# On new physics in

###### Abstract

Motivated by the recent measurement of the dimuon asymmetry by the DØ collaboration, which could be interpreted as an enhanced decay rate difference in the neutral -meson system, we investigate the possible size of new-physics contributions to . In particular, we perform model-independent studies of non-standard effects associated to the dimension-six current-current operators with as well as . In both cases we find that for certain flavour or Lorentz structures of the operators sizable deviations of away from the Standard Model expectation cannot be excluded in a model-independent fashion.

###### Keywords:

Mostly Weak Interactions: B-Physics, CP violation, Rare Decays, Beyond Standard ModelFLAVOUR(267104)-ERC-66

IPPP/14/29

DCPT/14/58

## 1 Introduction

The experimental measurements of the like-sign dimuon asymmetry by the DØ collaboration in 2010 and 2011 Abazov:2010hv ; Abazov:2010hj ; Abazov:2011yk triggered much interest in the flavour-physics community. If the value of the measured asymmetry is interpreted solely as a CP-violating effect in mixing of neutral mesons (),

(1) |

then using and the experimental number from 2011 Abazov:2011yk deviates by from the latest Standard Model (SM) prediction Lenz:2011ti . One finds, however, that new-physics contributions in transitions alone cannot explain the large central value of the like-sign dimuon asymmetry, since such a large enhancement would violate model-independent bounds (see e.g. Dobrescu:2010rh ; Lenz:2012mb ).

In Borissov:2013wwa the interpretation of the like-sign dimuon asymmetry within the SM was revisited. It was found that (1) should be modified to take into account previously neglected contributions proportional to the decay rate differences and . These arise from interference of -meson decays with and without mixing. The modified result reads

(2) |

Numerical values for the coefficients and can be extracted from Borissov:2013wwa and Abazov:2013uma . The sign of is such that a positive value of gives a negative contribution to and turns out to be negligible. It follows that the measured like-sign dimuon asymmetry is not simply proportional to the semi-leptonic asymmetries , as assumed in Abazov:2010hv ; Abazov:2010hj ; Abazov:2011yk .

Very recently the DØ collaboration presented a new measurement Abazov:2013uma of the coefficients , and ( was neglected) and more importantly of the inclusive single-muon charge asymmetry and the asymmetry . The result is

(3) |

If one uses (2) as a starting point and assumes the SM value for , then the new measurement can be used to extract the following result for CP violation in mixing

(4) |

which differs by from the SM prediction Lenz:2011ti . The result in (4) is considerably smaller than the value presented in 2011 Abazov:2011yk . The reason for the noticeable shift of the central value in is that in Abazov:2011yk (as well as in all other previous experimental and theoretical analyses) the contribution proportional to in (2) was neglected.

Stronger statements can be obtained from the data in Abazov:2013uma , if different regions for the muon impact parameter (denoted by the index ) are investigated separately instead of averaging over them, as done to get the values in (3). It is then possible to extract individual values for , and from the measurements of the and . One finds Abazov:2013uma

(5) |

The result differs from the combined SM expectation for the three observables by . If one instead assumes that the semi-leptonic asymmetries and are given by their SM values, then the decay rate difference measured by Abazov:2013uma using (2) is

(6) |

which differs by from the SM prediction.

All in all, the new DØ measurements still differ from SM expectations at the level of and part of this tension could be related to an anomalous enhancement of the decay rate difference . This raises the question to what extent can be enhanced by beyond the SM effects, without violating other experimental constraints. In fact, this is an interesting question in its own right. While potential new-physics contributions to the related quantity have been studied in detail (see e.g. Dighe:2007gt ; Dighe:2010nj ; Bauer:2010dga ; Bai:2010kf ; Oh:2010vc ; Alok:2010ij ; Bobeth:2011st ; Goertz:2011nx ) and turned out to be strongly constrained by different flavour observables — at most enhancements of are allowed experimentally — possible new-physics effects in have received much less attention. An exception is the article Gershon:2010wx which emphasises that a precision measurement of would provide an interesting window to new physics.

The main goal of this paper is to close the aforementioned gap by performing model-independent studies of two types of new-physics contributions to that could in principle be large. As a first possibility we consider new-physics contributions due to current-current operators with , allowing for flavour-dependent and complex Wilson coefficients. We study carefully the experimental constraints on each of the coefficients that arise from hadronic two-body decays such as , the inclusive decay and the dimension-eight contributions to Boos:2004xp as extracted from . Our analysis shows that large deviations in and are currently not ruled out, if they are associated to the current-current operator involving two charm quarks. We emphasise that our general model-independent framework covers the case of violations of the unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. As a second possibility we analyse the constraints on new-physics contributions from operators of the form . We show that since the existing constraints imposed by tree-level and loop-level mediated -meson decays such as and are quite loose, sizable modifications of and are possible also in this case, in particular if they arise from vector operators. From a purely phenomenological point of view it thus seems much easier to postulate absorptive new physics in the -meson system rather than in the -meson system.

Our paper is organised as follows. In Section 2 we briefly set our notation and collect the experimental values and SM predictions for the mixing quantities. In Section 3 we illustrate the principle differences between and . We turn to the aforementioned model-independent new-physics studies in Sections 4 and 5, while Section 6 contains our conclusions. In Appendix A we give the SM result for , including a detailed breakdown of theoretical uncertainties. The input values employed in our numerical calculations are summarised in Appendix B.

## 2 Mixing formalism

Mixing phenomena in the -meson system, with , are related to the off-diagonal elements of the complex mass matrix and decay rate matrix . We choose the three physical mixing observables as the mass difference , the width difference and the flavour specific (or semi-leptonic) CP asymmetries . The general expressions of these observables are

(7) |

with the mixing phase . The above equations are valid up to corrections of , which is smaller than in the SM for the -meson systems Lenz:2011ti .

To study new-physics effects it is convenient to parameterise the general expressions for the mixing matrices in such a way that the SM contributions are factored out. We employ the notation introduced in Lenz:2006hd ; Lenz:2011zz and write

(8) | ||||||

The observables are then modified with respect to their SM predictions according to

(9) |

where the mixing phase is given by

(10) |

Quantity | |||||||
---|---|---|---|---|---|---|---|

SM | Lenz:2011ti | Lenz:2011ti | |||||

exp. | Amhis:2012bh | HFAG | Amhis:2012bh | HFAG | |||

SM | Lenz:2011ti | — | |||||

exp. | Amhis:2012bh | HFAG | — | ||||

Abazov:2013uma | DØ | — | |||||

Aaij:2014owa | LHCb | — | |||||

SM | see (73) | Lenz:2011ti | |||||

exp. | Amhis:2012bh ; Abazov:2013uma | Amhis:2012bh | HFAG | ||||

SM | Lenz:2011ti | Lenz:2011ti | |||||

exp. | Lees:2013sua | BaBar | Aaij:2013gta | LHCb | |||

Abazov:2012hha | DØ | Abazov:2012zz | DØ | ||||

SM | Lenz:2011ti | Lenz:2011ti | |||||

exp. | — | Aaij:2013gta | LHCb | ||||

— | Abazov:2012zz | DØ | |||||

, | SM | Charles:2004jd | Charles:2004jd | ||||

exp. | Amhis:2012bh | HFAG | Amhis:2012bh | HFAG | |||

— | Aaij:2013oba | LHCb |

We now compare the SM results with the experimental measurements, contrasting the situations in the -meson and -meson systems. For this purpose we have collected the SM results of the mixing observables and the corresponding experimental measurements in Table 1. Concerning , the central values agree very nicely, but the experimental errors are much smaller than the theoretical ones, which are dominated by hadronic uncertainties. The experimental numbers are the current world averages obtained by the HFAG collaboration Amhis:2012bh and incorporate various measurements performed at experiments from ALEPH to LHCb that are all consistent with each other.

In the case of the width difference, the SM prediction of agrees quite well with experimental world average Amhis:2012bh that combines measurements from LHCb Aaij:2013oba , ATLAS Aad:2012kba , CDF Aaltonen:2012ie and DØ Abazov:2011ry . The very good agreement between theory and experiment shows that the Heavy Quark Expansion (HQE) used to calculate works to an accuracy of or better. The SM prediction for itself was not explicitly given in Lenz:2011ti but can be extracted from the numerical code used in that work. For this reason, we give a short description of the components that go into it and also a detailed analysis of different sources of uncertainties in Appendix A. While the theory treatment of is in complete analogy to , experimental measurements are much more challenging due to its small size, which follows from the hierarchy of the CKM matrix elements involved. The bounds on given in Table 1 stem from the latest HFAG average and two more recent investigations from DØ Abazov:2013uma and LHCb Aaij:2014owa . The measurements of are still subject to large errors and significant deviations from the SM calculation are not yet ruled out. We return to this in a moment.

The experimental situation for the semi-leptonic CP asymmetries is more complicated, because the current HFAG values and Amhis:2012bh are based on the traditional interpretation (1) of the dimuon asymmetry through CP violation in mixing alone, whereas the correct interpretation seems to be the one given in (2). We therefore quote in Table 1 separately the direct measurements in semi-leptonic decays from BaBar Lees:2013sua , DØ Abazov:2012hha ; Abazov:2012zz and LHCb Aaij:2013gta . These results for semi-leptonic CP asymmetries are now complemented by values based on the interpretation (2) of the dimuon asymmetry given in (5). Obviously, the experimental measurements of the semi-leptonic asymmetries do little to restrict possible new-physics contributions.

Alternatively, one might also compare the SM predictions for the mixing phases
with experimental constraints via the relation . Because of the large
uncertainty in , the experimental numbers do not provide
any stringent constraint on at present.^{1}^{1}1If one takes the CL
range of the DØ value for the semi-leptonic asymmetry, then small negative
values of are excluded. For the phase the confidence
level (CL) ranges are given in Table 1.

In addition to the three mixing observables , and , the phase of affects time-dependent CP asymmetries of neutral -meson decays. Neglecting CP violation in mixing and corrections of , the interference between mixing and decay,

(11) |

gives rise to the direct and the mixing-induced CP asymmetries and , respectively. Within these approximations, especially is sensitive to the new physics phase of

(12) |

where and is the hadronic amplitude of the ()-meson decay. In cases where the hadronic amplitude is dominated by a single weak phase, such as in the SM, can be cleanly related to the corresponding CKM parameters. This is for instance the case for decays which are dominated by the tree transition or the gluonic penguin () transitions.

The two well-known examples of mediated tree decays are and , both of which measure the relative phase between the hadronic decay amplitude and . In this case one has

(13) |

with for and for . The dominant SM amplitude contributes and , respectively. New-physics effects can change either the phase of see (8), whose contribution is denoted by , the phase of or both. The new-physics contributions to the hadronic decay amplitude can be further subdivided into the ones from penguin contributions and the new-physics phase appearing in the tree-level amplitude. The SM penguin contributions are expected to be quite small (see e.g. Faller:2008gt ). This general parametrisation implies that the are actually dependent on both weak and strong phases specific to the final state . In the current work we are only interested in new physics that does not introduce visible effects in the penguin sector and furthermore consider only interactions that affect the tree-level decay, but not . In consequence, we set and to zero in (13).

The lesson to learn from the above comparisons is that with the exception of , the mixing observables in the -meson system are rather loosely constrained experimentally. There is thus in principle plenty of room for beyond the SM contributions to and , which depend besides also strongly on the non-standard effects in . Within the SM, is determined by operators of dimension six, predominantly those arising from tree-level -boson exchange with . Beyond the SM, other contributions are conceivable, however the masses of should allow for the formation of intermediate on-shell states in order to affect . Below we will study constraints on such absorptive contributions within a model-independent, effective field theory framework for the two cases and .

A further important point to keep in mind is that is related to a product of two operators, while is determined from operators that are obtained by integrating out high-virtuality particles. In the effective field theory approach the two types of operators are independent of one another. Motivated by this observation we will study the case where new physics manifests itself to first approximation only in terms of interactions of the form , which change (i.e. ), while the leading dimension-six operators are assumed to remain SM-like. In this way we can get an idea of how large the maximum effects in can be. Of course, in explicit new-physics models the Wilson coefficients of both types of operators will generically be modified, which can lead to correlated effects in and , thereby reducing the possible shifts in . Studying such correlations requires to specify a concrete model, which goes beyond the scope of this work. However, even in our approach, the dimension-six operators give rise to dimension-eight operators due to operator mixing, and consequently, induce also new-physics contributions to (i.e. ). We will study these effects in Section 4 and show that they are phenomenologically harmless for the effective interactions considered in our analysis.

## 3 Comparison of and

The first important observation is that is triggered by the
CKM-suppressed decay , whose inclusive branching ratios reads
(based on the numerical evaluation in Krinner:2013cja ),
while receives
the dominant contribution from the CKM-favoured decay , which
has an inclusive branching ratio of
Krinner:2013cja . This means that a relative modification of by 100% shifts the total -quark decay rate
^{2}^{2}2We do not distinguish here between the total -meson decay rates
, and and
the total -quark decay rate , because the measured differences
are smaller than the current theoretical uncertainites.
by around 1% only, while a 100% variation in
results in an effect of roughly 25% in the same observable. Large enhancement
of the decay rate can therefore be hidden in the hadronic
uncertainties of , while this is not possible in the case of
the decay rate.

Second, the CKM structure of and are notably different within the SM. Separating into the individual contributions with only internal charm quarks () or up quarks () and one internal charm quark and one up quark (), one can write the SM contribution to as follows

(14) |

with and Lenz:2011ti

(15) |

The results for the coefficients relevant in the -meson system are obtained from the latter numbers by a simple rescaling with the factor . Notice that the three values in (15) are quite similar, which implies that phase-space effects are not very pronounced.

Combining the formulae (14) and (15), we obtain the following numerical expressions

(16) |

From the first line of the result for , we see that in the -meson system there is a partial cancellation between the individual contributions, because the two relevant CKM factors are of similar size in this case, i.e. with denoting the Cabibbo angle. In the case of , on the other hand, the result is fully dominated by the contribution due to , since while . The observed partial cancellation leads again to the feature that a modification in will have a much larger effect in , compared to the effect of a similar modification in in . For instance, a shift in leads to an almost modification of , while a change of results in a shift of only.

Another way of looking at the mixing systems is to investigate the ratio . Using the unitarity of the CKM matrix, i.e. , we find the SM expression

(17) |

where the numerical coefficients are identical for the -meson and -meson system within errors. The relevant CKM factors are and . It follows that the real part of and thus is dominated by the first term in the square bracket of (17), which encodes the contribution to involving charm quarks only. The situation is quite different for the imaginary parts of that arise from the second and third term and determine the size of . The fact that in the SM the semi-leptonic CP asymmetry in the -meson sector is about 20 times larger than the one in the -meson system and has opposite sign is hence a simple consequence of .

The structure of and also allows one to draw some general conclusions on how new
physics can modify and . Consider for
instance the violation of CKM unitarity , a property known from beyond the SM
scenarios (see e.g. delAguila:2000rc ; Casagrande:2008hr ; Eberhardt:2010bm )
in which heavy fermions mix with the SM quarks and/or new charged gauge bosons
mix with the boson.^{3}^{3}3See also Botella:2014qya for a recent
discussion of a similar point. In such models the relation (17) would
receive a shift that can be approximated by

(18) |

Given our imperfect knowledge of some of the elements of the CKM matrix, deviations of the form are not excluded phenomenologically. From (18) we then see that such a pattern of CKM unitarity violation can lead to a relative enhancement of by up to , while in the case of the relative shifts can be at most. Depending on the phase of the new contribution in (18) could hence affect and in a significant way, while leaving , , and unchanged within hadronic uncertainties. In fact, in the next two sections we will see that it is possible to find certain effective interactions that support the general arguments presented above.

## 4 New physics in : current-current operators

In the following we derive model-independent bounds on the Wilson coefficients of so-called current-current operators. We write the part of the effective weak Hamiltonian involving these operators as

(19) |

with the Fermi constant. The current-current operators are then defined as

(20) |

where projects onto left-handed fields and denote colour indices.

In the SM, the coefficients are real and depend neither on
the quark content nor on whether the transition is or
. On the other hand, a generic new-physics model will give rise to
different contributions to each non-leptonic decay channel and one must consider
carefully the constraints on the (complex) coefficients
individually. While some ingredients needed for such a study have been mentioned
previously in the literature, see for instance Bauer:2010dga , it has yet
to be carried out in any detail. The goal of this section is to fill this gap
by deriving bounds on the , and coefficients,
which multiply the operators governing ,
and transitions, respectively.^{4}^{4}4The coefficient
governs transitions, which are severely CKM
suppressed in the SM. Since currently the most stringent bounds on such a
coefficient come from itself, we exclude it from the
analysis. We structure this section by discussing the coefficients
in turn, examining not only the constraints but also their
implications for deviations in from the SM expectation. To do
so, we first write

(21) |

Then, generalising the expressions from Section 3, we obtain

(22) |

Here and in the remainder of the section we use a notation where the Wilson coefficients are to be evaluated at the scale unless otherwise specified. Given this expression, it is straightforward to calculate the ratio using (9).

While it is the coefficients which appear in low-energy observables such as (4), it is important to keep in mind that these are obtained from the matching coefficients at the new-physics scale through renormalisation-group (RG) evolution. In what follows, we will always present bounds on the coefficients at the scale for convenience. The leading-logarithmic (LL) evolution connecting the coefficients at the two scales can be written as

(23) |

where

(24) |

and should be evaluated in the five-flavour theory. Throughout our work, in deriving the constraints on the individual coefficients from a given observable, we work under the assumption of “single operator dominance” and consider only changes in the coefficients one at a time. E.g. to set constraints on we fix .

The dominant effect of the modifications of the current-current sector considered here is to change from its SM value. However, double insertions of operators also give dimension-eight contributions to . It turns out that these are completely negligible numerically with the exception of the contributions from the operator. We thus postpone a more detailed discussion of these contributions to Section 4.3.

### 4.1 Bounds on up-up-quark operators

We begin our analysis by deriving constraints on from decays.^{5}^{5}5Constraints on the real and imaginary parts of can also be derived from studies of the isospin asymmetry in Lyon:2013gba . The obtained bounds would benefit greatly from better measurements of the decay. QCD factorisation provides a tool for
calculating various observables in these decays to leading power in
Beneke:2003zv . The reliability of the
factorisation predictions is a subject of debate, and in the following we
consider only observables which can be argued to be under theoretical control
within this approach.

The process is to an excellent approximation a pure tree decay and thus provides strong constraints on the magnitudes and phases of the Wilson coefficients . A particularly clean probe of tree amplitudes is provided by the ratio Bjorken:1988kk

(25) |

which by construction is free of the uncertainty related to and the form factor . Here denotes the rate, is the semi-leptonic decay spectrum differential in the dilepton invariant mass , evaluated at . The numerical values of the pion decay constant and of are collected in Table 3.

Ignoring small electroweak penguin amplitudes the ratio
measures the magnitude of the sum of the coefficients of
the tree amplitudes Beneke:2003zv . These coefficients are currently known
to next-to-next-to-leading order (NNLO) in perturbation theory
Bell:2009fm ; Beneke:2009ek . Working to NNLO in the SM, but to leading
order (LO) in the new-physics contributions in
(21), we obtain the expressions^{6}^{6}6Note that
are interchanged with respect to the common definition in the literature
Beneke:2003zv , to comply with our choice of operator basis.

(26) |

The given SM values correspond to the central values for presented in Bell:2009fm . Inserting (26) into (25), one obtains the approximation

(27) |

where and we have restored the theoretical error of the SM prediction for as given in Bell:2009fm . The corresponding experimental value is

(28) |

which has been derived in Bell:2009fm based on the information given in Amhis:2012bh ; Ball:2006jz . Combining the theoretical expectation (27) with the experimental determination allows to directly constrain the magnitude and phase of . In order to turn this constraint into individual bounds on , we use (23) along with the assumption of single operator dominance explained after (24). In Figure 1 we show the parameter ranges (blue circular bands) that are allowed at 90% CL using this procedure. We see that effects in are allowed if they leave the magnitude SM-like.

A very effective way to constrain the phases of the Wilson coefficients in addition to their magnitude is to study mixing-induced CP asymmetries (12) in transitions. On the other hand, the direct CP asymmetries, , are suppressed by powers of and/or in QCD factorisation and are difficult to predict quantitatively. The mixing-induced CP asymmetries are directly proportional to in the SM, in the limit where the Wilson coefficients of the QCD penguin operators are ignored. When the Wilson coefficients are allowed to be complex, this SM relation is altered to include the phase of the coefficients, even at LO in . The situation is more complicated when penguin corrections are included. However, the penguin-to-tree ratio is of in and of in Beneke:2003zv , so that the constraints given by these observables can still be considered relatively clean.

We can evaluate the indirect asymmetries in the sectors at
NLO in QCD factorisation using the formulae given in
Beneke:2003zv .^{7}^{7}7We have also studied the indirect asymmetry
. The constraints are qualitatively similar to those from
and do little to cut down the allowed parameter space, so we
exclude them from Figure 1 for simplicity. We find

(29) |

with

(30) |

and

(31) |

with

(32) |

We have defined and the central values correspond to the default parameter choice employed in Beneke:2003zv , apart from for which we use Charles:2004jd . The result depends very strongly on the angle , for which we use , in line with the current world average Amhis:2012bh obtained from and related processes. The quoted error is derived from the procedure used in Beneke:2003zv and is dominated by the dependence on . We have also included in the error the range of values obtained in scenarios S2 to S4 of that work. The current experimental results are Amhis:2012bh

(33) |

and Lees:2013nwa

(34) |

The 90% CL constraints imposed by are displayed in brown in the panels of Figure 1 and those by in red. For the case of one sees that combining the restrictions from with those from the two indirect CP asymmetries singles out three allowed regions, one of which is the SM-like solution. The restriction on is much weaker than that on , which can be understood evaluating (23) numerically to find

(35) |

The quantity is multiplied by a small coefficient, so that constraints from quantities such as these indirect CP asymmetries, which only depend on this combination of coefficients, are roughly 10 times stronger for than for , when the method of varying only one coefficient at a time is used. While the quantity still cuts out a very small portion of the allowed parameter space for , it turns out that offers no further constraint on and has therefore been omitted from the figure.

Part of the remaining parameter space can be eliminated by the quantity , which is the ratio of branching ratios of and . The extension of the QCD factorisation formalism necessary to describe these decays to NLO has been derived in Beneke:2006hg ; Bartsch:2008ps . The results for these decays depend on the parameters , analogous to (26) and as in the sector these are known to NNLO from Bell:2009fm ; Beneke:2009ek . The decays also receive contributions from QCD and electroweak penguin coefficients, which have only been calculated up to the NLO level. Combining all known corrections, one finds

(36) |

The corresponding experimental value is Amhis:2012bh

(37) |

The 90% CL constraints from