Coarse graining scattering
We carry out an analysis of scattering in the , and channels in configuration space up to a maximal center-of-mass energy GeV. We separate the interaction into two regions marked by an elementarity radius of the system; namely, a long distance region above which pions can be assumed to interact as elementary particles and a short distance region where many physical effects cannot be disentangled. The long distance interaction is described by chiral dynamics, where a two-pion-exchange potential is identified, computed and compared to lattice calculations. The short distance piece corresponds to a coarse grained description exemplified by a superposition of delta-shell potentials sampling the interaction with the minimal wavelength. We show how the so constructed non-perturbative scattering amplitude complies with the proper analytic structure, allowing for an explicit N/D type decomposition in terms of the corresponding Jost functions and fulfilling dispersion relations without subtractions. We also address renormalization issues in coordinate space and investigate the role of crossing when fitting the scattering amplitudes above and below threshold to Roy-equation results. At higher energies, we show how inelasticities can be described by one single complex and energy dependent parameter. A successful description of the data can be achieved with a minimal number of fitting parameters, suggesting that coarse graining is a viable approach to analyze hadronic processes.
pacs:12.38.Gc, 12.39.Fe, 14.20.Dh
Hadronic interactions at low and intermediate energies are typically characterized by a combination of elementary and composite particle features. While at long distances hadrons behave as elementary particles and their interactions can be described in terms of purely color singlet degrees of freedom, at short distances their composite character becomes manifest in terms of quark and gluon fields in the fundamental and adjoint representations of the color group, respectively. The relevant scale separating between this dual description marks the onset of a confinement scale and we expect it to be of the order of the hadron size, which generally is found to be about 1 fm. While the hadronic dynamics can be organized quite often as a long distance perturbative hierarchy with an increasing number of exchanged particles, it is by itself incomplete; some further either ab initio or phenomenological information reflecting the underlying quark-gluon structure is needed to provide a full description of the scattering process.
The way how this separation is visualized in the complex energy plane is not completeley straightforward. Traditionally, and within a genuinely hadronic picture, one appeals to Mandelstam analyticity Mandelstam:1958xc (), i.e. the assumption that a scattering amplitude can be expressed by double dispersive integrals in terms of double-spectral density functions, where the integration ranges extend over those regions in the Mandelstam plane where the corresponding double-spectral functions have non-vanishing support Hoferichter:2015hva (). This viewpoint is ultimately grounded in the Mandelstam conjecture, which holds in lowest order in the coupling constant in quantum field theory Mandelstam:1958xc (); Mandelstam:1959bc () or to all orders within a non-relativistic context in potential scattering Blankenbecler:1960zz (), and, which, in the scattering case, has been rigorously proved in a finite domain Martin:1965jj (); Martin:1966 (). It is noteworthy that under this same assumption an equivalent local and energy dependent optical potential of non-relativistic form was derived many years ago by Cornwall and Ruderman cornwall1962mandelstam (); omnes1965optical (). For a balanced review on these issues at the textbook level see, for e.g., nussenzveig1972causality (); goldberger2004collision (). The existence of a finite analyticity domain suggests in turn the very existence of a finite cut-off on a purely hadronic basis but without an explicit reference to the underlying quark-gluon dynamics and in particular to the confinement scale, so that the cut-off may be determined phenomenologically from data.
Pion-pion scattering is the simplest reaction in QCD mediated by strong interactions involving the lightest hadrons. Tight theoretical constraints based on analyticity, crossing, unitarity, chiral symmetry and Regge behavior can be imposed (see for e.g. martin1976pion () for an early review). The machinery of effective field theories (EFT) Weinberg:1978kz () and in particular its implementation in Chiral Perturbation Theory (PT) Gasser:1983yg () has enabled as a consequence, the most precise theoretical extraction of the S-wave scattering lengths to date with about an order of magnitude more precision than the experiment Colangelo:2000jc (); Ananthanarayan:2000ht (); Colangelo:2001df (); Caprini:2003ta (); Pelaez:2004vs (); Kaminski:2006yv (); Kaminski:2006qe (); GarciaMartin:2011cn (), an unprecedented case in strong interactions, where invariably just the opposite situation happens. A historic overview is given in Gasser:2009zz (). Along these lines, the most precise -scattering analyses to date have been obtained in Colangelo:2001df (); GarciaMartin:2011cn (). The latter corresponds to a description up to GeV, obtained by fitting the available experimental data from and decays while imposing as further constraints Roy and Roy-like equations, and with statistical uncertainties satisfying the necessary normality requirements of the residual distributions Perez:2015pea (), (see for e.g. RuizdeElvira:2012mbw (); Pelaez:2015qba () for reviews). We stress that despite all these tight mathematical constraints, most of its non-perturbative setup rests upon the validity of the Mandelstam conjecture Mandelstam:1958xc (); Mandelstam:1959bc (), a result which, as already mentioned, has not yet been rigorously proven since it was first proposed in 1958. This tacit assumption will also be made throughout our work.
In the present paper we invoke the equivalent local and energy dependent optical potential approach suggested long ago in cornwall1962mandelstam (); omnes1965optical () to describe scattering in coordinate space. In order to do so, we consider a relativistic Schrödinger equation and define a potential to describe the interaction by matching the field theoretical result to an equivalent quantum mechanical problem in perturbation theory. Phenomenological precursors of scattering analyses in coordinate space were prompted in au1963pion (); au1964pi () within the boundary condition model of strong interactions feshbach1964boundary (). Equivalent coordinate space potentials using the Mandelstam representation or the Bethe-Salpeter equation as a starting point were also proposed to all orders in balazs1965low (); balazs1965lowII (); Balazs:1969bt (). As it will become clear below, it is remarkable if not surprising that so little work on scattering has been conducted within this approach as compared to more popular momentum space methods. Our work fills this gap by implementing Wilsonian ideas inspired by recent developments in the NN case NavarroPerez:2011fm (); Perez:2013jpa (); Perez:2013oba (). These NN investigations had as a consequence a selection of the largest np and pp database up to energies about pion production threshold of mutually consistent data. Our present investigation within is in a sense of exploratory character and it pretends also to provide some training playground with an eye put on the more compelling case, where the selection of the currently existing database is largely needed (see for e.g. Matsinos:2016fcd (); RuizdeElvira:2017stg (); RuizArriola:2017kqs () and references therein).
At short distances, where the interaction is non perturbative, we will assume a complete ignorance of the strong interaction behavior and consider a coarse graining of the interaction instead, very much in the spirit of the work done in NavarroPerez:2011fm (); Perez:2013jpa (); Perez:2013oba () for the NN case. The basic idea is to separate the interaction into an inner and outer region at a given separation distance, , located at about some elementarity radius. This radius is defined so that at larger distances pions behave effectively as point-like particles. We will assume that in this long distance regime their interactions are ruled by chiral symmetry, and hence they become calculable within PT. Thus, for , we will construct a chiral potential with the correct low-energy analytic properties by matching both quantum mechanical and field theoretical scattering amplitudes in perturbation theory 111 This is similar to the unitarization method based on the Bethe-Salpeter equation Nieves:1998hp (); Nieves:1999bx ().. On the contrary, the inner region, , is regarded as unknown and sampled with the minimal de Broglie wave length determined by the maximum energy we want to describe. This corresponds to a coarse graining of the short range piece and, in its simplest realization, the inner potential will be written as a superposition of equidistant delta-shell interactions. A key issue is to confidently determine the numerical value of the separation scale , since, as noted in Perez:2013cza (); Fernandez-Soler:2017kfu () and we will see below, the combination will fix the total number of independent fitting parameters. The longest range interaction corresponds to a -exchange which is , so that a naive estimate suggests fm 222Details here are important. The extra factor 2 is to ensure that is really negligible. This is confirmed by our analysis below., a number which will be corroborated by our numerical analysis.
While the potential approach has been explained in great detail in previous works within the NN context (see for instance Fernandez-Soler:2017kfu ()), it is unconventional within the scattering folklore. Thus, we will assume no previous knowledge from the side of the reader and for the sake of completeness we will briefly go through all the important issues along the paper. Moreover, scattering is characterized because at resonance energies relativistic effects cannot be ignored. For instance, for the prominent case of the -meson . Unlike the NN case, a new important aspect in the discussion is related to crossing symmetry, which actually intertwines the , and channels 333Crossing for NN relates the two-pion exchange interaction with the production channel. This implies an exponentially suppressed effect in the potential and hence having little practical relevance.. In addition, the current extraordinary precision achieved theoretically in extracting the S-waves scattering lengths or the lightest resonance pole parameters Caprini:2005zr (); GarciaMartin:2011jx (); Masjuan:2014psa (); Caprini:2016uxy () provides a great confidence on the theoretical ideas supporting these benchmarking extractions. The fact that the coarse graining approach works for NN scattering in a regime where relativistic and inelastic effects become important, such as pp scattering up to GeV Fernandez-Soler:2017kfu (), suggests extending the method to other hadronic reactions under similar operating conditions 444We remind that within such a context the methods based in analyticity, dispersion relations and crossing are currently considered to be, besides QCD, the most rigorous framework. We stress again that such an approach is based on the validity of the double spectral representation of the four-point function conjectured by Mandelstam..
Finally, for the sake of completeness let us mention that lattice calculations are naturally formulated in coordinate space. These calculations attack the problem on the finite lattice spacing and the finite volume in two different fashions: either an (energy dependent) potential is determined and the Schrödinger equation is solved subsequently in the continuum or, alternatively, the energy level shifts are determined on the lattice and converted into phase-shifts by means of Luscher’s formula Luscher:1990ux (). Actually, in a pioneering work Beane:2011sc () , the S-wave scattering phase shifts from Lattice QCD have been determined. Later on, scattering has been studied from and flavors in Bulava:2016mks () and Helmes:2015gla (), respectively. Connected and Disconnected Contractions have also been analyzed in Acharya:2017zje (). In addition, the channel has also been studied within the potential approach Yamazaki:2015nka (). A comparison between potential and Luscher’s approaches has been undertaken in Kurth:2013tua () for the case, with rather similar results. We remind that both methods have potential drawbacks. On the one hand, the potential method uses interpolating fields which may distort the physics at short distances, and we will explicitly show that in a chiral expansion such potential presents a short distance singularity, which evades the conventional solutions of the Schrödinger equation. On the other hand, the current applicability of this Luscher’s method Luscher:1990ux () requires the interaction to sharply vanish at the edges of the volume (in the relative coordinate), a fact that has been often ignored in momentum space treatments (see for e.g. Doring:2011vk (); Doring:2012eu ()) but needs to be established for the case. Our analysis below supports this assumption.
The paper is organized as follows. In Section II, we provide a general and brief field theoretical overview of scattering to fix our notation in a way that our problem can be easily formulated. In Section III, we show our choice for a quantum mechanical description in terms of a complex local and energy dependent optical potential. We analyze the long-range contributions within PT in Section IV, where an expression for the potential is obtained from the discontinuities of the scattering amplitude in the -channel. This requires introducing a short distance cut-off to handle the strong short distance power divergences of the chiral potential, an issue which we discuss at length in Section V. In Section VI, we analyze the concept of effective elementarity in order to display in two examples how the elementarity radius depends on the particular process. In Section VII, we address the problem of coarse graining interactions with and without the long-range contributions. The analytical properties of the scattering amplitude and the relation of our approach with the N/D method is discussed in Section VIII. The implementation of inelasticities within a coarse grained perspective is explained in Section IX. We also analyze some aspects concerning low energy constants and the number of parameters in Section X. Finally, in Section XI we summarize our main results and provide some outlook for future work.
Ii Formalism for scattering
We start by summarizing the relevant formulae for scattering to fix our notation and to provide a proper perspective of our subsequent analysis. A comprehensive presentation at the textbook level can be seen in martin1976pion () and also in the lecture Yndurain:2002ud (). More recent upgrades can be consulted in RuizdeElvira:2012mbw (); Pelaez:2015qba ()
For a pion state with , the relativistically invariant scattering amplitude can be written as
with , and the standard choice of Mandelstam variables. If we take and , with , in the Cartesian basis we obtain
where stands for the amplitude. This amplitude is the the only independent one thanks to isospin, crossing and Bose-Einstein symmetries, and . Denoting as the isospin combination with well defined isospin in the -channel, one has
where is the -channel scattering angle, MeV the pion mass, the Legendre polynomials and is the partial-wave projection of the scattering amplitude with isospin and total angular momentum . Thus, for waves fulfilling the relation one has
the phase factor and the scattering phase shift. The in-elasticity for and the unitarity condition for the partial wave amplitude reads in the elastic region
Of course, for one has absorption and inelastic processes such as take place at and GeV for and , respectively, as well as and at , etc.
In our discussion we will also use the quantum mechanical amplitude defined by
with the CM momentum. For elastic scattering one has , so that at low energies one has the threshold expansion
with and the lowest threshold parameters. An equivalent way of representing the low energy behavior is
or by an effective range expansion
where and is the effective range, which is generally positive (see below). Usually, the expansion (9) works for small scattering lengths, such as whereas (10) works for large scattering lengths, such as (see, e.g. , Adhikari:1983jb (); Adhikari:2018ukk () for a discussion)
ii.2 Anatomy of the interaction
The purpose of the present paper is to coarse grain the unknown pieces of the interaction in configuration space. It is thus important to gather some features emerging from comprehensive studies over the last decades Colangelo:2000jc (); Ananthanarayan:2000ht (); Colangelo:2001df (); Caprini:2003ta (); Pelaez:2004vs (); Kaminski:2006yv (); Kaminski:2006qe (); GarciaMartin:2011cn (). According to these findings the partial wave expansion in (3) is decomposed into two contributions: the low energy contribution described by means of a partial pave (PW) expansion to finite order and the high energy contribution assumed to be given by the leading Regge trajectories,
which accounts for the long and short distance behavior of the scattering amplitude respectively.
A standard quantum mechanical argument based on the impact parameter provides in the semi-classical limit and for an interaction of finite range , the number of necessary partial waves 555These arguments provide in addition a justification for analyticity Omnes:1966pp (); Kugler:1966zz ().. The impact parameter is defined as with the CM momentum and the orbital angular momentum, which in our case equals the total angular momentum . The quantization condition for the angular moment yields for . For a finite range, the maximal impact parameter where scattering happens is . Thus, for a maximum CM momentum , the maximum angular momentum for which the phase shift is compatible with zero within uncertainties is
For a maximum energy GeV, corresponding to GeV, it was found in GarciaMartin:2011cn () that waves beyond are vanishingly small for scattering. Therefore, one obtains from (12) a range fm. This simple estimate will be explicitly exploited below as an educated guess.
Low energies close to threshold are encoded by the threshold parameters, see (8) and (10). The S-wave scattering lengths are fm and fm whereas for the P-wave we have GarciaMartin:2011cn (). These are unnaturally small numbers compared with our above estimate of the range of the interaction, and the elementarity radius, (see the discussion in Section VI). While the behavior of the isotensor S-wave resembles a repulsive core, with positive effective range , the effective range in the isoscalar S-wave and isovector P-wave are negative, and , respectively. For waves the Wigner causality bound Wigner:1955zz () (see also Phillips:1996ae ()) restricts the maximum value of the effective range by the inequality
which for implies . The positivity of the effective range is not implied by this condition, and is usually violated in the presence of resonances. This requires some unconventional shape for the S-wave potential as we will see.
ii.3 Chiral Perturbation theory
The scattering amplitude can be computed perturbatively in Quantum Field Theory and in particular in PT as a sum of Feynmann diagrams in an expansion in , with the pion weak decay constant in the chiral limit. In the partial waves basis the expansion can schematically be written as
where . To one loop order, they were first computed in Weinberg:1978kz (); Gasser:1983yg () and the relevant non-polynomial contributions are reproduced for completeness in appendix A. Explicit analytical expressions for the corresponding partial wave amplitudes are displayed in Nieves:1999bx (). They obey the perturbative unitarity relation
At lowest order (LO) in the chiral expansion the threshold parameters are unnaturally small, a fact naturally accommodated by PT with pions coupled derivatively.
ii.4 Unitarization vs Crossing
The requirement of crossing is a fundamental one which stems from the local character of Quantum Field Theories. Chiral Perturbation Theory implements this symmetry at any order in the chiral expansion. The problems with perturbation theory, however, are on the one hand the lack of exact unitarity given by (6) and on the other hand the impossibility of describing outstanding non-perturbative features such as the generation of resonances, which emerge as poles of the scattering amplitude on unphysical Riemann sheets. Within a PT framework, many methods have been proposed (see for instance Nieves:1999bx (); Guo:2012ym (); Guo:2012yt () and references therein) based on imposing exact unitarity while matching perturbation theory at low energies. Most of them are nothing but algebraic tricks or functional solutions to a set of a priori conditions. As such, unitarization methods are not unique but strongly driven by experimental information, which explains partly their success. The Bethe-Salpeter method discussed at length in Nieves:1999bx () preserves an identification of Feynman diagrams but it is not free from field reparameterizations or off-shell ambiguities. In addition, they violate crossing symmetry, which, in general, is only fulfilled order by order, although these violations can be statistically not-significant Nieves:2001de ().
In Sections IV and VII, we will propose yet a new method based on first defining an equivalent quantum mechanical problem and, more importantly, on coarse graining the interaction. Of course, above the inelastic threshold one may wonder what condition should be imposed instead of just (6) 666Usually the coupled channel unitarity condition is implemented instead. Typically analyses within such a setup leave out the “small” multiple production channels, , see e.g. Ledwig:2014cla () and works cited therein.. We will extend the coarse graining idea to the case with inelasticities.
Iii Quantum mechanics Formalism
iii.1 Relativistic equation
At the maximum CM energy we will be considering in this work , relativity and inelasticities are crucial physical ingredients since firstly and secondly we can produce up to pions as well as one and pair in the final state. From a field theoretical point of view, this could be solved by using a multichannel Bethe-Salpeter equation for the several , , , , and coupled channels, but it would be an extremely difficult task, which has never been accomplished to our knowledge. Even in the simplest elastic case the off-shell ambiguities are present for calculations with a truncated kernel Nieves:1998hp (); Nieves:1999bx (). In order to grasp the nature of the ambiguities, consider for instance the case of scattering, in the elastic regime. The Bethe-Salpeter (BS) equation reads,
where , , and . is the free pion propagator and and stand for the two-particle irreducible kernel or potential and the scattering amplitude, respectively. The factor comes from the scattering of identical particles.
While the BS equation has been the subject of extensive research for a given potential, the main point of Nieves:1998hp (); Nieves:1999bx () was the flexible interpretation of the BS equation within PT or more generally within EFT. Indeed, while the potential can be organized as a power series with reference to the same expansion of the scattering amplitude , it can be done only in at on-shell mass scheme, i.e. for
Thus, there is an inherent ambiguity in the definition and form of the potential, which has no consequences perturbatively but become relevant in the solution of the BS equation (LABEL:eq:BS) where the off-shellness enters explicitly. This was mended in Nieves:1998hp (); Nieves:1999bx () by invoking an on-shell scheme, namely considering only on shell intermediate states, i.e. and , so that the on-shell amplitude depends only on the on-shell potential . Unfortunately, it also gives rise to pathologies in the coupled channel case producing spurious singularities due to an improper treatment of the crossed-channel exchanges Ledwig:2014cla (). The present paper pretends to address crossed-channel exchanges without invoking the on-shell scheme.
iii.2 Invariant mass and equivalent Schrödinger equation
We will follow here the invariant mass formulation Allen:2000xy () 777These authors wondered if there was a way to promote non-relativistic fits of NN scattering to a relativistic formulation without refitting parameters. The answer is in the affirmative by just reinterpreting the CM momentum by its relativistic counterpart., already used for NN scattering with an optical potential Arriola:2016bxa (); Fernandez-Soler:2017kfu (). This is the simplest way of retaining relativity without solving a BS equation but with a phenomenological optical potential that we review here for completeness. The idea is to write the total squared mass operator as
where represents the (invariant) interaction, which can be determined in the CM frame by matching in the non-relativistic limit to a non-relativistic potential . This yields for scattering after quantization , with . Thus, the relativistic equation can be written as , with the CM momentum, i.e. as a non-relativistic Schrödinger equation
This corresponds to the simple rule that one may effectively implement relativity by just promoting the non-relativistic CM momentum to the relativistic CM momentum. This minimal relativity ansatz is as good as the more fundamental one based on the Bethe-Salpeter equation as long as we use scattering data to determine the corresponding potential rather than an ab initio determination (see Ref. Nieves:1999bx () for an in-depth discussion).
To take into account the inelasticity within the mass-squared construction, we assume a local and energy-dependent phenomenological potential, , which could be obtained by fitting inelastic scattering data. Due to causality, the optical potential in the -channel satisfies a dispersion relation for each CM radial distance of the form cornwall1962mandelstam ()
where is the first inelastic threshold and is an energy independent component. The complete potential includes also the crossed channel component. The simple looking equation (19), together with the fixed- dispersion relation (20), incorporates the necessary physical ingredients present in any theoretical approach: relativity and inelasticity consistent with analyticity.
iii.3 Isospin and exchange potential
The incorporation of isospin into the game is straightforward. Rotational, isospin and particle exchange invariance requires the representation of the potential to be given by
where and stand for the direct and exchange potential pieces, respectively. is the particle exchange operator, which implements the Bose-Einstein symmetry and that can be factorized as . Moreover, for states with a well defined total isospin , we can use the relation with , so that . In addition, for angular momentum eigenstates , so that . Therefore, in the isospin basis the potential can be decomposed as
where we have introduced the projection operators
fulfilling the orthogonality relations .
In the partial wave representation, the exchange symmetry of the potential is preserved by just solving the Schrödinger equation for the direct potential for the allowed channels with . In addition, for a spherically symmetric potential we have the usual factorization of the wave function galindo2012quantum ()
where are the spherical harmonics and is the reduced wave function, fulfilling the radial Schrödinger equation
where is the central potential with isospin . This equation is indeed regular at the origin 888We are assuming that at short distances the centrifugal barrier dominates, i.e. . Nevertheless, chiral potentials diverge as , as it is discussed below, and require special treatment if extended to the origin.
and it satisfies the asymptotic scattering condition at infinity
Thus, the partial wave expansion for the quantum mechanical scattering amplitude with isospin I in the CM system is defined by:
iii.4 Inverse scattering problem
Although we will be determining the potentials from fits to phase shifts, it is worth reminding that the inverse scattering problem allows one to determine a local and continuous potential directly from scattering data by solving for each partial wave either the Gelfand-Levitan or Marchenko equations (see for e.g. Chadan:1977pq () for a review). It can be shown that for holomorphic S-matrix functions both methods yield the same local potential. While usually the discussion is conducted within a non-relativistic setup, according to our discussion above, the analysis can directly be overtaken and interpreted at the relativistic level.
This inverse scattering approach was adopted in Sander:1997br (), where a holomorphic S-matrix was used to parameterize the scattering data. In that work it was found that the S and P-wave potentials have a range around fm with strengths between GeV. Quite remarkably they also found a barrier in the isoscalar S-wave and a repulsive core in the isotensor S-wave. While this is a very insightful and mathematically rigorous approach, this method requires exact knowledge of the phase shifts at all energies. In practice, a meromorphic function is fitted up to a maximum energy corresponding to a maximum momentum . As we will see below, this puts in practice a limitation to the resolution with which the potential may be determined, so that a suitable coarse graining makes sense.
Iv The Chiral local potential
In this section we outline the perturbative matching procedure between quantum mechanic (QM) and quantum field theory (QFT) calculations in order to determine the local and energy dependent chiral potential. The connection between the QFT and QM scattering amplitudes is given by
The potential appearing in this equation will be determined in perturbation theory. For the quantum mechanical problem we have the Born series
where , and are the initial and final CM momenta, respectively, and is the momentum transfer. The potential is directly defined from the two-particle irreducible states included in the scattering amplitude. We will define the potential through the -channel exchanges of the amplitude, so that crossing symmetry will be incorporated exactly when simmetryzing the partial wave expansion 999 We have checked that one can either work either in the particle or the isospin basis, the resulting potential is the same.. Moreover, in a coordinate space description, contact terms are irrelevant as long as the field theoretical potential is not extended to the origin since
with a generic polynomial in . Thus, any polynomial part of the scattering amplitude gives a vanishing contribution to the long-range piece of the potential. Therefore, we will analyze only the effect of pion loop contributions on the -channel.
iv.1 Leading order
We will first discuss the lowest non-trivial order since it provides just contact terms. In the Born approximation, i.e. just taking the first term in (IV), the scattering amplitude just becomes the Fourier transform of the potential galindo2012quantum ()
where is the momentum transfer. This equation can be inverted to give
so in the Born approximation, the scattering amplitudes can be related to the potential by:
where denotes the disconnected part of the amplitude, i.e. contact terms and -channel exchange. In the same way, the potential (defined in spatial coordinates) is defined from the disconnected part of the amplitude by:
Using the PT lowest order amplitudes Gasser:1983yg () we get
These algebraic manipulations are purely formal, and in fact is unspecified what is the meaning of solving the wave equation with these highly singular potentials. Already at this level, we can see the need of introducing a regularization 101010There is a conservation of difficulty principle here, one could stay in momentum space in which case the potential is well defined, but the scattering equation is UV divergent..
iv.2 Next-to-leading order
The NLO contribution becomes more cumbersome. Firstly, we take the potential to be expanded as
so we get the matching condition
Due to the Dirac delta functions in the LO potential (IV.1), we have a divergence for the real part of , albeit it can be absorbed in the real part of the NLO potential . Besides, the non-polynomial pieces in amplitude corresponding to the t-channel exchange can generally be written as
where is a polynomial in both and , which analytical expression can be read from (110), and denotes the one-loop function. In order to integrate this amplitude, we will take advantage of the analytic structure of the loop function , which is analytic in the whole complex plane but for a cut above with a discontinuity, , with the phase-space factor defined in (5). Thus, up to subtractions one finds the dispersion relation
where C.T. is a subtraction constant that can be fixed by setting the value of . Likewise we have
Thus, taking into account the Yukawa integral
and the inversion of (38), the NLO potential becomes
where and the spectral function is defined as
with polynomials in and of fourth degree (see Appendix A) and C.T. map into contact terms, which are distributions at the origin and hence vanish elsewhere. All necessary integrals can be obtained from the general integral valid for ,
with the modified Bessel function of order and the Euler’s Gamma. Polynomials in can be generated from derivation with respect to . The chiral potentials obtained directly from the spectral representation (43), read then
From a more general point of view, these potentials play for the -system the role of relativistic van der Waals interactions (see Feinberg:1989ps () for a review in the atomic case) and hence display their characteristic features: they are attractive and diverge at short distances as , i.e.
and have the expected exponentially suppressed long distance behavior , namely
In Fig. 1 we show the threshold combination and, as we can see, they are attractive at all distances. The energy dependence generates a repulsive effect for increasing values of , i.e.
On the lattice, the energy dependence of the potential is generated from the Nambu-Bethe wave function Kurth:2013tua (). As already stated in the introduction, the potential in the channel has been computed on the lattice by the HAL QCD collaboration Kurth:2013tua (); Kawai:2017goq () for fm on a lattice and with a pion mass MeV. For these pion masses the value the chiral potentials in (IV.2) become smaller than for the physical case depicted in Fig. 1. In addition, the HAL QCD lattice potential presents a repulsive core below fm. This is a feature one can not obtain using the chiral potentials in (IV.2) which display strong short distance singularities. This fact already suggests that they can not be used at arbitrary short distances. At this point it is worth stressing that both the lattice as well as the present approach based on chiral perturbation theory assume point-like sources, a unrealistic feature. In the next sections we analyze this topic in more detail.
The renormalization of non-perturbative amplitudes is a tricky matter, particularly with the highly power-divergent kernels deduced from PT (see for e.g. Nieves:1998hp (); Nieves:1999bx () for a discussion within the Bethe-Salpeter framework in momentum space). The chiral potential deduced in coordinate space by a perturbative matching procedure presents an energy dependence. In this section we show how the scattering amplitude stemming from the iteration of the two-pion exchange (TPE) chiral potentials in (IV.2) can be renormalized from a coordinate space point of view if the energy dependence is ignored by taking, say, the threshold value . Hence, we will implement as renormalization conditions the scattering amplitude at threshold. We will see that, while this is a mathematically viable approach, it fails phenomenologically. Furthermore, the consideration of energy dependence will prevent a sensible non-perturbative renormalization procedure. For large values of the coordinate space cut-off , the results will not be affected by taking either or .
One of the advantages of the energy dependent coordinate space representation of the potential is that off-shell and field reparameterization ambiguities manifest as contact interactions at the origin. Thus, they reflect the cut-structure of the amplitude, which is hence unambiguously defined. This is unlike their momentum space counterpart, where both polynomial and cut contributions are treated on equal footing Nieves:1999bx ().
On the other hand, a difficulty with the chiral potentials in the previous section is that they become singular at short distances. Hence, the solution of the Schrödinger equation is not well defined in a conventional sense, since the short distance behavior is not dominated by the centrifugal barrier and the regular solution given in (26) is not suitable. An early review on the subject can be found in Frank:1971xx (). Singular potentials are commonplace within EFT and finite solutions exist in a renormalization sense within well specified conditions, as discussed at length in the NN scattering case PavonValderrama:2005gu (); PavonValderrama:2005wv (); PavonValderrama:2005uj (). Applications for -scattering Arriola:2007de (); RuizArriola:2008cy () and atom-atom scattering Cordon:2009wh (); RuizArriola:2009wi () are well documented by now (see for e.g. RuizArriola:2007wm () for a sucint and pedagogical presentation). While these renormalized solutions represent theoretically a viable solution to the problem, we will consider here a more phenomenological interpretation by introducing a short distance cut-off , which value reflects short distance effects not taken into account in the derivation of the chiral potential 111111The renormalization procedure would correspond to take while keeping scattering lengths fixed. This consistent choice assumes point-like hadrons.. This leaves undefined the short distance dynamics.
The energy dependence of the potential takes into account retardation effects. This can be seen as follows; if the potential is given as a function of the difference of two space-time causally related events then we have