Introduction

During 2011 the LHCb experiment [1] at CERN collected 1.0 fb−1 of \(\sqrt{s} = 7 ~\mathrm{TeV} \) pp collisions. Due to the large production cross-section, in the LHCb acceptance [2], with the comparable number for charm production about 20 times larger [3, 4], these data provide unprecedented samples of heavy flavoured hadrons. The first results from LHCb have made a significant impact on the flavour physics landscape and have definitively proved the concept of a flavour physics experiment in the forward region at a hadron collider.

The physics objectives of the first phase of LHCb were set out prior to the commencement of data taking in the “roadmap document” [5]. They centred on six main areas, in all of which LHCb has by now published its first results: (i) the tree-level determination of γ [6, 7], (ii) charmless two-body B decays [8, 9], (iii) the measurement of mixing-induced CP violation in \(B ^{0}_{ s } \rightarrow { J / \psi } \phi\) [10], (iv) analysis of the decay \(B ^{0}_{ s } \rightarrow \mu ^{+} \mu ^{-} \) [1114], (v) analysis of the decay B 0K ∗0 μ + μ  [15], (vi) analysis of \(B ^{0}_{ s } \rightarrow \phi\gamma\) and other radiative B decays [16, 17].Footnote 1 In addition, the search for CP violation in the charm sector was established as a priority, and interesting results in this area have also been published [18, 19].

The results demonstrate the capability of LHCb to test the Standard Model (SM) and, potentially, to reveal new physics (NP) effects in the flavour sector. This approach to search for NP is complementary to that used by the ATLAS and CMS experiments. While the high-\(p_{\rm T}\) experiments search for on-shell production of new particles, LHCb can look for their effects in processes that are precisely predicted in the SM. In particular, the SM has a highly distinctive flavour structure, with no tree-level flavour-changing neutral currents, and quark mixing described by the Cabibbo–Kobayashi–Maskawa (CKM) matrix [20, 21] which has a single source of CP violation. This structure is not necessarily replicated in extended models. Historically, new particles have first been seen through their virtual effects since this approach allows one to probe mass scales beyond the energy frontier. For example, the observation of CP violation in the kaon system [22] was, in hindsight, the discovery of the third family of quarks, well before the observations of the bottom and top quarks. Crucially, measurements of both high-\(p_{\rm T}\) and flavour observables are necessary in order to decipher the nature of NP.

The early data also illustrated the potential for LHCb to expand its physics programme beyond these “core” measurements. In particular, the development of trigger algorithms that select events inclusively based on properties of b-hadron decays [23, 24] facilitates a much broader output than previously foreseen. On the other hand, limitations imposed by the hardware trigger lead to a maximum instantaneous luminosity at which data can most effectively be collected (higher luminosity requires tighter trigger thresholds, so that there is no gain in yields, at least for channels that do not involve muons). To overcome this limitation, an upgrade of the LHCb experiment has been proposed to be installed during the long shutdown of the LHC planned for 2018. The upgraded detector will be read out at the maximum LHC bunch-crossing frequency of 40 MHz so that the trigger can be fully implemented in software. With such a flexible trigger strategy, the upgraded LHCb experiment can be considered as a general purpose detector in the forward region.

The Letter of Intent for the LHCb upgrade [25], containing a detailed physics case, was submitted to the LHCC in March 2011 and was subsequently endorsed. Indeed, the LHCC viewed the physics case as “compelling”. Nevertheless, the LHCb Collaboration continues to consider further possibilities to enhance the physics reach. Moreover, given the strong motivation to exploit fully the flavour physics potential of the LHC, it is timely to update the estimated sensitivities for various key observables based on the latest available data. These studies are described in this paper, and summarised in the framework technical design report for the LHCb upgrade [26], submitted to the LHCC in June 2012 and endorsed in September 2012.

In the remainder of this introduction, a brief summary of the current LHCb detector is given, together with the common assumptions made to estimate the sensitivity achievable by the upgraded experiment. Thereafter, the sections of the paper discuss rare charm and beauty decays in Sect. 2, CP violation in the B system in Sect. 3 and mixing and CP violation in the charm sector in Sect. 4. There are several other important topics, not covered in any of these sections, that can be studied at LHCb and its upgrade, and these are discussed in Sect. 5. A summary is given in Sect. 6.

Current LHCb detector and performance

The LHCb detector [1] is a single-arm forward spectrometer covering the pseudorapidity range 2<η<5, designed for the study of particles containing b or c quarks. The detector includes a high precision tracking system consisting of a silicon-strip vertex detector surrounding the pp interaction region, a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about 4 Tm, and three stations of silicon-strip detectors and straw drift tubes placed downstream. The combined tracking system has a momentum resolution Δp/p that varies from 0.4 % at 5 GeV/c to 0.6 % at 100 GeV/c, and an impact parameter resolution of 20 μm for tracks with high transverse momentum. Charged hadrons are identified using two ring-imaging Cherenkov detectors. Photon, electron and hadron candidates are identified by a calorimeter system consisting of scintillating-pad and preshower detectors, an electromagnetic calorimeter and a hadronic calorimeter. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers. The trigger consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage which applies a full event reconstruction.

During 2011, the LHCb experiment collected 1.0 fb−1 of integrated luminosity during the LHC pp run at a centre-of-mass energy \(\sqrt {s} = 7 ~\mathrm{TeV} \). The majority of the data was recorded at an instantaneous luminosity of \(\mathcal{L}_{\rm inst} = 3.5 \times10^{32}~\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\), nearly a factor of two above the LHCb design value, and with a pile-up rate (average number of visible interactions per crossing) of μ∼1.5 (four times the nominal value, but below the rates of up to μ∼2.5 seen in 2010). A luminosity levelling procedure, where the beams are displaced at the LHCb interaction region, allows LHCb to maintain an approximately constant luminosity throughout each LHC fill. This procedure permitted reliable operation of the experiment and a stable trigger configuration throughout 2011. The hardware stage of the trigger produced output at around 800 kHz, close to the nominal 1 MHz, while the output of the software stage was around 3 kHz, above the nominal 2 kHz, divided roughly equally between channels with muons, b decays to hadrons and charm decays. During data taking, the magnet polarity was flipped at a frequency of about one cycle per month in order to collect equal sized data samples of both polarities for periods of stable running conditions. Thanks to the excellent performance of the LHCb detector, the overall data taking efficiency exceeded 90 %.

Assumptions for LHCb upgrade performance

In the upgrade era, several important improvements compared to the current detector performance can be expected, as detailed in the framework TDR. However, to be conservative, the sensitivity studies reported in this paper all assume detector performance as achieved during 2011 data taking. The exception is in the trigger efficiency, where channels selected at hardware level by hadron, photon or electron triggers are expected to have their efficiencies double (channels selected by muon triggers are expected to have marginal gains, that have not been included in the extrapolations). Several other assumptions are made:

  • LHC collisions will be at \(\sqrt{s} = 14 ~\mathrm{TeV} \), with heavy flavour production cross-sections scaling linearly with \(\sqrt{s}\);

  • the instantaneous luminosityFootnote 2 in LHCb will be \(\mathcal{L}_{\rm inst} = 10^{33}~\mathrm{cm} ^{-2}\,\mathrm{s}^{-1}\): this will be achieved with 25 ns bunch crossings (compared to 50 ns in 2011) and μ=2;

  • LHCb will change the polarity of its dipole magnet with similar frequency as in 2011/12 data taking, to approximately equalise the amount of data taken with each polarity for better control of certain potential systematic biases;

  • the integrated luminosity will be \(\mathcal{L}_{\rm int} = 5~\mbox{fb}^{-1}\) per year, and the experiment will run for 10 years to give a total sample of 50 fb−1.

Rare decays

Introduction

The term rare decay is used within this document to refer loosely to two classes of decays:

  • flavour-changing neutral current (FCNC) processes that are mediated by electroweak box and penguin type diagrams in the SM;

  • more exotic decays, including searches for lepton flavour or number violating decays of B or D mesons and for light scalar particles.

The first broad class of decays includes the rare radiative process \(B ^{0}_{ s } \rightarrow \phi \gamma \) and rare leptonic and semileptonic decays \(B^{0}_{(s)} \rightarrow \mu ^{+} \mu ^{-} \) and B 0K ∗0 μ + μ . These were listed as priorities for the first phase of the LHCb experiment in the roadmap document [5]. In many well motivated new physics models, new particles at the TeV scale can enter in diagrams that compete with the SM processes, leading to modifications of branching fractions or angular distributions of the daughter particles in these decays.

For the second class of decay, there is either no SM contribution or the SM contribution is vanishingly small and any signal would indicate evidence for physics beyond the SM. Grouped in this class of decay are searches for GeV scale new particles that might be directly produced in B or D meson decays. This includes searches for light scalar particles and for B meson decays to pairs of same-charge leptons that can arise, for example, in models containing Majorana neutrinos [2729].

The focus of this section is on rare decays involving leptons or photons in the final states. There are also several interesting rare decays involving hadronic final states that can be pursued at LHCb, such as B +K π + π +, B +K + K + π  [30, 31], \(B ^{0}_{ s } \rightarrow \phi \pi ^{0} \) and \(B ^{0}_{ s } \rightarrow \phi\rho^{0}\) [32]; however, these are not discussed in this document.

Section 2.2 introduces the theoretical framework (the operator product expansion) that is used when discussing rare electroweak penguin processes. The observables and experimental constraints coming from rare semileptonic, radiative and leptonic B decays are then discussed in Sects. 2.32.4 and 2.5 respectively. The implications of these experimental constraints for NP contributions are discussed in Sects. 2.6 and 2.7. Possibilities with rare charm decays are then discussed in Sect. 2.8, and the potential of LHCb to search for rare kaon decays, lepton number and flavour violating decays, and for new light scalar particles is summarised in Sects. 2.92.10 and 2.11 respectively.

Model-independent analysis of new physics contributions to leptonic, semileptonic and radiative decays

Contributions from physics beyond the SM to the observables in rare radiative, semileptonic and leptonic B decays can be described by the modification of Wilson coefficients \(C^{(\prime)}_{i}\) of local operators in an effective Hamiltonian of the form

$$ {\mathcal{H}}_{\text{eff}} = - \frac{4G_F}{\sqrt{2}} V_{tb}V_{tq}^*\frac{e^2}{16\pi^2} \sum _i\bigl(C_i O_i + C^\prime_i O^\prime_i\bigr) + \text{h.c.}, $$
(1)

where q=d,s, and where the primed operators indicate right-handed couplings. This framework is known as the operator product expansion, and is described in more detail in, e.g., Refs. [33, 34]. In many concrete models, the operators that are most sensitive to NP are a subset of

(2)

which are customarily denoted as magnetic (\(O_{7}^{(\prime)}\)), chromomagnetic (\(O_{8}^{(\prime)}\)), semileptonic (\(O_{9}^{(\prime)}\) and \(O_{10}^{(\prime)}\)), pseudoscalar (\(O_{P}^{(\prime)}\)) and scalar (\(O_{S}^{(\prime)}\)) operators.Footnote 3 While the radiative b decays are sensitive only to the magnetic and chromomagnetic operators, semileptonic bqℓ + decays are, in principle, sensitive to all these operators.Footnote 4

In the SM, models with minimal flavour violation (MFV) [35, 36] and models with a flavour symmetry relating the first two generations [37], the Wilson coefficients appearing in Eq. (1) are equal for q=d or s and the ratio of amplitudes for bd relative to bs transitions is suppressed by |V td /V ts |. Due to this suppression, at the current level of experimental precision, constraints on decays with a bd transition are much weaker than those on decays with a bs transition for constraining \(C_{i}^{(\prime)}\). In the future, precise measurements of bd transitions will allow powerful tests to be made of this universality which could be violated by NP.

The dependence on the Wilson coefficients, and the set of operators that can contribute, is different for different rare B decays. In order to put the strongest constraints on the Wilson coefficients and to determine the room left for NP, it is therefore desirable to perform a combined analysis of all the available data on rare leptonic, semileptonic and radiative B decays. A number of such analyses have recently been carried out for subsets of the Wilson coefficients [3843].

The theoretically cleanest branching ratios probing the bs transition are the inclusive decays BX s γ and BX s + . In the former case, both the experimental measurement of the branching ratio and the SM expectation have uncertainties of about 7 % [44, 45]. In the latter case, semi-inclusive measurements at the B factories still have errors at the 30 % level [44]. At hadron colliders, the most promising modes to constrain NP are exclusive decays. In spite of the larger theory uncertainties on the branching fractions as compared to inclusive decays, the attainable experimental precision can lead to stringent constraints on the Wilson coefficients. Moreover, beyond simple branching fraction measurements, exclusive decays offer powerful probes of \(C_{7}^{(\prime)}\), \(C_{9}^{(\prime)}\) and \(C_{10}^{(\prime)}\) through angular and CP-violating observables. The exclusive decays most sensitive to NP in bs transitions are BK γ, \(B ^{0}_{ s } \rightarrow \mu^{+}\mu^{-}\), B + μ and BK μ + μ . These decays are discussed in more detail below.

Rare semileptonic B decays

The richest set of observables sensitive to NP are accessible through rare semileptonic decays of B mesons to a vector or pseudoscalar meson and a pair of leptons. In particular the angular distribution of BK μ + μ decays, discussed in Sect. 2.3.2, provides strong constraints on \(C_{7}^{(\prime)}\), \(C_{9}^{(\prime)}\) and \(C_{10}^{(\prime)}\).

Theoretical treatment of rare semileptonic BMℓ + decays

The theoretical treatment of exclusive rare semileptonic decays of the type BMℓ + is possible in two kinematic regimes for the meson M: large recoil (corresponding to low dilepton invariant mass squared, q 2) and small recoil (high q 2). Calculations are difficult outside these regimes, in particular in the q 2 region close to the narrow \(c \overline { c } \) resonances (the J/ψ and ψ(2S) states).

In the low q 2 region, these decays can be described by QCD-improved factorisation (QCDF) [46, 47] and the field theory formulation of soft-collinear effective theory (SCET) [48, 49]. The combined limit of a heavy b-quark and an energetic meson M, leads to the schematic form of the decay amplitude [50, 51]:

(3)

which is accurate to leading order in Λ QCD/m b and to all orders in α S . It factorises the calculation into process-independent non-perturbative quantities, BM form factors, ξ, and light cone distribution amplitudes (LCDAs), ϕ B(M), of the heavy (light) mesons, and perturbatively calculable quantities, C and T which are known to \({ \mathcal{O} } ( \alpha _{S} ^{1})\) [50, 51]. Further, in the case that M is a vector V (pseudoscalar P), the seven (three) a priori independent BV (BP) form factors reduce to two (one) universal soft form factors ξ ⊥,∥ (ξ P ) in QCDF/SCET [52]. The factorisation formula Eq. (3) applies well in the dilepton mass range, 1<q 2<6 GeV2.Footnote 5

For BK + , the three K spin amplitudes, corresponding to longitudinal and transverse polarisations of the K , are linear in the soft form factors ξ ⊥,∥,

(4)

at leading order in Λ QCD/m b and α S . The \(C_{\perp,\parallel}^{L,R}\) are combinations of the Wilson coefficients \(\mathcal{C}_{7,9,10}\) and the L and R indices refer to the chirality of the leptonic current. Symmetry breaking corrections to these relationships of order α S are known [50, 51]. This simplification of the amplitudes as linear combinations of \(C_{\perp,\parallel}^{L,R}\) and form factors, makes it possible to design a set of optimised observables in which any soft form factor dependence cancels out for all low dilepton masses q 2 at leading order in α S and Λ QCD/m b [5355], as discussed below in Sect. 2.3.2.

Within the QCDF/SCET approach, a general, quantitative method to estimate the important Λ QCD/m b corrections to the heavy quark limit is missing. In semileptonic decays, a simple dimensional estimate of 10 % is often used, largely from matching of the soft form factors to the full-QCD form factors (see also Ref. [56]).

The high q 2 (low hadronic recoil) region, corresponds to dilepton invariant masses above the two narrow resonances of J/ψ and ψ(2S), with q 2≳(14–15) GeV2. In this region, broad \(c\overline{c}\)-resonances are treated using a local operator product expansion [57, 58]. The operator product expansion (OPE) predicts small sub-leading corrections which are suppressed by either (Λ QCD/m b )2 [58] or α S Λ QCD/m b [57] (depending on whether full QCD or subsequent matching on heavy quark effective theory in combination with form factor symmetries [59] is adopted). The sub-leading corrections to the amplitude have been estimated to be below 2 % [58] and those due to form factor relations are suppressed numerically by \(C_{7} / C_{9} \sim{ \mathcal{O} } (0.1)\). Moreover, duality violating effects have been estimated within a model of resonances and found to be at the level of 2 % of the rate, if sufficiently large bins in q 2 are chosen [58]. Consequently, like the low q 2 region, this region is theoretically well under control.

At high q 2 the heavy-to-light form factors are known only as extrapolations from light cone sum rules (LCSR) calculations at low q 2. Results based on lattice calculations are being derived [60], and may play an important role in the near future in reducing the form factor uncertainties.

Angular distribution of B 0K ∗0 μ + μ  and \(B ^{0}_{ s } \rightarrow \phi \mu ^{+} \mu ^{-} \) decays

The physics opportunities of BVℓ + (=e,μ, V=K ,ϕ,ρ) can be maximised through measurements of the angular distribution of the decay. Using the decay BK (→) + , with K on the mass shell, as an example, the angular distribution has the differential form [61, 62]

(5)

with respect to q 2 and three decay angles θ l , θ K , and ϕ. For the B 0 (\(\overline{ B }{} ^{0} \)), θ l is the angle between the μ + (μ ) and the opposite of the B 0 (\(\overline{ B }{} ^{0} \)) direction in the dimuon rest frame, θ K is the angle between the kaon and the direction opposite to the B meson in the K ∗0 rest frame, and ϕ is the angle between the μ + μ and K + π decay planes in the B rest frame. There are twelve angular terms appearing in the distribution and it is a long-term experimental goal to measure the coefficient functions J i (q 2) associated with these twelve terms, from which all other BK (∗) + observables can be derived.

In the SM, with massless leptons, the J i depend on bilinear products of six complex K spin amplitudes \(A_{\bot,\|,0}^{L,R}\),Footnote 6 such as

$$ J_{1s} = \frac{3}{4} \bigl[\bigl|{ A_\perp^L } \bigr|^2 + \bigl|{ A_\parallel^L } \bigr|^2 + \bigl|{ A_\perp ^R } \bigr|^2 + \bigl|{ A_\parallel^R } \bigr|^2 \bigr]. $$
(6)

The expressions for the eleven other J i terms are given for example in Refs. [54, 63]. Depending on the number of operators that are taken into account in the analysis, it is possible to relate some of the J i terms. The full derivation of these symmetries can be found in Ref. [54].

When combining B and \(\overline{ B }{} \) decays, it is possible to form both CP-averaged and CP-asymmetric quantities: \(S_{i} =(J_{i} + \bar {J_{i}})/[d(\varGamma+ \bar{\varGamma})/d q^{2} ]\) and \(A_{i} = (J_{i} - \bar {J_{i}})/[d(\varGamma+ \bar{\varGamma})/d q^{2} ]\), from the J i [53, 54, 6266]. The terms J 5,6,8,9 in the angular distribution are CP-odd and, consequently, the associated CP-asymmetry, A 5,6,8,9 can be extracted from an untagged analysis (making it possible for example to measure A 5,6,8,9 in \(B ^{0}_{ s } \rightarrow \phi \mu ^{+} \mu ^{-} \) decays). Moreover, the terms J 7,8,9 are T-odd and avoid the usual suppression of the corresponding CP-asymmetries by small strong phases [64]. The decay B 0K ∗0 μ + μ , where the K ∗0 decays to K + π , is self-tagging (the flavour of the initial B meson is determined from the decay products) and it is therefore possible to measure both the A i and S i for the twelve angular terms.

In addition, a measurement of the T-odd CP asymmetries, A 7, A 8 and A 9, which are zero in the SM and are not suppressed by small strong phases in the presence of NP, would be useful to constrain non-standard CP violation. This is particularly true since the direct CP asymmetry in the inclusive BX s γ decay is plagued by sizeable long-distance contributions and is therefore not very useful as a constraint on NP [67].

Strategies for analysis of B 0K ∗0 + decays

In 1.0 fb−1 of integrated luminosity, LHCb has collected the world’s largest samples of B 0K ∗0 μ + μ (with K ∗0K + π ) and \(B ^{0}_{ s } \rightarrow \phi \mu ^{+} \mu ^{-} \) decays, with around 900 and 80 signal candidates respectively reported in preliminary analyses [68, 69]. These candidates are however sub-divided into six q 2 bins, following the binning scheme used in previous experiments [70]. With the present statistics, the most populated q 2 bin contains ∼300B 0K ∗0 μ + μ candidates which is not sufficient to perform a full angular analysis. The analyses are instead simplified by integrating over two of the three angles or by applying a folding technique to the ϕ angle, ϕϕ+π for ϕ<0, to cancel terms in the angular distribution.

In the case of massless leptons, one finds:

$$ \frac{d\varGamma^\prime}{d\phi} = \frac{\varGamma^\prime}{2\pi} ( 1 + S_3 \cos2\phi+ A_{9} \sin2\phi ), $$
(7)