Faddeev approach to the reaction

at

Analysis of the J-PARC experiment (E31)

- Realistic calculations made possible

- Predictions of L(1405) peak by chiral interactionscompared with preliminary data.

(Calculations for 2-step and higher-order processes)

An enhancement in recently found at J-PARC

so-called quasi-bound KNN state ?

3HeK pn- L

K d n-

1 GeV/cKP

K. Miyagawa

Okayama University of Science

K d n-

Outline

0n 1 GeV/cKP

( , )d K n -

Promising candidate which pins down the interaction

and property

Experiment E31 at J-PARK

Motivation

- Direct access to the resonance region via spectra

- Hadronic probe (induced by )

- Good-quality data

interaction is constrained from chiral SU(3) and data around the thresholdbut models show large differences in actual predictions

0n 1 GeV/cKP

KN -

1405( )L

1405( )L

K-

KN

( I=1 amp, ) 1405( )L

-

0 0 -

( , )d K n -

Σ+π−

Σ−π+

0n 1 GeV/cKP J-PARC E31Inoue et al., HNP2017, Osaka

KN threshold

Jido, Oset, sekihara Eur.Phys.J. A42, 257 (2009)

Analyses

Calculations of 2 step processes

Faddeev calculations

Ohnishi, Ikeda, Hyodo, Weise Phys.Rev. C93,025207 (2016)

Miyagawa, Haidenbauer Phys.Rev. C85,065201 (2012)

Miyagawa, Haidenbauer

Kamano, Lee: Inclusion of higher partial waves up to F7/2, Phys.Rev. C94,065205 (2016)

Kamano, Nakamura, Lee, Sato, Phys.Rev. C90,065204 (2014)

Comprehensive Analyses of data up to W=2.1 GeVK p-

Zhang, Tulpan, Shrestha, Manley, Phys.Rev. C88,035204 (2013)

Back ground

Calculation forf 2 step processes,

First E31 data came out in 2015, 2016

Situation changed completely by

KN KN

PWA byKSU (Kent State Univ.)

Coupled Channel

Cross sections of comparable magnitude to E31 data were obtained.

0

5

10

15

20

25

30

0 200 400 600 800 1000 1200

S1/2

P1/2

P3/2

D3/2

D5/2

F5/2

F7/2

G7/2

Full

S 1/2

0

2

4

6

8

10

0 200 400 600 800 1000 1200

S1/2

P1/2

P3/2

D3/2

D5/2

F5/2

F7/2

G7/2

Full

Zhang, Tulpan, Shrestha, Manley, Phys.Rev. C88,035204 (2013) Partial Wave Analysis by KSU group

K n K n- -

0K p K n

- total cross section

s (

mb

)s

(m

b)

Full

FullS 1/2

LABP (MeV c)

LABP (MeV c)

total cross section

Combine the KSU amplitudes up to G 7/2 and amplitudes from S-wave chiral potentials

Our analysis

KN KN

KN

2-step processes

Faddeev calculations for higher-order processes

- “New form” which avoids moving singularities

extended to the relativistic case

- boost of 2-body t-matrices based on a Poincare invariant few-body model (Keister, Polyzou)

(Witala, GlÖckle)

Special kinematics of E31 and Reaction mechanism

K-

n

p

n

p

K-

K d n-

1 GeV/c

0n 1 2 GeV/c.

The energies related to the two-process

, are well separated K Kn n- -

K p -

-

K-

d

K

n

N

, KNt

2 step processes

0N N NK dK Kn Kt G t, ,pp y

, NKN Kt

N

0

0

-

K-

d

0K

n, KN

t

2 step processes

, NKN Kt

p

n

K-

d

K-

n

, KNt

, NKN Kt

p

n

-

0

0

-

-

0

0

-

0N N NK dK Kn Kt G t, ,pp y2 processes

K-

d

K

n

N

, KNt

Our calculations for the 2 step processes

0N N NK dK Kn Kt G t, ,pp y

, NKN Kt

Partial-Wave Amplitudes by KSU

Up to 7 2/G

Cieply, Smejkal 2012

Ohnishi, Ikeda, Hyodo, Weise 2016

Oset, Ramos, Bennhold 2002

1 2 GeV. / c

Chiral Unitary models

KSU (1st ) + Cieply( 2nd )

( , )d K n - 1 GeV/cKP - 0n

KN KN KN

ds

/dM

d

Wn

[mb

/MeV

sr]

0

5

10

15

20

25

30

35

1390 1410 1430 1450 1470 1490

tfac3_correct

tfac3_correct

tfac3_correct

[MeV]M

-0 0

-

KN threshold 1434.6

Comparison with E31 preliminary data

n -

n -

pure I =0

1 GeV/cK

P - 0n ( , )d K n -

0

1

2

3

4

1370 1390 1410 1430 1450 1470 1490

t(1)-sqrt(2/3)*t(0)

- t(1)-sqrt(2/3)*t(0)

-sqrt(2/3)*t(0)

Cieply

0.00E+00

2.00E-02

4.00E-02

6.00E-02

8.00E-02

1400 1420 1440 1460 1480 1500

2

t πΣ,KN G0 t KN,KN ΦdΣ2 processes

t πΣ,KN 2 G0 t KN,KN Φd

2

K－

p → π+ Σ－

K－

p → π－ Σ+

K－

p → π0 Σ0

K－

p → K0bar n

K－

n → K－

n

Pronounced peaks: and QFS1405( )L

Contributions from higher partial-wave KN KNd

s/d

M

d

Wn

[mb

/MeV

sr]

[MeV]M

0

5

10

15

20

25

30

35

1390 1410 1430 1450 1470 1490

up to j=7/2

up to j=5/2

up to j=3/2

( )KN KN

Contributions from higher partial-wave KN KN

up to j=7/2

up to j=5/2

up to j=3/2

[MeV]M

S wave

up to j=1/2

up to j=3/2

( )KN KN

0

5

10

1390 1410 1430 1450 1470 1490

0

5

10

15

20

25

30

35

1390 1410 1430 1450 1470 1490

( , )d K n - 1 GeV/cK

P -

0n

pure I =0

0K nK p

-

Comparison of the three chiral potentials

Cieply

Oset-Ramos

Ohnish

n - n -

0 0n

IncludeI =0, 1

0( , )pd K - - 1 GeV/c

KP - 0n

pure I =1

Cieply

Oset-Ramos

Ohnish

Reaction with outgoing proton 0

K pd - -

Comparison with the results by Kamano-Lee

( , )d K n -

0( , )pd K - -

Ours (Cieply)

Kamano

Average of andns - ns -

Cieply

Oset-Ramos

Ohnish

( I ： isospin of )

average

Faddeev calculation for 3 step processes

K-

d

n

N

NN

N

K-

dn

N

N

K

KK

NN NN

KN nK

up to 3j

1 3

023 1

23P-( ) ( )( ) ( )

iG ty f y

23P -

3 coupled equations

K

N

1 2 3

023( ) ( ) ( )( ) ( )

iG ty f y y

2 3 1

0 13G t( ) ( ) ( )( ) ( )y y y

3 2 1

0 12G t( ) ( ) ( )( ) ( )y y y

Impose antisymmetry condition, when 2 and 3 are identical

3 3 1

012G t

23P

( ) ( ) ( )( ) ( )y y y-

1 2 3( ) ( ) ( )y y y

1 3123

P-( ) ( )( )y y

Total wave function is divided to 3 components

For particles 1,2 and 3

Faddeev equations

1

2

3

1

2

3

1

2

3

N

( )K NN

1 2

0 2323 1NNKNN KNi NG t PKNN( ) ( )( ) ( ) ( )= + -

{}

3 3 1

0 23

2 1

12

12

KNN KN KN KNN KNN

KN N N

G K N t P

t

N( ) ( ) ( )

,

( ) ( )

,

( ) ( ) ( )

( ) ( )S SS

= - +

+ +

{}

3 3 1

0 23

2 1

12

12

KN KNN KNNN

N N

G t P

t

N( ) ( ) ( )

,

( ) ( )

,

( ) ( ) ( )

( ) ( )

S S

S S S S

= - +

+ +

S

{ }1 3 20 23N N N N NG tN( ) ( ) ( ),( ) ( ) ( )S S S S S= +S

2 3 1

0 13N N N NG tN( ) ( ) ( )( ) ( ) ( )S S S+S=

1 --- meson

2,3 --- baryons

5 coupled equations: 2 are for , the other 3 for

Faddeev equations for KNN N- S

NSKNN

2323 23 23N N N Ntot

N NV V V P, , ,( ) ( ) ( )S S S S S S= -

Equations for partial breakup amplitudes

We solve the equations for breakup amplitudes

2

2

2

2 31

1E E qpq p

E E q Ep dp p

ipT t ( )| | ( )|

( ) ( )aa a a

e

2

2 2

p q

p qq q p

qdq R q q p p q

q p, ,( )( , , ) |

r

a ar

a

ar 3

T

t

p

qpp

q3T

Technicalities

f p q( )

Simple subtraction as in the 2-body problem can be used for integrationp

q

p

2qr

2qr

integralq

3 3

2

1

Witala, Glöckle, Eur. Phys. A37,87 (2008)

We extend it to the relativistic case

Faddeev calculation for 3 step processes

K-

d

n

N

NN

N

K-

dn

N

N

K

KK

NN NN

KN nK

up to 3j

Inclusion of the 3rd step with interactionKN

20n

0n

10n

ds

/dM

d

Wn

[mb

/MeV

sr]

[MeV]M

0

5

10

15

20

25

30

35

1390 1410 1430 1450 1470 1490

2step

2step

2step

3step

3step

3step

0

2

4

6

8

10

1390 1410 1430 1450 1470 1490

c

0.0

0.4

0.8

1.2

1390 1410 1430 1450 1470 1490

2 step

3 step

0

5

10

15

20

25

30

35

1400 1430 1460 14900

5

10

15

20

25

1400 1430 1460 14900

2

4

6

8

10

1400 1430 1460 1490

0

1

2

3

4

1400 1430 1460 1490

0.0

0.4

0.8

1.2

1400 1430 1460 1490

0n

20n

10n

5n (Lab)

(Lab)

15n (Lab)

(Lab)

Summary of the analysis of E31

● Calculation for 2step and 3step processes

● For 1st step Kbar N→ Kbar N, PWA amplitudes by KSU

For 2nd step , Chiral unitary model by Cieply, Ohnishi, Oset-Ramos.

● The 3rd steps give almost no effects at , but change lineshapes at

● Accurate data would pin down the Kbar N→ π Σ amplitude.

d (K－ , n) πΣ P K－=1 GeV/c θn = 0°

● Overall magnitude of the cross sections are well reproduced, But for all three potentials

- peaks are too large in magnitude- poor agreement in lineshape for (interference between I=0 nad 1) n -

1405( )L

Θ n = 0° Θ n =2 0°

T.Yamaga JPS 2017J-PARC E31Kawasaki et al., MENU2016

Σ+π−

Σ−π+

E15

An enhancement in

recently found at J-PARC

3HeK pn- L

dK n-

~40 Mev below the threshodKN

so-called quasi-bound state ?KNN

KNN threshold

K-

d

K

n

N

, KNt

E31 vs E15

, NKN Kt

N

K-

3 He

K

n

N

, NKN Kt

N

The data suggests similar reaction mechanism.

Full treatment of this coupled system is neededKNN N-

θn = 0°