Can scalar leptoquarks explain the fDs puzzle?

Ilja Dorsner,1, 2, ∗ Svjetlana Fajfer,1, 3, † Jernej F. Kamenik,1, 4, ‡ and Nejc Kosnik1, §

1J. Stefan Institute, Jamova 39, P. O. Box 3000, 1001 Ljubljana, Slovenia2Faculty of Natural Sciences, Zmaja od Bosne 33-35, 71000 Sarajevo, Bosnia and Herzegovina

3Department of Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia4INFN, Laboratori Nazionali di Frascati, Via E. Fermi 40 I-00044 Frascati, Italy

(Dated: October 30, 2018)

Motivated by the disagreement between experimental and lattice QCD results on the Ds decayconstant we systematically reinvestigate role of leptoquarks in charm meson decays. We considerscalar leptoquarks that transform as a weak interaction triplet, doublet, or singlet in a modelindependent approach, and also argue that in a particular SU(5) GUT model these leptoquarkstates, contained in the 45-dimensional Higgs representation, could be safe against proton decaybounds. Using the current experimental measurements in τ , kaon and charm sectors, we find thatscalar leptoquarks cannot naturally explain the Ds → µν and Ds → τν decay widths simultaneously.While any contributions of the triplet leptoquarks are already excluded, the singlets could onlycontribute significantly to the Ds → τν width. Finally, a moderate improvement of the experimentalupper bound on the D0 → µ+µ− decay width could exclude the doublet contribution to the Ds →µν, while present experimental data limits its mass to be below 1.4 TeV. Possible new signaturesat present and near future experiments are also briefly discussed.

PACS numbers: 13.20.Fc,12.10.Dm,12.15.Ff

I. INTRODUCTION

Leptoquark states are expected to exist in various ex-tensions of the Standard Model (SM). They were first in-troduced in the early grand unification theories (GUTs)in the seventies [1, 2]. Scalar leptoquarks are expected toexist at TeV scale in extended technicolor models as wellas in models of quark and lepton compositeness. Scalarquarks in supersymmetric models with R-parity viola-tion (RPV) may also have leptoquark-type Yukawa cou-plings [32].

Recently, discrepancies between the experimental mea-surements of leptonic decay modes of Ds mesons [3–6]and the lattice results for the relevant fDs decay con-stant [7–9] have stimulated many analyses. One intrigu-ing indication is that the central measured and predictedvalues for the fDs differ by more than 10 % with a com-bined significance of Ds → τν and Ds → µν channelsof roughly 2.3σ [10], while the corresponding values forfD are in perfect agreement. In Ref. [11] the idea ofscalar leptoquarks has been revived to explain the miss-ing decay widths. Some implications of this suggestionhave been further explored using semileptonic [12, 13]and rare charm decays [14].

Generally, leptoquarks which also couple to diquarksmediate fast proton decay and are therefore required tobe much above the electroweak scale [15], making themuninteresting for other low energy phenomena. “Gen-uine” leptoquarks on the other hand, couple only to pairs

∗Electronic address:ilja.dorsner@ijs.si†Electronic address:svjetlana.fajfer@ijs.si‡Electronic address:jernej.kamenik@ijs.si§Electronic address:nejc.kosnik@ijs.si

of quarks and leptons, and may thus be inert with respectto proton decay. In such cases, proton decay boundswould not apply and leptoquarks may produce signaturesin other low-energy phenomena. In this article we setout to study whether scalar leptoquarks can naturallyaccount for the fDs puzzle and at the same time complywith all other measured flavor observables.

We consider all possible renormalizable leptoquark in-teractions with SM matter fields consistent with the SMgauge symmetry. One can construct such dimension-four operators using leptoquarks which are either sin-glets, doublets or triplets under the SU(2)L. If we fur-thermore require that such leptoquarks contribute to lep-tonic decays of charged mesons at tree level, we are leftwith three possible representation assignments for theSU(3)c × SU(2)L × U(1)Y gauge groups: (3,3,−1/3),(3,2,−7/6) and (3,1,−1/3). Only the weak doubletleptoquark is “genuine” in the above sense. However,using a concrete SU(5) GUT model where the relevantleptoquarks are embedded into the 45-dimensional Higgsrepresentation (45H), we demonstrate how leptoquarkcouplings to matter can arise and in particular, how thedangerous couplings to diquarks – both direct and indi-rect [15] – can be avoided.

II. GENERAL CONSIDERATIONS

In our analysis we will assume the mass eigenstateswithin a leptoquark weak multiplet to be nearly degen-erate. While large mass splittings within a weak mul-tiplet may be considered unnatural, more importantly,they are also tightly constrained by electroweak preci-sion observable T [16]. Consequently, one genericallygets correlations between semileptonic charged currents

arX

iv:0

906.

5585

v2 [

hep-

ph]

7 J

ul 2

009

2

and (lepton) flavor violating neutral currents, which rep-resent important constraints on any leptoquark scenariotrying to resolve the Ds leptonic widths puzzle. Also,we focus on observables mediated by the relevant lepto-quark couplings at tree level since these already involveprocesses forbidden in the SM at tree level, i.e., flavorchanging neutral currents (FCNCs) and lepton flavor vi-olation (LFV) processes. Finally, since the present fDsdeviation is of mild significance, we require all the mea-sured constraints to be satisfied within one standard de-viation (at 68 % C.L.) except upper bounds, for whichwe use published 90 % C.L. limits. We consider a lepto-quark explanation of the fDs discrepancy as natural, ifboth Ds and D leptonic decay widths can be obtainedclose to their measured central values.

After the electroweak (EW) symmetry breaking,quarks and leptons acquire their masses from their re-spective Yukawa interactions. These induce in the phys-ical (mass eigen-)basis a CKM and PMNS rotations be-tween the upper and the lower components of the fermiondoublets. Consequently, it is impossible to completelyisolate leptoquark mediated charged current interactionsto a particular quark or lepton generation in the left-handed sector irrespective of the initial form of the lepto-quark couplings to SM matter fields, unless there is somespecial alignment with the right-handed quark sector. Tosee this, we denote as X(′) a 3× 3 arbitrary Yukawa ma-trix in the weak basis, and write down flavor structure ofinteraction of the quark and lepton doublet parts

Qwq Xq` =(uwq dwq )Xq` = (uq d′q)(U

†X)q`, (1a)

X ′q`Lw` =X ′q`(νw` ew` )T = (X ′E)q`(ν′` e`)T , (1b)

where fields with w superscript are in the weak basis,whereas d′ = VCKMd and ν′ = VPMNSν. The uni-tary matrices U,D,E, and N rotate the fields from massto weak basis and are unphysical per se, so we absorbthem in redefinition of the couplings (e.g. YLQ ≡ U†Xon the quark and Y ′LQ ≡ X ′E on the lepton side) andconsider them as free parameters. In what follows, wewill use the convention where all the remaining rotationsare assigned to down-type quark (VCKM = U†D) andneutrino (VPMNS = E†N) sectors, and the quark mass-eigenstates are defined as

(Q1, Q2, Q3) =(u c td′ s′ b′

),(d′ s′ b′

)=(d s b

)V TCKM.

It is obvious in this notation that, even if the YLQ matrixhad all rows, except for the q-th one, set to zero, whichwould correspond to leptoquark coupling only to uq, onewould still get non-zero couplings to all three left-handeddown-quarks. Same rationale holds true for the leptonsector due to VPMNS , but since the neutrino flavors arenot tagged in present experiments, the respective decaywidths are summed over all neutrino flavors. Whenever amass-eigenstate antineutrino νi is produced in a reaction,its amplitude includes, according to Eq. (1), a factor of∑j Y′qjLQV

jiPMNS for leptoquark interaction, or V liPMNS if

the neutrino was produced in W`ν vertex. In any case,when one sums the rates for all neutrino species

∑i=1,2,3

|Ai|2 ∼∑

i=1,2,3

V jiPMNSVli∗PMNS = δjl,

it becomes evident that in the summed rate all the neu-trino indices are replaced by the lepton flavors. This isequivalent to the absence of mixing in the lepton dou-blets.

The above considerations are more general and similarin spirit to the ones recently discussed in Ref. [17] for thecase of K−K and D−D mixing. In fact, any new physicscoupling to SM fermionic weak doublets exhibits similarkind of correlations, and contributions to charged cur-rent transitions cannot be isolated to a particular quarkgeneration.

Another important particularity of the fDs puzzle isthat it affects a Cabibbo favored c→ s transition. Con-sequently, the hierarchy of correlations with other pro-cesses is largely determined by the CKM mixing hierar-chy. In particular, the mixing of the third generationwith the first two is much smaller than the mixing ofthe first two generations among each other. Therefore,for our purposes, it is often a good approximation, tocompletely neglect effects of the third generation in thequark sector. Then we can parameterize a generic lepto-quark coupling in the weak basis using a common (real)prefactor and a rotation angle

Y q`LQ = y`LQ(sinφ, cosφ).

In addition, the only CKM rotation is due to the Cabibboangle and there is no SM CP violating phase (d′ =cos θcd+sin θcs, s′ = − sin θcd+cos θcs and Vus = −Vcd =sin θc = 0.225). The absence of SM phases is not criticalfor our purposes, since we only consider CP conservingquantities, and since the relevant SM amplitudes in ourconsidered processes have approximately the same weakphase even in the full three generation case. The lepto-quark couplings themselves, however, could in principlehave arbitrary new phases. These could be important inprocesses with two or more interfering amplitudes con-tributing, at least one of those being due to the lepto-quarks. We deal with this possibility on a case by casebasis. Finally, in all the scenarios considered we havechecked explicitly that the two generation approximationis valid by performing numerical leptoquark parameterscans including the full CKM structure and a full setof possible leptoquark couplings with arbitrary phases.In this case, (semi)leptonic B decays B → τν and espe-cially B → Dτν can be used to put additional constraintson the leptoquark parameters relevant to the Ds → τνwidth. Numerically however, these constraints turn outnot be competitive with the others due to presently lim-ited experimental precision.

3

III. TRIPLET LEPTOQUARK (3,3,−1/3)

The triplet leptoquark can in principle couple to di-quarks and thus destabilize the proton, so one has tocheck in an underlying model if that is indeed the case.The allowed leptoquark interaction Lagrangian consistsof a single term

L3 = Y ij3 Qci iτ2 τ ·∆∗3 Lj + h.c. , (2)

where Qc = −QTC−1, C = iγ2γ0 and τ are the Paulimatrices. The 3×3 coupling matrix Y3 is arbitrary in thebottom-up approach. On the other hand, its entries maybe related to other parameters in an UV embedding of theeffective theory. In the concrete SU(5) model analyzedin the Appendix A, the above couplings are due to thecontraction of 10 and 5 with 45∗H – also responsible forgiving masses to the down quarks and charged leptons. Adifferent contraction of 10 and 10 with 45H couples thetriplet to diquarks. The latter term can be consistentlyset to zero in the supersymmetric version of the model,thus sufficiently suppressing proton decay.

As already mentioned in Ref. [11], the triplet lepto-quarks cannot by themselves account for deviations inDs → `ν for both τ and µ in the final state due to con-straints coming from LFV tau decays, such as τ → η(′)µand τ → φµ. Numerically, the τ → ηµ decay turns outto be most constraining. The triplet leptoquark contri-bution can be written as

Γ(3)τ→ηµ =

∣∣∣∑q=u,d,s χqYqτ3 Y qµ∗3 fqη

∣∣∣2512πm4

∆3

m3τ

[1−

(mη

mτ

)2]3/2

,

where we have neglected the muon mass. The weightχq = 1 is for q = u and 2 for q = d, s comes froman additional

√2 factor in the interaction terms with

∆3(t3 = ±1) states. Decay constants of η meson fqη aredefined as in [18]. As explained in the previous section,the couplings Y3 contain an additional VCKM rotationfor the down-type quarks

Y q`3 ≡{

Y q`3 ; q = u, c, t,(V TCKMY3)q` ; q = d, s, b.

Thus, the upper bound on τ → ηµ decay widthdirectly constrains the products Y sτ3 Y sµ∗3 , Y dτ3 Y dµ∗3

and Y uτ3 Y uµ∗3 (=(cos θcY dτ3 + sin θcY sτ3 )(cos θcYdµ∗3 +

sin θcYsµ∗3 ) in the two-generations approximation). On

the other hand, the relative contributions to the Ds lep-tonic widths (see Eq. (3)) in the two generations approxi-mation are Y sµ3 (Y sµ∗3 −tan θcY

dµ∗3 ) for the muon channel

and Y sτ3 (Y sτ∗3 − tan θcY dτ∗3 ) for the tau channel. Bothcannot be sizable and at the same time agree with thebounds that come from the τ → ηµ decay width [33]. Inscenario of triplet leptoquarks therefore, one of the mea-sured leptonic channels Ds → `ν would necessarily haveto be a measurement artifact. We will consider both pos-sibilities separately.

If the leptoquarks have sizable coupling Y sτ3 (implyingY sµ3 ∼ 0 by the τ → ηµ decay width) we can obtain anon-zero contribution to the Ds → τν decay width due tothe interference term between the SM and the leptoquarkamplitude in

Γ(3)Ds→τν = ΓSMDs→τν

∣∣∣∣1 +δτ3

4√

2GF

∣∣∣∣2 ,where the SM width is

ΓSMDs→τν =G2Fm

2τ |Vcs| 2f2

DsmDs

8π

[1−

(mτ

mDs

)2]2

,

and the relative contribution of the triplet leptoquarksreads

δτ3 ≡Y cτ∗3 Y sτ3

Vcsm2∆3

. (3)

An important observation is that the leptoquarks in thisscenario contribute to the same effective operator as theSM and thus exhibit the same helicity suppression. Inthe two generations approximation, the relative tripletcontribution simplifies to δτ3 = (yτ3 )2 cosφ sinφ(tan θc +cotφ)/m2

∆3. Reproducing the measured branching ratio

Br(Ds → τν) = 0.0561(44) [10] while using the mostprecise lattice input fDs = 241(3) MeV [7] would re-quire

√δτ3 ≈ 0.002 GeV−1. On the other hand the Y uτ3

coupling of leptoquarks is constrained by precise mea-surement of the lepton flavor universality ratio πµ/τ ≡Br(τ → πν)/Br(π → µν) = 0.1092(7) [19, 20]. Lepto-quarks contribute to semileptonic tau decays in the form

Γ(3)τ→πν = ΓSMτ→πν ×

∣∣∣∣∣1 +1

4√

2GF

[Y uτ∗3 Y dτ3

Vudm2∆3

]∣∣∣∣∣2

, (4)

where

ΓSMτ→πν =G2Fm

3τ |Vud| 2f2

π

16π

[1−

(mπ

mτ

)2]2

.

In the two generations approximation the term in squarebrackets can be written as (yτ3 )2 sinφ cosφ(− tan θc +tanφ)/m2

∆3. To exactly satisfy both the πµ/τ value and

explain leptonic Ds → τν excess we need either

(a) tanφ ≈ tan θc, i.e., leptoquarks couple only to s

but not to d quark (Y dτ3 ≈ 0), or

(b) sinφ ≈ 0, i.e., leptoquarks couple only to c but notto u quark (Y uτ3 = 0).

However, in the limit (a) one must have a sizable cou-pling Y uτ3 due to CKM rotation which results in rela-tive contribution of size δτ3 to the Cabibbo suppressedsemileptonic tau decays τ → Kν (of the form (4) withappropriate flavor replacement d → s and π → K).These are measured in agreement with the SM at the

4

3 % level [34] (in particular the ratio Kµ/τ ≡ Br(τ →Kν)/Br(K → µν) = 0.0109(4) [19, 20]) [35].

In the other limit, (b), one must have sizable couplingY dτ3 and thus gets a relative contribution scaling as δτ3to the D → τν decay width. Currently only an upperbound exists for this channel Br(D → τν) < 1.2× 10−3

at 90 % C.L. [4]. Even more importantly, one gets a non-vanishing contribution to the rare K+ → π+νν decay.Since the triplet leptoquark contributes with the sameeffective operator as the SM, its contribution can be ob-tained by simply replacing the λtXt product in the mas-ter formula of [21] with

λtXt → λtXt +√

2π∑Y s`3 Y d`∗3

GFαem sin θWm2∆3

.

This process is measured to have Br(K+ → π+νν) =(17.3+11.5

−10.5) × 10−11 [22, 23]. It constraints thesum of leptoquark coupling combinations Y s`3 Y d`∗3 =(y`3)2 cos θc cosφ(tanφ−tan θc)(1+tan θc tanφ) and fixesvery accurately tanφ = tan θc or tanφ = − cot θc. Theconstraint applies to all lepton flavors since it is in-clusive with respect to neutrino flavor [36]. The com-bined impact of all these constraints on the triplet lep-toquark contribution to Ds → τν is shown on the firstplot in Fig. 1. One observes that the combination ofthe strong bounds coming from K+ → π+νν combinedwith Br(τ → Kν)/Br(K → µν) completely excludes thetriplet leptoquarks from explaining the Ds → τν excess.

In the opposite scenario where the leptoquarks coupleto muons the situation is similar to the tau case withtwo differences: (1) the D → µν decay width has alreadybeen measured and the Br(D → µν) = 3.8(4)× 10−4 [4]agrees perfectly with the SM prediction using the mostprecise lattice QCD value of fD = 208(4) MeV [7]; (2) anadditional constraint comes from the FCNC decay KL →µ+µ− as it receives contributions from leptoquarks of theform

Γ(3)KL→µ+µ− =

f2KmKm

2µ

√1− 4m2

µ

m2K

64π

∣∣∣Y sµ3 Y dµ∗3

∣∣∣2m4

∆3

.

The requirement that such leptoquark contributions donot exceed presently measured Br(KL → µ+µ−) =6.84(11)× 10−9 [19] produces a bound equivalent to theexisting one coming from K+ → π+νν. Combining thesetwo additional constraints with the rest also clearly dis-favors a triplet leptoquark explanation of the Ds → µνexcess, as shown on the bottom plot in Fig. 1.

IV. DOUBLET LEPTOQUARK (3,2,−7/6)

The doublet leptoquarks are innocuous as far as protondecay is concerned. The allowed dimension four interac-tions in this case are

L2 = Y ij2LQi iτ2∆∗2 ej + Y ij2R ui∆†2Lj + h.c. . (5)

KΜ�ΤKΜ�ΤKΜ�Τ

Ds®ΤΝDs®ΤΝDs®ΤΝ

D®ΤΝD®ΤΝ

ΠΜ�ΤΠΜ�Τ

K+®Π+ΝΝ

0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

y3Τ�mD3

@GeV-1D

Φ

KΜ�ΤKΜ�ΤKΜ�Τ

Ds®ΜΝDs®ΜΝDs®ΜΝDs®ΜΝDs®ΜΝDs®ΜΝDs®ΜΝDs®ΜΝDs®ΜΝ

D®ΜΝD®ΜΝ

ΠΜ�ΤΠΜ�ΤΠΜ�ΤΠΜ�Τ

KL®Μ+ Μ-

K+®Π+ΝΝ

0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

y3Μ�mD3

@GeV-1D

Φ

Figure 1: Combined bounds on the triplet leptoquark param-eters in the two-generation limit in the tau (upper plot) andmuon (lower plot) sectors. All bands represent 68% C.L. ex-clusion intervals, except the upper bound on D → τν whichis taken at 90% C.L.. The K+ → π+νν and KL → µ+µ−

constraints can only be satisfied on the two horizontal dashedlines. Within the green bands, the Ds → `ν excess can beaccounted for.

In the particular SU(5) model, the term proportional toY2R stems from the contraction of 10 and 5 with 45∗Hwhile the Y2L term is due to 10 and 10 being contractedwith 45H .

In this scenario the same states couple left-handedquarks to right-handed leptons and vice versa. Con-sequently, only the product of both couplings can con-tribute to the Ds leptonic widths through the interfer-

5

ence with the SM

Γ(2)Ds→`ν = ΓSMDs→`ν

∣∣∣∣1− δ`24√

2GF

∣∣∣∣2 , (6)

δ`2 ≡m2Ds

m`(mc +ms)Y c`∗2R Y s`2L

V ∗csm2∆2

.

Again, the couplings of left-handed down- and up-typequarks are misaligned

Y q`2L ≡ (V †CKMY2L)q` for q = d, s, b.

Note that the doublet leptoquark contribution exhibitsno helicity suppression. Thus, explaining both muon andtau leptonic partial widths of Ds requires vastly differentleptoquark couplings. On the other hand, now one alsohas to take into account the strict bound coming from thedecay D0 → µ+µ−. In the doublet leptoquark model,this mode receives potential contributions from severalcoupling combinations

ΓD0→µ−µ+ =f2Dm

3D

512π

√1−

4m2µ

m2D

[(1−

4m2µ

m2D

)A+B

],

(7)

where A and B contain the couplings of doublet lepto-quarks

A =m2D

m4∆2

(mc +mu)2

∣∣Y uµ2R Ycµ∗2L − Y

cµ∗2R Y uµ2L

∣∣2 ,B =

1m4

∆2

∣∣∣∣ mD

mc +mu

(Y uµ2R Y

cµ∗2L + Y cµ∗2R Y uµ2L

)+

2mµ

mD

(Y cµ∗2R Y uµ2R + Y cµ∗2L Y uµ2L

)∣∣∣∣2 .Two combinations involve Y uµ2R which is in conjunc-tion with Y sµ2L and Y sτ2L constrained through preci-sion kaon and tau lepton flavor universality testssimilarly as in the triplet scenario. In addition,this coupling does not contribute to the Ds → µνwidth (6). The remaining two combinations can berewritten by using the Cabibbo rotation in terms ofY cµ2RY

uµ∗2L = Y cµ2R(cos θcY

dµ∗2L +sin θcY

sµ∗2L ) and Y cµ2LY

uµ∗2L =

(cos θcYsµ2L − sin θcY

dµ2L )(cos θcY

dµ∗2L + sin θcY

sµ∗2L ). The

D → µν width measurement constrains directly thesize of Y dµ2L . In absence of this coupling, D0 → µ+µ−

would receive dominant contributions from just two non-interfering leptoquark amplitudes

A ≈ m2D

m4∆2

(mc +mu)2

∣∣∣Y cµ2R Ysµ∗2L sin θc

∣∣∣2 ,B ≈ sin2 θc

m4∆2

∣∣∣∣ mD

mc +muY cµ∗2R Y sµ2L +

2mµ

mD

(Y sµ2L

)2

cos θc

∣∣∣∣2 ,where A can be related to Ds → µν decay width con-tribution (it is proportional to sin2 θc cos2 θc|δµ2 |2). The

Ds®ΜΝDs®ΜΝDs®ΜΝ

D®ΜΝD®ΜΝ

D0®Μ+ Μ-D0®Μ+ Μ-

KL®Μ+ Μ-

0.0001 0.0002 0.0003 0.0004 0.0005-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

y2Μ�mD2

@GeV-1D

Φ

Figure 2: Combined D0 → µ+µ− and D → µν bounds on thedoublet leptoquark parameters in the two-generation limit inmuon sector as explained in the text. D → µν band represent68% C.L. exclusion interval, while the upper bound on D0 →µ+µ− is taken at 90% C.L.. The KL → µ+µ− constraint canonly be satisfied on the two horizontal dashed lines. Withinthe green band, the Ds → `ν excess can be accounted for.

term proportional to A alone yields Br(D0 → µ+µ−) ≈8.3 × 10−7 for the central value of Ds → µν decaywidth. Recently, an improved experimental limit ofBr(D0 → µ+µ−) < 4.3 × 10−7 at 90 % C.L. was putforward by CDF [24]. It is evident that this introducessome tension between explaining the Ds excess and notspoiling the agreement in the D case. Due to the mod-erate significance of the Ds discrepancy, this tension isnot yet conclusive as can be seen on Fig. 2, where weplot the combined constraints in the φ–yµ2 plane. Fig. 2is generated in the following way. We first parameter-ize Y sµ2L = yµ2L cosφ and Y dµ2L = yµ2L sinφ. We then setY uµ2R = 0 while Y cµ2R = y2R. Finally, we vary yµ2L and yµ2Rwhile keeping the product yµ2 =

√yµ2Ly

µ2R fixed and use

the best fit value to determine the allowed region. Weinclude the Br(KL → µ+µ−) constraint which is alsorelevant in this case

Γ(2)KL→µ+µ− =

∣∣∣Y sµ2L Ydµ∗2L

∣∣∣2 f2Km

5K

√1− 4m2

µ

m2K

(1− 2m2

µ

m2K

)256πm4

∆2(md +ms)

2 .

While both central values for D and Ds leptonic widthsclearly cannot be reproduced by the doublet leptoquarkcontribution, a future improvement of the bound onD0 → µ+µ− is clearly sought after to reach a definiteconclusion on this scenario.

Opposed to the triplet leptoquark case, the verdict onthe Ds → τν contribution of the doublet leptoquarkis still far from conclusive. Firstly, because the corre-sponding D leptonic mode has not been measured. Sec-

6

ondly, because there are presently no strong experimentalbounds on FCNCs in the up quark sector involving onlytau leptons or only neutrinos (doublet leptoquark doesnot contribute to s→ dνν transitions). We note in pass-ing that, provided the doublet leptoquarks are to explainboth tau and muon final state excesses, there is an impor-tant bound coming from the aforementioned τ → η(′)µdecays. In this scenario they constrain the following com-bination of parameters δLFV2 = |Y sτ2LY

∗sµ2L |/m2

∆2appear-

ing in

Γ(2s)τ→ηµ =

∣∣∣Y sτ2L Ysµ∗2L

∣∣∣2mτ (fsη )2m4η

(1− m2

η

m2τ

)2

128πm2sm

4∆2

,

where we have only considered the s-quark contributionand have neglected the muon mass. On the other hand,explanation of Ds leptonic excesses requires nonzero val-ues for δµ,τ2 . Finally, in order not to spoil perturbativetreatment of the couplings, none of the couplings shouldexceed a value of roughly |Y ij2L,R| <

√4π. Then, one can

combine the above inequalities to yield a robust upperbound on the doublet leptoquark mass:

m∆2 <√

4π δLFV2 /|δµ2 δτ2 | .

Taking the present Br(τ → ηµ) < 6.5×10−8 at 90 % C.L.bound [19] and the central values for the two leptonicdecay widths, one obtains a value of roughly 1.4 TeV,which is certainly within the LHC reach [25].

V. SINGLET LEPTOQUARK (3,1,−1/3)

This state was originally proposed to explain the Ds

leptonic width puzzle in Ref. [11]. On the other hand,singlet leptoquarks are notorious for their mediation ofproton decay. However, as in the case of the leptoquarktriplet, one can demonstrate in concrete SU(5) embed-ding (see Appendix A) that the dangerous couplings donot necessarily appear. Similarly to triplets, weak sin-glets can also couple to pairs of SM matter weak dou-blets. However, now also couplings to pairs of singletsare possible resulting in the dimension four interactionLagrangian with two terms

L1 = Y ij1LQci iτ2∆∗1Lj + Y ij1Ru

ci∆∗1ej + h.c. . (8)

In this generic effective theory description clear correla-tions among different charged and neutral current flavorobservables, present in the triplet case, are somewhatdiluted by the presence of the second interaction termwhich modifies the singlet leptoquark contribution to theDs leptonic width

Γ(1)Ds→τν

ΓSMDs→τν=

∣∣∣∣∣1 +1

4√

2GFm2∆1

×

[Y cτ∗1L Y sτ1L

Vcs

]−m2Ds

(Y cτ∗1R Y sτ1L

)V ∗csmτ (mc +ms)

∣∣∣∣∣2

, (9)

where Y q` are defined as in the triplet leptoquark sce-nario. The second term in Eq. (8) can come from theSU(5) embedding without causing any conflict with thebounds on proton decay lifetime even if the leptoquark isvery light, whereas the presence of the first term wouldrequire some fine tuning in order for the leptoquark notto couple to diquarks (see Appendix A). Note that if thefirst term is absent, then the singlet leptoquark cannotcontribute to the Ds leptonic decay width. If the secondterm is absent the analysis is analogous to the tripletleptoquark scenario, with the exception that the singletdoes not contribute to KL → µ+µ−. Such is for examplethe case of the RPV minimal supersymmetric SM, wherethe interaction term of a right-handed down squark toquark and lepton doublets is present and corresponds tothe first term in (8), while the second term is absent.

From the triplet scenario we know that K+ → π+ννforces the Y `1L couplings to be diagonal in the down-typequark basis and in particular Y d`1L ≈ 0. Also relevant isthe constraint from the lepton flavor universality ratioBr(τ → Kν)/Br(K → µν) which receives relative lepto-quark contributions of the form (9) with suitable flavorreplacement (c→ u), while the tau semileptonic width isgiven as

Γ(1)τ→Kν

ΓSMτ→Kν=

∣∣∣∣∣1 +1

4√

2GF

×

[Y uτ∗1L Y sτ1L

Vus

]−mτ

(Y uτ∗1R Y sτ1L

)V ∗us(mu +ms)

∣∣∣∣∣2

.

Remaining constraint is the rare decay D0 → µ+µ−

which in this case is of the form (7) with A and B

A =m2D

m4∆1

(mc +mu)2

∣∣Y uµ1R Ycµ∗1L − Y

cµ∗1R Y uµ1L

∣∣2 ,B =

1m4

∆1

∣∣∣∣ mD

mc +mu

(Y uµ1R Y

cµ∗1L + Y cµ∗1R Y uµ1L

)− 2mµ

mD

(Y cµ∗1R Y uµ1R + Y cµ∗1L Y uµ1L

)∣∣∣∣2 .The remaining relevant free parameters can correspond-ingly be chosen as an overall coupling magnitude δ andtwo angles (φ, ω), defined through Y sµ1L = yµ1 sinω, Y cµ1R =yµ1 cosω cosφ and Y uµ1R = yµ1 cosω sinφ. The value of yµ1 isbounded from above by the condition of perturbativity(yµ1 <

√4π). Together with existing direct experimen-

tal searches for second generation leptoquarks [26, 27]this gives an additional constraint on the possible size ofthe leptoquark contributions to the Ds leptonic width.By performing a numerical fit of (yµ1 , ω, φ) to these con-straints we obtain the result, that the experimental valuefor Br(Ds → µν) cannot be reproduced within one stan-dard deviation without violating any of the other con-straints, thus excluding the singlet leptoquark as a nat-ural explanation of the Ds → µν puzzle.

7

As in the doublet case, the lack of experimental infor-mation on up-quark FCNCs involving only tau leptonsleaves the verdict on the singlet leptoquark contributionto theDs → τν decay width open. What is certain is thatdue to the K+ → π+νν constraint any such contributionhas to be aligned with the down-type quark Yukawas suchthat Y dτ1 ≈ 0 can be ensured. In addition, the secondterm in Eq. (8) needs to be present and sizable to avoidthe bounds coming from Br(K → µν)/Br(τ → Kν).

VI. CONCLUSIONS

Scalar leptoquarks cannot naturally explain both en-hanced Ds → `ν decay widths due to existing constraintscoming from precision kaon, tau, and D meson observ-ables. The triplet leptoquark is excluded from contribut-ing to any of the widths. Sizable contributions due tosingle right-handed down squark exchange in RPV su-persymmetric models are also excluded, while a genericleptoquark singlet is definitely excluded only from ex-plaining the Ds → µν width. The doublet contribu-tion to this process is still technically allowed, while an

improvement in the search for D0 → µ+µ− could verysoon also completely exclude it. For the Ds → τν onlythe triplet (and RPV) explanations are already excluded,while the possible doublet explanation of both widths re-quires its mass to lie below 1.4 TeV and will certainlyalso be probed with direct leptoquark production at theLHC. Possible future signatures of a scenario where lep-toquarks are responsible for the Ds → τν width couldalso be Br(J/ψ → τ+τ−) at the level of 10−11, probablybeyond the reach of BESIII [28], and also Br(t→ cτ+τ−)at the level of 10−5, close to the limiting sensitivity of theLHC [29].

Acknowledgments

This work is supported in part by the European Com-mission RTN network, Contract No. MRTN-CT-2006-035482 (FLAVIAnet), the Marie Curie International In-coming Fellowship within the 6th European CommunityFramework Program (I.D.) and by the Slovenian Re-search Agency.

[1] H. Georgi and S. L. Glashow, Phys. Rev. Lett. 32, 438(1974).

[2] J. C. Pati and A. Salam, Phys. Rev. D10, 275 (1974).[3] K. Abe et al. (Belle), Phys. Rev. Lett. 100, 241801

(2008), 0709.1340.[4] B. I. Eisenstein et al. (CLEO), Phys. Rev. D78, 052003

(2008), 0806.2112.[5] J. P. Alexander et al. (CLEO), Phys. Rev. D79, 052001

(2009), 0901.1216.[6] P. U. E. Onyisi et al. (CLEO), Phys. Rev. D79, 052002

(2009), 0901.1147.[7] E. Follana, C. T. H. Davies, G. P. Lepage, and

J. Shigemitsu (HPQCD), Phys. Rev. Lett. 100, 062002(2008), 0706.1726.

[8] C. Bernard et al., PoS LATTICE2008, 278 (2008),0904.1895.

[9] B. Blossier et al. (2009), 0904.0954.[10] B. Jabsley (2009), taken from the unofficial

HFAG average shown at the Charm’09 workshop,Leimen/Heidelberg, May 20-22, 2009.

[11] B. A. Dobrescu and A. S. Kronfeld, Phys. Rev. Lett. 100,241802 (2008), 0803.0512.

[12] R. Benbrik and C.-H. Chen (2008), 0807.2373.[13] A. S. Kronfeld, PoS LATTICE2008, 282 (2008),

0812.2030.[14] S. Fajfer and N. Kosnik, Phys. Rev. D79, 017502 (2009),

0810.4858.[15] S. Weinberg, Phys. Rev. D22, 1694 (1980).[16] E. Keith and E. Ma, Phys. Rev. Lett. 79, 4318 (1997),

hep-ph/9707214.[17] K. Blum, Y. Grossman, Y. Nir, and G. Perez (2009),

0903.2118.[18] T. Feldmann, P. Kroll, and B. Stech, Phys. Rev. D58,

114006 (1998), hep-ph/9802409.

[19] C. Amsler et al. (Particle Data Group), Phys. Lett.B667, 1 (2008).

[20] A. Pich, Nucl. Phys. Proc. Suppl. 181-182, 300 (2008),0806.2793.

[21] F. Mescia and C. Smith, Phys. Rev. D76, 034017 (2007),0705.2025.

[22] S. S. Adler et al. (E787), Phys. Rev. Lett. 88, 041803(2002), hep-ex/0111091.

[23] A. V. Artamonov et al. (E949), Phys. Rev. Lett. 101,191802 (2008), 0808.2459.

[24] CDF, public note 9226 (2008).[25] V. A. Mitsou, N. C. Benekos, I. Panagoulias, and T. D.

Papadopoulou, Czech. J. Phys. 55, B659 (2005), hep-ph/0411189.

[26] A. Abulencia et al. (CDF), Phys. Rev. D73, 051102(2006), hep-ex/0512055.

[27] V. M. Abazov et al. (D0), Phys. Lett. B636, 183 (2006),hep-ex/0601047.

[28] D. M. Asner et al. (2008), 0809.1869.[29] D. Jana (ATLAS) (2008), 0810.3428.[30] H. Georgi and C. Jarlskog, Phys. Lett. B86, 297 (1979).[31] V. Cirigliano and I. Rosell, JHEP 10, 005 (2007),

0707.4464.[32] For a recent review c.f. pages 452-455 in [19][33] Phases of Y ij3 could in principle conspire to yield a small

contribution to Γτ→ηµ, however, such solutions cannot atthe same time reproduce other related LFV decay limits,such as τ → η′µ, τ → πµ or τ → φµ.

[34] At this level of precision the main theoretical uncertaintyin the SM comes from electromagnetic corrections, whichhave to be taken into account [31].

[35] A complementary triplet leptoquark contribution to therare decay D0 → µ+µ− presently gives a weaker con-straint and is not included in Fig. 1

8

[36] Possible phases in Y ij3 cannot invalidate this bound, sincetau LFV decays require the Y i`3 couplings to different `lepton flavors to be of different orders of magnitude.

Appendix A: SU(5) EMBEDDING

We now demonstrate i) how natural it is for the weaktriplet, doublet and singlet leptoquark interaction termsto arise in renormalizable SU(5) model, and ii) how plau-sible it is for them to be light enough to play role in flavorphysics phenomena.

In SU(5), an ith (i = 1, 2, 3) generation of the SMmatter fields comprises 10i(= (1,1, 1)i ⊕ (3,1,−2/3)i ⊕(3,2, 1/6)i = (eCi , u

Ci , Qi)) and 5i(= (1,2,−1/2)i ⊕

(3,1, 1/3)i = (Li, dCi )), where Qi = (ui di)T and Li =(νi ei)T . The up quark (down quark and charged lep-ton) masses originate from the contraction of 10i and 10j(5j) with 5- and/or 45-dimensional Higgs representation.(Observe that 10×10 = 5⊕45⊕50 and 10×5 = 5⊕45.)Only these two representations contain component thatis both electrically neutral and an SU(3)c singlet thatcan thus obtain phenomenologically allowed vacuum ex-pectation value (VEV). Actually, both are needed in arealistic renormalizable setting on purely phenomenolog-ical grounds.

The most general renormalizable set of Yukawa cou-pling contractions with 5H and 45H is

V = Y ij5∗10αβi 5αj5∗Hβ + Y ij5 εαβγδε10αβi 10γδj 5εH

+ Y ij45∗10αβi 5δj45∗ δHαβ + Y ij45 εαβγδε10αβi 10ζγj 45δεHζ ,

where Greek indices are contracted in the SU(5) space.Relevant fermion mass matrices are

MD =(Y T5∗v

∗5 + 2Y T45∗v

∗45

)/√

2, (A1a)

ME = (Y5∗v∗5 − 6Y45∗v

∗45) /√

2, (A1b)

MU =[4(Y T5 + Y5)v5 − 8(Y T45 − Y45)v45

]/√

2, (A1c)

where 〈55H〉 = v5/

√2, 〈4515

H1〉 = 〈4525H2〉 = 〈4535

H3〉 =v45/√

2 and |v5|2 + |v45|2 = v2 (v = 247 GeV). Y5∗ , Y45∗ ,Y5 and Y45 are arbitrary 3× 3 Yukawa matrices.

If only 5-dimensional (45-dimensional) Higgs represen-tation were present one would have MT

E = (−3)MD. Ascenario with only one Higgs representation would henceyield mτ/mb = mµ/ms = me/md at the GUT scale,which is in conflict with what is inferred from experi-mental observations. This is why both 5H and 45H areneeded at renormalizable level. (Note, since mµ/ms > 1whereas mτ/mb ≈ 1 at the GUT scale this would suggestthat the the Y 22

45∗ entry is enhanced compared to otherentries of Y45∗ [30].)

Conveniently enough, the 45-dimensional Higgs rep-resentation 45H(= (∆1,∆2,∆3,∆4,∆5,∆6,∆7) =

(8,2, 1/2) ⊕ (6,1,−1/3) ⊕ (3,3,−1/3) ⊕ (3,2,−7/6) ⊕(3,1,−1/3) ⊕ (3,1, 4/3) ⊕ (1,2, 1/2) contains a weaktriplet (∆3), doublet (∆4), and singlet (∆5) leptoquarkswe are interested in whereas the 5-dimensional Higgs rep-resentation, 5H , contains a singlet leptoquark only. So,these leptoquark states must be present in any renormal-izable theory based on SU(5).

The most stringent constraints on leptoquark massesand their couplings to matter originate from limits onpartial proton decay lifetimes. In that respect only ∆4

is innocuous enough since it does not directly mediateproton decay. (It cannot couple to a quark-quark pair.)It is also practically impossible for it to be a part ofthe process that destabilizes proton through mixing withthe Higgs doublet and some other state that couples toa quark-quark pair since (3,2, 1/6) – the only suitablecandidate – is not part of either 5- or 45-dimensionalHiggs representation (or 24-dimensional representation).It is thus phenomenologically possible for ∆4 to be lightand have couplings to the matter fields of the form givenin Eq. (5) in SU(5). In that case Y2L = −21/2[Y †45− Y ∗45]and Y2R = Y †45∗ .

It is also possible to have ∆3 that couples to the quark-lepton pairs and no proton decay. In particular, the 10-5-45∗H contraction yields a lepton-quark pair couplingswith ∆3 of the form given in Eq. (2): Y3 = Y †45∗ . On theother hand, the 10-10-45H contraction yields couplingsof ∆3 to a quark-quark pair only. Clearly, if only oneof these two possible contractions is present there wouldnot be a tree level proton decay due to ∆3. In the formercase there would not be proton decay due to the mixingof ∆3 with the Higgs doublet and some other states eithersince (3,1, 2/3) – which would be a suitable candidate –is not part of either 5- or 45-dimensional representation.

Finally, ∆5 could also be coupled to matter in a man-ner that renders proton stable contrary to the usual ex-pectation. Namely, the 10-10-45H contraction yieldscouplings to a lepton-quark pair only. (This should becompared to the 10-10-5H contraction that generatesboth the lepton-quark and quark-quark type of couplingssimultaneously for the singlet leptoquark in 5H .) Thiscontraction yields the second term in Eq. (8): Y1R =21/2[Y †45 − Y ∗45]. The 10-5-45∗H contraction, on the otherhand, yields not only the second term in Eq. (5), i.e.,Y1L = −21/2Y †45∗ , but also the quark-quark couplingswhich would lead to proton instability. If only 10-10-45H contraction is present proton could be stable andaccordingly ∆5 could be light. Interestingly enough, itis possible to have a scenario in which there would notbe any leptoquark induced proton decay. The necessarycondition for this to happen would be the absence of the10-5-45∗H contraction.