Higgs Mediated FCNC’s in Warped Extra Dimensions
In the context of a warped extra-dimension with Standard Model fields in the bulk, we obtain the general flavor structure of the Higgs couplings to fermions. These couplings will be generically misaligned with respect to the fermion mass matrix, producing large and potentially dangerous flavor changing neutral currents (FCNC’s). As recently pointed out in [arXiv:0906.1542], a similar effect is expected from the point of view of a composite Higgs sector, which corresponds to a 4D theory dual to the 5D setup by the AdS-CFT correspondence. We also point out that the effect is independent of the geographical nature of the Higgs (bulk or brane localized), and specifically that it does not go away as the Higgs is pushed towards the IR boundary. The FCNC’s mediated by a light enough Higgs (specially their contribution to ) could become of comparable size as the ones coming from the exchange of Kaluza-Klein (KK) gluons. Moreover, both sources of flavor violation are complementary since they have inverse dependence on the 5D Yukawa couplings, such that we cannot decouple the flavor violation effects by increasing or decreasing these couplings. We also find that for KK scales of a few TeV, the Higgs couplings to third generation fermions could experience suppressions of up to while the rest of diagonal couplings would suffer much milder corrections. Potential LHC signatures like the Higgs flavor violating decays or , or the exotic top decay channel , are finally addressed.
Introducing a warped extra-dimension in such a way as to create an exponential scale hierarchy between the two boundaries of the extra dimension RS1 () has generated a lot of attention in the recent years as a novel approach to solve the hierarchy problem. By placing the Standard Model (SM) fermions in the bulk of the extra dimension it was then realized that one can simultaneously address the flavor hierarchy puzzle of the SM a (); bulkSM (). The electroweak precision tests put important bounds on the scale of new physics but by introducing custodial symmetries RSeff () one can have it around few TeV RSeff (); AgashePGB (); EWPTmodel (); Agashezbb ().
In this paper we will study the class of models in which all the SM fields are in the bulk and the hierarchies in masses and mixings in the fermion sector are explained by small overlap integrals between fermion wave functions and the Higgs wave function along the extra dimension. This scenario can lead to the observed fermionic masses without any hierarchies in the initial 5D Lagrangian, so that our fundamental 5D Yukawa couplings have no structure and are all of the same order. Another interesting feature of these models is that the contributions to low energy observables coming from the exchange of heavy KK states will be suppressed by the so called “RS GIM” mechanism AgashePerezSoni (); hs (). In spite of it, it was still found that processes push the mass of the KK excitations to be above TeV Weiler (); BlankedeltaF2 (); Fitzpatrick:2007sa (); Davidson:2007si (), making it very hard to produce and observe them at the LHC kkgluon (); Agashe:2007ki (). These bounds coming from flavor violation in low energy observables can be avoided by introducing additional flavor symmetries Fitzpatrick:2007sa (); Davidson:2007si (); Cacciapaglia:2007fw (); Csakifp (). Another way to relax these low energy constraints is to promote the Higgs to be a 5D bulk field (instead of being brane localized). In this situation the bounds from could allow masses of the lowest KK gluon to be as low as TeV, although combining this result with the bounds from dipole moment operators pushes back the KK scale to be above TeV Agashe2site (); Gedalia:2009ws (). A similar tension was found in the lepton sector in Agashe:2006iy ().
It has recently been pointed out that in the context of a composite Higgs sector of strong dynamics, one generically expects some amount of flavor changing neutral currents (FCNC’s) mediated by the Higgs Agashe:2009di () (from an effective field theory point of view see also the earlier works Buchmuller:1985jz (); delAguila:2000aa (); Babu:1999me (); Giudice:2008uua ()). In the 5D picture, the presence of KK fermion states will actually produce a misalignment between the Higgs Yukawa couplings and the SM fermion masses, giving rise to tree-level flavor violating couplings of the Higgs to fermions. The induced FCNC’s are strongly constrained by various low energy experiments; if these constraints are somehow evaded, interesting signals at the LHC could also be generated.
The possibility of a flavor misalignment between the Higgs Yukawa matrices and the fermion mass matrices in the context of 5D warped scenarios was first briefly mentioned in Agashe:2006wa (), although it was not until NeubertRS () where a detailed analysis of the flavor structure of the couplings of the Higgs (brane localized) was first performed. There, the effects on the flavor violating Higgs couplings were found to be small (except for third generation quarks), with the (hidden) assumption that the contribution from a specific type of operators is negligible. In a more general Higgs context (bulk or brane localized), all the sources of Higgs flavor violation were then pointed out in BlankedeltaF2 (); Gori (), including the previously neglected operators, although no analysis on the overall size of the Higgs FCNC’s was performed. Moreover, in the limit of a brane localized Higgs, the effects of the larger sources of flavor are claimed to become negligible, and so it is again found that Higgs mediated FCNC’s are highly suppressed in the case of a brane Higgs.
In this work, we show that the induced misalignment in the Higgs couplings is generically large and phenomenologically important in both bulk and brane localized Higgs scenarios. The main cause for this result is the effect of the originally neglected operators which, due to a subtlety in the treatment of the brane localized Higgs, ends up surviving in the brane limit and giving rise to important misalignments between the Higgs Yukawa couplings and the fermion mass matrices.
The outline of the paper is as follows: in section II we review the model independent argument such that (TeV suppressed) higher order effective operators in the Higgs sector can lead to potentially large Higgs FCNC’s. This is then applied to the 5D RS model, first in the mass insertion approximation in order to quickly estimate the size of the corrections. In section III we proceed with a more precise calculation of the Higgs Yukawa couplings in the case of one fermion generation, and for a bulk Higgs scenario. The deviation in the Yukawa couplings is quite insensitive to how much the Higgs is localized near the IR brane; this result is confirmed in section IV by doing a 5D computation for the case of an exactly IR localized Higgs field, and it seems at odds with the mass insertion approximation which suggests that the corrections to the flavor violating Higgs couplings should vanish in the brane Higgs limit. This apparent contradiction is addressed and resolved in that same section. In section V we extend our results to the case of three generations and then in section VI, we give an estimate of the expected overall size of the Yukawa coupling matrices. We also argue that the couplings of the Higgs to third generation fermions might be significantly suppressed. These estimates are confirmed in section VII by the results of our numerical scan. Finally, section VIII is devoted to the study of phenomenological implications of Higgs mediated flavor violations, where we discuss low energy bounds arising from processes as well as interesting collider signatures.
Ii Flavor misalignment estimate
From an effective field theory approach it is easy to write the lowest order operators responsible for generating a misalignment in flavor space between the Higgs Yukawa couplings and the SM fermion masses. For simplicity we focus on the down quark sector and write the following dimension 6 operators of the 4D effective Lagrangian Buchmuller:1985jz (); delAguila:2000aa (); Babu:1999me (); Giudice:2008uua (); Agashe:2009di ():
where and are the fermionic doublets and singlets of the SM, with , and being complex coefficients and are flavor indices; is the cut-off or the threshold scale of the effective Lagrangian. Upon electroweak symmetry breaking (EWSB), these operators will give a correction to the fermion kinetic terms and to the fermion mass terms. Calling the original Yukawa couplings, the corrected fermion mass and kinetic terms become:
where GeV is the Higgs electroweak , i.e. , with being the physical Higgs scalar. On the other hand, the induced operators involving two fermions and one physical Higgs become:
From Eq.(2) it is clear that one has to redefine the fermion fields to canonically normalize the new kinetic terms and then perform a bi-unitary transformation to diagonalize the resulting mass matrix. These fermion redefinitions and rotations will not in general diagonalize the couplings from Eq. (3) and therefore, we will obtain tree-level flavor changing Higgs couplings, with a generic size controlled by .
In the warped extra dimensions scenarios that we are interested in, we can estimate easily the size of this type of misalignments between the Higgs Yukawa couplings and the SM fermion masses by using the insertion approximation in KK language. The 5D spacetime we consider takes the usual Randall-Sundrum form RS1 ():
with the UV (IR) branes localized at () and with being the curvature scale of the AdS space. We are interested here in the flavor structure of the Yukawa couplings between the Higgs and the fermions. However, it is instructive to first consider the case of only one generation and study the (potentially large) corrections induced to the single Yukawa coupling. One can then easily generalize to three generations and find the misalignment between the fermion mass matrix and the Yukawa couplings matrix.
We will focus on the down-quark sector of a simple setup in which we consider the 5D fermions , . They contain the 4D SM doublet and singlet fermions respectively with a 5D action
where and are the 5D fermion mass coefficients and is the bulk Higgs field localized towards IR brane. The wavefunctions of the fermion zero modes are determined by their corresponding 5D mass coefficients. To obtain a chiral spectrum, we choose the following boundary conditions for
Then, only and will have zero modes, with wavefunctions:
where we have defined and the hierarchically small parameter , which is generally referred to as the “warp factor”. Thus, if we choose , then the zero modes wavefunctions are localized towards the UV brane; if , they are localized towards the IR brane. The wavefunctions of the KK modes are all localized near the IR brane. Note that the wavefunctions of the KK modes and vanish at the IR brane due to their boundary conditions. The Yukawa couplings of the Higgs with fermions (zero modes or heavy KK modes) are set by the overlap integrals of the corresponding wavefunctions. For a bulk Higgs localized near the IR brane, the zero-zero-Higgs, zero-KK-Higgs, KK-KK-Higgs Yukawa couplings are given approximately by
where is the dimensionless 5D Yukawa coupling, and we ignored factors in the equations above. The SM fermions are mostly zero mode fermions with some small amount of mixing with KK mode fermions. Therefore, we can use the mass insertion approximation to calculate the masses and Yukawa couplings of SM fermions. This is shown in Fig. 1, where , are zero modes of doublet and singlet fermions respectively and , , , are KK mode fermions. From the Feynman diagram in Fig. 1 we see that the SM fermion mass is given by
where is the Higgs vev and we assume that all KK fermion masses are of the same order ().
The 4D effective Yukawa couplings of SM fermions can be calculated using the same diagram. However in the second diagram of Fig. 1, we have to set two external to their vev while the other one becomes the physical Higgs , and there are three different ways to do this. Thus we obtain the 4D Yukawa couplings
We see that the SM fermion masses and the 4D Yukawa couplings are not universally proportional; indeed there is a shift with respect to the SM prediction of .
We thus define the shift as
and it is easy to see that the contribution of the diagrams of Fig. 1 to is
There is yet another source of shift between masses and Yukawa couplings coming this time from the corrections to the kinetic terms. This is the contribution which was pointed out and carefully computed in NeubertRS (), and as wee will see later, in agreement with our own results for that specific term. As shown in Fig. 2, the kinetic term for the fermion mode receives a correction induced by the mixing with KK fermion modes
After redefining fields so that their kinetic term is canonical, there will be a new contribution to the shift between masses and Yukawa couplings given by
Similarly, the correction to the kinetic term of gives the contribution
Adding all terms together, we find the total fermion mass-Yukawa shift
If we extend to the case of three generations, we can see that this shift between SM fermion masses and Yukawa couplings produces a misalignment in flavor space between these. This misalignment will lead to flavor violating Higgs couplings once the fermion mass matrix is diagonalized.
For the first two generation quarks, we need to reproduce their small masses. Therefore, for these first two generations, the shift coming from the correction to kinetic terms (Fig. 2) is negligible and the correction coming from the diagrams in Fig. 1 will dominate. However, for the third generation, all effects are comparable. It is interesting to point out that the expression (Eq. 19) (valid for one generation) is always positive, which leads to a reduction in the 4d effective Yukawa couplings compared to the SM ones.
ii.1 Brane Higgs subtlety
Finally, we must mention that there is a subtlety in the case of an exactly brane localized Higgs. As pointed out in BlankedeltaF2 (); Gori (), since the wavefunctions of and vanish at TeV brane (due to Dirichlet boundary conditions), their couplings to a brane localized Higgs should also vanish. This means that the second diagram in Fig. 1 should give no contribution to the fermion mass-Yukawa shift (or at best a highly suppressed one). We would then expect to be left with only the correction coming from the kinetic term (Fig. 2), which as stated above is negligible for light quarks. We observe, however, that upon EWSB, the wavefunctions and become discontinuous at the brane location Csaki:2003sh (), with the jump of the wavefunctions being proportional to the brane Higgs vev . This discontinuity requires some sort of regularization of the brane location, meaning that the couplings of and with the brane Higgs would be infinitesimally small, but non-zero. But we note that in the second diagram of Fig. 1, one has to sum over infinite KK modes and even though each KK mode will give an infinitesimally small contribution, the sum of infinite terms can lead to a finite (non-zero) result (and as it turns out, this is what happens, as shown explicitly in Appendix C for this mass insertion approximation).
This brane Higgs issue is avoided in NeubertRS () because the authors did not include in their brane action any operator of the type . By avoiding these, the contribution to the shift coming from the diagrams of Fig. 1 is simply not present (except for highly suppressed corrections of order which are safe to ignore).
Iii 5D calculation: Bulk Higgs Scenario
In this section we perform a 5D calculation in order to evaluate more precisely the shift between Yukawa couplings and masses of SM fermions. We start by working with a single fermion generation for clarity but will later extend our results to the three generations case.
To proceed, we will need to solve for the wavefunctions of SM fermions along the fifth dimension in the bulk Higgs bulkhiggs (); bulkhiggs1 () scenario. This corresponds to including the contribution of all KK modes of the mass insertion approximation, and not just the lightest ones. As we will see, the most important shift does not go away as we push the Higgs profile towards the IR brane. In the bulk Higgs scenario, the Higgs comes from a 5D scalar with the following action bulkhiggs ()
where is the 5D mass for Higgs in unit of . The boundary potentials and give the boundary conditions for the Higgs wavefunction. We can choose these boundary conditions such that the profile of the Higgs vev takes the simple form
where is the SM Higgs vev. This nontrivial vev is localized towards the IR brane solving the Planck-weak hierarchy problem. Nevertheless we will treat the brane Higgs case separately later to review possible subtleties inherent to its localization by a Dirac delta function.
After writing the 5D fermions in two component notation, and , we perform a “mixed” KK decomposition as
where correspond to the light 4D SM fermions and the include the rest of heavy KK fermion fields. are the corresponding profiles of the 4D SM fermions and which verify the Dirac equation
with being the 4D SM down-type quark mass (the analysis can be carried out for up-type quarks in similar fashion).
The four profiles and must verify the coupled equations coming from the equations of motion.
where the denotes derivative with respect to the extra coordinate and is 5D Yukawa coupling. Even if one knows the analytical form of the nontrivial Higgs vev , solving analytically this system of equations might still be quite hard. Nevertheless it is simple to find the misalignment between Higgs Yukawa couplings and fermion masses based on the previous equations. To proceed, let us first multiply Eq. (29) by and the conjugate of Eq. (30) by , and then subtract them. One obtains
We can now multiply by and integrate the whole expression between and and obtain
The boundary conditions for the profile are chosen to be Dirichlet at both boundaries, i.e. , which means that the last term of Eq. (34) identically vanishes. Moreover, canonical normalization of the SM d-quark imposes the extra constraint
We can therefore rewrite Eq. (34) as
Note that this identity is exact, but also that each profile and depend on the mass . In the zero mode approximation, the profiles with Dirichlet boundary conditions, and vanish, and the identity can be expressed as
which agrees with the intuition that fermion mass is mostly generated by the 5D Yukawa couplings between the 5D Higgs and the zero mode fermion profiles. From the action in Eq. (5) we also extract the 4D Yukawa coupling of the Higgs field (the lightest KK mode of the 5D Higgs) and the SM down type quark.
where is the profile of the physical Higgs field. It is easy to show that the Higgs vev solution is related to the profile of the physical light Higgs (lightest KK mode) by
so for a light enough Higgs field both profiles and are proportional to each other. For a moderately heavy physical Higgs, there will be a misalignment between the profiles of the Higgs vev and the physical Higgs, leading to a misalignment between fermion masses and Yukawa couplings. However, this effect can actually be decoupled if the Higgs is pushed towards the IR brane (by increasing the parameter ). In this case, the Higgs vev profile will be more and more aligned with that of the physical Higgs, so that they become identical in the brane Higgs limit. This source of Higgs flavor violating couplings will be controlled by the parameter and for the sake of clarity we will ignore its effects in the rest of the paper because, as we discuss in Appendix B, they are numerically small and can be decoupled by pushing the Higgs towards the IR brane.
We can then compute the shift between the fermion mass and the Yukawa coupling as
This identity shows that the shift has to be relatively small since it vanishes in the zero mode approximation.
To proceed further, we will use a perturbative approach such that we assume that where is the SM Higgs vev. Thus, once we know the analytical form of the vev profile (see Eq. (21)) we can solve perturbatively the system of coupled equations (29-32)111It would be interesting to use this perturbative technique in the context of fermion flavor in soft-wall scenarios Batell:2008me (); Delgado:2009xb (); Aybat:2009mk () given that the setup is quite similar; we will leave this analysis for future studies..
with the constants and fixed by canonical normalization of the kinetic terms giving
Equipped with the solutions from Eqs. (41) to (44) one can evaluate perturbatively the shift defined in Eq. (40). For simplicity, we present here the results for UV localized fermions (). The general results for both UV and IR localized fermions are presented in Appendix A. We find that the main contribution to the shift coming from the last term in Eq. (40) can be written as
This result corresponds to the one we estimated earlier by using the insertion approximation (see Eq. (15)).
The first term in Eq. (40) gives a subleading contribution to the shift
Even if the fermion mass is small, the large warp factor will overcome most of the suppression, rendering the shift to be of the order . The shift is generally on the percent level with respect to fermion masses, but a misalignment of this order in the Higgs Yukawa couplings should introduce strong constraints due to FCNC’s.
iii.1 Pushing the Higgs from the bulk to the brane
Note that in the limit, the profile of the Higgs vev tends to become brane localized, as well as the light physical Higgs and the rest of Higgs KK modes. In this limit, the shift produced between the fermion mass and the Yukawa coupling, coming from the diagrams of Fig. 1, reduces to
and in particular we see that the effect does not decouple (i.e. it is non-zero). The fact that the expected misalignment is more or less independent on the localization of the Higgs is one of our main results since the bounds and predictions that we will extract can then be considered a general feature of RS models with fields in the bulk (and a Higgs scalar localized near or at IR brane)222An interesting exception to these results in the Higgs sector, proposed in Agashe:2009di (), would be to eliminate the Higgs as a fundamental scalar and consider the fifth component of a gauge field as playing the Higgs role in EWSB.. The shift coming from the corrections to the fermion kinetic terms (Fig. 2) becomes in the limit
in agreement with the results found in NeubertRS () (for a brane Higgs scenario).
Maybe it can be useful to discuss the validity of the limit starting from a bulk Higgs scenario. Let’s first look at the mass spectrum in this case. The Higgs profile is given by Eq. (111) and to find its mass eigenvalues one has to satisfy the appropriate boundary conditions at the IR brane bulkhiggs ()
This will lead to one light mode (i.e. SM Higgs) and a tower of heavy modes with masses proportional to , and so in the limit all the KK Higgs excitations are decoupled from the low energy spectrum. This means that in this limit we can treat Higgs field as an effective four dimensional field, and thus it corresponds to the brane Higgs scenario. As mentioned earlier (and in Appendix B), the misalignment caused by a difference in profiles between the Higgs physical field and its vev (and which we have neglected) will also disappear, as one can interpret that specific misalignment as a result of the mixing between SM Higgs and the heavy Higgs KK modes, which is controlled by .
Let us now look on the couplings of fermions to the Higgs in this limit. For the zero modes we will get:
where ; similarly one can look at the couplings of two KK fermions to the Higgs and in this case one finds its dependence to be . Naively both couplings do vanish in the limit. But if the 5D couplings scale as then these couplings will have a finite limit given by the usual brane Higgs results. One can argue whether we can scale the 5D Yukawas as because such large Yukawas should violate perturbativity of the theory, but as was shown above the couplings of the Higgs to the KK fermions are still . One can see that only the KK excitations of the Higgs will have couplings with KK fermions , but their masses are and they are completely decoupled from the spectrum. So we conclude this discussion by stressing that it is consistent to consider the limit with and it coincides with the usual brane Higgs scenario.
Iv 5D calculation: Brane Higgs Scenario
We argued in Section II that one might expect that the major contribution to the misalignment vanishes in the brane Higgs case since the odd KK modes , have vanishing wavefunctions on the IR brane. We also briefly mentioned that in the mass insertion approximation, one actually might need to sum the infinite tower of fermion KK modes to obtain a non-vanishing contribution (see Appendix C for details). However, without invoking that explanation, we just saw that in the limit, approaches a nonzero value of same numerical order as the case. Since the limit of bulk Higgs corresponds to a brane localized Higgs, there seems to be a counter-intuitive subtlety. In this section we try to address and resolve this point in a more precise way, by performing the 5D calculation of the shift for the specific scenario of a brane Higgs.
For brane Higgs, we can write the Yukawa couplings in the Lagrangian as
Here we choose the convention with . Note that compared to the bulk Higgs case, the Yukawa couplings an are independent and both . However, they should be of the same order due to the philosophy of flavor anarchy and naturalness. We can do KK decomposition as before, then the equations satisfied by the wavefunctions are
Notice that the odd wavefunctions and vanish at the IR brane. But the delta functions in equations above give a jump for and at the IR brane, which makes their values at IR brane ambiguous Csaki:2003sh (). To remove this ambiguity, we “regularize” the delta in the following way
This regularization is in a way similar to treating the Higgs as a bulk field and then taking the limit , although without apparent divergences coming from taking to be large. In any case one could also perform other regularization methods to remove the wavefunction ambiguities at the IR brane333For example, we could have chosen instead to move the delta function location from to , and enforce the usual boundary conditions on the fields at . Then, at the very end, we would take the limit Csaki:2003sh (). In that case we find (59) where we have used the step function for and for . Inserting this into Eq. (40) we obtain the same misalignment as in Eq. (65), namely .
Now we can easily impose Dirichlet boundary conditions for the profiles at IR brane
Integrating equations of motion (Eq. 54) from () will lead to
For the rectangular potential profiles will drop to zero linearly in the region , so the profiles near the IR brane can be approximated by
From our previous discussion, the main contribution to the misalignment between SM fermion masses and Yukawa couplings come from the second term of Eq.( 40), so plugging in the odd wavefunctions from Eq.(63), we get
On the other hand, to leading order in Higgs vev, the SM fermion mass is given by
Therefore, the misalignment can be expressed as
As advertised before, this result agrees with the one obtained in the previous section for the bulk Higgs scenario, once we take (Eq. 47). We again stress that this result shows that upon careful derivation, the misalignment obtained does not vanish in the particular case of a Brane localized Higgs. The main difference though, is the appearance of the independent couplings , which in the bulk Higgs case are forced to be equal to by 5D general covariance. These couplings are not necessary for generating fermion masses, and so it is technically possible to set their values as small as necessary to suppress the obtained misalignment. Nevertheless this seems to go against the main philosophy of our approach which is to assume the value of all dimensionless 5D parameters of order one.
Again, the fact that is non zero in the brane Higgs case is hard to understand in the mass insertion approximation since the contribution from each KK fermion (see Fig. 1) seems to be vanishing. In Appendix C we show that to resolve this point we need to sum up all the KK modes of the mass insertion approximation, as already mentioned before.
The subleading contribution to the misalignment between SM fermion masses and Yukawa coupling can be calculated in a similar way as in the previous section, and the result is (for UV localized fermions)