Rescaling mechanism and effective symmetry from the ideal cancellation of the S parameter in custodial models
RRescaling mechanism and effective symmetry from the ideal cancellationof the S parameter in custodial models Donatello Dolce
University of Camerino, Piazza Cavour 19F, 62032, Italy.E-mail: [email protected] (Dated: October 25, 2018)
Abstract
We present a model independent analysis of custodial corrections to the S parameter. The neg-ative contributions coming from direct corrections, i.e. the corrections associated to effects of newphysics in the fermionic sector, can be used to eliminate unwanted positive oblique contributionsto S . By means of such an ideal cancellation among oblique and direct corrections the electroweakphysics can be made insensitive to custodial new physics. The Lagrangian analysis of the ideal can-cellation reveals a possible sizable redefinition of the effective electroweak energy scale with respectto models with only oblique corrections. We then infer from general principles the full expressionfor the effective custodial operator responsible in the gauge sector for the contribution to S . Weshow that the ideal cancellation exactly eliminates all the custodial corrections, including those inthe non-abelian and longitudinal terms. Indeed, provided redefinitions of the physical parameters,the standard model Lagrangian can be equivalently defined modulo custodial corrections in thegauge and fermionic sectors. Thus the ideal cancellation can be regarded as an effective symmetry.Finally we investigate the theoretical origin of this effective symmetry in terms of the effective gaugestructure induced by custodial new physics. a r X i v : . [ h e p - ph ] A p r NTRODUCTION
The discovery of the Standard Model (SM) Higgs-like boson at LHC with mass M H (cid:39)
126 GeV represents a fundamental milestone to clarify the nature of the ElectroWeak (EW)gauge symmetry breaking and the origin of the elementary particle masses. Remarkablythis seems to complete the SM picture. On the other hand it implies reconsiderations ofsome of the most investigated extensions of the Standard Model (SM) proposed to curethe unsatisfactory theoretical aspects of the Brout-Englert-Higgs mechanism (commonlyknown as the Higgs mechanism). In particular such a light SM Higgs boson implies fine-tunings or rescaling mechanisms to make the physics at the EW scale insensitive to possiblenew physics sectors.Effects of new physics at the Z -pole are summarized by a set of EW parameters [1].The most relevant quantities parametrizing the EW observables are T , U and S [2] or,in an alternative notation, (cid:15) , (cid:15) and (cid:15) [3]. The Peskin-Takeuchi parameters T and U can be generally made small or vanishing by requiring custodial symmetric new physics,i.e. characterized by a symmetry breaking pattern SU L (2) ⊗ SU R (2) → SU D (2), where SU D (2) is the diagonal subgroup. Custodial extensions of the SM [4–8] are protectedfrom new physics corrections to the T and U precision parameters, but they are typicallyaffected by large positive oblique corrections to the S parameter [4, 9, 10]. This createtensions with respect to the experimental bounds. The oblique contributions to the EWparameters are associated to corrections to the tree-level transverse vacuum polarizationamplitudes of the SM gauge bosons. However, if new physics is allowed in the gaugesector, corresponding new physics contributions can be assumed in the fermionic sector.In principle the contributions coming from the fermionic sector, known as vertex or direct corrections, can be as big as those in the gauge sector.In this paper we present a model independent analysis of the oblique and direct con-tributions to the S parameter by introducing, within the framework of the SM of the EWinteractions, possible dimension six custodial operators in both the gauge and fermionic2ectors [9, 11–13]. In particular we assume a SM Higgs sector. As we will show in sec.(I), direct corrections give negative contributions to the S parameter. They can be thereforefine-tuned with the oblique contributions to conciliate the new physics corrections withthe experimental bounds [14–19]. We define ideal cancellation the condition that exactlycancels each other out the oblique and direct corrections of new physics.As known from extra-dimensional or moose models, the ideal cancellation of the S parameter implies that new possible vectorial resonances are fermiophobic so that theirmasses can be lower than expected in models with only oblique corrections, [16, 20, 21]. Forinstance, in extra-dimensional theories (or moose models), the ideal delocalization meansthat the fermionic Kaluza-Klein towers in the bulk (or along a moose) are proportional tothe delocalization of the vectorial Kaluza-Klein tower, in particular to the W delocalization.Since the new vectorial resonances, such as Z (cid:48) and W (cid:48) , are orthogonal to the SM Z and W , upon redefinitions of the physical quantities the ideal cancellation implies that thecoupling of these new vectorial to the SM fermions are strongly suppressed. Similarly,these considerations can be extended to technicolor and composite models.The standard analysis of oblique and direct corrections is based on the evaluation ofthe so-called ∆ parameters. This method must be however carefully used. Being basedon the ratio of the vector boson masses M W /M Z it does not reveal possible redefinitionof the effective EW scale, i.e. of the Fermi coupling. Indeed the direct contributions canlead to redefinitions of effective vector boson masses, Yukawa couplings and parameters inthe Higgs sector, with respect to the case in which only oblique corrections are assumed.In sec.(III), to see these rescalings and the exact condition which relates the oblique anddirect corrections in the ideal cancellation, we will perform the explicit Lagrangian analy-sis through the redefinition of the SM parameters. Both the extended parameters regionallowed by the ideal cancellation and the redefinition of the EW scale have important conse-quences in model building: physics at the Z -pole can be made insensitive to custodial newphysics sectors and the bounds of new physics can be in principle lowered. Furthermore,3he operator responsible for the direct custodial corrections investigated in this paper playsan important role in curing the tensions of some extensions of the SM in reproducing thecorrect Top quark mass and Zb ¯ b coupling, [13, 22, 23].In sec.(IV) we will infer from general principles, namely the requirement of custodialinvariance and the elimination of heavy modes from the Equations of Motion (EoM) inthe derivation of the effective theory, the correct form of the operator ˆ O W B responsiblein the gauge sector for the contributions to the S parameter. In this way we will findthat, besides the exact elimination of the corrections of new physics in the transversecomponents of the vector bosons, the ideal cancellation exactly eliminates also all thespurious corrections in the longitudinal and non-abelian terms of the effective Lagrangian.Thus, as long as new physics can be neglected in the EoM, the SM Lagrangian turns outto be equivalently defined modulo custodial oblique and direct operators (the concept ofideal cancellation can be also extended to the Higgs sector, [19]), provided redefinitions ofthe physical parameters. In analogy with the case of custodial symmetry protecting T and U , the ideal cancellations can be thought of as an “effective symmetry” (i.e. a symmetryvalid at low energy) of the SM Lagrangian protecting the S parameter.In sec.(IV A) we will investigate the possible theoretical origin of the ideal cancellation.It is known that in extra-dimensional, holographic, moose, composite and technicolor mod-els the ideal cancellation is related to a fine-tuning of the direct coupling of SM fermionsto the new physics sector. We will find that the origin of the custodial oblique and directcorrections can be associated to modifications of the covariant derivatives in the gauge andfermionic sectors. At the level of covariant derivative, the ideal cancellation means thatthese modifications must be the same in both the gauge and fermionic sectors. This sug-gests that the ideal cancellation has a possible interpretation in terms of the “universality”of gauge structure among the effective vectorial and fermionic sectors.4 . ELECTROWEAK CORRECTIONS OF CUSTODIAL NEW PHYSICS The contributions of new physics in extensions of the SM of EW interactions are mainlysummarized by three EW parameters T , U and S [2], or alternatively the (cid:15) , (cid:15) and (cid:15) parameters [3]. These corrections are inferred from the effective Lagrangian through three∆ parameters: ∆ r W is related to the M W and M Z ratio, ∆ ρ is related to the neutral andcharged current interactions, and ∆ k takes into account the difference between effectivevalue of the Weinberg angle, ˜ θ , and its physical value θ : L fermionsZ = − es θ c θ (cid:18) ρ (cid:19) [ J µ L − s θ (1 + ∆ k ) J µem ] Z µ ,s θ = (1 + ∆ k ) s θ , (cid:32) − M W M Z (cid:33) M W M Z = πα ( M Z ) √ G F M Z (1 − ∆ r W ) , (1)We define the following reduced EW parameters [1, 3],ˆ T = ∆ ρ , ˆ U = c θ ∆ ρ + s θ c θ ∆ r W − s θ ∆ k , ˆ S = c θ ∆ ρ + c θ ∆ k . (2)In an alternative notation these parameters correspond respectively to the (cid:15) , (cid:15) and (cid:15) parameters [3].In custodial models the parameters T and U are protected from new physics contri-butions. Notice, however, that besides the contribution of T , U and S , the reduced pa-rameters defined in (2) may have contributions from radiative corrections as well as fromhigher derivative corrections V , W , X , Y and Z . A non-SM Higgs sector typically impliescorrections to T and S , however in this paper we will work by assuming a SM Higgs sector,i.e. that new physics does not contribute to the SM particle masses. In general, higherderivative contributions of the same type can be consistently neglected with respect tothe lower derivative ones as they are suppressed by factors ( M W / Λ) where Λ is the new5hysics scale. The remaining lower derivative relevant parameters W and Y preserve thecustodial symmetry. They can be relevant in theories with new or composite vector bosons[9]. Nevertheless, we will work under the assumption of ideal cancellation and this impliesthat new or composite resonances are fermiophobic so that their effective contributionsto W and Y are suppressed or vanishing [20]. The scope of this paper is to investigatethe S parameter. The detailed Lagrangian analysis of the other EW precision parame-ters will be given in future publications and our analysis will be at tree-level. Thus thereduced EW parameters (2) are related to the ordinary EW parameters by the relations: U = − s θ ˆ U /α (cid:39) −
119 ˆ U , T = ˆ T /α = 129 ˆ T and S = 4 s θ ˆ S/α ∼
119 ˆ S , [9]. The S related values assumed in this paper are (cid:15) = (4 . ± . × − (experimental bounds) andˆ S = (0 . ± . × − (global fit with a light Higgs) [9]. A. Oblique corrections
The so-called oblique corrections to the EW parameters are directly inferred from thetransverse components of vectorial fields. If new vectorial resonancies are sufficiently heavywith respect to the mass of the external particles in Feynman diagrams the longitudinalpart of the two point functions are suppressed by factors ( m f /m V ) . The expansion in p of the tree-level transverse vacuum polarization amplitudes Π V V ( p ) (cid:39) Π V V (0) +˙Π V V (0) p + O ( p ), where V and V are generic SM gauge bosons, leads to the followingdefinitions of the oblique EW corrections [2]:ˆ T Obl = −
12 Π
W W (0) − Π ±± (0)Π W W (0) , ˆ U Obl = − ( ˙Π W W (0) − ˙Π ±± (0)) , ˆ S Obl = gg (cid:48) ˙Π(0) W Y , (3)Custodial extensions of the SM, such as custodial holographic, extra-dimensional,moose, composite-Higgs, Little Higgs, and technicolor models, are typically affected bypositive contributions to the S parameter, whereas T and U are protected by the custodial6ymmetry. In the literature, e.g. [9, 11, 12, 24, 25], the oblique contributions to S arecommonly associated to the operator O W B = (
U σ a U † ) F aµν ( ˜ W ) ˜ B µν (cid:59) −
12 ˜ g (cid:48) ˜ g ˆ S Obl F µν ( ˜ W ) ˜ B µν . (4)where ˆ S Obl is positive, U is the chiral field transforming under SU L (2) ⊗ SU R (2) as U → LU R † , F aµν ( ˜ W ) = ˜ W aµν + ˜ g(cid:15) abc ˜ W bµ ˜ W cν and ˜ W aµν = ∂ µ ˜ W aν − ∂ ν ˜ W aµ . This operator generatesanomalous trilinear vector boson couplings [12]. However, in sec.(IV A) we will infer fromgeneral principles the correct effective operator associated to custodial corrections, namelyˆ O W B , see (28). This indeed differs from O W B in the non-abelian terms. Since in this andin the next section we only consider the transverse components of the vector bosons such adifference in the operator O W B can be neglect for the moment. Thus, in momentum space,the effective Lagrangian encoding oblique corrections of custodial new physics is L EffObl ( p , ˜ v ) = L SM ( p , ˜ v ) + p ˜ g (cid:48) ˜ g ˆ S Obl ˜ W µ ˜ B µ . (5)The effective electroweak scale ˜ v and the effective Weinberg angle ˜ θ s.t. tan ˜ θ = ˜ g (cid:48) / ˜ g ,determine the effective vector boson mass spectrum associated to (5):˜ M W = ˜ g ˜ v , ˜ M Z = (cid:113) ˜ g + ˜ g (cid:48) ˜ v m f = 1 √ λ f ˜ v , (7)and the Higgs boson mass is determined by the SM Higgs potential parameter ˜ λ and theEW scale ˜ M H = (cid:112) λ ˜ v . (8)In other words we are assuming that the SM masses are generated by the SM Higgsmechanism, i.e new physics does not contribute to the SM mass spectrum. This means7hat the contribution of new physics to the vacuum polarization amplitudes is supposed tobe zero at leading order in p , i.e. Π V V (0) ≡
0, and that the Higgs field has no couplingsto the new physics sector. In case of alternative gauge symmetry breaking mechanismsthe parameters ˜ λ f and ˜ λ are model dependent (with negligible higher order correctionsassociated to the effective EW parameters ˜ g, ˜ g (cid:48) and ˜ v ). As exemplification, we will mentionlater some model dependent cases. B. Direct corrections
Besides the oblique corrections it is possible to introduce corrections affecting the ver-tices with the fermions. This are the so-called direct or vertex corrections. Indeed, theparameters (2) encode general corrections to neutral and charged currents. Thus they areaffected by direct couplings of the SM fermions to new physics, FIG.(1). Π XY J Y J X ∼ − & YX g Y Yg X X FIG. 1. A diagrammatic representation of the oblique and direct contributions to a current-currentinteraction.
A possible universal direct contribution to the S parameter allowed by the custodialsymmetry is given by the operator [12, 13, 16, 18] C = Tr [ ¯ ψ L γ µ ( iD µ U ) U † ψ L ] (cid:59) c γ µ ¯ ψ L (cid:18) ˜ g ˜ W aµ σ a − ˜ g (cid:48) ˜ B µ σ (cid:19) ψ L = c ˜ e sin ˜ θ cos ˜ θ ˜ Z µ ¯ ψ L γ µ σ ψ L + c θ √ W − µ ¯ ψ Ld γ µ ψ Lu (9)The sign of c is determined by the kinetic term and it is positive. C is for instance associ-ated to the universal delocalization of left-handed fermions in custodial extra-dimensional8odels or holographic models [15, 18] and, through dimensional deconstruction [26], to di-rect couplings of standard model fermions to the non-standard gauge symmetries of moosemodels [16, 17] or similar strongly-interacting and composite models [14, 25, 27]. Further-more, as noted in [13], this operator applied to the third quark family is important to curethe Zb ¯ b coupling in emergent EWSB models.When both oblique and direct custodial corrections are considered the effective La-grangian is L EffT ot ( p , ˜ v ) = L SM ( p , ˜ v ) + O W B + C . (10)Next we will investigate the EW corrections of this effective custodial Lagrangian. II. STANDARD ANALYSIS OF THE S CORRECTIONS
The standard analysis of the EW corrections associated to the effective Lagrangian (10)passes through the evaluation of the ∆ parameters (1). The explicit calculation of the bothoblique and direct contributions yields∆ ρ (cid:39) r W (cid:39) − c − ˆ S Obl ) ; ∆ k (cid:39) − c − ˆ S Obl cos 2˜ θ . (11)From this we find that ˆ U ∼ T ∼ r W and ∆ k cancel each other out in (2). This result was expected from the fact that bothoblique and direct contributions, (4) and (9), preserve the custodial symmetry, and ˆ U andˆ T , as well as the ρ parameter, are protected by the custodial symmetry.The ˆ S parameter is in general non vanishing in custodial models. From (2) and (11) wefind ˆ S (cid:39) ˆ S Obl − c . (12)so that the direct contribution to the ˆ S parameter is negative: ˆ S Dir (cid:39) − c .9n the case of fine-tuning between oblique and direct corrections c (cid:39) ˆ S Obl , (13)all the ∆ parameters are suppressed. Thus the ˆ S ∼ U (cid:39) T (cid:39) S (cid:39) . (14)We define ideal cancellation the condition that exactly cancel the oblique and directcontributions to the ˆ S parameter. In custodial extra-dimensional or moose models, this cor-responds to the “ideal delocalization” of left-handed fermions along the extra-dimensionalbulk [15, 18, 19] or along a moose [16, 17], or other ideal couplings of fermions to custodialextensions of the EW gauge group in composite-Higgs, little-Higgs and technicolor models[14, 27].In extra-dimensional models this requires that the left-handed fermions bulk profile isrelated to the W profile. According to the bulk EoM, the two profiles at zero order in p behave as f L ( z ) ∼ z c ) β L and h W ( z ) ∼ α W + β W z respectively, where: z = e ky /k is theconformal parametrization of the warped extra-dimension y ; β L , α W and β W are constantsdetermined by the boundary conditions; c = M bulk /k is the ratio between the fermionbulk mass and the curvature of the AdS metric. Thus the condition of ideal delocalization h W ( z ) ∼ ( kz ) − f L ( z ) is of difficult realization: it is necessary to have a flat fermionicprofile, namely c = − /
2, and to arrange the boundary conditions in such a way that α W ∼ β W ∼ β L /k . Notice that, concerning the fine-tuning (13), if the correctionsin the gauge and in the fermionic sectors come from the same new physics they can give,in principle, contributions of the same order in the effective Lagrangian. In such a standard analysis of the EW corrections the effect of possible custodial directcorrections to the vector boson mass spectrum (such that ρ = 1) is typically neglected, i.e. As the ideal cancellation related corrections to the gauge sector with those of the fermionic sector, itwould be particularly interesting to study this ideal cancellation in supersymmetric models (or in termsof the BRST symmetry). v associated tothe mass spectrum (6) of models with only oblique corrections with respect to the physicalEW energy v . We will find that this can lead to sizable corrections in the case of idealcancellation.In sec.(IV), we will find that the assumption of ideal cancellations represents an effectivesymmetry of the SM lagrangian. That is, the gauge and fermionic sectors of the SMLagrangian can be defined modulo oblique and direct terms, ˆ O W B and C respectively.Such an “effective symmetry” is indeed exact as soon as we consider the general form ofthe oblique operator ˆ O W B , (28) derived through the elimination of the heavy d.o.f. fromthe EoM.
III. LAGRANGIAN ANALYSIS OF THE S CORRECTIONS
To perform the Lagrangian analysis of the EW corrections we explicitly act in the effec-tive Lagrangian containing both the oblique and direct corrections (10) with redefinitionsof fields and couplings. In momentum space, neglecting for the moment the longitudinaland the non-abelian terms, (10) is L EffT ot ( p , ˜ v ) = p W aµ ˜ W aµ + p B µ ˜ B µ + p ˜ g (cid:48) ˜ g ˆ S Obl ˜ W µ ˜ B µ − ˜ v (cid:20) ˜ g ˜ W a σ a − ˜ g (cid:48) ˜ B σ (cid:21) + ¯ ψ L γ µ (cid:18) p µ − ˜ g ˜ W aµ σ a − ˜ g (cid:48) B − L B µ (cid:19) ψ L + ¯ ψ R γ µ (cid:18) p µ − ˜ g (cid:48) ˜ B µ σ − ˜ g (cid:48) B − L B µ (cid:19) ψ R − c γ µ ¯ ψ L (cid:18) − ˜ g ˜ W aµ σ a g (cid:48) ˜ B µ σ (cid:19) ψ L + L SMHiggs + L SMY ukawa . (15)11e apply the following redefinitions of the gauge fields and couplings,˜ W aµ σ a → W aµ σ a − ˆ S Obl ˜ g (cid:48) ˜ g ˜ B µ σ , ˜ g → g − c . (16)This diagonalizes the kinetic gauge terms. Besides this, we normalize the ˜ B µ kinetic termwith the (unphysical) redefinitions ˜ B µ → B µ / (cid:113)
32 ˜ g (cid:48) ˜ g ˆ S Obl and ˜ g (cid:48) → g (cid:48) (cid:113)
32 ˜ g (cid:48) ˜ g ˆ S Obl .The effect of these transformations is to transfer the oblique corrections to the fermionicsector and to the vector boson mass term L EffT ot ( p , ˜ v ) = p W aµ W aµ + p B µ B µ + ˜ v (cid:20) g − c W aµ σ a − g (cid:48) (1 + ˆ S Obl ) B µ σ (cid:21) + ¯ ψ L γ µ (cid:20) p µ − gW aµ σ a − g (cid:48) (cid:16) c − ˆ S Obl + ˆ S Obl c (cid:17) σ B µ − g (cid:48) B − L B µ (cid:21) ψ L + ¯ ψ R γ µ (cid:18) p µ − g (cid:48) B µ σ − g (cid:48) B − L B µ (cid:19) ψ R + L SMHiggs + L SMY ukawa . (17)Thus we find that the ideal cancellation among oblique and direct corrections in thecustodial effective Lagrangian is given by the following exact condition c ≡ ˆ S Obl S Obl , (18)so that ˆ S ≡
0, ˆ U ≡ T ≡
0. This result is in agreement with that obtained at the firstorder in ˆ S Obl through the standard analysis (12, 13). Indeed the condition (18) cancels theanomalous term in the left-handed neutral current. Furthermore it implies a factorizationof the corrections in the vector boson mass term L EffT ot ( p , ˜ v ) = p W aµ W aµ + p B µ B µ + (1 + ˆ S Obl ) ˜ v (cid:20) gW aµ σ a − g (cid:48) B µ σ (cid:21) + ¯ ψ L γ µ (cid:18) p µ − g ˜ W aµ σ a − g (cid:48) B − L B µ (cid:19) ψ L + ¯ ψ R γ µ (cid:18) p µ − g (cid:48) ˜ B µ σ − g (cid:48) B − L B µ (cid:19) ψ R + L SMHiggs + L SMY ukawa (19)Through the ideal cancellations the oblique and direct corrections are completely can-celled and the effective Lagrangian (15), i.e. (10), is reduced to the SM Lagrangian L EffT ot ( p , ˜ v ) = L SM ( p , ˜ v ) + ˆ O W B + C ≡ L SM ( p , v ) , (20)12 .00 0.01 0.02 0.03 0.04 0.05 0.06 0.07230235240245 S (cid:96) Obl v (cid:142) (cid:61) (cid:72) G (cid:142) F (cid:76) (cid:45) (cid:144) (cid:64) G e V (cid:68) FIG. 2. 90% and 99% CL contour plot for the rescaling of the effective EW energy scale (andFermi coupling) resulting from the ideal cancellation in the parameter ˆ S (red region) and (cid:15) (grayregion). ˜ v ≡ ( ˜ G F √ − = v/ (1 + ˆ S Obl ) is the effective value calculated in custodial models by onlyconsidering oblique contributions, v ≡ ( G F √ − = 246 .
22 GeV (green line) is the physical valueobtained by also considering direct corrections. with the following redefinition of the effective EW scale ˜ v = √ ˜ G F √ , i.e. of the effectiveFermi coupling ˜ G F , see FIG.2, ˜ v → v S Obl = v (1 − c ) . (21)To reproduce the correct physical spectrum in the Higgs and fermionic sectors the cor-responding effective parameters ˜ λ and ˜ λ f of custodial models with only oblique correctionsmust be rescaled as ˜ λ → λ (1 + ˆ S Obl ) and ˜ λ f → λ f (1 + ˆ S Obl ) when the ideal cancellation isconsidered. The rescaling of the effective fermion and Higgs masses ˜ m f = √ λ f (1 + ˆ S Obl )˜ v and ˜ M H = √ λ (1 + ˆ S Obl )˜ v with respects to their physical values is shown in FIG.3. For in-stance, an universal contribution C implies that the effective value of the quark Top massobtained by only considering the effect of oblique corrections must be higher to obtain,after considering the effect of the direction corrections to the EW scale, the experimentalvalue m t . This could compromise the benefit to introduce C to cure the Zb ¯ b coupling.13 .00 0.01 0.02 0.03 0.04 0.05172174176178180182 S (cid:96) Obl m (cid:142) t (cid:64) G e V (cid:68) S (cid:96) Obl M (cid:142) H (cid:64) G e V (cid:68) FIG. 3. 90% and 99% CL contour plots for the rescaling of the effective Top and Higgs masses, leftand right figure respectively, resulting from the ideal cancellation in the parameter ˆ S (red regions)and (cid:15) (gray regions). ˜ m t and ˜ M H are the effective values calculated in custodial models by onlyconsidering oblique contributions, m t = 173 . ± . M H = 125 . ± . The effect of the rescaling in terms of the oblique and direct parameters ˆ S Obl and c ofthe effective W mass, ˜ M W = ˜ v (cid:113) g (1 − c ) + g (cid:48) (1 + ˆ S Obl ) , with respect to its physical valueof M W is shown in FIG.4.Similarly, by assuming SM Higgs couplings with the effective fields ˜ W aµ and ˜ B µ , theredefinitions (16) implies the following modification with respect to the physical fields in theSM Higgs sector: g ( H + 2 H ˜ v )(1 + ˆ S Obl ) ( W + W − + Z / c θ ). Thus we have the followingrescalings for the SM Higgs couplings ˜ g Hff → g Hff (1 + ˆ S Obl ), ˜ g HV V → g HV V (1 + ˆ S Obl ),˜ g HHV V → g HHV V (1 + ˆ S Obl ) and ˜ g HHH → g HHH / (1 + ˆ S Obl ).Notice however that in phenomenological models new physics typically contributes tothe SM masses. These contributions may be compensated by the rescaling associated to theideal cancellation. As exemplification we consider the case of a custodial Randall-Sundrummodel with brane localized Higgs [10, 22] (the phenomenology of this model is in factinvestigated at higher order corrections of new physics). In this case the effective EW scale˜ v , determined through the Fermi constant [22], is related to the Higgs vacuum expectation14alue v H by the relation ˜ v (cid:39) v H (cid:104) − ˜ M W M KK (1 − L ) (cid:105) where v H is the v.e.v. of the Higgsboson, M KK is the mass scale of the low-lying Kaluza-Klein excitations and L = ln kM KK .The oblique contribution to the S parameter is ˆ S Obl = ˜ g M KK (1 − L ) — the correctionsto T associated to new physics contributions to the Higgs sector will be investigate infuture papers, see also sec.(IV B). Hence, upon ideal cancellation, the contribution of directcorrections in the fermionic sector implies the following redefinition of the effective EWenergy scale v = ˜ v (1 + ˆ S Obl ) = v H (cid:104) ˜ M W M KK (1 − L ) (cid:105) .In sec.(IV A) we will extend the ideal cancellation to the full effective Lagrangian includ-ing the cancellation on the non-abelian and transverse terms. Indeed, the ideal cancellationis an interesting “effective symmetry” of the SM, i.e. the SM Lagrangian can be definedmodulo oblique and direct corrections. Thus the unwanted oblique and direct contributionsof new physics can be separately greater than the actual experimental bounds on S .As well-known the ideal cancellation yields an extended parameter space to phenomeno-logical models [14–16, 20], allowing for a lower energy scale of new physics (especially inRandall-Sundrum models due to the warped factor). In addition to this, provided thatthe new physics modes are sufficiently heavy to be eliminated in the EoM, the Lagrangiananalysis of the ideal cancellation shows that this bound to new physics is further reduced bythe redefinition of the effective EW energy scale ˜ v with respect to v , described in FIG.(2). IV. IDEAL CANCELLATION AS “EFFECTIVE SYMMETRY”
So far we have investigated the ideal cancellation by considering only the transversecomponents of the gauge bosons. We now generalize our analysis to the longitudinaland non-abelian terms. We continue to assume that the custodial new physics sector issufficiently heavy with respect to the SM particles to be neglected in the EoM. Assuming aSM Higgs mechanism, the corrections of new physics are encoded in custodial dimension-six operators in the gauge and fermionic sectors. We now infer the general form of these15 M Z S (cid:96) Obl c FIG. 4. 90% and 99% CL bounds for the rescaling of the effective Z mass ˜ M Z (blue lines) withrespect the physical value M Z = 91 . ± .
002 GeV (green line) in terms of the oblique and directcorrections, ˆ S Obl and c respectively, to the ˆ S (red region) and (cid:15) (gray region) parameters. operators from considerations about the custodial symmetry and the elimination of theheavy modes from the EoM. In particular we find that the resulting operator ˆ O W B in thegauge sector coincides with O W B in the transverse terms but it has a different form in thenon-abelian terms. Thus, the general operator ˆ O W B gives the same contribution of O W B to the S parameter. Furthermore, contrarily to the operator O W B , the general operatorˆ O W B is exactly eliminated by the fermionic operator C upon ideal cancellation. That is,the cancellation occurs in the full effective Lagrangian (including non-abelian terms) givingback exactly the SM Lagrangian with rescaled EW energy.The operator O W B , (4), commonly used in literature, e.g. [9, 11, 12, 24], as sourceof the oblique corrections to S generates anomalous trilinear corrections of the type ∼ ˆ S Obl ˜ g (cid:48) ˜ W + µ ˜ W − ν ˜ B µν , [12]. By following the same line of the Lagrangian analysis in sec.(III),we find that the transformation of ˜ W in (16) eliminates these anomalous corrections.However the redefinition of ˜ g generates corrections to the non-abelian terms which are notcancelled by the condition of ideal cancellation (18). The residual corrections of O W B in thetrilinear and quartic vector boson couplings are respectively g trilin = (1 + ˆ S Obl ) g SMtrilin and16 quart = (1 + ˆ S Obl ) g SMquart . These residual corrections can be relevant in the analysis of thelongitudinal components associated to the elastic scattering amplitude
W W → W W andin turns in the unitarization of the theory. Moreover, by considering the corrections in theSM Higgs couplings described in the previous section, they imply a further enhancementof H → γγ decay rate with respect to the enhanced H → V V decays whereas H → ¯ f f is unchanged. This effect could be useful in interpreting possible excesses in the Higgsto vector bosons decay, especially excesses in H → γγ . These spurious corrections in thenon-abelian terms are exactly eliminated by the ideal cancellation as soon as we considerthe general custodial operator ˆ O W B . A. Exact cancellation of all spurious corrections
Phenomenological models typically involve extended EW gauge symmetries such as incomposite, little Higgs, moose, technicolor models, or bulk gauge symmetries in extra-dimensional models. The low energy corrections to the gauge sector are obtained by elim-inating from the EoM the related heavy modes in terms of the effective SM fields ˜ W aµ and ˜ B µ . In custodial models, if we denote the new physics gauge bosons V iµ ∈ G ⊇ SU L (2) ⊗ SU R (2), with generic coupling g V and generators T i , their elimination from theEoM leads to an effective solution of the following general SU D (2) custodial invariant form g (cid:48)(cid:48) V iµ T i (cid:59) h ( p ) (cid:18) − ˜ g ˜ W a σ a g (cid:48) ˜ B σ (cid:19) ∈ SU D (2) , (22)where h ( p ) is a function encoding the new physics corrections.By assuming direct couplings of the left-handed SM fermions with the new physicssector, in custodial models we have the following modification of the corresponding SMcovariant derivative˜ D µ ψ L = (cid:20) ∂ µ − i ˜ g ˜ W µ σ a − i ˜ g (cid:48) ˜ B µ B − L ic (cid:18) ˜ g ˜ W µ σ a − i ˜ g (cid:48) ˜ B µ σ (cid:19)(cid:21) ψ L (23) Notice that in extra-dimensional phenomenology the holographic prescription is equivalent to the elim-ination of the KK modes in terms of the SM vector bosons, i.e. the so called source fields [18]. Indeedthe holographic approach is often used in extra-dimensional theories to derive the effective Lagrangian. c describes the overlap between the fermions and the new physics vectorbosons. By comparison with (9) we see that actually (22) correctly generates the directoperator C investigated in sec.(I B).Similarly we now derive the general form of the custodial operator ˆ O W B in the gaugesector. In the field strength, the elimination of the heavy modes from the EoM induces thefollowing modification of the SM covariant derivative˜ D µ = ∂ µ − i ˜ g ˜ W µ σ a − i ˜ g (cid:48) ˜ B µ + i ˆ S Obl S Obl (cid:18) ˜ g ˜ W aµ σ a − ˜ g (cid:48) ˜ B µ σ (cid:19) , (24)where ˆ S Obl encodes the coupling of the heavy vector bosons with the SM ˜ W a and ˜ B .The field strength associate to the effective custodial theory is therefore directly inferredfrom the modified covariant derivative (24) as[ ˜ D µ , ˜ D ν ] = − ig F aµν σ a − i ˜ g (cid:48) ˜ B µν . (25)where, for convenience, we have parameterized ˜ g → g (1 + ˆ S Obl ) — this avoids a furthernormalization of ˜ W . Notice that upon the condition of ideal cancellation this choice cor-responds to the reparametrization in (16).Thus the effective ˜ W field strength encoding custodial new physics corrections in thegauge sector has the form F aµν σ a F aµν ( ˜ W ) σ a S Obl S Obl ∆ G aµν σ a S Obl ˜ g (cid:48) ˜ g σ B µν (26)where ∆ G ± µν = − i ˜ g ( ˜ W − µ ˜ W + ν − ˜ W − ν ˜ W + µ )∆ G µν = ∓ i ˜ g ( ˜ W ν ˜ W ± µ − ˜ W µ ˜ W ± ν ) ∓ i ˜ g (cid:48) ( ˜ B ν ˜ W ± µ − ˜ B µ ˜ W ± ν ) (27)The explicit derivation of these non-abelian custodial corrections in the gauge kinetic termsis given in [28] for the specific cases of a moose and an extra-dimensional custodial model.Thus, (upon unphysical normalizations of ˜ B and ˜ g (cid:48) ) the general operator ˆ O W B encoding18ustodial corrections in the kinetic gauge terms isˆ O W B = O W B −
12 ˆ S Obl S Obl (cid:18) ˆ S Obl ˜ g (cid:48) ˜ g ∆ G µν ˜ B µν + 2Tr[ F aµν ( ˜ W )∆ G aµν ]+ ˆ S Obl S Obl
Tr[∆ G aµν ∆ G aµν ] (cid:33) (28)Its bilinear terms are equivalent to O W B so that both O W B and ˆ O W B generate the sameoblique correction to S . This confirms our model independent derivation of custodialcorrections in terms of the elimination of the heavy fields from the EoM. At the same timethis shows that in general the custodial corrections to the gauge kinetic terms differ in thenon-abelian terms from the commonly used operator O W B .The general derivation of the custodial corrections in terms of redefinitions of the co-variant derivatives is particularly convenient. In this way it is in fact straightforward to seefrom the covariant derivatives (23) and (24) that the redefinition of ˜ W and ˜ g given (16)exactly cancel all the new physics corrections in the gauge invariant terms of the effectiveLagrangian. The only remaining corrections are in the spontaneously broken gauge termssuch as the vector boson mass term. In particular this implies the redefintion of the EWenergy scale (21). Furthermore, from (23) and (24) we immediately see that the conditionof ideal cancellations (18) means, in terms of gauge invariance, that the coefficients of theextra gauge terms must be exactly the same in the two covariant derivatives. We willdiscuss further this interesting aspect in the next section.Thus, the condition of ideal cancellation exactly eliminates all the new physics contribu-tions in the effective Lagrangian giving back the SM Lagrangian with a possible redefinitionof the EW scale, and thus of the parameters in the Higgs and Yukawa sectors, L EffT ot (˜ v ) = L SM (˜ v ) + ˆ O W B + C ≡ L SM ( v ) . (29)Upon the condition of ideal cancellation and rescaling of the effective EW energy ˜ v , thegauge and fermionic sectors of SM Lagrangian can be defined modulo custodial corrections,namely ˆ O W B and C . We conclude that, on the basis of general principles, the ideal19ancellation represents a new “ effective symmetry ” of the SM Lagrangian, i.e. a symmetryvalid in the low energy approximation and protecting the custodial effective Lagrangianfrom corrections with respect to the SM lagrangian. B. On the theoretical meaning of the ideal cancellation
It is worthwhile investigating the possible theoretical origin of the “effective symmetry”associated to the ideal cancellation. Here we study the origin of the ideal cancellationin terms of gauge invariance. Though the ideal cancellation is a very model dependentissue, our considerations about the modification of the covariant derivative allows us aninteresting model independent analysis.In extra-dimensional (including gauge-Higgs unification) or composite and moose mod-els, the direct couplings of fermions to new physics can be associated to delocalizations ofthe fermions through a Wilson line of the fifth component of the bulk gauge fields or of thedirect product of σ model scalar fields, e.g. [16, 28]. Through the eliminations of the newphysics d.o.f. from the EoM in the derivation of the effective theories, such a delocalizationcan be described in terms of the SM fields. For instance the direct correction C , i.e. thecovariant derivative (23), can be obtained from the effective Wilson line ψ L ( x ) → e − iπ aDir ( x ) σa ψ L ( x ) (30)with the custodial invariant contribution π aDir ( x ) σ a − c (cid:90) x dx (cid:48) µ (˜ g ˜ W µ σ a − ˜ g (cid:48) ˜ B µ σ ⇔ C . (31)Similarly, the oblique corrections in the gauge sector (24), namely the operator ˆ O W B ,can be associated to an analogous custodial effective Wilson line with modulated phase π aObl ( x ) σ a − ˆ S Obl S Obl (cid:90) x dx (cid:48) µ (˜ g ˜ W µ σ a − ˜ g (cid:48) ˜ B µ σ ⇔ ˆ O W B . (32)Even though the coefficients of the Wilson lines responsible for the oblique and directcorrections are in general different in effective models, we see that the condition of ideal20ancellation is related to the effective gauge invariance of the theory. For instance, gaugeinvariance implies that the SM couplings are such that g V ff = g V V V : the gauge couplingsin the fermionic covariant derivative and in the field strength must be the same. In termsof the coefficients of the oblique and direct corrections this implies the ideal cancellationcondition (18). Roughly speaking the ideal cancellation can be regarded as the gauging ofan extra custodial group SU D (2) with reduced couplings by factors ˆ S Obl
1+ ˆ S Obl , or equivalently c , with respect to the SM EW couplings in both the gauge and fermion sectors.In case of ideal cancellation, and as long as the new physics d.o.f. are sufficientlyheavy, the pure custodial corrections to the gauge and fermionic sector of the SM can bein principle much higher that the actual experimental bounds on S . This can imply sizableredefinitions of the effective parameters of the models, such as EW scale ˜ v , couplings andmass spectrum, and thus of the allowed scale for new physics.This paper is exclusively dedicated to the study of the S parameter in purely custodialeffective theories. Nevertheless, as we will see in future papers, our analysis can be extendedto the other EW parameters by considering the generalization of the ideal cancellation ofother possible operators among the different sectors of the SM lagrangian [12].Notice that if we allow direct coupling of the Higgs field to the custodial new physics, thecovariant derivative of the Higgs boson turns out to have a gauge structure similar to thoseof the field strength and of the left-handed fermions, (23) and (24). The coefficient c U infront of the non-standard contribution would be given by the overlap between the Higgs andthe vector boson profiles. In the Higgs sector, the dimension six operator responsible forthe custodial corrections is in fact | U † D µ U | (cid:59) c U (cid:16) ˜ g ˜ W aµ σ a − ˜ g (cid:48) ˜ B µ σ (cid:17) . By generalizingthe concept of ideal cancellation to the Higgs sector we can assume ideal direct coupling ordelocalization of the Higgs field such that this coefficient is the same of the other sectors,[19]. Hence the redefinitions (16) exactly reabsorb the possible corrections in the Higgssector, obtaining the SM Lagrangian with possible rescaling of the EW energy. An idealdelocalization of the Higgs profile could also play an interesting role in protecting the T CONCLUSIONS
The discovery of the light SM-like Higgs-boson at LHC demands for new mechanics tomake the EW physics insensible to new physics. In this paper we show that large obliquecontributions to the S parameter can be exactly eliminated from the SM Lagrangian byintroducing direct custodial corrections in the fermionic sector [14–18]. The main fermionicoperator investigated in this paper is, for instance, introduced in model building to curetensions in the Zb ¯ b coupling [13, 22, 23]. The ideal cancellation among oblique and directcorrections turns out to be associated to possible sizable redefinition of the effective EWenergy scale with respect to models with only oblique corrections, allowing for an extendedparameter space for custodial new physics. In turns this implies redefinitions of the vectorboson masses, Yukawa parameters, Higgs mass and couplings. From the general principleof custodial invariance and from the elimination of the heavy modes from the EoM, we haveinferred the exact form ˆ O W B of the custodial corrections of new physics in the gauge sec-tor. Upon ideal cancellation such custodial corrections in the gauge sector can be exactlyeliminated by direct corrections in the fermionic sector. Besides the elimination of correc-tions in the transverse components of the gauge bosons, this also eliminates the correctionsin the longitudinal and non-abelian terms. Thus, as long as the new vectorial modes canbe neglected in the effective theory, the SM Lagrangian turns out to be defined modulocustodial corrections in the gauge and fermionic sector (provided possible rescalings in theHiggs and Yukawa sector). Similarly to the custodial symmetry protecting the effective22heory from corrections to the T and U parameters, the ideal cancellation can be thereforethought of as an ”effective symmetry” protecting the SM from custodial corrections. It istherefore important to investigate the possible theoretical origin of such an “effective sym-metry”. In general, this implies a fine-tuning between the couplings of new physics withthe SM gauge and fermionic sectors. We have found, however, that such a fine-tuning hasa simple justification in terms of the universality, among the gauge and fermionic sectors,of the effective gauge structure associated to the new physics contributions. ACKNOWLEDGMENTS
I would like to thank Prof. A. Pomarol, Prof. D. Dominici, Prof. S. de Curtis, Prof.M. Neubert and Prof. T. Gherghetta for fruitful discussions. This work was partiallysupported by the Angelo della Riccia Foundations.
Appendix A: Ideal cancellation without rescaling
It is possible to study other custodial fermionic operators with negative contributionsto the S parameter, so that it is possible to define new conditions of ideal cancellation thatexactly reduces the effective Lagrangian to the SM one. As in the case studied in the text,the cancellation in the non-abelian terms is possible by introducing the general custodialoperator ˆ O W B . Here we will investigate a universal custodial operator in the fermionicsector acting on the hypercharge. In this case however the ideal cancellation will not implya rescaling of the effective EW scale with respect to its physical value.The correction that we want to investigate is C Y = c Y i ¯ ψγ µ ˜ g (cid:48) ˜ B µ (cid:20) − σ − ( B − L )2 (cid:21) ψ (A1)with positive defined c Y and ψ = ψ L + ψ R . By assuming an extended electroweak gaugesymmetry SU L (2) × SU R (2) × U B − L (1) where the EW symmetry is recovered through23he breaking SU R (2) × U X (1) → U Y (1), this correction can be associated to the operator C Y = Tr [ ¯ ψγ µ ( iD µ Σ Y )Σ † Y ψ ] where X transforms as Σ Y → R Σ Y X where X ∈ U Y (1) and T L = − T R . The standard analysis of the EW constrains yields ˆ S = ˆ S Obl − c Y , U ∼ T ∼
0, so that the ideal cancellation of the oblique and direct corrections is c Y ∼ ˆ S Obl :ˆ S ∼ , ˆ U ∼ , ˆ T ∼ . (A2)This result can be checked at the Lagrangian level. The effective Lagrangian containingboth ˆ O W B and C Y is reduced to the SM Lagrangian by imposing the following redefinition(upon unphysical redefinition of ˜ B and ˜ g (cid:48) normalizing the kinetic term) of the effectivefields and couplings ˜ W aµ σ a → W aµ σ a − ˆ S Obl ˜ g (cid:48) ˜ g B µ σ , ˜ g (cid:48) → g (cid:48) c Y .In this way, by following the line of the Lagrangian analysis in sec.(III), we find thatthe ideal cancellation is given by the condition c Y ≡ S Obl (A3)so that the effective Lagrangian is reduced to the SM Lagrangian L EffT ot = L SM + ˆ O W B + C Y ≡ L SM (A4)Notice that in this case there is no rescaling of ˜ v . In fact the contribution comingfrom the redefinition of ˜ W a in the vector boson mass term is completely reabsorbed bythe redefinition of ˜ g (cid:48) . Thus we have obtained another instance of “effective symmetry”of the SM lagrangian: upon ideal cancellation the SM Lagrangian can be defined modulooperators ˆ O W B and C Y in the gauge and fermionic sector, respectively. [1] Particle Data Group
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