1. Regge Description of 𝜋𝑁 Scatterings at Forward Angles
Byung-Geel Yu
Korea Aerospace Univ.
in collaboration with K.-J. Kong
APCTP-JLab Joint workshop
APCTP, July 1-4, 2018

2. Outline I. Motivation and objectives II. Reggeized model
Overview: 𝜋𝑁 reaction most fundamental to understand strong interaction from QCD origin
Investigate 𝜋𝑁 charge exchange (CEX) and elastic reactions
Establish scattering amplitude beyond resonances up to 𝑠 ≈22 GeV ( 𝑝 𝐿 ≈250 GeV/c) based on Lagrangian formulation for hadron interactions (in communication with Effective Lagrangian Approach)
II. Reggeized model
Reggeized Born amplitudes for 𝜋𝑁 CEX and elastic scatterings
Reggeon, Elastic Regge cuts, & Pomeron exch. in quark- Pomeron coupling picture.
III. Results & summary
Numerical results in cross sections & polarization observables

3. I. Overview : features of cross sections
𝜎 𝑇 (𝜋𝑁→𝑎𝑙𝑙)= 𝜎 𝑒𝑙 + 𝜎 𝑖𝑛𝑒𝑙
Δ ++ (1232)
𝜎 𝑒𝑙 =𝜎(𝜋𝑁→𝜋𝑁)
𝜎 𝑖𝑛𝑒𝑙 = 𝜎 𝐷 + 𝜎 𝐷𝐷 +…
Δ ++ (1905)
Resonance structure in low 𝑃 𝐿 , nonresonant diffraction at high 𝑃 𝐿
𝜎~ 1 𝑠 𝐼𝑚 [𝑀(𝑡=0)] 2 ~ 𝑠 𝛼 0 −1
𝛼 0 −1<0 :𝑅𝑒𝑔𝑔𝑒𝑜𝑛 𝛼(0)≈0.5
𝛼 0 −1>0 :𝑃𝑜𝑚𝑒𝑟𝑜𝑛 𝛼 0 ≈1+𝜖
Regge description for cross sections over resonances
𝑀~ 𝑠 𝛼𝑡+𝛼(0)
Δ 0 (1232)
𝑁(1675)

4. Total and elastic cross sections
Existing models : Total reactions in the t-ch. helicity Reggeons with coupling strength (residues) fit to exp. data
Mathieu (2015), Nys (2018) from JPAC
Huang (2008)
𝑀 𝑥 ~ 𝛽 13 (𝑡) 𝛽 24 (𝑡) ℛ 𝑥 (𝑠,𝑡)
Present work : CEX and elastic reaction in the Reggeized Born terms with interaction Lagrangians and coupling constants common in other hadron reactions, e.g., photoproductions
𝑀 𝑥 ~ ℒ 𝑥𝐾𝐾 (0) Π 𝑥 ℒ 𝑥𝑁𝑁 (0) ℛ 𝑥 (𝑠,𝑡)

5. 𝜋𝑝 Cross sections at high energies
Total crs. section
Elastic crs. section
A 𝜋 ± 𝑝 =ℙ+ 𝑓 2 ∓𝜌
𝑀 𝜋 ± 𝑝 =𝜎+ 𝑓 2 +ℙ∓𝜔∓𝜌
𝜌 large enough to make difference
𝜌, 𝜔 small & make no difference,
V. Mathieu, et al., PRD92, (2015)

6. II. Reggeized Born term model
Assume s-channel helicity conserved
𝑥-Reggeon in t-ch.
𝑀 𝐵 = ℒ 𝑥𝜋𝜋 (0) Π 𝑥 𝑞−𝑘 𝑡− 𝑚 𝑥 2 ℒ 𝑥𝑁𝑁 (0) ⟶
𝑀 𝑅 = ℒ 𝑥𝜋𝜋 (0) Π 𝑥 (𝑞−𝑘) ℒ 𝑥𝑁𝑁 (0) ℛ 𝑥 (𝑠,𝑡)
reggeization
ℛ 𝑥 𝑠,𝑡 = 𝜋 𝛼 ′ 𝜉+ 𝑒 −𝑖𝜋𝛼 𝑡 Γ 𝛼 𝑡 +1−𝐽 𝑠𝑖𝑛𝜋𝛼(𝑡) ( 𝑠 𝑠 0 ) 𝛼 𝑡 −𝐽 ⟶
( 𝑡− 𝑚 𝑥 2 ) −1
trajectory
𝛼 𝑡 = 𝛼 ′ 𝑡+𝛼(0)
phase
(−1 ) 𝐽 + 𝑒 −𝑖𝜋𝛼 ; nEXD
1, 𝑒 −𝑖𝜋𝛼 ; EXD (𝜌, 𝑎 2 ), (𝜔, 𝑓 2 )
𝑥-𝜑 elastic Regge-cut
𝑀 𝑅 𝑐 = ℒ 𝑥𝜋𝜋 0 Π 𝑥 𝑞−𝑘 ℒ 𝑥𝑁𝑁 0 { ℛ 𝑥 𝑠,𝑡 + 𝐶 𝜑 𝑒 𝑑 𝜑 𝑡 𝑒 − 𝑖𝜋 𝛼 𝑐 2 ( 𝑠 𝑠 0 ) 𝛼 𝑐 −1 }
𝛼 𝑐 𝑡 = 𝛼 𝑥 ′ 𝛼 𝜑 ′ 𝛼 𝑥 ′ +𝛼 𝜑 ′ 𝑡+ 𝛼 𝑥 0 + 𝛼 𝜑 (0)

7. Quark-Pomeron coupling picture
Donnachie-Landshoff (DL) NPB244, , Laget NPA581,
- Pomeron couples to an individual quark in the hadron rather than the hadron as a whole.
- Cpling strength of Pomeron to a hadron is determined by the radius of hadron (FF)
Isoscalar photon of C=+1 →ℙ𝑞𝑞~ 𝐹 ℎ (𝑡) 𝛽 𝑞 𝛾 𝜇
𝑑𝜎(𝜋𝑝) 𝑑𝑡 = 1 4𝜋 𝑠 2 | 2 𝛽 𝑞 𝐹 𝜋 (−𝑖 𝛼 ′ 𝑠 ) 𝛼 ℙ 𝑡 (3 𝛽 𝑞 ′ 𝐹 1 ) | 2
Pichowsky PRD56,
- Current quark propagation from DSE of QCD 𝑠 𝑓 −1 (𝑙)=𝛾∙𝑙𝐴 𝑙 2 +𝐵( 𝑙 2 ) - quark-meson vertex BS amplitude Γ 𝜋 𝑘 =𝑖 𝛾 5 𝐶( 𝑘 2 ) Γ 𝜋 𝑘 =𝑖 𝛾 5 𝐶( 𝑘 2 )
- But, current quark propagation at high E ⇒ free quark propagation, and on-shell approximation for quark loops with hadron form factors are a good approximation

8. Off-shell quark loop (𝑙=−𝑞/2)
Soft Pomeron Exch.
Pseudoscalar cpl. at 𝜋𝑞𝑞 vertex
Cpl.const. from G-T relation at quark level
𝑓 𝜋𝑞𝑞 2 𝑚 𝑞 = 1 2 𝑓 𝜋 𝑔 𝐴
Γ 𝜋 (𝑘)=𝑖 𝑓 𝜋𝑞𝑞 𝛾 5 ,
Off-shell quark loop (𝑙=−𝑞/2)
Vector meson photo.
Pomeron amp.
𝜋𝑁 scatt.
𝑀 ℙ =𝑖 2𝐹 𝜋 𝑡 𝛽 𝑞 𝑓 𝜋𝑞𝑞 2 2 𝑚 𝜋 2 2𝑚 𝑞 2 − 𝑚 𝜋 2 2 −𝑡 𝐹 ℙ𝑞𝑞 𝑡 (3 𝐹 1 𝑡 𝛽 𝑞 ′ ) 𝑢 (𝑝′)𝛾∙ 𝑘+𝑞 𝑢 𝑝 ℛ ℙ (𝑠,𝑡)
Form Factors
𝐹 𝜋 𝑡 =( 1−𝑡/ Λ 2 ) −1
Pion EM form factor
𝐹 ℙ𝑞𝑞 𝑡 = 2𝜇 𝜇 𝑀 𝑞 2 −𝑡
𝜇 0 2 =1.1 𝐺𝑒𝑉 2 fixed
𝐹 1 𝑡
Nucleon isoscalar F.F. fixed

9. Singularity of Pomeron at low energy from quark loop
4 𝑚 𝑞 2 = 𝑚 𝜋 2 leads to a singularity, in principle, and the unwanted divergence near threshold significantly in KN case.
Pion form factor with cutoff Λ having energy dependence to control the range of suppression.
𝐹 𝜋 𝑡 =( 1−𝑡/ Λ 2 ) −𝑛
Λ 𝑊 = 𝑘 𝜇 (𝑊− 𝑊 𝑡ℎ )

10. III. CEX & Elastic Cross sections
𝜋 − 𝑝→ 𝜋 0 𝑛 charge exchange reaction
- Only 𝜌-exch. allowed
- DCS shows a dip at NWSZ of 𝛼 𝜌 𝑡 =0
- Dip-filling mechanism
- Interference with another 𝜌 for P
𝑀=− 2 [𝜌 775 +𝜌(1450)+𝜌-cuts]
𝜌-Reggeon 𝐼 𝐺 𝐽 𝑃𝐶 = 1 + ( 1 −− )
𝑀 𝜌 = 𝑔 𝜌𝜋𝜋 (𝑞+𝑘 ) 𝜇 (− 𝑔 𝜇𝜈 + 𝑄 𝜇 𝑄 𝜈 / 𝑚 𝜌 2 ){ 𝑔 𝜌𝑁𝑁 𝑣 𝛾 𝜈 + 𝑔 𝜌𝑁𝑁 𝑡 4𝑀 𝛾 𝜈 , 𝛾∙𝑄 } ℛ 𝜌 (𝑠,𝑡)
Elastic Regge cuts 𝜌- 𝑓 2 & 𝜌-𝕡
𝑀 𝑅 𝑐 = ℒ 𝑥𝜋𝜋 0 Π 𝑥 𝑄 ℒ 𝑥𝑁𝑁 0 { ℛ 𝑥 𝑠,𝑡 + 𝐶 𝜑 𝑒 𝑑 𝜑 𝑡 𝑒 − 𝑖𝜋 𝛼 𝑐 2 ( 𝑠 𝑠 0 ) 𝛼 𝑐 −1 }
𝛼 𝑓 𝑡 =0.9𝑡+0.53,
𝛼 𝕡 𝑡 =0.9𝑡+0.46

11. Good to test 𝜌-trajectory & cpl. const. 𝜌-trajectory
𝛼 𝜌 𝑡 =0.9𝑡+0.46
𝛼 𝜌 𝑡 =0.8𝑡+0.55
𝜌-cpl.const.
𝑔 𝜌𝑁𝑁 𝑣 =2.6, 𝜅 𝜌 =3.7 (VMD)
𝑔 𝜌𝑁𝑁 𝑣 =3, 𝜅 𝜌 =6.2 (NN)
𝜌- 𝑓 2 & 𝜌-𝕡 cuts
Another 𝜌 for polarization
𝜌 1450 , (1 −− )
Trajectory from Rel. quark model
D. Ebert, et al., PRD79, (2018)
𝛼 𝜌 𝑡 =𝑡−1.23
cpl.const. from the fit
𝐺 𝜌(1450)𝑁𝑁 𝑣 =40, 𝐺 𝜌(1450)𝑁𝑁 𝑡 =−75

13. - Energy-dependence of crs. sec. in good agreement with data
- Background contribution to 𝑁 ∗ study 2 3
F. Huang, et al., EPJA40, 77 (2009)

14. 𝜋 ± 𝑝→ 𝜋 ± 𝑝 Elastic Scattering
- Mesons of natural parity meson with 2𝜋 decay
𝑀 𝜋 ± 𝑝 =𝜎±𝜌+ 𝑓 2 ∓𝜔 +ℙ
Scalar meson 𝜎 ( 0 ++ ) - lightest meson at low energy - Uncertain due to large decay width, 𝑚 𝜎 ~ Γ 𝜎 - Treated as a self-energy term
𝜎-Reggeon
𝛼 𝜎 𝑡 =0.7(𝑡− 𝑚 𝜎 2 ) phase=1/2(1+ 𝑒 −𝑖𝜋𝛼 )
chiral partner 𝑔 𝜎𝑁𝑁 ≈ 𝑔 𝜋𝑁𝑁

15. Significant role of 𝑓 2 Vector meson 𝜔(782) 0 − ( 1 −− )
B.-G.Yu, et al., PLB701, 332 (2011)
Vector meson 𝜔(782) 0 − ( 1 −− )
𝛼 𝜔 𝑡 =0.9𝑡+0.44
Tensor meson 𝑓 2 (1275) 0 + ( 2 ++ )
𝛼 𝑓 𝑡 =0.9𝑡+0.53
Significant role of 𝑓 2
Determination of cpl. const. 𝑔 𝑓 2 𝑁𝑁 and phase
(𝜔− 𝑓 2 ) EXD pair ⇒ constant phase “1” for both reactions
𝑓 2 -Reggeon

16. Coupling constants, etc …
𝑔 𝜑𝜋𝜋 estimated from exp. decay width
𝛽 𝑢 = 𝛽 𝑑 =2.07 𝐺𝑒𝑉 −1 , 𝛽 𝑠 =1.6 𝐺𝑒𝑉 −1
𝑓 𝜋𝑞𝑞 =2.65 obtained from 𝑔 𝐴 =1.25, 𝑓 𝜋 =93.1 MeV

17. 𝜋𝑝 Elastic cross sections
Determine Pomeron trajectory from data at p=100, 200 GeV/c
Isocalar Pomeron only at P=100, 200 GeV/c in both reactions.
Slope 0.12 is flatter than 0.25 from photoproductions, Total crs. sec.
Data at lower momenta insensitive to cutoff mass 𝜇
𝛼 ℙ 𝑡 =0.12𝑡+1.06
(𝛼 ℙ 𝑡 =0.25𝑡+1.08)

18. 𝜋𝑝 Elastic cross sections
- Dominance of 𝑓 2 and Pomeron, and possibly 𝜎 near threshold.
- Role of self-energy term for 𝜎
- 𝜌, 𝜔 minor roles.
- Polarization sensitive to 𝑓 2 and Pomeron
- Mirror symmetry well-reproduced

19. Summary
- Roles of meson exch. are investigated in 𝜋𝑁 CEX and elastic reactions up to 250 GeV/c.
- Unique roles of vector meson 𝜌(775) in the 𝜋𝑁 CEX are studied.
- Soft Pomeron in the quark-Pomeron coupling picture is constructed for 𝜋𝑁 elastic scatterings and applied successfully only with one parameter 𝜇.
- Dominance of isoscalar channel, 𝑓 2 (1275) and Pomeron over 𝑃 𝐿 ≈2 GeV/c in 𝜋𝑁 elastic scatterings.
- Polarizations for 𝜋𝑁 reactions are well reproduced.
- Present Model offers a useful tool to explore 12 GeV region and provides mesonic background contributions good enough to analyze baryon resonances while communicating with Lagrangian formulation.

20. Backup

21. I. Overview : features of cross sections
𝜎 𝑇 (𝜋𝑁→𝑎𝑙𝑙)= 𝜎 𝑒𝑙 + 𝜎 𝑖𝑛𝑒𝑙
Δ ++ (1232)
𝜎 𝑒𝑙 =𝜎(𝜋𝑁→𝜋𝑁)
𝜎 𝑖𝑛𝑒𝑙 = 𝜎 𝐷 + 𝜎 𝐷𝐷 +…
Δ ++ (1905)
By optical theorem, total and elastic crs.sec. are related as
𝜎 𝑇 = 1 2𝑘𝑊 𝐼𝑚[ 𝑀 𝑒𝑙 (𝑠,0)]
By Pomeranchuck theorem, particle and antiparticle crs. sec. are equal at high E
𝜎 𝑇 𝑎𝑏 ≈ 𝜎 𝑇 ( 𝑎 𝑏) as 𝑠→∞
Δ 0 (1232)
Baryon resonances in low 𝑃 𝐿 , nonresonant diffraction at high 𝑃 𝐿
𝑁(1675)

22. Overview : Features of cross sections
Energy-dep. from Regge Theory
Amp.
𝑀~𝑓 𝑡 𝑠 𝛼 𝑡
DCS
𝑑𝜎 𝑑𝑡 ~ 1 𝑠 2 |𝑀| 2 ~ 𝑠 2𝛼 𝑡 −2
TCS
𝜎~ 1 𝑠 𝐼𝑚 [𝑀(𝑡=0)] 2 ~ 𝑠 𝛼 0 −1
𝛼 0 −1<0 :𝑅𝑒𝑔𝑔𝑒𝑜𝑛 𝛼(0)≈0.5
𝛼 0 −1>0 :𝑃𝑜𝑚𝑒𝑟𝑜𝑛 𝛼 0 ≈1+𝜖
Exch. of trajectory 𝛼 𝑡 = 𝛼 ′ 𝑡+𝛼(0)
BDS
No resonances in 𝜎 𝐾 + 𝑝 repulsive s-wave phase shift below 𝑃 𝐿 ~1 GeV/c.
𝐾 𝑁 B.S. in the 𝜋Σ continuum in
𝜎 𝐾 − 𝑝 subthreshold.
cpled ch.

23. 𝜋𝑝 Cross sections at high energies
Total crs. section
Elastic crs. section
𝑀 𝜋 ± 𝑝 =𝜎+ 𝑓 2 +ℙ∓𝜔∓𝜌
A 𝜋 ± 𝑝 =ℙ+ 𝑓 2 ∓𝜌
𝜌, 𝜔 small & make no difference,
Slope of Pomeron 𝛼 𝕡 𝑡 different
𝜌 large enough to make difference
Pomeron for elastic crs.sec.
Pomeron for tot. crs. sec.

24. 𝐾𝑝 Cross sections at high energies
Total crs. section
Elastic crs. section
𝐴( 𝐾 ± 𝑝)=∓𝜌±𝜔+ 𝑓 2 + 𝑎 2 +ℙ
𝑀 𝐾 ± 𝑝 = 𝑓 0 + 𝑎 0 ∓𝜙+ 𝑓 2 + 𝑎 2 +ℙ
𝜌, 𝜔 large enough to make difference
𝜑 small & make no difference
Slope of Pomeron 𝛼 𝕡 𝑡 different

25. KN CEX & Elastic reactions
Elastic amp.
𝑀 𝐾 ± 𝑝 = 𝑓 0 + 𝑎 0 ∓𝜙+ 𝑓 2 + 𝑎 2 +ℙ
𝑀 𝐾 ± 𝑛 = 𝑓 0 − 𝑎 0 ∓𝜙+ 𝑓 2 − 𝑎 2 +ℙ
Isospin relations between amp.
𝑀 𝐾 − 𝑝→ 𝐾 0 𝑛 =𝑀 𝐾 − 𝑝→ 𝐾 − 𝑝 −𝑀( 𝐾 − 𝑛→ 𝐾 − 𝑛)
𝑀 𝐾 + 𝑛→ 𝐾 0 𝑝 =𝑀 𝐾 + 𝑝→ 𝐾 + 𝑝 −𝑀( 𝐾 + 𝑛→ 𝐾 + 𝑛)
CEX amp.
𝑀 𝐾 − 𝑝→ 𝐾 0 𝑛 =𝑀 𝐾 + 𝑛→ 𝐾 0 𝑝 =2( 𝑎 0 + 𝑎 2 )
- Only isovector scalar meson 𝑎 and tensor meson 𝑎 2 (1320) exch.
- Good for testing tensor meson 𝑎 2 (1320)
- Data show no evidence of dip: complex phases for scalar & tensor mesons

26. 𝐾 − 𝑝→ 𝐾 0 𝑛, 𝐾 + 𝑛→ 𝐾 0 𝑝 CEX
- Dominance of tensor meson 𝑎 2 (1320) over 𝑃 𝐿 ≈2 GeV/c
- Consistency with data obtained by 3x 𝑔 𝑎 0 𝑁𝑁 for first 4 data points
- 𝑔 𝑎 0 𝑁𝑁 =21.7 from 𝛾𝑝→𝜙𝑝
- Data show equality of both processes: same amp. for both reactions

27. 𝐾 ± 𝑝→ 𝐾 ± 𝑝 Elastic reactions
- Pomeron exch. universial with 𝜋𝑁 reaction.
- 𝑓 𝐾𝑞𝑞 =0.998 fit to data
- Tips from ratio at 𝑃 𝐿 =250 GeV/c
𝜎 𝑒𝑙 ( 𝐾 + 𝑝) 𝜎 𝑒𝑙 ( 𝜋 + 𝑝) = ≈ | 𝑓 𝐾𝑞𝑞 2 𝑚 𝐾 2 𝛽 𝑠 | 2 | 𝑓 𝜋𝑞𝑞 2 𝑚 𝜋 2 𝛽 𝑑 | 2
𝑓 𝐾𝑞𝑞 ≈0.3 𝑓 𝜋𝑞𝑞 ≈0.8
𝛼 ℙ 𝑡 =0.12𝑡+1.06

28. 𝐾 + 𝑝 elastic cross section at low energy
Repulsive s-wave phase shift
𝑀 𝑛𝑢𝑐𝑙 = 8𝜋𝑊 4𝑀𝑀′ 1 𝑘 𝑒 𝑖 𝛿 𝑙 𝑠𝑖𝑛 𝛿 𝑙
Parameterize s-wave:
𝛿 𝑙 𝑝 =𝐴 𝑝 2 +𝐵𝑝+𝐶
match point
𝛿 𝑙 𝑝 =𝐷 𝑒 −(𝑝− 𝑝 0 )/ 𝑚 0
𝑝 0 =1.5 𝐺𝑒𝑉/𝑐
s-wave + Pomeron in the absence of resonance
𝐹 𝐾 𝑡 =( 1−𝑡/ Λ 2 ) −1
Λ 𝑊 = 𝑘 (𝑊− 𝑊 𝑡ℎ )

29. 𝐾𝑝 Elastic total cross sections
- Repulsive s-wave phase shift below 1 GeV/c in 𝐾 + 𝑝 elastic process.
- Complicated reaction mechanism below 1 GeV/c in 𝐾 − 𝑝 elastic process.
- 𝜙 vector meson contribution negligible: Data show equality of both processes 𝐾 + 𝑝 and 𝐾 − 𝑝 except for low momentum region
- Dominance of 𝑓 2 (1275) and Pomeron over 𝑃 𝐿 ≈2 GeV/c