Dyson–Schwinger approach to pion–nucleon scattering using time-ordered perturbation theory

From the late 1970s through to the early 1990s, much effort was devoted to modeling the coupled πNN − NN system.^1–7 For practical reasons, almost all these works used the non-relativistic framework of Time-Ordered Perturbation Theory (TOPT) and implemented the so-called one-pion approximation (OPA) whereby all states containing two or more pions were neglected. The OPA allowed for the use of Faddeev-like equations for the coupled πNN − NN system but contained an inconsistency in that the nucleons in NN states were only partially dressed.⁸ In this respect, we note that in the OPA, nucleons in πNN states contain no explicit dressing at all, while the πN input to these equations (consisting of the dressed πNN vertex and a “background” πN t matrix) allows for a fully dressed nucleon in the pole-term. This inconsistency was finally resolved by relaxing the OPA and reformulating the πNN − NN equations in terms of convolution integrals, which resulted in all nucleons in the model (including the nucleons in the πN and NN input to the model) being fully dressed.^9,10 Unfortunately, no numerical work on the resulting consistent formulation has been carried out so far. Here, we would like to make a modest contribution to the future solution of the convolution equations by constructing the πN input to these equations. This input consists of separable πN t matrices that describe πN scattering data, and because all nucleons are fully dressed, it necessarily involves the solution of Dyson–Schwinger equations.

In the process of carrying out this work, it became apparent that the constructed πN equations and their solution can also be useful from a pedagogical point of view. In particular, we have in mind that the study of Relativistic Quantum Field Theory (RQFT) forms one of the most difficult challenges in a physics student’s education. Yet some of the most fundamental concepts encountered in RQFT, such as particle creation and annihilation, self-energy, renormalization, and associated equations such as the Dyson–Schwinger equations, can be easily demonstrated in the non-relativistic framework of Time-Ordered Perturbation Theory (TOPT) where little more than a knowledge of standard quantum mechanics is needed and, in particular, where the substantial extra machinery of a relativistically covariant approach is completely avoided. The current paper is able to provide just such a demonstration on the example of a simple yet field-theoretically complete model of pion–nucleon (πN) scattering where phenomenological separable potentials and a phenomenological bare πNN vertex are used to fit experimental πN phase shifts. The equations of the model are conveniently represented in a diagrammatic form in Fig. 2 of Sec. IV and have their origin in the works that described the coupled NN − πNN system using few-body equations;^1,2,4–6 however, all these works involved the OPA where all states of two or more pions are neglected. Our innovation, therefore, is to implement a complete dressing of all nucleon propagators appearing in the model, a task that involves the numerical solution of the Dyson–Schwinger equations [as represented by Figs. 2(b)–2(d)] and which, in the process, demonstrates some of the main features of RQFT presented in a simple non-relativistic setting. To enhance the pedagogical aspects of this paper, we endeavor to give a more detailed presentation of our work than may normally be expected.

The pion–nucleon model we consider involves only pion and nucleon degrees of freedom. To take into account that pions can get created and absorbed by the nucleon, we shall use a Hilbert space $H$ that is a direct sum of subspaces, each consisting of states with a different number of pions; thus,

Each of these subspaces contain free particle states that shall be labeled by the momenta of the particles (to save on notation, we suppress spin and isospin labels); thus, $|p〉∈HN$⁠, $|kp〉∈HπN$⁠, $|k1k2p〉∈HππN$⁠, etc., where p is the momentum of the nucleon and k, k₁, k₂, … are pion momenta. This multi-pion plus one-nucleon system is then described within a second quantization approach¹¹ where the Hamiltonian H is expressed in terms of annihilation and creation operators, $aπ,aπ†$ for pions and $aN,aN†$ for nucleons, defined such that

where |0⟩ is the vacuum state, and satisfying the commutation and anti-commutation relations

In particular, the free Hamiltonians for pions and nucleons are given by

where

with m_π being the mass of the pion and m being the mass of the nucleon. Note that for simplicity of presentation, we have used non-relativistic kinematics for the nucleon and relativistic kinematics for the pion (necessitated by the relatively small mass of the pion). Such a “semi-relativistic” choice of kinematics should be sufficiently accurate to describe πN scattering for the center of mass (c.m.) energies below m + 3m_π, being approximately the experimental onset of inelasticity, and will provide a convenient connection to related works using such kinematics.^12,13

Because the interactions that we shall use contribute to the physical mass m of a nucleon, the rest mass used for the nucleon in Eq. (4b), the so-called bare mass m₀, needs to be chosen appropriately (by contrast, our interactions do not contribute to the physical mass of the pion). We note that the free particle states are eigenstates of the total free Hamiltonian H₀,

so that

The full Hamiltonian is then specified as

where H_I is the interaction Hamiltonian. We assume that H_I is due to strong interactions only; thus, parity is conserved in our model. In this work, we shall assume a model interaction Hamiltonian of the form

where

In the above (and below), ω_π is to be understood as meaning ω_π(k). This interaction Hamiltonian describes the elementary processes πN ↔ N whose amplitude is given by the real function $F0(p,p′)=F̄0(p′,p)$ and which we shall refer to as the bareπNN vertex. It will be assumed that F₀(p, p′) is invariant under rotations and under space inversion so that angular momentum and parity are conserved in the resulting description of πN scattering.

With the model thus defined, we can now define the Green function G that shall be used to describe pion–nucleon scattering as

In a similar way, we define the single nucleon Green function g, also called the dressed nucleon propagator, as

the bare nucleon propagatorg₀ as

and the Green function for the process N → πN, $f̃$⁠, as

To help evaluate these Green functions, it is useful to note the commutation relations

which follow from Eqs. (3), (4), and (9). In a similar way, it follows that

where p + k = p′. Thus,

Despite the relative simplicity of the quantum field theoretic pion–nucleon model introduced above, there is no known way to calculate any of the Green functions of Eqs. (11) and (12), or (14) exactly. We therefore follow the usual path taken in both quantum mechanics and quantum field theory and introduce a perturbation expansion. In our case, this means expanding the Green function operator 1/(E⁺ − H) into a series around the free Green function operator 1/(E⁺ − H₀),

This expansion defines time-ordered perturbation theory (TOPT) and has the feature that momentum matrix elements of each term of the expansion can be represented graphically by a “perturbation diagram.” For example, if one evaluates this expansion for the matrix element defining the dressed nucleon propagator g(E) of Eq. (12), one obtains a series whose graphical representation is given in Fig. 1. The diagrammatic rules that relate these diagrams to mathematical expressions can be found by evaluating the momentum matrix elements of just the first three terms of Eq. (18). With the help of Eq. (17), one obtains

where p″ = p − k so that

where

Comparing Fig. 1 and Eq. (20), it is evident that a solitary solid line corresponds to the bare nucleon propagator g₀, every open-circle N → πN (πN → N) vertex in Fig. 1 corresponds to the bare vertex function $f0(f̄0)$ defined in Eq. (21), every intermediate state of solid (nucleon) and dashed (pion) lines corresponds to a propagator whose mathematical expression is given by the appropriate matrix elements of 1/(E⁺ − H₀), and every closed loop corresponds to an integral over the pion momentum.

As the Green function G(p′, k, p, E) describing πN scattering can be equated to a complete sum of perturbation diagrams, this provides the opportunity to rearrange this sum so as to express it completely in terms of useful quantities such as potentials, t matrices, and other Green functions, thereby generating scattering equations of a similar nature to those found in quantum mechanics (e.g., the Lippmann–Schwinger equation). Such a rearrangement leads to a set of coupled equations for πN scattering illustrated in Fig. 2. First derived by Mizutani and Koltun using Feshbach projection operators,¹ these equations were later derived in the same context of TOPT as used here⁴ and also in the context of RQFT.¹⁴ Here, we shall give a brief derivation following the arguments used in Ref. 4.

We start by writing the Green function G(p′, k, p, E) in the operator form as

where G₀(E) is the “dressed πN propagator” consisting of all the disconnected diagrams of G(E) and t(E) is the πN → πN t matrix, which is defined by this equation. Note that each quantity in Eq. (22) is an operator acting in subspace $HπN$⁠, with G(E) being specifically defined such that

It is evident that the term G₀(E)t(E)G₀(E) in Eq. (22) consists of all possible connectedπN → πN diagrams and that t(E) consists of the same diagrams but with no attached initial- and final-state πN propagators, colloquially referred to as diagrams with “chopped legs.”

Further progress can be made by defining a “background” πN t matrix t⁽¹⁾(E) as the sum of all diagrams of t(E) that have one or more pions in every intermediate state, as one can then write

where $f(E)(f̄(E))$ is the “dressed vertex” consisting of all possible N → πN (πN → N) chopped-leg diagrams with at least one pion in every intermediate state. Similarly, one can define t matrix t⁽²⁾(E) as the sum of all diagrams of t(E) that have two or more pions in every intermediate state, in which case we can write Lippmann–Schwinger-like equations

where v(E) ≡ t⁽²⁾(E) plays the role of a πN → πN potential. Using a similar argument, one can obtain the equations

where f₀(E) ≡ f⁽²⁾(E) $f̄0(E)≡f̄(2)(E)$ is the “bare vertex” consisting of all possible N → πN (πN → N) chopped-leg diagrams with at least two pions in every intermediate state. Finally, one can similarly write

where

is the nucleon “self-energy” or “dressing” term consisting of all diagrams of g(E) with at least one pion in every intermediate state but with chopped legs. The set of equations consisting of Eqs. (24)–(28) are illustrated in Fig. 2 and provide an exact and useful way of expressing the πN t matrix t(E).

In the context of RQFT, Eq. (27) [illustrated in Fig. 2(d)] is known as the Dyson equation, while the coupled set of equations [Eq. (25a), Eq. (26a), and Eq. (27)], illustrated in Figs. 2(b)–2(d), are known as the Dyson–Schwinger (DS) equations, and that is how we shall refer to the TOPT versions of these equations here. A feature of the Dyson equation is the fact that the dressed nucleon propagator g is expressed in terms of the self-energy term Σ, which itself is expressed in terms of g via the πN propagator G₀(E). Such self-referencing also occurs for the background t matrix t⁽¹⁾ and the dressed vertex f in the DS equations, a feature that makes these equations embody a lot of physics even in the case where the bare vertex f₀ and background potential v are modeled phenomenologically, as will be the case in Sec. V.

Here, we shall follow an often used procedure where the bare πNN vertex f₀ and the “background” πN potential v are modeled by energy-independent parameterized phenomenological functions; however, unlike in all such previous models,^1,2,4–6,13 we shall not be using the approximation where the exact πN propagator G₀(E, k, p), defined as

is modeled by the pole term 1/[E⁺ − E_N(p) − m − ω_π(k)]; rather, we shall retain its full exact form, which in the model specified by the Hamiltonian of Eqs. (4) and (9) is given by

As mentioned above, it is this exact form for G₀ that gives the DS equations the property of retaining a rich amount of physics despite what may be lost by taking phenomenological forms for f₀ and v.

It is convenient to solve the Dyson–Schwinger set of equations, given in the operator form in Eqs. (25)–(28), in the center of mass (c.m.) of the πN system so that k + p = k′ + p′ = 0, in which case Eq. (30) can be expressed as

where

In order to reduce the dimension of these equations from 3 to 1, we shall work in the partial wave representation using the basis states

where t₁ = 1 is the isospin of the pion, t₂ = 1/2 is the isospin of the nucleon, s = 1/2 is the spin of the nucleon, l specifies the πN relative orbital angular momentum, t is the total isospin, and j is the total angular momentum. By construction, the model Hamiltonian of Eq. (8) is invariant under rotations, which implies that all matrix elements using the above partial wave basis states will not depend on the magnetic quantum numbers m_j and m_t. Similarly, the model Hamiltonian is chosen to be invariant under space inversion, thus ensuring parity is conserved in our model, which, in turn, leads to πN partial wave matrix elements that preserve the value of l. We shall refer to the partial wave specified by the quantum numbers {ljt} using the usual (for πN scattering) spectroscopic notation of the form “L_{(2t) (2j)}.” Because the nucleon has quantum numbers t = j = 1/2 and the pion has an intrinsic parity of −1, it follows that the first term on the right hand side (RHS) of Eq. (24), the so-called nucleon pole term, contributes only in the P₁₁ partial wave. Likewise, the background πN t matrix t⁽¹⁾ appearing in the expression for the dressed πNN vertices of Eq. (26) is the one in the P₁₁ partial wave. Thus, restricting the discussion to πN scattering in the P₁₁ partial wave, we can write the numerical form of the partial wave equations corresponding to Fig. 2 as

where partial wave labels have been omitted to save on notation. For scattering in partial waves other than P₁₁, essentially the same equations would apply, the only differences being that the pole term in Eq. (34a) would not appear, and one would need to distinguish the t⁽¹⁾ appearing in Eq. (34a) from the P₁₁ partial wave t⁽¹⁾ appearing in Eq. (34c).

In order to help solve these equations numerically, it is useful to know the analytic structure of the dressed nucleon propagator g(E). As a basic requirement of the theory, g(E) must have a pole at the physical nucleon mass m. To ensure this, the bare mass is set to the value m₀ = m − Σ(m) so that

and therefore,

where Z is the wave function renormalization constant given by

where the prime indicates a derivative with respect to E. To evaluate Σ′(m), one can use the identity

which can be easily proved using the operator form of Eq. (34). In this work, we do not consider nucleon resonances in the P₁₁ channel as these lie beyond the energies that we consider. Thus, in our model, there are no further poles in the complex energy plane.

Besides a nucleon pole, it can be shown that g(E) also contains a cut starting at E = m + m_π and extending to +∞ and that this analytic structure implies the following “dispersion relation:”⁹ 

As we shall see, this relationship between g(E) and its imaginary part will prove very useful in the numerical solution of the DS equations. This expression for g(E) is also convenient for the evaluation of Eq. (38) as

To keep this model as simple as possible, we choose a separable form for the partial wave potential v,

where h(k) is a phenomenological form factor and λ specifies the sign of the potential (for the P₁₁ partial wave under consideration here, λ = −1). The use of separable potentials to describe the strong interactions of a pion and a nucleon has a long and rich history. In 1967, Varma¹⁵ used such potentials to perform the first three-body calculation of the πNN system; since then, they have been used extensively to investigate effects of pion absorption,^1,2,4–6,16 pion–nucleus scattering,^12,17–20 three-body forces,²¹ pion photoproduction,^22,23 and dibaryons,²⁴ and also to facilitate solutions of relativistic equations.^25,26 The motivation for their widespread use is their convenience, often leading to significant simplifications in both analytic and numerical work. Although short-range interactions are naturally separable at energies close to resonance poles, they can also be well approximated through the use of low-rank separable potentials at energies away from scattering poles, as long as the underlying realistic potential is energy independent.^27–33 For πN scattering at the considered energies (0 < T_π < 390 MeV), much of the low-energy energy dependence of the underlying interaction is due to the nucleon pole term, which only contributes to the P₁₁ partial wave. As we treat the pole term separately, it is reasonable to assume that the P₁₁ background term and the rest of the partial wave interactions can be well approximated with separable potentials.³⁴ Nevertheless, the neglect of residual energy dependence and the lack of crossing symmetry in our model³⁵ suggests that the separable potentials developed in this work would be most useful as input to calculations that do not rely critically on the accuracy of the off-shell behavior of the πN interaction.

In the case of a separable potential, Eq. (34b) can be solved algebraically, giving also a separable form for the background πN t matrix,

where

Defining the four dressing terms

where ϕ₁(q) ≡ f₀(q) and ϕ₂(q) ≡ h(q), the DS equations can be conveniently expressed as the set of three equations,

which determine the dressed nucleon propagator g(E), together with the additional two equations

that determine the consequent dressed πNN vertex f and πN t matrix t.

In our approach, modeling πN scattering with Eq. (34) begins by choosing parametrized analytic functions for the form factors f₀(k) and h(k). These form factors need to fulfill the requirement of providing a momentum cutoff that ensures finite values for the integrals defining the Σ functions of Eq. (44), and they must behave linearly with k in the limit of low momenta in order to be consistent with the l = 1 nature of a P₁₁ partial wave amplitude. We shall follow a previous work where separable potentials were used to model πN scattering and choose the following analytic forms:¹³ 

where C₀, C₁, C₂, β₁, β₂, Λ are free parameters and the powers n₀, n₁, n₂, n₃ are integers that can be chosen to change the functional form of the form factors.

For any given set of parameters, the first task is to solve the DS equations in the form of Eq. (45) for the dressed nucleon propagator g(E). We do this by following an iterative procedure where an approximation to g(E) is used in Eq. (44) to calculate all the functions Σ_ij(E), which are then used to calculate a new (and hopefully more accurate) version of g(E) using Eq. (45). The process is repeated until convergence for g(E) is achieved. By construction, the resulting propagator g(E) satisfied the DS equations and can then be used to generate the dressed vertex f and the πN t matrix t using Eq. (46).

To carry out the integral in Eq. (44) numerically, we use Gaussian quadratures, and to avoid the singularity coming from the pole of g(E), we rotate the integration contour from the positive real axis into the fourth quadrant of the complex q plane. However, a practical problem remains in carrying out these integrals because in order to generate a propagator g(E) at any iteration, one needs to know the previous iteration’s propagators $g(E−ωqi)$ for each of the rotated quadrature points q_i. Thus, the number of energies at which one needs to know g quickly escalates as the iteration proceeds. To get around this problem, we make use of the fact that the dressed nucleon propagator g(E), at each step of the iteration, satisfies the dispersion relation of Eq. (39). This allows us to evaluate Eq. (44) as

which requires knowledge of g(E) only at a number of fixed values of E corresponding to the Gaussian integration points ω_i used to evaluate the ω integral in Eq. (48). The iterative process thus proceeds according to the following steps:

1. For any given set of form factor parameters, begin the iteration by generating the “non-Dyson” dressed nucleon propagator g⁽⁰⁾(E) defined by Eq. (45) but where the nucleon in the πN propagator used in the dressing terms Σ_ij is not explicitly dressed, that is, by using

   $Σij(E)→∫0∞dqq2ϕi(q)ϕj(q)E+−ωq−m.$

   (49)

   It is just this g⁽⁰⁾(E) that has been used in previous works^13,36 to model πN scattering.

2. Having constructed the “zeroth iteration” of g(E) as above, we now use this g in Eq. (48) to generate new dressing terms Σ_ij(E).

3. Using these newly constructed Σ_ij(E)’s in Eq. (45) generates the next iteration of the dressed nucleon propagator g(E).

After constructing the dressed nucleon propagator g(E) as prescribed by the 3 steps outlined above, we now repeat steps 2 and 3 over and over, thus generating successive iterations of g(E), denoted as g⁽¹⁾(E), g⁽²⁾(E), g⁽³⁾(E), …, until the values of g^(r)(E) converge according to the criterion $[g(r)(ωi)−g(r−1)(ωi)]/g(r)(ωi)<ε$ for all ω integration points ω_i, where ɛ is some chosen tolerance value. We have found that the iterated dressed propagators g^(r)(E) converge for all considered models using a convergence tolerance of ɛ = 10⁻⁴ and that, correspondingly, the resulting numerical values of the converged g(E) functions are stable to at least 5 significant figures with respect to variations in the number of quadrature points used for all integrals and in the contour rotation angle used for all the q momentum integrals. We note that the convergence of g(E) also provides a self-consistency check that no spurious resonance poles have been generated by the assumed separable potentials.

With the Dyson–Schwinger equations of Eq. (34) solved in this way, it is interesting to compare the resulting fully dressed “Dyson” nucleon propagator g(E) with the “non-Dyson” one where the coupled equations of Fig. 2 are solved in the “one-pion approximation” where nucleon dressing in πN states is neglected. We present this comparison in Fig. 3 for the case where the parameters of the input bare πNN vertex and background πN potential are those of model M1 in Ref. 13. For ease of comparison, we have plotted the corresponding real and imaginary values of (E − m)g(E)/Z, being the renormalized nucleon propagators with the nucleon pole term factored out. As can be seen, there is a substantial difference between the two propagators, suggesting the importance of retaining nucleon dressing in πN states.

To obtain a variety of models of nucleon dressing, we have carried out fits to the WI08 P₁₁ πN phase shifts³⁷ (for pion laboratory energies up to 390 MeV) by using the functional forms of Eq. (47) for a number of choices of the integers n₀–n₃ and for a range of cutoff values Λ for the bare πNN vertex function f₀(k). Each such fit was constrained to reproduce the πNN coupling constant $fπNN2=0.079$ in the way described in Ref. 13. The parameters of four such fits are given in Table I with the corresponding values of (E − m)g(E)/Z plotted in Fig. 4. Unsurprisingly, the large number of parameters in this model allows one to fit πN data equally well for a wide range of cutoff parameters Λ. Although this flexibility of the model can be viewed as one of its weaknesses, it does allow one to accommodate the wide variety of πNN vertex cutoffs, in the range 300 < Λ < 2200 MeV, used in the literature.^38–48 

Finally, we show that Eq. (34), which use the fully dressed Dyson propagators, are able to be used to fit all s - and p -wave πN phase shifts for pion laboratory kinetic energies in the range 0 < T_lab < 390 MeV. To demonstrate this explicitly, we have chosen the M7 model of Table I whose cutoff parameter is Λ = 800 MeV, a value suggested by an investigation of quantum chromodynamic sum rules.⁴⁹ For the non-P₁₁ partial wave πN potentials, we use the separable forms of Thomas¹² whose form factors are parameterized as

for s-waves and

for p -waves. These form factors were used by Thomas to describe pion–deuteron scattering in a calculation using semi-relativistic kinematics.¹² 

Our partial wave phase shift fits using the M7 model for the Dyson propagator g(E) are shown in Fig. 5 with the corresponding parameters listed (under the rows labeled M7) in Table II. Equally good fits to all the phase shifts can be obtained using the other models for g(E) (M9, M8, and M6) with the corresponding parameters being given in Table I for the P₁₁ and Table II for the other partial waves. It should be noted, however, that equally good fits can also be obtained using non-Dyson propagators. Thus, even though we are able to demonstrate the importance of nucleon dressing, the large number of parameters in our model does not allow us to identify any features of the phase-shift data that may prefer the “Dyson” over the “non-Dyson” dressed nucleon propagators.

In summary, we have presented a simple but field-theoretically complete model of πN scattering that demonstrates a number of basic features of RQFT such as particle absorption and creation, self-energy, renormalization, and the use of Dyson–Schwinger equations but without the inherent complexity of a relativistically covariant approach. To do this, we used the non-relativistic framework of TOPT and exploited the simplicity of separable potentials to demonstrate how the DS equations can be solved in order to describe experimental data. It is worth noting that the model presented, together with the resulting fits to πN phase shifts as recorded in Fig. 5 and Tables I and II, can be of direct practical use in models of pion–multi-nucleon systems, such as the convolution model of the πNN system,^9,10 where a complete dressing of all nucleons is required.

A.N.K. was supported by the Shota Rustaveli National Science Foundation (Grant No. FR17-354).

The data that support the findings of this study are available from the corresponding author upon reasonable request.