SLIDE 1 Pion scattering and electro-production on nucleons in the resonance region in chiral quark models
München | 16 June 2011
- M. Fiolhais, P. Alberto, U of Coimbra
- S. Širca, U of Ljubljana
P11: EPJA 38 (2008) 271 P11: EPJA 42 (2009) 185 S11: EPJA 47 (2011) 61
1
SLIDE 2 Motivation for study of P11 and S11 resonances
- Large width of **** P11(1440) “Roper”; existence of *** P11(1710) unclear;
difficult to identify directly (in cross-sections)
- Atypical behaviour of ImTπN in J =I =1/2 partial wave
- Level ordering (parity inversion) of P11(1440) wrt. S11(1535) on Lattice
- Many competing explanations of the Roper in models, e.g.
q3g hybrid ⊲ Li, Burkert PRD 46 (1992) 70 qqqqq admixtures ⊲ Li, Riska PRC 74 (2006) 015202 dynamical generation by N +σ ⊲ Krehl++ PRC 62 (2000) 025207 ⊲ Döring++ NPA 829 (2009) 170
- Two negative-parity resonances:
resonance M [MeV] Γ [MeV] decays S11(1535) 1535 150 πN 35–55 % , ηN 45–60 % 2πN < 10 % , (πR) S11(1650) 1655 165 πN 60–95 % , ηN 3–10 % , KΛ 3–11 % 2πN 10–20 % , ρ 4–12 % , ∆ 1–7 %
2
SLIDE 3 Present work
- A coupled-channels approach that includes many-body states
- f quarks and mesons in the scattering formalism
- Calculate scattering and electro-production amplitudes
within the same framework
- Investigate whether quark+meson description is sufficient
i.e. no exotic degrees of freedom involved
- Baryons treated as composite particles
→ coupling constants and cut-offs of form-factors computed from the underlying model, not fitted → smaller number of free parameters
- Physical resonances appear as linear combinations of bare resonances
- Bare quark-meson and quark-photon vertices are strongly modified
by meson loops and mixing of resonances
- K-matrix real & symmetric → S-matrix unitary
3
SLIDE 4 Reminder: ∆(1232) in quark models with pion cloud
Helicity and electro-production amplitudes for γ∗N → ∆(1232) → Nπ A1/2 E1+/M1+ S1+/M1+
- M1+ is (∼ 50 % pion cloud) + (∼ 50 % quarks)
- E1+ is (∼ 100 % pion cloud)
Golli++ PLB 373 (1996) 229
4
SLIDE 5 Coupled-channel K-matrix formalism
Our model
Golli, Širca / EPJA 38 (2008) 271 Golli, Širca, Fiolhais / EPJA 42 (2009) 185
The meson field linearly couples to the quark core; no meson self-interaction
H = Hquark +
lmt(k)almt(k) +
- V lmt(k)almt(k) + V lmt(k)† a†
lmt(k)
- V lmt(k) induces also radial excitations of the quark core,
e.g. 1s → 2s, 1s → 1p1/2, 1s → 1p3/2, . . . transitions
For example: V(k) from Cloudy Bag Model
V s→s
1mt(k) =
1 2f k2
ωs ωs − 1 j1(kRbag) kRbag
3
σ i
mτi t
(p−wave pions) V s→2s
1mt (k) =
1 2f k2
(ω2s − 1)(ωs − 1) j1(kRbag) kRbag
3
σ i
mτi t
(p−wave) V
s→p1/2 1=0,t (k) =
1 2f k2
(ωp1/2 + 1)(ωs − 1) j0(kRbag) kRbag
3
τi
t
(s−wave) V
s→p3/2 2mt
(k) = 1 2f k2
(ωp3/2 − 2)(ωs − 1) j2(kRbag) kRbag
3
Σi
2mτi t
(d−wave)
5
SLIDE 6 Constructing the K-matrix
Aim: include many-body states of quarks (and mesons) in the scattering formalism (Chew-Low type approach) Construct the K-matrix in spin-isospin (JI) basis: KJI
M′B′ MB = −π
kMW Ψ MB
JI (W)||VM′(k)||ΨB′
by using principal-value (PV) states |Ψ MB
JI (W) =
kMW
JI − P H − W [V(kM)|ΨB]JI
Ψ MB(W)|Ψ M′B′(W ′) = δ(W − W ′)δMB,M′B′(1 + K2)MB,MB
dressed states
6
SLIDE 7 Ansatz for the channel PV states
|Ψ MB
JI =
kMW [a†(kM)| ΨB]JI +
cMB
R |ΦR
+
dk χM′B′MB(k, kM) ωk + EB′(k) − W [a†(k)| ΨB′]JI
Above the meson-baryon (MB) threshold:
KM′B′ MB(k, kM) = π
kMW
kM′W χM′B′ MB(k, kM) 2π decay through intermediate hadrons (∆(1232), N(1440); σ, ρ, . . .), e.g.
πN → B∗ → π∆ → ππN , πN → B∗ → σN → (2π)N Solve Lippmann-Schwinger eqs for χ; the solution has the form
χM′B′MB(k, kM) = −
cMB
R V M′ B′R(k) + DM′B′MB(k, kM)
free meson (defines the channel) bare (genuine) baryons (3q) meson “clouds” with amplitudes χ dressed background vertex part
7
SLIDE 8 Solving the coupled equations
The dressed vertices satisfy:
V M
BR(k) = V M BR(k) +
- M′B′
- dk′ KMB M′B′(k, k′) V M′
B′R(k′)
ω′
k + EB′(k′) − W
rMBR = V M
BR(k)
V M
BR(k)
Similarly, for the background part of the amplitude:
DM′B′ MB(k, kM) = KM′B′ MB(k, kM) +
- M′′B′′
- dk′ KM′B′ M′′B′′(k, k′)DM′′B′′ MB(k′, kM)
ω′
k + EB′′(k′) − W
The coefficients cMB
R′ of the quasi-bound states satisfy a set of equations:
ARR′(W) cMB
R′ (W) = V M BR(kM)
ARR′ = (W − M0
R)δRR′ +
B′R(k) V M′ B′R′(k)
ωk + EB′(k) − W
8
SLIDE 9 Calculating the K-matrix
To solve the set of equations, diagonalize A to obtain U, along with the poles of the K- matrix, and wave-function normalization Z: UAUT = ZR(W)(W − MR) ZR′(W)(W − MR′) ZR′′(W)(W − MR′′)
As a consequence, ΦR mix:
| ΦR =
URR′|ΦR
1
URR′VBR′
Solution for the K-matrix:
KMB,M′B′ = π
kMW
kM′W
R
BR
V M′
B′R
(MR − W) + DMB,M′B′ resonant background
Solution for the T matrix:
TMB,M′B′ = KMB,M′B′ + i
TMB,M′′B′′KM′′B′′,M′B′
9
SLIDE 10 Pion electro-production: including the γN channel
Only the strong TMB,M′B′ appears on the RHS:
TMB,γN = KMB,γN + i
TMB,M′B′KM′B′,γN KM′B′,γN = −π
kγW
JI
- Vγ
- ΨN
- Choosing a resonance, R = N∗, the principal-value state can be split into
the resonant and background parts. Then MMB γN =
γ N
ωMEB 1 πVBN∗
helicity amplitude
N∗ (W)|˜
Vγ|ΨN TMB MB
MB γN
+ M
(bkg) MB γN
The resonant state takes the form: |Ψ (res)
N∗ (W) =
1
| ΦN∗ −
dk
BN∗(k)
ωk + EB − W [a†(k)|ΨB]JI
bare quark contrib meson cloud contrib
10
SLIDE 11 Underlying quark model
Cloudy Bag Model extended to pseudo-scalar SU(3) octet L(quark−meson)
CBM
= − i 2f qγ5λaqφaδ
a = 1, 2, . . . , 8 Parameters: Rbag = 0.83 fm f π = 76 MeV f K = 1.2 f π f η = f π
1.2 f π Similar results for 0.75 fm < Rbag < 1.0 fm Free parameters: bare masses of the resonant states
11
SLIDE 12
Pion scattering in P11 partial wave πN → MB
Channels: πN, π∆, σN, πR (preliminary: ηN, KΛ)
Parameters of the σN-channel: gσNR = 1 , mσ = 450 MeV , Γσ = 550 MeV
Thin lines: only N(1440) included ((1s)2(2s)1) Thick lines: N(1710) added ((1s)1(2p)2) with gπNN(1710) = 0 gσNN(1710) ≈ gσNN(1440)
12
SLIDE 13
Pion scattering in S11 partial wave πN → MB
Allow single-quark excitations (1s → 1p1/2 and 1s → 1p3/2)
Φ(1535) = −sin ϑs|481/2 + cos ϑs|281/2 Φ(1650) = cos ϑs|481/2 + sin ϑs|281/2 ϑs is a free parameter (≈ −30◦) Myhrer, Wroldsen / Z. Phys. C 25 (1984) 281
13
SLIDE 14
Pion scattering in S11 partial wave
Inelastic Channels
S11 contribution dominates P11, P13 negligible P11, P13 contributions sizeable not yet included in our calculation PWA results on S11 : P11 : P13 uncertain
14
SLIDE 15
Helicity amplitudes for γp → N(1440)
[10−3 GeV−1/2]
15
SLIDE 16
P11 transverse photo-production amplitudes γN → Nπ0
[10−3/mπ] [10−3/mπ]
16
SLIDE 17
Helicity amplitudes γp → S11(1535), S11(1650) A1/2(Q2)
[10−3 GeV−1/2]
S1/2(Q2)
[10−3 GeV−1/2]
17
SLIDE 18
S11 transverse amplitudes γp → pπ0
[10−3/mπ] [10−3/mπ]
18
SLIDE 19
Eta, kaon photoproduction
work in progress
19
SLIDE 20 Summary
- Using a single set of parameters we reproduce the main features of pion-
and photon-induced production of π, η, and K mesons in P11 and S11 partial waves.
- Importance of the meson cloud:
– it enhances the bare baryon-meson couplings; – it improves the behaviour of the helicity amplitudes at low Q2.
- Enhancement of couplings stronger for P11 and P33 than in the case of
S11 resonances which are dominated by quark-core contributions.
20
SLIDE 21
Spare slides
21
SLIDE 22 Lippmann-Schwinger equation for the K-matrix
χNN
JT (k, k0) = −
cN
B (W)VNB(k) + KNN(k, k0) +
JT (k′, k0)
ω′
k + EN(k′) − W
+
M∆ (k, k′)ˆ
χ∆N
JT (k′, k0)
ω′
k + E∆(k′) − W
ˆ χ∆∆
JT (k, k1) = −
ˆ c∆
B (W, M) V M′ ∆B(k) + K∆∆ M′M(k, k1) +
M′M∆(k, k′)ˆ
χ∆∆
JT (k′, k1)
ω′
k + E∆(k′) − W
+
M′ (k, k′)ˆ
χN∆
JT (k′, k1)
ω′
k + EN(k′) − W
ˆ χ∆N
JT (k, k0) = −
cN
B (W)V m ∆B(k) + K∆N M (k, k0) +
M (k, k′)χNN JT (k′, k0)
ω′
k + EN(k′) − W
+
MM∆(k, k′)ˆ
χ∆N
JT (k′, k0)
ω′
k + E∆(k′) − W
ˆ χN∆
JT (k, k1) = −
ˆ c∆
B (W, M)VNB(k) + KN∆ M (k, k1) +
M∆ (k, k′)χ∆∆ JT (k′, k1)
ω′
k + E∆(k′) − W
+
χN∆
JT (k′, k1)
ω′
k + EN(k′) − W
(W − M0
B)cN B (W) = VNB(k0) +
ˆ χ∆N
JT (k, k0) V∆B(k)
ωk + E∆(k) − W +
χNN
JT (k, k0) VNB(k)
ωk + EN(k) − W (W − M0
B)ˆ
c∆
B (W, M) = V∆B(k1) +
χN∆
JT (k, k1) VNB(k)
ωk + EN(k) − W +
ˆ χ∆∆
JT (k, k1) V∆B(k)
ωk + E∆(k) − W
22
SLIDE 23 Solving the Lippmann-Schwinger equation: separable kernels
1 ωk + ω′
k − ω0 + EB(¯
k) − EN(k0) ≈ ω0 + EB(¯ k) − EN(k0) (ωk + EB(¯ k) − EN(k0))(ω′
k + EB(¯
k) − EN(k0)) ¯ k2 ≈ (k0 + k1)2 ≈ k2
0 + k2 1 ,
EB(¯ k) + EN(k0) − ω0 ≈ 2MB KNN(k, k′) =
f
Bi NN
MBi EN (ω0 + εN
i )
VBiN(k′)VBiN(k) (ω′
k + εN i )(ωk + εN i )
KN∆
M (k, k′) =
f
Bi N∆
MBi E (ω1 + εN
i )
VBiN(k′)VBi∆(k) (ω′
k + εN i )(ωk + ε∆ i (M)) = K∆N M (k′, k)
K∆∆
M′M(k, k′) =
f
Bi ∆∆
MBi E′ (ω′
1 + ε∆ i (M))
VBi∆(k) (ωk + ε∆
i (M))
VBi∆(k′) (ω′
k + ε∆ i (M′))
εN
i = M2 Bi − M2 N − m2 π
2EN , ε∆
i (M) = M2 Bi − M2 − m2 π
2E ,
23
SLIDE 24
Form factors of S, P and D-wave mesons-quark interaction
Determined by the bag radius Rbag = 0.83 fm Equivalent dipole momentum cut-off: ΛS = 510 MeV/c, ΛP = 550 MeV/c, ΛD = 550 MeV/c
24
SLIDE 25 π-quark vertex: S, P, and D-wave pions
V π
l=0,t(k) =
1 2fπ
(ωp1/2 + 1)(ωs − 1) 1 2π k2 √ωk j0(kR) kR
3
τt(i) Psp(i) V π
1mt(k) =
1 2fπ ωs (ωs − 1) 1 2π 1 √ 3 k2 √ωk j1(kR) kR
3
τt(i) ×
ωp1/2(ωs − 1) ωs(ωp1/2 + 1) S
[1
2]
1m(i) +
2ωp3/2(ωs − 1) 5ωs(ωp3/2 − 2) S
[3
2]
1m(i)
2mt(k) =
1 2fπ
(ωp3/2 − 2)(ωs − 1) √ 2 2π k2 √ωk j2(kR) kR
3
τt(i) Σ
[1
2 3 2]
2m (i)
Psp =
|smjp1/2mj| S
[3
2]
1m = √ 15 2
j
C
3 2mj 3 2m′ j1m|p3/2mjp3/2m′
j|
S
[1
2]
1m =
j
C
1 2mj 1 2m′ j1m|p1/2mjp1/2m′
j|
Σ
[1
2 3 2]
2m =
C
1 2ms 3 2mj2m|smsp3/2mj| 25
SLIDE 26 η-quark and K-quark vertex (S-wave)
V η(k) = 1 2fη
(ωp1/2 + 1)(ωs − 1) 1 2π k2 √ωk j0(kR) kR
3
λ8(i) Psp(i) V K
t (k) =
1 2fK
(ωp1/2 + 1)(ωs − 1) 1 2π k2 √ωk j0(kR) kR ×
3
(Vt(i) + Ut(i)) Psp(i) t = ±1
2, V±t = (λ4 ± iλ5)/
√ 2 U±t = (λ6 ± iλ7)/ √ 2 fη = fπ
fη = 1.2 fπ fK = 1.20 fπ.
26
SLIDE 27 ρ-quark vertex (S = 1
2, S-wave and S = 3 2, D-wave) V ρ
l=0mt(k) =
1 2fρ
(ωs − 1) 1 2π k2 √ωk j0(kR) kR
τt(i) × √ 8 3
ωp1/2 + 1 Σ
[1
2]
1m + 3
ωp3/2 − 2 Σ
[1
2 3 2]
1m (i)
l=2mt(k) =
1 2fρ
(ωp3/2 − 2)(ωs − 1) 1 2π 1 3 k2 √ωk j2(kR) kR
3
τt(i)Σ
[1
2 3 2]
1m (i)
fρ = 200 MeV Σ
[1
2]
1m =
C
1 2ms 1 2mj1m|smsp1/2mj|
Σ
[1
2 3 2]
1m =
C
1 2ms 3 2mj1m|smsp3/2mj| 27
SLIDE 28 P11(1440) q3 or q3g ?
Weber / PRC 41 (1990) 2783 Capstick, Keister / PRD 51 (1995) 3598 Cardarelli++ / PLB 397 (1997) 13 Aznauryan / PRC 76 (2007) 025212 Li, Burkert, Li / PRD 46 (1992) 70
- sign change of A1/2
- evidence for Roper as radial excitation of 3q
- nonzero S1/2, hybrid q3g picture ruled out
28
SLIDE 29 P11(1440) and S11(1535) on the Lattice
- close to chiral limit, effects of χSB important
- level ordering should change with mq
Heavy q: 1st radial above 1st orbital excitation chiral limit: reversed levels Bern-Graz-Regensburg / PRD 70 (2004) 054502 PRD 74 (2006) 014504 →
“... do not attempt a chiral extrapolation of our data ... numbers seem to approach the experimental data reasonably well” “... the Roper’s leading Fock component is a 3-quark state”
0.00 0.40 0.80 1.20
(mπ)
2 [GeV] 2 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
MN [GeV]
λ
(1), a = 0.148fm
λ
(2), a = 0.148fm
λ
(3), a = 0.148fm
λ
(1), a = 0.119fm
λ
(2), a = 0.119fm
λ
(3), a = 0.119fm
positive parity
N(938) N(1440) N(1710)
Kentucky / PLB 605 (2005) 137
“...+ and − parity excited states of the nucleon tend to cross over as the quark masses are taken to the chiral limit. Both results at the physical pion mass agree with the exp values ... seen for the first time in a lattice QCD calculation” “...a successful description of the Roper resonance depends not so much on the use of the dynamical quarks ... most
- f the essential physics is captured by using light quarks”
0.5 1 1.5 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Masses (GeV) mπ
2(GeV2)
Nucleon S11(1535) Roper
1.3 1.5 1.7 0.1 0.2
MR/MN
1.3 1.5 1.7 0.1 0.2
MR/MN
29
SLIDE 30
P11(1440) on the Lattice
Adelaide/JLab / PLB 679 (2009) 418 — quenched, FLIC fermion action
“A lower lying Roper state is observed that approaches the physical Roper state. To the best of our knowledge, the first time this state has been identified at light quark masses using a variational approach.”
30
SLIDE 31 Lattice N → P11(1440) EM transition form-factors
P11 VΜ p p VΜ P11 Exp. 1 1 2 3 4 0.2 0.2 0.4 Q2GeV2 F1
pP11
P11 VΜ p p VΜ P11 Exp. PDG 1 1 2 3 4 0.8 0.4 0.4 Q2GeV2 F2
pP11
P11 VΜ n n VΜ P11 1 1 2 3 4 0.10 0.05 0.05 Q2GeV2 F1
nP11
P11 VΜ n n VΜ P11 PDG 1 1 2 3 4 0.60 0.30 0.30 0.60 Q2GeV2 F2
nP11
- quenched, mπ = 720 MeV (!)
Lin++ / PRD 78 (2008) 114508
31
SLIDE 32
SAID PWA of πN scattering in P11 channel
ImT ← ImT − |T|2 ReT SAID FA02 MBW = (1468 ± 4.5) MeV, Γ/2 = (180 ± 13) MeV Mpole = 1357 − i 80 MeV (I RS) 1385 − i 83 MeV (II RS)
32
SLIDE 33 Ansaetze for the channel states More Detail
πN channel:
|Ψ πN
JT (W) =
k0W [a†
π(k0)|ΨN(k0)]JT +
cN
B (W)|ΦB
+ dk χNN
JT (k, k0)
ωk + EN(k) − W [a†
π(k)|ΨN(k)]JT +
dk χ∆N
JT (k, k0, M′)
ωk + E′(k) − W [a†
π(k)|
Ψ∆(M′)]JT + dk χmN
JT (k, k0)
ωk + EN(k) − W [a†
m(k)|ΨN(k)]JT
|Ψ π∆
JT (W, M) =
k1W [a†
π(k1)|
Ψ∆(M)]JT +
c∆
B (W, M)|ΦB
+ dk χN∆
JT (k, k1, M)
ωk + EN(k) − W [a†
π(k)|ΨN(k)]JT +
dk χ∆∆
JT (k, k1, M′, M)
ωk + E′(k) − W [a†
π(k)|
Ψ∆(M′)]JT
mN channel (e.g. m = σ):
|Ψ mN
JT (W, µ) =
kmW [a†
m(km)|ΨN(km)]JT +
cm
B (W, µ)|ΦB
+
dk χmm
JT (k, km, µ, µ′)
ωk + EN(k) − W [a†
m′(k)|ΨN(k)]JT +
dk χNm
JT (k, km)
ωk + EN(k) − W [a†
π(k)|ΨN(k)]JT
unmodified (free) π bare baryons (3q) meson “clouds” with amplitudes χ
33
SLIDE 34 Kinematics of two-pion decay
πN → B∗ → π∆ → ππN πN → B∗ → mN → (2π)N
(k ) π
2
(k ) π
2
(k ) π
1
N B* B* π ∆ (Μ) (k )
1
(µ) m N
ω1 = W − E = W 2 − M2 + m2
π
2W , ωµ = W − EN = W 2 − M2
N + µ2
2W , k1 =
1 − m2 π ,
E =
1
kµ =
µ − µ2
EN =
N + k2 µ .
The intermediate ∆ state
Ψ∆(M′) = δ(M − M′) | Ψ∆(M)≈ K √ 1 + K2
ωk + EN(k) − M [a†(k)|ΦN]
3 2 3 2 −
ωk + E∆(k) − M [a†(k)|Φ∆]
3 2 3 2
√ 1 + K2 ≈ 1 π (1
2Γ∆)2
(M∆ − M)2 + (1
2Γ∆)2 34
SLIDE 35 Calculating the K matrix Details, part 1
Connection between K-matrix elements and pion aplitudes Above π threshold : KπNπN(W) = π ω0EN(k0) k0W χNN
JT (k0, k0)
Above 2π threshold : Kπ∆πN(W, M) = π
k0k1W 2 χ∆N
JT (k1, k0, M)
KπNπ∆(W, M) = π
k0k1W 2 χN∆
JT (k0, k1, M)
Kπ∆π∆(W, M′, M) = π
1E(k′ 1)
k1k′
1W 2
χ∆∆
JT (k′ 1, k1, M′, M)
Above m(2π) threshold : KmNmN(W) = π ωmEN(km) kmW χmm
JT (km, km)
The form of amplitudes χ χNN
JT = −
cN
R(W)VNR(k) + DNN(k, k0)
dressed background vertex part χB′B
JT = −
cB
R(W, M)V M′ B′R(k) + DB′B M′M(k, k1)
etc.
35
SLIDE 36 Calculating the K matrix Details, part 2
δΨ|H − E|Ψ ⇒
- Lippmann-Schwinger equation for χ
VNR = VNR +
ω′
k + EN(k′) − W
+
MB′ (k, k′)V MB′ B′R(k′)
ω′
k + EB′(k′) − W
V M
BR = V M BR +
M (k, k′)VNR(k′)
ω′
k + EN(k′) − W
+
MMB′(k, k′)V MB′ B′R(k′)
ω′
k + EB′(k′) − W
- System of linear eqs for coefficients cH
R′ of the bare 3q states (H ∈ {πN , πB , σB })
ARR′(W) cH
R′(W, mH) = V M HR(kH)
ARR′ = (W − M0
R)δRR′ +
MB′ B′R(k)V MB′ B′R′(k)
ωk + EB′(k) − W
36
SLIDE 37 Calculating the K matrix Details, part 3
R are not eigenstates of Hamiltonian and therefore they mix: | ΦR =
Diagonalize A to obtain U, the poles of the K matrix, and wave-function normalization Z UAUT = D , D = ZR(W)(W − MR) ZR′(W)(W − MR′) ZR′′(W)(W − MR′′) Pion amplitudes pertaining to physical resonances H χH′H =
1 ZR(W)(MR − W)
URR′VHR′ Solution for the K matrix KHH′ = KHH′(resonant) + KHH′(background) = πNHNH′
VH′R ZR(W)(MR − W) + DHH′ T matrix and scattering matrix S same structure for all channels T = K + i TK S = I + 2 i T
37
SLIDE 38 Pion electro-production
p’ p γ π θπ
Formally, the K matrix acquires a new channel, γN Because the EM interaction is considerably weaker than the strong interaction, we assume KγN γN ≪ KγN πN ≪ KπN πN (and similarly for other channels). The Heitler-like equation for the electro-production amplitudes then reduces to TγNπN(W) = KγNπN(W) + i
- TπNπN(W)KγNπN(W) + T πNπ∆(W, ¯
M)KγNπ∆(W, M) + T πNmN(W, ¯ µ)KγNmN(W, µ)
- The T matrix for electro-production is related to the electro-production amplitudes by
T (JT)
γNπN = iπ
1 √ 2π
3
- m
- k0kγ MN(W, MJ, MT, t, kγ, µ) Y1m(ˆ
r) C
JMJ
1 2ms1mCTMT 1 2 1 21t
MγNπN(W) = MK
γNπN(W) + i
γNπN(W) + T πNπ∆(W, ¯
M)MK
γNπ∆(W, M) + · · ·
γNπN(W) + M (bkg) γNπN(W) 38
SLIDE 39 Evaluation of matrix elements
The resonant part of amplitude for a chosen R = N∗: M(res)
γNπN =
γ N
ω0EN 1 πVNN∗ Ψ (res)
N∗ (W)|˜
Vγ|ΨN TπNπN =
γ N
ω0EN 1 πVNN∗ A(γN → N∗)
TπNπN The background part obeys the equation in which the resonance pole is absent: M
(bkg) γNπN = M K (bkg) γNπN + i
K (bkg) γNπN + T πNπ∆M K (bkg) γNπ∆ + T πNmNM K (bkg) γNmN
- The helicity amplitude AN∗ for the electro-excitation of the resonance is proportional to
the transition electromagnetic form factor: AN∗ ≡ Ψ (res)
N∗ (W)|˜
Vγ|ΨN For example: MK
γNπN(W) = −
k0 Ψ πN
N∗ (W)|˜
Vγ(µ, kγ)|ΨN ˜ Vγ(µ, kγ) = e0
The resonant state takes the form: |Ψ (res)
N∗ (W) =
1
VNN∗(k) ωk + EN(k) − W [a†(k)|ΨN]JT − · · ·
SLIDE 40 Resonances and quark-model wave-functions S11
resonance M [MeV] Γ [MeV] decays S11(1535) 1535 150 πN 35–55 % , ηN 45–60 % 2πN < 10 % , (πR) S11(1650) 1655 165 πN 60–95 % , ηN 3–10 % , KΛ 3–11 % 2πN 10–20 % , ρ 4–12 % , ∆ 1–7 % Φ(1535) = −sin ϑs|481/2 + cos ϑs|281/2 = c1
A
+ c1
P
1 + c1 P′
2
Φ(1650) = cos ϑs|481/2 + sin ϑs|281/2 = c2
A
+ c2
P
1 + c2 P′
2
c1
A = 1
3(2cos ϑs − sin ϑs) , c2
A = 1
3(cos ϑs + 2sin ϑs) , ϑs = −30◦ Myhrer, Wroldsen / Z. Phys. C 25 (1984) 281
40
SLIDE 41 Separation of amplitudes into resonant and background parts
Because the K matrix elements contain poles, it convenient to separate the amplitudes as MK
H =
γ N
k0W g(W) KNH Ψ (res)
N∗ (W)|˜
Vγ|ΨN + M
K (bkg) H
H = πN, π∆, mN M
K (bkg) H
= −
γ N
k0W
(bkg) NH
Ψ (res)
N∗ (W)|˜
Vγ|ΨN +
kHW
N Ψ (n.p.) N∗
|˜ Vγ|ΨN + Ψ H (non−res)
N∗
|˜ Vγ|ΨN
- The resonant part takes the form
M(res)
πN =
γ N
k0W g(W) Ψ (res)
N∗ (W)|˜
Vγ|ΨN TπNπN =
γ N
k0W g(W) AN∗ TπNπN The background part obeys the EQ in which the resonance pole is absent: M
(bkg) πN
= M
K (bkg) πN
+ i
K (bkg) πN
+ T πNπ∆M
K (bkg) π∆
+ T πNmNM
K (bkg) mN
SLIDE 42 Phase shifts P11, P33
...................... resonant term only, no background ...................... πN and π∆ channels σN (σ∆) channel included : ...................... Born approximation, gπNR = 1.68 gquark
πNR , gπN∆ = 1.40 gquark πN∆
Phase shift
20 40 60 80 100 120 140 160 180 200 1100 1200 1300 1400 1500 1600 1700 1800
50 100 150 200 1100 1200 1300 1400 1500 1600 1700 1800
P11 P33
gπNR gquark
πNR
= 1.0 , gπN∆ gquark
πN∆
= 1.0 gπNR gquark
πNR
= 1.0 , gπN∆ gquark
πN∆
= 1.0
42
SLIDE 43 Resonant and background contributions to P11 phase shift
50 100 150 200 1000 1100 1200 1300 1400 1500 1600 1700 1800 δ [deg] W [MeV] total res
crossed
reson.
π N N N π *
π N N N π
u-channel
N N π ∆ π 43
SLIDE 44 Contributions to the proton transverse amplitude
π γ N N N * π γ N N N N N π γ ∆
N N π π γ N N π γ ω
resonant nucleon pole crossed pion pole
44
