End Matter

Appendix E $D^{0} - \bar{D^{0}}$ Mixing

https://doi.org/10.1093/oso/9780198503996.005.0005

Pages

480–484

Published:

July 1999

Appendix E

E.1 Theoretical expectations

The mesons $D^{0} \sim c \bar{u}$ and $\bar{D^{0}} \sim \bar{c} u$ have lifetime $(4.15 \pm 0.04) \times 10^{- 13} s$ ⁠, corresponding to

Γ = (1.586 \pm 0.015) \times 10^{- 12} GeV .

(E.1)

They decay faster than $B_{d}^{0}$ and $\bar{B_{d}^{0}}$ ⁠, in spite of their mass $1.8645 \pm 0.0005 GeV$ being much smaller than the mass of the latter mesons; this is because the decay of the charm quark is not suppressed by small CKM-matrix elements, contrary to the decay of the bottom quark, which is suppressed by $∣ V_{cb} ∣ ≪ 1$ and $∣ V_{ub} ∣ ≪ 1$ ⁠.

In the standard model (SM) $D^{0} - \bar{D^{0}}$ mixing should be tiny: one expects $x ≪ 1$ and $∣ y ∣ ≪ 1$ ⁠. If x and $∣ y ∣$ are very small, as predicted and as, to some extent, experimentally confirmed, then $D^{0}$ and $\bar{D^{0}}$ practically do not oscillate into and from each other while decaying; the linear superposition of $D^{0}$ and $\bar{D^{0}}$ which is created at production time is identical with the one to be found at decay time. This has important consequences in the theoretical analysis of some decays of the $B^{0} - \bar{B^{0}}$ systems, as seen in particular in Chapters 36 and 37.

A good measure of mixing is $F = (x^{2} + y^{2}) / (2 + x^{2} - y^{2})$ ⁠; this quantity tends to 1 when either $x \to \infty$ or $∣ y ∣ \to 1$ ⁠.

In the SM $D^{0} - \bar{D^{0}}$ mixing receives three main contributions: from box diagrams, from dipenguin diagrams, and from long-distance effects. In this section we review briefly each of these contributions and its expected size, and turn afterwards to $D^{0} - \bar{D^{0}}$ mixing in extensions of the SM.

E.1.1 The box diagrams

The box diagrams for $D^{0} - \bar{D^{0}}$ mixing are analogous to the ones for $K^{0} - \bar{K^{0}}$ mixing, which were analysed in Appendix B. One has charm and up quarks in the external lines, and any of the three down-type quarks in the internal fermion lines. One might expect the bottom quark to dominate, because the function $S_{0} (x)$ (see eqn B.16) grows with x. However, the loops with bottom quarks end up being negligible, for two reasons: firstly, the bottom quark is not that heavy; secondly, its contribution is very much suppressed by the small CKM-matrix elements $V_{cb}$ and $V_{ub}$ in the vertices. Hence, only the strange and down quarks contribute effectively and, due to the GIM suppression, the effective Hamiltonian is proportional to (Datta and Kumbhakar 1985; Donoghue et al. 1986c)

\frac{{(m_{s}^{2} - m_{d}^{2})}^{2}}{m_{W}^{2} m_{c}^{2}},

(E.2)

and thus very small. In the computation in Appendix B one could neglect the masses and momenta of the external s and d quarks. In the box diagrams for $D^{0} - \bar{D^{0}}$ mixing, on the other hand, the masses of the internal s and d quarks are small compared to the mass of the external charm quarks, and the latter cannot be neglected. An extra operator $(\bar{c} γ_{R} u) (\bar{c} γ_{R} u)$ then appears in the effective Hamiltonian, and its matrix element must be evaluated.

One may use the vacuum-insertion approximation to estimate the matrix elements. One obtains $∣ M_{12} ∣ \sim 10^{- 17}$ to $10^{- 16} GeV$ ⁠, corresponding to $F \sim 10^{- 10}$ to $10^{- 8}$ ⁠, which is extremely small.

E.1.2 Dipenguin diagrams

The contributions of dipenguin diagrams to $K^{0} - \bar{K^{0}}$ mixing and to $B^{0} - \bar{B^{0}}$ mixing are negligible (Donoghue et al. 1986d; Eeg and Picek 1987, 1988), and one may be tempted to neglect them in $D^{0} - \bar{D^{0}}$ mixing too. However, Petrov (1997) has claimed that in the latter case they yield a short-distance contribution to $M_{12}$ not much smaller than, and with the opposite sign to, the contribution from the box diagrams.

E.1.3 Long-distance contributions

The exact evaluation of the box diagrams and dipenguin diagrams is not so important because $D^{0} - \bar{D^{0}}$ mixing is probably (Wolfenstein 1985) dominated by long-distance effects, i.e., by intermediate hadronic states—not quarks—in the $D^{0} \leftrightarrow \bar{D^{0}}$ transitions. In order to understand why this is so, one may make a comparison with the $K^{0} - \bar{K^{0}}$ and $B_{d}^{0} - \bar{B_{d}^{0}}$ systems. In the latter system the three up-type quarks $α$ couple with CKM-matrix factors $V_{αb} V_{αd}^{*}$ of the same order of magnitude $λ^{3}$ ⁠; the top quark being very heavy, it overwhelms the contributions from the light quarks; then, the low-energy physics of hadrons, i.e., the long-distance effects, are irrelevant. In $K^{0} - \bar{K^{0}}$ mixing the charm quark competes with the top quark because, in spite of being much lighter, it couples with $V_{cs} V_{cd}^{*} \sim λ$ ⁠, while the coupling of the top quark $V_{ts} V_{td}^{*} \sim λ^{5}$ ⁠. Thus, the couplings of light hadrons to $K^{0}$ and $\bar{K^{0}}$ are relatively strong, and one therefore expects relevant long-distance contributions to $M_{12}$ ⁠, cf. § 17.6. In $D^{0} \to \bar{D^{0}}$ transitions the important intermediate quarks are the light s and d quarks; it can then be expected that light hadrons couple strongly to $D^{0}$ and $\bar{D^{0}}$ ⁠, from which large long-distance contributions should follow.

The long-distance contributions are non-perturbative and we cannot compute them from first principles. Donoghue et al. (1986c) have evaluated the contributions of intermediate states with two charged pseudoscalar mesons— $π^{+} π^{-}, K^{+} K^{-}, π^{+} K^{-},$ and $K^{+} π^{-}$ ⁠, for which some experimental data are available; they have obtained $F \sim 10^{- 8}$ ⁠. However, there are other intermediate states—with two vector mesons, or one pseudoscalar and one vector meson, as well as with one, three, four, … mesons. It is likely that these intermediate states yield contributions of the same order of magnitude as the one studied by Donoghue et al. (1986c), and moreover it is likely that all those contributions have different signs and partially cancel each other, in such a way that one may guess that the sum of all of them ends up giving $F \sim 10^{- 8}$ ⁠.

A different approach to the long-distance contributions is based on heavy-quark effective theory. This approach was pioneered by Georgi (1992) and followed by Ohl et al. (1993); they obtained $F \sim 10^{- 10}$ ⁠. This is much smaller than the estimate by Donoghue et al. (1986c).

It should be pointed out that both these approaches concentrate on the long-distance contribution to the dispersive part of the $\bar{D^{0}} \to D^{0}$ transition amplitude, $M_{12}$ ⁠. The absorptive part, $Γ_{12}$ ⁠, remains unchecked, and might be larger than $M_{12}$ ⁠. The original estimate of Wolfenstein (1985) was that F might be as large as $10^{- 4}$ due to the long-distance contributions; this estimate seems to stay on firm ground for $Γ_{12}$ (Le Yaouanc et al. 1995).

Golowich and Petrov (1998) have suggested that the rich spectrum of resonances with masses between 1.6 and 2.1 GeV may give important contributions to $D^{0} - \bar{D^{0}}$ mixing. In a partly phenomenological analysis they obtained $F \sim 10^{- 8}$ ⁠, and found that $∣ Γ_{12} / M_{12} ∣$ might be larger than unity.

E.1.4 $D^{0} - \bar{D^{0}}$ mixing beyond the standard model

The fact that the SM predicts $D^{0} - \bar{D^{0}}$ mixing to be so small means that there is a large window of opportunity to check extensions of the SM via a possible large $D^{0} - \bar{D^{0}}$ mixing. Various extensions of the SM may lead to large mixing (Burdman 1995; Nir 1996). In particular,

•

A fourth generation would contribute to $M_{12}$ through box diagrams with intermediate $b'$ quarks. With $∣ V_{u b^{'}} V_{c b^{'}} ∣ \sim λ^{3}$ and $m_{b^{'}} \sim 100 GeV$ the current experimental limit on mixing is saturated (Burdman 1995).

•

Vector-like singlet quarks of charge 2/3 lead to flavour-changing couplings of the Z boson with the up-type quarks. These couplings generate a potentially large $D^{0} - \bar{D^{0}}$ mixing at tree level.

•

Multi-Higgs-doublet models without flavour conservation similarly lead to $D^{0} - \bar{D^{0}}$ mixing at tree level. A neutral scalar with mass $\sim 100 GeV$ and coupling to $\bar{c} u$ with strength $\sim \sqrt{m_{c} m_{u}} / υ$ saturates the experimental bound (Burdman 1995).

•

Multi-Higgs-doublet models include charged scalars, which enter box diagrams for $D^{0} - \bar{D^{0}}$ mixing similar to the SM boxes but with one or both $W^{\pm}$ replaced by charged scalars. This, too, may easily saturate the experimental bound.

One or more of these mechanisms may be simultaneously operative. Thus, various viable theoretical ideas lead to values of F within reach of current or planned experiments.

E.2 Experimental results

Using eqns (9.14) and (9.15) one has

\begin{matrix} Γ [P^{0} (t) \to f] & = e^{- Γ t} {|\frac{q}{p}|}^{2} {|{\bar{A}}_{f}|}^{2} {{|{\bar{λ}}_{f}|}^{2} - Γ t (x Im ​ {\bar{λ}}_{f} + y Re {\bar{λ}}_{f}) \\ + \frac{Γ^{2} t^{2}}{4} [{|{\bar{λ}}_{f}|}^{2} (y^{2} - x^{2}) + y^{2} + x^{2}] + O (Γ^{3} t^{3})}, \\ Γ [\bar{P^{0}} (t) \to \bar{f}] & = e^{- Γ t} {|\frac{p}{q}|}^{2} {|A_{\bar{f}}|}^{2} {{|λ_{\bar{f}}|}^{2} - Γ t (x Im λ_{\bar{f}} + y Re λ_{\bar{f}}) \\ + \frac{Γ^{2} t^{2}}{4} [{|λ_{\bar{f}}|}^{2} (y^{2} - x^{2}) + y^{2} + x^{2}] + O (Γ^{3} t^{3})} . \end{matrix}

(E.3)

Therefore, when x and y are small,¹¹²

\begin{array}{l} Γ [P^{0} (t) \to f] & \approx e^{- Γ t} {|\frac{q}{p}|}^{2} {|{\bar{A}}_{f}|}^{2} {{|{\bar{λ}}_{f}|}^{2} - Γ t (x Im {\bar{λ}}_{f} + y Re {\bar{λ}}_{f}) \\ + \frac{Γ^{2} t^{2}}{4} (y^{2} + x^{2})} \Leftarrow |{\bar{A}}_{f}| ≫ |A_{f}|, \\ Γ [\bar{P^{0}} (t) \to \bar{f}] & \approx e^{- Γ t} {|\frac{p}{q}|}^{2} {|A_{\bar{f}}|}^{2} {{|λ_{\bar{f}}|}^{2} - Γ t (x Im λ_{\bar{f}} + y Re λ_{\bar{f}}) \\ + \frac{Γ^{2} t^{2}}{4} (y^{2} + x^{2})} \Leftarrow |A_{\bar{f}}| ≫ |{\bar{A}}_{\bar{f}}| . \end{array}

(E.4)

In particular, for flavour-specific decay modes,

\begin{matrix} Γ [P^{0} (t) \to \bar{o}] \approx e^{- Γ t} {|\frac{q}{p}|}^{2} {|{\bar{A}}_{\bar{o}}|}^{2} \frac{Γ^{2} t^{2}}{4} (y^{2} + x^{2}), \\ Γ [\bar{P^{0}} (t) \to o] \approx e^{- Γ t} {|\frac{p}{q}|}^{2} {|A_{o}|}^{2} \frac{Γ^{2} t^{2}}{4} (y^{2} + x^{2}) . \end{matrix}

(E.5)

When mixing is small $F \approx (x^{2} + y^{2}) / 2$ ⁠. This is the quantity $r_{mix}$ that experimentalists strive to measure.

Equations (E.5) have been used by the E791 Collaboration (1996) to set an experimental limit on $D^{0} - \bar{D^{0}}$ mixing. The E791 Collaboration (1996, 1998) has used the decays $D^{* +} \to π^{+} D^{0}$ and $D^{* -} \to π^{-} \bar{D^{0}}$ to identify the flavour of the neutral-D meson at production time. They have then compared the ‘right-sign’ decays

D^{0} \to K^{-} l^{+} ν_{l} and \bar{D^{0}} \to K^{+} l^{-} {\bar{ν}}_{l}

(E.6)

with the ‘wrong-sign’ decays

D^{0} \to K^{+} l^{-} {\bar{ν}}_{l} and \bar{D^{0}} \to K^{-} l^{+} ν_{l},

(E.7)

where l may be either e or $μ$ ⁠. They have used the fact that, according to eqns (E.5), the time-evolution of the wrong-sign decays should be given, when xand y are small, by $Γ^{2} t^{2} exp (- Γ t)$ ⁠. They have obtained the 90%-confidence-limit $r_{mix} < 5.0 \times 10^{- 3}$ ⁠.

Later, the E791 Collaboration (1998) has observed the ‘wrong-sign’ decays

D^{0} \to K^{+} π^{-} (π^{+} π^{-}) and \bar{D^{0}} \to K^{-} π^{+} (π^{+} π^{-}),

(E.8)

and has compared them to the ‘right-sign’ decays

D^{0} \to K^{-} π^{+} (π^{+} π^{-}) and \bar{D^{0}} \to K^{+} π^{-} (π^{+} π^{-}),

(E.9)

where the notation $(π^{+} π^{-})$ indicates the possible presence of an extra pair of charged pions in the final state. The right-sign decays are proportional to ${∣ V_{ud} V_{cs} ∣}^{2} \sim 1$ ⁠, while the wrong-sign decays are proportional to ${∣ V_{us} V_{cd} ∣}^{4} \sim λ^{4} \sim 2.5 \times 10^{- 3}$ ⁠. Thus, in this case the wrong-sign decays are not really forbidden, rather they are ‘doubly Cabibbo-suppressed’, i.e., their decay amplitudes are suppressed by two powers of the Cabibbo angle. Then, in eqns (E.4), with $P^{0} \equiv D^{0}, \bar{P^{0}} \equiv \bar{D^{0}}, f \equiv K^{+} π^{-}$ or $K^{+} π^{-} π^{+} π^{-}$ ⁠, and $\bar{f} \equiv K^{-} π^{+}$ or $K^{-} π^{+} π^{+} π^{-}$ ⁠, one expects ${∣ {\bar{λ}}_{f} ∣}^{2}$ and ${∣ λ_{f} ∣}^{2}$ to be $\sim 2.5 \times 10^{- 3}$ ⁠.

One must be careful to distinguish the different decay-time dependences:

•

Mixing-induced decays— ${(Γ t)}^{2} exp (- Γ t)$ ⁠;

•

Doubly-Cabibbo-suppressed decays— $exp (- Γ t)$ ⁠;

•

Interference terms— $(Γ t) exp (- Γ t)$ ⁠.

Carefully taking this into account,¹¹³ the E791 Collaboration (1998) obtained the 90%-confidence-limit $r_{mix} < 8.5 \times 10^{- 3}$ ⁠, comparable to the bound extracted from semileptonic decays.

The Particle Data Group (1996) refers to the pre-1996 limits on $D^{0} - \bar{D^{0}}$ mixing. Those experimental searches too have used either the semileptonic or the $K^{\pm} π^{\mp}$ (together with $K^{\pm} π^{\mp} π^{+} π^{-}$ ⁠) decays of the neutral-D mesons. They have obtained results which were either weaker or less general than the ones by the E791 Collaboration (1996, 1998).

E.3 Conclusions

•

The SM predicts $r_{mix} ≲ 10^{- 8}$ ⁠.

•

Experiment can at present only guarantee that $r_{mix} < 10^{- 2}$ ⁠.

•

Various models beyond the SM could saturate or even exceed the experimental bound.

Notes

¹¹²

The product $Γ t$ must remain of order 1 lest the factor $exp (- Γ t)$ renders the decays unobservable.

¹¹³

The importance of the interference terms was emphasized by Blaylock et al. (1995); Wolfenstein (1995); Browder and Pakvasa (1996); Liu (1996). Earlier it bad been usual to neglect terms with time-dependence $(Γ t) exp (- Γ t)$ ⁠.

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Contents

Appendix E $D^{0} - \bar{D^{0}}$ Mixing

Appendix E

E.1 Theoretical expectations

E.1.1 The box diagrams

E.1.2 Dipenguin diagrams

E.1.3 Long-distance contributions

E.1.4 $D^{0} - \bar{D^{0}}$ mixing beyond the standard model

E.2 Experimental results

E.3 Conclusions

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Contents

Appendix E D0−D0¯ Mixing

Appendix E

E.1 Theoretical expectations

E.1.1 The box diagrams

E.1.2 Dipenguin diagrams

E.1.3 Long-distance contributions

E.1.4 D0−D0¯ mixing beyond the standard model

E.2 Experimental results

E.3 Conclusions

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Appendix E $D^{0} - \bar{D^{0}}$ Mixing

E.1.4 $D^{0} - \bar{D^{0}}$ mixing beyond the standard model