Did one observe couplings of right-handed quarks to W ?
aa r X i v : . [ h e p - ph ] S e p Did one observe couplings of right-handed quarksto W ?
Jan STERN ∗ Institut de Physique Nucléaire, Orsay, FranceE-mail: [email protected]
I consider non standard EW couplings of light right-handed quarks to W and Z suggested in asystematic non decoupling bottom-up low-energy effective theory approach to possible extensionsof the Standard Model. New experimental tests in K L m decays based on recent measurements andscalar form factor analysis are discussed. A successful NLO fit to the standard set of Z-pole andother NC data is presented as well. KAON International ConferenceMay 21-25 2007Laboratori Nazionali di Frascati dell’INFN, Rome, Italy ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ ight-handed currents
Jan STERN
1. Introduction
In the Standard Model (SM), right handed fermions do not couple to W and their couplings toZ are proportional to the electric charge. Compelling tests of this feature exist for leptons, whereasfor quarks available tests are less conclusive due to the interference with non perturbative QCDeffects. Another characteristics of the right-handed sector of the SM is a rather complicated andapriori unexplained spectrum of weak hypercharges. (Since the seventies, the latter has motivatedleft-right symmetric extensions of the SM [1] which shed a new light on the EW couplings of righthanded fermions.) None of the above features of the SM follow from the EW symmetry S EW = SU ( ) W × U ( ) Y , as long as the latter is spontaneously broken: Indeed, with the help of agentsof Symmetry Breaking (Higgs fields), it is possible to construct S EW invariant couplings of righthanded fermions to W. This fact suggests to look for eventual modifications of the right-handedcouplings as a conceivable signal of a non standard EW symmetry breaking. Model independenttests of EWSB require first of all a “bottom-up” Effective Theory approach which starts fromthe known vertices of the SM and step by step in a low-energy expansion controlled by a powercounting orders possible non standard effects according to their importance at low energies. Next,it should be specified how the lepton - quark universality could be naturally broken at subleadingorders to escape strong experimental constraints concerning leptons.Such a class of LEETs has been proposed three years ago [2] and further developed and com-pleted later [3]. In this talk, I am going to review the characteristic feature of this class: Theappearance at NLO of couplings of right handed quarks to W and modification of their couplingsto Z. Then I will comment on first attempts to confront these predictions with experiment [4].
2. Not quite Decoupling EW Low Energy Effective Theory (LEET)
In its minimal version, the LEET contains the naturally light particles of the SM: SU(2) x U(1)gauge fields, chiral fermions (including right-handed neutrinos) and the triplet of GBs. For smallmomenta p ≪ p F W = L W ∼ TeV , the effective Lagrangian is written as a low-energy expansion L eff = (cid:229) d ≥ L d , L d = O ([ p / L W ] d ) , (2.1)where the infrared dimension of a local operator, d = n d + n g + n f / p d ), where d = + L + (cid:229) v ( d v − ) . (2.2)The LEET is renormalizable order by order in the LE expansion, provided at each order, all termsallowed by symmetries are effectively included in (2.1). In particular, the symmetry of the LEET S nat ⊃ S EW must prevent all “unwanted “ non standard vertices to appear already at the leading order O ( p ) . In a bottom-up approach, the higher symmetry S nat is unknown apriori (it is the remnantof the not quite decoupled high energy sector of the theory), but it can be inferred requiring that2 ight-handed currents Jan STERN the leading order L of the LEET coincides with the Higgs-free part of the SM Lagrangian. I referto [3], where it is shown that the minimal solution of this condition reads S nat = (cid:2) SU ( ) G L × SU ( ) G R × U ( ) B − LG B (cid:3) elem × [ SU ( ) G L × SU ( ) G R ] comp . (2.3)The Goldstone boson matrix S ( x ) ∈ SU ( ) (needed to give masses to W and Z) transforms accord-ing to a different local chiral symmetry S ( x ) → G L ( x ) S ( x ) [ G R ( x )] − (2.4)than the chiral fermion doublets and the elementary gauge fields coupled to fermions y L / R → G L / R exp (cid:20) − i B − L a (cid:21) y L / R . (2.5)The most general Lagrangian of dimension d = S nat reads L (cid:0) p (cid:1) = F W (cid:10) D m S † D m S (cid:11) + i y L g m D m y L + i y R g m D m y R − (cid:10) G L mn G mn L + G R mn G mn R (cid:11) − G B mn G mn B . (2.6)It contains several gauge fields not observed at low energies E < L W , no fermion masses and agauge boson mass term which has no obvious connection with the SM. Nevertheless, the aboveLagrangian reduces to the one of the SM upon imposing S nat - invariant constraints eliminatingthe redundant gauge fields through pairwise identification of different gauge factors up to a gaugetransformation. (Notice that these constraints break the accidental L-R symmetry present in (2.6).)Example of such constraints is G L , m = X g L G L , m X − + i X ¶ m X − (2.7)which replaces SU ( ) G L × SU ( ) G L by its diagonal subgroup (identified with the SM weak isospin)and a scalar object X which is a (constant) multiple of a SU ( ) matrix, and is called “spurion”.Similarly, one identifies up to a gauge G R , m ∼ g R G R , m ∼ g B G B , m t /
2. We then remain with thegauge fields of the SM, receiving standard masses and mixing through the first term in (2.6) andcoupled in the standard way to fermions. In addition, we now have three SU ( ) valued spurions X , Y and w X ( x ) = x W L ( x ) , W L ( x ) ∈ SU ( ) , Y = h W R , W R ∈ SU ( ) , w = z W B , W B ∈ SU ( ) , (2.8)populating the coset space S nat / S EW = SU ( ) . To maintain invariance under S nat , the spurionshave to transform as X → G L X G − L , Y → G R Y G − R , w → G R w G − B . (2.9)Consequently, the constraints selecting S EW = SU ( ) W × U ( ) Y of the SM as the maximal subgroupof S nat that is linearly realized at low energies can be equivalently written as3 ight-handed currents Jan STERN D m X = , D m Y = , D m w = x , h and z which are exterior to the SM and whose magnitude is not fixed bythe LEET. They will be considered as small expansion parameters describing effects beyond theSM. The physical origin of spurions satisfying the constraints (2.10) can be understood as resultingfrom a particular non decoupling limit of an ordinary Higgs mechanism in which both Higgs bosonsand some combinations of gauge fields become very massive. Massive gauge fields decouple,whereas heavy Higgs fields reduce to non propagating spurions, defining a non linear realization ofthe symmetry S nat / S EW .Spurions are needed to write down S nat invariant fermion masses. Consequently, the latter willbe suppressed with respect to the scale L W by powers of spurion parameters x and h . The leastsuppressed mass - the top mass - will be proportional to the product xh ∼ m top / L W = O ( p ) , d ∗ = d + ( n x + n h ) . (2.11)This suggests to extend the low-energy power counting to spurions introducing the chiral dimension d ⋆ defined above. This guarantees that both the fermion mass term and the lagrangian (2.6) have d ⋆ = d by d ⋆ .The third spurion w breaks B-L, which is thus predicted to be a part of the LEET. Conse-quently, the parameter z ≪ x ∼ h naturally accommodates the smallness of Lepton number viola-tion and of the Majorana masses.
3. Next to Leading Order (NLO)
The NLO consists of all S nat invariant operators of the chiral dimension d ⋆ = X or Y : O L = ¯ y L X † g m S D m S † X y L , (3.1)for left handed fermions, whereas for right handed fermions one has O a , bR = ¯ y R Y † a g m S † D m S Y b y R . (3.2)where a , b ∈ [ U , D ] , label covariant projections on Up and Down components of right handed dou-blets. These operators already carry their respective suppression factors, they are O ( p x ) and O ( p h ) respectively. The full d ⋆ = L NLO = r L O L ( l ) + l L O L ( q ) + (cid:229) a , b r a , bR O a , bR ( l ) + (cid:229) a , b l a , bR O a , bR ( q ) (3.3)where r and l are dimensionless low-energy constants which should be of order one (unless sup-pressed by an additional symmetry). The NLO couplings still respect the family symmetry . On the4 ight-handed currents Jan STERN other hand, at this subleading order, the lepton - quark universality could be broken, i.e. r = l bythe existence of additional reflection symmetry n R → − n R which does not exist for quarks. Sucha symmetry is not obstructed by the LO couplings to gauge fields (at LO, n R decouples). It allowsthe right handed neutrino to get a small Majorana mass of the order O ( z h ) , i.e. of a compa-rable size to left handed Majorana mass O ( z x ) and to the strength of LNV. On the other hand,the reflection symmetry n R → − n R forbids the Dirac neutrino-mass and could provide a naturalexplanation of the observed smallness of neutrino masses. A corollary of this “anti see-saw” mech-anism [3] of suppression of neutrino masses is the suppression of charged leptonic right-handedcurrents i.e. r UDR =
4. Couplings to W
Let us concentrate on couplings of fermions to W. Using the matrix notation in the familyspace U = ( u , c , t ) T , D = ( d , s , b ) T , N = ( n e , n m , n t ) T , L = ( e , m , t ) T and using the mass - diagonalbasis, the couplings to W up to and including NLO become L W = e ( − x r L ) √ s (cid:8) ¯N L V MNS g m L L + ( + d ) ¯U L V L g m D L + e ¯U R V R g m D R (cid:9) W + m + h.c . (4.1) V L and V R are two independent unitary mixing matrices resulting (as in SM) from the diagonaliza-tion of quark masses. The (small) spurionic parameters d = ( r L − l L ) x and e = l UDR h describethe chiral generalization of the CKM mixing induced by RHCs. Notice, in particular, that effectiveEW couplings in the vector and axial channels (more directly accessible than V L and V R ) V i jeff = ( + d ) V i jL + e V i jR + NNLO , A i jeff = − ( + d ) V i jL + e V i jR + NNLO , (4.2)need not to be unitary. The signal of RHCs can be detected as V i jeff = − A i jeff , i.e. comparing vectorand axial vector transitions.A particular attention should be payed to light quarks u , d , s for which the chirality breakingeffects are tiny. In this sector all EW effective couplings can be expressed in terms of d and threeparameters e ns = e Re (cid:16) V udR V udL (cid:17) , e s = e Re (cid:16) V usR V usL (cid:17) , V udeff = . ( ) ≡ cos ˆ q (4.3)where V udeff is determined from 0 + → + nuclear transitions [5]. Using further the unitarity of V L and neglecting | V ubL | , all light quark effective couplings can be expressed as | V udeff | = cos ˆ q | A udeff | = cos ˆ q ( − e ns ) | V useff | = sin ˆ q ( + d + e ns sin ˆ q )( + e s − e ns ) | A useff | = sin ˆ q ( + d + e ns sin ˆ q )( − e s − e ns ) . (4.4)5 ight-handed currents Jan STERN
The genuine spurion parameters d and e are expected to be at most of order few percent. Since | V usL | ≪ | V udL | ∼ V R is unitary, one should have | e NS | < e . On the other hand, theparameter e S measuring RHCs strangeness changing transitions can be enhanced if the mixinghierarchy for right handed light quarks is inverted, V udR < V usR . In this case | e S | could be as large as4 . e . Clearly, this question should be decided experimentally.
5. The stringent test of RHCs: Scalar K m form factor shape Model independent bounds on V + A couplings of light quarks to W are extremely difficult tofind, since they require an accurate control of QCD chiral symmetry breaking contributions whencomparing hadronic matrix elements of vector and axial vector currents. One such test (neverconsidered before) has been identified in Ref [7]. It is based on the Callan Treiman low - energyTheorem already discussed in the talk by E. Passemar [6]. The (normalized) scalar K L m form factor f ( t ) f ( t ) = f K p − S ( t ) f K p − + ( ) = f K p − + ( ) (cid:18) f K p − + ( t ) + t D K p f K p − − ( t ) (cid:19) , f ( ) = . (5.1)where D K p = m K − m p , satisfies C ≡ f ( D K p ) = F K + F p + f K p − + ( ) + D CT , C = B exp r + D CT . (5.2)Here, D CT = − . × − is a tiny correction which has been estimated in one loop ChPT. Inthe absence of RHCs, the value C of the scalar form factor at the Callan Treiman point can bedirectly expressed in terms of measured branching fractions ( K l / p l , K l ) and V ud giving [5] B exp = . ± . rr = (cid:12)(cid:12)(cid:12) A udeff V useff V udeff A useff (cid:12)(cid:12)(cid:12) = + ( e S − e NS ) . (5.3)Hence, in the presence of RHCs the Callan Treiman theorem yieldsln C = . ± . + ˜ D CT + ( e s − e ns ) = . ± . + D e , (5.4)with ˜ D CT = D CT / B exp .An accurate physically motivated parametrization of the scalar form-factor f ( t ) has been pro-posed [7] which allows to determine the parameter ln C from the measured K L m decay distributions.The corresponding measurement is particularly delicate, since the experimental t - distribution isnot easy to reconstruct from the data. Furthermore, different experiments have access to differentdecay distributions which do not have the same sensitivity to ln C and to the shape of the vectorform factor. There exists a relation between ln C and the slope parameter l [6] but it is not enoughprecise to reduce the determination of ln C to existing (controversial) determinations of the slope l assuming the linear t-dependence of the scalar form factor [8, 9, 10] or at most injecting infor-mation about its curvature [11]. Recently, NA48 collaboration has published the result of a directdetermination of ln C based on the dispersive representation of f ( t ) [10]ln C exp = . ± . , D e = − . ± .
014 (5.5)6 ight-handed currents
Jan STERN
Other analysis of K m decay distributions from KLOE [12] and KTeV [13] based on the dispersiverepresentation of the two form factors are underways. They should clarify the experimental situ-ation and provide an independent cross check of the NA48 result [10]. Awaiting an independentdispersive analysis of existing data samples, one should stress that the result (5.5) indicates a 5 s deviation from the SM prediction. In particular, if the discrepancy would have to be explainedwithin QCD, the ChPT estimate of D CT would have to be underestimated by a factor 20. On theother hand, within the class of LEET defined above the interpretation of the result (5.5) as a mani-festation of couplings of right handed quarks to W is unambiguous. It amounts to a determinationof the spurion parameter 2 ( e S − e NS ) . Its size can be understood as a result of enhancement of V usR relative to the suppressed V usL . Beyond our LEET framework, other interpretations might beconceivable. For example a subTeV charged scalar coupled to scalar densities ¯ us and ¯ mn couldinterfere with our analysis. We prefer to stay within the class of minimal LEET defined above andask how does the same non standard operator (3.2) affect the couplings of right handed quarks toZ.
6. Couplings to Z Non standard couplings to Z contained in the NLO Lagrangian (3.3) are suppressed by thesame two spurion parameters x (LHCs) and h (RHCs) as in the case of couplings to W discussedin Section 4. Hence, despite the apriori unknown “order one” prefactors r and l , it is possible torelate orders of magnitude of non standard CC and NC couplings. In the left - handed sector wehave altogether two NLO parameters: d = x ( r L − l L ) and x r L , whereas in the right - handedsector there are three new parameters denoted e e , e U , e D and proportional to the spurion h . Measurement Fit ( | O meas − O fit | ) σ meas Γ Z [GeV] 2.4952(23) 2.4943 σ had [nb] 41.540(37) 41.569 R e A lFB A l ( P τ ) 0.1465(32) 0.1485 R b R c A bFB A cFB A b A c A l (SLD) 0.1513(21) 0.1485 Br ( W → lν ) 0.1084(9) 0.10891 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) 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Figure 1:
Pull for the Z pole observables in the full fit
We have performed the NLO fit to the usual set of Z - pole pseudo-observables displayedin Fig. 1 including the lepton branching fraction of W (particularly sensitive to the parameter d ) as well as spin asymmetries measured at SLD. The fit is described in details in [4]. It has c / do f = . / d ≡ x ( r L − l L ) = − . ( ) , x r L = . ( ) and e e ≡ h r DDR = − . ( ) . 7 ight-handed currents Jan STERN
The most important NLO modification of couplings to Z turns out to occur for right handed quarks: e U ≡ h l UUR = − . ( ) e D ≡ h l DDR = − . ( ) . The correlations can be found in [4].Two comments are in order. First, the most important NLO non standard couplings to Z seemto occur for right handed quarks . Their size compares well with the couplings of right handedquarks to W as suggested by the K L m dispersive Dalitz plot analysis [10]. Next, the fit is of a verygood quality as illustrated in Fig. 1 in terms of “pulls”. In particular, the b-quark forward backwardasymmetry A bFB and R b are both well reproduced without modifying the flavour universality of NCEW couplings. The long standing “puzzle of b-asymmetries” has apparently gone thanks to themodified right-handed couplings of D type quarks to Z. F K / F p and f + ( ) The low-energy QCD quantities F K , F p , f + ( ) . . . are defined independently of EW interactionsin terms of QCD correlation functions and they are accessible to ChPT and lattice studies. On theother hand their experimental values extracted from semileptonic branching fractions depend onthe presumed EW vertices via the effective EW couplings (4.4). Fixing experimental values of V udeff (4.3) and of the semi leptonic branching ratios, F K , F p , f + ( ) . . . become unique functions ofspurion parameters e NS , e S and d . One has (cid:18) F K + F p + (cid:19) = (cid:18) ˆ F K + ˆ F p + (cid:19) + ( e s − e ns ) + ˆ q ( d + e ns ) , | f K p − + ( ) | = h ˆ f K p − + ( ) i − ( e s − e ns ) + ˆ q ( d + e ns ) , (7.1)where the hat indicates the corresponding values extracted from semi leptonic branching fractions( assuming SM couplings e NS = e S = d = F K + / ˆ F p + = . ( ) , ˆ f K p − + ( ) = . ( ) . (7.2)In fig. 2 are displayed lines of constant values of F K / F p and f + ( ) as a function of spurion pa-rameters. One notes that F K / F p significantly decreases compared with the value 1.22 often used asinput in ChPT. On the other hand, f + ( ) is not very constrained despite the Callan Treiman rela-tion. Finally, nothing prevents the effective vector mixing matrix V e f f to be nearly unitary withoutany fine tuning. One has | V udeff | + | V useff | = + ( d + e ns ) + ( e s − e ns ) sin ˆ q . (7.3)The contribution of e S , the only parameter which might be enhanced above 0.01 is suppressed by sin ˆ q .In conclusion, we have presented and motivated a new low-energy test of non standard EWcouplings of right handed quarks not considered before. Did one observe couplings of right handedquarks to W in K L m decay ? The final answer requires a more complete and dedicated experimentalanalysis. It also deserves a particular effort despite its difficulty.I thank V. Bernard, M. Oertel, and E. Passemar for a valuable collaboration. This work hasbeen supported in part by the EU contract MRTN-CT-2006-035482 (Flavianet).8 ight-handed currents Jan STERN -0.005 0 0.005-0.1 -0.05 0 d + e N S e S - e NS )F K /F p = 1.051.101.151.201.25 f +K p - (0) = .94.95.96.97.98.99 1 Figure 2:
Lines of constant values for F K + / F p + and f K p + ( ) in the plane d + e ns and ( e s − e ns ) as resultingfrom Eqs (7.1) and (7.2). The vertical band indicates the range suggested by the NA48 measurement [10].The SM point e = d = is also shown. References [1] R.N. Mohapatra and J.C. Pati,
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