How many pure numbers relate to elementary particles?

[ribbie]

Can someone give a list of all the pure numbers that relate to elementary particles (such as the fine-structure constant)? How many such numbers are there? Is it likely that more will be discovered?

Thanks

 

[maverick_starstrider]

I suppose that depends on what you mean by pure numbers. Do you just mean dimensionless (unitless)? In which case you could pretty much construct as many as you wanted, any ratio will be unitless. Ratio of muon mass to electron mass, electron mass to proton mass, charge of an electron to charge of proton, etc. You could think them up all day. As for the "important" ones wikipedia has the list at the fine structure constant, the proton-to-electron mass ratio, the coupling constant of the strong force and the gravitational coupling constant

 

[the_house]

Can someone give a list of all the pure numbers that relate to elementary particles (such as the fine-structure constant)? How many such numbers are there? Is it likely that more will be discovered?

Thanks

Like maverick_starstrider, I'm not sure what exactly you mean by pure numbers, but let's assume you mean free parameters in the standard model of particle physics.

See the table http://en.wikipedia.org/wiki/Standard_Model#Construction_of_the_Standard_Model_Lagrangian"

Besides the fine structure constant that you mention, there are 2 more coupling constants, making a total of 3. Then there are masses for all of the fermions, making 12 total (the neutrino masses aren't in the wikipedia table, since the original "standard model" had massless neutrinos, but now we know they have mass.) Then there are 3 CKM mixing angles and 1 phase, plus 3 neutrino mixing angles and a corresponding phase. There's also a possible QCD vacuum angle, as well as extra parameters for the Higgs mechanism (according to the wikipedia page, there are 2 parameters for a standard model Higgs).

If you prefer dimensionless numbers, you can just take all of the dimensionfull quantities above and divide them by Lambda_QCD.

 

[Parlyne]

Like maverick_starstrider, I'm not sure what exactly you mean by pure numbers, but let's assume you mean free parameters in the standard model of particle physics.

See the table http://en.wikipedia.org/wiki/Standard_Model#Construction_of_the_Standard_Model_Lagrangian"

Besides the fine structure constant that you mention, there are 2 more coupling constants, making a total of 3. Then there are masses for all of the fermions, making 12 total (the neutrino masses aren't in the wikipedia table, since the original "standard model" had massless neutrinos, but now we know they have mass.) Then there are 3 CKM mixing angles and 1 phase, plus 3 neutrino mixing angles and a corresponding phase. There's also a possible QCD vacuum angle, as well as extra parameters for the Higgs mechanism (according to the wikipedia page, there are 2 parameters for a standard model Higgs).

If you prefer dimensionless numbers, you can just take all of the dimensionfull quantities above and divide them by Lambda_QCD.

The neutrino mixing matrix may have as many as three phases, depending on whether or not neutrinos are Majorana fermions.

Additionally, it would be more conventional to divide the dimensionful parameters by the Higgs vacuum expectation value (vev), since all fundamental particle masses are defined (at tree level) as the product of the particle's (dimensionless) coupling to the Higgs multiplied by the Higgs vev.

Also, $\Lambda_{QCD}$ is usually defined in terms of the running of the strong coupling. And, the value it's usually run from (given the data on hand) is the value from LEP data, taken at the Z-pole. Thus, what we really know involves some couplings and the ratio of $\Lambda_{QCD}$ to the Higgs vev.

 

[the_house]

Thanks for the corrections. You're right on all counts.

I was on the right track at least, though, right? :)

 

[Parlyne]

Thanks for the corrections. You're right on all counts.

I was on the right track at least, though, right? :)

Definitely the right track. Some of this is just a matter of convention or history. For instance, the fact that people usually cite the fine structure constant and the Weinberg angle rather than the SU(2)_L and U(1)_Y coupling constants.

 

[PhilDSP]

Definitely the right track. Some of this is just a matter of convention or history. For instance, the fact that people usually cite the fine structure constant and the Weinberg angle rather than the SU(2)_L and U(1)_Y coupling constants.

Can you recommend a good resource that clearly explains the theory relating the Fine Structure Constant and Weinberg angle to the SU(2)_L and U(1)_Y coupling constants?

And in general, a resource that shows what factors of geometry and groups are necessary to arrive at the Standard Model?

 

[Parlyne]

Can you recommend a good resource that clearly explains the theory relating the Fine Structure Constant and Weinberg angle to the SU(2)_L and U(1)_Y coupling constants?

And in general, a resource that shows what factors of geometry and groups are necessary to arrive at the Standard Model?

This is generally covered in the major field theory texts to one degree or another. The treatment in the later chapters of Peskin and Schroeder is pretty good.

If you're looking for something a little cheaper, Howard Georgi has a book on the standard model available for download from his personal website.