Crossing symmetric potential model of pion-nucleon scattering
CCrossing symmetric potential model ofpion-nucleon scattering
B. Blankleider ∗ , A. N. Kvinikhidze † and T. Skawronski ∗ ∗ School of Chemical and Physical Sciences, Flinders University, South Australia † Razmadze Mathematical Institute, Republic of Georgia
Abstract.
A crossing symmetric π N scattering amplitude is constructed through a complete attach-ment of two external pions to the dressed nucleon propagator of an underlying π N potential model.Our formulation automatically provides expressions also for the crossing symmetric and gauge in-variant pion photoproduction and Compton scattering amplitudes. We show that our amplitudes areunitary if they coincide on-shell with the amplitudes obtained by attaching one pion to the dressed π NN vertex of the same potential model. Keywords:
Pion-nucleon scattering, Crossing symmetry, Pion photoproduction, Gauge invariance,Unitarity
PACS:
1. INTRODUCTION
The π N scattering amplitude t is often described using the set of equations [1, 2, 3] t = f g ¯ f + t b , t b = v + t b G v , (1a)¯ f = ¯ f + ¯ f G t b , f = f + t b G f , (1b) g = g + g Σ g , Σ = ¯ f G f , (1c)where f ( f ) is the dressed (bare) N → π N vertex, ¯ f ( ¯ f ) is the dressed (bare) π N → N vertex, g ( g ) is the dressed (bare) nucleon propagator, G is the renormalized discon-nected π N propagator, and t b ( v ) is the "non-pole" π N t-matrix (potential) with the poleterm f g ¯ f ( f g ¯ f ) removed. Although these equations provide an exact description infull field theory, their main feature is that they allow one to preserve unitarity whenmaking models for the potential v and bare vertex f . However, like all potential models,these equations suffer from a lack of crossing symmetry, a property whose importancehas been emphasized for more than 50 years [4, 5, 6].Similarly, the pion photoproduction amplitude t γ is often descsribed by a set ofequations that essentially result from Eq. (1a) and Eq. (1b) by replacing the initial pionwith a photon [7, 8, 9]: t γ = f g ¯ f γ + t γ b , t γ b = v γ + t b G v γ , (2a)¯ f γ = ¯ f γ + ¯ f G t γ b , (2b)where ¯ f γ ( ¯ f γ ) is the dressed (bare) γ N → N vertex, and t γ b ( v γ ) is the pion photopro-duction amplitude (Born term) with the pole term f g ¯ f γ ( f g ¯ f γ ) removed. Once again a r X i v : . [ nu c l - t h ] A p r he feature of these equations is that they respect unitarity. This time, however, theseequations suffer not only from a lack of crossing symmetry, but also from the breakingof manifest gauge invariance (because the photon is not coupled to all places in the un-derlying field theory). We shall refer to Eqs. (1) and Eqs. (2) as the standard description.In this paper we present new equations for π N scattering, pion photoproduction,and Compton scattering, that are based on the potential model of Eqs. (1), but thatpreserve crossing symmetry and manifest gauge invariance. Our approach is based onthe idea of coupling external pions and photons to all possible places in the dressedpropagator g of Eq. (1c), and is achieved using the gauging of equations method [10, 11].Like the standard description, our approach is exact in full field theory; however, justopposite to the standard description, when models are made for the potential v andbare vertex f , our approach preserves crossing symmetry and gauge invariance at theexpense of unitarity. The lack of built-in unitarity is not surprising since our approacheffectively sums the full perturbation series in a way that is different from the usualmethod of iterating a kernel. Nevertheless, we show that our amplitudes will satisfyunitarity whenever the crossing symmetric π N amplitude coincides, on-shell, with theone obtained by attaching one pion to the dressed π NN vertex f of Eq. (1b).
2. SINGLE-GAUGED AMPLITUDE
In Refs. [10, 11] we introduced a technique for attaching an external photon to allpossible places (vertices, propagators, potentials, etc.) within a strongly interactingsystem described by dynamical equations. The completeness of the attachment ledto the gauge invariance of the resulting electromagnetic currents. Here we use thesame technique to also attach first one external pion, and then in the next section, asecond external pion, in order to achieve our goal of deriving a crossing symmetric π N scattering amplitude.We begin by applying our gauging technique to the π N Green function G generatedby the non-pole potential v : G = G + G vG . (3)Denoting by G µ the 5-point function resulting from a complete attachment of an externalpion to G , Eq. (3) is "gauged" to obtain G µ = G µ + G µ vG + G v µ G + G vG µ (4)which is easily solved to get G µ = G Λ µ G , Λ µ = Λ µ + v µ (5)where Λ µ ≡ G − G µ G − is a vertex function derived by attaching an external pion to thedisconnected π N propagator G . Note that G = g π g N where g π is the pion propagator For simplicity of presentation, we ignore any terms that cannot be obtained by the attachment of externalpions or photons. Such contributions, if present, are gauge invariant and crossing symmetric on their own,and can therefore be separately added to our derived amplitudes. We shall use "gauging" to mean the process of attaching any external particle, not just a gauge boson.
IGURE 1.
The dressed π NN vertex function Γ µ resulting from a complete pion attachment to thedressed nucleon propagator g defined by Eq. (1c). and g N is the renormalized nucleon propagator (in exact field theory g N = g / Z where Z is the renormalization constant; however, in model calculations one usually takes g N tobe the bare propagator with physical nucleon mass). As G-parity conservation forbids athree-pion vertex, Λ µ = Γ µ N g − π where Γ µ N ≡ g − N g µ N g − N . Thus Λ µ = Γ µ N g − π + v µ . (6)The gauged potential v µ can be constructed phenomenologically, or derived by gauginga specific model for v . Similarly, the gauging of the dressed nucleon propagator g ofEq. (1c), gives the 3-point function g µ : g µ = g Γ µ g , Γ µ = Γ µ + Σ µ (7)where Γ µ is the dressed π NN vertex function, Γ µ ≡ g − g µ g − is the bare vertex func-tion, and Σ µ is the gauged dressing. Gauging Σ = ¯ f G f then leads to a simple intuitiveexpression for the π NN dressed vertex function: Γ µ = Γ µ + ¯ f µ G f + ¯ f G f µ + ¯ f G µ f + ¯ f G v µ G f , (8)which is illustrated in Fig. (1).The main results of this section come from the gauging of the dressed π NN vertrex f : ( G f g ) ν = ( G f g ) ν = G ν f g + G f ν g + G f g ν = G Λ ν G f g + G f ν g + G f g Γ ν g . (9)Cutting off the external legs immediately gives the amplitude T ν ≡ G − ( G f g ) ν g − : T ν = ( + t b G )( Λ ν G f + v µ G f + f ν ) + f g Γ ν . (10)If superscript ν corresponds to an external photon, then T ν √ Z is the properly normal-ized manifestly gauge invariant pion photoproduction amplitude, originally derived inRef. [12] using a more involved approach. If superscript ν corresponds to an externalpion, then T ν √ Z is a "hybrid" π N scattering amplitude where the initial state pion isdue to gauging and the final state pion is due to the original standard description. In a IGURE 2.
In exact field theory, relations connecting the bare π NN vertex f and "non-pole" π N potential v of the standard description, Eqs. (1), to the corresponding gauged quantities f µ and v µ of thesingle-gauged description - see the first two of Eqs. (14). similar way, vertex ¯ f can be gauged to obtain the hybrid amplitude ¯ T µ where the final state pion is due to gauging. One can express these amplitudes in a "pole plus non-pole"form analogous to Eq. (1a): T ν = f g Γ ν + T ν b , T ν b = ( + t b G ) V ν , V ν = f ν + Λ ν G f , (11a)¯ T µ = Γ µ g ¯ f + ¯ T µ b , ¯ T µ b = ¯ V µ ( + G t b ) , ¯ V µ = ¯ f µ + ¯ f G Λ µ . (11b)It follows that amplitudes T ν b and ¯ T µ b satisfy non-standard Bethe-Salpeter equations T ν b = V ν + vG T ν b , ¯ T µ b = ¯ V µ + ¯ T µ b G v (12)where the kernel and inhomogeneous terms involve π N potentials of different origin.The amplitudes are clearly not crossing symmetric.We can now express Eq. (8) in the following two ways analogous to Eq. (1b): Γ ν = ¯ F ν + ¯ f G V ν , ¯ F ν = Γ ν + ¯ f ν G f , (13a) Γ µ = F µ + ¯ V µ G f , F µ = Γ µ + ¯ f G f µ , (13b)where Eq. (13a) and Eq. (13b) are to be used to describe pion (or photon) absorption andcreation vertices, respectively. In exact field theory, Γ µ can be identified with function f / √ Z of the standard description. Eqs. (13) then imply the following identities (in exactfield theory) relating the standard description quantities f and v (typically the modelinputs) to their gauged counterparts: f / √ Z = F µ , v / √ Z = ¯ V µ ; ¯ f / √ Z = ¯ F ν , v / √ Z = V ν . (14)The first two of Eqs. (14) are illustrated in Fig. 2. For ease of presentation, we discuss unitarity within the framework of time orderedperturbation theory for which G ( E + ) − G ( E − ) = − π i δ ( E − H ) (15)here H is the free Hamiltonian and E ± = E ± i ε . We consider only 2-body unitarityand thus restrict the discussion to energies E below the two-pion threshold. In this energyregion the potential v and bare vertex f are real, and the standard description of Eqs.(1) will therefore satisfy the following unitarity relations: t − t † = t † δ t , t b − t † b = t † b δ t b , (16a) f − f † = t † b δ f , ¯ f − ¯ f † = ¯ f † δ t b , (16b) g − g † = g † ( Σ − Σ † ) g , Σ − Σ † = ¯ f † δ f , (16c) G − G † = ( + G †0 t † b ) δ ( + t b G ) , (16d)where δ is shorthand for − π i δ ( E − H ) and where identical quantities with and withouta dagger represent the same functions of E − and E + , respectively (i.e., a dagger does notmean Hermitean conjugate, but rather, T ≡ T ( E + ) and T † ≡ T ( E − ) ). Applying theserelations to Eqs. (11) and Eqs. (13) one obtains the analogous unitarity relations for thehybrid amplitudes: T ν − T ν † = t † δ T ν , T ν b − T ν b † = t † b δ T ν b , (17a)¯ T µ − ¯ T µ † = ¯ T µ † δ t , ¯ T µ b − ¯ T µ b † = ¯ T µ b † δ t b , (17b) Γ µ − Γ µ † = ¯ T µ b † δ f , Γ ν − Γ ν † = ¯ f † δ T ν b . (17c)In the case of gauging with photons, Eq. (17a) provides just the usual statement ofWatson’s theorem for the gauge invariant pion photoproduction amplitude T ν . However,in the case of gauging with pions, Eq. (17a) differs from the usual statement of unitarityfor the hybrid amplitude in that a T ν † has been replaced with a t † from the standarddescription. Thus the only way for a hybrid π N amplitude T ν to be unitary is for it tocoincide, on-shell, with the standard amplitude t .
3. DOUBLE-GAUGED AMPLITUDE
To obtain a crossing symmetric π N scattering amplitude, we attach two external pions tothe dressed nucleon propagator g ; that is, we first gauge g to obtain g µ as in Eq. (7), andthen gauge Eq. (7) to obtain g µν with indices ν and µ denoting the initial- and final-statepion, respectively. Thus g µν = gT µν g , T µν = Γ µ g Γ ν + Γ ν g Γ µ + Γ µν + Σ µν (18)where ZT µν is the properly normalized crossing symmetric π N amplitude. In Eq. (18) Γ µν = (cid:0) g − g µ g − (cid:1) ν = g − g µν g − − Γ µ g Γ ν − Γ ν g Γ µ , (19a) Σ µν = ¯ f µν G f + ¯ f G f µν + ¯ f G Λ µν G f + A µν + A νµ (19b)where Λ µν = Γ µν N g − π + v µν , (20a) A µν = ¯ V µ GV ν = ¯ V µ G T ν b = ¯ T µ b G V ν . (20b)ne can thus write the crossing symmetric T µν in terms of pole and non-pole parts as T µν = Γ µ g Γ ν + T µν b , T µν b = V µν + ¯ T µ b G V ν , (21a) V µν = Γ ν g Γ µ + Γ µν + ¯ f µν G f + ¯ f G f µν + ¯ f G Λ µν G f + A νµ = ( ¯ F ν ) µ + Γ ν g Γ µ + ¯ f G ( V ν ) µ . (21b)We note that the second of Eqs. (21a) still has the basic structure of a Bethe-Salpeterequation although neither the two non-pole π N potentials V µν and V ν , nor the two non-pole t matrices T µν b and ¯ T µ b are of the same origin. It is also important to note thatEqs. (21) apply also to the cases where either one or both of the superscripts µ and ν refer to the gauging by photons. That is, these equations provide a unified, crossingsymmetric description of pion-nucleon elastic scattering, pion photoproduction, andCompton scattering. Morever, the electromagnetic amplitudes are due to a completeattachment of photons and are therefore manifestly gauge invariant. In the crossing symmetric formulation, the π N potential V µν of Eq. (21b) is realbelow two-pion threshold, as is the hybrid potential V ν . It is thus straightforward toobtain the 2-body unitarity relations from Eq. (21a) and the unitarity relations for thehybrid amplitudes, Eqs. (17). One obtains T µν − T µν † = ¯ T µ † δ T ν , T µν b − T µν b † = ¯ T µ b † δ T ν b . (22)As for the hybrid case, these have the same form as usual unitarity relations, and differfrom them only in that they contain π N t matrices of different origin. The task ofachieving exact 2-body unitarity for the crossing symmetric amplitudes is therefore anumerical one - the standard potential model of Eqs. (1), and its parameters, need tobe adjusted so as to ensure that the single- and double-gauged amplitudes coincide on-shell. Finally, it is worth pointing out that 3-body unitarity could be obtained in a similarfashion by gauging a standard Faddeev-like description of the ππ N system [13]. REFERENCES
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