# Email from Richard Jones (06/14/2021)

Follow-up mail, 06/15/2021, from Richard:

Following up on our discussion today re. a generator for nucleon inelastic Bethe-Heitler processes: to summarize, based on Lorentz symmetry, single-photon-exchange leads to a factorization in the cross section for electron scattering from a general nuclear target: dsigma ~ L_mu_nu * W_mu_nu

where L_mu_nu = <i| j_nu |f> <f| j_mu |i> and W_mu_nu = <I| J_nu |F> <F| J_mu |I>

where |i> , |f> are the initial and final leptons and |I>, |F> are the initial and final nuclear states

If you sum over all nuclear final states (inclusive electron scattering), the sum{ |F> <F| } can be replaced with a Lorentz scalar operator P, and W_mu_nu = <I| J_nu P J_mu |I> that is a known operator product sandwiched between free nucleon states.

Lorentz symmetry implies the W_mu_nu can only contain two Lorentz tensors: W_mu_nu = W_1(q,nu) <I| g_mu_nu |I> + W_2(q,nu) <I| sigma_mu_nu |i>

hence all the nuclear structure related to inclusive electron scattering is contained in these two structure factors W_1 and W_2. this translates directly into a similar factorization for a process with a real photon, where L_mu_nu becomes <f+| A_slash \gamma_mu |f-> <f-| A_slash \gamma_nu |f+>

Based on this, the total rate for inelastic nucleon pair production can be calculated, without regard to specific final states. But if we want an event generator, we need to break the sum over |F> down into individual exclusive nuclear final states because the entire final state needs to be simulated, not just the lepton part. For that, we need a separate W_1 and W_2 for each exclusive final state, and these will depend on the final nuclear state kinematics in more ways than just q^2 and nu.

---

Simon,

I have implemented the Bernard (G4BetheHeitler5DModel) polarization-dependent pair production in hdgeant4 now. I have also verified that it gives the same results for forward pairs and triplets as my own built-in generator that includes a more complete treatment of the process. This Bernard generator produces unweighted events, so it is better for inclusive pair samples than my own weighted generator in that way, but it is unable to generate pairs with higher pair masses because it is only good for an infinite mass target.

I think we need a meeting to discuss how to incorporate inelastic processes. The reason is that there are many things that this can mean, depending on the context.

- "inelastic atomic" -- this means pair production from a bound electron in the target, with the recoil carried away by a free electron. I call this the triplet process.
- "inelastic nuclear" -- this means pair production from a proton bound in a nucleus, with the recoil carried away by a knock-out proton, with the A-1 system being a spectator
- "inelastic nucleon" -- this means pair production from a proton target, bound or not, with the recoil carried away by a multi-hadron system, eg. nucleon + pions.

Case #1 is already covered by my generator, and also by the Bernard generator option I have just added to hdgeant4.

Case #2 could easily be added, but I don't think it is relevant to this experiment with a hydrogen target.

Case #3 is complicated. If this is what you want, we should discuss it. Some scheme for specifying the target electromagnetic current operator will need to be adopted. You CANNOT just take the standard pair production 5D differential cross section from a proton target (infinite mass) and simply replace the Dirac form factors F1 and F2 by inelastic form factors W1 and W2. That might give non-polarization inclusive yields that look like real data because the inelastic form factors have been derived from such measurements, but it will not give predictive power for polarization observables. Here are some possibilities for how option #3 might be implemented.

- assume factorization, replace the nucleon vector current with a quark current, and sample from a PDF at some approximate scale. The inclusive recoil system could be generated as an N-pion + nucleon final state with the appropriate value of W corresponding to the generated x.
- implement the final state explicitly as a sum over a ladder of N*'s, compute the electromagnetic currents for each J-value (1/2, 3/2, 5/2, etc.) from Lorentz symmetry, and model the dozens of transition form factors for each N* resonance from some model or database of values.
- model the nucleon as a nucleon core + pion cloud, and implement the inelastic part as incoherent scattering from the pion cloud with a spectator core.

I really doubt that #1 would make any sense for W in the resonance region, and even at W > 2 GeV^2 it would only work for the very small fraction of pairs with qrecoil > 1GeV/c. One might make an argument from duality for why one might extend this below W=2GeV. Option #2 sounds like a lot of work. Option #3 seems like what Tsai is describing in his section with Eqs. B58, 59. That treatment given by W1, W2 in B58,B59 is not going to give reliable polarization observables unless you restrict the process to near qrecoil<<1GeV/c where the spin structure of the target is unresolved. Because of this, option #1 and #3 are complementary. However, if qrecoil<<1GeV/c then the elastic processes in Cases #1 and #2 will dominate over Case #3.

So unless one is specifically interested in exclusive inelastic final states, or events in the tail to qrecoil > 1GeV/c, leaving out Case #3 and covering Case #1 and #2 should be sufficient to get a good description of the inclusive pair population, shouldn't it?

-Richard Jones

On Thu, May 27, 2021 at 1:47 PM Richard Jones <richard.t.jones@uconn.edu> wrote: Simon,

If you think that the Tsai's cross-sections are just a subset of your more general treatment involving both space- and time-like photon exchange, as well as polarization --- and hence plugging Tsai into the Monte Carlo is a waste of time --- perhaps you can at least let us know how the elastic "condition" in the Monte Carlo (as claimed by Mark) can be relaxed, or at least extract the inelastic form factors from my code if you find them useful.

Let me have a look at your code and see if there are parts there I can pick out and incorporate into my generator. It is not a waste of time to plug in Tsai, and I plan to do that as well, although I think you will see that the approximation used in that calculation makes it less useful for GlueX than what I have created. We are having a collaboration meeting this week, but starting next week I hope to get a look an the extension to my BHgen facility.

-Richard Jones

On Thu, May 27, 2021 at 1:22 PM Širca, Simon <simon.sirca@fmf.uni-lj.si> wrote:

- Message sent from a system outside of UConn.*

Dear Richard,

it is great that you were able to allocate some time for this!

Indeed, the Tsai's treatment of the Bethe-Heitler process involves only space-like photon exchange. In his paper Tsai gives various formulas for the elastic and inelastic form-factors that enter the cross-section expressions, but I have replaced the latter by the modern parameterizations of W1 and W2 by Christy and Bosted.

Immediately the question popped up: what range of Q2 and nu of the virtual photon is covered within a typical GlueX acceptance? It is at this point that Mark told me that the Bethe-Heitler events in the Hall D Monte Carlo are currently implemented such that the Q2 and nu are always elastic. This keeps on puzzling me. Surely the inelastic domain must be reached at least for some events?

In the attachment please find my code fragments. They are in the form of a stand-alone C++ program at the moment, with some Numerical Recipes routines. Sorry; if you wish, I can try and incorporate all this stuff into a class, but I reckon you would be faster in implementing this, especially because you would know how to plug this into the geant framework.

The main code is pbh6.cc, and the test code that runs it is testpbh.cc, all very simple. See README.

If you think that the Tsai's cross-sections are just a subset of your more general treatment involving both space- and time-like photon exchange, as well as polarization --- and hence plugging Tsai into the Monte Carlo is a waste of time --- perhaps you can at least let us know how the elastic "condition" in the Monte Carlo (as claimed by Mark) can be relaxed, or at least extract the inelastic form factors from my code if you find them useful.

I am super happy that you are willing to look into this and potentially to plug it into the Hall D Monte Carlo, so please let me know if you need more information from me.

With kind regards,

Simon

--

prof. dr. Simon Sirca Faculty of Mathematics and Physics, University of Ljubljana Jadranska ulica 19 1000 Ljubljana, Slovenia Tel: +386 1 4766-574, Fax: +386 1 2517-281

________________________________________
From: Richard Jones <richard.t.jones@uconn.edu>
Sent: Wednesday, May 26, 2021 12:57 PM
To: Širca, Simon
Subject: Re: Incorporating Mo/Tsai BH cross-section into geant4 / Hall D sim (fwd)

Simon,

I now get back to you on this question after a long delay, sorry about that.

For the REGGE experiment, we would like to have circularly polarized photons and I imagine that the proper XS to include would be the one given by Tsai (1974), second attachment, Eqs. (2.1) and (2.3), which I have coded in C++ with all the form-factors for the elastic and inelastic (resonance region and DIS) cases.

This sounds good. I currently have the full polarization-dependent QED differential cross section currently implemented in hdgeant4. There are 4 form factors needed for the elastic process, two for the space-like scattering and two for the time-like exchange diagrams. Currently the form factors for the proton (hydrogen target) are parameterizations of F1 and F2 that I found on the NIST web site for the space-like part, and I have set the time-like form-factor to zero because I don't know what to put in for that. But the hooks are all there, and I can import any model you have for these form factors. At the kinematics I have looked at for GlueX, the time-like contribution is very small, but I am not sure this is true in general.

I understand that geant4 "somehow" (by you) gets ported into the Hall D simulation software one way or another. My question is: can Bernard's work be used in Hall D software and adapted to our needs? If not, can my C++ code for Tsai's six-fold XS which is what I believe is relevant for us, be plugged into the Hall D simulation and HOW?

In my scheme I do not make any approximations, I just compute the full QED amplitude by doing the Dirac traces in C++ and summing the amplitudes from all of the diagrams. I don't see how implementing Tsai's formula can offer any advantage in this regard, but I can certainly include it as an alternative during the build process. Sure, why not?

My comprehension would be that a lab photon comes in, the proton or whichever target is at rest, then one moves to the CMS, the e+ and e- momenta/angles are sampled according to some strategy, then the virtual photon is exchanged with the target, eating away some energy/momentum from one of the leptons, then one goes back to LAB and the XS weight a la Tsai is applied. Is this picture wrong? Mark tells me that event generation in the present version of the Hall D simulation is NOT implemented in this manner.

That is more or less correct. This description sounds like you are considering only the diagrams with the space-like photon exchange, whereas my algorithm covers all of the diagrams, and all possible polarization states for the incoming and outgoing electrons/positrons/nucleons, and not just the photon polarization. But that is the general strategy of course, yes. I am not sure what Mark was referring to... I have never given a talk on this so maybe he misunderstood what the code is doing?

To make the question even shorter: can we make use of Tsai's expression (2.1) in the Hall D simulation? As far as I can see, this is all that is needed to go beyond the purely elastic regime implemented presently.

Sure, why not? Would you like me to do this by coding it myself, or do you have a C++ class that you would recommend to me?

-Richard Jones

On Wed, May 12, 2021 at 5:01 AM Simon Sirca <simon.sirca@fmf.uni-lj.si<mailto:simon.sirca@fmf.uni-lj.si>> wrote:

- Message sent from a system outside of UConn.*

Dear Richard,

did you perhaps have a chance to look into this? (Please see below.)

With kind regards,

Simon

--

prof. dr. Simon Sirca Faculty of Mathematics and Physics University of Ljubljana Jadranska ulica 19 Tel: +386 1 4766-574 1000 Ljubljana, Slovenia (EU) Fax: +386 1 2517-281

Forwarded message ----------

Date: Tue, 16 Mar 2021 13:44:09 +0100 From: "Sirca, Simon" <simon.sirca@fmf.uni-lj.si<mailto:simon.sirca@fmf.uni-lj.si>> To: Richard Jones <richard.t.jones@uconn.edu<mailto:richard.t.jones@uconn.edu>> Cc: Alexandre Deur <deurpam@jlab.org<mailto:deurpam@jlab.org>>, Mark-Macrae Dalton <dalton@jlab.org<mailto:dalton@jlab.org>>,

Justin Stevens <jrsteven@jlab.org<mailto:jrsteven@jlab.org>>

Subject: Incorporating Mo/Tsai BH cross-section into geant4 / Hall D sim

Dear Richard,

a specific version of a Bethe-Heitler event generator for linearly polarized photons and e+e- production has been recently (since release 10.7) implemented into geant4: see attached paper by Bernard and

/home/sirca/geant4.10.07.p01/source/processes/electromagnetic/standard/include/G4BetheHeitler5DModel.hh /home/sirca/geant4.10.07.p01/source/processes/electromagnetic/standard/src/G4BetheHeitler5DModel.cc

For the REGGE experiment, we would like to have circularly polarized photons and I imagine that the proper XS to include would be the one given by Tsai (1974), second attachment, Eqs. (2.1) and (2.3), which I have coded in C++ with all the form-factors for the elastic and inelastic (resonance region and DIS) cases.

I understand that geant4 "somehow" (by you) gets ported into the Hall D simulation software one way or another. My question is: can Bernard's work be used in Hall D software and adapted to our needs? If not, can my C++ code for Tsai's six-fold XS which is what I believe is relevant for us, be plugged into the Hall D simulation and HOW?

My comprehension would be that a lab photon comes in, the proton or whichever target is at rest, then one moves to the CMS, the e+ and e- momenta/angles are sampled according to some strategy, then the virtual photon is exchanged with the target, eating away some energy/momentum from one of the leptons, then one goes back to LAB and the XS weight a la Tsai is applied. Is this picture wrong? Mark tells me that event generation in the present version of the Hall D simulation is NOT implemented in this manner.

To make the question even shorter: can we make use of Tsai's expression (2.1) in the Hall D simulation? As far as I can see, this is all that is needed to go beyond the purely elastic regime implemented presently.

With kind regards,

Simon

--

prof. dr. Simon Sirca Faculty of Mathematics and Physics, University of Ljubljana Jadranska ulica 19 1000 Ljubljana, Slovenia Tel: +386 1 4766-574, Fax: +386 1 2517-281