Generic constraints on extra-fermion(s) from the Higgs global fit

We study the fit of the Higgs boson rates, based on all the collider data, in the effective framework for any Extra-Fermion(s) [EF]. The best-fit results are presented in a generic formalism allowing to apply those for the test of any EF scenario. The variations of the fit with each one of five fundamental parameters are described.

In our framework, all the Higgs couplings receiving corrections can be written in the following effective Lagrangian, which allows to work out the current Higgs phenomenology at the LHC and Tevatron colliders : where Y t,b,τ are the SM Yukawa coupling constants of the associated fermions in the mass eigenbasis, v is the Higgs vacuum expectation value, the subscript L/R indicates the fermion chirality and the tensor fields in the hγγ and hgg coupling terms are respectively the electromagnetic and gluon field strengths. The c t,b,τ parameters -taken real for simplicity -are defined such that the limiting case c t,b,τ → 1 corresponds to the SM; deviations from unity of those parameters can be caused by mixings of EF (like t states,. . . ) with the SM fermions. Neglecting the mixings with the first two SM flavors, one gets, −Y t,b,τ = m t,b,τ /v [the minus sign is due to the sign taken in front of the Yukawa couplings in Eq. (1)], where m t,b,τ are the final masses generated after EW symmetry breaking. The EF mixing effect on the Yukawa couplings enters via the c t,b,τ parameters. Summing over the dominant loop contributions, the coefficients of the dimension-five operators in Eq. (1) can be written as, where m c (m W ) is the charm quark (W ± -boson) mass, C(r) is defined for the color representation, r, by Tr(T a r T b r ) = C(r)δ ab [T a denoting the eight generators of SU ( The terms proportional to c t , c b and c τ account for the contributions from the fermionic triangular loops involving respectively the top, bottom quark and tau lepton Yukawa coupling. The dimensionless c gg and c γγ quantities -vanishing in the SM -parametrize the EF loop-exchange contributions to the hgg and hγγ couplings.

C. Higgs rate modifications
Within the present context, let us write explicitly certain Higgs rates, normalized to their SM prediction, which will prove to be useful in the following. The expression for the cross section of the gluon-gluon fusion mechanism of single Higgs production, over its SM prediction, reads as (for the LHC or Tevatron), The expression for the ratio of the diphoton partial decay width over the SM expectation is, The ratios for the partial decay widths into the bottom quark and tau lepton pairs as well as for the cross section of Higgs production in association with a top pair (LHC or Tevatron) are given by, Let us make a comment related to the mass insertion in the triangular loops of fermions inducing the hγγ and hgg couplings. Strictly speaking, a factor t , equal to the ratio of the sign of m t in the SM over sign(m t ) in the EF scenario, should multiply c t in Eq.  For a better understanding of the above parametrization, we finally provide the examples of expressions for the c gg and c γγ quantities, in the case of the existence of a t quark [same color number and electromagnetic charge as the top], an exotic (possibly vector-like) q 5/3 quark with electromagnetic charge 5/3 and an additional lepton (colorless), in terms of their physical Yukawa couplings and mass eigenvalues : The dots stand for any other EF loop-contributions. The mass assumption made in Footnote [1] leads to real A[τ (m f )] functions and thus real c gg , c γγ values, for real masses and Yukawa coupling constants, as appears clearly in the two above expressions. It will turn out to be instructive to express the ratio of these parameters in the simplified scenario where a new single q quark is affecting the Higgs couplings; denoting its electromagnetic charge as Q q and assuming the q to have the same color representation as the top quark, this ratio reads as : This ratio takes indeed a simple form that will be exploited in Section III D. In particular, notice that c γγ | t = c gg | t . Clearly, q should have non-vanishing Yukawa couplings to satisfy Eq. (9), otherwise c γγ | q = c gg | q = 0. In the specific case of a vector-like q L/R , this one could for example constitute a singlet under the SU(2) L gauge group and have a Yukawa coupling with another q R/L state of same Q q charge but embedded in a SU(2) L doublet; then the heaviest q L/R mass eigenstate, composed of q L/R and q L/R , could decouple from the Higgs sector so that the orthogonal q

III. THE HIGGS RATE FITS
A. The data All the Higgs rates which have been measured at the Tevatron and LHC [for √ s = 7 and 8 TeV] have been summarized in Ref. [9]. The latest experimental values can be found in Ref. [7,8]. Generically, the measured observables are the signal strengths whose theoretical predictions read as, where the p-exponent labels the Higgs channel defined by its production and decay processes, the s-subscript represents the squared of the energy [we will note √ s = 1.96, 7, 8 in TeV] of the realized measurement, the c-subscript stands for the experimental collaboration (CDF and D0 at the Tevatron, ATLAS or CMS at LHC) having performed the measurement and i is an integer indicating the event cut category considered. σ hqq is the predicted cross section for the Higgs production in association with a pair of light SM quarks and σ hV is for the production in association with a gauge boson [V ≡ Z 0 , W ± bosons]; their s-subscript indicates the energy and in turn which collider is considered. gg→h , for the gg → h reaction example, is the experimental efficiency including the (kinematical) selection cut effects. In order to analyze the fit of the Higgs boson data from colliders within the effective theory described above, we assume gaussian error statistics and we use the χ 2 function, where the sum is taken over all the different channel observables and µ p s,c,i | exp are the measured central values for the corresponding signal strengths. δµ p s,c,i are the uncertainties on these values and are obtained by symmetrizing the provided errors below and above the central values : We observe on Fig.(1)[a,b,c] that a c t variation of amount, δc t , leads in a good approximation to a translation (no shape modification) of −δc t along both the c γγ and c gg axes, for each one of the three best-fit regions. It is particularly clear in Fig.(1)[b] where a large δc t is exhibited. This strong parameter interdependence implies that in order to determine experimentally the c γγ and c gg quantities, it is crucial to determine as well the c t Yukawa correction.
Concerning the c b variation (for fixed c t = c τ = 1), we first explain the impact of the c b increase on the typical c γγ , c gg values -starting from the best-fit domains around the best-fit point, {c b = 2.08; c gg = 0.66; c γγ = −1.09}, in Fig.(1)[c] -and the reasons why huge values up to c b 50 could still agree with present Higgs rate fits. For such a c b increase, the strengths are reduced via Γ h→bb , a reduction which has to be compensated by a σ gg→h increase through a c gg enhancement to conserve a satisfactory χ 2 (or equivalently here, ∆χ 2 ). This explains the shift of the considered best-fit domains, around {c b = 2.08; c gg = 0.66; c γγ = −1.09} in Fig.(1)[c], to higher c gg values in the plot [d] where c b = 10 (still with c t = 1). This necessary compensation between the Γ h→bb and σ gg→h increases also guarantees the stability of diphoton rates (there is also a significant gluon-gluon fusion contribution in the three dijet-tagged final states) letting the χ 2 at the same level, without c γγ modifications -explaining nearly identical c γγ values for the studied regions in Fig.(1)[c] and [d]. The Γ h→bb increase leads to enhancements of the strengths without major consequences on the fit; a c b increase up to ∼ 50 [leading to Γ h→bb 5 GeV] would still leave existing domains at 68.27%C.L. since in the theoretical limit, c b → ∞, B h→bb tends obviously to a finite value compatible with data : B h→bb → 1.
What is the experimental impact of the above c b analysis ? The present experimental results do not prevent c b from taking extremely large values -due in particular to Higgs rate compensations. In order to put a more stringent experimental upper limit on it, one could of course if possible improve the accuracies on the signal strengths involving σ gg→h and Γ h→bb . A new possibility to measure c b (or equivalently the bottom Yukawa coupling constant) would be to investigate the processes,qq → hbb and gg → hbb (orbb → h and bg → hb), followed by the decay, h →bb. Indeed, here both the production and decay rates should increase with c b (Γ h→bb being the dominant partial width) so that compensations should not occur; then too large c b values would be experimentally ruled out. This Higgs production in association with bottom quarks could have significant cross sections for high LHC luminosities and enhanced c b values compared to the SM as the present fit points out. The sensitivity to such a reaction relies deeply on the b-tagging capability. This reaction suffers from large QCD backgrounds but new search strategies have been developed for such a bottom final state topology, as in Ref. [10].

D. Effective EF scenarios
In this last section, we apply the above constraints from the Higgs rate fit to examples of simple scenarios where a unique EF state significantly affects the Higgs interactions. For instance, a b state [same color representation and electromagnetic charge as the bottom quark], that could be a light custodian top-partner in warped/composite frameworks, would lead to a ratio in Eq.  The other example of EF candidate able to be mixed with SM quarks is the t state, possibly constituted e.g. by a light top-partner in little Higgs models. For a dominant t state, the ratio of Eq. (9) tends to one which corresponds to the straight line on Fig.(1)[b]. Since a t field can mix with the top quark, c t = 1, but in the context of a single t one should have, c b = c τ = 1, as in Fig.(1)[b]. The predicted t line crosses two 95.45%C.L. regions e.g. for, c t = 0.5, as well as two 68.27%C.L. regions exclusively in the range, c t ∼ 1.1 ↔ 2.6 (above ∼ 2.6 the region sizes decrease as explained in previous section).
For a single extra-lepton with charge, Q = −1, potentially mixed with the τ -lepton, the parameters, c b = c t = 1, c gg = 0 [see Eq. (7)], are fixed and there remain two free effective parameters, namely c γγ and c τ . The best-fit regions for such a two-dimensional fit are presented in Fig.(2). The two best-fit points shown in this figure correspond to, χ 2 min = 52.54. It would also be possible theoretically that the new t and b particles do not mix with the SM top and bottom quarks. It would be the case also for additional q quarks with exotic electric charges. For illustration, let us first concentrate on the components of possible extensions of the SM quark multiplets under SU(2) L , as in warped/composite frameworks where SM multiplets are promoted to representations of the custodial symmetry [11][12][13][14][15][16][17][18]. The charges for such q components obey the relation, Y q = Q q − I q 3L (Y ≡ hypercharge, I 3L ≡ SU(2) L isospin). We will consider the electric charges of smallest absolute values, Q q = −1/3, 2/3, −4/3, 5/3, −7/3 and 8/3, keeping in mind that the naive perturbative limit on the electric charge reads as, |Q q | 4π/α 40 (α ≡ fine-structure constant). The q states are in the same color representation as the SM quarks.
In the case of the presence of such a q quark, unmixed with SM quarks, while c t = c b = c τ = 1, one has c γγ = 0 and c gg = 0 if the q state possesses non-zero Yukawa couplings. The best-fit domains for a twodimensional fit keeping the fixed parameters, c t = c b = c τ = 1, are shown in Fig.(2) together with the four best-fit points associated to, χ 2 min = 55.04. On this plot, we also represent the theoretically predicted regions in the cases of a single q quark with electric charge Q q : these regions are the straight lines defined by Eq. (9). All the predicted lines -whatever is the Q q charge -cross the SM point which is reached in the decoupling limit, c γγ → 0, c gg → 0. The first result is that the upper-left best-fit regions, around c γγ ∼ 8, c gg ∼ −1.8, cannot be explored in single q models [no line can reach it]. We also observe on Fig.(2) that the predicted line being the closest to a best-fit point is for, Q q = −7/3. This result means that, among any possible SM multiplet extension component, the fit prefers the q −7/3 state compared for example to a t or q 5/3 state. This prediction is independent of the Y q Yukawa coupling constants, the q mass values and the q representations under SU(2) L . Now we determine the physical parameters corresponding typically to an overlap between a given line in   8)] to c γγ , c gg quantities giving rise to the best ∆χ 2 values in the case of one free effective parameter, say c gg (related to c γγ through the fixed ratio c γγ /c gg | q ∝ Q 2 q ). In Fig.(3), we also illustrate the case of a single additional lepton (colorless) without significant mixing to SM leptons [c τ = 1], as may be justified by exotic Q charges or the large mass difference between the SM and extra-leptons. Here we choose, Q = −1, being quite common for extra-lepton scenarios. There is, again, a unique free effective parameter, c γγ , since c gg = 0.
Concerning the constraints on the signs, as shown in Fig.(3) based on the present Higgs data, the sign,Ỹ q < 0 [leading to c γγ < 0], is preferred at 68.27%C.L. [except with absolute charges, |Q q | 7, i.e. in a range close to the |Q q | pert. limit as illustrated in Fig.(2)] for any single extra-quark as it creates a constructive interference with the W ± -boson exchange increasing the diphoton rates. The specific sign configuration,Ỹ q < 0, is selected by the two relevant best-fit points which pin down, c γγ < 0, as obtained for extra-quarks in Fig.(2). This predicted condition means that the Yukawa coupling constant [−Y q in our conventions] must have a sign opposite to m q which could be written, Related to this condition, there are comments on the configuration denoted as dysfermiophilia in the literature. As described at the end of Section II C, strictly speaking the c t,b,τ parameters entering Eq. (4)-(5) -whose values are generally given in best-fit plots such as the present ones in Fig.(1) -should in fact be understood as being, in our conventions of Lagrangian (1), and similarly for b,τ c b,τ ; here the EF-exponent indicates that the parameter is considered within the context of an EF model (and remind that m t , Y t are in the SM). Therefore, the dysfermiophilia property of increasing, Γ h→γγ /Γ SM h→γγ , via changing the top Yukawa sign is in fact relying on the possibility to have, t c t < 0, or equivalently, sign(−Y EF t /m EF t ) < 0. This makes sense as it is the sign of, −Y EF t /m EF t , which has a physical meaning and appears in Γ h→γγ [see Eq. (8) for an analogy with the t -loop]. The other comment is that the dysfermiophilia possibility of having, t c t < 0, can indeed gives rise to an acceptable agreement with the Higgs data (see e.g. Fig.(1)[d]) but it is not necessary to achieve a good agreement (c.f. Fig.(2) where t c t = 1) since the constructive interference with the W ± -loop increasing the diphoton rates can be realized with an EF-loop inducing, c γγ < 0. Hence the above condition (12) can be called an extra-dysfermiophilia as it is exactly the same as for the top quark transposed to an EF. Besides, this condition (12) leads to a decrease of, σ gg→h /σ SM gg→h , for a single EF [see Eq. (7)] through negative c gg values [c.f. Fig.(2)].

IV. CONCLUSIONS
We have studied the behaviour of the global Higgs fit with the variations of fundamental parameters in the simplified scenario with extra-fermions and have learnt in particular that the determination of the Higgs couplings is difficult and sometimes suffers from correlations. We have also shown that the global fit allows quite simply to constrain the electric charge (and color) of EF. The extra-dysfermiophilia has also been clearly defined and pointed out as a present general prediction (for more details, see Ref. [19]).