Renormalization Group Evolution

Alpaca implements the running of the ALP couplings and the matching to the low-energy Lagrangian (i.e. with $W^\pm$, $Z$ and $t$ integrated out) as described in ¹.

The following code exemplifies the running between the scale $\mu= 1000\,\mathrm{GeV}$ down to $\mu= 5\,\mathrm{GeV}$, using the method match_run:

from alpaca import ALPcouplings

couplings_high = ALPcouplings({'cqL': 1.0}, scale=1000, basis='derivative_above')
couplings_low = couplings_high.match_run(scale=5, basis='VA_below')

The method match_run takes the following arguments:

• scale [float]: energy scale of the final couplings. Must be smaller than the initial scale.
• basis [str]: basis of the final couplings, as described here.

• integrator [str optional, default 'scipy']: Method used to numerically integrate the RG equations. The available options are

  • scipy: Solves the initial value problem given by
  \[\frac{d c_i(\mu)}{d \log \mu} = \frac{1}{16\pi^2} \gamma_{ij}(\mu) c_j(\mu)\,;\qquad c_i(\mu = \Lambda)=c_i^0\,,\]

  using the Runge-Kutta integrator implemented by scipy.

  • leadinglog: In the expression above, by approximating $\gamma_{ij}(\mu)$ as a constant (i.e. by neglecting the running of the SM parameters), an approximate solution can be found as
  \[c_i(\mu) \approx \left(\delta_{ij} + \frac{\gamma_{ij}}{16\pi^2}\log\frac{\mu}{\Lambda} \right)c_j(\Lambda)\,,\]

  which can be computed efficiently as a matrix multiplication.

  • no_rge: Does not change the values of the couplings, and just modifies the scale.

• beta [str optional, default full]: Controls the expression of the $\beta$ functions used for the running:

  • ytop: discards all the terms proportional to the Yukawa couplings other than $y_t$.
  • full: retains the full dependency on all the Yukawa couplings.
• matching_scale [float, default 100]: Energy where the matching between the full ALP Lagrangian and the low-energy Lagrangian is performed.
• match_2loops [bool, default False]: whether or substitute the 1-loop effective gauge couplings in the 1-loop matching expression.

References

1. M. Bauer, M. Neubert, S. Renner, M. Schnubel, A. Thamm: “The Low-Energy Effective Theory of Axions and ALPs”. JHEP 04 (2021), 063. arXiv:2012.12272 [hep-ph] ↩︎