Theory of two-pion photo- and electroproduction off the nucleon
A field-theoretical description of the electromagnetic production of two pions off the nucleon is derived and applied to photo- and electroproduction processes, assuming only one-photon exchange for the latter. The developed Lorentz-covariant theory is complete in the sense that all explicit three-body mechanisms of the interacting system are considered based on three-hadron vertices. The modifications necessary for incorporating -meson vertices for are discussed. The resulting reaction scenario subsumes and surpasses all existing approaches to the problem based on hadronic degrees of freedom. The full three-body dynamics of the interacting system is accounted for by the Faddeev-type ordering structure of the Alt-Grassberger-Sandhas equations. The formulation is valid for hadronic two-point and three-point functions dressed by arbitrary internal mechanisms — even those of the self-consistent nonlinear Dyson-Schwinger type (subject to the three-body truncation) — provided all associated electromagnetic currents are constructed to satisfy their respective (generalized) Ward-Takahashi identities. It is shown that coupling the photon to the Faddeev structure of the underlying hadronic two-pion production mechanisms results in a natural expansion of the full two-pion photoproduction current in terms of multiple dressed loops involving two-body subsystem scattering amplitudes of the system that preserves gauge invariance as a matter of course order by order in the number of (dressed) loops. A closed-form expression is presented for the entire gauge-invariant current with complete three-body dynamics. Individually gauge-invariant truncations of the full dynamics most relevant for practical applications at the no-loop, one-loop, and two-loop levels are discussed in detail. An approximation scheme to the full two-pion amplitude for calculational purposes is also presented. It approximates, systematically, the full amplitude to any desired order of expansion in the underlying hadronic two-body amplitude. Moreover, it allows for the approximate incorporation of all neglected higher-order mechanisms in terms of a phenomenological remainder current. The effect and phenomenological usefulness of this remainder current is assessed in a tree-level calculation of the reaction.
pacs:25.20.Lj, 13.75.Gx, 13.75.Lb, 25.30.Rw
The experimental study of double-pion production off the nucleon has a fairly long history, with some of the earliest experiments going back to more than half a century PH54 (); FC57 (); FM67 (); Hauser67 (); ABBHHM68 (); ABBHHM69 (); BCGG72 (). In the last two decades, with the availability of sophisticated experimental facilities at MAMI in Mainz, GRAAL in Grenoble, ELSA in Bonn, and the CLAS detector at Jefferson Lab (JLab), the emphasis of experiments with both real and virtual photons is clearly on using this reaction as a tool to study and extract the properties of excited baryonic states that form at intermediate stages of the reaction BMAA95 (); HABK97 (); ZABB97 (); ZABB99 (); KAAB00 (); WABH00 (); LABH01 (); KAAB04 (); ABBB03 (); CLAS-02 (); GDH-A2-03 (); CLAS-05 (); GDH-A2-05 (); GDH-A2-07 (); AABB07 (); CLAS-09b (); CB-A2-09 (); CLAS-09 (); SAPHIR05 (); AABB08 (); CB-ELSA-07c (); CBELSA/TAPS-08 (); CBELSA/TAPS-08c (); CBELSA-08 (); CBELSA-07 (); CLAS07b (); CBELSA/TAPS-09 (); CBELSA/TAPS-14 (); CBELSA/TAPS-15 (); CBELSA/TAPS-15b (); CBELSA/TAPS-15c (); CLAS-17 (). For comprehensive accounts on the pre-2013 activities in double-meson photo- and electro-production processes in particular, and on baryon spectroscopy in general, we refer to Refs. KR09 (); CR13 ().
Baryon spectroscopy has long been plagued by the so-called missing resonance problem IK77 (); KI80 (), which refers to resonances predicted by nonrelativistic quark models but not found in scattering experiments. One of the possible explanations for this problem is that those resonances may dominantly undergo sequential decays rather than direct decays into . An integral part of a comprehensive baryon-spectroscopy program, therefore, is the determination of sequential decay modes of baryons, in addition to direct one-step decays. To understand sequential decays, it is essential to investigate the production of two (or more) mesons. Indeed, analyses of some experiments in two-pion and photoproduction processes provide evidence for sequential decays of and resonances WABH00 (); CBELSA-07 (); CBELSA/TAPS-15c (); CBELSA-07b (); AABB08 (); ABKN11 ().
As the database for two-meson photo- and electroproduction increases, the need for more complete theoretical descriptions of such processes will increase as well to help in understanding their reaction dynamics. Theoretically, the study of double-pion electroproduction off the nucleon is a challenging problem because, unlike single-pion production, its correct description needs to combine baryon and meson degrees of freedom on an equal footing because the two pions in the final state can come off a decaying intermediate meson state, and not just off intermediate baryons as a sequence of two single-pion productions. This therefore requires accounting for all competing internal photo-subprocesses like, for example, the baryonic and the purely mesonic in a consistent manner.
Several groups have theoretically studied two-meson photo- and electro-productions employing a variety of approaches. The Bonn-Gatchina group has performed a multi-channel partial-wave analysis of the existing two-pion and photoproduction data CBELSA-07b (); ABKN11 () by extending its single-channel photoproduction partial-wave analyses. Double-pion photoproduction near threshold is described by chiral perturbation theory BKMS94a (); BKM95b (); BKM96 () and the 2004 data from MAMI on photoproduction off the proton KAAB04 () seem to be consistent with its predictions. Unitary chiral perturbation theory has been applied in the analyses of and photoproduction DOS05 (); DOS06 (); DOM10 (). In photoproduction, cross sections as well as spin-observables and were computed. The results are in good agreement with the existing data of Refs. AABB08 (); CBELSA/TAPS-08c (); CBELSA/TAPS-09 ().
At present, the most detailed model calculation of two-pion photoproduction is that of the EBAC/ANL-Osaka group KJLMS09 (). It is an extension of their dynamical coupled-channels approach for single pseudoscalar-meson production developed over recent years MSL06 () by describing the basic two-meson production mechanisms as isobar-type approximations obtained by attaching the vertices for , , and transitions to the corresponding single-meson production amplitudes, viz., , , and amplitudes, respectively, obtained in the dynamical coupled-channels approach MSL06 (). This model includes the hadronic channel KJLMS08 (), and the , , and resonances are found to be relevant to two-pion photoproduction up to .
The majority of existing model calculations of two-meson photo- and electro-photoproduction processes are based on straightforward tree-level effective Lagrangian approaches. In photoproduction, these models have been applied to two-pion GO94 (); GO95 (); NOVR00 (); NO01 (); Roca04 (); OHT97 (); HOT98 (); HKT02 (); FA05 (); MRAB01 (); MRAB03 (), and JOH01 () productions. Two-pion photoproduction in nuclear medium has been also studied within tree-level approximations ROV02 (); HOT97 (). In the strangeness sector, the photoproduction has been investigated within the tree-level effective Lagrangian approach ONL04 () as well as the photoproduction NOH06 (); MON11 (). The latter calculation includes a generalized four-point contact current to keep the resulting amplitude gauge invariant. The and electroproduction reactions were studied in a similar framework APSTW05 (); KCL07 (). A variation of the tree-level approximation in the analyses of two-pion electroproduction is adopted in Refs. MBLEFI08 (); CLAS-12 (); MBCE15 (). For the theoretical description of -production observables, we refer to Ref. RO04 () and references therein. Because of their simplicity, tree-level approximations are widely used in the analyses of two-meson photo- and electro-production processes. Despite their simplicity, they often provide insights into dominant aspects of the reaction mechanism in a more transparent way than more involved approaches.
The purpose of the present work is to derive two-meson photoproduction amplitudes which include the full miscroscopic details contained in the tree- and four-point hadronic vertices and thus offer the theoretical framework for exploiting the underlying reaction dynamics in a detailed and systematic manner beyond simple tree-level models. The derivation proceeds analogous to single-meson photoproduction, based on the field-theoretical approach of Haberzettl Haberzettl97 (), where the photoproduction amplitude is obtained by attaching a photon to the full three-point hadronic vertex using the Lehmann-Symanzik-Zimmermann (LSZ) reduction LSZ55 () which allows to express the full photoproduction amplitude in term of the gauge-derivative procedure proposed in Ref. Haberzettl97 (). For the two-meson case, we attach the photon to the full four-point hadronic vertex, whose microscopic structure is described in a nonlinear three-body Faddeev-type approach. The gauge-derivative device provides a very convenient tool to identify and link all relevant microscopic reaction mechanisms in a consistent manner. Similar to the single-meson photoproduction amplitude, the resulting two-meson photoproduction amplitude is analytic, covariant, and (locally) gauge-invariant as demanded by the generalized Ward-Takahashi identity Ward50 (); Takahashi57 (). Local gauge invariance, in particular, is important in electromagnetic processes because it requires consistency of all contributing mechanisms. Its violation may thus point to missing mechanisms, as was demonstrated for the bremsstrahlung reaction which is one of the most basic hadron-induced processes. In Refs. NH09 (); HN10 (), it was shown how to solve the long-standing problem of describing the high-precision KVI data by including in the model a properly constructed interaction current that obeys the generalized Ward-Takahashi identity required by local gauge invariance.
Since particle number is not conserved in meson dynamics, the full two-meson photoproduction amplitude as described here is highly nonlinear, thus making truncations unavoidable in practical calculations. However, to help with the incorporation of higher-order contributions, we present a scheme that expands the amplitude in powers of the underlying two-body hadronic -matrix elements and, in addition, provides a procedure for accounting for neglected higher-order contributions in a phenomenological manner. In principle, at least, the approximation can be refined to any desired accuracy. Local gauge invariance is maintained at each level of the approximation.
A preliminary account of a part of the main results of the present work can be found in the conference proceedings of Ref. HKO12 (). The present paper is organized as follows. In the subsequent Sec. II, we recapitulate some features of the theory of single-pion production off the nucleon of Ref. Haberzettl97 () so that we can establish the relevant techniques and tools to tackle the double-pion production problem. Then in Sec. III, using the basic topological properties of the process, we derive a formulation of the hadronic two-pion production process that incorporates all relevant degrees of freedom and all possible final-state mechanisms of the dressed system. We do this by employing the Faddeev-type Faddeev60 (); Faddeev () three-body Alt-Grassberger-Sandhas equations AGS67 () to sum up the corresponding multiple-scattering series. The actual electromagnetic production current is then constructed in Sec. IV, by applying the gauge derivative Haberzettl97 () to couple the (real or virtual) photon to the hadronic process found in Sec. III. (This procedure is sometimes referred to as ‘gauging’ of the underlying hadronic mechanisms.) We show that the resulting closed-form expression for the complete current satisfies the generalized Ward-Takahashi identity and thus is locally gauge invariant. We also show that the full current can be decomposed in a systematic manner into a sum of contributions that are directly related to topologically distinct hadronic two-pion production mechanisms of increasing complexity and that each of these partial currents is gauge invariant separately. This finding is important from a practical point of view because it allows one, to a certain extent, to separate the technical issue of maintaining gauge invariance from the question of how complex the reaction mechanisms must be to describe the physics at hand. In Sec. V, an approximation scheme to the full two-meson photoproduction amplitude is presented based on the expansion in powers of the underlying two-body hadronic interactions. Section VI contains the application of the approximation scheme in the lowest order to describe the reaction. Finally, we present a summarizing assessment and discussion in the concluding Sec. VII. Two appendices then provide additional material. Appendix A discusses the incorporation of four-meson vertices like and Appendix B provides a derivation of generic phenomenological contact currents for arbitrary hadronic transition that satisfy local gauge-invariance constraints, with specific applications to four- and five-point contact currents needed for the present formalism.
Ii Foundation: The Problem
A necessary prerequisite to understanding the photoproduction of two pions is to understand the photoproduction of a single pion off the nucleon. To this end, we recapitulate here some features of the theoretical formulation of that process following the field-theoretical treatment of Ref. Haberzettl97 (). This will also help us establish some of the necessary tools for the description of two-pion production.
The basic topological structure of the single-pion production current was given a long time ago GG54 () by observing how the photon can couple to the underlying hadronic single-pion production process . As shown in Fig. 1, there are two distinct types of contributions, respectively called class A and class B in Ref. GG54 (). Class A contains the three contributions , , and coming from the external legs of the vertex that have poles in the Mandelstam variables , , and , and class B is the non-polar contact-type current originating from the interaction of the photon with the interior of the vertex. The full current , therefore, can be written as
as indicated in Fig. 1. This structure is based on topology alone and therefore independent of the details of the individual current contributions.
These details matter, of course, if one wishes to derive the currents for practical applications. In general, an electromagnetic current for a hadronic process is defined by first employing minimal substitution for the connected part of the hadronic Green’s functions and then taking the functional derivative with respect to the electromagnetic four-potential , in the limit of vanishing . The current is then obtained by removing the propagators of the external hadron legs from this derivative in an LSZ reduction procedure LSZ55 (). The gauge-derivative procedure of Ref. Haberzettl97 () provides a formally equivalent method that is much simpler to handle in practice because it essentially amounts to the simple recipe of attaching a photon line to any topologically distinct feature of a hadronic process expressed in terms of Feynman diagrams and summing up the corresponding contributions to obtain the full current.
For the single-pion photoproduction process at hand, the connected part of the free Green’s function is given by , where is the vertex shown on the left-hand side of Fig. 1, is the propagator of the incoming nucleon leg and
is the product of the outgoing nucleon and pion propagators and , respectively, written here with generic momenta for the nucleon and the pion, where the sum of the respective four-momenta is the fixed available total momentum. Within a loop integration, this free propagator would correspond to a convolution integration of these momenta, as indicated by “” here. The LSZ expression for the photoproduction current may now be written as Haberzettl97 ()
where is the short-hand notation for the gauge derivative introduced in Ref. Haberzettl97 (), with indicating the Lorentz index of the incoming photon. Being a derivative, the product rule applies and we obtain
where in the last step
were used Haberzettl97 (), which relate the corresponding gauge derivatives to the nucleon current operator of the incoming nucleon, the interaction current for the interior of the vertex , and the dual-current contribution of the free system,
Note here that with the external momenta of the photo-process given as in Fig. 1, namely,
it was not necessary to write out the momentum dependence of any of the elements of the preceding equations because it can easily be found explicitly by knowing that the photon carries a momentum into the element to which it is attached.
ii.1 Gauge invariance
Gauge invariance as the manifestation of symmetry is of fundamental importance for any photo-process because it provides a conserved (on-shell) current and thus implies charge conservation. The requirement of local gauge invariance PS (), in particular, implies the very existence of the electromagnetic field and thus is of fundamental importance for the formulation of consistent reaction dynamics of photo-processes, which goes beyond the mere on-shell constraint of charge conservation.
where the four-momenta are those shown in Fig. 1 and the vertices are the vertex functions in the specific kinematic situations described by the Mandelstam variables in the figure. The charge operators for the initial and final nucleons are represented by and , respectively, and is the charge operator for the outgoing pion. The inverse propagators here ensure that this four-divergence vanishes for matrix elements with all hadron legs on-shell and thus provides a conserved current. The generalized WTI as such, however, is an off-shell constraint, thus providing a continuous dynamical link between the transverse and longitudinal regimes. This is analogous to the usual single-particle Ward-Takahashi identities Ward50 (); Takahashi57 () for the nucleon current,
and for the pion current,
which are also off-shell relations. Note that the validity of these two equations, which apply to the currents associated with the external legs in Fig. 1, and the generalized WTI of Eq. (9) immediately imply that the four-divergence of the interaction current is given by
In fact, given the usual single-particle WTIs of Eqs. (10) and (11), Eqs. (9) and (12) are equivalent formulations of gauge invariance of the photoproduction amplitude, with one condition implying the respective other. However, for practical purposes, in particular, in a semi-phenomenological approach, the interaction-current condition (12) is actually a more versatile tool because it lends itself very easily to phenomenological recipes that help ensure gauge invariance Haberzettl97 (); HBMF98a (); HNK06 (); HHN11 (); HWH15 (). The fact that all of these four-divergences are off-shell relations and therefore remain valid within whatever context the corresponding currents appear will be of immediate and direct relevance for two-pion production-current considerations in Sec. IV.
To facilitate the investigation of gauge invariance for the two-pion production case later on, we will now expand the meaning of the charge operators of particle . We first note that the charge operators appearing in all of the preceding relations only act on the isospin dependence within the vertices , i.e., their placements before or after a vertex cannot be changed, but otherwise they can appear anywhere in an equation. In all of the preceding equations, however, the charge operators have always been placed at the locations where the momentum of the particular particle line increases by the momentum of the incoming photon. Therefore, following Ref. Haberzettl97 (), we define the operator which injects the photon momentum into the equation where it is placed as well as having the role of the charge operator . We can then omit all explicit momenta in the equations because they can be recovered unambiguously from knowing the given external momenta of the process at hand. We can even go further to introduce Haberzettl97 ()
where the summation is taken to be context-dependent, i.e., wherever is placed in an equation, the sum extends over all particles that appear in that place in the equation. We may then write the generalized WTI of Eq. (9) equivalently and very succinctly as
i.e., as a commutator of and the connected Green’s function . Here, appearing on the left of subsumes the outgoing pion and nucleon, and on the right only comprises the incoming nucleon. The physical current on the left is amended with the propagators and of the incoming and outgoing particles, respectively, similar to the external propagators in the Green’s function . For the interaction current, the formulation equivalent to Eq. (12) is
where the propagators and are single-particle Green’s functions for the nucleon and the pion, respectively, in complete analogy to Eq. (14).
The structures of all equations here are similar: For a physical current, the four-divergence of the current, with propagators attached to its external legs, is expressed as a commutator of with the corresponding (connected) Green’s function. For an interaction current describing only the interaction with the interior of a hadronic process, the four-divergence is given by the commutator of with the underlying hadronic process. This finding is generic and holds true irrespective of how complicated the photo-process at hand actually is. The device will prove to be invaluable for investigating the gauge invariance of the two-pion production process.
ii.2 Dressing propagators and vertices
In the preceding discussion, we have not touched upon the question if, and if yes, to what extent, the propagators of the nucleon and pion and the vertex need to be dressed. As far as gauge invariance is concerned, the answer is very simple: for gauge invariance to hold true any degree of dressing that ensures the validity of Eqs. (16) for the propagators and of Eq. (15) for the interaction current is sufficient. Local gauge invariance, therefore, only requires that the single-particle and the interaction currents be constructed consistently with each other by keeping the overall structure of the production current depicted in Fig. 1. Besides that, it does not demand or imply any particular degree of dressing.
Even the simplest example, where the nucleon and pion propagators and their currents as well as the vertex are essentially bare, satisfies the generalized WTI of Eq. (9), as long as the masses are physical and the interaction current is the well-known Kroll-Ruderman current KR54 (). The key to maintaining gauge invariance, therefore, is consistency among all ingredients of a particular formulation of the reaction dynamics. Exploiting this consistency requirement in cases where gauge invariance does not follow — which nearly always is the case as soon as one introduces any kind of dressing mechanisms — is found to be indeed a powerful tool for constraining the interaction current by ensuring the validity of Eq. (12) Haberzettl97 (); HBMF98a (); HNK06 (); HHN11 ().
We will find, in Sec. IV, similar consistency constraints for the present problem of two-pion production. However, to understand better the structure of the problem, we need to look in more detail at some of the features of the dressing mechanisms resulting from the theoretical treatment of single-pion photoproduction of Ref. Haberzettl97 (), since the underlying field theory for both single-pion and two-pion production is the same. The full dressing mechanisms of single-pion production originate from the Dyson-Schwinger-type structure that governs the pion-nucleon scattering problem whose equations are summarized diagrammatically in Figs. 3 and 4. There is no need here to recapitulate all features of the treatment of Ref. Haberzettl97 () providing these structures. Relevant for the problem at hand is only the fact that the bare vertex from the underlying interaction Lagrangian is dressed by the non-polar part of the full matrix, i.e.,
depicted in Fig. 4(b). Here, solves the Bethe-Salpeter-type equation,
shown in Fig. 3(d), whose non-polar driving term is given in the lowest order by the -channel exchange of Fig. 3(e). At higher orders, also contains nonlinear contributions where the full itself is dressed by loops, as shown in the example of Fig. 5. (See also Ref. Haberzettl97 ().) In principle, therefore, everything in Eq. (18) is dressed fully by the nonlinear Dyson-Schwinger mechanisms.
where is the (bare) Kroll-Ruderman current and is the five-point interaction current resulting from applying the gauge derivative to Eq. (18), i.e.,
Here, is the five-point interaction current (whose lowest order is shown in Fig. 6) obtained by coupling the photon to all elements of the driving term . We see here that the internal dressing structure of the interaction current in Eq. (19) is quite complex; it contains, in particular, the full hadronic final-state interaction in terms of the non-polar scattering matrix . One can use Eq. (20) to bring Eq. (19) into a form better suited for practical applications, but there is no need to pursue this here (for more details; see Ref. Haberzettl97 () for formal derivations and Refs. HNK06 (); HHN11 () for practical aspects).
What we do need for the present purpose, however, is the proof that satisfies the usual gauge-invariant constraint of an interaction current. This proof was given already in Eq. (72) of Ref. Haberzettl97 (), but we repeat it here because it will introduce the general techniques of handling such four-divergences that we will need later on. For this purpose, let us restrict to be given only by the -channel exchange shown in Fig. 5. We emphasize that neglecting higher orders is done here only to simplify the derivation. In general, the proof will go through for any possible mechanism at any order Haberzettl97 (). For a simple -channel exchange, we may write as
where the indices and on the dressed vertices indicate whether the corresponding pion leg is an initial or a final particle for the process. The current resulting from coupling the photon to is then given by the three diagrams shown in Fig. 6, i.e.,
We note here that, because the photon couples into the fully dressed vertices of the -channel exchange (21), the currents and are the full four-point interaction currents of Eq. (19), with and indicating the direction of the pion leg. This type of nonlinearity is a natural and unavoidable consequence of the fact that particle number is not conserved in any process involving mesons. Using the four-divergences of Eqs. (15) and (16), we obtain
was used, which follows from the definition of and the WTIs of Eq. (16). Both four-divergences of and , therefore, produce the generic structure associated with interaction currents discussed at the end of the preceding subsection. For this generic result to hold, it is irrelevant whether we are dealing with four-point currents like or five-point currents like or . The result (24), in particular, will be relevant for the gauge-invariance proof of the two-pion photoproduction current given in Sec. IV in the context of Eq. (64).
ii.3 Topologically analogous problem:
The underlying field theory of single pion photoproduction just discussed above Haberzettl97 () contains pions, nucleons, and photons as explicit degrees of freedom. The resulting topological structure is complete in the sense that even if in actual practical applications one needs to expand the meaning of “pion” and “nucleon” to generically stand for all possible mesons and baryons, respectively, this structure does not change. The situation is different for two-pion production processes because, as we will discuss in more detail in Sec. III, two pions can be produced not only sequentially off baryons but also directly through the decay of mesons, and this will add topological features to the problem that cannot be expressed in the generic picture of pions and nucleons alone with their interaction being described by the vertex. In the following, therefore, we need to introduce the meson as an additional generic meson degree of freedom that can decay into two pions, i.e.,
As with pions and nucleons, in an actual application, “rho” can then be expanded to subsume all mesons that can decay into two pions.
As far as the interaction with photons is concerned, we now also need to consider the photon-induced process,
as being on par with the reaction. Topologically, the production current for this reaction has the structure depicted in Fig. 7, which is in complete analogy to the pion production of the nucleon shown in Fig. 1 because both types of processes are based on the interaction of the photon with a hadronic three-point vertex.
The hadronic final-state interaction of the system for this process can be depicted in a structure similar to Fig. 3, with all external lines being pions and the primary interaction being given by the vertex. Relevant for the following, in particular, is the fact that one can also split the full matrix into a pole part and a non-pole part whose lowest-order driving term is a -channel exchange as depicted in Fig. 8(a). The same is true for any meson-meson scattering problem whose basic interaction is described in terms of a bare three-meson vertex. Figure 8(b) shows the corresponding non-polar driving terms for .
As we shall see, the details of the underlying meson-meson scattering problem does not matter for the following. What matters is only the generic topological structure of the production current shown in Fig. 7 and the fact that non-polar contributions to the scattering amplitude satisfy a Bethe-Salpeter-type equation of the generic structure given in Eq. (18) that is driven at lowest order by non-polar -channel exchanges, like the ones shown in Fig. 8. All other details can be left to be worked out in an actual application.
Iii Hadronic two-pion production
We now turn to the problem of the production of two pions off a nucleon. Before looking at the photon-induced process, we first consider all possible hadronic transitions
including all possible dressing mechanisms. We will then derive the associated photoproduction current by attaching the photon in all possible ways to the dressed hadronic process. This is done in complete analogy to how the single-pion-production current is obtained from the fully dressed vertex as visualized in Fig. 1.
Describing the process within the generic field-theory framework of pions, rho mesons, and nucleons, there are three basic interaction vertices that are relatively easy to deal with, namely the three-hadron vertices , , and . These interactions provide the basic sequential production mechanisms shown in Figs. 9(a) and 9(b). However, there exist also multi-pion processes where a meson decays into three or more pions that cannot be resolved experimentally as being due to a sequence of three-meson interactions. For the meson, for example, the dominant decay mode is . Hence, one of the simplest examples of two-meson production due to a four-meson interaction is depicted in Fig. 9(c) showing an intermediate vertex where one of the pions is subsequently absorbed by the nucleon.
It should be clear that the full dynamical treatment of processes initiated by the three-pion vertex requires at least a four-body treatment of the intermediate system. In general, any process initiated by an -pion meson vertex would require employing the dynamics of at least an -body system. Such treatments clearly are beyond the scope of what is at present practically possible, and we will deal with this complication by, at first, ignoring multi-pion vertices like the one depicted in Fig. 9(c). We will restrict, therefore, the present formulation to the three-body dynamics of the system that is initiated by the two types of processes depicted in Figs. 9(a) and 9(b) based solely on three-hadron interactions. As we shall see, this does not exclude incorporating processes initiated by multi-pion vertices like the one in Fig. 9(c) at some later stage because all photo-processes that can be related to independent hadronic production mechanisms can be treated independently. Hence, we may safely ignore such multi-pion processes now, and we will revisit the problem later, in Appendix A.
For the time being, therefore, we only consider the two basic processes initiated by the two bare transitions depicted in Figs. 9(a) and 9(b). Figure 10 shows the first few terms of higher-order loop corrections of the basic processes. In the figure, we have omitted all contributions that can be subsumed in the dressing mechanisms of individual three-point vertices. In other words, the diagrams shown in Fig. 10 depict the first few contributions of the multiple-scattering series describing the three-body final-state interaction (FSI) within the system.
Inspecting the diagrams in Fig. 10 and noting that the -channel exchanges appearing there are the beginnings of the two-body multiple-scattering series,
it is a simple exercise to sum up all contributions up to the level of two dressed loops, i.e., the internal particle propagators and vertices in the resulting diagrams shown in Fig. 11 are fully dressed, and all meson-baryon and meson-meson FSI scattering processes are described by non-polar scattering matrices because all -channel pole contributions are accounted for in fully dressed sequential two-meson vertices. In drawing Fig. 11, we have relaxed the restriction to nucleons, pions, and rho mesons, and allowed the graphs to subsume all possible meson and baryon states that may contribute to the process of . The diagrams are grouped into no-loop (NL), one-loop (1L) and two-loop (2L) contributions in increasing complexity of the hadronic final-state interactions mediated by non-polar amplitudes.
We could now attach the photon to the hadronic diagrams in Fig. 11 and derive the corresponding production currents. The explicit results given in Sec. IV.2 presumably will be sufficient for most, if not all, practical purposes. For the fundamental theoretical understanding of the process, however, it would be interesting to derive a closed-form expression for the entire two-pion photo-process similar to what is possible for the single-pion production. And one would like to do so in a manner that maintains gauge invariance. To this end, we note that after the first interaction in the 1L graphs of Fig. 11, the system loses its memory about which of the two NL graphs of Fig. 11 was responsible for its initial creation and only retains the memory about the last two-body interaction, i.e., whether it was a or a reaction. Ignoring for the moment nonlinear effects that allow the creation of an arbitrary number of pions, all subsequent interactions, therefore, are governed by the dynamics of a three-body system. [We add here parenthetically that apart from the generic implications of a Dyson-Schwinger-type framework which is tantamount to having infinitely many mesons, the multi-pion aspect will also enter the picture through the driving-term’s nonlinearities discussed in the context of Eq. (35); see also Fig. 13.]
iii.1 Alt-Grassberger-Sandhas equations
The solution of the non-relativistic quantum-mechanical three-body scattering problem was given by Faddeev Faddeev60 (); Faddeev (). One of the most decisive aspects of the Faddeev approach is the manner in which the information about the sequence of interactions percolates through the system such that all interactions at all orders are possible, but double-counting of sequential interactions within the same two-body subsystem of the three particles is precluded, thus making the solutions unique. This basically is just an “accounting” problem and as such also valid in a relativistic context.111For relativistic Faddeev-type treatments of three-quark systems see, e.g., Ref. Eichmann2016 () and references therein. We may therefore translate the structure of the Faddeev equations to the present problem by (1) simply assuming covariant relativistic kinematics, (2) realizing that the proper counterparts of the non-relativistic two-body -matrices are the corresponding non-polar scattering matrices because non-relativistic potentials correspond to non-polar driving terms, and (3) allowing for non-trivial nonlinearities of the type analogous to those for the problem depicted in Fig. 5.
The particular variant of the Faddeev approach we will use in the present work are the Alt-Grassberger-Sandhas (AGS) equations AGS67 (); Sandhas72 () because they are given in terms of transition operators that are symmetric in their initial and final cluster configuration and thus can be applied to the present problem requiring only minor modifications related to relativistic kinematics and the fact that the particle number is not conserved.222The original Faddeev equations Faddeev60 (); Faddeev (), by contrast, correspond to a Green’s function description of the scattering process that contains unwanted disconnected contributions Sandhas72 () that need to be removed to be useful for the present context.
First, to organize the information, we assume that the pions are distinguishable and label them as and . (The indistinguishability of pions can easily be taken care of when calculating observables by appropriately symmetrizing the amplitudes.) Accordingly, we introduce three two-cluster channels by grouping the three particles as
Each () three-body configuration, therefore, consists of a two-body subsystem and a single-particle spectator. The indices , , , etc. may also refer to the two-body subsystem of these two-cluster configurations.
with where describes the transition from an initial two-cluster configuration to the final configuration . The equation is depicted in Fig. 12. For each two-body subsystem within the intermediate configurations , the full interaction is given by the corresponding non-polar scattering matrix of the two-body subsystem of that has to be extended into the three-body space such that the propagation of the single spectator particle within is unaffected. Hence, we may write in a generic manner,
where denotes the restriction to the two-body subspace within the cluster, and is a generic notation for the single-particle spectator propagator within the cluster. We thus have
where on the left-hand side describes the free intermediate propagation of the three particles within the system, i.e.,
which is the straightforward three-body extension of the two-body of Eq. (2), whereas within the brackets denotes the two-body restriction as given in Eq. (2). The meaning of within the present three-body context, therefore, is simply as a two-body expression convoluted with the spectator propagator of the free third particle that is unaffected by the two-body interaction .
The driving terms of Eq. (31) are given as
is the anti-Kronecker symbol that vanishes if the initial and final two-body groupings of the system are the same. The elements describe the nonlinear dressing of in the manner depicted in Fig. 13, in analogy to the nonlinear dressing mechanisms shown in Fig. 5. It is crucial here that this dressing happens around , i.e., the loop particle must connect particles of the initial and final two-body systems to avoid double-counting with the mechanisms described by Fig. 5 or with higher-order iterations of contributions. Nonlinearities, like , are absent from the original AGS equations AGS67 (); Sandhas72 () because they assume the particle number to be conserved. For three-body processes involving pions, however, terms like this one are necessary in principle (even if it is very difficult to calculate in practice) because internally infinitely many pions may contribute.
We emphasize that there are limits to the three-body treatment of the system even if one takes into account nonlinear dressings of the driving terms of the kind shown in Fig. 12. For example, if the loop particle for the last graph in Fig. 12 is the nucleon, the AGS amplitude enclosed by the loop is a three-meson amplitude and thus outside the scope of the three-body treatment of the system. Moreover, in general, depending on how many mesons one considers to be created at intermediate stages, much more complicated -body-type nonlinearities will result. It is possible in this way to recover some of the complexities of the problem associated with multi-pion vertices discussed in connection with the mechanism of Fig. 9(c), for example. We will consider additional three-body-force-type mechanisms associated with such processes in more detail in Appendix A. In general, of course, the actual calculation of such higher-order contributions in practical applications is quite challenging, to say the least, and we will, therefore, limit the detailed derivations in the following to the “pure” Faddeev contribution , and only mention in passing the ramifications of including nonlinearities in the driving term. Suffice it to say that the present formulation is consistent and correct for the system of two explicit pions and one nucleon where each of the particles may be fully dressed by any mechanism compatible with three-body dynamics.
Before we implement the AGS approach for the present problem, it is convenient to introduce a short-hand notation by defining operator-valued matrices according to
This permits us to write the AGS equation (31) as a matrix equation in the form of
which has the familiar Lippmann-Schwinger (LS) form of all scattering problems. Note in this context that the three-body dressing mechanism depicted in Fig. 13 corresponds to the dressing of , i.e., exactly analogous to the dressing of depicted in the right-most diagram of Fig. 5 for the two-body problem.
iii.2 Three-body Faddeev treatment of hadronic two-pion production
Following the reasoning that the primary dynamics of the system beyond the one-loop level is given by three-body dynamics, the multiple-scattering series providing the final-state interactions within the system can be summed up in terms of the three-body transition operators of the AGS approach, and we immediately find that the hadronic two-pion production can be described by three components () given by
where describes the three basic production mechanisms shown in Fig. 14. The second term here is only present because of nonlinearities like those depicted in Fig. 13; it is absent in a standard three-body treatment. Expanding the right-hand side to second order in produces
where the first three terms correspond precisely to the structure up to two loops shown in Fig. 11, with the terms here corresponding to the NL, 1L, and 2L graph groups of that figure. The lowest-order nonlinear effects contained in the last explicit term here are of second order in , like the preceding term, but they are of third order in the (dressed) loop structure, as shown in Fig. 15.
Defining formal three-component vectors with elements
we may rewrite Eq. (39) as
where the matrix I provides
One easily verifies that I is indeed the inverse of the matrix whose elements are the anti-Kronecker symbols. Here, 1 is the unit matrix of the three-body system with elements and
In summary, the present description of the process is given by
The “vertex” constructed in this manner provides a complete description of the reaction dynamics at the three-body level of the dressed system (subject to the general limitations of three-body dynamics discussed earlier).
Iv Attaching the Photon
Using the LSZ reduction, the full double-pion production current is given in terms of the gauge derivative by
where describes the incoming nucleon propagator and is the outgoing propagation of the free system. Hence, we have
where describes the current of the incoming nucleon. Here, is the three-body generalization of of Eq. (6), viz.
i.e., it subsumes the three currents of the outgoing legs analogous to what is depicted in Fig. 2 for the two-body case. The five-point interaction current contains all mechanisms where the photon is attached to the interior of the hadronic two-pion production mechanisms given by Eq. (44).
and the WTI of Eq. (10) for the nucleon current, the four-divergence of reads
which shows that the four-divergence of the interaction current , in analogy to Eq. (15), must be given by
to produce the generalized WTI,
This provides a conserved current in the usual manner when all external hadrons are on-shell. More explicit form of this result will be given later in Eq. (77).
iv.1 Proof of gauge invariance
To verify Eq. (50), let us define
as the vector whose components provide according to Eq. (44) as
Taking the gauge derivative of the matrix relation (42), the interaction-current component vector is given as
where the elements of ,
are the interaction currents associated with the elementary processes depicted in Fig. 14, and
is the current associated with the kernel of the LS equation (38). The current matrix reads
where, using Eq. (32), we obtain
which is the three-body extension of the two-body interaction current . The current of the spectator particle within the three-body cluster is represented by . The negative sign of this term is crucial for avoiding double-counting of spectator contributions. In Eq. (59), for example, it cancels out one of the spectator currents in (59), as shown in Fig. 16.
In detail, the AGS-kernel matrix is given by
If we neglect the nonlinearities , we have
One may use here for the entire three-body system, even though Eqs. (24) and (25) only contain the corresponding operator for the two-body subsystem, because the spectator contribution of , along the lower line on the right-hand side of Fig. 17, cancels between the two terms on the right-hand side of Eq. (64) since no interaction takes place along that line. One can show that the result (64) remains true even if the nonlinearities are taken into account. The proof requires tedious calculations and is not very illuminating; it will be omitted here.
To evaluate the four-divergence of , we use the four-divergence (64) and the fact that the current satisfies the generic relations of any interaction-type current, i.e.,333This will be proved explicitly in the next section in the context of Fig. 18.
and then we easily find
which upon using Eq. (53) immediately verifies the validity of Eq. (50) as stipulated. Hence, the current of Eq. (46) constructed with the help of the hadronic mechanisms (42) is indeed (locally) gauge invariant, and its generalized WTI is given by Eq. (51).
We can now write down the closed-form equation,
for the full two-pion photoproduction current , where
is the two-body component of in Eq. (56) that contains the full three-body final-state interactions of the problem. For practical applications, this presumes that the full two-pion production mechanisms of Eq. (39) can be calculated. In view of their complexity, this cannot be done easily in practice. One can show, however, that one can expand the full current in contributions of increasing complexity, similar to the NL, 1L, and 2L contributions in Fig. 11, which satisfy independent WTIs of their own. Maintaining local gauge invariance, therefore, is not predicated on being able to calculate the full current .
iv.2 Expanding the two-pion production current
To see how one may expand the full current, we define
which implies, formally, that M of Eq. (42) can be written as
Note that, without the nonlinearities , the matrix elements of the AGS kernel are just given by , as seen from Eq. (61). The expansion (70), therefore, provides the three-body multiple-scattering series of the final-state interactions within the system as a sequence of two-body interactions . One can show very easily, by the same techniques used in verifying the gauge invariance of the full current that the same is true order by order by coupling the photon to .
For the NL graphs of , whose components are shown in Fig. 14, the two-pion currents depicted in Fig. 18 are gauge invariant as a matter of course because the corresponding gauge-invariant subprocess currents indicated by in the diagrams trivially add up to make each of the NL and NL currents in Fig. 18 gauge invariant separately. This can be found immediately by taking the four-divergence of each current. These are simple examples for something which is generally true: Coupling the photon to topologically independent hadronic processes (like the two distinct processes summed up in the NL contributions of Fig. 11) will produce naturally independent gauge-invariance constraints. This means that each component of is gauge invariant separately.444Note that Fig. 18 only shows topologically different currents, i.e., no distinction is made for graphs that differ only by numbering the pions. Since the components of are given by sums of NL currents, this also implies an explicit proof for the gauge-invariance relation (65).
To investigate the gauge invariance of higher-order currents, we only need to look at the properties of the interaction-type currents because the contributions resulting from the four external legs of any current are trivial. We must show, therefore, that each of such currents satisfies a constraint similar to Eq. (50). We write
and find for the current ,
which gives its four-divergence as
This indeed is the generic result for an interaction-type current. With
we thus find