Doublet–triplet splitting problem
In particle physics, the doublet–triplet (splitting) problem is a problem of some Grand Unified Theories, such as SU(5), SO(10), and formula_1. Grand unified theories predict Higgs bosons (doublets of formula_2) arise from 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 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?' Doublet–triplet splitting and the μ-problem. In 'minimal' SU(5), the way one accomplishes doublet–triplet splitting is through a combination of interactions formula_3 where formula_4 is an adjoint of SU(5) and is traceless. When formula_4 acquires a vacuum expectation value formula_6 that breaks SU(5) to the Standard Model gauge symmetry the Higgs doublets and triplets acquire a mass formula_7 Since formula_8 is at the GUT scale (formula_9 GeV) and the Higgs doublets need to have a weak scale mass (100 GeV), this  requires formula_10. So to solve this doublet–triplet splitting problem requires a tuning of the two terms to within one part in formula_11. This is also why the mu problem of the MSSM (i.e. why are the Higgs doublets so light) and doublet–triplet splitting are so closely intertwined. Solutions to the doublet-triplet splitting. The missing partner mechanism. One solution to the doublet–triplet splitting (DTS) in the context of supersymmetric formula_12 proposed in and is called the missing partner mechanism (MPM). The main idea is that in addition to the usual fields there are two additional chiral super -fields formula_13 and formula_14. Note that formula_15 decomposes as follows under the SM gauge group: which contains no field that could couple to the formula_2 doublets of formula_18 or formula_19. Due to group theoretical reasons formula_12 has to be broken by a formula_21 instead of the usual formula_22, at least at the renormalizable level. The superpotential then reads After breaking to the SM the colour triplet can get super heavy, suppressing Proton decay, while the SM Higgs does not. Note that nevertheless the SM Higgs will have to pick up a mass in order to reproduce the electroweak theory correctly . Note that although solving the DTS problem the MPM tends to render models Non-perturbative just above the GUT scale. This problem is addressed by the "Double missing partner mechanism". Dimopoulos–Wilczek mechanism. 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, formula_4 acquires a vacuum expectation value of the form formula_25. formula_26 and formula_27 give masses to the Higgs doublet and triplet, respectively, and are independent of each other, because formula_4 is traceless for any values they may have. If formula_29, 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. 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. Higgs representations in Grand Unified Theories. In SU(5): In SO(10): Proton decay. Non-supersymmetric theories suffer from quartic radiative corrections 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 MSSM; there it is not enough simply to require the triplet to have a GUT scale mass.