Remarks on beyond leading order
Introduction
Pion production in nucleon–nucleon (NN) collisions is subject of theoretical and experimental investigations already since the 1960s—for a review of the history of the field see Ref. [1]. However, when new high precision data became available due to advanced accelerator technology in the beginning of the 1990s it became clear that all phenomenological studies performed so far were not capable of describing the data. Several mechanisms were proposed to cure the problem; however, no clear picture emerged [2].
There was the hope that chiral perturbation theory (ChPT) could resolve the issue. As the effective field theory for the Standard Model at low energies it should provide a framework to investigate the reactions in a field-theoretically consistent way. In a first attempt a scheme proposed by Weinberg to study elastic and inelastic pion reactions on nuclei [3] was applied to investigate also pion production in NN collisions. However, in doing so up to next-to-leading order (NLO) the discrepancy between the calculations and data became even worse [4]. In addition, loop contributions, formally of order NNLO, gave even larger effects [5], [6] shedding doubts on an applicability of chiral perturbation theory to .
In parallel, already in Refs. [7] it was stressed that the large momentum transfer, typical for meson production in NN collisions, needs to be taken care of in the power counting. This idea was further developed in Refs. [8], [9]. The appropriate expansion parameter for therefore is where denotes the threshold momentum for pion production in NN collisions. M and are the masses of the nucleon and pion, respectively. Here the leading-order (LO) scales as and subleading orders NnLO scale as . For the most recent developments for the reaction within chiral perturbation theory we refer to Refs. [10].
Thus in the reactions one is faced with a two-scale expansion, since both as well as appear explicitly in the expressions. For tree-level diagrams this does not cause any problem. To perform the power counting for loop integrals, however, a rule has to be given what scale to assign to the components of the loop momentum. After subtraction of the nucleon mass M, the residual energy of each external nucleon at threshold is , whereas the corresponding momentum is of order . One therefore would be tempted to take over this scaling also for the loop momentum. On the other hand, the new power counting is based on two scales, , and the pions in loops are off-shell. Therefore there is no reason why the scaling of the pion energies in loops should be different from the scaling of the pertinent three-momenta. In Appendix E of Ref. [2] it is shown that for all diagrams that do not have a two-nucleon cut, each component of the loop momentum should be counted of order of the largest external momentum in the loop. The argument there is based on the observation that in time ordered perturbation theory (TOPT) there is no ambiguity for the order assignment of energies since it is a 3-dimensional theory in the first place. On the other hand, the leading order of a given TOPT amplitude should agree to that of the corresponding Feynman amplitude. This allows to identify the proper scale for the energy of the loop momentum. The assignment was checked by explicit calculations in Refs. [5], [9]. As a result, all components of the loop momentum in diagrams (a) and (b) of Fig. 1 scale as , but those of diagrams (c) and (d) scale as and are therefore suppressed.1 One further consequence of the presence of two scales in the problem is that the individual loops no longer contribute to only a single order, but each loop contributes to infinitely many orders, since appears as the argument of non-analytic functions. The power counting only identifies the lowest order where the particular loop starts to contribute [9].
In Ref. [11] it was shown that the sum of all diagrams of Fig. 1 is independent of the choice of the pion field. However, based on the scheme developed in Refs. [8], [9] only diagrams (a) and (b) contribute at NLO whereas diagrams (c) and (d) start to contribute not until order N4LO (see Table 11 of Ref. [2]). The main purpose of this Letter is to investigate the consistency of these two statements.
As we go along we also need to discuss the appropriate choice of the nucleon propagator in the heavy baryon formulation. This is done in Section 3. Section 4 contains our conclusions. Moreover, for clarification two appendices are added, one is about reparameterizations of the chiral matrix U, the other is about the expansion of the nucleon propagator.
Section snippets
Dependence on the pion field to NLO
The Lagrangian relevant for our study may be written as [12] Here denotes a trace in the isospin-space, Ψ is the relativistic spinor of the nucleon, is its covariant derivative containing the Weinberg–Tomozawa term [13] and other terms, is the axial-vector constant, the pion decay constant. Furthermore, are the chiral vielbein and the mass term, respectively, with , where is the quark mass
Beyond leading order
We found that to NLO the sum of diagrams (a) and (b) (and—trivially—the sum of diagrams (c) and (d)) of Fig. 1 is invariant under the choice of the pion field. All terms that depend on the pion field vanish to this order. In this section we investigate the pion-field dependent terms of the diagrams shown in Fig. 1 to NNLO. Please note that there are several additional diagrams contributing to this order that are potentially pion-field dependent—one example being the so-called football diagrams
Conclusions
Of course, the fact that there is a cancellation of the summed α-dependent terms of the four diagrams of Fig. 1 does not come as a surprise, see Ref. [11], since these terms would cancel also in a relativistic calculation of the type [12] where the nucleon propagator (11) is replaced by the non-expanded covariant form (B.1) and where the terms appearing in the vertices (9), (10) are replaced by their covariant Dirac-analogs . In fact, this cancellation between the α-dependent
Acknowledgements
We thank Fred Myhrer for fruitful discussions at an early stage of this research.
References (28)
Phys. Rep.
(2004)Phys. Lett. B
(1992)- et al.
Phys. Lett. B
(1999) - et al.
Phys. Lett. B
(2001) - et al.
Phys. Rev. C
(1996)et al.Phys. Rev. C
(2000) - et al.
Nucl. Phys. B
(1992) Adv. Nucl. Phys.
(2003)- et al.
Nuovo Cimento
(1955)et al.Nuovo Cimento
(1957) - et al.
Nucl. Phys. A
(1996) - et al.
Phys. Lett. B
(1996)