# 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), ${\displaystyle E_{6}}$. Grand unified theories predict Higgs bosons (doublets of ${\displaystyle SU(2)}$) 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 ${\displaystyle \mu }$-problem

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

${\displaystyle \int d^{2}\theta \;\lambda H_{\bar {5}}\Sigma H_{5}+\mu H_{\bar {5}}H_{5}}$

where ${\displaystyle \Sigma }$  is an adjoint of SU(5) and is traceless. When ${\displaystyle \Sigma }$  acquires a vacuum expectation value

${\displaystyle \langle \Sigma \rangle ={\rm {{diag}(2,2,2,-3,-3)f}}}$

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

${\displaystyle \int d^{2}\theta \;(2\lambda f+\mu )H_{\bar {3}}H_{3}+(-3\lambda f+\mu )H_{\bar {2}}H_{2}}$

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

${\displaystyle \mu \sim 3\lambda f\pm 100{\mbox{GeV}}}$ .

So to solve this doublet–triplet splitting problem requires a tuning of the two terms to within one part in ${\displaystyle 10^{14}}$ . 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, ${\displaystyle \Sigma }$  acquires a vacuum expectation value of the form

${\displaystyle \langle \Sigma \rangle ={\mbox{diag}}(i\sigma _{2}f_{3},i\sigma _{2}f_{3},i\sigma _{2}f_{3},i\sigma _{2}f_{2},i\sigma _{2}f_{2})}$ .

${\displaystyle f_{2}}$  and ${\displaystyle f_{3}}$  give masses to the Higgs doublet and triplet, respectively, and are independent of each other, because ${\displaystyle \Sigma }$  is traceless for any values they may have. If ${\displaystyle f_{2}=0}$ , 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):

${\displaystyle 5\rightarrow (1,2)_{1 \over 2}\oplus (3,1)_{-{1 \over 3}}}$
${\displaystyle {\bar {5}}\rightarrow (1,2)_{-{1 \over 2}}\oplus ({\bar {3}},1)_{1 \over 3}}$

In SO(10):

${\displaystyle 10\rightarrow (1,2)_{1 \over 2}\oplus (1,2)_{-{1 \over 2}}\oplus (3,1)_{-{1 \over 3}}\oplus ({\bar {3}},1)_{1 \over 3}}$

## Proton decay

Dimension 6 proton decay mediated by the triplet Higgs ${\displaystyle T(3,1)_{-{\frac {1}{3}}}}$  and the anti-triplet Higgs ${\displaystyle {\bar {T}}({\bar {3}},1)_{\frac {1}{3}}}$  in ${\displaystyle SU(5)}$  GUT

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.