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PHYSICAL REVIEW D 100, 115014 (2019)

Correlating tauonic B decays with the electric dipole moment via a scalar leptoquark

Andreas Crivellin* Paul Scherrer Institut, CH–5232 Villigen PSI, Switzerland and Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH–8057 Zürich, Switzerland † Francesco Saturnino Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern, CH-3012 Bern, Switzerland

(Received 27 May 2019; published 6 December 2019)

In this article we investigate the correlations between tauonic B decays (e.g., B → τν, B → DðÞτν, B → πτν) and electric dipole moments (EDMs), in particular the one of the neutron, in the context of the S1 scalar leptoquark. We perform the matching of this model on the effective field theory taking into account the leading renormalization group effect for the relevant observables. We find that one can explain the hints for new physics in b → cτν transitions without violating bounds from other observables. Even more interesting, it can also give sizable effects in B → τν, to be tested at BELLE II, which are correlated to (chromo)electric dipole operators receiving mτ=mu enhanced contributions. Therefore, given a deviation from the (SM) expectations in B → τν, this model predicts a sizable neutron EDM. In fact, even if new physics has CP conserving real couplings, the Cabibbo-Kobayashi-Maskawa matrix induces a complex phase and already a 10% change of the B → τν branching ratio (with respect to the SM) will lead to an effect observable with the n2EDM experiment at Paul Scherrer Institut.

DOI: 10.1103/PhysRevD.100.115014

I. INTRODUCTION decouples with the NP scale which is a priori unknown, unless new , or at least deviations from the SM in In the past four decades, the Standard Model (SM) of other precision observables, are found. physics has been extensively tested and its pre- In this respect, tauonic B decays are very promising dictions were very successfully confirmed, both in high channels for the (indirect) search for NP, especially in the energy searches as well as in low energy precision experi- light of the observed tensions between the SM predictions ments. However, it is well known that the SM cannot be the and experiments above the 3σ level [10]. These decays ultimate theory describing the fundamental constituents of involve both down-type and charged of the and their interactions. For example, it cannot third generation (i.e., bottom quarks and leptons) which accommodate for the observed matter–antimatter asymme- are, due to their mass, very special and distinct from the try in the universe: For satisfying the Sakharov conditions of the first two generations.1 In fact, to explain these [1] the amount of CP violation within the SM is far too anomalies, TeV scale NP with order one couplings to the third small [2–7]. Therefore, additional sources of CP violation generation is required. Note that the tensions in b → cτν are required and such models in general lead to non- transitions are supported by b → uτν data (i.e., B → πτν and vanishing electric dipole moments of neutral fermions. B → τν) and the forthcoming measurements of both b → cτν Thus, EDMs are very promising places to search for and b → uτν processes by LHCb and BELLE II will be able physics beyond the SM (see e.g., Ref. [8,9] for a recent to confirm (or disprove) the presence of NP in these decays. review). However, the effect of new physics (NP) in EDMs Therefore, it is very interesting to investigate the possible impact of models which can give sizable effects in tauonic * B decays and EDMs. In this paper we choose the scalar andreas.crivellin@.ch † 2 [email protected] leptoquark S1 SUð ÞL singlet which couples to SM fermions via the Lagrangian Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to 1In group theory language, the SM possesses a global Uð3Þ5 the author(s) and the published article’s title, journal citation, flavor symmetry which is broken by the third generation Yukawa and DOI. Funded by SCOAP3. couplings to Uð2Þ5 [11].

2470-0010=2019=100(11)=115014(9) 115014-1 Published by the American Physical Society ANDREAS CRIVELLIN and FRANCESCO SATURNINO PHYS. REV. D 100, 115014 (2019)

Here, we work in the down basis, meaning that the CKM matrix V appears in the couplings to left-handed up-type quarks. We denote the mass of the LQ by M and neglect its couplings to the SM Higgs which have a negligible phenomenological impact. The most relevant classes of observables in our model are b → sνν and b → cðuÞτν transitions as well as EDMs, D0 − D¯ 0 mixing, and Z-ττ as well as W-τν couplings which we consider now in more FIG. 1. Feynman diagram showing the contribution of our detail. model to the dipole operators of Eq. (10). The cross denotes the chirality flip by the tau mass which leads to the crucial mτ=mu enhancement. A. b → sνν For b → sνν transitions we follow the conventions of L λL c τ λR cl Φ† Ref. [44], ¼ð fiQfi 2Li þ fiuf iÞ 1 þ H:c: ð1Þ 4G Here, L (Qc) is the (charge conjugated ) SUð2Þ Hνν ¼ − pffiffiffiF V V ðCfi Ofi þ Cfi Ofi Þ; L eff tdk tdj L;jk L;jk R;jk R;jk doublet, l (uc) the charged lepton (charge conjugated up 2 α fi ¯ μ quark) singlet, and f, i are flavor indices. This model is O ¼ ½d γ P d ½ν¯ γμð1 − γ5Þν ; ð2Þ LðRÞ;jk 4π j LðRÞ k f i theoretically well motivated since S1 is present within the R-parity violating minimal supersymmetric model (MSSM) in the form of right-handed down squarks [12–16].2 and obtain, already at tree level, the contribution This leptoquark (LQ) is a prime candidate for providing pffiffiffi 2 π λLλL the desired correlations between tauonic B decays and CfiNP ¼ jf ki : ð3Þ L;jk 4G V V α M2 EDMs. It possesses couplings to left- and right-handed F tdk tdj quarks which is a necessary requirement for generating EDMs at the one-loop level [18,19]. It also contributes to Here the most relevant decays are B → KðÞνν for which → τν – b c at tree level [20 39] and gives a very good fit to CSM;fi ≈−1.47=s2 δ and branching ratios, normalized by – L;sb W fi data (including polarization observables) [40 43] since it the corresponding SM predictions, read generates vector, scalar, and tensor operators. Similarly, it contributes to b → uτν transitions, in particular to B → τν, 1 X3 jCfi j2 where the situation becomes especially interesting. As we Rνν¯ ¼ L;sb : ð4Þ KðÞ 3 SM;ii 2 will see, in this case the model leads to mτ=mu enhanced f;i¼1 jCL;sb j CP violating effects in (chromo)electric dipole operators (see Fig. 1) which are even present for real NP parameters This has to be compared to the current experimental limits νν¯ 3 9 νν¯ 2 7 due to the large phase contained in the Cabibbo-Kobayashi- RK < . and RK < . [45] (both at 90% C.L.). The Maskawa (CKM) element Vub. future BELLE II sensitivity for B → KðÞνν¯ is 30% of the This paper is structured as follows: In the next section SM branching ratio [46]. we will calculate the contributions to the relevant observ- ables and discuss their experimental status. Section III b → c u τν presents our phenomenological analysis before we con- B. ð Þ clude in Sec. IV. For tauonic B decays we define the effective Hamiltonian as II. OBSERVABLES AND CONTRIBUTIONS 4G Hτν ¼ pffiffiffiF V ðCf Of þ Cf Of þ Cf Of Þ; In this section we discuss our setup, calculate the eff 2 ufb VL VL SL SL TL TL predictions for the relevant observables, and discuss their current experimental situation and future prospects. with the operators given by After electroweak symmetry breaking, the Lagrangian in Eq. (1) decomposes into components uf ¯ γμ τγ¯ ν OVL ¼ uf PLb μPL τ; u † f ¯ τ¯ ν LEW λR ¯ c l λL ¯ c l −λL ¯ c ν Φ OSL ¼ ufPLb PL τ; eff ¼ð fiufPR i þVfj jiufPL i fidfPL iÞ 1 þH:c: uf ¯ σμν τσ¯ ν OTL ¼ uf PLb μνPL τ: ð5Þ 2Note that in the minimal R-parity violating MSSM the uf 1 coupling to charged conjugated fields in Eq. (1) is absent. For In the SM CVL ¼ and our NP matching contributions at an analysis of EDM constraints within this setup see Ref. [17]. tree level are given by

115014-2 CORRELATING TAUONIC B DECAYS WITH THE NEUTRON … PHYS. REV. D 100, 115014 (2019) pffiffiffi 2 V λLλL theory prediction for B → τν crucially depends on V . uf ufi i3 33 ub CVL ¼ 8 2 ; While previous lattice calculations resulted in rather small GFVufb M pffiffiffi values of Vub, recent calculations give a larger value (see λRλL u u − 2 3 33 Ref. [66] for an overview). However, the measurement is C f ¼ −4C f ¼ f : ð6Þ SL TL 8 2 still above the SM prediction by more than 1σ, as can be GFVufb M π Br½B→πτν seen from the global fit [67].InRð Þ¼Br½B→πlν there is Taking into account the QCD effects of Ref. [47] to the also a small disagreement between theory [68] and experi- matching, the one-loop EW and two-loop QCD renormal- ment [69] which does not depend on Vub, once more ization group equation (RGE) for the scalar and tensor pointing towards an enhancement. Therefore, even though operators [48,49] can be taken consistently into account. the b → uτν results are not significant on their own, they Numerically, this RGE evolution is given by point in the same direction as b → cτν (i.e., towards an      enhancement with respect to the SM) and thus strengthen uf 1 75 −0 29 uf 1 C ðmbÞ . . C ð TeVÞ the case for NP in tauonic B decays. SL ≈ SL ; uf 0084 uf 1 CT ðmbÞ . CT ð TeVÞ C. EDMs for a matching scale of 1 TeV. Finally, the, ratios RðDðÞÞ¼ For EDMs the relevant Hamiltonian in our case is → ðÞτν Br½B D with l ¼fμ;eg in terms of the Wilson coef- Br½B→DðÞlν HnEDM u u u u uτ uτ ficients at the b scale are given by [42] eff ¼ Cγ Oγ þ Cg Og þ CT OT ; ð9Þ

R D ð Þ ≃ 1 c 2 1 54ℜ 1 c c with j þCVLj þ . ½ð þCVLÞCSL RSMðDÞ c 2 c c c 2 þ1.09jC j þ1.04ℜ½ð1þC ÞC þ0.75jC j ; u μν SL VL T T Oγ ¼ eu¯σ PRuFμν; RðD Þ c 2 c c u ¯σμν a a ≃j1þC j −0.13ℜ½ð1þC ÞC Og ¼ gsu PRuT Gμν; R D VL VL SL SMð Þ uτ μν O ¼ u¯σμνP uτσ¯ P τ: ð10Þ 0 05 c 2 −5 0ℜ 1 c c 16 27 c 2 T R R þ . jCSLj . ½ð þCVLÞCT þ . jCTj : ð7Þ At the high scale we find the matching contributions (depicted in Fig. 1) Similarly, for b → uτν transitions we have

2 u 2 λLλR Br½B → τν m C V1j j3 13 1 u − B SL Cuτ ¼ ; → τν ¼ þ CVL : ð8Þ T 8 2 Br½B SM mbmτ M u mτVub L R 2 2 Cγ ¼ − λ λ ð4 þ 3 logðμ =M ÞÞ; The corresponding formula for B → πτν can be found in 96π2M2 33 13 Ref. [50]. However, here the effect of scalar and tensor mτV Cu ¼ − ub λLλR : ð11Þ operators is much smaller, making the theoretically very g 64π2M2 33 13 clean B → τν decays the primary place to search for them. Combining the experimental measurements of b → cτν transitions from LHCb [51–53], Belle [54–58], and BABAR Note that we only get up-quark contributions since we 3 1σ ðÞ do not have (at the one-loop level) CP violating couplings [59,60], one finds a combined tension of . in RðD Þ u 3 → Ψτν to down-type quarks. Importantly, note that our effect in Cγ [10]. However, note that here the Bc J= measure- u ment of LHCb [63], which also lies significantly above the and Cg is parametrically enhanced by mτ=mu, making a SM prediction, is not included.4 In b → uτν transitions, the sizable effect in EDMs possible. This enhancement of the dipole operators also allows us to safely neglect the effects

3 of charm quarks, four- operators, and of the In Ref. [61] it was shown that uncertainties from meson Weinberg operator otherwise relevant for LQs [19]. exchanges between initial and final states might be bigger than the estimated SM uncertainty, which could alleviate the tension in Next, we use the one-loop RGE to evolve these Wilson RðDðÞÞ. On the other hand, recent improvements in form factor coefficients of Eq. (11) down to the neutron scale. Here, calculations [62] lower the SM prediction and increase the combining and adjusting the results of Ref. [70] and tension. These two effects are not included in Ref. [10] but will Ref. [71] to our case we obtain5 not change the result significantly. 4 See Refs. [64,65] for an analysis including Bc → J=Ψτν before the latest BELLE update [58]. 5For the same RGE in a different operator basis see Ref. [72].

115014-3 ANDREAS CRIVELLIN and FRANCESCO SATURNINO PHYS. REV. D 100, 115014 (2019)

0 1 0 α 10 1 ¯ CF s DD 0 0 0 μ uτ 00 uτ H ¼ C Q ;Q¼½u¯ αγμP cα½u¯ βγ P cβ; CT 2π CT eff 1 1 1 R R d B C B α 4 α CB C μ u B − mτ sCF CF s C u @ Cγ A ¼ @ 2π2 2π 3π A@ Cγ A: dμ and find at the high scale u α 10 −12 u Cg sð CF Þ Cg 00 4π ðλR λRÞ2 C0 ¼ 13 23 ; ð16Þ The solution to this differential equation can be written in 1 128π2M2 terms of an evolution matrix in the form 0 from the one-loop matching. The evolution of C1 was ⃗ ⃗ calculated in Refs. [80,81] and yields approximately [82] CðμlÞ¼Uðμl; μhÞCðμhÞð12Þ 0 3 ≈ 0 8 0 1 with C1ð GeVÞ . C1ð TeVÞ: ð17Þ 0 1 4 The matrix element for the D-meson mixing is given by η3β0 00 B C B 4 16 14 2 C μ μ − η3β0 η3β0 η3β0 − 1 1 Uð l; hÞ¼@ mτX 3 ð Þ A; ð13Þ ¯ 0 0 μ 0 μ 2 hD jQ1ð ÞjD i¼ B1ð ÞmDfD; ð18Þ 2 3 00 η3β0 where B1ðμÞ¼0.75 at the scale μ ¼ 3 GeV [83]. The mass 33 − 2f αsðμhÞ difference in the D-meson system is given by β0 ¼ ; η ¼ ; ð14Þ 3 αsðμlÞ Δ 2 ¯ 0 HDD¯ 0 ≡2 mD ¼ Re½hD j eff jD i Re½M12: ð19Þ and Further, we write 4 4   3β0 β0 η ðη − 1Þβ0 μl X ¼ 2 log ; ð15Þ 2Im½M12 8π logðηÞ μh sin ϕ12 ¼ − : ð20Þ ΔmD where f is the number of active quark flavors. The final evolution matrix is obtained by running with the appro- The averages of the experimental values read [84,85] priate numbers of flavors from the LQ scale down to 1 GeV. 0 001 −1 0 008 Finally, the effects in the neutron and EDMs are . < jM12j½ps < . ; given by [73] −3.5 < ϕ12½° < 3.3; 212 u u fD ¼ MeV; ð21Þ dn=e ¼ −ð0.44 0.06ÞIm½Cγ − ð1.10 0.56ÞIm½Cg ; u u dp=e ¼ð1.48 0.14ÞIm½Cγ þð2.6 1.3ÞIm½Cg ; at 95% C.L. At a high luminosity LHC (HL-LHC) the sensitivity to ϕ12 could be improved down to the SM in terms of the Wilson coefficients evaluated at 1 GeV. The expectation of ≈0.17° [86]. neutron and proton EDMs then enter atomic ones, most importantly in mercury and deuteron (see Ref. [73] for E. W → τν and Z → ττ details). Virtual corrections with top quarks and LQs modify On the experimental side, dHg [74] gives currently slightly better bounds than the neutron EDM, while the couplings of gauge to charged leptons, in particular one of the proton and the deuteron is not measured to the tau. Parametrizing the interactions as yet. However, d and d will be very precisely known p D g2 g2 −L ffiffiffi ΛW τγ¯ μ ν − τγ¯ μ ΛV − ΛAγ τ from future experiments [75,76] and concerning d ¼ p 3 ð PL iWμ Þþ ð 5Þ Zμ n 2 i 2c there will be soon an improvement of one order of w magnitude in sensitivity compared to the current limit of with 3.6 × 10−26 ecm [77,78] from the n2EDM experiment at Paul Scherrer Institut (PSI) [79]. Therefore, we will focus W LQ V;A V;A V;A Λ ¼ δ3 þ Λ ; Λ ¼ Λ þ Δ ; on d in our phenomenological analysis. 3i i 3i SM LQ n 1 1 ΛV − 2 2 ΛA − SM ¼ þ sw; SM ¼ ; ¯ 2 2 D. D0 − D0 mixing ¯ 2 To describe D0 − D0 mixing we use the effective the LQ effects at q ¼ 0 (the contributions proportional to Hamiltonian mass are suppressed) are given by

115014-4 CORRELATING TAUONIC B DECAYS WITH THE NEUTRON … PHYS. REV. D 100, 115014 (2019)

L R L FIG. 2. Left: preferred regions in the λ33–λ23 plane from b → cτν data for M ¼ 1 TeV. Here, both the case of λ23 ¼ 0 and the one L L taking the maximally allowed value of λ23 from B → K νν are shown. A good fit to data requires jλ33j ≈ 1 in both cases. Note that our model is compatible with LHC searches for monotaus and with Bc lifetime constraints which exclude the dark pink and gray regions. L Right: green (blue) regions indicate where B → τν is enhanced (suppressed) by 10%–40% w.r.t. the SM for M ¼ 1 TeV, λ33 ¼ 1 and L λ13 ¼ 0. The dark red region is excluded by the neutron EDM and the dark red contour denotes the n2EDM sensitivity. The orange ¯ contour shows the HL-LHC sensitivity to CP violation in D0 − D0 mixing which is nicely complementary to EDM searches.     2 2 LQ Ncmt L L mt large rates of τ → μðeÞγ. Similarly, our model cannot Λ ¼ 3V3 λ V λ 1 þ 2 log ; 3i 192π2M2 h h3 3k ki M2 address the b → sμþμ− anomalies if one aims at a sizable    2 2 effect in tauonic B decays [99]. Therefore, we will disregard L L L Ncmt mt Δ ¼ V3 λ 3 V3 λ 3 1 þ log ; (i.e., set to zero) the couplings to and . LQ l l a a 32π2M2 M2    Couplings to top-quarks affect τ → μνν [100] and Z → N m2 m2 τþτ− ΔL ≈−0 0006 λL 2 ΔR −λRλR c t 1 t [101].Hereweseethat . j 33j and LQ ¼ 33 33 2 2 þ log 2 ; ð22Þ LQ L 2 32π M M Λ33 ≈−0.0008jλ33j (for M ¼ 1 TeV) is compatible with experiments for jλL j < 1. Note that we improve the agree- ΔV −ΔL −ΔR ΔA ΔR − ΔL 33 with LQ ¼ LQ LQ and LQ ¼ LQ LQ. This leads ment in Z → ττ data while slightly worsening τ → lνν data, W LQ to jΛ33j¼j1 þ Λ33 j. Experimentally, the averaged modi- which is already a bit away from the SM prediction. τν τ → μνν λR λR λL fication of the W- coupling extracted from and Thus, we are left with 13, 23, and i3 as free parameters τ → eνν decays reads (averaging the central value but with for studying the effect in tauonic B decays and the unchanged error) [87,88] correlations with EDMs. In the following we will set M ¼ 1 TeV which is also well compatible with the latest direct ΛW exp ≈ 1 002 0 0015 6 j 33 j . . ; ð23Þ search results of CMS for third generation LQs [102,103]. Let us now turn to b → cτν processes, where effects of which provides a better constraint than data of W decays. the order of 10% compared to the corresponding tree-level Concerning Z → ττ the axial vector coupling is much SM amplitude are required. Since our model can give better constrained that the vectorial one [87,88] [according to Eq. (3)] tree-level effects in B → KðÞνν ΛA ΛA 1 0019 0 0015 exp= SM ¼ . . ; ð24Þ decays (which are loop suppressed in the SM), these contributions must be suppressed. Since the bottom cou- ΛA ΛA 1 2ΔL − 2ΔR with = SM ¼ þ LQ LQ. pling to taus should be sizable, the coupling to strange quarks is tightly bound. We show the preferred regions, → III. PHENOMENOLOGY according to the updated global fit of Ref. [42], from b cτν processes in the left plot of Fig. 2. These regions are Looking at the phenomenological consequences of our L L shown for λ23 ¼ 0 but also the possible impact of λ23 ≠ 0, model, note that couplings to muons or electrons are obviously not necessary to obtain the desired effects in 6 tauonic B decays. Even though our S1 model can in principle More sophisticated analysis of LHC data can be found in – account for the anomalous magnetic moment of the Refs. [104 106]. However, since for t-channel exchange the – effective field theory limits are in general stronger than the ones [26,89 98] (or [98]) via a mt=mμ enhanced effect, in the UV complete model, we will use for simplicity the results this is not possible in the presence of large couplings to tau of Ref. [107] in the following which show that 1 TeV is leptons since also here mt enhanced effects generate too compatible with data.

115014-5 ANDREAS CRIVELLIN and FRANCESCO SATURNINO PHYS. REV. D 100, 115014 (2019)

→ νν 2 taking its maximally allowed values from B K ,is neutron) in a model with a scalar LQ SUð ÞL singlet which depicted. Note that our model is not in conflict with the Bc can be identified with the right-handed down squark in the lifetime [108,109] (in fact, it is even compatible with the R-parity violating MSSM. We found that in order to explain L 10% limit of Ref. [110]) nor with direct LHC searches for the intriguing tensions in b → cτν data, λ33 must be sizable monotaus [107]. So far we worked with real parameters in and also a coupling to right-handed charm quarks and tau- ðÞ R order to maximize the effect in RðD Þ. However, even for leptons (λ23) is required. In this setup, the model gives a complex couplings the effect in nuclear and atomic EDMs very good fit to data and is compatible with b → sνν would be strongly suppressed since only up and down observables, LHC searches, and Bc lifetime constraints. quarks contribute directly to these observables. Extending this analysis to b → uτν transitions, in particular → τν R Therefore, let us now turn to b u where couplings to B → τν, again right-handed couplings to up quarks (λ13) up quarks are obviously needed. Here, even for real are required to have a sizable effect. This leads to very couplings an effect in the neutron EDM is generated due important mτ=mu enhanced effects in (chromo)electric to the large phase of Vub. This effect could only be avoided dipole operators generating in turn EDMs of λRλL for Arg½ 13 33¼Arg½Vub. However, since there is no and . In particular, even for real couplings of the LQ (obvious) symmetry which could impose this relation, such to fermions, the large phase of Vub generates a sizable a configuration would be fine-tuning. This can be seen from contribution to the neutron EDM. In fact, this effect should the right plot in Fig. 2, where we show the predictions for already be observable in the n2EDM experiment at PSI, → τν → τν Br½B =Br½B SM as a function of the absolute assuming that, within our model, B → τν is enhanced (or R L value and the phase of λ13 for λ33 ¼ 1 (as preferred by b → suppressed) by around 10% with respect to the SM. cτν data). The dark red contour lines denote the n2EDM sensitivity, showing that a 10% effect in B → τν with respect to the SM will lead to an observable effect in the neutron ACKNOWLEDGMENTS EDM within our model. Finally, taking λR ¼ −0.1,as 23 The work of A. C. is supported by a Professorship Grant preferred by b → cτν (see left plot of Fig. 2), CP violation (PP00P2_176884) of the Swiss National Science in D0 − D¯ 0 mixing is generated. Here the red contour Foundation. The work of F. S. is supported by the Swiss denotes the future HL-LHC sensitivity which is comple- National Foundation under Grant No. 200020_175449/1. mentary to the region covered by EDM searches. We thank Dario Müller for collaboration in the early stages of this article and Christoph Greub for useful comments on IV. CONCLUSIONS the manuscript. We are grateful to David Straub and Jason In this article we studied the interplay between tauonic B Aebischer for reminding us of the importance of Z-ττ and meson decays and EDMs (in particular the one of the W-τν couplings.

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