Dynamical Symmetry Breaking and Negative Cosmological Constant
aa r X i v : . [ h e p - ph ] S e p September 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree
International Journal of Modern Physics Ac (cid:13)
World Scientific Publishing Company
Dynamical Symmetry Breaking and Negative Cosmological Constant
Wei Lu
Manhasset, NY 11030, [email protected]
Received Day Month YearRevised Day Month YearIn the context of Clifford functional integral formalism, we revisit the Nambu-Jona-Lasinio-type dynamical symmetry breaking model and examine the properties of thedynamically generated composite bosons. Given that the model with 4-fermion interac-tions is nonrenormalizable in the traditional sense, the aim is to gain insight into thedivergent integrals without resorting to explicit regularization. We impose a restrictionon the linearly divergent primitive integrals, thus resolving the long-standing issue ofmomentum routing ambiguity associated with fermion-antifermion condensations. Theremoval of the ambiguity paves the way for the possible calculation of the true ratio ofHiggs boson mass to top quark mass in the top condensation model. In this paper, we alsoinvestigate the negative vacuum energy resulted from dynamical symmetry breaking andits cosmological implications. In the framework of modified Einstein-Cartan gravity, it isdemonstrated that the late-time acceleration is driven by a novel way of embedding theHubble parameter into the Friedmann equation via an interpolation function, whereasthe dynamically generated negative cosmological constant only plays a minor role for thecurrent epoch. Two cosmic scenarios are proposed, with one of which suggesting that theuniverse may have been evolving from an everlasting coasting state towards the acceler-ating era characterized by the deceleration parameter approaching -0.5 at low redshift.One inevitable outcome of the modified Friedmannian cosmology is that the directlymeasured local Hubble parameter should in general be larger than the Hubble parame-ter calibrated from the conventional Friedmann equation. This Hubble tension becomesmore pronounced when the Hubble parameter is comparable or less than a characteristicHubble scale.
Keywords : Clifford functional integral; dynamical symmetry breaking; vacuum energy;modified Friedmannian cosmologyPACS numbers:11.30.Qc, 12.60.Rc, 04.50.Kd, 95.36.+x
1. Introduction
The discovery of the Higgs boson
1, 2 has renewed the interest in possible explana-tions for the electroweak naturalness problem: the perturbative quantum correctionstend to draw the mass of a fundamental Higgs boson towards higher scales. Oneway of dealing with the naturalness problem is to regard the Higgs sector as aneffective description of the low energy physics represented by a composite boson. Itis conjectured that the fundamental Higgs boson can be replaced by a dynamically eptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree Wei Lu generated composite boson as the result of fermion-antifermion condensation. Thetop condensation model has been extensively studied along this line of thinking,motivated by the proximity of top quark mass scale and the electroweak symmetrybreaking scale.One of the challenges facing the top condensation model (for a review see Ref. 7)is to account for the vast range of fermion masses which span five orders of mag-nitude. Dimensionless ratios between parameters appearing in a physical theorycannot be accidentally small. This naturalness principle is elegantly defined by ’tHooft: a quantity should be small only if the underlying theory becomes more sym-metric as that quantity tends to zero. Weakly broken symmetry ensures that thesmallness of a parameter is preserved against possible perturbative disturbances.With a view toward explaining the fermion mass hierarchies in the context of com-posite electroweak Higgs bosons, we proposed the extended top condensation modelin our previous work. In addition to top quark condensation, the model involvestau neutrino and tau lepton condensations as well. The approach is based on theframework of Clifford algebra Cℓ , (note that there is an interesting connectionbetween Clifford algebra C l (6) and octonions via left-action maps ), wherebystandard model fermions are represented by algebraic spinors. There are two globalchiral symmetries on top of the local gauge symmetries. The chiral symmetries aredynamically broken by 4-fermion condensations. In accordance with the naturalnessprinciple, the chiral symmetries play a pivotal role in establishing the relative mag-nitudes of 4-fermion condensations, and consequently giving rise to fermion masshierarchies.That being said, there are still subjects related to the top condensation modelthat have not been explored in our earlier paper. Particularly, for compositeHiggs boson models with nonrenormalizable 4-fermion interactions, there is a long-standing issue of the momentum routing ambiguity associated with the fermionbubble diagram. When a Feynman integral is convergent or logarithmically diver-gent, the integral is independent of the momentum routing parameter, because theparameter can be shifted away by a translation of the integration variable. Whenit comes to integrals that are more than logarithmically divergent, one should pro-ceed with caution because the seemingly harmless momentum shifting changes theintegral values. One quintessential example is the triangle diagrams of the Adler-Bell-Jackiw (ABJ) anomaly,
13, 14 where the integrals are linearly divergent. Theambiguity is fixed by enforcing the vector Ward identity, at the expense of the axialWard identity.The Feynman integral corresponding to the fermion bubble diagram is quadrat-ically divergent
12, 15 and one shall seek a different way of removing the momentumrouting ambiguity. This paper is an attempt of addressing this issue without go-ing through explicit regularization. Via straightforward algebraic manipulations,we can show that momentum shifting changes a quadratically divergent integral by,eptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree
Symmetry breaking and cosmological constant amongst others, I µlin = Z d l (2 π ) l µ ( l − m ) . (1)This primitive integral is independent of external momentum. Since the integrandof I µlin is odd in l , one might suppose that it vanishes upon symmetrical integration.Considering that I µlin is linearly divergent, we can no longer make the cavalier as-sumption that I µlin can be discarded. Analogous to the situation of the ABJ anomaly,the surface terms could spoil the identity I µlin = 0. The solution to this issue is torequire that the physical outcome of a model should not depend on the ill-definednon-Lorentz-invariant primitive integrals which are more than logarithmically di-vergent. In other words, integrals like I µlin should not show up in the final calculationresults. This condition pins down the value of the momentum routing parameter,thus removing the routing ambiguity.With the ambiguity out of the way, we turn to the other aspects of the compositeboson model and its cosmological implications. It is argued that the bare fermionmass and the bare cosmological constant Lagrangian terms are prohibited by in-voking global symmetries. The dynamically generated effective masses and negativevacuum energy could therefore be naturally small, due to the protection from weaklybroken symmetries. The vacuum energy-related quartically divergent primitive in-tegral is deemed as a separate parameter, in addition to the logarithmically andquadratically divergent primitive integrals. As a consequence, the vacuum energycould be decoupled from the emergent mass scale. It’s also shown that the late-timecosmic acceleration can be realized in the framework of modified Friedmannian cos-mology, even in the presence of a small negative cosmological constant arisingfrom dynamical symmetry breaking.The present paper is in a sense a continuation of our previous research on the ex-tended top condensation model. Rather than analyzing the dynamical electroweaksymmetry breaking in its full extent, we will investigate the basic Nambu-Jona-Lasinio-type (NJL-type) model. The goal is to contemplate the viability and cos-mological consequences of a minimal dynamical symmetry breaking model withnonrenormalizable 4-fermion interactions. We hope that the lessons learned herecould shed some light on the future study of more sophisticated models such as theextended top condensation model.This paper is structured as follows: Section 2 introduces the 4-fermion interac-tion model and Clifford functional integral formalism. In section 3, we study diver-gent integrals, the momentum routing ambiguity, and the negative vacuum energy.In section 4, we examine the cosmological implications of a negative cosmologicalconstant based on the modified Friedmann equation, and explore the possibilityof time-varying emergent quantities. In the last section we draw our conclusions.Throughout this paper, we adopt the Planck units: c = ~ = G = 1.eptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree Wei Lu
2. The Dynamical Symmetry Breaking Model with 4-fermionInteractions2.1.
Clifford algebra and the fermion Lagrangian
Encouraged by the successful employment of Clifford algebra Cℓ , in the extendedtop condensation model, we are going to use Clifford algebra extensively in thispaper. Clifford algebra, also known as geometric algebra, is a powerful mathematicaltool with various applications in physics. Instead of Cℓ , , here we will focuson the familiar Cℓ , , which is also called Dirac algebra or spacetime algebra. TheClifford algebra Cℓ , is defined by the vector basis { γ µ ; µ = 0 , , , } satisfying γ µ γ ν + γ ν γ µ = 2 η µν , (2)where η µν = diag (1 , − , − , − ψ ( x ) is a linear combination of all 2 = 16 basis elements ofClifford algebra Cℓ , ψ ( x ) = ψ ( x ) + ψ µ ( x ) γ µ + 12 ψ µν ( x ) γ µ γ ν + ψ µ ( x ) iγ µ + ψ ( x ) i, (3)where ψ µν ( x ) = − ψ νµ ( x ) and the unit pseudoscalar i = γ γ γ γ squares to −
1, anticommutes with Clifford-odd elements, and commutes withClifford-even elements. Due to the fermion nature, the 16 linear combination coeffi-cients such as ψ ( x ), ψ µ ( x ), etc. are real Grassmann numbers. It is worth noting thatthe algebraic spinor as expressed in eq. 3 should not be confused with a bispinor,which is effectively bosonic and can be also expanded in terms of the 16 elementsof Cℓ , . The interested readers shall refer to Refs. 17–22 and especially section 4.1in Ref. 23 for detailed expositions on the mapping between an algebraic spinor anda conventional column spinor.Spinors with left (right) chirality correspond to Clifford-odd (even) multivectors ψ = ψ L + ψ R ,ψ L = 12 ( ψ + iψi ) ,ψ R = 12 ( ψ − iψi ) . According to the conventional column representation of fermions, each Diracfermion has 4 components which correspond to 4 complex or 8 real Grassmannnumbers. So there is an issue of reconciling the column spinor with the algebraicspinor endowed with N = 16 degrees of real Grassmann freedom. This issue isresolved by the seminal paper by Hestenes. It is pointed out that the algebraicspinor of Cℓ , can be identified with the neutrino-electron isospin doublet. Theeptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree Symmetry breaking and cosmological constant spinor ψ represents a pair of orthogonal left ideals ψ = ψ ν + ψ e , where ψ ν ( ψ e ) corresponds to the isospin up (down) projection of ψψ ν = ψ γ γ ,ψ e = ψ − γ γ . The algebraic spinor of Cℓ , is capable of accommodating Lagrangians which areboth Lorentz invariant and electroweak gauge invariant (see Ref. 24 for details).Here we are concerned with the NJL-type model with 4-fermion interactions.The fermion Lagrangian can be written as L = ˆ i (cid:10) ¯ ψ /∂ψ (cid:11) − N g (cid:0) (cid:10) i ¯ ψψ (cid:11) + (cid:10) i ¯ ψiψ (cid:11) (cid:1) , (4)where /∂ = γ µ ∂ µ (we adopt the summation convention for repeated indices in thispaper), γ µ = η µν γ ν , N = 16, g is the 4-fermion coupling constant , h . . . i stands forClifford-scalar part of the enclosed expression, and the Dirac conjugate ¯ ψ is definedas ¯ ψ = ψ † γ . Hermitian conjugate ψ † takes the form ψ † = γ ˜ ψγ , (5)where reversion of ψ , denoted ˜ ψ , reverses the order in any product of Clifford vectors.Note that the Hermitian conjugate in eq. (5) is defined specifically for Cliffordalgebra Cℓ , . For Clifford algebra Cℓ , , the Hermitian conjugate would assume adifferent definition.
9, 10
In the context of connecting Clifford algebra with the con-ventional matrix formalism, the interested readers are encouraged to consult section5.1 in Ref. 23 for a general and enlightening discussion concerning the distinctionbetween reversion (which acts on Clifford algebra valued objects) and Hermitianconjugate (which acts on the matrices representing Clifford numbers).The kinetic term ˆ i (cid:10) ¯ ψ /∂ψ (cid:11) involves the mathematical imaginary number ˆ i , whichis different from Clifford algebra Cℓ , pseudoscalar i . The imaginary number ˆ i commutes with all Clifford algebra elements. The ˆ i in the kinetic term is consistentwith the fact that a self-energy loop diagram would yield an imaginary correctionto the fermion propagator, since loop integrals would pick up an extra ˆ i via propercontour integral on the complex plane (or equivalently Wick rotation of time axis).Note that ˆ i does not show up in the multi-fermion interaction term, and the samegoes for Yang-Mills and gravitational interactions.The massless Lagrangian is invariant under vector U V (1) global transformation ψ L ⇒ ψ L e αi = e − αi ψ L , (6a) ψ R ⇒ ψ R e αi = e αi ψ R , (6b)eptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree Wei Lu and axial U A (1) global transformation ψ L ⇒ e βi ψ L = ψ L e − βi , (7a) ψ R ⇒ e βi ψ R = ψ R e βi . (7b)The individual Lagrangian terms (cid:10) i ¯ ψψ (cid:11) and (cid:10) i ¯ ψiψ (cid:11) are not invariant underthe axial transformation, albeit they are invariant in aggregation. If we add a baremass term m ˆ i (cid:10) i ¯ ψψ (cid:11) to the Lagrangian, the axial symmetry would be spoiled. The crux of the dynamicalsymmetry breaking mechanism is to induce an axial-symmetry-breaking effectivemass term via interactions.Note that we can write down an NJL-type Lagrangian involving electron ψ e orneutrino ψ ν only. The outcome of the forthcoming sections will not be qualitativelydifferent. All one has to do is to change the value of N from N = 16 to N = 8,where N is the degrees of real Grassmann freedom. And for that matter, we shallregard ψ as a generic spinor not necessarily tied to a specific fermion. For example, ψ could be top lepton, whose fermion-antifermion condensation is of concern in theextended top condensation model. Clifford functional integral and Schwinger-Dyson equation
To quantize the Lagrangian (4) in the framework of Clifford functional integralformalism, we have to leverage Clifford functional calculus. For our purpose here,the spinor ψ ( x ) in (3) is re-expressed as ψ ( x ) = ψ a ( x )ˆ γ a , (8)where index a runs from 1 to N , with N = 16, and each ψ a ( x ) is real-Grassmannvalued. The operators ˆ γ a span all the 16 basis elements of Clifford algebra Cℓ , ˆ γ = 1 , ˆ γ = γ , ˆ γ = γ , · · · , ˆ γ = i. We are interested in a Clifford functional derivative δ/δψ ( x ) suitable for thespinor ψ ( x ) defined in (8), δδψ ( x ) ≡ ˆ γ a δδψ a ( x ) , (9)where ˆ γ a is the inverse of ˆ γ a so that ˆ γ a ˆ γ a = 1 for each a , and δ/δψ a ( x ) followsthe standard definition of Grassmann functional derivative. Note that we stick withthe convention of always applying the Clifford functional derivative to the left ofa functional. In the same vein, the Clifford functional derivative δ/δ ¯ ψ ( x ) against¯ ψ ( x ) can also be defined.eptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree Symmetry breaking and cosmological constant There are a few useful functional derivative properties: δδψ ( x ) ψ ( y ) = N δ ( x − y ) ,δδψ ( x ) iψ ( y ) = 0 ,δδψ ( x ) h ψ ( y ) F i = δδψ ( x ) D ˙ ψ ( y ) F E + δδψ ( x ) D ψ ( y ) ˙ F E = δ ( x − y ) F + δδψ ( x ) D ψ ( y ) ˙ F E , where F is any Clifford functional and the dot on ˙ ψ ( y ) or ˙ F denotes functionalderivative performed on the designated element only. The second property stemsfrom the fact that the Clifford-odd portion and the Clifford-even portion of thederivative cancel out.Now we are ready for the quantization of the Lagrangian (4). The generatingfunctional Z [ η ] can be represented as the Clifford functional integral Z [ η ] = Z D ψe ˆ i R d x L ( x )+ R d xd y h ¯ η ( x ) ψ ( x ) i h η ( y ) ¯ ψ ( y ) i . (10)It is understood that Z [ η ] is required to be normalized to Z [0] = 1. The Grassmann-odd sources η ( x ) and ¯ η ( x ) are valued in the same Clifford space as ψ ( x ) and ¯ ψ ( x ).Hence their respective Clifford functional derivatives δ/δη ( x ) and δ/δ ¯ η ( x ) can bedefined in the same fashion.The combination of ¯ η ( x ) and η ( y ) in the source term h ¯ η ( x ) ψ ( x ) i (cid:10) η ( y ) ¯ ψ ( y ) (cid:11) should be deemed as a bilocal union. Therefore, η and ¯ η , and for that matter func-tional derivatives δ/δη and δ/δ ¯ η , should always appear in pairs. It’s worth men-tioning that bilocal sources have been employed by two-particle irreducible (2PI)effective actions and approximation schemes to go beyond the standard pertur-bative quantum field theory.Rather than treating ψ ( x ) and ¯ ψ ( x ) as independent variables as in many text-books, we regard them as dependent variables. The same logic applies to η ( x ) and¯ η ( x ). The extra 1 / Z D ψ δδψ ( x ) F = 0 , (11a) Z D ψ δδ ¯ ψ ( x ) F = 0 , (11b)where F is any functional.eptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree Wei Lu
A specific application of the property (11), Z D ψ δδ ¯ ψ ( x ) [ e ˆ i R d z L ( z )+ R d zd z ′ h ¯ η ( z ) ψ ( z ) i h η ( z ′ ) ¯ ψ ( z ′ ) i ¯ ψ ( y )] = 0 , yields the Schwinger-Dyson (SD) equation N δ ( x − y ) Z [ η ] = /∂ x δδ ¯ η ( x ) δδη ( y ) Z [ η ] − ˆ i gN h δδ ¯ η ( x ) i δδη ( y ) (cid:28) δδ ¯ η ( x ) i δδη ( x ) Z [ η ] (cid:29) + i δδ ¯ η ( x ) i δδη ( y ) (cid:28) i δδ ¯ η ( x ) i δδη ( x ) Z [ η ] (cid:29) i + ε Z d zη ( x ) δδη ( y ) (cid:28) ¯ η ( z ) δδ ¯ η ( z ) Z [ η ] (cid:29) , (12)where ε = 1 in the source term is a dummy parameter. It’s for book keeping purposewhen we seek an approximate solution in the next subsection.As mentioned earlier, functional derivatives δ/δη and δ/δ ¯ η should always showup in couples. In the above SD equation, it’s understood that the closest ones arepaired up. Bilocal source approximation
The SD equation (12) is an exact functional-differential equation. In the presenceof interactions, solving the SD equation is notoriously hard. The path well troddenis to find a perturbative solution, under the assumption that a certain couplingconstant is small. Here we follow a non-perturbative iterative scheme dubbed asbilocal source approximation,
26, 27 which effectively treats ε in the bilocal sourceterm (the last term in the SD equation (12)) as a series expansion parameter sothat Z = Z + Z ε + Z ε + · · · . (13)The equation of the zeroth-order approximation is N δ ( x − y ) Z = /∂ x δδ ¯ η ( x ) δδη ( y ) Z − ˆ i gN h δδ ¯ η ( x ) i δδη ( y ) (cid:28) δδ ¯ η ( x ) i δδη ( x ) Z (cid:29) + i δδ ¯ η ( x ) i δδη ( y ) (cid:28) i δδ ¯ η ( x ) i δδη ( x ) Z (cid:29) i . (14)It’s equivalent to the self-consistent Hartree mean-field approximation.The solution to the zeroth-order equation is readily obtained as (normalized to Z [0] = 1) Z [ η ] = e − R d p (2 π )4 h i ¯ η ( p ) S ( p ) η ( p ) i , (15)eptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree Symmetry breaking and cosmological constant where η ( p ) = R d xη ( x ) e ip · x and p · x = p µ x µ . The fermion Feynman propagator S ( p ) is given by a S ( p ) = 1 /p − m + ˆ iǫ , (16)where /p = p µ γ µ , and the dynamically generated mass m satisfies the gap equation m = ˆ ig Z d p (2 π ) h S ( p ) i = ˆ ig Z d p (2 π ) mp − m . (17)In this paper, we concentrate on the case where the nonzero mass solution isenergetically favored over the zero mass one (more on the dynamically generatednegative vacuum energy in the next section). The existence of fermion-antifermioncondensation is reflected in the nonzero value of R d p (2 π ) h S ( p ) i . The emergent massdynamically breaks the axial global symmetry (7). It’s tantamount to adding aneffective mass term m ˆ i (cid:10) i ¯ ψψ (cid:11) to the fermion Lagrangian (4).Note that the zeroth-order functional equation allows for a “complex” mass(scalar plus pseudoscalar) m = | m | e ϑi = | m | cos( ϑ ) + i | m | sin( ϑ ) , which appears in the fermion propagator (16). Nevertheless, it can be rotated intoto a real mass via redefining the spinor (a U A (1) transformation (7)) ψ ⇒ e − ϑ i ψ. So without loss of generality, we will focus on real mass only.With the goal of studying the bosonic bound state properties of fermion-antifermion condensation, we need to go beyond the zeroth-order approximationand turn to the functional equation of the first-order approximation
N δ ( x − y ) Z = /∂ x δδ ¯ η ( x ) δδη ( y ) Z − ˆ i gN h δδ ¯ η ( x ) i δδη ( y ) (cid:28) δδ ¯ η ( x ) i δδη ( x ) Z (cid:29) + i δδ ¯ η ( x ) i δδη ( y ) (cid:28) i δδ ¯ η ( x ) i δδη ( x ) Z (cid:29) i + Z d zη ( x ) δδη ( y ) (cid:28) ¯ η ( z ) δδ ¯ η ( z ) Z (cid:29) . (18) a The Lorentz-invariant Feynman propagator depends on the proper contour integral on the com-plex plane prescribed by ˆ iǫ . For the rest of the paper, we do not explicitly write down ˆ iǫ inpropagators for the sake of brevity. eptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree Wei Lu
According to the above equation, the first-order generating functional can becalculated as b Z [ η ] = Z [ η ] n Z d l (2 π ) h i ¯ η ( l ) S ( l ) η ( l ) i Z d l ′ (2 π ) h i ¯ η ( l ′ ) S ( l ′ ) η ( l ′ ) i + ˆ i Z d p (2 π ) D s ( p ) h Z d l (2 π ) h S s i i + ˆ i Z d p (2 π ) D p ( p ) h Z d l (2 π ) h S p i i + · · · o , (19a)where S s = i ¯ η ( l − p S ( l − p S ( l + p η ( l + p S p = i ¯ η ( l − p S ( l − p iS ( l + p η ( l + p · · · terms are related to the first-order corrections to the fermion propagator S ( p ), which we will not elaborate in this paper.The effective composite boson propagators c D s ( p ) and D p ( p ) in the scalar andpseudoscalar channels are D s ( p ) = 1 N g − − Π s ( p ) , (20a) D p ( p ) = 1 N g − − Π p ( p ) , (20b)where the bubble functions Π s ( p ) and Π p ( p ) in the scalar and pseudoscalar channelsare Π s ( p ) = ˆ i Z d l (2 π ) h S ( l + p ) S ( l ) i , (21a)Π p ( p ) = ˆ i Z d l (2 π ) h iS ( l + p ) iS ( l ) i , (21b)which correspond to the fermion bubble diagram in the context of fermion-antifermion condensation.
12, 15
The scalar and pseudoscalar propagators D s ( p ) and D p ( p ) are re-summations toinfinite order chains of fermion bubble diagrams. Similar leading order calculationgoes by different names such as random-phase approximation, Bethe-Salpeter T-matrix equation, or 1/N expansion. The collective modes of the composite bosonscan be determined by the poles of D s ( p ) and D p ( p ). b As an only exception to the rule of pairing up closest η and ¯ η , ¯ η ( l ) and η ( l ′ ) (¯ η ( l ′ ) and η ( l )) arepaired up in eq. (19a). c They are generally known as fermion-antifermion channel T-matrices. We call them effectivecomposite boson propagators conditional on the existence of fermion-antifermion condensation. eptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree
Symmetry breaking and cosmological constant The integrals in the gap equation (17) and eq. (21) are quadratically divergent.The analysis of divergent integrals is the focus of the next section.
3. Divergent Integrals3.1.
Implicit regularization
The calculations of quantum field theory are plagued by divergent integrals, whichneed to be regularized at intermediate steps. After renormalization, a finite andregularization-scheme independent result can be obtained for renormalizable theo-ries. The fermion contact interactions render the NJL-type model nonrenormaliz-able. Unlike the renormalizable theories, an NJL-type model depends on the formof regularization chosen, hence the regularization procedure is regarded as an in-tegral part of the definition of the model. In the literature, the NJL model hasbeen presented with many schemes: non-covariant 3-momentum cutoff, covariant 4-momentum cutoff in Euclidean space, and proper time regularization, among others(see Ref. 28 for a review).In this paper, we will adopt the technique of implicit regularization.
29, 30
Thebelow identity 1( l + p ) − m = 1( l − m ) − p + 2 p · l ( l − m ) [( l + p ) − m ] , (22)is applied repeatedly to the integrand of a divergent integral so that the divergentparts are isolated in primitive integrals that are independent of external physicalmomentum (i.e. p ). Only the convergent integrals are allowed to involve externalmomentum in the denominator. Because the convergent integrals are separated fromthe divergent ones, the finite parts can be integrated free from effects of regulariza-tion.On the other hand, granted that the divergent primitive integrals are indepen-dent of external momentum, they can be treated as finite quantities as a resultof unspecified regularization. It will be shown later that there are three Lorentz-invariant divergent integrals I log , I quad , and I quar . They are regarded as free param-eters of the model that shall be fixed (or constrained, see details in later subsections)by physical quantities, such as the dynamically generated fermion mass, compositeboson mass, and vacuum energy. Explicit regularization is therefore bypassed.As explained in the introduction section, we exercise extra caution while han-dling non-Lorentz-invariant divergent primitive integrals. We diverge from the stan-dard implicit regularization approach
29, 30 when linearly or more than linearly di-vergent non-Lorentz-invariant integrals (such as I µlin in eq. (1)) are concerned. Wedeem these integrals as ill-defined which should not appear in the final physicalresults, as opposed to throwing some away based on symmetrical integration argu-ment or stipulating the others via enforcing the consistency conditions (equivalentto discarding surface terms).eptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree Wei Lu
Gap equation
For the NJL-type model with 4-fermion interactions, the first Lorentz-invariantprimitive integral we encounter is I quad = Z i d l (2 π ) m − l , (23)which appears in the gap equation (17). This quadratically divergent integral isindependent of external momentum. Note that I quad is real and positive, since thecontour integral (with implicit regularization) on the complex plane (or equivalentlyWick rotation of time axis) would pick up an extra imaginary number ˆ i , cancelingout the ˆ i in the denominator.In the context of the conventional regularization scheme with a cutoff scale M ,the gap equation implies that the coupling constant g has to be fine-tuned to avalue just slightly larger than a critical value g crit ∼ M − , in order to establishthe hierarchy between M and the much smaller fermion mass m ≪ M . However,given that no explicit regularization is required in our calculation, the notion ofcutoff scale M or critical coupling g crit ( M ) is of no relevance here. We take theview that one is agnostic of the values of the coupling constant g and integral I quad .Individually, they are left as arbitrary, insofar as they jointly meet the condition gI quad = 1 , (24)which is dictated by the gap equation (17).In accordance with ’t Hooft’s technical naturalness principle, once the gapequation is satisfied for a symmetry breaking scale of order m , it’s ensured thatthe smallness of m is preserved against possible higher order disturbances, due tothe protection from the weakly broken axial symmetry. There is no need for furtherfine-tuning. Momentum routing ambiguity and composite boson mass
Another quadratically divergent integral concerns the bubble functions Π s ( p ) andΠ p ( p ) in eq. (21) corresponding to the fermion bubble diagram. To underscore theambiguity in momentum routing, we re-express the bubble functions Π s ( p ) andΠ p ( p ) in the scalar and pseudoscalar channels asΠ s ( p ) = ˆ i Z d l (2 π ) h S ( l + (1 − α ) p ) S ( l − αp ) i , (25a)Π p ( p ) = ˆ i Z d l (2 π ) h iS ( l + (1 − α ) p ) iS ( l − αp ) i , (25b)where α is an arbitrary parameter not determined by the theory. Unlike the case of convergent or logarithmically divergent integrals, the seem-ingly innocuous momentum shifting changes the integral values which are linearly ormore than linearly divergent. The central question is whether the momentum rout-ing parameter α is a spurious artifact arising from the approximation scheme or it’september 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree Symmetry breaking and cosmological constant indeed a phenomenologically consequential parameter that needs to be ascertainedone way or another.Let’s begin with straightforward algebraic manipulation of Π p ( p ) in the pseu-doscalar channelΠ p ( p )=ˆ i Z d l (2 π ) * i /l + (1 − α ) /p − m i /l − α/p − m + =ˆ i Z d l (2 π ) ( l + (1 − α ) p ) · ( l − αp ) − m [( l + (1 − α ) p ) − m ][( l − αp ) − m ]= 12ˆ i Z d l (2 π ) n l − αp ) − m + 1( l + (1 − α ) p ) − m − p [( l + p ) − m ][ l − m ] o . Note that we have shifted momentum routing in the last term since it’s allowedfor a logarithmically divergent integral. Now we apply the identity (22) repeatedlyto the integrands above, so that divergent integrals are isolated in terms that areindependent of external momentum p . Collecting all the terms, we getΠ p ( p ) = I quad + (2 α − iI µlin p µ + I log p + 1 − α + 2 α π p + · · · , (26)where · · · stands for finite integrals of order O ( p ) (and up) and will be neglectedhereafter. The quadratically divergent primitive integral I quad is given by (23) andit satisfies the condition (24). The linearly divergent primitive integral I µlin is givenby (1). The logarithmically divergent primitive integral is defined as I log = Z i d l (2 π ) m − l ) . (27)As discussed in the introduction section, despite the fact that the integrandof I µlin is odd in l , I µlin may not necessarily vanish upon symmetrical integration.It’s imperative that the physical outcome of a model should not depend on theill-defined non-Lorentz-invariant primitive integrals like I µlin which are more thanlogarithmically divergent. In other words, Π p ( p ) should depend on Lorentz-invariant p only. Therefore, we are compelled to impose the condition(2 α − iI µlin p µ = 0 . (28)This condition fixes the value of the α parameter to α = 12 , thus removing the momentum routing ambiguity. With that, the composite bosonpropagator D p ( p ) in the pseudoscalar channel is calculated as (utilizing the gapeptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree Wei Lu equation as well) D p ( p ) ≈ − N ( I log + π ) p = − NI log (1 + ∆) p , (29)where ∆ = 164 π I log . The term (64 π ) − p stems from the symmetric momentum routing (given that α = 1 /
2) of ˆ i Z d l (2 π ) n l − p/ − m + 1( l + p/ − m o = 164 π p + 2ˆ i Z d l (2 π ) l − m . Historically, such an extra term (proportional to p ) is either treated as ambiguous or totally absent in most NJL-type calculations. Of particular note is that thepseudoscalar composite boson mass m p is still zero, unchanged by the extra term,so that the Nambu-Goldstone requirement is satisfied.In the same fashion, the composite boson propagator D s ( p ) in the scalar channelcan be deduced as D s ( p ) ≈ − NI log [(1 + ∆) p − m ] . (30)In the presence of the extra term ∆ p (stemming from the momentum routing fixedat α = 1 / m . The ratiobetween the scalar composite boson mass m b and the fermion mass m is given by m b m ≈ √ , (31)therefore the composite boson mass m b is less than 2 m . Similar to the case ofthe fermion mass m , the dynamically generated composite boson mass (which isconstrained by m b < m at the level of approximation) is also protected by theweakly broken axial symmetry, because both masses are contingent on dynamicalsymmetry breaking.We take the view that the model’s predictability depends on the emergentquantities conditional on dynamical symmetry breaking. The measurements of thefermion mass m and the scalar composite boson mass m b fix the parameters of themodel. Unlike the quadratically divergent counterpart I quad , the logarithmically di-vergent primitive integral I log is dimensionless and can be inferred from the ratio m b /m . The integral I log is thus the only determinable divergent primitive integral,whereas we only have partial knowledge of the dimensionful integral I quad via theconstraint (24).When it comes to the top quark condensation model, one phenomenologicalproblem is related to the conventional prediction of the Higgs-top mass ratio. Sinceeptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree Symmetry breaking and cosmological constant the 2012 discovery, the Higgs boson is known to be lighter than the top quark. Thetop condensation model appears to fail since it gives too heavy Higgs mass comparedwith top quark mass. After factoring in the effects from the standard model gaugeinteractions and the non-leading-order corrections, the issue accompanying the topcondensation model is alleviated but remains unresolved. If we take into accountthe momentum routing-related contributions along the lines presented in this paper,the updated calculation may possibly lead to a Higgs-top mass ratio that matcheswith measurements. Dynamically generated vacuum energy
As long as quantum field theory is concerned, the absolute value of vacuum energyis normally irrelevant, because it’s only the difference of energies that matters.However, gravity is coupled to the energy of the vacuum. The vacuum energy’scontribution to the cosmological constant Λ leads to measurable effects.For a fundamental Higgs boson, the electroweak symmetry breaking gives riseto a vacuum energy of the order ρ vac ∼ υ , where υ is the electroweak scale. It is exorbitantly large (10 times too large)compared with the commonly accepted estimation of Λ. The cosmological constantproblem is perceived as the most severe problem in physics (see Ref. 31 for a review).In the context of dynamical symmetry breaking induced by multi-fermion in-teractions, one might wonder whether the vacuum energy’s contribution to thecosmological constant is more tractable. Before answering this question, we wouldlike to present an argument against a bare cosmological constant Lagrangian term.Let’s introduce the basic building blocks of the curved spacetime in terms of Lorentzgauge theory of gravity (see Refs. 32, 33 for reviews), e = e a γ a = e aµ dx µ γ a , (32a) ω = 14 ω ab γ ab = 14 ω abµ dx µ γ ab , (32b)where vierbein e and spin connection ω are Clifford-valued 1-forms, ω abµ = − ω baµ ,and γ ab = ( γ a γ b − γ b γ a ) /
2. The covariant derivative of the spinor field ψ ( x ) isdefined by Dψ = ( d + ω ) ψ, (33)where the spin connection ω is essential in maintaining the local Lorentz covarianceof Dψ . The spin connection curvature 2-form is expressed as R = dω + ω = 14 R ab γ ab = 14 ( dω ab + η cd ω ac ω db ) γ ab , (34)where outer products between differential forms are implicitly assumed.eptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree Wei Lu
The spinor kinetic, 4-fermion interaction, gravity, and bare cosmological constantLagrangian terms are of the forms L spinor − kinetic ∼ ˆ i (cid:10) ¯ ψie Dψ (cid:11) , (35a) L − fermion ∼ (cid:10) ie (cid:11) ( (cid:10) i ¯ ψψ (cid:11) + (cid:10) i ¯ ψiψ (cid:11) ) , (35b) L gravity ∼ (cid:10) ie R (cid:11) , (35c) L bare − CC ∼ (cid:10) ie (cid:11) . (35d)We propose a global transformation d e ⇒ e θ ˆ i e, (36a) ω ⇒ e − θ ˆ i ω, (36b) d ⇒ e − θ ˆ i d, (36c) ψ ⇒ e − θ ˆ i ψ, ¯ ψ ⇒ e − θ ˆ i ¯ ψ. (36d)Note that ¯ ψ transforms in the same way as ψ , since Dirac conjugate has no bearingon the imaginary number ˆ i . It follows that the Lagrangian terms of L spinor − kinetic , L − fermion , and L gravity are invariant, whereas L bare − CC transforms as L bare − CC ⇒ e θ ˆ i L bare − CC . If we enforce the θ transformation symmetry, then the bare cosmological constantterm would be precluded. Therefore, we only allow an effective cosmological con-stant, which arises from the 4-fermion term (35b) conditional on dynamical sym-metry breaking of the θ symmetry.For the flat spacetime fermion Lagrangian (4), the vacuum potential energy canbe calculated as the expectation value of the 4-fermion interaction term V = Z D ψ N g ( (cid:10) i ¯ ψψ (cid:11) + (cid:10) i ¯ ψiψ (cid:11) ) e ˆ i R d x L ( x ) ≈ N g (cid:16) (cid:28) δδ ¯ η ( x ) i δδη ( y ) (cid:28) δδ ¯ η ( x ) i δδη ( x ) Z (cid:29)(cid:29) + (cid:28) i δδ ¯ η ( x ) i δδη ( y ) (cid:28) i δδ ¯ η ( x ) i δδη ( x ) Z (cid:29)(cid:29) (cid:17)(cid:12)(cid:12)(cid:12) η ( x )=0 = − N m gI quar , (37)where the zeroth-order generating function Z ( η ) (15) is utilized. The quarticallydivergent primitive integral I quar = Z Z i d l (2 π ) i d l (2 π ) m − l )( m − l ) , (38) d One might define “vierbein dimension” as [ e ] = 1 and [ ω ] = [ d ] = [ ψ ] = [ ¯ ψ ] = −
1. Then the barecosmological constant Lagrangian term is “dimensionfull”, whereas the other Lagrangian termsare “dimensionless” . eptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree
Symmetry breaking and cosmological constant is real and positive. It’s the third primitive integral, in addition to the logarithmi-cally and quadratically divergent ones I log and I quad . According to the conventionalwisdom, one might erroneously expect that I quar = I quad . (39)The underlying assumption is that the order of integration d l and d l does notmake a difference and one can integrate individually. This sort of reasoning is prob-lematic for quartically divergent integrals, since the change of integral (and/or dif-ferential) order is generally not permitted for integrals that are more than logarith-mically divergent.As a rule of thumb, relationships involving multiplication of multiple diver-gent integrals such as eq. (39) should be avoided. Instead, the integral I quar shallbe treated as an independent quantity, unrelated to I quad or I log . Consequently,the presumably small value of vacuum energy can be detached from the emergentfermion/boson mass scale. Note that this small value is preserved against possibleperturbations, thanks to the protection from the weakly broken θ symmetry (36).We know that there could be a couple of condensations in the extended topcondensation model. And there are also quark-antiquark condensations (manifestedas mesons) induced by the strong interaction in QCD. Similar to the situation of thequadratically divergent counterpart, we are agnostic of the individual magnitudeof each coupling constant g k (one can only determine the ratios between g k ) oreach integral I quar ( m k ) (associated with the dynamically generated fermion mass m k ). They are left as arbitrary, albeit in aggregation they are constrained by thedynamically induced cosmological constantΛ = 8 π X k V k = − πN X k m k g k I quar ( m k ) . (40)In accordance with expectation, the vacuum with dynamical symmetry breakingis energetically favored over a symmetric one, since a negative effective vacuumpotential (37) is generated.
4. Cosmology with a Negative Cosmological Constant4.1.
Einstein-Cartan equations and Friedmannian cosmology
As we learned from last section, dynamical symmetry breaking yields an effectivecosmological constant which could be sufficiently small. However, a disturbing factis that it’s negative, hence of the “wrong sign”. It is widely believed that a smalland positive cosmological constant can account for the observation that the expan-sion of the universe is accelerating.
34, 35
This picture of the ΛCDM model is basedon Einstein’s theory of general relativity. In this subsection, we will briefly walkthrough the derivations of gravity equations and the Friedmannian cosmology. Inlater subsections, we will discuss how to reconcile a negative cosmological constantwith the accelerated expansion of the universe.eptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree Wei Lu
As mentioned earlier, gravity can be formulated as a Lorentz gauge theory interms of the vierbein (or tetrad/co-frame) e and the spin connection ω (32). Thegauge approach to gravity is also known as Einstein-Cartan gravity. The spin con-nection ω , associated with the local Lorentz group SO(1, 3) e , plays the role of thegauge fields in Yang-Mills theory.The Einstein-Cartan action of gravity (with a nonzero cosmological constant Λ)is of the form S gravity = 18 π Z (cid:28) i ( e R − Λ4! e ) (cid:29) , (41)where R = dω + ω is the spin connection curvature 2-form (34).Field equations are obtained by varying the total action (gravity plus matter)with the fields e and ω independently. The resultant Einstein-Cartan equations read18 π ( Re + eR − Λ3! e ) = T i, (42a)18 π ( T e − eT ) = 12 S i, (42b)where T is energy-momentum current 3-form, S is spin current 3-form, and T istorsion 2-form T = de + ωe + eω = T a γ a = ( de a + η bc ω ab e c ) γ a . (43)When the spin-current S is zero, the second Einstein-Cartan equation (42)amounts to enforcing the zero-torsion condition T = de + ωe + eω = 0 , (44)which can be used to express the spin connection ω in terms of the vierbein e . In thiscase, the remaining (first) Einstein-Cartan equation can be shown to be equivalentto the ordinary Einstein field equations for gravity with a cosmological constant.The reason of taking this detour to Einstein-Cartan gravity will become clearin later subsections. For now, let’s turn to cosmology. On the large scale, the spa-tially homogeneous and isotropic universe is depicted by the Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) metric ds = dt − a ( t ) (cid:18) dr − kr + r d Ω (cid:19) , (45)where Ω = dθ + sin θdφ and a ( t ) is the scale factor of the universe normalizedto a ( t ) = 1 at present day t . The constant curvature k takes the value k = 0, k >
0, or k < e In fact, the gauge group is the double cover of Lorentz group, namely Spin(1, 3). eptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree
Symmetry breaking and cosmological constant the zero-torsion condition), H ≡ (cid:18) ˙ aa (cid:19) = 8 π ρ + 13 Λ − ka , (46)where H is the Hubble parameter, ρ is the energy density, and dot stands for cosmictime derivative.The Friedmann equation shall be supplemented with the cosmological equationof state, which implies that the energy densities of the different constituents of theuniverse ρ = P w ρ w scale with a ( t ) as ρ w ( a ) ∼ a − w ) , (47)where the equation of state parameter w is 0 for non-relativistic matter (includingvisible matter and cold dark matter) and 1 / w = − − / k terms, respectively. Note that the cosmological constant is nowadays termed darkenergy,
36, 37 given that its value may evolve with cosmic time.As per the scaling equation (47), the cosmological constant Λ shall eventuallydominate over the other decaying components of the universe. Therefore, we arepersuaded of the need for a positive Λ, which is capable of driving the late-timecosmic acceleration by virtue of its negative pressure. Such a narrative of Friedman-nian cosmology hinges on the accuracy of general relativity. In the next subsection,however, we will investigate a challenge to Newton’s Law of gravitation and generalrelativity. See Ref. 38 for a review of a wide range of problems with the ΛCDMmodel and general relativity.
Modified Newtonian dynamics
Newton’s Law of gravitation, as the non-relativistic weak-field limit of general rel-ativity, is contradicted by the observation that the visible matter of spiral galax-ies cannot possibly account for the gravitational pull responsible for the galacticrotation curves. Extra dark matter is thus postulated to make up for the massdiscrepancy.Curiously, the deviation from Newton’s Law of gravitation only occurs when theacceleration is below a universal scale (in Planck units) a ≈ − . This phenomenon, which is observed in a vast array of galaxies, would necessitatedramatic fine-tuning of the dark matter distribution.As an alternative to the dark matter hypothesis, a modification of Newtoniandynamics (MOND) is proposed to link the Newtonian acceleration a N from thevisible matter to the true acceleration a µ (cid:18) aa (cid:19) a = a N , (48)eptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree Wei Lu where µ ( x ) is an interpolation function µ ( x ) → x ≫ µ ( x ) → x for x ≪ . (49)In the weak acceleration limit ( a ≪ a ) of MOND, an object would circulate aroundmass M with the velocity v = a M, which is independent of the radius. This result of MOND agrees well with theobserved behavior in galaxies known as the baryonic Tully-Fisher relation. In light of MOND theory’s success on the galactic scale, one might wonderwhat’s MOND theory’s implication for the ΛCDM model. To this end, we have togo beyond the non-relativistic theory of MOND. Most of the theoretical attempts(see Ref. 41 for a review) concentrate on modifying the Einstein field equations,or equivalently modifying the first Einstein-Cartan equation. In an earlier paper of ours, we took the road less traveled by: changing the second Einstein-Cartanequation, which amounts to altering the zero-torsion condition in the absence ofspin current.Let’s rephrase Newtonian gravity as the non-relativistic weak-field limit of thefirst and second Einstein-Cartan equations ∂ i ω i = 4 πρ, (50a) ∂ i e − ω i = 0 , (50b)where the spin current S and cosmological constant Λ terms are assumed to bezero, and ρ is mass density. In the parlance of Newtonian gravity, the Newtoniangravitational acceleration a N is a iN = − ∂ i V N ≡ − ∂ i e = − ω i , where the Newtonian gravitational potential V N satisfies ∇ V N = 4 πρ. Now let’s add one term to the second equation of (50) ∂ i e − (1 + p a /a N ) ω i = 0 , (51)where a N = q ( ω i ω i ) . (52)The additional term p a /a N ω i is negligible if a N ≫ a . In other words, Newtoniangravity is recovered in the limit a N ≫ a . On the other hand, this additional termbecomes consequential when a N is comparable or less than a . One can write downthe relationship between the Newtonian gravitational acceleration a iN = − ω i andthe true gravitational acceleration a i = − ∂ i e as(1 + p a /a N ) a N = a . (53)eptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree Symmetry breaking and cosmological constant Expressing a N in terms of a via inverting the above equation, we arrive at theMOND equation (48), with the interpolation function µ ( x ) defined in (49). There-fore, MOND is the result of modifying the second Einstein-Cartan equation in itsnon-relativistic weak-field limit. Modified Friedmannian cosmology
To introduce a relativistic counterpart of the modification along the lines of thelast subsection, we realize that one has to break the local Lorentz gauge symmetrywhile retaining the local spacial rotation symmetry. Hence we resort to the followingpartial vierbeins and partial spin connection e S = e j γ j = e jµ dx µ γ j , (54a) e T = e γ = e µ dx µ γ , (54b) ω T = 14 ( ω j γ j + ω j γ j ) = 12 ω j µ dx µ γ j , (54c)where j = 1 , ,
3. These fields have the preferable property of transforming like avector under local spacial rotations.The relativistic version of the modified second Einstein-Cartan equation can bewritten as π ( ˜ T e − e ˜ T ) = 12 S i. (55)The modified torsion 2-form ˜ T is defined by˜ T = T + ∆ T Schw + ∆ T F LRW , (56)where T is the original torsion 2-form (43). The additional terms ∆ T Schw and∆ T F LRW are ∆ T Schw = p a / | z Schw | ( ω T e S + e S ω T ) , (57a)∆ T F LRW = p h / | z F LRW | ( ω T e T + e T ω T ) , (57b)where z Schw = 4! e ( ω T e S + e S ω T )2 e ,z F LRW = 4! e ( ω T e T + e T ω T )2 e , and the magnitude of a multivector | M | is defined as | M | = q h M † M i . The modified torsion breaks the Lorentz symmetry, while it is covariant underspacial rotations. In the absence of spin current S , which is the case studied in thecurrent paper, the modified second Einstein-Cartan equation yields T + (∆ T Schw + ∆ T F LRW ) = 0 , (58)eptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree Wei Lu which means a change to the usual zero-torsion condition T = 0, since ∆ T Schw +∆ T F LRW is non-zero in general.The first term of the torsion modification ∆ T Schw (57a) is relevant for theSchwarzschild metric. The factor of p a / | z Schw | in eq. (57) reduces to p a /a N in the non-relativistic weak-field limit, hence eq. (51) is recovered from eq. (58).Therefore, the modification is indeed a relativistic parent theory of MOND.The interesting part of the torsion modification is the second term∆ T F LRW (57b), which bears striking resemblance to the first term ∆ T Schw (57a).One only has to replace e S and a with e T and 3 h , respectively. Unlike ∆ T Schw , itturns out that ∆ T F LRW is actually relevant for the FLRW metric and h is a char-acteristic Hubble scale, which is a new parameter independent of the characteristicMOND acceleration scale a . That being said, it is assumed that h is of the sameorder of a (in Planck units) h ≈ − . The torsion modification terms ∆ T Schw (57a) and ∆ T F LRW (57b) can be phe-nomenologically viewed as “dark torsion”. They might also be interpreted as “darkspin current” (if the modification terms are moved to the right-hand side of themodified Einstein-Cartan equation (55)), as an alternative to dark matter.With the help of the FLRW metric (45), one can show that the modified secondEinstein-Cartan equation (55), in conjunction with the original first Einstein-Cartanequation (42a), leads to the modified Friedmann equation H F = 8 π ρ + 13 Λ − ka , (59a)where (1 + r h H F ) H F = H. (59b)Here H F is the Friedmann Hubble parameter and H ≡ ˙ a/a is the true Hubbleparameter. The Friedmann Hubble parameter H F can be determined via eq. (59b)as H F = µ (cid:18) Hh (cid:19) H, (60)with the interpolation function specified in eq. (49). Consequently, the modifiedFriedmann equation can be reformulated as (cid:20) µ (cid:18) Hh (cid:19) H (cid:21) = 8 π ρ + 13 Λ − ka . (61)The modified Friedmann equation (61) parallels the MOND equation (48) in thesense that the former replaces Hubble parameter H with µ ( H/h ) H in Friedmannequation, while the latter replaces acceleration a with µ ( a/a ) a in Newtoniandynamics.eptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree Symmetry breaking and cosmological constant According to this modified Friedmannian cosmology (MFC), the characteristicHubble scale h marks the boundary between the validity domains of Friedmanniancosmology and MFC. For large Hubble parameter H ≫ h (the Friedmannianregime), one has H F ≈ H . Therefore, the modified Friedmann equation (61) isreduced to the usual Friedmann equation (46). On the other hand, in the limit ofsmall Hubble parameter H ≪ h (the deep MFC regime), H F is given by H F ≈ Hh H. The modified Friedmann equation then reads1 h (cid:18) ˙ aa (cid:19) = 8 π ρ + 13 Λ − ka , (62)which departs from the conventional Friedmannian cosmology.Given the relationship (59b) (equivalently the interpolation relationship (60)),the Friedmann Hubble parameter H F is generally smaller than the Hubble param-eter H . If one attempts to calibrate the Hubble parameter via the conventionalFriedmann equation (46), it’s actually H F that is inferred, which differs from thetrue Hubble parameter H = ˙ a/a . This discrepancy could be manifested in the “ H tension” between CMB-predicted value of the Hubble parameter in concert withΛCDM (corresponding to H F ( t )) and the local measurements from supernovae (which prefer a higher value, corresponding to H = H ( t )).Considering that MOND negates the need for dark matter in galactic systems, anatural question would be whether MFC, as a relativistic parent theory of MOND,can explain the cosmological mass discrepancies without invoking cold dark matter(CDM). Our hypothesis is that MFC could potentially reduce CDM’s percentagein the total mass-energy budget of the universe. Nevertheless, we leave open thepossibility that there might still be some remaining CDM constituents. The primecandidates are the 4-fermion condensations (which are electroweak singlets) in theextended top condensation model and the sterile (right-handed) neutrinos whichare endowed with seesaw-scale (believed to be much higher than the electroweakscale) Majorana masses. MFC scenario one: coping with negative cosmological constant
Now we are ready for the reconciliation of the dynamically generated negative cos-mological constant Λ with the accelerated expansion of the universe in the frame-work of MFC. Let’s study a flat universe ( k = 0) after the radiation-dominatedepoch. The predominant mass-energy density components are thus matter and anegative cosmological constant Λ <
0. These components can be rewritten as8 π ρ + 13 Λ = ¯ ρ ( a − − λ ) . (63)where ρ includes both visible and dark matter, and the negative Λ is reparametrizedby the positive parameter λ = − ρ Λ >
0. We assume that the magnitude of theeptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree Wei Lu dynamically generated Λ (40) is considerably smaller than the mass density of thecurrent epoch, hence λ ≪ . The modified Friedmann equation (61) can be cast in the form˙ a + V ( a ) = 0 , (64)where for H ≫ h (the Friedmannian regime) V ( a ) = − ¯ ρ ( a − − λa ) , (65)and for H ≪ h (the deep MFC regime) V ( a ) = − h √ ¯ ρ p ( a − λa ) . (66)This reformulation benefits from the Newtonian interpretation of a mass m = 2(with kinetic energy T = m ˙ a / a ) moving in the potential V ( a ) subject to theconstraint of zero total energy.Given that the Hubble parameter is large ( H ≫ h ) at the early stage of themass-dominated epoch, potential (65) implies that (neglecting the small λ term)˙ a ∼ t − , H ≈ t − , q ≈ , where q = − ¨ aa ˙ a is the deceleration parameter. The positive value of q = 1 / H ∼ t − even-tually enters the regime H ≪ h . Consequently, according to (66), we have (ne-glecting the small λ term)˙ a ∼ t , H ≈ t − , q ≈ − . The universe is therefore accelerating, characterized by the negative q ≈ − /
4. Wegenerally believe that the Hubble parameter of the current epoch is of the value H = H ( t ) . h . Thus we have already entered the MFC regime manifested by the late-time cosmicacceleration.That being said, the small and negative cosmological constant will in the endcatch up with the declining matter density and become significant in the far fu-ture. The acceleration is going to give way to deceleration at the critical scale a crit determined by dV ( a ) da | a = a crit = 0 , eptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree Symmetry breaking and cosmological constant which yields a crit = 2 − λ − . (67)Further down the road, the expansion of the universe will grind to a halt at themaximum scale a max derived from V ( a ) | a = a max = 0 , which implies a max = λ − ≈ . a crit . (68)After reaching this maximum scale a max , the universe will trace back and embarkon contraction with a decreasing scale factor. Note that both a crit and a max aredetermined using the deep MFC version of potential (66). MFC scenario two: coasting universe yielding to acceleration
As mentioned in earlier section, according to ’t Hooft’s technical naturalness prin-cipal, the small scales of the dynamically generated masses and vacuum energycould be protected by the weakly broken global symmetries. Nevertheless, technicalnaturalness does not answer the question of why a physical quantity is small inthe first place. The stronger naturalness in the sense of Dirac requires that thereshall be no unexplained large (or small) numbers in nature. Dirac suggested, in hislarge number hypothesis, that very large (or small) dimensionless ratios should beconsidered as variable parameters pertaining to the state of the universe.Inspired by Dirac’s hypothesis, we propose the notion of Planck naturalness: anemergent quantity, which varies with cosmic time, shall be of order 1 at Planck time t p = 1 and its current large (or small) value is determined by the age of the universe t ≈ (in Planck units).Two points should be clarified regarding Planck naturalness. First of all, wewill not speculate about the universe before Planck time t p , which belongs to therealm of quantum gravity. And from an effective field theory point of view, aninfinite number of terms allowed by symmetry requirements should be included ina generalized action of the world. The gravity and Yang-Mills actions are the firstfew order terms that are relevant in the low-energy limit. Therefore, Planck time t p is the furthest point at which we may give limited credence to the low-energyeffective field theory.Secondly, by emergent quantities we mean dynamically generated quantities,such as the fermion and composite boson masses, along with the negative vacuumenergy. In comparison, the Planck units (speed of light c , reduced Planck constant ~ ,and gravitational constant G ) are fixed. As such, the Planck units are the standardyardsticks to measure the variability of the emergent dimensionful quantities.eptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree Wei Lu
Rather than dwelling upon individual emergent quantities, we will focus on thetotal mass-energy density ¯ ρ tot = 8 π ρ + 13 Λ − ka , which is the sum of the right hand side of the modified Friedmann equation (61).That is to say, we regard every constituent of ¯ ρ tot as emergent and time-varying.Note that ¯ ρ tot is assumed to be positive, albeit the contribution from the dynami-cally generated time-varying Λ (dark energy) is negative and the contribution fromthe curvature term could be either positive or negative depending on the sign of k .Let’s suppose that the time-varying mass, radiation, and dark energy densitiessomehow work in concert, so that the aggregation of them (plus a presumably non-zero curvature term) collectively scales with a as¯ ρ tot = 1 t a = 1¯ a , (69)where ¯ a = t a is the re-scaled scale factor. The equation of state parameter w for¯ ρ tot is − /
3. This sort of cosmic fluid was first proposed in Ref. 45 and the termK-matter has been coined, because the w = − / − k/a (with a negative k ) in the empty Milne universe.If we for a moment forget about MFC and assume that the conventional Fried-mann equation (46) are more or less accurate for the K-matter universe, we wouldarrive at ¯ a = t, ¯ ρ tot = t − , H = t − , q = 0 . (70)This coasting universe solution, characterized by the deceleration parameter q = 0,has the appealing property of satisfying Planck naturalness: the dependence of¯ ρ tot ( t ) on time does not involve additional large (or small) parameter and it isof the order ¯ ρ tot ( t p ) = t − p = 1 at Planck time.Recent years have witnessed a renewed interest in this w = − / Ht = 1 . (71)For the current epoch, we do have approximately H t ≈ t ). In the context of ΛCDM,the equality is an uncanny coincidence because it is only true at present time. Forthe w = − / w = − / that the opposite sides of the cosmoshave remained causally connected to each other from the very first moments of theuniverse.Notwithstanding its merits, the eternally coasting universe is disfavored at lowredshift. The fact that the universe is currently in a phase of accelerated expansionhas been firmly established from the observations of supernovae,
34, 35 which indicateseptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree
Symmetry breaking and cosmological constant that the present value of the deceleration parameter q ( t ) = q is markedly less thanzero q <
0, contrary to the claim of q = 0 for the eternally coasting universe.Now let’s suppose that the universe has entered the MFC regime (i.e. H . h )at low redshift , whereas the w = − / h (cid:18) ˙¯ a ¯ a (cid:19) = 1¯ a , (72)which leads to ¯ a ∼ t , ¯ ρ tot ∼ t − , H ≈ t − , q ≈ − . (73)Hence at low redshift, the deceleration parameter q should have been evolving fromthe coasting q = 0 towards the accelerating q = − /
2. This unique behavior of q isdistinguishable from either the ΛCDM model or the eternally coasting model, thusit could be verified via further data analysis.Moreover, the Hubble parameter has been transitioning from H = t − (theFriedmannian regime) to H ≈ t − (the deep MFC regime), which suggests that2 > H t >
1, rather than H t = 1. The inequality of t > /H implies thatthe commonly quoted cosmic age (approximately 1 /H ) might be an underestima-tion. In this regard, we would like to draw attention to the observation that 9extremely old globular clusters are older than the widely accepted cosmic age.As a last note, we shall mention that there is an unexplained coincidence: thecharacteristic Hubble scale h of MFC and the characteristic acceleration scale a of MOND are of the same order as 1 /t h ≈ a ≈ t ≈ − . The underlying reason for this coincidence is unknown. One possibility is that h and a are emergent and time-varying as well. But we just leave it at that, withoutfurther elaborating the implications.
5. Conclusions
In the context of Clifford functional integral formalism, we revisit the NJL-typemodel with nonrenormalizable 4-fermion interactions. The model leads to a bosonicbound state as the result of the fermion-antifermion condensation. The goal ofthe paper is to gain insight into the divergent integrals without going throughexplicit regularization. It is argued that the physical outcome of the model shouldnot depend on the ill-defined non-Lorentz-invariant primitive integrals which aremore than logarithmically divergent. This condition removes the momentum routingambiguity associated with the fermion-antifermion condensation. A resultant extraterm shifts the pole of the scalar bosonic channel away from 4 m , where m is thedynamically generated fermion mass.eptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree Wei Lu
We present an alternative view on nonrenormalizable models conditional on dy-namical symmetry breaking: a model’s predictability shall depend on the emergentquantities such as the fermion and boson masses along with the dynamically gen-erated negative vacuum energy, whereas the absolute magnitude of the couplingconstant is not measurable. We argue that the bare fermion mass and the bare cos-mological constant Lagrangian terms are prohibited by enforcing the global axialand θ symmetries. The dynamically generated effective masses and vacuum energycould thus be protected by the weakly broken symmetries. The vacuum energy-related quartically divergent primitive integral is regarded as a separate parameterof the model independent of the quadratically and logarithmically divergent prim-itive integrals. The vacuum energy is therefore decoupled from the emergent massscale.In the presence of a small negative cosmological constant arising from dynamicalsymmetry breaking, it’s demonstrated that the late-time cosmic acceleration can beaccommodated in the framework of modified Friedmannian cosmology (MFC). MFCis originated from modifying the second Einstein-Cartan equation, which amounts toaltering the zero-torsion condition. A characteristic Hubble scale h demarcates theboundary between the validity domains of the Friedmannian cosmology and MFC.One prediction of MFC is that the Hubble parameter calibrated from the conven-tional Friedmann equation is in general smaller than the local Hubble parameterinferred from supernovae observations. This “Hubble tension” is more pronouncedwhen the Hubble parameter is comparable or less than the characteristic Hubblescale h .We propose two cosmic evolution scenarios, with one of which based on thepremise that the total mass-energy density of the the universe may behave like theK-matter characterized by the equation of state parameter w = − /
3. It follows thatthe universe could be coasting with a linearly increasing scale factor for the cosmicera when the Friedmannian cosmology is applicable. It has the appealing feature ofsatisfying Planck naturalness: the total mass-energy density is of order 1 at Plancktime. Our view is that the universe may have already entered the MFC regime( H . h ). Consequently, the deceleration parameter q should have been evolvingfrom the coasting q = 0 towards the accelerating q = − / Acknowledgments
I am grateful to Matej Pavsic and Ziqi Yan for helpful correspondences.
References
1. ATLAS Collaboration, (G. Aad et al. ), Phys. Lett. B , 1 (2012).2. CMS Collaboration, (S. Chatrchyan et al. ), Phys. Lett. B , 30 (2012).3. Y. Nambu, in
New Theories in Physics, Proceedings of the XI International Symposium eptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree
Symmetry breaking and cosmological constant on Elementary Particle Physics , eds. Z. Ajduk, S. Pokorski, and A. Trautman (WorldScientific, Singapore, 1988), p. 1.4. V. A. Miransky, M. Tanabashi and K. Yamawaki, Phys. Lett. B , 177 (1989).5. W. J. Marciano,
Phys. Rev. Lett. , 2793 (1989).6. W. A. Bardeen, C. T. Hill and M. Lindner, Phys. Rev. D , 1647 (1990).7. G. Cvetic, Rev. Mod. Phys. , 513 (1999).8. G. ’t Hooft, NATO Adv. Study Inst. Ser. B Phys. , 135 (1980).9. W. Lu, Int. J. Mod. Phys. A , 1750159 (2017).10. W. Lu, Adv. Appl. Clifford Algebras , 145 (2011).11. C. Furey, Phys. Lett. B , 84 (2018).12. R. Willey,
Phys. Rev. D , 2877 (1993).13. S. L. Adler, Phys. Rev. , 2426 (1969).14. J. S. Bell and R. Jackiw,
Il Nuovo Cimento A , 47 (1969).15. Y. Nambu and G. Jona-Lasinio, Phys. Rev. , 345 (1961).16. W. Lu,
Modified Einstein-Cartan gravity and its implications for cosmology ,arXiv:1406.7555 [gr-qc].17. D. Hestenes,
Space-Time Algebra , (Gordon and Breach, New York, 1966).18. D. Hestenes and G. Sobczyk,
Clifford algebra to geometric calculus: a unified languagefor mathematics and physics , (Kluwer Academic Publishers, Dordrecht, 1984).19. M. Pavsic,
The Landscape of Theoretical Physics: A Global View. From Point Particlesto the Brane World and Beyond, in Search of a Unifying Principle , (Kluwer AcademicPublishers, Dordrecht, 2001).20. P. Lounesto,
Clifford algebras and spinors , (Cambridge University Press, Cambridge,2001).21. C. Doran and A. Lasenby,
Geometric Algebra for Physicists , (Cambridge UniversityPress, Cambridge, 2003).22. J. Vaz Jr. and R. da Rocha Jr.,
An Introduction to Clifford Algebras and Spinors ,(Oxford University Press, Oxford, 2016).23. M. Pavsic,
Int. J. Mod. Phys. A , 5905 (2006).24. D. Hestenes, Found. Phys. , 153 (1982).25. J. M. Cornwall, R. Jackiw, and E. Tomboulis, Phys. Rev. D , 2428 (1974).26. V. E. Rochev, J. Phys. A , 3671 (1997).27. R. G. Jafarov and V. E. Rochev, Central Eur. J. Phys. , 367 (2004).28. S. P. Klevansky, Rev. Mod. Phys. , 649 (1992).29. O. A. Battistel, A. L. Mota, and M. C. Nemes, Mod. Phys. Lett. A , 1597 (1998).30. O. A. Battistel, G. Dallabona, and G. Krein, Phys. Rev. D , 065025 (2008).31. S. Weinberg, Rev. Mod. Phys. , 1 (1989).32. F. W. Hehl, P. Von Der Heyde, G. D. Kerlick and J. M. Nester, Rev. Mod. Phys. ,393 (1976).33. A. Randono, Gauge gravity: a forward-looking introduction , arXiv:1010.5822 [gr-qc].34. Supernova Search Team, (A. G. Riess et al. ), Astron. J. , 1009 (1998).35. Supernova Cosmology Project, (S. Chatrchyan et al. ), Astrophys. J. , 565 (1999).36. P. J. E. Peebles and B. Ratra,
Rev. Mod. Phys. , 559 (2003).37. E. J. Copeland, M. Sami, and S. Tsujikawa, Int. J. Mod. Phys. D , 1753 (2006).38. P. Bull et al. , Phys. Dark Univ. , 56 (2016).39. M. Milgrom, Astrophys. J. , 365 (1983).40. R. B. Tully and J. R. Fisher,
Astron. Astrophys. , 661 (1977).41. B. Famaey and S. McGaugh, Living Rev. Rel. , 10 (2012).42. A. G. Riess et al. , Astrophys. J. , 56 (2016).43. P. A. R. Ade et al. , Astron. Astrophys. , A13 (2016). eptember 17, 2019 1:5 WSPC/INSTRUCTION FILE AmbiguityFree Wei Lu
44. P. A. M. Dirac,
Nature , 323 (1937).45. E. W. Kolb,
Astrophys. J. , 543 (1989).46. F. Melia and A. Shevchuk,
Mon. Not. Roy. Astron. Soc. , 2579 (2012).47. J. Ma et al. , Astron. J. , 4884 (2009).48. S. Wang et al. , Astron. J. , 1438 (2010).49. S. Wang, X. -D. Li, and M. Li,
Phys. Rev. D , 103006 (2010).50. F. Melia, Astron. Astrophys.553