The N(1440) Roper resonance in the nuclear model with explicit mesons

We show that the N(1440) Roper resonance naturally appears in the nuclear model with explicit mesons as a structure in the continuum spectrum of the physical proton which in this calculation is made of a bare nucleon dressed with a pion cloud


Introduction
The N(1440) Roper resonance is a relatively broad nucleon resonance with the mass of about 1440 MeV and the width of about 350 MeV [1].Although its nature is still debated (see [2,3,4,5] and references therein) one line of thought is that it consists of a quark core augmented by a meson cloud.This concurs well with the nuclear model with explicit mesons (MEM) where the physical nucleon is made of a bare nucleon-the quark coredressed with a meson cloud [6,7].One might therefore expect that in MEM the Roper resonance should somehow reveal itself in the continuum spectrum of the physical proton.
In this contribution we investigate the continuum spectrum of the physical proton within one-pion MEM in the hope to identify the Roper resonance and establish the parameters of MEM that are consistent with the tabulated mass and width of the resonance.The method we use is calculation of the strength-function of a certain fictional reaction, where the proton is excitated from the ground state into continuum, with the subsequent fit of the calculated strength-function with a Breit-Wigner distribution.

The physical nucleon in MEM
The MEM is a nuclear interaction model, based on the Schrodinger equation, where the nucleons do not interact with each other via a potential but rather emit and absorb mesons [6,7].The mesons are treated explicitly on the same footing as the nucleons.
The physical nucleon in MEM is represented by a superposition of states where the bare nucleon is surrounded by different number of (virtual) mesons.In one-pion approximation the physical nucleon is a superposition of two states: the bare nucleon, and the bare nucleon surrounded by one pion.The corresponding wave-function of the physical nucleon, Ψ N , is a two-component vector, where ψ 0 is the (wave-function of the) state with the bare nucleon and no pions, ψ 1 -the state with a bare nucleon and one pion, ⃗ R is the center-of-mass coordinate of the system, and ⃗ r is the coordinate between the bare nucleon and the pion.
The Hamiltonian H that acts on this wave-function is a matrix, where K N , K π are kinetic energy operators for the bare nucleon and the pion, m N and m π are masses of the bare nucleon and the pion, and W and W † are pion emission and absorption operators (also called the nucleon-pion coupling operators).
The corresponding Schrodinger equation is given as with the normalization condition where E is the energy of the system and V is the normalization volume.
The simplest W -operator that is consistent with conservation of isospin, angular momentum, and parity can be written as where ⃗ σ is the vector of Pauli matrices that act on the spin of the nucleon, ⃗ τ is the isovector of Pauli matrices that act on the isospin of the nucleon 1 , and where F (r) is a (short-range) form-factor.The dimension of W is E/ √ V , therefore it might be of convenience to choose where f (r) is normalized such that in which case the strength factor S w has the dimension of energy and one can (hopefully) meaningfully compare form-factors of different shapes 2 .
1 The isospin factor ⃗ τ⃗ π is given as where π 0 , π + , and π − are the physical pions and where the τ -matrices are given as 2 The Gaussian form-factor normalized according to ( 9) is given as 3 The semi-relativistic Schrodinger equation for the bare proton dressed with a pion We shall search for the wave-function of the physical proton in the center-of-mass system in the form where p is the proton isospin state, and where ↑ is the spin-up state, and where the dimensionless constant c 0 and the function ϕ(r) are to be found by solving the Schrodinger equation.The normalization condition is given as With the ansatz (11) the Schrodinger equation ( 3) turns into the following system of equations for the constant c 0 and the function ϕ(r), It is of advantage to introduce the "radial" function u(r), with the simple boundary condition at the origin, and the normalization condition The Schrodinger equation then becomes In the center-of-mass frame the nucleon and the pion have equal momenta with opposite signs, −⃗ p and ⃗ p. Their (relativistic) kinetic energies are therefore given as The momentum as a quantum-mechanical operator in coordinate space is given as Correspondingly the kinetic energies operators are With these kinetic energies the Schrodinger equation turns into the following system of integro-differential equations, where the function is representable by a Maclaurin series 4 .The single action of the ∇ 2 operator on ⃗ r r 2 u(r) is given as where the action of the operator D is defined as Repeating the action suggests that for any non-negative integer n Therefore for any function f (x) that can be represented as a power series the following holds true, Inserting this into (24) gives, finally, the sought semi-relativistic radial Schrodinger equation for the physical proton in one-pion MEM, with the boundary condition u(0) = 0.
4 Real symmmetric eigenproblem The system of equations (31) can be solved numerically by discretizing the r-variable and using the finite-difference approximations for the derivatives and integrals.
Let us introduce a regular grid, where ∆r is the grid spacing.The integral in the first equation in (31) can then be approximated as Introducing the auxiliary tilde-variables, we turn the system of equations (31) into the following form, where the numbers f K ( D) ij are the matrix elements of the matrix representation of the operator f K ( D) on the grid; and where the ordinary matrix normalization condition applies, The system of equations (35) with the normalization condition (36) can be rewritten as a real symmetric matrix eigenproblem, where the Hamiltonian matrix H is given as The D operator on the grid can be built using the second order central finite-difference approximation for the second derivative, Now the matrix f K ( D) can be built in the following way: if λ i and v i are the eigenvalues and eigenvectors of the D operator, then the matrix representation of any Maclaurin expandable operator f ( D) is given as where V is the matrix of eigenvectors v i and where f (λ) is a diagonal matrix with diagonal elements f (λ i ).
The Hamiltonian matrix built this way has the box boundary conditions inbuilt, where R max = (n + 1)∆r.Diagonalization of this Hamiltonian matrix produces one state with the energy below m N (this state is the physical proton5 -the bound state of the bare proton and the pion) and n discretized-continuum states with energies above the pion emission threshold 6 .These states decay into a nucleon and a pion, and the Roper resonance, if any, must be lurking somewhere there.An example of the radial eigenfunctions is shown on Figure 1: the ground state is a localised bound state of the bare nucleon and the pion; the "below-resonance" continuum wave-function shows little presence at the shorter distances; the "on-resonance" wavefunction has a much larger amplitude in the inner region.

The strength-function of a fictional transition
A resonance is usually identified as a peak in a reaction cross-section with an approximately Breit-Wigner shape.The Roper resonance decays largely (55-75% [1]) into a nucleon and a pion, therefore it should be possible to observe the resonance in a reaction where our dressed proton is excited from the ground state into a continuum spectrum state which decay exactly into a nucleon and a pion.Let us consider a reaction where the proton undergoes a transition, caused by certain operator X, from the ground state Ψ 0 to a state in the continuum Ψ i (which then decays into a nucleon and a pion).The amplitude of this quantum transition is given, in the Born approximation, by the matrix element Since a resonance should not depend on the way the state Ψ i is populated, the particular form of the X operator should be unimportant (as long as the matrix element is not identically zero) and one can just as well use a fictional operator.We have chosen the following matrix element, as experimentation shows that this one produces the best looking strength-functions (see below).The cross-section of an excitation reaction into the continuum spectrum states with energies E ± ∆E 2 is determined by the so called strength-function, S(E), which is defined as Table 1: The results of the exploratory calculations: S w and b w are the strength and the range parameters of the Gaussian form-factor (50), M and Γ are the mass and the width of the resonance from the Breit-Wigner fit (48) to the strength-function (47), B is the binding energy of the virtual pion, and ) is the contribution of the state with the virtual pion to the total norm of the physical proton wave-function.In the box-discretized approximation that we use here the strength-function of a transition into a discretized-continuum state i with the energy E i is given as where ∆E i = E i+1 − E i .One can determine the parameters of a resonance, the mass and the width, by applying a Breit-Wigner fit to the strength-function [8].Unfortunately the Roper resonance is broad and is located close to threshold making it necessary to use a width that is energydependent.We shall use the following phenomenological parametrization of the strengthfunction (cf.[9]), θ is the step function, E th is the threshold energy, and where the mass M , width Γ, and power p are the fitting parameters (p ∼ 1).

Results
We use the Gaussian form-factor in the normalized form (10), where the strength, S w , and the range, b w , are the model parameters that are varied to reproduce the given mass M and width Γ of the resonance.
Exploratory calculations show that it is possible to reproduce the tabulated mass of the Roper resonance, 1440 Mev, and any given width within the tabulated limits, 250-450 MeV, with a relatively narrow range of the model parameters as indicated in Table 1 and on Figure 2.

Conclusion
We have shown that in the nuclear model with explicit mesons (MEM) the Roper resonance appears as a structure in the continuum spectrum of the physical proton (in one-pion MEM the physical proton is made of a bare nucleon dressed with a pion).We have established the range of the model parameters (the strength and the range of the nucleon-pion coupling operator) that reproduce the tabulated mass and width of the resonance.With these parameters the pion component in the physical proton takes about 10% of the norm of the wave-function and reduces the mass of the bare proton by about 5%.These numbers are much smaller then the corresponding estimates from the pion photo-production cross-section calculation [7].