When is a triangle not a triangle? Believe it or not, this peculiar riddle encapsulates much of the latest research in particle physics. The triangle in question is the “unitarity triangle”, an abstract diagram representing the interactions between different types of quarks – the fundamental constituents of neutrons and protons. A vital part of the Standard Model of particle physics, the unitarity triangle provides clues about the subtle difference between matter and antimatter, helping to reveal why the universe did not vanish into nothingness immediately after the Big Bang.

By rights, an equal amount of matter and antimatter should have been created in the Big Bang. But the universe we see today consists entirely of matter, implying that there must be some difference between the behaviour of particles and of their corresponding antiparticles. The Standard Model incorporates such a difference via a phenomenon called charge–parity (CP) violation – a broken symmetry of nature that was discovered experimentally in the 1960s. It is this phenomenon that links the unitarity triangle to the real world, since the extent of CP violation – and hence the degree to which matter and antimatter differ – is proportional to the area of the triangle.

However, the amount of CP violation predicted by the Standard Model can only account for about a billionth of the matter–antimatter imbalance we see in the universe. In fact, this is just one of several problems with the Standard Model; others being that it does not explain gravity or the identity of “dark matter”. Hence, physicists believe that, despite its many successful predictions, the Standard Model will eventually be overthrown by a more general theory. One way to test such theories is by measuring the properties of the unitarity triangle: according to the Standard Model, it is perfectly triangular; but if new particles exist, then the triangle will be “broken”.

Since the discovery of CP violation, various experiments have measured its effects on particles called neutral kaons, but such measurements do not relate directly to the properties of the unitarity triangle. However, in the last few years researchers at huge particle physics experiments called BaBar and Belle in the US and Japan, respectively, have made precise measurements of the angles and the lengths of the sides of the unitarity triangle by studying the behaviour of different particles called B-mesons. The results so far show that the diagram is very close to triangular. But there is still room for uncertainty, and refining our knowledge of the unitarity triangle with future experiments is a crucial way to finally take physics beyond the Standard Model.

Violating symmetry

In the Standard Model, all matter is made up of quarks and leptons, each of which comes in six different types or “flavours”. Both quark and lepton flavours fall into three generations of two particles each: for quarks these are up and down, charm and strange, and top and bottom; while the lepton generations each contain a charged lepton – electron, muon and tau – and a correspondingly named neutrino. Each quark and lepton also has an antimatter partner. All atoms are constructed from just the particles of the first generation – up and down quarks (which combine to form protons and neutrons) and electrons. One of the fundamental questions in particle physics is why there should be two, and only two, additional generations with similar properties but greater mass.

To complicate this picture of the subatomic world, the flavour of a quark can change as a result of the weak interaction. For example, nuclear beta decay – in which a neutron decays into a proton, an electron and an electron antineutrino – is the result of a down quark in the neutron converting to an up quark by emitting a “W-boson”, one of the particles that carry the weak interaction. Because W-bosons have a charge of ±1 (in units of the electron’s charge), such conversions can occur only between “up-type” quarks (up, charm, top), which have a charge of +2/3 and “down-type” quarks (down, strange, bottom), which have a charge of –1/3.

The probability of each of the nine possible quark flavour changes is described using a 3×3 matrix called the Cabibbo–Kobayashi–Maskawa (CKM) mixing matrix. The rows of the matrix correspond to the uptype quarks, the columns to the down-type quarks and the square of each element of the matrix gives the probability of that particular flavour change occurring (see below). Meanwhile, the weak interactions of antiquarks are governed by the complex conjugate of the CKM matrix – the matrix formed by keeping the real part of each element the same but reversing the sign of the imaginary part. Therefore, if the CKM matrix contained no imaginary elements, quarks and antiquarks would have exactly the same weak interactions.

However, experiments show that this is not the case due to CP violation. The letters C and P represent symmetry operations: C, for charge conjugation, means exchanging every particle in a system with its antiparticle; and P, for parity, means reversing all three spatial coordinates. Until the 1960s physicists thought that if both the C and P operations were carried out on a system simultaneously, the new system would behave in exactly the same way as the original. However, in 1964 Jim Cronin and Val Fitch at Princeton University observed that the decays of particles called long-lived neutral kaons, consisting of down and strange quarks and antiquarks, did not respect CP symmetry. This discovery, which earned Cronin and Fitch the Nobel prize, was a shock, and it took 10 years before Makoto Kobayashi and Toshihide Maskawa of Kyoto University in Japan proposed a solution in the form of the CKM matrix.

At the time of Cronin and Fitch’s discovery only quarks from the first two generations were known, which only allows for a 2×2 mixing matrix with real-number elements – following the 1963 proposal of Nicola Cabibbo. Kobayashi and Maskawa’s idea was to introduce a third generation of quarks, leading to a 3×3 matrix that contains complex numbers and therefore accommodates CP violation. Predicting the existence of two new quarks was a radical step, but the gamble paid off in 1977 when the bottom quark was discovered at Fermilab in the US. Indeed, the CKM matrix was a well-established part of the Standard Model even before the discovery of the top quark (the up-type version of the bottom quark), also at Fermilab, in 1995.

The nine elements of the CKM matrix are not all independent. For example, since an up quark must convert into one of the three down-type quarks, so the three associated probabilities must sum to one. In fact, the CKM matrix can be expressed in terms of just four independent parameters: three real “mixing angles” and one imaginary “phase” that leads to CP violation. The unitarity triangle is simply a way of representing these numbers in a diagram, first proposed by James Bjorken and Cecilia Jarlskog in 1988, where the angles (labelled α, β and γ) and the lengths of the sides (R_u and R_t) correspond to certain combinations of elements of the CKM matrix (see figure 5). The height of the triangle depends on the imaginary phase, so it is precisely because the CKM matrix includes complex numbers that the unitarity triangle is a triangle, not a 1D line.

Although Kobayashi and Maskawa had successfully predicted the third generation of quarks, to confirm that this was indeed the correct explanation of CP violation researchers had to show that the unitarity triangle really is triangular by determining its angles and sides. This can be done by measuring carefully chosen decays of particles in which the changes of quark flavour that occur, and hence the elements of the CKM matrix involved, correspond to one of the parameters of the triangle.

Critically for these experimental measurements, CP violation is expected to be much larger for particles containing bottom quarks than it is for the kaons in which it was first observed. That is, particles containing bottom quarks can behave very differently to those containing bottom antiquarks. This can be seen, for example, in the different rates at which B-mesons and antimesons (combinations of a bottom quark or antiquark with an up or down quark or antiquark) decay to give certain products. By studying these “CP asymmetries” in the B-meson system, researchers can therefore pin down the properties of the unitarity triangle.

A production line for bottom quarks

Only a tiny fraction of B-mesons undergo the interesting decays for which large CP asymmetries are predicted. In order to constrain the unitarity triangle, we therefore need to produce huge numbers of B-mesons, which calls for a particle collider known as a “B-factory”. In these machines, electrons and positrons are smashed into each other with just the right energy to create a particle called the Y(4S), which consists of a bottom quark and antiquark, and which decays almost immediately into a B-meson and antimeson. Crucially, the electron and positron beams in these colliders do not have the same energy, so that conservation of momentum ensures that the B-mesons produced are moving. This allows researchers to determine the time the particles take to decay, and hence their decay rate, from the distance that they travel before they do so. Two such asymmetric machines were completed in 1999 at the Stanford Linear Accelerator Center (SLAC) in California and KEK in Japan.

The primary objective of these B-factories was to measure the angle β of the unitarity triangle, so the target rate of electron–positron collisions was set by the estimated number of B-mesons needed to measure this parameter accurately. Serendipitously, at the time those estimates were made, the top quark was expected to be much lighter than it was subsequently found to be. This miscalculation, together with competition between SLAC and KEK, pushed the machines to achieve extremely high collision rates. Over a billion B-meson pairs have now been produced by the two B-factories, and their accompanying particle detectors – BaBar at SLAC and Belle at KEK – have reaped the benefits of the extra data.

The specific process that allows β to be measured is the asymmetry in the decay rates of B⁰-mesons (a bottom antiquark and a down quark) and B⁰-antimesons (a bottom quark and a down antiquark) to two other mesons, K_s (the short-lived neutral kaon) and J/ψ (a charm quark and antiquark). This asymmetry arises because a B⁰-meson can decay to a J/ψ and a K_s either directly or after first “oscillating” into a B⁰ – and vice versa for a B⁰. Oscillation is a phenomenon whereby a neutral B-meson can transform into its antimeson by momentarily creating a particle–antiparticle pair of “virtual” top quarks – the uncertainty principle allows the much heavier top quarks to be created but only for a very short time. Due to quantum-mechanical interference between the direct and oscillating decay routes, B⁰-mesons on average live slightly longer before decaying to J/ψ and K_s than do B⁰. Because of the particular elements of the CKM matrix that correspond to the flavour changes in the oscillation and direct decay, the size of this asymmetry is proportional to sin2β.

To observe this asymmetry, researchers first need to identify large numbers of decays resulting in J/ψ K_s, which is made more difficult by the fact that this particular decay only occurs once in every 2000 decays (see figure 2). We then need to know whether it was the B⁰ or the B⁰ from the initial collision that decayed to the J/ψ and K_s. This is done by looking at how the other meson produced in the electron–positron collision decayed to determine whether it was a B⁰ or a B⁰ – an operation known as “flavour tagging”. Finally, to determine how long the B-mesons lived, we must measure the position at which they decayed. It was these requirements to be able to tag flavour and measure decay positions accurately that drove the design of the BaBar and Belle detectors (see below).

The BaBar and Belle teams published their first results on the B⁰ → J/ψ K_s decay in 2001. As predicted by the Standard Model, they saw a large asymmetry in the decay rates of the B⁰ and B⁰ – enabling researchers to determine the angle β of the unitarity triangle. The teams have continued to refine these measurements, and the current world average is β = 21.2° ± 1.0°. The fact that the angle is non-zero means that CP violation definitely occurs in the B-meson system, and the amount agrees with previous studies of the much smaller CP-violation effects in the kaon system.

In order to define a triangle, however, it is necessary to measure at least two angles. It was initially thought that sin2α could be measured in much the same way as sin2β but using B⁰ → π⁺π^– decays instead of B⁰ → J/ψ K_s. Since pions consist of up and down quarks and antiquarks, different elements of the CKM matrix are involved compared with the latter decay, and hence a different angle can be measured. However, unlike the B⁰ → J/ψ K_s decay, the asymmetry in the B⁰ → π⁺π^– decay is not simply proportional to sin2α, so further calculation is necessary to extract the angle correctly.

Similarly, while it has been known for some time that the third angle, γ, can be obtained by measuring CP asymmetries in B → DK decays (where D is a meson containing charm quarks), it is hard to obtain precise constraints on the parameter because only about one in a million B-mesons decay to the relevant states. Nevertheless, impressive progress has been made in measuring both α and γ, thanks to close co-operation between theorists and experimentalists to devise new analysis techniques. The current world average values are α = 92° ± 7° and γ = 82° ± 20°. Within error, the three angles do indeed sum to 180°, but reducing their uncertainties remains of paramount importance to test for signs of “new physics”.

Sizing up the sides

What about the sides of the unitarity triangle? By definition the bottom side has a length of one, but the other two sides must be measured experimentally to test whether their lengths are consistent with the angles. This is known as “overconstraining” the triangle. If the Standard Model is correct, all the measurements will be consistent with a single triangle, while any deviation would indicate new physics at work. While the measurements of the angles rely on CP asymmetries between B-mesons and antimesons, the sides can be determined just from the rates of appropriate decays. This may sound less challenging, but considerable complications arise for measuring both sides.

The left-hand side of the triangle depends on the element of the CKM matrix that relates up and bottom quarks, V_ub. There are two strategies to try to measure its value: the first, known as “inclusive”, requires the rate for B-mesons to decay into any particle containing an up quark (plus a lepton and a neutrino) to be measured; the second, “exclusive”, technique requires just the rate for one particular such decay. In both cases, some theoretical input is necessary in order to extrapolate the measurements into values of V_ub, which is now measured as |V_ub| = (4.09 ± 0.26) × 10^–3. This gives the length of the left-hand side of the triangle as R_u = 0.381 ± 0.014.

The measurement of the other side of the unitarity triangle presents yet another complication. The crucial input to determine this side is the CKM element V_td, which governs the rate of transitions from a top quark to a down quark. Yet the energies in the B-factories are far too low to produce the extremely massive top quarks. Furthermore, the small numbers of top quarks that have been produced at higher-energy colliders decay predominantly to bottom quarks. This makes it almost impossible to observe their much rarer decays to lighter quarks in any existing or proposed experiment.

However, all is not lost. As mentioned earlier, virtual top quarks are created in the oscillation of B-mesons, and it turns out that the rate of B⁰–B⁰ oscillations, which is described by the parameter Δm_d, is governed by V_td. It is extremely challenging for theorists to calculate the constants that relate V_td to Δm_d, but by considering the rate of oscillations of another neutral meson called the B⁰_s (a bottom antiquark and a strange quark), given by Δm_s, progress has been made. In particular, a technique called lattice quantum chromodynamics has recently enabled the constants relating the ratio |V_td|/|V_ts| to Δm_d/Δm_s to be obtained. Since the values of Δm_d and |V_ts| are known, the only missing piece of information is then Δm_s.

Measuring Δm_s is difficult because the B⁰_s–B⁰_s oscillations are very fast. However, last year the CDF experiment at Fermilab was able to make a remarkably precise measurement of Δm_s of about 2.75 THz (see Physics World July 2006 pp32–34, print version only). This allowed the length of the final side of the triangle to be measured as R_t = 0.904 ± 0.028. By overlaying all the measurements of the angles and sides on a single diagram, we are now able to examine whether they are consistent with a single triangle (see figure 3). So far, everything matches up with a triangle with a height of 0.344 ± 0.016.

The hunt for new physics

The precise measurements of the unitarity triangle are a triumph for the B-factories and the Standard Model, but they have nevertheless left many physicists downhearted. The reason is that despite the successes of the Standard Model, it has long been known that it cannot be the full story. As well as not being able to account for most of the matter–antimatter asymmetry in the universe, it does not explain the dark matter and dark energy thought by cosmologists to make up most of the universe, nor does it include gravity. It is fervently hoped that results from the Large Hadron Collider (LHC), starting up later this year at CERN, will provide evi dence of physics beyond the Standard Model. Yet physicists are impatient, and measurements of bottom quarks could provide clues before then – for example, if more precise measurements showed that the angles and sides of the unitarity triangle did not exactly match up.

One inconsistency in the triangle might be revealed by comparing the value of β determined from B⁰ → J/ψ K_s decays with measurements of the same parameter in different decays, such as those in which a bottom quark converts to a strange quark. These decays occur via virtual-particle “loops” in which the bottom quark decays into a virtual W-boson and an up-type quark, which then recombine to form a strange quark. If heavier, as-yet unknown particles exist, they could also be produced fleetingly in these loops, leading to measurable effects in rates of these decays. Recent measurements from BaBar and Belle suggest that there may indeed be new particles appearing in the loops that mediate such decays. At present, the experimental uncertainties are too large for us to be certain, but this is a tantalizing hint that theories beyond the Standard Model, such as supersymmetry, may be in action in the real world.

As the B-factories continue to refine their measurements, the LHC draws nearer to switching on. While about a million B-mesons per day are produced at BaBar and Belle, the LHC will produce many more than this number every second. In particular, the LHCb experiment, one of four detectors positioned around the LHC collider, will allow much more precise measurements of parameters such as γ via dedicated studies of the wealth of particles containing bottom quarks (see Physics World October 2006 pp37–39; print version only). However, because each proton–proton collision at the LHC will produce hundreds of particles, in stark contrast to the electron–positron collisions at the B-factories that produce only a B-meson and antimeson, it will be challenging to identify the decays of interest.

As well as producing more particles containing bottom quarks, a crucial advantage of the LHC is that it can produce a wider variety of these particles, in particular the B⁰_s-meson, which is not produced by the B-factories. This will allow a more precise measurement of Δm_s and, more importantly, the first study of CP violation in B⁰_s-mesons. For example, the LHCb will study the decay B⁰_s → J/ψ Φ, which is analogous to the B⁰ → J/ψ K_s decay measured at the B-factories. Unlike the B⁰ decay, the CP asymmetry in this decay is predicted by the Standard Model to be rather small. Hence, if a significant amount of CP violation is observed, it will provide a clear sign of new physics.

It might seem that the observation of new physics would mean the end of the story for B physics and the unitarity triangle. But in fact, such a discovery would require much work to be done to determine the detailed properties of the new particles, and establish whether their CP-violation effects can make up the shortfall in our understanding of the matter dominance in the universe. For this, researchers are proposing a “Super Flavour Factory” – an electron–positron collider similar to the existing B-factories but with a rate of collisions at least 100 times greater. Such an experiment would allow us to precisely constrain the unitarity triangle – even though by then it may no longer actually be a triangle – as well as study for the first time processes that are particularly sensitive to the properties of new physics, such as decays of the tau lepton.

So, when is a triangle not a triangle? As far as we can tell from the impressive results obtained by the researchers at BaBar and Belle, not yet! But we know that there must be CP violation beyond that explained by the Standard Model, and we hope that new particles that may account for this will soon be observed at the LHC. This suggests that more precise studies will reveal a chink in the triangle, opening a rich new field of research to pin down the properties of the new physics.

Box: The CKM matrix and the unitarity triangle

The Standard Model of particle physics contains six “flavours” of quark – up (u), down (d), charm (c), strange (s), top (t) and bottom (b) – that are divided into three generations of “up-type” quarks (u, c and t) with a charge of +2/3 and “down-type” quarks (d,s and b) with a charge of –1/3. Quarks can change flavour from an up-type to a down-type or vice versa via the weak interaction. The probability for each of these flavour changes occurring is described by a 3×3 matrix with complex-number elements called the Cabibbo–Kobayashi–Maskawa (CKM) matrix, V_CKM. For example, the square of the element V_ud gives the probability for an up quark to convert to a down quark. The corresponding probabilities for antiquark conversions are given by the complex conjugate of this matrix, V^*_CKM. The fact that these two matrices are not identical (because they contain complex numbers) explains the phenomenon of CP violation, whereby quarks and antiquarks behave slightly differently.

Since the weak interactions have to conserve probability – that is an up-type quark must convert to one of the three down-type quarks but nothing else – the CKM matrix must be “unitary”. Mathematically, a unitary matrix has the property that when it is multiplied by its Hermitian conjugate, V^†_CKM (the transpose of its complex conjugate), the result is the unit matrix. This relationship leads to a number of equations relating the CKM matrix elements. Some represent the conservation of probability directly – i.e. the squares of the elements in each row or column must sum to give one – but there are others where three complex numbers sum to zero, such as: V^*_ubV_ud + V^*_cbV_cd + V^*_tbV_td = 0

The latter relationships can be represented graphically on an “Argand diagram”, which has axes that correspond to real and imaginary numbers. On such a diagram, a complex number is represented by a line, and so the sum of several complex numbers can be represented by starting one line where the previous one ends, much like adding up vectors. In the unitarity equations of the CKM matrix, three complex numbers sum to zero, so the three lines join up to form a triangle known as a “unitarity triangle”. There are six unitarity triangles in all – one for each pair of columns or rows in the CKM matrix – and the height of each triangle corresponds to the amount of CP violation in the transitions of the quarks involved.

Most of the triangles are very elongated, corresponding to very small CP-violation effects. However, the triangle describing the transitions of bottom and down quarks is predicted to have sides that are approximately the same length as each other. Due to the importance of this triangle, it is usually known as the unitarity triangle. The unitarity triangle is conventionally drawn with the length of one side rescaled to unit length and rotated to lie along the real axis. The lengths of the other two sides and the three angles correspond to combinations of the elements of the CKM matrix and must be determined experimentally.

Box: BaBar and Belle

A million B-mesons per day are produced in electron–positron collisions at SLAC in the US and KEK in Japan. The decay products of the B-mesons are examined by huge particle detectors comprising many subsystems – BaBar (figure 6) at SLAC and Belle (figure 7) at KEK.

To determine the positions at which the B-mesons decay requires precise measurements of the paths of the decay products, which is achieved with silicon detectors located as close as possible to the collision point. It is also important to determine the type of particles produced in the decays. Both experiments use innovative particle-identification techniques that take advantage of the “Cerenkov radiation” produced when a charged particle passes through a material at a speed greater than the speed of light in that material. The angle at which this light is emitted depends on the mass of the particle allowing, for example, pions to be distinguished from kaons.

At a Glance: Unitarity triangle

• The unitarity triangle is an abstract diagram representing how the six known quarks can transform into one another via the weak force
• It is a vital component of the Standard Model of particle physics because it explains the subtle difference between matter and antimatter known as CP violation
• The Standard Model predicts that the diagram is perfectly triangular, while more general proposed theories such as supersymmetry suggest that it is not
• Researchers at particle colliders known as B-factories in the US and Japan have recently made precise measurements of the angles and sides of the unitarity triangle
• Although these measurements are consistent with the diagram being triangular, particle physicists continue to hunt for deviations that could reveal new physics
• Future machines, such as the Large Hadron Collider and a proposed Super Flavour Factory, will allow even more precise measurements of the triangle to test for physics beyond the Standard Model

More about: Unitarity triangle

P Harrison 2003 “Nature’s flawed mirror” Physics World July pp27–31
T Nakadaira 2004 “B-factories score direct hit on CP violation” Physics World October pp21–22
The Heavy Flavour Averaging Group
The CKMFitter collaboration
The UTFit collaboration