Flavor anomalies meet the LHC

. Currently there are several discrepancies between the standard model (SM) prediction and experimental value in the ﬂavor observables, e.g. R D ∗ , BR( ¯ B → D + K − ), R K ( ∗ ) and muon anomalous magnetic moment. If we interpret the deviation as a hint for the physics beyond the SM, O (1)TeV new particles are often implied. Therefore it is natural to search those new particles at the large hadron colliders. In this note we interplay the ﬂavor anomalies and collider physics.


Introduction
In recent decades, we have performed high precision verification of the Standard Model (SM) and the long awaited scalar (Higgs boson) was found in 2012.In the meantime many physical observables have been measured with the high accuracy and their SM predictions have been also sharpened.As a result we can test not only the SM but also new physics (NP) beyond the SM, by comparing the model predictions with the experimental results.Most of the results suggest that the SM describes our nature very well, while we also find some measurements deviated from the SM predictions by more than 3 − 4 σ.Those flavor anomalies are one of the hot topics of the particle physics [1].In this note we discuss the following flavor anomalies in R D ( * ) = BR( B → D ( * ) τν)/BR( B → D ( * ) ν) where = e, µ, non leptonic two body B decays (b → cu q puzzle), and muon anomalous magnetic moment (muon g − 2).There are discrepancies in b → sµ μ, however, the implied NP scale is O (10) TeV.Therefore collider physics is less relevant to b → sµ μ, and is not discussed here. 1 The semi-tauonic B-meson decays have been intriguing processes to measure the lepton flavor universality (LFU) which is one of the most important prediction of the SM.The ratio R D ( * ) has been measured by B-factories for more than 10 years and the world average deviates from the SM prediction [5].Very recently the LHCb collaboration reported the new result which is again off from the SM prediction [6].The current status is given as R WA D =0.358 ± 0.028 , R WA D * = 0.285 ± 0.013 , (1) R SM D =0.248 ± 0.001, R SM D * = 0.289 ± 0.004, e-mail: igurosyuhei@gmail.compreprint number: TTP22-066, P3H-22-107 1 For the interested readers, see, Refs.[2][3][4] which discuss the collider phenomenology.
with the experimental correlation of −0.27 and the current significance is about 4σ [7].Within the SM the tree level W boson exchange describes the transition and the required size of the shift in R D ( * ) is 10-20%.Because of the size of the required shift, we can safely focus on the operators up to dimension 6 and then the variety of the possible new mediator particles is limited.Currently there are only two categories of the scenarios available.One is a charged Higgs (H − ) and the other is a leptoquark (X).In this proceedings, we will discuss the LHC phenomenology of the those scenarios.Since the 4-fermi interaction to enhance the R D ( * ) automatically generates τν final state we focus on the τν(+b) signature.Although we focus on the collider phenomenology in this note, it is known that polarization observables of the B → D ( * ) τν transition have a potential to discriminate the those new physics scenarios [8].
In addition to the semileptonic decays, there are coherent deviations between the SM prediction based on the QCD factorization and experimental measurements in hadronic two body decays [9].It is worthwhile to mention that three independent experiments agree well within the uncertainty [10].Those deviations are known in so-called color allowed decays of B → DP and B → D * P transitions where P denotes a pseudoscalar meson.Those decays are also mediated by the tree level W boson exchange, and the comparison between the experimental average and the theoretical predictions are given as [11] EPJ Web of Conferences 289, 01007 (2023) https://doi.org/10.1051/epjconf/202328901007FCCP2022

BR( B0
where V cb = 0.0397 is used and transition form factors are taken from Ref. [6].If we use the inclusive V cb which is larger than than the above V cb by about 5%, the deviations coherently get further larger.The higher order corrections are estimated to be O(1)% at most and thus not large enough to fill the gap.The naive size of the discrepancies are 3∼5 σ.To reproduce the measured branching ratio about 10∼20% negative shift at the amplitude level is necessary.Even if we consider the contribution from the final state quasi elastic rescattering, the global fit including color suppressed decays found that similar size of the negative shift is still necessary [11].It is worth mentioning that the similar tendency is also observed in B → DV modes where V denotes a vector meson, however, the large uncertainty do not allow a clear cut conclusion at this stage.Since the mediator needs to couple to light quarks the LHC production cross section could be large.
Another long standing anomaly is known in the muon anomalous magnetic moment (α µ ).The 2021 Fermi-lab result [12] is consistent with the Brookhaven result [13] and deviates from the SM prediction of the white paper (WP) [14] by more than 4σ, Although the recent lattice results are more consistent with the experimental result, there is tension with e + e − → π + π − data [15].If we take this deviation as a hint of the NP, the implied NP scale is also up to O(1) TeV even with the chirality enhancement.Since there is no complete solution to settle the problems and the LHC will accumulate the data regardless of the muon g − 2 situation, it would be important to propose the independent cross check for the new physics contribution at the high energy colliders.
In the following sections we will introduce the model set up and discuss the LHC sensitivity to probe the interesting scenarios.

R D ( * ) anomaly vs. LHC
The recent LHCb result favors larger (smaller) deviation in R D * (R D ) compared to the previous world average. 2 This shift has a great impact on the H − interpretation.Therefore we will update the result in Ref. [16,17] based on the updated world average.
After integrating out the new particle freedom in general we obtain the following effective Lagrangian with where projection operators, P L = (1 − γ 5 )/2 and P R = (1 + γ 5 )/2 are defined.The NP contribution is encoded in the Wilson coefficients (WCs) of C X , normalized by the SM factor of 2 The SM corresponds to C X = 0 for X = V L,R , S L,R , and T in this notation.

Charged scalar
It is well known that the charged Higgs explanation of the discrepancy easily enhances the decay width of the B − c → τν decay.all fermions.The upper limits on BR(B − c → τν) ≤ 30% and 10% are derived from the lifetime of the B c meson [18] and the LEP data [19], respectively.However, it was pointed out that the large charm mass uncertainty and energy dependence of the fragmentation function relax the bound as large as BR(B − c → τν) ≤ 60% [20].Recently the reevaluation in Ref. [21] found BR(B − c → τν) ≤ 63% based on the B c lifetime.
Testing the charged Higgs scenario at the LHC is easier than that for the other scenarios since the larger coupling is necessary to enhance R D * compared to the other scenarios due to small coefficients. 3In Ref. [22] authors reinterpreted the bound on the heavy resonance reported by the CMS to derive the constraint on the charged Higgs scenario.It was found that the result with 36 fb −1 can already exclude the explanation with m H − ≥ 400 GeV where the data is available.In light of the relaxed bound from the B c decay, Ref. [16] revisited the light mass window 180 GeV≤ m H − ≤ 400 GeV with the τν constraint based on Run 1 data, the stau constraint and low mass di-jet constraint at the LHC.However, it turned out that this mass region is difficult to fully cover the interesting parameter region even at the high luminosity LHC.More recently it is shown that requiring an additional b-tagged jet in a low mass τν resonance search, which is not performed experimentally, can enhance the signal to back ground ratio [17].In a conventional τν resonance search, the W boson tail consists of a dominant background.By the requiring the additional b-jet, we can suppress the valence quark contribution.For instance ug → W * b → τν + b contribution is suppressed by the factor of |V ub | 2 .Furthermore the j → b miss-tagged contribution in gu → W * j → τν + j is suppressed by the small miss tagging rate of O(10 −3 ).As a result, the huge background reduction of a factor of O(100) 3 See, Eq. (2.4) of Ref. [8] for instance.is observed while the signal reduction is found to be of a factor.
In Fig. 1, the sensitivity with the current luminosity and the projected sensitivity with the HL-LHC are shown in blue solid and dashed lines on the R D vs R D * plane.The area above the line can be probed and the dashed line is omitted on the right since the line almost degenerates with the SM prediction.The charged Higgs mass is assumed to be 250 (400) GeV on the left (right) panel.The world average of the R D ( * ) data at 1, 2 and 3σ are shown by the solid, dashed and dotted contours.The grey shaded region is out of the prediction within the model.The SM prediction is indicated by a yellow star, and the horizontal magenta solid (dashed) line corresponds to BR(B c → τν) = 63 (30)% as a comparison.
The figure shows that the collider prospect with 139 fb −1 of the data can cover the broader range and judge the 1 σ explanation of the anomaly within the model.
Different from the charged Higgs case, a leptoquark predicts the non resonant τν signature and the high p T region is sensitive to the new physics effect since the smaller background contribution is expected there.The significant leptoquark mass dependence is pointed out even with the tchannel mediator because the large momentum exchange, corresponding to the large Mandelstem variable t, is necessary to generate the high p T leptons [36].Furthermore it is shown that selecting the negatively charged τ can improve the sensitivity.This is because that the density of the up quark is larger than that of down quark in a proton.Also the additional b-tagged helps to enhance the sensitivity to leptoquark scenarios by 30-40% [37].It is noted that the ττ(+b) final state also provides the similar sensitivity [38].
In the following we focus on the R 2 and U 1 leptoquark scenarios as a demonstration.The R 2 scenario predicts C S 2 = +4C T at the leptoquark mass scale where C S 2 and C T are Wilson coefficients of a scalar and tensor operators which describes the b → cτν transition.In Fig. 2, the constraint and prospect in τν and τν + b modes are shown with m R 2 = 2 TeV(left) and m R 2 = 4 TeV(right).The regions outside the blue and red lines can be probed with the τν and τν + b signature, respectively, where the solid and dashed lines correspond to the prospect based on 139 fb −1 and 3 ab −1 of the data.The magenta line shows the bound with the previous CMS result with 36 fb −1 derived in Ref. [36].It is noted that the CMS found the smaller event number compared to their expectation, and hence it results in the stringent constraint.The darker and lighter gray shaded regions are constrained by BR(B c → τν) ≤ 0.6 and BR(B c → τν) ≤ 0.3.The R D and R D * anomalies are explained at 1 σ in the blue and green shaded regions, respectively.The combined fits at 1 and 2 σ are shown in orange and yellow, respectively.
It is found that the LHC sensitivity of the τν search is marginal with the Run 2 full data to probe the R D ( * ) explanation depending on the mass, while that of the τ ± ν + b search is enough to probe the 1σ parameter region in both m R 2 = 2 TeV and m R 2 = 4 TeV cases.Therefore, it is concluded that requiring an additional b-jet is significant to test a leptoquark scenario in light of the R D ( * ) anomaly.
Furthermore, the LHC is sensitive to the U(2) flavor symmetry based U 1 leptoquark scenario.Recently, the UV completion of the model is widely discussed [23][24][25][26][27][28][29][30][31][32][33][34][35].The U(2) flavor symmetry can naturally suppress the interaction between the LQ and 1st generation fermions and evade the stringent bound from K L → e μ.The sce-nario predicts C V L = 2e iφ R C S R at the LQ mass scale where φ R is free parameter of the model.In fact, unlike the scalar LQ scenarios, flavor constraints can be suppressed or avoided.Therefore, the LHC search is significant to probe the model.
In Fig. 3, the R D ( * ) -favored region is compared with the LHC sensitivities and flavor constraints for M LQ = 1.5 TeV (left) and 4 TeV (right) in the U(2)-U 1 scenario on the (C V 1 , φ R ) plane.The region in the right-hand side of the vertical (solid/dashed) lines is probed or constrained by the LHC searches.The orange (yellow) region is favored by the measured R D ( * ) at the 1 σ (2 σ) level.Note that the best fit is given at φ R ±0.42π, implying 94i.Similar to the R 2 LQ model, imaginary component is favored to be large.From this figure, we found that the R D ( * ) -favored region can be fully (mostly) probed by τ ± ν + b with 3 ab −1 of the data for M LQ > 4 TeV (< 4 TeV).
The study for the S 1 leptoquark scenario can be found in Refs.[36,37].For the S 1 LQ and G2HDM, single top + di-τ final state would provide the further important cross check of the scenario.
In section summary, it is clear that if the current size of the discrepancy persists even in the future, the interesting new physics scenarios can be probed at the LHC.

b → cu q puzzle vs. LHC
The B meson 2-body hadroic decays are described by the following effective Lagrangian, with the left-handed current-current operators in the CMM basis [39], where q = d, s.T a is the SU(3) C generator, and V is the Cabibbo-Kobayashi-Maskawa matrix.The suppression of the BR is favored in B → DP, B → D * P and B → DV modes.
Furthermore in a parton level b → cu s and b → cu d require the similar destructive effects.Therefore, if we take those coherent discrepancies as a hint of the NP, it is natural to consider that the new physics has similar flavor and Lorentz structure as the SM.Given that the extension of the gauge sector, adding an additional S U(2) group would be interesting.We consider an extended electroweak gauge group SU(2) 1 ×SU(2) 2 ×U(1) Y with heavy vector-like fermions produces heavy gauge bosons, W ± and Z , interacting with the left-handed SM fermions with a non-trivial flavor structure [40,41].These flavor structures are controlled by the number of generations of the vector-like fermions and mixings between the SM fermions and vector-like fermions.The heavy gauge boson interactions are where u L , d L are the mass eigenstates, and a coupling g i j is defined in the d L basis.In the following, we will take Integrating out W contribution is obtained as New physics contributions to the Wilson coefficients, C q,NP 1 and C q,NP 2 , become involved at the new physics scale Λ.These values are modified by the renormalizationgroup (RG) evolution from Λ down to the hadronic scale m b .
It is found that a universal destructive shift in the SM contributions is favored in the b → cūq puzzle, which corresponds to where C SM 2 (m b ) 1.In order to generate a desired shift in both b → cūd and b → cūs, a SM-like flavor structure in (Vg) 1q is required, and hence g 11 should be non zero.Also another non-zero entry of g 33 or g 23 is necessary to produce C q,W 2 .Three scenarios are considered in order: • scenario A, g 11 × g 33 0 and g 23 = 0, • scenario BR, g 11 × g 23 0 and g 33 = 0, The resultant parameter space and the relevant constraints are shown in Fig. 4 for each scenario.
In scenario A, C q,W 2 is proportional to V cb and thus the large couplings and/or lower NP scale are necessary.For instance when TeV is necessary to fit the data.Even if the flavor violating coupling g 23 is vanishing, there is W-W box contribution to ∆M.Also b → sγ and K → ππ as well as LHC searches can constrain the model parameter space since new gauge bosons considerably couple to valence quarks.More precisely, di-jet, t t and single top searches set upper limit on the couplings as long as the particle width does not exceed the experimental assumptions.Currently the maximum width to mass ratio considered in di-jet and t t are 55% and 30%.It is noted that the result of single top search is reported based on narrow width approximation EPJ Web of Conferences 289, 01007 (2023) https://doi.org/10.1051/epjconf/202328901007FCCP2022 Figure 5. Representative Feynman diagrams that contribute to the µ ± µ ± τ ∓ τ ∓ signal at the LHC.The left corresponds to electroweak pair production channel.The middle and right diagrams correspond to the single production process where φ denotes A or H.In addition, there are also those obtained by exchanging µ and τ which are included in our numerical calculation.(NWA).Once the particle width gets larger and exceeds the value that experimentally assumed, we can not apply the cross section limit directly and more dedicated analysis is necessary.It is found that if one allows the broad width regime Otherwise the allowed shift is less than sub percent.
In scenario BR, non zero g 23 induces ∆M s at tree level.Also di-jet resonance searches constrains the model.As a result, the possible deviation is C NP 2 (m b )/C SM 2 (m b ) −0.01 even if one allows the broad width regime.
In scenario C, we will demonstrate how large deviation could be possible within the broad width regime, otherwise it is obvious that the possible shift is less than 1%.Interestingly the cancellation between W box and tree Z contributions occurs and the stringent constraint from ∆M s can be relaxed.Consequently It is noted again that the more dedicated collider analysis is necessary to check this parameter region.More specifically, the cross section limit as a function of the minimal m j j would be helpful where m j j is the invariant mass of a pair of the jets.In any case collider constraints are crucial to the new physics interpretation of the discrepancy.Other new physics scenarios are also studied in Ref. [42] and the available mass window in a charged Higgs scenario is pointed out.However, the low mass dijet + initial state radiation jet search, for instance Ref. [43], would be enough to judge the window mass region.

Muon g − 2 anomary vs. LHC
It is known that the discrepancy is of the same order as the electroweak contribution, i. e. a new O(100) GeV weakly coupled particle can explain the discrepancy.This fact implies that in order to explain the discrepancy in terms of NP, some enhancement mechanism in the NP contribution to a µ is necessary to evade the current LHC bound.
A popular method to enhance the contribution is the introduction of a new flavor-violating particle.The dipole operator underlying g − 2 requires a chirality flip, which corresponds to the muon mass within flavor-conserving scenarios.A one-loop contribution involving a µτ lepton flavor-violating (LFV) particle is instead enhanced by a factor of m τ /m µ 17 [44][45][46][47].This mechanism can lift the mass scale of the new particle by more than a factor of four.However, LFV interactions are stringently constrained and easily spoil the model if the particle also has other couplings.Therefore one needs to ensure the absence of flavor-diagonal couplings for the m τ enhanced muon g − 2 solution to be viable.This specific coupling alignment can be realized by a discrete Z 4 flavor symmetry within a 2HDM [48].See Tab. 1 for the charge assignment.The gauge charge assignments of other SM fields, e. g. quarks, are the same as in the SM, and they trivially transform under Z 4 . 4In this model the g − 2 contribution is proportional to the µτ LFV Yukawa couplings (ρ µτ e and ρ τµ e ) and the mass difference of the additional neutral scalars which is proportional to the potential coupling λ 5 .
In the Z 4 limit the LFV τ decays are not allowed and thus it is not easy to test the scenario at the intensity frontier.Since the new scalars are quark-phobic, their production cross section at the LHC is not large.However, the unique coupling structure predicts that the neutral scalars decay into µ ± τ ∓ .Previously we pointed out the smokinggun signature of a µ ± µ ± τ ∓ τ ∓ final state via electroweak scalar pair production (left of Fig. 5) with a special focus on the case where all Yukawa and scalar potential couplings are smaller than one [45].We argued that the full Run 2 data set can test the model up to 500 GeV scalar mass thanks to the very unique double µτ LFV resonance nature of the signal events.However, if we accept relatively large coupling of O(1), the model can still explain the discrepancy with 1 TeV scalars.
Therefore we extend previous study with the perturbative unitarity constraint which gives an upper limit on the additional scalar mass as and evaluate the search potential [51]. 5 Fig. 6 (left) summarize the available parameter space.We use MadGraph5_aMC@NLO [49] with NNPDF3.1luxQED[50] to calculate the signal cross sections.To account for the minimal kinematic cuts, 7, and ∆R ≥ 0.1 are imposed.We assume focus on the hadronic decaying τ.Our signal of same sign µ ± µ ± and τ ∓ τ ∓ pairs of which µ ± τ ∓ forms a resonance is very distinctive.Specifically, due to the resonance structure, the final-state leptons are very energetic.Therefore we can safely assume the SM background (SMBG) to be negligible.Based on the Poisson statistics, we evaluated the sensitivity at 95% confidence level.
The cyan region in Fig. 6 show the HL-LHC sensitivity based on the EW pair production.The sensitivity is asymmetric in λ 5 , since λ 5 ≥ 0 corresponds to m H ≥ m A and thus the production cross section will be smaller compared to m H ≤ m A .The HL-LHC data of 3 ab −1 would be sensitive to mass scales of m A 960 GeV.It is worth mentioning that the pair production channel is powerful since once the neutral scalars are produced they dominantly decay into µτ, as long as a sizable τ-mass enhanced contribution to a µ is postulated.Nevertheless there is a mass gap between the sensitivity and the theoretical upper limit of Eq. ( 18).The loss of sensitivity for larger m A mainly comes from two factors: the contributing coupling constant is a weak gauge coupling which is independent of ∆a µ and the production cross section is suppressed by the heaviness of the pair-produced scalars.
One possible way to extend the LHC reach to our model is to include the single heavy scalar production channels corresponding to the middle and right diagrams 5 We required that the theory remains perturbative up to at least 5 TeV.If we require the theory to be perturbative up to 30 TeV, the upper limit is given as m A ≤ 1250 GeV of Fig. 5. Especially for O(1) TeV lepto-philic particles the inclusion of the photon initiated process is important [46].There are two terms in the single scalar production amplitude to generate µ ± µ ± τ ∓ τ ∓ events.One is proportional to (|ρ µτ e | 2 + |ρ τµ e | 2 ) which vanishes in the m A = m H limit. On the other hand the term proportional to ρ µτ e ρ τµ e does not disappear in this limit.Since the heavy mass scenario requires a large product of the LFV Yukawa couplings, the second contribution will be important for O(1) TeV scalars.
In Fig. 6 (right) the magenta region shows the combined sensitivity.The magenta regions can additionally be covered by including also the single scalar production.We note that there is also a non-resonant signal contribution which comes from t-channel A/H exchange.Since the lepton p T in this case is generally small and we are interested in the high p T region where the BG is negligible, this contribution is separated and subtracted to evaluate the sensitivity.We find that the inclusion of the single production process can improve the experimental reach by 130 and 60 GeV for |λ 5 | 1 and 2, respectively, when the Yukawa couplings are large.This still leaves a gap between the experimental reach and theoretical upper limit.
In order to further boost the sensitivity to the largemass scenario it is important to increase the center of mass energy from √ s = 14 TeV to, for instance, 27 TeV [52].Again the reach of the µ ± µ ± τ ∓ τ ∓ channel can be extended by including the single production process.As a result all the available parameter region can be probed with the high energy LHC with 3 ab −1 of the data.
It is noted that adding Z 4 charged singlet scalar could solve the dark matter problem [53], µ ± µ ± τ ∓ τ ∓ channel can also cover the interesting parameter space.
After the SM-like Higgs boson discovery there is no established NP signal at the LHC.On the other hand there are several statistically large discrepancies between the SM prediction and measurement known in the flavor physics.If we take those deviations as a hint of the NP, O(1) TeV new particles are implied.It is expected that the LHC plays a very important role in independently confirming/excluding the those scenarios.To improve the search potential and probe interesting scenarios, additional b-jet tagging technique, mediator mass dependence in t-channel processes and distinctive µ ± µ ± τ ∓ τ ∓ final state are discussed.We found that the future LHC have great potential and allow us to judge new physic interpretations.

Figure 1 .
Figure 1.The impact of the τν + b search on the charged Higgs with m H − = 250 GeV (left) and m H − = 400 GeV (right).The basic figures are taken from Ref. [17] and the new experimental world average is overlaid with a cyan contour.See the main text for the detail.

Figure 2 .
Figure 2. The impact of the τν and τν + b searches on the R 2 leptoquark scenario where m R 2 = 2 TeV (left) and m R 2 = 4 TeV (right) are assumed.The figure is taken from Ref. [37].See the main text for the detail.

EPJ 30 -Figure 3 .
Figure 3.The impact of the τν and τν + b searches on the U(2)-U 1 leptoquark scenario where m U 1 = 1.5 TeV (left) and m U 1 = 4 TeV (right) are assumed.The figure is taken from Ref. [37].See the main text for the detail.

4 EPJFigure 4 .
Figure 4. Contours of C NP 2 (m b )/C SM 2 (m b ) are presented in black.The figure taken from Ref.[41].The puzzle can be explained at 2σ level in the yellow bands.The blue and orange shaded regions are excluded by the di-jet and tt searches at 95% CL, respectively.The regions above the dashed lines are excluded by the single t searches in the NWA.Furthermore, the gray, red, green, and purple shaded regions are constrained by K → ππ, ∆M s , ∆M d , and b → sγ, respectively.The dotted line indicates Γ V /m V and the red-hatched regions represent Γ V /m V > 100%.Left: scenario A. g 33 = −g 11 is taken.Middle: scenario BR. g 23 = −0.01(MV /TeV) is taken.Right: scenario C. M V = 1 TeV and g 11 = −3.6 is fixed which already results in Γ V /m V ≥ 52%.

Figure 6 .
Figure 6.The black contours show the value of ρ µτ e ρ τµ e to explain the central value of α µ in the m A vs. λ 5 plane.The figure is taken from Ref. [51].The blue region is excluded by the lepton flavor universality of τ decays, BR(τ → µνν)/BR(τ → eνν).The orange region corresponds to Γ φ /m φ ≥ 30%.The purple contours depict the cutoff scale of the model.On the right panel the collider sensitivities are overlaid.The cyan region can be tested with 3 ab −1 of the electroweak pair production, while the magenta region can additionally be probed by including the single production channel.The blue and orange regions and purple contours are the same as in the left.∆a µ = 2.51 × 10 −9 is fixed and |ρ µτ e | = |ρ τµ e | is assumed in the figure.

Table 1 .
Relevant field content and charge assignment of the model.The notation of SM gauge quantum numbers is given as (SU(3) c , SU(2) L ) U(1) Y .H 1 is a SM-like doublet and H 2 is an additional doublet which includes H, A and H ± .