Physics Letters B 705 (2011) 388–392

Contents lists available at SciVerse ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Fermions and Goldstone bosons in an asymptotically safe model

F. Bazzocchi a,b, M. Fabbrichesi a, R. Percacci a,b, A. Tonero a,b,∗, L. Vecchi c

a INFN, Sezione di Trieste, Italyb SISSA, via Bonomea 265, 34136 Trieste, Italyc LANL, Los Alamos, NM 87545, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 18 May 2011Received in revised form 6 October 2011Accepted 12 October 2011Available online 15 October 2011Editor: A. Ringwald

We consider a model in which Goldstone bosons, described by a SU(N) chiral nonlinear σ model, arecoupled to an N-plet of colored fermions by means of a Yukawa interaction. We study the one-looprenormalization group flow and show that the non-Gaussian UV fixed point, which is present in thepurely bosonic model, is lost because of fermion loop effects unless N is sufficiently large. We thenadd four-fermion contact interactions to the Lagrangian and show that in this case there exist severalnon-Gaussian fixed points. The strength of the contact interactions, predicted by the requirement thatthe theory flows towards a fixed point in the UV, is compared to the current experimental bounds.This toy model could provide an important building block of an asymptotically safe model of the weakinteractions.

© 2011 Elsevier B.V. All rights reserved.

1. Motivations

Although the nonlinear σ model (NLσ M) is usually seen—because of its perturbative nonrenormalizability—as a mere low-energy effective field theory, it could also be renormalizable in anonperturbative sense, namely at a nontrivial (non-Gaussian) fixedpoint (FP) of the renormalization group (RG) flow [1] (see [2] forthe application of this idea in the context of gravity); in this case,the model is said to be asymptotically safe (AS).

As appealing as this approach may seem, it has the disad-vantage that perturbation theory can only be a rough guide, andtherefore many results are necessarily only suggestive rather thanconclusive. If in spite of this serious drawback one considers theRG one-loop β-functions, or some resummation thereof, it is easyto see that a nontrivial FP is indeed present in the NLσ M [3,4].It persists when one couples the Goldstone bosons to gauge fields[5] and, in some cases, when one adds terms with four derivativesof the Goldstone bosons [6]. While these results do not prove theexistence of the FP, they are suggestive and we use them to jus-tify our rather phenomenological approach in which the one-loopresults are assumed to hold at least qualitatively in the nonpertur-bative solution and the consequences of such an assumption arethen worked out.

Whether the physical system described by the NLσ M is actu-ally on a RG trajectory leading to such a FP or not depends on theinitial conditions at some given energy scale and the experimental

* Corresponding author at: SISSA, via Bonomea 265, 34136 Trieste, Italy.E-mail address: tonero@sissa.it (A. Tonero).

0370-2693/$ – see front matter © 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2011.10.029

constraints on the coefficients in the operator expansion. Whereasit appears that this is not the case for the chiral NLσ M of stronginteractions (for which the Goldstone bosons are the pions), thisis still an open question in the case of the Goldstone bosons ofthe spontaneous breaking of electroweak interactions. This ques-tion will eventually be answered by the experimental data.

In this work we address a more specific problem and reconsiderthe chiral NLσ M to study what happens to the FP of Ref. [4] whenthe Goldstone bosons are coupled to fermions. This is an importantquestion in view of utilizing the model in a realistic theory of theweak interactions.

As we shall see, a one-loop computation shows that the in-clusion of fermion fields drastically modifies the asymptotic prop-erties of the NLσ M. The Yukawa coupling does not have a UVfixed point and instead grows with the energy. The Goldstone bo-son coupling hovers about the FP of the uncoupled model—as longas the Yukawa coupling size is small enough to be negligible—butis eventually pushed away by the increasingly large Yukawa cou-pling.

We interpret this instability as a signal of new physics as-sociated to the standard model (SM) fermions, and in particularthe top quark, and model it by means of four-fermion contactinteractions—somewhat along the lines of what is done in nuclearphysics in describing the nucleon potential. The fermionic contentof the model is left unchanged and identical to that of the SM.The model thus extended admits a number of nontrivial FPs andcan be used in modeling the interaction between Goldstone bosonsand fermions in a AS theory.

We also show how current experimental bounds on quark con-tact interactions already limit the choice of RG trajectories which

F. Bazzocchi et al. / Physics Letters B 705 (2011) 388–392 389

can be used. As the bounds are improved, or the new interactionsare discovered, they will provide a test of the model.

2. Fermions and Goldstone bosons

We consider a SU(N)-valued scalar field U = exp(i f πa Ta)

where πa are the Goldstone boson fields, tr Ta Tb = δab/2 and f isthe Goldstone boson coupling, which in the SM case can be iden-tified with 2/υ , where υ = 246 GeV is the Higgs VEV. We limitourselves to the lowest order term in the NLσ M Lagrangian, whichreads

Lσ = − 1

f 2Tr

(U †∂μU U †∂μU

). (1)

This model is invariant under separate SU(N)L and SU(N)R trans-formations acting on U by left and right multiplication respec-tively.

We couple the Goldstone bosons to left- and right-handedfermions ψ ia

L and ψ iaR carrying the fundamental representation

of SU(N)L and SU(N)R respectively (corresponding to the indicesi = 1, . . . , N), and also the fundamental representation of a colorgroup SU(Nc) (corresponding to the indices a = 1, . . . , Nc). In thereal world the latter group is gauged; here we merely retain it as aglobal symmetry to count fermionic states. We couple the fermionsin a chiral invariant way to the U field by adding to the NLσ M La-grangian the fermion kinetic and the Yukawa terms:

Lψ2 = ψ̄L iγ μ∂μψL + ψ̄R iγ μ∂μψR − 2h

f

(ψ̄ ia

L U ijψjaR + h.c.

), (2)

where we have explicitly written out the group indices in the in-teraction.

In a more realistic model, the Yukawa coupling should distin-guish among the fermion components and there should be morethan one family, as it is the case in the SM, but this is not es-sential for the present discussion. Indeed, we find that the onlyphenomenologically relevant fermionic contribution comes fromthe top quark, so h in (2) will be viewed as the top Yukawa cou-pling. We also neglect the strong, weak and electromagnetic gaugecouplings (after having checked that they do not alter our conclu-sions) and for this reason the derivatives in Eqs. (1)–(2) are notcovariant.

Using a sharp cutoff regularization, we obtain the followingone-loop RG equations for f̃ = kf and h:

d f̃

dt= f̃ − N

64π2f̃ 3 + Nc

4π2h2 f̃ , (3)

dh

dt= 1

16π2

(4Nc − 2

N2 − 1

N

)h3 + 1

64π2

N2 − 2

Nh f̃ 2. (4)

The variable t is defined as log k/k0 where k is the renormaliza-tion group energy scale and k0 is a conventional reference energy,which we take to be k0 = υ . These β-functions have been obtainedby assuming that the mass of the fermions is much smaller thanthe cut-off scale; this sets a condition k > h/ f for the validity ofthese equations.

The system of Eqs. (3)–(4) admits a number of possible UV fixedpoints. There is a formal Gaussian FP f̃ = 0, h = 0 which is outsidethe domain of our approximation. In the following we study onlyRG trajectories for energy scales larger than υ , which correspondsto f̃ > 2. There is also a nontrivial FP at h∗ = 0, f̃ ∗ = 8π/

√N for

which f̃ is a relevant (UV-attractive) direction and h is marginallyirrelevant. Requiring that it is reached in the UV implies the trivi-ality of the Yukawa coupling at all scales. We therefore reject thischoice as uninteresting.

Fig. 1. Running of f̃ and h (rescaled by a factor of 10 to fit the figure) for N = 2,Nc = 3. The AS behavior of f̃ (represented in the figure by the blue dashed line)is destabilized around t = 4.5 (that is, around 22 TeV) due to the increasingly largeYukawa coupling contribution (green dashed line) which at about the same valuebecomes strongly coupled, that is larger than 2π . The asymptotically safe behavior(continuous lines) is recovered after the introduction of the four-fermion contactinteractions, as discussed in Section 3. (For interpretation of the references to colorin this figure legend, the reader is referred to the web version of this Letter.)

A physically interesting trajectory requires nonvanishing hand f̃ . If h is treated as a t-independent constant, the β-functionfor f̃ has a zero at f̃ ∗ = 4

√(4π2 + Nch2)/N , which is a deforma-

tion of the one appearing in the pure bosonic model. The existenceof a nontrivial FP for the coupled system thus hinges on the exis-tence of a nontrivial zero in the β-function of h. This requires thatthe first term in the right-hand side of Eq. (4) be negative, whichis true for N > 2Nc .

Unfortunately this condition is not satisfied for the phenomeno-logically most important case N = 2, Nc = 3. This is illustrated bythe dashed curves in Fig. 1, for the initial condition f̃ 0 = 2 andh0 = 0.7 at t = 0 (thus mimicking the top-quark value). The firstterm on the right-hand side of Eq. (3) is initially dominant, lead-ing to linear growth of f̃ . The second term then grows in absolutevalue and at some point nearly balances the first one, leading to anapproximate FP behavior in some range of energies. Eventually h,whose β-function is everywhere positive, becomes large and thethird term dominates leading to a Landau pole and the loss of AS.The scale at which destabilization occurs is very sensitive to theinitial conditions and for the Yukawa couplings corresponding tolight fermions no destabilization takes place up to very large ener-gies.

We conclude that our model is not AS in the case N = 2,Nc = 3, in the one-loop approximation. For it to be AS, either theone-loop approximation must break down, or else new physical ef-fects must enter in the fermion sector at some energy scale.

It is interesting to compare this behavior to similar models. Ifthe color symmetry was gauged there would be an additional con-tribution proportional to g2

s h to Eq. (4), which in this case wouldbecome

dh

dt= 1

16π2

(4Nc − 2

N2 − 1

N

)h3

+ 1

16π2

(N2 − 2

4Nf̃ 2 − 3

N2c − 1

Ncg2

s

)h. (5)

The new term does not change the behavior shown in Fig. 1.In the linear σ model and therefore also in the SM the quadrat-

ically divergent term proportional to f̃ 2h is absent: it is canceledby diagrams containing loops of the Higgs field (see Fig. 2). In thiscase the Yukawa coupling is perturbative up to very high scales [7].A study of the linear version of the model in the context of func-tional renormalization has been presented in [8] for QCD. Anotherstrictly related model is the linear σ model coupled to one right-handed and NL left-handed fermions, studied in [9]. Our Goldstone

390 F. Bazzocchi et al. / Physics Letters B 705 (2011) 388–392

Fig. 2. The quadratically divergent diagrams contributing to the beta-function of h in the linear σ model. The fermion, Goldstone boson and Higgs propagators are representedby solid, broad and narrow dashed lines respectively. Remember that the Goldstone–Higgs coupling involves derivatives. In the linear model the divergent parts of thesediagrams cancel exactly. In the nonlinear model the last two diagrams are absent and the remaining two give the second term in Eq. (4).

modes are contained in their scalar sector, with the VEV υ = 2/ fcorresponding to the minimum of the scalar potential. There it wasfound that the scalar potential and the Yukawa coupling admit aFP for 1 � NL � 57. Our results differ due both to the differentfermion content and to the nonlinear boson–fermion coupling.

3. Four-fermion interactions

New physics associated to the SM fermions might restoreasymptotic safety. In this section we will discuss a class of nontriv-ial UV FPs with asymptotically free Yukawa couplings that emergeonce short-range interactions among the fermions are included.

We now restrict ourselves to the case N = 2 and add to the La-grangian a complete set of SU(2)L × SU(2)R invariant four-fermionoperators. Requiring P invariance, all possible chiral invariant op-erators, up to Fierz reorderings, are given by the following La-grangian:

Lψ4 = λ1(ψ̄ ia

L ψjaR ψ̄

jbR ψ ib

L

) + λ2(ψ̄ ia

L ψjbR ψ̄

jbR ψ ia

L

)+ λ3

(ψ̄ ia

L γμψ iaL ψ̄

jbL γ μψ

jbL + ψ̄ ia

R γμψ iaR ψ̄

jbR γ μψ

jbR

)+ λ4

(ψ̄ ia

L γμψ ibL ψ̄

jbL γ μψ

jaL + ψ̄ ia

R γμψ ibR ψ̄

jbR γ μψ

jaR

). (6)

The coefficients λi have inverse square mass dimension. We do notinclude in the Lagrangian in Eq. (6) operators defined by taking thesquare of the Yukawa term in Eq. (2) because they are higher orderfrom the point of view of chiral perturbation theory.

Strictly speaking, only the third SM fermion generation requiresnew physics to emerge at relatively low scales, since four-fermionoperators involving the first two generations can be suppressed bymuch larger scales without spoiling our AS scenario. Yet, in dis-cussing the experimental bounds on our model, see Section 4, wewill be as conservative as possible and consider the scenario inwhich (6) also involve the first generation. We thus tacitly assumethat the operators in (6) are consistent with FCNC bounds.

The symmetries imposed on Eq. (6) make this Lagrangian theminimal choice, and the one we will study here. More general setsof operators may well be relevant depending on the symmetries ofthis new sector; the SM group SU(2)×U (1) being the first instancecoming to mind. The analysis is in this case more cumbersome be-cause more operators must be included, but no distinctive featuresare expected to arise.

The operators in Eq. (6) are similar to those discussed in top-quark condensation models. In these and other models of compos-ite quarks only operators with vector current structure that is iso-and color singlet are usually considered. Here the full set of op-erators are taken into account since their couplings mix in the RGevolution equations. A discussion of the four-fermion Lagrangian inEq. (6), and its FP, as a model of chiral symmetry breaking can befound in [10]. The special role played in that context by the op-erator associated to λ1 has been recently emphasized in [11]. Inthese approaches it is assumed that when one flows from the FPin the UV towards the IR, the couplings λ̃i become stronger andeventually trigger the formation of a condensate. In our approach

the couplings λ̃i become weaker towards the IR and the breakingof chiral symmetry is due to the Goldstone bosons, which are re-garded as fundamental degrees of freedom.

We write the system of coupled β-function equations for thedimensionless variables f̃ , h and λ̃i = λik2. In these variables, wefind the following RG equations:

d f̃

dt= f̃ − 1

32π2f̃ 3 + Nc

4π2h2 f̃ ,

dh

dt= 1

16π2

[4Nc − 3 + 16

f̃ 2(Nc λ̃1 + λ̃2)

]h3

+ 1

64π2

[f̃ 2 − 16(Nc λ̃1 + λ̃2)

]h,

dλ̃1

dt= 2λ̃1 − 1

4π2

[Nc λ̃

21 + 3

2λ̃1λ̃2 − 2λ̃1λ̃3 − 4λ̃1λ̃4

],

dλ̃2

dt= 2λ̃2 + 1

4π2

[1

4λ̃2

1 + 4λ̃1λ̃3 + 2λ̃1λ̃4 − 3

4λ̃2

2

+ 2(2Nc + 1)λ̃2λ̃3 + 2(Nc + 2)λ̃2λ̃4

],

dλ̃3

dt= 2λ̃3 + 1

4π2

[1

4λ̃1λ̃2 + Nc

8λ̃2

2 + (2Nc − 1)λ̃23

+ 2(Nc + 2)λ̃3λ̃4 − 2λ̃24

],

dλ̃4

dt= 2λ̃4 + 1

4π2

[1

8λ̃2

1 − 4λ̃3λ̃4 + (Nc + 2)λ̃24

]. (7)

In the equations for the coefficients λ̃i we have neglected contri-butions coming from the Yukawa terms which are proportional toh2 f̃ 2, h2λ̃i or h2λ̃2

i / f̃ 2. These terms are negligible in the UV be-cause we will select fixed points for which the Yukawa couplingapproaches zero. Some of them just become comparable to theleading terms for k ∼ υ , where we stop the flow.

Notice that only the operators proportional to λ1 and λ2 con-tribute to the β-function for h. The other two operators do notcontribute because of their chiral properties. Moreover, it is cru-cial that the combination 16 (Nc λ̃1 + λ̃2) at the FP of λ̃1 and λ̃2be different from zero and larger than f̃ 2 because otherwise therewould be no UV FP.

From here on we fix Nc = 3. We begin by observing that the β-functions of the λ̃i form a closed subsystem. The numerical studyof these equations reveals the presence of 16 real fixed points.Their coordinates are given in the first four columns in Table 1.They are listed in order of decreasing trace of the stability ma-

trix∂β

λ̃i

∂λ̃ j, from the most UV-repulsive, the Gaussian FP fp0, to

the most UV-attractive fp4. The coefficients of the operators re-lated to UV-repulsive directions are completely determined by theAS condition. Hence, the FPs with a low number of UV-attractivedirections are the most predictive ones, and hence the most phe-nomenologically appealing. The number in the name of the FP is

F. Bazzocchi et al. / Physics Letters B 705 (2011) 388–392 391

Table 1Values of the coefficients λ̃i∗ for the 16 FPs discussed in the text. f̃ ∗ = 17.78, h∗ = 0for all FPs. The FP fp1c is boxed.

λ̃1 λ̃2 λ̃3 λ̃4 εh

fp0 0 0 0 0 0.5fp1a 0 −28.71 −7.18 0 1.22fp1b 0 0 7.85 −9.51 0.5fp1c 0 25.61 −4.27 0 −0.15fp1d 25.80 −1.77 0.19 −1.15 −1.42fp2a 13.41 20.10 −3.80 −0.24 −1.03fp2b 20.86 −3.56 7.04 −8.94 −1.00fp2c 0 −36.55 2.34 −13.92 1.43fp2d 0 0 −15.79 0 0.5fp2e 37.17 −37.36 −8.43 −1.65 −1.38fp2f −2.92 32.59 4.67 −12.04 −0.10fp3a 0. 31.67 4.67 −12.06 −0.30fp3b 19.95 −8.59 −15.27 −0.36 −0.80fp3c 31.22 −44.52 0.73 −13.38 −0.74fp3d −4.87 1.54 −5.42 −20.10 0.83fp4 0 0 −5.42 −20.13 0.5

the number of relevant (UV-attractive) directions. These values canthen be used to find the zeroes of βh , and these in turn are usedto find the zeroes of β f̃ .

We are interested in the sixteen FPs for the complete systemwith h∗ = 0; this requirement implies f̃ 2∗ = 32π2. At each of thesesixteen FPs the direction f̃ is always a relevant one, with eigen-value −0.45; the direction h is also an eigendirection, with eigen-value

εh = ∂βh

∂h

∣∣∣∣∗ = 1

64π2

(f̃ 2∗ − 16(Ncλ̃1∗ + λ̃2∗)

). (8)

The numerical values of εh are listed in the last column of Ta-ble 1. The seven FPs with εh > 0 are physically uninteresting sincethe requirement of flowing to one of them in the UV implies thath(t) = 0 at all scales. The other nine FPs of the fermionic sectorfor which εh < 0 also admit a FP with nonzero h. We are notinterested here in these additional nontrivial FPs, but restrict ourattention to those with h∗ = 0. For the λ̃i , the condition of flow-ing to fpnx in the UV yields 4 − n predictions. The values of f̃ andh remain always free parameters, to be fixed by comparison withthe experiment.

We have studied numerically the trajectories emerging from theFPs in the directions of the relevant eigenvectors. Some of themlead to divergences, others flow to other FPs. Here we shall con-sider only the single renormalizable trajectory that ends at fp1c inthe UV. This is a natural choice as it is the most predictive (theone with the smallest number of relevant directions) among theFPs with εh < 0.

In order to select the fine-tuned initial conditions in the IR thatguarantee AS at fp1c, we have first solved numerically the flowequations of the fermionic subsystem for decreasing t , starting atan initial point λ̃i∗ + 10−8 vi , where vi is the relevant eigenvector.This trajectory is attracted towards fp0 after roughly 20 e-foldings,so the four-fermion couplings can be taken arbitrarily small by se-lecting the IR value of t appropriately. We then shift t such thatthis value is zero, in accordance with our convention that t = 0corresponds to the scale k0 = υ . Now we pick a trajectory for thewhole system by fixing the initial values of the λ̃i to agree with theones find by this method, while the initial value of f̃ is 2k0/υ = 2and the initial value for h is 0.7. These agree with the initial valuesof the trajectory discussed in Section 2.

The result is shown in Fig. 1 (continuous curves) and Fig. 3. Wesee that for small t the couplings f̃ and h behave as in the modelof Section 2, with f̃ and h both increasing. At some point, how-

Fig. 3. Running of the λ̃i for the FP fp1c. λ̃1 and λ̃4 are equal to zero at all energies.

ever, λ̃2 becomes sizable and then the last term in the right-handside of the second equation in (7) pulls h towards zero. The trajec-tory is therefore characterized by a crossover from the IR regimewhere the fermionic interactions are mainly of Yukawa type, withIR free four-fermion interactions, and the UV regime dominatedby the contact interactions, with UV free Yukawa coupling. This issimilar to the behavior discussed in [12], except that here we donot bosonize the contact interactions: we see that in the presenceof the Goldstone bosons the crossover is automatic.

4. Experimental constraints

In order to select a realistic trajectory among the various possi-bilities one would need to compare with experimental results. Themodel is consistent with EW precision measurements, as discussedin [13], where the values of the EW parameters S and T are com-puted for the gauged NLσ M; the addition of the fermion contactinteractions does not modify this result.

Unfortunately the current bounds on contact interactions havebeen published only for the case in which a single operator,namely that proportional to our λ3 is present [14]. This is ratherunrealistic, given the RG mixing, but no need for a more detailedstudy was felt necessary.

In this section we aim to show how the experimental boundsare in principle already able to tell us something about the size ofthe new interactions. Here we take, conservatively but rather un-realistically, the current experimental bound on λ3 and enforce iton all coefficients as if they were contributing in the same mannerto the partonic cross sections. In addition we assume that all threegenerations contribute with identical coefficients λ̃i .

The experimental bound is a lower bound of the so-called con-tact interaction scale Λ which is related to the coefficient λ̃3 bythe identity

λ̃3(k) = 2π

Λ2k2. (9)

A bound on the value of Λ translates into a curve in the k, λ̃3plane because of the energy dependence in Eq. (9). As a crude es-timate, we decide to impose the experimental bounds at the quark-level effective energy scale of the LHC, namely keff = √

s/(2 × 3) =1.17 TeV, where the factor of 2 comes from the sharing of the en-ergy with the gluons and the factor of 3 from the assumed equalenergy partition among the three valence quarks. As mentionedbefore, for simplicity we will impose the same constraint to theabsolute value of all four coefficients λ̃i .

As an example of the kind of constrains we can thus obtain,we take the most recent published bound [15] of Λ = 9.5 TeVfor an integrated luminosity of 36 pb−1 and values of λ̃i on aRG trajectory leading to the FP discussed in the previous section,namely fp1c. Of course, the FPs with fewer relevant deformations

392 F. Bazzocchi et al. / Physics Letters B 705 (2011) 388–392

Fig. 4. A zoom at low-energy of Fig. 3 for the values (continuous curves) of theabsolute values of the four-fermion operator coefficients λ̃i . The dashed curve is thecorresponding size for the contact interaction for the bound at Λ = 9.5 TeV. Thedot at t = 1.55 on the dashed curve represents the bound at the reference value ofthe quark-level effective energy of 1.17 TeV. The values of the λ̃i must lie below thedashed curve for the experimental bound to be satisfied.

lead to more predictions and therefore will be easier to disprove.As we can see from Fig. 4, the bound is satisfied. A future boundof Λ = 30 TeV is expected as the integrated luminosity of the LHCcome close to 100 fb−1 [16].

In comparing with the experimental data, one may worrywhether the power-law running of h and υ may imply potentiallyimportant mass corrections to the cross sections. We comparedthe RG running of the masses for the solution above and verifiedthat it does not deviate from the logarithmic one of dimensionalregularization for more than 10%, at least below the scale so farexplored, namely 600 GeV.

References

[1] S. Weinberg, Critical phenomena for field theorists, Lectures presented at Int.School of Subnuclear Physics, Ettore Majorana, Erice, Sicily, Jul 23–Aug 8,1976.

[2] S. Weinberg, in: S.W. Hawking, W. Israel (Eds.), General Relativity: An EinsteinCentenary Survey, Cambridge University Press, 1979, pp. 790–831;M. Niedermaier, M. Reuter, Living Rev. Relativity 9 (2006) 5;R. Percacci, in: D. Oriti (Ed.), Approaches to Quantum Gravity: Towards aNew Understanding of Space, Time and Matter, Cambridge University Press,2009.

[3] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford UniversityPress, 2002.

[4] A. Codello, R. Percacci, Phys. Lett. B 672 (2009) 280, arXiv:0810.0715 [hep-th].[5] M. Fabbrichesi, R. Percacci, A. Tonero, O. Zanusso, Phys. Rev. D 83 (2011)

025016, arXiv:1010.0912 [hep-ph].[6] P. Hasenfratz, Nucl. Phys. B 321 (1989) 139;

R. Percacci, O. Zanusso, Phys. Rev. D 81 (2010) 065012, arXiv:0910.0851[hep-th].

[7] H. Arason, D.J. Castano, B. Keszthelyi, et al., Phys. Rev. D 46 (1992) 3945.[8] D. Jungnickel, C. Wetterich, Phys. Rev. D 53 (1996) 5142.[9] H. Gies, M.M. Scherer, Eur. Phys. J. C 66 (2010) 387, arXiv:0901.2459 [hep-th];

H. Gies, S. Rechenberger, M.M. Scherer, Eur. Phys. J. C 66 (2010) 403,arXiv:0907.0327 [hep-th].

[10] H. Gies, J. Jaeckel, C. Wetterich, Phys. Rev. D 69 (2004) 105008, hep-ph/0312034.

[11] L. Vecchi, Phys. Rev. D 82 (2010) 045013, arXiv:1004.2063 [hep-th].[12] J.M. Schwindt, C. Wetterich, Phys. Rev. D 81 (2010) 055005, arXiv:0812.4223

[hep-th].[13] M. Fabbrichesi, R. Percacci, A. Tonero, L. Vecchi, Phys. Rev. Lett. 107 (2011)

021803.[14] E. Eichten, K.D. Lane, M.E. Peskin, Phys. Rev. Lett. 50 (1983) 811.[15] G. Aad, et al., ATLAS Collaboration, arXiv:1103.3864 [hep-ex].[16] See, for instance: M.L. Vazquez Acosta, ATLAS and CMS Collaborations, arXiv:

0709.2518 [hep-ph].