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| In [[particle physics]], the '''doublet–triplet (splitting) problem''' is a problem of ''some'' [[Grand unification theory|Grand Unified Theories]], such as [[Georgi-Glashow model|SU(5)]], [[SO(10) (physics)|SO(10)]], <math>E_6</math>. Grand unified theories predict [[Higgs boson]]s (doublets of <math>SU(2)</math>) arise from [[Group representation|representations]] of the unified group that contain other states, in particular, states that are triplets of color. The primary problem with these color triplet Higgs, is that they can mediate [[proton decay]] in [[Supersymmetry|supersymmetric]] theories that are only suppressed by two powers of GUT scale (i.e. they are dimension 5 supersymmetric operators). In addition to mediating proton decay, they alter [[gauge coupling unification]]. The doublet–triplet problem is the question 'what keeps the doublets light while the triplets are heavy?'
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| ==Doublet–triplet splitting and the <math>\mu</math>-problem==
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| In 'minimal' SU(5), the way one accomplishes doublet–triplet splitting is through a combination of interactions
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| <math> \int d^2\theta \; \lambda H_{\bar{5}} \Sigma H_{5} + \mu H_{\bar{5}} H_{5}</math>
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| where <math>\Sigma</math> is an adjoint of SU(5) and is [[traceless]]. When <math>\Sigma</math> acquires a vacuum expectation value
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| <math>\langle \Sigma\rangle = \rm{diag}(2, 2, 2, -3, -3) f</math>
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| that breaks SU(5) to the Standard Model gauge symmetry the Higgs doublets and triplets acquire a mass
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| <math> \int d^2\theta \; (2 \lambda f + \mu) H_{\bar{3}}H_3 + (-3\lambda f +\mu) H_{\bar{2}}H_2</math>
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| Since <math> f</math> is at the GUT scale (<math> 10^{16}</math> GeV) and the Higgs doublets need to have a weak scale mass (100 GeV), this requires
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| <math>\mu \sim 3 \lambda f \pm 100 \mbox{GeV}</math>.
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| So to solve this doublet–triplet splitting problem requires a tuning of the two terms to within one part in <math>10^{14}</math>.
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| This is also why the [[mu problem]] of the [[Minimal Supersymmetric Standard Model|MSSM]] (i.e. why are the Higgs doublets so light) and doublet–triplet splitting are so closely intertwined.
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| ===Dimopoulos–Wilczek mechanism===
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| In an SO(10) theory, there is a potential solution to the doublet–triplet splitting problem known as the 'Dimopoulos–Wilczek' mechanism. In SO(10), the adjoint field, <math>\Sigma</math> acquires a vacuum expectation value of the form
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| <math>\langle \Sigma \rangle = \mbox{diag}( i \sigma_2 f_3, i\sigma_2 f_3, i\sigma_2 f_3, i\sigma_2 f_2, i \sigma_2 f_2)</math>.
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| <math>f_2</math> and <math>f_3</math> give masses to the Higgs doublet and triplet, respectively, and are independent of each other, because <math>\Sigma</math> is [[traceless]] for any values they may have. If <math>f_2=0</math>, then the Higgs doublet remains massless. This is very similar to the way that doublet–triplet splitting is done in either higher-dimensional grand unified theories or string theory.
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| To arrange for the VEV to align along this direction (and still not mess up the other details of the model) often requires very contrived models, however.
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| ==Higgs representations in Grand Unified Theories==
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| In SU(5):
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| :<math>5\rightarrow (1,2)_{1\over 2}\oplus (3,1)_{-{1\over 3}}</math> | |
| :<math>\bar{5}\rightarrow (1,2)_{-{1\over 2}}\oplus (\bar{3},1)_{1\over 3}</math>
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| In SO(10):
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| :<math>10\rightarrow (1,2)_{1\over 2}\oplus (1,2)_{-{1\over 2}}\oplus (3,1)_{-{1\over 3}}\oplus (\bar{3},1)_{1\over 3}</math>
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| ==Proton decay==
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| [[Image:proton decay4.svg|left|frame|Dimension 6 proton decay mediated by the triplet Higgs <math>T (3,1)_{-\frac{1}{3}}</math> and the anti-triplet Higgs <math>\bar{T} (\bar{3},1)_{\frac{1}{3}}</math> in <math>SU(5)</math> GUT]]
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| Non-[[supersymmetric]] theories suffer from q­ratric [[radiative correction]]s to the mass squared of the electroweak Higgs boson (see [[hierarchy problem]]). In the presence of [[supersymmetry]], the triplet [[Higgsino]] needs to be more massive than the GUT scale to prevent proton decay because it generates dimension 5 operators in [[Minimal Supersymmetric Standard Model|MSSM]]; there it is not enough simply to require the triplet to have a [[GUT scale]] mass.
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| ==References==
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| * 'Supersymmetry at Ordinary Energies. 1. Masses AND Conservation Laws.' [[Steven Weinberg]]. Published in Phys.Rev.D26:287,1982.
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| * 'Proton Decay in Supersymmetric Models.' [[Savas Dimopoulos]], Stuart A. Raby, [[Frank Wilczek]]. Published in Phys.Lett.B112:133,1982.
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| * 'Incomplete Multiplets in Supersymmetric Unified Models.' Savas Dimopoulos, Frank Wilczek.
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| {{DEFAULTSORT:Doublet-triplet splitting problem}}
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| [[Category:Particle physics]]
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