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

2. 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)

3. 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,

4. 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.

5. 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

6. 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.

7. 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,

8. Derivation of the pD Cross Section
𝑑σ 𝑝𝐷 = 2𝜋 𝑝 𝑝 ∙ 𝑝 𝐷 𝑚 𝑝 2 𝑚 𝐷 𝜋 6 𝑑 𝑘 𝐸 1 𝑑 𝑘 𝐸 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 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.

9. 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.

10. 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 →∞

11. 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

12. 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,

13. 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.

14. Conclusion
Nuclear and off-shell corrections will be integrated into the CJ global PDF analysis.

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