Thank you for your ten minutes review on my 19 page paper anyway.

I understand people taken those courses are easily restricted to the old framework, don't even dare to think outside the mainstream.

Article Functions of the Gravitational Potential

The electroweak coupling constants (there are two of them), of course, depend on the energy-and they increase with energy. Their evolution, however, is known only in perturbation theory, so that sets the limit on how far we can trust their growth. Attempts to study their flow beyond perturbation theory, e.g. on the lattice are quite hard and haven't yet reached the same maturity level. It's a hard problem.

Thank you for your answer, Stam. Could you give me some reference links?

This is the basis of the Standard Model, there are countless lectures and books on the subject. One example is here: http://arxiv.org/abs/1210.5296 `Radiative Corrections in Precision Electroweak Physics: a Historical Perspective', or

http://www.ncp.edu.pk/docs/3PPW/GA3.pdf, `Electroweak Interactions in the Standard Model and Beyond'.

A substantial change of the gauge couplings of the electroweak sector of the SM in view of the decay properties of a Higgs-like particle into electroweak gauge bosons would contradict electroweak precision experiments at the Z resonance, or, as was pointed out by Stam, invalidate the perturbative evolution of these couplings from the Z mass up to ~ 126 GeV.

Guoliang, on Wikipedia

http://en.wikipedia.org/wiki/Landau_pole

you may find some answers to your question

and further literature references.

A simple answer could be: if weak coupling constant changes, then lifetimes of beta-decaying nuclei and of elementary particles decaying via W bosons would change significantly. Increase of the coupling constant would mean smaller lifetimes, and that means faster decays....

There isn't any doubt that the coupling constants *do* change with energy and it suffices to compare the measurements made at LEP, Tevatrron and LHC to see this. For instance, the fine structure constant changes from 1/137 at atomic scale energies to 1/128 at the Z mass, i.e. 90 GeV. This doesn't seem much-but it was measured.

And these results are in agreement with the calculations made from the renormalization group equations.

There are two different issues here, I think. 1) running coupling constant, which is a well known feature of QED and of weak interactions, 2) real absolute change of weak coupling constants g and g', which would cause masses of W and Z bosons to change, with consequent changes of beta-decay lifetimes, and neutrino interactions with matter.

Of course running coupling constants imply real change in their values.

However this change has different consequences on different processes. It will be much less visible in beta decay in ordinary atoms, for instance, because the energy of the process is controlled by the energy of the products. To observe the effect of the running coupling on beta decay one would need to observe neutrons at very high energies and measure their decay rate. From the dependence of the coupling on energy one could predict this change.

Similarly for any other process. All this is well known-and much has been measured. The radiative corrections to electroweak parameters have been instrumental in checking the consistency of the Standard Model-and were able to predict the value of the top mass two years before it was discovered.

The mass of the Z, measured at LEP or the LHC, is not equal to g v, with g the ``tree level'' value of the coupling constant and v the tree level value of the vev of the Higgs-and how it depends on energy is one way to check for new processes.

Cf. for instance, http://cds.cern.ch/record/217306/files/199103478.pdf.

To obtain agreement with experiment one *needs* to take into account the running of the couplings and of the other parameters with scale, cf. http://www.slac.stanford.edu/econf/C040802/papers/L009.PDF

Peter,

Bigger coupling constant means higher energy of a beta decay particle, should that means a longer half life of a beta decay according to special relativity time dilation?

Energy of Beta particle is determined by masses of decaying nucleus and daughter remnant nuleus, while anti-Neutrino si taking away part of this energy too. So: changing weak coupling constant would not change energy of beta particles. Only the probability of beta decay would be changed. Energy of emitted particles would remain the same. If my understanding of your question is correct...

The weak interaction is responsible for both the radioactive decay and nuclear fusion of subatomic particles. A bigger coupling constant should imply a stronger nuclear fusion and a more stable subatomic particle, which should have a longer half life.

In a usual sense Fusion means just joining separate nuclei together to create a bigger nucleus, for example D+D ---> He3 + n, or 15N+p ---> 16O + gamma, and thus weak interaction does not have to be involved. However, the fusion of p+p in Sun, known as 1) "pp" and 2) "pep" processes includes also weak interaction as a sub-process 1) p+p ---> He2 ---> (pn) + neutrino + e+, or subprocess 2) p+p+e ---> (pn) + neutrino, And therefore in this sense I agree with you: real change of the weak coupling constant would affect probability of "pep" and "pp" fusion process in the Sun.

In the Sun I'm not sure-wouldn't we've seen this in solar neutrino flux measurements? Perhaps in magnetars?

Should the electroweak coupling constant have an upper limit at 1? otherwise the weak force will be stronger than the nuclear force. In case of supernova explosion or forming a magnetar, it should be very close to the upper limit.

Once more: if the coupling constants of the electroweak sector become O(1) perturbation theory doesn't make sense, since the corrections become comparable to each other. In principle we can imagine using a lattice approach and studying the sector for any other values. There do exist some conceptual issues here to be resolved. But we do not, currently, control the evolution of these coupling constants with energy beyond perturbation theory. If we look at how they evolve with energy, e.g. http://hep-www.colorado.edu/~nlc/SUSY_Wagner/susy/node1.html, however,

we find that, extrapolating up to 10^(19) GeV, the electroweak couplings remain much less than 1, so perturbation theory should remain a good guide-assuming new degrees of freedom do not appear, of course. In that case we must take them into account, if and when they appear.

Above this scale we expect that gravitational interactions become significant and what happens then we do not, yet, know.

Electroweak theory predicted that the coupling constant can equal to 1/137, when the temperature reaches 10^15K, so unless there is an upper limit for the temperature, otherwise it can be bigger.

No, 1/137 is the value of the fine structure constant at atomic scale energies, i.e. about 1 eV. This energy corresponds to a temperature of k_B T, where k_B is Boltzmann's constant. If we put in the conventional units, we find that

1eV = 1.6 x 10^(-19) J = 1.38 x 10^(-23) T => T=10^4 K.

At 90 GeV=90 x 10^9 eV = 9 x 10^(13) K, the mass of the Z, its value is about 1/128 and it increases with energy, i.e. temperature.

Once more, cf. http://hep-www.colorado.edu/~nlc/SUSY_Wagner/susy/node1.html

(the graph there starts at around 100 GeV). As you can see, as the energy increases to 10^(19) GeV, nothing special happens.

However: if you take quarks and leptons and put them in contact with a heat bath at a given temperature, then interesting things can happen: you can have transitions to other phases. However this won't affect the value of the couplings as such.

All experiments so far have proved the fine structure constant is invariant, so I don't think the 1/128 at 90 Gev is the unified energy.

No, the fine structure constant does run with energy (cf. the graph in http://hep-www.colorado.edu/~nlc/SUSY_Wagner/susy/node1.html

1/128 is the *measured* value of the fine structure constant at the Z mass. Neither it, nor the Z mass is the `ùnified energy''. The scale of Grand Unification is around 10^(16) GeV.

I am talking about the electroweak unification not the grand unification.

When the temperature reaches about 2X10^11K, the electroweak coupling constant will be 1/137. At that point the weak force and the electromagnetic force will merge into a combine electroweak force, so the fine structure constant will disappear along with the electromagnetic force. The running coupling constant bigger than 1/137 cannot be called fine structure constant anymore.

2X10^11K is just about 10 MeV, which is larger than electron mass 0.511 MeV, so the fine structure constant actually "runs" a little away from 1/137: it is about 1/136.4 at 10 MeV. If you want to find an energy scale below which the fine structure constant doesn't run, the answer should be 0.511 MeV.

Whatismore, this energy scale is 4 orders of magnitude smaller than electroweak unification scale, which is about 100 GeV.

No, the running of the fine structure constant runs, also, below 0.511 MeV-in the Stndard Model renormalization doesn't depend on symmetry breaking, so the renormalization procedure can be realized in the symmetric phase, where the electron is massless. There an arbitrary scale must be introduced to define the theory, but

physical quantities don't depend on it. The translation of this statement into equations is that the coupling constant runs. It wouldn't run at a fixed point. The value of the fine structure constant at the electron mass, however is not such a fixed point. As far as we can control, the tbeta function for both coupling constants in the electroweak sector is positive-and the beta function for the strong coupling constant is negative.

Beta function (one-loop level) is zero when there is no charged particle below the energy scale. Why should electron be treated as a massless particle here? Electron mass is an important scale in QED, which I think cannot be neglected.

Of course the beta function of pure electromagnetism is zero, since pure electromagnetism is a free theory. However massless QED is not a free theory. And the beta function, in particular, does not depend on mass.

Jia-Shu,

Can you confirm that the electroweak coupling constant will be exactly 1/137 when the temperature is 1.945X10^11K? 0.511Mev equals 5.93X10^9K, quasars have temperature higher than this, but there is no evident showing the fine structure constant has changed in there. Over 100Mev seems to be the energy scale for a grand unification, because when the temperature reaches 2.287X10^12K, the coupling constant will be 1.

Could you give me some reference links to calculate the electroweak unification temperature?

No, fine structure constant runs about 0.5% away from 1/137 at 10 MeV.

No, 100 MeV is much smaller than grand unification scale, which is about 10^16 GeV.

No, fine structure constant is not 1 at 100 MeV. It will only run to 1/128 at m_Z=91 GeV, obviously far from 1.

Electroweak unification does not mean two couplings constants become equal. It just means that these two interactions both originates from SU(2)_L \times U(1)_Y gauge interaction.

If you want to know how couplings run between different energy scales, you can read QFT textbooks to learn about Renormalization Group Equations (RGE), especially how beta functions are calculated.

For electroweak symmetry breaking, you can start by reading wikipedia:

http://en.wikipedia.org/wiki/Standard_Model

Of course QFT knowledge is needed if you want to fully understand it.

You seems to be unfamiliar with energy scales, so I list some below:

\Lambda_{QCD}: about 200 MeV, below which QCD is unperturbative.

Electroweak symmetry breaking: 173 GeV (sometimes written as 246 GeV, a sqrt(2) factor larger), which is Higgs VEV.

GUT: about 10^16 GeV, three gauge coupling constants run to a same value here.

Plank scale: 10^19 GeV, where quantum gravity is important.

Thank you, Jia-shu. But how do you explain the fine structure constant seems to be invariant in very high temperature quasars?

As I said, fine structure constant runs about 0.5% away from 1/137 at 10 MeV. Astrophysical uncertainties are generally larger than this, so it is impossible to measure the small deviation from 1/137 by observing quasars.

Please check the link below: Over the 2 billion years, the change of alpha has therefore to be smaller than about 2 parts per 100 millions.

http://www.eso.org/public/news/eso0407/

That link is talking about arXiv:astro-ph/0402177.

This work is to measure the dependence of alpha on TIME, not on TEMPERATURE. They use absorption spectrum of atoms such as Mg and Fe to determine whether alpha has changed over time. As some absorption lines are insensitive to small alpha changes while others are sensitive, compare the relative shifts of different absorption lines can tell whether alpha is different. Time information can be obtained from red shifts of spectrums.

Atoms binding energy is much smaller than electron mass, so they can only survive in low temperature area, where alpha is just 1/137. This work uses atom absorption spectrum, so it only cares about alpha at energy scale lower than electron mass. To make it more clear, this work uses absorption lines such as Mg II 2797, which has wave length of 279.7 nm, corresponding to eV scale energy.

In a word, this work has nothing to do with detecting alpha at temperature as high as 10 MeV.

Thank you, Jia-Shu. Do you think the electroweak coupling constant has an upper limit at 1? What will happen if it is 1?

There is no upper-limit for alpha.

Mathematically it can run all the way to infinity. The energy scale where coupling constants become infinity is known as Landau pole.

Realistically, alpha is still less than 1 at GUT scale in Standard Model or MSSM, so there is no need to worry about landau pole. If you want to find what alpha will become beyond GUT scale, you need a complete GUT. Generally GUT is asymptotic free, so the gauge coupling will become smaller when energy is higher. After reaching the Planck scale, knowledge of quantum gravity is further needed.

Coupling constants larger than 1 just means perturbation method are going to be invalid. In fact, it is \alpha / 4\pi = 1 that indicates invalidity of perturbation.

Alpha equals the speed of electron divided by the speed of light in vacuum, a bigger than 1 alpha will violate the speed limit set by special relativity.

That is not true. Alpha only describes the strength of electromagnetic coupling.

The myth "alpha equals electron speed over c" originates from Bohr's hydrogen model, which is a NON-RELATIVISTIC model and uses E_k = mv^2 / 2 as kinetic energy of electron.

If alpha is larger than 1, then the ground state of hydrogen atom will have a relativistic electron, which cannot be described by Bohr's model or Schrodinger's equation. Of course, no matter how large is the electron's energy, its speed can never be large than c.

In fact, a proton is such a relativistic bound state of quarks, with strong coupling constant alpha_s >> 1. There is no way to solve relativistic bound states analytically like solving hydrogen atoms nowadays.

One consequence is that for a big alpha we cannot do

perturbation theory. So the QED theory, or electroweak

theory, both of them are perturbative field theories, will

break down. Then one has to develop an alternative

strongly interacting theory.

The highest temperature so far observed in the universe is about 500 billion K, which is in the center of a quasar. So the temperature or the coupling constant may have an upper limit for whatever reason.

Dear All, I read some literature regarding you question and find it very much interesting.

http://pdg.lbl.gov/2011/reviews/rpp2011-rev-standard-model.pdf

Guoliang,

The running of coupling constant has been measured in colliders with high accuracy. LEP, for example, had centre-of-mass energy of 200 GeV and measured values of three gauge coupling constants at M_Z=91 GeV. The result conincides with theory prediction, so there is no reason to worry about 'upper limit of couplings'.

For the 'highest temperature': about a year and a half ago, ALICE experiment produced 5.5 trillion K quark-gluon plasma, which is about 500 MeV.

Quasar tempererature of 500 biillion K is only 50 MeV, which is pretty low comparing with powerful colliders. If you are interested in the high energy frontier of particle physics, it is better to pay attention to colliders like LHC and ILC.

Jia-Shu,

I am more interesting in why supernova or hypernova is triggered at the temperature range from 2X10^11K to 2X10^12K. Can standard model give any explanation to this?

You should probably start a new question for SN, as this is not closely related to standard model.

Different nuclear fusion processes need different ignition temperature. A small star will cease nuclear fusion and becomes a carbon-oxygen white dwarf because temperature is not high enough to ignite carbon and oxygen fusion. A large star will have a core made of iron, which can never fuse.

When the mass of a dwarf star or the iron core of a large star becomes larger than Chandrasekhar limit, collapse will happen. The gravitational potential energy transforms into thermal energy, resulting in a temperature of order 10 MeV or higher. The collapse sometimes resulting in supernova blast.

What is the exact mechanism by which an implosion of a dying star becomes an supernova explosion? I think a gravitational theory combined with standard model are necessary to solve this puzzle.

Large amount of gravitational potential energy transforms into thermal energy. Hot matter radiates photons. Although photon energy only takes a small part of the total energy in a SN blast, SN photon is bright enough for observers on the Earth.

For core collapse SN, the hot core's energy is first transmitted to outer matter through neutrino radiation, so things are a little complex here. However, there are already successful numerical simulations of core collapse supernovae.

No,this is one of the unsolved problem in physics.

Actually I have proposed a cosmological model to solve this problem, please review my paper posted on RG, any comments will be welcome.

I have take a few glances at your paper. To be honest, you should take gaduate courses of general relativity, cosmology, astrophysics and particle physics before doing any serious research in these areas.

Thank you for your ten minutes review on my 19 page paper anyway.

I understand people taken those courses are easily restricted to the old framework, don't even dare to think outside the mainstream.

Article Functions of the Gravitational Potential

One should really think of two couplings for the electroweak theory.

Why I say two not one ? The reason is that electroweak theory

has the invariance SU(2)_L x U(1)_Y. And electric charge is

given by Q=T^3_L + Y/2. Now for the two factors we can talk

about two couplings at the scale m_Z, say alpha_2 and alpha_1.

Non Abelian theories are asymptotically free, but the U(1) coupling

can really grow bigger by the renormalization effects if there are

contributions to beta functions from fermions and scalars.