Factorization model for distributions of quarks in hadrons

We consider distributions of unpolarized (polarized) quarks in unpolarized (polarized) hadrons. Our approach is based on QCD factorization. We begin with study of Basic factorization for the parton-hadron scattering amplitudes in the forward kinematics and suggest a model for non-perturbative contributions to such amplitudes. This model is based on the simple observation: after emitting an active quark by the initial hadron, the remaining set quarks and gluons becomes unstable, so description of this colored state can approximately be done in terms of resonances, which leads to expressions of the Breit-Wigner type. for non-perturbative contributions to the distributions of unpolarized and polarized quarks in the hadrons. Then we reduce these formulae to obtain explicit expressions for the quark-hadron scattering amplitudes and quark distributions in K_T- and Collinear factorizations.


I. INTRODUCTION
QCD factorization, i.e. separation of perturbative and non-perturbative QCD contributions, proved to be an efficient instrument for describing hadron reaction at high energies. Being first applied to processes in the hard kinematics in the form of Collinear factorization [1], it was soon extended to cover the forward kinematic region, with DGLAP [2] used to account for perturbative contributions. Then, in order to be able to use BFKL [3], a new kind of factorization, K T -factorization was suggested in Ref. [4]. These kinds of factorization are usually illustrated by identical pictures. For instance, factorization of the DIS hadronic tensor W µν is conventionally depicted by the construction in Fig contrary, the lower blob is conventionally introduced from purely phenomenological considerations. Collinear and K T -factorizations operate with different parametrizations for momentum k of the connecting partons and as a result, they are described by different formulae. Collinear factorization assumes that while K T -factorization allows for the transverse momentum in addition: accounting therefore for one longitudinal and two transverse components of k. However as a matter-of-fact, k has four components: two of them are longitudinal and the other two are transverse. Accounting for the missing longitudinal component α (for definition of α see Eq. (3)) drove us to suggesting a new, more general factorization which we named in Ref. [5] Basic factorization. In contrast to K T -and Collinear factorizations, the analytic expressions in Basic factorization can be obtained from the graphs of the type of the one in Fig. 1 with applying the standard Feynman rules. It is worth reminding briefly our derivation of Basic factorization, for detail see Ref. [5]. Let us consider the Compton scattering amplitude off a hadron in the forward kinematics. It is depicted in Fig. 2. The blob in Fig. 2 denotes ensemble of perturbative and non-perturbative contributions. This blob can be expanded into an infinite series of terms, each of them is represented by two blobs connected with n parton lines, n = 2, 3, ... Considering only the simplest, two-parton state, we arrive to the graph similar to the one in r.h.s of Fig. 1 but without the s-cut and with the both blobs accommodating perturbative and non-perturbative contributions at the same time. The integration of the convolution in Fig. 1 over momentum k now runs over the whole phase space and it is expected to bring a finite result. However, the propagators of the connecting partons become singular at k 2 = 0 (we neglect quark masses). Besides, the upper blob may content IR-sensitive perturbative contributions ∼ ln n (2pk/k 2 ) (with n = 1, 2, ..) and besides yields the factor 2pq/k 2 when the upper blob includes unpolarized gluons. The only way to kill such IR singularity is to assume that the lowest, non-perturbative blob should tend to zero fast enough when k 2 → 0 . Doing so and repeating a similar procedure to regulate the UV singularity, we bring the convolution in Fig. 1 to agreement with the factorization concept: perturbative and non-perturbative contributions are located indifferent blobs. This is a new form of QCD factorization which we name Basic factorization.
We demonstrated in Ref. [5] that Basic factorization can be reduced step-by-step first to K T -and then to Collinear factorizations. In Ref. [5] we began with considering Basic factorization for Compton scattering amplitudes in the forward kinematics, where integration over momentum k of the connecting partons in Fig. 1 runs over the whole phase space. Confronting two obvious facts that, on one hand, the integration over k should yield a finite results and that, on the other hand, the perturbative part in Fig. 1 (the upper, perturbative blob and propagators of the connecting partons) is divergent in both the infra-red (IR) and ultra-violet (UV) regions, allowed us to impose theoretical restrictions on the lowest blob, which are necessary for the convolution in Fig. 1 to be finite. These restrictions led us to theoretical constraints on the fits for the parton distributions to the DIS structure functions in Collinear and K Tfactorizations. In particular, we predicted the general form of the fits in K T -factorization and excluded the factors x −a from the fits in both K T -and Collinear factorizations.
Another interesting object, where factorization is used, is distributions of partons in hadrons. In the present paper we examine their properties in IR and UV regions and suggest a simple resonance model for the non-perturbative contributions to the parton distributions. Our argumentation in favor of this model is as follows: after emitting an active quark by a hadron, the remains of the hadron, i.e. a set of quarks and gluons, acquires a color and therefore it becomes unstable. So, this colored state can be described in terms of resonances. We begin with considering amplitudes of the quark-hadron (QHA) and gluon-hadron (GHA) scattering in the forward kinematics. The Optical theorem relates such amplitudes to the parton distributions. Throughout the paper we use the standard Sudakov parametrization [6] for momentum k of the connecting partons: where momenta q ′ and p ′ are massless, p ′2 ≈ q ′2 ≈ 0, and they are made of the hadron momentum p and the parton momentum q: where In Sect. II we introduce the quark-hadron scattering amplitudes in the forward kinematics and examine their IR and UV behavior. In Sect. III we consider separately the unpolarized and spin-dependent quark-hadron amplitudes in Basic factorization and suggest a model for non-pertubative contributions to the amplitudes. This model involves a spinor structure accompanied by invariant amplitudes T (U) and T (S) . In Sect. III we specify the spinor structure of the nonperturbative contributions to the amplitudes and parton distributions. In Sect. IV we show how Basic factorization for the quark-hadron amplitudes and quark distributions in hadrons can be reduced to K T -and Collinear factorizations. In Sect. V we focus on a model for the invariant amplitudes T (U) and T (S) . The model is based on description of T (U) and T (S) in a quasi-resonant way and through the Optical theorem it easily leads to non-perturbative contributions to the parton distributions, with expressions of the Breit-Wigner kind both in Basic and in K T -factorizations. Finally, Sect. VI is for concluding remarks.

II. QUARK-HADRON AMPLITUDES
In the factorization approach, the quark-hadron amplitudes (QHA) A q are expressed through convolutions of perturbative amplitudes A (pert) and non-perturbative amplitudes T as shown in Fig. 3.
In the Born approximation A (pert) is depicted in Fig. 3 as a one-rung ladder. Adding more ladder rungs to it together with inclusion of non-ladder graphs and resumming all such graphs converts the Born amplitude into A (pert) .
Factorization of the quark-hadron amplitude.
In the present paper we do not consider mixing of quark and gluon ladder rungs, i.e. we consider the graphs where the vertical quark lines go from the bottom to the top without breaking. We begin consideration of the quark-hadron amplitudes A q in Basic factorization, studying the simplest case depicted in Fig. 4, where the perturbative contributions are accounted in the Born approximation and denote such distributions B q . In Basic factorization one can use the standard Feynman rules to write down the analytic expression corresponding to the graphs in Figs. 3,4. Doing so, we obtain that where we have used the standard notations: C F = (N 2 − 1)/(2N ) = 4/3 and α s is the QCD coupling. In Eq. (6) T q corresponds to the lowest blob in Fig. 3. It is altogether non-perturbative object. Throughout the paper we will address it as the primary quark-hadron amplitude 1 . Choosing the Feynman gauge, where d µν = g µν , for the virtual gluon and the Sudakov parametrization (3) for the quark momentum k, we rewrite Eq. (6) as follows: Throughout the paper, for the sake of simplicity, we will treat the external quarks with momentum q as on-shell ones, though our reasoning remains valid also when they are off-shell. Introducing the density matrix with q, m q and S q being the quark momentum, mass and spin respectively, we bring Eq. (7) to the following form: We stress that the replacement of Eq. (7) by Eq. (9) is not necessary for us but it allows us to carry out a more detailed consideration of A B q . In particular, we can consider separately the spin-dependent, B (spin) q and independent, B (unpol) q quark-hadron amplitudes in a simple way: In Eqs. (10,11) we have replaced the general primary amplitudeT q by more specific amplitudesT . In Eq. (10) we have neglected a contribution ∼ m inρ(q) compared to the contribution ∼q. Integrations in Eqs. (10,11) run over the whole phase space, where the explicit k-dependent terms in the integrands can have both IR and UV singularities. The only way to yield a finite result from the integrations is to impose appropriate constrains on the primary quark-hadron amplitudesT (unpol) q ,T (spin) q so that to kill the singularities.

A. IR and UV stability of QHA
The integrands in Eqs. (10,11) can become singular in the infra-red (IR) region, where k 2 ∼ 0, because of the denominators. In the frame of perturbative QCD IR singularities are regulated by introducing IR cut-offs. In our case this method cannot be used. We are left with the only way to kill these singularities: The primary quark-hadron amplitudesT q should become small at small k 2 : when k 2 → 0. Now let us consider the ultra-violet (UV) stability of the convolutions in Eqs. (10,11). The integration over α in Eqs. (10,11) runs from −∞ to ∞, so, at large |α| the integrands should decrease fast enough to guarantee UV stability. First of all we focus on the integration over α in Eq. (10). Taking into consideration that each factor in the denominator of Eq. (10) is ∼ α makes that the denominator to be ∼ α 3 . The term 2qk in the numerator depends on α because 2qk = w(β − x 1 α) and the factors k 2 andk are ∼ α, which makes 2qkk This divergence must be regulated by an appropriate decrease ofT (unpol) q at large |α|. The IR stability condition in Eq. (12) states thatT (unpol) q ∼ k 2 1+η at small k 2 but it can either disappear or be kept at large |α|. Therefore we have two options: (A) The factor k 2 1+η survives at large |α|. (B) The factor k 2 1+η disappears at large |α|. In the case (A), where IR and UV behaviors ofT should behave at large |α| as follows: with χ > η > 0. IR and UV behaviors ofT are disconnected in the case (B). It converts Eq. (14) into The first factor in Eq. (14) corresponds to the term k 2 1+η , while a contribution generating the asymptotic factor in the squared brackets has to be specified. We will do it in Sect. V. Now let us consider the spin-dependent amplitudes. In order to guarantee their IR stability, the primary spin-dependent amplitudeT (spin) q should also be ∼ (k 2 ) 1+η at small k 2 but the situation with its UV stability is more involved than in the unpolarized case. Indeed, the quark spin S q can be either in the plane formed by p and q, i.e. S q = S || q , or in the transverse space, where S q = S ⊥ q . Depending on it, there are the longitudinal spin-dependent amplitude, B || q and the transverse one, B ⊥ q . Now let us consider the term 2m q S q k in Eq. (11) for different orientations of the quark spin: When the spin is longitudinal, andk in the trace T r[kT q ] is also ∼ α. In contrast when the spin is transverse, and therefore S ⊥ q k does not depend on α. Then, this k ⊥ should be accompanied by another k ⊥ from the trace in order to get a non-zero result at integration over the azimuthal angle, i.e. The first term in the numerator of Eq. (11) does not depend on α, while the second term is ∼ α. It means that, withT || q dropped, the explicit α-dependence of A || q at large |α| coincides with the one in Eq. (13): and It follows from Eq. (18) states that the α -dependence of the amplitudeT || q at large |α| is identical to the one of T in the case (B). Each of Eqs. (20,22) consists of two factors, with the first factor corresponding to the term k 2 1+η . Contributions generating the factors in squared brackets will be specified in Sect. V.

III. MODELING THE SPINOR STRUCTURE OFTq
Our next step is to simplify the traces in Eqs. (10,11). In order to do it, we have to specify the spinor structure of the primary QHAT q . By definition,T q is altogether non-perturbative, so specifying its spinor structure can only be done on basis of phenomenological considerations. However, any model expression forT q should respect the integrability conditions in Eqs. (12,14,20,22). There is the well-known expression for the density matrix of an elementary fermion: where M and S are the fermion mass and spin. This expression drives us to approximateT q as follows: where p, S are the hadron momentum and spin respectively and T are scalar functions. Throughout the paper we will address them as invariant quark-hadron amplitudes. Substituting T q of Eq. (12) in Eqs. (10,11) and calculating the traces, we arrive at the following expressions: the perturbative amplitude in the Born approximation for the forward annihilation of unpolarized quark-quark pair. We have neglected contributions ∼ x 1,2 in the numerator of Eq. (26) and will do it in expressions for the spin-dependent amplitudes. These terms, if necessary, can easily be accounted for with more accurate implementation of Eq. (3) to Eqs. (26). Let us consider the structure of the integrand in Eq. (26) in more detail. The amplitude in the last brackets is entirely non-perturbative. It is suppose to mimic a transition from hadrons to quarks. The fraction in the middle corresponds to the convoluting the perturbative and non-perturbative amplitudes. The fraction in the first brackets corresponds to the perturbative amplitude for the forward scattering of quarks in the Born approximation. We explicitly wrote the factor ıǫ there to remind that this amplitude has the s-channel imaginary part. Doing similarly, we obtain an expression for the spin-dependent amplitudes: Let us consider Eq. (27) for different orientation of the hadron spin: (i) The hadron spin S is in the plane formed by momenta p and q, so for this case we use the notation S = S || . (ii) The hadron spin is transverse to this plane. We denote this case as S = S ⊥ .
Amplitude A q for the first case is given by the expression very close to the unpolarized amplitude: whereas the transverse amplitude is given by a different expression: Taking the s-imaginary part of Eq. (26), we arrive at the totally unintegrated, or fully unintegrated [8], distribution of unpolarized quarks in the hadron D unpol B q in the Born approximation: is the primary quark distribution of unpolarized quarks in the hadron, Ψ . This object is altogether non-perturbative. Applying the Optical theorem to Eq. (30), we arrive at the parton distributions beyond the Born approximation:

IV. REDUCTION OF BASIC FACTORIZATION TO CONVENTIONAL FACTORIZATIONS
Conventional forms of factorization are Collinear and k T -factorizations. In Ref. [5] we described reduction of Basic factorization to K T -and Collinear factorizations for the Compton scattering amplitudes and DIS structure functions without specifying the non-perturbative amplitudes T q . In this Sect. we show that these results perfectly agree with our assumption in Eq. (25) concerning structure of T q . We demonstrate that the parton distributions in both conventional factorizations can be obtained with step-by-step reductions of the expressions for D in Basic factorization. This reductions are the same for both the parton-hadron amplitudes and parton distributions, they are insensitive to spin and stands when the quarks are replaced by gluons. Because of that we consider such reductions for a generic parton-hadron distribution D in Basic factorization and skip unessential factors: where D (pert) stands for a perturbative contribution and Ψ is the altogether non-perturbative primary parton-hadron distribution.

A. Reduction to kT -factorization
In order to reduce Eq. (33) to k T -factorization, we have to perform integration with respect to α. However, this integration should not involve D (pert) , which, strictly speaking, is impossible because D (pert) depends on k 2 and thereby it depends on α: k 2 = wαβ − k 2 ⊥ . The only way out is to assume that the main contributions to Eq. (33) come from the region where i.e. k 2 ≈ −k 2 ⊥ . Let us notice that approximating ladder partons virtualities k 2 by their transverse momenta is wellknown. It is used in all available evolution equations, including DGLAP and BFKL, and now it allows us to convert Eq. (33) into an expression for the unintegrated (transverse momentum dependent [9]) parton distributions D KT in k T -factorization: with Φ being the primary (non-perturbative) k T -parton distribution: As we do not study perturbative parts of the parton distributions in the present paper, we have used in Eqs. (33,35) the same generic notation D (pert) for the perturbative contributions and will keep the same notation in the context of Collinear factorization.

B. Reduction to Collinear factorization
In order to reduce k T -factorization to Collinear factorization we should perform integration of Eq. (35) with respect to k ⊥ without integrating D (pert) . The only way to do it is to assume a peaked dependence of Ψ(wα, k 2 ⊥ ) on k 2 ⊥ with maximum at k 2 ⊥ = µ 2 as shown in Fig. 5. The number of such maximums can be unlimited.
We stress that Φ is non-perturbative, so typical values of µ must be of non-perturbative range, µ ∼ Λ QCD . The sharper is the peak, the higher is accuracy of the reduction. After the integration of Φ we arrive at Collinear factorization convolution: with µ being the intrinsic factorization scale and φ being the primary (non-perturbative) integrated parton distribution: In conventional scenario of Collinear factorization, the factorization scale µ is arbitrary, with typical values µ ∼ few GeV, i.e. µ > µ. The transition from the intrinsic scale µ to an arbitrary scale µ can be done with perturbative evolution, converting the primary parton distribution φ(µ) into the integrated parton distribution ϕ( µ) consisting of perturbative and non-perturbative contributions. It is easy to show that our reasoning remains true in the case when Φ has several maximums or an infinite series of them. This point was discussed in detail in Ref. [5], so we will not do it in the present paper. Instead, we focus on modeling invariant amplitudes T (U,S) introduced in Eq. (25).

V. MODELING THE INVARIANT QUARK-HADRON AMPLITUDES AND PRIMARY QUARK DISTRIBUTIONS
In this Sect. we suggest a model which mimics non-perturbative QCD contributions in the primary hadron-quark invariant amplitudes T (U,S) and in the primary quark distributions in all available forms of factorization. Once again we begin with consideration of the invariant amplitudes T (U,S) and then proceed to the quark distributions.

A. Resonance model for the primary quark-hadron invariant amplitudes
Amplitudes T (U,S) can be introduced in a model way only because QCD has not been solved in the non-perturbative region. All such models should satisfy several restrictions: (i): The IR stability conditions in Eq. (12) and the UV stability conditions should be respected. We remind that the UV stability conditions derived in Sect. II depend on UV-behavior of the factors regulating IR divergences . Namely,Eqs. (14,20,22) correspond to the case (A) while Eqs. (15,21,23) correspond to the case (B). In the present paper we focus on the most UV-divergent case (A), although our conclusions hold true for the case (B) as well.
where R U,S k 2 are supposed to behave as R U,S k 2 ∼ k 2 1+η at small k 2 . In addition, Eq. (25) contains parameters M 1,2,3,4 and Γ 1,2,3,4 . In terms of the Sudakov variables T (U,S) q are: We suggest that values of µ 2 j and Γ j should be within the non-perturbative scale domain, with M 2 j > Γ j . It is convenient to write T U,S as the sum of two resonances: , Now we define R U,S : where λ U,S and µ 2 U,S , (µ 2 U,S > 0) are parameters. We stress that specifying R U,S cannot be done unambiguously. We are going to consider this issue in detail in our subsequent paper. In the present paper we use R U,S of Eq. (43). It is easy to check now that the expressions for T where r = U, S. The upper limit of integration, α max should obey Eq. (34), so we choose According to Eq. (34), k 2 ≈ −k 2 ⊥ . The integration leads to the following expression for T (j) q (see Appendix D for detail): They depend on k ⊥ very slowly and they can be neglected at large k 2 ⊥ .

B. Primary quark distributions
The Optical theorem relates the s-channel imaginary parts of T (U,S) and T (U,S) to the primary quark distributions Ψ U,S in Basic factorization and to unintegrated (or ) quark distributions Φ U,S in k T -factorization respectively. So applying the Optical theorem, we obtain the following expression for the primary quark distribution Ψ r in Basic factorization: . and a similar expression for the primary quark distribution Φ U,S in k T -factorization: Obviously, the expressions in Eqs. (48,49) are of the Breit-Wigner type. Substituting Eq. (49) in Eq. (35) and integrating over k 2 ⊥ , we arrive at the quark parton distribution D (col) j in k T -factorization, where the non-perturbative contributions i.e. the unintegrated parton distributions are specified: Let us consider the k ⊥ -dependence in Eqs. (49,35) in more detail. Obviously, the structures of expressions for D  49), is symmetric with respect to replacement 1 ⇋ 2. Each term in the parentheses has a peaked form, with maximums at k 2 ⊥ = µ 2 1,2 . The less Γ 1,2 , the sharper the peaks are. We remind that R U,S ∼ (k 2 ⊥ ) 1+η at small k 2 ⊥ . By definition, see Eq. (41), µ 2 1,2 = M 2 1,2 − p 2 , so they can be either positive or negative while k 2 ⊥ cannot be negative. In any case the both terms in Φ U and Φ S contribute to D (kT ) U,S but a result of interference of the two peaks depends on values of the parameters. There are possible three particular cases: Case (i): both µ 2 1 and µ 2 2 are positive. In this the both maximums are within the integration region of Eq. (35) and interference of the two peaks generates various forms of Φ U (β, k 2 ⊥ ) ranging from the picture with two isolated peaks to a kind of plateau, depending on values of Γ 1,2 .
Case (ii): µ 2 1 > 0 and µ 2 2 < 0 or vice versa. Here the peak from the first term in Eq. (49) combines with a tail of the contribution of the second term whose maximum is beyond the integration region of Eq. (35). The resulting picture has a resemblance to the dual model combining a resonant and a constant term.
Case (iii): both µ 2 1 and µ 2 2 are negative. The both maximums now are out of the integration region, so tails of the peaks, taken by themselves, generate a form slow decreasing with growth of k 2 ⊥ . However, this slope is affected by an impact of R U . We remind that R U = 0 at k 2 ⊥ = 0.

C. Primary quark distributions in Collinear factorization
Performing integration over k 2 ⊥ in Eq. (35), we arrive at the parton distributions D (col) j in Collinear factorization. Presuming that parameter Γ j is small, we write the result of the integration in the following form (see Appendix C for detail): where the integration regions Ω 1 = Ω ′ 1 [0, w] and Ω 1 = Ω ′ 2 [0, w], with the subregions Ω ′ 1 , Ω ′ 2 being located around the maximums of the peaks. Formally, the both terms in Eq. (52) contribute to φ U at any signs of µ 2 1 , µ 2 2 , but in the limit of sharp peaks these contributions have different weights: At µ 2 1 > 0, µ 2 2 > 0 the both terms contribute equally: Mostly the first term contributes, when µ 2 1 > 0, µ 2 2 < 0: and vice versa. Finally, at µ 2 1 , µ 2 2 < 0 only tails of the both peaks contribute and therefore φ U is small and flat compared to the previous cases: Using the Mellin transform, we can rewrite Eq. (37) in terms of conventional Collinear factorization: where C U is a generic notation for the quark coefficient function, ϕ U (ω, µ 2 ) is the conventional parton distribution in the ω -space (momentum space). It is related to δq U (x, µ 2 ) by the Mellin transform and according to the conventional version of Collinear factorization, ϕ U (ω, µ 2 ) is fixed at an arbitrary chosen scale µ, (µ 2 > 0 and µ 2 > µ 2 1,2 ). E U (ω, q 2 , µ 2 ) is a generic notation for evolving the quark distribution ϕ U (ω, µ 2 ) from the factorization scale µ 2 to q 2 . E U (ω, q 2 , µ 2 ) involves anomalous dimensions and it is described by different expressions in different perturbative approaches (see Appendix E for detail). When µ 2 Combining Eqs. (57) and (58), integrating over β and remembering that at small x essential values of ω are small leads to the following expression for ϕ U (ω, µ 2 ):

VI. CONCLUSION
In the present paper we have considered the quark-hadron scattering amplitudes and distributions of polarized and unpolarized quarks in hadrons in the framework of the factorization concept where the both amplitudes and distributions are expressed through convolutions of the perturbative and non-perturbative components. We began with considering the quark-hadron amplitudes in Basic factorization where integration over momenta of connecting partons runs over the whole phase space and obtained the conditions for the factorization convolution to be stable both in IR and UV regions. Then we demonstrated how to reduce Basic factorization to K T -and Collinear factorizations. We suggested a Resonance Model for non-perturbative contributions to the unpolarized and spin-dependent partonhadron scattering amplitudes. This model is based on the simple argumentation: after emitting an active quark by a hadron, the remaining colored quark-gluon state cannot be stable and therefore it can be described by quasi-resonant expressions. We needed at least two resonances in Basic factorization and this remained true when Basic factorization was reduced to K T -factorization. Applying the Optical theorem to the Resonance Model provided us first with the expressions of the Breit-Wigner type for non-perturbative (primary) contributions to the quark distributions in Basic and K T -factorizations and then, after one more reduction, to the parton distributions in Collinear factorization. To conclude, let us notice that the Resonance Model can also be used for analysis of the non-singlet components of the DIS structure functions.

Appendix B: Projection operators for forward Compton amplitudes
The conventional of dealing with the forward Compton scattering amplitude A µν is, in the first place, to simplify their tensor structure. To this end, A µν is represented as an expansion of A µν into the series of simpler tensors, each multiplied by an invariant amplitude. Such tensors are called projection operators. Through the Optical theorem the invariant amplitudes are related to the DIS structure functions.
In the case of the unpolarized Compton scattering such an expansion looks as follows: where P (1) µν = −g µν + q µ q ν /q 2 , P (2) µν = (1/pq) p µ − q µ (pq/q 2 ) p ν − q ν (pq/q 2 ) (B2) are the projection operators and A 1 , A 2 are invariant amplitudes. According to the Optical theorem Similarly, for the polarized Compton scattering where P (3) µν = ıǫ µνλρ M q λ S ρ , P (4) µν = ıǫ µνλρ M q λ [S ρ − p ρ (qS/qp)] , with M and S being the hadron mass and spin respectively, and A 3,4 are spin-dependent invariant amplitudes. The Optical theorem states that All operators P (n) µν respect the electromagnetic current conservation: q µ P (n) µν = q ν P (n) µν = 0. It is convenient to introduce the longitudinal, S || and transverse, S ⊥ components of the spin, so that S ⊥ p = S ⊥ q = 0 and S || ρ = p ρ (qS/pq). In such terms Eq. (B4) can be written as follows: This expression is useful for practical attributing different terms in the spin-dependent A µν to proper invariant amplitudes. In the unpolarized case one can use the simple rule: expressions ∼ g µν contribute to A 1 while expressions ∼ p µ p ν /pq form A 2 . In contrast, the gauge invariance admits adding arbitrary terms ∼ q µ , q ν .

Appendix C: Convolutions involving the Breit-Wigner formula
Let us consider the following convolution: Replacing x by t, with t = (x − x 0 )/Γ, we convert Eq. (C1) into At small Γ, we can expand f (tΓ + x 0 ) in the power series and retain several terms: Substituting Eq. (C3) in (C2) and integrating (C2) yields The first term in Eq. (C4) corresponds to the well-known representation of the δ -function: