Genesis of electroweak and dark matter scales from a bilinear scalar condensate
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Genesis of electroweak and dark matter scales from a bilinearscalar condensate
Jisuke Kubo ∗ and Masatoshi Yamada † Institute for Theoretical Physics, Kanazawa University, Kanazawa 920-1192, Japan
The condensation of scalar bilinear in a classically scale invariant strongly inter-acting hidden sector is used to generate the electroweak scale, where the excitationof the condensate is identified as dark matter. We formulate an effective theoryfor the condensation of the scalar bilinear and find in the self-consistent mean fieldapproximation that the dark matter mass is of O (1) TeV with the spin-independentelastic cross section off the nucleon slightly below the LUX upper bound. PACS numbers: 11.30.Ly,11.15.Tk,12.60.Rc,95.35.+d ∗ [email protected] † [email protected] I. INTRODUCTION
Can we explain the origin of “mass without mass” [1]? Yes, a large portion of the baryonmass can be produced by dynamical chiral symmetry breaking (D χ SB) “from nothing” [2, 3].This nonperturbative mechanism, instead of the Brout-Englert-Higgs mechanism, can alsobe applied to trigger electroweak symmetry breaking [4, 5]. After the discovery of the Higgsparticle [6, 7], however, it is a fair assumption that fundamental scalars can exist. Since theHiggs mass term is the only term in the standard model (SM), that breaks scale invarianceat the classical level, we can thus ask where the Higgs mass term comes from. Even theHiggs mass term, too, may have its origin in a nonperturbative effect. In fact D χ SB in aQCD-like hidden sector has been recently used to induce the Higgs mass term in a classicallyscale invariant extension of the SM [8–11].In this paper we focus on another nonperturbative effect, the condensation of the scalarbilinear (CSB) [12, 13] (see also [14, 15]) in a strongly interacting hidden sector, to generatedirectly the Higgs mass term via the Higgs portal [16]. The main difference between twoclasses of models, apart from how a scale is dynamically generated, is that in the firstclass of models (with D χ SB) the scale generated in a hidden sector has to be transmittedto the SM via a mediator, e.g. a SM singlet scalar in the model considered in [8–11],while such a mediator is not needed in the second class of models (with CSB). This willbe an important difference if two classes should be experimentally distinguished. Anotherimportant difference is that the DM particles of the first class are CP -odd scalars, whilethey are CP -even scalars in the second class, as we will see.Our interest in the second class of models is twofold: first, because the discussion on howthe Higgs mass is generated in [16] is rather qualitative, we here formulate an effective theoryto nonperturbative breaking of scale invariance by the CSB. This enables us to performan approximate but quantitative treatment. Second, since only one flavor for the stronglyinteracting scalar field S is considered in [16] so that there is no dark matter (DM) candidate,we introduce N f flavors and investigate whether we can obtain realistic candidates of DM.The DM candidates in our scenario are scalar-antiscalar bound states, which are introducedas the excitation of the condensate in the self-consistent mean field approximation (SCMF)[17, 18]. Their interactions with the SM can be obtained by integrating out the “constituent”scalars. In this approximation we can constrain the parameter space of the effective theoryin which realistic DM candidates are present. II. THE MODEL AND ITS EFFECTIVE LAGRANGIAN
We start by considering a hidden sector described by an SU ( N c ) gauge theory with thescalar fields S ai ( a = 1 , . . . , N c , i = 1 , . . . , N f ) in the fundamental representation of SU ( N c ).The Lagrangian of the hidden sector is L H = −
12 tr F + ([ D µ S i ] † D µ S i ) − ˆ λ S ( S † i S i )( S † j S j ) − ˆ λ ′ S ( S † i S j )( S † j S i ) + ˆ λ HS ( S † i S i ) H † H, (1)where D µ S i = ∂ µ S i − ig H G µ S i , G µ is the matrix-valued gauge field, the trace is taken overthe color indices, and the parentheses in Eq. (1) stands for an SU ( N c ) invariant product.The SM Higgs doublet field is denoted by H . The total Lagrangian is L T = L H + L SM , andthe SM part, L SM , contains the SM gauge and Yukawa interactions along with the scalarpotential V SM = λ H ( H † H ) without the Higgs mass term.We assume that for a certain energy the gauge coupling g H becomes so strong that the SU ( N c ) invariant scalar bilinear forms a U ( N f ) invariant condensate [12, 13] h ( S † i S j ) i = h N c X a =1 S a † i S aj i ∝ δ ij . (2)This nonperturbative condensate breaks scale invariance, but it is not an order parameter,because scale invariance is broken by scale anomaly [19]. The breaking by anomaly is hardbut only logarithmic, which means basically that the coupling constants depend on theenergy scale [19]. Moreover, we should note that the mass term is not generated by theanomaly since the beta function of the mass is propotional to the mass itself, see e.g. [20]. The creation of the mass term from nothing can happen only by a nonperturbative effective,i.e. the condensate (2) is taken place. Therefore, the non-perturbative breaking due to thecondensation may be assumed to be dominant, so that we can ignore the breaking by scaleanomaly in the lowest order approximation to the breaking of scale invariance.Under this assumption the condensate is a good order parameter, and we would like toformulate an effective theory, which is an analog of the Nambu–Jona-Lasinio (NJL) theory In viewpoint of Wilsonian renormalization group, the classical scale invariance means that the bare massis exactly put on the critical surface [21]. Once this tuning is done, the renormalized mass keeps vinishingunder the renoramlization group transformation. Once the mass is dynamically generated, the scale anomaly contributes to the mass. [3] to D χ SB. The Lagrangian of the effective theory will not contain the SU ( N c ) gaugefields, because they are integrated out, while it contains the “constituent” scalar fields S ai ,for which we use the same symbol as the original scalar fields . Since the effective theoryshould describe the symmetry breaking dynamically, the effective Lagrangian has to beinvariant under the symmetry transformation in question: L eff = ([ ∂ µ S i ] † ∂ µ S i ) − λ S ( S † i S i )( S † j S j ) − λ ′ S ( S † i S j )( S † j S i )+ λ HS ( S † i S i ) H † H − λ H ( H † H ) , (3)with all positive λ ’s. This is the most general form which is consistent with the SU ( N c ) × U ( N f ) symmetry and the classical scale invariance, where we have not included the kineticterm for H in L eff , because it does not play any significant role as far as the effective theoryfor the CSB is concerned. Note that the couplings ˆ λ S , ˆ λ ′ S and ˆ λ HS in L H of (1) are not thesame as λ S , λ ′ S and λ HS in L eff , respectively. We emphasize that the effective Lagrangian (3)is scaleless, and describes the dynamics of scalar field S at slightly above the confinementscale, thus, the scalar condensate has not taken place yet. Therefore the mixing of multiplescales discussed in [21] does not appear. Using the effective Lagrangian (3), we attemptto approximately describe the genesis of scale by the original gauge theory (1) as the “non-perturbative” dimensional transmutation, `a la
Coleman–Weinberg. In the following, wedemonstrate this mechanism and present our formalism by considering first N f = 1 case.A. N f = 1 (with λ ′ S = 0)In the SCMF approximation, which has proved to be a successful approximation for theNJL theory [17], the perturbative vacuum is Bogoliubov-Valatin (BV) transformed to | B i ,such that h B | ( S † S ) | B i = f , where f has to be determined in a self-consistent way. Onefirst splits up the effective Lagrangian (3) into the sum L eff = L MFA + L I , where L I isnormal ordered (i.e. h B |L I | B i = 0), and L MFA contains at most the bilinears of S whichare not normal ordered. Using the Wick theorem ( S † S ) = :( S † S ) : + f , ( S † S ) = :( S † S ) : +2 f ( S † S ) − f , etc., we find L MFA = ( ∂ µ S † ∂ µ S ) − M ( S † S ) − λ H ( H † H ) + λ S f , Quantum field theory defined by (3) with the kinetic term for H is renormalizable in perturbation theory[22]. Although NJL model is also defined before the dynamical breaking of chiral symmetry, its Lagrangianhas a scale at which the Lagrangian is given. where M = 2 λ S f − λ HS H † H . To the lowest order in the SCMF approximation the “inter-acting ” part L I does not contribute to the amplitudes without external S s (the mean fieldvacuum amplitudes). We emphasize that, in applying the Wick theorem, only the SU ( N c )invariant bilinear product ( S † S ) = P N c a S a † S a has a nonzero (BV transformed) vacuumexpectation value. To compute loop corrections we employ the MS scheme, because dimen-sional regularization does not break scale invariance. To the lowest order the divergencescan be removed by renormalization of λ I ( I = H, S, HS ), i.e. λ I → ( µ ) ǫ ( λ I + δλ I ), and alsoby the shift f → f + δf , where ǫ = (4 − D ) /
2, and µ is the scale introduced in dimensionalregularization. The effective potential for L MFA can be straightforwardly computed : V MFA = M ( S † S ) + λ H ( H † H ) − λ S f + N c π M ln M Λ H , (4)where Λ H = µ exp(3 /
4) is chosen such that the loop correction vanishes at M = Λ H . ( V MFA with a term linear in f included but without the Higgs doublet H has been discussed in[23–25]. The classical scale invariance forbids the presence of this linear term.) Note herethat the scale Λ H is generated by the non-perturbative loop effect. To find the minimum of V MFA we look for the solutions of0 = ∂∂S a V MFA = ∂∂f V MFA = ∂∂H l V MFA ( l = 1 , . (5)The first equation gives 0 = ( S a ) † M = ( S a ) † (2 λ S f − λ HS H † H ), which has three solutions:(i) h S a i 6 = 0 and h M i = 0, (ii) h S a i = 0 and h M i = 0, and (iii) h S a i = 0 and h M i 6 = 0.The effective potential V MFA in the solution (i) has a flat direction, which corresponds to theend-point contribution discussed in [26]. In the flat direction (i.e. f = H = 0), V MFA = 0for any value of S a , so that the SU ( N c ) symmetry is spontaneously broken. If all theextremum conditions (5) are imposed for the solution (i), we obtain h f i = h ([ S a ] † S a ) i =(2 λ H /λ HS ) h H † H i along with C = 0 and h V MFA i = 0 [23]. Next we consider (ii) and findthat h S a i = h f i = h H i = 0 with h V MFA i = 0. The third solution (iii) can exist if C = 4 λ H λ S − λ HS > |h H i| = v h / λ HS C Λ H exp (cid:18) π λ H N c C − (cid:19) , h f i = 2 λ H λ HS |h H i| , Due to h M i = 0 there exists a tachyonic state, because the inequality of [24], 16 π / (2 N c λ S ) − ln[ h M i / Λ H exp( − / <
0, cannot be satisfied for a finite Λ H and a positive λ S . h V MFA i = − N c π Λ H exp (cid:18) π λ H N c C − (cid:19) < . Consequently, the solution (iii) presents the true potential minimum if (6) is satisfied (inthe energy region where (3) should serve as the effective Lagrangian). Self-consistencymeans that f = h B | ( S † S ) | B i is equal to h f i at the potential minimum in the mean fieldapproximation. The Higgs mass at this level of approximation becomes m h = λ HS Λ H C (cid:18) λ H λ S C + N c λ HS π (cid:19) exp (cid:18) π λ H N c C − (cid:19) . (7)In the small λ HS limit we obtain m h ≃ λ H |h H i| = 2 λ HS h f i , where the first equation isthe SM expression, and the second one is simply assumed in [16]. So the Higgs mass (7)contains the backreaction. The analysis above shows that the scale created in the hiddensector can be desirably transmitted to the SM sector. The reason that h V MFA i < f in V MFA , which can lift the h V MFA i into a positive direction [24, 25], while V MFA = 0remains in the flat direction [26].At this stage we would like to mention that Bardeen and Moshe [26] (and also others)pointed out the intrinsic instability inherent in (3) (which is related to its triviality) if oneregards (3) as a fundamental Lagrangian. We however discard this fundamental problem,because we assume that such a problem is absent in the original theory described by (1).B. N f > and dark matter Here we consider the case with N f > σ and φ α ( α = 1 , . . . , N f − h B | ( S † i S j ) | B i = f ij = h f ij i + Z / σ δ ij σ + Z / φ t αji φ α . (8)Here t α are the SU ( N f ) generators in the Hermitian matrix representation, and Z σ and Z φ are the wave function renormalization constants of a canonical dimension 2. The unbroken U ( N f ) flavor symmetry implies h f ij i = δ ij f and h φ α i = 0, where h σ i can be absorbed into f , so that we can always assume h σ i = 0. Furthermore, the flavor symmetry ensures thestability of φ α , i.e. they can be good DM candidates, because they are electrically neutraland their interactions with the SM sector are loop suppressed, as we will see. Note that σ and φ α in (8) are introduced as c-numbers without kinetic terms. However, their kineticterms will be generated through S a loop effects, and consequently we will reinterpret themas quantum fields describing physical degrees of freedom. The investigation of the vacuumstructure is basically the same as in the N f = 1 case. We are interested in the solution oftype (iii) of the previous case, i.e. f = 0 , |h H i| = v h / = 0, which is the true potentialminimum if G = 4 N f λ H λ S − N f λ HS + 4 λ H λ ′ S > m h = λ HS N f Λ H G (cid:18) λ H ( N f λ S + λ ′ S ) G + N c N f λ HS π (cid:19) × exp (cid:18) π λ H N c G − (cid:19) . (10)The SCMF Lagrangian L ′ MFA involving σ and φ α can now be written as L ′ MFA = ( ∂ µ S † i ∂ µ S i ) − M ( S † i S i )+ N f ( N f λ S + λ ′ S ) Z σ σ + λ ′ S Z φ φ α φ α (11) − N f λ S + λ ′ S ) Z / σ σ ( S † i S i ) − λ ′ S Z / φ ( S † i t αij φ α S j )+ λ HS S † S ) h (2 v h + h ) − λ H h (6 v h + 4 v h h + h ) , where M = 2( N f λ S + λ ′ S ) f − λ HS v h /
2, and Tr( t α t β ) = δ αβ /
2. Further, h is the Higgs fieldcontained in the Higgs doublet as H T = ( H + , ( v h + h + iχ ) / √ H + and χ are thewould-be Nambu-Goldstone fields. Linear terms in σ and h are suppressed in (11), becausethey will be canceled against the corresponding tadpole corrections.Using (11) and integrating out the constituent scalars S ai , we can obtain effective inter-actions among σ, φ and the Higgs h . We first compute their inverse propagators, up toand including one-loop order, to obtain their masses and the wave function renormalizationconstants: Γ αβφ ( p ) = Z φ δ αβ λ ′ S Γ φ ( p ) = Z φ δ αβ λ ′ S (cid:2) λ ′ S N c Γ( p ) (cid:3) , (12)Γ σ ( p ) = 2 Z σ N f ( N f λ S + λ ′ S ) (cid:2) N c ( N f λ S + λ ′ S )Γ( p ) (cid:3) , Γ hσ ( p ) = − Z / σ v h λ HS ( N f λ S + λ ′ S ) N f N c Γ( p ) , Γ h ( p ) = p − m h + ( v h λ HS ) N f N c (Γ( p ) − Γ(0)) , where m h is given in (10), the canonical kinetic term for H is included, andΓ( p ) = − π Z dx ln (cid:20) − x (1 − x ) p + M Λ H exp( − / (cid:21) . φ α φ β hhS + cross FIG. 1.
The interaction between DM and the Higgs h arises at the one-loop level. Diagrams ∝ λ HS ( v h /M ) are ignored, because λ HS ( v h /M ) ≪ λ HS . We have included neither the wave function renormalization constant for h (which is ap-proximately equal to 1 within the approximation here) nor the corrections to Γ h comingfrom the SM sector (which will only slightly influence our result).The DM mass is the zero of the inverse propagator, i.e.Γ αβφ ( p = m DM2 ) = 0 , (13)and Z φ (which has a canonical dimension 2) can be obtained from Z − φ = 2( λ ′ S ) N c ( d Γ /dp ) | p = m .The σ and Higgs masses are obtained from the zero eigenvalues of the h − σ mixing matrix.Strictly speaking, this mixing has to be taken into account in determining the renormal-ization constants (matrix) for σ and h . However, the mixing is less than 1% in a realisticparameter space so that we ignore the mixing for the renormalization constants. As wecan see from (12), the radiative correction to the inverse propagator is proportional to2 λ ′ S N c / π , so that the solution of (13) for a real positive p can exist if λ ′ S N c is sufficientlylarge. Therefore, if an upper limit of λ ′ S is set, there will be a minimum value of N c . Itturns out that the minimum N c is 3 for Γ φ ( p ) with N f = 2 to have a zero if 0 < λ ′ S < π .For a larger N f we need a larger N c : the minimum N c is 4 for N f = 3 for instance.The link of φ to the SM model is established through the interaction with the Higgs, whichis generated at one-loop as shown in Fig. 1. We use the s-channel momenta p = p ′ = ( m DM , )for DM annihilation, because we restrict ourselves to the s-wave part of the velocity-averagedannihilation cross section h vσ i . For the spin-independent elastic cross section off the nucleon σ SI we use the t-channel momenta p = − p ′ = ( m DM , ). In these approximations thediagrams of Fig. 1 yield the effective couplings κ s ( t ) δ αβ = δ αβ Γ φ h ( M , m DM , ǫ = 1( − , (14)whereΓ φ h ( M , m DM , ǫ ) = Z φ N c ( λ ′ S ) λ HS π Z dx Z − x dy (cid:2) M + m ( x ( x −
1) + y ( y − − ǫxy ) (cid:3) − , and we consider only the parameter space with m DM , m σ < M , because beyond that ourSCMF approximation will break down. Then we obtain h vσ i = 132 πm X I = W,Z,t,h ( m − m I ) / a I + O ( v ) , where m W,Z,t,h are the
W, Z , top quark and Higgs masses, respectively, and a W ( Z ) = 4(2) κ s ∆ h m W ( Z ) m m W ( Z ) − m m W ( Z ) ! ,a t = 24 κ s ∆ h m t ( m − m t ) , a h = κ s (cid:18) λ H ∆ h m W g (cid:19) with ∆ h = (4 m − m h ) − [ m h is the corrected Higgs mass which should be compared with m h of (10).] The DM relic abundance is Ωˆ h = ( N f − × ( Y ∞ s m DM ) / ( ρ c / ˆ h ), where Y ∞ is the asymptotic value of the ratio n DM /s ; s = 2890 / cm is the entropy density at present; ρ c = 3 H / πG = 1 . × − ˆ h GeV / cm is the critical density; ˆ h is the dimensionlessHubble parameter; M pl = 1 . × GeV is the Planck energy; and g ∗ = 106 .
75 + N f − Y ∞ we solve the Boltzmann equation dYdx = − . g / ∗ (cid:18) m DM M PL x (cid:19) h vσ i (cid:0) Y − ¯ Y (cid:1) numerically, where x is the inverse temperature m DM /T , and ¯ Y is Y in thermal equilibrium.The spin-independent elastic cross section off the nucleon σ SI can be obtained from [27] σ SI = 14 π κ t ˆ f m N m DM m h ! (cid:18) m DM m N + m DM (cid:19) , where κ t is given in (14), m N is the nucleon mass, and ˆ f ∼ . Before we scan the parameter space, we consider a representative point in the four-dimensional parameter space of the scalar couplings with N f = 2 and N c = 5: λ S = 1 . , λ ′ S = 5 . , λ HS = 0 . , λ H = 0 . , If the value ˆ f improved by the recent lattice simulation [29] is used, we obtain slightly smaller values(about 20%). m DM [TeV] -46 -45 -44 -43 s S I [ c m ] FIG. 2.
The spin-independent elastic cross section σ SI of DM off the nucleon as a function of m DM for N f = 2 , N c = 5 (red) and 8 (green) and for N f = 3 , N c = 6 (blue), where Ωˆ h is required to be consistentwith the PLANCK experiment at 2 σ level [33]. The black dashed line stands for the central value of theLUX upper bound [30]. which give f = 0 . , M = 1 .
08 TeV, m DM = 0 .
801 TeV, m σ = 1 .
98 TeV,Λ H = 0 .
501 TeV, Ωˆ h = 0 . σ SI = 1 . × − cm , κ s = 0 . κ t = 0 . m DM - σ SI plane the predicted area for various N f and N c . The predictedvalues of σ SI are just below the LUX upper bound (black dashed line) [30] and can be testedby XENON1T, whose sensitivity is O (10 − ) cm [31, 32]. If we increase N f , we have tosuppress Y ∞ , because Ωˆ h ∝ ( N f − Y ∞ , which requires a larger h vσ i , leading to a larger σ SI . III. SUMMARY
We have assumed that the SM without the Higgs mass term is coupled through a Higgsportal term with a classically scale invariant gauge sector, which contains N f scalar fields.Due to the strong confining force the gauge invariant scalar bilinear forms a condensate,thereby violating scale invariance. The Higgs portal term is responsible for the transmissionof the scale to the SM sector, realizing electroweak scalegenesis. We have formulated an1effective theory for the condensation of the scalar bilinear. The excitation of the condensateis identified as DM, where its scale is dynamically generated in the hidden gauge sector.Our formalism is simple and its application will be multifold. We have found that the DMmass is of O (1) TeV and the predicted spin-independent elastic cross section off the nucleonis slightly below the LUX upper bound and could be tested by the XENON1T experiment. Acknowledgements:
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