FASTERD: a Monte Carlo event generator for the study of final state radiation in the process e+e ππγ at DAΦNE − → 9 O. Shekhovtsovaa,b,1, G. Venanzonia, G. Pancheria 0 0 2 aINFN Laboratori Nazionale di Frascati, Frascati (RM) 00044, Italy n bNSC KIPT, Kharkov 61202, Ukraine a J 8 2 Abstract ] h p FASTERD is a Monte Carlo event generator to study the final state radiation - p both in the e+e− π+π−γ and e+e− π0π0γ processes in the energy region e → → of the φ-factory DAΦNE. Differential spectra that include both initial and final h [ state radiation and the interference between them are produced. Three different mechanisms for theππγ final state are considered: Bremsstrahlungprocess (both in 1 v the framework of sQED and Resonance Perturbation Theory), the φ direct decay 0 (e+e− φ (f ;f + σ)γ ππγ) and the double resonance mechanism (as 4 → → 0 0 → e+e− φ ρ±π∓ π+π−γ and e+e− ρ ωπ0 π0π0γ). Additional models 4 → → → → → → 4 can be incorporated as well. . 1 PACS: 13.25.Jx; 12.39.Fe; 13.40.Gp. 0 9 0 : Key words: Quantum electrodynamics (QED), e+e−-annihilation, hadronic cross v section, radiative corrections, low energy photon-pion interaction model i X r a PROGRAM SUMMARY Program Title: FASTERD Authors: G. Pancheri, O. Shekhovtsova, G. Venanzoni Journal Reference: Catalogue identifier: Licensing provisions: none Programming language: FORTRAN77 Computer: any computer with FORTRAN77 compiler Operating system: UNIX, LINUX, MAC OSX 1 Corresponding author Preprint submitted to Elsevier 28 January 2009 Keywords:Quantumelectrodynamics(QED),e+e−-annihilation,hadroniccrosssec- tion, radiative corrections, low energy photon-pion interaction model. PACS: 13.25.Jx; 12.39.Fe; 13.40.Gp. Classification: 11.1 External routines/libraries: MATHLIB, PACKLIB from CERN library Nature of problem: General parameterization of the γ∗ ππγ process; test mod- → els describing Bremsstrahlung process, the φ direct decay, double vector resonance mechanism. Solution method: Numerical integration of analytical formulae Restrictions: Only one photon emission is considered Running time 28 sec with standard input card (1e6 events generated) on a Intel Core 2 Duo 2. GHz with 1 GB RAM. 2 1 Introduction The anomalous magnetic moment of the muon (a ) is one of the most precise µ test of the Standard Model [1]. Theoretical predictions differ from the exper- imental result for more than 3σ [2]. The main source of uncertainty in the theoretical prediction comes from the hadronic contribution, a(had) [2]. This µ contribution cannot be reliably calculated in the framework of perturbative QCD (pQCD), because low-energy region dominates, but it can be estimated by dispersion relation using the experimental cross sections of e+e− annihila- tion into hadrons as an input [3]. About 70% of the hadronic part of the muon anomalous magnetic moment, a(had), comes from the energy region below 1 µ GeV and, due to the presence of the ρ-meson, the main contribution to a(had) µ is related with the π+π− final state. Experimentally, the energy region from threshold to the collider beam en- ergy is explored at the Φ-factory DAΦNE, PEP II and KEKB at Υ(4S)- − resonance using the method of radiative return (for a review see [4] and refer- ences therein). This method relies on the factorization of the radiative cross sectionintotheproductofthehadroniccrosssectiontimesaradiationfunction H(q2,θ ,θ ) known from Quantum Electrodynamics (QED) [5,6,7,8,9]. max min For two pions final state it means that, in the presence of only the initial state radiation (by leptons, ISR), the radiative cross section σππγ corresponding to the process e+(p )+e−(p ) π+(p )+π−(p )+γ(k), (1) + − 1 2 → can be written as dσππγ = dσππ(q2)H(q2,θ ,θ ) [4,7,9], where θ and max min min θ are the minimal and maximal azimuthal angles of the radiated photon, max q = p + p and the hadronic cross section σππ is taken at a reduced CM 1 2 energy. The final state (FS) radiation (FSR) is an irreducible background in radiative return measurements of the hadronic cross section [7,10] and spoils the factorization of the cross section. In any experimental setup the process of FSR cannot be excluded from the analysis. The KLOE experiment has devel- oped two different analysis strategies: the first one is with the photon emitted at small angle (θ < 15◦) and the other one is for the photon reconstructed at γ large angle (60◦ < θ < 120◦), being for both 50◦ < θ < 130◦. In the case of γ π the smallangle kinematics theFSR contribution canbe safely neglected, while for the large angle analysis it becomes relevant (upto 40% of ISR). The large angle analysis allows to scan the pion form factor down to the threshold [11]. Radiative corrections (RC’s) related to initial state radiation, i.e. the function H, can be safely computed in QED. For the FSR process the situation is dif- ferent. In the region below 2 GeV the pQCD is not applicable to describe FSR and calculation of the cross section relies on the low energy pion–photon inter- 3 e+(−p+) γ(k) π−(p2) π− π−(π0) + γ∗(q) + γ∗(Q) γ(k) e−(p−(cid:1)) π+(p1) (cid:2)π+ (cid:1)π+(π0) (a) (b) Fig. 1. actionmodel. Thus the measured FSRcross section gives anunique possibility to get very interesting information on the dynamics of interacting mesons and photons, to test the pion–photon interaction models and extract their param- eters [12]. In the case of neutral pions in the final states e+(p )+e−(p ) π0(p )+π0(p )+γ(k), (2) + − 1 2 → the ISR contribution is absent2, and the cross section is determined solely by the FSR mechanism. This process, together with asymmetries [13] and cross section in the charged channel, allows to extract information on pion-photon interaction and test effective models for FSR. For realistic experimental cuts on the angle and energy of the final particles the cross section cannot be evaluated analytically and one has to use Monte Carlo(MC)event generator.ThefirstMCdescribing thereaction(1)wasEVA [7]. EVA simulates both ISR and FSR processes for non zero angle emitted photon (θ > θ ).For FSR the sQED modelwas chosen. Afterwards the MC γ min PHOKHARAwas written to include different charged finalstate andradiative corrections to ISR [14]. The contribution of the φ–meson direct decay, that is relevant at the DAΦNE energy, was added, firstly in EVA [15] and then in PHOKHARA [13]. ThecomputercodeFASTERD3 presentedinthispaperisaMonteCarloevent generator written in FORTRAN that simulates both processes (1) and (2), where the hard photon γ(k) can be emitted by the leptons4 and/or the pions Fig. 1. This program was inspired by EVA. The present version of FASTERD includes different mechanisms for the ππγ production: final Bremsstrahlung5 in the framework of both Resonance Perturbation Theory and sQED, the con- 2 This statement is valid only if one neglects multi-photon emission 3 FinAL STatE Radiation at DAΦNE 4 Only for the charged channel 5 Only for the charged channel 4 tributions related to the φ intermediate state and the double vector resonance part (for details see Section 3). Up to now only the case of one photon emis- sion has been considered. The code contains two types of source files: (1) the main program fasterd.f, where the calculations are done, and (2) the input file cards fasterd.dat which defines the parameters for the generation. Both these files will be described in the following. As output of the program, the cross section of the process is evaluated and a PAW/HBOOK ntuple is pro- duced with the 4-momenta for each event of the outgoing particles. For the convenience of the user we have added a Makefile and the test output files for the Journal library. This paper is organized as follows. In Section 2 the main formulae for ISR and interference between ISR and FSR are presented. In Section 3 we give a general description of FSR process and present the FSR models that are included in our program. In Section 4 the spectra for different FSR models are compared with analytical results and with MC PHOKHARA. Also a pos- sible generalization applicable to a wider region is described in Section 4. In Section 5, we summarize the general procedure for calculating the spectrum. In Appendices A, B, C a short description of the input and output files is presented. 2 Initial state radiation models and pion form factor The cross section of the processes (1) and (2) can be written as 1 d3p d3p d3k dσ= C δ4(Q p p k) 1 2 M 2 2s(2π)5 12 − 1 − 2 − 8E E ω | | Z + − α3(s q2) 4m2 =C N M 2dq2dΩγdΩπ+, N = − 1 π (3) 12 | | 64π2s2 s − q2 ! where α is the fine structure constant, m is the pion mass, ω is the photon π energy, Q = p +p , s = Q2 and the invariant amplitude squared, averaged + − over initial lepton polarizations and summed over the photon polarizations6 is M 2 = M 2 + M 2 +2Re(M M∗ ). (4) | | | ISR| | FSR| ISR FSR M(ISR) (M(FSR)) corresponds to the ISR(FSR) production amplitude. The factor C = 1 for π0π0 in the final state and C = 1 for π+π−. 12 2 12 6 We use ǫ∗ǫ = g polar. ρ σ − ρσ P 5 For ISR process the invariant amplitude squared, averaged over initial lepton polarizations and summed over the photon polarizations, is 4 M(ISR) 2 = F (q2) 2R, (5) | | −q2| π | m2 χ2 +χ2 χ (q2 t ) χ (q2 t ) R = πF + 1 2 − 1 − 2 − 2 − 1 q2 t t 1 2 2m2χ χ 2m2χ χ (q2 t )2 +(q2 t )2 e 1 1 1 e 2 2 1 , F = − 1 − 2 , − t22 q2 − !− t21 q2 − ! t1t2 where m is the electron mass, χ 2p p , t = 2p k, t = 2p k. e 1,2 −,+ 2 1 − 2 + ≡ · − − The nonpoint-like behaviour of pions is determined by the form-factor (FF) F (q2) that is the function of the pion mass squared q2. In the case of the π neutral channel M(ISR) = 0. Four different parameterizations for the pion FF are considered: Ku¨hn-Santamaria (KS) [16], Gounaris-Sakurai (GS) [17], the RPT parametrization and an ”improved” version of Ku¨hn-Santamaria (see below) [18]. 2.1 Ku¨hn-Santamaria and Gounaris-Sakurai pion FF Based on the results of Ref. [16], the pion FF describing the ρ ω mixing and − the first excited ρ resonance (ρ′), can be written as F (q2) = Bρ1+1+αBαω +βBρ′, (6) π 1+β where for the KS parametrization [16] m2 BKS(q2) = r (7) r m2 q2 i√q2Γ (q2) r − − r and for the GS one [17] m2 +H(0) BGS(q2) = r , (8) r m2 q2 +H(q2) i√q2Γ (q2) r − − r with m2Γ dh H(q2)= ρ ρ p2(q2)(h(q2) h(m2))+(m2 q2)p2(m2) , p3(m2)" π − ρ ρ − π ρ dq2|q2=m2ρ# π ρ 6 2 p (q2) √q2 +2p (q2) 1 h(q2)= π ln π , p (q2) = q2 m2. π √q2 2m π 2 − π π q The energy dependence for the ρ mesons is taken in the form m2 p (q2) 3 Γ (q2) = Γ ρ π Θ(q2 4m2). (9) ρ ρ q2 pπ(m2ρ)! · − π Fortheω resonanceasimpleBreit-Wignerresonanceformwithconstantwidth was used for both parameterizations. As one can see the single resonance contribution is normalized to unity at q2 = 0forbothparameterizations(B (0) = 1)whereas therightnormalization r for the pion FF, F (0) = 1, is realized by the corresponding choice of the π parameters α, β. All model parameters (the mass and width of the resonances as well as the parameter α, β) are determined in the input file cards fasterd.dat. The KS pion FF parametrization corresponds to the function F pi ks whereas the GS one to F pi gs. 2.2 RPT parametrization for the pion FF The Resonance Perturbation Theory is based on Chiral Perturbation The- ory (χPT) with the explicit inclusion of the vector and axial–vector mesons, ρ (770)anda (1260).Whereas χPTgives correctpredictions forthepionform 0 1 factor at very low energy, RPT is the appropriate framework to describe the pion form factor at intermediate energies (E m ) [19]. According to the ρ ∼ RPT model the pion form factor, that describes the ρ ω mixing, can be − written as: F G q2 Π F (q2) = 1+ V V BKS(q2) 1 ρω BKS(q2) , (10) π f2 m2 ρ − 3m2 ω ! π ρ ω where q2 isthe virtuality ofthephoton,f = 92.4MeV andthe parameter Π π ρω describes the ρ-ω mixing. As before a constant width is used for the ω–meson. We also assume that the parameter Π is a constant and is related to the ρω branching fraction Br(ω π+π−): → Π 2 Br(ω π+π−) = | ρω| . (11) → Γ Γ m2 ρ ω ρ 7 As was mentioned above, the value of F and G , as well as the mass of V V the ρ and ω mesons (m and m , correspondingly), the parameter of the ρ-ω ρ ω mixing Π and the width of the ω meson are determined in the input file ρω cards fasterd.dat. The RPT parametrization corresponds to the function F pi rpt. Inclusion of the ρ′ meson modifies the form factor (10) as F G q2 Π F (q2)=1+ V V BKS(q2) 1 ρω BKS(q2) (12) π f2 m2 ρ − 3m2 ω ! π ρ ω + FV′GV′ q2 BKS(q2). f2 m′2 ρ′ π ρ 2.3 ”Improved” Ku¨hn-Santamaria InRef.[20]thepionFFwasestimatedintheframeworkofthedual-QCD Nc→∞ model. However this model assumes the resonances to be of the zero width. The authors of Ref. [18] included the final width of the ρ0 and the three lowest excited ρ′ states and obtained the following expression for the pion FF 3 m2 F (q2) = c BKS(q2)+ c n . (13) π n r nm2 s nX=0 nX>4 n − ThisparametrizationforthepionFFiscontainedinthefunctionF pi ks new. The numerical value for the parameters c and m (the mass of the excited ρ n n mesons) is taken from Ref. [18]. 3 Final state radiation models Using the underlying symmetry, like gauge invariance, charge-conjugation symmetry of the final particles and the photon crossing symmetry, it is pos- sible to write the FS tensor M(µν), that describes the γ∗ π+π−γ vertex, in F → terms of three gauge invariant tensors (see [21] and Ref. [23,24] therein): Mµν(Q,k,l) = τµνf +τµνf +τµνf , (14) F 1 1 2 2 3 3 τµν = kµQν gµνk Q, l = p p , 1 − · 1 − 2 τµν = k l(lµQν gµνk l)+lν(kµk l lµk Q), 2 · − · · − · τµν = s(gµνk l kµlν)+Qµ(lνk Q Qνk l). 3 · − · − · 8 The model dependence comes in only via the implicit form of the scalar func- tions f (we will call them structure functions). i Thus the FSR and interference (ISR FSR) part to the invariant amplitude ∗ squared (4) is 1 M 2= a f 2 +2a Re(f f∗)+a f 2 | FSR| s2 11| 1| 12 1 2 22| 2| (cid:20) +2a Re(f f∗)+a f 2 +2a Re(f f∗) , (15) 23 2 3 33| 3| 13 1 3 (cid:21) and 1 Re(M M∗ )= A Re(F (q2)f∗)+A Re(F (q2)f∗) (16) ISR FSR −4sq2 1 π 1 2 π 2 (cid:20) +A Re(F (q2)f∗) . 3 π 3 (cid:21) The form for the coefficients a and A can be found in Ref. [21] Eqs. (17), ik i (26), correspondingly. Here is the list of the FSR production mechanisms that are included in FAS- TERD: e+ +e− π+ +π− +γ Bremsstrahlung process (17) → e+ +e− φ (f ;f +σ)γ π +π +γ φ direct decay (18) 0 0 → → → e+ +e− (φ;ω′) ρπ π +π +γ Double resonance process (19) → → → e+ +e− (ρ/ρ′) ωπ0 π0 +π0 +γ Double resonance process(20) → → → Thus the total contribution to the functions f introduced in (14) is i (Brem) (φ) (vect) f = f +f +f (21) i i i i In the next section we present the models describing these processes. 3.1 Final state Bremsstrahlung UsuallythecombinedsQED VMDmodelisassumedfortheFSBremsstrahlung ∗ process [7,14]. In this case the pions are treated as point-like particles (the sQED model) and the total FSR amplitude is multiplied by the pion form fac- torF (s),thatisestimatedintheVMDmodel.UnfortunatelythesQED VMD π ∗ 9 model is an approximation that is valid for relatively soft photons and it can fail for high energy photons, i.e near the π+π− threshold. In this energy region the contributions to FSR, beyond the sQED VMD model, can be important. ∗ As mentioned in Section 2, RPT is supposed to be an appropriate model to describe the pion-photon interaction in the region about and below 1 GeV and this model is used to estimate the contributions beyond sQED VMD. ∗ (Brem) Using the sQED VMD model the structure functions f (see Eq.(21)) ∗ i are 2k QF (s) 2F (s) fsQED= · π , fsQED = − π , (22) 1 (k Q)2 (k l)2 2 (k Q)2 (k l)2 · − · · − · fsQED=0. (23) 3 In the framework of RPT the result is f(Brem) = fsQED +∆fRPT, (24) i i i where F2 2F G 1 1 ∆fRPT = V − V V + 1 f2 m2 m2 s i√sΓ (s) π (cid:18) ρ ρ − − ρ (cid:19) F2 (k l)2 (s+k Q)[4m2 (s+l2 +2k Q)] A 2+ · + · a − · , (25) −f2m2 D(l)D( l) 8D(l)D( l) π a(cid:20) − − (cid:21) F2 4m2 (s+l2 +2k Q) ∆fRPT = A a − · , (26) 2 −f2m2 8D(l)D( l) π a − F2 k l ∆fRPT = A · ,D(l) = m2 (s+l2 +2kPQ+4kl)/4.(27) 3 f2m2 2D(l)D( l) a − π a − For notations and details of the calculation we refer the reader to [21]. The functions ∆fRPT are calculated by the subroutine chpt rpt whereas fsQED i i are evaluated in the functions fsr1 sqed, fsr2 sqed, fsr3 sqed. We would like to mention here that the contribution of any model describing Bremsstrahlung FS process can be conveniently rewritten as in Eq. (24) and in the soft photon limit the results should coincide with the prediction of the sQED VMD model. ∗ 10