$R_{D^{(\ast)}}$ in custodial warped space

Flavor physics experiments allow to probe the accuracy of the Standard Model (SM) description at low energies, and are sensitive to new heavy gauge bosons that couple to quarks and leptons in a relevant way. The apparent anomaly in the ratios of the decay of $B$-mesons into $D$-mesons and different lepton flavors, $R_{D^{(\ast)}} = \mathcal B(B \to D^{(\ast)} \tau \nu)/ \mathcal B(B \to D^{(\ast)} \ell \nu )$ is particularly intriguing, since these decay processes occur at tree-level in the SM. Recently, it has been suggested that this anomaly may be explained by new gauge bosons coupled to right-handed currents of quarks and leptons, involving light right-handed neutrinos. In this work we present a well-motivated ultraviolet complete realization of this idea, embedding the SM in a warped space with an $SU(2)_L \otimes SU(2)_R \otimes U(1)_{B-L}$ bulk gauge symmetry. Besides providing a solution to the hierarchy problem, we show that this model, which has an explicit custodial symmetry, can explain the $R_{D^{(\ast)}}$ anomaly and at the same time allow for a solution to the $R_{K^{(\ast)}}$ anomalies, related to the decay of $B$-mesons into $K$-mesons and leptons, $R_{K^{(*)}} = \mathcal B(B\to K^{(*)} \mu \mu)/ \mathcal B(B \to K^{(*)} e e)$. In addition, a model prediction is an anomalous value of the forward-backward asymmetry $A^b_{FB}$, driven by the $Z\bar b_R b_R$ coupling, in agreement with LEP data.


Introduction
The Standard Model (SM) of particle physics provides an excellent description of all observables measured at collider experiments. The discovery of the Higgs boson [1,2] is an evidence of the realization of the simplest electroweak symmetry breaking mechanism, based on the vacuum expectation value (VEV) of a Higgs doublet. This mechanism provides a moderate breakdown of the custodial SU (2) R symmetry that affects the gauge bosons only at the loop level. The predictions of the SM are also in agreement with precision electroweak observables, which show only loop-size departures from the treelevel gauge predictions [3].
Flavor physics experiments allow to further probe the accuracy of the SM predictions. While studying SM rare processes, these experiments become sensitive to heavy new physics coupled in a relevant way to quarks and leptons. Recently, the BABAR [4,5], BELLE [6][7][8][9][10] and LHCb [11]  These decay processes occur at tree-level in the SM, and therefore can only be affected in a relevant way by either light charged gauge bosons, or heavy ones strongly coupled to the SM fermion fields. Currently, the measurements of these experiments seem to suggest a deviation of a few tens of percent from the SM predictions, a somewhat surprising result in view of the absence of any clear LHC new physics signatures, or other similar deviations in other flavor physics experiment.
In particular, the presence of new SU (2) gauge interactions affecting the left-handed neutrinos, which could provide an explanation of the new R D ( * ) anomaly, is strongly restricted by the measurement of the branching ratio of the decay of B-mesons into K-mesons plus invisible signatures by the BELLE collaboration B(B → Kνν) [12][13][14]. Recently, it was proposed that a possible way of avoiding these constraints was to assume that the new gauge interactions were coupled to right-handed currents and the neutrinos are therefore right handed neutrinos [15,16]. The right-handed neutral currents are then affected by right-handed quark mixing angles that are not restricted by current measurements, and provide the freedom to adjust the invisible decays to values consistent with current measurements.
In this work, we propose a well-motivated, ultraviolet complete, realization of the new gauge interactions coupled to the right handed currents, by embedding the SM in warped space, with a bulk gauge symmetry SU (2) L ⊗ SU (2) R ⊗ U (1) B−L [17][18][19][20]. This symmetry is broken to SU (2) L ⊗ U (1) Y in the ultraviolet brane, implying the absence of charged, W ± R , and neutral, Z R , gauge boson zero modes. Third generation quark and leptons are localized in the infrared-brane, where a Higgs bi-doublet provides the necessary breakdown of the SM gauge symmetry, giving masses to quarks and leptons. Although there have been previous works on the flavor structure of warped extra dimensions with a SU (2) L ⊗ SU (2) R × U (1) X bulk gauge symmetry (see, for example, Refs. [21,22]), those works put emphasis on rare Kaon and B-meson decays unrelated to R D ( * ) , that will also be analyzed in our work whenever relevant. Moreover, in the context of warped extra-dimensions, there has also been a recent analysis in Ref. [23] where lepto-quarks are introduced, and general results in composite Higgs models in Ref. [24].
In this work, similarly to the previous proposal by the authors of Refs. [15,16], the new SU (2) R gauge bosons provide an explanation of the R D ( * ) anomaly, and the freedom in the right-handed mixing angles allows to avoid the invisible B decay and B-meson mixing constraints. On the other hand, our model depicts unique, attractive special features such as having an explicit custodial symmetry that protects it from large deviations in precision electroweak observables, and providing a solution of the hierarchy problem through the usual warped space embedding. Finally, although it is not the main aim of this article, the left-handed KK gauge bosons may be used to provide an explanation of the R K * anomalies in the way proposed in Refs. [25][26][27].
Our study is organized as follows. In Sec. 2 we present the model in some detail. In Sec. 3 we explain the solution to the R D ( * ) anomaly. In Sec. 4 we discuss the existing experimental constraints on this model. In Sec. 5 we study the predictions of our model, including the forward-backward bottom asymmetry, the invisible decay of B mesons into K mesons, and the b → sµµ observables, including R K ( * ) . Finally we reserve Sec. 6 for our conclusions and App. A for some technical details on the KK modes.

The model
Our setup will be a five dimensional (5D) model with metric (with the mostly minus signs convention) g µν = exp(−ky)η µν , g 55 = −1, in proper coordinates, and two branes, at the ultraviolet (UV) y = 0, and infrared (IR) y = y 1 , regions, respectively [28]. The parameter k, close to the Planck scale, is related to the Anti de Sitter (AdS 5 ) curvature, and ky 1 has to be fixed by the stabilizing Goldberger-Wise (GW) mechanism [29] to a value of O (35), in order to solve the hierarchy problem.
where Y is the SM hypercharge with gauge boson B and coupling g Y , is done in the UV brane by boundary conditions. Therefore the 1 The 5D (g 5 ) and 4D (g 4 ) couplings are related by g 4 = g 5 / √ y 1 .
The SU (2) L ⊗ SU (2) R symmetry is unbroken in the IR brane, where all composite states are localized, such that the custodial symmetry is exact. In App. A we present some technical details leading to the wave function, mass and coupling of the n th KK modes for both (+, +) and (−, +) boundary conditions. It is shown there that the difference for the KK mode masses m n , and couplings, is tiny for the different boundary conditions, (+, +) and (−, +), and different electroweak symmetry breaking masses, and we will neglect it throughout this paper. In particular we will use the notation m 1 for the first KK mode mass of the different 5D gauge bosons after electroweak breaking:

The covariant derivative for fermions is
where g Y and g Z R are defined in terms of g R and g X as and the hypercharge Y and the charge Q Z R are defined by Electroweak symmetry breaking is triggered in the IR brane by the bulk Higgs bidoublet where the rows transform under SU (2) L and the columns under SU (2) R . We will denote their VEVs as H 0 2 ≡ v 2 / √ 2 and H 0 1 ≡ v 1 / √ 2, so that we will introduce the angle β as, We will find it useful to add an extra Higgs bi-doublet with Σ 0 = v Σ / √ 2, whose usefulness will be justified later on in this paper. After electroweak breaking, and rotating to the gauge boson mass eigenstates, one can re-write the covariant derivative as where θ L ≡ θ W is the usual weak mixing angle, the gauge boson Z µ L ≡ Z µ , and θ R is defined as Using g R and g Y , with g R > g Y , as independent parameters we can write As for fermions, left-handed (LH) ones are in SU (2) L bulk doublets as in the SM where the index i runs over the three generations. On the other hand, as SU (2) R is a symmetry of the bulk, right-handed (RH) fermions should appear in doublets of SU (2) R . However, as SU (2) R is broken by the orbifold conditions on the UV brane it means, for bulk right-handed fermions, that one component of the doublet must be even, under the orbifold Z 2 parity, and has a zero mode, while the other component of the doublet must be odd, and thus without any zero mode. We thus have to double the SM right-handed fermions in the bulk. The natural assignment is to assume in the bulk first and second (light) generation fermions: where only the unprimed fermions have zero modes, while third generation (heavy) fermions are localized on the IR brane and thus are in SU (2) R doublets as Then only the third generation RH fermions interact in a significant way with the field W R , and can give rise to a sizable R D ( * ) , as we will see. We define the KK modes for gauge bosons as normalized as where, from now on, we are switching to the notation where g L and g R are the 4D couplings, and G n are the overlapping integrals of the fermion zero-mode profiles, f L,R (y)f L,R (y), with the gauge boson KK mode ones, W n L,R (y). On the other hand, the neutral gauge bosons A, Z L and Z R interact with both chiralities, and we can thus define the 4D neutral current Lagrangian for KK modes as where for simplicity we have omitted the chirality indeces and G n f is the overlapping integral of zero modes fermion profiles, f 2 L,R (y), with the one of the (neutral) gauge boson KK modes. The 4D coupling of photons with fermions is defined as g Af f = sin θ L cos θ L Q, the couplings of fermions with Z R are given by 21) and with Z L by The 5D Yukawa couplings for RH quarks localized on the IR brane are where H = iσ 2 H * iσ 2 , and for the bulk RH quarks so that the 4D Yukawa matrices are given by and In the previous expressions the 4D Yukawa matrices Y u,d ij contain the 5D Yukawa matrices Y Q , Y Q ,Ŷ Q ,ˆ Y Q times the integrals overlapping the 5D profiles of the corresponding fermions with the profile of the Higgs acquiring vacuum expectation value, h(y) ∝ e αky .
where c L,R are the fermion bulk mass parameters and we have assumed that α > c L + c R . The parameter α has to be larger than two, to solve the hierarchy problem, and in our computations we will fix α = 2.
Similarly for RH leptons in the IR brane and for bulk RH leptons where we have added the bulk first and second generation right-handed neutrino doublets The Yukawa couplings for charged leptons are then given by and for neutrinos, by In the presence of a non-zero vacuum expectation value of the Σ field, we shall define where v = v 2 H + v 2 Σ . In the decoupling limit, H 1 = cos θ Σ cos βh−sin βH−sin θ Σ cos βH Σ and H 2 = cos θ Σ sin βh + cos βH − sin θ Σ sin βH Σ , while the neutral component of the Σ field, Σ 0 = sin θ Σ h + cos θ Σ H Σ . The SM-like Higgs boson is induced by excitations of the field h = sin θ Σ Σ 0 + cos θ Σ (cos βH 1 + sin βH 2 ), while the excitations induced by the orthogonal combinations H and H Σ are supposed to lead to heavy neutral states, decoupled from the low energy theory. Since quarks and leptons only couple to the field H, the masses are proportional to v H and therefore the Yukawa couplings must be enhanced by a factor (cos θ Σ ) −1 with respect to the value they would obtain in the absence of the Σ field.
In order to avoid strong constraints from lepton flavor violating processes, as e.g. µ → eγ, µ → eee, or µ − e conversion, we will assume that for charged leptons the interaction and mass eigenstate bases coincide, and therefore, hereafter, that the matrix Y e is diagonal. This can be obtained by imposing a U (1) 3 flavor symmetry in the lepton sector broken only by the tiny effects due to the neutrino masses [30].
For neutrinos propagating in the bulk, one can obtain realistic values of their masses by adopting one of the proposed solutions for theories with warped extra dimensions [31][32][33][34][35]. In our scenario, however, neutrinos localized on the IR brane, as is the case with the right-handed neutrinos ν τ,R , couple in a relevant way to the Higgs and tend to acquire masses of the same order as the charge lepton masses. This can be seen from the fact that the Yukawa couplings in Eq. (2.32) will provide a Dirac mass to the third generation neutrinos m DνL ν R + h.c.. Therefore, in order to obtain realistic masses we will assume a double seesaw scenario [36]. We shall first concentrate on the example of third generation neutrinos. In order to realize this mechanism, we will introduce a Higgs H R , transforming when its neutral, hyperchargeless, component gets a vacuum expectation value v R , as well as a localized fermion singlet (1, 1, 0), S L , which provides the Dirac mass Finally, we can also write down a Majorana mass term as M S L S L . Therefore the mass matrix in the basis (ν L , ν c R , S L ) can be written as (which is obviously massless in the limit where M = 0), and an approximate Dirac spinor This mechanism has been dubbed in the literature, double seesaw [36]. The double seesaw mechanism allows for acceptable masses for the left-and right-handed neutrinos without extreme fine-tuning of the Yukawa couplings. For instance, for m D 1 MeV, m D 100 MeV and M = O(1 KeV), one obtains a mostly left-handed neutrino of mass of order 0.1 eV, and an additional pseudo-Dirac neutrino, containing ν R , of mass of order 100 MeV. Such masses are enough to accommodate the value of R D ( * ) without any sizable kinematic suppression.
The above mechanism can be easily generalized to give mass to the three generations of neutrinos. As suggested before, we will consider in the bulk the two RH neutrino doublets N I R and add two singlets S I L , while the third generation right-handed leptons and the singlet S 3 L are as before localized in the IR brane. States transform under the flavor symmetry group U (1) 3 The quantum numbers in Tab. 1 lead to the off-diagonal entries in Eq. (2.34). In particular (m D ) ij , defined as is a diagonal matrix, while also the matrix (m D ) ij is diagonal as the bi-doublet H does not carry any lepton number. Moreover we will introduce the non-diagonal Majorana mass matrix for singlets as M ij S i L S j L which will constitute a soft breakdown of the global symmetry U (1) Le ⊗ U (1) Lµ ⊗ U (1) Lτ , by the small M mass matrix elements, leading to the neutrino mass matrix [36] which should describe the neutrino masses and PMNS mixing angles [3].

Generating R D ( * )
Only fermion doublets localized on the IR brane, with both non-vanishing components, will interact with W R . Then we can write the 4D charged current Lagrangian, Eq. (2.19), in the mass eigenstate fermion basis as where the matrix form has been used. The coupling matrix G n can be approximated by After integration of the KK modes we can write down the effective Lagrangian which has been normalized to the SM contribution, where the Wilson coefficient is given by where G 3 ≡ G 1 3 and m 1 are the coupling and mass of the first KK mode, and the pre-factor 1.45 takes into account the contribution of the whole tower.
The Wilson coefficient C τ contributes to the process b → cτν τ and thus to the ratio is the SM prediction [37][38][39][40], and the best fit value to experimental data is given by C τ 0.46 [16] 2 . Using this value there is a relation between the ratio V † u R 23 /V cb and the mass m 1 given by so that the element V † u R 23 as a function of sin θ R and the mass m 1 is given in Fig. 1. In principle the anomaly in the branching ratio B(B → D ( * ) τ RνR ) might give rise to a large contribution to the branching fraction B(D s → τν) 0.05 from the process sc → τ + R ν R , which is mediated by the KK modes W n R . However since c R and s R are in the bulk, and in different SU (2) R doublets, they couple to W n R only via mixing with the third generation quarks. This implies that this contribution is further suppressed by a factor (V d R ) 32 which, as we will see, is restricted to be small to satisfy the constraints on ∆m Bs . Thus, no significant contribution to the branching ratio B(D s → τ ν) is obtained.
Similarly, in this model one would also expect an excess in the observable . (3.8) The LHCb experiment has recently provided a result on this observable, showing an excess of the order of 2 σ above the SM expected value, R(J/Ψ) SM 0.25-0.28, Ref. [42,43], with large errors Theoretical analyses of this observable [44,45] confirm this anomaly and show it to be governed by the same operator as the one governing R D ( * ) . In our particular model, Given the value of R(J/Ψ) SM , the measured value of this ratio is about 2.6±1. Hence, the value of C τ obtained above to explain R D ( * ) can only slightly ameliorate this anomaly, and one should wait for more accurate experimental measurements of R(J/Ψ) before further discussion of this issue.

Constraints
In this section we will examine the main constraints in processes which are related to R D ( * ) , and where the strong coupling of the third generation RH quarks and leptons to KK modes plays a significant role. To do that one has to compute the mixing between the electroweak gauge bosons W ± L and Z L and the KK modes using the effective Lagrangian. We can easily compute the effective description of the Lagrangian, with mixing terms W L W n L,R and Z L Z n L,R , generated by the vacuum expectation values of the bulk Higgs bidoublets H and Σ as well as the Higgs doublet H R in the representation (1, 2), with VEV H R = (v R , 0) T , and with Q X = −1/2. These are induced from the kinetic terms in the 5D Lagrangian as where we are using the fact that T a L acts on the bi-doublets rows and T a R on the bi-doublets columns.
A straightforward calculation gives for the 4D quadratic Lagrangian for the gauge boson n-th KK modes , the first two terms provide the W L and Z L -masses, and we have introduced the function r h (α) which depends on the localization in the bulk of the h Higgs direction acquiring a vacuum expectation value. In fact for a Higgs localized in the IR brane, α → ∞, one gets r h 1, while for a Higgs localized towards the UV brane α ≤ 1 one gets r h 0. For α = 2 the Higgs is sufficiently localized towards the IR brane to solve the hierarchy problem, and we shall use this value in the rest of in this article, leading to a factor r h 0.68.
Another important effect for analyzing the relevant constraints, in the presence of composite, and partly composite, fermions f , is that in our model the effective operators are induced, with Wilson coefficients given by (4.5) In the above, we have introduced the function r f (c f ) as where G n f (c f ) is the overlapping integral of fermion zero mode profiles, for the given value of the c f parameter, and the gauge boson KK mode profile. In particular, for IR localized fermions, which could be considered as the limiting case where c f → −∞, it turns out that lim c f →−∞ r f (c f ) = 1. The Wilson coefficients trigger a one-loop modification of the Z Lf f couplings, through a top-quark loop diagram followed by emission of the Z L gauge boson [46],which in turn induces the modification of the corresponding Z Lf f coupling. In particular, for the relevant cases we will analyze here f As the τ R lepton is localized on the IR brane, and it couples strongly to the KK modes, the main constraint will be the modification of the coupling Z L τ R τ R , defined as where the term δg Z L τ R τ R is constrained by the global fit to the experimental data of Ref. [47] as The term δg Z L τ R τ R in Eq. (4.6) is generated at the tree level by the mixing Z n L,R Z L induced by the Higgs vacuum expectation value, and through radiative corrections using the effective operator where the first line comes from the contribution of the KK gauge bosons through mixing effects and the second line is the radiative contribution from the top quark loop 3 induced by the operator (4.8). The coupling h t is the SM top-Yukawa coupling, defined by  Figure 2: The region between the (brown) solid lines is allowed by the best fit to δg Z L τ R τ R for m 1 = 3 TeV.
In order to determine the KK-mode contribution we use the condition (3.4) on R D ( * ) and get the allowed region in the plane (sin θ Σ , sin θ R ) shown in Fig. 2, where we are assuming m 1 = 3 TeV. Fig. 2 shows that the constraint on δg Z L τ R τ R puts a lower bound on sin θ Σ , which is given by 12) and in particular excludes the value sin θ Σ = 0, i.e. it requires the introduction of the Higgs bi-doublet Σ.

Oblique observables
In these theories the T -parameter, defined as, is protected by the custodial symmetry in the bulk only in the case when tan β = 1 and sin θ Σ = 0.
In general, there may be relevant contributions to the precision electroweak observables induced by the mixing of the gauge boson zero modes with the KK modes, as given by Eq. (4.3), as well as loop corrections induced by top loop corrections. In fact in a similar way as the operator (4.4) is generated by exchange of (A n , Z n L , Z n R ) KK modes, the operator is generated by the mixing of Z L with KK modes in (4.3) followed by the exchange of (Z n L , Z n R ) KK modes coupled to the top quark. The radiative correction to the T parameter is obtained after closing the top-loop, and by emission of a Z L -gauge boson from it.
There are also loop contributions involving fermionic KK modes, but in a scenario in which the right handed third generation fermions are localized on the infrared brane, they strongly depend on the localization of the left handed third generation quarks (see, for example, Refs. [48][49][50]). In particular, these loop corrections are strongly suppressed when the left-handed third generation quarks are localized close to the IR brane, or in the presence of sizable quark brane kinetic terms. Moreover, unlike the mixing between gauge KK n-modes and gauge zero modes, which is enhanced for IR brane localized fermions by ∼ |G n 3 | = √ 2ky 1 , the mixing between fermion KK n-modes and fermion zero modes is ∼ G n 3 / √ ky 1 , so that the loop corrections to the T parameter are not volume-enhanced, while they are suppressed by the mass of the heavy fermions and by loop factors. Hence, in this work, we shall concentrate on the relevant corrections to flavor physics observables induced by the gauge boson mixing, and the inter-generational mixing of the right-handed quarks, as well as by the top loop corrections we have just described from the operator (4.14). These corrections to the precision electroweak observables are well defined within our framework, and are strongly correlated with our proposed solution to R D ( * ) .
We can easily compute the contributions to the T -parameter induced by the mixing of the zero mode gauge bosons with the KK modes by using the effective description of the Lagrangian, with mixing terms W L W n L,R and Z L Z n L,R , from Eq. (4.3), and at one-loop from the effective operator (4.14). Working to lowest order, O(v 4 ), in Higgs insertions, we obtain the result where the first two lines is the tree-level result and the third line the radiative correction induced at one-loop by the mixing between the tree-level (4.3) and one-loop (4.14) operators. Using now the expression fitting the value of R D ( * ) , we can obtain the allowed regions for the T parameter in the (sin θ Σ , sin θ R ) plane, fixing the values of m 1 and tan β. In Fig. 3, in addition to the δg Z L τ R τ R bounds from Fig. 2, we show the regions allowed by the T parameter experimental bounds at the 95% confidence level [3] T = 0.07 ± 0.12,   which, using the effective description of Eq. (4.3), can be cast as (4.21) where, as their tree level values is so small we are neglecting its crossing with the radiative corrections induced by the operator (4.14).
After applying the constraint from the R D ( * ) anomaly, fixing the value of the KK mass, m 1 = 3 TeV, and tan β = 2, the S and U countors are depicted in Fig. 4. It follows from this figure that the predicted values are consistent with the experimental constraint [3] S = 0.02 ± 0.10, U = 0.00 ± 0.09 (4.22) in all the parameter region. Similar small values of S and U are obtained for other values of tan β.

Flavor observables
New physics contribution to ∆F = 2 observables appears mainly from exchange of KK gluons. The leading flavor violating couplings of the KK gluons G n µ involving RH down and up quarks is given by After integrating out the gluon KK modes we obtain a set of ∆F = 2 dimension six operators. In particular, the most constrained operators are those given by where the Wilson coefficients are given by where (V † u R ) 23 is constrained by R D ( * ) , see Fig. 1. If, for simplicity, we assume real matrices V u R and V d R (no CP violation in the right-handed sector) the Wilson coefficients C sd , C cu , C bd and C bs are constrained from ∆m K , ∆m D , ∆m B d and ∆m Bs , respectively, as [51,52] C sd < 9 × 10 −7 TeV −2 , (4.29) C cu < 5.6 × 10 −7 TeV −2 , (4.30) Operators involving third generation quarks, although providing weaker bounds on the Wilson coefficients, are very constraining as they contain the element (V † d R ) 33 1. In particular the bounds on C bd and C bs , Eqs. (4.31) and (4.32), provide bounds on (V d R ) 31 and (V † d R ) 23 , respectively, as Using now the bounds in Eq. (4.33) we can bound the element C sd as which is a stronger bound than Eq. (4.29). Moreover, from the definition of C cu in Eq. (4.26) and the corresponding bound (4.30), we can fix an upper bound on the element (V u R ) 31 using the value of (V † u R ) 23 provided by R D ( * ) . The result is plotted in Fig. 5

Lepton flavor universality tests
There are two processes where lepton flavor universality has been tested to hold with a high accuracy. The first one is the ratio 1.02 [53]. In our model, the process Γ(b → cW * R → c ν ) = 0, for = (µ, e), since only the third generation leptons couple to W R . Hence, it follows that R The second process is which is constrained by experimental data to be R τ /µ µ = 1.0022 ± 0.0030 and R τ /e µ = 1.0060 ± 0.0030. It turns out that the contribution to these processes from W R , B(τ → ν τ W * R → ν τ ν ) and similarly B(µ → ν µ W * R → ν µ ν ) is negligible for the same reason as before, and hence the deviation of R τ / τ with respect to the SM values is also negligible, in good agreement with these measurements.

LHC bounds
The first neutral KK resonance X 1 (X = Z L , Z R , A) can be produced on-shell at LHC in Drell-Yan processes σ(bb → X 1 ), followed by decays X 1 → ff where f = τ R , b R , t R . The production cross-section times branching ratio can be written as where f (m 1 ) is the production cross-section for unit coupling obtained by MadGraph v5 [54]. Our model prediction for X σ(pp → X 1 ) × B(X 1 →f f ) is given by the upper, middle and lower solid lines of Fig. 6 for f = b R , t R , τ R , respectively. We compare them with the experimental 95% CL upper bounds from the corresponding processes, which are given by the dot-dashed (red), dashed (black) and dotted (blue) horizontal lines from the ATLAS experiment on σ × B(Z →tt) [55], σ × B(Z →bb) [56] and σ × B(Z → τ τ ) [57] for m Z = 3 TeV, respectively. As can be seen from Fig. 6 only the process σ × B(Z →tt) puts a significant bound on our model, of sin θ R 0.15 for m 1 = 3 TeV, as we are assuming.
In a similar way the first charged KK resonance W 1 R can be produced on-shell at the LHC in the process σ(bc → W 1 R ), followed by the decays W R → τ R ν τ R , t RbR , that assuming that there are no exotic fermions localized in the IR brane, yield branching ratios around 1/4 and 3/4, respectively. In our model the production cross sections times branching-ratio is where g(m 1 ) is the production cross-section for unit coupling obtained by MadGraph v5 [54] 4 . Our model prediction for σ(pp → W R ) × B(W R → τ R ν τ R ) is given in Fig. 7, from where it follows that the model prediction is below the ATLAS 95% CL experimental [58] by a factor of order of a few.   In the previous analyses we did not take into account the width of resonances. While the width (with respect to its mass m 1 ) of the KK photon A 1 is around ∼0.24, those of the other resonances depend on the angle sin θ R . For instance, in the range 0.35 sin θ R 0.5 the Z 1 L width varies between 0.05 and 0.08, while those of Z 1 R and W 1 R are generically O(1). For the case of broad resonances, as is the case of the Z 1 R and W 1 R resonances, we expect that the effect of the width can affect the production cross-section (due to possible KK mode superpositions) as well as the experimental bounds (due to the absence of a clear resonance). Recent ATLAS studies [55] show that bounds on the cross-sections for the case of broad resonances are affected by factors of order a few, while the cross-section predictions are also affected by similar factors. Hence, although a detailed experimental and theoretical analysis would be necessary to determine the precise bounds on the gauge boson KK mode masses, they are expected to be of the same order as the ones shown in Figs. 6 and 7.
Finally there are also strong constraints on the mass of KK gluons G 1 from the crosssection σ(pp → G 1 ) × B(G 1 →tt) from the ATLAS experimental analysis in Ref. [55]. As the resonance G 1 is a broad one, both the experimental results and the theoretical calculation of the production cross sections should be re-analyzed to get reliable bounds on the mass of the KK gluons. However, a simple way of relaxing the bounds is introducing brane kinetic terms for the SU (3) gauge bosons, in particular in the IR brane. This theory has been analyzed in Refs. [59,60], where it is shown that, even for small coefficients in front of the brane kinetic terms, the coupling of the KK modes G n to IR localized fermions decreases very fast while the mass of the modes m n increases. Both facts going in the same directions, the bounds on KK gluons can be easily avoided. As the strong sector does not interfere with the electroweak one SU (2) L ⊗ SU (2) R ⊗ U (1) X , the presence of brane kinetic terms will not affect our mechanism for reproducing the R D ( * ) anomaly. Moreover in the presence of brane kinetic terms for SU (3) gauge bosons the flavor bounds in Sec. 4.3 should be subsequently softened, an analysis that, to be conservative, we are not considering in this paper.

Predictions
In this section we will present some predictions of our theory consistent with the experimental value of R D ( * ) and all the previously analyzed experimental constraints.

The forward-backward asymmetry A b F B
We shall study the shifts in the couplings g Z L b L,R b L,R , parametrized as The shift of these couplings induce an anomalous modification of the forward-backward bottom asymmetry, conventionally defined as The currently measured value of and hence A b F B exhibits a ∼2.3 σ anomalous departure with respect to the SM prediction [61].
In our model the values of δg Z L b L b L and δg Z L b R b R are induced by the Z L Z n L,R mixing, in turn induced by the electroweak breaking, followed by the corresponding coupling , and by one-loop radiative corrections induced by the operators in Eq. (4.4). An analysis similar to that done in Sec. 4.1 yields the expressions where, again, the first lines in Eqs. (5.5) and (5.6) are the contributions from the gauge bosons KK modes through mixing effects, and the second lines come from the contribution of the radiative corrections induced by the operators Finally, the modification of the left-handed and right-handed bottom couplings to the Z gauge boson induce a modification of A b F B which, at linear order in δg Zb L,R b L,R is given by The shift δg Z L b L b L is constrained by electroweak precision data, to be [47] The region (5.8) constrains the available values of c b L , as shown in the left panel of Fig. 8, where we have fixed sin θ Σ = 0.72 and where the shaded area is excluded at the 95% CL. After fixing the condition to fit R D ( * ) , and using e.g. the value c b L = 0.35, for which δg Z L b L b L 4.7 × 10 −3 , we find that the 1 σ (2 σ) experimental value (5.4) is obtained between the dashed (dot-dashed) lines in Fig. 9, implying that the anomalous value of A b F B remains consistent with the explanation of the R D ( * ) anomaly, and the rest of electroweak and LHC constraints, for the parameter region near tan β = 2 ± 1, sin θ R 0.32 ± 0.08 and sin θ Σ 0.72 ± 0.02. Observe, however, that tan β close to one demands large values of the top-quark Yukawa coupling. As it is clear from Fig. 9, for somewhat larger values of tan β the corrections to the right-handed bottom coupling allow to reduce the current 2.3 σ anomaly on A b F B into a value that is about 1 σ away from the central experimental value.
Observe that this custodial symmetry model differs from the results obtained in an abelian gauge symmetry extension of the SM, where an explanation of the forwardbackward asymmetry demands the extra gauge bosons to be light, with masses below about 150 GeV, in order to induce small corrections to the T parameter [63].  Figure 9: The region between the solid lines is allowed by δg Z L τ R τ R (brown lines) and by T for m 1 = 3 TeV and tan β = 1 (black lines) tan β = 3 (blue lines) and tan β = 5 (red lines). Region between dashed (dot-dashed) lines encompasses the 1 σ (2 σ) interval for the anomaly in A b F B .

The processes B → Kνν and B
The R D ( * ) anomaly can in principle induce a large production in the process B → Kνν, i.e. b → sνν, mainly induced by the RH neutral current Lagrangian 6 where the couplings of Z R to RH quarks and leptons are given in Eq. (2.21). After integrating out the KK modes we get the effective Lagrangian where we are normalizing B(B → Kν R ν R ) to the SM value of B(B → Kν ν ), and the Wilson coefficient C νν is given by and where we have used that in the Wolfenstein parametrization V cb = −V ts = Aλ 2 , and V tb = 1. Now we can write the ratio where we have used the SM prediction C SM νν −6.4 [64]. Using the experimental bound R νν K < 5.2 at the 95% CL [14], one finds the bound |C νν | 23. However, after imposing the constraints coming from the flavor condition (4.33) on the matrix element (V † d R ) 23 , one easily obtains values that are well below the experimental bound, particularly for values of sin θ R > 0.2. This is shown in the left panel of Fig. 10, where we plot contours of constant R νν K in the plane (sin θ R , m 1 ) after using the bound for (V † d R ) 23 in Eq. (4.33). Lower values of R νν K may be obtained for smaller values of sin θ R by using the freedom on the value of (V † d R ) 23 , as shown in the right panel of Fig. 10, where we plot R νν K in the plane sin θ R , (V † d R ) 23 after fixing m 1 = 3 TeV.
This model predicts a strong τ τ production in the observable In our model this observable is dominated by the Wilson coefficient C τ RR such that where widely consistent with present experimental bounds from the BaBar Collaboration [65] which yield the 90% CL upper bound, R τ K < 10 4 .

R K ( * )
One of the general applications of our theory is that it generically predicts a value of R K ( * ) which can easily differ from its SM prediction [66,67]. The general effective operator Lagangian is written as We will find it convenient to work in the chiral basis for the operators O i such that with chiralities χ, χ = {L, R}, have Wilson coefficients defined as C χχ ≡ C SM χχ + ∆C χχ 7 . The SM predictions are given by while ∆C χχ are the contributions to the Wilson coefficients coming from New Physics. The prediction of R K ( * ) is given by where the upper signs correspond to R K and the lower signs to R K * and we have assumed that the polarization of the K * is close to p = 1, what is a good approximation in the relevant q 2 region associated with the R K * measurement [69]. The above equation, Eq. (5.20), shows the well known correlation (anti-correlation) of the corrections to R K and R K * associated to the left-(right-) handed currents. Therefore, considering the fact that both R K and R K * are suppressed with respect to the SM values, this leads to a preference of new physics effects involving left-handed currents. The experimental value of R K ( * ) departs from the SM prediction R K ( * ) 1 [70] by around 2.5 σ. Moreover global fits [71][72][73][74][75][76][77] where the first, second and third terms inside the square bracket comes from the contribution of the Z n L , A n and Z n R KK modes, respectively, and we are assuming [78] that V u L 1 and V d L V , the CKM matrix. Similarly, the prediction for ∆C µ RL is given by Observe that the combined contribution to ∆C µ LL from the Z n L and A n KK modes is considerably larger than the one from the Z n R KK modes. Recent global fits to experimental data [75] Fig. 12 shows the 1 σ (solid lines) and 2 σ contours of ∆C µ LL in the plane (c b L , c µ L ), where we have fixed sin θ R = 0.35. The values of c b L and c µ L are mainly constrained from δg Z L b L b L , as given in Eq. (5.6), and plotted in the left panel of Fig. 8, and from δg Z L µ L µ L as given by where, again, the first line in Eq. (5.26) denote the contributions from the gauge bosons KK modes through the mixing and the second line denote those from the radiative corrections induced by the operators The prediction for δg Z L µ L µ L is plotted in the right panel of Fig. 8, where we also have fixed sin θ Σ = 0.72, and where the white region is allowed at the 95% CL given the fitted value to experimental data [47] δg Z L µ L u L = (0.1 ± 1.2) × 10 −3 . The prediction for C µ RL is shown in the right panel of Fig. 12 in the plane (sin θ R , c µ L ) where we are already using the upper bound on (V † d R ) 23 from flavor observables, while the shaded region is excluded by δg Z L µ L µ L . We see that the values of ∆C µ RL in the region defined by Eq.

Conclusions
The experimental measurements of R D ( * ) show significant deviations from the SM values, a surprising result due to the tree-level nature of this process in the SM. Possible resolutions of this anomaly face significant constraints from the excellent agreement of flavor physics observables with the values predicted within the SM. In this work, we have presented an explicit realization of the solution to the R D ( * ) anomaly based on the contribution of right-handed currents of quarks and leptons to this process. The model is based on the embedding of the SM in warped space, with a bulk gauge symmetry SU (2) L ⊗ SU (2) R ⊗ U (1) B−L , with third-generation right-handed quarks and leptons localized on the infraredbrane, ensuring a large coupling of these modes to the charged gauge boson W n R KKmodes.
The right-handed SU (2) R gauge boson KK-modes provide the necessary contribution to R D ( * ) , due to relevant mixing parameters in the right-handed up-quark sector. This may be done without inducing large contributions to the B-meson invisible decays, or the Bmeson mixings, since these observables strongly depend on the down-quark right handed mixing angles, which do not affect R D ( * ) in any significant way within this framework. The mass of the lightest KK-mode tends to be of about a few TeV, and it is in natural agreement with current LHC constraints.
An important assumption within this model is that there is no mixing in the lepton sector. This can be ensured with appropriate symmetries, that must be (softly) broken in order to allow the proper neutrino mixing. We have presented a scenario, based on symmetries and a double seesaw mechanism, that allows for a proper description of the lepton sector of the model. The origin of the new parameters in the lepton sector remains, however, as one of the most challenging aspects of these (and many) scenarios. Aside of this question, beyond providing a resolution to the R D ( * ) anomaly, this model also provides a solution of the hierarchy problem, has an explicit custodial symmetry that implies small corrections to the precision electroweak observables, and allows a solution to the R K ( * ) anomalies mainly via the contribution of the SU (2) L KK modes. Moreover, the proposed model naturally predicts an anomalous value of the forward-backward asymmetry A b F B , as implied by LEP data, driven by the Zb R b R coupling.  are arbitrary constants. Notice that a constant f A (y) fulfills the (+, +) boundary conditions, and from Eq. (A.1) one finds that this corresponds to a zero mode. The (−, +) boundary conditions, however, do not lead to zero modes.
In the limit of large ky 1 , the Neumann boundary conditions in the IR brane lead to the following equations for the eigenvalues 0 = J 0 (m ++ ) + The function f A (y) grows with y, so that this integral is dominated by the regime close to y y 1 . In this regime the dominant contribution to the wave function is the term ∼ e ky J 1 (e ky−ky 1m ) in Eqs. In the second equality we have added a term whose integral is vanishing whenm is an eigenvalue of J 0 (m). To see this, let us note that e 2ky J 1 (e ky−ky 1m ) d dy J 1 (e ky−ky 1m ) = d dy 1 2 e 2ky J 0 (e ky−ky 1m )J 2 (e ky−ky 1m ) . (A.14) This implies that after integrating this term in