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2018 Closing the window on single leptoquark solutions to the B-physics anomalies A. Angelescu

D. Becirevic

D. A. Faroughy

O. Sumensari

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This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Faculty Publications, Department of Physics and Astronomy by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. JHEP10(2018)183 ) ∗ ( SM D 2 TeV ÷ > R 1 Springer ) ∗ ' ( exp D LQ R October 18, 2018 October 19, 2018 October 29, 2018 modes, such as ) by two orders September 6, 2018 : : : m : ∓ 2 `  1 Kµτ s` , d,e → Revised → Accepted Published model as well. Received B b ( 1 B U Published for SISSA by https://doi.org/10.1007/JHEP10(2018)183 -physics anomalies, viz. and O. Sumensari B c , which we re-examine in its minimal form (letting only [email protected] 1 , U D.A. Faroughy b [email protected] , 1 TeV can alone accommodate the above mentioned anomalies. ' . . To do so we combine the constraints arising both from the low- LQ 3 ) ∗ ( m SM K 1808.08179 D. Beˇcirevi´c, The Authors. a < R c Beyond Standard Model, Effective Field Theories

). Improving the experimental upper bound on We examine various scenarios in which the Standard Model is extended by a ) , ∗ ( [email protected] -physics anomalies exp K Kµτ R B → B Dipartamento di Fisica eVia Astronomia “G. Francesco Marzolo Galilei”, 8, Universit`adi 35121 Padova, Padova,E-mail: Italy [email protected] CNRS, Univ. Paris-Sud, Universit´eParis-Saclay, 91405J. Orsay, Stefan France Institute, Jamova 39, P.O. Box 3000,Istituto 1001 Nazionale Ljubljana, Fisica Nucleare, Slovenia SezioneI-35131 di Padova, Padova, Italy Department of Physics andLincoln, Astronomy, NE, University 68588, of U.S.A. Nebraska-Lincoln, Laboratoire de Physique Th´eorique(Bˆat.210), ( b c e d a Open Access Article funded by SCOAP of magnitude could compromise the viability of the minimal Keywords: ArXiv ePrint: is a vector leptoquark, knownleft-handed as couplings todirect have searches non-zero are complementary values). tofind the a We low-energy physics rather find constraints. stable thatB In lower particular, bound the we on limits the deduced lepton flavor from violating light leptoquark state toor/and solve for oneenergy or observables both and fromleptoquarks direct of searches mass at theThe LHC. only single We leptoquark find scenario that which can none provide a of viable the solution for scalar Abstract: A. Angelescu, Closing the window onthe single leptoquark solutions to JHEP10(2018)183 4 5 7 8 9 7 13 10 10 11 20 (1.1) larger is ) ∗ ( exp D R , } e,µ ∈{ l

) ) ) ¯ ν ¯ ∗ ν ( l τ ) ) D ∗ ∗ ( ( R D D ¯ ` 12 ` → → – 1 – B → B ( ( ). Since that time, Belle and LHCb measured the ¯ B ν and/or B pp τ ) ) 3 ∗ ∗ = ( ( 15 ) K ∗ 25 D 22 ( 20 ] in which they measured 12 R D 3 tails of 2 / → ) 6 6 , 3 3 3 3 R ∗ 2 / / / / 1 / / ( 10 T 7 1 7 2 2 1 1 B p ) ) ) ) ) K -mesons pointing towards LFUV. The first such indication was ) ) 1) ( ∗ 2 2 ( 1 3 , 3 1 R B B , , , , , , D 3 3 3 3 ¯ 3 ¯ 3 R and = (3 = ( = ( = ( = ( = ( = ( 1 2 2 1 + 2 1 3 1 3 and ` e U U S R R S U ] and observed a similar feature, namely that the measured ¯ − 1 ν 7 phenomenology s` c` – 3 T → → p 3.2.2 3.1.2 3.1.3 3.1.4 3.2.1 b b 3.1.1 6.1 Low-energy constraints 6.2 Results and predictions 4.1 Direct limits4.2 on pair-produced LQs Limits from high- 3.2 Vector leptoquarks 2.1 2.2 3.1 Scalar leptoquarks and found an excesssame in ratio [ Over the past several years wegin witnessed of a lepton growing flavor interest universality inhints violation theoretical in (LFUV), studies motivated of weak by the decays a ori- of reported number by of BaBar experimental in refs. [ 7 Summary and conclusion 1 Introduction 5 Which leptoquark model? 6 Revisiting 4 High- 3 Leptoquark models for Contents 1 Introduction 2 Effective theory description JHEP10(2018)183 2 1 = )” q A (1.4) (1.5) (1.2) (1.3) SM D − , R V ( 10 . × 0 between ) 2  A ) are commonly . Even though − 68 − ) l } . ∗ ]. Although the p ( V SM K + = 0 19 , + l e, µ, τ 1] . p < R 18 1 , ) ∈ { ∗ ( 0 = ( 045 ` ∗ . , exp K 2 [0 K q R ), , 25 . ¯ ν 0 ` , ) and ,R ∗ , but one should also keep in mind that )  ) ( ) ) values to reveal that the NP scale ∗ 31(2) ∗ ( 10 . ee D ( . µµ 71 ) ) decay has not yet been reached, the SM K ) . 0 ∗ SM D ∗ ∗ ( ( ( ¯ → ν = 0  ,D K < R = 0 ) K c` ∗ ]: B ∗ > R ) 71 ( ( D ) ) ∗ . ( ) → ¯ ¯ → ν exp K ν B → ∗ 17 exp K ( , . The experimentalists of LHCb measured ]. Apart from the reduction of a significant R b B R exp D B = 0 − ( ( 16 0 l 14 0 – 2 – R 6] 1 , – + B J/ψτ 0 J/ψµ B and 1 ,R ∗ . ”. sl ) 12 [1 K ∗ = → → ( ] R c c → SM D 2 2 ) b B excess with respect to (w.r.t.) the SM values, B 41(5) ,q ∗ ≡ ( . ( 2 ( 1 q larger than its SM value. > R ] ∗ [ K σ B B ) K 257(3) [ 8 ∗ R σ 15 = 0 . . ( = 2 exp D D than the values predicted in the SM [ R ≈ R = 0 ,R J/ψ ). They reported [ ∗ R 09 2 . SM D 0 R -physics anomalies  smaller B 75 σ . 5 . ]: ], and 2 8 = 0 11 6] . + , the value predicted in the Standard Model (SM). The most recent HFLAV – ≈ , ) 9 stands for the partial branching fraction comprising [1 K ∗ ( SM J/ψ 0 R (in units of GeV SM D B significance of LFUV in the tree-level 2 R 2 ≡ > R q Apart from the mass effects, different phase space, and moderate hadronic uncertain- Another indication of the LFUV came from the weak decays mediated by a flavor σ For shortness, we only write K 1 300(8) [ exp J/ψ R . R nario is to introduce the couplingsthat of means left-handed that fermions the to New new Physicsform, vector (NP) bosons. which effective operators In are will practice then beaffecting of fit the the with “( charged current the processes measured is very different from the one needed to explain the referred to as the “ ties, no other reason canother be words, found in in the order SMSM, to to one explain explain needs the (or above-mentioned to anomalies. merely invoke accommodate) a In scenario the of observed physics deviations beyond w.r.t. the SM. The simplest effective sce- which are experimental confirmation of theseneeded results to is increase still the lackingthe significance and of indications the the of further observedare LFUV deviations improvement in is w.r.t. do the the SM not SM, generatedphysics concern the by community. quantum fact only The loops, that observations the that stimulated a tree-level considerable decays activity in but the also flavor those that where and changing neutral current (FCNC), experimentalists of LHCb wereenvironment. They able measured to [ confirm the same tendency in another hadronic which again appears to be part of the systematic experimentalrates errors, lies the in advantage the of fact consideringin the that the ratio the fact of Cabibbo-Kobayashi-Maskawa that (CKM) decay case the factors with sensitivity cancel one to out of and hadronic thea uncertainties branching 5 fractions is alone much smaller than it is in the averages are [ which, when combined, give0 3 than JHEP10(2018)183 , , ) T p 4 µτ sµτ → → -physics s b B B bounds on ( B σ -physics anomalies, i.e. B , respectively. In section ) ∗ ( D , we briefly go through the list , the phenomenology of which is R 1 3 U , we remind the reader of a low-energy and to ]. 2 ) Since this kind of models can give rise to ∗ 21 ( 2 K decays) which can be modified w.r.t. the SM transitions, and we present the 1 R the low-energy and high energy constraints are – 3 – ¯ ν 5 c` µµ, ττ → . We will go through various LQ scenarios to examine b -physics anomalies, and for each of them we compute ) , where we also show that the LFV processes → ∗ ( B 6 ], a particular attention will be devoted to the vector LQ SM D pp and 22 > R s`` ) ∗ ( → exp D lepton tails at LHC, can provide complementary constraints and b ), could validate or discard the model in its minimalistic form. Be- R T µτ . p ) ∗ 7 ( K and/or ) → ∗ ( SM K ) for which we will show that the improved experimental bounds on B ] and could be viewed as its update. The new element is the fact that we combine ( 1 -channel LQ exchange. In section B U 20 t < R ) This paper is organized as follows. In section ∗ Research on combining the low-energy constraints to building a SUSY inspired model with the results ( 2 exp K of direct searches at the LHC has been reported in ref. [ 2 Effective theory description Effective theories provide anin which efficient the way short-distance to physics describe is encoded the in low-energy the physics so processes called Wilson coefficients, which survive all the constraints turns outanalyzed to in be more the detail vector in LQ together section with high- can be used tofindings validate in or section invalidate model in its minimalistic form. We summarize our we discuss the boundsBesides on LQ the Yukawa pair couplings production, arisingdistributions of from we lepton the also pairs direct comment (in by searches the on at LHC. the measurementscombined. of We will tails then show of that high- the only one that can accommodate both anomalies and effective description of the effective coefficients whichoperators accompany the relevant to hadronic matrix aof elements general the of NP single dimension-six scenario. LQthe solutions In effective to section Wilson the coefficients relevant to and/or fore we embark onmean the one details multiplet of (singlet, thissame doublet work, quantum or we numbers triplet) should of of the emphasize mass SM that gauge degenerate by group. LQ’s a which single carry LQ the we to ref. [ the updated constraints arisingdeduced from numerous from low-energy the physics direct observableslepton searches with flavor at those violation the (LFV) LHC. [ model ( In this paper we are goingas to a test the mediator possibility of ofR a NP single that scalar can or vector accommodateif leptoquark (LQ) one any of or them boththat remains of by plausible. the extending the In SMthen doing by permits so one to we test LQ will the involves model go the experimentally. least for In number a that of minimalistic respect, new the approach, parameters, current i.e. which paper is similar adjusts Yukawa couplings while keeping theanomalies. NP scale Another the possibility same isalso for allowed both to (such types build as of the a scalarin model and/or which tensor in most currents). which of One othermore of these Lorentz leptoquark the ideas structures most states, can popular are the be scenarios colored tested new are bosons those which couple based to on both the quarks introduction and of leptons. one or LFUV in the FCNC processes. In such a situation one needs to build a model in which one JHEP10(2018)183 . is µν . sµµ c 1 F (2.3) (2.1) (2.2) . ) → b , which , which b 0 i R } + h P 0, or by a O ) ). , µν b i i < ) 0 0 , 2 m sσ e, µ, τ  ` )) ( ) µµ (¯ 9 ) i 5 2 b µ µ ` C γ ( C ( ∈ { i 1 1 m ]. In the following 2 ¯ ¯ ` ` ` 2
O ≡ , 1 2 ( )( 24 )( 1 ` i i ) 0 b b ` C π O R R ))  µ (4 0 ( sP sP ]. Looking for NP in this i (¯  (¯ e/ ) 2 2 C 23 , with ) ) µ 2 2 + 2 ( , cf. ref. [ π π , e e ` 2 2 = ` ` (4 − 1 (4 1 ) + ( 1 ` 7 i ` µ 50) s` ) . process includes the next-to-next- = ( C = O ∗ i 0 (  2 2 → ` − ` O − 1 1 ` ) K , ` b S ` P + µ ) + ( O O 85 → i µ . s` ( 0 C 2 B ` h − 1 → . From this Hamiltonian it is straightforward The result of our fit, illustrated in figure 8 ( ` i , R b 3 O ∈ P – 4 – =7 X and ) i 0. µ 2 µµ ( 10 ↔ ` , < 2 1 ) C ` ) + L ` 1 2 , µ ` − i ` µµ P 10 ) ( 5 i 2 C → C ` γ = h O s ) correspond to the chirality flipped operators, µ µ − ) γ γ B µµ µ 9 1 1 = ( 2.1 ,S,P ¯ ¯ ` ` i ) C ∗ 10 C ( )( )( , µµ X 9 b b K C L L 6 =9 by NP couplings to electrons is disfavored by global analysis of the =1 i i decays. X P P R ) µ µ ]. ∗ + ¯ ( ν ( after replacing 28 ∗ K sγ sγ ts c` i – (¯ (¯ R V and 2 2 can be explained by a purely vector Wilson coefficient, O 25 ) are the Wilson coefficients, while the effective operators relevant to our ) ) tb → 2 2 ∗ µ + 2 π π V e e b ( ` K 2 F (4 (4 2 ` − 1 R 1 G ` √ i 4 = = s` and C 2 2 − ` ` and − 1 1 ` → ` ` 9 10 = K + b O O R ) and s` eff By assuming that the NP couplings to electrons are negligible, it has been established An explanation of µ 3 H ( i → left-handed combination, observables, cf. refs. [ The primed quantities in eq.are ( obtained from to compute the decay rateswe for will omit the dependence on the renormalization scalethat and take in addition to the electromagnetic penguin operator, C study are defined by 2.1 Since we will bewill concerned describe with the both effective Hamiltonian leptoncan for flavor be a conserving written generic and as LFV decay modes we context means to look for the non-SMto contributions the to the operators Wilson coefficients already corresponding the present SM in but thedefiniteness, SM, which in are we addition remind allowed tob on the those the reader that of Lorentz are symmetry the not present low-energy grounds. to effective In theories relevant this to section, both for terms of a number ofsuppressed effective (dimension-six) by operators, powers higher of dimensionand operators a effective being high theories energy forto-leading scale. the logarithmic corrections loop-induced and In the thebeen resummation SM made of potentially by the large means matching logarithms of has between the the renormalization full group equations [ are computed perturbatively, whereas the remaining long-distance physics is expressed in JHEP10(2018)183 , × µµ 10 C 23) .  0  = ) are the b ], and the 65 . µµ ) (2.4) 9 m L C ( ν 17 i , = (3 µ g . Darker (lighter) γ 16 [ L ≡ SM ) µµ 10 ` ) ) 1 i L C ∗ )( g ( ν µµ R R exp K d vs. ` → R µ )( γ s µµ 9 4 R R B ]. C d ( u ., L B ( c 29 . 0 u [ R V 9 )( g + h − µ should be lepton flavor universal ( i 10 ) R R ) + S L V L × g ν g ν . In order to describe the anomalies
µ } ) in the plane μμ µν 9 3 2 . . 10 γ σ ) + -1 0 L C µµ ,T L R +0 − ) ` ` ν 6 R . =C → )( R ( )( 0 . In this fit we used ` 9 L is particularly interesting because it is realized s L L – 5 – σ d )( d C B  µ ( ,S L µµ 10 ) 0 10 γ µν B . d R C L , agrees with the SM value σ 3 ( R -2 decays one necessarily needs to introduce the new u R − L u exp u V and ) )( ¯ ν = = )( = -C ∗ L )( µ 9 c` µµ V K ( µ g ∈ { µµ 9 exp C ( L R ) → C S T i , → s g g K ]. From now on, for notational simplicity, we will omit the µµ B b (1 + -3 ( R + + ) h 31 B ∗ → ( 0.5 0.0 2.0 1.5 1.0 ) accuracy. Blue dashed lines correspond to scenarios with ud

s σ

D -0.5 -1.0 V

10 symmetry which means that B

R (2 F C ( μμ Y σ G B 2 √ stand for a generic up- and down-type quarks, and and 2 U(1) d ¯ ν − × c` L = and . Low-energy fit to → ]. eff u b L 30 [ The possibility of having ”-superscript. Notice that the measured 9 4 − µµ observed in thebosonic exclusive fields above thethe electroweak SU(2) scale. Such an extended theory should10 also respect where effective NP couplings with 2.2 The most general low-energyators effective Lagrangian capable to involving all generate of a (semi-)leptonic the decay dimension-six via oper- charged currents reads theoretically clean in several LQ scenarios“ [ Figure 1 region is allowed to 1 while the black dot denotes the SM point. which deviates from the SM by almost 4 JHEP10(2018)183 T at g 5 30% T . (2.6) (2.7) (2.5) ]. In g 8 . , 4 2 ) 33 

, ¯ ν c which can ≈ − τ 32 = 2 B ) c L T m L S g → m S ) g c , but also other g L ) + B ∗ S ( ( b g and D and B m − R R ( T S τ R g . g S . The first solution we m , g ) we assume that NP only ( 14 ∗ L . ( i 68) S . g D g + ]. By using these theoretical 0 +8 described above, we conclude , R , ], and combine them with the L 14 ) V L 8 ∗ , since this possibility would require 14 ≈ V g ( . L have also been considered in the g . 1 D V L g ]. That relation is modified when S − T R form factors at non-zero recoil are 1 + ( g g

13) 31 . ∗ 2 ) we were able to constrain the values ∈ 0 , ) D ! b 0. Furthermore, plausible solutions are c 1.2 0 can accommodate 2 τ 09 2 B . m → ( > > m and/or m become (0 L L S ) given in ref. [ T L B V ∈ T g − } g S semileptonic form factors computed by means – 6 – g g g τ 1 4 − 4 ) computed in ref. [ 6

D e, µ − , and that its effect is negligible to the electron 2 ) cτν − b q ¯ ν 2 τ ( → = m 1 b ∈ { → m = cτ (

are indeed much larger than those involving muons so that the 2 l L L R | ) and thus accommodate ( L /A . In the same plot, we superimpose the limits derived B 0, which corresponds to an overall rescaling of the SM. V S S τ ) S cb → g g ¯ g ν 2 g 2 V l 2.4 q | > . ) b ]. Since the ( ) ≡ 2 F and ∗ ]. The low-energy fits to various combinations of effective 3 L ( ) but allowing the couplings to be mostly imaginary. These π b − V G b and Dπ D 10 45 1 8 c 6= 0, g , T 2 m T B R . After including in the renormalization group running the one- 9 T g ) ( b → f g b remains unchanged after setting the couplings to muons to zero. ( c 4 P ∗ m ) g B m  ∗ We use the ( ( D m = ) and T D , which is of the order of 1 TeV [ ]. More precisely, we consider the conservative limit c 2 g = 5 R = +4 µ → q B LQ ( ]. In particular, two specific scalar LQ scenarios predict range in this case reads τ 47 L , L 1 B S m S σ 44 g g 46 – /A ) = -meson lifetime, which is particularly efficient to constraint the pseudoscalar = ) , respectively [ c ¯ ν 2 34 b that not only the scenario with τ q µ B ( m 2 0 → A = c To determine the allowed values of the effective couplings B That assumption is a very good approximation to the realistic situation. AsWe we disregard shall the see, solution we with find that large and negative values of µ ( 5 6 B obtained for the couplings of leptoquarksphysics discussion to of excessively large NP couplings. By combining this constraint with thein low-energy figure fit to scenarios such as from which we derive that the relations at coefficients are shown infrom figure the contribution [ and the expression Other solutions involving theliterature coefficients [ the scale running down to loop electroweak corrections, in addition to the three-loop QCD anomalous dimensions, consider is the coefficient The allowed 1 of lattice QCDstill in not refs. available from [ LQCD,distribution we of consider the onesratios extracted from the measuredinputs angular and the experimental valuesof given effective in coefficients eq. ( in eq. ( other words, we arepotentially left contribute to with four effective coefficients, contributes to theand transition muon modes. at dimension-six, and as such it is irrelevant for the discussion that follows [ JHEP10(2018)183 - c = B and Q 1.0 ). In (3.1) 2 SM 3 χ V T . In the -physics b 0.5 B m ] and specify = 0.0 31 µ ) at b = 1 TeV, for purely -0.5 m T , compared to the SM ( , the matrices g ∗ µ L T S ] D g at R -1.0 L i and ` T i L g ) S and 4 , real ]. The Yukawa Lagrangian of -1.5 g L T , real , imag. T  T ., D , 51 c , where the electric charge, L . R Uν – = V Y = -4g =4g =4g g -2.0 ) ], which are updated in this paper L L L ) S S S 48 L ∗ g g L g + h ( S = [( , a weak triplet of scalar LQ states 34 is a component of the LQ triplet, and g j 3 i D / L k -2.5 3 5 1 1 L ) R 50 10 )

S 0.1 0.5 k

3 100 3 SU(2)

χ S , , 2 c k and ¯ 3 3), τ , ( T 2 2 ] , = ( – 7 – iτ at tree-level [ 1.0 L i ) and the third weak isospin component ( 3 2 SM and/or C i d is the only scalar particle that can simultaneously ∗ χ Y = 1 S is plotted against i ) ) by assuming Q 3 SM K ) ∗ b k ( S 2 0.5 ij L L m χ y K u ( † < R L R = S V ∗ 0.0 g 3 exp K S R L = [( ) b i -0.5 m Q ( and i Remarkably, g 3 7 -1.0 SM K / 3. 1 / 3) . < R -1.5 , 7. In the left panel, 1 . ¯ 3 is plotted against exp K = values for individual effective coefficients fits of 19 2 -2.0 ] L L R 2 χ S = ( V T U ≈ g g g χ 31 3 is one of the Pauli matrices ( . S k 2 SM -2.5 1 5 , is the sum of the hypercharge ( τ 50 10 χ 3 0.5 0.1

100

T

χ We follow a common practice and in this paper we consider the LQ’s belonging to the same multiplet 2 stands for a generic Yukawa matrix. We have neglected the LQ couplings to diquarks reads [ are the CKM and Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrices, respectively. 7 + 3 L where y to be mass degenerate. The first scenario wewith consider hypercharge 1 is with account for S Since the neutrino masses arewe insignificant can for set the phenomenology we study in3.1 this paper, Scalar leptoquarks 3.1.1 In this section weanomalies briefly by review introducing the a LQ singlethe mediator. models LQ proposed We by adopt their to the SM accommodateY notation quantum the of numbers, ref. (SU(3) [ the left-handed doublets, U with the most recentpossibility experimental of imaginary and couplings. theoretical inputs and extended by3 allowing the Leptoquark models for right panel, imaginary and real couplings.lifetime constraints, The see dashed text for regions details. correspond to the values excludedfindings are by in the agreement with the literature, cf. ref. [ Figure 2 value, JHEP10(2018)183 ) ). k 3.5 (3.5) (3.6) (3.2) (3.3) (3.4) = l . Notice ) ) in terms transition ) and the ∗ . ( 0 ` ¯ .,  νν SM K 3.1 ν c ∗ . ) decay, namely, K c` d` L + k < R . y ` + h → → c ( . ) − l b ∗ 3) cb cd ( / B s` V V exp K ( (1 + h 3 , clearly at odds with B R L S → + 3) ., 3) V / ∗ / b c g . ) L j (5 2 (2 2 ` , s` are not necessarily negative, L R R y ∗ C L i + h ( ), so that this scenario cannot
d` L indices have been omitted for u j at loop-level and accommodate L j y R j sl 3 L cb cs ` d,s ` L ) ij 2 S L y V V ∗ 2 3) ) B ( / R i ]. Otherwise one should devise an L i m L iτ K + m u and d y (4 3 bk L 2 31 ∗ ∗ R ij y S ij R ) s` ) L ) V , the weak doublet of scalar LQ’s with y L b` ( L 6 R em L j R i y y / y ` ( u α ( 7 2 − ) It is convenient to rewrite eq. ( ∗  ts C L i ij 0 L − 2 3) + ( πv V 8 d y , / b` L 2 3) tb system (∆ y 3 ij 3) L / − – 8 – − V / 3 y ( 3 (2 2 2 refer only to the LQ contribution, that is to be 2 S 2 (5 2 0 2 d,s S = ( = R v R ¯ m R B √ 10 2 4 , L j kl kl 10 9 L j R j − R ν − R j − ν ` C C ` i C L i 3) − 0 = d,s R i / Q L i is the only scalar LQ for which the proton stability is auto- u ¯ u B (1 u 3 = 2 ij R c` gives a negative contribution to ij ij ) S y ij ) ) R kl 3 ∗ 9 L ) L L b` L 2 S = R C L j y y y ∗ ν 2 m V y ( R V V y 0 + ( C L i cb ]. The most general Lagrangian describing the Yukawa interactions , clearly disfavored by the data. One can get around this problem, L d 6 V b` L ], and generate corrections to ) / 2 ( 4 ∗ y = ( 53 ij ( L 7 6. This scenario is known to be unsatisfactory at tree level because it 52 . 2 y √ 2 ) SM K / are the Yukawa matrices, and SU(2) v ∗ 2) R ( − + , − L R D = 7 y = > R R . Interestingly, = . Note that the terms depending on ) ) 3 ) ] for illustration of a Grand Unification scenario in which these couplings are absent in a Y ∗ ∗ ∗ L S ( ( = (3 ( V 48 L and ) in order to ensure the proton stability [ SM K K SM D g 2 can be written as, R L R 3.1 2 y < R > R R ) ) ∗ ∗ See ref. [ ( ( 8 exp exp K D concrete SU(5) set-up. with where simplicity. More explicitly, in termscan of be the electric written charge as eigenstates, the Lagrangian ( hypercharge leads to as proposed in ref.R [ matically preserved [ frequency of oscillations inaccommodate the 3.1.2 The second scenario we consider is We see that theR coupling but they are tightly constrained by other flavor limits, such as which is precisely theonce effective again scenario that needed inadded to this to accommodate discussion the SMSimilarly, we value also read which off isand the contribution find non-zero arising in from this the LQ lepton to the flavor conserving case ( from which one can easily extract the Wilson coefficients for the appropriate symmetry to suppress these couplingsby that experimental are limits known on to the be protonof tightly lifetime. charge constrained eigenstates as: in eq. ( JHEP10(2018)183 ) (3.7) (3.8) (3.9) (3.12) (3.11) (3.10) ∆ x − u u x x )( u ] x 4) log , 52 ) − 0 − 0 u ., u u c . x x , x via the following coeffi- ( u 0 u x ` 1)( , + h x ( ¯ ν , ∗ − F 3)
∗ cτ / 2 u ∗
bl L 1 e x 2 2 R  y , − bk 0, one finds [ R ( → ( 2 2 R ul L y . m ∗ b e , the weak doublet of scalar LQ’s y R sk ≡
L m + 6  y  sl / R 2 in this model and find, ) b` R ) R 1 0 Lj y k 2 R y ) y 0 u . This scenario can accommodate the ν ]. Therefore, this scenario can accom- ∆ + k em is, however, excluded due to the strin- u 2 L 2 0 x m ` x em y , α Ri ) R 52 c` L 2 − l ∗ ∗ s d 3 α − ( ts 0 − 0 y ∗ 2 m ∗ ∆ ∗ ts u u s` 0 ., ts V D ij L πv ∆ x V u c cb x V 2 V y tb . = = ( πv R x x V tb v tb V → ub 2 4 )( µ V + 2 )( V V b e 0 u R + h – 9 – 2 u ]. } 3) − = x and 4) log j 4) log / x − ) T L − = (2 2 ∗ − 55 − 2 ( g − u,c,t , e R 0 = ∆ kl ∆ K iτ tree u u ∈{ X 0 x 2 10 0 x x 54 R x ]. As before, we again identify the Wilson coefficients Lj ( , = 4 e kl 10 ( C R ` 0 u,u ∆ 1)( 57 L C u 1)( − 35 Ri Ri x S x d d − g − = = = for complex and even purely imaginary couplings, as it can , in conflict with experiment. Instead, if the LQ corrections , but not both. loop 0 ij ij L L ) ∆ ) u ∗ ) kl 9 y y kl ∗ ( x model, the potentially troublesome diquark couplings to LQ are ∗ ( 0 kl 10 x ( 9 ( C ), which would receive a contribution enhanced by a factor of − − ( SM K 2 D C 6, which couples to SM fermions through a single gauge invariant D C  − R / R = = 0 − µγ R 6 u 2 em / > R ] x or = e R 1 = 1 → ) u , and the loop function reads, ) ∗ L πα 56 kl ∗ x ( 9 , and in refs. [ 2) ( Y 2 W τ , K C ( √ 32 K 2 B R R /m 2 i ) = = (3 0 m u is a generic matrix of Yukawa couplings. Another appealing feature of this 2 = e , x L R i at the amplitude level, as discussed in ref. [ u y x x τ ( A simultaneous explanation of F /m t which can be eliminatedbeen by done, imposing for a example,arising suitable in from symmetry ref. the in tree [ a level contributions way to similar to what has where scenario is that, like the absent. Proton decay can still be generated in this scenario but by higher order operators Another important scenario to considerwith hypercharge is operator, namely, [ gent limits on m modate either 3.1.3 obtained by tree-level matching atobserved the scale deviations in be seen in figure Furthermore, this LQ state contributescients, to the transition where which leads to start at loop-level, which can be achieved by setting As mentioned above, the tree-level contribution to the Wilson coefficients amounts to JHEP10(2018)183 , ) ∗ ∗ ( SM K K R (3.14) (3.15) (3.13) (3.16) < R , ∗ ]. In that exp K kl 58
R ) do not con- L y · ., . This limitation c 3.11 † L . ¯ ν y cτ ] for accommodating we omitted the terms + h bs 54 3
i ∗ → ts † L S at tree-level, the effective y b V R j ] · ` + k tb ` 59 V L C R i − l . y , the weak singlet vector LQ, reads u , a weak singlet scalar LQ with 3 ∗ s` . 1 / 3 1
ij R . 2 / S 2 S 5 y but in conflict with )  c 1 1 c` R → . ) ) 2 S 1 y m t + b 1 , . The possibility of explaining x 0 m 2 , SM K ¯ 3 ( em + h  v , b` ¯ 3 L L j f , 1 1 y ν c` πα S = ( kl −
− < R cb = ( 2
C L i ∗ L 1 32 t V 1 v 1 1 d 1 L 2 t R j x 2 S 4 U 2 S y − S e exp K ij t L − m · V y m − x y m log R ∗ tl C R i † 0 L 
– 10 – = u y − b` L L 1 log y ij R T y − 3  bs ∗ y g L j t cb 2 ∗ tl
` x 4 ∗ V V
ts v † L + 4 C L i − y R V 1 u y · tk tb S
= = V L ij j L tk
y L L L y
V S L 2 ∗ ) = 1 + g R g t y 1 iτ ∗ V y 2 S x ( C V m 1 1 f Q 2 S 2 S 2 ]. Furthermore, the Yukawa interactions in eq. ( h em 1 v ij 3 L m m 3 2 t 2 t 56 / S y / and 2 πα 1 3. This model was deemed to be viable in ref. [ em em m m / = = receive contributions at loop-level, namely [ 1) 32 2 W 1) are generic Yukawa matrices. Like in the case of 1 πα πα , , S 8 8 − = 1 ¯ 3 R /m L y . The most general Yukawa Lagrangian of , in this scenario will be discussed in section 2 t 9(10) = = ) 1 Y ∗ ) S ( = (3 m C = ( ∗ ( kl kl 10 10 SM D m 1 and 1 D = C C U S t L R = , which will not interfere with the SM contributions and therefore can provide only + − x y > R R µ ) kl kl ν 9 9 ∗ 2 ( C C e R exp D 3.2 Vector leptoquarks 3.2.1 The first scenario ofwhich this received sort considerable we attention consider because is it can provide a simultaneous explanation where and/or Although this LQ does not contributecoefficients to the transition involving diquark couplings tothe LQ, proton which stability. mustscale By be integrating forbidden out by a the symmetry LQ to state protect we obtain that, at the matching where a small shift with respect to the3.1.4 SM predictions. Finally, one can also considerhypercharge a scenario with R cf. discussion in ref.tribute [ to the chargedcan current be processes, overcome such by ascase, introducing the scalar light transition and right-handed tensor neutrinosQ operators to will this be set-up generated [ through the gauge invariant operator which turned out to be in agreement with JHEP10(2018)183 1 → U and b ), we (3.17) (3.19) (3.20) (3.21) (3.18) P C 2.1 ]. Due to − 67 = [ by giving rise 3 S / 0 2 C ` ) ¯ ν , 3 , 0 c` 10 3 C → = , = ( b ., c 0  9 3 0, in agreement with the . 0 C U , , d` L > x ∗ ∗ + h L

. cb cd V c 3 1 bk L bk L . g V R j V model. Concerning the effective U 2 2 U ` x x 1 ]. The most general Lagrangian µ 1 + m m U + h 0 sl L sl L U 60 j µ x x s` L γ L x ) em em µ cb cs R i k ¯ α α 3 2 2 V d V ∗ ∗ . Switching on the right-handed couplings, ts ts U ) ij R ] for concrete examples. k πv πv + ∗ ∗ V V ( x τ
0 tb tb ( 66 SM K b` L b` L V V µ + – ), the only non-vanishing effective coefficient is x x – 11 – γ transitions [ j 1 − − i  U 2 L 61 0 ∗ c < R 2.4 , Q µ
= = 1 c` m ) ) ij L ∗ b` L U 53 ( kl kl cb L → 10 10 x µ x exp K V γ ] C C i b 2 = 1 R V x − − ¯ 31 2 U Q ( 3 2 U v , in a way similar to the 2 m = = ij L ) L v 2 and ∗ x ( kl kl 9 9 s K = = = C C R 1 ) amounts to L U V → . ) g 3 L ∗ 2.1 / b ( 2 SM D are the couplings. The contribution of the left-handed couplings to the 3) , ij R > R ) and matching at tree-level onto the effective Lagrangian in eq. ( x ) . However, since the latter Wilson coefficients are disfavored by the current 0 ∗ ( ) model. Furthermore, this scenario also contributes to = (3 ]. Importantly, however, this model is nonrenormalizable, which undermines exp D 3.20 P 1 and 3 R C U 60 U ij L , data, we will set the right-handed couplings to zero and call such a scenario the x = ( 20 s 0 [ A peculiarity of this scenario is the absence of contributions to the transition ) 6= 0, amounts to contributions to other Wilson coefficients, S ¯ ν → R C obtain that which can accommodate operators in the charged-current case ( gauge symmetry, this LQmost couples general only Lagrangian to being the [ left-handed SM fermion doublets, with the Using eq. ( its predictivity at the loop-levelmass unless is the explicitly specified, ultraviolet (UV) see mechanism refs. generating [ the 3.2.2 The last scenario we consider is the weak triplet of vector LQ’s, which is clearly acceptableobserved since the leading term implies sν as required by the observationx of ( b minimal to the effective coefficient effective Lagrangian ( consistent with the SMhanded gauge fermions, symmetry namely, allows couplings to both left-handed and right- where to the anomalies in JHEP10(2018)183 3 × (3.22) ), which µe anomalies, s → L → ¯ ℓ ℓ K b ( B ). . ¯ ν  0 d` L Kν x LQ cb → cd V V B ( + production at the LHC. The dashed B 0 ¯ ` s` L ` x cb cs → V V is negative, while the others are tightly pp 2 , + | 0 ¯ ∗ q q bτ L
b` L x | x b` L , this model cannot accommodate the deviation can be calculated. More precisely, if the 3  x 3 – 12 – as a viable explanation of the 3 ∗ 3 S 2 U 0  L U (a). In both ATLAS and CMS the searches for c` m x b` L ) conversion in Au nuclei and from x 3 cb L LQ LQ  e V 2 3 V x − 2 U ( 2 µ , have been made. The results of these searches lead to v 2 m ¯ ν model is generally nonrenormalizable. Nevertheless, under v 2 -channel exchange in t ¯ 3 qν − − U ) is unitary, UV-divergences appearing in loop-induced FCNCs = = ¯ ]. `, q L (a) (b) ¯ V q` 20 3.20 g q , the 1 U → are canceled through the GIM mechanism. However, the price to pay for scenario with unitary LQ, shown in figure phenomenology 3 † we list the most recent lower limits on the masses of second/third gener- 3 U LQ T U 1 † g g , can no longer be avoided. In turn, the presence of such couplings is in strong LQ from eq. ( p u which is given by because the term proportional to . (a) Representative Feynman diagram for LQ pair production via QCD interactions. L L → x ) V ∗ , or g ( ) d ¯ q D In table Similarly to , q R e ( this process in differentleptons, decay LQ channels intomodel second independent and/or bounds third on both generation the quarks mass and andation branching fractions scalar of and the vector LQ. LQs relevant to this work, for benchmark branching ratios set to 4.1 Direct limitsAn on efficient pair-produced way LQs to setthe limits LHC on LQs one is ofgg to the directly most search for significant them example at of hadron colliders. such a At processes is the pair production as conflict with LFV bounds from exclude the see discussion in ref. [ 4 High- certain circumstances, loops involving matrix mediated by having a unitary coupling matrix is that LQ couplings to first generation SM fermions, such We see that, similar to thein scenario with constrained by other flavor physics observables, such as again Figure 3 (b) Feynman diagram forpropagator LQ represents either a scalar or vector LQ state. JHEP10(2018)183 , ν ] V ] ] ] ] ] ] ] for a 74 µν , 1 75 75 75 72 71 70 69 [ [ [ [ [ [ [ G − -channel 73 than the µ , instead, t [ 1 1 1 1 1 1 1 κ V T − − − − − − − s / Ref. 1 p — − κg int ” denotes any jet L j 35.9 fb 35.9 fb 35.9 fb 36.1 fb 35.9 fb 35.9 fb 12.9 fb 36.1 fb L ⊃ − LQ with a † LQ ” indicate that no recast or − → tails of Drell-Yan processes -channel exchange of LQs to ¯ q t T q is a fundamental gauge boson p — µ V -collisions at 13 TeV to set limits on pp Vector LQ limits 1780 (1530) GeV 1810 (1540) GeV 1790 (1500) GeV 1780 (1560) GeV 1900 (1700) GeV 2110 (1860) GeV 1440 (1220) GeV 1550 (1290) GeV ]. If 76 ¯ ` ` → – 13 – 5). In the last column we display the value of the LHC pp 1 may arise in a UV theory where the vector LQ is . — ) the LQ-gluon interaction term, < µ = 0 | V ], respectively. κ β | 980 (640) GeV 900 (560) GeV 850 (550) GeV 80 1020 (820) GeV 1100 (800) GeV 1420 (950) GeV Scalar LQ limits [ 1400 (1160) GeV 1530 (1275) GeV tails of ¯ τ = 1 ( T τ 3 3 β p 3 3 3 3 3 3 ], a contribution arising from the ,U ,U . ]. In this regime, contributions to 3 1 1 ,U ,U ,U ,U ,U ,U 1 78 68 1 1 1 3 3 3 ] and , ,U ,U ,U -anomalies, could modify the tail of the differential cross section of 1 3 3 ,S ,S ,S 79 ) can be directly probed in the high- ,U ,U ,U 77 B [ 2 2 2 3 3 3 LQs ,U ,S ,S ¯ 2 2 2 µ = 1. The first assumption is in general true for LQ-fermion couplings of ,S ,S ,S µ, τ ,R ,R ,R µ 2 2 2 R 1 1 1 κ ,R ,R S S S R R R 1 1 = S S ` ( 1 or smaller [ . In the following we use LHC data from ¯ . Summary of the current limits from LQ pair production searches at the LHC. In the 5). These limits assume the following: (i) pair production is dominated by QCD in- ` ¯ . ` ¯ ` ¯ ¯ ¯ ¯ ¯ ν µ τ ¯ ν µ τ ¯ ¯ ν µ τ ` ∼ ) in the final state,given that these perform considerably better at high For the di-tau analysis we focused on the fully inclusive channel with hadronic taus ¯ ¯ t τ b τ t ν b ν ¯ ¯ ¯ t µ b µ ¯ → t t → decaying to t b b jj τ jj ν b jj µ = 1. The possibility of having Decays had 0 =1 (0 τ pp each LQ model. ForZ this we have recast two recent searches by ATLAS( at 36.1 fb 4.2 Limits fromAs high- shown in refs.pp [ at the LHC.accommodate In the particular, larger values of Yukawa couplings, that are often needed to depends on theof UV a origin new of non-abelianκ the gauge vector group LQ then [ a the composite gauge particle, symmetry therefore givingLQ completely presented rise fixes in to the table. LHC choice limits weaker than for the gauge vector teractions, and (ii) for vector LQsis ( taken with order lepton (where the amplitude isleading proportional compared to to QCD the induced squared production. LQ-fermion coupling) The are assumption sub- on the value of luminosity for each searchoriginating along from with a the charm experimentalsearch or references. in a this strange Note channel quark. that has “ been Entries performed marked with up to “ thisβ date. Table 1 first column we givesearch is the relevant. searched In final thevector next states LQs, two respectively, columns and for we in present the the current second limits on column the the mass for LQs scalar for and which this JHEP10(2018)183 , } for and from µµ ¯ 0 µ s, c, b q µ m assuming 6= ] with the ∈ { from QCD turning on → q q ¯ 85 int ` . [ ` 3 pp L had / 1 3 τ √ → S with ). After exporting l ¯ q 0 we recast the LFV q L q x for the di-tau search with a LQ exchanged channels. In order to 3.17 µτ ]. Final state hadronic ql L ) coupling versus mass 0 . Similar bounds of the x Delphes3 tot ql L 84 } → `` [ with LFV couplings have x m 0 ( pp → and Z µ, τ ql we have only considered, for L l y 0 1 q L ) coupling exclusively to second pp ∈ { ) and eq. ( 3 y results from the U eµ, eτ, µτ ` U ql L Pythia8 4 3.6 for initial sea quarks, y → by scaling the data and background ¯ and τ 0 and and 1 2 Z 1 } − R ¯ µ, τ U , µ 2 s, c, b R -channel process → t , ¯ 3 q . ∈ { = 300 fb – 14 – q ¯ S ` . We find, however, that the bounds on the LQs ` q , int 1 1 ). An upper bound at 95% C.L. on the number of − L S → 0 tot 2 fb qq m . tails are always weaker than the combined bounds from since these bounds are identical to the bounds for µτ 1 ] at 3 m S ] we generated for each LQ mediator a statistically significant T 83 87 [ p , 86 ]. Besides the current luminosity limits, we also estimated projected 81 ] for each LQ mediator ( MadGraph5 82 ] for the definition of [ 80 searches by ATLAS for scalar (vector) LQs in the Each line corresponds to the 95% upper limit for the process -channel may provide an additional handle for setting constraints on the flavor ¯ τ -channel Drell-Yan event samples t t 9 τ Besides producing deviations in the di-lepton tails, LQ mediators that couple simulta- For our simulations we created the Universal FeynRules Output (UFO) files using We did not present plots for search by ATLAS [ → 9 0 determine the sensitivity of theZ LHC to the extracted from the high- the flavor conserving di-muon and di-tau tails described above. neously to differently charged leptonsables may at also produce the measurable LHC. effectsin In in the LFV particular, observ- searching forstructure the of process these LQ models.been Existing presented searches by for the a massive LHC collaborations in the plane. each flavor coupling one atsame a order time, can with be extractedthe quark for flavor the violating coupling process products sample was subsequently showered and hadronized using taus and isolated muonsparameters where set reconstructed according and to smearedreach each experimental using of scenario. the In LHCpp order for to each illustrate LQ the state current we show in figure and third generation ofsimplicity, non-zero quarks left-handed and Yukawa matrices in leptons. eq. ( the For UFOs to set of and for vector (scalar) LQs at different masses in the 1–6 TeV (0.6–3 TeV) range. Each improve with more statistics.di-tau searches Additionally, are systematics well underis in insensitive control the to and tails the only massive of dominate LQs. the the lower di-muon bins and whereFeynRules the search as described in [ limits at a higherevents LHC with luminosity the of luminositythat ratio the and data the inwell background the for distribution uncertainties the tails with leading are backgroundsjet statistically such mis-tagging as dominated. since SM these This are Drell-Yan production assumption estimated using or holds experimental fake data from control regions that ground events above different thresholdthe values di-muon of search the invariant and mass(see the distributions ref. total [ transverse massallowed signal distribution events above each massing threshold the was Log-Likelihood extracted for ratio each with search by nuisance minimiz- parameters for the background uncertainties, leptonic tau decay channels. For each search we counted the number of observed and back- JHEP10(2018)183 is but ) at L -� -� V ql L ) g ∗ x 2.4 ( 2.4 ( since it D ) ql L ∗ R ( y 2.2 2.2 exp K R ] 2.0 ] 2.0 �� ��� ������ �� �� �� ��� ������ �� �� ). The subleading tails at 13 TeV with ��� 1.8 T [ 1.8 ��� 2.5 . This can be easily [ p ) � ∗ � ( � � 1.6 SM D 1.6 . On the other hand, this � � high- 10 ¯ ` ` 1.4 C > R 1.4 − ) → ∗ ( �� → ττ �� → μμ �� → ττ �� → μμ �� → ττ �� → μμ = 1.2 exp D pp �� → ττ �� → μμ �� → ττ �� → μμ �� → ττ �� → μμ to the constraints stemming both 1.2 9 , see also eq. ( R ) C 3 ∗ ( 1.0

SM D

1.0

model can accommodate

.

0.0 1 0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 4 3 2

� � R s � � 3

B S �� �� m satisfying , the -� – 15 – -� 10 , 2.4 2.4 9 and ∆ C 3.1.1 2.2 2.2 Kνν ] → 2.0 ] ), where we see that the sign of the leading term for 2.0 �� ��� ������ �� �� �� ��� ������ �� �� B based on the expressions derived above. 3.4 ) ��� 1.8 ∗ [ ��� 1.8 ( [ � exp D � � R � 1.6 1.6 � � : 3 1.4 / 1 1.4 ) and/or 3 , �� → ττ �� → μμ �� → ττ �� → μμ 1.2 ) ∗ ¯ 3 �� → ττ �� → μμ �� → ττ �� → μμ �� → ττ �� → μμ 1.2 ( exp K . The top panel (lower panel) shows current limits in the coupling vs mass plane for = ( 1.0 R

of data. The solid and dashed lines represent limits from di-tau and di-muon searches, 1.0 3

2.0 1.5 1.0 0.5 0.0 3.5 3.0 2.5 2 1 0 4 3

� � understood from eq. ( negative, thus further lowering theterms value in of this equation couldsuch in a principle situation provide would a be positivephysics in contribution limits, to conflict such with as tight constraints coming from other flavor S As already discussed inpredicts section the NP contribution to scenario cannot accommodate the anomalies in

1 �

� − �� �� • modate 5 Which leptoquark model? In this section we subjectfrom direct the and models indirect listed NP in searches. section We will then select LQ scenarios which can accom- Figure 4 several scalar LQ (vector36 LQ) fb models from LHCrespectively, searches for in different initiala quarks time. while turning one scalar (vector) LQ coupling JHEP10(2018)183 . ) ∗ ∗ ]. ∗ > ( 4 SM K exp K ) 73 D sµµ ∗ R (5.1) ( R D → > R R b ] and let and ∗ K ) to realize 88 , R exp K 3.6 R 55 , 54 , ]. . This situation can ] this model has also ), for two benchmark 5 . We should reiterate . The main challenge 35 52 58 ) cannot be obtained in ∗ 2 ( ) , , it predicts 3.1.2 ∗ K ( R D SM K ) this LQ scenario generates 1 TeV, which can turn = 0 R R 2.4 ' ]. The results of our fit are then ]. In ref. [ < R in such a way that they satisfy µ 52 and 56 K 10 ) . From this plot we see that a large , ∗ R , cf. section 9 ( ) 1 ∗ C K ( SM we use the allowed values of Wilson co- K R , y ) ∗ , and shown in figure ( (    < R 4 K 0 0 ) are loop induced and the NP Wilson coefficients R ∗ derived at LHC. This is illustrated in figure – 16 – ( tµ cµ L L K 10 y y , ]. The minimal flavor ansatz in eq. ( 9 ττ R , the non-zero NP Wilson coefficients in the 0 00 0 0 -pole observables [ 2 C 35 level. This is so because the couplings required to 5 TeV, cf. table Z is mostly imaginary, cf. figure    e . → R ), as well as the constraints arising from the limits on level. Therefore, if the central values of at the matching scale σ L = σ S pp (1), which is compatible both with low-energy observables 2.3 T g L g O y if model contributes to ) = 4 ∗ ( 2 L exp D are non-zero couplings. For the time being, we neglected the cou- R , as needed [ S 2 TeV and 1 R . g 10 tµ L y ) and from the C : : only at the 2 ¯ ν = 1 − 6 6 ν ) / / . While this is enough to explain 2 ) ∗ 1 7 because it plays no role in the discussion of ( ∗ = and ) ) to be of order R anomalies. A similar conclusion has been independently found in ref. [ ( ), due to the chiral enhancement by the top quark [ 0 10 , which disagrees with K is absent. As a result, 2 2 τ 9 ∗ s m C , , K cµ L
10 R µγ C 3 3 y − C bτ R → . To accommodate those one could proceed as in refs. [ → ) sµµ y → needed at low-energies is in tension with LHC limits, allowing then an explana- b ∗ = = = ( = ( ( B τ 0 SM 2 D 2 9 9 → ( ( cτ cµ L L e In a model with thedecay LQ state are thoseC corresponding to thewhich chirality is in flipped disagreement operators withbeen experimental and considered data [ to they account satisfy for the anomalies in charged currents. To this purpose, that a simultaneousthis explanation scenario of even both accommodate at each the of 2 B the observed anomalies would induceR too large a value for y and with upper limitsFrom on the point ofthe view combination of the effectivecompatible Lagrangian with ( tion of remain unchanged with more data,the this model will be excludedThe as above an flavor explanation pattern of isR clearly not satisfactory to explain the observed efficients specified in eq.B ( combined with limits derived in section masses, y where pling to in this scenario isin to the comply region with themodel LHC anomalies can limits, can accommodate which be measured are accounted particularly for. constraining To see the extent to which the such a scenario is given by At tree levelC the be avoided by choosingb a flavor structuresatisfy such that the tree-level contribution to R • • JHEP10(2018)183 . s ]. )] ) ∗ ( 20 → 2.3 D (5.2) b 3 R for both dilepton σ = 1.5 TeV 2 2 R ττ m to 1 ) 1 ∗ ( K LHC constraint [eq. ( can be induced by R cμ L 0 , and (ii) y since it generates the s`` 10 , 1 9 ) ∗ → ( C , b -1 SM D ), as discussed in ref. [ ¯ ν ]. It appears, however, that e = 0 but ) > R ∗ -2 1 TeV and if one wants to stay 59 ( R ) ∗ s`` D ≈ . On the other hand, this scenario ( 1 D 8) are plotted in dark (light) blue for 2 -3 . S → → R ]. The black line denotes the LHC limits m b 3(6 B . 52

0 3 2 1

(

-1 -2 -3 L y B

/ , y = 2 μ t , both being equally viable when it comes ) , as desired [ 2 ¯ T ν    , see figure χ g 10 µ ) 0 0 ) ∗ 4 C ∗ ( ] in which the model was shown to be viable if – 17 – ( -lepton, is allowed. − bµ sµ SM L D L − 3 τ D 89 y y = = , which excludes an explanation of with ∆ 0 00 0 0 1 → > R (right panel). We consider the 9 L = 1.2 TeV −    2 2 tµ ) S R C ∗ y B g ( m = ( . The main difficulty of this scenario is to face the LHC exp D T eV B L vs. 1 R y and = cµ ) LHC y L 3.1.3 ∗ model can only account for a very small enhancement in V ( cμ L 0 g y 2 µ/e D e R R -pole observables computed in [ data with 36 fb Z -1 : 3 µµ / 1 ) -2 → 1 model is a viable candidate to explain ]. The combination of these latter constraints is particularly efficient in this , pp ¯ 3 1 77 -3 S . Regions in plane = ( 2 TeV (left panel) and 1.5 . 1 3 2 1 0

-2 -3 -1 To evaluate the extent to whichanomalies, this we particle can consider improve the the minimal description flavor of ansatz, the compatible with A step beyond was(i) made a in larger ref. LQ [ massright-handed is LQ considered, couplings and to (ii) the a different coupling structure, with large effective couplings down to accommodating does not provide athe tree-level box-diagrams contribution satisfying to this is not a viable scenario if one works with limits on (i) recent limitstails on [ pair-produced LQs given inscenario, table so that S The it is necessary tomentioned postulate in the section existence of light right-handed neutrinos, as already L

y = 1

μ t • 2 R derived from masses. See text for details. Figure 5 m and the ones from the JHEP10(2018)183 , 2 3 ), ), . . π ¯ ¯ ν ν 2 4 τ 6= 0: 3 for (5.4) (5.3) √ < / Kν 4 TeV, = 11 cτ R can be 2 2 → . analysis y & χ s ) & → ∗ | 2 ( 2 min 1 D χ ( S D bµ B L χ = 11 ( y B R | m . B f . , o . We then perform . and d , and / ) 10    π . ∗ ( 4 2 TeV tτ cτ R R . 2 min 10 K y y √ 5 . If instead we focus on χ . The value C 2 TeV. To accommodate ) R ∗ , and (vi) the rare charm π ≈ − ( ¯ 4 ν . | ≈ 1 K µ 0 0 0 0 0 0 0 1 = √ S cτ R R S 9 y |    ]. These conclusions agree with → m ]. By performing a C | ≤ = . From this plot we see that one 59 20 -boson couplings to leptons mea- ij K L ]. We find we plot the regions with ∆ and y bµ L R Z | y 6 20 π 4 and , m ) must be extended to allow for √ vs. ¯ ν anomalies, 2, showing that the discrepancy can be 15 , (iv) . . µ 1 s 5.2 S 0 to satisfactorily accommodate the current ], much lower than the one we get from the we also show the bounds coming from the | ≈ B s`` → m | 6 bµ L 74 m ) , y y ), (ii) the experimental limits on = 21 bµ L ≈ − s → | ( – 18 – y | b    and D sµ 2.3 L 2 SM bτ sτ L L , are too weak, as expected, because the flavor fit χ sµ L y y 4 TeV [ y . 1 4.2 bµ sµ L L ), as described in ref. [ y y vs. > ). We should note that limits from the study of dilepton , the bound we deduce from the pair production searches | 1 1 0 00 0 0 µγ S S sµ π , y L 5.3 4 y m    m → to right-handed leptons will generate the combination of Wilson coefficients √ = τ bring in new constraints, such as those coming from ij R ( ≈ y L |  | τ B y . All the expression needed for that analysis, together with the ex- bµ L bµ L y y | µµ channel is ) and → ¯ µ 0 τν ¯ tµ ]. t D 8 in the planes then the flavor ansatz given in eq. ( . → ) 89 ∗ ( B , which is disfavored by the current data. D ( 10 them one would need large couplings, which is different from the originalref. proposal [ in ref. [ The couplings to B similar to the onemoderately reduced above if we one is conclude to remain that in the the range tension of in R flavor fit, given intails, eq. ( as described infavors couplings section mostly tovery the suppressed. third generation of quarks,So far for we which were the concerned PDFs with are the indicating that one needs athe systematic viability study of of this one-loopdirect scenario. corrections searches before In at assessing LHC figure whichBy are assuming weak in thein region the of parameters described above. is smaller thandecreased the but not SM fully value, accommodated.and In figure 6 needs large LQdata. masses and Such large large couplings are close to the nonperturbative regime perimental values/bounds can be foundthe in following ref. choice [ of parameters: where we have imposed the perturbativity limit a fit to the low-energyof observables modified Wilson by coefficients these given couplings, in(iii) namely, eq. (i) ( the the values experimentally establishedsured ∆ at LEP, (v) leptonicdecay decays where we keep only the couplings that contribute to C Note that the couplings + 10 9 C JHEP10(2018)183 , 2 ) ), 10 ∗ , ¯ R ν ( 9 7 ν SM D ) C ∗ ( -physics > R K 6 B ) ), which is ∗ is the only ( ¯ ν → ], and for D ν 1 ) . This can be R ∗ B ) U 5 , in agreement ( ∗ ( ) 91 ( , ∗ ( B . K SM D , since it predicts ) D 6 ∗ 90 ( , 4 R → [TeV] K > R 1 60 B S R ) ( ∗ ( and m B 3 D ) ∗ . The interesting features of R ( ) ∗ K ( in refs. [ 2 D R 3 R LHC S (right panel). Light (dark) blue regions 1 and allows to accommodate and 3 2 1 0

bµ

L )

-1 -2 -3 come from the Wilson coefficients L y ∗ L 1

y (

model is one of the best candidates to V

) S μ b g ∗ K ( 1 R K , from which we learn that U R 7 2 – 19 – ) from which we learn that the leading contribution ]. See text for details. 6 74 [ 3.22 (left panel) and is very small. tµ 5 sµ L ) ∗ y ( ] the minimal D is also a viable candidate to explain 60 R 4 3 , (ii) a positive sign of [TeV] 8). The black line in the right panel denotes the LHC exclusion limit U 1 . 10 S ]. A slightly non-minimalistic possibility is to build a model with C 3(6 m 3 . 60 − 2 : : , but it cannot provide an explanation of = 3 3 < ), and (iii) the absence of tree-level contributions to / / 9 10 2 2 ]. Note that our conclusions can also serve as a guideline for future studies 2 2 ) ) C C χ is plotted against 3 1 88 LHC 3.19 , , 1 − comes with the wrong (negative) sign. The subleading couplings to the strange S 3 3 1 L m V = = ( = ( 0. . g 0.4 0.2

9 in ref. [ 3 1

-0.2 -0.4

L to quark are constrained by thewhy the tight net experimental effect limits on on U Finally, the model C understood on the basis of eq. ( satisfying cf. eq. ( which is often aanomalies. major We obstacle will to discuss the this models scenario in built more to detail accommodate in the section U As discussed insimultaneously ref. explain the [ anomalies in this scenario are: (i) contributions to 3

y S Our findings are summarized in table s μ • • single LQ modelwith that findings can of simultaneously ref. accommodatetwo [ different scalar leptoquarks, asand explored for if one of the anomalies disappears. Figure 6 correspond to ∆ on pair produced LQs decaying into JHEP10(2018)183 . 1 U , cf. (6.3) (6.1) (6.2) L V (second g ) ∗ ( ), and find D R . 3.18 2 − 18] TeV . . 0 2 ) (first column), , , ∗ − ( ) ∗ D ( 12 . K = 0 R , we compare the result of the [0 TeV ∗ ) R R ∗      & 3 X ( ∈ − ) exp K ∗ ∗ ( ) R 10 K bτ L we will assume the following structure × R x ) ), with the expression eq. ( ∗ ( )( 41] ) SM D . 2.3 bτ L ∗ , x ( 1 x 1 ,    X X X D gives rise to the effective coefficient 2 U    cb > R R 83 – 20 – V . ) m bτ L sτ L ) ∗ ( ∗ x x [0 ( + ) D ) results in ∗ D ∈ ( ∗ ∗ bµ L sµ L sτ L R    R X X X K x x ∗ x 2.5
conversion on nuclei, the atomic parity violation and on R 0 00 0 0 cs to 1 (third column) without inducing other phenomenological prob- 3 bµ L e V / 2 U    and ) x 1 ( 2 ∗ ( − ) m U = ∗ D 2 2 3 1 = 3 1 ( 1) sµ L µ L f R S S U U , R R x SM K ∗ x
Model − bτ L and < R x ) = (3 1 ∗ observables, given in eq. ( ) ( 2 U means that the discrepancy can be alleviated, but not fully accommodated. ∗ 1 ( cτ K ∗ ) m K U R  L sµµ R V x → exchange to constrain the model parameters. We will also compare these ( ). To determine the region allowed by b ¯ 1 ν U πν ), which compared to eq. ( . Summary of the LQ models which can accommodate → 3.19 K ( A tree-level contributioneq. of ( B global fit to where the couplings towith the experimental first limits on generation are set to zero in order to avoid the conflicts 6.1 Low-energy constraints To satisfy both for the Yukawa matrices: In this section weWe discuss will in use more the detailfrom low-energy the the physics phenomenological observables status whichresults of receive with the the the scenario tree-level ones contributions the deduced LHC. from Furthermore, the we experimental will make bounds a based brief on comment direct concerning the searches loop at effects. column), and both lems. The symbol See text for details. 6 Revisiting Table 2 JHEP10(2018)183 ). → ¯ ), ν σ (6.4) K De → scenario. B ], which was ( 1 B U 93 / , ) ] ] ¯ ν 92 ] ] ] ] ] ] [ 28 ] ] 97 – 8 8 , 8 Dµ 99 98 48 92 92 92 [ [ [ [ [ [ [ [ Ref. − 25 92 decays, can however be [ [ → 10 τ × B 5 , including the decays 2 3 4 2 ( 4 − . − − − 5 8 3 B 10 8 − − 10 10 10 10 = × . < 10 10 ) ) × × × × 2 × × − 2.5 2.3 µ/e D exp 8 4 ( ( ) . . R 4 8 995(22)(39) Exp. value µφ . 670(67) 48(23) 50(23) 06(19) < < . 488(10) . . . 0 could be in conflict with constraints from . 4 5 5 1 → 2 1 018 TeV . U τ 0 ( ) ) ) ) ) B 1 < – 21 – U 2.6 2.6 2.6 2.6 2.6 | ) ) ) ) ) model, which is model dependent. We will simply mixing and LFU tests in sτ L 1 1 ), ( ), ( ), ( ), ( ), ( x s 6.4 2 U 3.18 3.19 3.19 3.18 U ( || ¯ ( ( ( ( B m sµ L Eqs. – 3.19 3.19 3.19 3.19 3.19 − x ( ( ( ( ( | s B , as well as the ratio ) , ¯ ) ) ν ) ) τ τν µν µγ ] it was shown that the leading-log renormalization group effects τν µφ Kµτ → cτν sµµ 96 . One-loop contributions to this transition as well as to many other → → → µ/e D → – e/µ K τ/µ K ¯ → ν s → s r r B τ R → ν → τ 94 B ) D D ( b ( ∗ b ], similar to the one we consider here. In this work, we will not consider B ( ( ( ( B Observable decays in the scenarios in which the dominant coupling is the one to the B B B 60 B and K τ ¯ ν → τ and B → Z ) . Tree-level observables considered in our phenomenological analysis and their correspond- s ( D , ¯ ν below, the synergy between thebounds tree-level coming constraints from from flavor direct physicsthe searches and parameter the at tree-level space the LHC ofelectroweak is this corrections already scenario. which sufficient must Of to beUV course, significantly systematically completion limit included this is in fact specified. the does models not in reduce which the importance of third generation fermions. Toflavor get structure around [ that difficultythe one constraints can induced allow by forthe the a unknown loop more UV effects general completion since ofassume they that the this could problem be isances sensitive taken of to care the divergences of by details to some of higher mechanism which order prevents in the appear- perturbation theory. Moreover, as we shall see butions to observables, such as important. In refs. [ induced by the effective operatorsleptonic related to We reiterate that an important feature of this scenario is the absence of tree-level contri- Other relevant constraints to thisµ scenario are listed inAnother table important constraint stemsoften from neglected in previous studies of this particular LQ model, and which gives (to 2 Table 3 ing experimental values (or limits), as well as their theoretical expressions for the JHEP10(2018)183 . 1 It 1. bτ L `` . − x 0 11 (6.5) & 5 TeV, 5 TeV, . . | forces sτ L = 1 = 1 ) . x ? | 1 1 ( U U 81 D . m m 2 R ), but assuming 1 ≈ − ) are shown in grey. ]. bτ L 1 − 60 (left panel) and between (300 fb sτ L 1 , x x − 03. Moreover, our projected 3 . from the best fit point. The − 0 ], it is interesting to note that σ and 10 96 & – bµ L | × = 0, we see that current LHC data . We select only the points which x 94 4 sτ L . π x sτ L 4 4 | x √ ≈ need to be different from zero. Moreover, For our analysis with fixed -function built from the flavor observables . These considerations have an important . In the plots presented in this section, the | sτ L 2 bµ L χ | ≤ x bτ L x ij L 4.2 | x – 22 – | 18, i.e. within 2 and . , x . In the second part, we repeat the same exercise 6 5 TeV. sµ L . 25 µφ x ≤ . 0 61 for , current and future LHC limits provide a lower bound can be obtained by these means [ . → = 1 | ≈ 7 2 min τ 1 sτ L χ , the correlation between the two LQ couplings entering our prediction for the correlation of two LFV observables, = 6 bµ L U x | 9 7 the correlations between − ), with the hatched black lines denoting the current experimen- m as a free parameter. and 2 min 8 χ 1 U , x Kµτ (par) mass not yet excluded by vector LQ pair production searches at the m 5 TeV. One observes that the experimental value of 2 2 . Kµτ sets a lower limit on ) that in order to explain the measured deviation with respect to the 1 − → χ will push this limit even further away from 0, implying ) transitions, both U ? → ( 10 = 1 1 B ≡ 3.18 ( exp D − 1 B B . We find U R sµµ ≈ − 3 m (right panel). The color code remains the same as before and the red (blue) → sµ L (par) 2 ), and enforce perturbativity, b x ) and sµ L χ x ) for ) final states, as detailed in section ]. The resulting parameter space will then be used to show our predictions for two , while 6.1 µφ τ | 75 , sτ L 3.19 µ → We finally show in figure Next, we show in figure We first show, in figure and x Setting other couplings to zero to get the dashed regions in these plots is the reason why some of the | = τ 11 ( bτ L ` B red points remain within thenot dashed in blue the rectangles plot (because for are these not points set the to other zero). couplings which are for simplicity that the couplingsis which clear are from not eq. present ( SM in in a the given plotas are discussed set in to zero. theon case of figure impact on the LFV decays, as we discuss below. derived within the leadingcomparable logarithm lower approximation bounds [ on x dashed lines correspond to the LHC limits obtained at 36 fb to be different frommeasurements zero, of thus low-energy bounding observables its allowexclude absolute for this value possibility, from imposing below.bound the for Even lower though bound 300 fb the Even though we opted to not consider the model dependent radiative bounds, which are points excluded by direct searchesFurthermore, based the on red current points LHC data areThe (36 those blue fb that points are are excluded those from that our would survive. projectionseq. to ( 300 fb We then perform ain random eq. scan over ( thesatisfy values ∆ of the fourselected left-handed points couplings are shown compared( with the limits deduced from the direct LHC searches in Scan of parameters with we first find a bestlisted fit in point table by minimizing a The results of our analysis willwhich be is presented the in lowest two parts.LHC In [ the first we set LFV processes, but this time treating 6.2 Results and predictions JHEP10(2018)183 ). µ, τ (right = L sµ ` x ( `` → pp ) on 1 . − 7 is plotted against bτ L x allowed by flavor constraints is plotted L bτ x . Blue points are allowed by all constraints, and . 1 L sτ 4.2 − x (left panel), and – 23 – L bµ x 5 TeV. Color code is the same as in figure . is plotted against = 1 , as discussed in section 1 1 sτ L U − x m 5 TeV. Gray points are excluded by current LHC data (36 fb . . Coupling . The correlation between the couplings = 1 1 U m Figure 8 panel), by assuming including the extrapolated LHC results to 300 fb Figure 7 for The future LHC sensitivitycurrent is data depicted to by 300 the fb red points, which were obtained by extrapolating JHEP10(2018)183 , ) 1 π 7 4 U − √ has a 10 . Kµτ 2 | = × ] and by → sτ L 2 99 x sτ L | B & x ( ) B = ) at the LHCb we get that this bτ L 1 2 fermion scattering cannot be higher Kµτ x − Kµτ 1 → anomaly can still be → U → m B The main result of that ( cτν B B ( 12 B → model: as can be seen in fig- b model. Color code is the same as 1 1 . We notice that, in order to be U 1 U U corresponds to the same couplings framework, because we only consider tree-level flavor m 1 1 U U 2 min ]. χ ) for the 100 . In other words, we get an absolute lower We now repeat the same analysis as before 7 µφ − also showcase the complementarity between → . is set to 1.5 TeV. As mentioned in the previous 10 1 dilepton (direct) searches in constraining the – 24 – τ (perturbativity limit). 1 10 U ( × U T B π p 5 m m transition [ 4 is minimized along a line in parameter space defined c & 2 √ ) χ → < b | ij L Kµτ x (right panel) depend on | , where we show how the LFV branching fraction ) → ∗ ( 10 SM D B ( , with B /R ) is plotted against 1 9 TeV for the ) This upper bound roughly corresponds to setting . Moreover the lower bounds on both quantities remain rather stable U ∗ to be a free parameter too. ( ≈ σ 1 . D 13 Kµτ 1 ). We see that lowering the upper bound on /m U R 7 U ), except that now ij L m − → x m 6.5 B ( (10 5 TeV. . B O . Current bounds on these two decays, as respectively established by BaBar [ . 12 ], are also shown. 7 93 ∼ . Interestingly, we see that the current LHC bounds lead to Finally, the plots presented in figure This scale invariance relationship holds inNote the that case this of value is similar to the upper bound derived from unitarity of 2 9 12 13 amplitudes, namely Λ flavor physics (indirect) and LHC high- constraints, and all the tree-level Wilson coefficients entering the analysis scale as (coupling/mass) (left panel) and able to explainthan both flavor anomaliesand within looking the for theexplained highest within value 2 of thewhen mass varying for which the Scan of parameters withbut varying by letting given in eq. ( by a constant analysis is shown in figure bound is improved to bound of and/or Belle II canparameter have a space. major impact on the model building by further restraining the tal bounds on these processes.paragraph, Again, the fact thatdramatic the impact LHC on sets the a amounture lower of bound LFV on predicted the bywhich absolute the remains rather value stable of lower bound for the LFV mode. With 300 fb Figure 9 in figure Belle [ JHEP10(2018)183 , 8 > < − ) ) ∗ ∗ ( (6.6) 25. ( 10 . ) thus K exp D 1 & R R ≤ ) of data at ) Kµτ ∗ 1 ( . We use the − SM D → Kµτ if one accounts `` B /R 7 → ( ) quite stable. The ∗ → (right panel) for the − B ( ) B ) ∗ D . ( ( ∗ ] as a good probe of 10 ( pp SM D 9 R B SM D . × 0 ≤ 5 /R 102 /R , ) ) ≈ 1 4 TeV. Going to projected ∗ ∗ . ( ∼ ( ) D ) D < 101 -physics anomalies, R R 1 B µτ U model. Kµτ m 1 → → U s ) and B B ( ( B B Kµτ → ) (left panel) and B . ( , we see that the direct searches can play a very – 25 – 7 ) B of data and for ∗ ( 1 Kµτ , SM D − , because their branching fractions are known to be 8 . → 1 /R , but the discussion would be completely equivalent if µτ ) B ∗ ∗ ( ≈ ( B K D ) ) Kµτ R ) is also stable but already superseded by the experimental µτ → . We find that none of the scalar LQs alone, with the mass ∗ ) → ∗ ( Kµτ B K already results in an absolute lower bound Kµτ B SM K spectrum of the differential cross section of ) → or → ∗ ( T → B p SM D < R B ( µτ ( B ) scenario altogether, or further corroborate its viability. Note that for ( B B ∗ ( B → 1 > R is plotted against exp K ) U s ∗ 1 ] R ( B U D 24 m R . leaves the lower bound on both 1 TeV, can provide a model of NP that accommodates simultaneously both kinds of 1 and/or and − ' ) ) As for the upper limit on ∗ ∗ ( ( model. Color code is the same as in figure LQ SM SM D K 1 m anomalies. To arrive to thatparameters conclusion arising we combined from a the number low-energythe of flavor direct constraints physics searches on observables at the with the model come LHC. those from coming Concerning from the the latter large- ones the most significant constraints the LHC the possible range of values for this7 ratio reduces to 1 Summary and conclusion In this work weR revisited the single LQ solutions to the This kind of LFVvalidity decay of modes a was specific also UV mentioned completion in of the refs. low-energy [ important role in further reducing the space of parameters and with 300 fb definiteness we focus on we discussed related via [ 300 fb upper bound on limit on this decay modebecomes established very by appealing BaBar. as Measuringdiscarding (bounding) the the improvement of the current upper bound can either help scenario as a possible explanationR of the flavor anomalies.that While bound accommodating gets shiftedfor upwards the by direct an LHC order searches of with 36 magnitude fb to Figure 10 U JHEP10(2018)183 , 5 − , 10 Decays × τ ¯ ν 8 . decays − 4 674896 and τ τ ) ¯ 5 TeV), and ◦ ν . ∗ < ( − τ D ) ∗ exp ( ) → D ¯ B → Kµτ ¯ ]. B → (2013) 072012 scenario in which only left- B 1 , a promising route for model ( SPIRE Young Researchers Programme U in which Yukawa couplings are B 2 IN D 88 1 ][ U 37468. m ◦ 2 TeV can indeed accommodate both ÷ 1 Phys. Rev. , ' – 26 – Measurement of an Excess of Evidence for an excess of LQ arXiv:1205.5442 m [ ]. We find that the new results from direct searches for any mass of 60 7 , in its minimal version, i.e. by allowing non-zero values − ) ]. ∗ ), which permits any use, distribution and reproduction in ( 10 SM K ]. -physics anomalies), see table ) of mass × 1 B SPIRE U 91 < R (2012) 101802 , IN few ) ∗ ][ ( 90 & , 109 exp K CC-BY 4.0 ) R 88 This article is distributed under the terms of the Creative Commons and Kµτ collaboration, J.P. Lees et al., collaboration, J.P. Lees et al., ) ∗ scenario. ( → 1 SM D U B ( arXiv:1303.0571 B > R BaBar Phys. Rev. Lett. BaBar and Implications for Charged Higgs[ Bosons ) Besides the scalar LQs we also considered the vector ones. The main difficulty in ∗ 690575. Work at University of Nebraska-Lincoln is supported by the Department of ( [1] [2] exp D ◦ Attribution License ( any medium, provided the original author(s) and source are credited. References Physics and Astronomy. A.A. wouldleptoquark like searches to at the thank LHC. Frankof Golf D.A.F the is for supported Slovenian insightful Research by discussions the Agency on under the grant N Open Access. Acknowledgments This project has receivedinnovation support programme under from the the Marie EuropeanN Sklodowska-Curie Union’s grant agreement Horizon N 2020 research and bound is superseded bywhich the can current be experimental improvedof both bound magnitude at can LHCb therefore andminimal Belle either II. exclude Improving or, that if bound observed, by corroborate two the orders validity of the observe a pronounced complementarity ofthose the obtained low-energy from (flavor physics) directmode constraints searches. with In particularkept we within find the the perturbativityhanded lower limits couplings bound and are on in allowed the the to LFV minimal take values different from zero. Notice that the upper this case is thatthe one loop has effects. toweak specify singlet By a vector focusing UV LQR completion only ( of on the theonly model to tree in the level order left-handedindeed observables, to couplings compute push [ we confirm the lower that bound the of the vector LQ to larger values (above 1 the models considered inconstraints this (including work. the Since nonebuilding of involving the leptoquarks scalar seems LQsbeen to can done alone be in satisfy combining refs. all two [ the scalar LQs, in a way that has most recent experimental results which we recast to obtain the bounds relevant to each of JHEP10(2018)183 , τ ∗ ) ¯ ν | ∗ K − cb D R τ ( V , | + R ∗ 08 ] ] ] and D ]. D 94 ]. D 95 Phys. Rev. K → , Eur. Phys. J. Erratum ibid. Phys. Rev. R -lepton 0 JHEP , τ [ , ¯ B , SPIRE and the standard SPIRE ]. IN ` IN ][ `ν ][ ) decays Phys. Rev. Phys. 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