Generalised Holographic Electroweak Symmetry Breaking Models and the Possibility of Negative S
aa r X i v : . [ h e p - ph ] J un Generalised Holographic Electroweak Symmetry Breaking Models and the Possibilityof Negative ˆ S Mark Round
Swansea University, School of Physical Sciences,Singleton Park, Swansea, Wales, UK (Dated: November 15, 2018)Within an AdS/CFT inspired model of electroweak symmetry breaking the effects of variousboundary terms and modifications to the background are studied. The effect on the ˆ S precisionparameter is discussed with particular attention to its sign and whether the theory is unitary whenˆ S <
0. Connections between the various possible AdS slice models of symmetry breaking arediscussed.
I. INTRODUCTION
Models of electroweak symmetry breaking are com-pared to experiment through the precision electroweakparameters [1–3]. The parameter of primary interest inHiggsless [4–6] and technicolor models [7–9] is ˆ S . Mucheffort has gone into understanding how one can producemodels where ˆ S can be tuned within experimentally ac-ceptable bounds. Recently studies pursued this end using5D models formulated on a slice of a space that asymp-totes towards AdS in the UV [4–6, 12, 15, 17, 25–32].In developing holographic models that produce an arbi-trarily small ˆ S parameter it is interesting to ask whetherˆ S can be made negative. It is known that the differencein vector and axial two-point functions is strictly posi-tive [10], suggesting that ˆ S is too (in a physically accept-able scenario). In addition, a unitary four-dimensionalmodel with negative ˆ S has never been found; except formodels where new Majorana fermions are added into theStandard Model [11]. This can lead to ˆ S < S rather than purely themechanism of electroweak symmetry breaking. We willnot be interested in the effects of matter on ˆ S . Ignoringthis possibility, the absence of a negative ˆ S model andthe result of Witten [10] suggest that it might not be notpossible to make ˆ S <
S <
S > S parameter showed regions whereˆ S <
S <
0, it is not apparent that this can be done withinthe context of a consistent effective theory. Further de-velopment of these ideas was made by studying a real-istic example. The authors of [15] studied an examplemodel and discussed how the result of a negative ˆ S could be understood in 4D. The effects on ˆ S by adding moregeneral terms into the Lagrangian was studied by usingSt¨uckelberg fields [16]. Other authors, with particular at-tention to the sign of ˆ S , have discussed symmetry break-ing by boundary terms and a numerical scan of possibleVEV profiles [17].In this paper we will focus on the Goldstone modes andwhether or not they are negative-norm states. This willbe done by computing the sign of the Goldstone kineticterm. This exercise will be carried out with the inclusionof all extra bulk and boundary terms (relevant to thediscussion of the sign of ˆ S ) that can be added to a basicholographic model of electroweak symmetry breaking. Asthis introduction has shown, some parts of this studyalready exist in the literature. We will include all the(leading-order) terms possible and discuss their interplayin a systematic and analytic way. II. FIVE DIMENSIONAL SU (2) × SU (2) ACTION
This work will study an SU (2) L × SU (2) R gauge sym-metry broken to the diagonal SU (2) by a bifundamentalscalar field, H , in five dimensions. The geometry is de-fined by the metric ds = g MN dx M dx N = ω ( y ) (cid:0) η µν dx µ dx ν − dy (cid:1) , (1)with η µν the Minkowski metric of signature (+ , − , − , − ).Here, y ∈ [ L , L ] is the 5 th dimension. The scales L and L represent the UV and IR cut-offs of the theory respec-tively. Lorentz indices running over the 5D manifold willbe denoted by uppercase Latin indices. Lowercase Greekindices run over the first four of the five co-ordinates.Requiring that the space asymptotes towards AdS givesthe constraint ω ( y ) → Ly as y → L . (2)If the gauge fields are denoted by L aM and R aM , then thematrix-valued fields L M = L aM σ a / R M = R aM σ a / σ a , a = 1 , ,
3, the Pauli matrices) can be used towrite out the possible Lorentz and gauge invariant oper-ators. The relevant operators are of the form m | H | , Tr | D M H | , Tr L MN L MN , Tr R MN R MN . (3)The field strength and covariant derivative are L MN = ∂ M L N − ∂ N L M − ig L [ L M , L N ] , (4) D M H = ∂ M H − ig L L M H + ig R HR M , (5)with g L ( g R ) the gauge coupling of the L ( R ) field. Lo-calised on the boundaries there are further relevant op-erators,Tr | D µ H | , m ′ Tr | H | , Tr L µν L µν , Tr R µν R µν (6)The operator HH † HH † is marginal on the boundaries.Gauge kinetic operators Tr L µν L µν and Tr R µν R µν when localised on the UV boundary act as countertermsto the UV boundary theory. The co-efficients of the termscan be dialed so as to remove spurious dependence uponthe UV cut-off 1 /L . This is the process of holographicrenormalisation. See [18, 19] for a discussion of holo-graphic renormalisation in this context.In addition to the operators already discussed theremight be higher-order corrections. The first correctionsto the bulk action are,Tr | H † D M H | , Tr L MN HR MN H † (7)which are the 5D analogues of the operators correspond-ing to the ˆ T and ˆ S electroweak precision parameters re-spectively [1–3]. It is the ˆ S operator that will be of pri-mary interest in this work. More formally, one couldargue that ˆ T can be suppressed by invoking a custodialsymmetry. In this work there is a custodial symmetrywhich allows us to neglect the ˆ T -like operator.In addition to the operators listed in eqs. (3) and (7)one can write potentials for both the scalar fields, theEinstein-Hilbert term, couplings of the vector and scalarfield to gravity and topological terms. In this work wewill not back-react the fields on the geometry in neglect-ing backreaction we work under the probe approxima-tion. Interactions of the gauge fields are suppressed bythe large- N c limit ( N c the degree of gauge group in thedual theory) which implies that g L ( R ) ∼ /N c and there-fore in this work the 3-point and 4-point boson verticeswill be set to zero. For a discussion of the large- N c limitsee [18].In summary the leading order bulk action for thismodel is S = Z √ gd xdy Tr (cid:20) − g MN g P Q [ L MP L NQ + R MP R NQ ]+ g MN ( D M H )( D N H ) † + m | H | (cid:21) (8)where the metric is defined in eq. (1). This is the simplestconstruction possible for this model. Additional termswill be discussed in later sections. To break the electroweak symmetry the scalar field ob-tains a VEV h H i = 12 f ( y ) e iπ ( x µ ,y ) , (9)where π = π a σ a / SU (2) × SU (2) → SU (2). f ( y ) is a VEV profileand is the unit matrix. Notice that the exponent doesnot include a factor of f ( y ) for convenience.It is useful to perform calculations in the basis of axial(A) and vector (V) fields defined as A M = g L L M − g R R M p g L + g R , (10) V M = g R L M + g L R M p g L + g R . (11)The bulk equations of motion for the gauge fields canbe written by first Fourier transforming in 4D and sep-arating the gauge field out into a y -dependent functionand a q dependent function, A µ ( x µ , y ) → a ( q , y ) ˆ A µ ( q ) , (12) V µ ( x µ , y ) → v ( q , y ) ˆ V µ ( q ) . (13)Using this notation the equations of motion are, (cid:20) ω ∂ y ω∂ y + q (cid:21) v ( q , y ) = 0 (14) (cid:20) ω ∂ y ω∂ y + q −
14 ( g L + g R ) ω f ( y ) (cid:21) a ( q , y ) = 0 (15)The IR boundary conditions, found by demanding thatthe field variations of the action eq. (8) vanish on the IRboundary, are − ω ( y ) ∂ y v ( q , y ) | y = L − ω ( y ) ∂ y a ( q y ) | y = L (cid:27) = IR boundary terms (16)For the case where there are no IR boundary terms in theaction the expressions above both vanish. Substitutingthe solutions of the equations of motion back into theaction leaves the 4D UV boundary theory with action S = Z ˆ A µ ( − q )Π A ( q ) P µν ˆ A ν ( q )+ ˆ V µ ( − q )Π V ( q ) P µν ˆ V ν ( q ) d q, (17)Π V ( q ) = ω ( y ) ∂ y v ( q , y ) v ( y ) (cid:12)(cid:12)(cid:12)(cid:12) y = L + UV terms , (18)Π A ( q ) = ω ( y ) ∂ y a ( q , y ) a ( y ) (cid:12)(cid:12)(cid:12)(cid:12) y = L + UV terms . (19)where P µν = η µν − q µ q ν /q and Π A,V are vacuum polar-isations. The expression ‘UV terms’ refers to contribu-tions to Π
A,V that originate from operators in the actionthat are localised on the UV boundary.In the small- L limit, there is a logarithmic diver-gence associated to Π V,A . This can be cured by addinga boundary counterterm of the form S C.T. = Z √ gd xdyδ ( y − L ) Zg µν g ρσ × (cid:20) −
12 Tr L µρ L νσ − R µρ R νσ (cid:21) (20)to the action ( S = S + S C.T ) and adjusting Z to removedependence upon the cut-off L . Choosing to renormaliseonly R means that the associated charged states decou-ple. In this way the lightest modes of the spectrum arethat of the standard model. Above these modes a fullcopy of the broken SU (2) × SU (2) is realised at each levelin the tower of states. For further discussion, includingthe gauge invariance of S C.T. see [18].
A. Precision Electroweak Parameters
The precision parameters ˆ S and W are defined asˆ S = gg ′ ddq Π W B ( q ) (cid:12)(cid:12)(cid:12)(cid:12) q =0 (21) W = 12 m W d d ( q ) Π W W (cid:12)(cid:12)(cid:12)(cid:12) q =0 (22)where Π W B and Π W ,W are elements in the vacuumpolarisations expressed using the ( W , B ) basis. Thecouplings g and g ′ are the SU (2) and U (1) couplingsin the standard model. The vacuum polarisations arenormalised so that ddq Π BB ( q ) (cid:12)(cid:12)(cid:12)(cid:12) q =0 = ddq Π + − ( q ) (cid:12)(cid:12)(cid:12)(cid:12) q =0 = 1 (23)Π + − (0) = m W (24)where ± refers to the W ± fields and m W is the W ± mass. B is the U (1) gauge field of the Standard Model. B. Elementary Example
In this section the simplest construction of an AdSmodel of electroweak symmetry breaking is studied. Thiswill provide a basis for further discussion, define somenotation and remind the reader of material available inthe literature [4–6, 12, 15, 17, 20–32].By using the effective action eq.(8) and a given back-ground ω the VEV profile f ( y ) can be found by the solv-ing the equations of motion for H . The simplest case is anAdS geometry which implies the VEV profile that solvesthe equations is schematically f ( y ) ∼ c y ∆ + c y − ∆ .Now pick just one of the two power-laws, our choice isthat f ( y ) = Υ (cid:18) yL (cid:19) ∆ and ω ( y ) = Ly , (25) Υ is the VEV and ∆ is the anomalous dimension of thecondensate. In this case the 5D mass of H is related tothe VEV profile through m = ∆(∆ − L . (26)The early sections of this paper will use eq. (25) as theVEV profile. In later sections contributions will be addedto the action that imply a different VEV profile.The computation of ˆ S proceeds by solving the equa-tions of motion for the profiles v ( q , y ) and a ( q , y ), sub-stituting into the vacuum polarisations given in Eqs. (18)& (19) and taking the small- L limit. The counterterm Z is dialed to remove the divergences that occur. Therenormalised Π V,A can be rotated back into the ( W , B )basis and ˆ S extracted.In general the equations of motion do not have closedform solutions. However, for the case of eq. (25), the vec-tor equation has a closed form solution. Solving eq. (14)with the choice of eq. (25) and expanding the resultingexpression for Π V ( q ) in powers of L one encounters adivergence that is cured by the counterterm Z if Z = L log L L + L ǫ . (27)In this case Π V is rendered free from spurious dependenceon the UV cut-off L but one introduces a new parameter ǫ . With the vacuum polarisation free from divergencesthe limit L → P ( q , y ) = ω ( y ) ∂ y log a ( q , y ) (28)which satisfies a differential equation derived from eq.(15),1 ω P ′ ( q , y ) + 1 ω P ( q , y ) + q −
14 ( g L + g R ) ω f ( y ) = 0 . (29)Expand P in powers of q so that P ( q , y ) = P ( y ) + q P ( y ) + . . . which satisfy the differential equations1 ω P ′ + 1 ω P −
14 ( g L + g R ) ω f = 0 , (30)1 ω P ′ + 2 1 ω P P + 1 = 0 . (31)For the case of eq. (25), where the boundary conditionis written as P ( q , L ) = 0, both equations have closedform solutions. The expressions that solve the equationsof motion can be found in Appendix A.A simple expression for ˆ S can obtained in the sameregime of ( g L + g R ) L Υ ≪ > S ≃ g L ǫ ∆ + 1∆ Υ L ≃
12 ∆ − m W L . (32)where the W mass m W = g L m Z / ( g L + g R ) has beenintroduced. Further details of this calculation can befound in the literature [4–6, 12, 15, 17, 20–32]. For thiswork the conclusion to draw is that ˆ S is positive definitein the most simple case we study, as already noted in[12].Now consider the same background and VEV profilebut with a modified action that no longer contains gaugefields. The SU (2) × SU (2) symmetry is now a globalsymmetry of the scalar field H . In doing this a theorydescribing the scalar field π is obtained. Recall that π was defined in eq. (9) as the scalar rotations about theVEV. Working in the unitary gauge of the full local the-ory, the pions are Higgsed away: only when the gaugecouplings are turned off are the pions manifest (and ina unitary gauge). This allows one to ensure that theproposed action for some model of electroweak symme-try breaking satisfies the conditions of a sensible theory– in this case that the theory is free from negative-normstates. S = Z √ gd xdyg MN Tr ( ∂ M H )( ∂ N H ) † + m Tr | H | = X a Z d xdyπ a [ − ω ( y ) f ( y ) η µν ∂ µ ∂ ν + ∂ y ω ( y ) f ( y ) ∂ y ] π a − Z d x ω ( y ) f ( y ) π∂ y π a (cid:12)(cid:12) y = L y = L + terms in f ( y ) only (33)Fourier transforming in 4D the pion field π a ( x, y ) =ˆ π a ( q ) σ ( q , y ) the equation of motion and boundary con-dition is,0 = (cid:2) f ( y ) ω ( y ) q + ∂ y ω ( y ) f ( y ) ∂ y (cid:3) σ ( q , y ) (34)0 = − ω ( y ) f ( y ) ∂ y σ ( q , y ) (cid:12)(cid:12) y = L . (35)The resulting boundary theory is X a Z d q ˆ π a ( q )Σ( q )ˆ π a ( − q ) , (36)Σ( q ) = 18 ω ( y ) f ( y ) ∂ y σ ( q , y ) σ ( q , y ) (cid:12)(cid:12)(cid:12)(cid:12) y = L . (37)For the simplest case it is instructive to demonstratethat the pion field is a positive-norm state. This willprovide a starting point for the discussion of more so-phisticated models later in the paper. The equation ofmotion for σ ( q , y ) can be solved and substituted into Σ.Expanding at small L and assuming ∆ = 1,Σ( q ) = "(cid:18) L L (cid:19) − + 14 + O ( L ) Υ L (∆ − L q . (38)There are two regimes to consider. If ∆ > L − term goes to zero as L →
0, and the resultingco-efficient of the q term is positive definite implyingthat the theory is healthy. If ∆ < L − term in Σ diverges as L → q co-efficient is negative indicating a negative-norm state. The theory issick when ∆ < V in eq.(B1). In the expression for Π V is a log-divergence. Ef-fectively the L − term in Σ when ∆ = 1 ‘becomes alogarithmic divergence’ when ∆ = 1. The expression forΣ is proportional to q log L /L . As L → >
1. This is illustratedby the negative sign found in Σ in the ∆ < trivial . Finally the casethat ∆ > >
1, the pion is positively normalisedor that ˆ S is positive. Any one of these three statementsimplies the remaining pair.Under the assumption that ∆ >
1, the pion field isa positive-norm state and ˆ S is positive. With the basiccase analysed we can begin adding additional terms tothe model. In doing we ask whether given ∆ > ′ ( q ) > S < III. BOUNDARY ACTIONS FOR SCALARS
Now consider eq. (8) with the addition of new piececonsisting of a boundary Higgs term S DH = Z √ gd xdy (cid:20) δ ( y − L ) λ UV + δ ( y − L ) λ IR (cid:21) × g µν Tr ( D µ H )( D ν H ) † (39)so that the action is now S + S DH .When λ IR = 0, and in the unitary gauge, S DH doesnot change the expression for ˆ S in terms of Lagrangianparameters. Though it will change the spectrum andso the numerical value of ˆ S when the values of physicalparameters are set. The constraint from requiring thatthe mass of the lowest lying axial state be positive is that λ UV > − L ∆ − (cid:18) L L (cid:19) − . (40)If Σ( q ) is computed in the case of a global symmetry,analogous to the computation in Sec. II B then a situa-tion can occur where Σ ′ ( q ) < q = 0. This imposesa constraint that for the pion to be positively normalised λ UV > − L ∆ − (cid:18) L L (cid:19) − . (41)The pion constraint is more restrictive than that fromcoming from the requirement of positive boson masses.Notice this means that it is possible to have a pion fieldthat is a negative-norm state but not see any pathologiesin the spectrum or ˆ S . In Sec. VI this result will re-occur.Now consider the case that λ UV = 0 in eq. (39) and λ IR is unspecified. In this case one is changing the IRboundary condition to P ( q , y ) = 18 λ IR ( g L + g R ) ω ( y ) f ( y ) (42)for the axial field. Resolving the equations of motion inthis case leads to different expressions for P and P thenby extension for ˆ S . Full details of the computation aregiven in App. A. Of interest to this discussion is thatprovided ( g L + g R ) L Υ ≪ S = ǫ L ∆ g L L Υ (cid:20) ∆ + 1 + 2 L (∆ − λ IR (cid:21) (43)By tuning λ IR one can produce ˆ S negative. The boundto produce ˆ S as positive is λ IR > − L − λ UV = 0 , λ IR unspecified. By adding eq.(39) to the action the IR boundary condition changes to (cid:2) ∂ y σ ( q , y ) − λ IR q σ ( q , y ) (cid:3) y = L = 0 . (45)Re-solving the equations of motion and computing thenew form of Σ givesΣ( q ) = L + 2(∆ − λ IR L (∆ −
1) Υ L q . (46)In order that the pion be a properly normalised state λ IR > − L ∆ − . (47)This bound is more restrictive than the region of negativeˆ S .In the case that neither λ IR or λ UV is constrained onemay try to dial λ IR so that ˆ S is negative and then dial λ UV so that the pion is nevertheless a positively nor-malised state. However this will not allow a healthy the-ory to be produced. If one dials λ IR so that ˆ S is negative FIG. 1: A plot of the λ UV with changing λ IR . The blueregion represents those points in parameter space wherethe pion field is a positive-norm state. The yellow regionmarks where λ UV is so small as to produce unphysicalgauge fields. The orange regions marks the points inparameter space where ˆ S is negative. The plot shows itis not possible to produce a healthy theory withnegative ˆ S by dialing scalar boundary term co-efficients.For this graph λ IR has been normalised with a factor of L to make it dimensionless and similarly a factor of L (cid:16) L L (cid:17) − has been included in λ UV .then in order to obtain a positively normalised pion onewould require that λ UV > − L − λ IR L ∆ − (cid:18) L L (cid:19) − . (48)If λ UV is made as small as possible, to saturate thebound on producing negative boson masses, then λ IR isstill bounded to be above the point at which ˆ S becomesnegative. This is illustrated in fig. (1).Therefore it is not possible to obtain ˆ S negative in aphysically acceptable theory by dialing a kinetic bound-ary term for the scalar field. IV. BOUNDARY ACTIONS FOR VECTORS
Now consider eq. (8) with the addition of a new termthat introduces an off-diagonal operator localised on theUV boundary. Such a term gives a direct contribution toˆ S and shifts the mass spectrum.In doing this one must be careful not to break fur-ther symmetry. Our model contains a bulk breaking of SU (2) L × SU (2) R → SU (2) V . In adding an off-diagonalterm to the UV boundary the mass basis on the UVboundary may no longer be that of the axial and vectorfields – the mass basis in the bulk. If this were to hap-pen then the theory need not contain a spectrum of masseigenstates with the quantum numbers of the standardmodel. For example adding the most obvious operator,of the form L µν HR ρσ H † , to the UV boundary would givea mass basis on the boundary that does not match thebulk mass basis.If L µν HR ρσ H † is re-written as part of another termthat ensures any mass is given to the axial fields only(so that no further symmetry breaking takes place)the symmetry breaking of the standard model is en-sured. This can be done by using the operator[ D µ , D ν ] H ([ D ρ , D σ ] H ) † . Define S ˆ S = Z √ gd xdyδ ( y − L ) λ ˆ S g µν g ρσ × Tr ([ D µ , D ρ ] H )([ D ν , D σ ] H ) † (49)so that the action we study is now S + S ˆ S .To compute ˆ S one takes what could be termed the‘bulk contribution’ originating from S and adds the di-rect contribution from S ˆ S . Combining the two contribu-tionsˆ S = ǫ g L ∆ + 1∆ Υ L + 12 g L g R λ ˆ S Υ (cid:18) L L (cid:19) LL . (50)Which implies that ˆ S is positive whenever λ ˆ S > −
18 ∆ + 1∆ ǫ g L g R LL (cid:18) L L (cid:19) . (51)By introducing ˆ S the masses of the axial states are ad-justed. In order that the gauge boson masses be posi-tively normalised requires that, λ ˆ S > − LL (cid:18) L L (cid:19) . (52)This result requires a well-behaved expansion in Υ ( g L + g R ) L . Notice that this bound does not directly constrainthe sign of ˆ S . The two bounds on λ ˆ S can be mutuallysatisfied to give a negative ˆ S in a healthy theory.The mechanism that is employed can be understood inthe following way. Using the action of S + S ˆ S , the pa-rameter ˆ S is schematically a bulk term plus a correctionproportional to λ ˆ S . Meanwhile the q -coefficient of theaxial vacuum polarisation has a log-divergence (which isrenormalised by the counterterm Z into an ǫ -dependentterm). One can push λ ˆ S very large and negative andstill leave a properly normalised gauge field because ofthe log-divergence. After renormalisation the divergence‘becomes the ǫ -term’ in the vacuum polarisation. It isthen expected that one may choose ǫ in such a way asto properly normalise the gauge field. Meanwhile ˆ S canbecome negative as one dials λ ˆ S and ǫ . Physically the scenario studied is of some higher the-ory that has a contribution to ˆ S . This is parameterizedby λ ˆ S in the effective description used here. If this highercontribution were to start out as negative then we haveshown that the final value of ˆ S can remain negative, asone would expect. It remains to be learnt whether or notit is possible to write a higher-energy description (i.e.moving beyond a simple 5D model) that can give fun-damental contributions to ˆ S that are negative. Somedistance towards this goal is achieved in later sections. V. RECOVERING THE HIGGSLESS MODELS
This section introduces an alternative logic to the ex-isting literature on how one can understand the rela-tionship between Higgsless models [4–6] and electroweaksymmetry breaking models based around AdS/QCD [12,18, 25–27, 31, 32].To explain the idea this section introduces, considerthe derivation of IR boundary conditions in this work ascompared to the Higgsless models [4–6] or [14].In this work the IR boundary conditions of bulk fieldsare obtained by demanding that the variation of the ac-tion vanish on the IR boundary. By placing terms lo-calised on the IR boundary the conditions imposed onthe bulk fields can be adjusted. In this way one can ar-range for whatever boundary conditions are appropriate.While in a Higgsless models one simply enforces a bound-ary condition on the gauge fields – the conditions are notthe result of some variation of the action vanishing.It is easy to write down an IR boundary action fora Higgsless model so that the conditions arrived at bydemanding the variation of the action vanish are the onesthat were being enforced. Going to the lengths of writingdown a boundary action gives two benefits.Firstly, as has been extensively made use of in thiswork, one has a pion field that can be studied. If thegauge field boundary conditions are not derived, thenone does not know what the consistent set of conditionsfor the remaining (i.e. non-gauge) fields are. As such,the bulk profiles are not fully known and the fields cannot be studied. The pion is an example of such a field.Secondly the IR boundary action motivates one tothink in an alternative way about AdS slice models thatis more general than previous thinking. In particular oneis able to connect a large class of models as being specialcases of a single model. This is the idea that we wish todiscuss further in this section.To illustrate how models formulated on an AdS slicecan be connected, begin with the simpler task of pro-ducing a Higgsless model from eq. (8). There are twodifferences between the starting point of eq. (8) and thegoal of a Higgsless model. Firstly there is no bulk break-ing in a Higgsless model and secondly the IR boundarycondition of the axial field must be changed.Add an IR boundary term for the scalar field of theform δ ( y − L ) λ IR | D µ H | (53)to the action eq. (8). Our aim in adding this term is toproduce the boundary condition A µ = 0 at the IR bound-ary; as is used in the Higgsless model. This boundarycondition is achieved by taking λ IR → ∞ . This leavesthe problem of the bulk scalar kinetic term, which mustbe switched off – but the IR boundary scalar kinetic mustbe kept.Therefore consider two limits. Firstly the VEV profile f is taken to zero — which can be achieved by dialingthe overall numerical co-efficient Υ. This removes thebulk breaking term. However, to prevent the boundaryterm | D µ H | from vanishing while taking Υ → ζ = λ IR Υ held fixed. Then the limit to produce a van-ishing axial field on the IR boundary is taken by ζ → ∞ .The resulting theory is of the same class of models asconsidered in the Higgsless scenario.The procedure of limits promotes a way of thinkingabout the model in eq. (8) and a Higgsless model. A Hig-gsless model is a generic model of electroweak symmetrybreaking formulated on a slice of AdS space where theIR bulk breaking is far stronger than any bulk breaking.Conversely, the model of eq. (8) is the case of a genericAdS model where bulk breaking is the most significantsymmetry breaking.An alternative to the limit procedure discussed here isto take ∆ → ∞ limit directly in eq. (8). This has theeffect of localising the bulk Higgs term to the IR wall.In fact, rather than thinking in terms of special caseswhere bulk breaking is small, as is being promoted inthis section, for the specific scenario of comparing themodel of eq. (8) and the Higgsless case one can alwaystake the limit ∆ → ∞ in any result derived from eq.(8). To see this is correct notice that the value ∆ = 1 isalways critical, above and below unity represent differentphysical scenarios. Therefore increasing ∆, when it isalready larger than one , will not lead to new phenomena.As a result one can take ∆ → ∞ without encounteringproblems and indeed obtains the correct results.Having seen an explicit example, the argument of thissection can be stated concisely. Given a specific AdS slice model, one can consider it as a particular point inparameter space of the most general model, that is onethat includes all possible bulk and boundary terms. InSec. II it was discussed how to build the most generalAdS slice model: one simply lists all possible operatorsthat can live in the bulk and on the boundary then writesa model using operators up to some order in q . Byconsidering various limits; for example where the bulkbreaking is sub-leading to the boundary breaking or thereis a significant off-diagonal term present in the bulk, onecan arrive at the specific models in the literature.As a further example the ‘effective metric’ scenarioof [14] is produced as a special case of the most generalAdS action one could write down. In this scenario a bulkterm of the form L MN HR MN H † is needed in addition to the boundary scalar kinetic term and the basic actioneq. (8). As before (see Sec. IV) an off-diagonal term isbest included by adding a term of the form λ ˆ S Tr ([ D M , D N ] H )([ D P , D Q ] H ) † (54)into the bulk action[ ? ]. The procedure of limits then fol-lows the previous reasoning. To remove the bulk kineticterm in eq. (8) take Υ → ζ = λ IR Υ and λ ˆ S Υ fixed. This leaves a model with the correctfield content and Lagrangian terms of [14]. In the limitthat ζ → ∞ the correct IR boundary condition is also ob-tained. Therefore this scenario is also a particular caseof the general AdS model one could begin with.Finally, it is instructive to further study the procedureof limits required to produce a Higgsless model and inparticular the spectrum. This section makes use of theability to study the pion field when one uses boundaryfield variations to obtain boundary conditions.Once again consider the action required to produce aHiggsless model, this is eq. (8) and an IR boundary scalarfield. In the limit of Υ → ζ = Υ λ IR fixed theaction becomes S Υ → = Z √ gd xdyζ ′ δ ( y − L ) g µν ( D µ e iπ )( D ν e iπ ) † − g MN g P Q
12 Tr (cid:0) L MP L NQ + R MP R NQ (cid:1) , (55) ζ ′ = 14 (cid:18) yL (cid:19) ζ. (56)The spectrum of neutral states for this theory, in the limitthat ζ → ∞ , is two light states, one vector and one axial.These states correspond to the Standard Model photonand Z . Above that is a tower of masses approximatelygiven by the zeros of J ν ( L q ) where ν = 1 for the axialbosons and ν = 0 for the vector bosons. For large val-ues of q the masses of the axial and vector bosons willform two towers of states each tower having the spacingbetween states of π/L . The two towers are offset – thelightest axial state that arises from the zeros of J ( L q )is heavier by π/ (2 L ) than the lightest vector state, thatoriginates from the zeros of J ( L q ).Using S Υ → the pion field can be studied by turningoff the gauge couplings. Solving the equation of motionfor σ and applying the IR boundary condition σ = 0 thatresults from taking ζ → ∞ gives ∂ y σ ( q , L ) σ ( q , L ) = s q → ∞ as ζ → ∞ (57)where s is the constant of integration. Solving for s requires that s → ∞ which causes Σ( q ) to diverge too.The pion profile has become a constant because of thelimit Υ →
0. After normalising the pion kinetic term tounity the pion will become a free field. For this reason,that the pion is free in the limit ζ → ∞ , to study the‘effective metric’ scenario of [14] one needs to use a largebut finite ζ and consider increasing ζ and the effects ithas on the pion and ˆ S . (For example, it may be that thepion is healthy at large and finite ζ in a region where ˆ S isnegative and increasing ζ does not change this. Thereforeone has evidence of a region in parameter space withnegative ˆ S and a pion that appears to be healthy.)Studying the pion field shows that by forcing A µ = 0one produces a theory with a free pion field. As a resultthe symmetry breaking induced by H can not be restoredat finite energy. This is reflected by the fact that thespectrum of the gauge bosons do not become degenerateat high energies, instead the spacing is constant at allenergies, illustrated in Fig. 2. --- --- --- --- ------ --- --- --- ------ --- --- --- ------ --- --- --- ------ --- --- --- ------ --- --- --- ------ --- --- --- ------ --- --- --- ------ --- --- --- ------ --- --- --- ------ --- --- --- ------ --- --- --- ------ --- --- --- ------ --- --- --- --- Ζ M a ss FIG. 2: The mass spectrum with increasing boundaryHiggs VEV ζ . The red lines represent the masses of theaxial states and the blue lines the masses of the vectorstates. As ζ increases the axial and vector masses donot become approximately degenerate until higher mass. VI. EFFECTIVE METRICS
This section contains a study of the effects of includinga bulk off-diagonal term. To produce a scenario similarto that of [14] we also include an IR boundary term.Therefore we add S = Z √ gd xdyg MN g P Q Tr (cid:18) − λ B Tr ([ D M , D P ] H )([ D N , D Q ] H ) † (cid:19) + δ ( y − L ) λ IR g µν Tr D µ H ( D ν H ) † . (58)to eq. (8) as the action we study in this section. We con-sider two cases, with and without the bulk Higgs kineticterm of eq. (8) switched on. A. The Limit of No Bulk Kinetic Term
Here the case that Tr | D M H | is switched-off in thebulk is studied. Doing this keeps the number of free parameters in the survey under control.The logic of this section is that: Given an action itis possible to produce a negative ˆ S by dialing the co-efficient of a bulk off-diagonal term (see [14] for exam-ples). However one must check that the pion field ishealthy, as has been focused on in this work. To controlthe study and not produce an overly complicated picturewith many free parameters we study ˆ S under the scenariothat the bulk breaking is weak. In fact we will neglect thebulk term altogether. When examining the pion, the bulkkinetic term of the Higgs should not be turned off. Theidea that allows one to connect statements about ˆ S in atheory with no bulk Higgs kinetic term and statementsabout the pion is that the limit one takes to remove thebulk Higgs field can be done smoothly and without affect-ing the sign of the pion kinetic term. This implies thatone is analysing a scenario where the pion is healthy, forexample, and one restricts the analysis of ˆ S to a partic-ular corner of parameter space where the bulk breakingis sub-leading.To switch the bulk kinetic term off, one takes the limitΥ → ζ = λ IR Υ and ζ B ( y ) = λ B ( y )Υ fixed (see Sec. V). In the axial and vector field basis theaction becomes Z d xdy − η MN η P Q
Tr [( ω ( y ) + ω ( y ) ζ B ( y )) V MP V NQ + ( ω ( y ) − ω ( y ) ζ B ( y ) A MP A NQ ] + 14 ζ (cid:18) yL (cid:19) × δ ( y − L ) η µν Tr ( D µ e iπ )( D ν e − iπ ) † . (59)Our study will be over a range of values for ζ . Of par-ticular interest is the limit ζ → ∞ where one obtainsa similar scenario to that found in [14]. A large nega-tive value of ζ leads to tachyonic axial masses. Introduce‘effective metrics’ defined as ω V ( y ) = ω ( y ) (cid:20) ζ B ( y ) y L (cid:21) , (60) ω A ( y ) = ω ( y ) (cid:20) − ζ B ( y ) y L (cid:21) . (61)Using this notation the action becomes, Z d xdy − η MN η P Q
Tr [ ω V ( y ) V MP V NQ + ω A ( y ) A MP A NQ ] + 14 ζ (cid:18) yL (cid:19) × δ ( y − L ) η µν Tr ( D µ e iπ )( D ν e − iπ ) † . (62)From this, and comparing to the equations in Sec. II Bone can read off the equations of motion for the axialand vector fields. In the case of the pion, the equation ofmotion is the same as in previous sections. However aswe will choose to specify the profiles ω A,V one must findthe metric by simultaneously solving the definitions of ω A,V for ω . Solutions to the axial and vector equations ofmotion, in the limit of no Higgs kinetic term are discussedin Appendix B.Discussing bulk off-diagonal terms analytically re-quires a survey of all functions that tend to 1 /y at small y . Handling such a space of functions and various inte-grals of them usually results in numerical surveys. Nev-ertheless one can make some progress if the metric pro-files ω X ( y ) are assumed to be a monotonically decreasingfunction of y (or more generally a decreasing function ofsufficiently modest variation).Under this assumption, first examine the scenario ζ =0 – where there is no electroweak symmetry breaking. Itwill be shown that it is not possible to obtain a negativecontribution to ˆ S given that the pion must be a positive-norm state. Then the result is extended to ζ = 0.Recall that in the case of ζ = 0 the pion vacuum po-larisation at order q is (written in terms of ω A,V ) Z L y ( ω V ( y ) − ω A ( y ))( ω V ( y ) + ω A ( y )) dy (63)For a positive-norm state we demand that this is posi-tive definite which can be ensured if ω V > ω A ∀ y . Whenapplying this constraint to ˆ S one can make progress byconsidering the cases of c and R dx/ω A ( x ) dominatingthe bracket c + R dx/ω A ( x ) (see eq. (A10) ). If c dom-inates the bracket then at leading order in 1 /c the signof ˆ S depends upon the difference ω V − ω A which is posi-tive definite by choice. Therefore only when the bracketin Π A is dominated by R dx/ω A ( x ) (and c is a correc-tion) can ˆ S become negative. In this case, the sign of ˆ S depends on whether Z dyω A ( y ) Z L y dxω A ( x ) > Z L y dx (cid:18) ω A ( x ) Z dyω A ( y ) (cid:19) (64)with ˆ S positive if the inequality is satisfied. Given a de-creasing function ω A , either monotonically or of sufficientrate, this inequality is satisfied and ˆ S is positive.When ζ = 0 one is changing the value of c . However ithas just been argued that a negative ˆ S only occurs whencorrections to ˆ S from c are suppressed relative to thecontribution of metric factors themselves to ˆ S (see previ-ous paragraph). In contrast, the line in ( λ IR , λ UV ) spacethat divides the two norm states of the pion, positive andnegative, is shifted as a leading order effect in c . As a re-sult as one increases c more and more of the ( λ IR , λ UV )plane consists of negative-norm pion states. Similarlythere are an increasing number of theories with negativeˆ S but the rate at which the number of theories increasesis lower that at which theories are ruled out – becausethe correction to ˆ S from changing c is sub-leading.In summary there are two cases to consider in under-standing the expressions relating to ˆ S . In either case itis not possible to arrive at a situation where the pion ishealthy and ˆ S is negative.By way of illustration a survey of metrics with warpfactors of the form ω X ( y ) = Ly cos o X y (65) - - - o V o A S `=- S `= FIG. 3: A scan over the ‘effective metric’ parameters o A,V producing values of ˆ S , indicated by colour. Ablack line marks the regions of space, marked ‘healthy’(‘unhealthy’), where the pion field is a positive(negative)-norm state.is given in fig. 3. The figure shows the ˆ S parameter as afunction of o A,V with the constraint from requiring thatthe pion be a positive-norm state added. The value ofthe o A is constrained to be positive and less than ap-proximately 1.5 (beyond this value the axial 2-point isnot defined due to the specific metric profile choice).The chosen form of metric pre-factor 1 /y cos o X y canbe thought of as corresponding to ∆ = 2 in the case ofthe earlier notation eq.(25). To see this consider the pionfield with a non-zero f , solve for ζ B λ B f = ω V ( y ) − ω A ( y ) ω V ( y ) + ω A ( y ) . (66)If the expression is Taylor expanded for small y then onehas that f ∝ y which implies that in the UV one isexamining a bulk profile that is dual to condensate withanomalous dimension of γ m = 3 − ∆ = 1, through theAdS/CFT dictionary.
1. Infinite VEV Boundary Higgs
Previous authors have considered the case that theboundary breaking term is infinite. As has been shownin sec. VI A the theories considered that have ˆ
S < S by some mechanismthen this will show up in the low energy description as afundamental contribution to ˆ S . B. W and Y Parameters
In principle, models based around eq. (8) have uncon-trolled contributions to the W and Y electroweak preci-sion parameters [18]. Having explored the possibility ofnegative ˆ S it is interesting to consider how these modelscontribute to the q -order precision parameters. Restric-tions on the W and Y parameters from experiment implyan upper bound of W, Y . − [3]. From na¨ıve dimen-sional analysis and applying the process of holographicrenormalisation [18] W ∼ O ( ǫ m W /M ρ ) where m W isthe W -mass and M ρ is the ρ mass. The parameter ǫ is the renormalisation parameter that scales as N . Themass of the ρ is limited by the ˆ S parameter. The valueof ǫ is constrained by requiring that the theory be atlarge- N (implying large ǫ ) and that W and Y be withinexperimental bounds (implying a small or moderate N ).The relationship between the bulk and boundary gaugecoupling can then be used to understand whether someconsistent phenomenological scenario agrees with the as-sumptions of the model. That is, given that the precisionparameters are of acceptable values for some choice of L and ǫ , is the bulk coupling perturbative. If a consistentpicture is not available then one can increase the IR scale.This lowers ˆ S and raise the ρ mass. The models discussedin this work have the potential to provide scenarios whereit is W and Y that are the most constraining of the pre-cision parameters, rather than ˆ S .Driving the IR cut-off to higher and higher values be-gins to imply fine tuning. For example it can be thoughtof as “natural” that the IR scale is of order 1 TeV be-cause it is related to the electroweak symmetry breakingscale but increasing it greatly above such a value couldbe seen as questionable.An example of this behaviour is given by, ω X = Ly cos o X y . (67)Using the reasoning that allowed us to connect an “ef-fective metric” to ∆ = 2, this second profile, here, cor-responds to ∆ slightly greater than one. In the simplecase of the VEV in eq.(25), it has been shown [18] thatchoosing a small ∆ constrains ǫ to be below one (aftersaturating the ˆ S bound) and thus providing an examplewhere W is the most constraining parameter.Whilst the example given is closely linked to existingmodels in the literature, it serves to highlight the factthat in choosing arbitrary profiles one will produce a largenumber of models where W and Y are the precision pa-rameters that most constrain the IR cut-off of the model. C. Inclusion of Bulk Higgs Kinetic Term
In the most general case, that there is an off-diagonalbulk term, a boundary breaking term and a bulk break-ing term, again one may try to play the two breakingschemes against one another and produce a negative ˆ S and a healthy theory (similar to Sec. III). This studywill be entirely numerical and have a number of free pa-rameters that would be hard to control. For this reasona detailed study of this most general case is beyond therealm of the present work. VII. CONCLUSIONS
This work has investigated the possibility of a nega-tive ˆ S parameter within models of electroweak symmetrybreaking formulated on a 5D slice of AdS. Attention hasbeen given to ascertaining whether a particular scenariothat gives a negative ˆ S is acceptable on physical grounds.In the simplest AdS model, which has been well-studied in the literature already, it was demonstratedthat the pion field associated to the symmetry breaking isa positive-norm state if and only if ˆ S is positive. In turnboth of these statements are equivalent to the statementthat the scaling of the condensate, ∆, is greater than one.To consider possibilities where ˆ S is negative additionaloperators were added to the UV and IR boundaries, thenfinally also to the bulk. This indeed allowed situationswhere ˆ S is negative. Then, within each scenario consid-ered, the norm states of each field was considered; whichlimited the parameter space. This implied that it wasnot possible to produce a negative ˆ S in a simple theoryby dialing Lagrangian parameters and requiring that thetheory be healthy.One particularly interesting scenario was the additionof a bulk off-diagonal term. In principle the most con-straining of the precision parameters of models whichhave off-diagonal contributions in the bulk can be the W and Y parameters. This is in contrast to the usualpicture that ˆ S must be made to fall within bounds bydialing the IR cut-off. Instead one would need to dialthe IR cut-off to make W and Y fall within experimentalbounds. On the question of whether a negative ˆ S couldbe produced in a theory containing healthy fields it wasfound that this could only happen if one considered asituation with a fundamental contribution to ˆ S that wasnegative. This would originate from a higher energy the-ory which the model studied in this work would be a lowenergy effective theory of.In conclusion this study has not found a consistent sce-nario that allows for ˆ S to be negative and all states tohave a positive-norm, without implying a fundamentaland negative contribution to ˆ S originating from outsidethe electroweak symmetry breaking. Our survey is nottotally complete, and it is possible that even higher orderterms to those considered, or an ‘effective metric’ profileexists that provides a healthy theory where ˆ S is negative.1We have demonstrated that this would be an atypicaltheory and that caution should be exercised to ensurethat the theory does not contain a pathology of somekind. In addition we have illustrated that frequently ap-parently healthy scenarios of negative ˆ S can be thoughtof as containing a fundamental contribution to ˆ S that islarge and negative. In this light, it seems unlikely thata generic scenario can occur where ˆ S is negative due tothe contribution of electroweak symmetry breaking. VIII. ACKNOWLEDGEMENTS
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Appendix A: Solution to the Equations of Motion
Here a solution to the equations of motion for P , orderby order in q are given. Recall the equations of motionfor P are 1 ω P ′ + 1 ω P −
14 ( g L + g R ) ω f = 0 , (A1)1 ω P ′ + 2 1 ω P P + 1 = 0 , (A2)1 ω P ′ + 1 ω (cid:0) P + 2 P P (cid:1) = 0 . (A3)2When using the choices of eq. (25) and for ∆ = 1 P ( y ) = L y I (cid:16) αy ∆ ∆ (cid:17) Γ(1 + 1 / ∆) + αy ∆ (cid:16)h I − (cid:16) αy ∆ ∆ (cid:17) + I +1 (cid:16) αy ∆ ∆ (cid:17)i Γ(1 + 1 / ∆) + 2 c I − (cid:16) αy ∆ ∆ (cid:17) Γ(1 − / ∆) (cid:17) I (cid:16) αy ∆ ∆ (cid:17) Γ(1 + 1 / ∆) + c I − (cid:16) αy ∆ ∆ (cid:17) Γ(1 − / ∆) (A4) P ( y ) = exp Z y − xL P ( x ) dx (cid:20) c + Z y x (cid:18) exp Z x zL P ( z ) dz (cid:19) dx (cid:21) (A5) α = 14 ( g L + g R ) L L Υ (A6)where the constants c , are found by imposing the boundary condition.In order to substitute P into the expression for P first expand in powers of y . P ( y ) = 12 α ∆ − y − + 1 c ( − − (cid:16) α ∆ (cid:17) − ( − c − − α ∆ (∆ + 1) (cid:16) α ∆ (cid:17) y + O ( y a ; a > y →
0, there is a term that diverges when ∆ < P ,a constant term and then higher terms that die away. Substituting the expansion of P into the expression for P P ( y ) = log L y + 12 L c ( − − (cid:16) α ∆ (cid:17) + 14∆ L α ∆ − O ([ L ∆1 α ] a ; a >
2) + O ( y ) (A8)Here, the constant of integration, c , has been solved for – because we are only interested in boundary conditionswhich lead to P ( L ) = 0 in the main text. The expansion in L ∆1 α is a more compact notation for the constraintmentioned in the text that Υ ( g L + g R ) L / ≪ V ( q ) = q Z L y ω ( y ) dy + q Z L y dx ω ( x ) "Z L x dzω ( z ) + O ( q ) (A9)Π A ( q ) = 1 c + R dyω ( y ) + q (cid:18) c + Z dyω ( y ) (cid:19) − Z L y dx " ω ( x ) (cid:18) c + Z dxω ( x ) (cid:19) + q (cid:18) c + Z dyω ( y ) (cid:19) − Z L y dx " P ( x ) ω ( x ) (cid:18) c + Z dxω ( x ) (cid:19) + O ( q ) (A10)with P defined as P = Π ′ A (0). Appendix B: Vacuum Polarisations
By solving the bulk equations of motion eq.(14) for thechoice of eq. (25) and substituting into the definition ofthe vacuum polarisation one obtains the unrenormalisedexpression for Π V . Then the counterterm Z is dialed togiveΠ V ( q ) = q (cid:20) − ǫ (cid:18) γ E + log 12 L √ q − π Y ( L √ q ) J ( L √ q ) (cid:19)(cid:21) (B1) for γ E the Euler-Mascheroni number and J, Y
Besselfunctions. In order to produce this expression a normal-isation of Π V has been chosen by choosing Π ′ V ( q ) = 1at q = 0.The expression for Π V allows the spectrum to be readoff. There is a massless photon and a tower of heavierstates with the same quantum numbers as the StandardModel photon. The first such state will be referred to asthe ρ in analogy with QCD and has a mass M ρ dependentupon ǫ . The ρ mass for particular values of ǫ can be given3as M ρ ≃ . L for ǫ small . L for ǫ = 1 . L for ǫ large (B2) To obtain the axial vacuum polarisation, Π A , the ex-pressions for P , are used. The counterterm Z removesthe log-divergence originating from P . The result for∆ > A = ǫ L (cid:20) c ( − − (cid:16) α ∆ (cid:17) + q (cid:18) L c ( − − (cid:16) α ∆ (cid:17) + 14∆2 L α ∆ − (cid:19) + O ( q ) (cid:21) (B3)Assume that the first zero of the vacuum polarisation canbe well approximated by the ratio of the first two terms inthe q expansion of Π A . Provided that ( g L + g R ) L Υ ≪ > Z mass is approximately m Z ≃
18 1∆ − ǫ Υ ( g L + g R ) L L ..