On the onedimensional reggeon model: eigenvalues of the Hamiltonian and the propagator
Abstract
The effective reggeon field theory in zero transverse dimension (“the toy model”) is studied. The transcendental equation for the eigenvalues of the Hamiltonian of this theory is derived and solved numerically. The found eigenvalues are used for the calculation of the pomeron propagator.
1 The onedimensional quantum reggeon model
In Ref. [10] this theory was considered in the “loopless” approximation, when one of the terms inside brackets in (4) is ignored. In this case the theory becomes integrable. However, in [11, 12] the evolution in rapidity of the hadron scattering amplitude in the framework of the onedimensional reggeon model was studied numerically and it was shown that the difference between the full theory and the “loopless” approximation becomes substantial at high \(\lambda \). Thus it is meaningful to develop methods that are not connected with the perturbation theory in \(\lambda \).
In [13] the eigenvalues of the Hamiltonian (4) were found for values of \(\mu /\lambda =1,3,5\) and the propagator at \(\mu /\lambda =5\) was calculated, although neither the calculational procedure nor the precision were reported. In this paper we concentrate on the theoretical problems related to the Hamiltonian in the complex plane and on the method of calculation of its eigenvalues and eigenfunctions. We enlarge the domain of the values of \(\mu /\lambda \) to include negative ones for comparison with the perturbative approach.
2 Eigenvalues problem
When \(\mu =0\), the finiteness of the norm does not exclude both asymptotics \(\psi \sim \psi _{3,4}\) on the ray \(x \in [0, +\infty )\) and on the ray \(x \in (\infty , 0]\). In this case the conditions (16) can be used as well, excluding the asymptotics \(\psi _4\), and all the eigenvalues are found to be positive. Furthermore, in Sect. 6 the completeness property of the eigenfunctions found was partially checked, and as a result the necessary solutions were not lost.
In [9] it was proven that the spectral representation of the Smatrix of the theory (4) exists and it is analytical in \(\mu \) on the entire real axis \(\mu \). The choice of the quantization conditions (16) for all values of \(\mu \) is to be understood in the sense of such an analytical continuation. The positiveness of the eigenvalues found indicates the correctness of this approach.
3 Biconfluent Heun equation
It is well known [14] that the asymptotics, defined by the Thomè series, is not reached uniformly in \(\text{ arg }(x)\). In our case, when the irregular singular point \(x=\infty \) has the rank \(R=3\), there exist \(R+1=4\) Stokes rays \(\text{ arg }(x)=\pi k/4\) (\(k=0,1,2,3\)), connecting singular points 0 and \(\infty \) and dividing the complex plane x in four sectors. In the general case, in any of these sectors and on any of the Stokes rays the asymptotics of the solution may be different. In the appendix it is shown that the asymptotics \(\psi \sim const\) is reached in the area \(\frac{\pi }{2}< \text{ arg }(x) < \frac{\pi }{2}\), i.e. on the ray \([0, + \infty )\) and two adjacent sectors.
4 Orthogonal scalar product
5 Numerical calculation of eigenvalues: the method and results
The numerical solutions of the equation \(T_{4}=0\) were found^{1} for all integer values of \(\mu /\lambda \) from \(11\) to \(+11\). These values were chosen to compare our solutions with the results of [11]. To check the smooth dependence of the solutions on \(\mu /\lambda \) in the vicinity of the peculiar value \(\mu =0\) we also found the solutions for \(\mu /\lambda =\pm 0.1, \pm 0.25, \pm 0.5\).
For the numerical calculation one can take a finite number \({\mathcal {T}}\) of terms in (44); thus \(T_{4}\) becomes a polynomial in \(\delta \). The convergence of solutions was checked by comparing the roots found for \({\mathcal {T}}50\) and \({\mathcal {T}}\) terms in the equation. It is worth to note that to find roots with large absolute values one has to increase simultaneously the number of terms \({\mathcal {T}}\) and the precision of the calculations. We chose the limit of absolute values of roots corresponding to \(E/\lambda <99\). For this purpose it is sufficient to take \({\mathcal {T}}=400\) and to use the precision of 55 digits in the calculation. The independence of the roots on the arbitrary number \({\mathcal {N}}\) was not checked; we always set \({\mathcal {N}}=50\). However, we checked that for all considered \(\mu /\lambda \) the equation has no real positive roots \(\delta \) of the same order corresponding to negative energies. Positive roots with much larger values and also complex roots may appear as an artifact of the finiteness of \({\mathcal {T}}\).
The plots of the universal ratios \(E_{\text {N}}/\lambda \) for the first 11 eigenvalues (interpolated by cubic splines) are shown in Fig. 1 for \(\mu /\lambda \) from \(5\) to \(+11\); the plots in Fig. 2 are for the first five eigenvalues for \(\mu /\lambda \) from \(11\) to \(+11\). These plots demonstrate the approximate double degeneracy of eigenvalues at \(\mu /\lambda \rightarrow +\infty \) noted in [9]. In the data files applied to this article one can find nontrivial (\(E\ne 0\)) ratios \(E/\lambda \) with 20digit precision.
6 An application: calculation of the propagator
 (1)

\(\mu =1\), \(\lambda =0.1\);
 (2)

\(\mu =1\), \(\lambda =1/3\);
 (3)

\(\mu =1\), \(\lambda =1\);
 (4)

\(\mu =0.1\), \(\lambda =1\);
 (5)

\(\mu =0\), \(\lambda =1\);
 (6)

\(\mu =1\), \(\lambda =0.1\).
To compare the results we calculated the propagator for the same values of the parameters \(\mu \) and \(\lambda \).
There are two sources of inaccuracy in this calculation. First, we know the eigenvalue with a finite precision, hence the equation \(T_{4}=0\) is fulfilled only approximately. In correspondence with (41), an eigenfunction determined using an approximate eigenvalue has a contribution of the growing function \(\psi _4\) with a very small coefficient. This contribution grows very rapidly, so at sufficiently large z the approximate eigenfunction differs significantly from the exact one. In calculation it manifests itself in solving of the recurrence relations for \(c^{[N]}_n\), where the cumulation of errors occurs which leads to bad convergence of (50). Second, one can take only a finite number of terms in the sum (50). Our calculation shows that the values of the “norms” depend on the number of terms (we got 1000, 2000, 5000, 10,000).
 (2)

\(L^2\) from 25 to 100, (3) \(L^2\) from 25 to 1200,
 (4)

and (5) \(L^2\) from 25 to 4000, (6) \(L^2\) from 10 to 30.
The condition \(P(0)=1\), which is a consequence of the completeness condition, is fulfilled with an error of 0.0025 in case (2) and less than \(10^{5}\) in cases (3)–(6). One can see that the larger the value of \(\mu /\lambda \), the smaller the interval of z for which the approximated eigenfunctions are in good coincidence with the exact ones.
The calculation shows that not all “norms” are positive. In the considered cases their signs alternate for sequential N. Perhaps, it can be explained by the behavior of the eigenfunctions (see Fig. 3) on the real axis x. For all solutions \(\psi '(0)=1\) is implied. The function \(\psi _{\text {N}}(x)\) has N zeros, including \(x=0\), and its asymptotics at large \(x>0\) is constant, so the sign of this constant is \((1)^{N1}\). At \(x\rightarrow \infty \) the function grows most rapidly, as \(\psi _{\text {N}}(x)\sim \exp (x^2+\beta x)/x\), and it has a negative sign. Hence, the main contribution to the “norm” (37) comes from the real axis x and the sign of this contribution, if the common “–” is taken into account, coincides with the sign of the constant.
However, case (1) with large positive \(\mu /\lambda \) differs radically from the others. Here it is impossible to select an interval of \(L^2\) for which all values of “norms” \(\langle {\bar{\psi }}_{\text {N}}\psi _{\text {N}}\rangle \) do not depend on the cutoff. We considered the condition \(P(0)=1\) and the alternation of signs of “norms” as a criterion of success. We tried many variants from \(L^2=0.5\) to \(L^2=100\), but we cannot find any acceptable result. For case (1) we also tried to use the resolution of the identity, analogous to (39) but induced by the scalar product (40) with the cutoff \(\xi <L\) for the integration over \(\xi \). We could reliably find only the first coefficient \(1/(\psi _1\psi _1)\) of the expansion, the others significantly depend on the cutoff.
In Fig. 5 plots of the propagator as a function of the rapidity y are shown in a logarithmic scale. The numeration of curves 2–5 coincides with that of the considered cases. In the calculation of (49) 10 eigenfunctions for the cases (2)–(5) and six eigenfunctions for case (6) were taken into account. The general appearance of the curves in Fig. 5 is in full agreement with the curves in Fig. 1 from [11]. The comparison of the values of the propagators with the original numerical data for Fig. 1 from [11] at integer y from 0 to 20 (plots are shown only for \(y<10\)) demonstrates that they coincide with a relative precision of less than 0.002 for all \(y>0\).
7 Conclusions
In the present work the onedimensional reggeon model was considered in its equivalent form of the quantum mechanics in imaginary time. Since the Hamiltonian of the model is nonHermitian, an indefinite scalar product with respect to which the eigenfunctions are orthogonal had to be introduced. This allows one to write the correct resolution of the identity and, hence, the spectral representation for the propagator. The similarity transformation is known to turn the Hamiltonian into a Hermitian form [7], which establishes the completeness of the basis of eigenfunctions and the reality of eigenvalues.
The choice of the quantization conditions in this model is not trivial [9]. The condition of finiteness of the norm in the Fock–Bargmann space can be imposed only for negative values of parameter \(\mu \), i.e. in the area of applicability of the perturbation theory. For \(\mu \ge 0\) the asymptotical conditions for the eigenfunctions have to be chosen in the same manner as in the case \(\mu <0\); then the energies are real and positive. In the case \(\mu \ge 0\) the asymptotical conditions do not lead to the finite Bargmann norm, but the nonpositive “norm” connected with the indefinite scalar product is finite for all cases.
The eigenfunction equation has the canonical form of the biconfluent Heun equation for solutions of which the asymptotical conditions are implied. By resolving these conditions, using the new method of [15], we can derive the equation which completely defines the eigenvalues. This equation is transcendental, because it contains an infinite sum of polynomials, so that we can solve it only numerically. For the chosen sets of parameters of the model (in fact, the only parameter is the ratio \(\mu /\lambda \)) we found several eigenvalues of the energy.
We used the found values of energies for a calculation of the pomeron propagator. In principle, knowing the eigenvalues and eigenfunctions, one can apply the spectral representation. The problem is that we express eigenfunctions as power series and their coefficients are defined with cumulative errors, even if we know the eigenvalues with high precision. These errors lead to bad convergence of the series for the scalar products appearing as coefficients of the spectral representation. To provide convergence of the series we introduced a cutoff into the integration which defines the scalar product. This allows one to calculate the propagator, excluding the case of large positive \(\mu /\lambda \), when the scalar products depend significantly on the cutoff. So this method gives satisfactory results only for values of \(\mu /\lambda \) not very large, just when the perturbative theory does not work. In this case we find full agreement with previous straightforward numerical calculations [11].
Footnotes
 1.
All numerical calculations and graphical plotting were carried out using Maple computer algebra system.
Notes
Acknowledgements
The authors are thankful to N.V. Antonov, M.V. Ioffe, M.V. Komarova, M.V. Kompaniets, V.N. Kovalenko for very useful discussions.
Supplementary material
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