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

Are the chiral based potentials KN really energy ‐ dependent ? J. Révai MTA Wigner RCP, Budapest, Hungary 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  K p in various reactions, just below the threshold. PDG:     E i (1405 25 ) i MeV 2 I     0 KN Theoretically: an quasi ‐ bound state in the system,   which decays into the channel KN Constructing any multichannel interaction – more or less reproducing the scarce and old experimental data – one of the first questions is : “What  (1405) kind of it produces?” KN 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,  (1405) that the observed is the result of interplay of two T ‐ matrix poles. J. Revai MESON2018, Krakow,7 ‐ 12 June 2018 2

The subject of the talk is to challenge this opinion 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): c    ij 0 0 q v q 2 ( q q ) i ij j i j 4 f  Multichannel interaction, isospin basis, the channels:         I 0 I 1            I 0   I 1   I 1 i 1,2,3,4,5 KN , KN , , ,     q ̶ meson c.m. momentum i 2 ;   0 2 0 q m q ( ) q ̶ meson c.m. energy m M ̶ meson(baryon) masses i i i i i i c ̶ SU(3) Clebsch ‐ Gordan coefficients ij f  ̶ pion decay constant 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 c 1    ij 0' 0' q v q ( q q )  i ij j i j 3 64 F F m m i j i j  0 2 2 2 2 ( ) q m q q         0' 0 0 i i i i q q q m ( ) q  i i i i i i 2 M 2 M 2 nonrel i i i    m M / ( m M ) i reduced mass in channel i i i i i    K  , , F i ‐ meson decay constants i ̶ regularization, using separable potential representation with suitable u q ( ) cut ‐ off factors ensuring the convergence of the integrals i i J. Revai MESON2018, Krakow,7 ‐ 12 June 2018 4

V Finally, the potential entering the LS equation ij    q T W ( ) q q V q q V q G q W ( ; ) q T W ( ) q dq i ij j i ij j i is s s s s sj j s s has the form     q V q u q ( ) q v q u q ( ) ( g ( ) q g ( q ) g ( ) q g ( q )) i ij j i i i ij j j j ij iA i iB j iB i iA j which is a two ‐ term multichannel separable potential with form ‐ factors    g ( ) q u q ( ); g ( ) q u q ( ) ( ) q iA i i i iB i i i i i c and coupling constant 1    ij  ij 3 64 F F m m i j i j G The non ‐ relativistic propagator has the form s  2 q 2         1 s s G q W ( ; ) ( W m M i ) ,     s s s s 2 2 2 ( k q i ) s s s     k 2 ( W m M ) s where is the on ‐ shell c.m. momemtum in channel . s s s s J. Revai MESON2018, Krakow,7 ‐ 12 June 2018 5

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  q q ( ) q ‐ dependence of the interaction, by replacing in by its on ‐ shell i i i k value : i      ( ) q ( ) k W M i i i i i This is the s.c. on ‐ sell factorization approximation and as its result, the coupling    (2 W M M ) constant acquires the familiar energy ‐ dependent factor , i j ij which turns out to be responsible for the appearance of a second pole in the    KN system. 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 of the on ‐ shell factorization. Some technical details of the solution: J. Revai MESON2018, Krakow,7 ‐ 12 June 2018 6

Introducing the concise matrix notations for the momenta and form ‐ factors:       g 0 q k ( ) 0   1 ( A B ) 1 1      q k    g    ( ) ;  A(B)       0 ( ) q k   0 g   n n nA B ( ) and also for the propagator    G q W ( ; ) 0 1 1    G    ( W )     0 G q W ( ; )  n n   ij q V q the matrix element our potential can be written as with ij  V g λ g + g λ g A B B A  n n n ( Bold face letters denote matrices, ‐ number of channels ). T The ‐ matrix has the form : T = g τ g + g τ g + g τ g + g τ g , A AA A A AB B B BA B B BB B J. Revai MESON2018, Krakow,7 ‐ 12 June 2018 7

  M  τ 1 n n 2 n 2 n where the matrices are submatrices of the matrix  1   -1 λ - g G( ) g - g G( ) g W W B A B B     τ τ     1  AB AA M   τ τ     BB BA -1 - g G( ) g λ - g G( ) g W W   A A A B The convergence of all Green’s function matrix elements can be ensured, if the ( ) cut ‐ off factors are of the s. c . dipole type: u q i 2    2   i  ( ) u q   i 2 2   q i J. Revai MESON2018, Krakow,7 ‐ 12 June 2018 8

In order to understand the nature of the on ‐ shell factorization approximation, g let us consider one of these matrix elements, containing : B   2   u q ( ) ( ) q dq        G g G g i i 2 BA B A    ij i 2 2 ij ij k q i i     On ‐ shell factorization means, that is replaced by and ( ) k W M i q ( ) i i i k taken out from the integral. It can be seen, that for real positive , when i the integrand is singular, this might have a certain justification, however, for k complex , which is the case, when complex pole positions are sought, the i approximation seems to be meaningless. J. Revai MESON2018, Krakow,7 ‐ 12 June 2018 9

T Performing this operation for all matrix elements, the on ‐ shell ‐ matrix can be written as: k T k = k g τ g k A A with τ = τ + τ γ + γτ + γτ γ . AA AB BA BB  Here we introduced the matrix of on ‐ shell functions     ( ) k 0 1 1    γ         0 ( k )  n n k g = k g γ . and used the fact, that on ‐ shell we have It can B A τ be shown, that can be written as    1     1  τ λγ + γλ G , AA  τ which coincides with the corresponding ‐ matrix of the on ‐ shell n n U factorized potential : J. Revai MESON2018, Krakow,7 ‐ 12 June 2018 10

  U = g λγ + γλ g A A Now we can compare the results obtained from the “full” WT potential V = g λ g + g λ g A B B A U and its on ‐ shell factorized, energy ‐ dependent counterpart . Both potentials depend on the same set of 7 adjustable parameters      , , , , , , , F F  1 2 3 4 5 K 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: V For any reasonable combination of the parameters, the “full” WT potential KN produces only one pole below and close to the thershold, which can  U (1405) be associated with the , while produces the familiar two poles: one close to the threshold and a second one, much lower and broader. J. Revai MESON2018, Krakow,7 ‐ 12 June 2018 11

The potential parameters were fitted to the available experimental data:  K p six low ‐ energy cross sections and three threshold branching ratios             0 ( K p ) ( K p )    ; R ;               n 0 0 0 ( K p ) ( K p , )            ( K p , )  R    c ( K p all inelastic channels)  1 s E and the level shift in kaonic hydrogen. Results of the fit: J. Revai MESON2018, Krakow,7 ‐ 12 June 2018 12

Low ‐ energy cross sections: