Electroweak Symmetry Breaking via QCD
EElectroweak Symmetry Breaking via QCD
Jisuke Kubo, ∗ Kher Sham Lim, † and Manfred Lindner ‡ Institute for Theoretical Physics, Kanazawa University, Kanazawa 920-1192, Japan Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany
We propose a new mechanism to generate the electroweak scale within the framework of QCD,which is extended to include conformally invariant scalar degrees of freedom belonging to a largerirreducible representation of SU(3) c . The electroweak symmetry breaking is triggered dynamicallyvia the Higgs portal by the condensation of the colored scalar field around 1 TeV. The mass of thecolored boson is restricted to be 350 GeV (cid:46) m S (cid:46) INTRODUCTION
With only the Standard Model (SM) Higgs like parti-cle discovered and no new particle beyond the SM beingfound, there is no evidence for any of the generally pro-posed solutions to the hierarchy problem. With the cur-rent measured Higgs mass and the top quark mass, theSM could even survive up to the Planck scale [1]. How-ever, one has to face the puzzle of why the electroweak(EW) scale is many orders of magnitude smaller than thePlanck scale. A possible solution for the hierarchy prob-lem is based on scale invariance, which is violated at thequantum level and hence a scale is introduced: The EWscale is generated dynamically by either the Coleman-Weinberg mechanism [2] or dimensional transmutationof a non-perturbatively created scale in a strongly cou-pled hidden sector [3]. Many of these attempts to gen-erate the EW scale radiatively rely on the Higgs por-tal λ HS S † SH † H , where the additional scalar field S (charged or neutral under a certain gauge group) ob-tains a vacuum expectation value (VEV) either directlyor indirectly. In this letter we propose a new non-perturbative mechanism to generate the EW symmetrybreaking (EWSB) scale. Though the hierarchy problembetween the EW scale and the Planck scale is not com-pletely solved in our proposed minimal model, which is aleast extension of the SM, our mechanism can be appliedto more realistic model building scenario in solving thehierarchy problem. Specifically EWSB is triggered bythe condensation of an additional scalar field S , whichbelongs to a larger representation of SU(3) c . In generalthe condensation of S , i.e. (cid:104) S † S (cid:105) (cid:54) = 0, takes place when C ( S ) α (Λ) (cid:38) , (1)with C representing the quadratic Casimir operator ofa certain representation R of S and α is the gauge cou- pling of the chosen non-abelian gauge group. The crucialpoint to notice here is that confinement (we through-out assume that the confinement scale is the same asthe condensation scale) can take place even if α is rel-atively small, provided that the representation of S islarge enough. QCD is a part of the SM and generates dy-namically an energy scale of O (1 GeV) by the gluon andquark condensates. However, we emphasize that thesescales are closely related to the fact that the quarks be-long to the fundamental representation of SU(3) c . There-fore, according to Eq. (1), if there exist colored degrees offreedom belonging to a larger representation of SU(3) c ,QCD can generate much higher energy scale in principle.In fact exotic quarks that are confined at higher energyscale have been considered in Refs. [4]. However most ofthese exotic fermions with EW charges cannot generatethe correct EW scale without large deviations from EWprecision tests. This situation will change if we considera colored EW singlet scalar field, as we will see in thenext sections. ELECTROWEAK SYMMETRY BREAKING BYSCALAR QCD
We assume that the SM with the new scalar QCD ex-tension is classically scale invariant and the EW scale isgenerated via the condensation scale of S . In fact, theSU(3) c sector of the SM itself before EWSB is scale in-variant, contrary to ordinary QCD with explicit massivequarks. The full Lagrangian is given as L = L SM ,m → + ( D µ,ij S j ) † ( D µik S k )+ λ HS H † HS † S − λ i (cid:2) ¯ S × S × ¯ S × S (cid:3) i , (2)where D µij = δ ij ∂ µ − ig s ( T R ) kij G µk and T R represents thegenerator for the representation R of SU(3) c . The term a r X i v : . [ h e p - ph ] S e p λ i denotes the quartic scalar coupling for the i -th invari-ant formed by the four tensor products of the S represen-tation. Due to classical scale invariance, the Lagrangianin Eq. (2) does not contain quadratic and cubic termsof S . Conventional scalar QCD would be quadraticallysensitive to an embedding scale and it would thereforenot solve the hierarchy problem. Note, however, thatour scenario is based on conformal QCD which shouldhave only logarithmic scale dependence. Note that ac-cidental U(1) symmetry appears for the S sector due tothe absence of cubic term and this has interesting phe-nomenology on its own if this U(1) is identified withthe U(1) Y hypercharge of the SM, which we will dis-cuss later. EWSB triggered by QCD is as follows: Thestrong coupling g s runs as usual from a finite value setat high energy (Planck or GUT) scale to the condensa-tion scale of S . The scalar condensate (cid:104) S † S (cid:105) forms whenEq. (1) is satisfied for O (1 TeV) where the small value of α s (Λ = 1 TeV) ≈ .
09 is compensated by the large C of S in higher representation. Note that the confinementscale is fixed once a representation for S is chosen, seeTable I. The condensate generates a scale which entersthe portal λ HS (cid:104) S † S (cid:105) H † H → λ HS Λ H † H, (3)and triggers the EWSB radiatively. The Higgs mass afterEWSB is determined by m h = 2 λ HS Λ , (4)and this in turn determines the value of Higgs quar-tic coupling λ h = λ HS Λ /v , with the Higgs VEV v = 246 GeV. The coupling λ HS is determined once theconfinement scale is fixed to be any value higher thanthe EW scale, as we require that confinement happensbefore EWSB. In general we have no upper bound on Λ,except that larger representation of S is required as α s decreases with higher value of Λ.The low energy QCD remains unaltered by our newadditional field as the coupling of higher representationof field S with the quarks in fundamental representationto form a singlet requires typically higher dimensionaloperators. It is important to remember that such con-densation takes place albeit the small coupling of α s atscales of O (TeV) due to a large C value for larger rep-resentation. As we can read off from Table I, (cid:48) is theunique representation for our purpose as it generates thedesired condensation scale at O (1 TeV).The phenomenology of this new scalar QCD extensionwith the representation of S being (cid:48) will now be dis-cussed in detail. First we can constrain the coupling λ i and λ HS from the requirement that all the scalar cou-plings do not hit a Landau pole or destabilize the vacuum.For the case of (cid:48) , we have 3 quartic couplings λ i dueto the existence of 3 invariants formed from the four ten-sor products of (cid:48) . The invariants formed by the tensor Rep ( R ) C ( R ) C ( R ) Λ (GeV) / / (cid:48) / / / Table I. Values of the quadratic Casimir and index for certainrepresentations of QCD. The approximate confinement scaleΛ for each representation is listed. products can be calculated with proper Clebsch-Gordancoefficients and subsequently one-loop beta functions forthe quartic couplings can be calculated [5]. To simplifyour calculation further we assume that the order of each λ i is roughly the same, i.e. λ i ≈ λ S / m S of S can be extractedfrom the Lagrangian. Notice that the bare m S of S doesnot exist in Eq. (2) due to scale invariance. The massterm can be approximately obtained from self-consistentmean field approximation [6] after confinement has takenplace, where the mean field serves as a back-reaction tothe field S and the mass is obtained from λ S S † S )( S † S ) → λ S (cid:104) S † S (cid:105) S † S = λ S Λ S † S. (5)The coupling λ S dictates directly m S = λ S Λ while themixing parameter λ HS determines m h . The large m S prevents the S field from obtaining non-zero VEV, hencecolor symmetry is not spontaneously broken. From therenormalization group equation (RGE) analysis we ob-tain the running of scalar couplings once the confinementscale is set. The measured m h fixes λ HS , while the mass m S ∼ λ S cannot be pushed arbitrarily high due to theemergence of Landau pole, yielding an upper bound on m S (cid:46) λ HS is relatively slow and it will only hitthe triviality bound when λ S hits the Landau pole, thissubsequently drives λ h to a Landau pole. We would liketo stress that other RGE scenarios maybe viable if theparameters λ i and the confinement scale Λ are variedindependently. In this letter we study only the simplestmodel to explain EWSB triggered by QCD. More realisticmodels should include dark matter and neutrino massesand their coupling to our new scalar could alter the highUV behaviour of the RGE of S significantly. The Lan-dau pole at 10 TeV may therefore be absent in a morerealistic model, or be a signal for non-perturbativity. COLLIDER PHENOMENOLOGY
The scalar S can change the Higgs production rate inthe gluon fusion channel due to λ HS . We have calcu-lated σ ( pp → H ) to the Next-to-leading (NLO) order Λ HS (cid:61) Λ HS (cid:61) Μ ATLAS (cid:137)Σ SM Μ CMS (cid:137)Σ SM m s (cid:64) TeV (cid:68) Σ (cid:72) pp (cid:72) gg (cid:76) (cid:45) (cid:62) H (cid:76)(cid:64) pb (cid:68) Figure 1. The Higgs production cross section from gluonfusion channel at NLO is calculated for different values of λ HS . The solid (dashed) curves represent the prediction of σ ( gg → H ) at √ s = 14 TeV (8 TeV). The combined signalstrength µ for ATLAS [7] and CMS [8] is shown where wehave assumed a SM-like BR. with this additional scalar. We followed the calculationof Ref. [9] and utilize the heavy scalar approximation.The MSTW2008 parametrization of parton density func-tion (PDF) [10] implemented in LHAPDF [11] has beenused in our computation with the factorization scale µ F and the renormalization scale µ R set to be equal to m h .We have utilized also the zero-width approximation forthe Higgs boson to simplify the calculation and the re-sulting production cross section is shown in Fig.1. Sinceour model does not modify the branching ratio (BR) ofthe SM Higgs (the decay H → γγ is modified with acci-dental symmetry, which we will discuss later on, but thisloop induced decay is very small compared to the tree-level decays), we can compare the signal strength µ times σ ( pp → H ) SM measured by ATLAS [7] and CMS [8] toour model’s prediction. The additional S field decreasesthe Higgs gluon fusion production rate, with almost halfthe rate for large λ HS (small Λ) and small m S . We ob-tain the suppression of ggH production rate as opposedto the enhancement due to the negative sign of λ HS .The condensate (cid:104) S † S (cid:105) has to be heavier than the Higgsto trigger the EWSB, therefore it will decay to Higgsparticles or two gluons. The scalar S can be producedat the LHC, with the dominating production channel gg → S ∗ i S j . The pair production of colored scalars withhigher dimensional representation at LO in the gluon fu-sion channel has been calculated in Ref. [12] and the re-sult for our case is given in Fig.2. The resulting particles S ∗ i S j will form two bound state pairs, with each pair de-caying predominantly to gg (2 jets) or to Higgs particles.Since the BR of H → b ¯ b dominates, we would expect al-most 70% for S ∗ S → jjjj in the total cross section. Thewidth of the band in Fig.2 represents the factorizationand renormalization scale dependence and the α s uncer-tainty from RGE with extra S contribution. In Fig.2 weplot the ATLAS exclusion limit on pair production of new s (cid:61) s (cid:61) m S (cid:64) TeV (cid:68) (cid:37) C L Σ (cid:72) pp (cid:45) (cid:62) S (cid:42) S (cid:76) (cid:137) BR (cid:64) pb (cid:68) Figure 2. The S pair production cross section from gluonfusion channel is calculated for different value of m S . The95% confidence level exclusion limit on σ × BR for √ s = 7 TeVby ATLAS is plotted. We assume 100% BR of (cid:104) S † S (cid:105) into twojets. color scalar decaying to four jets [13], where we have as-sumed 100% BR to four jets. m S (cid:46)
350 GeV is excludedat 95% confidence level and serves as our lower boundon m S . Combining this result with the upper bound dueto the triviality constraint above, the mass parameter ofthis model is very constrained, i.e.350 GeV (cid:46) m S (cid:46) . (6)The S field in Eq. (2) possesses an accidental U(1) sym-metry due to the absence of the cubic term as we haveimposed classical scale invariance in the Lagrangian. Apriori this U(1) is another global symmetry, but if it isidentified with the local U(1) Y of the SM, we would ob-tain more interesting phenomenology. For instance the H → γγ channel is enhanced by the additional S runningin the loop. Contrary to other scalar extension, enhance-ment of H → γγ is obtained instead of suppression dueto the minus sign of λ HS [14]. Strong enhancement ofsignal strength µ γγ for different values of m S can be ob-tained, with the result normalized to the SM predictionshown in Fig.3. The signal strength µ γγ can be only en-hanced by increasing the electric charge or λ HS of S tocompensate the suppression of production cross section.Compared to µ γγ ≈ .
65 (0 .
77) reported by ATLAS [15](CMS [8]) with the average µ γγ ≈ .
21, our model wouldrequire large electric charge to explain the large H → γγ anomaly. The large electric charge provides a possiblealternative to study the S particle via Drell-Yan produc-tion in linear collider. CONFINEMENT OF STRONGLY COUPLEDSCALAR FIELD
So far we have discussed the perturbative sector of thecolored scalar S . We restricted the non-perturbative as-pect of the model to the upscaling of the gap equation in Λ HS (cid:61) (cid:200) e (cid:200) (cid:61) Λ HS (cid:61) (cid:200) e (cid:200) (cid:61) Λ HS (cid:61) (cid:200) e (cid:200) (cid:61) Λ HS (cid:61) (cid:200) e (cid:200) (cid:61) m s (cid:64) TeV (cid:68) Μ ΓΓ (cid:61) Σ (cid:137) BR S (cid:43) S M (cid:144) Σ (cid:137) BR S M s (cid:61) Figure 3. The signal strength of H → γγ branching ratio withthe additional S contribution relative to the SM prediction areplotted for different values of electric charge e and λ HS of S .The large electric charge has to compensate the suppressionof production cross section for µ γγ enhancement. Eq. (1). Let us discuss a bit the physics in Eq. (1). Ananalytical way to understand confinement in the quarkssector of QCD is to calculate the scaling of the gap equa-tion from Dyson-Schwinger equation (DSE) (cid:1) − = (cid:1) − + (cid:1) + ..., (7)where we have utilized the rainbow-ladder approximationand only kept the leading order contribution to our anal-ysis. The diagram above resembles the DSE for quarkpropagator, which can be solved within certain trunca-tion scheme in order to obtain the critical value X in C ( S ) α (Λ) (cid:38) X, (8)for confinement to take place. However there are sub-tleties that one has to be careful when trying to extractthe exact bound of X . First the value X is gauge andtruncation scheme dependent. Different values rangingfrom 0 . π/ X will allow us to consider lower representation of S butin our analysis above we assume the conservative bound X > .
8. Second, the DSE for quark is linearizable withits linear form a Fredholm integral equation [16, 19, 20]as the wave function renormalization part and the self-energy part can be dealt separately in Landau gauge.However such privilege is not enjoyed by the scalar DSEas the integral equation F ( p ) = p + 3 C α s π (cid:34)(cid:90) p dq q p F ( q ) + (cid:90) ∞ p dq p F ( q ) (cid:35) , (9)is not linearizable, where we have denoted the function F ( k ) = Z ( k ) k +Σ ( k ). The main reason for such dif-ficulty is due to the lack of confinement order parameter for scalar QCD. Comparing to fermionic QCD, the orderparameter for confinement can be related to the degree ofchiral symmetry breaking. From the perturbative calcu-lation of the anomalous dimension of operator (cid:104) ¯ ψψ (cid:105) and (cid:104) S † S (cid:105) in the same representation, it can be shown that γ (cid:104) ¯ ψψ (cid:105) = γ (cid:104) S † S (cid:105) + O ( λ S ) . (10)Hence one can conjecture that the relevant order param-eter C α s at leading order for determining confinementshould be the same for both fermionic QCD and scalarQCD, which we have assumed. In fact it has been arguedthat the scaling property for scalar and quark propaga-tor in the infrared is identical [21]. This result can beverified in lattice QCD.Note that the QCD coupling becomes non-perturbativein the TeV regime even though the coupling is prettysmall. This stems from the large value of C which is re-sponsible for condensation. As a consequence, the exactevolution of α s cannot be precisely calculated in the TeVregime. However the coupling may become perturbativeagain at sufficiently small α s or high energy. A similarconclusion was made in Ref. [4]. Measuring α s at highenergy will provide an independent test for our model. CONCLUSION
With no signature of any SM extension at the LHCand in other searches, the notion of naturalness deservesto be reexamined and other ideas of explaining the EWscale should be considered. We discussed in this lettera scenario where conformal symmetry plays an essentialrole and where the EW scale is a consequence of quan-tum effects. The idea of mass scale generation from aquantum effect, so called dimensional transmutation, isalready implemented in the QCD sector of the SM. Wehave shown that it is possible to extend the success ofQCD and to explain the existence of the EW scale by in-cluding a new scalar particle belonging to (cid:48) of SU(3) c .The extension is rather minimal and moreover unique if X in Eq. (8) is greater than 0 .
8. The mass of this new col-ored boson is constrained such that it can be explored orruled out by the LHC. The signature of this colored scalarfield is comparatively clean. The accidental U(1) symme-try can also be probed in the H → γγ signal strength ifit is identified with the U(1) Y of the SM. Furthermore,with a non-zero hypercharge the new colored boson canbe directly produced at linear colliders, which will be ournext target to investigate. We leave the more detailed in-vestigation of non-perturbative aspect of this model andthe implication of EW phase transition to future work. Acknowledgements:
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