One-loop effects from spin-1 resonances in Composite Higgs models

We compute the 1-loop correction to the electroweak observables from spin-1 resonances in SO(5)/SO(4) composite Higgs models. The strong dynamics is modeled with an effective description comprising the Nambu-Goldstone bosons and the lowest-lying spin-1 resonances. A classification is performed of the relevant operators including custodially-breaking effects from the gauging of hypercharge. The 1-loop contribution of the resonances is extracted in a diagrammatic approach by matching to the low-energy theory of Nambu-Goldstone bosons. We find that the correction is numerically important in a significant fraction of the parameter space and tends to weaken the bounds providing a negative shift to the S parameter.


Introduction
The electroweak precision measurements performed at LEP, SLD and Tevatron have provided a powerful test of the Standard Model (SM) and set tight constraints on generic models of new physics. They represent a challenge especially for theories where electroweak symmetry breaking (EWSB) originates from new strong dynamics at the TeV scale. Composite Higgs models [1,2] are currently the most interesting representative of this class of theories, as they can accommodate naturally a light Higgs boson. The experimental information on universal corrections to the precision observables at the Z pole can be conveniently summarized in terms of the three parameters [3,4], whose measured value is of order a few × 10 −3 with an error of 10 −3 . A first important correction to the i in composite Higgs models arises as a consequence of the modified couplings of the Higgs to the W and Z bosons [5]. The largest effect comes in particular from the imperfect cancellation of the logarithmic divergence be- been computed in Ref. [6] and are small (of order a few×10 −5 (ξ/0.1)). Threshold corrections at the new physics scale m ρ are instead large, as resonance exchange can give a tree-level contribution to the i . In this case one naively expects shifts of order m 2 W /m 2 ρ , so that a per mille precision on the i implies a lower limit on m ρ at the 2 − 3 TeV level. Given the experimental accuracy, one-loop corrections from the resonances can also give an important contribution. Compared to the IR running they are subleading by a factor log(m ρ /m Z ), although this latter is numerically not very large in natural scenarios (e.g. log(m ρ /m Z ) 3.6 for m ρ = 3 TeV) and can be compensated by a multiplicity factor from the loop of resonances or simply by a numerical accidental enhancement. For example, one-loop corrections from fermionic resonances to 3 are enhanced by color and generation multiplicity factors [7,8], while those to 1 represent the leading effect from new physics if the strong dynamics is custodially symmetric [9,10,5,7].
Aim of this work is to compute the one-loop threshold corrections due to spin-1 resonances in composite Higgs models. These effects were studied in detail in the framework of stronglyinteracting Higgless models (with an SO(4)/SO(3) coset), for which computations exist both in the diagrammatic approach [11][12][13][14] and through the use of dispersion relations [15,16].
Previous analyses of composite Higgs models, on the other hand, included the contribution of spin-1 resonances only at the tree level, see for example Ref. [6] for a generalization of the Peskin-Takeuchi dispertion relation for the S parameter [17] to SO(5)/SO (4). In this paper we perform a calculation of these one-loop threshold effects in SO(5)/SO(4) composite Higgs theories by modeling the strong dynamics with a simple effective description including the Nambu-Goldstone (NG) bosons and the lowest-lying spin-1 resonances. These latter are assumed to be lighter and more weakly interacting than the other composite states at the cutoff. Although this working assumption might not be realized by the underlying strong dynamics, we expect our calculation to give a quantitative approximate description of the contributions from spin-1 resonances arising in full models. Our results represent a required step towards a complete one-loop analysis of precision observables in composite Higgs models including both fermionic and bosonic resonances. This paper is organized as follows. Section 2 discusses the effective Lagrangian for the NG bosons and the spin-1 resonances, highlighting the role of symmetries. The computation of the one-loop correction to the parameters from spin-1 resonances is illustrated in Section 3.
The heavy states are integrated out at a scale µ ∼ m ρ matching to the low-energy theory with only NG bosons. Our results are used to perform a fit to the electroweak observables in Section 4, where limits on the scale m ρ and the degree of Higgs compositeness ξ are derived. We draw our conclusions in Section 5. Finally, we collect in the Appendices some useful additional results: Section A discusses the two-site limit of the spin-1 Lagrangian; Sections B, D and F report formulas related to our calculation; a discussion of the one-loop renormalization of the spin-1 Lagrangian is given in Section C; while Section E provides an alternative derivation of the matching for the T parameter.

(2.3)
In particular, E µ = E L µ +E R µ transforms as a gauge field of SO(4) and can be used to define a covariant derivative ∇ µ = ∂ µ + iE µ as well as a field strength The SM electroweak vector bosons weakly gauge a subgroup SU (2) L × U (1) Y ⊂ SO (4) contained in SO (5), where the SO(4) is misaligned by an angle θ with respect to the unbroken SO (4). Hypercharge is identified with Y = T 3 R 0 , where T a L 0 , T a R 0 are the generators of SO(4) 3 . The derivative appearing in Eq. (2.2) is thus covariant with respect to local Although the EW gauging introduces an explicit breaking of the global SO(5) symmetry, the low-energy Lagrangian can still be expressed in an SO(5)-invariant fashion by introducing suitable spurions that 2 We denote with T a = {T a L , T a R } the generators of SO(4) ∼ SU (2) L × SU (2) R and with Tâ those of SO(5)/SO(4), normalized such that Tr[T A T B ] = δ AB . 3 They are related to the generators {T a , Tâ} through a rotation by an angle θ: T A 0 = r −1 (θ)T A r(θ), see Ref. [19]. encode the breaking. We will be mainly interested in custodially-breaking radiative effects induced by loops of the hypercharge field, while W µ will be treated as an external source. In this limit the explicit breaking of SO(5) can be parametrized in terms of a single spurion whose formal transformation rule is The part of the Lagrangian which describes the interactions among NG bosons can be organized in a derivative expansion controlled by ∂/Λ: where Λ 4πf is the cutoff of the effective theory and L (n) indicates terms with n derivatives.
Omitting for simplicity CP -violating operators, one has: 4 (2.10) and the dots stand for higher-derivative terms and O(p 4 ) operators involving χ. We adopted the basis of four-derivative SO(5)-invariant operators of Ref. [8] (see also Ref. [19]) but 4 Additional O(p 2 ) operators with two powers of the spurion are not linearly independent. Specifically, by using the identity ∇ µ χ = −i[d µ , χ] it is easy to show that: dropped the operator O 5 there appearing because it identically vanishes [20]. Among the terms with 6 derivatives we only list two operators that are relevant for our analysis: The operators O − 3 , O − 4 are odd under the action of the parity P LR exchanging the SU (2) L and SU (2) R groups inside the unbroken SO(4) [19]; all the other operators in Eqs. (2.8), (2.10) are P LR even. In particular, under P LR the spurion χ transforms as Spin-1 resonances will be described by vector fields ρ L µ = ρ a L µ T a L and ρ R µ = ρ a R µ T a R living in the adjoint of SO(4) ∼ SU (2) L × SU (2) R and transforming non-homogeneously under SO(5) global rotations: (2.15) 5 The operator O χ breaks explicitly SO(5) down to the gauged SO(4) . This can be easily seen by rewriting where the gauged SO(4) acts on the first four components of SO(5).

In the unitary gauge one has
, which is custodially symmetric. 6 In the unitary gauge (with gauge kinetic terms normalized as −W a µν W µν a /4g 2 , where the dots indicate terms with more than two gauge fields. By expanding in powers of the fields, at the level of dimension-6 operators, one has (2.14) We will assume that the Lagrangian that describes their interactions can also be organized in a derivative expansion controlled by ∂/Λ, so that physical quantities at E Λ are saturated by the lowest terms [19]. In order to estimate the coefficients of the operators appearing in the effective Lagrangian, we adopt the criterion of Partial UV Completion (PUVC) [19]. This premises that the coupling strengths of the resonances to the NG bosons and to themselves do not exceed, and preferably saturate, the σ-model coupling g * = Λ/f at the cutoff scale.
Under this assumption, neglecting for simplicity CP -odd operators, the leading terms in the Lagrangian are Among the operators involving χ, we have kept only those relevant for the present analysis.

Hidden local symmetry description
The above construction relies on describing the resonances in terms of massive vector fields, which propagate three polarizations. At energies m ρ E < Λ, however, the longitudinal and transverse polarizations behave differently (their interactions scale differently with the energy), and it is convenient to describe them in terms of distinct fields. Indeed, it is always possible to parametrize the longitudinal polarizations of massive spin-1 fields in terms of a set of eaten NG bosons 7 . In the case of the Lagrangian (2.16) the corresponding coset is SO(5) × SO(4) H /SO(4) d , which leads to 10 NG bosons transforming under the unbroken diagonal SO(4) d as π = (2, 2), η L = (3, 1) and η R = (1, 3) [19]. Their σ-model Lagrangian can be obtained by taking the limit g ρ → 0 with m ρ /g ρ fixed; transverse polarizations are then reintroduced by gauging the SO(4) H subgroup with vector fields ρ µ . It is convenient to parametrize the NG bosons in terms of U (π, η) = e i √ 2π/f e iη L /fρ L e iη R /fρ R [19], where η L (x) = 7 See for example Ref. [21]. η a L (x)X a L , η R (x) = η a R (x)X a R and, we recall, π(x) = πâ(x)Tâ 8 . It is thus straightforward to derive the CCWZ decomposition where d µ (π, η),d µ (π, η) and E µ (π, η) are obtained by projecting respectively along the generators Tâ, X a and Y a . Here d µ (π) and E µ (π) denote the uplift of the corresponding SO(5)/SO(4) functions to the 9 × 9 space (they have non-vanishing components in the 5 × 5 subspace). Notice that d µ (π, η) is just an (η-dependent) SO(4) d rotation of d µ (π).
Since SO(5) × SO(4) H /SO(4) d is not a symmetric space, hence no grading of the algebra exists, the d and E symbols will contain terms with both odd and even numbers of NG bosons in their expansion. In particular, and similarly ford R µ . In the unitary gauge η a = 0 one has (d r µ ) a = (E r µ (π) − ρ r µ ) a / √ 2 (r = L, R). It is thus easy to see that the kinetic terms of the NG bosons η are mapped into the ρ mass terms of Eq. (2.16), (2.21) 8 We denote the SO(5) × SO(4) H /SO(4) d (broken) generators by Tâ, X a = (T a − T a H )/ √ 2, where T a H are those of SO(4) H , and the SO(4) d (unbroken) generators by Y a = (T a + T a H )/ √ 2. We will consider their matrix representation on a 9 × 9 space, so that T a , Tâ and T a H act respectively on 5 × 5 and 4 × 4 subspaces. All the traces in this section and in the next one (Sections 2.1 and 2.2) will be 9 × 9 ones except where explicitly indicated.

Two-site model limit
While in general π, η L , η R form three irreducible representations of the unbroken group, in the gauge-less limit g ρ = g = g = 0 and for the special choice an unbroken SO(5) symmetry which allows one to align the vacuum to θ = 0 without loss of generality. This means that for g ρ → 0, with non-zero EW couplings, all EWSB effects must vanish in the two-site model. Indeed, the Higgs is a NG boson under both SO(5)'s, and both symmetries must be explicitly broken (hence the collective breaking) in order to generate any EWSB effect.
The authors of Ref. [22] also put forward a simple power counting argument showing that collective breaking lowers the superficial degree of divergence of EWSB quantities. This is easy to see by working in a renormalizable gauge and noticing that the NG bosons η interact with strength E/f ρ , while the gauge field ρ µ has coupling g ρ . In any 1PI diagram, replacing an internal η line with a ρ propagator lowers the degree of divergence by two unites. Indeed, if one focuses on the divergent part, the extra relative factor g 2 ρ f 2 ρ of the new diagram can only be compensated by a factor 1/Λ 2 , where Λ is the cutoff scale. Therefore, diagrams with loops of NG bosons alone (and no transverse gauge field ρ) carry the largest superficial degree of divergence. If they entail a breaking of the EW symmetry, then their sum will vanish in the two-site model, since one can set g ρ = 0 in their evaluation and by the previous argument the electroweak symmetry is exact in this limit. The original superficial degree of divergence is thus lowered. In particular, 1PI contributions to EWSB observables will be finite in the SO(5)/SO(4) theory (with both ρ L and ρ R ) for a ρ = 1/ √ 2 9 if they are at most logarithmically divergent in the general case.
This power counting argument was used in Ref. [22] to conclude that the S and T parameters are finite in the a ρ = 1/ √ 2 limit. In the case of the S parameter one can easily prove that for g ρ = 0 there is no local counterterm for 1PI divergent contributions to the W 3 µ B ν Green function that can be constructed in the two-site model compatibly with the SO(5) × SO(5) H symmetry, see Appendix A. Local operators built by including powers of the spurion g ρ can be generated at the cutoff scale through loops where both the heavier states and the ρ circulate. By power counting these effects are finite at the 1-loop level, and lead to a contribution to the S parameter that is suppressed by an additional factor (g ρ f /Λ) 2 = (g ρ /g * ) 2 compared to the naive estimate. They are thus subleading and can be neglected if g ρ g * . As discussed in Section 3.1, our calculation confirms that the 1PI divergence (hence the β-function of c + 3 ) vanishes for a ρ = 1/ √ 2. The S parameter is thus 9 Here and in the following we use the notation a ρ = 1/ √ 2 as a shorthand for a ρ L = a ρ R = 1/ √ 2.
calculable in terms of the renormalized g ρ and α 2 , which absorb the divergences associated to subdiagrams. Things work differently for the T parameter, however. It turns out that while the 1PI divergence to the W 1 W 1 − W 3 W 3 Green function vanishes according to the argument of Ref. [22], the β-function of c T does not vanish for a ρ = 1/ √ 2 and there is still a dependence on c T in the final result which enters through the cancellation of the subdivergences. This can be seen as follows.
First of all, we notice that in the theory above m ρ it is possible to embed O T into the  But how a non-vanishing c T is compatible with the fact that no EWSB occurs in the two-site model for g ρ = 0 ? In this limit, there is an after the EW gauging which gives 10 NG bosons. Four of these are eaten to give mass to the W a µ triplet and to the hypercharge, while the others remain massless and transform as a 2 1/2 (the composite Higgs doublet), and a 1 ±1 of the 10 . In particular, the unbroken global symmetry forces the W i to form a degenerate triplet. Compatibly with this, the operator in Eq. (2.24) does not lead to any splitting between W 3 and W 1,2 : the term W 3 µ W 3 µ contained in the expansion of O T is exactly canceled by a similar contribution from the other operators in the Lagrangian (2. 16) as a consequence of the relations (2.25). One has: Since no corresponding counterterm is contained in Eq. (2.24), any 1PI contribution to the Green function W 1 W 1 − W 3 W 3 must be finite, in agreement with the power counting argument of Ref. [22]. This is however not sufficient to conclude that the T parameter is finite, since non-1PI diagrams also contribute and can be divergent. 11 Our calculation in Appendix E indeed shows that a divergent contribution arises from subdiagrams through the 1-loop correction to the ρ propagator. The associated counterterm is contained in the operator (2.24), whose coefficient c T thus enters in the expression of the T parameter.
It is interesting to notice that the T parameter can also be extracted from the Green function π 3 π 3 , as done in Section 3.1, for which a 1PI divergent contribution does exist.
The corresponding counterterm (π 3 ) 2 is contained in Eq. (2.24), and it is not in clash with the argument of Ref. [22]. This is because π 3 appears in the linear combination of NG bosons, the one in parenthesis in the first line of Eq. (2.26), that is eaten to give mass to the hypercharge for g ρ = 0. 12 The π 3 π 3 Green function thus does not break the [SU (2) L × U (1) Y ] d symmetry and can be divergent.
Although it depends on c T , the T parameter can still be regarded as a calculable quantity in the two-site limit, up to g 2 ρ /g 2 * effects. This is because the operator (2.24) gives a custodially-breaking shift to the mass of the neutral ρ's, so that c T can be rewritten in terms of the difference of charged and neutral renormalized ρ masses. In this sense T , similarly to S, is calculable in terms of parameters related to the ρ, which can be fixed experimentally by measuring its properties. 11 We thank G. Panico and A. Wulzer for discussions on this point. 12 For θ = 0 the NG boson eaten by the hypercharge is η 3R , while the η aL are eaten to give mass to the W triplet.

Electroweak parameters at 1 loop
Oblique corrections to the electroweak precision observables at the Z-pole are conveniently described by the three parameters [3,4] defined in terms of the following vector-boson self energies: Here s W (c W ) denotes the sine (cosine) of the Weinberg angle and, according to the standard notation, the vacuum polarizations are decomposed as There are two kind of modifications to the self-energies (3.28) from new physics in our model.
The first is due to the virtual exchange of the spin-1 resonances, which at energies E ∼ m Z m ρ can be parametrized in terms of local operators of the effective Lagrangian (2.6).
The tree-level contribution of these local operators to physical observables is a pure shortdistance effect, while their insertion in 1-loop diagrams with light fields contains also a long-distance part. The second modification comes from the fact that the composite Higgs has non-standard couplings with the electroweak vector bosons. The bulk of the correction in this case is given by a logarithmically divergent part that can be easily computed in the low-energy theory with light fields [5]. Extracting the finite part instead requires fully recomputing the Higgs contribution to the vector boson self energies in Fig. 1, as pointed out in Ref. [6]. Since the Higgs boson is light, this is a long-distance effect. It is so even if contributions. 13 We have performed a calculation of the i at the 1-loop level including all the contributions mentioned above. We have used dimensional regularization and performed a minimal subtraction of the divergences (M S scheme). We choose to work in the Landau gauge for the elementary gauge fields, ∂ µ W i µ = 0 = ∂ µ B µ , since it conveniently preserves the custodial invariance of the strong sector and leads to massless (hence degenerate) NG bosons π 1,2,3 .
The one-loop contribution from the spin-1 resonances is computed through a matching procedure. We integrate out the ρ at a scale µ ∼ m ρ and match with a low-energy Lagrangian which has the same form of Eq. (2.6). Its coefficients will be denoted byc i (µ), where the tilde distinguishes them from the corresponding quantities in the full theory. By working in 13 The divergent part of the diagrams corresponds to a renormalization of the local operators of the effective Lagrangian, and it is thus a short-distance effect. The finite part is instead genuinely long distance.
such low-energy theory and defining the shifts to the epsilons to be ∆ i = i − SM i , we find The first term in each equation corresponds to the long-distance Higgs contribution of ,c 3B encode the short-distance contribution from the ρ and from cutoff states, and are in one-to-one correspondence with the coefficients S, T, W, Y defined in Refs. [17,23]. The latter are introduced through an expansion of the self energies (3.29) in powers of q 2 and parametrize the tree-level contribution from new heavy physics. At the tree level one can thus identifŷ . The naive estimate of W and Y is suppressed by a factor g 2 /g 2 ρ compared to that ofŜ andT [23]. We thus included their contribution (i.e. the contribution ofc 3W andc 3B ) only in 2 , where it gives the leading effect. At the oneloop level, the coefficientsc i acquire a dependence on the subtraction scale µ, as required to balance the variation of the logarithms in Eqs. (3.30)-(3.32). Indeed, the ∆ i are observables quantities and therefore independent of µ. The naive identities (3.33) thus require specifying 14 It can be found from the Higgs contribution in the SM by considering that the Higgs couplings to vector bosons are rescaled by a factor cos θ, so that i | Higgs = cos 2 θ SM i | Higgs , hence ∆ i | Higgs = − sin 2 θ SM i | Higgs . a value for the matching scale µ in order to be valid beyond the tree level. More in general, the contribution from new heavy physics cannot be simply parametrized in terms of four constant coefficients, as its proper description requires taking into account the RG evolution of the coefficientsc i (µ). We find that this evolution is described by the RG equations

Matching
The explicit contribution of the spin-1 resonances to thec i can be obtained by integrating them out and matching to the low-energy Lagrangian. We perform this matching at the 1loop level. This requires working out at the same time the renormalization of the Lagrangian for the ρ, in order to derive the RG evolution of its parameters. We considered two choices to fix the gauge invariance associated with the ρ field and checked that they both lead to the same result for physical quantities: the first is the unitary gauge, where the ρ is described by the Lagrangian (2.16); the second is the Landau gauge ∂ µ ρ a µ = 0, obtained by introducing the NG bosons η as discussed in Section 2.1. In the following we will report results for the unitary gauge, and collect formulas for the Landau gauge in Appendix C. Particularly relevant for our calculation is the running of g ρ and α 2 , since these parameters enter at tree level in the expression of the i . In the unitary gauge we find Finally, our formulas will include the contribution of both the ρ L and the ρ R . In case only one resonance is present in the theory,c + 3 andc T have the same expression for both ρ L and ρ R , whereas ρ L only generatesc 3W , and ρ R onlyc 3B . This follows from a simple symmetry argument. Given a theory with a ρ L , the case with a ρ R is obtained by performing a P LR transformation on the strong dynamics. The equality ofc + 3 andc T then follows from the invariance of the operators O + 3 and O T under such transformation. On the other hand, acting with P LR interchanges O 3W with O 3B , so that the expression ofc 3W in a theory with  a ρ L equals that ofc 3B in a theory with ρ R . We report the corresponding expressions in Appendix F for convenience.
Let us start discussing the matching forc + 3 . We make use of the two-point Green function W 3 µ B ν , in particular its derivative evaluated at q 2 = 0, and match its expression in the full and effective theories. We focus on the leading contribution in g 2 , thus considering diagrams where only the ρ and the NG bosons (i.e. no elementary gauge field) circulate in the loop.
These are the diagrams of Figs. 3, 4 and 5 for the full theory (ρ + NG bosons) and of Fig. 4 for the effective theory (only NG bosons). Neglecting diagrams with EW vector bosons circulating in the loop implies a relative error of order g 2 /g 2 ρ . Divergences from subdiagrams in the full theory are canceled by the addition of suitable counterterms. The remaining divergence is associated with the running of c + 3 between m ρ and Λ due to loops of ρ's and NG bosons. We find Notice that β c + 3 (hence the associated divergence) vanishes for a ρ L = a ρ R = 1/ √ 2, in agreement with the symmetry argument of Section 2.2. By matching the full and low-energy theories at a scale µ ∼ m ρ , we obtaiñ (3.38) Obviously, sincec + 3 contributes to an observable such as ∆ 3 (see Eq. (3.32)), its expression (3.38) is the same in any gauge. In fact, it turns out that even the β-function of c + 3 , Eq. (3.37), is gauge invariant at one loop. The argument goes as follows. When working at the 1-loop level, the logarithms that appear in the expression of an observable determine the running of the combination of the parameters giving the tree-level contribution. Since the expression of the observable is gauge invariant, also the RG evolution of such combination will be invariant. In the case of ∆ 3 , the tree-level contribution is given by the terms in the first line of Eq. (3.38). Furthermore, (1/2g ρ − α 2 g ρ ) 2 (for each ρ species) also has a gauge invariant running, since it gives the tree-level contribution to another observable: the pole residue of the ρ two-point function [24]. Working in the approximation in which 1-loop effects from α 1,2 are neglected, this in turn implies that (1/4g 2 ρ − α 2 ) has an invariant RG evolution, 15 hence the same follows for c + 3 . Clearly, when including the 1-loop contribution of α 2 or going to two loops, the running of c + 3 acquires a gauge-dependent part. Let us now turn toc T . In order to extract it, we make use of the two-point Green function of the π field, in particular we consider the custodially breaking combination π 1 π 1 − π 3 π 3 and compute its derivative at q 2 = 0. This gives access to the coefficient of the operator O T , as it follows from the expansion Tr[d  Fig. 6 for the low-energy theory of NG bosons. Only diagrams where an elementary B µ circulates contribute, as this latter gives the required breaking of custodial symmetry. As forc + 3 , we neglect diagrams with further insertions of EW vector bosons, since they are of higher order in g 2 . The corresponding relative error is 15 The running of the α 2 2 term is of the same order of the neglected terms.  of order g 2 /g 2 ρ . Since there are no divergent subdiagrams, the overall divergence in the full theory is associated with the running of c T between the scales Λ and m ρ . We find: Since c T gives the only tree-level contribution to ∆ 1 (see Eq. (3.41) below), its RG evolution is gauge invariant. One can see that β c T does not vanish for a ρ L = a ρ R = 1/ √ 2. This confirms the argument of Section 2.2, where it was noticed that a counterterm exists also in the SO(5)×SO(5) symmetric limit (see Eq. (2.24)), and no cancellation of the 1PI divergence of the π 1 π 1 − π 3 π 3 Green function is expected in this case. There is in fact a limit in which the divergence partly cancels, as already discussed in Ref. [15] for a Higgsless model.
Indeed, the diagram of Fig. 6 and the first two diagrams in Fig. 7 can be combined into one where B µ couples to the NG bosons through the effective vertex where the Bπâπb form factor denoted by the gray blob is equal to (3.40) In the limit a ρ L = a ρ R = 1 one obtains Vector Meson Dominance (VMD) for any value of θ, i.e. the form factor goes to 0 in the limit q 2 → ∞. Consequently, the diagram built with the effective vertex (i.e. the sum of the diagram in Fig. 6 and the first two of Fig. 7) is finite.
This does not imply, however, that the β-function of c T vanishes, since the last diagram of (3.41) Sincec T contributes to the observable ∆ 1 , this expression is gauge invariant.
Finally, we discuss the matching to extractc 3W andc 3B . We make use of the W µ W ν and B µ B ν Green functions, in particular we compute their second derivative evaluated at In fact, such contribution is required in order to properly match the IR divergence of the full and low-energy theories. The cancellation occurs ifc ± 3 andc ± 4 are set to the value they have at tree-level for α i = 0 (that is:c ± 3 = −1/8g 2 ρ L ∓ 1/8g 2 ρ R andc ± 4 = 0) when evaluating the diagram of Fig. 2; we will thus adopt this choice. 16 There are no divergences left after removing those from subdiagrams through the renormalization of the ρ mass and kinetic terms. This implies that the running of the coefficients c 3W and c 3B vanishes in the full theory between m ρ and Λ: This result is independent of the choice of gauge. Indeed, by matching the full and low-energy 16 When including the contribution of α 2 at the 1-loop level, as done in Appendix D, one should instead setc ± 3 = (−1/4g 2 ρ L + α 2L )/2 ± (−1/4g 2 ρ R + α 2R )/2, while including α 1 at the 1-loop level requires setting c ± 4 = (α 1L ± α 1R )/2. theories we obtaiñ The tree-level contribution to ∆ 2 comes from the combination of terms in the first line of the above equations. We already noticed that (1/g ρ − 2α 2 g ρ ) 2 has an invariant RG evolution at the 1-loop level; the same holds true for m ρ , since it gives the tree-level contribution to the pole mass. It thus follows that the RG evolution of c 3W and c 3B is also gauge invariant at one loop.

Fit to the EW observables
The results of the previous section can be used to perform a fit to the i . It is convenient to express the corrections ∆ i in terms of the parameters g ρ , α 2 and m ρ evaluated at the physical mass scale of the resonances m pole ρ . 17 This removes all the logarithms originating from subdivergences leaving only those associated with the running of O + 3 , O T , O 3W and O 3B . We will consider two benchmark scenarios: in the first (Scenario 1) both ρ L and ρ R are present with equal masses and couplings (as implied for example by P LR invariance); in the second (Scenario 2) only a ρ L is included. In either case the ∆ i can be written as In the case of 3 , the leading corrections come from the tree-level contribution (of order m 2 W /m 2 ρ ) and the IR running. Compared to the former, the latter effect is suppressed by a factor g 2 ρ /16π 2 but enhanced by log(m ρ /m Z ). The 1-loop ρ contribution is subleading because also suppressed by g 2 ρ /16π 2 and enhanced by the smaller logarithm associated with the running between Λ and m ρ . The contribution from cutoff physics encoded by c + 3 (Λ) can be estimated through Naive Dimensional Analysis (NDA) [25]. If the dynamics at the scale Λ is maximally strongly coupled one expects c + 3 (Λ) ∼ 1/16π 2 , which leads to a correction of  naively expects c + 3 (Λ) ∼ 1/g 2 * . For g ρ < g * < 4π this implies a correction larger than the 1-loop ρ contribution, though smaller than the tree-level one. Interestingly, in the two-site limit (Scenario 1 with a ρ = 1/ √ 2) the SO(5) × SO(5) H global invariance of the theory below the cutoff ensures c + 3 (Λ) = 0, since the corresponding operator vanishes. Notice that β c + Similar estimates of the various terms hold for ∆ 1 , except there is no tree-level correction due to custodial invariance, so that the largest effect comes from the IR running. In the case of ∆ 2 , the contribution from the ρ exchange (both at tree and loop level) is suppressed by a factor (g 2 /g 2 ρ ) compared to the one entering ∆ 1 and ∆ 3 . This is because the leading short-distance contribution in the low-energy theory arises at O(p 6 ) through the operators [10]. The RG evolution of these latter in turn proceeds through the 1-loop insertion of O(p 4 ) operators, as discussed in the previous section, implying that the IR running contribution to ∆ 2 is also suppressed by a factor (g 2 /g 2 ρ ). The only unsuppressed effect is the finite term from Higgs compositeness, which is however numerically small. The overall shift to 2 thus tends to be small and plays a minor role in the fit.
Besides the direct contributions to the ∆ i described above there is also an indirect one from the evolution of g ρ , m ρ and α 2 from the cutoff Λ down to the scale m ρ . This is a numerically large effect if the ∆ i are expressed in terms of the values of these parameters at the scale Λ. The running of g ρ , in particular, proceeds through a sizable and negative (for a ρ not too large) β-function, growing quickly in the IR. This implies that for moderately large values of g ρ at the cutoff scale, the gap Λ/m ρ cannot be too large otherwise g ρ would hit a Landau pole for µ > m ρ . For example, g ρ (Λ) = 3 gives a Landau pole at µ Λ/3.6 in the unitary gauge. Although the evolution of g ρ is gauge dependent, it gives a rough indication on how strongly coupled the theory of spin-1 resonances is. A more refined estimate could make use for example of the combination λ ≡ (1/g ρ −2α 2 g ρ ) −1 with gauge-invariant running.
Notice also that β gρ will in general receive contributions also from other resonances lighter than the cutoff, like for example the top partners, which could slow down the growth of g ρ in the IR and allow larger gaps.
In the following we analyze the constraints from the current electroweak data by constructing a χ 2 function using the fit of Refs. [26,27] to the ∆ i and their theoretical predictions in Eqs. (4.45)-(4.47). 18 These latter will be evaluated in terms of the values of the param- 18 We perform a 3-parameters fit by using Table 4 of Ref. [27] . We derive the limits by determining the isocurves of ∆χ 2 corresponding to 3 degrees of freedom. Considering that 2 does not vary much in our model (the new physics corrections is small), one could adopt a more conservative choice and derive the isocurves with 2 degrees of freedom. This would lead to slightly stronger constraints, without qualitatively affecting our conclusions. eters g ρ , m ρ and f at the scale µ = m ρ . In particular we use the identity g ρ = m ρ /(a ρ f ) (Eq. (2.21)) to rewrite g ρ in terms of a ρ and fix where ξ ≡ sin 2 θ and v = 246 GeV is the electroweak scale. This relation follows from the minimization of the Higgs potential generated by loops of heavy resonances. 19 The value of the remaining parameters c + 3 , c T , c 3W , c 3B is set to vanish at the scale Λ. For the case of c + 3 , whose β-functions is gauge dependent when including the contribution from α 1,2 at one loop, this condition is imposed in the unitary gauge. 20 Our results are expressed as 95% CL exclusion regions in the plane (m ρ (m ρ ), ξ). The left and right plots in Figure 8 show the limits respectively for Scenario 1 with a ρ (m ρ ) = 1/ √ 2 (two-site limit) and Scenario 2 with a ρ (m ρ ) = 1. Notice that the tree-level shift to 3 is the same in the two cases: (4.47)). In both cases we fix Λ = 3m ρ (m ρ ) and set α 2 (m ρ ) = a 2 ρ (1 − a 2 ρ )/(96π 2 ) log(m ρ /Λ). This one-loop value is chosen so that α 2 vanishes at the scale µ = Λ in the unitary gauge. The orange area represents the region allowed at 95% CL following from the full 1-loop results of Eqs. (4.45)-(4.47).
The dashed line shows instead the corresponding limit obtained by including the effect of the ρ at the tree level. The dotted blue lines are isocurves of constant g ρ (m ρ ), and the blue area corresponds to the region with g ρ (m ρ ) > 4π. As expected, the 1-loop ρ contribution is more important for larger values of g ρ , for which the tree-level shift to 3 is smaller. It gives a negative shift to 3 and a small correction to 1 , thus enlarging the allowed region. The numerical values are reported in Table 2 and compared to the shifts from the IR running and Higgs compositeness. The effect of including the new physics correction to 2 is small, except for g ρ 1.5 where it makes the bound on m ρ less strong (tail of the orange region at smaller values of m ρ and ξ). For small g ρ the 1-loop ρ contribution becomes less important and the limit almost coincides with the tree-level one. The interpretation of our results 19 If electroweak symmetry breaking is triggered by the contribution of a lighter set of resonances with mass m Ψ , for instance the top partners, the relation becomes f (m Ψ ) = v/ √ ξ. In this case f (m ρ ) can be derived by running from m Ψ . Notice that β f is gauge invariant at one loop, since f gives the tree-level correction to physical observables like the on-shell ππ → ππ scattering amplitude and the W mass. 20 Equivalently, one can fix c + 3 (m ρ ) so that c + 3 vanishes at µ = Λ in the unitary gauge. The condition formulated in this way at µ = m ρ is gauge independent.   for very large values of g ρ requires some caution: naively the perturbative expansion breaks down for g ρ 4π (blue region), but in practice higher-loop effects can become sizable earlier, invalidating our approximate result. For example, we find that the 1-loop correction to g ρ and to the pole mass m pole ρ becomes as large as the tree-level term already for g ρ ∼ 5 − 6. 21 Also notice that, as a consequence of fixing Λ/m ρ = 3, values g ρ > 4π/(3a ρ ) correspond to a cutoff scale Λ larger than its naive upper limit 4πf . The latter should not be interpreted as a sharp bound but rather as an indicative values suggested by NDA. Yet, the above estimate also suggests that perturbativity might be lost for g ρ somewhat smaller than 4π.
The plots of Figure 8 shows the limits for a benchmark choice of parameters. When these latter are varied, the results can change even significantly. Increasing the value of the gap Λ/m ρ amplifies the logarithmic term in the 1-loop ρ contribution. For values of the other parameters as in Fig. 8, the effect turns out to be small and tends to reduce the allowed region. Varying a ρ has a larger impact on the fit, since this parameter controls the size of the tree-level correction to 3 : smaller values of a ρ imply smaller ∆ 3 | tree , hence weaker bounds on m ρ . The value of a ρ also controls the size and the sign of the 1-loop ρ contribution. Table 2 shows for example how this changes when varying 0.5 < a ρ < 1.5. We find that in general the finite part is numerically comparable, if not larger, than the log term. For illustration we show in Figure 9 the limits obtained in Scenario 2 for a ρ = 0.5 (left plot) and a ρ = 1.5 (right plot). Finally, one could consider a scenario where α 2 is of order 1/g 2 ρ , leading to a cancellation in the tree-level contribution to 3 . A proper calculation of the ∆ i in this case requires including the 1-loop contribution from α 2 through the formulas of Appendix D, thus re-summing all powers of α 2 g 2 ρ . As an illustration, Figure 10 shows the limits obtained for α 2 g 2 ρ = 1/8 and 1/4 at the scale µ = m ρ , corresponding respectively to a 50% and 100% cancellation of the tree-level contribution to 3 . In the (extreme) case of a complete cancellation, the tail of the allowed region at large ξ and small m ρ is a result of the new physics contribution to 2 . It is indeed possible to compensate the positive (negative) shift to 3 ( 1 ) from the IR running with a sizable and negative ∆ 2 , due to the correlation in the 3-dimensional χ 2 function. For small g ρ such large and negative ∆ 2 is provided by the tree-level ρ exchange, thus leading to the narrow region extending up to ξ ∼ 0.5 and 21 It is because of the premature loss of perturbativity in the pole mass that we prefer to show the plots of The bounds that follow on m ρ and ξ from our analysis are quite severe. As already pointed out in previous studies, this is because the tree-level exchange of the ρ generally implies a large and positive ∆ 3 , while the IR running gives a positive ∆ 3 and a negative ∆ 1 . The combination of these effects brings the theoretical prediction far outside the 95% CL contour in the plane ( 3 , 1 ) unless ξ (m ρ ) is very small (large). This is illustrated by Figure 11, where the region spanned by varying m ρ and ξ is shown in red for a ρ = 0.5, 1, 1.5 in the case of Scenario 2. It is evident that an additional negative contribution to 3 or positive contribution to 1 , as for example coming from loops of fermionic resonances, can relax even significantly the bounds (see for example Refs. [7,8])

Conclusions
In this paper we have computed the 1-loop contribution to the electroweak parameters 1,2,3 arising from spin-1 resonances in a class of SO(5)/SO(4) composite Higgs theories. We performed our analysis by giving a low-energy effective description of the strong dynamics in terms of Nambu-Goldstone bosons and lowest-lying spin-1 resonances (ρ L and ρ R ), these latter transforming as an adjoint representation of the unbroken SO(4). We provided a classification of the relevant operators by including the custodially-breaking effects arising from the external gauging of hypercharge. A detailed discussion was given of the so-called 'hidden local symmetry' description of the spin-1 resonances, where their longitudinal polarizations are parametrized in terms of the NG bosons from a larger coset. This was useful to analyze a particular limit, noticed by Ref. [22], in which the theory acquires a larger SO(5) × SO(5)/SO(5) global symmetry and has a collective breaking mechanism. In particular, we reviewed the argument that shows how certain EWSB quantities enjoy an improved convergence in this limit, clarifying the role of divergent subdiagrams in the calculation of S and T .
The contribution of the ρ to the electroweak parameters was computed by performing a 1-loop matching to the low-energy theory of NG bosons. We used dimensional regularization and analyzed in detail the renormalization of the spin-1 Lagrangian and the RG evolution of its coefficients. We estimate a relative uncertainty in our calculation of order  47)). Although parametrically subdominant compared to the IR running and the tree-level contribution, we find it to be numerically important in a significant fraction of the parameter space, where the coupling strength g ρ is moderately large. Its effect is that of enlarging the allowed region providing a negative shift to 3 (see Table 2 and Figs. 8-10). The relative importance of the 1-loop contribution grows with g ρ . Although one would naively expect perturbativity to remain valid until g ρ ∼ 4π, the 1-loop correction becomes as important as the tree-level term already for g ρ ∼ 5 − 6 in several quantities, as for example the running of g ρ or the pole mass m pole ρ . This suggests that any limit extending to such large values of g ρ should be interpreted with caution. The contribution from cutoff states to the electroweak observables might also be important. Its naive estimate in the case of a fully strongly coupled dynamics at the scale Λ suggests that it is subleading compared to the 1-loop ρ contribution only by a factor log(Λ/m ρ ), which is not expected to be very large. In fact, the very existence of a gap Λ/m ρ 1 should be considered as a working hypothesis of our study, not necessarily realized by the underlying strong dynamics. In this sense our calculation should be regarded as a way, more refined than a simple estimate, to assess the contribution of the spectrum of resonances lying at the compositeness scale to the oblique parameters.
At the level of two derivatives and two powers of the hypercharge spurion g T 3 R 0 , there is one (SO(5) × SO(5) H )-invariant operator which can be constructed: Notice that the combinationŪ D µŪ † transforms asŪ D µŪ † → g(Ū D µŪ † )g † . In the η = 0 gauge, by defining χ(π) =Ū † (π, 0)g T 3 R 0Ū (π, 0), one has which coincides with the right-hand side of Eq. (2.24). On the other hand, at order g 0 ρ there is no operator with two EW field strengths and no derivative acting onŪ which can contribute to the S parameter. This is because there is no way to saturate the SO(5) H index ofŪ except in the trivial productŪŪ † = 1.

C One-loop renormalization of the spin-1 Lagrangian
Consistently with the 1-loop matching of the full and effective theories, one should also perform a 1-loop renormalization of the Lagrangian of spin-1 resonances. We first describe our procedure for the unitary gauge and then give the results also for the Landau gauge.
We will not specify the quantum numbers of the spin-1 resonance unless necessary since the same expressions hold for both ρ L and ρ R , there being no mixed renormalization at one loop.
Starting from the bare Lagrangian, we define renormalized fields and parameters as follows We thus obtain (C.59) From these expressions it follows Eq. (3.35) and The renormalized c i and α 2 are instead defined by The value of the counterterm ∆ α 2 is obtained by renormalizing the ρ µ A µ Green function.
We find ∆ α 2 = a 2 ρ (1 − a 2 ρ )/96π 2 , which leads to Eq. A similar procedure also applies in the Landau gauge with a few differences however.
First, another field is present, that of the NG bosons η, which needs to be renormalized.
Second, the ρ mass originates from the η kinetic term, and m ρ is defined in terms of f ρ according to Eq. (2.21). It is thus more convenient to include f ρ in the list of renormalized quantities and treat m ρ as a derived parameter. By defining (C.64) D One-loop contribution from α 2 When including the effect of α 2 at the 1-loop level, there arise the following additional contributions to the i : (D.67) The renormalization of the various parameters is also affected, in particular each β-function gets an additional contribution. We report the corresponding expressions in the unitary gauge:  , where r = L, R andρ r µ ≡ ρ r µ − E r µ . The contribution of these counterterms, however, cancels out when summing all the diagrams. The overall divergence of the W 1 W 1 − W 3 W 3 Green function is thus removed by the single counterterm (Tr[d µ χ]) 2 , as required to reproduce the calculation ofc T through π 1 π 1 − π 3 π 3 . By matching the low-energy and full theories one obtains Eq. (3.41). A further check of the calculation follows from the fact that in the limit a ρ L = a ρ R = 1/ √ 2 the counterterms combine into the (SO(5) × SO(5) H )-invariant operator of Eq. (2.24). In this limit the 1PI divergence vanishes, and the only divergent contribution to W 1 W 1 − W 3 W 3 comes from subdiagrams.

F Results for a single ρ
In a theory with a single spin-1 resonance, either ρ L or ρ R , the RG evolution and matching conditions for c + 3 and c T are respectively (neglecting 1-loop contributions from α 1,2 ) (F.82)