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models:speculative_models:grand_unified_theories [2017/12/18 16:11]
jakobadmin [Criticism]
models:speculative_models:grand_unified_theories [2018/05/22 14:23] (current)
jakobadmin [Criticism]
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-====== ​ Grand Unified ​Theories ​====== +====== ​ Grand Unified ​Models ​====== 
- + 
-<tabbox Why is it interesting?>​  +
- +
- +
  
-<​tabbox ​Layman+<​tabbox ​Intuitive
  
 <​blockquote>​ <​blockquote>​
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   ​   ​
-<​tabbox ​Student+<​tabbox ​Concrete
  
   * Grand Unified Theories by Graham Ross   * Grand Unified Theories by Graham Ross
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   * http://​users.physics.uoc.gr/​~rosen/​projects/​gutFive.pdf   * http://​users.physics.uoc.gr/​~rosen/​projects/​gutFive.pdf
    
-<​tabbox ​Researcher+<​tabbox ​Abstract
  
   * [[https://​arxiv.org/​abs/​0904.1556|The Algebra of Grand Unified Theories]] by John C. Baez, John Huerta   * [[https://​arxiv.org/​abs/​0904.1556|The Algebra of Grand Unified Theories]] by John C. Baez, John Huerta
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   ​   ​
-<​tabbox ​Examples+<​tabbox ​Why is it interesting?​
  
---> Example1# 
  
- +Despite the success of the [[models:​standard_model|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} \label{eq:​schematicgutbreaking} 
 +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: 
 + 
 +-->​Quantization of Electric Charge# 
 + 
 +In the standard model the electric charges of the various particles must be put in by hand and there is no reason why there should be any relation between the electron and proton charge. However from experiments it is known that $Q_{\text{proton}}+Q_{\text{electron}}= \mathcal{O}(10^{-20})$. In GUTs one multiplet of $G_{GUT}$ contains quarks and leptons. This way, GUTs provide an elegant explanation for the experimental fact of charge quantization.  
 + 
 +\begin{equation} 
 + ​\bar{5} = \begin{pmatrix} \nu_L \\ e_L \\ (d_R^c)_{\text{red}} \\ (d_R^c)_{\text{blue}} \, .\\ (d_R^c)_{\text{green}} \end{pmatrix}  
 +\end{equation} 
 + 
 +The standard model generators must correspond to generators of $G_{GUT}$. Thus the electric charge generator must correspond to one Cartan generator\footnote{The eigenvalues of the Cartan generators of a given gauge group correspond to the quantum numbers commonly used in particle physics. This is discussed in section \ref{sec:​easysu3intro}.} of $G_{GUT}$. In $SU(5)$ the Cartan generators can be written as diagonal $5\times 5$ matrices with trace zero\footnote{In $SU(5)$ is the set of $5 \times 5$ matrices $U$ with determinant $1$ that fulfil $U^\dagger U = 1$. For the generators $T_a$ this means $\text{det}(e^{i \alpha_a T_a})=e^{i \alpha_a \text{Tr}(T_a)} \stackrel{!}{=}1$. Therefore ​ $Tr(T_a) \stackrel{!}{=} 0$}. Therefore we have 
 + 
 +\begin{align} 
 +\text{Tr}(Q)&​= \text{Tr} \begin{pmatrix} Q(\nu_L) & 0 & 0 & 0 &0 \\  0 & Q(e_L) & 0 & 0 &0 \\ 0 & 0 & Q((d_R^c)_{\text{red}}) & 0 &0\\ 0 & 0 & 0 & Q((d_R^c)_{\text{blue}})&​0\\ 0 & 0 & 0 & 0 &​Q((d_R^c)_{\text{green}}) \end{pmatrix} \stackrel{!}{=} 0  \notag \\ 
 +&​\rightarrow Q(\nu_L) + Q(e_L) + 3Q(d_R^c) \stackrel{!}{=} 0 \notag \\ 
 +&​\rightarrow ​ Q(d_R^c) \stackrel{!}{=} -\frac{1}{3} Q(e_L) \, . 
 +\end{align} 
 + 
 +Analogously,​ we can derive a relation between $e_R^c$, $u_L$ and $u_R^c$. Thus $Q_{\text{proton}}+Q_{\text{electron}}= \mathcal{O}(10^{-20})$ is no longer a miracle, but rather a direct consequence of of the embedding of $G_{SM}$ in an enlarged gauge symmetry.
 <-- <--
  
---> ​Example2:#+--> ​The Standard Model Coupling Strengths# 
 + 
 +{{ :​theories:​speculative_theories:​unification-coupling-strengths.png?​nolink&​200|}} 
 + 
 +The standard model contains three gauge couplings, which are very different in strength. Again, this is not a real problem of the standard model, because we can simply put these values in by hand. However, GUTs provide a beautiful explanation for this difference in strength. A simple group $G_{GUT}$ implies that we have only one gauge coupling as long as $G_{GUT}$ is unbroken. The gauge symmetry $G_{GUT}$ is broken at some high energy scale in the early universe. Afterwards, we have three distinct gauge couplings with approximately equal strength. The gauge couplings are not constant, but depend on the energy scale. The RGEs for a gauge coupling depend on the number of particles that carry the corresponding charge. Gauge bosons have the effect that a given gauge coupling becomes stronger at lower energies and fermions have the opposite effect. The adjoint of $SU(3)$ is $8$-dimensional and therefore we have $8$ corresponding gauge bosons. In contrast the adjoint of $SU(2)$ is $3$-dimensional and thus we have $3$ gauge bosons. For $U(1)$ there is only one gauge boson. As a result for $SU(3)$ the gauge boson effect dominates and the corresponding gauge coupling becomes stronger at lower energies. For $SU(2)$ the fermion and boson effect almost cancel each other and thus the corresponding gauge coupling is approximately constant. For $U(1)$ the fermions dominate and the $U(1)$ gauge coupling becomes much weaker at low energies. ​ This way GUTs provide an explanation why strong interactions are strong and weak interactions are weak.  
  
-  
 <-- <--
  
 +--> Lightness of the Neutrino Masses #
 +
 +A quite generic implication of grand unification is small neutrino masses through the type-1 seesaw mechanism. Models based on the popular $SO(10)$ or $E_6$ groups contain automatically a right-handed neutrino $\nu_R$. As a result of the breaking chain this standard model singlet $\nu_R$ gets a superheavy mass $M$. After the last breaking step $G_{SM}\rightarrow SU(3)_C \times U(1)_Y$ the right-handed and left-handed neutrinos mix. This yields a suppressed mass of the left-handed neutrino of order $\frac{m^2}{M}$,​ where $m$ denotes a typical standard model mass. 
 +<--
 +
 +--> Matter-Antimatter Asymmetry #
 +
 +GUTs provide a natural framework to explain the observed matter-antimatter asymmetry in the universe. As already noted above a general implication of GUTs is that protons are no longer stable. Formulated differently,​ GUTs allow baryon number-violating interactions. This is one of three central ingredients,​ known as Sakharov conditions, needed to produce more baryons than antibaryons in the early universe. Thus, as D. V. Nanopoulos put it, "if the proton was stable it would not exist"​.
 +
 +<--
 +
 +----
 +
 +<​blockquote>​“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.”<​cite>​[[http://​inspirehep.net/​record/​1642530/​files/​1641483_174-182.pdf|Altarelli]]</​cite></​blockquote>​
 <tabbox FAQ> ​ <tabbox FAQ> ​
  
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   ​   ​
 <tabbox History> ​ <tabbox History> ​
 +  * [[http://​nautil.us/​issue/​46/​balance/​a-brief-history-of-the-grand-unified-theory-of-physics|A Brief History of the Grand Unified Theory of Physics]] by Lawrence Krauss
 <tabbox Criticism> ​ <tabbox Criticism> ​
 <​blockquote>"​Physics is littered with the corpses of dead unified field theories."​ <​blockquote>"​Physics is littered with the corpses of dead unified field theories."​
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 ]]</​blockquote>​ ]]</​blockquote>​
  
-<​blockquote>​“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/</​blockquote>​+<​blockquote>​“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/</​blockquote>​
  
  
 ---- ----
 +
 +**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   * https://​physics.stackexchange.com/​questions/​79064/​how-does-the-super-kamiokande-experiment-falsify-su5
 +
 +
  
 </​tabbox>​ </​tabbox>​
  
  
models/speculative_models/grand_unified_theories.1513609892.txt.gz · Last modified: 2017/12/18 15:11 (external edit)