The Art of Counting: a reappraisal of the HEFT expansion

TL;DR

The article addresses a foundational issue in HEFT by deriving a principled, observable-driven power-counting framework. It presents two distinct counting schemes—one with a single scale $v$ and another with an additional scale $f$, governed by NDA-inspired rules that connect Lagrangian content, amplitudes, and observable cross sections. The work delivers explicit master formulas for counting ($N_{ ext{HEFT}}$, $N_{ ext{HEFT}}^s$, $N_{ ext{HEFT}}^oldsymbol{ ext{ξ}}$) and provides practical guidance for truncation, basis reduction, and renormalization, reinforced by concrete examples. It clarifies how SMEFT and HEFT relate, how to treat operator normalizations, and how to incorporate UV-matching effects, thereby enabling consistent HEFT phenomenology and potential automation of HEFT calculations.

Abstract

We revisit the power counting of the Higgs Effective Field Theory (HEFT) from first principles, by requiring that predictions for physical observables follow a series expansion in small, dimensionless quantities. Depending on whether HEFT is formulated in terms of a unique low-energy scale $v$ or in terms of two scales $v<f$, this approach identifies two viable power counting rules that can accommodate any operator normalization choice. We provide quantitative prescriptions for the consistent truncation of HEFT operators, amplitudes and observable contributions and we illustrate our arguments with a number of examples.

Figures (10)

• Figure 1: One-loop diagrams contributing to $gg \rightarrow hh$ that contain a $C_{HG}$ coupling.
• Figure 2: HEFT scattering amplitudes can be classified according to a double expansion in QCD ($N_{g_s,\mathcal{M}}$) and $N_{\rm HEFT}^\mathcal{M}$ orders, which can take discrete values represented with red dots. The grey dashed lines are curves of constant $N_{\rm HEFT}^{s\mathcal{M}} = N_{\rm HEFT}^\mathcal{M}+N_{g_s,\mathcal{M}}$, which can be used as an alternative counting. Truncating a calculation at a certain $N_{\rm HEFT}^{s\mathcal{M}}$ selects values within a triangular region, as shown in yellow. Truncating in $N_{\rm HEFT}^\mathcal{M}$ selects a horizontal band (green), while truncating simultaneously in $N_{\rm HEFT}^\mathcal{M}$ and $N_{g_s,\mathcal{M}}$ identifies a rectangular region (orange).
• Figure 3: Left: the blue shaded region highlights the values of $(N_{g_s,\mathcal{M}}, N_{\rm HEFT}^\mathcal{M})$ achievable by diagrams containing insertions of an operator with fixed $N_\chi, \min_i(N_{\chi,i}-N_{g_s,i})$. We use the shorthand notation $\mathfrak{L} = 2L+n-2$. Right: HEFT operators can be characterized by their $N_\chi$ and $\min_i(N_{\chi,i}-N_{g_s,i})$, which take discrete values represented with blue dots. The gray shaded regions are forbidden. The grey dashed lines are curves of constant $\max_i N_{g_s,i}$. The yellow region shows how a cut on $N_{\rm HEFT}^{s,\mathcal{M}}\leq N_{\max}^s$ at the amplitudes level projects onto the operators space. The green hatched region shows the same for a double cut in $N_{\rm HEFT}^\mathcal{M}\leq N_{\max}$ and $N_{g_s,\mathcal{M}}\leq 2 O_{\alpha_s}$. If the $N_{g_s,\mathcal{M}}$ cut was removed, the green region would extend indefinitely to the right.
• Figure 4: Left: visualization of how "dressing" an operator $\mathcal{P}$ increases its $N_\chi$ and $\min_i(N_{\chi,i}-N_{g_s,i})$ orders. Right: graphic representation of the generic relations in Eqs. \ref{['eq.generic_relation_1']} (blue) and \ref{['eq.generic_relation_2']} (red). The solid lines trace the true relations, which involve dressed operators, while the dashed lines trace relations among naked operators.
• Figure 5: Left: graphical representation of the relation among operators $\mathcal{P}_1,\mathcal{P}_2$ defined in Eq. \ref{['eq.eomG_ex_operators']}. Right: representation of the amplitude orders at which the two naked operators can contribute, for tree-level (solid) and 1-loop (dashed) diagrams. The dots labeled with (a)--(d) indicate the orders of the diagrams in Fig. \ref{['fig.diagrams_qqqq']}.