Anomalies with an S1 Leptoquark from SO(10) Grand Unification
PHYSICAL REVIEW D 101, 015011 (2020)
Addressing the RD( ) anomalies with an S1 leptoquark from SO(10) grand unification
† ‡ Ufuk Aydemir ,1,2,* Tanumoy Mandal ,3,4, and Subhadip Mitra 5, 1Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, P. R. China 2School of Physics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, P. R. China 3Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala, Sweden 4Department of Physics and Astrophysics, University of Delhi, Delhi 110 007, India 5Center for Computational Natural Sciences and Bioinformatics, International Institute of Information Technology, Hyderabad 500 032, India
(Received 4 September 2019; published 22 January 2020)
Motivated by the RDð Þ anomalies, we investigate an SO(10) grand unification scenario where a charge −1=3 scalar leptoquark (S1) remains as the only new physics candidate at the TeV scale. This leptoquark along with the Standard Model (SM) Higgs doublet originates from the same ten-dimensional real scalar multiplet in the SO(10) grand unification framework taking its mass close to the electroweak scale. We explicitly show how the gauge coupling unification is achieved with only one intermediate symmetry- breaking scale at which the Pati-Salam gauge group is broken into the SM group. We investigate the phenomenological implications of our scenario and show that an S1 with a specific Yukawa texture can
explain the RDð Þ anomalies. We perform a multiparameter scan considering the relevant flavor constraints νν on R ð Þ , F D , Pτ D and R as well as the constraint coming from the Z → ττ decay and the latest D Lð Þ ð Þ Kð Þ ττ resonance search data at the LHC. Our analysis shows that a single leptoquark solution to the observed
RDð Þ anomalies with S1 is still a viable solution.
DOI: 10.1103/PhysRevD.101.015011
3 08σ I. INTRODUCTION (combined excess of . in RDð Þ ) based on the world averages as of spring 2019, according to the Heavy Flavor During the past few years, several disagreements Averaging Group [16], whereas, the observed R and R between experiments and the Standard Model (SM) pre- K K are both suppressed compared to their SM predictions dictions in the rare B decays have been reported by the [17,18] by ∼2.6σ. BABAR [1,2], LHCb [3–7] and Belle [8–11] collaborations. One of the possible explanations for the B-decay So far, these anomalies have been quite persistent. The anomalies is the existence of scalar leptoquarks whose most significant ones have been observed in the R ð Þ and D masses are in the few-TeV range [19–53]).1 Leptoquarks, R ð Þ observables, defined as, K which posses both lepton and quark couplings, often exist in the grand unified theories (GUTs) or in the Pati-Salam- B → Dð Þτν BRð Þ type models. Considering that the LHC searches, except for RDð Þ ¼ and BRðB → Dð ÞlνÞ these anomalies, have so far returned empty handed, if the BRðB → Kð Þμþμ−Þ leptoquarks indeed turn out to be behind these anomalies, it RKð Þ ¼ − : is likely that there will only be a small number of particles BRðB → Kð Þeþe Þ to be discovered. However, their existence in small num- Here, l ¼fe or μg and BR stands for branching ratio. The bers at the TeV scale would be curious in terms of its implications regarding physics beyond the Standard Model. experimental values of RD and RD are in excess of their SM predictions [12–15] by 1.4σ and 2.5σ, respectively, Scalars in these models mostly come in large multiplets and it would be peculiar that only one or a few of the components become light at the TeV scale while others *[email protected] † remain heavy. Mass splitting is actually a well-speculated [email protected] ‡ [email protected] subject in the literature in the context of the infamous doublet-triplet splitting problem in supersymmetric theo- Published by the American Physical Society under the terms of ries and GUTs. From this point of view, even the SM Higgs the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, 1See Refs. [54–61] for the vector-leptoquark solutions pro- and DOI. Funded by SCOAP3. posed to explain the B-decay anomalies.
2470-0010=2020=101(1)=015011(20) 015011-1 Published by the American Physical Society AYDEMIR, MANDAL, and MITRA PHYS. REV. D 101, 015011 (2020) in a GUT framework is troublesome if it turns out to be the symmetry mechanisms such as the utilization of Peccei- only scalar at the electroweak (EW) scale. Quinn (PQ) symmetry [68,76], other U(1) symmetries such In this paper, we consider a single scalar leptoquark, as the one discussed in Ref. [77], or a discrete symmetry S1ð3; 1; −1=3Þ, at the TeV scale, in the SO(10) GUT similar to the one considered in Ref. [22]. Operators framework [62–75]. This particular leptoquark was dis- leading to proton decay could also be suppressed by a cussed in the literature to be responsible for one (or specific mechanism such as the one discussed in Ref. [78] possibly both) of the RDð Þ and RKð Þ anomalies or they could be completely forbidden by geometrical [21,22,25,32]. (Later, however, it was shown in reasons [38]. In this paper, we adopt a discrete symmetry as Ref. [45] that a single S1 leptoquark can only alleviate suggested in Ref. [22], assumed to operate below the the RKð Þ discrepancy, but cannot fully resolve it.) intermediate symmetry-breaking scale even though it is Furthermore, it is contained in a relatively small multiplet not manifest at higher energies. that resides in the fundamental representation of the SO(10) Motivated by the possible existence of a single TeV-scale group, a real 10, together with a scalar doublet with the S1 in the SO(10) GUT framework, we move on to quantum numbers that allow it to be identified as the SM investigate the phenomenological implications of our Higgs. Therefore, S1 being the only light scalar entity other model. In Ref. [22], it was shown that a TeV-scale S1 than the SM Higgs doublet is justified in this scenario. A leptoquark can explain the RDð Þ anomalies while simulta- future discovery of such a leptoquark at the TeV scale could neously inducing the desired suppression in RKð Þ through be interpreted as evidence in favor of an SO(10) GUT. box diagrams. Since the most significant anomalies are 10 It has been argued in the literature that a real H in the seen in the RDð Þ observables, in this paper, we concentrate minimal SO(10) setup (even together with a real 120H or a mostly on scenarios that can accommodate these observ- complex 120H) is not favored in terms of a realistic Yukawa ables. Generally, a TeV-scale S1 requires one Yukawa sector [68]. On the other hand, it has been recently coupling to be large to accommodate the RDð Þ anomalies → discussed in Ref. [75] that a Yukawa sector consisting of [32,45]. This, however, could create a problem for the b sνν¯ transition rate measured in the Rνν observable. In the a real 10H, a real 120H and a complex 126H, can establish a K realistic Yukawa sector due to the contributions from the SM, this decay proceeds through a loop whereas S1 can contribute at the tree level in this transition. Therefore, the scalars whose quantum numbers are the same as the SM νν measurement of RK is very important to restrict the Higgs doublet. Thus, this is the scalar content we assign in 2 parameter space of S1. Some specific Yukawa couplings our model for the Yukawa sector. of S1 are also severely constrained from the Z → ττ decay The inclusion of S1 in the particle content of the model at [47] and the LHC ττ resonance search data [48]. Therefore, the TeV scale does not improve the status of the SM in it is evident that in order to find the R ð Þ -favored parameter terms of gauge coupling unification, which cannot be D space while successfully accommodating other relevant realized by the particle content in question. Fortunately, constraints, one has to introduce new d.o.f. in terms of new in the SO(10) framework, there are other ways to unify the couplings and/or new particles. Here, we consider a gauge coupling constants, in contrast to models based on specific Yukawa texture with three free couplings to show the SU(5) group which also contains such a leptoquark that a TeV-scale S1, consistent with relevant measurements, within the same multiplet as the SM Higgs. As we illustrate can still explain the R ð Þ anomalies. in this paper, inserting a single intermediate phase where D The rest of the paper is organized as follows. In Sec. II, the active gauge group is the Pati-Salam group, SUð4ÞC ⊗ we introduce our model. In Sec. III, we display the SUð2Þ ⊗ SUð2Þ , which appears to be the favored L R unification of the couplings for two versions of our model. route of symmetry breaking by various phenomenological In Sec. IV, we present the related LHC phenomenology bounds [71], establishes coupling unification as desired. In with a single extra leptoquark S1. We display the exclusion our model, the Pati-Salam group is broken into the SM limits from the LHC data and discuss related future gauge group at an intermediate energy scale M , while SO C prospects. We also study the renormalization group (RG) (10) is broken into the Pati-Salam group at the unification running of the Yukawa couplings. Finally in Sec. V, we end scale M . We consider two versions of this scenario, U our paper with a discussion and conclusions. depending on whether the left-right symmetry, so-called D parity, is broken together with SO(10) at M ,oritis U II. THE SO(10) MODEL broken at a later stage, at MC, where the Pati-Salam symmetry is broken into the SM gauge symmetry. In our SO(10) model, we entertain the idea that the SM Light color triplets, similar to the one we consider in this Higgs doublet is not the only scalar multiplet at the TeV paper, are often dismissed for the sake of proton stability since these particles in general have the right quantum 2One can avoid this conflict by introducing some additional numbers for them to couple potentially dangerous opera- degrees offreedom (d.o.f.), as shown in Ref. [30]. There, the authors tors that mediate proton decay. On the other hand, the introduced an S3 leptoquark in addition to the S1 to concomitantly νν proton stability could possibly be ensured through various explain RDð Þ and RKð Þ while being consistent with RK .
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1 15 scale, but it is accompanied by a leptoquark S1 ¼ u ð Þ ð Þ MU ¼ v10Y10 þ v126Y126 þðv120 þ v120 ÞY120; ð3; 1; −1=3Þ, both of which reside in a real ten-dimensional d ð1Þ ð15Þ representation, 10, of SO(10) group. The peculiar mass MD ¼ v10Y10 þ v126Y126 þðv120 þ v120 ÞY120; splitting among the components of this multiplet does not d ð1Þ ð15Þ M ¼ v10Y10 − 3v Y126 þðv − 3v ÞY120; occur, leading to a naturally light scalar leptoquark at the E 126 120 120 TeV scale. − 3 u ð1Þ − 3 ð15Þ MνD ¼ v10Y10 v126Y126 þðv120 v120 ÞY120; We start with a real 10H of SO(10) whose Pati-Salam and Mν ¼ v Y126; ð5Þ SM decompositions are given as R;L R;L
u d 10 ¼ð1; 2; 2Þ422 ⊕ ð6; 1; 1Þ422 where v ðv Þ are the presumed vacuum expectation values i i 1 1 1 of the neutral components of the corresponding bidoublets, 1 2 ⊕ 1 2 − ⊕ 3 1 − ¼ ; ; 2 ; ; 2 ; ; 3 contributing to the masses of the up quarks and the Dirac |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}321 |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}321 |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}321 neutrinos (the down quarks and the charged leptons), and H H S1 where ¯ 1 ⊕ 3; 1; ; ð1Þ u d 3 v10 ≡ v10 ¼ v10; |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}321 1 1 1 ð Þ u;ð Þ d;ð Þ S1 v120 ≡ v120 ¼ v120 ; ð15Þ ≡ u;ð15Þ d;ð15Þ where subscripts denote the corresponding gauge group v120 v120 ¼ v120 ; ð6Þ and we set Q ¼ I3 þ Y. The scalar content we assign for the Yukawa sector due to the reality condition of 10 and 120. The superscripts ϕ 1 2 2 consists of a real 10H, a real 120H, and a complex 126H, (1) and (15) denote the doublets residing in ð ; ; Þ422 which establishes a realistic Yukawa sector through mixing and Σð15; 2; 2Þ422 of 120, respectively. vR in the Majorana between the scalars whose quantum numbers are the same mass matrices given in Eq. (5) is the vacuum expectation as the SM Higgs doublet, as shown in Ref. [75]. Note that value of the SM singlet contained in ð10; 1; 3Þ422 and is responsible for the heavy masses of the right-handed 2 2 16 ⊗ 16 ¼ 10s ⊕ 120a ⊕ 126s; ð2Þ ∼ neutrinos (type-I seesaw), and vL vRv10=MGUT is 102 1202 where 16 is the spinor representation in which each family induced by the potential term H H and is responsible of fermions, including the right-handed neutrino, resides in. for the left-handed Majorana neutrino masses (type-II The subscripts s and a denote the symmetric and anti- seesaw) [75]. symmetric components. The Pati-Salam decompositions of For the symmetry-breaking pattern, we consider a 120 126 scenario in which the intermediate phase has the gauge and are given as 4 ⊗ 2 ⊗ symmetry of the Pati-Salam group SUð ÞC SUð ÞL 2 120 ¼ð1; 2; 2Þ422 ⊕ ð1; 1; 10Þ422 ⊕ ð1; 1; 10Þ422 SUð ÞR. Note that this specific route of symmetry breaking appears to be favored by various phenomenological bounds ⊕ 6 3 1 ⊕ 6 1 3 ⊕ 15 2 2 ð ; ; Þ422 ð ; ; Þ422 ð ; ; Þ422; [71]. We consider two versions of this scenario depending 126 ¼ð10; 3; 1Þ422 ⊕ ð10; 1; 3Þ422 ⊕ ð15; 2; 2Þ422 on whether or not the Pati-Salam gauge symmetry is accompanied by the D-parity invariance, a Z2 symmetry ⊕ 6 1 1 ð ; ; Þ422: ð3Þ that maintains the complete equivalence of the left and right The Yukawa terms are then given as sectors [65,66,79], after the SO(10) breaking. The sym- metry-breaking sequence is schematically given as
LY ¼ 16FðY1010H þ Y120120H þ Y126126HÞ16F þ H:c:; 10 ⟶MU ⟶MC ⟶MZ ð4Þ SOð Þ G422 ðor 422DÞ G321ðSMÞ G31; h210iðor h54iÞ h126i h10i where Y10 and Y126 are complex Yukawa matrices, sym- ð7Þ metric in the generation space, and Y120 is a complex antisymmetric one. where, we use the notation, The SM doublet contained in ϕð1; 2; 2Þ422 of 10 is mixed with other doublets accommodated in ϕð1; 2; 2Þ422 and ≡ 4 ⊗ 2 ⊗ 2 ⊗ G422D SUð ÞC SUð ÞL SUð ÞR D; Σð15; 2; 2Þ422 of the real multiplet 120 and Σð15; 2; 2Þ422 of ≡ 4 ⊗ 2 ⊗ 2 126, yielding a Yukawa sector consistent with the observed G422 SUð ÞC SUð ÞL SUð ÞR; fermion masses [75]. The fermion mass matrices for the up ≡ 3 ⊗ 2 ⊗ 1 G321 SUð ÞC SUð ÞL Uð ÞY; quark, down quark, charged leptons, Dirac neutrinos and ≡ 3 ⊗ 1 Majorana neutrinos are given as [75] G31 SUð ÞC Uð ÞQ: ð8Þ
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The first stage of the spontaneous symmetry breaking Pati-Salam breaking scale under which quarks and leptons occurs through the Pati-Salam singlet in the SO(10) transform with opposite parities whereas the leptoquark is multiplet 210, acquiring a vacuum expectation value assigned odd parity, i.e., ðq; l; S1Þ → ð q; ∓l; −S1Þ. Note (VEV) at the unification scale MU. This singlet is odd also that the inclusion of S1 can affect the stability of the under D parity and, therefore, the resulting symmetry group electroweak vacuum via loop effects. The relevant dis- is G422 in the first stage of the symmetry breaking. In the cussion can be found in Ref. [80]. second step, the breaking of G422 into the SM gauge group Evidently, the Lagrangian given in Eq. (9) together with G321 is realized through the SM singlet contained in the SM Lagrangian should be understood in the effective field theory context. The new and SM Yukawa couplings in Δ ð10; 1; 3Þ422 of 126, acquiring a VEV at the energy R the TeV-scale Lagrangian are induced from the original SO scale MC, which also yields a Majorana mass for the right- handed neutrino. The last stage of the symmetry breaking is (10) Yukawa couplings each of which is generated by a realized predominantly through the SM doublet contained linear combination of unification-scale operators and gets modified due to the mixing effects induced by the scalar in ϕð1; 2; 2Þ of 10. The mass scale of these Pati-Salam 422 fields that have Yukawa couplings to three chiral families multiplets is set as M , the energy scale at which the Pati- C of 16 . It is indeed this rich structure that enables the Salam symmetry is broken into the SM, while the rest of the F realization of a fermion mass spectrum consistent with the fields are assumed to be heavy at the unification scale M . U expected fermion masses of the SM model at the unification The only d.o.f., assumed to survive down to the electro- scale, as shown in Ref. [75]. As we will discuss later in weak scale, are the SM Higgs doublet and the color triplet, Sec. IV G, the modification to the SM RG running of the S1. We call this model A1. Yukawa couplings due to the inclusion of ΛL=R does not In the second scenario, which we call model A2, the first register strong changes in the fermionic mass spectrum and stage of the symmetry breaking is realized through the Pati- hence the main message of Ref. [75] is valid in our case, Salam singlet contained in 54, which acquires a VEV. This as well. singlet is even under D parity, and therefore, D parity is not broken at this stage with the SO(10) symmetry, and the III. GAUGE COUPLING UNIFICATION resulting symmetry group valid down to MC is G422. The rest of the symmetry breaking continues in the same way as In this section, after we lay out the preliminaries for one- in model A1. Consequently in model A2, we include one loop RG running and show that the new particle content at more Pati-Salam multiplet at MC, ΔLð10; 3; 1Þ, in order to the TeV scale does not lead to the unification of the SM maintain a complete left-right symmetry down to MC. The gauge couplings directly, we illustrate gauge coupling scalar content and, for later use, the corresponding RG unification with a single intermediate step of symmetry coefficients in each energy interval are given in Table I. breaking. Once the particle content at low energies is Finally, the relevant Lagrangian for phenomenological determined, there may be numerous ways to unify the analysis at low energy is given by gauge couplings, depending on the selection of the scalar content in SO(10) representations. In the literature, the † μ 2 2 2 2 canonical way to make this selection is through adopting a L ⊃ ðDμS1Þ ðD S1Þ − M jS1j − λjS1j jHj S1 minimalistic approach, allowed by the observational con- ΛL ¯ c τ ΛR ¯ c † þð Q i 2L þ uReRÞS1 þ H:c: ð9Þ straints. In the following, we pursue the same strategy while taking into account the analysis made in Ref. [75] for a where Q and L are the SM quark and lepton doublets (for realistic Yukawa sector. each family), ΛL=R are coupling matrices in flavor space and ψ c ¼ Cψ¯ T are charge-conjugate spinors. Notice the A. One-loop RG running absence of dangerous diquark couplings of S1 that would For a given particle content, the gauge couplings in an lead to proton decay. One way to forbid these couplings energy interval ½MA;MB evolve under one-loop RG is to impose a Z2 symmetry [22] that emerges below the running as
TABLE I. The scalar content and the RG coefficients in the energy intervals for model A1;2. Note that the ϕ fields, the Φ field, and one of the Σ fields originate from real SO(10) multiplets and thus the η ¼ 1=2 condition should be employed when necessary while determining the RG coefficients in Eq. (11).
Interval Scalar content for model A1 (A2) RG coefficients ϕ 1 2 2 2 Φ 6 1 1 1 7 9 67 67 II ð ; ; Þ × , ð ; ; Þ, ½a4;aL;aR ¼½2 ð2Þ; 2 ð 6 Þ; 6 Σð15; 2; 2Þ × 2, ΔRð10; 1; 3Þ, (and ΔLð10; 3; 1Þ for model A2) 1 2 1 3 1 − 1 −41 −19 125 I Hð ; ; 2Þ, S1ð ; ; 3Þ ½a3;a2;a1 ¼½ 6 ; 6 ; 18
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(a) (b)
FIG. 1. Running of the gauge couplings with the particle content of the SM and with the inclusion of S1. The vertical dotted line −1 ˜ −1 3 −1 correspondstotheelectroweakscaleMZ.Forα1 ,weplottheredefinedquantityα1 ≡ 5 α1 asrequiredbytheSO(10)boundaryconditions. Including the leptoquark in the particle content does not provide a significant modification to the SM RG running in favor of unification. X X 1 1 a M 2 − i B ; T ¼ Y ; ð12Þ 2 2 ¼ 8π2 ln ð10Þ gi ðMAÞ gi ðMBÞ MA f;s f;s
1 where the RG coefficients ai are given by [81,82] where Y is the Uð ÞY charge. The addition of S1 to the particle content of the SM does 11 2 X not help in unifying the gauge couplings as displayed in − ai ¼ 3 C2ðGiÞþ3 TiðRfÞ · d1ðRfÞ dnðRfÞ Fig. 1, where the RG running is performed with the Rf modified RG coefficients given in Table I in interval I, η X while interval II is irrelevant to this particular case. þ 3 TiðRsÞ · d1ðRsÞ dnðRsÞ; ð11Þ Rs Unification of the gauge couplings can be established through intermediate symmetry breaking between the and the full gauge group is given as G ¼ Gi ⊗ G1 ⊗ electroweak scale and unification scale, as we illustrate … ⊗ Gn. The summation in Eq. (11) is over irreducible in the next subsection with a single intermediate step of chiral representations of fermions (Rf) and irreducible symmetry breaking. representations of scalars (Rs) in the second and third η 1 2 terms, respectively. The coefficient is either 1 or = , B. Unification with a single intermediate scale depending on whether the corresponding representation is We start by labeling the energy intervals in between complex or (pseudo)real, respectively. djðRÞ is the dimen- symmetry-breaking scales ½MZ;MC and ½MC;MU with sion of the representation R under the group G ≠ . C2ðG Þ is j i i Roman numerals as the quadratic Casimir for the adjoint representation of the group Gi, and Ti is the Dynkin index of each representation ∶ M ;M ;G213 ; (see Table II). For the U(1) group, C2ðGÞ¼0 and I ½ Z C ðSMÞ
II∶ ½MC;MU ;G224 or G224D: ð13Þ TABLE II. Dynkin index Ti for various irreducible representa- tions of SU(2), SU(3), and SU(4). Our normalization convention in this paper follows the one adopted in Ref. [82]. Notice that The boundary/matching conditions we impose on the there are two inequivalent 15-dimensional irreducible represen- couplings at the symmetry-breaking scales are tations for SU(3).
Representation SU(2) SU(3) SU(4) MU∶ gLðMUÞ¼gRðMUÞ¼g4ðMUÞ; 2 1 −− 2 M ∶ g3ðM Þ¼g4ðM Þ;g2ðM Þ¼g ðM Þ; 1 − C C C C L C 322 1 1 2 1 45− 1 2 2 ¼ 2 þ 3 2 ; 35 5 g1ðMCÞ gRðMCÞ g4ðMCÞ 6 2 2 1 − 8423gLðMCÞ¼gRðMCÞðabsent in the G224 caseÞ; 165 15 10 2 2 3 1 1 1 35 ∶ 15 280 10; 2 4 MZ 2 ¼ 2 þ 2 : ð14Þ e ðMZÞ g1ðMZÞ g2ðMZÞ
015011-5 AYDEMIR, MANDAL, and MITRA PHYS. REV. D 101, 015011 (2020) TABLE III. The predictions of models A1 and A2. 2π 2π MU MC ¼ − a4 ln þ a3 ln : ð16Þ αU αsðMZÞ MC MZ Model A1 A2 log10ðMU=GeVÞ 17.1 15.6 log10ðMC=GeVÞ 10.9 13.7 Thus, once the RG coefficients in each interval are α−1 M M α U 29.6 35.4 specified, the scales U and C, and the value of U are uniquely determined. The results are given in Table III, and unification of the couplings is displayed in Fig. 2 for We use the central values of the low-energy data as the each model. boundary conditions in the RG running (in the MS scheme) As mentioned previously, we assume in this paper that [83,84]: α−1 ¼ 127.95, α ¼ 0.118, sin2 θ ¼ 0.2312 at the proton-decay-mediating couplings of S1 are suppressed. s W On the other hand, we do not make any assumptions M ¼ 91.2 GeV, which translate to g1 ¼ 0.357, g2 ¼ Z regarding the other potentially dangerous operators which 0.652, g3 ¼ 1.219. The coupling constants are all required to remain in the perturbative regime during the evolution could lead to proton decay. Thus, it is necessary to inspect from M to M . whether the predictions of our models displayed in Table III Z U are compatible with the current bounds coming from the The RG coefficients, ai, differ depending on the particle content in each energy interval, changing every time proton decay searches or not. The most recent and stringent symmetry breaking occurs. Together with the matching bound on the lifetime of the proton comes from the mode → þπ0 τ 1 6 1034 and boundary conditions, one-loop RG running leads to the p e , and is p > . × years [85]. As for the proton decay modes that are mediated by the super-heavy following conditions on the symmetry-breaking scales MU gauge bosons, which reside in the adjoint representation and MC: 45 τ ∼ 4 5 α2 of SO(10) , considering that p MU=mp U [86],we ≳ 1015.9 obtain MU GeV, which is consistent with predic- 2 3 − 8sin θWðMZÞ MC tions of both model A1 and model A2, within an order of 2π ¼ð3a1 − 5a2Þ ln αðM Þ M magnitude of the latter. Additionally, since MC is the scale Z Z at which the Pati-Salam symmetry breaks into the SM, it MU þð−5a þ 3a þ 2a4Þ ln ; determines the expected mass values for the proton-decay- L R M C mediating color triplets. From a naive analysis [71], it can 3 8 2π − 3 3 − 8 MC be shown that the current bounds on the proton lifetime ¼ð a1 þ a2 a3Þ ln ≳ 1011 αðMZÞ αsðMZÞ MZ require MC GeV, again consistent with the predic- tions of both model A1 and model A2, within an order of MU þð3a þ 3a − 6a4Þ ln ; magnitude of the former. Note that these bounds should L R M C be taken as order-of-magnitude estimates since, while ð15Þ obtaining them, we approximate the anticipated masses of the super-heavy gauge bosons and the color triplets as MX ≈ MU and MT ≈ MC, while it would not be unreason- where the notation on ai is self-evident. The unified gauge able to expect that these mass values could differ from the coupling αU at the scale MU is then obtained from corresponding energy scales within an order of magnitude.
(a) (b)
˜ −1 3 −1 FIG. 2. Running of the gauge couplings for models A1 and A2. Note that α1 ≡ 5 α1 . The discontinuity at MC in each plot is due to the boundary conditions given in Eq. (14).
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(a) (b) (c)
ð Þ FIG. 3. Leading-order parton-level Feynman diagrams responsible for the B → D τν decay (a) in the SM, and (b) in the S1 model. The parton-level process in the presence of S1 that would contribute to B → Kνν is shown in diagram (c).
IV. LOW-ENERGY PHENOMENOLOGY with a charge-conjugate quark of the ith generation and a lepton of the jth generation with chirality H. Without The existence of a TeV-scale charge −1=3 scalar lep- any loss of generality, we assume all λ’s are real in our toquark in a GUT framework is quite interesting from a collider analysis since the LHC data that we consider phenomenological perspective mainly for two reasons. are insensitive to their complex nature. Also, we only First, its existence is testable at the LHC. The direct- consider mixing among quarks [Cabibbo-Kobayashi- detection searches for scalar leptoquarks have been putting Maskawa (CKM) mixing] and ignore neutrino mixing exclusion bounds on S1 with different decay hypotheses (Pontecorvo-Maki-Nakagawa-Sakata mixing) completely [87,88]. Second, as mentioned earlier, such a leptoquark as all neutrino flavors contribute to the missing energy can offer an explanation of some persistent flavor anoma- and hence are not distinguishable at the LHC. The lies observed in several experiments. For example, if we couplings of S1 to the first-generation SM fermions are consider the anomalies observed in the B-meson semi- heavily constrained [32]. Hence, we assume λ1 ; λ 1 ¼ 0 in leptonic decays via charged currents (collectively these i i our analysis.3 show the most significant departure from the SM expect- The parton-level Feynman diagrams for the b → cτν ations), S1 can provide an explanation if it couples with τ → ð Þτν and neutrino(s) and b and c quarks. The direct LHC bounds decay (responsible for the B D decay) are shown in Fig. 3. In order to have a nonzero contribution in the RDð Þ on such a leptoquark are not very severe but as it has been ν τ pointed out in Ref. [48], the present LHC data in the pp → observables from S1, we need the b S1 and c S1 couplings ττ τν to be nonzero simultaneously. Minimally, one can start with = channels have actually put constraints on the S1 L L just a single free coupling: either λ23 or λ33. The coupling parameter space relevant for explaining the observed R ð Þ D λL λL τ ν anomalies. Here, using flavor data and LHC constraints, we 23 ( 33) directly generates the c S1 (b S1) interaction and ν τ obtain the allowed parameter space in our model. We also the other one, i.e., the b S1 (c S1) can be generated through point out some possible new search channels at the LHC. the CKM mixing among quarks. These two minimal On the flavor side, our primary focus is on the charged- scenarios were discussed in detail in Ref. [48]. For these current anomalies observed in the semileptonic B decays in two cases, the Lagrangian in Eq. (17) can be written explicitly as, the RDð Þ observables. Hence, as in Ref. [48], we focus on the interaction terms of S1 that could play a role to L c ¯ c c ¯ c † C1∶ L ⊃ λ23½c¯ τL − ðVcbb þ Vcss¯ þ Vcdd Þν S1 þ H:c: address the RDð Þ anomalies for simplicity. L ðλ33 ¼ 0Þ; ð18Þ A. The S1 model L c c c ¯ c † C2∶ L ⊃ λ ½ðV u¯ þ V c¯ þ V t¯ Þτ − b ν S þ H:c: The single TeV-scale S1 leptoquark that originates from 33 ub cb tb L 1 L the GUT model discussed in Sec. II transforms under ðλ23 ¼ 0Þ: ð19Þ the SM gauge group as ð3; 1; −1=3Þ. The low-energy interactions of S1 with the SM fields are shown in a In Ref. [48], it was shown that for C1, the RDð Þ -favored compact manner in Eq. (9). Below, we display the relevant parameter space is already ruled out by the latest LHC data. interaction terms required for our phenomenological On the other hand, C2 is not seriously constrained by the analysis, LHC data since this scenario is insensitive to the coupling L λ33. Only the pair-production searches, which are largely L ⊃ λL ¯ c τ λR ¯ cl † ½ ijQi ði 2ÞLj þ ijui Rj S1 þ H:c:; ð17Þ 3However, these couplings can be generated through the CKM where Qi and Li denote the ith-generation quark and lepton mixing. We refer the interested readers to Ref. [32] for various λH doublets, respectively and ij represents the coupling of S1 important flavor constraints in this regard.
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λL ττ νν insensitive to 33, in the tt and bb modes exclude MS1 (3) Tensor operator: up to 900 GeV [87] and 1100 GeV [88], respectively for a 100% BR in each decay mode. However, Ref. [47] O ¯σμν τσ¯ ν TL ¼½c PLb ½ μνPL : showed that the RDð Þ -favored parameter space in C2 is also ruled out by the electroweak precision data on the The operator O is SM-like and the other four operators → ττ VL Z decay. introduce new Lorentz structures into the Lagrangian. Note The above two minimal cases, C1 and C2, are the two that the operator O is identically zero, i.e., extremes. One can, however, consider a next-to-minimal TR λL λL situation, where both 23 and 33 are nonzero to explain O ¯σμν τσ¯ ν 0 TR ¼½c PRb ½ μνPL ¼ : ð22Þ RDð Þ anomalies being within the LHC bounds [48]. However, B → Kð Þνν decay results severely constrain such a scenario due to the tree-level leptoquark contribution The S1 leptoquark that we consider can generate only R OV ;S ;T . Hence, the coefficients of the other two oper- [see Fig. 3(c)]. References [32,45] indicated that a large λ23 L L L ators, namely, C and C remain zero in our model. In might help explain various flavor anomalies simultaneously VR SR while being consistent with other relevant experimental terms of the S1 parameters the Wilson coefficients can be results. expressed as, L L R In this paper, we allow λ23, λ33 and λ23 to be nonzero and λL λL 9 ffiffi 1 23 33 perform a parameter scan for a single S1 solution of the C ¼ p 2 ; > VL 2 2G V 2M > F cb S1 => RDð Þ anomalies. We locate the RDð Þ -favored parameter λL λR → ð Þνν → ττ ffiffi 1 33 23 space that satisfies the limits from B K and Z C − p 2 ð23Þ SL ¼ 2 2 2 ; > GFVcb MS > decays and is still allowed by the latest LHC data. A S1 can 1 ;> ττ C − 1 C also provide new final states at the LHC like þ jets and TL ¼ 4 SL : τ þ =ET þ jets in which leptoquarks have not been searched for before. These relations are obtained at the mass scale MS1 . However, running of the strong coupling constant down R ( ) S B. D with 1 to mb ∼ 4.2 GeV changes these coefficients substantially C In the SM, the semitauonic B decay is mediated by the except for VL which is protected by the QCD Ward C C left-handed charged currents and the corresponding four- identity. As a result, the ratio SL = TL becomes, Fermi interactions are given by the following effective Lagrangian: C C SL SL ¼ ρðm ;M Þ ¼ −4ρðm ;M Þ: ð24Þ C b S1 C b S1 TL m TL M 4GF μ b S1 L ¼ − pffiffiffi V ½c¯γ P b ½τγ¯ μP ντ : ð20Þ SM 2 cb L L The modification factor ρ can be obtained from Ref. [32], In the presence of new physics, there are a total of five four- and we display it in Fig. 4. In terms of the nonzero Wilson SM coefficients we can express the ratios r ð Þ R ð Þ =R Fermi operators that appear in the effective Lagrangian for D ¼ D Dð Þ the B → Dð Þτν decay [89], as [49],
4G L ⊃−pffiffiffiF V ½ð1 þ C ÞO þ C O þ C O 2 cb VL VL VR VR SL SR C O C O þ SR SR þ TR TR ; ð21Þ where the CX’s are the Wilson coefficients associated with the effective operators: (1) Vector operators:
O ¯γμ τγ¯ ν VL ¼½c PLb ½ μPL ; O ¯γμ τγ¯ ν VR ¼½c PRb ½ μPL :
(2) Scalar operators:
O ¼½cP¯ b ½τ¯P ν ; SL L L ρ FIG. 4. The ratio defined in Eq. (24), ðmb;MS1 Þ¼ O ¯ τ¯ ν C C S ¼½cPRb ½ PL : SL μ SL μ R C ð ¼ mbÞ= C ð ¼ MS1Þ, obtained from Ref. [32]. TL TL
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TABLE IV. Summary of the RDð Þ related inputs for our parameter scan. Observable Experimental average SM expectation Ratio Value
RD 0.340 0.027 0.013 [16] 0.299 0.003 [12] rD 1.137 0.101 RD 0.295 0.011 0.008 [16] 0.258 0.005 [16] rD 1.144 0.057 FLðD Þ 0.60 0.08 0.035 [10,11] 0.46 0.04 [92] fLðD Þ 1.313 0.198 0 21 τ −0 38 0 51þ . −0 497 0 013 τ 0 766 1 093 P ðD Þ . . −0.16 [90] . . [89] p ðD Þ . .
R 2 2 2 1 7.02 r ≡ D ≈ j1 þ C j þ 1.02jC j þ 0.9jC j 1 C 2 0 08 C 2 D SM VL SL TL fLðD Þ¼ j þ VL j þ . þ 2 j SL j R r 16ρ D D þ 1.49Re½ð1 þ C ÞC 4.37 VL SL − 0.24 − Re½ð1 þ CV ÞC ; ð31Þ 4ρ L SL þ 1.14Re½ð1 þ C ÞC ; ð25Þ VL TL RD 2 2 2 1 2 1.86 2 r ≡ ≈ j1 þ C j þ 0.04jC j þ 16.07jC j 1 C − 0 07 C D SM VL SL TL pτðD Þ¼ j þ VL j . þ 2 j SL j R r 16ρ D D − 0.11 Re½ð1 þ C ÞC 3.37 VL SL þ 0.22 þ Re½ð1 þ CV ÞC : ð32Þ 4ρ L SL − 5.12 Re½ð1 þ C ÞC : ð26Þ VL TL
With Eq. (24) one can simplify the above equations as, These two observables have the power to discriminate between new physics models with different Lorentz struc- 0 9 tures (see e.g., Ref. [91]). In Table IV, we list the bounds on 1 C 2 1 02 . C 2 rD ¼j þ V j þ . þ 2 j S j L 16ρ L the RDð Þ related observables that we include in our parameter scan. 1.14 þ 1.49 − Re½ð1 þ C ÞC ; ð27Þ VL SL 4ρ νν C. Constraint from R ( ) K 2 16.07 2 The SM flavor-changing neutral-current b → sνν¯ tran- r ¼j1 þ C j þ 0.04 þ jC j D VL 16ρ2 SL sition proceeds through a loop and is suppressed by the 5 12 Glashow-Iliopoulos-Maiani mechanism, whereas in our . − 0.11 − Re½ð1 þ CV ÞC ; ð28Þ model, S1 can mediate this transition at the tree level 4ρ L SL [see Fig. 3(c)]. Therefore, this neutral-current decay can ρ ρ heavily constrain the parameter space of our model. We where ¼ ðmb;MS1 Þ. There are two other observables define the following ratio: related to the RD —the longitudinal D polarization τ FLðD Þ and the longitudinal polarization asymmetry ð Þ ΓðB → K ννÞ PτðD Þ—which have recently been measured by the Belle Rνν ¼ : ð33Þ Kð Þ Γ → ð Þνν Collaboration [10,11,90]. In terms of the nonzero Wilson ðB K ÞSM coefficients in our model, FLðD Þ and PτðD Þ are expressed as [49], The current experimental 90% confidence limit (C.L.) νν 3 9 upper limits on the above quantities are RK < . and 1 νν νν FLðD Þ 2 2 R < 2.7 [93]. In terms of our model parameters, R is f D ≡ ≈ 1 C 0.08 C K Kð Þ Lð Þ SM fj þ VL j þ j SL j FL ðD Þ rD given by the following expression: 7 02 C 2 − 0 24 1 C C þ . j TL j . Re½ð þ VL Þ S L 2a λL λL a2 λL λL 2 − 4 37 1 C C νν 1 − 23 33 23 33 . Re½ð þ Þ g; ð29Þ R ð Þ ¼ 2 Re þ 4 ; ð34Þ VL TL K 3 3 MS1 VtbVts MS1 VtbVts 1 PτðD Þ 2 2 pffiffiffi pτðD Þ ≡ ≈ fj1 þ C j − 0.07jC j 2 2 SM SM SM VL SL where a ¼ 2π =ðe G jC jÞ with C ≈−6.38 [32]. Pτ ðD Þ rD F L L We use this constraint in our analysis and find that it − 1.86 × jC j2 þ 0.22 Re½ð1 þ C ÞC TL VL SL significantly restricts our parameter space. Note that this L L R − 3 37 1 C C constraint applies on λ23 and λ33 but not on λ23. In Fig. 5, . Re½ð þ VL Þ T g: ð30Þ L λL − λL νν 2 7 we show the regions in the 23 33 plane with RK < .
These equations can further be simplified as, for two different values of MS1 .
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output [94] model files from Ref. [48] in MADGRAPH5 [95]. We use the NNPDF23LO [96] parton distribution func- tions. Wherever required, we include the next-to-leading- order QCD K-factor of ∼1.3 for the pair production in our analysis [97].
1. Decay modes of S1 L L R For nonzero λ23, λ33 and λ23, S1 can decay to cτ, sν, tτ and bν states. CKM mixing among quarks enables decays to uτ and dν but we neglect them in our analysis as the off- diagonal CKM elements are small. The BRs of S1 to various decay modes vary depending on the coupling R L L strengths. If λ23 ≫ λ23; λ33, the dominant decay mode is L R L S1 → cτ, whereas for λ23 ≫ λ23; λ33,BRðS1 → cτÞ ≈ L BRðS1 → sνÞ ≈ 50%. On the other hand, when λ33 ≫ L R λ23; λ23, the dominant decay modes are S1 → tτ and S1 → bν with about a 50% BR in each mode. Since partial decay
widths depend linearly on MS1 , BRs are insensitive to the νν 2 7 1 FIG. 5. The region where RK < . for MS1 ¼ and 2 TeV. mass of S1.
2. Production of S D. Constraint from Z → ττ decay 1 Another important constraint comes from the Zττ At the LHC, S1 can be produced resonantly in pairs coupling measurements. The Z → ττ decay is affected or singly and nonresonantly through indirect production t S1 by the S1 loops as shown in Ref. [47]. The contribution ( -channel exchange process). Pair production: The pair production of S1 is dominated of S1 to the Zττ coupling shift (ΔκZττ) comes from a loop with an up-type quark (q) and an S1. The shift scales as the by the strong coupling and, therefore, it is almost model L=R 2 independent. The mild model dependence enters in the square of the S1tτ coupling (λ ) and m . Hence, the q3 q pair production through the t-channel lepton or neutrino dominant contribution comes from when q is the top quark exchange processes. However, the amplitudes of those implying that the Zττ coupling measurements can restrict λ2 L L R diagrams are proportional to and generally suppressed only λ33 but not λ23 or λ23. For instance, we see from λ λ for small values (for bigger MS1 and large values, this Ref. [47] that λL ≳ 1.4 can be excluded for M ∼ 1 TeV 33 S1 part could be comparable to the model-independent part of 2σ with confidence. We incorporate this bound into our the pair production). Pair production is heavily phase-space parameter scan. suppressed for large MS1 and we find that its contribution is very small in our recast analysis. Pair production can be E. LHC phenomenology and constraints categorized into two types depending on the final states: We now make a quick survey of the relevant LHC symmetric, where both leptoquarks decay to the same phenomenology of a TeV-range S1 that couples with τ, ν modes, and asymmetric, where the two leptoquarks decay and s and c quarks. For this discussion we compute all the via two different modes. These two types give rise to necessary cross sections using the universal FeynRules various novel final states.
Symmetric modes:
⌒ ⌒ ⌒ ⌒ → ⌒τ ⌒τ ≡ττ 2 τ τ ≡ ττ ⌒ν ⌒ν ≡2 ν ν ≡2 S1S1 c c þ j; t t tt þ ;ss j þ =ET;bb b þ =ET:
Asymmetric modes:
⌒ ⌒ → ⌒τ ⌒ν ≡τ 2 ⌒τ ν ≡τ ⌒τ τ ≡ττ S1S1 c s þ j þ =ET;cb þ b þ j þ =ET;ct þ t þ j; ⌒ ⌒ ⌒ ⌒ → ⌒ν ν ≡ ⌒ν τ ≡ τ ν τ ≡ τ S1S1 s b b þ j þ =ET;st t þ þ j þ =ET;bt t þ þ b þ =ET;
015011-10 … ADDRESSING THE RDð Þ ANOMALIES WITH AN PHYS. REV. D 101, 015011 (2020) where the curved connection over a pair of particles indicates that the pair is coming from a decay of S1. Searches for leptoquarks in some of the symmetric modes were already done at the LHC [87,88]. Leptoquark searches in some of the symmetric and most of the asymmetric modes are yet to be performed at the LHC. Single production: The single productions of S1, where S1 is produced in association with a SM particle, are fully model dependent as they depend on the leptoquark-quark-lepton couplings. These are important production modes for large couplings and heavier masses (since single productions receive less phase-space suppression than the pair production). Depending on the final states, single productions can be categorized as follows. Symmetric modes:
⌒ ⌒ ⌒ S1τX → τcτ þ τcτj ðþτcτjj þ Þ≡ ττ þ jets; ⌒ ⌒ ⌒ S1τX → τtτ þ τtτj ðþτtτjj þ Þ≡ ττ þ t þ jets; S ν → ν⌒ ν ν⌒ ν ν⌒ ν ≡ 1 X s þ s j ðþ s jj þ Þ =ET þ jets; ⌒ ⌒ ⌒ S ν → ν ν ν ν ν ν ≡ 1 X b þ b j ðþ b jj þ Þ =ET þ b þ jets:
Asymmetric modes:
τ → ν⌒ τ ν⌒ τ ν⌒ τ ≡ τ S1 X s þ s j ðþ s jj þ Þ =ET þ þ jets; ν → τ⌒ ν τ⌒ ν τ⌒ ν ≡ τ S1 X c þ c j ðþ c jj þ Þ =ET þ þ jets; ⌒ ⌒ ⌒ τ → ν τ ν τ ν τ ≡ τ S1 X b þ b j ðþ b jj þ Þ =ET þ þ b þ jets; ⌒ ⌒ ⌒ ν → τ ν τ ν τ ν ≡ τ S1 X t þ t j ðþ t jj þ Þ =ET þ þ t þ jets:
Here j stands for an untagged jet and “jets” means any TeV-scale S1 if λ is large, this destructive interference number (≥1) of untagged jets. These extra jets can be becomes its dominant signature in the leptonic final states. either radiation or hard (genuine three-body single pro- duction processes can have sizeable cross sections; see 3. Constraints from the LHC – Refs. [98 100] for how one can systematically compute The mass exclusion limits from the pair-production them). As single production is model dependent, the searches for S1 at the LHC are as follows. Assuming a relative strengths of these modes depend on the relative 100% BR in the S → tτ mode, a recent search at the CMS strengths of the coupling involved in the production as well detector has excluded masses below 900 GeV [87]. as the BR of the decay mode involved. Similarly, for a leptoquark that decays exclusively to bν Indirect production: Indirect production is the nonreso- or sνð≡jν) final states, the exclusion limits are at 1100 and nant process where a leptoquark is exchanged in the t 980 GeV [88], respectively. However, going beyond simple τ ν channel. With leptoquark couplings to and , this mass exclusions, we make use of the analysis done in basically gives rise to three possible final states: Ref. [48] for the LHC constraints. It contains the inde- ττ ττ τν τ νν ð þ jetsÞ, ð=ET þ þ jetsÞ and ð=ET þ jetsÞ. The pendent LHC limits on the three couplings shown in λ2 amplitudes of these processes are proportional to .So Eq. (17) as functions of M as well as a summery of λ4 λ S1 the cross section grows as . Hence, for an order-one , the direct-detection exclusion limits. indirect production has a larger cross section than other Apart from the processes with =ET þ jets final states, all production processes for large MS1 (see Fig. 2 of Ref. [48]). other production processes can have either ττ þ jets final τ → However, the indirect production substantially interferes states or =ET þ þ jets final states. Hence, the latest pp with the SM background process pp → Vð Þ → ll Z0 → ττ and pp → W0 → τν searches at the ATLAS (l ¼ τ=ν). Though the interference is Oðλ2Þ, its contribu- detector [102,103] were used to derive the constraints in L tion can be significant for a TeV-scale S1 because of the Ref. [48]. There we notice that the limits on λ23 from the τν large SM contribution (larger than both the direct produc- data are weaker that the ones obtained from the ττ data. The 4 R tion modes and the λ indirect contribution, assuming ττ data also constrain λ23. From the earlier discussion, it is λ ≳ 1). In general, the interference could be either con- clear that the interference contribution plays the dominant structive or destructive depending on the nature of the role in determining these limits. However, its destructive leptoquark species and its mass [101].ForS1, we find that nature means that in the signal region one would expect less the interference is destructive in nature [48]. Hence, for a events than in the SM-only predictions. Hence, the limits
015011-11 AYDEMIR, MANDAL, and MITRA PHYS. REV. D 101, 015011 (2020) pffiffiffiffiffiffi pffiffiffiffiffiffi range − 4π to 4π (i.e., jλj2=4π ≤ 1). We do not consider complex values for the couplings. In Fig. 7, we show the outcome of our scan with different two-dimensional pro- jections. In every plot we show two couplings and allow the third coupling to vary. In each of these plots we show the following. (1) The flavor ∪ EW (FEW) regions: The orange dots mark the regions favored by the RDð Þ observables within 95% C.L. while satisfying the available νν bounds on the F D , Pτ D and R observ- Lð Þ ð Þ Kð Þ ables (flavor bounds). In addition, these points also L satisfy the bound on λ33 coming from the Z → ττ decay within 95% C.L. [47] (electroweak bound). (2) The flavor ∪ EW ∪ LHC (FEWL) regions: As we L=R pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi take into account the limits on λ23 from the ATLAS L=R L 2 R 2 FIG. 6. Two-sigma exclusion limits on λ23 ¼ ðλ23Þ þðλ23Þ pp → ττ data from the 13 TeV LHC [48] along with as a function of MS1 as obtained in Ref. [48] from the ATLAS the previous constraints we obtain the regions pp → ττ data [102]. The colored region is excluded. marked by the green points. These are the points that survive all the limits considered in this paper. L R were obtained assuming that either λ23 or λ23 is nonzero at a From the plots we see that substantial portions of parameter χ2 regions survive after all the constraints. This implies that time or by performing a test of the transverse mass (mT) L the S1 model can successfully explain the R ð Þ anomalies. distributions of the data. As, for heavy S1, the limits on λ23 D R If one looks only at the R ð Þ anomalies, in principle, one and λ23 are dominantly determined by the interference of D λL λL the indirect production, they are very similar. We can can just set 23 and/or 33 to be large. But coupling values that make C [see Eq. (23)] big come into conflict with the translate these limits from the ττ data on any combination VL νν L R L=R R bound [see Eq. (34)]. This is why we do not see any of λ23 and λ23 in a simple manner assuming λ23 ¼ Kð Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L L L 2 R 2 point where both λ23 and λ33 are large in the first column of ðλ23Þ þðλ23Þ . In Fig. 6 we display the limits on λL=R λL=R as a function of M .4 Fig. 7. In addition, the LHC puts bounds on 23 [48] 23 S1 → ττ The LHC data is insensitive to λL as it was shown in whereas the Z data puts a complimentary bound on 33 λL λL → ττ Ref. [48]. This can be understood from the following 33 [47]. The restriction on 33 from the Z data can be 1 argument. First, the pair production is insensitive to this seen in Figs. 7(a) and 7(c) for MS1 ¼ TeV and Figs. 7(d) 2 coupling as we have already mentioned. Second, the single- and 7(f) for MS1 ¼ TeV, whereas from the middle production process pp → tτν via an S1 has too small a column [i.e., Figs. 7(b) and 7(e)] it is clear that the LHC ∼2 λL ∼ 1 1 prevents both λL and λR from taking large values simulta- cross section ( fb for 33 and MS1 ¼ TeV) to make 23 23 R L any difference at the present luminosity. Finally, there is no neously. From the λ23 vs λ33 plots [Figs. 7(c) and 7(f)]we ττ τ interference contribution in the and þ =ET channels as see that these two couplings take opposite signs mainly λL there is no t quark in the initial state. Hence, 33 remains because both RD and FLðD Þ prefer a positive CSL [see unbounded from these searches. Eqs. (28) and (31)].
F. Parameter scan G. RG running of the Yukawa couplings
To find the RDð Þ -favored regions in the S1 parameter and perturbativity space that are not in conflict with the limits on FLðD Þ, One of the questions raised by the introduction of new νν Pτ D , R , Z → ττ decay and the bounds from the LHC, ð Þ Kð Þ Yukawa couplings is whether the new model remains 1 we consider two benchmark leptoquark masses: MS1 ¼ perturbative up to high-enough energies. This is particu- L R and 2 TeV. We allow all three free couplings, λ23, λ23 and larly important in the GUT framework since the RG L λ33 to vary. For every benchmark mass, we perform a running of the gauge couplings is performed under the random scan over the three couplings in the perturbative assumption of perturbativity. Fortunately, with the latest LHC data, we have quite a large available parameter space 4 λL in the deep perturbative region, as can be seen in Fig. 7. Actually, for MS1 between 1 and 2 TeV, the limits on 23 are R While this suggests that the model is safe in terms of slightly stronger than those on λ23 because in the SM the Z boson couples differently to left- and right-handed τ’s. However, we perturbativity as long as the new Yukawa couplings are L ignore this minor difference and take the stronger limits on λ23 as small enough at the electroweak scale, it is still informative L=R the limits on λ23 to remain conservative. to investigate the RG running in detail, especially the case
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FIG. 7. Two-dimensional projections of the regions in the S1 parameter space allowed by the bounds on the RDð Þ , FLðD Þ, PτðD Þ and Rνν observables and the Z → ττ decay, i.e., the FEW regions (orange) (see Sec. IV F), and the LHC constraints in addition to all these Kð Þ 1 2 constraints, i.e., the FEWL regions (green) for MS1 ¼ TeV (upper panel) and MS1 ¼ TeV (lower panel). We assume all the couplings are real. in which the Yukawa couplings take larger initial values. Note that although we perform the Yukawa RG running 5 We address this issue in this part of the paper. based on the SM augmented with a TeV-scale leptoquark S1 The equations for the RG running of the Yukawa all the way up to the UV, these equations are prone to couplings are given in the Appendix. Results for various changes above the intermediate symmetry-breaking scale benchmark cases are displayed in Fig. 8. Among the S1 provided that there is one (as in the examples studied in mass values we have considered in this paper, the most Sec. III), mainly due to contributions from the running of 2 constrained parameter space is that of MS1 ¼ TeV. the scalars whose masses are around the intermediate scale. Therefore, we choose our benchmark values from the However, these effects are expected to be minor due to the parameter space for this mass value, displayed in Fig. 7. corresponding beta-function coefficients not being large L L Note that λ33ðMZÞ and λ23ðMZÞ cannot both be large and enough to significantly change the logarithmic RG running L R that λ33ðMZÞ and λ23ðMZÞ must have opposite signs, as can [71]. This is even more likely to be the case especially if L be seen in Fig. 7. When λ23ðMZÞ is taken in the interval this symmetry-breaking scale, for instance the scale in our ½−0.2; 0.1 , the system remains perturbative up to the grand scenario where the Pati-Salam symmetry is spontaneously 15 17 L unification scale, ∼10 –10 GeV, even when jλ33ðMZÞj, broken to the symmetry of the SM, is considerably close to R jλ23ðMZÞj ≈ 1, as displayed in the first six plots in Fig. 8. the scale of the SO(10) symmetry-breaking scale (as in the However, it deteriorates quickly for larger values of second case in Sec. III, namely model A2), suggesting that L R L jλ33ðMZÞj and jλ23ðMZÞj [Fig. 8(g)]. When jλ23ðMZÞj is the supposed modification in the Yukawa running is indeed L not an issue of concern since the slow logarithmic running large, the parameter space is quite limited for jλ33ðMZÞj, which is in the [0.10, 0.15] band (Fig. 7). In this case as would most likely not significantly alter the outcome in this L small interval. Threshold corrections due to the intermedi- well, the system is well behaved up to jλ23ðMZÞj, R ate scale are also known to be subleading to the one-loop jλ23ðMZÞj ≈ 1 [e.g., Fig. 8(h)], and the situation declines for the values above in that the perturbativity bound is running [75]. reached below the unification scale [e.g., Fig. 8(i)]. Furthermore, although we have studied some specific examples for gauge coupling unification, there are no
5 restrictions on the choice of the mass values of the high- A perturbativity analysis for the case of the Standard Model energy field content regarding which fields remain heavy augmented by a leptoquark S1 was done in Ref. [80], as well. Note that their Yukawa matrix is flavor diagonal, and hence at the unification scale and which ones slide through different from the one in this paper. the intermediate scale, as long as the gauge coupling
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(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
FIG. 8. Behavior of Yukawa couplings with various benchmark values at the EW scale. Thep labelsffiffiffiffiffi of the couplings are given in the first plot. The dashed horizontal lines denote the values of the assumed perturbativity bound, 4π. The dashed vertical line denotes the energy scale at which this bound is first reached. unification is realized (and as long as the intermediate scale pass, as long as the high-energy content of the theory is is high enough to evade the proton-decay constraints for the chosen such that the intermediate symmetry breaking terms not forbidden by any symmetry in the Lagrangian). occurs before the perturbativity bound is reached and such The main point of our case that we emphasize is that if a that the couplings remain in the perturbative realm up to the SO(10) theory is indeed the UV completion of the SM, unification scale. then one may naturally anticipate a TeV-scale leptoquark We comment in passing on the unification-scale impli- accompanying the SM Higgs field, and this could define cations of our model regarding the fermion mass spectrum. the field content up to very high energies. Beyond that, one It has been known in the literature that obtaining a realistic has the freedom to choose the high-energy particle content Yukawa sector in the SO(10) framework is not trivial. None and the corresponding potential terms in the Lagrangian of the single- and dual-field combinations of the scalar that lead to an appropriate symmetry-breaking sequence in fields 10H, 120H, and 126H yields GUT-scale relations which the Yukawa couplings remain in the perturbative between fermion masses consistent with the SM values realm above the intermediate symmetry-breaking scale. [68,75]. On the other hand, it was concluded in Ref. [75] With this in mind, even the other cases, displayed in that a Higgs sector consisting of a real 10H, a real 120H, and Figs. 8(g) and 8(i), that suffer from the perturbativity a complex 126H can provide a fermion mass spectrum problem at relatively low energies could arguably get a at the unification scale that matches the expected values
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16 TABLE V. Fermion masses at the unification scale MU ¼ 2 × 10 GeV that are selected for the sake of argument as the exponential midpoint of the unification scales of the high-energy scenarios discussed in Sec. III. Clearly, inclusion of S1 in the low-energy particle spectrum does not lead to significant changes in the fermion mass values at L R L the unification scale compared to the SM predictions, especially in the deep perturbative region in the ðλ33; λ23; λ23Þ parameter space.
L R L ðλ33; λ23; λ23Þ Fermion masses/ratios (0,0,0); SM ð0.5; −0.4; 0.1Þð0.8; −0.9; 0.1Þð0.1; −1.0; 1.0Þ
mt=mb 75.57 75.66 76.37 75.36 mτ=mb 1.63 1.74 2.31 4.28 mμ=ms 4.41 4.46 4.46 3.94 me=md 0.400 0.401 0.401 0.401 mt=GeV 79.32 79.52 83.46 78.14 mc=GeV 0.254 0.254 0.275 0.344 −3 mμ=ð10 GeVÞ 100.505 100.576 101.518 100.175 −3 me=ð10 GeVÞ 0.476 0.476 0.481 0.475 −3 mu=ð10 GeVÞ 0.469 0.464 0.469 0.462
obtained by the RG running of the SM with small threshold significant changes regarding the expected fermion masses corrections at the intermediate scale. Since this is the scalar at the unification scale and therefore the analysis of sector we adopt in this paper, it is informative to inspect Ref. [75], based on using the extrapolated SM values at whether the light leptoquark S1 can register significant the unification scale as input data in order to numerically fit changes to the expected fermion mass values at the them to the parameters of the Yukawa sector of their SO unification scale obtained via the SM RG running. (10) model, can be applied in our case as well. The results, obtained by using the equations given in the Appendix and the same values for the input parameters at V. SUMMARY AND DISCUSSIONS MZ as in Ref. [75], are given in Table V, some of which are displayed in terms of mass ratios relevant in the In this paper, we considered the scenario that there is a SO(10) framework. The unification scale is selected as single scalar leptoquark, S1, at the TeV scale, and it is the 16 color-triplet component of a real 10 of SO(10), which M ¼ 2 × 10 GeV, which is around the exponential U also contains a SU(2) doublet which is identified as the midpoint of the unification scales of our models A1 and SM Higgs. In this scenario, the leptoquark being the only A2 discussed in Sec. III; small numerical differences scalar entity other than the SM Higgs is natural; a depending on the M value in that range are not relevant U peculiar mass splitting between the components of 10 to our discussion here. Our values for the fermion masses at does not occur and the leptoquark picks up an electro- M in the SM case differ from the ones in Ref. [75] by U weak-scale mass together with the SM Higgs, as 1–5%, which might be due to a combination of effects expected. This is appealing because this leptoquark by coming from the running of the right-handed neutrinos itself can potentially explain the B-decay anomalies at the above the intermediate scale and the threshold corrections, 6 LHC [21,22,25,32]. both of which are ignored in our estimation. The results for The SO(10) grand unification framework is an appealing the leptoquark case are given for three benchmark points scenario, which has been heavily studied in the literature in the ðλL ; λR ; λL Þ parameter space that are consistent with 33 23 23 [62,64–75]. It unifies the three forces in the SM, explains the perturbativity analysis above. As expected, when all the quantization of electric charge, provides numerous dark three of the new Yukawa couplings are in the deep matter candidates, accommodates a seesaw mechanism for perturbative region, the differences from the SM values small neutrino masses, and justifies the remarkable can- remain insignificant. When the couplings get larger close to cellation of anomalies through the anomaly-free nature of unity, some deviations are observed yet they do not become the SO(10) gauge group. Moreover, the fermionic content substantial. Therefore, we conclude that the addition of S1 of the SM fits elegantly in 16 , including a right-handed to the particle content up to the TeV scale does not lead to F neutrino for each family. The particles in the SM (except probably the neutrinos) acquire masses up to the electro- 6We do not discuss neutrino masses in this paper but have no weak scale through interacting with the electroweak-scale reason to suspect that the outcome would be different than that in Higgs field, which is generally assigned to a real 10 . Ref. [75] particularly because their intermediate symmetry- H breaking scales at which the generation of neutrino masses Considering that the leptoquark S1 is the only other 10 through the seesaw mechanism occurs are quite close to the component in H, a TeV-scale S1, from this perspective, ones in our models A1 and A2. is consistent with the idea that it is the last piece of the
015011-15 AYDEMIR, MANDAL, and MITRA PHYS. REV. D 101, 015011 (2020) puzzle regarding the particle content up to the electroweak High Energy Physics, Chinese Academy of Sciences, under scale (modulo the right-handed neutrinos). Therefore, the Contract No. Y9291120K2; and the National Natural possible detection of S1, in the absence of any other new Science Foundation of China (NSFC) under Grant particles, at the TeV scale could be interpreted as evidence No. 11505067. T. M. is financially supported by the in favor of SO(10) grand unification. Royal Society of Arts and Sciences of Uppsala as a One obvious issue of concern is proton decay since the researcher at Uppsala University and by the INSPIRE leptoquark S1 possesses the right quantum numbers for it Faculty Fellowship of the Department of Science and to couple to potentially dangerous diquark operators. On Technology (DST) under Grant No. IFA16-PH182 at the the other hand, the proton stability could possibly be University of Delhi. S. M. acknowledges support from the ensured through various mechanisms [22,38,68,76–78]. Science and Engineering Research Board (SERB), DST, In this paper, we ensured the proton stability by assuming India under Grant No. ECR/2017/000517. a discrete symmetry that is imposed in an ad hoc manner below the Pati-Salam breaking scale. It would certainly be more compelling to realize a similar mechanism at the APPENDIX: RENORMALIZATION GROUP fundamental level although it seems unlikely that it RUNNING OF THE YUKAWA COUPLINGS would interfere with the bottom line of this work. The new Yukawa matrices in the Lagrangian given in Having a single S1 leptoquark at the TeV scale as the Eq. (9) are taken in our setup as only new physics remnant from our SO(10) GUT model, 0 1 0 1 we investigated how competent this S1 is at addressing 00 0 00 0 the R ð Þ anomalies while simultaneously satisfying other B C B C D ΛL → 00λL ΛR → 00λR relevant constraints from flavor, electroweak and direct @ 23 A and @ 23 A: L LHC searches. We adopted a specific Yukawa coupling 00λ33 00 0 texture with only three free (real) nonzero couplings viz. L L R ðA1Þ λ23, λ33 and λ23. We have found that this minimal consideration can alleviate the potential tension between νν Following Ref. [105] (or implementing the model in the R ð Þ -favored region and the R measurements. The D Kð Þ SARAH [106]), the corresponding one-loop RG equations → ττ λL ττ Z decay constrains 33 whereas the resonance can be found as λL search at the LHC puts complimentary bounds on 23 and λR . By combining these constraints with all the relevant 13 23 2 R 3 R 2 2 16π βλR ¼ 3ðλ23Þ − λ23 g1 þ 4g3 flavor constraints coming from the latest data on RDð Þ , 23 3 νν F D , Pτ D , R we have found that a substantial Lð Þ ð Þ Kð Þ R L 2 L 2 þ 2λ23ððλ23Þ þðλ33Þ Þ; ðA2Þ region of the RDð Þ -favored parameter space is allowed. Our multiparameter analysis clearly shows that contrary 2 5 9 to the common perception, a single leptoquark solution to 2 L 3 L yt 2 2 2 16π βλL ¼ 4ðλ33Þ − λ33 − þ g1 þ g2 þ 4g3 33 2 6 2 the observed RDð Þ anomalies with S1 is still a viable L R 2 L 2 solution. þ λ33ððλ23Þ þ 4ðλ23Þ Þ; ðA3Þ Evidently, by introducing new d.o.f. into our frame- work, one can, a priori, enlarge the allowed parameter 5 9 2 L 3 L 2 2 2 16π βλL ¼ 4ðλ23Þ − λ23 g1 þ g2 þ 4g3 region. For example, by considering some of the cou- 23 6 2 plings as complex or by choosing a complex (instead of a λL λR 2 4 λL 2 real) 10 representation of SO(10), which will introduce þ 23ðð 23Þ þ ð 33Þ Þ; ðA4Þ an additional S1 and another complex scalar doublet at 3 17 9 the TeV scale, one can relax our obtained bounds. We 16π2β 3 3 3 − 2 − 2 − 8 2 yt ¼ yt þ yt yt g1 g2 g3 also pointed out search strategies for S1 at the LHC using 2 12 4 symmetric and asymmetric pair- and single-production 1 λL 2 channels. Systematic studies of these channels are dis- þ 2 ytð 33Þ ; ðA5Þ cussed elsewhere [104]. 2 2 17 2 9 2 2 16π β ¼ y 3y − g1 − g2 − 8g3 ACKNOWLEDGMENTS yc c t 12 4 1 We thank Diganta Das for valuable discussions and R 2 L 2 þ ycððλ23Þ þðλ23Þ Þ; ðA6Þ collaboration in the early stages of the project. Work of 2 U. A. is supported in part by the Chinese Academy of 17 9 ’ 2 2 2 2 2 Sciences President s International Fellowship Initiative 16π βy ¼ yu 3yt − g1 − g2 − 8g3 ; ðA7Þ (PIFI) under Grant No. 2020PM0019; the Institute of u 12 4
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2 3 2 5 2 9 2 2 1 2 15 9 16π β ¼ y y − g − g − 8g þ y ðλL Þ ; 2 2 2 2 yb b t 1 2 3 b 33 16π β ¼ yμ 3y − g − g ; ðA12Þ 2 12 4 2 yμ t 4 1 4 2 ðA8Þ 15 9 5 9 1 16π2β 3 2 − 2 − 2 2 2 2 2 2 L 2 yτ ¼ yτ yt g1 g2 16π β ¼ y 3y − g1 − g2 − 8g3 þ y ðλ23Þ ; 4 4 ys s t 12 4 2 s 3 λR 2 λL 2 λL 2 ðA9Þ þ 2 yτðð 23Þ þð 23Þ þð 33Þ Þ; ðA13Þ 5 9 2 2 2 2 2 β ≡ μ dy 16π β ¼ y 3y − g1 − g2 − 8g3 ; ðA10Þ where y μ and the RG running of the gauge couplings yd d t 12 4 d are performed in the usual way according to Eq. (10). In the Yukawa running above, only the dominant terms are taken 2 2 15 2 9 2 16π βy ¼ ye 3yt − g1 − g2 ; ðA11Þ into account since the subleading ones are significantly e 4 4 suppressed and have no noticeable effects on our analysis.
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