Effective Field Theory Approach to Pionium

The various dynamical scales below the pion mass involved in $\pi^{+}$ $\pi^{-}$ atoms are sequentially integrated out using non-relativistic effective field theory techniques. This allows us to systematically organise the corrections to the energy levels and decay width. We present our results in terms of a single unknown constant which may be obtained by matching to the Chiral Lagrangian with electromagnetic interactions at two loops.


Introduction
Hadronic atoms have attracted much interest since long [1]. Typically there is an interesting interplay between strong and electromagnetic interactions. Whereas the latter are the responsible of the bound state formation, the former produce their decay. Although the treatment of electromagnetic interactions is based on solid theoretical grounds, this is not so for the strong interactions. Traditionally, the latter are modeled by various types of short range potentials [2]. Although this is usually enough to fit the available data, it would be desirable to have a more direct connection with what is believed to be the fundamental theory of strong interactions, namely QCD. This is becoming even more urgent since the current DIRAC experiment at CERN [3], which plans to measure the pionium decay width at 10% accuracy [4], is meant to extract the pure hadronic pion-pion scattering lengths, which may, in principle, be obtained from QCD.
It has become apparent during the last decade, that the most fruitful way to approach low energy strong interaction physics from QCD is not by direct calculations from this theory but going through intermediate effective field theories (EFT), which are equivalent to QCD in a particular range of energies. For instance the Chiral Lagrangian [5] is an EFT for pions, which is equivalent to QCD for energies below the rho mass. The EFTs typically depend on various unknown constants, which in principle may be obtained from the fundamental theory. In practise, this may sometimes be achieved, like for instance in the case of Non-Relativistic QED (NRQED) [6] where the constants can be determined order by order in α, but many times is beyond our current technical abilities, like in the case of the Chiral Lagrangian, which would require large lattice simulations with light dynamical quarks or yet-to-be-discovered alternative non-perturbative techniques with a good control on the mechanism of chiral symmetry breaking. In any case, if the number of constants is small enough, they can be phenomenologically obtained from available data and used later on to predict new results, as it is the case of the Chiral Lagrangian.
Pionium is a π + π − electromagnetic bound state of binding energy ∼ 2keV which decays strongly, basically to two π 0 , with a width Γ ∼ .6eV [7]. Clearly a QCD based analysis of this system should better start with the Chiral Lagrangian. However, the Chiral Lagrangian is a relativistic (manifestly Lorentz invariant) theory where electromagnetic bound state problems are difficult to handle (see [8,9] for direct approaches). Moreover, both the binding energy and the decay width are much smaller than the pion mass ∼ 140MeV , which suggests that a non-relativistic approach should be appropriated.
It is the aim of this work to present a non-relativistic approach to pionium based on a series of EFTs which are obtained from the Chiral Lagrangian coupled to electromagnetism after sequentially integrating out the various physical scales of the system until we reach the scale of the binding energy ∼ mα 2 /4. The first scale to be integrated out is the pion mass m. This produces a local non-relativistic EFT for pion pairs near threshold, much in the same way as NRQED is obtained from QED [6,10,11]. The next relevant scale in the problem is the mass difference between charged and neutral pions ∆m ∼ 5MeV . Integrating out this scale produces a local EFT with only charged pion fields in it. The next relevant scale is the typical relative momentum of pions in the bound state mα/2 ∼ 0.5MeV (soft). Integrating out this scale is, at lower orders of α, equivalent to calculating the electromagnetic potential between the two charged pions. The calculations in the latter EFT reduce to quantum mechanical ones. The main advantage of this approach is that there are well defined counting rules at any stage of the calculation, so that the size of any neglected term is easy to estimate. This is particularly important in order to extract more accurate values for the parameters of the Chiral Lagrangian from the improved measurement of the pionium decay width in the DIRAC experiment [3].
We distribute the paper as follows. In Section 2 we present the most general nonrelativistic effective field theory for pion pairs near threshold. The constrains due to Lorentz invariance are implemented and the lagrangian is reduced to its minimal form by local field redefinitions. In Section 3 the neutral pions are integrated out which gives rise to a non-relativistic theory of charged pions interacting with the electromagnetic field. In Section 4 we integrate out soft photons, which produce the electromagnetic potentials between the charged pions. In Section 5 we present the calculation of the bound state energies and decay widths. Section 6 is devoted to the discussion of our results. In Appendix A we discuss the realisation of Lorentz symmetry in non-relativistic theories. In Appendix B we display the local field redefinitions and the various reshuffling of constants carried out along the paper . In Appendix C we present a new way to regulate the Coulomb propagator in D space dimensions. 4 2 Non-relativistic lagrangian for pion pairs near threshold At relative momentum much smaller than the pion mass a non-relativistic description of pion pairs should be appropriated. In order to implement it, we shall write down a lagrangian organised in powers of 1/m in which any scale smaller than m is treated perturbatively. For the problem at hand the next relevant energy and momentum scales are ∆m and √ m∆m, its associated momentum, respectively. These scales are to be used to estimate the (maximum) size of each term.
The symmetries (exact and approximate) of the fundamental theory, namely the Chiral Lagrangian, must be incorporated. Let us consider first the internal symmetries. The Chiral Lagrangian is approximately invariant under (non-linear) chiral transformations, which are explicitely broken by the pion mass terms. Since the pion mass is a large parameter in the non-relativistic lagrangian, no algebraic constraints from chiral symmetry are expected to survive. All information about chiral symmetry will be hidden in the parameters of the lagrangian. The only remaining approximate internal symmetry will be isospin, which is explicitely broken by m u = m d and the e.m. interactions both at the quark and at the Chiral Lagrangian level. The size of the explicit breaking may be estimated from m π + − m π 0 ∼ 5MeV which is much smaller than the pion mass. Hence isospin symmetry is a good (approximate) symmetry for the non-relativistic lagrangian. In order to implement it we shall use the vector π where π + , π − and π 0 annihilate positive, negative and neutral pions respectively.
Concerning the space-time symmetries, Poincaré invariance (including the discrete symmetries) must also be implemented in the non-relativistic lagrangian. The translational and rotational part of the Poincaré group as well as the discrete symmetries are implemented in the standard way. The Lorentz subgroup requires the introduction of a non-linear realisation which is equivalent to impose the so called reparametrisation invariance [12]. This is discussed in Appendix A. The outcome is relatively simple for spin zero fields. Consider a composite spin zero field made out of tensor products of n π and m π † . Define w = n − m the weight of this field. If w = 0, all derivatives acting on this field must be introduced through the combination If w = 0, ∂ µ on this field can be introduced. The lagrangian must have all the Lorentz indices contracted in a formally Lorentz invariant way and D must be considered Lorentz invariant itself.
Having in mind the rules above, consider first the limit of exact isospin symmetry. We have Consider next the isospin breaking terms. These may be due to e.m. interactions at the quark level, e.m. interactions in the relativistic Chiral Lagrangian and m u = m d . The electromagnetic interactions at quark level have an isospin invariant piece which is absorbed in the constants (2.3). The e.m. isospin breaking pieces, both at quark level and in the Chiral Lagrangian, are proportional to T 3 , and so is the isospin breaking piece due to m u = m d . Hence, in order to incorporate isospin breaking effects in the non-relativistic lagrangian, it is enough to construct further invariants with the vectors Q ∼ (0, 0, e) and M ∼ (0, 0, m u − m d ), taking into account that Q must always appear in pairs. Although there is no extra difficulty in taking M into account, we shall ignore it here since, due to charge conjugation, it appears quadratically and turns out to be very small [13]. Then the e.m. isospin breaking terms read Before going on, let us discuss the general structure of the constants A i , B i , C i and δ i above. Let us call Z to any such a constant and z its dimension. Then the general form of Z will be where f ∼ 93MeV is the pion decay constant. The a i , i = −1, 0, 1, ... stand for pure strong interaction contributions. It is interesting to notice that spontaneous chiral symmetry breaking implies a −1 = 0 for Z = A 0 , δ i . Indeed in the limit f → ∞ (keeping m constant) the pions in the Chiral Lagrangian become free particles as far as the strong interactions is concerned. Hence, in this limit any EFT derived from the Chiral Lagrangian must not contain strong interactions. Then the subscript i = 0, 1, .. coincides with the number of loops at which the term a i receives contributions. We stopped at the number of loops which have been calculated so far [14]. b i , i = 1, 2, , ... stand for purely electromagnetic contributions and c i,j , i, j = 1, 2, , ... for mixed electromagnetic and strong contributions. We stop here at the orders which compare to the two loop purely strong contribution. b 1 may receive contributions from tree level annihilation graphs, c 1,1 from one loop graphs [15] and c 1,2 from two loop graphs yet to be calculated. For this discussion to apply to the constants C i and δ i of the isospin breaking terms Q must be counted as a dimension one object.
The lagrangian (2.3) and (2.4) contains higher time derivative terms. One can get rid of these terms by local field redefinitions. We can set A 0 = δ 2 = A 1 = A 2 = 0 by using local field redefinitions which maintain Lorentz symmetry explicit. However, the new lagrangian still contains time derivatives beyond the expected i∂ 0 . We can also get rid of the extra time derivatives by using again local field redefinitions, which cannot maintain Lorentz symmetry explicit anymore. The details of this are displayed in Appendix B. We finally obtain the lagrangian in the so called minimal form The new constants above are defined in formula (B.8) of Appendix B. Lorentz symmetry guarantees that the bilinear terms have the standard form including relativistic corrections. It also relates A 3 and A 5 in the two last terms to the remaining constants (see Appendix B). Unfortunately, the latter relations have no practical consequences because the two last terms are proportional to the center of mass momentum and hence irrelevant to our problem. The zero charge sector in terms of the pion field reads The new constants above are defined in formula (B.9) of Appendix B. Notice that since the origin of energies appears to be at the two charged pion threshold, the neutral pion shows a negative energy gap −∆m < 0. Notice also that the terms in the bilinear neutral pion lagrangian can be combined into the standard form.
Nevertheless, in order to keep the expansion systematic we shall not use the expression above.
The coupling to e.m. fields is done by promoting normal derivatives to covariant derivatives. None of the possible non-minimal couplings contributes at the order we are interested in and we will ignore them.
Before closing this section let us remark that we have assumed that the lagrangian (2.3) and (2.4) is hermitian. This is correct at the order we are interesting in. However, in general the hermiticity constraint must be relaxed. This is due to the fact that the π + π − atom may decay into degrees of freedom which do not appear in the non relativistic lagrangian, for instance to hard photons or hard electron-positron pairs. The non-hermitian pieces would be obtained in the matching to the Chiral Lagrangian at the same time as the hermitian ones, as it happens in NRQED [10,11,18].
Since the π 0 energy gap is negative, the integration will produce imaginary parts in R ′ cc and P ′ . By calculating the diagrams in Fig. 1 in dimensional regularisation (DR) we obtain where s = √ 2m∆m. R ′ cc and P ′ contain the leading corrections in ∆m/m and mα 2 /4∆m respectively.
The electromagnetic contributions to L ′ coming from the energy scale ∆m are neglegible, as well as the relativistic corrections ∼ ∇ 4 /8m 3 to the charge pions and the terms P cc and S cc in (2.7).

Integrating out the scale mα
The lagrangian in the previous section is almost identical to NRQED (for spin zero particles) plus small local interactions. In refs. [18] it was shown that we can integrate out next dynamical scale, namely, mα/2 in NRQED obtaining a further effective theory called potential NRQED (pNRQED) which contains the usual potential terms and only the ultrasoft degrees of freedom (∼ mα 2 /4) remain dynamical. We shall do the same here. The (maximum) size of each term in (3.1) is obtained by assigning mα to any scale which is not explicit. In fact, since we are only interested in O(α) corrections, only the Coulomb potential seems to be important, since the tranverse photons give rise to O(α 2 ) corrections. However, as pointed out in ref. [19], below the pion threshold there are further light degrees of freedom apart from the photon. In particular, the electron mass m e ∼ mα/2 and hence it must be integrated out here. This gives rise to a potential term which is only O(α) suppressed with respect to the Coulomb one. By calculating the diagrams in Fig. 2 we obtain where V vpc (k) is given in formula (10) of ref. [19]. The lagrangian above contains no further degree of freedom than the non relativistic charged pions and hence it is totally equivalent to standard quantum mechanics. We like better to stay within the lagrangian formalism and use the π − π + wave function field φ(x, X, t), where x and X are the relative and center of mass coordinates respectively, as introduced in [18].
The center of mass kinetic term has been dropped.

Quantum mechanical calculation
In order to calculate the corrections to the energy levels and decay width we shall consider the propagator of (4.3) and identify its pole. At the order we are interested in only the diagrams in Fig. 3 contribute. The diagrams in the first line of Fig. 3 correspond to first order perturbation theory and can be easily evaluated. They give rise to where Ψ n (x) is the Coulomb wave function.
The diagrams in the second line of Fig. 3 correspond to second order perturbation theory and are not so easily calculated. The second diagram gives a finite contribution δ V 1 Γ (2) n which for the ground state has been evaluated numerically in [19]. The first diagram has also been considered before [17]. However, since it is UV divergent a suitable regularisation and renormalisation scheme must be specified. The subtraction point dependence of the result will eventually cancel against the subtraction point dependences in the matching coefficients. The matching coeficients are to be found by matching the Chiral Lagrangian with e.m. interactions with (2.3) and (2.4) which requires a two loop calculation yet to be carried out. The matching calculation is most efficiently done at threshold using DR and MS (or MS) scheme for both UV and IR divergences [11]. This requires to use the same regularisation and renormalisation scheme when calculating in the effective theory. We have calculated in Appendix C the first diagram using DR and MS scheme so that our results can be readily applied once the above mentioned matching calculation is carried out. We obtain where ∆ n is given in formula (C.12) of Appendix C. Putting all this together, our final expressions for the energy and the decay width read where we have substituted R 00 , R cc , R 0c and S 0c by their tree level values in the corrections We have also dropped terms proportional to R 00 in (3.2) because they are suppressed by extra factors of m 2 /4πf 2 . ∆ χP T summarizes all the contributions to |R 0c | 2 beyond the one at tree level in the isospin symmetric limit, in particular those from pion and photon loops in the Chiral Lagrangian. δ V 1 E (1) n is given for n = 1 in (5.3) and δ V 1 Γ (2) n is only known for n = 1 [19].
Formulas (5.6) and (5.7) are exact up to next to leading order in α, ∆m/m and mα 2 /∆m, except for the optional substitutions mentioned above. Notice that only R 0c is needed beyond tree level (∆ χP T ). In ∆ χP T there should be a contribution ∼ log m/µ which cancels the µ dependence in ∆ n . This would arise from a two loop calculation involving photons which has not been carried out yet.

Discussion
We have presented an approach to pionium which consists of separating the various dynamical scales involved in the problem by using effective field theory techniques. The main advantadge of this approach is, apart from its simplicity, that error estimates can be carried out very easily. A few remarks concerning other approaches are in order. First of all, relativistic aproaches [8,9] , apart from being technically more involved, have all the scales in the problem entangled which makes very difficult to estimate errors or to gauge the size of a given diagram. We would like to emphasize that Lorentz symmetry, even though it is not linearly realised, it is implemented in our approach to the required order. Several non-relativistic approaches have appeared in the literature addressing particular aspects of the problem [16,17,19]. Our analysis shows that a coupled channel approach to pionium [17] is unnecessary because the ∆m is much larger than the bound state energy. It also shows that, although it is technically possible (trivial in fact) to make a resummation of bubble diagramsà la Lipmann-Schwinger, it does not make much sense doing it since there are higher derivative terms in the effective lagrangian, which have been neglected, which would give rise to contributions of the same order. In a way, our approach implements the remark of [16] related to the fact that neutral pion loops give rise to important contributions in the non-relativistic regime. We have supplemented this remark with a full theoretical framework and with relativistic corrections of the same order which had been overlooked.
We have refrained ourselves to write our final results in terms of physical amplitudes because no experimental information is available for them at threshold energies. Simple minded identifications may be dangerous. For instance, one may be tempted to identify R ′ cc with the strong scattering length. This is correct at leading order but not at the order we are interested in.
On the technical side we have worked out a new method to calculate the Coulomb propagator G c (0, 0; E) in DR. The expresions for G c (0, 0; E) when E → E n are easily obtained for any n. Using DR here it is not just a matter of taste. Eventually a two loop matching calculation is to be done in order to extract the parameters of the Chiral Lagrangian from the pionium width. These kind of calculations are only efficiently done in DR. Since the matching coefficients depend on the renormalisation scheme, it is important to have our calculation in DR in order to be able to use the outcome of such a matching calculation straight away.
While this paper was being written up ref. [20] appeared which deals with the same problem by similar techniques.
where m is the mass of φ and J h (x) is slowly varying (i.e. contains energy and momentum much smaller than m). From (A.2) and (A.3) one easily finds that for Lorentz transformations close to the identity In the non-relativistic regime Z(J) can be approximated to the desired order of accuracy by Then Z N R (J h , J † h ) must be invariant under the transformation (A.4). Invariance of the terms coupled to the sources implies the following transformations for h(x).
Hence S N R (h, h † ) must be constructed in such a way that it is invariant under (A.6). In order to do so, notice first of all that ∂ µ h(x) does not transform in a way similar to h(x). We would like to introduce a kind of covariant derivative. The following operator appears to be a succesful candidate We have under Lorentz transformations which upon the change x → Λx becomes Analogously, if we have C w (x) = (h † (x)) m (h(x)) n , w = n − m, we may define for w = 0 a generalisation of (A.7 ) Then D k C w (x) has the same transformation properties as C w (x). We call w the weight of the composite field C w (x). If w = 0 then ∂ µ C 0 (x) transforms as a usual Lorentz vector. From the discussion above the following rules can be inferred in order to built a Lorentz invariant non-relativistic effective theory for spin zero particles: (i) Write down all possible terms in the particle sector we are interested in with weight zero and no derivatives up to the desired order.
(ii) For each term, which is not already of the higher relevant order, insert Ds or ∂ µ s in all possible ways. All µ indices coming from the ∂ µ must be contracted in a Lorentz invariant way.
Applying the rules above we obtain the lagrangians (2.3) and (2.4). Recall also that for the particular case we are interested in the (minimal) supression of D is ∆m/m whereas the (minimal) suppresion of ∂ µ is ∆m/m.
Finally, let us mention that for practical purposes the rules that we have obtained are identical to those derived from the so called reparametrisation invariance [12] (see also [22]). Hence, it should be clear that reparametrisation invariance is nothing but a way to implement Lorentz symmetry in a non-relativistic theory. We believe that this point is important and has not been sufficiently stressed in the literature.

Appendix B : Local field redefinitions
The lagrangian given in formulas (2.3) and (2.4) contains higher time derivative terms whereas the usual non-relativistic lagrangians contain only a time derivative in the bilinear terms of each field. The latter is known as the minimal form of the lagrangian. In this Appendix we display the local field redefinitions which bring the lagrangian (2.3) and (2.4) to its minimal form. Let us only mention that local field redefinitions exploit the freedom we have in field theory to choose the interpolating field we wish, and refer the interested reader to the literature on the subject [12,21,23,24,25]. The price we pay for having the lagrangian in its minimal form is that Lorentz symmetry (reparametrization invariance) will not be explicit anymore. The constraints given by Lorentz symmetry will reduce to non-trivial relations between the parameters of the lagrangian in its minimal form.
We are retaining corrections up to the relative order (∆m/m) 3 2 . In order to reduce our operator basis, we will take advantadge of the fact that local field redefinitions can also be organised in powers of ∆m/m. The induced terms beyond the desired order as well as the terms which do not contribute to the two particle sector (six pion terms and beyond) will be neglected.
Let us first consider local field redefinitions which keep Lorentz symmetry explicit. We can get rid of the A 0 and δ 2 terms in (2.3) and (2.4) by The bilinear terms become and the following constants of the four pion terms get modified We can also get rid of the A ′ 1 and A ′ 2 keeping Lorentz invariance by making The remaining time derivatives in D and in the A 3 and A 4 terms can only be removed if we give up the explicit realisation of Lorentz symmetry which we have kept so far. Notice that the time derivatives in the A 4 term are higher order and can be dropped. The following field redefinition gets rid of the higher order time derivatives in the bilinear terms Finally the time derivatives induced by this redefinition in the four pion terms together with the remaining time derivatives in A 3 and A 4 can be removed by Putting all together, we obtain for the constants in the lagrangian (2.6) the following expressions in terms of the original constants Upon restricting the lagrangian (2.6) to the zero charge sector we obtain the lagrangian (2.7) the constants of which are related to the above ones according to  admits the following exact representation, which is a generalisation of that presented in [26] G c (x, y, E) = We obtain (D = 3 + 2ǫ ′ ) =: m 2 α∆ n 4π where we have used the MS renormalisation scheme and changed µ ′ by µ according to (C.6) so that the results above are in standard DR with MS scheme. Clearly the singular part is local, independent of the principal quantum number n, and can be absorbed in a renormalisation of R ′ cc . This result is in agreement with a recent DR calculation of the same object carried out in [28]. Finally formula (5.5) is obtained.