# Understanding the like-sign dimuon charge asymmetry in collisions

###### Abstract

The DØ collaboration has measured the like-sign dimuon charge asymmetry in collisions at the Fermilab Tevatron collider. The result is significantly different from the standard model expectation of CP violation in mixing. In this paper we consider the possible causes of this asymmetry and identify one standard model source not considered before. It decreases the discrepancy of the like-sign dimuon charge asymmetry with the standard model prediction, although does not eliminate it completely.

###### pacs:

13.20.HeFermilab-Pub-13-055-E

## I Introduction

The DØ Collaboration has measured D01 ; D02 ; D02a ; D03 the like-sign dimuon charge asymmetry and the inclusive muon charge asymmetry in collisions at a center-of-mass energy TeV at the Fermilab Tevatron collider. After subtracting the known background sources, the inclusive muon charge asymmetry is found to be compatible with zero while the like-sign dimuon charge asymmetry significantly deviates from the standard model (SM) prediction. This deviation is usually interpreted as the anomalous charge asymmetry of and semileptonic decays Grossman ; lenz .

In this paper we consider other possible sources of the like-sign dimuon charge asymmetry, taking into account the constraint that the inclusive muon asymmetry is consistent with zero. We identify one significant contribution which was not accounted for previously. In addition, we discuss all other sources of the dimuon charge asymmetry and show that the measurements of the DØ collaboration put strong constraints on them. After presenting available experimental results related to the dimuon charge asymmetry in Section II, we consider in Section III one by one the contribution of different processes into this asymmetry. Our results are summarised in Section IV.

## Ii Experimental results

We use the results of Ref. D03 and express them in a model independent way as the charge asymmetry of the inclusive muon sample , and the charge asymmetry of the like-sign dimuon sample . We follow the notations and definitions of Ref. D03 , where the asymmetries and are computed using the muons coming from the decays of and quarks and leptons, and from decays of short-living mesons such as , , , , . The origin of the asymmetries and may be the semileptonic charge asymmetry of and decays, as well as other -violating processes.

From the information presented in D03 we obtain

(1) | |||||

(2) |

These values are obtained, respectively, by multiplying the results given in Eqs. (34) and (35) by the coefficients and given in Eqs. (30) and (31) of Ref. D03 . An important observation which can be derived from these results is that is compatible with zero, while is significantly different from zero.

The composition of the inclusive muon sample is presented, for convenience, in Table 1 taken from Ref. D03 . The composition of the like-sign dimuon sample can be derived from the information given in Table 1 assuming that the two muons come from independent processes. This assumption can be applied because the requirement that the invariant mass of the two like-sign muons is greater than 2.8 GeV, which is used for selecting the dimuon pairs D03 , suppresses decays with the two muons originating from the same quark. Since the oscillation of the meson is found to be small PDG , the contribution of process in Table 1 to the like-sign dimuon sample is suppressed and, therefore, is neglected in the following discussion.

Process | Weight | |
---|---|---|

(non-osc) | ||

(osc) | ||

(non-osc) | ||

(osc) | ||

with or | ||

with or | ||

with or |

For our purposes it is convenient to classify the processes into three different categories denoted , and . Processes correspond to the decays of or to the final state containing two quarks and accessible to both and . At the quark level these decays correspond to the process or . Similarly, processes correspond to the decays of or to the final state containing two quarks and accessible to both and . These decays at the quark level are or . All other decays of hadrons producing two charm quarks are flavour specific and included in the group denoted . The weights of these processes are respectively , and . By definition,

(3) |

## Iii Contributions to the like-sign dimuon charge asymmetry

We consider in this section some SM processes producing the like-sign dimuon charge asymmetry , and the current experimental constraints of their contribution taken from PDG . Particles of physics beyond the SM may add new Feynman diagrams with loops in mixing, mixing, and in penguin decays. They are not discussed here.

The main source of the like-sign dimuon pairs produced in collisions is events. The background muons from and decays are excluded by definition from the asymmetries and , and are not discussed here. One of the hadrons from the pair decays to a “right sign” muon, i.e. to a muon with charge of the same sign as the charge of the initial or quark at the time of the collision. To obtain a like-sign dimuon event, the other -hadron must decay to a “wrong sign” muon.

### iii.1 violation in mixing of and mesons

The complex phase of the mass matrix of the system produces the charge asymmetry of the “wrong sign” semileptonic decays defined as

(4) |

The asymmetry is related to the phase as PDG

(5) |

The contributions of this asymmetry to the inclusive muon charge asymmetry and the like-sign dimuon charge asymmetry are

(6) | |||||

(7) |

where the coefficients , , and are defined in Ref. D03 :

(8) | |||||

(9) |

The SM predictions of the phases and asymmetries are lenz :

(10) | |||||

(11) | |||||

(12) | |||||

(13) |

Recently, the experimental measurements of both and became available. The measurements of are performed at hfag and by the DØ experiment asld-d0 :

(14) | |||||

(15) |

Our combination of these values gives

(16) |

The measurements of are performed by DØ asls-d0 and LHCb asls-lhcb collaborations:

(17) | |||||

(18) |

Our combination of these values gives

(19) |

Using the values (16) and (19) we obtain the allowed contributions to the inclusive muon and like-sign dimuon charge asymmetries from CPV in mixing of and mesons:

(20) | |||||

(21) | |||||

(22) | |||||

(23) |

and their sum

(24) | |||||

(25) |

For comparison, the SM prediction is

(26) |

In addition, the estimate of can be extracted from the measurement of violation in the decay. The corresponding phase is expected to change in the same way as the phase due to a new physics contribution Nierste ; lenz :

(27) | |||||

(28) |

The current world average value of is hfag :

(29) |

From these expressions we get the following estimate of from decay:

(30) |

However, we do not use the result (30) because Eq. (27) may be subject to large penguin corrections from new physics lenz2 .

Thus, the current experimental results constrain the contribution of CPV in mixing of and mesons to the measured like-sign dimuon charge asymmetry.

### iii.2 violation in interference of decay with and without mixing

The final states of the decay are accessible from both and . Therefore, the interference of decays to these final states with and without mixing results in CPV PDG . It turns out that this CPV produces a like-sign dimuon charge asymmetry. At the same time, its contribution is negligible in the inclusive muon charge asymmetry.

To demonstrate this, let us consider an example of the process producing a positive dimuon pair

(31) |

The state is a CP-even eigenstate accessible from both and mesons. The decay produces a “wrong sign” muon, and contributes to the like-sign dimuon sample. The decay produces a “right sign” muon, and therefore does not contribute to the like-sign dimuon sample. The number of positive and negative muons from the decay is the same, therefore there is no contribution to the inclusive muon charge asymmetry from this decay.

The decay rate of the meson that is initially produced as a is PDG

(32) |

The term proportional to is due to CPV in the interference of decays with and without mixing to , of the meson that is initially produced as . The term proportional to is due to the dirct CPV in decay. Neglecting the loop contributions to the decay amplitude, the coefficients and in the SM are expressed as Sanda

(33) | |||||

where is the eigenvalue of the final state , and is the angle of the Unitarity Triangle PDG . The loop diagrams can change this estimate by a few percent Xing1 ; Xing2 . For the final state . If is not a eigenstate, the coefficient may be non-zero, but experimentally for the decays considered here is negligible PDG . Therefore, we omit it in the following discussion. Integrating (32) we obtain the width of the decay to this final state

(34) | |||||

Now consider the CP-conjugate process producing a negative dimuon pair

(35) |

The decay rate of the meson that is initially produced as a is

(36) |

and the partial width is

(37) |

The like-sign dimuon charge asymmetry from this process is

(38) |

Numerically the absolute value of this asymmetry is large, because and PDG .

Let us now obtain the contribution of the decay channel with weight to the like-sign dimuon charge asymmetry . This weight takes into account both the branching fraction of decay and detector-related efficiency of muon reconstruction. The weights for the various processes have been defined in Section II. The probability that an initial quark produces a “right sign” muon is D01

(39) | |||||

The factor 0.5 in this expression corresponds to the statement that the number of positive and negative muons produced in the processes , and is the same. The probability that an initial quark produces a “wrong sign” muon is

(40) | |||||

The number of events is proportional to . The number of events is obtained by replacing by . The charge asymmetry from channels is then

(41) |

Thus, CPV in interference of decay with and without mixing produces a like-sign dimuon charge asymmetry, while it does not contribute to the inclusive muon charge asymmetry. The possible final states produced in the decay include , , , , , , , etc. For many of these final states the value is measured experimentally PDG and in all cases it is consistent with the SM value , which corresponds to the expectation of the dominance of the -even final states in the decay.

On the contrary, the contribution of CPV in the transition producing the eigenstates, like the decay, should not contribute to the like-sign dimuon charge asymmetry, because for each -even final state there should be the corresponding -odd final state. For example, the contribution from the decay is canceled by the decay .

The weight can be obtained by using the measured branching fraction of decay. For example, the weight can be computed using the following expression

(42) |

The coefficient by definition is normalised to the weight , which is proportional to . The value is the average branching fraction of the direct decay of all hadrons to muon weighted with the relative production rate of different hadrons at the hadron collider. Also by definition, the weight includes both decays and , hence the factors 0.5 in Eqs. (39) to (41), and the factor 2 in Eq. (42). The factor PDG is the fraction of plus in the admixture of -hadrons. The factor is the ratio of detector acceptances of muons from and decay. Muons from and mesons have different detector acceptances because they have different kinematic distributions.

Using the results of Ref. D03 and other experimental values from PDG we estimate the coefficient from the following expressions:

(43) |

Here is the branching fraction of hadron decays producing pair. We use the experimental value which is obtained from PDG . The quantity is the average branching fraction of the direct decay of all charmed hadrons to muon weighted with the relative production rate of different pairs of hadrons in hadron decay. Using the values of corresponding branching fractions for the -baryon mixture from Ref. PDG we obtain

(44) | |||

(45) |

From these expressions we obtain

This estimate of does not take into account different kinematic distributions for the various decays . Therefore, a simulation of the DØ detector is required to obtain a more accurate value for .

Using this observation, we estimate by several methods.

Estimate 1. Let us consider four measured decay channels PDG : with , ; with , ; with , (our guess); and with , . Using these numbers we obtain for the sum of these channels

(48) | |||||

Using this value and Eqs. (41) and (47), the contribution to from these 4 channels is

(49) |

Estimate 2. For this estimate we assume that the final states are mostly -even (), which is appropriate for final states and confirmed by the experimental measurements. Then

(50) |

Eq. (41) becomes approximately

(51) |

The factor takes into account the fact that the final state contains more mesons than the generic state, and that the semileptonic branching fraction of meson is about 2.7 times larger than that of all other charm hadrons. Using the known branching fractions of -meson decays, we estimate . Using the values from Table 1 we obtain the following estimate of from CPV in interference:

(52) |

This value gives an upper bound of the estimate, since in deriving it we assume that all states are -even, which is clearly not the case.

Estimate 3. In the SM the violation in mixing of neutral mesons is small and the mass eigenstates of the system coincide with eigenstates. In that case Dunietz

(53) |

where () is the width of decay to the -even (-odd) final states, respectively. Assuming that this difference is saturated by the transition, we obtain the following estimate:

(54) |

In the SM framework lenz . An alternative estimate of uses the SM relation given in Ref. PDG , page 1067. It results in a similar value . Substituting the expression (54) in Eqs. (41) and (47) we obtain

(55) |

Estimate 4. In this estimate we use the measured value PDG . We replace by and obtain

(56) |

All these estimates of are consistent. In the following, we use the estimate (55). Comparing it with the experimental result (2), we conclude that CPV in interference of decay with and without mixing accounts for a part of the observed like-sign dimuon charge asymmetry. The experimental uncertainty on keeps open the possibility of a substantially larger contribution from this source. We also note that CPV in interference of is much larger than the SM prediction for CPV in mixing of and given in Eq. (26).

In Ref. D03 the like-sign dimuon charge asymmetry is measured in several sub-samples of events with an additional selection according to the muon impact parameter. This selection effectively changes the contribution of muons coming from the oscillated decays. The estimate (55) after applying this selection is also modified and the contribution from CPV in interference can be enhanced by selecting the dimuon events with large muon impact parameter. Therefore, the study of the dependence of the asymmetry on the muon impact parameter can provide an additional insight on this source of CPV.

### iii.3 CPV in interference of decay with and without mixing

Again we present several estimates.

Estimate 1. Four channels of interest are with . PDG The CPV parameters have not been measured, but in the SM should be approximately

(57) |

Here, similarly to decays, we assume that these final states are mainly CP-even. For these 4 decay channels we obtain

(58) |

Estimate 2. We assume that the final states are mostly -even (), which is appropriate for final states. We take

(59) |

Then Eq. (41) becomes approximately

(60) | |||||

The absolute value of estimate (60) can be considered as an upper bound on the contribution to from this source, because some of the final states are not -even.

Estimate 3. It is known that the four decay channels do not exhaust the contributions to PDG . To obtain an estimate of the like-sign dimuon charge asymmetry from CPV in interference of we replace by . We use the experimental value ps and obtain

(61) |

which can be compared with (2).

In conclusion, CPV in interference of decay with and without mixing is suppressed by the small values of and .

### iii.4 Direct CPV in decay ( or ), followed by .

This type of CPV occurs due to the interference of the tree level and penguin diagrams with different strong phases and different weak phases. Let us consider, as an example, the decay , followed by . Its branching fraction is . The CP-violating asymmetry in this decay has not been measured, but should be less than because the penguin diagram is suppressed by one loop. The like-sign dimuon charge asymmetry from this channel,

(62) | |||||

is less than 0.0002%. Considering also decays with , which have a CP-violating asymmetry suppressed by one loop , we conclude that the direct violation in the decays measured so far have a negligible contribution to the like-sign dimuon charge asymmetry.

### iii.5 Direct CPV in semileptonic decays of and quarks.

In this subsection we assume that the like-sign dimuon charge asymmetry is due solely to CPV in direct semileptonic decays of charged and neutral hadrons containing or quarks. This type of violation vanishes in the lowest order due to the symmetry Gronau . The second order calculations give extremely small value for the asymmetry of the order of Gronau . Despite such a strong theoretical constraint, the possibility of large contribution from this source is discussed in Ref. Descotes . There is no direct experimental measurements of this CPV in semileptonic decays, and the only experimental limitation can be derived from Eqs. (1) and (2). Let () be the flavor averaged CP violating charge asymmetry of direct semileptonic decays of () quarks. The contribution of and to and are

(63) | |||||

and

(64) | |||||

If the only asymmetry is , then . If the only asymmetry is , then . Taking from (1), we obtain the following estimates for the contribution of direct CPV to :

(65) |

Thus, the “closure test” (1) begins to constrain the contributions of to the like-sign dimuon charge asymmetry .

## Iv Conclusions

Process | allowed | |
---|---|---|

A | Mixing of | |

A | Mixing of | |

B | Interference of | |

C | Interference of | |

D | CPV in decays | |

E | in decays | |

E | in decays |

We have considered several possible causes of the measured like-sign dimuon charge asymmetry , and obtained their experimental constraints. A summary is presented in Table 2. We find that standard model CP violation in interference of decays with and without mixing of to flavor non-specific states , followed by the decay , contributes

(66) |

to the like-sign dimuon charge asymmetry . CP violation in interference does not contribute to the inclusive muon charge asymmetry and therefore is compatible with the observation that is consistent with zero.

Among all other possible sources of the dimuon charge asymmetry, only the direct violation in semileptonic - and -hadron decays is not yet limited experimentally. It is very small in the SM, but, until experimentally measured, this source of dimuon charge asymmetry cannot be excluded. Our estimate of this source is derived from the DØ measurements (1) and (2). The exclusive measurement of this type of violation is required to improve this constraint.

Taking into account the additional SM source of dimuon charge asymmetry (66) identified in this paper, the combination of DØ measurements (1) and (2) becomes consistent with the SM expectation within 3 standard deviations. Still the difference between (2), and (26) and (66), , leaves some room for new physics violation in and mixing, in the interference of and decays with and without mixing, or in semileptonic decays of and hadrons. A deviation in the value of from the SM prediction could also contribute to the difference between the observed and expected like-sign dimuon charge asymmetry.

As D0 collaborators deeply involved in the dimuon asymmetry analysis, we thank the DØ Collaboration for inspiring, commenting, and supporting this work. We thank Michael Gronau and Jonathan L. Rosner for very useful comments and suggestions to our paper.

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