Quantum gauge models
without classical Higgs mechanism

Michael Dütsch,1 José M. Gracia-Bondı́a,2

Florian Scheck3 and Joseph C. Várilly4

1 Courant Research Center “Higher order structures in Mathematics”,
Mathematisches Institut, Universität Göttingen, 37073 Göttingen, Germany

2 Departamento de Fı́sica Teórica, Universidad de Zaragoza, Zaragoza 50009, Spain
3 Institut für Physik, Theoretische Elementarteilchenphysik,
Johannes Gutenberg-Universität, 55099 Mainz, Germany

4 Escuela de Matemática, Universidad de Costa Rica, 11501 San José, Costa Rica

Eur. Phys. J. C 69 (2010), 599–621

Abstract

We examine the status of massive gauge theories, such as those usually obtained by spon-
taneous symmetry breakdown, from the viewpoint of causal (Epstein–Glaser) renormalization.
The BRST formulation of gauge invariance in this framework, starting from canonical quanti-
zation of massive (as well as massless) vector bosons as fundamental entities, and proceeding
perturbatively, allows one to rederive the reductive group symmetry of interactions, the need for
scalar fields in gauge theory, and the covariant derivative. Thus the presence of higgs particles is
understood without recourse to a Higgs(–Englert–Brout–Guralnik–Hagen–Kibble) mechanism.
Along the way, we dispel doubts about the compatibility of causal gauge invariance with grand
unified theories.

“Sire, je n’avais pas besoin de cette hypothèse-là”

– Pierre Simon de Laplace

Contents

1 Introduction 2

2 The scheme of causal gauge invariance 5

3 Mass and interaction patterns 10

4 The relation between CGI and SSB 14

1



5 The CGI methods in practice 18

6 Conclusion 33

A On the Standard Model in CGI 34

B Derivation of the second main constraint 34

C Epistemological second thoughts 35

1 Introduction

With the start of the Large Hadron Collider (LHC) operation, the Higgs sector of the Standard
Model (SM) and Higgs’ mechanism of spontaneous symmetry breakdown (SSB) allegedly giving
rise to it [1–4] have become a topical issue [5].

The Higgs sector has unsatisfactory aspects, often discussed: the self-coupling terms appear to
be ad-hoc, unrelated to other aspects of the theory, not seeming to constitute a gauge interaction [6,
Sect. 22]. They raise the hierarchy problem [7,8]. Since most fields involved in the Higgs mechanism
are unobservable, the status question for it cannot be simply resolved by sighting of higgses1 in
the LHC. Arguably this is why, at the end of his Nobel lecture, Veltman wrote: “While theoretically
the use of spontaneous symmetry breakdown leads to renormalizable Lagrangians, the question of
whether this is really what happens in Nature is entirely open” [9]. This mistrust, also apparent
in [10], is more widespread than current orthodoxy would have us believe. The theoretical puzzles,
as well as present phenomenological ones, advise a new look at the scalar sectors of the SM and
grand unified theories (GUTs) within a quantum field theoretic framework.

In this introduction we first briefly summarize experimental information on the Higgs sector of
the standard model. Then the conclusions of the “reality check” on the Higgs mechanism worked
out in this paper are compared to the ones of a first reality check performed in the mid-seventies.
Finally we summarize the contents of the paper.

1.1 Phenomenological puzzles

Generally speaking, precision electroweak measurements were successful in pinning down new
features of the SM and its constituents, even before actual discoveries took place. Perhaps the best
example is provided by the top quark, whose mass could be estimated rather precisely from a global
analysis of all available electroweak data before it was found at [20]. To illustrate this point we quote
a global fit of all SM electroweak data [11], excluding the directly measured top mass, which yields
178.2 +9.8

−4.2 GeV. If the experimental value 172.4 +1.2
−1.2 GeV is included, one obtains the improved fit

value 172.9 +1.2
−1.2 GeV, not too different from the indirect determination.

The case of the Higgs boson at present is more complicated. The result of the standard fit
not taking into account direct searches for the higgs (i.e., the lower limits on the higgs mass 𝑀H
obtained at LEP and the Tevatron) is:

𝑀H = 80 +30
−23 GeV. (1)

1Following L. B. Okun, and for obvious grammatical reasons, we refer to a (physical) Higgs boson as higgs, with a
lower-case h.

2



The complete fit of all data, including the lower limits, gives the estimate

𝑀H = 116.4 +18.3
−1.3 GeV. (2)

The deviation of the central values in (1) and (2) from one another are due to contradictory tendencies
in the data. Most notable among these is the forward-backward asymmetry 𝐴

0,𝑏
FB into 𝑏-quarks whose

pull value in the complete fit is

1
𝜎meas

(
𝐴

0,𝑏
FB

���
fit
− 𝐴

0,𝑏
FB

���
meas

)
= 2.44,

the denominator 𝜎meas being the error in the measurement. This hadronic asymmetry, taken in
isolation, yields a tendency to rather high values of the higgs mass, while the leptonic asymmetries
in the case of the LEP data either agree with the value (2) or tend towards lower values of that mass
as is the case for the SLD data – consult Fig. 3 in [11]. In spite of the tension between leptonic and
hadronic asymmetries, no single pull value exceeds the 3𝜎 level. The present state of such fits, and
the influence of the (more than twenty) SM quantities on them, are well summarized in Table 1 and
Figs. 2 and 3 of [11]. The divergent influences of various electroweak data on the mass of the higgs
were noted already a few years ago [12]. They have been quantified in [13],emphasizing the strong
correlation between 𝐴

(0,𝑏)
FB and the predicted higgs mass.

The modest quality of the overall fit might be due to inconsistencies in the data and/or radiative
corrections, that might disappear when further progress is made. However, it might as well signal
that the scalar sector is considerably more complicated than in standard lore, leading to a reduction
of the standard higgs couplings. Consequently something could have been overlooked at LEP: for
instance, mixing with “hidden world” scalars [14] yields such a reduction, in particular for the
𝑍𝑍𝐻 coupling; and this could not be, and was not, ruled out by LEP2 for those relatively low
energies. Other scenarios shielding the higgs from detection have been discussed in the literature –
see [15, 16] as well as the illuminating remarks in [17]. Recent experiment has made the situation
even murkier: on Halloween night of 2008, ghostly (albeit rather abundant) multi-muon events at
Fermilab were reported by the CDF collaboration [18,19]. A possible explanation for them invokes
new light higgs-like particles coupling relatively strongly to the “old” ones, and less so to the SM
fermions and vector bosons [20, 21]. Strassler has cogently emphasized that “minimality” of the
scalar sector of the SM is just a theoretical prejudice [22]. (Yes, entia non sunt multiplicanda praeter
necessitatem. But, who ordered the muon?) The recent discovery of excess charge asymmetry (that
is, CP violation) in 𝑏-hadrons [23] points in the same direction [24]. Given this state of affairs, it
seems premature to draw any definitive conclusion.

1.2 Reality checks for SSB

Causal perturbation theory as developed by H. Epstein and V. Glaser (EG) and applied to QED by
Scharf and collaborators [25, 26] is not directly applicable a priori to non-Abelian gauge theories.
Indeed the EG method involves an expansion in terms of the coupling constant(s) whereas, as
is well known, gauge invariance in the non-Abelian case interrelates terms of different orders in
these couplings. Nevertheless, causal gauge invariance (CGI) interprets BRST symmetry as a
fundamental property of quantum gauge theory —in the spirit of [27] and [28, Sect. 3.3], providing
a canonical description of vector bosons, eliminating unphysical fields, and helping (through the

3



consistency relations it imposes) to reconstruct the gauge-invariant Lagrangian from a general
Ansatz.

In this context it may be useful to recall a half-forgotten chapter of the early history of gauge
theory, chiefly due to Bell, Cornwall, Levin, Llewelyn Smith, Sucher, Tiktopoulos and Woo in
the seventies: see [29, 30] and references therein. The connection between “tree-unitarity” (the
natural high-energy boundedness condition for 𝕊-matrix elements in the tree approximation) and
perturbative unitarity, leading to plausible renormalizability requisites for Lagrangians, was well
understood by then. All those papers started essentially without preconditions from Lagrangians
made out of massive vector bosons (MVB) as fundamental entities, and found that:

• First and foremost, the couplings of the vector bosons had to be of the gauge theory type,
governed by reductive symmetry groups (in physics parlance, “groups” often denote “Lie
algebras” in this paper).

• Furthermore, scalar fields necessarily entered the picture. The allowed theories so obtained
were essentially equivalent to (the phenomenological outcome of) SSB models, with one
general exception: “Abelian mass terms” were possible for the vector bosons.

The latter is understandable: QED with massive photons is a well-behaved theory. Whenever
the symmetry group possesses an invariant Abelian subgroup, one may add such terms. For our
purposes this second finding is not moot, since it suggests the description of spin-1 massive models
with the help of Stückelberg fields. After all, the SM contains an invariant Abelian subgroup and the
mass of the 𝑍 particle can be (though it need not be) thought to be of that type, see the discussion in
Appendix A. Stückelberg fields are of course unphysical. But they have a rightful place in quantum
field theory for the canonical description of MVB, already at the level of free fields [31–34].

Right afterwards the BRS revolution took hold, and the formalism for gauge symmetry changed
forever.2

1.3 Outline of the sequel

The book by Scharf [35], crowning a successful line of research [36,38,40–43] which in particular
establishes a consistent formulation of the SM without SSB [41], aimed to bring a fresh perspective
to the subject from the standpoint of CGI. In tune with it, with the earlier reality check, and with
the phenomenological SM Lagrangian, here we stop pretending we know the origin of mass, and
start without preconditions again from MVB as fundamental fields. That the reductive Lie algebra
structure then follows from CGI was recognized by Stora in [44], which constituted an important
motivation for this work.

In Section 2 we expose the theoretical underpinnings of our own reality check. There are actually
at least two CGI methods; both are expounded there. The first method is constructive. The second,
stemming from a theorem by one of us in [45], is useful rather to verify CGI.

Section 3 summarizes the first results of the theory. We report the outcome of that first method,
rendering the cubic coupling relations for CGI models, determined by BRST invariance at first and
second order.

2Some of the authors of that reality check openly suspected SSB as a formal recipe without physical meaning; others
were swayed by the remarkable “success rate” of the Higgs mechanism; some apparently remained agnostic. And so is
the case with the present writers.

4



The work on tree-unitarity invoked above seemed to certify every SSB-kind model as acceptable.
On the other hand, Ambauen and Scharf have claimed [46] that the CGI approach clashes with the
outcome of SSB for the Georgi–Glashow GUT. The matter deserved further investigation, all the
more so since their assertion is in contradiction with the second CGI method. As it turns out, CGI
produces constraints on the allowed patterns of masses and couplings. A certain obstruction put
forward by Scharf, sensible enough in some circumstances, was responsible for the rejection of the
Georgi–Glashow SU(5) and other scenarios. Next, in Section 4, we unravel this internal problem
in CGI by uncovering an oversight responsible for the mentioned rejections: there is no problem
with GUTs.

Properly reformulated, the obstruction is the germ of the general 𝑆-representation, that is, of
a derivation from first principles of the covariant derivative coupling, familiar in the standard
approaches. The theorem in [45] of course fits with our construction. The previous analysis allows
next to describe what is presently known to us on quartic terms in the Lagrangian from CGI.

Section 5, intended to familiarize the reader thoroughly with the workings of CGI, is made out
of examples. Some readers might prefer to go to this section before tackling the general aspects
expounded before. First we review an Abelian model in some detail. Next we examine a few slightly
more complicated models within CGI. We put aside the Abelian exception by dealing with simple
groups. Semisimplicity dictates the number of vector bosons (corresponding to Cartan’s classical
groups) for irreducible symmetry realizations. We consider allowed mass patterns for models with
one higgs, pondering first the simple but all-important case with only three gauge bosons, and next
CGI for higher-rank groups. We recall the corresponding SSB mindset: the choice of only one
higgs corresponds to mass patterns produced by the Higgs mechanism by vector realizations of the
gauge group. Then we look at BRST invariance and minimal coupling from CGI corresponding to
SSB with fields in the adjoint representation.

Section 6 dwells on our conclusions.
To put matters in perspective, in Appendix A we report on the SM from the angle of CGI.3

Some technical aspects of the machinery underlying this work are confined to Appendix B. Finally,
in Appendix C we amplify on the epistemological implications of the article.

2 The scheme of causal gauge invariance

2.1 The method in general

In the origin of causal perturbation theory [47] – the formulation of gauge symmetry and its preser-
vation in the process of renormalization was not taken into account. Besides other related methods to
deal with symmetries in that framework [48–51], causal gauge invariance is a systematic technique
to treat quantum gauge theories perturbatively by Epstein–Glaser renormalization. As pointed out
above, it was first broached by Scharf and collaborators for QED. A developed formulation was
found in the treatment of massless Yang–Mills theories [36]. It has been applied successfully also
to massive non-Abelian models, namely the SM [41, 42], spin-2 gauge fields and supersymmetry.
Quite recently, the method has been recruited to examine the issue of the (putative) decoupling of
ghosts in a class of noncovariant gauges [52].

Rather than follow the motivation of CGI in the books [26] and [35], we adopt a viewpoint

3We regret that its elegant formulation without SSB is so widely ignored.

5



inspired by (perturbative) algebraic quantum field theory. BRST invariance is input already in the
canonical description of vector bosons. The unphysical fields are eliminated by using the BRST
transformation 𝑠: the algebra of observables is obtained as its cohomology, implemented by the
nilpotent BRST charge 𝑄. The space of physical states can be described cohomologically as well –
see in particular [53].

The construction of 𝑄 in perturbative gauge field theory meets the problem that in general the
BRST charge 𝑄 changes when the interaction is switched on [53]. For theories with good infrared
behaviour like purely massive theories, Kugo and Ojima [54] showed that 𝑄 can be identified
with the incoming (free) BRST charge 𝑄in, which implements the BRST transformation 𝑠0 of the
incoming fields. That the 𝕊-matrix be well-defined on the physical Hilbert space of the free theory
amounts to the requirement [31, 43]:

lim
𝑔↑1

[
𝑄in,𝕊(𝑔𝜅𝐿1)

] ���
ker𝑄in

= 0. (3)

Here 𝕊(𝑔𝜅𝐿1) is the 𝕊-matrix corresponding to the interaction 𝑔(𝑥)𝜅 𝐿1(𝑥), an operator on the Fock
space of the incoming free fields. The local Wick polynomial 𝐿1 is the part of the total interaction
Lagrangian 𝐿tot =

∑∞
𝑛=1 𝜅

𝑛𝐿𝑛 linear in the coupling constant 𝜅 (notationally assumed unique for
simplicity). The function 𝑔 ∈ S(ℝ4) switches the coupling constant on and off; the adiabatic limit
𝑔 ↑ 1 has to be performed to obtain the physically relevant 𝕊-matrix. Now, 𝕊(𝑔𝜅𝐿1) is a formal
power series,

𝕊(𝑔𝜅𝐿1) = 1 +
∑︁
𝑛⩾1

𝑖𝑛𝜅𝑛

𝑛!

∫
𝑑𝑥1 · · · 𝑑𝑥𝑛 𝑔(𝑥1) · · · 𝑔(𝑥𝑛)𝑇𝑛

(
𝐿1(𝑥1) · · · 𝐿1(𝑥𝑛)

)
,

where the time ordered product 𝑇𝑛
(
𝐿1(𝑥1) · · · 𝐿1(𝑥𝑛)

)
is an operator-valued distribution.

The higher order terms of the interaction 𝐿𝑛 for 𝑛 ⩾ 2 – which are also local Wick polynomials
– are taken into account as local terms in 𝑇𝑛

(
𝐿1(𝑥1) · · · 𝐿1(𝑥𝑛)

)
: the latter will contain a term

𝑛!(−𝑖)𝑛−1 𝛿(𝑥1 − 𝑥𝑛, . . . , 𝑥𝑛−1 − 𝑥𝑛) 𝐿𝑛 (𝑥𝑛), (4)

which propagates to higher orders 𝑛′ > 𝑛 by the inductive machinery of Epstein–Glaser renormali-
zation [25].

To satisfy (3) to first order in 𝜅, one just searches for a local Wick polynomial 𝑃𝜈 (called a
“𝑄-vertex”) such that

𝑠0𝐿1(𝑥) ≡ [𝑄in, 𝐿1(𝑥)] = 𝜕𝜈𝑃𝜈 (𝑥). (5)
Turning to higher orders, we first note that if 𝑥𝑖 ≠ 𝑥 𝑗 for all 𝑖 < 𝑗 , there is a permutation 𝜋 such that
𝑥𝜋( 𝑗) ∩ (𝑥𝜋( 𝑗+1) +𝑉−) = ∅ for every 𝑗 , with 𝑉− being the solid backward lightcone. Hence, for such
configurations the time-ordered product can be written as a standard operator product:

𝑇𝑛
(
𝐿1(𝑥1) · · · 𝐿1(𝑥𝑛)

)
= 𝐿1(𝑥𝜋1) · · · 𝐿1(𝑥𝜋𝑛).

In view of

[𝑄in, 𝐿1(𝑥1) · · · 𝐿1(𝑥𝑛)] =
𝑛∑︁
𝑙=1

𝐿1(𝑥1) · · · [𝑄in, 𝐿1(𝑥𝑙)] · · · 𝐿1(𝑥𝑛)

=

𝑛∑︁
𝑙=1

𝜕𝜈𝑥𝑙

(
𝐿1(𝑥1) · · · 𝑃𝜈 (𝑥𝑙) · · · 𝐿1(𝑥𝑛)

)
, (6)

6



one generalizes (5) to higher orders by requiring that

𝑠0𝑇𝑛 (𝐿1(𝑥1) · · · 𝐿1(𝑥𝑛)) ≡
[
𝑄in,T𝑛 (𝐿1(𝑥1) · · · 𝐿1(𝑥𝑛))

]
=

𝑛∑︁
𝑙=1

𝜕𝜈𝑥𝑙 𝑇𝑛
(
𝐿1(𝑥1) · · · 𝑃𝜈 (𝑥𝑙) · · · 𝐿1(𝑥𝑛)

)
. (7)

Formulas (5) and (7) constitute the operator CGI conditions, enough to guarantee (3) if the adiabatic
limit exists. (In theories involving massless fields that limit is problematic, to be sure. For instance,
in QED the 𝕊-matrix contains infrared divergences, which cancel in the cross sections. In models
with confinement, the situation is worse, and a perturbative treatment is possible only for short
distances; an adequate description is the local construction of the observables [53] by using a
coupling 𝑔(𝑥)𝜅 with a compactly supported test function 𝑔. However, the CGI conditions (5) and (7)
are well defined even in models with bad infrared behaviour; in that case they can be justified by
deriving them from the conservation of the BRST current for non-constant coupling [49, 50].

Requirement (7) is a renormalization condition, which restricts also tree diagrams, see below.
Namely, if the sequence of time-ordered products {𝑇𝑛} is constructed inductively by causal per-
turbation theory [47], from (6) we conclude that (7) can be violated only in the extension to the
total diagonal Δ𝑛 ≡ { (𝑥1, . . . , 𝑥𝑛) ∈ ℝ4𝑛 : 𝑥1 = · · · = 𝑥𝑛 } of the 𝑇𝑛; that is, the extension from
S′(ℝ4𝑛 \ Δ𝑛) to S′(ℝ4𝑛) at the level of numerical distributions. Indeed there is violation, in that
causal splitting does not respect the divergences in general; however, CGI can be restored. That
gauge-invariant causal renormalization can be performed to all orders has been proved for QED [26]
and massless SU(𝑁) Yang–Mills theories [36–39].4

In summary, within CGI determination by BRST cohomology acts as a subsidiary physical rule.
This answers to the deeply rooted [57] need to amend Wigner representation theory of particles
for bosons with spin 1 in Fock space with a Krein structure. Recognition of this is an undoubted
merit of [35]. (Cherished positivity could be restored at the price of nonlocality; behind the veil,
string-localized potentials and new field theory phases might well lurk [58, 59]. We aver that CGI
and the addition of higgs-like fields (in the next subsection) is only a sufficient and not a necessary
condition for unitarity – consult [38,39] and [60] in this respect. A nonperturbative understanding of
field theory could restore unitarity in some other way. However, in appropriate contexts and hands,
perturbative methods have something to say about non-perturbative issues. An example is provided
by the exploitation of perturbative BRST invariance in the understanding of confinement [61] –
recently, the same approach has been applied by Nishijima and Tureanu to the study of the gauge
dependence of the Green’s functions in non-Abelian gauge theory [62].

2.2 The grubby machinery

As said above, (5) and (7) are already nontrivial when used for tree diagrams. CGI strongly restricts
the set of allowed models, determining the interaction 𝐿 =

∑∞
𝑛=1 𝜅

𝑛𝐿𝑛 to a great extent, independently

4One expects that the only obstructions to CGI stem from the usual anomalies of quantum field theory. Our general
analysis includes all gauge models which are known to be free of anomalies; but there are cases where CGI at tree
level applied to a general Ansatz for the Lagrangian generates a model exhibiting anomalies; this happens for the axial
anomaly [42]. The question may be examined quite generally by algebraic criteria developed in [55, 56] and [51].
The former authors showed in particular that gauge-invariant and gauge-fixed cohomologies in the EG framework are
equivalent.

7



of the infrared behaviour. Given the free theory, one makes a polynomial and renormalizable Ansatz
for 𝐿1. The CGI condition (5) determines most of the coefficients in this Ansatz, or yields relations
between them. Turning to higher-order tree diagrams, terms of the form (4) remain undetermined in
the inductive Epstein–Glaser construction. Then (7) determines the higher-order interaction terms
𝐿𝑛 and the as yet undetermined coefficients of 𝐿1. The process is constructive. The reductive Lie
algebraic structure and the need to add additional physical scalar fields (higgs fields) in massive
non-Abelian models are not to be put in; they follow from CGI. To be precise in the last respect:
for such a model with MVB and only the unphysical fermionic ghosts and Stückelberg fields, CGI
breaks down at order 𝜅2 [41]; however the inclusion of at least one additional physical scalar makes
CGI solvable. The “puzzling” [63, Preface] existence of fundamental scalars is demanded in our
framework.

The process terminates after a finite number of steps in renormalizable theories. Consider
spin-1 gauge models in 4-dimensional Minkowski space with an 𝐿1 trilinear in the fields whose
mass dimension is ⩽ 4. Then tree diagrams can give nontrivial constraints only up to third order.
Indeed, because CGI can be violated only in the extension of 𝑇𝑛 to Δ𝑛, a possible violation of (7)
must be of the form ∑︁

𝑎,O

𝐶𝑎,O 𝜕𝑎𝛿(𝑥1 − 𝑥𝑛, . . . , 𝑥𝑛−1 − 𝑥𝑛) O(𝑥1, . . . , 𝑥𝑛),

where O(𝑥1, . . . , 𝑥𝑛) denotes a normally ordered product of free fields, 𝑎 = (𝑎𝜇
𝑙
)𝜇=0,1,2,3
𝑙=1,...,𝑛−1 is a multi-

index and the 𝐶𝑎,O are suitable numbers. Power counting yields the restriction |𝑎 | + dimO ⩽ 5 with
|𝑎 | ≡ ∑

𝑙,𝜇 𝑎
𝜇

𝑙
. Since each vertex of 𝐿1 has three legs, it follows that a tree diagram to 𝑛-th order

satisfies dimO ⩾ 2 + 𝑛. Combining these two inequalities, one sees that CGI can be violated at
tree level only for 𝑛 ⩽ 3. In practice, most interesting information is concentrated at the first and
second-order orders; third-order CGI only refines a few coefficients in the Higgs sector.

We illustrate the construction of the time-ordered products at the tree level at second order, in
the case of spin-1 gauge fields, following essentially [41]; the bulk of calculations buttressing this
paper are of this type. Note that

𝑇2(𝑥, 𝑦)
��
tree := T2

(
𝐿1(𝑥)𝐿1(𝑦)

) ��
tree,

𝑇 𝜈
2/1(𝑥, 𝑦)

��
tree := T2

(
𝑃𝜈 (𝑥)𝐿1(𝑦)

) ��
tree,

as well as 𝑇 𝜈
2/2(𝑥, 𝑦)

��
tree = 𝑇 𝜈

2/1(𝑦, 𝑥)
��
tree, can be written as

𝑇2
��
tree = 𝑇2

��0
tree + 𝑁2, 𝑇 𝜈

2/1
��
tree = 𝑇 𝜈

2/1
��0
tree + 𝑁𝜈

2/1,

where𝑇2
��0
tree, 𝑇 𝜈

2/1

��0
tree include all terms not vanishing for 𝑥 ≠ 𝑦; these are the terms with the Feynman

propagator Δ𝐹
𝑚 or its derivatives 𝜕𝜇Δ𝐹

𝑚, 𝜕𝜈𝜕𝜇Δ𝐹
𝑚. We replace □Δ𝐹

𝑚 by −𝑚2 Δ𝐹
𝑚 + 𝛿; the −𝑚2 Δ𝐹

𝑚 term
belongs to 𝑇2

��0
tree and the 𝛿-term to 𝑁2. Expressions 𝑁2 and 𝑁2/1 are of the form

𝑁2(𝑥, 𝑦) = 𝜅2
∑︁

𝜙1𝜙2𝜙3𝜙4

𝐶𝜙1𝜙2𝜙3𝜙4 𝛿(𝑥 − 𝑦) :𝜙1𝜙2𝜙3𝜙4(𝑥):

and similarly for 𝑁2/1, where 𝐶𝜙1𝜙2𝜙3𝜙4 are 𝑐-numbers, and the sum runs over all kinds of free fields
𝜙1, . . . , 𝜙4 present in the model. In the framework of causal perturbation theory, such local terms

8



may be added to 𝑇2, if they respect power counting, Lorentz covariance, unitarity, ghost number,
etc. A glance at formula (4) indicates that 𝑁2(𝑥, 𝑦) = −2𝑖 𝛿(𝑥 − 𝑦) 𝐿2(𝑥), where 𝐿2 is a sum of
quartic terms.

In contrast to𝑇2
��0
tree and𝑇 𝜈

2/1

��0
tree, which are uniquely given in terms of 𝐿1 and 𝑃𝜈, the coefficients

in 𝑁2 and 𝑁2/1 are not yet determined. To prove CGI for second order tree diagrams, we have to
show that the as yet undetermined coupling parameters of 𝐿1 and the coefficients 𝐶𝜙1𝜙2𝜙3𝜙4 in 𝑁2
and 𝑁2/1 can be chosen in such a way that[

𝑄in,
(
𝑇2
��0
tree + 𝑁2

)
(𝑥, 𝑦)

]
= 𝜕𝑥𝜈

(
𝑇 𝜈

2/1
��0
tree + 𝑁𝜈

2/1
)
(𝑥, 𝑦) + [𝑥 ↔ 𝑦] . (8)

Since this condition holds by induction for 𝑥 ≠ 𝑦, we need only study the local contributions.
However, the splitting of a distribution into local and nonlocal parts in principle is not unique, and
some caution is called for. Recall that we replace (𝜕)□Δ𝐹

𝑚 by −𝑚2(𝜕)Δ𝐹
𝑚 + (𝜕)𝛿. Then for 𝑥 ≠ 𝑦

only terms ∼ Δ𝐹
𝑚, ∼ 𝜕𝜇Δ

𝐹
𝑚, ∼ 𝜕𝜈𝜕𝜇Δ

𝐹
𝑚 and ∼ 𝜕𝜈𝜕𝜇𝜕𝜆Δ

𝐹
𝑚 with no contraction of Lorentz indices

contribute to (8). Since these terms cancel for 𝑥 ≠ 𝑦, they cancel for 𝑥 = 𝑦 also. There remain only
terms ∼ 𝛿(𝑥 − 𝑦) and ∼ 𝜕𝛿(𝑥 − 𝑦). For spin-1 gauge theories, such (𝜕)𝛿-terms can be generated
only in the following ways.

(i) First of all,

[𝑄in, 𝑁2(𝑥, 𝑦)] = 𝜅2
∑︁

𝜙1𝜙2𝜙3𝜙4

𝐶𝜙1𝜙2𝜙3𝜙4 𝛿(𝑥 − 𝑦)

×
(
:[𝑄in, 𝜙1(𝑥)] 𝜙2(𝑥) · · ·: + :𝜙1(𝑥) [𝑄in, 𝜙2𝑥)] · · ·: + · · ·

)
.

(ii) From 𝑁𝜈
2/1(𝑥, 𝑦) = 𝐶𝜈 𝛿(𝑥 − 𝑦) 𝑀 (𝑥), where 𝑀 = :𝜙1𝜙2𝜙3𝜙4: , we obtain

𝜕𝑥𝜈𝑁
𝜈
2/1(𝑥, 𝑦) + [𝑥 ↔ 𝑦] = 𝐶𝜈 𝜕𝑥𝜈

(
𝛿(𝑥 − 𝑦) 𝑀 (𝑥)

)
+ [𝑥 ↔ 𝑦]

= 𝐶𝜈 𝛿(𝑥 − 𝑦) 𝜕𝜈𝑀 (𝑥). (9)

In addition, 𝜕𝑥𝜈𝑇 𝜈
2/1

��0
tree(𝑥, 𝑦) + [𝑥 ↔ 𝑦] also contains some (𝜕)𝛿-terms, generated due to the

propagator equation (□ + 𝑚2)Δ𝐹
𝑚 = 𝛿:

(iii) If 𝑃𝜈 = 𝑏 :𝜕𝜈𝜙 𝐹: + · · · and 𝐿1 = 𝑎 :𝜙 𝐸 : + · · · , then the contraction of 𝜕𝜈𝜙(𝑥) with 𝜙(𝑦)
gives a propagator −𝑖𝜕𝜈Δ𝐹

𝑚 (𝑥 − 𝑦), and on computing its divergence we find the contribution

𝜕𝑥𝜈𝑇
𝜈
2/1

��0
tree(𝑥, 𝑦) + [𝑥 ↔ 𝑦] = −2𝑖𝑏𝑎 𝛿(𝑥 − 𝑦) :𝐹 (𝑥) 𝐸 (𝑥): + · · · .

(iv) If 𝑃𝜈 is as before and 𝐿1(𝑦) = 𝑎𝜇 :𝜕𝜇𝜙(𝑦) 𝐸 (𝑦): +· · · , the contraction of 𝜕𝜈𝜙(𝑥) with 𝜕𝜇𝜙(𝑦)
gives a propagator 𝑖𝜕𝜈𝜕𝜇Δ𝐹

𝑚 (𝑥 − 𝑦). On computing the divergence we now obtain a 𝜕𝛿-term,
which we transform into a 𝛿-term by using the following identity

𝑀 (𝑦) 𝜕𝑥𝜈 𝛿(𝑥 − 𝑦) = 𝑀 (𝑥) 𝜕𝑥𝜈 𝛿(𝑥 − 𝑦) + 𝜕𝜈𝑀 (𝑥) 𝛿(𝑥 − 𝑦);

so that

𝜕𝑥𝜈𝑇
𝜈
2/1

��0
tree(𝑥, 𝑦) + [𝑥 ↔ 𝑦] = 𝑖𝑏𝑎𝜇 𝜕

𝜇𝛿(𝑥 − 𝑦) :𝐹 (𝑥) 𝐸 (𝑦): + [𝑥 ↔ 𝑦] + · · ·
= 𝑖𝑏𝑎𝜇 𝛿(𝑥 − 𝑦)

(
:𝐹 (𝑥) 𝜕𝜇𝐸 (𝑥): − :𝜕𝜇𝐹 (𝑥) 𝐸 (𝑥):

)
+ · · · . (10)

9



After the transformations (9) and (10), all terms remaining in (8) are of the form 𝑐1 𝛿(𝑥 −
𝑦) :𝜕𝜙1 𝜙2𝜙3𝜙4(𝑥): or 𝑐2 𝛿(𝑥 − 𝑦) :𝜙1𝜙2𝜙3𝜙4(𝑥): . These are linearly independent. Therefore, (8)
is equivalent to a system of 𝑐-number equations, obtained by equating the coefficients belonging to
the same Wick monomial.

2.3 The second CGI method

Basic model-building according to CGI uses tree-diagram calculations. In this connection, and
alternatively to the previous method, models satisfying CGI at tree level can also be obtained
by using that classical BRST invariance of the Lagrangian implies CGI for tree diagrams to all
orders [45]. The ideas behind this can be summarized thus: given a BRST-invariant free theory,
that is

𝑠0𝐿0 = 𝜕𝜇 𝐼
𝜇

0 =: 𝜕 · 𝐼0 for some local 𝐼0,
with 𝐿0 quadratic in the fields, seek deformations 𝐿0 → 𝐿tot =

∑∞
𝑛=0 𝜅

𝑛𝐿𝑛 and 𝑠0 → 𝑠 =
∑∞

𝑛=0 𝜅
𝑛𝑠𝑛

(with 𝐿𝑛, 𝑠𝑛 satisfying some obvious properties), such that

𝑠𝐿tot = 𝜕 · 𝐼 where 𝐼𝜇 =

∞∑︁
𝑛=0

𝜅𝑛𝐼
𝜇
𝑛 (11)

is some local power series. Here 𝐿tot is assumed to contain only first-order derivatives; have a
look at [27]. BRST invariance of the Lagrangian in this sense implies CGI for tree diagrams to all
orders by the following: in the case of a constant coupling 𝜅 formula (11) implies conservation of
the corresponding classical Noether (BRST) current: 𝜕 · 𝑗𝜅,class = 0. Replacing 𝜅 by 𝜅𝑔, for a test
function 𝑔, a generalized current conservation can be derived from (11):

𝜕 · 𝑗𝜅𝑔,class(𝑥) = 𝜕𝑔(𝑥) · 𝑃𝜅𝑔,class(𝑥), (12)

where 𝑃𝜅𝑔,class(𝑥) is that classical interacting vector field which agrees for 𝜅 = 0 with the𝑄-vertex 𝑃

of (5), more precisely with the corresponding classical (local) field polynomial. Since the classical
limit of an interacting perturbative quantum field is given by the contribution of connected tree
diagrams, current conservation (12) can be expressed as a tree diagram relation in perturbative
quantum field theory. Smearing out this relation with suitable test functions, conservation of the
BRST current goes over to commutation with the free BRST charge 𝑄in belonging to the conserved
Noether current of the symmetry 𝑠0𝐿0 = 𝜕 · 𝐼0. In this way the CGI relation (7) for tree diagrams
to all orders is obtained.

For theories with MVB one can use that, generically, models coming from SSB are classically
BRST invariant in the sense of (11); this will be spelled out in Section 5.4. Therefore they satisfy
CGI at tree level. Most likely, the two methods outlined here are equivalent, in the sense that the
sets of allowed models are the same. It remains that, whereas the first method amounts to a direct
(perchance tedious) search for the general solution of the CGI conditions for tree diagrams, we do
not know whether the second yields the most general solution as well.

3 Mass and interaction patterns

Consider a model with 𝑡 intermediate vector bosons 𝐴𝑎 in all, of which any may be in principle
massive or massless. Let there be 𝑟 massive ones (𝑎 = 1, . . . , 𝑟 , with masses 𝑚𝑎 > 0) and 𝑠 massless

10



(𝑎 = 𝑟 + 1, . . . , 𝑟 + 𝑠), so 𝑡 = 𝑟 + 𝑠. They are accompanied by 𝑧 physical scalar particles 𝜑𝑝 of
respective masses 𝜇𝑝. The free BRST transformation 𝑠0 ≡ [𝑄in, ·]∓ is a superderivation commuting
with partial derivatives, hence is given by its action on the basic fields:

𝑠0𝐴
𝜇
𝑎 = 𝜕𝜇𝑢𝑎, 𝑠0𝐵𝑎 = 𝑚𝑎𝑢𝑎, 𝑠0𝑢𝑎 = 0,

𝑠0�̃�𝑎 = −(𝜕 · 𝐴𝑎 + 𝑚𝑎𝐵𝑎), 𝑠0𝜑𝑝 = 0.

Here we let 𝐵𝑎 denote the Stückelberg field associated to the vector field 𝐴𝑎; in case 𝐴𝑎 is massless,
𝐵𝑎 drops out. The total bosonic interaction Lagrangian is of the form 𝐿int = 𝜅𝐿1 + 𝜅2 𝐿2. For 𝐿1 we
make the following Ansatz (with unknown coefficients 𝑓 ∗∗∗∗ in the terms below). Let 𝐿1 = 𝐿1𝐵+𝐿1𝜑,
the higgs-free cubic couplings being

𝐿1𝐵 = 𝐿1
1 + 𝐿2

1 + 𝐿3
1 + 𝐿4

1,

and

𝐿1𝜑 = 𝐿5
1 + 𝐿6

1 + 𝐿7
1 + 𝐿8

1 + 𝐿9
1 + 𝐿10

1 + 𝐿11
1

being the couplings involving physical scalars. In the Feynman gauge, the allowed higgs-free cubic
couplings are

𝐿1
1 = 𝑓𝑎𝑏𝑐

[
𝐴𝑎 · (𝐴𝑏 · 𝜕)𝐴𝑐 − 𝑢𝑏 (𝐴𝑎 · 𝜕�̃�𝑐)

]
;

𝐿2
1 = 𝑓 2

𝑎𝑏𝑐 (𝐴𝑎 · 𝐴𝑏)𝐵𝑐;
𝐿3

1 = 𝑓 3
𝑎𝑏𝑐

[
(𝐴𝑎 · 𝜕𝐵𝑐)𝐵𝑏 − (𝐴𝑎 · 𝜕𝐵𝑏)𝐵𝑐

]
;

𝐿4
1 = 𝑓 4

𝑎𝑏𝑐�̃�𝑎𝑢𝑏𝐵𝑐; (13)

and the remaining ones, involving higgses, are

𝐿5
1 = 𝑓 5

𝑎𝑏𝑝

[
(𝐴𝑎 · 𝜕𝜑𝑝)𝐵𝑏 − (𝐴𝑎 · 𝜕𝐵𝑏)𝜑𝑝

]
;

𝐿6
1 = 𝑓 6

𝑎𝑞𝑝

[
(𝐴𝑎 · 𝜕𝜑𝑝)𝜑𝑞 − (𝐴𝑎 · 𝜕𝜑𝑞)𝜑𝑝

]
;

𝐿7
1 = 𝑓 7

𝑎𝑏𝑝 (𝐴𝑎 · 𝐴𝑏)𝜑𝑝;

𝐿8
1 = 𝑓 8

𝑎𝑏𝑝�̃�𝑎𝑢𝑏𝜑𝑝;

𝐿9
1 = 𝑓 9

𝑎𝑏𝑝
𝐵𝑎𝐵𝑏𝜑𝑝;

𝐿10
1 = 𝑓 10

𝑎𝑝𝑞𝐵𝑎𝜑𝑝𝜑𝑞;
𝐿11

1 = 𝑓 11
𝑝𝑞𝑟𝜑𝑝𝜑𝑞𝜑𝑟 . (14)

As products of field operators, these monomials are understood to be normally ordered. Some
symmetry relations of the coefficients under exchange of indices are evident from the definition.
Because the dimension of the Lagrangian must be 𝑀4 in natural units, and the boson field dimension
is 1 in our formulation, the coefficients 𝑓 , 𝑓 3, 𝑓 5, 𝑓 6 are dimensionless, and 𝑓 2, 𝑓 4, 𝑓 7, . . . , 𝑓 11 have
dimension of mass. It is taken into account that CGI holds a term in 𝐵𝑎𝐵𝑏𝐵𝑐 to vanish. With that, the
formulas (13) and (14) give the most general trilinear and renormalizable Ansatz modulo divergence
terms and 𝑠0-coboundaries.

We list the determination of the couplings in terms of the 𝑓𝑎𝑏𝑐 and the pattern of masses imposed
by CGI at orders 𝜅 and 𝜅2, still essentially in the version of [35].

11



1. As repeatedly indicated, and like in Section 1.2, with independence of the masses CGI
unambiguously leads to gauge fields with real coupling parameters 𝑓𝑎𝑏𝑐 that are totally
antisymmetric and satisfy the Jacobi identity: that is, to generalized Yang–Mills theories on
reductive Lie algebras. This is remarkable.

2. When all 𝐴𝑎 are massless, there is no need to add physical or unphysical scalars for renormal-
izability, and only 𝐿1

1 (and later, the quartic coupling 𝐿1
2) survive. They of course coincide

respectively with the first- and second-order part of the usual Yang–Mills Lagrangian. In
particular: CGI gives rise to gluodynamics.

3. The relation
2𝑚𝑐 𝑓

2
𝑎𝑏𝑐 = (𝑚2

𝑏 − 𝑚2
𝑎) 𝑓𝑎𝑏𝑐 (15)

holds. Thus if 𝑚𝑐 = 0 and 𝑓𝑎𝑏𝑐 ≠ 0, then 𝑚𝑎 = 𝑚𝑏 necessarily. And if 𝑚𝑐 ≠ 0, then

𝑓 2
𝑎𝑏𝑐 = 𝑓𝑎𝑏𝑐

𝑚2
𝑏
− 𝑚2

𝑎

2𝑚𝑐

.

The useful relation between masses and structure constants:

(𝑚2
𝑏 − 𝑚2

𝑎)
∑︁

𝑐:𝑚𝑐=0
( 𝑓𝑎𝑏𝑐)2 = 0; (16)

follows directly from (15).

4. The relation
2( 𝑓 3

𝑏𝑐𝑎𝑚𝑎 − 𝑓 3
𝑎𝑐𝑏𝑚𝑏) = 𝑓𝑎𝑏𝑐𝑚𝑐 (17)

holds. From this, after multiplication by 𝑚𝑐 and cyclic permutation, one obtains the important
formula

𝑓 3
𝑎𝑏𝑐 = 𝑓𝑎𝑏𝑐 (𝑚2

𝑏 + 𝑚2
𝑐 − 𝑚2

𝑎)/4𝑚𝑏𝑚𝑐 . (18)
If either 𝑚𝑏 or 𝑚𝑐 vanishes, then 𝑓 3

𝑎𝑏𝑐
= 0.

5. 𝑓 4
𝑎𝑏𝑐

= 𝑓𝑎𝑏𝑐 (𝑚2
𝑐 − 𝑚2

𝑏
+ 𝑚2

𝑎)/2𝑚𝑐.

6. In the non-Abelian case, when some 𝐴𝑎 are massive, coefficients 𝑓 5 to 𝑓 11 cannot all vanish:
renormalizability asks for physical Higgs bosons. The 𝐿5

1 and 𝐿6
1 terms are the nub of the

problem. Reference [35] claims that 𝐿6
1 is just zero and that the coefficients of 𝐿5

1 are diagonal
in the sense that 𝑓 5

𝑎𝑏𝑝
= 𝐶5𝑝𝑚𝑎 𝛿𝑎𝑏, where the 𝐶5𝑝 (with dimension 𝑀−1) are independent

of 𝑎; but this is only warranted when there is a single higgs field, for which the 𝐿6
1 term is

absent. A relatively involved expression, given in the next subsection, ties this key coupling
with the structure constants and the masses.

7. 𝑓 7
𝑎𝑏𝑝

= − 𝑓 8
𝑎𝑏𝑝

= 𝑚𝑏 𝑓
5
𝑎𝑏𝑝

. This is 𝐶5𝑚
2
𝑎 𝛿𝑎𝑏 when 𝑧 = 1.

8. Because 𝑓 7 is obviously symmetric in the first two indices, so too is 𝑓 8. Now, the symmetry
for 𝑓 8 implies

𝑚𝑏 𝑓 5
𝑎𝑏𝑝 = 𝑚𝑎 𝑓 5

𝑏𝑎𝑝 . (19)

Note that 𝑓 5
𝑎𝑏𝑝

= 0 if 𝑚𝑎 = 0 or 𝑚𝑏 = 0: for 𝑚𝑏 = 0 this is clear (no Stückelberg field 𝐵𝑏),
and for 𝑚𝑎 = 0 it follows from (19).

12



9. 𝑓 9
𝑎𝑏𝑝

= −(𝜇2
𝑝/2𝑚𝑎) 𝑓 5

𝑎𝑏𝑝
for 𝑚𝑎 > 0; 𝑓 9

𝑎𝑏𝑝
= 0 if 𝑚𝑎 = 0. This is −1

2𝐶5𝑚
2
𝐻
𝛿𝑎𝑏 when 𝑧 = 1,

with 𝜇1 ≡ 𝑚𝐻 .

10. 𝑓 10
𝑎𝑝𝑞 =

𝜇2
𝑞−𝜇2

𝑝

𝑚𝑎
𝑓 6
𝑎𝑝𝑞 for 𝑚𝑎 > 0, with 𝑓 10

𝑎𝑝𝑞 = 0 if 𝑚𝑎 = 0. This vanishes when 𝑧 = 1.

11. The 𝑓 11
𝑝𝑞𝑟 are not determined, except (with the help of third-order tree graphs) in the case of

only one higgs; then 𝑓 11 = −1
2𝐶5𝑚

2
𝐻

.

3.1 The first CGI parameter constraint

In the preceding subsection we have listed all conditions coming from CGI for first and second-order
tree diagrams which determine directly the coupling parameters 𝑓 ∗∗∗∗ in terms of 𝑓𝑎𝑏𝑐 and the masses.
However, for second-order tree diagrams CGI gives further constraints relating the couplings and
the MVB masses. Using implicit summation on repeated indices, the first of those is

𝑓 5
𝑎 𝑗 𝑝 𝑓

5
𝑑𝑏𝑝 − 𝑓 5

𝑎𝑏𝑝 𝑓
5
𝑑𝑗 𝑝 =

𝑚2
𝑗
+ 𝑚2

𝑏
− 𝑚2

𝑐

2𝑚 𝑗𝑚𝑏

𝑓𝑑𝑎𝑐 𝑓𝑐𝑏 𝑗

+
(
𝑚2

𝑘
+ 𝑚2

𝑗
− 𝑚2

𝑑

𝑚 𝑗𝑚𝑘

𝑚2
𝑘
+ 𝑚2

𝑏
− 𝑚2

𝑎

4𝑚𝑏𝑚𝑘

𝑓𝑑𝑗 𝑘 𝑓𝑎𝑏𝑘 − [𝑎 ↔ 𝑑]
)
, (20)

if 𝑚𝑏, 𝑚 𝑗 > 0. The sum over 𝑐 is over all gauge bosons and the sum over 𝑘 runs only over massive
ones.

In particular, setting 𝑗 = 𝑎 and 𝑑 = 𝑏 ≠ 𝑎, one infers that

𝑓 5
𝑎𝑎𝑝 𝑓

5
𝑏𝑏𝑝 − 𝑓 5

𝑎𝑏𝑝 𝑓
5
𝑏𝑎𝑝

=
1

2𝑚𝑎𝑚𝑏

[ 𝑡∑︁
𝑐=1

(𝑚2
𝑎 + 𝑚2

𝑏 − 𝑚2
𝑐) ( 𝑓𝑎𝑏𝑐)2 +

∑︁
𝑘:𝑚𝑘≠0

(𝑚2
𝑎 − 𝑚2

𝑏
)2 − 𝑚4

𝑘

2𝑚2
𝑘

( 𝑓𝑎𝑏𝑘 )2
]
. (21)

On the one hand this relation allows us to compute 𝑓 5 from the masses and the structure constants;
on the other hand, since it is valid for any 𝑏 ≠ 𝑎, it implies direct relations between the masses and
the structure constants.5

It will help to reorganize (21), separating the massless from the massive bosons in the sum. If
𝑚𝑘 ≠ 0, the coefficient of ( 𝑓𝑎𝑏𝑘 )2/4𝑚𝑎𝑚𝑏𝑚

2
𝑘

is

2𝑚2
𝑘 (𝑚

2
𝑎 + 𝑚2

𝑏 − 𝑚2
𝑘 ) + (𝑚2

𝑎 − 𝑚2
𝑏)

2 − 𝑚4
𝑘 = (𝑚2

𝑎 + 𝑚2
𝑏 + 𝑚2

𝑘 )
2 − 4(𝑚2

𝑎𝑚
2
𝑏 + 𝑚4

𝑘 ).
Thus the main consequence of (20) can be written, for 𝑎 ≠ 𝑏 with 𝑚𝑏 ≠ 0, as:

4𝑚𝑎𝑚𝑏

𝑧∑︁
𝑝=1

����� 𝑓 5
𝑎𝑎𝑝 𝑓 5

𝑎𝑏𝑝

𝑓 5
𝑏𝑎𝑝

𝑓 5
𝑏𝑏𝑝

����� = 2(𝑚2
𝑎 + 𝑚2

𝑏)
∑︁

𝑑:𝑚𝑑=0
( 𝑓𝑎𝑏𝑑)2

+
∑︁

𝑘:𝑚𝑘≠0

( 𝑓𝑎𝑏𝑘 )2

𝑚2
𝑘

[
(𝑚2

𝑎 + 𝑚2
𝑏 + 𝑚2

𝑘 )
2 − 4(𝑚2

𝑎𝑚
2
𝑏 + 𝑚4

𝑘 )
]
. (22)

This first constraint and (15), with their respective consequences (22) and (16), restrict strongly the
masses of the gauge bosons; (subsections 5.2–5.3).

5We observe that Scharf writes the previous equation differently, since he mistakenly “derived” 𝑓 5
𝑎𝑏𝑝

= 0 when
𝑎 ≠ 𝑏.

13



4 The relation between CGI and SSB

The primary aim of this section is to work out explicitly the connection of model building by CGI
to the SSB approach. In particular we show that one obtains the covariant derivative of the scalar
fields, that is, the “minimal coupling” recipe. A related aim is to disprove the claim [46] about
standard GUT models not satisfying CGI at tree level, that would contradict our aforementioned
statement. Finally, we collect information on the 𝐿2 piece of the Lagrangian. We restate that:

• From CGI for spin-one particles, one is led to discover the gauge symmetry: the coupling
parameters 𝑓𝑎𝑏𝑐 in 𝐿1

1 are the structure constants of a reductive Lie algebra, and the other
couplings 𝑓 ∗∗∗∗ in 𝐿∗

1 are determined by the 𝑓𝑎𝑏𝑐 and the masses. Knowledge of this hidden
symmetry is of course very useful, but not needed a priori within CGI.

• In the opposite direction, i.e. postulating the underlying gauge symmetry, we expect that
models built by SSB be classically BRST invariant, and hence satisfy CGI at tree level to all
orders [45].

4.1 Reinterpreting the first constraint from CGI

Using (18), the main obstruction (20) is rewritten

𝑓 5
𝑎 𝑗 𝑝 𝑓

5
𝑑𝑏𝑝 − 𝑓 5

𝑎𝑏𝑝 𝑓
5
𝑑𝑗 𝑝 = 2 𝑓𝑑𝑎𝑐 𝑓 3

𝑐𝑏 𝑗 + 4 𝑓 3
𝑎 𝑗 𝑘 𝑓

3
𝑑𝑘𝑏 − 4 𝑓 3

𝑎𝑏𝑘 𝑓
3
𝑑𝑘 𝑗 . (23)

In view of (13) and (14) it is clear that 𝑓 3 and 𝑓 5 should be related. We introduce the notation

(𝐹𝑎)𝑏𝑐 = −2 𝑓 3
𝑎𝑏𝑐

for 𝑟 × 𝑟 skewsymmetric matrices 𝐹𝑎. If we provisionally assume that only one physical scalar is
present (𝑧 = 1), let 𝐺𝑎 be 𝑟 × 1 matrices (there are 𝑟 + 𝑠 of these) given by

(𝐺𝑎) 𝑗 = − 𝑓 5
𝑎 𝑗 ,

and form the (𝑟 + 1) × (𝑟 + 1) skewsymmetric matrices, for 𝑎 = 1, . . . , 𝑟 + 𝑠:

𝑆𝑎 =

(
𝐹𝑎 𝐺𝑎

−𝑡𝐺𝑎 0

)
, with 𝑡𝐺𝑎 being the transpose of 𝐺𝑎 .

Relation (23) corresponds to the left upper corner of the commutator bracket

[𝑆𝑎, 𝑆𝑑] = 𝑓𝑎𝑑𝑐 𝑆
𝑐 . (24)

Employing 𝑓 5
𝑎𝑏

= 𝐶5 𝑚𝑎 𝛿𝑎𝑏 and (17), one sees that the other corners of this bracket formula are
fulfilled, too. Thus equation (23) means that the 𝑓 3, 𝑓 5 taken together define a real skewsymmetric
matrix representation of the gauge group with dimensionless entries, for only one higgs.

14



4.2 How the covariant derivative arises from CGI

When more than one higgs is present, one should admit terms like (𝐴𝑎 · 𝜕𝜑𝑞)𝜑𝑝 − (𝐴𝑎 · 𝜕𝜑𝑝)𝜑𝑞,
and so we have done in (14). In this case a second constraint is found,

0 = 𝑓𝑎𝑏𝑐 𝑓
8
𝑑𝑐𝑝 − 𝑓 4

𝑑𝑏𝑘 𝑓
5
𝑎𝑘 𝑝 + 𝑓 4

𝑑𝑎𝑘 𝑓
5
𝑏𝑘 𝑝 + 2 𝑓 6

𝑎𝑝𝑣 𝑓
8
𝑑𝑏𝑣 − 2 𝑓 6

𝑏𝑝𝑣 𝑓
8
𝑑𝑎𝑣 .

Here 𝑎, 𝑏, 𝑑, 𝑝 are fixed; the summation indices are 𝑐 = 1, . . . , 𝑟 + 𝑠; 𝑘 = 1, . . . , 𝑟; and 𝑣 = 1, . . . , 𝑧.
The right hand side is the coefficient of the term [𝑢𝑎𝑢𝑏�̃�𝑑𝜑𝑝] (𝑥) 𝛿(𝑥 − 𝑦) on the right hand side of
the CGI condition (8) for 𝑛 = 2. We point out that the expression must be antisymmetric in 𝑎 ↔ 𝑏

because 𝑢𝑎𝑢𝑏�̃�𝑑𝜑𝑝 is. There is no contribution coming from [𝑄in, 𝑁2] – hence the zero on the
left-hand side – since quartic terms involving ghost fields 𝑢, �̃� are not admitted here. This can be
justified by the second CGI method.

We finally line up the following system of constraints:

4 𝑓 3
𝑎𝑑𝑘 𝑓

3
𝑏𝑘𝑒 − 4 𝑓 3

𝑏𝑑𝑘 𝑓
3
𝑎𝑘𝑒 − 𝑓 5

𝑎𝑑𝑣 𝑓
5
𝑏𝑒𝑣 + 𝑓 5

𝑏𝑑𝑣 𝑓
5
𝑎𝑒𝑣 = −2 𝑓𝑎𝑏𝑐 𝑓 3

𝑐𝑑𝑒 , (𝑚𝑑 > 0, 𝑚𝑒 > 0)

2 𝑓 3
𝑎𝑑𝑘 𝑓

5
𝑏𝑘 𝑝 − 2 𝑓 3

𝑏𝑑𝑘 𝑓
5
𝑎𝑘 𝑝 − 2 𝑓 5

𝑎𝑑𝑣 𝑓
6
𝑏𝑝𝑣 + 2 𝑓 5

𝑏𝑑𝑣 𝑓
6
𝑎𝑝𝑣 = − 𝑓𝑎𝑏𝑐 𝑓

5
𝑐𝑑𝑝 , (𝑚𝑑 > 0)

− 𝑓 5
𝑎𝑘 𝑝 𝑓

5
𝑏𝑘𝑞 + 𝑓 5

𝑏𝑘 𝑝 𝑓
5
𝑎𝑘𝑞 + 4 𝑓 6

𝑎𝑝𝑣 𝑓
6
𝑏𝑣𝑞 − 4 𝑓 6

𝑏𝑝𝑣 𝑓
6
𝑎𝑣𝑞 = −2 𝑓𝑎𝑏𝑐 𝑓 6

𝑐𝑝𝑞 . (25)

The first equation is the by now familiar basic constraint of Section 3.1; the second is the previously
displayed equation divided by −𝑚𝑑 . The derivation of these three constraints from CGI is found in
Appendix B.

Let us reintroduce the matrices (𝐺𝑎)𝑑𝑝 = − 𝑓 5
𝑎𝑑𝑝

, which are now 𝑟 × 𝑧, and introduce the 𝑧 × 𝑧

ones:
(𝐻𝑎)𝑝𝑞 = −2 𝑓 6

𝑎𝑝𝑞 .

Then the system (25) amounts to the triplet of matrix equations,

[𝐹𝑎, 𝐹𝑏] − 𝐺𝑎 𝑡𝐺𝑏 + 𝐺𝑏 𝑡𝐺𝑎 = 𝑓𝑎𝑏𝑐 𝐹
𝑐, (𝑟 × 𝑟)

𝐹𝑎𝐺𝑏 − 𝐹𝑏𝐺𝑎 + 𝐺𝑎𝐻𝑏 − 𝐺𝑏𝐻𝑎 = 𝑓𝑎𝑏𝑐 𝐺
𝑐, (𝑟 × 𝑧)

−𝑡𝐺𝑎𝐺𝑏 + 𝑡𝐺𝑏𝐺𝑎 + [𝐻𝑎, 𝐻𝑏] = 𝑓𝑎𝑏𝑐 𝐻
𝑐 . (𝑧 × 𝑧).

Putting it all together in the skewsymmetric (𝑟 + 𝑧) × (𝑟 + 𝑧) package:

𝑆𝑎 =

(
𝐹𝑎 𝐺𝑎

−𝑡𝐺𝑎 𝐻𝑎

)
=

(
−2 𝑓 3

𝑎∗∗ − 𝑓 5
𝑎∗★

𝑓 5
𝑎★∗ −2 𝑓 6

𝑎★★

)
,

this generalizes the gauge-group representation [𝑆𝑎, 𝑆𝑏] = 𝑓𝑎𝑏𝑐 𝑆
𝑐 mooted in (24). In fine, the key

couplings 𝑓 5, 𝑓 6 of the physical scalars are constrained by this algebraic relation in terms of the
“known” 𝑓 3 couplings.6

6Also within CGI, in an analogous way the coupling of vector bosons to fermions induces a gauge-group represen-
tation among the latter fields.

15



Moreover, we contend that the representation above is the one yielding the covariant derivative
on the scalar multiplets of the “minimal coupling” recipe, written in real form. More explicitly, let
𝜂 be a scalar multiplet assembled from the Stückelberg fields and the higgses by

𝜂𝑡 := (𝐵1, . . . , 𝐵𝑟 , 𝜑1 + 𝑣1, . . . , 𝜑𝑧 + 𝑣𝑧), (26)

where the field shifts 𝑣𝑝 are real numbers. We make two assertions. The first is a statement
about the corresponding hidden gauge symmetry; namely that the multiplet 𝜂 transforms with the
representation 𝑆𝑎 =: 𝑆(𝑇𝑎), with 𝑇𝑎 the generators of the gauge Lie algebra, and hence

𝐷𝜇𝜂 := (𝜕𝜇 + 𝜅𝐴
𝜇
𝑎𝑆

𝑎) 𝜂

is the covariant derivative of 𝜂. Our second claim is that the minimal coupling recipe holds true, in
the sense that, with a suitable choice of the 𝑣𝑝,

1
2
𝜕𝜂𝑡 · 𝜕𝜂 + 𝜅

2
(
(𝐴𝑎 · 𝜕𝜂𝑡 )𝑆𝑎𝜂 − 𝜂𝑡𝑆𝑎 (𝐴𝑎 · 𝜕𝜂)

)
− 𝜅2

4
(𝐴𝑎 · 𝐴𝑏)𝜂𝑡 [𝑆𝑎, 𝑆𝑏]+𝜂 (27)

agrees with what one obtains by the CGI method for the scalar-gauge Lagrangian, that is, besides
the kinetic terms of the 𝐵- and 𝜑-fields, the vector-boson mass term, plus an (𝐴 · 𝜕𝐵) term, plus
the trilinear couplings 𝐿2

1 + 𝐿3
1 + 𝐿5

1 + 𝐿6
1 + 𝐿7

1 of the gauge fields 𝐴 to the scalars (𝐵, 𝜑) considered
in (13) and (14), plus the quartic terms 𝐿2

2 + 𝐿3
2 + 𝐿4

2 defined in Section 4.3 right below. (Strictly
speaking, within CGI the shift of the fields by the 𝑣𝑝 is not required at second order. However,
it is convenient for our purposes. In examples, the “correct” choice of 𝑣𝑝 can be obtained from a
comparison with SSB: the fields 𝜑𝑝 + 𝑣𝑝 are the ones of the “unbroken” model with its full gauge
symmetry.) We routinely verify our assertions in the example models constructed by CGI in the
next section. Therefore our procedure derives within CGI the crucial piece 1

2 (𝐷𝜂)𝑡 · 𝐷𝜂 of the
Lagrangian. Indeed, this provides the crowning point of the construction.

4.3 On the quartic couplings

There are quartic terms

𝐿2 = 1
2 (𝐿

1
2 + 𝐿2

2 + 𝐿3
2 + 𝐿4

2 + 𝐿5
2 + 𝐿6

2 + 𝐿7
2),

with obvious symmetries as before, of the following form:

𝐿1
2 = ℎ1

𝑏𝑐𝑑𝑒 (𝐴𝑏 · 𝐴𝑑) (𝐴𝑐 · 𝐴𝑒), 𝐿5
2 = ℎ5

𝑎𝑏𝑐𝑑 𝐵𝑎𝐵𝑏𝐵𝑐𝐵𝑑 ,

𝐿2
2 = ℎ2

𝑎𝑏𝑐𝑑 (𝐴𝑎 · 𝐴𝑏)𝐵𝑐𝐵𝑑 , 𝐿6
2 = ℎ6

𝑎𝑏𝑝𝑞 𝐵𝑎𝐵𝑏𝜑𝑝𝜑𝑞,

𝐿3
2 = ℎ3

𝑎𝑏𝑐𝑝 (𝐴𝑎 · 𝐴𝑏)𝐵𝑐𝜑𝑝, 𝐿7
2 = ℎ7

𝑝𝑞𝑟𝑠 𝜑𝑝𝜑𝑞𝜑𝑟𝜑𝑠,

𝐿4
2 = ℎ4

𝑎𝑏𝑝𝑞 (𝐴𝑎 · 𝐴𝑏)𝜑𝑝𝜑𝑞 . (28)

A complete account of the permitted quartic terms would take us too far afield. For instance,
to answer the question of whether models are completely fixed in the general case by CGI and by
requiring that the number of higgs fields be as small as possible, one needs a complete study of the
tree-level third-order conditions, as well as to revisit some corners of the second-order conditions,

16



here unexplored. This is better left for another paper. We limit ourselves to reporting on what can be
gleaned from the foregoing and calculations analogous to those performed in the coming Section 5
and in Appendix B.

One finds from CGI ℎ1
𝑏𝑐𝑑𝑒

= −1
2 𝑓𝑎𝑏𝑐 𝑓𝑎𝑑𝑒 as thoroughly expected: it just yields the quartic part

in the Yang–Mills Lagrangian, irrespectively of masses.
Now, it is plain what 𝐿2

2, 𝐿3
2, 𝐿4

2 of formula (28) must be. Have a look back at (27). According to
our results on minimal coupling from CGI at second order, these terms in the interaction Lagrangian
are generated by suitable combinations not involving 𝑣 in−1

4 (𝐴𝑎 ·𝐴𝑏)𝜂𝑡 [𝑆𝑎, 𝑆𝑏]+𝜂. Therefore, taking
into account the factor 1

2 in the definitions, one finds:

• For the higgs-free term 𝐿2
2,

ℎ2
𝑎𝑏𝑐𝑑 = −2 𝑓 3

𝑎𝑐𝑘 𝑓
3
𝑏𝑘𝑑 − 2 𝑓 3

𝑏𝑐𝑘 𝑓
3
𝑎𝑘𝑑 +

1
2 𝑓

5
𝑎𝑐𝑣 𝑓

5
𝑏𝑑𝑣 +

1
2 𝑓

5
𝑏𝑐𝑣 𝑓

5
𝑎𝑑𝑣 .

Here and in the subsequent formulas we sum over repeated indices. This is symmetric under
𝑎 ↔ 𝑏 and 𝑐 ↔ 𝑑, as it should be.

• ℎ3
𝑎𝑏𝑐𝑝

= −2 𝑓 3
𝑎𝑐𝑘

𝑓 5
𝑏𝑘 𝑝

− 2 𝑓 3
𝑏𝑐𝑘

𝑓 5
𝑎𝑘 𝑝

+ 2 𝑓 5
𝑎𝑐𝑣 𝑓

6
𝑏𝑝𝑣

+ 2 𝑓 5
𝑏𝑐𝑣

𝑓 6
𝑎𝑝𝑣. This is symmetric in 𝑎, 𝑏.

• ℎ4
𝑎𝑏𝑝𝑞

= −2 𝑓 6
𝑎𝑝𝑣 𝑓

6
𝑏𝑣𝑞

− 2 𝑓 6
𝑏𝑝𝑣

𝑓 6
𝑎𝑣𝑞 + 1

2 𝑓
5
𝑎𝑘 𝑝

𝑓 5
𝑏𝑘𝑞

+ 1
2 𝑓

5
𝑏𝑘 𝑝

𝑓 5
𝑎𝑘𝑞

. This is symmetric under 𝑎 ↔ 𝑏

and 𝑝 ↔ 𝑞.

For the higgs-free term 𝐿5
2 we find, for 𝑎, 𝑏, 𝑐, 𝑑 ⩽ 𝑟:

ℎ5
𝑎𝑏𝑐𝑑 = −

𝜇2
𝑝

12

(
𝑓 5
𝑎𝑏𝑝

𝑓 5
𝑐𝑑𝑝

𝑚𝑎𝑚𝑐

+
𝑓 5
𝑎𝑐𝑝 𝑓

5
𝑏𝑑𝑝

𝑚𝑎𝑚𝑏

+
𝑓 5
𝑎𝑑𝑝

𝑓 5
𝑐𝑏𝑝

𝑚𝑎𝑚𝑐

)
.

This ought to be symmetric under exchanges of 𝑎, 𝑏, 𝑐, 𝑑, and indeed it is: the relations 𝑚𝑏 𝑓
5
𝑎𝑏𝑝

=

𝑚𝑎 𝑓
5
𝑏𝑎𝑝

save the day. For 𝐿6
2 we find:

ℎ6
𝑎𝑏𝑝𝑞 =

𝜇2
𝑝 + 𝜇2

𝑞

4𝑚𝑎𝑚𝑏

( 𝑓 5
𝑎𝑐𝑝 𝑓

5
𝑏𝑐𝑞 + 𝑓 5

𝑎𝑐𝑞 𝑓
5
𝑏𝑐𝑝) +

3
𝑚𝑎

𝑓 11
𝑝𝑞𝑢 𝑓

5
𝑎𝑏𝑢

+
2𝜇2

𝑢 − 𝜇2
𝑞 − 𝜇2

𝑝

𝑚𝑎𝑚𝑏

( 𝑓 6
𝑎𝑢𝑝 𝑓

6
𝑏𝑢𝑞 + 𝑓 6

𝑎𝑢𝑞 𝑓
6
𝑏𝑢𝑝) +

(𝑚2
𝑎 − 𝑚2

𝑏
) (𝜇2

𝑝 − 𝜇2
𝑞)

2𝑚𝑎𝑚𝑏𝑚
2
𝑘

𝑓𝑎𝑏𝑘 𝑓
6
𝑘 𝑝𝑞 .

This has the required symmetries under 𝑎 ↔ 𝑏, 𝑝 ↔ 𝑞; it is undetermined at second order, because
𝑓 11 is.

Finally, ℎ7 is undetermined at second order. CGI for third-order tree diagrams yields conditions
restricting ℎ7 and 𝑓 11 (via conditions on ℎ6), which in the case 𝑧 = 1 determine these parameters
uniquely; see the next subsection. The procedure was explained in [42, Sect. 5], with calculations
given in detail for the SM; consult [35] as well.

4.4 Quartic couplings for models with only one higgs

For the case 𝑧 = 1, with the 𝜑-index suppressed, we obtain:

17



(a) First,

ℎ2
𝑎𝑏𝑐𝑑 =

∑︁
𝑘:𝑚𝑘≠0

1
8𝑚𝑐𝑚𝑑𝑚

2
𝑘

(
𝑓𝑎𝑐𝑘 𝑓𝑏𝑑𝑘 (𝑚2

𝑐 + 𝑚2
𝑘 − 𝑚2

𝑎) (𝑚2
𝑑 + 𝑚2

𝑘 − 𝑚2
𝑏) + [𝑎 ↔ 𝑏]

)
+ 1

2
𝐶2

5 𝑚𝑎𝑚𝑏 (𝛿𝑎𝑐 𝛿𝑏𝑑 + 𝛿𝑎𝑑 𝛿𝑏𝑐).

(b) ℎ3
𝑎𝑏𝑐

= 2 𝑓 2
𝑎𝑏𝑐

𝐶5 = 𝑓𝑎𝑏𝑐𝐶5(𝑚2
𝑏
− 𝑚2

𝑎)/𝑚𝑐.

(c) ℎ4
𝑎𝑏

= 𝐶2
5𝑚

2
𝑎 𝛿𝑎𝑏.

(d) ℎ5
𝑎𝑏𝑐𝑑

= 1
3 (𝛿𝑎𝑏 𝛿𝑐𝑑 + 𝛿𝑎𝑐 𝛿𝑏𝑑 + 𝛿𝑎𝑑 𝛿𝑏𝑐) ℎ7 ; ℎ6

𝑎𝑏
= 2𝛿𝑎𝑏 ℎ7 ; ℎ7 = −1

4𝐶
2
5𝑚

2
𝐻

, independently of
indices ⩽ 𝑟.

This allows us to peek at the purely scalar sector with one higgs. Including its mass term, it
becomes

−1
2
𝑚2

𝐻𝜑
2 + 𝜅( 𝑓 9

𝑎𝑏
𝐵𝑎𝐵𝑏𝜑 + 𝑓 11𝜑3) + 𝜅2

2
(ℎ5

𝑎𝑏𝑐𝑑𝐵𝑎𝐵𝑏𝐵𝑐𝐵𝑑 + ℎ6
𝑎𝑏𝐵𝑎𝐵𝑏𝜑

2 + ℎ7𝜑4)

= −
𝑚2

𝐻

2

(
𝜑2 + 𝜅𝐶5

( 𝑟∑︁
𝑎=1

𝐵2
𝑎 + 𝜑2

)
𝜑 + 𝜅2

4
𝐶2

5

( 𝑟∑︁
𝑎=1

𝐵2
𝑎 + 𝜑2

)2
)

= −
𝜅2𝑚2

𝐻
𝐶2

5
8

(
2𝜑
𝜅𝐶5

+ 𝜑2 + | ®𝐵 |2
)2

=: −𝑉 (𝜑, ®𝐵). (29)

These formulas are correctly given in [35]. The potential exhibits a characteristic 𝑂 (𝑟 + 1) sym-
metry [64]. Leaving aside the Stückelberg fields, it has a minimum at 𝜑 = 0. Hence, the physical
higgs field can be realized in an ordinary Fock representation, with a unique vacuum and vanishing
vacuum expectation value.

5 The CGI methods in practice

The plan of this section is as follows. We first attack from the perspective of the first CGI approach
the simplest example one can think of – dealt with only summarily in [35]. We investigate next
models with several massive vector bosons, but one physical higgs (𝑧 = 1) only, using the first CGI
method as in Section 3.1. One may derive here the possible mass patterns of the gauge bosons by
taking only the consequences of equations (15) and (20) into account. Of course, to show that the
resulting models indeed satisfy CGI at tree level, one must verify all 𝑐-number identities expressing
(5) and (7) on that level. The solutions of those equations that we work out are compatible with
the CGI conditions at all orders. We finally look at causal gauge invariance for models with scalar
fields in the adjoint. All along, we flesh out the relation between CGI and SSB whose theoretical
underpinning was derived in the previous section.

5.1 The toy model

The case 𝑟 = 1, 𝑠 = 0, 𝑧 = 1 leads to an Abelian model in which all the terms 𝐿5
1 to 𝐿11

1 with the
higgs-like field 𝜑 appear, except 𝐿6

1. All contributions of the first group, 𝐿1
1 to 𝐿4

1, disappear. Also

18



𝐿1
2 and 𝐿2

2 vanish. The obstructions of Section 3.1 play no role here. This does not sound very
interesting; but it is instructive. Eleven contributions in all survive, we find that 𝐶5 = 1/𝑚 with 𝑚

being the mass of the spin 1 particle, and the resulting interaction Lagrangian reads

𝐿int(𝑥) = 𝜅𝑚(𝐴 · 𝐴)𝜑 − 𝜅𝑚�̃�𝑢𝜑 + 𝜅𝐵(𝐴 · 𝜕𝜑) − 𝜅𝜑(𝐴 · 𝜕𝐵) −
𝜅𝑚2

𝐻

2𝑚
𝜑3 −

𝜅𝑚2
𝐻

2𝑚
𝐵2𝜑

+ 𝜅2

2
(𝐴 · 𝐴)𝜑2 + 𝜅2

2
(𝐴 · 𝐴)𝐵2 −

𝜅2𝑚2
𝐻

8𝑚2 𝜑4 −
𝜅2𝑚2

𝐻

4𝑚2 𝜑2𝐵2 −
𝜅2𝑚2

𝐻

8𝑚2 𝐵4, (30)

where 𝑚𝐻 is the mass of the higgs field 𝜑.
For the derivation of (30), recall that 𝑇1 = 𝐿1 is given by the first line of (30). Assume that CGI

to first order (5) has already been put to work, yielding the first six terms on the right-hand side
in (30), except that the coefficient of the 𝜑3-coupling is undetermined. The 𝑄-vertex here is given
by

𝑠0𝑇1 = 𝜕 · 𝑃 with 𝑃 = 𝜅
(
𝑚𝑢𝜑𝐴 − 𝑢(𝜑 𝜕𝐵 − 𝐵 𝜕𝜑)

)
.

Next put to work CGI for second-order tree diagrams (8). As a rule, calculations of this kind are
elementary, but tedious. Unhurried readers are referred to the leisurely treatment in [65]. Also,
a technically more detailed version of this paper, containing the computations pertaining here in
particular, is available as hep-th/1001.0932v2.

5.1.1 The second CGI criterion and gauge independence

For the sake of training, we wish to verify the second CGI criterion in this example directly. So
far we have adhered to the Feynman gauge, whereby the masses of gauge and Stückelberg fields
coincide. To show that this restriction is not necessary, we proceed here in an arbitrary Λ-gauge à
la ‘t Hooft.

It is instructive to look first at the free model. The Stückelberg Lagrangian for a MVB is most
elegantly written [32, 33]:

𝐿Stue = 𝐿kin(𝐴) +
𝑚2

2

(
𝐴 − 𝜕𝐵

𝑚

)2
− Λ

2

(
𝜕 · 𝐴 + 𝑚

Λ
𝐵

)2
,

where Λ is the gauge-fixing parameter. The first two terms are manifestly gauge invariant by
𝛿𝐴 = 𝜕𝛼, 𝛿𝐵 = 𝑚𝛼; however, the last one (which is the gauge-fixing term 𝐿

gf
0 ) is gauge invariant

only if (□ + 𝑚2/Λ)𝛼 = 0.
For the zeroth order BRST transformation, the gauge-fixing parameter Λ appears only in 𝑠0�̃�:

𝑠0�̃� = −(Λ 𝜕 · 𝐴 + 𝑚𝐵)

and 𝑠0𝐴 = 𝜕𝑢, 𝑠0𝐵 = 𝑚𝑢, 𝑠0𝑢 = 0, 𝑠0𝜑 = 0 as in the Feynman gauge. Obviously 𝑠0 is nilpotent,
except maybe for 𝑠2

0�̃�; we return to this point below.
The first two terms in 𝐿Stue are 𝑠0-invariant, but for an unrestricted 𝑢-field this does not hold

for 𝐿gf
0 :

𝑠0𝐿
gf
0 =

(
𝜕 · 𝑠0𝐴 + 𝑚

Λ
𝑠0𝐵

)
𝑠0�̃� =

(
□ + 𝑚2

Λ

)
𝑢 𝑠0�̃�.

19



For this reason one introduces a ghost Lagrangian 𝐿
gh
0 such that 𝑠0(𝐿gf

0 + 𝐿
gh
0 ) is a divergence: with

𝐿
gh
0 = 𝜕�̃� · 𝜕𝑢 − 𝑚2

Λ
�̃�𝑢 = 𝜕�̃� · 𝑠0𝐴 − 𝑚

Λ
�̃�𝑠0𝐵, (31)

we indeed obtain
𝑠0(𝐿gf

0 + 𝐿
gh
0 ) = 𝜕 · (𝑠0�̃� 𝑠0𝐴) =: 𝜕 · 𝐼0.

Adding 𝐿
gh
0 and the kinetic and mass terms for the higgs to 𝐿Stue, the total free Lagrangian 𝐿0 takes

the form

𝐿0 = 𝐿kin(𝐴) +
𝑚2

2
(𝐴 · 𝐴) + 1

2
(𝜕𝐵 · 𝜕𝐵) − 𝑚2

2Λ
𝐵2 − Λ

2
(𝜕𝐴)2

+ 1
2
(𝜕𝜑 · 𝜕𝜑) −

𝑚2
𝐻

2
𝜑2 + 𝜕�̃� · 𝜕𝑢 − 𝑚2

Λ
�̃�𝑢 − 𝑚 𝜕 · (𝐴𝐵).

It is BRST invariant in the sense that 𝑠0𝐿0 = 𝜕 · 𝐼0.
Returning to nilpotence of 𝑠0, we see that 𝑠2

0�̃� vanishes modulo the free field equations:

𝑠2
0�̃� = −Λ 𝜕 (𝑠0𝐴) − 𝑚𝑠0𝐵 = −Λ□𝑢 − 𝑚2𝑢 = Λ

𝛿𝑆0
𝛿�̃�

,

where 𝑆0 is the action corresponding to 𝐿0. The equations of motion for the free vector field 𝐴 and
the Stückelberg field 𝐵 are seen to be

(□ + 𝑚2)𝐴 = (1 − Λ) 𝜕 (𝜕 · 𝐴); (□ + Λ−1𝑚2)𝐵 = 0.

Thus, if any other than the Feynman gauge Λ = 1 is chosen, the mass of the Stückelberg field
becomes 𝑚/

√
Λ; this is also the mass of the ghost fields 𝑢, �̃�, and of 𝜕𝐴.

Turning to the interacting sector, we need to verify BRST invariance for the terms (30) obtained
by the first CGI method. The interacting BRST transformation 𝑠 = 𝑠0 + 𝑠1 has an additional term
𝑠1 ∼ 𝜅, given by

𝑠1𝐵 = 𝜅 𝑢𝜑, 𝑠1𝜑 = −𝜅 𝐵𝑢, (32)
and zero for the other fields. Let us look immediately at the scalar sector. In this instance ®𝐵 of (29)
has a single component, and the point is that (32) guarantees that

𝑠𝑉 (𝜑, 𝐵) ∝ (𝑠0 + 𝑠1)
(
2𝑚𝜑

𝜅
+ 𝜑2 + 𝐵2

)2
= 0.

From 𝑠𝑢 = 0 and 𝑢𝑢 = 0 we obtain 𝑠2𝐴 = 𝑠2𝐵 = 𝑠2𝜑 = 0. For 𝐿gh we keep the form 𝐿gh =

𝜕�̃� · 𝑠𝐴 − 𝑚
Λ
�̃�𝑠𝐵 in (31). Note that this contributes the second term in (30), when Λ = 1. Still with

the new action 𝑆, we find that

𝑠2�̃� = −Λ 𝜕 (𝑠𝐴) − 𝑚𝑠𝐵 = Λ
𝛿𝑆

𝛿�̃�

vanishes on-shell, since only 𝐿gh contributes to 𝛿𝑆/𝛿�̃�. The gauge-fixing part is not modified, and
again

𝑠(𝐿gf + 𝐿gh) = (𝜕 · 𝑠𝐴 + 𝑚
Λ
𝑠𝐵)𝑠�̃� + 𝜕 (𝑠�̃�) · 𝑠𝐴 − 𝑚

Λ
𝑠�̃� 𝑠𝐵

= 𝜕 · (𝑠�̃� 𝑠𝐴) = 𝜕 · 𝐼0 .

20



In this particularly simple case, the vector 𝐼 has only components of degree zero.
We know that 𝑠𝐿kin(𝐴) = 0. The total Lagrangian reads

𝐿 = 𝐿kin(𝐴) +
𝑚2

2

(
𝐴 − 𝜕𝐵

𝑚

)2
+ 1

2
𝜕𝜑 · 𝜕𝜑 −

𝑚2
𝐻

2
𝜑2 + 𝐿gf + 𝐿gh + (𝐿int + 𝜅𝑚�̃�𝑢𝜑),

where 𝐿int is given by (30). It remains to verify BRST invariance of

𝐿 − 𝐿kin(𝐴) − 𝐿gf − 𝐿gh +𝑉 (𝜑, 𝐵) =: 𝐿𝜂 .

As discussed in Section 4, these terms can be grouped into a minimal coupling recipe. As an

example of the 𝑆-representation of that section, we have the sole 𝑆-matrix
(
0 −1
1 0

)
. Let us use it to

the purpose. With 𝜂 = (𝐵, 𝑚/𝜅 + 𝜑)𝑡 and 𝐷 = 𝜕 + 𝜅 𝐴 𝑆, we obtain

1
2
(𝐷𝜂)𝑡 · 𝐷𝜂 =

1
2

(
𝜕𝐵 − 𝜅(𝑚/𝜅 + 𝜑)𝐴

𝜕𝜑 + 𝜅𝐵𝐴

) 𝑡 (
𝜕𝐵 − 𝜅(𝑚/𝜅 + 𝜑)𝐴

𝜕𝜑 + 𝜅𝐵𝐴

)
= 1

2 𝜕𝐵 · 𝜕𝐵 + 1
2 𝜕𝜑 · 𝜕𝜑 + 1

2𝑚
2𝐴 · 𝐴 − 𝑚𝐴 · 𝜕𝐵

+ 𝜅
(
𝑚(𝐴 · 𝐴)𝜑 + 𝐵(𝐴 · 𝜕𝜑) − 𝜑(𝐴 · 𝜕𝐵)

)
+ 1

2𝜅
2 ((𝐴 · 𝐴)𝜑2 + (𝐴 · 𝐴)𝐵2) = 𝐿𝜂,

indeed providing the sought-for terms. Since the BRST variation of 𝜂 has the form of an infinitesimal
gauge transformation,

𝑠𝜂 =

(
𝑠𝐵

𝑠𝜑

)
=

(
𝑢(𝑚 + 𝜅𝜑)
−𝜅𝑢𝐵

)
= −𝜅𝑢 𝑆𝜂,

the covariant derivative satisfies 𝑠 𝐷𝜂 = −𝜅𝑢 𝑆𝐷𝜂, and hence 𝑠𝐿𝜂 = 0. We conclude that our toy
model is BRST invariant, thus causal gauge invariant on the tree level, and that the first [35] and
second [45] CGI criteria match for it.

5.1.2 Comparison with SSB

The model with one massive and no massless gauge boson we have been working with can obviously
be obtained by SSB of an U(1) ≃ O(2) gauge model. Let us employ instead of 𝜂 the complex field
Φ := 𝑖𝐵 + 𝑚/𝜅 + 𝜑. The real part of Φ is interpreted as a shifted higgs-like field 𝐻 = 1/𝜅𝐶5 + 𝜑 =

𝑚/𝜅 + 𝜑, and we rewrite (29) in terms of it, obtaining the quartic polynomial

𝑉 (Φ) = 𝑉0 −
𝜇2

2
Φ𝑡Φ + 𝜆

4
(Φ𝑡 Φ)2 =: 𝑉0 +𝑉mod(Φ),

where

𝑉0 =
𝑚2

𝐻
𝑚2

8𝜅2 ; 𝜇 =
𝑚𝐻√

2
; 𝜆 =

𝜅2𝑚2
𝐻

2𝑚2 .

In order to extract the SSB model from this, drop the constant term – this has “only” epistemological
and gravity-cosmological consequences [66]. Then seek the minimum of the potential 𝛿𝑉/𝛿Φ =

(−𝜇2+𝜆Φ𝑡 Φ)Φ = 0. Any solution of this can be “rotated” to a real value ⟨Φ⟩ = 𝜇/
√
𝜆 = 𝑚/𝜅 =: 𝑣.

Patently we have reconstructed the “Abelian Higgs model”, in which an initially massless vector
boson 𝐴 is held to acquire the mass 𝑚 = 𝜅𝑣. (A pity that we cannot switch off the interaction to see
whether 𝐴 was indeed massless.) The remaining scalar particle 𝜑, or higgs, corresponding to the
perturbation of Φ with respect to 𝑣, has a mass

√
2𝜆 𝑣, which is precisely 𝑚𝐻 .

21



5.2 SU(2) models with only one higgs

With three gauge fields, the only relevant Lie algebra entering the game is SU(2); this means we take
𝑓𝑎𝑏𝑐 = 𝜀𝑎𝑏𝑐, whereupon total antisymmetry implies the Jacobi identity. This is surely an important
case.

1. The case 𝑚1 = 𝑚2 = 𝑚3 = 0 is certainly possible, and then neither higgses nor Stückelberg
fields are necessary.

2. We see from (15) that if 𝑚3 = 0 then 𝑚1 = 𝑚2 must hold; the pattern 𝑚2 = 𝑚3 = 0, 𝑚1 ≠ 0 is
downright forbidden.

3. The case 𝑚3 = 0, 𝑚1 = 𝑚2 ≠ 0, after the necessary checks of all CGI tree-level conditions,
turns out to be possible with one higgs-like field.

4. Finally, if we assume that all masses are different from zero, then necessarily 𝑚1 = 𝑚2 = 𝑚3.
This last case, also after all necessary checks, turns out to be possible as well with one
higgs-like field.

Physically, the two cases just mentioned correspond respectively to the Georgi and Glashow
“electroweak” theory without neutral currents; and to the SU(2) Higgs–Kibble model. Both can be
thought of as a limit of the SM, in the second case by setting the Weinberg angle equal to zero and
dropping the decoupled photon field.

With respect to the pending checks, let us show first why 𝑚1 = 𝑚2 = 𝑚3 must hold when there
are three massive gauge fields. Indeed, equation (22) implies

4𝑚2
𝑎𝑚

2
𝑏𝑚

2
𝑐𝐶

2
5 = (𝜀𝑎𝑏𝑐)2 [(𝑚2

𝑎 + 𝑚2
𝑏 + 𝑚2

𝑐)2 − 4(𝑚2
𝑎𝑚

2
𝑏 + 𝑚4

𝑐)
]
,

where (𝑎, 𝑏, 𝑐) is any permutation of (1, 2, 3). Therefore,

𝑚2
1𝑚

2
2 + 𝑚4

3 = 𝑚2
2𝑚

2
3 + 𝑚4

1 = 𝑚2
3𝑚

2
1 + 𝑚4

2.

This yields

(𝑚2
1𝑚

2
2 + 𝑚4

3) − (𝑚2
2𝑚

2
3 + 𝑚4

1) = (𝑚2
3 − 𝑚2

1) (𝑚
2
1 − 𝑚2

2 + 𝑚2
3) = 0,

(𝑚2
2𝑚

2
3 + 𝑚4

1) − (𝑚2
3𝑚

2
1 + 𝑚4

2) = (𝑚2
1 − 𝑚2

2) (𝑚
2
2 − 𝑚2

3 + 𝑚2
1) = 0,

whose only all-positive solution is 𝑚1 = 𝑚2 = 𝑚3 =: 𝑚; and then 4𝑚6𝐶2
5 = 𝑚4 imposes 𝐶5 = 1/2𝑚.

Formula (16) is void here; the test (20) is cleared as well. This model has been exhaustively studied
in [31] and [49]. Note that actually 𝐿2

1 = 0 = 𝐿3
2 for it.

For the other MVB model with 𝑚3 = 0, equation (16) is clearly verified for all values of (𝑎, 𝑏).
As noted earlier, the equality 𝑚1 = 𝑚2 can already be deduced from (15), or from (16) alone. Now
we find 𝐶5 = 1/𝑚.

In both SU(2) cases with massive gauge bosons, the couplings are completely determined from
CGI, without SSB playing any role. It is nevertheless quite easy to identify the corresponding models

22



in the framework of the Higgs mechanism, with the help of the 𝑆-matrices. For the three-boson
model with one vanishing mass, the results for 𝑓 3 and 𝑓 5 here give

𝑆1 =
©­«
0 0 −1
0 0 0
1 0 0

ª®¬ ; 𝑆2 =
©­«
0 0 0
0 0 −1
0 1 0

ª®¬ ; 𝑆3 =
©­«
0 −1 0
1 0 0
0 0 0

ª®¬ ,
and clearly (24) holds. For the three-boson model with three equal masses, suppressing some null
entries in the notation,

𝑆1 =

©­­­«
0 −1

2
−1

2 0
0 1

2
1
2 0

ª®®®¬ ; 𝑆2 =

©­­­«
1
2 0
0 −1

2
−1

2 0
0 1

2

ª®®®¬ ; 𝑆3 =

©­­­«
0 −1

2
1
2 0

0 −1
2

1
2 0

ª®®®¬ .
5.3 SU(𝒏) models with only one higgs

For SU(3) take the basis of Gell-Mann matrices 𝑇𝑎 = 𝜆𝑎/2, for 𝑎 = 1, . . . , 8, normalized by
tr(𝑇𝑎𝑇𝑏) = 1

2 𝛿𝑎𝑏. The well-known structure constants, defined by [𝑇𝑎, 𝑇𝑏] = 𝑖 𝑓𝑎𝑏𝑐 𝑇𝑐, are

𝑓123 = 1; 𝑓147 = 𝑓246 = 𝑓257 = 𝑓345 = 1
2 ; 𝑓156 = 𝑓367 = −1

2 ; 𝑓458 = 𝑓678 =
√

3
2 ,

and 𝑓𝑎𝑏𝑐 = 0 in all cases not arising from these by permuting indices.
It is instructive to play with different mass patterns.

(i) Does the set of constraints allow an SU(3) model with all masses positive? If 𝑎 ≠ 𝑏 and
𝑓𝑎𝑏𝑘 ≠ 0 for exactly one value of 𝑘 , then (22) simplifies to

4𝑚2
𝑎𝑚

2
𝑏𝑚

2
𝑘𝐶

2
5 = ( 𝑓𝑎𝑏𝑘 )2 [(𝑚2

𝑎 + 𝑚2
𝑏 + 𝑚2

𝑘 )
2 − 4(𝑚2

𝑎𝑚
2
𝑏 + 𝑚4

𝑘 )
]
,

and, just as in subsection 5.2, invariance of 𝑚2
𝑎𝑚

2
𝑏
+𝑚4

𝑘
under permutations of 𝑎, 𝑏, 𝑘 implies

that 𝑚𝑎 = 𝑚𝑏 = 𝑚𝑘 . Applying this procedure for (𝑎, 𝑏, 𝑘) = (1, 2, 3), (1, 4, 7), (2, 4, 6) and
(2, 5, 7) shows that 𝑚1 = 𝑚2 = 𝑚3 = 𝑚4 = 𝑚5 = 𝑚6 = 𝑚7. However, it should be noted that
the cases (𝑎, 𝑏) = (1, 2) and (1, 4) respectively lead to

4𝑚4
1( 𝑓

5
11)

2 = ( 𝑓123)2 𝑚4
1 = 𝑚4

1,

4𝑚4
1( 𝑓

5
11)

2 = ( 𝑓147)2 𝑚4
1 = 1

4𝑚
4
1.

Therefore the inequality 𝑓123 ≠ ± 𝑓147 yields an impossibility: there is no all-massive SU(3)
model within our approach. Note that SU(2) escapes this sentence because all squared
structure constants are equal. The reader should be able to check that the same phenomenon
raises obstructions to several other putative SU(3) models.

(ii) This leads us to ponder the “natural” pattern:

𝑚1 = 𝑚2 = 𝑚3 = 0; 𝑚4, 𝑚5, 𝑚6, 𝑚7 ≠ 0.

Indeed, the “photons” 𝑚1, 𝑚2 force 𝑚4 = 𝑚5 = 𝑚6 = 𝑚7 =: 𝑚 ≠ 0 through use of (22). Then
𝑓 5
44 = 1

2 by just considering in this equation (𝑎, 𝑏) = (4, 6), say. By considering (𝑎, 𝑏) = (4, 5),

23



one obtains 𝑚8 = 2𝑚/
√

3, and after some work, it is checked that there is no contradiction in
this. Note that (𝑎, 𝑏, 𝑘) = (4, 5, 8) is not symmetrical with (𝑎, 𝑏, 𝑘) = (4, 8, 5), since in one
case there is a massless contribution ( 𝑓453 = 1

2 ), but not in the other. In conclusion: the model

𝑚1 = 𝑚2 = 𝑚3 = 0; 𝑚4 = 𝑚5 = 𝑚6 = 𝑚7 = 𝑚; 𝑚8 = 2√
3
𝑚 ≠ 0 (33)

solves the CGI mass conditions (15) and (20).

Turning to SU(4), we can take basis matrices {𝑇𝑎} extending those of SU(3), filled out with a
fourth row and column of zeroes, by {𝑇9, . . . , 𝑇15}, where

𝑇15 = 1
2
√

6
diag(1, 1, 1,−3),

so that {𝑇3, 𝑇8, 𝑇15} spans the Cartan subalgebra of diagonal matrices, and the off-diagonal ones are
the hermitian matrices given in terms of the matrix units 𝑒𝑖 𝑗 by7

𝑇1 + 𝑖𝑇2 = 𝑒12, 𝑇4 + 𝑖𝑇5 = 𝑒13, 𝑇6 + 𝑖𝑇7 = 𝑒23,

𝑇9 + 𝑖𝑇10 = 𝑒14, 𝑇11 + 𝑖𝑇12 = 𝑒24, 𝑇13 + 𝑖𝑇14 = 𝑒34.

We keep the normalization tr(𝑇𝑎𝑇𝑏) = 1
2 𝛿𝑎𝑏. The structure constants 𝑓𝑎𝑏𝑐 have the following

nonzero squares, with a hexadecimal labelling:

( 𝑓123)2 = 1; ( 𝑓458)2 = ( 𝑓678)2 = 3
4 ; ( 𝑓89𝐴)2 = ( 𝑓8𝐵𝐶)2 = 1

12 ; ( 𝑓8𝐷𝐸 )2 = 1
3 ;

( 𝑓147)2 = ( 𝑓156)2 = ( 𝑓19𝐶)2 = ( 𝑓1𝐴𝐵)2 = ( 𝑓246)2 = ( 𝑓257)2 = ( 𝑓29𝐵)2 = 1
4 ,

( 𝑓2𝐴𝐶)2 = ( 𝑓345)2 = ( 𝑓367)2 = ( 𝑓39𝐴)2 = ( 𝑓3𝐵𝐶)2 = ( 𝑓49𝐸 )2 = ( 𝑓4𝐴𝐷)2 = 1
4 ,

( 𝑓59𝐷)2 = ( 𝑓5𝐴𝐸 )2 = ( 𝑓6𝐵𝐸 )2 = ( 𝑓6𝐶𝐷)2 = ( 𝑓7𝐵𝐷)2 = ( 𝑓7𝐶𝐸 )2 = 1
4 ;

( 𝑓9𝐴𝐹)2 = ( 𝑓𝐵𝐶𝐹)2 = ( 𝑓𝐷𝐸𝐹)2 = 2
3 . (34)

Naturally, the structure constants for the Lie subalgebra SU(3) are a subset of those for SU(4). Thus
objections to putative models for SU(3) carry over to the SU(4) case. Nevertheless, the allowable
pattern for SU(3) given by (33) does have an analogue for SU(4). Let us assume that the bosons
labelled by the SU(3) subalgebra are massless, and that the new ones are massive:

𝑚1 = · · · = 𝑚8 = 0; 𝑚9, . . . , 𝑚15 ≠ 0.

The relation (16) together with (34) gives at once

𝑚9 = 𝑚10 = 𝑚11 = 𝑚12 = 𝑚13 = 𝑚14 =: 𝑚,

but remains silent about 𝑚15. Now we check this for consistency with (22). For 𝑎 ⩽ 8, 𝑏 ⩾ 9,
this relation always reduces to 0 = ((2𝑚2)2 − 4𝑚4)/𝑚2. Taking 𝑎 ≠ 𝑏 in the range {9, . . . , 14} we
typically obtain

4𝑚2( 𝑓 5
𝑎𝑎)2 = 4𝑚2

∑︁
𝑑⩽8

( 𝑓𝑎𝑏𝑑)2 + ( 𝑓𝑎𝑏𝐹)2(4𝑚2 − 3𝑚2
15). (35)

7In the standard notation for root vectors, the simple roots 𝛼, 𝛽, 𝛾 give 𝐸𝛼 = 𝑒12, 𝐸𝛽 = 𝑒23, 𝐸𝛾 = 𝑒34, 𝐸𝛼+𝛽 = 𝑒13,
𝐸𝛽+𝛾 = 𝑒24, 𝐸𝛼+𝛽+𝛾 = 𝑒14. Note that 𝛼 + 𝛾 is not a root of SU(4) since [𝐸𝛼, 𝐸𝛾] = 0.

24



In most cases, this reduces to ( 𝑓 5
𝑎𝑎)2 = 1

4 . When (𝑎, 𝑏) = (9, 10) or (11, 12) or (13, 14), we then get

𝑚2 = 1
3 (4𝑚

2) + 2
3 (4𝑚

2 − 3𝑚2
15),

that is, 3𝑚2 = 2𝑚2
15. When 𝑎 = 15 and 9 ⩽ 𝑏 ⩽ 14, the constraint (22) becomes 4𝑚2( 𝑓 5

𝐹𝐹
)2 =

(2𝑚4
15/3𝑚2) = 𝑚2

15, consistent with 𝑓 5
𝐹𝐹

/ 𝑓 5
𝑏𝑏

= 𝑚15/𝑚, as expected. To sum up, this mass pattern
is compatible with the CGI mass conditions (15) and (20), provided that

𝑚15 =
√︁

3/2𝑚.

No other pattern for SU(4) with one physical scalar seems to solve (15) and (20), although an
exhaustive search has not been performed.

Going to the general SU(𝑛) case, one can show likewise that for a theory with 𝑛2 − 1 vector
bosons and one physical scalar:

𝑚1 = 𝑚2 = · · · = 𝑚𝑛2−2𝑛 = 0;
𝑚 (𝑛−1)2 = · · · = 𝑚𝑛2−2 =: 𝑚 ≠ 0;

𝑚𝑛2−1 =

√︂
2(𝑛 − 1)

𝑛
𝑚.

The “odd man out” corresponds to the last Cartan matrix

𝑇𝑅 = 𝐶𝑛 =
1√︁

2𝑛(𝑛 − 1)
diag(1, . . . , 1,−(𝑛 − 1)),

while the previous ones become massless. With the labels 𝐷 = 𝑛2 − 2𝑛, 𝑃 = 𝑛2 − 3, 𝑄 = 𝑛2 − 2,
𝑅 = 𝑛2 − 1, then for (𝑛 − 1)2 ⩽ 𝑎 ⩽ 𝑛2 − 3 and 𝑏 = 𝑎 + 1, one finds that8∑︁

𝑑⩽𝐷

( 𝑓𝑎𝑏𝑑)2 = ( 𝑓𝑃𝑄𝐷)2 =
𝑛 − 2

2𝑛 − 2
, ( 𝑓𝑎𝑏𝑅)2 = ( 𝑓𝑃𝑄𝑅)2 =

𝑛

2𝑛 − 2
.

Thus, the analogue of (35) for the SU(𝑛) case is

𝑚2 =
2(𝑛 − 2) 𝑚2

𝑛 − 1
+ 𝑛

2(𝑛 − 1) (4𝑚
2 − 3𝑚2

𝑅),

yielding
𝑚2

𝑅 =
2𝑛 − 2

𝑛
𝑚2. (36)

It seems clear that the masses of the gauge particles organize themselves in SU(𝑛 − 1) multiplets,
concretely the fundamental one and a singlet.

Thus the translation of our allowed models into the SSB phraseology follows a well-trodden
path: in general, a vector representation for SU(𝑛) contains 2𝑛 real fields, of which 2𝑛 − 1 are
“eaten” to provide the longitudinal components for 2𝑛 − 1 “initially massless” gauge fields, leaving
𝑛2 − 1 − (2𝑛 − 1) = (𝑛 − 1)2 − 1 “photons” (corresponding to an SU(𝑛 − 1) as yet “unbroken”

8To compute 𝑓𝑃𝑄𝐷 and 𝑓𝑃𝑄𝑅, just evaluate the commutators [𝐶𝑛−1, 𝑒𝑛−1,𝑛] and [𝐶𝑛, 𝑒𝑛−1,𝑛].

25



symmetry), and the remaining one is the physical higgs field. In such a framework formula (36)
is well known [67, Sect. 84]. Things work out similarly for vector representations of O(𝑛); the
O(3) ≃ SU(2) case we have seen already in Section 5.2.

In conclusion, the CGI mass conditions (15) and (20) efficiently identify SSB-type models in
the vector representation. The reader will have no difficulty in writing the 𝑆-representations and
checking the commutation relations.9

5.4 Causal gauge invariance for SU(3) with fields in the adjoint

We finally turn to more involved models with the scalar fields in the adjoint representation, to
exemplify minimal coupling in the CGI framework for such models and to deal with the alleged
clash between CGI and SSB in [46].

Concerning GUT models, (most simply) two irreducible representations of pre-higgs particles
are needed for SSB to yield something recognizably akin to the SM; to wit, in [68] the adjoint
representation 24 and the complex fundamental representation 5 of SU(5). Thus the question
is whether a model with 24 vector bosons, of which 12 have identical nonzero mass and 12 are
massless, passes muster in CGI, allowing for 12 higgs-like fields. Naturally, one should seek to
answer the similar question for simpler models first. For SU(2), the model with 3 vector bosons, two
with identical nonzero mass and one massless, together with one higgs-like field, has been shown
in subsection 5.2 to pass muster in CGI. For SU(3) there would be in the adjoint representation 8
vector bosons, 4 of which have identical nonzero mass and 4 are massless, with 4 higgs-like fields.
We take up this case as the simplest proxy for our problem, embarking on this from the opposite
end to that of subsection 5.1: first we recall SSB for this model; then, in consonance with [45]
we verify BRST invariance for the resulting classical Lagrangian; this proves CGI at tree level.
Finally, we check the representation property (24) of the 𝑆-matrices, and substantiate our claim
in subsection 4.2 that minimal coupling follows from CGI. We deem the exercise important and
proceed in fastidious detail.

5.4.1 BRST invariance of the classical Lagrangian

An invariant potential for that representation is

𝑉 (Φ) = −𝜇2 trΦ2 + 𝜆(trΦ2)2 (37)

with Φ a traceless hermitian 3 × 3 matrix variable and 𝜆 > 0. (We are not striving for maximum
generality here, so we have suppressed a term in trΦ3. The usual trΦ4 term is missing since in
this somewhat degenerate case trΦ4 = 1

2 (trΦ
2)2 by the Cayley–Hamilton formula.) The pattern of

symmetry breaking is 𝑆𝑟𝑈 (2) × U(1)); a minimum of the potential 𝑉 (Φ) is of the form

⟨Φ⟩ = 𝑣 diag
(
1/2

√
3, 1/2

√
3,−1/

√
3
)
= 𝑣𝜆8/2,

where 𝑣 is to be determined such that 𝑉 (𝑣) := 𝑉 (𝑣𝜆8/2) = −𝜇2𝑣2/2 + 𝜆𝑣4/4 be minimal, see [7]
for instance. This gives 𝑣2

min = 𝜇2/𝜆; 𝑉 (𝑣min) = −𝜇4/4𝜆. From now on, we just write 𝑣 for 𝑣min.

9Group theory dictates that the sum of the squared 𝑓 5
𝑎𝑎 be equal to the Casimir for the corresponding representations,

respectively 2 and 3
4 for the SU(2) models just above. For every SU(𝑛) model of this type ( 𝑓 5)2 = 1

4 holds.

26



Also write 𝐴𝜇 ≡ 𝐴
𝜇
𝑎𝑇𝑎, Φ ≡ 𝜙𝑎𝑇𝑎, 𝑢 ≡ 𝑢𝑎𝑇𝑎, �̃� ≡ �̃�𝑎𝑇𝑎, using the Gell-Mann basis. One can easily

check that a shifted field 𝜑 is required only for the 𝜙8 component: 𝜙8 = 𝑣 + 𝜑.
The covariant derivative in the adjoint representation is of the form 𝐷𝜇 = 𝜕𝜇 − 𝑖𝜅 [𝐴𝜇, · ]; in

components,

𝐷
𝜇

𝑎𝑏
= 𝛿𝑎𝑏 𝜕

𝜇 − 𝜅 𝑓𝑎𝑏𝑐 𝐴
𝜇
𝑐 , and thus

𝐷𝜇Φ = 𝐷𝜇 (𝜙𝑒𝑇𝑒) = 𝜕𝜇𝜙𝑒 𝑇𝑒 − 𝑖𝜅 [𝐴𝜇
𝑎𝑇𝑎, 𝜙𝑏𝑇𝑏] = 𝜕𝜇𝜙𝑒 𝑇𝑒 + 𝜅 𝑓𝑎𝑏𝑐𝐴

𝜇
𝑎𝜙𝑏𝑇𝑐

= (𝜕𝜇𝜙𝑏 + 𝜅 𝑓𝑎𝑏𝑐𝐴
𝜇
𝑐 𝜙𝑎)𝑇𝑏 = (𝛿𝑎𝑏 𝜕𝜇𝜙𝑏 − 𝜅 𝑓𝑎𝑏𝑐𝐴

𝜇
𝑐 𝜙𝑏)𝑇𝑎 = (𝐷𝜇Φ)†.

The Lagrangian for bosonic scalar fields reads

𝐿Φ = tr(𝐷Φ · 𝐷Φ) −𝑉 (Φ),

where 𝑉 is given by (37) with 𝜇2 ∼ 𝜅0 and 𝜆 ∼ 𝜅2. To determine the mass spectrum of the gauge
fields, one collects the mass terms in tr(𝐷Φ · 𝐷Φ), with 𝜙8 replaced by 𝑣 + 𝜑, with the result that

𝜅2(𝐴𝑏 · 𝐴𝑑) 𝑓8𝑎𝑏 𝑓8𝑐𝑑 𝑣2 tr(𝑇𝑎 𝑇𝑐) =
3𝑣2𝜅2

8
(𝐴4 · 𝐴4 + 𝐴5 · 𝐴5 + 𝐴6 · 𝐴6 + 𝐴7 · 𝐴7). (38)

Hence 𝑚2
1 = 𝑚2

2 = 𝑚2
3 = 𝑚2

8 = 0, and

𝑚2 := 𝑚2
4 = 𝑚2

5 = 𝑚2
6 = 𝑚2

7 =
3𝜇2𝜅2

4𝜆
=

3𝑣2𝜅2

4
.

The potential 𝑉 (Φ) contains a mass term only for the field 𝜑, namely one can show that 𝑉 (Φ) =
𝑉0 + 𝜇2𝜑2 + (terms trilinear and quadrilinear in the fields), where 𝑉0 := 𝑉 (𝑣min). Hence 𝜑 is
the “polar” higgs field, in the direction of symmetry breakdown. The other three higgs fields are
massless, they are pseudo-Goldstone bosons in the precise sense of [69]. Next we collect all terms
∼ 𝜅0 in tr(𝐷Φ · 𝐷Φ). Since there appears a term

2 tr
[
(𝜕𝜇𝜙𝑎𝑇𝑎) (𝜅 𝑓8𝑏𝑐 𝐴𝜇

𝑐 𝑣𝑇𝑏)
]
=

2𝑚
√

3
𝜕𝜇𝜙𝑏 𝑓8𝑏𝑐 𝐴

𝜇
𝑐 = −2𝑚

√
3
𝜕𝜇𝜙𝑏 𝑓𝑎𝑏8 𝐴

𝜇
𝑎 ,

we introduce the notation
𝐵𝑎 =

2
√

3
𝑓𝑎𝑏8 𝜙𝑏,

that is, 𝐵4 = 𝜙5, 𝐵5 = −𝜙4, 𝐵6 = 𝜙7, 𝐵7 = −𝜙6, and 𝐵𝑎 = 0 for 𝑎 = 1, 2, 3, 8; later we shall see that
the 𝐵𝑎’s are the Stückelberg fields belonging to the massive vector bosons 𝐴𝑎 (for 𝑎 = 4, 5, 6, 7).
Therefore the remaining bosonic scalar fields 𝜑1 ≡ 𝜙1, 𝜑2 ≡ 𝜙2, 𝜑3 ≡ 𝜙3 are the massless physical
higgs fields. With that one obtains

tr(𝐷Φ · 𝐷Φ) = 1
2

∑︁
𝑎=1,2,3

𝜕𝜑𝑎 · 𝜕𝜑𝑎 +
1
2

∑︁
𝑎=4,5,6,7

𝜕𝐵𝑎 · 𝜕𝐵𝑎 +
1
2
𝜕𝜑 · 𝜕𝜑

+ 𝑚2

2

∑︁
𝑎=4,5,6,7

(𝐴𝑎 · 𝐴𝑎) − 𝑚
∑︁

𝑎=4,5,6,7
𝐴𝑎 · 𝜕𝐵𝑎 +𝑂 (𝜅).

27



We now assemble the last pieces of the Lagrangian. For the 𝑚𝐴 · 𝜕𝐵 terms to add up to a
divergence, choose for the gauge-fixing term in the Feynman gauge

𝐿gf = 𝐿
gf
0 = −1

2

∑︁
𝑎

(𝜕 · 𝐴𝑎 + 𝑚 𝐵𝑎)2

= −1
2

8∑︁
𝑎=1

(𝜕 · 𝐴𝑎)2 − 𝑚2

2

∑︁
𝑎=4,5,6,7

𝐵2
𝑎 − 𝑚

∑︁
𝑎=4,5,6,7

(𝜕 · 𝐴𝑎)𝐵𝑎 .

Due to 𝐹𝜇𝜈 ≡ 𝜕𝜇𝐴𝜈 − 𝜕𝜈𝐴𝜇 − 𝑖𝜅 [𝐴𝜇, 𝐴𝜈], the Yang–Mills Lagrangian

𝐿YM = −1
2

tr 𝐹𝜇𝜈𝐹𝜇𝜈 = −1
4
𝐹
𝜇𝜈
𝑎 𝐹𝑎 𝜇𝜈

is of the form 𝐿YM = 𝐿YM
0 + 𝜅 𝐿YM

1 + 𝜅2 𝐿YM
2 .

Next we introduce the BRST transformation and verify classical BRST invariance of the total
Lagrangian.10

For 𝐴𝜇 and Φ the 𝑠-operator is given by the infinitesimal gauge transformations

𝑠𝐴𝜇 = 𝐷𝜇𝑢 = 𝜕𝜇𝑢 − 𝑖𝜅 [𝐴𝜇, 𝑢], 𝑠Φ = 𝑖𝜅 [𝑢,Φ] .

For example, the latter gives

𝑠𝜑1 = −𝜅(𝑢2𝜑3 − 𝑢3𝜑2) −
𝜅

2
(𝑢4𝐵6 + 𝑢7𝐵5 + 𝑢6𝐵4 + 𝑢5𝐵7),

in general, 𝑠𝜑𝑎 = 𝑂 (𝜅) for 𝑎 = 1, 2, 3; whereas

𝑠𝐵4 = 𝑚 𝑢4 +
𝜅

2
(
𝑢6𝜑1 + 𝑢1𝐵7 − 𝑢7𝜑2 + 𝑢2𝐵6 + 𝑢3𝐵5 + 𝑢4𝜑3 +

√
3(𝑢4𝜑 + 𝑢8𝐵5)

)
,

in general, 𝑠𝐵𝑎 = 𝑚 𝑢𝑎 + O(𝜅) for 𝑎 = 4, 5, 6, 7;

𝑠𝜑 = −𝜅
√

3
2

∑︁
𝑎=4,5,6,7

𝑢𝑎𝐵𝑎 .

For the ghost fields 𝑢, we get 𝑠𝑢 = 𝑖 𝜅/2 [𝑢, 𝑢], as in the massless case, and for the antighosts �̃�,

𝑠�̃� = −(𝜕 · 𝐴 + 𝑚 𝐵).

With that the gauge fixing term can be written as 𝐿gf = −1
2
∑

𝑎 (𝑠�̃�𝑎)2, as usual in the Feynman
gauge.

For the nilpotence of 𝑠, we find that 𝑠2𝐴 = 0 and 𝑠2𝑢 = 0 are similar to the well-known massless
case. In the Lie superalgebra generated by the 𝐴 = 𝐴

𝜇
𝑎𝑇𝑎 and the 𝑢 = 𝑢𝑏𝑇𝑏, we can write

𝑠2𝐴 = 𝑠(𝐷𝑢) = 𝜕 (𝑠𝑢) − 𝑖𝜅 [𝑠𝐴, 𝑢] − 𝑖𝜅 [𝐴, 𝑠𝑢]

=
𝑖 𝜅

2
(
[𝑢, 𝜕𝑢] − [𝜕𝑢, 𝑢]

)
− 𝜅2

2
(
2 [[𝐴, 𝑢], 𝑢] − [𝐴, [𝑢, 𝑢]]

)
.

10The reader might ask: why bother? Are not all models generated by SSB automatically BRST invariant? Surely
yes. But there are few discussions of this in the literature: typically textbooks go at great length into the proof
that Yang–Mills theories are BRST invariant, and then resolutely tiptoe around the same question for “hidden local
symmetry”. Reference [70] furnishes an amusing example.

28



Since the bracket is symmetric between expressions of ghost number 1, the first two terms cancel;
and the second two terms also cancel by the same symmetry and the Jacobi identity, whereby [𝐴, ·]
is a derivation:

[𝐴, [𝑢, 𝑢]] = [[𝐴, 𝑢], 𝑢] + [𝑢, [𝐴, 𝑢]] = 2[[𝐴, 𝑢], 𝑢] .
The same identity takes care of 𝑠2Φ:

𝑠2Φ = 𝑖𝜅
(
[𝑠𝑢,Φ] − [𝑢, 𝑠Φ]

)
= −𝜅2

2
(
[[𝑢, 𝑢],Φ] − 2[𝑢, [𝑢,Φ]]

)
= 0.

Vanishing of 𝑠2�̃� is discussed further down.
The BRST transformation of 𝐴𝜇 and Φ has the form of an infinitesimal gauge transformation of

the unshifted field. It follows that 𝑠𝐿YM = 0 and 𝑠𝐿Φ = 0. We perform the explicit verification for
𝐿Φ by using:

𝑠(𝐷Φ) = 𝜕 (𝑠Φ) − 𝑖𝜅
(
[𝑠𝐴,Φ] + [𝐴, 𝑠Φ]

)
= 𝑖𝜅 𝜕 ( [𝑢,Φ]) − 𝑖𝜅 [𝜕𝑢,Φ] + 𝜅2 (−[[𝐴, 𝑢],Φ] + [𝐴, [𝑢,Φ]]

)
= 𝑖𝜅 [𝑢, 𝜕Φ] + 𝜅2 [𝑢, [𝐴,Φ]] = 𝑖𝜅 [𝑢, 𝐷Φ] .

Since 𝐷Φ = 𝜕Φ − 𝑖𝜅 [𝐴,Φ] has ghost number zero, it follows that

𝑠 tr(𝐷Φ · 𝐷Φ) = 𝑖𝜅 tr
(
[𝑢, 𝐷Φ] · 𝐷Φ + 𝐷Φ · [𝑢, 𝐷Φ]

)
= 𝑖𝜅 tr[𝑢, 𝐷Φ · 𝐷Φ] = 0.

Next we introduce the ghost Lagrangian, chosen in a such a way that 𝑠𝐿gf + 𝑠𝐿gh is a divergence:
one sets

𝐿gh =

8∑︁
𝑎=1

𝜕𝜇�̃�𝑎 𝑠𝐴
𝜇
𝑎 − 𝑚

∑︁
𝑎=4,5,6,7

�̃�𝑎 𝑠𝐵𝑎 = 𝐿
gh
0 + 𝜅𝐿

gh
1 ,

which yields indeed 𝑠(𝐿gf + 𝐿gh) = 𝜕 · 𝐼, with

𝐼 := −(𝜕 · 𝐴𝑎 + 𝑚 𝐵𝑎) 𝐷𝑎𝑏𝑢𝑏 = 𝐼0 + 𝜅 𝐼1. (39)

Summing up, the total Lagrangian

𝐿tot = 𝐿YM + 𝐿Φ + 𝐿gf + 𝐿gh = −𝑉0 + 𝐿0 + 𝜅 𝐿1 + 𝜅2𝐿2 (40)

is classically BRST invariant since
𝑠𝐿tot = 𝜕 · 𝐼 .

All terms of 𝐿2 come from 𝐿YM + 𝐿Φ. Notice that

𝐿0 =

8∑︁
𝑎=1

(
−1

4
(𝐴𝜇,𝜈

𝑎 − 𝐴
𝜈,𝜇
𝑎 ) (𝐴𝑎 𝜇,𝜈 − 𝐴𝑎 𝜈,𝜇) −

1
2
𝜕𝐴𝑎 · 𝜕𝐴𝑎

)
+ 𝑚2

2

∑︁
𝑎=4,5,6,7

𝐴𝑎 · 𝐴𝑎

+
8∑︁

𝑎=1
𝜕�̃�𝑎 · 𝜕𝑢𝑎 − 𝑚2

∑︁
𝑎=4,5,6,7

�̃�𝑎𝑢𝑎 − 𝑚
∑︁

𝑎=4,5,6,7
𝜕 · (𝐴𝑎 𝐵𝑎)

+ 1
2

( ∑︁
𝑝=1,2,3

𝜕𝜑𝑝 · 𝜕𝜑𝑝 +
∑︁

𝑎=4,5,6,7
(𝜕𝐵𝑎 · 𝜕𝐵𝑎 − 𝑚2 𝐵2

𝑎) + 𝜕𝜑 · 𝜕𝜑 − 2 𝜇2𝜑2
)

(41)

29



contains the divergence term −𝑚 𝜕 · (𝐴𝐵), which is irrelevant for the field equations, but contributes
to the BRST current 𝑗(0) of the free theory (see below).

We may go back now to nilpotence of 𝑠. The vanishing of 𝑠2�̃�𝑎 takes place on-shell: with
𝑆tot =

∫
𝑑4𝑥 𝐿tot(𝑥), the relation

𝑠2�̃�𝑎 =
𝛿 𝑆tot
𝛿 �̃�𝑎

= 0

follows from the Euler–Lagrange equations.

The BRST current of the free theory can be computed by the formula:

𝑗
𝜇

(0) = −
(

𝜕𝐿0
𝜕 (𝜕𝜇𝜑𝑖)

𝑠0𝜑𝑖 + 𝐼
𝜇

0

)
, (42)

where there is a summation over 𝜑𝑖 = 𝐴
𝜇
𝑎 , 𝑢𝑎, �̃�𝑎, 𝐵𝑎, 𝜑𝑝, 𝜑. For the present model it comes out that

𝑗
𝜇

(0) = (𝜕 · 𝐴𝑎 + 𝑚𝑎 𝐵𝑎)𝜕𝜇𝑢𝑎 − 𝑢𝑎 𝜕
𝜇 (𝜕 · 𝐴𝑎 + 𝑚𝑎 𝐵𝑎) − 𝜕𝜈 ((𝜕𝜈𝐴𝜇

𝑎 − 𝜕𝜇𝐴𝜈
𝑎)𝑢𝑎) + 𝑢𝑎 (□ + 𝑚2

𝑎)𝐴
𝜇
𝑎 ,

by inserting the explicit expressions in (39) for 𝐼0 and (41) for 𝐿0 into (42). As explained in [45], the
last two terms do not contribute to the nilpotent charge𝑄in – which is defined only for on-shell fields.
Hence, the superderivation [𝑄in, ·]∓ is defined as in [35], and the 𝐵𝑎 are indeed the Stückelberg
fields of this reference.

Since the second CGI criterion [45] is fulfilled, the model is causal gauge invariant at the tree
level to all orders.

5.4.2 Minimal coupling from CGI with fields in the adjoint

Let us now take stock of the CGI relation (24) for the adjoint SU(3) model. In order to avoid
misunderstandings, we remind the reader that each vector boson is entitled to its matrix, but there
are no rows or columns corresponding to the massless ones. The labelling of the entries is thus
given in the order: 4, 5, 6, 7 (for the MVB) and 1, 2, 3, 8 (for the “higgses”). That is to say, in the
notation of (26), we use the multiplet

𝜂 =

©­­­­­­­­­­­«

𝐵4
𝐵5
𝐵6
𝐵7
𝜙1
𝜙2
𝜙3

𝑣 + 𝜑

ª®®®®®®®®®®®¬
versus

©­­­­­­­­­­­«

𝜙1
𝜙2
𝜙3
𝜙4
𝜙5
𝜙6
𝜙7
𝜙8

ª®®®®®®®®®®®¬
=

©­­­­­­­­­­­«

𝜑1
𝜑2
𝜑3
−𝐵5
𝐵4
−𝐵7
𝐵6

𝑣 + 𝜑

ª®®®®®®®®®®®¬
(43)

of the previous subsection. To write down the matrices 𝑆𝑎, we read off 𝑓 3
∗∗∗, 𝑓 5

∗∗∗, 𝑓 6
∗∗∗ from

𝜅 𝑓𝑎𝑏𝑐 (𝐴𝑐 · 𝜕𝜙𝑏) 𝜙𝑎 in tr(𝐷Φ · 𝐷Φ); note that (40) contains no further contributions to these cubic

30



coupling coefficients. With that, we obtain

𝑆1 =

©­­­­­­­­­­­«

0 −1
2

1
2 0

0 −1
2

1
2 0

0
0 −1
1 0

0

ª®®®®®®®®®®®¬
.

Here we note at once the new relations: −2 𝑓 6
123 = 2 𝑓 6

132 = −1 (so 𝑓 6 is not zero). The contributions
in the upper left corner come from 𝑓 3

147 and such, that do not vanish. Similarly:

𝑆2 =

©­­­­­­­­­­­«

−1
2 0

0 −1
2

1
2 0
0 1

2
1

0
−1

0

ª®®®®®®®®®®®¬
; 𝑆3 =

©­­­­­­­­­­­«

0 −1
2

1
2 0

0 1
2

−1
2 0

0 −1
1 0

0
0

ª®®®®®®®®®®®¬
;

and the commutation relations of the SU(2) subgroup are clearly fulfilled.
The reader will easily write down the other basis matrices of the 𝑆-representation and check

the group property. The main remark is that, besides the nonvanishing of 𝑓 6
1∗∗, 𝑓 6

2∗∗, 𝑓 6
3∗∗, there are

nondiagonal values of 𝑓 5
∗∗𝑝, for example 𝑓 5

472 = − 𝑓 5
461 = 1

2 . Matters work in the same way as in
this example for any SU(𝑛) model in the adjoint representation for higher 𝑛. Look no farther for
the solution to the conundrum raised by [46]: the model coming by SSB of the representation 24,
responsible for “superstrong breaking” in the original GUT by Georgi and Glashow, is incompatible
with the obstructions given by Scharf [35]; but CGI does not exclude it.

We shall now verify that the scalar-gauge Lagrangian resulting from the first CGI method can be
expressed as the minimal coupling 1

2 (𝐷𝜂)𝑡 ·𝐷𝜂 = 1
2
∑

𝑏 (𝐷𝜂)𝑏 · (𝐷𝜂)𝑏. The former can be expressed
by the term

tr(𝐷Φ · 𝐷Φ) = 1
2

∑︁
𝑏

(𝐷𝜙)𝑏 · (𝐷𝜙)𝑏, (𝐷𝜙)𝑏 := 𝜕𝜙𝑏 − 𝜅 𝑓𝑏𝑎𝑐 𝐴𝑐 𝜙𝑎

obtained by SSB, since construction of the SU(3)-adjoint model by SSB agrees with what one
obtains by CGI (as shown in subsection 5.4). Then 1

2 (𝐷𝜂)𝑡 ·𝐷𝜂 = tr(𝐷Φ ·𝐷Φ) follows from noting

31



that the sets of covariant derivatives agree. For example, by direct calculation,

(𝐷𝜙)1 = 𝜕𝜑1 + 𝜅(𝐴2𝜑3 − 𝐴3𝜑2) +
𝜅

2
(𝐴4𝐵6 + 𝐴6𝐵4 + 𝐴5𝐵7 + 𝐴7𝐵5) = (𝐷𝜂)5,

(𝐷𝜙)4 = −𝜕𝐵5 −
𝜅

2
(𝐴1𝐵6 − 𝐴2𝐵7 + 𝐴3𝐵4 − 𝐴5𝜑3 − 𝐴6𝜑2 − 𝐴7𝜑1)

− 𝜅
√

3
2

(−𝐴5(𝜑 + 𝑣) + 𝐴8𝐵4) = −(𝐷𝜂)2,

(𝐷𝜙)8 = 𝜕𝜑 + 𝜅
√

3
2

(𝐴4𝐵4 + 𝐴5𝐵5 + 𝐴6𝐵6 + 𝐴7𝐵7) = (𝐷𝜂)8.

Agreement of other components follows by permutation.
It is instructive to write down term by term the equality 1

2 (𝐷𝜂)𝑡 ·𝐷𝜂 = scalar gauge Lagrangian
obtained by the first CGI method, see (27). The validity of the resulting equations is not limited to
the SU(3)-adjoint model.

The mass terms for vector gauge fields originate from (𝐴𝑎 · 𝐴𝑏) 𝑣𝑡 [𝑆𝑎, 𝑆𝑏]+𝑣, where 𝑣𝑡 :=
(𝑣1, . . . , 𝑣𝑧). Equating the corresponding coefficients, we find that

−𝜅2

4
𝑣𝑡

(
−𝑡𝐺𝑎𝐺𝑏 − 𝑡𝐺𝑏𝐺𝑎 + [𝐻𝑎, 𝐻𝑏]+

)
𝑣 = 𝛿𝑎𝑏 𝑚

2
𝑎 .

For the SU(3)-adjoint model, with the scalar multiplet as in (43), on inserting the above 𝑆-matrices,
this formula gives indeed 3

8𝑣
2𝜅2 ∑7

𝑘=4 𝐴𝑘 · 𝐴𝑘 =
1
2𝑚

2 ∑7
𝑘=4 𝐴𝑘 · 𝐴𝑘 , like in (38). Note that although

𝑓 6 ≠ 0, it does not contribute.
By the definition of the 𝑆-matrices, the couplings 𝐿3

1 + 𝐿5
1 + 𝐿6

1 must be contained in (𝐴𝑎 ·
𝜕𝜂𝑡 )𝑆𝑎𝜂 − 𝜂𝑡𝑆𝑎 (𝐴𝑎 · 𝜕𝜂). Indeed, we obtain

(𝐴𝑎 · 𝜕𝜂𝑡 )𝑆𝑎𝜂 − 𝜂𝑡𝑆𝑎 (𝐴𝑎 · 𝜕𝜂) = 𝐿3
1 + 𝐿5

1 + 𝐿6
1 − 𝜅 𝑓 5

𝑎𝑏𝑝 (𝐴𝑎 · 𝜕𝐵𝑏) 𝑣𝑝 .

The additional (𝐴 · 𝜕𝐵) term gives, as expected,

−𝜅 𝑓 5
𝑎𝑏𝑝 (𝐴𝑎 · 𝜕𝐵𝑏)𝑣𝑝 = −𝜅𝑣 𝑓 5

𝑎𝑏8(𝐴𝑎 · 𝜕𝐵𝑏) = −
√

3 𝜅𝑣

2

∑︁
𝑎:𝑚𝑎≠0

(𝐴𝑎 · 𝜕𝐵𝑎)

= −𝑚
∑︁

𝑎:𝑚𝑎≠0
(𝐴𝑎 · 𝜕𝐵𝑎).

Moreover, in (𝐴𝑎 · 𝐴𝑏)𝜂𝑡 [𝑆𝑎, 𝑆𝑏]+𝜂 there are trilinear terms corresponding to 𝐿2
1+𝐿

7
1. Equating

the pertinent coefficients, we get

𝑓 2
𝑎𝑏★ = −𝜅2

2
(𝐹𝑎𝐺𝑏 + 𝐹𝑏𝐺𝑎 + 𝐺𝑎𝐻𝑏 + 𝐺𝑏𝐻𝑎)𝑣;

𝑓 7
𝑎𝑏★ =

𝜅2

2
(𝑡𝐺𝑎 𝐺𝑏 + 𝑡𝐺𝑏 𝐺𝑎 − 𝐻𝑎𝐻𝑏 − 𝐻𝑏𝐻𝑎)𝑣.

The pattern exemplified above is typical for SU(𝑛) models with the scalar fields in the adjoint
representation. An even simpler example is provided by the three-boson case with two identical
masses and one “photon” of subsection 5.2. Proceeding as above, the reader should have no difficulty
in verifying that 1

2 (𝐷𝜂)𝑡 · 𝐷𝜂 equals the SSB-type expression tr(𝐷Φ · 𝐷Φ).

32



6 Conclusion

The Higgs sector of the SM, in the perspective of basic structures of gauge theories, plays a somewhat
ambiguous and enigmatic role. The massless and massive gauge bosons which are the carriers of the
fundamental forces belong to what might be termed radiation, in analogy to electrodynamics. Now,
by itself, the gauge boson sector of gauge theories of interest for physics defines a nontrivial theory.
Quarks and leptons, on the other hand, belong to the category matter which cannot “live on its own”
without the gauge sector. Indeed, a theory of quarks and leptons only is a theory of free particles
and, being untestable in experiment, is uninteresting. The Higgs sector’s place in this classification
is perhaps not as obvious as it may appear at first sight. Extensions of the Standard Model within
noncommutative geometry [71–74] view scalar fields as an integral part of the connection, i.e.
classify them in the sector of gauge bosons and hence place them in the category “radiation”. Its
alleged phenomenological role of providing masses for the fermions and some bosons of the model,
and its likely kinship with dark matter [15], in turn, might suggest that it rather represents another
form of “matter” beyond the ordinary one made out of quarks and leptons.

Be that as it may, the traditional description of the Higgs sector by means of the “hidden
symmetry” concept, however attractive it may seem from the standpoint of group theory, is still
a purely classical one. Classical and semi-classical mechanisms have their uses in quantum field
theory: no one will dispute that anomalies are a quantum phenomenon although they can be
described in purely classical terms [75, 76]. For conceptual clarity, nevertheless, one should cling
to root quantum explanations.

One may reckon, furthermore, that on the subject of this paper the panorama has been obscured
by much theoretical prejudice. The Higgs mechanism is burdened with giving masses to all matter
and force fields; a heavy load to carry indeed. Explicit mass terms for the vector bosons of
electroweak theory are said to be forbidden by gauge invariance. It ain’t so: these mass terms can be
accommodated in gauge theory by regarding the “swallowed” Higgs ghosts of lore as Stückelberg
fields. Also it is said that chirality of the fermions and gauge invariance in weak interactions requires
the Higgs mechanism to generate masses by Yukawa couplings. It ain’t so: one can use Dirac masses
for the fermions and derive chirality of couplings from causal gauge invariance [41, 42].

In conclusion, starting from the BRST description for MVB as fundamental objects [27,31,33,
35, 41], we have perturbatively performed a second reality check of the Higgs mechanism in the
spirit of causal gauge invariance, with the outcome that, reversing the dictum by Yang, interaction
dictates symmetry – fixing the models up to minute details.11 That vindicates the conclusions of
the historically first reality check [29, 30] as well. Beyond reestablishing the manifold aspects of
renormalizable gauge theories, the analysis in the path-breaking book [35] has been completed with
the causal derivation of minimal coupling. This allows now for a reliable list of renormalizable
couplings in BRST invariant models. The contention that there might be contradiction between
causal gauge invariance and some GUT models [46] has been laid to rest.

11A role for MVB as sources of symmetry, with very different intent, is found in [77].

33



Appendices

A On the Standard Model in CGI

Postulating four gauge bosons, one of which is massless, and one physical scalar, one is unerringly
led by CGI [41] to U(2) symmetry12 and the “phenomenological” boson sector of the SM. The
only alternatives allowed by CGI are limits of the SM in which one, three or all of the vector
particles decouple. In standard presentations the U(2) symmetry is said to be “broken”, among
other reasons, because there is only one conserved quantity, electric charge, instead of four. But
from our viewpoint symmetry is broken at the level of the free Lagrangian, due to different masses
(the residual equality of two masses reflects conservation of electric charge). This is to say that
there is a natural basis of the Lie algebra linked to the pattern of masses. The role of the constraint
(20) is precisely to pick out this basis. Little support comes from this quarter for the idea that the
SM as it stands is “imperfectly unified”.

Now, the CGI conditions can likewise be applied to the fermion sector. As hinted at earlier,
incompatibility of Dirac masses for fermions with gauge symmetry is just another popular miscon-
ception. The basic fermion-vector-boson vertices between carriers and matter in a gauge theory are
written:

𝜅(𝑏𝑎 �̄� /𝐴𝑎𝜓 + 𝑏′𝑎 �̄� /𝐴𝑎𝛾
5𝜓)

à la Bjorken and Drell, with �̄� the Dirac adjoint spinor and 𝑏, 𝑏′ appropriate coefficients. Taking for
the fermions the known ones – see [41,42] and [35, Sect. 4.7] – first-order gauge invariance already
determines some couplings: in particular, the photon has no axial vector couplings “because” there
is no Stückelberg field for it. At second order, contractions between the corresponding fermionic 𝑄-
vertex and the bosonic 𝐿1 and between the bosonic 𝑄-vertex and 𝐿𝐹

1 determine the matter couplings
completely (contractions between 𝐿𝐹

1 and its 𝑄-vertex contribute nothing). It is beautiful to behold
that couplings of the physical scalar to fermions are proportional to their mass, and that chirality of
the interactions need not to be brought from the outside, but is a consequence of CGI. As usual,
CGI at third order for tree graphs fixes the higgs potential [42]. Since the causal version of the
SM leads to the same phenomenological Lagrangian, excepting only that the vacuum expectation
value of the higgs field is zero, there is no way within pure particle physics to tell it apart from the
ordinary version. However, all the above springs just from the BRST treatment for free spin one
bosons and causal renormalization theory. This stands our approach in good stead in the face of
breakdown of symmetry.

B Derivation of the second main constraint

The three constraints (25) are crucial in this paper. All of them follow from (8), that is, CGI for
second-order tree diagrams by using the technique explained in subsection 2.2. The third constraint
is derived like the first, except that the roles of the 𝐵- and 𝜑-fields are reversed; and for the first one
no substantial deviation from reference [35] is required. Thus we focus on the second constraint.

12As remarked early on in [78], the true group of the electroweak interaction is U(2), not SU(2) × U(1).

34



Since only those terms of 𝑃𝜈 having a derivative 𝜕𝜈 on a field operator contribute, we list only
such terms:

𝑃𝜈 = 𝜅
(
𝑓𝑎𝑏𝑐 (−𝐴𝜇𝑎𝑢𝑏 𝜕

𝜈𝐴
𝜇
𝑐 + 1

2𝑢𝑎𝑢𝑏 𝜕
𝜈�̃�𝑐) − 2 𝑓 3

𝑎𝑏𝑐 𝑢𝑎𝐵𝑏 𝜕
𝜈𝐵𝑐

− 𝑓 5
𝑎𝑏𝑝 𝑢𝑎 (𝐵𝑏 𝜕

𝜈𝜑𝑝 − 𝜕𝜈𝐵𝑏 𝜑𝑝) − 2 𝑓 6
𝑎𝑝𝑞 𝑢𝑎𝜑𝑝 𝜕

𝜈𝜑𝑞 + · · ·
)
.

See (4.3.17) in [35] in this respect. The 𝑘-th term in this expression will be called 𝑃𝑘 (𝑘 = 1, . . . , 6)
henceforth. We omit the notation for normal ordering.

The second constraint is obtained from (8) by equating the coefficients of 𝛿(𝑥 − 𝑦) 𝑢𝑎𝑢𝑏�̃�𝑑𝜑𝑝.
A type (i) term 𝑁2 = 𝐶𝑎𝑏𝑑𝑝 𝛿 𝐵𝑎𝑢𝑏�̃�𝑑𝜑𝑝 would contribute. However, as indicated in subsec-

tion 4.2, there are no quartic terms containing ghost fields 𝑢, �̃�. Also, there are no type (ii) or
type (iv) terms, since the contributions of these terms are ∼ (𝜕𝜙1)𝜙2𝜙3𝜙4 and not ∼𝜙1𝜙2𝜙3𝜙4.

The following type (iii) terms contribute. For the contraction of 𝜕�̃� in 𝑃2 with 𝑢 in 𝐿8
1, we must

be careful with the sign, because of the many Fermi operators. In the region 𝑥0 > 𝑦0 we obtain

T2(𝑃𝜈
2 (𝑥) 𝐿

8
1(𝑦)) = 𝑃𝜈

2 (𝑥) 𝐿
8
1(𝑦) ∼ :𝑢𝑎𝑢𝑏 𝜕𝜈�̃�𝑐 (𝑥): :�̃�𝑑𝑢𝑐′𝜑𝑝 (𝑦):

= −𝑖 𝛿𝑐𝑐′ 𝜕𝜈𝑥Δ+
𝑚 (𝑥 − 𝑦) :𝑢𝑎 (𝑥)𝑢𝑏 (𝑥)�̃�𝑑 (𝑦)𝜑𝑝 (𝑦): + · · ·

and for 𝑦0 > 𝑥0,

T2(𝑃𝜈
2 (𝑥) 𝐿

8
1(𝑦)) = 𝐿8

1(𝑦) 𝑃
𝜈
2 (𝑥) ∼ :�̃�𝑑𝑢𝑐′𝜑𝑝 (𝑦): :𝑢𝑎𝑢𝑏 𝜕𝜈�̃�𝑐 (𝑥):

= −𝑖 𝛿𝑐𝑐′ 𝜕𝜈𝑥Δ+
𝑚 (𝑦 − 𝑥) :𝑢𝑎 (𝑥)𝑢𝑏 (𝑥)�̃�𝑑 (𝑦)𝜑𝑝 (𝑦): + · · ·

by using Wick’s theorem. Together these give a term

∼ −𝑖 𝜕𝜈𝑥
(
Δ+
𝑚 (𝑥 − 𝑦)𝜃 (𝑥0 − 𝑦0) + Δ+

𝑚 (𝑦 − 𝑥)𝜃 (𝑦0 − 𝑥0)
)
= −𝑖𝜕𝜈Δ𝐹

𝑚 (𝑥 − 𝑦).

On computing the divergence 𝜕𝑥𝜈 and adding the term with 𝑥 and 𝑦 exchanged, we find the contribution

−𝑖 𝑓𝑎𝑏𝑐 𝑓 8
𝑑𝑐𝑝 𝑢𝑎 (𝑥)𝑢𝑏 (𝑥)�̃�𝑑 (𝑥)𝜑𝑝 (𝑥) 𝛿(𝑥 − 𝑦).

Additional terms come from contracting 𝜕𝜑 in 𝑃6 with 𝜑 in 𝐿8
1 and 𝜕𝐵 in 𝑃5 with 𝐵 in 𝐿4

1.
These respectively read

−2𝑖( 𝑓 6
𝑎𝑝𝑣 𝑓

8
𝑑𝑏𝑣 − 𝑓 6

𝑏𝑝𝑣 𝑓
8
𝑑𝑎𝑣) 𝑢𝑎 (𝑥)𝑢𝑏 (𝑥)�̃�𝑑 (𝑥)𝜑𝑝 (𝑥) 𝛿(𝑥 − 𝑦)

and
−𝑖(− 𝑓 5

𝑎𝑘 𝑝 𝑓
4
𝑑𝑏𝑘 + 𝑓 5

𝑏𝑘 𝑝 𝑓
4
𝑑𝑎𝑘 ) 𝑢𝑎 (𝑥)𝑢𝑏 (𝑥)�̃�𝑑 (𝑥)𝜑𝑝 (𝑥) 𝛿(𝑥 − 𝑦),

respectively. On adding all terms and setting the resulting coefficient equal to zero, the second
constraint (25) follows.

C Epistemological second thoughts

Among the motivations of this article was the realization of how relatively poor a reputation SSB
enjoys among knowledgeable philosophers of science. In such quarters it is regarded as a non-
empirical device of little explanatory value. More precisely, Higgs’s argument is rightly seen as
possessing tremendous heuristic value in the context of discovery, but less so in the context of
justification.

35



“As the semi-popular presentations put it, ‘particles get their masses by eating the higgs
field.’ Readers of Scientific American can be satisfied with these just-so stories. But
philosophers of science should not be. For a genuine property like mass cannot be
gained by eating descriptive fluff, which is just what gauge is. (They) should be asking
. . . what is the objective (i.e., gauge invariant) structure of the world corresponding to
the gauge theory presented in the Higgs mechanism?”

This criticism by Earman is quoted in [79], which tries to explore the epistemological meaning of
SSB. Consult as well [80].

A final remark is in order. When constructing via CGI the Higgs potentials 𝑉 , a zero vacuum
expectation value emerges. Making this explicit is however noxious to the Higgs mechanism
interpretation. On which interpretation is preferable, we quote Kibble:

“It is perfectly possible to describe our model without ever introducing the notion
of SSB, merely by writing down the (phenomenological) Lagrangian. Indeed if the
physical world were described by this model, it is to the latter rather than to the former
to which we should be led by experiment. The only advantage of SSB is that it is easier
to understand the appearance of an exact symmetry than an approximate one” [81].

Such honesty is nowadays refreshing. It is all perhaps a matter of taste. Tastes change over time,
though; and to some the works of the “exact” symmetry are uglier than the refusal to deal with
unobservable fields.

Acknowledgments

MD was supported by the Deutsche Forschungsgemeinschaft through the Institutional Strategy of
the University of Göttingen. JMG-B is grateful to Luis J. Boya for calling attention to the paper [79]
and for sound counsel on representations of the classical Lie algebras. He is also indebted to Jean
Zinn-Justin and Marc Henneaux for illuminating conversations on the BRST invariance of SSB
models. His work was supported by DGIID–DGA (grant E24/2). FS thanks Stefan Tapprogge for
valuable information about the experimental Higgs searches. JCV acknowledges support from the
Vicerrectorı́a de Investigación of the University of Costa Rica.

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