A spectral representation for baryon to meson transition distribution amplitudes
aa r X i v : . [ h e p - ph ] N ov CPHT-RR051.0610, LPT-ORSAY 10-47
A spectral representation for baryon to meson transitiondistribution amplitudes
B. Pire , K. Semenov-Tian-Shansky , , L. Szymanowski CPhT, ´Ecole Polytechnique, CNRS, 91128, Palaiseau, France LPT, Universit´e d’Orsay, CNRS, 91404 Orsay, France Soltan Institute for Nuclear Studies, Warsaw, Poland.
Abstract
We construct a spectral representation for the baron to meson transition distribution am-plitudes (TDAs), i.e. matrix elements involving three quark correlators which arise in thedescription of baryon to meson transitions within the factorization approach to hard exclusivereactions. We generalize for these quantities the notion of double distributions introduced in thecontext of generalized parton distributions. We propose the generalization of A. Radyushkin’sfactorized Ansatz for the case of baryon to meson TDAs. Our construction opens the way tomodeling of baryon to meson TDAs in their complete domain of definition and quantitativeestimates of cross-sections for various hard exclusive reactions.
PACS numbers: 13.60.-r, 13.60.Le, 14.20.Dh . INTRODUCTION The concept of generalized parton distributions (GPDs) [1–4], which in the simplest(leading twist) case are non-diagonal matrix elements of quark-antiquark or gluon-gluonnon local operators on the light cone, has recently been extended [5, 6] to baryon to meson(and baryon to photon) transition distribution amplitudes (TDAs), non diagonal matrixelements of three quark operators between two hadronic states of different baryon number(or between a baryon state and a photon). Nucleon to meson TDAs are conceptually muchrelated to meson-nucleon generalized distribution amplitudes [7, 8] since they involve thesame non-local operators [9–12]. These objects are useful for the description of exclusiveprocesses characterized by a baryonic exchange such as backward electroproduction ofmesons [13–15] or proton-antiproton hard exclusive annihilation processes [16]. Nucleonto meson TDAs are also considered to be a useful tool to quantify the pion cloud inbaryons [17].Up to now TDAs between the states of unequal baryon number lacked any suitablephenomenological parametrization in the whole domain of their definition, as for examplein the framework of the quark model developed in [18]. The complete parametrizationshould properly take into account the fundamental requirement of Lorentz covariancewhich is manifest as the polynomiality property of the Mellin moments in the relevantlight-cone momentum fraction on the complete domain of their definition. For the caseof the GPDs an elegant way to fulfill this requirement consists in employing the spectralrepresentation. The corresponding spectral properties were established with the help ofthe alpha-representation techniques [19, 20]. Radyushkin’s factorized Ansatz based onthe double distribution representation for GPDs [21–24] became the basis for varioussuccessful phenomenological GPD models (see [25–29]).In this paper we address the problem of construction of a spectral representation ofbaryon to meson transition distribution amplitudes. We introduce the notion of quadrupledistributions and generalize Radyushkin’s factorized Ansatz for this issue. This allowsthe modeling of baryon to meson TDAs in the complete domain of their definition andquantitative rate estimates in various hard exclusive reactions.Similarly as the nucleon to meson TDAs factorize in backward meson electroproduc-tion, nucleon to photon TDAs may factorize in backward virtual Compton scattering [30].2he main part of the analysis performed in our paper can be directly applied to the nu-cleon to photon TDAs. But the modelling of the quadruple distribution has to accountfor the anomalous nature of a photon. The studies of the anomalous photon structurefunctions [31] and of the photon GPDs [32] show that taking it into account is a nontrivialtask which deserves separate studies.
2. BASIC DEFINITIONS AND KINEMATICS
Nucleon to meson transition distribution amplitudes also called in the literature asskewed DAs [5] and superskewed parton distributions [6] which extend the concept of usualgeneralized parton distributions arise e.g. in the description of meson electroproductionon the nucleon target [13–15]. For definiteness below we consider the case of nucleon topion transition distribution amplitudes ( πN TDAs for brevity) although our analysis isgeneral enough to be applied to other baryon-meson and also baryon to photon TDAs. πN TDAs arise in the description of backward pion electroproduction γ ∗ ( q ) + N ( p ) → N ′ ( p ) + π ( p π ) , (1)in the generalized Bjorken regime ( − q –large; q / (2 p · q ) kept fixed; − q ≫ − u ). Thefactorization theorem was argued for the process (1) in [5, 6] (see Fig. 1). The appropriatekinematics is described as follows [15]: P = 12 ( p + p π ) ; ∆ = p π − p ; u = ∆ ; ξ = − ∆ · n P · n , (2)where u denotes the transfer momentum squared between the meson and the nucleontarget and ξ is the skewness parameter. n and p are the usual light-cone vectors occurringin the Sudakov decomposition of momenta ( n = p = 0, n · p = 1). The light-conedecomposition of the particular vector v µ is given by v µ = v + p µ + v − n µ + v µT .The definition of πN TDAs can be symbolically written as [5, 6]:
Z " Y i =1 dz − i π e ix ( P · z )+ ix ( P · z )+ ix ( P · z ) × h π ( P + ∆ / | ǫ abc ψ aj ( z ) ψ bj ( z ) ψ cj ( z ) | N ( P − ∆ / i (cid:12)(cid:12) z + i = z ⊥ i =0 ∼ δ (2 ξ − x − x − x ) H j j j ( x , x , x , ξ, u ) . (3)3 DA DA ℓ ℓ k k M h P ( p ) P ′ ( p ) γ ⋆ ( q ) π ( p π ) FIG. 1: The factorization of the process γ ∗ + P → P ′ + π . The lower blob is thepion-nucleon transition distribution amplitude, M h denotes the hard subprocessamplitude, DA is the nucleon distribution amplitude.Here j , , stand for spin-flavor indices and a , b , c are color indices. The decompositionof the Fourier transform (3) of the matrix element of the three-local light-cone quarkoperator involves a set of independent spin-flavor structures multiplied by correspondinginvariant functions: πN TDAs.It is worth to mention that in order to preserve gauge invariance one has to insert thepath-ordered gluonic exponentials [ z i ; z ] along the straight line connecting an arbitraryinitial point z n and a final one z i n : h π | ǫ abc ψ a ′ j ( z ) [ z ; z ] a ′ ,a ψ b ′ j ( z ) [ z ; z ] b ′ b ψ c ′ j ( z )[ z ; z ] c ′ c | N i . (4)Throughout this paper we adopt the light-cone gauge A + = 0, so that the gauge link isequal to unity. Thus we do not show it explicitly in the definition (3).For the case of proton to π transition the decomposition of (3) over the independentspinor structures at the leading twist involves 8 independent terms. It reads [15]:4 F (cid:16) h π ( p π ) | ǫ abc u aα ( z n ) u bβ ( z n ) d cγ ( z n ) | P ( p , s ) i (cid:17) = δ (2 ξ − x − x − x ) × i f N f π h V pπ (/ pC ) αβ ( N + ) γ + A pπ (/ pγ C ) αβ ( γ N + ) γ + T pπ ( σ pµ C ) αβ ( γ µ N + ) γ + M − V pπ (/ pC ) αβ (/∆ T N + ) γ + M − A pπ (/ pγ C ) αβ ( γ /∆ T N + ) γ + M − T pπ ( σ p ∆ T C ) αβ ( N + ) γ + M − T pπ ( σ pµ C ) αβ ( σ µ ∆ T N + ) γ + M − T pπ ( σ p ∆ T C ) αβ (/∆ T N + ) γ i . (5) We make use of the notation F ( · ) = ( P · n ) R hQ i =1 dz i π i e ix ( P · z )+ ix ( P · z )+ ix ( P · z ) ( · ) p is the usual Dirac slash notation (/ p = p µ γ µ ), σ µν = [ γ µ , γ ν ] with σ pµ = p ν σ νµ , C is the charge conjugation matrix and N + is the large component of the nucleon spinor( N = (/ n / p + / p / n ) N = N − + N + with N + ∼ p p +1 and N − ∼ p /p +1 ). M stands for thenucleon mass, f π is the pion decay constant ( f π = 131 MeV) and f N is a constant withthe dimension of energy squared. All the 8 p → π TDAs V i , A i and T i are dimensionless.In this paper we concentrate on the dependence of the invariant functions V i , A i , T i multiplying the independent spin-flavor structures in (5) on the longitudinal momentumfractions x , x , x and skewness parameter ξ . Let us stress that our subsequent analysisis completely general: all invariant functions can be treated at the same footing. Forsimplicity in what follows we employ the same notation for all the invariant functions H ( x , x , x , ξ, t ) ≡ { V i , A i , T i } ( x , x , x , ξ, u ) . (6)A basic feature of model building cleverness is to fulfill fundamental requirements offield theory, such as general Lorentz covariance. In particular this requirement leads to theso-called polynomiality property of the Mellin moments in light-cone momentum fractions x , x , x of πN TDAs: Z dx dx dx δ (2 ξ − x − x − x ) x n x n x n H ( x , x , x , ξ, u ) ∼ (cid:18) i ddz − (cid:19) n (cid:18) i ddz − (cid:19) n (cid:18) i ddz − (cid:19) n [ h π ( P + ∆ / | ψ ( z ) ψ ( z ) ψ ( z ) | N ( P − ∆ / i ] (cid:12)(cid:12)(cid:12)(cid:12) z i =0 . (7)Indeed the x , x , x - Mellin moments of πN TDA are the form factors of the local twist-3three quark operators between nucleon and pion states. This leads to the appearance ofpolynomials in ξ at the right hand side of (7) . Naive counting gives n + n + n for the order of this polynomial. However, the problem of determina-tion of the highest possible power of ξ in (7) still lacks some analysis. This is a rather important questionsince it would allow to make the conclusion on the necessity of adding of D -term like contributions [33]to the spectral representation of πN TDAs (see discussion in Sec. 7). . SUPPORT PROPERTIES OF πN TDAS3.1. ERBL-like and DGLAP-like domains for πN TDAs
In order to specify the support properties of πN TDAs let us first consider the caseof the GPDs (see Fig. 2.a). Let x and x be the fractions (defined with respect toaverage nucleon momentum P = p + p ) of the light-cone momentum carried by quarkand antiquark inside nucleon ( x + x = 2 ξ ). In the so-called ERBL region both x and x are positive. The variable x is usually defined as x = x − x . (8)In the ERBL region x , x ∈ [0 , ξ ] and thus x ∈ [ − ξ, ξ ]. In the so-called DGLAP regioneither x is positive x ∈ [2 ξ, ξ ] and x is negative x ∈ [ − ξ,
0] or vice versa ( x is negative x ∈ [ − ξ,
0] and x positive x ∈ [2 ξ, ξ ]). These two DGLAP domainsresult in x ∈ [ ξ,
1] and x ∈ [ − , − ξ ] respectively. x x x ( b )( a ) x x p p p p π FIG. 2: Longitudinal momentum flow in the ERBL regime for GPDs (a) and πN TDAs (b).Now let us turn to the case of πN TDAs. Let x , x and x satisfying the constraint x + x + x = 2 ξ , with ξ ≥ P = p + p π . The convenient way to depict the support properties of πN TDAsis to employ barycentric coordinates (Mandelstam plane).First of all we identify the analogous of the ERBL domain, in which three longitudinalmomentum fraction carried by three quarks are positive. In the barycentric coordinatesthe ERBL-like region corresponds to the interior of the equilateral triangle with the height6 ξ (see Figure 3). It is natural to assume that the DGLAP-like domains are bounded bythe lines x = − ξ ; x = 0 ; x = 1 + ξ ; x = − ξ ; x = 0 ; x = 1 + ξ ; x = − ξ ; x = 0 ; x = 1 + ξ . (9)We are guided by the following requirements. • The complete domain of definition of πN TDA should be symmetric in x , x , x . • In the limiting case ξ = 1 this domain should reduce to the ERBL-like domainon which the nucleon DA is defined. In the barycentric coordinates the domain ofdefinition of the nucleon DA is equilateral triangle. • For any x i set to zero we should recover the usual domain of definition of GPDs forthe two remaining variables.Three small equilateral triangles correspond to DGLAP-like type I domains, whereonly one longitudinal momentum fractions is positive while two others are negative. Threetrapezoid domains correspond to DGLAP-like type II, where two longitudinal momentumfractions are positive and one is negative.The support properties (9) are invariant under the permutation of the longitudinalmomentum fractions x i . In the limit ξ → πN TDA is reduced to theERBL-like domain (the equilateral triangle) (see Fig. 4) and coincide with that of thenucleon distribution amplitude (DA). In fact this is natural since ξ = 1 corresponds tothe soft pion limit in which πN TDA reduces to the corresponding nucleon DA [15].In the limiting case ξ → πN TDA in the barycentric coordinates isgiven by the regular hexagon.
In order to describe πN TDA instead of x , x , x which satisfy x + x + x = 2 ξ , (10)7 x x FIG. 3: Physical domains for πN TDAs in the barycentric coordinates. x x x x x x FIG. 4: Physical domains for πN TDAs in the barycentric coordinates. Two limitingcases: ξ = 0 (left) and ξ = 1 (right).it is convenient to introduce the so-called quark-diquark coordinates. Let us stress thatwe do not imply any dynamical meaning to the notion of “diquark”. There are threedifferent possible choices depending on which quarks are supposed to form a “diquarksystem”: v = x − x w = x − x − x v = x − x w = x − x − x v = x − x w = x − x − x . (11)8e suggest to introduce the notations ξ ′ , ξ ′ and ξ ′ for the fraction of the longitudinalmomentum carried by the diquark: x + x ξ − w ≡ ξ ′ ; x + x ξ − w ≡ ξ ′ ; x + x ξ − w ≡ ξ ′ . (12)The variables x , x , x are expressed through the new variables (11) as follows: x = ξ + w ; x = v + ξ ′ ; x = − v + ξ ′ ; x = − v + ξ ′ ; x = ξ + w ; x = v + ξ ′ ; x = v + ξ ′ ; x = − v + ξ ′ ; x = ξ + w . (13) πN TDA in quark-diquark coor-dinates
Let us consider how the ERBL-like and DGLAP-like domains for πN TDA look like inquark-diquark coordinates. Throughout the rest of this section we employ the particularchoice of quark-diquark coordinates (11): v ≡ v = x − x w ≡ w = x − x − x ξ ′ ≡ ξ ′ = ξ − w . (14)The generalization for the alternative cases is straightforward.The ERBL-like and DGLAP-like domains for πN TDA in quark-diquark coordinates(14). are depicted on Figure 5. In these coordinates the ERBL-like region corresponds tothe central isosceles triangular domain. It is bounded by the lines v = − ξ ′ ( x = 0) ; v = ξ ′ ( x = 0) ; w = − ξ ( x = 0) . (15)DGLAP-like type I regions correspond to three smaller isosceles triangular domains. Fi-nally, three trapezoid domains correspond to DGLAP-like type II region.For w ∈ [ − , − ξ ] DGLAP-like region is bounded by v = 1 + ξ − ξ ′ ( x = 1 + ξ ) and v = − − ξ + ξ ′ ( x = 1 + ξ ) . (16)9 - Ξ Ξ w - - Ξ Ξ v - - Ξ Ξ - - Ξ Ξ FIG. 5: ERBL-like and DGLAP-like domains for πN TDA in quark-diquarkcoordinates (14). Three lines: w = − ξ and v = ± ξ ′ form the isosceles triangle whichcorresponds to ERBL-like region. Three smaller isosceles triangles correspond toDGLAP-like type I region. Three trapezoid domains correspond to DGLAP-like type IIregion.For w ∈ [ − ξ, ξ ] DGLAP-like region is bounded by v = − ξ − ξ ′ ( x = − ξ ) ; v = − ξ ′ ; v = ξ ′ ; v = 1 − ξ + ξ ′ ( x = − ξ ) . (17)For w ∈ [ ξ,
1] DGLAP-like region is bounded by v = − ξ − ξ ′ ( x = − ξ ) ; v = − ξ ′ ; v = ξ ′ ; v = 1 − ξ + ξ ′ ( x = − ξ ) . (18)One can easily check that for ξ ≥ w < − ξξ ′ > ξ ; and w > − ξξ ′ < ξ . (19) ξ ′ = ξ occurs on the line w = − ξ . Thus the whole domain of definition of πN TDA inquark-diquark coordinates depicted on Figure 5 can parameterized as follows: − ≤ w ≤ − | ξ − ξ ′ | ≤ v ≤ − | ξ − ξ ′ | . (20)10et us briefly summarize our result. • w ∈ [ − − ξ ] with v ∈ [ ξ ′ ; 1 − ξ ′ + ξ ] or v ∈ [ − ξ ′ − ξ ; − ξ ′ ] correspond toDGLAP-like type I domains. • w ∈ [ − − ξ ] and v ∈ [ − ξ ′ ; ξ ′ ] corresponds to DGLAP-like type II domain. • w ∈ [ − ξ ; ξ ] with v ∈ [ − ξ ′ ; ξ ′ ] corresponds to ERBL-like domain. • w ∈ [ − ξ ; ξ ] with v ∈ [ ξ ′ ; 1 − ξ + ξ ′ ] or v ∈ [ − ξ − ξ ′ ; − ξ ′ ] correspond toDGLAP-like type II domain. • w ∈ [ ξ ; 1] with v ∈ [ − ξ ′ ; 1 − ξ + ξ ′ ] or v ∈ [ − ξ − ξ ′ ; ξ ′ ] correspond to DGLAP-liketype II domain. • w ∈ [ ξ ; 1] with v ∈ [ ξ ′ ; − ξ ′ ] correspond to DGLAP-like type I domain.The Mellin moments of πN TDAs in x , x , x computed with the weight Z dx dx dx δ ( x + x + x − ξ ) (21)are the quantities of major theoretical importance. In the quark-diquark coordinates (14)the corresponding integrals can be rewritten as Z ξ − ξ dx dx dx δ ( x + x + x − ξ ) x n x n x n H ( x , x , x = 2 ξ − x − x )= Z − dw Z −| ξ − ξ ′ |− | ξ − ξ ′ | dv ( v + ξ ′ ) n ( − v + ξ ′ ) n ( w + ξ ) n H ( w, v, ξ ) . (22)
4. SPECTRAL REPRESENTATION FOR πN TDAS FROM THE SUPPORTPROPERTIES AND THE POLYNOMIALITY CONDITION
The double distribution representation [21–24] was found to be an elegant way toincorporate both the polynomiality property of the Mellin moments and the supportproperties of GPDs. In the framework of this representation the GPD H is given as a onedimensional section of the double distribution (DD) f ( α, β ): H ( x, ξ ) = Z − dβ Z −| β |− | β | dα δ ( x − β − αξ ) f ( β, α ) . (23)11he spectral representation (23) was originally recovered in the diagrammatical analysisemploying the α -representation techniques [19, 20]. The spectral conditions | β | ≤ | α | ≤ − | β | ensure the support property of GPD | x | ≤ | ξ | ≤ x which resides on the funda-mental field theoretic requirements (Lorentz covariance) is ensured by the fact that the x dependence of GPD in (23) is introduced solely through the integration path. In [34]it was pointed out that the relation between GPDs and DDs is the particular case of theRadon transform. It is worth to mention that the polynomiality property is well knownin the framework of the Radon transform theory as the Cavalieri conditions [35].Now we propose to invert the logic. From the pure mathematical point of view rep-resenting GPD as the Radon transform of a certain spectral density is the most naturalway to ensure polynomiality property. Postulating the polynomiality property of GPDand the support property | x | ≤ | β | ≤ | α | ≤ − | β | . Let us stressthat this does not provide the alternative derivation of (23) since there is no way to showindependently the support property | x | ≤ x as the building block for the spectral representation of multipartonic gen-eralizations of GPDs and in particular for πN TDAs. In order to derive the form of thespectral representation for πN TDA let us first consider the simple example of ordinaryGPDs.
We are going to treat the example of usual GPDs in a slightly unusual way which wefind more suitable for further generalization. Let us introduce the light-cone momentumfractions x and x of the average hadron momentum carried by the quark and antiquarkrespectively. The variables x and x satisfy the condition x + x = 2 ξ . The supportproperty in x , x is known to be given by − ξ ≤ x ≤ ξ ; − ξ ≤ x ≤ ξ . (24)12n order to write down the spectral representation for GPD we introduce two sets ofspectral parameters β , , α , . The momentum fractions x , are supposed to have thefollowing decomposition in terms of spectral parameters: x = ξ + β + α ξ ; x = ξ + β + α ξ . (25)The condition x + x = 2 ξ can be taken into account by introducing two δ -functions δ ( β + β ) δ ( α + α ). This allows us to write down the following spectral representationfor GPD H ( x , x = 2 ξ − x , ξ ): H ( x , x = 2 ξ − x , ξ )= Z Ω dβ dα Z Ω dβ dα δ ( x − ξ − β − α ξ ) δ ( β + β ) δ ( α + α ) F ( β , β , α , α ) . (26)Here by Ω , we denote the usual domains in the parameter space:Ω , = {| β , | ≤ | α , | ≤ − | β , |} ; (27)and F ( β , β , α , α ) is a certain quadruple distribution.The important advantage of the spectral representation (26) is that it is symmetricunder the interchange of the longitudinal momentum fractions x and x . Note thatthe spectral conditions (27) ensure the support properties (24) both in x and x . The( n , n )-th Mellin moments in x , x of H ( x , x = 2 ξ − x , ξ ) are polynomials of order n + n of ξ : Z ξ − ξ dx Z ξ − ξ dx δ (2 ξ − x − x ) x n x n H ( x , x = 2 ξ − x , ξ )= Z Ω dβ dα Z Ω dβ dα ( ξ + β + α ξ ) n ( ξ + β + α ξ ) n × δ ( β + β ) δ ( α + α ) F ( β , β , α , α ) = P n + n ( ξ ) . (28)Now we are about to show that the spectral representation (26) is equivalent to theusual Radyushkin’s representation (23) for GPDs in terms of double rather than quadrupledistributions. For this issue we can lift the two superfluous integrations employing the twodelta functions. In order to perform this in the astute way let us introduce the naturalspectral variables α ± , β ± : α ± = α ± α β ± = β ± β . (29)13t is also useful to to perform the related change of the variables in the ( x , x ) space inthe initial spectral representation (26). The corresponding natural variables are x − = x − x α − + β − ξ ; and x + = x + x ξ + α + + β + ξ . (30)Thus instead of using (26) we switch to the natural variables and consider: H ( x , x = 2 ξ − x , ξ )= 12 Z Ω dβ dα Z Ω dβ dα δ ( x − − β − − α − ξ ) δ ( β + ) δ ( α + ) F ( β , β , α , α ) . (31)The appropriate definition of the integration domain in (31) after the change of thevariables (29) require special attention. In particular, Z − dβ Z − dβ ... = 2 Z − dβ − Z −| β − |− | β − | dβ + ... . (32)Now since 1 − | β − | ≥ − | β − | ≤ β + can be easily liftedwith no influence on the integration domain in α + , α − . The problem of definition of theintegration domain in α + , α − in principle is reduced to change of the variables in theintegral Z a − a dα Z b − b dα δ ( α + α ) ... , (33)where a = 1 − | β + + β − | , b = 1 − | β + − β − | . It is much simplified due to the fact that β + = 0 and thus a = b ≡ − | β − | . This gives Z a − a dα Z a − a dα δ ( α + α ) ... = 2 Z a − a dα − Z a −| α − |− a + | α − | dα + δ ( α + ) ... . (34)Now the integral over α + can be trivially performed with the help of δ -function againproducing no additional restrictions for the integration domain in α − and β − . The finalresult reads H ( x , x = 2 ξ − x , ξ )= Z − dβ − Z −| β − |− | β − | dα − δ ( x − − β − − α − ξ ) 2 F ( β − , − β − , α − , − α − ) | {z } f ( β − , α − ) . (35)Certainly we just recovered the known Radyushkin’s result for the double distributionrepresentation of GPDs.Let us just make a short summary of the crucial points.14 We started from the spectral representation for H ( x , x = 2 ξ − x , ξ ) as the functionof the skewness parameter ξ and of two longitudinal momentum fractions x , x satisfying the condition x + x = 2 ξ . The form of this spectral representationensured the proper support properties in x , x as well as the polynomiality propertyof the corresponding Mellin moments in x and x . The spectral density was acertain quadruple rather than double distribution. • The constraint x + x = 2 ξ was taken into account by the introduction of two δ -functions restricting the integration domain in the space of spectral variables. • The two superfluous integrations can be lifted with the help of two δ -functions. Thisrequires the special attention to the integration domain in the space of spectralparameters. This problem can be most easily solved by switching to the set ofnatural variables b oth in the space of spectral parameters and x , x space. • In our toy exercise lifting the two integrations does not lead to any special restric-tions on the remaining spectral parameters α − , β − and we just recover the usualRadyushkin’s result for the double distribution representation of GPDs. • We find the spectral representation (26) which is symmetric under the exchange of x and x suitable for the generalization to the multiparton case. The analysis of πN TDAs with the help of the approach discussed above is presented in the nextsubsection. πN TDAs
We are now about to apply the ideas described in the previous section to the case of πN TDAs. Let us consider πN TDA H ( x , x , x = 2 ξ − x − x , ξ ) as a function oflight-cone momentum fractions x , x and x carried by three quarks. The three light-cone momentum fractions satisfy the condition x + x + x = 2 ξ . The support propertyin x , x , x is given by − ξ ≤ x ≤ ξ ; − ξ ≤ x ≤ ξ ; − ξ ≤ x ≤ ξ . (36)15n order to write down the spectral representation for H ( x , x , x = 2 ξ − x − x , ξ )we introduce three sets of spectral parameters β , , , α , , . The momentum fractions x , , are supposed to have the following decomposition in terms of spectral parameters: x = ξ + β + α ξ ; x = ξ + β + α ξ ; x = ξ + β + α ξ . (37)In order to satisfy this constrain we require that β + β + β = 0 ; α + α + α = − . (38)This allows to write down the following spectral representation for πN TDAs: H ( x , x , x = 2 ξ − x − x , ξ )= " Y i =1 Z Ω i dβ i dα i δ ( x − ξ − β − α ξ ) δ ( x − ξ − β − α ξ ) × δ ( β + β + β ) δ ( α + α + α + 1) F ( β , β , β , α , α , α ) . (39)By Ω i , i = { , , } we denote the usual domains in the parameter space:Ω i = {| β i | ≤ | α i | ≤ − | β i |} ; (40)and F ( β , β , β , α , α , α ) is now a sextuple distribution. The spectral conditions (40)ensure the support properties (36). Obviously, the ( n , n , n )-th Mellin moment in( x , x , x ) of πN TDA is a polynomial of order n + n + n of ξ : " Y i =1 Z ξ − ξ dx i δ ( x + x + x − ξ ) x n x n x n H ( x , x , x = 2 ξ − x − x , ξ )= P n + n + n ( ξ ) . (41)In complete analogy with the previously considered case of usual GPDs in order toproperly reduce the spectral representation in terms of sextuple distribution for πN TDAto that in terms of quadruple distribution we need to perform two integrations in " Y i =1 Z Ω i dβ i dα i δ ( β + β + β ) δ ( α + α + α + 1) ... (42)employing δ -functions and specify the integration limits in the remaining four integrals.This problem can be solved by introducing the appropriate natural variables.16et us start with the integral Z − dβ Z − dβ Z − dβ δ ( β + β + β ) . (43)In order to visualize the integration domain (43) it is natural to employ the barycentriccoordinates. In these coordinates the domain selected by the conditions | β i | ≤ i ∈{ , , } ) and β + β + β = 0 is represented by a regular hexagon (confer Fig 4). It isconvenient to single out three domains inside this hexagon: D : { β ≥ , β ≤ , β ≤ } ∪ { β ≤ , β ≥ , β ≥ } ; D : { β ≥ , β ≤ , β ≤ } ∪ { β ≤ , β ≥ , β ≥ } ; D : { β ≥ , β ≤ , β ≤ } ∪ { β ≤ , β ≥ , β ≥ } . (44)Obviously Z − dβ Z − dβ Z − dβ δ ( β + β + β ) = X i =1 Z D i dβ dβ dβ δ ( β + β + β ) . (45)Now in order to get rid of one of three integrations in (45) we should switch to thenatural coordinates. There are three possible choices of the natural coordinates in (45).For the moment we are going to adopt the coordinates ρ = β − β σ = β − β − β . (46)The constrained triple integral (43) can be then rewritten as Z − dσ Z − | σ | − | σ | dρ ... . (47)In principle in a completely analogous way one may also employ the coordinates ρ = β − β σ = β − β − β ρ = β − β σ = β − β − β Z − dσ i Z − | σi | − | σi | dρ i ... . (49)17ow let us address the problem of computation of the constrained triple integral over α i in (39): Z a − a dα Z b − b dα Z c − c dα δ ( α + α + α + 1) ... , (50)where we introduced the notations a ≡ − | β | ; b ≡ − | β | ; c ≡ − | β | ; (51)( a ≥ a ≤ b ≥ b ≤ c ≥ c ≤ ω = α ; ν = α − α β i ∈ D ∪ D ∪ D theconstrained integral (50) can be rewritten as Z −| β |−| β |− | β | dω Z −| β |− ω − | β | + ω dν ... . (53)Now let us put all together and write down the spectral representation for πN TDAs interms of quadruple distributions. The important observation is that once we have chosenthe variables σ , ρ and ω , ν to perform the constrained integration in β , β , β and α , α , α respectively the natural variables on which πN TDAs depends are w = x − x − x , v = x − x . (54)Expressing the β i and α i through σ , ρ , ω , ν the two delta functions in the definition(39) can be traded for δ ( x − ξ − β − α ξ ) δ ( x − ξ − β − α ξ ) | x + x + x =2 ξ = δ ( w − σ − ω ξ ) δ ( v − ρ − ν ξ ) . (55)Note that at the level of delta functions we achieved the “factorization” of w and v dependencies on the spectral parameters. There are two additional possible choices: ω = α ; ν = α − α and ω = α ; ν = α − α . πN TDAs in terms of quadrupledistributions: H ( w , v , ξ )= Z − dβ dβ dβ δ ( β + β + β ) Z −| β |− | β | dα Z −| β |− | β | dα Z −| β |− | β | dα δ ( α + α + α + 1) δ ( x − ξ − β − α ξ ) δ ( x − ξ − β − α ξ ) F ( β , β , β , α , α , α )= Z − dσ Z − | σ | − | σ | dρ Z −| ρ − σ |−| ρ + σ |− | σ | dω Z −| ρ + σ |− ω − + | ρ − σ | + ω dν δ ( w − σ − ω ξ ) × δ ( v − ρ − ν ξ ) F ( σ , ρ , ω , ν ) , (56)where F ( σ , ρ , ω , ν ) ≡ F ( ρ − σ , − ρ − σ , σ , ν − ω , − ν − ω , ω ) . (57)Employing three possible sets of natural spectral parameters one can write down threeequivalent spectral representations in terms of three sets of quark-diquark coordinates w i , v i with i = 1 , , H ( w i , v i , ξ )= Z − dσ i Z − | σi | − | σi | dρ i Z −| ρ i − σi |−| ρ i + σi |− | σ i | dω i Z −| ρ i + σi |− ωi − + | ρ i − σi | + ωi dν i δ ( w i − σ i − ω i ξ ) × δ ( v i − ρ i − ν i ξ ) F i ( σ i , ρ i , ω i , ν i ) , (58)where F ( σ , ρ , ω , ν ) is defined in (57) and F ( σ , ρ , ω , ν ) ≡ F ( σ , ρ − σ , − ρ − σ , ω , ν − ω , − ν − ω F ( σ , ρ , ω , ν ) ≡ F ( − ρ − σ , σ , ρ − σ , − ν − ω , ω , ν − ω . (59)The spectral representation (58) for πN TDA in terms of quadruple distribution is themain result of our paper. However this form of the result is still not very convenient forpractical applications. In the next section we demonstrate that the spectral representation(58) satisfies the support properties of πN TDAs established in Sec. 3. We also derivethe explicit expressions for πN TDAs in the ERBL-like and DGLAP-like type I and IIdomains. 19 . SUPPORT PROPERTIES OF πN TDAS AND THE SPECTRAL REPRESEN-TATION
In order to make our formulas more compact in what follows we omit the indice i forthe quark-diquark coordinates w i and v i , spectral parameters σ i , ρ i , ω i , ν i and the spectraldensities F i . Our subsequent analysis equally applies for all i = 1 , , πN TDA in (58) satisfiesthe support properties which were established in Sec. 3: − ≤ w ≤ − | ξ − ξ ′ | ≤ v ≤ − | ξ − ξ ′ | (60)with ξ ′ defined in (12). In particular this allows to check that ( N − n, n )-th ( N ≥ n ≥ πN TDA in ( w, v ) indeed satisfy the polynomiality property: Z − dw Z −| ξ − ξ ′ |− | ξ − ξ ′ | dv w N − n v n H ( w, v, ξ ) = P N ( ξ ) , (61)where P N ( ξ ) is a polynomial of order N in ξ . Case ξ = 0 Let us first consider the case ξ = 0. Employing the first delta function we get σ = w for − ≤ w ≤ ξ = 0.Once the integral over σ is performed the dependence on v is introduced through Z − | w | − | w | dρ δ ( v − ρ ) ... . (62)The result of this integral is non-zero only for − | w | ≤ v ≤ − | w | , (63)that is precisely the second condition (60) for ξ = 0. Case < ξ ≤ Let us now show that the spectral representation (56) possesses the desired supportproperties for arbitrary value of ξ ∈ (0; 1] . The final result for ξ ∈ [ −
1; 0) is presented in the Appendix B. w dependence in (58) is introduced through the expression Z − dσ Z −| ρ − σ |−| ρ + σ |− | σ | dω δ ( w − σ − ωξ ) ... . (64)From the inequalities (A8), (A11) and (A14) it follows that − | σ | ≤ − | ρ − σ | − | ρ + σ | ≤ − | σ | . (65)Thus in (64) we are integrating only over some part of the familiar “GPD square” | ρ | ≤ − | σ | . This guarantees the vanishing of πN TDA for | w | >
1. One can in the usualway perform the integration over ω introducing the additional θ -function to take into theaccount the unusual upper limit in the integral over ω : θ (1 − | ρ − σ | − | ρ + σ | − w − σξ ) ≡ θ ( ... ) . (66)For ξ > H ( w, v, ξ ) = For w ∈ ( −∞ ; −
1) : 0 ;For w ∈ [ − − ξ ] :1 ξ Z w + ξ − ξw − ξ ξ dσ Z − | σ | − | σ | dρ Z −| ρ + σ |− w − σ ξ − + | ρ − σ | + w − σ ξ dν δ ( v − ρ − νξ ) θ ( ... ) F ( σ, ρ, w − σξ , ν ) ;For w ∈ [ − ξ ; ξ ] :1 ξ Z w + ξ ξw − ξ ξ dσ Z − | σ | − | σ | dρ Z −| ρ + σ |− w − σ ξ − + | ρ − σ | + w − σ ξ dν δ ( v − ρ − νξ ) θ ( ... ) F ( σ, ρ, w − σξ , ν ) ;For w ∈ [ ξ ; 1] :1 ξ Z w + ξ ξw − ξ − ξ dσ Z − | σ | − | σ | dρ Z −| ρ + σ |− w − σ ξ − + | ρ − σ | + w − σ ξ dν δ ( v − ρ − νξ ) θ ( ... ) F ( σ, ρ, w − σξ , ν ) ;For w ∈ (1; ∞ ) : 0 . (67)Now we are about to perform the integration over ν with the help of the last remaining δ -function. The resulting domain of integration in σ and ρ is defined by the inequalities − | σ | ≤ ρ ≤ − | σ |
12 + | ρ − σ | + w − σ ξ ≤ v − ρξ ≤ − | ρ + σ | − w − σ ξ ; (69)1 − | ρ − σ | − | ρ + σ | ≥ w − σξ ; (70)as well as the integration limits in σ depending on the value of w (see (67)).It can be shown that for ξ ≥ ρ ≤ σ v + ξ ′ ξ for v ≥ − ξ ′ ; ρ ≤ σ v + ξ ′ − ξ for v ≤ − ξ ′ (71)together with ρ ≥ − σ v − ξ ′ − ξ for v ≥ ξ ′ ; ρ ≥ − σ v − ξ ′ ξ for v ≤ ξ ′ . (72)Analogously the inequality (70) for ξ ≥ ρ ≤ σ ξ + ξ ′ ξ for v ≥ | ξ ′ | ; ρ ≥ − σ ξ − ξ ′ ξ for v ≤ −| ξ ′ | ; σ ≥ w − ξ ξ for v ≥ − ξ ′ v ≤ ξ ′ ; σ ≥ w − ξ − ξ for v ≤ − ξ ′ v ≥ ξ ′ . (73)The last step is to match the integration domain defined by the inequalities (68), (71),(72) and (73) with the explicit w -dependent limits of integration in σ (67). There are 9possibilities: { w ∈ [ − − ξ ] , w ∈ [ − ξ ; − ξ ] , w ∈ [ ξ ; 1] }⊗ { v ∈ ( −∞ ; −| ξ ′ | ] , v ∈ [ −| ξ ′ | ; | ξ ′ | ] , v ∈ [ | ξ ′ | ; ∞ ) } . (74)Let us consider in details the case w ∈ [ − − ξ ] ; v ∈ [ ξ ′ ; ∞ ) . (75)The integration domain in ( σ, ρ ) plane is defined by the intersection of a domain specifiedby the inequalities (68), (71), (72), (73): ρ ≥ − σ v − ξ ′ − ξ ; ρ ≤ σ v + ξ ′ ξ ; ρ ≤ σ ξ + ξ ′ ξ ; | ρ | ≤ − | σ | w − ξ ξ ≤ σ ≤ w + ξ − ξ . (77)22he domain defined by the inequalities (76) and (77) is presented on Fig. 6. By the thicksolid lines we show the borders of the domain defined by the first two inequalities (76).The thin solid line is the border of the domain defined by the inequality ρ ≤ − σ . Thedashed line is the border of the domain defined by the inequality (73): ρ ≤ σ ξ + ξ ′ ξ . Theshaded area corresponds to the resulting domain of integration in (67) for − ≤ w ≤ − ξ and ξ ′ ≤ v ≤ − ξ ′ + ξ . w -Ξ +Ξ I v Ξ-Ξ ’ M -Ξ w +Ξ -Ξ ΣΡ FIG. 6: The domain of integration in ( σ, ρ ) plane in eq. (67) for − ≤ w ≤ − ξ and ξ ′ ≤ v ≤ − ξ ′ + ξ defined by the inequalities (76) and (77). See explanations in the text.The abscissa of the apex of this triangular domain is σ = 2( vξ − ξ ′ )1 − ξ . (78)One may check that for v = ξ ′ the abscissa of the apex coincides with the left boundaryof the strip (77): 2( vξ − ξ ′ )1 − ξ (cid:12)(cid:12)(cid:12)(cid:12) v = ξ ′ = w − ξ ξ , (79)while for v = 1 − ξ ′ + ξ it coincides with the right boundary of the strip (77):2( vξ − ξ ′ )1 − ξ (cid:12)(cid:12)(cid:12)(cid:12) v =1 − ξ ′ + ξ = w + ξ − ξ . (80)For v > − ξ ′ + ξ the apex of the triangular domain lies on the right of the strip (77)and hence has empty intersection with it. This makes the double integral (67) vanish for v ≥ − ξ ′ + ξ and ensures the desired support property of H ( w, v, ξ ).23he third inequality in (76) does not further restrict the domain since the apex of thetriangular domain belongs to the line ρ = σ ξ + ξ ′ ξ and the triangular domain lies to theright of this line for 0 ≤ ξ ≤
1. The two inequalities (68) also do not impose additionalrestriction for the domain. Indeed one may check that the ρ = σ ξ + ξ ′ ξ intersects with ρ = 1 − | σ | at σ = w + ξ − ξ .The eight remaining cases (74) can be considered according to this pattern in a com-pletely analogous way. One may check that the quadruple integral (58) for H ( w, v, ξ ) for ξ ≥ • For w and v outside the domain w ∈ [ −
1; 1] and v ∈ [ − | ξ − ξ ′ | ; 1 − | ξ − ξ ′ | ] theintegral vanishes. • For w ∈ [ − − ξ ] and v ∈ [ ξ ′ ; 1 − ξ ′ + ξ ] (DGLAP-like type I domain): H ( w, v, ξ ) = 1 ξ Z w + ξ − ξ vξ − ξ ′ )1 − ξ dσ Z σ + v + ξ ′ ξ − σ + v − ξ ′ − ξ dρ F ( σ, ρ, w − σξ , v − ρξ ) . (81) • For w ∈ [ − − ξ ] and v ∈ [ − ξ ′ ; ξ ′ ] (DGLAP-like type II domain): H ( w, v, ξ ) = 1 ξ Z w + ξ − ξw − ξ ξ dσ Z σ + v + ξ ′ ξ − σ + v − ξ ′ ξ dρ F ( σ, ρ, w − σξ , v − ρξ ) . (82) • For w ∈ [ − − ξ ] and v ∈ [ − ξ ′ − ξ ; − ξ ′ ] (DGLAP-like type I domain): H ( w, v, ξ ) = 1 ξ Z w + ξ − ξ − vξ + ξ ′ )1 − ξ dσ Z σ + v + ξ ′ − ξ − σ + v − ξ ′ ξ dρ F ( σ, ρ, w − σξ , v − ρξ ) . (83) • For w ∈ [ − ξ ; ξ ] and v ∈ [ ξ ′ ; 1 − ξ + ξ ′ ] (DGLAP-like type II domain): H ( w, v, ξ ) = 1 ξ Z w + ξ ξ vξ − ξ ′ )1 − ξ dσ Z σ + v + ξ ′ ξ − σ + v − ξ ′ − ξ dρ F ( σ, ρ, w − σξ , v − ρξ ) . (84) • w ∈ [ − ξ ; ξ ] and v ∈ [ − ξ ′ ; ξ ′ ] (ERBL-like domain): H ( w, v, ξ ) = 1 ξ Z w + ξ ξw − ξ ξ dσ Z σ + v + ξ ′ ξ − σ + v − ξ ′ ξ dρ F ( σ, ρ, w − σξ , v − ρξ ) . (85) • w ∈ [ − ξ ; ξ ] and v ∈ [ − ξ − ξ ′ ; − ξ ′ ] (DGLAP-like type II domain): H ( w, v, ξ ) = 1 ξ Z w + ξ ξ − vξ + ξ ′ )1 − ξ dσ Z σ + v + ξ ′ − ξ − σ + v − ξ ′ ξ dρ F ( σ, ρ, w − σξ , v − ρξ ) . (86)24 w ∈ [ ξ ; 1] and v ∈ [ − ξ ′ ; 1 − ξ + ξ ′ ]: the result coincides with (84) as it certainlyshould be since this is the part of the same DGLAP type II domain. Note thatthis makes H ( w, v, ξ ) a smooth function for w = ξ as it should be since this line( w i = ξ ⇔ x i = 2 ξ ) does not correspond to any change of evolution properties of H ( w, v, ξ ). • w ∈ [ ξ ; 1] and v ∈ [ ξ ′ ; − ξ ′ ] (DGLAP-like type I domain): H ( w, v, ξ ) = 1 ξ Z w + ξ ξw − ξ − ξ dσ Z σ + v + ξ ′ − ξ − σ + v − ξ ′ − ξ dρ F ( σ, ρ, w − σξ , v − ρξ ) . (87) • w ∈ [ ξ ; 1] and v ∈ [ − ξ − ξ ′ ; ξ ′ ]: the result again coincides with (86) since thisis the part of the same DGLAP-like type II domain.
6. RADYUSHKIN TYPE ANSATZ FOR πN TDAS
In this section we discuss what could be a possible approach for modelling of quadrupledistributions F ( σ, ρ, ω, ν ) occurring in the spectral representation (58).Employing the analogy with the case of usual GPDs one may assume that the profile of F ( σ, ρ, ω, ν ) in ( σ, ρ ) space is determined by the shape of the function f ( σ, ρ ) to which πN TDA is reduced in the limit ξ →
0. For the moment we put aside the complicated andinteresting problem of the rigorous physical meaning of this limit. It will be discussedelsewhere. Thus, we suggest to employ the following factorized Ansatz for quadrupledistributions: F ( σ, ρ, ω, ν ) = f ( σ, ρ ) h ( σ, ρ, ω, ν ) , (88)where h ( σ, ρ, ω, ν ) is a profile function normalized according to: Z −| ρ − σ |−| ρ + σ |− | σ | dω Z −| ρ + σ |− ω − | ρ − σ | + ω dν h ( σ, ρ, ω, ν ) = 1 . (89)A possible model is to exploit further the analogy with the standard Radyushkin Ansatzfor the double distributions [24] and to assume that the ( ω, ν ) profile of h ( σ, ρ, ω, ν ) isdetermined by the shape of the asymptotic form of the nucleon distribution amplitude:Φ as ( y , y , y ) = 154 y y y . (90)25he DA (90) is defined for y , , ∈ [0; 2] such that y + y + y = 2.In terms of quark-diquark variables ˜ ω = 1 − y − y and ˜ ν = y − y Φ as reads:Φ as (˜ ω, ˜ ν ) = 154 (˜ ω + 1)(˜ ν + 1 − ˜ ω − ˜ ν + 1 − ˜ ω . (91)Note that Z dy dy dy δ (2 − y − y − y )Φ as ( y , y , y ) ≡ Z − d ˜ ω Z − ˜ ω − − ˜ ω d ˜ ν Φ as (˜ ω, ˜ ν ) = 1 (92)Φ as (˜ ω, ˜ ν ) is defined for − ≤ ˜ ω ≤ − − ˜ ω ≤ ˜ ν ≤ − ˜ ω , (93)while h ( σ, ρ, ω, ν ) is defined for − | σ | ≤ ω ≤ − | ρ − σ | − | ρ + σ | ; − − ω | ρ − σ | ≤ ν ≤ − ω − | ρ + σ | . (94)Thus it makes sense to employ the following substitution of the variables:˜ ω = ω + (cid:0)(cid:12)(cid:12) ρ − σ (cid:12)(cid:12) + (cid:12)(cid:12) ρ + σ (cid:12)(cid:12) − | σ | (cid:1) − (cid:0)(cid:12)(cid:12) ρ − σ (cid:12)(cid:12) + (cid:12)(cid:12) ρ + σ (cid:12)(cid:12) + | σ | (cid:1) ;˜ ν = (1 − ˜ ω )2 2 ν − (cid:12)(cid:12) ρ − σ (cid:12)(cid:12) + (cid:12)(cid:12) ρ + σ (cid:12)(cid:12) − ω − (cid:12)(cid:12) ρ − σ (cid:12)(cid:12) − (cid:12)(cid:12) ρ + σ (cid:12)(cid:12) . (95)This results in the following expression for the profile function h ( σ, ρ, ω, ν ): h ( σ, ρ, ω, ν ) = 1516 (cid:0) ν − ω − (cid:12)(cid:12) ρ − σ (cid:12)(cid:12)(cid:1) (cid:0) − ν − ω − (cid:12)(cid:12) ρ + σ (cid:12)(cid:12)(cid:1) (1 − | σ | + ω ) (cid:0) − (cid:0)(cid:12)(cid:12) ρ − σ (cid:12)(cid:12) + (cid:12)(cid:12) ρ + σ (cid:12)(cid:12) + | σ | (cid:1)(cid:1) . (96)One may check that the profile function (96) satisfies the normalization condition (89).It is extremely interesting to note that in terms of the initial spectral parameters α , α , α and β , β , β satisfying α + α + α = − β + β + β = 0 the profile function(96) can be rewritten in the very symmetric form: h ( β , β , β ; α , α , α ) | P i βi =0 P i αi = − = 154 Q i =1 (1 + α i − | β i | )(1 − ( | β | + | β | + | β | )) (cid:12)(cid:12)(cid:12) P i βi =0 P i αi = − . (97)26he inverse transformation (95) reads ω = ˜ ω (cid:18) − (cid:16)(cid:12)(cid:12)(cid:12) ρ − σ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ρ + σ (cid:12)(cid:12)(cid:12) + | σ | (cid:17)(cid:19) − (cid:16)(cid:12)(cid:12)(cid:12) ρ − σ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ρ + σ (cid:12)(cid:12)(cid:12) − | σ | (cid:17) ; ν = ˜ ν (cid:18) − (cid:16)(cid:12)(cid:12)(cid:12) ρ − σ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ρ + σ (cid:12)(cid:12)(cid:12) + | σ | (cid:17)(cid:19) + 12 (cid:16)(cid:12)(cid:12)(cid:12) ρ − σ (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) ρ + σ (cid:12)(cid:12)(cid:12)(cid:17) . (98)This allows to easily compute the integrals occurring in the calculation of ( N − n, n )-thMellin moments ( N ≥ n ≥
0) in ( w, v ) of πN TDAs: Z −| ρ − σ |−| ρ + σ |− | σ | dω Z −| ρ + σ |− ω − | ρ − σ | + ω dν ω N − n ν n h ( σ, ρ, ω, ν ) . (99)In principle one may also think of a more intricate profile function. In fact any partic-ular function Φ(˜ ω, ˜ ν ) normalized according to Z − d ˜ ω Z − ˜ ω − − ˜ ω d ˜ ν Φ(˜ ω, ˜ ν ) = 1 (100)will define some profile function h ( σ, ρ, ω, ν ) after the substitution (95) . E.g. takingΦ(˜ ω, ˜ ν ) ∼ (˜ ω + 1) b (˜ ν + − ˜ ω ) b ( − ˜ ν + − ˜ ω ) b would lead to the natural generalization ofthe b parameter dependent Radyushkin profile familiar for usual GPDs.It is interesting also to consider the most simple possible profile with no distortion in( ω, ν ) directions: Φ(˜ ω, ˜ ν ) = δ (˜ ω ) δ (˜ ν ) . (101)Contrary to the case of usual GPDs for which the counterpart of the profile (101) leadsto ξ -independent Ansatz the resulting πN TDA preserves the minimal necessary ξ de-pendence. Indeed ˜ ω = 0 and ˜ ν = 0 does not imply ω = 0 and ν = 0 and hence the ξ -dependence introduced through two δ -functions in (58) is preserved and generates theproper ξ -dependent domain of definition for the resulting πN TDA (20). Unfortunatelythe model with the profile (101) turns out to be pathological since it leads to πN TDAswhich are not continuous at the cross-over lines v = ± ξ ′ and w = − ξ separating ERBL-like and DGLAP-like type I, II domains. This makes impossible the calculation of theamplitude of the hard exclusive process in question given by convolution of πN TDA However, one has to make certain assumptions on the endpoint behavior of the function f ( σ, ρ ) towhich πN TDA is reduced in the limit ξ → πN TDA at the cross-over lines v = ± ξ ′ and w = − ξ .For the moment as a toy model we are going to employ the factorized Ansatz (88)with the profile function (96). It is a good point now to discuss a possible model for thefunction f ( σ, ρ ) that is the second ingredient of the factorized Ansatz (88). In the limit ξ → πN TDA reduces to this function: H ( w, v, ξ = 0) = f ( w, v ) . (102)The requirements of convergence of integrals (81)–(87) for πN TDA impose somerestriction on the behavior of the function f ( σ, ρ ) on the border of its domain of definition.It turns out that f ( σ, ρ ) should vanish at least as a certain power of the relevant variablesat the borders of its domain of definition. Thus for the function f ( σ, ρ ) we suggest thefollowing simple form: f ( σ, ρ ) = θ ( − ≤ σ ≤ θ ( − | σ | ≤ ρ ≤ − | σ | × (cid:0) − σ (cid:1) (cid:18) ( ρ − − σ (cid:19) (cid:18) ( ρ + 1) − σ (cid:19) . (103)In terms of the initial spectral parameters β i satisfying P i β i = 0 (103) can be rewrittenas: f ( β , β , β ) | P i β i =0 = 4047 Y i =1 θ ( | β i | ≤ − β i ) (cid:12)(cid:12) P i β i =0 . (104)The function f ( σ, ρ ) vanishes on the border of its domain of definition and is normalizedaccording to Z − dσ Z − | σ | − | σ | f ( σ, ρ ) = 1 . (105)Let us stress that we employ the normalization (105) only for our toy model. Advancedmodelling of πN TDAs aiming the quantitative description of the physical observableswould certainly require more complicated form of f ( σ, ρ ).The normalization for the nucleon to pion TDAs can be derived either from the softpion limit or from the lattice calculations of several first Mellin moments of πN TDAsor from the comparison with the results of [18]. On the other hand it can be computed28onsidering the light baryon exchange contributions into the Mellin moments of πN TDAsusing the phenomenological values say of g πNN and g πN ∆ couplings. The normalizationcan also in principle be established directly form the experimental measurements of thecross-section once the scaling behavior would be found reasonable.On Figure 7 we show the results of the calculation of the contribution H ( w , v , ξ ) ≡ H ( w, v, ξ ) as a function of w and v for different values of ξ computed with the help of thefactorized Ansatz (88) with the profile (96) and f ( σ, ρ ) given by the toy model (103).Note that for ξ = 1 the TDA H ( w, v, ξ ) does not vanish at the corners of its domainof definition. This is potentially dangerous since this may lead to the break up of thefactorization property of the hard exclusive process in question. Fortunately this problemis an artefact of our oversimplified toy model (104) for the forward limit of πN TDA. Itwas checked that taking f ( σ, ρ ) that vanishes quadratically at the borders of the domainof definition f ( β , β , β ) | P i β i =0 = 44103167 Y i =1 θ ( | β i | ≤ − β i ) (cid:12)(cid:12) P i β i =0 (106)leads to a vanishing πN TDA at the corners of its domain of definition for ξ = 1.On Figure 8 we show πN TDA H ( x , x , x , ξ ) for ξ = 0 . x , x and x ( x + x + x = 2 ξ ) in thebarycentric coordinates. By thick solid lines we show the continuation of the edges of theequilateral triangle which form the ERBL-like domain cf. Fig. 3.29IG. 7: The contribution into πN TDA H ( w , v , ξ ) ≡ H ( w, v, ξ ) as a function of w and v for different values of ξ computed using the factorized Ansatz (88) with the profilefunction (96) and f ( σ, ρ ) given by (103). 30IG. 8: πN TDA H ( x , x , x , ξ ) as a function of x , x and x ( x + x + x = 2 ξ ) for ξ = 0 . cf. Fig. 3.
7. CONCLUSIONS
The non-perturbative part of hard processes involving hadrons is encoded in variousuniversal partonic distributions (parton distribution functions, fragmentation functions,distribution amplitudes and their generalizations). Waiting for a complete understandingof the dynamics of quark and gluon confinement in hadrons, one should model thesedistributions in agreement with general requirements of the underlying field theory suchas Lorentz invariance and causality. Spectral representation of hadronic matrix elementsoffers an elegant way to address this program. The double distribution representation forGPDs became the basis for various successful phenomenological GPD models.In this paper we introduced the notion of quadruple distributions and constructedthe spectral representation for the transition distribution amplitudes involving three par-ton correlators which arise in the description of baryon to meson transitions. We alsogeneralized Radyushkin’s factorized Ansatz for the case of quadruple distributions andprovided an explicit expression for the corresponding profile function. Analogously to thecase of GPDs the shape of the corresponding profile function is supposed to be fixed bythe asymptotic form of the nucleon distribution amplitude. Our model also requires the31nowledge of nucleon to meson TDAs in the forward limit as input quantities. Contrarilyto the GPD case, the nucleon to meson TDAs suffer from the fact that there is no illu-minating forward limit. This problem requires further investigation. For a moment, wesuggest to employ a simple shape of nucleon to meson TDAs in the forward limit assumingthat they are fixed by their behavior at the borders of their domain of definition. Ourconstruction opens the way to quantitative modeling of baryon-meson and baryon-photonTDAs in their complete domain of definition.Let us emphasize that for the moment we have not included any D -term like contribu-tions to the spectral representation of the nucleon to meson TDAs in terms of quadrupledistributions. Indeed the results of [36] and of Chapter 3.8 of [27] give us confidence thatthe eventual D -term like contributions to TDAs can be included by means of comple-menting the spectral density in (39) with additional terms proportional to powers of ξ .The subsequent analysis can be performed according to the same pattern.Let us also point out that our method can be generalized for the case of 4-quarkcorrelators important for the description of higher twist contributions. Acknowledgements
We are thankful to Igor Anikin, Jean-Philippe Lansberg, Anatoly Radyushkin andSamuel Wallon for many discussions and helpful comments. K.S. also acknowledges muchthe partial support by Consortium Physique des Deux Infinis (P2I). This work was sup-ported by the Polish Grant N202 249235.
A. A USEFUL CONSTRAINED INTEGRAL
Let us consider the constrained triple integral I ( a, b, c ) = Z a − a dα Z b − b dα Z c − c dα δ ( α + α + α + 1) f ( α , α , α ) , (A1)where a ≥ a ≤ b ≥ b ≤ c ≥ c ≤
1. We introduce the natural coordinates ω and ν : α = ν + − − ω α = − ν + − − ω α = ω . (A2)32n the natural coordinates ω and ν the integration in (A1) is over the intersection ofthree stripes: − c ≤ ω ≤ c ; − a + 1 + ω ≤ ν ≤ a + 1 + ω − b − ω ≤ ν ≤ b − ω . (A3)One may check that for a ≥ b the integral (A1) can be rewritten as I ( a, b, c ) = Z − − a + b − − a − b dω θ ( ω + c ) θ ( c − ω ) Z a + ω − b − ω dν f ( ω , ν )+ Z − a − b − − a + b dω θ ( ω + c ) θ ( c − ω ) Z b − ω − b − ω dν f ( ω , ν )+ Z − a + b − a − b dω θ ( ω + c ) θ ( c − ω ) Z b − ω − a + ω dν f ( ω , ν ) . (A4)Analogously for b ≥ a the integral (A1) can be rewritten as I ( a, b, c ) = Z − a − b − − a − b dω θ ( ω + c ) θ ( c − ω ) Z a + ω − b − ω dν f ( ω , ν )+ Z − b − a − − b + a dω θ ( ω + c ) θ ( c − ω ) Z a + ω − a + ω dν f ( ω , ν )+ Z − a + b − b − a dω θ ( ω + c ) θ ( c − ω ) Z b − ω − a + ω dν f ( ω , ν ) . (A5)In order to be able to perform the integral (A1) we need to specify the intersection ofthree stripes (A3). The results (A4) and (A5) are obtained for arbitrary positive a , b and c . Let us now take into the account that a = 1 − | β | ; b = 1 − | β | ; c = 1 − | β | (A6)with | β i | ≤ β + β + β = 0. • Let us first consider the case when β i s belong to the domain D (44). In this domainwe have | β | = | β | + | β | and thus a = b + c −
1. So in the domain D the followinginequalities are respected:0 ≤ a ≤ , ≤ b ≤ , ≤ a − b + 1 ≤ . (A7)33ne may check that these inequalities result in c ≥ − a + b ; − c = − b − a ; and b ≥ a . (A8)Thus employing (A5) we get I ( a, b, c ) | D = Z − a + b − c dω Z b − ω − a + ω dν f ( ω , ν ) . (A9) • Analogously, in the domain D we have | β | = | β | + | β | and thus b = a + c − ≤ a ≤ , ≤ b ≤ , ≤ b − a + 1 ≤ c ≥ − a + b ; − c = − a − b ; and a ≥ b . (A11)Employing (A4) we get I ( a, b, c ) | D = Z − a + b − c dω Z b − ω − a + ω dν f ( ω , ν ) . (A12) • Finally, let us consider the case when β i belong to the domain D . In this domainwe have | β | = | β | + | β | and hence c = a + b −
1. Thus in the domain D thefollowing inequalities are respected:0 ≤ a ≤ , ≤ b ≤ , ≤ a + b − ≤ . (A13)One may check that in this domain − c ≥ − b − a ; − c ≥ − a − b ; c = − a + b . (A14)Thus independently of a ≥ b or a ≤ b the integral over the intersection of threestripes (A4) or (A5) is again reduced to I ( a, b, c ) | D = Z − a + b − c dω Z b − ω − a + ω dν f ( ω , ν ) . (A15)34 . CASE ξ < For completeness in this Appendix we present the result for πN TDA H ( w, v, ξ ) inthe ERBL-like and DGLAP-like type I and II domains for the case − ≤ ξ < e.g. for ¯ N N → πγ ∗ in the forward region [16]. • For w and v outside the domain − ≤ w ≤ − | ξ − ξ ′ | ≤ v ≤ − | ξ − ξ ′ | the integral vanishes. • For w ∈ [ − ξ ] and v ∈ [ ξ ′ ; 1 − ξ ′ + ξ ] (DGLAP-like type II domain): H ( w, v, ξ ) = 1 ξ Z vξ − ξ ′ )1 − ξ w + ξ − ξ dσ Z − σ + v − ξ ′ − ξσ + v + ξ ′ ξ dρ F ( σ, ρ, w − σξ , v − ρξ ) . (B16) • For w ∈ [ − ξ ] and v ∈ [ − ξ ′ ; ξ ′ ] (DGLAP-like type I domain): H ( w, v, ξ ) = 1 ξ Z w − ξ ξw + ξ − ξ dσ Z − σ + v − ξ ′ ξσ + v + ξ ′ ξ dρ F ( σ, ρ, w − σξ , v − ρξ ) . (B17) • For w ∈ [ − ξ ] and v ∈ [ − ξ ′ − ξ ; − ξ ′ ] (DGLAP-like type II domain): H ( w, v, ξ ) = 1 ξ Z − vξ + ξ ′ )1 − ξ w + ξ − ξ dσ Z − σ + v − ξ ′ ξσ + v + ξ ′ − ξ dρ F ( σ, ρ, w − σξ , v − ρξ ) . (B18) • For w ∈ [ ξ ; − ξ ] and v ∈ [ − ξ ′ ; 1 − ξ ′ + ξ ] (DGLAP-like type II domain) the resultcoincides with (B16). • w ∈ [ ξ ; − ξ ] and v ∈ [ ξ ′ ; − ξ ′ ] (ERBL-like domain): H ( w, v, ξ ) = 1 ξ Z w − ξ − ξw + ξ − ξ dσ Z − σ + v − ξ ′ − ξσ + v + ξ ′ − ξ dρ F ( σ, ρ, w − σξ , v − ρξ ) . (B19) • w ∈ [ ξ ; − ξ ] and v ∈ [ − ξ ′ − ξ ; ξ ′ ] (DGLAP-like type II domain): the resultcoincides with (B18). • w ∈ [ − ξ ; 1] and v ∈ [ − ξ ′ ; 1 − ξ + ξ ′ ] (DGLAP-like type I domain): H ( w, v, ξ ) = 1 ξ Z vξ − ξ ′ )1 − ξ w + ξ ξ dσ Z − σ + v − ξ ′ − ξσ + v + ξ ′ ξ dρ F ( σ, ρ, w − σξ , v − ρξ ) . (B20)35 w ∈ [ − ξ ; 1] and v ∈ [ ξ ′ ; − ξ ′ ] (DGLAP-like type II domain): H ( w, v, ξ ) = 1 ξ Z − vξ + ξ ′ )1 − ξ w + ξ ξ dσ Z − σ + v − ξ ′ − ξσ + v + ξ ′ − ξ dρ F ( σ, ρ, w − σξ , v − ρξ ) . (B21) • w ∈ [ ξ ; 1] and v ∈ [ − ξ − ξ ′ ; ξ ′ ] (DGLAP-like type I domain): H ( w, v, ξ ) = 1 ξ Z w − ξ − ξw + ξ ξ dσ Z − σ + v − ξ ′ ξσ + v + ξ ′ − ξ dρ F ( σ, ρ, w − σξ , v − ρξ ) . (B22) [1] D. Mueller, D. Robaschik, B. Geyer, F.M. Dittes, and J. Horejsi, Fortschr. Phys. , 101(1994).[2] A. V. Radyushkin, Phys. Lett. B (1996) 417 [arXiv:hep-ph/9604317].[3] X. D. Ji, Phys. Rev. D (1997) 7114 [arXiv:hep-ph/9609381].[4] J. C. Collins, L. Frankfurt and M. Strikman, Phys. Rev. D (1997) 2982 [arXiv:hep-ph/9611433].[5] L. L. Frankfurt, P. V. Pobylitsa, M. V. Polyakov and M. Strikman, Phys. Rev. D , 014010(1999) [arXiv:hep-ph/9901429].[6] L. Frankfurt, M. V. Polyakov, M. Strikman, D. Zhalov and M. Zhalov, “Novel hard semiex-clusive processes and color singlet clusters in hadrons,” in Newport News 2002, ExclusiveProcesses at High Momentum Transfer , edited by A. Radyushkin and P. Stoler; WorldScientific, Singapore, 2002, pp.361-368 [arXiv:hep-ph/0211263].[7] V. M. Braun, D. Y. Ivanov, A. Lenz and A. Peters, Phys. Rev. D , 014021 (2007);[8] V. M. Braun, D. Y. Ivanov and A. Peters, Phys. Rev. D , 034016 (2008).[9] A. V. Radyushkin, arXiv:hep-ph/0410276.[10] A. V. Efremov and A. V. Radyushkin, Theor. Math. Phys. , 97 (1980) [Teor. Mat. Fiz. , 147 (1980)].[11] G. P. Lepage and S. J. Brodsky, Phys. Rev. D , 2157 (1980).[12] V. L. Chernyak and A. R. Zhitnitsky, Phys. Rept. , 173 (1984).[13] B. Pire and L. Szymanowski, Phys. Lett. B , 83 (2005) [arXiv:hep-ph/0504255].[14] B. Pire and L. Szymanowski, PoS HEP2005 , 103 (2006) [arXiv:hep-ph/0509368].
15] J. P. Lansberg, B. Pire and L. Szymanowski, Phys. Rev. D , 074004 (2007) [Erratum-ibid.D , 019902 (2008)] [arXiv:hep-ph/0701125].[16] J. P. Lansberg, B. Pire and L. Szymanowski, Phys. Rev. D , 111502 (2007)[arXiv:0710.1267 [hep-ph]].[17] M. Strikman and C. Weiss, Phys. Rev. C , 042201 (2010) [arXiv:1004.3535 [hep-ph]].[18] B. Pasquini, M. Pincetti and S. Boffi, Phys. Rev. D , 014017 (2009) [arXiv:0905.4018[hep-ph]].[19] A. V. Radyushkin, Theor. Math. Phys. , 1144 (1984) [Teor. Mat. Fiz. , 284 (1984)].[20] A. V. Radyushkin, Phys. Lett. B , 179 (1983).[21] A. V. Radyushkin, Phys. Rev. D , 5524 (1997) [arXiv:hep-ph/9704207].[22] A. V. Radyushkin, Phys. Lett. B , 81 (1999) [arXiv:hep-ph/9810466].[23] A. V. Radyushkin, Phys. Rev. D , 014030 (1998) [arXiv:hep-ph/9805342].[24] I. V. Musatov and A. V. Radyushkin, Phys. Rev. D , 074027 (2000) [arXiv:hep-ph/9905376].[25] K.Goeke, M.V.Polyakov and M.Vanderhaeghen, Progr. Part. Nucl. Phys. Vol.47, 401,(2001) [arXiv:hep-ph/0106012].[26] M. Diehl, Phys. Rept. , 41 (2003) [arXiv:hep-ph/0307382].[27] A.V. Belitsky and A. V. Radyushkin, Phys. Rept. , 1 (2005), [arXiv:hep-ph/0504030].[28] S. Boffi and B. Pasquini, Riv. Nuovo Cim. , 387 (2007) [arXiv:0711.2625 [hep-ph]].[29] M. Vanderhaeghen, P. A. M. Guichon and M. Guidal, Phys. Rev. D , 094017 (1999)[arXiv:hep-ph/9905372].[30] B. Pire and L. Szymanowski, Phys. Rev. D , 111501 (2005) [arXiv:hep-ph/0411387].[31] E. Witten, Nucl. Phys. B (1977) 189.[32] S. Friot, B. Pire and L. Szymanowski, Phys. Lett. B (2007) 153 [arXiv:hep-ph/0611176].[33] M. V. Polyakov and C. Weiss, Phys. Rev. D , 114017 (1999) [arXiv:hep-ph/9902451].[34] O. V. Teryaev, Phys. Lett. B , 125 (2001) [arXiv:hep-ph/0102303].[35] I.M. Gelfand, M.I. Graev and N.Ya. Vilenkin, Generalized functions (Academic Press, N.Y.-London, 1966), Vol. 5.[36] A. V. Belitsky, D. Mueller, A. Kirchner and A. Schafer, Phys. Rev. D , 116002 (2001)[arXiv:hep-ph/0011314]., 116002 (2001)[arXiv:hep-ph/0011314].