# Calculating production cross-section of muonic-pair

Dear Experts,

In my simulation gold ions collide with tungsten target. In some events muonic pairs are born. After that i calculate their invariant mass. And i want to rescale that distribution to obtain cross-section. What steps i should take? For example i know luminosity of experiment. Can i just obtain cross section like number of pairs/luminosity?

Dear Semyon,

My apologies for the delay.

In some events muonic pairs are born. (…)

From which processes do you observe muon pairs in FLUKA and how do you identify them?

The only implemented mechanism I can think of now is a photon producing the pair (you can intercept these events in USDRAW filtering ICODE==237), i.e. not as direct product from an Au interaction but as a result of a subsequent photon interaction elsewhere in the shower.

Let me know if you spot obvious counterexamples in your simulation.

Another thing of course is if you care about muon production (mu+ and/or mu-, but not in pairs). Clarify as/if needed.

In my simulation gold ions collide with tungsten target (…) and i want to rescale that distribution to obtain cross-section.

Reading your post literally, one understands you have a Au ion beam on a W (probably thick) target (i.e. the W is at rest, as opposed to starting with an Au-W beam-beam collision with SPECSOUR as source term, see below). Your comment later on the luminosity makes me doubt, though.

Under these allegedly (thick) fixed-target circumstances, your desired final state (muon pair production) is fed by different interaction mechanisms, e.g.:

• The Au ion knocking out a sufficiently energetic delta ray, which might later emit a Bremsstrahlung photon, which might produce the muon pair.

• The Au ion undergoing a nuclear interaction with W, during which neutral pions might be produced, which produce the photons that eventually produce the muon pair

(Depending on the ion energy some other processes might become relevant)

Sure, you can always score a yield of muon-pair-production events in your geometry and scale it by a pertinent cross section to have the right physical units, but that’s not advisable for thick targets, since
you’re almost guaranteed to obtain a formally questionable quantity:

• since your desired events (muon pair production) are fed by different interaction mechanisms of the primary Au ion (delta ray production, nuclear inelastic interaction, etc) you’d have to carefully keep track of their respective interaction cross sections. Scaling by the inverse luminosity as you suggest, while dimensionally appealing, tacitly assumes the integrated cross section for a (typically just one!) clearly specified interaction type. In your case it’s probably the Au-W nuclear inelastic interaction (?), so you’d not properly account for processes a la Au → delta ray → photon → muon pair, rendering the final quantity very questionable.

• you’d have the original Au interaction (nuclear inelastic, delta-ray production, etc) somewhere in your sample, and the muon pair would be produced elsewhere in the sample / simulation geometry as the shower develops. This renders your eventually scored “cross section” strongly geometry-dependent, which if one is feeling permissive could be fine for internal consumption as a coarse measure of occurrence, but it would be essentially useless everywhere else: ordinarily one gives cross sections as a measure of the occurrence of a particular final state arising as a direct result from an individual interaction of a well-specified interaction type. You’d instead obtain a convolution of many ingredients for the resulting quantity to be acceptable as a cross section (contributions from different interaction mechanisms of different particle species at different positions, occurrence and energy degrading of involved shower particles in a geometry-dependent way, etc).

With the info at hand, I’d see two possibilities for now:

1. be content with a yield (not a cross section) from your specific thick-target/geometry situation. You count the number of muon pairs (optionally resolved as a distribution as a function of whichever
kinematic variables you need, dividing by the corresponding bin widths), divide by the primary event weight, so as to report a meaningful and formally clean quantity for your experimental setup.

2. instead of a thick target situation you are actually starting from a Au-W collision (e.g. a la SPECSOUR in FLUKA), which would align with your luminosity comment, but not with your “Au beam on W” one. In this case at least the source interaction term (nuclear inelastic, electromagnetic dissociation, etc) is clearly isolated, so the scaling from a yield up to a quantity involving units of mb can be easily done in a more meaningful way. You’d still suffer from the dependence on your specific geometry and from a convolution of various interaction types of various particles in various positions, though.

With kind regards,

Cesc