Probe a family non-universal Z boson effects ′ ¯ in B φµ+µ decay s − → 1 1 0 Qin Changa,c, Yin-Hao Gaob 2 ∗ n a a Department of Physics, Henan Normal University, Xinxiang, Henan 453007, P. R. China J 6 b Henan Institute of Science and Technology, Xinxiang 453003, P. R. China ] h c Institute of Particle Physics, Huazhong Normal University, Wuhan, Hubei 430079, P. R. China p - p e Abstract h [ 1 Motivated by the recent measurement on (B¯ φµ+µ−) by CDF collaboration, we s v B → 2 study the effects of a family non-universal Z′ boson on rare semileptonic B¯ φµ+µ− s 7 → 2 decay. In our evaluations, we analyze the dependences of the dimuon invariant mass 1 ′ . spectrum and normalized forward-backward asymmetry on Z couplings and show that 1 0 ′ theseobservablesarehighlysensitivetonewZ contributions. Threelimitingscenariosare 1 1 ′ presented in the detailed analyses. Numerically, within the allowed ranges of Z couplings : v i under the constraints from B¯ B mixing, B πK, B¯ (X ,K,K∗)µ+µ− decays X s s d s − → → r and so on, (B¯ φµ+µ−) and A(L)(B¯ φµ+µ−) could be enhanced by about 96% a B s → FB s → and 17%(133%) respectively at most by Z′ contributions. However, (B¯ φµ+µ−) is s B → hardly to be reduced. Furthermore, the zero crossing in A (B¯ φµ+µ−) spectrum at FB s → low dimuon mass always exists. Keywords: B-physics, Rare decays, Beyond Standard Model ∗ Corresponding author. [email protected] 1 1 Introduction Rare B decays induced by the flavor-changing neutral current (FCNC) occur at looplevel in the Standard Model (SM) and thus proceed at a low rate. They can provide useful information on the parameters of the SM and test its predictions. Meanwhile, they offer a valuable possibility of an indirect search of new physics (NP) for their sensitivity to the gauge structure and new contributions. Experimentally, the fruitful running of BABAR, Belle and Tevatron in the past decade providesaveryfertilegroundfortesting SMandprobingpossible NPeffects. Asparticle physics is entering the era of LHC, B physics has attracted much more attention. s Recently, CDF collaboration has reported the first observation of the rare semileptonic B¯ φµ+µ− decay and measured its branching fraction to be [1] s → (B¯ φµ+µ−) = [1.44 0.33(stat.) 0.46(syst.)] 10−6 CDF. (1) s B → ± ± × Theoretically, many evaluations for B¯ φµ+µ− decay have been done within both SM and s → various NP scenarios (for example, Refs. [2, 3]). The SM prediction for SM(B¯ φµ+µ−) ( s B → ∼ 1.65 10−6(QCDSR)[2], forexample)agreeswell withCDFmeasurement (1.44 0.57) 10−6 for × ± × large experimental error. If more exact measurement on B¯ φµ+µ− is gotten by the running s → LHC-b and future super-B, the possible NP space will be strongly constrained or excluded. So, ′ it is worth evaluating the effects of the possible NP, such as a family non-universal Z boson, on B¯ φµ+µ− decay. s → ′ A new family non-universal Z boson could be naturally derived in certain string construc- ′ tions [4], E models [5] and so on. Searching for such an extra Z boson is an important mission 6 in the experimental programs of Tevatron [6] and LHC [7]. The general framework for non- ′ universal Z model has been developed in Ref. [8]. Within such model, FCNC in b s and → ′ d transitions could be induced by family non-universal U(1) gauge symmetries at tree level. Its effects on b s transition have attracted much more attention and been widely studied. → ′ Interestingly, the behavior of a family non-universal Z boson is helpful to resolve many puzzles in B decays, such as “πK puzzle” [9, 10], anomalous B¯ B mixing phase [11, 12] and (u,d,s) s s − mismatch in A (B K∗µ+µ−) spectrum at low q2 region [13, 14]. FB → Within a family non-universal Z′ model, B¯ φµ+µ− decay involves b s Z′ and s → − − µ µ Z′ couplings, which have been strictly bounded by the constraints from B¯ B mixing, s s − − − 2 B πK(∗), ρK, B¯ X µµ, K(∗)µµ decays and so on [10, 12, 13]. So, it is worth evaluating d s → → the effects ofa non-universal Z′ bosononB¯ φµ+µ− decay andchecking whether such settled s → values of Z′ couplings are permitted by CDF measurement on (B¯ φµ+µ−). s B → Our paper is organized as follows. In Section 2, we briefly review the theoretical framework for b sl+l− decay within both SM and a family non-universal Z′ model. In Section 3, the → effects of a non-universal Z′ boson on B¯ φµ+µ− decay are investigated in detail. Our s → conclusions are summarized in Section 4. Appendix A and B include all of the theoretical input parameters. 2 The theoretical framework for b sl+l decays − → In the SM, neglecting the doubly Cabibbo-suppressed contributions, the effective Hamiltonian governing semileptonic b sℓ+ℓ− transition is given by [15, 16] → 10 4G F ∗ = V V C (µ)O (µ). (2) Heff − √2 tb ts i i i=1 X Here we choose the operator basis given by Ref. [15], in which e2 e2 O = (d¯γ P b)(¯lγµl), O = (d¯γ P b)(¯lγµγ l). (3) 9 µ L 10 µ L 5 g2 g2 s s Wilson coefficients C can be calculated perturbatively [17, 18, 19, 20], with the numerical i results listed in Table 1. The effective coefficients Ceff, which are particular combinations of 7,9 C with the other C , are defined as [15] 7,9 i 4π 1 4 20 80 Ceff = C C C C C , 7 α 7 − 3 3 − 9 4 − 3 5 − 9 6 s 4π 4π Ceff = C +Y(q2), Ceff = C , (4) 9 α 9 10 α 10 s s in which Y(q2) denotes the matrix element of four-quark operators and given by 4 1 4 64 Y(q2) = h(q2,m ) C +C +6C +60C h(q2,m ) 7C + C +76C + C c 1 2 3 5 b 3 4 5 6 3 − 2 3 3 1 (cid:0) 4 64(cid:1) 4 6(cid:0)4 64 (cid:1) h(q2,0) C + C +16C + C + C + C + C . (5) 3 4 5 6 3 5 6 −2 3 3 3 9 27 (cid:0) (cid:1) ′ We have neglected the long-distance contribution mainly due to J/Ψ and Ψ in the decay chain B¯ φΨ(′) φl+l−, which could be vetoed experimentally [1]. For the recent detailed s → → discussion of such resonance effects, we refer to Ref. [21]. 3 Table 1: The SM Wilson coefficients at the scale µ = m . b C (m ) C (m ) C (m ) C (m ) C (m ) C (m ) Ceff(m ) Ceff(m ) Y(q2) Ceff(m ) 1 b 2 b 3 b 4 b 5 b 6 b 7 b 9 b − 10 b 0.284 1.007 0.004 0.078 0.000 0.001 0.303 4.095 4.153 − − − − − Althoughtherearequite alot ofinteresting observables insemileptonic b sℓ+ℓ− decay, we → shall focus only on the dilepton invariant mass spectrum and the forward-backward asymmetry in this paper. Adopting the same convention and notation as [22], the dilepton invariant mass spectrum and forward-backward asymmetry for B¯ φℓ+ℓ− decay is given as s → dΓφ G2 α2m5 A 2 mˆ2 uˆ(sˆ)2 = F Bs V∗V 2 uˆ(sˆ) | | sˆλ(1+2 ℓ)+ E 2sˆ dsˆ 210π5 | ts tb| 3 sˆ | | 3 ( 1 uˆ(sˆ)2 uˆ(sˆ)2 + B 2(λ +8mˆ2(sˆ+2mˆ2))+ F 2(λ +8mˆ2(sˆ 4mˆ2)) 4mˆ2 | | − 3 φ ℓ | | − 3 φ − ℓ φ (cid:20) (cid:21) λ uˆ(sˆ)2 uˆ(sˆ)2 + C 2(λ )+ G 2 λ +4mˆ2(2+2mˆ2 sˆ) 4mˆ2 | | − 3 | | − 3 ℓ φ − φ (cid:20) (cid:18) (cid:19)(cid:21) 1 uˆ(sˆ)2 Re(BC∗)(λ )(1 mˆ2 sˆ) −2mˆ2 − 3 − φ − φ (cid:20) uˆ(sˆ)2 + Re(FG∗)((λ )(1 mˆ2 sˆ)+4mˆ2λ) − 3 − φ − ℓ # mˆ2 mˆ2 2 ℓ λ Re(FH∗) Re(GH∗)(1 mˆ2) + ℓsˆλ H 2 ; (6) − mˆ2 − − φ mˆ2 | | φ φ ) (cid:2) (cid:3) d φ G2 α2m5 AFB = F Bs V∗V 2 sˆuˆ(sˆ)2 dsˆ − 28π5 | ts tb| Re(C effCeff∗)VA + mˆbRe(C effCeff∗) VT (1 mˆ )+A T (1+mˆ ) , (7) × 9 10 1 sˆ 7 10 2 − φ 1 1 φ (cid:20) (cid:21) (cid:16) (cid:17) with s = q2 and sˆ = s/m2 . Here the auxiliary functions A,B,C,E,F and G, with the Bs explicit expressions given in Ref. [22], are combinations of the effective Wilson coefficients in Eq. (4) and the B φ transition form factors, which are calculated with light-cone QCD sum s → rule approach in Ref. [23] and given in Appendix B. From the experimental point of view, the normalized forward-backward asymmetry is more useful, which is defined as [22] d ¯ d dΓ FB FB A = A / . (8) dsˆ dsˆ dsˆ 4 ′ A new family non-universal Z boson could be naturally derived in many extension of SM. ′ ′ One of the possible way to get such non-universal Z boson is to include an addition U (1) gauge symmetry, which has been formulated in detail by Langacker and Plu¨macher [8]. Under ′ the assumption that the couplings of right-handed quark flavors with Z boson are diagonal, the Z′ part of the effective Hamiltonian for b sl+l− transition can be written as [11] → 2G BLBL BLBR HeZf′f(b → sl+l−) = − √2FVtbVt∗s − VsbV∗ll(s¯b)V−A(¯ll)V−A− VsbVl∗l (s¯b)V−A(¯ll)V+A +h.c.. (9) tb ts tb ts h i With the assumption that no significant RG running effect between MZ′ and MW scales, ′ ′ Z contributions could be treated as modification to wilson coefficients, i.e. C (M ) = 9,10 W CSM(M )+ C′(M ). As a result, Eq. (9) could also be reformulated as 9,10 W △ 9 W 4G Z′ (b sl+l−) = FV V∗[ C′O + C′ O ]+h.c., (10) Heff → − √2 tb ts △ 9 9 △ 10 10 with g2 BL C′(M ) = s sb SLR, SLR = (BL +BR), △ 9 W −e2 V∗V ll ll ll ll ts tb g2 BL C′ (M ) = s sb DLR, DLR = (BL BR). (11) △ 10 W e2 V∗V ll ll ll − ll ts tb BL and BL,R denote the effective chiral Z′ couplings to quarks and leptons, in which the sb ll off-diagonal element BL can contain a new weak phase and could be written as BL eiφLs. sb | sb| ′ To include Z contributions, one just needs to make the replacements 4π Ceff C¯eff = C′ +Y(q2) , 9 → 9 α 9 s 4π Ceff C¯eff = C′ , (12) 10 → 10 α 10 s in the formalisms relevant to B¯ φℓ+ℓ−. s → 3 Numerical analyses and discussions With the relevant theoretical formulas collected in Section 2 and the input parameters summa- rized in the Appendix, we now proceed to present our numerical analyses and discussions. In Table 2, we present our theoretical predictions for integrated branching fraction and forward-backward asymmetry of B¯ φµ+µ− decay. Within the SM, we again find our pre- s → diction SM(B¯ φµ+µ−) = 1.46 10−6 is perfectly consistent with CDF measurement s B → × 5 Table 2: Predictions for (B¯ φµ+µ−)[ 10−6] and A (B¯ φµ+µ−)[ 10−2] within the s FB s B → × → × ′ SM and the non-universal Z model. Exp. [24] SM S1 S2 Scen. I Scen. II Scen. III 1.44 0.57 1.46 0.10 2.47 1.18 1.40 0.27 2.86 1.26 1.92 B ± ± ± ± L — 0.34 0.04 0.56 0.27 2.61 0.19 0.64 0.28 0.44 B ± ± ± H — 0.29 0.02 0.51 0.25 1.26 0.08 0.59 0.26 0.39 B ± ± ± A — 25.6 1.2 19.4 10.9 24 0.03 29.9 26.6 8.9 FB ± ± ± AL — 5.7 0.6 6.0 7.4 0.09 0.02 13.3 6.9 1.4 FB ± ± ± AH — 34.1 0.2 22.5 12.9 0.07 0.01 35.0 34.8 13.1 FB ± ± − ± 0.30 0.6 -6D 0.25 0.4 0 +-L(cid:144)@®ΦΜΜ´Bds1s 000...112050 SScceenSnaMarrioioIIII +-LH®ΦΜΜABsFB -000...022 ScenarSioMI SScceennaarriiooIIIII BH 0.05 ScenarioII d -0.4 0.00 0 5 10 15 20 0 5 10 15 20 s@GeV2D s@GeV2D HaL HbL Figure 1: Dimuon invariant mass distribution and normalized forward-backward asymmetry of the B¯ φµ+µ− decay within SM and three limiting scenarios. s → (1.44 0.57) 10−6. The forward-backward asymmetry for B¯ φµ+µ− decay is eval- s ± × → uated at 25%, which hasn’t be measured by the experiment. In addition, in Table 2, ∼ we also calculate their results L,H and AL,H at both low (1GeV2 < s < 6GeV2) and B FB high (14.4GeV2 < s < 25GeV2) integration regions, which are sufficiently below and above ′ the threshold for charmonium resonances J/ψ,ψ respectively. The dimuon invariant mass dis- tribution and forward-backward asymmetry spectrum are shown in Fig. 1. As Fig. 1(b) shows, similar to the situation in B¯0 K∗µ+µ− decay, the zero crossing exist in A spectrum at FB → s 3 GeV2, whose position is well-determined and free from hadronic uncertainties at the 0 ∼ leading order in α [17, 22, 25]. In B¯0 K∗µ+µ− decay, the A spectrum measured by Belle s FB → collaboration [26] indicates that there might be no zero crossing, which presents a challenge to 6 ′ Table 3: The inputs parameters for the Z couplings [12, 13]. BL ( 10−3) φL[◦] SLR( 10−2) DLR( 10−2) | sb| × s µµ × µµ × S1 1.09 0.22 72 7 2.8 3.9 6.7 2.6 ± − ± − ± − ± S2 2.20 0.15 82 4 1.2 1.4 2.5 0.9 ± − ± − ± − ± the SM in low s region. If the future measurement on A (B¯ φµ+µ−) spectrum presents a FB s → similar result as the one in B¯0 K∗µ+µ− decay, it will be a significant NP signal. → Within a family non-universal Z′ model, the Z′ contributions to B¯ φµ+µ− decay involve s → four new Z′ parameters BL , φL, SLR and DLR. Combining the constraints from B¯ B | sb| s µµ µµ s − s mixing, B πK(∗) and ρK decays, BL and φL have been strictly constrained [10, 12]. After → | sb| s having included the constraints from B¯ X µµ, Kµµ and K∗µµ, as well as B µµ decays, d s s → → we have also gotten the allowed ranges for SLR and DLR in Ref. [13]. For convenience, we µµ µµ recollect their numerical results in Table 3, in which S1 and S2 correspond to UTfit collabora- tion’s two fitting results for B¯ B mixing [27]. Our following evaluations and discussions are s s − ′ ′ based on these given ranges for Z couplings. With the values of Z parameters listed in Table 3 as inputs, we present our predictions for the observables in the third and fourth columns of Table 2. As illustrated in Fig. 2, integrated branching fraction for B¯ φµ+µ− is sensitive to the Z′ s → contributions. Obviously, B¯ φµ+µ− is enhanced by the Z′ contributions with large negative s → SLR, DLR and φL. Moreover, compared Fig. 2 (a,b) with (c,d), we find the effects of solution µµ µµ s S1 is more significant than the one of S2. So, for simplicity, we just pay our attention to the ′ solution S1 in the following. As Fig. 2 shows, the Z contributions with a small negative weak phase φL are helpful to reduce (B¯ φµ+µ−). However, because the range φL > 65◦ is s B s → s − excluded by the constraints from B¯ B mixing and B πK decays [10, 12], (B¯ φµ+µ−) s s s − → B → ′ is hardly to be reduced so much by Z contributions. In order to see the Z′ effect on A (B¯ φµ+µ−) explicitly, with Y(q2) being excluded, FB s → we can rewrite Re(C¯effC¯eff∗) and Re(C¯effC¯eff∗) in Eq. (7), which dominates A (B¯ φµ+µ−) 9 10 7 10 FB s → 7 -6+-L@D®ΦΜΜ´10 223...050 S1:ÈBsLbÈ=ΦΦΦ1sssLLL.0===9---´67715290ëëë-:::3,DLΜΜR=0 -6+-L@D®ΦΜΜ´10 223...050 S1:ÈBsLbÈ=ΦΦΦ1sssLLL.0===9---´67715290ëëë:::-3,SΜLΜR=0 BHBs 1.5 BHBs 1.5 1.0 1.0 -6 -4 -2 0 -9 -8 -7 -6 -5 -4 SLR@´10-2D DLR@´10-2D ΜΜ ΜΜ (a) (b) 3.0 S2:ÈBsLbÈ=2.20´10-3,DLΜΜR=0 3.0 S2:ÈBsLbÈ=2.20´10-3,SΜLΜR=0 ΦL=-78ë: ΦL=-78ë: -6@D´10 2.5 ΦΦsssLL==--8826ëë:: -6@D´10 2.5 ΦΦsssLL==--8826ëë:: -LΜ 2.0 -LΜ 2.0 + + Μ Μ Φ Φ ® ® Bs 1.5 Bs 1.5 BH BH 1.0 1.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 -3.0 -2.5 -2.0 SLR@´10-2D DLR@´10-2D ΜΜ ΜΜ (c) (d) Figure 2: The dependence of (B¯ φµ+µ−) on SLR and DLR within their allowed ranges in s µµ µµ B → S1 and S2 with different φL values. The black dashed line corresponds to the SM result. s (a) (b) Figure 3: The dependence of A (B¯ φµ+µ−) on SL,R and DL,R at s = 1.5GeV2 (a) and FB s → ud ud s = 15GeV2 (b) with BL = 1.09( 10−3), φL = 72◦ (S1) and the central values of the other | db| × s − theoretical input parameters. The blue planes correspond to SM results. in high and low s regions respectively, as 2 4π Re(C¯effC¯eff∗) Re(C¯eff)Re(C¯eff∗)+ Im( C′)Im( C′ ), (13) 9 10 ≃ 9 10 α △ 9 △ 10 (cid:18) s(cid:19) Re(C¯effC¯eff∗) Re(C¯eff)Re(C¯eff∗). (14) 7 10 ≃ 7 10 8 Combining Eq. (11) and Eq. (14), due to the tiny Z′ contribution to Ceff, the only solution to 7 enhance A in low s region is a larger negative DLR, which also can be found in Fig. 3(a). In FB µµ high s region, as Fig. 3 (b) shows, A could be reduced significantly and enhanced a bit by FB ′ Z contributions. ′ Based on the analyses above, in order to evaluate the exact strength of Z effects, our following analyses can be divided into three limiting scenarios: Scenario I In order to get the maximum (B¯ φµ+µ−), within the allowed ranges for Z′ couplings listed s B → in Table 3, we choose a set of extreme values BL = 1.31 10−3,φL = 79◦,SLR = 6.7 10−2,DLR = 9.3 10−2 Scen. I, (15) | sb| × s − µµ − × µµ − × named Scenario I. With the central values of the other theoretical input parameters, we get (B¯ φµ+µ−) = 2.86 10−6, which is 2.5σ larger than CDF result (1.44 0.57) 10−6. s B → × ± × Compared with the SM prediction 1.46 10−6, we find (B¯ φµ+µ−) could be enhanced by s × B → ′ about 96% at most by Z contributions. This scenario is the most helpful solution to moderate the discrepancy for A (B¯ FB s → K∗µ+µ−) between SM prediction and experimental data in low s region [13, 14]. As Fig. 3 (a) shows, we find Scenario I also provides the most helpful solution to enhance A (B¯ φµ+µ−) FB s → in low s region. Compared with the SM results, we find A(L)(B¯ φµ+µ−) could be enhanced FB s → by about 17%(133.3%) at most. However, in the high s region, the effect of Scenario I on A (B¯ φµ+µ−), as Fig. 3 (b) shows, is not significant. FB s → In addition, due to the strong constraints on DLR from B¯ X µµ decay, the much larger µµ d → s value DLR > 9.3 10−2 is forbidden [12], which means the sign of Re(C¯effC¯eff∗) can hardly | µµ| × 7 10 ′ be flipped by Z contributions [13]. So, as Fig. 1 (b) shows, the zero crossing in A spectrum FB also exists and moves to s 1GeV2 point in this scenario. 0 ∼ Scenario II From Fig. 2, one may find that (B¯ φµ+µ−) can hardly be reduced by Z′ contributions so s B → much within the allowed Z′ parameters’ ranges. The most minimal value of (B¯ φµ+µ−) s B → 9 appears at BL = 1.31 10−3,φL = 65◦,SLR = 2 10−2,DLR = 4 10−2 Scen. II, (16) | sb| × s − µµ − × µµ − × named Scenario II. In this scenario, compared with SM prediction, we find (B¯ φµ+µ−) s B → ′ ′ could be reduced just by about 14% at most by Z contributions. Due to the small Z contri- butions, its effect on A (B¯ φµ+µ−) is also tiny. FB s → Scenario III As Fig. 3 (b) shows, A (B¯ φµ+µ−) would be reduced rapidly in high s region when SLR FB s → µµ is enlarged. So, we present a limiting scenario for the minimal AH (B¯ φµ+µ−), FB s → BL = 1.31 10−3,φL = 65◦,SLR = 1.1 10−2,DLR = 9.3 10−2 Scen. III, (17) | sb| × s − µµ × µµ − × named Scenario III. Compared with SM prediction, A(H)(B¯ φµ+µ−) is reduced by about FB s → 62% (62%). However, as Fig. 3 (b) shows, in the low s region, A is just enhanced a bit. FB So, this scenario also leads to the minimal A (B¯ φµ+µ−) 8.9%, which is 65% smaller FB s → ∼ than SM prediction. While, in this scenario, our prediction (B¯ φµ+µ−) = 1.92 10−6 also s B → × agrees with CDF measurement within 1σ. So, although Scenario III presents a strange effects on A spectrum, it is not excluded by current measurement either. Moreover, different from FB Scenario I, zero crossing in A spectrum moves to positive side in this scenario. FB 4 Conclusion In conclusion, motivated by recent measurement on (B¯ φµ+µ−) by CDF Collaboration, s B → after revisiting B¯ φµ+µ− decay within SM, we have investigated the effects of a family s → ′ ′ non-universal Z boson with the given Z couplings. Our conclusions can be summarized as: Branching fractionandforward-backward asymmetryforB¯ φµ+µ− decayaresensitive s • → ′ ′ to Z contributions. All of the Z couplings listed in Table 3 survive under the constraint from (B¯ φµ+µ−) measured by CDF within errors. s B → We present three limiting scenarios: (B¯ π−K+) and A(L)(B¯ φµ+µ−) could • B s → FB s → ′ be enhanced by about 96% and 17%(133%) at most by Z contributions (Scenario I); 10