[Paper Review] Basics of D0-\bar D0 Mixing

Patricia Ball|ArXiv.org|Mar 22, 2007

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper provides a theoretical overview of $D^0$--$\bar{D}^0$ mixing, analyzing its sensitivity to new physics due to small Standard Model (SM) contributions dominated by long-distance QCD effects. It explains that experimental evidence from BaBar and Belle for $y \sim 10^{-2}$ exceeds SM predictions, indicating significant non-perturbative contributions, and highlights CP violation in mixing as a clean probe for new physics due to its extreme suppression in the SM.

ABSTRACT

Complementing the presentations, at this conference, of the first experimental evidence for $D$ mixing found at BaBar and Belle, I discuss the theoretical status of $D$ mixing.

Motivation & Objective

• To provide a theoretical framework for understanding $D^0$--$\bar{D}^0$ mixing in light of new experimental evidence.
• To clarify why $D$ mixing is highly sensitive to new physics due to the suppression of short-distance SM contributions.
• To analyze the role of long-distance QCD effects, particularly phase-space suppression in SU(3) multiplets, in generating the observed mixing parameters.
• To emphasize CP violation in mixing as a theoretically clean signal for new physics, given its extreme suppression in the SM.
• To assess the limitations of current theoretical approaches—both inclusive and exclusive—for predicting $x$ and $y$ in $D$ mixing.

Proposed method

• Adapts the formalism of $B$ mixing to $D$ mixing using the Schrödinger equation with non-diagonal $M$ and $\Gamma$ matrices.
• Defines the mixing parameters $x = \Delta M / \Gamma$ and $y = \Delta \Gamma / (2\Gamma)$, normalized to the average width $\Gamma$.
• Uses the ratio of wrong-sign to favored decay rates to extract $x'$ and $y'$, incorporating $|q/p|$ and strong phase effects.
• Analyzes box diagrams with internal $s$ and $d$ quarks, showing that SM contributions are suppressed by GIM and CKM factors, yielding $x_{\text{box}} \sim 10^{-5}$.
• Applies the exclusive approach by summing over intermediate hadronic states, particularly emphasizing phase-space suppression in SU(3) multiplets as a source of large $y$.
• Derives a dispersion relation linking $x$ and $y$, predicting $|x| \sim 0.1\%$ to $1\%$ for $y \sim 1\%$, though with higher model dependence.

Experimental results

Research questions

• RQ1 Why is $D^0$--$\bar{D}^0$ mixing so small in the Standard Model, and what mechanisms suppress it?
• RQ2 How can the observed experimental value of $y \sim 10^{-2}$ be explained given the small SM prediction of $x_{\text{box}} \sim 10^{-5}$?
• RQ3 To what extent do long-distance QCD effects, particularly phase-space suppression in SU(3) multiplets, explain the large observed mixing parameters?
• RQ4 Why is CP violation in $D$ mixing a particularly clean signal for new physics, despite hadronic uncertainties?
• RQ5 Can the inclusive or exclusive theoretical approaches accurately predict $x$ and $y$, and what are their limitations?

Key findings

• The experimental values of $y \sim 10^{-2}$ from BaBar and Belle are significantly larger than the SM box diagram prediction of $x_{\text{box}} \sim 10^{-5}$, indicating dominant long-distance contributions.
• The exclusive approach, which sums over hadronic intermediate states with phase-space suppression, can naturally generate $y \sim 10^{-2}$, explaining the experimental data.
• Phase-space suppression breaks SU(3) symmetry in multiplets, preventing full cancellation and leading to large effective mixing parameters.
• The inclusive approach, relying on quark-hadron duality, is insensitive to threshold effects and thus fails to reproduce the observed $y$ values.
• A dispersion relation suggests $|x|$ should be between $0.1\%$ and $1\%$ for $y \sim 1\%$, with $x$ and $y$ expected to be of opposite sign, though this is model-dependent.
• CP violation in mixing, characterized by $A_M$ and $\phi$, remains a theoretically clean probe for new physics, as it is strongly suppressed in the SM.