Doublet–triplet splitting problem

Doublet–triplet splitting and the μ {displaystyle mu } -problem

In 'minimal' SU(5), the way one accomplishes doublet–triplet splitting is through a combination of interactions

∫ d 2 θ λ H 5 ¯ Σ H 5 + μ H 5 ¯ H 5

where Σ is an adjoint of SU(5) and is traceless. When Σ acquires a vacuum expectation value

⟨ Σ ⟩ = d i a g ( 2 , 2 , 2 , − 3 , − 3 ) f

that breaks SU(5) to the Standard Model gauge symmetry the Higgs doublets and triplets acquire a mass

∫ d 2 θ ( 2 λ f + μ ) H 3 ¯ H 3 + ( − 3 λ f + μ ) H 2 ¯ H 2

Since f is at the GUT scale ( 10 16 GeV) and the Higgs doublets need to have a weak scale mass (100 GeV), this requires

μ ∼ 3 λ f ± 100 GeV .

So to solve this doublet–triplet splitting problem requires a tuning of the two terms to within one part in 10 14 . 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.

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, Σ acquires a vacuum expectation value of the form

⟨ Σ ⟩ = diag ( i σ 2 f 3 , i σ 2 f 3 , i σ 2 f 3 , i σ 2 f 2 , i σ 2 f 2 ) .

f 2 and f 3 give masses to the Higgs doublet and triplet, respectively, and are independent of each other, because Σ is traceless for any values they may have. If f 2 = 0 , 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.