Chirality and neutrinos, a student first approach

Does Nature care about left-handed and right-handed configurations? Are left and right just two versions of the same thing? Although at first we may think that a particular orientation is irrelevant the fact is that Nature may distinguish between left-handed and right-handed systems. This results in different behaviors and properties. In this short presentation we will approach this phenomenon and how it affects the neutrinos.


Introduction
shows two photos: Catarina and Catarina as seen on a mirror. Can we take the real Catarina from her mirror image? What about the cuckoo clocks from Figure 2? A detailed observation of Figure 1 will not help us to tell the mirror from the real image. The same is not the case for the cuckoo. We can see that the cuckoo clock on the right has the numbers displayed counterclockwise and we can read at the bottom of the display. The clock on the right is the mirror image.
Both of Catarina's pictures are possible representations of a girl. This makes it impossible to tell the mirror from the real image. Although we can certainly make a cuckoo clock with everything reversed, a counterclockwise clock is not the pattern.
The fact that we build clocks the way we do, the clockwise way, is basically a question of choice. We have chosen to do so. What about Nature? Does Nature also choose a particular orientation? Well, the surprising answer is that it, many times, does!  Figure 2. Two views of a cuckoo clock: real and mirror image. Can we tell the real from the mirror image?

Helicity
Let's consider a spinning body in movement and project the spinning on the direction of movement (we only care about the spinning in the direction of movement). Helicity is the projection of the spin along the direction of movement: it is right-handed if it is paralell to the movement; left-handed otherwise. Figure 3 illustrates the concept of helicity. Figure 3. The direction of the spinning is defined by the right hand rule: when holding the body with the right hand with the fingers following the spinning the thumb points in the direction of the spinning. The left figure shows a body moving to the right (with momentum p represented by the green arrow) and spinning in the same direction of its movement: it has right-handed helicity. The figure on the right shows a body also moving to the right (green arrow) but with the spinning in the opposit direction: it has left-handed helicity.
If we are moving in the same direction as the spinning body we can see the body moving forward, if it is going faster, or backwards, if it is going slower, or not moving at all if if we are moving at the same speed. The spinning does not change though. Helicity will then be right-handed or left-handed 1 , depending on the frame of reference. Helicity is not an intrinsic property! IOP Publishing doi:10.1088/1742-6596/1558/1/012014 3 Elementary particles, like electrons and neutrinos, are considered pointlike and can not spin like a soccer ball but nevertheless they have an intrinsic spin and present right-handed of lefthanded helicity depending on the frame of reference. Although the spin is an intrisic property of a particular particle its helicity, as described above, depends on the frame of reference, as long as we can move faster than the particle 2 . Is there anything related to helicity that is invarint and does not depend on the frame of reference?
If I rotate like a rotisserie chicken (Figure 4) observers in different frames of reference will disagree regarding my helicity but there will be no doubts about which one is my right hand and which one is my left hand. 3. Chirality Figure 5 shows a screw, a nail, and their image as seen on a mirror. The screw can not be superimposed on its mirror image since they have threads with different handness: right-hand in the screw and left-hand in the mirror image. The nail, however, can be superimposed on its mirror image. An object that can not be superimposed on its mirror image is said to be chiral. The screw from Figure 5 is a chiral object.
Surely we can make screws of any orientation, right or left-handed; therefore, in principle, we could not identify the mirror image as a not real screw. We basically choose to build right-handed or left-handed screws. Nature can do the same and, surprisingly, in many instances decides for just one of the two. Many molecules are chiral and found only in one of the configurations. The well known B-DNA and A-DNA helix come only in a right-handed version while the Z-DNA comes only as a left-handed helix! Figure 6 shows the structure of the DNA where its orientation is visible. Figure 6 also depicts a chiral molecule. A mirror image of a B-DNA would be right-handed but it simply does not exist. The mirror image of the molecule is also not found in Nature.
We can not place a particle in front of a mirror to find out if it is chiral. However, particles belong to the quantum realm where a particle is represented by a wave function depending on the space and time coordinates, ψ(x, y, z, t). Figure 7 shows the effect of reversing the axis of a (x, y, z) frame: it produces its mirror image changing the system from right-handed to left-handed. We can then perform an operation that would provide the particle mirror image by simply reversing the space variables, ψ(−x, −y, −z, t). It comes out that the neutrino is chiral and only the left-handed version is observed in nature: Nature provides us only with 2 If the particle is massless then no frame of reference can move faster. In this case, helicity will not vary.  Figure 5. The screw can not be superimposed on its mirror image (the screw threads make a right-handed helix while the mirror image has a left-handed helix).
The nail can be superimposed on its mirror image (both are identical).
B-DNA and A/DNA are always a right-handed helix while the Z-DNA is always a left-handed helix.
Some molecules are also chiral. The molecule on the right can not be superimposed on its mirror image (figure from E. Generalic, https://glossary.periodni.com/glossary.php?en=chiral+molecule). Nature does have preferences.
neutrinos with left-handed chirality. The neutrino antiparticle, the antineutrino, comes only in its right-handed chirality version. One may ask, and so what? Figure 7. A (x, y, z) right-handed frame of reference (left) has its axis reversed resulting in a (x , y , z ) left-handed frame of reference (center). The (x , y , z ) is rotated (right) so we can see that it is the mirror image of the original (x, y, z). Reversing the axis produces the system mirror image as seen on a mirror placed perpendicular to z.

Chirality and interactions
We know of four interactions: strong interaction; electromagnetic interaction; weak interaction; gravitational interaction. The fundamental particles are quarks, affected by all interactions, charged leptons (electron, muon and tau), affected by the electromagnetic and the weak interaction, and the neutral leptons (the electron neutrino, the muon neutrino and the tau neutrino) that are affected only by the weak interaction 3 . In all cases, the interaction is mediated by the exchange of a boson: gluons for the strong interaction; photons for the electromagnetic interaction; and W± and Z 0 for the weak interaction. The interactions have very different intensities: the strong interaction is about 10 3 times more intense than the electromagnetic interaction and the electromagnetic interaction is about 10 5 times more intense than the weak interaction. Table 1 shows the fundamental particles and their electric charge, Table 2 shows which interactions affect each fundamental particle and Table 3 shows the relative intensity of the interactions and their associated bosons.  When two quarks come close enough they may interact strongly by the exchange of a gluon, electromagnetically by the exchange of a photon, or weakly by the exchange of one of the weak bosons. When a quark and an electron get close enough they may interact only electromagnetically or weakly, since the electron is not affected by the strong interaction. When two electrons get close enough they may interact electromagnetically, by exchanging a photon, or weakly, by exchanging a weak boson. When a neutrino is involved, the only possible interaction is the weak interaction by the exchange of one of the weak bosons. Since the interactions have very different intensity, the strongest one will be favored when two particles interact.
The weak interaction may happen through two different processes: charged current, when a W is exchange; and neutral current, when a Z 0 is exchanged. Since the W carries electric charge,  exchanging a W changes the charge of the interacting particle. A d quark has a fractional electric charge of −1/3 and aū antiquark has an electric charge of −2/3. Together they form a π − with charge −1. The π − (dū) decay when the dū annihilates 4 with the emission of a W − that carries the electric charge. The W − may then materialize as aν l µ l pair. The decay process happens through the weak interaction.

Particle mass and chirality
In the standard model all particles are created massless and left-handed and antiparticles are also created massless but right-handed. Particles acquire mass when interacting with the Higgs field. Once created, the particle starts interacting with the Higgs field. As more a particle interacts, more mass it acquires. If a particle does not interact with the Higgs field then it remains massless. Each interaction with the Higgs produces mass and keeps everything else unchanged ... except for the chirality that is reversed at each interaction. A particle then acquires mass and develops a right-handed chirality component (or a left-handed component for antiparticles). The right-handed and left-handed components will not necessarily be equal and as more energy the particle acquires, "bigger" the right-handed component. The electron we observe is a mix of left-handed and right-handed electron 5 . Figure 8 illustrates the process.  about chirality; however, the weak interaction does! The weak boson will couple only to a lefthanded particle or to a right-handed antiparticle. The weak interaction is chiral and depends on the particle chirality.
In the standard model, neutrinos are massless and only come with left-handed chirality (and antineutrinos come with right-handed chirality) 6 .

Pion decay and chirality
All decay processes must conserve several properties like, among others, electric charge, energy and momentum. The pion decay, mentioned at the end of section 4, may occur as π − → µ −ν µ or π − → e −ν e as shown in Figure 9. The muon mass (105.66 MeV) is about 200 times the electron mass. This leaves more energy available for the π − → e −ν e channel (139.06 MeV versus 33.91 MeV as shown in the figure). As far as energy is concerned the π − → e −ν e process should be favored; however, it is heavily suppressed happening only in 0.01 % of the time (one out of 10,000 decays). What is intervening in this process to so heavily disfavor the π − → e −ν e decay? Figure 9. Pion decay through weak interaction. Top: π − → µ −ν µ . Bottom: π − → e −ν e . Since the muon mass is much bigger than the electron's, there is much more energy available for the π − → e −ν e decay; however, it happens only in 0.01 % of the time. It is highly disfavored. Figure 10 shows the process π − → l −ν l as a sequence of steps as seen in the pion rest frame. This is a two body decay process; therefore, momentum conservation implies that the l and thē ν l leave in opposite directions and with the same momentum. Antineutrinos are always righthanded; therefore, chirality equals helicity (spin is in the same direction as the momentum, as shown by the red arrow). Angular momentum conservation implies that the l must also be right-handed (the π − has spin zero). Here is where chirality plays a role. For a charged lepton l chirality is not equal to helicity 7 . A lepton l is born massless and with left-handed chirality. It acquires mass by interacting with the Higgs field. As more interaction with the Higgs field, bigger the mass. Each bump with Higgs not just gives mass to the particle but also changes its chirality and the particle will start presenting a right-handed component. The particle we 6 The observed phenomenon of neutrino oscillation points to the fact that neutrinos do have mass: very small but non zero. This poses a problem for the standard model and the non observation of right-handed neutrinos excludes the Higgs mechanism as the way a neutrino acquires mass. This is currently one of the main subjects in neutrino physics. 7 Helicity and chirality are the same only for massless particles. Massive particles move below the speed of light and we can move faster than the particle and invert the helicity. observe will be a combination of two components: one with left-handed chirality and the other with right-handed chirality. Figure 10. Pion decay into a charged l lepton and its associated antineutrinoν l .
As mentioned at the beginning of this section and shown in Figure 9, there is much more energy available for the π − → e −ν e process than for the π − → µ −ν µ process and all this extra energy will imply that the electron will come out with more speed than the muon. As higher the speed, bigger the right-handed component. The weak boson couples only to left-handed particle (or right-handed antiparticles) and, since the electron will be "more" right-handed than the muon, the process will be suppressed (the weak interaction cares only for a particle's lefthanded component or antiparticle's right-handed component and the muon beats the electron on left-handness). Notice that had the electron no mass, then the process π − → e −ν e would not happen at all; it would be forbidden because a massless particle has only left-handed chirality and left-handed helicity (since for massless particles chirality equals to helicity) and conservation of angular momentum would prevent the process from happening.

Conclusions
The weak interaction is chiral and differentiates right-handed particles from left-handed ones. Neutrinos are always left-handed and antineutrinos are always right-handed and this reflects in the way they interact. Neutrino oscillation shows that neutrinos must have mass, even if a tiny one, what is in direct conflict with the Standard Model. Some questions are yet to be answered like the origin of the neutrino mass, the possible existence of right-handed neutrinos, heavy neutrinos and sterile neutrinos that would be affected only by gravity and that do not fit in the standard model. All this points to physics beyond the Standard Model. These are all important topics that are subject of intense research in the field of neutrino physics. I want to thank the organizers of the XVII Meeting of Physics for the invitation and Victor Cunha for valuable comments and suggestions.