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Prof. Sachindeo Vaidya (CHEP, IISc) | PDF

Previous: A Short Break from Renormalisation

Next: Lecture 16

We want to know how our parameters \(\lambda\) (coupling const.) changes with scale \((\mu)\).

Here \(\lambda_{\text{ren.}}\) = experimentally measured.

Mass independent renormalisation ---

\[ \lambda_0 \;=\; \mu^{2\epsilon}\left(\lambda + \sum_{k=1}^{\infty}\frac{a_k(\lambda)}{\epsilon^k}\right) \]

\(\lambda_0\) is the bare coupling and does not know about \(\mu\), so

\[ 0 \;=\; \mu\frac{\partial}{\partial\mu}(\text{RHS}) \;=\; 2\epsilon\left(\lambda + \sum_k \frac{a_k(\lambda)}{\epsilon^k}\right) + \mu\frac{\partial\lambda}{\partial\mu}\left(1 + \sum_k \frac{a_k'(\lambda)}{\epsilon^k}\right) \]

\(\lambda\) & \(\dfrac{\partial\lambda}{\partial\mu}\) are analytic at \(\epsilon = 0\). So, comparing
powers of \(\epsilon\),

\[ \mu\frac{\partial\lambda}{\partial\mu} \;=\; -2\epsilon\lambda \;-\; 2a_1(\lambda) \;+\; 2\lambda\,a_1'(\lambda) \]

in the \(\epsilon\to 0\) limit

\[ \beta(\lambda) \;=\; \lim_{\epsilon\to 0}\ \mu\frac{\partial\lambda}{\partial\mu} \;=\; -2\left(1 - \lambda\frac{\partial}{\partial\lambda}\right)a_1(\lambda) \]

and the higher \(a_k\) are fixed recursively,

\[ \left(1 - \lambda\frac{d}{d\lambda}\right)a_{k+1}(\lambda) \;=\; a_k'(\lambda)\left(1 - \lambda\frac{d}{d\lambda}\right)a_1(\lambda) \]

With

\[ a_1(\lambda) \;=\; \frac{3}{2}\frac{\lambda^2}{16\pi^2} + O(\lambda^3) \]

we get

\[ \mu\frac{\partial\lambda}{\partial\mu} \;=\; \frac{3\lambda^2}{16\pi^2}\left(1 + O(\lambda^3)\right) \]

The running coupling & the Landau pole ---

Integrate this equation:

\[ \lambda(\mu) \;=\; \lambda_s\ \frac{1}{1 - \dfrac{3}{16\pi^2}\lambda_s\ln\dfrac{\mu}{\mu_s}} \ ;\qquad \text{where}\quad \lambda_s \equiv \lambda(\mu_s) \]

\(\lambda\) increases with increase in \(\mu\).

If we start with small \(\lambda_s\) (as we should) then, (recall, perturbation theory is valid
for \(\lambda \ll 1\)) we exit the regime of validity of perturbation theory, i.e.

\[ \frac{3}{16\pi^2}\,\lambda_s\,\ln\frac{\mu}{\mu_s} \;\ll\; 1 \]

For short distance (large \(\mu\)), we will need more & more terms in the perturbative expansion
of \(\tilde{\Gamma}^{(n)}\).

Conversely, for large distance (\(\sim\) small \(\mu\)) perturbation theory becomes more reliable.

For

\[ \mu \;=\; \mu_s\,\exp\left(\frac{16\pi^2}{3\lambda_s}\right) \]

\(\lambda\) blows up. This is called the \underline{Landau point} (or pole).

Scenarios for \(\beta(\lambda)\) ---

'''(1)''' \(\beta(\lambda)\) stays positive for large \(\lambda\) (ex. \(\beta = \frac{3}{16\pi^2}\lambda^2\)).
Then \(\lambda\) keeps increasing with \(\mu\) (depends on sign of \(\beta'\)). If \(\beta\) blows up
for some \(\lambda\), then \(\lambda\) itself is \(\infty\).

'''(2)''' \(\beta(\lambda)\) starts off positive for small \(\lambda\), then turns over and crosses
the axis at \(\lambda = \lambda_F\) : \(\beta(\lambda_F) = 0\).

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\(\lambda_F \equiv\) fixed pt. (the coupling does not change with \(\mu\))

\[ \beta \;=\; \mu\frac{\partial\lambda}{\partial\mu} \;=\; (\lambda - \lambda_F)\,\beta'(\lambda_F) + \cdots \]

If \((\lambda - \lambda_F) < 0\) & \(\beta'(\lambda_F)\) is also \(< 0\), then

\[ \mu\frac{\partial\lambda}{\partial\mu} \;>\; 0 \qquad \text{where } (\mu > 0) \quad\Rightarrow\quad \frac{\partial\lambda}{\partial\mu} \;>\; 0 \] \[ \Rightarrow\qquad \text{as } \mu\uparrow\ \Rightarrow\ \lambda\uparrow \qquad (\text{for }\lambda < \lambda_F) \]

i.e. \(\lambda \longrightarrow \lambda_F\) as \(\mu\) increases.

\(\lambda_F\) is called a \underline{UV fixed pt.}

If \(\lambda_F \ll 1\), we should be able to access this fixed pt. within perturbation theory.

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'''(3)''' The pt. \(\lambda = 0\) is a fixed pt. with \(\beta'(\lambda) > 0\)

\[ \Rightarrow\quad \mu\frac{\partial\lambda}{\partial\mu} \;>\; 0 \]

\(\Rightarrow\) \(\lambda\) increases as the distance decreases, so \(\lambda = 0\) is called an
\underline{IR fixed point}.

If instead \(\beta(\lambda)\) starts off negative for small \(\lambda\), decreasing monotonically
with \(\ln\mu\), then perturbation theory gets better at short distances, & \(\lambda\) is driven to
zero. This is \underline{asymptotic freedom} (free theory) --- a property of QCD.

Yang--Mills theory is an example of this behaviour.

'''(4)''' \(\beta(\lambda)\) starts \(-\)ve, becomes \(+\)ve & crosses the \(x\)-axis.

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\(\beta'(\lambda_F) > 0\); so \(\lambda_F\) is an IR fixed point.

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Behaviour of \(m\) as a function of \(\mu\) ---

Recall:

\[ b_0\left(\lambda,\frac{m}{\mu},\epsilon\right) \;=\; 1 + \frac{1}{2}\left(\hat{\lambda}F_1 + \hat{\lambda}^2F_2\right) + \hat{\lambda}^2 H_2 + O(\lambda^3) \]

and \(b_1(\ \ )\) likewise. Then

\[ \mu\frac{\partial m^2}{\partial\mu} \;=\; 2\lambda\, m^2\, \frac{db_1}{d\lambda} \]

We defined

\[ \gamma_m\left(\lambda,\frac{m}{\mu},\epsilon\right) \;=\; \frac{1}{2}\,\mu\frac{\partial}{\partial\mu}\ln m^2 \] \[ \gamma_m(\lambda) \;=\; \lambda\,\frac{db_1}{d\lambda} \]

For \(\varphi^4\) theory:

\[ \gamma_m(\lambda) \;=\; \frac{\lambda}{16\pi^2} + \frac{7}{32}\left(\frac{\lambda}{16\pi^2}\right)^2 + O(\lambda^3) \]

Recursion relations & the anomalous dimension ---

\[ \lambda\frac{db_{k+1}}{d\lambda} \;=\; b_k\,\lambda\frac{db_1}{d\lambda} \;-\; \frac{db_k}{d\lambda}\left(1 - \lambda\frac{d}{d\lambda}\right)a_1(\lambda) \qquad (k = 1,2,3,\ldots) \] \[ \mu\frac{\partial}{\partial\mu}\ln Z_\varphi \;=\; -2\lambda\frac{dC_1}{d\lambda} \qquad\Longleftrightarrow\qquad \gamma_d(\lambda) \;=\; -\lambda\frac{dC_1}{d\lambda} \]

So again for \(\lambda\varphi^4\) theory,

\[ \gamma_d(\lambda) \;=\; \frac{1}{12}\left(\frac{\lambda}{16\pi^2}\right)^2 + O(\lambda^3) \]

Integrating the RG equation ---

In the mass independent renormalisation scheme we can integrate the RG equation:

\[ \left(-s\frac{\partial}{\partial s} + \beta\frac{\partial}{\partial\lambda} + \left(\gamma_m(\lambda) - 1\right)m\frac{\partial}{\partial m} + d_k - n\,\gamma_d(\lambda)\right)\tilde{\Gamma}^{(n)}(s p_i; m,\lambda) \;=\; 0 \]

\(\bar{\lambda}(s)\), \(\bar{m}(s)\) : scale dependent variables

\[ s\frac{\partial}{\partial s}\bar{\lambda}(s) \;=\; \beta\left(\bar{\lambda}(s)\right) \] \[ s\frac{\partial}{\partial s}\bar{m}(s) \;=\; \bar{m}(s)\left(\gamma_m(\bar{\lambda}(s)) - 1\right) \]

\((s=1)\) (no scaling): \(\bar{\lambda}(1) = \lambda\), \(\bar{m}(1) = m\).

\[ \tilde{\Gamma}^{(n)}(sp;m,\lambda,\mu) \;=\; s^{d_n}\, \tilde{\Gamma}^{(n)}\left(sp;\bar{m}(s),\bar{\lambda}(s),\mu\right) \cdot\exp\left(-n\int_1^{s}\frac{ds'}{s'}\,\gamma_d\left(\bar{\lambda}(s')\right)\right) \]

Under a change of scale of external momenta, the Green's function or vertex function changes,
not just by a factor \(s^{d_n}\), but also depends on \(\gamma_d\).

\(\left\{\text{sine--Gordon},\ \text{Thirring theory (in 2d)}\right\}\)

----------------- \(\times\) -----------------

An aside in \(1+1\) dimensions ---

\[ \mathcal{L} \;=\; \frac{1}{2}(\partial\varphi)^2 \;\underbrace{-\ \frac{m^2\varphi^2}{2}\ -\ \frac{g}{4!}\varphi^4 \ -\ g_6\varphi^6\ -\ g_8\varphi^8 + \cdots}_{m^2\left(1 - g\cosh\varphi\right)} \]

\(\downarrow\)

integrable model! \(\longrightarrow\) 2-particle amplitudes factorize.

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