# How does the electron mass run with energy?

1. Dec 29, 2008

### franoisbelfor

Renormalization in QED implies the running of charge and mass. But does the physical electron mass increase or decrease with energy? And what value is reached at the Planck energy? Somewhere I read that the mass change was 16%, but all books I on QED I searched do not give numbers. (Assuming the strong and weak interactions are neglected.)

Does anybody know a reference ?

François

2. Dec 29, 2008

### blechman

From a "straightforward" calculation (as straightforward as these calculations can be!):

$$\frac{dm_e}{d\log\mu} = \left(\frac{3e^2}{8\pi^2}\right)m_e$$

to one loop. So as you might expect from screening arguments, the mass increases in the UV.

3. Dec 29, 2008

Staff Emeritus
To add, you cannot neglect the weak interaction in the UV, as QED is the IR limit of part of the electroweak theory.

4. Dec 29, 2008

### blechman

quite right. the formula I gave is ONLY for QED at one loop - that means photon, electron and positron, NOTHING else. Strictly speaking, this is only correct in the UV up to the muon mass.

To get an accurate and correct number that describes the universe that we live in, of course you must go much further, including the other fermions (leptons and quarks), and then match to EW theory above the W boson mass, if you want to push the calculation up to the Planck scale (assuming that there's a desert from the weak to Planck scales).

5. Dec 30, 2008

### franoisbelfor

Ok, if one assumes hat the 511keV are measured at $$\mu=$$ 1 eV, and uses $$\left(\frac{3e^2}{8\pi^2}\right)=2.7*10^{-4}$$ one gets a electron mass at the Planck energy (10^28 eV) that is (10^28)^(2.7*10^-4)=1.018 times
higher than at 1 eV. Only 2% difference!

Is that correct? (This neglects weak and strong interaction effects, of course.)

François

6. Dec 30, 2008

### blechman

you have to be more careful:

(1) nothing runs below the electron mass itself, so you don't want to start the running at 1 eV, but at $m_e(m_e)=$511 keV.

(2) you also have to take into account the (very important!) point that e^2 is not a constant but also runs, so the diffEQ is a little more complicated than what you naively wrote down. This will enhance the mass a bit.

But it is true that the effect is not very large... a few percent at most -- QED remains quite weak up to the Planck scale.

And of course, this is in a fictitious universe where there are no other forces or particles other than the electron, positron and photon. In other words, a homework problem, not a true physics result!

7. Dec 30, 2008

### franoisbelfor

Thank you very much! This was very illuminating. Even after recalculation, the effect
still is in the 2% range.

Thank you again!
François

8. Dec 30, 2008

### franoisbelfor

How does the NEUTRINO mass run with energy?

In a separate thread I was told very clearly that even at Planck energy, the electron mass is renormalized (by QED alone) only by a few percent. The renormalization equation is quite simple.

Now, what happens to the *neutrino* mass at Planck energy? Obviously, being neutral, the neutrino mass is not normalized by QED (well, at least not at first order), but by the weak interaction.

Are there any numbers available on how neutrino masses run up to the Planck energy?
(Assuming, of course, that the standard model is valid throughout, i.e., no susy, no technicolor, etc.) Are the renormalization equations as simple as for QED? Are there any books/papers on the issue?

Thank you for any hint!

François