= Solution
Use $\hbar=c=1$ and normalize the interaction of <quartic scalar field theory> as $\lambda\phi^4/4!$. For a renormalized $n$-point <connected correlation function>, adopt $\phi_B=Z_\phi^{1/2}\phi_R$ and $\gamma_\phi=\tfrac12\mu\,d\log Z_\phi/d\mu$. Independence of bare quantities from the <renormalization scale> gives the <Callan-Symanzik equation>
$$
\left[\mu\partial_\mu+\beta(\lambda)\partial_\lambda+\gamma_{m^2}(\lambda)m^2\partial_{m^2}+n\gamma_\phi(\lambda)\right]G_R^{(n)}=0,
$$
where $\beta=\mu\,d\lambda/d\mu$ and $\gamma_{m^2}=\mu\,d\log m^2/d\mu$, with bare parameters fixed. A convention using proper vertices reverses the field-normalization term, so the choice of Green's function must be stated.
Solve it by characteristics: set $\mu(s)=\mu_0e^s$, $d\lambda(s)/ds=\beta(\lambda(s))$, and $d\log m^2(s)/ds=\gamma_{m^2}(\lambda(s))$. At fixed external momenta,
$$
G_R^{(n)}(\mu(s),\lambda(s),m^2(s))=G_R^{(n)}(\mu_0,\lambda_0,m_0^2)\exp\left[-n\int_0^s\gamma_\phi(\lambda(v))\,dv\right].
$$
This is the <characteristic solution of the multiplicative Callan-Symanzik equation>. If the field anomalous dimension vanishes, the function is constant along the running-parameter characteristics. At a massless fixed point with constant anomalous dimension, the normalization factor is a power of $\mu/\mu_0$. Choosing the scale close to a characteristic external momentum avoids large perturbative logarithms.
For the supplied <minimal subtraction scheme> relation in <dimensional regularization> with $d=4-\epsilon$, let $a=3/(16\pi^2)$ and write $\lambda_B=\mu^\epsilon[\lambda+a\lambda^2/\epsilon+O(\lambda^3)]$. Differentiating at fixed bare coupling gives
$$
0=\epsilon\left(\lambda+\frac{a\lambda^2}{\epsilon}\right)+\beta(\lambda)\left(1+\frac{2a\lambda}{\epsilon}\right)+O(\lambda^3).
$$
A consistent expansion yields $\beta=-\epsilon\lambda+a\lambda^2+O(\lambda^3)$. The $-\epsilon\lambda$ term must be retained while it multiplies the pole; setting $\epsilon=0$ prematurely loses the finite result. In four dimensions the <one-loop quartic scalar beta function> is
$$
\boxed{\beta(\lambda)=\frac{3\lambda^2}{16\pi^2}+O(\lambda^3).}
$$
The one-loop two-point diagram is a momentum-independent <tadpole diagram>, so there is no field wavefunction renormalization at this order and $\gamma_\phi=O(\lambda^2)$. Differentiating the supplied bare-mass relation in the same way gives $\gamma_{m^2}=\lambda/(16\pi^2)+O(\lambda^2)$.
Integrating $d\lambda/d\log\mu=a\lambda^2$ gives the <running coupling>
$$
\boxed{\lambda(\mu)=\frac{\lambda(\mu_0)}{1-\dfrac{3\lambda(\mu_0)}{16\pi^2}\log(\mu/\mu_0)}.}
$$
For positive coupling it grows toward the ultraviolet and decreases toward the infrared: the theory is not asymptotically free. Its one-loop <Landau pole> lies at $\mu_L=\mu_0\exp[16\pi^2/(3\lambda(\mu_0))]$. Perturbation theory fails before reaching that pole, so the pole in this approximation alone does not prove the existence of an exact physical divergence. The corresponding one-loop <running mass> satisfies
$$
\boxed{m^2(\mu)=m^2(\mu_0)\left[\frac{\lambda(\mu)}{\lambda(\mu_0)}\right]^{1/3}.}
$$
It follows from $d\log m^2/d\lambda=1/(3\lambda)$. The mass-independent subtraction scheme keeps these beta functions independent of the mass; physical low-energy matching can still require threshold treatment. Running parameters resum the leading logarithms within the regime of small coupling.
Back to article page
