Grand unification
Grand unification refers to gauge field theories in which the strong, weak, and electromagnetic interactions are described by a simple gauge group, with the quarks and leptons and their antiparticles combined in the same multiplets. They therefore relate the known microscopic interactions as different aspects of a simpler unified theory, in much the same way that electricity and magnetism were seen in the Maxwell theory to be aspects of a simpler electromagnetic interaction. Grand unified theories (GUTs) typically predict proton and bound neutron decay, though with an extremely long lifetime (e.g., much larger than \(10^{30}\) years) that has not yet been observed. They predict that the (distance-dependent) interaction strengths of the known interactions should become equal at short distance scales, lead to partially successful relations between quark and lepton masses, may be associated with small neutrino masses, may have implications for cosmology, and may lead to new gauge interactions that survive to low energies. Grand unified theories do not incorporate quantum gravity, but the even more ambitious superstring theories often include many of the aspects of grand unification, often involving grand unified groups in a higher-dimensional space that are broken to the standard \(SU(3) \times SU(2) \times U(1)\) model in the effective four-dimensional theory that is valid below the string scale.
Motivations
The standard model is a theory of the known microscopic interactions, combining the \(SU(2) \times U(1)\) electroweak theory of the weak and electromagnetic interactions with quantum chromodynamics (QCD), which is the modern theory of the strong interactions based on the \(SU(3)\) color group. The standard model is a gauge theory, which means in part that the interactions are mediated by the exchange of (apparently) massless spin-\(1\) gauge bosons. In addition, the interactions of the gauge bosons with spin-\(1/2\) and spin-\(0\) fields and with each other are dictated by an underlying gauge invariance. Once the gauge group and representations are specified, the gauge interactions are uniquely determined up to one gauge coupling constant per group factor. The gauge invariance does not allow elementary mass terms for the gauge bosons, suggesting long range forces. In fact, the strong interactions are short range due to the self-interactions of the \(SU(3)\) gluons (gauge bosons) and non-perturbative (strong coupling) effects at long distance scales, which prevent the colored quarks and gluons from propagating freely. The weak interactions are also short-ranged, but in this case it is because the associated gauge bosons acquire masses by some sort of dynamical or spontaneous breaking of the underlying gauge symmetry in the ground state (vacuum) of the theory. The simplest mechanism is the Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism, which predicts the existence of a (still unobserved) spin-\(0\) Higgs particle. Some of the properties of the fundamental interactions are indicated in Figure 1.
Problems with the standard model
The standard model is a mathematically-consistent renormalizable field theory. When combined with classical general relativity it successfully describes an enormous amount of observational data from accelerator, collider, non-accelerator, astrophysical, and cosmological experiments and observations. It has been stringently tested quantitatively and has made many successful predictions. Except possibly for the mechanism for electroweak symmetry breaking it is undoubtedly an approximately correct description of nature down to a distance scale 1/1000^{th} the size of the atomic nucleus. Nevertheless, the standard model is almost certainly not the ultimate description of nature. As stated, the electroweak symmetry breaking mechanism has not yet been established. In addition, there is more than one way to incorporate the observed small neutrino masses, the standard model does not contain satisfactory explanations of the observed dark matter and dark energy of the Universe, and does not include a quantum mechanical description of gravity. The more important problem, however, is that the standard model is extremely complicated and arbitrary; the properties of the strong, weak, and electromagnetic interactions are very different from each other; and the standard model requires an extreme fine-tuning of several parameters.
One way to illustrate the complications is that the standard model involves 20 free parameters if one ignores neutrino mass. Including the latter, there are another 7 or 9 parameters, depending on whether the neutrinos are Dirac (lepton number conserving) or Majorana (lepton number violating). Not included in the count are the electric charge assignments of the quarks and leptons, which are inputs rather than predictions. There are also a number of specific problems. These include:
- The gauge symmetry problem: the standard model is very complicated. It involves three distinct gauge groups with three gauge couplings. The electroweak part is chiral , i.e., it acts differently on left and right-handed fermions (spin antiparallel or parallel to momentum, respectively), leading to parity \((P)\) and charge conjugation \((C)\) violation, while QCD is non-chiral. Only the electroweak part is spontaneously broken. Charge quantization, i.e., the equality of the magnitudes of the proton and electron electric charges, leading to the electrical neutrality of ordinary matter, is put into the theory but not explained. These issues suggest the existence of some underlying unification of the interactions, such as in a grand unified and/or superstring theory.
- The fermion problem: The first family of quarks and leptons \((u, d, e^-, \nu_e)\ ,\) is sufficient to describe matter under ordinary circumstances. However, two heavier repetitions (families), \((c, s, \mu^-, \nu_\mu)\) and \((t, b, \tau^-, \nu_\tau)\) have been observed in laboratory and cosmic ray experiments. The rationale for the existence of these families, the large hierarchy of their masses \((m_t > 10^5 m_{e^-})\ ,\) and the pattern of weak interaction transitions between the particles of definite mass (known as mixing) is not explained. The origin of the much smaller neutrino masses and even whether they are Dirac or Majorana is not understood. Complex phases associated with quark mixing can account for the tiny amount of \(CP\) violation observed in the laboratory, but is not sufficient to explain the baryon asymmetry, i.e., the observed excess of matter over antimatter in the universe. The fermion spectrum suggests some form of family symmetry, perhaps combined with grand unification and possibly derivable from an underlying superstring theory. Alternatively, the fermion mass hierarchy could be associated with wave function overlaps if there are additional small or warped dimensions of space.
- The Higgs/hierarchy problem: The simplest model of electroweak symmetry breaking involves the existence of a spin-\(0\) Higgs particle, with mass less than \(\sim 1000\) GeV \(\equiv\) \(1\) TeV. However, radiative corrections (loop diagrams) tend to push the Higgs mass up to the next higher physical scale, such as the quantum gravity (Planck) scale, \(M_P = G_N^{-1/2} \sim 10^{19}\) GeV, unless there is an incredibly fine-tuned cancellation between the initial (tree-level) value and the radiative contributions. The problem is aggravated in GUTs, where it is difficult for even the tree-level value to be much smaller than the unification scale. One possible solution is to invoke supersymmetry, broken at the TeV scale, which enforces a cancellation of the dominant loop corrections between the ordinary particles and their superpartners of a different spin (but still leaves the tree-level problem for supersymmetric GUTs). Another possibility is an alternative electroweak symmetry breaking mechanism, perhaps involving strong dynamics at the TeV scale. Yet another is that the fundamental scale of nature is much smaller than \(M_P\), which could occur if there are additional large or warped dimensions of space.
- The gravity problem: The standard model does not relate gravity to the microscopic interactions, and does not by itself lead to a consistent theory of quantum gravity. A promising possibility is superstring theory, which yields a finite theory of quantum gravity as well as the other interactions. However, superstring theory appears to have an enormous landscape of possible vacua, each leading to different low energy physics (some of which may include supersymmetry at the TeV scale), with no known principle to select one over the others. A related difficulty is that electroweak symmetry breaking contributes to the energy of the vacuum and acts as a cosmological constant, with a value some 50 orders of magnitude larger than the observed dark energy of the universe. Grand unified and superstring theories typically give a much larger contribution, some 120 orders of magnitude too large. Thus an incredibly fine-tuned cancellation between these and possible other contributions appears needed. One possibility is some sort of anthropic solution, perhaps associated with the landscape of superstring vacua and some form of eternal inflation.
These difficulties suggest that the standard model should be part of a more fundamental underlying theory, such as grand unification and/or superstring theory, perhaps combined with supersymmetry broken at the TeV scale.
The unification of gauge couplings
In the standard model (SM) or its minimal supersymmetric extension (MSSM), the gauge couplings (analogous to the electric charge \(e\) in electrodynamics) of the \(SU(3)\ ,\) \(SU(2)\ ,\) and \(U(1)\) groups are independent quantities, denoted \( g_3\ ,\) \(g_2=g\ ,\) and \(g_1 = \sqrt{5/3}g'\ ,\) respectively (the \(\sqrt{5/3}\) is due to an alternative normalization useful for low energy studies). Due to higher-order corrections, such as those in Figure 2, each of these is actually a function of the typical energy scale \(Q\) relevant to the process (\(Q \sim 1/r\ ,\) where \(r\) is the distance between two charged particles). This running of the gauge couplings is described by the renormalization group equations \[\tag{1} \frac{1}{\alpha_i (Q^2)} = \frac{1}{\alpha_i (M_Z^2)} - 4 \pi b_i \ln\frac{Q^2}{M_Z^2}, \]
where \(\alpha_i \equiv g_i^2/4\pi\ ,\) \(M_Z\) is the mass of the electroweak \(Z\) boson, and the coefficients are \[\tag{2} \left(\begin{array}{c}b_1\\b_2\\b_3\end{array}\right) =\frac{1}{16\pi^2} \left(\begin{array}{c}\ \frac{41}{10} \\-\frac{19}{6}\\-7\end{array}\right)_{SM} \text{ or }\quad \frac{1}{16\pi^2} \left(\begin{array}{c}\ \frac{33}{5}\\ \ 1\\-3\end{array}\right)_{MSSM}. \]
The difference is due to the contributions of the superpartners in the MSSM to the higher-order corrections.
In many grand unified theories the three gauge couplings are predicted to meet at some high energy unification (GUT) scale \(M_X\ ,\) above which the three interactions are unified and GUT symmetry breaking can be ignored. By around 1990 the gauge couplings at \(M_Z\) had been determined very precisely, allowing a test of this gauge coupling unification. As can be seen in Figure 3 the three gauge couplings do not quite meet when extrapolated using the SM model expression. However, the unification works quite well in the MSSM, with \(M_X \sim 3 \times 10^{16}\) GeV, motivating not only grand unification (or some superstring theories with similar properties) but also that supersymmetry emerges at around \(M_Z\ .\) This successful pictures survives when higher-order corrections to the running are included and somewhat higher superpartner masses (e.g., \(\sim 1\) TeV) are allowed, but can be modified for more complicated embeddings of the standard model into the GUT or superstring theory, or if there are additional thresholds (new particle mass scales) between \(M_Z\) and \(M_X\ .\)
Grand unified theories
The \(SU(4)_c \times SU(2)_L \times SU(2)_R\) model of Pati and Salam (1974) achieved a partial unification of quarks and leptons. The \(SU(4)_c\) group extends QCD to include a fourth color associated with the leptons, so that, for example, the three colors of \(u\) quark would be related to the \(\nu_e\) by the symmetry and a new interaction. The \(SU(4)_c\) symmetry was assumed to be spontaneously broken to \(SU(3) \times U(1)_{B-L}\) at a sufficiently high scale, where \(U(1)_{B-L}\) is associated with baryon number (\(B\)) - lepton number (\(L\)). The electroweak \(SU(2)_L \times SU(2)_R\times U(1)_{B-L}\) group is a left-right symmetric (parity conserving) version of the SM, eventually broken to \(SU(2)\times U(1)\ .\) Extensions of the model involved extended electroweak groups.
The Georgi-Glashow (1974) \(SU(5)\) model was the first full unification of \(SU(3) \times SU(2) \times U(1)\) into a simple group.
The (properly normalized) strong, weak, and electromagnetic gauge couplings were expected to unify at the scale \(M_X\) of \(SU(5\)) breaking
(Georgi, Quinn, and Weinberg (1974), Buras et al. (1977)), which worked reasonably well given the experimental precision of the inputs at the time,
suggesting \(M_X \sim 10^{14-15}\) GeV. Quarks, leptons, and their antiparticles are combined in the same \(SU(5)\) multiplets (each family is, however, still reducible),
leading to charge quantization. There are also new interactions mediated by superheavy \(X\) and \(Y\) gauge bosons (with \(M_Y \sim M_X\)) that connect quarks with antileptons and with antiquarks. These lead to the prediction of proton decay and also the decay of neutrons bound into nuclei that would otherwise be stable, as shown in Figure 4.
A typical decay mode is \(p \rightarrow e^+ \pi^0\ ,\) with an expected lifetime
\[\tag{3}
\tau_p \sim \frac{M_X^4}{\alpha^2m_p^5}\rightarrow 10^{29} \text{ yr}\]
for \(M_X \sim 10^{14}\) GeV. Detailed calculations allowed a somewhat longer lifetime, but nevertheless the original model
was eventually excluded by dedicated searches for proton decay in large underground experiments (to shield from cosmic rays). For example, the SuperKamiokande experiment
in Japan obtained a partial lifetime \( \tau_p > 1.3\times 10^{34}\) yr for the \(e^+ \pi^0\) mode.
In the supersymmetric version of \(SU(5)\) the lifetime from \(X\) and \(Y\) exchange is much longer (\(\sim 10^{38}\) yr)
because of the larger unification scale \(M_X \sim 3 \times 10^{16}\) yr, but there are other mechanisms involving the exchange of superpartners that lead to faster decays into
modes such as \(\bar{\nu} K^+\ .\) The original (simplest) version of the \( SU(5)\) model also made predictions relating quark and charged lepton masses, which are only partially successful. For general reviews of grand unification, see Langacker (1981), Hewett and Rizzo (1989).
The \(SU(5)\) model
\(SU(5)\) is the group of unitary \( 5 \times 5\)-dimensional matrices with unit determinant. The classical \(5\)-dimensional defining representation of the associated Lie algebra consists of \(24\) traceless, Hermitian matrices \(L^i=\lambda^i/2,\, i=1\cdots 24\ ,\) defined in analogy with the \( 2 \times 2\) Pauli matrices. It is convenient to choose a convention in which the upper left \( 3 \times 3\) block corresponds to the \(SU(3\)) (color) subgroup (with indices \(a,b=1,2,3\) denoted by \(\alpha\) and \(\beta\)). The lower right \( 2 \times 2\) block (with \(a,b=4,5\) denoted by \(r\) and \(s\)) corresponds to the electroweak \(SU(2)\) subgroup, while the off-diagonal blocks are associated with new gauge interactions that connect quarks with antiquarks and antileptons. Like the standard model, \(SU(5)\) has four diagonal generators. Three are associated with \(SU(3) \times SU(2)\ ,\) while the fourth is the normalized hypercharge generator \(Y_1=\text{Diag}(-2\ -2\ -2\quad 3 \quad 3)/\sqrt{60}\ .\) This is related to the traditional SM \(U(1)\) generator (hypercharge operator) \(Y=Q-T^3\ ,\) where \(Q\) and \(T^3\) are respectively the electric charge and third component of weak isospin, by \(Y_1\equiv \sqrt{\frac{3}{5}} Y\ .\) The extra \(\sqrt{\frac{3}{5}}\) ensures the correct normalization for the \(SU(5)\) generators, \(\text{Tr}(L^i L^j)= \delta^{ij}/2\ ,\) and leads to a relation between the two electroweak gauge couplings \(g\) and \(g'\) (which are arbitrary in the SM).
The representations for scalars, fermions, and gauge fields
\(SU(5)\) has \(24\) Hermitian gauge fields, \(A^i, i=1 \cdots 24\ ,\) transforming under the adjoint (\(24\)) representation. Under the \(SU(3) \times SU(2)\times U(1)\) SM subgroup, the \(24\) decomposes as \[\tag{4} \underbrace{(8,1,0)} _{G_\alpha^\beta} + \underbrace{(1,3,0)} _{W^\pm,W^0} + \underbrace{(1,1,0)} _{B} + \underbrace{(3,2^*,-{\frac 5 6})} _{A_\alpha^r} + \underbrace{(3^*,2,+{\frac 5 6})} _{A_r^\alpha},\]
where the numbers refer to the \(SU(3)\) and \(SU(2)\) representations and to the eigenvalue \(y=q-t^3\) of the weak hypercharge operator \(Y\ .\) \(A\) can be written as a \(5 \times 5\) matrix \(A = \sum _{i=1} ^{24} A^i {\lambda^i}/{\sqrt{2}}\ :\) \[\tag{5} A =\left( \begin{array}{ccc|cc} G^1_1 - \frac{2 B}{\sqrt{30}} & G^2_1 & G^3_1 & \bar{X}_1 & \bar{Y}_1 \\ G^1_2& G^2_2 - \frac{2 B}{\sqrt{30}} & G^3_2 & \bar{X}_2 & \bar{Y}_2 \\ G^1_3& G^2_3 & G^3_3 - \frac{2 B}{\sqrt{30}} & \bar{X}_3 & \bar{Y}_3 \\ \hline X^1 & X^2 & X^3 & \frac{W^0}{\sqrt{2}} + \frac{3 B}{\sqrt{30}} & W^+ \\ Y^1 & Y^2 & Y^3 & W^- & -\frac{W^0}{\sqrt{2}} + \frac{3 B}{\sqrt{30}} \end{array} \right).\]
The \(ab\) element of \(A\) is often written in tensor notation as \(A_{ab} \rightarrow A_a^b=(A_b^a)^\dagger\ ,\) where the upper index refers to the anti-fundamental (\(5^\ast\)) representation, and the lower to the fundamental (\(5\)). The diagonal entries are related by \(\sum_a A_a^a=0\ .\) In Equations (4) and (5), \(W^\pm=(W^1\mp i W^2)/\sqrt{2}\ ,\) \(W^0=W^3\ ,\) and \(B\) are the \(SU(2)\times U(1)\) gauge bosons, and \(G_{\alpha}^{\beta}\) are the \(SU(3)\) gauge fields (gluons), with \(G^\alpha_\alpha=0\ .\) The \(12\) new fields \[\tag{6} A_4^\alpha \equiv X^\alpha \ [3^\ast, 2, q_X = {\scriptstyle\frac 4 3} ], \qquad A_\alpha^4 \equiv \bar{X}_\alpha \ [3, 2^\ast, q_{\bar{X} }= - {\scriptstyle\frac 4 3} ], \qquad A_5^\alpha \equiv Y^\alpha \, \ [3^\ast, 2, q_Y = {\scriptstyle\frac 1 3} ], \qquad A_\alpha^5 \equiv \bar{Y}_\alpha \, \ [3, 2^\ast, q_{\bar{Y} }= - {\scriptstyle\frac 1 3} ] , \]
carry both \(SU(3)\) and \(SU(2)\) indices, with the representations shown along with the electric charge \(q\ .\)
For spin-\(1/2\) fermions is convenient to work in terms of left-chiral particles and antiparticles, where chirality coincides with helicity (spin with respect to the direction of momentum) for massless particles. Thus, left-chiral implies that the spin is opposite the direction of momentum for a relativistic particle. The right-chiral fields are not independent, but are related by a \(CP\) transformation. Each of the three families of fermion fields consists of \(15\) left-chiral particles and antiparticles, as well as the possible addition of an \(SU(5)\)-singlet right-handed neutrino field \(\nu^c_L\ .\) These transform under the reducible representation \(5^\ast + 10\ ,\) where the \(5^\ast\) is the anti-fundamental, and the \(10\) is the antisymmetric product of two \(5\)'s. The \(SU(3) \times SU(2)\times U(1)\) decomposition is \[\tag{7} \underbrace{5^*} _{\chi^a} \rightarrow \underbrace{(3^*,1,{\frac 1 3})} _{\chi^\alpha} + \underbrace{(1,2^*,-{\frac 1 2})} _{\chi^r}, \qquad \underbrace{10} _{\psi_{ ab}=-\psi_{ ba} } \rightarrow \underbrace{(3^*,1,-{\frac 2 3})} _{\psi_{ \alpha \beta}} + \underbrace{(3,2,{\frac 1 6})} _{\psi_{\alpha r}} + \underbrace{(1,1,1)} _{\psi_{ 45}},\]
and the field content is \[\tag{8} \chi^a = \left( d^{c1}\ d^{c2}\ d^{c3}\ e^-\, -\nu_e \,\right)_L^T, \qquad \psi_{ab}=\frac{1}{\sqrt{2}} \left( \begin{array}{ccc|cc} 0 & u^{c3} & -u^{c2} & -u_1 & - d_1 \\ -u^{c3} & 0 & u^{c1} & -u_2 & -d_2 \\ u^{c2} & -u^{c1} & 0 & -u_3 & -d_3 \\ \hline u_1 & u_2 & u_3 & 0 & -e^+ \\ d_1 & d_2 & d_3 & e^+ & 0 \end{array} \right)_L.\]
In Equation (8), the superscript \(T\) refers to the transpose, while the subscript \(L\) refers to left-chiral.
To break the \(SU(5)\) gauge symmetry down to \(SU(3) \times SU(2)\times U(1)\) one introduces a real adjoint representation of spin-\(0\) Higgs fields \(\Phi = \sum_{i=1}^{24}\Phi^i {\lambda^i}/{\sqrt{2}}\ ,\) which have the same quantum numbers as the gauge fields in Equation (5), as well as a complex Higgs fundamental multiplet \({H}_a = (\mathcal{H}_\alpha \quad \phi^+ \quad \phi^0)^T\ .\) The SM Higgs doublet \( \left(\begin{array}{c} \phi^+ \\ \phi^0 \end{array}\right)\ ,\) which transforms as \((1,2,\frac{1}{2})\ ,\) breaks \(SU(2) \times U(1)\) down to the electromagnetic \(U(1)_Q\) subgroup. Its \(SU(5)\) partner \(\mathcal{H}_\alpha = (3,1,-\frac{1}{3})\) is a color triplet with charge \(q_\mathcal{H}=-\frac{1}{3}\ .\)
The gauge and Yukawa interactions
The fermion kinetic energy terms and gauge interactions are described by the \(SU(5)\) Lagrangian density \[\tag{9} \mathcal{L}_f = \bar \chi_{a}i \left(\, \!\not\!\!D \chi \right)^a+ \bar \psi^{ab}i\left(\, \!\not\!\!D \psi \right)_{ab},\]
where the gauge covariant derivatives \(\!\not\!\!D\) are defined by \[\tag{10} \left( D_\mu \chi \right)^a= \partial_\mu \chi^a-i \frac{g_5}{\sqrt{2}} (A_\mu )_b^a \chi^b, \qquad \left( D_\mu \psi \right)_{ab} = \partial_\mu \psi_{ab}+ i \frac{g_5}{\sqrt{2}} (A_\mu )_a^c \psi_{cb} + i \frac{g_5}{\sqrt{2}} (A_\mu )_b^d \psi_{ad}. \]
\(g_5\) is the \(SU(5)\) gauge coupling constant. Similar expressions hold for the gauge covariant derivative of \(\Phi\) and \(H_a\ .\)
\(\mathcal{L}_f\) includes the SM fermion gauge interactions, with the additional constraint that \(g_3=g_2=g_1=g_5\ .\) Thus, the gauge couplings are expected to unify at scales \(Q\) above which \(SU(5)\) breaking can be ignored, as in Figure 2. Equation (9) implies additional gauge interactions involving the \(X\) and \(Y\) gauge fields, \[\tag{11} \begin{align} \mathcal{L}_f^{XY}=&-\underbrace{ \frac{ g_5}{\sqrt 2} \left[ \bar{e} _{R}^{+} \not \!{X}^\alpha d_{R\alpha} +\bar{e} _{L}^{+} \not \!{X}^\alpha d_{L\alpha} - \bar \nu^c_R \not \!{Y}^\alpha d_{R\alpha} - \bar{e} _{L}^{+} \not \!{Y}^\alpha u_{L\alpha} \right]}_{\text{leptoquark vertices}} \\ &\qquad - \underbrace{ \frac{ g_5}{\sqrt 2} \left[ \epsilon ^{\alpha \beta \gamma} \bar{u} _{L \gamma }^{c} \not \! \bar{X}_\alpha u_{L\beta} + \epsilon^{\alpha \beta \gamma} \bar{u} _{L \gamma}^{c} \not \! \bar{Y}_\alpha d_{L\beta} \right]}_{\text{diquark vertices}} + \text{Hermitian conjugate}. \end{align}\]
The first set of interactions are known as leptoquark vertices because they connect quarks to antileptons. The second (diquark) vertices connect quarks to antiquarks. The \(X\) and \(Y\) bosons can mediate proton decay by the diagrams in Figure 4, which involve both leptoquark and diquark vertices. The decays violate both baryon number (\(B\)) and lepton number (\(L\)), but conserve \(B-L\ .\)
The Yukawa interactions between the fermions and the Higgs field \(H_a\) are given by \[\tag{12} \mathcal{L}_{Yuk} = \gamma _{mn} \ \chi^{aT} _{m} \mathcal{C}\, \psi _{ab}\ H^{\dagger b } \ + \Gamma _{mn}\ \epsilon ^{abcde} \, \psi ^{T} _{mab}\ \mathcal{C}\, \psi _{ncd} \ H_e + \kappa_{mn} \nu^{cT}_{mL} \mathcal{C} \chi^a_n H_a+\text{Hermitian conjugate},\]
where \(m\) and \(n\ ,\) which run from \(1\) to \(3\ ,\) label the fermion families; \(\mathcal{C}\) is the charge conjugation matrix; \(\Gamma_{mn}\) is an arbitrary symmetric matrix; \(\gamma_{mn}\) and \(\kappa_{mn}\) are arbitrary matrices; and \(\epsilon ^{abcde}\) is the totally antisymmetric tensor with \(\epsilon ^{12345}=-1\ .\) The fermions will acquire mass only after \(H_5\) acquires a symmetry breaking vacuum expectation value, so the fermion fields in Equation (12) are actually weak eigenstates, i.e. linear combinations of the fields of definite mass (mass eigenstates). (The gauge interactions in Equation (9) take the same form whether expressed in terms of weak eigenstates or mass eigenstates.) The \(SU(5)\) symmetry does not allow any fermion couplings to \(\Phi\ .\)
Symmetry breaking
The most general renormalizable \(SU(5)\)-invariant potential for the adjoint Higgs field \(\Phi\) is \[\tag{13} V(\Phi) = \frac{\mu^2}{2}\ \text{Tr}(\Phi^2) +\frac{a}{4} \left[ \text{Tr}(\Phi^2) \right]^2 + \frac{b}{2} \text{Tr}(\Phi^4)+ \frac{c}{2} \text{Tr}(\Phi^3),\]
where \(\text{Tr}(\Phi^2)\equiv \sum_{ab=1}^5 \Phi^a_b \Phi^b_a\ .\) Vacuum stability implies the constraint \(a> -7b/15\ .\) The model is often simplified by imposing an invariance under \(\Phi \rightarrow -\Phi\ ,\) so that \(c=0\ .\) For \(\mu^2 < 0, \) \(\Phi\) will acquire a nonzero vacuum expectation value \(\langle \Phi \rangle\ ,\) breaking \(SU(5)\) to a subgroup. \(\langle \Phi \rangle\) can always be chosen to be diagonal by performing an \(SU(5)\) transformation. For \(b>0, c=0\) the minimum of \(V\) is at \[\tag{14} \langle \Phi \rangle= \text{Diag}(\nu_{\Phi}\ \nu_{\Phi} \ \nu_{\Phi} \, -\frac{3}{2}\nu_{\Phi}\, -\frac{3}{2}\nu_{\Phi}), \qquad \nu_{\Phi}^2 = \frac{-2\mu^2}{15a+7b}.\]
The upper left \(3\times 3\) and lower right \(2 \times 2\) blocks of \(\langle \Phi \rangle\) are proportional to the identity matrix, so that \(SU(5)\) is spontaneously broken to \(SU(3) \times SU(2) \times U(1)\ ,\) where the \(U(1)\) is associated with the diagonal matrix \(\text{Diag}(2\ 2\ 2\ -3\ -3)\ .\) When \(\Phi\) is replaced by \(\langle \Phi \rangle\ ,\) its gauge interactions become effective mass terms for \(X\) and \(Y\ ,\) with masses at the GUT scale \[\tag{15} M_X = M_Y = \sqrt{\frac{25}{8}}g_5 \nu_{\Phi}.\]
\(12\) of the components of \(\Phi\) are eaten to become the longitudinal spin-components of the \(X\) and \(Y\ ,\) while the other \(12\) are of little phenomenological interest because they do not couple to fermions.
At this stage the \(SU(3) \times SU(2) \times U(1)\) gauge bosons are still massless. To further break the symmetry to \(SU(3) \times U(1)_Q\) one must include the Higgs multiplet \(H_a\ .\) The scalar potential becomes \[\tag{16} V(\Phi,H) = \frac{\mu^2}{2}\ \text{Tr}(\Phi^2) +\frac{a}{4} \left[ \text{Tr}(\Phi^2) \right]^2 + \frac{b}{2} \text{Tr}(\Phi^4)+ \frac{c}{2} \text{Tr}(\Phi^3) + \frac{\mu_5^2}{2}H^\dagger H\ +\frac{\lambda}{4}\left( H^\dagger H \right)^2 + \alpha H^\dagger H\ \text{Tr} \left( \Phi^2 \right) + \beta H^\dagger \Phi^2 H + \delta H^\dagger \Phi H,\]
where a contraction of \(SU(5)\) indices is implied. Imposing the \(\Phi \rightarrow -\Phi\) symmetry requires \(c=\delta=0\ .\) \(V(\Phi,H)\) must be simultaneously minimized with respect to \(\langle \Phi \rangle\) and \(\langle \phi^0 \rangle = {\nu}/{\sqrt{2}}\ .\) One expects that Equation (14) will still hold approximately, with \( \nu_\Phi \sim 10^{15} \text{ GeV}\) because of the approximate gauge unification and to ensure a sufficiently long proton lifetime (at least compared to the limits that existed when the theory was proposed). However, the observed \(SU(2)\times U(1)\) breaking scale is \(\nu = 246 \text{ GeV} \sim 10^{-13}\nu_\Phi\ ,\) from the \(W\) and \(Z\) masses. This enormous GUT hierarchy between the two scales requires an incredibly precise fine-tuning between the parameters in \(V(\Phi,H)\ ,\) and in some sense is worse than the Higgs/hierarchy problem of the SM because it already enters at tree level. A second fine-tuning (involving the parameters \(\beta\) and \(\delta\)) is required to split the color triplet \(\mathcal{H}_\alpha\) in \(H_a\) from the Higgs doublet \((\phi^+ \ \phi^0)^T\) (the doublet-triplet problem), i.e., one requires \(M_\mathcal{H} \gtrsim 10 ^{14} \text{ GeV} \gtrsim 10 ^{12} M_\phi\) because the \(\mathcal{H} \) can itself mediate proton decay. These two fine-tuning problems are aggravated by the fact that they must take into account not only the lowest order terms in \(V(\Phi,H)\) but also the higher-order radiative corrections.
Implications
Gauge unification
In the standard model the weak angle \(\theta_W\ ,\) defined by \(\sin^2 \theta_W \equiv \frac{g'^{2}}{g^2 + g'^{2}}\ ,\) is a critical parameter in the prediction of the properties of the two neutral electroweak gauge bosons \[\tag{17} A = \cos \theta_W B + \sin \theta_W W^0, \qquad Z= -\sin \theta_W B + \cos \theta_W W^0,\]
where \(A\) is the (massless) photon and \(Z\) is the heavy neutral boson predicted by the \(SU(2)\times U(1)\) model. It also enters in the predictions for the masses of the \(W\) and \(Z\) and for the interactions of the \(Z\ .\) Although \(\sin^2 \theta_W \sim 0.23 \) has been precisely measured, its value is not predicted by the SM due to the fact that it involves two gauge groups with two gauge couplings.
In \(SU(5)\) however, \(SU(2)\) and \(U(1)\) are both embedded in \(SU(5)\ ,\) which has a single gauge coupling \(g_5\ .\) Including an appropriate normalization factor (as described in Section The gauge and Yukawa interactions), one expects \(g=\sqrt{5/3}g'=g_5\) at energy scales for which \(SU(5)\) breaking can be ignored, so that \[\tag{18} \sin^2 \theta_W =\frac{3/5}{1+3/5}=\frac 3 8.\]
Below, \(M_X\ ,\) however, \(SU(5)\) is broken and the individual couplings run according to Equation (2) (Georgi, Quinn, and Weinberg (1974)). To an excellent approximation one can ignore the effects of the Higgs doublet. Generalizing Equation (2) to \(F\) fermion families rather than \(3\) one finds \[\tag{19} b_1=\frac{F}{12\pi^2}, \quad b_2=- \frac{1}{16 \pi^2} \left[ \frac{22}{3}-\frac{4F}{3}\right], \quad b_3 = -\frac{1}{16\pi^2}\left[ 11 - \frac{4F}{3}\right].\]
These can be solved, using Equation (18) as a boundary condition, to obtain \[\tag{20} \frac{\alpha(M_Z^2)}{\alpha_s(M_Z^2)}= \frac{3}{8}\left[ 1-\frac{11\alpha(M_Z^2)}{2\pi} \ln\frac{M_X^2}{M_Z^2} \right] , \qquad \hat s_{Z}^2 (M_Z^2) = \frac 1 6 + \frac 5 9 \frac{\alpha(M_Z^2)}{\alpha_s(M_Z^2)},\]
which is independent of \(F\) to this order. In Equation (20), \(\alpha_s \equiv g_3^2/4\pi\) is the strong fine structure constant, \(\alpha\equiv e^2/4\pi\) with \(e\equiv g/\sin \theta_W\) is the electromagnetic fine structure constant, and \(\hat s_{Z}^2 (M_Z^2)\) is the running \(\sin^2 \theta_W\) evaluated at \(M_Z\sim 91\) GeV. Taking the experimental values \(\alpha_s(M_Z^2) \sim 0.12\) and \(\alpha(M_Z^2)\sim 1/128\) (which is increased from the value \(\sim 1/137\) at low energy), one predicts \(\sin^2 \theta_W (M_Z^2) \sim 0.20\) and \(M_X\sim 10^{15}\) GeV. These were reasonable first approximations to the value of \(\sin^2 \theta_W \sim 0.23 \) inferred from the weak neutral current data in the early 1970's and to the lower limit on \( M_X\) at the time from proton decay. However, as discussed in Sections The unification of gauge couplings and Grand unified theories, these predictions are not consistent with current experimental constraints. Nevertheless, the results are suggestive that some alternative version of grand unification (such as including supersymmetry at the TeV scale) or string theory might be more successful.
Fermion masses
The fermions acquire mass when the Higgs field \(\phi^0\) acquires a vacuum expectation value \({\nu}/{\sqrt{2}}\ .\) In the standard model, the Yukawa couplings of \(\phi^0\) that lead to masses for the up-type quarks, the down-type quarks, the charged leptons, and neutrinos are all independent. In \(SU(5)\ ,\) however, those involving the down-type quarks and the charged leptons are both given by the first term in Equation (12), which leads to the effective mass terms \[\tag{21} \mathcal{L}_{de} = - \bar d_L M^d d_R -\bar e^+_L M ^{eT} e^+_R +h.c.= - \bar d_L M^d d_R -\bar e_L M^e e_R+h.c.\ ,\]
where \(M^d = M ^{eT} = \nu \gamma^\dagger/2\ .\) This implies that the masses are related by \[\tag{22} m_d = m_e, \qquad m_s=m_\mu, \qquad m_b=m_\tau,\]
apparently in strong disagreement with the experimental values. However, like the gauge couplings, these are actually running quantities when higher-order corrections are included, and Equation (22) refers to the values at \(M_X\ .\) At lower energies the leading-order prediction becomes (Buras et al. (1977)) \[\tag{23} \ln \left[ \frac{m_d(Q^2)}{m_e(Q^2) }\right] = \underbrace{\ln \left[ \frac{m_d(M_X^2)}{m_e(M_X^2) }\right] }_{0} + \frac{4}{11-4F/3} \ln \left[ \frac{\alpha_s(Q^2)}{\alpha_5(M_X^2)}\right] + \frac{3}{4F}\ln \left[ \frac{\alpha_1(Q^2)}{\alpha_5(M_X^2)}\right],\]
where \(F\) is the number of families. The \(\alpha_s\) term dominates, and implies (for small \(F\)) that the quarks are heavier than the leptons at low energy. For \(F=3\) and \(m_\tau\sim 1.7\) GeV one finds \(m_b\sim 5\) GeV, in reasonable agreement with experiment. However, the first two relations in (22) imply that \({m_\mu}/{m_e}= {m_s}/{m_d}\) for all \(Q^2\ ,\) which disagrees by an order of magnitude from the observed values (\(m_\mu/m_e \sim 200\) and \(m_s/m_d\sim 20\)). This difficulty persists for other grand unified groups, although it can be resolved in models with a more complicated Higgs sector, such as introducing an additional Higgs \(45\)-plet with family symmetries to restrict the couplings of the \(5\) and the \(45\) (Georgi and Jarlskog (1979)). The second and third terms in Equation (12) lead to a mass matrix for the up-type quarks and a Dirac mass matrix for the neutrinos. Like in the standard model, these are not restricted by \(SU(5)\ ,\) except that the up mass matrix is symmetric. Majorana mass terms for the \(SU(5)\)-singlet \(\nu^c_L\) fields, which would lead to a neutrino mass seesaw model, can be added by hand as bare terms or by introducing an \(SU(5)\)-singlet Higgs field.
Cosmology
One of the outstanding issues in cosmology is to explain the observed excess of matter over antimatter. At present the ratio of number densities of baryons to photons is around \(n_B/n_\gamma \sim 6 \times 10^{-10}\ ,\) corresponding to a baryon number to entropy ratio of \(n_B/s \sim 9 \times 10^{-11}\ ,\) with a much small density \(n_{\bar B}\) of anti-baryons consistent with secondary production mechanisms. This implies that there was a tiny asymmetry \((n_B-n_{\bar B})/s \sim 9 \times 10^{-11}\) in the very early universe. As the universe cooled, the antibaryons annihilated, leaving the tiny baryon excess. Any initial asymmetry would have been eliminated if the universe underwent an early period of rapid inflation, suggesting that the baryon asymmetry was created dynamically. This would require the three Sakharov conditions of \(B\) violation, \(CP\) violation, and non-equilibrium (or \(CPT\) violation). These were apparently fulfilled in the \(SU(5)\) model by the out of equilibrium decays of the superheavy colored partners \(\mathcal{H}_\alpha\) of the Higgs bosons. The total lifetimes of the \(\mathcal{H}_\alpha\) and their charge conjugates \(\mathcal{H}_\alpha^c\) would have to be equal by \(CPT\ ,\) but the \(CP\) violation would allow the partial-rate asymmetries \[\tag{24} \Gamma( \mathcal{H}\rightarrow q^cq^c)\ne \Gamma( \mathcal{H}^c\rightarrow qq), \qquad \Gamma( \mathcal{H}\rightarrow q\ell)\ne \Gamma( \mathcal{H}^c\rightarrow q^c\ell^c)\ ,\]
leading to a baryon asymmetry which could apparently account for the observations in non-minimal versions of \(SU(5)\ .\) However, it was subsequently understood that the SM has non-perturbative vacuum tunneling sphaleron effects that would wipe out the produced asymmetry provided it had a conserved \(B-L\) (as in \(SU(5\))). Though unsuccessful in detail, these investigations led to other more viable possibilities for explaining the asymmetry, such as leptogenesis (involving \(L\) violation in the decays of superheavy Majorana neutrinos found in some theories of neutrino mass, with the \(L\) asymmetry partially converted to a \(B\) asymmetry by sphalerons), or electroweak baryogenesis (in which the sphalerons are associated with the production of the asymmetry during the phase transition associated with electroweak symmetry breaking).
Beyond \(SU(5)\)
Larger groups
\(SU(5)\) can be embedded into a larger group, such as \(SO(10)\) or \(E_6\ .\) In \(SO(10)\) each fermion family transforms as an irreducible \(16\)-dimensional representation \(\psi_{16}\ ,\) which contains the reducible \(5^\ast+ 10\) of \(SU(5)\) as well as the (now required) \(SU(5)\)-singlet right-handed neutrino \(\nu^c_L\ .\) \(SO(10)\) has an additional diagonal generator compared to the SM or \(SU(5)\) (i.e., it is rank \(5\)). The breaking pattern \(SO(10)\rightarrow SU(5)\times U(1)_\chi\) therefore allows for an additional neutral gauge boson, the \(Z_\chi\ ,\) which could be almost as light as the SM gauge bosons, e.g., at the TeV scale. \(SO(10)\) has other symmetry breaking patterns, including the Pati-Salam group and flipped \(SU(5)\) (which involves an alternative identification of the particles in the \(16\)-plet). Fermion masses can be generated by adding a \(10\)-dimensional Higgs representation \(\phi_{10}\ .\) The \({10}\) decomposes as \(5+ 5^\ast\) under \(SU(5)\ ,\) implying that \(\phi_{10}\) actually contains two distinct Higgs doublets. These play the roles of the \(H\) and \(H^\dagger\) of \(SU(5)\ ,\) and can generate masses for the \((u,\nu)\) and \((d,e)\ ,\) respectively. However, the \(SO(10)\) symmetry allows only a single Yukawa interaction (up to fermion family indices), of the form \(\psi_{16} \psi_{16} \phi_{10}\ ,\) which leads to disastrous mass relations. More realistic models can be obtained by including additional Higgs multiplets, including high-dimensional ones such as \(120\) or \(126\ .\) The \(126\) also allows couplings that can generate Majorana neutrino masses, such as a GUT-scale mass for \(\nu^c_L\ ,\) which leads to a small Majorana mass for the \(\nu_L\) due to mixing (the seesaw model). \(SO(10)\) models are therefore frequently combined with family symmetries to generate detailed models of neutrino, quark, and charged lepton masses. However, large representations such as \(126\) are unlikely to emerge from an underlying superstring construction. An alternative is to replace them by higher-dimensional operators to generate fermion masses.
\(E_6\) is an even larger group which emerges from some superstring constructions. It is of rank 6, and contains the subgroup \(SO(10) \times U(1)_\psi\) (an alternative breaking is to \(SU(3)_c\times SU(3)_L \times SU(3)_R\)). Each fermion family is assigned to an irreducible \(27\)-plet, which decomposes as \(16 + 10 + 1\) under \(SO(10)\ .\) The \(16\) contains an \(SO(10)\) family, the \(1\) is an additional SM singlet which can break the \(U(1)_\psi\) symmetry when it obtains a vacuum expectation value. The \(10\) contains new predicted exotic fermions \[\tag{25} 10 = \left( \begin{array}{c} E^0 \\ E^- \end{array} \right)_L + \left( \begin{array}{c} E^0 \\ E^- \end{array} \right)_R + {D}_L + {D}_R\ ,\]
which can also be given masses when the \(U(1)_\psi\) is broken. The \((E^0, E^-)_{L,R}\) are color-singlet fields that transform as \(SU(2)\) doublets, and can be thought of as heavy leptons. Similarly, the \(D_{L,R}\) are heavy down-type (charge -\(1/3\)) quarks. The two additional \(U(1)\) factors in \(E_6\) (\(U(1)_\chi\) and \(U(1)_\psi\)) and their associated exotic fermions are often used as examples of new physics that could possibly be present at the TeV scale or even lower, and are often considered outside of the original \(E_6\) context.
Supersymmetry
Supersymmetry refers to possible relations between the spectrum and interactions of fermions (half-integer spin particles) and bosons (integer spin particles). It can be viewed as a space-time extension of the Poincare (Lorentz plus translational invariance) group, involving new anti-commuting dimensions. Under reasonable assumptions it is the unique extension of the usual Poincare and internal symmetries of field theory. Supersymmetry provides a possible route to unify gravity with the other interactions through superstring theory. If supersymmetry exists in nature it must be broken. For the connection with gravity it would suffice for the breaking scale to be very large. However, as mentioned earlier, there would be a number of advantages to a low breaking scale, e.g., a TeV, including the Higgs/hierarchy problem, gauge coupling unification, and the existence of a plausible dark matter candidate in some versions.
It is straightforward to construct supersymmetric versions of the standard model (the MSSM), or of \(SU(5)\) and the larger grand unified groups (Dimopoulos and Georgi (1981), Raby (2009)). In each case each particle (spin-\(0\ ,\) \(1/2\ ,\) or \(1\)) is accompanied by a predicted superpartner, which differs in spin by \(1/2\) unit. In addition to this doubling of the spectrum one must introduce two distinct Higgs doublets, \(h_u =\left(\begin{array}{c} h_u^+ \\ h_u^0 \end{array}\right)\) and \(h_d= \left(\begin{array}{c} h_d^0 \\ h_d^- \end{array}\right)\ ,\) as well as their spin-\(1/2\) superpartners. Their \(SU(5)\) analogs are \[\tag{26} {H}_{ua} = \left( \begin{array}{c} \mathcal{H}_\alpha \\ h_u^+ \\ h_u^0 \end{array} \right), \qquad {H}_{d}^a = \left( \begin{array}{c} \mathcal{H}^{c\alpha} \\ h_d^0 \\ h_d^- \end{array} \right),\]
which transform as \(5\) and \(5^\ast\ ,\) respectively. \(h_u\) and \(h_d\) can have Yukawa couplings which generate masses for the \((u,\nu)\) and \((d,e)\ ,\) respectively (similar to \(SO(10)\)). The second Higgs multiplets are needed because supersymmetry forbids coupings involving \(H^\dagger\ ,\) such as in (12), as well as for anomaly cancellation.
An especially striking effect of low scale supersymmetry is that the superpartners modify the renormalization group equations (Equation (2)), allowing a much better unification of the gauge couplings than in the non-supersymmetric case, as can be seen in Figure 3. The unification is not perfect. For example, using \(\alpha(M_Z^2)\) and \(\hat s_{Z}^2 (M_Z^2)\) as inputs, exact unification would predict a strong coupling \(\alpha_s(M_Z^2) \sim 0.13\ ,\) about \(10\%\) higher than the observed value. However, the discrepancy can be accounted for by small corrections, such as small differences between the masses of the GUT-scale particles (with masses near \(M_X\)), or differences between the superpartner masses and \(M_Z\ .\) The successful prediction of the ratio of \(m_b/m_\tau\) and the problems with the masses of the lighter particles are similar in the supersymmetric case.
One also sees in Figure 3 that the unification scale is higher in the supersymmetric case, \(M_X\sim 3 \times 10^{16}\) GeV rather than \(10^{14-15}\) GeV. Since the lifetime for the proton to decay through the diagrams in Figure 4 scales as \(M_X^4\) (Equation (3)), this implies a much longer lifetime into modes such as \(e^+\pi^0\ ,\) considerably longer than experimental limits. However, as commented in Section Grand unified theories there are additional decay mechanisms involving the superpartners that scale as \(M_X^2\ ,\) leading to faster decays into different final states, such as \(\bar \nu K^+\ .\) The minimal versions of supersymmetric \(SU(5)\) and \(SO(10)\) are already excluded by the non-observation of such decays, while non-minimal versions should allow observable proton decay rates in future experiments. There are also (unrealistic) versions of low-scale supersymmetry in which new interactions of the superpartners would lead to rapid proton decay, with a rate that is not suppressed by powers of \(M_X\ .\)
Extra dimensions and strings
As far as we are aware, there are three dimensions of space and one of time. In particular, the space dimensions are large or infinite in size. However, it is possible that there are additional space dimensions that we cannot readily perceive, perhaps because they are are compactified (curled up) into a tiny circle or other manifold, highly warped by gravitational effects, or because some dynamical principle causes us to be stuck in a limited domain of the new dimensions. Considerable theoretical activity has been directed towards such possibilities, e.g., in connection with the fermion or Higgs/hierarchy problems, or in connection with gravity (superstring theories require additional dimensions for a consistent formulation).
In orbifold GUTs the grand unification is present in a higher-dimensional space. The GUT symmetry may be broken by boundary conditions in the extra dimensions, so that our apparent four-dimensional world has a lower symmetry, e.g., of the SM or MSSM. Orbifold GUTs may retain the desirable features of grand unification, such as gauge coupling unification, third family Yukawa relations, etc., while avoiding such difficulties as the doublet-triplet problem, too rapid proton decay, and the need for large Higgs representations.
Superstring theories incorporate quantum gravity, and are therefore more ambitious than the SM, MSSM, or grand unification. There are actually a large number of string theories, which may be thought of as different points in an enormous landscape of string vacua. Many of these include underlying grand unification symmetries. They may compactify into an effective four-dimensional GUT, although it is difficult to generate the adjoint and other large Higgs multiplets introduced in many non-string motivated models. They may also lead to versions of orbifold GUTs, or compactify directly to the SM or MSSM, or to an extended version, with limited memory of the underlying GUT. Constructions may retain simple MSSM-type gauge unification, or the unification may be modified (and complicated) by the effects of new particles and/or by the string scale gauge coupling boundary conditions. The fermion families or the elements of the families may have different origins in the construction, breaking or modifying GUT Yukawa relations and possibly leading to family nonuniversal couplings to new neutral gauge bosons. Other classes of string theories usually do not involve a full underlying GUT, but they often descend to four dimensions using a Pati-Salam group.
References
- Buras, A.J.; Ellis, J.R.; Gaillard, M.K. and Nanopoulos, D.V. (1978). Aspects of the grand unification of strong, weak and electromagnetic interactions. Nucl. Phys. B135: 66-92. inSpire
- Dimopoulos, S. and Georgi, H. (1981). Softly Broken Supersymmetry and \(SU(5)\). Nucl.Phys. B193: 151. inSpire
- Georgi, H. and Glashow, S. (1974). Unity Of All Elementary Particle Forces. Phys. Rev. Lett. 32: 438. inSpire
- Georgi, H. and Jarlskog, C. (1979). A New Lepton - Quark Mass Relation in a Unified Theory. Phys. Lett. B86: 297. inSpire
- Georgi, H.; Quinn, H.R. and Weinberg, S. (1974). Hierarchy of interactions in unified gauge theories. Phys. Rev. Lett. 33: 451-454. inSpire
- Hewett, J.L. and Rizzo, T.G. (1989). Low-energy phenomenology of superstring inspired \(E(6)\) models. Phys. Rept. 183: 193. inSpire
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Further reading
- Langacker, P. (2010). The standard model and beyond. Taylor and Francis, Boca Raton, FL. ISBN 978-1-4200790-6-7. Especially chapter 8.
- Mohapatra, R.N. (2003). Unification and supersymmetry: the frontiers of quark-lepton physics. Springer, Berlin.
- Nakamura, K. et al. (2010). Particle Data Group. J. Phys. G37: 075021. See the review on Grand Unified Theories by S. Raby.
- Ross, G.G. (1985). Grand unified theories. Westview Press, Reading, MA.