The method of unitary clothing transformations in the theory of nucleon–nucleon scattering
The clothing procedure, put forward in quantum field theory (QFT) by Greenberg and Schweber, is applied for the description of nucleon–nucleon (–) scattering. We consider pseudoscalar ( and ), vector ( and ) and scalar ( and ) meson fields interacting with spin ( and ) fermion ones via the Yukawa–type couplings to introduce trial interactions between ”bare” particles. The subsequent unitary clothing transformations (UCTs) are found to express the total Hamiltonian through new interaction operators that refer to particles with physical (observable) properties, the so–called clothed particles. In this work, we are focused upon the Hermitian and energy–independent operators for the clothed nucleons, being built up in the second order in the coupling constants. The corresponding analytic expressions in momentum space are compared with the separate meson contributions to the one–boson–exchange potentials in the meson theory of nuclear forces. In order to evaluate the matrix of the – scattering we have used an equivalence theorem that enables us to operate in the clothed particle representation (CPR) instead of the bare particle representation (BPR) with its huge amount of virtual processes. We have derived the Lippmann–Schwinger(LS)–type equation for the CPR elements of the –matrix for a given collision energy in the two–nucleon sector of the Hilbert space of hadronic states and elaborated a code for its numerical solution in momentum space.
1 Introductory remarks
We know that there are a number of high precision, boson–exchange models of the two nucleon force , such as Paris Lacom80 (), Bonn MachHolElst87 (), Nijmegen Stocks94 (), Argonne WirStocksSchia95 (), CD Bonn Mach01 () potentials and a fresh family of covariant one–boson–exchange (OBE) ones GrossStad08 (). Note also successful treatments based on chiral effective field theory OrdoRayKolck94 (); EpelGloeMeiss00 (), for a review see Epel05 ().
In this talk, we would like to draw attention to the first application of unitary clothing transformations SheShi01 (); KorCanShe07 () in describing the nucleon-nucleon (–) scattering. Recall that such transformations , being aimed at the inclusion of the so–called cloud or persistent effects, make it possible the transition from the bare–particle representation (BPR) to the clothed–particle representation (CPR) in the Hilbert space of meson–nucleon states. In this way, a large amount of virtual processes induced with the meson absorption/emission, the –pair annihilation/production and other cloud effects can be accumulated in the creation (destruction) operators for the ”clothed” (physical) mesons and nucleons. Such a bootstrap reflects the most significant distinction between the concepts of clothed and bare particles.
In the course of the clothing procedure all the generators of the Poincaré group get one and the same sparse structure on SheShi01 (). Here we will focus upon one of them, viz., the total Hamiltonian
where free part and interaction depend on creation (destruction) operators in the BPR , i.e., referred to bare particles with physical masses KorCanShe07 (), where they have been introduced via the mass–changing Bogoliubov–type UTs. To be more definite, let us consider fermions (nucleons and antinucleons) and bosons (–, –, –, –mesons, etc.) interacting via the Yukawa-type couplings for scalar (s), pseudoscalar (ps) and vector (v) mesons (see, e.g., MachHolElst87 ()). Then, using a trick prompted by the derivation of Eq. (7.5.22) in WeinbergBook1995 () to eliminate in a proper way the vector–field component , we have with
with the boson fields and the fermion field , where the tensor of the vector field included. The mass (vertex) counterterms are given by Eqs. (32)–(33) of Ref. KorCanShe07 () (the difference - where a primary interaction is derived from replacing the ”physical” coupling constants by ”bare” ones).
The corresponding set involves operators for the bosons, for the nucleons and for the antinucleons. In their terms, e.g., we have the free pion and fermion parts
and the primary trilinear interaction
with the three-legs vertices. Here () the pion (nucleon) energy with physical mass , the fermion polarization index.
We have tried to draw parallels with that field–theoretic background which has been employed in boson–exchange models. First of all, we imply the approach by the Bonn group MachHolElst87 (); Mach01 (), where, following the idea by Schütte Schutte (), the authors started from the total Hamiltonian (in our notations),
with the boson-nucleon interaction
2 Analytic expressions for the quasipotentials in momentum space
As shown in SheShi01 (), after eliminating the so-called bad terms111By definition, they prevent the bare vacuum () and the bare one–particle states () to be eigenstates. from the primary Hamiltonian can be represented in the form,
The free part of the new decomposition is determined by
while contains only interactions responsible for physical processes, these quasipotentials between the clothed particles, e.g.,
In accordance with the clothing procedure developed in SheShi01 () they obey the following requirements:
i) The physical vacuum (the lowest eigenstate) must coincide with a new no–particle state , i.e., the state that obeys the equations
ii) New one-clothed-particle states etc. are the eigenvectors both of and ,
iii) The spectrum of indices that enumerate the new operators must be the same as that for the bare ones .
iv) The new operators satisfy the same commutation rules as do their bare counterparts , since the both sets are connected to each other via the similarity transformation
with a unitary operator to be obtained as in SheShi01 ().
It is important to realize that operator is the same Hamiltonian . Accordingly [10,11] the – interaction operator in the CPR has the following structure:
where the symbol denotes the summation over nucleon spin projections, , etc.
where the mass of the clothed boson (its physical value) and . In the framework of the isospin formalism one needs to add the factor in the corresponding expressions.
At this point, our derivation of the vector-boson contribution (21) is to be specifically commented. Actually, it is the case, where for a Lorentz–invariant Lagrangian it is not necessarily to have ”… the interaction Hamiltonian as the integral over space of a scalar interaction density; we also need to add non–scalar terms to the interaction density …” (quoted from p.292 of Ref. WeinbergBook1995 ()). Let us recall that the density in question has the property,
where the operators realize a unitary irreducible representation of the Poincaré group in the Hilbert space of states for free (non–interacting) fields.
By definition, the first clothing transformation () eliminates all interactions linear in the coupling constants, viz.,
Following Ref.SheShi01 () we have
if . Here
where is the Lorentz scalar.
The corresponding interaction operator in the CPR (12) can be written as
where we have kept only the contributions of the second order in the coupling constants, so
We point out that all quantities in the r.h.s. of Eq.(25) depend on the new creation(destruction) operators . In particular, it means that in the standard Fourier expansions of the fields involved in the definitions of and one should replace the set by the set . Thus, there is an essential distinction between (), on the one hand, and the first(second) integral in the r.h.s. of Eq.(5), on the other hand.
For this exposition we do not intend to derive all interactions between the clothed mesons and nucleons, allowed by formula (25). Our aim is more humble, viz., to find in the r.h.s. of Eq.(25) terms of the type (17), responsible for the – interaction. Meanwhile, in case of the vector mesons we encounter an interplay between the commutator and the integral (26). Indeed, after a simple algebra we find
where the first term has the structure of Eq.(17) with the coefficients by (21). At the same time the second term completely cancels the non–scalar operator . The latter may be associated with a contact interaction since it does not contain any propagators (cf. the approach by the Osaka group TamuraSato88 ()), being expressed through the . In other words, the first UCT enables us to remove the non–invariant terms directly in the Hamiltonian. In our opinion, such a cancellation, first discussed here, is a pleasant feature of the CPR.
Moreover, as it was shown in Ref.SheShi01 (), for each boson included the corresponding relativistic and properly symmetrized – interaction, the kernel of integral equations for the – bound and scattering states, is determined by
where we have separated the so–called direct
terms. For example, the one–pion–exchange contribution can be divided into the two parts:
to be depicted in Fig.1, where the dashed lines correspond to the following Feynman–like ”propagators”:
on the left panel and
3 The field–theoretic description of the elastic N–N scattering
3.1 The –matrix in the CPR
In order to evaluate the – scattering amplitude for the collision energy we will regard a field operator that meets the equation
and whose matrix elements on the energy shell can be expressed through the phase shifts and mixing parameters.
Unlike nonrelativistic quantum mechanics (NQM) in relativistic QFT the interaction does not conserve the particle number, being the spring of particle creation and destruction. The feature makes the problem of finding the – scattering matrix much more complicated than in the framework of nonrelativistic approach.
Such a general field–theoretic consideration can be simplified with the help of an equivalence theorem ShebekoFB17 () according to which the matrix elements in the Dirac (D) picture, viz.,
are equal to the corresponding elements
of the matrix in the CPR once the UCTs
obey the condition
The operator in the CPR satisfies the equation
and the matrix
where ( ) are the ( ) eigenvectors, may be evaluated relying upon properties of the new interaction .
If in Eq.(36) we approximate by , then initial task of evaluating the CPR matrix elements can be reduced to solving the equation
3.2 The –matrix equation and its angular–momentum decomposition
For practical applications one prefers to work with the corresponding –matrix that meets the set of equations
with and , where the operation means the summation over nucleon polarizations and the integration over nucleon momenta. The kernel of Eq.(39) is
The subsequent calculations are essentially simplified in the center–of–mass system (c.m.s) in which
Here the quantum numbers are the individual spin (isospin) projections.
Accordingly Eq. (27)
Following a common practice we are interested in the angular–momentum decomposition of Eq.(40) assuming a nonrelativistic analog of relativistic partial wave expansions (see Werle66 () and refs. therein) for two–particle states. For example, the clothed two–nucleon state (the so–called two–nucleon plane wave) can be represented as
where , and are, respectively, total angular momentum, spin and isospin of the pair, being the eigenvalues of the operators , and . Here
where () the orbital (spin) momentum of the fermion field,
where are the Pauli spinors. For brevity, we do not show the isospin operator .
The corresponding eigenvalue equations look as
Doing so, we have introduced the vectors222For a moment, the isospin quantum numbers are suppressed.
One should note that in our case the separable ansatz
often exploited in relativistic quantum mechanics (RQM) (see, e.g., Werle66 () and KeisPoly91 ()) does not work. However, one can employ the similarity transformation 333Sometimes it is convenient to handle the operators and their adjoints that meet covariant relations
with the Wigner rotation (e.g., for rotations ) and the property of the physical vacuum to be invariant with respect to unitary transformations in the CPR (some details can be found in a separate paper).
The use of expansion (43) gives rise to the well known JST representation, in which