Doublet–triplet splitting problem

Solutions to the doublet-triplet splitting

The missing partner mechanism

One solution to the doublet–triplet splitting (DTS) in the context of supersymmetric ${\displaystyle SU(5)}$ 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 ${\displaystyle Z_{50}}$ and ${\displaystyle Z_{\overline{50}}}$. Note that ${\displaystyle \mathbf{50}}$ decomposes as follows under the SM gauge group:

${\displaystyle \mathbf{50}\to (\mathbf{1},\mathbf{1},-2)+(\mathbf{3},\mathbf{1},-\frac{1}{3})+(\overline{\mathbf{3}},\mathbf{2},-\frac{7}{6})+(\mathbf{6},\mathbf{1},\frac{4}{3})+(\overline{\mathbf{6}},\mathbf{3},-\frac{1}{3})+(\mathbf{8},\mathbf{2},\frac{1}{2})}$

which contains no field that could couple to the ${\displaystyle SU(2)}$ doublets of ${\displaystyle H_{\overline{5}}}$ or ${\displaystyle H_5}$. Due to group theoretical reasons ${\displaystyle SU(5)}$ has to be broken by a ${\displaystyle \mathbf{75}}$ instead of the usual ${\displaystyle \mathbf{24}}$, at least at the renormalizable level. The superpotential then reads

${\displaystyle W_{MPM}=y_1{H}_{\overline{5}}{H}_{75}{Z}_{50}+y_2{Z}_{\overline{50}}{H}_{75}{H}_5+m_{50}{Z}_{50}{Z}_{\overline{50}}.}$

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, ${\displaystyle \mathrm{\Sigma }}$ acquires a vacuum expectation value of the form

${\displaystyle ⟨\mathrm{\Sigma }⟩=\text{diag}(i{\sigma }_2{f}_3,i{\sigma }_2{f}_3,i{\sigma }_2{f}_3,i{\sigma }_2{f}_2,i{\sigma }_2{f}_2)}$.

${\displaystyle f_2}$ and ${\displaystyle f_3}$ give masses to the Higgs doublet and triplet, respectively, and are independent of each other, because ${\displaystyle \mathrm{\Sigma }}$ 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):

${\displaystyle 5\to (1,2{)}_{\frac{1}{2}}\oplus (3,1{)}_{-\frac{1}{3}}}$

${\displaystyle \overline{5}\to (1,2{)}_{-\frac{1}{2}}\oplus (\overline{3},1{)}_{\frac{1}{3}}}$

In SO(10):

${\displaystyle 10\to (1,2{)}_{\frac{1}{2}}\oplus (1,2{)}_{-\frac{1}{2}}\oplus (3,1{)}_{-\frac{1}{3}}\oplus (\overline{3},1{)}_{\frac{1}{3}}}$

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 Minimal Supersymmetric Standard Model; there it is not enough simply to require the triplet to have a GUT scale mass.