Lepton flavor violating μ→eγ and μ−e conversion in unparticle physics
aa r X i v : . [ h e p - ph ] N ov Lepton flavor violating µ → eγ and µ − e conversion in unparticlephysics Gui-Jun Ding a and Mu-Lin Yan a,ba Department of Modern Physics,University of Science and Technology of China,Hefei, Anhui 230026, China b Interdisciplinary Center for Theoretical Study,University of Science and Technology of China,Hefei, Anhui 230026, China
Abstract
We have studied lepton flavor violation processes µ → eγ and µ − e conversion in nuclei inducedby unparticle. Both Br( µ → eγ ) and µ − e conversion rate CR( µ − e, Nuclei) strongly dependon the scale dimension d U and the unparticle coupling λ ff ′ K (K=V, A, S, P). Present experimentalupper bounds on Br( µ → eγ ), CR( µ − e, Ti) and CR( µ − e, Au) put stringent constraints on theparameters of unaprticle physics. The scale dimensions d U around 2 are favored for the unparticlescale Λ U of O (10 TeV) and the unparticle coupling of O (10 − ). CR( µ − e, Nuclei) is proportionalto Z A / Z for the pure vector and scalar couplings between unparticle and SM fermions, thispeculiar atomatic number dependence can be used to distinguish unparticle from other theoreticalmodels. . INTRODUCTION Scale invariance proves to be a very powerful concept in physics. At low energy, the scaleinvariance is explicitly broken by the masses of particles, and it is manifestly broken bythe Higgs potential in the standard model(SM). However, there may exist a scale invariantsector at a much higher scale, e.g., above TeV scale. Motivated by the Banks-Zaks theory[1],recently Georgi suggested that a scale invariant sector with nontrivial infrared fixed-pointmay appear, which weakly couples to the SM[2, 3]. At low energy scale, this sector is matchedonto the so called ”unparticle” with non-integral scale dimension d U . For simplicity, mostliteratures so far have assumeed that scale invariance remains until the low energy scale.Unparticle is very peculiar from the view of particle physics, it looks like a non-integralnumber d U of invisible massless particles, this leads to peculiar energy and momentumdistributions, through which unparticle may be detected in high energy colliders [2]. Un-particle doesn’t have a definite invariant mass, instead a continuous mass spectrum, whichcan be represented by an infinite tower of massive particles from the perspective of particlephysics[4]. Moreover, the unparticle two-point correlation function has an unusual phase inthe time-like region, which can produce interesting interference patterns between the theamplitude of S-channel unparticle exchange and that of SM processes[3].Despite of the complexities of the scale invariant sector, we can use the effective theoryto deal with its low energy behavior. The unparticle operator can have different lorentzstructures: scalar O U , vector O µ U , tensor O µν U or spinor. However, so far there is no principleto constrain the interactions between the SM fields and unparticle. The rich unparticlephenomenological implications have been extensively studied[2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37,38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50] in particle physics, astrophysics, cosmology,gravity and so on.In the minimal version of SM, lepton flavor violating(LFV) interactions are strictly for-bidden. In the minimal extension of SM in order to accomodate the present data on neutrinomasses and mixings, the LFV processes, such as ℓ i → ℓ j γ ( i = j ) and µ − → e − e + e − are verystrongly suppressed due to tiny neutrino masses and unitarity of the mixing matrix(MNSmatrix). In particular, the branching ratio for µ → eγ amounts to at most 10 − , to be com-pared with the present experimental upper bound 1 . × − [51]. However, most extensions2f the SM predict LFV, and some of them predict LFV at much higher rates, which maybe in conflict with the existing experimental bounds. LFV provides an unique insight intothe nature of new physics beyond SM[52], and various LFV processes have been consideredin many scenarios of new physics beyond SM, such as the see-saw model with or withoutGUT[53], supersymmetry[54], Z ′ model[55] and so on. Three kinds of LFV processes areusually discussed: LFV radiative decays ℓ i → ℓ j γ ( i = j ), µ → e like processes and µ − e conversion in nuclei. Unparticle induced µ − → e − e + e − and other cross symmetry relatedprocesses such as e + + e − → e + + µ − have been considered[11, 14, 18]. In this work , wewill consider LFV radiative decay µ → eγ and µ − e conversion in nuclei.Besides the great theoretical interests, there has been a lot of theoretical efforts in de-tecting LFV processes at CERN LEP and B-factories. The current experimental bound onthe LFV radiative decay µ → eγ is as follows[51]Br( µ → eγ ) < . × − , CL = 90% (1)For µ − e conversion in heavy nuclei, the most stringent constraints arise for Titanium andGold, respectively with CR( µ − e , Ti) < . × − [56] and CR( µ − e , Au) < × − [57].Several experiments have been designed to explore LFV with much higher sensitivity thanpresently available. In particular, the MEG experiment at PSI will detect µ → eγ downto the 10 − − − level in the very near future[58]. Concerning the challenging µ − e conversion in heavy nuclei, the J-PARC experiment PRISM/PRIME is expected to reach asentsitivity of O (10 − )[59], i.e. an improvement by six orders of magnitude relative to thepresent upper bound.Motivated by the future considerable progress in experimental measuements, studying µ → eγ and µ − e conversion in unparticle physics are of great theoretical interests. Thepaper is organized as follows. In Section II, we review the baisc aspects of unparticle physics.In Section III, we calculate the LFV radiative decays µ → eγ . µ − e conversion rates innuclei are considered in Section IV. Finally we present our conclusions and some discussions. II. THE MODEL
As was suggested by Georgi[2], we shall assume that at a very high energy scale, theworld consists of the SM sector and the so-called Banks-Zaks ( BZ ) sector with non-trivial3nfrared(IR) fixed point, and the two sectors interact with each other via the exchange ofparticles with a large mass scale M U >> U , the interactions betweenthese two sectors may be described by the effective non-renormalizable Lagrangian,1 M d BZ + d SM − U O SM O BZ (2)which is analogous to the four-fermion interactions in SM, where O SM and O BZ are respec-tively local operators constructed from the SM fields and the BZ fields. The BZ theory hasIR fixed point around an energy scale Λ U ∼ BZ sector under-goes dimensional transmutation and the scale invariant unparticle sector emerges. The BZ operator O BZ matches onto the unparticle operator O U , and the interactions between theunparticle and the SM fields generally have the form L eff = λ Λ d U + d SM − U O SM O U (3)where λ = C U ( Λ U M U ) d BZ + d SM − and C U is the Wilson-like coefficient function. The lowestorder effective interactions between the unparticle and the SM fermion fields are as follows L int = λ ff ′ V Λ d U − U f γ µ f ′ O µ U + λ ff ′ A Λ d U − U f γ µ γ f ′ O µ U + λ ff ′ S Λ d U − U f f ′ O U + λ ff ′ P Λ d U − U f iγ f ′ O U (4)Here f and f ′ denote SM fermions(leptons or quarks), and they should have same electriccharges. We note that both the third and the fourth term are absent, if we require thatthe effective Lagragian L int is consistent with the SM symmetry with unparticle being SMsinglet. The unparticle operators have been set to be hermitian, and O µ U is assumed to betransverse ∂ µ O µ U = 0. The couplings between the SM fermion fields and unparticle are quitearbitrary, it can be flavor conserving or changing. Moreover, there is no any correlation inthe transitions among three generations for flavor changing processes. In Ref.[8], the authorsintroduced BZ charges for the SM particles at very high energy scale, then tree level flavorchanging neutral current(FCNC) can be induced by rediagonalizations of the SM fermionmass matrices. Under the Fritzsch ansatz of the mass matrices, the FCNC effects werefound to associated with the mass ratios q m i m j /m . Scale invariance fixes the two-pointcorrelation function of unparticle, by dispersion relation, the two-point correlation functionis determined to be[3, 5] Z d x e iP · x h | T ( O U ( x ) O †U (0)) | i = iA d U π Z ∞ s d U − P − s + iǫ ds = iA d U d U π ) ( − P − iǫ ) d U − (5)4here the normalization factor A d U is chosen to be A d U = 16 π / (2 π ) d U Γ( d U + )Γ( d U − d U ) (6)and the complex function ( − P − iǫ ) d U − is defined to be( − P − iǫ ) d U − = ( | P | − iǫ ) d U − , P < | P | + iǫ ) d U − e − id U π , P > O µ U Z d x e iP · x h | T ( O µ U ( x ) O ν †U (0)) | i = iA d U d U π ) ( − P − iǫ ) d U − ( − g µν + P µ P ν /P ) (8)We note that the dispersion representation of the unparticle two-point correlation functionis very useful, if unparticle appear in the loop, e.g. the unparticle induced lepton anomalousmagnetic momentum and LFV radiative decay µ → eγ in the following. III. LFV RADIATIVE DECAYS (cid:1) (a) (cid:1) (b) (cid:1) (c) FIG. 1: The Feynman diagrams contributing to µ → eγ The diagrams for the LFV µ → eγ are shown in Fig.1. Generally, the amplitude for µ → eγ can be written as M ( µ → eγ ) = ε µ ∗ u e ( p e )[ iq ν σ µν (A + B γ ) + γ µ (C + D γ ) + q µ (E + F γ )] u µ ( p µ ) (9)where q µ and ε µ are respectively the photon momentum and polarization, A , B , ..., F areinvariant amplitudes. The electromagnetic gauge invariance requires the above amplitude isinvariant under ε µ → ε µ + q µ , then we have,C = D = 0 (10)5ince the photon is on shell q = 0 and transverse ε µ q µ = 0, µ → eγ is a magnetic transition M ( µ → eγ ) = ε µ ∗ u e ( p e )[ iq ν σ µν (A + B γ )] u µ ( p µ ) (11)It is easy to calculate of the corresponding radiative decay widthΓ( µ → eγ ) = m µ π ( | A | + | B | ) (12)where we have neglected the final state electron mass. Using Γ( µ → eν e ν µ ) = m µ G / π ,here G F is the Fermi constant, this can be converted into the branching ratioBr( µ → eγ ) = Γ( µ → eγ )Γ( µ → eν e ν µ ) = 24 π m µ G ( | A | + | B | ) (13)We note that the couplings between unparticle and photon such as O µα U O U να F νµ also con-tribute to µ → eγ , However, its contribution is highly suppressed compared with those inFig.1. Using the dispersion representation of the unparticle two-point correlation function,it is straightforward, albeit some lengthy, to work out these unparticle induced radiativedecay µ → eγ amplitude. In fact, we only need to consider Fig.1(b), since the contributionof Fig.1(a) and Fig.1(c) are proportional to ε µ ∗ u e ( p e ) γ µ u µ ( p µ ) or ε µ ∗ u e ( p e ) γ µ γ u µ ( p µ ). A S = eA d U d U π ) − i (4 π ) X a = e, µ, τ λ eaS λ aµS (Λ U ) d U − Z dxdydz δ ( x + y + z − xzm e + 2 yzm µ +2( x + y ) m a ] e − i ( d U − π [ xzm e + yzm µ − ( x + y ) m a ] d U − z − d U (14) A P = eA d U d U π ) − i (4 π ) X a = e, µ, τ λ eaP λ aµP (Λ U ) d U − Z dxdydz δ ( x + y + z − xzm e + 2 yzm µ − x + y ) m a ] e − i ( d U − π [ xzm e + yzm µ − ( x + y ) m a ] d U − z − d U (15) A V = eA d U d U π ) − i (4 π ) X a = e, µ, τ λ eaV λ aµV (Λ U ) d U − Z dxdydz δ ( x + y + z − { [ − z (1 − x ) m e − z (1 − y ) m µ + 8 zm a ] e − i ( d U − π [ xzm e + yzm µ − ( x + y ) m a ] d U − z − d U +2 [2 y ( m a − m e ) + 2 x ( m a − m µ ) − (1 + z ) m a + 2 z ( xm e + ym µ ) + z (1 − x ) m e + x (1 − z ) m e + z (1 − y ) m µ + y (1 − z ) m µ ] e − i ( d U − π [ xzm e + yzm µ − ( x + y ) m a ] d U − × z − d U / (2 − d U ) + [2 y ( m a − m e )( xzm e + (1 − y )(1 − z ) m µ + ym a m µ ) +2 x ( m a − m µ )( yzm µ + (1 − x )(1 − z ) m e + xm e m a ) + 2 xy ( m e + m µ )( m a − m e ) × ( m a − m µ )] e − i ( d U − π [ xzm e + yzm µ − ( x + y ) m a ] d U − z − d U } (16)6 A = eA d U d U π ) − i (4 π ) X a = e, µ, τ λ eaA λ aµA (Λ U ) d U − Z dxdydz δ ( x + y + z − { [ − z (1 − x ) m e − z (1 − y ) m µ − zm a ] e − i ( d U − π [ xzm e + yzm µ − ( x + y ) m a ] d U − z − d U +2 [ − y ( m a + m e ) − x ( m a + m µ ) + (1 + z ) m a + 2 z ( xm e + ym µ ) + z (1 − x ) m e + x (1 − z ) m e + z (1 − y ) m µ + y (1 − z ) m µ ] e − i ( d U − π [ xzm e + yzm µ − ( x + y ) m a ] d U − × z − d U / (2 − d U ) + [ − y ( m a + m e )( xzm e + (1 − y )(1 − z ) m µ − ym a m µ ) − x ( m a + m µ )( yzm µ + (1 − x )(1 − z ) m e − xm e m a ) + 2 xy ( m e + m µ )( m a + m e )( m a + m µ )] e − i ( d U − π [ xzm e + yzm µ − ( x + y ) m a ] d U − z − d U } (17) B r( e ) d u V A S P exp.
FIG. 2: Variation of the branching ratios Br( µ → eγ ) with the scale dimension d U . V, A, Sand P respectively denote the branching ratios for the pure vector, axial vector, scalar and pseu-doscalar couplings between unparticle and SM fermions. The horozontal line indicates the presentexperimental bounds for Br( µ → eγ ). We have taken λ V = λ A = λ S = λ P = 0 . κ = 3 andΛ U = 10 TeV. where the subscript denotes the contribution from the corresponding interactions betweenunparticle and the SM fermions, B S , B P , B V and B A equal zero. If both vector couplingand axial vector coupling between the unparticle and fermions(or scalar coupling and pseu-7oscalar coupling) exist simultaneously, B would be non-zero. In Eq.(14)-Eq.(17), there isthe factor [ xzm e + yzm µ − ( x + y ) m a ] d U − with a = e, µ, τ ( or [ xzm e + yzm µ − ( x + y ) m a ] d U − ).It is well-defined if xzm e + yzm µ − ( x + y ) m a >
0, whereas [ xzm e + yzm µ − ( x + y ) m a ] d U − =exp( i ( d U − π )[ − xzm e − yzm µ + ( x + y ) m a ] d U − if xzm e + yzm µ − ( x + y ) m a <
0. Notethat A V and A A are computed for a transverse vector unparticle operator O µ U , both g µν and P µ P ν /P parts in the unparticle two-point correlation function contribute to the decayamplitude.In Fig.2 we present the variation of the branching ratio Br( µ → eγ ) as a function ofthe scale dimension d U respectively for the pure vector, axial vector, scalar, pseudoscalarcouplings between unparticle and the SM fermions. For simplicity, we assume that theunparticle couplings with the SM fermions are universal λ ff ′ K = λ K , f = f ′ κλ K , f = f ′ (18)where κ > µ → eγ ) decreases with d U in theconsidered range, and it strongly depends on the scale dimension d U . There is little differencebetween Br( µ → eγ ) in the pure vector coupling case and that in the pure axial vectorcoupling case. The same is true for the pure scalar coupling and the pseudoscalar coupling.From Eq.(13) and Eq.(14)-Eq.(17), we can see Br( µ → eγ ) is proportional to 1 / (Λ U ) d U − .The Λ U dependence of Br( µ → eγ ) for the pure vector coupling case is shown in Fig.3.From Fig.3, we find that Br( µ → eγ ), for d U = 1 . µ → eγ ) favors the scaledimension d U close to 2 for Λ U of O (10 TeV) and the unparticle couplings of O (10 − ). IV. µ − e CONVERSION IN NUCLEI
The µ − e conversion in nuclei is described by the Feynman diagram presented in Fig.4,it means the following exotic process µ − + (A , Z) → e − + (A , Z) (19)8 B r( e ) u (TeV) d u =1.1 d u =1.3 d u =1.5 d u =1.7 d u =1.9 exp. FIG. 3: Br( µ → eγ ) as a function of the unparticle scale Λ U for various scale dimension d U inthe pure vector coupling case. The horozontal line indicates the present experimental bound forBr( µ → eγ ). We have taken λ V = 0 . κ = 3. (cid:1) q µ q e FIG. 4: Feymann diagram for µ − e conversion in nuclei It violates the conservation of lepton flavor number L e and L µ by one unit, but conserve thetotal lepton number L . The µ − e conversion rate is usually expressed byCR( µ − e, X ) = Γ( µ + X → e + X )Γ( µ + X → capture) (20)where Γ( µ + X → capture) is the µ capture rate of the nuclei X . A very detailed calculationof the µ − e conversion rate in various nuclei has been performed in[60], using the methods9eveloped by Czarnecki et al. [61]. It has been emphasized in [60] that the atomic numberdependence of the conversion rate can be used to distinguish between different theoreticalmodels of LFV.We will calculated the µ − e conversion rates in nuclei using the general model-independentformulae of both [60] and [61]. For the nucleon numbers relevant for µ − e conversionexperiments, the rate for the coherent process dominate over the incoherent excitations ofthe nuclear system, and the rate of the coherent conversion process over the nocoherent onesis enhanced by a factor approximately equal to the number of nucleons in nucleus. Explicitcalculations based on nuclear models [62] show that the ratio between the coherent rate andthe total µ − e conversion rate for nuclei as Ti can be as large as 90%.For coherent µ − e conversion, only vector coupling and scalar coupling between thequarks and unparticle do contribute, and the contributions of axial vector and pseudoscalarcouplings are negligible. For the pure vector coupling between SM fermions and unparticle,the four fermion effective interaction, which describes coherent µ − e conversion, is given by L V µ − e conv = λ eµ V λ qq V A d U d U π ) 1Λ U ( − q Λ U ) d U − eγ µ µ qγ µ q (21)For the pure scalar coupling case, L S µ − e conv = λ eµ S λ qq S A d U d U π ) 1Λ U ( − q Λ U ) d U − eµ qq (22)In Eq.(21) and Eq.(22), q is the momentum transfer in the µ − e conversion process(q ≃− m µ ), which is much smaller than the scale associated with the structure of the nucleon,and we can neglect the q dependence in the nucleon form factors. The above effectiveLagrangian at the quark level is then converted to the effective Lagrangian at the nucleonlevel, by means of the approximate nucleon form factors[52, 60]. The matrix elements of thequark current for the nucleon N = p, n can be written as, h p | q Γ K q | p i = G ( q, p )K p Γ K p h n | q Γ K q | n i = G ( q, n )K n Γ K n (23)where Γ K = 1 , γ µ respectively for K = S , V. The numerical values of the relevant G K are asfollows[52] G ( u, p )V = G ( d, n )V = 2 , G ( d, p )V = G ( u, n )V = 1 , G ( s, p )V = G ( s, n )V = 0G ( u, p )S = G ( d, n )S = 5 . , G ( d, p )S = G ( u, n )S = 4 . , G ( s, p )S = G ( s, n )S = 2 . s state, the final formula for the µ − e conversionrate for the pure vector coupling between SM fermions and unparticle, relative to the muoncapture rate, is given byCR( µ − e, Nucleus) = p e E e m µ α Z F π Z [ λ eµ V λ qq V A d U d U π ) 1Λ U ( m µ Λ U ) d U − ] | Z X q G ( q, p )V +N X q G ( q, n )V | capt (25)For pure scalar coupling case, it isCR( µ − e, Nucleus) = p e E e m µ α Z F π Z [ λ eµ S λ qq S A d U d U π ) 1Λ U ( m µ Λ U ) d U − ] | Z X q G ( q, p )S +N X q G ( q, n )S | capt (26)where Z and N are the numbers of proton and neutron in nucleus, while Z eff ia an effectiveatomic charge, obtained by averaging the muon wavefunction over the nuclear density[62]. F p is the nuclear matrix element and Γ capt denotes the total muon capture rate. m µ is the muonmass, p e and E e is the momentum and energy of the electron. Since P q G ( q, p )V = P q G ( q, n )V = 3and P q G ( q, p )S = P q G ( q, n )S = 11 .
9, the µ − e conversion rate is proportional to Z A / Zwith the atomic number A = Z + N, which can distinguish unparticle from other theoreticalmodels.In Fig.5, we display the predicted µ − e conversion rates for Al, Ti, Sr, Sb, Au and Pbas a function of the scale dimension d U in the case of vector coupling between unparticleand SM fermions. The values of the relevant parameters for these nuclei, Z eff , F p and Γ capt have been collected in Table I[60]. Here the universal couplings between unparticle and SMfermions are assumed as we have done in µ → eγ . We clearly see that the µ − e conversionrates in nuclei CR( µ − e, Nucleus) are sensitive to the scale dimension d U , and they decreasewith d U as well, which is obvious from Eq.(25) and Eq.(26), since (m µ / Λ U ) d U − dominatesthe d U dependence in the plot range, and m µ / Λ U is a small quantity. Moreover, the presentexperimental bound on CR( µ − e , Ti) and CR( µ − e , Au) favor d U near 2 for the inputparameters in this plot. The same conclusion has been found from LFV radiative decay µ → eγ . 11 .2 1.3 1.4 1.5 1.6 1.7 1.8 1.91E-161E-151E-141E-131E-121E-111E-101E-91E-81E-71E-61E-51E-41E-30.010.1110100 CR ( e , N u c l e i ) d u Al Ti Sr Sb Au Pb CR exp ( e,Ti) CR exp ( e,Au)
FIG. 5: µ − e conversion rates for various nuclei as a function of the scale dimension d U for thevector coupling between unparticle and SM fermions. The horozontal lines denote the presentexperimental bounds for CR( µ − e , Ti) and CR( µ − e , Au). We have taken λ V = 0 . κ = 3 andΛ U = 10TeV. V. SUMMARY
Since LFV processes are sensitive probes to new physics beyond SM, we have explored thepeculiar aspects of unparticle physics in µ → eγ and µ − e conversion in nuclei, where vector,axial vector, scalar, pseudoscalar couplings between unparticle and SM fermions are consid-ered. The difference between the branching ratio Br( µ → eγ ) in the pure vector couplingcase and that in the pure axial vector coupling case is small, the same is true for scalar cou-pling and pseudoscalar coupling. Only pure vector coupling and scalar coupling contributeto µ − e conversion in nuclei, which is proportional to Z A / Z, which can be used to dis-tinguish unparticle from other theoretical models. Both Br( µ → eγ ) and CR( µ − e, Nuclei)are sensitive to the scale dimension d U and the unparticle coupling λ ff ′ K (K=V,A,S,P), and12 ABLE I: The value of Z eff , F p and Γ capt for various nucleis, which is taken from [60]. AZ Nucleus Z eff F p Γ capt (GeV) Al 11.5 0.64 4 . × − Ti 17.6 0.54 1 . × − Sr 25.0 0.39 4 . × − Sb 29.0 0.32 6 . × − Au 33.5 0.16 8 . × − Pb 34.0 0.15 8 . × − the present data on Br( µ → eγ ), CR( µ − e, Ti) and CR( µ − e, Au) put stringent constraintson the parameters of unparticle stuff. The scale dimensions d U near 2 are favored for theunparticle scale Λ U of O (10 TeV) and the unparticle coupling of O (10 − ). The interactionsbetween unparticle and SM fermions can also lead to LFV µ → e − e + e − and cross symmetryrelated processes such as e + + e − → e + + µ − , detailed analyses of these processes have beenperformed[11, 14, 18]. Future dedicated LFV measurments MEG experiment and J-PARCexperiment PRISM/PRIME would provide important clues to understanding the nature ofunparticle.Unparticle associated with conformal hidden sector may exist, and it has very distinctivephenomenologies. Unparticle may weakly couples to the SM field so that we are able toexplore the peculiar properties of unparticle, However, whether observable effects can beproduced strongly depends on how weakly the unparticle interacts with ordinary matter. Sofar there is no principle to constraint and organize the interactions between the SM particlesand unparticles, therefore there are many freedoms in the present phenomenological studiesof unparticle. It would be enlightened and interesting to build an explicit model, wherehidden sector with strict or broken scale invariance is realized and it connects to the SMfields via a connector sector. These issues lie outside the scope of the this work, and will beconsidered elsewhere[63]. 13 CKNOWLEDGEMENTS
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