Charm Mixing and CPviolations  Theory
Abstract
This document describes my talk at the Flavor Physics and CP Violation Conference held at Taiwan National University (5/5/085/9/08). I begin by commenting on the most recent experimental compilation of mixing data, emphasizing the socalled ‘strong phase’ issue. This is followed by a review of the theory underlying charm mixing, both Standard Model and New Physics. The mechanism of Rviolation is used to illustrate the methodology for New Physics contributions and the relation of this to rare decays is pointed out. Finally, I address the subject of CPviolating asymmetries by describing some suggestions for future experimental studies and a recent theoretical analysis of New Physics contributions.
I Introduction
We are in the year following the momentous announcement of
experimental evidence for 
mixing (1); (2); (3); (4).
Ii Current Status of Charm Mixing
Let us consider four points regarding the current experimental situation as summarized in Ref. (5).

Experimental evidence for charm mixing has improved. At the CHARM 2007 Workshop (Cornell University 8/5/078/8/07) the euphoria of the moment provoked me to point out (6) that in light of the Physical Review Letters criteria of ‘observation’ () or ‘evidence’ (to), the thenexisting determination of amounted to merely a ‘measurement’ (). The current values (for the ‘no CPviolation’ fit)
(1) are at the level of ‘evidence’, and indeed in a plot of , the point is excluded by (5).

The current data set contains no evidence for CPviolation (hereafter CPV) in charm mixing. We will consider the corresponding situation for charm decays in Sect. IV.

Weeding out theoretical descriptions: Due to the heretofore uncertain status of charm mixing, I have been reluctant to discard various theoretical descriptions. However, in view of the C.L. values in the CPVallowed fit to charm mixing, for and for , I now feel that models having are no longer tenable.

The strong phase is not ‘very large’. The noCPV determination yields whereas in the CPVallowed fit the C.L. values are for . This developing topic is detailed in the following subsection.
ii.1 The Strong Phase
In the field of charm mixing, the ‘strong phase’ is defined as the relative phase between the and decay amplitudes. It appears in wrongsign transitions because the final state occurs both via doubly Cabibbo suppressed (DCS) decays and mixing followed by a Cabibbo favored (CF) decay. As a consequence, the parameters
(2) 
appear in the analysis. In the world of flavor SU(3) invariance, one has .
There is no way to completely avoid the presence of . For example, the time dependent rate for wrongsign events in decay,
(3) 
depends explicitly on and . There is no fundamental physics in ; it is a detail of the strong interactions. From the viewpoint of electroweak physics (and more specifically of charm mixing), it is an irritant, a nuisance.
Let me briefly review two rather different theoretical attempts to determine . Although neither is a firstprinciples application of QCD, both are well thoughtout calculations.

Nearby Resonances: In Ref. (7), the CF and DCS amplitudes, respectively called and , are each expressed as a sum of tree and resonance contributions. The latter involves the weak transition of the initial into a resonance whose mass is nearby that of the (8). The propagates and then decays strongly into the final state. The relative phase between tree and resonance components arises from the phase of the propagator,
(4) Straightforward algebra then relates the strong phase to the resonance phase . In view of the vanishing of in the SU(3) limit, it is convenient to plot as a function of an SU(3)breaking parameter ,
(5) At the time Ref. (7) was written, one had . This led to speculation among some that was quite large, although the large uncertainty in allows no such conclusion. A more recent evaluation gives . The uncertainty is now rather smaller and so is the central value.

Phenomenological Analysis: In Ref. (9), a study of seven CF and DCS modes is carried out in a model based on a traditional factorization and quark diagram approach. SU(3) breaking is incorporated largely via the decay constants and . A formula for is derived in terms of the branching ratios , and .
From the above two models, we should not be suprised to find (both approaches predict only the magnitude ). In other words, the strong phase is not expected to be ‘very large’, a result in accord with the phenomenological analysis of Ref. (10) (whose determinaton gives a result consistent with zero).
There has been recent progress in measuring experimentally. Consider the reaction chain (11)
(6) 
where refers to some final state. The pair will have . Define the CP eigenstates
(7) 
The approach under discussion has the two remarkable features of emphasizing quantum correlations and of directly involving the CPeigenstates. The transition rate for producing the final state obeys
(8) 
The minus sign (since ) in the first of the rate equations is due to the quantum nature of the process and the second of the rate equations exhibits the explicit presence of the CPeigenstates . Not all final states are optimal for determining the strong phase. It is best to choose one of the final states as a CP eigenstate and the other a pair, e.g. as in
(9) 
where the dependence on is explicit. Measurements based on this approach are presented in Refs. (12); (13), which give . By further including external measurements of charm mixing parameters, an alternate measurement of is obtained, yielding . Presumably, future experimental studies will be able to reduce present uncertainties, e.g. for a discussion of measuring the strong phase at BESIII see Ref. (14).
Iii The Origin of Charm Mixing
In principle, charm mixing can arise from the Standard Model (SM) and/or from New Physics (NP). We shall cover both in this talk, but consider the NP possibility in greater detail.
iii.1 Standard Model
Theoretical estimates of charm mixing have been performed using either quark or hadron degrees of freedom. We shall discuss each of these in turn.
Quark Degrees of Freedom
To my knowledge, the earliest attempt of this type is continaed in Ref. (15). These days, the usual approach (like that used in mixing) is to express the mixing matrix element as a sum of local operators ordered according to dimension (operator product expansion or simply OPE) (16). At a given order in the OPE, the mixing amplitude is expanded in QCD perturbation theory. Finally matrix elements of the various local operators are determined. It is a peculiarity of charm mixing that the various mixing amplitudes are most conveniently characterized by expanding in the small parameter .
A full implementation of this program is daunting because the number of local operators increases sharply with the operator dimension (e.g. has two operators, has fifteen, and so on) (17). The matrix elements of the various local operators are unknown and can be only roughly approximated in model calculations. In principle, QCD lattice determinations would be of great use, but they are not generally available at this time (18).
An analysis of and in the leading order in the OPE has been carried out through in Ref. (19). The result through is . These small values are due in part to severe flavor cancellations (the leading terms in the expansion for and respectively are and at order and and at order .
Evidently, the quark approach as implemented via the OPE has been seen as not the way to understand charm mixing. It involves a triple expansion (in operator dimension , QCD coupling and parameter ) which is at best slowly convergent. One longstanding beacon of hope has been the suggestion in Ref. (20) that sixquark operators whose Wilson coefficients suffer only one power of suppression might give rise to . Even this effect (whose estimated size is problematic due to uncertainties in matrix element evaluation) is too small.
Hadronic Degrees of Freedom
Let us restrict our attention to the following exact relation for the width difference where
We can calculate by inserting intermediate states between the weak hamiltonian densities . Of course, knowledge of the matrix elements is required. This method yielded an entirely reasonable estimate for , where the number of large matrix elements turns out to be quite limited (21).
By contrast, for charm mixing the number of contributing matrix elements is quite large.Perhaps the most comprehensive analysis to date for charm is the phenomenological evaluation based on factorization given in Ref. (22). The result thus obtained is too small.
This unfortunate circumstance shows how delicate this sum over many contributions seems to be. What then is one to do, given that the hadron approach appears tied to the issue of matrix element evaluation? Perhaps it is best to rely more on charm decay data and less on the underlying theory. The earliest work of this type (23); (24) focussed on the intermediate states in Eq. (LABEL:had1). In the flavor SU(3) limit, exact cancellations reduce the contribution from this subset of states to zero. However, SU(3) breaking was already known to be significant in individual charm decays, and based on data available at that time, these references concluded that ’ might be large’.
A modern version of this approach exists (25), but with the above argument essentially turned on its head. A main new ingredient is the realization that SU(3) breaking occurs at second order in charm mixing (26). Can it be that all twoparticle and threeparticle sectors (such as ) contribute very little to charm mixing due to flavor cancellations? Perhaps, but this argument cannot be continued to the fourparticle intermediate states because decay into fourkaon states is kinematically forbidden. In fact, Ref. (25) claims that these very sectors can generate . A dispersion relation calculation using charm decay widths as input can be used to estimate , but this contains an additional layer of model dependence.
I believe the claim that SM contributions produce values for and at the level is not unreasonable. At the same time, however, compared to SM predictions for kaon, and mixing, the status of charm mixing is decidely ‘fuzzy’.
iii.2 New Physics
The LHC era is about to begin. Yet, what we will learn from LHC data is still highly uncertain. This is in stark contrast with SM expectations at the time LEP came on line. Our own recent work on has tried to be biasfree by allowing for a variety of extensions to the Standard Model (27),
1] Extra gauge bosons (LR models, etc)
2] Extra scalars (multiHiggs models, etc)
3] Extra fermions (little Higgs models, etc)
4] Extra dimensions (split fermion models, etc)
5] Extra global symmetries (SUSY, etc).
NP contributions to charm mixing can affect as well as . We do not consider the former in this talk, but instead refer the reader to Refs. (28); (29).
The strategy for calculating the effect of NP on mixing is, for the most part, straightforward. One considers a particular NP model and calculates the mixing amplitude for as a function of the model parameters. If the mixing signal is sufficiently large, constraints on the parameters are obtained. For all we know, the observed mixing signal is a product of both SM and NP contributions. In general we will not know the relative phase between the SM and NP amplitudes, as depicted in Fig. 1, or even the precise value of the ‘fuzzy’ SM component. This affects how NP constraints are treated, as shown later in a specific example.
We now turn to the issue of NP and , as based on the work in Ref. (27), which studied a total of 21 NP models. These are listed in Table 1.
Model 

Fourth Generation 
Singlet Quark 
Singlet Quark 
Little Higgs 
Generic 
Family Symmetries 
LeftRight Symmetric 
Alternate LeftRight Symmetric 
Vector Leptoquark Bosons 
Flavor Conserving TwoHiggsDoublet 
Flavor Changing Neutral Higgs 
FC Neutral Higgs (ChengSher ansatz) 
Scalar Leptoquark Bosons 
Higgsless 
Universal Extra Dimensions 
Split Fermion 
Warped Geometries 
Minimal Supersymmetric Standard 
Supersymmetric Alignment 
Supersymmetry with RPV 
Split Supersymmetry 
Of these 21 NP models, only four (split SUSY, universal extra dimensions, leftright symmetric and flavorchanging twohiggs doublet) are ineffective in producing charm mixing at the observed level. This has several causes, e.g. the NP mass scale is too large, severe cancellations occur in the mixing signal, etc. This means that 17 of the NP models can produce charm mixing. For these, we can get constraints on masses and mixing parameters.
R Violations and Mixing
We cannot review all 17 NP models here, so we shall conentrate on just one of them, the case of Rparity violating (RPV) supersymmetry. RPV contributes to mixing via box amplitudes, as displayed in Fig. 2. Each box diagram is seen to contain four vertices in which quarks, squarks and leptons interact.
Fig. 3 provides a brief summary about this topic. The quantum number R distinguishes between particles of the SM and their supersymmetric partners (‘sparticles’). Rparity need not be conserved and in Fig. 3 we display an Rviolating lagrangian that is relevant to charmmixing. The coupling strength is and the indices are generation labels. The amplitude for appearing in Fig. 3 is part of the boxdiagram which mediates the charm mixing. Note that it is proportional to the product . Incidentally, to an experimentalist who has dealt with leptonantilepton pairs, this amplitude has the unexpected feature that the pair come from distinct vertices and not from a photon or a Z.
We refer the reader to Ref. (27) for details regarding the calculation of the RPV contributions to mixing. The end result of the analysis is the set of constraints displayed in Fig. 4. There, is plotted as a function of the product of the RPV couplings . We take , with and 2000 GeV corresponding to the solid, green dashed, red dotted, and blue dasheddot curves, respectively. The experimental bounds are as indicated, with the yellow shaded region depicting the region that is excluded. The bound cited in Ref. (27) for the RPV couplings is
(11) 
or using the updated mixing value of Eq. (1) we find
(12) 
The Rare Decay
It is often profitable for phenomenological studies to encompass various physical processes at the same time. The following is a case in point. It should be clear that the squarkexchange diagram at the bottom of Fig. 3 contributes not only to vertices in the mixing amplitude but also to rare transitions such as , , etc (30). In fact, the current experimental limit (31)
implies the constraint
(13) 
This ilustrates how charm mixing and charm rare decays are both of interest to RPV phenomenology. They are roughly competitive at present. However, the limit from charm isnot going to change much whereas that from can continue to improve.
Iv CPV Asymmetries
Since there is no existing evidence for CPV in the charm sector, it is natural to look to the future. We consider two topics of this type, first, possible experimental strategies for detecting CPV signals and next, a survey of NP models and their CPV asymmetries. For existing literature on the subject, I recommend a recent discussion by Petrov (32) and a treatment of basic CPV formalism applied to charm by Xing (33).
iv.1 Future Strategies
I briefly review two papers, each involving a facility planned for future operation.

Super Bfactory: A suggestion for work at a super Bfactory is to probe charm mixing and CPV using coherent events from decays (34). The point is that the large boost factor in the rest frame () would allow a precise determination ofthe proper time interval between the two decays. Thus for a final state , one would measure
(14) Symmetric and asymmetric factories are considered and various CPV asymmetries discussed. A yield of Dpairs per year is estimated.

Charm Factory: Ref. (35) considers the decay modes obtained by running on the and resonances at a Charm factory. Note that the Cparity values differ for these two states, with and . The production of final states for such experiments would be coherent, and one defines the quantities
(15) The authors conclude that it is favorable to measure decays of correlated ’s the various states by running on the , with a candidate CPV observable being .
iv.2 Calculating CPV Asymmetries
An interesting analysis of CPviolations in the singlyCabibbosuppressed transitions
as recently been carried out in Ref. (36). Final states which are both CP eigenstates () and nonCP eigenstates () are considered.
Let us restrict our attention to the timeintegrated CPV asymmetries of a final state which is a CP eigenstate (),
(16) 
Such an asymmetry can receive contributions from decay, mixing and interference,
(17) 
The ‘direct’ component (i.e. from decay) is generally expressed as
(18) 
where the phases and arise respectively from CPV and QCD.
What are the experimental prospects for measuring such CPV asymmetries? There can, in principle, be both SM and NP components. As with charm mixing, there can be both shortdistance and longdistance SM contributions, with the latter subject to less suppression than the former. However, due to uncertainties in estimating the longdistance component it is hard to be very precise about the actual size of SM asymmetries. At any rate, it is concluded in Ref. (36) that the SM cannot generate CPV asymmetries in the SCS sector much larger than .
At the time at which the work of Ref. (36) was carried out, the scale of experimental limits on the CPV asymmetries was roughly . This would appear to present a wide window of opportunity for observation of NP effects. However, some uptodate (as of 1/31/08) experimental limits from the Charm Heavy Flavor Averaging Group (37) are exhibited in Table 2. In arranging this Table, I selected only those limits whose uncertainties are less than . We see that the window is not as large today!
Asymmetry  Mode  Value 

I leave it to the reader to study the various technical details present in Ref. (36). However, some general patterns of expected NP behavior are:

Some supersymmetric models can give whereas models with minimal flavor violation cannot.

Only the SCS decays probe gluonic penguin amplitudes. Thus any large CPV asymmetries arising from this source would be unlikely for CF and DCS decays.

CPV asymmetries as large as would be expected from NP theories which contribute via loop amplitudes but not tree amplitudes (tree amplitudes tend to be constrained by mixing constraints).
V Conclusions
As we enter the LHC era, our field will require the resources to pursue both discovery and precision options. The discovery option will be carried out at the LHC. If, as anticipated, New Physics is revealed, perhaps (i) the signature will be so striking that a specific NP model is clearly identified, or (ii) the situation will be unclear for quite some time (e.g. some of the NP degrees of freedom might remain beyond the LHC reach). In either case, it will be important to carry out the precision option. For the case (i) above, we need to check and verify the LHC results, whereas for case (ii) observing the pattern of rare effects should help clarify the LHC findings. This will require the participation of LHCB and superflavor factories. How many such facilities will become operable only time will tell. One can only hope. Now onto a summary of the main topics:
Charm mixing and experiment:
The data on mixing allow us at long last to claim (in the sense of PRL discovery criteria) ‘evidence’ for determinations of and a true ‘observation’ of mixing. The quality of the mixing signal now can rule out theoretical descriptions predicting charm mixing at the level. There has been real progress on the issue of the strong phase difference between the and amplitudes. We expect that improved sensitivity in the quantum correlation approach will provide a more accurate measure of .
Charm mixing and Standard Model theory:
There is little change in our previous understanding of this subject. The quark approach which is carried out in the OPE has, to date, yielded . Even the most optimistic prediction for using this method predicts a mixing signal an order of magnitude too small. We have here a very slowly convergent process which nobody has yet been able to conquer. More promising is the hadron approach which might in fact get the magnitudes right, but is hampered by theoretical uncertainties. Nonetheless, we are still able to conclude that the observed mixing might well be a consequence of SM physics.
Charm mixing and New Physics theory:
The comprehensive study in Ref. (27) of 21 possible NP contributions to charm mixing shows that the observed mixing might also well be a consequence of beyondSM physics! Further progress on this front will presumably require input from LHC data for selecting among NP possibilities. We have pointed out how the interrelated phenomenologies of charm mixing and rare charm decays allows for a more systematic probe of NP parameter spaces.
Studies involving CPviolations in charm:
With the observation of charm mixing, the study of CPviolations in charm has taken its place at the forefront of research in this field. Given the expectation that CPviolating SM asymmetries should be less than and that some NP models can exceed this value, there should be a real window of opportunity to aim at. However, this window has begun to close. One is left wondering as usual – where is the New Physics?
Acknowledgements.
This work was supported in part by the U.S. National Science Foundation under Grant PHY–0555304. I wish to thank the Taiwan particle physics community for their fine job in organizing and carrying out FPCP 2008.Footnotes
 Throughout the talk I will often refer to  mixing as ‘charm mixing’ or simply ‘ mixing’.
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