Standard Model Extended by a Heavy Singlet: Linear vs. Nonlinear EFT
LLMU-ASC 35/16August 2016
Standard Model Extended by a Heavy Singlet:Linear vs. Nonlinear EFT
G. Buchalla, O. Cat`a, A. Celis and C. Krause
Ludwig-Maximilians-Universit¨at M¨unchen, Fakult¨at f¨ur Physik,Arnold Sommerfeld Center for Theoretical Physics, D–80333 M¨unchen, Germany
Abstract
We consider the Standard Model extended by a heavy scalar singlet in differentregions of parameter space and construct the appropriate low-energy effective fieldtheories up to first nontrivial order. This top-down exercise in effective field theoryis meant primarily to illustrate with a simple example the systematics of the linearand nonlinear electroweak effective Lagrangians and to clarify the relation betweenthem. We discuss power-counting aspects and the transition between both effectivetheories on the basis of the model, confirming in all cases the rules and proceduresderived in previous works from a bottom-up approach. a r X i v : . [ h e p - ph ] A ug Introduction
The discovery of the Higgs boson at the LHC together with the absence (so far) ofnew-physics states has triggered a renewed interest in effective field theories (EFTs) atthe electroweak scale. In the last years, there has been a surge of papers reassessingdifferent technical and conceptual aspects (completeness of operators [1, 2], aspects ofpower counting [3, 4], etc.), and a program to carry out the one-loop renormalizationof the EFTs has emerged [5–8]. This has been paralleled by an increasing interest inexploiting the potential of EFTs as a phenomenological tool for indirect searches of newphysics at the LHC [9–13]. One of the main goals of the recent developments is to getthe formalism ready for the level of scrutiny required at the LHC in the forthcomingRun II and III (see, e.g., [14] for an updated review).The main virtue of an EFT approach is that it is general and model-independent.Once (i) the symmetries and the particle content relevant at the scale of interest and(ii) the nature of the underlying dynamics are specified, the resulting set of operatorsrepresents the most general way in which deviations caused by ultraviolet (UV) physicscan be parametrized. If the UV physics is known, one can construct the EFT by inte-grating out the heavy degrees of freedom. This is sometimes referred to as a top-down approach. EFTs of this sort are typically useful to simplify calculations at low scales.More challenging are those situations where the ultraviolet physics is unknown. Such bottom-up
EFTs heavily rely on (i) symmetry arguments for the build-up of operatorsand (ii) power counting both in order to organize the expansion and to estimate thetypical size of the operator coefficients. By comparing the estimated sizes of operatorswith their experimental bounds one is thus sensitive to indirect effects from new physics.In the electroweak sector, there are two different (bottom-up) EFTs one can build.They both are invariant under the Standard Model gauge symmetry and have the sameparticle content. However, they fundamentally differ in the assumed nature of the dy-namics responsible for electroweak symmetry breaking. As a result, the very nature ofthe EFT expansion, i.e. its power counting, is different. If the underlying dynamics isweakly coupled, new-physics effects decouple and the expansion is in canonical dimen-sions of the fields. In contrast, if the underlying dynamics is strongly coupled (around theTeV scale), new-physics effects do not decouple and the expansion is topological (i.e., inthe number of loops), or equivalently in the chiral dimensions of fields and couplings [3].These two EFTs are normally termed linear and nonlinear, in reference to the re-alization of the electroweak gauge symmetry. In the former, the scalar sector is mostconveniently assembled as an electroweak doublet field Φ( x ), while in the latter it isconvenient to split the Goldstone modes and the Higgs scalar and represent them withthe fields U ( x ) and h ( x ), respectively. Obviously the choice of variables is a matter ofconvention: physics certainly should not depend on how the scalar degrees of freedomare parametrized. The choice of variables simply makes the power counting associatedwith each EFT more transparent.In this paper we would like to show this difference in power counting explicitly from atop-down approach, using a simple UV-complete toy model and integrating out its heavy1egrees of freedom. This model should be rich enough to possess, depending on the valuesof its parameters, a decoupling and nondecoupling regime while still being perturbative.We examine the simplest model that exhibits these features, namely the Standard Modelextended with a heavy real scalar field endowed with a Z symmetry [15–28]. If the heavyfield acquires a nontrivial vacuum expectation value, this model can be recast as a SO (5)linear sigma model both spontaneously and explicitly broken down to SO (4). We showexplicitly how, depending on the sizes of the different parameters, integrating out theheavy scalar generates either a nonlinear EFT (with a pseudo-Goldstone Higgs) or alinear EFT (with a Standard Model Higgs), leading to expansions in either chiral orcanonical dimensions.From a phenomenological viewpoint, this scalar model is far from being realistic asan extension of the Standard Model. On the one hand, current experimental Higgs dataseverely constrain its parameter space [19, 26], especially in the nondecoupling regime.On the other hand, a realistic strongly-coupled sector is likely to be more sophisticated,with a confining phase giving rise to an infinite set of resonances, much like what happensin QCD. However, even in QCD the (linear) sigma model, while not phenomenologicallyrealistic, is still useful to the extent that it illustrates the systematics of the correspondinglow-energy expansion, chiral perturbation theory (ChPT). In this paper, we follow asimilar strategy for the electroweak sector. The value of the toy model is therefore notits phenomenological viability, but the fact that it illustrates in a simple and explicitway how the linear and nonlinear EFTs are related.Interestingly, the scalar toy model not only clarifies the origin of the different powercountings, but also shows that in certain settings the transition between a nonlinear anda linear EFT is not a discrete choice but a continuous one. In particular, there is awell-defined limit, in which the Standard Model is recovered. This supports the claim[12, 13] that using a nonlinear EFT at the LHC is the right framework to determine thenature of the Higgs boson from experimental data.This paper is organized as follows: In Sections 2 and 3 we describe the toy model andwork out its couplings in the nonlinear Higgs representation. In Section 4 we integrateout the heavy scalar in the nondecoupling regime. We work out the effective Lagrangianat tree level up to next-to-leading order (NLO) and find a particular version of theelectroweak chiral Lagrangian (EWChL). In Section 5 we repeat the same steps in theweakly-coupled regime and end up with the Standard Model extended by dimension-6operators. We also examine the transition between the two different regimes. Section 6is devoted to the decoupling limit of the general, model-independent chiral Lagrangian.Expanding this nonlinear EFT for small values of ξ = v /f , the ratio of scalar vacuumexpectation values, to O ( ξ n ), one recovers the expansion of the linear EFT to operatorsof dimension d = 2 n + 4. We do this explicitly for the leading-order (LO) chiral La-grangian through O ( ξ ). We summarize our conclusions in Section 7. Technical detailsare relegated to the Appendix. 2 Model
We consider an extension of the Standard Model (SM) with the Higgs doublet Φ by a realscalar gauge singlet S . Imposing a Z symmetry under which S → − S , the Lagrangianfor the scalar sector reads [15–28] L = ( D µ Φ) † ( D µ Φ) + ∂ µ S∂ µ S − V (Φ , S ) (1)with V (Φ , S ) = − µ † Φ − µ S + λ † Φ) + λ S + λ † Φ S (2)Requiring the potential to be bounded from below and to have a stable minimum implies λ , λ > , λ λ − λ > v + h √ U (cid:32) (cid:33) , S = v s + h √ U = exp(2 iϕ a T a /v ) is the Goldstone-bosonmatrix. The vevs are given by µ = λ v + λ v s , µ = λ v + λ v s (cid:32) hH (cid:33) = (cid:34) cos χ − sin χ sin χ cos χ (cid:35) (cid:32) h h (cid:33) (6)with tan(2 χ ) = 2 λ vv s λ v s − λ v (7)Without loss of generality we may restrict the range of χ to − π/ ≤ χ ≤ π/
2. Themasses of the scalar bosons are M h,H = 14 (cid:104) λ v + λ v s ∓ (cid:112) ( λ v − λ v s ) + 4( λ vv s ) (cid:105) (8)with M h ≡ m < M H ≡ M by convention.The full parameter space of the model in (1) is spanned by the five values of µ , µ , λ , λ and λ . Equivalently, we may express those in terms of the physical quantities m , v , M , f ≡ (cid:112) v + v s and χ , or m, v, r ≡ m M , ξ ≡ v f , ω ≡ sin χ (9)3he two sets of parameters are related through λ = 2 M f r + ω (1 − r ) ξλ = 2 M f − ω (1 − r )1 − ξλ = 2 M f (1 − r ) (cid:115) ω (1 − ω ) ξ (1 − ξ ) (10)together with (5). After fixing v = ( √ G F ) − / = 246 GeV and m = 125 GeV in (9),we are left with r , ξ and ω , parametrizing the dynamics beyond the SM. Apart from theresonance mass M , which sets the scale of new-particle thresholds, and which we assumeto be in the TeV range, this dynamics is essentially governed by the two parameters ξ and ω , where ξ, ω ∈ [0 , SO (5) symmetry. Under this symmetry the four real compo-nents of Φ and S transform in the fundamental representation. This limit is physicallymotivated as the Higgs mass m is then protected by the pseudo-Goldstone nature of thefield h , which is of interest in particular in the strongly-coupled scenario [29].In the strict SO (5) symmetric limit, we have λ = λ = λ ≡ λ = 2 M /f , r = 0and ω = ξ . Also in this limit µ = µ = M . We parametrize deviations from the exactsymmetry by r and δ ≡ ω/ξ −
1. We denote byΣ ≡ Φ † Φ + S (11)the square of the scalar multiplet in the fundamental representation of SO (5). We thendecompose the potential (2) as V ≡ V + V into an SO (5) invariant part, V = − µ + λ (12)and terms that explicitly break the SO (5) symmetry, V = µ − µ S + λ + λ − λ S + λ − λ S (13)The three SO (5)-breaking couplings in (13) correspond to the three different SO (5)-breaking, SO (4)-symmetric operators of dimension less or equal to four that respect the Z symmetry of the model: S , S , and Σ S . All three are governed by the SO (5)-breaking operator S ≡ n T Σ, where n T = (0 , , , ,
1) is the spurion that breaks SO (5)while preserving SO (4). Note that sin χ ≡ sgn( χ ) √ ω . In the following, we sometimes write sin χ = √ ω for simplicity,dropping the sgn( χ ), which has to be included for negative χ . SO (5) breaking, the case of particular interest to us, we require r , δ (cid:28) r and δ , we find, using (5) and (10), µ − µ = M δ − ξ ) λ + λ − λ = 2 M f rξ (1 − ξ ) λ − λ = − M f (cid:18) rξ + δ − ξ ) (cid:19) (14)The requirement r , δ (cid:28) λ i . Similarly, µ − µ remains of order v for large M .Counting parameters, we observe that we can group the five couplings of the originalpotential (2) into the two SO (5)-symmetric couplings in (12) and the three SO (5)-breaking couplings in (13). The former correspond to M and f , the latter to r , δ and ξ .Out of these three, r and δ control the (small) SO (5) breaking, whereas ξ is naturally oforder unity. The last property reflects the degeneracy of vacua in the strict SO (5) limit,which is lifted by the small explicit symmetry breaking triggered by r and δ .For the construction of a low-energy EFT by integrating out high mass scales, we aremainly interested in the following two basic scenarios, depicted in Figs. 1(b) and 1(c):I) strongly-coupled regime (nonlinear EFT) | λ i | ∼ < π , m ∼ v ∼ f (cid:28) M ⇒ ξ, ω = O (1) (15)II) weakly-coupled regime (linear EFT) λ i = O (1) , m ∼ v (cid:28) f ∼ M ⇒ ξ, ω (cid:28) M ≈ πf , corresponding to | λ i | ≈ π . In thiscase, a simple description of the dynamics in terms of a resonance H would cease to bevalid. We assume that the λ i remain somewhat below, in a regime where perturbationtheory is still a sufficiently reliable approximation.We will show that integrating out M in case I) leads to a nonlinear EFT, organizedby a power counting in chiral dimensions. We will also demonstrate that integrating out M ∼ f in case II) gives rise to a linear EFT, organized in terms of canonical dimensions. Following the notation of [2], we write the complete Lagrangian of the SM extended bya scalar singlet as L = L + L hH (17)5 M (cid:41) λf (cid:41) ξv (a) generic singlet model E Mfv (b) nonlinear EFT
E Mfv (c) linear EFT
Figure 1: Schematic picture of the different possible hierarchies. Further details aregiven in the main text.where L = − (cid:104) G µν G µν (cid:105) − (cid:104) W µν W µν (cid:105) − B µν B µν + ¯ qi (cid:54) Dq + ¯ (cid:96)i (cid:54) D(cid:96) + ¯ ui (cid:54) Du + ¯ di (cid:54) Dd + ¯ ei (cid:54) De (18)and the scalar sector is given, in terms of the physical fields h and H , by L hH = 12 ∂ µ h∂ µ h + 12 ∂ µ H∂ µ H − V ( h, H )+ v (cid:104) D µ U † D µ U (cid:105) (cid:18) cv h + 2 sv H + c v h + s v H + 2 scv hH (cid:19) − v (cid:0) ¯ qY u U P + r + ¯ qY d U P − r + ¯ (cid:96)Y e U P − η + h . c . (cid:1) (cid:104) cv h + sv H (cid:105) (19)Here U = exp(2 iϕ a T a /v ) is the Goldstone-boson matrix; q = ( u L , d L ) T and (cid:96) = ( ν L , e L ) T are the left-handed doublets; u = u R , d = d R and e = e R the right-handed singlets; and r = ( u R , d R ) T , η = ( ν R , e R ) T . We suppress generation indices. The coefficients arecos χ ≡ c , sin χ ≡ s (20)The full scalar potential reads V ( h, H ) = 12 m h + 12 M H − d h − d h H − d hH − d H − z h − z h H − z h H − z hH − z H (21)with d = m vv s [ s v − c v s ] 6 = − m + M vv s sc [ sv + cv s ] d = 2 M + m vv s sc [ cv − sv s ] d = − M vv s [ c v + s v s ] z = − v v s (cid:2) m ( s v − c v s ) + M s c ( sv + cv s ) (cid:3) z = sc v v s ( sv + cv s ) (cid:2) m ( s v − c v s ) + M sc ( cv − sv s ) (cid:3) z = − sc v v s (cid:2) m (6 sc ( sv + cv s ) − vv s ) + M (6 sc ( cv − sv s ) + 2 vv s ) (cid:3) z = sc v v s ( cv − sv s ) (cid:2) M ( c v + s v s ) + m sc ( sv + cv s ) (cid:3) z = − v v s (cid:2) M ( c v + s v s ) + m s c ( cv − sv s ) (cid:3) (22)We emphasize that (19) represents the complete, renormalizable model, expressedhere in terms of nonlinear coordinates U for the electroweak Goldstone fields. In this section we integrate out the heavy scalar mass eigenstate H at tree level in thestrongly-coupled limit defined in (15), including leading and next-to-leading order terms.We show that the resulting EFT takes the form of the electroweak chiral Lagrangianwith a light Higgs [2, 3, 30]. To leading order the scalar sector of this Lagrangian can,in general, be written as [2, 3] L Uh,LO = v (cid:104) D µ U † D µ U (cid:105) (1 + F U ( h )) + 12 ∂ µ h∂ µ h − V ( h ) − v (cid:104) ¯ q (cid:32) Y u + ∞ (cid:88) n =1 Y ( n ) u (cid:18) hv (cid:19) n (cid:33) U P + r + ¯ q (cid:32) Y d + ∞ (cid:88) n =1 Y ( n ) d (cid:18) hv (cid:19) n (cid:33) U P − r + ¯ (cid:96) (cid:32) Y e + ∞ (cid:88) n =1 Y ( n ) e (cid:18) hv (cid:19) n (cid:33) U P − η + h . c . (cid:105) (23)to be supplemented by the usual gauge and fermion terms of the unbroken SM (18).We start from the full theory in (17) and follow the procedure outlined in [2]. Thepart of this Lagrangian that depends on H reads L H = 12 H ( − ∂ − M ) H + J H + J H + J H + J H (24)7here the J i are given by J = d h + z h + v (cid:104) D µ U † D µ U (cid:105) (cid:18) sv + 2 scv h (cid:19) − sJ f J = d h + z h + s (cid:104) D µ U † D µ U (cid:105) J = d + z h , J = z (25)with J f ≡ ¯ qY u U P + r + ¯ qY d U P − r + ¯ (cid:96)Y e U P − η + h . c . (26)To perform the EFT expansion, we make the dependence of the J i on the heavy mass M explicit by writing J i ≡ M J i + ¯ J i (27)and similarly d i ≡ M d i + ¯ d i , z i ≡ M z i + ¯ z i (28)for the coefficients in the potential (21). The J i are pure polynomials in h .We integrate out the heavy field H at tree level by solving its equation of motion( − ∂ − M + 2 J ) H + J + 3 J H + 4 J H = 0 (29)and inserting the solution into the Lagrangian (19). We can solve (29) order by order inpowers of 1 /M by expanding H = H + H + H + . . . , H l = O (1 /M l ) (30)Inserting (30) into (29) and keeping only the terms of O ( M ) yields an (algebraic)equation for H : J + ( − J ) H + 3 J H + 4 J H = 0 (31)Retaining the terms of O (1) gives an equation that determines H as a function of H .The solution reads H = ( − ∂ + 2 ¯ J ) H + ¯ J + 3 ¯ J H + 4 ¯ J H M (1 − J − J H − J H ) (32)Proceeding to higher orders in 1 /M , the H l , l ≥
2, can be successively computed.As a first step, we obtain H from (31). Since the coefficients J i depend on no otherfield than h , the solution H will also have this property. It is convenient to find H ( h )as an infinite series in powers of h H ( h ) = ∞ (cid:88) k =2 r k h k (33)8nserting (33) into (31), we obtain for the first few coefficients r k r = d r = d d r = d d + d d r = d d + 3 d d d (34)In Appendix A, we derive a closed-form solution for H ( h ) to all orders in h . We alsoshow there that only one solution of the cubic equation (31) is relevant. This solutionstarts at order h , as anticipated in (33).To obtain the leading-order effective Lagrangian, we insert H = H + H into (19)and expand the expression, retaining terms of O ( M ) and O (1). Terms with H vanishat this order due to the equation of motion for H . We show in Appendix A that ingeneral all terms of O ( M ) cancel up to an irrelevant constant. The leading-order scalarLagrangian then becomes L hH,LO = 12 ( ∂h ) − m h + d h + ¯ z h + 12 ( ∂H ) + ¯ J H + ¯ J H + ¯ J H + ¯ J H + v (cid:104) D µ U † D µ U (cid:105) (cid:18) cv h + c v h (cid:19) − vJ f (cid:16) cv h (cid:17) (35)where H = H ( h ). The kinetic term for h has acquired the form L h,kin = 12 ( ∂h ) + 12 ( ∂H ) = 12 ( ∂h ) (1 + F h ( h )) with F h ( h ) = (cid:18) dH ( h ) dh (cid:19) (36)The field redefinition [2]˜ h = (cid:90) h (cid:112) F h ( s ) ds = h (cid:18) r h + 32 r r h + O ( h ) (cid:19) (37)brings (36) to its canonical form L h,kin = ( ∂ ˜ h ) / h in (35) in favour of ˜ h using (37) and dropping the tilde in the end, thescalar-sector Lagrangian takes the form of (23). Together with the gauge and fermionkinetic terms, this is an electroweak chiral Lagrangian including a light Higgs boson.Specifically, the general functions in (23) are F U ( h ) = 2 c (cid:18) hv (cid:19) + (cid:20) c − s c vv s (cid:21) (cid:18) hv (cid:19) − v s s c ( vs + v s c ) (cid:18) hv (cid:19) + O ( h ) V ( h ) = m v (cid:34) (cid:18) hv (cid:19) + c v s − s v v s (cid:18) hv (cid:19) − s c ( sv + cv s ) − s v + c v s )24 v s (cid:18) hv (cid:19) s c ( sv + cv s ) v s (cid:20) − s ) − cv s − svcv s + sv (cid:21) (cid:18) hv (cid:19) + O ( h ) (cid:35) (38)and Y f + ∞ (cid:88) n =1 Y ( n ) f (cid:18) hv (cid:19) n = Y f (cid:34) c (cid:18) hv (cid:19) − s c vs + v s c v s (cid:18) hv (cid:19) − s c vs + v s c v s (4 vsc + v s (1 − s )) (cid:18) hv (cid:19) + O ( h ) (cid:35) (39)To leading order in the SO (5) limit ( ω → ξ ) these expressions become F U ( h ) = 2 (cid:112) − ξ (cid:18) hv (cid:19) + (1 − ξ ) (cid:18) hv (cid:19) − ξ (cid:112) − ξ (cid:18) hv (cid:19) + O ( h ) V ( h ) = m v (cid:34) (cid:18) hv (cid:19) + 1 − ξ √ − ξ (cid:18) hv (cid:19) + 11 − ξ (cid:18) − ξ + 76 ξ (cid:19) (cid:18) hv (cid:19) − ξ (1 − ξ )2 √ − ξ (cid:18) hv (cid:19) + O ( h ) (cid:35) (40)and Y f + ∞ (cid:88) n =1 Y ( n ) f (cid:18) hv (cid:19) n = Y f (cid:34) (cid:112) − ξ (cid:18) hv (cid:19) − ξ (cid:18) hv (cid:19) − ξ (cid:112) − ξ (cid:18) hv (cid:19) + O ( h ) (cid:35) (41)We can extend the derivation to include the NLO terms of O (1 /M ) in the effectiveLagrangian L eff = L LO + ∆ L NLO + O (cid:18) M (cid:19) , L LO = L + L Uh,LO (42)Using (32), we find∆ L NLO = (cid:2) ( − ∂ + 2 ¯ J ) H + ¯ J + 3 ¯ J H + 4 ¯ J H (cid:3) M (1 − J − J H − J H ) (43)The effective Lagrangian ∆ L NLO contains operators that modify the leading-order La-grangian (23) as well as a subset of the next-to-leading operators of [2]. In the notationof [2], the NLO operators generated by (42) are O D , O D , O D ; O ψS , O ψS , O ψS , O ψS , O ψS , O ψS (44)10nd their hermitean conjugates, together with 4-fermion operators coming from thesquare of the Yukawa bilinears contained in ¯ J . The 4-fermion operators that arise havethe same structure as those in the heavy-Higgs model discussed in [31], which are O F Y , O F Y , O F Y , O F Y , O F Y , O F Y , O ST , O ST , (45) O LR , O LR , O LR , O LR , O LR , O LR , O LR , O LR , O LR , O LR , O LR , O LR and their hermitean conjugates, but they are now multiplied by functions F i ( h/v ).We discuss several important aspects of these results. • The solution for H ( h ) in the limit (15) contains terms to all orders in h , withcoefficients of O (1), since ξ , ω = O (1) (see Appendix A). Upon integrating outthe heavy scalar, the function H ( h ) enters the various terms in the effective La-grangian. The singlet-model thus gives an explicit illustration of how the all-orderpolynomial functions F ( h ) are generated in the strong-coupling limit of the under-lying scalar sector. They are characteristic for the nonlinear EFT. • The leading-order Lagrangian, (23) with (38) and (39), is of O (1) in the 1 /M expansion. The next-to-leading order terms in (43) are of O (1 /M ). However,the corresponding nonlinear EFT of the singlet model is organized by chiral di-mensions [3], rather than by canonical dimensions. This is expected on generalgrounds and is further elaborated in the following items. • It is easy to check that all terms of L LO in (23), with (40) and (41), including thegauge and fermion kinetic terms, carry chiral dimension 2. Note that the mass m of the light Higgs counts with one unit of chiral dimension. The smallness of m can be understood as arising from an approximate SO (5) symmetry, where thesmall parameters of explicit SO (5) breaking act as weak couplings carrying chiraldimension. • The NLO terms in (43) have chiral dimension 4, consistent with the chiral counting.Since we integrate out the heavy scalar at tree level, the contributions shown in (43)have a suppression by v /M . There are additional contributions to ∆ L NLO fromone-loop diagrams of the full model, which are suppressed by a factor of 1 / π .In the strong-coupling limit M ∼ < πf . Then both factors are parametrically ofcomparable size: v /M ∼ > ξ/ π ≈ / π . • As mentioned above, the limit we consider here has a heavy mass M that stayssomewhat below the nominal strong-coupling value 4 πf . In that way, the picture The terms O LR , O LR , O LR and O LR had been missed in the discussion of the heavy-Higgsmodels in [2, 31]. The assignment of chiral dimensions is 0 for bosons, and 1 for each derivative, weak coupling orfermion bilinear.
11f the heavy resonance as an elementary field in the full theory is still a reason-able approximation. A very similar limit was considered previously in the contextof integrating out a heavy SM Higgs to obtain a (Higgsless) electroweak chiralLagrangian as the low-energy EFT in [32–35]. • The results derived here at tree level remain stable under radiative corrections.We demonstrate this explicitly for the one-loop effective potential in Appendix B.There, we show that the one-loop corrections to the Higgs potential are at mostof order M / (16 π f ) in the case of a weakly broken SO (5) symmetry. This issmaller than one for large, but still sufficiently perturbative couplings. In thenominal strong-coupling case, the loop corrections would become of order unity.The potential would then no longer be calculable, as expected.We end this section with an illustration of how the nonlinear EFT reproduces thefull-theory result in the strong-coupling limit, taking the process hh → hh as an example.In the full theory, the amplitude for hh → hh is given by M = M + M , where − i M =24 z − d (cid:20) s M − M + 1 t M − M + 1 u M − M (cid:21) (46)is the local contribution, from the quartic interaction and from H -boson exchange, and − i M = − d (cid:20) s M − m + 1 t M − m + 1 u M − m (cid:21) (47)is the nonlocal term from the exchange of h . M is identical in the full theory and inthe low-energy EFT. We therefore concentrate on M in the following. In the heavy- H limit, we have 1 s M − M = − M (cid:16) s M M + · · · (cid:17) (48)Since the Mandelstam variables satisfy s M + t M + u M = 4 m , we find − i M =24 z + 12 d M + 16 d m M (49)Fully expanded in the strong-coupling limit, this gives − i M = m v v s (cid:0) s c ( sv + cv s ) − s v + c v s ) (cid:1) (50)which coincides with the amplitude from the local h -term of the nonlinear EFT in (38).12 Linear EFT and comparison with nonlinear EFT
We now consider the weakly-coupled limit (16) of the singlet model. We integrate outthe heavy field S , while retaining the doublet Φ. This is consistent, even though S and Φ are not the physical fields. The key point is that the mixing is subleading anddiagonalization is not needed, as opposed to the nonlinear EFT case. The resultingLagrangian can be expanded in canonical dimensions. The dominant corrections comefrom terms of dimension six [1, 36]. For the singlet model, this was discussed already in[37, 38]. Starting from (1) and (2), and rewriting S = ( v H + H s ) / √
2, we find L = ( D µ Φ) † ( D µ Φ) + (cid:18) µ − λ v H (cid:19) Φ † Φ − λ † Φ) + 12 ∂ µ H s ∂ µ H s − M s H s − λ v H † Φ H s − λ † Φ H s − λ v H H s − λ H s (51)where we identify µ = M s = λ v H J i are now constants andfunctions of (Φ † Φ) that can be read off from (51). As in the previous section, we solvethe equation of motion order by order in powers of 1 /M s . Keeping in mind that v H /M s = O (1) because of (52), we find H s = − λ v H M s Φ † Φ + O (cid:18) M s (cid:19) (53)and L = ( D µ Φ) † ( D µ Φ) + (cid:18) µ − λ M s λ (cid:19) Φ † Φ − (cid:18) λ − λ λ (cid:19) (Φ † Φ) + 14 λ λ M s ∂ µ (Φ † Φ) ∂ µ (Φ † Φ) + O (cid:18) M s (cid:19) (54)in agreement with [37]. Out of the two custodially symmetric scalar operators of di-mension six in the SM, only ∂ µ (Φ † Φ) ∂ µ (Φ † Φ) appears. The second operator, (Φ † Φ) , isabsent.At low energies, the doublet develops a vev,Φ = v + h √ U (cid:32) (cid:33) (55)13nd h is identified with the light scalar discovered at the LHC. In the broken phase, thedimension-6 correction in (54) translates to L NLO = α ∂ µ h∂ µ h (cid:18) hv (cid:19) (56)with α ≡ λ λ vv H ˙= χ (57)To first order in v/v H , α is equal to the mixing angle χ in (7). We remove the term in(56) by a field redefinition of h [2]. The complete EFT Lagrangian including terms oforder 1 /M s then becomes (with L from (18)) L = L + 12 ∂ µ h∂ µ h + v (cid:104) D µ U † D µ U (cid:105) (cid:32) − α ) hv + (1 − α ) (cid:18) hv (cid:19) − α (cid:18) hv (cid:19) − α (cid:18) hv (cid:19) (cid:33) − m h − m v (cid:34)(cid:18) − α (cid:19) (cid:18) hv (cid:19) + (cid:18) − α (cid:19) (cid:18) hv (cid:19) − α (cid:18) hv (cid:19) − α (cid:18) hv (cid:19) (cid:35) − v (cid:2) ¯ qY u U P + r + ¯ qY d U P − r + ¯ lY e U P − η + h . c . (cid:3) ×× (cid:32) − α hv − α (cid:18) hv (cid:19) − α (cid:18) hv (cid:19) (cid:33) (58)We observe that all Higgs couplings in (58) are reduced with respect to their Standard-Model values.In Section 4 we performed the matching of the SM extended by a heavy scalar singletto the leading order of the nonlinear EFT by integrating out the heavy degree of freedomat tree level. We showed that such a low-energy EFT is the result of integrating outthe heavy field when the theory approaches a strongly-coupled regime. In the presentsection, we carried out a matching of the theory to the linear EFT through operatorsof dimension six by integrating out the heavy scalar in the weakly-coupled regime. Wenow compare these two scenarios further, based on the discussion in Section 2.As stressed previously, the character of the low-energy EFT is dictated by the under-lying dynamics. In the model at hand, the difference between weak and strong coupling,and the respective EFTs, is connected to the size of the parameters ξ and ω = sin χ ,where ω quantifies the admixture of the doublet and singlet components in the phys-ical scalar fields. When the theory approaches the strongly-coupled regime, we have ξ , ω = O (1). The heavy mass eigenstate that is integrated out then has a significantdoublet component (see also [39] for a similar observation in a different context). In the14igure 2: Left: Allowed values in the plane { ξ, χ } for M = 1 TeV when imposing tree-levelperturbative unitarity conditions in the full theory. The lines correspond to the SO (5) limitand perturbations around it. Right: Illustration of the resulting Higgs couplings to massivevector bosons and the decorrelation from the linear EFT at dimension six. weakly-coupled regime the mixing angle χ shows instead a typical decoupling behaviorbetween the two mass scales of the theory, ω ∼ v /f , and ξ , ω (cid:28) O ( ω, ξ ), we reproduce theresults of the linear expansion given in (58). We emphasize that in the limit of smallmixing angle the linear EFT through operators of dimension six provides, in particular,a correct description of the leading mixing effects in single-Higgs and multiple-Higgsinteractions.When the mixing angle χ ∼ v/f becomes large, indicating the onset of a strongly-coupled regime, the linear expansion starts to fail and the nonlinear character of thelow-energy EFT becomes manifest. In this scenario, the deviations of the Higgs prop-erties from the SM are generically of O (1) and correspond to a resummation in χ and ξ . Another typical feature of the nondecoupling behavior is the decorrelation betweenthe linear and quadratic Higgs couplings to massive vector bosons [40]. The latter arelinearly correlated in the linear EFT at dimension six, as seen from (58). Such a cor-relation is not present in the leading order of the nonlinear EFT, as shown in (38). Inorder to illustrate the size of such effects within the perturbative domain of the fulltheory, we fix M = 1 TeV and scan the remaining ( ξ, χ ) parameter space of the model,imposing tree-level perturbative unitarity bounds for all two-to-two processes involving { W + W − , ZZ, hh, hH, HH } [18]. Figure 2 shows the resulting linear and quadratic Higgscoupling to massive vector bosons (conveniently normalized) in the nonlinear EFT fromthis scan. For comparison, it also shows the correlations obtained between these twocouplings near the SO (5)-symmetric limit and within the linear expansion at dimensionsix. The gap between the two regions in Figure 2 (right) originates from the regions of15arameter space in which χ is close to zero. The Higgs couplings have to be close totheir SM values in this case. O ( ξ ) In the present section we consider the general, model-independent electroweak chiral La-grangian. We assume that a decoupling limit exists, in which the chiral Lagrangian re-duces to the renormalizable Standard Model. Deviations from this limit are parametrizedby a quantity ξ ≡ v /f , defined in terms of the scale f of the new strong dynamics.Corrections at O ( ξ n ) enter through operators of canonical dimension 2 n + 4 [2, 41]. Weconnect this general picture with the singlet model at the end of the section.We start from the electroweak chiral Lagrangian at leading order and perform thematching to the linear expansion up to O ( ξ ) along the lines of [2, 41]. We neglectterms of O ( ξ/ π ) from higher orders in the chiral expansion, which can be justified aslong as ξ (cid:29) ξ/ (16 π ). We write the Higgs sector of the LO effective Lagrangian in thedimensional expansion as L = L + L + L (59)Here L d contains those operators of chiral dimension 2 that have canonical dimension d .The corresponding terms can be expressed in terms of the Goldstone matrix U and theHiggs singlet h .At chiral dimension 2 and canonical dimension 4, we have the Higgs sector of theStandard Model: L = 12 ∂ µ h∂ µ h + µ v (cid:18) hv (cid:19) − λv (cid:18) hv (cid:19) − v (cid:18) hv (cid:19) ¯Ψ Y (0)Ψ U P
Ψ + v (cid:104) D µ U † D µ U (cid:105) (cid:18) hv (cid:19) (60)The SM at dimension 6 [1] contains exactly three operators that contribute with chiraldimension 2. These are κ (Φ † Φ) (including two weak couplings κ ), ∂ µ (Φ † Φ) ∂ µ (Φ † Φ),and the modified Yukawa terms, here generically written as ¯Ψ L Y ΦΨ R Φ † Φ. This gives L = − λa v ξ (cid:18) hv (cid:19) + a ξ∂ µ h∂ µ h (cid:18) hv (cid:19) − vξ ¯Ψ ˆ Y U P Ψ (cid:18) hv (cid:19) (61)In a similar way, we construct all operators of canonical dimension 8 and chiral dimension2 and obtain L = − λb v ξ (cid:18) hv (cid:19) + b ξ ∂ µ h∂ µ h (cid:18) hv (cid:19) − vξ ¯Ψ ˆ Y U P Ψ (cid:18) hv (cid:19) (62)We define a and b with an additional factor of λ to obtain a convenient normalization.The Lagrangian of (59) has to be matched to the leading order chiral Lagrangian. In16rder for the kinetic term to be of the form ∂ µ h∂ µ h/
2, without any other factors, wehave to redefine h : h → h (cid:110) − ξ a (cid:18) hv + h v (cid:19) + ξ a (cid:32)
38 + hv + 1312 (cid:18) hv (cid:19) + 1324 (cid:18) hv (cid:19) + 13120 (cid:18) hv (cid:19) (cid:33) − ξ b (cid:32)
12 + hv + (cid:18) hv (cid:19) + 12 (cid:18) hv (cid:19) + 110 (cid:18) hv (cid:19) (cid:33) (cid:111) + O ( ξ ) (63)The parameter v describes the physical vev. We find it by requiring the linear termin the potential (after the redefinition above) to vanish. We find: v = (cid:114) µ λ (cid:18) − a ξ + ξ (cid:18) a − b (cid:19) + O (cid:0) ξ (cid:1)(cid:19) (64)The quadratic term of the potential should be given by the physical Higgs mass m . Thiscondition, together with (64) enables us to express the bare quantities µ and λ of (60)in terms of the physical quantities v and m , and the coefficients a i , b i : µ = m (cid:0) ξ ( a − a ) + ξ (2 a − a a − b + b ) + O (cid:0) ξ (cid:1)(cid:1) λ = m v (cid:0) ξ ( a − a ) + ξ (cid:0) a − a a − b + b (cid:1) + O (cid:0) ξ (cid:1)(cid:1) (65)The Lagrangian then acquires the following form: L = 12 ∂ µ h∂ µ h − V ( h ) + v (cid:104) D µ U † D µ U (cid:105) (1 + F U ( h )) − v ¯Ψ (cid:32) Y Ψ + (cid:88) n =1 Y ( n )Ψ (cid:0) hv (cid:1) n (cid:33) U P
Ψ(66)with V ( h ) = 12 m h + 12 m v (cid:34)(cid:0) ξ (cid:0) a − a (cid:1) + ξ (cid:0) − a a + a + 4 b − b − a (cid:1)(cid:1) (cid:18) hv (cid:19) + (cid:0) + ξ (cid:0) a − a (cid:1) + ξ (cid:0) − a a + a + 8 b − b − a (cid:1)(cid:1) (cid:18) hv (cid:19) + (cid:0) ξ ( a − a ) + ξ (cid:0) − a a − a + a + 7 b − b (cid:1)(cid:1) (cid:18) hv (cid:19) + (cid:0) ξ ( a − a ) + ξ (cid:0) − a a − a + a + b − b (cid:1)(cid:1) (cid:18) hv (cid:19) ξ (cid:0) − a a + a + b − b (cid:1) (cid:18) hv (cid:19) + ξ (cid:0) − a a + a + b − b (cid:1) (cid:18) hv (cid:19) (cid:35) (67) F U ( h ) = (cid:0) − a ξ + ξ (cid:0) a − b (cid:1)(cid:1) (cid:18) hv (cid:19) + (cid:0) − a ξ + ξ (cid:0) a − b (cid:1)(cid:1) (cid:18) hv (cid:19) + (cid:0) − ξ a + ξ (cid:0) a − b (cid:1)(cid:1) (cid:18) hv (cid:19) + (cid:0) − ξ a + ξ (cid:0) a − b (cid:1)(cid:1) (cid:18) hv (cid:19) + ξ (cid:0) a − b (cid:1) (cid:18) hv (cid:19) + ξ (cid:0) a − b (cid:1) (cid:18) hv (cid:19) (68) (cid:88) n =1 Y ( n )Ψ (cid:0) hv (cid:1) n = (cid:16) Y Ψ + ξ (cid:16) Y − a Y Ψ (cid:17) + ξ (cid:16) a Y Ψ − a ˆ Y − b Y Ψ + 4 ˆ Y (cid:17)(cid:17) hv + (cid:16) ξ (cid:16) Y − a Y Ψ (cid:17) + ξ (cid:16) a Y Ψ − a ˆ Y − b Y Ψ + 10 ˆ Y (cid:17)(cid:17) (cid:18) hv (cid:19) + (cid:16) ξ (cid:16) Y − a Y Ψ (cid:17) + ξ (cid:16) a Y Ψ − a ˆ Y − b Y Ψ + 10 ˆ Y (cid:17)(cid:17) (cid:18) hv (cid:19) + ξ (cid:16) a Y Ψ − a ˆ Y − b Y Ψ + 5 ˆ Y (cid:17) (cid:18) hv (cid:19) + ξ (cid:16) a Y Ψ − a ˆ Y − b Y Ψ + 5 ˆ Y (cid:17) (cid:18) hv (cid:19) (69)where Y Ψ = Y (0)Ψ + ξ ˆ Y + ξ ˆ Y (70)Comparing (67), (68) and (69) with the results for F U ( h ), V ( h ) and the Yukawa termsin the singlet model, displayed in (40) and (41), we find agreement to second order in ξ with a = b = ˆ Y = ˆ Y = 0, a = b = 1, Y Ψ = Y f .In relation to our previous discussion we make the following observations. Since (58)contains contributions through dimension six, it induces a pattern of coefficients that isexpected from the O ( ξ ) expansion of the chiral Lagrangian. Indeed, (58) can be obtainedfrom (67), (68) and (69) by neglecting terms of O ( ξ ) and identifying a ξ = α . This re-sult could have been anticipated also from the analysis in [41]. The decorrelation betweenthe linear and quadratic Higgs couplings to massive vector bosons appears at dimensioneight. This is in agreement with the discussion in [40]. Additional correlations at the O ( ξ ) level are also present in the Yukawa sector ( h and h couplings) and in the scalarpotential ( h and h couplings), though these seem less interesting phenomenologically.18 Conclusions
We have studied a simple extension of the Standard Model where new physics is limitedto a heavy real scalar singlet endowed with a Z symmetry. This model has been used inthe past, e.g., for searches of dark matter. Here we use it as a (UV-complete) toy modelto illustrate, by explicit construction, how the different effective field theories at theelectroweak scale, the so-called SM-EFT and EWChL, arise. These two EFTs possessthe same degrees of freedom and symmetries, yet they have very different systematics:SM-EFT is an expansion in canonical dimensions while EWChL is an expansion in loopsor chiral dimensions. The toy model allows us to show in a transparent way why thisdifference in power counting occurs, and helps to substantiate by way of example anumber of statements about both EFTs. • Dynamics of the EFTs . The model depends on three free parameters: the heavymass M , the mixing angle ω and the vev ratio ξ . In scenarios where M is largeand ω, ξ (cid:28)
1, the heavy scalar scales as H ∼ O ( M − )( v + h ) and the resultingEFT is organized in inverse powers of M (SM-EFT). In contrast, if ω, ξ ∼ O (1),then H ∼ O (1) f ( h ), with f ( h ) an (untruncated) function of h . This correspondsto a nondecoupling regime and the EFT is then organized in chiral dimensions(EWChL). Generically, theories that exhibit nondecoupling effects lead to EFTsgoverned by chiral dimensions, while theories with only decoupling effects admitEFTs based on an expansion in canonical dimensions. • Relation between the EFTs . The model shows that the choice of EFT dependsonly on the size of the parameters. The transition between EWChL and SM-EFTis therefore a smooth one, as can be shown by further expanding EWChL for small ω, ξ . This conclusion holds as long as there is a well-defined decoupling limit. • ξ expansion . In a bottom-up EFT the ξ dependence is hidden in the Wilsoncoefficients and cannot be determined from power counting. One can neverthelessuncover this ξ dependence in EWChL starting from operators in SM-EFT [41].Here we have extended this procedure to the leading-order EWChL at O ( ξ ) andcompared it explicitly with the toy model expanded at the same order. We find aconsistent matching, which validates the procedure adopted in [41]. • Naturalness . The toy model at hand (in the nondecoupling regime) admits anembedding into an SO (5) model spontaneously broken down to SO (4). In thatcase, and if explicit breaking of the SO (5) symmetry is small, the Higgs can beinterpreted as a pseudo-Goldstone boson and is therefore naturally light, m ∼ f .Its precise value cannot be computed in perturbation theory unless one assumesthat M/ (4 πf ) (cid:46) SO (5) limit, fine-tuning is required to build a hierarchy between the Higgs andthe heavy scalar. 19 cknowledgements This work was performed in the context of the ERC Advanced Grant project FLAVOUR(267104) and was supported in part by the DFG grant BU 1391/2-1, the DFG clusterof excellence EXC 153 ’Origin and Structure of the Universe’ and the Munich Institutefor Astro- and Particle Physics (MIAPP). A.C. is supported by a Research Fellowshipof the Alexander von Humboldt Foundation.
A Exact solution for H ( h ) We integrate out the heavy field H with mass M at tree level by solving its equation ofmotion. In the strong-coupling limit (15) the leading-order term H ( h ), of order O (1) inthe 1 /M expansion, follows from solving the equation of motion at O ( M ). We achievedthis in Section 4 through a series expansion of H in powers of h . Here we obtain anexact, analytic solution for the function H ( h ).Retaining only the terms of order M , sufficient for the computation of H ( h ), theLagrangian of the full singlet model simplifies to (see (1)) L M = µ φ + µ S − λ φ − λ S − λ φ S (A.1)where φ ≡ Φ † Φ and λ = 2 M f ωξ , λ = 2 M f − ω − ξ , λ = 2 M f (cid:115) ω (1 − ω ) ξ (1 − ξ ) (A.2) µ = M (cid:18) ω + (cid:114) ωξ (cid:112) (1 − ω )(1 − ξ ) (cid:19) , µ = M (cid:18) − ω + (cid:114) − ω − ξ (cid:112) ωξ (cid:19) (A.3)Expanding φ and S around their vevs and using (6), we write φ = 1 √ f (cid:112) ξ + √ − ωh + √ ωH ) S = 1 √ f (cid:112) − ξ − √ ωh + √ − ωH ) (A.4)Defining next R ≡ (cid:114) ωξ φ + (cid:114) − ω − ξ S (A.5)the Lagrangian (A.1) becomes L M = M (cid:16)(cid:112) ωξ + (cid:112) (1 − ω )(1 − ξ ) (cid:17) R − M f R (A.6)20he resulting equation of motion for H reads ∂ L ∂H = ∂ L ∂R ∂R ∂H = 0 (A.7)The relevant solution H ( h ), inserted back into L , describes the effect of integrating out H at tree level. This is equivalent to matching all possible tree diagrams with internal H lines to an effective Lagrangian for h .The Lagrangian for H has the form of (24). A diagram with only internal H linescontains, in general, a number V n of vertices J n H n ( n = 1 , . . . , P H -field propagators,and L loops. Combining the well-known topological identities2 P = V + 2 V + 3 V + 4 V L = P − ( V + V + V + V ) + 1 (A.8)for the number of H -lines attached to vertices and the number of loops, respectively, oneobtains L = V + V − V L = 0), this implies V = V + 2 V + 2 (A.10)Since V , V ≥
0, we find V ≥
2. This means that the effective Lagrangian, obtainedfrom (24) by integrating out H at tree level, has to start at order ( J ) . Equivalently,the solution of the equation of motion for H has to start at O ( J ). To order M , relevantfor H , this implies that H ( h ) = O ( h ).This consideration eliminates the solution for H ( h ) of0 = ∂R ∂H = f + (cid:112) ω (1 − ω ) (cid:18)(cid:114) ωξ − (cid:114) − ω − ξ (cid:19) h + (cid:18) ω (cid:114) ωξ + (1 − ω ) (cid:114) − ω − ξ (cid:19) H (A.11)and one of the solutions of the equation ∂ L /∂R = 0, quadratic in H , which can also bewritten as R = f (cid:16)(cid:112) ωξ + (cid:112) (1 − ω )(1 − ξ ) (cid:17) (A.12)The remaining solution of (A.12) is H ( h ) = f + (cid:16)(cid:113) ω (1 − ω ) ξ − (cid:113) ω (1 − ω ) − ξ (cid:17) h (cid:113) ω ξ + (cid:113) (1 − ω ) − ξ × (cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) − (cid:16)(cid:113) ω ξ + (cid:113) (1 − ω ) − ξ (cid:17) (cid:16)(cid:113) ω (1 − ω ) ξ + (cid:113) ω (1 − ω )1 − ξ (cid:17) h (cid:16) f + (cid:16)(cid:113) ω (1 − ω ) ξ − (cid:113) ω (1 − ω ) − ξ (cid:17) h (cid:17) − (A.13)21s expected, H ( h ) = O ( h ). All coefficients of h n in H ( h ) are polynomial in √ ω = sin χ and √ − ω = cos χ .Expanded in powers of h , (A.13) agrees with the result for H obtained in Section 4through order h . In the SO (5) limit, where ω = ξ , (A.13) becomes H = f (cid:34)(cid:115) − h f − (cid:35) (A.14)As a byproduct of our derivation, we can show explicitly that the terms of order M cancel out in the effective Lagrangian, as it has to be the case. This property is notimmediately obvious from the full theory in (21), where the coefficients carry O ( M )contributions. From (A.12) we see that the solution for H fulfills R ( h, H ( h )) = const .Therefore, when H ( h ) is inserted back into (A.6), the M -terms in the Lagrangianreduce to a field-independent constant. This demonstrates the absence of a nontrivial O ( M ) piece in the effective theory. B The scalar effective potential to one loop
We consider the one-loop effective potential of the scalar sector defined in (1) and (2),when the heavy field is integrated out. The result illustrates the parametric impact ofradiative corrections within the model, in particular in the strong-coupling limit.We start from the scalar Lagrangian of the model in terms of the mass eigenstates,given by L = − h∂ h − H∂ H − m h − M H − V (B.1)where V are the cubic and quartic terms of V ( h, H ) in (21). Following the background-field methods described in [34], we split the fields into a background component, denotedby a hat, and a fluctuating part h → ˆ h + h , H → ˆ H + H (B.2)For the one-loop computation, we need the part of V quadratic in the fluctuating fields.It reads − V , = Ah + BH + 2 ChH (B.3)where A =3 d ˆ h + d ˆ H + 6 z ˆ h + 3 z ˆ h ˆ H + z ˆ H B = d ˆ h + 3 d ˆ H + z ˆ h + 3 z ˆ h ˆ H + 6 z ˆ H C = d ˆ h + d ˆ H + 32 z ˆ h + 2 z ˆ h ˆ H + 32 z ˆ H (B.4)22he Lagrangian terms of second order in the fluctuating fields h and H then become L = − h∂ h − H∂ H − m h − M H + Ah + BH + 2 ChH (B.5)We next isolate the dependence on M that is still hidden in the coefficients d i and z i in (B.4). Following Appendix A, the full M dependence takes the form of L M in (A.6).The terms of second order in h and H are L ,M = 12 ∂ L M ∂h (cid:12)(cid:12)(cid:12)(cid:12) ˆ h + 12 ∂ L M ∂H (cid:12)(cid:12)(cid:12)(cid:12) ˆ H + ∂ L M ∂h∂H (cid:12)(cid:12)(cid:12)(cid:12) ˆ hH (B.6)where the subscript ˆ after an expression indicates that its field variables are taken attheir background values. The second derivatives are ∂ L M ∂h = ∂ L M ( ∂R ) (cid:18) ∂R ∂h (cid:19) + ∂ L M ∂R ∂ R ∂h ∂ L M ∂H = ∂ L M ( ∂R ) (cid:18) ∂R ∂H (cid:19) + ∂ L M ∂R ∂ R ∂H ∂ L M ∂h∂H = ∂ L M ( ∂R ) ∂R ∂h ∂R ∂H + ∂ L M ∂R ∂ R ∂h∂H (B.7)where R is defined in (A.4) and (A.5). We then have ∂R ∂H = f + (cid:18)(cid:114) − ωξ ω − (cid:114) ω − ξ (1 − ω ) (cid:19) h + (cid:18)(cid:114) ωξ ω + (cid:114) − ω − ξ (1 − ω ) (cid:19) H∂R ∂h = (cid:18)(cid:114) ωξ (1 − ω ) + (cid:114) − ω − ξ ω (cid:19) h + (cid:18)(cid:114) − ωξ ω − (cid:114) ω − ξ (1 − ω ) (cid:19) H (B.8)Evaluated with background fields to leading order in M , the expressions in (B.7) simplifybecause ∂ L M ∂R (cid:12)(cid:12)(cid:12)(cid:12) ˆ= 0 , ∂ L M ( ∂R ) (cid:12)(cid:12)(cid:12)(cid:12) ˆ= − M f = const. (B.9)from (A.6) and (A.12). Defining α ≡ (cid:18) ∂R ∂h (cid:19) ˆ , β ≡ (cid:18) ∂R ∂H (cid:19) ˆ (B.10)we obtain L ,M = − M f (cid:0) α h + β H + 2 αβhH (cid:1) (B.11)This result gives an explicit expression for the M -dependent terms contained in (B.5).23e now write the second-order Lagrangian in (B.5) as L = −
12 ( h, H ) (cid:32) ∆ + a cc ∆ + b (cid:33) (cid:32) hH (cid:33) ≡ −
12 ( h, H ) K (cid:32) hH (cid:33) (B.12)Here ∆ ≡ ∂ + m (B.13)and a ≡ − A + M f α , b ≡ − B − m + M f β , c ≡ − C + M f αβ (B.14)where ¯ A , ¯ B and ¯ C are, respectively, the functions A , B and C of (B.4) without the M -pieces. The latter are made explicit in (B.14).We obtain the one-loop effective action S eff from the path integral (cid:90) D h D H exp (cid:20) i (cid:90) d x L (cid:21) = Det (cid:0) iK δ (4) ( x − y ) (cid:1) − / = exp( iS eff ) (B.15)It follows that S eff = i (cid:0) Det(
K δ (4) ( x − y )) (cid:1) = i (cid:0) ln K δ (4) ( x − y ) (cid:1) (B.16)We write [34]ln K δ (4) ( x − y ) = (cid:90) d p (2 π ) ln K ( x, ∂ x ) e ip ( x − y ) = (cid:90) d p (2 π ) e ip ( x − y ) ln K ( x, ∂ x + ip ) (B.17)and find Tr (cid:0) ln K δ (4) ( x − y ) (cid:1) = (cid:90) d x (cid:90) d p (2 π ) tr (ln K ( x, ∂ x + ip )) (B.18)Here the trace Tr is taken over both space-time indices and the matrix K , the trace tronly over K . We use a similar convention for the determinant symbols Det and det.Inserting (B.18) into (B.16), we obtain the one-loop effective Lagrangian L eff = i (cid:90) d p (2 π ) tr (ln K ( x, ∂ x + ip )) = i (cid:90) d p (2 π ) ln (det K ( x, ∂ x + ip )) (B.19)where det K = ∆(∆ + a + b ) (cid:20) ab − c ∆(∆ + a + b ) (cid:21) (B.20)In the following, we specialize to the effective potential with constant backgroundfields. The derivatives ∂ x of K in (B.19) can then be dropped and ∆ → − p + m . Upto an irrelevant constant, the effective Lagrangian becomes L eff = L eff, + L eff, (B.21)24ith L eff, = i (cid:90) d p (2 π ) ln( p − ( a + b + m )) (B.22) L eff, = i (cid:90) d p (2 π ) ∞ (cid:88) n =1 ( − n +1 n (cid:18) ab − c ( p − m )( p − ( a + b + m )) (cid:19) n (B.23)We assume the model has an SO (5) symmetry in the scalar sector, which is weaklybroken, as discussed at the end of Section 2. With the parameter δ = ω/ξ − (cid:28)
1, wefind α + β = ( f + ˆ H ) + ˆ h + O ( f δ ) = 2 R + O ( f δ ) (B.24)The equation of motion (A.12) gives R = f / O ( f δ ). This implies α + β = f + O ( f δ ) (B.25)The field ˆ H = ˆ H (ˆ h ) is understood to be expressed as a function of ˆ h from solving thee.o.m., as shown in Appendix A.We then find for the parameters in (B.22) and (B.23) a + b + m = M − A − B + O ( M δ ) = M + O ( v ) (B.26)and ab − c = − M f (cid:0) (2 ¯ B + m ) α + 2 ¯ Aβ − Cαβ (cid:1) + 2 ¯ A (2 ¯ B + m ) − C (B.27)The leading term of a + b + m in the limit (15) is just M , while the remaining ˆ h -dependent terms are only of order v : ¯ A and ¯ B are of this order by definition, and M δ = O ( v ) because of (14). The term ab − c has a leading, field-dependent part ∼ M and subleading contributions of order v . Note that terms of order M present in ab and c cancel in the difference.Using (B.26), we rewrite L eff, in (B.22), up to a constant, as L eff, = i (cid:90) d p (2 π ) ∞ (cid:88) n =1 ( − n +1 n (cid:18)
2( ¯ A + ¯ B ) + O ( M δ ) p − M (cid:19) n (B.28)The dominant corrections in (B.23) and (B.28) in the strong-coupling limit (15)arise from the first term in the sums with n = 1. Relative to the tree-level potentialin (38), they are of order M / (16 π f ), up to logarithms ln M/m . Further terms aresubleading, of order 1 / (16 π ) · ( v /M ) k , with k ≥
0. The one-loop corrections to theeffective potential in (B.23) and (B.28) are still divergent, requiring renormalization ofthe leading-order parameters.In the regime of large, but still perturbative couplings, as discussed in Section 2,the parameter M / (16 π f ) is smaller than unity and the tree level potential remainsa meaningful approximation. In the case of genuine strong coupling, M / (16 π f ) ≈ M / (16 π f ) ≈ / (16 π ), and theloop corrections are of the usual perturbative size.The resulting consistent picture of the one-loop corrections to the effective potentialin the limit (15) relies on the approximate SO (5) symmetry of the scalar model, as we seefrom (B.26) and (B.27). The corresponding role of the light Higgs as a pseudo-Goldstoneboson is also illustrated by considering the limit of an exact SO (5) symmetry. In thatcase δ = r = 0, and we find R = (( f + H ) + h ) / α = ˆ h , and β = f + ˆ H . The equationof motion then fixes α + β = f = const. , see (B.25). 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