###### Abstract

We have recently seen new upper bounds for , a key decay to search for physics beyond the Standard Model. Furthermore a non-vanishing decay width difference of the system has been measured. We show that affects the extraction of the branching ratio and the resulting constraints on the New Physics parameter space, and give formulae for including this effect. Moreover, we point out that provides a new observable, the effective lifetime , which offers a theoretically clean probe for New Physics searches that is complementary to the branching ratio. Should the branching ratio agree with the Standard Model, the measurement of , which appears feasible at upgrades of the LHC experiments, may still reveal large New Physics effects.

^{†}

^{†}preprint: Nikhef-2012-nnn

Nikhef-2012-006

Probing New Physics via the Effective Lifetime

Kristof De Bruyn , Robert Fleischer , Robert Knegjens ,

Patrick Koppenburg , Marcel Merk , Antonio Pellegrino , Niels Tuning

Nikhef, Science Park 105, NL-1098 XG Amsterdam, The Netherlands

Department of Physics and Astronomy, Vrije Universiteit Amsterdam,

NL-1081 HV Amsterdam, The Netherlands

Abstract

We have recently seen new upper bounds for , a key decay to search for physics beyond the Standard Model. Furthermore a non-vanishing decay width difference of the system has been measured. We show that affects the extraction of the branching ratio and the resulting constraints on the New Physics parameter space, and give formulae for including this effect. Moreover, we point out that provides a new observable, the effective lifetime , which offers a theoretically clean probe for New Physics searches that is complementary to the branching ratio. Should the branching ratio agree with the Standard Model, the measurement of , which appears feasible at upgrades of the LHC experiments, may still reveal large New Physics effects.

April 2012

## I Introduction

Thanks to the Large Hadron Collider (LHC) at CERN we have entered a new era of particle physics. One of the most promising processes for probing the quark-flavor sector of the Standard Model (SM) is the rare decay . In the SM, it originates only from box and penguin topologies, and the CP-averaged branching ratio is predicted to be buras ()

(1) |

where the error is fully dominated by non-perturbative QCD effects determined through lattice studies. The most stringent experimental upper bound on this branching ratio is given by at the 95% confidence level (C.L.) LHCb-Bsmumu ().

In the presence of “New Physics” (NP), there may be additional contributions through new particles in the loops or new contributions at the tree level, which are forbidden in the SM (see Ref. buras () and references therein).

A key feature of the -meson system is – mixing. This quantum mechanical effect gives rise to time-dependent oscillations between the and states. In contrast to the system, we expect a sizable difference between the decay widths of the light and heavy mass eigenstates LN ().

Performing a time-dependent analysis of , the LHCb collaboration has recently reported LHCb-Mor-12 (), which represents the current most precise measurement of this observable.

As we pointed out in Ref. BR-Bs (), the sizable complicates the extraction of the branching ratios of -meson decays, leading to systematic biases as large as that depend on the dynamics of the decay at hand.

In the case of the channel, the comparison of the experimentally measured branching ratio with the theoretical prediction (1) is also affected by this effect, which has so far been neglected in the literature.

It can be included through a measurement of the effective lifetime. As is a rare decay, it turns out that this observable offers another sensitive probe for NP that is theoretically clean and complementary to the branching ratio.

## Ii The General Amplitudes

The general low-energy effective Hamiltonian for the decay can be written as

(2) |

Here is Fermi’s constant, the are elements of the Cabibbo–Kobayashi–Maskawa (CKM) matrix, is the QED fine structure constant, the , are Wilson coefficients encoding the short-distance physics, while the

(3) | |||||

are four-fermion operators with and is the -quark mass. The are obtained from the by making the replacements . Only operators resulting in non-vanishing contributions to are included in (2). In particular the matrix elements of operators involving the vector current vanish.

This notation is similar to Ref. APS (), where a model-independent analysis of NP effects in transitions was performed. In the SM, as assumed in (1), only is non-vanishing and given by the real coefficient . An outstanding feature of is the sensitivity to (pseudo-)scalar lepton densities, as described by the and operators. Their Wilson coefficients are still largely unconstrained and leave ample space for NP.

The hadronic sector of the leptonic decay can be expressed in terms of a single, non-perturbative parameter, the -meson decay constant buras ().

For the discussion of the observables in Section III, we go to the rest frame of the decaying meson and distinguish between the and helicity configurations, which we denote as with . In this notation, and are related to each other through a CP transformation:

(4) |

where is convention-dependent. We then obtain

(5) |

where is the mass, and , and

(6) |

(7) |

The and carry, in general, non-trivial CP-violating phases and . However, in the SM, we simply have and (see also Ref. APS ()). The factor in (5) originates from using the operator relation and (4) in the leptonic parts of the four-fermion operators.

## Iii The Observables

For the observables discussed below we need the

(8) |

amplitude. Inserting again into the matrix elements of the four-fermion operators and using both (4) and , we obtain

(9) |

which should be compared with (5). We observe that

(10) |

Following the formalism to describe – mixing discussed in Ref. RF-habil (), we consider the observable

(11) |

Here we have taken into account that the – mixing phase is cancelled by the CKM factors in (5) and (9), and that the convention-dependent phase is cancelled through (9), whereas simply cancels in the amplitude ratio. We notice the relation

(12) |

The observables contain all the information for calculating the time-dependent rate asymmetries RF-habil ():

(13) |

Here is the mass difference of the heavy and light mass eigenstates, and

(14) |

where is the mean lifetime; the numerical value corresponds to the results of Ref. LHCb-Mor-12 (). CP asymmetries of this kind were considered for decays (neglecting ) in various NP scenarios in Refs. HL (); DP (); CKWW ().

The observables entering (13) are given as follows:

(15) |

(16) |

(17) |

It should be emphasized that due to (12) and do not depend on the helicity of the muons and are theoretically clean observables.

Since it is difficult to measure the muon helicity, we consider the rates

(18) |

and obtain then the CP-violating rate asymmetry

(19) |

where the terms (15) cancel because of the factor.

It would be most interesting to measure (19) since a non-zero value immediately signaled CP-violating NP phases. Unfortunately, this is challenging in view of the tiny branching ratio and as tagging, distinguishing between initially present and mesons, and time information are required. An expression analogous to (19) holds also for decays.

In practice, the branching ratio

(20) |

is the first measurement, where the “untagged” rate

(21) |

is introduced BR-Bs (); DFN (). The branching ratio (20) is extracted ignoring tagging and time information. As shown in Ref. BR-Bs (), due to the sizable width difference, the experimental value (20) is related to the theoretical value (calculated in the literature, see, e.g., Refs.buras (); APS ()) through

(22) |

where

(23) |

The terms in (22) were so far not taken into account in the comparison between theory and experiment.

## Iv The Effective Lifetime

With more data available, the decay time information can be included in the analysis. As we pointed out in Ref. BR-Bs (), the effective lifetime

(26) |

allows the extraction of

(27) |

yielding

(28) |

We emphasize that it is crucial to the above equations that in (17) indeed does not depend on the helicities of the muons, i.e. .

Effective lifetimes are experimentally accessible through the decay time distributions of the same samples of untagged events used for the branching fraction measurements, as illustrated by recent measurements of the and lifetimes tauExp () by the CDF and LHCb collaborations: both attained a 7% precision with approximately 500 events, while an even larger sample of events can be collected by the LHC experiments, assuming the Standard Model value of the branching fraction. Although a precise estimate is beyond the scope of this article, we believe that the data samples that will be collected in the planned high-luminosity upgrades of the CMS and LHCb experiments Upgrade () can lead to a precision of 5% or better.

## V Constraints on New Physics

In order to explore constraints on NP, we introduce

(29) |

where we have used (17) and (22). Using (1) and the upper bound LHCb-Bsmumu () yield , neglecting the theoretical uncertainty from (1). In the case of , fixes a circle in the – plane. For non-zero values, gives ellipses dependent on the phases . As these phases are in general unknown, a value of results in a circular band. We obtain the upper bounds . As does not allow us to separate the and contributions, there may still be a large amount of NP present, even if the measured branching ratio is close to the SM value.

The measurement of and the resulting observable allows us to resolve this situation, as

(30) |

fixes a straight line through the origin in the – plane. In Fig. 1, we show the current constraints in the – plane, and illustrate also those corresponding to (30). In Fig. 2, we illustrate the situation in the observable space of the – plane. It will be interesting to complement these model-independent considerations with a scan of popular specific NP models.

Let us finally note that the formalism discussed above can also straightforwardly be applied to decays where the polarizations of the leptons can be inferred from their decay products CKWW (). This would allow an analysis of (13), where non-vanishing observables would unambiguously signal the presence of the scalar term. Unfortunately, these measurements are currently out of reach from the experimental point of view.

## Vi Conclusions

The recently established width difference implies that the theoretical branching ratio in (1) has to be rescaled by for the comparison with the experimental branching ratio, giving the SM reference value of . The possibility of NP in the decay introduces an additional relative uncertainty of originating from .

The effective lifetime offers a new observable. On the one hand, it allows us to take into account the width difference in the comparison between theory and experiments. On the other hand, it also provides a new, theoretically clean probe of NP. In particular, may reveal large NP effects, especially those related to (pseudo-)scalar densities of four-fermion operators originating from the physics beyond the SM, even in the case that the branching ratio is close to the SM prediction.

The determination of appears feasible with the large data samples that will be collected in the high-luminosity running of the LHC with upgraded experiments and should be further investigated, as this measurement would open a new era for the exploration of at the LHC, which may eventually allow the resolution of NP contributions to one of the rarest weak decay processes that Nature has to offer.

## Acknowledgements

This work is supported by the Netherlands Organisation for Scientific Research (NWO) and the Foundation for Fundamental Research on Matter (FOM).

## References

- (1) A. J. Buras, PoS BEAUTY 2011, 008 (2011) [arXiv:1106.0998 [hep-ph]].
- (2) R. Aaij et al. (LHCb Collaboration), arXiv:1203.4493 [hep-ex]; S. Chatrchyan et al. (CMS Collaboration), arXiv:1203.3976 [hep-ex]; T. Aaltonen et al. (CDF Collaboration), Phys. Rev. Lett. 107, 239903 (2011) [Phys. Rev. Lett. 107, 191801 (2011)] [arXiv:1107.2304 [hep-ex]]; V. M. Abazov et al. (D0 Collaboration), Phys. Lett. B 693, 539 (2010) [arXiv:1006.3469 [hep-ex]]; G. Aad et al. (ATLAS Collaboration), arXiv:1204.0735 [hep-ex].
- (3) A. Lenz and U. Nierste, arXiv:1102.4274 [hep-ph].
- (4) R. Aaij et al. (LHCb Coll.), LHCb-CONF-2012-002.
- (5) K. De Bruyn, R. Fleischer, R. Knegjens, P. Koppenburg, M. Merk and N. Tuning, Phys. Rev. D 86, 014027 (2012) [arXiv:1204.1735 [hep-ph]].
- (6) W. Altmannshofer, P. Paradisi and D. M. Straub, arXiv:1111.1257 [hep-ph].
- (7) R. Fleischer, Phys. Rept. 370, 537 (2002) [hep-ph/0207108].
- (8) C.-S. Huang and W. Liao, Phys. Lett. B 525, 107 (2002) [hep-ph/0011089].
- (9) A. Dedes and A. Pilaftsis, Phys. Rev. D 67, 015012 (2003) [hep-ph/0209306].
- (10) P. H. Chankowski, J. Kalinowski, Z. Was and M. Worek, Nucl. Phys. B 713, 555 (2005) [hep-ph/0412253].
- (11) I. Dunietz, R. Fleischer and U. Nierste, Phys. Rev. D 63, 114015 (2001) [hep-ph/0012219].
- (12) T. Aaltonen et al. (CDF Collaboration), Phys. Rev. D 84, 052012 (2011) [arXiv:1106.3682 [hep-ex]]; R. Aaij et al. (LHCb Collaboration), Phys. Lett. B 707, 349 (2012) [arXiv:1111.0521 [hep-ex]].
- (13) CMS Collaboration, CERN-LHCC-2011-006; LHCb Collaboration, CERN-LHCC-2012-007.