Antiproton--proton annihilation into pion pairs within effective meson theory
aa r X i v : . [ h e p - ph ] D ec Physics Letters B 00 (2015) 1–11
PhysicsLetters B
Antiproton–proton annihilation into pion pairs within e ff ectivemeson theory Y. Wang a, ∗ , Yu. M. Bystritskiy b , E. Tomasi-Gustafsson c a Univ Paris-Sud, CNRS / IN2P3, Institut de Physique Nucl´eaire, UMR 8608, 91405 Orsay, France b Joint Institute for Nuclear Research, Dubna, Russia c CEA,IRFU,SPhN, Saclay, 91191 Gif-sur-Yvette, France
Abstract
Antiproton–proton annihilation into light mesons is revisited in the few GeV energy domain, in view of a global description ofthe existing data. An e ff ective meson model is developed, with mesonic and baryonic degrees od freedom in s , t , and u channels.Regge factors are added to reproduce the proper energy behavior and the forward and backward peaked behavior. A comparisonwith existing data and predictions for angular distributions and energy dependence are done for charged and neutral pion pairproduction.c (cid:13) Keywords:
1. Introduction
Large experimental and theoretical e ff orts have been going on since decades in order to understand and classifyhigh energy processes driven by strong interaction. We revisit here hadronic reactions at incident energies above1 GeV, focusing on two body processes.Antiprotons are a very peculiar probe, due to the fact that scattering and annihilation may occur in the sameprocess, with definite kinematical characteristics. We discuss the annihilation reaction of antiproton-proton into twocharged or neutral pions and the crossed channels of pion-proton elastic scattering. These reactions have been studiedin the past, mostly at lower energies, in connection with data from LEAR and FermiLab (for a review, see [1]). Atlow energies the annihilation into light meson pairs is dominated by few partial waves and the angular distributionsshow a series of oscillations. Data are usually given in terms of Legendre polynomials. This regime was studied withthe aim to look for resonances in the ¯ pp system. A change of behavior appears around √ s = t or u , respectively ( s , t and u are standard kinematical Mandelstam variables of binary process). The di ff erential crosssection at large momentum transfer and the integrated cross section show a power-law behavior as a function of theenergy. At larger energies, the total cross section becomes asymptotically constant and reaches a regime where d σ/ dt is function only of t and is independent on s . ∗ Chinese CSC Scholar
Email address: [email protected] (E. Tomasi-Gustafsson) Physics Letters B 00 (2015) 1–11 The most exhaustive data on neutral pion (and other neutral meson) production have been published by the Fer-miLab E760 collaboration in the energy range (2 . ≤ √ s ≤ .
4) GeV[2]. Charged pion production data are scarce,and do not fill with continuity a large angular or energy range [3, 4, 5]. According to the foreseen performances ofthe experiment PANDA at FAIR, a large amount of data related to light meson pairs production from ¯ pp annihilationis expected in next future. The best possible knowledge of light meson production is also requested prior to the ex-periment, as pions constitute an important background for many other channels making timely the development of areliable model.Few calculations exist in the literature. A baryon ( N and ∆ ) t -channel exchange model was developed by [6]with applicability below 1 GeV beam momentum. A model was recently developed at larger energies, includingmeson exchanges in s − channel, which qualitatively reproduces selected sets of angular distributions [7]. However,the authors warns against application to neighboring energies, eventually related to a specific extrapolation of Reggetrajectories in the region t < s , t , and u channels, applicable in the energy range(2 ≤ √ s ≤ .
5) GeV , that is the accessible domain for the PANDA experiment at FAIR. It is known that first orderBorn diagrams give cross sections much larger than measured, as Feynman diagram assume point-like particles. Formfactors are added in order to take into account the composite nature of the interacting particles at vertexes. Their formis, however, somehow arbitrary, and parameters as coupling constants or cuto ff are adjusted to reproduce the data.A ”Reggeization” of the trajectories is added to reproduce the very forward and very backward scattering angles.Therefore this class of models should be considered as an e ff ective way to take into account microscopic degrees offreedom and quark exchange diagrams.Our aim is to build a model with minimal ingredients, to calculate the basic features of neutral and charged pionproduction in the energy range that will be investigated by the future experiment PANDA at FAIR. To get maximumprofit from the available data, we consider also existing π ± p elastic scattering data, and apply crossing symmetry inorder to compare the predictions based on the annihilation channel, at least in a limited kinematical range. The mainrequirement is that the model should be able to reproduce charged and neutral pion production from annihilation, and π ± p elastic scattering without readjustment of the parameters.
2. Formalism
We consider the annihilation reaction: ¯ p ( p ) + p ( p ) → π − ( k ) + π + ( k ) . (1)in the center of mass system (CMS). The notation of four momenta is shown in the parenthesis. The followingnotations are used: q t = ( − p + k ), q t = t , q u = ( − p + k ), q u = u and q s = ( p + p ), q s = s , s + t + u = M N + m π , M N ( m π ) is the nucleon(pion) mass. The useful scalar product between four vectors are explicitly written as:2 p k = k p = M N + m π − u , p k = k p = M N + m π − t , p p = s − M N , k k = s − m π , p = p = M N = E − | ~ p | , k = k = m π = ε − | ~ k | . (2)The general expression for the di ff erential cross section in the CMS of reaction (1) is: d σ d Ω = π s β π β p |M| , d σ d cos θ = E β p β π d σ dt , (3)where M is the amplitude of the process, β p ( β π ) is the velocity and E ( ε ) is the energy of the proton(pion) in CMS.The phase volume can be transformed as d Ω → π d cos θ due to the azimuthal symmetry of binary reactions. Thetotal cross section is : σ = Z |M| π s | ~ p || ~ k | d Ω , (4)where | ~ p | and | ~ k | are the initial and final momenta (moduli) in CMS.2 Physics Letters B 00 (2015) 1–11 Annihilation Scattering¯ p ( p ) + p ( p ) → π − ( k ) + π + ( k ) π − ( − k ) + p ( p ) → π − ( k ) + p ( − p ) s a = ( p + p ) s s = ( − k + p ) t a = ( p − k ) t s = ( − k − k ) u a = ( p − k ) u s = ( p − k ) s a = E = M + | ~ p a | ) s s = m + M + E ′ ǫ ′ + | k s | σ ( a ) = |M ( a ) | π s | ~ k a || ~ p a | σ ( s ) = |M ( s ) | π s | ~ k s || ~ p s | Table 1. Correspondence between variables in the crossed scattering (s) and annihilation (a) channels.
Crossing symmetry relates annihilation and scattering cross sections. Crossing symmetry states that the amplitudesof the crossed processes are the same. This means that the matrix element M ( s , t ) is the same at corresponding s and t values, but the variables span di ff erent regions of the kinematical space. In order to find this correspondence,kinematical replacements between variables should be done, as indicated in Table 1. Note that the coe ffi cients 1 / / S π + S p +
1) and (2 S ¯ p + S p +
1) for the scattering andannihilation channels, respectively, where S π , S p and S ¯ p are the spin moduli of the corresponding initial particles.The incident momentum in the annihilation channel, corresponding to the invariant s is: | ~ p a | = p s / − M . From theequality s a = s s , the CMS momentum for π − p scattering, | k s | , is evaluated at the same s value: | ~ k s | = s h m − m ( M + s ) + ( M − s ) i . (5)Then the cross sections for the two crossed processes are related by: σ a = | ~ k s | | ~ p a | σ s . (6)If the scattering cross section is measured at a value s s = s di ff erent from s a = s , at small t values one may rescalethe cross section, using the empirical dependence: σ s ≃ const · s − .
3. Formalism
The formulas written above are model independent, i.e., they hold for any reaction mechanism. In order tocalculate M , one needs to specify a model for the reaction. In this work we consider the process (1) within theformalism of e ff ective meson lagrangians.The following contributions to the cross section for reaction (1) are calculated, as illustrated in Fig. (1): • baryon exchange: – t -channel nucleon (neutron) exchange, Fig. 1.a, – t -channel ∆ exchange, Fig. 1.b, – u -channel ∆ ++ exchange , Fig. 1c; • s -channel ρ -meson exchange, Fig. 1d.The total amplitude is written as a coherent sum of all the amplitudes: M = M n + M ∆ + M ∆ ++ + M ρ . (7)In case of charged pions, the dominant contribution in forward direction is N exchange, whereas ∆ ++ mostly contributeto backward scattering. We neglect the di ff erence of masses between the nucleons as well as between di ff erent charge3 Physics Letters B 00 (2015) 1–11 q p p + π - π n (a) p + π - π p (b) ∆ pp ρ - π + π (c) ++ ∆ (d) pp + π - π Figure 1. Feynman diagrams for the reaction ¯ p + p → π − + π + within e ff ective meson Lagrangian approach. states of the pion and of the ∆ . Large angle scattering is driven by s-channel exchange of vector mesons, with thesame quantum numbers as the photon. We limit our considerations to ρ -meson exchange. The expressions for theamplitudes and their interferences are detailed in the Appendix. Coupling constants are fixed from the known decaysof the particles if it is possible, otherwise we use the values from e ff ective potentials as [20]. Masses and widths aretaken from [21].The e ff ects of strong interaction in the initial state coming from the exchange by vector and (pseudo)scalar mesonsbetween proton-antiproton are essential and e ff ectively lead to the Regge form of the amplitude. The t and u diagramsare modified by adding a general Regge factor R x (where x = t , u ) with the following form: R ( x ) = ss ! α ( x ) − , α ( x ) = + r α s π x − M M ; (8)where s ≃ can be considered a fitting parameter [22] and r α s /π ≃ . s = . and r α s /π = . ∆ (1232) and others).The trajectory for the ∆ resonance is known to be di ff erent from the nucleon. The slope parameter is fixed in this caseas r α s /π = .
4. As for excited resonances like N ∗ (1440) they belong to a daughter Regge trajectory which is powersuppressed compared to the leading one. Omitting these contributions involves an estimated uncertainty of 10%.A form factor of the form: F = / ( x − p N , ∆ ) , was introduced in the N p ¯ p and N p ¯ ∆ vertexes, with p N = . p ∆ = ρ NN vertex includes the proton structure in the vector current form with two form factors (FF) F ρ , : Γ µ ( q s ) = F ρ ( q s ) γ µ ( q s ) + i M N F ρ ( q s ) , σ µν q ν s , (9)where σ µν = i γ µ γ ν − γ ν γ µ ] is the antisymmetric tensor. Due to the isovector nature of the ρ , the ρ NN is similar tothe electromagnetic vertex γ NN . However the two form factors F ρ , ( q s ) are di ff erent from the proton electromagneticones. Due to the freedom of the choice, we do not attempt any rearrangement, but prefer to fix the form, the constantsand the parameters of F ρ , ( s ) according to [23, 24, 20] as: F ρ ( s ) = g ρ NN Λ Λ + ( s − M ρ ) , F ρ ( s ) = κ ρ F ρ ( s ) , (10)with normalization F ρ ( M ρ ) = g ρ NN , where the constant g ρ NN corresponds to the coupling of the vector meson ρ withthe nucleon ( g ρ NN / (4 π ) = .
55) , κ ρ = . ρ , and Λ = .
911 is an empirical cut-o ff .To take into account the composite nature of the pion, in principle, a monopole type ρππ form factor may beintroduced: F ρππ = A π / ( s − A π ) , where A π is a parameters to be adjusted on the data. In the present case F ρππ was setto one.The diagrams for neutral pion pair production are illustrated in Fig. 2 , where we consider proton and ∆ + exchangein t − channel and ρ -meson exchange in s − channel. The calculated amplitudes are symmetrized, to take into accountthe identity of the final particles. 4 Physics Letters B 00 (2015) 1–11 q p p o π o π p (a) p o π o π p (b) + ∆ pp (cid:10) o π o π (c) ρ Figure 2. Feynman diagrams for di ff erent exchanged particles for the reaction ¯ p + p → π + π .
4. Comparison with existing data and Discussion
The following procedure was applied, in order to reproduce the collected data basis. The data, from Ref. [2] onneutral pion angular distributions, were first reproduced at best, with particular attention to the s dependence of thecross section and the parameters were fixed. Besides the form factors listed above, we introduced 10% renormalization N ρ and a mild s -dependent relative phase φ ρ = φ + φ s of the ρ diagram, with φ = φ = . . ≤ √ s ≤ . θ = s -channelcontributions. Moreover, the used form of Regge parametrization is not expected to work properly at low energies.Therefore the Regge factor was set to be unity for s < s .The s dependence of the model is shown in Fig. 4, and compared to the experimental data and to the s − predictionfrom quark counting rules [25, 26] for cos θ = . ∆ ++ exchange has to be introduced in u channel, to account for the asymmetric forward / backward production. The introduction of the ∆ ++ diagram allows toreproduce the backward angles for the charged pion data. As we use the same mass and couplings for the di ff erentcharged states of the ∆ , the same form factors parameters for ∆ + , , ++ are taken, not requiring any additional parameters.The angular dependence for the reaction ¯ p + p → π − + π + , for di ff erent value of the total CMS energy √ s areshown in Fig. 5 (a-d).The agreement is satisfactory, taking into account that no rearrangement of the parameters wasdone. They correspond to very backward angles, and are also well reproduced by the model.The results for the crossed channels π ± elastic scattering are also reported in Fig. 5 (e-f), where data for thedi ff erential cross section span a small very forward or very backward angular region, bringing an additional test of themodel.The angular distribution for √ s = ρ s-channel exchange, whereas n exchange in t channel dominates forward angles followed by ∆ exchange. ∆ ++ represent the largest contribution forbackward angles. The interferences are also shown. Their contribution a ff ects the shape of the angular distribution,some of them being negative in part of the angular region.
5. Conclusions
An model, built on e ff ective meson Lagrangian, has been build in order to reproduce the existing data for twopion production in proton-antiproton annihilation at moderate and large energies. Form factors and Regge factors areimplemented and parameters adjusted to the existing data for neutral and charged pion pair production. Coupling5 Physics Letters B 00 (2015) 1–11 (k)(i) (j) (l)(a) (b) (c) (d)(e) (f) (g) (h)(m) (n) (o) (p)(q) (r) (s) (t)(u) (v) (w) θ σ cos θ d / d c o s [ nb ] Figure 3. Angular distributions for the reaction ¯ p + p → π + π for di ff erent values of √ s : (a) √ s = .
911 GeV, (b) √ s = .
950 GeV, (c) √ s = .
975 GeV, (d) √ s = .
979 GeV, (e) √ s = .
981 GeV, (f) √ s = .
985 GeV, (g) √ s = .
990 GeV, (h) √ s = .
994 GeV, (i) √ s = .
005 GeV,(j) √ s = .
050 GeV, (k) √ s = .
095 GeV, (l) √ s = .
524 GeV, (m) √ s = .
526 GeV, (n) √ s = .
556 GeV, (o) √ s = .
591 GeV, (p) √ s = . √ s = .
613 GeV, (r) √ s = .
616 GeV, (s) √ s = .
619 GeV, (t) √ s = .
621 GeV, (u) √ s = .
686 GeV, (w) √ s = .
274 GeV. Data aretaken from Ref. [2], the curve is the prediction of the present model. ] s [GeV5 10 15 20 / d t [ nb ] σ d -1 Model -8 s =0.0125 θ cos Figure 4. s -dependence for the reaction ¯ p + p → π + π for the central region (cos θ = . s − prediction from Ref. [25, 26] (red dashed line). Data are taken from Ref. [2]. Physics Letters B 00 (2015) 1–11 -1 -0.5 0 0.5 1200040006000 -1 -0.5 0 0.5 1500010000 -1 -0.5 0 0.5 11000020000 -1 -0.5 0 0.5 1500010000 -1 -0.5 0 0.5 1200040006000 -1 -0.5 0 0.5 12000400060008000 (b)(a) (d)(c) (f)(e) cos θ d σ / d c o s θ [ nb ] Figure 5. (color online) Angular dependence for the reaction ¯ p + p → π − + π + , for di ff erent value of the total CMS energy √ s : (a) √ s = .
362 GeVfrom Ref. [5], (b) √ s = .
627 GeV from Ref. [27], (c) √ s = .
680 GeV from Ref. [4], (d) √ s = .
559 GeV from Ref. [28]. The correspondingdata from the elastic reactions π + p → π + p are also reported: (e) √ s = .
463 GeV from Ref. [29], (f) √ s = .
747 GeV from Ref. [30]. constants are fixed from the known properties of the corresponding decay channels. The agreement with a large set ofdata is satisfactory for the angular dependence as well as the energy dependence of the cross section. At large anglesthe model follows naturally the expected behavior from quark counting rules.A comparison with data from elastic π ± p → π ± p , using crossing symmetry prescriptions shows a good agreementalso at very forward and backward angles, within the uncertainty. Discussion about validity of crossing symmetry canbe found in Refs. [5, 29, 27]. The present results verify that crossing symmetry works at least at backward angles,where one diagram is dominant.This model can be extended to other binary channels, with appropriate changes of constants. The implementationto MonteCarlo simulations for predictions and optimization to coming experiments is also foreseen.
6. Acknowledgments
Thanks are due to D. Marchand and A. Dorokhov, for useful discussions and interest in this work. One of us(Yu.B) acknowledges kind hospitality at IPN Orsay, in frame of JINR-IN2P3 agreement.
7. Appendix
The relevant formulas for the amplitudes and their interferences are given below.
The amplitude for nucleon exchange is written as: M N = g π NN q t − M N ¯ v ( p )( − ˆ q t + M N ) u ( p ) . (11)where u ( p )(¯ v ( p )), are the four-component spinors of the proton(antiproton), which obey the Dirac equation. Thematrix element squared for the diagram corresponding to neutron exchange, Fig. 1a is written as: |M N | = g π NN ( q t − M N ) T r (cid:2) ( ˆ p − M N )( − ˆ q t + M N )( ˆ p + M N )( − ˆ q t + M N ) (cid:3) Physics Letters B 00 (2015) 1–11 θ cos-1 -0.5 0 0.5 1 [ nb ] θ / d c o s σ d Totaln ∆ ++ ∆ρ ∆ n ++ ∆ n ++ ∆ ∆ ρ n ρ ∆ ρ ++ ∆ θ cos-1 -0.5 0 0.5 1 [ nb ] θ / d c o s σ d Figure 6. (color online) cos θ -dependence for the reaction ¯ p + p → π − + π + for √ s = ff erent componentsare illustrated: n exchange (yellow thick short dash line) dominates at forward angle, followed by ∆ ( read thick dotted line) ∆ ++ (green thick dash-dotted line) represent the largest contribution for backward angles, ρ channel (blue thick long dash line) dominates at large angles, The interferencesare n ∆ (thin black short-dash line) , n ∆ ++ (thin red dotted line), ∆ ∆ ++ (green thin short dash-dotted line), n ρ (blue thin long-dashed dotted line) , ∆ ρ (blue thin dash-dotted line), ∆ ++ ρ (blue thin long dash line). visible at large angles. Data are taken from Ref. [4] . = − g π NN ( t − M N ) h m + ( M N − t )( M N − s − t + m π ) i , (12)with q t = k − p = p − k , q t = t . ∆ The specific ingredients for ∆ exchange, Fig. 1b, are related to the spin 3 / ∆ -resonance . For the ∆ -spin-vector, U ∆ , we take the expression from [31]: where the density matrix is P µν = U ∆ µ ( p ∆ ) ¯ U ∗ ∆ ν ( p ∆ ) = − g µν + γ µ γ ν + γ µ P ν − γ ν P µ M ∆ + P µ P ν M ∆ . (13)and: a ) i (2 π ) ˆ q t + M ∆ q t − M ∆ P µν , b ) − i (2 π ) g ∆ π N k µ (14)are the expressions for a) the ∆ propagator and b) the vertex ∆ → π N . M ∆ = . ± ∆ resonance, (i.e., the mass averaged over ∆ -multiplet), and g ∆ π N is the coupling constant for the vertex ∆ → π N .The matrix element for the diagram Fig. 1.b is: M ∆ = − g ∆ π N t − M ∆ ¯ v ( p )( ˆ q t + M ∆ ) P µν ( q t ) u ( p ) k µ k ν . (15)Squaring the amplitude one has |M ∆ | = g ∆ π N ( t − M ∆ ) k µ k ν k α k β T r h ( ˆ p − M N )( ˆ q t + M ∆ ) P µν ( q t )( ˆ p + M N ) ˜ P αβ ( q t )( ˆ q t + M ∆ ) i . (16)In order to find the value of g ∆ N π coupling constant we consider the decay width of ∆ into nucleon and pion: Γ ∆ = | ~ p | π M ∆ |M ( ∆ → N π ) | , (17)and using the experimental values of the decay width Γ ∆ = ± g ∆ N π = . ± . − . 8 Physics Letters B 00 (2015) 1–11 ∆ ++ The diagram in Fig. 1c corresponds to π − emitted at backward angle involves the exchange of ∆ ++ and can beobtained from t -exchange 1b with the replacements: t ↔ u and k ↔ k . ∆ ∆ − N interference Re [ M ∗ N M ∆ ] = Re g π NN g ∆ π N ( t − M N )( t − M ∆ ) T r h ( ˆ p + M N )( − ˆ q t + M N )( ˆ p + M N ) ˜ P µν ( q t )( ˆ q u + M ∆ ) i k µ k ν , (18)with q u = k − p = p − k , q u = u . ∆ ++ − N interference Re [ M ∗ N M ++∆ ] = Re g π NN g ∆ π N ( u − M ∆ )( t − M N ) T r h ( ˆ p + M N )( − ˆ q t + M N )( ˆ p + M N ) ˜ P µν ( q u )( ˆ q u + M ∆ ) i k ν k µ . (19) ∆ ++ − ∆ interference Re [ M ∗ ∆ M ++∆ ] = Re g ∆ π N ( t − M ∆ )( u − M ∆ ) T r h ( ˆ p − M N )( ˆ q t + M N ) P µν ( q t )( ˆ p + M N ) ˜ P αβ ( q u )( ˆ q u + M ∆ ) i k µ k ν k α k β . (20) ρ meson The largest contribution to meson exchange in s-channel, Fig 1d, is given by the ρ -meson, with ∼ ρ - propagator and b) the ρππ vertex we take a ) − i (2 π ) g µν − ( q µ s q ν s ) / m ρ q s − m ρ + i p q s Γ ρ ( q s ) , b ) − ig ρππ ( k − k ) ν (2 π ) , q s = p + p = k + k , (21)where g µν is the symmetric tensor, and q s = s . The matrix element is written as: M ρ = g ρ pp g ρππ [ s − m ρ + i √ s Γ ρ ( s )] [¯ v ( p ) Γ µ ( q ) u ( p )]( k − k ) ν g µν − q µ q ν m ρ , (22)Squaring the amplitude one gets: |M ρ | = g ρ NN g ρππ [ s − m ρ + i √ s Γ ρ ( s )] ( k − k ) ν ( k − k ) β g µν − ( q s ) µ ( q s ) ν m ρ g αβ − ( q s ) α ( q s ) β m ρ T r (cid:2) ( ˆ p − M N ) Γ µ ( q s )( ˆ p + M N ) Γ α ( q s ) (cid:3) . (23)The coupling constant g ρ → ππ is found from the the experimental value of the total width Γ for the decay ρ → ππ : Γ ( ρ ) = ± ≈ Γ = g ρππ π m ρ ( m ρ − m π ) / , (24)where we added a factor 4 / ρ meson and fourpossible charged decays. Inverting Eq. (24), using the experimental value for the decay width one can get thefollowing value of the coupling constant: g ρππ = . ± . Physics Letters B 00 (2015) 1–11 ρ − ρ interference Re [ M ∗ N M ρ ] = Re g π NN g ρππ g ρ NN [ s − m ρ + i √ s Γ ρ ( s )]( t − M N ) T r h ( ˆ p − M N ) Γ µ ( q s )( ˆ p + M N ) ˜ P αβ ( q t )( − ˆ q t + M N ) i k α k β ( k − k ) ν g µν − q µ q ν m ρ . (25) ∆ − ρ interference Re [ M ∆ m ∗ ρ ] = Re g ρ NN g ρππ g ∆ ρ N [ s − m ρ + i √ s Γ ρ ( s )]( t − M ∆ ) T r h ( ˆ p − M N ) Γ µ ( q s )( ˆ p + M N ) ˜ P αβ ( q t )( − ˆ q t + M ∆ ) i k α k β ( k − k ) ν g µν − q µ q ν m ρ . 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