Mass Creation from Extra Dimensions

Transcription

1 Journl of oern Physics, 04, 5, Publishe Online April 04 in SciRes. ss Cretion from Extr Dimensions Do Vong Duc, Nguyen ong Gio Institute of Physics, Hnoi, Vietnm Hung Vuong University, Ho Chi inh City, Vietnm Emil: Receive rch 04; revise 9 rch 04; ccepte 5 April 04 Copyright 04 by uthors n Scientific Reserch Publishing Inc. This work is license uner the Cretive Commons Attribution Interntionl icense (CC BY. Abstrct In this work we consier mechnism for mss cretion bse on the perioicity conition ictte from the compctifiction of extrimensions. It is shown tht the existence n the compctifiction of extrimensions re the origin for creting prticle mss in orinry 4-imensionl spce-time. ss of Higgs prticles themselves woul be lso originte from the geometric topology of extrimensions. Keywors Origin of ss, ss of Higg Prticles, Extr Dimensions. Introuction The existence of spce-time extrimensions hs been subject of intensive reserch stuy uring the lst eces []-[3]. The topology of extrimensions, especilly their compctifiction, plys crucil role in mny physicl spects, mostly in the construction of vrious moels of unifie theory of interctions, such s superstring theory, extene generl reltivity, n so on [4]-[7]. It is worth noting, on the other hn, tht it is in such pproches the prticle mss lwys remins problem of ctul chrcters. In this work we propose mechnism for mss cretion through the compctifiction of spcetime extrimensions. The crucil rgument is the propose perioicity conition ictte from the compctifiction of extr imensions. The originl fiel functions epen on ll spce-time coorinte components incluing those for extrimensions, the orinry fiel functions, orinry 4-imensionl spce-time consiere s effective fiel functions obtine by integrtion of the originl ones over extr spce-time. In Section, we present some generl principles relte to the compctifiction of extrimensions. How to cite this pper: Duc, D.V. n Gio, N.. (04 ss Cretion from Extr Dimensions. Journl of oern Physics, 5,

2 In Section 3, mechnism for mss cretion is trete.. Perioicity Compctifiction Conition or simplicity let us begin with the cse of one extr imension. Denote the 5-imensionl coorinte vector by x with =, 5. The Greek inices, υ, will be use s conventionl 4-imensionl orentz inices (0,,, n 3. We o not irectly cre from the extr imensions is topologiclly compctifie, but inste specific perioicity conition is put on the fiel functions epening on extr imensions, nmely (, 5 5 ( (, x x + = f x x ( where f ( is some prmeter function epening on the compctifiction length. The conition ( correspons to the eqution: With the reltions: In generl we cn put x f 5 ( ( = ( ( x g x = e lg ( g ln f π ni n Z ( = ( + ( ( ρ ( iθ e ( f = (4 g( = ln ρ ( + i( θ ( + π n or neutrl fiel, f is to be rel n therefore θ = 0, n = 0 The perioicity conition ( cn be generlize for the cse of rbitrry number of extr imensions in the following mnner or convenience we enote the extr imension coorintes x, x,, x + 4+ by y x, =,,, n write + =, ( ( (, (, The perioicity conition ( is now generlize to be: n the corresponing Eqution ( becomes: with the reltions: x x y x y (5 ( (, ( (, xy + = f xy (6 y ( ( ( ( xy, = g xy, ( ( ( ( f = f e 3. Effective iel Eqution n ss ( iθ ( ( ( ( ( ln ( ( g = f + i θ ( + π n The generl proceure of our tretment is s follows. We strt from the (4 + imensionl orentz invrint Kinetic grngin (x, y n the ction for the fiel (x, y efine s ( (3 (7 (8 478

3 where ( = ( ( ( 4 xxy (, S S y y S y y y y y n the integrl is performe over the whole extr spce time. The principle of miniml ction for S(y then gives the Euler-grnge eqution ( ( ( ( xy xy, xy, = 0 xy,, ( (9 (0 which in turn les to the eqution of Klein-Goron type: or the effective fiel efine s ( m ( x + = 0 ( ( y ( x, y x ( or illustrtion let us consier in more etils the cses of sclr, spinor n vector fiels. 3.. Sclr iel The free neutrl sclr fiel ( xy, Φ is escribe by the grngin xy (, = Φ( xy, Φ( xy, Φ Φ + = Φ Φ ( xy, ( xy, ( xy, ( xy, ( where, b is inkonski metric for extr imensions: y By inverting (7 into (, we obtin: An from here the eqution or the effective fiel With b 0 if b = if = b timelike if = b spcelike ( ( ( ( xy (, = Φ( xy, Φ ( xy, + g Φ xy, (3 = ( mφ + Φ ( x = 0 (4 ( x ( y ( xy, Φ = Φ ( ( ( m = Φ g (5 It is worth nothing tht the squre mss m Φ is positive if ll the extr imensions re spce-live, n cn be negtive if there exist time-live extr imensions. or chnge sclr fiel inste of ( we tke 479

4 + (, = Φ (, Φ(, xy xy xy + + = Φ Φ + Φ Φ = ( xy, ( xy, ( xy, ( xy, (6 An inste of (3 we hve: + Φ + = ( ( ( xy, = Φ xy, Φ xy, + g Φ xy, Φ xy, (7 An from here the eqution the sme Eqution s (4 with: 3.. Spinor iel Φ Φ ( ( ( ( ( ( In (4 + imensionl spce-time, the spinor fiel is escribe by the live grngin m = g (8 4+ i i + 4 xy (, = ψ ( xy, Γ ψ ( xy, ψγ ψ + ψγ ψ = component function ( xy, ψ with (9 where Γ enote (4 + Dirc mtrices obeying the nt commuttion reltions: { Γ Γ } = { 4+ } { }, υ υ Γ, Γ = 0 Γ, Γ = b b (0 ψ ψ +Γ 0 By inverting ( ( ( ψ xy, = g ψ xy, ( ψ ( ψ xy, = g ψ xy, ( * ψ ( Into (9 we obtin: i ( ( ( ( 4 xy, = ψ xy, Γ ψ xy, Im gψ ( ψγ + ψ ( An from here the eqution ( 4+ iγ Im gψ ( Γ ψ ( xy, = 0 3 = By cting from the left both sies of this eqution by b= ( b 4 ( υ iγ Im g An tking into ccount the reltions (0 we hve: υ ψ b Γ + b ( ( gψ ( ψ ( xy Im, = 0 (4 = 480

5 An hence ( ( Im ( ψ mψ g We note tht m ψ > 0 if ll the extr imensions re spce-like, be negtive if there exists time-like extr imension 3.3. Vector iel = (5 m ψ = 0 if ll We restrict ourselves to the cse = l n consier the neutrl vector fiel V (, conition An in corresponence (, ( (, V ( g ψ re rel, n m ψ cn xy stisfying the perioicity V xy+ = f V xy (6 The free vector fiel V (, where V xy gv V xy y V (, = ( (, ( = e ( f g xy is escribe by the grngin V N υ 5 xy (, = N = ( υ = V V + V V V V 4 ( υ υ V V υ υ υ V V (7 (8 By inverting (7 into (8 we hve: ( x, y = υ 55 ( V 5 V5 gv ( V V gv ( VV υ + 5 (9 4 Now we efine new physicl vector fiel W by putting W V V (30 ( 5 g V Expresse in terms of W, the grngin (9 hs the form: ( x, y = G G g ( WW 4 G W W υ υ υ υ υ 55 V (3 The grngin (3 les to the eqution: which mens tht the effective vector fiel ( gv ( = ( W x y W x, y ( = ( 0 Hs squre mss 48

6 W 55 V ( m = g (33 It s positive or negtive epening upon whether the extr imension is spce-like or time-like. 4. Conclusion n Discussion In this work we hve propose mechnism for the cretion of prticle mss. The key ie is tht the mss is originte from the compctifiction of extr imensions followe by the perioicity conition for the prticle fiels. It is worth noting tht ccoring to the mechnism the existence of tchyon hving negtive squre mss is closely relte to the existence of time-like extr imensions. In this work we hve consiere originlity for mss cretion, which is originte from the compctifiction of extrimensions. It is shown tht the mss spectrum is completely etermine by some functions of compctifiction length n closely relte to the metric of extrimensions. The key ie is tht the existence n the compctifiction of extrimensions re the origin for creting prticle mss in orinry 4-imensionl spce-time. In this connection one might think tht the mss of Higgs prticles themselves woul be lso originte from the geometric topology of extrimensions. The problem of whether there exists some mechnism llowing the extrimensions to crete Higgs prticles woul be problem of significnt mening. It might be lso tht the prticles coul cquire mss through ifferent mechnisms, incluing those relte to extrimensions. This coul le to thinking tht the probbility for experimentlly fining Higgs prticles is much smller thn the vlue theoreticlly obtine when Higgs mechnism is consiere s the only one for mss cretion. References [] Cs ki, C. (004 TASI ectures on Extrimensions n Brnes. hep_ph [] Rnll,. n Schwrz,.D. (00 Quntum iel Theory n Unifiction in A55. JHEP, 0, 003. [3] Sunrum, R. (005 To the ifth Dimension n Bck. TASI. [4] Green,.B., Schwrz, J.H. n Witten, E. (987 Superstring Theory. University Press, Cmbrige. [5] Brink,. n Henneux,. (988 Principles of String Theory. Plenum Press, New York. [6] Crroll, S.. (997 ecture Notes on Generl Reltivity. University of Cliforni, Okln. [7] urln, G., et l. (997 Superstrings, Super Grvity n Unifie Theories Worl Scientific. 48