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

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

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)

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)

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, 074004 (2015)

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)

Quark-Pomeron coupling picture Donnachie-Landshoff (DL) NPB244, 322 1984, Laget NPA581, 397 1995 - 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, 1644 1997 - 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

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 3 5 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 0 2 2𝜇 0 2 + 𝑀 𝑞 2 −𝑡 𝜇 0 2 =1.1 𝐺𝑒𝑉 2 fixed 𝐹 1 𝑡 Nucleon isoscalar F.F. fixed

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 )

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

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 + (1 −− ) Trajectory from Rel. quark model D. Ebert, et al., PRD79, 114029 (2018) 𝛼 𝜌 𝑡 =𝑡−1.23 cpl.const. from the fit 𝐺 𝜌(1450)𝑁𝑁 𝑣 =40, 𝐺 𝜌(1450)𝑁𝑁 𝑡 =−75

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

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 500 0 + ( 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

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

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

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)

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

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.

Backup

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)

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.

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.

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

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 0 980 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

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

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) = 2.76 3.3 ≈ | 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

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 493.67 (𝑊− 𝑊 𝑡ℎ )

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