Spontaneous Symmetry Breaking in Non Abelian Gauge Theories

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Spontaneous Symmetry Breaking in Non Abelian Gauge Theories

Transcript Of Spontaneous Symmetry Breaking in Non Abelian Gauge Theories

Spontaneous Symmetry Breaking in Non Abelian Gauge Theories
Michael LeBlanc
Abstract
This term paper explores the phenomenon of spontaneous symmetry breaking in systems with non-abelian gauge symmetry. The most famous example comes from the standard model of particle physics, wherein symmetry breaking is responsible for the masses of the W and Z bosons and the corresponding short range of the weak nuclear force. The electroweak theory will be discussed as well as a more ambitious (but less successful) attempt to unify the strong, weak and electromagnetic forces in which symmetry breaking also plays a key role.
1 Introduction
The history of the principle of gauge invariance [9] is a long and storied one that stretches far back into the nineteenth century, beginning with the observation that the electromagnetic scalar and vector potential were underdetermined and thus could be subjected to somewhat arbitrary additional constraint equations (gauge conditions). In 1926, Fock discovered that a charged particle’s wave equation was invariant under a position and time-dependent phase transformation φ → exp(iθ(￿x, t))φ provided that the electromagnetic potentials were properly adjusted by a gauge transformation to compensate. The idea that the electromagnetic coupling can be looked at as a consequence of local phase invariance, which traces back to a speculation made by Weyl in 1928, stands at the center of particle physics today.
In the mid fifties, Yang and Mills [16] showed that the gauge principle could be generalized from phase (U (1)) to isospin (SU (2)) transformations. The main difficulty associated with this extension is that the isospin transformations do not commute with one another, thus their theory is a termed non-abelian gauge theory, in contrast with the abelian electromagnetism. Their approach is easily generalized from SU (2) to any compact lie group, therefore gauge theories have the allure of associating to an abstract symmetry group of one’s choosing a unique theory of interacting matter and gauge fields.
Notwithstanding this allure, these theories at first seemed unsuitable for describing fundamental interactions since they involved massless gauge bosons which had not been observed and were deemed unfit for describing the short-ranged nuclear forces. It turns out this problem can be avoided in two ways: the bosons can become massive due
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to spontaneous symmetry breaking or the bosons can not be observed in the particle spectrum due to confinement. Both of these possibilities are realized in the weak and the strong force, respectively, but it is the first that will be the main subject of this paper. The outline is as follows:
First, I will give a brief mathematical introduction to theories with spontaneously broken continuous symmetries, beginning with Goldstone’s U (1) model and the abelian Higgs model. This is really nothing more than the Ginzburg-Landau theory learned in class, but it will give me a chance to establish relativistic notation in a familiar context. Then I will generalize to non abelian gauge theories, keeping the mathematical overhead to a minimum.
After the formalism is established I will explore three different theories with different symmetry breaking patterns. The first is SU (2) → U (1), which is a simple model for electroweak symmetry breaking with no Z boson. This model is not correct, but it is still interesting. For instance, it has magnetic monopole solutions which are an example of topological defects as a consequence of symmetry breaking. The analogue from class would be flux tubes in the U (1) theory.
The second example is SU (2) × U (1)Y → U (1)EM . This is the celebrated GlashowWeinberg-Salam model. I will discuss matter content and the experimental successes of this model. I will also briefly discuss some of its shortcomings as well as alternative symmetry breaking mechanisms to the standard model Higgs.
The third example is SU (5) → SU (3)×SU (2)×U (1). This is Georgi and Glashow’s minimal Grand Unified Theory (GUT) attempting to unify the strong, weak and electromagnetic forces.

2 Symmetry Breaking in Gauge Theories

2.1 U(1) theory
For our first example of a theory with a con￿tinuous symmetry [5], we consider a complex scalar field φ with relativistic action S = d4x L where

L = (∂µφ)∗∂µφ − V (|φ|2).

The action is invariant under transformations of the form φ → eiθφ, so we say the

theory has internal U (1) symmetry. The equation of motion (Euler-Lagrange equation)

derived from this action is

∂2φ + φ ∂V = 0, ∂|φ|2

which becomes the nice linear Klein Gordon equation for V = m2|φ|2. The Hamiltonian

is

￿

H = d3x |∂tφ|2 + |∇￿ φ|2 + V (|φ|2).

Notice that the lowest energy field configuration is a constant that minimizes V .
To establish a symmetry breaking minimum, we select a quartic potential V = λ2 (|φ|2 − v2/2)2 where v is a real number. The minimum field configuration then

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becomes

the

degenerate

circle

|φ|2

=

v2 2

.√We

can

then

expand

the

Lagrangian

about

the arbitrarily selected minimum φ0 = v/ 2 in terms of new fields η(x) and θ(x) where

φ(x) = √1 (v + η(x))eiθ(x)/v and obtain
2

L = 1 ∂µη∂µη + (1 + η/v)2 ∂µθ∂µθ − λv2 η2 + O(η3).

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2

2

Our new fields obey linearized equations of motion

(∂2 + m2η)η = 0;

∂2θ = 0 (exact)

where m2η = λv2 and the θ field is massless, the first example of a Goldstone mode. Now, return to the original version of the Lagrangian and consider local U (1) trans-
formations of the form φ → eiθ(x)φ where the transformation parameter is a function
of space-time. Now, the derivative terms in the action are not invariant, but we can
compensate by replacing the derivatives with covariant derivatives Dµ ≡ ∂µ − ieAµ and having the vector field Aµ compensate by transforming Aµ → Aµ + 1e ∂µθ. This is an example of the principle mentioned above: a global symmetry can be promoted to a
local symmetry with the addition of a vector field and the form of the interactions are determined by this process. Notice that the transformation on Aµ is just a gauge trans-
formation from electricity and magnetism, so we’ll identify it with an electromagnetic
field and write the Lagrangian:

L = (Dµφ)∗Dµφ − V (|φ|2) − 1 Fµν F µν 4
where Fµν ≡ ∂µAν − ∂νAµ is the electromagnetic field tensor, and the last term was chosen to generate Maxwell’s vacuum equations when φ = 0. Notice that the term (and in fact Fµν itself) is gauge invariant, so its addition to the Lagrangian does not break the local symmetry.
We’re in position to see the abelian Higgs mechanism first derived in this form in [6]. Consider the same potential as before and expand again about the same φ0, we get

L = 12 ∂µη∂µη + (1 + 2η/v)2 (∂µθ − evAµ)(∂µθ − evAµ) − m22η η2 − 14 F µν Fµν + O(η3).
We can perform a gauge transformation Aµ → Aµ + e1v ∂µθ and obtain
L = 12 ∂µη∂µη − m22η η2 + v22e2 AµAµ − 14 F µν Fµν + higher order terms.
The linearized equation of motion for the electromagnetic field is therefore ∂µF µν = −e2v2Aν. The Lorenz gauge condition ∂µAµ = 0, which can be assumed from the outset, is not ruined by the gauge transformation since ∂2θ = 0. Therefore the equation of motion becomes
(∂2 + m2γ)Aµ = 0
where m2γ = e2v2. The electromagnetic field has obtained a mass.

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We’ve put the Higgs mechanism we learned about in class into the language of relativistic field theory. The main idea is that since a Goldstone boson in a gauge theory is an excitation along an unphysical degree of freedom, it cannot correspond to a physical particle. However, removing the boson from the theory amounts to a choice of gauge. We’ve now used up our option to choose a gauge that’s normally used to make the polarization transverse; the longitudinal polarization is now a physical degree of freedom and the gauge boson can have a mass. This means we are left with a massive η particle (Higgs), a massive gauge boson and no Goldstone boson.

2.2 Yang-Mills Theory

We will now generalize to scalar theories invariant under SU (N ) transformations φ →
exp(iθaτa)φ [16] Here, φ = (φ1, . . . φN ) is a complex N-tuple and τa are generator
matrices, which act as basis vectors for the SU (N ) at the identity. In the most common SU (2) case, the index a goes from one to three and τa = 12 σa where σa are the Pauli matrices. The generators can be chosen [2] to obey tr(τa) = 0, tr(τaτb) = 12 δab and [τa, τb] = ifabcτc where the fabc is a real, antisymmetric tensor. In the familiar SU (2) case, fabc = ￿abc.
We can take a similar Lagrangian for the scalar field as above L = |∂φ|2 − V (|φ|2)
(where Lorentz and SU (N ) index sums are implied) which has the required invariance.
To promote the symmetry to a local symmetry, we must replace the derivatives with
covariant derivatives Dµ = ∂µ − igAµ. The gauge field must transform like Aµ → exp(iθaτa)(Aµ + g1 ∂µθbτb) exp(−iθaτa) so that Dµφ → exp(iθaτa)Dµφ. Notice that the gauge field is now an N × N matrix. It can be taken to be traceless and Hermitian
(notice the gauge transformation doesn’t change this property), so it can be written in
terms of its components Aµ = Aaµτa.
All that remains is to come up with dynamics for the gauge field alone, the analogue of the Maxwell Lagrangian − 14 (Fµν)2. It turns out [10] that the key property of the field tensor that generalizes is that −ieFµν = [Dµ, Dν]. In the non-abelian case have [Dµ, Dν] = −ig(∂µAν − ∂νAµ − g2[Aµ, Aν]. The second term, −g2[Aµ, Aν] = −ig2AaµAbνf abcτ c was not present in the U (1) case. The analogue of the field tensor can then be defined Fµν = Fµaντ a where

Fµaν = ∂µAaν − ∂ν Aaµ + gf abcAbµAcν .

The Lagrangian for a non abelian gauge field coupled to a scalar N-tuple is then defined

as

L

=

(Dµφ)†Dµφ



V

(φ†φ)



1 F aµν F a
µν

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The final term is gauge invariant since it can be written − 12 tr(F µνFµν) and the field tensor transforms like Fµν → exp(iθaτa)Fµν exp(−iθaτa). This term can be justified

not only because it looks similar to the Maxwell Lagrangian, but also since it is the

only renormalizable, Lorentz-and-gauge–invariant operator that obeys time-reversal

symmetry.

The gauge field part of the Lagrangian can be expanded to

− 1 (∂µAaν



∂ν Aaµ)2



1 gf abc(∂µAaν



∂ν Aaµ)AbµAcν



1 g2(f abcAb Ac )2.
µν

4

2

4

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This Lagrangian has quartic and cubic terms in the Aaµ so unlike the maxwell Lagrangian, it contains interactions whose forms are fixed by symmetry. There is no mass term of the form m2(Aaµ)2 so the gauge fields are massless and a term of such a form would violate gauge invariance just like in the U (1) case.

Symmetry Breaking Before we discuss symmetry breaking, we will make one

slight generalization. There is no reason why the matter fields need to transform

under the fundamental representation of SU (N ). In fact, everything we did above

holds for fields transforming under an arbitrary representation. The generators will

change (perhaps to matrices of a different size), but they will still be traceless and

Hermitian and obey the commutation relations (these are properties of special unitary

transformations in general).

If the matter field’s potential is chosen so that spontaneous symmetry breaking

occurs, we can see how the gauge bosons will obtain mass [8]. Lets, for instance, take

phi in the fundamental representation of SU (2). Let’s say there is a potential as before of V (φ†φ) = λ (φ†φ − v2 )2. This yields the ground state condition φ†φ = v2/2. We can
2 ￿ ￿2 choose φ0 = √12 v0 . Plugging φ = φ0 this into the term |Dµφ|2 = |(∂µ −igAaµτ a)φ|2,

we get a term quadratic in the gauge fields g2AaµAbµφ†0τ aτ bφ0 = 12 (m2)abAaµAbµ. The

quantity

(m2)ab = 2g2φ†0τ (aτ b)φ0

is known as the mass matrix for the gauge bosons. Diagonalizing it give the linear
combinations of gauge fields that have definite masses. Notice we took the symmetric part of the matrix since it’s multiplied by the symmetric AaµAbµ. In our current example,

(m2)ab = 2g2φ†0 14 δabφ0 = 14 (gv)2δab.

In this example the mass matrix is diagonal so each of the three gauge bosons A1µ, A2µ and A3µ hs a mass mA = gv/2.
We will see below that the symmetry group that leaves φ0 invariant is important. In the present example and the U (1) example above the classical minimum had no residual symmetry and so we say the symmetry breaking patterns are SU (2) → nothing and U (1) → nothing. We will see that the lack of residual symmetry means that all of the gauge fields get mass and that breaking patterns with residual symmetry have gauge particles that remain massless.

3 SU (2) → U (1)
The first example we will discuss in detail was introduced by Georgi and Glashow in 1972 [3] as a model of electroweak symmetry breaking. In this case, our Higgs field transforms under the three-dimensional vector representation of SU (2). This representation is familiar as the defining representation of SO(3). The Hermitian, traceless

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generator matrices are given by





00 0

 0

τ 1 =  0 0 −i  τ 2 =  0

 0i





0 −i 0

0 0  τ3 =  i 0 0  .

0i 0

−i 0 0

000

In other words, (τ a)bc = −i￿abc. Noticing that this is a real representation, w can pick

a Lagrangian

L

=

1 (Dµφ)2



λ (φ†φ



v2)2



1 F aµν F a .
µν

2

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4



0

We can pick an average value φ0 =  0  . Notice that this value of φ0 has residual

v

symmetry: it is invariant under rotations about the z axis. This also follows from the

fact that iτ 3φ0 = 0.

We can immediately compute the mass matrix for this theory

(m2)ab = g2φ†0τ aτ bφ0 = v2(τ a)3c(τ b)c3 = −v2￿a3c￿bc3 = (gv)2(δab − δa3δb3).

So we see m1 = m2 = gv and m3 = 0. One of our bosons is massless and the other two

are massive. These can be interpreted as a photon and two massive bosons mediating

the weak force.

We can parameterize the fluctuations of the real scalar field about its ground state

by





0

φ = exp(iθ1(x)τ 1 + iθ2(x)τ 2)  0 .

v + η(x)

Just like in the U (1) case, a gauge transformation can then be performed to eliminate the exponential pre-factor, retaining only the longitudinal fluctuation. The Lagrangian then becomes

L = 1 (∂µη)2− λv2 η2− 1 F aµν F a + g2v2 (A1µA1 +A2µA2 )− λ η4− λv η3+ 2ηv + η2 (A1µA1 +A2µA2 ).

2

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µν 2

µ

µ8

4

2

µ

µ

√ This has a massive scalar Higgs field η with mass λv2, two massive and one massive

gauge field, as well as Higgs self-interactions, gauge-gauge interactions from the pure

Yang-Mills term and interactions between the Higgs and gauge bosons.

Monopoles The SU (2) → U (1) model is not the correct theory of electroweak
symmetry breaking. The most obvious problem is the omission of the Z boson and the corresponding neutral current weak interactions. However, the theory has one interesting feature that is not present in the correct GWS theory: monopole solutions discovered independently by ’t Hooft and Polyakov in 1974 [14]. The idea is that if the massless generator is identified with a photon, one can find stable solutions of the classical field equations with magnetic charge. This is a direct consequence of spontaneously broken gauge symmetry as we shall see.

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The problem is similar to magnetic flux tubes in a superconductor. The feature is that the phase of the order parameter winds around as you transit a circle enclosing the defect. In this three-dimensional analogue, we take the Higgs field to obey
φ￿ = vrˆ

at spatial infinity. Normally, the fact that the field is changing at infinity would cause
the energy of this solution to diverge thanks to the gradient term, but we can arrange the gauge field to cancel this and cause Dµφ￿ → 0 at infinity by putting

Aa = 1 ￿bij rj i e r2

(this is a static solution so the fields are time independent and Aa0 can be taken to vanish.)

Now, we need an electromagnetic field tensor. This is given by the gauge invariant

definition

Fµaν φa ￿abcφa(Dµφ)b(Dν φ)c

Gµν = |φ| −

g|φ|3

.

In a small region at spatial infinity where the Higgs field can be taken to be constant,

Gµν = ∂µA3ν − ∂ν A3µ.

Plugging

in

the

asymptotic

values

gives

Fij

=

￿ij k rk gr3

which

corresponds

to

a

radial

magnetic field

B￿ = rˆ . gr2

Since this result for the charge of the monopole is not analytic in g, this is a non-

perturbative property.

The energy of this solution (mass of the monopole is

￿￿

￿

E = d3x 1 (Fij)2 + 1 (Diφ)2 + V > 4πv/g = 4πm/g2 ≈ 137m

4

2

where m is the mass of one of the gauge bosons. So the monopole’s mass is a couple orders of magnitude above the mass scale associated with the symmetry breaking.

4 SU (2) × U (1) → U (1)[15]
The SU (2) → nothing breaking pattern mentioned above yields no massless gauge bosons and so cannot incorporate the photon, but if the gauge symmetry is extended to include an additional U (1) factor, the φ0 we choose will remain invariant under a one parameter group of transformations and we will have residual symmetry and one massless boson. Matter fields are classified by which representation of SU (2) they transform under and what hypercharge they carry under the U (1) transformation. For instance if the field transforms ψ → exp(iY θ)ψ under a U (1) rotation of θ, the hypercharge is given by that Y value.
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The Higgs field is a complex double that transforms as (2, 1/2), which is code for it
transforming under the two-dimensional defining representation of SU (2) and having Y = ￿1/2. We ￿will again set a potential that forces |φ0|2 = v/2 and arbitrarily set φ0 = 0√ . Then the covariant derivative can be worked out explicitly
v/ 2

￿

￿

0

iv ￿ gA1 − igA2 ￿

Dµφ0 = (∂µ−igAaµτ a−ig￿Y Bµ)φ0 = −i/2(Aaµσa+Bµa)

√ v/ 2

=− √ 22

µ

µ

g￿Bµ − gA3µ

which yields

(Dµφ0)†Dµφ0 = v2 (g2(A1 )2 + g2(A2 )2 + (gA3 − g￿Bµ)2).

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µ

µ

µ

This is our first example of a mass matrix with off-diagonal terms:

 g2 0 0

 0

m2 = v42  00 g02 g02 −g0g￿ 

0 0 −gg￿ (g￿)2

The determinant of this matrix is zero, showing that there is a massless boson. The first two components both have mass, as well as the linear combination gA3µ − g￿Bµ. The conventional choices for the orthogonal linear combinations that diagonalize m2
are

W± =

1 √

(A1 ± A2 ),

µ

2µ µ

Zµ = ￿ 1 (gA3µ − g￿Bµ), g2 + g￿2

Aµ = ￿ 1 (g￿A3µ + gBµ) g2 + g￿2

￿ which have mass eigenvalues MW = gv/2, MZ = g2 + g￿2v/2 and MA = 0. The gauge field part of the covariant derivative can be rewritten:

gAa τ a+g￿Y B

=

g √

(W +τ ++W −τ −)+ ￿

1

Z (g2τ 3−g￿2Y )+ ￿ gg￿ A (τ 3+Y )

µ

µ



µ

g2 + g￿2 µ

g2 + g￿2 µ

with τ ± = τ 1 ± τ 2 This shows that the electric charge of a particle is given by e(τ 3 + Y ) where e = √ gg￿ . In other words, the particle’s charge depends on its weak isospin
g 2 +g ￿2
and hypercharge. For instance, the upper component of the Higgs doublet has electric
charge 1/2 + 1/2 = 1 and the bottom component has charge −1/2 + 1/2 = 0.

Experiments[1] Generally, particle masses and coupling constants are input pa-
rameters of a theory and can’t be predicted. However, the situation is somewhat different in a theory with spontaneously broken symmetry. Since the masses of the bosons are derived from the same input parameters, the theory actually makes concrete predictions about the relationship of the masses in terms of experimentally measured quantities.
The Weinberg mixing angle θW is defined by tan(θW ) = g￿/g. This angle which is a function of the couplings can be measured by experiment. Due to radiative corrections,

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its value changes with the energy scale involved and its most precisely measured value is sin2(θW ) = .2313 at the energy 91.2 GeV (the mass of the Z boson). We have the formula MW = MZ cos θW which agrees well with the experimental values of MZ = 91.19 and MW = 83.98. The independent parameters measured to high precision are the Z boson mass, the Fermi constant GF and the fine structure constant α, from which θW can be derived. The predicted value of MW agrees with experiment very well when radiative corrections are taken into account.
The GWS theory’s prediction of weak neutral currents was verified in 1973, and the W and Z bosons were directly discovered ten years later (both in experiments at CERN). This and the aforementioned high-precision agreements of the mass relationships with theory make the GWS theory an experimental success to be sure.
However, the theory predicts another particle, the famous Higgs boson whose mass depends on the unknown parameter λ (the Higgs self coupling), so cannot be predicted from the known quantities above. As of right now the particle has not been discovered, though it’s mass has been narrowed down substantially and a promising program is underway at the LHC.

Fermions The parity-violating nature of the weak interaction means that the inter-

action treats left-handed and right-handed particle fields differently. In fact, only the

left-handed fields

￿￿￿￿

ν

u

eL

dL

transform nontrivially under the SU (2) group. The formula Q = τ 3+Y implies that the neutrino-electron doublet has Y = −1/2 to make the neutrino neutral and the electron have Q = −1. Similarly, the quark doublet has Y = +1/6. The right-handed fields are all arranged in separate singlet representations where the hypercharge is necessarily equal to the electric charge of the particle. For instance, the right-handed down-quark transforms as (1, −1/3).
The impact of symmetry breaking on fermions is to give them mass. Explicit mass terms are forbidden in chiral gauge theories since they mix left-handed and right-handed fields. To get around this problem, the masses are introduced in by Yukawa couplings to the Higgs boson. For instance, for the lepton field, we have the termλee¯LφeR which, for φ = φ0 gives an electron mass term (but not a neutrino mass). There is a similar, but slightly more complicated Yukawa coupling for the quark fields that gives them their masses. These Yukawa couplings make up a large portion of the free parameters of the standard model.

Alternatives Despite the success of the GWS theory, it has some shortcomings
that have led some to suggest alternatives to the model suggested above. Although the experimental evidence suggests that the symmetry breaking pattern above occurs, there’s no guarantee that it is driven by a fundamental scalar field. One popular alternative is Technicolor [12] wherein the symmetry breaking field is dynamically generated, as it is in superconductivity. The condensate driving the symmetry breaking may be from known or hypothetical particles. One appeal of this mechanism over the fundamental scalar is that it avoids hierarchy problem, which is that radiative

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corrections to the Higgs mass have very large natural values which must be precisely cancelled in order to have the Higgs mass on the order of the electroweak scale.

5 SU (5) → SU (3) × SU (2) × U (1)

The GWS model shows the promise of non abelian gauge theories for unifying the fundamental forces. It would be aesthetically nice if the standard model was the lowenergy phenomenology of a larger gauge theory, a Grand Unified Theory. The energy scales could be separated by symmetry breaking. Georgi and Glashow proposed a theory to that effect in 1974 [4] based on the SU (5) gauge group. The Higgs field required transforms under the adjoint representation of SU (5) and as such can be represented by a 5 × 5 traceless hermitian matrix. If the vacuum value is

φ0 ∝ vdiag[2, 2, 2, −3, −3],

￿

￿

this value commutes with an SU (3) subgroup

Ta 0 00

where Ta are the 3 × 3 SU (3)

generators as well as an SU (2) subgroup and a one-parameter U (1) subgroup cor-

responding to the generator proportional to φ0. Since, in the adjoint representation, the mass matrix is proportional to g2tr[τ a, φ0][τ b, φ0] [10], the commuting generators

correspond to massless gauge fields. In this case there are twelve massless fields, cor-

responding to the generators of SU (3) × SU (2) × U (1) and twelve massive generators

whose mass scale MGUT ≈ 1015 GeV is set by parameters in a symmetry breaking potential of the form [11]

V (φ) = m2tr(φ2) + a(tr(φ2))2 + btr(φ4).

This adjoint Higgs particle only accomplishes the separation of the GUT scale.

Another Higgs field, transforming as a 5 (fundamental) provides electroweak symmetry

breaking.

A single generation of the standard model fits perfectly into two irreducible representations of SU (5), the ¯5 and 10

 d1   d2   d3  e
νR

 0

 u¯3 −u¯2 u1 −d1

 −u¯3 0 u¯1 −u2 −d2   u¯2 −u¯1 0 −u3 −d3   u1 u2 u3 0 −e¯ 

d1 d2 d3 e¯ 0 L

This perfect fit could be improved aesthetically in only one way: by unifying into a single irreducible representation. In fact, the 16 dimensional representation of Spin(10) works [7], provided that one additional matter particle, the sterile right-handed neutrino, is introduced.
The massive gauge bosons in the SU (5) theory mix the quark and lepton fields, leading to the prediction of proton decay. The proton lifetime has an experimental lower bound of around 1034 years [1] from the Super-K experiment, which is above the theoretical prediction of 1032 years of SU (5).

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SymmetryTheoryMassLagrangianRepresentation