Michel parameters in radiative muon decay
PPrepared for submission to JHEP
Michel parameters in radiative muondecay
A.B. Arbuzov, a,b, T.V. Kopylova ba Bogoliubov Laboratory for Theoretical Physics, JINR, Dubna 141980, Russia b Dubna State University, Dubna 141982, Russia
E-mail: [email protected]
Abstract:
Radiative muon and tau lepton decays are described within the model-independent approach with the help of generalized Michel parameters. The exactdependence on charged lepton masses is taken into account. The results are relevantfor modern and future experiments on muon and tau lepton decays.
Keywords:
Electroweak interaction, Tau Physics, Beyond Standard Model Corresponding author. a r X i v : . [ h e p - ph ] A ug ontents F , G , and H
5B Generalized Michel parameters 6
Studies of muon decay are one of the keystones in particle physics. This decay isalmost a pure weak-interaction process. That allows us to use it for high-precisiontests of the Standard Model (SM) and searches for new physics, see e.g. the review [1].According to the principle of lepton universality that is adapted in the SM,the pure leptonic modes of tau lepton decays are described by the same Feynmandiagrams as the corresponding muon decays. The only difference is coming fromthe fact that the masses of charged leptons are changed. As concerning modelsbeyond the SM, some of them predict violation of lepton universality for tau leptons,assuming that the third generation is more strongly coupled to some “new physics”than the first two ones [2, 3]. This motivates experimental searches for new physicsin tau decays at various high-energy machines including the LHC, see e.g. [4, 5]. Inparticular, a new method to probe magnetic and electric dipole moments of tau leptonusing precise measurements of the differential rates of radiative leptonic decays athigh-luminosity B -factories was proposed in [6]. Note that a high precision has beenalready reached in measurements of radiative leptonic tau decays at the B -factories,see e.g. [7].The precision of the Michel parameters [8–10] definition from tau decays [11–15] is not yet competing with the one achieved in muon decays. Nevertheless, highstatistics of tau lepton observation at the B -factories Belle and BaBar, and at theLHC provides good perspectives for studies of various tau lepton decay modes.– 1 – Preliminaries and notation
Within the SM, muon decays are described by interactions of vector currents formedby left fermions. Meanwhile, many models beyond the SM predict contributions ofother kinds. Since the energy scale of new physics is (most likely) higher than theelectroweak scale, the corresponding contributions can be parameterized by contactfour-fermion interactions with different currents and coupling constants. The matrixelement of the muon decay can be presented in the general form [16] M = 4 G √ (cid:88) γ =S , V , T (cid:15),ω =R , L g γ(cid:15)ω (cid:104) ¯ l (cid:15) | Γ γ | ν l (cid:105) (cid:104) ¯ ν τ | Γ γ | τ ω (cid:105) , (2.1)see for details the Particle Data Group review [17] and references therein. Here index γ denotes the type of the interaction: scalar (S), vector (V) or tensor (T); and Γ γ are × matrices defined in terms of the Dirac matrices: Γ S = 1 , Γ V = γ µ , Γ T = 1 √ σ µν = i √ γ µ γ ν − γ ν γ µ ) . (2.2)The indices ω and (cid:15) denote the chiralities of the initial and final charged leptons,respectively. For a given pair ( ω , (cid:15) ) the chiralities of neutrinos are uniquely deter-mined. Tensor interactions can contribute only for opposite chiralities of the chargedleptons. This leads to the existence of 10 complex coupling constants, g γ(cid:15)ω . The Stan-dard Model predicts g VLL = 1 and all others being zero. Choosing the arbitrary phaseby defining g VLL to be real and positive leaves 19 real numbers to be determined bythe experiment. As long as one is interested in the relative strengths of the couplings,it is convenient to require the following normalization condition: N ≡ (cid:16) | g SLL | + | g SLR | + | g SRL | + | g SRR | (cid:17) + (cid:16) | g VLL | + | g VLR | + | g VRL | + | g VRR | (cid:17) + 3 (cid:16) | g TLR | + | g TRL | (cid:17) = 1 . (2.3)This restricts the allowed ranges of the coupling constants to | g S | ≤ , | g V | ≤ ,and | g T | ≤ √ . The overall normalization can be incorporated into G which thenaccounts for deviations from the Fermi coupling constant G Fermi . We have to notethat the precision of the muon life time measurement is much higher than the oneobtained for definition of G Fermi via other parameters of the SM, i.e. α QED , M Z , M W etc. Unfortunately, for this reason the extremely precise measurement of G Fermi doesnot provide any valuable test of models beyond the SM.In this paper we provide a parameterization of radiative muon (tau) decays interms of the generalized Michel parameters [9, 10] which are certain bilinear com-binations of the coupling constants g γ(cid:15)ω . Our aim is to take into account the exactdependence on the charged fermion masses.– 2 – Radiative muon decay
At the Born level the decay µ − ( p µ ) → e − ( p e ) + ν µ ( k ) + ¯ ν e ( k ) + γ ( p γ ) (3.1)was considered by many authors since late fifties [9, 18, 19]. Here we follow thenotation adapted in the review [1] and represent the differential width of this decayin the form d Γ( µ ± → e ± ¯ ννγ ) dx dy d Ω e d Ω γ = Γ α QED π β e y (cid:20) F ( x, y, d ) ∓ β e P µ cos θ e G ( x, y, d ) ∓ P µ cos θ γ H ( x, y, d ) (cid:21) , Γ = G m µ π , d = 1 − β e cos θ eγ , β e = (cid:115) − m e E e , (3.2)where Ω e,γ are solid angles of the observable final state particles; θ e and θ γ are theangles between the muon spin and the electron and photon momenta, respectively; P µ is the muon polarization degree; θ eγ is the angle between the electron and photonmomenta; x and y are energy fractions of electron and photon, respectively, x ≡ E e /m µ and y ≡ E γ /m µ .Functions F , G , and H are polynomials in the electron to muon mass ratio: F ( x, y, d ) = (cid:88) k =1 (cid:18) m e m µ (cid:19) k F ( k ) , F ≡
F, G, H. (3.3)We computed these functions in the tree-level approximation. Analytical calcula-tions were performed with the help of the
FORM computer language [20]. The explicitexpression for these functions are given in Appendix A below. These functions de-pend on the generalized Michel parameters ρ , η , ¯ η , ξ , δ , κ , α , and β , see Appendix B.The dependence on ¯ η , κ , α , and β provides important additional information aboutthe structure of weak interactions with respect to the studies of non-radiative muonand tau decays, see [21]. Note that the dependence on α , and β appear only interms suppressed by the first power of the mass ratio r ≡ m e /m µ ≈ · − (or m µ /m τ ≈ · − for the τ → µν ¯ ν decay). The presence of the contributions pro-portional to the first power of the mass ratio (and in general in odd powers) is anon trivial effect. Presumably, it is related to a hidden spin flip. An analogous effecttakes place in the one-loop corrections to polarized muon decay spectrum: expansionof the exact result [22] for these corrections contains some terms which are linear inthe electron to muon mass ratio.The contributions suppressed by the electron to muon mass ratio can be im-portant for modern high-precision experiments on muon and especially tau lepton– 3 –adiative decays. As one can see from (A.1), (A.2), and (A.3), the most importanthigher order effect is given by function F (1) ( x, y, d ) since it is the only one being linearin the charged lepton mass ratio. Moreover, this function depends on the generalizedMichel parameters α and β which do not appear in the lowest approximation in r .In addition, this function contains terms proportional to d − which provide a con-siderable enhancement in the kinematical domain of collinear radiation of photonsalong the electron direction of motion. Thus one can hope that future experimentson radiative muon and tau decays can provide additional information on the Michelparameter values.We verified our results by comparison with ones the existing in the literature.We have checked that in the case of V − A interactions our results completely agreewith the formulae given in [1]. For the case of general interactions in the limit m e (cid:28) m µ , we performed a comparison with the results of [18]. Here the agreementis not complete. Namely, we have the coincidence for functions F ( x, y, d ) | m e → and G ( x, y, d ) | m e → . But we encountered a small discrepancy for H ( x, y, d ) | m e → , whichcan be removed after the following modification in one term in the Appendix of [18]: F T γ = − x y (1 − y ) + 2 x y ] → F T γ = − x y (1 − y ) + 2 x y ] . (3.4)Moreover, one more obvious misprint in [18] has to be corrected in eq. (4b): [∆ + (2 /µ x )] → [∆ + (2 m e /µ x )] . (3.5)Note that the expression in square brackets above is a certain approximation ofquantity d given in (3.2).The NLO QED radiative corrections to this process (for the case of the pure V − A interactions) were recently calculated in [23] with taking into account the exactdependence on the final state charged lepton mass. Earlier, the one-loop correctionsto radiative muon decay were also considered in [24, 25]. Besides the QED effects,radiative corrections to the W boson propagator within the Standard Model becomenumerically relevant [26]. The processes of five-body leptonic decays of muons andtau leptons can be also used to get an additional information about the structure ofweak interactions, see e.g. [27]. Our results can be used in the analysis of high-precision experimental data on τ and muon radiative decays in order to extract the generalized Michel parameters.The exact dependence on the final charged lepton mass is taken into account. Thecorresponding effect for certain cases is linear in the ratio of the final and initialcharged lepton masses. So it becomes numerically important first of all for studiesof the τ → µ ¯ ννγ decay. A correction to the result of paper [18] is also made.– 4 – cknowledgments We are grateful to D. Epifanov and S. Eidelman for statement of the problem anduseful discussions and to E. Akzhigitova for the help with cross checks of the results.
A Functions F , G , and H Here we give the explicit formulae for functions F , G , and H which appear in (3.2): F (0) = 32 ρ (cid:18) x y + 12 xy + 4 y + 8 x − y − x − xy d − x y − x y − xy − xy − x y − x xy + x + xyd (cid:0) x x y + xy + 34 xy − x (cid:1) − x y d y ) (cid:19) + ¯ η xy (cid:18) y + 4 x − xd − xyd − (cid:19) + 48 y + 2 xy + 2 x − y − x − xy − x yd + 96 x y + 112 x y + 56 xy + 48 x ( x + y −
1) + 16 xy − xy + 4 xyd (cid:0) x − xy (1 + y ) − x − x y (cid:1) + 6 x y d (2 + y ) ,F (1) = β (cid:18) x ( x + y − d − y d + 4 xy − x ( xy + x + y − − x y d + 6 x yd (cid:19) + α (cid:18) x (1 − x − y ) d + 6 x ( xy + x + y − − x yd (cid:19) ,F (2) = 16 ρ (cid:18) x + y )(3 − x − y )3 xd + 18 y − x + 20 xy − x − y d + 2 x y − xy x + 2 xy + 8 x − x yd (2 + y )2 (cid:19) + 16¯ η (cid:18) − xy − y d (cid:19) + 192( x + y + 2 xy − x − y ) xd + 96 x − y − xy + 96 yd − x y + 16 xy − x − xy + 12 x yd (2 + y ) ,F (3) = 96 η (cid:18) x + y − xd + 2( x − y ) d − x (cid:19) ,F (4) = 64 ρ x + 6 y − xyd xd + 96 xyd − y − xxd ,F (5) = η (cid:18) − xd (cid:19) , (A.1) G (0) = 8 ξ (cid:18) x + 6 x y − x − xyd + 4 xy d − x y − x y − x − x y + 2 x + x yd (2 + y )2 (cid:19) + 32 δξ (cid:18) xyd − xy d + 2 x d − x yd − x d − x + 5 x y
6+ 11 x y x x y − x yd y ) (cid:19) + 8 κξxy (cid:18) x − d (cid:19) , – 5 – (2) = 8 ξ (cid:18) − x − y ) d + 2 xyd − x d + 2 x + x y (cid:19) + 16 δξ (cid:18) x + 4 y − d − xyd + 20 x d − x − x y (cid:19) ,G (4) = ξ d − δξ d ,G (1) = 0 , G (3) = 0 , G (5) = 0 , (A.2) H = 8 ξ (cid:18) y ( x + y − y − x ) d + 14 xy d − xy − x y + 2 xy − x y + x y + 76 x y d + x y d − x y d − x y d (cid:19) + 8 δξ (cid:18) y (3 x + 3 y − x − y )3 d − xy d + 409 xy + 8 x y x y + 4 xy − x y + x y d − x y d − x y d + 23 x y d (cid:19) + 8 κξ (cid:18) xy d + 2 xy + 3 x y − xy − x y d (cid:19) ,H (2) = 8 ξ (cid:18) y (1 − y − x ) xd + 14 y d + 2 xyd − yd − xy x y − x y d (cid:19) + 8 δξ (cid:18) y (4 y + 4 x − xd − y d − xyd + 8 yd + 20 xy − x y x y d (cid:19) + 16 κξ (cid:18) − xy − y d (cid:19) ,H (4) = 32 ξ (cid:18) yxd − yd (cid:19) + 320 δξ (cid:18) y d − y xd (cid:19) ,H (1) = 0 , H (3) = 0 , H (5) = 0 . (A.3) B Generalized Michel parameters
For the sake of completeness, we give here the explicit expressions for the generalizedMichel parameters defined by Kinoshita and Sirlin [9, 10], see also [17] for details,via the coupling constants g γ(cid:15)ω which enter (2.1): ρ = 3 b + 6 c
16 = 34 + 34 · (2 c − a )16 , η = α − β , ¯ η = a + 2 c ,κξ = a + 2 c , δξ = 6 c − b , ξ = 14 c − a − b , – 6 – = 8Re (cid:18) g VRL ( g S ∗ LR + 6 g T ∗ LR ) + g VLR ( g S ∗ RL + 6 g T ∗ RL ) (cid:19) , β = − g VRR g S ∗ LL + g VLL g S ∗ RR ) ,a = 16( | g VRL | + | g VLR | ) + | g SRL + 6 g TRL | + | g SLR + 6 g TLR | ,b = 4( | g VRR | + | g VLL | ) + | g SRR | + | g SLL | ,c = 12 | g SRL − g TRL | + 12 | g SLR − g TLR | ,a = 16( | g VRL | − | g VLR | ) + | g SRL + 6 g TRL | − | g SLR + 6 g TLR | ,b = 4( | g VRR | − | g VLL | ) + | g SRR | − | g SLL | ,c = 12 | g SRL − g TRL | − | g SLR − g TLR | . (B.-5)Note that we used above the normalization condition (2.3), which can be re-writtenalso as a + 4 b + 6 c = 16 . (B.-4)In the limit of the pure V − A interactions we have ρ = δ = , ξ = 1 , η = ¯ η = κ = α = β = 0 . References [1] Y. Kuno and Y. Okada,
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