Deepgen: an Event Generator Framework to Study Polarized Deep Exclusive Electro- and Photo-Production Processes

DEEPGen: an event generator framework to study polarized Deep Exclusive Electro- and Photo-production processes

Marie Boër,∗ Temple University, Philadelphia, PA, USA.

December, 2018

Abstract

We present in this note the framework of the DEEPGen event generator, initially developed to study deeply exclusive electro- and photo-production of lepton pairs. The main purpose of this event generator is to perform impact studies and predictions for reactions accessing Generalized Parton Distributions. We provide in this version the following reactions: Double Deeply Virtual Compton Scattering + Bethe-Heitler, Timelike Compton Scattering + Bethe-Heitler and Deeply Virtual Compton Scattering + Bethe-Heitler. Generated events contain various unpolarized and polarized weighting options, and several cross section decompositions into sub-processes. Various beam and target options allow for realistic predictions in the context of experiments at the Jefferson Laboratory.

Contents

1 Introduction 3

2 Deeply Virtual Compton scattering processes 4 2.1 Reactions ...... 4 2.2 Kinematics and cross sections ...... 8 2.2.1 TCS+BH ...... 8 2.2.2 DDVCS+BH ...... 8 2.2.3 DVCS+BH ...... 9

3 Principle of the generator and main options 10 3.1 Reactions and running modes ...... 10 3.2 Events generation ...... 12 3.3 Events weighting ...... 12 3.4 Test versions and binning optimizations ...... 13 3.5 Flags in TCS ...... 13

∗contact: [email protected]

1 4 Beam and polarization options 15 4.1 Beam proﬁle ...... 15 4.2 Electron induced TCS photon ﬂux ...... 15 4.2.1 Bremsstrahlung photons ...... 15 4.2.2 Quasi-real photons ...... 16 4.3 Asymmetries and polarization dilution factors ...... 17

5 User files 18 5.1 Input files ...... 18 5.2 Output files ...... 18 5.2.1 TCS Tree (SIM_Tree) ...... 22 5.2.2 DVCS Tree (SIM_Tree) ...... 24 5.2.3 DDVCS Tree (SIM_Tree) ...... 25 5.2.4 Additional information: Dump_Tree ...... 26

6 Analysis 28 6.1 Generator units ...... 28 6.2 Normalization ...... 28 6.3 Results ...... 29

7 How to? 33 7.1 Binaries ...... 33 7.2 Running the code ...... 33

8 Conclusion 34

9 Aknowledgements 34

2 1 Introduction

This event generator is intended to study exclusive Compton-like reactions, where the main process of interest (Deeply Virtual Compton Scattering = DVCS, Timelike Compton Scattering = TCS, Double Deeply Virtual Compton Scattering = DDVCS...) is interfering with an associated Bethe- Heitler (BH) process. The main purpose of these studies is to access Generalized Parton Distribu- tions (GPDs), contained in the DVCS or TCS or DDVCS parametrization. Due to BH dominance in the reactions, generating events in a standard way (LUND, HEP... formats), with a unique event weighting being proportional to the total unpolarized cross section and the counting rates, could bias predictions in terms of physics impact. Indeed, BH is non sensitive to GPDs and the most interesting regions for physics studies are not necessarily the one where the predicted counting rates are the largest. In addition to total cross sections (polarized and unpolarized), we provide weights for a "Bethe-Heitler only" scenario, and similarly for other processes that enter the reaction.

We developed this generator is to perform impact studies. Therefore, we included options to generate simultenaously several observables in addition to the unpolarized cross sections. We propose beam and/or target polarization options and provide event weighting for all different scenarios.

Historically starting the developement of this event generator to study DDVCS and TCS process, we optimized it for efﬁciently calculating weights as a function of many independent variables: unpolarized DDVCS+BH depends on 7 independent variables, unpolarized TCS+BH depends on 5 independent variables. It is possible to use tables of pre-calculated cross sections to extrapolate to a given kinematic with minimal running time per events up to 4 dimensions. Above 5 dimensions, the reading of tables and processing time to interpolate events is drastically slowed down. We could reduce the dimension of tables by cutting out the phase space. However, for our purpose, we need to generate events in a broad phase space as a function of all the independent variables. To reduce the dimension of our tables and have a more efﬁcient generation, we applied kinematic cuts at both the table preparation and the generator level, rejecting the large phase space area which is kinematically inaccessible. Since we want to keep the same framework accessible for various JLab experiments, we calculated the cross sections in a 4π acceptance, at accessible kinematics.

First, we briefly present the reactions and their interests, supported by the current version of the event generator. Then, we present the way the generator is running and the main options. We explain the various polarization options, and provide our calculations for the polarization and flux. Then, we provide a documentation on input and output files. Eventually, we show example of generated events and explain how to interpret the results.

Regularly updated information and binary ﬁles are accessible from the wiki page [1].

3 γ (q') γ (q) out in

hard soft

N (p) N (p')

Figure 1: General diagram for the Deeply Virtual Compton Scattering process. Incoming and outgoing photons can be real or virtual, with at least one photon having large virtuality (typically larger than 1 GeV) to ensure a hard scale.

2 Deeply Virtual Compton scattering processes

2.1 Reactions Deeply Virtual Compton-like processes, where virtual and/or real photons are scattered off quarks of the nucleon, provide access to its partonic structure. In these processes, at least one hard scale is provided by the invariant mass of one of the photons. It allows a factorized approach between a hard perturbative part, calculable from QED, and a non perturbative part [2], as shown on diagram 1. The general case where the 2 photons carry a high invariant mass (typically at a scale larger than 1 GeV) is called Double Deeply Virtual Compton Scattering (DDVCS, Fig. 4). Deeply Virtual Compton Scattering (DVCS), corresponds to the case of an outgoing real photon and an incoming spacelike photon, provided by a lepton beam (Fig. 2). Timelike Compton Scattering (TCS) corresponds to the case of an incoming real photon. A timelike virtual photon is emitted and de- cays into a lepton pair (Fig. 3). The soft part of these processes can be parametrized by Generalized Parton Distributions (GPDs). We refer to [3, 4, 5, 6] for details about the GPD formalism.

All these processes have an interference with an associated Bethe-Heitler (BH) process, having the same ﬁnal state, but insensitive to the GPDs. The Bethe-Heitler associated to DVCS corresponds to a photon emission from the lepton beam (Fig. 5). The Bethe-Heitler associated to TCS corresponds to a splitting of the incoming photon in the nucleon ﬁeld into a lepton pair (Fig. 6). Two kinds of Bethe-Heitler diagrams interfere with DDVCS: one behaves like that associated with DVCS, where the lepton pair comes from the radiation of a virtual photon from the lepton beam (Fig. 7, top panel), while the other one behaves like that associated with TCS, where the lepton pair comes from a splitting of the virtual photon interacting with the nucleon (Fig. 7, bottom panel).

4 e' (k') e' (k') e (k) e (k)

γ γ γ∗(q) (q') γ∗(q) (q') hard hard soft soft x+ ξ x- ξ x+ ξ x- ξ

N (p) N' (p') N (p) N' (p')

t t

Figure 2: Leading order and leading twist diagrams for DVCS.

e- (k) e - (k)

e+ (k') e + (k')

γ (q) γ∗(q') γ (q) γ∗(q') hard hard soft soft x+ ξ x- ξ x+ ξ x- ξ

N (p) N' (p') N (p) N' (p')

t t

Figure 3: Leading order and leading twist diagrams for TCS.

5 e' (k') - e (k) µ (l)

µ+ ∗ (l') γ ∗ 1 (q) γ 2 (q')

hard soft x+ξ x-ξ

N (p) N' (p')

Figure 4: Leading order and leading twist diagram for DDVCS. Cross diagram is not represented.

e (k) e' (k') e (k) e' (k')

γ∗ γ∗

N (p) N' (p') N (p) N' (p')

Figure 5: Bethe-Heitler process with the same ﬁnal state than DVCS.

6 (k’) (l')

(l)

+ - + (k) -

e e µ µ

N' (p') e' (k')

N’ (p’) (q') (q)

∗ ∗ 1 2 (q’) γ γ

∗

γ (q) e (k) γ N (p) N (p) 7

(k) (k’)

- +

e e (l')

(l)

- +

N’ (p’)

µ µ N' (p') e' (k') (q’)

∗

γ (q')

∗ 2

γ (q) ∗ 1 γ Figure 6: Bethe-Heitler process with the same final state than TCS. (q) γ N (p) e (k) N (p) Figure 7: Bethe-Heitlerbehaves like process the with DVCS-associated the one,one. same the Cross final diagrams diagram state are on than not the represented. DDVCS. right The behaves diagram like on the the TCS left associated We refer to [7, 8] for the DVCS formalism, to [9] for TCS and to [10] for DDVCS. These articles are not an exhaustive list of articles on these processes: they refer to the formalism and the calculations we are using in the present work. We use calculations based on [11], using GPDs from VGG model [7, 8] and Form Factors from [12, 13] (proton) and [14, 15] (neutron). As GPD H is dominant in cross sections, we generated DVCS and TCS cross sections only using GPD H. We generated DDVCS cross sections only using GPD H from the imaginary part of its Compton Form Factor, as defined in [10, 11]. This choice avoid bias in predictions coming from parametrization and model choice: GPD models predictions are close when using only GPD H, while they can be drastically different if other GPDs are included. Other versions of the event generator support GPD parametrization including GPD H˜ , E and/or E˜.

2.2 Kinematics and cross sections 2.2.1 TCS+BH Fig. 8 is a scheme of the TCS reaction where we deﬁne several angles. According to our notations, these angles are: • φ is the azimuthal angle between the lepton pair and the reaction plane,

• θ is the polar angle of the electron compared to the virtual photon direction in the photon- proton CM frame,

• Ψs is the angle between the incoming photon spin direction and the reaction plane, in case of a linearly polarized photon,

• φs and θs are the azimuthal and the polar angle corresponding to the proton spin compared ◦ to the proton direction axis (z-axis in this notation) and the reaction plane. {θs = 0 } corre- ◦ ◦ sponds to a longitudinal target polarization, {θs = 90 , φs = 0 } corresponds to a transverse ◦ ◦ polarization along the x-axis, and {θs = 90 , φs = 90 } corresponds to a transverse polarization along the y-axis. The TCS unpolarized cross section depends on 5 independent variables, which we choose to be the φ and θ angles deﬁned above, the incoming photon energy (Eγ), the outgoing photon virtuality (Q’2 = q’2, where q’ is the outgoing photon momentum) and the momentum transfer squared t (t = p − p0, where p and p’ are respectivelly the incoming and outgoing proton momenta). The transversely target polarized TCS also depends on φS, and the linearly beam polarized TCS depends on ΨS.

2.2.2 DDVCS+BH Fig. 9 is a scheme of the DDVCS reaction, where we deﬁne our notations for the angles involved in the process:

• φL is the azimuthal angle between the plane formed by the incoming and scattered beam lepton, and the reaction plane (γin, γout). It plays the same role as the DVCS φ angle.

• φ is the azimuthal angle between the decay lepton pair and the reaction plane (it plays the same role as the TCS φ angle),

• θ is the polar angle of the electron compared to the virtual photon direction in the photon- proton CM frame,

8 y x

z e- (k) Φ Φ e- (k) Ψ ⤴ ⤴

γ (q) ⤴ x ⤵ y θ γ* (q’) N’ (p’)

N’ (p’) e+ (k’) z' t γ N CM N (p) e- e+ CM e+ (k’)

Figure 8: Scheme of the TCS reaction in nucleon-photon center of mass frame (left panel) and in the ﬁnal photon rest frame (right panel). Several angles are displayed, and their denomination is described in the text. The red arrow on the photon and the nucleon indicate the directions of their possible polarization vector.

DDVCS+BH unpolarized cross section depends on 7 independent variables, which we choose to be these 3 angles, and the 4 following invariants:

• t: Mandelstam variable, momentum transfer squared,

• Q2: virtuality of the initial photon,

• Q’2: virtuality of the outcoming photon,

2 • x = Q M ν bj 2Mpν , is the Bjorken variable, with p the mass of the nucleon and the energy loss by 0 the lepton beam (ν = Ee − Ee = Eγin ). We provide cross sections for DDVCS+BH in the eP → e0P 0µ+µ− channel to avoid having anti- symmetrization terms if we would use electrons for the ﬁnal lepton pair. Current version of the event generator doesn’t provide event weighting for the case of a ﬁnal electron-positron pair.

2.2.3 DVCS+BH Fig. 10 is a scheme of the DVCS reaction where we define various angles. The azimuthal angle φLH between the lepton scattering plane and the photon’s scattering plane allows for separating BH and DVCS through angular modulation studies. At fixed beam energy, the DVCS cross section 2 depends on 4 independent variables, which we choose to be φLH , xbj, t and Q , as defined for the DDVCS. Transversely polarized nucleon cross sections also depend on φS (5 independent variables at fixed beam energy).

9 y x z e- (k’) - - ΦL μ (e ) (l) ΦCM

θL γ*in (q) γ* (q’) e- (k) out N’ (p’) μ+ (e+) (l’) t N (p) γ* N CM

Figure 9: Scheme of the DDVCS reaction in the nucleon + incoming photon center of mass frame. Several angles are displayed, and their denomination is described in the text. The red arrow on the nucleon indicate the directions of its possible polarization vectors.

3 Principle of the generator and main options

3.1 Reactions and running modes The reactions implemented in the current version of the generator, named "process" in the following, are:

1. γN → γ∗ (large Q02) N → e+e−N (N = proton or neutron) = TCS + BH. Generator options: unpolarized beam and nucleon, circularly or linearly polarized beam, longitudinally or transversely polarized nucleon. Beam and/or target can be polarized. - Option "process" = "tcs" = generates weighted events, - Option "ps_eephoto_ﬁx" (select the photon beam option) generated unweighted events.

2. eN → γ∗(Q2 ∼ 0)N → e+e−N(e0) (N = proton or neutron) = TCS + BH from quasi-real photon beam + bremsstrahlung in the target. - Option "tcs" = generate weighted events. - Option "ps_eephoto_ﬁx" (select the electron beam option) generates unweighted events.

3. eP → γ∗ (large Q2) P → e0µ+µ−P = DDVCS + BH - Option "ddvcs" = generate events and calculate cross sections, - Option "ps_eeel_ﬁx" generates phase-space.

4. eN → γ∗ (large Q2)N→ e0N 0γ (N = proton or neutron) = DVCS + BH. - Option "dvcs" = generate events and calculate cross sections, - Option "ps_vcs_ﬁx" generates phase-space.

10 e (k) Φ γ (q’) ⤴

γ* (q) ⤵ x y t N (p) z N’ (p’) γ* N CM e (k')

Figure 10: Scheme of the DVCS reaction in nucleon- virtual photon center of mass frame (left panel) and in the ﬁnal photon rest frame (right panel). Several angles are displayed, and their denomination is described in the text. The red arrow on the photon and the nucleon indicate the directions of their possible polarization vector.

11 3.2 Events generation The event generator is built from two independent sources codes. We use the ﬁrst one to generate tables of cross sections as a function of kinematic invariants and angular variables. If the same format is used, the tables can be generated from a different model than the one used in the present version of the generator. The second source code provides the calculation of the kinematics and four-vectors of particles involved in the process, and read the tables for event weighting. A phase-space generation mode for fast acceptance studies generates events without reading the cross sections table. The generation steps as follows:

• The user input ﬁle ("process.input") contains the kinematic range to generate, polarization options, target mode, number of events to generate, and output format options.

• A set of kinematic variables is randomly generated within these kinematic limits. We generate flat distributions in all dimensions. If this set of variables corresponds to a kinematic that is not achievable (by instance -t is below -tmin), the event is rejected and is not acounted in the total number of generated events defined by the user. Remark: final generated distributions are not flat for the unweighted events, due to the kinematic cuts. The actual number of events initially generated in a fixed hypercube is provided to normalize the cross sections.

• According to the generated set of kinematic variables, invariants and four vectors of incoming and outgoing particles are calculated in different frames. At ﬁrst, the beam particle is thrown along the z-axis at x=y=0.

• Particles are boosted from the lab frame to the (virtual) photon + target nucleon center of mass frame. The outgoing photon is emitted at this step. In case of TCS+BH or DDVCS+BH reactions, particles are boosted into the outgoing virtual photon rest frame, where the lepton pair can be deﬁned from the randomly generated polar and azimuthal angles (φ, θ).

• All particles are boosted back to the lab frame. In case of TCS+BH events, an arbitrary rotation of all vectors is applied to ensure there is no favored azimuthal direction for the virtual photon emission.

• In case of DDVCS+BH or DVCS+BH events, this rotation corresponds to the initial azimuthal angle (φLH ) between the lepton beam and scattered lepton plane, and the photon’s plane.

• Events are weighted according to their kinematics, from reading all the nearest points in the table. A ROOT Tree is filled as well as a HEP file, depending on the user’s choice of output file format.

3.3 Events weighting Tables in text format are generated in a linear equidistant binning in all the variables we choose for calculating the cross sections, as deﬁned in section 2.2. Only kinematically accessible bins are printed. The tables contain polarized cross sections and the decomposition of the cross section in sub-processes. The generator code reads the table once and ﬁll the relevant information in a C++ map. Kinematics are retrieved thanks to a unique line indexation, corresponding to the kinematic bin. When the kinematics of an event are generated, we call its corresponding bin previous and next value from the map, in each dimension. It requires 2N readings from the map and linear interpolations to get the event cross section, where N is the number of independent variables. Comparing results from directly calculated cross sections and the interpolated one, we found a

12 percent level systematic uncertainty introduced by this method. However, the gain in time and practicality for the generator is large, and this error is lower than the uncertainties introduced by the choice of a particular GPD model.

In the case of DDVCS, we perform an interpolation in 7 variables (only beam polarized and unpolarized cross sections are provided). In the case of TCS, the interpolation of unpolarized, beam polarized and longitudinnaly polarized cross sections is performed for 5 independent variables. The problem depends on 6 variables in case of a transverselly polarized target or a linearly polarized photon, but we decided to reduce the problem to 4 dimensions by not interpolating in beam energy and φS (or ΨS). The impact of this approximation is negligible. In case of unpolarized, beam polarized and longitudinally polarized nucleon DVCS, we perform interpolation in 4 variables. The interpolation is done for 5 variables if the target is transversely polarized.

3.4 Test versions and binning optimizations To estimate the binning and interpolation effects due to using a pre-calculated table of cross sections rather than calculations for each events, we created a version that get weights from the table and calculates them directly at the same time. We consider that binning is optimal at a percent level accuracy of the results after interpolation. The optimal binning has to be balanced with the size of the table, the main factor of potential slow running time of the code. We also had to balance the number of bins in each dimension to keep more points in variables in which cross sections are fast-variating, and where kinematic cuts can occur (such as t). We don’t provide the version directly calculating the cross sections for reason of running efﬁciency and because of its large usage on the cluster. The other major reason of not using direct calculations is detailed in the next paragraph.

We decided to start calculating the tables for azimuthal angles starting at at 3◦ rather than 0◦. We made this choice to avoid having a table based on particular kinematic points such as φ = 0, φ = 90... Indeed, for such angles, BH can present peaks and fast varying cross sections. Addition- ally, asymmetries can be minimal or maximal. Introducing this shift in angle allow for calculating the cross sections around these speciﬁc points, and then interpolating results from 8 (DVCS) to 14 (DDVCS) values around these points to the given kinematic (if one wants the result at φ = 0◦ by instance). It gives more accurate results and smooth distributions. Indeed, as we discussed in the introduction, cross sections at φ = 0◦ can in some case be orders of magnitude away from cross sections at φ = 0.1◦. Thus, calculating directly cross sections for each event would not give more accurate results in these regions, but on the contrary will create spikes, unless an "inﬁnite" amount of statistics is generated (in some cases, the need to solve these points can be > 105 events in a very tiny bin). We actually noticed that the angular region affected by these effects around BH peaks can be large at some kinematics (E, t...).

In DDVCS, to avoid missinterpretations of the GPDs in regions where Q02 ∼ Q2, we binned the tables in such way that the virtualities can never be equal.

3.5 Flags in TCS For TCS process, we attribute a ﬂag for each events as decribed in [16] and corresponding to a kinematic dependent cut in (θ, φ). A ﬂag set at 0 means that the event can be kept. If it is set to 1, the events are close to the near-singularity Bethe-Heitler peaks, where an increased precision

13 is needed for accurate results. The event has to be rejected from the analysis. Indeed, in this region, very high experimental statistics are needed to solve the fast increasing cross section of the BH. Similarly, for accurate calculations, we would need to increase the precision of our integral calculations. It would require an unrealistic amount of time to produce our cross section tables. For most experiments, the outgoing particles are out of acceptance in this region due to their very forward momenta, and therefore it doesn’t worth to have accurate results in the regions that will be rejected in the analysis and the simulations. As explained previously, we avoid spikes in generated distributions by interpolations over a table, rather than calculating cross sections for each event.

In case of TCS induced by an electron beam, as explained in Section 4.2, there are 2 sources of photons: from bremsstrahlung in the target (real photons), and from interactions in the target (quasi-real photons). Depending on the probability for each kinematic, we set a ﬂag (VirtualFlag), equal to 1 if the photon is quasi-real. The main difference for an event from a quasi-real and a real photon is the incident angle of the incoming photon, and the Q2 correction in kinematics in case of a virtual photon.

14 4 Beam and polarization options

4.1 Beam profile DVCS and DDVCS events are generated at fixed beam energy. TCS events are generated at fixed electron energy (user defined) in case of quasi-real photoproduction and flat in the given beam energy range in the case of real photoproduction. In this case, to get a realistic beam profile and normalization, the beam energy distribution can be rescaled at the analysis level by the probability of a given photon energy. By instance, if the real photon is provided by a radiator with no favored energy or cuts, the beam profile function is a normalized Bremsstrahlung distribution. Event weights have to be multiplied by this function. For quasi-real TCS, the generator is providing two sets of event weights: one corresponds to differential cross sections ("cross...") and the other one corresponds to the differential cross sections multiplied by the photon flux factor, as described in the next section. The user does not have to rescale the beam energy distribution if this option is used.

4.2 Electron induced TCS photon ﬂux If no real photon beam is available, TCS can be measured from very low virtuality (quasi-real) photons. In a ﬁxed target experiment, an electron beam can provide 2 sources of photons: from bremsstrahlung in the target (real photons), or quasi-real photons from inelastic interactions. Usu- ally, one assume that photons are quasi-real up to a virtuality of Q2 ' 0.3 GeV. Both sources of quasi-real and real photons from an electron beam are of the same order of magnitude at JLab energy, for typical target dimensions.

To calculate the equivalent photon ﬂux when using an initial electron beam, we assume the factorized approach for cross sections [17]:

σγ = f(γ → e) σe (1) where σγ, e are respectively the cross sections from initial photon or electron induced reactions. f(γ → e) is the probability of a photon emission from an electron beam, or equivalent flux factor. It depends on the energies of the electron beam and the photon. It also depends on target parameters (target length, composition...). The equivalent flux factor is the sum of a bremsstrahlung equivalent flux factor (fB) and quasi real photons equivalent flux factor (fqr).

4.2.1 Bremsstrahlung photons

The radiation length X0 for electrons crossing a target made of atoms (A, Z) is [18, 19, 20]:

1 2 NA 2 0 = 4α re Z [Lrad − f(Z)] + ZLrad ρ, (2) X0 A where α is the electromagnetic constant, ρ is the target density, NA is Avogadro number, and 2 −1 2 −1 0 4αre NA = 716.408A (g .cm ) . Lrad and Lrad are radiation length parameters which could be found in [20] and f(Z) is the following parametrized sum (with a = αZ) [21]:

1 f(Z) = a2 + 0.20206 − 0.0369 a2 + 0.0083 a4 − 0.002 a6. (3) (1 + a2)

15 Generator’s code material radiation length (g.cm−2)

1001 H2 (liquid) 890.4 1002 He 94.32 1003 LiH 79.61 1006 C 19.32 1007 NH3 40.87 1011 dry air 30.39 1013 Al 8.897 1026 Fe 1.757 1082 Pb 0.5612 1083 lead-glas 1.265

Table 1: Generator’s internal codes and target materials. Values of radiation lengths are from [23]

The Bremsstrahlung cross section for high energy electrons which we use here as equivalent ﬂux factor reads [22] dσ 1 4 4 1 f = = 4α r2 ( − y + y2)(Z2[L − f(Z)] + ZL0 ) + (1 − y)(Z2 + Z) , (4) B dν ν e 3 3 rad rad 9 where ν is the emitted photon energy and y = ν/Ee.

For some target elements, we used the radiation length from [23] instead of making the above calculation. The number of emitted photons for a given energy ν and a given target length L becomes: L 1 1 4 4 2 fB = − y + y (5) 2 X0 ν 3 3 We summarize in table 1 the internal codes (user deﬁnded) and target materials which are cur- rently handled by the generator.

For the angular distribution of real bremsstrahlung photon we follow the sampling method proposed in [24]: v 1 u Θˆ θ (B) = u , (6) γ t ˆ 1 Ee 1 − Θ + 2 (π Ee) where Θˆ a random number between 0 and 1.

4.2.2 Quasi-real photons The Quasi-real photon equivalent ﬂux factor reads [25]

2 2 α 1 y Qmax fqr = (1 − y + ) log 2 − (1 − y) (7) E π y 2 Qmin where y m2 Q2 = e (8) min 1 − y

16 Figure 11: Probability of generating a photon from an 11.5 GeV electron beam in a 40 cm LH2 target. The bremsstrahlung photon ﬂux is displayed in black and the quasi-real photon ﬂux is displayed in red.

2 2 and Qmax is the maximal limit allowed by the user. We set a ﬁx Qmax cut at 0.3 GeV for the event generator. This has a minimal effect on results since low Q2 photons are largely dominant.

We follow the formalism of DDVCS reaction for the angular distribution of quasi-real low Q2 photons. The scattering angle of the photons is 2 −1 −Q − 2Eeν θγ(qr) = π − cos . (9) p 2 2 2Ee ν + Q We display Fig. 11 the probability of generating a bremsstrahlung photon (black) in a 40 cm LH2 target, and the probability of generating a quasi-real photon, as a function of its energy. The initial electron energy is 11.5 GeV.

4.3 Asymmetries and polarization dilution factors We introduced polarization dilution factors in calculations of polarized cross sections. The target polarization is assumed to be a constant provided by the user, as well as the DVCS and DDVCS beam polarization. The TCS photon polarization is assumed to be constant if the beam is linearly polarized. If the beam is circularly polarized, we calculate the polarization transfer from the electron beam, assuming the photon is always initiated from an electron, either in "quasi-real" TCS, or in real TCS (photon produced in a radiator). The polarization transfer depends on the electron and photon energy, as [17, 25] (4 − y) P (qr) = D y , (10) 4 − 4y + 3y2 where D is the degree of polarization of the initial electron. The polarization transfer factor as a function of the photon energy for an initial 11.5 GeV electron is displayed Fig. 12.

17 Figure 12: Polarization transfer as a function of the photon energy for an 11.5 GeV polarized electron beam.

5 User ﬁles

5.1 Input files In this section we present the user input file and how user options should be set. For versions using grids, the kinematic limits where cross sections have been calculated are indicated in following tables and in the input files from the generator.

The TCS input file options are presented Table 2, with the name of variables that can be modi- fied, the typical values that should be set, the recommendations and few explanations. Similarly, we present the DVCS input file options Table 3 and the DDVCS input file options Table 4.

5.2 Output files The standard output of the generator is a ROOT file, named DEEPGen_(index).root. The "index" is an arbitrary number provided by the user. It contains two ROOT Tree: SIM_Tree contains information about particle momenta in lab frame, kinematics, event weights, flags, and unique event number. Dump_Tree contains the same 4-vectors but they are also provided in other reference frames. It also contains the user input options, the phase-space factor and some useful information. This Tree is only filled for the first 50 events and is needed only for retrieving some useful information. The user doesn’t generally need it to perform analysis.

HEP files can be generated associated with a "kin" file containing other information. However, we recommend generating only files in the ROOT format. Any format with any weighting option can be printed in a loop reading the Tree. We detail in the following subsections the contents of the Tree for the different processes.

18 Variable name usage limits (grid) default value other recom- mandations Number of limit size of out- 10000 limit to 50000 for events to gener- put file memory ate Beam type real photon (0) 0 or 1 0 or 1 initial electron (1) Photon energy cross section [5, 11.5] GeV 11 less than elec- range tron if quasi-real Beam energy (if for photon flux [∼ 5, 11.5] GeV 11 > max(Eγ) electron beam) θγ(max) bremsstrahlung - 0 photon cone for angle max bremsstrahlung flux lepton type electron (1) 1 or 2 1 kinematic only, muon (2) no muons in cross sections Target lenght bremsstrahlung - 15 cm only electron mode Target composi- bremsstrahlung material (1,1) or 1001 only electron tion (A,Z) and EPA mode Target = p (1) or cross section 1, 2 1 n (2) Beam polariza- pol. cross sec- [0, 1] 0.8 electron po- tion dilution tions larization or factor linearly pol. photon Beam pol. vector polarized cross 0 (circular) 1 (x- 0 set 0 if unpolar- direction sections axis) 2 (y-axis) or ized 3 (45◦) Target polariza- polarized cross 0 (unpolarized), 3 set 0 if unpolar- tion direction sections 1 (x-axis), 2 (y- ized axis), 3 (z-axis) Target dilution polarized cross 0 to 1 0.7 factor section -t Mandelstam cross section [.04, 2.04] variable Q’2 outgoing photon cross section [.09, 9.2] GeV2 virtuality ◦ ◦ ◦ ◦ θCM azimuthal angle [30 , 150 ] [30 , 130 ] of decay leptons 2 2 Qmax quasi-real pho- 0 to 0.3 0.3 low Q domi- tons max. nate Output (0) ROOT, (2) 0, 1, 2 0 recommend only HEP, (1) both ROOT

Table 2: User’s input ﬁle parameters for TCS-type events generation. All units are GeV.

19 Variable name usage limits (grid) default value other recom- mandations Number of limit size of out- 10000 limit to 50000 for events to gener- put ﬁle memory ate Beam energy cross section 11 GeV 11 GeV only 11 GeV in this version lepton pair electrons (1) or 1 or 2 2 only 2 in this muons (2) version Target compo- luminosity 1,1 (1,1) disabled sition and size (A,Z) Target = p (1) or cross section 1, 2 1 only proton n (2) Beam polariza- polarized cross [0, 1] 0.8 polarization = tion dilution sections longitudinal factor Target polariza- polarized cross 0 (unpolarized), 3 set 0 if unpolar- tion direction sections 1 (x-axis), 2 (y- ized axis), 3 (z-axis) Target dilution polarized cross 0 to 1 0.7 factor section -t Mandelstam [.05, 2.05] variable xbj Bjorken variable [0.05, 0.5] Q2 initial photon [0.7, 5.7] virtuality y limit phase [0, 1] [0, 1] only for efﬁ- space ciency Output (0) ROOT, (2) 0, 1, 2 0 recommend only HEP, (1) both ROOT

Table 3: User’s input ﬁle parameters for DVCS event generation. All units are GeV

20 Variable name usage limits (grid) default value other recom- mandations Number of limit size of out- 10000 limit to 50000 for events to gener- put ﬁle memory ate Beam energy cross section 11 GeV 11 GeV only 11 GeV in this version lepton pair electrons (1) or 1 or 2 2 only 2 in this muons (2) version Target compo- luminosity 1,1 (1,1) disabled sition and size (A,Z) Target = p (1) or cross section 1, 2 1 only proton n (2) Beam polariza- pol. cross sec- [0, 1] 0.8 polarization = tion dilution tions longitudinal factor -t Mandelstam [.04,.84] variable xbj Bjorken variable [.1,.3] Q2 initial photon [0.9, 4.5] virtuality Q’2 ﬁnal photon vir- [1.7, 5.3] tuality ◦ ◦ ◦ ◦ θCM decay leptons [30 , 150 ] [30 , 130 ] Output (0) ROOT, (2) 0, 1, 2 0 recommend only HEP, (1) both ROOT

Table 4: User’s input ﬁle parameters for DDVCS event generation. All units are GeV

21 5.2.1 TCS Tree (SIM_Tree) General information:

- ALV_minus_lab = 4-vector or outgoing lepton in lab frame (all 4-vector "ALV_xxx" are arrays containing: E, px, py, pz)

- ALV_plus_lab = outgoing anti-lepton in lab frame

- ALV_gamma_in = incoming photon in lab frame

- ALV_Recoil_lab = recoil proton in lab frame

- Egamma = energy of incoming photon

- Qp2 = ﬁnal photon virtuality

- tt = t Mandelstam

- ttmin = minimal t allowed at this kinematic

- Phi_CMV = azimuthal angle of decay leptons plane versus reaction plane

- Theta_CMV = polar angle of decay electron versus reaction plane and virtual photon direction

- phi_beam = azimuthal angle of electron and incoming photon versus x-axis in case of quasi real beam, or arbitrary rotation of the reaction plane in case of real photoproduction

- CosThetagg = angle between the 2 photons in the gamma-target CM frame

- FlagSing = if =1, this event is in the near-singularity region and has to be rejected for proper reconstruction (cut in θ, φ)

- thetamin_nocut = minimal θ above the near-singularity region for this given E, t, Q’2. Has to be associated with φ value to apply a 2-D cut

- EventNumber = event number

- TrueEventNumber = actual number of events generated in hyper cubical space (for normalization)

Only if events are weighted:

- target_spindir = spin orientation of target (+1 or -1)

- beam_spindir = spin orientation of beam (+1 or -1)

- polbeamdeg = beam polarization transfer times initial electron polarization rate (dilution factor)

- poltargetdeg = dilution factor of target polarization

- W_tot_unpol = actual total unpolarized cross section with ﬂux factor for quasi real beam

- W_BH = B-H "only" unpolarized cross section with ﬂux factor for quasi real beam

22 - W_TCS = TCS "only" unpolarized cross section with ﬂux factor for quasi real beam

- W_tot_pol = beam + target actual polarized cross section (quasi real beam + ﬂux / dilution factors)

- W_tot_pol_beam = beam only polarized cross section (quasi real photon + ﬂux / dilution factor)

- W_tot_pol_target = target polarized cross section (quasi real photon + ﬂux / dilution factor)

- BSA = approximate single beam spin asymmetry at this kinematic (no dilution)

- TSA = approximate single target spin asymmetry (no dilution)

- BTSA = approximate double beam and target spin asymmetry (no dilution)

Only in case of incoming electron (quasi-real photoproduction):

- ALV_el_in = incoming electron in lab frame (case of quasi-real beam only)

- ALV_el_out = scattered beam electron in lab frame (case of quasi-real beam only)

- Q2 = incoming photon virtuality in case of quasi real beam

- theta_gamma = polar angle of incoming photon in case of quasi real beam

- theta_beam = scattered electron polar angle in case of quasi real beam

- yy = electron beam energy loss

- epsilon = transverse ﬂux

- VirtualFlag = if 0 the incoming photon is real (real beam or bremsstrahlung in target), if 1 the incoming photon is quasi real

- Flux_qr = quasi real equivalent ﬂux

- Flux_bmr = bremsstrahlung in the target equivalent ﬂux

- cross_tot_unpol = unpolarized cross section for real photon beam

- cross_BH = Bethe-Heitler "only" unpolarized cross section for real photon beam

- cross_TCS = TCS "only" unpolarized cross section for real photon beam

- cross_tot_pol = beam + target polarized cross section (real photon / dilution factors)

- cross_tot_pol_beam = beam only polarized cross section (real photon / dilution factor)

- cross_tot_pol_target = target polarized cross section (real photon / dilution factor)

Only in case of linearly polarized photon:

- Psi_s = angle between reaction plane and spin direction of linearly polarized incoming photon (option linear polarization only)

23 - cross_BH_x = polarized differential cross section for BH if the beam spin is oriented "in plane" for the given kinematic

- cross_BH_y = polarized differential cross section for BH if the beam spin is oriented "tangent to plane" for the given kinematic

Only in case of transversely polarized target:

- phi_s = target spin azimuthal angle versus x+z plane in lab frame

- theta_s = target spin polar angle versus z-axis in lab frame

5.2.2 DVCS Tree (SIM_Tree) Particle momenta

- ALV_minus_lab = 4-vector or outgoing lepton in lab frame (all 4-vector "ALV_xxx" are arrays containing: E, px, py, pz)

- ALV_plus_lab = outgoing anti-lepton in lab frame

- ALV_gamma_in = incoming virtual photon in lab frame

- ALV_Recoil_lab = recoil proton in lab frame

- ALV_gamma_out_lab = outgoing real photon in lab frame

- ALV_el_in = electron beam incoming in lab frame

General:

- Q2 = incoming photon virtuality in case of quasi real beam

- theta_beam = scattered electron polar angle in case of quasi real beam

- phi_beam = azimuthal angle of electron and incoming photon versus x-axis

- yy = electron beam energy loss

- Xbj = Bjorken variable

- tt = t Mandelstam

- ttmin = minimal t allowed at this kinematic

- Phi_LH = angle between leptonic (beam, scattered) and reaction plane

- CosThetagg = angle between the 2 photons in the gamma-target CM frame

- epsilon = longitudinnal transverse ﬂux factor

- EventNumber = event number

- TrueEventNumber = actual number of events generated in hyper cubical space (for normalization)

24 Only if events are weighted:

- target_spindir = spin orientation of target (+1 or -1)

- beam_spindir = spin orientation of beam (+1 or -1)

- polbeamdeg = beam polarization transfer times initial electron polarization rate (dilution factor)

- poltargetdeg = dilution factor of target polarization

- W_tot_unpol = actual total unpolarized cross section

- W_BH = B-H "only" unpolarized cross section

- BSA = approximate single beam spin asymmetry at this kinematic (no dilution)

- TSA = approximate single target spin asymmetry (no dilution)

- BTSA = approximate double beam and target spin asymmetry (no dilution)

- W_DVCS = DVCS "only" unpolarized cross section

Only if transversely polarized target: phi_s = target spin orientation relative to the reaction plane (photons)

5.2.3 DDVCS Tree (SIM_Tree) General:

- ALV_minus_lab = 4-vector or outgoing lepton in lab frame (all 4-vector "ALV_xxx" are arrays containing: E, px, py, pz)

- ALV_plus_lab = outgoing anti-lepton in lab frame

- ALV_gamma_in = incoming photon in lab frame

- ALV_Recoil_lab = recoil proton in lab frame

- ALV_el_in = incoming electron in lab frame (case of quasi-real beam only)

- ALV_el_out = scattered beam electron in lab frame (case of quasi-real beam only)

- Q2 = incoming photon virtuality in case of quasi real beam

- theta_beam = scattered electron polar angle in case of quasi real beam

- phi_beam = azimuthal angle of electron and incoming photon versus x-axis

- yy = electron beam energy loss

- Xbj = Bjorken variable

- Qp2 = ﬁnal photon virtuality

- tt = t Mandelstam

25 - ttmin = minimal t allowed at this kinematic

- Phi_LH = angle between leptonic (beam, scattered) and reaction plane

- Phi_CMV = azimuthal angle of decay leptons plane versus reaction plane

- Theta_CMV = polar angle of decay electron versus reaction plane and virtual photon direction

- CosThetagg = angle between the 2 photons in the gamma-target CM frame

- epsilon = transverse ﬂux

- EventNumber = event number

- TrueEventNumber = actual number of events generated in hyper cubical space (for normalization)

Only if events are weighted:

- polbeamdeg = beam polarization transfer times initial electron polarization rate (dilution factor)

- beam_spindir = spin orientation of beam (+1 or -1)

- W_tot_unpol = actual total unpolarized cross section

- W_BH = B-H "only" unpolarized cross section

- W_DDVCS = DDVCS "only" unpolarized cross section

- BSA = approximate single beam spin asymmetry at this kinematic (no dilution)

5.2.4 Additional information: Dump_Tree This tree is filled only for the first 50 events of the run, and for the last one. It can be used for checks and for general information. Depending on which process is generated, some information may be irrelevant or the values are not filled.

- indexrun = run index, i.e. number of the ﬁle given as an input when running the code

- param_initﬁle = printout of input parameters as given in the user input ﬁle, in the same order (array)

- EventNumber = event number

- TrueEventNumber = number of events actually generated at this step (for normalization purpose)

- phase_space = phase space factor according to the user input ﬁle

- ALV_minus_lab = 4-vector or outgoing lepton- in lab frame (all 4-vector "ALV_xxx" are arrays containing: E, px, py, pz)

- ALV_plus_lab = outgoing lepton+ in lab frame

26 - ALV_gamma_in = incoming photon in lab frame

- ALV_el_in = incoming electron in lab frame (case of quasi-real beam if tcs, or ddvcs)

- ALV_Recoil_lab = recoil proton in lab frame

- ALV_el_out = scattered beam electron in lab frame (case of quasi-real beam if tcs, or ddvcs)

- ALV_gamma_out_lab = outgoing (real) photon in lab frame

- ALV_minus_CMV = lepton- in virtual photon center of mass frame

- ALV_plus_CMV = lepton+ in virtual photon center of mass frame

- ALV_minus_CMeP = lepton- in photon-proton center of mass frame

- ALV_plus_CMeP = lepton+ in photon-proton center of mass frame

- ALV_gamma_CMeP = incoming photon in photon-proton center of mass frame

- ALV_Recoil_CMeP = recoil proton in photon-proton center of mass frame

- ALV_Virtual_CMeP = ﬁnal virtual photon in photon-proton center of mass frame

- ALV_Virtual = ﬁnal virtual photon in lab frame

- ALV_Target_CMeP = incoming proton in photon-proton center of mass frame

- Q2 = incoming photon virtuality in case of tcs quasi real beam or ddvcs

- theta_beam = scattered electron polar angle in case of tcs quasi real beam or ddvcs

- phi_beam = azimuthal angle of scattered electron and photon plane in case of tcs quasi real beam or ddvcs

- yy = beam electron energy loss

- Xbj = valid for ddvcs

- Qp2 = ﬁnal photon virtuality

- tt = t Mandelstam

- ttmin = minimal t at this kinematic

- Phi_CMV = azimuthal angle of decay lepton plane versus reaction plane

- Phi_LH = azimuthal angle of scattered electron plane versus virtual photon (reaction) plane for ddvcs

- Theta_CMV = polar angle of decay lepton minus versus virtual photon orientation

27 6 Analysis

6.1 Generator units The default unit for all momenta, invariant mass... is GeV (GeV/c...) and the default unit for angle is radians. For the different reactions, the units are: • DVCS+BH and VCS+BH weights are expressed in nb.GeV −4rad−1. • TCS+BH weights are expressed in pb.GeV −4.rad−2. • DDVCS+BH weights are expressed in pb.GeV −6.rad−3. • Phase space factors for the different reactions are expressed in the inverse of the unit for the given reaction (see next paragraph).

6.2 Normalization Generated cross sections can be normalized according to the dimension of the generated phase space and the actual number of generated events in this hypercube ("TrueEventNumber"). The user needs to sum the "TrueEventNumber" of the last event of each file, and calculate the constant phase space factor, such as: • "process" = "tcs": event weights are unpolarized or polarized cross sections, dσ/(dtdQ02dθdφ), Full Phase Space (PS) = ∆|t|∆Q02∆θ∆φ. • "process" = "dvcs": event weights are unpolarized or polarized cross sections, 2 dσ/(dxbjdtdQ dφLH ), 2 PS = ∆xbj∆|t|∆Q ∆φLH . • "process" = "ddvcs": event weights are unpolarized or polarized cross sections, 2 02 dσ/(dxbjdtdQ dQ dφLH dθdφ), 2 02 PS = ∆xbj∆|t|∆Q ∆Q ∆φLH ∆θ∆φ. By definition, for the variable A, ∆A = A(max) − A(min) from the user input file. The weights are not differential in beam energy for any of the reactions. For providing analysis results, the beam energy range (or fixed value) has to be taken into account. By default, azimuthal angles are generated such that ∆φ = ∆φLH = 2π.

The integrated cross section in a given bin where N events have been reconstructed at the analysis level is PN W ∗ PS σbin = i i , Pﬁles (11) j (T ) where "T" is the "TrueEventNumber" of the last event of each generated ﬁles used in the analysis. PS is the constant phase space factor. W is the event weight for the unpolarized or polarized process considered (W _tot_unpol is proportional to the counting rates). To get the counting rates of an experiment, this number has to be multiplied by the integrated luminosity and an acceptance factor.

Approximation of spin asymmetries are provided (BSA...). No normalization is needed. The PN averaged asymmetry in a kinematic bin is i Wi/Nb, where Nb is the number of events in the bin.

28 6

5 Q2 (GeV2) 4

1 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Xbj

2 Figure 13: DVCS+BH phase space as a function of xbj and Q , weighted by the unpolarized cross section. Note that no cut is applied on the generated events. Units in Q2 are GeV2.

6.3 Results This section contains a few distributions of unweighted and weighted generated events for the different reactions. The following examples correspond to kinematic distributions and to generated spin asymmetries.

2 We display Fig. 13 the correlation between xbj and Q for generated DVCS+BH events at an 11 GeV beam energy (the weighting is proportional to the total cross section). The DVCS+BH beam spin asymmetry, longitudinally polarized target spin asymmetry and corresponding double spin asymmetry are shown Fig. 14, as a function of the azimuthal angle φLH , in the kinematic bin: 2 2 2 0.2 < xbj < 0.25, 4 < Q < 5 GeV , −0.6 < t < −0.5 GeV .

We display Fig. 15 the phase space for the TCS+BH reaction with a photon beam from 5 to 11.4 GeV, coming from an 11.5 GeV electron. The φ and θ dependence of the beam spin asymmetry is shown Fig 16. Additional recent examples of generated TCS+BH can be found in [16, 26].

2 02 Fig. 17 shows the correlation between xbj, Q and Q for generated DDVCS+BH events. We also display Fig. 18 the DDVCS+BH beam spin asymmetry as a function of φCM and φLH in a nar- 2 2 2 2 2 row kinematic bin: .1

29 0.8 0.2 0.6 0.15 0.4 0.1

beam asym. 0.2 0.05 0 0 − −0.2 0.05

target asym, longitudinal −0.1 −0.4 −0.15 −0.6 −0.2 1 2 3 4 5 6 1 2 3 4 5 6 Phi_LH Phi_LH 0

−0.1

−0.2

−0.3

−

beam + target L asym. 0.4

−0.5 1 2 3 4 5 6 Phi_LH

Figure 14: DVCS+BH generated spin asymmetries from a polarized electron beam (top left panel), longitudinally polarized target (top right panel), polarized beam+longitudinally polarized target 2 2 (bottom panel). The beam energy is set at 11 GeV and 0.2 < xbj < 0.25, 4 < Q < 5 GeV , 2 −0.6 < t < −0.5 GeV . Asymmetries are displayed as a function of φLH (rad.).

30 Figure 15: TCS+BH generated phase space as a function of Q02, E (photon) and t.

0.3 0.2 0.1 beam asym 0 −0.1 −0.2

−0.3 2.22.4 1.82 1.41.6 6 5 4 3 1 1.2 Phi_CMV 2 1 0 0.8 Theta_CMV

02 Figure 16: TCS+BH beam spin asymmetry as a function of φ, for 5

31 2 02 2 Figure 17: Unweighted DDVCS+BH like events as a function of xbj, Q and Q . Units are GeV .

Figure 18: DDVCS+BH beam spin asymmetry as a function of φCM and φLH (units are radians).

32 7 How to?

7.1 Binaries Binaries are available to the JLab user community. The directory can be copied in the user home directory. Examples of generated ﬁles are available with the corresponding input ﬁles.

Binaries: /work/halla/solid/mboer/public/Generator_publicversion/version4

Example files (root file + used input files + log files): /work/halla/solid/mboer/public/Generator_publicversion/examples/version4/ (directory TCS, DVCS, DDVCS for each reaction).

7.2 Running the code Go to the directory where the generator binary ﬁles are copied, then:

1. run "set.csh".

2. Modify the parameters in the (process).input ﬁle.

3. run "./DEEPGen (process) (index) (seed)"

Run "DEEPGen" to see the available option list.

The processes (see section 2.1) are labelled: (process) = dvcs, tcs, ddvcs, ps_eeel_fix, ps_eephoto_fix, ps_vcs_fix. The label (index) is the run number index of the output file, defined by the user. The label (seed) is a random seed, only needed in batch mode.

33 8 Conclusion

We presented the framework of the DEEPGen event generator. The ﬁrst version was developed in 2015 to perform physics impact studies for future TCS and DDVCS measurements. Past versions of the event generator have been used to develop future JLab TCS and DDVCS experiments [26, 27, 28, 29, 30].

Current version of the event generator support generation of the DVCS+BH, TCS+BH and DDVCS+BH reactions. We provide several beam and target polarization options. The cross sections are provided for the full unpolarized reaction, and also for sub-processes. This approach enable studies of the optimal phase space regions for measurements at JLab. Spin options allow for directly generating observables of interest in addition to the unpolarized counting rates.

We regularly update the generator with new options available from the webpage. Options to study other reactions and to use different models are under development.

9 Aknowledgements

The author would like to thanks M. Guidal, R. Paremuzyan, I. Hrivnacova, C. Diarra, R. Lustrat and A. Luboz. This work was initially developed with the technical support of the IPN Orsay IT division, using the clusters of IPN Orsay (in 2015) and of Jefferson Laboratory (in 2017 and 2018). This work is supported by the US Department of Energy awards DE-SC0016577 and DE-FG02- 94ER4084.

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