Cross section and mass

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Ranku
In the sentence "WIMP-nucleon cross sections of 1.2x10-47cm2 at 1 TeV/c2 WIMPs", there is a relationship between cross section and mass. Is there a general formula that relates the two quantities, in that if there is a certain cross section that means it will be associated with a certain mass?

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In the sentence "WIMP-nucleon cross sections of 1.2x10-47cm2 at 1 TeV/c2 WIMPs", there is a relationship between cross section and mass. Is there a general formula that relates the two quantities, in that if there is a certain cross section that means it will be associated with a certain mass?
Source?

In general cross sections depends on mass for various reasons. Available phase space for particles that are produced. Some particles, like the higgs boson, have interaction "strenghts" which depends on the masses of the involved particles.

There is no general formula which applies to all processes.

Ranku and vanhees71
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There is a certain (known) amount of dark matter in the universe. The heavier each individual particle is, the fewer particles you need. If you need fewer particles, you need a lower production cross-section. That's all they are saying.

vanhees71, Astronuc and malawi_glenn
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In the sentence "WIMP-nucleon cross sections of 1.2x10-47cm2 at 1 TeV/c2 WIMPs", there is a relationship between cross section and mass. Is there a general formula that relates the two quantities, in that if there is a certain cross section that means it will be associated with a certain mass?
How Are Maximum Possible Cross-Sections Of Interaction Calculated For Different Possible Dark Matter Particle Masses In Direct Dark Matter Detection Experiments?

Their Assumptions About Dark Matter Near Earth

Direct Dark Matter Detection experiment scientists have modeled how many dark matter particles should have passed through their detector's area in the time period during which they gathered data.

This is based upon the estimated dark matter density (mass per volume) in the vicinity of the Earth, divided by different possible dark matter particle masses, and adjusted for plausible velocities of the dark matter particles relative to the detector.

The assumptions about the local dark matter mass density are based upon models of mass density of the inferred dark matter halo of the Milky Way in the vicinity of Earth, which in turn is based upon the dynamics of the Milky Way galaxy relative to a Newtonian, no dark matter assumption.

The assumptions about the velocity of dark matter particles are based upon galaxy dynamics and the amount of observed structure in the universe (which support a conclusion of warm or cold dark matter, that are defined by the mean velocity of the dark matter particles -- lower velocities would produce way too many satellite galaxies and subhalos, while higher velocities would prevent galaxies from forming at the rates that they are observed to form).

One paper from the LUX dark matter detection experiment for example, modeled what a signal ought to look by making the following assumption (on page four of this paper) about the velocities of the dark matter particles and the Earth, in plain text, and about the estimated local dark matter density in the vicinity of Earth, which I have put in bold:

Nuclear-recoil energy spectra for the WIMP signal are derived from a standard Maxwellian velocity distribution with v0= 220 km/s, vesc= 544 km/s, ρ0= 0.3 GeV/cm3, average Earth velocity during data taking of 245 km/s, and a Helm form factor
Using Their Assumptions To Make Calculations

Then, the scientists determine based upon the actual data and the uncertainty in their data, the maximum number of events in excess of the predicted number of background events that they expect to see over the course of the experiment at different strengths of nuclear recoil in the protons and neutrons in the detector.

The amount of nuclear recoil in a signal event of a collision between a proton or neutron and a dark matter particle of a particular mass and assumed velocity can be related to the mass of the dark matter particle in the rare cases where it does interact with a nucleon.

More massive dark matter particles should give rise to larger average nucleon recoils but should happen less frequently (once backgrounds are subtracted out), while less massive dark matter particles should give rise to smaller average nucleon recoils but should happen more frequently (once backgrounds are subtracted out).

The more precise the nuclear recoil event detections are and the more precisely and accurately the background expectation nuclear recoil events can be modeled, the smaller the potential signal that can be due to experimental uncertainty alone will be, and the smaller the amount of the maximum dark matter cross-section which isn't just due to experimental uncertainty will be.

The maximum possible cross-section of interaction at a given hypothetical dark matter particle mass is then calculated based upon the maximum possible dark matter signal divided by the number of dark matter particles that passed through the area of their detector during the course of the experiment if the particles had that mass and produced the appropriate nuclear recoil in the cases where they did interact with the protons and neutrons in the detector.

The data are then analyzed for a wide range of dark matter particle masses hypotheses, doing a statistical hypothesis test at each possible dark matter particle mass.

This is then converted to appropriate units and displayed in a chart like this one:

From this 2015 paper (chosen at random).

Real Direct Detector Exclusion Curves Analyzed

The constraints on the cross-section of interaction at low masses gets much weaker because the backgrounds get bigger when you approach the masses of nucleons themselves making it hard to distinguish nucleon-nucleon collisions giving rise to nuclear recoils in a given mass range from dark matter-nucleon collisions for dark matter particles of similar mass.

The uncertainties at high masses arises, in part, from the fact that for a high enough dark matter particle mass, the predicted number of dark matter particles to pass through the detector during the experiment is so low that even with a fairly high cross-section of interaction it becomes more and more plausible that there would be no signal events at all during your data taking period as a matter of simple random chance.

The sweet spot in between where the cross-section is constrained to be smallest experimentally is where the backgrounds are small and well controlled, but the number of dark matter particles that are expected to pass through the detector during the course of data taking is still fairly large.

Real Collider Exclusions Curves Analyzed

The constraints from CMS (one of the LHC collider experiments) has a different shape because it is making assumptions about the energy scales at which dark matter particles could be produced at different masses in its collisions compared to missing energy at different energy scales that could be due to dark matter particles, after subtracting out the expected neutrino background (and other sources of missing energy like charged particle detector inefficiency).

All missing energy can be detected at CMS, even though the source of the missing energy has to be inferred in light of the experiment's models of the missing energy producing backgrounds.

So, the cross-section is estimated based upon what cross-section would be necessary to produce enough dark matter particles of a specific mass to account for all potentially non-background missing energy, rather than based upon the number of dark matter particles that could have interacted with a detector.

This approach turns out to be much less sensitive to the mass of the hypothetical dark matter particle.

Calibration

You can also test your direct dark matter detector to see that you haven't missed something important in your design by exposing it to a stream of ordinary particles with a known cross-section of interaction and known energies from a source that emits those particles.

For example, you could put an exactly measured amount of a very pure substance that experiences beta decay at a known rate with a known range of energies to see if the electrons and neutrinos that the beta decay from the source produces and sends through your detector matches what your model says it should in addition to your estimated backgrounds.

If it does, you can be confident that your modeling is experimentally confirmed and that dark matter particles with the properties you are testing for should behave the way that you expect them to if dark matter particles with those properties exist.

What Scientists Have Learned So Far

The state of the art direct dark matter detection experiments, like LUX, have determined for dark matter particle mass ranges from below 1 GeV to up to about 1,000 GeV that if dark matter particles in this mass range exist and have any interactions with ordinary protons and neutrons at all, that the strength of this interaction described by the cross-section of interaction, is on the order of millions to billions of times weaker than the cross-section of interaction of a neutrino (which is the most feebly interacting kind of ordinary matter definitively determined to exist in the Standard Model so far).

In and of itself, that isn't a huge blow to the general idea of a dark matter particle hypothesis. One important class of proposed dark matter particles (which are called "sterile") has a zero cross-section of interaction with ordinary matter, and the LambdaCDM "standard model of cosmology" assumes that dark matter is, at a minimum, "nearly collisionless".

But the state of the art results are a big problem for theories like supersymmetric WIMP dark matter particle proposals, where the model you are using assumes that your dark matter particle candidates will have a particular, calculable, non-zero cross-section of interaction.

For example, if your dark matter particle candidate has a mass of 1 GeV to 1000 GeV and interacts with other particles only via the weak force and gravity, and has the same weak force cross-section of interaction as a neutrino does (which was precisely what scientists were expecting at first in the 1980s when both dark matter and supersymmetry were brand new promising ideas), your ship has sailed and you get a participation medal, but no Nobel prize, because that possibility was ruled out long ago.

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Ranku and vanhees71
Ranku
Source?
It's from a report.

Ranku
If you need fewer particles, you need a lower production cross-section
Does lower cross section naturally follow from having fewer heavy particles?

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The other way around. If the cross-section is low, you don;t make so many.

vanhees71
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The other way around. If the cross-section is low, you don;t make so many.
If cross-section is low, does that imply lesser number of collisions with the target nucleon?

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That's what a cross-section is.

malawi_glenn and vanhees71
Ranku
So how to reconcile
There is a certain (known) amount of dark matter in the universe. The heavier each individual particle is, the fewer particles you need. If you need fewer particles, you need a lower production cross-section. That's all they are saying.
with this:
More massive dark matter particles should give rise to larger average nucleon recoils but should happen less frequently (once backgrounds are subtracted out), while less massive dark matter particles should give rise to smaller average nucleon recoils but should happen more frequently (once backgrounds are subtracted out).

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Why do you think they dont say the same thing?

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So how to reconcile

with this:
The statements are apples and oranges.

The first statement is about the number of dark matter particles that go through the detector.

If there is 0.3 GeV per cc of DM in the vicinity of the universe, then you will have three times as many particles pass through your detector is the DM particle has a mass of 100 MeV as you will if it has a mass of 300 MeV. The cross-section is a function of the number of signal events per particle passing through the detector.

The second statement is about what happens in the rare individual events when a DM particle of a given speed (by assumption DM is moving at the same average velocity regardless of its mass) actually interacts with a proton or a neutron. It is not a statement about the cross-section itself, which only depends upon the number of events per particle passing through the detector.

If the DM particle has a mass of 100 MeV the proton or neutron will receive a smaller bump than it will if the DM particle has a mass of 300 MeV. So, the amount of recoil resulting from DM interactions in your excess events helps you estimate the mass of the DM particle that is causing the excess events. Big recoils in the excess events means more massive DM particles, while small recoils in the excess events means less massive DM particles.

Of course, the recoil size data isn't very meaningful, and is basically just playing games with the power of your experiment in light of its measurement uncertainties, if you don't have any statistically significant excess events at all, which has been basically what has happened so far in every single direct detection experiment.

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mfb
Ranku
Why do you think they don't say the same thing?
A bit of context here: Cross-section is generally understood in terms of probability or frequency of interaction between particles. It can, however, be also loosely thought of in terms of the 'size' of particles, since after all the unit of cross-section is area: cm -2.
However equating cross-section with size of particles seems to create a contradiction, since quote 1 in post #10 "There is a certain (known) amount of dark matter in the universe. The heavier each individual particle is, the fewer particles you need. If you need fewer particles, you need a lower production cross-section. That's all they are saying" associates heavier particles with smaller cross-section, and which would be equivalent to smaller-sized particles, while quote 2 in #10 "More massive dark matter particles should give rise to larger average nucleon recoils but should happen less frequently (once backgrounds are subtracted out), while less massive dark matter particles should give rise to smaller average nucleon recoils but should happen more frequently (once backgrounds are subtracted out)" associates lighter particles with larger cross-section, which would be equivalent to larger-sized particles, both of which seem physically counter intuitive.
In terms of frequency of interactions, heavier particles have smaller cross-sections because they are fewer, while lighter particles have larger cross-sections because they are numerous. Thus, instead of directly equating cross-section with size of particles, it would be more appropriate to inversely associate lower cross-section, heavier, fewer particles with larger-sized particles, and larger cross-section, lighter, numerous particles with smaller-sized particles.

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This is impossible to read. Ever heard of paragraphs?

Ranku
This is impossible to read. Ever heard of paragraphs?
Politeness of tone is also appreciated, along with paragraphic ease of reading.

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Politeness of tone is also appreciated, along with paragraphic ease of reading.
You are the one asking questions and want to get help.

Here, I made it for you:
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A bit of context here: Cross-section is generally understood in terms of probability or frequency of interaction between particles. It can, however, be also loosely thought of in terms of the 'size' of particles, since after all the unit of cross-section is area: cm-2.

However equating cross-section with size of particles seems to create a contradiction, since quote 1 in post #10
There is a certain (known) amount of dark matter in the universe. The heavier each individual particle is, the fewer particles you need. If you need fewer particles, you need a lower production cross-section. That's all they are saying.
associates heavier particles with smaller cross-section, and which would be equivalent to smaller-sized particles, while quote 2 in #10
More massive dark matter particles should give rise to larger average nucleon recoils but should happen less frequently (once backgrounds are subtracted out), while less massive dark matter particles should give rise to smaller average nucleon recoils but should happen more frequently (once backgrounds are subtracted out).
associates lighter particles with larger cross-section, which would be equivalent to larger-sized particles, both of which seem physically counter intuitive.

In terms of frequency of interactions, heavier particles have smaller cross-sections because they are fewer, while lighter particles have larger cross-sections because they are numerous. Thus, instead of directly equating cross-section with size of particles, it would be more appropriate to inversely associate lower cross-section, heavier, fewer particles with larger-sized particles, and larger cross-section, lighter, numerous particles with smaller-sized particles.

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To answer your question, Vanadium wrote "heavier", not "bigger" as is "bigger size". It seems to me that you have misunderstood this concept. After all, the DM particles are thought of being point-like anyway.

Ranku
To answer your question, Vanadium wrote "heavier", not "bigger" as is "bigger size". It seems to me that you have misunderstood this concept. After all, the DM particles are thought of being point-like anyway.
Of course I understand ‘heavier’ is not ‘bigger’.

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Of course I understand ‘heavier’ is not ‘bigger’.
Then why are you referring to sizes all the time? You are the only one doing so in this thread.

Ranku
Then why are you referring to sizes all the time? You are the only one doing so in this thread.
Because SM particles are assigned sizes, even though SM particles can also be seen as point-like.
I am trying to relate SM to DM particle size. I am wondering how tight is the correlation between size and mass of an SM particle. Suppose a DM particle were to be of similar size to an SM particle, would their mass be similar as well?

malawi_glenn
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Because SM particles are assigned sizes,
Are they?

vanhees71
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weirdoguy
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Forbes is not a good source for particle physics...

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I wrote weight, not size.

Forbes is not a very good source for particle physics.

The omniparagraph is hard to read.

Cross-section does not mean size. I can take two nuclei of similar size and the neutron cross section of one can be many thousands (possibly millions - I'd have to look it up) of times bigger than the other. So long as you insist cross-section means "size" you will not get a satisfactory answer to your question.

My point is simple - if you have 10 pounds of fruit and apples weigh 4 ounces and oranges weigh 8, I have either 40 apples or 20 oranges. I need more apples than oranges, and thus a higher apple production rate, There is nothing more profound than that.

malawi_glenn, Ranku and vanhees71
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Forbes is not a good source for particle physics...
Ethan Siegel is.

Ranku
So long as you insist cross-section means "size" you will not get a satisfactory answer to your question.
I am not insisting they are - I am trying to explore the correlation between the two, if any, since the unit of cross-section is cm-2, which is intriguing.

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Ethan Siegel is.
No, in particular not his (or any other) popular science articles.

Ranku
No, in particular not his (or any other) popular science articles.

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I am not insisting they are - I am trying to explore the correlation between the two, if any, since the unit of cross-section is cm-2, which is intriguing.
Moment of force has the same unit as mechanical work (Nm), what are the correleation between those two?

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Ethan Siegel is not a reliable source. On multiple occasions he has made an error, I reported it to him, he acknowledged the error, and then didn't correct it. He writes "sciency entertainment" not science.

In any event, I think this thread can be closed. You asked a question, got an answer, and did not accept the answer. What purpose is there in going around again and again and again?

ohwilleke, vanhees71 and malawi_glenn
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Ethan Siegel is not a reliable source. On multiple occasions he has made an error, I reported it to him, he acknowledged the error, and then didn't correct it. He writes "sciency entertainment" not science.

In any event, I think this thread can be closed. You asked a question, got an answer, and did not accept the answer. What purpose is there in going around again and again and again?
As a matter of fact I also got into an argument with him about an error, and while he acknowledged the error privately, he didn’t correct it in his blog.
Anyways, that’s besides the point. I didn’t say I didn‘t accept the answer…I am just exploring the issue. I was about to request closing the thread myself, since someone else is extending it unnecessarily.

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... since someone else is extending it unnecessarily.

In this case, I thank everybody for their participation and lock the thread.

vanhees71