Confluent Heun functions in gauge theories on thick braneworlds
aa r X i v : . [ h e p - t h ] S e p Confluent Heun functions in gauge theories on thick braneworlds
M. S. Cunha † and H. R. Christiansen †∗† Grupo de F´ısica Te´orica, State University of Ceara (UECE),Av. Paranjana 1700, 60740-903 Fortaleza - CE, Brazil ∗ Universidade Estadual Vale do Acara´u,Av. da Universidade 850, 62040-370 Sobral - CE, Brazil
Abstract
We investigate the propagation modes of gauge fields in an infinite Randall-Sundrum scenario.In this model a sine-Gordon soliton represents our thick four-dimensional braneworld while anexponentially coupled scalar acts for the dilaton field. For the gauge-field motion we find a differ-ential equation which can be transformed into a confluent Heun equation. By means of anotherchange of variables we obtain a related Schrodinger equation with a family of symmetric rational( γ − ωz ) / (1 − z ) potential functions. We discuss both results and present the infinite spectrumof analytical solutions for the gauge field. Finally, we assess the existence and the relative weightsof Kaluza-Klein modes in the present setup. Keywords: Extra-dimensions, Sine-Gordon, Dilaton, Kaluza-Klein, . INTRODUCTION One of the main purposes of superstring theory is the inclusion of all the relevant fieldsof Nature together in one single Lagrangian. Field theoretic scenarios inspired in such atheory put in contact gauge and matter fields with metric degrees of freedom, altogetherdefined in some extra-dimensional space [1]. Extra-dimensions, combined with the influenceof the gravitational field, modify nontrivially all sectors so that gauge forces must be provento remain the same in the usual four-dimensional (4D) subspace or predict new physics insome consistent way. Localization in the gauge sector is expected to hold and the effective4D electromagnetic force must be mediated by massless photons as usual. On the otherhand, higher-dimensional spaces help solving fundamental problems such as the hierarchygap between the Planck and gauge-coupling scales in the Standard model [2].Also inspired in String theory is the use of branes to represent our Universe. In Stringtheory, gauge modes are deposited on D-branes from open strings ending on them, so weexpect gauge fields in field-theoretic models to have finite localized modes on stringy topo-logical defects of lower dimensionality. Actually, to obtain finite eigenstates in 4D it hasbeen shown that to fulfill this task we need not just gravity but also a dilaton [3, 4], a fieldalready predicted in String theory.In the present paper, we describe gauge fields in a warped five-dimensional bulk witha dilaton and a brane defect that mimics the ordinary world. Both brane and dilatonconfigurations are geometrically consistent solutions of a two scalar world action in a curved5D space-time where the field potential is of sine-Gordon type.We show a relationship between the fundamental parameters of the 5D theory which iscrucial to determine the dynamics of the fields both in the bulk and ordinary space. Indeed,for different choices of a parameter defined by the quotient of some power of the sine-Gordonfrequency-amplitude and the 5D Planck mass, the equations of motion of the gauge field canbe completely different. Notably, in Ref. [5] we have been able to find the whole spectrumof a theory involving both Maxwell and Kalb-Ramond fields for a particular value of thisparameter. As we will see, there exists a minimal value for the dilaton coupling constantabove which the finiteness of the action is assured and it is directly related to the localizationof gauge fields.In what follows we analytically obtain the propagation modes (massless and massive) of2 gauge theory in a background of the sine-Gordon type that results in new equations ofmotion. We show that the dynamics of the quantum mechanical system associated withthe problem is given by a simple (rational) potential function and that the solutions to theSchrodinger equation are of the Mathieu type (with a power-law factor). In a more generalcase we obtain the exact spectrum given by the set of confluent Heun functions and showthat Kaluza-Klein states are strongly suppressed in ordinary space.The paper is organized as follows. In the next Section, we present the geometrical back-ground. In Section III we introduce the action for the 5D gauge field coupled to warpedgravity and a dilaton background and derive the 5D equations of motion. In Section IV weobtain the quantum analog problem showing explicitly the quantum-mechanical Schrodingerpotential. The eigenvalue spectrum is computed and graphically shown. Next, in SectionsV and VI we discuss the general problem and draw our conclusions. Other recent resultsabout thick braneworlds can be found in e.g. [6].
II. GEOMETRICAL BACKGROUND
Our framework is a five-dimensional space-time embedding a four-dimensional membranealso called thick brane. The (space-like) extra-dimension is assumed infinite and the branewill be dynamically obtained as a solution to the Einstein equations for gravity coupled toa pair of scalar fields. One of these scalars represents a domain wall defect (the thick brane)while the other is the dilaton. The dilaton, together with the warping of the fifth dimension,happens to be crucial in the gauge theory that will be developed and makes more clearthe stringy origin of the theory. Since gauge field theory is conformal [7] all the informationcoming from the warping of the 4D metric is automatically lost. As a consequence the photonis non-normalizable in the four-dimensional space unless the gauge coupling is dynamicallymodified. Indeed, the exponential coupling of both the dilaton and the 5D warping to thegauge field conveniently modifies the scaling properties and the zero-mode becomes localized[3].The five-dimensional world action which determines the background is S B = Z d x dy p − det G MN [2 M R −
12 ( ∂ Φ) −
12 ( ∂ Π) − V (Φ , Π)] , (1)where M is the Planck mass in 5D, and R is the Ricci scalar. The solution for Φ represents the3orld membrane and the corresponding field solution for Π will be the dilaton configurationconsistent with the metric and the kink. As usual we adopt Latin capitals on the bulk andGreek lower case letters on 4D.We next adopt the following ansatz for the metric ds = e y ) η µν dx µ dx ν + e y ) dy , (2)where Λ and Σ depend just on the fifth coordinate, y , and diag( η ) = ( − , , , ′ ) + 12 (Π ′ ) − e y ) V (Φ , Π) = 24 M (Λ ′ ) , (3)12 (Φ ′ ) + 12 (Π ′ ) + e y ) V (Φ , Π) = − M Λ ′′ − M (Λ ′ ) + 12 M Λ ′ Σ ′ , and Φ ′′ + (4Λ ′ − Σ ′ )Φ ′ = e ∂ V ∂ Φ , Π ′′ + (4Λ ′ − Σ ′ )Π ′ = e ∂ V ∂ Π . (4)where the prime means derivative with respect to y .By means of a supergravity motivated functional W (Φ) defined byΦ ′ = d W d Φ (5)[8] the system of differential equations can be more easily handled. This method is alsoapplicable to non-supersymmetric domain walls [9, 10] as the present one.First, we consider the action in the absence of gravity (and no dilaton) in order to obtainan expression for Φ. Then, we put this solution into the equations of motion (3) and (4).The standard sine-Gordon Lagrangian reads L SG = − ∂ Φ − V (Φ) (6)with V (Φ) = 1 b (1 − cos( b Φ)) . The free parameter b signals bulk-symmetries δ Φ → nπ/b ( n ∈ Z ) among the vacua ofthis theory. Solutions interpolating vacua are possible and, assuming they depend only on y , one-solitons read Φ( y ) = 4 b arctan e y . (7)4hese functions kink on our 4D-world slice, namely at y ∼ V for the general action (1), viz. V (Φ , Π) = exp (Π / √ M ) (cid:18)
12 ( d W d Φ ) − M W (Φ) (cid:19) . (8)Taking into account eq.(5), the superpotential functional results W (Φ) = − b cos( b V (Φ , Π) = − e (Π / √ M ) (cid:18) b sin b M b cos b (cid:19) . (10)If we now conveniently write the Hamiltonian `a la Bogomol’nyi, we can detect the followingrelations among the warping functions, the dilaton and the superpotentialΠ = −√ M Λ , Σ = Λ / , Λ ′ = −W / M . (11)Finally, totally solving the equations of motion, the dilaton field is given byΠ( y ) = 1 √ M b ln cosh y, (12)and Λ = − M b ln cosh y, Σ = − M b ln cosh y. (13)The relation between Π and Φ allows also writing V as V (Φ) = − b (sin b / M b (cid:18) M b −
1) cos b (cid:19) , (14)which fully shows its dependence on b and M (see Fig. 1).As it happens with dilaton configurations related to D-brane solutions, functions such as(7) and (12) are singular when | y | → ∞ . However, since the metric vanishes exponentiallyand both dilaton and warp factors operate under an exponential coupling, the model is keptfree of divergences.The warping functions amount to a shift in the effective four-dimensional Planck scale,which remains finite with the following definition M P ≡ M Z ∞−∞ dy e y )+Σ( y ) . (15)5 - - - - - - - - - - - - - - Figure 1. Family of background potential functionals V (Φ) (eq.14) for different values of a ≡ / M b : even a = 2 , , , ,
10 (solid line), odd a = 1 , , , , a . Using the consistency relation (11) just found, the action reads S B ∼ Z dy e y )+ Λ( y )+ λ √ M Λ( y ) S (4) (16)where S (4) is the remaining of the action integrated in 4D. According to the solutionΛ( y ) = 2 a ln sechy (c.f. eq.(11) - eq.(13)) the 5D factor results finite provided c ≡ (17 +2 λ √ M ) / >
0, namely λ > − √ M = λ .Studying the fluctuations of the metric about the above background configuration, itis possible to see that this model supports a massless zero-mode of the gravitational fieldlocalized on the membrane even in the presence of the dilaton. In order to prove the stabilityof the background solution, we would have to show that there are no negative mass solutionsto the equations of motion of a perturbation h µν of the metric. Actually, a gravitationalKaluza-Klein spectrum appears, starting from zero and presenting no gap. This can beeasily seen after an appropriate change of variables and decomposition of the gravitationalfield, and a subsequent supersymmetric type expression of the Schrodinger type operatorresulting from the equation of motion (see [3, 11] for details). The issue of the coupling ofthese massive modes to the brane has been analyzed in detail in [12].6 II. GAUGE FIELD ACTION IN A WARPED SPACE WITH DILATON
Let us consider the following 5D action where a five-dimensional electromagnetic field A N is coupled to the dilaton [13] in a warped space-time S g = Z dy d x p − det G AB e − λ Π (cid:26) − F MN F MN (cid:27) (17)where F MN = ∂ [ M A N ] .Assuming that the gauge field energy density should not strongly modify the geometricalbackground, we can study the behavior of the propagating modes in the background of thetopological configuration studied in the last Section. In general, most of the attempts tostabilize 5D brane worlds by means of a scalar field in the bulk do not take into account theback-reaction of the scalar field on the background metric [3, 11, 12, 14] and those in orderto compute the scalar back-reaction on the metric were unsuccessful except in a few specialcases [9, 15].The factor exp(Σ − λ Π / A M √− G ∂ M ( G MR G NP F RP √− Ge − λ Π( y ) ) = 0 (18)where diag G MN = ( e η µν , e ). For this, we adopt the following gauge choice A =0 , ∂ µ A µ = 0 and separate the fifth from the ordinary coordinates as follows A µ ( x, y ) = a µ ( x ) u ( y ) . (19)Now, from eq.(18) we just get[ (cid:3) + 1 u f ∂ ( f ∂ u )] a µ = 0 . (20)Note that the warped metric and the dilaton deform the solutions of this differential equationby means of the factor f ( y ) ≡ e − λ Π / multiplying u ( y ) and u ′ ( y ). A full Kaluza-Kleinspectrum results from the solution of the general case ∂ ( f ∂ u ) = − m f u, (21)7here m is an arbitrary constant representing the 4D squared boson mass of the vectorgauge field. It means that a µ = a µ (0) e ipx with p = − m .By expanding eq.(20), we obtain the most general y -dependent equation of motion forthe modified sine-Gordon potential (14) derived from action (17) in a form which exhibitsits dependence on a = 1 / M b , and cu ′′ ( y ) + a (1 − c ) tanh y u ′ ( y ) + m sech a y u ( y ) = 0 , (22)where y ∈ ( −∞ , ∞ ) as already stated. Looking back at the definition of the auxiliaryconstants we get the explicit dependence of the solutions on the original parameters b , λ and M .Below, we will discuss the possible values of m as resulting from an eingenvalue problemrelated to the equation of motion (22). Indeed, there exists a Schrodinger like equationequivalent to eq.(22) with a potential function which concentrates all the richness implicit inthe complicated equation (22). Note that the particular solution u ( y ) =constant representsthe m = 0 photon state of the 5D theory. Since this solution satisfies eq.(22) for any valueof a and c , any member of the family of problems has a guaranteed localized zero-mode. See[5] for details.Localization of gauge-field modes in the ordinary space can be established by verifyingthat the corresponding 5D action is finite. From eq.(19) one has F µν = f µν u ( y ), where f µν = G µα G νβ f αβ , so that for a gauge mode A sol .M the relevant part of eq.(17) reads S g [ A sol .M ] = Z dy u ( y ) e y )+Σ( y ) − λ Π( y ) / Z d x f µν f µν . (23)Using the field solutions found in eq.(7) and the equations thereafter, the fifth dimensionfactor will remain finite for each mode u ( y ) growing below e ac at infinity. Thus, any finitesolution is a physically acceptable Kaluza-Klein state (as we have seen above, to have afinite 5D Planck mass and background action S B we already need c >
0, i.e. λ > λ ).It is known that by means of a transformation u ( y ) = e − α Λ / U ( z ) , dzdy = e − β Λ (24)we can turn eq.(22) into a Schrodinger-like equation in the variable z (see e.g. [2, 3]). Ingeneral, the existence of an analog Schrodinger equation is useful to give us a feeling of the8hysical profile of the solutions of the original problem, as e.g. parity and eigenvalues. With α = c − / β = − / U and have a puremass term as usual. The resulting differential equation reads precisely (cid:20) − d dz + V a ( z ) (cid:21) U ( z ) = m U ( z ) , where V a ( z ) = e − Λ / ( α Λ ′′ − γ Λ ′ ) and γ = α ( − α ). In a few cases the last expressioncan be inverted after exact integration in order that an analytical expression for the analognon-relativistic potential comes about. In Ref.[5] we have solved the a = 2 case and found V ( z ) = − α (cid:2) − (2 α −
1) tan z (cid:3) . In this paper, for a = 4 we find V ( z ) = ( γ − ωz )(1 − z ) , where γ and ω are constants and we shall analyze it in what follows. IV. THE QUANTUM ANALOG
In the present case we can turn eq.(22) into a Sturm-Liouville problem by means of z = tanh y, (25) u ( y ) = (cosh y ) c − U ( z ) (26)(see eq.(24)). Now, we have a related Schrodinger equation defined in the z variable (cid:20) − d dz + V ( z ) (cid:21) U ( z ) = m U ( z ) , (27)with V ( z ) = 1 − c (1 − z ) (cid:2) − c ) z (cid:3) . (28)We can see that the potential function diverges at z = ± { U ( z = ±
1) = 0 , U ′ ( z = ±
1) finite } which must be in order tomatch finite u ( y ) solutions to eq. (22) at y → ±∞ . After solving the quantum analog wehave to transform back variables and functions to check the finiteness and continuity of theoriginal solution u ( y ) in order to be physically acceptable.9e now better introduce the variable θ by means of z = cos θ (29)which results in equation U ′′ ( θ ) − cot( θ ) U ′ ( θ ) − − c sin θ (cid:2) − c ) cos θ (cid:3) U ( θ ) = − m sin θ U ( θ ) (30)for the analog wave function U ( θ ) with U ( θ = π,
0) = 0.According to the arguments of localization seen in the previous section, physically ac-ceptable solutions require c >
1. The c=1/4 case
Equation (30) gets strongly simplified for the value c = 1 /
4. In this case we obtain1sin θ ddθ (cid:18) θ U ′ ( θ ) (cid:19) + m sin θ U ( θ ) = 0 (31)with solutions U (1) ( θ ) = U sin( m cos( θ )) (32) U (2) ( θ ) = U cos( m cos( θ )) , (33)which in terms of the original variable and function read u (1) ( y ) = u sin( m tanh y ) (34) u (2) ( y ) = u cos( m tanh y ) , (35)(see Figs. 2, 3). The zero-mode, m = 0, is then related to u ( y ) = u as already mentioned.Since V ( z ) is an even function (in this case trivial), solutions must have definite parity.Besides, the potential divergence at z = ± m = nπ while symmetric solutions correspond to m = (2 n +1) π/ n ∈ N (or simply m = ( n + 1) π/ n even for symmetric solutions and odd for theantisymmetric ones). 10 igure 2. Plot of sin( m tanh( y )), for c = 1 / m = π (black line), m = 2 π (long-dashed blue line),and m = 3 π (dashed red line)).Figure 3. Plot of cos( m tanh( y )), for c = 1 / m = π/ m = 3 π/ m = 5 π/
2. Other analytical solutions
If we perform the transformation U ( θ ) = sin κ θ M ( θ ) , (36)in place of eq.(30) we get the following problem for M ( θ ) M ′′ ( θ ) + (2 κ −
1) cot( θ ) M ′ ( θ )+ (cid:20) κ ( κ −
2) cot ( θ ) − κ + 1 − c sin θ [1 + 2(1 − c ) cos θ ] (cid:21) M ( θ )= − m sin θ M ( θ ) . (37)Now, we can choose a convenient power for the last transformation, κ = 1 /
2, in order to11urn this into M ′′ ( θ ) − (cid:20) cot ( θ ) (cid:18) c − c + 1516 (cid:19) + 38 − c (cid:21) M ( θ ) = − m sin θ M ( θ ) (38)which, for 4 c − c + = 0, is known as the Mathieu differential equation M ′′ ( θ ) + (cid:0) c − / m sin θ (cid:1) M ( θ ) = 0 , (39)with c = 3 / c = 5 /
3. The case c = 5 / In this case, Eq. (39) results in M ′′ ( θ ) + (cid:18) m − m θ ) (cid:19) M ( θ ) = 0 (40)whose analytic solutions are the general Mathieu functions Table I. List of the first 21 values of m s (symmetric solutions) and m a (antisymmetric ones) for c = 5 / m s m a −− . . . . . . igure 4. Symmetric and antisymmetric eigenfunctions U in z space for c = 5 / m = 0 (black line,symmetric), m = 2 . m = 4 . m = 5 . m = 7 . M (1) ( θ ) = M c (cid:18) m , m , θ (cid:19) (41) M (2) ( θ ) = M s (cid:18) m , m , θ (cid:19) , (42)which, in terms of the z variable, result in the analog wave-functions U ( z ) U (1) ( z ) = (1 − z ) / M c (cid:18) m , m , arccos( z ) (cid:19) (43) U (2) ( z ) = (1 − z ) / M s (cid:18) m , m , arccos( z ) (cid:19) (44)(see [16] for details about Mathieu functions).As mentioned above, the boundary conditions of the present problem are { U ( z = ±
1) =0 , U ′ ( z = ±
1) finite } , related to finite u ( y ) solutions to the original equation, recallingthat y ∈ ( −∞ , ∞ ). The first set of solutions, U (1) ( z ), is not physically interesting be-cause the derivatives of these functions are divergent at the boundary. The reason is that M c (arccos( z )) cannot be zero at z = 1 for any value of m . The second set, on the otherhand, has physically acceptable solutions for a discrete set of values of m , the (twenty) firstof which we show in Table I. These solutions are symmetric or antisymmetric, as expected(see Fig. 4). Note the presence of a zero mode.13 igure 5. Symmetric solutions u (2) ( y ) (Eq. (46)) for c = 5 / m = 0 (dash-dotted black line), m = 4 . m = 7 . m = 10 . u (2) ( y ) for c = 5 / m = 2 . m =5 . m = 8 . m = 12 . In terms of y , we have u (1) ( y ) = cosh y M c (cid:18) m , m , arccos(tanh y ) (cid:19) (45) u (2) ( y ) = cosh y M s (cid:18) m , m , arccos(tanh y ) (cid:19) , (46)where the set u (1) ( y ) diverges when y → ±∞ , as due from the comments above, so we justkeep the solutions u (2) ( y ) (see Figs. 5 and 6).14 . The case c = 3 / Now, Eq. (39) reads M ′′ ( θ ) + (cid:18) m − m θ ) (cid:19) M ( θ ) = 0 (47)with solutions given by M (1) ( θ ) = M c (cid:18) m , m , θ (cid:19) (48) M (2) ( θ ) = M s (cid:18) m , m , θ (cid:19) , (49)corresponding to U (1) ( z ) = (1 − z ) / M c (cid:18) m , m , arccos( z ) (cid:19) (50) U (2) ( z ) = (1 − z ) / M s (cid:18) m , m , arccos( z ) (cid:19) . (51)in the z space with the boundary conditions already seen. Table II. List of the first 20 values of m s and m a for c = 3 / m s m a −− −− . . . . . . igure 7. Symmetric solutions U (2) ( z ) for c = 3 / m = 1 . m =4 . m = 7 . m = 13 . U (2) ( z ) for c = 3 / m = 2 . m =5 . m = 9 . m = 15 . As we discussed in the previous ( c = 5 /
8) case, only the second set of solutions isphysically relevant and just for a discrete (infinite) sequence of m eigenvalues. For suchvalues solutions have definite parity according to V ( z ), Eq. (28), (see Figs. (7) and (8)).Note that the solutions to the Schrodinger equation (the analytical expressions (50) and(51)) are not compatible with a zero-mode for the z -boundary conditions given above. Ac-tually, as a general result, for any value excluded from the sequence starting in Table II, U (1) ( z ) -eq. (50)- has divergent derivatives at z = ± U (2) ( z ) -eq. (51)- is not evensymmetric. For this reason the zero-mass solutions of eq. (47), M ( θ, m = 0) ∈ { cons ., θ } ,16 igure 9. Mathieu function at the boundary ( θ = π ) as a function of the mass for c = 3 / M s (cid:16) m , m , θ = π (cid:17) . At the other boundary, M s (cid:16) m , m , θ = 0 (cid:17) = 0. do not correspond to valid solutions of the Schrodinger problem. In Fig.9 we can see all thefirst mass values of the sequence which nullify the Mathieu functions M s at the boundary.These values, also listed in Table II, guarantee finite derivatives of U (2) ( z ). The absence ofthe zero mode in this list indicates a limitation of the Schrodinger analogue approach. Wewill come again to this point in the next Section. In terms of y we have u (1) ( y ) = M c (cid:18) m , m , arccos(tanh y ) (cid:19) (52) u (2) ( y ) = M s (cid:18) m , m , arccos(tanh y ) (cid:19) , (53)of which u (2) represents the only non-divergent set of solutions, as we illustrate in Figs. (10)and (11) for the first quantum values of m . V. THE CONFLUENT HEUN EQUATION
We now investigate our original problem by relaxing the quantum analog condition. Inorder to obtain the general solution of Eq. (22) we perform the following change of variable x = tanh y. (54)It maps the y space to x ∈ ( − ,
1) and we will eventually transform it back in order tocome into the original space and variables. Note that Eq. (22) is symmetric under a paritytransformation and thus the differential equation admits even as well as odd parity solutions,as it should. Now, Eq. (22) becomes 17 ′′ ( x ) + (2 − ˜ c ) xx − u ′ ( x ) + m (1 − x ) a − u ( x ) = 0 . (55)which is an homogeneous second-order linear differential equation with polynomial coeffi-cients provided a is even. Here ˜ c = a (1 − c ) so that ˜ c = ( −∞ , a ). Now, Eq.(55) looks morefamiliar if we change x into zu ′′ ( z ) + (cid:18) / z + 1 − ˜ c/ z − (cid:19) u ′ ( z ) + m u ( z ) z = 0 . (56) Figure 10. Symmetric solutions u (2) ( y ), eq.(53), for c = 3 / m = 1 . m = 4 . m = 7 . m = 13 . u (2) ( y ), eq.(53), for c = 3 / m = 2 . m = 5 . m = 9 . m = 15 . z = 0 ,
1, and an irregular one at z = ∞ . This is known as a Confluent Heun equation[17–19].We can compare Eq. (56) with the canonical non-symmetrical general form of the con-fluent Heun equation as given in [18–20] Hc ′′ ( z ) + (cid:18) α + β + 1 z + γ + 1 z − (cid:19) Hc ′ ( z )+ " [ δ + α ( β + γ + 2)] z + η + β + ( γ − α )( β + 1) z ( z − Hc ( z ) = 0 , (57)whose solutions around z = 0 are denoted by H (1) = Hc ( α, β, γ, δ, η ; z ) (58) H (2) = z − β Hc ( α, − β, γ, δ, η ; z ) . (59)In general, there are two linearly independent local series solutions around each singularpoint. In the region of interest, z <
1, we look for a regular local solution around z = 0which is defined by the Heun series as Hc ( z ) = ∞ X n =0 d n z n . (60)Here the constants d n (with d − = 0 and d = 1) are determined by the three-term recurrencerelation [21] A n d n = B n d n − + C n d n − , (61)where A n = 1 + βn → − n (62) B n = 1 + − α + β + γ − n + η + ( α − β − γ ) / − αβ/ βγ/ n → − ˜ c/ − / n + ˜ c/ / − m / n (63) C n = 1 n (cid:18) δ + α ( β + γ )2 + α ( n − (cid:19) → m n . (64)By comparing Eqs. (56) and (57), it is easy to identify α = 0, β = − / γ = − ˜ c/ δ = m /
4, and η = ˜ c/ / − m /
4. Then the solutions of Eq. (22) are given by u (1) ( y ) = Hc (cid:18) , − , − ˜ c , m ,
14 + ˜ c − m y (cid:19) (65) u (2) ( y ) = tanh y Hc (cid:18) , , − ˜ c , m ,
14 + ˜ c − m y (cid:19) (66)19or arbitrary values of ˜ c (or c ), namely of the dilaton coupling constant. The conditionsthese Heun u ( y ) solutions must obey to be acceptable are the original ones, i.e. finitenessand continuity in the whole space. Table III. List of first values of m s and m a for ˜ c = − m s m a −− . . . . . . - - - - - - - - (a) Figure 12. Symmetric solutions of the Heun equation (˜ c = −
30) for m = 0 (black dash-doted), m = 8 . m = 13 . m = 17 . m = 20 . m = 24 . A noteworthy point in the present approach is that now, depending on ˜ c , the mass values m can be quantized, as we saw in Sect. IV, or not, as we will explain in what follows.After a lengthy numerical exam, we found clear evidence that for ˜ c ≤ λ ≥ λ ≡ / λ ) all the mass spectra are discrete. For ˜ c ∈ (0 , , λ < λ < λ , on the otherhand, the corresponding spectra start with a zero mode and grow continuously. This sharp20 - - - - - - - - - (a) Figure 13. Antisymmetric solutions of the Heun equation (˜ c = −
30) for m = 5 . m = 11 . m = 15 . m = 18 . contrast may be traced back to eq. (22) where the second term of the differential equationflips precisely with the sign of ˜ c . Note that for any well-behaved solution u ( y ), the thirdterm of eq. (22) can be disregarded at infinity. The remainder differential equation can beeasily solved showing that, for ˜ c >
0, solutions are always convergent to zero and for ˜ c ≤ c > m while, otherwise, only a discrete sequence of masses allow for finite solutions atthe border.It should be mentioned that for small values of ˜ c the solutions stabilize quickly. On thecontrary, for ˜ c . −
10 the numerical calculation is more difficult and more digits are neededin the mass precision to stabilize solutions at large values of y . For example, for ˜ c = − λ = 0, more than thirty significant digits were necessary in the massspectrum to find the solutions as shown in Figs. 12 and 13. In Table III we listed the firstvalues of the mass up to the eighth decimal place.As awaited, for the cases studied in the previous Section we find again the same results.Note however that the zero-mode in the ˜ c = 1 ( c = 3 /
8) case now appears explicitly.This was expected since there exists an analytical m = 0 solution to Eq. (22), namely u ( y ) = e arctan( e y ) + e , which must be present in a full approach. Furthermore, for ˜ c = 1the Heun solution indicates that Table II would not only start from zero but would alsobe continuously filled in as mentioned above. In Figs. 14 and 15 we can see finite analytic21eun solutions, given by eq. (65) and eq. (66), for some arbitrary values of m besides thequantum-mechanical analog ones. Another way to see it is by means Fig. 16 and Fig. 17where m has been fixed arbitrarily to one of the eigenvalues of ˜ c = 1 and ˜ c is then varied.In the ˜ c = − c = 5 /
8) case the spectrum still obeys quantized values as given in Table I.
Figure 14. Symmetric solutions Eq. (65) for ˜ c = 1; m = 0 (dash-dotted black line), m = 1 . m = 4 (dot green line), m = 4 . m = 7 (dashed blackline), and m = 7 . c = 1; m = 0 (solid black line), m = 2 . m = 3 . m = 5 (long-dashed black line), m = 5 . m = 5 . Thus, although Mathieu functions have been sufficient to characterize a part of the spec-trum of the Schrodinger analog of our problem, we actually need to consider confluent Heunfunctions to cover all the cases. In other words, even when we achieved fully analyticalsolutions of the quantum analog differential equation, the spectra appeared just discrete not22 igure 16. Symmetric Heun solutions, Eq. (65), for m = 4 . c :˜ c = 0 . c = 1 (long-dashed blue line), ˜ c = 1 . c = 2 (dashedgreen line), and ˜ c = 3 (dotted black line).Figure 17. Antisymmetric Heun solutions, Eq. (66), for m = 2 . c :˜ c = 0 . c = 1 (long-dashed blue line), ˜ c = 1 . c = 2 (dashedgreen line), and ˜ c = 3 (dotted black line). revealing that some of them could be eventually continua.The set of confluent Heun functions therefore provide all the possible physical solutionsof the actual problem in the 5D space. This was not apparent from the Hamiltonian pointof view which assumes the Sturm-Liouville operator H = h − d dz + V ( z ) i to represent thephysical situation. 23 I. FINAL REMARKS AND CONCLUSION
In order to physically assess massive modes, one can evaluate the variation of the effectivegauge coupling as a function of the Kaluza-Klein masses. Actually, KK contributions cannot be significant as compared with the Coulomb potential because the coupling of massivemodes to (fermion) matter on the brane develops a Yukawa type potential in the non-relativistic limit. To show that this is a decreasing function of m we should evaluate thecoefficients that multiply the relevant sector of the four-dimensional action ∼ Z dy e Σ( y ) − λ Π( y ) / u m =0 ( y ) + X n u m n ( y ) ! Z d xf µν f µν . (67)However, in order to simplify this computation we can assume that the coupling with thebrane takes place exactly on the 4D ordinary space-time, namely at y = 0. It is precisely atthis value of y where the relevant physical effects should be much stronger. For simplicitylet us consider the series of the quantum analog eigenvalues which serves as a discreterepresentative of the continuum. Thus, the effective 4D electrostatic potential would read V ( r ) ∼ q q c r + X n e − m n r r u m n (0) ! (68)where q , q are two test charges separated a distance r in ordinary 3D space and the Kaluza-Klein masses m are numbered with n in ascending order. See Fig. 14 where the first u (1)even ( y )modes are fully displayed, and Fig. 18 where the first and the tenth modes are compared.See Fig. 19 to appreciate the first 10 values at the origin. This, together with the negativeexponential factor, essentially decouples the massive modes from the physics on the domainwall. Far from the membrane, all massive modes become constants like the zero-mode is,and as a consequence the 5D phenomenology results completely modified from ordinary 4Delectromagnetism. See e.g. Refs.[2, 12] for the study of this issue in the case of gravity.In this paper we have studied bulk and four-dimensional gauge propagation modes in awarped extra-dimensional space with a dilaton field. We have set up a sine-Gordon thickmembrane which bounces at the extra-coordinate origin. A five-dimensional metric wasdynamically generated consistently with the soliton brane and the dilaton background. Insuch a framework we studied the solutions of a five-dimensional gauge field.24 igure 18. First and tenth even KK eigenmodes exhibit their relative weights.Figure 19. Sequence of the first KK values of u (0) for c = 3 / First, we have found the exact quantum-mechanical analog of our original five-dimensionalstringy problem. We have shown that the corresponding Schrodinger potential function isa quotient of simple second- and fourth-order polynomials that we could solve analytically.We next obtained the exact quantum-mechanical analog eigenspectrum and used it as aguide to analyze eventually the general solution. A localized zero-mode corresponding tothe ordinary photon was guaranteed for a dilaton coupling constant above λ . In general, wehave found that the gauge-field dynamics are analytically given by confluent Heun functionswhich we have displayed for several representative cases. Furthermore, in contrast to thequantum analog results, in the general approach the mass of the gauge-field modes can bearbitrary for λ ∈ ( λ , λ ). In any case, we have shown that the Kaluza-Klein gauge spectrumis strongly attenuated on the brane as compared to the zero-mode of the theory. On the25ther hand, we observed that in the bulk, far from the brane, the amplitude of an infinitetower of massive modes gets progressively relevant. Interestingly, the quantum-mechanicaldiscrete mass eigenfunctions are completely decoupled in that region. [1] J.Polchinski, String Theory, vols. 1 & 2, (Cambridge University Press, Cambridge, England,1998).[2] L. Randall and R. Sundrum, Phys. Rev. Lett. , 4690 (1999); idem Phys. Rev. Lett. 83(1999) 3370.[3] A. Kehagias and K. Tamvakis, Phys. Lett. B , 38 (2001).[4] D. Youm, Nucl. Phys. B 589, 315 (2000); Phys. Rev. D64, 127501 (2001).[5] H. R. Christiansen, M. S. Cunha, M. K. Tahim, Phys. Rev. D , 085023 (2010).[6] Y.-X. Liu, et al. JHEP 06, 135 (2011); Chun-E Fu, Yu-Xiao Liu, Heng Guo. Phys. Rev. D 84,044036 (2011); Y.-X. Liu, et al. arXiv:1102.4500; R.R. Landim, et al., arXiv:1105.5573.[7] G. Dvali, M. Shifman, Phys. Lett. B
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