We consider quantum inverse scattering with singular potentials and calculate the sine-Gordon model effective potential in the laboratory and centre-of-mass frames. The effective potentials are frame dependent, but closely resemble the zero-momentum potential of the equivalent Ruijsenaars–Schneider model.

## 1. Introduction and motivation

In recent years, advances in lattice QCD techniques made it possible to measure and study the forces between nucleons. A major success was the first-principles calculation of the two-nucleon potential by the HAL QCD collaboration [1–3], which was later extended to nucleon–hyperon interactions [4,5] and to the study of three-baryon forces [6]. Three-neutron (and higher) interactions are crucial to determining the correct nuclear equation of state, which is used in the calculation of the mass and radius of neutron stars. The gravitational wave signals expected from in-spiraling neutron star systems are sensitive to the resulting mass–radius relation.

The HAL QCD method [1] is based on measuring the Nambu–Bethe–Salpeter (NBS) wave function $\Psi E(x)$ of a two-nucleon state which satisfies (in the centre-of-mass frame) the “Schrödinger equation”

The problem of energy dependence can be studied in some ($1+1$)-dimensional integrable models [10]. The Ising model and the $O(3)$ nonlinear $\sigma $-model were studied, and it was found that at low energies the energy-dependent $UE(x)$ can be approximated well by its zero-momentum limit (corresponding to the case where the relative momentum of the two-particle state vanishes). The problem was also studied in the sine-Gordon (SG) model [11]. In the semiclassical limit an energy-independent effective potential was constructed that exactly reproduces the semiclassical time delays for all energies. This could be compared to the zero-momentum potential, which is explicitly known in this model from its equivalent Ruijsenaars–Schneider (RS) formulation [12,13].

In this paper we continue to study the notion of effective potential in the integrable (analytically solvable) SG model in ($1+1$) dimensions. We model the way the phenomenological potential was determined from scattering experiments: we require that the quantum mechanical effective potential exactly reproduces the (analytically known) scattering phase shifts at all energies. The price we have to pay is that the effective potential is frame dependent. We will construct the effective potential in the laboratory frame of the scattering process and also in the centre-of-mass frame of the two particles. We will compare them to each other and to the zero-momentum potential known from the RS formulation of the model.

The paper is organized as follows. In Sect. 2 we define the notion of effective potential for relativistic scattering. Section 3 is a review of quantum mechanical inverse scattering in one dimension. We generalize known results for the case of singular potentials. In Sects. 4 and 5 we calculate the effective potential for soliton–soliton scattering in the SG model in the laboratory and centre-of-mass frames, respectively. Section 6 is a short summary of the results and contains our conclusions. Some technical details and examples can be found in the appendices, together with a summary of the scattering phase shifts in the SG model.

## 2. Effective potentials

We will study the one-dimensional scattering of two identical particles of mass $m$ (with positions $x1$, $x2$ and momenta $p1$, $p2$), whose interaction has a strong repulsive core which does not allow the particles to come close to each other. If, initially, particle 1 is to the left of particle 2 then $x2>x1$ at all times. Initially, $p1>p2$:

Asymptotically, for $(x2\u2212x1)\u2192\u221e$, the two-particle wave function $\Phi (x1,x2)$ is a superposition of free waves:

For relativistic scattering, the “S-matrix” $S(p1,p2)$ is a function of the relative rapidity of the particles:

For non-relativistic scattering we can use a quantum mechanical description with a potential depending on the relative distance of the particles. The Hamilton operator has the form

Our aim is to find a suitable effective potential $U(x)$ that, by solving the corresponding nonrelativistic Schrödinger equation, leads to the physical, i.e. relativistic, scattering S-matrix as function of the momenta of the particles. Thus we require

There are, however, two important special cases where exact identification is possible. In the laboratory (fixed target) frame of the scattering we can require

The problem we have to solve in both cases is to find the potential $q(x)$ in Eq. (12) if the corresponding S-matrix $S(k)$ is given. We are interested in potentials with a strong repulsive core, which means that $q(x)$ has to be singular when the relative distance $x$ approaches zero. This leads us to the mathematical problem of quantum inverse scattering with singular potentials, which is discussed in the next section.

## 3. Quantum inverse scattering with singular potentials

Quantum inverse scattering, the problem of finding the potential from scattering data, is a classical problem in quantum mechanics. It has been completely solved in the one-dimensional case [14–16] both for the entire line and the half-line cases. The latter case is more important because the same mathematical problem emerges for three-dimensional spherically symmetric potentials after partial wave expansion. Here we will also be interested in this case, because we consider strongly repulsive potentials. The details of the reconstruction procedure depend on the class of the potentials, and the simplest case is that of regular potentials [17]. We will proceed along the lines presented in Ref. [17], with some modifications necessary due to the singular core of our potentials.

We will consider the Schrödinger equation on the half-line $x\u22650$,

### 3.1. Direct scattering

For any given $k$, we will need three special solutions of the differential equation (19). The physical solution $\phi (x,k)$ is defined by its regular behavior near the origin,

Since the second-order differential equation (19) has only two linearly independent solutions, any of the above solutions can be expressed as linear combinations of the other two. For example, the Jost solution can be written as

^{1}It can be shown that $f(k)$ can alternatively be defined by the linear combination

It is possible to show that the large-$k$ behavior of the Jost function is

The physical solutions $\phi (x,k)$ satisfy the completeness relation

Another important object in inverse scattering theory is the transformation kernel $A(x,y)$. It is defined as the unique solution of the Goursat problem

### 3.2. Inverse scattering

Starting from the completeness relation (35), by acting on it with the inverse of the unitary operator $A^$, one can derive the most important equation of inverse scattering, the Marchenko integral equation. We have followed the steps presented in Ref. [17] for regular potentials. In our case with singular potential one has to be careful because, unlike for regular potentials, $\delta (\u221e)=0$ here. The result is that $A(x,y)$ satisfies the Marchenko equation

Quantum inverse scattering now proceeds in three steps. The first step is to calculate $F(x)$ using the scattering data $S(k)$ in Eqs. (42) or (43). The second step is to solve Eq. (41) for $A(x,y)$. The third and final step is to use

## 4. Sine-Gordon effective potential in the laboratory frame

In this section we carry out the three steps of quantum inverse scattering to determine the effective SG potential that exactly reproduces the SG soliton–soliton scattering in the laboratory frame (case I). The SG S-matrix is given in Appendix B.

For simplicity, we deal with integer $p$ only. Using the identification (17) and the SG S-matrix (B8), we have

The first step is to calculate $F(x)$. For the above S-matrix, Eq. (43) is easily evaluated with the help of the residue theorem, and we obtain

The next step is to solve the Marchenko equation for $A(x,y)$. For $F(x)$ given by Eq. (46) we have to solve

The solution of this algebraic problem turns out to be very simple. We can rearrange Eq. (52) to the matrix form

It is easy to see that for $p=2$ we have $D^=H1$. We have calculated the reduced determinant for $p=3,4,5$ using Mathematica. For $p=3$,

From the above formulas it is clear how $D^$ can be constructed from the variables $sj$, $Hj$, and $Cj$ in general. Since our calculation is algebraic, it must also be valid for the case discussed in Appendix A, since the corresponding S-matrix is also of the form (45), with $sm=m$. It is a very nontrivial check on our result that in this case Eq. (57) is equal to

The small-$x$ expansion of Eq. (57) takes the form

We have compared the (integrated) laboratory frame effective potential and the (integrated) zero-momentum potential in Figs. 1 and 2 for $p=3,4$, respectively.

## 5. Sine-Gordon effective potential in the centre-of-mass frame

In this section we calculate the SG effective potential in the centre-of-mass frame. Again, we restrict our attention to integer $p$. Using Eqs. (18) and (B8), we have

We calculated $FII(x)$ numerically for $p=2,3$, and by discretizing the integrals solved the corresponding Marchenko equations numerically. The results are shown in Figs. 4 and 3. For comparison, we also show in these plots the corresponding LAB frame (integrated) effective potentials. It can be seen that the frame dependence is weak: both effective potentials have the same qualitative features and are close to each other. The expected $1/x$ short-distance behavior is also reproduced. We can conclude that the notion of effective potential makes sense in this model.

## 6. Summary and conclusion

The phenomenological potential in nuclear physics has a limited range of applicability because the very notion of a potential used in the Schrödinger equation is a nonrelativistic concept which is meaningful and valid (approximately) only below the $\pi $-production threshold. The NBS potential as measured by the original HAL QCD method [1] is energy dependent (although this energy dependence is moderate at low energies). An alternative possibility is to define [2,3] an energy-independent but nonlocal “potential.”

($1+1$)-dimensional integrable models are useful because the analogous problems can be studied more explicitly. Moreover, since there is no particle production in integrable models, the two-particle description remains valid at all energies. It is possible to define an effective potential that is energy independent and reproduces the scattering data exactly. The price one has to pay for energy independence is that due to the relativistic nature of the problem this effective potential becomes frame dependent.

In this paper we studied the effective potential in the SG model. We calculated the effective potential algebraically in the laboratory frame and numerically in the centre-of-mass frame using inverse scattering techniques. Our results are summarized in Fig. 5, where the LAB- and COM-frame effective potentials are compared and the zero-momentum potential (obtained from the equivalent Ruijsenaars–Schneider formulation of the model) is also shown. The three potentials are qualitatively very similar and also close numerically. Our conclusion is that [at least in this ($1+1$)-dimensional toy model] in spite of the problems discussed above the effective potential remains a useful concept.

## Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11575254), the Major State Basic Research Development Program in China (No. 2015CB856903), and by the Hungarian National Science Fund OTKA (under K116505). J. B. would like to thank the CAS Institute of Modern Physics, Lanzhou, where most of this work has been carried out, for their hospitality.

#### Appendix A. Scattering and inverse scattering for the $1/sinh2x$ potential

To illustrate the steps of direct and inverse scattering, we take the solvable potential

Using the well-known linear relations between the hypergeometric functions of argument $z$ and argument $1\u2212z$ we can read off the coefficients defined by Eq. (25). In this example they turn out to be

The simplest nontrivial case is $p=2$. The corresponding S-matrix is

#### Appendix B. The sine-Gordon S-matrix

The SG model is perhaps the most studied two-dimensional integrable field theory. Its spectrum and S-matrix is exactly known from its bootstrap solution [18–20]. Moreover, an equivalent relativistic quantum mechanical description exists, the Ruijsenaars–Schneider model [12,13].

The SG field theory Lagrangian is^{2}

The full S-matrix of the model (scattering among solitons, antisolitons, breathers) is completely known [18–20], but in this paper we only need the soliton–soliton scattering S-matrix. Here there are no bound states and it is given by the formula

The RS model [12,13] is an integrable relativistic quantum mechanical model whose dynamics and S-matrix are completely equivalent to that of the SG field theory. From the RS description it is possible to read off the corresponding zero-momentum potential [11,13]. In our conventions, it reads (after restoring the constants $\u210f$, $c$)

#### Appendix C. Determinant solution

Let us recall Eq. (55), the set of equations we have to solve for $bm$ written in matrix form:

## References

^{1}For the case of regular potentials $\phi (0,k)=0$, $\phi \u02dc(0,k)=1$ and $f(k)$ is simply given by $f(0,k)$.

^{2}Here we use the $\u210f=c=1$ system of units as usual in relativistic quantum field theory.