Charged pion decay mediated by a non-standard scalar boson
M. M. Guzzo, L. J. F. Leite, S. W. P. Novelo, O. L. G. Peres, V. Pleitez
CCharged pion decay mediated by a non-standard scalar boson
M. M. Guzzo ID , ∗ L. J. F. Leite ID , † S. W. P. Novelo ID , ‡ and O. L. G. Peres ID § Instituto de F´ısica Gleb Wataghin - UNICAMP, 13083-859, Campinas-SP, Brazil
V. Pleitez ID ¶ Instituto de F´ısica Te´orica, Universidade Estadual Paulista,R. Dr. Bento Teobaldo Ferraz 271,Barra Funda, S˜ao Paulo - SP, 01140-070Brazil. (Dated: March 1, 2021)A model-independent analysis of a non-standard scalar contribution to the pion decay in a leptonand a neutrino is presented. We discuss the necessity of such scalars in many models beyond thestandard model to conclude that an entirely new range of masses and couplings associated withthe new scalar, entirely apart from standard contributions alone, is compatible with experimentalresults. The new range for the coupling constant and mass of the charged scalar is a consequenceof suitable cancellations of standard and non-standard contributions to the pion decay.
I. INTRODUCTION
In most possible new physics beyond the StandardModel (BSM), even when considering their minimal ver-sions, a richer scalar sector is encountered than the sim-ple neutral Higgs present in the particle spectrum of theStandard Model (SM) [1]. Even in the context of theSU(2) × U(1) gauge symmetry, nothing limits the num-ber of scalar fields, though at least a single doublet isnecessary for the usual spontaneous symmetry breakingpattern. Thus, one cannot rule out the possibility thatextra scalars, heavier than the observed Higgs (or lighter,but with sufficiently weak couplings), do exist. More-over, many mechanisms to generate neutrino masses re-quire additional scalars [2, 3]. Such particles can also beused, in principle, to solve some anomalies in high-energyexperiments, like those in B-meson decays [4, 5] or themuon anomalous magnetic moment [6, 7], for example.However, if those extra scalars exist, they possibly con-tribute to several well-known processes and must agreewith the related experimental results. Among others,this would be the case of a pion decaying into a chargedlepton and a neutrino. Such a process has an astonishingagreement between the experimental results and the SMtheoretical calculations, often used as a hallmark of theweak interactions’ V-A structure. Its helicity suppressionexplains the dominant decay in muons (99.99%) and notelectrons. Therefore, strong constraints on new physicsare expected in this decay.This article performs a model-independent analysis ofa new scalar contribution to the pion decay in a chargedlepton and a neutrino, taking into account some hy-potheses discussed in the text. Interestingly enough, due ∗ guzzo@ifi.unicamp.br † lfleite@ifi.unicamp.br ‡ wnovelo@ifi.unicamp.br § orlando@ifi.unicamp.br ¶ [email protected] to some suitable cancellations of the standard and non-standard contributions, this range includes an entirelynew set of such parameters, which is utterly differentfrom those coming from standard contributions alone.We organize our article in the following way: in SectionII, we present the phenomenological approach used inthis analysis, and in Section III, a detailed presentationof our procedures and results are done. In Section IV,our conclusions are shown. II. SCALAR CONTRIBUTION TO π ± → l ± ν In the usual electroweak standard model (SM) withmassive neutrinos, the charged pion decay can be de-scribed by the effective Lagrangian − L W = G F √ J α U li (cid:96) l γ α P L ν i + H.c. , (1)where G F = √ g W (cid:14) m W is the tree level Fermi con-stant ( g W and m W are, respectively, the SU (2) couplingconstant and mass for the standard W -boson), U = ( U li )is the PMNS mixing matrix [8, 9], P L = (1 − γ )/2 isthe left-handed chiral projector, and the neutrino fields ν i ( i = 1 , ,
3) are mass eigenstates. An implicit sum-mation in l = e, µ is assumed (as well as in i ), since thedecay in τ -leptons is kinematically forbidden. Finally,the weak current for the pion, J α , is given by the matrixelement (cid:10) (cid:12)(cid:12) J α (cid:12)(cid:12) π − ( q ) (cid:11) = V ud (cid:10) (cid:12)(cid:12) uγ α γ d (cid:12)(cid:12) π − ( q ) (cid:11) = if π q α V ud , (2)with f π the pion decay constant, and V the CKM mixingmatrix.We propose a new non-standard interaction, couplingthe charged leptons and the left-handed active neutri-nos, as well as u − and d − type quarks, and mediated bya novel charged scalar, η ± . As such, this new interac-tion can, in principle, contribute to π ± → lν . Assuming m η (cid:29) m π , where m η is the mass of the new scalar boson a r X i v : . [ h e p - ph ] F e b and m π is the charged pion mass, we have the followingeffective interaction − L η = 1 √ m η Y li j (cid:96) l P L ν i + H.c. , (3)with Y = ( Y li ) being an effective coupling matrix betweenthe pion and the leptonic sector. The pseudo-scalar pionmatrix element is given by [10, 11] (cid:10) (cid:12)(cid:12) j (cid:12)(cid:12) π − ( q ) (cid:11) = (cid:10) (cid:12)(cid:12) uγ d (cid:12)(cid:12) π − ( q ) (cid:11) = i ˜ f π , (4) where the constants ˜ f π and f π are related by [10]˜ f π = f π m π m u + m d , (5)with m u and m d being the respective bare quark masses.At tree-level, the two contributions given in Eqs. (1)and (3) combines coherently yielding, in the pion restframe, the total decay rateΓ( π − → l − ν [ γ ]) = R l (cid:88) i =1 G F f π m π π (cid:18) − m l m π (cid:19) × (cid:40) | V ud | | U li | m l + 1 g W | Y li | (cid:18) m W m η (cid:19) (cid:18) m π ( m u + m d ) (cid:19) − g W Re( U li Y ∗ li V ud ) (cid:18) m W m η (cid:19) (cid:18) m π m u + m d (cid:19) m l (cid:41) , (6)where terms proportional to the neutrino masses wereneglected, and, for each charged lepton state, we summedover all the active neutrino mass states.In Eq. (6), the first term inside the curly brackets cor-responds to the usual SM contribution, the second termcomes purely from the new scalar, and the third is the in-terference term between them. The multiplicative factor R l accounts for radiative corrections mostly due to theemission of soft photons [12–15] and is given explicitly inAppendix A.A more interesting quantity is the ratio of branchingratios r e/µ = Γ( π − → e − ν [ γ ])Γ( π − → µ − ν [ γ ]) , (7)since its experimental uncertainty is smaller than the in-dividual decay rates.Using Eq. (6), the ratio r e/µ can be written as r e/µ = R e R µ (cid:18) m π − m e m π − m µ (cid:19) e µ , (8)where∆ l = (cid:88) i =1 (cid:40) | Y li | g W m l | V ud | (cid:18) m W m η (cid:19) (cid:18) m π ( m u + m d ) (cid:19) − U li Y ∗ li V ud ) g W m l | V ud | (cid:18) m W m η (cid:19) (cid:18) m π m u + m d (cid:19)(cid:41) , (9)quantifies the presence of new physics beyond the SM.If ∆ l = 0 (for both l = e, µ ), or ∆ e = ∆ µ , the ra-tio r e/µ gives the same result as the SM, hence no newphysics could be probed in π + → l + ν decay in thosescenarios. Of course, such a situation would imply anextreme fine-tuning of the corresponding matrix entries,since ∆ l = ∆ l ( m l ), and m e (cid:54) = m µ . To reduce the number of free parameters and makevisualization of the results possible, we make henceforththe assumption that the elements of the coupling matrix Y = ( Y li ) are real and universal, that is, Y li = y η , (10)for all l = e, µ and i = 1 , ,
3, with y η a real number.This assumption means that we consider in our analysisthat the new scalar couples identically to all the massiveneutrino fields and for both electrons and muons. Fur-thermore, with this assumption, we are guaranteed tohave ∆ e (cid:54) = ∆ µ .Finally, we note two direct consequences from the as-sumption (10). First, the summation over | Y li | reducesto (cid:88) i =1 | Y li | = 3 y η . (11)And last, Re( U li Y ∗ li V ud ) = y η V ud Re( U li ) , (12)which carries a dependence on the usual Dirac CP phaseof the PMNS matrix, usually denoted by δ . However, wepoint out that, even though this dependence exists, noviolation of CP occurs, since the dependence comes asthe real part of the entries of the PMNS matrix, which isproportional to cos( δ ). In other words, the substitution U li → U ∗ li 1 , going from neutrinos to anti-neutrinos, doesnot modify the decay rate calculated in Eq. (6). Notice that no violation occurs even when the couplings Y li arecomplex. But in this case, the substitution U li → U ∗ li , going fromneutrinos to anti-neutrinos, is accompanied by Y li → Y ∗ li . III. DISCUSSION
With the ratio r e/µ calculated in Eq. (8), our next stepis to determine the allowed values for our new parameters y η and m η consistent with the experimental results frompion decay. In Subsection III A we introduce our statisti-cal analysis based on a χ -test, and in Subsection III B wepresent the allowed region in parameter space obtainedwith this test, together with a discussion of the results. A. Statistical analysis
First, we define a χ -function χ ( y η , m η ; δ ) = r e/µ ( y η , m η ; δ ) − r (exp) e/µ σ r ( y η , m η ; δ ) , (13)which is a function of the coupling constant y η and themass m η of the new scalar field η ± . We regard the DiracCP phase δ as a fixed parameter throughout this analysis,emphasized using a semi-colon to separate it from theother arguments.In Eq. (13), r e/µ is the ratio of branching ratios givenin (8), r exp e/µ = 1 . × − its experimental value [16], σ r the total uncertainty associated with this ratio givenby the usual error propagation formula σ r = (cid:118)(cid:117)(cid:117)(cid:116)(cid:88) x (cid:18) ∂r e/µ ∂x (cid:19) σ x + (cid:16) σ (exp) r (cid:17) , (14)and σ exp r = 2 × − is the experimental error of r exp e/µ .Table I lists the experimental values for th x quantitiesused in calculating Eqs. (8) and (14). We note that theradiative correction factor R l ( l = e, µ ) depends on thefollowing numerical parameters m ρ and C i ( i = 1 , , χ -function (13) with re-spect to m η and y η . Then, we obtain the allowed regionapplying the condition χ ( m η , y η ; δ ) = χ ( δ ) + P, (15)over all point ( m η , y η ) in parameter space. For two freeparameters at 1 σ confidence level, we have P = 2 .
30 [17].We repeat the above analysis for the four principalvalues of δ = 0, π , π /2 and 3 π /2 . Not surprisingly, thevalues of χ (best fits) found are all consistent with nonew interactions, that is compatible with y η ≈ δ . Such a result is expected since the SM cal-culations already agree quite well with the experimental Hence, the δ -dependence of χ in Eq. (15) is weak. TABLE I. Experimental quantities used in the calculation ofthe ratio r e/µ given in Eq. (8) with the respective uncertaintiesin parenthesis [16]. x Experimental value α .
297 352 569 3(11) × − G F .
166 378 7(6) × − MeV m W . m Z . m µ .
658 374 5(24) MeV m e .
510 998 946 1(31) MeV m π .
570 61(24) MeV m u . m d . m ρ . ( θ ) 0 . ( θ ) 0 . ( θ ) 2 . × − | V ud | .
974 20(21) f π . . data, giving strong constraints to contributions that donot suppress helicity, like the scalar mediator we considerhere. Nevertheless, as we shall see in the next subsection,a notable region of the parameter space is permitted bythe experimental results. B. Results
With the above χ -analysis, the total allowed region,constrained from the experimental values for the ratio r e/µ as a function of the new scalar coupling constant y η and mass m η is given in Fig. 1. The results show massesbetween 1 TeV and 20 TeV. The lower bound is inspiredby constraints of charged scalar bosons from Refs. [16, 18]in the context of the Minimal Supersymmetric StandardModel. Other limits may be possible for different models. FIG. 1. Total allowed region for the coupling constant y η (inunits of g W ) and mass m η at 1 σ confidence level, for arbitraryvalues of the CP phase δ . Figure 1 shows us two distinct allowed regions in pa-rameter space. The first region corresponds to a negligi-ble coupling constant, y η ∼
0, and is compatible with nonew physics. Such a region was expected, once the SMresult already agrees reasonably well with the experimen-tal data. The second region, with a parabolic shape, iscompletely non-trivial and corresponds to a suitable can-cellation of the non-standard terms. To understand thisentirely new and unexpected region, we recall Eqs. (8)and (9). In those equations, the quantifier of new physics,∆ l ( l = e, µ ), is composed of a sum corresponding to thepure scalar and interference terms. If those terms haveopposite signs, a cancellation may occur for some valuesof the free parameters, m η and y η , within the experi-mental error. Setting ∆ l = 0 and using Eq. (10), thissituation corresponds to y η g W = (cid:34) (cid:88) i =1 Re( U li ) V ud ( m u + m d ) m π m l m W (cid:35) m η , (16)which is a parabola in the m η - y η space. That is theexact behavior appearing in Fig. 1. More precisely, theparabolic region in Fig. 1 corresponds to the cancellationof the electron contribution, that is ∆ e = 0, as can beseen in Fig. 2 below.In Fig. 2, we show the same region as in Fig. 1, butnow with curves corresponding to some chosen values ofthe effective coupling constant˜ G η √ y η m η = 1 . × − y η / g W (cid:0) m η
20 TeV (cid:1) G F √ G η measures the strength of thenew scalar interaction in Eq. (3), and Eq. (17) comparedit to the Fermi constant G F . The non-trivial allowed re-gion has typical values of 10 − G F < ˜ G η < × − G F ,hence, weaker than the electroweak strength.Finally, in Fig. 3, we show the explicit dependence ofCP phase δ for the allowed region, while in Fig. 1 thisregion is shown for any value of the CP phase. We cansee a significant change in the allowed region with thedifferent values of δ in the branch compatible with y η (cid:54) = 0,while the SM compatible region ( y η ∼
0) does not varywith it.
IV. CONCLUSIONS
In this paper, we studied the contribution of a scalarinteraction in the context of the charged pion decay, in amodel-independent manner, introduced via the effectiveLagrangian in Eq. (3).Using a statistical analysis based on a χ -test, pre-sented in Section III A, we were able to determine anallowed region for a real and universal effective coupling y η and mass m η associated with the new scalar proposed.As Figures 1-2 show, at 1 σ confidence level, a region com-patible with a new non-standard interaction is possible FIG. 2. Total allowed region for the coupling constant y η (inunits of g W ) and mass m η at 1 σ confidence level, for arbitraryvalues of the CP phase δ . The black curves corresponds to∆ e = 0, as given in Eq. (16), or, equivalently, Γ e = Γ SM e ,where Γ e is the decay rate in electrons. The constant ˜ G η isdefined in Eq. (17).FIG. 3. Allowed region for the coupling constant y η (in unitsof g W ) and mass m η , at 1 σ confidence level, for four chosenvalues of the CP phase δ . due to a suitable cancellation in the pure scalar and in-terference terms. Furthermore, although this decay con-serves CP, Fig. 3 shows that the allowed region for theparameters of the new scalar particle is slightly differentfor different values of the Dirac CP phase δ present inthe PMNS matrix.Even though the above calculations were made inde-pendently of a model, we can still interpret them differ-ently. We assumed only experimentally known leptonicdegrees of freedom; as such only three left-hand neutrinoswere considered. As such only three left-hand neutrinoswere considered. Other choices would be possible as, forexample, right-handed neutrinos. In such a case, new pa-rameters may have to be introduced, like a new couplingconstant matrix Y R = ( Y Rli ) as in Eq. (3), that couplesthe pion state with the aforementioned right-handed neu-trinos.At last, we mention a few possible models where ourresults may be applied. The Standard Model enlargedwith a scalar triplet, corresponding to a Type-II Seesawmechanism [2, 3], is such an example. The addition of adimension-five effective operator [19] in a two-Higgs dou-blet model is another possibility. In both these cases,neutrinos are pure Majorana particles, and the PMNSis an exact unitary matrix [20, 21]. Another importantexample is the Minimal 3-3-1 model [22], where the neu-trinos again can be taken as pure Majorana particles. Forother models, like the Left-Right-Symmetric Model withcomplex triplets [23], or any model that implements theType-I Seesaw mechanism, our results do not directly ap-ply, and additions may be necessary. Also, the existenceof charged scalars can be associated with non-standardneutrino interactions at source as described by Refs. [24].However, even in the models where the results are appli-able, caution should be taken. In most of them, morethan one charged scalar exist, and extra mixing anglesin the scalar sector have to be taken into account. Thecase of a specific model deserves unique analysis and goesbeyond the scope of this paper.
ACKNOWLEDGMENTS
M.M.G., V.P., and O.L.G.P. are thankful for the sup-port of FAPESP funding Grant 2014/19164-6. M.M.G.is supported by CNPq grant 304001/2017-1. O.L.G.P.is supported by FAPESP funding Grant 2016/08308-2, FAEPEX funding grant 2391/2017 and 2541/2019,CNPq grant 306565/2019-6. S.W.P.N. and L.J.F.L.are grateful for the support of CNPq under grant140727/2019-1 and 131548/2019-0, respectively. Thisstudy was financed in part by the Coordena¸c˜ao deAperfei¸coamento de Pessoal de N´ıvel Superior - Brasil(CAPES) - Finance Code 001.
Appendix A: Radiative Corrections
Usually, experiments that measure the pion decay π ± → lν are unable to distinguish between this processand the radiative decay with low energy photons, that is π ± → lνγ , where γ is a soft photon. Therefore, radiativecorrections are needed, even in a tree-level calculation,since low-energy photons can be generated in the decay, which can not be detected. We follow here the computa-tions of [13] with the definition R l = (cid:20) απ ln (cid:18) m Z m ρ (cid:19)(cid:21) × (cid:34) − απ (cid:32)
32 ln (cid:18) m ρ m π (cid:19) + C + C m l m ρ ln (cid:32) m ρ m l (cid:33) + C m l m ρ (cid:19)(cid:21) (cid:20) απ F (cid:18) m l m π (cid:19)(cid:21) , (A1)where, F ( x ) = 3 ln( x ) + 13 − x − x ) − − x − x ) x ln( x ) − (cid:18) x − x ln( x ) + 1 (cid:19) ln (cid:0) − x (cid:1) + 2 (cid:18) x − x (cid:19) L (1 − x ) , (A2)and L ( z ) = (cid:90) z ln(1 − t ) t d t . (A3)The first term inside square brackets in Eq. (A1) comesfrom the short distance Z -boson exchange. To dis-tinguish between short- and long-distance loop correc-tions, we use the ρ -meson mass, m ρ = 775 . C , C , and C account for the internal structureof the pion. Following the discussion in Ref. [13], we setthese constants to be C = 0 ± . , (A4a) C = 3 + 23 (cid:18) −
74 0 . (cid:19) m ρ π f π , (A4b) C = 0 ± . (A4c)The constant C is model-dependent. Following [13],we set it to zero and consider a conservative uncertaintyin the calculations, such that it can be applied to a varietyof models. [1] J. L. Rosner, Am. J. Phys. , 302 (2003), arXiv:hep-ph/0206176.[2] T. P. Cheng and L.-F. Li, Phys. Rev. D , 2860 (1980).[3] J. Schechter and J. W. F. Valle, Phys. Rev. D , 2227(1980). [4] A. Celis, M. Jung, X.-Q. Li, and A. Pich, JHEP (2013), 54.[5] S. Schacht and A. Soni, JHEP (2020), 163,arXiv:2007.06587 [hep-ph].[6] A. Czarnecki and W. J. Marciano, Phys. Rev. D64 , et al. (Muon g-2), Phys. Rev. D73 , 072003(2006), arXiv:hep-ex/0602035 [hep-ex].[8] B. Pontecorvo, Sov.Phys.JETP , 429 (1957).[9] Z. Maki, M. Nakagawa, and S. Sakata, Prog. Theor. Phys. , 870 (1962).[10] V. Bernard and U.-G. Meissner, Ann. Rev. Nucl. Part.Sci. , 33 (2007), arXiv:hep-ph/0611231.[11] B. A. Campbell and A. Ismail (2008) arXiv:0810.4918.[12] T. Kinoshita, Phys. Rev. Lett. , 477 (1959).[13] W. J. Marciano and A. Sirlin, Phys. Rev. Lett. , 3629(1993).[14] A. Sirlin, Rev. Mod. Phys. , 573 (1978), [Erratum:Rev.Mod.Phys. 50, 905 (1978)].[15] V. Cirigliano and I. Rosell, JHEP (10), 005;V. Cirigliano, G. Ecker, H. Neufeld, A. Pich, and J. Por-toles, Rev. Mod. Phys. , 399 (2012), arXiv:1107.6001[hep-ph].[16] P. A. Zyla et al. (Particle Data Group), Progress of Theoretical and Experimental Physics ,10.1093/ptep/ptaa104 (2020), 083C01.[17] F. James, MINUIT Function Minimization and ErrorAnalysis: Reference Manual Version 94.1 (1994).[18] M. Aaboud et al. (ATLAS), JHEP (2018), 139,arXiv:1807.07915 [hep-ex].[19] S. Weinberg, Phys. Rev. Lett. , 1566 (1979).[20] G. Senjanovic, Nucl. Phys. B , 334 (1979).[21] H. Diaz, V. Pleitez, and O. P. Ravinez, Phys. Rev. D , 075006 (2020), arXiv:1908.02828 [hep-ph].[22] R. Foot, O. F. Hernandez, F. Pisano, and V. Pleitez,Phys. Rev. D , 4158 (1993), arXiv:hep-ph/9207264.[23] R. N. Mohapatra and G. Senjanovic, Phys. Rev. D ,165 (1981).[24] A. Falkowski, M. Gonz´alez-Alonso, and Z. Tabrizi, JHEP (2019), 173, arXiv:1901.04553 [hep-ph]; JHEP11