Mass creation from extra-dimensions 1 2 Dao Vong Duc , Nguyen Mong Giao 1Institute of Physics, Hanoi 2Hung Vuong University, Ho Chi Minh 3 1 Abstract 0 2 In this work we consider a mechanism for mass creation based on the periodicity con- y dition dicated from the compactification of extradimensions. It is also shown that the a M existence of Tachyon having negative square mass is closely related to time-like extradi- mensions. 9 Keyword: Mass creation, Extra-dimensions ] h 1. Introduction p The existence of space-time extradimensions has been a subject of intensive research study - n during the last decades [1−3]. e g TheTopologyofextradimension, especially their compactification playacrucial roleinmany . s physical aspects, mostly in the construction of various models of Unified theory of interactions, c i such as Superstring theory, Extended General Ralativity, and so on [4−7]. s y It is worth noting, on the other hand, that is such approaches the particles mass always h remains a problem of actual characters. p [ Inthisworkweproposeamechanismformasscreationthroughthecompactificationofspace- timeextradimensions. Thecrucial argument istheproposedperiodicityconditiondictatedfrom 2 v the compactification of extradimensions. 5 The original field functions depend on all space-time coordinate components including those 0 4 for extradimensions, the ordinary field functions ordinary 4-dimensional space-time are consid- 1 eredaseffectivefieldfunctionsobtainedbyintegrationoftheoriginalonesoverextraspace-time. . 1 In section 2, we present some general principles related to the compactification of extradi- 0 mensions. 3 1 In section 3, a mechanism for mass creation is treated. : v 2. Periodicity campactification condition i X For simplicity let us begin with the case of one extra dimension. Denote the 5-dimensional r coordinate vector by xM with M = µ,5. The Greek indices µ, ν,... will be use as conventional a 4-dimensional Lorentz indices (0,1,2, and 3). We do not directly care from the extra dimen- sions is topologically compactified, but instead a specific periodicity condition is put on the field functions depending on extra dimensions, namely F(xµ,xς +L) = f (L).F(xµ,xς) (1) L Where f (L) is some parameter function depending on the compactification lenth L. L The condition (1) corresponds to the equation: ∂ F(xM) = g (L).F(xM) (2) ∂xς L 1 With the relations: f (L) = elgM(L) (3) L g (L) = 1 [lnf (L)+2πni](n ∈ Z) L L L In general we can put f (L) = ρ (L).eiθF(L) (4) L F g (L) = 1 [lnρ (L)+i(θ (L)+2πn] L L F F For neutral field, F+ = F , f (L) is to be real and therefore θ = 0, n = 0 . L F The periodicity condition (1) can be generalized for the case of arbitrary number of extra dimensions in the following manner. Forconveniencewedenotetheextradimensioncoordinatesx5,x6,··· ,x4+d byyα ≡ x4+α,,α = 1,2,··· ,d and write F(xM) ≡ F (xµ,yα) ≡ F(x,y) (5) The periodicity condition (1) is now generalized to be: F (x,yα +Lα) = f(α)(Lα).F(x,y) (6) F and the corresponding equation (2) becomes: ∂ F(x,y) = g(α)(L ).F(x,y) (7) ∂yα F α with the ralations: f(α)(Lα) = f(α)(Lα).eiθF(α)(L ) F F α 1 g(α)(L ) = lnf(α)(Lα)+i(θ(α)(Lα)+2πn) (8) F α Lα F F h i 3. Effective field equation and mass The general procedure of our treatment is as follows. We start from the (4+d) dimensional Lorentz invariant Kinetri Lagrangian L(x,y) and the action for the field F(x,y) defined as S = S(y)(dy) (9) Z S(y) ≡ d4x.L(x,y) Where (dy) ≡ dy1.dy2...dyd and the general is performed over the whole extra space time. R The principle of minimal action for S(y) then gives the Euler-Lagrange equation ∂L(x,y) ∂L(x,y) −∂ = 0 (10) µ ∂F(x,y) ∂(∂ .F(x,y)) µ Which in turn leads to the equation of Klein-Gordon type: (✷+m2)F(x) = 0 F For the effective field defined as F(x) ≡ (dy)F(x,y) (11) Z For illustration let us consider in more details the cases of scalar, spinor and vector fields. 2 3.1. Scalar field The free neutral scalar field Φ(x,y) is described by the Lagrangian 1 L (x,y) = ∂MΦ(x,y).∂ Φ(x,y) M 2 d (12) 1 ≡ ∂MΦ(x,y)∂ Φ(x,y)+ ς ∂ Φ(x,y).∂ Φ(x,y) M aa a a 2 ( ) a=1 X Where ∂ ≡ ∂ ,ς is a MinKonski metric for extra dimensions: a ∂ya ab 0, ifa 6= b ς = 1, ifa = b−timelike ab  −1, ifa = b−spacelike  By inverting (7) into (12), we obtain: d 1 L(x,y) = ∂MΦ(x,y).∂ Φ(x,y)+ ς g(a)(L ) 2.Φ2(x,y) (13) M aa a 2 ( ) a=1 X (cid:0) (cid:1) And from here the equation (✷+m2)Φ(x) = 0 (14) Φ For the effective field Φ(x) = (dy)Φ(x,y) Z With m2 = −Σ ς g(a)(L ) 2 (15) Φ a aa a It is worth nothing that the squared mass(cid:0)m2 is po(cid:1)sitive if all the extra dimensions are φ space-live, and can be negative if there exists time-live extra dimensions. For changed scalar field instead of (12) we take L(x,y) = ∂MΦ+(x,y).∂ Φ(x,y) M d (16) = ∂µΦ+(x,y).∂ Φ(x,y)+ ∂aΦ+(x,y).∂ Φ(x,y) µ a a=1 X And instead of (13) we have: L(x,y) = ∂MΦ+(x,y).∂ Φ(x,y) M d 2 (17) + ς g(a)(L ) .Φ+(x,y).Φ(x,y) aa Φ a Xa=1 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) And from here the same equation as (14) with: 2 m2 = −Σ ς g(a)(L ) (18) Φ a aa Φ a (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 3 3.2. Spinor field 4+d In(4+d)-dimensionalspace-time, thespinorfieldisdecribedbya2 componentfunction 2 ψ (x,y) with the free Lagrangian a d i i L(x,y) = ψ(x,y).ΓM∂ ψ(x,y) ≡ ψΓµ∂ ψ + ΨΓa+4∂ ψ (19) M µ a 2 2 ( ) a=1 X Where ΓM denote (4+d) Dirac 24+d x 24+d matrices obeying the ant commutation rela- 2 2 tions: ΓM,Γν = 2ςµν (20) (cid:8)ΓM,Γ4(cid:9)+a = 0 Γ(cid:8)4+a,Γ4+b (cid:9)= 2ςab (cid:8) ψ ≡ ψ +(cid:9)Γ0 By inverting ∂ ψ(x,y) = g(a)L .ψ(x,y) (21) a ψ a ∂ ψ(x,y) = g(a)L .ψ(x,y) a ψ a Into (19) we obtain: i L(x,y) = ψ(x,y).ΓM∂ ψ(x,y)−Img(a)(L )ψΓ4+aψ (22) 2 M ψ a And from here the equation d iΓM∂ − img(a)(L ).Γ4+a ψ(x,y) = 0 (23) µ ψ a ! a=1 X By acting from the left both sides of this equation by d iΓυ∂ − img(b)(L ).Γ4+b υ ψ b b=1 X And taking into account the relations (20) we have: d 2 ✷− ς img(a)(L ) ψ(x,y) = 0 (24) aa ψ a ( ) Xa=1 (cid:16) (cid:17) And hence 2 m2 = − ς img(a)(L ) (25) ψ aa ψ a Xa (cid:16) (cid:17) We note that m2 > 0 if all the extra dimensions are space-like, m2 = 0 if all g(a) are real, ψ ψ ψ and m2 can be negative if there exists time-like extra dimension. ψ 3.3. Vector field We restrict ourselves to the case d=1 and consider the neutral vector field V (x,y) sat- M isfying the periodicity condition V (x,y +L) = f (L).V (x,y) (26) M V M 4 And in correspondence ∂ V (x,y) = g (L).V (x,y) (27) M V M ∂ y f (L) = eLg (L) V V The free vector field V (x,y) is described by the Lagrangian M −1 L (x,y) = F FMN MN 4 −1 = (F Fµυ +2F Fµς) (28) µυ µς 4 −1 1 = F Fµυ − ς (∂ V .∂µV +∂ V .∂ Vµ −2∂ V .∂ Vµ) µυ ςς µ ς ς ς µ ς µ ς ς 4 2 Where F ≡ ∂ V −∂ V µυ µ υ υ µ F ≡ ∂ V −∂ V µς µ ς ς µ By inverting (27) into (28) we have: −1 1 L(x,y) = F Fµυ − ς ∂ V .∂µV +g2(L)V Vµ −2g (L)∂ V Vµ (29) 4 µυ 2 ςς µ ς ς V µ V µ ς Now we define a new physical vect(cid:0)or field W by putting (cid:1) µ 1 W ≡ V − ∂ V (30) µ µ µ ς g (L) V Expressed in terms of W , the Lagrangian (29) has the form: µ 1 1 L(x,y) = − G Gµυ − ς g2(L)W Wµ (31) 4 µυ 2 ςς V µ G ≡ ∂ W −∂ Wµ µυ µ υ υ The Lagrangian (31) leads to the equation: ✷−ς g2(L) W = 0 (32) ςς V µ Which means that the effective(cid:0)vector field (cid:1) L W (x) = dy.W (x,y) µ µ Z0 Has squared means m2 = −ς g2(L) (33) W ςς V It’s positive or negative depending upon whether the extra dimension is space-like or time-like. 4. Conclusion In this work we have proposed a mechanism for the creation of particle mass. The key idea is that the mass is originated from the compactification of extra dimensions followed by the periodicity condition for the particle fields. It isworth nothing thataccording to themechanism the existence of tachyon having negative squared mass is closely related to the existence of time-like extra dimensions. 5 References [1] C.Cs’ aki, ASI Lectures on Extradimensions and Branes, help ph 0404096. [2] L.RandallandM.D.Schwarz (2001),QuantumFieldTheoryandUnificationinAd55, JHEF, 0111,003 [3] R.Sundrum (2005), To the fifth Dimension and Back TASI [4] M.B.Green, J.H.Schwarz, E.Witten (1987), Superstring Theory, Cambridge, University Press. [5] L.Brink, M.Henneaux (1988), Principles of String Theory, Plenum Press, New York. [6] S.M.Carroll (1997), Lecture Notes on General Relativity, University of California. 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