Effective model for K N interactions
including the L = 1 partial wave

Vojtěch Krejčiřı́kTheoretical Research Division, RIKEN Nishina Center, Wako, Saitama, Japan

HHIQCD2015, Kyoto, March 3
based on A. Cieplý, V.K., arXiv:1501.06415 [nucl-th] (2015)
V.K., Phys. Rev. C 86, 024003 (2012)
1 / 21Vojtěch Krejčiřı́k

Effective model for K N interactions

introduction
strange S = −1 meson-baryon interactions
strongly coupled multichannel systemchannelthreshold [MeV]::πΛ1250

πΣ
1330

KN1435

ηΛ1660ηΣ
1740Λ(1405) in the S-wave and isospin I = 0 channel
Σ(1385) in the P-wave and isospin I = 1 channel

resonances are present below the K N threshold2 / 21Vojtěch Krejčiřı́kEffective model for K N interactions

KΞ1810state of the art — S-wave

O(q 1 )-constactO(q 2 )-directO(q 2 )-crossedO(q 2 )-constact

constructed from 2nd order chiral Lagrangianparameters fit to scattering data, threshold branching ratios, kaonic hydrogenICS2012 - Cieplý, Smejkal, NPA 881 (2012)

IIHW2011 - Ikeda, Hyodo, Weise, NPA 881(2012)

I

MM2013 - Mai, Meißner, NPA 890 (2013)

IGO2013 - Guo, Oller, PRC 87 (2013)

courtesy of A. Cieplý
3 / 21Vojtěch Krejčiřı́k

Effective model for K N interactions

state of the art — S-wave

Λ(1405) resonance generated dynamically TWO POLES



relative agreement on the position of the K N
related pole, the position of the πΣ relatedpole much less constrained πΣ MASS SPECTRUM

different experiments (CLAS, ANKE, HADES)
exhibit slightly different curves
depends on the reaction mechanism

courtesy of A. Cieplý
4 / 21

Vojtěch Krejčiřı́k

Effective model for K N interactions

Why bother with the P-wave?

things get entangled in the nuclear medium

A. Scordo et al. [AMADEUS Collaboration], arXiv:1304.7149 [nucl-ex].
5 / 21

Vojtěch Krejčiřı́k

Effective model for K N interactionseffective model — general formulation
three independent partial waves:

L = 0, J = 21.......

0+

L = 1, J = 21.......

1−L = 1, J = 23.......

1+

as long as the interaction is parity and time reversal invariant!

——————————————————————————————————

potentials in the separable form:l±ll±lV(ai)→(bj)(p, p0 ) = g(ai)(p) v(ai)→(bj)
g(bj)
(p0 )
l±v(ai)→(bj)is a coupling matrix where the inter-channel dynamics is encoded.

,0g(ai)(p) = 1

6 / 21p2
1+ 2α(ai)!

Vojtěch Krejčiřı́k,
1g(ai)(p) = pp2
1+ 2α(ai)Effective model for K N interactions!3/2

effective model — general formulation
three separate Lippmann-Schwinger Equations:
F 0+ = V 0+ + V 0+ G0 F 0+F 1− = V 1− + V 1− G1 F 1−F 1+ = V 1+ + V 0+ G1 F 1+——————————————————————————————————

Green function in the vacuum:lG(ck)(√1s) = − 22πZ2g l (q)d q 2p(ck) − q 2 + i
3Green function in the nuclear medium:lG(ck)(√1s, ρ) = − 22πZΩ(ck ) (ρ)

2g l (q)√d q 2p(ck ) − q 2 − Π(ck ) ( s, p, ρ) + i3√Ω(ck ) (ρ) integration domain determined by the Pauli blocking, and Π(ck ) ( s, p, ρ) selfenergy

7 / 21

Vojtěch Krejčiřı́kEffective model for K N interactions

potentials
S-WAVE POTENTIAL



Weinberg-Tomozawa only
parameters TW1 determinedby Cieplý, and Smejkal (2012)8 / 21

Vojtěch Krejčiřı́k

Effective model for K N interactions

S-wave potential
free parameters:




 

meson decay constant fπ , one range parameter α ———————————————————————————————————

experimental data:kaonic hydrogen (SIDDHARTA)
IIshiftwidth∆EN (1s) = 283 ± 36 (stat.) ± 6 (syst.)eV
Γ(1s) = 541 ± 89 (stat.) ± 22 (syst.)eV

K − p threshold branching ratiosIσ(K − p→π + Σ− )γp = σ(K − p→π− Σ+ )Rc =σ(K − p→charged)σ(K − p→all)σ(K − p→π 0 Λ)

Rn = σ(K − p→neutral)

low energy K − p cross sections———————————————————————————————————results of the S-wave fit:

fπ = 113 MeV, α = 701 MeV
A. Cieplý, J. Smejkal, Nucl. Phys. A 881 (2012).
9 / 21Vojtěch Krejčiřı́k

Effective model for K N interactions

potentials
S-WAVE POTENTIAL



Weinberg-Tomozawa only
parameters TW1 determinedby Cieplý, and Smejkal (2012)



P-WAVE POTENTIAL

in principle four different contributions
10 / 21

Vojtěch Krejčiřı́k

Effective model for K N interactions

P-wave potential

∼ E1∼ E11∼ E−∆1∼ E+∆Σ(1385) dominates the P-wave physics!
———————————————————————————————————How to determine the meson-baryon-baryon coupling?

Follow the large Nc limit of QCD!!

is one option.

at large Nc , octet and decuplet baryons are created equal(rigorously speaking, the belong to the same ground state multiplet)

11 / 21

Vojtěch Krejčiřı́k

Effective model for K N interactions

P-wave potential
key formal features of large Nc χPT:

consistency relations; the contracted SU(2Nf ) symmetry
practical meaning:

relative couplings given by CG coefficients∂ i φa D 0 i a E
B | σ λ |B ∼fπ

8 ⊗ 8 = 1 ⊕ 8 ⊕ 8 ⊕ 10 ⊕ 10 ⊕ 27vanishing πN scattering determines the strength+
12 / 21+Vojtěch Krejčiřı́k+
Effective model for K N interactions= 0

effective model — summary of the P-wave addition

diagrams determined by scaling — Σ(1385) dominantcoupling matrix determined using large Nc motivation
bare mass of the Σ(1385) determined to match its physical mass
free parameters determined by the S-wave analysis

iNO ADDITIONAL FREEDOM !!
.

13 / 21

Vojtěch Krejčiřı́k

Effective model for K N interactions

vacuum results

K − p → . . . total cross sections

the effect of the P-wave is marginal

14 / 21

Vojtěch Krejčiřı́k

Effective model for K N interactions

vacuum results

K − p → K − p differential cross section0

K − p → K n differential cross section

0

the slope direction well reproduced, the slope angle better for K n transition

15 / 21

Vojtěch Krejčiřı́k

Effective model for K N interactions

nuclear medium effectsπΣ amplitude in the S-wave
dashed free space, dot-dashed +Pauli blocking, full +selfenergies
1.5

I=0, S-wave

1.0Im FπΣ [fm]Re FπΣ [fm]

1.5

0.5
0.0

1.0

0.5

-0.5
1300

1350

1400
1450
s1/2 [MeV]

1500

0.01300

1350

1400
1450
s1/2 [MeV]Λ(1405) dissolves in the nuclear medium and shifts its position

16 / 21

Vojtěch Krejčiřı́k

Effective model for K N interactions

1500

nuclear medium effectsπΣ amplitude in the P-wave
dashed free space, dot-dashed +Pauli blocking, full +selfenergies0.1

I=1, P-wave
Im FπΣ [fm]Re FπΣ [fm]

0.2
0.0

0.1

-0.1
1300

1350

14001450s [MeV]1500

1/20.0
1300

1350

1400
1450
s1/2 [MeV]

effects of the nuclear medium on Σ(1385) are quite limited

17 / 21

Vojtěch Krejčiřı́k

Effective model for K N interactions

1500

nuclear medium effectsπΣ mass distribution
dashed free space, dot-dashed +Pauli blocking, full +selfenergies
200

events150

100

50013501400
1450
s1/2 (MeV)1500

the in-medium peak is much narrower than in the vacuum

18 / 21Vojtěch Krejčiřı́k

Effective model for K N interactions

nuclear medium effects

threshold branching ratios

32101350

14001450
s1/2 [MeV]−+1500−p→π Σ )γp = σ(Kσ(K − p→π − Σ+ )0.6
Rn [π0 Λ / (all neutral)]

0.8

Rc ( charged / all )

0.9

4-

γ p (π+ Σ / π- Σ+)dashed free space, dot-dashed +Pauli blocking, full +selfenergies
5

0.5
0.4

0.7

0.3

0.6

0.2

0.5
0.413501400
1450
s1/2 [MeV]−1500p→charged)
Rc = σ(Kσ(K − p→all)

0.1

0.01350

Vojtěch Krejčiřı́k−

15000σ(K p→π Λ)
Rn = σ(K− p→neutral)

see the peak in the γp energy dependence

19 / 21

14001450
s1/2 [MeV]

Effective model for K N interactions

nuclear medium effects

isospin decompositionpi ∼12hπ− +Σ | ∼ −ss| K̄ NiI=0 −112s41hπΣ |I=12ss
11+ −hπ Σ| ∼ −hπΣ |I=0 −hπΣ |I=1323hπΣ |I=0 +5| K̄ NiI=13-−γ p (π+ Σ / π- Σ+)

s| K

210

γp =


T0 +
T0 −

√32

√3213501400
1450
s1/2 [MeV]

1500



2 T12 T1



0denominator suppression due to T1 -T0 interference, peak at K n threshold
20 / 21Vojtěch Krejčiřı́kEffective model for K N interactions

conclusions

the effective model for K N was extended toinclude P-wave physics
reasonable description of the vacuum data;
suitable for the in-medium analysis
future – more involved EFT backed model
lattice QCD calculation

work was supported by RIKEN programs for junior scientists

21 / 21Vojtěch Krejčiřı́k

Effective model for K N interactions