TASI 2009 Lectures – Flavor Physics
The standard model picture of flavor and CP violation is now experimentally verified, hence strong bounds on the flavor structure of new physics follow. We begin by discussing in detail the unique way that flavor conversion and CP violation arise in the standard model. The description provided is based on a spurion, symmetry oriented, analysis, and a covariant basis for describing flavor transition processes is introduced, in order to make the discussion transparent for non-experts. We show how to derive model independent bounds on generic new physics models. Furthermore, we demonstrate, using the covariant basis, how recent data and LHC projections can be applied to constrain models with an arbitrary mechanism of alignment. Next, we discuss the various limits of the minimal flavor violation framework and their phenomenological aspects, as well as the implications to the underlying microscopic origin of the framework. We also briefly discuss aspects of supersymmetry and warped extra dimension flavor violation. Finally we speculate on the possible role of flavor physics in the LHC era.
- 1 Introduction
- 2 The standard model flavor sector
- 3 Spurion analysis of the SM flavor sector
- 4 Covariant description of flavor violation
- 5 Model independent bounds
- 6 Minimal flavor violation
- 7 Supersymmetry
- 8 Extra Dimensions
- 9 High Flavor Physics Beyond the SM
- 10 Conclusions
Flavors are replications of states with identical quantum numbers. The standard model (SM) consists of three such replications of the five fermionic representations of the SM gauge group. Flavor physics describes the non-trivial spectrum and interactions of the flavor sector. What makes this field particularly interesting is that the SM flavor sector is rather unique, and its special characteristics make it testable and predictive. 111This set of lectures discusses the quark sector only. Many of the concepts that are explained here can be directly applied to the lepton sector. Let us list few of the SM unique flavor predictions:
It Contains a single CP violating parameter.222The SM contains an additional flavor diagonal CP violating parameter, namely the strong CP phase. However, experimental data constrains it to be smaller than , hence negligibly small.
Flavor conversion is driven by three mixing angles.
To leading order, flavor conversion proceeds through weak charged current interactions.
To leading order, flavor conversion involves left handed (LH) currents.
CP violating processes must involve all three generations.
The dominant flavor breaking is due to the top Yukawa coupling, hence the SM possesses a large approximate global flavor symmetry (as shown below, technically it is given by ).
In the last four decades or so, a huge effort was invested towards testing the SM predictions related to its flavor sector. Recently, due to the success of the B factories, the field of flavor physics has made a dramatic progress, culminated in Kobayashi and Maskawa winning the Nobel prize. It is now established that the SM contributions drive the observed flavor and CP violation (CPV) in nature, via the Cabibbo-Kobayashi-Maskawa (CKM) [1, 2] description. To verify that this is indeed the case, one can allow new physics (NP) to contribute to various clean observables, which can be calculated precisely within the SM. Analyses of the data before and after the B factories data have matured [3, 4, 5, 6], demonstrating that the NP contributions to these clean processes cannot be bigger than of the SM contributions [7, 8].
Very recently, the SM passed another non-trivial test. The neutral meson system (for formalism see e.g. [9, 10, 11, 12, 13] and refs. therein) bears two unique aspects among the four neutral meson system (): (i) The long distance contributions to the mixing are orders of magnitude above the SM short distance ones [14, 15], thus making it difficult to theoretically predict the width and mass splitting. (ii) The SM contribution to the CP violation in the mixing amplitude is expected to be below the permil level , hence mixing can unambiguously signal new physics if CPV is observed. Present data [17, 18, 19, 20, 21, 22, 23, 24] implies that generic CPV contributions can be only of of the total (un-calculable) contributions to the mixing amplitudes, again consistent with the SM null prediction.
We have just given rather solid arguments for the validity of the SM flavor description. What else is there to say then? Could this be the end of the story? We have several important reasons to think that flavor physics will continue to play a significant role in our understanding of microscopical physics at and beyond the reach of current colliders. Let us first mention a few examples that demonstrate the role that flavor precision tests played in the past:
The smallness of led to predicting a fourth quark (the charm) via the discovery of the GIM mechanism .
The size of the mass difference in the neutral Kaon system, , led to a successful prediction of the charm mass .
The size of led to a successful prediction of the top mass (for a review see  and refs. therein).
This partial list demonstrates the power of flavor precision tests in terms of being sensitive to short distance dynamics. Even in view of the SM successful flavor story, it is likely that there are missing experimental and theoretical ingredients, as follows:
Within the SM, as mentioned, there is a single CP violating parameter. We shall see that the unique structure of the SM flavor sector implies that CP violating phenomena are highly suppressed. Baryogenesis, which requires a sizable CP violating source, therefore cannot be accounted for by the SM CKM phase. Measurements of CPV in flavor changing processes might provide evidence for additional sources coming from short distance physics.
The SM flavor parameters are hierarchical, and most of them are small (excluding the top Yukawa and the CKM phase), which is denoted as the flavor puzzle. This peculiarity might stem from unknown flavor dynamics. Though it might be related to very short distance physics, we can still get indirect information about its nature via combinations of flavor precision and high measurements.
The SM fine tuning problem, which is related to the quadratic divergence of the Higgs mass, generically requires new physics at, or below, the TeV scale. If such new physics has a generic flavor structure, it would contribute to flavor changing neutral current (FCNC) processes orders of magnitude above the observed rates. Putting it differently, the flavor scale at which NP is allowed to have a generic flavor structure is required to be larger than TeV, in order to be consistent with flavor precision tests. Since this is well above the electroweak symmetry breaking scale, it implies an “intermediate” hierarchy puzzle (cf. the little hierarchy [28, 29] problem). We use the term “puzzle” and not “problem” since in general, the smallness of the flavor parameters, even within NP models, implies the presence of approximate symmetries. One can imagine, for instance, a situation where the suppression of the NP contributions to FCNC processes is linked with the SM small mixing angles and small light quark Yukawas [4, 5]. In such a case, this intermediate hierarchy is resolved in a technically natural way, or radiatively stable manner, and no fine tuning is required.333Unlike, say, the case of the electroweak parameter, where in general one cannot associate an approximate symmetry with the limit of small NP contributions to .
2 The standard model flavor sector
The SM quarks furnish three representations of the SM gauge group, : , where stand for weak doublet, up type and down type singlet quarks, respectively. Flavor physics is related to the fact that the SM consists of three replications/generations/flavors of these three representations. The flavor sector of the SM is described via the following part of the SM Lagrangian
where with being a covariant derivative, , within the SM with a single Higgs (however, the reader should keep in mind that at present, the nature and content of the SM Higgs sector is unknown) and are flavor indices.
If we switch off the Yukawa interactions, the SM would possess a large global flavor symmetry, ,444At the quantum level, a linear combination of the diagonal ’s inside the ’s, which corresponds to the axial current, is anomalous.
Inspecting Eq. (1) shows that the only non-trivial flavor dependence in the Lagrangian is in the form of the Yukawa interactions. It is encoded in a pair of complex matrices, .
2.1 The SM quark flavor parameters
Naively one might think that the number of the SM flavor parameters is given by real numbers and imaginary ones, the elements of . However, some of the parameters which appear in the Yukawa matrices are unphysical. A simple way to see that (see e.g. [30, 31, 32] and refs. therein) is to use the fact that a flavor basis transformation,
leaves the SM Lagrangian invariant, apart from redefinition of the Yukawas,
where is a unitary rotation matrix. Each of the three rotation matrices contains three real parameters and six imaginary ones (the former ones correspond to the three generators of the group, and the latter correspond to the remaining six generators of the group). We know, however, that physical observables do not depend on our choice of basis. Hence, we can use these rotations to eliminate unphysical flavor parameters from . Out of the 18 real parameters, we can remove 9 () ones. Out of the 18 imaginary parameters, we can remove 17 (3) ones. We cannot remove all the imaginary parameters, due to the fact that the SM Lagrangian conserves a symmetry.555More precisely, only the combination is non-anomalous. Thus, there is a linear combination of the diagonal generators of which is unbroken even in the presence of the Yukawa matrices, and hence cannot be used in order to remove the extra imaginary parameter.
An explicit calculation shows that the 9 real parameters correspond to 6 masses and 3 CKM mixing angles, while the imaginary parameter corresponds to the CKM celebrated CPV phase. To see that, we can define a mass basis where are both diagonal. This can be achieved by applying a bi-unitary transformation on each of the Yukawas:
which leaves the SM Lagrangian invariant, apart from redefinition of the Yukawas,
The difference between the transformations used in Eqs. (3) and (4) and the ones above (5,6), is in the fact that each component of the weak doublets (denoted as and ) transforms independently. This manifestly breaks the gauge invariance, hence such a transformation makes sense only for a theory in which the electroweak symmetry is broken. This is precisely the case for the SM, where the masses are induced by spontaneous electroweak symmetry breaking via the Higgs mechanism. Applying the above transformation amounts to “moving” to the mass basis. The SM flavor Lagrangian, in the mass basis, is given by (in a unitary gauge),
where the subscript NC stands for neutral current interaction for the gluons, the photon and the gauge bosons, stands for the charged electroweak gauge bosons, is the physical Higgs field, GeV, and is the CKM matrix
In general, the CKM is a unitary matrix, with 6 imaginary parameters. However, as evident from Eq. (7), the charged current interactions are the only terms which are not invariant under individual quark vectorial field redefinitions,
The diagonal part of this transformation corresponds to the classically conserved baryon current, while the non-diagonal, , part of the transformation can be used to remove 5 out of the 6 phases, leaving the CKM matrix with a single physical phase. Notice also that a possible permutation ambiguity for ordering the CKM entries is removed, given that we have ordered the fields in Eq. (7) according to their masses, light fields first. This exercise of explicitly identifying the mass basis rotation is quite instructive, and we have already learned several important issues regarding how flavor is broken within the SM (we shall derive the same conclusions using a spurion analysis in a symmetry oriented manner in Sec. 3):
Flavor conversions only proceed via the three CKM mixing angles.
Flavor conversion is mediated via the charged current electroweak interactions.
The charge current interactions only involve LH fields.
Even after removing all the unphysical parameters, there are various possible forms for the CKM matrix. For example, a parameterization used by the particle data group , is given by
where and . The three are the three real mixing parameters, while is the Kobayashi-Maskawa phase.
2.2 CP violation
The SM predictive power picks up once CPV is considered. We have already proven that the SM flavor sector contains a single CP violating parameter. Once presented with a SM Lagrangian where the Yukawa matrices are given in a generic basis, it is not trivial to determine whether CP is violated or not. This is even more challenging when discussing beyond the SM dynamics, where new CP violating sources might be present. A brute force way to establish that CP is violated would be to show that no field redefinitions would render a Lagrangian real. For example, consider a Lagrangian with a single Yukawa matrix,
where is a scalar and is a fermion field. A CP transformation exchanges the operators
but leaves their coefficients, and , unchanged, since CP is a linear unitary non-anomalous transformation. This means that CP is conserved if
This is, however, not a basis independent statement. Since physical observables do no depend on a specific basis choice, it is enough to find a basis in which the above relation holds.666It is easy to show that in this example, in fact, CP is not violated for any number of generations.
Sometimes the brute force way is tedious and might be rather complicated. A more systematic approach would be to identify a phase reparameterization invariant or basis independent quantity, that vanishes in the CP conserving limit. As discovered in [34, 35], for the SM case one can define the following quantity
and the SM is CP violating if and only if
It is trivial to prove that only if the number of generations is three or more, then CP is violated. Hence, within the SM, where CP is broken explicitly in the flavor sector, any CP violating process must involve all three generations. This is an important condition, which implies strong predictive power. Furthermore, all the CPV observables are correlated, since they are all proportional to a single CP violating parameter, . Finally, it is worth mentioning that CPV observables are related to interference between different processes, and hence are measurements of amplitude ratios. Thus, in various known cases, they turn out to be cleaner and easier to interpret theoretically.
2.3 The flavor puzzle
Now that we have precisely identified the SM physical flavor parameters, it is interesting to ask what is their experimental value (using ) :
where corresponds to the magnitude of the entry in the CKM matrix, is the CKM phase, only uncertainties bigger than 10% are shown, numbers are shown to a 2-digit precision and the entries involve indirect information (a detailed description and refs. can be found in ).
Inspecting the actual numerical values for the flavor parameters given in Eq. (2.3), shows a peculiar structure. Most of the parameters, apart from the top mass and the CKM phase, are small and hierarchical. The amount of hierarchy can be characterized by looking at two different classes of observables:
Hierarchies between the masses, which are not related to flavor converting processes – as a measure of these hierarchies, we can just estimate what is the size of the product of the Yukawa coupling square differences (in the mass basis)
Hierarchies in the mixing which mediate flavor conversion – this is related to the tiny misalignment between the up and down Yukawas; one can quantify this effect in a basis independent fashion as follows. A CP violating quantity, associated with , that is independent of parametrization [34, 35], , is defined through
where . We see that even though is of order unity, the resulting CP violating parameter is small, as it is “screened” by small mixing angles. If any of the mixing angles is a multiple of , then the SM Lagrangian becomes real. Another explicit way to see that and are quasi aligned is via the Wolfenstein parametrization of the CKM matrix, where the four mixing parameters are , with playing the role of an expansion parameter :
Basically, to zeroth order, the CKM matrix is just a unit matrix !
As we shall discuss further below, both kinds of hierarchies described in the bullets lead to suppression of CPV. Thus, a nice way to quantify the amount of hierarchies, both in masses and mixing angles, is to compute the value of the reparameterization invariant measure of CPV introduced in Eq. (14)
This tiny value of that characterizes the flavor hierarchy in nature would be of order 10% in theories where are generic order one complex matrices. The smallness of is something that many flavor models beyond the SM try to address. Furthermore, SM extensions that have new sources of CPV tend not to have the SM built-in CP screening mechanism. As a result, they give too large contributions to the various observables that are sensitive to CP breaking. Therefore, these models are usually excluded by the data, which is, as mentioned, consistent with the SM predictions.
3 Spurion analysis of the SM flavor sector
In this part we shall try to be more systematic in understanding the way flavor is broken within the SM. We shall develop a spurion, symmetry-oriented description for the SM flavor structure, and also generalize it to NP models with similar flavor structure, that goes under the name minimal flavor violation (MFV).
3.1 Understanding the SM flavor breaking
It is clear that if we set the Yukawa couplings of the SM to zero, we restore the full global flavor group, In order to be able to better understand the nature of flavor and CPV within the SM, in the presence of the Yukawa terms, we can use a spurion analysis as follows. Let us formally promote the Yukawa matrices to spurion fields, which transform under in a manner that makes the SM invariant under the full flavor group (see e.g.  and refs. therein). From the flavor transformation given in Eqs. (3,4), we can read the representation of the various fields under (see illustration in Fig. 1)
The flavor group is broken by the “background” value of the spurions , which are bi-fundamentals of . It is instructive to consider the breaking of the different flavor groups separately (since are bi-fundamentals, the breaking of quark doublet and singlet flavor groups are linked together, so this analysis only gives partial information to be completed below). Consider the quark singlet flavor group, , first. We can construct a polynomial of the Yukawas with simple transformation properties under the flavor group. For instance, consider the objects
Under the flavor group transform as
Thus, are adjoints of and singlets of the rest of the flavor group [while are flavor singlets]. Via similarity transformation, we can bring to a diagonal form, simultaneously. Thus, we learn that the background value of each of the Yukawa matrices separately breaks the down to a residual group, as illustrated in Fig. 2.
Let us now discuss the breaking of the LH flavor group. We can, in principle, apply the same analysis for the LH flavor group, , via defining the adjoints (in this case we have two independent ones),
However, in this case the breaking is more involved, since are adjoints of the same flavor group. This is a direct consequence of the weak gauge interaction, which relates the two components of the doublets. This actually motivates one to extend the global flavor group as follows. If we switch off the electroweak interactions, the SM global flavor group is actually enlarged to 
since now each doublet, , can be split into two independent flavors, , with identical gauge quantum numbers . This limit, however, is not very illuminating, since it does not allow for flavor violation at all. To make a progress, it is instructive to distinguish the neutral current interactions from the charged current ones, as follows: The couplings are flavor universal, which, however, couple up and down quarks separately. The couplings, , link between the up and down LH quarks. In the presence of only couplings, the residual flavor group is given by777To get to this limit formally, one can think of a model where the Higgs field is an adjoint of and a singlet of color and hypercharge. In this case the Higgs vacuum expectation value (VEV) preserves a gauge symmetry, and the would therefore remain massless. However, the will acquire masses of the order of the Higgs VEV, and therefore charged current interactions would be suppressed.
In this limit, even in the presence of the Yukawa matrices, flavor conversion is forbidden. We have already seen explicitly that only the charged currents link between different flavors (see Eq. (7)). It is thus evident that to formally characterize flavor violation, we can extend the flavor group from , where now we break the quark doublets to their isospin components, , and add another spurion,
Flavor breaking within the SM occurs only when is fully broken via the Yukawa background values, but also due to the fact that has a background value. Unlike , is a special spurion in the sense that its eigenvalues are degenerate, as required by the weak gauge symmetry. Hence, it breaks the down to a diagonal group, which is nothing but . We can identify two bases where has an interesting background value: The weak interaction basis, in which the background value of is simply a unit matrix888Note that the interaction basis is not unique, given that is invariant under a flavor transformation where and are rotated by the same amount – see more in the following.
and the mass basis, where (after removing all unphysical parameters) the background value of is the CKM matrix
Now we are in a position to understand the way flavor conversion is obtained in the SM. Three spurions must participate in the breaking: and . Since is involved, it is clear that generation transitions must involve LH charged current interactions. These transitions are mediated by the spurion backgrounds, (see Eq. (23)), which characterize the breaking of the individual LH flavor symmetries,
Flavor conversion occurs because of the fact that in general we cannot diagonalize simultaneously and , where the misalignment between and is precisely characterized by the CKM matrix. This is illustrated in Fig. 3, where it is shown that the flavor breaking within the SM goes through collective breaking  – a term often used in the context of little Higgs models (see e.g.  and refs. therein). We can now combine the LH and RH quark flavor symmetry breaking to obtain the complete picture of how flavor is broken within the SM. As we saw, the breaking of the quark singlet groups is rather trivial. It is, however, linked to the more involved LH flavor breaking, since the Yukawa matrices are bi-fundamentals – the LH and RH flavor breaking are tied together. The full breaking is illustrated in Fig. 4.
3.2 A comment on description of flavor conversion in physical processes
The above spurion structure allows us to describe SM flavor converting processes. However, the reader might be confused, since we have argued above that flavor converting processes must involve the three spurions, and . It is well known that the rates for charge current processes, which are described via conversion of down quark to an up one (and vise a versa), say like beta decay or transitions, are only suppressed by the corresponding CKM entry, or . What happened to the dependence on ? The key point here is that in a typical flavor precision measurement, the experimentalists produce mass eigenstate (for example a neutron or a meson), and thus the fields involved are chosen to be in the mass basis. For example, a process is characterized by producing a meson which decays into a charmed one. Hence, both and participate, being forced to be diagonal, but in a nonlinear way. Physically, we can characterize it by writing an operator
where both the and quarks are mass eigenstate. Note that this is consistent with the transformation rules for the extended gauge group, , given in Eqs. (25) and (26), where the fields involved belong to different representations of the extended flavor group.
The situation is different when FCNC processes are considered. In such a case, a typical measurement involves mass eigenstate quarks belonging to the same representation of . For example, processes that mediate oscillation due to the tiny mass difference between the two mass eigenstates (which was first measured by the ARGUS experiment ), are described via the following operator, omitting the spurion structure for simplicity,
Obviously, this operator cannot be generated by SM processes, as it is violates the symmetry explicitly. Since it involves flavor conversion (it violates number by two units, hence denoted as and belongs to class of FCNC processes), it must have some power of . A single power of connects a LH down quark to a LH up one, so the leading contribution should go like (). Hence, as expected, this process is mediated at least via one loop. This would not work as well, since we can always rotate the down quark fields into the mass basis, and simultaneously rotate also the up type quarks (away from their mass basis) so that . These manipulations define the interaction basis, which is not unique (see Eq. (27)). Therefore, the leading flavor invariant spurion that mediates FCNC transition would have to involve the up type Yukawa spurion as well. A naive guess would be
where it is understood that is evaluated in the down quark mass basis (tiny corrections of order are neglected in the above). This expression captures the right flavor structure, and is correct for a sizeable class of SM extensions. However, it is actually incorrect in the SM case. The reason is that within the SM, the flavor symmetries are strongly broken by the large top quark mass . The SM corresponding amplitude consists of a rather non-trivial and non-linear function of , instead of the above naive expression (see e.g.  and refs. therein), which assumes only the simplest polynomial dependence of the spurions. The SM amplitude for is described via a box diagram, and two out of the four powers of masses are canceled, since they appear in the propagators.
3.3 The SM approximate symmetry structure
In the above we have considered the most general breaking pattern. However, as discussed, the essence of the flavor puzzle is the large hierarchies in the quark masses, the eigenvalues of and their approximate alignment. Going back to the spurions that mediate the SM flavor conversions defined in Eqs. (21) and (23), we can write them as
where in the above we took advantage of the fact that are small. The hierarchies in the quark masses are translated to an approximate residual RH flavor group (see Fig. 5), implying that RH currents which involve light quarks are very small.
We have so far only briefly discussed the role of FCNCs. In the above we have argued, both based on an explicit calculation and in terms of a spurion analysis, that at tree level there are no flavor violating neutral currents, since they must be mediated through the couplings or . In fact, this situation, which is nothing but the celebrated GIM mechanism , goes beyond the SM to all models in which all LH quarks are doublets and all RH ones are singlets. The boson might have flavor changing couplings in models where this is not the case.
Can we guess what is the leading spurion structure that induces FCNC within the SM, say which mediates the decay process via an operator ? The process changes quark number by one unit (belongs to class of FCNC transitions). It clearly has to contain down type LH quark fields (let us ignore the lepton current, which is flavor-trivial; for effects related to neutrino masses and lepton number breaking in this class of models see e.g. [44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54]). Therefore, using the argument presented when discussing (see Eq. (32)), the leading flavor invariant spurion that mediates FCNC would have to involve the up type Yukawa spurion as well
The above considerations demonstrate how the GIM mechanism removes the SM divergencies from various one loop FCNC processes, which are naively expected to be log divergent. The reason is that the insertion of is translated to quark mass difference insertion. It means that the relevant one loop diagram has to be proportional to (). Thus, the superficial degree of divergency is lowered by two units, which renders the amplitude finite.999For simplicity, we only consider cases with hard GIM, in which the dependence on mass differences is polynomial. There is a large class of amplitudes, for example processes that are mediated via penguin diagrams with gluon or photon lines, where the quark mass dependence is more complicated, and may involve logarithms. The suppression of the corresponding amplitudes goes under the name soft GIM . Furthermore, as explained above (see also Eq. (37)), we can use the fact that the top contribution dominates the flavor violation to simplify the form of
where we have added a one loop suppression factor and an expected weak scale suppression. This rough estimation actually reproduces the SM result up to a factor of about 1.5 (see e.g. [43, 55, 56, 57]).
We thus find that down quark FCNC amplitudes are expected to be highly suppressed due to the smallness of the top off-diagonal entries of the CKM matrix. Parameterically, we find the following suppression factor for transition between the th and th generations:
where for the case one needs to simply square the parametric suppression factors. This simple exercise illustrates how powerful is the SM FCNC suppression mechanism. The gist of it is that the rate of SM FCNC processes is small, since they occur at one loop, and more importantly due to the fact that they are suppressed by the top CKM off-diagonal entries, which are very small. Furthermore, since
in most cases the dominant flavor conversion effects are expected to be mediated via the top Yukawa coupling.101010This is definitely correct for CP violating processes, or any ones which involve the third generation quarks. It also generically holds for new physics MFV models. Within the SM, for CP conserving processes which involve only the first two generations, one can find exceptions, for instance when considering the Kaon and meson mass differences, .
We can now understand how the SM uniqueness related to suppression of flavor converting processes arises:
RH currents for light quarks are suppressed due to their small Yukawa couplings (them being light).
Flavor transition occurs to leading order only via LH charged current interactions.
To leading order, flavor conversion is only due to the large top Yukawa coupling.
4 Covariant description of flavor violation
The spurion language discussed in the previous section is useful in understanding the flavor structure of the SM. In the current section we present a covariant formalism, based on this language, that enables to express physical observables in an explicitly basis independent form. This formalism, introduced in [58, 59], can be later used to analyze NP contributions to such observables, and obtain model independent bounds based on experimental data. We focus only on the LH sector.
4.1 Two generations
We start with the simpler two generation case, which is actually very useful in constraining new physics, as a result of the richer experimental precision data. Any hermitian traceless matrix can be expressed as a linear combination of the Pauli matrices . This combination can be naturally interpreted as a vector in three dimensional real space, which applies to and . We can then define a length of such a vector, a scalar product, a cross product and an angle between two vectors, all of which are basis independent111111The factor of in the cross product is required in order to have the standard geometrical interpretation , with defined through the scalar product as in Eq. (38).:
These definitions allow for an intuitive understanding of the flavor and CP violation induced by a new physics source, based on simple geometric terms. Consider a dimension six -invariant operator, involving only quark doublets,
where is some high energy scale.121212This use of effective field theory to describe NP contributions will be explained in detail in the next section. Note also that we employ here a slightly different notation, more suitable for the current needs, than in the next section. is a traceless hermitian matrix, transforming as an adjoint of (or for two generations), so it “lives” in the same space as and . In the down sector for example, the operator above is relevant for flavor violation through mixing. To analyze its contribution, we define a covariant orthonormal basis for each sector, with the following unit vectors
Then the contribution of the operator in Eq. (39) to processes is given by the misalignment between and , which is equal to
This result is manifestly invariant under a change of basis. The meaning of Eq. (41) can be understood as follows: We can choose an explicit basis, for example the down mass basis, where is proportional to . transitions are induced by the off-diagonal element of , so that . Furthermore, is simply the combined size of the and components of . Its size is given by the length of times the sine of the angle between and (see Fig. 6). This is exactly what Eq. (41) describes.
Next we discuss CPV, which is given by
The above expression is easy to understand in the down basis, for instance. In addition to diagonalizing , we can also choose to reside in the plane (Fig. 7) without loss of generality, since there is no CPV in the SM for two generations. As a result, all of the potential CPV originates from in this basis. is the square of the off-diagonal element in , , thus Im is simply twice the real part ( component) times the imaginary part ( component). In this basis we have and , this proves the validity of Eq. (42).
An interesting conclusion can be inferred from the analysis above: In addition to the known necessary condition for CPV in two generation 
we identify a second necessary condition, exclusive for processes:
These conditions are physically transparent and involve only observables.
4.2 Three generations
4.2.1 Approximate limit of massless light quarks
For three generations, a simple 3D geometric interpretation does not naturally emerge anymore, as the relevant space is characterized by the eight Gell-Mann matrices131313We denote the Gell-Mann matrices by , where . Choosing this convention allows us to keep the definitions of Eq. (38).. A useful approximation appropriate for third generation flavor violation is to neglect the masses of the first two generation quarks, where the breaking of the flavor symmetry is characterized by . This description is especially suitable for the LHC, where it would be difficult to distinguish between light quark jets of different flavor. In this limit, the 1-2 rotation and the phase of the CKM matrix become unphysical, and we can, for instance, further apply a rotation to the first two generations to “undo” the 1-3 rotation. Therefore, the CKM matrix is effectively reduced to a real matrix with a single rotation angle, , between an active light flavor (say, the 2nd one) and the 3rd generation,
where and are the corresponding CKM mixing angles. The other generation (the first one) decouples, and is protected by a residual symmetry. This can be easily seen when writing and in, say, the down mass basis
where stands for a non-zero real entry. The resulting flavor symmetry breaking scheme is depicted in Fig. 5, where we now focus only on the LH sector.
An interesting consequence of this approximation is that a complete basis cannot be defined covariantly, since in Eq. (46) clearly span only a part of the eight dimensional space. More concretely, we can identify four directions in this space: and from Eq. (40) and either one of the two orthogonal pairs
Note that corresponds to the conserved generator, so it commutes with both and , and takes the same form in both bases141414The meaning of these basis elements can be understood from the following: In the down mass basis we have , , and . The alternative diagonal generators from Eq. (48) are and . It is then easy to see that commutes with the effective CKM matrix, which is just a 2-3 rotation, and that it corresponds to the generator, , after trace subtraction and proper normalization.. There are four additional directions, collectively denoted as , which transform as a doublet under the CKM (2-3) rotation, and do not mix with the other generators. The fact that these cannot be written as combinations of stems from the approximation introduced above of neglecting light quark masses. Without this assumption, it is possible to span the entire space using the Yukawa matrices [60, 61, 62]. Despite the fact that this can be done in several ways, in the next subsection we focus on a realization for which the basis elements have a clear physical meaning.
It is interesting to notice that a given traceless adjoint object in three generations flavor space has an inherent symmetry (that is, two identical eigenvalues) if and only if it satisfies
In this case it must be a unitary rotation of either or its permutations , which form an equilateral triangle in the plane (see Fig. (8)).
As before, we wish to characterize the flavor violation induced by in a basis independent form. The simplest observable we can construct is the overall flavor violation of the third generation quark, that is, its decay to any quark of the first two generations. This can be written as
which extracts in each basis.
4.2.2 No limit – complete covariant basis
It is sufficient to restore the masses of the second generation quarks in order to describe the full flavor space. A simplifying step to accomplish this is to define the following object: We take the -th power of , remove the trace, normalize and take the limit . This is denoted by :