THE LARGE HADRON COLLIDER AT CERN
We now come to the largest collider ever built, the LHC at CERN. It took 25 years of planning and construction to build this accelerator, which was paid for by a consortium of countries.
The accelerator tunnel, 27 km in circumference, lies between 50 m and 175 m beneath the earth’s surface, straddling the French–Swiss border. This concrete-lined tunnel is 3.8 m wide, and was constructed between 1983 and 1988 for the LEP machine. The LHC is an intersecting ring collider, which contains two parallel beam pipes that intersect at four points, and each of the 27-km-long, circular beam pipes contains a proton beam. These two beams travel in opposite directions around the ring. They are kept in circular orbits by means of 1,232 dipole magnets, and an additional 392 quadrupole magnets focus the beams, maximizing the chances of collisions between the protons at the four intersecting points. Most of the magnets weigh more than 27 tons each. It requires 96 tons of liquid helium to keep the copper-clad niobium–titanium magnets at their operating temperature of 1.9 K (−271.25°C). This amazing cooling system comprises the largest cryogenic facility in the world, which is kept at liquid helium temperatures.
When the LHC is running at maximum energy, once or twice a day the protons will be accelerated from 450 GeV to 7 TeV. To accomplish this acceleration, the magnetic fields of the superconducting dipole magnets are strengthened from 0.54 to 8.3 tesla, which is a unit of measure of the strength of magnetic fields. When they are fully accelerated, the protons will be moving extremely close to the speed of light—namely, at about 0.999999991 times c, the speed of light. This incredible speed is merely 3 m/s slower than the speed of light. The protons go around the 27-km ring in less than 90 microseconds, completing 11,000 revolutions per second (see Figures 2.7 and 2.8 for diagrams of the LHC and its detectors).
Figure 2.7 Diagram of the LHC showing the positions of the four detectors: ALICE (A large ion collider experiment), ATLAS, LHCb (large hadron collider beauty), and CMS. © CERN
The LHC is actually a chain of accelerators. The protons, which represent the “hadrons” in the machine’s name, are prepared before being ejected into the main accelerator by a series of systems that increase their energy. The linear accelerator 2 (LINAC 2), a linear accelerator at the beginning of the LHC, generates 50-MeV protons and feeds them into the proton synchrotron booster, which accelerates them to 1.4 GeV and then injects them into the PS, where they are accelerated to 26 GeV. The SPS is then used to increase the energy of the protons to 450 GeV, and at last these protons are injected into the main ring.
In September 2008, the LHC was finally switched on and shot 2,808 bunches of protons around the ring. Each bunch contained 100 billion protons, and they collided 40 million times per second inside four detectors. The interactions between the two proton beams take place at intervals never shorter than 25 nanoseconds (a nanosecond is 10−9 seconds, or 0.000000001 second). The detectors are situated at the intersection points in the accelerator ring: the ATLAS detector, the CMS detector,6 the LHCb (large hadron collider beauty) detector, and the ALICE (a large ion collider experiment) detector. The circulating beam energies of the protons rise to an energy of 7 TeV, and when they collide, they generate a combined energy of 14 TeV, making the LHC the most powerful accelerator ever built7 (Figure 2.9 shows the ATLAS detector).
Figure 2.8 Drawing of the LHC above and below ground. © CERN
However, nine days after the machine was switched on, one of the superconducting magnets blew up because an unstable cable connection evaporated during a high-current test. One of the serious problems in dealing with this accident was interrupting the machine, which was running at a temperature of 1.9 K, using 130 tons of liquid helium. Fixing the magnet was a sensitive operation that took a long time.
As we recall, one of the important elements in an accelerator is the luminosity, or intensity, of the colliding beams. At full luminosity at the LHC, the particle collision rate is about 1034/cm2/second. That creates a lot of debris to sort through! Currently, CERN is upgrading the LHC to produce much higher luminosities and collision rates, which will enhance the possibility of finding new forces and particles. This upgrade of the LHC, to be completed in 2015, will be called the high-luminosity LHC (HL-LHC).
Figure 2.9 The ATLAS detector. Note the person standing at the base, for scale. © CERN for the benefit of the ATLAS Collaboration
The main experimental program at the LHC is based on proton–proton collisions. However, one month is set aside every year to perform heavy-ion collisions, mainly with lead ions. They are first accelerated by the LINAC 3 linear accelerator, and the low-energy ion ring is also used as an ion storage system. This experiment aims to produce the quark–gluon plasma that is thought to be the initial stage of matter in the early universe. The detector used to study heavy ion collisions is the ALICE detector.
The main purpose of the LHC is to probe the constituents of the standard model of particle physics and to look for exotic phenomena such as mini black holes, extra dimensions of space, supersymmetric particles, and especially the Higgs boson. If the current suggestions of the Higgs boson are confirmed by the HL-LHC, that discovery will reveal the mechanism underlying electroweak symmetry breaking. It will reveal the mechanism that generates the masses of the elementary particles, if indeed a Higgs-type mechanism is responsible for this. If the new boson discovered at the LHC turns out not to be the standard-model Higgs boson, however, then physicists hope that the LHC will be able to identify another mechanism responsible for the symmetry breaking in the electroweak theory.
Figure 2.10 © Edward Steed, New Yorker Magazine
The LHC is also investigating the nature of dark matter, the dominant and invisible substance postulated to exist throughout the universe to account for the data that show stronger gravity in galaxies, clusters of galaxies, and cosmology than is predicted by Einstein’s general relativity theory. The dark matter particles that the LHC is looking for are stable supersymmetric partners of the standard-model particles. So far, no supersymmetric particles have been found at the LHC, including any that could be identified as dark matter.
The building of the LHC was accomplished by an extraordinary consortium of countries contributing to the $9 billion cost of the project. With the machine running for some time, it was hoped that a major new discovery such as the Higgs boson or detection of a supersymmetric particle could be announced. Experimentalists at CERN worried that no new major discovery would be made before the proton–proton collisions ended in late 2012, when the machine would turn to heavy-ion collisions and a two-year period of maintenance and upgrading. In view of the 2011/2012 financial crisis in Europe, such a null outcome could mean that the funding of the machine would be reduced, perhaps leading to another superconducting “supercollider” (SSC)-like setback for particle physics. However, with the discovery of a new boson at 125 GeV, announced on July 4, 2012, with properties consistent with a Higgs boson, a new era of particle physics has begun, and the future running of the LHC seems secured.
3
Group Theory and Gauge Invariance
To understand particle physics, it is necessary to explain some of the underlying mathematics; otherwise, a deeper insight into the subject is not possible. Many of the basic features of particle physics are mathematically abstract. Because particle physics—as opposed to gravity or other macroscopic physical phenomena—takes place at extremely small distance scales, we use quantum mechanics and quantum field theory to explore the interactions of particles and fields.
SYMMETRY AND MATHEMATICAL GROUPS
A notion that has played a dominant role in classical physics as well as quantum mechanics and particle physics is symmetry. We can describe fairly easily what is known as the symmetry of objects. Take, for example, a solid cube. Every cube has the same features—namely, six square faces and eight corners. All the faces and corners look the same. If you draw an axis through the cube and rotate it 90 degrees around this axis to return where you
began, then you get back the same object—an identical face of the cube. This symmetry property of a cube is called an invariance, and the rotation is a transformation on the cube. The pyramid is another object with faces and corners that are the same as the faces and corners of other pyramids. These objects belong to the class of regular polyhedrons.
However, in classical physics and particle physics, we have to broaden our understanding of symmetry or lack of symmetry, because we are not dealing with the symmetry of physical objects. Instead, we are considering something much more abstract—the symmetry or lack of symmetry of the laws of nature. Humans have a desire to see symmetries in nature. From ancient astronomers up to Johannes Kepler during the 16th and 17th centuries, astronomers insisted on the orbits of the planets being circular, because the circle and sphere were perfect symmetric shapes. However, as Kepler discovered, the orbits of the planets are not circular orbits; they are ellipses. That is, they do not exhibit rotational invariance (i.e., symmetry). The elliptical orbits of the planets depend on a direction in space—the major axis of the ellipse points in a certain direction—and therefore they cannot be rotationally invariant. Isaac Newton discovered from the equations of motion contained in his laws of mechanics and gravity that both circular and elliptical orbits were solutions of his equations. However, if the solutions of Newton’s equations of motion were purely circular orbits, then these solutions would describe orbits that are symmetric—independent of direction in space.
Physicists often talk about “broken symmetries,” particularly in particle physics. In terms of the previous example, the elliptical orbits of the planets are considered broken symmetries compared with the concept of circular orbits. To understand a broken symmetry, we have to define the initial meaning of a symmetry before we break it. This is done mathematically through what is called group theory. The mathematical system of transformations that can reveal the symmetry or invariance of a law of nature is called a group. Group theory was developed first by mathematicians, who were not concerned with applying their results to the laws of nature. Only later did physicists adopt the mathematical results to understand symmetry in nature.
The mathematical notion of groups was invented primarily by Evariste Galois, a young French mathematical genius who, during his short lifetime in the 19th century, discovered the properties of groups while investigating solutions of algebraic equations. Norwegian mathematician Niels Henrik Abel, in his similarly short lifetime, at age 19 derived the important result that there is no algebraic solution for the roots of quintic, or fifth-order, equations, or indeed of any polynomial equation of degree greater than four, if one uses just algebraic operations. In theoretical physics, the adjective Abelian, derived from Abel’s name, has become so commonplace in the physics literature that the initial “A” is now often written in lowercase. Galois proved independently the same result as Abel. To carry out the proof, they both invented the mathematics of group theory.
Another Norwegian mathematician, Sophus Lie, born in 1842, developed the mathematics of group theory further and invented what we call Lie groups (pronounced “Lee” groups). A Lie group describes a continuous set of transformations, in contrast to a discrete or discontinuous set of transformations. For example, the rotation of the cube through 90 degrees can be done in a continuous way, from zero to 90 degrees, or in a discontinuous way by breaking up the transformation into intervals of, for example, 20 degrees, then 30 degrees, and finally 40 degrees. Let’s investigate these concepts—abelian and nonabelian groups, group theory, and Lie groups.
A mathematical “group” combines pairs of elements in a set using an associated operation or rule. For example, the operation could be addition; you choose a pair of whole integers and add them together to get another whole integer. The resulting number then belongs to the original set, or group, of whole integer numbers. In other words, whole numbers form a mathematical group. Equivalent operations on a pair of elements would include subtraction, multiplication, and division, whether the elements are whole numbers or fractions. However, there are restrictions on the definition of a group.
A group must satisfy four criteria or axioms of group theory. The first is that the elements must be “closed” when the operation on a pair of elements is performed. In other words, the elements close in on themselves; they cannot become anything through the given operation other than members of the group to which they belong. For example, if you add 4 and 6, you get 10. You don’t get 10½ or 10.7. Ten is a whole number that belongs to the original set of whole numbers.
The second axiom is the “associative rule.” It says that if you have three elements, a, b, and c, you can combine them either as (a + b) + c or as a + (b + c) and they will give the same answer. Similarly, you can combine (a × b) × c or a × (b × c) and you will get the same answer; that is, it doesn’t matter what the order of the operations is. One might consider that combining b + c first and then adding a could give a different answer, but it doesn’t. In physics, the associative rule has not been considered of great significance, since most of the physical applications of group theory assume that the associative rule is always true.
The third axiom of group theory is that the group has an “identity” element for a given operation. This means, for example, that the identity “I” is such that I × a = a. This rule says that for two elements, a and b, we have a × b = I, where I is the identity element.
This leads to the fourth axiom of group theory, that there exists an “inverse” for every element of the group, using the concept of group identity. That is, b is equal to I/a, where b = I/a defines the inverse of the operation a × b. Thus, this fourth axiom states, in effect, that every element in the set or group must have an inverse.
An important characteristic of group theory is that if you take two elements, a and b, and multiply them together, then a × b may or may not be equal to b × a. This is called the rule of commutation of elements. Indeed, matrices in mathematics do not necessarily commute when multiplied together. A matrix is a mathematical abstraction consisting of a block of symbols in rows and columns. When Heisenberg developed his first quantum mechanics, he did not understand that when deriving experimental spectral lines from his new quantum theory, he was using matrices that did not commute. His professor, Max Born, pointed out this important fact when Heisenberg described his new quantum mechanics to him. Here is where Niels Henrik Abel left his indelible mark on mathematics and physics; the groups for which the commutative law holds—that is, for which a × b = b × a—are called abelian groups, whereas those that do not satisfy the commutative law are called nonabelian groups.
As we will discover in the following chapters, our understanding of the standard model of particle physics depends critically on the mathematics of group theory—and on one of the two basic kinds of groups in group theory. We will find that a significant part of the standard model is based on nonabelian groups, those in which the elements of the group do not commute. These groups have a continuous set of elements and are a type of Lie group. The elements within them could be representations of particles or particle fields. On the other hand, groups containing a finite or discrete set of elements are commonly used in chemistry and studies of crystals and crystallography.
DEEPER INTO GROUPS
Let us consider some special cases of continuous groups, such as the group of rotations known as O(2). This is the rotation group of a two-dimensional plane, which leaves invariant a circle inscribed on a sheet of paper. Visualize an arrow pointing from the center of the circle to the perimeter of the circle. Like a combination lock, this arrow can be rotated clockwise or counterclockwise through an infinite number of angles about the fixed central point. The order of the operations of rotation is not important; that is, the order of rotations commutes. Therefore O(2) is an abelian group.
Another continuous group is U(1), in one complex dimension. This dimension is described by a complex number a + ib, where i is the imaginary number equal to the square roo
t of −1. U(1) is the group of complex numbers for which the product of two complex numbers is also a complex number, and hence they form a group under ordinary complex multiplication. These two continuous groups, O(2) and U(1), are, in fact, the same because the complex number a + ib is actually composed of two real numbers, a and b, such that the one complex dimension is equivalent to the two “real” dimensions, a and b, of O(2). U(1) is designated the unitary abelian group in one dimension.
We now step up the number of dimensions to three, so that we have three axes x, y, and z. From these three-dimensional Cartesian coordinates, we can form the continuous group O(3). Picture a rectangular box, or a book, aligned with the x-, y-, and z-axes. We can rotate this object about the x-, y-, or z-axis. It now turns out that a 90-degree rotation around the x-axis, the width of the book, followed by a 90-degree rotation around the y-axis, the height of the book, does not give the same orientation as rotating 90 degrees around the z-axis, the thickness of the book, and then another 90 degrees around the y-axis. It follows that the operations of rotation do not commute. This means that O(3) is a nonabelian group.
An interesting feature of the group O(3) is that we can use a finite number of elements to generate an infinite number of resulting elements, a feature that places this group in the class of Lie groups. Thus, in a Lie group, instead of considering the infinite number of rotations of the box around the x-, y-, and z-axes, we consider just the finite number of “generators” that produce the infinite number of elements of the group. In O(3), there are three generators—namely, the rotations measured by the angles θ x, θy, and θ z which are related to the three axes x, y, and z. The reason that O(3) is a Lie group is because the rotations are not discrete; they are continuous.
Cracking the Particle Code of the Universe Page 8
