---
abstract: 'An analytical relation between center of mass momenta in a nonrelativistic and a relativistic two-nucleon Schrödinger equation is proposed which allows to analytically rewrite the two Schrödinger equations into each other.  As a consequence a NN potential occurring in the relativistic Schrödinger equation can be gained from a nonrelativistic one by an analytical procedure.  The S-matrices in the two equations are exactly identical and therefore the two-nucleon phase shifts.' 
address: 'Institut für Theoretische Physik II, Ruhr-Universität Bochum, D-44780 Bochum, Germany'
author:
- 'H. Kamada[^1] and W. Glöckle'
title: ' A Momentum Transformation Connecting a NN Potential in the Nonrelativistic and the Relativistic Two-Nucleon Schrödinger Equation '
---

Few-nucleon equations and their numerical solutions are mostly carried out in a nonrelativistic framework.  Fitting NN potentials to NN data incorporates to some extent relativistic features into the non-relativistic framework, in other words relativistic features contained in the data are absorbed in the potential parameters of the nonrelativistic Schr/“odinger equation.  This however is not sufficient, especially relativistic effects in systems with more than two particles are not accessible in this manner.  As a first step towards a relativistic framework the NN system in its center of mass frame should have the correct form [@bak; @fong], which requires that the operator for the kinetic energy is formed out of square roots.  The present day so called realistic NN forces are fitted to NN data together with the nonrelativistic form of the kinetic energy [@cdbonn; @nijm; @av18].  More precisely a mixed procedure has been applied.  These potential models were fitted to the phaseshifts and amplitudes of the Nijmegen partial- wave analysis [@NEWREF1], whereby relativistic relations between the c.m.  momenta and the kinetic energy and between the differential cross section and the scattering amplitude have been used.  Clearly consistent steps would be desireable and to refrain altogether from using the nonrelativistic NN Schrödinger equation.  The question addressed in this paper is, whether the potentials have to be refitted to the data once the relativistic form of the kinetic energy is used instead of the nonrelativistic one.  A refitting of the NN potential has been undertaken e.g. in [@car].  In earlier work [@glo; @coe] approximate relations among the potentials in the nonrelativistic and relativistic Schrödinger equations were introduced.  In Ref[@kei] relations among the squares of the interacting and the free mass operators are used to find formal relations between the relativistic and nonrelativistic Schrödinger equation.  However these formal relations do not allow to write the potential in the relativistic Schrödinger equation explicitely in terms of the potential in the nonrelativistic Schrödinger equation.  We want to propose here a procedure providing an exact analytic relation between those potentials.  Before, however, we would like to emphasize that we are not generating a NN potential which has all the correct relativistic features built in ( like an effective potential derived from field theory in the Hamiltonian formalism [@NEWREF2]) but simply establish a formal connection among potentials in the relativistic and nonrelativistic forms of the Schr/”odinger equation.  The potentials are treated as “ black boxes” simply as functions of certain momenta with no reference to whether the potentials had originally some relativistic background or not. 

In practice few-nucleon problems are mostly solved in a partial wave representation.  Therefore we formulate that transformation among the potentials for given angular momentum states.  Let $l$, $s$ and $j$ denote the orbital, total spin and total angular momenta of the NN system, then for a given $j$ and $s$ the nonrelativistic Schrödinger equation in momentum space reads

$$\begin{aligned}
\left( { p^2 \over m} + 2m \right)  \psi_l (p)  +
\sum _{l'} \int_0 ^\infty dp'  {p'}^2  V_{l l '} (p,p') \psi_{l '} (p
') = E \psi_l (p) 
\label{eq1}\end{aligned}$$

Here $m$ is the nucleon mass and the sum over $l'$ is present or not depending on $s$ and $j$. In Eq.  (\[eq1\]) the rest masses $2m$, which are usually absorbed into the definition of the energy, are still contained.  The relativistic version of the two-nucleon Schrödinger equation is given by [@bak; @fong; @kei] $$\begin{aligned}
2\sqrt{m^2 + p^2} \phi_l (p) + \sum_{l'} \int _0^\infty dp' {p'}^2 
U_{l l'} (p,p') \phi_{l '} (p')  = E \phi_l (p) 
\label{eq2}\end{aligned}$$ The question is, if one can find the potential $U$ in Eq.  (\[eq2\]), once the potential $V$ in the nonrelativistic Schrödinger equation Eq.  (\[eq1\]) is given. 

We propose the use of two types of momenta and denote the momenta in the nonrelativistic Eq.  (\[eq1\]) by $q$. We define the following relation between the momenta q in Eq.  (\[eq1\]) and p in Eq.  (\[eq2\]): $$2 \sqrt{ m^2 + p^2 } \equiv 2m + { { q ^2 } \over m } 
\label{eq3}$$ This relation can be solved either for p or for q : $$p = q \sqrt{ 1 + { {q^2} \over { 4 m ^2 } } } 
\label{eq4}$$ and $$q = \sqrt{2m} \sqrt{ E_p - m } 
\label{eq5}$$ where we define $E_p= \sqrt{m^2 + p^2}$. For small values of $p$ or $q$ one obtains $$p \approx 
q ( 1 + { { q^2 } \over { 8 m ^2 } } ) 
\label{eq6}$$ and $$\begin{aligned}
q \approx
p (  1 - { { p^2 } \over { 8 m ^2 } } )
\label{eq7}\end{aligned}$$ whereas for large values of $p$ and $q$ one gets $$\begin{aligned}
p \approx { { q^2 } \over { 2m} } 
\label{eq8}\end{aligned}$$ With these definitions it is possible to rewrite Eq.  (\[eq1\]) into Eq.  (\[eq2\]) and vice versa.  Using Eqs.  (\[eq3\]) and (\[eq4\]) one obtains $$\begin{aligned}
p^2 dp &=& q^2 dq \sqrt{ 1 + { { q^2} \over { 4 m^2 } } } ( 1 + { { q^2} 
\over { 2 m^2 } } ) \cr
&\equiv&  q^2 dq h^2(q)
\label{eq9}\end{aligned}$$ with $$h(q) \equiv \sqrt{ \left( 1 + \frac {q^2}{2 m^2} \right) 
\sqrt{ 1 + \frac {q^2}{4 m^2}} } 
\label{eq10}$$ Then Eq.  (\[eq2\]) turns into $$\begin{aligned}
\left( 2m + { {q^2 } \over { m }}  \right) \phi_l (p) 
+ \sum_{l'} \int _0 ^\infty dq' {q'}^2 h^2 (q') U _{l l'} (p,p') \phi_{l'}
(p') = E \phi_l (p) 
\label{eq11}\end{aligned}$$ If we define $$\begin{aligned}
\psi_l(q) \equiv h(q) \phi_l(p) 
\label{eq12}\end{aligned}$$ and $$\begin{aligned}
V_{l l'}(q,q') \equiv h(q) U_{l l'} (p,p') h(q') 
\label{eq13}\end{aligned}$$ we arrive at $$\begin{aligned}
\left( 2m + { {q^2 } \over m} \right) \psi_l (q) + \sum _{l'}
\int _0 ^\infty dq' {q'}^2 V_{l l'} (q,q') \psi_{l'} (q') 
= E \psi_{l} (q)
\label{eq14}\end{aligned}$$ which is the nonrelativistic Eq.  (\[eq1\]).  Thus Eqs.  (\[eq12\]) and (\[eq13\]) provide the desired relations between the wave functions and the potentials.  Explicitly written , the potential $U$ results from $V$ via $$\begin{aligned}
U_{l l'} (p,p') &=& h^{-1}(q) V_{l l'}(q,q')h^{-1} (q') 
\cr
&=& { { \sqrt{2} m } \over \sqrt{ E_p \sqrt{ 2m^2 + 2m E_p}} } 
V_{l l'} (\sqrt{2m}\sqrt{E_p -m} , \sqrt{2m}\sqrt{E_{p'}-m} ) 
    { { \sqrt{2} m } \over \sqrt{ E_{p'} \sqrt{ 2m^2 + 2m E_{p'}}} } 
\label{eq15}\end{aligned}$$ and the potential $V$ results from $U$ via $$\begin{aligned}
V_{l l'} ( q , q' ) &=& 
\sqrt{ ( 1 + { { q^2 } \over { 2 m^2 } } ) \sqrt{ 1 + {{ q^2 } \over { 4 m^2}}}
}
\cr
&~& U_{l l'} ( \sqrt{ q^2    + { { q^4 } \over { 4 m^2 }  } } , 
               \sqrt{ {q'}^2 + { { {q'}^4 \over { 4 m^2} }} } ) 
\sqrt{ ( 1 + { { {q'}^2 } \over { 2 m^2 } } ) 
\sqrt{ 1 + {{ {q'}^2 } \over { 4 m^2}} } }
\label{eq16}\end{aligned}$$ Correspondingly, the relativistic wave function expressed in terms of the nonrelativistic one is given by $$\begin{aligned}
\phi_l(p) = { { \sqrt{2} m} \over { \root 4 \of { 2m ^2 + 2m E_p }
\sqrt{ E_p } }  } \psi_l ( \sqrt{ 2m } \sqrt{ E_p -m } ) 
\label{eq17}\end{aligned}$$ The transformation of the wave function given in Eq.  (\[eq12\]) conserves the norm.  We have $$\begin{aligned}
\int _0 ^\infty dp p^2 \phi_l ^2 (p) = \int _0 ^\infty dq q^2 h^2 (q) 
\phi_l^2 (p)  =  \int _0 ^\infty dq q^2 \psi_l ^2 (q ) 
\label{eq18}\end{aligned}$$

Let us now consider the Lippmann Schwinger equations for the partial wave projected half-shell t-matrices.  The relativistic version is given by $$\begin{aligned}
T_{l l'} (p,p') = U_{l l'} ( p,p') + 
\sum_{l''}\int _0 ^\infty dp'' {p''}^2 U_{l l''} (p,p'') { 1 \over 
{ 2E_{p '} - 2E_{p''} + i \epsilon } } T_{l'' l'} (p'',p') 
\label{eq19}\end{aligned}$$ Using Eqs.  (\[eq9\]) and (\[eq13\]) and defining $$\begin{aligned}
t_{l l'} (q ,q') = h(q) T_{l l'} (p,p') h(q') 
\label{eq20}\end{aligned}$$ we obtain from Eq.  (\[eq19\]) $$\begin{aligned}
t_{l l'} (q,q') = V_{l l'} (q, q') + \sum _{l''} \int _{0 } ^{\infty} 
dq'' {q''} ^{2 } V_{l l''} (q,q'') { 1 \over { { {q{ '}}^{2} \over m} - 
{{q''}^{2} \over m } + i\epsilon  }  } t_{l'' l'} (q'' , q' ) 
\label{eq21}\end{aligned}$$ This is the standard nonrelativistic Lippmann Schwinger equation. 

Eq.  (\[eq20\]) provides also a relation between the phase shifts evaluated through the relativistic and nonrelativistic Lippmann Schwinger equations.  The unitarity relation resulting from Eqs.  (\[eq19\]) and (\[eq21\]) are given by $$\begin{aligned}
Im T_{l l} (p, p ) = 
- { {\pi p E_{p} } \over 2 } \sum _{l'} \vert T_{l l'} (p
, p) \vert ^{2 } \end{aligned}$$ and $$\begin{aligned}
Im t_{l l} (q , q) = 
-{ { \pi q m } \over 2 } \sum _{ l'} \vert t _{l l'}
(q , q) \vert ^{2 } \end{aligned}$$ As a consequence the S-matrices given by $$\begin{aligned}
S_{l l'} ( p)  \equiv \delta_{l l'} - i\pi p E_{p} T _{l l'} (p, p)\end{aligned}$$ and $$\begin{aligned}
s_{l l'}(q)  \equiv \delta_{l l'} - i \pi q m t_{l l'} (q,q) \end{aligned}$$ are unitary.  Using Eq.  (\[eq20\]) and the relations (\[eq3\]) and (\[eq4\]) one obtains $$\begin{aligned}
s _{l l'} (q) 
&=& \delta_{l l'} - i \pi q m h(q) T_{l l'} (p, p) h(q)
\cr
&=& \delta_{l l'} - i \pi q m \sqrt{ 1 + {{ q^{2} } \over {4 m^{2}} } } 
( 1 + {{q^{2}} \over {2m^{2}}})    T _{l l'} (p ,p) 
\cr
&=&  \delta_{l l'} - i \pi 
q m { p \over q } { 1 \over {2m}  } ( 2m +{ {q^{2}}
  \over m} ) T _{l l'} (p, p) 
\cr
&=&\delta_{l l'} - i \pi p E_{p} T_{l l'} (p , p) =S_{l l'} (p)
\label{eq26}\end{aligned}$$ From Eq.  (\[eq26\]) we see that the S-matrices are identical, and as a consequence the scattering phase shifts and mixing parameters are identical.  Though the momenta q and p for the nonrelativistic and relativistic S-matrices are different, the related energies are the same, namely $E=2 m + q^2/m = 2 \sqrt{m^2 + p^2}$.

In Fig.  1 we show the relation between the momenta $p$ and $q$ as given in Eq.  (\[eq4\]).  In Fig.  2 we display the s- and d-wave components of the deuteron wave function.  We show these wave function components as a function of their respective momenta, $p$ in the relativistic case and $q$ in the nonrelativistic case.  The underlying NN potential was arbitrarily chosen as the CD Bonn potential [@cdbonn].  The minima for the s-and d-wave components are shifted towards larger momenta in the relativistic case.  The effects are large above about 5 $fm^{-1}$.

Summarizing, we have shown that NN potentials fitted to NN data in a nonrelativistic framework can be analytically rewritten by a scale transformation in the momenta such that they lead to the same S-matrix when used in a relativistic Schrödinger equation.  There is no need for refitting parameters when the nonrelativistic operator of kinetic energy is replaced by the relativistic square root expressions. 

The potential $U$ can be used in few-nucleon systems with $ A > 2 $ as done in the studies [@car; @glo]. 

Acknowledgment: This work was supported by the Research Contract No.  41324878 (COSY-044) with the Forschungszentrum Jülich, Germany.  The authors would like to thank Ch.  Elster for critically reading the manuscript. 

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[^1]: present address: Institut für Kernphysik, Fachbereich 5 der Technischen Hochschule Darmstadt, D-64289 Darmstadt, Germany
