The pionnucleon scattering lengths from pionic deuterium ^{†}^{†}thanks: This research is part of the EU Integrated Infrastructure Initiative Hadron Physics Project under contract number RII3CT2004506078. Work supported in part by DFG (SFB/TR 16, “Subnuclear Structure of Matter”).
Abstract
We use the framework of effective field theories to discuss the determination of the wave scattering lengths from the recent highprecision measurements of pionic deuterium observables. The theoretical analysis proceeds in several steps. Initially, the precise value of the piondeuteron scattering length is extracted from the data. Next, is related to the wave scattering lengths and . We discuss the use of this information for constraining the values of these scattering lengths in the full analysis, which also includes the input from the pionic hydrogen energy shift and width measurements, and throughly investigate the accuracy limits for this procedure. In this paper, we also give a detailed comparison to other effective field theory approaches, as well as with the earlier work on the subject, carried out within the potential model and multiple scattering framework.
pacs:
36.10.Gv and 12.39.Fe and 13.75.Cs and 13.75.Gx1 Introduction
The Pionic Hydrogen collaboration at PSI PSI1 ; PSI2 has performed high precision measurements of the strong interaction shift and width of the state of pionic deuterium from the ray transition. The (complex) piondeuteron scattering length was extracted from these measurements with the use of the leadingorder Deser formula Deser
(1.1) 
where is the Coulomb binding energy in the state, and denotes the Bohr radius (in these formulae, stands for the reduced mass of the system). The most recent measurement of the piondeuteron scattering length by the Pionic Hydrogen collaboration at PSI PSI2 yields
Performing experiments to determine is usually justified by the possibility of extracting independent information about the wave isoscalar () and isovector () scattering lengths. What makes this enterprise particularly interesting is the fact that in the multiple scattering theory . If one could accurately evaluate the higherorder terms in this expression, then a precise measurement of would enable one to constrain the value of , which is in general a rather delicate task. The reason for this is that, since is much smaller than the isospinodd scattering length , a very high accuracy is needed in order to determine from the measurements of the linear combinations of and . This is exactly the case in the experiments measuring the pionic hydrogen energy shift and width, which enable one to determine the combinations and , respectively. Thus, the measurement of contributes a complementary piece of information about the scattering lengths, which can be used in the complex theoretical analysis finally aimed at the determination of both and PSI3 . We wish to mention here that these scattering lengths are quantities of fundamental importance in lowenergy hadronic physics, since they test the QCD symmetries and the exact pattern of the chiral symmetry breaking. Moreover, since the knowledge of these scattering lengths places a constraint on the interactions at low energy, it also affects our understanding of more complicated systems where interaction serves as an input, e.g. interaction, nucleus scattering, threenucleon forces, etc.
Expressing the scattering length in terms of the parameters characterizing the underlying pionnucleon and nucleonnucleon dynamics is one of the classical problems in conventional nuclear physics based on the potential scattering formalism, see, e.g. Afnan ; Faldt ; Mizutani ; Thomas ; Deloff ; Baru ; Loiseau ; Hanhart ; Wilkin . Note, however, that the experiments on pionic deuterium will be used to extract the scattering lengths in QCD and not in any potential model. In other words, using the latter in order to establish the relation between and scattering lengths introduces a theoretical error in the analysis of the experimental data, whose magnitude is very hard to control.
In recent years, the problem of a very low energy piondeuteron scattering has been studied within the framework of effective field theories (EFTs). The method originates from the seminal paper of Weinberg Weinberg , where chiral Lagrangians have been systematically applied for the description of interactions of pions with nuclei. In this paper, by using the chiral Lagrangian, one calculates the set of diagrams contributing to the “irreducible transition kernel” for the pion scattering on two nucleons, and the result is then sandwiched between “realistic” deuteron wave functions in order to evaluate the scattering amplitude of the pion on the deuteron. In actual calculations carried out in Ref. Weinberg , the phenomenological deuteron wave function for the Bonn potential model has been used. We wish to note that this last step, in general, can be justified within the framework of the effective field theories, only if the particular process which one is going to describe is dominated by the longrange mechanisms, e.g. by onepion exchange. On the other hand, when the calculations within such a “hybrid” approach are pursued in higher orders in Chiral Perturbation theory (ChPT), the kernels grow faster with a large momenta and probe shorter distances. Moreover, this shortdistance behavior is not necessarily correlated to that of the phenomenological wave function. From this we conclude that in order to obtain a systematic description of the piondeuteron system, it is preferable to use the deuteron wave functions and the transition kernels, evaluated within the same fieldtheoretical setting – in this case, no specific conjecture about the dominance in the unobservable quantities, like the kernel or the wave function is needed. For the latest work within the hybrid approach, see e.g. Kaiser ; Doring .
Further development of the approach based on chiral Lagrangians (see, e.g. Kaiser ; Doring ; BBLM ; Borasoy ; Beane ; Bernard ), has followed different paths. In the paper Borasoy one has used the framework with perturbative pions, whereas the authors of Ref. Beane make use of the socalled Heavy Pion EFT (HP EFT) with the dibaryon field. The latter approach is quite close to the one used in the present work. The technique used in Refs. Borasoy ; Beane has the advantage that one may easily construct the deuteron wave function in a closed form, since the lowestorder nucleonnucleon interactions are described by a contact fournucleon vertex. The central problem in both the papers is related to the calculation of one particular diagram, describing double scattering of pions. These calculations lead to a very strong scale dependence near – a natural choice of the scale parameter in this sort of the effective theories. Since, on the other hand, this dependence must be canceled by a contribution from the lowenergy constant (LEC), which describes pointlike interactions of four nucleons and two pions, we easily conclude that the magnitude of this LEC can not be small. In the absence of any information about the actual value of this LEC apart from naive dimensional orderofmagnitude estimates based on the naturalness arguments, one may finally conclude that the theoretical uncertainty in the relation of and scattering lengths should be very large.
On the other hand, the results obtained in Ref. Bernard seem not to be in agreement with those of Refs. Borasoy ; Beane . The method which is used in Ref. Bernard , is a systematic extension and elaboration of Weinberg’s original proposal, where both the transition kernel and the deuteron wave function are constructed in ChPT (note also, that the systematic derivation of the unitary and the energyindependent potentials within this framework has been discussed recently in Ref. Krebs ). The approach uses cutoff regularization to deal with the potentials that are growing for a large threemomenta. The typical scales for the cutoff mass are somewhat smaller than the hadronic scale in QCD (depending on the order in ChPT in which the calculations are carried out). The results of the calculations are dependent, which is a reminiscent of the scaledependence in the dimensional regularization scheme. The bulk of this dependence should be canceled by an analogous dependence in the LECs, and the remainder, which is an artifact of the nonperturbative formalism used, should be of a higher order in ChPT. In Ref. Bernard , the cutoff dependence of the scattering length has been studied, with the LECs assumed to vanish. In a remarkable contrast with Refs. Borasoy ; Beane , the dependence of the results in Ref. Bernard turns out to be very mild, thereby concluding that the LECs must have a weak cutoff dependence. If one could interpret the cutoff dependence as an estimate of the uncertainty of the method, then the results of Ref. Bernard would amount to a rather accurate prediction of the scattering length within the framework of ChPT.
The present situation which was described above is unacceptable from the point of view of both the theory and the phenomenological analysis of the data. From the theoretical point of view, the calculations carried out in Ref. Bernard , clearly indicate that the diagrams in which the virtual pions are emitted or annihilated, are strongly suppressed. This phenomenon originates from the infrared enhancement of a certain class of the diagrams in the Weinberg scheme, as well as the threshold suppression of the diagrams containing pseudoscalar vertices. In order to accommodate the above feature in the theory with nonperturbative pions, in Ref. Bernard a novel counting, inspired by the HP EFT, has been enforced on top of the conventional ChPT Lagrangian. Stated differently, this means that simpler effective theories, which were used in Refs. Borasoy ; Beane , are physically adequate for the problem considered. How can it then be that using a simpler theory, we get an answer which contradicts the answer obtained in Weinberg’s framework Bernard , the very approach one starts from? From the point of view of phenomenology, the existing conflicting predictions, on the one hand, do not encourage the experimentalist’s efforts to measure the piondeuteron scattering length to a better accuracy and on the other hand, suggest that the values of the scattering lengths extracted from the analysis of the data should be taken with a grain of salt.
The aim of the present paper is to perform a thorough investigation of piondeuteron scattering at threshold within the framework of lowenergy effective theories. In particular, we plan to clearly establish the limits of accuracy for extracting scattering lengths from the measured scattering lengths. We also perform a detailed study of the abovementioned discrepancy between the results obtained within the HP EFT and in the Weinberg approach. Moreover, the investigation of this subtle question, in our opinion, is by itself very informative and sheds light on many peculiar aspects of the effective field theories in general.
The complex problem, which we are going to consider in this paper, naturally falls into several subproblems, which are characterized by distinct momentum scales. Consequently, instead of trying to describe everything at once, it is convenient to construct a tower of effective field theories, matched one to another, each designed for one particular momentum scale.

At the momentum scales , the charged pion and the deuteron form an atom, whose observables are measured by the experiment. The characteristic distances in such an atom – hundreds of fm – are much larger than the deuteron size, and the binding energy in the ground state, which almost coincides with the Coulomb binding energy . Stated differently, at these energies the deuteron can not be resolved as a composite particle, and the effective theory, which describes the atom, contains the deuteron field (not the nucleon fields) as an elementary degree of freedom. The hard momentum scale in this effective theory is given by the average value of the threemomentum of the nucleons bound within the deuteron, , where stands for the nucleon mass. The expansion parameter in this theory is given by the ratio of the scales . The output from the calculations within this effective theory is relation which connects the measured energy shift of the bound state to the scattering amplitude at threshold in the nexttoleading order in isospin breaking. In its turn, the latter at the leading order in isospin breaking coincides with the scattering length . , is much smaller, than the binding energy of the deuteron

Extracting the scattering length from the pionic deuteron one next has to find the relation of this quantity to the wave scattering lengths and . In order to achieve this goal, we have to construct another effective field theory, in which the independent degrees of freedom are the pion and the nucleon fields, whereas the deuteron emerges as a bound state of the proton and the neutron. The characteristic momentum scale in this theory is defined by the binding momentum . Furthermore, a careful analysis of the results of Ref. Bernard provides us with an important clue: The processes, in which the virtual creation and annihilation of pions takes place, are suppressed as compared to the processes where this does not occur (although both processes may formally have the same chiral order). Note that these processes naturally come together in the conventional relativistic QFT. This fact clearly indicates that the most economic way to describe scattering at threshold is to design an effective field theory, in which the pion creation and annihilation processes are explicitly excluded – all vertices in the Lagrangian contain equal number of ingoing and outgoing pions and nucleons. The whole information about these processes is, however, not lost: it is included in the pertinent LECs of such an effective theory. Moreover, it is also clear that for such small energies, one can treat kinematical relativistic factors as perturbations both for pions and for nucleons.
The calculations of the deuteron properties in the abovedescribed theory, which will be referred to as the heavypion effective theory hereafter, dramatically simplify and can be performed analytically. The output of the calculations is the quantity , expressed in terms of the threshold parameters of the and scattering. The hard scale in such a theory is given by the pion mass , and the expansion parameter is given by the ratio of scales . The matching to the previous effective theory is performed for the scattering amplitude at threshold: this quantity must be the same in both theories.
Note also that from now on we neglect all isospinbreaking effects (one could not do this at the earlier step, because the pionic deuterium is created predominately by Coulomb interactions.). In this approximation, the threshold scattering amplitude coincides with the scattering length . If needed, the isospinbreaking effects can be turned on later.

The simplicity of the calculations in the HP EFT comes at the cost of the large size of the LECs. Since the hard scale of the theory is determined by , this is also the scale that enters in the estimate of the size of the (unknown) LECs in the assumption that these LECs have the natural size (note that some LECs might be parametrically enhanced as compared to the value which is expected on the purely dimensional grounds, see below). On the other hand, if the calculations are done in ChPT, the naturalsize LECs are suppressed by a higher scale rather than . Thus, the rationale for performing calculations in the Weinberg framework can be formulated as follows. In these calculations, one “resolves” the dynamics of the system at the scales from up to the scale , which is the energy range where the interactions in the system of few pions and nucleons are predominately determined by (multi)pion exchanges. One may then assume that the bulk contribution to the HP LECs comes from the momentum region between and and can be expressed in terms of pion loops, which are calculated in the Weinberg scheme.
If we suppose that such a scheme is realized, we arrive at the effective theory, where the characteristic momenta are of order and the hard scale is given by . The expansion parameter, in the absence of other scale, is given by the ratio of two scales . The matching to the HP EFT is performed for the matrix element of the process , that determines a particular LEC of the HP Lagrangian. One of the objectives of the present paper is to find out whether doing the calculations in the Weinberg framework and performing the matching to the HP EFT enables one to indeed reduce the uncertainty related to the choice of LECs.
The organization of the paper follows the abovedescribed scheme of “nested” effective field theories. Namely, in section 2 we consider the precise extraction of the scattering length from the experimental data on the pionic deuterium. Then, in section 3, we construct the systematic heavypion effective theory (HP EFT) in order to calculate the scattering lengths in terms of the threshold parameters of and interactions. In order to establish the connection to ChPT in the Weinberg scheme, in section 4 we perform the matching of the threshold amplitudes in both theories. We also provide a numerical analysis and discuss the question of accuracy. A detailed comparison to the existing approaches is carried out in section 5. Finally, section 6 contains our conclusions.
2 Pionic deuterium
In the experiment at PSI PSI1 ; PSI2 , one measures the energy of ray transition, deducing the strong shift of the pionic deuterium in the state and the scattering length from this measurement. At the first step, in order to obtain the strong shift, one has to subtract the socalled “electromagnetic shift” from the full measured value, where the former is calculated in the accuracy that matches the experimental precision. At the next step, the scattering length should be extracted from the strong shift by means of the Desertype formula (1.1). If required for accuracy considerations, the latter relation can also be generalized to include nexttoleading order isospinbreaking corrections.
To the best of our knowledge, complete calculations of the electromagnetic shift in the pionic deuterium are not available in the literature, except the results contained in table 1 of Ref. PSI1 , where different contributions are given without a derivation. The investigations in Ref. Irgaziev are not complete – as the authors themselves note, they do not include all isospinbreaking corrections at nexttoleading order. In order to have a complete and transparent fieldtheoretical treatment of the pionic deuterium problem at all levels, we find it appropriate here to rederive the expression for the full energy shift at order , , and to check (at least, numerically) the results given in table 1 of Ref. PSI1 .
The method, which will be used in our calculations, is analogous to the one applied recently to describe Bern1 ; Bern2 ; Bern4 ; Schweizer , Bern3 ; Zemp , Schweizer and Bonn1 atoms. In this section, we display only the final results of the calculations – the necessary details are provided in appendix A. The full binding energy in a given stationary state of the pionic deuterium depends on the principal quantum number , on the orbital quantum number and on the total angular momentum . For a given (except ) the total angular momentum takes the values . This splitting, which is explicitly evaluated in appendix A, is tiny. The following averaged value is relevant for the analysis of the experimental data
(2.1) 
Up to the nexttoleading order in isospin breaking, the full energy shift of the state can be separated in what is called “electromagnetic” and “strong“ parts. In order to simplify the comparison to the existing results, the former is additionally split by hand in different pieces. Finally, at this order one obtains
(2.2) 
In the above formula, we have chosen the same naming scheme as in Ref. PSI1 . Note that in this paper individual contributions are not specified explicitly, so the identification, which is given below and in table 1, is performed by analogy with the pionic hydrogen case Sigg . Our explicit expressions are given below
(2.3) 
where denotes the mass of the deuteron. An explicit expression for the vacuum polarization contribution is given in Ref. Eiras , see Eq. (3) of that paper.
In order to be able to compare with the existing results, our numerical calculations have been performed for the same values of the input parameters as in Ref. PSI1 . We take the deuteron binding energy to be , and the charge radii of the deuteron and of the pion are taken to be equal and , respectively. The calculations were performed for the value of the charged pion mass . In addition, the calculation of the finitesize correction has been performed by using the latest data for the charge radii new1 and , see new2 and references therein (the result changes slightly). The results of our calculations and the comparison to the results of Ref. PSI1 are given in table 1. Note that we have not calculated higherorder (nexttonexttoleading) isospinbreaking corrections that are given in the last two entries of this table. The results of the calculations from Ref. PSI1 in these cases should be taken at face value. It can be immediately seen from the table that our calculations completely confirm the results of Ref. PSI1 at nexttoleading order – the agreement between the two columns is perfect.
After having subtracted the calculated electromagnetic contributions from the measured transition energy, one finally arrives at the strong shift, which is related to the scattering length. Since in the states the strong shift is proportional to and is thus tiny, the measurement of the quantity yields directly the strong shift in the state. In nexttoleading order in isospin breaking, the strong shift for the states with is given by
(2.4) 
where the quantity denotes the threshold scattering amplitude in the presence of photons, which is obtained from the conventional amplitude by subtracting all singular contributions at threshold (see Bern1 ; Bern2 ; Bern4 ; Schweizer ; Bern3 ; Zemp ; Bonn1 for more details and definitions). The normalization of this quantity is chosen so that in the absence of the isospinbreaking effects, it reduces to the scattering length
(2.5) 
where the ellipses stand for terms vanishing at and . These terms can be in principle evaluated in ChPT in a systematic manner, in analogy with more simple cases of Meissner_piN ; Mojzis and Meissner_NN scattering. Further, the quantity stands for the vacuum polarization correction to the strong shift ( denotes the correction of the Coulomb wave function at the origin due to the vacuum polarization effects). This correction was evaluated in Ref. Eiras only for the ground state. However, the approach used in this paper can be straightforwardly generalized for the radially excited states. Finally, it is interesting to note that the dependence of the correction term in Eq. (2) is universal, since shortrange effects are the same in all atomic states. For this reason, even potential models (see, e.g. Ericson:2004ps ) agree with our result in what concerns the difference of the correction terms in the states with a different .
Calculated corrections  Ref. PSI1  This work 
to [eV]  
Point nucleus (KleinGordon)  
Nuclear and finite size  
new1 ; new2  
Vacuum polarization  
Relativistic recoil  
Higher order radiative  –  
corrections  
Nuclear polarization  – 
To summarize, in this section we have checked the validity of the procedure which is used for the theoretical analysis of the pionic deuterium data at PSI. The calculated electromagnetic shift agrees very well to the one given in Ref. PSI1 . Further, we have obtained the general expression for the (complex) strong energy shift of the pionic deuterium in the nexttoleading order in isospin breaking, in terms of the threshold scattering amplitude. This relation should in principle be used to replace the lowestorder formula (1.1) in the data analysis. Note however, that the Coulomb correction which is explicitly displayed in Eq. (2) (second term in the brackets), is of order of , if one replaces by Eq. (2.5) and uses the value of the scattering length given in Eq. (1). Note also that is an expected order of magnitude for the vacuum polarization contribution in Eq. (2), see Ref. Eiras . Since there are no obvious reasons for having an anomalously large isospinbreaking correction in the quantity either (see e.g. PSI1 ; PSI2 and references therein), in the following we do not consider isospinbreaking corrections to the energy shift at all and concentrate on the lowestorder relation Eq. (1.1). If it turns out that the determination of the scattering lengths from the analysis of the pionic deuterium data can be performed at a few percent level that requires the inclusion of the isospinbreaking effects in Eq. (1.1), one can always go back to the relation Eq. (2).
3 Heavypion effective theory for scattering
3.1 The Lagrangian
The findings of Ref. Bernard , as well as the earlier work on the subject (see, e.g. Weinberg ) serve as a clear indication of the fact that the chiral counting is not the most suitable one to be applied for the description of lowenergy scattering. Most straightforwardly, this can be visualized by comparing the contributions from the individual diagrams, depicted in Fig. 2 of Bernard , which is reproduced here, in Fig. 1. The contribution from the diagrams 1b+1c is by two orders of magnitude smaller than the contribution from 1a, although all three diagrams emerge at the same chiral order. The reason for this difference is that, in contrary to 1a, the diagrams 1b+1c describe processes with the virtual pion emission/absorption (additional suppression at a small momenta is caused by the presence of the vertices in the diagrams 1b and 1c).
The above discussions lead to the conclusion that it will be convenient to describe the scattering at threshold in a framework in which the absorption and emission of hadrons does not appear explicitly at the level of Feynman diagrams, but is included in the couplings of the effective Lagrangian. In this manner, we arrive at the theory which must be in a spirit similar to HP EFT (see Beane and references therein). Below we dwell on some differences which exist between the approach used here and in Ref. Beane .

The HP EFT of Ref. Beane uses the notion of the dibaryon field, whereas we work in terms of the elementary nucleon constituents and sum up all interactions in the subsystems. After substituting the expression for the couplings of the Lagrangian in terms of the observables (coefficients in the effective range expansion), the “deuteron propagator” in the present paper coincides with the dibaryon propagator of Ref. Beane in the limit of vanishing effective range. We opt to work in terms of nucleon field in order to make the comparison with ChPT in the Weinberg picture more transparent.

It has been argued that the technique based on the introduction of the dibaryon field enables one to effectively sum up all potentially large contributions coming from the large scattering length and the effective range, whereas higherorder terms can be treated perturbatively. The results of the present paper are obtained under an additional assumption that the effectiverange term is small, leading to some technical simplifications. This assumption is, however, not critical – the inclusion of the effectiverange term is straightforward in our approach and does not affect the conclusions.

In Ref. Beane the authors have studied one particular diagram that corresponds to the doublescattering contribution the the scattering length. In this paper, a systematic expansion of the quantity is performed in the small parameter , up to and including terms of order . At this order, there are additional diagrams apart from the one mentioned above.

The most important question is the convergence of the series for the scattering length. We believe that there are (indirect) indications which testify in favor of the convergence. First of all, at the momenta , the pionless effective theory gives still a reasonable description of the sector. As was mentioned, this fact is in agreement with the observation made in Ref. Bernard that the “modified power counting” in the scattering length works much better than the original chiral counting^{1}^{1}1 And vice versa, one may treat the HP EFT, as the systematic fieldtheoretical realization of the counting , which is heuristically implemented in the “modified power counting” of Ref. Bernard .. Yet another justification of the method is provided by the wellknown fact that in the Faddeev approach, the multiplescattering series for the threshold scattering amplitude are rapidly convergent, since the scattering lengths are much smaller than the deuteron radius (see e.g. Doring and references therein).
After these preliminary remarks, let us consider the Lagrangian of our theory. By construction, the Lagrangian does contains only vertices with the same number of the incoming and outgoing pions and nucleons. Restricting ourselves to the nonderivative couplings, one may easily write down the most general form of this Lagrangian (in addition, we omit below also all threebody nonderivative terms which contain , and/or the pair in the state: such terms do not contribute to the threshold scattering amplitude at the accuracy we are working)
(3.1)  
where the ellipses stand for the omitted threebody terms, as well as for the higherorder terms in the derivative expansion. The nonrelativistic pion and nucleon fields are defined as , where , and . Further, and denote the projection operators onto the and states, respectively
(3.2) 
where and are the Pauli matrices in the spin and isospin space, respectively. Note that we have not introduced an elementary deuteron field in the Lagrangian. In our approach, the deuteron emerges as a boundstate pole in the Green functions after the nonperturbative resummation of the lowestorder fournucleon vertex.
In the above Lagrangian, stand for the effective lowenergy couplings. These should be determined from matching of the various observables. Using dimensional regularization for calculating the loops enormously simplifies these calculations: as it is well known, all twoparticle bubbles vanish at threshold and the results of the treelevel matching in the twoparticle sectors remain intact. For example, the constants are related to the scattering lengths through
(3.3) 
and this relation remains unaffected by loop corrections. As concerning the constant , we find it more convenient to perform the matching in the channel for the deuteron binding energy, and not for the scattering length in the system. The difference between these two methods shows up at higher orders.
3.2 The deuteron
In the twonucleon sector of the HP EFT, there is no trace of pionnucleon interactions: scattering is described completely in terms of contact fournucleon interactions. The only possible loop diagrams are the channel bubbles containing the vertices with the couplings . At higher orders, one should also include the derivative fournucleon vertices.
Consider the following connected fourpoint function in dimensions
(3.4)  
where all spin and isospin indices have been suppressed, and the CM and relative momenta are defined as
(3.5) 
The centerofmass (CM)
frame corresponds to
, and in physical space.
At the energy , the fourpoint function Eq. (3.4) develops a boundstate pole corresponding to the deuteron
where the deuteron wave function is defined as
(3.7) 
and the sum in Eq. (3.2) runs over the polarizations of the deuteron spin.
On the other hand, one may evaluate the 4point function Eq. (3.4) with the use of the Lagrangian Eq. (3.1) that amounts to the resummation of the geometrical series corresponding to the channel bubbles with fournucleon vertices. Further, since at the leading order the deuteron is a purely state, we may put in order to get the deuteron pole. As the result of this resummation, one gets
(3.8) 
where, , and
(3.9) 
with . In the CM frame the denominator in Eq. (3.2) develops a pole at . This gives (in dimensions)
(3.10) 
Finally, in the CM frame the behavior of the Green function near the deuteron pole is given by
(3.11) 
where the deuteron wave function renormalization constant is given by
(3.12) 
3.3 Piondeuteron scattering
The piondeuteron scattering amplitude can be extracted from the 6point connected Green function
(3.13) 
where the CM and relative momenta of nucleon pairs are again given by Eq. (3.2), and denote the pion momenta. Near the mass shell , , the sixpoint function (3.3) develops the double deuteron pole. Since we are interested only in the threshold scattering amplitude, we may take from the beginning and . Then, in the vicinity of the pole, one has
(3.14)  
with
(3.15) 
The residue of the quantity on the pion mass shell yields the threshold scattering amplitude
(3.16) 
On the other hand, in the theory with the Lagrangian Eq. (3.1) the 6point Green function for vanishing 3momenta can be given in the following form
(3.17) 
The quantity in Eq. (3.3) corresponds to the “truncated” Green function, and the factor emerges after the resummation of the bubbles in the state before the first and after the last interaction of the pion with one of the nucleons (see Fig. 2). Finally, denotes the sum of all diagrams in which the virtual scattering of the pion on one of the nucleons occurs before (or after) all interactions, or in which the first (or last) interaction happens in the state (the corresponding vertex is proportional to ). This class of the diagrams does not develop a double deuteron pole, and contributes only to the regular part of the Green function. Consequently, from the comparison of Eq. (3.3) to Eqs. (3.14), (3.3) and (3.16) one may read of the scattering amplitude at threshold
(3.18) 
where
(3.19) 
Hence, the prescription for calculating the threshold scattering amplitude is formulated as follows: in the connected 6point Green function Eq. (3.3) omit all Feynman diagrams, where the very first or very last interaction occurs between the pion and nucleon, or between pair in the state. In the remaining diagrams, resum all initial and finalstate bubbles and write the final result in a form of Eq. (3.3); read off the quantity ; perform the massshell limit, let all 3momenta vanish, multiply by the normalization factor , given by Eq. (3.19) and get the threshold scattering amplitude .
3.4 Leading order
At the lowest order in the expansion parameter , there is a single contribution to the quantity defined by Eq. (3.3), which is depicted in Fig. 3. At threshold, this contribution equals to
(3.20) 
Substituting this result into Eq. (3.18), using Eqs. (2.5), (3.3) and (3.19) and expanding , at the leading order we finally obtain
(3.21) 
which of course coincides with the wellknown result. Here we only wish to note that our result is valid at all orders in the chiral expansion for the scattering length . On the other hand, if one works in the Weinberg scheme, one has to identify the contributions to the quantity order by order in the chiral expansion Bernard .
3.5 Nexttoleading order
Since the quantity turns out to be very small, the corrections exceed in magnitude the leadingorder result. In the HP EFT, four diagrams depicted in Fig. 4 contribute at nexttoleading order. At threshold, we get
(3.22) 
Note that performing (formally) the limit in the quantity , one gets . In this limit, it is possible to relate the quantity to the average of the operator between the deuteron wave functions in the potential that corresponds to pointlike interaction . In the same normalization as in Ref. Bernard one obtains
(3.23) 
and the standard expression for the doublescattering contribution in the limit (see e.g. Bernard ) is reproduced.
The counting of the above diagrams proceeds as follows. According to Eq. (3.10), the coupling counts like , and the couplings count like . Further, after integrating over the timelike components , one may rescale , , , with . Each propagator of a pion or a nucleon counts as and the “virtual deuteron propagator” counts as . With these counting rules, it is straightforward to ensure that (modulo logarithms). Furthermore, since the constant cancels the ultraviolet divergences in the diagrams 4a,b,c, it must count at the same order in . This fixes . Note also that the contributions proportional to the coupling constant ( intermediate states) have been dropped from altogether at this order. This is related to the orthogonality of the projectors and given in Eq. (3.2).
The fact that the quantity in Eq. (3.5) is proportional to simplifies the calculations considerably. Since is very small, we shall systematically neglect in all expressions, and thus assume . Evaluating the remaining integrals in dimensional regularization and carrying out the renormalization in a standard manner, we finally arrive at the following expression for the scattering length at the nexttoleading order
(3.24)  
where denotes the renormalized coupling constant
(3.25) 
and denotes the scale of dimensional regularization. Further, , denote the integrals over Feynman parameters which depend on the dimensionless variable and emerge from . These integrals are evaluated in appendix B. Here, we only give their approximate values
(3.26) 
At present, the numerical value of the counterterm is not known. For this reason, one has to include this unknown quantity completely in the theoretical error and to estimate the uncertainty that emerges already at nexttoleading order. Most easily, this can be done by using the renormalization group equation for the scale dependence of , which at this order reads
(3.27) 
We use the following procedure to estimate the uncertainty. Since the hard scale of the theory is of order of , we may set at a some scale and study the dependence of the plot in the plane, which emerges from Eq. (3.24) at a given (experimental) value of . This plot is given in Fig. 5. Varying in a “reasonable” range, we may thus visualize the error that is caused by the scale dependence. Here, we wish to note that the scale dependence is of course not the only possible source of the theoretical uncertainty in general. In order to have a reliable estimate of the error (in the case of the weak scale dependence) one has, in addition, to use dimensional arguments to estimate the size of the LECs. However, in the case of a strong scale dependence, as in the example considered here, additional arguments are not needed.
The results which are displayed in Fig. 5 are in a qualitative agreement with the findings of Refs. Borasoy ; Beane . Note that these results are obtained at the nexttoleading order in HP EFT. It is unlikely that taking into account higherorder terms will reduce the uncertainty due to the scale dependence. On the contrary, from the comparison with e.g. Eq. (18) of Ref. Beane , one may conclude that numerically the most important corrections due to the effective range at this order amount to the multiplication of the loops by the deuteron wavefunction renormalization factor , where is the parameter related to the effective range in the channel. This effect leads to further amplification of the ambiguity related to the scale dependence.
The interpretation of the results displayed in Fig. 5 is unambiguous. The experimental data together with the above theoretical analysis constrain the wave scattering lengths within the band which – for large values of – also intersects with the common area