Radiative corrections to WL WL scattering in composite Higgs models

The scattering of longitudinally polarized electroweak bosons is likely to play an important role in the elucidation of the fundamental nature of the Electroweak Symmetry Breaking sector and in determining the Higgs interactions with this sector. In this paper, by making use of the Equivalence Theorem, we determine the renormalization properties of the electroweak effective theory parameters in a model with generic Higgs couplings to the W and Z bosons. When the couplings between the Higgs and the electroweak gauge bosons deviate from their Standard Model values, additional counterterms of O(p^4) in the usual chiral counting are required. We also determine in the same approximation the full radiative corrections to the WL WL->ZL ZL process in this type of models. Assuming custodial invariance, all the related processes can be easily derived from this amplitude.


I. INTRODUCTION
Much of the current theoretical work concerning the LHC implications for the Electroweak Symmetry Breaking sector (EWSBS) focus on the deviations of the Higgs boson couplings to the electroweak gauge sector rather than the self-couplings of the gauge bosons themselves 1 .
Yet, any deviations of the former from their Standard Model (SM) values turn out to have implications for the latter; they are intimately intertwined at loop level and should be understood together, as unitarity considerations demand. We seek in the present paper to provide a consistent framework for future studies of both in the scattering of longitudinally polarized electroweak gauge bosons.
In a previous paper [1] we have already examined the implications of unitarity in the scattering of longitudinally polarized electroweak gauge bosons when-in addition to the usual SM lagrangian with a light scalar state (the Higgs particle with M H ≃ 125 GeV [2,3])-one includes an EWSBS assumed to be strongly interacting. This sector can be described at energies, M 2 H < s < (4πv) 2 by an Electroweak Chiral Effective lagrangian (EChL) [4]. In [1] we included a set of O(p 4 ) operators to describe the strongly interacting EWSBS but assumed that the couplings between the Higgs and the electroweak gauge bosons were indistinguishable from the values that they take in the SM. The main purpose of the present work is to relax this hypothesis.
A general chiral lagrangian with a nonlinear realization of the SU(2) L ×SU(2) R symmetry up to O(p 4 ) terms and including a light Higgs is Here, the U field contain the three Goldstone bosons associated to the breaking of the global 1 Anomalous four-gauge-boson couplings have not been measured yet in LHC experiments at the moment of writing this paper group to the custodial sub-group SU (2) the w being the three Goldstone boson fields 2 . The matrix U transforms as U → LUR † under the action of the global group SU(2) L × SU(2) R . The covariant derivative is defined as The Higgs field h is a gauge and SU(2) L × SU(2) R singlet. The vacuum expectation value v ≃ 250 GeV gives the right dimensions to the exponent in U. The terms L GF and L FP in Eq. (1) correspond to the gauge-fixing and Faddeev-Popov pieces respectively, whereas the includes a complete set of CP -even, local, Lorentz and gauge invariant operators, fourdimensional operators O i constructed with the help of the field U, covariant derivatives and the SU(2) L × U(1) Y field strengths W µν and B µν . A complete list can be found in [4] and also in [1]. While we will still restrict ourselves to a small subset of all possible general couplings we study those that are experimentally accessible now or in the near future.
In Eq. (1)  indicate that a and b are not too far from these SM values [6], but at present deviations from these SM values cannot be excluded. In [1] we assumed that the extended EWSBS would manifest itself only through the appearance of non-zero values for the a i O(p 4 ) coefficients 2 We shall denote by z the neutral Goldstone boson. w ± = (w 1 ∓ w 2 )/ √ 2.
but a and b (as well as d 3 and d 4 ) were assumed to be very close to 1. This is the most conservative hypothesis. However, even if a ≃ b ≃ 1, if the EWSBS is such that O(p 4 ) operators are present unitarity violations reappear at large energies in a way apparently similar to what happens in models that were copiously studied in the past [7] in the context of a very heavy Higgs or Higgsless theories.
In [1] we calculated the scattering amplitudes using the longitudinal components of the vector bosons themselves as external states, rather than the corresponding Goldstone bosons 3 as it is customarily done when one takes advantage of the Equivalence Theorem [8]. The reason to do so is that at the energies being now explored at the LHC, corrections to the Equivalence Theorem can be of some relevance [9].
We enforced unitarity through the use of the Inverse Amplitude Method [10]. We found that, even when including a light SM Higgs boson of mass M H = 125 GeV, the unitarity analysis predicts the appearance of dynamical resonances in much of the parameter space of the higher-order coefficients. Their masses extend from as low as 300 GeV to nearly as high as the cutoff of the method of 4πv ≃ 3 TeV, with rather narrow widths typically of order 1 to 10 GeV. In the absence of these resonances virtually all parameter space of the anomalous couplings could be excluded. However, we also showed that the actual signal strength of these resonances, when compared with current Higgs search data, is such that they are not currently being probed in LHC Higgs search data. Yet, if anomalous vector boson couplings exist, the resulting dynamical resonances they predict should definitely be observable with future LHC data.
The study in [1] therefore showed that there is a direct connection -also when a light Higgs is present-between anomalous four gauge boson couplings and the underlying structure of dynamical resonances in the scalar and vector channels. This emphasizes the importance of measuring these couplings (currently not yet observed at the LHC) to elucidate the fundamental nature of the EWSBS. These measurements have to go hand-in-hand with the search for the putative additional resonances, bearing in mind that their peak heights and widths bear little resemblance to the Higgs signal (in the scalar sector) or even to what is expected in previously studied strongly interacting theories (particularly in the vector 3 In [1] we treated the tree-level and the imaginary part of the one-loop exactly, but we actually had to resort to the Equivalence Theorem for the real part of the one-loop correction in order to keep the calculation manageable. channel). The reason being that the unitarization of the scattering amplitudes with a light Higgs profoundly changes the resonance structure with respect to the Higgs-less (or a very heavy Higgs) scenario in extended scenarios of EWSBS. The situation could be also more intriguing if the hypothesis of setting a and b to their SM values, namely a = b = 1 is relaxed as unitarity violations are already apparent at tree-level.
Before the phenomenological analysis however, the case a = 1 and b = 1 requires a complete new study of the radiative corrections, including a detailed study of the divergences and counterterms in this new scenario. This is part of the present work. We will also present a complete calculation of the one-loop W L W L → Z L Z L scattering amplitude (and by extension, upon use of custodial symmetry, of all four longitudinal electroweak gauge boson couplings). The one-loop calculation will be done by making use the Equivalence Theorem [8], where the longitudinal components are replaced by the corresponding Goldstone bosons.
This approximation is enough to derive the counterterms relevant for the process being discussed. The calculation is done in the non-linear realization, discussed above, as this is the natural language in composite Higgs models. Note that although S-matrix elements are independent of the particular parametrization, renormalization constants need not be.
Finally we mention that when computing electroweak gauge boson scattering amplitudes by making use of the Equivalence Theorem approximation, particularly if the calculation is done in the gauge where the Goldstone bosons are massless, some subtleties appearing in a complete calculation are not present. For instance, the results are automatically custodially invariant as one is assuming g = g ′ = 0. Crossing symmetry is also easily implemented by the usual exchanges of the Mandelstam variables. Therefore it is particularly simple to reproduce all amplitudes from the ww → zz one and, accordingly, only higher dimensional operators that are manifestly custodially invariant are needed when moving away from the SM. However, in a full calculation of the corrections, new non custodially invariant operators would be required as counterterms.
Furthermore crossing symmetry (although obviously still holding) is harder to implement (see e.g. the discussion in [1]). We emphasize once more that none of this affects the determination of the counterterms derived in this paper.

II. LAGRANGIAN AND COUNTERTERMS
The lagrangian in Eq. (1) will be our starting point. The parameters there have to be considered as renormalized quantities. We trade µ for M 2 H using M 2 H ≡ (µ 2 + 3v 2 λ). We will use a renormalization scheme where the relation M 2 H = 2λv 2 that holds true at tree level remains true for renormalized quantities.
Next we have to consider the counterterm lagrangian. This will be We have included the possible higher-order terms from the two O(p 4 ) operators that are relevant for W L W L scattering in the custodial limit, namely L 4 and L 5 (see e.g. [1] for details). We omit the pieces that are not relevant for W L W L scattering. In the treatment of this paper non-custodial O(p 4 ) operators are not needed.
The counterterm lagrangian needs some explanation. To begin with, we have not introduced counterterms for d 3 and d 4 as they affect mostly the renormalization of the Higgs self-interactions of which there is no experimental information at present. Their renormalization should not affect the counterterms that interest us most, namely those directly related to W L W L scattering, such as δa 4 and δa 5 . Secondly, there are additional δv 2 counterterms coming from the third line of Eq. 5 that depend on the number of factors of v in the different terms of the U expansion. For instance, terms like will have no corresponding counterterm because they contain no factor of v. On the other hand, terms with more than two w fields will result in counterterms. For example, consider one term contributing to the four-point interaction In addition there are wave function renormalization constants for the Higgs field, Z H , and for the Goldstone boson fields, Z w . Note that there is no mass term (and no corresponding counterterm) for the Goldstone bosons as we shall consistently work in the 't Hooft-Landau gauge, where Goldstone bosons are strictly massless. The renormalization conditions we will employ are that (i) the tadpoles vanish at one loop, (ii) the mass parameters are the on-shell masses, (iii) and that the relation λ = M 2 H /(2v 2 ) is now true of the renormalized quantities, rather than the bare ones. We also note that condition (ii) only ends up effecting the Higgs mass counterterm, as the Goldstone bosons will remain massless independent of any corrections to the two-point function.
As indicated in the introduction we shall make use of the Equivalence Theorem to determine the counterterms and the W L W L scattering amplitude rather than using the actual gauge degrees of freedom. As far as the counterterms are concerned, this procedure is good enough to give the correct renormalization of the parameters a, b, a 4 and a 5 that parametrize the EWSBS and thus the departures from the SM result. As for the finite pieces of the amplitude, the use of the Equivalence Theorem is just an approximation 4 that becomes better for s ≫ M W . A complete calculation using the gauge degrees of freedom is just too complicated for the present purposes and it is available numerically only for the SM [11].
The tree level calculation is fairly straightforward and comes from the sum of the two diagrams as in the usual linear realization case, albeit with different couplings: the wwzz 4-pt diagram, and the s-channel Higgs exchange diagram. These diagrams are shown in Combined they give which obviously reduces to the same value as the linear case for the SM (a = 1). Note that in the following the assumption that p 2 i = 0 is already made when presenting the amplitude. This expression shows clearly the ∼ s 2 growth of the tree-level amplitude as s ≫ M 2 H if a = 1 signaling the breakdown of unitarity already at tree-level when one moves away from the SM.
In the following, the classification of diagrams roughly follows the conventions given in ref. [12], but of course the calculation is completely different as the non-linear realization is used in the present paper and additional topologies of the diagrams do appear. Single diagram includes contributions from internal h, w ± , and z loops. We labelled by (a) the subdiagrams for the h loops and by (b) the combined ones for w ± and z loops.
Here, we will present the radiative corrections to the process grouped in several classes.
There are the Higgs self-energy corrections to the diagram in Fig.2 and the vertex corrections in Fig.3. Then we have some irreducible diagrams that following [12] we classify as bubbles (in Fig.4), triangles (in Fig.5) and boxes (in Fig.6). In addition we have two new type of diagrams that appear only in the non-linear realization and thus have no counterpart in ref. [12]. We have called them five-field (in Fig.7) and six-field (in Fig.8)  The two-point diagrams given in [12] correspond to −iΠ(s) and are plotted in Fig. 2.
Their contribution to the tree-level diagram w + w − → h → zz can be parametrized as and for d 3 = d 4 = 1 we have The scalar functions A 0 and B 0 are described in the appendix and both are ultraviolet divergent. Note that the calculation includes the counterterm for δM 2 H (last line).

B. hw + w − and hzz vertex corrections
The three-point diagrams given in [12] correspond to the hww/hzz vertex correction iΓ 3 , which is also related to the one-loop corrections to the Higgs decay width to ww/zz. The total correction is the same for both the hww and hzz vertices, although the actual set of diagrams is slightly different for each in the non-linear representation, as there is a 4-w coupling but no 4-z coupling. We draw in Fig.3 diagrams for the case of the hw + w − vertex.
Replacing appropriately w's by z's lines, we get the diagrams for the hzz vertex. In this case, however, we only have z internal loops in Fig.3(b). The rest of the diagrams are the same, however, the total correction can be given as twice the correction to any one vertex to give We then have (for d 3 = d 4 = 1) the total contribution Note the inclusion of the counterterms for the parameter a (describing departures from the SM hww and hzz couplings in the non-linear realization) and for the scale v 2 . The (finite) scalar function C 0 is described in the appendix.

C. Bubble diagrams
The bubble diagrams are given in Fig. 4 and their contributions for d 3 = d 4 = 1 sum up Note the inclusion here of the counterterms for the O(p 4 ) coefficients a 4 and a 5 and δa 5 here, but this is simply a choice.

D. Triangle diagrams
The triangle diagrams are given in Fig. 5 and their contributions give (for d 3 = d 4 = 1) the total result The box diagrams are depicted in Fig. 6 and their contributions differ only in the exchange The scalar function D 0 is also described in the appendix.
Finally, there is a single diagram here in which two Higgs legs connect to the central wwzz four-point vertex and then connect to each other to form a single closed loop. As with the five-field case, it is again necessary to ensure the calculation is complete to O((M H /v) 4 ) and similarly has no linear-calculation counterpart. This is given in Fig. 8. It gives

A. Tadpoles
The one-loop tadpole diagram and counterterm are given in Fig. 9. For M w = 0, and when assuming the relationship λ = M 2 H 2v 2 for the renormalized quantities, there is a single contributing diagram to the Higgs tadpole at one-loop: a Higgs loop deriving from a three-Higgs coupling. This gives a value of the tadpole (with external leg removed) of From the counterterm lagrangian Eq. (5) the contribution from the tadpole counterterm is Therefore, to meet our renormalization condition for vanishing tadpoles at one-loop, we must have When all Higgs tadpoles are appropriately canceled, there are only mixed Higgs/Goldstone boson loops, a Higgs loop, and w/z loops (which are zero when the w/z are massless). Any divergences which appear due to the wave-function renormalization of the external fields must be canceled by something in the remainder of this amplitude. We shall see later that this is easily achieved with the renormalization of v 2 , which is also a global factor multiplying the tree-level contribution. In fact from the mere requirement of finiteness of the amplitude after including the one loop diagrams, we can derive only a condition on the combination 2δZ w − δv 2 . Therefore the renormalization condition on the wave function has to be imposed separately and this consists in requesting the unit residue condition on the external legs.
The two-point function for the Goldstone bosons in Fig. 10 gives the following which verifies Π w (0) = 0 for all a and b, and therefore the Goldstone bosons stay massless, as they should 5 .
The wavefunction renormalization factor is then In the SM case, this is finite and matches the value given by ref. [13] Z SM but in general it is divergent. This divergence is canceled against contributions from δv 2 when the corresponding contribution to the one-loop amplitude is placed in the complete calculation. The one-loop contribution to the amplitude w + w − → zz from wave-function renormalization is

C. Higgs boson wave-function renormalization
The contributions to the Higgs two-point function can be derived from Sec. IV A, while the counterterm contribution is simply The on-shell condition for the Higgs mass requires Independent of this condition and the counterterm, we have the wavefunction renormalization factor of (now setting d 3 = d 4 = 1) This is divergent in the SM case and only becomes finite for a = 0. When the one-loop correction to iΓ(h → w + w − ) is performed and all external wavefunction renormalizations are included (i.e. both Z H and Z w ), all divergences cancel for arbitrary a and b when using the appropriate values for the counterterms given in Section VI. This is a good check on this value of Z H . It should also be noted that the SM value for Z H does not match that given in ref. [13]; this is a result of the nonlinear nature of the calculation.
The complete, renormalized decay width for the Higgs boson into Goldstone bosons is for arbitrary a and b, where δv 2 is a finite renormalization, not fixed by our conditions. For a = b = 1 and δv 2 = − 1 2 (the value used in refs. [12] and [13]), this reproduces the known SM result

VI. DIVERGENCES AND DETERMINATION OF THE COUNTERTERMS
Here we give the pieces of each individual diagram proportional to ∆ ǫ = 2 ǫ − γ E + log 4π + log µ 2 M 2 H . We give the results in the case d 3 = d 4 = 1 but it is quite straightforward to restore these factors for each individual diagram if so desired. These factors appear only in the radiative corrections to two-and three-point functions.

M (a)
2−pt ∼ from the full amplitude for a = b = 0), so once again plays no part. Finally, the δv 2 term is finite. Therefore only δa 4 and δa 5 are needed to remove the one-loop divergences from the Goldstone boson scattering amplitudes, which is what one would expect in the EChL approach.

VII. FINAL RESULT AND CONCLUSIONS
Finally, the complete one-loop amplitude iM loop (w + w − → zz) (for arbitrary a and b and rendered finite by using the counterterms in Eq. 41) is given by the following Here the functionsĀ 0 andB 0 are the corresponding scalar integral functions with the divergences removed (see appendix). The amplitude as written above has been grouped by scalar loop integrals. In the SM limit (a = b = 1), this simplifies quite a bit iM loop SM = i 1 4πv 2 2 as they simply contribute as overall factors to vertex and self-energy corrections. None of those diagrams behave as ∼ s 2 (or as t 2 or u 2 ) for large values of s and they therefore do not contribute to δa 4 and δa 5 that are totally independent of d 3 and d 4 .
It would be interesting to extend the present study to other low energy constants of the effective theory parametrizing the EWSBS. In particular a 1 and a 2 correspond to operators that contribute to the triple gauge boson vertex that has been recently measured for the first time at the LHC [14]. The renormalization of d 3 and d 4 would eventually be of interest too, but their relevance for comparison with experiment is still well ahead.
We have also presented a full one-loop calculation using the Equivalence Theorem approximation (and taking the masses of the Goldstone bosons to vanish, i.e. in the 't Hooft-Landau gauge) of the W L W L → Z L Z L in the general case with generic couplings of the Higgs to the electroweak gauge bosons. This calculation should be quite useful in precise comparisons of measurements of the four gauge boson coupling (not yet measured at the LHC) to theoretical predictions. Its knowledge is also very relevant in connection with unitarity analysis such as the one done in [1] and the prediction of new resonances originating from the EWSBS.
As emphasized in the introduction, the search for such resonances has to go hand-in-hand with accurate measurements of the four gauge boson couplings. Almost any deviation of these coefficients from their SM values would lead to unitarity violations at high energies and thus require additional resonances to restore it. In a forthcoming publication we will study in detail the issue of unitarity and extend the results of [1] to the case where the tree-level O(p 2 ) parameters a and b depart from their SM values. Both the determination of the counterterms and the full calculation of the real part of the scattering amplitude derived in this preparatory paper are necessary ingredients for such an analysis.
In conclusion, we have successfully provided a one-loop theory of Goldstone boson scattering in the context of an extended EWSBS where the Higgs is allowed to have arbitrary couplings. The coefficients a and b describing the coupling of the Higgs to the W and Z gauge bosons are currently of great interest to SM fits but their treatment so far has only been of tree-level studies. If a and b are not exactly equal to one some O(p 4 ) operators with running coefficients are required for a consistent treatment at one loop. Their running has been determined in this work. The results smoothly connect to the SM and are, we believe, completely general.