Leptoquark explanation of h→μτ and muon (g−2)
aa r X i v : . [ h e p - ph ] J a n KIAS-P15052
Leptoquark explanation of h → µτ and muon ( g − Seungwon Baek ∗ and Kenji Nishiwaki † School of Physics, KIAS, 85 Hoegiro, Seoul 02455, Korea
We consider lepton flavor violating Higgs decay, specifically h → µτ , in a lepto-quark model. We introduce two scalar leptoquarks with the SU (3) c × SU (2) L × U (1) Y quantum numbers, (3 , , /
6) and (3 , , / τ − → µ − γ , is very strong when only one leptoquark contribution isconsidered. However, we demonstrate that significant cancellation is possible be-tween the two leptoquark contributions. We show that we can explain the CMS(ATLAS) excess in h → µτ . We also show that muon ( g −
2) anomaly can also beaccommodated.
PACS numbers: 12.60.Fr, 12.60.-i, 13.85.Qk ∗ Electronic address: [email protected] † Electronic address: [email protected]
I. INTRODUCTION
Leptoquarks (LQs) are scalar particles which carry both baryon and lepton numbers [1].They appear in gauge theories with “unified” gauge groups, such as Pati-Salam model, SU (5) grand unification, etc.Since LQs are strongly interacting particles which can decay semileptonically, their massesare strongly bounded by the LHC experiments, such as ATLAS and CMS. For the third-generation scalar LQs, the ATLAS group excludes the mass in the range m LQ <
625 GeVand 200 GeV < m LQ <
640 GeV at 95% confidence level (C.L.) based on their 8 TeV data,assuming 100% branching fractions into bν τ and tν τ , respectively [2]. On the other hand,the CMS group had reported various 8 TeV bounds at 95% C.L. on m LQ as m LQ >
740 GeV, m LQ >
650 GeV and m LQ >
685 GeV with assumptions of 100% branching fractions into bτ , tν τ and tτ , respectively [3, 4].We note that the CMS excess of eejj and eνjj [5] can also be interpreted as a signalof the first generation LQ with mass about 650 GeV. An example of detailed study of LQmodels for the excess can be found in [6].In the standard model (SM), lepton flavor violating (LFV) Higgs decay channels areabsent at tree level and highly suppressed by small neutrino masses and the GIM mechanismat loop level. Therefore, once they are observed with sizable branching fractions, theyindicate a clear signal of new physics beyond the SM. The CMS collaboration reported theLFV Higgs decay branching fraction, using the 19.7 fb − of √ s = 8 TeV, B ( h → µτ ) = (0 . +0 . − . )% , (I.1)which deviates 2 . σ from zero [7]. Here, µτ means the inclusive final state consisting of µ + τ − and µ − τ + . Although recent ATLAS measurement, using the 20.3 fb − of √ s = 8 TeV, B ( h → µτ ) = (0 . ± . , (I.2)does not show a significant deviation from the SM [8], it is at least consistent with theCMS result. If confirmed by the future data at LHC Run II which can probe down to ∼ − , it would be a clear signal requiring new physics beyond the SM. There are severalmodel-independent [9–21] and also model-dependent studies [22–38] to accommodate thisdeviation.The theoretical and experimental sensitivity of the anomalous magnetic moment of themuon, i.e. ( g − µ , has reached to probe the electroweak scale. State of the art calculations inthe SM cannot explain the experimental result, and there is about 3 σ discrepancies betweenthem [39, 40]: ∆ a µ = a exp µ − a SM µ = (299 ±
90 to 394 ± × − , (I.3)which also calls for new physics models.In this paper we consider a LQ model as an explanation of the LFV Higgs decay, h → µτ and muon ( g −
2) anomaly. Considering the proton decay constraints, only two types of SU (2) L doublet leptoquarks are favored. We assume both of them are realized in nature.We first show that a strong constraint from τ − → µ − γ can be alleviated significantly dueto cancellations between the top and bottom quark contributions. We show that there isallowed parameter space to accommodate the h → µτ anomaly and ( g − µ . A smoking gunsignal which distinguishes our model from other models would be the direct LQ productionat colliders. A promising signature at the LHC is the pair production of LQs decaying intoa quark and a lepton, where the decay pattern is so characteristic. Especially, componentswith +2 / Y and Y later, should be relatively light and coupleto the bottom quark, the electron and the muon for the explanation of the excess in h → µτ and the muon ( g −
2) with circumventing the bound from τ − → µ − γ . Therefore, the smokinggun final states in our model are bb τ − τ + and bb µ − µ + .The paper is organized as follows. In Sec. II, we introduce our model. In Sec. III, weconsider τ − → µ − γ constraint. In Sec. IV, we consider h → µτ signal. In Sec. V, we showthat we show that we can accommodate ( g − µ . In Sec. VI, we summarize and conclude. II. THE MODEL
Among the possible LQs which have renormalizable interactions with the SM fermions,only R and e R in the notation of [1] do not have problem with the constraint from the protondecay within renormalizable perturbation theory [6, 41]. They are in the representation R (3 , , / , e R (3 , , / , (II.1) Note that similar discussions in the context of LQs are found in [30, 36]. in the SM gauge group SU (3) C × SU (2) L × U (1) Y . Assuming both of them exist in nature at renormalizable level, they interact with quarksand leptons via the interaction Lagrangian L = − λ iju u iR R T ǫ L jL − λ ije e iR R † Q jL − λ ijd d iR e R T ǫ L jL + h.c., (II.2)where we have suppressed color indices and ǫ ( ≡ iσ ) is the two-by-two antisymmetric matrixwith ǫ = 1. The scalar potential is given by V = µ H | H | + µ | R | + e µ | e R | + λ H | H | + λ | R | + e λ | e R | + λ HR | H | | R | + e λ HR | H | | e R | + λ H R † HH † R + e λ H e R † HH † e R + (cid:16) λ mix R † H e R ǫ H + h.c. (cid:17) , (II.3)where H (1 , , /
2) is the SM Higgs doublet. R and e R fields can be decomposed into SU (2) L components, R = VY , e R = e Y e Z . (II.4)After the Higgs gets vacuum expectation value (vev), v ( ≃
246 GeV), we can write H = √ ( v + h ) , (II.5)in the unitary gauge. Then, the masses of V and e Z are given by m V = µ + 12 λ HR v , m e Z = e µ + 12 e λ HR v + 12 e λ H v . (II.6)The mass terms of Y and e Y are written as L mass ( Y, e Y ) = − (cid:16) Y † e Y † (cid:17) µ + λ HR v + λ H v λ mix v λ mix v e µ + e λ HR v Y e Y . (II.7) In the case of non- SU (2) L -doublet LQs, we can write down gauge-invariant dimension-four operatorsgenerating rapid proton decay. The SU (2) L doublet ones do not allow such dangerous operators atrenormalizable level. However, as discussed in [6, 41], constraints from dimension-five effective operators(generating proton decay) are still severe, where m LQ should be greater than around 10 TeV even whenthe cutoff scale is equal to the Planck scale. A remedy for reducing m LQ is to introduce a new symmetryprohibiting the operators. In this paper, we do not consider constraints from the proton decay caused byhigher dimensional operators. The mass eigenstates, Y , Y (with the electromagnetic charge +2 /
3) are mixture of Y and e Y with mixing angle α Y , Y e Y = c Y s Y − s Y c Y Y Y ≡ O Y Y , (II.8)where c Y = cos α Y , s Y = sin α Y .As we will see, large α Y and large mass splitting between V and Y i are favored to satisfythe experimental constraints and also to enhance h → µτ . Concretely speaking, the relation m Y i ∼ m V / τ − → µ − γ naturally with sizablecouplings, which are required for explanations of the excess in h → µτ .Here, we look into the mass matrix in Eq. (II.7) and discuss whether we can realize themass hierarchy as m Y i ∼ m V / α Y in our setup. A key point is thatthe (1 ,
1) component of the mass matrix is rephrased as m V + λ H v . Then, when thefollowing relations are realized, m V , λ H v > λ mix v , e µ + 12 e λ HR v , (II.9)and a cancellation occurs between m V and λ H v with a negative λ H , the relation m Y i ∼ m V / i = 1 ,
2) can be realized. In addition, if the off-diagonal terms are comparable withdiagonal ones, a large mixing angle in α Y is expected. For example, if the (1,1) and (2,2)components are of similar size with ∼ O (1) TeV and λ mix ∼
10, we get m Y , & . α Y , which can obviously avoid the current direct search bound on thirdgeneration LQs. However, when m V is multi TeV, a realization of such cancellation between m V and λ H v would get to be nontrivial within perturbative λ H . To further enhancemass difference between V and Y i and/or the mixing angle α Y , we can implicitly assumeadditional contributions via higher dimensional operators such as φ Λ R † HH † R , φ Λ R † H e R ǫ H, (II.10)where φ is a new singlet with a large vev as h φ i > Λ.Although there is no apparent symmetry which leads to the mass ratio m Y i ∼ m V / τ µV ( Y i ) t ( b ) FIG. 1: Feynman diagrams for τ − → µ − γ . The photon line can be attached to any chargedparticles, and there are four possibilities. Finally, we comment on the case with a small mixing angle α Y , which corresponds tothe possibility that no sizable cancellation occurs between the terms m V and λ H v . Evenin this case, (at least) one mass eigenstate can be light as the relation m Y i ∼ m V / τ − → µ − γ . To make matters worse, asdiscussed in Sec. IV, such hierarchical couplings are inappropriate for explaining the excessin h → µτ . III. τ − → µ − γ In this section, we consider the constraints from the charged lepton flavor violating pro-cesses. Since we are interested in 2 ↔ τ − → µ − γ decay. Our study can be applied to other LFVs, such as µ − → e − or τ − → e − transitions,similarly. However, we assume they are sufficiently suppressed by small LFV couplings.The effective Hamiltonian for τ − → µ − γ is written as H eff = C γR µ L σ µν F µν τ R + C γL µ R σ µν F µν τ L , (III.1)where C γR,L are Wilson coefficients and F µν (= ∂ µ A ν − ∂ ν A µ ) is the photon field strengthtensor.The Feynman diagrams for τ − → µ − γ are shown in Fig. 1. We note that in our model,the chirality flip appearing in (III.1) can occur inside the loop. Therefore the amplitudescan be proportional to m t or m b instead of small masses from the external lines, m τ or m µ .This is the main reason that this LFV process becomes a very strong constraint in ordinarythird generation LQ models.The Wilson coefficients C γR,L can be calculated from the diagrams in Fig. 1: C γR = N c e π m V "(cid:16) λ e λ ∗ e m µ + λ ∗ u λ u m τ (cid:17)(cid:16) I ( x ) + 53 J ( x ) (cid:17) + λ ∗ u λ ∗ e m t (cid:16) I ( x ) + 53 J ( x ) (cid:17) + X j =1 , N c e π m Y j "(cid:16) λ e λ ∗ e O j m µ + λ ∗ d λ d O j m τ (cid:17)(cid:16) − I ( y j ) + 23 J ( y j ) (cid:17) + λ ∗ d λ ∗ e O j O j m b (cid:16) − I ( y j ) + 23 J ( y j ) (cid:17) ,C γL = N c e π m V "(cid:16) λ ∗ u λ u m µ + λ e λ ∗ e m τ (cid:17)(cid:16) I ( x ) + 53 J ( x ) (cid:17) + λ e λ u m t (cid:16) I ( x ) + 53 J ( x ) (cid:17) + X j =1 , N c e π m Y j "(cid:16) λ ∗ d λ d O j m µ + λ e λ ∗ e O j m τ (cid:17)(cid:16) − I ( y j ) + 23 J ( y j ) (cid:17) + λ e λ d O j O j m b (cid:16) − I ( y j ) + 23 J ( y j ) (cid:17) , (III.2)where N c = 3 is the color factor, x = m t /m V , and y i = m b /m Y i . The loop functions areobtained to be I ( x ) = 2 + 3 x − x + x + 6 x log x − x ) ,J ( x ) = 1 − x + 3 x + 2 x − x log x − x ) ,I ( x ) = − x − x − x − x ) ,J ( x ) = 1 − x + 2 x log x − x ) . (III.3)The branching ratio of τ − → µ − γ is then B ( τ − → µ − γ ) = τ τ ( m τ − m µ ) πm τ (cid:0) | C γR | + | C γL | (cid:1) , (III.4)where τ τ = 87 . µ m is the lifetime of τ . The current experimental bound is [42] B ( τ − → µ − γ ) < . × − . (III.5)This corresponds to | C γR | + | C γL | < (cid:18) . × − GeV (cid:19) . (III.6) m Y H GeV L a H m Y = a m Y L H Λ u (cid:144) Λ d L * H sin Α Y = (cid:144) L m Y H GeV L a H m Y = a m Y L H Λ u (cid:144) Λ d L * H sin Α Y = L m Y H GeV L a H m Y = a m Y L H Λ u (cid:144) Λ d L * H sin Α Y = L FIG. 2: Contour plots for ( λ u /λ d ) ∗ which is required for exact cancellations of τ − → µ − γ in( m Y , m Y )-plane through Eq. (III.7). From top-left to bottom, the sine of the mixing angle sin α Y is chosen as 1 / √
2, 0 . .
2, respectively. m V is set as m V = 6 m Y and m Y is formulated as m Y = a m Y by use of the factor a , where the range [1 . , .
0] is considered in the three plots.
For the discussion of τ − → µ − γ , we assume C γL = 0 for simplicity. If we consider a singleleptoquark contribution from V , B ( τ − → µ − γ ) gives lower mass bound 0 . , . , ,
42 TeV,for λ e = 0 . , . , . ,
1, respectively, where we took λ u = 0 .
35. With these parameters,we obtain too small contribution to h → µτ as was noticed in [30].Since we introduce both R and e R , we can have diagrams with the chirality flip insidethe b -quark loop, which generate terms proportional to m b . The Y i − b contributions arenaively expected to be smaller than V − t contribution by factor m b /m t ∼ /
35. However,since C γR,L are proportional to m f /m ( f = t, b ) as can be seen in (III.2), if m Y i ∼ m V / t and b contributions can occur naturally. Note that a nonzero mixingbetween Y and e Y is mandatory for a natural cancellation since in the limit sin α Y → m b turn out to be zero. Neglecting small termsproportional to m τ or m µ , an exact cancellation in C γR occurs when the following conditionis held, λ ∗ u λ ∗ d = − X i =1 O i O i m b m t (cid:18) m V m Y i (cid:19) − I ( y i ) + J ( y i )+ I ( x ) + J ( x ) . (III.7)In Fig. 2, the values of the ratio ( λ u /λ d ) ∗ which are required for exact cancellations of τ − → µ − γ are shown as ( m Y , m Y )-planes through Eq. (III.7) with the three choices of thesine of the mixing angle sin α Y as 1 / √
2, 0 . . Here, m V is set as m V = 6 m Y and m Y is formulated as m Y = a m Y by use of the factor a , where the range [1 . , .
0] is consideredin the three plots. Note that in the case that m Y and m Y are completely degenerated,the two contributions being proportional to m b are exactly canceled out between them andno cancellation mechanism works in τ − → µ − γ . Here, almost all the shown regions inFig. 2 (where sin α Y is greater than 0 . λ u /λ d ) ∗ are greaterthan 0 .
05, which means that we can adjust naturally the two couplings for realizing thecancellation. However, as we will see in the following section, when the ratio ( λ u /λ d ) ∗ getsto be small, it is hard to explain the excess of h → µτ .The Wilson coefficient C γR can be rewritten in terms of the ratio in (III.7), which we willdefine as ( λ ∗ u /λ ∗ d ) cancel , C γR ≃ N c e π m V λ ∗ u λ ∗ e m t (cid:16) I ( x ) + 53 J ( x ) (cid:17)" − λ ∗ d λ ∗ u (cid:18) λ ∗ u λ ∗ d (cid:19) cancel . (III.8)This equation shows again that, if λ ∗ d /λ ∗ u = ( λ ∗ d /λ ∗ u ) cancel , C γR = 0. We can considera deviation from the exact cancellation by introducing δ in such a way that λ ∗ d /λ ∗ u =( λ ∗ d /λ ∗ u ) cancel (1 − δ ). Then we can take δ as a degree of required tuning for cancellation Note that the sign of sin α Y is not important. We can compensate a negative sign by flipping the sign ofthe coupling λ u or λ d . FIG. 3: The blue region shows the 2 σ favored region for h → µτ in ( m V , λ conv )-plane, where λ conv ≡ | λ u λ e | . The black contours indicate degrees of the fine tuning defined around Eq. (III.8)in percentage terms. in τ − → µ − γ . In Fig. 3, the black lines show a constant contour plot of δ in ( m V , λ conv ( ≡| λ u λ e | ))-plane in percentage terms when we take the upper limit on C γR (with C γL = 0)in Eq. (III.6). The plot shows that we need fine-tuning at the level of 0 .
1% to explain theexcess of h → µτ consistently. IV. h → µτ The lepton flavor violating Higgs decay is evaluated from the Feynman diagrams shownin Fig. 4. The divergence in diagram Fig. 4 (a) cancels those in Fig. 4 (c), (d), and the totalresult is finite, generating the dimension-four effective operators H eff ( h → µτ ) = hµ ( C R P R + C L P L ) τ + H.c. (IV.1)The dimensionless effective couplings C R,L are calculated to be C R = − λ ∗ u λ ∗ e N c m t π v " I acd ( r t , r h ) + I a ( r t , r h ) − λ HR v m V I b ( r t , r h ) τ µV ( Y i ) t ( b ) τ µV ( Y i ) t ( b ) τ µV ( Y i ) t ( b ) τ µV ( Y i ) t ( b ) τ µV ( Y i ) t ( b ) h hh h ( a ) ( b )( c ) ( d ) FIG. 4: Feynman diagrams of one-loop correction for H − µ − τ vertex. − λ ∗ d λ ∗ e N c m b π v ( X i =1 O i O i h I abc (0 , s i ) + I a (0 , s i ) i − X i,j =1 O i O j ( O T Λ O ) ji v m Y b I b ( s ij , s j ) ) , (IV.2) C L = − λ e λ u N c m t π v " I acd ( r t , r h ) + I a ( r t , r h ) − λ HX v m V I b ( r t , r h ) − λ e λ d N c m b π v ( X i =1 O i O i h I abc (0 , s i ) + I a (0 , s i ) i − X i,j =1 O i O j ( O T Λ O ) ji v m Y b I b ( s ij , s j ) ) , (IV.3)with r t = m t /m V , r h = m h /m V , s i = m h /m Y i , s ij = m Y i /m Y j , andΛ ≡ λ HR λ mix λ mix e λ HR . (IV.4)Note that the coupling combinations, λ ∗ u λ ∗ e and λ ∗ d λ ∗ e in C R ; λ e λ u and λ e λ d in C L ,are also found in the terms in C γR and C γL for describing primary contributions to τ − → µ − γ ,respectively. But here, no sizable cancellation emerges between terms being proportional2to m t and m b when we adjust parameters for realizing the cancellation in τ − → µ − γ . Weignore the apparently irrelevant terms being proportional to m τ or m µ , which arise fromchirality flips in the external lines. The loop functions are I acd ( r t , r h ) = − − Z [ dx ] log h x + (1 − x ) r t − x x r h − iε i + Z dx log h x + (1 − x ) r t − iε i ,I a ( r t , r h ) = Z [ dx ] x x r h − r t x + (1 − x ) r t − x x r h ,I b ( r t , r h ) = Z [ dx ] 11 − x + x r t − x x r h ,I b ( s ij , s j ) = Z [ dx ] 1 x s ij + x − x x s j , (IV.5)where R [ dx ] ≡ R dx R dx R dx δ (1 − x − x − x ) and ε represents an infinitesimalpositive value. The form of the partial width Γ h → µ − τ + is described by use of the Wilsoncoefficient C R and C L in Eq. (IV.1) asΓ h → µ − τ + = ¯ β πm h (cid:2) ( m h − m µ − m τ ) (cid:0) | C R | + | C L | (cid:1) − m µ m τ ( C R C ∗ L + C L C ∗ R ) (cid:3) , (IV.6)with the kinetic factor ¯ β = s − m µ + m τ ) m h + ( m µ − m τ ) m h , (IV.7)while that of the conjugated process Γ h → µ + τ − is straightforwardly obtained by the replace-ments C R → C ∗ R and C L → C ∗ L . The inclusive width Γ h → µτ is simply defined asΓ h → µτ = Γ h → µ − τ + + Γ h → µ + τ − . (IV.8)We use the value Γ SM h = 4 .
07 MeV in m h = 125 GeV reported by the LHC Higgs CrossSection Working Group [43] for evaluating B ( h → µτ ) in our model.In the following analysis, as we did in the τ − → µ − γ in Sec. III, we adopt the assumptionof C L = 0. Among many terms in (IV.2), the two terms in the first line, i.e. , the top-quarkcontribution in Fig. 4 (a) dominates and we ignore the bottom-quark contributions in thefollowing numerical estimation. In Fig. 3, we show the 2 σ range to explain the excess in When X is real, the relation log [ X ± iε ] = log [ | X | ] ± iπθ ( − X ) with the Heaviside theta function θ isuseful. h → µτ shown in Eq. (I.1) in ( m V , λ conv )-plane, where λ conv ≡ | λ u λ e | . We set λ HR = 1,which is the coupling of the subleading term in Eq. (IV.2) with the suppression factor v /m V . Here, an upper limit on λ conv is estimated as ( λ u λ e ) | max = (( λ u /λ d ) λ d λ e ) | max ≃ . · π · π ≃
40, where 0 .
25 means a typical maximal value of the ratio ( λ u /λ d ) shown inFig. 2 (when m V is multi TeV and sin α Y = 1 / √ π comes from perturbative regimein λ d and λ e .Combining Fig. 2 and Fig. 3, we can see that it is possible to explain the excess shownin Eq. (I.1) in our scenario. At first, we will remember the relation in the LQ’s masses, m Y i ∼ m V / λ u and λ d in τ − → µ − γ . When werequest the (exact) cancellation in τ − → µ − γ , as shown in Fig. 2, the ratio ( λ u /λ d ) ∗ shouldbe smaller than unity. Considering a typical scale of m V is more than a few TeV through therelation m Y i ∼ m V / m LQ , as a rough estimation, λ conv needsto be larger than around ten. Taking into account the bound via perturbativity λ e < π ,roughly speaking, λ u should be greater than one through the definition of λ conv . Followingthis property, we should think about the property of the ratio ( λ u /λ d ) ∗ . Roughly, greaterthan 0 . λ u > λ d isstill perturbative ( λ d < π ). This means that the mixing angle α Y should be large to someextent since when α Y becomes far from the maximal case, the region with ( λ u /λ d ) ∗ > . As an example, we can satisfy τ − → µ − γ constraint, with ( m V , m Y , m Y ) =(3 . , . , .
6) TeV and sin α Y = 1 / √ λ u /λ d ≈ .
15. If we take λ d ≈ λ e ≈
4, we get λ conv ≈
6, which can explain the central value shown in Eq. (I.1). V. ( g − µ The anomalous magnetic moment of the muon has been measured to 0.5 ppm level [44], a exp µ = 116 592 080(63) × − . (V.1) It is possible to modify the mass relation m Y i ∼ m V / λ u and λ d . When m V is heavier than the case following m Y i ∼ m V /
6, the top contribution in τ − → µ − γ decreases and the ratio ( λ u /λ d ) ∗ can get to be large, which means that larger λ u would be realizable.On the other hand, however, a large m V suppresses the process h → µτ . a SM µ = 116 591 785(61) × − . (V.2)The discrepancy ∆ a µ = a exp µ − a SM µ = (295 ± × − (V.3)is believed to come from new physics contributions.However, we should also keep in mind that there is a possibility that the discrepancy(or part of it) comes from underestimated uncertainties in hadronic part, for example, inhadronic light-by-light scattering. Lattice calculations [46–48] as well as calculations usingdispersion relations [49–51] will reduce the hadronic uncertainties in the future.In our model the leptoquark contribution to ( g − µ is given by∆ a µ = − N c m µ π m V " m µ (cid:16) (cid:12)(cid:12) λ e (cid:12)(cid:12) + (cid:12)(cid:12) λ u (cid:12)(cid:12) (cid:17)(cid:16) I ( x ) + 53 J ( x ) (cid:17) + Re( λ u λ e ) m t (cid:16) I ( x ) + 53 J ( x ) (cid:17) − X j =1 , N c m µ π m Y j " m µ (cid:16) (cid:12)(cid:12) λ e (cid:12)(cid:12) O j + (cid:12)(cid:12) λ d (cid:12)(cid:12) O j (cid:17)(cid:16) − I ( y j ) + 23 J ( y j ) (cid:17) + Re( λ d λ e ) O j O j m b (cid:16) − I ( y j ) + 23 J ( y j ) (cid:17) , (V.4)the loop functions are given in (III.3). We notice that, if we set m µ,τ → C γR in Eq. (III.2)and inside the square brackets in ∆ a µ in Eq. (V.4), ∆ a µ is exactly proportional to C γR as∆ a µ = − m µ e λ e λ e C γR , (V.5)where we assumed all the couplings are real. If we use the current upper bound of C γR in(III.6), we get ∆ a µ ≈ − (66 . × − ) λ e λ e . (V.6)Therefore we see that, if − . λ e /λ e . −
2, we can explain the muon ( g − µ with ± σ accuracy.Since in our case the Yukawa couplings to explain h → µτ and ( g − µ are rather large,one may expect higher order diagrams such as Barr-Zee type two-loop diagrams [52] mayenhance ( g − µ as in the case of MSSM with large tan β [53, 54]. In our estimate, the5 µ µ t V V t t µ γ FIG. 5: A two-loop Barr-Zee type diagram for ( g − µ . dominant two-loop diagram is shown in Fig. 5. Other diagrams, such as the one with LQsrunning inside the loop, are suppressed, for example, by small muon mass, and we do notconsider them. Although the diagram in Fig. 5 may look comparable with the one-loopdiagrams due to large λ u λ e , we still need chirality flip inside the muon line in the fermionictriangle loop. Concretely, the diagram is estimated to be suppressed at least by ∼ π λ u λ e m µ m t ∼ − (V.7)compared to the one-loop diagram. VI. CONCLUSION
In this paper, we considered the recent CMS excess in h → µτ and the muon ( g − R (3 , , /
6) and e R (3 , , /
6) that are free from proton decay problems atrenormalizable level. The constraints from lepton flavor violating process τ − → µ − γ canbe evaded by a natural cancellation between leptoquark contributions with some tuning on λ u and λ d , where their orders can be the same. When the cancellation is realized, sizablecouplings contributing to h → µτ are allowed and then we give a reasonable explanation onthe excess. The ( g − µ anomaly is also explained. Finally, we mention that various kindsof other anomalies in flavor physics have been reported [55–60]. Giving a more exhaustiveexplanation in the context of leptoquarks would be an important task [61].6 Acknowledgments
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