Invitation to a Contemporary Physics (2004)

which contains SU(5) as a subgroup. We will discuss these two cases separately below.

There are 8 + 3 + 1 = 12 gauge bosons in the SM group SU(3)C × SU(2)L × U(1)Y . A GUT contains all those gauge bosons, and more. Below µGUT, the GUT group breaks down into the SM group, presumably triggered by the formation of a vacuum condensate that contains a GUT quantum number but no SM quantum number. The gauge bosons not in the SM will then interact with the condensate to get a mass, of the order of µGUT (possibly multiplied by some coupling constants).

There are 15 left-handed fermions per generation in the SM. Those in the first

¯

generation are the u and d quarks, and the left-handed u¯ and d anti-quarks, each coming in three colors, making a total of 12. Then, there are two left-handed leptons, e− and νe, as well as one left-handed anti-electron, e+, making a grand total of 15. There is no anti-neutrino with a left-handed helicity in the SM. These 15 fermions must fit into one or more multiplets in the GUT group. Since they are massless in the SM at high temperature, they are expected to be massless in the GUT theory at high temperature as well. Neutrino oscillation experiments (Sec. 9.6.3) indicate that right-handed neutrinos νR might exist. If so, we should include νR = (¯ν)L in the GUT multiplets as well.

9.7.1.1SU(5)

The multiplets of SU(5) are specified by four non-negative integers (see Sec. 9.4.2): (p1p2p3p4). In particular, the fundamental multiplets are 5 = (1000) and 5 = (0001), and the adjoint (gauge-boson) multiplet is 24 = (1001).

In addition to the 12 SM bosons, there are 12 additional ones which we will call X and X. X is a color triplet (3) and an isospin doublet. X, its antiparticle, is a conjugate color triplet (3 ) and an isospin doublet. The emission of X can change a quark into a lepton, and that will cause the proton to decay, say into e+π0. The decay rate is severely limited by the ultra short range of the X-exchange force, because its mass is very large. Experimentally, the proton has not been found to decay; its lifetime into e+π0 is larger than 1032 years. This experimental limit places considerable constraint on the details of the SU(5) theory.

Their CP partners are right-handed fermions, and they are fitted into the multiplets 5 and 10 .

We can use Table 9.2 to compute the sum of Y 2 for the members of 5, or 5 . It is 3 × (2/3)2 + 2 × (1)2 = 10/3. The corresponding additive quantum

number Q1, which is correctly normalized, namely, whose square sum is 12 , is then

Q1 = 3/20 Y . The coupling constant g1 so that g = g1 = g2 = g3 at µ = µGUT is

defined by g1Q1 = gY (Y/2). Hence, g1 = 5/3 gY , as previously claimed.

376 Elementary Particles and Forces

Remember from Sec. 9.5.3 that the Weinberg angle is defined by tan θW = gY /gI,

running of coupling constants between these two energy scales takes care of this di erence.

SU(5) breaks down into the SM group by forming a vacuum condensate. There are many ways of doing that, each giving di erent results for masses and other relations. Unfortunately, none of them seems to be both successful and simple.

9.7.1.2SO(10)

The vector defining the quantum number of SO(10) is 45 dimensional, which is also the dimension of its gauge multiplet. The (color, isospin) content of this multiplet is given by

45 = (3, 1) + (1, 2) + (1, 1) + (3, 2) + (3 , 2) + (3 , 1) + (3, 2)

+ (1, 1) + (3, 1) + (3 , 2) + (1, 1) + (1, 1) . (9.4)

The first line is the 24 of SU(5), so it contains the 12 SM gauge bosons and the 12 X,

¯

X bosons. The second line contains 20 more bosons in the 10 and 10 multiplet of SU(5). The last line is an additional singlet in SU(5). After the vacuum condensate is formed, the SM bosons remain massless but the other 33 bosons must gain a large mass.

Since SO(10) contains SU(5) as a subgroup, it is also possible that SO(10) is conserved at an even higher energy scale, and then it breaks down to SU(5) before µGUT. However, it does not have to do it that way.

There is a 16 multiplet in which all the 15 left-handed fermions and νR = (¯ν)L are fitted in. This is an appealing feature of SO(10) over SU(5), in that the righthanded neutrinos suggested by neutrino oscillations are automatically included, and that all the fermions are contained in a single multiplet, not two as is the case in SU(5).

The details of the outcome again depend on the scalar particle contents, and how the vacuum condensate is formed.

9.7.2Supersymmetry

The symmetries in SM and GUT produce multiplets of particles of the same spin and statistics. Supersymmetry (SUSY) is a hypothetical symmetry whose multiplets contain an equal number of bosons and fermions.

Since bosons have integral spins and fermions have half integral spins, a SUSY multiplet necessarily involves particles of di erent spins and statistics.

With this symmetry, every SM particle is doubled up with a partner of the opposite statistics. The partner of a lepton of a given chirality is a spin-0 particle called a slepton, the partner of a gauge particle is a spin-12 particle called a gaugino,

the partner of a Higgs boson is a spin-12 particle called a Higgsino, and the partner of the graviton is a spin-32 particle called the gravitino.

None of these SUSY partners have yet been found. If SUSY is correct, these partners must have high masses, which means that SUSY is broken. There are many suggestions as to how SUSY breaking takes place, but none have been universally accepted.

There are theoretical reasons to believe these new particles to have masses in the TeV (1,000 GeV) region, so there is hope that they may be found in the Large Hadron Collider (LHC), under construction at CERN, or some other high-energy accelerators in the future.

With no SUSY partners found, why do people think that SUSY might exist? The reason is primarily because exact SUSY brings with it many nice properties, and some of those are preserved at least qualitatively even when SUSY is broken.

What then, are the nice properties of SUSY that warrant the introduction of this whole zoo of unobserved particles?

Recall that quantum mechanics allows the appearance and disappearance of virtual particles. In a SUSY theory, the probability amplitudes of creating and annihilating a virtual boson-antiboson pair is opposite to that of a fermion-antifermion pair, because of their opposing statistics. As a result, when the two amplitudes are added together, they cancel each other as if no virtual pairs were present, at least for some purposes.

The mass of a particle is a ected by its surrounding Yukawa cloud, as discussed in Sec. 9.5.5. Without virtual pairs, there is no cloud, the mass will not be a ected by interactions and it will not be renormalized.

Similarly, the energy of the vacuum is not shifted in the presence of a SUSY interaction. In fact, it can be shown that the energy of the vacuum must be zero in that case.

However, SUSY is broken, so the cancelation between the boson-antiboson pairs and the fermion-antifermion pairs is not complete. The Yukawa cloud is back, though reduced in strength compared to the one without SUSY. Mass shifts are back, but in reduced amounts. Vacuum energy is shifted, but not by as much as without SUSY. Above the SUSY breaking scale µSUSY, exact SUSY is restored, so the shifts should somehow depend on this scale.

The mass shift of a fermion is fairly insensitive to this scale, but not so for a spin-0 boson. If this scale is too high, the mass of the Higgs boson H0 will be shifted too much for us to understand why we can still keep it among the SM particles. This is known as the hierarchy problem. For the H0 mass to be tolerable, this scale cannot be too high, and that implies the SUSY particles will make their appearances in the TeV region.

Broken SUSY also helps the unification of coupling constants. As mentioned in the last subsection, the three lines in Fig. 9.5a do not intersect at a single point in light of the precision data now available. The introduction of SUSY partners

around 1 TeV increases the number of virtual pairs, and reduces the slopes of the lines from 1 TeV on. With this change, the three lines now do meet at a point, as seen in Fig. 9.5b, so the possibility of grand unification is restored.

There is another very attractive feature of SUSY, broken or not. We know that the universe is full of cold dark matter, which is some five times more abundant than the usual matter (see Sec. 10.3.4). If the lightest SUSY partner is stable, then it qualifies quite well as a cold dark matter candidate. They are therefore being searched for in the dark matter around us, though so far none have been detected.

9.7.3 Extra Dimensions

A cherished dream of Albert Einstein was to unite the theory of gravity and electromagnetism. Although he never succeeded in his lifetime, a novel attempt in that direction by Theodor Kaluza in 1919, later improved by Oskar Klein in 1926, pioneered an important idea that is still being explored today.

Mathematically, Einstein’s gravity is described by a two-index object gµν, which is symmetric in µ and ν (a symmetric tensor). Electromagnetism is described by a one-index object Aµ (a vector). The indices µ and ν are either 1, 2, 3, or 4, indicating in that order the component along the x-axis, the y-axis, the z-axis, and the time-axis. Kaluza suggested that if there was an extra spatial dimension, then fivedimensional gravity would automatically contain a four-dimensional gravity and a four-dimensional electromagnetism. The reason is simple. Five-dimensional gravity is described by gMN , where M, N now range over 1, 2, 3, 4, and 5. Namely, µ and 5. The gµν components of gMN describe four-dimensional gravity, the gµ5 = g5µ components describe four-dimensional electromagnetism. In addition, it also contains a four-dimensional scalar field g55.

Klein modified this idea by assuming the extra dimension to be a circle, rather than an infinite line used in Euclidean geometry.

Klein’s theory possesses a rotational symmetry in the fifth dimension. Since the extra dimension gives rise to electromagnetism, the conserved quantum number for which symmetry is the electric charge. We may now regard the electric charge to be on the same footing as energy, momentum, and angular momentum, since all of them are symmetries of the five-dimensional spacetime.

More spatial dimensions have since been introduced. For example, the so-called superstring theory requires six extra spatial dimensions, and the M theory requires seven. If the internal space (extra dimensions) possesses a larger symmetry, one might imagine the other SM quantum numbers also to be a consequence of the symmetry. Whether that is the true origin of internal quantum number is presently unclear.

What is the size R of these extra dimensions? We have never seen it in daily life, so it must be small. We can actually do better than that by giving an estimate of how small it must be. Let Np = (pNe, Npi) be the momentum of a particle, with Npe its components in the usual four-dimensional space, and pNi its components in the extra

dimensions. The uncertainty principle allows internal momentum pi to be either 0, or some discrete multiples of /R. According to the theory of special relativity,

m2 ≡ M2 as the mass square of an excited state, separated from m2 by an amount (pi/c)2 = (n /Rc)2, for some integer n. Current accelerators can produce and detect masses of the order of 1 TeV, and none of these excited particles have been seen. From that we can estimate the size of the extra dimensions to be smaller thanc/(1 TeV) 10−19 m.

In the last few years, inspired by the discovery of branes in superstring theory, another possibility was proposed in which the extra dimension could be as large as 0.1 mm.

In this braneworld scenario, it is assumed that none of the SM particles can escape the confines of our four-dimensional spacetime, known as a 3-brane. Only SM singlets like the graviton and the right-handed neutrino νR may roam freely in the extra dimensions. Since the SM particles are confined to four dimensions, they have no Kaluza–Klein excited states, so their absence cannot be used to limit the size of the extra dimensions. We must somehow use gravity, or the right-handed neutrino to do it.

The gravitational force between two particles of masses m1 and m2 follows an inverse square law, Gm1m2/r2, where G is the Newtonian gravitational coupling constant. As explained in Sec. 9.3.2, this inverse square law originates from the surface area 4πr2 for a sphere of radius r. In a spatial dimension of 3 + n, the surface area of the sphere is proportional to r2+n. The gravitational force in that case is G m1m2/r2+n, with G being the gravitational coupling constant for the higher-dimensional theory. This r dependence is correct for r < R, the size of the extra dimension. For r > R, the sphere with radius r will be squashed to a size R along the extra dimensions, so the corresponding surface area will be proportional to Rnr2. The force law for r > R is then κG m1m2/(Rnr2), where κ is a calculable geometrical factor. So even in the presence of extra dimensions, the inverse square law is restored for distances r > R. This property can be used to measure R, simply by reducing r until a deviation from the inverse square law is found. Current experiments detect no such deviation down to about 0.1 mm, so R cannot be larger than that. However, a size almost as large as 0.1 mm is not yet ruled out.

By comparing the inverse square law in four dimensions, and in higher dimensions in the region r > R, we see that G = GRn/κ. So G can become fairly large for large R or n. If R = 0.1 mm and n = 2, G will become su ciently large to render gravity strong at the TeV scale, or an equivalent distance of c/(1 TeV) 10−19 m, making gravitational radiations and other e ects possibly detectable in the next round of accelerators.

Another way to probe such extra dimensions is through neutrino oscillations. As discussed in Sec. 9.6.3, neutrino oscillations require the neutrinos to have a finite

mass, which in turn suggests the possible presence of a right-handed neutrino νR, which is an SM singlet. If so, this neutrino can roam in the extra dimensions, and be detected through neutrino oscillation experiments. The unusual properties of the neutrinos, with tiny masses and large mixings, might be attributed to the fact that νR can see the extra dimensions but other fermions cannot. Unfortunately, there are as yet an insu cient amount of data either to support or to refute this idea.

9.7.4 Superstring and M Theories

The fundamental entity in a superstring theory is not a particle, but a string. Its various vibrational and rotational modes give rise to particles and their excited states. Supersymmetry is incorporated into the formulation. Without it, a tachyon appears, which is a particle with a negative mass square. This is not allowed in a stable theory, hence supersymmetry.

A supersymmetric theory of particle physics contains an equal number of bosons and fermions. Similarly, a superstring theory brings in a fermionic string as the partner of an ordinary (bosonic) string. A bosonic string is just like a violin string, but it is a rather harder to get an intuitive understanding of what a fermionic string is, because there are no analogs in daily life. But it can be written down mathematically.

The lowest superstring modes have zero mass. The excited modes have higher masses and possibly higher spins. The mass squares are separated by constant gaps, given by a parameter usually designated as 1/α . In the limit α → 0, only massless states remain, and string theory reduces to the usual quantum field theory.

There are two kinds of fundamental strings, open or closed. An open string has two ends, which are free to flap around. A closed string does not. Later on, we shall discuss strings whose two ends are tied down, but that necessitates the introduction of objects which the string is tied down to. These objects are called D-branes.

There is a simple property of a vibrating string which is not shared by a vibrating membrane (a drum head), or a vibrating solid. This property is responsible for the magical features of the string theory, which we will describe.

A pulse propagating along a (closed or infinite) string does not change its shape, whereas a pulse propagating in a membrane or a solid does. Imagine now an additive quantum number, like the electric charge, carried by the pulse. Since the shape of the pulse does not change, the charge carried in any segment of the pulse will still be conserved. In this way a string generates an infinite number of conserved quantities, each for a di erent segment of the pulse. This infinite number of conserved quantities corresponds to a huge symmetry, known as the conformal symmetry. It is this symmetry that allows very nice things to happen in the string theory.

For example, it tells us how strings interact, by joining two strings into one, or by splitting one string into two. This symmetry guarantees such interactions to be

unique, and specified only by a coupling constant gs, telling us how often such acts happen, and nothing else.

This description of string interaction is useful only when gs is not too large. In other words, it is a perturbative description. We do not quite know how the string behaves when the coupling is strong, but hints are available, some of which will be discussed later.

This conformal symmetry present in the classical string is destroyed by quantum fluctuation unless the underlying spacetime is ten-dimensional. Thus there are six extra spatial dimensions in a consistent quantum superstring theory. To preserve supersymmetry, this compact six-dimensional space must be a special kind of mathematical object known as Calabi–Yau manifolds.

We saw in the Kaluza–Klein theory (Sec. 9.7.3) that a single object in a fivedimensional spacetime (the gravitational field) can decompose into several objects in a four-dimensional spacetime (the gravitational, electromagnetic, and scalar fields5). This one-to-many correspondence is again there when we view the ten-dimensional superstring in four dimensions at the present energy. The detailed correspondence depends on what Calabi–Yau space we choose, and how to break down the internal and super symmetries to get to the Standard Model. There is a great deal of arbitrariness in such a process, which we do not know how to fix in perturbative string theory. It is partly for that reason that superstring theory has yielded no experimentally verifiable predictions so far. It is also why we must find a way to deal with the non-perturbative aspects of the theory.

The internal symmetry group of superstrings is also restricted by our desire to preserve quantum conformal symmetry. A careful analysis shows that there are only five types of consistent superstrings. They are, an open string which is called Type I, whose internal symmetry is SO(32), two closed strings called Type IIA and Type IIB, and two ‘heterotic’ strings whose internal symmetry is either SO(32) or E8 × E8. We have not explained what the groups SO(32) and E8 × E8 are. Su ce it to know for our purpose that they give rise to well-defined vectorial quantum numbers and multiplets, and the multiplicity of the adjoint (gauge) multiplet of both is 496.

Each of these five superstrings has its own spectrum. Since 1/α is likely to be much larger than the available experimental energy, it is the zero mass particles in each case that are of the most interest. In this regard, all particles in the SM are assumed to be massless in the high energy limit at which the string theory is supposed to operate.

The spectra are supersymmetric; every boson has a fermionic partner. In what follows we shall describe only the bosonic part. It should also be pointed out that the Type I string also has a closed string sector, obtained by gluing two open strings together end to end. In other words, a closed string is present in all five superstrings.

5A classical field may be thought of as the wave function of a single particle, and a quantum field may be thought of as a collection of many particles.

Now onto the massless spectra. The gravitational tensor gµν (µ, ν = 1, 2, . . . , 10) is contained in every superstring. So is an antisymmetric tensor Bµν , and a scalar φ called the dilaton.

In addition, the Type I and the SO(32) heterotic string theory all contain an SO(32) gauge field Aµ, and the E8 × E8 heterotic string contains an E8 × E8 gauge field Aµ.

Type IIA also contains totally antisymmetric tensors with 1, 3, 5, and 7 indices, and Type IIB also contains totally antisymmetric tensors with 2, 4, 6, and 8 indices. These are known as the Ramond–Ramond (RR) potentials. Besides this, these two theories are also di erent in the fermionic sector. The fermions in a IIA theory have both (ten-dimensional) chiralities, whereas the fermions in a IIB theory have only one.

We know what the gravitational and gauge fields are, albeit only in fourdimensional spacetime. We have also encountered the scalar fields, which are objects needed to form vacuum condensates. What generates these anti-symmetric objects, Bµν and the RR potentials in a Type II theory?

A p-dimensional object, when traced through time, generates a (p + 1)- dimensional spacetime manifold called a world volume. There is a natural way to couple mathematically an antisymmetric tensor with (p + 1) indices to a (p + 1)- dimensional world volume. With this coupling, we may regard the p-dimensional object to be the source of the antisymmetric tensor with (p + 1) indices. The presence of these antisymmetric tensors therefore suggests the presence of these p-dimensional objects in a superstring theory.

For Bµν, the one-dimensional source is just the string itself.

For the RR potentials with (p + 1) indices (p is even for IIA, and odd for IIB), the source must be some p-dimensional object, called a p-brane. It can be shown that the 1-brane cannot be the original string, it has to be something else.

We have introduced these p-branes to be the source of the RR potentials. But what are they? They are certainly not present in the original formulation of the perturbative string theory. If they are really there, they must come from the nonperturbative sector of the string theory. We may therefore regard the RR potentials as something that can guide us to an understanding of the elusive non-perturbative string theory.

In what follows we will try to motivate the existence of p-branes in another way: through T-duality.

There are certain excited modes in the closed string which are not present in an open string, nor a particle. That happens, for example, when one of the extra dimensions (say, the 9th) is a circle. A closed string can wrap around the circle, one or more times, like a rubber band. To do so, it has to stretch, and that costs it a certain amount of energy. This energy (actually c times the momentum along the 9th dimension) depends on the length of stretch, and the sti ness of the string. It is equal to wR/ cα , where w is the number of times the string is wrapped around the circle of radius R, whose circumference is 2πR. The proportionality constant

1/2π cα , which has the correct dimension of energy/distance (because 1/α has the dimension of square energy) tells us how sti the string is. We may actually use this relation to define the parameter 1/α used in measuring the mass square gaps of the excited states.

Winding is an excitation mode unique to closed strings. There are excitation modes common to everything, obtained by having a non-zero momentum component along the 9th direction. This Kaluza–Klein momentum is given by the uncertainty principle to be n /R, for some integer n. The mass square of the excited state of a closed string thus receives a contribution (n c/R + wR/ cα )2 from the 9th dimension. This expression remains the same if we make the swap n ↔ w, R ↔ ( c)2α /R ≡ R . Thus, from an energetic point of view, there is no way to distinguish R from R , provided we also interchange n and w. This is known as

T -duality.

Note that supersymmetry has not been mentioned in the discussion. T -duality is true for any closed bosonic string. Note also that the duality is between two perturbative strings, with radii R and R , respectively. The non-perturbative e ect

That leaves the open string, which does not have a winding mode. So does it have a T -duality symmetry? If so, what is dual to the Kaluza–Klein momentum n/R?

An open string always has a closed string sector, which obeys T -duality. So it is very hard to imagine that the duality does not apply to the open string sector as well. In order to see what n/R is dual to, we go back to the closed string for guidance.

In a closed string, a pulse traveling in one direction will do so forever, because there are no ends to reflect it. We are particularly interested in the pulse along the 9th direction. The amplitude for the pulse traveling to the right will be denoted fR(τ − σ), and the amplitude for the pulse traveling to the left will be denoted by fL(τ + σ). The parameter σ, with values between 0 and π, is a scale painted on the string to tell us where we are on the string. The parameter τ is just time. The total amplitude is therefore f(τ, σ) = fR(τ − σ) + fL(τ + σ).

Under a T -duality transformation, n ↔ w. This can be shown to be equivalent to flipping the sign of the right-moving amplitude. In other words, under this transformation, f(τ, σ) → f (τ, σ) = −fR(τ − σ) + fL(τ + σ). This now is a definition of T -duality which we can apply even to the open string.

The result is the following. The T -dual of an open string whose two ends are free to move in space, is an open string whose two ends are separated in the 9th dimension by a fixed distance 2πnR . Actually, further consideration shows that we can alter this distance, but whatever it is, the 9th coordinates of the two ends are fixed when the string vibrates and moves. There is no restriction on the other coordinates for the ends.

384 Elementary Particles and Forces

We can now imagine having two parallel 8-branes, each having its 9th coordinate fixed. The dual of the open string is now ‘tied’ to these two branes as it vibrates. These branes are called D-branes, or in this case D8-branes, because ends with a fixed (9th) coordinate are said to obey the Dirichlet boundary condition.

So far we have talked about the T -duality when one extra dimension is compactified into a circle. Similar discussions apply when a > 1 dimensions are so compactified. In that case the ends of the dual open string are tied down to two parallel D(9 − a)-planes.

These branes turn out to be the same ones that give rise to the RR potentials in a Type II theory. Since open strings end on them, you can pull the strings to deform them. They are indeed dynamical objects, though they do not appear originally in a perturbative string theory. They should be present in a full string theory including non-perturbative contributions.

An open string whose two ends are stuck to the same D-brane may have the middle o the brane when it is excited. However, such a configuration may be decomposed into a string lying completely in the D-brane, and a closed loop above it, as shown in Fig. 9.6. If we associate the open-string vibrations in a D3-brane with SM particles, then we are led to the braneworld scenario discussed in Sec. 9.7.3: that SM particles are always confined to our three-dimensional space. The closed

(a)

(b)

Figure 9.6: (a) A string with two ends restricted to a brane; (b) the same string viewed as the composite of two strings: an open string lying completely in the brane, and a closed string above it in the bulk.