Measuring $V_{cb}$ at electron-proton colliders

Technical Summary: Measuring $V_{cb}$ at Electron-Proton Colliders

Problem and Motivation
The Cabibbo-Kobayashi-Maskawa (CKM) matrix element $|V_{cb}|$ is a fundamental parameter in flavor physics, governing semileptonic $B$ meson decays and entering predictions for rare processes and CP violation. Current determinations rely on exclusive ($B \to D^{(*)}\ell\nu$) and inclusive ($B \to X_c\ell\nu$) decays, which exhibit a persistent tension of approximately $2.5\sigma$ (2026 PDG averages: $39.5 \pm 0.5$ vs. $42.0 \pm 0.5$ in units of $10^{-3}$). Further improvements in these low-energy measurements are limited by experimental and theoretical systematic uncertainties. A determination of $|V_{cb}|$ utilizing a distinct set of systematics is therefore highly valuable.

This paper investigates the potential for an independent, high-energy determination of $|V_{cb}|$ via bottom-tagged charged-current (CC) deep-inelastic scattering (DIS) at future electron-proton colliders: the Electron-Ion Collider (EIC), the Large Hadron-electron Collider (LHeC), and the FCC-eh. The signal process is the partonic transition $e^- c \to \nu_e b$, where the cross-section is directly proportional to $|V_{cb}|^2$. The primary challenge lies in distinguishing this CKM-suppressed signal from the significantly larger backgrounds of light-flavor and anticharm CC DIS events, as well as top-quark production at higher energies.

Methodology
The study employs a statistical analysis based on the ratio of $b$-tagged event counts to flavor-inclusive CC DIS event counts within kinematic bins of the DIS variables $x$ and $y$. This ratio observable is chosen to cancel common multiplicative factors, such as luminosity and polarized beam factors, thereby reducing systematic uncertainties.

• Colliders and Scenarios: The analysis considers specific beam energy and luminosity configurations for the EIC (10/18 GeV electrons, 275 GeV protons), LHeC (50 GeV electrons, 7 TeV protons), and FCC-eh (60 GeV electrons, 42.5 TeV protons). Both unpolarized and polarized electron beams are considered.
• Signal and Backgrounds: The signal is the $c \to b$ transition. Backgrounds include $u \to b$ (CKM suppressed but valence-enhanced), light-flavor CC DIS, anticharm CC DIS, and single-top production (at LHeC/FCC-eh). Neutral-current (NC) DIS backgrounds where the scattered electron is lost are estimated to be negligible based on HERA data but are assessed for potential impact.
• Flavor Tagging: The analysis assumes specific $b$-tagging efficiencies and mistag rates for charm and light quarks, motivated by recent LHC studies (e.g., CMS UParT v2 and ParticleNet algorithms). Crucially, the analysis propagates uncertainties in these tagging efficiencies via Gaussian "pull" parameters.
• Theoretical Inputs: The calculation is performed at the Born level with on-shell bottom quark mass ($m_b = 4.75$ GeV). The analysis incorporates projected Parton Distribution Function (PDF) uncertainties for the future colliders (EIC and LHeC projections) as well as current PDF sets (PDF4LHC21) for comparison. All other CKM matrix elements are floated in the fit with their PDG uncertainties, and no unitarity constraints are imposed.
• Statistical Framework: A profile likelihood ratio test statistic is constructed using the ratio of tagged to inclusive counts in $(x, y)$ bins. Nuisance parameters include PDF variations, CKM elements, top normalization, and tagging calibration uncertainties.

Key Results
The projected relative uncertainties on $|V_{cb}|$ vary significantly across the collider scenarios:

• EIC: The sensitivity is limited, with projected relative uncertainties ranging from 20% to 30%, depending on the energy stage and polarization. While not competitive with current precision, this would provide an independent probe in a different kinematic regime.
• LHeC: With 1 ab$^{-1}$ of integrated luminosity, the LHeC could achieve a relative uncertainty of 0.89% (unpolarized) and 0.79% (with $P_e = -0.8$). This precision is competitive with current low-energy determinations.
• FCC-eh: With 2 ab$^{-1}$, the projected uncertainty improves to 0.70% (unpolarized) and 0.67% (polarized).

Sensitivity Drivers and Systematics
The paper identifies specific experimental requirements necessary to realize these sensitivities:

1. Tagging Calibration: For high-energy colliders (LHeC/FCC-eh), the dominant systematic uncertainty arises from the calibration of the $b$-tagging efficiency ($\delta_b$). A 1% uncertainty in $\delta_b$ translates to roughly a 0.5% uncertainty in $|V_{cb}|$. The analysis shows that improving the calibration precision (e.g., halving the uncertainty widths) yields a more significant gain in $|V_{cb}|$ precision than further improving the rejection power of the tagger alone.
2. Kinematic Thresholds: At the EIC, the sensitivity is highly sensitive to the missing transverse momentum ($p_T$) threshold; raising the threshold from 5 GeV to 20 GeV degrades the uncertainty from ~22% to ~63%. Conversely, high-energy colliders are robust against higher $p_T$ thresholds (up to 20 GeV), allowing for better experimental control without significant loss of sensitivity.
3. Shape Uncertainties: Uncertainties in the kinematic shape of mistag rates (specifically the $y$-dependence) must be controlled at the few-percent level to preserve sub-percent precision at the LHeC and FCC-eh.
4. PDFs: While projected PDFs improve the precision compared to current sets, the impact of PDF uncertainties is sub-dominant to tagging calibration uncertainties at the LHeC and FCC-eh.

Significance and Claims
The paper claims that the LHeC, operating as a bridge project between the HL-LHC and FCC-ee, has the potential to provide a high-energy determination of $|V_{cb}|$ with a precision competitive with low-energy measurements (~0.8%). The primary significance of such a measurement lies in its independence: it probes $|V_{cb}|$ through a different process (CC DIS) with substantially different dominant systematic uncertainties (tagging calibration vs. hadronic form factors or inclusive modeling).

The author emphasizes that their results are sensitivity forecasts based on Born-level calculations and specific assumptions about tagging performance. They identify precise flavor-tag calibration and control of the kinematic response as the critical experimental requirements. The study serves as a motivation for more detailed experimental analyses that must address residual theory uncertainties (higher-order QCD/EW corrections), detector effects, and the consistent treatment of heavy-flavor production and jet definitions. The paper does not claim that these sensitivities are guaranteed but rather establishes the performance targets required to achieve them.