TWOPEG-D: An Extension of TWOPEG for the Case of a Moving Proton Target
CCLAS12 Note 2017-014
TWOPEG-D: An Extension of TWOPEG for the Case ofa Moving Proton Target
Iu. Skorodumina ,a , G.V. Fedotov , ,b , R.W. Gothe Department of Physics and Astronomy, University of South Carolina, Columbia, SC 29208. Ohio University, Athens, Ohio 45701 Skobel’tsyn Institute of Nuclear Physics, Moscow State University, Moscow, 119991, RussiaE-mail: a [email protected], b [email protected] Abstract
A new TWOPEG-D version of the event generator TWOPEG was developed. This newversion simulates the quasi-free process of double-pion electroproduction off the proton thatmoves in the deuteron target. The underlying idea is the equivalence of the moving protonexperiment performed with fixed laboratory beam energy to the proton at rest experimentconducted with effective beam energy different from the laboratory one. This effective beamenergy differs event by event and is determined by the boost from the Lab system to theproton rest frame. The Fermi momentum of the target proton is generated according tothe Bonn potential. The specific aspects of the deuteron target data analysis are discussed.The plots that illustrate the performance of TWOPEG-D are given. The link to the code isprovided. The generator was tested in the analysis of the CLAS data on electron scatteringoff the deuteron target. a r X i v : . [ phy s i c s . d a t a - a n ] D ec LAS12 Note 2017-014
Contents Q versus W distribution boundaries . . . . . . . . . . . . . 12 LAS12 Note 2017-014
Chapter 1Introduction
During the last decades great efforts have been performed in laboratories all over the worldin order to experimentally investigate exclusive reactions of meson electroproduction off theproton. This investigation is typically carried out by detailed analyses of the experimentaldata with the final goal of extracting various observables.By now exclusive reactions off the free proton have been studied in quite detail, anda lot of information about different observables for various exclusive channels have beenaccumulated [1]. Meanwhile the exclusive reactions off the deuteron, being less investigated,start to attract more and more scientific attention, thus causing a strong demand to developeffective tools for their analysis. For this purpose a reliable Monte-Carlo simulation of theprocess of meson electroproduction off the deuteron target should be elaborated.This note presents the successful attempt to simulate the quasi-free process of double-pion electroproduction off the proton that moves in the deuteron. The note introducesthe TWOPEG-D event generator, which is an extension of the TWOPEG that is the eventgenerator for double-pion electroproduction off the free proton [2]. In TWOPEG-D the Fermimomentum of the target proton is generated according to the Bonn potential [3] and thennaturally merged into the specific kinematics of double-pion electroproduction.The basic idea that underlies TWOPEG-D consists in the equivalence of the movingproton experiment performed with fixed laboratory beam energy to the proton at rest exper-iment conducted with effective beam energy different from the laboratory one. This effectivebeam energy differs event by event and is determined by the boost from the Lab system tothe proton rest frame and hence depends on the Fermi momentum of the target proton.TWOPEG-D does not simulate effects of final state interactions (FSI) due to their com-plexity and not fully understood nature, thus claiming only the ability to imitate the quasi-free process of double-pion electroproduction off moving protons. Beside that, other effectsthat are intrinsic to experiments on the bound nucleon (such as the off-shellness of the targetnucleon, possible modifications of the reaction amplitudes in the nuclear medium, etc.) areignored in TWOPEG-D due to their minor significance.The note is organized in the following way. Section 2 describes the specific featuresof a deuteron target experiment, which originate from the fact that the target proton is4
LAS12 Note 2017-014 in motion and cause difficulties during the data analysis. This section also outlines somemethods for overcoming these difficulties and demonstrates the essential need for a properMonte-Carlo simulation of the reaction under investigation. Section 3 gives the details of theevent generation process and describes the multi-stage procedure of calculating the momentaof the final particles in the Lab frame. The specifity of obtaining the cross section weightis given in Sect. 4, while the details of managing with the simulation of the radiative effectsare presented in Sect. 5. The final Section 6 contains the link to the repository, where theTWOPEG-D code is located.It needs to be mentioned that here the reaction is assumed to occur off the proton thatmoves in the deuteron, but the whole procedure can also be used for any type of the nucleonmotion. For instance, if a nucleon moves inside any nucleus other than deuteron, the Bonnpotential should be changed to an appropriate potential of the nucleon-nucleon interaction.Beside that, the procedure can be simply generalized for any exclusive channel.It also should be emphasized that TWOPEG-D was especially developed to be used inthe analyses of data, where the experimental information of the target proton momentum isinaccessible, and one is forced to work under the target-at-rest assumption. If the qualityof the experimental data allows to avoid the target-at-rest assumption, it is appropriate tostart with the conventional free proton TWOPEG for the Monte-Carlo simulation.The user is strongly encouraged to read firstly the note with the detailed descriptionof TWOPEG [2], which sketches the kinematics of double-pion electroproduction off theproton, describes the method of event generation with weights, illustrates the quality of thedata description, provides details on simulating the radiative effects, etc. This particularnote should be treated as an addendum to the report [2], since it is fully devoted to thesimulation of the effects related to the target motion and no material from the report [2] istherefore repeated here. 5
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Chapter 2Specifity of the data analysis off amoving proton
During the deuteron target data analysis one encounters specific issues that are completelyexternal to the free proton data analysis. Those of them that originate solely from the factof initial proton motion are sketched below.
For the process of double-pion electroproduction off the proton (as for any other exclusiveprocess) the reaction invariant mass can in general be determined in two ways, i.e. eitherfrom the initial particle four-momenta ( W i ) or from the final particle four-momenta ( W f )as Eqs. (2.1) and (2.2) demonstrate . W i = (cid:113) ( P p + P γ v ) (2.1) W f = (cid:113) ( P π + + P π − + P p (cid:48) ) (2.2)Here P π + , P π − , and P p (cid:48) are the four-momenta of the final state hadrons, P p is the four-momentum of the initial proton and P γ v = P e − P e (cid:48) the four-momentum of the virtual photonwith P e and P e (cid:48) the four-momenta of the incoming and scattered electrons, respectively.To determine W f , all final hadrons should be registered, while for the calculation of W i it is sufficient to register the scattered electron. In the analyses of exclusive reactions, thelatter opportunity allows to use event samples with one unregistered final hadron, whosefour-momentum is reconstructed via the missing mass technique. This approach allows toincrease the analyzed statistics (sometimes significantly). Although the scattered electron is treated as a final particle, here it is classified as “initial”, since itdefines the virtual photon, which in turn is attributed to the initial state. In electron scattering experiments W i is distorted by the radiative effects, which electrons undergo. Thedetector resolution also contributes to the difference between experimental values of W i and W f . LAS12 Note 2017-014
The situation complicates for reactions that happen off the proton that moves as in thedeuteron. The motion of the target proton is concealed from the observer and usually is notmeasured. If all particles in the final state are registered, one can restore the informationabout the momentum of the target proton via the energy-momentum conservation , howeverif one of the final hadrons is not registered this information turned out to be totally lost.Therefore the value of W i given by Eq. (2.1) turns out to be ill-defined, since P p is not known.This brings us to the choice to either demand the registration of all final hadrons to determine W f , which reduces the flexibility of the analysis, or to work under a so-called “target-at-restassumption”, which assumes the initial proton to be at rest. In the last approach the valueof W i appears to be smeared.As a consequence of this smearing, all extracted observables, which depend on the valueof W , turned out to be convoluted with a function that is determined by the Fermi motion ofthe initial proton [4]. To retrieve the non-smeared observables, a correction that unfolds thiseffect should be applied. In order to develop this correction, one needs to simulate properlythe investigated exclusive process off the moving proton.The simulation of W -smearing in TWOPEG-D is described in Sect. 3. In order to pick out the exclusive reaction, it is a common practice to perform a so-called“exclusivity cut” as a final step of the event selection. This is a cut on the missing mass, whichis calculated via the energy-momentum conservation from the four-momenta of registeredparticles and reflects the mass spectrum of the unregistered part. For example, for thereaction ep → e (cid:48) p (cid:48) π + X , where the scattered electron and final p and π + are registered, themissing mass squared of the unregistered part X is determined in the following way, M X [ π − ] = [ P e + P p − P e (cid:48) − P p (cid:48) − P π + ] . (2.3)The investigation of the distribution of the quantity M X [ π − ] allows to judge the admixtureof any types of background as well as the reliability of the entire event selection. A properlychosen position of the exclusivity cut allows at least to suppress the background contributionor even eliminate it completely and to get rid of the non-physical events.The missing mass is generally subject to the smearing due to the detector resolution.However, if the target proton moves as in the deuteron, the missing mass is also subject toFermi smearing due to the inevitability to work under the target-at-rest assumption thatoriginates from the incomplete knowledge about the target motion as well as the final hadronstate.If the data analysis includes the estimation of the detector efficiency (for example withthe goal to extract a cross section), then the exclusivity cut should also be applied to thereconstructed Monte-Carlo events. In order to calculate the efficiency correctly, the Monte- In general the target proton momentum can also be reconstructed by measuring the spectator nucleon. LAS12 Note 2017-014
Carlo distributions should match the experimental ones as well as possible. Fermi smearing ofthe experimental distributions demands that the simulation should reproduce it. Therefore,the effects of the Fermi motion should be properly included into the Monte-Carlo simulation .Section 3 describes the method that is used in TWOPEG-D for the simulation of theparticle four-momenta and gives examples of smeared missing mass distributions. For universality purposes the observables are usually extracted in the center-of-mass system(CMS). This implies the transformation of the four-momenta of all particles from the labo-ratory system (Lab) to the CMS and the subsequent calculation of all kinematical variablesfrom these transformed four-momenta. The description of the kinematical variables for thereaction of double-pion electroproduction off the free proton is given in [2, 5, 6].The CMS system is uniquely defined as the system, where the initial proton and thephoton move towards each other with the Z CMS -axis pointing along the photon direction andthe net momentum equal to zero. However, the procedure of the Lab to CMS transformationdiffers depending on the specifity of the reaction’s initial state.Figure 2.1 illustrates three options for the experimental specification of the initial state: • The reaction off the free proton induced by the real photons (upper illustration inFig. 2.1). In this case the CMS axis orientation is the same for all reaction eventsand coincides with that in the Lab system. To transform from the Lab to the CMS, itis sufficient to just perform the boost along the Z -axis. • The reaction off the free proton induced by the virtual photons (left bottom illustrationin Fig. 2.1). In this case the CMS axis orientation is different for each reaction eventand is specified by the direction of the scattered electron. To transform from the Labto the CMS, one needs to perform two rotations to situate the X -axis in the electronscattering plane and to align the Z -axis with the virtual photon direction. Then theboost along the Z -axis can be performed. The analysis report [6] gives the detaileddescription of the Lab to CMS transformation for this case. • The reaction off the moving proton induced by the virtual photons (right bottom illus-tration in Fig. 2.1). In this case the CMS axis orientation is again different for eachreaction event and is specified by both the scattered electron and the target protondirections. To transform from the Lab to the CMS, one needs to perform the transitionto the auxiliary system first, where the proton is at rest, while the incoming electron In addition to Fermi smearing, the experimental missing mass distributions are subject to distortionsdue to FSI effects [4], which can hardly be simulated. This fact increases the importance of the reliablesimulation of Fermi smearing for the proper dealing with the FSI contributions during the data analysis. The fourth option of the reaction off the moving proton induced by the real photons is also possible. LAS12 Note 2017-014 Photoproduction off the free protonElectroproduction off the free proton Electroproduction off the moving proton
CMS
Figure 2.1:
The illustration of three options for the experimental specification of the initial state. moves along the Z -axis. This transition is determined by the momentum of the targetproton. Then the standard procedure described in the previous step can be applied.Therefore, the need to transform properly from the Lab to the CMS brings us again tothe necessity to be aware of the initial proton momentum for each reaction event. If theexperiment neither provides the registration of spectator nucleons nor the registration ofall final state particles, the correct transformation can not be performed and the extractedobservables will lack accuracy. This systematic effect should either be estimated or correctedfor. For this purpose the proper simulation of the investigated reaction off the moving protonshould be developed. Electron scattering off the moving proton performed with the beam energy E beam is equiv-alent to that off the proton at rest conducted with the effective beam energy (cid:101) E beam . Thiseffective beam energy is determined by the boost from the Lab system to the proton restframe and thus depends on the Fermi momentum of the target proton and differs event by9 LAS12 Note 2017-014 event. Therefore, the experiment off the moving proton with the fixed electron beam energycorresponds to that off the proton at rest performed with the altered beam energy.The virtual photoproduction cross section σ v , being decomposed into the combinationof the structure functions, has a specific dependence on the beam energy – the structurefunctions themselves do not depend on the beam energy, while the dependence is explicitlyfactorized by the coefficients in front of them . These coefficients incorporate the informationabout the virtual photon polarization – they are expressed via the quantities ε T , ε L or theircombinations, where ε T , ε L are the degrees of transverse and longitudinal polarization of thevirtual photon, respectively.The quantities ε T and ε L can be determined according to the following relations , ε T = (cid:32) Q · |−→ P γ v | · [ −→ P e × −→ P e (cid:48) ] (cid:33) − and (2.4) ε L = Q ν ε T , (2.5)where −→ P γ v and ν are the three-momentum and energy of the virtual photon, respectively,while −→ P e and −→ P e (cid:48) are the three-momenta of the incoming and scattered electrons, respec-tively.Eq. (2.4) gives the general formula for the transverse virtual photon polarization [7].In the particular case, when the incoming electron moves along the Z -axis, this formula isreduced to the well-known expression given by Eq. (2.3) of the report [2], which in turncan be rewritten in the following way to demonstrate the dependence on the beam energyexplicitly, ε T = 11 + Q + ν )4 E beam ( E beam − ν ) − Q , (2.6)where the energy of the virtual photon ν is fixed for a given W and Q .Figure 2.2 illustrates the dependence of ε T on the beam energy given by Eq. (2.6). Theupper bunch of the solid curves corresponds to the fixed Q = 0 . , while the lowerbunch of the dashed curves stands for Q = 1 GeV . Different colors indicate different fixedvalues of W . The higher the beam energy is, the closer the curves are to unity and to eachother. For the case of the unpolarized electron beam and the π + π − p final state this decomposition is given byEq. (2.5) of the report [2]. ε T and ε L given by Eqs. (2.4) and (2.5) are invariant under the coordinate axis transformation, but notinvariant under the Lorentz boost. LAS12 Note 2017-014 (GeV) beam E T e = 0.3 GeV Q W = 1.40 GeV W = 1.45 GeVW = 1.50 GeV = 1 GeV Q W = 1.40 GeV W = 1.45 GeVW = 1.50 GeV
Figure 2.2:
The dependence of ε T on the beam energy given by Eq. (2.6) for the case, whenthe incoming electron moves along the Z -axis in the proton rest frame. The upper bunch of thesolid curves corresponds to Q = 0 . , while the lower bunch of the dashed curves stands for Q = 1 GeV . Different colors indicate different fixed values of W : 1.4 GeV (black), 1.4 GeV (blue),and 1.5 GeV (green). The red line shows the position of unity. (GeV) beam E ) - ( G e V v G -3 · = 0.25 GeV Q W = 1.4 GeV W = 1.5 GeVW = 1.6 GeV = 0.3 GeV Q W = 1.4 GeV W = 1.5 GeVW = 1.6 GeV
Figure 2.3:
The dependence of Γ v on the beam energy for the case, when the incoming electronmoves along the Z -axis in the proton rest frame. The upper bunch of the solid curves correspondsto Q = 0 .
25 GeV , while the lower bunch of the dashed curves stands for Q = 0 . . Differentcolors indicate different fixed values of W : 1.4 GeV (black), 1.5 GeV (blue), and 1.6 GeV (green). LAS12 Note 2017-014
On top of that, the electroproduction cross section is connected to the virtual photopro-duction one via the virtual photon flux Γ v , which is also beam energy dependent, as Eq. (2.2)of the report [2] demonstrates . Figure 2.3 illustrates the dependence of the virtual photonflux on the beam energy. The upper bunch of the solid curves again corresponds to the fixed Q = 0 .
25 GeV and the lower bunch of the dashed curves to Q = 0 . . Differentcolors indicate different fixed values of W .In the proton at rest experiments the conventional practice is to determine ε T , ε L , and Γ v in the Lab frame. For the consistency, in the experiments off moving proton these quantitiesshould be defined in the proton rest frame, where the incoming electron has the altered effec-tive beam energy (cid:101) E beam . This circumstance convolutes the extracted cross section with thedependencies of the quantities ε T , ε L , and Γ v on the beam energy, hence further complicatingthe interpretation of the result and its comparison with the cross section of the proton atrest experiment. Although this systematic effect seems not to be significant, it neverthelessshould be estimated or corrected for. This can be performed using the proper Monte-Carlosimulation of the reaction under investigation.Section 3 describes the calculation of the effective beam energy in TWOPEG-D, whileSection 4 estimates the influence of the beam energy alteration on the cross section. Q versus W distribution bound-aries In electron scattering experiments the fixed beam energy imposes kinematical limits on themaximal achievable values of W and Q . The kinematical limitations are usually morestrongly restricted by the experimental conditions. One of the experimental restrictionscomes from the geometrical limitations of the polar angle of the scattered electron. Theboundary of the Q versus W distribution is then determined by Q = 2 E beam sin θ e (cid:48) (cid:0) E beam m p − W + m p (cid:1) m p + 2 E beam sin θ e (cid:48) , (2.7)where m p is the proton mass and θ e (cid:48) is the polar angle of the scattered electron in the Labframe.Figure 2.4 shows the boundary curves determined by Eq. (2.7) for E beam = 2 GeV andthree values of θ e (cid:48) , i.e. θ mine (cid:48) = 20 ◦ (dashed blue), θ maxe (cid:48) = 50 ◦ (dashed magenta), and θ e (cid:48) = 180 ◦ (solid black). The last curve stands for the maximal achievable limit of the Q versus W distribution.Beside that, the experimental coverage can be restricted due to the limitation on theminimal detectable energy of the scattered electron E mine (cid:48) . For this case the boundary curveis given by the following relation, This formula was derived under the assumptions of the incoming electron moving along the Z-axis andthe target proton being at rest [8]. LAS12 Note 2017-014 Q = m p + 2 m p ( E beam − E mine (cid:48) ) − W . (2.8)Figure 2.4 also shows the boundary curve given by Eq. (2.8) for the case E mine (cid:48) = 0 .
46 GeV(dotted red).
W (GeV) ) ( G e V Q mine' q = 50 deg maxe' q = 0.46 GeV mine' E = 2 GeV beam E Figure 2.4:
The margins of the Q versus W distribution for an experiment conducted with 2GeV beam energy. The solid black curve shows the maximal achievable boundary and is given byEq. (2.7) with θ e (cid:48) = 180 ◦ . The dashed blue and magenta curves stand for the edges due to thelimitation of the polar angle of the scattered electron. They are given by Eq. (2.7) for θ mine (cid:48) = 20 ◦ and θ maxe (cid:48) = 50 ◦ , respectively. The dotted red curve shows the edge due to the limitation on theminimal detectable energy of the scattered electron and is given by Eq. (2.8) for E mine (cid:48) = 0 .
46 GeV.
The edges of the Q versus W distribution given by Eqs. (2.7) and (2.8) are beam energydependent. As written above, the experiment off the moving proton with fixed beam energyis equivalent to that off the proton at rest performed with altered effective beam energy.Therefore, the distribution edges, being sharp and distinct in the proton at rest experiment,become blurred in the experiment off the moving proton.Let’s consider a moving proton experiment conducted with 2 GeV beam energy andassume the deviation of the effective beam energy from this value to be ±
250 MeV. Thissituation is illustrated in Fig. 2.5, where the maximal achievable boundaries are shown forthree choices of the beam energy: 2 GeV (solid black curve), 1.75 GeV (dashed blue curve),and 2.25 GeV (dashed magenta curve). The region between the two dashed curves shows thescope of the expected blurring. The boundaries caused by the experimental restrictions (thedashed and dotted curves in Fig. 2.4), being also beam energy dependent, are subject to theanalogous blurring.The event yield in the blurring region suffers from the depletion of events (compared tothat for the case of fixed beam energy and sharp disrtribution edge). To estimate this effect,13
LAS12 Note 2017-014
W (GeV) ) ( G e V Q = 2 GeV beam E = 1.75 GeV beam
E = 2.25 GeV beam E Figure 2.5:
The illustration of blurring of the maximal achievable limit of the Q versus W dis-tribution. The curves are given by Eq. (2.7) for the case θ e (cid:48) = 180 ◦ and three choices of beamenergy. one should know the function that describes the alteration of the effective beam energy.This function is in turn determined by the target proton momentum distribution. The crosssections extracted in the blurring region need a special correction, otherwise they will sufferfrom underestimation. This correction requires either experimental knowledge on initialproton momentum for each reaction event or the proper Monte Carlo simulation of theblurring effect.Note that Eqs. (2.7) and (2.8) as well as Figs. 2.4 and 2.5 assume the value of W to bethe true value of the invariant mass of the final hadron system given by Eq. (2.2). Only inthis case the boundary blurring takes place. If the smeared value of W , calculated underthe target-at-rest assumption, is used instead, the distribution edges are not subject to thisblurring because the fixed value of the laboratory beam energy is used in calculations.14 LAS12 Note 2017-014
Chapter 3The event generation procedure
For each event the values of all kinematical variables W , Q , S , S , cosθ h , φ h , α h aregenerated randomly exactly in the same way as it is described in Sect 3.1 of the report [2]. ϕ F θ F z yx −→ p F ep Lab
Figure 3.1:
The initial conditions of the reaction in the Lab frame. The incoming electron scattersoff the proton that moves with the momentum −→ p F . The simulation of the initial proton motion is performed under the following assumptions. • The Lab frame no longer corresponds to the system, where the target proton is at rest.The target proton moves in the Lab frame with the Fermi momentum as it is shown inFig. 3.1. The axis orientation in the Lab frame is the following: Z lab – along the beam, Y lab – up, and X lab – along [ (cid:126)Y lab × (cid:126)Z lab ].15 LAS12 Note 2017-014 • The generated value of W is treated as the smeared one calculated from the initialparticle four-momenta according to Eq. (2.1) under the target-at-rest assumption (seeexplanation in Sect. 2). Hereinafter this generated value is denoted as W sm . Theboundaries of the generated Q versus W sm distribution are set according to Eqs. (2.7)and (2.8) with E beam defined in the Lab frame. • The generated value of Q is treated as the actual Q value of the reaction. • The four momentum of the incoming electron in the Lab frame is P Labe = (0 , , E beam , E beam ) , (3.1)where E beam is the energy of the incoming electron beam that is given as an inputparameter. • The four-momentum of the scattered electron is defined in the Lab frame exactly inthe same way as it is done in the report [2] (see Eqs. (3.2) here as well as Eqs. (3.2) inthe report [2]). ν = W sm + Q − m p m p E e (cid:48) = E beam − νθ e (cid:48) = acos (cid:18) − Q E beam E e (cid:48) (cid:19) P Labe (cid:48) = ( E e (cid:48) sinθ e (cid:48) cosϕ e (cid:48) , E e (cid:48) sinθ e (cid:48) sinϕ e (cid:48) , E e (cid:48) cosθ e (cid:48) , E e (cid:48) ) . (3.2)Here ν is the virtual photon energy in the Lab frame, m p the target proton mass, and E e (cid:48) and θ e (cid:48) the scattered electron energy and polar angle, respectively. W sm , Q , and ϕ e (cid:48) are the generated reaction invariant mass, the photon virtuality, and the azimuthalangle of the scattered electron, respectively.The electron defined by Eqs. (3.2) imitates the actual scattered electron experimentallyregistered after the reaction of double-pion electroproduction off the moving proton hashappened.The components of the initial proton three-momentum p Fx , p Fy , and p Fz are generatedrandomly according to the Bonn potential [3]. The four-momentum of the initial proton inthe Lab frame is then determined by P Labp = ( p Fx , p Fy , p Fz , (cid:113) m p + [ p Fx ] + [ p Fy ] + [ p Fz ] ) . (3.3)The actual value of the invariant mass of the final hadron system is then determined by The algorithm of generating the initial proton three-momentum is coded in the subroutine fermi bonn.cxx . The determination of W true according to Eq. (3.4) distorts the flatness of the unweighted event distribu-tion of W true . This question is addressed in Sect. 4. LAS12 Note 2017-014 W true = (cid:113) ( P Labp + P Labγ v ) , (3.4)where P Labp is the four-momentum of the moving initial proton defined by Eq. (3.3) and P Labγ v = P Labe − P Labe (cid:48) the four-momentum of the virtual photon with P Labe and P Labe (cid:48) the four-momenta of the incoming and scattered electrons defined by Eqs. (3.1) and (3.2), respectively.The components of the initial proton three-momentum are generated under the condition W true > . W is illustrated in Fig. 3.2, which shows the unweighteddistribution of W true for the fixed value of W sm = 1 . W sm ± σ to illustrate the distribution’s spread. It is seen that the majority of eventsdeviates from the value W sm = 1 . / ndf c – – – (GeV) true W · / ndf c – – – Figure 3.2:
The unweighted distribution of W true for the fixed value of W sm = 1 . W sm ± σ to illustrate the distribution’s spread. The example is given for E beam = 2 GeV and 0.4 GeV < Q < . The generated values of the kinematical variables should be used to obtain the four-momentaof all final particles in the Lab frame. For the case of the free proton target the recipe for thisis described in Sect. 3.2 of the report [2] . However, it can not be straightforwardly used forthe case of the reaction off the moving proton. Therefore the following multistage methodhas been developed.I. The four-momenta of the initial particles should be transformed from the Lab frameto the specific system, where the target proton is at rest, while the incoming electron17
LAS12 Note 2017-014 moves along the Z -axis. This system hereinafter is denoted as “quasi-Lab”. The initialconditions of the reaction in the quasi-Lab frame imitate those existing in the Lab framein the case of the free proton experiment. This circumstance determines the name choice“quasi-Lab” that was assigned to this system .II. The procedure described in Sect. 3.2 of the report [2] should be applied in order toobtain the four-momenta of the final particles in the quasi-Lab frame.III. The four-momenta of the final particles should be transformed from the quasi-Labsystem to the conventional Lab frame .Each step of this method is described below in more details. I. Obtaining the initial particle four-momenta in the quasi-Labframe
The four-momenta of the initial particles defined by Eqs. (3.1) to (3.3) should be transformedfrom the Lab frame to the quasi-Lab. This transition is performed via three steps, which areschematically shown by the green arrows in Fig. 3.3. These steps are described below.1. The first step is the transformation from the Lab to the auxiliary system, which isdenoted in Fig. 3.3 as “System 1” and represents the frame that has its Z -axis along thetarget proton momentum. This transformation is performed through a set of rotationsof the coordinate axis as described below.Firstly the polar θ F and azimuthal ϕ F angles of the moving initial proton should be cal-culated in the Lab frame. These angles are marked in Fig. 3.1 and defined by Eq. (3.5). θ F = acos p Fz (cid:113) [ p Fx ] + [ p Fy ] + [ p Fz ] (cid:101) ϕ F = acos | p Fx | (cid:113) [ p Fx ] + [ p Fy ] ϕ F = (cid:101) ϕ F , if p Fx > and p Fy > π − (cid:101) ϕ F , if p Fx < and p Fy > (cid:101) ϕ F + π, if p Fx < and p Fy < π − (cid:101) ϕ F , if p Fx > and p Fy < The transformation of the initial particle four-momenta to the quasi-Lab frame is coded in the subroutine fermi rot.cxx . The transformation of the final particle momenta from the quasi-Lab frame to the Lab system is codedin the subroutine fermi anti rot.cxx . LAS12 Note 2017-014
Lab e −→ p f System e x y z z x e x qLab z qLab y p p p pe −→ p f z Lab y Lab θ s e . Rotation . Boost . Rotationβ F ϕ F , θ F System System p at rest ) ( quasi - Lab ) Figure 3.3:
Schematical representation of the transformation from the Lab frame to the specificsystem, where the target proton is at rest, while the incoming electron moves along the Z -axis. Thissystem is denoted as “quasi-Lab”. The transformation proceeds via three steps, which are shownby the green arrows. The X lab -axis is rotated by the angle ϕ F in the XY -plane (around the Z lab -axis) toforce the Fermi momentum to lay in the XZ -plane. This rotation translates the axis Y lab to Y and transforms the four-momentum as P (cid:48) = P · R ϕ F ( ϕ F ) with R ϕ F ( ϕ F ) = cosϕ F − sinϕ F sinϕ F cosϕ F . (3.6)Then one should rotate the Z lab -axis by the angle θ F in the XZ -plane in order totranslate the axis Z lab to Z and direct it along the Fermi momentum. This rotationtransforms the four-momentum as P (cid:48)(cid:48) = P (cid:48) · R θ F ( θ F ) with R θ F ( θ F ) = cosθ F sinθ F
00 1 0 0 − sinθ F cosθ F
00 0 0 1 . (3.7)As it is sketched in Fig. 3.3, the incoming electron, being transformed into the “System1”, turns out to be located in the XZ -plane. In all derivations the energy is assumed to be the last component of the four-momentum and the four-momentum to be a row vector. LAS12 Note 2017-014
2. After that the boost from the “System 1” to the proton rest frame, which is denotedin Fig. 3.3 as the “System 2” should be performed. The boost transforms the four-momentum as P (cid:48)(cid:48)(cid:48) = P (cid:48)(cid:48) · R boost ( β ) with R boost ( β ) = γ − γβ − γβ γ , β = (cid:113) [ p Fx ] + [ p Fy ] + [ p Fz ] (cid:113) m p + [ p Fx ] + [ p Fy ] + [ p Fz ] , and γ = 1 (cid:112) − β , (3.8)where β is the magnitude and Z -component of the three-vector −→ β = (0 , , β ).In “System 2” the incoming electron is still located in the XZ -plane.3. Finally, one should rotate the axis of the proton rest frame (“System 2”) to find oneselfin the quasi-Lab frame (“System 3”), which has its Z qLab -axis along the incomingelectron. For that purpose the polar angle of the incoming electron in the “System 2”should be defined by θ s e = acos p ez (cid:113) [ p ex ] + [ p ey ] + [ p ez ] , (3.9)where p ex , p ey , and p ez are the corresponded components of the incoming electron mo-mentum in the “System 2”.Then the Z -axis should be rotated with the angle θ s e in the XZ -plane in order to betranslated into Z qLab , which is directed along the incoming electron momentum. Thisrotation transforms the four-momentum as P (cid:48)(cid:48)(cid:48)(cid:48) = P (cid:48)(cid:48)(cid:48) · R θ s e ( θ s e ) with R θ s e ( θ s e ) = cosθ s e − sinθ s e
00 1 0 0 sinθ s e cosθ s e
00 0 0 1 . (3.10)After all manipulations the four-momenta of the initial particles are written in the quasi-Lab system in the following way, P qLabp = (0 , , , m p ) ,P qLabe = (0 , , (cid:101) E qLbeam , (cid:101) E qLbeam ), and P qLabe (cid:48) = ( E qLe (cid:48) sinθ qLe (cid:48) cosϕ qLe (cid:48) , E qLe (cid:48) sinθ qLe (cid:48) sinϕ qLe (cid:48) , E e (cid:48) qL cosθ qLe (cid:48) , E qLe (cid:48) ) , (3.11)where (cid:101) E qLbeam is Z -component of the incoming electron momentum in the quasi-Lab frame,while E qLe (cid:48) , θ qLe (cid:48) , and ϕ qLe (cid:48) are the energy and spatial angles of the scattered electron in thequasi-Lab frame, respectively. 20 LAS12 Note 2017-014
As it is seen from Eqs. (3.11), in the quasi-Lab system the target proton is at rest, theincoming electron moves along the Z qLab -axis, while the scattered electron has a certain knownorientation. Thus, the initial conditions of the reaction in the quasi-Lab system perfectlyimitate those existing in the Lab system in the case of the free proton experiment. / ndf c – – – (GeV) qLbeam E~ · / ndf c – – – Figure 3.4:
The distribution of the effective beam energy (cid:101) E qLbeam defined in the quasi-Lab frame.The solid vertical line shows the value of the beam energy in the Lab frame E beam = 2 GeV. Thered curve stands for the Gaussian fit, while the dashed vertical lines mark the values E beam ± σ to illustrate the distribution’s spread. The example is given for 1.3 GeV < W sm < < Q < . (cid:101) E qLbeam in Eqs. (3.11) is a so-called effective beam energy of the incoming electron in thequasi-Lab system, which does not coincide with the usual E beam that is defined in the Labsystem and given as an input parameter. This effective beam energy is unique for each eventand determined by the generated Fermi momentum.The distribution of the effective beam energy (cid:101) E qLbeam is shown in Fig. 3.4. The solid verticalline shows the value of the beam energy in the Lab frame E beam = 2 GeV. The distribution isalmost symmetric with respect to that line . The red curve stands for the Gaussian fit, whilethe dashed vertical lines mark the values E beam ± σ to illustrate the distribution’s spread.It is seen that for the majority of events the effective beam energy (cid:101) E qLbeam deviates from thefixed laboratory value within 200 MeV.As it was discussed in Sect. 2.5, the alteration of the effective beam energy causes theblurring of the kinematically achievable limits of W true and Q . TWOPEG-D automaticallytakes into account this effect, since the calculation of W true according to Eq. (3.4) considersthe effective beam energy (cid:101) E qLbeam .Note that the actual invariant mass of the final hadron system W true as well as thephoton virtuality Q , being Lorentz invariant, are not subject to any changes during thetransformation described above. The minor asymmetry of the distribution comes from the imposed restriction W true > . LAS12 Note 2017-014
II. Obtaining the final hadron four-momenta in the quasi-Lab frame
The four-momenta of the final hadrons in the quasi-Lab frame are calculated by exactlythe same procedure that is described in Sect. 3.2 of the report [2] for the case of the freeproton experiment. The procedure should be used as a “black box” with the following threemodifications of its input parameters. • One should use the true value of the invariant mass of the final hadron system W true de-fined by Eq. (3.4) instead of the generated value W sm , which is assumed to be smeared. • Instead of the true beam energy of the experiment E beam , which is defined in the Labframe, the effective and for each event unique beam energy (cid:101) E qLbeam from Eqs. (3.11)should be used. • Instead of the generated azimuthal angle of the scattered electron ϕ e (cid:48) , which is assumedto be given in the Lab frame, one should use ϕ qLe (cid:48) from Eqs. (3.11), which is defined inthe quasi-Lab frame. III. Obtaining the final particle four-momenta in the Lab frame
Once the four-momenta of the final particles are obtained in the quasi-Lab frame, they shouldbe transformed into the conventional Lab frame. For this purpose they should undergo alltransformations shown in Fig. 3.3 in the reverse order. Thus the rule of the four-momentumtransformation from the quasi-Lab to Lab is P Labi = P qLabi · R θ s e ( − θ s e ) · R boost ( − β ) · R θ F ( − θ F ) · R ϕ F ( − ϕ F ) , (3.12)where P Labi and P qLabi denote the four-momenta of the particle i in the Lab and quasi-Labframes, respectively. The index i corresponds to p (cid:48) , π + , π − , and e (cid:48) . Transformation matrices R θ s e , R boost , R θ F , and R ϕ F are defined by Eqs. (3.10), (3.8), (3.7), and (3.6), respectively.Figure 3.5 demonstrates the distributions of the quantities M π − ] (left), M (middle),and P [0] (right), which are defined in the following way, M π − ] = [ P Labe + P p − P Labe (cid:48) − P Labp (cid:48) − P Labπ + ] ,M = [ P Labe + P p − P Labe (cid:48) − P Labp (cid:48) − P Labπ + − P Labπ − ] , and P [0] = |−→ P Labe + −→ P p − −→ P Labe (cid:48) − −→ P Labp (cid:48) − −→ P Labπ + − −→ P Labπ − | , (3.13)where P Labe and P Labe (cid:48) are the four-momenta of the incoming and scattered electrons givenby Eqs. (3.1) and (3.2), respectively. P Labp (cid:48) , P Labπ + , and P Labπ − are the four-momenta of thefinal hadrons determined by the method described above, while P p = (0 , , , m p ) is the four-momentum of the target proton under the target-at-rest assumption. The vectors indicatethe corresponding three-momenta. 22 LAS12 Note 2017-014 ) (GeV ] - p [2 M -0.02 0 0.02 0.04 0.0600.51 ) (GeV [0]2 M -0.04 -0.02 0 0.02 0.0400.51 (GeV) [0] P Figure 3.5:
The distributions of the quantities M π − ] (left), M (middle), and P [0] (right), whichare defined under the target-at-rest assumption by Eqs. (3.13) and are therefore Fermi smeared.The dashed vertical line in the left plot corresponds to the pion mass squared. The example is givenfor E beam = 2 GeV, 1.3 GeV < W sm < < Q < . Equation set (3.13) defines M π − ] , M , and P [0] under the target-at-rest assumption inorder to imitate the conditions of the real experiment, where the target proton momentummay be not known. The distributions in Fig. 3.5 demonstrate therefore Fermi smearing. Thequantity P [0] shown in the right plot, being the missing momentum of the target proton, isdistributed according to the Bonn potential [3].23 LAS12 Note 2017-014
Chapter 4Obtaining the weights
The weight for each event is determined by exactly the same procedure that is describedin Sect. 4 of the report [2]. The weight factor is calculated according to Eq. (4.7) from thatsection with the following three modifications. • Instead of the generated value W sm , which is assumed to be smeared, the true value W true defined by Eq. (3.4) should be used for picking up the cross section. • To combine the structure functions into the full virtual photoproduction cross section,one should use the values of ε T and ε L calculated in the quasi-Lab frame according toEqs. (2.4) and (2.5). See the discussion in Sect. 2.4. • To obtain the electroproduction cross section from the virtual photoproduction one(mode F flux = 1), the virtual photon flux Γ v should also be calculated in the quasi-Labsystem using the effective beam energy (cid:101) E qLbeam introduced by Eqs. (3.11) and the valueof ε T calculated according to Eq. (2.4) in the quasi-Lab frame.Event distributions that illustrate this procedure are shown in Fig. 4.1. The plot (a)shows the comparison of two weighted event distributions, i.e. the W distribution producedby TWOPEG for the case of free proton (green curve) is compared with the W sm distributionproduced by TWOPEG-D (blue curve). The blue curve demonstrates the expected blurringof the resonance structure caused by Fermi smearing. The plot (b) compares the same greencurve from free proton TWOPEG with the W true distribution produced by TWOPEG-D(purple curve) and reveals their intrinsic consistency.Figure 4.1 (b) requires further clarifications. As it is written in Sect. 3.1, TWOPEG-Dflatly generates W sm , while W true is calculated according to Eq. (3.4) and therefore loses theflatness of generation, being affected by the generation of the Fermi momentum. This isillustrated in Fig. 4.1 (c), which shows the unweighted TWOPEG-D distributions of W sm generated in a range 1.3 GeV < W sm < W true (purple curve). While the former distribution is flat, the latter is not: it drops abruptlyat the edges and has a plateau in the middle. This behavior is quite justified, since eachvalue of W true can correspond to the sequence of W sm symmetrically scattered into a certain24 LAS12 Note 2017-014 range (see Fig. 3.2). The Fermi momentum distribution forces most of the W sm values tobe located in the vicinity of W true with a deviation of 50-100 MeV, while wider deviationsare significantly less probable. Hence, the plateau values of the W true distribution manage tocollect the majority of the corresponded W sm values within whole generated range, while theedge values of W true fail to achieve it. To saturate the edge regions of the W true distribution,the values of W sm should be generated in a wider range, as it is demonstrated in Fig. 4.1 (d).To produce this plot, W sm was generated in a range 1.25 GeV < W sm < W true distribution in a range 1.3 GeV < W true < W true distribution shown in Fig. 4.1 (d).The comparison presented in Fig. 4.1 (b) demonstrates that the convolution of the crosssection with the dependencies of the quantities ε T , ε L , and Γ v on the beam energy (see thediscussion in Sect. 2.4) has an insignificant influence on it. The explanation for that is thefollowing. Due to the fact that the Fermi momentum is directed isotropically, the effectivebeam energy turned out to be spreaded symmetrically around the actual beam energy withthe deviation of ∼
200 MeV for the majority of events (see Fig. 3.4). Thus in the limit of highstatistics this effect drops out assuming the linear dependence of ε T , ε L , and Γ v on the beamenergy. The actual dependence of these quantities on the beam energy is demonstrated inFigs. 2.2 and 2.3 and although it is non-linear, in any ∼
400 MeV-wide beam energy intervalits non-linearity is not pronounced. Therefore, the influence of this effect on the cross sectiondrops out to first order and is negligible in higher orders.It needs to be mentioned that TWOPEG-D was especially developed to be used in theanalyses of data, where the experimental information of the target proton momentum is in-accessible, and one is forced to work under the target-at-rest assumption. The flat generationof W sm serves this purpose best. If the quality of the experimental data allows to avoid thetarget-at-rest assumption, one can start with the conventional free proton TWOPEG for theMonte-Carlo simulation. The validity of this proposal is justified by the comparison shownin Fig. 4.1 (b). 25 LAS12 Note 2017-014
W (GeV) · free proton smeared moving proton, W (GeV) · free proton non-smeared moving proton, W (GeV) · < 1.9 GeV sm sm W true W W (GeV) · < 2 GeV sm sm W true W (a) (b)(c) (d) Figure 4.1: (a)
The comparison of two weighted event distributions, i.e. the W distribution pro-duced by TWOPEG for the case of free proton (green curve) is compared with the W sm distributionproduced by TWOPEG-D (blue curve). (b) The comparison of two weighted event distributions, i.e. the W distribution produced byTWOPEG for the case of free proton (green curve) is compared with the W true distribution pro-duced by TWOPEG-D (purple curve). See text for more details. (c) The unweighted TWOPEG-D distributions of W sm generated in a range 1.3 GeV < W sm < W true (purple curve). (d) The unweighted TWOPEG-D distributions of W sm generated in a range 1.25 GeV < W sm < W true (purple curve).The examples are given for E beam = 2 GeV and 0.4 GeV < Q < . LAS12 Note 2017-014
Chapter 5Managing with radiative effects
For the simulation of the radiative effects the procedure described in Chapter 7 of thereport [2] was used. However, the task of combining this procedure with the simulation of thetarget motion is not straightforward. The following two methods were therefore developedand tested.
In this approach the simulation of the radiative effects was done first, while the simulationof the target motion is performed after that, using the radiated values of (cid:102) W and (cid:102) Q as wellas radiated four-momenta of the incoming and scattered electrons as a starting point.This method is implemented into the free proton TWOPEG (which also has a comple-mentary moving target mode) and executed under the options F fermi = 1 and F rad = 1 or 2. In this approach the simulation of the radiative effects is merged with that of the targetmotion in the following way . • The Fermi momentum is generated and the true value of the final hadron systeminvariant mass W true is calculated according to Eq. (3.4). • The nonradiated cross section in Eqs. (7.2) to (7.5) are taken for the true value W true .To combine transverse and longitudinal structure functions into the full virtual photo-production cross section and to convert it then to the electroproduction one, the valuesof ε T , ε L , and Γ v were calculated in the quasi-Lab frame. • The factor R radsoft in Eq. (7.2) as well as the integrals given by Eqs. (7.3) and (7.4)are calculated in the Lab frame. Eqs. (7.2) to (7.5) are given in the report [2]. LAS12 Note 2017-014 • The maximal allowed energy of the radiated photon ( ω inimax and ω finmax given by Eqs. (7.3)and (7.4)) is restricted by the demand to produce a pion pair. This restriction is imposedin the quasi-Lab frame, although the values of ω inimax and ω finmax are calculated in the Labsystem.This method is implemented into the TWOPEG-D version of the event generator, whichalways works in the moving target mode ( F fermi = 1).These two methods turned out to give almost indistinguishable missing mass and mo-mentum distributions and very similar weighted W distributions. Nevertheless, the secondapproach is thought to be the preferential one and, therefore, is recommended. ) (GeV ] - p [2 M -0.02 0 0.02 0.04 0.0600.51 (GeV) [0] P W (GeV)
Figure 5.1:
The comparison of the event distributions produced by TWOPEG-D with radiativeeffects (orange curves) and without (blue curves). The left and middle plots correspond to thequantities M π − ] and P [0] , respectively, which were calculated according to Eqs. (3.13). The right plotshows the comparison of the weighted W sm distributions. The example is given for E beam = 2 GeV,1.3 GeV < W sm < < Q < . Figure 5.1 shows the comparison of the event distributions produced by TWOPEG-Dwith radiative effects (orange curves) and without (blue curves). The left and middle plotscorrespond to the quantities M π − ] and P [0] , respectively, which are given by Eqs. (3.13).These quantities were calculated assuming (like in experiment) that P Labe and P Labe (cid:48) are notaffected by the radiative effects, while P Labp (cid:48) , P Labπ + , and P Labπ − , on the contrary, take them intoaccount. The right plot shows the comparison of the weighted W sm distributions.28 LAS12 Note 2017-014
Chapter 6Conclusions and code availability
As an extension of TWOPEG [2] the version TWOPEG-D that simulates the quasi-freeprocess of double-pion electroproduction off a moving proton was developed.TWOPEG-D is available as: • a separate program TWOPEG-D that works for the case of moving protons only (themode F fermi = 1 is fixed).It can be downloaded at: https://github.com/gleb811/twopeg d.git • a part of free proton TWOPEG using the mode F fermi = 1.It can be downloaded from the same place as specified in the report [2](i.e., https://github.com/JeffersonLab/Hybrid-Baryons/).With the option F rad = 0 (without radiative effects) these two editions produce identicalresults. However, in the mode F rad = 1 or 2 (with radiative effects) they differ, i.e. the firstedition employs the advanced method of merging the radiative effect with the target motion,while the second edition merges them by the naive method, as it is described in more detailsin Sect. 5.The specifications of building and running TWOPEG-D are the same as for the freeproton TWOPEG. They are described in Sect. 8 of the report [2].The performance of TWOPEG-D was tested during the analysis of CLAS data on electronscattering off the deuteron target (the part of the “e1e” run period) [9], where it has beenused for the efficiency evaluation and the corrections due to the radiative effects and Fermimotion of the target proton. For that purpose TWOPEG-D was run in a mode that keptBOS output, which was then passed through the standard CLAS packages GSIM and recsis.In this data analysis TWOPEG-D has proven itself as an effective tool for simulating effectsof the target motion for the reaction of double-pion electroproduction off protons.29 LAS12 Note 2017-014
Bibliography [1] “CLAS Physics Database,” http://clas.sinp.msu.ru/cgi-bin/jlab/db.cgi .[2] Iu. Skorodumina, G. V. Fedotov, et al. , “TWOPEG: An Event Generator forCharged Double Pion Electroproduction off Proton,”
CLAS12-NOTE-2017-001 , 2017,arXiv/1703.08081[physics.data-an].[3] R. Machleidt, K. Holinde, and C. Elster, “The Bonn Meson Exchange Model for theNucleon Nucleon Interaction,”
Phys. Rept. , vol. 149, pp. 1–89, 1987.[4] Yu. A. Skorodumina, E. N. Golovach, R. W. Gothe, B. S. Ishkhanov, E. L. Isupov,V. I. Mokeev, and G. V. Fedotov, “Investigating of the exclusive reaction of π + π pairelectroproduction on a proton bound in a deuteron,” Bull. Russ. Acad. Sci. Phys. , vol. 79,no. 4, pp. 532–536, 2015. [Izv. Ross. Akad. Nauk Ser. Fiz. 79, No. 4, 575 (2015)].[5] E. Byckling and K. Kajantie,
Particle Kinematics . Jyvaskyla, Finland: University ofJyvaskyla, 1971.[6] G. V. Fedotov et al. , “Analysis report on the ep → e (cid:48) pπ + π − reaction in the CLAS detectorwith a 2.039 GeV,” CLAS-Analysis 2017-101 , 2017.[7] K. Schilling and G. Wolf, “How to analyze vector meson production in inelastic leptonscattering,”
Nucl. Phys. , vol. B61, pp. 381–413, 1973.[8] Iu.. A. Skorodumina, V. D. Burkert, E. N. Golovach, R. W. Gothe, E. L. Isupov, B. S.Ishkhanov, V. I. Mokeev, and G. V. Fedotov, “Nucleon resonances in exclusive reactionsof photo- and electroproduction of mesons,”
Moscow Univ. Phys. Bull. , vol. 70, no. 6,pp. 429–447, 2015. [Vestn. Mosk. Univ., No. 6, 3 (2015)].[9] Iu. Skorodumina, “Investigation of Exclusive π + π − Electroproduction off theBound Proton in Deuterium in the Resonance Region with CLAS,” wiki page,https://clasweb.jlab.org/wiki/index.php/User:Skorodumwiki page,https://clasweb.jlab.org/wiki/index.php/User:Skorodum