Grand Unified Models

Prerequisites: Standard Model of Particle Physics, Representation Theory. Full path (33 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?
• FAQ
• History
• Criticism

Intuitive

The similarities between the vacuum of space and low-temperature phases of matter are legendary in physics. Not only are phases static,uniform quantum states, but their most subtle internal motions are physically indistinguishable om elementary particles very generally.3This is one of the most astonishing facts in science, and something students always find upsetting and difficult to believe. But they even­tually become convinced a er looking at enough experiments, for the evidence is plenti l and consistent. In fact, the more one studies the mathematical descriptions of cold phases, the more accustomed one gets to using the parallel terminologies of matter and space inter­changeably. Thus instead of a phase of matter we speak of a vacuum.Instead of particles we speak of excitations. Instead of collective mo­tions we speak of quasiparticles. The pre x "quasi" turns out to be a vestige of the historical battles over the physical meaning of these ob­jects and conveys no meaning. In private conversations one drops the pretense and refers to the objects as particle

[…]

The vacuum, when cooled from very high temperatures, is thought to undergo stepwise events called unification transitions, in which the known forces of nature split off sequentially from their fundamental parent. Similarly, holmium metal, a rare earth element, when cooled from very high temperature, first condenses into a liquid at 2993 degrees Kelvin, then solidifies at 1743 degrees, then develops a special kind of spiraling magnetism at 130 degrees, then tils the spiral to make weak ferromagnet at 20 degrees. Between 130 degrees and 20 degrees the pitch of the spiral varies continuously, as though it were a rubber screw being stretched along its axis. With each of these transitions the "forces" between electrons in the metal transmitted by various elastic distortions of the ordered state split off from the fundamental parent in beautiful and apt analogy with what occurs in the vacuum. The temperatures required to see the unification transitions of the vacuum cannot be achieved in the laboratory, or even in the centers of the largest stars, so the evidence for unification is indirect, but it nicely parallels what one would find in the tilted spiral magnet if one's experiments could reach only long length and time scales. One of the strongest of these bits of evidence is renormalizability, an effect that causes the accessible measurements to be simple and redundant (one measurement predicts another) but that at the same time is incapable of revealing anything about the forces at the top of the hierarchical tree.

[…]

There are numer­ous experimental indications that the vacuum emerges in a hierarchy of phase transitions in which the various forceS differentiate from each other….

If renormalizability of he vacuum is caused by proximity to phase transitions th the search for an ultimate theory would been doomend on two counts: it would not predict anything even if you found it, and it could not be falsified.

page 112 in "A different Universe" by Robert Laughlin

Concrete

• Grand Unified Theories by Graham Ross
• Grand Unification with and without Supersymmetry by Ed Witten
• Grand Unification with and Without Supersymmetry And Cosmology Implications by C Kounnas et. al.
• http://users.physics.uoc.gr/~rosen/projects/gutFive.pdf

Abstract

A grand unified theory is a Yang-Mills theory with a compact gauge group $G_{GUT}$, usually simple so that there is a single gauge coupling, together with a homomorphism $G_{SM} \to G_{GUT}$ and a unitary representation $V$ of $G_{GUT}$ whose restriction to $G_{SM}$ is the Standard Model fermion representation. Here the gauge group that acts faithfully on the Standard Model fields is

$$G_{SM} = \big(SU(3) \times SU(2) \times U(1)\big)/\mathbb{Z}_6 ,$$

since a $\mathbb{Z}_6$ subgroup of $SU(3)\times SU(2)\times U(1)$ acts trivially on every known particle. $G_{SM}$ is isomorphic to $S(U(3)\times U(2))$, the block diagonal matrices in $SU(5)$, which gives the minimal embedding $G_{SM} \subset SU(5)$ of Georgi and Glashow. The hypercharge generator is the traceless matrix

$$Y = \text{diag}\left(-\tfrac{1}{3},-\tfrac{1}{3},-\tfrac{1}{3},\tfrac{1}{2},\tfrac{1}{2}\right)$$

acting on the fundamental $5$, in the convention $Q = T_3 + Y$.

Fermions. Write all fermions of one generation as left-handed Weyl spinors. Under $SU(5)$ they fill the reducible representation $\bar 5 \oplus 10$, with $10 = \Lambda^2 5$:

$$\bar 5 \to (\bar 3,1)_{1/3} \oplus (1,2)_{-1/2}, \qquad 10 \to (\bar 3,1)_{-2/3} \oplus (3,2)_{1/6} \oplus (1,1)_{1},$$

which are $d^c$, $L$, $u^c$, $Q$ and $e^c$. Tracelessness of $Y$ on each irreducible representation is what fixes the charge relations discussed in "Why is it interesting?". The gauge anomaly of $SU(5)$ cancels within a generation, since the cubic anomaly coefficients satisfy $A(10) = -A(\bar 5)$.

Adding a singlet $\nu^c$ gives 16 states per generation, and these form one irreducible representation of a larger group. $SU(5) \times U(1) \subset Spin(10)$, and the 16 dimensional chiral spinor of $Spin(10)$ decomposes as

$$16 \to 10_{1} \oplus \bar 5_{-3} \oplus 1_{5} .$$

Concretely, the Dirac spinor of $Spin(10)$ can be realised on the exterior algebra $\Lambda \mathbb{C}^5$ of dimension $2^5 = 32$, and the chiral spinor $16$ is the even part $\Lambda^0 \oplus \Lambda^2 \oplus \Lambda^4$, which under $SU(5)$ is $1 \oplus 10 \oplus \bar 5$. Baez and Huerta organise the embeddings $G_{SM} \subset SU(5) \subset Spin(10)$ and $G_{SM} \subset SU(4)\times SU(2) \times SU(2)$ (Pati-Salam) $\subset Spin(10)$ into a commuting square of group homomorphisms that are compatible with the representations on $\Lambda \mathbb{C}^5$.

Couplings. A simple $G_{GUT}$ has one gauge coupling. The Standard Model couplings are its restrictions, which requires the generators to be normalised in the same way, $\text{Tr}_V(T^a T^b) \propto \delta^{ab}$. On the $5$ one finds $\text{Tr}(Y^2) = \frac{5}{6}$ and $\text{Tr}(T_3^2) = \frac{1}{2}$, so the correctly normalised $U(1)$ generator is $\sqrt{3/5}\,Y$ and $g_1 = \sqrt{5/3}\, g'$. Equal couplings at the unification scale give $\sin^2\theta_W = \frac{g'^2}{g^2 + g'^2} = \frac{3}{8}$ there. The measured value at $M_Z$ is obtained only after renormalisation group running, which is how the unification scale is estimated.

Symmetry breaking and proton decay. $G_{GUT}$ must be broken to $G_{SM}$, in minimal $SU(5)$ by a Higgs field in the adjoint $24$, and then to $SU(3)\times U(1)_{Q}$ by a $5$ containing the Standard Model Higgs doublet. The gauge bosons in $\mathfrak{su}(5)/\mathfrak{g}_{SM}$, the $(3,2)_{-5/6}$ and its conjugate, carry both colour and weak charge and connect quarks with leptons. Integrating them out produces baryon number violating dimension-6 operators suppressed by $1/M_X^2$, so the proton lifetime scales as $\tau_p \sim M_X^4/(\alpha_{GUT}^2 m_p^5)$. Larger candidate groups, such as $E_6$ with its $27$, follow the same pattern with more representations to place.

• The Algebra of Grand Unified Theories by John C. Baez, John Huerta
• Group Theory for Unified Model Building by R. Slansky
• Model Building and Phenomenology in Grand Unified Theories by Tomas E. Gonzalo Velasco
• http://jakobschwichtenberg.com/resources-that-helped-me-understand-grand-unified-theories/

Why is it interesting?

Despite the success of the standard model, there are several reasons to believe that it is not the final description of nature at the most fundamental level. For example, there is no explanation in the standard model why $Q_{\text{proton}}+Q_{\text{electron}}= \mathcal{O}(10^{-20})$. The success of the standard model, tells us that nature likes gauge theories. Thus in order to find answers to questions left unanswered by the standard model, we follow the gauge idea further. The basic hypothesis of grand unified theories (GUTs) is that we embed the standard model gauge group in a simple group $G_{GUT} \supset G_{SM}$. This way the strong and electroweak interactions are unified and we are left with a single gauge coupling constant. In present day colliders, we do not observe effects of a $G_{GUT}$ structure and thus we assume the unified gauge symmetry is broken at some high energy scale $$\begin{equation} G_{GUT} \stackrel{M_{GUT}}{\rightarrow} \ldots \stackrel{M_I}{\rightarrow} G_{SM} \stackrel{M_Z}{\rightarrow} SU(3)_C \times U(1)_Q \, , \end{equation}$$ where the dots indicate possible intermediate scales between $G_{GUT}$ and $G_{SM}$. Here are some of the "mysteries" of the standard model that can be resolved by a GUT:

“GUTs are the most attractive conjecture for the large scale picture of particle physics. GUT is not the Standard Model (SM), is beyond the SM, but is the most standard physics beyond the SM. Most of us think that there should be something like a GUT.”Altarelli

FAQ

History

• A Brief History of the Grand Unified Theory of Physics by Lawrence Krauss

Criticism

"Physics is littered with the corpses of dead unified field theories." Freeman Dyson

"The bad news is that those people who went off to detect proton decay never found it! It became clear in the mid-1980s that the proton lifetime was at least $10^{32}$ years or so, much larger than what the SU(5) theory most naturally predicts. Of course, if one is desperate to save a beautiful theory from an ugly fact, one can resort to desperate measures. For example, one can get the SU(5) model to predict very slow proton decay by making the grand unification mass scale large. Unfortunately, then the coupling constants of the strong and electroweak forces don't match at the grand unification mass scale. This became painfully clear as better measurements of the strong coupling constant came in." from This Week's Finds in Mathematical Physics (Week 119) by John Baez

“People can construct models with higher symmetries and stand on their nose and try to avoid proton decay,” Nanopoulos said. “OK, you can do it, but … you cannot show it to your mother with a straight face.” https://www.quantamagazine.org/20161215-proton-decay-grand-unification/

Major Questions still unsolved in GUTs

• The Hierarchy Problem, i.e. why $M_W/M_{GUT} <10^{-12}$.
• The strong CP problem.
• The cosmological constant problem.
• The absence of gravity.
• The family repetition problem, i.e. why are there three families.
• The fermion mass spectrum.

GUTs have nothing to say on these matters.

• https://physics.stackexchange.com/questions/79064/how-does-the-super-kamiokande-experiment-falsify-su5

Path to this page

1. Calculus
2. Imaginary Numbers
3. Linear Algebra
4. Functional
5. Group Theory
6. Hilbert Space
7. Legendre Transformation
8. Manifold
9. Newton's Second Law
10. Vector Calculus
11. Coulomb's law
12. Fiber Bundles
13. Newtonian Mechanics
14. Representation Theory
15. Symmetry
16. Variational Calculus
17. Configuration Space
18. Euler-Lagrange Equations
19. Maxwell Equations
20. Special Relativity
21. Lagrangian Formalism
22. Phase Space
23. Classical Field Theory
24. Gauge Symmetry
25. Hamiltonian Formalism
26. Symmetry Breaking
27. Gauge Models
28. Goldstone's theorem
29. Schrödinger Equation
30. Canonical Quantum Mechanics
31. Higgs Mechanism
32. Canonical Quantum Field Theory
33. Standard Model of Particle Physics