*2.7. FT-IR Spectroscopic Analysis*

Glucans and pectins were analysed by FT-IR using a Bruker ALPHA FT-IR spectrometer equipped with a A241/D reflection module in 4000–400 cm−1 range, by following the standard KBr method. The sample was ground with KBr in the 1:40 ratio, and the resulting powder was hard-pressed into tablets. All experiments were carried out in triplicate at a spectral resolution of 4 cm−1 .


For the decay B0 → J*/ψ*KS the *CP* eigenvalue of the final state is *ηJ/ψ*KS = −1. Due to opposite CP eigenvalue of the KL the *CP* eigenvalue of the decay B0 → J*/ψ*KL is *ηJ/ψ*KL = +1. Both statements are strictly true only if one ignores the very small CP violation in the *K*0*K*0 mixing. The decay B0 → J*/ψ*KL is interesting as one expects exactly the same magnitude of the *CP* violation as for B0 → J*/ψ*KS but an opposite time behavior due to the opposite *CP* eigenvalue.

Experimentally the decay B0 → *J/ψK*0 *s* is easy to access. The *J/ψ* meson decays into two leptons and can be easily triggered and selected. Additional requirements on the displacement of the B vertex from the primary vertex allow an almost background-free reconstruction even at a hadron collider. The measurement has been extended to include other (*cc*¯) resonances for which the same *CP* behavior as for the golden mode is expected.

At the B-factories the time-dependent *CP* asymmetry is measured as function of the time difference *t* = *t*2 − *t*1. The most precise determination of the quantity sin *(*2*β)* has been performed by the Belle experiment using about 32,000 signal candidates with an average signal purity of 79% [98],

$$
\sin(2\beta) = 0.667 \pm 0.023 \pm 0.012 \dots
$$

The time-dependent CP asymmetry for the CP-odd and CP-even final states as measured by Belle are shown in Fig. 9.30. The BABAR experiment has recorded less signal candidates (15,481 candidates with a purity of 76% ) and obtained [99],

$$
\sin\left(2\beta\right) = 0.687 \pm 0.028 \pm 0.012\,,
$$

with a slightly larger statistical error than the Belle measurement.

The LHCb collaboration has also performed the measurement of sin *(*2*β)* by analysing the time dependent *CP* violation in B0 → J*/ψ*KS and B0 → *ψ(*2S*)*KS decays with 52,000 and 8000 signal candidates, respectively [100, 101]. As for the measurement of the *CP* violation in mixing the observed time dependent asymmetry has to be corrected for the production asymmetry between B0 and B0 of about 1%.

**Fig. 9.30** Flavor-tagged *t* distributions and raw CP asymmetries for the Belle measurements of sin 2*β*. The top plots show the B0 → *(cc)*¯ KS (*ηf* = −1) measurement and the bottom plots show the measurement for B0 → J*/ψ*KL (*ηf* = +1) The distributions are background subtracted. Figure adapted from [98]

The resulting value sin *(*2*β)* = 0*.*760 ± 0*.*034 [101] is slightly larger than the ones measured at the B-factories. It is important to notice that despite the much larger number of signal candidates used for the LHCb measurement the total error is comparable with the ones of Belle and BABAR. The reason is the significantly larger mistag probability at a hadron collider which leads to a reduction of the statistical power of the events. Using all available measurements of sin *(*2*β)* an average value of

$$\sin\left(2\beta\right) = 0.691 \pm 0.017$$

is obtained [87] which corresponds to a value of the phase *β* of,

$$
\beta = (21.9 \pm 0.7)^\circ \text{ or } \beta = (68.1 \pm 0.7)^\circ \text{,}
$$

where the ambiguity is not resolved.

#### **<IP_ADDRESS> B0 s Meson**

Analogue to the measurement of the CKM phase *β* in the B0 system, the measurement of a corresponding *CP* violating phase *βs*, defined in Sect. 9.7, can be performed in the B0 s system, assuming that the B0 s oscillation can be resolved. The *golden* channel used is the decay B0 s → J*/ψφ*. Due to its higher mass the B0 s system cannot be produced at the *ϒ(*4S*)* resonance and *βs* was therefore not accessible at the B-factories.

The two Tevatron experiments CDF and D0 have performed explorative studies of the decay B0 s → J*/ψφ* [102, 103] and also first measurements [104, 105] but the results have been limited by the number of recorded signal events. The first significant measurement of the phase6 *φs* = −2*βs* has been presented by the LHCb experiment in 2011 [106].

In the decay B0 s → J*/ψφ* the final-state is composed of two vector particles and angular momentum conservation allows for the relative angular momentum values *L* = 0*,* 1 or 2. The final-state *J/ψφ* is thus not a pure *CP*-state but a linear combination of *CP*-even and *CP*-odd eigenstates depending on the relative angular momentum *L* of the two vector mesons: *ηCP* = *ηCP (J/ψ)ηCP (φ)(*−1*)L* = *(*−1*)L*. The measurement of the time dependent CP asymmetry requires a separate treatment of the CP-odd and CP-even states which can be achieved statistically by analyzing the angular distribution of the final-state particles *J/ψ* → *μ*+*μ*− and *φ* → *K*+*K*−. To measure the angular distribution of the final state particles it is common to use the helicity basis. The three decay angles necessary to describe the decay are denoted by *(θK, θμ, ϕh)* and are defined in Fig. 9.31. The polar angle *θK* (*θμ*) is the angle between the *K*+ (*μ*+) momentum and the direction opposite to the B0 s momentum in the *K*+*K*− (*μ*+*μ*−) centre-of-mass system. The azimuthal angle between the *K*+*K*− and *μ*+*μ*− planes is *ϕh*.

The analysis of the B0 s → J*/ψφ* events is further complicated by a small fraction of non-resonant B0 s → J*/ψ*K+K− decays (*CP*-odd) which interfere with the *φ* meson decaying to two charged kaons. These non-resonant events need to be considered. Effectively one therefore studies B0 s decays to *J/ψK*+*K*−.

6In the Standard Model the phase *φs* determined by the measurement of the time-dependent *CP* asymmetry in the decay B0 s → J*/ψφ* equals in leading order to −2*βs* with *βs* being defined in Sect. 9.7. However, in case of new physics effects the two phases could differ. In the context of this textbook these differences are ignored.

**Fig. 9.31** Definition of helicity angles as discussed in the text. Figure taken from [107]

In the absence of *CP* violation in mixing the *CP*-odd and *CP*-even decays of the B0 s meson correspond to the heavy and light mass eigenstates. Thus separating statistically the polarization amplitudes in an angular analysis allows the measurement of the decay width *H* and *L* and thus the measurement of the different lifetimes of the two B0 s mass states.

With a data-sample of 96,000 B0 s → J*/ψ*K+K− events LHCb has performed a simultaneous analysis of the measured decay time and the three decay angles to determine the time-dependent *CP* asymmetry and the related *CP* phase *φs* [108]. Figure 9.32 shows the decay time distribution as well as the distributions of the three decay angles. One clearly sees how the *CP*-odd and *CP*-even components can be statistically separated using the decay angles. The decay time distribution shows the different decay behavior of the two *CP* components with the *CP*-even component decaying visibly faster. For the average decay width *s* and for the decay width difference *s* one obtains [108],

$$
\Gamma\_s = (\Gamma\_L + \Gamma\_H)/2 = 0.6603 \pm 0.0027 \pm 0.0015 \,\text{ps}^{-1},
$$

$$
\Delta\Gamma\_s = \Gamma\_L - \Gamma\_H = 0.0805 \pm 0.0091 \pm 0.0032 \,\text{ps}^{-1}.
$$

The simultaneous determination of the time-dependent *CP* asymmetry results in the following value for the *CP* violating phase *φs*,

$$
\phi\_s = -0.058 \pm 0.049 \pm 0.006 \dots
$$

The time-dependent *CP* violation in the B0 s system is thus very small and within the sensitivity of the measurement no *CP* violation has been seen. The phase *φs* can also be measured in the mode B0 s → J*/ψπ*+*π*−. The J*/ψπ*+*π*− final-state has been shown to be an almost pure (97.7%) CP-odd final state [109] and the CP violating phase can thus be extracted without the complication of an angular analysis. A much smaller signal sample with 27,000 events results into a comparable precision for *φs*, *φs* = −0*.*070±0*.*068±0*.*008 [110]. The analysis of these events however, requires the knowledge *s* as external input. Performing a combined analysis of the two decay channels [108] LHCb reports a value of *φs* of

$$
\phi\_s = -0.010 \pm 0.039 \dots
$$

**Fig. 9.32** Decay-time and helicity-angle distributions for B0 s → J*/ψ*K+K− decays (data points). The solid blue line shows the total signal contribution, which is composed of *CP*-even (long-dashed red), *CP*-odd (short-dashed green) and S-wave (dotted-dashed purple) contributions. Figure taken from [108]

The ATLAS and CMS collaborations have also performed measurements of the phase *φs* and of the width difference *s* [111, 112]. The overall experimental situation is summarized in Fig. 9.33 where also the theoretical expectation is shown. It should be noted that there is no ab-initio theory prediction for the phase *φs*. The prediction results from the measurement of sin *(*2*β)* and exploits the unitarity of the CKM matrix, see also Sect. 9.7. The average value for *φs*, *φs* = −0*.*021 ± 0*.*031 [130], agrees very well with the expectation *φs* = −0*.*0370 ± 0*.*0006 [113].

#### *9.6.6 Direct CP Violation*

B meson decays offer a broad decay phenomenology and are an ideal system to look also for large direct *CP* violation. Topologically very different tree-level and penguin decay amplitudes can result into the same final state and their interference can cause significant *CP* asymmetries. Considering the B meson decay *B* → *f*

**Fig. 9.33** 68% confidence level contours in the *φs*-*s* plane, showing the measurements from CDF, D0, ATLAS, CMS and LHCb together with their combination (white ellipse). The thin black line represents the Standard Model predictions. Figure taken from [130]

and its *CP* conjugates *B*¯ → *f*¯, direct *CP* violation or *CP* violation in the decay is measured as time-integrated asymmetry of the observed signal yields:

$$A\_{CP} = \frac{\Gamma(\mathcal{B} \to f) - \Gamma(\bar{\mathcal{B}} \to \bar{f})}{\Gamma(\mathcal{B} \to f) + \Gamma(\bar{\mathcal{B}} \to \bar{f})}$$

For a given B decay *B* → *f* , direct *CP* violation can only arise if at least two interfering decay amplitudes *A*1 and *A*2 exist and if these decay amplitudes carry different weak and strong phases, *ϕi* and *δi*. The total decay amplitude *Af* is given as sum

$$A\_f = |A\_1| \, e^{\varphi\_1 + \delta\_1} + |A\_2| \, e^{\varphi\_2 + \delta\_2} \dots$$

For the amplitudes of the charge conjugated process *A*¯ *f*¯ the weak phases change sign while the strong phases stay unchanged,

$$
\bar{A}\_{\bar{f}} = |A\_1| \, e^{-\varphi\_1 + \delta\_1} + |A\_2| \, e^{-\varphi\_2 + \delta\_2} \,.
$$

The expected signal yields depend on the interference of the amplitudes and the observable *CP* asymmetry is determined by the difference of the interference terms. The *CP* asymmetry is a function of the weak and strong phases as well as a function of the ratio *R* of the decay amplitudes,

$$A\_{CP} = \frac{2\sin(\phi\_1 - \phi\_2)\sin(\delta\_1 - \delta\_2)}{R + R^{-1} + \cos(\phi\_1 - \phi\_2)\cos(\delta\_1 - \delta\_2)} \text{ with } R \equiv \left| \frac{A\_1}{A\_2} \right| \,. \tag{9.211}$$

To observe large asymmetries the interfering amplitudes should have different weak and strong phases. While the weak phases, and in particular the weak phase difference, are determined by the CKM parameters, the theoretical determination of the strong phases is difficult. The B meson are subject to both, short and longdistance QCD effects which cannot be treated perturbatively.

In the following, two classes of B decays exhibiting large direct *CP* violation will be discussed: B decays to charmless final-states, i.e. to final-states without D mesons, and decays to charmed final-states of the type *B* → *DK*.

# **<IP_ADDRESS> Direct** *CP* **Violation in Charmless B Decays**

As sketched in Fig. 9.34 charmless two-body B decays such as *B*0 → *K*+*π*− can proceed via two topologically very different classes of decay amplitudes, tree and penguin processes, which carry different weak and in generally also different strong phases. Large direct *CP* violation is therefore expected for these decays.

First experimental evidence for direct *CP* violation in B meson decays was reported by the Belle collaboration in 2004 for the decay *B*0 → *π*+*π*− [114]. As the neutral B meson decays here to a *CP* eigenstate the formalism of (9.194) and (9.207) has to be applied, i.e. the observable time dependent asymmetry can be parametrized by the coefficient *Adir ππ* describing the effect of direct *CP* asymmetry and the coefficient *Amix ππ* to describe the effect of indirect *CP* violation through interference of diagrams with and without mixing. Belle observed a deviation of the coefficient *Adir ππ* from zero, i.e. direct *CP* violation, at the level of 3*.*2*σ*. The result suggested large interference effects between the relevant tree and penguin diagrams and was confirmed by subsequent measurements. Today, the most recent value of this coefficient is *Adir ππ* = 0*.*32 ± 0*.*04 [87] and confirms the early hints for direct *CP* violation in *B*0 → *π*+*π*−.

The two B-factory experiments, BABAR and Belle, also studied *B*0 → *K*+*π*− decays where large penguin contributions were expected. Both collaborations performed time-independent analyses and reported the observation of large direct *CP* violations for *B*0 → *K*+*π*− decays ( Babar: *ACP* = −0*.*133 ± 0*.*030 ± 0*.*009 [115]; Belle: *ACP* = −0*.*101±0*.*025±0*.*005 [116]). Both measurements represent the first doubtless observation of large direct *CP* violation in the B0 system. It is

**Fig. 9.34** Tree-level (left) and penguin (right) contribution to the charmless two-body decay *B*0 → *K*+*π*−

**Fig. 9.35** Invariant mass spectra of (**a**, **b**) *B*0 → *K*+*π*− the recorded decays and zoomed in (**c**, **d**) to show the *B*0 *s* → *K*−*π*+ decays. Panels (**a**) and (**c**) represent the *K*+*π*− invariant mass, whereas panels (**b**) and (**d**) represent the *K*−*π*+ invariant mass. The results of a mass fit describing the signals are overlaid. The main components contributing to the fit model are also shown. Figure taken from [118]

useful to remember that direct *CP* violation in the neutral kaon system is many orders smaller (O(10−6)). It is the effect of the large weak phases entering the B0 decay amplitudes which produces the much larger direct *CP* violation.

Already with the very first data-set recorded in 2011 and corresponding to only 0.35 fb−1 of data, the LHCb experiment has repeated the measurement of *ACP* for the channel *B*0 → *K*+*π*− [117]. LHCb's excellent resolution of the reconstructed *K*±*π*∓ invariant mass allows to distinguish between a decaying B0 and B0 s and thus also probed the decay *B*0 *s* → *K*−*π*+. The measurement of *ACP (B*0 *s* → *K*−*π*+*)* = 0*.*27 ± 0*.*08 ± 0*.*02 provided the first evidence (3*.*3*σ* ) for direct *CP* violation in the B0 s system. The measurement was repeated with more data and the observed direct *CP* violation *ACP (B*0 *s* → *K*−*π*+*)* = 0*.*27 ± 0*.*04 ± 0*.*01 [118] confirmed the earlier result. Figure 9.35 shows the invariant mass distribution for the recorded *B*0 *(s)* → *Kπ* decays. Clear differences in the number of recorded decays for the B and the anti-B decays are observed.

**Fig. 9.36** Tree-level Feynman diagrams of the decays *B*− → *D*0*(D*0 *)K*−. The left diagram implies a *b* → *ucs*¯ transition, and is strongly suppressed by the small value of |*Vub*|. The right diagram proceeds via a transition *b* → *cus*¯

#### **<IP_ADDRESS> Direct** *CP* **Violation in** *B* **→** *DK* **Decays and Measurement of CKM Phase γ**

While direct *CP* violation in B decays is large and experimentally easy accessible, the theoretical interpretation is non-trivial. The calculation of the non-perturbative hadronic effects and the resulting strong phases is difficult. A way to use direct *CP* asymmetries to constrain the CKM parameters is therefore a simultaneous determination of the weak (CKM) phases and the hadronic nuisance parameters including the strong phases. This approach is followed to determine the CKM phase *γ* as defined in Sect. 9.7. Ignoring higher order terms, *γ* is in good approximation equal to the phase of the CKM matrix element *Vub* (*Vub* = |*Vub*| *e*−*iγ* ).

The phase can be probed using *B*− → *D*0*(D*0 *)K*− decays with either a *b* → *ucs*¯ (*D*0 *K*−) or *b* → *cus*¯ (*D*0*K*−) tree-level quark transition as depicted in Fig. 9.36. The two decays can be described by three parameters: *rB* (≈ *O(*0*.*1*)*), the absolute value of the ratio of both amplitudes; *δB* the strong phase difference; and the CKM phase *γ* . In order to observe the *CP* violating interference between the two amplitudes the *D*0 and the *D*0 mesons emerging in case of the two different amplitudes should decay into a common final state *fD*, i.e. *D, D*¯ → *fD*. The size of the *CP* violating interference which provides the sensitivity to *γ* is proportional to *rB*.

The possibility of observing direct *CP* violation in *B*− → *DK*− decays was first discussed in the 1980s [119, 120] using decays of the *D(D)*¯ to neutral kaons and pions. Since then, several methods have been proposed which can be grouped according to the choice of the final state.

#### 9 Symmetry Violations and Quark Flavour Physics 605

Simultaneous fits to several observables, *CP* asymmetries or ratios of suppressed to favored modes, allow the determination of the decay parameters including the CKM phase *γ* . The main issue with all methods is the small overall branching fraction of the observable decays which range from 5 × 10−5 to 5 × 10−9 [127]. The precise determination of the CKM phase *γ* therefore requires a very large datasample. The B factories have pioneered different methods to determine *γ* , however the achieved overall statistical precision on *γ* was limited. Combining the different methods and quoting a single result for *γ* , BABAR and Belle report the following values [127]:

$$\begin{array}{l} \gamma = (69 \pm 17)^{\circ} \text{ (BABAR)}, \\ \gamma = (68 \pm 14)^{\circ} \text{ (Belle)}. \end{array}$$

The breakthrough towards a precision determination of the phase *γ* came when the LHCb experiment was able to measure even very rare doubly Cabibbosuppressed decays such as *B*− → *π*−*K*+ *D K*− and its charge-conjugated counter-parts with sufficiently high statistics to observe *CP* asymmetries. Figure 9.37 shows as an example the decay *B*− → *π*−*K*+ *D K*− as well as the decay *B*− → *π*−*K*+ *D π*− which also provides sensitivity to *γ* together with the corresponding charge conjugated decays [128]. The observed yields clearly signal direct *CP* violation for both channels. LHCb has analysed *B*+ → *DK*+ decays with a multitude of different decay modes of the neutral D, *B*+ → *D*∗*K*+ and *B*+ → *DK*∗+ decays and *B*0 → *DK*∗ decays. In addition time-dependent analyses of *B*0 *s* → *D*∓ *s K*± and *B*0 → *D*∓*π*± are performed and measure combinations of the CKM phases *γ* and *βs* or *β*, respectively. The combination of the different LHCb measurements [129] finally result into a value of

$$\chi = (\text{74.0}\_{-\text{5.8}}^{+\text{5.0}})^{\circ} \text{ (LHCb)},\tag{9.212}$$

in perfect agreement with the early measurements by BABAR and Belle. For the average of all *γ* measurements one finds [87],

$$\gamma = (\Im \mathfrak{J}\_{-\mathfrak{F}, \mathbb{I}}^{+4, 2})^\circ . \tag{9.213}$$

**Fig. 9.37** Invariant mass distributions of selected *B*± → *π*±*K*∓ *D h*± decays, separated by charge. The dashed pink line left of the signal peak shows partially reconstructed *B*0 *s* → *K*+*π*− *D K*−*π*+ decays, where the bachelor pion is missed. For the two channels, one clearly observes a yield difference, i.e. *CP* asymmetry, between the two charge-conjugated channels. Figure taken from [128]

#### **9.7 Weak Quark Mixing and the CKM Matrix**

The observation of direct *CP* violation in the neutral K meson and the confirmation of large *CP* violating effects in the neutral B meson system provided the experimental evidence of weak quark-mixing as the primary source of *CP* violation in the hadron sector. A multitude of precision measurements confirmed the prediction of the quark-mixing paradigm of the Standard Model [57] and led to the Nobel prize for Kobayshi and Maskawa. Despite this success of experimental and theoretical quark flavor physics we know today that other sources of *CP* violation must exists to explain the baryon asymmetry of our universe.

#### *9.7.1 Quark-Mixing Matrix*

Historically quark-mixing was introduced by Cabibbo [56] to explain the different coupling strength of hadronic currents in weak decays of neutrons and pions compared to strangeness changing processes, such as the decay of K mesons and *Λ* hyperons. Cabibbo postulated—expressed in today's notation—that the weak eigenstates of the then known quarks with charge −1*/*3 were not the flavor eigenstates *d* and *s* but a linear combination, rotated by an angle *θ*, the Cabibbo angle:

$$d\_c = d\cos\theta + \mathbf{s}\sin\theta \,\,. \,\tag{9.214}$$

The GIM mechanism [131], introduced to cancel the KL → *μ*+*μ*− amplitude, required the orthogonal state

$$\mathbf{s}\_c = -d \begin{array}{c} \sin \theta + \mathbf{s} \ \cos \theta \end{array} , \tag{9.215}$$

and an additional charge +2*/*3 quark, the charm quark, complementing the two quark doublets.

We know today that three generations of up-type quarks *ui* (*i* = 1*,* 2*,* 3: u-, c-, t-quark) with electrical charge +2*/*3 and three generations of down-type quarks *di* (*i* = 1*,* 2*,* 3: d-, s-, b-quark) with electrical charge −1*/*3 exist. In the Standard Model, the masses and mixing of quarks arise from the Yukawa interaction with the Higgs condensate which couples the left-handed quark fields *uL* and *dL* to the right-handed quark fields *uR* and *dR* (for better readability the generation index is suppressed). After spontaneous breaking of the electroweak symmetry the Yukawa terms give rise to masses and mixing:

$$\mathcal{L}\_{\text{Yukawa}} = -\frac{v}{\sqrt{2}} \left( \bar{d}\_L \mathbf{Y\_d} d\_R + \bar{\mu}\_L \mathbf{Y\_u} \mu\_R \right) + h.c.$$

The Yukawa matrices **Yd** and **Yu** are complex 3 × 3 matrices in generation or flavor space and do not need to be diagonal. Indeed, in the Standard Model they are not, and as a consequence the flavor states are not equal to the mass eigenstates of the quarks. The mass eigenstates *u*˜*L,R* and *d*˜ *L,R* are obtained by unitary transformations of the above quark flavor states: *u*˜*A* = **VA***,***u***uA* and *d*˜ *A* = **VA***,***d***dA* (with chirality index *A* = *L, R* for left and right-handed quark fields, and suppressed generation indices). The unitary matrices **VA***,***u** and **VA***,***d** diagonalize the Yukawa matrices and one obtains the diagonal quark mass matrices for the up- and down-type quarks,

$$\mathbf{M}\_{\mathbf{u}} = \text{diag}(m\_{\boldsymbol{u}}, m\_{\boldsymbol{c}}, m\_{\boldsymbol{t}}) = \frac{\boldsymbol{\upsilon}}{\sqrt{2}} \mathbf{V}\_{\mathbf{L}, \mathbf{u}} \mathbf{Y}\_{\mathbf{u}} \mathbf{V}\_{\mathbf{R}, \mathbf{u}}^{\dagger},$$

$$\mathbf{M}\_{\mathbf{d}} = \text{diag}(m\_{\boldsymbol{d}}, m\_{\boldsymbol{s}}, m\_{\boldsymbol{b}}) = \frac{\boldsymbol{\upsilon}}{\sqrt{2}} \mathbf{V}\_{\mathbf{L}, \mathbf{d}} \mathbf{Y}\_{\mathbf{d}} \mathbf{V}\_{\mathbf{R}, \mathbf{d}}^{\dagger}.$$

The quark masses will appear as usual Dirac mass terms in the above Yukawa part of the Lagrangian:

$$\mathcal{L}\_{\text{Yukawa}} = -\bar{\tilde{d}}\_L \mathbf{M}\_{\mathbf{d}} \tilde{d}\_R - \bar{\tilde{u}}\_L \mathbf{M}\_{\mathbf{d}} \tilde{u}\_R + h.c.$$

If the up-type and and down-type Yuakawa matrices **Yu** and **Yd** cannot be diagonalized simultaneously by the same transformations there is a net effect of the change of the quark basis. The charged current terms of the Standard Model Lagrangian combining left-handed up and down-type quarks therefore get a flavor structure imprinted which is described by the Cabibbo Kobayshi Maskawa (CKM) quark-mixing matrix

$$\mathbf{V\_{CKM}} = \mathbf{V\_{L,u}} \mathbf{V\_{L,d}^{\dagger}} \cdot$$

The charge current terms expressed in the mass eigenstates have the form

$$\mathcal{L}\_{\mathcal{CC}} = -\frac{g}{\sqrt{2}} \left( \bar{\tilde{\mu\_L}} \gamma^\mu W\_\mu^+ \mathbf{V\_{CKM}} \tilde{d}\_L + \bar{\tilde{d}}\_L \gamma^\mu W\_\mu^- \mathbf{V\_{CKM}}^\dagger \tilde{u}\_L \right) \dots$$

Here, the matrix element *(***VCKM***)ij* connects a left-handed up-type quark of the *i*th generation to a left-handed down-type quark of the *j* th generation. The matrix elements are therefore expressed using flavor indices:

$$\mathbf{V\_{CKM}} = \begin{pmatrix} V\_{ud} \ V\_{us} \ V\_{ub} \\ V\_{cd} \ V\_{cs} \ V\_{cb} \\ V\_{td} \ V\_{ts} \ V\_{tb} \end{pmatrix}$$

For a non-diagonal CKM matrix the charged quark currents are thus inter-generation flavor changing currents. In analogy to the flavor eigenstates introduced by Cabibbo it is usual to absorb the CKM matrix by introducing for the down-type quarks the weak quark eigenstates *d*˜ = **VCKM***d*˜ *L*.

Since it is the product of unitary matrices the CKM matrix itself is unitary, i.e. **VCKMVCKM**† = **1**, and its elements are in general complex. The number of parameters of a general unitary 3 × 3 matrix is nine, three rotation angles and six phases. By rephasing the quark mass eigenstates *q*˜ → *eiαq q*˜ one can remove five phases, corresponding to the five independent phase differences between the quarks, and leaving one *CP* violating phase *δ*. The usual parametrization of the CKM matrix uses the three rotation angles *θ*12, *θ*23, *θ*13 and the phase *δ*:

$$\mathbf{V\_{CKM}} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & c\_{23} & s\_{23} \\ 0 & -s\_{23} & c\_{23} \end{pmatrix} \begin{pmatrix} c\_{13} & 0 \ s\_{13} \mathbf{e}^{-i\delta} \\ 0 & 1 & 0 \\ -s\_{12} \mathbf{e}^{i\delta} & 0 & c\_{13} \end{pmatrix} \begin{pmatrix} c\_{12} & s\_{12} & 0 \\ -s\_{12} & c\_{13} & 0 \\ 0 & 0 & 1 \end{pmatrix} \tag{9.216}$$

$$= \begin{pmatrix} c\_{12}c\_{13} & s\_{12}c\_{13} & s\_{13}\mathbf{e}^{-i\delta} \\ -s\_{12}c\_{23} - c\_{12}s\_{23}s\_{13}\mathbf{e}^{i\delta} & c\_{12}c\_{23} - s\_{12}s\_{23}s\_{13}\mathbf{e}^{i\delta} & s\_{23}c\_{13} \\ s\_{12}s\_{23} - c\_{12}c\_{23}s\_{13}\mathbf{e}^{i\delta} & -c\_{12}s\_{23} - s\_{12}c\_{23}s\_{13}\mathbf{e}^{i\delta} & c\_{23}c\_{13} \end{pmatrix},$$

where *sij* = sin *θij* and *cij* = cos *θij* . in the Standard Model, the phase *δ* is responsible for all *CP* violating phenomena in quark-flavor changing processes.

#### 9 Symmetry Violations and Quark Flavour Physics 609

The rotation angles are defined and labeled in a way which relates to the mixing of two specific generations. In the limit *θ*23 = *θ*13 = 0 the third generation decouples, and the situation reduces to the usual Cabibbo mixing of the first two generations, with *θ*12 identified as the Cabibbo angle. The angles *θ*12, *θ*23, *θ*13 can all be chosen to lie in the first quadrant, i.e. *sij , cij >* 0, by appropriate redefinition of the quark field phases.

From measurements it is known that 1 *s*12 *s*23 *s*13 . It is therefore common to use a parametrization of the CKM matrix that emphasizes this hierarchy. In the Wolfenstein parametrization one defines

$$\lambda = s\_{12} = \frac{|V\_{us}|}{\sqrt{|V\_{ud}|^2 + |V\_{us}|^2}}, \quad A\lambda^2 = s\_{23} = \lambda \left| \frac{V\_{cb}}{V\_{us}} \right|, \quad A\lambda^3 (\rho + i\eta) = s\_{13}e^{i\delta} = V\_{ub}^\*, \quad i\delta = \frac{1}{2}$$

where *λ* is the sine of the Cabibbo angle (sin *θ* ≈ 0*.*22) and the real numbers *A*, *ρ* and *η* are of order unity. With these parameters the CKM matrix can be expressed in powers of *λ* and takes the convenient form

$$V = \begin{pmatrix} 1 - \lambda^2/2 & \lambda & A\lambda^3(\rho - \mathrm{i}\eta) \\ -\lambda & 1 - \lambda^2/2 & A\lambda^2 \\ A\lambda^3(1 - \rho - \mathrm{i}\eta) & -A\lambda^2 & 1 \end{pmatrix} + \mathcal{O}(\lambda^4) \,. \tag{9.217}$$

As the definition of *(ρ* + *iη)* depends on the phase convention, one often introduces the parameters *(ρ*¯ + *iη)*¯ , defined by

$$A\lambda^3 \left(\rho + i\eta\right) = \frac{A\lambda^3(\bar{\rho} + i\bar{\eta})\sqrt{1 - A^2\lambda^4}}{\sqrt{1 - \lambda^2}\left[1 - A^2\lambda^4(\bar{\rho} + i\bar{\eta})\right]}.\tag{9.218}$$

This definition ensures that *(ρ*¯ + *iη)*¯ = −*(VudV* ∗ *ub)(VcdV* ∗ *cb)* is phase convention independent and the CKM matrix written in the parameters *λ*, *A*, *ρ*¯ and *η*¯ is unitary to all orders in *λ*. 7

The elements of the CKM matrix are fundamental parameters of the Standard Model and need to be experimentally determined. The unitarity condition of the CKM matrix imposes a set of relations between the matrix elements:

$$\sum\_{i} V\_{ij} V\_{ik}^{\*} = \delta\_{jk} \quad \text{with} \quad jk = \{dd, ds, db, ss, sb, bb\} \tag{9.219}$$

$$\sum\_{j} V\_{lj} V\_{kj}^{\*} = \delta\_{lk} \text{ with } ik = \{ \mu u, \mu c, \mu t, \, cc, ct, tt \} \tag{9.220}$$

7To *O(λ*2*)* one finds *ρ*¯ = *ρ(*1 − *λ*2*/*2*)* .

**Fig. 9.38** Sketch of the Unitarity Triangle (UT). Figure taken from [133]

The six vanishing combinations (*j* = *k, i* = *k*) describe triangles in the complex plane. The area of all triangles is given by half of the Jarlskog invariant *J* [132],

$$J \cong \left[ V\_{lj} V\_{kl} V\_{il}^\* V\_{kj}^\* \right] = J \sum\_{m,n=1}^3 \epsilon\_{ikm} \epsilon\_{jln} \,, \tag{9.221}$$

where one representation of Eq. (9.221) reads for instance *J* = & *VusVcbV* ∗ *ubV* ∗ *cs* . Expressed in the parameters of the standard CKM representation one finds *J* = −*c*12 *c*13 *c*23 *s*12 *s*13 *s*23 sin *δ*. A nonvanishing CKM phase and hence *CP* violation requires *J* = 0.

While four of the six unitarity triangles are degenerated and rather slim, only two triangles have approximately equal sides, of which one is usually referred as the *Unitarity Triangle* (UT),

$$V\_{ud}V\_{ub}^{\*} + V\_{cd}V\_{cb}^{\*} + V\_{ld}V\_{tb}^{\*} = 0 \ . \tag{9.222}$$

Commonly one normalizes the triangle basis to unity by dividing each side by *VcdV* ∗ *cb* to obtain a triangle with vertices exactly at *(*0*,* 0*)*, *(*0*,* 1*)* and the apex at *(ρ,*¯ *η)*¯ . The UT is sketched in Fig. 9.38. As can be seen from Fig. 9.38 the angles of the UT are

$$\alpha = \arg \left( -\frac{V\_{td} V\_{tb}^\*}{V\_{ud} V\_{ub}^\*} \right) \tag{9.223}$$

$$\beta = \arg\left(-\frac{V\_{cd}V\_{cb}^\*}{V\_{td}V\_{tb}^\*}\right) \tag{9.224}$$

$$\gamma = \arg\left(-\frac{V\_{ud}V\_{ub}^\*}{V\_{cd}V\_{cb}^\*}\right) \tag{9.225}$$

Often a different naming convention, also shown in Fig. 9.38, is used to label the three angles. The UT angles are constraint by the *CP* observables discussed in Sect. 9.6:

$$
\beta = (21.9 \pm 0.7)^{\circ} \tag{9.226}
$$

$$\boldsymbol{\gamma} = (\mathbf{73.5}\_{-\mathbf{5.1}}^{+4.2})^\circ \tag{9.227}$$

The measurement of the angle *α* has not been discussed in Sect. 9.6. It is measured using the observation of time-dependent *CP* violation in the decays *B*0 → *ππ*, *ρρ*, and *πρ* by BABAR and Belle. The average of this measurements result into [133, 134],

$$
\alpha = (84. \mathbf{5}\_{-5.2}^{+5.9})^\circ \,. \tag{9.228}
$$

The sum of three angles, *α* + *β* + *γ* = *(*180 ± 7*)* ◦, is consistent with the expectation and represents a first test of the unitarity of the CKM matrix. An additional constraint of the CKM phases and thus of the UT angles comes from the measurement of time-dependent *CP* violation in B0 s decays and the measurement of the phase *βs* (with *φs* = −2*βs*, see Sect. 9.6),

$$\beta\_s = \arg \left( -\frac{V\_{ts} V\_{tb}^\*}{V\_{cs} V\_{cb}^\*} \right) = (0.60 \pm 0.89)^\circ. \tag{9.229}$$

In addition to the measurements of *CP* violation in the B meson systems also the measurements of *CP* violation in the *K*0*K*¯ 0 mixing, |*K*| = *(*2*.*233±0*.*015*)*×10−3 (see Sect. 9.5) provides information about the CKM matrix. The measurement of |*K*| can be translated into an approximate hyperbolic constraint on the apex (*ρ*¯, *η*¯) of the UT [135].

The sides of the unitarity triangle are accessible by measuring decay rates and mixing frequencies. In the following the experimental determination of the CKM elements as well as a test of the unitarity of the CKM matrix is discussed.

#### *9.7.2 Determination of the CKM Matrix Elements*

1. |*Vud* |. Precise determinations of |*Vud* | are available from nuclear beta decays, from the decay of the free neutron and from semileptonic pion decays (*π*+ → *π*0*e*+*ν*). The most precise value results from an analysis of superallowed 0+ → 0+ nuclear beta decays which are pure vector transitions. The measurements of the transition energies, the partial branching fractions, and the half-lives of the parent nuclei together with radiative and isospin-symmetry-breaking corrections allow the determination of the corrected *Ft*-value, from which, by using the muon life-time, |*Vud*| can be determined. The average of the fourteen most precise determinations yield [136],

$$|V\_{\rm ud}| = 0.97417 \pm 0.00021 \,\text{J}$$

where the error is dominated by the theoretical uncertainty stemming from the nuclear structure and radiation correction.

The theoretical uncertainties in extracting a value of |*Vud* | from neutron decays are significantly smaller than those for the superallowed decays, however the value depends on the ratio between axial-vector and vector couplings (*gA* ≡ *G*A*/G*V) and on the neutron lifetime. Using the most recent measurements [133] the following value for |*Vud* | is obtained,

$$|V\_{\rm ud}| = 0.9763 \pm 0.0016\,,$$

with the error dominated by the *gA* uncertainty.

An alternative approach is the measurement of the very small (*O(*10−8*)*) branching ratio of the pion beta-decay *π*+ → *π*0*e*+*ν*. The value normalized by a very precise theoretical prediction for *π*+ → *e*+*ν* and yields [137]

$$|V\_{ud}| = 0.9749 \pm 0.0026\dots$$

The error here stems mainly from the measurement of the rare process.

2. |*Vus*|. Earlier measurements of |*Vus*| from kaon decays have used KL → *πeν* to extract the product of |*Vus*| and the form factor |*Vus*|*f*+*(*0*)* at *q*2 = 0. The most recent data provide enough experimental constraints to also use decays to muons as well as decays of KS and K±. Averaging results from KL → *πeν*, KL → *πμν*, K± → *π*0*e*±*ν*, K± → *π*0*μ*±*ν* and KS → *πeν* yields the value |*Vus*|*f*+*(*0*)* = 0*.*2165 ± 0*.*0004 [133]. Lattice QCD calculations of *f*+*(*0*)* have been carried out for different numbers of quark flavors. The form-factor average, *f*+*(*0*)* = 0*.*9704±0*.*0032, of the (2+1)-flavor lattice calculations [138] is in good agreement with a classical calculation [139]. With this value one obtains,

$$|V\_{us}| = 0.2231 \pm 0.0008\dots$$

The lattice calculation of the ratio of kaon and pion decay constant, *fK/fπ* = 1*.*1933 ± 0*.*0029 [138], allows the determination of the ratio |*Vus/Vud* | from *K* → *μν* and *π* → *μν* decays. Using the precise measurement of the *K* → *μν* branching fraction by the KLOE collaboration [140] results in |*Vus*| = 0*.*2253 ± 0*.*0007.

An alternative determination of |*Vus*| uses hadronic *τ* decays to strange hadrons. The average of the measured inclusive and exclusive branching fractions yields |*Vus*| = 0*.*2216 ± 0*.*0015 [134].

3. |*Vcd* |. First determinations of |*Vcd* | came from neutrino scattering experiments. The difference of the ratio of double-muon to single-muon production for neutrino and antineutrino scattering is depending on the charm production cross section and thus on |*Vcd* | 2, as well as on the semileptonic (muonic) branching ratio of the produced charm mesons, *B*¯ *μ*. The method was first used by the CDHS group [141] but has also been applied by CCFR [142] and by CHARM II [143]. Averaging the results is complicated, also because *B*¯ *μ* is an effective quantity which depends on the specific neutrino beam characteristics. One finds *B*¯ *μ*|*Vcd* | 2 = *(*0*.*463 ± 0*.*034*)* × 10−2 [144] and using the average value of *B*¯ *μ* = 0*.*087 ± 0*.*005 one obtains [133]

$$|V\_{cd}| = 0.230 \pm 0.011\dots$$

Similar to |*Vus*|, |*Vcd* | can also be extracted from semileptonic (*D* → *π ν*) and leptonic (*D*+ → *μ*+*ν*) charm decays. Also here, QCD lattice calculations are used to determine the relevant form factors *f Dπ* + *(q*2 = 0*)* and *fD* [138]. Using the average of the branching fraction measurements from BABAR, Belle, BES III and CLEO-c for *D* → *π ν* [134] results into [133]

$$|V\_{cd}| = 0.2140 \pm 0.0029 \pm 0.0093,$$

where the first uncertainty is experimental and the second stems from the theoretical uncertainty of the form factor calculation. The measurements of the leptonic branching ratio *D*+ → *μ*+*ν* by BES III and a CLEO-c results into [134]

$$|V\_{cd}| = 0.2164 \pm 0.0050 \pm 0.0015 \,,$$

where also here the first error is experimental while the second describes the uncertainty of the form factor calculation. For the average of the three different determinations of |*Vcd* | the Particle Data Group [133] quotes a value of

$$|V\_{cd}| = 0.218 \pm 0.004\dots$$

4. |*Vcs*|. Measurements of semileptonic decays of D mesons to kaons *D* → K *ν* as well as the measurement of leptonic decays of *Ds* mesons, *D*+ *s* → *μ*+*ν*, together with the corresponding form factors from lattice calculations allow the determination of |*Vcs*|. Branching fraction measurements have been performed by Belle, BABAR, CLEO-c and BES III, and are averaged in [134]. From the semileptonic measurements one obtains |*Vcs*| = 0*.*967 ± 0*.*025 where the error is dominated by the theoretical uncertainty of the form factor calculations. The average of the leptonic measurements results into |*Vcs*| = 1*.*006 ± 0*.*019 where the dominating uncertainty is experimental. For the average of both values the Particle Data Group [133] reports a values of

$$|V\_{cs}| = 0.997 \pm 0.017.$$

5. |*Vcb*|. This matrix element is determined from semileptonic B decays to *D* or *D*∗ mesons. Two experimentally and theoretically different approaches are used. The *inclusive* approach measures the inclusive semileptonic decay rate to any charmed final state together with the moments of the leptonic energy and the hadronic invariant mass spectra. An operator product expansion within the Heavy Quark Effective Theory (HQET) allows the calculation of decay rates and the energy and mass spectra in dependence of expansion parameters *αs* and the inverse of the heavy quark mass. The simultaneous measurement of several distributions over-constrains the physical parameters and allows a determination of |*Vcb*|. An analysis and an averaging of existing measurements is performed in [145] and leads to |*Vcb*| = *(*42*.*2 ± 0*.*8*)* × 103 with uncertainties arising mainly from higher-order perturbative and non-perturbative corrections.

In the *exclusive* approach semileptonic B decays to exclusive channels containing *D* and *D*∗ mesons are studied. In the infinite quark mass limit with *mb , mc Λ*Q*CD*, heavy quark symmetry predicts that all form-factors are given by a single function which depends on the product of the four-velocities *v* of the *B* and *v* of the *D(*∗*)* (mesons) and are normalized at the point of maximum momentum transfer to the lepton system (*v* · *v* = 1). The matrix element |*Vcb*| is obtained from an extrapolation to this point. The precise determination of normalization and the shape of the form factor function requires additional corrections calculated using HQET. Reference [145] quotes |*Vcb*| = *(*41*.*9 ± 2*.*0*)* × 103 as the average exclusive value of |*Vcb*|.

The inclusive and the exclusive determination of |*Vcb*| are in agreement and [145] performs the average of both values:

$$|V\_{cb}| = (42.2 \pm 0.8) \times 10^3 \,\text{J}$$

6. |*Vub*|. Similar to |*Vcb*| also |*Vub*| is determined analysing inclusive and exclusive semileptonic B decays with a *b* → *u ν*¯ transition. The determination of |*Vub*| from inclusive decays however suffers experimental and theoretical difficulties. The total inclusive decay rate is hard to measure due to the large background from CKM-favored *b* → *c ν*¯ transitions. Therefore, strong kinematic cuts are introduced to suppress these background contributions. The restriction to tight kinematic regions however complicates the theoretical description. The calculation of partial decay rates in the various kinematic regions requires the introduction of non-perturbative distribution functions—the so called shape functions—to describe the effect of hadronic physics. At leading order there is only a single shape function which can be determined using inclusive *B*¯ → *Xsγ* decays. Subleading effects are considered using different theoretical models based on Heavy Quark Expansion (HQE). A recent summary of the |*Vub*| values extracted within different models from measurements by BABAR, BELLE and CLEO can be found in [145]. All calculations give similar values for |*Vub*| and similar error estimates. As average [145] quotes

$$|V\_{ub}| = (4.49 \pm 0.1 \mathcal{S}\_{-0.17}^{+0.16}) \times 10^{-3} \text{ J}$$

To consider different theoretical treatments the authors assign an additional quadratic error of ±0*.*17 × 10−3.

The determination of |*Vub*| from exclusive decays, such as *B* → *π ν* decays, suffers experimentally from very small signal yields and requires the theoretical determination of the corresponding form factors. Lattice form factor calculations are available for the high *q*2 regions. So called light-cone QCD sum rules (LCSR) are applicable for the low *q*2 region. A simultaneous fit to experimental *B* → *π ν* data and lattice results as function of *q*2 together with additional constraints from LCSR results into

$$|V\_{ub}| = (3.67 \pm 0.09 \pm 0.12) \times 10^{-3} \text{ [134]}.$$

The inclusive and exclusive determinations of |*Vub*| are largely independent and the large discrepancy between both methods remains a puzzle. Inflating the errors to account for this discrepancy [145] quotes an average of

$$|V\_{ub}| = (3.94 \pm 0.36) \times 10^{-3}$$

The LHCb experiment has used the ratio of the two baryonic decays *Λb* → *pμ*−*ν*¯ and *Λb* → *Λcμ*−*ν*¯ to extract the ratio |*Vub/Vcb*| = 0*.*083 ± 0*.*006 [146]. The *q*2-dependent form factor ratio had to be taken into account to consider the different kinematical ranges of the two decays. Using the above average for |*Vcb*| one obtains

$$|V\_{ub}| = (3.50 \pm 0.26) \times 10^{-3} \text{ J}$$

7. |*Vt b*|. An experimental determination of |*Vt b*| without assuming unitarity is possible using the production of single-top quarks. Single-top quark production cross-sections have been measured by the Tevatron experiments CDF and D0, and at LHC by ATLAS and CMS. Using these measurements [133] quotes the value

$$|V\_{lb}| = 1.019 \pm 0.025\dots$$

8. |*Vt d* | and |*Vt s*|. The two CKM elements |*Vt d*| and |*Vt s*| are expected to be very tiny and tree-level decays of the top-quark to a *d*-quark or *s*-quark will be very difficult to measure. However, both elements are accessible through the measurements of the mixing frequency of B0 and B0 s mesons (see Sect. 9.6). Using the most recent lattice QCD results for the hadronic factors, *fBd B*ˆ*Bd* = *(*219±14*)* MeV and *fBs B*ˆ*Bs* = *(*270±16*)* MeV [138], together with the world averages for the mixing frequencies *Δmd* and *Δms* results into [133]

$$|V\_{td}| = (8.1 \pm 0.5) \times 10^{-3} \quad \text{and}$$

$$|V\_{ls}| = (39.4 \pm 2.3) \times 10^{-3} \text{ ,}$$

where the uncertainties are dominated by the lattice uncertainties of the hadronic factors. Several uncertainties are reduced when calculating the ratio

$$\xi = \left( f\_{B\_{\rm s}} \sqrt{\hat{B}\_{B\_{\rm s}}} \right) / \left( f\_{B\_{\rm d}} \sqrt{\hat{B}\_{B\_{\rm d}}} \right) = 1.239 \pm 0.046 \text{ J}$$

The ratio |*Vt d*|*/*|*Vt s*| is therefore stronger constraint [133],

$$|V\_{td}|/|V\_{ts}| = 0.210 \pm 0.001 \pm 0.008 \dots$$

Using the independent measurements of the CKM elements the unitarity of the CKM matrix can be checked. One obtains for the first two rows [133],

$$\begin{aligned} \left| \left| V\_{ud} \right| ^2 + \left| V\_{us} \right| ^2 + \left| V\_{ub} \right| ^2 &= 0.9994 \pm 0.0005 \,, \\ \left| \left| V\_{cd} \right| ^2 + \left| V\_{cs} \right| ^2 + \left| V\_{cb} \right| ^2 &= 1.043 \pm 0.034 \,, \end{aligned} $$

which agrees well with the unitarity assumption. In addition, the direct measurement of |*Vt b*| leaves little room for mixing of the top into unknown other states.

# *9.7.3 Global Analysis and Test of the Unitarity of the CKM Matrix*

The available information on the magnitude and the phases of the CKM elements can be analysed by a global fit to all CKM parameters. The most precise information is obtained imposing Standard Model constraints such as the unitarity of the CKM matrix and the existence of exactly three quark generations. Input parameters of the global analysis are, beside the experimental measurements, also theoretically determined hadronic parameters with sometimes large errors.

Different approaches exist to combine the experimental data and to treat the experimental and theoretical errors. The CKM-Fitter group [135, 147] is using a frequentist's framework based on a *χ*2 analysis. In these fits the frequentist's treatment is also applied to the theoretical errors. The UTfit group [148] uses a baysian approach for all errors. The two different statistical approaches lead to very similar results and here only the results of [135] are presented.

Figure 9.39 shows the experimental constraints in the *(ρ,*¯ *η)*¯ plane. Indicated is the unitarity triangle with its angles *α*, *β* and *γ* . The different measurements clearly limit the apex of the triangle to a small (dashed) region. For the Wolfenstein parameters introduced in Eq. (9.217) the global fit [133, 135] gives

$$
\lambda = 0.22453 \pm 0.00044 \,, \qquad A = 0.836 \pm 0.015 \,, \qquad (9.230)
$$

$$
\bar{\rho} = 0.122^{+0.018}\_{-0.017}, \qquad \qquad \qquad \bar{\eta} = 0.35 \mathfrak{F}^{+0.012}\_{-0.011}. \tag{9.231}
$$

**Fig. 9.39** Experimental constraints of the unitarity triangle. Shaded areas correspond to 95% C.L. Figure taken from [133]

Alternatively one can give the fit results of the magnitudes of all nine elements of the CKM matrix:

$$\mathbf{V\_{CKM}} = \begin{pmatrix} 0.97446 \pm 0.00010 & 0.22452 \pm 0.00044 & 0.00365 \pm 0.00012 \\ 0.22438 \pm 0.00044 & 0.97359^{+0.00010}\_{-0.00010} & 0.04214 \pm 0.00076 \\ 0.00896^{+0.00024}\_{-0.00023} & 0.04133 \pm 0.00074 & 0.999105 \pm 0.000032 \end{pmatrix} \tag{9.232}$$

Figure 9.39 represents an impressive confirmation of the CKM paradigm which describes the flavor transition and the *CP* violation in the quark sector. The global fit currently does not point to deviations from the Standard Model picture. However, the effect of *New Physics* might be small. Over-constraining measurements of *CP* asymmetries, mixing and flavor changing decays will put further bounds on possible *New Physics* contributions.

#### **9.8 Conclusion**

Nearly 40 years after the discovery of *CP* violation, the nature of the phenomenon has been clarified experimentally. In the K meson system, *CP* violation has been discovered in the mixing (*ε*), in the decay (*ε /ε*, direct *CP* violation), and in the interference between mixing and decay (&*m ε*); in the B meson system, *CP* violation in the interference between mixing and decay (sin 2*β*) as well as *CP* violation in the decay have been observed. While finishing this article, the LHCb collaboration has also reported the first observation of *CP* violation in the decay of neutral D mesons to *K*+*K*− and *π*+*π*− (*O(*1*.*5 × 10−3*)*) [205]. All observations are consistent8 with the model of Kobayashi and Maskawa where the 3 × 3 mixing matrix of six quarks has one non-trivial complex phase *δ* = *(*71*.*0 ± 0*.*3*)* ◦. This leads to complex weak coupling constants of quarks, and to *T* and *CP* violation.

In cosmology, *CP* violation together with a large departure from thermal equilibrium via a first-order electroweak phase transition [206] and a baryon number violation by instanton processes [207] could explain the observed baryon asymmetry. The source would be the asymmetric interactions of quarks and antiquarks with the Higgs field [208]. The size of the observed *CP* violation in the quark sector, expressed by the Jarlskog determinant, however is insufficient to explain the matter asymmetry by several orders of magnitude [209]. Moreover, the low mass of the Higgs boson disfavours baryogenesis during the electroweak phase transition [210]. A way out of this dilemma is the hypothesis that *CP* violation in the neutrino sector could cause a lepton asymmetry. (*B*+*L*)-violating processes before the electroweak phase transition could then convert the lepton asymmetry into a baryon asymmetry. Leptogenesis would proceed through the production of heavy Majorana neutrinos in the early Universe [211]. While the experimental proof of leptogenesis is difficult, the observation of *CP* violation in the neutrino sector and the exploration of the nature of the neutrino (Dirac or Majorana fermion) could establish strong hints.

In quark flavor physics the focus has changed after the successful establishment of the Kobayashi-Maskawa paradigm. Today, the precise measurement of flavor changing processes and *CP* asymmetries are used as tools to test the Standard Model predictions. Tiny deviations between observations and theory predictions could hint to additional quantum corrections modifying the size and the phase of flavor changing amplitudes. The origin of these additional corrections could be new heavy particles at mass scales much higher than the energies accessible by today's collider experiments. Experimental results on neutral meson mixing already constrain the ratio between mass scale and the coupling strength of new particles to values above 102 to 104 TeV, where the strongest bounds come from the neutral K meson system [212]. The next generation of precision quark-flavor experiments will

8For the D mesons the theoretical predictions still have large uncertainties. For K mesons recent theoretical results on the value of the *CP* violating amplitude ' \$ */*% both from dual QCD [213] and from lattice QCD [214] deviate from the experimental value reported in Sect. 9.5, thus further clarification is needed.

thus open a window to *New Physics* complementary to the direct searches at highest energies.

## **References**

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# **Chapter 10 The Future of Particle Physics: The LHC and Beyond**

**Ken Peach**

# **10.1 2017 Update**

I have been asked to submit a revised version of this chapter, published almost a decade ago. However, I think that it is better to leave the historical record as it was—this was an article written in its time and for its time. If I was writing this article today, I would call it "The Future of Particle Physics—Beyond the LHC", in recognition of the fact that, when originally written, the LHC was still under construction and now it has been operating for several years. The other key event which informed the original article was the recently-developed European Strategy for Particle Physics, adopted by the CERN Council in July 2006; the strategy was updated in 2013 and formally adopted in May of that year [1]; as I write, the process of updating the strategy is under way.

The intervening decade has seen significant progress, most spectacularly in the observation of the Higgs scalar [3] at 125.10 ± 0.14 GeV/c2 [2] which completes the Standard Model of the particles and their interactions (excluding gravity) which dominate the local region of the universe. This means that all of the free parameters of the Standard Model are now known with reasonable precision, which can be chosen as (see Table 10.1):

K. Peach (-)

John Adams Institute for Accelerator Science, Oxford, UK e-mail<EMAIL_ADDRESS>
H. Schopper (ed.), *Particle Physics Reference Library*, https://doi.org/10.1007/978-3-030-38207-0\_10

**Table 10.1** Parameters of the Standard Model [2]

7. the weak coupling constant GF; and

8. the Cabibbo-Kobayashi-Maskawa (CKM) parameters in the Wolfenstein parametrization λ, A, ρ and η.

In addition, two other useful quantities are given—MW and sin2θW—which can be written in terms of the other parameters, as shown in the last line of Table 10.1.

This is a total of 18 parameters which can only (so far) be determined by experiment. However, once determined, they describe the observed interactions with remarkable precision; for example, the muon anomalous magnetic moment (averaged between μ+ and μ−) is (11,659,209 1 ± 54stat ± 33syst) × 10−11, to be compared with the theoretical expectation of (11,659,180 3 ± 1EW ± 49Had) × 10−11, where the first error in the theoretical calculation comes from electroweak corrections and the second comes from a combination of low-order (42) and higher order (26) hadronic corrections [4]. The difference between these is (288 ± 80) × 10−11 (3.5 standard deviations) and it is, of course, hotly debated whether this is statistically significant and, if it is, what that significance might be. There is a new experiment at Fermilab [5] which aims to reduce the total error in the muon anomalous magnetic moment to 14 × 10−11, which, even without improvements to the theoretical calculations, would (if the central value of the discrepancy remained unchanged) increase the significance to more than 5 standard deviations.

Within the Standard Model, because the neutrinos have only one helicity state, they must be strictly massless. However, the phenomenon of neutrino oscillations, now well established, requires the neutrinos to have mass—essentially, the oscillation is a beating in the propagation through time of the different mass eigenstates that form the flavour eigenstates. The past decade has seen enormous progress in

**Table 10.2** Neutrino Oscillation Parameters [6, 7] a

aThe 3-sigma limits have been converted into 1-sigma asymmetric errors

the determination of the neutrino oscillation parameters (see Table 10.2). Although the mathematics of the flavour mixing is similar in the quark and neutrino sectors, the phenomenology is very different. In the quark sector, the flavour oscillations described by the CKM matrix—take place in either mixing or decay of bound states. In the neutrino sector—described by the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix—the flavour oscillations take place in vacuum, although there is an additional term for neutrino transmission in matter to take account of the different interaction cross-sections of the electron neutrinos, and the muon and tau neutrinos (and, of course, their antineutrinos). The main uncertainty is whether the third neutrino mass is heavier than the other two (normal hierarchy) or lighter (inverted hierarchy), with an emerging preference [8] for normal hierarchy. Most impressive is the evidence that the CP-violating phase angle in the PMNS matrix (δ) is non-zero –0/2π is excluded at the 3-standard deviation level, and π at 2.5 (1.5) standard deviations for the normal (inverted) hierarchy. The best limits on the absolute neutrino mass scale come from cosmology, giving an upper limit on mν of 0.183 eV at the 95% confidence level [7]. There are essentially two ideas for extending the Standard Model to include finite neutrino masses. The first notes that, since the neutrino has no electric charge, it is possible to construct a masslike term from the left-handed neutrino and right-handed-antineutrino fields, which changes its nature to that of a Majorana particle [9]. (If the neutrinos are indeed Majorana particles, there are two additional phase angles (α and β) in the PMNS matrix which are, however, extremely difficult to measure.) The second postulates the existence of a heavy right-handed neutrino which, in combination with a very high mass scale (like the GUT scale), produces through the "see-saw" mechanism one light left-handed neutrino (<< 1 eV/c2) and a heavy right-handed neutrino (>> 1012 GeV/c2)—as the left-handed neutrino becomes lighter the right-handed neutrino becomes heavier through the "see-saw". One consequence of these is that there are interactions that allow a change of lepton number by 2 units, thus enabling neutrinoless double-β decay to occur. (For a recent review, see [10].)

The other area where the Standard Model obviously fails is in the Dark Sector— Dark Matter and Dark Energy. The cosmological and astronomical evidence for these is overwhelming (see reference [11] for a recent review) with a recipe for the current state of the Universe containing about 70% of Dark Energy, 25% Dark Matter and 5% "normal" matter, the only part addressed by the Standard Model. From a particle physics perspective, Dark Matter is more tractable. First, there is the challenge of directly detecting it using large volume, sensitive detectors deep underground. The past decade has seen the limits steadily improved (see reference [12] for a recent review), while the small number of claimed signals have still not been independently confirmed. There is an ongoing programme of experiments at the major underground laboratories which should improve these limits by several order of magnitude over the next few years. Second, there are straightforward extensions of the Standard Model which provide candidates for Dark Matter. Unfortunately, there is no reliable estimate for the mass scale at which these new particles might appear. Such evidence might come from the Dark Matter searches themselves. The other place where a mass scale might be forthcoming is in the analysis of the deficiencies of the Standard Model. For example, Supersymmetry was proposed to address some hierarchy problems within the model, and also to assist in Grand Unification, both of which pointed to a TeV energy scale for appearance of new particles. However, so far these have failed to materialize at the LHC; unfortunately, the parameter space for Supersymmetry is extensive and so, while this might be disappointing it may not be conclusive. Dark Energy, on the other hand, is still mysterious and it is unclear where it fits in to the global picture; its understanding might require the reconciliation of quantum mechanics with general relativity.

The "open questions within the Standard Model" identified in the original article have been largely resolved, although there are still details to be addressed, for example the transition from perturbative to non-perturbative QCD. However, the "open questions beyond the Standard Model" remain, refined somewhat; for example, the "pattern of fermion masses" might nowadays be cast as "understanding the Yukawa couplings of the fundamental fermions to the Higgs". The improvements in the knowledge of the neutrino oscillation parameters do not explain how the mixing comes about, nor how the neutrinos derive their (tiny) masses. Again, there has been much progress in the measurement of the various cosmological parameters, including those which either determine or are determined by particle physics, but the "open questions in particle physics and cosmology" are still far from resolved. For example, the arguments for Grand Unification of the strong and electroweak forces remain intact (see Fig. 10.1) even though there is, as yet, no direct evidence for the existence of Supersymmetric particles. A second area which points to new physics is to note that, of the 18 parameters that define the Standard Model, 11 are directly related to mass (the quarks, the charged leptons, the Z and the Higgs), which becomes 12 if the Fermi constant is replaced (as it can be) by the mass of the W. Given this, there are three mass hierarchies to be explained. First, within a generation, what determines the relative masses of the charged lepton, the downtype quark (charge −1/3) and the up-type quark (charge +2/3) (and why is the up-quark lighter than the down quark, while the charm and top quarks are heavier than the strange and bottom quarks)? Second, what determines the large (several

**Fig. 10.1** The evolution of the fundamental forces as a function of energy in the standard model (left) and a particular set of parameters in the Minimal Supersymmetric Standard Model (right). (Adapted from [13])

orders of magnitude) mass differences between the generations of the same type of fundamental fermion. Third, what determines the masses of the heavy field bosons (Higgs and the W and Z)? (The other gauge bosons—the photon and the gluon remain massless, protected by an unbroken gauge symmetry.) The origin of the mixing matrix and CP-violation in the quark sector is still unresolved, as are the masses and oscillation parameters in the neutrino sector.

The detection of Gravitational Waves from the merger of two black holes [14] and the subsequent detection of gravitational waves from a binary neutron star inspiral [15] open the door to a new era of observational astronomy of matter under extreme conditions, and may also provide insight into the long-standing issue of the incompatibility of general relativity with quantum mechanics. Resolving this is likely to require understanding of physics at the Planck scale.

Given this, the main conclusions of the "Way Forward" are largely unchanged. Clues to the physics that must exist beyond the Standard Model could come directly from the observation of new states of matter (e.g. Supersymmetry), either at the LHC or at a new energy frontier machine, or indirectly from precision measurements of Standard Model parameters, for example at a Linear Collider built to produce large numbers of Higgs particles under clean conditions. Alternatively, ultrahigh precision measurements of low energy parameters (g-2, μ → e conversion, neutrinoless double β decay *...*) as well as new astrophysical observations (Dark Matter, gravitational waves) could give strong pointers toward the new physics. The tremendous progress in the neutrino sector from, particularly, T2K and Noνa probably means that the arguments in favour of a neutrino factory have weakened somewhat, but the arguments for the other facilities discussed (including the luminosity upgrade for the LHC) remain and have strengthened.

A recent publication by the T2K collaboration (*Nature* 580, 339–344 (2020)) reports that "The 3*σ* confidence interval for *δ*CP, which is cyclic and repeats every 2ϕ, is [−3.41, −0.03] for the so-called normal mass ordering and [−2.54, −0.32] for the inverted mass ordering".

## **References**

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**2. The MetaFlex framework: integrating metacognition and psychological flexibility**

The integration of metacognition and psychological flexibility provides a promising framework for understanding and enhancing individuals' cognitive and emotional processes, as well as their overall well-being. Metacognition, which involves the awareness and regulation of one's own cognitive processes, can complement psychological flexibility, which focuses on adaptively responding to thoughts, emotions, and experiences in line with personal values. Recent studies support the notion that combining these two constructs can yield synergistic benefits.

Research by Sugiura and Sugiura [1, 4] found that individuals with higher levels of metacognitive awareness demonstrated greater psychological flexibility. Metacognitive awareness facilitated the ability to recognize and regulate unhelpful thinking patterns, enabling individuals to flexibly respond to challenging situations and align their behavior with their values. Similarly, a study by de Bruin et al. [3] highlighted the positive relationship between metacognitive skills and psychological flexibility, emphasizing that metacognition can enhance individuals' ability to adaptively respond to their thoughts and emotions. Moreover, a study by Gloster et al. [24, 25] examined the role of metacognitive beliefs in the context of psychological flexibility. The findings indicated that individuals with more flexible metacognitive beliefs, characterized by a balanced view of their thoughts and emotions, exhibited higher levels of psychological flexibility. These results suggest that metacognitive beliefs can influence individuals' ability to engage in valued actions, despite the presence of challenging thoughts or emotions.

Additionally, studies have shown that interventions targeting metacognitive processes and psychological flexibility can lead to positive outcomes. For instance, a randomized controlled trial by Gillanders et al. [26] evaluated the effectiveness of a metacognitive therapy intervention that incorporated elements of psychological flexibility. The intervention resulted in improved psychological flexibility and reductions in symptoms of anxiety and depression.

Recent studies provide empirical support for the integration of metacognition and psychological flexibility. Metacognition enhances individuals' awareness of their cognitive processes and beliefs, allowing for more flexible responses to thoughts and emotions. This integration has been associated with improved psychological flexibility and positive outcomes. Further research and interventions combining these constructs hold promise for enhancing cognitive and emotional well-being and promoting personal growth.

## **2.1 Exploring the synergistic relationship between metacognition and psychological flexibility**

Metacognition and psychological flexibility are two distinct constructs that play complementary roles in enhancing cognitive and emotional processes. Metacognition involves the awareness and regulation of one's own cognitive processes, including thinking, memory, and problem-solving. On the other hand, psychological flexibility refers to the ability to adaptively respond to thoughts, emotions, and experiences in a way that aligns with personal values and goals. While these constructs operate at different levels of cognitive processing, recent research suggests that they have a synergistic relationship that can enhance overall well-being and personal growth.

The integration of metacognition and psychological flexibility allows individuals to develop a deeper understanding of their cognitive and emotional experiences, as well as the ability to respond to them in a flexible and adaptive manner. By cultivating metacognitive awareness, individuals become more attuned to their thoughts, emotions, and beliefs, allowing them to recognize unhelpful thinking patterns and biases. This heightened self-awareness enables individuals to disengage from automatic and rigid cognitive processes, promoting psychological flexibility.

Studies have demonstrated a positive relationship between metacognition and psychological flexibility. For instance, research by Levin et al. [27] found that metacognitive awareness was associated with greater psychological flexibility in individuals with anxiety disorders. The study revealed that those who exhibited higher levels of metacognitive awareness were better able to engage in valued actions, despite experiencing anxious thoughts and emotions. This suggests that metacognition can enhance individuals' ability to respond flexibly to challenging internal experiences (**Table 1**).

Moreover, metacognition can facilitate the application of psychological flexibility strategies. For example, the process of cognitive defusion, which involves creating distance from unhelpful thoughts and beliefs, can be strengthened through metacognitive awareness. By recognizing thoughts as mental events rather than accurate representations of reality, individuals can defuse from them more effectively and choose responses that align with their values. The integration of metacognition and psychological flexibility has practical implications for interventions and therapies. Combining metacognitive strategies with Acceptance and Commitment Therapy (ACT) or cognitive-behavioral therapy (CBT) interventions can provide individuals with a comprehensive toolkit for enhancing cognitive and emotional well-being. Interventions that target metacognitive processes, such as metacognitive therapy, can be integrated with techniques that promote psychological flexibility, such as values clarification and mindfulness. This integrated approach allows individuals to develop metacognitive skills that support flexible responses to their thoughts and emotions, ultimately leading to improved mental health outcomes.

### **Table 1.**

*Relationship between metacognition and psychological flexibility as MetaFlex framework [27].*

*Perspective Chapter: The MetaFlex Framework – Harnessing Metacognition to Foster… DOI: http://dx.doi.org/10.5772/intechopen.113920*

The relationship between metacognition and psychological flexibility is synergistic, as they mutually enhance and support each other. Metacognitive awareness provides individuals with the ability to recognize and regulate their cognitive processes, while psychological flexibility allows for adaptive responses to thoughts and emotions in alignment with personal values. Future research and interventions can further explore and leverage this synergistic relationship to promote well-being, personal growth, and effective coping strategies in various contexts.

### **2.2 Role of metacognition in enhancing psychological flexibility and promoting growth**

Metacognition plays a crucial role in enhancing psychological flexibility and promoting personal growth. When integrated with psychological flexibility, metacognition can facilitate adaptive responses to internal experiences, promote resilience, and foster personal growth. Metacognition enables individuals to become aware of their thoughts and beliefs, including their accuracy, biases, and impact on emotions and behaviors. This awareness allows individuals to recognize unhelpful or negative thinking patterns and cognitive distortions. By identifying and challenging these patterns, individuals can cultivate psychological flexibility and develop more adaptive ways of thinking. Metacognition supports cognitive defusion, a process central to psychological flexibility. Cognitive diffusion involves creating distance from and changing the relationship with one's thoughts, allowing individuals to observe them as mental events rather than absolute truths. Metacognitive awareness helps individuals recognize the transient nature of thoughts and reduces their impact on emotions and behaviors, enabling more flexible and adaptive responses.

Metacognition encourages self-reflection, introspection, and the ability to take different perspectives. Through metacognitive processes, individuals can reflect on their thoughts, emotions, and experiences, gaining insights into their own cognitive and emotional processes. This self-reflection promotes self-understanding, empathy, and a broader perspective, all of which contribute to psychological flexibility and personal growth. Metacognition involves monitoring and regulating emotions, which are integral components of psychological flexibility. By being aware of their emotional states, individuals can identify when emotions are driving their thoughts and behaviors. This awareness allows individuals to choose responses aligned with their values rather than being driven by reactive emotions. Metacognitive strategies, such as emotion regulation techniques and mindfulness, can enhance emotional selfregulation and support psychological flexibility.

Metacognition facilitates learning from experiences and adjusting cognitive and behavioral strategies accordingly. Through metacognitive processes, individuals can reflect on past experiences, evaluate their effectiveness, and make moral and mature adjustments for future situations [28]. Popandopulo et al. [29, 30] highlight the iterative process of self-reflection and learning enhances adaptive responses, fosters personal growth, and promotes the development of more effective coping strategies. Research supports the link between metacognition and psychological flexibility. Studies have found that metacognitive interventions, such as metacognitive therapy or mindfulness-based approaches, enhance psychological flexibility and reduce psychological distress [31, 32]. The integration of metacognitive strategies within Acceptance and Commitment Therapy (ACT) has also shown promising results in promoting psychological flexibility [27].

Research supports the importance of metacognition and psychological flexibility in promoting growth and well-being. For example, a study by Sauer et al. (2019) found that metacognitive awareness was positively associated with psychological well-being and adaptive coping strategies. Individuals with higher levels of metacognitive awareness demonstrated greater emotional regulation and resilience in the face of stressors.

Similarly, research by Kashdan et al. [33] highlighted the benefits of psychological flexibility for well-being. The study showed that psychological flexibility was associated with higher levels of life satisfaction, positive emotions, and overall psychological well-being. It also found that psychological flexibility played a protective role against the negative impact of stress on mental health.

Interventions that target metacognition and psychological flexibility have also shown promise in promoting growth and well-being. For instance, metacognitive therapy and Acceptance and Commitment Therapy (ACT) interventions incorporate strategies to enhance metacognition and psychological flexibility, respectively. These interventions have been effective in reducing symptoms of anxiety, depression, and other mental health disorders while promoting personal growth and adaptive functioning.

Metacognition plays a crucial role in enhancing psychological flexibility and promoting personal growth. By cultivating metacognitive awareness, individuals can develop a deeper understanding of their cognitive and emotional processes, challenge unhelpful thoughts and beliefs, regulate emotions, and engage in adaptive responses. The integration of metacognition with interventions and therapies can support individuals in developing greater psychological flexibility, resilience, and personal growth. Metacognition and psychological flexibility are essential for promoting growth and well-being. Metacognition enables individuals to develop selfawareness, challenge unhelpful thinking patterns, and improve cognitive functioning. Psychological flexibility empowers individuals to accept their internal experiences, make choices aligned with their values, and respond adaptively to life's challenges. By cultivating these constructs, individuals can enhance their resilience, cope effectively with stressors, and pursue meaningful and fulfilling lives.

## **2.3 Core components of the MetaFlex framework**


*Appendix D.2. Transfer in the Dictator Game*

The distribution of the *dictator game* transfer can be found in Table A1 and is illustrated in Figure A2.

**Table A1.** Frequency of Dictator Game Transfer by Intuitive (Reflective) Individuals.

**Figure A2.** Frequency of Transfer by Intuitive (Reflective) Individuals.