Terascale Precision Tests

• Terascale Precision Tests are high-precision analyses of electroweak observables that confront Standard Model predictions by examining loop-induced corrections at the TeV scale.
• The SMEFT framework is employed to incorporate heavy new physics via dimension-6 operators, enabling constraints on models up to tens of TeV indirectly.
• Advanced multi-loop techniques and differential equation methods achieve per-mille accuracy, leveraging vast statistics from future Tera-Z runs at FCC-ee and CEPC.

Terascale Precision Tests refer to the rigorous confrontation of theoretical predictions of the Standard Model (SM) and its extensions with high-precision electroweak measurements, primarily at the energy frontier near and above the Tera-electronvolt (TeV) scale. Such tests are sensitive to quantum effects from heavy states that are not kinematically accessible but leave indirect signatures through loop-induced corrections and effective operators. The modern paradigm leverages advanced theoretical frameworks, notably the Standard Model Effective Field Theory (SMEFT), and exploits enormous experimental statistics expected at future facilities like the FCC-ee or CEPC’s proposed "Tera-Z" runs, to probe new physics far beyond direct search reach—up to tens of TeV—by squeezing subtle deviations from SM expectations at the per-mille level.

1. Fundamentals of Precision Electroweak Tests

Terascale precision tests operate on the principle that virtual effects of heavy particles manifest as small shifts in electroweak precision observables (EWPOs). These include the $W$ and $Z$ mass, effective weak mixing angle $\sin^2\theta_{\rm eff}$, $Z$ partial widths, forward-backward and left-right asymmetries, and the oblique parameters $S, T, U$ defined via gauge-boson self-energies. Core predictions within the SM involve on-shell renormalization, including mass and field renormalization, mixing terms such as $\delta Z_{AZ}$, and gauge-invariant pole definitions for unstable particles. Radiative corrections are primarily from self-energies (oblique corrections), vertex adjustments, and box diagrams, with input parameters $\alpha, G_F, M_Z, m_t, M_H, \Delta\alpha_{\rm had}$ forming the backbone of theoretical predictions (Freitas, 2020).

The precision with which EWPOs are measured directly constrains possible extensions of the SM. For instance, a shift in the $M_W$ is mapped to the oblique parameters by

$$\delta M_W \simeq \frac{\alpha M_W}{2(c_W^2-s_W^2)} \left( -\frac{1}{2} S + c_W^2 T + \frac{c_W^2 - s_W^2}{4s_W^2} U \right),$$

with similar mappings for $\sin^2\theta_{\rm eff}$, thus allowing fits that translate experimental accuracy into exclusion bounds for new physics (Freitas, 2020).

2. SMEFT Approach and Operator Sensitivity at the Tera-Z Pole

The SMEFT formalism systematizes the effects of heavy new physics at scales $M \gg m_Z$ via dimension-6 operators:

$$\mathcal{L}_{\rm SMEFT} \supset \sum_i \frac{C_i}{\Lambda^2} Q_i \quad (\Lambda \sim M),$$

where relevant Higgs and gauge sector operators include $Q_{H\Box}$, $Q_H$, $Q_{HW}$, $Q_{HB}$, as well as pure-gauge SILH operators such as $$\mathcal O_T, \mathcal O_{B+W}, \mathcal O_{2W}, \mathcal O_{2B}$$ (Maura et al., 2024). Key features at Tera-Z are:

• Some operators modify Higgs and gauge boson two- and three-point functions, entering $Z$-pole observables exclusively at one-loop (NLO) or two-loop (NNLO), in contrast to off-pole processes (e.g., $e^+ e^- \to ZH$) where they may appear at leading order.
• Pure-gauge operators define the oblique parameters $(\hat S, \hat T, \hat W, \hat Y)$, historically bounded by high-energy runs but at FCC-ee/CEPC Tera-Z, the colossal event count compensates for loop and energy suppression (Maura et al., 2024).

The SMEFT matching and RGE effects are exemplified by one-loop mixing formulas, e.g.,

$$\dot C_{HD} \supset \frac{80}{3}g_1^2 y_h^2 C_{H\Box}, \qquad [\dot C_{Hl}^{(3)}]_{pr} \supset \frac{g_2^2}{6}C_{H\Box}\delta_{pr},$$

and by the fact that $Q_H$ enters $Z$-pole observables only at two loops, shifting $\hat S$ and $\hat T$:

$$10^6\,\hat S = -0.71\,C_H - 0.25\,C_H^2, \quad 10^6\,\hat T = +1.2\,C_H + 0.36\,C_H^2.$$

3. Statistical Power and Sensitivity to High Mass Scales

The Tera-Z concept exploits up to $10^{12}$ $Z$ decays, offering an order-of-magnitude gain in statistical precision over LEP. The effective sensitivity to new physics is enhanced by trading the loop suppression factor $1/16\pi^2$ or the energy suppression $m_Z^2/E^2$ for $\sqrt{N_Z/N_{\rm run}}$ statistical power. This yields indirect reach to mass scales $M \sim 10$–$30$ TeV for weak couplings, as the minimal detectable shift $\delta O \sim \frac{g^2}{16\pi^2}\frac{m_Z^2}{M^2}$ can be resolved for $\delta O \gtrsim 10^{-5}$ (Maura et al., 2024).

Gaussian global fits to SMEFT operators combining on- and off-pole data deliver $1\sigma$ intervals at $\Lambda = 1$ TeV such as: $$C_H = \pm 1.09, \quad C_{H\Box} = \pm 0.052, \quad C_{HW} = \pm 0.020, \quad C_{HB} = \pm 0.034,$$ with corresponding effective probe scales up to $\Lambda_{\rm eff}(C_{HWB}) \simeq 111$ TeV, $\Lambda_{\rm eff}(C_{HD}) \simeq 57$ TeV (at 2$\sigma$) (Maura et al., 2024).

4. Advanced Computational Methodologies

Terascale precision tests require multi-loop, multi-scale theoretical predictions with at least eight significant digits accuracy. A semi-numerical differential equation (DE) approach is employed (Dubovyk et al., 2022):

• Master integrals $I(\epsilon, x)$ (dimensional regularization with $\epsilon = (4-D)/2$) satisfy a linear DE system derived via IBP, solved as a power series around Euclidean kinematic points where boundary integrals are computed by high-precision sector decomposition.
• Analytical continuation and accuracy cross-checks (e.g., multiple boundary points) are built into the pipeline.
• Typical precision reaches $\geq 8$ digits for three-loop self-energies and two-loop box integrals, with run times on the order of hours to days for sector decomposition and series transport.

These methods underpin both the calculation of SM expectations and the matching/RGE for new physics effects, providing the required theoretical fidelity for precision EW fits.

5. Model-Specific Precision Constraints

Recent global fits and model analyses illustrate the power of Terascale tests:

• Third-family quark-lepton unification in the non-universal 4321 gauge model: One-loop SMEFT matching, tree-level shifts in $W \to \tau\nu$ and $Z \to \nu_\tau\nu_\tau$, and RG-enhanced colored-vector contributions yield $\gtrsim 2\sigma$ improvement over the SM fit. Benchmark parameters point to $\Lambda_U \simeq 1.6$ TeV, $Y_+ \simeq 0.36$–$0.65$, with strong constraints from lepton-flavor-universality (LFU) and high-$p_T$ tails at LHC requiring $\Lambda_U \gtrsim 1.25$–$1.6$ TeV (Allwicher et al., 2023).
• Twin Higgs/composite Higgs scenarios: The separation between colored-partner masses $m_*$ and $v$ is controlled by strong coupling $g_*$. Agreement with EW data with mild tuning $\Delta \simeq 5$–$20\%$ can be achieved for $m_* \sim 3$–$5$ TeV, lying above LHC reach but within 100 TeV machine capability. Key observables ($$\Delta \widehat S, \Delta \widehat T, \delta g_{Lb}$$) remain at the $10^{-3}$–$10^{-4}$ level, matching experimental sensitivity (Contino et al., 2017).

For simplified UV scenarios, Tera-Z bounds surpass HL-LHC limits. Real singlet scalars, weakly interacting massive particles (WIMPs), and custodial quadruplets receive indirect exclusions up to $M \sim 1$–$2$ TeV or higher, matching or exceeding direct search capabilities (Maura et al., 2024).

6. Future Prospects and Theoretical Implications

The ongoing and planned experimental programs centered on Tera-Z runs at FCC-ee/CEPC and next-generation colliders anticipate substantial advances:

• Precision reaches tens of TeV for new physics scales through one-loop and two-loop indirect effects.
• Enhanced constraints on SMEFT Wilson coefficients, including those which only contribute at NLO or NNLO. Several single-operator bounds in the Warsaw basis already reach $\Lambda_{\rm eff} \gtrsim 50$ TeV at 2$\sigma$.
• The comprehensive on-shell and off-shell SMEFT global fits leverage the full statistical and theoretical power of the datasets, with future collider runs expected to probe residual tuning levels $\Delta \lesssim 1\%$.

A plausible implication is that the accuracy frontier may overtake the energy frontier in sensitivity to certain classes of new physics, "anticipating" direct observability at higher-energy machines by exploiting quantum fluctuations in electroweak observables at the $Z$ pole. The principle that "accuracy complements energy" underpins current strategies for probing physics beyond the Standard Model (Maura et al., 2024).

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Benchmark SMEFT Operator Constraints at Tera-Z (FCC-ee/CEPC) (Maura et al., 2024)

Operator	1$\sigma$ interval at $\Lambda$=1 TeV	$\Lambda_{\rm eff}$ (TeV) at 2$\sigma$
$C_H$	$\pm 1.09$	1
$C_{H\Box}$	$\pm 0.052$	9.2
$C_{HW}$	$\pm 0.020$	22
$C_{HB}$	$\pm 0.034$	17
$C_{HWB}$	—	111
$C_{HD}$	—	57

This high-precision regime serves as a critical discriminator of theoretical models and forms the backbone of contemporary Terascale precision tests.