Dispersion Relations for Electroweak Observables in Composite Higgs Models

We derive dispersion relations for the electroweak oblique observables measured at LEP in the context of $SO(5)/SO(4)$ composite Higgs models. It is shown how these relations can be used and must be modified when modeling the spectral functions through a low-energy effective description of the strong dynamics. The dispersion relation for the parameter $\epsilon_3$ is then used to estimate the contribution from spin-1 resonances at the 1-loop level. Finally, it is shown that the sign of the contribution to the $\hat S$ parameter from the lowest-lying spin-1 states is not necessarily positive definite, but depends on the energy scale at which the asymptotic behavior of current correlators is attained.


Introduction
Theories with strong electroweak symmetry breaking are severely constrained by the electroweak precision observables measured at LEP, SLC and Tevatron. Large corrections to vector boson polarizations, especially those encoded by the Peskin-Takeuchi S parameter [1], were the most severe problem of Technicolor theories [2], together with flavor, before the discovery of a light Higgs boson. To date, electroweak tests set the strongest constraints on composite Higgs theories [3,4], and this is even more true for their recent Twin Higgs realizations [5][6][7][8][9]. However, while corrections to electroweak observables can be naively estimated to be generally large, their precise determination in the context of strongly-interacting dynamics is a challenge. A first-principle approach based on a non-perturbative method such as lattice gauge theories is possible but demanding in terms of theoretical efforts and computational power (see for example Refs. [10] for calculations of the S parameter on the lattice).
Simpler, though less rigorous approaches include a variety of perturbative methods like the inclusion of chiral logarithms, effective models of the lowest-lying resonances, and the large-N expansion. Especially powerful in this sense is the 5-dimensional perturbative approach of holographic theories, which allows one to effectively resum the corrections of a whole tower of states, the Kaluza-Klein excitations, neglecting smaller effects from string modes.
An alternative strategy consists in making use of dispersion relations to express an observable as the integral over the spectral functions of the strong dynamics. Extracting the spectral functions from experimental data thus leads to a result which is, at least in principle, free from theoretical ambiguities. The most successful application of this idea is perhaps the determination of the correction from the electromagnetic vacuum polarization due to QCD to the muon g − 2 [11], though equally famous is the estimate of the S parameter in Technicolor theories made by Peskin and Takeuchi in their seminal paper [1] (where they also compute the chiral coefficient l 5 using the dispersive formula first derived by Gasser and Leutwyler [12]). Although the most powerful use of dispersion relations is in conjunction with experimental data, in the absence of the latter one can make models of the spectral functions based on theoretical considerations. Computing the spectral functions through a low-energy effective theory of resonances leads in fact to the same result obtained by a more conventional diagrammatic technique, though the dispersive approach can simplify the calculation and gives a different viewpoint.
The first application of dispersion relations to composite Higgs theories was given in Ref. [13] by Rychkov and Orgogozo, who derived a dispersion formula for the parameter 3 defined by Altarelli and Barbieri [14]. A dispersive 1-loop calculation of the S parameter was later performed by Ref. [15] (see Appendix B therein). The aim of this paper is to give an alternative derivation and extend the work of Ref. [13] by obtaining spectral representations for the electroweak parametersŜ, W and Y of Ref. [16]. We will focus on SO(5)/SO (4) models as simple though representative examples of composite Higgs theories; the extension to other cosets is straightforward. We will then use the dispersion formula for 3 to estimate the contribution from spin-1 resonances at O(m 2 W /16π 2 f 2 ) by computing the spectral functions in a low-energy effective theory. The result will be shown to coincide with the one we obtained in Ref. [17] through a diagrammatic calculation. The different viewpoint offered by the dispersive approach will allow us to clarify an issue on the positivity ofŜ raised in Ref. [13].
The paper is organized as follows. in Section 2 we review the definition of 3 by distinguishing between long-and short-distance contributions. Short-distance contributions, in particular, will be parametrized in terms ofŜ, W , Y and X. We derive expressions forŜ, W and Y in terms of two-point current correlators of the strong dynamics, which can be used for a non-perturbative computation on the lattice. Section 2. 2 Dispersion relation for 3 We start by deriving the dispersion relation for the 3 parameter in the context of SO(5)/SO (4) composite Higgs theories. Our analysis will be similar to that of Ref. [13], although it differs in the way in which short-and long-distance contributions from new physics are parametrized. In this respect our approach is closer to the original work of Peskin and Takeuchi [1], where the S parameter is defined to include only short-distance effects from the new dynamics.

Short-and long-distance contributions to 3
It is well known that universal corrections to the electroweak precision observables at the Z-pole can be described by three parameters [14]. In this paper we are mainly interested in the 3 parameter, which can be expressed as [18] in terms of the vector-boson self energies Here s W (c W ) denotes the sine (cosine) of the Weinberg angle and we have followed the standard convention decomposing the self energies (for canonically normalized gauge fields) We consider scenarios in which the new physics modifies only the self energies, i.e. its effects are oblique. The form of the non-oblique vertex and box corrections in Eq. (2.1) is thus irrelevant to our analysis, since these cancel out when considering the new physics correction It is useful to distinguish between a short-and a long-distance contribution to ∆ 3 . Heavy states with mass m * m Z affect only the short-distance part. This latter can be expressed as the contribution of local operators, and is generated also by loops of light (i.e. Standard Model (SM)) particles. We define it to be where ∆ē i ≡ē i −ē SM i and It is convenient to express ∆ 3 | SD in terms of the parametersŜ, W , Y and X defined in Ref. [16]: (2.7) The S parameter originally introduced by Peskin and Takeuchi in Ref. [1] is related toŜ bŷ S = (α em /4s 2 W )S. The long-distance correction to 3 arises from loops of light particles only, as a consequence of their non-standard couplings. We define where x h = m 2 h /m 2 Z and the function f 3 is given by [17] (2.10) Additional long-distance effects arise from the top quark and are further suppressed by at least a factor ζ 2 t , where ζ t is the degree of compositeness of the top quark. They will be neglected in the following.
From Eqs. (2.4), (2.6) and (2.8) we find resonances, m * ∼ g * f , one could in principle get large modifications to the Higgs couplings for f ∼ v while still having a mass gap provided g * g. In fact, current experimental data on Higgs production at the LHC disfavor large shifts and constrain (v/f ) 2 0.1 at 95% C.L. [19] (see also Refs. [20][21][22] for previous theoretical fits

Dispersion relations for the short-distance contributions
We are now ready to derive the dispersion relations forŜ, W and Y in terms of the spectral functions of the strongly-interacting dynamics. We start by consideringŜ.
The strong dynamics is assumed to have a global SO(5) invariance spontaneously broken to SO(4) ∼ SU (2) L × SU (2) R . The elementary W µ and B µ fields gauge an SU (2) L × U (1) Y subgroup contained into an SO(4) misaligned by an angle θ with respect to the unbroken SO(4) (see Refs. [23,17] for details). They couple to the following linear combinations of where T A (θ) are the SO(5) generators, while T a (0) are the generators of the gauged SO(4) .
Using the expressions for the generators given in Appendix A of Ref. [23] (see especially Eq. (88) therein), we find where J a L µ , J a R µ are the SO(4) ∼ SU (2) L × SU (2) R currents (a L , a R = 1, 2, 3) and Jî µ the SO(5)/SO(4) ones (î = 1, 2, 3, 4). We assume that these currents are conserved in the limit in which the strong dynamics is taken in isolation, i.e. when the couplings to the elementary fields are switched off. This is for example the case of holographic composite Higgs models [24]. The generalization to the case in which the strong dynamics itself contains a small source of explicit SO(5) breaking is discussed in Appendix A. By working at second order in the interactions (2.12) (i.e. at second order in the weak couplings), the vectorboson self energies in Eq. (2.7) can be expressed in terms of two-point current correlators.
The corresponding contribution toŜ and to the other oblique parameters W, Y, X is gauge invariant (see the detailed discussion in Ref. [1]). TheŜ parameter, in particular, gets a which we will retain in our calculation. Notice that since this term is not of the form of a two-point current correlator of the strong dynamics in isolation, it was not included by Peskin and Takeuchi in their estimate of S in Ref. [1]. 5 In the limit in which the strong sector is taken in isolation, i.e. for unbroken SO (5) symmetry, the Fourier transform of the Green functions of two conserved currents can be decomposed as:
where (P T ) µν ≡ (η µν − q µ q ν /q 2 ). Any other two-point current Green function vanishes by SO(5) invariance. By using its definition in Eq. (2.7), together with Eqs. (2.12), (2.14) and (2.16), the parameterŜ can be expressed in terms of the correlators Π ij as: denotes the expression of Π 3B obtained by replacing the strong dynamics with 6 The IR divergence is completely removed if the strong dynamics contains a small breaking of the SO (5) symmetry giving the Higgs boson a mass. It is shown in Appendix A that, even in this case, it is useful to rewrite Eq. (2.17) as discussed below to explicitly extract the Higgs chiral logarithm. 7 A possible alternative strategy is to define the correlators Π ij by including the explicit breaking of SO (5) due to the coupling of the strong dynamics to the elementary fermions, in particular to the top quark. The resulting formula, however, is less convenient to computeŜ by means of non-perturbative tools such as lattice field theory. We thank Slava Rychkov for drawing our attention on the importance of working with two-point current correlators defined in terms of the strong sector in isolation.
the linear SO(5)/SO(4) model and is computed for a non-vanishing Higgs mass. The mass of the scalar η is an arbitrary parameter which can be taken to be of the order of the mass of the heavy resonances of the strong sector, m η ∼ m * . In this way the Higgs chiral logarithm is fully captured by δŜ LSO5 , and the first term in parenthesis in Eq. (2.19) can be evaluated setting the Higgs mass to zero (the relative error that follows is of order m 2 h /m 2 * and can be thus neglected). The IR singularities exactly cancel out in the difference of correlators in parenthesis, since the linear model by construction coincides with the strong dynamics in the infrared. Equation At this point we can make use of the dispersive representation of the correlators Π ij . This is obtained by inserting a complete set of states in the T -product of the two currents and defining The spectral functions ρ ij and (ρ ij − ρ ij ) encode, respectively, the contribution of spin-1 and spin-0 intermediate states; they are real and positive definite. Current conservation implies ρ ij =ρ ij , while from analyticity and unitarity it follows that The (n + 1)-subtracted dispersive representation thus reads (for a given q 2 0 ) In the full theory of strong dynamics, the asymptotic behavior of the 8 One has P 0 (q 2 ) = Π ij (0) and Notice that Π LL (0) and Π RR (0) vanish if the strong dynamics is considered in isolation.
linear combination is controlled by the scaling dimension, ∆ ≥ 1, of the first scalar operator entering its OPE (see the discussion in Ref. [13]): Π 1 (s) ∼ s 1−∆/2 . One can thus write a dispersion representation for Π 1 with just one subtraction (setting n = 0 in Eq. (2.23)), which in turn implies an unsubtracted dispersive representation forŜ. Using the explicit expression of Π LSO5

3B
(0) we obtain: This result generalizes the dispersion formula derived by Peskin and Takeuchi in Ref. [1] for Technicolor to the case of SO(5)/SO(4) composite Higgs theories. The dispersive integral accounts for the contribution from heavy states (of O(m 2 Z /m 2 * )), while the chiral logarithm due to Higgs compositeness is encoded by δŜ LSO5 . The dependence on m η cancels out when summing this latter term with the dispersive integral.
Let us now turn to W , Y and X. In our class of theories the contribution of heavy particles to X is of O(m 4 Z /m 4 * ) and will be neglected (it is of the same order as the uncertainty due to our definition of short-and long-distance parts in ∆ 3 ). The contribution of heavy particles to W and Y is instead of O[(m 2 Z /m 2 * )(g 2 /g 2 * )] and will be retained. Finally, the contribution to W , Y and X from the diagrams of Fig. 1 involving light particles only is not suppressed and must be fully included. For X we find where the dots indicate O(m 4 Z /m 4 * ) terms generated by the exchange of heavy particles. In the case of W and Y , it is straightforward to derive a dispersion relation by following a procedure analogous to that discussed forŜ. 9 By neglecting terms of order O(m 4 W /m 4 * ), we obtain 10 : The first term in each equation encodes the contribution from the heavy resonances and is ). The second terms come from the difference between the SO(5)/SO(4) linear model and the SM (they are the analogous to Eq. (2.20)), while δW Zh and δY Zh are the contributions from the Zh loop in Fig. 1: By putting together the expressions ofŜ, W , Y , X, and of the long-distance part Eq. (2.9), we obtain a dispersive formula for ∆ 3 : (2.32) The second and third terms encode the contribution from the heavy resonances and are, ]. When modeling the spectral functions 10 The O(m 4 W /m 4 * ) neglected terms give a contribution to W which can be written as follows: (2.28) The additional contribution to Y has the same form provided one exchanges LL ↔ RR and g ↔ g .
-as we will do in the next section-in terms of the lowest-lying resonances of the strong dynamics, these contributions arise from the tree-level exchange of massive spin-1 states.
We neglected terms of O(m 4 Z /m 4 * ) (arising in particular from our definition of short-and long-distance contributions) and of O[(m 2 W /16π 2 f 2 )(g 2 /g 2 * )] (arising from the expansion in powers of the weak couplings required to obtain a formula in terms of current correlators).
Equation (2.32) should be compared to the analogous result previously derived by Rychkov and Orgogozo in Ref. [13]. The expression given there also relies on an expansion in g 2 and does not include the heavy-particle contribution to W and Y (the last term of our Eq. (2.32)).
Rychkov and Orgogozo also define the dispersive integral to comprise the contribution of the heavy states only, but do not perform any subtraction to remove the NG boson contribution.
Rather, the integration over light modes is done explicitly and in an approximate way. Their procedure implies a relative uncertainty of order m h /m * , which follows in particular from neglecting the Higgs mass and the contribution of the heavy states in the evaluation of the low-energy part of the dispersive integral. In our case the relative uncertainty implied by our definition of short-and long-distance parts is smaller and of order (m Z /m * ) 2 . Within their accuracy, the two results coincide. transforming as a (3, 1) of the SO(4) ∼ SU (2) L × SU (2) R global symmetry. We will thus compute the spectral functions in the effective theory and integrate them to obtainŜ, W and Y , hence ∆ 3 , through their dispersion relations. In this case, the spectral integrals are generically divergent in the ultraviolet, since the effective description is approximately valid at low energy but not adequate for momenta larger than the cutoff scale. In other words, the dispersion relations derived in the previous section need to be modified in order to be used in the effective theory. Let us see how.
By considering the gauge fields A µ as external sources for the currents, any two-point current correlator can be expressed as the second derivative of an effective action W [A] with respect to the source: Notice however that the low-energy action S IR will not depend on the source only through its coupling to the low-energy conserved current J IR µ , but will contain non-minimal interactions. At quadratic order in the source, we can write is also a conserved current, and the dots stand for higher-derivative local terms. The Green functions J µ J ν can thus be computed in terms of the two-point functions of the effective currentsJ µ . The coefficients c i are arbitrary in the effective theory and can be chosen to cancel the UV divergences arising in J µJν . 11 Performing a Fourier transformation one has whereΠ ij is the two-point current correlator in the effective theory and ∆(q 2 ) = k (q 2 ) k c k denotes the local counterterms.
It is always possible to expressΠ ij (q 2 ) as an integral over a contour in the complex plane that runs below and above its branch cut on the real axis (where the imaginary part ofΠ ij is discontinuous) and then describes a circle of radius M 2 counterclockwise. We thus obtain where C M 2 denotes the part of the contour over the circle, andρ ij ( is the spectral function of the currentsJ µ . Since the value of M is arbitrary (as long as q 2 is inside the contour), the dependence on M 2 cancels out in Eq. (3.39). IfΠ ij (q 2 )/q 2 → 0 for |q 2 | → ∞, it is possible to take the limit M 2 → ∞ so that the integral on the circle vanishes. In this case one obtains a dispersion relation for Π ij (q 2 ) in terms ofρ ij similar to the one valid in the full theory, except for the appearance of the local term. In general, however, the correlatorΠ ij is not sufficiently well behaved at infinity, and M must be kept finite. IfΠ(q 2 ) ∼ (q 2 ) 1+k at large q 2 , both the dispersive integral and the integral over the circle scale as (M 2 /m 2 * ) k , where m * is the mass of the resonances included in the low-energy theory. Also,Π ij generally requires a regularization to be defined and contains divergences which are removed by the counterterm ∆ ij . The dispersive integral, on the other hand, is convergent sinceρ ij is finite (after subdivergences are removed).
A particularly convenient way to defineΠ ij (q 2 ) is through dimensional regularization.
Upon extending the theory to D dimensions, indeed, its asymptotic q 2 behavior arising at the radiative level can be arbitrarily softened. For example, the 1-loop contribution toΠ ij 11 The value of c 0 can be adjusted to ensure that the contributions to the two-point correlator from the tree-level exchange of, respectively, one NG boson and one spin-1 resonance are transverse. A simple way to enforce the Ward identity is in fact demanding that the effective action S IR [ϕ IR , A] be invariant under local SO(5) transformations under which the source A µ transforms as a gauge field. We thank Massimo Testa for a discussion on this point. Notice also that adding the pure source terms in Eq. (3.36) corresponds to a redefinition of the T * product of two currents. scales like (q 2 ) 1+n− /2 at large q 2 , where n is some integer and ≡ 4 − D. It is thus possible to choose sufficiently large and positive ( > 2n), such that the contribution to the integral on the circle from 1-loop effects vanishes when taking the limit M 2 → ∞. In doing so, the dispersive integral (now with its upper limit extended to infinity) becomes singular for → 0. The divergence is thus transferred from the integral over the circle to the dispersive integral, and the 1/ poles are still removed by the counterterm ∆ ij . The same argument goes through after including higher-loop contributions. The large-q 2 behavior of the treelevel part ofΠ ij , on the other hand, cannot be softened through dimensional continuation.
If thusΠ ij scales like (q 2 ) 1+n at tree level, with n > 0, it is not possible to take the M 2 → ∞ limit in Eq. (3.39) (unless one performs n additional subtractions). The case with n = 0 is special, in that M 2 can be sent to infinity but the integral over the circle tends to a constant and does not vanish. Assuming thatΠ ij (q 2 ) grows no faster than q 2 in D dimensions, one can thus derive the following dispersion relation: This is the formula that we will use in the next section to computeŜ, W and Y .
We conclude by noticing that another approach is also possible to derive a dispersion relation in the effective theory. One could use Eq. (3.38) and approximate Im[Π ij (q 2 )] Im[Π ij (q 2 )] for q 2 Λ 2 . Substituting ρ ij (s) =ρ ij (s) + O(s/Λ 2 ) in the dispersion relation of the full theory, one thus obtains The value of M can be conveniently chosen to be much larger than the mass of the resonances m * , so as to fully include their contribution to the dispersive integral, and much smaller than the cutoff scale Λ, as required forρ ij to give a good approximation of the full spectral function. With this choice, the last two terms in Eq. (3.42) encode the contribution from the cutoff dynamics. Comparing with Eq. (3.39), it follows that (3.43)

One-loop computation of ∆ 3
Having discussed how the dispersion relations are modified in the effective theory, we now put them to work and perform an explicit calculation of ∆ 3 . Our goal is thus computing the spectral functionsρ ij of the currentsJ µ in the effective theory with NG bosons and a spin-1 resonance ρ L . The dynamics of the spin-1 resonance will be described by the effective Lagrangian of Ref. [17] (see Eqs. (2.6) and (2.16) therein), the notation of which we follow.
The SU (2) L , SU (2) R and SO(5)/SO(4) components ofJ µ read, respectively: where g ρ is the resonance's coupling strength, a ρ ≡ m ρ /(g ρ f ) and the ellipses denote terms with higher powers of the fields or terms that are not relevant for the present calculation.
The last term in Eq. . . . . It is however possible to simplify the calculation by noticing the following. We want to derive an expression for theŜ parameter at order g 0 ρ , by expanding for g ρ /4π small. Since the contribution from the tree-level exchange of the ρ L is of order 1/g 2 ρ , our result will include terms that appear at the 1-loop level in a diagrammatic calculation ofŜ. The role of treeand loop-level effects in the dispersive computation, on the other hand, is subtler. Consider for example the contribution to the ππ state coming from the exchange of a ρ L , i.e. that of the second diagram in the first row of Fig. 3. The vertex with the current is of order 1/g ρ , while that with the two NG bosons is of order g ρ . The diagram, and thus its contribution to 12 Notice that a different basis was used in Ref. [17] where Q 2 = Tr[ρ µν L E L µν ]. The definition adopted in this paper is more convenient for our discussion. 13 The exchange of one NG boson contributes only to the spectral functionρ BB and is thus irrelevant to our calculation. the parameterŜ, is naively of O(g 0 ρ ). There is however an enhanced contribution of O(1/g 2 ρ ) that comes from the kinematic region s ∼ M 2 ρ in the dispersive integral (2.26), where M ρ is the pole mass of the ρ L . To see this, notice that the small g ρ limit coincides with a narrow-width expansion. The Breit-Wigner function that follows from the square of the ρ L propagator can be thus expanded as where Γ ρ is the decay width of the ρ L . The left-hand side is of O(g 2 ρ ) for s away from M 2 ρ , but the delta-function term in the right-hand side is of O(g 0 ρ ). The contribution to the dispersive integral at the ρ L peak is thus enhanced compared to the naive counting. As a consequence, the leading contribution to theŜ parameter from the ππ final state is of order 1/g 2 ρ , and in fact corresponds to the tree-level correction of the diagrammatic calculation.
Loosely speaking, we can say that whenever the ρ L goes "on shell", the order in powers of g ρ is lowered by two units. This has two consequences. The first is that the leading contribution from the 3π and 4π states can be captured by replacing them, respectively, with the states πρ L and ρ L ρ L obtained by treating the ρ L as an asymptotic state. This approximation is sufficient to extractŜ at O(g 0 ρ ) and simplifies considerably the calculation. The second consequence is that, in the calculation of the ππ contribution, 1-loop corrections to the vertices and to the ρ L propagator should be included for s M 2 ρ , as they contribute at O(g 0 ρ ). In other words, 1-loop corrections to the spectral functions need to be retained (only) near the ρ L peak.
The Feynman diagrams relative to the calculation of the spectral functionsρ LL ,ρ RR and ρ BB are shown in Fig. 3 in terms of the relevant final states ππ, ρ L ρ L and πρ L . We work in the unitary gauge for ρ L , choosing dimensional regularization and an on-shell minimal subtraction scheme [17] to remove the divergences of the 1-loop contributions. While the calculation ofρ RR andρ BB is straightforward, it is worth discussing in some detail how the 1-loop corrections have been included inρ LL . As already stressed, we need to consider 1-loop effects only at the ρ L peak, for s ∼ M 2 ρ . The first and third diagrams in the first row of Fig. 3 can thus be evaluated at tree level. The second diagram gets 1-loop corrections in the vertex with the current (light blue blob with a cross), the ρ L propagator (dark blue box) and the ρ L ππ vertex (light blue blob). By decomposing each of these three terms into a longitudinal and a transverse part, the contribution of the diagram to the matrix element of the current between the vacuum and two NG bosons can be written as: where P µν T = (η µν − q µ q ν /q 2 ), P µν L = q µ q ν /q 2 and q = p 1 + p 2 . The spectral functioñ ρ LL is extracted by squaring this matrix element, integrating over the two-particle phase space and finally projecting over the transverse part (see Eq. For the transverse terms we use the following approximate expressions, we make use of its resummed expression near the ρ L pole in terms of the pole mass M ρ , total decay width Γ ρ and pole residueZ ρ . Finally the vertex V (q 2 ) is expressed in terms of the decay width Γ ρ . We report the analytic formulas for Π (1L) Jρ , M 2 ρ ,Z ρ and Γ ρ in Appendix B. Notice that a tree-level expression for Γ ρ is sufficient to reach the O(g 0 ρ ) precision we are aiming for in the spectral function. Adding the contribution of the first diagram in the first row of Fig. 3 and inserting the total matrix element in Eq. (2.21), one finds the following result for the spectral functioñ ρ (ππ) whereρ RR is given in Eq. (B.67). Away from the ρ L peak the 1-loop corrections can be neglected, and the second term in the absolute value in Eq. (4.52) is of order g 0 ρ , like the first one. At the peak, on the other hand, this second term develops an O(1/g 2 ρ ) contribution. This can be identified by using Eq. (4.47) to expandρ (ππ) LL (s) as a distribution. One has: Here Z L is the pole residue of the two-point current correlator: It is of order 1/g 2 ρ and, being an observable, is RG invariant. The function f LL denotes instead the O(g 0 ρ ) continuum (which receives a contribution from both the NG bosons and the ρ L ).  Fig. 4 for the following benchmark choice of parameters: m ρ (m ρ ) = 2 TeV, g ρ (m ρ ) = 3, a ρ = 1 and α 2 (m ρ ) = 0 (here m ρ (µ), g ρ (µ) and α 2 (µ) are the running parameters, see Ref. [17]). 14 One can notice the following. The functionsρ LL (s) andρ RR (s) become constant and equal for s → 0 (in D = 4). This constant tail corresponds to the NG boson contribution to the spectral functions; it gives rise to the IR logarithmic singularity in theŜ parameter that is eventually canceled by the subtraction in Eq. (2.26).
Having set α 2 = 0, the spectral functions tend to a constant also for s → ∞. This gives rise to a UV logarithmic divergence in the spectral integral forŜ which can be regulated by extending the theory to D dimensions (Notice that one should consistently extend both the spectral functions and also the subtraction term in Eq. (2.26)). The divergence is canceled by the local counterterm generated by the operator O + 3 = Tr[(E L µν ) 2 +(E R µν ) 2 ]. The correlator Π 1 thus obeys a dispersion relation of the form (3.40), , and the dots indicate local terms with higher powers of q 2 . For α 2 = 0 the contribution from the integral on the circle vanishes, C 1 = 0, when extending the theory to D dimensions. For non-vanishing α 2 , on the other hand,Π 1 (q 2 ) grows like q 2 in any dimension (as a consequence of its tree-level behavior) and one finds C 1 = −4α 2 2 g 2 ρ . Using the expressions of the spectral functions we can derive our final expression forŜ.
We find: (4.56) Notice that the term proportional to C 1 cancels the α 2 2 part in the first term. The parameters W and Y obey the same dispersion relations of the full theory, Eqs. (2.29) and (2.30), with ρ ij replaced by the spectral functions of the effective theoryρ ij . All contributions from the integrals on the circle, in this case, can be made to vanish through dimensional continuation. The contact terms to be added in the effective theory are generated by the operators O 2W = (∇ µ E L µν ) 2 and O 2B = (∇ µ E R µν ) 2 . Their contribution is naively of O[(m 2 W /m 2 ρ )(g 2 /16π 2 )], i.e. of higher order in our approximation, and will be thus neglected. Furthermore, since we are interested in the leading correction of O[(m 2 W /m 2 ρ )(g 2 /g 2 ρ )] from the ρ L , the integral in Eq. (2.29) can be computed by retaining only the delta function in the expansion ofρ LL in Eq. (4.53) (while that in Eq. (2.30) is negligible). We thus find: 15 Using Eqs. (4.56), (4.58) and (4.59), together with Eq. (2.9), we obtain our final formula for ∆ 3 : which also come from the delta function in the expansion ofρ LL . 16 The O[(m 2 W /m 2 ρ )(g 2 /g 2 ρ )] contribution from W and Y was neglected in Ref. [17], see Eq. (4.47) therein.
where ∆ 1 ≥ 1 is the scaling dimension of the first scalar operator contributing to its OPE. If this condition is enforced on Eq. (4.55) by neglecting the higher-derivative terms denoted by the dots, one obtains c + 3 = C 1 /8 = −α 2 2 g 2 ρ /2, where from now on we focus on the tree-level contribution neglecting the O(1/16π 2 ) radiative corrections. This relation implies that the last term of Eq. (4.56) identically vanishes, giving the positive definite expression derived in Ref. [13]:Ŝ = (g 2 sin 2 θ/4g 2 ρ )(1 − 2α 2 g 2 ρ ) 2 . Now, the higher-derivative terms in Eq. (4.55) are suppressed by corresponding powers of the cutoff scale Λ. As such they become important at energies E ∼ Λ. Neglecting them when enforcing the asymptotic behavior is in fact equivalent to requiring that this latter is attained at energies E ∼ M ρ through the exchange of the ρ L , while the cutoff states have no effect.
In this sense, the correction coming from c + 3 should be regarded as characterizing part of the ρ L contribution rather than encoding the effect of the cutoff states. Requiring that the asymptotic behavior be obtained at the scale M ρ , as effectively done in Ref. [13], thus leads to a positiveŜ.
There is, on the other hand, the possibility that the correct asymptotic behavior is recovered only at energies E ∼ Λ as the effect of the higher-derivative terms. That is to say, it can be enforced by the exchange of the cutoff states rather than by the lighter resonance ρ L . In this case it is reasonable to assume c + 3 < 1/g 2 ρ , as suggested by its naive estimate, so thatŜ = (g 2 sin 2 θ/4g 2 ρ )(1 − 4α 2 g 2 ρ ) up to smaller corrections. This expression is not definite positive, as previously noticed. It is a result consistent with the properties of the underlying strong dynamics and in fact plausible to some degree. Indeed, the behavior of the correlators in the deep Euclidean could be determined by the dynamics at or beyond the cutoff scale, while theŜ parameter is saturated in the infrared and as such gets its leading contribution from the lightest modes. A simple model with three spin-1 resonances is discussed in Appendix C which illustrates this possibility with an explicit example.
The tree-level value of theŜ parameter can then be tuned to be small or may even become negative for α 2 of order 1/g 2 ρ . While such large values are not expected from a naive estimate if α 2 is generated by the physics at the cutoff scale (in this case one would expect α 2 ∼ f 2 /Λ 2 or smaller), they are consistent with the request of the absence of a ghost in the low-energy theory [23]. Having α 2 ∼ 1/g 2 ρ , on the other hand, affects the naive estimate of c + 3 . For non-vanishing α 2 , the 1-loop correction toΠ 1 (0) is quadratically divergent, which implies c + 3 (Λ) ∼ (Λ 2 /m 2 ρ )(α 2 2 g 4 ρ )/16π 2 . For α 2 ∼ 1/g 2 ρ and setting Λ = g * f one has c + 3 (Λ) ∼ g 2 * /(16π 2 g 2 ρ ). This can be as large as the tree-level contribution from the ρ L exchange if g * ∼ 4π. Such enhancement of the 1-loop contribution from the cutoff dynamics originates from the increased coupling strength through which the transverse gauge fields interact with the composite states. In particular, the ππW ρ L vertex gets an energy-growing contribution of order gg ρ (α 2 g 2 ρ )E 2 /m 2 ρ . For α 2 ∼ 1/g 2 ρ , this translates into a coupling strength squared of order gg * (g * /g ρ ) at the cutoff scale, which is a factor (g * /g ρ ) stronger than the naive estimate based on the Partial UV Completion (PUVC) criterion [23]. This is precisely the enhancement factor appearing in the estimate of c + 3 . We thus conclude that while for α 2 ∼ 1/g 2 ρ it is possible to make the tree-level value ofŜ small or even negative, this is at the price of increasing the naive size of the unknown contribution from the cutoff states.
Such a contribution becomes of order 1/g 2 ρ if g * ∼ 4π, making theŜ parameter in practice incalculable in the effective theory.
As a final remark we notice that when including the 1-loop corrections, the asymptotic behavior of the full theory is not attained at M ρ even for α 2 = 0. In fact, one hasΠ 1 (q 2 ) ∼ . Setting a 2 ρ equal to 1 or 5/2 (and α 2 = 0) thus gives a model of the strong dynamics where the asymptotic behavior of Π 1 is enforced by the exchange of the ρ L , and the dispersive integral of theŜ parameter in the effective theory is convergent in D = 4. In a low-energy theory with both ρ L and ρ R , one has thatΠ 1 (q 2 )/q 2 vanishes at infinity for a 2 ρ L = a 2 ρ R = 1/2 or 3 (and α 2L = α 2R = 0). The choice a 2 ρ L = a 2 ρ L = 1/2, in particular, corresponds to a two-site model limit in which the global symmetry is enhanced to SO(5) × SO(5) → SO(5) [17]. The finiteness of theŜ parameter in this case follows as a consequence of the larger symmetry. [25,17] In this paper we have derived dispersion relations for the electroweak oblique parameters in the context of SO(5)/SO(4) composite Higgs theories. We have distinguished between long-and short-distance contributions to 3 , and obtained a dispersion relation for each of the parametersŜ, W and Y characterizing the short-distance part (Eqs. (2.26), (2.29) and (2.30)). Our analysis generalizes the dispersion relation written by Peskin and Takeuchi for the S parameter in the case of Technicolor [1]. We thus derived a dispersion relation for 3 (Eq. (2.32)), extending the work of Rychkov and Orgogozo [13]. Our formula (2.32) agrees with their result and further reduces the relative theoretical uncertainty to order m 2 h /m 2 * , where m * is the mass scale of the resonances of the strong sector. This is to be compared with the O(m h /m * ) relative uncertainty of Ref. [13]. We also discussed how the dispersion relations can be used and get modified in the context of a low-energy effective description of the strong dynamics. Making use of dimensional regularization we provided a definition of the otherwise divergent spectral integrals, pointing out the importance of the contribution from the integral on the circle in the case in which the two-point correlators of the effective theory do not die off fast enough at infinity. We utilized our formula to perform the dispersive calculation of 3 at the 1-loop level in a theory with a spin-1 resonance ρ L . We pointed out that 1-loop corrections need to be retained only at the ρ L peak to obtain 3 at the O(g 0 ρ ) level. This considerably simplified our calculation and conveniently reproduced the result of the diagrammatic computation that we performed in Ref. [17]. The dispersive approach is particularly suitable to clarify the connection between the positivity of theŜ parameter and the UV behavior of two-point current correlators, as first suggested by Ref. [13]. We argued that if the behavior dictated by the OPE in the deep Euclidean is enforced at the scale M ρ through the exchange of the light resonances, then theŜ parameter is positive definite in agreement with the expectation of Ref. [13]. It is possible, on the other hand, that the UV behavior is recovered only at the cutoff scale as an effect of the heavier resonances, while the leading contribution to theŜ parameter is still saturated by the lowest lying modes. In this caseŜ can be negative if the ρ L dynamics is characterized by a large kinetic mixing with the gauge fields of order α 2 ∼ 1/g 2 ρ .
A Generalization to the case of strong dynamics with small SO(5) breaking In deriving our dispersion relations we have assumed that the strong dynamics in isolation is SO (5) BB (q 2 ) .

(A.61)
Any two-point function with one SO(5)/SO(4) and one SO(4) current vanishes due to P R invariance. As a consequence of the SO(5) breaking, in particular, Π LR does not vanish and must be included in the definition of Π 3B when deriving Eq. (2.17): where δŜ LSO5 is still defined by Eq. (2.20) and computed at the physical Higgs mass. Similarly, the dispersion relations for W and Y are: The formula for ∆ 3 finally reads: The functionρ RR receives a contribution from the intermediate states χχ and χh, where χ 1,2,3 ≡ π 1,2,3 and h = π 4 . We find: ρ RR (q 2 ) =ρ LL andρ BB . This corresponds to including α 2 only at the tree level in a diagrammatic calculation, see Ref. [17].
For completeness, we also report the expression for the ρ L pole mass squared M 2 ρ , the pole residueZ ρ , the decay width Γ ρ (tree-level expression), and the 1-loop vertex correction Consider a low-energy theory with three spin-1 resonances transforming, respectively, as a (3, 1) (the ρ L ), a (1, 3) (ρ R ) and a (2, 2) (ρ B ) of SU (2) L × SU (2) R . We will assume for the moment that their masses are all of the same order and accidentally (much) lighter than the cutoff scale. The Lagrangian characterizing the ρ R is defined in Ref. [17] and can be obtained from that of the ρ L through an obvious L ↔ R exchange. The ρ B is instead described by where ρ B µν ≡ ∇ µ ρ B ν − ∇ ν ρ B µ and f − µν is the component of the dressed field strength along the broken SO(5)/SO(4) generators [23]. A simple calculation shows that in the deep Eu-clideanΠ LL (q 2 )/q 2 4α 2 2L g 2 ρ L ,Π RR (q 2 )/q 2 4α 2 2R g 2 ρ R andΠ BB (q 2 )/q 2 4α 2 2B g 2 ρ B , where the L, R, B subindices are used to denote the parameters of the corresponding resonances.
The asymptotic behavior Π LL (q 2 ) ∼ Π RR (q 2 ) ∼ Π BB (q 2 ) ∼ γ q 2 , where γ is a constant proportional to the central charge of the OPE, is thus reproduced by the correlators in the effective theory if Under this condition,Π 1 (q 2 )/q 2 → 0 for |q 2 | → ∞, and the integral on the circle vanishes (i.e. C 1 = 0 in this model). The contribution toŜ from the tree-level exchange of the resonances, as obtained through the dispersion integral, thus readŝ where the second equality follows from Eq. (C.78). The expression in the last line coincides with the result of the diagrammatic calculation, where the tree-level exchange of the ρ B gives no contribution toŜ. 17 Notice that althoughŜ is obtained through a dispersive integral it is not positive definite, because the contribution from the spectral function ρ BB comes with a negative sign in Eq. (2.26).
Now consider the limit in which the resonance ρ B is much heavier than the other two and has a mass m ρ B ∼ g * f m ρ L ∼ m ρ R ∼ g ρ f . The scale m ρ B acts as a cutoff for the effective theory with just ρ L and ρ R . In such a low-energy description the leading O(1/g 2 ρ ) contribution to theŜ parameter is fully accounted for by the exchange of the light resonances (last line of Eq. (C.79)), and no anomalously large coefficient for the dimension-6 operators is generated by the cutoff dynamics. The result from the diagrammatic calculation is reproduced by the dispersive approach only after adding the contribution of the integral on the circle at infinity.
WhileŜ is not positive definite, the correct asymptotic behavior of the two-point current correlators is recovered at the cutoff scale through the exchange of the ρ B , as a consequence of Eq. (C.78). The latter can be satisfied for α 2L ∼ α 2R ∼ 1/g 2 ρ and α 2B ∼ 1/(g ρ g * ).