# Understanding crossing symmetry: inverse beta decay

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• peguerosdc
In summary, crossing symmetry in particle physics allows for particles to be "crossed" over to the other side of the equation by turning them into their antiparticles, resulting in a possible reaction. This is demonstrated in the Compton scattering and inverse beta decay equations. The arrow direction in the inverse beta decay reaction is not reversed because both directions of the reaction satisfy conservation laws, but the energy requirements may be different.
peguerosdc
TL;DR Summary
When applying cross symmetry to ##\nu + n \rightarrow e^- + p##, why in the result ##\bar{\nu} + p \rightarrow e^+ + n## the arrow is pointing to the right?
Hi! This is a very very noob question, but I am starting to get into particle physics and I don't understand the application of crossing symmetry in the inverse beta decay.

Crossing symmetry says (from Griffiths) that, in a reaction "any of these particles can be 'crossed' over to the other side of the equation, provided it is turned into its antiparticle, and the resulting interaction will also be allowed". And the Compton scattering is mentioned as an example.

Then, why another common example is the inverse beta decay used to detect the neutrino? The beta decay reaction is:

$$\nu + n \rightarrow e^- + p$$

And the inverse beta decay reaction is:

$$\bar{\nu} + p \rightarrow e^+ + n$$

But if you just do crossing symmetry to the beta decay equation, I understand that you should get the reverse reaction (with the arrow pointing to the other side):

$$e^+ + n \rightarrow \bar{\nu} + p$$

That is, I am:
• crossing ##e^-## from the right to the left as ##e^+##
• crossing ##\nu## from the left to the right as ##\bar{\nu}##
• leaving ##n## where it is
• leaving ##p## where it is
So, why is the arrow not pointing to the left in the correct inverse beta decay reaction?

Thanks!

If ##a \to b## is a possible reaction then ##b \to a## is a possible reaction, too (assuming you have enough energy). They are closely linked. Similarly, ##\overline a \to \overline b## is a possible reaction, which you can convert to ##b \to a##. Here a and b can be any set of particles.

peguerosdc
mfb said:
If ##a \to b## is a possible reaction then ##b \to a## is a possible reaction, too (assuming you have enough energy).

@mfb thanks for your reply! From what I read in Griffiths', due to energy conservation if ##m_a > m_b##, (where ##m## stands for mass) then I would need to supply enough energy to make it up in the reaction ##b \rightarrow a##. Is this what you mean?

And, starting from a given reaction (assuming it is possible i.e. it complies with conservation laws), is this the only requirement to see it going in the opposite direction?

If the sum of masses is larger in a then the reverse reaction will need to have the particles colliding with some kinetic energy, yes. If it's smaller then you need that kinetic energy for a->b.

Conservation laws always say "x stays the same". That's true for both directions, which means both directions always satisfy the conservation laws (which means they are possible). The only catch is the energy. In e.g. ##e^+ e^- \to \gamma \gamma## the photons will always have at least 511 keV in the center of mass frame, but you don't write it down explicitly. In ##\gamma \gamma \to e^+ e^-## the photons need at least 511 keV in the center of mass frame to make the reaction possible. If you try to collide two visible light photons you are not reversing the reaction. You are starting with a condition that cannot be the final state of the reverse direction.

vanhees71 and peguerosdc
Got it! Your example helped me a lot. Thank you!

## 1. What is crossing symmetry in the context of inverse beta decay?

Crossing symmetry is a fundamental principle in particle physics that states that the laws of physics should remain unchanged when the roles of the initial and final states of a reaction are interchanged. In the case of inverse beta decay, this means that the same physical laws should apply regardless of whether a neutron and an electron are colliding to form a proton and a neutrino, or if a proton and a neutrino are colliding to form a neutron and an electron.

## 2. How does crossing symmetry affect the outcome of inverse beta decay?

Crossing symmetry is important in inverse beta decay because it allows us to make predictions about the behavior of particles in a reaction based on our understanding of similar reactions. This symmetry allows us to use the known properties of particles in one direction of the reaction to make predictions about the properties of particles in the opposite direction.

## 3. What are the implications of crossing symmetry for the study of particle physics?

Crossing symmetry is a powerful tool in the study of particle physics because it allows us to make connections between seemingly different reactions and particles. This symmetry helps us to understand the fundamental laws of nature and make predictions about the behavior of particles in a wide range of reactions.

## 4. Are there any exceptions to crossing symmetry in inverse beta decay?

While crossing symmetry is a fundamental principle, there are some cases where it may not hold true. This could be due to experimental limitations or the presence of unknown particles. However, these exceptions are rare and do not affect the overall validity of crossing symmetry in inverse beta decay.

## 5. How is crossing symmetry tested and verified in the study of inverse beta decay?

Crossing symmetry is tested and verified through experimental observations and theoretical calculations. Scientists use particle accelerators and detectors to study the outcomes of inverse beta decay and compare them to predictions based on crossing symmetry. If the results match, this provides evidence for the validity of crossing symmetry in this reaction.

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