ehlers University of Minnesota, Morris Mentor: Wally Melnitchouk Alberto Accardi

Drell-Yan Process Two hadrons: proton (p) and nucleon (N). Quark (𝑞) from one and antiquark ( 𝑞 ) from the other annihilate. Virtual photon becomes lepton-antilepton pair ( ℓ − , ℓ + ). Other hadrons ( 𝑋 𝑝 , 𝑋 𝑁 ) produced are not observed. N is either another proton (p) or a neutron (n)

Drell-Yan Process Individual quark flavor distributions can be probed at high energies. Deep inelastic scattering (DIS) measures sums of quark and antiquark distributions. 𝑑σ 𝑝𝑁 𝑑 𝑥 𝑝 𝑑 𝑥 𝑁 = 4𝜋 𝛼 2 9 𝑀 ℓℓ 2 𝑞 𝑒 𝑞 2 𝑞 𝑥 𝑝 𝑞 𝑥 𝑁 + 𝑞 𝑥 𝑝 𝑞 𝑥 𝑁 , where 𝑥 𝑝 = 𝑝 𝑞 𝑝 + / 𝑝 𝑝+ and 𝑥 𝑁 = 𝑝 𝑞 𝑁 + / 𝑝 𝑁+ Ratio of pD to pp cross sections ≈1+ 𝑑 𝑥 𝑁 𝑢 𝑥 𝑁 ( 𝑥 𝑝 ≫ 𝑥 𝑁 ) R. S. Towell et al., Phys. Rev. D 64, 052002

Motivation for using Deuterons Uncover the internal structure of the neutron. Free neutrons are unstable. Deuteron is composed of one proton & one neutron. Weak nuclear binding. Easy place to start examining nuclear effects.

Goals Compute the nuclear effects on the proton- deuteron cross section ( 𝑑σ 𝑝𝐷 ). Earlier analyses use 𝑑σ 𝑝𝐷 ≈ 𝑑σ 𝑝𝑝 + 𝑑σ 𝑝𝑛 Examined in DIS, very little attention in DY. Derive a relation between 𝑑σ 𝑝𝐷 and 𝑑σ 𝑝𝑁 that accounts for: Nuclear binding Fermi motion (internal nucleon motion) Nucleon off-shell corrections

Proton-Deuteron DY Process One scattered nucleon (N), one spectator nucleon (S). p & N are involved in the Drell-Yan process. N is now an internal line; not observable.

Energy of the Struck Nucleon Because N is not observable, it does not obey the on-mass shell relation 𝑝 𝑁 2 = 𝑝 𝑁0 2 − 𝑝 𝑁 2 = 𝑚 𝑁 2 . However, the spectator nucleon S is on-shell. 𝑝 𝑁 + 𝑝 𝑆 = 𝑝 𝐷 = (𝑚 𝐷 ,0, 0, 0) in D rest frame. 𝑝 𝑆 =− 𝑝 𝑁 𝑝 𝑆0 = 𝑝 𝑆 2 + 𝑚 𝑆 2 = 𝑝 𝑁 2 + 𝑚 𝑆 2 𝑝 𝑁0 = 𝑚 𝐷 − 𝑝 𝑆0 = 𝑚 𝐷 − 𝑝 𝑁 2 + 𝑚 𝑆 2 A previous analysis1 used time-ordered perturbation theory, where N is on-shell but energy is not conserved. 1H. Kamano and T.-S. H. Lee, Phys. Rev. D 86, 094037

Derivation of the pD Cross Section 𝑑σ 𝑝𝐷 = 2𝜋 4 4 𝑝 𝑝 ∙ 𝑝 𝐷 2 + 𝑚 𝑝 2 𝑚 𝐷 2 1 2𝜋 6 𝑑 𝑘 1 2 𝐸 1 𝑑 𝑘 2 2 𝐸 2 1 𝑞 4 𝑓 𝜇𝜐 𝑘 1 , 𝑘 2 𝐹 𝜇𝜈 𝑝𝐷 𝑝 𝑝 , 𝑝 𝐷 ,𝑞 Definition of a scattering cross section for the Drell-Yan Process 𝐹 𝜇𝜈 𝑝𝐷 𝑝 𝑝 , 𝑝 𝐷 ,𝑞 = 𝑑 𝑝 𝑁 Ψ 𝐷 𝑝 𝑁 2 𝑝 𝐷0 𝑝 𝑁0 𝐹 𝜇𝜈 𝑝𝑁 ( 𝑝 𝑝 , 𝑝 𝑁 ,𝑞) Deuteron hadron tensor in terms of the nucleon hadron tensor. 𝐹 𝜇𝜈 𝑝𝑁 has no transverse momentum or off-shell dependence. No final state interactions between the spectator nucleon S and hadronic debris XN.

Results 𝑑σ 𝑝𝐷 𝑑 𝑥 𝑝 𝑑 𝑥 𝑁 ( 𝑥 𝑝 , 𝑥 𝑁 )= 𝑁=𝑝,𝑛 𝑑𝑦 𝑓 𝑦 𝑑σ 𝑝𝑁 𝑑 𝑥 𝑝 𝑑 𝑥 𝑁 ( 𝑥 𝑝 , 𝑥 𝑁 𝑦 ) pD cross section in terms of the light-cone convolution formula. 𝑓 𝑦 = 𝜋 𝑚 𝑁 𝑚 𝐷 − 𝑚 𝑁 𝑦 𝑑 𝑝 𝑁⊥ 2 Ψ 𝐷 𝑝 𝑁 2 𝑝 𝑆0 𝑝 𝑝 ∙ 𝑝 𝑁 2 − 𝑚 𝑝 2 𝑝 𝑁 2 𝑝 𝑁0 𝑝 𝑝 where y= 𝑝 𝑁+ 𝑝 𝐷+ 𝑚 𝐷 𝑚 𝑁 = 𝑝 𝑁0 + 𝑝 𝑁𝑧 𝑚 𝑁 is the fraction of nucleon light-cone momentum in the deuteron.

Results Steeply peaked near y= 1, or 𝑝 𝑁+ ≈ 1 2 𝑝 𝐷+ Quickly approaches zero as y deviates from 1 DIS smearing function is very similar. Both have factors that approach 1+ 𝑝 𝑁𝑧 𝑝 𝑁0 in their high energy limits. 𝑝 𝑝 , 𝑄 2 →∞

Results Ratio is 𝑑σ 𝑝𝐷 2 𝑑σ 𝑝𝑁 using a test function for 𝑑σ 𝑝𝑁 . Sharp increase near 1 is because 𝑑σ 𝑝𝑁 =0 at 𝑥=1. Only about 1% correction from 𝑥= 0.4 to 𝑥=0.5

Results Ratio is 𝑑σ 𝑝𝐷 2 𝑑σ 𝑝𝑝 𝑝 𝑝 =800 GeV 𝑥 𝑝 =0.35 𝑄 2 =54 GeV 2 CTEQ5m PDFs Experimental data use similar parameters. Off-shell corrections make a large contribution. R. S. Towell et al., Phys. Rev. D 64, 052002

Results Ratio is 𝑑σ 𝑝𝐷 2 𝑑σ 𝑝𝑝 𝑝 𝑝 =120 GeV 𝑥 𝑝 =0.6 𝑄 2 =54 GeV 2 CTEQ5m PDFs Kinematics for the Fermilab E-906/ SeaQuest experiment. Smearing function contributes primarily at large x.

Conclusion Nuclear and off-shell corrections will be integrated into the CJ global PDF analysis. http://www.jlab.org/CJ

DIS Comparison 𝑝 𝑝 = 20 MeV for Drell-Yan. 𝛾=1.6 in DIS, where 𝛾= 1+ 4 𝑥 𝑁 2 𝑚 𝑁 2 𝑄 2