How Can Point Particles Have Cross Sections?

In summary: I don't even know where to start. They have no spatial extent? If they didn't have cross sections, then they would simply fly past one another and not interact at all (i.e. the probability of an interaction would be infinitesimal). Is there any explanation as to how point particles like electrons or quarks or neutrinos can have non-zero cross sections?I don't know if standard particle physics is enough to answer this question. If concepts relating to QFT or other areas of physics are required, then the Mentors are free to move this thread to the appropriate forum.
  • #1
Nathan Warford
23
1
This is a topic that I have tried researching and I have not been able to find any meaningful information about it. The standard model of particle physics describes particles as point-like objects with no spatial extent. What I can't wrap my head around is how true point particles can participate in interactions. If particles did not have cross-sections, then they would simply fly past one another and not interact at all (i.e. the probability of an interaction would be infinitesimal). Is there any explanation as to how point particles like electrons or quarks or neutrinos can have non-zero cross sections?

I don't know if standard particle physics is enough to answer this question. If concepts relating to QFT or other areas of physics are required, then the Mentors are free to move this thread to the appropriate forum.
 
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  • #2
Counter examples: Gravity and electrostatics. Objects can interact with each other without "touching". Certainly the Earth does not need to hit the Sun in order to keep its orbit.
 
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  • #3
Orodruin said:
Counter examples: Gravity and electrostatics. Objects can interact with each other without "touching". Certainly the Earth does not need to hit the Sun in order to keep its orbit.
Gravity doesn't really apply because elementary particles are so light that gravity has a negligible affect on them. Gravity is better explained by General Relativity anyway.

As for electrostatics, the way I understand the phenomenon is that charged particles like electrons exchange virtual photons. At the subatomic level, emission and absorption of photons are the actual interactions that take place. At least that's the sense that I get based on my limited experience with Feynman Diagrams.
 
  • #4
Nathan Warford said:
Gravity doesn't really apply because elementary particles are so light that gravity has a negligible affect on them.
I am not talking about elementary particles here. It was an example of how things could interact without "touching".

Nathan Warford said:
As for electrostatics, the way I understand the phenomenon is that charged particles like electrons exchange virtual photons. At the subatomic level, emission and absorption of photons are the actual interactions that take place. At least that's the sense that I get based on my limited experience with Feynman Diagrams.
Virtual photons are not small balls that are thrown between electrons. The cross sections you are talking about are e+e- cross sections.
 
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  • #5
Orodruin said:
I am not talking about elementary particles here. It was an example of how things could interact without "touching".
Still, I would like to keep this discussion in the realm of particle physics. If gravity turns out to actually be important to the discussion, maybe the thread should be moved to the Beyond the Standard Model forum?
Orodruin said:
The cross sections you are talking about are e+e- cross sections.
Don't assume to know what I'm talking about. I'm asking about all types of particle interactions. These include electrostatic repulsion, electrostatic attraction, beta decay, electron capture, strong nuclear interactions, particle-antiparticle production, particle-antiparticle annihilation, basically any interaction that can take place between elementary particles. The interactions I'm interested in all seem to be quantum mechanical in nature, so maybe the thread should be moved to the Quantum Physics forum?
Orodruin said:
Virtual photons are not small balls that are thrown between electrons.
I was not a physics major, so I probably don't understand electrostatic interactions correctly. Could you explain to me how particle interactions actually work?

Also my initial question has not been answered yet. How are point particles able to interact even though they don't have finite spatial extent?
 
  • #6
Nathan Warford said:
How are point particles able to interact even though they don't have finite spatial extent?

Why should that even be necessary? I could ask "how are particles able to interact even though they aren't blue?" or "how are particles able to interact even though they aren't chocolate flavored." Do you have any better reason than "It seems that they should"?
 
  • #7
Nathan Warford said:
Still, I would like to keep this discussion in the realm of particle physics. If gravity turns out to actually be important to the discussion, maybe the thread should be moved to the Beyond the Standard Model forum?

Don't assume to know what I'm talking about. I'm asking about all types of particle interactions. These include electrostatic repulsion, electrostatic attraction, beta decay, electron capture, strong nuclear interactions, particle-antiparticle production, particle-antiparticle annihilation, basically any interaction that can take place between elementary particles. The interactions I'm interested in all seem to be quantum mechanical in nature, so maybe the thread should be moved to the Quantum Physics forum?

I was not a physics major, so I probably don't understand electrostatic interactions correctly. Could you explain to me how particle interactions actually work?

Also my initial question has not been answered yet. How are point particles able to interact even though they don't have finite spatial extent?

You are missing the point of people trying to GUIDE you to the answer. So obviously, a more direct approach is necessary.

Take for example, electrons. They are, as far as we know, point particles. However, they also have CHARGE, and when something has charge, they give off this thing called "electric field". When another charge gets close, this charge senses the electric field of the first charge. So already, these two charges are INTERACTING with one another even at a distance.

This interaction is what causes point particles to affect each other. It has nothing to do with physical dimension. This is the point that people were trying to get across to you by using gravitational field as an example that you might be familiar with. Obviously, you didn't get the connection.

Zz.
 
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  • #8
Nathan Warford said:
Don't assume to know what I'm talking about.
Insulting people who are trying to help you and guide you to an answer is not a very constructive approach to take. Gravity might not be what you are trying to understand but the underlying principles are exactly the same. However, as you seem to be quite stubborn in not wanting to think about what people tell you, I am out of this conversation.
 
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  • #9
One should think of "cross section" as one of those terms that physicists have adopted from common language and applied to a situation that is only vaguely reminiscent of their original use.

Shoot a beam of something (call them "particles" for convenience) through a thin target of thickness Δx, containing n "target particles" per unit volume (the "number density" of target particles). If you start with Nin beam particles, Nout of them emerge on the other side, the difference ΔN = Nin - Nout having been scattered or absorbed. It turns out that ΔN is proportional to Nin, Δx and n. We can turn this into an equation by inserting a proportionality constant σ:

ΔN = NinσnΔx

Analyzing the units of the other quantities shows that σ must have units of area, which leads to the term "cross section". Many textbooks have diagrams showing the target particles as acting like little circular targets with area σ, but you should consider this as only a heuristic visualization device. Some beam particles that pass through one of the "target circles" do not interact at all, and some that pass outside all the "target circles" do interact.
 
  • #10
Unless I have completely misunderstood something, even a point interaction where the potential energy of a two particle system is a Dirac delta function ##\delta^3 (\mathbf{r}_1 - \mathbf{r}_2)##, can have a nonzero scattering cross section. So in other words, even particles of zero size and zero interaction range can "collide" with each other in quantum mechanical systems.
 
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  • #11
Orodruin said:
Insulting people who are trying to help you and guide you to an answer is not a very constructive approach to take.
You're right. I should have chosen my words differently. I apologize.
ZapperZ said:
Take for example, electrons. They are, as far as we know, point particles. However, they also have CHARGE, and when something has charge, they give off this thing called "electric field". When another charge gets close, this charge senses the electric field of the first charge. So already, these two charges are INTERACTING with one another even at a distance.

The thing is that there can also be interactions involving particles that don't have electric charge. A photon doesn't have an electric charge, but it can still be absorbed or scattered by matter particles like electrons. An electron can also interact with the uncharged neutrino to create a W boson. So interaction through electric fields can't be the whole story.
jtbell said:
Shoot a beam of something (call them "particles" for convenience) through a thin target of thickness Δx, containing n "target particles" per unit volume (the "number density" of target particles). If you start with Nin beam particles, Nout of them emerge on the other side, the difference ΔN = Nin - Nout having been scattered or absorbed. It turns out that ΔN is proportional to Nin, Δx and n. We can turn this into an equation by inserting a proportionality constant σ:

ΔN = NinσnΔx

Analyzing the units of the other quantities shows that σ must have units of area, which leads to the term "cross section".
This is a much more useful response. Thank you. I guess I'm just trying to understand the basic interactions between the beam particles and the target particles, whatever types of particles they may be.
hilbert2 said:
Unless I have completely misunderstood something, even a point interaction where the potential energy of a two particle system is a Dirac delta function ##\delta^3 (\mathbf{r}_1 - \mathbf{r}_2)##, can have a nonzero scattering cross section. So in other words, even particles of zero size and zero interaction range can "collide" with each other in quantum mechanical systems.
This is also a useful response. "Dirac delta function" sounds interesting. Give me some time to look it up.
 
  • #12
Nathan Warford said:
You're right. I should have chosen my words differently. I apologize.The thing is that there can also be interactions involving particles that don't have electric charge. A photon doesn't have an electric charge, but it can still be absorbed or scattered by matter particles like electrons. An electron can also interact with the uncharged neutrino to create a W boson. So interaction through electric fields can't be the whole story.

Again, you are missing the point!. I used electrons and charge and electric field as an example of how interaction can take place at a distance without things "touching", meaning the physical "size" of something doesn't matter! This means that for other particles, the mechanism of interaction can be anything - EM interaction, weak interaction, strong interaction, gravity, etc... etc.. Heck, in a superconductor, two electrons even exchange virtual phonons (not photons) to cause them to "attract"!

You are looking at the trees without realizing that we are describing the forest. We are trying to explain the GENERAL concept via using a specific example, but you seem to not be able to get past the specific example and not seeing the general concept! If this still doesn't make the connection for you, then I give up.

BTW, light or photons have electric field and interacts with charge particles. It is how we accelerate charge particles in particle accelerators, and why it can interact with electric/magnetic dipoles.

Zz.
 
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  • #13
Let me ask again - I could ask "how are particles able to interact even though they aren't blue?" or "how are particles able to interact even though they aren't chocolate flavored." Do you have any better reason than "It seems that they should"? Note that the physical extent of a particle does not in general appear in the QM calculation of scattering amplitudes; for example, the total scattering cross-sections of two electrons and two grapefruits carrying one extra electron each are the same - and they are both infinite, as it happens.
 
  • #14
ZapperZ said:
Take for example, electrons. They are, as far as we know, point particles.
I read that and thought to myself - yes, that's true. But I thought again. What does the extent of a particle mean? Their electric presence is a Potential Well so the 'electric size' is a bit meaningless. Any 'size' could be assessed by its likely scattering collision with a neutral particle (neutron?). Size would only be a valid description for an unbound electron because, in a bound state, we would be stuck with a probability density function, which would be of atomic size.
I looked at this wiki link which presents an argument based on "electrostatic self energy" but isn't that making a lot of assumptions?
 
  • #15
sophiecentaur said:
I read that and thought to myself - yes, that's true. But I thought again. What does the extent of a particle mean?

I can turn around and ask you by what you mean as "... the extent of a particle..."

We know that if an electron has any "internal structure", (i.e. it has a non-zero size) then one of the possible outcome of that is the presence of an electric dipole moment when it is in a high gradient field. I'm sure you are well aware of at least a couple of different recent experiments that could not detect such effect.

Zz.
 
  • #16
Nathan Warford said:
Gravity doesn't really apply because elementary particles are so light that gravity has a negligible affect on them. Gravity is better explained by General Relativity anyway.

As for electrostatics, the way I understand the phenomenon is that charged particles like electrons exchange virtual photons. At the subatomic level, emission and absorption of photons are the actual interactions that take place. At least that's the sense that I get based on my limited experience with Feynman Diagrams.
Virtual photons are no localized particles. Even real photons are not localized in a naive sense. You cannot even define a position observable for them! Virtual photons are rather symbolizing the electromagnetic interaction and are depicted as internal lines of Feynman diagrams. Feynman diagrams, however, are not pictures of "what's really going on" but very clever notations of quite complicated mathematics to evaluate the only truly observable facts about collisions of particles, i.e., the socalled S-matrix elements, which are used to calculate cross sections that can be measured in experiment and compared to the predictions of the calculations.
 
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  • #17
Nathan Warford said:
The standard model of particle physics describes particles as point-like objects with no spatial extent. What I can't wrap my head around is how true point particles can participate in interactions.

Regarding your question, I find this text "Are electrons pointlike/structureless?" by Arnold Neumaier very helpful: https://arnold-neumaier.at/physfaq/topics/pointlike.html
 
  • #18
Nathan Warford said:
This is a topic that I have tried researching and I have not been able to find any meaningful information about it. The standard model of particle physics describes particles as point-like objects with no spatial extent.
I don't know, where you got this wrong idea from. The Standard Model of particle physics is entirely based on relativistic quantum field theory, and describes the strong and the electro-weak interaction in terms of local gauge symmetry based on the group ##\text{SU}(3)_{\text{color}} \times \text{SU}(2)_{\text{WISO}} \times \text{U}(1)_{\text{Y}}##, which is higgsed to ##\text{SU}(3)_{\text{color}} \times \text{U}(1)_{\text{em}}##.

The measurable quantities are defined in terms of abstract correlation functions that define ##S##-matrix elements, i.e., probability amplitudes for transitions between asymptotically free in-states (usually two particles in a scattering experiment) to aysmptotically free out-states (any kind of many-particle state). This is for vacuum quantum field theory describing scattering experiments with a few particles (although the final state of pp collisions at the LHC can have very many particles in the final state, indeed).

Another application is relativistic many-body physics, as needed in the description of the hot and dense medium created in heavy-ion collisions (done also at the LHC, where Pb-Pb collisions at the highest so far available energies are smashed together creating a hot and dense fireball, most likely forming a quark-gluon plasma for a few fm/c). Also here the observables finally are asymptotic free states, i.e., "particles in a detector".

A mental picture of elementary particles, particularly in the relativistic realm, as microscopic billard balls is flawed! The only chance to really understand, what elementary particles are, according to the up-to-date state of research is to get used to the quite abstract formulation as a relativistic QFT. Popular-science media usually do not give an adequate description.
 
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  • #19
ZapperZ said:
I can turn around and ask you by what you mean as "... the extent of a particle..."
Fair enough question. I was referring to the "cross section" of the thread title. The dimension / size / cross section that you refer to in your reply is only in terms of a sort of derived quantity and that's ok with me but, as the above post says:
vanhees71 said:
A mental picture of elementary particles, particularly in the relativistic realm, as microscopic billard balls is flawed!
So the "cross section" is a quantity on its own and it cannot be thought of in the form of two arrows and a number beside a picture of an electron, equivalent to the height of an IKEA bookcase.
 
  • #20
sophiecentaur said:
What does the extent of a particle mean?

Suppose you modeled electrons as an object of radius r and of constant charge density. You could calculate electron-electron scattering as a function of momentum transferred and you will discover that it's a function of r. From there you compare with scattering data and see if your data supports r > 0. If not, you set a limit on how small r could have been and you would have see it. That's your limit.

In real life, things are a little different because what I wrote is computationally horrid. But that's the general idea.
 
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  • #21
Vanadium 50 said:
I wrote is computationally horrid.

And grammatically horrid.
 
  • #22
Vanadium 50 said:
Suppose you modeled electrons as an object of radius r and of constant charge density. You could calculate electron-electron scattering as a function of momentum transferred and you will discover that it's a function of r. From there you compare with scattering data and see if your data supports r > 0. If not, you set a limit on how small r could have been and you would have see it. That's your limit.

In real life, things are a little different because what I wrote is computationally horrid. But that's the general idea.
I suppose that, at some point, applying the normal inverse square law forces to predict the scattering pattern falls down and that will be equivalent to a 'flat bottom' to the potential well pattern. You could call that the 'size'.
 
  • #23
sophiecentaur said:
I suppose that, at some point, applying the normal inverse square law forces to predict the scattering pattern falls down and that will be equivalent to a 'flat bottom' to the potential well pattern. You could call that the 'size'.
Can you point to this place?
(Electron-positron cross section)

LeR5G.png
 

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  • #24
mfb said:
Can you point to this place?
(Electron-positron cross section)

View attachment 221421
I can't fathom what the graph shows. What are the axis titles and units? In particular "mb"? I would have moaned at my students for leaving out / assuming such information - and how about a title? :smile:
 
  • #25
mb = millibarns = 10^-31 m^2
 
  • #26
sophiecentaur said:
I can't fathom what the graph shows. What are the axis titles and units? In particular "mb"? I would have moaned at my students for leaving out / assuming such information - and how about a title? :smile:
Millibarns, a unit of cross section, and the x-axis is the center of mass energy in GeV.
The title was written above the image.

I didn't make the graph, and that was the nicest-looking I found with a quick search.
 
  • #27
mfb said:
I didn't make the graph,
But you made the post and included the image. I imagine you wanted to get some message across with it. If you don't actually want to help then why bother posting?
What is the x-axis and what would the units be? kWh / Joules / Bthu?
I can see there are some peaks and an 'inverted well' shape. It may all be blindingly obvious to you but it may not be to all your readers.
Sorry to be grumpy but I am on this thread in the hope of getting to understand the subject (as well as having a friendly conversation).
 
  • #28
sophiecentaur said:
What is the x-axis and what would the units be? kWh / Joules / Bthu?
GeV, see above.
The scale doesn't matter here. The main point is that the plot is full of features, but none of them looks like what you were looking for.
Your approach doesn't work at all.
 
  • #29
mfb said:
GeV, see above.
Oh yes, I see now.
mfb said:
Your approach doesn't work at all.
I was just trying to interpret the suggestion (?) in Post #20 from Vanadium. So how can you find or estimate the 'size' from that data in your post? Could you inject just a tad of positive into your reply? Saying how something could be done is much more useful than saying another thing just won't work. I know there are a lot of people on PF who know more than I (or even you) do in some areas but what do we post here for?
 
  • #30
sophiecentaur said:
I was just trying to interpret the suggestion (?) in Post #20 from Vanadium. So how can you find or estimate the 'size' from that data in your post?
You cannot, that is the point. If electrons would have something like a classical size you could, but they do not have this.
 
  • #31
I don't quite understand what sophiecentaur is saying, but here's some data that shows what a non-pointlike behavior would look like:

mass_signal_elec.png


The red lines on the far right is roughly (and this is oversimplifed) what a ~1.5 x 10-20 m electron size would look like.
 

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  • #32
What you're missing in your description of a "cross-section" is that it is an interaction cross-section. The interaction being through some field whether it is the electromagnetic field, the weak field, or the chromodynamic field. That's how point particles have cross-sections.
 
  • #33
alantheastronomer said:
What you're missing in your description of a "cross-section" is that it is an interaction cross-section. The interaction being through some field whether it is the electromagnetic field, the weak field, or the chromodynamic field. That's how point particles have cross-sections.
I think that's as much as we can expect and I can cope with that idea. It's actually all we get from any cross section measurement. As far as I remember, talk of an electron cross section seems to be in terms of an upper limit - as in "it must be smaller than this value".
mfb said:
You cannot, that is the point. If electrons would have something like a classical size you could, but they do not have this.
I could understand that the cross section would be different for different interactions - eg for charged and non charged particles but that could be the same for proton / proton cross section and proton / neutron cross section and how it could relate to the KE of an electron. My (mis?)conception is based on a though experiment of what would happen for two beams of electrons fired at one another. The distribution of the scattered beams would depend on the KE, the higher the KE, the narrower would be the 'shadow' of an emerging beam. But you are telling me that would fall down once the energies got high enough. (is that what the graph you posted shows?)
 
  • #34
You get various bumps in the spectrum from the production of new particles at their mass, you get steps from additional reactions that become possible above some thresholds, and similar effects. None of these things have anything to do with a size of an electron.
 
  • #35
mfb said:
You get various bumps in the spectrum from the production of new particles at their mass, you get steps from additional reactions that become possible above some thresholds, and similar effects. None of these things have anything to do with a size of an electron.
But I have always been referring to scattering - which implies direction, not the energies involved. (As with Rutherford scattering and the size of the atom)
 
<h2>1. How can point particles have cross sections?</h2><p>Point particles, also known as elementary particles, are subatomic particles that have no measurable size or volume. However, they do have a property called cross section, which represents the probability of a particle interacting with other particles. This is due to the fact that point particles have a certain amount of energy and momentum, which allows them to interact with other particles despite their lack of size.</p><h2>2. What is the significance of cross sections for point particles?</h2><p>Cross sections are important for understanding the behavior and interactions of point particles. They help us calculate the likelihood of particles colliding or interacting with each other, and they also provide information about the strength of these interactions.</p><h2>3. How do scientists measure the cross sections of point particles?</h2><p>There are several methods for measuring the cross sections of point particles, including scattering experiments and particle accelerators. In scattering experiments, particles are fired at a target and the resulting interactions are measured and analyzed. Particle accelerators, on the other hand, use electromagnetic fields to accelerate particles and collide them at high energies, allowing scientists to observe and measure their cross sections.</p><h2>4. Can point particles with different cross sections still interact with each other?</h2><p>Yes, point particles with different cross sections can still interact with each other. The cross section is just one factor that determines the probability of interaction. Other factors, such as the particles' energies and the strength of the interaction, also play a role.</p><h2>5. How do cross sections of point particles relate to the concept of quantum mechanics?</h2><p>The concept of cross sections for point particles is closely related to quantum mechanics, which is the branch of physics that studies the behavior of particles at the subatomic level. In quantum mechanics, particles are described as both particles and waves, and cross sections represent the probability of a particle interacting with other particles as a wave. This probabilistic nature of particles is a fundamental aspect of quantum mechanics.</p>

1. How can point particles have cross sections?

Point particles, also known as elementary particles, are subatomic particles that have no measurable size or volume. However, they do have a property called cross section, which represents the probability of a particle interacting with other particles. This is due to the fact that point particles have a certain amount of energy and momentum, which allows them to interact with other particles despite their lack of size.

2. What is the significance of cross sections for point particles?

Cross sections are important for understanding the behavior and interactions of point particles. They help us calculate the likelihood of particles colliding or interacting with each other, and they also provide information about the strength of these interactions.

3. How do scientists measure the cross sections of point particles?

There are several methods for measuring the cross sections of point particles, including scattering experiments and particle accelerators. In scattering experiments, particles are fired at a target and the resulting interactions are measured and analyzed. Particle accelerators, on the other hand, use electromagnetic fields to accelerate particles and collide them at high energies, allowing scientists to observe and measure their cross sections.

4. Can point particles with different cross sections still interact with each other?

Yes, point particles with different cross sections can still interact with each other. The cross section is just one factor that determines the probability of interaction. Other factors, such as the particles' energies and the strength of the interaction, also play a role.

5. How do cross sections of point particles relate to the concept of quantum mechanics?

The concept of cross sections for point particles is closely related to quantum mechanics, which is the branch of physics that studies the behavior of particles at the subatomic level. In quantum mechanics, particles are described as both particles and waves, and cross sections represent the probability of a particle interacting with other particles as a wave. This probabilistic nature of particles is a fundamental aspect of quantum mechanics.

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