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

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

based on: V.K.: Phys. Rev. C 86, 024003 (2012)
A. Cieplý, V.K.: prepared for publication
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Vojtěch Krejčiřı́k

K N interactions with L=1

introduction
K N interaction from broader perspective
part of a strongly coupled multichannel systemchannelthreshold [MeV]

::πΛ
1250

πΣ
1330

KN1435ηΛ
1660

ηΣ1740KΞ
1810

resonances are present between πΣ and K N thresholds
Λ(1405) in the S-wave and isospin I = 0 channel
Σ(1385) in the P-wave and isospin I = 1 channel
Due to the presence of resonance, conventional chiral effective
perturbation theory is not applicable!
(One of) Alternative solution(s) is to construct effective potentials andsolve the Lippmann-Schwinger equation!
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Vojtěch Krejčiřı́k

K N interactions with L=1

state of the art — S-wave

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

potentials constucted from 2nd order chiral Lagrangianparameters fit to scattering data, threshold branchich ratios, kaonic hydrogenI

CS2012 - Cieplý, Smejkal, NPA 881 (2012)

I

IHW2011 - Ikeda, Hyodo, Weise, NPA 881 (2012)

I

MM2013 - Mai, Meißner, NPA 890 (2013)

IGO2013 - Guo, Oller, PRC 87 (2013)

courtesy of A. Cieplý
3 / 16Vojtěch Krejčiřı́kK N interactions with L=1

state of the art — S-wave

Λ(1405) resonance generated dynamicallyin the L=1, I=0 channel
 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,AMADEUS) exhibit slightly differnet curves

courtesy of A. Cieplý
4 / 16Vojtěch Krejčiřı́kK N interactions with L=1

effective model — general formulation
three independent partial waves:

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

0+

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

1−

L = 1, J = 23.......

1+——————————————————————————————————and three separate Lippmann-Schwinger Equations:
F 0+/1−/1+ = V 0+/1−/1+ + V 0+/1−/1+ G0/1 F 0+/1−/1+
as long as the interaction is parity invariant!

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Vojtěch Krejčiřı́kK N interactions with L=1

effective model — general formulation
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——————————————————————————————————

Green function in the vacuum:√1lG(ck) ( s) = −2π 2Zd3 q

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

K N interactions with L=1

effective model — potentials
S-WAVE POTENTIAL



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



P-WAVE POTENTIAL

in principle four different contributions
7 / 16Vojtěch Krejčiřı́kK N interactions with L=1

effective model — P-wave

∼ 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!!
here, octet and decuplet baryons are created equal(belong to the same ground state multiplet)

8 / 16Vojtěch Krejčiřı́k

K N interactions with L=1

effective model — P-wave∂ i φa D 0 i a E
B | σ λ |Bfπ

vertices determined by the contracted spin-flavor
symmetry (read Clebsch-Gordan coefficients)
8 ⊗ 8 = 1 ⊕ 8 ⊕ 8 ⊕ 10 ⊕ 10 ⊕ 27
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Vojtěch Krejčiřı́k

K N interactions with L=1

effective model — summary of the P-wave addition

diagrams determined by scaling — Σ(1385) dominantvertex functions 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 FREE PARAMETERS !!
.

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Vojtěch Krejčiřı́k

K N interactions with L=1

vacuum results

K − p → . . . total cross sections
K − p → π0 ΛK − p → π 0 Σ0K − p → π − Σ+K − p → π + Σ−K −p → K −p

K −p → K n

the effect of P-wave is marginal

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Vojtěch Krejčiřı́k

K N interactions with L=10

vacuum results

K − p → K − p differential cross section

0K − p → K n differential cross section

0

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

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Vojtěch Krejčiřı́k

K N interactions with L=1

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

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Vojtěch Krejčiřı́k

K N interactions with L=11500

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

0.1

I=1, P-wave
Im FπΣ [fm]Re FπΣ [fm]

0.2
0.0

0.1

-0.1
1300

1350

1400
1450
s [MeV]

1500

0.0
1300

1350

1/2

1400
1450
s1/2 [MeV]

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

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Vojtěch Krejčiřı́k

K N interactions with L=11500

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

150

100

50013501400
1450
s1/2 (MeV)1500

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

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Vojtěch Krejčiřı́k

K N interactions with L=1

conclusions

an extension of the effective model for K N to
include P-wave physics was formulated
reasonable description of experimental data inthe vacuum; L=0 p.w. is dominantsuitable for the analysis of nuclear-medium
impacts

work was supported by RIKEN programs for junior scientists

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Vojtěch Krejčiřı́k

K N interactions with L=1