Extra-dimensional models on the lattice and the Layer Phase. Petros Dimopoulos

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1 Extra-dimensional models on the lattice and the Layer Phase Petros Dimopoulos Frascati, 7/6/008

2 OUTLINE Introductory hints on the extra dimensional theories Models with extra dimension on the lattice; inclusion of anisotropic couplings in the action Pure U() model in D and the Layer phase D anisotropic Abelian-Higgs model: The Higgs Layer phase D SU()-Higgs model in the adjoint representation Remarks & Conclusions

3 Extra dimensions As far as it is known, there is not any fundamental principle for which the space-time should be 3+ dimensional Hence, it might be legitimate to assume extra spatial dimensions. It can not be denied that, up to now, the theoretical exploration of extra dimensional models is rather speculative than based on well established facts. Nevertheless, the construction of the extra dimensional models is guided by the requirement of physical consistency in order to show possible ways on how the extra dimensions can be revealed (or why are hidden) and to connect the extra-dimensional models with a more fundamental theory in order to solve long-standing theoretical problems.

4 Kaluza-Klein proposal (90, 96): Consider a five-dimensional flat space with the fifth dimension compactified The photon resides in the extra dimension The unification of gravity with electromagnetism can be achieved Extra dimensions à la Kaluza-Klein are hidden as a result of their size : l ~ 0-33 cm. Need for energies as high as the Planck energy to make them detectable.

5 Large Extra dimensions The idea of models with Large Extra dimensions is connected with the Brane world picture The ordinary matter (with exception of gravitons) is localised to a 3-D submanifold (brane) embedded in a fundamental multi-dimensional space (bulk) It is assumed that the brane picture is an effective model representation. It comes from a high-energy theory which appears at a fundamental scale M * The large extra dimensions should be so large as to non contradict the observations For compactified extra dimensions their size can be as big as a fraction of millimeter

6 Large Extra dimensions (continue) One of the main motivations for the Brane picture used in higher dimensional models is to give an answer to the : Hierarchy Problem Why M Pl >> M EW? Possible answer: The fundamental scale is M * ~ TeV in a theory defined in a 4+n space ADD model Arkani-Hamed, Dimopoulos & Dvali Phys.Lett.B 998 RS models Randall & Sundrum Phys.Rev.Lett. 998

7 ADD model Consider a flat brane (no brane tension) embedded in a 4+n space-time Standard matter resides on the brane Gravity spreads in the bulk The extra dimensions are compactified The metric does not depend on the extra coordinates Define a unique fundamental scale M * Volumetric scaling: M Pl = M n+ * V δ It is sufficient to assume n and R~0. mm for getting M * TeV

8 RS model Consider a five-dimensional anti de Sitter space the extra dimension obeys to z = z + π & z -z (S /Z orbifolded) negative cosmological constant in the bulk Assume the existence of two branes resided along the extra dimension and having opposite value of tension. Matter is assumed on the branes while the gravity can be spread in the bulk It is shown that the metric admitted for this set-up takes the form: ds = ω (z) η µν dx µ dx ν dz (with ω (z) = e The extra dimension becomes warped as a price of maintaining a 4D Lorentz space at every point of the extra dimension (non-factorizable geometry). The model provides an exponential hierarchy expressed by the distance which separates the two branes along the extra coordinate: -k z ) M Pl ~ e kzc M *

9 (Non) Renormalisability The couplings defined in extra dimensional theories are of negative mass dimension The current attitude is to make the conjecture to treat these theories as effective theories for energy scales up to /n E ~ /g (n is the mass dimension of the coupling) For bigger energies the effective theory has to be replaced by a more fundamental theory.

10 (4+)-dimensional pure U() gauge model

11 The gauge field in extra dimensional space What about a possible extension of the gauge field in the bulk? Is there any possibility of gauge localisation on the brane in the RS set-up? The answer given in terms of analytical work is negative Pomarol, Phys.Lett.B000 Davoudias, Hewett & Rizzo, Phys.Lett.B000 But since the localisation of the gauge field on the brane may involve strong coupling dynamics along the extra dimensions, the non-perturbative methods are necessary for giving an answer.

12 Abelian Gauge field in the RS background kz Assume the RS metric : ds = e ( dx dx dx dx ) dz 0 3 Consider a -dim abelian gauge model in the background of the RS metric: S gauge = 4g = d 4 d xdz 4g x g F F µν MN F κλ F η KQ µκ η g νλ MK g NQ g e kz F µ F ν η µν In the Euclidean space the gauge action reads: E k x S = d x F F + e F F µ, ν =,...,4 4g g gauge µν κλ µ µ We get a model with two different couplings, one defined on the 4D subspace and the second (bigger ) along the extra dimension. P.D., K. Farakos, A. Kehagias and G. Koutsoumbas NPB 00

13 On the lattice we define the pure U() Wilson gauge action with anisotropic couplings: S gauge = β ( ReU (x)) + β ( ReU (x)) µν µ x, µ < ν 4 x, µ 4 Link variables: Plaquettes: iasaµ ia { U e, U e A = } µ U = = M U U µν µ (x) = U (x) = U µ µ (x)u (x)u ν (x + a (x + a s s µˆ)u µˆ)u µ µ (x + a (x + a s νˆ)u ˆ)U ν (x) (x) The Fu-Nielsen proposal for dimensional reduction from lattice anisotropic models ( couplings β β ) Fu and Nielsen NPB 984, 98 Phase diagram becomes richer: A new phase appears

14 LAYER PHASE (I) Confining phase For β 0 and β =0 phase diagram of the 4D isotropic model 4D Coulomb phase Confining phase For β = β phase diagram of the D isotropic model D Coulomb phase Confining phase 4D Coulomb phase For β β β phase diagram of the D anisotropic model Layer phase Confining extra dimension D Coulomb phase

15 LAYER PHASE (II) The potential between heavy test charges is closely connected with the Wilson loops: W W W W µν µν µν µ (L, L ) exp (L, L ) exp (L, L ) exp (L, L ) exp [ σl L ] [ τ(l + L )] [ τ(l + L )] [ σ (L + L )] (Confining (Coulomb phase, phase, Layer phase µ, ν µ, ν ) ) Phase Diagram from Mean Field analysis D-Coulomb Confining phase Fu and Nielsen NPB 984, 98

16 Phase Diagram Monte Carlo Study Basic order parameters 4D (or space) plaquette : extra dimension (or transverse) plaquette : Pˆ Pˆ S 6N 4N x, µ < ν 4 x, µ 4 cos (F cos (F µ µν (x)) (x)) P = S PˆS P = Pˆ Behaviour of the Plaquette in the two limits of the coupling values P = β/ + O( β Dβ ) + O( β ) [ β<<, strong coupling ] [ β>>, weak coupling ] D is the dimension of the space

17 β =.0 Confining D Coulomb Phase Transition ~ [-(/β)] : D-Coulomb phase ~ β /: Confining phase P.D.,Farakos, Kehagias and Koutsoumbas NPB 00 Hulsebos, Korthals-Altes and Nicolis NPB 99 Big hysteresis loop Two-state signal st order PT

18 Confining Layer Phase Transition There is very strong evidence that the 4D Confining Coulomb phase transition is first order and takes place at β=.033() (β =0). [Arnold, Bunk, Lippert and Schilling 003] What does it happen as β > 0? We do an extensive study of the PT for two values of β =0.0 and 0. while we let β variable.

19 Confining Layer Phase Transition (continue) β =0. As the lattice volume becomes bigger the transition becomes steeper For β<<, P β/ S : Confining phase For β>>, P S (-/4β) : 4D-Coulomb phase For all β, P β /=0. (constant) extra dimension is confined!!

20 Identification of the phases using the helicity modulus The helicity modulus is an order parameter; it gives the response of the system to an external elecrtomagnetic flow. It is defined through the second derivative of the free energy: [Vettorazzo and de Forcrand, NPB 004 ] F( Φ ) h ( β ) = Φ Φ= 0 β =0. In the confinement phase the system does not feel any changes due to the external flux. On the contrary in the Coulomb phase the system reveals a response. h(β) 0 in the Coulomb phase h(β) = 0 in the confining phase The space-like and the transverse-like helicity modulus are: Helicity modulus in the transverse direction, h T takes zero value for all values of β. h ( β ) = (β cos(f )) (β sin(f )) S µν µν (Lµ Lν ) P P h = ( β ) (β cos(fµ )) (β sin(fµ )) (L L ) µ P P Confinement along the extra dimension

21 Confining Layer Phase Transition (continue) Distribution of the space-like plaquette: Two-state signal Susceptibility of the space-like plaquette S(Pˆ S ) V Pˆ S Pˆ S A net increase with the volume appears

22 Confining Layer Phase Transition (continue) The relative analysis shows that: The transition between the Confining phase and the Layer phase is of st order (though weak) as it happens to be the phase transition between the confining and the Coulomb phases for the 4D model. The transition points of the 4D gauge coupling almost lie on the horizontal line the initial point of which is critical value of the pure 4D model. Hence the Layer phase is completely distinguished from the confining one. P.D., Farakos, Vrentzos Phys.Rev.D 006

23 Layer D Coulomb Phase Transition β=.4 As the system passes to the D-Coulomb phase the extra fifth dimension ceases to be confined : h passes from zero to a non-zero value

24 Layer D Coulomb Phase Transition (continue) The susceptibility of the tranverse-like plaquette is: S(Pˆ ) V Pˆ Pˆ An increase of the susceptibility peaks, though weak, with the volume is observed. δ S ~ V, δ < max

25 Layer D Coulomb Phase Transition (continue) A finite scaling analysis assumes: β (L) = β (+ C L S = C + C max 0 L γ/ν -(/v) ) ν 0.7() β = 0.308(3) γ =.4(44) = δ = 0.44() Evidence for a nd PT * Volumes bigger than V=4 have to be used in order to confirm this evidence

26 It can be shown numerically on the lattice: how it can be identified a phase (Layer) which is coulombic on the four-dimensional subspace while it exhibits confinement along the extra dimension The Layer phase is stable and it is well separated from the confinement phase and the five-dimensional Coulomb phase the potential between two test charges in the layer phase is that of a 4D Coulomb interaction (~/r) and it is distinguishable from the potential of the D Coulomb phase (which goes as ~/r ) ( Farakos and Vrentzos, Phys.Rev.D 008)

27 (4+)-dimensional Abelian Higgs model

28 D anisotropic U()-Higgs model Assume a U()-Higgs model in five dimensions in the RS background A general metric of the RS type (warped extra dimension) is written as: ds = ω (z) [ dx dx ] dz ( ω 0 for z ) (z) 0 Write the action of a five-dimensional Abelian Higgs model (in this background): S = = = S + gauge 4 g d xdz d S d xdz scalar x 4 g * ΜΝ [ D Φ D Φ g V ( Φ )] * µν 4 * 4 [ D Φ D Φ η a ( z) D Φ D Φ a ( z) V ( Φ )] µ g F ( Μ, Ν = 0,...,4 and µ, ν = 0,...,3 ) F ΜΝ µν ν F F ΚΛ κλ η g µκ ΜΚ η g νλ ΝΛ + d x a ( z) F g z µ z g F ν η M µν + Ν

29 Make the rescaling α(z)φ =ϕ and consider a quartic scalar potential. The scalar action reads: S scalar = d 4 xdz * µ * * * [ D ϕ D ϕ a (z)d ϕ D ϕ M(z) ϕ ϕ λ( ϕ ϕ) ] µ [ ] + a (z) [ a (z)] with M(z) = a (z) m + z z Translate the action on the lattice and we get: S Lattice = S = β + β gauge + β + x h (we set : + S ( cosu (x)) + β ( cosu (x)) µν g µ g x, µ < ν 4 * [ ϕ (x) U (x) ϕ (x + aµˆ) ] [ ϕ (x) U (x) ϕ (x + aµˆ) ] L µˆ L L µˆ L h x, µ < ν 4 a * [ ϕ (x) U (x) ϕ (x + aˆ) ] [ ϕ (x) U (x) ϕ (x + aˆ) ] L L L L * m ϕ (x) ϕ (x) + β / x L 3/ scalar L ϕ = ϕ ) L L ˆ R * ( ϕ (x) ϕ (x)) L x,µ L The lattice couplings obey certain equations which depend on the warp factor: R β g = a (x )βg βh = a (x )β h λ = a M (x ) = βh 4β Due to the assumed function of the warp factor, the couplings for both the gauge and the scalar fileds along the extra dimension are strongly coupled. a ˆ β h m L

30 We adopt a simplified version for the lattice model under study: the couplings along the extra dimension do not depend on the extra coordinate but they are allowed to get different values from the couplings defined on the four dimensional subspace. The lattice action reads: S Lattice = = S β β h β gauge x h + S g x, µ < ν 4 scalar ( cos U (x) ) + β ( cos U (x) ) µν g µ * Re 4ϕ L (x) ϕ L (x) x, µ x µ 4 * ϕ (x)u ϕ (x + aµˆ ) * * Re [ 4ϕ ] L (x) ϕ L (x) ϕ L (x)u ϕ L (x + aˆ ) ˆ x * * [( β 4β β ) ϕ (x) ϕ (x) + β ( ϕ (x) ϕ (x) ) ] R h h L L R L L L µˆ L an extension of the initial proposal of Fu and Nielsen to the Abelian Higgs model

31 For the full phase diagram we need to study five couplings (!) β, β, β, β g g h h and β R We can make a reasonable compromise : keep small: β h << take β g < in order to have the possibility to get a confinement phase in the four-dimensional subspace. Choose different values for β. As decreases the phase R βr transitions become stronger. Study the phase diagram in terms of β g and β h

32 Order Parameters 4D (or space) plaquette) : extra dimension (or transverse) plaquette : 4D (or space) link: extra dimension (or transverse) link : Higgs field measure squared: P S P 6N 4N x, µ < ν 4 x, µ 4 cos (F cos (F µν µ (x)) (x)) L S cos (χ(x µˆ) A + + µˆ 4N x, µ 4 L cos (χ(x ˆ) A + + N x, µ 4 R = ρ (x) N x ˆ - χ(x)) - χ(x)) [ define: ϕ = ρ(x) exp(iχ (x)) ] L

33 The Phase Diagram β = 0. β = 0. β R g h = 0.00 C: Confining phase D Coulomb phase H : five dimensional Higgs phase H 4 : four dimensional Higgs phase broken symmetry on 4D subspace confinement along the extra dim

34 Confining - H 4 phase transition β = 0. β = g g 0. P does not feel the phase transition It remains in the confined regime The same happens to L R passes from a value ~ (unbroken) to values > (broken phase)

35 Confining-H phase transition β = 0. 8 β 0. = g g Both P and P S do feel the phase transition R passes from a value ~ (unbroken) to values > (broken phase) Big hysteresis loop: st order PT

36 Interesting features of the D Abelian-Higgs model with anisotropic couplings There is a stable phase with broken symmetry on the four-dimensional subspace and confinement along the extra-fifth dimension. Strong evidence can be provided that for a certain value of the scalar self-coupling (β R =0. () ) the transition from the strongly coupled phase to the layer Higgs phase becomes nd order. The separation of H 4 from the H phase seems to be a crossover (for a definite conclusion bigger volumes are needed)

37 (4+)-dimensional SU()- Higgs model in the adjoint representation

38 D SU()-Higgs in the adjoint representation It is a toy-model based on the Georgi-Glashow model The SU() symmetry breaks to U() massless photon massive W-bosons massive scalar neutral field In the Higgs phase The 3D model is equivalent with the SU() pure model at finite temperature after dimensional reduction due to the integration of the heavy modes It presents two phases (confining + Higgs) [ Hart, Philipsen, Stack and Teper Phys.Lett.B 997 ] A first numerical study of the 4D model can be found in [ Mitrjushkin and Zadorozhny, Phys.Lett.B 986 ]

39 The lattice model The action is : S D Lattice + β h + β h = β x,µ x + ( β g x, µ < ν 4 R Tr Tr 4β + [ Φ (x)] Tr[ Φ(x)U (x) Φ(x + µ)u (x)] + [ Φ (x)] Tr[ Φ(x)U ] (x)φ (x + ˆ)U (x) h β ) h x TrU µν Tr (x) + β g µ x,µ [ ] [ ] Φ (x) + β Tr Φ (x) R x TrU µ µ (x) The Links are defined by: U = iga µ iga µ = e U e The gauge potential and the matter fields are represented by x Hermitian matrices: a Aµ = Aµ σa and Φ = Φ a σ a (σ a : Pauli matrices) Five couplings: β, β, β, β g g h h and β R

40 Order Parameters 4D (or space) plaquette) : P S 6N x, µ < ν 4 TrU µν (x) extra dimension (or transverse) plaquette : P 4N x, µ 4 TrU µ (x) 4D (or space) link: L S = 4N x, µ 4 Tr + [ Φ(x)U (x)φx) + µ)u (x)]/ Tr[ Φ (x)] µ µ extra dimension (or transverse) link : + [ Φ(x)U + ] [ ] (x)φx) ˆ)U (x) / Tr Φ L = Tr (x) N x Higgs field measure squared: R = N x Tr [ Φ (x)]

41 The phase diagram of the model with isotropic couplings Take: βg = β g, βh = β h The phase diagram in the parametric-space of β g and β h for β R =0.0 is: New Higgs phases of the D model H S : five dimensional Higgs phase with the U() symmetry in the confining regime H C : five dimensional Higgs phase with the U() symmetry in the Coulomb regime The pure SU() transition point, β g ~.63 [ Creutz, Phys.Rev.Lett. 979 ]

42 Relationship of the Layer phase and the dimensionality A lattice model defined in D=d+n dimensions can provide a Layer phase as long as the d-dimensional model can be found in at least two phases one of which is the Coulomb phase. Fu and Nielsen NPB 984, 98 Therefore it is reasonable that a 4D layer phase can exist in the D anisotropic U() gauge model since the phase diagram of the four-dimensional model consists of two phases : confining and Coulomb. On the contrary, the D anisotropic SU() gauge model can not furnish a layer phase since it is known that the 4D SU() gauge model does not have a well defined Coulomb phase. However it can exist a D layer phase of SU() in a 6D space. But, what does it happen when the scalar field is added?

43 The phase diagram of the model with anisotropic couplings For β = 0 and β = 0 g h the model becomes four-dimensional. It has a confining phase (for β h small) and a four-dimensional Higgs phase (for β h big). The U() symmetry that survives in the Higgs phase shows a phase transition between the strong coupled and the weak phase (following the β value). g The parameter β R controls the strength of the phase transitions Switch on β and : β g β h Keep h small (=0.0). Vary β g in the interval for which the confined phase can be present (β g <.63). Give the Phase Diagram in terms of β h and β g

44 SU() Confining phase The Phase Diagram β g =. D Coulomb phase (for SU()) H : five dimensional Higgs phase U() unbroken in D H 4 : four dimensional Higgs phase U() unbroken in 4D space SU() confinement along the extra dim

45 H 4 H phase transition β g =. P S ~-(/4β g ) P S ~-(/β g ) & P S ~ P P ~β g /4 Restoration of the D dynamics in the Higgs phase L ~small For both H 4 and H the order parameter R >> O()

46 Confining - H 4 phase transition β g =.

47 Confining - H phase transition β g =.

48 The Higgs phase H 4 is located in a four-dimensional subspace embedded in a five-dimensional space. A massless photon is found to be localized on the four-dimensional subspace The confinement along the extra dimension due to the SU() interaction serves for the screening of the extra dimension. H 4 is a four-dimensional Layer phase formed in a five-dimensional bulk It shares the characteristics of the analytical solution of the Dvali-Shifman model (Phys.Lett.B 997) for the localisation of a massless photon on a two-dimensional wall out of which the interaciton is confined by the SU() interaction.

49 Remarks & Conclusions (D+)-dimensional gauge models with anisotropic couplings can reveal a new kind of D-dimensional phase which we call Layer. It is a D-dimensional Coulomb phase accompanied by confinement along the extra dimension. The necessary condition for the layer phase formation is that the D-dimensional gauge model must already have two distinct phases. Hence the minimum dimensionality is D=4 for the pure U() model and D= for the pure SU() model. Extra dimensional lattice gauge models with anisotropic couplings can be inspired in an extra dimensional space described by the RS metric.

50 The study of the D-Abelian Higgs model with anisotropic couplings shows that a Layer phase exists in the broken phase: we get a set of 4-dimensional subspaces in the Higgs phase which do not comunicate due to confinement along the extra direction. The D anisotropic SU()-Higgs model in the adjoint representation shows two main features: The inclusion of the scalar field is responsible for the formation of a 4 dimensional layer (higgs) phase in a model with non-abelian dynamics The confinement along the extra dimension is of the non-abelian type The existence of the layer phase in the phase diagram of (4+)D lattice gauge models with anisotropic couplings supports the conjecture of an effectively four-dimensional world embedded in a bulk of extra dimensions.

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