Identifying a Z ′ behind b→sℓℓ anomalies at the LHC
IIdentifying a Z (cid:48) behind b → s(cid:96)(cid:96) anomalies at the LHC M. Kohda and T. Modak
Department of Physics, National Taiwan University, Taipei 10617, Taiwan
A. Soffer
Tel Aviv University, Tel Aviv, 69978, Israel
Recent b → s(cid:96)(cid:96) anomalies may imply the existence of a new Z (cid:48) boson with left-handed Z (cid:48) bs and Z (cid:48) µµ couplings. Such a Z (cid:48) may be directly observed at LHC via b ¯ s → Z (cid:48) → µ + µ − , and itsrelevance to b → s(cid:96)(cid:96) may be studied by searching for the process gs → Z (cid:48) b → µ + µ − b . In this paper,we analyze the capability of the 14 TeV LHC to observe the Z (cid:48) in the µ + µ − and µ + µ − b modes basedon an effective model with major phenomenological constraints imposed. We find that both modescan be discovered with 3000 fb − data if the Z (cid:48) bs coupling saturates the latest B s − ¯ B s mixinglimit from UTfit at around 2 σ . Besides, a tiny right-handed Z (cid:48) bs coupling, if it exists, opens up thepossibility of a relatively large left-handed counterpart, due to cancellation in the B s − ¯ B s mixingamplitude. In this case, we show that even a data sample of O (100) fb − would enable discoveryof both modes. We further study the impact of a Z (cid:48) bb coupling as large as the Z (cid:48) bs coupling. Thisscenario enables discovery of the Z (cid:48) in both modes with milder effects on the B s − ¯ B s mixing, butobscures the relevance of the Z (cid:48) to b → s(cid:96)(cid:96) . Discrimination between the Z (cid:48) bs and Z (cid:48) bb couplingsmay come from the production cross section for the Z (cid:48) b ¯ b final state. However, we do not find theprospect for this to be promising. I. INTRODUCTION
Bottom-quark transitions of b → s have been of greatinterest as a means for studying physics beyond the stan-dard model (SM) since the observation of the decay B → K ∗ γ by the CLEO collaboration [1]. Involving aflavor-changing neutral current (FCNC), such processesare possible in the SM only at loop level, providing uniquesensitivity to new physics (NP).The high production rate and detection efficiency forbottom hadrons at the LHCb experiment have enabledprecision tests that probe physics at high energy scales.Based on the Run-1 data, LHCb measurements of severalobservables related to b → s(cid:96) + (cid:96) − ( (cid:96) = e or µ ) transitionsare in tension with SM predictions. The most notablediscrepancies are found in the P (cid:48) [2] angular-distributionobservable for the B → K ∗ µ + µ − decay [3, 4], and inthe lepton-flavor universality observables R K ≡ B ( B + → K + µ + µ − ) / B ( B + → K + e + e − ) [5] and R K ∗ ≡ B ( B → K ∗ µ + µ − ) / B ( B → K ∗ e + e − ) [6]. Moreover, measureddifferential branching ratios for exclusive b → sµ + µ − de-cays such as B → K µ + µ − , B + → K + µ + µ − , B + → K ∗ + µ + µ − [7], B → K ∗ µ + µ − [8], B s → φµ + µ − [9] andΛ b → Λ µ + µ − [10] are consistently lower than SM predic-tions in the dimuon-invariant mass range below the J/ψ threshold. ATLAS and CMS are also capable of studying b → sµ + µ − transitions. They have performed angularanalyses for B → K ∗ µ + µ − with 8 TeV data [11, 12],where the measured P (cid:48) by ATLAS supports the discrep-ancy found by LHCb while the measurements by CMSare in agreement with SM predictions. Belle [13] reportsan angular analysis for B → K ∗ (cid:96) + (cid:96) − , finding mild ten-sion in P (cid:48) in the muon mode, but not in the electronmode. The measurements will be significantly improvedwith more data collected by LHC, as well as with theupcoming Belle II experiment [14]. While the statistical significance of each discrepancyis not large enough, there is excitement about the possi-bility that their combination might suggest the presenceof NP. To investigate this possibility, global-fit analysesbased on the effective Hamiltonian formalism have beenperformed by several groups (see Refs. [15–22] for stud-ies that include the recent R K ∗ [6] result). These fitsfind that the tensions in the b → s(cid:96)(cid:96) observables canbe simultaneously alleviated to a great extent by a NPcontribution in a single Wilson coefficient. Most authorsfind this to be the coupling between the left-handed (LH) b → s current and either the LH or vector muon current. In particular, the findings of the global-fit analyses mo-tivate studying a new gauge boson, Z (cid:48) , with FCNC in-teractions. Many phenomenological studies of such a Z (cid:48) have been performed (see, e.g., [24–78]).Produced at LHC via b ¯ s → Z (cid:48) and undergoing the de-cay Z (cid:48) → µ + µ − , the Z (cid:48) may be discovered in dimuonresonance searches. Such a discovery by itself, however,would not reveal the relevance of the observed resonanceto the b → s(cid:96)(cid:96) anomalies. Comparison with searches inthe electron mode can test lepton-flavor universality inthe Z (cid:48) couplings, but one should also establish the cou-pling to the b → s current. In principle, this can bedone with the decay Z (cid:48) → b ¯ s . However, this decay issuppressed relative to Z (cid:48) → µ + µ − due to the B s − ¯ B s mixing constraint, as we discuss below, and its detec-tion suffers from overwhelming QCD background. Onthe other hand, the Z (cid:48) bs coupling can induce the pro-cess gs → Z (cid:48) b . Therefore, this coupling can be exploredthrough production modes of the Z (cid:48) at LHC. However, we note the existence of a fit to other B → K ∗ µ + µ − observables [23], which indicates a NP contribution in the right-handed b → s current. a r X i v : . [ h e p - ph ] J un In this paper, we investigate the prospects for directobservation of the Z (cid:48) , as well as determination of the fla-vor structure of its couplings at LHC, with √ s = 14 TeV.We argue that achieving this dual goal requires mea-suring the cross sections of both pp → Z (cid:48) + X and pp → Z (cid:48) b + X , where X refers to additional activityin the pp collision. We employ an effective-model de-scription of the Z (cid:48) couplings, and assume that it is thesource of all the NP required to alleviate the tensions in b → sµ + µ − . We focus mainly on the role of the Z (cid:48) bs coupling in the Z (cid:48) production processes pp → Z (cid:48) + X and pp → Z (cid:48) b + X . The constraints on the leptonic cou-pling of the Z (cid:48) are much weaker than those on the Z (cid:48) bs coupling. Therefore, we use Z (cid:48) → µ + µ − as the main dis-covery mode. As the LH Z (cid:48) bs coupling is accompaniedby a LH Z (cid:48) bb coupling in many UV complete models(see, e.g. Refs. [31, 51, 58, 61]), we also study scenar-ios in which the Z (cid:48) bb coupling is of the same order asthe Z (cid:48) bs coupling. We note that a larger Z (cid:48) bb couplingimplies larger cross sections and easier discovery of the Z (cid:48) at LHC, but obscures the role of the Z (cid:48) in b → s(cid:96)(cid:96) .Therefore, this case is not the focus of our work. We alsoconsider the process pp → Z (cid:48) b ¯ b + X for discriminationbetween the Z (cid:48) bs and Z (cid:48) bb couplings, where the lattercoupling uniquely contributes via gg → Z (cid:48) b ¯ b .We emphasize that our purpose is different from thatof existing studies, which focus on discovery and/or con-straint of the Z (cid:48) rather than testing its role in b → s(cid:96)(cid:96) .Recent studies on the impact of existing dimuon reso-nance searches on a Z (cid:48) motivated by the b → s(cid:96)(cid:96) anoma-lies can be found, e.g., in Refs. [64, 71]. Ref. [71] alsostudies the future sensitivity at LHC, exploiting the useof additional b jets for background suppression. How-ever, it targets a Z (cid:48) bb coupling that is much larger thanthe Z (cid:48) bs coupling. The future sensitivities at LHC anda 100 TeV pp collider are studied in Ref. [72] with anextrapolation of existing ATLAS limit.We will show that the cross sections for b ¯ s → Z (cid:48) and gs → Z (cid:48) b are limited by a rather tight constraint on the Z (cid:48) bs coupling, which originates from the B s − ¯ B s mixing.This severely restricts the discovery potential of the Z (cid:48) ,unless the Z (cid:48) bb coupling is comparable or larger thanthe Z (cid:48) bs coupling. As the B s − ¯ B s mixing constraint isindirect, the actual limit on the Z (cid:48) bs coupling dependson the details of the UV-complete model. In particular, anonzero right-handed (RH) Z (cid:48) bs coupling can drasticallychange the constraint, due to the large ( V − A ) ⊗ ( V + A )term [79] in the B s − ¯ B s mixing amplitude. Althoughthere is no strong indication of a RH b → s current in themajority of the global-fit analyses, even a tiny RH Z (cid:48) bs coupling would allow for a large LH Z (cid:48) bs coupling due tothe cancellation. This would significantly boost the Z (cid:48) production cross sections. We investigate the discoverypotential in this case as well.This paper is organized as follows. In Sec. II, we intro-duce the effective model for our collider study. In Sec. III,we evaluate existing phenomenological constraints on therelevant couplings of the Z (cid:48) boson to quarks and leptons. This is carried out for two representative Z (cid:48) mass val-ues, m Z (cid:48) = 200 and 500 GeV. In Sec. IV, we study thesignal and background cross sections for the three pro-cesses of interest, pp → Z (cid:48) + X , Z (cid:48) b + X and Z (cid:48) b ¯ b + X ,given the coupling constraints. We then proceed to esti-mate the signal significances for the full integrated lumi-nosity of the High-Luminosity LHC (HL-LHC) program, L = 3000 fb − . In Sec. V, we discuss the impact of atiny but nonzero RH Z (cid:48) bs coupling, which allows discov-ery with smaller integrated luminosities. Summary anddiscussions are given in Sec. VI. II. EFFECTIVE MODEL
We describe the Z (cid:48) couplings to the SM fermions withthe effective Lagrangian L ⊃ − Z (cid:48) α (cid:2) g Lbb ¯ bγ α P L b + g Lbs (cid:0) ¯ bγ α P L s + ¯ sγ α P L b (cid:1) + g Lµµ (¯ µγ α P L µ + ¯ ν µ γ α P L ν µ ) + g Rµµ ¯ µγ α P R µ (cid:3) , (1)where P L,R = (1 ∓ γ ) /
2, and g Lbb , g Lbs and g L,Rµµ arecoupling constants. For simplicity, since no significant CP violation has been observed in the relevant observ-ables, we take the couplings to be real. In addition to theLH Z (cid:48) bs coupling and LH and RH Z (cid:48) µµ couplings moti-vated by the b → s(cid:96) + (cid:96) − global fits, we introduce the LH Z (cid:48) bb coupling predicted in many UV complete models.We take g Lµµ to also be the coupling to the muon neu-trino, as required by the SU(2) L gauge symmetry, andsince observables containing neutrinos give meaningfulconstraints on the Z (cid:48) parameters, as shown below. TheRH Z (cid:48) bs coupling is not included at this stage, as it isdiscussed only in Sec. V.Since we take m Z (cid:48) (cid:29) m b , we integrate out the Z (cid:48) toobtain its contributions to the effective Hamiltonian for b → sµ + µ − transitions, given by∆ H eff = N (cid:2) C NP9 (¯ sγ α P L b )(¯ µγ α µ )+ C NP10 (¯ sγ α P L b )(¯ µγ α γ µ ) (cid:3) + h . c ., (2)where N = − αG F √ π V tb V ∗ ts , and C NP9 = g Lbs g Vµµ N m Z (cid:48) , C NP10 = g Lbs g Aµµ N m Z (cid:48) , (3)are the Wilson coefficients, with the vector and axial-vector muon couplings defined by g Vµµ ≡ g Rµµ + g Lµµ , g Aµµ ≡ g Rµµ − g Lµµ . (4)With the parametrization of the Cabibbo-Kobayashi-Maskawa (CKM) matrix in Ref. [80], C NP9 , are treatedas real to a good approximation.Motivated by the global-fit analyses, we consider twopossibilities for the chiral structure of the muon cou-plings: a vector coupling and a LH coupling. For eachof these, we extract constraints on the couplings fromglobal-fit analyses presented in Ref. [15]. The first anal-ysis uses all available b → s(cid:96)(cid:96) data from LHCb, ATLAS,CMS and Belle. This yields the following two scenarios:(i) Vector coupling ( g Lµµ = g Rµµ ) and, hence, C NP10 = 0.In this case, the best fit value is Re C NP9 = − . σ ) range of − . ≤ Re C NP9 ≤ − . . (5)(ii) LH coupling ( g Rµµ = 0), so that C NP9 = − C NP10 .For this scenario, the best fit value is Re C NP9 = − Re C NP10 = − .
62, with a 2 σ range − . ≤ Re C NP9 = − Re C NP10 ≤ − . . (6)The authors of Ref. [15] also present results whentaking into account only lepton-flavor universality ob-servables, such as R K ( ∗ ) measured by LHCb [5, 6] andthe differences Q and Q [81] between angular observ-ables in B → K ∗ µ + µ − and B → K ∗ e + e − , measured byBelle [13]. This leads to the following two scenarios:(i’) In the case of vector coupling, the best-fit value isRe C NP9 = − .
76, with − . ≤ Re C NP9 ≤ − . σ interval.(ii’) For the LH-coupling case, the best-fit value is foundto be Re C NP9 = − Re C NP10 = − .
66, while at 2 σ range − . ≤ Re C NP9 = − Re C NP10 ≤ − . III. ALLOWED PARAMETER SPACE
In Fig. 1 we show the various constraints on g Lbs vs. g Vµµ in scenario (i), for the representative Z (cid:48) -mass val-ues of m Z (cid:48) = 200 and 500 GeV. The relevant inputs andconstraint calculation methods are described in the re-mainder of this section.Before embarking on this detailed description, we notethat, as seen in Fig. 1, the limits on g Vµµ are much weakerthan those on g Lbs . Therefore, the Z (cid:48) is likely to decayprimarily to leptons, so that its decays into quarks canbe ignored. The leptonic-decay dominance simplifies thediscussion. In fact, it is also essential for direct observa-tion of the Z (cid:48) at LHC, since searches with Z (cid:48) → b ¯ b, b ¯ s suffer from overwhelming QCD background. Thus, in thescenario (i), where g Lµµ = g Rµµ (cid:54) = 0, the dominant branch-ing ratios are B ( Z (cid:48) → µ + µ − ) (cid:39) , B ( Z (cid:48) → ν µ ¯ ν µ ) (cid:39) . (7) In scenario (ii), each of these branching ratios is 50%.To incorporate the results of b → s(cid:96)(cid:96) global fits, weuse Eq. (3) to convert the 2 σ constraints of Eq. (5) tothe space of g Lbs vs. g Vµµ . The result is given by the bluehyperbolae in Fig. 1As an example of the impact of the choice of scenarioamong those listed in Sec. II, we note that in scenario (i’),the resulting values of g Vµµ are generically higher thanthose shown in Fig. 1, and have a wider range. Althoughthis somewhat changes the Z (cid:48) → µ + µ − branching ratio,Eq. (7) is still satisfied. Therefore, the results we obtainin Section IV are not affected.The most important constraint on the Z (cid:48) bs couplingcomes from the B s mixing. With the tree-level Z (cid:48) ex-change contribution, the total B s − ¯ B s mixing amplituderelative to the SM one (see, e.g. Ref. [82]) is given by M M SM12 = 1 + (cid:0) g Lbs (cid:1) m Z (cid:48) (cid:18) g π v ( V tb V ∗ ts ) S (cid:19) − , (8)where S (cid:39) . g is theSU(2) L gauge coupling, v (cid:39)
246 GeV, and a commonQCD correction factor is assumed for the SM and NPcontributions. The mass difference ∆ m B s = 2 | M | isprecisely measured, at the per mill level [80], while thecalculation of M suffers from various sources of uncer-tainty. One of the dominant uncertainties is the CKMfactor, with an uncertainty of ∼
5% [83, 84]. The otheris the hadronic matrix element, obtained from the lat-tice. The average of N f = 2 + 1 lattice results compiledby FLAG in 2016 [85] implied a ∼
12% uncertainty in M SM12 . Recently, the situation was greatly improved withthe advent of the accurate estimate by the Fermilab Lat-tice and MILC collaborations [86], which pushes downthe uncertainty of the FLAG average to ∼ M with NP as | M /M SM12 | = 1 . +0 . − . . Onthe other hand, the Summer 2016 result [84] by UTfit,which includes the result of Ref. [86], constrains NP witha better precision: | M /M SM12 | = 1 . ± . g Lbs to be real, the Z (cid:48) contribution always enhances | M | . If one takes these uncertainties to be Gaussian,these results imply | M /M SM12 | < .
32 [83] or 1 .
25 [84] at2 σ for the CKMfitter and UTfit results, respectively. Inthis paper, we explore NP contributions to | M /M SM12 | at the level of up to 30%. Excluding larger contributionsleads to the gray-shaded regions in Fig. 1. Future im-provements in lattice calculations and measurements ofthe CKM parameters would tighten the constraint [87].In Fig. 1, we illustrate the impact of possible future im-provements by the vertical dotted lines for deviation of M from SM by 15% or 5%.We note that while the CKMfitter [83] and UT-fit [84] results are tolerant to a NP contribution thatenhances | M | , there are studies that find the SM pre-diction of ∆ m B s = 2 | M | to be larger than the mea-sured value, slightly favoring NP that reduces | M | . In - - - - - g bs L × g μμ V m Z ' =
200 GeV % % % % % % B s mixing ( %) B s mixing ( %) - - - - - g bs L × g μμ V m Z ' =
500 GeV % % % % % % B s mixing ( %) B s mixing ( %) C = - B s mixingMuon g - → K (*) νν FIG. 1. Constraints on the vector Z (cid:48) µµ coupling g Vµµ = g Lµµ = g Rµµ vs. the LH Z (cid:48) bs coupling g Lbs (in units of 10 − ) inscenario (i), for m Z (cid:48) = 200 GeV [left] and 500 GeV [right]. The red dots show the benchmark points discussed in Section IV.See the main text for details. particular, a recent study [77], which adopts the 2017FLAG result [85], finds the SM prediction to be 1 . σ above the measured value. Their result can be read as | M /M SM12 | (cid:39) . ± .
06, which allows an enhancementby NP only up to ∼
1% at 2 σ . The rather small un-certainty is in part due to a smaller uncertainty of 2.1%assigned to the CKM factor. Addressing the discrepancyamong the theoretical calculations is beyond the scope ofthis paper. Instead, the 1% vertical dotted lines are alsoshown in Fig. 1 for illustrating the impact of the resultby Ref. [77].The nonzero LH Z (cid:48) µµ coupling implies also the ex-istence of a Z (cid:48) νν coupling, due to SU(2) L . Therefore,constraints are also set by B → K ( ∗ ) ν ¯ ν . The effectiveHamiltonian for b → sν ¯ ν is [88] H νeff = N (cid:88) (cid:96) = e,µ,τ C (cid:96)L (¯ sγ α P L b ) [¯ ν (cid:96) γ α (1 − γ ) ν (cid:96) ] + h . c ., (9)where C µL = C SM L + C NP L and C (cid:96)L = C SM L ( (cid:96) = e, τ )are lepton-flavor dependent Wilson coefficients. The SMcontribution is lepton-flavor universal and is given by C SM L = − X t /s W with X t = 1 . ± .
017 [89]. The Z (cid:48) contribution is given by C NP L = g Lbs g Lµµ N m Z (cid:48) . (10)Normalizing B → K ( ∗ ) ν ¯ ν branching ratios by the SMones and defining R νK ( ∗ ) ≡ B ( B → K ( ∗ ) ν ¯ ν ) / B ( B → K ( ∗ ) ν ¯ ν ) SM , we obtain R νK = R νK ∗ = 23 + 13 (cid:12)(cid:12)(cid:12)(cid:12) C SM L + C NP L C SM L (cid:12)(cid:12)(cid:12)(cid:12) . (11) Combining the charged and neutral B meson decays, thetightest limits [90] are set by Belle [91], who find R νK < . , R νK ∗ < . R νK ∗ , and is shown by the cyan lines in Fig. 1.The allowed region fully contains the blue hyperbolaefavored by b → s(cid:96)(cid:96) The Z (cid:48) bs and Z (cid:48) bb couplings induce Z (cid:48) production atLHC via b ¯ s → Z (cid:48) and b ¯ b → Z (cid:48) . Hence, with Z (cid:48) → µ + µ − ,these couplings are constrained by dimuon resonancesearches at LHC.We use the results from ATLAS, performed with36.1 fb − at 13 TeV [92] and extract [93] the 95% credi-bility level limit: σ ( pp → Z (cid:48) + X ) B ( Z (cid:48) → µ + µ − ) <
42 fb(9 fb) for m Z (cid:48) = 200 GeV (500 GeV). We calculatethe Z (cid:48) production cross section at leading order (LO)using MadGraph5 aMC@NLO [94] with the NN23LO1parton distribution function (PDF) set [95]. As the AT-LAS search does not veto additional activity in the event,we include also the processes gs → Z (cid:48) b , gb → Z (cid:48) b and gg → Z (cid:48) b ¯ b, Z (cid:48) b ¯ s in the cross-section calculation. We de-fer the more detailed discussion about Z (cid:48) production atLHC to Section IV. From the cross sections and the AT-LAS limits, we find (cid:113) | g Lbs | + 0 . | g Lbb | < .
004 ( m Z (cid:48) = 200 GeV) , (cid:113) | g Lbs | + 0 . | g Lbb | < .
011 ( m Z (cid:48) = 500 GeV) (13)in scenario (i). In scenario (ii), where the Z (cid:48) couplesto the LH muon current, the limits are weakened by anoverall factor of 2 / √ B ( Z (cid:48) → µ + µ − ).As long as | g Lbb | (cid:46) | g Lbs | , which is our scenario of in-terest, these limits are significantly weaker than thosefrom the B s − ¯ B s mixing. Hence, they are not shownin Fig. 1. For flavor universal models, Z (cid:48) masses below m Z (cid:48) (cid:46) − . Z (cid:48) such as described by Eq.(1), escapes the detec-tion and could emerge in the future runs of LHC.Muon pair production in the scattering of a muon neu-trino and a nucleus N , known as neutrino trident produc-tion, tightly constrains Z (cid:48) µµ and Z (cid:48) ν µ ν µ couplings [96].The ratio between the total ν µ N → ν µ N µ + µ − cross sec-tion and its SM prediction is given by [31] σσ SM = 1 + (cid:104) s W + 2( g Vµµ ) v m Z (cid:48) (cid:105) s W ) , (14)in scenario (i) with m Z (cid:48) (cid:38)
10 GeV. The measurement [97]by the CCFR collaboration is in a good agreement withSM, and implies σ/σ SM = 0 . ± .
28. We show theresulting 2 σ upper limits on | g Vµµ | by the horizontal solidred lines in Fig. 1.The couplings of the Z boson with the muon and muonneutrino are modified by Z (cid:48) -loop contributions, whichcan lead to violation of the lepton-flavor universality in Z decays. In scenario (i), the vector and axial-vector Zµµ couplings relative to the SM-like
Zee couplings aregiven by [31, 50] g V µ g V e (cid:39) g Aµ g Ae (cid:39) g Vµµ ) π Re (cid:2) K ( m Z /m Z (cid:48) ) (cid:3) , (15)where K ( m Z /m Z (cid:48) ) is a loop function given in Ref. [98],and its real part is taken to match the convention ofRef. [99]. Here, the lepton-flavor universality in the SMcase is exploited. Similarly, normalized Zνν couplingsare given by g V ν g Ae = g Aν g Ae (cid:39) − (cid:40) g Lµµ ) π Re (cid:2) K ( m Z /m Z (cid:48) ) (cid:3)(cid:41) , (16)where the factor of 1 / Z → ν µ ¯ ν µ is affected by the Z (cid:48) among thethree neutrino modes.The Z couplings were very precisely measured at SLCand LEP. Relevant results from the average of 14 elec-troweak measurements are g V e = − . ± . g Ae = − . ± . g V µ = − . ± . g Aµ = − . ± . g V ν = g Aν = 0 . ± . Z coupling ratios, wetake only the one which is most sensitive to the effectof the Z (cid:48) , i.e. g Aµ /g Ae = 1 . ± . σ upper limits on | g Vµµ | are shown by the horizontal reddashed lines in Fig. 1.Nonzero values of g Lbs or g Lbb can alter the
Zbb and
Zss couplings at one loop. Taking the b and s quarks to be massless, we find that the Z (cid:48) loop with a nonzero g Lbs modifies the LH
Zbb and
Zss couplings g Lb and g Ls relative to their SM values g SM Lb and g SM Ls in the sameway as the LH Zµµ coupling, but with the replacement g Lµµ → g Lbs : g Lb g SM Lb (cid:39) g Ls g SM Ls (cid:39) g Lbs ) π Re (cid:2) K ( m Z /m Z (cid:48) ) (cid:3) . (17)The RH counterparts remain unchanged. The effect ofthe Z (cid:48) loop can be constrained by comparing the mea-sured value g Lb = − . ± . g Ls = − . ± . g SM Lb = − . +0 . − . ( g SM Ls = − . +0 . − . ), derivedfrom the SM Z -pole fit [99]. Since g Lb is more preciselymeasured than g Ls , we use g Lb to extract the limit on g Lbs . Adding the errors in g Lb and g SM Lb in quadrature aftersymmetrizing the g SM Lb errors, we find the 2 σ upper limit | g Lbs | (cid:46) .
34 (0 .
67) for m Z (cid:48) = 200 (500) GeV. These lim-its are much weaker than the ones obtained from the B s mixing, and we do not display them in Fig. 1. A similarconclusion can be made for the g Lbb coupling as well.If both g Lbs and g Lbb are nonzero, an FCNC decay Z → b ¯ s , which is absent in the SM at tree-level, is in-duced by the one-loop Z (cid:48) contribution. A preliminaryresult by DELPHI [100] sets the 90% CL upper limit R b(cid:96) = (cid:80) q = d,s σ ( e + e − → b ¯ q + ¯ bq ) /σ ( e + e − → hadrons) ≤ . × − at the energy scale of the Z mass. Using B ( Z → hadrons) (cid:39)
70% [80], one may rewrite the limitas (cid:80) q = d,s B ( Z → b ¯ q + ¯ bq ) (cid:46) . × − . Since the Z (cid:48) -loop-induced LH Zbs coupling is suppressed by the factor g Lbs g Lbb / (16 π ), the DELPHI limit is relevant only if both g Lbs and g Lbb are O (1) and the Z (cid:48) mass is not far from the Z mass. Since the B s -mixing constraint on g Lbs is muchtighter, and we concentrate on the case | g Lbb | (cid:46) | g Lbs | , theimpact on B ( Z → b ¯ s + ¯ bs ) is generically far below theDELPHI limit in the scenarios considered in this paper.The Z (cid:48) one-loop contribution to the muon anomalousmagnetic moment a µ = ( g µ − / a µ = ( g Vµµ ) π m µ m Z (cid:48) , (18)where scenario (i) and m Z (cid:48) (cid:29) m µ are assumed. Thedifference between the measured value of a µ and its SMprediction is (2 . ± . × − [102]. Assigning this dif-ference to Eq. (18) yields the dark yellow 2 σ regions inFig. 1. Since the 2 σ constraints from the neutrino tridentcross section are tighter, the Z (cid:48) does not solve the tensionin the muon g −
2. At the 3 σ level, the g µ − g µ − Furthermore, the measured value of g Ls in Ref. [99] is obtainedunder the assumption of g Ls = g Ld . This is not valid in our case,since g Ld receives no correction from Z (cid:48) at one loop. b ¯ s Z ′ µ − µ + FIG. 2. A Feynman diagram for the process b ¯ s → Z (cid:48) → µ + µ − . s s b Z ′ g s Z ′ b bg ( a ) ( b ) FIG. 3. Feynman diagrams for gs → Z (cid:48) b . We have discussed so far the constraints in scenario (i),summarized in Fig. 1 on the g Vµµ vs. g Lbs plane. FromFig. 1, we find the constraint on g Lbs is very stringent dueto the B s − ¯ B s mixing, while the constraint on g Vµµ ismuch weaker. With the 30% NP effects allowed in the B s − ¯ B s mixing amplitude, the former constraint is | g Lbs | (cid:46) . . × − for m Z (cid:48) = 200 (500) GeV, imposing thelower limit on the Z (cid:48) µµ coupling | g Vµµ | (cid:38) .
031 (0 . b → s(cid:96)(cid:96) hyperbolae. This validates the numer-ical values of the branching ratios in Eq. (7) to a goodapproximation, if | g Lbb | is not too large compared to | g Lbs | .The qualitative feature is the same in scenario (ii),where the Z (cid:48) µµ coupling is of LH, as the B s − ¯ B s mixingconstraint does not change. Some of the other observ-ables would give slightly different constraints on the Z (cid:48) µµ coupling, but the effect is minor to our study. The onlynotable difference from scenario (i) is the slight changein the value of B ( Z (cid:48) → µ + µ − ) from Eq. (7). A nonzeroaxial-vector Z (cid:48) µµ coupling can make B s → µ + µ − rele-vant, but the latest LHCb result [103] does not excludethe allowed regions for the b → s(cid:96)(cid:96) anomalies. IV. DISCOVERY AND IDENTIFICATION OFTHE Z (cid:48) AT LHC
Having determined the constraints on the Z (cid:48) couplings,we proceed to study signatures for direct production ofthe on-shell Z (cid:48) in pp collisions with a center-of-mass en-ergy of √ s = 14 TeV. The goal of this study is to as-certain the LHC potential for both discovery of the Z (cid:48) and determination of the flavor structure of its couplings.Therefore, motivated by the tensions in b → s(cid:96)(cid:96) , we fo- cus on the role of the Z (cid:48) bs coupling g Lbs . If this is thedominant coupling to the quark sector, the Z (cid:48) will beprimarily produced via the parton-level process b ¯ s → Z (cid:48) shown in Fig. 2. With the decay Z (cid:48) → µ + µ − , the Z (cid:48) may be discovered in the conventional dimuon resonancesearches.Such a discovery of a dimuon resonance, however, doesnot necessarily imply the existence of the Z (cid:48) bs coupling.In general, the process pp → Z (cid:48) + X may be facilitatedby coupling to other quarks, particularly flavor-diagonalcouplings. To test for dominance of the Z (cid:48) bs coupling,we propose to also search for pp → Z (cid:48) b + X (see Fig. 3for typical parton-level processes). A Z (cid:48) bb coupling, pre-dicted in many models (e.g., Refs [31, 51, 58, 61]), mo-tivated by the b → s(cid:96)(cid:96) anomalies, also contribute to pp → Z (cid:48) + X and pp → Z (cid:48) b + X via the parton levelprocesses b ¯ b → Z (cid:48) and gb → Z (cid:48) b . Since a Z (cid:48) bb couplingalso leads to pp → b ¯ bZ (cid:48) + X , via, for example, gg → Z (cid:48) b ¯ b ,measuring the cross section for pp → Z (cid:48) b ¯ b + X may facil-itate to discriminate the Z (cid:48) bb coupling from Z (cid:48) bs . Sim-ilarly, the production process pp → Z (cid:48) b ¯ s + X , occurringdue to gg → Z (cid:48) b ¯ s , can in principle help probe the Z (cid:48) bs coupling.We explore these signatures using the effective La-grangian of Eq. (1) with scenario (i), i.e., with a vector Z (cid:48) µµ coupling. For each of the two Z (cid:48) mass values stud-ied, we fix the couplings to the benchmark points shownby the red dots in Fig. 1: (cid:12)(cid:12) g Lbs (cid:12)(cid:12) = 0 . , (cid:12)(cid:12) g Vµµ (cid:12)(cid:12) = 0 .
04 ( m Z (cid:48) = 200 GeV) , (cid:12)(cid:12) g Lbs (cid:12)(cid:12) = 0 . , (cid:12)(cid:12) g Vµµ (cid:12)(cid:12) = 0 . m Z (cid:48) = 500 GeV) . (19)These values are selected such that g Lbs leads to a 15%enhancement in the B s − ¯ B s -mixing amplitude M . Thevalue of g Vµµ is then chosen so as to lie in the range givenby Eqs. (5) and (3). We note that one may take a larger | g Lbs | (with a smaller | g Vµµ | ), which would enlarge the pp → Z (cid:48) + X and pp → Z (cid:48) b + X cross sections by up to a factorof two, with the B s − ¯ B s -mixing constraint saturated at2 σ , i.e. a ∼
30% enhancement in M .For the Z (cid:48) bb coupling, we study three cases for eachbenchmark point. The baseline case is g Lbb = 0, whichrestricts assumptions about the Z (cid:48) couplings to the min-imum needed to explain the b → s(cid:96)(cid:96) anomalies. In addi-tion, we also explore the cases g Lbb = g Lbs and g Lbb = 2 g Lbs ,to study the impact of a nonzero g Lbb . These choices ofquark couplings satisfy the dimuon resonance search lim-its in Eq. (13) and maintain the Z (cid:48) branching ratios inEq. (7).In the following subsections, we mainly focus on thediscovery potential of the Z (cid:48) in the production processes pp → Z (cid:48) + X , pp → Z (cid:48) b + X , and pp → Z (cid:48) b ¯ b + X , withthe Z (cid:48) always decaying to µ + µ − . We use Monte Carloevent generator MadGraph5 aMC@NLO [94] to generatesignal and background samples at LO with the NN23LO1PDF set [95]. The effective Lagrangian of Eq. (1) is im-plemented in the FeynRules 2.0 [104] framework. Thematrix elements for signal and background are gener-ated with up to two additional jets and interfaced withPYTHIA 6.4 [105] for parton showering and hadroniza-tion. Matching is performed with the MLM prescrip-tion [106]. The generated events are passed into theDelphes 3.3.3 [107] fast detector simulation to incorpo-rate detector effects based on ATLAS. A. Observation of pp → Z (cid:48) + X Several SM processes constitute background for pp → Z (cid:48) + X , where we remind the reader that the Z (cid:48) de-cays into µ + µ − . The dominant background is due to theDrell-Yan (DY) events, pp → Z/γ ∗ + X . The pp → t ¯ t events with semileptonic decay of both top quarks is thenext largest background. Smaller backgrounds arise from pp → W t and
V V , where V ≡ W, Z . Background mayalso arise from leptons produced in heavy-flavor decays orfrom jets faking leptons. These background sources arenot well modeled by the simulation tools, and we ignorethem, assuming that they can be reduced to subdom-inant level with lepton quality cuts without drasticallyimpacting the results of our analysis.We scale the LO cross sections obtained by Mad-Graph5 aMC@NLO as follows. The DY cross sectionis normalized to a NNLO QCD+NLO EW cross sectionby a factor of 1 .
27, obtained with FEWZ 3.1 [108] inthe dimuon-invariant mass range m µµ >
106 GeV. Wenormalize the LO pp → t ¯ t and pp → W t cross sec-tions to NNLO+NNLL cross sections by 1 .
84 [109] and1 .
35 [110], respectively. The pp → W W , pp → W Z and pp → ZZ cross sections are normalized to NNLO QCDby 1 .
98 [111], 2 .
07 [112] and 1 .
74 [113], respectively. Wedo not apply correction factors to the signal cross sections throughout this paper.We select events that contain at least two oppositelycharged muons. The transverse momentum of each muonis required to satisfy p Tµ >
50 GeV, and its pseudorapid-ity must be in the range | η µ | < .
5. The two muonsmust satisfy ∆ R µµ = (cid:113) ∆ φ µµ + ∆ η µµ > .
4, where∆ φ µµ and ∆ η µµ are the separations in azimuthal an-gle and pseudorapidity between the muons. Finally, werequire the invariant mass of the two muons to satisfy | m µµ − m Z (cid:48) | < m cut , where m cut = 4 GeV and 16 GeVfor m Z (cid:48) = 200 GeV and m Z (cid:48) = 500 GeV, respectively.These values are chosen so as to maximize the naive localsignificance of the no-signal hypothesis, S l = N S / √ N B ,where N S and N B are the expected signal and back-ground yields.The invariant mass cut | m µµ − m Z (cid:48) | < m cut is not real-istic for a discovery scenario, in which one does not knowthe true mass m Z (cid:48) . However, the value of S l thus ob-tained is a rough estimate of the one that will be foundby the more sophisticated analysis that will eventuallybe performed with the full LHC data. One is actually in-terested in the global significance S g , which accounts forthe probability to obtain the given value of S l anywherein the dimuon-invariant mass range. Rigorous methodsfor estimating S g exist [114]. However, at this level ofapproximation, it is sufficient to use the crude estimate P g = P l m cut m range , (20)where P g and P l are the χ probabilities correspondingto S g and S l , respectively, and m range is the size of therange of m µµ values explored in the analysis. Since crosssections drop to negligible levels at high m µµ , it is rea-sonable to take m µµ ∼ g Lbb . The correspond-ing values of the local significance S l for an integratedluminosity L = 3000 fb − are summarized in Table II.Inserting the values of S l into Eq. (20), we conclude thatthe global significance will likely be greater than 5 σ forthe case g Lbb = 2 g Lbs , allowing separate discovery by AT-LAS and CMS. For | g Lbb | ≤ | g Lbs | , the global significancewill be under 5 σ . Whether the 5 σ mark will be passedby combining ATLAS and CMS results is beyond theprecision of our rough estimate.A larger | g Lbs | can enhance the significance in eachbenchmark scenario. For the scenario of m Z (cid:48) = 200 (500)GeV with g Lbb = 0, taking | g Lbs | = 0 . . | g Vµµ | = 0 .
031 (0 . σ , which may imply a global significance of 5 σ .In this case, the B s − ¯ B s mixing amplitude | M | is alsoenhanced from the SM one by 25% (28%), but still withinthe nominal 2 σ allowed range, as discussed in Sec. III. m Z (cid:48) (GeV) σ signal (fb) σ background (fb) g Lbb = 0 g Lbb = g Lbs g Lbb = 2 g Lbs DY t ¯ t W t V V
200 1.0 1.3 2.2 170 41 4.1 5.1500 0.27 0.33 0.50 14 4.3 0.5 1.0TABLE I. Cross sections for the signal process pp → Z (cid:48) + X with Z (cid:48) → µ + µ − , and the dominant backgrounds after the eventselection for the benchmark points defined in Eq. (19) with the three choices for g Lbb . The combined cross sections for
W W , W Z and ZZ backgrounds are denoted together as V V . m Z (cid:48) (GeV) Local significance g Lbb = 0 g Lbb = g Lbs g Lbb = 2 g Lbs
200 3.7 4.9 8.3500 3.3 4.1 6.5TABLE II. Local significance S l = N S / √ N B for discovery of the process pp → Z (cid:48) + X with Z (cid:48) → µ + µ − , with an integratedluminosity of 3000 fb − , given the signal and background cross sections shown in Table I. m Z (cid:48) (GeV) σ signal (fb) σ background (fb) g Lbb = 0 g Lbb = g Lbs g Lbb = 2 g Lbs
DY + b DY + c DY + j t ¯ t W t V V
200 0.17 0.22 0.37 1.3 1.0 0.22 5.6 0.8 0.5500 0.043 0.049 0.10 0.15 0.048 0.028 0.26 0.08 0.064TABLE III. Cross sections for the signal process pp → Z (cid:48) b + X with Z (cid:48) → µ + µ − , and the dominant backgrounds after theevent selection for the benchmark points defined in Eq. (19) with the three choices for g Lbb . m Z (cid:48) (GeV) Local significance g Lbb = 0 g Lbb = g Lbs g Lbb = 2 g Lbs
200 3.0 3.9 6.6500 3.0 3.4 7.2TABLE IV. Local significance S l = N S / √ N B for discovery of the process pp → Z (cid:48) b + X with Z (cid:48) → µ + µ − , with an integratedluminosity of 3000 fb − , given the signal and background cross sections shown in Table III. B. Observation of pp → Z (cid:48) b + X The main SM background for pp → Z (cid:48) b + X is pp → t ¯ t . The second-largest background is Drell-Yan with atleast one additional b jet, labeled as DY + b . Smallercontributions arise from DY + c , pp → W t , pp → V V ,and DY + j , where j stands for a jet from a gluon or a u , d , or s quark. We normalize the cross sections for DY + b ,DY + c , and DY + j to NNLO QCD by 1 .
83 [115]. Thecorrection factors for pp → t ¯ t , pp → W t , and pp → V V are taken to be the same as in Section IV A.We select simulated events that contain at least twoopposite-charge muons. The muons are required to be inthe pseudorapidity range | η µ | < .
5, have minimal trans-verse momenta of p Tµ >
50 (60) GeV for m Z (cid:48) = 200 (500)GeV, and be separated by ∆ R µµ > .
4. Jets are recon-structed using the anti- k T algorithm with radius param-eter R = 0 .
5. It is assumed that a b -tagging algorithmreduces the efficiency for c jets and light jets by factorsof 5 and 137, respectively [116]. Its efficiency for b jetsis calculated in Delphes, accounting for the p T and η dependence. The leading b jet is required to have trans-verse momentum p Tb >
30 GeV with | η b | < .
5, and itsseparation from each of the two leading muons must sat- isfy ∆ R bµ > .
4. We reject events that have a second b -tagged jet with p Tb >
30 GeV, slightly increasing thelocal significance. The missing transverse energy mustbe less than 40 GeV, in order to reduce pp → t ¯ t and pp → W t backgrounds. Finally, we apply the optimizeddimuon-invariant mass cut | m µµ − m Z (cid:48) | < m Z (cid:48) = 200 (500) GeV.The resulting cross sections are shown in Table III,and the corresponding local signal significances with 3000fb − are summarized in Table IV. The local signifi-cances are slightly smaller than the corresponding onesin Table II, except in the case of m Z (cid:48) = 500 GeV and g bb = 2 g Lbs . Thus, we conclude that, like pp → Z (cid:48) + X , theprocess pp → Z (cid:48) b + X is likely be discovered at S g > g Lbb ≥ g Lbs in our benchmark points. By scaling thevalues in Table IV, we observe that a local significanceof 6 σ can be attained with | g Lbs | (cid:38) . . m Z (cid:48) = 200 (500) GeV, even if g Lbb = 0, at the cost of a ∼
30% enhancement in | M /M SM12 | . C. Observation of pp → Z (cid:48) b ¯ b + X and pp → Z (cid:48) b ¯ s + X The dominant SM backgrounds for pp → Z (cid:48) b ¯ b + X are pp → t ¯ t , pp → W t and DY + b or c jets. pp → m Z (cid:48) (GeV) σ signal (fb) σ background (fb) g Lbb = 0 g Lbb = g Lbs g Lbb = 2 g Lbs
DY + h.f. jets t ¯ t W t
200 0.00018 0.0025 0.0094 0.2 1.5 0.5500 0.00008 0.0006 0.0026 0.07 0.27 0.17TABLE V. Cross sections for the signal process pp → Z (cid:48) b ¯ b + X with Z (cid:48) → µ + µ − , and the dominant backgrounds after theevent selection for the benchmark points defined in Eq. (19) with the three choices for g Lbb . m Z (cid:48) (GeV) Local significance g Lbb = 0 g Lbb = g Lbs g Lbb = 2 g Lbs
200 0.007 0.1 0.35500 0.006 0.05 0.2TABLE VI. Local significance S l = N S / √ N B for discovery of the process pp → Z (cid:48) b ¯ b + X with Z (cid:48) → µ + µ − , with an integratedluminosity of 3000 fb − , given the signal and background cross sections shown in Table V. V V gives a negligible contribution. We adopt the samecorrection factors for the background cross sections andfollow the same event selection criteria as in Section IV C,and in addition require the subleading b jet to have p Tb >
30 GeV, | η b | < .
5, and to be separated from the leading b jet and each of the two leading muons by ∆ R > . Z (cid:48) masses.The resulting cross sections are shown in Table V. The g Lbb = 0 cases have tiny but nonzero cross sections, dueto production via b ¯ s → Z (cid:48) g ∗ , with g ∗ → b ¯ b . By con-trast, a nonzero g Lbb induces the less suppressed process gg → Z (cid:48) b ¯ b . The corresponding local signal significancesfor 3000 fb − are given in Table VI. As the local signifi-cances are much less than 1, we conclude that observationof this process is not possible at LHC within the rangeof couplings explored here.In general, | g Lbb | may take a larger value, up to the limitof Eq. (13), namely, | g Lbb | = 0 .
007 (0 . m Z (cid:48) = 200(500) GeV with g Lbs ∼
0. With these values, we esti-mate the cross section of pp → Z (cid:48) b ¯ b + X to be 0.098 fb(0.055 fb) after the event selection cuts. This correspondsto a local significance of around 3 . σ (4 . σ ) for an inte-grated luminosity of 3000 fb − . Thus, a global signifi-cance of 5 σ is not likely. We note, however, that sincewe used the same QCD correction factors for the back-ground cross sections as in the pp → Z (cid:48) b + X case, thereis a greater uncertainty on these cross sections.Generally, the cross section for pp → Z (cid:48) b ¯ b + X isstrongly suppressed by the 3-body phase space. Sincethe same suppression applies for pp → Z (cid:48) b ¯ s + X , one ex-pects the cross section for this process to be small as well.Moreover, the process pp → Z (cid:48) b ¯ s + X would also sufferfrom light-jet backgrounds which make the discovery notpossible, given the B s − ¯ B s mixing constraint. V. IMPACT OF THE RIGHT-HANDED Z (cid:48) bs COUPLING
In this section, we study an impact of a tiny butnonzero RH Z (cid:48) bs coupling by adding the following terms to the effective Lagrangian in Eq. (1):∆ L = − g Rbs (cid:0) ¯ bγ α P R s + ¯ sγ α P R b (cid:1) Z (cid:48) α . (21)The resulting additional contributions to b → sµ + µ − aredescribed by effective operators as in Eq. (2), with P L replaced by P R , and with C NP9 and C NP10 replaced by theWilson coefficients C (cid:48) = g Rbs g Vµµ N m Z (cid:48) , C (cid:48) = g Rbs g Aµµ N m Z (cid:48) . (22)There is no significant indication for nonzero C (cid:48) , in themajority of the b → s(cid:96)(cid:96) global fit analyses. However,even a tiny g Rbs can drastically affect the B s − ¯ B s mixing,which is now given by [31] M M SM12 = 1 + 1 m Z (cid:48) (cid:2) ( g Lbs ) − . g Lbs g Rbs + ( g Rbs ) (cid:3) × (cid:18) g π v ( V tb V ∗ ts ) S (cid:19) − , (23)calculated with the hadronic matrix elements inRef. [117]. The large negative coefficient of the g Lbs g Rbs term, which is partly due to renormalization group ef-fects [79], means that a small value of g Rbs allows for alarge g Lbs , due to cancellation between the terms. In Fig. 4we show the B s − ¯ B s -mixing constraint on g Rbs vs. g Lbs ,when | M | is allowed to change by up to 30% of its SMvalue. Reducing the allowed NP contribution to | M | ,say, to 15%, would narrow the width of the tilted-cross-shaped allowed region in Fig. 4. However, it would notchange the conclusion, namely, that a large value of | g Lbs | is allowed. What now becomes the most significant limiton g Lbs is the ATLAS dimuon resonance search [92], shownby the solid lines, assuming Eq. (7).The cancellation in M requires g Lbs g Rbs >
0. This im-plies (Re C NP9 )(Re C (cid:48) ) >
0, contrary to the best-fit val-ues for C NP9 and C (cid:48) , e.g., Refs. [15, 16]. However, thecancellation requires only g Rbs ∼ . g Lbs or C (cid:48) ∼ . C NP9 .While the fits favor C NP9 ∼ − - - - g bs L × g b s R × m Z ' =
200 GeV
15 10 - - - g bs L × g b s R × m Z ' =
500 GeV
FIG. 4. Constraints on g Rsb vs. g Lsb (in units of 10 − ) for m Z (cid:48) = 200 GeV [left] and 500 GeV [right], when allowing the B s − ¯ B s -mixing amplitude to deviate by up to 30% from its SM prediction. The gray regions are excluded. The solid linesshow the exclusion by the ATLAS dimuon resonance search [92] for B ( Z (cid:48) → µ + µ − ) (cid:39) /
3. The red dots show our benchmarkpoints. C (cid:48) = 0, they cannot exclude a small negative C (cid:48) . In-deed, the point ( C NP9 , C (cid:48) ) = ( − . , − .
1) is at the bor-der of the 1 σ ellipse in Ref. [15] with the assumption C NP10 = C (cid:48) = 0.To illustrate the impact of such a possibly large Z (cid:48) bs coupling on the Z (cid:48) discovery potential, we consider sce-nario (i) with the following benchmark points for m Z (cid:48) =200 and 500 GeV respectively (corresponding to the dotsin Fig. 4): (cid:12)(cid:12) g Lbs (cid:12)(cid:12) = 0 . , (cid:12)(cid:12) g Rbs (cid:12)(cid:12) = 0 . , (cid:12)(cid:12) g Vµµ (cid:12)(cid:12) = 0 . , (cid:12)(cid:12) g Lbs (cid:12)(cid:12) = 0 . , (cid:12)(cid:12) g Rbs (cid:12)(cid:12) = 0 . , (cid:12)(cid:12) g Vµµ (cid:12)(cid:12) = 0 . . (24)Both points correspond to ( C NP9 , C (cid:48) ) = ( − . , − . g Rbs on other constraints, e.g. B → K ( ∗ ) ν ¯ ν , are negligible. Taking g Lbb = 0, we find that the pp → Z (cid:48) + X and pp → Z (cid:48) b + X cross sections are highlyenhanced compared to the ones in Sec. IV, while the rel-atively large g Lbs values lead to a non-negligible Z (cid:48) → b ¯ s branching ratio, slightly reducing B ( Z (cid:48) → µ + µ − ) from ∼
66% to ∼
52% and ∼
55% respectively for 200 and 500GeV Z (cid:48) . m Z (cid:48) (GeV) Local significance µ + µ − + X µ + µ − b + X
200 9.3 7.6500 7.7 6.9TABLE VII. Local significance N S / √ N B for discovery of theprocess pp → Z (cid:48) + X → µ + µ − + X and pp → Z (cid:48) b + X → µ + µ − b + X with the integrated luminosity of 300 fb − , ob-tained by rescaling the results of Table II and IV to the bench-mark points defined in Eq. (24) for g Lbb = 0.
Taking account of these effects and rescaling the sig-nificances obtained in Sec. IV, we show in Table VII the local significances for discovery of pp → Z (cid:48) + X and pp → Z (cid:48) b + X with the integrated luminosity of 300 fb − ,for the two benchmark points defined above. The resultssuggest that both processes can be discovered with theRun-3 dataset, even if the Z (cid:48) bb coupling vanishes. Alarger (cid:12)(cid:12) g Lbs (cid:12)(cid:12) in this scenario enhances the cross section for pp → Z (cid:48) b ¯ s + X process, but the discovery would still bebeyond the reach of the HL-LHC. VI. SUMMARY AND DISCUSSIONS
Observed tensions in b → s(cid:96)(cid:96) measurements can beexplained by a new Z (cid:48) boson that couples to the left-handed b → s current as well as to muons. In this paper,we have studied the collider phenomenology of such a Z (cid:48) based on an effective model introduced in Eq. (1).For this purpose, we first estimated phenomenologicalconstraints on the Z (cid:48) bs and Z (cid:48) µµ couplings for the rep-resentative masses m Z (cid:48) = 200 and 500 GeV. The mostimportant constraint is the B s mixing, which tightly con-strains the LH Z (cid:48) bs coupling g Lbs . For fixed values of g Lbs and m Z (cid:48) , the allowed coupling to muons is determinedby global fits to b → s(cid:96)(cid:96) data, up to the value allowedby the constraint from the neutrino trident production,where the Z (cid:48) ν µ ν µ coupling is related to Z (cid:48) µµ coupling bythe SU(2) L symmetry. We also introduced the Z (cid:48) bb cou-pling g Lbb , which is mildly constrained by the dimuon res-onance search at LHC. The resulting couplings are suchthat the Z (cid:48) decays mostly to µ + µ − and ν µ ¯ ν µ , with thetwo branching ratio values mildly depending on whetherthe muon coupling is vector-like or left-handed.Given the coupling constraints, we explored the ca-pability of the 14 TeV LHC to discover the Z (cid:48) and todetermine the flavor structure of its couplings. For the1sake of this dual goal, we studied the two processes pp → Z (cid:48) + X → µ + µ − + X and pp → Z (cid:48) b + X → µ + µ − b + X ,where the former may be induced by b ¯ s → Z (cid:48) and/or b ¯ b → Z (cid:48) and the latter by gs → Z (cid:48) b and/or gb → Z (cid:48) b . Weconsidered two representative Z (cid:48) masses of 200 and 500GeV with three scenarios for the Z (cid:48) bb coupling: g Lbb = 0, g Lbs or 2 g Lbs . For g Lbb = 0, we found that discovery of pp → Z (cid:48) + X and pp → Z (cid:48) b + X (with about 5 σ globalsignificance) with 3000 fb − data requires a large Z (cid:48) bs coupling, so that the B s mixing amplitude M is en-hanced by ∼
30% or more relative to the SM expecta-tion. This corresponds roughly to the 2 σ upper limitsof the global analyses [83, 84] for the CKM parameters.With a nonzero g Lbb , discovery in both the modes is pos-sible without such a drastic effect on the B s mixing; inparticular, for g Lbb = 2 g Lbs , discovery is possible with a ∼
15% deviation in the M . For further discriminationbetween the Z (cid:48) bs and Z (cid:48) bb couplings, we also studied theprocess pp → Z (cid:48) b ¯ b + X → µ + µ − b ¯ b + X , predominantlyarising from Z (cid:48) bb coupling. However, we found it to benot promising even with 3000 fb − integrated luminos-ity, due primarily to three-body phase-space suppression.The same conclusion applies to pp → Z (cid:48) b ¯ s + X , whichgives direct access to the Z (cid:48) bs coupling.The discovery potential of the Z (cid:48) is rather limited dueto the B s mixing constraint. The B s mixing constraint,however, is only indirect and is susceptible to the detailsof the UV completion of the effective model. In particu-lar, we illustrated that the existence of a tiny but nonzeroright-handed Z (cid:48) bs coupling g Rbs accommodates a large LH Z (cid:48) bs coupling due to the cancellation in the B s mixingamplitude, without conflicting with the b → s(cid:96)(cid:96) globalfits. In this case we found that discovery in both the pp → Z (cid:48) + X and pp → Z (cid:48) b + X processes may occureven with O (100) fb − integrated luminosity.Comments on the subtlety of the implementation ofthe B s mixing constraint are in order (see also Sec. III).As mentioned above, a 30% enhancement in the B s mix-ing amplitude M by NP roughly corresponds to the2 σ upper limits by the latest global analyses of CKM-fitter [83] and UTfit [84]. This may look rather toler-ant, in view of the recent progress [86] in the estima-tion of the hadronic matrix element by lattice, whichlead to ∼
6% uncertainty in M SM12 [85]. This is becausethe central values of | M /M SM12 | are greater than unityin Ref. [83] and Ref. [84], while the Z (cid:48) contribution al-ways enhances | M | relative to the SM value under theassumption of a real-valued g Lbs with g Rbs = 0. On theother hand, a recent study [77] finds the SM predictionof ∆ m B s = 2 | M | to be 1.8 σ above the measured value,favoring | M /M SM12 | smaller than unity. This is oppo-site to the results by CKMfitter [83] and UTfit [84], al-though both UTfit (Summer 2016 result) and Ref. [77]take into account the recent lattice result [86]. If theresult of Ref. [77] is the case, the Z (cid:48) contribution mayenhance | M | only up to ∼
1% so that g Lbs is stronglyconstrained. In this case the estimated signal signifi-cances at LHC would shrink down to insignificant values for | g Lbb | (cid:46) | g Lbs | , unless a tiny RH Z (cid:48) bs coupling exists forthe cancellation in M and/or g Lbs is close to pure imagi-nary so that it gives a negative contribution in M . Thelatter implies a nearly imaginary C NP9 and would needa dedicated global analysis of b → s(cid:96)(cid:96) observables, asdiscussed in Ref. [77]. In any case, a consensus amongthe different groups seems to be still missing for the pre-diction of M in the SM, and a better understandingwould be required for its calculation. At the same time,improvements in lattice calculations and determinationsof CKM parameters will also facilitate a more precise SMprediction for M .Although we considered Z (cid:48) bb and Z (cid:48) bs couplings tobe the only couplings to the quark sector, the Z (cid:48) mayalso couple to other quarks in general. For instance, if anon zero Z (cid:48) cc coupling exists, the process pp → Z (cid:48) c + X can be induced at LHC. Such a process can mimic the pp → Z (cid:48) b + X signature if the final state c -jet getsmisidentified as b -jet. This possibility can not be ex-cluded yet as pointed out in Ref. [118], where a proce-dure to disentangle pp → Z (cid:48) c + X and pp → Z (cid:48) b + X is discussed with the simultaneous application of both c -and b -tagging. We also remark that our estimation ofthe signal significances ignored various experimental un-certainties and the QCD corrections to the signal crosssections.For illustration, we focused on m Z (cid:48) = 200 and 500GeV. In general, heavier Z (cid:48) are possible. However, dueto the fall in the parton luminosity with the resonancemass, the achievable significances are lower than those of200 GeV and 500 GeV in both the pp → Z (cid:48) and pp → Z (cid:48) b processes.Our results illustrate three possible scenarios for theLHC discovery and identification of a Z (cid:48) that might bebehind the b → s(cid:96)(cid:96) anomalies. The first one is the casewith the minimal assumption, where the LH Z (cid:48) bs cou-pling is the only coupling to the quark sector. In thiscase, the discovery of the pp → Z (cid:48) + X → µ + µ − + X and pp → Z (cid:48) b + X → µ + µ − b + X processes may occur withthe full HL-LHC data, but should be accompanied by a ∼
30% or larger enhancement in the B s mixing, whichcan be tested following future improvements in the esti-mation of the B s mixing. The second one is the case witha tiny but nonzero RH Z (cid:48) bs coupling such that the B s mixing remains SM-like due to the cancellation of the Z (cid:48) effects. In this case, the discovery of the two modes mayoccur with Run-3 data (or perhaps even Run-2 data);this scenario predicts a nonzero RH b → s current, with C (cid:48) ∼ . C NP9 , which can be tested with improvements in b → s(cid:96)(cid:96) measurements by ATLAS, CMS, LHCb and BelleII. In particular, precise measurements of R K and R K ∗ by LHCb with Run-2 or further dataset may pin downthe chiral structure of the b → s current. The third sce-nario is the case with a flavor-conserving Z (cid:48) bb couplingmuch larger than Z (cid:48) bs . In this case, the two modes maybe discovered with Run-3 data without a significant ef-fect in the B s mixing and RH b → s current, but the roleof the observed resonance in b → s(cid:96)(cid:96) is obscured.2 Note added:
While revising the manuscript we no-ticed that the CMS 13 TeV 36 fb − result [119] for aheavy resonance search in the dilepton final state is nowavailable. We find that the extracted upper limits [120]on g Lbb from Ref. [119] are comparable to those from AT-LAS [92] and do not change the conclusion of our results.
ACKNOWLEDGMENTS
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If the Z (cid:48) width is narrow,the R σ can be interpreted as the limits on σ ( pp → Z (cid:48) + X ) · B ( Z (cid:48) → µ + µ − ), with the multiplication of the SMprediction of σ ( pp → Z + X ) B ( Z → µ + µ − ) = 1928 ..