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Kaluza-Klein Theories

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Kaluza-Klein Theories Kaluza-Klein theories • Add extra space-time • The 5-D general coordinate dimensions to unify gravity invariance broken in ground and electromagnetism state ➝ ordinary gravity in (T. Kaluza 1921) 4-D plus an Abelian gauge field • The fifth dimension is • U(1) gauge symmetry appears compact, periodic, very small associated with coordinate (O. Klein 1926) transformations on circle • Start with a theory of Einstein • Parameters of the two gravity in 5-D. One of the theories connected, since they dimensions is compactified in have same origin a very small circle • Tower of massive scalars M 2 n/R2 KK / 113 Kaluza theory • Consider a 5D theory only with gravity, the gravitons are hMN M,N = µ, 5 • Correspond to fluctuations around flat space gMN = ⌘MN + hMN which decompose into a spin 2 h µ ⌫ , identified with a graviton, a spin 0, h µ 5 identified with a photon, and a scalar h55 • The first proposal did not include massless charged matter particles 114 faster as r increases, F 1/r2+d, than the gauge forces, F 1/r2. Of course, we know that at ⇠ ⇠ very large distances gravity lives in 4D, since we know that Newton’s law reproduces very accurately, for example, the orbits of the planets. This means that the extra dimensions must be compact with a compactification radius R. At distances larger than R we will have a 4D theory with Newton’s law: m m F = G 1 2 , (52) N r2 where GN is the observed Newton’s constant. Matching Eqs. (51) and (52) at r = R one gets G G = grav . (53) N Rd Therefore large R implies a small GN . In other words, 4D gravity must be weaker than the other interac- tions if its field lines spread over large extra dimensions. The larger the extra dimensions, the weaker is gravity. This is a very interesting possibility that, as we will see below, has spectacular phenomenological implications. Several years later Randall and Sundrum found a different reason to have extra dimensions [43]. If the extra dimensions were curved or ‘warped’, gravitons would behave differently than gauge bosons and this could explain their different couplings to matter. Below we will discuss these two scenarios in more detail. Let us first explain the situation in the old Kaluza–Klein picture. 9 Kaluza–Klein theories As we said before, Kaluza was one of the first to consider theories with more than four dimensions in an attempt to unify gravity with electromagnetism. Klein developed this idea in 1926 using a formalism that is usually called Kaluza–Klein reduction [44]. Although their initial motivation and ideas do not seem to be viable, the formalism that they and others developed is still useful nowadays. This is the one that will be considered below. y xµ Kaluza-KleinFig. 14: Compactification theories on S1 For• simplicity,The action we will in start 5D, with y is a 5Dfifth-dimension. field theory of scalars. Only The scalars action is given by 4 2 2 2 4 S5 = d xdyM @µφ + @yφ + g5 φ , (54) − ⇤ | | | | | | Z h i where by y we refer to the extra fifth dimension.y = Wey have+2⇡ extractedR a universal scale M in front of the y compactified in a circle ⇤ action in•This order corresponds to keep the to the 5D identification field with the of samey with mass-dimensiony +2⇡R. In such as a in case, 4D. we Let can us expand now consider the 5D complex that the y y +2⇡R Thisfifth corresponds dimensionscalar field is to in compact the Fourier identification and series: flat. We of willwith consider that. it In has such the a topology case, we of can a circle expandS1 theas in 5D Fig. complex14. scalar field• inExpand Fourier series: the complex scalar in a Fourier series 1 iny/R (n) (0) iny/R (n) φ(x, y)= e φ 25(x)=φ (x)+ e φ (x) , (55) 1 iny/R (n) (0) iny/R (n) φ(x, y)= ne= φ (x)=φ (x)+ n=0e φ (x) , X1 X6 (55) n= n=0 that inserted in Eq. (54) andX1 integrated over y gives X6 that inserted inIntegrating Eq. (54) and over integrated y gives over y gives (0) (n) • S5 = S4 + S4 (56) (0) (n) S = S + S 2/r (56) masslesswhere scalar 5 4 4 1/r (0) 4 (0) 2 2 (0) 4 0 where S4 = d x 2⇡RM @µφ + g5 φ , (57) − ⇤ | | | | Z h i (0) 2 1 (n) 4 4 (0) 2 (2n) 2(0) 4 nFig. 17: The(n Kaluza) 2 Klein tower of massive states due to an extra S dimension. Masses mn = n /r grow S = d x 2⇡RM @ φ + g φ , | | 4 S4 = d x 2⇡RMµ @µφ5 + linearly withφ the fifth dimension’s+ quartic wave number n Z. couplings . (58)(57) − − ⇤ | ⇤ | | | | | R | | −2 Z Z n=0 h X6 ⇣i ⌘ 4 µ µ µ tower of 2 =2⇡ r d x @ ' (x ) @ ' (x )⇤ + ... =2⇡r + ... (n) 4 (n) 2 n (n) 2 0 µ 0 S4D Z massiveS 4modes= d x 2⇡RM @µφ + φ + quartic couplings(0) . (58) We see that the above action⇤ corresponds to115 a 4D theoryThis with means that a the massless 5D action reduces scalar to one 4D actionφ forand a massless a scalar tower field plus of an infinite sum − | | R of| massive scalar| actions in 4D. If we are only− interested in energies smaller than the 1 scale, we may Z (n) n(0)=0 (n) r massive modes φ . The field φX6 will be referred to⇣ as⌘ theconcentrate zero-mode, only on the action while of the masslessφ mode.will be referred to as Kaluza–Klein (KK) modes. 3.2 Compactification of a Vector Field in 5 Dimensions(0) Vector fields are decomposed in a completely analogousφ way: AM = Aµ,A4 = φ . Consider the We see that the above action corresponds to a 4D theory with a massless scalar and{ } a{ tower} of action This(n) reduction of a(0) 5D theory to a 4D theory allows one to treat 5D theories(n) as1 4D field theories. = d5x F F MN (46) massive modes φ . The field φ will be referred to as the zero-mode, whileS5φD willg2 MN be referred to as This is very useful since we know much more about 4D theories than 5D theories.Z At5D low energies (large with a field strength Kaluza–Klein (KK) modes. F = @ A @ A (47) distances) we know that massive states in 4D theories can be neglected. ThereforeMN M theN − N effectiveM theory implying (0) Thisat energies reduction below of a1 5D/R is theory described to a by 4D the theory zero-mode allows Eq. one (57 to). Aftertreat 5Dnormalizing theories@M @ A φ@ asM @ 4D,A we=0 field. obtain theories. from (48) M N − N M This is veryEq. useful (57) since we know much more about 4D theoriesIf we now than choose 5D a gauge, theories. e.g. the transverse gauge: At low energies (large M M (0) @ AM =0,A0 =0 @ @M AN =0, (49) 4 (0) 2 2 (0) 4 ) distances) we know that massive statesS4 in= 4D theoriesd x @µ canφ be+ neglected.g4 φ , Therefore the effective theory(59) − | | then this obviously| becomes| equivalent to the scalar(0) field case (for each component AM ) indicating an at energies below 1/R is described by the zero-modeZ Eq. (57infinite). After tower of massive normalizing states for each masslessφ state in 5D., weIn order obtain to find the 4D effective from action we h once again plug this intoi the 5D action: Eq. (57) where the 4D self-coupling is given by 2 S5D 7! S4D g 2⇡r 2⇡r 2 5 = d4x F µ⌫ F + @ ⇢ @µ⇢ + ... (0) g = . 2 (0) (0)µ⌫ 2 µ 0 0 4 4 (0) 2 2 (0) 4 g5D g5D (60) S4 = d x @µφ2⇡RM+ g4 φ , Z ✓ ◆ (59) F µ⌫ ⇢ − | | ⇤ Therefore| we end| up with a 4D theory of a massless gauge particle (0), a massless scalar 0 from the massless Kaluza-Klein state of φ and infinite towers of massive vector and scalar fields. Notice that the This equation tells us that the strengthZ of theh interaction of the zero-modei decreases as the radiusMN in-µ⌫ where the 4D self-coupling is given by gauge couplings of 4- and 5 dimensional actions (coefficients of FMNF and Fµ⌫F ) are related by creases. If R is large, the scalar is weakly coupled. 1 2⇡r 2 2 = 2 . (50) g g4 g5 The general features described aboveg2 = for a 5D5 scalar. will also hold for gauge fields and gravity.(60) 4 2⇡RM 24 After Kaluza–Klein reduction, we will have a 4D theory⇤ with a massless gauge field and a graviton: This equation tells us that the strength of the interaction2 of the zero-mode decreases as the radius in- 4 M 1 MN creases. If R is large, the scalar is weaklyd xdyM coupled.⇤ + 2 F FMN − ⇤ 2 R 4g5 The general features describedZ above for a 5D scalar will also hold for gauge fields and gravity. 4 2 (0) 2⇡R (0) µ⌫ (0) After Kaluza–Klein reduction, we= will haved xM a 4D⇡RM theory with+ a massless2 F F gaugeµ⌫ + field... and a graviton: − ⇤ ⇤ R 4g Z 5 4 M 21 (0)1 1 (0) µ⌫ (0) 4 d x + MN2 F Fµ⌫ + ... , (61) d⌘xdyM 16⇤⇡GN+R F4g FMN − Z ⇤ 2 R 4g2 4 Z 5 where (0) is the 4D scalar-curvature containing the zero-mode2⇡R (massless) graviton, and F (0) is the R 4 2 (0) (0) µ⌫ (0) = d xM ⇡RM + 2 F Fµ⌫ + ..
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