Antinucleon-nucleon interaction in chiral effective field theory
Results of an exploratory study of the antinucleon-nucleon interaction within chiral effective field theory are reported. The antinucleon-nucleon potential is derived up to next-to-next-to-leading order, based on a modified Weinberg power counting, in close analogy to pertinent studies of the nucleon-nucleon interaction. The low-energy constants associated with the arising contact interactions are fixed by a fit to phase shifts and inelasticities provided by a recently published phase-shift analysis of antiproton-proton scattering data. The overall quality of the achieved description of the antinucleon-nucleon amplitudes is comparable to the one found in case of the nucleon-nucleon interaction at the same order. For most -waves and several -waves good agreement with the antinucleon-nucleon phase shifts and inelasticities is obtained up to laboratory energies of around 200 MeV.
Keywords:Chiral Lagragians, Antinucleon-nucleon interaction
The antinucleon-nucleon () interaction has been studied quite extensively in the past Dover80 ; Dover82 ; Cote ; Timmers ; Hippchen ; Mulla ; Mull ; Entem06 ; Bennich , not least because of the wealth of data collected at the LEAR facility at CERN, cf. the reviews Rev1 ; Rev2 ; Rev3 . The majority of those investigations has been performed in the traditional meson-exchange framework where the -parity transformation is exploited to connect the elastic part of the interaction with the dynamics in the nucleon-nucleon () system. Annihilation processes are described either by a simple optical potential (which is often assumed to be spin- as well as energy-independent) Dover80 ; Dover82 ; Hippchen ; Mull or in terms of a coupling to a small number of effective two-body annihilation channels Cote ; Timmers ; Bennich .
In the last two decades chiral effective field theory (EFT) has become a standard tool in the studies of the interaction at low energies. This developement was initiated by two seminal papers by Weinberg Wei90 ; Wei91 in which he proposed that EFT and the power-counting rules associated with it should be applied to the potential rather than to the reaction amplitude. The reaction amplitude is then obtained from solving a regularized Lippmann-Schwinger equation for the derived interaction potential. His suggestion is based on the observation that diagrams with purely nucleonic intermediate states are strongly enhanced and, therefore, not amenable to a perturbative treatment. However, they can be taken into account and they are actually summed up to infinite order when solving the Lippmann-Schwinger equation. The chiral potential contains pion exchanges and a series of contact interactions with an increasing number of derivatives. The latter represent the short-range part of the force and are parametrized by low-energy constants (LECs), that need to be fixed by a fit to data. For reviews we refer the reader to the recent Refs. Epelbaum:2008ga ; Machleidt:2011zz . Presently the most refined calculations extend up to next-to-next-to-next-to-leading order (NLO) Entem:2003ft ; Epe05 and they yield a rather accurate description of the phase shifts up to laboratory energies of 250-300 MeV.
Naturally, the success of chiral EFT in the sector provides a strong motivation to apply the same approach also to the interaction. First and most important for the practical implementation, recently an update of the Nijmegen partial-wave analysis (PWA) of antiproton-proton () scattering data Timmermans has been published. For the new PWA Zhou2012 the resulting phase shifts and inelasticities are explicitly given and can be readily used for applying the chiral EFT approach to the interaction in the very same way as it has been done for the system.
A further incentive for exploring the feasibility of investigating the system within chiral EFT comes from the expected increase in interest in the interaction in the future due to the Facility for Antiproton and Ion Research (FAIR) in Darmstadt whose construction is finally on its way. Among the various project planned at this site is the PANDA experiment PANDA which aims to study the interactions between antiprotons and fixed target protons and nuclei in the momentum range of 1.5-15 GeV/c using the high energy storage ring HESR.
Finally, chiral EFT could be a very powerful tool to analyze data from recent measurements of the invariant mass in the decays of , mesons, etc., and of the reaction . In several of those reactions a near-threshold enhancement in the mass spectrum was found Bai ; Aubert ; Aubert1 ; BES12 and this enhancement could allow one to extract information on the interaction at very low energies BuggFSI ; Zou ; Sibirtsev05 ; Loiseau ; JH06 ; JH06a ; Entem ; Dedonder ; JH12 .
In the present paper we report on results of an exploratory study of the antinucleon-nucleon interaction within chiral EFT. In our application of chiral EFT to the interaction we follow exactly the approach used by Epelbaum et al. Epe05 ; Epe04 ; Epe04a in the case. It is consistent with the scheme originally proposed by Weinberg except that one aims for an energy-independent representation of the chiral potential Epe98 . For the time being we restrict ourselves to an evaluation of the potential up to next–to–next–to–leading order (NNLO). At leading order (LO) the potential is given by one–pion exchange (OPE) and two contact terms without derivatives. At next–to–leading order (NLO) contributions from the leading two–pion exchange (TPE) diagrams as well as seven more contact operators arise. Finally, at NNLO one gets contributions from the subleading TPE with one insertion of dimension two pion–nucleon vertices. Once the potential is established it has to be inserted into a regularized scattering equation in order to obtain the reaction amplitude. For the regularization we follow again closely the procedure adopted by Epelbaum et al. Epe05 ; Epe04a and others Entem:2003ft , in their study of the interaction and introduce a momentum-dependent exponential regulator function.
For investigations of the interaction within EFT based on other schemes see Refs. Chen2010 ; Chen2011 , where the Kaplan-Savage-Wise resummation scheme Kaplan is employed. These authors considered the interaction up to NLO. There have been also attempts to compute specific annihilation channels in chiral EFT Tarasov .
The present paper is structured as follows: The effective potential up to NNLO is described in Section 2. We start with a brief review of the underlying power counting and then provide explicit expressions for the contributions from pion exchange and for the contact terms. We also discuss how we treat the annihilation processes. Finally, we introduce the Lippmann-Schwinger equation that we solve and the parameterization of the S-matrix that we use. In Section 3 we indicate our fitting procedure and then we present the results achieved at NLO and at NNLO. Phase shifts and inelasticites for -, -, and - waves, obtained from our EFT interaction, are displayed and compared with those of the phase-shift analysis. Furthermore, predictions for -wave scattering lengths are given. A summary of our work and an outlook on future investigations is given in Section 4.
2 Chiral potential at next-to-next-to-leading order
where is the number of loops in the diagram, is the number of derivatives or pion mass insertions, and the number of internal nucleon fields at the vertex under consideration. The LO potential corresponds to and consists of two four-nucleon contact terms without derivatives and of one-pion exchange. There are no contributions at order due to requirements from parity conservation and time-reversal invariance. At NLO () seven new contact terms (with two derivatives) arise, together with loop contributions from (irreducible) two-pion exchange. Finally, at NNLO () there are additional contributions from two-pion exchange resulting from one insertion of dimension two pion-nucleon vertices, see e.g. Ref. Bernard:1995dp . The corresponding diagrams are summarized in Fig. 1.
The structure of the interaction is practically identical and, therefore, the potential given in Refs. Epe05 ; Epe04a can be adapted straightforwardly for the case. For the ease of the reader and also for defining our potential uniquely we provide the explicit expressions below.
2.1 Pion exchange
In line with Epe05 we adopt the following expression for the one-pion exchange potential
where is the transferred momentum defined in terms of the final () and initial () center-of-mass momenta of the baryons (nucleon or antinucleon). Obviously here relativistic corrections to the static one-pion exchange potential have been taken into account. As in the work Epe05 we take the larger value instead of in order to account for the Goldberger–Treiman discrepancy. This value, together with the used MeV, implies the pion-nucleon coupling constant which is consistent with the empirical value obtained from and data deSwart ; Bugg and also with modern determinations utilizing the GMO sum rule Baru:2011bw . For the nucleon (antinucleon) and pion mass we use the isospin-averaged values MeV and MeV, respectively. Note that the contribution of one-pion exchange to the interaction is of opposite sign as that in the case. This sign difference arises from transforming the vertex to the vertex via charge conjugation and a rotation in the isospin space and is commonly referred to as -parity transformation.
The two-pion exchange potential calculated using spectral function regularization Epe05 is given at NLO by
and at NNLO by
The NLO and NNLO loop functions and are given by
For the LECs and we adopt the central values from the –analysis of the system Paul : GeV, GeV. For the constant the value GeV is used, which is on the lower side but still consistent with the results from Ref. Paul . Note that slightly different values are employed in the partial-wave analysis Zhou2012 , namely GeV, GeV and GeV. These values are also consistent with the recent determination in Krebs:2012yv .
2.2 Contact terms
The spin-dependence of the potentials due to the leading order contact terms is given by Epe00
where the parameters and are low-energy constants (LECs) which need to be determined in a fit to data. At NLO, the spin- and momentum-dependence of the contact terms reads
where () are additional LECs. The average momentum is defined by . When performing a partial-wave projection, these terms contribute to the two –wave (, ) potentials, the four –wave (, , , ) potentials, and the - transition potential in the following way Epe05 :
with and . There are no additional contact terms at NNLO.
Note that the Pauli principle is absent in case of the interaction. Accordingly, each partial wave that is allowed by angular momentum conservation occurs in the isospin and in the channel. Therefore, there are now twice as many contact terms as in .
The main new feature in the interaction is the presence of annihilation processes. The system annihilates into a multitude of channels, where the decay to 4 to 6 pions is dominant in the low-energy region of scattering Rev1 . The threshold energy of those channels is in the order of 700 MeV while the threshold is at 1878 MeV. Therefore, one does not expect that annihilation introduces a new scale into the problem. Accordingly, there should be no need to modify the power counting when going from to because the momenta associated with the annihilation channels should be, in average, much larger than those in the system itself. This conjecture is supported by the fact that phenomenological models of the interaction can describe the bulk properties of annihilation very well by simple energy-independent optical potentials of Woods-Saxon or Gaussian type Dover80 ; Dover82 ; Hippchen ; Mull . The ranges associated with those interactions are of the order of 1 fm or less. The above considerations suggest that annihilation processes are primarily tied to short-distance physics and, therefore, can be and should be simply incorporated into the contact terms which anyway are meant to parameterize effectively the short-range part of (elastic) and/or scattering.
Nonetheless we want to emphasize that the above arguments are of pragmatical nature and not fundamental ones. There are definitely annihilation channels that open near the threshold. Specifically, there are indications that a sizeable part of the annihilation into multipion channels proceeds via two-meson doorway modes like or , and some of those have nominal thresholds close to that of scattering. On the other hand, according to empirical information the actual branching ratios into individual two-body channels are typically of the order of 1% Mull only and, therefore, they do not have any noticeable impact on the description of the bulk properties of annihilation. In fact, all the two-body annihilation channels together – as far as they have been measured – yield only about 30% of the total annihilation cross section at the threshold which is a strong evidence for the dominance of annihilation into 3 or more (uncorrelated) pions.
The study of scattering in EFT in Refs. Chen2010 ; Chen2011 followed the above arguments and took into account annihilation by simply using complex LECs in Eqs. (11)-(18). However, this prescription has an unpleasant drawback – it does not allow one to impose sensible unitarity requirements on the resulting scattering amplitude. With unitarity requirements we mean a condition that guarantees that for each partial wave its contribution to the total cross section is larger than its contribution to the integrated elastic cross section. In case of strict two-body unitary like for scattering below the pion production threshold these two quantities are, of course, identical.
Since we want an approach that manifestly fulfils unitarity constraints we treat annihilation in a different way. We start out from the observation that unitarity requires the annihilation potential to be of the form
where is the sum over all open annihilation channels, and is the propagator of the intermediate state . Note that Eq. (19) is exact under the assumption that there is no interaction in and no transition between the various annihilation channels. Performing an expansion of up to NNLO analoguous to the interaction and evaluating formally the sum and integral in Eq. (19) yields a contribution from the unitarity cut that can be written as
where stands for the , , , and partial waves. For the coupled partial wave we get
In those expressions the parameters and are real. Thus, for each partial wave we essentially recover the structure of the potential that follows from the contact terms considered above, with the same number of free parameters. However, in Eqs. (20)–(21) the sign of as required by unitarity is already explicitly fixed and does not depend on the sign of the parameters and anymore. Moreover, and most importantly, we see that a term proportional to arises in the waves at NLO and NNLO from unitarity constraints and it has to be included in order to make sure that unitarity is fulfilled at any energy.
Note that, in principle, there is also a contribution from the principal-value part of the integral in Eq. (19). However, it is real and, therefore, its structure is already accounted for by the standard LECs in Eqs. (11)–(18).
Finally we would like to add that in practice the treatment of annihilation via Eqs. (20)–(21) corresponds to the introduction of an effective two-body annihilation channel with a threshold significantly below the one of so that the center-of-mass momentum in the annihilation channel is already fairly large and its variation in the low-energy region of scattering considered by us is negligible.
2.3 Scattering equation
In the actual calculation a partial-wave projection of the interaction potentials is performed, as described in detail in Ref. Epe05 . The reaction amplitudes are obtained from the solution of a relativistic Lippmann-Schwinger (LS) equation:
Here, , where is the on-shell momentum. Like in the case we have either uncoupled spin-singlet and triplet waves (where ) or coupled partial waves (where ). We solve the LS equation in the isospin basis, i.e. for and separately, and we compare the resulting phase shifts with those in Ref. Zhou2012 that are likewise given in the isospin basis. It should be said, however, that for a comparison directly with data a more refined treatment is required. Then one should solve the LS equation in particle basis and consider the coupling between the and channels explicitly. In this case one can take into account the mass difference between () and () and, thereby, implement the fact that the physical thresholds of the and channels are separated by about 2.5 MeV, and also one can add the Coulomb interaction in the channel. The potential in the LS equation is cut off with a regulator function,
The relation between the – and on–the–energy shell –matrix is given by
The phase shifts in the uncoupled cases can be obtained from the –matrix via
For the –matrix in the coupled channels () we use the so–called Stapp parametrization Stapp
In case of elastic scattering the phase parameters in Eqs. (25) and (26) are real quantities while in the presence of inelasticites they become complex. Because of that, in the past several generalizations of these formulae have been proposed that still allow one to write the -matrix in terms of real parameters Arndt ; Zhou2012 . We follow here Ref. Bystricky and calculate and present simply the real and imaginary parts of the phase shifts and the mixing parameters obtained via the above parameterization. Note that with this choice the real part of the phase shifts is identical to the phase shifts one obtains from another popular parameterization where the imaginary part is written in terms of an inelasticity parameter , e.g. for uncoupled partial waves
Indeed, for this case which implies that since because of unitarity. Since our calculation implements unitarity, the optical theorem
is fulfilled for each partial wave, where .
For the fitting procedure and for the comparison of our results with those by Zhou and Timmermans we reconstructed the -matrix based on the phase shifts listed in Tables VIII-X in Ref. Zhou2012 and on the formulae presented in Sect. VII of that paper and then converted them to our convention specified in Eqs. (25) and (26).
In the fitting procedure we follow very closely the strategy of Epelbaum et al. in their study of the interaction Epe05 ; Epe04a . In particular, we consider the same ranges for the cutoffs, namely for the cutoff in the LS equation values of = 450–600 MeV at NLO and = 450–650 MeV at NNLO while for the spectral function regularization variations we consider values in the range = 500–700 MeV. For any combination of the cutoffs and , the LECs and are fixed from a fit to the - and -waves and the mixing parameter of Ref. Zhou2012 for laboratory energies below 125 MeV ( MeV/c). The numerical values of the LECs are compiled in Tables 1 (NLO) and 2 (NNLO) for a selected combination of the cutoffs. The values for in the isospin case found in the fitting procedure turned out to be very small and, therefore, we set them to zero.
Our results are displayed and compared with the PWA Zhou2012 in Figs. 2-6. The bands represent the variation of the obtained phase shifts and mixing parameters with the cutoff. Those variations can be viewed as an estimate for the theoretical uncertainty. Thus, in principle for the same variation of the cutoff those bands should become narrower and narrower when one goes to higher order. However, as argued in Ref. Epe04a , in practice one has to be careful in the interpretation of the bands, specifically for the transition from NLO to NNLO. Since the same number of contact terms are present in the interactions at NLO and NNLO one rather should expect variations of similar magnitude. In particular, for reasons discussed in Epe04a the cutoff variation underestimates the uncertainty for the NLO results. In any case one has to keep in mind that, following Ref. Epe04a , we use a larger cutoff region at NNLO than for the NLO case.
Let us now discuss the individual partial waves. Results for the channel can be found in the upper part of Fig. 2. Obviously, the phase shift for isospin (we use here the spectral notation ) is very well described up to fairly high energies – even at NLO – and likewise the inelasticity, presented in terms of the imaginary part of the phase shift. Moreover, the dependence on the cutoff is very small. In the channel the situation is rather different. Here we observe a sizeable cutoff dependence of the results for energy above 150 MeV. This has to do with the fact that the PWA suggests a resonance-like behavior of the phase in this region. Since this resonance lies in an energy region where we expect our results to show increasing uncertainties, based on the experience from the case Epe04a , it is not surprising that it is difficult to reproduce this structure quantitatively. Nevertheless, there is a visible improvement when going from NLO to NNLO and at the latter order the empirical phase shifts already lie within the error bands of theory.
We want to emphasize that this improvement is entirely due to inclusion of the subleading two-pion exchange potential, since as already stressed above no new contact terms arise at NNLO and thus the number of adjustable parameters is the same at NLO and NNLO. Also, it should be said that the NLO result, shown here up to MeV, exhibits a similar trend like the one for NNLO at higher energies, i.e. the phases reach a maximum and then become more negative again.
The situation for the partial wave is similar, see. Fig. 2 (lower part). Also here the phase shifts are well reproduced while in the case there is an even larger cutoff dependence than in the . Obviously also the amplitude of the PWA Zhou2012 exhibits a resonance-like behavior. Its reproduction requires a potential that is repulsive at large separations of the antinucleon and nucleon but becomes attractive for short distances. Since there is only a single LEC up to NNLO for waves, the magnitude and range of such an attraction cannot be adequately accounted for. For improvements one has to wait for a NLO calculation.
Results for the and partial waves are shown in Fig. 3. In general, the description improves when going from NLO to NNLO. Specifically for the two channels and the the results at NNLO agree with those of the PWA within the uncertainty bands for energies up to 150 MeV and often even up to 250 MeV. An exception is the partial wave where the phase shift can only be described up to 50 MeV or so. Similar to the , the PWA yields a negative phase at low energies which tends towards positive values at larger energies Zhou2012 and one encouters the same difficulty as discussed above.
In Fig. 4 one can find our results for the coupled – partial wave. Here the -wave phase shifts (and also the inelasticity) are satisfactorily described over the whole energy range considered with uncertainties comparable to those observed for the interaction Epe04a . There is a larger cutoff dependence in the waves and the mixing parameter , specifically for . However, one has to keep in mind that there is no LEC up to NNLO for the waves. The exhibits the trend of turning from negative to positive values at higher energies which cannot be described in an NNLO calculation, as discussed above.
The situation in the – channel is displayed in Fig. 5. In general our results agree with those of the PWA up to about 200 MeV within the uncertainty. Stronger deviations are visible again for those phases which show a resonance-like behavior like, e.g., the .
At last, in Fig. 6 the and phase shifts are presented. There are no LECs in those partial waves up to NNLO and, thus, our results are genuine predictions. The potential consists only of one- and two-pion exchange and, consequently, there is no contribution to annihilation. Thus, and we do not show this quantity.