In particle physics

Particle physics

Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

, the doublet-triplet (splitting) problem is a problem of some Grand Unified Theories

Grand unification theory

The term Grand Unified Theory, often abbreviated as GUT, refers to any of several similar candidate models in particle physics in which at high-energy, the three gauge interactions of the Standard Model which define the electromagnetic, weak, and strong interactions, are merged into one single...

, such as SU(5)

Georgi-Glashow model

In particle physics, the Georgi–Glashow model is a particular grand unification theory proposed by Howard Georgi and Sheldon Glashow in 1974. In this model the standard model gauge groups SU×SU×U are combined into a single simple gauge group -- SU...

, SO(10)

SO(10) (physics)

In particle physics, one of the grand unified theories is based on the SO Lie group....

, . Grand unified theories predict Higgs boson

Higgs boson

The Higgs boson is a hypothetical massive elementary particle that is predicted to exist by the Standard Model of particle physics. Its existence is postulated as a means of resolving inconsistencies in the Standard Model...

s (doublets of ) arise from representations

Group representation

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...

 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

Proton decay

In particle physics, proton decay is a hypothetical form of radioactive decay in which the proton decays into lighter subatomic particles, such as a neutral pion and a positron...

 in supersymmetric

Supersymmetry

In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...

 theories that are only suppressed by two powers of GUT scale (ie 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 -problem

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



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



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



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

.

So to solve thus doublet-triplet splitting problem requries a tuning of the two terms to within 1 part in .
This is also why the mu problem

Mu problem

In theoretical physics, the μ problem is a problem of supersymmetric theories, concerned with understanding the parameters of the theory.The supersymmetric Higgs mass parameter μ appears as the following term in the superpotential: μHuHd. It is necessary to provide a mass for the fermionic...

 of the MSSM

Minimal Supersymmetric Standard Model

The Minimal Supersymmetric Standard Model is the minimal extension to the Standard Model that realizes supersymmetry, although non-minimal extensions do exist. Supersymmetry pairs bosons with fermions; therefore every Standard Model particle has a partner that has yet to be discovered...

 (i.e. why are the Higgs doublets so light) and doublet-triplet splitting are so closely intertwined.

The Dimopoulos–Wilczek mechanism

In an SO(10) theory, there is potential solution to the doublet-triplet splitting problem known as the 'Dimopoulos–Wilczek' mechanism. In SO(10), the adjoint field, acquires a vacuum expectation value of the form

.

and give masses to the Higgs doublet and triplet, respectively, and are independent of each other, because is traceless for any values they may have. If , 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 quadratric radiative corrections to the mass squared of the electroweak Higgs boson (see hierarchy problem

Hierarchy problem

In theoretical physics, a hierarchy problem occurs when the fundamental parameters of some Lagrangian are vastly different than the parameters measured by experiment. This can happen because measured parameters are related to the fundamental parameters by a prescription known as renormalization...

). In the presence of supersymmetry

Supersymmetry

In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...

, the triplet Higgsino

Higgsino

In particle physics, a Higgsino, symbol , is the hypothetical superpartner of the Higgs boson, as predicted by supersymmetry. The Higgsino is a Dirac fermion and that is a weak isodoublet with hypercharge half under the Standard Model gauge symmetries...

 needs to be more massive than the GUT scale to prevent proton decay because it generates dimension 5 operators in MSSM

Minimal Supersymmetric Standard Model

The Minimal Supersymmetric Standard Model is the minimal extension to the Standard Model that realizes supersymmetry, although non-minimal extensions do exist. Supersymmetry pairs bosons with fermions; therefore every Standard Model particle has a partner that has yet to be discovered...

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