A The Number of Different Adjoints in a \Delta F=2 Operator

Covariant Description of Flavor Conversion in the LHC Era


A simple covariant formalism to describe flavor and CP violation in the left-handed quark sector in a model independent way is provided. The introduction of a covariant basis, which makes the standard model approximate symmetry structure manifest, leads to a physical and transparent picture of flavor conversion processes. Our method is particularly useful to derive robust bounds on models with arbitrary mechanisms of alignment. Known constraints on flavor violation in the and systems are reproduced in a straightforward manner. Assumptions-free limits, based on top flavor violation at the LHC, are then obtained. In the absence of signal, with 100 fb of data, the LHC will exclude weakly coupled (strongly coupled) new physics up to a scale of 0.6 TeV (7.6 TeV), while at present no general constraint can be set related to processes. LHC data will constrain contributions via same-sign tops signal, with a model independent exclusion region of 0.08 TeV (1.0 TeV). However, in this case, stronger bounds are found from the study of CP violation in mixing with a scale of 0.57 TeV (7.2 TeV). In addition, we apply our analysis to models of supersymmetry and warped extra dimension. The minimal flavor violation framework is also discussed, where the formalism allows to distinguish between the linear and generic non-linear limits within this class of models.

1 Introduction

The standard model (SM) has a unique way of incorporating CP violation (CPV) and suppressing flavor changing neutral currents (FCNCs). In fact, the way the SM flavor symmetry is broken to allow flavor conversion is quite intriguing. It can be described in the language of collective breaking [1, 2], a term commonly used in the Little Higgs literature (see e.g [3] and refs. therein). Inter-generation transitions require the presence of non-universal Yukawa couplings for both up and down quarks, a non-vanishing weak coupling and a non-trivial CKM matrix. The lightness of the first two generation masses and the approximate alignment between the Yukawa matrices further suppress FCNC transitions involving the first two generation. This is manifest in particular in processes which are characterized by hard GIM, such as ones involving CPV. Inclusive third generation processes are further simplified due to the presence of an approximate residual symmetry [4], only broken by the mass differences of the light quarks.

New types of microscopic dynamics with a different flavor breaking machinery typically give rise to deviation from the SM approximated selection rules, and hence can be distinctively distinguished from the SM. Till today no deviation from the SM predictions related to quark flavor violation has been observed1. This probably implies that new physics (NP) searches should focus on SM extensions which, if not flavor blind, share some of the structure and properties described above.

Regarding the first two generations, models which do not include some sort of degeneracies or flavor alignment (that is, when NP contributions are diagonal in the quark mass basis) are bounded to a high energy scale. Moreover, contributions involving only quark doublets cannot be simultaneously aligned with both the down and the up mass bases, hence even alignment theories are constrained by measurements. However, the hierarchy problem is not triggered by the light quarks, but rather by the large top Yukawa, where almost any natural NP model consists of an extended top sector. In addition, within the SM, the top dominates the CP violating transitions, and dials the amount of custodial symmetry breaking. Ironically, the top sector is the least experimentally explored, and at present model independent bounds on its flavor violating couplings are rather poor.

In this work, we elaborate on a basis independent formalism for studying flavor constraints in the quark sector, that was recently introduced by us in [4] (see also [6] for related work about algebraic flavor invariants). Apart from yielding a simple, symmetry driven, manner to understand the SM way of breaking flavor and CP, it also provides a straightforward method to study generic forms of NP flavor violation and derive model independent bounds (focusing on the left-handed quark sector and assuming -invariant NP contributions). We start with a two generations analysis, where a natural geometric interpretation can be applied. It allows us to straightforwardly reproduce known results [7]. We then consider the three generations case, where a dramatic improvement in the measurements related to the top sector is expected at the LHC. Thus, it is rather interesting to asses, before the data is analyzed, what is the potential impact of the projected sensitivity on beyond the SM searches. Our formalism makes manifest the SM approximate symmetry, due to the lightness of the first two generation masses, for the up and down quark sectors. In this limit, the SM actually posses a residual symmetry, which is automatically incorporated by our formalism. Under this symmetry, the massless first two generations break into an “active” one, which interacts with the heavy state, and a non-interactive “sterile” state. This description is useful, not only conceptually, but also when considering top and jet physics at the LHC, which in practice cannot distinguish between light quark jets. The combination of data from the down and the up sectors is used to robustly constrain models including arbitrary mechanisms of alignment.

The analysis is based on the SM flavor group for quarks:


where , and stand for quarks doublets, up-type singlets and down-type singlets, respectively. As mentioned, is broken within the SM only by the Yukawa interactions. Therefore, we can treat the Yukawa matrices and as spurions, which transform as and , respectively, under the flavor group. In order to attain a covariant geometric picture, we need to construct objects out of the Yukawa matrices which transform in the same way. These are simply and , which are in the representation. Since the trace of these matrices does not affect flavor changing processes, it is useful to remove it, and work with and , both of which are adjoints of . For simplicity of notation, we denote these objects as


As shown below, we can use these SM spurions of flavor violation to construct a covariant basis. This basis turns out to physically describe the flavor violation of the SM, as well as of NP.

This paper is organized as follows: The two generations case, for which a geometric formalism is devised, is discussed in Sec. 2. The covariant description for third generation flavor violation is given in Sec. 3. In Sec. 4 we use our formalism to constrain NP models in an assumption-free manner, based on third generation decays. Sec. 5 similarly deals with processes involving the third generation quarks. For the latter two sections, current experimental data is used for the down sector constraints, while the up sector bounds are mostly based on LHC prospects. Secs. 6 and 7 present concrete examples for the application of the analysis to supersymmetry and warped extra dimension, respectively. Finally, we conclude in Sec. 8.

2 Two Generations

We start with the simpler two generations case, which is actually very useful in constraining new physics, as a result of the richer experimental 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-independent2:


These definitions allow for an intuitive understanding of the flavor and CP violation induced by a new physics source. Consider a dimension six -invariant operator, involving only quark doublets,


where is some high energy scale and is the Wilson coefficient. is a traceless hermitian matrix, transforming as an adjoint of (or for two generations), so it “lives” in the same space as and .3 In the down sector for example, the operator above is relevant for flavor violation through mixing. To analyze its contribution, we define a covariant basis for each sector, with the following unit vectors


Then the contribution of the operator in Eq. (4) 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. (6) 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 1). This is exactly what Eq. (6) describes.

Figure 1: The contribution of to mixing, , given by the solid blue line. In the down mass basis, corresponds to , is and is .

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. 2) 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. (7).

Figure 2: CP violation in the Kaon system induced by . is twice the product of the two solid orange lines, which are the projections of on the and axes. Note that the angle between and is twice the Cabibbo angle, .

The weakest unavoidable bound coming from measurements in the and systems was derived in [7] using a specific parameterization of . In the covariant bases defined in Eq. (5), can be written as


and the two bases are related through


while remains invariant. Plugging Eqs. (8) and (9) into Eqs. (6) and (7), we obtain explicit results. It is then easy to see that in the parameterization employed in [7], is equal to , is equal to etc., therefore their results coincide with ours.

An interesting conclusion can be inferred from the analysis above: In addition to the known necessary condition for CPV in two generation [7]


we identify a second necessary condition, exclusive for processes:


The strength of these conditions is that they involve only the basic physical ingredients and , and they can be clearly identified from the geometric interpretation. Note, however, that this new condition in Eq. (11) is only applicable to either the down or the up sector, while the known condition in Eq. (10) is universal.

3 Three Generations

3.1 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 matrices4. 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  [1]. 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. 3.

Figure 3: The flavor symmetry breaking pattern for the left-handed sector: The group is broken by each of and to an approximate ; including the fact that these two objects are not aligned (that is, the CKM matrix is not trivial), the symmetry is broken into baryon number and an approximate for the light quarks combination that effectively decouples.

An interesting consequence of this approximation is that a complete basis cannot be defined covariantly, since in Eq. (13) clearly span only a part of the eight dimensional space. More concretely, we can identify four directions in this space: and from Eq. (5) 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 bases5. 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 [8, 9, 10]. 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. (4)).

Figure 4: The three unit-length diagonal traceless matrices with an inherent symmetry. and were schematically added (their angle to the axis is actually much smaller than what appears in the plot).

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.

3.2 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 :


and we similarly define . Once we take the limit the small eigenvalues of go to zero and the approximation assumed before is formally reproduced. As before, we compose the following basis elements:


which are again identical to the previous case. The important observation for this case is that the symmetry is now broken. Consequently, the generator, , does not commute with and anymore (nor does , which is different from only by normalization and a shift by , see Eqs. (14) and (15)). It is thus expected that the commutation relation (where now contains also the strange quark mass) would point to a new direction, which could not be obtained in the approximation used before. Further commutations with the existing basis elements should complete the description of the flavor space.

We thus define


In order to understand the physical interpretation, note that does not commute with , so it must induce flavor violation, yet it does commute with . The latter can be identified as a generator of a symmetry for the bottom quark (it is proportional to diag(0,0,1) in its diagonal form, without removing the trace), so this fact means that preserves this symmetry. Therefore it must represent a transition between the first two generations of the down sector.

We further define


which complete the basis. All of these do not commute with , thus producing down flavor violation. commutes with , so it is of the same status as . The last two elements, , are responsible for third generation decays, similarly to and . More concretely, the latter two involve transitions between the third generation and what was previously referred to as the “active” generation (a linear combination of the first two), while mediate transitions to the orthogonal combination. It is of course possible to define linear combinations of these four basis elements, such that the decays to the strange and the down mass eigenstates are separated, but we do not proceed with this derivation. It is also important to note that this basis is not completely orthogonal. An explicit decomposition of all the covariant objects in a specific basis can be found in Appendix B.

An instructive exercise is to decompose in this covariant “down” basis, since is a flavor violating source within the SM. Focusing only on the dependence on the small parameters and (and omitting for simplicity factors such as the Wolfenstein parameter ), we have


This shows the different types of flavor violation in the down sector within the SM. It should be mentioned that the and projections of vanish when the CKM phase is taken to zero, and also when either of the CKM mixing angles is zero or . Therefore these basis elements can be interpreted as CP violating, together with . As an example, notice that a transition in the down sector, represented by the projection to , can either occur via mixing with the third generation at the order or among the first two generations only at the order . Yet CPV in this transition can only be generated through the latter type of contribution at , as can be seen from the projection (recall again that these are not the and mass eigenstates, but instead the “active” and “inactive” generations, after a rotation has been applied). Analogously, a transition occurs at whether it is CP conserving or CP violating, as inferred from the projections.

In the rest of the paper we use the description based on the approximate symmetry, rather than the full basis, whenever possible.

4 Third Generation Transitions

We now use measurements from the down and the up sectors to derive a model independent bound on the corresponding NP scale, based on the overall flavor violating decay of the third generation quarks. We focus on the following operator


which contributes at tree level to both top and bottom decays [11]6. Note that as in the two generations case, we only deal with the left-handed sector, where down and up contributions are related. We omit an additional operator for quark doublets, , which induces bottom decays only at one loop, but in principle it should be included in a more detailed analysis.

The experimental constraints we use are [13, 14, 15]


where the latter corresponds to the prospect of the LHC bound in the absence of signal for 100 fb. We adopt the weakest limits on the coefficient of the operator in Eq. (23), , derived in [11]:


and define .

The NP contribution can be decomposed in the covariant bases as


The length of is denoted, based on the definition in Eq. (3), by


The weakest bound is obtained, for a fixed , by finding a direction of that minimizes the contributions to and , thus constituting the “best” alignment. However, since commutes with , as discussed above, it does not contribute to third generation decay (Eq. (17)) in neither sectors. On the other hand, any component of may also generate flavor violation among the first two generations (when their masses are switched back on), which is more strongly constrained. Specifically, the bound that stems from the case of , derived in Appendix C, is


where the latter is for . This is stronger than the limit given below for other forms of , hence this does not constitute the optimal alignment. To conclude this issue, all directions that contribute to first two generations flavor and CPV at , that is , and , are not favorable in terms of alignment, as discussed in Appendix C.

The induced third generation flavor violation, after removing these contributions, is then given by


and in order to see this in a common basis, we express as


with as defined in Eq. (12). From this it is clear that contributes the same to both the top and the bottom decay rates, so it should be set to zero for optimal alignment. Thus the best alignment is obtained by varying , defined by


Here we use , which is the coefficient of , instead of , since the former does not produce flavor violation among the first two generations to leading order (up to ).

We now consider two possibilities: (i) complete alignment with the down sector; (ii) the best alignment satisfying the bounds of Eq. (25), which gives the weakest unavoidable limit. Note that we can also consider up alignment, but it would give a stronger bound than down alignment, as a result of the stronger experimental constraints. The bounds for these cases are [4]


as shown in Fig. 5, where in parentheses we give the strong coupling bound, in which the coefficient of the operators in Eqs. (4) and (23) is assumed to be . Note that these are weaker than the bound in Eq. (28).

Figure 5: Upper bounds on as a function of , coming from the measurements of flavor violating decays of the bottom and the top quarks, assuming TeV.

It is important to mention that the optimized form of generates also decay at higher order in , which might yield stronger constraints than the top decay. In Appendix C it is shown that the resulting bound from the former is actually much weaker than the one from the top. Therefore, the LHC is indeed expected to strengthen the model independent constraints.

5 Third Generation Transitions

In the previous section we used decays of third generation quarks to obtain the weakest model independent constraint. Here we do the same using processes. For simplicity, we only consider complete alignment with the down sector


as the constraints from this sector are much stronger. This generates in the up sector top flavor violation, and also mixing at higher order. Yet there is no top meson, as the top quark decays too rapidly to hadronize. Instead, we analyze the process (related to mixing by crossing symmetry), which is most appropriate for the LHC. It should be emphasized, however, that in this case the parton distribution functions of the proton strongly break the approximate symmetry of the first two generations. The simple covariant basis introduced in Sec. 3.1, which is based on this approximate symmetry, cannot be used as a result. Furthermore, this LHC process is dominated by , so we focus only on the operator involving up (and not charm) quarks. We verified numerically that indeed the charm contribution to this process is smaller by an order of magnitude.

The production of same-sign tops was studied in the literature in the context of different models (see e.g. [16, 17, 18] and refs. therein). The simplest way observe it at the LHC and distinguish it from production, is based on the dilepton mode, in which two same-sign (mostly positive sign) leptons are produced from the top quarks. However, the branching ratio of this mode is only about 5%, and there are several types of SM backgrounds, such as . We therefore choose to adopt a realistic assumption of 1% efficiency for detecting same-sign tops at the LHC [16], including b-tagging efficiency and the necessary cuts to isolate the signal7. In any case, our conclusions are only mildly sensitive to this assumption, as explained below.

In order to estimate the prospect for the LHC bound on same-sign tops production, we calculated the cross section using MadGraph/MadEvent [19], as a (or ) channel process mediated by a heavy vector boson, the mass of which is identified with . The resulting cross section for the LHC with center of mass energy of 14, 10 and 7 TeV, and for the Tevatron, is given by


respectively8. This was matched onto the operator in Eq. (4). We then used the fact that the cross section times the integrated luminosity must be lower than 3 for a 95% exclusion, in the absence of signal [20]. Adding the assumption of 1% signal efficiency, we find


for 100 fb at a center of mass energy of 14 TeV. The experimental constraint from CPV in the system is [21, 22]


The contribution of to these processes is calculated by applying a CKM rotation to Eq. (33). CPV in the system is then given by , and describes . Note that we have


with as the CKM matrix elements. The resulting bounds are


for and


for mixing. It should be mentioned that the bound in Eq. (38) depends on the quartic root of the cross section that was evaluated above, thus it is only mildly sensitive to that calculation and to the efficiency assumption. Interestingly, the bound that stems from the Tevatron with 5 fb (assuming that same-sign top pairs were searched for and not detected) is weaker than Eq. (38) by a factor of 17.

The limits in Eqs. (38) and (39) can be further weakened by optimizing the alignment between the down and the up sectors, as in the previous section. Since this would only yield a marginal improvement of about 10%, we do not analyze this case in detail.

To conclude, we learn that for processes, the existing bound is stronger than the one which will be obtained at the LHC for top quarks, as opposed to case considered above.

6 Supersymmetry

The analysis presented above uses a model-independent language via effective field theory. Here we apply our results to two SM extensions – supersymmetry (SUSY) and the Randall-Sundrum (RS) model of a warped extra dimension (in the next section).

We focus now on both and left-handed processes within supersymmetric extensions of the SM. The idea as in the above is to provide robust bounds, which could be applied even to SUSY alignment models [23] (for a possible connection with bounds from EDM see [24]). The analysis of transitions is more involved as follows. The relevant contributions to the left-handed operators is driven by the squark doublets mass matrix, which transforms as an adjoint of the minimal supersymmetric standard model (MSSM) flavor group. However, the contributions to top and bottom decays are induced by different operators in the effective Hamiltonian, hence our treatment above does not apply. Instead we rederive the relevant bounds on the squark mass matrix explicitly.

Given the large number of parameters involved in flavor changing processes, it is often convenient to use the mass insertion (MI) formalism. The mass insertions are defined in the so-called super CKM basis. In this basis all the neutral gaugino couplings are flavor diagonal, and the charged quark-squark mixing angles are equal to the CKM angles. In general, the squarks mass matrices are not diagonal in the Super CKM basis. Flavor violation is induced by the off diagonal elements, and can be parameterized in terms of the ratios


where are the off-diagonal elements of the mass squared matrix that mixes flavors for both left- and right-handed scalars (), and where indicates the average squark mass.

6.1 Top Decay

In the computations of the branching ratio we follow the analysis of [25] (see also [26]). We first work in the basis where the squarks mass matrix is diagonal. In this basis the diagrams relevant for the process are shown in Fig. (6).

Figure 6: Feynman diagrams for the process . The dashed lines represent squarks exchange.

The effective vertex relevant for FCNC can be parameterized as


where . The form factors can be in general written as


where the indices identify the squarks mass eigenstates and is the matrix that diagonalizes the squarks mass matrix. Moreover, it can be shown that analogous expressions are valid for and from Eq. (41).

Given that we perform a model independent analysis, we choose the model parameters in order to obtain a robust bound on . Thus we set in the squarks mass matrix. In this case the only contribution to comes from


where , and correspond respectively to the top, gluino and squarks masses. Furthermore we introduce the definitions and


The complete expression for the form factors, including the expressions for the decay into photons and gluons can be found in [25]. The quantities and in (44) are given by


where the divergent parts in the above integrals cancel in the final result. In the following we work in the approximation of quasi-degeneracy for the squarks, thus moving to the super-CKM basis. Expanding in terms of the mass insertions , we arrive at the following expression for the part of the form factors contributing to flavor violation


with given by


and where we use


In terms of the form factors, the expression for the branching ratio, normalized to , is given by


Evaluating Eq. (50) at and , we get the following bound for


6.2 Bottom Decay

We now move to discuss the transition. The branching ratio is given in terms of the Wilson coefficients by [11]


The contributions to flavor violation coming from the MSSM can be derived in the MI approximation. In order obtain a robust bound on , we neglect the chargino contributions, which depend on additional parameters, such as , etc.. Under this assumption, the explicit values for the MSSM contributions to with and are


and vanishes, see [27]9 (and similar Refs. [28]).

Combining Eqs. (52) and (53) with the experimental bound in Eq. (24), we obtain the following bound


6.3 Best alignment

As a result of the large difference between the top and bottom bounds, Eqs. (51) and (54), the best alignment scenario is practically equivalent to alignment with the down sector. In this case, and is simply proportional to the squarks mass squared difference multiplied by . Taking for concreteness (appropriate for models with only weak degeneracy [29]), the bound is then