# Solving Peskin Equation 12.66 Problem

• physichu
In summary, the conversation discusses the relationship between the momentum ##p## and the 3-momentum magnitude ##p## in the context of a massless theory. It also explains how this is used in the Callan-Symanzik equation to solve for the behavior of the Green's function. The conversation also mentions the use of dimensional analysis and leading-log resummation in this context.
physichu
I have a problem with that equation. I understand (dont know if I'm right) that ##p = - M##. But than, isn't ##g\left( { - {{{p^2}} \over {{M^2}}}} \right)## just equal ##g\left( { - 1} \right)##?

And my bigest problem: in 12.66

##\left[ {p{\partial \over {\partial p}} - \beta \left( \lambda \right){\partial \over {\partial \lambda }} + 2 - 2\gamma \left( \lambda \right)} \right]{G^{\left( 2 \right)}}\left( p \right) = 0##

Where dose the free 2 (after the ##\beta ## term) comes from?

From looking at 12.78 and 12.84 I realize it's the inverse mass dimantionalty of ##G##, But how did it got ther?

Any help would be apriciated :)

there is a transition I think from the 4-momentum ##p## to a 3-momentum magnitude ##p## in 12.66?

No, by definition Peskin/Schroeder sets ##p=\sqrt{-p^2}## for space-like ##p##, i.e., ##p^2<0##, where you evaluate the Green's functions (corresponding to Euclidean field theory) first. The key is (12.65), which makes use of dimensional analysis: For a massless theory the mass-dimension (-2) quantitiy ##G^{(2)}## must be proportional to ##1/p^2## times a dimensionless function of ##p##. Since for a massless theory the only way to get a dimensionless function like this, it must be a function of ##p^2/M^2##, where ##M## is the renormalization scale. Thus the two-point Green's function must be of the form (12.65):
$$G^{(2)}(p^2,M^2)=\frac{\mathrm{i}}{p^2} g(-p^2/M^2).$$
Now you can indeed express the derivative wrt. ##M## by a derivative with respect to ##p##, just knowing this functional form. In the original Callan-Symanzik equation you need the derivative
$$M \partial_M G^{(2)}=\frac{2\mathrm{i}}{M^2} g'(-p^2/M^2).$$
In the last step, I used (12.65). On the other hand from this ansatz you get
$$p \partial_p G^{(2)}=-\frac{2 \mathrm{i}}{p^2} g(-p^2/M^2)-\frac{2 \mathrm{i}}{M^2} g'(p^2/M^2)=-2 G^{(2)}-M \partial_M G^{(2)}.$$
In the last step used again (12.65) and our result for ##M \partial_M G^{(2)}##. From this you get
$$M \partial_M G^{(2)}=-p\partial_p G^{(2)}-2 G^{(2)}.$$
Now substituting this into the Callan-Symanzik equation and flip the sign on the left-hand side gives (12.66).

That's a fine trick to get the behavior of the Green's function by solving this RG flow equation, given the functions ##\beta## and ##\gamma##. This goes beyond perturbation theory despite the fact that ##\beta## and ##\gamma## are given only perturbatively. It's a kind of leading-log resummation (see Weinberg, The quantum theory of fields, vol. 2 for a nice explanation of this issue).

physichu
My Hero :)

Just so I understand, I was supposed to get to that by myself?

physichu said:
I was supposed to get to that by myself?

Ehmmm, is that a question...? you will either have to derive the things that seem non-trivial to you [in order to understand them], or you will have to just accept them as they are [if you don't care about understanding them]

## 1. What is the Peskin Equation 12.66 Problem?

The Peskin Equation 12.66 Problem is a mathematical equation that describes the dynamics of a quantum mechanical system. It is commonly used in particle physics and quantum mechanics to study the behavior of particles at a subatomic level.

## 2. What is the importance of solving this problem?

Solving the Peskin Equation 12.66 Problem allows us to gain a better understanding of the behavior and interactions of particles at a quantum level. It can also help us make predictions about the behavior of particles in different physical scenarios.

## 3. What are the challenges in solving this problem?

One of the main challenges in solving the Peskin Equation 12.66 Problem is the complexity of the equation itself. It involves multiple variables and parameters, making it difficult to find exact solutions. Additionally, the equations of quantum mechanics are often non-linear, making it challenging to apply traditional mathematical techniques.

## 4. How is this problem typically approached?

There are various approaches to solving the Peskin Equation 12.66 Problem, including numerical methods and approximation techniques. Some researchers also use computer simulations to model the behavior of particles and gather data for solving the equation.

## 5. What are the potential applications of solving this problem?

The solutions to the Peskin Equation 12.66 Problem can have a wide range of applications, including improving our understanding of fundamental particles and their interactions, developing new technologies, and advancing our understanding of the universe.

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