Standard Electroweak Interactions in Gravitational Theory with Chameleon Field and Torsion
aa r X i v : . [ g r- q c ] A p r Standard Electroweak Interactionsin Gravitational Theory with Chameleon Field and Torsion
A. N. Ivanov ∗ and M. Wellenzohn
1, 2, † Atominstitut, Technische Universit¨at Wien, Stadionallee 2, A-1020 Wien, Austria FH Campus Wien, University of Applied Sciences, Favoritenstraße 226, 1100 Wien, Austria (Dated: April 23, 2015)We propose a version of a gravitational theory with the torsion field, induced by the chameleonfield. Following Hojman et al.
Phys. Rev. D , 3141 (1976) the results, obtained in Phys. Rev. D90, 045040 (2014), are generalised by extending the Einstein gravity to the Einstein–Cartan gravitywith the torsion field as a gradient of the chameleon field through a modification of local gaugeinvariance of minimal coupling in the Weinberg–Salam electroweak model. The contributions of thechameleon (torsion) field to the observables of electromagnetic and weak processes are calculated.Since in our approach the chameleon–photon coupling constant β γ is equal to the chameleon–matter coupling constant β , i.e. β γ = β , the experimental constraints on β , obtained in terrestriallaboratories by T. Jenke et al. (Phys. Rev. Lett. , 115105 (2014)) and by H. Lemmel et al. (Phys. Lett. B , 310 (2015)), can be used for the analysis of astrophysical sources of chameleons,proposed by C. Burrage et al. (Phys. Rev. D , 044028 (2009)), A.-Ch. Davis et al. (Phys. Rev.D , 064016 (2009) and in references therein, where chameleons induce photons because of directchameleon–photon transitions in the magnetic fields. PACS numbers: 03.65.Pm, 04.62.+v, 13.15.+g, 23.40.Bw
I. INTRODUCTION
The chameleon field, the properties of which are analogous to a quintessence [1, 2], i.e. a canonical scalar fieldinvented to explain the late–time acceleration of the Universe expansion [3–5], has been proposed in [6–8]. In orderto avoid the problem of violation of the equivalence principle [9] a chameleon mass depends on a mass density ρ of alocal environment [6–8]. The self–interaction of the chameleon field and its interaction to a local environment with amass density ρ are described by the effective potential V eff ( φ ) [6–8, 10–12] V eff ( φ ) = V ( φ ) + ρ e βφ/M Pl , (1)where φ is a chameleon field, β is a chameleon–matter field coupling constant and M Pl = 1 / √ πG N = 2 . × eVis the reduced Planck mass [13]. The potential V ( φ ) defines self–interaction of a chameleon field.As has been pointed out in Ref. [10–12], ultracold neutrons (UCNs), bouncing in the gravitational field of the Earthabove a mirror and between two mirrors, can be a good laboratory for testing of a chameleon–matter field interaction.Using the solutions of equations of motion for a chameleon field, confined between two mirrors, there has been foundthe upper limit for the coupling constant β < . × [12], which was estimated from the contribution of a chameleonfield to the transition frequencies of the quantum gravitational states of UCNs, bouncing in the gravitational fieldof the Earth. For the analysis of the chameleon–matter field interactions in Refs.[10–12] the potential V ( φ ) of achameleon–field self–interaction has been taken in the form of the Ratra–Peebles potential [14] (see also [6, 7]) V ( φ ) = Λ + Λ n φ n , (2)where Λ = p Λ H M = 2 . × − eV [15] with Ω Λ = 0 . +0 . − . and H = 1 . × − eV are therelative dark energy density and the Hubble constant [13], respectively, and n is the Ratra–Peebles index. Therunaway form Λ n /φ n for φ → ∞ is required by the quintessence models [1, 2]. Such a potential of a self–interactionof the chameleon field allows to realise the regime of the strong chameleon–matter coupling constant β ≫ [10–12].Recently [16] some new chameleon–matter field interactions have been derived from the non–relativistic approx-imation of the Dirac equation for slow fermions, moving in spacetimes with a static metric, caused by the weak ∗ Electronic address: [email protected] † Electronic address: [email protected] gravitational field of the Earth and a chameleon field. The derivation of the non–relativistic Hamilton operator of theDirac equation has been carried out by using the standard Foldy–Wouthuysen (SFW) transformation. There has beenalso shown that the chameleon field can serve as a source of a torsion field and torsion–matter interactions [17]–[24].A relativistic covariant torsion–neutron interaction has been found in the following form [16] L T ( x ) = i g T T µ ( x ) ¯ ψ ( x ) σ µν ←→ ∂ ν ψ ( x ) , (3)where T µ is the torsion field, ψ ( x ) is the neutron field operator, A ( x ) ←→ ∂ ν B ( x ) = A ( x ) ∂ ν B ( x ) − ( ∂ ν A ( x )) B ( x ) and σ µν = i ( γ µ γ ν − γ ν γ µ ) is one of the Dirac matrices [25]. In the non–relativistic limit we get δ L T ( x ) = i g T ~ T · ¯ ψ ( x )( ~ Σ × ~ ∇ ) ψ ( x ) + . . . = i g T ~ T · ϕ † ( x )( ~σ × ~ ∇ ) ϕ ( x ) + . . . , (4)where ϕ ( x ) is the operator of the large component of the Dirac bispinor field operator ψ ( x ) and ~ Σ = γ ~γγ is thediagonal Dirac matrix with elements ~ Σ = diag( ~σ, ~σ ) and ~σ are the 2 × g T ~ T is equal to g T ~ T ( x ) = − γ m βM Pl ~ ∇ φ ( ~r ) , (5)where γ = 1 for the Schwarzschild metric of a weak gravitational field [16]( see also [26–28]).Following [18] we introduce the torsion field tensor T αµν as follows f T αµν = δ αν f ,µ − δ αµ f ,ν , (6)where f ,µ = ∂ µ f and f = e φ H . Such an expression one obtains from the requirement of local gauge invariance of theelectromagnetic field strength [18] (see also section III). According to [16], the scalar field φ H can be identified witha chameleon field φ as φ H = βφ/M Pl . As a result, we get T αµν = βM Pl ( δ αν φ ,µ − δ αµ φ ,ν ) = g αλ βM Pl ( g λν φ ,µ − g λµ φ ,ν ) = g αλ T λµν , (7)where δ αν = g αλ g λν and g νλ and g λα are the metric and inverse metric tensor, respectively. The torsion tensor field T αµν is anti–symmetric T αµν = −T ανµ .For the subsequent analysis we need a definition of the covariant derivative V µ ; ν of a vector field V µ in the curvespacetime. It is given by [29–31] V µ ; ν = V µ,ν − V α Γ αµν , (8)where Γ αµν is the affine connection, determined by [18]Γ αµν = { αµν } − g ασ ( T σµν − T µσν − T νσµ ) = { αµν } + βM Pl g ασ ( − g σν φ ,µ + g µν φ ,σ ) , (9)where { αµν } are the Christoffel symbols [29–31] { αµν } = 12 g αλ ( g λµ,ν + g λν,µ − g µν,λ ) (10)and T αµν is the torsion tensor field [17, 18] T αµν = Γ ανµ − Γ αµν = βM Pl ( δ αν φ ,µ − δ αµ φ ,ν ) . (11)We introduce the contribution of the torsion field in agreement with Hojman et al. [18] (see Eq.(38) of Ref.[18]).Having determined the affine connection we may introduce the Riemann–Christoffel tensor R αµνλ or curvaturetensor as [29–31] R αµνλ = Γ αµν,λ − Γ αµλ,ν + Γ αλϕ Γ ϕµν − Γ ανϕ Γ ϕµλ , (12)which is necessary for the definition of the Lagrangian of the gravitational field in terms of the scalar curvature R [29] related to the Riemann–Christoffel tensor R αµνλ by [29–31] R = g µλ R αµαλ = g µλ R µλ . (13)Here R µλ is the Ricci tensor [29–31]. Following [18] and skipping intermediate calculations one may show that thescalar curvature R is equal to R = R + 3 β M g µν φ ,µ φ ,ν , (14)where the curvature R is determined by the Riemann–Christoffel tensor Eq.(12) with the replacement Γ αµν → { αµν } .The paper is organised as follows. In section II we consider the chameleon field in the gravitational field withtorsion (a version of the Einstein–Cartan gravity), caused by the chameleon field. We derive the effective Lagrangianand the equations of motion of the chameleon field coupled to the gravitational field (a version of the Einstein gravitywith a scalar self–interacting field). In section III we analyse the interaction of the chameleon (torsion) field with theelectromagnetic field, coupled also to the gravitational field. Following Hojman et al. [18] and modifying local gaugeinvariance of the electromagnetic strength tensor field we derive the torsion field tensor T αµν in terms of the chameleonfield (see Eq.(7). In section IV we analyse the torsion (chameleon) - photon interactions in terms of the two–photondecay φ → γ + γ of the chameleon and the photon–chameleon scattering γ + φ → φ + γ . We show that the amplitudesof the two–photon decay and the photon–chameleon scattering are gauge invariant. In order words we show that thereplacement of the photon polarisation vectors by their 4–momenta leads to the vanishing of the amplitudes of thetwo–photon decay and the photon–chameleon scattering. In section V we investigate the Weinberg–Salam electroweakmodel [13] without fermions. We derive the effective Lagrangian of the electroweak bosons, the electromagnetic fieldand the Higgs boson coupled to the gravitational and chameleon field. Such a derivation we carry out by meansof a modification of local gauge invariance. In section VI we include fermions into the Weinberg–Salam model andderive the effective interactions of the electroweak bosons, the Higgs field and fermions with the gravitational andchameleon field. In section VII we calculate the contributions of the chameleon to the charge radii of the neutronand proton. We calculate the contributions of the chameleon to the correlation coefficients of the neutron β − –decay n → p + e − + ¯ ν e + φ with a polarised neutron and unpolarised proton and electron. In addition we calculate the crosssection for the neutron β − –decay φ + n → p + e − + ¯ ν e , induced by the chameleon field. In section VIII we discussthe obtained results and perspectives of the experimental analysis of the approach, developed in this paper, and ofobservation of the neutron β − –decay, induced by the chameleon. II. TORSION GRAVITY AND EFFECTIVE LAGRANGIAN OF CHAMELEON FIELD
The action of the gravitational field with torsion, the chameleon field and matter fields we define by [6, 7] S g , ch = Z d x √− g L [ R , φ ] + Z d x p − ˜ g L m [˜ g µν ] , (15)where the Lagrangian L [ R , φ ] is given by L [ R , φ ] = 12 M R + 12 (1 − β ) φ ,µ φ ,µ − V ( φ ) . (16)Here φ ,µ = ∂φ/∂x µ and φ ,µ = ∂φ/∂x µ and V ( φ ) is the potential of the self–interaction of the chameleon field Eq.(2).The matter fields are described by the Lagrangian L m [˜ g µν ]. The interaction of the matter field with the chameleonfield runs through the metric tensor ˜ g µν in the Jordan–frame [6, 7, 32, 33], which is conformally related to theEinstein–frame metric tensor g µν by ˜ g µν = f g µν (or ˜ g µν = f − g µν ) and √− ˜ g = f √− g with f = e βφ/M Pl [6, 34].The factor e βφ/M Pl can be interpreted also as a conformal coupling to matter fields [6, 7]. Using Eq.(14) we transcribethe action Eq.(15) into the form S g , ch = Z d x √− g (cid:16) M R + L [ φ ] (cid:17) + Z d x p − ˜ g L m [˜ g µν ] , (17)where the contribution of the torsion field to the scalar curvature is absorbed by the kinetic term of the chameleonfield. The Lagrangian L [ φ ] is equal to L [ φ ] = 12 φ ,µ φ ,µ − V ( φ ) . (18)The total Lagrangian in the action Eq.(17) is usually referred as the Lagrangian in the Einstein frame, where g µν as well as g µν is the Einstein–frame metric such as φ ,µ φ ,µ = g µν φ ,µ φ ,ν and ✷ φ = (1 / √− g )( √− g φ ,µ ) ; µ =(1 / √− g )( √− g g µν φ ,ν ) ; µ [34] (see also [6, 7]).Varying the action Eq.(17) with respect to φ ,µ and φ we arrive at the equation of motion of the chameleon field ∂∂x µ δ ( √− g L [ φ ]) δφ ,µ = δ ( √− g L [ φ ]) δφ + δ ( √− ˜ g L m ) δφ , (19)Using the Lagrangian Eq.(18) we transform Eq.(19) into the form1 √− g ∂∂x µ ( √− g φ ,µ ) = − V ′ φ ( φ ) − f ′ φ f ˜ g αλ √− ˜ g δ ( √− ˜ g L m ) δ ˜ g αλ , (20)where V ′ φ ( φ ) and f ′ φ are derivatives with respect to φ . Since by definition [6, 7] the derivative2 √− ˜ g δ ( √− ˜ g L m ) δ ˜ g αλ = ˜ T αλ (21)is a matter stress–energy tensor in the Jordan frame, Eq.(20) takes the form ✷ φ = − V ′ φ ( φ ) − f ′ φ f ˜ T αα , (22)where ˜ T αα = ˜ g αλ ˜ T αλ . For a pressureless matter ˜ T αα = ˜ ρ , where ˜ ρ is a matter density in the Jordan frame, related toa matter density in the Einstein frame ρ by ˜ ρ = f − ρ [6] we get ✷ φ = − V ′ φ ( φ ) − ρ f ′ φ , (23)where we have set f ˜ T αα = ρ [6]. Then, V ′ φ ( φ ) − ρ f ′ φ coincides with the derivative of the effective potential of thechameleon–matter interaction V eff ( φ ) with respect to φ , given by Eq.(1) for f = e βφ/M Pl . III. TORSION GRAVITY WITH CHAMELEON AND ELECTROMAGNETIC FIELDS
In this section we analyse the interactions of the torsion (chameleon) field with the electromagnetic field. Theaction of the gravitational field, the chameleon field, the matter fields and the electromagnetic field is equal to S g , ch , em = Z d x √− g (cid:16) M R + 12 φ ,µ φ ,µ − V ( φ ) (cid:17) − Z d x p − ˜ g ˜ g αµ ˜ g βν F αβ F µν + Z d x p − ˜ g L m [˜ g µν ] , (24)where ˜ g µν = f g µν . Since √− ˜ g = f √− g , we get that √− ˜ g ˜ g αµ ˜ g βν = √− g g αµ g βν . The term √− ˜ g L m [˜ g µν ] describesan environment where the chameleon field couples to the electromagnetic field.Following then Hojman et al. [18] we define the electromagnetic strength tensor field F µν in the gravitational andtorsion field F µν = A ν ; µ − A µ ; ν = A ν,µ − A µ,ν − A α T αµν = F µν − A λ T λµν , (25)where F µν = A ν,µ − A µ,ν and A µ is the electromagnetic 4–potential. According to Hojman et al. [18], under a gaugetransformation the electromagnetic potential transforms as follows A µ → A ′ µ = A µ + c µα (Φ)Λ ,α , (26)where c µα (Φ) is a functional of the scalar field Φ, which we identify with the chameleon field Φ = βφ/M Pl , i.e. c µα (Φ) → c µα ( φ ) = δ µα e βφ/M Pl , and Λ is an arbitrary gauge function. The gauge invariance of the electromagneticfield strength imposes the constraint [18] c να ( φ ) ,µ − c µα ( φ ) ,ν − c σα ( φ ) T σµν = 0 . (27)This gives the torsion tensor field given by Eq.(6) and Eq.(7). Substituting Eq.(25) into Eq.(24) we arrive at theexpression S g , ch , em = Z d x √− g (cid:16) M R + 12 φ ,µ φ ,µ − V ( φ ) − F µν F µν + 12 g αµ g βν F µν A σ T σαβ − g αµ g βν A σ A λ T σαβ T λµν (cid:17) + Z d x p − ˜ g L m [˜ g µν ] . (28)Using the definition of the torsion tensor field Eq.(7) we arrive at the action S g , ch , em = Z d x √− g (cid:16) M R + 12 φ ,µ φ ,µ − V ( φ ) − F µν F µν − βM Pl F µν ( A µ φ ,ν − A ν φ ,µ ) − β M ( A µ φ ,ν − A ν φ ,µ )( A µ φ ,ν − A ν φ ,µ ) (cid:17) + Z d x p − ˜ g L m [˜ g µν ] (29)Thus, because of the torsion–electromagnetic field interaction the chameleon becomes unstable under the two–photondecay φ → γ + γ and may scatter by photons γ + φ → φ + γ with the chameleon–matter coupling constant β/M Pl .These reactions are described by the effective Lagrangians L φγγ = −√− g βM Pl F µν A µ φ ,ν (30)and L φφγγ = − √− g β M ( A µ φ ,ν − A ν φ ,µ )( A µ φ ,ν − A ν φ ,µ ) == − √− g β M ( A µ A µ φ ,ν φ ,ν − A µ A ν φ ,µ φ ,ν ) . (31)For the application of the action Eq.(29) with chameleon–photon interactions to the calculation of the specific reactionsof the chameleon–photon and chameleon–photon–matter interactions we have to fix the gauge of the electromagneticfield. We may do this in a standard way [25] S g , ch , em = Z d x √− g (cid:16) M R + 12 φ ,µ φ ,µ − V ( φ ) − F µν F µν + f ( A µ ; µ , A ν , φ ) (cid:17) + Z d x p − ˜ g L m [˜ g µν ] , (32)where f ( A µ ; µ , A ν , φ ) is a gauge fixing functional and the divergence A µ ; µ is defined by [30, 31] A µ ; µ = A µ,µ + ˜Γ µµν A ν . (33)The affine connection ˜Γ αµν we have to calculate for the Jordan–frame metric ˜ g µν = f g µν [6, 32]. We get˜Γ αµν = { αµν } + δ αµ f − f ,ν = { αµν } + βM Pl δ αµ φ ,ν (34)such as ˜Γ ανµ − ˜Γ αµν = T αµν (see Eq.(7)). As a result, the divergence A µ ; µ is equal to [30, 31] A µ ; µ = 1 √− g ∂ ( √− g A µ ) ∂x µ + 4 βM Pl φ ,ν A ν . (35)Since a gauge condition should not depend on the chameleon field, we propose to fix a gauge as follows f ( A µ ; µ , A ν , φ ) = − ξ (cid:16) A µ ; µ − βM Pl φ ,ν A ν (cid:17) , (36)where ξ is a gauge parameter. Now we are able to investigate some specific processes of chameleon–photon interactions. IV. CHAMELEON–PHOTON INTERACTIONS
The specific processes of the chameleon–photon interaction, which we analyse in this section, are i) the two–photondecay φ → γ + γ and ii) the photon–chameleon scattering γ + φ → φ + γ . The calculation of these reactions we carry outin the Minkowski spacetime. For this aim in the interactions Eq.(30) and Eq.(31) we make a replacement g µν → η µν ,where η µν is the metric tensor in the Minkowski space time with only diagonal components (+1 , − , − , − g → det { η µν } = − FIG. 1: Feynman diagram for the φ → γ + γ decay.FIG. 2: Feynman diagram for the amplitude of the photon–chameleon (Compton) scattering A. Two–photon φ → γ + γ decay of the chameleon For the calculation of the two–photon decay rate of the chameleon we use the Lagrangian Eq.(30). The Feynmandiagram of the amplitude of the two–photon decay of the chameleon is shown in Fig. 1. The analytical expression ofthe amplitude of the φ → γ + γ decay is equal to M ( φ → γ γ ) = − βM Pl (( ε ∗ · ε ∗ )( k · k ) − ( ε ∗ · k )( ε ∗ · k )) , (37)where ε ∗ j ( k j ) and k j for j = 1 , ε ∗ j ( k j ) · k j = 0. Skipping standard calculations we obtain the following expression for the two–photondecay rate of the chameleon Γ( φ → γ γ ) = β M m φ π , (38)where m φ is the chameleon mass, defined by [11] m φ = Λ p n ( n + 1) (cid:16) βρnM Pl Λ (cid:17) n +22 n +2 (39)as a function of the chameleon–matter coupling constant β , the environment density ρ and the Ratra–Peebles index n . B. Photon–chameleon γ + φ → φ + γ scattering The Feynman diagrams of the amplitude of the photon–chameleon scattering are shown in Fig. 2. The contributionsof the diagrams in Fig. 2 are given by M ( a ) ( γ φ → φ γ ) = β M (cid:16) k µ ( ε · p ) − ε µ ( k · p ) (cid:17) D µα ( q ) (cid:16) k α ( ε ∗ · p ) − ε ∗ α ( k · p ) (cid:17)(cid:12)(cid:12)(cid:12) q = p + k = p + k + β M ε µ p ν (cid:16) q µ q α D νβ ( q ) − q µ q β D να ( q ) − q ν q α D µβ ( q ) + q ν q β D µα ( q ) (cid:17) ε ∗ α p β (cid:12)(cid:12)(cid:12) q = p + k = p + k + β M (cid:16) k µ ( ε · p ) − ε µ ( k · p ) (cid:17)(cid:16) q α D µβ ( q ) − q β D µα ( q ) (cid:17) ε ∗ α p β (cid:12)(cid:12)(cid:12) q = p + k = p + k + β M ε µ p ν (cid:16) q µ D ν α ( q ) − q ν D µα (cid:17)(cid:16) k α ( ε ∗ · p ) − ε ∗ α ( k · p ) (cid:17)(cid:12)(cid:12)(cid:12) q = p + k = p + k , (40) M ( b ) ( γ φ → φ γ ) = β M (cid:16) k µ ( ε ∗ · p ) − ε ∗ µ ( k · p ) (cid:17) D µα ( q ) (cid:16) k α ( ε · p ) − ε α ( k · p ) (cid:17)(cid:12)(cid:12)(cid:12) q = p − k = p − k + β M ε ∗ µ p ν (cid:16) q µ q α D νβ ( q ) − q µ q β D να ( q ) − q ν q α D µβ ( q ) + q ν q β D µα ( q ) (cid:17) ε α p β (cid:12)(cid:12)(cid:12) q = p − k = p − k − β M (cid:16) k µ ( ε ∗ · p ) − ε ∗ µ ( k · p ) (cid:17)(cid:16) q α D µβ ( q ) − q β D µα ( q ) (cid:17) ε α p β (cid:12)(cid:12)(cid:12) q = p − k = p − k − β M ε ∗ µ p ν (cid:16) q µ D ν α ( q ) − q ν D µα (cid:17)(cid:16) k α ( ε · p ) − ε α ( k · p ) (cid:17)(cid:12)(cid:12)(cid:12) q = p − k = p − k , (41) M ( c ) ( γ φ → φ γ ) = β M (cid:16) − ε ∗ · ε )( p · p ) + ( ε ∗ · p )( ε · p ) + ( ε ∗ · p )( ε · p ) (cid:17) , (42) M ( d ) ( γ φ → φ γ ) = 2 n ( n + 1)( n + 2) βM Pl Λ n +4 φ n +3min ( ε ∗ · k )( ε · k ) − ( ε ∗ · ε )( k · k ) m φ − q − i (cid:12)(cid:12)(cid:12) q = k − k = p − p , (43)where ε and ε ∗ are the photon polarisation vectors in the initial and final states of the photon–chameleon scattering.They depend on the photon momenta ε ( k ) and ε ∗ ( k ) and obey the constraints ε ( k ) · k = ε ∗ ( k ) · k = 0. Thechameleon field mass m φ is defined by Eq.(39). The vertex of φ interaction is defined by the effective Lagrangian L φφφ = n ( n + 1)( n + 2)6 Λ n +4 φ n +3min φ . (44)Here φ min is the minimum of the chameleon field, given by [10, 11] φ min = Λ (cid:16) nM Pl Λ βρ (cid:17) n +1 , (45)where ρ is the density of the medium in which the chameleon field propagates. The photon propagator D αβ ( q ) isequal to D αβ ( q ) = 1 q + i (cid:16) g αβ − (1 − ξ ) q α q β q (cid:17) . (46)One may show that the amplitudes M ( a ) ( γ φ → φ γ ) and M ( b ) ( γ φ → φ γ ) do not depend on the longitudinal part ofthe photon propagator. As a result the amplitudes M ( a ) ( γ φ → φ γ ) and M ( b ) ( γ φ → φ γ ) can be transcribed into theform M ( a ) ( γ φ → φ γ ) = β M q + i (cid:16) ( ε ∗ · p )( ε · p )( k · k ) − ( ε ∗ · p )( ε · k )( k · p ) − ( ε ∗ · k )( ε · p )( k · p ) + ( ε ∗ · ε )( k · p )( k · p ) (cid:17)(cid:12)(cid:12)(cid:12) q = p + k = p + k + β M q + i (cid:16) ( ε ∗ · q )( ε · q )( p · p ) − ( ε ∗ · p )( ε · q )( p · q ) − ( ε ∗ · q )( ε · p )( p · q ) + ( ε ∗ · ε )( p · q )( p · q ) (cid:17)(cid:12)(cid:12)(cid:12) q = p + k = p + k + β M q + i (cid:16) ( ε ∗ · q )( ε · p )( k · p ) − ( ε ∗ · q )( ε · p )( k · p ) − ( ε ∗ · k )( ε · p )( p · q ) + ( ε ∗ · ε )( k · p )( p · q ) (cid:17)(cid:12)(cid:12)(cid:12) q = p + k = p + k + β M q + i (cid:16) ( ε ∗ · p )( ε · q )( k · p ) − ( ε ∗ · p )( ε · k )( p · q ) − ( ε ∗ · p )( ε · q )( k · p ) + ( ε ∗ · ε )( k · p )( p · q ) (cid:17)(cid:12)(cid:12)(cid:12) q = p + k = p + k (47)and M ( b ) ( γ φ → φ γ ) = β M q + i (cid:16) ( ε ∗ · p )( ε · p )( k · k ) − ( ε ∗ · k )( ε · p )( k · p ) − ( ε ∗ · p )( ε · k )( k · p ) + ( ε ∗ · ε )( k · p )( k · p ) (cid:17)(cid:12)(cid:12)(cid:12) q = p − k = p − k + β M q + i (cid:16) ( ε ∗ · q )( ε · q )( p · p ) − ( ε ∗ · q )( ε · p )( p · q ) − ( ε ∗ · p )( ε · q )( p · q ) + ( ε ∗ · ε )( p · q )( p · q ) (cid:17)(cid:12)(cid:12)(cid:12) q = p − k = p − k − β M q + i (cid:16) ( ε ∗ · p )( ε · q )( k · p ) − ( ε ∗ · p )( ε · q )( k · p ) − ( ε ∗ · p )( ε · k )( p · q ) + ( ε ∗ · ε )( k · p )( p · q ) (cid:17)(cid:12)(cid:12)(cid:12) q = p − k = p − k − β M q + i (cid:16) ( ε ∗ · q )( ε · p )( k · p ) − ( ε ∗ · k )( ε · p )( p · q ) − ( ε ∗ · q )( ε · p )( k · p ) + ( ε ∗ · ε )( k · p )( p · q ) (cid:17)(cid:12)(cid:12)(cid:12) q = p − k = p − k . (48)The total amplitude of the photon–chameleon scattering is defined by the sum of the amplitudes Eq.(40) M ( γ φ → φ γ ) = X j = a,b,c,d M ( j ) ( γ φ → φ γ ) . (49)Now let us check gauge invariance of the amplitude of the photon–chameleon scattering Eq.(42). As we have foundalready the amplitudes M ( a ) ( γ φ → φ γ ) and M ( b ) ( γ φ → φ γ ) do not depend on the longitudinal part of the photonpropagator, i.e. on the gauge parameter ξ . Then, according to general theory of photon–particle ( γ h ) interactions[25], the amplitude of photon–particle scattering should vanish, when the polarisation vector of the photon either inthe initial or in the final state is replaced by the photon momentum. This means that replacing either ε → k or ε ∗ → k one has to get zero for the total amplitude Eq.(49). Since one may see that the amplitude M ( d ) ( γ φ → φ γ ) isself–gauge invariant, one has to check the vanishing of the sum of the amplitudes, defined by the first three Feynmandiagrams in Fig. 2, i.e. ˜ M ( γ φ → φ γ ) = X j = a,b,c M ( j ) ( γ φ → φ γ ) . (50)Replacing ε → k we obtain M ( a ) ( γ φ → φ γ ) (cid:12)(cid:12)(cid:12) ε → k = β M (cid:16) − ( ε ∗ · p )( k · p ) + ( ε ∗ · k ) q (cid:17)(cid:12)(cid:12)(cid:12) q = p + k = p + k ,M ( b ) ( γ φ → φ γ ) (cid:12)(cid:12)(cid:12) ε → k = β M (cid:16) − ( ε ∗ · p )( k · p ) + ( ε ∗ · k ) q (cid:17)(cid:12)(cid:12)(cid:12) q = p − k = p − k ,M ( c ) ( γ φ → φ γ ) (cid:12)(cid:12)(cid:12) ε → k = β M (cid:16) − ε ∗ · k )( p · p ) + ( ε ∗ · p )( k · p ) + ( ε ∗ · p )( k · p ) (cid:17) . (51)Because of the relation q (cid:12)(cid:12)(cid:12) q = p + k = p + k + q (cid:12)(cid:12)(cid:12) q = p − k = p − k = 2( p · p ) (52)the sum of the amplitudes Eq.(51) vanishes, i.e.˜ M ( γ φ → φ γ ) (cid:12)(cid:12)(cid:12) ε → k = X j = a,b,c M ( j ) ( γ φ → φ γ ) (cid:12)(cid:12)(cid:12) ε → k = 0 . (53)The same result one may obtain replacing ε ∗ → k . Thus, the obtained results confirm gauge invariance of theamplitude of the photon–chameleon scattering, the complete set of Feynman diagrams of which is shown in Fig. 2.Of course, because of the smallness of the constant β /M < − barn / eV , estimated for β < . × [12], thecross section for the photon–chameleon scattering is extremely small and hardly plays any important cosmologicalrole at low energies, for example, for a formation of the cosmological microwave background and so on [55, 56].Nevertheless, the observed gauge invariance of the amplitude of the photon–chameleon scattering is important forthe subsequent extension of the minimal coupling inclusion of a torsion field to the Weinberg–Salam electroweakmodel [25] in the Einstein–Cartan gravity. One of the interesting consequences of the observed gauge invariance ofthe chameleon–photon interaction might be unrenormalisability of the coupling constant β/M Pl by the contributionsof all possible interactions. This might mean that the upper bound on the chameleon–matter coupling constant β < . × , measured in the qBounce experiments with ultracold neutrons [12], should not be change by takinginto account the contributions of some other possible interactions.In this connection the results, obtained in this section, can be of interest with respect to the analysis of the contribu-tions of the photon–chameleon direct transitions in the magnetic field to the cosmological microwave background [55].The effective chameleon–photon coupling constant g eff , introduced by Davis, Schelpe and Shaw [55], in our approachis equal to g eff = β/M Pl . Using the experimental upper bound β < . × we obtain g eff < . × − GeV − . Thisconstraint is in qualitative agreement with the results, obtained by Davis, Schelpe and Shaw [55]. The experimentalconstraints on the chameleon–matter coupling β < . × ( n = 1), β < . × ( n = 2), β < . × ( n = 3)and β < . × ( n = 4), measured recently by H. Lemmel et al. [57] using the neutron interferometer, place morestrict constraints of the astrophysical sources of chameleons, investigated in [51]–[55]. V. TORSION GRAVITY AND WEINBERG–SALAM ELECTROWEAK MODEL WITHOUTFERMIONS
In this section we investigate the Weinberg–Salam electroweak model without fermions in the minimal couplingapproach to the torsion field (see [18]), caused by the chameleon field [16].According to Hojman et al. [18], in the Einstein–Cartan gravity with a torsion field, induced by a scalar field, thecovariant derivative of a charged (pseudo)scalar particle with electric charge q should be equal to D µ = ∂ µ − i q f − A µ (54)with f = e βφ/M Pl . Using such a definition of the covariant derivative one may calculate the electromagnetic fieldstrength tensor F µν as follows [25] F µν = − iq f [ D µ , D ν ] = f (cid:16) ∂ µ (cid:16) f − A ν (cid:17) − ∂ ν (cid:16) f − A µ (cid:17)(cid:17) = A ν,µ − A µ,ν − A α T αµν , (55)where the torsion tensor field T αµν is given by Eq.(11). In this section we discuss the Weinberg–Salam electroweakmodel [25] in the Einstein–Cartan gravity with a torsion field, caused by the chameleon field. Below we consider theWeinberg–Salam electroweak model without fermions.The Lagrangian of the Weinberg–Salam electroweak model of the electroweak bosons and the Higgs field with gauge SU (2) × U (1) symmetry, determined in the Minkowski space–time, takes the form [25] L ew = − ~A µν · ~A µν − B µν B µν + (cid:16) ∂ µ Φ − i g ′ Y w B µ Φ − i g ~I w · ~A µ Φ (cid:17) † × (cid:16) ∂ µ Φ − i g ′′ Y w B µ Φ − i g ~I w · ~A µ Φ (cid:17) − V (Φ † Φ) , (56)where g ′ and g are the electroweak coupling constants and ~A µ and B µ are vector fields and Φ is the Higgs boson field.Then, Y w and ~I w = ~τ w are the weak hypercharge and the weak isopin, respectively: ~τ w = ( τ w , τ w , τ w ) are the weakisospin 2 × τ aw τ bw = δ ab + i ε abc τ cw and tr( τ aw τ bw ) = 2 δ ab [25]. The weak hypercharge Y w and the thirdcomponent of the weak isospin I w are related by Q = I w + Y w /
2, where Q is the electric charge of the field in unitsof the proton electric charge e [25]. In the Weinberg–Salam electroweak model the Higgs boson field Φ possesses theweak isospin I w = 1 / Y w = 1. The field strength tensors ~A µν and B µν are equal to ~A µν = ∂ µ ~A ν − ∂ ν ~A µ + g ~A µ × ~A ν ,B µν = ∂ µ B ν − ∂ ν B µ . (57)The Higgs boson field Φ and its vacuum expectation value are given in the standard form [25]Φ = (cid:16) Φ + Φ (cid:17) , h Φ i = 1 √ (cid:16) v (cid:17) , (58)0where Φ = ( v + ϕ ) / √ ϕ is a physical Higgs boson field. The potential energy density V (Φ † Φ) has also thestandard form [25] V (Φ † Φ) = − µ Φ † Φ + κ (Φ † Φ) (59)with µ > κ > v = µ /κ . The vacuum expectation value v is related to the Fermi coupling constant G F by √ G F v = 1, where G F = 1 . × − MeV − [13]. The covariant derivative of the Higgs field is given by[25] D µ = ∂ µ − i g ′ B µ Φ − i g ~τ · ~A µ . (60)Using the covariant derivative Eq.(60) we may calculate the commutator [ D µ , D ν ] and obtain the following expression[ D µ , D ν ] = − i g ′ ( ∂ µ B ν − ∂ ν B µ ) − i g ~τ · ( ∂ µ ~A ν − ∂ ν ~A µ + g ~A µ × ~A ν ) = − i g ′ B µν − i g ~τ · ~A µν , (61)where ~A µν and B µν are the field strength tensors Eq.(57). Under gauge transformations A µ → Ω A µ = Ω A µ Ω − + 1 ig ∂ µ Ω Ω − ,B µ → Λ B µ = B µ + ∂ µ Λ , (62)where Ω and Λ are the gauge matrix and gauge function, respectively, and A µ = ~τ · ~A µ , the field strength tensors A µν = ~τ · ~A µν and B µν transform as follows [25] A µν → Ω A µν = Ω A µν Ω − ,B µν → Λ B µν = B µν . (63)In the Einstein–Cartan gravity with a torsion field in the minimal coupling approach the covariant derivative Eq.(60)should be taken in the following form D µ = ∂ µ − i g ′ f − B µ − i g f − ~τ · ~A µ . (64)For the definition of field strength tensors ~A µν and B µν , extended by the contribution of a torsion field, we proposeto calculate the commutator [ D µ , D ν ]. The result of the calculation is[ D µ , D ν ] = − i g ′ f − B µν − i g f − ~τ · ~ A µν , (65)where the field strength tensors ~ A µν and B µν are equal to ~ A µν = ~A ν,µ − ~A µ,ν + g f − ~A µ × ~A ν − ~A α T αµν , B µν = B ν,µ − B µ,ν − B α T αµν , (66)where the torsion tensor field T αµν is given in Eq.(11). Thus, the Lagrangian of electroweak interactions in theEinstein–Cartan gravity with a torsion field in the minimal coupling constant approach and the chameleon field,coupled through the Jordan metric ˜ g µν = f g µν , takes the form [34] L ew √− g = − ~ A µν · ~ A µν − B µν B µν + f (cid:16) ∂ µ Φ − i g ′ f − B µ Φ − i g f − ~τ · ~A µ Φ (cid:17) † × (cid:16) ∂ µ Φ − i g ′ f − B µ Φ − i g f − ~τ · ~A µ Φ (cid:17) − f V (Φ † Φ) , (67)where the factor f comes from √− ˜ g = f √− g . The physical vector boson states of the Weinberg–Salam electroweakmodel are [25] W ± µ = 1 √ A µ ∓ iA µ ) ,Z µ = sin θ W B µ − cos θ W A µ ,A µ = cos θ W B µ + sin θ W A µ , (68)1where W ± µ and Z µ are the electroweak W –boson and Z –boson fields and A µ is the electromagnetic field, respectively,and θ W is the Weinberg angle defined by tan θ W = g ′ /g . The electromagnetic coupling constant e as a function of thecoupling constants g and g ′ is given by e = gg ′ / p g + g ′ = g sin θ W = g ′ cos θ W [25]. In terms of the electroweakboson fields W ± µ and Z µ , the electromagnetic field A µ , the Higgs boson field ϕ and the chameleon field φ , coupled tothe gravitational field with torsion, the Lagrangian of the electroweak interactions takes the following form L ew √− g = − A µν F µν − Z µν Z µν − W + µν W − µν + 12 M W W + µ W − µ + 12 M Z Z µ Z µ + 12 f − i g W + µν h sin θ W (cid:16) W − µ A ν − A µ W − ν (cid:17) − cos θ W (cid:16) W − µ Z ν − Z µ W − ν (cid:17)i − f − i g W − µν h sin θ W (cid:16) W + µ A ν − A µ W + ν (cid:17) − cos θ W (cid:16) W + µ Z ν − Z µ W + ν (cid:17)i − f − i g (cid:16) sin θ W A µν − cos θ W Z µν (cid:17)(cid:16) W − µ W + ν − W − ν W + µ (cid:17) − f − g (cid:16) W − µ W + ν − W − ν W + µ (cid:17)(cid:16) W + µ W − ν − W + ν W − µ (cid:17) − f − g h sin θ W (cid:16) W + µ A ν − A µ W + ν (cid:17) − cos θ W (cid:16) W + µ Z ν − Z µ W + ν (cid:17)i × h sin θ W (cid:16) W − µ A ν − A µ W − ν (cid:17) − cos θ W (cid:16) W − µ Z ν − Z µ W − ν (cid:17)i + 12 g M W ϕ W + µ W − µ + 18 g ϕ W + µ W − µ + 12 g cos θ W M Z ϕ Z µ Z µ + 18 g cos θ W ϕ Z µ Z µ + 12 f ϕ ,µ ϕ ,µ − f M ϕ ϕ − f κ v ϕ − f κ ϕ , (69)where M W = g v / M Z = M W /cos θ W and M ϕ = 2 κ v are the squared masses of the W ± –boson, Z –boson andHiggs boson field, respectively, A µν , Z µν and W ± µν are the strength field tensors of the electromagnetic, Z –boson and W ± –boson fields. They are equal to A µν = A ν,µ − A µ,ν − A σ T σµν , Z µν = Z ν,µ − Z µ,ν − Z σ T σµν , W ± µν = W ± ν,µ − W ± µ,ν − W ± σ T σµν . (70)Now we are able to extend the obtained results to fermions. VI. TORSION GRAVITY WITH CHAMELEON FIELD AND WEINBERG–SALAM ELECTROWEAKMODEL WITH FERMIONSA. Dirac fermions with mass m in the Einstein–Cartan gravity, coupled to the chameleon field through theJordan metric ˜ g µν = f g µν The Dirac equation in an arbitrary (world) coordinate system is specified by the metric tensor g µν ( x ). It definesan infinitesimal squared interval between two events ds = g µν ( x ) dx µ dx ν . (71)The relativistic invariant form of the Dirac equation in an arbitrary coordinate system is [26]( iγ µ ( x ) ∇ µ − m ) ψ ( x ) = 0 , (72)where γ µ ( x ) are a set of Dirac matrices satisfying the anticommutation relation γ µ ( x ) γ ν ( x ) + γ ν ( x ) γ µ ( x ) = 2 g µν ( x ) (73)and ∇ µ is a covariant derivative without gauge fields. For an exact definition of the Dirac matrices γ µ ( x ) and thecovariant derivative ∇ µ we follow [26] and use a set of tetrad (or vierbein) fields e ˆ αµ ( x ) at each spacetime point x defined by dx ˆ α = e ˆ αµ ( x ) dx µ . (74)2The tetrad fields relate in an arbitrary (world) coordinate system a spacetime point x , which is characterised by theindex µ = 0 , , ,
3, to a locally Minkowskian coordinate system erected at a spacetime point x , which is characterisedby the index ˆ α = 0 , , ,
3. The tetrad fields e ˆ αµ ( x ) are related to the metric tensor g µν ( x ) as follows: ds = η ˆ α ˆ β dx ˆ α dx ˆ β = η ˆ α ˆ β [ e ˆ αµ ( x ) dx µ ][ e ˆ βν ( x ) dx ν ] = [ η ˆ α ˆ β e ˆ αµ ( x ) e ˆ βν ( x )] dx µ dx ν = g µν ( x ) dx µ dx ν . (75)This gives g µν ( x ) = η ˆ α ˆ β e ˆ αµ ( x ) e ˆ βν ( x ) . (76)Thus, the tetrad fields can be viewed as the square root of the metric tensor g µν ( x ) in the sense of a matrix equation[26]. Inverting the relation Eq.(74) we obtain η ˆ α ˆ β = g µν ( x ) e µ ˆ α ( x ) e ν ˆ β ( x ) . (77)There are also the following relations e µ ˆ α ( x ) e ˆ βµ ( x ) = δ ˆ β ˆ α ,e µ ˆ α ( x ) e ˆ αν ( x ) = δ µν ,e µ ˆ α ( x ) e ˆ βµ ( x ) = η ˆ α ˆ β ,e ˆ αµ ( x ) = η ˆ α ˆ β e ˆ βµ ( x ) ,e ˆ αµ ( x ) e ˆ αν ( x ) = g µν ( x ) . (78)In terms of the tetrad fields e µ ˆ α ( x ) and the Dirac matrices γ ˆ α in the Minkowski spacetime the Dirac matrices γ µ ( x )are defined by γ µ ( x ) = e µ ˆ α ( x ) γ ˆ α . (79)A covariant derivative ∇ µ we define as [26] ∇ µ = ∂ µ − Γ µ ( x ) . (80)The spinorial affine connection Γ µ ( x ) is defined by [26]Γ µ ( x ) = i σ ˆ α ˆ β e ν ˆ α ( x ) e ˆ βν ; µ ( x ) , (81)where σ ˆ α ˆ β = i ( γ ˆ α γ ˆ β − γ ˆ β γ ˆ α ) and e ˆ βν ; µ ( x ) is given in terms of the affine connection Γ αµν ( x ) e ˆ βν ; µ ( x ) = e ˆ βν,µ ( x ) − Γ αµν ( x ) e ˆ βα ( x ) ,e ˆ βν ; µ ( x ) = e ˆ βν,µ ( x ) − Γ αµν ( x ) e ˆ βα ( x ) . (82)In the Einstein gravity the affine connection Γ αµν ( x ) is equal to Γ αµν ( x ) = { αµν } (see Eq.(10)). Specifying thespacetime metric one may transform the Dirac equation Eq.(72) into the standard form i ∂ψ ( t, ~r ) ∂t = ˆH ψ ( t, ~r ) , (83)where ˆH is the Hamilton operator. For example, for the static metric ds = V dt − W ( d~r ) , where V and W arespatial functions, one may show that the Hamilton operator is given by (see [16, 35])ˆH = γ mV − i VW γ ~γ · (cid:16) ~ ∇ + ~ ( ∇ V )2 V + ~ ( ∇ W ) W (cid:17) . (84)In the approach to the Einstein–Cartan gravity with torsion, developed above, the Lagrangian of the Dirac field ψ ( x )with mass m , coupled to the chameleon field through the Jordan metric ˜ g µν = f g µν , is equal to L m = p − ˜ g ¯ ψ (cid:16) i ˜ g µν ˜ γ µ ( x ) ˜ ∇ ν − m (cid:17) ψ = p − ˜ g ¯ ψ (cid:16) i ˜ e µ ˆ α γ ˆ α ˜ ∇ µ − m (cid:17) ψ, (85)3where the tetrad fields in the Jordan–frame and the Einstein–frame are related by˜ e ˆ αµ = f e ˆ αµ , ˜ e µ ˆ α = f − e µ ˆ α (86)and the covariant derivative is equal to˜ ∇ µ = ∇ µ = ∂ µ − Γ µ ( x ) , Γ µ ( x ) = i σ ˆ α ˆ β e ν ˆ α ( x ) e ˆ βν ; µ , ˜Γ αµν = { αµν } + δ αµ f − f ,ν = { αµν } + βM Pl δ αµ φ ,ν . (87)As a result we get L m = √− g f ¯ ψ (cid:16) f − iγ µ ( x ) ∇ µ − m (cid:17) ψ. (88)Below we apply the obtained results to the analysis of the electroweak interactions of the neutron and proton. B. Electroweak model for neutron and proton, coupled to chameleon field through the Jordan metric ˜ g µν = f g µν For the subsequent application of the results, obtained below, to the analysis of the contribution of the chameleonfield to the radii of the neutron and the proton and to the neutron β − –decay we defined the electroweak model forthe following multiplets N L = (cid:16) p L n L (cid:17) , p R , n R ,ℓ L = (cid:16) ν eL e − L (cid:17) , e − R , (89)where ψ L = (1 − γ ) ψ and ψ R = (1 + γ ) ψ . Such a model is renormalisable also because of the vanishing of thecontribution of the Adler–Bell–Jackiw anomalies Q p + Q e = 0, where Q p = +1 and Q e = − e [36].The fermion states Eq.(89) have the following electroweak quantum numbers: N L : ( I w = , Y w = 1), p R : ( I w =0 , Y w = 2), n R : ( I w = 0 , Y w = 0) and ℓ L : ( I w = , Y w = − e − R : ( I w = 0 , Y w = − I w and weak hypercharge Y w are related by Q = I w + Y w /
2. In the Einstein–Cartan gravitywith the torsion field and the chameleon field, coupled to matter field through the Jordan metric ˜ g µν = f g µν , theLagrangian of the fermion fields Eq.(89), coupled to the vector electroweak boson fields and the Higgs boson field, isequal to L ewf √− g = f ¯ N L h f − iγ µ ( x ) (cid:16) ∂ µ − i g ′ f − B µ − i g f − ~τ · ~A µ − Γ µ (cid:17)i N L + f ¯ p R h f − iγ µ ( x ) (cid:16) ∂ µ − i g ′ f − B µ − Γ µ (cid:17)i p R + f ¯ n R h f − iγ µ ( x ) (cid:16) ∂ µ − Γ µ (cid:17)i n R + f ¯ ℓ L h f − iγ µ ( x ) (cid:16) ∂ µ + i g ′ f − B µ − i g f − ~τ · ~A µ − Γ µ (cid:17)i ℓ L + f ¯ e − R h f − iγ µ ( x ) (cid:16) ∂ µ + i g ′ f − B µ − Γ µ (cid:17)i e − R . (90)The masses of the neutron and electron one may gain by virtue of the following interactions with the Higgs field Φ δ L hne √− g = − f κ n (cid:16) ¯ N L Φ n R + ¯ n R Φ † N L ) − f κ e (cid:16) ¯ ℓ L Φ e − R + ¯ e − R Φ † ℓ L ) == − f m n ¯ nn − f κ n √ nn ϕ − f m e ¯ e − e − − f κ e √ e − e − ϕ, (91)4where κ n and κ e are the input parameters, defining the neutron and electron masses m n = κ n v/ √ m e = κ e v/ √ v + ϕ ) / √ − = (Φ + ) † . Using the Higgs field ¯Φ one gets δ L hp √− g = − f κ p (cid:16) ¯ N L ¯Φ p R + ¯ p R ¯Φ † N L ) = − f m p ¯ pp − f κ p √ pp ϕ, (92)where m p = κ p v/ √ L ew √− g = f ¯ p ( x ) iγ µ ( x ) (cid:16) ∂ µ − i f − e A µ ( x ) + i f − g θ W (1 − θ W ) Z µ ( x ) − Γ µ ( x ) (cid:17) (cid:16) − γ (cid:17) p ( x )+ f ¯ p ( x ) iγ µ ( x ) (cid:16) ∂ µ − i f − e A µ ( x ) + i f − g cos θ W sin θ W Z µ ( x ) − Γ µ ( x ) (cid:17) (cid:16) γ (cid:17) p ( x )+ f ¯ n ( x ) iγ µ ( x ) (cid:16) ∂ µ − i f − g θ W Z µ ( x ) − Γ µ ( x ) (cid:17) (cid:16) − γ (cid:17) n ( x )+ f ¯ n ( x ) iγ µ ( x ) (cid:16) ∂ µ − Γ µ ( x ) (cid:17) (cid:16) γ (cid:17) n ( x )+ f g √ p ( x ) γ µ ( x ) (cid:16) − γ (cid:17) n ( x ) W + µ ( x ) + f g √ n ( x ) γ µ ( x ) (cid:16) − γ (cid:17) p ( x ) W − µ ( x )+ f ¯ ν e ( x ) iγ µ ( x ) (cid:16) ∂ µ + i f − g θ W Z µ ( x ) − Γ µ ( x ) (cid:17) (cid:16) − γ (cid:17) ν e ( x )+ f ¯ e − ( x ) iγ µ ( x ) (cid:16) ∂ µ + i f − e A µ ( x ) − i f − g θ W (1 − θ W ) Z µ ( x ) − Γ µ ( x ) (cid:17) (cid:16) − γ (cid:17) e − ( x )+ f ¯ e − ( x ) iγ µ ( x ) (cid:16) ∂ µ + i f − e A µ ( x ) + i f − g cos θ W sin θ W ) Z µ ( x ) − Γ µ ( x ) (cid:17) (cid:16) γ (cid:17) e − ( x ) − f m p ¯ p ( x ) p ( x ) − f m n ¯ n ( x ) n ( x ) − f m e ¯ e − ( x ) e − ( x ) − f κ p √ p ( x ) p ( x ) ϕ ( x ) − f κ n √ n ( x ) n ( x ) ϕ ( x ) − f κ e √ e − ( x ) e − ( x ) ϕ ( x ) . (93)We note that for the calculation of Γ µ ( x ) we have to use the affine connection, given by Eq.(87). VII. CONTRIBUTION OF CHAMELEON FIELD TO CHARGE RADII OF NEUTRON AND PROTONAND TO NEUTRON β − –DECAY The electroweak model in the Einstein–Cartan gravity with the torsion and chameleon fields, analysed above, isapplied to some specific processes of electromagnetic and weak interactions of the neutron and proton to the chameleonfield. In this section we calculate the amplitudes of the electron–neutron and electron–proton low–energy scatteringwith the chameleon field exchange and define the contributions of the chameleon field to the charge radii of theneutron and proton. We calculate also the contribution of the chameleon field to the energy spectra of the neutron β − –decay and the lifetime of the neutron. The calculations we carry out in the Minkowski spacetime replacing metrictensor g µν in the Einstein frame by the metric tensor of the Minkowski spacetime g µν → η µν . The Lagrangian ofthe electromagnetic and electroweak interactions of the neutron, the proton, the electron and the electron neutrino,coupled to the torsion field and the chameleon field in the Minkowski spacetime, is given by L ew = f ¯ p ( x ) n iγ µ (cid:16) ∂ µ − i f − e A µ ( x ) + i f − g θ W (1 − (1 − θ W ) γ ) Z µ ( x ) − Γ µ ( x ) (cid:17) − m p f o p ( x )+ f ¯ n ( x ) n iγ µ (cid:16) ∂ µ − i f − g θ W (1 − γ ) Z µ ( x ) − Γ µ ( x ) (cid:17) − m n f o n ( x )+ f ¯ ν e ( x ) iγ µ (cid:16) ∂ µ + i f − g θ W Z µ ( x ) − Γ µ ( x ) (cid:17) (cid:16) − γ (cid:17) ν e ( x )+ f ¯ e − ( x ) n iγ µ (cid:16) ∂ µ + i f − e A µ ( x ) − i f − g θ W ((1 − θ W ) − γ ) Z µ ( x ) − Γ µ ( x ) (cid:17) − m e f o e − ( x )+ f g √ p ( x ) γ µ (cid:16) − γ (cid:17) n ( x ) W + µ ( x ) + f g √ n ( x ) γ µ (cid:16) − γ (cid:17) p ( x ) W − µ ( x )5 FIG. 3: Feynman diagram for the contribution of the torsion–chameleon field to the squared charge radius of the neutron + f g √ ν e ( x ) γ µ (cid:16) − γ (cid:17) e − ( x ) W + µ ( x ) + f g √ e − ( x ) γ µ (cid:16) − γ (cid:17) ν e ( x ) W − µ ( x ) + . . . , (94)where the ellipsis denotes the interactions with the Higgs field. Then. m p = 938 . m n = 939 . m e = 0 . f = e βφ/M Pl in powers of the chameleon field and keeping only the linear terms we arrive at the followinginteractions L int = − β m p M Pl ¯ p ( x ) p ( x ) φ ( x ) − β m n M Pl ¯ n ( x ) n ( x ) φ ( x ) − β m e M Pl ¯ e − ( x ) e − ( x ) φ ( x )+ e ¯ p ( x ) γ µ p ( x ) A µ ( x ) − g θ W ¯ p ( x ) γ µ (cid:16) − (1 − θ W ) γ (cid:17) p ( x ) Z µ ( x ) − e ¯ e − ( x ) γ µ e − ( x ) A µ ( x ) + g θ W ¯ p ( x ) γ µ (cid:16) (1 − θ W ) − γ (cid:17) e − ( x ) Z µ ( x )+ g θ W ¯ n ( x ) γ µ (cid:16) − γ (cid:17) n ( x ) Z µ ( x ) − g θ W ¯ ν e ( x ) γ µ (1 − γ ) ν e ( x ) Z µ ( x )+ g √ p ( x ) γ µ (cid:16) − γ (cid:17) n ( x ) W + µ ( x ) + g √ n ( x ) γ µ (cid:16) − γ (cid:17) p ( x ) W − µ ( x )+ g √ ν e ( x ) γ µ (cid:16) − γ (cid:17) e − ( x ) W + µ ( x ) + g √ e − ( x ) γ µ (cid:16) − γ (cid:17) ν e ( x ) W − µ ( x ) , (95)where we have omitted the interactions with the Higgs field, which do not contribute to the processes our interest.The contribution of the terms, containing Γ µ ( x ), which in the Minkowski spacetime is equal to Γ µ = i σ µν ( ℓnf ) ,ν ,can be transformed into total divergences and omitted. A. Contributions of chameleon to squared charge radius of neutron r n The torsion (chameleon) contribution to the squared charge radius of the neutron is defined by the Feynman diagramin Fig. 3. The chameleon–neutron interaction is defined by (see Eq.(95)) L nnφ ( x ) = − β m n M Pl ¯ n ( x ) n ( x ) φ ( x ) . (96)The Lagrangian of the q φγγ interaction, given by Eq.(30) for √− g = 1, can be transcribed into the form L φγγ ( x ) = − βM Pl F µν ( x ) F µν ( x ) φ ( x ) , (97)where we have omitted the total divergence. The analytical expression for the Feynman diagram in Fig. 3 is equal to M ( e − n → e − n ) = e β m n M Z d q (2 π ) i h ¯ u ( k ′ e ) γ µ m e − ˆ k e − ˆ q − i γ ν u ( k e ) i ( η σλ q ϕ − η σϕ q λ ) D σν ( q ) × ( η αλ ( q + k e − k ′ e ) ϕ − η αϕ ( q + k e − k ′ e ) λ ) D αµ ( q + k e − k ′ e ) 1 m φ − ( k n − k ′ n ) [¯ u ( k ′ n ) u ( k n )] , (98)where m φ is the chameleon mass Eq.(39) as a function of the chameleon–matter coupling constant β , the environmentdensity ρ and the Ratra–Peebles index n , and D αβ ( Q ) is the photon propagator Eq.(46). Substituting the photon6propagators D αµ ( q ) and D σν ( q + k e − k ′ e ), taken in the form of Eq.(46), into Eq.(98) one may show that the integranddoes not depend on the gauge parameter ξ , i.e. the integrand is gauge invariant.Measuring the electric charge on the neutron in the electric charge of the proton for the calculation of the neutronelectric radius we have to compare Eq.(98) to the amplitude M ( e − n → e − n ) = − e q e p r n [¯ u ( k ′ e ) u ( k e )][¯ u ( k ′ n ) u ( k n )] , (99)where e q = − e p and e p are the electric charges of the electron and proton, respectively, and r n is the squared chargeradius of the neutron. The product [¯ u ( k ′ e ) u ( k e )][¯ u ( k ′ n ) u ( k n )] is equivalent to the product [¯ u ( k ′ e ) γ u ( k e )][¯ u ( k ′ n ) γ u ( k n )]in the low–energy limit.From the comparison of Eq.(98) with Eq.(99) the contribution of the chameleon to the squared charge radius ofthe neutron can be determined by the following analytical expression r n = 6 β mm e m φ M Z d q (2 π ) i h ¯ u ( k e ) γ µ m e − ˆ k e − ˆ q − i γ ν u ( k e ) i ( η µν q − q µ q ν ) 1( q + i , (100)where we have set k ′ e = k e and k ′ n = k n . Merging denominators by using the Feynman formula1 A B = Z xdx [ Ax + B (1 − x )] (101)we arrive at the following expression r n = 6 β mm e m φ M Z dx x Z d q (2 π ) i ¯ u ( k e ) γ µ ( m e + ˆ k e + ˆ q ) γ ν u ( k e )[ m e (1 − x ) − ( q + k e (1 − x )) ] ( η µν q − q µ q ν ) . (102)Making use a standard procedure for the calculation of the integrals Eq.(102), i.e. i) the shift of the virtual momentum q + k e (1 − x ), ii) the integration over the 4–dimensional solid angle and iii) the Wick rotation, we arrive at the expression r n = − β m e mm φ M Z dx x Z d q (2 π ) − x ) q − m e (1 − x ) [ q + m e (1 − x ) ] = − π β m e mm φ M ℓn (cid:16) Mm e (cid:17) , (103)where M is the ultra–violet cut–off. For numerical estimates we set M = M Pl [29]. This gives r n = − π β m φ m e mM ℓn (cid:16) M Pl m e (cid:17) . (104)According to [37], the squared charge radius of the neutron can be defined by the expression r n = 3 b ne mα , (105)where α = 1 / .
036 and b ne are the fine–structure constant and the electron–neutron scattering length, respectively.For the experimental values of the electron–neutron scattering lengths b ne = ( − . ± . ± . × − fm and b ne = ( − . ± . ± . × − fm, measured from the scattering of low–energy electrons by Pb and
Bi[37], respectively, we get ( r n ) exp = − . and ( r n ) exp = − . [37], respectively. The theoreticalvalue of the squared charge radius of the neutron is r n = − . × − β m φ fm = − . × − β nn +1 n ( n + 1) (cid:16) nM Pl Λ ρ m (cid:17) n +2 n +1 fm , (106)where the chameleon mass is measured in meV. From the comparison to the experimental values we obtain β ≥ [1 . × n ( n + 1)] n +1 n (cid:16) ρ m nM Pl Λ (cid:17) n +2 n for Pb ,β ≥ [1 . × n ( n + 1)] n +1 n (cid:16) ρ m nM Pl Λ (cid:17) n +2 n for Bi , (107)7 n I Β M n I Β M FIG. 4: The lower bound of the chameleon–matter coupling constant β from the experiments on the electron–neutron scatteringlength for the scattering of the slow neutron by lead (left) and bismuth (right), respectively. The shaded area is excluded. respectively. One may use Eq.(107) for the estimate of the lower bound of the chameleon–matter coupling constant β . Since the experiments on the measuring of the electron–neutron scattering length have been carried out for liquidlead and bismuth [37] with densities ρ Pb = 10 .
678 g / cm and ρ Bi = 10 .
022 g / cm , respectively, in Fig. 4 we plot thelower bound of the chameleon–matter coupling β at which the contributions of the chameleon field are essential.The minimal lower bound β ≥ is ten orders of magnitude larger compared to the value β < . × , measuredrecently in the qBounce experiments [12]. As a result, the contribution of the chameleon field to the electron–scatteringlength in the environment of the liquid lead and bismuth is negligible. Thus, in order to obtain a tangible contributionof the chameleon to the electron–neutron scattering length b ne or the squared charge radius of the neutron r n , theexperiments should be carried out in the environments with densities of order ρ ∼ − g / cm or even smaller. B. Contributions of the chameleon field to the squared charge radius of the proton r p The results obtained above for the squared charge radius of the neutron can be applied to the analysis of thecontribution of the chameleon (torsion) to the squared charge radius of the proton. Since the interaction of thechameleon field with the proton is described by the Lagrangian Eq.(96) with the replacement n ( x ) → p ( x ), where p ( x ) is the operator of the proton field, the contribution of the chameleon field to the squared charge radius of theproton r p is defined by Eqs.(104) and (45) with the replacement r n → δr p , m n → m p and m e → m µ , where m µ is themass of the µ − –meson [13].The contribution of the chameleon field to the charge radius of the proton has been recently investigated by Braxand Burrage [38]. According to Brax and Burrage [38], the contribution of the chameleon field may solve the so–called“the proton radius anomaly” [39–42]. As has been shown in [43]–[47] the Lamb shift ∆ E s → p of the muonic hydrogen,calculated in QED with the account for the nuclear effects, can be expressed in terms of the charge radius of theproton r p ∆ E s → p = 209 . − . r p + 0 . r p , (108)where ∆ E s → p and r p are measured in meV and fm, respectively. The charge radius of the proton r p = 0 . r p = 0 . r p = 0 . δr p , is equal to δE s → p = ( − . . r p ) δr p = − . δr p , (109)where we have set r p = 0 . µ p scattering, is equal to (see Eq.(104)) δr p = − π β m φ m µ m p M ℓn (cid:16) M Pl m µ (cid:17) = − . × − β m φ , (110)where m µ = 105 . m p = 938 . m φ and δr p are measured in meV and fm , respectively. Substituting Eq.(110) into Eq.(109) we express the correction to the8 n I Β M FIG. 5: The chameleon–matter coupling constant β as a function of the index n , fitting the experimental value δE s − p =∆ E s − p − ∆ E s − p | r p =0 . = 0 .
311 meV, where ∆ E s − p = 206 . Lamb shift of the muonic hydrogen in terms of the parameters of the chameleon field theory and the matter density ρ m . We get δE s → p = 3 . × − β m φ = 5 . × − β nn +1 n (( n + 1) (cid:16) nM Pl Λ ρ m (cid:17) n +2 n +1 (111)with δE s → p = 0 .
311 meV [42]. In Fig. 5 we plot the coupling constant β as a function of the Ratra–Peebles index n at the environment density ρ m ≈
10 g / cm [6]. One may see that for δE s → p = 0 .
311 meV the lower bound of thechameleon–matter coupling constant β , at which the contribution of the chameleon is tangible, is β ≥ . This isseven orders of magnitude larger compared to the recent experimental upper bound β < . × [12]. C. Contribution of the chameleon field to the neutron β − –decay In this section we investigate the neutron β − –decay, caused by the interaction with the chameleon. This meansthat we investigate two reactions: i) the neutron β − –decay with an emission of the chameleon n → p + e − + ¯ ν e + φ and ii) the chameleon induced neutron β − –decay φ + n → p + e − + ¯ ν e . Since formally these two reactions are relatedby k φ → − k φ , where k φ is a 4–momentum of the chameleon, we give the calculation of the amplitude of the neutron β − –decay n → p + e − + ¯ ν e + φ with an emission of the chameleon. Neutron β − –decay with the chameleon particle in the final state n → p + e − + ¯ ν e + φ The calculation of the amplitude of such a decay we use the following effective interactions L eff = − β m n M Pl ¯ n ( x ) n ( x ) φ ( x ) − β m p M Pl ¯ p ( x ) p ( x ) φ ( x ) − G F √ V ud n [¯ p ( x ) γ µ (1 + λγ ) n ( x )] + κ M ∂ ν [¯ p ( x ) σ µν n ( x )] o [¯ e − ( x ) γ µ (1 − γ ) ν e ( x )] , (112)where G F = 1 . × − MeV − is the Fermi coupling constant, V ud = 0 . λ = − . κ = κ p − κ n = 3 . κ p = 1 . κ n = − . M = ( m n + m p ) / β − –decay n → p + e − + ¯ ν e + φ are shown in Fig. 6. Forthe calculation of the analytical expression of the decay amplitude we follow [59] and carry out it in the rest frameof the neutron, keeping the contributions of the terms to order 1 /M . The energy spectrum and angular distributionof the neutron β − –decay with polarised neutron and unpolarised proton and electron we may write in the following9 FIG. 6: Feynman diagrams of the neutron β − –decay with an emission of the chameleon n → p + e − + ¯ ν e + φ . general form d λ nφ d Γ = G F | V ud | m n β M M | M ( n → p e − ¯ ν e φ ) | F ( E e , Z = 1) Φ( ~k e , ~k ν , ~k φ ) (2 π ) δ (4) ( k n − k p − k e − k ν − k φ ) , (113)where F ( E e , Z = 1) is the relativistic Fermi function, describing the final–state electron–proton Coulomb interaction[11], d Γ is the phase–volume of the decay final state d Γ = d k p (2 π ) E p d k e (2 π ) E e d k ν (2 π ) E ν d k φ (2 π ) E φ (114)and k j for j = n, p, e, ν and φ is a 4–momentum of the neutron and the decay particles, respectively. The factorΦ( ~k e , ~k ν , ~k φ ) is the contribution of the phase–volume, taking into account the terms of order 1 /M . Following [59] oneobtains Φ( ~k e , ~k ν , ~k φ ) = 1 + 3 M (cid:16) E e − ~k e · ~k ν E ν (cid:17) + 3 M (cid:16) E φ − ~k φ · ~k ν E ν (cid:17) + 2 M E e E φ − ~k e · ~k φ E ν . (115)The last two terms define the deviation from the phase–volume factor, calculated in [59] for the neutron β − –decay n → p + e − + ¯ ν e . Taking into account the phase–volume factor Φ( ~k e , ~k ν , ~k φ ) we may carry out the integration overthe phase–volume of the n → p + e − + ¯ ν e + φ decay, neglecting the contribution of the kinetic energy of the proton.Then, | M ( n → p e − ¯ ν e φ ) | is the squared absolute value of the decay amplitude, summed over the polarisation ofthe decay electron and proton. The analytical expression of the amplitude M ( n → p e − ¯ ν e φ ) is defined by M ( n → p e − ¯ ν e φ ) = h ¯ u p ( k p , σ p ) 1 m p − ˆ k p + ˆ k φ − i O ( − ) µ u n ( k n , σ n ) ih ¯ u e ( k e , σ e ) γ µ (1 − γ ) v ¯ ν e ( k ν , + 12 ) i + h ¯ u p ( k p , σ p ) O (+) µ m n − ˆ k n + ˆ k φ − i u n ( k n , σ n ) ih ¯ u e ( k e , σ e ) γ µ (1 − γ ) v ¯ ν e ( k ν , + 12 ) i , (116)where u j ( k j , σ j ) for j = n, p and e are the Dirac bispinors of fermions with polarisations σ j , v ¯ ν e ( k ν , +1 /
2) is the Diracbispinor of the electron antineutrino and O µ is defined by [59] O ( ± ) µ ( k p , k n ) = γ µ (1 + λγ ) + i κ M σ µν ( k p − k n ± k φ ) ν . (117)In the accepted approximation the amplitude Eq.(115) can be defined by the expression M ( n → p e − ¯ ν e φ ) = 2 E φ [¯ u p ( k p , σ p ) O µ u n ( k n , σ n )] h ¯ u e ( k e , σ e ) γ µ (1 − γ ) v ¯ ν e ( k ν , + 12 ) i == 2 E φ [¯ u p ( k p , σ p ) O u n ( k n , σ n )] h ¯ u e ( k e , σ e ) γ (1 − γ ) v ¯ ν e ( k ν , + 12 ) i − E φ [¯ u p ( k p , σ p ) ~ O u n ( k n , σ n )] · h ¯ u e ( k e , σ e ) ~γ (1 − γ ) v ¯ ν e ( k ν , + 12 ) i (118)where O µ is given by O µ = γ µ (1 + λγ ) + i κ M σ µν ( k p − k n ) ν − k φµ M − i λ M σ µν γ k νφ (119)0and the matrices O and ~ O , taken to order 1 /M , are equal to O = − E φ M + λ M ( ~σ · ~k φ ) λ + κ M ( ~σ · ~k p ) − λ + κ M ( ~σ · ~k p ) − − E φ M + λ M ( ~σ · ~k φ ) (120)and ~O = λ~σ (cid:16) − E φ M (cid:17) − ~k φ M + i κ M ( ~σ × ~k p ) ~σ (cid:16) − κ M E (cid:17) − i λ M ( ~σ × ~k φ ) − ~σ (cid:16) κ M E (cid:17) − i λ M ( ~σ × ~k φ ) − λ~σ (cid:16) E φ M (cid:17) − ~k φ M + i κ M ( ~σ × ~k p ) , (121)where E = (( m n − m φ ) − m p + m e ) / m n − m φ ) is the end–point energy of the electron–energy spectrum. Thematrices O and ~O are defined to order 1 /M only [59] . For the calculation of the amplitude of the β − –decay of theneutron we use the Dirac bispinorial wave functions of the neutron and the proton u n ( ~ , σ n ) = √ m n (cid:16) ϕ n (cid:17) , u p ( ~k p , σ p ) = p E p + m p ϕ p ~σ · ~k p E p + m p ϕ p , (122)where ϕ n and ϕ p are the Pauli spinor functions of the neutron and proton, respectively. For the energy spectrum andangular distribution of the neutron β − –decay with polarised neutron and unpolarised proton and electron we maywrite in the following general form d λ nφ d Γ = 32 m p E e E ν G F | V ud | E φ β M M F ( E e , Z = 1) (2 π ) δ (4) ( k n − k p − k e − k ν − k φ ) (1 + 3 λ ) ζ ( E e ) × n a ( E e ) ~k e · ~k ν E e E ν + A ( E e ) ~ξ n · ~k e E e + B ( E e ) ~ξ n · ~k ν E ν + K n ( E e ) ( ~ξ n · ~k e )( ~k e · ~k ν ) E e E ν + Q n ( E e ) ( ~ξ n · ~k ν )( ~k e · ~k ν ) E e E ν + D ( E e ) ~ξ n · ( ~k e × ~k ν ) E e E − − λ λ E e M (cid:16) ( ~k e · ~k ν ) E e E ν − k e E e (cid:17) + 1 M
11 + 3 λ F φ o , (123)where k e = p E e − m e is the absolute value of the electron 3–momentum and ~ξ n is the unit polarisation vector ofthe neutron | ~ξ n | = 1. The correlation coefficients ζ ( E e ), a ( E e ), A ( E e ), B ( E e ), K n ( E e ), Q n ( E e ) and D ( E e ) can takenfrom [59] at the neglect of the radiative corrections. The correction coefficient F φ we may represent in the followingform F φ = F (1) φ + F (2) φ + F (3) φ , (124)where the correlation coefficients F ( j ) φ for j = 1 , , O and ~ O given by Eqs.(119) and (120),ii) the dependence of the 3–momentum of the proton ~k p on the 3–momentum of the chameleon particle, caused bythe 3–momentum conservation ~k p = − ~k e − ~k ν − ~k φ (see Eqs.(A.16) and (A.17) in Appendix A of Ref.[59]) and iii)the contributions of the phase–volume factor Eq.(115), respectively. The analytical expressions of these correlationcoefficients are equal to F (1) φ = (1 + 3 λ ) E φ − (1 − λ ) E φ ~k e · ~k ν E e E ν + 2 λ E φ ~ξ n · ~k e E e − λ E φ ~ξ n · ~k ν E ν + 2 λ ( ~ξ n · ~k φ )( ~k e · ~k ν ) E e E ν − λ ( ~ξ n · ~k e )( ~k ν · ~k φ ) E e E ν − λ ( ~ξ n · ~k ν )( ~k e · ~k φ ) E e E ν , (125) F (2) φ = ( λ − κ + 1) λ ) ~k e · ~k φ E e + ( λ + 2( κ + 1) λ ) ~k ν · ~k φ E ν + (2 κ + 1) λ ( ~ξ n · ~k φ ) − λ ( ~ξ n · ~k φ )( ~k e · ~k ν ) E e E ν − ( λ + ( κ + 1) λ + ( κ + 1)) ( ~ξ n · ~k e )( ~k ν · ~k φ ) E e E ν + ( λ − ( κ + 1) λ + ( κ + 1)) ( ~ξ n · ~k ν )( ~k e · ~k φ ) E e E ν (126)1and F (3) φ = h (cid:16) E φ − ~k φ · ~k ν E ν (cid:17) + 2 E e E φ − ~k e · ~k φ E ν ih (1 − λ ) ~k e · ~k ν E e E ν − λ (1 + λ ) ~ξ n · ~k e E e − λ (1 − λ ) ~ξ n · ~k ν E ν i . (127)Now we may integrate over the phase–volume of the n → p + e − + ¯ ν e + φ decay. First of all we integrate over the3–momentum of the proton ~k p . As has been mention above, we make such an integration at the neglect of the kineticenergy of the proton. Then, since one can hardly observe the dependence of the energy spectrum and the angulardistribution on the direction of the 3–momentum of the chameleon, we make the integration over the 3–momentumof the chameleon ~k φ . The obtained energy spectrum and angular distribution is d λ nφ ( E e , E ν , E φ , ~k e , ~k ν , ~ξ n ) dE e dE ν dE φ d Ω e d Ω ν = G F | V ud | π E φ β M M F ( E e , Z = 1) δ ( E − E e − E ν − E φ ) k e E e E ν (1 + 3 λ ) ζ φ ( E e ) × n a φ ( E e ) ~k e · ~k ν E e E ν + A φ ( E e ) ~ξ n · ~k e E e + B φ ( E e ) ~ξ n · ~k ν E ν + K n ( E e ) ( ~ξ n · ~k e )( ~k e · ~k ν ) E e E ν + Q n ( E e ) ( ~ξ n · ~k ν )( ~k e · ~k ν ) E e E ν + D ( E e ) ~ξ n · ( ~k e × ~k ν ) E e E ν − − λ λ E e M (cid:16) ( ~k e · ~k ν ) E e E ν − k e E e (cid:17)o . (128)The correlation coefficients ζ φ ( E e ), a φ ( E e ), A φ ( E e ) and B φ ( E e ) are equal to ζ φ ( E e ) = ¯ ζ ( E e ) + E φ M ,a φ ( E e ) = ¯ a ( E e ) + a (cid:16) E e E ν (cid:17) E φ MA φ ( E e ) = ¯ A ( E e ) + (cid:16) − λ + A (cid:16) E e E ν (cid:17)(cid:17) E φ M ,B φ ( E e ) = ¯ B ( E e ) + (cid:16) − λ + B (cid:16) E e E ν (cid:17)(cid:17) E φ M , (129)where a = (1 − λ ) / (1 + 3 λ ), A = − λ (1 + λ ) / (1 + 3 λ ) and B = − λ (1 − λ ) / (1 + 3 λ ) [58](see also [59]). Thecorrelation coefficients ¯ ζ ( E e ), ¯ a ( E e ), ¯ A ( E e ) and ¯ B ( E e ) are calculated in [59] by taking into account the contributionsof the weak magnetism and the proton recoil to order 1 /M but without radiative corrections.The rate of the decay n → p + e − + ¯ ν e + φ diverges logarithmically at E φ →
0. We regularise the logarithmicallydivergent integral by the chameleon mass m φ . As result we get λ nφ = G F | V ud | π β π M M f φ ( E , Z = 1) , (130)where f φ ( E , Z = 1) is the Fermi integral f φ ( E , Z = 1) = Z E m e dE e E e p E e − m e ( E − E e ) h ℓn (cid:16) E − E e m φ (cid:17) −
32 + 13 E − E e M i ¯ ζ ( E e ) F ( E e , Z = 1) , (131)where ¯ ζ ( E e ) is given by (see Eq.(7) of Ref.[59])¯ ζ ( E e ) = 1 + 1 M
11 + 3 λ h − λ (cid:16) λ − ( κ + 1) (cid:17) E + (cid:16) λ − κ + 1) λ + 2 (cid:17) E e − λ (cid:16) λ − ( κ + 1) (cid:17) m e E e i . (132)In Fig. 7 we plot the electron–energy spectrum ρ φ ( E e ) of the neutron β − –decay with an emission of the chameleon,defined by ρ φ ( E e ) = E e p E − m e ( E − E e ) h ℓn (cid:16) E − E e m φ (cid:17) −
32 + 13 E − E e M i ¯ ζ ( E e ) F ( E e , Z = 1) f ( E , Z = 1) , (133)where f ( E , Z = 1) is the Fermi integral calculated in [59], and compare it with the electron–energy spectrum ρ β − c ( E e )of the neutron β − –decay calculated in [59] (see Eq.(D-59) of Ref.[59]). The chameleon mass m φ is determined at thelocal density ρ ≃ . × − g / cm = 5 . × − MeV . This is the density of air at room temperature and pressure P ≃ − mbar [60, 61]. For β < . × we obtain λ nφ < × − s − and the branching ratio BR nφ < . × − .2 x x x x x x x x x x E e @ MeV D f Φ H E ,Z = L E e @ MeV D Ρ H E e L FIG. 7: (left) The values of the Fermi integral f φ ( E , Z = 1) as a function of the index n for n = 1 , , . . . ,
10. (right) Theelectron–energy spectra ρ φ ( E e ) (continuous lines) and ρ β − c ( E e ) (dashed line) of the neutron β − –decay with and without anemission of a chameleon, respectively. The densities ρ φ ( E e ) depend slightly on the index n and are represented by only onecontinuous blue line. FIG. 8: Feynman diagrams for the reaction φ + n → p + e − + ¯ ν e Neutron β − –decay φ + n → p + e − + ¯ ν e , induced by the chameleon field For the calculation of the amplitude M ( φ n → p e − ¯ ν e ) of the induced neutron β − –decay φ + n → p + e − + ¯ ν e wemay use the amplitude Eq.(116) with the replacement k φ → − k φ , where k φ is a 4–momentum of the chameleon. TheFeynman diagrams for the chameleon–induced neutron β − –decay is shown in Fig. 8. The cross section for the inducedneutron β − –decay is σ φ n → p e − ¯ ν e ( E φ ) = 14 m n E φ Z | M ( φ n → p e − ¯ ν e ) | (2 π ) δ (4) ( k n + k φ − k p − k e − k ν ) d k p (2 π ) E p d k e (2 π ) E e d k ν (2 π ) E ν , (134)where the contribution of the electron–proton final–state Coulomb interaction is not important and neglected. Usingthe results, obtained in previous subsection, we get σ φ n → p e − ¯ ν e ( E φ ) = (1 + 3 λ ) G F | V ud | π E φ β M M Z E + E φ m e dE e E e p E − m e ( E + E φ − E e ) . (135)Integrating over E e we obtain σ φ n → p e − ¯ ν e ( E φ ) = (1 + 3 λ ) G F | V ud | π β M M ( E + E φ ) E φ ( − m e ( E + E φ ) − m e ( E + E φ ) ! × s − m e ( E + E φ ) + 152 m e ( E + E φ ) ℓn E + E φ m e + s ( E + E φ ) m e − !) . (136)Using the results, obtained in [62] (see also [63]), we may analyse the quantity λ φn = Z ∞ dE φ Φ ch ( E φ ) σ φ n → p e − ¯ ν e ( E φ ) , (137)3which defines the number of transitions n → p e − ¯ ν e per second, induced by the chameleon, where Φ ch ( E φ ) is thenumber of solar chameleons per eV · s · cm , normalised to 10 % of the solar luminosity per unite area L ⊙ / πR ⊙ =3 . × eV s − cm − = 0 . [62], where L ⊙ = 2 . × eV s − and R ⊙ = 6 . × cm are thetotal luminosity and the radius of the Sun [13]. Following [62] we obtain that λ φn < − s − for β < . × [12]. VIII. CONCLUSION
We have developed the results, obtained in [16], where there was shown that the chameleon field can serve also asa source for a torsion field and low–energy torsion–neutron interactions, where the torsion field is determined by agradient of the chameleon one. Following Hojman et al. [18] we have extended the Einstein gravitational theory withthe chameleon field to a version of the Einstein–Cartan gravitational one with a torsion field. For the inclusion of thetorsion field we have used a modified form of local gauge invariance in the Weinberg–Salam electroweak model withminimal coupling and derived the Lagrangians of the electroweak and gravitational interactions with the chameleon(torsion) field.Gauge invariance of the torsion–photon interactions has been explicitly checked by calculating the amplitudes ofthe two–photon decay of the torsion (chameleon) field φ → γ + γ and the photon–torsion (chameleon) scattering γ + φ → φ + γ or the Compton photon–torsion (chameleon) scattering. Unlike the Compton–scattering, wherephotons scatter by free charged particles with charged particles in the virtual intermediate states, in the photon–torsion (chameleon) scattering a transition from an initial ( γ φ ) state to a final ( γ φ ) goes through the one–virtualphoton exchange (see Fig. 2a and Fig. 2b) and the local L γγφφ interaction (see Fig. 2c). The Feynman diagram Fig. 2dis self–gauge invariant due to the local L γγφ interaction. Gauge invariance has been checked directly by a replacementof the one of the polarisation vectors of the photons in the initial and final state by its 4-momentum. Since in thesereactions the coupling constant of the photon–torsion (chameleon) interaction is g eff = β/M Pl , in analogy with gaugeinvariance of photon–charge particles interactions, where electric charge is a coupling constant - unrenormalisableby any interactions, one may assert that the coupling constant g eff = β/M Pl should be also unrenormalisable byany interactions. This may place some strict constraints on possible mechanisms of the chameleon–matter couplingconstant β/M Pl screening [64, 65]. In this connection the Vainstein mechanism, leading the screening of the couplingconstant β/M Pl by the factor 1 / √ Z , where Z > β /M < − barn / eV , estimated for β < . × [12], the cross section for thephoton–chameleon scattering is extremely small and hardly may play any important cosmological role at low energies,for example, for a formation of the cosmological microwave background and so on. However, since in our approachthe coupling constant β γ /M Pl is fixed in terms of the coupling constant β/M Pl , the recent measurement of the upperbound β < . × can make new constraints on the photon–chameleon oscillations in the magnetic field of thelaboratory search for the chameleon field [67–69].In our approach the effective chameleon–photon coupling g eff = β/M Pl is equal to g eff = β/M Pl < . × − GeV − ,where we have used the experimental upper bound of the chameleon–matter coupling constant β < . × [12].The obtained upper bound g eff < . × − GeV − is in qualitative agreement with the upper bounds, estimatedby Davis, Schelpe and Shaw [55]. Then, the constraints on β : β < . × ( n = 1), β < . × ( n = 2), β < . × ( n = 3) and β < . × ( n = 4), measured recently by H. Lemmel et al. [57] using the neutroninterferometer, may be used for more strict constraints on the astrophysical sources of chameleons, investigated in[51]–[55].Using the photon–torsion (chameleon) interaction we have estimated the contributions of the chameleon field to thecharged radii of the neutron and proton. All tangible contributions can appear only for β ≫ . This, of course, is notcompatible with recent experimental data β < . × by Jenke et al. [12]. The branching ratio for the production ofthe chameleon in the neutron β − –decay n → p + e − + ¯ ν e + φ is extremely small Br( n → p e − ¯ ν e φ ) < . × − . In turn,the half–life of the neutron T ( φ n )1 / = ℓn /λ φn , caused by the chameleon induced neutron β − –decay φ + n → p + e − + ¯ ν e ,is extremely large T ( φ n )1 / = ℓn /λ φn > × yr. Of course, because of the neutron life–time τ n = 880 . .
1) s [13],being in agreement with the recent theoretical value τ n = 879 . .
1) s [59], the chameleon induced neutron β − –decaycannot be observed by a free neutron. The experiment, which can give any meaningful result, can be organised in away, which is used for the detection of the neutrinoless double β decays [70]. For example, it is known that the isotope Ge is both stable with respect to non–exotic weak, electromagnetic and nuclear decays and are neutron–rich. Itis unstable only with respect to the neutrinoless double β − –decay [70]. The experimental analysis of the low–boundon the half–life of Ge has been carried out by the GERDA Collaboration Agostini et al. [71, 72] by measuring theenergy spectrum of the electrons. The experimental lower bound has been found to be equal to T / > × yr4(90 % C . L . ). We would like also to mention the experiments on the proton decays, carried out by Super–Kamiokande[76, 77]. For specific modes of the proton decay, i.e. p → e + π , p → µ + π and p → νK + , for the half–life of the protonthere have been found the following lower bounds: T / > . × yr, T / > . × yr and T / > . × yr,respectively.Since the lifetime of the neutron τ n = 880 . .
1) s [13], one cannot analyse experimentally the chameleon–induced β − –decay on free neutrons. For the experimental investigation of the chameleon–induced β − –decay one may proposethe following reaction φ + Cd → In + e − + ¯ ν e , (138)where Cd is an atom with a stable nucleus in the ground state with spin (parity) J π = 0 + . Since In is an atomwith a nucleus in the ground state with spin (parity) J π = 1 + , the reaction Eq.(138) is determined by the Gamow–Tellertransition Cd → In for chameleon energies E φ ≥ .
088 MeV. The calculation of the threshold energies in weakdecay of heavy atoms with the account for the contribution of the electron shells can be found in [78].We have to note that the atom
In is unstable under the electron capture (EC) and β − decays with the branches56 % and 44 %, respectively [79]. Since the EC decay of In is not observable in the experiment on the chameleon–induced β − –decay, one may observe the β − –decay of In. However, the electron energy spectrum of
In is restrictedby the end–point energy E = 0 .
673 MeV and can be distinguished from the energy spectrum of the electron, appearingin the final state of the reaction Eq.(138).The main background for the chameleon–induced β − –decay Eq.(138) is the reaction ν e + Cd → In + e − , (139)caused by solar neutrinos with energies E ν e ≥ .
088 MeV. Because of the threshold energy the reaction Eq.(139) canbe induced by only the solar B and hep neutrinos [13]. Since the hep –solar neutrino flux is approximately 1000 timesweaker in comparison to the B–solar neutrino flux [13], the reaction Eq.(139) should be induced by the B–solarneutrinos.Concluding our analysis of standard electroweak interactions in the gravitational theory we would like to discussthe results, obtained recently by Obukhov et al. [73]. There, the behaviour of the Dirac fermions in the Poincar´egauge gravitational field including a torsion was analysed. The Hamilton operator of the spin–torsion interaction hasbeen derived [73]. In a weak gravitational field and torsion field approximation, which we develop in this paper, sucha spin–torsion interaction takes the form H spin − tors = −
14 ( ~ Σ · ~T + γ T ) , (140)where ~ Σ = γ ~γγ and γ = iγ γ γ γ are the Dirac matrices [25]. Then, T and ~T are the time and spatialcomponents of the axial torsion vector field T α = ( T , ~T ), defined by T α = − ε αβµν T βµν , (141)where T βµν is the torsion tensor field and ε αβµν is the totally antisymmetric Levi–Civita tensor ε = 1 [25]. Usingthe experimental data [74] and [75] on the measurements of the ratio of the nuclear spin–precession frequencies ofthe pairs of atoms ( Hg , Hg) [74] and ( He , Xe) [75] with nuclear spins and parities ( J π = − , J π = − ) and( J π =
12 + , J π =
12 + ), respectively, Obukhov, Silenko and Teryaev [73] have found the strong new upper bound on theabsolute value of the torsion axial vector field ~T . They have got | ~T || cos Θ | < . × − eV . (142)In the approach, developed in our paper, the tensor torsion field T βµν is equal to T βµν = ( β/M Pl )( g βν ∂ µ φ − g βµ ∂ ν φ )(see Eq.(7)). Multiplying such a tensor torsion field by the totally antisymmetric Levi-Civita tensor ε αβµν we get T α = 0. Thus, the upper bound of the absolute value of the axial vector torsion field Eq.(142), obtained by Obukhov,Silenko and Teryaev [73], does not rule out a possibility for a torsion field to be induced by the chameleon field as itis proposed in our paper. IX. ACKNOWLEDGEMENTS
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