# Standard Model and Prospects

Jean Iliopoulos

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Jean Iliopoulos. Standard Model and Prospects. 10th Symposium on Large TPCs for Low-Energy Rare Event Detection, Dec 2021, Paris, France. pp.012001, 10.1088/1742-6596/2502/1/012001. hal-04126369

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Standard Model and Prospects

To cite this article: J Iliopoulos 2023 *J. Phys.: Conf. Ser.* **2502** 012001

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# Standard Model and Prospects

J Iliopoulos

LPENS Ecole Normale Supérieure, 24 rue Lhomond, 75005, Paris, France

E-mail: jean.iliopoulos@phys.ens.fr

**Abstract.** We review the present status of the Standard Model in High Energy Physics, its great successes as well as the questions which are left unanswered. We briefly indicate some theoretical ideas and some crucial experiments which may show the way to obtain a theory Beyond the Standard Model.

## 1. The Standard Model

The Standard Model in particle physics is a quantum field theory describing in a unified framework all experimentally measurable interactions among elementary particles. It has two fundamental properties:

(i) It is invariant under gauge transformations implied by an internal symmetry $U(1) \times SU(2) \times SU(3)$ spontaneously broken to $U(1)_{em} \times SU(3)$.

(ii) It is renormalisable.

Let me first note that we do not have exact solutions for any non-trivial quantum field theory, and that of the Standard Model is not an exception. In fact, we do not even know whether such solutions exist. All information we have comes from calculations of the first low orders in a perturbation expansion, or from numerical simulations in a truncated finite space-time lattice. As a result, we have only a superficial understanding of these two fundamental properties which appear to be profound and mysterious. We are fond of them for both reasons. The first property, namely the invariance under gauge, i.e. local, transformations, gives a deep relation between geometry and dynamics. It shows that the gauge principle determines, to a large extent, the forces among the elementary particles. It seems to apply to all known interactions, including gravitation. The second, which at first sight sounds purely technical, puts severe limits on the possible forms of the interactions, but it does not apply to gravitation. Both have been discussed extensively during the last decades and they hardly need any explanations.

**Gauge invariance** The $SU(3)$ part, which remains unbroken, refers to strong interactions. It is known as “quantum chromodynamics”, or QCD. In the standard notation the Lagrangian density is written as:

$$ \mathcal{L}_{\text{QCD}} = -\frac{1}{2} \text{Tr} \mathcal{G}_{\mu\nu} \mathcal{G}^{\mu\nu} + \sum_{f=1}^{6} \sum_{i,j=1}^{3} \bar{q}_{f}^{i} (i \not D_{ij} - m_{f} \delta_{ij}) q_{f}^{j} \quad (1) $$

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Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
where $q_f^i(x)$ is the Dirac field of the $i$-th color of the $f$-th quark flavour with mass $m_f$. The matrix valued field strength $G_{\mu\nu}(x)$ contains the eight gluon gauge fields $G_\mu^\alpha(x)$, $\alpha = 1, ..., 8$. At the non-perturbative level we must add to (1) an extra term, conveniently written as $\frac{i\theta}{64\pi^2} \text{Tr} \epsilon_{\mu\nu\rho\sigma} G^{\mu\nu} G^{\rho\sigma}$, with a new coupling constant $\theta$. This term induces CP violation at the level of strong interactions and it will be discussed later.

The electromagnetic and the weak interactions are described by the group $U(1) \times SU(2)$ spontaneously broken to $U(1)_\text{em}$. They involve the gauge bosons $W_\mu^\pm$, $Z_\mu$ and the photon $A_\mu$.

The power of gauge invariance appears already in these equations. If we forget for the moment the phenomenon of spontaneous symmetry breaking, we see that gauge invariance determines the interactions almost entirely: they are all due to the exchange of spin-one bosons and we have just one coupling constant for each factor of the group. (Strictly speaking we need an extra assumption in order to enforce the uniqueness of the coupling constant for the $U(1)$ group). Extending the principle of gauge invariance to diffeomorphisms tells us that gravitational interactions are due to the exchange of a spin-two boson.

**Renormalisation** Our views on this question have been profoundly influenced by Kenneth Wilson and I will come back to this point shortly. Superficially, this property appears to be related to our ability to compute higher order terms in the perturbation expansion. It sounds purely technical and we wouldn't expect Nature to attach any particular meaning to it. Yet, the opposite seems to be true. In a four-dimensional space-time we have the following list of renormalisable quantum field theories: (i) $\phi(x)^3$, (ii) $\phi(x)^4$, (iii) Yukawa interactions, (iv) scalar and/or spinor electrodynamics, (v) Yang-Mills theories with scalar and/or spinor fields. All other conceivable interactions are non-renormalisable. The amazing fact is that Nature uses all these renormalisable interactions, and only them, in order to describe the weak, electromagnetic and strong interactions among elementary particles. They are the interactions of the Standard Model. An immediate consequence is that the effective coupling strengths of all interactions depend in a well-defined and calculable way on the energy scale through the renormalisation group equations. This dependence has been verified experimentally. Note also that gravitational interactions are not among the renormalisable ones.

## 1.1. The Standard Model has been enormously successful.

The Lagrangian of the Model contains a certain number of arbitrary parameters. Ignoring for the moment non-perturbative effects and assuming that the masses and mixings of the neutrinos belong to physics beyond SM, we still count: (i) 3 gauge coupling constants. (ii) Two parameters of the scalar potential $\lambda$ and $\mu^2$. (iii) Three Yukawa coupling constants for the three lepton families. (iv) Six Yukawa coupling constants for the three quark families. (v) Four parameters of the *KM* matrix. Some remarks:

* All but the three gauge coupling constants come from the scalar sector of the theory.

* All these parameters have been measured independently, so the model makes a large number of predictions with no arbitrary constants.

* This set of parameters appears to be irreducible in the sense that any relation among them which would reduce their number seems to be incompatible with renormalisation.

Over the last decades an enormous amount of work, both experimental and theoretical, has been devoted to the identification, computation, and experimental verification of the predictions of the Standard Model. The overall picture, as it is presented in the Tables and Figures of the Particle Data Group, is very impressive. Cross sections, branching ratios, the renormalisation group evolutions of coupling constants and structure functions as well as many other quantities, show a remarkable agreement between theory and experiment. In addition, QCD lattice
calculations give a very good description of the low lying hadronic spectrum. In Figures 1, 2 and 3 we show some of these results. Given this great success, we should no more call it “The Standard Model” but rather “The Standard Theory”. The few cases which show some tension between theory and experiment will be briefly discussed below. In this section I want to point out several cases in which the theory, often combined with high precision measurements, led to successful predictions of new physics. Some examples:

(i) The discovery of weak neutral currents by Gargamelle in 1972-1973.
Not only their existence, but also their strength and their properties, were predicted by the Model.

(ii) The discovery of charmed particles at SLAC in 1974.
Their presence was essential to ensure the absence of strangeness changing neutral currents, ex. $K^0 \to \mu^+ + \mu^-$. Their characteristic property is to decay predominantly into strange particles.

(iii) A necessary condition for the consistency of the Model is that $\sum_i Q_i = 0$ inside each family. When the $\tau$ lepton was discovered the b and t quarks were predicted with the right electric charges. The t-quark was seen at LEP through its effects in radiative corrections before its actual discovery at Fermilab.

(iv) The discovery of the W and Z bosons at CERN in 1983.
The characteristic relation of the Standard Model with an isodoublet BEH mechanism $m_Z = m_W / \cos\theta_W$ is checked with very high accuracy (including radiative corrections).

(v) The final touch: the discovery of the BEH scalar at CERN.

<figure><img src="image_6.png" /><figcaption>**Figure 1.** A global fit of various quantities computed and measured [1].</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 2.** The running of the strong interaction effective coupling constant [2].</figcaption></figure>

## 2. Beyond the Standard Theory

Given this impressive success... What does *Beyond* mean? Or, what is wrong with the Standard Theory? I will list here a few physics questions to which the Standard Theory offers no answers.
<figure><img src="image_8.png" /><figcaption>Figure 3. Some low-lying hadronic states computed by lattice QCD simulations.[3]</figcaption></figure>

I will put them in three classes: (i) Theoretical questions (ii) Phenomenological questions, and (iii) Experimental questions. The trouble is that I do not know the answer to any of them, so this section will raise many questions, but it will offer very few answers.

## 2.1. Theoretical questions

Typical examples include: (i) Why does the model contain three, apparently very similar, families? (ii) Why is the gauge group given by the product $U(1) \times SU(2) \times SU(3)$? Is it the remnant of a Grand Unification? (iii) Why are the fermion masses spread over such a wide spectrum? (iv) Why quantum gravity is not included? (v) Why is the BEH scalar so light? All these questions, and many others you can think of, have been raised and discussed extensively over the years, with no satisfactory answer so far. Here I will only present the last one.

Let us consider a quantum field theory described by a Lagrangian density $\mathcal{L}$ written in terms of a set of fields $\Psi_i(x)$, $i = 1, \dots, N$. Even if $\mathcal{L}$ is in agreement with all known data, we want to argue that $\mathcal{L}$ should be viewed as an effective theory valid in a certain energy range. It contains the degrees of freedom which are relevant in that range. No theory can be considered as valid at all energies, because we cannot claim that we know the physics, and the relevant degrees of freedom, at arbitrarily high energies. For the Standard Model today this energy range is up to a scale $M$ on the order of a few TeV. Following Wilson, let us integrate over all degrees of freedom which are heavier than $M$. The result will be an effective theory $\mathcal{L}_{\text{eff}}$ written in terms of the degrees of freedom $\psi_j$, $j = 1, \dots, N'$ which are lighter than $M$. Let us write the most general form of $\mathcal{L}_{\text{eff}}$. It will be an infinite sum of monomials made out of the fields $\psi_j$ and their derivatives restricted only by symmetry requirements. Since integrating over the heavy degrees of freedom does not break any symmetry, every term in $\mathcal{L}_{\text{eff}}$ will be invariant under whichever symmetries we had assumed in $\mathcal{L}$.

$$ \mathcal{L}_{\text{eff}} = \sum_{i} C_{i}(g, M) O_{i}, \qquad (2) $$

where $O_i$ are monomials in the fields $\psi$ and $C_i(g, M)$ are numerical coefficients which depend on the parameters of the original theory, denoted collectively by $g$, and the scale $M$. If the
original theory was renormalisable, the c-number functions $C_i$ can be computed order by order in perturbation. But even if we do not compute them, we can find their dependence on $M$ from dimensional analysis. Let $d_i$ be the dimension of the operator $O_i$. Then, assuming $M$ to be much larger than any other mass parameter in the theory, we have that $C_i \sim M^{4-d_i}$. This dependence will be corrected by terms which are at most logarithmic in $M$ and are not important for this argument. We thus obtain three classes of operators $O_i$:

* An infinite class of operators with $d_i > 4$. The corresponding coefficients behave like negative powers of $M$ and can be neglected for large $M$. We call them *irrelevant operators*.

* Operators having $d_i = 4$. The coefficient functions are independent of $M$. We call them *marginal operators*.

* Operators with dimension smaller than 4. The coefficient functions behave like positive powers of $M$ and we call them *dominant operators*.

Let us apply this reasoning starting from the Lagrangian of the Standard Model. We must choose $M$ to be larger than a few TeV, the scale already explored by LHC, but otherwise the choice is arbitrary. In the spirit of Wilson, $M$ represents the scale up to which the Standard Model can be trusted. If $M$ is large enough the irrelevant operators die away and can be neglected. The marginal operators are precisely the dimension four terms in the Lagrangian of the Standard Model. There are only two dominant operators: the unit operator $1$ with dimension zero and the mass operator of the BEH scalar $\phi^2$ with dimension two. The coefficient of the first behaves like $M^4$ and that of the second like $M^2$. The first gives a constant term in the Lagrangian which, in the absence of gravitation, can be dropped. If we include gravitational interactions this term is known as *the cosmological constant*. The second gives a correction to $m_H^2$ which grows like $M^2$. The coefficient $C_{\phi^2}$ contains coupling constants and, generically, it is expected to be of order $\alpha \sim 10^{-2}$. In the Standard Model $C_{\phi^2}$ turns out to be even larger because of the large value of the coupling constant between the BEH scalar and the $t$ quark. This would mean that $M$ cannot exceed much the TeV scale. This is the puzzle: what keeps the mass of the scalar boson at 125 GeV? This problem is known in the literature as the problem of *naturalness* and it is common to all theories which contain elementary scalar fields. Unless specifically tuned, a quantum field theory is unable to sustain naturally a light scalar. By “specifically tuned” we mean theories for which the parameters satisfy certain relations ensuring that $C_{\phi^2}$ turns out to be very small, or even zero. Supersymmetry is an example. So, the real question is: why LHC has seen no signs of new physics?

## 2.2. Phenomenological questions

In this category I put questions raised by observations which do not quite fit the predictions of the Standard Theory. I will consider three examples: (i) the evidence for the existence of dark matter, (ii) the failure to detect axions, and (iii) the lack of theoretical understanding of the neutrino masses and oscillations.

**Dark matter** A special session of this Conference was dedicated to this subject, so I will only say a few words. Several astrophysical observations seem to indicate that the gravitational field in the Universe is much stronger than what we expect from the entire visible mass distribution. The evidence is either direct, coming from the rotational curves of various galaxies, or indirect, such as the attractive field required in models of galaxy formation. Particle physicists want to interpret these observations as evidence for the existence of a new kind of elementary particle, which is neutral, interacts weakly with ordinary matter, and has been created, together with all other particles, during the big bang. It forms the “dark matter”. A rough estimation gives for the ratio visible/dark matter the value 0.2, which means that dark matter by far exceeds the amount of “visible” matter in the Universe.
Most proposals aiming to extend the Standard Theory predict the existence of such particles as dark matter candidates and a considerable experimental effort has been devoted to a direct detection, with no success so far. Some of these experiments, running or planned, have been presented in this Conference. The trouble is that we have no reliable way to estimate the masses of such “particles”. If they are very light they must be abundantly produced in order to create the required gravitational field, if they are heavy they should be rare. In fact, the entire spectrum, from fractions of eV all the way to massive black holes, has been considered. Dark matter detection remains an open and active research field.

Before leaving this subject let me mention an alternative explanation. All observational evidence assumes the validity of classical gravitational theory (newtonian or relativistic) at very large distances. This has prompted theorists to propose “modified theories of gravity” in which the $1/r^2$ law is changed for $r$ larger than a typical galactic size. I will not review such theories here because, to my taste, none imposes itself by simplicity or aesthetic beauty. They all concede too much sacrifice of elegance to expediency.

Axions As I said earlier, when a certain class of non-trivial field configurations are included in the calculation of the QCD functional integral, the action changes with the addition of a new term

$$S_{\text{QCD}}^{\text{eff}} = S_{\text{QCD}}^{0} + \frac{i\theta}{32\pi^2} \int d^4x \operatorname{Tr} \tilde{G}_{\mu\nu}G^{\mu\nu} \quad \text{with} \quad \tilde{G}_{\mu\nu} = \frac{1}{2}\epsilon_{\mu\nu\rho\sigma}G^{\rho\sigma} \qquad (3)$$

with $S_{\text{QCD}}^0$ the action corresponding to the Lagrangian density given in eq. (1). It is a strange term with several peculiar properties.

First, it violates parity and CP. Thus, the absence of such effects in strong interactions puts very severe limits on the magnitude of the constant $\theta$. We could be tempted to set $\theta = 0$, but, since the Standard Model does violate CP, this term would be generated at higher orders. This is the strong CP problem.

Second, we would expect naively this new term to give a vanishing contribution because it is easy to show, using the equations of motion, that

$$\operatorname{Tr} \tilde{G}_{\mu\nu}G^{\mu\nu} = \partial^{\mu}R_{\mu} \qquad R_{\mu} = 2\epsilon_{\mu\nu\rho\sigma}\operatorname{Tr}\left(A^{\nu}\partial^{\rho}A^{\sigma} + \frac{2}{3}A^{\nu}A^{\rho}A^{\sigma}\right) \qquad (4)$$

which means that the new term in (3) is the integral over a four-divergence. The reason why this conclusion is incorrect is that the non-trivial configurations of the gauge field that gave rise to this term do not vanish at infinity, which means that the integral over a four-divergence is not necessarily equal to zero.

Third, we can show that this term vanishes at the limit of exact chiral invariance. In fact we can prove a stronger result. Let us consider the covariant Dirac operator $\mathcal{D}$ which appears in eq. (1) and study its eigenvalues $\lambda_n$ and eigenfunctions $\phi_n$, i.e. the equation $\mathcal{D}\phi_n = \lambda_n\phi_n$. We are interested in particular in the “zero modes” of $\mathcal{D}$, i.e. the eigenfunctions with zero eigenvalues, $\lambda_n = 0$. Since $\mathcal{D}$ and $\gamma_5$ anticommute, in the subspace of the zero modes we can look for eigenfunctions common to both. $\gamma_5^2 = 1$, so its eigenvalues are $\pm 1$. Let us denote the number of the corresponding eigenvalues by $n_{\pm}$ and set $\nu = n_{+} - n_{-}$. $\nu$ is called winding number, but I will not explain the origin of this terminology. It is however clear that at the limit of exact chiral symmetry $\nu = 0$. The result, which I will not prove here, can be written as

$$n_{+} - n_{-} = \nu = -\frac{1}{32\pi^2} \int d^4x \operatorname{Tr} \tilde{G}_{\mu\nu}(x)G^{\mu\nu}(x). \qquad (5)$$

The surprising conclusion is that, no matter what kind of field configurations we are using to compute the integral in (5), the result is always an integer. In other words, all we deed in
eq. (3) was to add a term to the action which is just a constant integer. Standard field theory books do not tell us how to handle such terms since they do not contribute in any obvious way to the equations of motion.

Were chiral symmetry exact, $\nu$ would have been zero and the strong CP problem would have been absent. This would have been the case if at least one quark flavour had vanishing mass. Quark masses, especially the light ones, are not directly measurable, but all phenomenological studies-chiral expansions, lattice simulations, etc-seem to indicate that none is massless. The Peccei-Quinn solution consists in enlarging the symmetry of the Standard Model to include a new $U(1)$ chiral symmetry by the addition of a new doublet of scalar fields. It is an approximate global symmetry broken spontaneously since this new doublet also acquires a non-zero vacuum expectation value. Following S. Weinberg and F. Wilczek, we conclude that the theory now contains an almost massless ("almost", because the symmetry is not exact) Goldstone boson, the axion.

The detailed properties of this hypothetical particle are model-dependent and in the dedicated session of this Conference we heard all the experimental program specifically designed to detect axions. So far the phenomenological question remains unanswered: granted that an axion is predicted, why no axion has been detected?

### Neutrino masses and oscillations
It is an exciting field but I do not have much to say on it. It is a data driven subject in which theorists have not played any significant role. A very rich and diverse experimental program addresses some among the fundamental physics questions related to neutrinos. An incomplete list involves

(i) The origin of neutrino masses.

(ii) How are neutrino masses ordered (oscillation experiments)

(iii) What is the absolute neutrino mass scale (β-decay spectrum, cosmic surveys)

(iv) Do neutrinos and anti-neutrinos oscillate differently (oscillation experiments)

(v) Are there additional neutrino types and interactions (oscillation experiments, cosmic surveys)

(vi) Majorana or Dirac (neutrinoless double beta decay)

Some personal remarks:

* Substantial improvement in precision could be expected during the coming years.

* The significance of such improvements is not easy to judge.

* The non-vanishing of all mixing angles offers exciting possibilities for CP violation in the lepton sector.

* The neutrino mass matrix does not look at all like the CKM one.

* So far no real illumination came from leptons to be combined with the quark sector for a more complete theory of flavour.

* The trouble is that I do not see how this could change!

* In the last 25 years 4 Nobel Prizes were attributed in experimental neutrino physics (1988, 1995, 2002 and 2015). There is no reason to assume that the series is terminated.

### 2.3. *Experimental questions*

There are a couple of experimental measurements that seem to disagree significantly with the predictions of the Standard Theory. I will consider two such problems: (i) the muon $g-2$, and (ii) some decay properties of B-hadrons.
Muon $g-2$ Fermilab has already published their Run1 results. They agree with the previous Brookhaven measurements and, since both are dominated by statistics, they can be combined to give a value

$$a_{\mu}(Exp) = 116592061(41) \times 10^{-11}.$$

We can decompose the theoretical calculation in four parts, see Table 1. Part I is the pure QED contribution which we can safely assume that it is known with the required precision, as shown in the Table. Part II contains the weak interaction contributions with the Z boson replacing the photon. Again, we do not expect any uncertainties in the calculation, and we can assume that the value given in the Table is correct with very small errors. Part III contain the hadronic corrections. The most important part seems to be given by the hadronic corrections to the photon propagator. In the Table we give the results for the leading order (LO) calculation, as well as the next-to-leading order (NLO) and the next-to-next-to-leading order (NNLO) corrections. The LO is very large, but it has been computed by two groups independently. Finally, Part IV contains the hadronic contribution to the scattering of light by light. It has been estimated in model calculations, as well as in lattice simulations. The combined result is shown in the Table, together with the latest measurement. As we can see, the discrepancy is of order $4.2 \sigma$.

<table><caption>**Table 1.** Comparison between theory and experiment for the muon $g-2$.</caption><thead><tr><th>Contribution</th><th>Value×10<sup>10</sup></th><th>Uncertainty×10<sup>10</sup></th></tr></thead><tbody><tr><td>QED</td><td>11658471.895</td><td>0.008</td></tr><tr><td>EW Corrections</td><td>15.4</td><td>0.1</td></tr><tr><td>HVP (LO)</td><td>693.5</td><td>4.3</td></tr><tr><td>HVP (NLO)</td><td>-9.84</td><td>0.06</td></tr><tr><td>HVP (NNLO)</td><td>1.24</td><td>0.01</td></tr><tr><td>HLbL</td><td>10.5</td><td>2.6</td></tr><tr><td>Total SM Th</td><td>11659183</td><td>5.0</td></tr><tr><td>Present exp. value</td><td>11659206.1</td><td>4.1</td></tr><tr><td>Fermilab target</td><td>—</td><td>1.6</td></tr></tbody></table>

However, a new theoretical estimation of HVP-LO was published recently based on ab initio lattice QCD simulations with the result $a_{\mu}(HVP - LO) = (707 \pm 5.5) \times 10^{-10}$. If this new value is considered the discrepancy between theory and experiment becomes insignificant. It is obvious that this new result should be checked independently and, if confirmed, the origin of the difference between the dispersion relations and the lattice calculations should be understood. In addition, there are proposals for dedicated experiments to settle the issue. Notice also that the HVP value is the hadronic contribution to the photon propagator which controls the running of $\alpha_{QED}$. Therefore it affects many other calculations and fits and its precise knowledge is very important.

*Decay properties of B-hadrons* Three experimental collaborations, BaBar at SLAC, BELLE in Japan and LHCb at CERN, have measured the decay properties of B-mesons, i.e. mesons containing one b quark. Among the physics questions they address is that of lepton universality. A basic assumption of the Standard Model is that the three charged leptons, to wit the electron, the muon, and the tau, have identical properties and their only difference is their mass. The
<figure><img src="image_9.png" /><figcaption>Figure 4. $R(D)$ plotted against $R(D^*)$ showing the experimental measurements and the Standard Model prediction. [2]</figcaption></figure>

three experiments attempted to test this assumption in the semi-leptonic decays of B-mesons and they measured ratios of branching ratios of the form:

$$R(D) = \frac{BR(\bar{B} \rightarrow D + \tau + \bar{\nu}_{\tau})}{BR(\bar{B} \rightarrow D + l + \bar{\nu}_{l})}, \quad R(D^{*}) = \frac{BR(\bar{B} \rightarrow D^{*} + \tau + \bar{\nu}_{\tau})}{BR(\bar{B} \rightarrow D^{*} + l + \bar{\nu}_{l})} \quad (6)$$

where $l$ stands for either $\mu$, or $e$, and $D$ ($D^*$) is the $0^-$ ($1^-$) charmed meson. Theoretically these ratios are expected to be different from 1, because the tau is much heavier than the other two leptons. This has two effects: a trivial kinematic effect due to a different phase space, and a more subtle dynamic one, because the matrix elements of the hadronic weak current $\langle D | h_{\lambda} | \bar{B} \rangle$ should be computed at different values of the momentum transfer. This last effect has been estimated using a combination of model calculations and lattice simulations. The results are:

$$R(D)^{\text{Th}} = 0.299 \pm 0.003, \quad R(D^{*})^{\text{Th}} = 0.252 \pm 0.003. \quad (7)$$

The comparison with the measured values is given in Figure 4. Since the experimental errors in the two measurements are correlated, it is more meaningful to plot the two ratios one against the other. We see that there is a discrepancy between theory and experiment of order $3\sigma$.

In the same line, in the decays $B \rightarrow K(K^*) + l^+ + l^-$ LHCb finds that the ratio of produced muons versus electrons is too small, especially if you plot the number of events as a function of the lepton pair invariant mass. If we combine all experiments the discrepancy reaches several $\sigma$, but this may not be the right thing to do. This subject is right now a very active field of research and it is probable that the situation will evolve considerably in the near future. For the moment no clear theoretical conclusion can be drawn from these effects in view of the uncertainties in the theoretical calculations and the difficulty to evaluate very precisely possible experimental biases.

### 3. Standard Model and Prospects

It is the title of the talk that was assigned to me. As you have seen, I spoke much more about the Standard Model than about “Prospects”. When LHC was commissioned we were all convinced that New Physics was around the corner! Ten years later we have to admit that no corner has been found. Is it because we missed it? I cannot see how could this happen. Probably Nature has been more subtle and the corner is a bit further away. How much? There are two ways to find out: (i) brute force, build a larger machine. It should, and hopefully will, be done but
it will take too long for me to learn the answer. (ii) Try to be also subtle and discover small and neglected effects. Large TPCs may help. You may not learn everything but you will know better what to look for. And, in any case, this is my only hope to learn something.

References

[1] J. Haller, A. Hoecker, R. Kogler, K. Mönig, T. Peiffer and J. Stelzer [The Gfitter Group]. Eur. Phys. J. C 78 (2018) 675. https://doi.org/10.1140/epjc/s10052-018-6131-3 [arXiv:1803.01853 [hep-ph]]

[2] P.A. Zyla et al. [Particle Data Group], Prog. Theor. Exp. Phys. 2020 (2020) no.8, 083C01 and 2021 update https://doi.org/10.1093/ptep/ptaa104

[3] A. S. Kronfeld, Annu. Rev. Nucl. Part. Sci. 62 (2012) pp. 265-284 https://doi.org/10.1146/annurev-nucl-102711-094942