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| {{Multiple issues|
| | Garry is when I'm called and I really like it. Researching cryptography is the hobby Let me never stop doing. The job I've been occupying for years is a human resources tool. Maine is the only place I've been residing back in. If you want to get more information check out his website: http://simpleit.edublogs.org/ |
| {{expert-subject|date=April 2008}}
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| {{cleanup|date=November 2007}}
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| {{refimprove|date=July 2009}}
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| [[File:Georgi-Glashow charges.svg|300px|right|thumb|The pattern of [[weak isospin]]s, [[weak hypercharge]]s, and strong charges for particles in the Georgi-Glashow model, rotated by the predicted [[weak mixing angle]], showing electric charge roughly along the vertical. In addition to [[Standard Model]] particles, the theory includes twelve colored X bosons, responsible for [[proton decay]].]]
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| In [[particle physics]], the '''Georgi–Glashow model''' is a particular [[grand unification theory]] (GUT) proposed by [[Howard Georgi]] and [[Sheldon Glashow]] in 1974. In this model the [[standard model]] gauge groups [[SU(3) × SU(2) × U(1)]] are combined into a single [[simple Lie group|simple]] gauge group -- [[special unitary group|SU(5)]]. The unified group SU(5) is then thought to be [[Spontaneous symmetry breaking|spontaneously broken]] to the standard model subgroup at some high energy scale called the [[grand unification energy|grand unification scale]].
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| Since the Georgi–Glashow model combines [[lepton]]s and [[quark]]s into single [[irreducible representation]]s, there exist interactions which do not conserve [[baryon]] number, although they still conserve [[B-L]]. This yields a mechanism for [[proton decay]], and the rate of proton decay can be predicted from the dynamics of the model. However, proton decay has not yet been observed experimentally, and the resulting lower limit on the lifetime of the proton contradicts the predictions of this model. However, the elegance of the model has led particle physicists to use it as the foundation for more complex models which yield longer proton lifetimes.
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| (For a more elementary introduction to how the representation theory of Lie algebras are related to particle physics, see the article [[Particle physics and representation theory]].)
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| This model suffers from the [[doublet-triplet splitting problem]].{{clarify|date=March 2012}}
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| == Breaking SU(5) ==
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| SU(5) breaking occurs when a [[Scalar field theory|scalar field]], analogous to the [[Higgs field]], and transforming in the [[adjoint endomorphism|adjoint]] of SU(5) acquires a [[vacuum expectation value]] proportional to the [[weak hypercharge]] generator{{Why?|date=August 2010}}
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| :<math>\frac{Y}{2}=\operatorname{diag}\left(-1/3, -1/3, -1/3, 1/2, 1/2\right)</math>
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| When this occurs SU(5) is [[spontaneous symmetry breaking|spontaneously broken]] to the [[subgroup]] of SU(5) commuting with the group generated by ''Y''. This unbroken subgroup is just the [[standard model]] group: [SU(3)×SU(2)×U(1)_Y]/'''Z'''<sub>6</sub>.
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| Under the unbroken subgroup the adjoint '''24''' transforms as
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| :<math>24\rightarrow (8,1)_0\oplus (1,3)_0\oplus (1,1)_0\oplus (3,2)_{-\frac{5}{6}}\oplus (\bar{3},2)_{\frac{5}{6}}</math>
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| giving the [[gauge boson]]s of the standard model. See [[restricted representation]].
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| The standard model [[quark]]s and [[lepton]]s fit neatly into representations of SU(5). Specifically, the left-handed [[fermion]]s combine into 3 generations of <math>\mathbf{\bar{5}}\oplus\mathbf{10}\oplus\mathbf{1}</math>. Under the unbroken subgroup these transform as
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| :<math>\bar{5}\rightarrow (\bar{3},1)_{\frac{1}{3}}\oplus (1,2)_{-\frac{1}{2}}</math> (d<sup>c</sup> and l)
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| :<math>10\rightarrow (3,2)_{\frac{1}{6}}\oplus (\bar{3},1)_{-\frac{2}{3}}\oplus (1,1)_1</math> (q, u<sup>c</sup> and e<sup>c</sup>)
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| :<math>1\rightarrow (1,1)_0</math> (ν<sup>c</sup>)
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| giving precisely the left-handed [[fermion]]ic content of the standard model, where for every [[Generation (particle physics)|generation]] d<sup>c</sup>, u<sup>c</sup>, e<sup>c</sup> and ν<sup>c</sup> stand for anti-[[Standard Model#Particles of Ordinary Matter|down-type quark]], anti-[[Standard Model#Particles of Ordinary Matter|up-type quark]], anti-[[Standard Model#Particles of Ordinary Matter|down-type lepton]] and anti-[[Standard Model#Particles of Ordinary Matter|up-type lepton]], respectively, and q and l stand for [[quark]] and [[lepton]].
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| Note that fermions transforming as a '''1''' under SU(5) are now thought to be necessary because of the evidence for [[neutrino oscillation]]s. Actually though, it is possible for there to be only left-handed neutrinos without any right-handed neutrinos if we could somehow introduce a tiny [[Majorana]]{{dn|date=December 2013}} coupling for the left-handed neutrinos.
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| Since the [[homotopy group]]
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| :<math>\pi_2\left(\frac{SU(5)}{[SU(3)\times SU(2)\times U(1)_Y]/\mathbb{Z}_6}\right)=\mathbb{Z}</math>
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| this model predicts [['t Hooft-Polyakov monopole]]s.
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| These monopoles have quantized Y magnetic charges. Since the electromagnetic charge Q is a linear combination of some SU(2) generator with Y/2, these monopoles also have quantized magnetic charges, where by magnetic here, we mean electromagnetic magnetic charges.
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| ==Minimal supersymmetric SU(5)==
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| {{Expand section|date=April 2008}}
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| ===Spacetime===
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| The N=1 superspace extension of 3+1 Minkowski spacetime.
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| ===Spatial symmetry===
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| N=1 SUSY over 3+1 Minkowski spacetime without R-symmetry.
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| ===Gauge symmetry group===
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| SU(5)
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| ===Global internal symmetry===
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| '''Z'''<sub>2</sub> (matter parity)
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| ===Matter parity===
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| To prevent unwanted couplings in the [[supersymmetry|supersymmetric]] version of the model, we assign a '''Z'''<sub>2</sub> [[matter parity]] to the chiral superfields with the matter fields having odd parity and the Higgs having even parity. This is unnecessary in the nonsupersymmetric version, but then, we can't protect the electroweak Higgs from quadratic radiative mass corrections. See [[hierarchy problem]]. In the nonsupersymmetric version the action is invariant under a similar '''Z'''<sub>2</sub> symmetry because the matter fields are all [[fermion]]ic and thus must appear in the action in pairs, while the Higgs fields are [[boson]]ic.
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| ===Vector superfields===
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| Those associated with the SU(5) gauge symmetry
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| ===Chiral superfields===
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| As complex representations:
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| {|
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| |-
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| |label||description||multiplicity||SU(5) rep||<math>\mathbb{Z}_2</math> rep
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| |-
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| |Φ||GUT Higgs field||1||'''24'''||+
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| |-
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| |H<sub>u<sub>||electroweak Higgs field||1||'''5'''||+
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| |-
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| |H<sub>d</sub>||electroweak Higgs field||1||'''<math>\bar{5}</math>'''||+
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| |-
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| |'''<math>\bar{5}</math>'''||matter fields||3||'''<math>\bar{5}</math>'''||-
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| |-
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| |'''10'''||matter fields||3||'''10'''||-
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| |-
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| |N<sup>c</sup>||sterile neutrinos||???||'''1'''||-
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| |}
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| ===Superpotential===
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| A generic invariant [[renormalizable]] [[superpotential]] is a (complex) <math>SU(5)\times\mathbb{Z}_2</math> invariant cubic polynomial in the superfields. It is a linear combination of the following terms:
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| <math>
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| \begin{matrix}
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| \Phi^2&\Phi^A_B \Phi^B_A\\
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| \Phi^3&\Phi^A_B \Phi^B_C \Phi^C_A\\
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| H_d H_u&{H_d}_A H_u^A\\
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| H_d \Phi H_u&{H_d}_A \Phi^A_B H_u^B\\
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| H_u \mathbf{10}_i\;\mathbf{10}_j&\epsilon_{ABCDE} H_u^A \mathbf{10}^{BC}_i \mathbf{10}^{DE}_j\\
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| H_d \mathbf{\bar{5}}_i\;\mathbf{10}_j&{H_d}_A \mathbf{\bar{5}}_{Bi} \mathbf{10}^{AB}_{j}\\
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| H_u \mathbf{\bar{5}}_i N^c_j&H_u^A \mathbf{\bar{5}}_{Ai} N^c_j\\
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| N^c_i N^c_j&N^c_i N^c_j\\
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| \end{matrix}
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| </math>
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| The first column is an Abbreviation of the second column (neglecting proper normalization factors), where capital indices are SU(5) indices, and i and j are the generation indices.
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| The last two rows presupposes the multiplicity of N<sup>c</sup> is not zero (i.e. that a [[sterile neutrino]] exists). The coupling H<sub>u</sub> 10<sub>i</sub> 10<sub>j</sub> has coefficients which are symmetric in i and j. The coupling N<sup>c</sup><sub>i</sub>N<sup>c</sup><sub>j</sub> has coefficients which are symmetric in i and j. Note that the number of [[sterile neutrino]] [[Generation (particle physics)|generation]]s need not be three, unless the SU(5) is embedded in a higher unification scheme such as [[SO(10) (physics)|SO(10)]].
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| ===Vacua===
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| The vacua correspond to the mutual zeros of the F and D terms. Let's first look at the case where the VEVs of all the chiral fields are zero except for Φ.
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| ====The Φ sector====
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| <math>W=Tr[a\Phi^2+b\Phi^3]</math>
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| The F zeros corresponds to finding the stationary points of W subject to the traceless constraint <math>Tr[\Phi]=0</math>. So,
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| <math>2a\Phi+3b\Phi^2=\lambda \mathbf{1}</math>
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| where λ is a Lagrange multiplier.
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| Up to an SU(5) (unitary) transformation,
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| <math>
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| \Phi=\left\{
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| \begin{matrix}
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| \operatorname{diag}(0,0,0,0,0)\\
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| \operatorname{diag}(\frac{2a}{9b},\frac{2a}{9b},\frac{2a}{9b},\frac{2a}{9b},-\frac{8a}{9b})\\
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| \operatorname{diag}(\frac{4a}{3b},\frac{4a}{3b},\frac{4a}{3b},-\frac{2a}{b},-\frac{2a}{b})
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| \end{matrix}
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| \right.
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| </math>
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| The three cases are called case I, II and III and they break the gauge symmetry into SU(5), <math>[SU(4)\times U(1)]/\mathbb{Z}_4</math> and <math>[SU(3)\times SU(2)\times U(1)]/\mathbb{Z}_6</math> respectively (the stabilizer of the VEV). | |
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| In other words, there at least three different superselection sections, which is typical for supersymmetric theories.
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| Only case III makes any [[phenomenology (science)|phenomenological]] sense and so, we will focus on this case from now onwards.
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| It can be verified that this solution together with zero VEVs for all the other chiral multiplets is a zero of the [[F-term]]s and [[D-term]]s. The matter parity remains unbroken (right up to the TeV scale).
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| ====Decomposition====
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| The gauge algebra '''24''' decomposes as <math>\begin{pmatrix}(8,1)_0\\(1,3)_0\\(1,1)_0\\(3,2)_{-\frac{5}{6}}\\(\bar{3},2)_{\frac{5}{6}}\end{pmatrix}</math>. This '''24''' is a real representation, so the last two terms need explanation. Both <math>(3,2)_{-\frac{5}{6}}</math> and <math>(\bar{3},2)_{\frac{5}{6}}</math> are complex representations. However, the direct sum of both representation decomposes into two irreducible real representations and we only take half of the direct sum, i.e. one of the two real irreducible copies. The first three components are left unbroken. The adjoint Higgs also has a similar decomposition, except that it is complex. The [[Higgs mechanism]] causes one real HALF of the <math>(3,2)_{-\frac{5}{6}}</math> and <math>(\bar{3},2)_{\frac{5}{6}}</math> of the adjoint Higgs to be absorbed. The other real half acquires a mass coming from the [[D-term]]s. And the other three components of the adjoint Higgs, <math>(8,1)_0</math>, <math>(1,3)_0</math> and <math>(1,1)_0</math> acquire GUT scale masses coming from self pairings of the superpotential, aΦ<sup>2</sup>+b<Φ>Φ<sup>2</sup>.
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| The sterile neutrinos, if any exists, would also acquire a GUT scale Majorana mass coming from the superpotential coupling ν<sup>c2</sup>.
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| Because of matter parity, the matter representations <math>\bar{5}</math> and '''10''' remain chiral.
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| It's the Higgs fields 5<sub>H</sub> and <math>\bar{5}_H</math> which are interesting.
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| {|
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| |-
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| |<math>5_H</math>
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| |<math>\bar{5}_H</math>
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| |-
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| |<math>
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| \begin{pmatrix}
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| (3,1)_{-\frac{1}{3}}\\
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| (1,2)_{\frac{1}{2}}
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| \end{pmatrix}
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| </math>
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| |<math>
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| \begin{matrix}
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| \_\_\_\\
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| ???
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| \end{matrix}
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| </math>
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| |<math>
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| \begin{pmatrix}
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| (\bar{3},1)_{\frac{1}{3}}\\
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| (1,2)_{-\frac{1}{2}}
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| \end{pmatrix}
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| </math>
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| |}
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| The two relevant superpotential terms here are <math>5_H \bar{5}_H</math> and <math><24>5_H \bar{5}_H</math>. Unless there happens to be some [[fine tuning]], we would expect both the triplet terms and the doublet terms to pair up, leaving us with no light electroweak doublets. This is in complete disagreement with phenomenology. See [[doublet-triplet splitting problem]] for more details.
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| ====Fermion masses====
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| See [[Georgi-Jarlskog mass relation]].
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| ==Lee Smolin's view of SU(5)==
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| In his book ''[[The Trouble with Physics]]'', Smolin states:
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| {{quote|After some twenty-five years, we are still waiting. No protons have decayed. We have been waiting long enough to know that SU(5) grand unification is wrong. It's a beautiful idea, but one that nature seems not to have adopted. Page 64.
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| Indeed, it would be hard to underestimate the implications of this negative result. SU(5) is the most elegant way imaginable of unifying quarks with leptons, and it leads to a codification of the properties of the standard model in simple terms. Even after twenty-five years, I still find it stunning that SU(5) doesn't work. Page 65.
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| {{cite book |last=Smolin |first=Lee|authorlink=Lee Smolin|year=2007 |title=[[The Trouble with Physics]]}}
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| }}
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| ==Popular culture==
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| When the filmmaker Sandy Bates (played by [[Woody Allen]]) in the 1980 Woody Allen film [[Stardust Memories]] launches a depressive soliloquy with the quote, "Did anybody read on the front page of The Times that matter is decaying?", this was almost certainly a reference to the Georgi–Glashow model, given the film's period, the importance of the Georgi–Glashow model at the time and the many contemporary layperson articles in circulation about some of the model's most striking consequences, particularly its mechanism for proton decay. An actual New York Times article<ref>Physics sometimes takes G.U.T.s, New York Times, September 19, 1982</ref> appeared two years later, fulfilling Allen's blackly humorous foreshadowing of a world whose news was so baleful that the mainstream media were systematically reporting its material demise.
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| ==References==
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| <references />
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| * Howard Georgi and Sheldon Glashow, ''Unity of All Elementary-Particle Forces'', Physical Review Letters, '''32''' (1974) 438.
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| *{{cite arXiv |author=[[John Baez|J.C. Baez]], J. Huerta |eprint=0904.1556 |title=The Algebra of Grand Unified Theories |year=2009 |class=hep-th }}
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| {{DEFAULTSORT:Georgi-Glashow model}}
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| [[Category:Particle physics]]
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