Taming the $\epsilon_K$ in Little Randall Sundrum Models

The Randall Sundrum (RS) models receive significant constraints from the neutral Kaon system. The CP violating observable $\epsilon_K$, in Randall Sundrum scenario, requires the lightest KK gluon to be heavier than $\sim$ 24 TeV. The constraint is even stronger in the Little Randall Sundrum models (LRS), $\gtrsim$ 32 TeV. The LRS models are motivated for their possible visibility at the Large Hardon Collider (LHC). We show that the stringent constraints from K-physics can be relaxed in the LRS models, in the presence of the Brane Localised Kinetic Terms (BLKT). In particular, for a range of values, a UV BLKT could significantly modify the lightest KK gluon wave function such that the limit can reduces to 5 TeV. We also show that such a relaxation of the constraints can also be achieved by imposing flavour symmetries {\`a la } Minimal Flavour Protection.


Introduction 2 Recap of the little RS model
To make the present paper self contained, we briefly recap the Little RS model in this section. The extra-dimensional set up is the same as the Randall Sundrum model [8] with the background metric defined on a M 4 × S 1 /Z 2 orbifold with a negative bulk cosmological constant. This geometry is defined by the line element ds 2 = g M N g M N = e −2ky η µν dx µ dx ν + dy 2 (1) where M,N are 5 dimensional space-time indices, η µν = diag(−1, +1, +1, +1) and 0 ≤ y ≤ L. The warp factor k, is set such that kL ∼ 7 with the fundamental scale M 5 ∼ O(10 3 TeV) [12] much lower than the UV scale in the RS case. We assume the bulk to be populated by gauge fields and fermions transforming under the adjoint representation and fundamental representations of the Standard Model gauge group SU (3)×SU (2) L ×U (1) Y respectively [10]. The Higgs field, which transforms as a doublet under the weak gauge group, is assumed to be localized on the IR brane. This stabilises the Higgs vacuum expectation value to H = M 5 e −kL ∼ O(1 TeV). The fermionic content includes three copies of the left handed quark doublets, Q i , i = 1, 2, 3, and three copies of right handed singlets, q i = u i , d i . Since the Clifford algebra in 5-dimension, given by five 4 × 4 Gamma matrices (Γ M ), is non-reducible, the fermion representations in this geometry have 4 complex degrees of freedom. Hence, on breaking the 5-dimensional Lorentz group down to 4-dimensions via compactification these fermions become vector like under the Weyl representation of the 4-dimensional Clifford algebra. The orbifolding ensures that the unwanted set of chiralities are projected out from the lowest modes.
The five dimensional fermionic action for doublet( Q) and singlet( q) quarks with bulk mass terms is given as, where m Q and m q are the bulk masses for the doublet and singlet quark fields respectively; i, j are generational indices and Ỹ (5) u,d ij are the five dimensional Yukawa matrices. In the above equation, D M represent the covariant derivative in 5-dimensions. The following boundary conditions at the orbifold fixed points (y = 0, y = L), ensure that only the correct chiral projections for the lowest modes of the respective fields survive at the boundaries.
Here l, r stand for left and right chiral fields under the 4-dimensional chiral projection operator and +(−) stands for the Neumann (Dirichlet) boundary conditions.
The Kaluza-Klein decomposition of a generic fermion field (Q) is given by where Q (n) l,r (x) stands for the corresponding four dimensional chiral KK modes and f l,r (y) are profiles of these modes in the bulk set to satisfy the ortho-normality conditions The normalized zero mode profile for doublets and singlets with bulk mass parameter c Q i = m Q i /k and c q i = −m q i /k respectively is computed to be, where, In this paper we are not interested in the higher KK-modes of fermions, as their contribution will always be suppressed by ∼ O( H /M KK ), where M KK is the compactification scale. Inserting the zero mode profile into the Yukawa action given in Eq.
(2), we obtain the effective 4D Yukawa coupling relevant for the SM fermion masses and mixings as, where O(1) Yukawa (Y (5) u,d ) parameters entering the mass matrices are defined as: The transformation from the quark flavour eigenbasis u l,r , d l,r to the mass eigenbasis u l,r , d l,r will then be given by performing rotation of unitary mixing matrices U l,r and D l,r as With this the CKM matrix is given by It should be noted that there is no reason for bulk fermionic masses to be diagonal or real. Since we are not assuming Minimal Flavour Violation(MFV), the Unitary matrices that diagonalise these bulk mass terms do not diagonalise the five dimensional Yukawa Lagrangian.
Gauge couplings To derive the bulk wave profile and coupling of the gauge boson with fermion bilinear it suffices to describe a U (1) gauge group in the bulk of the AdS. The generalisation to non-abelian gauge fields is straight forward. The five dimensional action for such a gauge field is given as where the field strength tensor F M N = ∂ M A N − ∂ N A M and g −2 5 is the 5D gauge coupling. In the unitary gauge, the vector field can be Fourier expanded as where A (n) µ (x) are the 4D gauge field KK modes and f (n) A (y) are profiles of these modes in the bulk. The equations of motion are given as by On integrating the above equation and demanding the zero KK mode be non vanishing, we arrive at the boundary condition (δA µ ∂ y A µ ) 0,L = 0. For canonical kinetic terms, the orthonormality condition is Solving Eq.(16) for m n = 0, we find that the zero-mode profile of the gauge boson is flat and is given as For the higher KK modes ( m n = 0) the solution is given in terms of the Bessel J and Y functions and is of the form: A are the the normalisation constants. The coefficients b n A are determined at the boundaries as Equating b (n) A y=L we get the spectrum m n = x n ke −kL where x n are the roots of this equation. The coupling of the zero mode fermions with the gauge KK modes is set by the overlap integral [20] g (n) l,r (c Q,q ) = g 5 L 0 dy e −3ky f (n) where f l,r and f

(n)
A are given by Eq. (9) and Eq. (19) respectively. Note that this function is quite sensitive to the UV scale and changing the scale could significantly modify the low energy phenomenology. Fig.(1) shows the difference between the coupling of KK-1 gluon in RS and Little RS and it could be seen that the coupling is enhanced by a factor 3 in the Little RS in comparison to RS for the fermion with the same 'c' value. This could be understood from the fact that, since the fundamental scale is much smaller, the UV localized fermions have bigger overlap with the composite gauge boson states 1 . This has important implications for LHC phenomenology and K as will be demonstrated shortly.  21) is that it is non-universal in generational space. This is evident when we explicitly write these couplings down as: After electroweak symmetry breaking, unitary transformations (D l,r , U l,r ) of Eq.(12) are used to go to mass eigen basis.This non-universality in couplings leads to flavour violation in these models.
Phenomenological constraints on Little RS The original RS model is envisaged to solve the Higgs mass hierarchy problem. This soon led to phenomenological issues. Localising the entire SM to 3-brane located at IR predicted large flavour violating and proton decay currents. On allowing SM fields, except for the Higgs, to propagate into the bulk solved such large contributions. As an added benefit, doing so also relaxed the large S-parameter contribution to the Electro-weak observables [10,18,21,22]. On the other hand, the T-parameter contribution remained large. And for a 125 GeV Higgs, the lower limit on mass of the lightest KK excitation turns out to be ∼ 13.6 TeV [22]. Thus, making this model irrelevant for the current LHC searches. Assuming a truncated space, Little RS (kL ∼ 7) brings down this correction and lowers the limit of the compactification scale to ∼ 4 TeV [12]. Another important expectation from Little RS model comes from its accessibility at the colliders. At LHC, the production of the lightest KK gluon and the Z boson in s-channel is via the annihilation of light quarks, which are typically UV localized. Fig.(1) clearly shows that the coupling of first KK gluon with light quark bilenears (bulk mass parameter c 1.0) are larger in Little RS in comparison with RS. This leads to enhanced production cross-section for this process. Recent searches were carried out at ATLAS collaboration [23] which looks for a heavy particles that decay into top-quark pair at 13 TeV LHC with an integrated luminosity of 36.1f b −1 . And the data rules out the masses smaller than ∼ 3.6 TeV at 95% CL for the KK gluon with a branching fraction of 30%. In this analysis, the strong coupling of light fermions to the KK gluon modes were set to −0.2g s , where g s is the strong coupling of the SM gluon. They considered the process pp → tt with top decaying in to t → bW with events selected requiring single charged isolated lepton, jets, missing transverse energy (or p T ). SM background processes were reduced by requiring the jets identified as likely to contain b-hadrons. We use this result to constrain our model.
While these limits hold for the RS model, a straight forward interpolation to Little RS is only approximately valid. Still, a direct comparison to the limit given by ATLAS will give a conservative limit on mass of the lightest KK gluon in our model. For the analysis, we will use couplings and branching fractions demanded by the bulk mass parameters of our model given in Table 3. These parameters were chosen such that they would satisfy the CKM matrix and will be useful for our further analysis of K . To compute the cross section of the process q q → g (1) → tt, we used CalcHEP 3.7.5 [24] with NNPDF2.3 with QED corrections. Since we did not mention a specific K-factor, CalcHEP uses a version of this function which always returns 1. With these considerations we find that the mass of the first KK partner of gluon to be 4.2 TeV (as shown in Fig.(2). And we will use this limit for rest of our analysis.

K in Little RS
Other than the hadron collider and LEP limits mentioned in the previous section, observables from the neutral meson mixing, especially the CP-Violation observable ( K ) of K 0 −K 0 system, has been shown to constraint the UV scale of RS [19,25,26,27,28,29] and Little RS [26,14] models significantly. As shown in Fig.(1), the light fermions couple to the first KK mode of gluons more strongly in the case of Little RS compared to RS. Thus understanding this constraint in the Little RS setup is imperative. In this section, we recall the important results of ∆F = 2 process in warped geometry. Note that from Eq.(21) the coupling of the bulk gauge boson KK states to fermion bilinears are dependent on the bulk mass parameters of the fermions. This renders the couplings family non-universal in the gauge basis. On rotating to the mass basis, defined by D l,r and U l,r given in the Appendix, the couplings become flavour non-diagonal. In principle all the gauge KK states would contribute to this process, but the dominant contribution comes from the tree level exchange of the lightest gluon KK state. On integrating out the new physics with mass M g , the effective Hamiltonian for ∆F = 2 becomes [28], where T a are the generators of the QCD gauge group, Latin indices i, j, k, l, m denote the fermion generation index and Greek indices represent the colour indices. Couplingsg l,r are defined as: ) x BR(g (1) t,T) [pb] M g (TeV) Observed 95% CL upper limit LO KK gluon cross section Figure 2: The observed cross-section 95% CL upper limit [23] on the g 1 signal and the theoretical prediction for the production cross-section times branching ratio of g 1 → tt at corresponding masses where g l (c Q ) and g r (c d ) are the gluon couplings defined in Eq. (21). Using the unitarity of D l,r , the above expression could be further expanded to understand the ∆F = 2 transitions in first and second generations. For this process, we only require the off diagonal elements which represent the flavour violating couplings and are given as, Adopting the usual parametrization of new physics effects in Kaon oscillation [30,31] we can write the model independent effective Hamiltonian as In the above equation, the first set of operators C a contain left handed chiral states and the operatorsC a contain right chiral states. And the operators O are, Since the CP-Violating K parameter in terms of the Effective Hamiltonian Eq.(27) is given as, Im(Wilson coeff.) Bound (TeV) Im(C 1 ) 1.5 × 10 3 Im(C 4 ) 1.6 × 10 4 Im(C 5 ) 1.4 × 10 4 only the imaginary parts of the Wilson coefficients contribute and the bound on the scale M g is summarised in Table (1) The model independent bound from K is strongest on the Wilson coefficient C 4 due to (as compared to C 1 ) chiral enhancement of the hadronic matrix element and the RG running from the new physics scale to the hadronic scale [30,32,33]. With the above formalism in place, comparison of the constraint coming from K in RS and Little RS model would be rendered simple since the difference between these models arise through the off diagonal couplingsg 12 l,r in the respective scenarios. For numerical study, we fit the bulk quark mass parameters to obtain the spectrum in MS scheme computed at 3 TeV while keeping the 5D Yuakwa anarchic (0.1 ≤ |Y (5) u,d ij | ≤ 3). We refrain from discussing the procedure here, but the detailed methodology for the numerical analysis can be found in Appendix A.
In Fig.(3), we present our result showing the dependence of the cumulative distribution functions (CDF) for the number of states satisfying the K observable (|Im(C sd 4 )|) with varying KK gluon mass scale M g for both RS and Little RS. It can be seen that the average value of K becomes consistent with the measurement only for M g ≥ 24 TeV for RS and M g ≥ 32 TeV for Little RS. From Eq.(21) and Eq.(29) one could infer that the bounds on the operators C 1,4,5 crucially depends on the value of g 5 and the fermion bilinear overlap with the first KK gluon wavefunction. Reducing this overlap can be achieved by using Brane Localized Kinetic Terms. This possibility has been mentioned earlier in Refs. [25,27] within the context of RS models.

Brane Localized Gauge Kinetic Terms
To understand the impact of Brane Localized Kinetic Term (BLKT) on the Wilson coefficients given in Eq. (29), lets start by considering the U(1) gauge field in the warped background. This could easily be generalized to the case of non-Abelian fields, since we are only interested in operators in the fermion bilinears. The generalisation to the gauge 5-dimensional action given in Eq. (14), including the BLKT could be written as [17,18] where, l IR and l U V are the strengths of the localized kinetic terms at IR and UV branes respectively. Here, we have chosen the convention where both the 5-dimensional gauge field A µ (x, y) and g −2 5 have dimensions of mass. In the presence of BLKT, the equation of motion derived from Eq.(31) becomes, And the ortho-normality condition is, The solution to the Eq.(32) is given by (34) where (N

(n)
A ) is the normalisation constant for the n th mode and zero mode wavefunction is given by, Integrating the equation of motion at the fixed points y = 0 and y = L yields the modified boundary conditions Demanding that the solution Eq.(34) should satisfy the boundary conditions, the b (n) where m n = x n ke −kL , and x n are the roots of the master equation obtained by imposing b A | IR . Now, we try to understand the implications of BLKTs on the KK-spectrum. The dependence of the first root, x 1 , on BLKT is shown in the Fig.(4(a)) for the Little RS geometry with warp factor k = 10 3 TeV. We have considered four cases of BLKTs (i) kl U V = 0, kl IR = 0, (ii) kl U V = kl U V = 0, (iii) kl U V = -kl U V = 0 and (iv) kl U V = 0, kl IR = 0. As can be seen the lowest root itself can modify by roughly a factor between 2 and 3 for the case (ii), when both BLKTs are switched on, with the same sign 2 . A bigger change is however expected in the coupling of KK-gluon and the fermion bilinear since BLKT modifies the bulk gauge field wave profile as shown in Eq. (34) and hence the overlap of the gauge boson with the fermion wave profile. With that, the gauge coupling becomes, This modification could have significant impact on flavour observable. Following the discussion in Sec.3, we plot the relevant coupling for the K process as a function of BLKT strength. To this extent we define where g (1) l,r (c) is given in Eq. (39). Fig.(4(b)) displays the difference in coupling strengths of first KK partner of the gluon, ∆g 1 g 0 , with fermions localized with the bulk mass parameters c 1 = 1.2 and c 2 = 0.8. In the figure we have explicitly shown the scenarios with kl IR = 0, kl U V = 0 and vice versa. The values of ∆g 1 g 0 for (i) kl IR = kl U V = 0 (ii) kl IR = −5,kl U V = 0 and (iii) kl IR = 0,kl U V = −5 are given in Table 2. This clearly shows that UV BLKT is significantly better in reducing the couplings and considerably relaxes the bound in Little RS. For the full numerical analysis, we have used the fermion masses given in Table 5 and scanned over the bulk mass parameter, 'c', with central values given in Table 3. This was done while imposing the anarchic condition on Yukawa, 0.1 ≤ |Y (5) u,d ij | ≤ 3. Fermion masses were fit as detailed in the Appendix[A]. The Cumulative distribution function and Probability distribution function satisfying the constraint on Im(C 4 ) is presented in Fig.(5). The plots clearly show the drastic reduction on the constraints while imposing BLKTs. The result is even more spectacular for a UV BLKT. The bounds for BLKT∼ 5 are summarised in Table 4. This clearly shows that the UV BLKT is significantly better in reducing the couplings and thus relaxes the bounds. It is clear that the BLKTs achieve this by bringing the couplings of the two light flavours closer to each other. Thus it is only natural that such a thing could also happen by imposing a flavour symmetry which we will explore in the next section.  Table 3: Central values 'c' parameter of the fermions used to obtain the fits in Table 4.

Minimal Flavour Protection
Minimal Flavour Protection (MFP) [19] was introduced to suppress the chiral enhanced New Physics contributions to the ∆F = 2 Hamiltonian. These are typically contained in C 4 and C 5 Wilson coefficients, while the dominant Standard Model contributions are contained in C 1 . In MFP, the singlet down sector is assumed to transform as triplet under a global U (3) group with all other fields transforming as singlets.
Hence the bulk mass term of fermions, taken to be diagonal in the flavour basis, become where Q i and u i , d i are the respective doublet and singlet quark fields in five dimensions. Note that the U (3) symmetry makes sure that the bulk mass parameter c d is the same for all three generations of the singlet down sector. Hence unlike the scenario discussed in the previous section, the hierarchy in singlet down sector wave profiles, given in Eq.(59), vanishes. Due to which the couplings of n th KK-gluon with the right handed chiral zero mode bilinears, as given in Eq. (21), become where i, j denote the generation index and g Hence this paradigm expects a conserved chiral symmetry with a consequence of significantly suppressing the contributions from C 4 and C 5 . Its application to RS model is discussed in literature [19]. On the other hand, in Little RS, though the most important contribution comes again from the C 4 coefficient, the bound is more stringent than in RS as shown in Fig.(3). And the lowest allowed mass of the first KK excited gluon is M g Phenomenologically we expect both these mechanisms to have significantly different implications especially for electroweak precision observables and LHC signatures. Before we close two comments are in order: (a) It would be definitely interesting to see the implications of the BLKTs and flavour symmetries on systems other than s − d like b − d or b − s. In some cases, BLKTs on the gluon field might not be sufficient. (b) There could be other mechanisms like Minimal Flavour Violation (MFV) [36] which could as well suppress the New Physics contributions in a similar manner as MFP [37]. This would be interesting to explore as well.
Another line of research testing the b − s-transitions is the B-anomalies [38]. Several models, leptoquarks, Z , .., with New Physics scale of few TeV have been constructed in order to explain the so called Banomalies [39,40,41,42,43,44]. If B-anomalies are confirmed, due to the low scale of New Physics required, this will be kind of a revolution compared to the traditional paradigm of Minimal Flavour Violation where a gap is expected between the scale of the solution of the Naturalness problem and the MFV scale [38]. Our scenarios depart from MFV and seem suitable to address B-anomalies: we plan to pursue the set-ups studied here by looking for flavour signatures.

A Flavour Parameters and Numerical Analysis
In this section we analyze the independent flavour parameters in the quark sector and discuss the full parameter scan of c values of quark, which could fit the CKM matrices and quark masses under experimental uncertainties. To understand the flavour parameters in RS model, we follow the approach of Ref. [11]. In our model we got 3 × 3 complex matrices of 5-D Yukawa couplings Y (5) u,d , each contains 9 real and 9 complex parameters. In RS model we get additional flavour parameters through 3 × 3 Hermitian bulk mass matrices, c Q,u,d . This brings in additional 18 real parameters and 9 complex phases. For Numerical scan it is convenient to work in the basis where the bulk mass matrices c Q,u,d are diagonal and comprise of 9 real parameters. The remaining 18 real parameters and 10 physical phases are then collected in the 5D Yukawa coupling matrices Y (5) u,d . We use the parameterisation adopted in the [45], where the bulk mass matrices are real and diagonal. We derive the parameterisation of the unitary matrices U l,r and D l,r generalizing the usual CKM parameterisation, as product of three rotations and introducing a complex phases in each of them [46], this parameterisation ensures the information about only the physical parameters. We therefore aim to derive a parameterisation of the RS flavour sector in terms of the SM quark masses, the CKM parameters, and the parameters of the new flavour mixing matrices D l , U r and D r . Neglecting the mixing with fermionic KK modes and approximating the Higgs field to be exactly localised on the IR brane, we can write