Strong dynamics behind the formation of the 125 GeV Higgs boson
aa r X i v : . [ h e p - ph ] M a r Strong dynamics behind the formation of the
GeV Higgs boson
M.A. Zubkov ∗ University of Western Ontario, London, ON, Canada N6A 5B7
We consider the scenario, in which the new strong dynamics is responsible for the formation of the
GeV Higgs boson. The Higgs boson appears as composed of all known quarks and leptons of the StandardModel. The description of the mentioned strong dynamics is given using the ζ - regularization. It allows toconstruct the effective theory without ultraviolet divergences, in which the /N c expansion works naturally.It is shown, that in the leading order of the /N c expansion the mass of the composite h - boson is given by M h = m t / √ ≈ GeV, where m t is the top - quark mass. PACS numbers:
I. INTRODUCTION
In the present paper we suggest the scenario, in which the recently discovered
GeV h - boson [1, 2] is composite.According to the suggested model it is composed of all fermions of the Standard Model (SM). The scale of the hiddenstrong dynamics is supposed to be of the order of several TeV. We suppose, that the W and Z boson masses aredetermined by the condensate of the
GeV h - boson according to the Higgs mechanism [3, 4]. All Dirac fermionmasses are determined by the h - boson as well. In the present paper we do not consider the Majorana masses at alland assume, that neutrinos have extremely small Dirac masses [5]. The low energy effective theory contains the four -fermion interaction [8] between all SM fermions. Our model differs from the conventional models with four - fermioninteractions [6], in which the top quark is condensed. The model of top - quark condensation was first suggested in[7] (and developed later in [9–13]). The idea, that the Higgs boson may be composed of the known SM fermionswas discussed even earlier, in [14], together with the certain preon models (however, in [14] there was no emphasisin the dominating role of the top quark and the compositeness of the Higgs boson was considered together with thecompositeness of quarks and leptons). There are two main aspects, in which the model discussed in the present paperdiffers from the conventional models of top - quark condensation:1. The four - fermion interaction is non - local. It contains the form - factors G that correspond to the interactionbetween the composite Higgs boson and the pair fermion - antifermion. The formfactors depend on three scalarparameters of the dimension of mass squared: q , p , k , where p and k are the 4 - momenta of the fermion andanti - fermion while q = p − k is the 4 - momentum of the composite scalar boson. G ( p, k ) = g ( q , p , k ) are different for different fermions at large distances (both time - like and space - like), i.e. if at least one of thequantities | q | , | p | , | k | is much smaller, than the Electroweak scale M Z ≈ GeV. However, at small space- like and time - like distances (i.e. for | q | ∼ | p | ∼ | k | ∼ M Z ) those form - factors become equal for allSM fermions. Unlike for the other fermions the mentioned form - factor for the top - quark is assumed to beindependent of momenta.2. Our model with the four - fermion interaction is not renormalizable. Therefore, it is the effective theory onlyand its output strongly depends on the regularization scheme. Contrary to the approach of the mentioned abovepapers on the top - quark condensation models we do not use the conventional cutoff regularization. We use zetaregularization [15, 16]. It allows to construct the effective theory without any ultraviolet divergencies.The /N c expansion is a good approximation within our effective theory. This is because the dangerous ultravioletdivergences are absent. This is well - known, that these divergences break the /N c expansion in the NJL modelsdefined in ordinary cutoff regularization. For example, in [17] it is shown, that when the ultraviolet cutoff is muchlarger, than the generated masses, the next to leading order /N c approximation to various quantities does not givethe corrections smaller, than the leading one. However, this occurs because of the terms that are formally divergentin the limit of infinite cutoff. Therefore, in the regularization, that is free from divergences the next to leading ap-proximation is suppressed by the factor /N c compared to the leading approximation. Moreover, for the calculation ∗ on leave of absence from ITEP, B.Cheremushkinskaya 25, Moscow, 117259, Russia of various quantities related to the extra high energy processes (the processes with the characteristic energies muchlarger, than m t ) the next - to leading order approximation is suppressed by the factor /N total , where N total = 24 is the total number of the Standard Model fermions (the color components are counted as well, so that we have N total = 3 generations × (3 colors + 1 lepton) × ).Zeta regularization [15, 16] provides the natural mechanism of cancelling the unphysical divergent contributions tovarious physical quantities in the field theories. For example, being applied to quantum hydrodynamics it provides theabsence of the unphysical divergent contributions to vacuum energy due to the phonon loops. The latter are knownto be cancelled exactly by the physics above the natural cutoff of the hydrodynamics [18]. If zeta regularization isapplied to quantum hydrodynamics, those divergences do not appear from the very beginning. The cancellation ofthe mentioned divergences in quantum hydrodynamics occurs due to the thermodynamical stability of vacuum [18].That’s why zeta regularization is the natural way to incorporate the principle of the thermodynamical stability ofvacuum already at the level of low energy approximation (i.e. at the level of the hydrodynamics). It was suggested toapply the similar principles of stability to subtract divergences from the vacuum energy in the quantum theory in thepresence of gravity [19] and to subtract quadratic divergences from the contributions to the Higgs boson masses in theNJL models of composite Higgs bosons [20–23]. Now we understand, that the zeta regularization allows to implementsuch principles of stability to various theories in a very natural and common way. That’s why this regularization mayappear to be not only the regularization, but the essential ingredient of the quantum field theory.The approach presented in the given paper allows to derive the relation M H = m t / on the level of the leadingorder of the /N c expansion. In the simplest models of the top - quark condensation the mass of the composite Higgsboson is typically related to the mass of the top quark by the relation M H = 4 m t . The resulting mass ≈ GeVcontradicts to the recent discovery of the
GeV h - boson. It is worth mentioning, that there were attempts toconstruct the theory that incorporates the existence of several composite Higgs bosons M H,i related to the top - quarkmass by the Nambu Sum rule P M H,i = 4 m t . Those models admit the appearance of the GeV h - boson andpredict the existence of its Nambu partners [21–23]. The approach of the present paper predicts the only compositeHiggs boson with the observed mass around m t / √ ≈ GeV.The paper is organized as follows. In Section II we define the model under consideration. In Section III we presentthe effective action and evaluate physical quantities (the practical calculations using zeta regularization are placed inAppendix). In Section IV we end with the conclusions.
II. THE MODEL UNDER CONSIDERATIONA. Notations
SM fermions carry the following indices:1. Weil spinor indices. We denote them by large Latin letters
A, B, C, ... SU (2) doublet indices (isospin) both for the right - handed and for the left - handed spinors are denoted bysmall Latin letters a, b, c, ... For the left - handed doublets index a may be equal to and . For the right -handed singlets of the Standard Model this index takes values U, D, N, E that denote the up quark, down quark,neutrino and electron of the first generation and the similar states from the second and the third generations.3. Color SU (3) indices are denoted by small Latin letters i, j, k, ...
4. The generation indices are denoted by bold small Latin letters a , b , c , ... Besides, space - time indices are denoted by small Greek letters µ, ν, ρ, ...
Let us introduce the following notations: L = (cid:18) u L d L (cid:19) , L = (cid:18) c L s L (cid:19) , L = (cid:18) t L b L (cid:19) R U = u R , R U = c R , R U = t R R D = d R , R D = s R , R D = b R (II.1)and L = (cid:18) ν L e L (cid:19) , L = (cid:18) ν µL µ L (cid:19) , L = (cid:18) ν τL τ L (cid:19) R N = ν R , R N = ν µR , R N = ν τR R E = e R , R E = µ R , R E = τ R (II.2) B. The action
In our consideration we neglect gauge fields. The partition function has the form: Z = Z D ¯ ψDψe iS , (II.3)By ψ we denote the set of all Standard Model fermions. The action S = R d x ¯ ψi∂γψ + S I contains two terms. Thefirst one is the kinetic term. The second one is related to the hidden strong interaction between the SM fermions, thatresults in the appearance of the additional four - fermion non - local interaction term S I = 1 M I X a Z d x d x d y d y d z (cid:16) ¯ L a b ( x ) R a U ( x ) G a U ( x − z, x − z ) + ¯ L a b ( x ) R a N ( x ) G a N ( x − z, x − z )+ ¯ R a D ( x ) L a c ( x ) ǫ bc G a D ( x − z, x − z ) + ¯ R a E ( x ) L a c ( x ) ǫ bc G a E ( x − z, x − z ) (cid:17)(cid:16) ¯ G a D ( y − z, y − z ) ¯ L a d ( y ) R a D ( y ) ǫ bd + ¯ G a E ( y − z, y − z ) ¯ L a d ( y ) R a E ( y ) ǫ bd + ¯ G a U ( y − z, y − z ) ¯ R a U ( y ) L a b ( y ) + ¯ G a N ( y − z, y − z ) ¯ R a N ( y ) L a b ( y ) (cid:17) , (II.4)Here M I is the dimensional parameter (it will be shown below that it’s bare value is negative and is around M I ≈− M Z ≈ − [90GeV] ). Functions G a a ( y − z, y − z ) for a = U, D, N, E are the form - factors mentioned in theintroduction defined in coordinate space ( ¯ G a a means complex conjugation, ǫ ab is defined in such a way, that ǫ = − ǫ = 1 ). In momentum representation we have S I = VM I X a X q = p − k = p ′ − k ′ (cid:16) ¯ L a b ( p ) R a U ( k ) G a U ( p, k ) + ¯ L a b ( p ) R a N ( k ) G a N ( p, k )+ ¯ R a D ( p ) L a c ( k ) ǫ bc G a D ( p, k ) + ¯ R a E ( p ) L a c ( k ) ǫ bc G a E ( p, k ) (cid:17)(cid:16) ¯ G a D ( p ′ , k ′ ) ¯ L a d ( k ′ ) R a D ( p ′ ) ǫ bd + ¯ G a E ( p ′ , k ′ ) ¯ L a d ( k ′ ) R a E ( p ′ ) ǫ bd + ¯ G a U ( p ′ , k ′ ) ¯ R a U ( k ′ ) L a b ( p ′ ) + ¯ G a N ( p ′ , k ′ ) ¯ R a N ( k ′ ) L a b ( p ′ ) (cid:17) , (II.5)Here V is the four - volume. Fermion fields in momentum space are denoted as ψ ( q ) = V R d xe − iqx ψ ( x ) whilethe form - factors in momentum space are defined as G a a ( p, k ) = R d xd ye − ipx + iky G a a ( x, y ) for a = U, D, N, E .According to our supposition all functions G a a ( p, k ) = g a a ( q , p , k ) vary between fixed values much smaller thanunity at small momenta and unity at large values of | q | , | p | , | k | : G a a ( p, k ) = g a a ( q , p , k ) → | q | , | p | , | k | ∼ M Z ∼ [90 GeV] ); G a a ( p, k ) = g a a ( q , p , k ) → κ a a ( | q | , | p | , | k | ≪ M Z ) | G a a ( p, k ) | = | g a a ( q , p , k ) | ≪ | q | ∼ M Z ; | p | , | k | ≪ M Z ;except for top quark) (II.6)Here M Z ≈ GeV is the Electroweak scale, κ a a are the dimensionless coupling constants that are all muchsmaller than unity except for the top quark. Because of the isotropy of space - time all G a a are the functionsof the invariants p , k , and q = ( p − k ) only. Besides, in order to provide the left - right symmetric massmatrix (see below) we require G a a ( x, y ) = ¯ G a a ( y, x ) that leads to G ( p, k ) = ¯ G ( k, p ) and to g a a ( q , p , k ) =¯ g a a ( q , k , p ) . In principle, we may consider the real - valued form - factors G ( x, y ) = G ( y, x ) that would resultin G a a ( p, k ) = R d xd ye − ipx + iky G a a ( x, y ) = R d xd ye − ipy + ikx G a a ( x, y ) = ¯ G a a ( k, p ) = G a a ( − k, − p ) . Since G ( k, p ) = G ( − k, − p ) = g (( p − k ) , k , p ) we would arrive at the real - valued form - factors in momentum space.We imply that Eq. (II.6) is valid for the complex - valued p , q , k in the upper half of the complex plane. Besides,we require, that for the the top quark G t ( p, k ) ≡ G U ( p, k ) ≈ , (II.7)for any p, k . We may choose, for example, g a a ( q , p , k ) = q + α a a M I q + γ a a M I × p + β a a M I p + λ a a M I × k + β a a M I k + λ a a M I (II.8)with some constants α, β, γ, λ such that α a a β a a β a a γ a a λ a a λ a a = κ a a while α a a , β a a , γ a a , λ a a ≪ , and α U = β U = γ U = λ U = 1 . Wemay also interpolate the Form - factors as follows (for all fermions except for the top - quark): g a a ( q , p , k ) = (cid:26) | q | , | p | , | k | > M κ a a << (cid:12)(cid:12)(cid:12)(cid:12) (II.9)with a dimensional parameter M such that M ≪ M Z (where M Z is the Z - boson mass). For the top - quark we set G U ( p, k ) = g U ( q , p , k ) = κ U = 1 . Let us introduce the auxiliary scalar SU (2) doublet H . The resulting actionreceives the form S = Z d x h ¯ ψi∂γψ − | H | M I (cid:3) − X a Z d x d x d z h(cid:16)h ¯ L a b ( x ) R a U ( x ) G a U ( x − z, x − z ) H b ( z ) + ¯ L a b ( x ) R a N ( x ) G a N ( x − z, x − z ) H b ( z ) i + h ¯ L a c ( x ) R a D ( x ) ¯ G a D ( x − z, x − z ) ¯ H b ( z ) ǫ bc + ¯ L a c ( x ) R a E ( x ) ¯ G a E ( x − z, x − z ) ¯ H b ( z ) ǫ bc i + ( h.c. ) (cid:17)i In momentum space we have: S = V X p h ¯ ψ ( p ) pγψ ( p ) − H + ( p ) H ( p ) M I i − V X a X q = p − k h ¯ L a b ( p ) R a U ( k ) G a U ( p, k ) H b ( q ) + ¯ L a b ( p ) R a N ( k ) G a N ( p, k ) H b ( q ) + ( h.c. ) i − V X a X q = p − k h ¯ L a c ( k ) R a D ( p ) ¯ G a D ( p, k ) ¯ H b ( q ) ǫ bc + ¯ L a c ( k ) R a E ( p ) ¯ G a E ( p, k ) ¯ H b ( q ) ǫ bc + ( h.c. ) ii , (II.10)where H ( q ) = V R d xH ( x ) e − iqx .The important property of the functions g a a ( q , p , k ) is that the Wick rotation from the positive values of p , q , k to negative values of p , q , k (i.e. positive values of Euclidean momenta squared) keeps the relation G a a ( p, k ) = g a a ( q , p , k ) → | q | , | p | , | k | ∼ M Z ∼ [90 GeV] ) (II.11)This will allow us to evaluate loop integrals using the Euclidean expressons (see Appendix). It is worth mentioning,that the invariance of Eq.(II.11) under the Wick rotation is seen directly in the form of the Form - factors of Eq. (II.8). C. Fermion masses
In the following we fix Unitary gauge H = ( v + h, T , where v is vacuum average of the scalar field. We denotethe Dirac - component spinors in this gauge corresponding to the SM fermions by ψ a a . We omit angle degrees offreedom to be eaten by the gauge bosons. In this gauge the propagator Q a a , a = U, D, N, E of the SM fermion hasthe form: [ Q a a ] − = ˆ pγ − vG a a ( p, p ) (II.12)For each fermion except for the top - quark there is the pole of this propagator at | p | ≪ M Z that gives the fermionmass m a a ≈ vG a a (0 , ≈ vκ a a (II.13)For the top quark we have m U = v . The value of v is to be calculated using the gap equation that is the extremumcondition for the effective action as a function of h . The effective action of Eq. (II.10) can be rewritten as S = Z d x (cid:16) ¯ ψ ( x )( i∂γ − M ) ψ ( x ) − ¯ ψ ( x ) h ˆ G h ψ i ( x ) − M I ( v + h ( x )) (cid:17) , (II.14)where M is the mass matrix. It is diagonal, with the diagonal components m a a ( a = U, D, N, E ; a = , , ) given by κ a a v . We use here the following notation for the h - dependent operator ˆ G h : [ ˆ G h ξ ]( x ) = Z d zd x ξ ( x ) G ( x − z, x − z ) h ( z ) (II.15)This operator is hermitian as follows from the condition G ( x, y ) = ¯ G ( y, x ) . We imply, that the h - boson of Eq. (II.10)is responsible for the formation of masses of W and Z bosons. Here and below we omit indices a , a for G and κ . Thesum over these indices is implied in the following expressions. G and κ are considered as the diagonal matrices withthe diagonal elements G a a and κ a a correspondingly.The interaction at the momenta of fermions | p | , | k | ∼ [90 GeV] and the momentum of the Higgs boson | q | = | ( p − k ) | ∼ [90 GeV] gives rise to the U ( N total ) (with N total = 24 ) symmetric interaction between the real - valuedHiggs field excitations h and the SM fermions. The interaction at momenta | ( p − k ) | ≈ corresponds to the Higgsfield condensate. The existing poles of fermion propagators give the masses of all SM quarks and leptons. The valueof v is to be determined through the requirement, that δδh S eff ( h ) = 0 at h = 0 , where S eff is the effective action(obtained after the integration over the fermions). III. EFFECTIVE LOW ENERGY ACTION AND THE EVALUATION OF PHYSICAL QUANTITIES.A. Effective action for the h - boson. Gap equation and Higgs boson mass. The effective action for the theory with action Eq. (II.14) as a function of the field h is obtained as a result ofthe integration over fermions. In zeta regularization this effective action is calculated in Appendix up to the termsquadratic in h . This corresponds to the leading /N c approximation.The total one - loop effective action for the h - boson receives the form: S [ h ] = Z d x h − M I ( v + h ) − ˆ Cv ( v + h ) + h ( x ) Z h ( w (ˆ p )ˆ p − M H ) h ( x ) i , (III.1)where w ( p ) ≈ , ( | p | ∼ M H ) w ( p ) ≈ / , ( | p | ≪ M H ) Z h ≈ N total π log µ m t M H = 4 N c m t /N total ;ˆ C = N c π log µ m t (III.2)According to our supposition κ t = 1 , and the mass of the top - quark is given by v = m t . Here we neglect theimaginary part of the effective action that is much smaller, than the real part for the momenta squared of the h - bosonsmaller, than m t . For p ≥ m t the decay of the h - boson into the pair t ¯ t should be taken into account through theimaginary part of effective action.Recall, that the resulting expressions for various quantities in zeta - regularization contain the scale parameter µ . Weidentify this parameter with the typical scale of the interaction that is responsible for the formation of the compositeHiggs boson. Below it will be shown, that the value of µ consistent with the observed masses of Higgs, W, and Z -bosons is µ ≈ TeV.The vacuum value v of H satisfies gap equation δδh S [ h ] = 0 (III.3)Then we get the negative value for the bare mass parameter of the four - fermion interaction: M I = − N c π m t log µ m t (III.4)This negative value M I ≈ − M Z means, that the bare four - fermion interaction at the high energy scale µ ≫ m t ofEq. (II.4) is repulsive. However, the appearance of the vacuum average means, that this repulsive interaction is subjectto finite renormalization: at low energies ≪ µ , where the Higgs boson mass is formed, the renormalized interactionbetween the fermions is attractive. One can see, that there is only one pole of the propagator for the field h . As a resultthe Higgs mass is indeed given by M H ≈ N c m t /N total = m t / ≈
125 GeV
B. W and Z boson masses. The new interaction scale µ . We may consider Eq. (III.1) as the low energy effective action at most quadratic in the scalar field. There existsthe effective action written in terms of the field H . It should contain the λ | H | term in order to provide the nonzerovacuum average h H i = ( v, T . Comparing the coefficients at the different terms of this effective action with that ofEq. (III.1) we arrive at S [ H ] = Z d x (cid:16) H + ( x ) Z h w ( − D ) (cid:16) − D (cid:17) H ( x ) − Z h | H | − v ) (cid:17) (III.5)Here the gauge fields of the SM are taken into account. As a result the usual derivative ∂ of the field H is substitutedby the covariant one D = ∂ + iB with B = B aSU (2) σ a + Y B U (1) (where Y = − is the hypercharge of H ). Functions w, Z h are defined in Appendix. They depend strongly on the particular choice of the form - factors G a a and satisfy theconditions of Eq. (III.2).In order to calculate gauge boson masses we need to substitute into Eq. (III.5) the SU (2) ⊗ U (1) gauge field B instead of iD and ( v, T instead of H . As a result we obtain the effective potential for the field B : V B ≈ m t Z h w ( k B k ) k B k (III.6)Here we denote k B k = (1 , B (1 , T = (cid:16) ( B SU (2) − B U (1) ) + [ B SU (2) ] + [ B SU (2) ] (cid:17) . Up to the terms quadraticin B we get: V (2) B ≈ η k B k (III.7)The renormalized vacuum average of the scalar field η should be equal to ≈ GeV in order to provide theappropriate values of the gauge boson masses. In order to evaluate the gauge boson masses in the given model wecannot neglect nontrivial dependence of w ( k B k ) on B in Eq. (III.6). To demonstrate this let us evaluate the typicalvalue of B that enters Eq. (III.6). Our calculation follows the classical method for the calculation of fluctuations inquantum field theory and statistical theory (see, for example, volume 5, paragraph 146 of [30]). The typical value of B is given by the average fluctuation hk ¯ B k i within the four - volume Ω of the linear size ∼ M Z , where M Z is the Z -boson mass, while k ¯ B k = | Ω | R Ω k B k d x . The direct estimate gives hk B k i ≈ − ∂∂ | Ω | M Z log R dBe −| Ω | M Z k B k = | Ω | M Z ∼ M Z . (We calculate the integral in space - time of Euclidean signature and then rotate back the final result.)As a result in Eq. (III.6) we may substitute w ( M Z ) ≈ instead of w ( k B k ) . This gives S B ≈ η k B k ≈ N total π m t log µ m t k B k k B k ∼ M Z ≈ [90 GeV] (III.8)That’s why we arrive at the following expression for ηη ≈ Z h v = 2 N total π m t log µ m t (III.9)From here we obtain µ ∼ TeV.
C. Branching ratios and the Higgs production cross - sections
The Higgs boson production cross - sections and the decays of the Higgs bosons are described by the effectivelagrangian [27]: L eff = 2 m W η h W + µ W − µ + m Z η h Z µ Z µ + c g α s πη h G aµν G aµν + c γ απη h A µν A µν − c b m b η h ¯ bb − c c m c η h ¯ cc − c τ m τ η h ¯ τ τ. (III.10)Here G µν and A µν are the field strengths of gluon and photon fields, η the conventionally normalized vacuum averageof the scalar field η ≈
246 GeV . This effective lagrangian should be considered at the tree level only and describesthe channels h → gg, γγ, ZZ, W W, ¯ bb, ¯ cc, ¯ τ τ . The fermions and W bosons have been integrated out in the termscorresponding to the decays h → γγ, gg , and their effects are included in the effective couplings c g and c γ (the similarconstants in the SM are given by c g ≃ . , c γ ≈ − . , see [27]). The effective lagrangian Eq. (III.10) is similarto that of the Standard Model. This results from our initial requirement, that the Form - factors G a a ( p, k ) ≪ for | ( p − k ) | ∼ M Z ; | p | , | k | ≪ M Z for all fermions except for the top quark. This reduces considerably the rates ofthe direct decays of the Higgs boson to light fermions. However, unlike the SM the values of c b , c c , c τ may differ fromunity and are given by c b ≈ g ( M H , m b , m b ) /κ D ,c c ≈ g ( M H , m c , m c ) /κ U c b ≈ g ( M H , m τ , m τ ) /κ E (III.11)The similar decay constants may be defined for all SM fermions: c f a a ≈ g ( M H , m f , m f ) /κ a a . We assume, that theseconstants are not much larger than unity, so that the decays into the heavy fermions dominate like in the SM.The consideration of constants c g and c γ is more involved. Those constants contain the fermion loops. Let usconsider for the definiteness the constant c g (the consideration of c γ is similar). The contribution of each fermion(other than the top - quark) is given by the loop integral. The typical value of momentum circulating within the loopis of the order of the largest of the two external parameters: M H and m f . As a result we cannot neglect the form -factor G ( p, k ) ∼ g ( M H , M H , M H ) for the light fermions compared to that of the top quark. Within this integral thereare three fermion propagators and two vertices proportional to γ µ . We arrive at the expression that is proportional to m f that results in δc ( f ) g = m f M H δr ( f ) g . We may estimate quantity δr ( f ) g as follows. It is given by the dimensionlessloop integral. For the definiteness let us consider the expression for the Form - factors given by Eq. (II.9). Weinsert G ( p, k ) ∼ g ( M H , M H , M H ) ≈ into expression for the loop integral. However, while evaluating this loopintegral we are to consider the region of momenta squared bounded from below by M . Besides, we should omit thecontribution of the part of this expression related to the decay of the Higgs boson into the intermediate state composedof two light fermions. Such a contribution is related to the imaginary part of the fermion propagator proportional to δ ( p − m f ) . Therefore, during the evaluation of such a contribution related to the intermediate fermion states on massshell one should substitute the value of the form - factor g ( M H , m f , m f ) ≪ instead of g ( M H , M H , M H ) ≈ .Thus, this contribution is suppressed for the light fermions. The value g ( M H , M H , M H ) ≈ is to be substituted intothe remaining (the main) part of the amplitude, and we should omit the imaginary - valued expressions during thecalculation. For the rough evaluation of r ( f ) g we use the expression for A f ( τ ) given in Eq. (2.5) of [27]. We substituteinto the expression the value τ = M H M instead of τ = M H m f and multiply the resulting expression by the two factors: m t M ≈ M H √ M = 2 √ √ τ (takes into account the vertex g ( M H , M H , M H ) ≈ instead of M /m t ) and √ τ (takesinto account that we deal with δr ( f ) g instead of δc ( f ) g ): δr ( f ) g ≈ √ τ (cid:16) τ − τ − h log (cid:12)(cid:12)(cid:12) p − /τ + 1 p − /τ − (cid:12)(cid:12)(cid:12)i (cid:17) (III.12)As it was explained above, we omitted the imaginary part of the logarithm. One can see, that the value δr g = δr ( f ) g calculated in this way does not depend on the fermion flavor. This value ranges between ∼ − at M = 10 GeV and ∼ at M = 35 GeV. We should notice that this is very rough evaluation that should be considered as the order ofmagnitude estimate only. The output from this estimate is that the expression for the quantity r g depends strongly onthe particular form of the Form - factors, is negative and is of the order of ∼ − . Nevertheless, we feel this instructiveto give the estimate for c g using the particular form of r g of Eq. (III.12). Namely, the contributions of the top - quark,bottom quark and charm quark dominate. The ratio c g /c ( SM ) g (where c ( SM ) g is the SM value) ranges from ∼ − . for M = 10 GeV to ∼ +1 . for M = 35 GeV. We conclude, therefore, that the contribution of the light fermionsdepends strongly on the particular form of the From - factors g a a . However, there exists the possibility, that the valueof the gluon fusion cross - section matches the present experimental constraints (see Fig. 12 of [28] and Fig. 47 of[29]). In particular, the values c g /c ( SM ) g ∼ ± . give the same cross - section for the process gg → h as the StandardModel.The contribution of the light fermions to the decay constant c γ may be roughly estimated in the similar way. Now,the b, c - quarks and τ - lepton dominate, and their contribution is given by δc ( b,c,τ ) γ = (cid:16) / m b + 3(2 / m c + m τ (cid:17) M H δr g . In the Standard Model value c ( SM ) γ the contribution of the W - boson dominates. The modelingexpression for r g of Eq. (III.12) gives the value of c γ /c ( SM ) γ that ranges from ∼ . for M = 10 GeV to ∼ . for M = 35 GeV. One can see, that the constant c γ also depends on the particular form of the Form - factors.There certainly exists the choice of the form - factors such that the resulting expressions for c g , c γ match the presentexperimental constraints given by Fig. 12 of [28] and Fig. 47 of [29]. There is a hint in the present experimentalresults. The best fit to the ratio | c γ /c ( SM ) γ | reported by ATLAS is ∼ . while the best fit reported by CMS is ∼ . .At the same time our rough estimate described above gives this ratio within the interval (1 . , . . The best fit to theratio | c g /c ( SM ) g | reported by ATLAS is ∼ . , the best fit reported by CMS is ∼ . while our estimate gives this ratiowithin the interval (0 . , . .We conclude, that in the model suggested here the branching ratios of Higgs decay into the light fermions, thebranching ratio for the decay into two photons and the production cross - section for the process that goes through thegluon fusion gg → h depend strongly on the particular form of the Form - factors of Eq. (II.5). In the two latter casesthe dependence on the particular form of the Form - factors and the potential deviation from the SM value is the mostdramatic. Nevertheless, in this subsection we demonstrated, that the values of the branching ratios and productioncross - sections can be made matching the present experimental constraints (given, for example, in [28] and [29]) by acertain choice of the Form - factors. IV. CONCLUSIONS AND DISCUSSION
In this paper we suggest, that there exists the hidden strong dynamics behind the formation of the
GeV Higgsboson. According to our scenario it is composed of all known SM fermions. The corresponding interaction has theform of the non - local four - fermion term of Eqs. (II.4), (II.5). The scale of the given interaction between the SMfermions is of the order of µ ≈ TeV. The form - factors entering this term of the action depend on the values ofthe fermion momenta p, k . At large values | ( p − k ) | , | p | , | k | ∼ [90 GeV] all form - factors are equal to unity,and the interaction with real - valued excitation of the Higgs boson becomes identical for all SM fermions. In theopposite limit of small momenta the symmetry is lost, and the form - factors are different for different fermions. Thisallows to obtain the observed hierarchy of masses for quarks and leptons. The Higgs production cross - section andthe branching ratios of the Higgs boson decay depend strongly on the particular form of the Form - factors. The mostdramatic dependence is encoded in the constants c g , c γ of effective lagrangian Eq. (III.10) which are very sensitive tothe new physics. These constants are related to the cross section of the Higgs boson production that goes through thegluon fusion and the branching ratio for the decay of the Higgs boson into two photons. A certain choice of the Form- factors gives the observable branching ratios and production cross - sections that do not deviate significantly fromthe SM values, and the present model matches existing experimental constraints. The further investigation of the LHCdata may give more information and more strong constraints on the experimentally allowed dependence of the Form -factors g a a of Eq. (II.5) on the values of momenta.In our model the condensate of the composite h - boson is responsible for the fermion masses. This distinguishes itfrom the ones, in which the GeV h - boson is responsible for the masses of W and Z while there are the sources ofthe fermion masses different from the h - boson (see, for example, [24–26]). It is worth mentioning, that in the usualmodel of top - quark condensation [7], the interaction scale was extremely high Λ ∼ GeV while in our case theinteraction scale is µ ∼ TeV.In order to define the effective theory with the interaction term of Eqs. (II.4), (II.5) we use zeta regularization thatgives the finite expressions for the considered observables. Bare interaction term at the scale of µ is repulsive. How-ever, the loop corrections provide the appearance of the condensate for the composite scalar field. This distinguishesour effective theory defined using zeta regularization from the models with the four - fermion interaction defined usingthe usual cutoff regularization. Gap equation appears as an extremum condition for the effective action considered as afunction of the field h (represents the excitations of the composite scalar field above its condensate). The value of theHiggs boson mass M H ≈ m t / follows from the symmetry of the interaction term that takes place at small distances(i.e. large momenta | ( p − k ) | , | p | , | q | ∼ M Z ≈ [90 GeV] ). At these values of momenta we have the effectiveinteraction term between the composite Higgs boson and the SM fermions of the form: S I = − V X a X q = p − k h ¯ L a b ( p ) R a U ( k ) H b ( q ) + ¯ L a b ( p ) R a N ( k ) H b ( q ) + ( h.c. ) i (IV.1) − V X a X q = p − k h ¯ L a c ( k ) R a D ( p ) ¯ H b ( q ) ǫ bc + ¯ L a c ( k ) R a E ( p ) ¯ H b ( q ) ǫ bc + ( h.c. ) ii , ( | ( p − k ) | , | p | , | k | ∼ M Z ) All SM fermions enter this interaction term in an equal way. This form of the interaction term points out, that theultraviolet completion of the Standard Model preserves the symmetry of Eq. (IV.1) already at the values of fermionand scalar boson momenta of the order of Electroweak scale M Z ∼ GeV. (Recall, that this implies the lightfermions to be far out of the mass shell.) We suppose, that the appearance of the effective interaction of the form ofEq. (IV.1) may be checked already using the existing data. At least, we may observe the regime of approaching theeffective interaction between the fermions and the Higgs boson to that of Eq. (IV.1).Finally, we would like to remark, that the interaction term of the specific form Eq. (II.4) suggested here may betoo idealized. The actual interaction between the fermions that leads to the formation of their masses may be morecomplicated. However, we expect that if Nature chooses the pattern of the composite Higgs boson suggested here, theinteraction between the composite Higgs boson and the SM fermions should anyway have the form of Eq. (IV.1) atsmall enough distances. This is, probably, the main output of the present paper: we suggest to look carefully at theexisting experimental data in order to confirm or reject the existence of the interaction of the form of Eq. (IV.1).0
Acknowledgements
The author is grateful to V.A.Miransky for useful discussions and careful reading of the manuscript, and toG.E.Volovik, who noticed the relation M H ≈ m t / soon after the discovery of the GeV Higgs boson. The authoris benefited from discussion of the model with Yu. Shylnov, and especially from the discussions with Z.Sullivan of theexperimental constraints on the Higgs boson cross sections and the decay branching ratios. The work is supported bythe Natural Sciences and Engineering Research Council of Canada.
Appendix. Application of zeta - regularization technique to the calculation of various correlators.A. Evaluation of the fermion determinant
Here we use the method for the calculation of various Green functions using zeta - regularization developed in [16].In order to calculate the fermion determinant we perform the rotation to Euclidean space - time. It corresponds to thechange: t → − ix , ¯ ψ → i ¯ ψ, γ → Γ , γ k → i Γ k ( k = 1 , , . (The new gamma - matrices are Euclidean ones.) Theresulting Euclidean functional determinant has the form: Z [ h ] = det h ˆ P Γ + iM + i ˆ G h i (IV.2) = Z D ¯ ψDψ exp (cid:16)Z d x ¯ ψ h ˆ P Γ + i ( M + ˆ G h ) i ψ (cid:17) Here ψ involves all SM fermions, M is the fermion mass matrix that originates from the condensate of h . The operator ˆ G h depends on the functions G ( x, y ) and h ( x ) and is given by Eq. (II.15). The transformation ψ → Γ ψ, ¯ ψ → − ¯ ψ Γ results in Z [ h ] = det h ˆ P Γ + i ( M + ˆ G h ) i = det h ˆ P Γ − i ( M + ˆ G h ) i . Therefore, Z [ h ] = (cid:16) det h ˆ P Γ + i ( M + ˆ G h ) ih ˆ P Γ − i ( M + ˆ G h ) i(cid:17) / (IV.3)We get the fermionic part of the Euclidean effective action: S f [ h ] = −
12 Tr log h ˆ P + M + (2 M ˆ G h + ˆ G h − Γ[ ∂, ˆ G h ]) i (IV.4)We denote A = ˆ P + M , and V = 2 M ˆ G h + ˆ G h − Γ[ ∂, ˆ G h ] . It is worth mentioning, that in momentum space wehave [[ ∂, ˆ G h ] ξ ]( P ) = i X Q = P − K Qξ ( K ) g ( − Q , − P , − K ) h ( Q ) (IV.5)In zeta - regularization we have: S f [ h ] = 12 ∂ s s ) µ s Z dtt s − Tr exp h − ( A + V ) t i (IV.6)At the end of the calculation s is to be set to zero. Here the dimensional parameter µ appears. We identify thisdimensional parameter with the working scale of the interaction that is responsible for the formation of composite h -boson. Further we expand: Tr exp h − ( A + V ) t i = Tr (cid:16) e − At + ( − t ) e − At V + ( − t ) Z due − (1 − u ) At V e − uAt V + ... (cid:17) (IV.7)1 B. One - point function
The part of the effective action that produces the one - point Green function of the field h is equal to S (1) f [ h ] = Z d xh ( x ) C, (IV.8)where C = ∂ s Γ( s + 1)Γ( s ) Tr (cid:16) − M µ s Z d Q (2 π ) g (0 , − Q , − Q )( Q + M ) s +1 (cid:17) = − π ∂ s X a ,a κ a a [ m a a ] − s )+1 s − µ s = 28 π X a ,a κ a a [ m a a ] (log µ [ m a a ] + 1) ≈ − N c π m t (cid:16) − log µ m t (cid:17) (IV.9)where Tr is over the spinor and flavor indices, N c = 3 is the number of colors. In the first row we take into account,that the momentum circulating within the fermion loop is of the order of the fermion mass m a a . On the languageof zeta regularization the interaction scale becomes the parameter entering all expressions to give the dimensionlesscombinations m a a /µ . One can easily check that the typical values of | Q | are indeed of the order of [ m a a ] as follows.We use the standard expression for the loop integral: J = Z d Q (2 π ) µ s ( Q + M ) s +1 = 116 π M (cid:16) µ M (cid:17) s Γ( s − s + 1) (IV.10)The typical value of momentum | Q | = p Q circulating within the loop may be estimated as J ∼ Q µ s ( Q + M ) s +1 ∼ µ s Q − s ∼ M (cid:16) µ M (cid:17) s . From here we derive Q ∼ M . This distinguishes dramatically zeta regularization fromthe usual cutoff regularization, where the typical value of momentum circulating within the divergent fermion loopbecomes of the order of the ultraviolet cutoff instead of the natural internal parameter of the theory (such as the fermionmass).As a result we substitute g (0 , − Q , − Q ) ≈ κ according to our basic supposition about the Formfactors g . C. Two - point Green function in Euclidean region
The part of the effective action that produces the two - point Green functions of the field h is equal to S (2) f [ h ] = V Z d P (2 π ) h ( P )Π( − P ) h ( − P ) , (IV.11)where Π = Π (1) + Π (2) + Π (3) . Below we calculate these three terms separately. In this subsection we shall beinterested in the values of Π( − P ) in the Euclidean region P ∼ M H that correspond to the space - like four -momentum of the Higgs field in space - time of Minkowsky signature. This will be required for the estimate of thevalues of the form - factors g a a in what follows. In the next subsection we shall discuss the analytical continuation ofthe obtained results to the negative values of P that correspond to the time - like four - momenta of the field h . Π (1) corresponds to the term in V proportional to ˆ G h squared. Π (1) ( − P ) = 12 ∂ s Γ( s + 1)Γ( s ) Tr (cid:16) − Z d Q (2 π ) | g ( − P , − Q , − ( P + Q ) ) | ( Q + M ) s +1 µ s (cid:17) ≈ − π ∂ s [ m t ] − s ) s − ≈ π [ m t ] (log µ [ m t ] + 1) (IV.12)2where Tr is over the spinor and flavor indices. Here the top quark dominates the sum, which follows from the propertiesof the Form - factors g because within the integral for the light fermions the typical values of momenta dominate thatare much smaller, than m t . To see this we recall that | g ( − P , − ( P + Q ) , − Q ) | ≤ for each fermion. Thereforethe absolute value of the contribution of each fermion with mass m f is not larger, than ∼ π m f (log µ m f + 1) . Π (2) corresponds to V and to the term in V proportional to − Γ[ ∂, ˆ G h ] . Π (2) ( − P ) = ∂ s s ) Z dtdut s +1 Tr (cid:16) Γ µ Γ ν P µ P ν Z d Q (2 π ) | g a a ( − P , − Q , − ( P + Q ) ) | e − t ( Q u +( P + Q ) (1 − u )+ M ) µ s (cid:17) = ∂ s s ) Z dtdut s +1 X a ,a (cid:16) P Z d Q (2 π ) | g a a ( − P , − ( Q + uP − P ) , − ( Q + uP ) ) | e − t ( P u (1 − u )+ Q +[ m a a ] ) µ s (cid:17) ≈ ∂ s Γ( s + 2)Γ( s ) Z du X a ,a (cid:16) P Z d Q (2 π ) | g a a ( − P , − ( Q + uP − P ) , − ( Q + uP ) ) | ( P u (1 − u ) + Q + [ m a a ] ) s +2 µ s (cid:17) (IV.13)We applied the transformation Q → Q − (1 − u ) P . In the fermion loop in Π (2) the typical values of circulatingmomentum Q are of the order of P . This may be demonstrated in the similar way as done in Sect. IV B. Namely,in the region Q ∼ P ∼ M H also the typical values of ( Q + uP − P ) and ( Q + uP ) are of the order of ∼ M H .Therefore, we substitute g by unity into the integral for light fermions, and have the integral of the form J ( − P ) = Z d Q (2 π ) µ s ( Q + P u (1 − u ) + [ m a a ] ) s +2 ≈ π M µ s (cid:16) P u (1 − u ) + [ m a a ] (cid:17) − s Γ( s )Γ( s + 2) ≈ π M µ s (cid:16) P u (1 − u ) (cid:17) − s Γ( s )Γ( s + 2) (IV.14)Here we omit the fermion masses because for the light fermions m is much smaller, than P ∼ M H . On the otherhand J may be expressed through the typical value of momentum Q as J ∼ µ s Q s . From here we derive the typicalvalue of momentum Q ∼ P u (1 − u ) ∼ P .This justifies the supposition, that all form - factors G may be substituted by unity in the expressionfor Π (2) . The other regions of momenta Q , where the values of the form - factors g are much smallerthan do not contribute essentially to the overall integral. Thus, we substitute g a a ( − P , − ( − Q − uP + P ) , − ( Q + uP ) ) ≈ g ( − M H , − M H , − M H ) ≈ for all fermions. This gives the estimate Π (2) ( − P ) ≈ π P a ,a (cid:16) P R du log µ P u (1 − u )+[ ˜ m a a ] (cid:17) . Here ˜ m U = m t while the other values of mass parameters differ fromthe masses of the light fermions. Instead we should substitute the values of mass parameters such that for | Q | ≫ ˜ m a a we have g a a ( M H , Q , Q ) ≈ while for | Q | ≪ ˜ m a a we have g a a ( M H , Q , Q ) ≪ (the final answer will not dependon the particular values of these mass parameters). Finally, we arrive at Π (2) ( − P ) ≈ N total π P log µ m t (IV.15)The last term corresponds to V and to the contribution of M ˆ G h in V . It is evaluated for any values of P in theway similar to that of Π (1) : Π (3) ( − P ) = ∂ s s ) Z dtdut s +1 Tr (cid:16) M Z d Q (2 π ) | g a a ( − P , − Q , − ( P + Q ) ) | e − t ( Q u +( P + Q ) (1 − u )+ M ) (cid:17) ≈ N c m t π Z du log µ P u (1 − u ) + m t ≈ N c m t π log µ m t (IV.16)3for µ ≫ m t .Total expression for the function Π( − P ) for P ∼ M H and µ ≫ m t has the form: Π( − P ) ≈ ( N total P + 4 N c m t ) 116 π log µ m t + 18 π N c m t (log µ m t + 1) (IV.17)Here m t ≈ GeV is the top - quark mass, N total = 2(3 N c + 3) = 24 is the total number of SM quarks and leptons,while N c = 3 is the number of colors. D. Two - point Green function in space - time of Minkowsky signature
Let us consider the two - point correlation function for the Higgs boson in Minkowsky space - time Π( p ) =Π ′ ( p ) + i Π ′′ ( p ) that depends on the external momentum of the Higgs boson p . It is related to the action inMinkowsky space - time as S = − R d xh ( x )Π(ˆ p ) h ( x ) . This quantity may be represented as the Euclidean loopintegral for the external Euclidean momentum P . For the simplicity we set P = 0 and P = P .The consideration of the parts Π (1) , Π (3) is the most simple one. The top - quark dominates in the resulting expres-sions for all values of P . The result does not depend on momentum and is given by Eq. (IV.12) and Eq. (IV.16).In consideration of Π (2) we notice, that for P ∼ M H the fermion Euclidean momenta Q, P + Q with ( Q + P ) ∼ Q ∼ P dominate. Then we substitute into the loop integral g a a ( − P , − Q , − ( P + Q ) ) = 1 . The resulting real -valued expression Π (2) ( − P ) depends on the external Euclidean momentum P > . However, we shall be interestedin the complex - valued expression for negative values P = − p ∼ − M H . In order to construct the analyticalcontinuation of the expression Π (2) ( − P ) with P ∼ + M H to P ∼ − M H we should choose the line in the complexplane of P that connects points M H and − iM H , and along which the analytical continuation is to be performed.We choose the sector of the circle with | P | = M H . While moving along this line for P ∼ e iφ M H we rotate in theintegral Q → e iφ | Q | . It occurs, that after this rotation the values of the momenta | Q | ∼ M H (and as a consequence | P + Q | ∼ M H ) remain dominant in the real part Π (2) ′ of Π (2) , then we may simply substitute into the approximateanalytical Euclidean expression Π (2) ( − P ) (calculated with g a a ( − P , − Q , − ( P + Q ) ) = 1 ) the value P ∼ − M H ,take its real part and obtain the needed expression for Π (2) ′ ( p ) with p ∼ M H in space of Minkowsky signature. Inthe limit, when the new strong interaction scale given by µ is much larger, than all fermion masses we combine thethree terms Π (1) , Π (2) , Π (3) and arrive at: Π ′ ( p ) ≈ − ( p − m t N c N total ) Z h + N c π m t log µ m t for | p | ∼ M H (IV.18)with Z h ≈ N total π log µ m t (IV.19)We may check ourselves and estimate expression for Π( p ) for the form - factors of the form of Eq. (II.8). Thedominant contribution to Π ′ is given by Eq. (IV.18) while the sub - dominant terms are given by the residues atthe positions of the poles of Eq. (II.8). The resulting sub - dominant contributions are suppressed by the ratios ∼ p λ a a M I /M H ≪ .For the imaginary part Π ′′ of Π the situation is different: say, for iP = M H − i in the integral over Q for Π ′′ thevalues Q = − m f and ( Q + P ) = − m f dominate (this corresponds to the decay of the Higgs boson into the pairfermion - antifermion with masses m f and - momenta q = ( p + q ) = + m f ). As a result in order to calculate theimaginary part of Π( p ) we cannot simply substitute P ∼ − M H into the Euclidean expression. The consideration ismore involved and we are to substitute g a a ( M H , m f , m f ) instead of g a a ( − P , − Q , − ( P + Q ) ) = 1 . This gives Π ′′ ( p ) ≈ π X a ,a ( p − m a a ] ) | g a a ( M H , [ m a a ] , [ m a a ] ) | θ ( p − m a a ] ) (IV.20)4Since all values κ a a ≪ for all fermions except for the top quark, we may neglect the imaginary part of Π at p < m t .Notice, that the result for the one - point function obtained above as well as the expressions for Π (1) , Π (3) are validfor all possible values of the momentum of the field h . At the same time the term proportional to p in Eq. (IV.18)(that originates from Π (3) ) is valid for the values of momenta of the order of the Higgs boson mass | p | ∼ M H . Inorder to obtain the expression for the effective action of the Higgs boson field h for all values of momenta p we needto modify this expression. The local effective action for the field h will result in the following expression for Π( p ) valid for all values of momentum p : Π( p ) ≈ − Z h ( w ( p ) p − M H ) + N c π m t log µ m t , (IV.21)where Z h = N total π log µ m t , while w ( p ) ≈ for | p | ∼ M H , and M H ≈ N c m t /N total (IV.22)We may easily estimate the values of w ( p ) for | p | ≪ M H . The expressions for Π (1) and Π (3) remain the samewhile the expression for Π (2) is given by the integral considered in the previous subsection with the typical values Q ≪ M H instead of Q ∼ M H . That’s why we substitute g a a (0 , , ≈ κ a a instead of g a a ( − P , − Q , − ( P + Q ) ) =1 . This leads to w ( p ) ≈ N c N total = 1 / | p | ≪ M Z The expression of w ( p ) for the intermediate values of momenta depends on the particular form of the form - factors g . 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