SM Electroweak Symmetry Breaking
| Defining the broken phase fields | Bringing the Lagrangian to the broken phase |
| Specifying the symmetry breaking pattern |
Matchete has functionality to facilitate partially automated symmetry breaking of theories. By detailing the symmetry-breaking pattern and the decomposition of symmetric-phase gauge eigenstates into their constituent broken fields, Matchete can transform the full symmetric-phase Lagrangian to the corresponding broken phase one. This is the primary way to obtain a renormalizable theory of heavy vectors. While much easier than doing these manipulations by hand, the process can be somewhat complicated; this tutorial therefore aims to cover the details of the SM EWSB as a useful example of the necessary steps.
Loading this model file defines all symmetric-phase fields and gauge groups, following the usual behavior of LoadModel.
To get the broken phase, one must first define the gauge symmetry and fields of the broken phase Lagrangian, along with any new couplings that appear:
The broken phases also contains massive vectors, as the result of some of the original gauge fields acquiring masses. The is charged whereas is neutral:
The vector-like fermion fields of the SM broken phase are (we have used gothic letters to distinguish them from the right-handed symmetric-phase fields)
where the neutrino is massless and left-handed, with no mass-generating mechanism in the SM. This also defines diagonal fermion masses, with which we parameterize the broken-phase Lagrangian (to be continued). The Higgs VEV is a real scalar coupling set by
The weak mixing angle is used in the broken phase. With the DefineCoSinePair function, we obtain a coupling representing both the cosine and the sine of the angle. Simplification functions will employ trigonometric identities to simplify combinations in expressions:
Finally, the miss-alignment of down- and up-quarks in the left-handed quark doublets is parametrized with 3×3 unitary CKM mixing matrix:
In the crucial second step, we must instruct Matchete on how symmetry breaking occurs within the SM gauge group. Matchete does not automatically determine a minimum of the potential; the user will have to parametrize the scalar fields with a consistent VEV. All branching rules of the original gauge representation will have to be specified by the user.
| SetSymmetryBreakingPattern[originalGroup, stabilityGroup] | sets the overall symmetry-breaking pattern. |
| RepresentationDecomposition[originalRepresentation, decomposition] | defines the branching rules for a representation of the original symmetry group in terms of representations of the stability group. |
| CGDecomposition[cg, decomposition] | instructs Matchete on how a CG from the original group decomposes in terms of components from the stability group. |
| FieldDecomposition[field, decomposition] | instructs Matchete on how a field charged under the original symmetry decomposes into components of the stability group. |
| SetSSBReplacements[rules] | sets replacement rules to be used implicitly when going to the broken phase of a Lagrangian. |
Groups and representations
At the highest level one must specify what groups are part of the symmetry breaking. This gives a handle for what representations, CGs, and fields should be broken from the original Lagrangian. It also determines what is considered valid inputs to sequential functions.
Since the original symmetry contains two group factors, we simply wrap them in a List. The 
color group is not part of the symmetry breaking; it exists both in the symmetric and broken phases. As such we don't need to include it in SetSymmetryBreakingPattern.
color group is not part of the symmetry breaking; it exists both in the symmetric and broken phases. As such we don't need to include it in SetSymmetryBreakingPattern.Next, one should specify the branching rules for all representations of the original symmetry group. One can think of this as defining a new, convenient basis for the original representation that explicitly take the form of a direct sum of irreducible representations of the stability group. This means that the ordering of the decomposition is largely arbitrary: different choices corresponds to different basis choices. The only issue is that one needs to be consistent with the basis in CGDecomposition and FieldDecomposition afterwards.
Only two non-trivial 
representations – the fundamental and adjoint – appear in the SM. The stability group is Abelian, all its representations are one-dimensional. The branching rule for the fundamental representation (dimension two), therefore, has two charged one-dimensional components. The EM charges of the decomposition are determined by 
and we have
representations – the fundamental and adjoint – appear in the SM. The stability group is Abelian, all its representations are one-dimensional. The branching rule for the fundamental representation (dimension two), therefore, has two charged one-dimensional components. The EM charges of the decomposition are determined by and we have
Something interesting happens for the adjoint representation. In the original basis all components of the adjoint are real. However, the charge ±1 components of the branching rule are necessarily complex. This does not violate the total dimension, since the charged components form a conjugate pair (they are each others conjugate) rather than two independent complex components. To identify such a conjugate pair embedded in a real representation of the original group, they must immediately follow each other, as in
indicating that 3L∼(1)em⊕(-1)em⊕1. The Singlet keyword is used to indicate a component which is trivial w.r.t. the stability group. With these restrictions, {U1em[1], Singlet, U1em[-1]} would be and invalid branching rule for the adjoint, whereas, e.g., {Singlet, U1em[1], U1em[-1]} is another valid rule.
N.b. The rotation of the adjoint basis needed to embed complex pairs is non-orthogonal and does not preserve the Kronecker delta (two-index invariant) of the representation. The new two-index invariant have diagonal blocks (
) for the conjugate pairs. In this example this ensures that the contraction of two adjoint indices leads to contraction of the U1em[1] and U1em[-1] representations. For instance 
in the broken phase, because the implicit delta contraction of the unbroken phase becomes the new two-index invariant.
| 0 | 1 |
| 1 | 0 |
in the broken phase, because the implicit delta contraction of the unbroken phase becomes the new two-index invariant. The hypercharge component of the original symmetry group also changes in the breaking. All hypercharges become EM charges, and from the rule 
, it becomes clear that a unit-charge of hypercharge becomes a unit of EM charge (no normalization necessary). The syntax for this identification is
, it becomes clear that a unit-charge of hypercharge becomes a unit of EM charge (no normalization necessary). The syntax for this identification is Note that the normalization of the electromagnetic and hyper-charges is purely a matter of convention. With different normalization, we would not necessarily have unit charge going in to another unit charge.
Clebsch-Gordan coefficients
Having determined the branching rules for the representations of the broken symmetry group, and thereby fixing a basis for these representations, we need to inform Matchete what the original Clebsch-Gordan coefficients (
CG
s) look like in the chosen basis. This should be done for each CG appearing in the original Lagrangian (also implicit in, e.g., covariant derivatives). Exceptions are that Kronecker deltas (
del
) are constructed automatically, while structure constants (
fStruct
) are derived automatically when decomposition of any generator is specified.
The machinery for specifying CGs may seem unfamiliar to some users (the price of generality), due to the implicit way indices are handled when the stability group is non-Abelian; however for EWSB, things will look somewhat familiar.
The generator of the fundamental 
representation, has one adjoint, one fundamental, and one anti-fundamental index. The branching rules for these original representations have 3, 2, and 2 components, respectively. The CGDecomposition of the generator is a 3×2×2 tensor, where, e.g., the (3,1,1) element is the third component of the decomposition of the adjoint index and the first components of both fundamental and anti-fundamental decomposition. The ordering is determined by the ordering of indices in the original CG, which must be specified with open indices to define the substitution. For simplicity, we need only specify the non-zero components with an ArrayRules format:
representation, has one adjoint, one fundamental, and one anti-fundamental index. The branching rules for these original representations have 3, 2, and 2 components, respectively. The CGDecomposition of the generator is a 3×2×2 tensor, where, e.g., the (3,1,1) element is the third component of the decomposition of the adjoint index and the first components of both fundamental and anti-fundamental decomposition. The ordering is determined by the ordering of indices in the original CG, which must be specified with open indices to define the substitution. For simplicity, we need only specify the non-zero components with an ArrayRules format:The user may wonder, why we did not enter the ordinary Pauli matrices above. The reason is that the decomposition of the adjoint index was rotated from the basis used in the Pauli matrices to the basis of 
. Note that it is the second component of the adjoint index that contracts with 
because in the new basis, the two-index invariants is off-diagonal as per above. The decomposition of the generator automatically constructs a compatible decomposition for the structure constant.
. Note that it is the second component of the adjoint index that contracts with because in the new basis, the two-index invariants is off-diagonal as per above. The decomposition of the generator automatically constructs a compatible decomposition for the structure constant. The two-index invariant of the 
fundamental (the Levi-Civita tensor) decomposes into a 2×2 tensor under the representation branching rules. Anti-symmetry of the original tensor along with normalization immediately gives
fundamental (the Levi-Civita tensor) decomposes into a 2×2 tensor under the representation branching rules. Anti-symmetry of the original tensor along with normalization immediately givesDecomposition of the gauge eigenstates
Finally, one must also specify how the symmetric-phase fields decompose in terms of eigenstates of the stability group. Fields that carry indices of the original symmetry group, decompose according to the branching rules of the correspond representations.
The singlet gauge field 
of hypercharge is identified as a mixture of the EM gauge field 
and the heavy 
in the broken phase. Since 
does not carry any representation of the original group, it decomposes like a scalar:
of hypercharge is identified as a mixture of the EM gauge field and the heavy in the broken phase. Since does not carry any representation of the original group, it decomposes like a scalar:All terms in the decomposition must carry the same unbroken index (the Lorentz index, μ, in this case) of the original field. The normalization looks a little different than what is found in most text books. The reason is the canonical normalization of the gauge fields in Matchete, which uses 
on gauge kinetic terms while the massive vectors are assumed to be unit normalized (there is no single symmetry-protected coupling associated with these fields).
on gauge kinetic terms while the massive vectors are assumed to be unit normalized (there is no single symmetry-protected coupling associated with these fields).The gauge kinetic terms are 
in Matchete's canonical normalization. The mixing of the fields are obtained by rescaling the fields before and after the unitary rotation to the mass basis: 
.
in Matchete's canonical normalization. The mixing of the fields are obtained by rescaling the fields before and after the unitary rotation to the mass basis: .The triplet gauge field 
has an adjoint 
index, which decomposes into three component following the branching rules. Accordingly, the field decomposes into a 3-dimensional vector:
has an adjoint index, which decomposes into three component following the branching rules. Accordingly, the field decomposes into a 3-dimensional vector:The first two components are associated with the conjugate pair with EM charge ±1 in the branching rule of the adjoint index. They are not independent DOFs, but rather a single complex field and its conjugate (
), consistent with three real DOFs of the original triplet.
), consistent with three real DOFs of the original triplet. The Higgs doublet 
contains the VEV (v[]) and the real scalar 
in addition to the would-be Goldstone bosons. In our parametrization, where the lower component has EM charge 0 (isospin -1/2 and hypercharge +1/2), we let
contains the VEV (v[]) and the real scalar in addition to the would-be Goldstone bosons. In our parametrization, where the lower component has EM charge 0 (isospin -1/2 and hypercharge +1/2), we let The fermion fields do not change much, but we have to identify the chiral fields of the symmetric phase with the broken-phase vector-like fermions (and the neutrinos). Color and flavor indices are preserved to the broken phase:
The left-handed quarks are a little subtle in that the up and down quark mass eigenstates are embedded differently in the it. In the case of down-alignment, we take the down component to be aligned with its mass basis. Thus, the up-component is misaligned by the CKM matrix. We simply introduce the unitary CKM matrix with dummy index contractions in flavor space, setting
Convenient coupling replacements
A final touch is to define substitutions for the couplings of the symmetric-phase Lagrangian in terms of parameters more convenient for the broken phase. Using SetSSBReplacements is a good way of organizing this substitution as it will perform the substitutions as part of going to the broken phase and do various trigonometric simplifications along the way.
In the SM, we substitute the original gauge couplings in terms of the weak angle and the electromagnetic coupling constant. The Yukawa coupling matrices are substituted in favor of the masses and the CKM element.
| ToBrokenPhase[lagrangian] | expands out the lagrangian in the broken phase using decompositions of fields and CGs defined with the SB pattern. |
| GetVacuumConditions[lagrangian] |
get conditions on the couplings, which ensure that a broken phase Lagrangian is expanded around a vacuum.
|
| ImplementVacuumConditions[lagrangian] | implements the vacuum conditions on a broken phase Lagrangian. |
| GaugeFixLagrangian[lagrangian] |
gauge-fixes the Lagrangian, adding appropriate gauge-fixing and ghost terms.
|
Functions used to bring a Lagrangian to the broken phase and ensure that it is expanded around the tree-level vacuum.
Having instructed Matchete on how to decompose all fields and CGs in the same consistent basis of the original representations, we can now substitute all this into the original symmetric phase Lagrangian. With ToBrokenPhase, Matchete will automatically convert all dummy index contractions of broken-phase representations into tensor contractions and carry out the summation according to specifications.
We take the symmetric-phase SM (LSM) to the broken phase with (showing only a small part of the full Lagrangian with Short)
This step uses a minimum of simplification methods, leaving the Lagrangian w/o running GreensSimplify or similar functions. The Lagrangian now contains only broken-phase fields as expected.
Note that the resulting broken phase Lagrangian still has a tadpole term for the Higgs boson . Additionally, it looks like there are mass terms for the would-be Goldstone bosons χ and χ0. This happens because v is still a free parameter; we have not yet made use of the fact that this is identified with the VEV of the Higgs field, a parameter which minimizes the potential.
So far we have not made any assumptions about the Lagrangian now being at the proper vacuum. While we have parametrized the VEV v as part of the Higgs doublet, nothing has been specified about its value. The result is that ToBrokenPhase returns a Lagrangian, which is not necessarily 'healthy'. Conditions on the parameters must be obtained to ensure the absence of tadpole terms, canonical normalization of kinetic terms, absence of mass mixing (among heavy fields), an absence of non-GB scalar kinetic mixing with massive vectors, and an absence of GB mass terms.
We provide the function GetVacuumConditions, which will collect all the conditions on the parameters that should be fulfilled for the broken-phase Lagrangian to be at the vacuum and in a mass basis. Simple methods are implement to attempt to find a common solution among all the conditions.
In the broken phase of the SM, we get a condition for removing the Higgs boson tadpole and two for the GB mass terms (which should, of course, vanish as well). Here a common "Solution" for all conditions was identified automatically. We may also read of the masses for the massive vectors and the radial scalar modes, in "MassSubstitutions".
ImplementVacuumConditions simplifies the Lagrangian, under the assumption that all conditions can be simultaneously verified. It is up to the user to check and verify that the conditions can actually be simultaneously satisfied.
Implementing the vacuum condition on the broken-phase Lagrangian removes the Higgs tadpole and the GB masses. With default options it also substitutes the mass expressions for the massive vectors and radial scalar modes in terms of the pre-defined mass couplings (M, M, M in this case):
We have now gotten all the way to the true vacuum in the broken phase of the SM. This concludes the electroweak symmetry breaking in the SM.
All the steps needed to get the broken-phase SM have been implemented in the model file "SM+breaking", which can be loaded with the LoadModel["SM+Breaking"] method as per-usual. This returns the symmetric-phase Lagrangian, but all definitions have been setup, so one can call ToBrokenPhase directly after loading the model.
Gauge-fixing the Lagrangian
Gauge-fixing a broken-phase Lagrangian can be rather complicated: it depends on the "broken" part of the Lie algebra from the original symmetry group. We therefore provide the GaugeFixLagrangian function, which gauge-fixes Lagrangians either to the Rξ (default) or unitary gauge.
For instance the Rξ gauge-fixing terms for the broken-phase SM read (we have subtracted the original Lagrangian, so as to only show the fixing terms)
Most Matchete functions work exclusively on gauge-invariant (un-fixed) Lagrangians, but in particular FeynmanRules can produce Feynman rules also for gauge-fixed Lagrangians.