Post on 18-Aug-2018
Photoproduction of pseudoscalar mesonsin isobar and Regge formalism
HASPECT meeting, Krakow, May 30 – June 1, 2016
Petr Bydzovsky, Nuclear Physics Institute, Rez, Czech Republic
Outline:
Introduction – description of electro- and photoproduction
Isobar model – coments on formalism, reaction mechanism
in π, η, K photoproduction
Regge and Regge-plus-resonance models
Results for kaon photoproduction
P. Bydzovsky (NPI Rez) Photoproduction of pseudoscalar mesons Krakow,May 30 -June 1,2016 1 / 23
Description of the electromagnetic production
electroproduction of pseudoscalar mesons (M = π, η, K) on nucleons (N)
e + N −→ e′ +M+ B
electromagnetic interaction is well known and the coupling constant
is small: α = e2
2π
.= 1
137 1 → a one-photon approximation is justified
the leptonic and hadronic parts are separated in opa
Mfi = u(p′e, λ′)(−ieγµ) u(pe, λ)
−gµνq2J ν(pN, σ, q, ε, pB, σ
′, pM)(−ie)
the transition current J ν describes photoproduction by virtual photons
γ(v) + N −→ M+ B
with q2 < 0, a longitudinal polarization
photoproduction (q2=0) is not the limit case (q2→ 0) of electroproduction
P. Bydzovsky (NPI Rez) Photoproduction of pseudoscalar mesons Krakow,May 30 -June 1,2016 2 / 23
kinematics in the laboratory frame: Ee, E′e, θe, θM, ΦM
the unpolarized cross section: ∝ 14 m2
e
∑λλ′σσ′ |Mfi |2 = 1
4e4
q4 Lsµν W νµ
d5σ
dE′e dΩ′e dΩM= Γ
[dσT
dΩM+ ε
dσL
dΩM+ ε
dσTT
dΩMcos 2ΦM+
√2ε(1+ε)
dσTL
dΩMcos ΦM
]
virtual-photon flux: Γ(pe, p′e, θe) polarization: ε =
(1 + 2 q2
−q2 tan2 θe
2
)−1
dσx
dΩM(Q2, s, t), Q2 = −q2, s = (q + pN)2, t = (q− pM)2
laboratory frame (~pN = 0):
Φ
θeθ
p
p'γ
q
p
pB
Scattering (Leptonic) Plane
Reaction (Hadronic) Plane
z
xy
e
e
M
M
M
^
^
P. Bydzovsky (NPI Rez) Photoproduction of pseudoscalar mesons Krakow,May 30 -June 1,2016 3 / 23
Photoproduction by real photons (q2=0)
γ + N −→ M+ B in the resonance region√
s < 2.6 GeV.
Which degrees of freedom are relevant: quarks and gluons ↔ hadrons?
non perturbative regime of QCD → effective description via baryons andmesons completed with form factors (short-distance physics)
effective hadronic Lagrangian in lowest order → tree-level approximation
many opened channels, e.g. π (≈ 1.07 GeV), η (≈ 1.49 GeV), K (≈ 1.61 GeV)
→ coupling of the channels via final-state meson-baryon interactions (FSI)
multi-channel ↔ single-channel analysis ?
unitary (coupled-channel) approach – in some cases FSI is not well known
γ + p −→ π0 + p −→ K+ + Λ
−→ η + p −→ K+ + Λ
−→ K+ + Λ −→ K+ + Λ
→ the single-channel approach – FSI included in effective parameters
P. Bydzovsky (NPI Rez) Photoproduction of pseudoscalar mesons Krakow,May 30 -June 1,2016 4 / 23
Isobar model – phenomenological description in resonance region
single-channel approach
I violation of unitarity – can be partially restored by means of energydependent widths of nucleon resonances in the propagator
I effective coupling constants (FSI)
effective hadronic Lagrangian: L = Lo + Lint
I baryon, meson and photon fieldsI possible resonances in the intermediate states – “missing resonances”I couplings of the fields in Lint
I short-distance physics (frozen degrees of freedom) → form factorsI gauge invariance: PV coupling, hadron form factors → a contact termI tree-level approximation → s-, t-, and u-channel Feynman diagramsI free parameters determined in a data analysis
reaction mechanism – invariant amplitude: Mfi =Mbgr +Mres
I Mbgr : non resonant contributions: Born, t- and u-channel diagramsI Mres : resonant contributions: N∗ and ∆ s-channel diagrams
P. Bydzovsky (NPI Rez) Photoproduction of pseudoscalar mesons Krakow,May 30 -June 1,2016 5 / 23
Dominant resonances in pion and eta photoproduction
0.2 0.3 0.4 0.5 0.6 0.7 0.8Eγ [GeV]
0
5
10
15
20
25
30
dσ /
dΩ [
µb/s
r]
Born termsBorn + ∆(M1)
γ + n --> π- + p
θ = 90o
Mbgr – Born terms: exchanges ofn in s channel, p in u channel,and π− in t channel
Mres is dominated by ∆(1232) 32
+
with a branching ratio to Nπ ≈ 100%
γN −→ ∆ −→ πN
0.7 0.8 0.9 1 1.1 1.2Eγ [GeV]
0
0.4
0.8
1.2
1.6
dσ /
dΩ [
µb/s
r]
Born termsBorn + S11(1535)
γ + p --> η + p
θ = 90o
Mbgr – Born terms: exchanges ofp in s and u channels;– other contributions: ρ, ω in t channel
Mres : N∗(1535) 12
−(42% to Nη)
other states, e.g.: N∗(1710) 12
+(10-30%)
N∗(1650) 12
−(5-15%), N∗(1720) 3
2
+(4%)
P. Bydzovsky (NPI Rez) Photoproduction of pseudoscalar mesons Krakow,May 30 -June 1,2016 6 / 23
No dominant resonance in kaon photoproduction
many N∗ with a reasonable branching ratio to K Λ can contribute
Mres in the Kaon-MAID model: S11(1650) (3–11%), P11(1710) (5–25%),P13(1720) (1–15%), D13(1895) (missing resonance – not seen in π and η channels)
Born terms (p, Λ, Σ0, and K+) suppressed by the hadron form factors
Mres in the BS1 model: S11(1535), S11(1650), F15(1680), P13(1720),F15(1860), D13(1875), P13(1900), F15(2000)Mbgr : Born terms, K∗(892), K1(1270), Λ(1520), Σ(1660), Σ(1750),Λ(1800), Λ(1890), Σ(1940);
1.2 1.6 2 2.4Eγ [GeV]
0
0.05
0.1
0.15
0.2
dσ /
dΩ [
µb/s
r]
SAPHIRCLASKaon-MAIDBorn + K* + K1
γ + p --> K+ + Λ
θ = 90o
1,2 1,6 2 2,4Eγ [GeV]
0
0,05
0,1
0,15
0,2
dσ /
dΩ [
µb/s
r]
full BS1 modelbackground
γ + p --> K+ + Λ
θ = 90o
P. Bydzovsky (NPI Rez) Photoproduction of pseudoscalar mesons Krakow,May 30 -June 1,2016 7 / 23
Consistent description of higher-spin resonances
contributions of resonances with spin> 12 important: ∆(1232), N∗(1720),..
the Rarita Schwinger propagator for spin 3/2
Sµν(p) =/p + m
p2 −m2P3/2µν −
2
3m2(/p + m)P1/2
22,µν +1√3m
(P1/212,µν + P1/2
21,µν)
allows nonphysical contributions of the lower spin components
the nonphysical contributions can be removed by a suitable form of Lint
[V. Pascalutsa, P.R.D 58 (1998) 096002; T. Vrancx et al, P.R.C 84 (2011) 045201]
invariance of Lint under local U(1) gauge transformation of the R.-S. field
⇒ transverse interaction vertexes:
V Sµ pµ = V EM
µ pµ = 0
⇒ removes all nonphysical contributions:
V Sµ P
1/2,µνij V EM
ν = 0N B
MγS
µν(p)
VEMµ V
Sµ
P. Bydzovsky (NPI Rez) Photoproduction of pseudoscalar mesons Krakow,May 30 -June 1,2016 8 / 23
the gauge-invariant coupling results in high momentum-power dependence;it is good: regularizes the resonance contributionbut it has also a drawback: too big growth of the cross section⇒ strong form factors (multidipole, Gauss) are introduced
P. Bydzovsky (NPI Rez) Photoproduction of pseudoscalar mesons Krakow,May 30 -June 1,2016 9 / 23
Using symmetries to constrain models
SU(3)f symmetry can be used to determin (constrain) some couplingconstants: SU(3)-invariant Lagrangian for baryon and meson octets
L(BBM) = ga Sp([BB]M) + gs Sp(BBM)
[BB] and BB is antisymmetric and symmetric part, respectively.
Parameters: gs − ga =√
2 gπNN (well known) and α ≡ gs
gs−ga
from SU(6) α =35
or experimental value: 0.644± 0.009.
⇒ gKΛN = − 1√3
gπNN (3− α), gKΣN = −gπNN (1− 2α), ...
If we take 20% breaking of the SU(3) symmetry → constraints
−4.4 ≤ gKΛN√4π≤ −3 0.8 ≤ gKΣN√
4π≤ 1.3
crossing symmetry: description of processes in crossed channels,
e.g. Γ(K−p −→ γ Λ) in K+Λ photoproduction: s ↔ u
P. Bydzovsky (NPI Rez) Photoproduction of pseudoscalar mesons Krakow,May 30 -June 1,2016 10 / 23
Isobar models
MAID – the interactive Web page of Mainz University(www.kph.kph.uni-mainz.de/MAID/)
I models for photo- and electroproduction on NI pion: unitary isobar modelI eta: isobar model – Breit-Wigner forms for N∗ contributions;
effective Lagrangian approach for the non resonant part of amplitudeI kaon: isobar model (Kaon-MAID) – effective Lagrangian approach;
hadron form factors; no hyperon exchanges in the u channel
other isobar models
I eta photoproduction: Benmerrouche at al (95)– an effective Lagrangian; Born terms and N∗(1/2,3/2), ρ, and ωexchanges; hadron form factors
I kaon production: Adelseck-Saghai (90), Williams-Ji-Cotanch (92),Saclay-Lyon (96), Gent isobar (01), Maxwell (07), Skoupil-B. (16)
P. Bydzovsky (NPI Rez) Photoproduction of pseudoscalar mesons Krakow,May 30 -June 1,2016 11 / 23
Regge theory for γ + N −→ M+ B
high-energy approach – economically takes into account “infinite” numberof exchanges of high-spin and large-mass particles in the t (u) channel
the main idea: the angular momentum → a complex continuous variable
Regge trajectory – a family of hadrons with the same internal quantumnumbers, except the spin → a linear dependence between the spin and m2
α(t) = α0 + α1t, for t = m2, α(t) equals the particle spin
0 1 2 3 4 5
t [GeV2]
0
1
2
3
4
5
α(t)
ρ(770)ω(782)
a2(1320)
b1(1235)
π
π2(1670)
ρ3(1690)
a4(2040)
ρ5(2350)
f2(1270)
π, ρ, ω trajectories f4(2050)
ω3(1670)
0 1 2 3 4 5 6 7
t [GeV2]
0
1
2
3
4
5
6
α(t)
K*(892)
K*2(1430)
K*3(1780)
K*4(2045)
K*5(2380)
K1(1270)
K+
K2(1770)
K3(2320)
K4(2500)
K and K*
trajectories
P. Bydzovsky (NPI Rez) Photoproduction of pseudoscalar mesons Krakow,May 30 -June 1,2016 12 / 23
Regge theory
two signatures (ζ = ±1) – a mathematical requirement due to convergenceof a contour integral ⇒ two amplitudes, propagators, trajectories
assumption of strong degeneracy: trajectories for even and odd spinscoincide and residua of both signatures are equal, β+(s, t) = β−(s, t)
validity is seen in the data: dσ/dt should be a smooth function of energy
if dσ/dt reveals dips ⇒ the assumption is not valid
the Regge propagator for degenerate trajectories
PRegge(s, t) =(s/s0)α(t)
sinπ α(t)
π α′
Γ(1 + α(t))
1
e−iπα(t)
undetermined relative phase makes constant or rotating phase→ have to be fixed in a data analysis
in the lowest mass pole, t → m20, PRegge(s, t) −→ 1
t−m20
P. Bydzovsky (NPI Rez) Photoproduction of pseudoscalar mesons Krakow,May 30 -June 1,2016 13 / 23
Regge theory
the amplitude M = PRegge(s, t) β(s, t)
determination of the residue β(s, t): near the lowest pole, t → m20
PRegge(s, t)× β(s, t) −→ β(s, t)
t −m20
⇐⇒ t−channel Feynman diagram
the “first materialisation” of the trajectory
a formal approach – “reggeisation”: the Feynman propagatorin the t-channel exchange is substituted with the Regge propagator
gauge invariance: the π± and K+ t-channel exchanges are not gaugeinvariant, Mµ
t qµ 6= 0 but together with the s- or u-channel exchangesthe invariance is restored: (Mµ
t +Mµs(u)) qµ = 0
⇒ the s- or u-channel proton exchange have to be added
β(s, t)
t −m20
⇐⇒ (Mµt (meson) +Mµ
s(u)(proton) ) εµ
P. Bydzovsky (NPI Rez) Photoproduction of pseudoscalar mesons Krakow,May 30 -June 1,2016 14 / 23
Regge-plus-resonance model – hybrid Regge-isobar approach
for kaon photoproduction [L.De Cruz et al, Phys.Rev.C86 (2012) 015212]
Invariant amplitude:
M =Mbgr (Regge) +Mres(isobar)
the background part – exchanges of degenerate K+ and K∗ trajectories
Mbgr = (MKFeyn +Mp,elec
Feyn ) (t −m2K )× PK
Regge +MK∗
Feyn (t −m2K∗)× PK∗
Regge
Regge propagators with the rotating phase; materialisation of K+ and K∗
⇒ only 3 parameters of Mbgr fixed by high-energy data (W>2.6 GeV)
Mres : s-channel Feynman diagrams for N∗ with multidipole hadron f. f.
P. Bydzovsky (NPI Rez) Photoproduction of pseudoscalar mesons Krakow,May 30 -June 1,2016 15 / 23
Regge and Regge-plus-resonance models
StrangeCalc – the interactive Web page of Gent University(http://rprmodel.ugent.be/calc/)
I models for photo- and electroproduction on NI pionsI kaons: RPR-2007, RPR-2011
Gent Regge model for kaon photoproduction – Regge2011;fit to CLAS data above the resonance region (W> 2.5 GeV)
Guidal et al (97) – Regge model for pion and kaon photoproduction;pion: π and ρ trajectories for charged-pion production,
ρ, ω and b1(1235) trajectories for neutral-pion productionkaon: Regge97 model, K+ and K∗ trajectories,
SU(3)f symmetry for gKΛN and gKΣN ;fit to SLAC high-energy data
Wen-Tai Chiang and Shin Nan Yang (03) – Regge modelfor eta and eta’ photoproduction – ρ and ω trajectories
P. Bydzovsky (NPI Rez) Photoproduction of pseudoscalar mesons Krakow,May 30 -June 1,2016 16 / 23
Results for kaon photoproduction – isobar and Regge models
0
0,1
0,2
0,3
0,4
0,5
CLAS 2010CLAS 2005
0
0,1
0,2
0,3
0,4
dσ /
dΩ [
µb/s
r] Saclay-LyonKaon-MAIDBS1RPR fit
0 30 60 90 120 150 180θ
K
c.m. [deg]
0
0,1
0,2
0,3
0,4
W = 1.805 GeV
W = 2.005 GeV
W = 2.205 GeV
angular dependence of the cross section in p(γ, Κ+)Λ
P. Bydzovsky (NPI Rez) Photoproduction of pseudoscalar mesons Krakow,May 30 -June 1,2016 17 / 23
0
0,1
0,2
0,3
0,4
0,5
LEPS (0.85)
Saclay-LyonKaon-MAIDBS1RPR fit
1,6 1,8 2 2,2 2,4W [GeV]
0
0,1
0,2
dσ /
dΩ [
µb/s
r]
CLAS 2010CLAS 2005Adelseck-Saghai
1,6 1,8 2 2,2 2,4
cos θK
c.m. = 0.8 cos θ
K
c.m. = 0.5
cos θK
c.m. = 0.2
cos θK
c.m. = -0.4
energy dependence of the cross section in p(γ, K+)Λ
P. Bydzovsky (NPI Rez) Photoproduction of pseudoscalar mesons Krakow,May 30 -June 1,2016 18 / 23
Very small kaon angles
lack of very-forward kaon-angle data (CLAS: −0.9 < cos θK < 0.9)
models are not well determined below θK ≈ 30
this region is crucial for the calculations of hypernucelus-production crosssections (in DWIA, contributions from larger kaon angles are suppressedby nucleus-hypernucleus transition form factors)
0 30 60 90 120 150 180
θK
c.m. [deg]
0
0,1
0,2
0,3
0,4
0,5
0,6
dσ /
dΩ
[µb/
sr]
Brown72BS1BS2E94-107CLASLEPSSAPHIRRPR-2007RPR-1SLARPR-2H2
p(γ, K+)Λ
W = 2.24 GeV
Elab
γ = 2.21 GeV
DWIA 1 1,5 2 2,5 3
Eγlab
[GeV]
0
0,2
0,4
0,6
0,8
1
dσ /
dΩ [
µb/s
r]
BleckmannBrownE94-107SLASLKMH2RPR-1GiessenBS2BS1
θK
c.m. = 6
op(γ, K+)Λ
P. Bydzovsky (NPI Rez) Photoproduction of pseudoscalar mesons Krakow,May 30 -June 1,2016 19 / 23
Angular dependence of the hypernucleus cross section
electroproduction of 16NΛ (Elabγ = 2.21 GeV, θlab
e = 6o) as predictedby Kaon-MAID and Saclay-Lyon models → factor 10 different magnitudes
generally a steeply decreasing dependence, the slope depends on the spin (J)
the angular behaviour depends also on the spin structure of the elementaryamplitude at small θK → more detailed information on the amplitude
6 7 8 9 10 11 12 13
θKe
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
nb
/sr2 /G
eV
16O(e,e’K+)16NΛ Ei=3.654 GeV, Ef=1.445 GeV, θe=6
E=0.03 MeV, J=1- state SLA1
KMAID 10x
6 7 8 9 10 11 12 13
θKe
0.0
0.5
1.0
1.5
2.0
2.5
3.0
nb
/sr2 /G
eV
16O(e,e’K+)16NΛ Ei=3.654 GeV, Ef=1.445 GeV, θe=6
J=1+, 2+ states at E approx. 11 MeV
J=2+
J=1+
SLA1
KMAID 10x
P. Bydzovsky (NPI Rez) Photoproduction of pseudoscalar mesons Krakow,May 30 -June 1,2016 20 / 23
The very forward angular dependence above the resonance region
inconsistency of CLAS and SLAC data
0
0.1
0.2
0.3
0.4
dσ
/dΩ
[µ
b/s
r]
Regge97
RPRRegge2011
0 20 40 60 80θ
K
0
0.1
0.2
0.3
0.4
dσ
/dΩ
[µ
b/s
r]
0 20 40 60 80θ
K
0 20 40 60 80θ
K
0 20 40 60 80θ
K
CLAS09 CLAS09 CLAS09
SLAC69 SLAC69 SLAC69 SLAC69
Elab
γ = 2.19 GeV E
lab
γ = 2.90 GeV
Elab
γ = 5 GeV E
lab
γ = 8 GeV E
lab
γ = 11 GeV E
lab
γ = 16 GeV
Elab
γ = 3.81 GeV
CLAS09
Elab
γ = 1.74 GeV
W = 2.04 GeV
W = 2.24 GeV
W = 2.52 GeV W = 2.84 GeV
W = 3.20 GeV W = 3.99 GeV W = 4.64 GeV W = 5.56 GeV
P. Bydzovsky (NPI Rez) Photoproduction of pseudoscalar mesons Krakow,May 30 -June 1,2016 21 / 23
Summary
in the resonance region, photoproduction of π, η, and K mesons can besuccessfully described by an effective Lagrangian on tree-level (isobar model);
data above the resonance region reveal a smooth energy dependence andcan be described using the Regge formalism;
a hybrid isobar-Regge model can be used to describe data both in theresonance region and above this region, e.g., up to 16 GeV in p(γ,K)Λ;
predictions of isobar and Regge-plus-resonance models for the cross sectionsin p(γ,K)Λ still significantly differ for small kaon angles due to lack of data;
the ample world data (CLAS, SAPHIR, LEPS,..) on p(γ,K)Λ cover mainlythe region θK > 20 which is not enough to fully determine the angulardependence of the elementary cross section;
new good quality data at very small θK (and Q2) are needed to testthe models which then will provide more reliable predictions of the crosssections in electroproduction of hypernuclei;
P. Bydzovsky (NPI Rez) Photoproduction of pseudoscalar mesons Krakow,May 30 -June 1,2016 22 / 23
Thank you
P. Bydzovsky (NPI Rez) Photoproduction of pseudoscalar mesons Krakow,May 30 -June 1,2016 23 / 23