KN really energy dependent ? J. Rvai MTA Wigner RCP, Budapest, - - PowerPoint PPT Presentation

Are the chiral based potentials really energy‐dependent ?

• J. Révai

MTA Wigner RCP, Budapest, Hungary

KN

Few Body Systems 59(2018)49

• J. Revai

MESON2018, Krakow,7‐12 June 2018 1

(1405) 

The is one of the basic objects of strangeness nuclear physics. Experimentally: a well‐pronounced bump in the missing mass spectrum in various reactions, just below the threshold. PDG: Theoretically: an quasi‐bound state in the system, which decays into the channel Constructing any multichannel interaction – more or less reproducing the scarce and old experimental data – one of the first questions is : “What kind of it produces?” At present it is believed, that theoretically substantiated interactions (potentials) – apart from the phenomenological ones – can be derived from the chiral perturbation expansion of the SU(3) meson‐baryon Lagrangian. For these interactions the widely accepted answer to the above question is, that the observed is the result of interplay of two T‐matrix poles.

  K p



(1405 25 ) 2 E i i MeV     I  KN      KN (1405)  KN (1405) 

• J. Revai

MESON2018, Krakow,7‐12 June 2018 2

Starting point Lowest order Weinberg‐Tomozawa (WT) meson‐baryon interaction term of the chiral SU(3) Lagrangian (from the basic paper E. Oset, A. Ramos NPA 635(1998) 99):

2 2 ; i i i

q m q  

Multichannel interaction, isospin basis, the channels:

     

1 1 1

1,2,3,4,5 , , , ,

I I I I I

i KN KN   

    

            

i

q

̶ meson c.m. momentum

i

q

̶ meson c.m. energy

2 (

) 4

ij i ij j i j

c q v q q q f   

̶ meson(baryon) masses

( )

i i

m M

ij

c

̶ SU(3) Clebsch‐Gordan coefficients

f

̶ pion decay constant

The subject of the talk is to challenge this opinion

• J. Revai

MESON2018, Krakow,7‐12 June 2018 3

Dynamical framework: (a) relativistic: BS equation, relativistic kinematics (b) non‐relativistic: LS equation, non‐relativistic kinematics Our choice is (b), having in mind application for N>2 systems In practical calculations the original interaction is used with certain modifications: ̶ normalization + rel. correction to meson energies reduced mass in channel ‐ meson decay constants ̶ regularization, using separable potential representation with suitable cut‐off factors ensuring the convergence of the integrals

0' 0' 3 2 2 2 2 0'

1 ( ) 64 ( ) ( ) 2 2 2 / ( )

ij i ij j i j i j i j i i i i i i i i i i i i i i i i i i

c q v q q q F F m m q m q q q q q m q M M m M m M                  ( )

i i

u q i

 

, ,

i

F i K  

nonrel

• J. Revai

MESON2018, Krakow,7‐12 June 2018 4

Finally, the potential entering the LS equation has the form which is a two‐term multichannel separable potential with form‐factors and coupling constant

( ) ( ; ) ( )

i ij j i ij j i is s s s s sj j s s

q T W q q V q q V q G q W q T W q dq  

ij

V ( ) ( ) ( ( ) ( ) ( ) ( ))

i ij j i i i ij j j j ij iA i iB j iB i iA j

q V q u q q v q u q g q g q g q g q     ( ) ( ); ( ) ( ) ( )

iA i i i iB i i i i i

g q u q g q u q q   

The non‐relativistic propagator has the form where is the on‐shell c.m. momemtum in channel .

s

G

2 1 2 2

2 ( ; ) ( ) , 2 ( )

s s s s s s s s s

q G q W W m M i k q i    



        2 ( )

s s s s

k W m M     s

3

1 64

ij ij i j i j

c F F m m    

• J. Revai

MESON2018, Krakow,7‐12 June 2018 5

( ) ( )

i i i i i

q k W M     

In order to simplify the solution of the dynamical equation ‐‐ both in (a) and (b) approaches ‐‐ a commonly used approximation is to remove the inherent ‐ dependence of the interaction, by replacing in by its on‐shell value :

q

i

q ( )

i i

q 

i

k

This is the s.c. on‐sell factorization approximation and as its result, the coupling constant acquires the familiar energy‐dependent factor , which turns out to be responsible for the appearance of a second pole in the system.

ij



(2 )

i j

W M M   KN   

The multichennel two‐term separable form of the potential allows an exact solution of the LS equation both with the full and on‐shell factorized WT interactions thus offering a possibility to check the validity and/or consequences

• f the on‐shell factorization.

Some technical details of the solution:

• J. Revai

MESON2018, Krakow,7‐12 June 2018 6

1 1

( ; ) ( ) ( ; )

n n

G q W W G q W          G     

Introducing the concise matrix notations for the momenta and form‐factors:

1 ( ) 1 1 ( )

( ) ( ) ; ( )

A B n n nA B

g q k q k g                    

A(B)

q k g          

and also for the propagator the matrix element our potential can be written as with

ij

 ij

q V q 

A B B A

V g λ g + g λ g

( Bold face letters denote matrices, ‐ number of channels ). The ‐ matrix has the form :

n n  n T ,

A AA A A AB B B BA B B BB B

T = g τ g + g τ g + g τ g + g τ g

• J. Revai

MESON2018, Krakow,7‐12 June 2018 7

where the matrices are submatrices of the matrix

τ n n  2 2 n n 

1

M 

1 1

W W M W W

 

                 

• 1

B A B B AB AA BB BA

• 1

A A A B

λ - g G( ) g

• g

G( ) g τ τ τ τ

• g

G( ) g λ - g G( ) g

The convergence of all Green’s function matrix elements can be ensured, if the cut‐off factors are of the s. c . dipole type:

( )

i

u q

2 2 2 2

( )

i i i

u q q          

• J. Revai

MESON2018, Krakow,7‐12 June 2018 8

In order to understand the nature of the on‐shell factorization approximation, let us consider one of these matrix elements, containing :

B

g

 

 

2 2 2

( ) ( ) 2

i i ij i ij ij i

u q q dq k q i        



BA B A

G g G g 

On‐shell factorization means, that is replaced by and taken out from the integral. It can be seen, that for real positive , when the integrand is singular, this might have a certain justification, however, for complex , which is the case, when complex pole positions are sought, the approximation seems to be meaningless.

( )

i q

 ( )

i i i

k W M   

i

k

i

k

• J. Revai

MESON2018, Krakow,7‐12 June 2018 9

Performing this operation for all matrix elements, the on‐shell ‐ matrix can be written as: with

T

A A

k T k = k g τ g k .

AA AB BA BB

τ = τ + τ γ + γτ + γτ γ

Here we introduced the matrix of on‐shell functions and used the fact, that on‐shell we have It can be shown, that can be written as which coincides with the corresponding ‐ matrix of the on‐shell factorized potential :

1 1

( ) ( )

n n

k k            γ      .

B A

k g = k g γ τ

 

 

1 1

,

 

 

AA

τ λγ + γλ G n n  τ U 

• J. Revai

MESON2018, Krakow,7‐12 June 2018 10

 

A A

U = g λγ + γλ g

Now we can compare the results obtained from the “full” WT potential and its on‐shell factorized, energy‐dependent counterpart . Both potentials depend on the same set of 7 adjustable parameters which have to be determined by fitting to experimental data. But before proceeding to the discussion of fit results we make our most important statement:

A B B A

V = g λ g + g λ g U

1 2 3 4 5

, , , , , , ,

K

F F



    

For any reasonable combination of the parameters, the “full” WT potential produces only one pole below and close to the thershold, which can be associated with the , while produces the familiar two poles:

• ne close to the threshold and a second one, much lower and broader.

V KN (1405)  U

• J. Revai

MESON2018, Krakow,7‐12 June 2018 11

The potential parameters were fitted to the available experimental data: six low‐energy cross sections and three threshold branching ratios and the level shift in kaonic hydrogen. Results of the fit:

K p



( ) ( ) ; ; ( ) ( , ) ( , ) ( all inelastic channels)

n c

K p K p R K p K p K p R K p              

             

                1s E 

• J. Revai

MESON2018, Krakow,7‐12 June 2018 12

Low‐energy cross sections:

Branching ratios and :

E 

ϒ Rc Rn ΔE(eV) U 2.35 0.664 0.194 302 ‐ 294 i V 2.32 0.671 0.202 350 – 279 i Exp 2.36±0.04 0.664±0.011 0.189±0.015 (283±36) ‐ (271±46) i The obtained parameter values (in MeV) : Fπ FK β1 β2 β3 β4 β5 U 107 109 1247 1622 919 959 443 V 80.8 132 1094 960 516 537 629 Pole positions (MeV): z1 z2 U 1428 – 35 i 1384 – 62 i V 1425 – 21 i ̶

• The fits could be probably further improved to obtain better parameter

sets, however, this was not the main aim of the work.

• More or less equal quality fits can be obtained for both potentials

and , but for very different parameter values. This means, that can not be considered as an approximaon to ̶ they are basically different interactions.

• While the on‐shell properties of the corresponding T‐matrices can be

made similar by suitable choice of the parameters, this is not true for their analytical behavior in energy, which governs the pole positions.

U V U V

• J. Revai

MESON2018, Krakow,7‐12 June 2018 15

Conclusions

• It was shown, that the energy‐dependence of the WT term of the

interaction, derived from the chiral SU(3) Lagrangian, and responsible for the appearance of a second pole in the region, follows from the on‐shell factorization approximation.

• Without this approximation a new, chiral based, energy independent

potential was derived, which supports only one pole in the region of the resonance.

• The widely accepted “two‐pole structure” of the state thus

becomes questionable.

• In the calculations for systems the use of the new potential

avoids the difficulties arising from the energy‐dependence of the interactions.

KN (1405)  KN (1405)  (1405)  2 N  KN

• J. Revai

MESON2018, Krakow,7‐12 June 2018 16

KN    Most recent and accurate information on the line‐shape are the CLAS photoproduction data. For their analysis at present we have the semi‐phenomenological formula of Roca and Oset . It contains a few adjustable parameters and the T‐matrix elements of the

• potential. With its help we calculated the pure missing

mass spectrum for the unchanged

• potential. The preliminary results

for the lowest energy bins: I    V (1405)   The complete analysis of CLAS data (including the charged states) is our next task