Skip to main content
Physics

Return to Answer

added 7 characters in body
Source Link
MadMax
MadMax
  • 4.5k
  • 11
  • 34

I don't even understand what do we mean with resumming

Resummation literally means re-summation.

The one-loop $$ \Pi(p^{2})\propto log(p^{2}/\mu^{2}) $$ is the first non-constant term if we expand $\alpha(\mu)$: $$ \alpha(\mu)=\frac{\alpha_{EM}}{1-\frac{1}{3\pi}\alpha_{EM}log(\mu^{2}/m^{2})} \\ =1 + \frac{1}{3\pi}\alpha_{EM}^2log(\mu^{2}/m^{2}) + \dots $$

If you re-sum all the omitted $\dots$ terms, you get the full $\alpha(\mu)$.

Note that RG equation is not the only way to arrive at resummation. There are myriads of other ways to arrive at the same resummed formula. Just to name a few alternatives to RG:

  • Truncated Schwinger–Dyson equation
  • Large N approximation
  • Random-phase approximation
  • Bethe-Salpeter T-matrix approximation

All of the above alternative methodologies involve re-summing a (geometric) series of loop diagrams. And actually resummation has a very long history: Landau found the Landau pole by resumming the geometric series by brute force, a lot earlier than the invention of RG equation.

I don't even understand what do we mean with resumming

Resummation literally means re-summation.

The one-loop $$ \Pi(p^{2})\propto log(p^{2}/\mu^{2}) $$ is the first non-constant term if we expand $\alpha(\mu)$: $$ \alpha(\mu)=\frac{\alpha_{EM}}{1-\frac{1}{3\pi}\alpha_{EM}log(\mu^{2}/m^{2})} \\ =1 + \frac{1}{3\pi}\alpha_{EM}^2log(\mu^{2}/m^{2}) + \dots $$

If you re-sum all the omitted $\dots$ terms, you get the full $\alpha(\mu)$.

Note that RG equation is not the only way to arrive at resummation. There are myriads of other ways to arrive at the same resummed formula. Just to name a few alternatives to RG:

  • Truncated Schwinger–Dyson equation
  • Large N approximation
  • Random-phase approximation
  • Bethe-Salpeter T-matrix approximation

I don't even understand what do we mean with resumming

Resummation literally means re-summation.

The one-loop $$ \Pi(p^{2})\propto log(p^{2}/\mu^{2}) $$ is the first non-constant term if we expand $\alpha(\mu)$: $$ \alpha(\mu)=\frac{\alpha_{EM}}{1-\frac{1}{3\pi}\alpha_{EM}log(\mu^{2}/m^{2})} \\ =1 + \frac{1}{3\pi}\alpha_{EM}^2log(\mu^{2}/m^{2}) + \dots $$

If you re-sum all the omitted $\dots$ terms, you get the full $\alpha(\mu)$.

Note that RG equation is not the only way to arrive at resummation. There are myriads of other ways to arrive at the same resummed formula. Just to name a few alternatives to RG:

  • Truncated Schwinger–Dyson equation
  • Large N approximation
  • Random-phase approximation
  • Bethe-Salpeter T-matrix approximation

All of the above alternative methodologies involve re-summing a (geometric) series of loop diagrams. And actually resummation has a very long history: Landau found the Landau pole by resumming the geometric series by brute force, a lot earlier than the invention of RG equation.

added 7 characters in body
Source Link
MadMax
MadMax
  • 4.5k
  • 11
  • 34

I don't even understand what do we mean with resumming

Resummation literally means re-summation.

The one-loop $$ \Pi(p^{2})\propto log(p^{2}/\mu^{2}) $$ is the first non-constant term if we expand $\alpha(\mu)$: $$ \alpha(\mu)=\frac{\alpha_{EM}}{1-\frac{1}{3\pi}\alpha_{EM}log(\mu^{2}/m^{2})} \\ =1 + \frac{1}{3\pi}\alpha_{EM}^2log(\mu^{2}/m^{2}) + \dots $$

If you re-sum all the omitted $\dots$ terms, you get the full $\alpha(\mu)$. 

Note that RG equation is not the only way to arrive at resummation. There are myriads of other ways to arrive at the same resummationresummed formula. Just to name a few alternatives to RG:

  • Truncated Schwinger–Dyson equation
  • Large N approximation
  • Random-phase approximation
  • Bethe-Salpeter T-matrix approximation

I don't even understand what do we mean with resumming

Resummation literally means re-summation.

The one-loop $$ \Pi(p^{2})\propto log(p^{2}/\mu^{2}) $$ is the first non-constant term if we expand $\alpha(\mu)$: $$ \alpha(\mu)=\frac{\alpha_{EM}}{1-\frac{1}{3\pi}\alpha_{EM}log(\mu^{2}/m^{2})} \\ =1 + \frac{1}{3\pi}\alpha_{EM}^2log(\mu^{2}/m^{2}) + \dots $$

If you re-sum all the omitted $\dots$ terms, you get the full $\alpha(\mu)$. Note that RG equation is not the only way to arrive at resummation. There are myriads other ways to arrive at the same resummation formula. Just to name a few alternatives to RG:

  • Truncated Schwinger–Dyson equation
  • Large N approximation
  • Random-phase approximation
  • Bethe-Salpeter T-matrix approximation

I don't even understand what do we mean with resumming

Resummation literally means re-summation.

The one-loop $$ \Pi(p^{2})\propto log(p^{2}/\mu^{2}) $$ is the first non-constant term if we expand $\alpha(\mu)$: $$ \alpha(\mu)=\frac{\alpha_{EM}}{1-\frac{1}{3\pi}\alpha_{EM}log(\mu^{2}/m^{2})} \\ =1 + \frac{1}{3\pi}\alpha_{EM}^2log(\mu^{2}/m^{2}) + \dots $$

If you re-sum all the omitted $\dots$ terms, you get the full $\alpha(\mu)$. 

Note that RG equation is not the only way to arrive at resummation. There are myriads of other ways to arrive at the same resummed formula. Just to name a few alternatives to RG:

  • Truncated Schwinger–Dyson equation
  • Large N approximation
  • Random-phase approximation
  • Bethe-Salpeter T-matrix approximation
Source Link
MadMax
MadMax
  • 4.5k
  • 11
  • 34

I don't even understand what do we mean with resumming

Resummation literally means re-summation.

The one-loop $$ \Pi(p^{2})\propto log(p^{2}/\mu^{2}) $$ is the first non-constant term if we expand $\alpha(\mu)$: $$ \alpha(\mu)=\frac{\alpha_{EM}}{1-\frac{1}{3\pi}\alpha_{EM}log(\mu^{2}/m^{2})} \\ =1 + \frac{1}{3\pi}\alpha_{EM}^2log(\mu^{2}/m^{2}) + \dots $$

If you re-sum all the omitted $\dots$ terms, you get the full $\alpha(\mu)$. Note that RG equation is not the only way to arrive at resummation. There are myriads other ways to arrive at the same resummation formula. Just to name a few alternatives to RG:

  • Truncated Schwinger–Dyson equation
  • Large N approximation
  • Random-phase approximation
  • Bethe-Salpeter T-matrix approximation