Basic concepts of Monte Carlo and reconstruction. 1

Now that I described in detail how a meson wave-function is a superposition of its quark-contents through the latter’s wave-function as a solution to a differential equation known as a Schrodinger’s equation its time to describe a little more how exactly this is worked out in an experimental facility.

Since I gave the {K-zero + anti-K-zero} wave-function as a production facility for the K-long or the K-short mesons which are produced at the same time from such a wave-function but not at equal rates and with same mass but not with equal lifetime, here is a specific example of how this occurs in nature.

Reconstruction of

from the simulation of a two body decay.

A simulation is a “mimicry” of the actual reality so that we could understand better whats going on even if we may not have access to all the real information and also we may not simply have such information to start with in a given situation and we need to have a well-informed guess about how to go about understanding such physical situations..

A simulation is like a rehearsal before a drama. Its not actual but it has a factual proximity to the actual drama. And a two body decay is a jargon for processes in particle or nuclear physics usually, like one given:

Here the particle or the wave-function state in the left, decays into two particles as shown and each of these two particles or wave-function states [also called as state-vectors] are called final-state-particles. This is why its a two body decay. Because one body goes into two.

But these two may in turn decay into other possible particles and the further produced particles into more and more like in a nuclear chain reaction. This is whats called a topography as explained in my adjacent article written prior to this [link] A topography is an actually-connected chain of decay-channels. So the final-ends of this chain are now called final-state particles. In other words all the particles that do not decay at all in a topography are called final state particles. eg here Klong and pi-zero are the final state particles.

But given this topography can be extended to more, eg the Klong can decay into three pions [] and the pi-zero can decay into two gammas [gamma =  is a photons or light quantum with  various energy possibilities] So with such an extended topography the final state particles are now: {,,,} The is a charge-less particle but the may or may not be charged. Similarly the neutral charmed-meson Dzero as shown here may not be the actual initial particle from which everything emanates. The Dzero might in turn emit from its possible parents. Not everything is possible from everything else, like a fish does not produce shrimps but fish, elementary particles can produce only particular kind of other elementary particles. This is decided by what we call the physical laws of nature or simply laws of nature. eg the laws of nature here that are involved are the ones that are studied in Quantum Mechanics, Relativity and ordinary motion laws. [no emotion laws, sorry, love ain’t cutting our hearts]

The Dzero usually comes from what are called other D’s which could be charged or neutral or resonances. Resonances are very-very-short-lived particles and can decay into other particles. The parents from which daughter-particles are emitted are always heavier than the daughter particles. This is a physical law of nature called “energy conservation”. The D’s which are heavier than Dzero can come from particles such as B mesons. I have described in much detail what are mesons, in the preceding article [link].

Eventually such reactions or channels are induced by whats called a virtual particle. A virtual particle is one which is not real. It has to be consistent with the nature’s laws as the real particles. But the virtual ones actually do not exist. Which is why one of their application is in Monte Carlo. It approximates the role of actual particles. In other words the virtual particles actuate or simulate the role of real particles. I have described the meaning of virtual particles in another article of mine.  [link]

Now we have a simple problem to handle. In reality its quite quite complicated on which I worked for several years. But here is the very basic idea. We have a Dzero that decays into a Klong and a pi-zero. All three are mesons. But we are not worried about this at this point. This process occurs in nature. We have information about this process from two sources. One: what we know about other related processes, eg the properties of Dzero, Klong and pizero in different situations rather than ours. eg we know about Dzero in a decay: {D* -to- Dzero + pizero} which does not concern us directly. But what we have done for decades is when we learned something about any particles or processes we kept such information.

This rigorous list of information regarding properties of particles and how they behave when they pass through matter etc is called a Monte Carlo. In colloquial terminology you will hear this MC being replaced by Standard Model but thats like saying “Physics” each time Sachin hits a boundary. Harsha Bhogle goes “Here Sachin could have changed his Physics and made it into a six”. Sachin hears and says “yeah, try it yourself”. In the same sense we say Standard Model or [more precise: Standard Model Monte Calo] predicts this or that property. Standard Model is vast and it is basically Quantum Mechnaics itself in its content and consistencies. So since this process occurs in nature and we have useful information about such we have one source of information.

The other source of information is direct. Its from a set of experimental facilities that tell us exactly what is happening when this process is occurring in our machine. Our machine is an accelerator and a detector. Its like TV center is the accelerator and your TV is the detector. You see whats broadcast. Similarly we see on our detector this process if it occurs among 100 others that can occur in the detector. This is called real-experiment or experimental data. The previous source was Monte Carlo or simulation data.

Then we match the simulation data with the real data. This gives us in-depth understanding of the actual process and any chance to update our knowledge from the past regarding this and other processes. That means since there are literally 100s or  even 1000s of decay processes in nature of various kinds and we study all these in the particle accelerators spread across the world, information gained here regarding this or that process comes in handy for there for this or that process.

Since there are two sources of studying this process {D* -to- Dzero + pizero} we see that the second one viz the one where we observe this process actually in our detector there may not be all the information we want. Such partial knowledge about some processes are recorded only in special cases. There are two types of particles that suffer such discrimination. The Klong’s and the neutrinos. The Klong’s are not completely constructed in the detector in which I worked, because it does not change the important goal with which this detector was constructed. But the Klong’s might be fully knowable in other detectors for other physics purposes. Also the neutrinos are often not completely knowable in experimental situations which is why they are known by employing whats known as “missing mass information method”. When a particle can’t be reconstructed fully one can also call it a “partial reconstruction”. But such methods can be overall useful despite whether such inabilities are optional or compulsions. So we reconstruct the Klong’s by employing a similar technique.

This is so because the Klong’s partial information is available from the detector. The Klong is known only for what direction it moves. Its total path of movement is not known. Hence one has to employ other method which in this case is called a “Dzero mass constraint method”.  This is because the Klong comes from a parent which is a Dzero. The Dzero therefore decides some information regarding the Klong. The other information is from the pi-zero.

Why we are talking about a constraint here? See what really happens in nature is a process occurs. Like electricity come to your wiring and lits the bulbs. [not lit your fire] Similarly a current/channel of a particle passes through your detector. The detector is like a TV or a mobile. It registers some information if thats available. And in case of Klong only what direction it moved is known as an average. In other words you know where it might have started when produced and where it ends up. But we don’t know what path the Klong moved, hence we don’t know how fast the Klong was moving.

The pi-zero and the Dzero on the other hand even if not positive or negative charges are known: how fast they moved, what path they took registers as a line. In particle detectors charges are known better since there are Electric and Magnetic Fields placed around the detector to guide these charges according to their charge. A field is a region where the energy of the magnet and the charges and currents are experienced. Its like touching the wire if there is electricity in it will shock you but if you are a little off you would not be shocked. But with high voltage devices one might get a shock from a bit far away. In other words these high voltage devices create a region where its energy is experienced. That region and the energy is called a Field and there are many kinds of fields.

In case of a particle detector we use electric and magnetic fields to divert charges in a particular way so that we would know something about the charge. So Klong and Dzero and Pi-zero being neutral particles the field does not work on them. But these fields might influence other particles that were produced. eg in some other processes there are many charged pions that were produced. Then some of these pions [pi mesons: pi-zero, pi-plus, pi-minus etc] might actually have come from say a Kshort meson. A Kshort often goes into pi-plus and pi-minus which will move opposite to each other in archs because there is a magnetic field. Then its known which is a pi-minus and which is a pi-plus because plus and minus charges always move in opposite direction and its known from the direction of the Magnetic field which is positive and which is negative. That would mean we would know where the Kshort was produced how fast and which path it moved. Thats all thats necessary.

So all we need to know in a particle detector for the purpose of this simple description is whats the speed of the say the Dzero particle. Whats its energy? These information are available from the path it moves in and how it reacts with the detector parts. The speed is often called a momentum when multiplied with a mass, hence we know the momentum and energy of any particle except Klong. These are called the 4-vectors because energy is one variable and momentum or speed is 3 variables. These are called Lorentz Vectors and a basic work in Physics by Einstein.

All particles in our detector at Japan where I worked will have 4-vector information available to us from the detector, except Klong and neutrinos. So what one does is one adds the 4-vectors information of one particle with another and one gets 4-vector information about another particles. eg If you have a particle A and one particle B then their 4-vectors add up to A+B. Then this 4-vector information can be matched to a particle called C. That means if A and B were to come from C, then it would be known by adding over all particles from the detector. What we will have is a list of particles that are possibilities and their mass and energy will be distributed like a bell-shaped curve on top of a mountain or heap like structure. The bell is called a signal and the heap is called a background or noise. This way we know if we have signal of something we are looking for.

For the Klong a 4-vector is not available. Even a 3-vector is not available. Whats available is the direction of the Klong which is a 3-vector whose magnitude is not known.

So Dzero{4-vector} -to- Klong{3-vector direction} + pizero{4vector}.

Usually 4 vector is known for all 3 particles in a two-body-decay.

Here what we do is when we add arbitrary particles by assuming them to be either Klong or pi-zero what we obtain is a Dzero but only with partial information because Klong is partially known. Also when we add arbitrarily because our list of particles known in the detector to be pi-zero or this or that particle is not perfect in other words there is impurity. An impure sample leads to more and more unclear signal.

These are dealt properly by various methods which I am not going into any details here. But the partially known Dzero is now therefore a sample which has among other impurities has the actual Dzero’s known as Monte Carlo true-Dzero if we  know the Dzero from Monte Carlo source rather than as I explained the 2nd source called experimental data. In both cases MC and Experiment, one will have Dzeros which will have partial knowledge.

In both cases what we therefore do is employ some of our knowledge about Dzero from prior data or MC which are standard knowledge. This is called a nominal value of Dzero mass. We know what mass Dzero is. We know that its about twice as heavy as a Hydrogen atom which as I explained in preceding article 4 times as heavy as the Klong meson. 2000 times as heavy as the electron and 14 times as heavy as the pi mesons. So when the Dzero mother produces the Klong and the pions the latter can move really fast. When the pions move slow its called slow-pion constraint because then Klong takes away most energy and it becomes easy to identify clearly the Klongs because the mass difference between D* and Dzero is about same as pizero. So, such very slow pions would come only from D* and Dzero connection. Sherlock Holmes would have become a particle physicist in this century.

When we assume that the arbitrary particles were all a Dzero by assigning it a mass of 2.01 GeV which is what is ~2000 times heavier than an electron we have a way to separate what is a Klong and whats not, in other words we reject fake results of adding arbitrary particles and have better purity. This also solves the problem that Klong isn’t known completely. This is called a Dzero mass constraint because we fixed the hypothetical-Klong and hypothetical-pizero to have come from the Dzero mass. This then we can match in MC and data and see if we get good result or not.

I devised a very simple hand weaving Monte Carlo code by writing out everything from first principles and employing Physics properties to predict what would happen if a Dzero decays into a Klong and pizero. This way I generated 2400 events and when I reconstructed them I could determine the Dzero and I could match this with the generated information and lo-and-behold they all turn to be correct. In other words I had a mini-facility which has no connection with actual facilities like the detector where I worked. This mini facilitiy is called a Toy Monte-Carlo.

This was one of my first lessons in Particle Physics. This was 2000 summer when I learned C++ and later implemented these codes into a mini monte carlo program that I developed by recognizing that in a two-body decay process there are always 8 relativistic unknowns and 6 equations hence two parameters are to be input by other considerations. Hence I generate random numbers in pairs which simulate the direction of the Klong. Remember direction of Klong is what is known from the detector.

We feed the {theta, phi} by renormalizing them as cosines of Klong direction. Then the Einstein’s laws for the motion of the 3 particles in a two body supply 6 solutions and we have 8 unknowns solvable this way. Each time you give the computer a set of {theta, phi} you get the direction of a Klong and by adding this to the 4-vector of pizero you get the partial info of Dzero, where by assuming the Dzero mass to be its nominal value one gets the signal for the Dzero. When one adds the Dzero to its sibling pi-plus or pi-minus one gets the D*plus or D*minus and thats what we are looking for as a signal.

This was a two step check. In step one you generate Klong direction and use that and Dzero mass constraint to get a solution for Klong’s momentum. Since this is a quadratic equation one gets two solutions only one of which may be a good solution. One can also keep both solutions and get an efficient answer to the problem and be careful about howto be consistent about it for all computations. But once we have a solution for the unknown momentum of Klong we can add the 4-vectors of Klong and pi-zero and resulting particle would be a Dzero.

This scheme as you could see is only a flip-flop and works to check if this works and my toy Monte Carlo proved that it does. But in real situations this Dzero mass constraint is used to recover the Klong 4-vector which also means we have the Dzero full-info now. But we input some Dzero info as given {its mass} hence its only trivial. But we can add this Dzero to other released particles in the topography and see the signal in terms of D* particle.

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