Semileptonic decays of light quarks beyond the Standard Model
Abstract
We describe nonstandard contributions to semileptonic processes in a model independent way in terms of an invariant effective lagrangian at the weak scale, from which we derive the lowenergy effective lagrangian governing muon and beta decays. We find that the deviation from Cabibbo universality, , receives contributions from four effective operators. The phenomenological bound provides strong constraints on all four operators, corresponding to an effective scale TeV (90% CL). Depending on the operator, this constraint is at the same level or better then the Z pole observables. Conversely, precision electroweak constraints alone would allow universality violations as large as (90% CL). An observed at this level could be explained in terms of a single fourfermion operator which is relatively poorly constrained by electroweak precision measurements.
I Introduction
Precise lifetime and branching ratio measurements Amsler et al. (2008) combined with improved theoretical control of hadronic matrix elements and radiative corrections make semileptonic decays of light quarks a deep probe of the nature of weak interactions Marciano (2008); Antonelli et al. (2008). In particular, the determination of the elements and of the CabibboKobayashiMaskawa (CKM) Cabibbo (1963); Kobayashi and Maskawa (1973) quark mixing matrix is approaching the 0.025% and 0.5% level, respectively. Such precise knowledge of and enables tests of Cabibbo universality, equivalent to the CKM unitarity condition^{1}^{1}1 contributes negligibly to this relation. , at the level of or better. Assuming that new physics contributions scale as , the unitarity test probes energy scales on the order of the TeV, which will be directly probed at the LHC.
While the consequences of Cabibbo universality tests on Standard Model (SM) extensions have been considered in a number of explicit (mostly supersymmetric) scenarios Barbieri et al. (1985); Marciano and Sirlin (1987); Hagiwara et al. (1995); Kurylov and RamseyMusolf (2002), a modelindependent analysis of semileptonic processes beyond the SM is missing. The goal of this investigation is to analyze in a modelindependent effective theory setup new physics contributions to low energy chargedcurrent (CC) processes. The resulting framework allows us to assess in a fairly general way the impact of semileptonic processes in constraining and discriminating SM extensions. We shall pay special attention to purely leptonic and semileptonic decays of light hadrons used to extract the CKM elements and .
Assuming the existence of a mass gap between the SM and its extension, we parameterize the effect of new degrees of freedom and interactions beyond the SM via a series of higher dimensional operators constructed with the lowenergy SM fields. If the SM extension is weakly coupled, the resulting TeVscale effective lagrangian linearly realizes the electroweak (EW) symmetry and contains a SMlike Higgs doublet Buchmuller and Wyler (1986). This method is quite general and allows us to study the implications of precision measurements on a large class of models. In particular, the effective theory approach allows us to understand in a modelindependent way (i) the significance of Cabibbo universality constraints compared to other precision measurements (for example, could we expect sizable deviations from universality in light of no deviation from the SM in precision tests at the Z pole?); (ii) the correlations between possible deviations from universality and other precision observables, not always simple to identify in a specific model analysis.
This article is organized as follows. In Section II we review the form of the most general weak scale effective lagrangian including operators up to dimension six, contributing to precision electroweak measurements and semileptonic decays. In Section III we derive the lowenergy ( GeV) effective lagrangian describing purely leptonic and semileptonic CC interaction. We discuss the flavor structure of the relevant effective couplings in Section IV. In Section V we give an overview of the phenomenology of and beyond the SM, and derive the relation between universality violations and other precision measurements at the operator level. Section VI is devoted to a quantitative analysis of the interplay between the universality constraint and other precision measurements, while Section VII contains our conclusions.
Ii Weak scale effective lagrangian
As discussed in the introduction, our aim is to analyze in a modelindependent framework new physics contributions to both precision electroweak observables and beta decays. Given the successes of the SM at energies up to the electroweak scale GeV, we adopt here the point of view that the SM is the lowenergy limit of a more fundamental theory. Specifically, we assume that: (i) there is a gap between the weak scale and the scale where new degrees of freedom appear; (ii) the SM extension at the weak scale is weakly coupled, so the EW symmetry is linearly realized and the lowenergy theory contains a SMlike Higgs doublet Buchmuller and Wyler (1986). Analyses of EW precision data in nonlinear realizations of EW symmetry can be found in the literature Appelquist and Wu (1993); Longhitano (1980); Feruglio (1993); Wudka (1994). In the spirit of the effective field theory approach, we integrate out all the heavy fields and describe physics at the weak scale (and below) by means of an effective nonrenormalizable lagrangian of the form:
(1)  
(2) 
where is the characteristic scale of the new physics and are local gaugeinvariant operators of dimension built out of SM fields. Assuming that righthanded neutrinos do not appear as lowenergy degrees of freedom, the building blocks to construct local operators are the gauge fields , corresponding to , the five fermionic gauge multiplets,
(3) 
the Higgs doublet
(4) 
and the covariant derivative
(5) 
In the above expression are the GellMann matrices, are the Pauli matrices, are the gauge couplings and is the hypercharge of a given multiplet.
In our analysis we will not consider operators that violate total lepton and baryon number (we assume they are suppressed by a scale much higher than de Gouvea and Jenkins (2008)). Under the above assumptions, it can be shown Buchmuller and Wyler (1986) that the first corrections to the SM lagrangian are of dimension six. A complete set of dimensionsix operators is given in the pioneering work of Buchmüller and Wyler (BW) Buchmuller and Wyler (1986)^{2}^{2}2In the original list of BW there are eighty operators, but it can be shown that it can be reduced to seventyseven (see Appendix A).. Truncating the expansion at this order we have
(6) 
For operators involving quarks and leptons, both the coefficients and the operators carry flavor indices. When needed, we will make the flavor indices explicit, using the notation for fourfermion operators.
The above effective lagrangian allows one to parameterize nonstandard corrections to any observable involving SM particles. The contribution from the dimension six operators involve terms proportional to and , where is the vacuum expectation value (VEV) of the Higgs field and is the characteristic energy scale of a given process. In order to be consistent with the truncation of (1) we will work at linear order in the above ratios.
We are interested in the minimal subset of the BW basis that contribute at tree level to CPconserving electroweak precision observables and beta decays. Upon imposing these requirements (see Appendix A) we end up with a basis involving twentyfive operators. In selecting the operators, flavor symmetries played no role (in fact at this level the coefficients can carry any flavor structure). However, in order to organize the subsequent phenomenological analysis, it is useful to classify the operators according to their behavior under the flavor symmetry of the SM gauge lagrangian (the freedom to perform transformations in family space for each of the five fermionic gauge multiplets, listed in Eq. 3).
ii.1 invariant operators
The operators that contain only vectors and scalars are
(7) 
There are eleven fourfermion operators:
(8)  
(9)  
(10)  
(11)  
(12) 
Some comments are in order. In principle, in order to avoid redundancy (see discussion in Appendix A), one must discard either or . However, here we have followed the common practice to work with both operators and consider only flavor structures factorized according to fermion bilinears. Moreover, we use the structure in operators (10), instead of their Fierz transformed , that BW use. They are related by a factor ().
There are seven operators containing two fermions that alter the couplings of fermions to the gauge bosons:
(13)  
(14)  
(15)  
(16) 
Finally, there is one operator that modifies the triple gauge boson interactions
(17) 
ii.2 Non invariant operators
Three are three fourfermion operators
(18)  
(19) 
and one operator with two fermions
(20) 
which gives rise to a right handed charged current coupling.
The twentyone invariant operators contribute to precision EW measurements (see Ref. Han and Skiba (2005)), whereas only nine of the twentyfive operators contribute to the semileptonic decays, including all four breaking operators.
We conclude this section with some remarks on our convention for the coefficients of the “flavored” operators: (i) in those operators that include the h.c. in their definition, the flavor matrix will appear in the h.c.part with a dagger; (ii) for the operators and , because of the symmetry between the two bilinears, we impose ; (iii) in order to ensure the hermiticity of the operators (8)(12) we impose . None of these conditions entails any loss of generality.
Iii Effective lagrangian for and quark decays
Our task is to identify new physics contributions to lowenergy CC processes. In order to achieve this goal, we need to derive from the the effective lagrangian at the weak scale (in which heavy gauge bosons and heavy fermions are still active degrees of freedom) a lowenergy effective lagrangian describing muon and quark CC decays. The analysis involves several steps which we discuss in some detail, since a complete derivation is missing in the literature, as far as we know.
iii.1 Choice of weak basis for fermions
At the level of weak scale effective lagrangian, we can use the invariance to pick a particular basis for the fermionic fields. In general, a transformation leaves the gauge part of the lagrangian invariant while affecting both the Yukawa couplings and the coefficients of dimension six operators involving fermions. We perform a specific transformation that diagonalizes the downquark and charged lepton Yukawa matrices and and puts the uptype Yukawa matrix in the form , where is the CKM matrix. The flavored coefficients correspond to this specific choice of weak basis for the fermion fields.
iii.2 Electroweak symmetry breaking: transformation to propagating eigenstates
Once the Higgs acquires a VEV the quadratic part of the lagrangian for gauge bosons and fermions becomes nondiagonal, receiving contributions from both SM interactions and dimension six operators. In particular, the NP contributions induce kinetic mixing of the weak gauge bosons, in addition to the usual mass mixing. Therefore the next step is to perform a change of basis so that the new fields have canonically normalized kinetic term and definite masses.
Let us first discuss the gauge boson sector. We agree with the BW results on the definition of gauge field mass eigenstates and on the expressions for the physical masses (Ref. Buchmuller and Wyler (1986), section 4.1). However, we find small differences from their results in the couplings of the and to fermion pairs, which can be written as (ref. Buchmuller and Wyler (1986), section 4.2):
(21)  
(22)  
(23)  
Here the ’s and ’s are matrices in flavor space. In the case of the charged current we find (BW do not have the in and )
(24)  
(25)  
(26) 
where we have introduced the notation
(27) 
In the case of the neutral current ( coefficients) we obtain the same results as BW except for the following replacement:
(28) 
for .
Finally, we need to diagonalize the fermion mass matrices. With our choice of weak basis for the fermions, the only step that is left is the diagonalization of the upquark mass matrix, proportional to the Yukawa matrix , where is the CKM matrix. This can be accomplished by a transformation of the fields:
(29) 
As a consequence, the charged current and neutral current couplings involving up quarks change as follows:
(30) 
Similarly, appropriate insertions of the CKM matrix will appear in every operator that contains the field.
iii.3 Effective lagrangian for muon decay
The muon decay amplitude receives contributions from gauge boson exchange diagrams (with modified couplings) and from contact operators such as , , . Since we work to first order in , we do not need to consider diagrams contributing to with the “wrong neutrino flavor”, because they would correct the muon decay rate to . After integrating out the and , the muon decay effective lagrangian reads:
(31) 
where is the uncorrected W mass and
(32)  
(33) 
represent the correction to the standard structure and the coupling associated with the new structure, respectively.
iii.4 Effective lagrangian for beta decays:
The lowenergy effective lagrangian for semileptonic transitions receives contributions from both W exchange diagrams (with modified Wfermion couplings) and the fourfermion operators , , , . As in the muon case, we neglect lepton flavor violating contributions (wrong neutrino flavor). The resulting lowenergy effective lagrangian governing semileptonic transitions (for a given lepton flavor ) reads:
(34)  
where
(35)  
(36)  
(37)  
(38)  
(39) 
In Eqs. (3539) the repeated indices are not summed over, while the index is.
Iv Flavor structure of the effective couplings
So far we have presented our results for the effective lagrangian keeping generic flavor structures in the couplings (see Eqs. 32, 33, and 35 through 39). However, some of the operators considered in the analysis contribute to flavor changing neutral current (FCNC) processes, so that their flavor structure cannot be generic if the effective scale is around : the offdiagonal coefficients are experimentally constrained to be very small. While it is certainly possible that some operators (weakly constrained by FCNC) have generic structures, we would like to understand the FCNC suppression needed for many operators in terms of a symmetry principle. Therefore, we organize the discussion in terms of perturbations around the flavor symmetry limit.
If the underlying new physics respects the flavor symmetry of the SM gauge lagrangian, no problem arises from FCNC constraints. The largest contributions to the coefficients are flavor conserving and universal. Flavor breaking contributions arise through SM radiative corrections, due to insertions of Yukawa matrices that break the symmetry. As a consequence, imposing exact symmetry on the underlying model does not seem realistic. A weaker assumption, the Minimal Flavor Violation (MFV) hypothesis, requires that is broken in the underlying model only by structures proportional to the SM Yukawa couplings Chivukula and Georgi (1987); Hall and Randall (1990); Buras et al. (2001); D’Ambrosio et al. (2002), and by the structures generating neutrino masses Cirigliano et al. (2005a). We will therefore organize our discussion in several stages:

first, assume dominance of invariant operators;

consider effect of breaking induced within MFV;

consider the effect of generic nonMFV flavor structures.
In order to proceed with this program, for the relevant operators we list below the flavor structures allowed within MFV. The notation is as follows: we denote by the diagonal Yukawa matrices; represents the diagonal light neutrino mass matrix; denotes the CKM matrix, while is the PMNS Maki et al. (1962) neutrino mixing matrix; is the Higgs VEV and is the scale of lepton number violation, that appears in the definition of MFV in the lepton sector (we follow here the “minimal” scenario of Ref. Cirigliano et al. (2005a)). With this notation, the leading “leftleft” flavor structures in the quark and lepton sector read:
(40)  
(41) 
We use Greek letters for the lepton flavor indices, while for the quark flavor indices, and we neglect terms with more than two Yukawa insertions. Moreover, we denote by , , and the numerical coefficients of that multiply the appropriate matrices in flavor space. For the operators that have a nonvanishing contribution in the limit, we find:
(42)  
(43)  
(44)  
(45)  
(46) 
For the operators that vanish in the limit of exact symmetry, we find:
(47)  
(48)  
(49)  
The coefficient of the tensor operator, has an expansion similar to the one of .
Except for the top quark, the Yukawa insertions typically involve a large suppression factor, as . In the case of SM extensions containing two Higgs doublets, this scaling can be modified if there is a hierarchy between the vacuum expectation values of the Higgs fields giving mass to the up or downtype quarks, respectively. In this case, for large the Yukawa insertions scale as:
(50)  
(51)  
(52) 
V Phenomenology of and : overview
Using the general effective lagrangians of Eqs. (31) and (34) for charged current transitions, one can calculate the deviations from SM predictions in various semileptonic decays. In principle a rich phenomenology is possible. Helicity suppressed leptonic decays of mesons have been recently analyzed in Ref. Filipuzzi and Isidori (2009). Concerning semileptonic transitions, several reviews treat in some detail decay differential distributions Herczeg (2001); Severijns et al. (2006). Here we focus on the integrated decay rates, which give access to the CKM matrix elements and : since both the SM prediction and the experimental measurements are reaching the subpercent level, we expect these observables to provide strong constraints on NP operators.
and can be determined with high precision in a number of channels. The degree of needed theoretical input varies, depending on which component of the weak current contributes to the hadronic matrix element. Roughly speaking, one can group the channels leading to into three classes:

semileptonic decays in which only the vector component of the weak current contributes. These are theoretically favorable in the Standard Model because the matrix elements of the vector current at zero momentum transfer are known in the () limit of equal light quark masses: . Moreover, corrections to the symmetry limit are quadratic in Behrends and Sirlin (1960); Ademollo and Gatto (1964). Superallowed nuclear beta decays (), pion beta decay (), and decays belong to this class. The determination of from these modes requires theoretical input on radiative corrections Marciano and Sirlin (1986, 2006); Cirigliano et al. (2002, 2004, 2008) and hadronic matrix elements via analytic methods Hardy and Towner (2008), Leutwyler and Roos (1984); Bijnens and Talavera (2003); Jamin et al. (2004); Cirigliano et al. (2005b), or lattice QCD methods Becirevic et al. (2005); Dawson et al. (2006); Boyle et al. (2008); Lubicz et al. (2009).

semileptonic transitions in which both the vector and axial component of the weak current contribute. Neutron decay () and hyperon decays (, ….) belong to this class. In this case the matrix elements of the axial current have to be determined experimentally Cabibbo et al. (2003).

Leptonic transitions in which only the axial component of the weak current contributes. In this class one finds meson decays such as but also exclusive decays such as . Experimentally one can determine the products and . With the advent of precision calculations of in lattice QCD Aubin et al. (2004); Beane et al. (2007); Follana et al. (2008); Blossier et al. (2009); Bazavov et al. (2009), this class of decays provides a useful constraint on the ratio Marciano (2004).
Currently, the determination of is dominated by superallowed nuclear beta decays Hardy and Towner (2008), while the best determination of arises from decays Antonelli et al. (2008). Experimental improvements in neutron decay and decays, as well as in lattice calculations of the decay constants will allow in the future competitive determinations from other channels. In light of this, we set out to perform a comprehensive analysis of possible new physics effects in the extraction of and .
As outlined in the previous section, we start our analysis by assuming dominance of the invariant operators. These are not constrained by FCNC and can have a relatively low effective scale . In the limit the phenomenology of CC processes greatly simplifies: all receive the same universal shift (coming from the same short distance structure). As a consequence, extractions of from different channels (vector transitions, axial transitions, etc.) should agree within errors. Therefore, in this limit the new physics effects are entirely captured by the quantity
(53) 
constructed from the elements extracted from semileptonic transitions using the standard procedure outlined below. We now make these points more explicit.
v.1 Extraction of and contributions to in the limit
If we assume invariance, only the SM operator survives in the muon decay lagrangian of Eq. (31), with ^{3}^{3}3We disagree with the result of BW on the sign of .
(54) 
Therefore, in this case the effect of new physics can be encoded into the following definition of the leptonic Fermi constant:
(55) 
where . Similarly, in the symmetry limit, only the SM operator survives in the effective langrangian for semileptonic quark decays of Eq. (34), with coupling:
(56) 
As in the muon decay, the new physics can be encoded in a (different) shift to the effective semileptonic (SL) Fermi constant:
(57) 
The value of extracted from semileptonic decays is affected by this redefinition of the semileptonic Fermi constant and by the shift in the muon Fermi constant , to which one usually normalizes semileptonic transitions. In fact one has
(58)  
So in the limit a common shift affects all the (from all channels). The only way to expose new physics contributions is to construct universality tests, in which the absolute normalization of matters. For light quark transitions this involves checking that the first row of the CKM matrix is a vector of unit length (see definition of in Eq. (53)). The new physics contributions to involve four operators of our basis and read:
(59) 
In specific SM extensions, the are functions of the underlying parameters. Therefore, through the above relation one can work out the constraints of quarklepton universality tests on any weakly coupled SM extension.
v.2 Beyond
Corrections to the limit can be introduced both within MFV and via generic flavor structures. In MFV, as evident from the results of Section IV, the coefficients parameterizing deviations from are highly suppressed. This is true even when one considers the flavor diagonal elements of the effective couplings, due to the smallness of the Yukawa eigenvalues and the hierarchy of the CKM matrix elements. As a consequence, in MFV we expect the conclusions of the previous subsections to hold. The various CKM elements receive a common dominant shift plus suppressed channeldependent corrections, so that Eq. (59) remains valid to a good approximation. In other words, both in the exact limit and in MFV, probes the leading coefficients of the four operators .
In a generic nonMFV framework, the channeldependent shifts to could be appreciable, so that would depend on the channels used to extract . Therefore, comparing the values of and (or their ratios) extracted from different channels gives us a handle on breaking structures beyond MFV. We will discuss this in a separate publication, where we will analyze the new physics contributions to the ratios , , , and from both inclusive and exclusive channels. In summary, we organize our analysis in two somewhat orthogonal parts, as follows:

In the rest of this work we focus on the phenomenology of and its relation to other precision measurements. This analysis applies to models of TeV scale physics with approximate invariance, in which flavor breaking is suppressed by a symmetry principle (as in MFV) or by the hierarchy

In a subsequent publication we will explore in detail the constraints arising by comparing the values of () extracted from different channels. These constraints probe the breaking structures, to which other precision measurements (especially at high energy) are essentially insensitive.
Classification  Standard Notation  Measurement  Reference 
Atomic parity  Weak charge in Cs  Wood et al. (1997)  
violation ()  Weak charge in Tl  Vetter et al. (1995)  
DIS  nucleon scattering from NuTeV  Zeller et al. (2002)  
nucleon scattering from CDHS and CHARM  Blondel et al. (1990); Allaby et al. (1986)  
nucleon scattering from CCFR  McFarland et al. (1998)  
 scattering from CHARM II  Vilain et al. (1994)  
Zline  Total width  Coll. (2003, 2006)  
(lepton and  hadronic cross section at pole  Coll. (2003, 2006)  
light quark)  Ratios of lepton decay rates  Coll. (2003, 2006)  
Forwardbackward lepton asymmetries  Coll. (2003, 2006)  
pol  Polarized lepton asymmetries  Coll. (2003, 2006)  
bc  Ratios of hadronic decay rates  Coll. (2003, 2006)  
(heavy quark)  Forwardbackward hadronic asymmetries  Coll. (2003, 2006)  
Polarized hadronic asymmetries  Coll. (2003, 2006)  
LEPII Fermion  Total cross sections for  Coll. (2003, 2006)  
production  Forwardbackward asymmetries for  Coll. (2003, 2006)  
eOPAL  Differential cross section for  Abbiendi et al. (2004)  
WL3  Differential cross section for  Achard et al. (2004)  
MW  W mass  Coll. (2003, 2006); Abazov et al. (2004)  
Hadronic charge asymmetry  Coll. (2003, 2006) 