We have performed a global analysis of muon decay measurements to establish
model-independent limits on the space-time structure of the muon decay matrix
element. We find limits on the scalar, vector and tensor coupling of right- and
left-handed muons to right- and left-handed electrons. The limits on those
terms that involve the decay of right-handed muons to left-handed electrons are
more restrictive than in previous global analyses, while the limits on the
other non-standard model interactions are comparable. The value of the Michel
parameter eta found in the global analysis is -0.0036 \pm 0.0069, slightly more
precise than the value found in a more restrictive analysis of a recent
measurement. This has implications for the Fermi coupling constant G_F.Comment: 5 pages, 3 table

C. A. Gagliardi

D. P. Stoker

N. J. Williams

R. E. Tribble

W. Fetscher

English

arXiv

Gagliardi, C. A.

Tribble, R. E.

Williams, N. J.

OAKTrust Digital Repository (Texas A&M Univ)

arXiv:hep-ph/0509069v1  7 Sep 2005Global analysis of muon decay measurementsC.A. Gagliardi, R.E. Tribble, and N.J. Williams∗Cyclotron Institute, Texas A&M University, College Station, TX 77843(Dated: March 24, 2018)We have performed a global analysis of muon decay measurements to establish model-independentlimits on the space-time structure of the muon decay matrix element. We find limits on the scalar,vector and tensor coupling of right- and left-handed muons to right- and left-handed electrons. Thelimits on those terms that involve the decay of right-handed muons to left-handed electrons aremore restrictive than in previous global analyses, while the limits on the other non-standard modelinteractions are comparable. The value of the Michel parameter η found in the global analysis is−0.0036±0.0069, slightly more precise than the value found in a more restrictive analysis of a recentmeasurement. This has implications for the Fermi coupling constant GF .PACS numbers: 13.35.Bv, 14.60.Ef, 12.60.CnI. INTRODUCTIONMuon decay, µ→ eνν¯, is an excellent laboratory to in-vestigate the weak interaction. The energy and angulardistributions of the electrons emitted in polarized muondecay are specified by the muon decay parameters ρ, η,ξ, and δ [1, 2], conventionally referred to as the Michelparameters. The additional decay parameters ξ′ and ξ′′determine the longitudinal polarization of the outgoingelectrons, and the parameters α, β, α′, and β′ determinethe transverse polarization [2]. Radiative muon decay,µ → eνν¯γ, provides access to another decay parameter,η¯ [3]. These 11 decay parameters, which are not all in-dependent, together with the muon lifetime provide acomplete description of muon decay if the neutrinos arenot observed.The decay parameters are related to the underlyingspace-time structure of muon decay by the most gen-eral, local, derivative-free, Lorentz-invariant transitionmatrix element, which can be written in terms of helicity-preserving amplitudes as [4]:M =4GF√2∑γ=S,V,T ;ǫ,µ=R,Lgγǫµ〈e¯ǫ|Γγ |νe〉〈ν¯µ|Γγ |µµ〉. (1)In this equation, GF is the Fermi coupling constant, de-termined from the muon lifetime, and the gγǫµ specifythe scalar, vector and tensor couplings between µ-handedmuons and ǫ-handed electrons. Deviations from this ex-pression due to the non-local nature of the standardmodel weak interaction are O(m2µ/m2W ) and, thus, arenegligible compared to our current experimental knowl-edge of the muon decay parameters. Equation (1) is par-ticularly convenient because the standard model, withpure V − A coupling, implies gVLL = 1 and all the othercoupling constants are zero. In contrast, additional cou-pling constants are non-zero in many extensions to thestandard model. For example, gVRR, gVLR, and gVRL are∗Present address: Langston University, Langston, OK 73050.non-zero in left-right symmetric models [5]. For recentreviews of muon decay, see [6, 7, 8].The coupling constants gγǫµ represent 18 independentreal parameters, so it is not possible to determine all ofthem from the existing muon decay observables. A globalanalysis of all the existing muon decay measurementscan nonetheless set stringent model-independent exper-imental limits on the non-standard model contributions[4, 9, 10, 11] that can then be compared to model calcula-tions such as those in [12], which predict that gS,V,TLR andgS,V,TRL should be very small based on their contributionsto neutrino masses. Recently, new measurements havebeen performed for the Michel parameters ρ [13] and δ[14] and for the transverse polarization parameters η, η′′,α′, and β′ [15]. (η and η′′ parametrize the momentum-dependence of the CP -allowed transverse polarization ina manner different from, but equivalent to, α and β. SeeEq. (5) below.) Each of these new results is a factor of∼ 2.5 more precise than the previous best measurement[16]. This makes a new global analysis timely. In thispaper, we present such an analysis to establish updatedlimits on possible non-standard model contributions tomuon decay.II. FIT PROCEDURESeveral different parametrizations for muon decay havebeen introduced over the years. One common version[2], which is based on the expression for the muon decaymatrix element in charge-retention order, describes themuon decay observables in terms of 10 constants: a/A,a′/A, b/A, b′/A, c/A, c′/A, α/A, β/A, α′/A, β′/A. Eachof these 10 constants is a bilinear combination of thecoupling constants gγǫµ [8]. The normalization factor, A =a+ 4b + 6c = 16, is fixed by adjusting GF to reproducethe muon lifetime. The 9 remaining linearly independentterms may be determined from the experimental valuesof the muon decay parameters. This procedure was usedin the global analysis performed in [9].An alternative parametrization [4] utilizes a different2set of bilinear combinations of the coupling constants:QRR =14|gSRR|2 + |gVRR|2,QLR =14|gSLR|2 + |gVLR|2 + 3|gTLR|2,QRL =14|gSRL|2 + |gVRL|2 + 3|gTRL|2,QLL =14|gSLL|2 + |gVLL|2,BLR =116|gSLR + 6gTLR|2 + |gVLR|2,BRL =116|gSRL + 6gTRL|2 + |gVRL|2,Iα =14[gVLR(gSRL + 6gTRL)∗ + (gVRL)∗(gSLR + 6gTLR)]= (α+ iα′)/2A,Iβ =12[gVLL(gSRR)∗ + (gVRR)∗gSLL]= −2(β + iβ′)/A. (2)These bilinear combinations satisfy the constraints [4]:0 ≤ Qǫµ ≤ 1, where ǫ, µ = R,L,0 ≤ Bǫµ ≤ Qǫµ, where ǫµ = RL,LR,|Iα|2 ≤ BLRBRL,|Iβ |2 ≤ QLLQRR, (3)and the normalization condition,QRR +QLR +QRL +QLL = 1. (4)The advantages of this parametrization are that it con-tains the maximum number of positive semi-definite bi-linear combinations of the gγǫµ and that the Qǫµ aredirectly interpretable as the total probabilities for µ-handed muons to decay into ǫ-handed electrons [4]. Fur-thermore, experimentally it is found that QLL is closeto unity. Thus, if Eq. (4) is used to eliminate QLL, the9 remaining variables are all close to zero. The bilinearcombinations QRR, QLR, QRL, BLR, BRL, Iα, Iβ wereadopted in the global analyses presented in [4, 10, 11].For convenience, in the present analysis we haveadopted a hybrid set of independent variables: QRR,QLR, QRL, BLR, BRL, α/A, β/A, α′/A, β′/A. Themuon decay parameters may be expressed in terms ofthese variables as [6, 8]:ρ =34+14(QLR +QRL)− (BLR +BRL),ξ = 1− 2QRR − 103QLR +43QRL +163(BLR −BRL),ξδ =34− 32QRR − 74QLR +14QRL + (BLR −BRL),ξ′ = 1− 2QRR − 2QRL,ξ′′ = 1− 103(QLR +QRL) +163(BLR +BRL),TABLE I: Experimental measurements included in the globalanalysis.Parameter Value Referenceρ 0.7518 ± 0.0026 [16]0.75080 ± 0.00105a [13]δ 0.7486 ± 0.0038 [11]0.74964 ± 0.00130 [14]Pµξ 1.0027 ± 0.0085b [18]Pµξδ/ρ 0.99787 ± 0.00082b [19, 20]ξ′ 1.00 ± 0.04 [16]ξ′′ 0.65 ± 0.36 [21]η¯ 0.02 ± 0.08 [16]α/A 0.015 ± 0.052c [9]β/A 0.002 ± 0.018c [9]η 0.071 ± 0.037d [15]η′′ 0.105 ± 0.052d [15]α′/A -0.047 ± 0.052e [9]-0.0034 ± 0.0219f [15]β′/A 0.017 ± 0.018e [9]-0.0005 ± 0.0080f [15]a Correlated with η in [13]. See text.b Fit assumes Pµ = 1. See text.c ρ(α/A, β/A) = −0.894 in [9].d ρ(η, η′′) = +0.946 in [15].e ρ(α′/A, β′/A) = −0.894 in [9].f ρ(α′/A, β′/A) = −0.893 in [15].η¯ =13(QLR +QRL) +23(BLR +BRL),η = (α− 2β)/A,η′′ = (3α+ 2β)/A. (5)We have computed the joint probability density func-tion of the 9 independent variables using Monte Carlointegration techniques, in a manner similar to that de-scribed in [9]. For each experimental input, we haveassumed that the corresponding probability distributiontakes the form of a one- or two-dimensional Gaussian,truncated to the allowed parameter region if necessary.When statistical and systematic uncertainties have beenquoted separately, we have added them in quadrature.III. INPUT VALUESEach of the previous muon decay global analyses[4, 9, 10, 11] utilized the same input parameters – ρ, δ,Pµξδ/ρ, ξ′, ξ′′, α/A, β/A, α′/A, β′/A. For each of thesequantities, the previous analyses adopted the single mostprecise experimental measurement that was available atthe time.We have utilized a different philosophy. We includeall of the accepted muon decay parameter measurementsthat are reported in [16], with two exceptions. When[16] determines a decay constant from a single input, we3have included it. When [16] computes an average from aset of previous measurements, we have utilized this aver-age value in our fits. The exceptions involve the valuesadopted in [16] for η and the transverse polarization pa-rameters α/A, β/A, α′/A, and β′/A. The adopted valuesfor these five parameters are all derived from the globalfit in [9]. Rather, for the transverse polarization parame-ters, we have used the experimental results [9] that wereinputs to that previous global analysis. Meanwhile, wehave not included the previous measurement of η in [17].Its uncertainty is a factor of 30 larger than the uncer-tainty in η derived from our fit to the rest of the existingdata.In addition to the measurements reported in [16], wehave included the recent measurements of the Michel pa-rameters ρ [13] and δ [14] and of the transverse polariza-tion parameters η, η′′, α′/A, and β′/A [15]. The completelist of inputs is given in Table I.Three of the input values require special consideration.In [13], the value obtained for ρ assumed that η is givenby the accepted value, −0.007, and a contribution of± 0.00023 was assigned to the uncertainty in ρ associ-ated with the ± 0.013 uncertainty in the accepted valuefor η [16]. We have performed two separate global fits. Inone fit, we included the ± 0.00023 in the uncertainty forρ, then treated the measurement as uncorrelated with η.In the other fit, we included the dependence of the mea-sured value of ρ on η [13] and reduced the uncertainty inρ accordingly. The two fits give very similar results. Theresults for the latter are reported here.References [18] and [19, 20] report measurements ofPµξ and Pµξδ/ρ, respectively. In each case, Pµ repre-sents the polarization of the muon in pion decay. This isknown to be > 0.99682 with 90% confidence [19, 20, 22].We have assumed Pµ = 1 when including the Pµξ mea-surement in our fits. In contrast, [19, 20] specifically men-tion the existence of possible unknown sources of muondepolarization prior to the decay and, therefore, chooseto interpret the measurement of Pµξδ/ρ as a lower limit.In recognition of this, we have performed two separatefits. In one fit, we assumed the probability distributionfor ξδ/ρ is Gaussian as given in Table I, which is equiv-alent to assuming Pµ = 1. In the other, we treated theprobability distribution as Gaussian when ξδ/ρ is lessthan the central value and constant when ξδ/ρ is greaterthan the central value. We report results from the for-mer fit because it finds slightly less restrictive limits forQRR, QLR, and BLR, while the remaining parametersare unchanged between the two fits.IV. RESULTSTable II shows the results of the global fits. 90% con-fidence level upper limits (lower limit for QLL) are givenfor the positive semi-definite quantities. For complete-ness, mean and r.m.s. values are also specified for thesevariables, even though the corresponding probability dis-TABLE II: Results of the global fit. For the positive semi-definite quantitites, 90% confidence level upper limits (lowerlimit for QLL) are given, and for completeness, the mean andr.m.s. values from the fit are also given in parentheses.Parameter Fit Result (×103)QRR < 1.14 (0.60 ± 0.38)QLR < 1.94 (1.22 ± 0.53)BLR < 1.27 (0.72 ± 0.40)QRL < 44 (26 ± 13)BRL < 10.9 (6.4 ± 3.3)QLL > 955 (973 ± 13)α/A 0.3 ± 2.1β/A 2.0 ± 3.1α′/A -0.1 ± 2.2β′/A -0.8 ± 3.2tributions are far from Gaussian. In contrast, the outputprobability distributions for α/A, β/A, α′/A, and β′/Aare very close to Gaussian.Only a subset of the input quantities play a role in con-straining each of the fit parameters. QRR is constrainedprimarily by the measurement of Pµξδ/ρ. QLR and BLRare constrained by the measurements of ρ, δ, and Pµξδ/ρ.QRL is constrained by the measurement of ξ′. BRL isconstrained by the combination of ξ′, ρ, and δ. α/Aand α′/A are constrained primarily by the requirement|Iα|2 ≤ BLRBRL. Finally, the large reduction in theuncertainties for α/A and α′/A obtained when applyingthe Iα constraint significantly reduces the correlations be-tween α/A and β/A and between α′/A and β′/A. The fitfinds ρ(α/A, β/A) = −0.19 and ρ(α′/A, β′/A) = −0.20.The correlations dominate the uncertainties in β/A andβ′/A quoted in Table I, so the result is that the trans-verse polarization measurements become far more pre-cise. Overall, the constraints in Eq. (3) play a crucialrole in reducing the uncertainties for a number of thefitted parameters, as was noted in [9].The fit results may be used to determine limits onthe coupling constants gγǫµ. These are given in TableIII. Limits are also given for certain linear combina-tions of scalar and tensor interactions. Muon decay mea-surements alone are unable to separate contributions ofgSLL from the standard model term, gVLL. Other experi-mental results, such as the branching ratio for π → eν,argue that the scalar contribution must be very small,but this requires assumptions that go beyond the model-independent experimental limits that we are exploringhere. In [4] it was noted that the rate for inverse muondecay, νµe− → µ−νe, may be used to distinguish betweengSLL and gVLL. In Table III, the limits quoted for gSLL andgVLL come from [8], which utilized the inverse muon decaymeasurements in [23].Table III shows that the present limits on |gSLR|, |gVLR|,and |gTLR| are significantly better than those in [8], whicharises from the inclusion of the new measurements of ρ4TABLE III: 90% confidence limits on the muon decay cou-pling constants in Eq. (1) are compared to the previous ac-cepted values from [8]. Limits are also given for certain scalar-tensor interference combinations.Ref. [8] Present work|gSRR| < 0.066 < 0.067|gVRR| < 0.033 < 0.034|gSLR| < 0.125 < 0.088|gVLR| < 0.060 < 0.036|gTLR| < 0.036 < 0.025|gSRL| < 0.424 < 0.417|gVRL| < 0.110 < 0.104|gTRL| < 0.122 < 0.104|gSLL| < 0.550 < 0.550|gVLL| > 0.960 > 0.960|gSLR + 6gTLR| < 0.143|gSLR + 2gTLR| < 0.108|gSLR − 2gTLR| < 0.070|gSRL + 6gTRL| < 0.418|gSRL + 2gTRL| < 0.417|gSRL − 2gTRL| < 0.418and δ. This was already noted in [14], where calculationsbased on the measurements of ρ, δ, and Pµξδ/ρ gave aslightly more restrictive limit for |gSLR| and a less restric-tive limit for |gVLR|. The new limits for |gSRR| and |gVRR|in Table III are slightly less restrictive than their previ-ously accepted values [8]. In [8], these limits came fromcalculations in [11] that were performed before an errorin the reported value of Pµξδ/ρ [19] had been discovered.The present fit utilizes the corrected, less restrictive valueof Pµξδ/ρ [20]. The new results for |gSRL| and |gVRL| arelittle different from the previously accepted values sincethe primary experimental input that constrains these, ξ′,has not been remeasured since [21].The Michel parameter η merits special discussion be-cause it has an impact on GF . GF is typically calcu-lated from the measured muon lifetime assuming a pureV −A interaction. This leads to the quoted precision of∆GF /GF = 9× 10−6 [16]. However, a non-zero value ofη would change the muon decay phase space so that [24]:GF ≈ G(V−A)F (1− 2ηme/mµ). (6)Thus, the ± 0.013 uncertainty in the accepted valueof η [16] leads to an additional uncertainty in GF of∆GF /GF = 1.3× 10−4. The recent measurement of thetransverse polarization of the electrons emitted in muondecay [15] included two analyses of the results. The gen-eral (model-independent) analysis led to the values of η,η′′, α′/A, and β′/A quoted in Table I. A second, re-stricted analysis assumed that muon decay can be de-scribed with only two coupling constants, gVLL and gSRR.Equations (2) and (5) imply that this assumption re-quires α/A = α′/A = 0 and η′′ = −η. The restrictedanalysis concluded η = −0.0021 ± 0.0070 ± 0.0010 andβ′/A = −0.0013 ± 0.0035 ± 0.0006. As noted above,the allowed ranges of α/A and α′/A are severely con-strained by other muon decay data in the present globalanalysis. Thus, we find a model-independent result,η = −0.0036± 0.0069, with slightly better precision thanthat of the model-dependent restricted analysis in [15].This reduces the contribution of η to the uncertainty inGF to ∆GF /GF = 6.7× 10−5. Note that all of the newmeasurements [13, 14, 15] play important roles in reduc-ing the uncertainty in η.V. CONCLUSIONWe have performed a new global analysis of all the ex-isting data on muon decay that do not include observa-tion of the outgoing neutrinos, including recent measure-ments of the Michel parameters ρ and δ and the muon de-cay transverse polarization parameters η, η′′, α′/A, andβ′/A. The global analysis finds that the upper limits onthe coupling constants |gSLR|, |gVLR|, and |gTLR| are morerestrictive than the previous accepted values. It also findsthat η = −0.0036±0.0069, which is nearly a factor of twomore precise than the previous accepted value.AcknowledgmentsWe thank colleagues in the TWIST Collaboration formany useful discussions regarding muon decay limits.This work was supported in part by the U.S. National Sci-ence Foundation Research Experiences for Undergradu-ates program, grant number 0354098, and by the U.S. De-partment of Energy, grant number DE-FG03-93ER40765.[1] L. Michel, Proc. Phys. Soc. London, Sect. A 63, 514(1950); C. Bouchiat and L. Michel, Phys. 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Global analysis of muon decay measurements

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Global analysis of muon decay measurements

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