- #1
tomdodd4598
- 138
- 13
- TL;DR Summary
- A question about the renormalisation and running of dimensionful coupling constants.
I am still rather new to renormalising QFT, still using the cut-off scheme with counterterms, and have only looked at the ##\varphi^4## model to one loop order (in 4D). In that case, I can renormalise with a counterterm to the one-loop four-point 1PI diagram at a certain energy scale. I can choose an on-shell point ##\{s={ \sigma }{ \mu }^{ 2 },\quad t={ \tau }{ \mu }^{ 2 },\quad u={ \upsilon }{ \mu }^{ 2 }\}## (where ##s,t,u## are the Mandelstam variables) at which to define a physical coupling ##\lambda\equiv\lambda(\mu)##, yielding a four-point amplitude of $$\tilde { \Gamma } \left( { p }_{ 1 },...,{ p }_{ 4 } \right) \approx \lambda +a{ \lambda }^{ 2 }\ln { \left[ \frac { stu }{ \sigma \tau \upsilon { \mu }^{ 6 } } \right] }.$$ We can also get a beta function from this using $$\lambda \left( \mu +d\mu \right) \approx \lambda \left( \mu \right) +a{ \lambda \left( \mu \right) }^{ 2 }\ln { \left[ \frac { { \left( \mu +d\mu \right) }^{ 6 } }{ { \mu }^{ 6 } } \right] },$$ $$\cdots$$ $$\mu \frac { d\lambda }{ d\mu }=\beta \left( \lambda \right) \approx6a{ { \lambda }^{ 2 } }$$ (I believe the constant ##a## can be found to be ##\frac { 1 }{ 32{ \pi }^{ 2 } }##).
Moving on to ##\varphi^3## (still in 4D), there's a couple of things that I'm a little unsure about. First of all, the theory is super-renormalisable, and so the beta function should not be a marginal one; in fact, it's quoted in this answer to be $$\beta (g)\approx -g-\frac { 3g^{ 3 } }{ 256\pi ^{ 3 } }.$$
The first question is: how is this expression dimensionally consistent? The coupling ##g## has units of mass. Are we implicitly defining a 'dimensionless' coupling such as ##g(\mu )=\mu \cdot { g }_{ dimless }(\mu )##? If so, the first term in the beta function becomes clear, as we then also have ##g(\mu +d\mu )=(\mu +d\mu )\cdot{ g }_{ dimless }(\mu +d\mu )##.
The second question is: I imagine that I should be renormalising the three-point function in the case of ##\varphi^3##, similar to how I renormalised the four-point function in the case of ##\varphi^4##. Is this the right thing to do, and if so, how do I choose a renormalisation point? Unlike the case of a four-point function, there is no on-shell choice for the momenta of the incoming/outgoing particles. I could just choose an off-shell point, but still don't quite understand the legitimacy of doing this from an 'experimental' point of view.
Moving on to ##\varphi^3## (still in 4D), there's a couple of things that I'm a little unsure about. First of all, the theory is super-renormalisable, and so the beta function should not be a marginal one; in fact, it's quoted in this answer to be $$\beta (g)\approx -g-\frac { 3g^{ 3 } }{ 256\pi ^{ 3 } }.$$
The first question is: how is this expression dimensionally consistent? The coupling ##g## has units of mass. Are we implicitly defining a 'dimensionless' coupling such as ##g(\mu )=\mu \cdot { g }_{ dimless }(\mu )##? If so, the first term in the beta function becomes clear, as we then also have ##g(\mu +d\mu )=(\mu +d\mu )\cdot{ g }_{ dimless }(\mu +d\mu )##.
The second question is: I imagine that I should be renormalising the three-point function in the case of ##\varphi^3##, similar to how I renormalised the four-point function in the case of ##\varphi^4##. Is this the right thing to do, and if so, how do I choose a renormalisation point? Unlike the case of a four-point function, there is no on-shell choice for the momenta of the incoming/outgoing particles. I could just choose an off-shell point, but still don't quite understand the legitimacy of doing this from an 'experimental' point of view.