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Rare decays of b hadrons
|Patrick Koppenburg et al. (2016), Scholarpedia, 11(6):32643.
|revision #190467 [link to/cite this article]
Physics studies fundamental interactions and their effects. At the most basic level, particle physics aims to describe the fundamental blocks of matter and their interactions. A century of research has led to the Standard Model of Particle Physics. It relies on firm theoretical grounds unifying quantum mechanics, special relativity and field theory, and is successful at describing all phenomena measured in particle interactions, whether at low or very high energies. Yet, it has an empirical character with many parameters that need to be determined by experiment, and it is incomplete as it does not account for gravity, does not explain the baryon asymmetry in the Universe and does not provide a candidate for dark matter. The Standard Model is therefore believed to be an approximation of a more complete theory that is currently unknown (just like Newton's laws are an approximation of General Relativity). The primary goal of research in particle physics is to find this more complete theory. In the following "New Physics" is used as a catch-all for any contribution, whether particle or coupling, not included in the Standard Model.
Rare decays of hadrons containing a heavy "beauty" (also called "bottom") quark, denoted b hadrons, provide a powerful way of exploring yet unknown physics. Small contributions from virtual new particles that are too heavy to be produced at colliders may lead to measurable deviations from the expected properties in the Standard Model. See the box for an example of a virtual particle.
Virtual Particles
Fundamental particle interactions are mediated by force carriers: The photon for the electromagnetic interaction, the gluon for the strong interaction and the W± and Z bosons for the weak interaction. In the nuclear beta decay a neutron decays to a proton, an electron and a neutrino (Figure 1). This weak interaction is mediated by a virtual W− boson, sometimes denoted W−*. Here virtual means that the process violates energy and momentum conservation for a very short time, as allowed by Heisenberg's Uncertainty Principle. The mass of the W− boson is about 80GeV/c2, while the neutron-proton mass difference is considerably less: 1.3 MeV/c2. It is said that the W− boson is "off-shell".
The study of rare decays is an active field within flavour physics, the field of research studying transitions of quarks or leptons from one species (or "flavour") to another. This article focuses on rare decays of hadrons containing b quarks. The most prevalently produced b hadrons are the B0 meson composed of a $\overline b$ anti-quark and a d quark, the B+ ($\overline b$ u) and Bs0 ($\overline b$ s) mesons, as well as the Λb0 (bud) baryon. Their masses are in the range 5 to 6 GeV/c2, which is about six times that of the proton, but well below the mass of the W boson of 80 GeV/c2. The corresponding antiparticles $\overline B^0$, B−, $\overline B^0_s$ and $\overline \Lambda_b^0$ are obtained by replacing all quarks by anti-quarks and vice-versa. The study of CP violation involves investigation of differences in the behaviour of particles and antiparticles, and is the subject of a dedicated review. The inclusion of charge conjugate processes is implied throughout this document.
Hadrons with b quarks decay most of the time via a b → cW−* transition, where the asterisk indicates the W boson is virtual. The transitions b → uW−* also occur, but are less likely. These two transitions are called "tree decays" as the process involves a single mediator, the W− boson. An example of a tree d → uW−* transition is shown in Figure 1.
This article describes mostly transitions involving more complicated processes. The quark transitions b → d and b → s do not happen at tree level in the Standard Model as the Z boson does not couple to quarks of different flavour.
Processes like the rare decay Bs0 → μ+μ− proceed via loops as shown in Figure 2 (sometimes referred to as penguins, a word coined by John Ellis - see Izlar, Kelly (2013)). Such processes are rare as the probability of a transition rapidly decreases with the number of electroweak vertices: two in the case of a tree decay, three or four for a loop. Also, the heavier the virtual particles involved, the more suppressed the decay. In the following, decays with probabilities in the range 10−4 to 10−10 are discussed. Some of the most interesting decays are described in the Section Key players. They all have in common the following features:
Suppressed decay amplitudes, as predicted by the Standard Model, which may potentially be of the same size as New Physics amplitudes.
Sufficiently precise Standard Model predictions for their decay rate, or any other observable of interest.
Experimental precision which potentially allows disentangling the Standard Model contribution from other contributions.
A historical example, the decay KL0 → μ+μ−, is described in this box.
A Historical Example: KL0 → μ+μ−
The KL0 → μ+μ− decay is forbidden at tree level and had an important role in opening the field of rare decays in the 1960s. Its unexpected non-observation allowed the prediction of the then unknown charm quark by Glashow, Iliopoulos and Maiani (GIM mechanism) in 1970. The idea of the GIM mechanism is that this decay only occurs via loops, one involving the u quark and the other the c quark (Figure 3). The amplitudes of the two loops are of opposite sign, causing complete cancellation in the limit mc − mu = 0. The non-observation of this decay could be explained by adding a new particle to the theory, the c quark, which was eventually discovered in 1974. This is an example of an observation of New Physics mediated by a new virtual particle. The KL0 → μ+μ− branching fraction is now measured to be (6.84 ± 0.11)×10−9. Nowadays there is a great deal of interest in the Bs0-counterpart of this decay: Bs0 → μ+μ−, discussed in Section The decays Bs0 → μ+μ− and B0 → μ+μ−.
A special category of rare decays is those forbidden in the Standard Model, like lepton- or baryon-number violating decays. In their case the Standard Model prediction is effectively zero, but other models may predict non-zero rates. Any observation would be a sign of New Physics.
Given the impressive success of the Standard Model, New Physics amplitudes are known to be small. Therefore, any search for potentially observable deviations from Standard Model predictions will be facilitated if the Standard Model amplitudes are also suppressed, which is the case in rare decays. Studies of such decay modes require large data samples to produce enough of the relevant particles. This is referred to as the intensity frontier, as opposed to the energy frontier aiming at producing and studying heavy particles on-shell. The main experiments are briefly described in Experiments.
Key players
The description of the process of the formation of hadrons out of quarks and gluons, called hadronisation, is difficult and leads to large theoretical uncertainties. Theoretically favoured are thus decays to purely leptonic final states, such as the decay Bs0 → μ+μ− (Section The decays Bs0 → μ+μ− and B0 → μ+μ−). There is also interest in the charged counterparts of these decays, notably B+ → l+ν, where l+ is any lepton, e+, μ+, τ+ (Section Other leptonic decays). They are generated by a charged W+ current, but have interesting theoretical connections to decays that are induced by loops.
Radiative b → sγ and electroweak penguin b → sl+l−, and $b\to s \nu \overline \nu$ decays are also of great interest. These are quark-level transitions, which cannot be measured directly as the quarks form immediately hadrons. In experiments exclusive decays are detected, and the inclusive decay is the sum of all contributions. For instance the decay b → sγ was first observed by its exclusive contribution B → K*γ (Section The decay b → sγ).
Exclusive decays are experimentally favoured, but come with larger theoretical uncertainties. The decay B0 → K*0l+l− is a well-known example. While the decay rate is hard to compute precisely, observables describing angular distributions of the decay products can be more precisely predicted (Section The decays b → sl+l−).
Among forbidden decays, lepton flavour violating decays of b and c hadrons, like B(s)0 → e±μ∓ or B+ → K+e±μ∓, or of leptons, like μ+ → e+γ, τ+ → μ+μ+μ− or τ+ → μ+γ are actively being searched for. Rare charm hadron decays are also being studied, but the experimental sensitivity is presently not sufficient to reach the very low rates predicted in the Standard Model. Finally, research in rare kaon decays is ongoing, though mainly at different experiments than those studying rare charm or beauty hadron decays. These channels are not further discussed in this article.
This section describes briefly the theoretical framework that is commonly used to study rare decays. Its main goal is to define some vocabulary which is commonly used in publications on rare decays. It may be skipped by readers mostly interested in experimental results.
The common theoretical approach to rare decays is model independent. In flavour physics and in particular in rare decays studies, the underlying physics is parametrised in terms of an effective Hamiltonian describing the transition amplitude of an initial state I to a final state F following Fermi's Golden Rule (Dirac, Paul A.M. (1927), Fermi, Enrico (1949)). The partial decay width is written as
$$\Gamma(I \to F) = \frac{2\pi}{\hbar}\left|\left<F|{\cal H}_\text{eff}|I\right>\right|^2\times\text{phase-space}.$$
Experimentally, the branching fraction ${\cal B}$ is measured rather than the decay width. They are related by
$${{\cal B}(I \to F) = \frac{\Gamma(I \to F)}{\Gamma(I,\text{total})}},\quad \Gamma(I,\text{total}) = \frac{1}{\tau_I},$$
where τI is the lifetime of particle I (and natural units with c = ℏ = 1 are used).
The Standard Model prediction for any particular transition can be inferred from a calculation of the effective Hamiltonian derived from the Standard Model Lagrangian. This Hamiltonian is parametrised in terms of a sum of operators ${\cal O}_i$ and Wilson coefficients Ci
$${\cal H}_\text{eff} = -\frac{G_F}{\sqrt{2}}\sum\limits_i V_\text{CKM}C_i{\cal O}_i,$$
where VCKM stands for some product of Cabibbo-Kobayashi-Maskawa matrix elements that describe the probability of given transitions between different quark flavours. The operators encompass the information about the Lorentz structure and the Wilson coefficients encode the effects of higher energy scales. In the case of the Standard Model these are the effects of the W, Z bosons and top quarks, which are effectively removed from the theory and incorporated in the coefficients.
Any I → F decay can be described by this effective Hamiltonian, usually with many terms $<F|{\cal O}_i|I>=0$. Thus studying a set of decays will give various constraints on the effective Hamiltonian, permitting global fits to Wilson coefficients. This is briefly discussed in Section Wilson coefficient fits. This procedure does not simplify the computation of the amplitudes, as the matrix elements $\left<F|{\cal O}_i|I\right>$ contain the most difficult parts of the calculation. It provides however a common language that is not dependent on the considered New Physics model.
In particular, calculations of decay rates of exclusive decays with hadrons in the final state (B0 → K*0μ+μ− for example) are difficult and part of our lack of knowledge needs to be parametrised in heuristic quantities that describe the hadronisation, like form-factors and decay constants. They can be calculated in lattice QCD and, in many cases, can also be determined experimentally. Their discussion is beyond the scope of this document.
The operators ${\cal O}_{1,2}$ describe the V − A structure of weak decays and first-order corrections. For example, the W boson having been absorbed into the C1 and C2 coefficients, the nuclear beta decay $n \to p e \overline \nu$ is represented by a four-fermion operator as shown in Figure 4. This is how Enrico Fermi first described the process in 1934 (Fermi, Enrico (1934)). The operators ${\cal O}_{3-6,8}$ describe loops involving gluons. They are not of interest for this article.
Of most interest in rare decays are the suppressed operators ${\cal O}_{7}$, ${\cal O}_{9}$ and ${\cal O}_{10}$. The operator ${\cal O}_{7}$ dominates the radiative decay b → sγ giving a decay width
$$\Gamma(b \to s \gamma) = \frac{G_F^2\alpha_{\rm EM}m_b^5}{32\pi^4}|V_{ts}^\ast V_{tb}|^2 |C_{7}|^2 + \text{corrections},$$
where $\alpha_{\rm EM}$ is the electromagnetic constant, mb the b quark mass, and Vij are parameters of the CKM matrix. A measurement of the b → sγ branching fraction thus provides a direct constraint on C7.
The operators ${\cal O}_{9}$ and ${\cal O}_{10}$ dominate b → qll transitions, with ${\cal O}_{9}$ corresponding to a vector and ${\cal O}_{10}$ to an axial current. Finally the decays B → l+l− are, in the Standard Model, dominated by operator ${\cal O}_{10}$, with a decay rate which can be written as
$${\Gamma}(B \to l^+l^-) = \frac{G_F^2M_W^2m_B^3f_B^2}{8\pi^5}|V_{tb}^\ast V_{tq}|^2\frac{4m_\ell^2}{m_B^2}\sqrt{1-\frac{4m_\ell^2}{m_B^2}} |C_{10}|^2 + \text{corrections},$$
for B = B0, Bs0 (and q = d, s) with fB the B decay constant and Vij CKM matrix elements. The Bs0 → μ+μ− branching fraction thus provides a constraint on C10. Other operators, labelled ${\cal O}_\text{P}$ and ${\cal O}_\text{S}$, which are negligible in the Standard Model, could also contribute to this decay.
If the V − A structure of weak interactions is not assumed, new primed operators with flipped helicities appear, most notably ${\cal O}_7'$, and its Wilson coefficient C7′ which generate a right-handed photon in b → sγ decays.
For a comprehensive review of the effective Hamiltonian used to study rare decays, see Buchalla, Gerhard; Buras, Andrzej J and Lautenbacher , Markus E (1996) and Buras, Andrzej J and Fleischer, R (1998). A more pedagogical introduction can be found in Chapter 20 of Branco, G C; Lavoura, L and Silva, J P (1999). Standard Model expectations of Wilson coefficients and operators have been calculated at next-to-leading order or better.
There exist many theories beyond the Standard Model providing predictions for Wilson coefficients. Often these values depend on unknown parameters of the theory, as masses of yet unseen new particles. This is particularly the case for supersymmetry, a well-motivated extension of the Standard Model.
There are essentially two families of experiments studying b hadrons:
- B factories
are experiments based at e+e− colliders operating most of the time at a collision energy near 10.6 GeV, corresponding to the mass of the $\Upsilon$(4S) resonance, the lightest meson decaying to two B mesons. ARGUS (at DESY, Germany), CLEO at Cornell (Ithaka, USA), BaBar at SLAC (Stanford, USA), Belle and its successor Belle II at KEK (Tsukuba, Japan) are notable examples of such experiments.
- Hadron collider experiments
operate at a pp or $p \overline p$ collider with centre-of-mass energies of several TeV. CDF and D0 were located at Fermilab's Tevatron (Batavia, USA). ATLAS, CMS and LHCb presently operate at CERN's LHC (Geneva, Switzerland).
Hadron colliders have the advantage of much larger production rates: the production cross-section of b quarks is a factor 500000 larger at the LHC than at a B factory. The advantage of the B factories is cleanliness. Collision events with a produced $\Upsilon$(4S) resonance are easy to identify, allowing for high efficiencies and low background levels. In such events only two B mesons are produced, making the reconstruction of the full collision event possible. In a typical LHC collision only one in hundred collisions produce a b quark pair and the two b hadrons are surrounded by hundreds of other particles. Efficient background fighting techniques are thus essential but have a cost in terms of efficiencies. Excellent background rejection is achieved precise vertexing and exploiting the b-hadron flight distance. The physics programme is also somewhat different: B factories have only access to B0 and B+ mesons (and Bs0 mesons when operating at the $\Upsilon$(5S) resonance), while hadronic collisions produce all b hadrons, including the Bs0 meson, the Bc+ meson (composed of a c quark and an $\overline b$ anti-quark), as well as the Λb0 and $\Xi$b baryons.
Short history of b-quark physics
After the discovery of the b quark at Fermilab through the observation of mesons formed by a b and an $\overline b$ anti-quark in 1977 and of the B meson at Cornell , searches for rare decays of b hadrons rapidly took pace. The first limit on the decay B0 → μ+μ− was set by the CLEO collaboration in 1985, the start of a long quest during which the sensitivity was improved by six orders of magnitude, as illustrated in (See Section The decays Bs0 → μ+μ− and B0 → μ+μ−).
The CLEO and ARGUS experiments were located at e+e− colliders operating at the $\Upsilon$(4S) resonance. The same concept was employed and improved by the BaBar and Belle experiments in the first decade of the 21st century. If Cornell was initially able to produce few tens of $B\bar{B}$ pairs per day, the PEP-II and KEKB accelerators at SLAC and KEK achieved a daily rate of one million $B\bar{B}$ pairs. In the meantime, experiments at CERN's LEP e+e− collider and at Fermilab's Tevatron proton-anti-proton collider used higher energy collisions to produce and study all b-hadron species. All the above-mentioned experiments have terminated their programme but most still exploit their data set to produce new results. Belle and the associated accelerator complex has been undergoing a major upgrade and has recently come back as the Belle II experiment.
Until Belle II reaches its design luminosity, the leadership in b physics is taken by the LHCb experiment. Important contributions also come from the ATLAS and CMS experiments. These three experiments exploit LHC data collected in proton-proton collisions at centre-of-mass energies of 7 (2010–11), 8 (2012) and 13 TeV (2015–18).
ATLAS and CMS are detectors optimised for high-energy processes, such as the discovery of the Higgs boson. They also perform b-physics research, most effectively in decays of b hadrons to pairs of muons. This distinct signature allows for efficient selection of these decays during the online filtering phase where a large reduction of the recorded collision rate is required, which is difficult to achieve for decays to electrons or hadrons.
The LHCb experiment on the contrary is optimised for the physics of hadrons containing b and c quarks. It is a single-arm forward detector designed to exploit the relatively large $b\bar{b}$ production in LHC proton-proton collisions in the forward direction. It includes a tracking system surrounding a dipole magnet whose polarity can be reversed, silicon sensors coming as close as 8 mm to the proton beam and a particle identification system based on Cherenkov radiation. The high-resolution silicon system exploits the typical b-hadron flight distances of a few millimetres before their decay to select them. This sets requirements on the number of pp collisions per bunch crossing, defining an upper limit to the total collision rate at which the experiment can operate. Consequently, the luminosity is decreased compared to ATLAS and CMS.
Main experimental results
This section presents the main recent experimental results and their interpretation. It starts with a more historical section on the decay b → sγ which had (and still has) an important role in the development of the field.
The decay b → sγ
In the Standard Model the decay b → sγ occurs dominantly via a loop involving the top quark and the W boson (Figure 6). It has played a very important role in flavour physics from the 1980s. At the time it was the dependency of the branching fraction on the then unknown top quark mass that was the driving force behind the theoretical calculations and the experimental searches. When $B^0-\overline B^0$ mixing was (at the time surprisingly) observed in 1987, it became clear that the top quark was very heavy. The top quark was eventually discovered at the Tevatron in 1995 and its mass measured, which determined the Standard Model decay rate of b → sγ to be a few 10−4.
The first observation of the b → sγ decay actually preceded the top quark observation. In 1993 the CLEO collaboration reported a signal of the exclusive decay B → K*0γ with a branching fraction of (4.5 ± 1.5 ± 0.9)×10−5 (Ammar, R et al. (1993), Figure 7), where the first uncertainty is statistical and the second systematic. This opened the quest for the inclusive decay b → sγ, i.e. the sum of all exclusive contributions. The branching fraction of this decay is more precisely calculable than its individual exclusive components, like B0 → K*0γ, allowing for more precise comparisons of experimental and theoretical results.
Many experiments located at e+e− colliders have performed measurements using different methods. The total rate of any B meson to a photon plus anything (where the photon is not caused by an electromagnetic decay, e.g. π0 → γγ or η → γγ) can be measured by a sum of exclusive decay modes. A fully inclusive approach is also possible but more challenging. Only the photon and properties of the rest of the collision event are used to separate signal and backgrounds, mostly originating from non-B decays. At the B factories they are determined from data taken at centre-of-mass energies below the $\Upsilon$(4S) resonance mass, decays to π0 and η which are vetoed and modelled from data, and mis-identified photons which are modelled from simulation. This method has the disadvantage of larger backgrounds, but has the advantage not to rely on any modelling of the composition of the hadronic state. It thus comes with smaller theoretical uncertainties. A typical result for the measured photon energy spectrum in inclusive b → sγ decays is shown in Figure 8. The integral of the spectrum gives the decay rate. The width is related to the momentum of the b quark in the B meson, which can be seen as a two-body system of a b and a light quark, similar to a hydrogen atom.
The world-average measured branching fraction for a photon with an energy above 1.6 GeV in the B-meson rest-frame is (3.49 ± 0.19)×10−4, which can be compared with the latest theoretical calculation of (3.36 ± 0.23)×10−4. The very good agreement of these two values sets very strong constraints on New Physics models, in particular supersymmetry. Although the total branching fraction seems to indicate no discrepancy with the Standard Model prediction, small contributions from New Physics may still occur. The b → sγ decay rate is essentially a measurement of the Wilson coefficient C7 (see Section Theory).
In the Standard Model, the left-handed chirality structure of the weak interactions makes the photon emitted in b → sγ decays mainly left-handed. It is interesting to also probe right-handed contributions (sensitive to the Wilson coefficient C7′), which requires determination of the polarisation of the photon. This is challenging as the helicity (or chirality) of the photon cannot be measured directly in the detector. Several methods have been proposed, none of which provides a strong constraint so far. The first and so far only measurement of a non-zero photon polarisation uses the decay B → K0π−π+γ , but the interpretation in terms of the photon chirality is still unclear. The most stringent constraints come from global fits to Wilson coefficients (see Section Wilson coefficient fits).
A promising way of searching for right-handed contributions to $b\to s\gamma$ decays is the study of time-dependent $CP$ violation in transitions to flavour eigenstates. For instance in the decay $B^0\to K^{*0}\gamma$ with $K^{*0}\to K_{\rm S}^0\pi^0$, the amplitude of $B^0\to K^{*0}\gamma$ may interfere with that of the same decay following $B^0$ mixing, $\overline{B}^0\to \overline{K}^{*0}\gamma$. But this is only the case if the photons have the same helicities, and so the left-handed component of the $B^0$ decay will interfere with the right-handed component of the $\overline{B}^0$ decay. This process has been studied at the BaBar and Belle experiments, and the $B^0_s$ counterpart decay $B_s^0\to \phi\gamma$ at LHCb. No significant $CP$ asymmetry is observed to date.
The decays Bs0 → μ+μ− and B0 → μ+μ−
The rare decay Bs0 → μ+μ− proceeds in the Standard Model by a box-type diagram involving the W and Z bosons and the t quark (Figure 9). The most recent Standard-Model determination of its branching fraction is (3.57 ± 0.17)×10−9, where the uncertainty is dominated about equally by CKM matrix elements and the Bs0 decay constant. In Standard Model extensions, the branching fraction of Bs0 → μ+μ− could be enhanced, in particular in models containing additional Higgs bosons (Figure 9, right). The decay Bs0 → μ+μ− and the even more suppressed decay B0 → μ+μ− have been searched for over three decades, with most recent results from the Tevatron and the LHC ().
The first observation was reported in 2014 jointly by the CMS and LHCb collaborations. The CMS and LHCb LHC Run 1 data sets, which had been already published separately, were combined in a joint fit to the data of both experiments. The result was published in Nature (Khachatryan, Vardan et al. (2015)), which is unusual in high energy physics. The fit to the invariant mass distribution of the two-muon system is shown in Figure 10. The result of the combination is an observation of the Bs0 → μ+μ− decay and a small excess over the background for B0 → μ+μ−. The first measured branching fraction ℬ(Bs0 → μ+μ−)=($2.8{{\:}^{+{\:}0.7}_{-{\:}0.6}}) \times 10^{−9}$ came out consistent with the Standard Model prediction.
For the even more suppressed B0 decay, the result is ℬ(B0 → μ+μ−)=$(3.9{{\:}^{+{\:}1.6}_{-{\:}1.4}}) \times 10^{−10}$, which is slightly, but not significantly, larger than the Standard Model prediction of (1.06 ± 0.09)×10−10. LHCb, ATLAS and CMS updated their Run 1 result, adding data obtained in 2015 and 2016. The obtained $B^0_s\to\mu^+\mu^-$ branching fractions and $B^0\to\mu^+\mu^-$ limits are all consistent.
Other leptonic decays
Just as Bs0 → μ+μ− and B0 → μ+μ− are theoretically clean decays, so are their counterparts with neutrinos. The challenge is on the experimental side. The decay $B^0 \to \nu \overline \nu$ is traditionally labelled as "B0 → invisible" as there is no way to experimentally determine the number of neutrinos (or if there were any at all). In the Standard Model the branching fraction is vanishing as it is helicity-suppressed by a factor (mν/mB0)3. Helicity suppression occurs because of the B0 meson is spinless, so the two spin-1/2 neutrinos must have opposite spins. For massless neutrinos this would be impossible as neutrinos are always left-handed and antineutrinos right-handed. Only the minute mass of neutrinos (rarely) allows opposite-spin neutrinos to be emitted.
Searches have been performed by the B factory experiments using the full reconstruction technique (also referred to as "on the recoil"). One B meson from the $B \overline B$ pair is fully reconstructed and the other is required to leave no trace in the detector. The branching fraction is limited to be less than 2.4 × 10−5 at 90% confidence level, by the BaBar experiment.
Decays to one charged lepton, B+ → l+ν, are similarly helicity suppressed, with the strength of this suppression depending on the mass of the charged lepton. These are tree decays where the $\overline b$ and u quarks in the B+ meson annihilate, but rare because of the helicity suppression. Contributions with the W+ mediator replaced by a charged Higgs boson could enhance or suppress the branching fraction. These decays have all been searched for by the B factories using the full reconstruction technique describe above.
The B+ → τ+ντ decay, where the suppression is the weakest, has a predicted branching fraction of ${\cal B}^\text{SM}=(0.76{{\:}^{+{\:}0.08}_{-{\:}0.06}})\times 10^{-4}$ in the Standard Model and a measured rate of (1.14 ± 0.27)×10−4, which are in agreement. The more suppressed decays B+ → e+ν and B+ → μ+ν have not been observed yet, with limits on their branching fraction around 10−6.
The decay Bs0 → e+e− is out of reach in the foreseeable future. Due to the low electron mass, it is even more helicity-suppressed than Bs0 → μ+μ−. Moreover the study of final states involving electrons at hadron colliders is difficult due to the lower reconstruction efficiency and the poorer mass resolution (Figure 16). When passing through matter, electrons radiate a significant amount of energy by bremsstrahlung. This affects the reconstructed momentum and thus smears all derived quantities, like the invariant mass of the two-electron system.
The expected rate of $B_s^0\rightarrow\tau^+\tau^-$ is considerably larger, but the decay is experimentally challenging due to the difficult tau lepton reconstruction and associated large backgrounds. LHCb published a search for the decay $B_s^0\rightarrow\tau^+\tau^-$, but with a sensitivity still far from the Standard Model expectation. Belle II are likely to perform improved searches of such decays in the near future.
The decays b → sl+l−
The family of decays b → sl+l− (l = e, μ) is a laboratory of New Physics studies on its own. In the Standard Model these decays are induced by a loop diagram similar to that of b → sγ (but with a Z component) and a box diagram (Figure 11). The amplitudes corresponding to these diagrams interfere, which causes complex phenomenology.
The exclusive decay B0 → K*0l+l−, with K*0 →K+π−, provides a rich set of observables with different sensitivities to New Physics, and for which theoretical predictions are available. These observables are affected by varying levels of uncertainties related to the calculation of quantum chromodynamical effects. Yet, selected ratios of observables benefit from cancellations of uncertainties, thus providing a cleaner test of the Standard Model. The best known example is the lepton forward-backward asymmetry, explained in more details in the box.
This interesting picture is complicated by a dependence on q2, the dilepton mass squared (Figure 12). At very low q2, B0 → K*0l+l− behaves like B0 → K*γ, with a slightly off-shell photon decaying to two leptons. The physics is dominated by the ${\cal O}_7$ operator, as discussed in Section The decay b → sγ.
Above, there is an interference of the amplitudes controlled by the ${\cal O}_{9}$ and ${\cal O}_{10}$ operators, related to the Z loop and W box diagrams, respectively. This "low-q2" region between 1 and 6 GeV2/c4 is the most interesting and theoretically cleanest. Beyond this, non-suppressed $c \overline c$ contributions (Figure 11, right) make the picture more complicated and theoretical estimates are less reliable. The observation of high mass resonances above the ψ(2S) meson by the LHCb collaboration ( Aaij, R et al., (2013) ) is an indication that a lot of care is needed when interpreting the high-q2 region.
The $A_{\mathrm{FB}}$ and $P_5'$ asymmetries
In the decay B0 → K*0μ+μ−, followed by K*0 → K+π−, the direction of the four outgoing particles can be described by three angles, shown in . The forward-backward asymmetry AFB is defined as the relative difference between the number of positive and negative leptons going along the direction of the B0 meson in the rest frame of the two-lepton system. This corresponds to an asymmetry in the distribution of the θl angle. Similarly, the K*0 polarisation fraction FL depends on the angle θK, defined analogously to θl.
Other asymmetries can be constructed from the other angles or combinations of them. The P5′ asymmetry is based on the angles θK and ϕ. It is defined as the relative difference between the number of decays in the regions in red and blue in Figure 14, divided by $\sqrt{F_\text{L}(1-F_\text{L})}$. Quantities based on several angles are more difficult to measure than single-angle ones as they require a better understanding of the reconstruction efficiencies depending on the kinematics of the outgoing particles.
The differential decay width with respect to the dilepton mass squared q2, the forward-backward asymmetry AFB, and the longitudinal polarisation fraction FL of the K*0 resonance have been measured by many experiments with no significant sign of deviations from the Standard Model expectation. The most recent measurement of AFB by the LHCb experiment is shown in (left). LHCb also studied other angular asymmetries. In particular a local deviation of the P5′ observable (see the box) from the Standard Model expectation is observed around q2 ∼ 5 GeV2/c4, see (right, figure from Aaij, R et al. (2016)). Belle, ATLAS and CMS subsequently presented data that are consistent with the LHCb results.
This deviation triggered a lot of interest among theorists, with interpretation articles being quickly submitted to journals. It is not clear yet if the discrepancy in P5′ is a statistical fluctuation, is due to under-estimated theoretical uncertainties, or is the manifestation of a new vector current beyond the Standard Model.
Similar measurements have been made in the decays B → Kll, Bs0 → ϕμ+μ−, Λb0 → Λμ+μ− and pKb0 → Λμ+μ−. The angular observables are consistent with the Standard Model, but there is some tension in the branching fraction measurements, which are on the low side compared to the expectation.
The decay family $b\to s \nu \overline \nu$ is theoretically cleaner than its charged-lepton counterpart b → sl+l−. There are no interferences from $c \overline c$ loops as those do not annihilate to neutrino pairs. The main difficulty is on the experimental side and only B factory experiments have attempted looking at such decays using the full reconstruction technique. None have been found and the most stringent limits on the decay rates of $B^0 \to K^* \nu \overline \nu$ and $B^+ \to K^+ \nu \overline \nu$ are at the 10−5 level.
Lepton universality tests
The above-mentioned \(b \to s l^+ l^-\) measurements have been reported assuming that muons and electrons behave the same way. This assumption, called lepton universality, is built into the Standard Model and has been extensively tested, most notably at LEP experiments. The only Standard Model particle that has different couplings to leptons is the Higgs boson, which couples proportionally to mass.
Yet, surprisingly, the lepton universality ratio $R_K=\frac{{\cal B}(B^+ \to K^+ \mu^+ \mu^-)}{{\cal B}(B^+ \to K^+ e^+e^-)}$ has been measured to be $0.745^{+0.090}_{-0.074}\pm0.036$ by the LHCb experiment (Aaij, R et al. (2014)) in the $1<q^2<6$ GeV\(^2/c^4\) range, which corresponds to a $2.6\sigma$ tension with unity. The Standard Model prediction for this ratio is unity within $10^{-3}$ as all hadronic uncertainties cancel in the ratio. Figure 16 shows the mass peaks of \(B^+ \to K^+ \mu^+ \mu^-\) and \(B^+ \to K^+ e^+e^-\), highlighting the effect of bremsstrahlung affecting electron reconstruction. A similar deviation is seen by BaBar in the low-$q^2$ region, although with larger uncertainties. Belle do not report the low and high $q^2$ values separately, getting a result consistent with unity.
The ratio $R_{K^*}$ is defined in analogy. LHCb published two measurements in bins of $q^2$, which each differ by about $2\sigma$ from the Standard Model expectation of approximately unity. These results are much more precise than those previously determined by the Belle and BaBar collaborations.Similar decays can be used to perform tests of lepton universality in \(B^0_s \to \phi l^+ l^-\), \(\Lambda^0_b \to \Lambda l^+ l^-\), \(\Lambda^0_b \to p K^- l^+ l^-\), which are all accessible by the LHCb experiment, but some will have very limited yields. These measurements are complementary, as the different spins of the hadronic component probe different New Physics couplings. Also, the angular distributions described in Section The decays b → sl+l− should be investigated separately for decays to electrons and to muons. Belle reported separately the values of the $P_5'$ asymmetry, but no discrepancies were observed given the small available data sample.
Given the hints of lepton flavour universality violation between muons and electrons, it seems natural to wonder if such an effect can be seen in processes involving the third generation (τ) lepton. This has been tested in $B \to D^{(\ast)}\tau \overline \nu_\tau$ decays, comparing to the same decay with muons or electrons instead. Unlike the decays described above, the Standard Model contribution to this decay is not suppressed (and does not match the definition of a rare decay). It proceeds via a tree-level b → cW− transition, with the W− decaying to a lepton and a neutrino. The expectation is that the rates for the decays involving electrons, muons and tau leptons differ only due to phase-space effects (plus small effects due to form factors). The ratios of the rates of $b\rightarrow c\tau\bar{\nu}$ to $b\rightarrow c\ell\bar{\nu}$ ($\ell=\mu,e$) measured by BaBar, Belle and LHCb come out larger than expected, with an average deviating by approximately $4\sigma$ from the Standard Model predictions. This could indicate the presence of new couplings preferring tau leptons.
Wilson coefficient fits
This section briefly describes some constraints on Wilson coefficients. It relies on Section Theory.
Several groups have performed model-independent fits of Wilson coefficients, using most of the experimental results presented above. The fits differ by the set of experimental results used, the statistical treatment of uncertainties and choices of form factors. Another major difference is the level of trust of computations of quark loops (most notably $c \overline c$ loops) incorporated in the fit. Depending on these choices, the determined tension with the Standard Model ranges from one to several standard deviations.
In all cases, the New Physics scenario which is preferred changes the value of the C9 coefficient (adding a non-zero term C9NP). This term could then be different depending on the flavour of the involved leptons (introducing C9eNP and C9μNP), thus breaking lepton universality, see Figure 18. The data are not conclusive yet, but a tension with the Standard Model point at (0, 0) is visible. The significance of this tension depends on the assumed theory uncertainties.
Another popular model is to assume that the weak interaction V − A structure holds in New Physics and thus to impose C9NP = −C10NP. The data are consistent with such a hypothesis, but again it is too early to draw conclusions.
Right-handed components are also added in the fits, in particular using asymmetries in b → sl+l− decays that are sensitive to such effects. Presently there is no evidence for any significant need for right-handed currents.
There is a plethora of model-dependent interpretations of these findings. The deviations can be accommodated by supersymmetry, models with new vector bosons, two Higgs doublets, scalar interactions or leptoquarks.
At the risk of stating the obvious, rare decays have the advantage of being rare. This ensures that the experimental precision will stay dominated by statistical uncertainties, and thus will not run into a limit imposed by irreducible systematic uncertainties. The theoretically cleanest measurements, like the lepton-universality ratios RXs and the ratio of B0 → μ+μ− to Bs0 → μ+μ− branching fractions will continue to be of interest as more data are acquired at the LHC and by Belle II. The future will tell us if the deviations from expectations hold and tell us something new about Nature.
Other measurements, like branching fractions (for instance b → sγ) have already reached the theoretical precision and more work is needed on this side to allow more precise comparisons of experimental values and Standard Model predictions. Finally, asymmetries in B0 → K*0l+l− are in between. If the presently measured central values stay while the uncertainties reduce, we may soon be in the situation of having to understand a very significant deviation with predictions based on the Standard Model. More investigations of theory uncertainties are needed before any conclusion can be reached.
The Run 2 of the Large Hadron Collider is now ongoing and the amount of rare b-hadron decays collected by the LHC experiments will increase rapidly. Also, the Belle II experiment will start in 2018. Improved measurements of the processes described in this article, and also new complementary measurements, will become available and will lead to improved precision which will be useful in global fits.
- Aaij, R et al. (2013). Observation of a resonance in $B^+\to K^+\mu^+\mu^-$ decays at low recoil, Phys. Rev. Lett. 111(112003): . doi:10.1103/PhysRevLett.111.112003.
- Aaij, R et al. (2014). Test of lepton universality using $B^+\to K^+\ell^+\ell^-$ decays, Phys. Rev. Lett. 113(151601): . doi:10.1103/PhysRevLett.113.151601.
- Aaij, R et al. (2016). Angular analysis of the $B^0\to K^{\ast0}\mu^+\mu^-$ decay using $3$~fb$^{-1}$ of integrated luminosity, JHEP 02: 104. doi:10.1007/JHEP02(2016)104.
- Aaboud, M et al. (2015). Study of the rare decays of $B^0_s$ and $B^0$ into muon pairs from data collected during the LHC Run 1 with the ATLAS detector, : . arXiv:1604.04263
- Ammar, R et al. (1993). Evidence for penguins: First observation of $B \to K^*(892)\gamma$, Phys. Rev. Lett. 71(5): 674-678. doi:10.1103/PhysRevLett.71.674.
- Branco, G C; Lavoura, L and Silva, J P (1999). CP Violation, Int. Ser. Monogr. Phys., Volume 103. Clarendon Press, Oxford. ISBN 978-0198716754
- Buchalla, Gerhard; Buras, Andrzej J and Lautenbacher , Markus E (1996). Weak decays beyond leading logarithms, Rev. Mod. Phys. 68: 1125-1144. doi:10.1103/RevModPhys.68.1125.
- Buras, Andrzej J and Fleischer, R (1998). Quark mixing, CP violation and rare decays after the top quark discovery, Adv. Ser. Direct. High Energy Phys. 15: 65-238.
- Dirac, Paul A.M. (1927). The quantum theory of the emission and absorption of radiation, Proceedings of the Royal Society of London A: Mathematical,Physical and Engineering Sciences 114(767): 243-265. doi:10.1098/rspa.1927.0039.
- Fermi, Enrico (1934). Versuch einer Theorie der $\beta$-Strahlen. I, Z. Phys. 88(3-4): 161-177. doi:10.1007/BF01351864.
- Fermi, Enrico (1949). Nuclear Physics - A Course Given by Enrico Fermi at the University of Chicago, Int. Ser. Monogr. Phys., Volume 103. University of Chicago Press, Chicago. ISBN 978-0226243658
- Khachatryan, Vardan et al. (2015). Observation of the rare $B_s^0 \to \mu^+\mu^-$ decay from the combined analysis of CMS and LHCb data, Nature 522:(68): 68-72. doi:10.1038/nature14474.
- Limosani, A et al. (2009). Measurement of Inclusive radiative \(B\)-meson decays with a photon energy threshold of 1.7 GeV, Phys. Rev. Lett. 103(241801): . doi:10.1103/PhysRevLett.103.241801.
- Izlar, Kelly (2013). The march of the penguin diagrams, Symmetry Magazine : .
- Capdevila, Bernat et al. (2015). Patterns of New Physics in $b\to s\ell^+\ell^-$ transitions in the light of recent data, : . arXiv:1704.05340


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What You'll Learn
By the end of the class you will:
- Know how to make pasta dough from scratch
- Know how to roll and cut two types of fresh pasta
- Be able to create simple seasonal pasta sauces
About This Class
Pasta is a pantry staple and every cook should have a repertoire of go-to recipes. In this class, you'll explore both fresh and dried pasta, learn how to make accompanying sauces, and understand why certain shapes of pasta are paired with certain sauces.
Everyone will make a batch of fresh pasta dough and hand rolled orecchiette. Be prepared to come away with plenty of tricks that will help your pasta dishes go from good to outstanding.
Homemade Tagliatelle, Classic Carbonara, and Hand-Shaped Orecchiette
- Wine included
- Full meal served in class
- Work in teams
- Suitable for vegetarians
- Ages 18+
The average duration of our cooking classes is 4 hours, and they are broken down into three segments: Introductory lecture or demo, hands-on cooking, and eating.
The class will begin with a short discussion or demo from your instructor. After, you can expect to work either on a team or individually to prepare the class menu. As a group, your class will prepare the selected menu in its entirety. In most classes, not every student will make every dish.
Learning happens throughout the class, and your instructor will often call out key techniques as they happen in each recipe.
Most classes conclude by sitting down as a group to eat what you've prepared, followed by a discussion of each recipe. You will leave with a copy of the recipe packet and a working knowledge of every dish that was made so you can recreate them at home.
Doors open 15 minutes before the class start time.
You do not need to bring anything with you to class, unless otherwise noted in the class description. The kitchens at SFCS are fully equipped and aprons, recipes, equipment, food and wine are all provided for you.
In most classes, you will be sitting down to eat a complete meal that you have prepared along with your classmates. Since this typically occurs towards the end of the class period, an appetizer or small plate is usually prepared early on in class. However, you're welcome to additionally have a snack before you arrive or bring one with you.
Please note, for our Baking & Pastry classes, a light meal will be provided for you.
We suggest wearing casual clothing you can easily move around in, and highly recommend wearing comfortable closed-toe shoes. Long hair should be pulled back.
Cancellation Policy
Kirsten Goldberg
Growing up in the Bay Area, Kirsten was exposed to an extraordinary array of foods. This early exposure instilled in her a great curiosity and passion for regional cuisines. At 18 years old, Kirsten studied cooking under the tutelage of Catherine Pantsios before working in restaurants such as Mario Batali's famous Babbo in New York's West Village and Nancy Oaks' Boulevard in San Francisco. Kirsten has worked in restaurant kitchens in Europe and traveled extensively in South East Asia, where she took cooking courses and hung out in the food markets of every city she visited. Kirsten is currently the chef-instructor of the Professional Culinary Program at San Francisco Cooking School.
Marc Baltes
Marc graduated from the Culinary Institute of America in Hyde Park, New York, then was so impressed with the food on a visit to San Francisco that he decided to make it his home. He worked in the kitchens of Zuni Café, Quince and Pizzaiolo, then became head chef of Boot and Shoe Service in 2011, a position he held until 2017 when he joined SFCS full time as a chef instructor.

subset: commoncrawl


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The castle, about 15 miles north east of Fort William, was used to train elite commandos from Britain and the US as well as France, the Netherlands, Norway, Czechoslovakia, Poland and Belgium from 1942 to 1945.
The grounds are now being excavated. Earlier excavations have uncovered live ammunition, including bullets for captured German weapons.
Instructors included Dutchman Cpl Martien van Barneveld, the tall figure wearing a beret pictured in a snow storm at the centre. A veteran of the Battle of Arnhem, he was known to his Dutch recruits as the Big Man of Achnacarry.
More than 25,000 men passed through Achnacarry from when the commando centre opened in 1942 and until it closed in 1946.
Photo's: Commando Veterans Archive


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Regular price $8.00
A strand of vibrant Kantha textile beads is finished with a multicolored thread tassel for a trendy and fun bracelet. It is perfect for layering or wearing alone. The recycled sari beads measure ¼" and the bracelet stretches to fit most wrists. (Colors and patterns will vary.) Sustainably handmade by women artisans in India who are working to improve their lives through fair trade.


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How To Play Loaded Slots January 3, 2017
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We ship worldwide.
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Please wait until our stated time frames. If your order has still not arrived please contact us.
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You can but please be quick, all orders placed before 2pm are dispatched on the same working day.
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We are very sorry if an item you are interested in is out of stock, please contact us and we will add you to the waiting list and we will notify you when the item is back in stock.
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All of our parcels are sent via Royal Mail service. See below for full pricing list.
Next Day Delivery - £5.49
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This week's story is a nostalgia piece for me. It's about the special allure of a club raceday back in 1974-1975 time frame. This will bring back memories from the folks who were there back in the day, and I hope it will give the younger folks and non-racers a bit of the feeling of the special excitement of a raceday. See the page WE. Raceday!
Monthly Archives: January 2016
The First Two Years
I'm working on a new story for the blog but it's giving me trouble — maybe I can pound it onto shape for next week. In the meantime here's something that might be of interest to a few people. See the page WF. Blog Stats for some numbers from the first two years of this blog, Jan. 2014 to Dec. 2015.
New Year, New Blog Page
Happy 2016 to everyone. I'm starting the year with a page about something Joe Gresh said in the Motorcyclist magazine's book review of Racing the Gods. See the page WG. A Terrible Gift.


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A SADD(Students against Drunk Driving) party gets out of hand.
Drunken teenagers at a party funded by a high school's anti-drink-driving group have trashed a rural Southland hall.
Bottles were thrown at passing cars and into a children's playground during the out-of-control party, organised by high school pupils using money from the St Peter's College SADD (Students Against Driving Drunk) committee, last Friday.
Damage to the Waimumu Hall, near Gore, has disgusted locals and prompted police to warn parents against providing alcohol to young people going to parties, The Southland Times reported today.
Waimumu-Te Tipua Hall Society president Duncan Falconer said yesterday the hirer had given an assurance there would be parental supervision at the party.
"Had we known that would not eventuate we would have gone down (to the hall) and turned people away."
Damage would run into thousands of dollars and those responsible would be held accountable, Mr Falconer said.
The extensive damage inside the hall included panels and doors kicked in, broken windows, and dents in the ceiling.
Mr Falconer said bottles were thrown at passing cars and into a neighbouring playground.
I see these responsible young people are concerned for driver safety. Kind of reminds me of a line from the Munsters. Grandpa is speaking to Eddie as he gives him boxing lessons. "Your father is so opposed to violence that if he saw what we're doing, he would kill me."
Hat tip- Don Surber


text: As anglers, we know that there's no such thing as a "bad day of fishing". Even when things don't go our way, we have fun and make the best of being out on the water. A lot of the time we go to our local pond or lake just to get ourselves a fix, and for many anglers who do this, they'll usually fish off a dock or the bank. For these anglers, float fishing is the ideal way to enjoy both the adventure of fishing while taking in the sights and sounds of nature. Each location offers a unique fishing experience and float fishing allows you to enjoy each to its fullest! In this blog, we're going to take a look at some mistakes to avoid while float fishing.


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Leggings - Tribal Flatten It
by Tribal
Save $-75.00
Flatten It technology means a little bit of tummy control with a shelf liner across the abdomen in these super popular pull on leggings with the leg opening noted as slim. Totally smooth without pockets at all means a very nice, slender look. The smooth, pull on waistband is wide.
These are a stable stretch jersey knit of 86% cotton and 14% spandex for lots of stretch.
Misses Inseam is 28" long
Colour: Black


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LiveMusicNow musicians Mini concert
On Thursday 15th March pupils in Lower and Middle school had a visit from LiveMusicNow musicians Heather and Rosanna. Heather is a harp player and Rosanna plays the flute.
They started their performance with some Disney songs closely followed by classical music extracts, a piece of Spanish music and even Ed Sheeran.
During The Swan by Saint-Saens, pupils had to close their eyes and try to follow the sound of the music with their nose. It was great to see the concentration from pupils as they listened carefully to what they could hear. The pupils thoroughly enjoyed the concert and were keen to ask questions at the end.


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Image: Longchang stone arch stone arch - Guan - "the parents of the people" Niu Shumei de Zheng Fang by Zhangzhugang
Neijiang Tourist Attractions
Neijiang Incompete Tourist Attractions
Have you been here? You can help us by writing a review of these locations.
- Neijiang Tzu Memorial Hall (内江张大千纪念馆; Nèi Jiāng Zhāng Dà Qiān Jì Niàn Guǎn)
- Longshan Neijiang-owned medium-heavy (内江资中重龙山; Nèi Jiāng Zī Zhōng Zhòng;chóng Lóng Shān)
- Neijiang Longchang ancient lake scenic spot (内江隆昌古湖景区; Nèi Jiāng Lóng;lōng Chāng Gǔ Hú Jǐng Qū)
Neijiang Tourist information
Neijiang is a prefecture level city within Sichuan province in China.


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Move a vCenter Server or Database to a New EVC Enabled Cluster with Nexus 1000v
I recently came across an issue regarding moving a server that holds our vCenter database to a new cluster. The big issue was that moving this server to the EVC (Enhanced vMotion Compatibility) cluster meant not using Virtual Infrastructure. As you can see, this took a little preparation before I dived right into it. Not using vCenter meant I couldn't just drag and drop the servers into the new cluster. To make things more discombobulated, we had just configured Nexus on our primary cluster as well. This could have attributed to the amount of time I spent trying to fix the issue. I tried everything from removing the server from inventory and adding it back in from another host. I did this by connecting directly to a host instead of the vCenter server directly. If you have never done it then it is a great fallback tool that helps you get around in case of disaster.
When I tried to re-add the server to inventory on a host in the new cluster I started getting error messages. When I tried connecting the NIC under "Edit Settings", I got the error invalid configuration for device "0". Remember, I am trying to link it with a cluster that is configured with a Distributed Switch (aka. Nexus) instead of the Virtual Switches. After talking with support for a few hours they told me to just clone it from one cluster to the other. I couldn't believe it was that simple, but it worked perfectly! It took about an hour for the clone to take place. The server was a Windows 2008, so I did not even have to keep it shut down in order to clone it. They never helped me understand why my method did not work since it was actually quicker. I did hear one of them say the once added in the cluster, the server didn't have permission to add the NIC. That is because I was connected directly to the ESX host and not vCenter Server.
The error actually came back after the host was migrated. This time I moved the same server off of the host it was origionally cloned to. This brought back "Invalid configuration for device "0"". This time I was not able to migrate it anywhere else in the cluster. vCenter was down and I wasn't going to clone it somewhere else. The resolution was to create a vSwitch on the host and connect an extra NIC to it. Connect the server client to the switch and then move it back to the Nexus. This must be a bug with our current version of Nexus.


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The Animal Kingdom may be divided into creatures which one can feed and creatures which one cannot feed. Animals which one cannot feed are nearly always unsatisfactory; and the grasshopper is no exception. Anyone who has tried feeding a grasshopper will agree with me.
Yet he is one of the most interesting of British creatures. The Encyclopædia Britannica is as terse and simple as ever about him. "Grasshoppers," it says, "are specially remarkable for their saltatory powers, due to the great development of the hind legs; and also for their stridulation, which is not always an attribute of the male only." To translate, grasshoppers have a habit of hopping ("saltatory powers") and chirping ("stridulation").
It is commonly supposed that the grasshopper stridulates by rubbing his back legs together; but this is not the case. For one thing I have tried it myself and failed to make any kind of noise; and for another, after exhaustive observations, I have established the fact that, though he does move his back legs every time he stridulates, his back legs do not touch each other. Now it is a law of friction that you cannot have friction between two back legs if the back legs are not touching; in other words the grasshopper does not rub his back legs together to produce stridulation, or, to put it quite shortly, he does not rub his back legs together at all. I hope I have made this point quite clear. If not, a more detailed treatment will be found in the Paper which I read to the Royal Society in 1912.
Nevertheless I have always felt that there was something fishy about the grasshopper's back legs. I mean, why should he wave his back legs about when he is stridulating? My own theory is that it is purely due to the nervous excitement produced by the act of singing. The same phenomenon can be observed in many singers and public speakers. I do not think myself that we need seek for a more elaborate hypothesis. The Encyclopædia Britannica, of course, says that "the stridulation or song in the Acridiidæ is produced by friction of the hind legs against portions of the wings or wing-covers," but that is just the sort of statement which the scientific man thinks he can pass off on the public with impunity. Considering that stridulation takes place about every ten seconds, I calculate that the grasshopper must require a new set of wings every ten days. It would be more in keeping with the traditions of our public life if the scientific man simply confessed that he was baffled by this problem of the grasshopper's back legs. Yet, as I have said, if a public speaker may fidget with his back legs while he is stridulating, why not a public grasshopper? The more I see of science the more it strikes me as one large mystification.
But I ought to have mentioned that "the Acridiidæ have the auditory organs on the first abdominal segment," while "the Locustidæ have the auditory organ on the tibia of the first leg." In other words one kind of grasshopper hears with its stomach and the other kind listens with its leg....


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Recent studies suggest an ounce of milled flaxseed in our daily diet for six months can be beneficial to health. Flaxseed has high levels of Omega-3s and studies show it reduces blood pressure, especially for patients with hypertension. It also appears flax seed can neutralize age-associated inflammation in the body. Hard to believe, but maybe an ounce of flaxseed could save your life!
It's equally difficult to imagine half an ounce (0.3 oz to be exact) of xylitol for six months can eliminate plaque on teeth, prevent the transmission of cavity-bacteria from one generation to the next, and stop gum disease and cavities. These problems seem too big and the solution too small and easy!
The most effective time to enjoy xylitol is immediately after meals to prevent the damage that normally happens after eating or drinking. Acidity from eating grows plaque and damages teeth. Xylitol instantly makes the mouth alkaline, makes plaque less sticky, and helps strengthen teeth.
Xylitol provides a simple option for a huge health problem. Why not use xylitol for 6 months and see how it works for you and your family?
Potent Antihypertensive Action of Dietary Flaxseed in Hypertensive Patients
Dietary supplementation with flaxseed oil lowers blood pressure in dyslipidaemic patients
Mutans Streptococci Dose Response to Xylitol Chewing Gum
My passion is educating people about oral health. For decades I've worked with patients of all ages, in different countries, in high and low-risk communities, and from every walk of life. I've seen the same results over and over again –people who transformed their oral health – sometimes in days or weeks, even after they had suffered years of pain and frustration. It's never too late, and my ways will give you the key to a lifetime of improved oral health.
Interview with Howard Farran: The Benefits of Xylitol
Your Teeth and Travel
Have Healthier Teeth & Gums by Following These 5 Wellness Points!
Xylitol for Seniors: Keeping Teeth Young and Ageless
Meet Your Dental Plaque
Granular Xylitol – the Perfect Sugar Alternative
Tooth Brushing Tip
Are Cavities About Zip Code?


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5711 Autumn Trail, Celina, 75009
Click for Map
Price $769,900
Beds:5 Baths:4.1 Garage:3
1.88 acres East of Preston Road, just 5 minutes from Celina High School. Surrounded by horse friendly rural neighbors and mature native trees. 5 bedroom plan includes office and XL game room with closets. Covered patio overlooks large back yard and includes builtin outdoor fireplace. Master retreat includes super size walk in closet and additional tornado shelter safe room. Gas heat, high efficiency HVAC, on demand hot water are all standard feat
Broker : Austin C Entrop
Agent : Austin Entrop
All information is from sources believed to be reliable and is subject to change of price, rental, commission or other conditions, prior sale, lease or financing or withdrawal without notice. No representation is made as to the accuracy of any description. All measurements and square footage are approximate and all information should be confirmed by customer.


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First Time User? Sign Up Now
First Time User? Enroll now.
Home > Care & Treatment > Imaging & Radiology > Mammography > Kay Yow North Hills Event
UNC REX Mobile Mammography is offering 3D screening mammograms at the Kay Yow Cancer Fund North Hills Event on May 11th!
All appointments must be confirmed by May 6th.
Register online and choose "Kay Yow Fund North Hills Event"
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Passionate Beauty Push-Up Bra #10067
Be confident and stay attractive with our Passionate Beauty Push-Up Bra! It has sexy beauty back straps that let you wear it however you want to – regular or crossover. It has automatic stretch adjustment and let you manage this for optimal comfort you need!It's a perfect choice for women with different breast types and there are sexy corners cup that can match your dress easily and express your style!
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