In memory of Ivan T. Todorov (1933-2025)

1 Introduction↩︎

It is widely agreed that gauge invariance is extremely successful as a criterium to select classical Lagrangians for the Standard Model interactions. But several questions remain unanswered: Why gauge invariance in the first place? Is there any operational meaning of gauge invariance when gauge transformations do not affect observables? Can the notorious trouble with canonical quantization of gauge potentials (indefinite metric, unphysical degrees of freedom) be avoided?

Ever since Jordan [1] and Dirac [2], physicists and philosophers of science (e.g., [3]–[6]) have worried about these and related questions. Concerning the observable content of classical gauge-invariant Lagrangians, an inspiring discussion (and an extended guide to the literature) can be found in [7], see Remark 14 and Sect. 5.

We want to contribute a new point of view to the discussion. It is “genuinely quantum”, namely, it is ultimately the need for a Hilbert space formulation of the quantum field theory (not addressed in [7]) that forces renormalizable interactions to be what they are.¹ Gauge invariance rather emerges as a (possibly hidden) feature of admissible Hilbert space interactions.

The meaning of “admissible”, and the first sketch of “hidden gauge invariance” in Item 5 below require some explanations of the setup. It proceeds from an “autonomous approach” to particle interactions.

1.0.0.1 1. The autonomous approach.

The “autonomous approach” to the interactions of particles ([9]–[13]) results in a highly constrained method to select admissible interaction densities, to be used in quantum perturbation theory. It explains several pertinent physical features of the Standard Model, where gauge theory falls short, see the quoted literature and Remark 3, Remark 11, and App. 7.

“Admissible” means, first of all, that the interactions densities are renormalizable and are defined on the physical Hilbert space of the particles. (The decisive, most restrictive conditions will be stated below.) For particles of helicity or spin 1, this means that one cannot use local vector potentials: Covariant quantization of massless vector potentials \(A_\mu(x)\) requires state spaces of indefinite metric, while the massive Proca vector field \(B_\mu(x)\) does not admit power-counting renormalizable interactions because its short-distance (UV) fluctuations are too strong (the two-point function has scaling degree 2).

The autonomous approach is committed to quantum principles (Hilbert space, covariance and locality). The Hilbert space and covariance principles provide substitutes for the problematic fields \(A_\mu\) and \(B_\mu\), given by “mildly non-local” (“string-localized”) potentials, see Item 2 below. However, the S-matrix must not depend on this non-locality. This highly nontrivial condition replaces a “gauge principle” to select the admissible interactions.

1.0.0.2 2. String-localization.

String-localized potentials for massless vector bosons (“photons”) and massive vector bosons (MVBs) are defined by the same formula \[\begin{align} \label{AF} A_\mu(c,x) := I_c^\nu(F_{\mu\nu})(x) \equiv \int d^4y \, c^\nu(y)F_{\mu\nu}(x+y), \end{align}\tag{1}\] where \(F_{\mu\nu}\) is the local Maxwell tensor defined on the Fock space over the unitary Wigner representation of helicity \(\pm1\) in the massless case, and \(F_{\mu\nu}:= \partial_\mu B_\nu-\partial_\nu B_\mu\) on the Wigner Fock space of spin 1 in the massive case, satisfying the free field equations. For massive vector bosons, one defines also \[\begin{align} \label{phic} \phi(c,x):= I_c^\nu(B_\nu)(x)\equiv \int d^4y\, c^\nu(y) B_\nu(x+y). \end{align}\tag{2}\] The function \(c^\mu(y)\) is required to satisfy \[\begin{align} \label{pac} \partial_\mu c^\mu(y)=\delta(y), \end{align}\tag{3}\] which entails the identity for the integral transform \(I^\nu_c(f)(x) = \int dy\, c^\nu(y)f(x+y)\) \[\begin{align} \label{Ipa} I_c^\nu(\partial_\nu f) = \partial_\nu I_c^\nu(f) =-f. \end{align}\tag{4}\] The identity (4 ) in turn, together with the field equations \(\partial_\kappa F_{\mu\nu}+cycl.=0\) for the local field tensor in both cases (massless and massive) implies \[\begin{align} \label{AcF} \partial_\mu A_\nu(c,x)-\partial_\nu A_\mu(c,x)= F_{\mu\nu}(x). \end{align}\tag{5}\] In the massive case, it also follows \[\begin{align} \label{paphi} A_\mu(c)=B_\mu+\partial_\mu\phi(c). \end{align}\tag{6}\]

The potentials \(A_\mu(c)\) have UV dimension 1 and serve as Hilbert space counterparts of gauge potentials, admitting power-counting renormalizable interactions² among each other and with scalar fields and spinor currents. We denote by \(L_{\rm int}(c,x)\) the interaction densities.

There are many functions \(c^\mu\) satisfying (3 ), but their supports necessarily extend to infinity. Yet, they may (but need not) be narrow cones, which explains the terminology “string-localized”.

Under infinitesimal changes \(\delta c^\mu\) of the function \(c^\mu\), one has \(\delta_c(F)=\delta_c(B)=0\) and \[\begin{align} \label{deltac} \delta_c(A_\mu(c,x)) = \partial_\mu w(\delta c,x), \quad \delta_c(\phi(c,x))= w(\delta c,x) , \end{align}\tag{7}\] where \(\phi(c)\) exists only if \(m>0\). The explicit form \(w\equiv w(\delta c)= -I^\nu_c(\delta_c A_\nu(c))\) of the “infinitesimal variation field” [12] will never be needed.

1.0.0.3 3. String-localization and string-independence.

String-localization is a problem because locality of the perturbation theory is at stake with a non-local interaction density. This is the origin of the most important constraints on admissible interactions: the resulting S-matrix \[\begin{align} \label{SL} S_{L_{\rm int}(c)}= Te^{i\int dx\, L_{\rm int}(c,x)} \end{align}\tag{8}\] (also with source terms for observable fields) must be independent of the string function \(c^\mu\). We write this postulate as \[\begin{align} \label{SI} \delta_c(S_{L_{\rm int}(c)}) \stackrel!= 0. \qquad \sl (String-independence) \end{align}\tag{9}\] A more ambitious demand is that there exists another manifestly string-independent and local interaction density \(K_{\rm int}\) such that \[\begin{align} \label{SLSK} S_{L_{\rm int}(c)} \stackrel!= S_{K_{\rm int}}. \qquad\sl (Equality of S-matrices) \end{align}\tag{10}\] The postulate (9 ) is truly autonomous: The fields that may appear in \(L_{\rm int}(c)\) are dictated by quantum principles, and the postulate selects those \(L_{\rm int}(c)\) as admissible, for which the string-localization has no effect. In particular, there is no reference to a “gauge principle”.

For (10 ) in contrast, one has in mind a local density \(K_{\rm int}\) to compare with, which may be provided by the standard local (gauge theory) approaches to particle interactions. It is possibly not defined on a Hilbert space (massless case), or non-renormalizable (massive case) or both.

We emphasize that S-matrices (8 ) involve only the interaction density \(L_{\rm int}\) (a Wick polynomial in free quantum fields). A “free Lagrangian” \(L_0\) is not needed, and not present in the autonomous approach, because the free fields are not quantized by “canonical quantization”, but directly constructed from the Wigner representations [15]. (Yet, there will appear some \(L_0\) in the sequel of the paper (notably 4 and Sect. 3), but it is just a tool to simplify the conditions for (9 ) and (10 ), reducing a tedious explicit computation to a simple invariance argument.)

We also emphasize that we analyze (9 ) and (?? ) only at tree-level, since we are interested in necessary conditions to fix the interactions. The validity at loop level remains open, but we expect that it can be imposed as a renormalization condition.

1.0.0.4 4. Obstructions.

The constraints on \(L_{\rm int}(c)\) (and \(K_{\rm int}\)) arising from (9 ) and (10 ) can be analysized recursively order by order in perturbation theory. The basic mechanism is that “total derivatives do not contribute” to the integrals in the perturbative expansion of the S-matrix. The “initial” first-order condition on \(L_{\rm int}(c)=gL_1(c) + \frac{1}{2}g^2L_2(c)+\dots\) (\(g\) is a coupling constant) is therefore that the \(c\)-dependence of \(L_1(c)\) is a total derivative, Eq. (15 ), respectively \(L_1(c)-K_1\) is a total derivative, Eq. (24 ). (For these features, the derivative in \(\delta_c(A)=\partial w\) is instrumental.) The problem is that time-ordering does not commute with derivatives, which results in “obstructions” at higher orders, see Sect. 2. The latter must be cancelled, by adding higher-order interactions \(L_2(c),\dots\), determined (“induced”) by the obstructions that they must cancel. In principle, this might not be possible; but if they exist, the obstructions are said to be “resolved”. Remarkably, several admissible cubic interaction densities \(L_1(c)\) can be identified, describing self-interactions and interactions with scalar fields and spinor currents. Resolvability at second order typically constrains parameters present in \(L_1(c)\), and then \(L_2(c)\) exists. Moreover, higher \(L_n(c)\) (\(n\geq3\)) are not needed to resolve higher order obstructions, and \(L_{\rm int}(c)\) is renormalizable. It also happens that for resolvability, one has to add another piece to \(L_1(c)\), and then proceed. E.g., self-interactions of massive vector bosons necessarily require a coupling to a scalar field.

Going beyond the recursive method, we shall formulate in Sect. 2 new necessary and sufficient conditions for (9 ) and for (10 ) (Prop. 1 and Prop. 2) at all perturbative orders at tree level.

Even more remarkably, when the particle content is specified (including the masses), all the interactions \(L_{\rm int}(c)\) – distinguished by string-independence – exhibit a strong similarity with the known interactions of the Standard Model (electroweak, QCD) [9]. We shall give a precise meaning to “strong similarity” in Sect. 3: namely, there appears a “hidden” relation between \(L_{\rm int}(c)\) and a gauge-invariant classical Lagrangian, as illustrated in Item 5 below.

Together with systematic properties of the obstructions, this relation is responsible that the conditions of Prop. 1 and Prop. 2 are satisfied at all perturbative orders (see Item 6 below and Sect. 3, and Sect. 4 that this structure prevails in all SM interactions).

1.0.0.5 5. Hidden gauge invariance.

To give an idea of “hidden gauge invariance”, we present here a simple example. For details, see Sect. 4, Item 3. Mutatis mutandis, the same structure is realized in several physically important models, including all bosonic SM interactions, see Sect. 4.

The unique autonomous string-localized interaction between one massive vector boson and one massive scalar particle [16], [17] is the string-localized version of the Abelian Higgs model [18]: \[\begin{align} \label{} \notag L_{\rm int}(c) &=& mg A_\mu(c)(B^\mu H +\phi(c)\partial^\mu H) + \large{\frac{g^2}{2}}A_\mu(c)A^\mu(c) (H^2+m^2\phi(c)^2)\\ && - \large{\frac{1}{2}} m_H^2 \big(\large{\frac{g}{m}} H(H^2+m^2\phi(c)^2) + \large{\frac{g^2}{4m^2}}(H^2+m^2\phi(c)^2)^2\big). \end{align}\tag{11}\] Here and everywhere below, Wick ordering is understood but will never be written.

“Hidden gauge invariance” is the identity, valid in the Wick algebra, \[\begin{align} \label{hidden} L_0 + L_{\rm int}(c)= L[A(c),\Phi(c)] \qquad with\quad \Phi(c):=v+H+im\phi(c) \quad (v=\large{\frac{m}{g}}), \end{align}\tag{12}\] where \(L_0=-\frac{1}{4} F_{\mu\nu}F^{\mu\nu} + \frac{1}{2}m^2 B_\mu B^\mu + \frac{1}{2}\partial_\mu H\partial^\mu H -\frac{1}{2}m_H^2 H^2\) is the free Lagrangian of the massive Proca field \(B\) and the Higgs scalar \(H\) (lifted to a Wick polynomial), and \[\begin{align} \label{LAPhi-Intro} L[A,\Phi] =-\large{\frac{1}{4}} F_{\mu\nu}F^{\mu\nu} +\large{\frac{1}{2}} (D_\mu\Phi)^*D^\mu\Phi -\large{\frac{m_H^2}{8v^2}}(\Phi^*\Phi-v^2)^2 \end{align}\tag{13}\] is the (classical) gauge-theoretic Lagrangian of a massless gauge potential \(A\) coupled to a complex scalar field \(\Phi\) with a double-well potential.

\(L[A,\Phi]\) is manifestly gauge-invariant as a functional of \(A\) and \(\Phi\). Thus, the autonomous interaction \(L_{\rm int}(c)\) enjoys a hidden gauge invariance, revealed via (12 ). The massless gauge potential \(A\) in (13 ) needs not be quantized, because in (12 ), the quantum fields \(A(c)\) and \(\Phi(c)\) are inserted into \(L[A,\Phi]\).

The gauge invariance of \(L_0+L_{\rm int}(c)\) is exact and unbroken: The same identity (12 ) would hold with every \(U(1)\)-transform \(A(c)+\partial\alpha\), \(e^{ig\alpha}\Phi(c)\) of \(A(c)\), \(\Phi(c)\), see Remark 7.

1.0.0.6 6. The new role of gauge invariance.

Standing alone, a hidden gauge invariance as in (12 ) would be a “nice-to-have” feature of \(L_{\rm int}(c)\), but useless. Instead, the point is that the postulate of string-independence (9 ) at all perturbative orders can be formulated as the invariance of \(L_0+L_{\rm int}(c)\) under a derivation \(\delta_c+\omega_Q\), where \(\delta_c\) is the string variation and \(\omega_Q\) is a map induced by the obstructions of the model (Prop. 1) – and that this derivation turns out to act like an infinitesimal gauge transformation with string-dependent operator-valued gauge parameters³ on the string-localized fields \(A(c)\) and \(\Phi(c)\) in (analogues of) (12 ). In this way, hidden gauge invariance secures (9 ) at all orders, Prop. 6.

Similarly, Prop. 2 reformulates the equality of S-matrices (10 ) in terms of a field homomorphism \(e^{\omega_U}\), induced by the obstructions of the model (Prop. 2). This transformation takes string-localized fields to gauge transforms (again with string-dependent operator-valued parameters) of local fields in a gauge-invariant \(K_{\rm int}\). In this way, hidden gauge invariance also secures (10 ), Prop. 8.

Hidden gauge invariance and the nontrivial properties of obstruction maps “acting like gauge transformations” as just outlined, will be shown to prevail in all autonomous SM interactions, which consequently provide an equivalent description of the SM.

Gauge invariance, that is never imposed in the autonomous approach, is therefore not a “first principle”, but an emerging feature, that serves as a sufficient condition to make all obstructions resolvable, precisely in the physically interesting cases. An emphasis is that there is no “Higgs mechanism”: not as a theoretical trick in the sense that gauge potentials are quantized as massless, and “turn massive” by spontaneous symmetry breaking; even less as a dynamical physical process. MVBs are massive from the outset, with autonomous renormalizable interactions given in terms of the string-localized quantum potentials (1 ) associated with the Maxwell and Proca fields.

1.0.0.7 7. Plan of the paper.

In Sect. 2, we briefly sketch the “new organization” of obstructions and their cancellation, that leads to the present results, and point out where it departs from the earlier treatments. We then derive the new formulation (Prop. 1) of the condition for string-independence (9 ) in terms of an “obstruction map” \(\omega_Q\). We also present the new formulation (Prop. 2) of the condition for equality of S-matrices (10 ) in terms of another obstruction map \(\omega_U\), but we refer to [19] for the proof.

In Sect. 3, we show how hidden gauge invariance is instrumental to secure (9 ) and (10 ).

In Sect. 4, we address the hidden gauge invariance and ensuing string-independence of all autonomous interactions of the Standard Model.

2 Obstruction maps and the resolution of obstructions↩︎

2.0.0.1 1. Obstructions.

Perturbative QFT proceeds in terms of time-ordered products of interaction densities, which by Wick’s theorem can be expressed in terms of time-ordered correlation functions (propagators) of free fields. Propagators for derivatives of fields can be defined “kinematically”: \(\langle 0\vert T[\partial\varphi(x)\chi(y)]\vert 0\rangle:= \partial^x\langle 0\vert T[\varphi(x)\chi(y)]\vert 0\rangle\) – except when equations of motion allow to express \(\partial\varphi\) in terms of other fields, whose propagators are defined independently. In this case, time-ordering does not commute with derivatives. For Wick polynomials \(Y\) and \(X\), we define the “obstruction”⁴ \[\begin{align} \label{Om} O_\mu(Y(y),X(x)):= \big(T[\partial_\mu Y(y)X(x)] - \partial^y_\mu T[ Y(y)X(x)]\big) \big\vert^{\rm tree}. \end{align}\tag{14}\]

By Wick’s theorem, the difference in the bracket can be expanded into Wick products multiplied with products of \(k\) propagators. The tree-level contribution has only terms with \(k=1\). Since we are interested in necessary conditions on interactions, the tree-level analysis is justified as a first condition, while the extension to \(k>1\) (“loops”) will require renormalization. We adopt here the attitude that the preservation of the structures established at tree level should be imposed as a renormalization condition at loop level, that hopefully might fix infinitely many renormalization parameters at all orders. We shall not delve deeper into this issue.

By the restriction to tree level, obstructions for Wick products can be computed “factorwise” in terms of two-point obstructions \(O_\mu(\varphi(y),\chi(x))\) of linear fields, which can be worked out for the relevant linear fields of a model, see App. 6. Two-point obstructions of local fields can only be derivatives of \(\delta(y-x)\), while for string-localized fields, string-integrals over derivatives of \(\delta(x-y)\) may appear.

We therefore assume that \(O_\mu(Y(y),X(x))\) are known for each model, except that there may be some free parameters in possibly non-kinematic propagators of some linear fields of higher UV dimension (such as the Proca field or the derivative of a scalar field).

Let us now study the appearance of obstructions in perturbation theory. Let \(\boldsymbol{\delta}\) some infinitesimal variation of the linear fields, extended as a derivation to the first-order interaction \(L_1\). Assume there is a vector-valued field \(Q_1^\mu\) such that \(L_1\) and \(Q_1\) satisfy the “initial condition” \[\begin{align} \label{LQ1} \boldsymbol{\delta}L_1(x)=\partial_\mu Q_1^\mu(x). \end{align}\tag{15}\] Then the first order of the S-matrix \(S_{L_{\rm int}}\) is automatically invariant:⁵ \[ig\, \boldsymbol{\delta}\int dx\, L_1(x)= \int dx \, \partial_\mu Q^\mu_1(x) =0.\] However, the second order is not invariant: \[\begin{align} \label{O2} \large{\frac{-g^2}{2}}\boldsymbol{\delta}\iint dy \, dx\, T[L_1(y)L_1(x)] &=& \large{\frac{-g^2}{2}}\iint dy\, dx\, T[\partial^y_\mu Q^\mu_1(y)L_1(x)] + (x\leftrightarrow y) \notag \\ &=& \large{\frac{-g^2}{2}}\int dx\int dy\, O_\mu(Q^\mu_1(y),L_1(x)) + (x\leftrightarrow y)\qquad \end{align}\tag{16}\] after subtraction of an integral over a total derivative. In previous work, we have demanded that this obstruction can be cancelled (“resolved”) by a higher-order interaction \(\frac{1}{2}g^2 L_2\), such that the second-order contribution to \(\boldsymbol{\delta}S_{L_{\rm int}}\) is another derivative of the form \[\begin{align} \label{resold} i O_\mu(Q^\mu_1(y),L_1(x)) + (x\leftrightarrow y) + \boldsymbol{\delta}L_2(x)\cdot \delta(x-y)\stackrel! = \large{\frac{1}{2}}\partial_\mu^xQ_2^\mu(x,y)+(x\leftrightarrow y). \end{align}\tag{17}\] The obstruction would determine \(L_2\) and \(Q_2\) – without a guarantee that they exist.

2.0.0.2 2. The new organization of obstructions.

The pattern for the cancellation of obstructions is not unique. (16 ) is as well cancelled if \[\begin{align} \label{resnew} 2i \int dy\, O_\mu(Q^\mu_1(y),L_1(x)) + \boldsymbol{\delta}L_2(x) \stackrel! = \partial_\mu Q^\mu_2(x). \end{align}\tag{18}\] This has the advantage that terms in \(O_\mu(Q^\mu_1(y),L_1(x))\) that are total \(y\)-derivatives need not be considered. In fact, such terms do appear in some models. With the symmetrized pattern (17 ), they contribute to \(Q_2\), where they may jeopardize the resolvability of the obstruction at the next order. With the integrated pattern (18 ), they are systematically discarded. Moreover, \(Q_n\) defined recursively as in (18 ) will have a single argument \(x\), unlike in (17 ).

Therefore, we adopt the new pattern (18 ) and extend it to higher orders. We define the “integrated obstruction map” for vector-valued Wick polynomials \(Y^\mu\) \[\begin{align} \label{omega} \omega_Y (X)(x) := i \int dy \, O_\mu(Y^\mu(y),X(x)) \end{align}\tag{19}\] on Wick polynomials \(X\), and rewrite (18 ) as \[\begin{align} \label{ILQ2} 2\omega_{Q_1}(L_1)(x) + \boldsymbol{\delta}L_2(x) \stackrel!= \partial_\mu Q^\mu_2(x). \end{align}\tag{20}\]

The following proposition will pave the way to string-independence (when \(\boldsymbol{\delta}=\delta_c\) is the string variation), but is formulated in a context-independent way with an unspecified infinitesimal variation \(\boldsymbol{\delta}\) of the fields.

Proof: Because \(\boldsymbol{\delta}\) is a derivation, we have \[\begin{align} \label{dS} \boldsymbol{\delta}S_{L_{\rm int}} = \boldsymbol{\delta}T [e^{i\int dx\, L_{\rm int}(x)}] &=& i \int dy\, T [\boldsymbol{\delta} L_{\rm int}(y) e^{i\int dx\, L_{\rm int}(x)}]. \end{align}\tag{21}\] Therefore, (?? ) inserted into (21 ) implies \(\boldsymbol{\delta}S_{L_{\rm int}} =0\) by Lemma 3.

Conversely, if \(\boldsymbol{\delta} S_{L_{\rm int}}=0\), then by (21 ) and again Lemma 3, it holds for arbitrary \(Q\) \[\begin{align} \label{XX} i \int dy\, T [(\boldsymbol{\delta}L_{\rm int}+\omega_Q(L_{\rm int}) -\partial_\mu Q^\mu)(y) e^{i\int dx\, L_{\rm int}(x)}] =0. \end{align}\tag{22}\]

Now, expand (22 ) in \(g\). If \(Q\) satisfies (?? ) until order \(g^{n-1}\), the exponential does not contribute to order \(g^n\) of (22 ), which then asserts that \(\boldsymbol{\delta}L_n + \omega_Q(L_{\rm int})^{(n)}\) is a total derivative, where the \(n\)-th order contribution \(\omega_Q(L_{\rm int})^{(n)}\) involves only \(Q_\nu\) (\(\nu<n\)). This re-defines \(Q_n\), so as to satisfy (?? ) at order \(g^n\). Thus, (?? ) implies (?? ) with \(Q\) a power series in \(g\). \(\square\)

The expansion of (?? ) gives back the first- and second-order conditions (15 ) and (20 ), and at third order \[\begin{align} \label{ILQ3} 3 \omega_{Q_2}(L_1) + 3 \omega_{Q_1}(L_2) + \boldsymbol{\delta}L_3 \stackrel != \partial_\mu Q^\mu_3, \end{align}\tag{23}\] etc. These conditions recursively determine \(L_n\) and \(Q_n\) – provided they exist. If they exist for all \(n\), then \(\boldsymbol{\delta}S_{L_{\rm int}}=0\). The point of Prop. 1 is, however, that one can find \(Q\) that solves (?? ) at all orders “in one stroke”, see Remark 10. In particular, proving (?? ) (or (?? ) below) with \(L_3=0\) includes proving (23 ) with \(L_3=0\).

The analogous proof of Prop. 2 (along similar lines) was given in [19]:

In the perturbative expansion of (?? ), the “initial” first-order condition is \[\begin{align} \label{LV1} L_1-K_1=\partial_\mu U^\mu_1. \end{align}\tag{24}\]

We present two simple corollaries to Prop. 1 and Prop. 2, that are interesting because their assumptions turn out to be satisfied in many models that we are going to study, including all bosonic interactions of the Standard Model. It is part of the scheme how hidden gauge invariance secures string-independence and coinciding S-matrices.

Proof: The proof of (i) trivial. For (ii), use that \(F(\omega_U)\circ\omega_U=e^{\omega_U}-\mathrm{id}\) in Prop. 1. \(\square\)

3 Hidden gauge invariance at work↩︎

3.0.0.1 1. String-independence.

The admissible autonomous interactions \(L_{\rm int}(c)\) involving massless and massive vector bosons and their obstruction maps \(\omega_Q\) all enjoy some remarkable properties that we call “hidden gauge invariance”. This is unexpected because \(L_{\rm int}(c)\) were determined by imposing string-independence (9 ) in lowest order of perturbation theory, without assuming any gauge invariance. See the comments in the Conclusions, Sect. 5.

Hidden gauge invariance – if it prevails – is most efficient to prove string-independence at all orders. While the first order (15 ) is an initial condition and the second order obstruction (20 ) has to be resolved by hand in order to determine \(L_{\rm int}(c)=gL_1(c)+\frac{1}{2}g^2L_2(c)\), all higher orders are taken care of by hidden gauge invariance (including the proof that all higher \(L_n(c)=0\)).

We present the argument in the setup of Prop. 1 independent of the nature of the derivation \(\boldsymbol{\delta}\) (the string variation \(\delta_c\) in the case of string-localized QFT (sQFT)). The properties of \(L_{\rm int}\) and \(Q\) consist of several parts:

1. It holds \(\partial_\mu Q^\mu=-(\boldsymbol{\delta}+\omega_Q)(L_0)\) for some quadratic Wick polynomial \(L_0\).

2. There are a classical Lagrangian \(L[A,\Phi]\), and quantum fields \(\widehat A\) and \(\widehat\Phi\) (functions of the fields in \(L_{\rm int}\)) such that it holds\[\begin{align} \label{hGI-LQ} (L_0+L_{\rm int})(x) = L[\widehat A, \widehat\Phi](x). \end{align}\tag{25}\]

3. \(L[A,\Phi]\) is invariant under infinitesimal gauge transformations \(\delta_\lambda\) with arbitrary gauge parameters \(\lambda\).

4. There exist Wick-algebra-valued, field-dependent gauge parameters \(\lambda\) such that \[\begin{align} \label{sGT} (\boldsymbol{\delta}+\omega_Q)(\widehat A) = \delta_{\lambda}(\widehat A) ,\quad (\boldsymbol{\delta}+\omega_Q)(\widehat\Phi) = \delta_{\lambda}(\widehat\Phi). \end{align}\tag{26}\]

Proof: By (P1) and 4(i), (?? ) is equivalent to the condition (?? ). On the other hand, \[(\boldsymbol{\delta}+\omega_Q)(L_0+L_{\rm int}) \stackrel{\mathrm{(P2)}}= (\boldsymbol{\delta}+\omega_Q)(L[\widehat A,\widehat\Phi])\stackrel{\mathrm{(P4)}}= \delta_{\lambda}(L[\widehat A,\widehat\Phi]) \stackrel{\mathrm{(P3)}}=0.\] In the last step, it was used, that gauge invariance of \(L[A,\Phi]\) makes no specific assumptions about the nature of the fields and the gauge parameters. Thus, condition (?? ) is fulfilled. \(\square\)

3.0.0.2 2. Equality of S-matrices.

We now show how hidden gauge invariance secures equality of S-matrices (10 ), where \(L_0\) from (P1) and \(K_{\rm int}\) are functions of \(\boldsymbol{\delta}\)-invariant fields only. (The latter is actually not needed – it only fixes the idea that the interaction \(K_{\rm int}\) to compare with is manifestly \(\boldsymbol{\delta}\)-invariant.) In sQFT, this means that \(L_0\) and \(K_{\rm int}\) are functions of local fields only.

We have to assume that the fields involved in the interactions \(L_{\rm int}\) and \(K_{\rm int}\) are defined on the same Fock space, so that (10 ) is meaningful. In theories with massless vector bosons, the Hilbert space of \(L_{\rm int}\) must be embedded into the indefinite state space of the gauge theory interaction \(K_{\rm int}\). This requires a minor modification of Prop. 2 due the appearance of null fields, cf.[19]. Prop. 8 below can be adapted accordingly. Further subtleties in the massless case arise due to a logarithmic IR divergence. These will be briefly sketched in Remark 11. We ignore these subtleties here and present the argument in a way which is litterally applicable only when there are no massless vector bosons.

On top of (P1)–(P2) (determining \(L_0\) and \(L[A,\Phi]\)), we assume

1. There exists a mediator field \(U^\mu\) such that \(\partial_\mu U^\mu =-\omega_U(L_0)\).

2. There exist (\(\boldsymbol{\delta}\)-invariant) quantum fields \(A_0\) and \(\Phi_0\) such that \(L_0+K_{\rm int}\) can be written as \[\begin{align} \label{hGI-LV} (L_0+K_{\rm int})(x) =L[A_0,\Phi_0](x). \end{align}\tag{27}\]

3. \(L[A,\Phi]\) is invariant under arbitrary finite gauge transformations \(\alpha_\gamma\).

4. There exist Wick-algebra-valued, field-dependent gauge parameters \(\gamma\) such that \[e^{\omega_U}(\widehat A) = \alpha_\gamma(A_0), \quad e^{\omega_U}(\widehat \Phi) = \alpha_\gamma(\Phi_0).\]

Again, these strong assumptions are only justified because they are satisfied for the abelian Higgs model, and with appropriate modifications also for models with massless vector bosons (YM and electroweak).

Proof: By (P5) and 4(ii), (?? ) is equivalent to the condition (?? ). On the other hand, the remaining assumptions and the fact that \(e^{\omega_U}\) is a Wick algebra homomorphism, allow to conclude \[\begin{align} \label{ILQ-LV} e^{\omega_U}(L_0+L_{\rm int}) &\stackrel{\mathrm{(P2)}} =& e^{\omega_U}\big(L[\widehat A,\widehat\Phi]\big) = L\big[e^{\omega_U}(\widehat A), e^{\omega_U}(\widehat\Phi)\big] \stackrel{\mathrm{(P8)}}= L(\alpha_\gamma(A_0),\alpha_\gamma(\Phi_0)) \notag \\ &\stackrel{\mathrm{(P7)}}=& L[A_0,\Phi_0] \stackrel{\mathrm{(P6)}}= L_0+K_{\rm int}. \end{align}\tag{28}\] This is (?? ). Then 4(ii) implies the equality of S-matrices. \(\square\)

4 Standard model interactions↩︎

All Standard Model interactions involve massless particles of helicity \(\pm 1\) (photons, gluons) and/or massive vector bosons of spin 1 (MVBs: \(W\) and \(Z\)-particles).

We begin with the case of Yang-Mills theory (only massless vector bosons). We then turn to the abelian Higgs model (one MVB, no photon), which is instructive for the more general case: the electroweak interactions, treated last.

4.0.0.1 1. Yang-Mills.

Autonomous interaction densities \(L_{\rm int}(c)\) for self-interactions of any finite number of massless particles of helicity \(\pm1\) were classified in [10]. String-independence requires that the vector fields \(A_a(c)\) and their field tensors \(F_a\) form adjoint multiplets of a real Lie algebra \({\mathfrak{g}}\) with completely antisymmetric structure constants \(f_{abc}\).⁶ For adjoint field multiplets, we write \(([X,Y])_a:= \sum_{ab}f_{abc}X_bY_c\), and \(\langle X\vert Y\rangle := \sum_aX_aY_a\), hence \(\langle [X,Y]\vert Z\rangle=\langle X\vert[Y,Z]\rangle\).

The autonomous \(L_{\rm int}(c)\) was found to be (Wick ordering is understood, as always) \[\begin{align} \label{L-YM} L_{\rm int}(c) = \large{\frac{g}{2}} \big\langle F^{\mu\nu}\big\vert [A_\mu(c),A_\nu(c)]\big\rangle-\large{\frac{g^2}{4}} \big\langle [A^\mu(c),A^\nu(c)]\big\vert [A_\mu(c),A_\nu(c)]\big\rangle. \end{align}\tag{29}\] These are the cubic and quadratic terms of the classical YM Lagrangian \[\begin{align} \label{GA} L[A]= -\large{\frac{1}{4}} \langle G_{\mu\nu}[A]\vert G^{\mu\nu}[A] \rangle, \qquad G_{\mu\nu}[A]\equiv\partial_\mu A_\nu-\partial_\nu A_\mu - ig [A_\mu,A_\nu], \end{align}\tag{30}\] with the classical field \(A\) replaced by the quantum field \(A(c)\) in the Wick algebra.

In other words, the “hidden” gauge invariance (properties (P2) and (P3)) is quite manifest with \(L_0 = -\frac{1}{4}\langle F\vert F\rangle\) and \(\widehat A=A(c)\). (A field \(\Phi\) as in (25 ) is not needed for Yang-Mills.) We verify also the properties (P1) and (P4).

(P1) is easy: For any “skew-inert” field \(\rho_\nu\) (i.e., \(O_{[\mu}(\rho_{\nu]}(y),X(x))=0\) for all fields \(X\)), and \[\begin{align} \label{Qrho} Q^\mu = \big\langle F^{\mu\nu}\big\vert\rho_\nu\big\rangle, \end{align}\tag{31}\] it holds ([19]) \[\begin{align} \label{omega-YM} \omega_Q(A_\mu(c)) = \rho_\mu+ \partial_\mu I_c^\nu(\rho_\nu), \quad \omega_Q(F_{\mu\nu}) = \partial_{[\mu}\rho_{\nu]}. \end{align}\tag{32}\] Because \(\delta_c(F)=0\) and \(\partial_\mu F^{\mu\nu}=0\), (P1) follows: \[\begin{align} \label{dQ-YM} (\delta_c+\omega_Q)(L_0) = -\big\langle F^{\mu\nu}\big\vert \partial_\mu\rho_\nu\big\rangle=-\partial_\mu Q^\mu. \end{align}\tag{33}\] Of course, \(\rho_\nu\) must be further specified so that \(Q_\mu\) fulfills also (P4). This is done by

Proof: The proof will show that \(\lambda\) given by (?? ) is unique to satisfy (26 ), while \(\rho_\nu\) is unique only up to a derivative.

\(\lambda\) is inert by the same argument for inertness of \(\gamma\) in [19]; namely, it is legitimate to choose kinematic propagators for the expansion coefficients \(\lambda_{n}\). Then, the skew-inertness of \(\rho_\nu\) follows as in [19].

With (7 ) and (32 ), the claim (26 ) is the assertion that \[\begin{align} \label{pawGT} \delta_c(A_\mu(c)) + \omega_Q(A_\mu(c)) = \partial_\mu w + (\rho_\mu + \partial_\mu I_c(\rho)) &\stackrel!=& \partial_\mu\lambda + g[\lambda,A_\mu(c)], \end{align}\tag{34}\] which implies the second equation in (26 ) because both \(\delta_c\) and \(\omega_Q\) respect the exterior derivative, see (32 ). Applying \(-I_c^\mu\) to (34 ) and using (4 ), gives the implicit equation for \(\lambda\) \[\begin{align} \label{wGT} \lambda -gI_c([\lambda,A(c)]) \stackrel!= w. \end{align}\tag{35}\] (35 ) is solved by the power series \(\lambda\) in (?? ). Plugging (35 ) into (34 ), yields \[\begin{align} \label{rhoGT} (\delta_\mu^\nu + \partial_\mu I_c^\nu)(\rho_\nu) = (\delta_\mu^\nu + \partial_\mu I_c^\nu)(g[\lambda,A_\nu(c)]), \end{align}\tag{36}\] which is obviously solved by \(\rho_\nu\) in (?? ). \(\square\)

By Lemma 1 also (P4) is fulfilled, and by Prop. 6 the S-matrix is automatically string-independent at all orders. For the equality of S-matrices, see Remark 11.

4.0.0.2 2. QCD.

We add a minimal coupling \(\widetilde{L}_{\rm int}(c)= g\langle A_\mu(c)\vert j^\mu\rangle\) to the Yang-Mills interaction, where \((j_a^\mu)_a\) is an adjoint multiplet of conserved quark currents.⁸ Property (P1) is not fulfilled in this case.⁹ Yet, one can establish string-independence directly with Prop. 1.

The abelian case (QED, \(\lambda=w\)) is of course included.

Proof: Because \(L_{\rm int}(c)\) and \(Q\) separately satisfy the condition (?? ), we have to consider only the additional obstructions arising from the fermionic couplings: \[\begin{align} \label{ILQ-QCD} \omega_{\widetilde{Q}}(L_{\rm int}) + \omega_Q(\widetilde{L}_{\rm int}) + \omega_{\widetilde{Q}}(\widetilde{L}_{\rm int}) - \partial_\mu \widetilde{Q}^\mu\stackrel!=0. \end{align}\tag{37}\] Using (32 ), (7 ), and the inertness of \(\lambda\), as well as \(\omega_{\langle \lambda\vert j\rangle}(\langle A(c)\vert j\rangle) = - \langle [\lambda,A(c)]\vert j\rangle\) ([19]), we see that \[\begin{align} \label{lala} g\big\langle \partial_\mu w + \rho_\mu + \partial_\mu I_c(\rho) -\partial_\mu\lambda - [\lambda,A_\mu(c)]\big\vert j^\mu\big\rangle\stackrel!=0 \end{align}\tag{38}\] must vanish. This is true by (35 ). \(\square\)

4.0.0.3 3. The abelian Higgs model.

We present the abelian Higgs model (AHM) as a simple prototype for nontrivial hidden gauge invariance. The general structure is the same in the electroweak interaction (Item 4 below), but the computations are most transparent in the AHM.

Free massive vector bosons are described by the local Proca field \(B_\mu\) satisfying the Klein-Gordon equation with mass \(m\) and \[\begin{align} \label{eom-Proca} \partial_\mu B^\mu=0, \quad \partial_\mu F^{\mu\nu}=-m^2 B_\mu, \quad F_{\mu\nu}=\partial_{[\mu}B_{\nu]}. \end{align}\tag{39}\] The string-localized fields are \(A_\mu(c)\) and \(\phi(c)\) as in Sect. 1 with equations of motion \[\begin{align} \label{eom-AHM} \partial_{[\mu}A_{\nu]}(c)=F_{\mu\nu}, \quad \partial_\mu A^\mu(c) = -m^2 \phi(c), \quad \partial_\mu\phi(c) = A_\mu(c)-B_\mu. \end{align}\tag{40}\] The free Higgs field is a canonical scalar field \(H\) of mass \(m_H\).

\(B\), \(F\), and \(H\), are string-independent local fields, while the string variations of \(A(c)\) and \(\phi(c)\) were given in (7 ).

The autonomous interaction density \(L_{\rm int}(c)\) for the interaction of a single massive vector boson with a scalar was determined in [16], [17]. The improvements of the present paper, outlined in Sect. 2, change the computation in several ways. First, the integration in (19 ) deletes total derivatives in the variable \(y\), while they survived in [17] due to the symmetrization in \(x\) and \(y\). Thus, \(Q_2\) in [17] is now absent. In [17], because the presence of \(Q_2\) would have prevented the resolution of higher-order obstructions, the authors were forced to adopt nontrivial renormalizations of propagators \(\langle 0\vert T[BB]\vert 0\rangle\) and \(\langle 0\vert T[\partial H\partial H]\vert 0\rangle\) (\(c_B=c_H=-1\)). These are now obsolete. Instead, we work with kinematic propagators (\(c_B=c_H=0\)). As a consequence, some extra terms \(L_2^*(c)\) must be included into the second-order interaction \(L_2(c)\). They can also be found in [17].

To conclude: The cubic interaction \(L_1(c)\) and \(Q^\mu=gQ^\mu_1\) are the unique solution to the initial condition (15 ) with the given field content, and the quartic part \(L_2(c)\) (including \(L_2^*(c)\)) is determined by string-independence at second order (20 ) with kinematic propagators. \(Q^\mu\) satisfies (?? ) in the abelian case \(\lambda=w\).

Proof: \(L_0\) is determined by \(Q^\mu\) “in the autonomous way”, i.e., via \(\partial_\mu Q^\mu = - (\delta_c+\omega_Q)(L_0)\) in 4(i) and (P1), as follows: The obstruction map \(\omega_Q(\cdot)\) is computed from the two-point obstructions given in Table ¿tbl:tb:2pt? and (60 ) in App. 6. This gives Table ¿tbl:tb:omQ-AHM?.

\begin{table}[h] \[ \begin{tabular}{@{}l||c|c|c|c|c|c|@{}} & $F$ & $A(c)$ & $B$ & $\phi(c)$ & $H$ & $\pa H$ \\ \hline\hline $ \omega_Q(\cdot)$ \halfquad & \halfquad $0$ \halfquad & \halfquad $0$ \halfquad & \halfquad$-\frac gm \pa (wH)$ \halfquad & \halfquad $\frac gm \, wH$\halfquad & \halfquad$-mg\,w\phi$\halfquad & \halfquad $-mg\, \pa(w\phi)$ \halfquad \mystrut{11}{6} \\ \hline \end{tabular} \]

\end{table}

Then, the equations of motion and Table ¿tbl:tb:omQ-AHM? immediately imply \(\partial_\mu Q^\mu = - \omega_Q(L_0)\) where \(L_0\) given in (?? ). Because \(\delta_c(L_0)=0\), (P1) is fulfilled.

Notice that \(L_0\) is the sum of the free Proca and scalar Lagrangians. Next, let \[\begin{align} \label{LAPhi-AHM} L[A,\Phi] = -\large{\frac{1}{4}}\big\langle F_{\mu\nu}[A]\big\vert F^{\mu\nu}[A]\big\rangle + (D_\mu\Phi)^*D^\mu\Phi - V(\Phi^*\Phi), \end{align}\tag{41}\] where \(A\) and \(\Phi\) are a vector field and a complex scalar field with covariant derivative \(D_\mu\Phi = (\partial_\mu-ig A_\mu)\Phi\), and \(V\) is the double-well potential: \[\begin{align} \label{V} V(\Phi^*\Phi) = \large{\frac{m_H^2}{8v^2}}(\Phi^*\Phi-v^2)^2. \end{align}\tag{42}\] Then, (P2) and (P3) are fulfilled with \[\begin{align} \label{AcPhic-AHM} \widehat A_\mu := A_\mu(c), \quad \widehat\Phi := v+H + im\phi(c), \end{align}\tag{43}\] provided the parameter \(v\) in (41 ) is chosen to be \[\begin{align} \label{} v=\large{\frac{m}{g}}. \end{align}\tag{44}\] Clearly \(L[A,\Phi]\) is invariant under infinitesimal and finite gauge transformations \[\begin{align} \label{GT-ab} \delta_\lambda (A) = \partial\lambda, \quad \delta_\lambda(\Phi) = ig\lambda \,\Phi, \quadresp.\quad \alpha_\gamma (A) = A+\partial\gamma, \quad \alpha_\gamma(\Phi) = e^{ig\gamma}. \end{align}\tag{45}\]

In the verification of (25 ), the crucial detail is the field equation \(\partial\phi(c)=A(c)-B\), see (6 ): \[\begin{align} \label{} \notag (\partial- ig \widehat A)\widehat\Phi &=& \partial H + im \partial\phi(c) -ig A(c) (v+H+im\phi(c)) \\ \notag &=& \partial H + mg A(c)\phi(c)- i\big(m B + g A(c) H \big). \end{align}\tag{46}\] This, together with \(F[\widehat A] = F\) and \(\widehat\Phi^*\widehat\Phi = (v+H)^2+m^2\phi(c)^2\), yields (25 ).

There remains (P4). From Table ¿tbl:tb:omQ-AHM?, we read off the action of \((\delta_c+\omega_Q)(\cdot)\). We compute \[\begin{align} \label{} (\delta_c+\omega_Q)(\widehat A_\mu) = \partial_\mu w, \quad (\delta_c+\omega_Q)(\widehat\Phi) = - mg\,w\phi(c) +im(w+\large{\frac{g}{m}}\,w H) =igw \,\widehat\Phi. \notag \end{align}\tag{47}\] In view of (45 ), (P4) is fulfilled with \(\lambda=w\).

With (P1)–(P4) fulfilled, the S-matrix is string-independent at all orders by Prop. 6. \(\square\)

Recall Remark 7 that the vacuum expectation \(\langle 0\vert\widehat\Phi\vert 0\rangle=v\) does not mean that there is spontaneous symmetry breaking.

Proof: We do not need to know the functions \(\beta(H,\phi)\) and \(\eta(H,\phi)\) to establish (P5). Namely, (?? ) together with Table ¿tbl:tb:2pt? and the inertness of \(\phi\) and \(H\) yields Table ¿tbl:tb:omU-AHM?.

\[\begin{array}{@{}l||c|c|c|c|c|@{}} & A(c) & B & \phi(c) & H & \pa H \\ \hline\hline \omega_U(\cdot) \halfquad & \halfquad 0 \halfquad & \halfquad -m^{-2}\pa \beta(H,\phi(c)) \halfquad & \halfquad m^{-2}\beta(H,\phi(c)) \halfquad & \halfquad -\eta(H,\phi(c)) \halfquad & \halfquad -\pa\eta(H,\phi(c)) \halfquad \mystrut{12}{7} \\ \hline \end{array}\]

Then, the equations of motion and Table ¿tbl:tb:omU-AHM? immediately imply (P5): \[\partial_\mu U^\mu = \partial_\mu \beta \cdot B^\mu +\partial_\mu\eta \cdot \partial^\mu H - \eta\cdot m_H^2H = -\omega_U(L_0) .\]

In the verification of (P6), the crucial part is again the covariant derivative: \[(\partial_\mu-igA_{0,\mu})\Phi_0= (\partial_\mu-igB_\mu)(v+H) = \partial_\mu H - imB_\mu(1+\large{\frac{g}{m}}H).\] (P7) is obviously satisfied. For the nontrivial property (P8), we need a Lemma.

Proof: In view of Table ¿tbl:tb:omU-AHM? and \(v=\frac{m}{g}\), (?? ) becomes the differential equation \[\begin{align} \label{U-diff} e^{m^{-2}\beta(H,\phi) \partial_\phi -\eta(H,\phi)\partial_H}(1+\large{\frac{g}{m}}H + ig\phi) = e^{ig\phi}(1+\large{\frac{g}{m}}H). \end{align}\tag{48}\] At each order \(g^{n+1}\), the left-hand side is \((\frac{1}{m^2}\beta_n\partial_\phi -\eta_n\partial_H)(\frac{1}{m} H+i\phi) \equiv - \frac{1}{m}\eta_n+\frac{i}{m^2}\beta_n\) plus terms involving \(\beta_\nu\) and \(\eta_\nu\) (\(\nu<n\)). This allows to recursively solve (48 ) for \(\eta_{n}\) and \(\beta_n\). \(\square\)

Now, on top of (?? ), \(\omega_U(A(c))=0\) implies \(e^{\omega_U}(\widehat A) = A(c)=B+\partial\phi(c)=A_0+\partial\phi(c)\). Clearly, the right-hand sides are gauge transforms of \(\Phi_0\) and \(A_0\) with \(\lambda = \phi(c)\), that is, (P8).

With (P5)–(P8) fulfilled, the S-matrices of \(L_{\rm int}(c)\) and of \(K_{\rm int}\) automatically coincide (at tree-level) at all orders by Prop. 8. This concludes the proof of Prop. 13. \(\square\)

Notice that \(K_{\rm int}\) is a non-renormalizable interaction. Remember our attitude, expressed in Sect. 2, that the preservation of the properties (P1)–(P8) should be imposed as a renormalization condition at loop level, that would fix infinitely many renormalization parameters at all orders.

4.0.0.4 4. The electroweak interactions.

The most general autonomous interaction density \(L_{\rm int}(c)\) for the self-interaction of any (finite) number of massless and massive vector bosons and a single scalar Higgs particle was determined in [12], by imposing string-independence at first and second order. At third order, some parameters in \(L_1(c)\) and \(L_2(c)\) were fixed, and \(L_3(c)=0\). (Generalizations with several Higgs are possible, but at least one Higgs must must be present.)

Because fields \(B(x)\) and \(\phi(c,x)\) do not exist for the photon, we introduce indices \({\underline a}\) running only over the MVBs. The fields are therefore the local field tensors \(F_{a,\mu\nu}\), the vector potentials \(A_{a,\mu}(c)\) and their variation fields \(w_a\) for all vector bosons in a mass eigenbasis, and the local Proca fields \(B_{{\underline a},\mu}\) and the fields \(\phi_{\underline a}(c)\) for the massive vector bosons (MVBs).

As in Yang-Mills, the self-couplings of vector bosons are described by completely antisymmetric structure constants \(f_{abc}\) of a Lie algebra \({\mathfrak{g}}\) of compact type. We use the same notations as in Item 1, also when the fields may be massive. But the masses lead to further constraints: in particular, the adjoint action \(\mathrm{ad}_{\tau_b}\) of the “massless generators” must “preserve the mass”, i.e., \(m_b=0\) and \(f_{abc}\neq 0\) implies \(m_a=m_c\). As a consequence, the massless generators generate a Lie subalgebra \({\mathfrak{h}}\subset{\mathfrak{g}}\), and \(\mathrm{ad}_{\mathfrak{h}}\) preserves the subspace \({\mathfrak{m}}\subset{\mathfrak{g}}\) spanned by the “massive generators” \(\tau_{\underline a}\).

String independence at second order imposes more constraints, see [12].

For the same reason as explained before Prop. 12 in the abelian Higgs model, one must include terms \(L_2^*(c)\) into \(L_2(c)\) when working with the new condition (18 ). These were displayed in [12]. But the resolvability of the third-order obstruction has to be re-done. While this would be a tedious task to do “by hand”, hidden gauge invariance comes to assist. Namely, we shall show that for the autonomous string-localized interaction \(L_{\rm int}(c) = gL_1(c)+ \frac{1}{2}{g^2}L_2(c)\), there exists \(Q^\mu\), such that the properties (P1)–(P4) are fulfilled, securing the resolution of obstructions at all orders. In particular, this confirms \(L_3(c)=0\).

The electroweak interaction is the unique solution to all constraints with one photon and three MVBs (up to cases where one particle decouples) [12]. We label the vector bosons in a mass eigenbasis by indices \(A,1,2,Z\), where \(1\) and \(2\) are coupled by the photon: \(f_{A12}\neq0\), hence \(m_1=m_2=:m_W\). We may normalize the structure constants such that¹⁴ \[\begin{align} \label{fabc} f_{A12}=\sin\theta, \quad f_{12Z}=\cos\theta \end{align}\tag{49}\] with an angle \(\theta\) to be determined. \(f_{AZi}\) (\(i=1,2\)) must vanish because \(m_W\neq m_Z\), see above.

With this input, the first and second orders of the interaction \(L_{\rm int}(c)\) and \(Q^\mu\) are uniquely determined by imposing (15 ) and (20 ) with kinematic propagators. As it turns out, the inclusion of \(L_2^*(c)\) and the addition of \(L_0\) determined “in the autonomous way” via (P1) simplify the structure of \(L_0+L_{\rm int}(c)\): it becomes a “sum of squares”, of which \(L_0\), \(gL_1(c)\), \(\frac{1}{2}g^2L_2(c)\) are the quadratic, cubic, and quartic terms, respectively: \[\begin{align} \label{Ltot-EW} L_0+L_{\rm int}(c)\! &\!\!\!=\!\!\!&\!-\large{\frac{1}{4}} \big\langle G_{\mu\nu}(c)\big\vert G^{\mu\nu}(c)\big\rangle +\large{\frac{1}{2}}\sum\nolimits_{\underline a}\!\!m_{\underline a}^2 \big(B _{\underline a}+ C_{{\underline a}}[A(c)] \big)^2 \notag \\ && \! +\large{\frac{1}{2}}\big(\partial H + C_H[A(c)]\big)^2 - \large{\frac{1}{2}}m_H^2\big(H + \large{\frac{gK}{2}}(H^2 + \sum\nolimits_{\underline a}\!\! m_{\underline a}^2\phi_{\underline a}(c)^2)\big)^2 , \qquad \quad \end{align}\tag{50}\] where \[\begin{align} \label{Ca} C_{{\underline a},\mu}[A(c)] = \large{\frac{g}{m_{\underline a}^2}}\large{\frac{\partial L_1(c)}{\partial B^\mu_{\underline a}}} &=& g\sum\nolimits_{b{\underline c}}\large{\frac{m_{\underline c}}{m_{\underline a}}}\gamma_{{\underline a} b{\underline c}}A_{b,\mu}(c)\phi_{\underline c}(c) + gK A_{{\underline a},\mu}(c) H, \notag \\ C_{H,\mu}[A(c)] = g\large{\frac{\partial L_1(c)}{\partial(\partial^\mu H)}} &=& gK\sum\nolimits_{\underline a}m_{\underline a}^2A_{{\underline a},\mu}(c)\phi_{\underline a}(c). \end{align}\tag{51}\] The coefficients \(\gamma_{{\underline a}b{\underline c}}\) are only defined for massive \({\underline a}\) and \({\underline c}\): \[\begin{align} \label{ga} \gamma_{{\underline a}b{\underline c}} := \large{\frac{m_{\underline a}^2-m_b^2+m_{\underline c}^2}{2m_{\underline a}m_{\underline c}}} \cdot f_{{\underline a}b{\underline c}} =-\gamma_{{\underline c}b{\underline a}},\qquadin particular\quad \gamma_{{\underline a}b{\underline c}}=f_{{\underline a}b{\underline c}} \quadfor\quad b=A.\quad \end{align}\tag{52}\] Moreover, string independence requires that \(m_Z\geq m_W\) and \[\begin{align} \label{spec} \large{\frac{m_W}{m_Z}}=\cos\theta \quad and \quad K=\large{\frac{1}{2m_W}}. \end{align}\tag{53}\] The masses of the particles fix all parameters in (50 ), (51 ) (except the coupling constant).

\(L_0\) in (50 ) is seen to be the sum of the free Maxwell Lagrangian for the photon, free Proca Lagrangians for the massive vector bosons, and the free Lagrangian of the scalar Higgs particle. The specific form of the quadratic “shifts” (51 ) in (50 ) was “autonomously” fixed by the initial first-order condition (15 ), and already anticipates the hidden gauge invariance, see (55 ).

To exhibit the hidden gauge invariance, it is convenient to define \[\begin{align} \label{ggv} g_1:= \large{\frac{1}{2}}\tan\theta\cdot g, \qquad g_2:= g, \qquad v:= \large{\frac{2m_W}{g}}. \end{align}\tag{54}\]

Proof: We insert into \(L[A,\Phi]\) the string-localized quantum fields \(\widehat A\) and \(\widehat\Phi\). Because the passage from \((\widehat A_i)_i\) to \((A_a(c))_a\) just amounts to an orthogonal change of basis of the Lie algebra \(\mathfrak{u}(1)\oplus\mathfrak{su}(2)\), the term \(-\frac{1}{4}\big\langle G[A(c)]\big\vert G[A(c)]\big\rangle\) equals the first term in (50 ). Because \(\widehat\Phi^*\widehat\Phi-v^2= 2v(H + \frac{1}{2v}(H^2 + \sum_{\underline a}m_{\underline a}^2 \phi_{\underline a}(c)^2)\) and \(\frac{1}{2v}=\frac{g}{2}K\), the term \(-V(\widehat\Phi^*\widehat\Phi)\) equals the last term in (50 ).

Working out the vector-valued doublet \(i\big(g_1\widehat A_0\mathbf{1}+g_2\sum_i\widehat A_i\frac{1}{2}\sigma_i\big)\widehat\Phi\) in (?? ) with the above specifications, we find¹⁵ \[\begin{align} \label{igAPhic-EW} i\large{\frac{g}{2}}\big(\tan\theta \widehat A_0\mathbf{1}+\sum\nolimits_i \widehat A_i\sigma_i\big)\widehat\Phi &=& \begin{pmatrix}m_W(iA_1(c)+A_2(c)) \\ -im_ZA_Z(c) \end{pmatrix}\notag \\ &+& \begin{pmatrix}m_W(i C_1[A(c)]+C_2[A(c)]) \\ C_H[A(c)] - i m_Z C_Z[A(c)]\end{pmatrix}. \quad \end{align}\tag{55}\] Adding \(\partial\widehat\Phi\) to the first term and using (6 ), one sees that the real and imaginary parts of the components of \(D^{[\widehat A]}\widehat\Phi\) are \(m_{\underline a}(B_{\underline a}+C_{\underline a}[A(c)])\) and \(\partial H+C_H[A(c)]\).

Thus, the term \(\large{\frac{1}{2}} \big\vert D^{[\widehat A]}\widehat\Phi\big\vert^2\) equals the second and third terms in (50 ). This proves (P2) and (P3).

In order to verify (P1), we need the equations of motion and Table ¿tbl:tb:omQ-EW?, which follows from (?? ) and Table ¿tbl:tb:2pt?. Thus, we have to work out \[\begin{align} \label{} \notag \partial Q + \omega_Q(L_0) = \large{\frac{\partial Q}{\partial \lambda_a}}\partial\lambda_a + \large{\frac{\partial Q}{\partial F_{\underline a}}}(-m_{\underline a}^2 B_{\underline a}) + \large{\frac{\partial Q}{\partial\phi_{\underline a}(c)}}(A_{\underline a}(c)-B_{\underline a}) + \large{\frac{\partial Q}{\partial H}}\partial H +\large{\frac{\partial Q}{\partial(\partial H)}}(-m_H^2 H) \\ \notag -\large{\frac{1}{2}}\big\langle F\big\vert\partial\wedge\rho\big\rangle + m_{\underline a}^2B_{\underline a} (\rho_{\underline a}-\large{\frac{1}{m_{\underline a}^2}}\partial\big(\large{\frac{\partial Q}{\partial B_{\underline a}}}\big)) + \partial H \big(-\partial\big(\large{\frac{\partial Q}{\partial(\partial H)}}\big)\big) - m_H^2 H \big(-\large{\frac{\partial Q}{\partial(\partial H)}}\big). \end{align}\tag{56}\] There are no contributions to \(\partial Q\) arising from \(\large{\frac{\partial Q}{\partial A}}\sim \eta_{\mu\nu} F^{\mu\nu}=0\) and from \(\large{\frac{\partial Q}{\partial B}}\) because \(\partial B=0\). A straightforward computation yields \(\partial Q + \omega_Q(L_0)=0\).

\[\begin{array}{@{}l||c|c|c|c|c|c|@{}} & F_a & A_a(c) & B_a & \phi_a(c) & H & \pa H \\ \hline\hline \omega_Q(\cdot) \, & \, \pa\wedge\rho_a \, & \, \rho_a + \pa I(\rho_a) \, & \, \rho_a - \frac1{m_a^2}\pa \big(\frac{\pa Q}{\pa B_a}\big)\, & \, I_c(\rho_a) + \frac1{m_a^2} \frac{\pa Q}{\pa B_a}\, & \, - \frac{\pa Q}{\pa(\pa H)}\, & \, -\pa \big(\frac{\pa Q}{\pa(\pa H)}\big)\, \mystrut{14}{8} \\ \hline \end{array}\]

In order to verify (P4), we have to add \(\delta_c(\cdot)\) to the entries of Table ¿tbl:tb:omQ-EW? and work out \[\begin{align} \label{} \notag (\delta_c + \omega_Q)(\widehat A_\mu) &\stackrel!=& \delta_\lambda(\widehat A_\mu) = \partial_\mu\lambda + g[\lambda,\widehat A_\mu], \\ \notag (\delta_c + \omega_Q)(\widehat\Phi) &\stackrel!=& \delta_\lambda(\widehat\Phi) = ig \pi(\lambda)\widehat\Phi = -i\big(g_1\lambda_0\mathbf{1}+\large{\frac{g_2}{2}}\sum\nolimits_i\lambda_i\sigma_i\big)\widehat\Phi, \end{align}\tag{57}\] where \(\pi\) is the vector representation of \(\mathfrak{su}(2)\) and the representations of charge \(-1\) of \(\mathfrak{u}(1)\), and \(\lambda_0=\cos\theta\cdot \lambda_A-\sin\theta\cdot \lambda_Z\), \(\lambda_3=\sin\theta\cdot \lambda_A+\cos\theta\cdot \lambda_Z\).

The first condition is the same as (34 ), solved by (?? ). The right-hand side of the second condition is the same as (55 ) with \(A_a(c)\) replaced by \(\lambda_a\). Thus, it remains to check \[(\delta_c + \omega_Q)\begin{pmatrix}-m_W(i\phi_1(c) +\phi_2(c)) \\ v+H - im_Z\phi_Z(c)\end{pmatrix} \stackrel != -\begin{pmatrix}m_W(i\lambda_1+\lambda_2) \\ -im_Z\lambda_Z \end{pmatrix}- \begin{pmatrix}m_W(i C_1[\lambda]+C_2[\lambda]) \\ C_H[\lambda] - i m_Z C_Z[\lambda]\end{pmatrix},\] which is obviously satisfied by Table ¿tbl:tb:omQ-EW? and (35 ). This proves (P4). With (P1)–(P4) established, the S-matrix is string-independent at all orders by Prop. 6. \(\square\)

One may as well verify (P5)–(P8) with \(A_0=A\) the massless gauge potential and \(\Phi_0=\begin{pmatrix}0\\[-2mm] v+H\end{pmatrix}\). The mediator field \(U\) has a Yang-Mills part, such that \(e^{\omega_U}(A(c)) = \alpha_{\gamma(c)}(A) = e^{i\gamma(c)}(A-ig^{-1}\partial) e^{-i\gamma(c)}\) with \(i\gamma(c):=\sum_a\gamma_a(c)\tau_a\) as in [19], and an MVB part to be determined similarly as in Lemma 2, such that \(e^{\omega_U}(\Phi(c))= \pi(W(c))\Phi(c)\). Then Prop. 8 applies, and consequently, \(S_{L_{\rm int}(c)}=S_{K_{\rm int}}\).

4.0.0.5 5. Quark and lepton couplings: The chirality theorem.

Given \(L_{\rm int}\) as in (50 ), we want to couple fermionic vector and axial currents of the form \(j^\mu=\overline{\psi} \gamma^\mu T\psi\) and \(j^{\mu5}=\overline{\psi} \gamma^\mu\gamma^5T'\psi\), where \(\psi\) is a Fermi multiplet with a diagonal mass matrix \(M\), and \(T\) and \(T'\) are hermitean \(N\times N\) coupling matrices to be determined.

The currents are generally not conserved: \[\begin{align} \label{cnc} \partial_\mu j^\mu(T)= -i S([T,M]) \qquad \partial_\mu j^{\mu5}(T')= i S^5([T',M]_+) \end{align}\tag{58}\] where \(S(X)= \overline{\psi} X\psi\) and \(S^5(X')=\overline{\psi}\gamma^5X'\psi\) are scalar and pseudoscalar fields, and \([\cdot,\cdot]_+\) is the matrix anti-commutator.

For an arbitrary Lie algebra \({\mathfrak{g}}\), the most general coupling separately satisfying the initial first-order condition (15 ) is \[\begin{align} \label{wtL} \notag \widetilde{L}_1(c) &=& \sum\nolimits_a\!\!A_{a,\mu}(c)\big(j^\mu(T_a)+ j^{\mu5}_a(T'_a)\big) -i \sum\nolimits_{\underline a}\!\!\phi_{\underline a}(c)\big(S([T_{\underline a},M])-S^5([T'_{\underline a},M]_+)\big)+ \\ &&+ H\big(S(V)+S^5(V')\big). \end{align}\tag{59}\] Since for “photons” (\(m_a=0\)), \(\phi_a\) do not exist, we have to require \([T_a,M]=0\) and \([T'_a,M]_+=0\) if \(m_a=0\). In particular, photons can couple only to mass-diagonal vector currents and to massless axial currents. For QCD with massless gluons and massive quarks, this means that the couplings are non-chiral (\(T'=0\)) and the quark masses are color-independent.

The string-independent terms in the second line are not necessary for (15 ), but will be needed to cancel higher-oder obstructions, see Prop. 17. Renormalizable higher-order interactions \(\widetilde{L}_n(c)\) (\(n>1\)) do not exist, hence \(\widetilde{L}_{\rm int}(c) = g\widetilde{L}_1(c)\).

For the particle content of the electroweak interactions with the specifications given in Sect. 4, the cancellation of all second-order obstructions was imposed in [11] to determine the matrices \(T_a\), \(T'_a\), \(V\) and \(V'\) for fermion doublets (\(N=2\)), under the a priori assumption (“charge conservation”) that \(T_A\) and \(T_Z\) and \(T'_Z\) are diagonal. In App. 7, we show that this assumption can be dropped, because it also follows from the cancellation conditions.

The main result of [11] is that the coupling of the \(W\)-bosons is necessarily maximally chiral, that is, \(T'_i=\varepsilon T_i\) (\(i=1,2\)) where \(\varepsilon\) must be a sign. (The physical sign is \(\varepsilon=-1\), corresponding to the “\(V-A\)” structure of the weak interaction.) A second result is that \(\widetilde{L}_{\rm int}\) coincides with the local Standard Model interaction \(\widetilde{K}_{\rm int}\) (with indefinite metric) found by gauge theory with spontaneously broken symmetry – up to a total derivative as in (24 ) that “carries away” unphysical degrees of freedom and UV-divergences that make \(\widetilde{K}_{\rm int}\) non-renormalizable [11].

In particular, \(V'=0\) (the Higgs coupling is non-chiral), and \(V=-KM\) corresponds to the proportionality between Yukawa couplings and fermion masses.

In App. 7, we re-prove the chirality result in the new setting. The main point of this re-working is that the cancellation of obstructions at second order as in [11] actually implies (?? ) at all orders, see Remark 10.

5 Conclusion↩︎

We hope to have made a contribution to elucidate the role of gauge invariance – given that gauge transformations do not affect observables, and given the trouble conjured up by canonical quantization of gauge fields. This includes a “quantum answer” to the “classical question” raised by François after an epistemic discussion in [7]:

[\(\ldots\)] it remains to determine what constitutes the proper context of justification for the electroweak theory. [\(\ldots\)] Is there a principle that would make the theory something other than a raw fact?

The question naturally arose when the author had characterized the \(SU(2)\) gauge symmetry of the electroweak interactions as being “artificial”¹⁶ (i.e., devoid of an operational meaning) because it can be “erased” by a local transformation of gauge-dependent fields into gauge-invariant fields; while only the \(U(1)\) gauge symmetry is “substantial”. Our answer is

Yes, there is such a principle. And it is well known: it is the need of a Hilbert space in quantum theory.

Let us quote from [5], pointing out that the “Higgs mechanism” has no explanatory, but an important heuristic value “in the context of discovery”. Lyre writes:

**[\(\ldots\)] it is almost impossible to invent or to discover \(\mathcal{L}_{\rm GSW}'''\) from scratch. It is, instead, more than convenient to have some ‘guiding story’ [\(\ldots\)].

Here, \(\mathcal{L}_{\rm GSW}'''\) refers to the non-renormalizable local Glashow-Salam-Weinberg Lagrangian with the correct masses after SSB. One would not guess \(L_{\rm int}(c)\) in (50 ), either, which yields the same S-matrix. But the recursive method of the autonomous approach via cancellation of obstructions allows to derive it from scratch [11], [12], without a heuristic, but physically misleading “guiding story” (as if SSB were a physical process).

The autonomous approach to particle interactions originally set out as a potential “alternative to gauge theory” [26] altogether. Now, after many improvements, it developped into a more mature scheme in which gauge invariance does plays a role. However, it is “downgraded” from a “principle” that determines the SM interactions, to a property that automatically comes with the autonomous selection of string-localized interactions \(L_{\rm int}(c)\). It signals the consistency of a model with fundamental quantum principles, and conversely can be used to prove consistency. In the presence of massive vector bosons, it is “hidden”.

No classical gauge-invariant Lagrangian has to be quantized. Instead, already quantized free fields in the Wigner representations of physical particles are perturbed by string-localized quantum interactions, that are constrained by the condition that the S-matrix is string-independent. In order that this condition is fulfilled, the (hidden) gauge invariance of \(L_{\rm int}(c)\) is instrumental: it secures invariance under the derivation \(\delta_c+\omega_Q\) (the condition for string-independence), which acts like a (field-dependent) gauge transformation on the string-localized fields. With techniques exemplified for the abelian Higgs model, one can also show that the S-matrices are the same as in local approaches.

Specifically, in the fermionic sector of the electroweak interactions, the autonomous interaction matches the local but non-renormalizable interactions of massive vector bosons with the fermions, up to a total derivative that “carries away” unphysical degrees of freedom and UV-divergences responsible for non-renormalizability.

But the autonomous strategy is a complete converse of the textbook narrative, according to which one must introduce a scalar doublet with an inverted mass term in order to break a postulated chiral gauge symmetry with massless vector bosons. There, the ratio of coupling constants for \(\mathfrak{u}(1)\) and \(\mathfrak{su}(2)\) defines the Weinberg angle, which in turn determines the ratio of the masses that the vector bosons acquire via SSB. In order to give masses to the fermions, Yukawa couplings to the scalar doublet have to be introduced by hand. In contrast, in the autonomous approach, the masses can be prescribed arbitrarily (except that \(m_W\) cannot exceed \(m_Z\)), the free particles are quantized on their Wigner Hilbert spaces, and this input determines a unique string-localized interaction (50 ) (with only one coupling constant) whose S-matrix is string-independent. A scalar doublet never appears (except in the hidden gauge invariance used as a tool to prove string-independence), and chirality of fermions and their Yukawa couplings to the Higgs, proportional to their masses, arise by necessity for string independence.

It is true that many explicit calculations, notably in the electroweak interaction, are very much reminiscent of familiar manipulations of classical Lagrangians. But the autonomous selection criterium is genuinely quantum, referring to the absence of obstructions against string-independence of the quantum S-matrix, that in turn arise through time-ordered quantum two-point functions.

All assertions in this paper are understood to hold only at tree level. This is sufficient to determine the interactions, and to show that they coincide with the SM interactions. For the full perturbation theory, we suggest that the preservation of the structures established at tree level should be imposed as a renormalization condition at loop level, that with the help of [14] might fix infinitely many renormalization parameters at all orders. Except for (unpublished) case studies that reveal auspicious cancellations, this task remains open.

The technical main result of this paper: that an underlying (in the present setting possibly “hidden”) gauge invariance secures the cancellation of obstructions at all orders, is in parallel with an analogous result obtained in the BRST setting [27], where gauge invariance secures the existence of a deformed BRST operator at all orders [28].

6 Basic obstruction theory↩︎

By Wick’s theorem, the obstructions \(O_\mu(Y(y),X(x))\) defined in (14 ) are Wick derivations in both arguments. In particular, acting on linear fields \(\chi\), the obstruction map \(\omega_Y(\cdot)\) defined in (19 ) for vector-valued Wick polynomials \(Y^\mu\) is evaluated “factorwise” in the integrand: \[\begin{align} \label{factorwise} \omega_Y(\chi(x)) = \int d^4y\, \sum\nolimits_\varphi\large{\frac{\partial Y^\mu}{\partial\varphi}} \cdot iO_\mu(\varphi(y),\chi(x)), \end{align}\tag{60}\] where the sum extends over all linear fields in \(Y^\mu\), and it extends to Wick polynomials \(X(x)\) as a derivation: \[\begin{align} \label{deriv} \omega_Y(X(x)) = \sum\nolimits_\chi\large{\frac{\partial X}{\partial\chi}} \cdot \omega_Y(\chi(x)). \end{align}\tag{61}\] Thus, all obstructions can be computed from two-point abstructions (14 ) for linear fields.

The relevant two-point obstructions needed in the paper, computed with kinematic propagators (\(c_B=c_H=c_F=0\)), can be found in [12], [17], [19]: All obstructions \(O(w,\cdot)\), \(O(\phi(c),\cdot)\), \(O(H,\cdot)\) are zero (\(w\), \(\phi\), and \(H\) are “inert”). Obstructions \(O(A(c),\cdot)\) appear only in the antisymmetrized form \(O_{[\mu}(A(c)_{\nu]},\cdot)\), which also vanishes (\(A\) is “skew-inert”). The remaining two-point obstructions are listed in Table ¿tbl:tb:2pt?.

\[\begin{array}{@{}l||c|c|c|c|c|c|@{}} &F^{\ka\la}(x) & A^\ka(c,x) & B^\ka(x) & \phi(c,x) & H(x) &\pa^\ka H(x) \\ \hline\hline \mystrut{12}{7} iO_\mu(F^{\mu\nu}(y),\cdot) & \delta_\nu^{[\ka]}\pa_y^{\la]}\delta_{xy}& ( \delta_\nu^\ka-\pa_y^\ka I_\nu)\delta_{xy} & \delta_\nu^\ka\delta_{xy} & I_\nu\delta_{xy} & 0& 0 \\ \hline \mystrut{12}{7} iO_\mu(B^\mu(y),\cdot) & 0 & 0 & m^{-2}\pa_y^\ka\delta_{xy} & m^{-2}\delta_{xy} & 0& 0 \\ \hline \mystrut{12}{7} iO_\mu(\pa^\mu H(y),\cdot) & 0 & 0 & 0 & 0 & -\delta_{xy} & \pa_y^\ka \delta_{xy} \\ \hline \end{array}\]

The Master Ward Identity [29] for “\(n\)-field obstructions” is an identity at tree-level, and can be imposed as a renormalization condition at loop level. After integration, it becomes¹⁷ \[\begin{align} \label{MWI} \omega_Y(X_1(x_1),\dots,X_n(x_n)) &:=& i\int d^4y\,\big(T[\partial^y_\mu Y^\mu(y)X_1(x_1)\dots X_n(x_n) - \partial^y_\mu T[Y^\mu(y)\ldots]\big) \notag \\ &=& \sum\nolimits_{k=1}^n T[X_1(x_1)\dots \omega_Y(X_k)(x_k)\dots X_n(x_n)]. \end{align}\tag{62}\]

Proof: When the exponential is expanded, the first equality is – term by term – the definition in (62 ), where the integral over a derivative vanishes. The second equality is the resummation of the exponential in the right-hand side of (62 ). \(\square\)

7 Electroweak fermion couplings↩︎

Starting with an arbitrary Lie algebra \({\mathfrak{g}}\) and fermion multiplets of size \(N\), we study the constraints on the fermionic coupling matrices \(T_a^{(\prime)}\), \(V^{(\prime)}\) in (59 ) imposed by the condition (?? ) for \(L_{\rm int}+\widetilde{L}_{\rm int}\) and \(Q + \widetilde{Q}\). As for QCD (see Sect. 4, Item 2), the method using hidden gauge invariance fails, because property (P1) fails. Therefore, we proceed by a direct analysis.

The initial first-order condition (15 ) for \(\widetilde{L}_1(c)\) requires the relations \[\begin{align} \label{TA} [T_a,M]=[T'_a,M]_+=0\qquadif\quad m_a=0 \end{align}\tag{63}\] and determines \(\widetilde{Q}_1\). In view of Remark 10, let \(\lambda\) as in Lemma 1, and \[\begin{align} \label{wtQ} \widetilde{Q}^\mu = g\sum\nolimits_a\lambda_a\large{\frac{\partial L_1(c)}{\partial A_{a,\mu}(c)}}= g\sum\nolimits_a\lambda_a\big(j^\mu(T_a)+ j^{\mu5}(T'_a)\big). \end{align}\tag{64}\] By Prop. 15, the bosonic interaction \(L_{\rm int}\) in (50 ) and \(Q\) in (?? ) separately solve (?? ). Thus, we have to consider only the additional obstructions from the fermionic couplings: \[\begin{align} \label{ILQ-f} \omega_{\widetilde{Q}}(L_{\rm int}) + \omega_Q(\widetilde{L}_{\rm int}) + \omega_{\widetilde{Q}}(\widetilde{L}_{\rm int}) - \partial_\mu \widetilde{Q}^\mu\stackrel!=0. \end{align}\tag{65}\] The first term is zero because \(\lambda\) is inert. \(\omega_Q\) in the second term acts only on \(A\), \(\phi\), and \(H\), as given in Table ¿tbl:tb:omQ-EW?. For \(\omega_{\widetilde{Q}}(\widetilde{L}_{\rm int})\), we need Table ¿tbl:tb:jj? (see [11] and [19]).

\[\begin{array}{@{}l||c|c|c|c|@{}} & j^\mu(X) & j^{\mu5}(X) & S(X) & S^5(X) \\ \hline\hline \mystrut{12}{7} \omega_{\la j(T)} &-i \la j^\mu([T,X])&-i\la j^{5\mu}([T,X])&-i\la S[T,X]& -i\la S^5[T,X]\\ \hline \mystrut{12}{7} \omega_{\la j(T)} &-i g\la j^{5\mu}([T,X]) &-i\la j^\mu([T,X])&i\la S^5[T,X]_+&i\la S[T,X]_+\\ \end{array}\]

Thus, all terms in (65 ) are of the form (inert fields) times \(Aj,Aj^5, HS, HS^5, \phi S,\) or \(\phi S^5\), which must vanish separately. This yields six conditions on top of (63 ): \[\begin{align} \label{}\notag \sum\nolimits_a\big(\rho_a+\partial I(\rho_a) + \partial w_a-\partial\lambda_a\big) j(T_a) -ig\sum\nolimits_{ab}\lambda_a A_b j([T_a,T_b]+[T'_a,T'_b]) =0, \\ \notag \sum\nolimits_a\big(\rho_a+\partial I(\rho_a) + \partial w_a-\partial\lambda_a\big) j^5(T'_a) -ig\sum\nolimits_{ab}\lambda_a A_b j^5([T_a,T'_b]+[T'_a,T_b]) =0, \\ \notag \sum\nolimits_a i\lambda_a H S\big(-K [T_a,M]- [T_a,V]+[T'_a,V']_+\big) =0, \\ \notag \sum\nolimits_a i\lambda_a H S^5\big(K[T'_a,M]_+ +[T'_a,V]_+ - [T_a,V']\big) =0, \\ \notag \sum\nolimits_{{\underline a}b}\lambda_b\phi_{\underline a}S\big(\sum\nolimits_{\underline c}\large{\frac{im_{\underline a}}{m_{\underline c}}} \gamma_{{\underline a}b{\underline c}} [T_c,M] -Km_{\underline a}^2\delta_{{\underline a}b}V - [T_b,[T_{\underline a},M]]+[T'_b,[T'_{\underline a},M]_+]_+\big)=0, \\ \notag \sum\nolimits_{{\underline a}b}\lambda_b\phi_{\underline a}S^5\big(-\sum\nolimits_{\underline c}\large{\frac{im_{\underline a}}{m_{\underline c}}} \gamma_{{\underline a}b{\underline c}} [T'_{\underline c},M]_+ -Km_{\underline a}^2\delta_{{\underline a}b}V' + [T_b,[T'_{\underline a},M]_+]+[T'_b,[T_{\underline a},M]]_+\big)=0, \end{align}\tag{66}\] where terms involving underlined indices \({\underline a}\) are to be suppressed if \(m_a=0\).

In the last two lines, we have omitted a term with prefactor \(I(\rho_a)+w_a-\lambda_a\), which vanishes by (35 ). Likewise by (34 ), in the first two lines we have \(\rho_a+\partial I(\rho_a) + \partial w_a-\partial\lambda_a = gf_{abc}\lambda_b A_c\).

After these preparatory cancellations, all remaining obstructions are of the form \(\lambda\) times \(j^{(5)}\) or \(HS^{(5)}\) or \(\phi S^{(5)}\). They exactly coincide with the second-order obstructions, with just \(w\) replaced by \(\lambda(g)=w+O(g)\), as announced in Remark 10. In particular, string-independence at second order already implies the same at all orders.

The conditions can be re-written as matrix relations: \[\begin{align} \tag{67} i\big([T_a,T_b]+ [T'_a,T'_b]\big) &=& \sum\nolimits_c f_{abc} \cdot T_c, \\ \tag{68} i\big([T_a,T'_b]+[T'_a,T_b]\big) &=& \sum\nolimits_c f_{abc}\cdot T'_c, \\ \tag{69} - [T_a,V]+[T_a',V']_+ &=& K[T_a,M], \\ \tag{70} - [T'_a,V]_++[T_a,V'] &=& K[T'_a,M]_+,\qquad \\ \tag{71} \sum\nolimits_{\underline c}\large{\frac{im_{\underline a}}{m_{\underline c}}} \gamma_{{\underline a}b{\underline c}} [T_{\underline c},M] -[T_b,[T_{\underline a},M]]-[T'_b,[T'_{\underline a},M]_+]_+&=& Km_{\underline a}^2\delta_{{\underline a}b}V, \\ \tag{72} \sum\nolimits_{\underline c}\large{\frac{im_{\underline a}}{m_{\underline c}}} \gamma_{{\underline a}b{\underline c}} [T'_{\underline c},M]_+ - [T_b,[T'_{\underline a},M]_+]-[T'_b,[T_{\underline a},M]]_+&=& -Km_{\underline a}^2\delta_{{\underline a}b}V'. \end{align}\] (67 ) \(\pm\) (68 ) assert that both \(T_a^\pm:= T_a\pm T'_a\) are representations of the Lie algebra \({\mathfrak{g}}\): \[\begin{align} \label{pipma}i[T^\pm_a,T^\pm_b] = \sum\nolimits_c f_{abc} \cdot T^\pm_c \qquad\Leftrightarrow\qquad iT_a^\pm = \pi^\pm(\tau_a). \end{align}\tag{73}\] (69 ) \(\pm\) (70 ) can be written as intertwining relations between \(\pi^+\) and \(\pi^-\): \[\begin{align} \label{intertwine} (KM+V\pm V')\pi^\pm(\tau)=\pi^\mp(\tau)(KM+V\pm V'). \end{align}\tag{74}\]

We now specify \({\mathfrak{g}}=\mathfrak{u}(1)\oplus\mathfrak{su}(2)\) for the electroweak interactions with structure constants \(f_{A12}=\sin\theta\), \(f_{12Z}=\cos\theta\) both \(\neq0\). We consider Fermi doublets, \(N=2\), in a mass eigenbasis with masses \(\mu_2>\mu_1\). (The case of equal masses \(\mu_1=\mu_2\) requires a separate analysis.)

Let \(T^\pm_0:=\cos\theta \,T_A^\pm\,-\sin\theta\, T_Z^\pm=-i\pi^\pm(\tau_0)\) and \(T^\pm_3:=\sin\theta \,T_A^\pm+\cos\theta\, T_Z^\pm\). Then \[\begin{align} \label{pipmi} T_0^\pm = -i\pi^\pm (\tau_0),\qquad T_i^\pm=-\large{\frac{1}{2}}\pi^\pm(\sigma_i) \qquad (i=1,2,3), \end{align}\tag{75}\] where \(\tau_0\) is the (imaginary) generator of \(\mathfrak{u}(1)\). \(\pi^+\) and \(\pi^-\) are (each) either trivial on \(\mathfrak{su}(2)\) or unitarily equivalent to the identical representation. In the latter case, \(T^\pm_0\) is a multiple of \(\mathbf{1}\).

The next Lemma asserts that “charge conservation” in the interaction vertices, that was assumed a priori in [11], is actually a consequence of the autonomous approach that does not assume any kind of a priori symmetry, including charge conservation.

Proof: \([T_A,M]=0\) in (63 ) implies that \(T_A\) is diagonal. If \(\mu_1>0\), then \([T'_A,M]_+=0\) implies \(T'_A=0\). If \(\mu_1=0\), it implies that \(T'_A\) is a multiple of \(P^\perp\).

If \(\pi^+\) and \(\pi^-\) are both nontrivial on \(\mathfrak{su}(2)\), then both \(T^\pm_0\) must be multiples of \(\mathbf{1}\). If, say, \(\pi^+\) is nontrivial and \(\pi^-\) is trivial on \(\mathfrak{su}(2)\), then \(T^+_0\) is a multiple of \(\mathbf{1}\) and \(T^-_3=0\). If \(\pi^+\) and \(\pi^-\) are both trivial on \(\mathfrak{su}(2)\), then both \(T^\pm_3=0\). In each case, we have four linearly independent combinations of \(T_A\), \(T_A'\), \(T_Z\), and \(T'_Z\) that are diagonal.

The last statement follows from (71 ) and (72 ) with \(a=b=Z\). \(\square\)

With the assumption of “charge conservation” fulfilled by Lemma 4, there is no need to repeat the detailed determination of the coupling matrices \(T_a\) and \(T'_a\) done in [11]. Nevertheless, we want to reproduce here the central results of [11], but in a more “algebraic” way.

When we discard the physically uninteresting case that the W-bosons do not couple to fermions, the next Proposition is the autonomous prediction of maximal chirality of the weak interactions [11]: it is another necessary condition for string-independence.

The condition \(\mu_1\neq\mu_2\) is crucial. If \(\mu_1=\mu_2\), Lemma 4 does not hold. Then, \(T'_a=0\) for all \(a\) and \(V=V'=0\) is a solution, and \(\pi^+=\pi^-\) may be nontrivial on \(\mathfrak{su}(2)\).

Proof of Prop. 16: Assume that \(\pi^\pm\) are both nontrivial on \(\mathfrak{su}(2)\). Then \(T^\pm_0\) are both multiples of \(\mathbf{1}\), hence \(T_0=\cos\theta \,T_A-\sin\theta\, T_Z\) and \(T'_0=\cos\theta \,T'_A-\sin\theta\, T'_Z\) are multiples of \(\mathbf{1}\).

By Lemma 4, \(T^\pm_3\) are diagonal, hence both multiples \(\frac{1}{2}\sigma_3\), but may differ by a sign: \(T_3^+=\pm T_3^-\).

We first exclude the case \(T_3^+=T_3^-\). Then \(T'_3=0\), hence along with \(T'_A\), also \(T'_Z\) and \(T'_0\) would be multiples of \(P^\perp\). But \(T'_0\) is a multiple of \(\mathbf{1}\), hence \(T'_0=0\), hence also \(T'_A=T'_Z=0\).

Because \(T_Z\) is diagonal and \(T'_Z=0\), (71 ) and (72 ) with \(a=b=Z\) would imply that \(V=V'=0\). But this conflicts with (74 ) because \(M\) is not a multiple of a unitary.

The case \(T_3^+=-T_3^-\), hence \(T_3=0\) and \(T'_3=\pm \frac{1}{2}\sigma_3\), is easily excluded if \(\mu_1>0\): Namely, if \(T'_A=0\) and \(T'_0\) is a multiple of \(\mathbf{1}\), then also \(T'_3\) must be a multiple of \(\mathbf{1}\).

If \(\mu_1=0\), the conditions \(T'_0=a\mathbf{1}\) and \(T'_A=bP^\perp\) determine \(T'_Z=\pm\frac{1}{2\cos\theta}\begin{pmatrix}\cos2\theta &\\[-2mm]&1\end{pmatrix}\). On the other hand, the intertwiner in (74 ) is diagonal by Lemma 4, and cannot switch the sign of \(\sigma_3\), unless \(KM+V=V'=0\). The insertion of \(T'_Z\) and \(V=-KM\) into (71 ) with \(a=b=Z\) gives a numerical contradiction with \(K=\frac{1}{2m_W}\) from (53 ).

Finally, if both representations \(\pi^\pm\) are trivial on \(\mathfrak{su}(2)\), then \(T_i=T'_i=0\) (\(i=1,2,3\)). \(\square\)

String-independence does not decide whether \(\pi^+\) or \(\pi^-\) is nontrivial on \(\mathfrak{su}(2)\). Nature chooses \(\pi^-\) (the “\(V-A\)” structure of the weak interaction).

The vanishing of \(V'\) (Lemma 4) means that the Higgs couples non-chirally to the fermions. \(S(V)=-KS(M)\) asserts that the Yukawa couplings are proportional to the Fermi masses.

Proof of Prop. 17: Because \(\pi^+\) and \(\pi^-\) are inequivalent as representations of \(\mathfrak{su}(2)\), (74 ) implies \(KM+V\pm V'=0\). \(\square\)

Just for the sake of completeness, we present the final result of [11]: \(T'_A=0\) because the neutrino was assumed massive, cf. (63 ), and \[T_i=-T'_i = -\large{\frac{1}{4}} \sigma_i \quad (i=1,2), \quad T_A = \begin{pmatrix}0& \\ &s \end{pmatrix}, \quad T_Z= \large{\frac{1}{4c}}\begin{pmatrix}-1 & \\ & 1-4s^2 \end{pmatrix}, \quad T'_Z= \large{\frac{1}{4c}}\sigma_3,\] where \(s\equiv \sin\theta\), \(c\equiv \cos\theta\). Conditions (67 )–(72 ) can be verified one by one by elementary matrix calculus. The coupling matrix \(T_A\) asserts that the electric unit of charge is \(e=g\,\sin\theta\). The eigenvalues of \(T^\pm_0=\cos\theta\, T_A-\sin\theta\, T^\pm_Z\) are the hypercharges times \(\frac{g_1}{g}=\frac{1}{2}\tan\theta\). The displayed solution is unique, except that one may add a multiple of \(\mathbf{1}\) to \(T_A\), Thus, also quark doublets are covered.

Acknowledgments. We thank J. Mund, J.M. Gracia-Bondía and B. Schroer for valuable comments on earlier versions of the paper.

References↩︎

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1. The answer given in [8], based on the observation that a covariant Lorentz transformation law for a gauge potential in the Wigner Hilbert space of helicity \(\pm1\) fails by a derivative term, so that the QED interaction \(A_\mu j^\mu\) is Lorentz invariant only up to a total derivative, has quite some similarity to our more comprehensive answer.↩︎

2. That power-counting is applicable as a criterium for renormalizability of string-localized interactions, is a consequence of work by Gass [14].↩︎

3. Strictly speaking, such transformations should not be called “gauge transformations”. See [7].↩︎

4. When \(Y=j^\mu\) is a conserved current, the obstruction means the violation of a Ward identity.↩︎

5. Throughout, we assume sufficiently rapid decay of the integrands such that the integral over a total derivative vanishes. This assumption is true for all \(L(c)\) and \(Q\) of interest, which have UV scaling dimension \(\geq 4\) and \(\geq 3\), respectively.↩︎

6. This means that \({\mathfrak{g}}\) is the Lie algebra of a subproup of some unitary matrix group. The generators \(\tau_a\) are anti-selfadjoint matrices. Our conventions for \(\mathfrak{u}(1)\) and for \(\mathfrak{su}(2)\) will be \(i\tau_0=\mathbf{1}\) and \(i\tau_j=\frac{1}{2} \sigma_j\), such that \(f_{ijk}=\varepsilon_{ijk}\).
   In [19], we rather used a convention with self-adjoint generators \(\tau'_a=-i\tau_a\) such that \(i[\tau'_a,\tau'_b]=\sum_cf_{abc}\tau'_c\).
   The present reformulation in terms of adjoint field multiplets avoids the somewhat weird notion of “Lie-algebra-valued quantum fields” \(X = \sum_a \tau'_a X_a\) in [19].↩︎

7. However, it differs from the gauge transformation in [19], whose action on the string-localized potential is induced from the gauge transformation \(\delta_\lambda(A)=\partial\lambda+g[\lambda,A]\) of the local gauge potential \(A\) via \(A(c)=A+\partial I_c(A)\).↩︎

8. With a general ansatz for currents \(j^\mu_a = \overline{\psi} \gamma^\mu T_a\psi\) coupled to massless vector bosons, string-independence at first and second order requires that the matrices \(T_a\) must commute with the fermionic mass matrix (i.e., quark masses are color-independent) and satisfy \(i[T_a,T_b]=\sum_cf_{abc}T_c\), hence \(iT_a=\pi(\tau_a)\) are a representation of the Lie algebra, see App. 7. Consequently, \(j_a\) form an adjoint multiplet.↩︎

9. This would require \(\omega_{Q+\widetilde{Q}}(\widetilde{L}_0) = -g\langle \partial\lambda\vert j\rangle\). The “natural” candidate for \(\widetilde{L}_0\) would be the free Dirac Lagrangian, but that one vanishes as a quantum field. 4 seems to apply only for bosonic models.↩︎

10. We don’t have a general proof of this feature. But for general arguments for the first and second orders, see [12], where the argument for \(Q_2\) has to be adapted to the integrated reformulation of the present paper: in particular, with kinematic propagators, the term \(Q_2\vert_\delta\) in [12] is absent. (The prefactor of the sum in [12] is a typo. It should be \(+2\).)↩︎

11. The term \(-\frac{1}{4} FF\) is not needed because it is separately invariant under \(\delta_c\) and \(\omega_Q\), and separately gauge invariant. We include it here for the sake of congruence with nonabelian models, where it is not invariant under \(\omega_Q\).↩︎

12. The couplings of \(\frac{m^2}{2}B^2\) to \((1+\frac{g}{m} H)^2-1\) in (?? ) and to \(1-(1+\frac{g}{m} H)^{-2}\) in [17] are related by a renormalization group transformation taking \(c_B=0\) to \(c_B=-1\). The argument in [17] can be extended to all orders.↩︎

13. The real and imaginary parts of (?? ) give the dressings of \(H\) and \(\phi(c)\) separately. They were first conjectured in [24] by the perturbative evaluation of string-independence until \(g^5\). Prop. 13 shows the conjecture to be correct at all orders.
    Observe that (?? ) takes complex polar to Cartesian coordinates. It could be an appealing challenge to analytically understand the interpolating flow \(e^{t\omega_U}\) between polar and Cartesian, and the field \(U^\mu\) as its generator.↩︎

14. (49 ) defines \(\theta\). The present convention differs from the normalization in [12] by a factor of \(2\).↩︎

15. The first term arises from \(\begin{pmatrix}0\\[-2mm]v\end{pmatrix}\) in \(\widehat\Phi\). The “strange” mass factors in \(\gamma_{{\underline a}b {\underline c}}\) (52 ) appearing in \(C_{\underline a}\) in the second term arise from the Weinberg rotation. E.g., the matrix \(g_1 A_0(c)\mathbf{1}+ \frac{g_2}{2}A_3(c)\sigma_3\) has the \(1\)-\(1\)-component \(\frac{g}{2}[2\sin\theta A_A+\frac{2\cos^2\theta-1}{\cos\theta}A_Z] =g[\sin\theta A_A+\frac{2m_W^2-m_Z^2}{2m_W^2}\cos\theta A_Z]=-g [\gamma^A_{12} A_A+\gamma^Z_{12} A_Z]\).↩︎

16. See also [5], [18], [25] and Remark 14.↩︎

17. Without integration, there is a subtlety when there are derivatives of \(\delta\)-functions in the two-point obstructions. This is detailed in [29], see also [24]. The subtlety is ineffective after integration, because an integration by parts in (19 ) precisely takes care of it.↩︎