Veltman, renormalizability, calculability

In his talk at Higgs Hunting $2015$ ¹¹1 https://indico.ijclab.in2p3.fr/event/2722/contributions/5996/ Martinus Veltman said

It is clear that he meant “strictly renormalizable” but it is evident that the main emphasis is on Observable results can be calculated and compared with experiment. In order to understand the situation before $1971$, it is interesting to observe [1] that most of the papers on the subject came to the same conclusion “ $\dots\,$ it is concluded that all theories based on simple Lie groups are unrenormalizable”. All difficulties in the Yang-Mills theories disappeared in $1971$ and the theories became fully renormalizable, that is all occurring infinities could be absorbed in the available free parameters. Theories with a Yang-Mills structure were now renormalizable theories and a precise model for the weak interactions existed already (although it had received virtually no attention). The phase of “precise calculations” started, extending the pioneering work of Berends, Gastmans and collaborators [2, 3, 4] to the full Standard Model [5, 6, 7, 8, 9] and continued till the most recent successes [10, 11, 12, 13]. For a complete list of references, see Ref. [14].

To summarize a long journey, we can say that the transition was from the Fermi theory to the standard model (SM); for that, Veltman [15, 16] had to convince the community that the weak interactions were some form of a Yang-Mills theory. As he wrote: I could not have done that without the knowledge of experimental physics that I had acquired at CERN. It is worth noting that the journey never contemplated the extension of the Fermi theory with the inclusion of even higher operators; therefore, the SM represented the beyond-Fermi physics. Comparing with the present: we are now in a beyond-SM desert looking for alternative paths, although the regulative ideal of an ultimate theory remains a powerful aesthetic ingredient.

Almost equally important, and a landmark through the whole region of “strict renormalizability”, is the work on quantum gravity: “In case of gravitation interacting with scalar particles, divergencies in physical quantities remain” [17, 1]. It is possible that at some very large energy scale, all nonrenormalizable interactions disappear. This seems unlikely, given the difficulty with gravity. It is possible that the rules change drastically. It may even be possible that there is no end, simply more and more scales [18].

The last step in the renormalization procedure is the connection between renormalized quantities and physical observables. Since all quantities at this stage are UV-free, we term it finite renormalization. Note that the absorption of UV divergencies into local counterterms is, to some extent, a trivial step (except for the problem of overlapping divergencies [19]); finite renormalization, instead, requires more attention. For example, beyond one loop one cannot use on-shell masses but only complex poles for all unstable particles [20, 21, 22, 23]. The complete formulation of finite renormalization is beyond the goal of this work. However, let us show some examples where the concept of an on-shell mass can be employed. Suppose that we renormalize a physical observable $\mathrm{F}$,

$$\mathrm{F}=\mathrm{F}_{{\scriptscriptstyle{\mathrm{B}}}}+g^{2}\,\mathrm{F}_{1{\scriptscriptstyle{\mathrm{L}}}}(\mathrm{M}^{2})+g^{4}\,\mathrm{F}_{2{\scriptscriptstyle{\mathrm{L}}}}(\mathrm{M}^{2})\hskip 1.42262pt,$$

(1)

where $M$ is some renormalized mass which appears at one and two loops in $\mathrm{F}_{1{\scriptscriptstyle{\mathrm{L}}}}$ and $\mathrm{F}_{2{\scriptscriptstyle{\mathrm{L}}}}$ but does not show up in the Born term $\mathrm{F}_{{\scriptscriptstyle{\mathrm{B}}}}$. In this case we can use the concept of an on-shell mass identifying $M=M_{{\scriptscriptstyle{\mathrm{O}}}{\scriptscriptstyle{\mathrm{S}}}}$ for the two-loop term and performing a finite mass renormalization at one loop,

$$\mathrm{M}^{2}=\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{O}}}{\scriptscriptstyle{\mathrm{S}}}}\,\Bigl{\{}1\,+\,\frac{g^{2}}{16\,\pi^{2}}\,\Bigl{[}\,\mathrm{Re}\,\Sigma^{(1)}_{{\scriptscriptstyle{\mathrm{M}}}}\mid_{p^{2}=-\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{O}}}{\scriptscriptstyle{\mathrm{S}}}}}-\,\delta Z^{(1)}_{{\scriptscriptstyle{\mathrm{M}}}}\Bigr{]}\,\Bigr{\}}=\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{O}}}{\scriptscriptstyle{\mathrm{S}}}}+g^{2}\,\Delta\mathrm{M}^{2}\hskip 1.42262pt,$$

(2)

where $M_{{\scriptscriptstyle{\mathrm{O}}}{\scriptscriptstyle{\mathrm{S}}}}$ is the on-shell mass and $\Sigma$ is extracted from the required one-particle irreducible Green function. It is worth noting that Eq.(2) is still meaningful (no dependence on gauge parameters) and will be used inside the one-loop result,

$$\mathrm{F}=\mathrm{F}_{{\scriptscriptstyle{\mathrm{B}}}}+g^{2}\,\mathrm{F}_{1{\scriptscriptstyle{\mathrm{L}}}}(\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{O}}}{\scriptscriptstyle{\mathrm{S}}}})+g^{4}\,\Bigl{[}\mathrm{F}_{2{\scriptscriptstyle{\mathrm{L}}}}(\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{O}}}{\scriptscriptstyle{\mathrm{S}}}})+\mathrm{F}^{\prime}_{1{\scriptscriptstyle{\mathrm{L}}}}(\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{O}}}{\scriptscriptstyle{\mathrm{S}}}})\,\Delta\mathrm{M}^{2}\Bigr{]},$$

(3)

where

$$\mathrm{F}^{\prime}_{1{\scriptscriptstyle{\mathrm{L}}}}(\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{O}}}{\scriptscriptstyle{\mathrm{S}}}})\,=\,\frac{\partial\mathrm{F}_{1{\scriptscriptstyle{\mathrm{L}}}}(\mathrm{M}^{2})}{\partial\mathrm{M}^{2}}|_{\mathrm{M}^{2}=\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{O}}}{\scriptscriptstyle{\mathrm{S}}}}}\hskip 1.42262pt.$$

(4)

If we focus on renormalization, we can safely state that all the necessary ingredients are available. Here the crucial point is to connect a set of input experimental data (an input-parameter set, hereafter IPS) to the free parameters of the theory:

• –

  mass renormalization involves the calculation of self-energies;

• –

  renormalization of coupling constants requires additional elements, which depend on the choice of the observables in the IPS.

The most-obvious selection of an IPS is based on the choice of those data which are known with the best experimental precision, e.g. the electromagnetic coupling constant, the Fermi coupling constant and the $\mathrm{Z}\,-$boson mass.

Before the advent of the LEP operations we had very few options for the IPS [24, 25]. To give an example, we consider a first approximation where the lowest order expressions are compared with a set of low-energy data points, electric charge, neutral currents and $\mu$-decay:

$$g^{2}=4\,\pi\alpha(0)\hskip 1.42262pt,\qquad\mathrm{M}^{2}=\frac{g^{2}\sqrt{2}}{8\mathrm{G}_{\mathrm{F}}}\hskip 1.42262pt,\qquad\mathrm{s}_{{}_{\theta}}^{2}\quad{\mbox{from}}\quad R=\frac{\sigma_{{\overline{\nu}}e}}{\sigma_{\nu e}}\hskip 1.42262pt.$$

(5)

The values for $g^{2},\mathrm{M}^{2}$ and $\mathrm{s}_{{}_{\theta}}^{2}$, correct in the lowest order, will subsequently be used in the expressions for the radiative corrections. In the next order we replace

$$g^{2}\to g^{2}\left(1+\delta g^{2}\right)\hskip 1.42262pt,\qquad\mathrm{M}^{2}\to\mathrm{M}^{2}\left(1+\delta\mathrm{M}^{2}\right)\hskip 1.42262pt,\qquad\mathrm{s}_{{}_{\theta}}^{2}\to\mathrm{s}_{{}_{\theta}}^{2}\left(1+\delta\mathrm{s}_{{}_{\theta}}^{2}\right)\hskip 1.42262pt.$$

(6)

The counter-terms are chosen to compensate precisely for the radiative corrections for $e$–$\mu$ scattering, $\mu$-decay and the ratio $R$. Having determined these quantities we may proceed to making predictions. The ratio $R$ is $R\left(\mathrm{s}_{{}_{\theta}}\right)$. We use the fact that it does not depend on $g^{2}$ or $\mathrm{M}^{2}$ at lowest order and at zero energy and momentum transfer. Thus

$$R_{0}\left(\mathrm{s}_{{}_{\theta}}^{2}+\mathrm{s}_{{}_{\theta}}^{2}\delta\mathrm{s}_{{}_{\theta}}^{2}\right)=R_{0}\left(\mathrm{s}_{{}_{\theta}}^{2}\right)+R^{\prime}_{0}\,\mathrm{s}_{{}_{\theta}}^{2}\delta\mathrm{s}_{{}_{\theta}}^{2}\hskip 1.42262pt,$$

(7)

with $R^{\prime}=dR/d\mathrm{s}_{{}_{\theta}}^{2}$. The one-loop radiative corrections to $R$ will be some $R_{1}$, then $\delta\mathrm{s}_{{}_{\theta}}^{2}$ is fixed by

$$\delta\mathrm{s}_{{}_{\theta}}^{2}=-\frac{R_{1}}{R^{\prime}_{0}\mathrm{s}_{{}_{\theta}}^{2}}\hskip 1.42262pt.$$

(8)

The rest is standard and gives $e^{4}\to e^{4}(\,1+\delta e^{4})$ and

$$\delta g^{2}=\frac{1}{2}\delta e^{4}-\delta\mathrm{s}_{{}_{\theta}}^{2}\hskip 1.42262pt,\qquad\delta\mathrm{M}^{2}=\frac{1}{2}\left(\delta_{\mu}-\delta^{\rm em}_{\mu}\right)+\delta g^{2}\hskip 1.42262pt.$$

(9)

Let us denote by $\mathrm{s}^{2}_{\nu e}$ the low-energy weak mixing angle defined through some $R_{{\rm exp}}$. Then we can derive masses for the vector bosons in the low-energy convention. They are given by

$$\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{W}}}}=\frac{\pi\alpha}{\sqrt{2}\mathrm{G}_{\mathrm{F}}\mathrm{s}^{2}_{\nu e}}\hskip 1.42262pt,\qquad\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}=\frac{\pi\alpha}{\sqrt{2}\mathrm{G}_{\mathrm{F}}\mathrm{s}^{2}_{\nu e}c^{2}_{\nu e}}\hskip 1.42262pt.$$

(10)

Let us consider the amplitude for $\overline{\nu}_{\mu}e^{-}\to\overline{\nu}_{\mu}e^{-}$. In real life many different contributions should be considered, but to illustrate some of the relevant points in the procedure it is enough to limit the calculation to contributions from the heavy quark doublet, ($t$–$b$), to the $\mathrm{Z}\mathrm{Z}$ and $\mathrm{Z}\gamma$ transitions.

In this case, we obtain

$$A\left(\overline{\nu}_{\mu}e^{-}\to\overline{\nu}_{\mu}e^{-}\right)=\left(\frac{ig}{4\,\mathrm{c}_{{}_{\theta}}}\right)^{2}\gamma^{\mu}\,\gamma_{+}\otimes\gamma_{\mu}\,\bigl{(}a+b\,\gamma^{5}\bigr{)}\hskip 1.42262pt,$$

(11)

where $\gamma^{\alpha}\otimes\gamma^{\alpha}={\overline{\nu}}_{\mu}\gamma^{\alpha}\overline{\nu}_{\mu}{\overline{e}}\gamma^{\alpha}e$, etc. and

$$a=\Bigl{[}4\,\mathrm{s}_{{}_{\theta}}^{2}-1-\frac{g^{2}\mathrm{s}_{{}_{\theta}}^{2}}{4\,\pi^{2}}\,\frac{\Sigma_{{\scriptscriptstyle{\mathrm{Z}}}\gamma}(p^{2})}{p^{2}}\Bigr{]}\,\Delta_{{\scriptscriptstyle{\mathrm{Z}}}}(p^{2})\hskip 1.42262pt,\qquad b=-\Delta_{{\scriptscriptstyle{\mathrm{Z}}}}(p^{2})\hskip 1.42262pt,$$

(12)

with a propagator $\Delta_{{\scriptscriptstyle{\mathrm{Z}}}}$ given by

$$\Delta^{-1}_{{\scriptscriptstyle{\mathrm{Z}}}}(p^{2})=p^{2}+\mathrm{M}^{2}_{{}_{0}}-\frac{g^{2}}{16\,\pi^{2}\mathrm{c}_{{}_{\theta}}^{2}}\,\Sigma_{{\scriptscriptstyle{\mathrm{Z}}}{\scriptscriptstyle{\mathrm{Z}}}}(p^{2})\hskip 1.42262pt,$$

(13)

where $\Sigma_{{\scriptscriptstyle{\mathrm{Z}}}{\scriptscriptstyle{\mathrm{Z}}}}$ is the $\mathrm{Z}\,$-boson self-energy and $\Sigma_{{\scriptscriptstyle{\mathrm{Z}}}\gamma}$ is the corresponding transition. Then the total cross-section $\sigma_{\nu e}$ can be computed and the data point $R=\sigma_{{\overline{\nu}}e}/\sigma_{\nu e}$ used:

$$R=\frac{\xi_{\nu e}^{2}-\xi_{\nu e}+1}{\xi_{\nu e}^{2}+\xi_{\nu e}+1}\hskip 1.42262pt,\qquad\xi_{\nu e}=\frac{a}{b}\hskip 1.42262pt,$$

(14)

where we assume the approximation of zero momentum transfer. Subtracting the terms involving UV poles and introducing a mass scale $\mu$ we define the counter-term for $\mathrm{s}_{{}_{\theta}}$ and fix ${\mathrm{s}_{{}_{\theta}}}^{\mbox{$\overline{\scriptscriptstyle MS}$}}$ to first order in $\alpha$. The whole renormalization procedure amounts to throwing away infinities. If we subtract the terms involving UV poles then the $\overline{MS}$ redefinition of the parameters is obtained (including $\mathrm{s}_{{}_{\theta}}$), but we could as well assign any finite value to $1/{\bar{\varepsilon}}$ (to be defined in Eq.(20)) and check for the independence of the physical quantities of $1/{\bar{\varepsilon}}$. From the $\mathrm{t}$–$\mathrm{b}$ quark doublet we obtain

	$\displaystyle{\mathrm{s}_{{}_{\theta}}^{2}}^{\mbox{$\overline{\scriptscriptstyle MS}$}}$	$\displaystyle=$	$\displaystyle\mathrm{s}^{2}_{\nu e}+\frac{\alpha}{12\,\pi}\,\Bigl{[}2\,\left(1-\frac{8}{3}\,\mathrm{s}^{2}_{\nu e}\right)\,\ln\frac{m^{2}_{t}}{\mu^{2}}+\left(1-\frac{4}{3}\,\mathrm{s}^{2}_{\nu e}\right)\,\ln\frac{m^{2}_{b}}{\mu^{2}}\Bigr{]}\hskip 1.42262pt,$
	$\displaystyle\mathrm{s}^{2}_{\nu e}$	$\displaystyle=$	$\displaystyle\frac{1-\xi_{\nu e}}{4}\hskip 1.42262pt.$		(15)

As expected, there are no terms quadratic in the quark masses.

At this point we are ready to make predictions. Starting from $p^{2}=0$ we can introduce an effective $p^{2}$-dependent weak mixing angle, etc.

There is an important lesson to learn: there are Lagrangian parameters (e.g. $\mathrm{s}_{{}_{\theta}}$), input parameters (e.g. $\xi_{\nu e}$) and predictions (or “pseudo-observables” ). In the SM there is no one-to-one correspondence between Lagrangian parameters and input parameters. It was different in the older days of QED where only two Lagrangian parameters are present, $e$ and $\mathrm{m}_{\mathrm{e}}$. In this case one uses $\mathrm{m}_{\exp}\,(1+\Delta\mathrm{m}_{e})$ reflecting some vage intuition about the physical meaning of the bare mass. The strategy to prescribe precisely what a Lagrangian parameter is offers a problem when there is no unique experimental quantity that can play the role of defining the parameter.

At LEP $\mathrm{s}^{2}_{\nu e}$ will be replaced by a $\sin\theta^{\mathrm{f}}_{{\mbox{\scriptsize eff}}}$, related to the vector and axial couplings of the $\mathrm{Z}\,$-boson.

One way to explain renormalization is to say that infinities are unobservable and can thus be absorbed into the parameters of the Lagrangian. What about potentially large effects in the renormalized theory? When are they observable? For instance the $m^{2}_{\mathrm{t}}\,$-terms at one-loop, which are there and show up in physical observables. Conversely, the $\mathrm{M}_{{\scriptscriptstyle{\mathrm{H}}}}$ dependence of one-loop radiative corrections was another seminal contribution of Veltman and it was described by the screening theorem [26, 27]: the one-loop $\mathrm{M}_{{\scriptscriptstyle{\mathrm{H}}}}$-dependence in physical observables is only logarithmic. Terms proportional to $\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{H}}}}$ are unobservable at the one-loop level (they start at two loops) and can be absorbed into the parameters of the SM Lagrangian, apart from the case of one-loop diagrams with external (on-shell) Higgs-lines. The $\rho\,$-parameter [28, 29, 30] was born.

A comment is needed for the “original” $\rho\,$-parameter, defined as the ratio of the $\mathrm{W}$ and $\mathrm{Z}$ masses squared divided by $\mathrm{c}_{{}_{\theta}}^{2}$. We start by introducing a bare quantity: $\rho_{0}=\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{W}}}}/(\mathrm{c}_{{}_{\theta}}^{2}\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}})$. The quantities appearing in this relation must be related to experimental data. There is no ambiguity in what is meant by an experimental mass. There remains the experimental $\mathrm{c}_{{}_{\theta}}^{2}$ which should be extracted from data (at the time, low-energy data). In the original formulation, where everything was extracted from low-energy data $(p^{2}=0)$, a $\rho$ was introduced as

$$\rho=1+\frac{g^{2}}{16\,\pi^{2}\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{W}}}}}\,\Bigl{[}\Sigma_{{\scriptscriptstyle{\mathrm{W}}}{\scriptscriptstyle{\mathrm{W}}}}(0)-\Sigma_{{\scriptscriptstyle{\mathrm{Z}}}{\scriptscriptstyle{\mathrm{Z}}}}(0)\Bigr{]}=1+\frac{g^{2}}{16\,\pi^{2}}\,\Delta\rho\hskip 1.42262pt,$$

(16)

where the $1$ in the r.h.s. of the equation is actually $\rho_{\rm bare}$ in the SM. The $\rho$-parameter of Eq.(16) is finite and numerically very close to the experimental one, in any scheme where the counterterms are prescribed. The main correction is due to the top quark mass. Usually we do not attach any particular relevance to bare parameters, only the renormalized Lagrangian predicts meaningful — measurable — quantities. However, the $\rho$-parameter of the SM plays a special role. It is finite because of a residual symmetry which is nothing but the usual isospin invariance. Individual components in Eq.(16) are by themselves infinite, but the combination occurring in this equation for the $\rho$-parameter is finite, as it should be.

The Nobel Prize in Physics $1999$ was awarded “for elucidating the quantum structure of electroweak interactions in physics”’ which was crucial for LEP. At LEP we had the SM with one missing ingredient, therefore the strategy was: test the SM hypothesis versus $\mathrm{M}_{{\scriptscriptstyle{\mathrm{H}}}}$, introduce Pseudo-Observables (generalizations of the $\rho$ parameter [26]), fit them and derive limits on the Higgs boson mass [31].

Ideally, the strategy should have been to combine the results of the LEP experiments at the level of the measured cross-sections and asymmetries - a goal that has never been achieved because of the intrinsic complexity, given the large number of measurements with different cuts and the complicated structure of the experimental covariance matrices relating their errors. This reflects computing limitations at the time.

What the experimenters did [31] was just collapsing (and/or transforming) some “primordial quantities” (say number of observed events in some pre-defined set-up) into some “secondary quantities” (the POs) which are closer to the theoretical description of the phenomena. The practical attitude of the experiments was to stay with a fit from “primordial quantities” to POs (with a SM remnant) for each experiment, and these sets of POs were averaged. The result of this procedure are best values for POs. The extraction of Lagrangian parameters, was based on the LEP-averaged POs. The PO-strategy was made possible thanks to high-precision QED calculations and tools, from the pioneering work of Ref. [32](for an update see Ref. [33]) up to the Monte Carlos that were instrumental at LEP in extracting the realistic observables; for instance KKMC, the most advanced MC for $2\,$-fermion production [34] (including second order QED with resummation, initial-final state interference and spin polarizations, for an update see Ref. [35]).

After the LHC Run 1, the SM has been completed, raising its status to that of a full theory [36]. Despite its successes, this SM has shortcomings vis-à-vis cosmological observations. At the same time, there is presently a lack of direct evidence for new physics phenomena at the accelerator energy frontier. From this state of affairs arises the need for a consistent theoretical framework in which deviations from the SM predictions can be calculated.

Theoretical physics suffers from some inherent difficulties: great successes during the $2$0th century, increasing difficulties to do better, as the easier problems get solved. The lesson of experiments from $1973$ to today is that it is extremely difficult to find a flaw in the SM ²²2Note however the long-standing tension between experiment and SM prediction in the anomalous magnetic moment of the muon, recently reaffirmed by the Fermilab experiment, T. Albahri et al. Muon $g-2$, Phys. Rev. Accel. Beams (2021).: maybe the SM includes elements of a truly fundamental theory. The conventional vision is: some very different physics occurs at Planck scale, the SM is just an effective field theory (EFT). What about the next SM? A new weakly-coupled renormalizable model? A tower of EFTs? Of course there is a different vision: is the SM close to a fundamental theory?

We need a consistent theoretical framework in which deviations from the SM (or next-SM) predictions can be calculated. Such a framework should be applicable to comprehensively describe measurements in all sectors of particle physics: LHC Higgs measurements, past electroweak precision data (EWPD), etc. Here we outline the strategy:

• •

  Consider the SM augmented with the inclusion of higher dimensional operators (say theory $\mathrm{T}_{1}$); not strictly renormalizable. Although workable to all orders, $\mathrm{T}_{1}$ fails above a certain scale, $\Lambda_{1}$.

• •

  Consider any beyond-standard-model (BSM) model that is strictly renormalizable and respects unitarity ($\mathrm{T}_{2}$); its parameters can be fixed by comparison with data, while masses of heavy states are presently unknown. $\mathrm{T}_{1}\not=\mathrm{T}_{2}$ in the UV but they must have the same IR behavior.

• •

  Consider now the whole set of data below $\Lambda_{1}$. $\mathrm{T}_{1}$ should be able to explain them by fitting Wilson coefficients, $\mathrm{T}_{2}$ adjusting the masses of heavy states (as SM did with the Higgs mass at LEP) should be able to explain the data. Goodness of both explanations is crucial in understanding how well they match and how reasonable is to use $\mathrm{T}_{1}$ instead of the full $\mathrm{T}_{2}$.

• •

  Does $\mathrm{T}_{2}$ explain everything? Certainly not, but it should be able to explain something more than $\mathrm{T}_{1}$.

• •

  We could now define $\mathrm{T}_{3}$ as $\mathrm{T}_{2}$ augmented with (its own) higher dimensional operators; it is valid up to a scale $\Lambda_{2}$.

• •

  Continue.

This prompts the important question whether there is a last fundamental theory in this tower of EFTs that supersede each other with rising energies ³³3Kuhlmann, Meinard, “Quantum Field Theory”, The Stanford Encyclopedia of Philosophy (Winter 2018Edition), Edward N. Zalta (ed.). Some people conjecture that this deeper theory could be a string theory, i.e. a theory which is not a field theory any more. Or should one ultimately expect from physics theories that they are only valid as approximations and in a limited domain? To summarize: experiments occur at finite energy and measure $\mathrm{S}^{\mathrm{eff}}(\Lambda)$ (an effective $\mathrm{S}\,$-matrix); whatever QFT should give low energy $\mathrm{S}^{\mathrm{eff}}(\Lambda)\,,\;\forall\,\Lambda<\infty$, i.e. there is no fundamental scale above which $\mathrm{S}^{\mathrm{eff}}(\Lambda)$ is not defined. However $\mathrm{S}^{\mathrm{eff}}(\Lambda)$ loses its predictive power if a process at $E=\Lambda$ requires an infinite number of renormalized parameters [37, 38, 39].

To summarize: before LHC we had the SM, a weakly coupled, strictly renormalizable (a theory with $n$ Lagrangian parameters, requiring $n$ data points, requiring $n$ calculations; the $(n+1)\,$th calculation is a prediction) theory with one unknown, $\mathrm{M}_{{\scriptscriptstyle{\mathrm{H}}}}$. The strategy was: test data against predictions vs. $\mathrm{M}_{{\scriptscriptstyle{\mathrm{H}}}}$. At LHC, after the discovery, with a lack of direct evidence for new physics phenomena we have all the ingredients required to asses (in)consistency of the SM against data.

We briefly review the SMEFT Lagrangian [40, 41, 42]: consider the standard model, described by a Lagrangian $\mathcal{L}^{(4)}_{\mathrm{\scriptscriptstyle{SM}}}$ with a symmetry group $\mathrm{G}=SU(3)\,\times\,SU(2)\,\times\,U(1)$. The SMEFT extension is described by a Lagrangian

$$\mathcal{L}_{\mathrm{\scriptscriptstyle{SMEFT}}}=\mathcal{L}^{(4)}_{\mathrm{\scriptscriptstyle{SM}}}+\sum_{d>4}\,\sum_{i}\,\frac{\mathrm{a}^{d}_{i}}{\Lambda^{4-d}}\,\mathcal{Q}^{(d)}_{i}\hskip 1.42262pt,$$

(17)

where $\Lambda$ is the cutoff of the effective theory, $\mathrm{a}^{d}_{i}$ are Wilson coefficients and $\mathcal{Q}^{(d)}_{i}$ are $\mathrm{G}\,$-invariant operators of mass-dimension $d$ involving the $\mathcal{L}^{(4)}_{\mathrm{\scriptscriptstyle{SM}}}$ fields. In this work we will use the so-called “Warsaw basis” [43].

Unconventional approach to EFT: derivative-coupled field theories are known to develop ghosts [44]. The EFT option [45] replaces the original $\mathcal{L}$ with some $\mathcal{L}_{{\mbox{\scriptsize eff}}}$ truncated at some order in the $\Lambda\,$-expansion; the “dangerous” terms are substituted by using the equations of motion where, for instance, we neglect terms of $\mathcal{O}(\Lambda^{-2})$. We assume that $\mathcal{L}_{{\mbox{\scriptsize eff}}}$ will be replaced by a well-behaved $\mathcal{L}^{\prime}$ at some larger scale, therefore justifying a truncated perturbative expansion; the EFT does not have ghosts while remaining within its regime of validity.

A renormalizable theory is determined by a fixed number of parameters; once these are determined (after finite renormalization) we can make definite predictions at a fixed accuracy. An EFT theory requires at higher and higher energies more and more counterterms; the asymptotic expansion in $\mathrm{E}/\Lambda$ may break down completely above some scale. Given a truncated expansion, we still have a large family of UV-complete theories with these low order terms, which have different behavior at higher energies. Note that the notion of UV completion adopted here [48] is the claim that a theory is “formally” predictive up to all (possible) high energies, but we do not include the additional criterion that the theory be a final, unified “theory of everything”.

Therefore, it is crucial to prove that our EFT is closed under renormalization, order-by-order in the asymptotic expansion, although the number of counterterms will grow with the order (as mentioned above, the predictive power is lost at scales approaching the cutoff). For any given process the amplitude can be written as follows:

$\displaystyle\mathrm{A}$

$\displaystyle=$

$\displaystyle\sum_{n=\mathrm{N}}^{\infty}\,g^{n}\,\mathrm{A}^{(4)}_{n}+\sum_{n=\mathrm{N}_{6}}^{\infty}\,\sum_{l=1}^{n}\,\sum_{k=1}^{\infty}\,g^{n}\,g^{l}_{4+2\,k}\,\mathrm{A}^{(4+2\,k)}_{n\,l\,k}\hskip 1.42262pt,$

(18)

where $g$ is the $SU(2)$ coupling constant and $g_{4+2\,k}=1/(\sqrt{2}\,\mathrm{G}_{\mathrm{F}}\,\Lambda^{2})^{k}=g^{k}_{6}$, where $\mathrm{G}_{\mathrm{F}}$ is the Fermi coupling constant and $\Lambda$ is the scale around which new physics (NP) must be resolved. For each process, $N$ defines the $\mathrm{dim}=4$ LO (e.g. $N=1$ for $\mathrm{H}\to\mathrm{V}\mathrm{V}$ etc. but $N=3$ for $\mathrm{H}\to\gamma\gamma$). $N_{6}=N$ for tree initiated processes and $N-2$ for loop initiated ones. Here we consider single insertions of $\mathrm{dim}=6$ operators, which defines the so-called NLO SMEFT. To be more precise, we define a NLO SMEFT amplitude as the one containing SMEFT vertices inserted in tree-level SM diagrams, tree-level (SMEFT-induced) diagrams with a non-SM topology, SMEFT vertices inserted in one-loop SM diagrams, and one-loop (SMEFT-induced) non-SM diagrams.

The amplitude can be rewritten as

	$\displaystyle\mathrm{A}$	$\displaystyle=$	$\displaystyle g^{N}\,\mathrm{A}^{(4)}_{\mathrm{\scriptscriptstyle{LO}}}\bigl{(}\{p\}\bigr{)}+g^{N}\,g_{{}_{6}}\,\mathrm{A}^{(6)}_{\mathrm{\scriptscriptstyle{LO}}}\bigl{(}\{p\}\,,\,\{\mathrm{a}\}\bigr{)}+\frac{g^{N+2}}{16\,\pi^{2}}\,\mathrm{A}^{(4)}_{\mathrm{\scriptscriptstyle{NLO}}}\bigl{(}\{p\}\bigr{)}$		(19)
		$\displaystyle+$	$\displaystyle\frac{g^{N+2}\,g_{6}}{16\,\pi^{2}}\,\mathrm{A}^{(6)}_{\mathrm{\scriptscriptstyle{NLO}}}\bigl{(}\{p\}\,,\,\{\mathrm{a}\}\bigr{)}\hskip 1.42262pt,$

where $\{p\}$ is the set of SM parameters and $\{\mathrm{a}\}$ the set of Wilson coefficients. Counterterms are introduced using

$$\Delta_{\mathrm{UV}}=\frac{2}{4-\mathrm{d}}-\gamma-\ln\pi-\ln\frac{\mu^{2}_{\mathrm{R}}}{\mu^{2}}=\frac{1}{\bar{\varepsilon}}-\ln\frac{\mu^{2}_{\mathrm{R}}}{\mu^{2}}\hskip 1.42262pt,$$

(20)

where $\mathrm{d}$ is the space-time dimension, $\gamma$ is the Euler-Mascheroni constant, the loop measure is $\mu^{4-d}\,d^{d}q$ and $\mu_{\mathrm{R}}$ is the renormalization scale. The counterterms are defined by

$$\mathrm{Z}_{i}=1+\frac{g^{2}}{16\,\pi^{2}}\,\bigl{(}d\mathrm{Z}^{(4)}_{i}+g_{6}\,d\mathrm{Z}^{(6)}_{i}\bigr{)}\,\Delta_{\mathrm{UV}}\hskip 1.42262pt.$$

(21)

With field/parameter counterterms we can make UV finite (at $\mathcal{O}(g^{2}\,g_{6})$) all self-energies and transitions and the corresponding Dyson resummed propagators. Of course we have to prove cancellation of UV poles for all Green’s functions. This means that we need to make the SMEFT $\mathrm{S}\,$-matrix UV (and IR) finite, including $\mathrm{dim}=6$ operators and, at least, $\mathrm{dim}=8$ operators (truncation uncertainty). The verification of any claim with explicit computations is of importance. The role of symmetry is crucial. The best way to understand the connection between UV poles and symmetry is given by the background-field-method [49, 17]. Let us give an example of the complexity of proving cancellation of UV poles in any EFT. Consider a scalar theory

	$\displaystyle\mathcal{L}(\phi_{\mathrm{c}}+\phi)$	$\displaystyle=$	$\displaystyle\mathcal{L}(\phi_{\mathrm{c}})+\phi_{i}\,\mathcal{L}^{\prime}_{i}(\phi_{\mathrm{c}})+\frac{1}{2}\,\partial_{\mu}\,\phi_{i}\,\partial_{\mu}\,\phi_{i}$		(22)
		$\displaystyle+$	$\displaystyle\phi_{i}\,\mathrm{N}^{\mu}_{ij}\,\partial_{\mu}\,\phi_{j}+\frac{1}{2}\,\phi_{i}\,\mathrm{M}_{ij}(\phi_{\mathrm{c}})\,\phi_{j}+\mathcal{O}(\phi^{3})+\mbox{tot. der.}$

($\mathcal{L}^{\prime}_{i}(\phi_{\mathrm{c}})=0$). All one loop diagrams are generated by $\mathcal{L}_{2}(\phi)$, the part quadratic in $\phi$.

$\displaystyle\mathcal{L}_{2}(\phi)$

$\displaystyle\to$

$\displaystyle-\frac{1}{2}\,\left(\partial_{\mu}\,\phi\right)^{2}+\phi\,\mathrm{N}^{\mu}\,\partial_{\mu}\,\phi+\frac{1}{2}\,\phi\,\mathrm{M}\,\phi\hskip 1.42262pt.$

(23)

The counter-Lagrangian is given by

	$\displaystyle\Delta\,\mathcal{L}$	$\displaystyle=$	$\displaystyle\frac{1}{8\,\pi^{2}\,(d-4)}\,\Bigl{[}a_{0}\,M^{2}+a_{1}\,\left(\partial_{\mu}\,\mathrm{N}_{\nu}\right)^{2}+a_{2}\,\left(\partial_{\mu}\,\mathrm{N}_{\mu}\right)^{2}+a_{3}\,\mathrm{M}\,\mathrm{N}^{2}$		(24)
		$\displaystyle+$	$\displaystyle a_{4}\,\mathrm{N}_{\mu}\,\mathrm{N}_{\nu}\,\partial_{\mu}\,\mathrm{N}_{\nu}+a_{5}\,\left(\mathrm{N}^{2}\right)^{2}+a_{6}\,\left(\mathrm{N}_{\mu}\,\mathrm{N}_{\nu}\right)^{2}\Bigr{]}\hskip 1.42262pt.$

However, define $X=\mathrm{M}-\mathrm{N}^{\mu}\,\mathrm{N}_{\mu}$ and see that $\mathcal{L}$ is invariant under ’t Hooft transformation ($\mathrm{H}$) , $\Lambda$ antisymmetric

$\displaystyle\phi^{\prime}=\phi+\Lambda\,\phi\hskip 1.42262pt,\quad\mathrm{N}^{\prime}_{\mu}=\mathrm{N}_{\mu}-\partial_{\mu}\,\Lambda+\bigl{[}\Lambda\,,\,\mathrm{N}_{\mu}\bigr{]}\hskip 1.42262pt,\quad X^{\prime}=X+\bigl{[}\Lambda\,,\,X\bigr{]}$

(25)

Therefore $\Delta\,\mathcal{L}$ also will be invariant ($\mathrm{Tr}\,X$ is invariant)

	$\displaystyle\Delta\,\mathcal{L}$	$\displaystyle=$	$\displaystyle\frac{1}{\varepsilon}\,\mbox{Tr}\,\left(a\,X^{2}+b\,Y^{\mu\nu}\,Y_{\mu\nu}\right)\hskip 1.42262pt,$
	$\displaystyle Y_{\mu\nu}$	$\displaystyle=$	$\displaystyle\partial_{\mu}\,\mathrm{N}_{\nu}-\partial_{\nu}\,\mathrm{N}_{\mu}+\bigl{[}\mathrm{N}_{\mu}\,,\,\mathrm{N}_{\nu}\bigr{]}\hskip 1.42262pt.$		(26)

and $Y$ transforms as $X$. The counter-Lagrangian is made of products of objects transforming as $X$ and of $\mathrm{dim}=4$. As a consequence of the $\mathrm{H}\,$-invariance the number of counterterms goes from $7$ to $2$. Any approach to SMEFT violating invariance is doomed to failure [50]. An EFT (e.g. SMEFT) including $\mathrm{dim}=6,8$ operators will contain a term (for $\mathrm{dim}=6$ see Ref. [51])

	$\displaystyle\frac{1}{2}\,\partial_{\mu}\,\phi_{i}\;\;\mathrm{g}^{\mu\nu}_{ij}(\phi_{\mathrm{c}})\;\;\partial_{\nu}\,\phi_{j}$
	matrix-valued metric tensor		(27)

We should pay attention to the fact that in the SM $\mathrm{g}^{\mu\nu}_{ij}\propto\delta^{\mu\nu}\,\delta_{ij}$ while in quantum gravity (QGR) it remains diagonal only in the $ij$ indices. Therefore, in SMEFT we will have matrix-valued Riemann tensors, i.e. more invariants for the counter-Lagrangian $\Delta\,\mathcal{L}$, i.e. $\mathrm{Tr}(X\,\mathrm{R}),\mathrm{Tr}\,(\mathrm{R}^{2})\,\dots$; as a consequence, EFT is computationally more complex than QGR [52]. We should remember that the name of the game is to have the full $\Delta\,\mathcal{L}$, not the counterterms for one or two processes. If $\mathcal{L}$ is invariant under a group $\mathrm{G}$ then the relation between the $\mathrm{G}$ transformation and the $\mathrm{H}$ one is crucial in proving closure under renormalization (not the same as strict renormalizability). In other words the $\mathrm{H}\,$-invariant counterterms of Eq.(26) must also be $\mathrm{G}\,$-invariant.

To give an example of Eq.(27) we consider a Lagrangian

$$\mathcal{L}=-\partial_{\mu}\Phi^{*}\,\partial_{\mu}\Phi-\mathrm{m}^{2}\,\Phi^{*}\Phi-\frac{1}{2}\,g^{2}\,\bigl{(}\Phi^{*}\Phi\bigr{)}\hskip 1.42262pt,$$

(28)

where $\Phi=\bigl{(}\phi_{1}+\mathrm{v}+i\,\phi_{2}\bigr{)}/\sqrt{2}$. We introduce $v=\mathrm{M}/g$ and $\mathrm{m}^{2}=\beta-\mathrm{M}^{2}/2$ and add

$$\mathcal{Q}^{(8)}=\Bigl{(}\frac{gg_{6}}{\mathrm{M}^{2}}\Bigr{)}^{2}\,\partial_{\mu}\Phi^{*}\,\partial_{\mu}\Phi\,\Box\,\bigl{(}\Phi^{*}\Phi\bigr{)}\hskip 1.42262pt.$$

(29)

After splitting $\phi_{i}=\phi_{i\mathrm{c}}+\phi_{i}$ we obtain that the contribution of the operator of $\mathrm{dim}=8$ to the metric tensor is $\delta g=\delta_{1}g+\delta_{2}g+\mathcal{O}(g^{3})$, where

$$\delta_{1}g^{\mu\nu}=\frac{gg^{2}_{6}}{\mathrm{M}}\,\phi_{1\mathrm{c}}\,\delta^{\mu\nu}\,\mathrm{I}\hskip 1.42262pt,$$

(30)

$$\delta_{2}g^{\mu\nu}=\frac{g^{2}g^{2}_{6}}{\mathrm{M}^{4}}\left(\begin{array}[]{ll}(g_{11}+g_{22})\,\delta^{\mu\nu}+4\,\partial^{\mu}\phi_{1\mathrm{c}}\,\partial^{\nu}\phi_{1\mathrm{c}}&4\,\partial^{\mu}\phi_{1\mathrm{c}}\partial^{\nu}\phi_{2\mathrm{c}}\\ 4\,\partial^{\mu}\phi_{12\mathrm{c}}\partial^{\nu}\phi_{1\mathrm{c}}&(g_{11}+g_{22})\,\delta^{\mu\nu}+4\,\partial^{\mu}\phi_{2\mathrm{c}}\,\partial^{\nu}\phi_{2\mathrm{c}}\end{array}\right)$$

Here we will summarize the main steps in the renormalization procedure for the SMEFT. Field/parameter counterterms are not enough to make UV finite the Green’s functions with more than two legs. A mixing matrix among Wilson coefficients is needed:

$$\mathrm{a}_{i}=\sum_{j}\,\mathrm{Z}^{{\scriptscriptstyle{\mathrm{W}}}}_{ij}\,\mathrm{a}^{\mathrm{ren}}_{j}\hskip 1.42262pt,\quad\mathrm{Z}^{{\scriptscriptstyle{\mathrm{W}}}}_{ij}=\delta_{ij}+\frac{g^{2}}{16\,\pi^{2}}\,d\mathrm{Z}^{{\scriptscriptstyle{\mathrm{W}}}}_{ij}\,\Delta_{\mathrm{UV}}\hskip 1.42262pt.$$

(31)

We can start by computing the amplitude for the (on-shell) decay $\mathrm{H}\to\gamma\gamma$ and fix a few entries in the mixing matrix; we can continue with the $\mathrm{H}\mathrm{Z}\gamma$ and $\mathrm{H}\mathrm{Z}\mathrm{Z}$ amplitudes. When we arrive at $\mathrm{H}\mathrm{W}\mathrm{W}$ we find that the $\mathrm{dim}=4$ part can be made UV finite but for the $\mathrm{dim}=6$ part there are no Wilson coefficients left free so that the UV finiteness follows from gauge cancellations. We then continue with the decay of the Higgs boson (and of the $\mathrm{Z}$ boson) into fermion pairs and make all the corresponding amplitudes UV finite.

At this point we are left with the universality of the electric charge. In QED there is a Ward identity telling us that $e$ is renormalized in terms of vacuum polarization, Ward-Slavnov-Taylor identities allow us to generalize the argument to the full SM [12, 53].

We can give a quantitative meaning to the the previous statement by saying that the contribution from vertices (at zero momentum transfer) exactly cancel those from (fermion) wave function renormalization factors. Therefore, we need to compute the vertex ${\overline{f}}f\gamma$ (at $q^{2}=0$) and the $f$ wave function factor in SMEFT, proving that the WST identity can be extended to $\mathrm{dim}=6$; this is non trivial since there are no free Wilson coefficients in these terms (after the previous steps); (non-trivial) finiteness of LEP processes follows.

Finite renormalization in SMEFT; let us recall that the renormalization procedure comprises the specification of the gauge-fixing term including, together with the corresponding FP Lagrangian, the choice of the regularization scheme — nowadays dimensional regularization — the prescription for the renormalization scheme and the choice of a input parameter set. For the SM parameters we use on-shell renormalization which requires the choice of some input parameter set (IPS); for the Wilson coefficients we use the $\overline{MS}$ scheme. Therefore, the final answer will contain “universal” logarithms (renormalization group) but also “non-universal” logarithms which depend on the choice of the IPS. The choice of the IPS should be such that the effect of “non-universal” logarithms is minimal, e.g. $\alpha(0)$ should be avoided. Once we work with NLO SMEFT, non-local effects (e.g. normal-threshold singularities or even anomalous thresholds [54]) will also show up [55, 56, 45] for all those observables where the light masses are small compared to the scale at which we test the process and much smaller than the cutoff $\Lambda$.

Finally, the infrared/collinear part of the one-loop virtual and of the real corrections shows double factorization and the total is finite [57] at $\mathcal{O}(g^{4}\,g_{6})$.

In this Section we will use the so-called “Warsaw basis” [43]; note however that we will rescale the Wilson coefficients: in front of an operator $\mathcal{Q}^{(\mathrm{d})}_{i}$ of dimension $\mathrm{d}$ containing $n$ fields we will write $g^{n-2}\,\mathrm{a}^{\mathrm{d}}_{i}/\Lambda^{\mathrm{d}-4}$, where $g$ is the $SU(2)$ coupling constant. Our goal is to consider the SMEFT with its anomalies [37, 60, 61, 62]; in these calculations a certain amount of $\gamma\,$-matrix manipulation is unavoidable and we must specify the regularization scheme to be used, in particular, we must specify how to treat $\gamma^{5}$. We will use the scheme developed by Veltman in Ref. [63], which is based on the work of Refs. [64, 65]. Therefore, $\gamma^{\mu},\gamma^{5}$, and $\varepsilon^{\mu\nu\alpha\beta}$ are formal objects where

	$\displaystyle\bigl{\{}\gamma^{\mu}\,,\,\gamma^{\nu}\bigr{\}}=2\,\delta^{\mu\nu}\,\mathrm{I}$		$\displaystyle\mathrm{Tr}\,\mathrm{I}=4\hskip 1.42262pt,$
	$\displaystyle\delta^{\mu\nu}=\delta^{\bar{\mu}\bar{\nu}}+\delta^{\hat{\mu}\hat{\nu}},\;\;\delta^{\bar{\mu}\bar{\mu}}=4,$		$\displaystyle\delta^{\hat{\mu}\hat{\mu}}=\mathrm{d}-4,$
	$\displaystyle\delta^{\mu\alpha}\,\delta^{\alpha\nu}$	$\displaystyle=$	$\displaystyle\delta^{\bar{\mu}\bar{\alpha}}\,\delta^{\bar{\alpha}\bar{\nu}}+\delta^{\hat{\mu}\hat{\alpha}}\,\delta^{\hat{\alpha}\hat{\nu}}\hskip 1.42262pt,$		(35)

where $\mathrm{d}$ is the space-time dimension. As described in Ref. [63] we have the following relation

	$\displaystyle\mathrm{Tr}\,\bigl{(}\cdots\,\gamma^{\alpha}\,\cdots\,\gamma^{\beta}\,\cdots\bigr{)}$	$\displaystyle=$	$\displaystyle\mathrm{Tr}\,\bigl{(}\cdots\,\gamma^{\bar{\alpha}}\,\cdots\,\gamma^{\bar{\beta}}\,\cdots\bigr{)}$		(36)
		$\displaystyle+$	$\displaystyle\mathrm{Tr}\,\bigl{(}\cdots\,\gamma^{\hat{\alpha}}\,\cdots\,\gamma^{\hat{\beta}}\,\cdots\bigr{)}\hskip 1.42262pt,$

where the dots indicate strings of four-dimensional gamma matrices and also $\gamma^{5}$. The second trace in the r.h.s. of Eq.(36) is computed according to the following rules:

1. 1.

   move all the $\gamma^{\hat{\mu}}$ matrices to the right using $\gamma^{\hat{\mu}}\,\gamma^{\bar{\nu}}=-\gamma^{\bar{\nu}}\,\gamma^{\hat{\mu}}$,

2. 2.

   for a trace containing an odd number of $\gamma^{5}$ matrices use $\gamma^{\hat{\mu}}\,\gamma^{5}=\gamma^{5}\,\gamma^{\hat{\mu}}$,

3. 3.

   for a trace containing an even number of $\gamma^{5}$ matrices use $\gamma^{\hat{\mu}}\,\gamma^{5}=-\gamma^{5}\,\gamma^{\hat{\mu}}$.

As a consequence we obtain

$$\mathrm{Tr}\,\bigl{(}\cdots\,\gamma^{\hat{\alpha}}\,\gamma^{\hat{\beta}}\,\cdots\bigr{)}=\mathrm{Tr}\,\bigl{(}\cdots\bigr{)}\,\mathrm{Tr}\,\bigl{(}\gamma^{\hat{\alpha}}\,\gamma^{\hat{\beta}}\bigr{)}\hskip 1.42262pt,$$

(37)

where the first trace in the r.h.s. only contains four-dimensional quantities while $\mathrm{Tr}\,\bigl{(}\gamma^{\hat{\alpha}}\,\gamma^{\hat{\beta}}\bigr{)}=\mathrm{d}-4$ etc.

Using a $\gamma^{5}$ which does not anticommute with the other Dirac matrices leads to “spurious anomalies” which violate gauge invariance (spoiling renormalizability) and we must impose the relevant Ward-Takahashi (Refs. [66, 67]) and Slavnov-Taylor (Refs. [68, 69, 70]) identities (for a review see Ref. [71]). The problems of course are related to the existence of the ABJ anomaly (Refs. [72, 73, 74]), which cancels in the SM. The goal of this Section is to study the ABJ anomaly in the SMEFT extension of the SM.