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Adam Davis | USLHC | USA

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Looking Forward to 2015: Analysis Techniques

With 2015 a few weeks old, it seems like a fine time to review what happened in 2014 and to look forward to the new year and the restart of data taking. Along with many interesting physics results, just to name a few, LHCb saw its 200th publication, a test of lepton universality. With protons about to enter the LHC, and the ALICE and LHCb detectors recording muon data from transfer line tests between the SPS and LHC (see also here), the start of data-taking is almost upon us. For some implications, see Ken Bloom’s post here. Will we find supersymmetry? Split Higgs? Nothing at all? I’m not going to speculate on that, but I would like to review two techniques which played a key role in two results from LHCb and a few analysis techniques which enabled them.

The first result I want to discuss is the \(Z(4430)^{-}\). The first evidence for this state came from the Belle Collaboration in 2007, with subsequent studies in 2009 and in 2013. BaBar also searched for the state, and while they did not see it, they did not rule it out.

The LHCb collaboration searched for this state, using the specific decay mode \(B^0\to \psi’ K^{+} \pi^{-} \), with \(\psi’\) decaying to two muons. For more reading, see the nice writeup from earlier in 2014. As in the Belle analyses, which looked using muons or electrons in the final \(\psi’\) state, the trick here is to look for bumps in the \(\psi’ \pi^{-}\) mass distribution. If a peak appears which is not described  by the conventional 2 and 3 quark states, mesons and baryons, we know and love, it must be from a state involving a \(c \overline{c}d\overline{u}\) quark combination. The search is performed in two ways: a model-dependent search, which looks at the \(K\pi\) and \(\psi’\pi\) invariant mass and decay angle distributions, and a “model independent” search which looks for structure induced in the \(K\pi\) system induced by a resonance in the \(\psi’\pi\) system and does not invoke any exotic resonances.

At the end of the day, it is found in both cases that the data are not described without including a resonance for the \(Z(4430)^-\).

Now, it appears that we have a resonance on our hands, but how can we be sure? In the context of the aforementioned model dependent analysis, the amplitude for the \(Z(4430)^{-}\) is modeled as a Breit-Wigner amplitude, which is a complex number. If this amplitude is plotted in the imaginary plane as a function of the invariant mass of the resonance, a circular shape is traced out. This is characteristic of a resonance. Therefore, by fitting the real and imaginary parts of the amplitude in six bins of \(\psi’\pi\) invariant mass, the shape can be directly compared to that of an exected resonance. That’s exactly what’s done in the plot below:

The argand plane for the Z(4430)- search. Units are arbitrary.

The argand plane for the Z(4430)- search. Units are arbitrary.

What is seen is that the data (black points) roughly follow the outlined circular shape given by the Breit-Wigner resonance (red). The outliers are pulled due to detector effects. The shape quite clearly follows the circular characteristic of a resonance. This diagram is called an Argand Diagram.


Another analysis technique to identify resonances was used to find the two newest particles by LHCb:

Depiction of the two Xi_b resonances found by the LHCb Collaboration. Credit to Italic Pig (http://italicpig.com/blog/)

Depiction of the two Xi_b resonances found by the LHCb Collaboration. Credit to Italic Pig

Or perhaps seen as


Xi_b resonances, depicted by Lison Bernet.

Xi_b resonances, depicted by Lison Bernet.

Any way that you draw them, the two new particles, the \(\Xi_b’^-\) and \(\Xi_b^{*-}\) were seen by the LHCb collaboration a few months ago. Notably, the paper was released almost 40 years to the day that the discovery of the \(J/\psi\) was announced, sparking the November Revolution, and the understanding that mesons and baryons are composed of quarks. The \(\Xi_b’^-\) and \(\Xi_b^{*-}\) baryons are yet another example of the quark model at work. The two particles are shown in \(\delta m \equiv m_{candidate}(\Xi_b^0\pi_s^-)-m_{candidate}(\Xi_b^0)-m(\pi)\) space below.

Xi_b'^- and Xi_b^{*-} mass peaks shown in delta(m_candidate) space.

\(\Xi_b’^-\) and \(\Xi_b^{*-}\) mass peaks shown in \(\delta(m_{candidate})\) space.

Here, the search is performed by reconstructing \(\Xi_b^0 \pi^-_s\) decays, where the \(\Xi_b^0\) decays to \(\Xi_c^+\pi^-\), and \(\Xi_c^+\to p K^- \pi^+\). The terminology \(\pi_s\) is only used to distinguish between that pion and the other pions. The peaks are clearly visible. Now, we know that there are two resonances, but how do we determine whether or not the particles are the \(\Xi_b’^-\) and \(\Xi_b^{*-}\)? The answer is to fit what is called the helicity distributions of the two particles.

To understand the concept, let’s consider a toy example. First, let’s say that particle A decays to B and C, as \(A\to B C\). Now, let’s let particle C also decay, to particles D and F, as \(C\to D F\). In the frame where A decays at rest, the decay looks something like the following picture.

Simple Model of A->BC, C->DF

Simple Model of \(A\to BC\), \(C\to DF\)

There should be no preferential direction for B and C to decay if A is at rest, and they will decay back to back from conservation of momentum. Likewise, the same would be true if we jump to the frame where C is at rest; D and F would have no preferential decay direction. Therefore, we can play a trick. Let’s take the picture above, and exactly at the point where C decays, jump to its rest frame. We can then measure the directions of the outgoing particles. We can then define a helicity angle \(\theta_h\) as the angle between the C flight in A’s rest frame and D’s flight in C’s rest frame. I’ve shown this in the picture below.

Helicity Angle Definition for a simple model

Helicity Angle Definition for a simple model

If there is no preferential direction of the decay, we would expect a flat distribution of \(\theta_h\). The important caveat here is that I’m not including anything about angular momentum, spin or otherwise, in this argument. We’ll come back to that later. Now, we can identify A as the \(\Xi_b’\) or \(\Xi_b^*\) candidate, C as the \(\Xi_b^0\) and D as the \(\Xi_C\) candidates used in the analysis. The actual data are shown below.

Helicity angle distributions for the Xi_b' and Xi_b* candidates (upper and lower, respectively).

Helicity angle distributions for the \(\Xi_b’ \)and \(\Xi_b*\) candidates (upper and lower, respectively).

While it appears that the lower mass may have variations, it is statistically consistent with being a flat line. Now the extra power of such an analysis is that if we now consider angular momentum of the particles themselves, there are implied selection rules which will alter the distributions above, and which allow for exclusion or validation of particle spin hypotheses simply by the distribution shape. This is the rationale for having the extra fit in the plot above. As it turns out, both distributions being flat allows for the identification of  the \(\Xi ‘_b^-\) and the \(\Xi_b^{*-}\), but do not allow for conclusive ruling out of other spins.

With the restart of data taking at the LHC almost upon us (go look on Twitter for #restartLHC), if you see a claim for a new resonance, keep an eye out for Argand Diagrams or Helicity Distributions.


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  • The outliers are pulled due to detector effects.” If you have exactly three points, you exactly define a circle. One fails to see how three accepted points not within a resonance would display differently. Enlighten me.

  • AdamDavis

    If the three points lie on a line, there is no definition of a circle. If you don’t have a resonance, then there is no phase motion across the Argand diagram, and you do not expect a circular shape. That is the power of the Argand diagram.

  • Given statistical and systematic “noise,” you have three non-colinear points. Absent a radial boundary condition, that defines a qualifying circle. Given a radial boundary condition, you arbitrarily exclude major empirical falsifications (if they exist). Math alone is not enough.

    It is obvious, by common sense and high school geometry, that a triangle’s three interior angles sum to exactly 180°. This excludes Earth’s surface, cooling towers, and general relativity. Perhaps the model is incomplete.