Muon Anomalous Magnetic Moment and Lepton Flavor Violating Tau Decay in Unparticle Physics
aa r X i v : . [ h e p - ph ] O c t Muon Anomalous Magnetic Moment and Lepton Flavor ViolatingTau Decay in Unparticle Physics
Andi Hektor, ∗ Yuji Kajiyama, † and Kristjan Kannike ‡ National Institute of Chemical Physics and Biophysics, Ravala 10, Tallinn 10143, Estonia
Abstract
We study effects of unparticle physics on muon g − g − O (0 . . . .
1) without conflict with experimental bounds of LFV tau decay if thescaling dimension of unparticle operator d U > ∼ . PACS numbers: 13.35.Dx, 13.40.Em, 14.80.-j ∗ [email protected] † yuji.kajiyama@kbfi.ee ‡ [email protected] . INTRODUCTION Scale or conformal invariance of a quantum field theory requires that particles included in thattheory be massless. In the Standard Model (SM), scale invariance is broken by the mass parameterin the Higgs sector and by running of gauge couplings. However, this does not forbid the existenceof scale invariant hidden sector. Recently, motivated by Banks and Zaks [1], Georgi suggested [2]that there may exist a scale invariant hidden sector of unparticles U coupled to the Standard Model(SM) at TeV scale. The theory at high energy contains both the SM fields and so-called Banks-Zaks( BZ ) fields of a theory with with a non-trivial infrared fixed point, interacting via messenger fields ofhigh mass. At the TeV scale, BZ fields are mapped to effective scale invariant unparticle operatorsinteracting with the SM fields. An intriguing property of unparticles is their non-integral scalingdimension d U . They behave like d U number of massless invisible particles.Since the principle to constrain the interactions of unparticles with SM sector is still unknown,there are many possibilities of interactions that preserve Lorentz structure: unparticles that areSM gauge singlets [3], have baryon number [4] or gauge quantum numbers [5]. Moreover, LeptonFlavor Violating (LFV) as well as Conserving (LFC) interactions are possible. The unparticle physicsbased on these interactions has rich phenomenological implications, and it has been studied by manyauthors for collider signature of unparticles [6], neutral meson mixing system [7], muon anomalousmagnetic moment ( g −
2) [8, 9, 10, 11, 12], LFV processes [13, 14, 15, 16, 17, 18, 19], and so on.Experimental results [20] for muon anomalous magnetic moment a µ = ( g − µ / a µ = a exp µ − a SM µ = 29 . . × − , (1)with a discrepancy of 3 . σ . There have been many attempts to explain this discrepancy by newphysics (see [22, 23, 24] and references therein).In this paper, we investigate the muon g − g − µµ U coupling [8, 9, 10, 11, 12] and tau loop fromLFV τ µ U coupling. These couplings generate LFV tau decay processes τ → µ at tree level and τ → µγ at one-loop level as well. If the discrepancy Eq. (1) is saturated by unparticles, one canconstrain the coupling constants and the scaling dimension d U without conflicting the experimentalbound on LFV tau decays.This paper is organized as follows: in Section II, we give a brief introduction of unparticles. InSection III, unparticle mediated muon g − d U that iscompatible with both muon g − I. UNPARTICLES
Unparticles are scale invariant objects originating from a hidden BZ sector with a non-trivialinfrared fixed point. This BZ sector is assumed to interact with the SM sector by exchanging veryheavy particles at a high scale M U . The effective operator of that interaction has the form1 M d SM + d BZ − U O SM O BZ , (2)where O SM( BZ ) is an operator constructed by fields of the SM ( BZ ) sector with mass dimension d SM( BZ ) . Renormalization effects in the BZ sector induce dimensional transmutation at the scaleΛ U ∼ BZ fields match onto unparticle operators O U , and effective interac-tions with the SM sector are written as C U Λ d BZ − d U U M d SM + d BZ − U O SM O U = λ Λ d SM + d U − U O SM O U , (3)where C U is a coefficient fixed by the matching condition and λ = C U (Λ U /M U ) d SM + d BZ − . Althoughin principle the form of unparticle operator O U is determined by the theory in the hidden sector, thelatter is yet unknown, and only Lorentz invariance constrains the unparticle operators.In this paper we consider LFV interactions between SM fields and unparticles of scalar ( S ) andpseudo-scalar ( P ) type. L = λ Sij Λ d U − U ¯ ℓ i ℓ j O U + λ Pij Λ d U − U ¯ ℓ i iγ ℓ j O U , (4)where ℓ i ( i = e, µ, τ ) denotes charged lepton of i th generation, and we assume unparticle scale Λ U =1 TeV throughout this paper. The coupling constant λ Sij and λ Pij are assumed to be real.Propagators of scalar unparticle of momentum P is derived from the principle of scale invarianceas [2, 8] iA d U d U π (cid:0) − P − iǫ (cid:1) d U − , (5)where the normalization factor A d U is A d U = 16 π / (2 π ) d U Γ( d U + 1 / d U − d U ) . (6)In this paper, we consider only the region 1 < d U < < d U )[2, 25, 26] and convergence condition ( d U < II. MUON ANOMALOUS MAGNETIC MOMENT
In this section we consider unparticle contributions to muon g − a µ of betweenexperimentally measured muon g − a µ by (pseudo)scalar unparticles are generatedby one-loop diagram (Fig. 1), and the results are∆ a Sµ = − X j = e,µ,τ (cid:12)(cid:12) λ Sµj (cid:12)(cid:12) π (cid:18) m j Λ U (cid:19) d U − Z d U √ r j Z dzF S ( z, d U , r j ) , (7)∆ a Pµ = + X j = e,µ,τ (cid:12)(cid:12) λ Pµj (cid:12)(cid:12) π (cid:18) m j Λ U (cid:19) d U − Z d U √ r j Z dzF P ( z, d U , r j ) , (8)where j = ( e, µ, τ ) denotes the flavor of internal charged leptons, r j = m µ /m j , Z d U = A d U / (2 sin d U π )and functions under Feynman parameter integrals are defined as F S ( z, d U , r j ) = z − d U (1 − z ) d U (1 + √ r j z )(1 − r j z ) d U − , (9) F P ( z, d U , r j ) = z − d U (1 − z ) d U (1 − √ r j z )(1 − r j z ) d U − . (10)Contribution from pseudoscalar interactions ∆ a Pµ is obtained by replacing m j with − m j in ∆ a Sµ fromthe chirality structure. One can easily verify that Eqs. (7)–(8) reduce to the formulae of [9] in thecase of flavor-blind interactions when r j = 1.The contribution from scalar interactions ∆ a Sµ to Eq. (1) is positive for all j = e, µ, τ . Thecontribution from pseudoscalar interactions ∆ a Pµ has the same sign as ∆ a Sµ for j = e because of r j = e ≫
1. However, for j = µ, τ the contribution of ∆ a Pµ to muon g − λ Pµj = 0 , ( j = µ, τ ) in the following analysis. The treatment ofthis ( µ, e ) LFV coupling is discussed below.Fig. 2 shows ∆ a µ as a function of d U calculated from Eq. (7) with various values of λ Sµτ,µµ,µe with Λ U = 1 TeV. The solid, thick-solid, dashed, thick-dashed and dotted curve correspond to( λ Sµτ , λ
Sµµ , λ
Sµe ) = (1 , , , (10 − , , , (0 , , , (0 , − ,
0) and (0 , , a µ at d U = 1, they aredecreasing with larger d U , but diverge at d U = 2, while the dotted curve for d U < . λ Sµe (dotted curve) is negative for d U < . d U > . λ Sµe = 1. Moreover, this( µ, e ) LFV coupling must be suppressed by experiments of µ → eγ and µ − e conversion in nuclei [14],and µ → e decay process [15] ( λ Sµe = 0 is consistent with these experiments). In fact, at d U = 1 . a µ , upper bound of λ Sµe is 10 − from µ → eγ and 10 − from µ → e for λ µµ = λ ee = 10 − . Such small (or vanishing) ( µ, e ) LFV coupling can not play significantrole here. Therefore, we neglect this coupling in the following analysis.4 IG. 1: One-loop diagram of muon g − U . Fig. 3 shows the consistent region of λ Sµτ (left) and λ Sµµ (right) at Λ U = 1 TeV under the assumptionthat both LFC and LFV couplings simultaneously contribute to ∆ a µ , and it is in the bound of Eq. (1).For λ Sµτ , λ Sµµ is only a free parameter, and vice versa . For both couplings, λ S s have to be small forthe region of small d U , but they can be of O (1) for relatively large d U . They must be extremelysmall when d U is closer to 2, and there is no solution at d U = 2 because ∆ a µ diverges at this point.Since we have assumed that both λ Sµτ and λ Sµµ contribute, these allowed regions are not independentof each other. The relation between these two couplings are shown in Fig. 4. These figures representthe allowed region of λ S s in the λ Sµτ − λ Sµµ plane with d U = (1 . , .
9) (left) and d U = (1 . , .
8) (right).In the left panel, the region surrounded by thick-solid (solid) curves correspond to d U = 1 . . d U , and weemphasize that both or at least one coupling of λ Sµτ ( µµ ) have to be of O (0 . . . .
1) for large d U ( > ∼ . IV. LFV TAU DECAY
In this section, we investigate the LFV τ decay processes generated by the same unparticle inter-actions as those of ∆ a µ . We discuss τ → µ and τ → µγ . Since the constraints of couplings λ Sµτ ( µµ ) obtained from the consistency of ∆ a µ in the previous section tolerate large LFV couplings, our nexttask is to make certain that this does not conflict with the experimental bound of LFV tau decays[21, 27] BR ( τ → µ ) < . × − , BR ( τ → µγ ) < . × − . (11)Here, we will find regions of couplings which are consistent with both ∆ a µ and LFV tau decays.First we consider τ → µ LFV tau decay [15, 16]. This decay mode mediated by unparticleoperators occurs at tree-level (Fig. 5, left). 5
IG. 2: ∆ a µ from the couplings with scalar unparticles as a function of the scaling dimension d U withΛ U = 1 TeV. The solid, thick-solid, dashed, thick-dashed and dotted curves correspond to ( λ Sµτ , λ
Sµµ , λ
Sµe ) =(1 , , , (10 − , , , (0 , , , (0 , − ,
0) and (0 , , d U = 2. The hori-zontal lines are the upper and lower value of Eq. (1).FIG. 3: Allowed region of λ Sµτ (left) and λ Sµµ (right) from the condition of ∆ a µ . The decay rate of τ → µ derived in [15] is d Γ ds sin θdθ = 12 π √ s s(cid:18) − ( m τ − m µ ) s (cid:19) (cid:18) − ( m τ + m µ ) s (cid:19)r − m µ s X spin |M| , (12)where θ is angle between three-momenta p and p , s = ( p − p ) and its integral range is 4 m µ ≤ s ≤ ( m τ − m µ ) . Amplitude M is X spin |M| = 4 (cid:12)(cid:12) λ Sµτ (cid:12)(cid:12) (cid:12)(cid:12) λ Sµµ (cid:12)(cid:12) (cid:2) p · p )( p · p ) | F | + 4( p · p )( p · p ) | F | − { ( p · p )( p · p ) + ( p · p )( p · p ) − ( p · p )( p · p ) } Re( F F ∗ )] , (13)6 IG. 4: Allowed region of the coupling constants in the λ Sµτ − λ Sµµ plane with d U = (1 . , .
9) (left) and d U = (1 . , .
8) (right). In the left panel, the region surrounded by thick-solid (solid) curves corresponds to d U = 1 . . d U = 1 . . τ → µ decay mediated by unparticles (left) and τ → µγ at one-looplevel (right). For τ → µ , there also exists the u -channel diagram by exchanging external muons of momenta p and p . where F = Z d U Λ d U − U (cid:0) − ( p − p ) − iǫ (cid:1) d U − ,F = Z d U Λ d U − U (cid:0) − ( p − p ) − iǫ (cid:1) d U − . (14)LFV unparticle interactions can generate other τ → ℓ decay processes in general, and τ → eµµ is the one of them which contains the coupling responsible for muon g −
2. This process contains λ Sµe as well as λ Sµτ . However, as mentioned in the previous section, we have neglected this ( µ, e ) LFVcoupling because it is strongly suppressed by other LFV processes. τ → µγ process is the other decay mode by the same unparticle interactions at one-loop level7Fig. 5, right). Decay Rate of this process is [14]Γ = m τ π |A| (15)where the amplitude A is given by A = − X j = e,µ,τ ie (4 π ) λ Sµj λ Sjτ Z d U U ) d U − G Sj ( d U ) , (16)and functions G Sj ( d U ) are G Sj ( d U ) = Z dxdydzδ ( x + y + z − z − d U (cid:2) − xzm τ − yzm µ + ( x + y ) m j (cid:3) d U − [ xzm τ + yzm µ + ( x + y ) m j ] . (17)While three leptons j = ( e, µ, τ ) can exist in the loop, we consider only the case of muon virtualparticle because we are interested in tau decays generated by the same couplings as those of muon g −
2. Contributions from other virtual particles depend on different combination of λ S , such as λ Sτe λ Sµe for electron loop and λ Sττ λ Sµτ for tauon loop. Moreover, τ → eγ is also possible LFV taudecay mediated by unparticles. However this process also contains unknown, or more suppressedparameters λ See,τe,ττ . These couplings may be constrained by other processes, but we neglect thesehere because all of them must be small or can be zero.If there are couplings of unparticle with photons [19],1Λ d U U (cid:16) λ γ F µν F µν + λ ˜ γ ˜ F µν F µν (cid:17) O U (18)these operators also generate τ → µγ at one-loop level. However, these operators are more suppressedby the factor m τ / Λ U ∼ − than Eq. (16), and contain unknown parameters λ γ (˜ γ ) . Therefore, weagain neglect these interactions.In the next subsection, we perform numerical calculation of these LFV tau decay processes, andverify that there exist regions which do not conflict with experiments. Numerical Calculation
Now we are ready to find whether unparticle can explain muon g − τ → µ (left) and τ → µγ (right) as a function of d U . Inthese figures, λ Sµτ and λ Sµµ are generated independently and randomly in the region allowed by g − λ Sµτ,µµ > . λ Sµτ , λ
Sµµ ) give the BR below the8
IG. 6: Branching ratio of τ → µ (left) and τ → µγ (right) by scalar unparticle operators with λ Sµµ,µτ > . d U > ∼ . experimental bound of τ → µ if d U > ∼ .
6, while it is enough suppressed for almost all d U for theone-loop process τ → µγ . These results are not changed for larger Λ U , because the dependence of λ S / Λ d U − U is the same for all phenomena.Fig. 7 shows the final region in the λ Sµτ − λ Sµµ plane at d U = (1 . , .
9) (left) and d U = (1 . , . d U = 1 . d U = 1 . τ → µ experiment Eq. (11) are superimposed on Fig. 4. Dark and light shadedareas represent the combined allowed region of all experiments for d U = 1 . , . d U = 1 . , . × − for τ → µ and 2 × − for τ → µγ , and dotted lines are obtained from BR ( τ → µ ) < × − .Comparing the Fig. 4, the regions in which both couplings are large vanish, and those in whicheither of them is large remain. These regions are compatible with both muon g − O (0 . . . .
1) for the case of large d U .In the case of both the couplings are small, which is favored in the point of view of LFV tau decay,the scaling dimension d U has to be small in order to obtain appropriate value of muon g −
2. FutureLFV tau decay experiments will restrict the allowed regions.The situation λ Sµτ = 0 is also a solution. The coupling λ Sµµ can give a desired value of ∆ a µ even if λ Sµτ = 0, and in this case LFV tau decays of our consideration can not occur.We conclude that LFV coupling λ Sµτ does not have to be zero or extremely suppressed, it can beof O (0 . . . . λ Sµµ < ∼ − and d U > ∼ .
6. 9
IG. 7: Consistent region in the λ Sµτ − λ Sµµ plane at d U = (1 . , .
9) (left) and d U = (1 . , .
8) (right). Leftlower areas of each diagonal line representing the allowed region from τ → µ experiment are superimposedon Fig. 4, and dotted lines are obtained from expected future experiments. Shaded areas represent thecombined allowed region of all present experiments. V. CONCLUSIONS
We have studied muon anomalous magnetic moment and lepton flavor violating tau decay τ → µ and τ → µγ generated by scalar unparticle interactions. Since the principle to determine unparticleinteractions is still unknown, both lepton flavor violating and conserving interactions may exist,and these interactions can simultaneously generate both phenomena. We have found that scalarunparticles explain the discrepancy of experimental value of muon g − g − d U . On the other hand, when both couplings exist, thesecouplings and the scaling dimension are constrained by LFV tau decay. In the case of large scalingdimension ( d U > ∼ . λ Sµτ need not be small. It can be of O (1) if LFC coupling λ Sµµ is enough small, and vice versa . Acknowledgments
The authors would like to thank M. Raidal for useful discussions. This work is supported by theESF grant No. 6190, and postdoc contract 01-JD/06 (Y.K.). [1] T. Banks and A. Zaks, Nucl. Phys.
B196 , 189 (1982).[2] H. Georgi, Phys. Rev. Lett. , 221601 (2007); H. Georgi, Phys. Lett. B650 , 275 (2007).[3] S.-L. Chen and X.-G. He, Phys. Rev.
D76 , 091702 (2007).
4] X.-G. He and S. Pakvasa, Phys. Lett.
B662 , 259 (2008).[5] Y. Liao, Eur. Phys.J.
C55 , 483 (2008). G. Cacciapaglia, G. Marandella and J. Terning, JHEP ,070 (2008).[6] M. Duraisamy, arXiv:0705.2622 [hep-ph]; N. Greiner, Phys. Lett.
B653 , 75 (2007); P. Mathews and V.Ravindran, Phys. Lett.
B657 , 198 (2007); K. Cheung, W.-Y. Keung and T.-C. Yuan, Phys. Rev.
D76 ,055003 (2007); A. T. Alan and N. K. Pak, arXiv:0708.3802 [hep-ph]; S. Majhi, Phys .Lett.
B665 ,44(2008); M. C. Kumar, P. Mathews, V. Ravindran and A. Tripathi, Phys. Rev.
D77 , 055013 (2008);A. T. Alan, N. K. Pak, A. Senol, arXiv:0710.4239 [hep-ph]; K. Huitu and S. K. Rai, Phys. Rev.
D77 ,035015 (2008); O. Cakir and K. O. Ozansoy, eur. Phys. J.
C56 , 279 (2008); T. Kikuchi, N. Okada andM. Takeuchi, Phys. Rev.
D77 , 094012 (2008); J. L. Feng and A. Rajaraman, Phys. Rev. , 075007(2008); C.-F. Chang and K. Cheung and T.-C. Yuan, Phys.Lett. , 291 (2008); V. Barger, Y. Gao,W.-Y. Keung, D. Marfatia and V. N. Senoguz, Phys. Lett. B661 , 276 (2008); H.-F. Li, H.-L. Li, Z.-G.Si and Z.-J. Yang, arXiv:0802.0236 [hep-ph]; B. Sahin, arXiv:0802.1937 [hep-ph].[7] X.-Q. Li and Z.-T. Wei, Phys. Lett.
B651 , 380 (2007); R. Mohanta and A. K. Giri, Phys. Rev.
D76 ,075015 (2007); A. Lenz, Phys. Rev.
D76 , 065006 (2007); S.-L. Chen, X.-G. He, X.-Q. Li, H.-C. Tsaiand Z.-T. Wei, arXiv:0710.3663 [hep-ph]; C.-H. Chen, C. S. Kim and Y. W. Yoon, arXiv:0801.0895[hep-ph].[8] K. Cheung, W.-Y. Keung and T.-C. Yuan, Phys. Rev. Lett. , 051803 (2007).[9] Y. Liao, Phys. Rev. D76 , 056006 (2007).[10] M. Luo and G. Zhu, Phys. Lett.
B659 , 341 (2008).[11] C.-H. Chen and C.-Q. Geng, Phys. Lett.
B661 , 118 (2008).[12] T.-I Hur, P. Ko and X.-H. Wu, Phys. Rev.
D76 , 096008 (2007).[13] D. Choudhury, D. K. Ghosh and Mamta, Phys. Lett.
B658 , 148 (2008).[14] G.-J. Ding and M.-L. Yan, Phys. Rev.
D77 , 014005 (2008).[15] T. M. Aliev, A. S. Cornell and N. Gaur, Phys. Lett.
B657 , 77 (2007).[16] M. Giffels, J. Kallarackal, M. Kr¨amer, B. O’Leary and A. Stahl, Phys. Rev.
D77 , 073010 (2008).[17] C.-D. L¨u, W. Wang and Y. -M. Wang, Phys. Rev.
D76 , 077701 (2007).[18] E. O. Iltan, Eur. Phys. J.
C56 , 105 (2008);
C56 , 113 (2008).[19] E. O. Iltan, arXiv:0801.0301[hep-ph].[20] G. W. Bennett et al. [Muon G-2 Collaboration], Phys. Rev. D (2006) 072003[21] W.-M. Yao et al. [Particle Data Group], J. Phys. G33 , 1 (2006).[22] J. P. Miller, E. de Rafael and B. L. Roberts, Rept. Prog. Phys. , 795 (2007).[23] A. Czarnecki and W. J. Marciano, Phys. Rev. D64 , 013014 (2001).[24] F. Jegerlehner, Acta Phys. Polon.
B38 , 3021 (2007).[25] G. Mack, Commun. Math. Phys. , 1 (1977).[26] B. Grinstein, K. Intriligator and I. Z. Rothstein, Phys. Lett. B662 , 367 (2008).
27] K. Abe et al. [Belle Collaboration], arXiv:0708.3272 [hep-ex]; Y. Miyazaki et al. [Belle Collaboration],Phys. Lett.
B660 , 154 (2008); B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. , 041802(2005); Phys. Rev. Lett. , 251803 (2007).[28] A. G. Akeroyd et al . [SuperKEKB Physics Working Group], KEK report , [arXiv:hep-ex/0406071];M. Bona et al. , arXiv:0709.0451 [hep-ex]., arXiv:0709.0451 [hep-ex].