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P. J. Rijken

Publications and source records attributed to P. J. Rijken.

7 recordsLinked to original sources
arXiv2026

Novel method for evaluating the eigenvalues of the Heun differential equation with an application to the Breit equation

Eigenvalues of the Breit equation, in which only the static Coulomb potential is considered, have been found. Over the past decades several authors have analyzed the Breit equation to obtain numerically or by approximation an estimation of the energy levels. Various approaches have been used and no determination of the energy levels currently exists that is directly based on the second order Heun differential equation derived. The aim of this work is to provide a method of calculation that can be used to numerically calculate the energy levels for various spin states to high accuracy. From the Breit equation, we derive the corresponding second-order Heun differential equation and continued fraction from which the eigenvalues can be determined very accurately. Next, we present a novel method based on the Green function method, which leads to a semi-infinite determinant from which we are able to obtain the numerical values of the eigenvalues by direct calculation. Using suitable numerical methods for the direct calculation of the continued fraction and the semi-infinite determinant, we show that both methods are consistent within 25 digits of accuracy. We show that the correct energy levels for the Dirac equation follow from our results by a suitable mapping of the variables. The results are in total agreement with earlier calculations found in the literature and extend this by several digits of additional accuracy. The condition on the determinant giving the energy levels provides a rich structure that is promising in extending the results of this work.

quant-ph↗
arXiv1996

$O(α_s^2)$ Contributions to the asymmetric fragmentation function in $e^+e^-$ annihilation

The order α_s^2 contributions to the coefficient functions corresponding to the asymmetric fragmentation function $F_A(x,Q^2)$ in $e^+e^-$ annihilation are calculated. From this calculation we infer that the order $(α_s/4π)^2$ correction to the flavour asymmetry sum rule is non vanishing and amounts to $-12β_0C_Fζ(3)$. We also study the effect of the higher order QCD corrections on $F_A(x,Q^2)$ and compare them with the OPAL data. The latter put a strong constraint on the valence part of the fragmentation densities $D_q^H(x,μ^2)$.

hep-ph↗
arXiv1996

Higher order QCD corrections to the transverse and longitudinal fragmentation functions in electron-positron annihilation

We present the calculation of the order $α_s^2$ corrections to the coefficient functions contributing to the longitudinal ($F_L(x,Q^2)$) and transverse fragmentation functions ($F_T(x,Q^2)$) measured in electron-positron annihilation. The effect of these higher order QCD corrections on the behaviour of the fragmentation functions and the corresponding longitudinal ($dσ_L(x,Q^2)/dx$) and transverse cross sections ($dσ_T(x,Q^2)/dx$) are studied. In particular we investigate the dependence of the above quantities on the mass factorization scale ($M$) and the various parameterizations chosen for the parton fragmentation densities $D_p^H(x,M^2)$ ($p=q,g$; $H=π^\pm, K^\pm, P, \bar{P}$). Our analysis reveals that the order α_s^2 contributions to $F_L(x,Q^2)$ are large whereas these contributions to $F_T(x,Q^2)$ are small. From the above fragmentation functions one can also compute the integrated cross sections $σ_L$ and $σ_T$ in an independent way. The sum $σ_{tot} = σ_L + σ_T$, corrected up to order \alphastwo, agrees with the well known result in the literature providing us with an independent check an our calculations.

hep-ph↗
arXiv1996

$O(α_s^2)$ Contributions to the longitudinal fragmentation function in $e^+\,e^-$ annihilation

We present the order $α_s^2$ contributions to the coefficient functions corresponding to the longitudinal fragmentation function $F_L(x,Q^2)$. A comparison with the leading order $α_s$ result for $F_L(x,Q^2)$ shows that the corrections are large and vary from 44\% to 67\% in the region $0.01 < x < 0.9$ at $Q^2=M_Z^2$. Our calculations also reveal that the ratio of the longitudinal and total cross section $σ_L/σ_{\rm tot}$ amounts to 0.054. This number is very close to the most recent value obtained by the OPAL collaboration which obtained $0.057\pm 0.005$.

hep-ph↗
arXiv1995

Heavy flavor contributions to the Drell-Yan cross section

We investigate the effect of heavy flavor contributions to vector boson ($V$ = $γ$, $Z$, $W$) production which is described by the Drell-Yan mechanism. All reactions with bottom and top quarks ($Q_i$ = $b$, $t$) in the final state, like $q_1 + \bar{q}_2\rightarrow V + Q_1 + \bar{Q}_2$ and $g + g\rightarrow V + Q_1 + \bar{Q}_2$, are considered. This study also includes the virtual contributions containing heavy flavor loops which were not taken into account earlier in the literature. Our analysis reveals that the above corrections to the Drell-Yan cross section are very small. Only at energies characteristic for the LHC they are of the same order of magnitude as the order \alphastwo~QCD contributions due to light quark and gluon subprocesses calculated earlier in the literature.

hep-ph↗
arXiv1994

Order $α_s^{2}$ contributions to the Drell-Yan cross section at fixed target energies

{}From the literature one infers that the bulk of the order $α_{s}$ corrections to the Drell-Yan cross sections $dσ/dm$ and $m^{3}d^{2}σ/dmdx_{F}$ is constituted by the soft plus virtual gluon part of the coefficient function. In the case of $dσ/dm$ it can be shown that at fixed target energies the effect of the exact order $α_{s}^{2}$ corrected coefficient function is very well approximated by its soft plus virtual gluon part. Since the complete order $α_{s}^{2}$ contribution to the coefficient function is missing we have to assume that the same approximation also holds for $m^{3}d^{2}σ/dmdx_{F}$. It appears that the discrepancy between the exact order $α_{s}$ corrected cross section and the massive lepton pair data taken at fixed target experiments can be partially explained by including the order $α_{s}^{2}$ soft plus virtual gluon part of the coefficient function.

hep-ph↗