Quark-Hadron Duality in DIS Form Factors and. Drell-Yan Antiquark Flavor Asymmetries

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1 Quark-Hadron Duality in DIS Form Factors and Peter Ehlers University of Minnesota, Morris University of Washington Mentor: Wally Melnitchouk

2 Table of Contents 1 Quark-Hadron Duality in DIS Form Factors 2

3 Motivation for investigating Quark-Hadron duality Quark-Hadron duality tells us about the transition from hadronic degrees of freedom at low energies to quark and gluon degrees of freedom at high energies.

4 Motivation for investigating Quark-Hadron duality Quark-Hadron duality tells us about the transition from hadronic degrees of freedom at low energies to quark and gluon degrees of freedom at high energies. There has been considerable interest in quark-hadron duality at JLab. W. Melnitchouk, R. Ent, and C. Keppel, Phys. Rept. 406 (2005)

5 Motivation for investigating Quark-Hadron duality G. Domokos, S. Kovesi-Domokos, and E. Schonberg, Phys. Rev. D 3 (1971) This paper served as an early attempt to a scaling structure function built up from an oscillator-like resonance spectrum of n resonances, depending only on x B.

6 Motivation for investigating Quark-Hadron duality G. Domokos, S. Kovesi-Domokos, and E. Schonberg, Phys. Rev. D 3 (1971) This paper served as an early attempt to a scaling structure function built up from an oscillator-like resonance spectrum of n resonances, depending only on x B. Revisiting this calculation is useful to bring the notation up to date and to discover exactly what properties of the resonance model lead to a scale-independent structure function.

7 Structure Functions The structure functions W 1 and W 2 are defined in terms of the hadronic tensor W µν. W µν = ( g µν q ) µq ν q 2 W 1 + 1m ( 2 p µ p q ) ( q 2 q µ p ν p q q 2 q ν where p and q are the four-momenta of the incoming hadron and virtual photon, respectively. ) W 2

8 Structure Functions The structure functions W 1 and W 2 are defined in terms of the hadronic tensor W µν. W µν = ( g µν q ) µq ν q 2 W 1 + 1m ( 2 p µ p q ) ( q 2 q µ p ν p q q 2 q ν where p and q are the four-momenta of the incoming hadron and virtual photon, respectively. ) W 2 The structure functions obtained in the paper by Domokos et al. were also obtained in an earlier paper by Bjorken and Walecka. The two papers use different methods of derivation to achieve the same result. J. D. Bjorken and J. D. Walecka, Annals of Physics 38, (1966)

9 Structure Functions W1 JnN = M n q 5 2J+1 (E + m)pn 2 ( ) (2J + 1)!! 2J J+3/2 m 2 (2J)!! 2J 1 G2 2 + G 2 3, W2 JnN = m2 Q 2 Mn q 2 2 W 1 JnN + M n q 2J 1 (E + m)pn 2 (2J + 1)!! 2 J+3/2 G 2 (2J)!! 1, where m is the initial nucleon mass, M n is the mass of the resonance, q is the CM frame momentum of the two incoming particles, Q 2 = q 2 is the virtuality of the incoming photon, E is the CM frame energy of the incoming nucleon, J is the final spin of the excited state nucleon, G i is an invariant form factor, and P N is a parity-dependent factor equivalent to { 1, N = 1 P N = q E+m, N = 1.

10 Elastic Case W 1/2,0,N 1 = m3 q 2 (E + m)pn 2 2 ( Q 2 W 1/2,0,N 2 = ) G 2 3δ (ν Q2, 2m ) q 2 W 1/2,0,N 1 + m(e + m)p N 2 2 ) ) = δ (ν Q2 2m (The 2mδ ( (p + q) 2 Mn 2 from the previous slide for brevity) G 2 1 ) δ (ν Q2. 2m factors were omitted

11 Elastic Case W 1/2,0,N 1 = m3 q 2 (E + m)pn 2 2 ( Q 2 W 1/2,0,N 2 = ) G 2 3δ (ν Q2, 2m ) q 2 W 1/2,0,N 1 + m(e + m)p N 2 2 ) ) = δ (ν Q2 2m (The 2mδ ( (p + q) 2 Mn 2 from the previous slide for brevity) By defining τ = Q2 W 1/2,0,N G 2 1 ) δ (ν Q2. 2m factors were omitted, the structure functions can be simplified to 4m 2 ) 1 = 4m 6 τ(1 + τ) 2 PNG 2 2 3δ (ν Q2 2m ( ) W 1/2,0,N W 1/2,0,N ) 1 2 = + m 2 (1 + τ)p τ NG 2 1 δ (ν Q2 2m

12 Elastic Case By relating the previous structure function expressions to known ones, ) W 1/2,0,N 1 = τg 2 Mδ (ν Q2, 2m W 1/2,0,N 2 = G2 E + τg2 M δ 1 + τ (ν Q2 2m ), we find that the elastic electric and magnetic form factors are related to G 1 and G 3 by G M = 2m 3 (1 + τ)p N G 3, G E = m(1 + τ)p N G 1.

13 Form Factors in the Bjorken Limit The full structure function W 1 is given by W 1 = JnN W JnN 1 = Ef(Π + ), mef(µ 2 1/2 + µ2 3/2 ) 0 B n M n (1 + Q2 (mr) 2 ) 4 dn, M 2 n where f is a free parameter, Π and are the isospin 1 2 and 3 2 contributions, respectively, µ I is the magnetic moment of the nucleon with isospin I, r is the effective radius of the nucleus, and B n is a Breit-Wigner factor.

14 Form Factors in the Bjorken Limit The full structure function W 1 is given by W 1 = JnN W JnN 1 = Ef(Π + ), mef(µ 2 1/2 + µ2 3/2 ) 0 B n M n (1 + Q2 (mr) 2 ) 4 dn, M 2 n where f is a free parameter, Π and are the isospin 1 2 and 3 2 contributions, respectively, µ I is the magnetic moment of the nucleon with isospin I, r is the effective radius of the nucleus, and B n is a Breit-Wigner factor. From here, the limit as Q 2 is taken and the variable ω = 2p q+m2 is defined, which is equivalent to 1 Q 2 x B in the high Q 2 limit.

15 Form Factors in the Bjorken Limit Finally, we can make a change of variables to u = dn = Q 2 du so that W 1 M 2 n m 2 Q 2 with ω f ( ) 4 2 ω 1 (µ2 1/2 + µ2 3/2 ) 1 Q 2 B n 1 + r2 du. 0 u u

16 Form Factors in the Bjorken Limit Finally, we can make a change of variables to u = dn = Q 2 du so that W 1 M 2 n m 2 Q 2 with ω f ( ) 4 2 ω 1 (µ2 1/2 + µ2 3/2 ) 1 Q 2 B n 1 + r2 du. 0 u u The other structure function F 2 = νw 2 has similar properties, as it is proportional to W 1 in the high Q 2 limit. The final result of Domokos et al. is that F 2 f(µ 2 1/2 + µ2 3/2 ) (ω 1) 3 (ω 1 + m 2 r 2 ) 4. when B n is replaced with a delta function.

17 Breit-Wigner factor The Breit-Wigner factor is used to replace the delta function δ ( s Mn 2 ) Bn = 1 Γ 0 Mn 2 π (s Mn) (Γ 0 Mn) 2 2, where Γ 0 is some initial resonance width and s = (p + q) 2.

18 Breit-Wigner factor The Breit-Wigner factor is used to replace the delta function δ ( s Mn 2 ) Bn = 1 Γ 0 Mn 2 π (s Mn) (Γ 0 Mn) 2 2, where Γ 0 is some initial resonance width and s = (p + q) 2. When converting to the variable u and taking the high Q 2 limit, this becomes B n = 1 Γ 0 m 2 u πq 2 (ω 1 m 2 u) 2 + (Γ 0 m 2 u) 2 so that Q 2 B n becomes independent of Q 2. In fact, any function in place of B n that goes with 1 in the high Q 2 limit will have this Mn 2 property.

19 Summary The structure functions given in Domokos et al. were updated to modern language. Their relation to the elastic electric and magnetic form factors was established.

20 Summary The structure functions given in Domokos et al. were updated to modern language. Their relation to the elastic electric and magnetic form factors was established. The 1 dependence of the Breit-Wigner factor B Mn 2 n was shown to be important in making the structure functions scale independent at high Q 2.

21 Summary The structure functions given in Domokos et al. were updated to modern language. Their relation to the elastic electric and magnetic form factors was established. The 1 dependence of the Breit-Wigner factor B Mn 2 n was shown to be important in making the structure functions scale independent at high Q 2. More research into the scale independence is to be done.

22 Motivation for New Project From a paper by Jen-Chieh Peng et al. (arxiv: )

23 Motivation for New Project From a paper by Jen-Chieh Peng et al. (arxiv: ) Want to investigate apparent discrepancy in the data sets.

24 Motivation for New Project From a paper by Jen-Chieh Peng et al. (arxiv: ) Want to investigate apparent discrepancy in the data sets. Identified two problems with how the NMC data are used here

25 d ū plots and comparisons In particular, the paper used the leading order relation F p 2 (x) F 2 n (x) = x ( ) u(x) + ū(x) d(x) d(x) 3 to derive d(x) ū(x) = u(x) d(x) 3 x (F p 2 (x) F 2 n (x)) = 1 ( u v (x) d v (x) 3 ) 2 x (F p 2 (x) F 2 n (x))

26 d ū plots and comparisons In particular, the paper used the leading order relation F p 2 (x) F 2 n (x) = x ( ) u(x) + ū(x) d(x) d(x) 3 to derive d(x) ū(x) = u(x) d(x) 3 x (F p 2 (x) F 2 n (x)) = 1 ( u v (x) d v (x) 3 ) 2 x (F p 2 (x) F 2 n (x)) Also want to look at the effects of nuclear smearing and off-shell corrections.

27 d ū plots and comparisons

28 d ū plot errors Also calculated these relations with errors using CJ12 PDF grids, as detailed in Owens et al. (Sec. II E, Phys. Rev. D 87, (2013)).

29 DY addition to CJ code Also added a routine to the CJ14 fitting package to compute nuclear smearing and off-shell corrections for the Drell-Yan deuteron cross-section. The corrections come from Ehlers et al. At the GeV scale, the smearing is approximately the same as that of DIS at γ = 1, and the fmkp off-shell model is valid for both DIS and Drell-Yan. P. J. Ehlers, A. Accardi, L. Brady, and W. Melnitchouk, Phys. Rev D 90, (2014)

30 DY addition to CJ code

31 Summary The amount by which the LO equation F p 2 (x) F 2 n(x) = x ( ) 3 u(x) + ū(x) d(x) d(x) fails to hold at NLO was quantified.

32 Summary The amount by which the LO equation F p 2 (x) F 2 n(x) = x ( ) 3 u(x) + ū(x) d(x) d(x) fails to hold at NLO was quantified. Error calculations on the above equation solved for d ū were performed.

33 Summary The amount by which the LO equation F p 2 (x) F 2 n(x) = x ( ) 3 u(x) + ū(x) d(x) d(x) fails to hold at NLO was quantified. Error calculations on the above equation solved for d ū were performed. Drell-Yan smearing and off-shell routines were added into the CJ14 fitting package.

34 Summary The amount by which the LO equation F p 2 (x) F 2 n(x) = x ( ) 3 u(x) + ū(x) d(x) d(x) fails to hold at NLO was quantified. Error calculations on the above equation solved for d ū were performed. Drell-Yan smearing and off-shell routines were added into the CJ14 fitting package. Various other theoretical calculations were performed using CJ14.