SU(2)W × U(1)Y≡ Csp (2)

should be realized nonlinearly. This means thatthe symmetry (2) is not manifest in the low-energy spectrum but it constrains interaction ver-tices. The effective Lagrangian should exhibit thesymmetry Snat and should be organized in a sys-tematic low-energy expansion in powers of mo-menta

Leff =∑d≥2

Ld (3)

where Ld = O(pd) in the low-energy limit.( Ingeneral, the chiral dimension d �= D . A conciseΛD−4

OD (1)

where D denotes the mass (ultraviolet) dimensionof the operator OD. Using the UV dimension Das the sole indication of the degree of suppressionof a non standard operator is related to the em-phasize put on renormalizability. However, it isnot very practical: At the NLO order, there are80 independent gauge invariant D=6 operators [6]and a finer classification of new operators mightbe needed .

1.2. A not quite decoupling alternativeEven if below the scale Λ the heavy particles

decouple, their interaction may be such that somehigh energy symmetries Snat beyond SU(2)W ×U(1)Y survive at low energies and constrain theLEET. Since below the scale Λ the set of effec-tive fields - typically the SM fields - is too smallto span a linear representation of Snat, the extrasymmetry relevant at low energy

Snat

Chiral Dynamics beyond the Standard Model

Jan Sterna

aTheory Group, IPN-Orsay, Universite de Paris Sud XI, 91406 ORSAY, France

The SM Lagrangian whithout physical scalars is rewritten as the LO of a Low-Energy Effective Theory invariantunder a higher non linear symmetry Snat ⊃ SU(2)W × U(1)Y . Soft breaking of Snat defines a hierarchy of nonstandard effects dominated by universal couplings of right handed quarks to W. The interface of correspondingEW tests with non perturbative QCD aspects is briefly discussed.

1. ELECTROWEAK LOW-ENERGY EF-FECTIVE THEORY

A systematic search of Physics beyond theStandard Model is usually formulated in theframework of a Low-Energy Effective theory(LEET). One expects that at very high ener-gies E > Λ there exist new (gauge) particlesand that their interaction is governed by new (lo-cal) symmetries not contained in the SM gaugegroup SU(2)W × U(1)Y . In order to specify theLEET which would describe the Physics belowthe scale Λ in a systematic low-energy expansion,one has to identify the characteristic propertywhich makes from the SM the unique and pre-cise effective description of low-energy phenom-ena. The most popular approach is provided bythe

1.1. Decoupling ScenarioOne assumes that below the scale Λ, it is possi-

ble to ignore (integrate out) the heavy states andforget the corresponding (gauge) symmetries be-yond SU(2)W × U(1)Y , i.e. to remain with thedegrees of freedom and symmetries of the SM.The latter is then identified as the most generalSU(2)W × U(1)Y invariant Lagrangian that canbe constructed out of the SM fields and is renor-malizable order by order in powers of cou-pling constants. The effective Lagrangian thenreads

Leff = LSM +∑D>4

1

Nuclear Physics B (Proc. Suppl.) 174 (2007) 109–114

0920-5632/$ – see front matter © 2007 Published by Elsevier B.V.

www.elsevierphysics.com

doi:10.1016/j.nuclphysbps.2007.08.101

review of infrared power counting can be foundin [3] , [5].) The resulting non decoupling LEETis renormalized and unitarized order by or-der in the momentum expansion (3), follow-ing and extending the example of Chiral Pertur-bation Theory (ChPT), [1] [2]. Under these cir-cumstances there is no particular reason to startthe LE expansion around a renormalizable limit:In its minimal version, the LEET may containall the observed particles of the SM without thephysical Higgs scalar, unless such a light scalaris found experimentally. On the other hand, thepresence of three GB fields Σ ∈ SU(2) is cru-cial to generate the masses of W and Z by thestandard Higgs mechanism. The standard gaugeboson mass term coincides with the Goldstoneboson kinetic term

Lmass =14F 2

W Tr(DμΣ†DμΣ). (4)

(This becomes manifest in the physical gaugeΣ = 1). An important difference between thedecoupling and non decoupling LEET concernsthe scale Λ above which the effective descriptionbreaks down. Whereas in the decoupling case Λis essentially independent of the low-energy dy-namics itself ( the latter is renormalizable) andit cannot be estimated apriori, the consistency ofthe loop expansion in the decoupling case requires

Λ ≈ ΛW = 4πFW ≈ 3TeV. (5)

1.3. What is Snat?The experimental fact that at the leading order

of the low-energy expansion (3) one meets thebare SM interaction vertices should now followfrom the higher symmetry

Snat ⊃ SEW ≡ SU(2)W × U(1)Y (6)

and not from the requirement of renormalizabil-ity as in eq (1) above. Indeed, as shown in ref[4] , given the set of SM fields observed at lowenergy and their standard transformation prop-erties under SEW , one can contruct several in-variants of SEW carying the leading infrared di-mension d = 2 that are not present in the SM La-grangian. Such operators are not observed and itis a primary role of the symmetry Snat to suppressthem. This presumes that Snat is a non linearly

realized (hidden) symmetry of the SM itself andit should be possible to infer it from the knownSM interaction vertices. This turns out to be thecase [4] , [5], provided one sticks to the part L′

SM

of the SM Lagrangian that does not contain thephysical Higgs scalar nor the Yukawa couplingsto fermions. Proceeding by trial and error onecan show that the condition L2 = L′

SM admits aunique minimal solution

Snat = [SU(2) × SU(2)]2 × U(1)B−L. (7)

This provides a guide of constructing the LEETbelow the scale ΛW and it might indicate whichnew heavy particles could be expected well abovethis scale.

1.4. SpurionsIt is conceivable that at ultrahigh energies the

symmetry Snat is linearly realised via 4 SU(2)and one U(1) gauge fields with 9 of them acquir-ing a mass > ΛW . As the energy decreases belowΛW , this linearly realized symmetry is reducedending up with SEW = SU(2)W × U(1)Y spanedby the electroweak gauge bosons. This reduc-tion may be viewed as a pairwise identificationof two independent SU(2) factors in (7) up to agauge transformation Ω(x). Denoting the cor-responding SU(2)I × SU(2)II connections by AI

μ

and AIIμ respectively, the identification amounts

to the constraint

AIμ = ΩAII

μ Ω−1 + iΩ∂μΩ−1 (8)

This constraint is covariant under the originalsymmetry SU(2)I × SU(2)II provided the gaugetransformation Ω(x) is promoted to a field trans-forming with respect to the SU(2)I × SU(2)II

group as the bifundamental representation

Ω(x) → VI(x)Ω(x)VII (x)−1. (9)

The field Ω(x) ( more precisely its multiple) is aremnant of the reduction procedure and we call itspurion: It does not propagate since its covari-ant derivatives vanishes DμΩ(x) = 0 by virtue ofeq.(8). There exists a gauge in which each spu-rion reduces to a constant multiple of the unitematrix. After the reduction of the original sym-metry Snat to SEW , one remains with the 4 SM

J. Stern / Nuclear Physics B (Proc. Suppl.) 174 (2007) 109–114110

gauge fields and with 3 SU(2) valued scalar spu-rions populating the coset space

Csp =Snat

SEW= [SU(2)]3 (10)

and transforming as bifundamental representa-tions of the symmetry group Snat , as illustratedby the above example,c.f.(9). We refer the readerto [5], where more details of this construction,quantum number matching and unicity are givenand discussed.

1.5. SummaryBefore the symmetry Snat is reduced via spuri-

ons, the content of the LEET consists of two dis-connected sectors: First the elementary sectorgoverned by the gauge group [SU(2)L×SU(2)R×U(1)B−L]el that acts on a set of (elementary)chiral fermion doublets transforming as [1/2,0;B-L] and [0,1/2;B-L] respectively, (similarly as inLR symmetric models [7]) . Second, the com-posite sector containing three GBs Σ ∈ SU(2)arising from the spontaneous breakdown of the“composite” symmetry [SU(2)L ×SU(2)R]c sim-ilarly to what happens in QCD. The covariantconstraints reducing Snat → SEW first identify(up to a gauge) the elementary and the compos-ite SU(2)L. This gives rise to a single SU(2)W

, the one of the SM , and to a spurion X trans-forming as [1/2,1/2] with respect to the composite× elementary SU(2)L. Analogously, in the righthanded sector, one identifies the two SU(2)R

groups (spurion Y) and finally, the resulting rightisospin is identified with U(1)B−L (embeded intoSU(2)) (spurion Z). What remains is U(1)Y

where the hypercharge Y/2 = I3R + (B − L)/2 as

required by the SM. Notice that the spurion Y ad-mits a gauge invariant decomposition Y = Yu+Yd

making appear projectors of the up and the downcomponents of right handed doublets.

Hence, the structure of the SM vertices leads tothree SU(2) valued spurions which in the physicalgauge reduce to three parameters X → ξ , Y → ηand Z → ζ. They are external to the SM andthey parametrize the small explicit breaking ofthe symmetry Snat. The fact that the spurion Znecessarily carries a non zero value of B−L meansthat the lepton number violation is an unavoid-able byproduct of the above construction [5]: Its

strength is tuned by the parameter ζ � ξ , η � 1and it cannot be anticipated within the LEETalone.

2. LEADING EFFECTS BEYOND THESTANDARD MODEL

The whole construction of Leff can now be de-viced in three steps:

• One collects all invariants under Snat madeup from the corresponding 13 gauge fields,GB field Σ, chiral fermions and the threespurions .

• One orders them according to the increasingchiral dimension d and to the number ofspurion insertions.

• One imposes the invariant constraints be-tween gauge fields and spurions eliminatingthe redundant degrees of freedom (c.f. (8)).

By construction, the leading term with d = 2 andno spurion insertion coincides with the higgslessvertices of the SM with massive W and Z andmassless fermions. The genuine effects beyondSM are identified with terms explicitly containingspurions. The latter naturally appear in a hier-archical order given by the power of the (small)spurionic parameters ξ , η (we can consistentlydisregard the tiny spurion ζ and LNV processes.)Despite the present unability of the LEET for-malism to anticipate the actual size of individualnon standard effects, we might well be able to or-der these effects i.e.,to predict their relativeimportance.

2.1. Fermion Masses and power countingSpurions are needed to construct a Snat invari-

ant fermion mass term: At leading order suchterms are proportional to the operators

Lfm = ΨLX †ΣYaΨR (11)

where a = u , d. The chiral dimension of these op-erators is d = 1, (1/2 for each fermion field) andthey are furthermore suppressed by the spurionicfactor ξη. The mass term of the heaviest fermion(the top quark) with the chirally protected mass

J. Stern / Nuclear Physics B (Proc. Suppl.) 174 (2007) 109–114 111

should count at low energies as O(p2). This sug-gests the following counting rule for the spurionparameters

ξη ∼ mtop/ΛW ∼ O(p) (12)

attributing to both ξ and η the chiral dimensiond = 1/2. This leaves space for the existence ofmuch lighter fermions with the mass term con-taining additional powers of ξ and η.

On the other hand, the understanding of thesmallness of neutrino masses within thepresent LEET framework represents an alterna-tive to the see-saw mechanism. The Majoranamass term necessarily involves the ΔL = 2 spu-rion Z which brings in a new (tiny) scale relatedto LNV. Yet, one has to find the symmetry thatsuppresses the neutrino Dirac mass of the type(11). In [4] it has been proposed to associatethis latter suppression with the discrete reflectionsymmetry

νiR → −νi

R. (13)

This is possible since at the leading order the righthanded neutrino does not carry any gauge chargeand the symmetry (13) does not prevent the righthanded neutrino to develope its own (small) Ma-jorana mass. This mechanism of suppression ofneutrino masses has yet another consequence. Itforbids the charged right-handed leptoniccurrents despite the fact that νR remains light.This corollary will take its importance later.

2.2. The full NLOSofar, all terms had the total chiral dimension

d = 2 , including (in the case of the mass term)the spurion factor according to the counting rule(12). This is characteristic of the LO. There isno d = 3 term not containing spurions. On theother hand there are two and only two opera-tors quadratic in spurions with the total chiraldimension d = 3. They represent non standard(i.e. containing spurions) Snat invariant couplingsof fermions to gauge bosons. In the case of lefthanded fermions the unique such operator reads

OL = ΨLX †ΣγμDμΣ†XΨL, (14)

whereas in the right handed sector the corre-sponding operator has four components (a, b =

u, d) that are separately invariant under Snat

Oa,bR = ΨRY†

aΣ†γμDμΣYbΨR. (15)

Both operators are suppressed by a quadraticspurion factor: In view of the rule (12), theycount as O(p2ξ2) ∼ O(p3) and O(p2η2) ∼ O(p3),repectively. It is remarkable that the operators(14) and (15) cannot be generated by loops. Thisfollows from the Weinbergs power counting for-mula

d = 2 + 2L +∑

v

(dv − 2) (16)

originally established in the case of ChPT [1] andsubsequently extended to the case of LEET con-taining massive vector particles, chiral fermions[3] as well as spurions [4]. Eq (16) represents thechiral dimension of a connected Feynman diagramcontaining L loops and the vertices v of a chiraldimension dv ≥ 2. In the low-energy expansionthe tree diagrams with a single insertion of a ver-tex (14) or (15) are more important than any loopcontribution or than higher order purely bosonictrees. Accordingly, the NLO operators (14) and(15) are universal in the flavour space. Loops,oblique corrections , flavour dependence (and re-lated FCNC), only come at the NNLO. The twooperators (14) and (15) thus describe the poten-tially most important effects beyond the SM, pre-dicted in the present framework of a non decou-pling LEET. Unfortunately, it is not straightfor-ward to translate this qualitative prediction intoa more quantitative statement (see [8], [11]). Theoperators (14) and (15) both contain couplings offermions to W and to Z. The latter involves toomany apriori unknown LECs, especially in theright handed sector (15). For this reason, we firstconsider

2.3. Couplings of quarks to W at NLOIn the physical gauge and in a flavour basis in

which both the mass matrix of u and d quarks arediagonal, the LO + NLO of a universal chargedcurrent coupled to W as defined by eqs (14) and(15) reads

LCC = g[lμ+

12U

(Veffγμ+Aeffγμγ5

)D

]Wμ+hc,(17)

J. Stern / Nuclear Physics B (Proc. Suppl.) 174 (2007) 109–114112

where lμ stands for the standard leptonic chargedV − A current not affected at NLO. The ma-trix notation is used in the flavour (family) space:UT = (u, c, t) , DT = (d, s, b), whereas Veff andAeff are complex 3 × 3 effective EW couplingmatrices. At NLO they take the form

V ijeff = (1 + δ)V ij

L + εV ijR (18)

Aijeff = −(1 + δ)V ij

L + εV ijR , (19)

where VL = LuL†d and VR = RuR†

d with Lu,Ru

and Ld ,Rd denoting the pairs of unitary ma-trices that diagonalize the masses of the u-typeand d-type quarks respectively. The parametersδ = O(ξ2) and ε = O(η2) originate from the spu-rions. Their magnitude is estimated to be of atmost a fraction of per cent[5]. Hence at NLO, theLEET predicts two major non standard effectsconcerning the coupling of quarks to W:

i) The existence of direct couplings of righthanded quarks to W ( keeping in mind that thediscrete symmetry (15) forbids similar couplingsfor leptons.)

ii) The chiral generalization of CKM unitarityand mixing.

At the LO one has δ = ε = 0 and one recoversthe CKM unitarity of the SM: Veff = −Aeff =VCKM . At the NLO, the two distinct matrices VL

and VR are both unitary (for the same reason as inthe SM) but the effective vector and axial-vectormatrices V and A which are accessible in semi-leptonic transistions are not unitary anymore.

3. INTERFACE OF THE EW AND QCDEFFECTIVE COUPLINGS

A measurement of effective EW couplings V ijeff

and Aijeff requires an independent knowledge of

the involved non perturbative QCD parameterslike the decay constants Fπ ,FK , FD, FB or thetransition form factors such as fK0π−

+ (0) . . .. Thelongstanding problem of an accurate extractionof the CKM matrix element V us and the re-lated test of the “CKM unitarity” illustrates thispoint and it becomes even more accute in thepresence of non standard EW couplings, such asRHCs. The unfortunate circumstance is that the

most accurate experimental information on QCDquantities mentioned above ,in turn come fromsemi leptonic transitions of the type P → lν andP ′ → Plν where P = π, K, D, B and, conse-quently, the result of their measurement dependson (apriori uknown) EW couplings (18) , (19) .Finding an exit from this circular trap is a ma-jor task of phenomenological Flavor Physics.

3.1. Non Standard EW Parameters in theLight Quark Sector

Let us first concentrate on light quarks u, d,and s. For them the SM loop effects simulat-ing RHCs are strongly suppressed by at least twopowers of mass.

Since V ubL is negligible and

Vudeff = 0.97377(26) ≡ cosθ (20)

is very precisely known [9] from nuclear 0+ → 0+

transitions , the light quark effective couplingsVua

eff , Auaeff , a = d, s can be expressed in terms

of three non standard effective EW parameters:the spurion parameter δ defined in (20),(21) andtwo RHCs parameters εNS and εS defined as

εV udR = εNScosθ, εV us

R = εSsinθ, (21)

where,upon neglecting V ubL , we have denoted

V udL = cosθ = cosθ(1 − δ − εNS), V us

L = sinθ (22)

In the limit of SM all spurion NS parameters van-ish , δ = εNS = εS = 0 and all light quark EWeffective couplings are fixed by the experimentalvalue of Vud

eff .

3.2. Fπ ,FK ,f+(0) and Vuseff

Here Fπ , FK and fK0

+ (0) stand for radiativelycorrected genuine QCD quantities defined asresidues of GB poles in two-point and three-pointfunctions of axial and vector currents. These arethe quantities that are subject to ChPT and/orlattice studies. It is further understood thatall isospin breaking effects due to md − mu areincluded. From the experimentally well knownbranching ratios [10] K(l2(γ))/π(l2(γ)) one caninfer AusFK/AudFπ and from the rate of K0

l3 wecan extract the value of |fK0

+ (0)Vus|. Assum-ing the Standard Model couplings of quarks to

J. Stern / Nuclear Physics B (Proc. Suppl.) 174 (2007) 109–114 113

W this is sufficient to extract very accurate val-ues that the corresponding QCD quantities wouldtake in a world with vanishing spurion parametersδ, εNS, εS . The latter are denoted by a hat. Theyread

Fπ = (92.4±0.2)MeV, FK/Fπ = 1.182±0.007,(23)

fK0

+ (0) = 0.951 ± 0.005 (24)

Here the errors merely reflect the experimentaluncertainties in the measured branching ratios.In the presence of NS couplings of quarks to Wthe values of genuine QCD quantities extractedfrom semileptonic BRs are modified. Neglectinghigher powers of spurion parameters δ and ε onegets using Eqs (21) and (22)

F 2π = F 2

π (1 + 4εNS) (25)

(FK

Fπ

)2

=

(FK

Fπ

)21 + 2(εS − εNS)

1 + 2

sin2 θ(δ + εNS)

(26)

[fK0π−+ (0)

]2

=[fK0π−+ (0)

]2 1 − 2(εS − εNS)1 + 2

sin2 θ(δ + εNS)

(27)

The 3 NLO EW parameters δ , εNS and εS

can be constrained using independent informa-tions on QCD quantities Fπ , FK or fK0π

+ . Suchinformation can originate from lattice simulationsor from ChPT based measurements. As a matterof example let us mention the possible determi-nation of Fπ from the π0 → 2γ partial width orfrom precision ππ scattering experiments, whichare independent from the standard determinationbased on the πl2 decay rate. Despite a presentlack of accuracy, this way may eventually providea measurement of εNS through Eqs (23) and (25).

The non standard EW parameters allow to in-fer the NLO deviation from the unitarity of theeffective mixing matrix Veff , which is relevant inthe description of Kl3 decays. One finds[Vud

eff

]2+

[Vuseff

]2 = 1+2(δ+εNScos2θ+εSsin2θ)(28)

Due to the presence of RHCs , flavour mixingeffects in FK/Fπ and in fK0π

+ are no more relatedas in the SM.

3.3. Enhancement of εS ?As already pointed out the genuine spurion pa-

rameters δ and ε can hardly exceed 0.01. Onthe other hand , the unitarity of the right-handedmixing matrix VR implies

|εNS |2cos2θ + |εS |2sin2θ ≤ ε2. (29)

Since we live close to the left-handed word, onehas sinθ ∼ 0.22, reflecting the well known hier-archy of left handed flavour mixing. This in turnimplies that |εNS | < ε remains tiny. On the otherhand , |εS | ≤ 4.5ε and it can indeed be enhancedto a few percent level , provided the hierar-chy in right-handed flavour mixing is in-verted ,i.e. V ud

R � V usR . In [8] it has been shown

that a stringent test involving the EW couplingεS − εNS can be deviced in KL

μ3 decay, (see [11]).On the other hand, it seems rather difficult tofind another clean manifestation of RHCs drivenby εS and the reamining 2 NLO EW constantsare presumably too small to be reliably detected.This remains true even if the preceeding discus-sion is extended to processes involving the shortdistance rather than chiral QCD dynamics.

REFERENCES

1. S. Weinberg, Physica A96 , 327 , (1979).2. J. Gasser and H. Leutwyler, Nucl.Phys. B250,

465 (1985).3. J. Wudka, Int. J. Mod. Phys. A9 , 2301,

(1994).4. J. Hirn and J. Stern, Eur. Phys. J. C 34 (2004)

,447.5. J. Hirn and J. Stern, Phys. Rev. D73, (2006)

, 056001.6. W. Buchmueller and D. Wyler , Nucl. Phys.

B268,(1986), 621.7. R. N. Mohapatra , J.C. Pati, Phys. Rev. D11,

(1975), 2558; G. Senjanovic ,R. N. Mohapa-tra, Phys. Rev. D12 (19750, 1502.

8. V. Bernard, M. Oertel, E. Passemar and J.Stern, Phys. Lett B638, (2006) ,480.

9. W. J. Marciano and A. Sirlin, Phys. Rev.Lett. 96, (2006),032002.

10. F. Ambrosino et al (KLOE), Phys. Lett.B632, (2006), 43

11. M. Oertel , these proceedings.

J. Stern / Nuclear Physics B (Proc. Suppl.) 174 (2007) 109–114114