Photoproduction of pi+ pi- pairs in a model with tensor-pomeron and vector-odderon exchange

We consider the reaction (gamma p) to (pi+ pi- p) at high energies. Our description includes dipion production via the resonances rho, omega, rho-prime and f2, and via non-resonant mechanisms. The calculation is based on a model of high energy scattering with the exchanges of photon, pomeron, odderon and reggeons. The pomeron and the C=+1 reggeons are described as effective tensor exchanges, the odderon and the C=-1 reggeons as effective vector exchanges. We obtain a gauge-invariant version of the Drell-Soeding mechanism which produces the skewing of the rho-meson shape. Starting from the explicit formulae for the matrix element for dipion production we construct an event generator which comprises all contributions mentioned above and includes all interference terms. We give examples of total and differential cross sections and discuss asymmetries which are due to interference of C=+1 and C=-1 exchange contributions. These asymmetries can be used to search for odderon effects. Our model is intended to provide all necessary theoretical tools for a detailed experimental analysis of elastic dipion production for which data exist from fixed target experiments, from HERA, and are now being collected by LHC experiments.


Introduction
Photoproduction of π + π − pairs on protons at high energies, that is the reaction has been studied for a long time, both in theory and experiment. For reviews of topics relevant for this reaction see, for instance, [1][2][3], and for model calculations [4][5][6], for example. The reaction (1.1) has been investigated by fixed target experiments [7][8][9][10][11][12][13][14][15] and by the experiments H1 and ZEUS at HERA [16][17][18]. It can also be studied by the LHC experiments in ultra-peripheral pp and Ap collisions; cf. [19][20][21][22]. Indeed, first results for the reaction (1.1) from such collisions have been presented in [23]. The purpose of our paper is to give detailed formulae for the reaction (1.1) as obtained in the recently proposed model [24]. This model is constructed to describe soft high-energy reactions including Regge behaviour of the amplitudes. The pomeron and the charge conjugation C = +1 reggeons are described as effective tensor-exchange objects, the odderon and the C = −1 reggeons as effective vector-exchange objects. Due to these properties the amplitudes obtained in the model [24] naturally satisfy the rules of quantum field theory. In particular, they exhibit the correct behaviour under crossing and under charge conjugation. The reaction (1.1) offers the interesting possibility to find odderon effects. Let us, therefore, first discuss the status of the odderon. The odderon, the charge conjugation C = −1 counterpart of the C = +1 pomeron, was introduced on theoretical grounds in [25,26]. More than forty years after its introduction the odderon is seen in theoretical papers, but not yet clearly in experiments. For a review see [27]; for a general discussion of the pomeron and the odderon in high-energy reactions and QCD see [2]. Various reactions have been proposed in order to look for odderon effects. We are concerned here with two of these proposals. One, suggested in [28][29][30][31][32], is to study the exclusive photoproduction of C = +1 mesons and in particular the production of the f 2 ≡ f 2 (1270) meson. The other is to look for certain asymmetries of the final state caused by the interference of C-even and C-odd exchanges. Various asymmetries of this kind have been described in [33][34][35][36][37][38][39][40]. For a review of these and other observables suited for odderon searches see [27].
We shall discuss now the relevant production mechanisms and diagrams for the reaction (1.1) for high energies, s m 2 p , and for π + π − invariant mass, m π + π − , from threshold to m π + π − ≈ 2 GeV. The kinematic quantities for this reaction are collected in appendix A. In this kinematic region we expect to see the production of the ρ meson, including ρ-ω interference effects, and the resonant production of the f 2 and ρ ≡ ρ(1450) mesons. The exchanges to be considered are: the pomeron P, the reggeons f 2R , a 2R , ω R , ρ R , the photon γ and -if it exists -the odderon O. In addition to resonance production, the π + π − continuum production is considered. The diagrams for all these subprocesses are shown in figure 1. In the following these diagrams are evaluated in the model presented in [24] and we shall give explicit formulae which can serve as basis for experimental analyses. The photoproduction of a neutral meson by photon exchange is called the Primakoff effect [41]. Our diagram 1(c) corresponds to this effect with f 2 as the neutral meson. We see from figure 1 that exchanges with C = +1 (P, f 2R , a 2R ) and C = −1 (ω R , ρ R , γ, O) contribute to the reaction (1.1). This leads to asymmetries in the π + π − distributions, as we shall discuss at length below. Such asymmetries have been proposed for odderon searches in [34][35][36][37][38][39][40]. But, as we see from figure 1, odderon exchange is not the only C = −1 exchange contribution. Thus, a careful analysis of all contributing C = +1 and C = −1 exchanges is necessary in order to assess the rôle of the odderon. In this paper we present such a description as is a prerequisite for an experimental search for odderon effects in the reaction (1.1).
Before we do this we give an argument why the photoproduction of f 2 mesons may be particularly sensitive to odderon effects. We show in figure 2 a QCD diagram which contributes and is specific to f 2 production. Via loops the photon γ can couple to three gluons and the f 2 can couple to two gluons, resulting with proper arrangement of the gluon lines in a three-gluon exchange (the simplest perturbative representation of the odderon) with the proton. We note that this type of diagram does not exist for the photoproduction of π 0 and a 2 mesons on which rather strict experimental limits exist; see [42,43]. The non-observation of odderon exchange in these reactions was discussed in [44], and it was ¡ ρ, ω, ρ ρ, ω, ρ P, f2R, a2R π + (k1) Figure 1. Diagrams for π + π − photoproduction: (a) vector-meson ρ(770), ω(782), ρ (1450) production; (b), (c), (d) f 2 production via reggeon, photon (Primakoff effect), and odderon exchanges, respectively; (e) non-resonant π + π − production via pomeron and f 2R reggeon exchanges; (f) nonresonant π + π − production via ρ R and γ exchanges. The diagrams (a) and (e) correspond to C = +1 exchange, the diagrams (b), (c), (d), and (f) to C = −1 exchange.
shown that chiral symmetry implies a strong suppression of π 0 photoproduction via odderon exchange [45]. Our paper is organised as follows. In section 2 we give analytic expressions for the diagrams of figure 1. In section 3 we present numerical results of total and differential cross sections and discuss the π + π − asymmetries. Section 4 contains our conclusions. Kinematical relations and a list of the effective propagators and vertices are given in appendix A and appendix B, respectively. In appendix C we discuss the behaviour of the differential cross section for t → 0. Appendix D deals with the determination of the Monte Carlo weights for our event generator.

Matrix elements, cross sections and asymmetries
We define the matrix element M µ s ,s for the reaction (1.1) as M µ s ,s (k 1 , k 2 , p , q, p) µ = π + (k 1 ), π − (k 2 ), p(p , s )|T |γ(q, ), p(p, s) . (2.1) Here k 1 , k 2 , p , q and p are the four-vectors of the involved particles, is the photon's polarisation vector, and s and s are the spins of the incoming and outgoing proton, respectively. Details of the kinematics of the reaction are discussed in appendix A. The matrix element (2.1) gets contributions from all diagrams shown in figure 1: The diagrams (a) and (e) correspond to C = +1 exchange, the diagrams (b), (c), (d) and (f) to C = −1 exchange. Gauge invariance requires In our calculation we find that this gauge-invariance relation holds for each subclass of diagrams (a) to (f) separately: The cross section for (1.1) assuming unpolarised particles in the initial state and no observation of polarisations in the final state reads R can only depend on the variables indicated, see (A.10), where the Mandelstam variables s and t denote the squared center of mass energy and the squared momentum transfer of the reaction, respectively. The four-vector of the dipion system is defined by the four-vector sum of the π + and π − as k = k 1 + k 2 . We can split R from (2.6) into R = R + + R − where the parts R + and R − are even and odd, respectively, under the simultaneous sign change of the last two arguments: (2.7) R + contains the squares of C = +1 and C = −1 exchange amplitudes, R − contains the interference terms of the C = +1 and C = −1 exchanges. We get Observables which are odd under the transformation (2.7) are, therefore, particularly suitable to measure the interference of C = +1 and C = −1 exchanges, and thus allow to study possible odderon contributions. Examples of such observables are discussed in section 3.
We shall now calculate the amplitudes corresponding to the diagrams of figure 1. The effective propagators and vertices needed for these calculations are mostly taken from [24]. To make the present paper self-contained we list the propagators and vertices needed here in appendix B. We consider it an asset of our approach that given these propagators and vertices we can use the standard rules of QFT to obtain the amplitudes. This guarantees, for instance, that all gauge-invariance and charge-conjugation properties of the amplitudes are automatically satisfied. We shall only present the final results in the following. These explicit expressions are the basis for the construction of our event generator for the reaction (1.1).

Vector-meson production
Here we discuss the diagrams of figure 1(a), that is, the production of the vector-mesons ρ, ω and ρ which then decay into π + π − . For ρ and ω we consider both PV V (pomeron) and f 2R V V (reggeon) couplings (V = ρ, ω) whereas for the rather small ρ contribution only the Pρ ρ coupling is considered. We include strong-isospin violating effects for the (ρ, ω) propagator and for the ω → π + π − decay but assume absence of such violations for the couplings of pomeron and reggeons to vector mesons. With the expressions for the propagators, vertices and form factors from appendix B we then obtain (2.10) where for i = 0, 2 and For a 2R reggeon exchange in figure 1(a) we have four contributions. The photon can turn into a ρ which, upon a 2R exchange, turns into a ω; and we can have the rôles of ρ and ω exchanged. Both, the final ω and ρ, can then, by propagator mixing, go to V ∈ {ρ, ω} which decays to π + π − . The amplitude taking into account all these processes is Here and in the following M 0 = 1 GeV is used in various places for dimensional reasons. All quantities occurring here and their definitions can be found in table 1 of appendix B. We note that for the ρ meson, apart from the known mass m ρ and the width Γ ρ , only the combinations µ,s ,s . Thus, only these two combinations can be determined by studying the ρ contribution to the reaction (1.1).

Production of f 2 by reggeon exchange
Here the corresponding diagram is shown in figure 1(b). We get with the expressions for the propagators and vertices from appendix B We find for i = 0, 2 and for V = ρ, ω For the definition of the parameters occurring here we refer again to table 1 in appendix B.

Production of f 2 by photon exchange
Here we calculate the amplitude corresponding to the diagram of figure 1(c). Using the expressions for the propagators and vertices from appendix B we get the following: (2.21) Here the N (i) µ,s ,s (i = 0, 2) are taken from (2.19) and we have defined for i = 0, 2: and

Production of f 2 by odderon exchange
Now we come to the diagram of figure 1(d) describing f 2 production via odderon exchange. We get here, using the formulae from appendix B, Here again the N (i) µ,s ,s (i = 0, 2) are as in (2.19) and we have 2.5 Non-resonant production of π + π − by pomeron and f 2R exchange The diagrams of figure 1(e) describe the non-resonant production of a π + π − pair by exchange of the pomeron P and the f 2R . This type of process was first discussed by Söding [46] following a suggestion by Drell [47,48]. However, typically it is difficult to maintain gauge invariance in calculations of this Drell-Söding term.
In our approach the effective vertices are derived from coupling Lagrangians. In this formalism we include the coupling to photons by the minimal substitution rule, i. e. derivatives are replaced by corresponding covariant derivatives. Thereby we are guaranteed to have gauge invariance automatically. For the pomeron exchange contribution to nonresonant π + π − production we start from the Pππ Lagrangian in equation (7.3) of [24]. We note that for writing down such a Pππ Lagrangian it is essential to consider the pomeron exchange as an effective tensor exchange. The minimal substitution gives then the Lagrangian (B.66) which fixes the Pγπ + π − contact term in figure 1(e). Using analogous arguments the f 2R γπ + π − contact term in figure 1(e) is fixed. With the formulae of appendix B we obtain M (e) µ,s ,s = i 2 Note that all three diagrams in figure 1(e) come with the same energy dependence, that is with the same energy variable s in the Regge factor. In Regge language this corresponds to saying that the propagation of the pion between its couplings to the photon and to the pomeron or f 2R in the first two diagrams of figure 1(e) is part of the impact factor. This ensures the gauge invariance of the expression (2.26) and is in agreement with Regge factorisation. Due to its lower spin the t-channel propagation of the pion over a sizeable range in rapidity is strongly suppressed with respect to pomeron and f 2R exchange.
2.6 Non-resonant production of π + π − by ρ R and photon exchange From the diagrams of figure 1(f) we get the following amplitude for non-resonant π + π − production via ρ R and photon exchange: As it must be, the amplitudes corresponding to C = +1 exchanges are odd under the exchange of the pion momenta, k 1 ↔ k 2 ; see (2.10)-(2.16) and (2.26 In agreement with Regge factorisation the energy dependence of all three diagrams of figure 1(f) is the same. This is analogous to the case of pomeron and f 2R exchange, see section 2.5.
This concludes our presentation of the amplitudes corresponding to the diagrams of figure 1(a-f). For further analysis an event generator is built, based on the following techniques. For the evaluation of the matrix elements the FeynCalc package [49] (version 8.2.0) is used and the results are cross-checked with a direct implementation based on the template-library ltensor [50]. Based on that, a weight and the kinematic variables of an event are determined using a standard Monte Carlo (MC) method with pre-sampling, see appendix D. The output of the event generator is stored in a standard format of the ROOT-package [51] and used to derive (partially integrated) differential cross sections as a function of various variables.

Results
In this section we present selected results for cross sections, angular distributions and asymmetries. For the numerical evaluation of the formulae presented above we use a set of default parameters for the model [24]. These default parameters are given in table 1 of appendix B. Many of the parameters are well constrained while we can only guess the values of some others. In the present study we do not make any attempt to fit the parameters to data. That would be desirable in general but is not the scope of the present paper. Instead, we concentrate here on studying the qualitative features of the reaction (1.1) in the model. We find that some asymmetries seem particularly promising for detailed experimental studies as they allow to enhance interesting effects by choosing suitable cuts on the data. In particular, we point out promising ways to look for effects of the elusive odderon.

Cross sections
We start with the total cross section for the reaction (1.1) which we integrate here from the dipion mass threshold to 1.5 GeV (as is done for example in the experimental study [18]): In figure 3 we show the cross section as function of W γp . Also the cross sections for various subclasses of diagrams according to figure 1 are shown. The dominant contribution to the total cross section comes from ρ 0 production. Total cross section data for γp → ρ 0 p are shown for illustration. They are taken from figure 10 of [18] and are defined in the same mass range as given in (3.1). For W γp 5 GeV our model describes the essential features of the data. At very low W γp -values an agreement with data is not expected as further processes, not included in our high-energy model, contribute to the cross section. We emphasise again that our model curves are just examples and not fits. Note that our model contains all contributions to the reaction (1.1) given by the diagrams of figure 1, not just the ρ contribution, which nonetheless dominates the total cross section.
The total cross section σ (γp → π + π − p) as a function of the center-of-mass energy W γp . The cross section is integrated over 2m π ≤ m π + π − ≤ 1.5 GeV and −1 GeV 2 ≤ t ≤ 0. The full model and individual contributions from vector meson production, non-resonant processes, and f 2 production are shown. The reggeon contributions comprise f 2R and a 2R in case of vector meson, and ρ R and ω R in case of f 2 production. High energy data for σ γp → ρ 0 p from H1 [17] and ZEUS [16,18] at HERA as well as fixed target data, referenced in [18], are shown for illustration.
As a second example we show in figure 4 the differential cross section dσ/dt (γp → π + π − p) as function of |t|. We integrate over m π + π − as indicated in (3.1) and choose W γp = 30 GeV. Again, in addition to the full differential cross section also the various contributions from individual subprocesses are shown. The production of ρ mesons is dominant for all values of t, followed by non-resonant π + π − production and the -by more than three orders of magnitude -smaller f 2 production. The cross section of all contributions falls approximately exponentially as function of |t| for |t| 0.1 GeV 2 . We note that the Primakoff contribution to f 2 production, figure 1(c), has a singularity for |t| → 0. This singularity is, of course, never reached since for reaction (1.1) there is a minimal value for |t|, |t min | > 0 (see (A.12)). The singularity is driven by the photon propagator and behaves as 1/|t|. In contrast, the reggeon contributions from ρ R , ω R and odderon exchange, figures 1(b) and 1(d), show a dip for |t| → 0. An explanation for this different behaviour is given in appendix C.
Also note that with the chosen default parameters odderon exchange is the dominant contribution to f 2 production for |t| 0.1 GeV 2 which can be exploited in an experimental search to enhance a possible odderon signal. Reggeon contributions to f 2 production are negligible at W γp = 30 GeV or higher values.  The differential cross section dσ/dt (γp → π + π − p) as function of |t|. The cross section is integrated over the range 2m π ≤ m π + π − ≤ 1.5 GeV and given for fixed W γp = 30 GeV. In addition to the full model results also contributions from the main diagrams are shown, see figure 3 for explanations.
In figure 5 we show the differential cross section dσ/dm π + π − (γp → π + π − p) as function of the π + π − invariant mass m π + π − for the range from threshold to m π + π − = 2 GeV, and enlarged for the ρ mass region. The differential cross sections are shown for W γp = 30 GeV integrated in the range −1 GeV 2 ≤ t ≤ 0. The contributions of various subclasses of diagrams are also shown as well as the dominant contributions from interferences in the ρ mass region. The resonance structure of the ρ peak is clearly visible. Note that the shape of the ρ(770) peak from the diagrams of figure 1(a) alone is rather symmetric. The skewing of the ρ shape caused by the interference of the diagrams of figures 1(a) and 1(e),(f) (non-resonant contributions) is clearly exposed. We see here the Drell-Söding mechanism [46][47][48] at work. Note that the skewing depends crucially on the choice of the ρ form factor parameterisation, which was here implemented according to equation (B.85). Furthermore, the effect of the ρ-ω interference, the steep falloff at the top of the ρ 0 peak, is clearly visible.
In the mass region of the ρ meson, 1.2 GeV m π + π − 1.6 GeV, a distortion of the skewed ρ line-shape due to the interference of the ρ diagram in figure 1(a), and to a lesser extent the f 2 diagrams in figures 1(b, c, d), with the ρ meson and the non-resonant contribution is visible. However, clear resonance peaks in the m π + π − distribution due to f 2 and ρ do not show up for the chosen default parameters.  Figure 5. Differential cross sections dσ/dm π + π − (γp → π + π − p) as function of m π + π − for fixed W γp = 30 GeV and integrated over the range −1 GeV 2 ≤ t ≤ 0.

Angular distributions
In the following we study the pion angular distribution in the π + π − rest system. For illustration we choose as reference system here the proton-Jackson system, see figure 10 in appendix A, in which the polar angle θ k 1 ,p of the π + is measured with respect to the incoming proton direction. This and other reference systems are discussed in detail in appendix A. As the decay-angle distribution is mass dependent and mainly driven by the spin of the resonance in case of decays we study the angular distribution in the ρ and the f 2 mass regions separately.
In figure 6(a) the differential distribution dσ/d cos θ k 1 ,p (γp → π + π − p) in the ρ mass region, 0.45 GeV ≤ m π + π − ≤ 1.1 GeV, is shown as function of cos θ k 1 ,p . The distribution shows a typical ∝ sin 2 θ k 1 ,p behaviour as is expected for photo-produced vector mesons if (approximate) s-channel helicity conservation [52] holds. 1 (For the general formalism of helicity amplitudes in this context see [53,54].) In figure 6(b) the same distribution is shown in the f 2 mass region 1.1 GeV ≤ m π + π − ≤ 1.35 GeV. In addition to the dominant ρ contribution, which exhibits again the typical ∝ sin 2 θ k 1 ,p behaviour, the small f 2 contributions show more features in the angular distribution as it is expected for a J = 2 resonance. The interference of the C-even and C-odd exchange contributions leads to a small asymmetry of the cos θ k 1 ,p distribution as shown in figure 6(c). This asymmetry comes dominantly from the interference between the ρ and f 2 production diagrams. The asymmetry is partially cancelled by the interference of the f 2 resonance with non-resonant π + π − production but a net asymmetry remains if all Figure 6. Differential cross section dσ/d cos θ k1,p (γp → π + π − p) as function of the cosine of the polar angle θ k1,p in the proton-Jackson system for fixed W γp = 30 GeV and integrated over the range −1 GeV 2 ≤ t ≤ 0. The full model and the dominant contributions for the mass regions 0.45 GeV ≤ m π + π − ≤ 1.1 GeV (ρ mass region) and 1.1 GeV ≤ m π + π − ≤ 1.35 GeV (f 2 mass region) are shown in (a) and (b),(c), respectively. (c) shows in addition to the full differential cross section the dominant interference terms on a linear scale. All contributions are explained in the legend; for R + and R − see (2.8) and (2.9). diagrams are included. The resulting charge asymmetries are discussed in more detail in the following.

Charge asymmetries
Let us now turn to asymmetries in the π + π − rest system. As already noted in section 2 the interference of diagrams with exchange of C = +1 and C = −1 objects is signalled by an asymmetry under see the discussion following (2.7) and (2.27). We note first that P-invariance tells us that the distributions of the π + momentum must be symmetric under a reflection on the reaction plane, which in the π + π − system is given by the plane spanned by the momenta of the incoming proton and the outgoing proton. We turn, therefore, to charge asymmetries which are defined with respect to specific directions (axes) in the reaction plane. In the literature many different definitions of reference systems can be found, which are used to study asymmetries. A summary of the different definitions is given in appendix A.
We start our discussion of charge asymmetries in the proton-Jackson system, for which the θ k 1 ,p distributions were shown in the previous section. It is convenient to define a θ k 1 ,p dependent charge asymmetry: We also define a total charge asymmetry: using the definitions In figure 7(a) the total charge asymmetry defined in the proton-Jackson system is shown as function of the invariant mass of the π + π − system. A negative asymmetry of a few percent is visible in the f 2 resonance region for the chosen parameter values. This asymmetry is mainly due to the interference of the f 2 resonance with the high mass tail of the ρ resonance. For higher masses, m π + π − > 1. 35 GeV, an asymmetry with opposite sign is visible in the model, which is mainly due to the interference of the f 2 resonance with the non-resonant π + π − production with C = +1 pomeron exchange. Another but smaller positive asymmetry is visible just above the π + π − production threshold which is mainly due to the interference of C = +1 and C = −1 exchange diagrams of non-resonant π + π − production.
We are particularly interested in the asymmetry contribution from the odderon. In figure 7(b) the total charge asymmetry is shown as function of t in the f 2 mass region 1.1 GeV ≤ m π + π − ≤ 1.35 GeV. A small negative asymmetry, almost constant in t, is generated by the Primakoff contribution. In contrast, a strong increase of the absolute value of the negative charge asymmetry approximately linear in |t| is predicted by odderon exchange with the chosen model parameters. As most events are located at low |t|, the experimental sensitivity to the odderon exchange diagram can be significantly enhanced by requiring a minimum |t|-cut, or better by measuring the approximately linear increase of the absolute value of the charge asymmetry as function of |t|.
Finally we compare the charge asymmetry in different systems. Figure 8  shows the charge asymmetry as function of cos θ k 1 ,γ in the photon-Jackson system, in which the polar angle θ k 1 ,γ is defined with respect to the incoming photon; see appendix A. For the photon-Jackson system it is interesting to note that (1) in the central region, cos θ k 1 ,γ 0.6, the asymmetry contributions due to Primakoff and odderon exchange are of similar size and (2) that the asymmetry changes sign in the polar region, cos θ k 1 ,γ 0.6. At large cos θ k 1 ,γ the large positive asymmetry is dominated by the contribution from odderon exchange. Therefore, the photon-Jackson system offers the opportunity to study interference effects due to Primakoff and odderon exchange diagrams separately by analysing different cos θ k 1 ,γ regions. Note that due to the sign change the total asymmetry is strongly reduced (after integration over cos θ k 1 ,γ ) in the photon-Jackson system and that a possible odderon signal can be experimentally overlooked if only total asymmetries are studied. In the same context it should be remarked that limited detector acceptances in experimental searches might affect the measurement of angular distributions and thus the sensitivity to charge asymmetries.
The full asymmetry information of the π + π − system can be exploited by investigating the asymmetry for all solid angles. For this study we define the angle α, which describes in the reaction plane the azimuthal angle of the π + with respect to the incoming proton beam direction, and the elevation angle β; see figure 11 and (A.20) in appendix A.
In figure 9 the charge asymmetry of the π + π − final state is shown as function of α and β in the f 2 mass region. By construction, see (3.6), the relation A(α, β) = −A(α + π, −β) holds. From P-invariance follows A(α, β) = A(α, −β). The asymmetry distribution exhibits two dipoles: one at α ≈ 0 (−π) related to the incoming proton direction and a second one at α ≈ −π/2 (+π/2) which is broader in both α and β. For illustration, lines of constant polar angle θ k 1 ,p are shown, along which the asymmetry was integrated to calculate the differential asymmetries for figure 8(a). The presence of complex structures in the asymmetry distribution, which are generally integrated out in one-dimensional projections, suggests to exploit the full twodimensional information for asymmetry measurements in order to increase sensitivity and to separate the different asymmetry sources.

Conclusions
In this article we have presented a study of exclusive photoproduction of π + π − pairs on protons, γp → π + π − p, in the framework of a comprehensive model for soft high-energy reactions. We have considered π + π − production via the ρ, ω, ρ and f 2 resonances as well as production of non-resonant π + π − pairs. Taking into account photon, pomeron, odderon and reggeon exchanges we have obtained analytic expressions for all contributing diagrams. We have calculated the total and various differential cross sections, angular distributions and asymmetries for a set of default parameters of the model. We emphasise that in the present paper no attempt has been made to fit data. The purpose of our paper is to provide all necessary theoretical tools for such a comparison with data. Our methods can easily be extended to exclusive electroproduction e + p −→ e + π + + π − + p (4.1) at low to moderate values of Q 2 . Another extension is to central production of π + π − pairs in peripheral pp collisions p + p −→ p + π + + π − + p . This reaction will be discussed in a forthcoming paper [55]. Central production of scalar and pseudoscalar mesons with techniques similar to the ones presented here has been discussed in [56].
In summary, we have presented a study of the reaction γp → π + π − p in an explicit model for soft high-energy scattering that includes the exchanges of pomeron, odderon, photon, and reggeons. In particular, the model incorporates a gauge-invariant version of the Drell-Söding mechanism which is responsible for the skewing of the ρ meson shape. We have paid particular attention to the effects of the elusive odderon in the photoproduction of pion pairs. The odderon is expected to contribute to f 2 meson photoproduction as first suggested in [28,29], and to asymmetries in the angular distribution of the π + π − pairs as suggested in [34][35][36][37][38][39][40]. Our results indicate that the corresponding observables appear indeed very promising for an odderon search. More generally, we hope that our results can be used as guidance for the experimental study of interesting effects in the photoproduction of pion pairs.

Acknowledgments
The authors would like to thank P. Lebiedowicz, M. Guzzi, C. Royon, R. Schicker, and A. Szczurek for useful discussions. The work of C. E. was supported by the Alliance Program of the Helmholtz Association (HA216/EMMI).

A Kinematics
Here we discuss kinematic relations for the reaction where we consider a real or virtual photon, γ or γ * , with polarisation vector . In the final state p stands for a proton or a diffractively excited proton, for instance the resonance N (1520). The spin indices of p and p are denoted by s and s , respectively. We set and we have, in general, For the elastic photoproduction reaction (1.1) we have, of course, We denote the space-part of the four-vector p by p, etc. From the five momentum vectors of the particles in the reaction (A.1) we can form 15 scalar products and one parity-odd (P-odd) invariant I P . As the P-odd invariant we can choose where we use the convention 0123 = +1 for the totally antisymmetric symbol µνρσ . This parity-odd variable cannot enter the cross section calculation for unpolarised particles since we consider a process where only the P-conserving strong and electromagnetic interactions contribute. Not all 15 scalar products are independent. We have energy-momentum conservation q + p = k 1 + k 2 + p (A.6) and the mass-shell conditions We take as independent variables All scalar products can be expressed in terms of the variables in (A.8). We get s , t , k 2 , p · (k 1 − k 2 ) , q · (k 1 − k 2 ) , (A. 10) and the sign of I P . Equivalently we can chose where θ, φ are the polar and azimuthal angles of k 1 in the π + π − rest system. We also note that t ≤ t min < 0 with where w(x, y, z) = (x 2 + y 2 + z 2 − 2xy − 2xz − 2yz) We are interested in the angular distribution of the π + in the centre-of-mass system of the π + π − pair. There we have Various reference systems are commonly used; see figure 10 for illustration. We list here the proton and photon-Jackson systems, see [57,58], which are used in section 3 to illustrate cross sections and asymmetries. In addition we also mention the Collins-Soper [59] and the helicity system [60]. For the proton-Jackson system we set (cf. p. 125 of [60]) withp = p/|p|,q = q/|q| and p, q the three-momenta of the initial proton and the γ ( * ) in the π + π − rest system. The corresponding polar and azimuthal angles of the π + are denoted by θ k 1 ,p and φ k 1 ,p , respectively, and we have cos θ k 1 ,p =k 1 ·p. For the photon-Jackson system we set (cf. p. 125 of [60]) The polar and azimuthal angles of the π + in the photon-Jackson system are denoted by θ k 1 ,γ and φ k 1 ,γ , respectively, and we have cos θ k 1 ,γ =k 1 ·q.
We mention two more systems of reference, the Collins-Soper and the helicity systems. The unit vectors of the CS system are chosen as e 1,CS =p +q |p +q| , and the unit vectors of the helicity system are chosen as  Figure 11. Definition of the angles α and β in the π + π − rest frame. As reference the proton-Jackson system is chosen. We have 0 ≤ α < 2π and −π/2 ≤ β ≤ π/2. This property, and the fact that P-invariance with respect to reflection at the reaction plane holds, motivates to define two new angles α and β. These angles describe the direction of the π + in the π + π − rest frame and are defined as cos β cos α =k 1 · e 3,p =k 1 ·p , where α = 0 is aligned to the incoming proton direction. Note that α and β can also be interpreted as the spherical coordinates of the π + in the π + π − rest frame, with e 2,p having the rôle of the z-and e 3,p the rôle of the x-axis. Transformation from one system i to another system j from (A.15) to (A.18) can then be represented by rotations: Reflections with respect to the reaction plane are described by a transformation of the elevation angle β → −β and P-invariance requires The following relations between the angles α, β and θ k 1 ,p , φ k 1 ,p in the proton-Jackson system hold:

B Propagators and vertices
In this appendix we collect the propagators and vertices needed for the evaluation of the diagrams of figure 1. Here the propagators and vertices involving the pomeron, the odderon and the reggeons are to be understood as effective propagators and vertices. Most of the relations listed in the following are taken over from [24]. We reproduce them here in order to make the present paper self-contained.
The numerical values of coupling constants and other parameters quoted in the following are to be considered as default values. Adjustments of these parameters should come from detailed comparisons with experiment.
All our vertices respect the standard crossing and charge-conjugation (C) relations of quantum field theory.
• vector-mesons V = ρ 0 , ω Since we include in our calculations ρ 0 -ω interference effects we rely here on the analysis of the γ-ρ-ω propagator matrix as given in [61]. But here we are only interested in the ρ-ω, the strong-interaction, part of the 3 × 3 propagator matrix studied there. We get the following from appendix B of [61], setting e = 0 in the relations there, for V , V ∈ {ρ, ω}: Here and in the following ρ is understood as ρ 0 . The longitudinal parts ∆ (V ,V ) L (k 2 ) never enter in our calculations and, thus, need not be discussed further. For the transverse parts we use matrix notation. With k 2 = s we set We have from [61] where all these relations are derived and discussed at length: Re B ωω (m 2 ω ) = 0 , Here V. P. means the principal value prescription. Explicitly we get the following. For s > 4m 2 R(s, m 2 ) = 1 96π 2 and for s < 0 Furthermore, we have from [61] B ωω (s) = g 2 ωKK s R(s, m 2 B ρω (s) = g ρππ g ωππ s R(s, m 2 π ) − R(m 2 ρ , m 2 π ) + iI(s, m 2 π ) . (B.16) In [61] fits to the pion electromagnetic, weak, and πγ transition form factors were made. These values are quite consistent with the corresponding ones from (B.17). We recall that we are only quoting default values for our calculations here and for this purpose it makes no difference if we take those from (B.17) or from (B.18). The definitions of the coupling constants g ρππ and g ωππ through the corresponding vertices are given in (B.55) below. The ρ-ω propagator matrix as defined above should be a good model for |s| 15 GeV 2 ; see section 4 of [24].
we make a simple ansatz as suggested in [63] The values for m ρ and Γ ρ listed in [62] are We note that in [64] -contrary to what is written there -a different form for the ρ propagator is used, not the one from [63] which we reproduce here in (B.20), (B.21). The Breit-Wigner ansatz made in Eqs. (4), (5) of [64] corresponds to replacing in (B.20) √ s Γ ρ (s) by m ρ Γ ρ (s). We shall not use such a form for the ρ propagator.

Vertices
In this section we list the vertices which are needed for discussing the reaction (1.1). We use as in [24] the following rank-four tensor functions The vertices read as follows.
• γpp (see (3.26)-(3.32) of [24]) The proton's Dirac and Pauli form factors and the dipole form factor are denoted by F 1 , F 2 , and G D , respectively; see for instance chapter 2 in [3]. In many hadronic vertices we have, realistically, to introduce form factors. For simplicity we use for the proton for momentum transfer squared t < 0 in general the Dirac form factor F 1 (t) (B.50). For mesons we use for t < 0 a simple parametrisation of the pion's electromagnetic form factor see (3.34) of [24]. Further form factors are introduced and discussed below.

(B.59)
A convenient form for the F (f 2 ππ) form factor is The parameter Λ f 2 is estimated to be in the range 1 to 4 GeV.