Field-To-Particle Transition and Nonminimal Particles in Sigma Model, Dilaton Gravity and Gauged Supergravity

25 October 2001

Physics Letters B 519 (2001) 111–120 www.elsevier.com/locate/npe

Field-to-particle transition and nonminimal particles in sigma model, dilaton gravity and gauged supergravity

Konstantin G. Zloshchastiev

Department of Physics, National University of Singapore, Singapore 117542, Republic of Singapore and Department of Theoretical Physics, Dnepropetrovsk State University, Dnepropetrovsk 49050, Ukraine Received 19 July 2001; received in revised form 19 August 2001; accepted 28 August 2001 Editor: T. Yanagida

Abstract

The field-to-particle transition formalism is applied to the sigma model, and then to the 2D dilaton gravity and gauged supergravity 0-brane solutions. This approach yields the method of the consistent quantization in the vicinity of the nontrivial vacuum induced by a field solution, as well as the recipe of the nontrivial dynamical dimensional reduction which takes into account field fluctuations. It is explicitly shown that in all cases the end product is the so-called nonminimal point particle—an object whose action depends on the world-line curvature. Such objects are suspected to be common at early stages of evolution of the Universe.  2001 Elsevier Science B.V. All rights reserved.

PACS: 04.60.Kz; 04.65.+e; 11.25.Sq; 11.27.+d; 12.60.Jv

1. Introduction plain bosonic string in the N-dimensional space–time. The remainder of that section is devoted to quantiza- The studies of the field-to-particle transition grew tion of the obtained effective action as a constrained up from the old program of constructing a theory mechanics with higher derivatives. There we discuss which does not contain matter as an external postu- the recent state of the theory and calculate the quantum lated entity but consider fields as sources of matter corrections to the mass of such a field/string-induced and the particles as special field configurations. How- particle [1]. ever, if one tries to fit a field solution into the particle Further, the 2D gravity [2], which can be viewed interpretation, one encounters eventually the problem also as a dimensional reduction of the 3D BTZ black of how to deal with the field fluctuations. Then it is hole and spherically symmetric solution of 4D dilaton necessary to correctly handle the circumstance that a Einstein–Maxwell gravity, seems to be a good place field solution has an infinite number of degrees of free- for demonstration of the approach on classical and dom, unlike a particle. In Section 2 we demonstrate the quantum level. field-to-particle transition for the soliton-like solutions Finally, in Section 4 we demonstrate how gauged of the N-component 2D sigma model or, equivalently, supergravity 0-brane solutions can be also described in terms of the mechanics of nonminimal particles with rigidity. The latter is invariant under the transforma- E-mail address: [email protected] (K.G. Zloshchastiev). tions of the proper parameter s, so one may recall an-

0370-2693/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII:S0370-2693(01)01087-5 112 K.G. Zloshchastiev / Physics Letters B 519 (2001) 111–120 other diff-invariant candidates for the role of a 0-brane coinciding with the total energy up to the sign and boundary theory, e.g., the generalized conformal me- Lorentz factor γ (below the boundaries ρi will be chanics of a probe 0-brane [3,4]. However, the probe omitted for brevity). mechanics by construction takes into account neither Let us change to the set of the collective coordinates the field fluctuations near the brane solution nor aris- {σ0 = s,σ1 = ρ} such that ing zero modes, therefore, the nonminimal terms could m = m + m = not arise there in principle. x x (s) e(1)(s)ρ, Xa(x, t) Xa(σ), (2.6) where xm(s) turn out to be the coordinates of a (1+1)- 2. Effective nonminimal-particle action from dimensional point particle, em (s) is the unit spacelike sigma model (1) vector orthogonal to the world-line. Hence, the initial action can be rewritten in new coordinates as In this section, we will construct the nonlinear effective action of the sigma model in the vicinity S[X]= L(X)∆d 2σ, of a localized soliton solution, and then consider its quantum aspects. In fact, here we will describe 1 a b L(X) = σab∂−X ∂+X − U(X), the procedure of the correct transition from field to 2 √ particle degrees of freedom. 1 ∂xm ∂± = ∂ ± ∂ ∆ = = x˙2 ( − where ∆ s ρ , det ∂σk 1 2.1. General formalism ρk), whereas k is the curvature of a particle world-line ε x˙mx¨n = mn Let us begin with the action of the following k √ , (2.7) x˙2 3 N-component 2D sigma model where εmn is the unit antisymmetric tensor. This S[X]= L(X) d2x, (2.1) new action contains the N redundant degrees of freedom which eventually lead to appearance of the where the Lagrangian is given by so-called “zero modes”. To eliminate them we must constrain the model by means of the condition of 1 a ν b L(X) = σab∂ν X ∂ X − U(X), (2.2) vanishing of the functional derivative with respect to 2 field fluctuations about a chosen static solution, and in × where σ is a constant N N matrix, Greek indices run result we will obtain the required effective action. from 0 to 1, and a, b = 1,...,N. The corresponding So, the fluctuations of the fields Xa(σ) in the equations of motion are (s) neighborhood of the static solution Xa (ρ) are given ν b by the expression σab ∂ ∂νX + Ua(X) = 0, (2.3) = ∂U(X) = X (σ) = X(s)(ρ) + δX (σ). (2.8) where we have defined Ua(X) ∂Xa , Uab(X) a a a ∂2 U(X) ∂Xa∂Xb . Suppose, we have the solution which depends Substituting them into the expression (2.7) and con- on the single combination of initial variables, e.g., the (s) sidering the static equations of motion for Xa (ρ) we Lorentz-invariant one: have (below we will omit the superscript “(s)”atX’s (s) = (s) − = − 2 for brevity assuming that all the values are taken on Xa (ρ) Xa γ(x vt) ,γ 1/ 1 v , (2.4) (s) the solutions Xa (ρ)): possessing localized Lagrangian density in the sense that the mass integral over the domain of applicability 2 ∆ a b S[δX]= d σ 2L + σab∂−δX ∂+δX 2 is finite a b a b ρ2 − U δX δX + ∆ σ X δX ab ab µ =− L X(s) dρ < ∞, (2.5) + O δX3 + S,˘ (2.9) ρ1 K.G. Zloshchastiev / Physics Letters B 519 (2001) 111–120 113 where S˘ are surface terms (see below for details), describing the nonminimal point-particle with curvature, where µ is the value of the mass integral above, 1 a b L =− σabX X − U, and 2 ρ2 and prime means the derivative with respect to ρ. (0) 1 a b µ α = σabX f dρ = . (2.16) Extremizing this action with respect to δXa one can 2 2 2caˆ obtain the system of equations in partial derivatives for ρ1 field fluctuations: ˘ Let us restore the surface terms S in the action −1 a ac b ∂s ∆ ∂s − ∂ρ ∆∂ρ δX + ∆σ UcbδX (2.15). In general case they are nonzero and give a contribution: a 2 2 + X k x˙ + O δX = 0, (2.10) S =− ds x˙2 µ + α k + α k2 + α k3 , which is the constraint removing redundant degrees eff 1 2 3 of freedom, besides we have denoted the inverse ma- where α =ˇα |ρ2 , αˇ ≡ σ Xa f b,α = α(0) + trix as σ with raised indices. Supposing δXa(s, ρ) = 1 1 ρ1 1 ab 2 2 − ˇ |ρ2 =−1 a b|ρ2 k(s)fa(ρ), in the linear approximations ρk 1 α3 (ρα1) ρ1 ,α3 2 ρσabf f ρ1 .However,the (which naturally guarantees also the smoothness of a natural requirement of vanishing of field fluctuations world-line at ρ → 0, i.e., the absence of a cusp in this at the spatial boundaries leads to the canceling of the point) and O(δX2) = 0 we obtain the system of three terms containing k and k3, as. Thus, below we will ˘ ordinary derivative equations regard the action (2.15), i.e., (2.9) at S = 0, as a basic one. It is straightforward to derive the corresponding 1 d 1 dk √ √ + ck = 0, (2.11) equation of motion in the Frenet basis x˙2 ds x˙2 ds 1 d 1 dk 1 a ac a b a √ √ + − 2 = −f + σ Ucb − cδ f + X = 0, (2.12) c k k 0, (2.17) b x˙2 ds x˙2 ds 2 where c is the constant of separation. Searching for a hence one can see that Eq. (2.11) was nothing but this solution of the last subsystem in the form (we assume equation in the linear approximation with respect to that we can have several c’s and thus no summation curvature, as was expected. over a in this and next formula) Thus, the only problem, which yet demands on resolving is the determination of the concrete (eigen) = + 1 fa ga Xa, (2.13) value of the constant(s) c. It is equivalent to the caˆ eigenvalue problem for the system (2.14) under some we obtain the homogeneous system chosen boundary conditions. If one supposes, for − a + ac − a b = instance, the finiteness of g at infinity then the c’s g σ Ucb caˆ δb g 0. (2.14) spectrum turns out to be discrete. Moreover, it often a Strictly speaking, the explicit form of g (ρ) is not happens that c has only one or two admissible values significant for us, because we always can suppose if the effective potential is steep (e.g., of exponential integration constants to be zero thus restricting our- type). In any case, the exact value of c is necessary selves by the special solution. Nevertheless, the ho- hence the system (2.14) should be resolved as exactly mogeneous system should be considered as the eigen- as possible. Let us consider it more closely. The value problem for caˆ ’s (see below). After resolving main problem is the functions ga are mixed between this eigenvalue problem and substituting the found equations. To separate them, let us recall that there = functions δXa kfa and eigenvalues caˆ back in the exist N − 1 orbit equations, whose varying resolves action (2.9), we can rewrite it in the explicit zero-brane the separation problem. Let us demonstrate this for form (omitting surface terms) the most important for us case N = 2. Considering (class) (fluct) Eq. (2.13), the varying of a single orbit equation yields Seff = S + S eff eff δX2 X g2 =− ˙2 + (0) 2 = 2 = , (2.18) ds x µ α2 k , (2.15) δX1 X1 g1 114 K.G. Zloshchastiev / Physics Letters B 519 (2001) 111–120 hence the system (2.14) at N = 2 can be separated into where α =−c/µ and f>0 is some function which the two independent equations depends on a concrete gauge: e.g., f = 1 + q2

corresponds to the proper-time gauge. This function Xa −g + − cˆ g = 0, (2.19) does not affect the Hamiltonian picture but appears in a X a a a the final Schrödinger equation (h¯ = 1), if one uses Xa = σ acU Xb . In this form it is much cb ˜ 2 l(l + 1) µ − Mf(r ) easier to resolve the eigenvalue problem. Therefore, −Ψ (r)+ + Ψ(r)= 0, the two independent parameters for the action (2.15), r2(1 + r2) α(1 + r2) µ and c , can be determined immediately by virtue of a where r = q/ −q2, f˜ = f/ −q2, M is the total Eqs. (2.5) and (2.19). mass, l is the spin eigenvalue, a non-negative integer 2.2. Hamiltonian structure and Dirac quantization (half-integer? see the corresponding discussion and criticism by Plyushchay [7] of the Pavšic’sˇ paper [6]). The constructing of Hamiltonian and quantum pic- At second, there is no clear understanding what is the tures of the nonminimal theories is a separate large difference between theories with positive and negative problem. From the particle-with-rigidity action parameters α and µ, in particular, it is unclear at which values of those parameters we have the discrete 1 M S =−µ ds x˙2 1 + k2 , (2.20) spectrum of and at which ones we have the 2c continuous spectrum. In fact, it is also caused by and definition of the world-line curvature one can see the first problem, i.e., by the nonuniqueness of the that we have the theory with higher derivatives. Be- Schrödinger equation in coordinate representation. In sides, the Hessian matrix constructed from the deriv- this Letter we will follow the approach [9] because 2 M = ∂ Leff , its eigenvalue structure is well-studied and confirmed atives with respect to accelerations, ab ∂xä∂x¨b appears to be singular that reveals the presence of the both by numerical methods and by comparison with constraints on phase variables of the theory. The con- others approaches. Namely, it was shown that the first- structing of Hamiltonian formalism and quantization order (with respect to Planck constant) corrections to of the quadratic-curvature model has been performed, the bare mass µ are given by the equation reperformed or improved by many people [5–9], see 2ε = B(B − 1) + O h¯ 2 , (2.21) Chapter IVA of Ref. [9] for a recent state of the the- √ ory and details. Summarizing those results one can where ε = 8µ2∆/c, ∆ = 1 − M/µ, B = 8 µM/c; say that, despite the Hamiltonian picture of phase vari- all the values should be real (B should be positive) ables and constraints is more or less understood, some to ensure the existence of bound states and hence to questions, such as the reliable (unique) obtaining of guarantee the stability of the quantum nonminimal the mass spectrum of such a nonminimal particle, re- particle. Further, Eq. (2.21) can be rewritten as main to be open. √ √ √ At least, the two following problems seem to be 4 − √ c 1 ∆ c 2 presented here. At first, there appears no uniquely ∆ − √ 8 1 − ∆ − = O h¯ . (2.22) µ 4 2 µ defined Schrödinger equation and hence it is known no unique spectrum of eigenvalues. This is caused by Suppose first that c and µ are positive. Then the the gauge freedom which arises at the constructing ground-state eigenvalue of M is given by ∆ = 0, i.e., of the Hamiltonian by virtue of the Dirac approach. M0 = µ. We have deal with small oscillations around When considering constraints on the phase variables a ground state hence expanding the Eq. (2.22) in the we eventually have the following set of constraints (the Taylor series in the vicinity of the ground state ∆0 = 0 definitions of Ref. [7] are used) we eventually obtain the desired correction + − 2 + 2 + 2 ≈ M a2(a2 − 16) Pq q µ α p (pr) 0, = 1 − 4 + O h¯ 2∆2 ,c>0, 4 − − Pq µ a 64a 256 √ + f q2 ≈ 0, −P 2 − M ≈ 0, −P 2 (2.23) K.G. Zloshchastiev / Physics Letters B 519 (2001) 111–120 115 √ √ where a ≡ 2 8 − c/µ. Note, here unlike Ref. [9] dilaton gravity described by the action we are not needed in further approximations like [ ] O(c/µ2) etc. SDG g,φ √ Now, what is happening when c<0? The first im- = 1 2 − + 1 2 + 2 d x g D(φ)Rg 2 (∂φ) V(φ) , pression is that imaginary terms appear in Eq. (2.22) 2k2 and break bound states. However, numerical simula- (3.1) tions show that it is not so. To demonstrate it analyti- where R is the Ricci scalar with respect to metric cally let us look at Eq. (2.21) and recall that our theory g , D and V are arbitrary functions. By virtue of is defined up to the sign of µ (at least), therefore, the µν the Weyl rescaling transformation g¯ = Ω2(φ)g , substitution µ =−|µ| and c =−|c| (keeping M pos- µν µν φ¯ = D(φ), where Ω is such that dD d lnΩ = 1 , we itive) changes nothing in Eq. (2.21) (note, the root in dφ dφ 4 obtain the action with only one function of dilaton, the R.H.S, of this equation is defined up to a sign), and V( φ)¯ = V(φ(φ))Ω¯ −2(φ(φ))¯ , eventually we have instead of Eq. (2.22) the following one ¯ 1 2 ¯ ¯ SDG[¯g,φ]= d x −¯g φRg¯ + V(φ) . (3.2) √ √ 2k2 | | 4 ¯ − | | 2 ¯ c ∆ 1 ¯ c 2 ∆ − √ 8 ∆ − 1 − = O h¯ , In the conformal coordinates g¯ = e2u¯η this action |µ| 4 2 |µ| µν µν can be written in the form (2.1) where N = 2and (2.24) = 2 = − − where ∆¯ = 1 + M/|µ|. Note, now the ground state S k2SDG,σab offdiag( 1, 1), (3.3) belongs to the lower continuum M =−|µ| but the 1 ¯ 0 Xa ={φ,¯ u¯},U=− e2uV( φ),¯ (3.4) excited-state eigenvalue M must be positive. In fact, 2 it means that we can impose the near-ground-state and, therefore, we can apply to dilaton-gravitational 2 approximation O(∆ ) ≈ 0 (which was used to obtain systems all the machinery of the previous section. Eq.√ (2.23)) only if |µ|→0. Then, keeping M/µ and Namely, knowing the localized solution (if it exists, of |c|/µ finite, we obtain course) it is supposed to perform the transition from the Liouville coordinates to the collective coordinates M a2(a2 − 16) =−1 − 4 + O h¯ 2∆2 ,c<0, of the solution, get rid of zero modes, obtain the effec- |µ| a4 − 64a − 256 tive one-dimensional point-particle action with non- (2.25) minimal corrections, and then quantize it in a standard √ √ way as a mechanical 1D system. In practice, however, where a ≡ 2 8 − |c|/|µ| and the values of c and it is enough to know the localized dilaton-gravity solu- µ are such that M is positive (which is possible fol- tion and then everything is (almost) automatic due to lowing the simple numerical estimations). However, the analogy given by Eqs. (3.3), (3.4). if |µ| is not small then the total mass is given by Thus, every localized solution of the dilaton grav- Eq. (2.24) resolved with respect to the total mass M. ity can be described in terms of the quantum mechan- Thus, we have ruled out the expressions for the ics of nonminimal particles. That discloses the way of first-order corrections to the total mass of a quantum uniform “Hamiltonization” and consistent nonpertur- particle with rigidity. It should be pointed out that bative quantization of any given dilaton gravity in the this approach is indeed nonperturbative because all the vicinity of the solution as a region with nontrivial vac- formulae above cannot be obtained by virtue of the uum. perturbation theory assuming small 1/c or µ.

4. Supergravity 0-brane solutions as nonminimal 3. Nonminimal particles in dilaton gravity particles

In this section we demonstrate how the above- Here we demonstrate on classical and quantum mentioned formalism can be also applied to the 2D levels the examples of how the 0-brane solution of 116 K.G. Zloshchastiev / Physics Letters B 519 (2001) 111–120

ˇ 2 the gauged supergravity above can be described in µ 2 2 2 + dr +ˇµ dΩ − , (4.6) terms of the one-dimensional mechanics with rigidity r D 2 if certain (very natural and evident, such as the and it is possible to perform the Freund–Rubin com- convergence of necessary integrals) requirements of − pactification on SD 2 of the action (4.3) to obtain the the field-to-particle transition formalism are satisfied. following effective gauged supergravity action √ 4.1. Dilatonic brane = 1 2 − δφ + 2 + S 2 d x g e Rg γ(∂φ) Λ , (4.7) 2k2 The supergravity action in the Einstein frame is = D−3 2 D−2 − 2 given by where Λ µˇ D−3 ∆ . After the Weyl rescal- −γ/δ2 δφ √ ing gµν = Φ g¯µν (Φ = e ) it takes the form of 1 D e 4 2 S = d x −G RGe − (∂φ) the action (3.2): 2k2 D − 2 D 1 1 2 1−γ/δ2 − 2aφ 2 S = d x −¯g ΦRg¯ + ΛΦ , (4.8) e F2 , (4.1) 2 2 · 2|!| 2k2 hence in conformal gauge it also can be written as where φ is D-dimensional dilaton, F2 = dA is the M field strength of the 1-form potential A = AM dx , the special case of the sigma model (2.1), following where index M runs from 0 to D − 1. Eqs. (2.2)–(2.4). Thus, 0-brane solution of the gauged Applying the electric–magnetic duality to this ac- supergravity above can be described in terms of the tion one can interpret the action of the dilatonic quantum mechanics of nonminimal particles as well. 0-brane as being magnetically charged under the (D − In turn it means that we can construct the Hamiltonian 2)-form field strength FD−2: and quantum theories which properly take into account √ the fluctuations of all the acting fields including 1 D e 4 2 gravity. S = d x −G RGe − (∂φ) 2k2 D − 2 The static solutions of the system described by the D 1 latter action are well-known [11] − −2aφ 2 e FD−2 , (4.2) √ √ 2(D − 2)|!| 1 < ds2 =− x Λ − 2 ΛM dT 2 which after the Weyl rescaling to the dual frame < e − 2a φ d √ √ −1 G = e D−3 G can be rewritten in the form 1 MN MN + x Λ < − 2 ΛM dx2, (4.9) < 1 D d δφ 2 √ Sd = d x −G e R d + γ(∂φ) γ 2k2 G Φ = Λx, <≡ 2 − , (4.10) D δ2 − 1 2 where M is the diffeomorphism-invariant parameter FD−2 , (4.3) 2(D − 2)|!| associated with the mass of solution: √ − − where δ =−a D 2 , γ = D 1 δ2 − 4 . In the dual Λ 1 D−3 D−2 D−2 M =− Λ(∇Φ)2 + Φ< . (4.11) frame the 0-brane solution is 2 < 2 − 4 2 = D−3 − ∆ 2 + 2 + 2 2 If the black hole is formed then the event horizon dsd H H dt dr r dΩD−2 , (4.4) √ √1 √M 1/< − appears at x = Φ / Λ = 2< , and the φ a(D 2) H H Λ Λ e = H 2∆ ,FD−2 =∗(dH ∧ dt), (4.5) surface gravity (= 2πTH ) and entropy are given by − a2 = + ˇ D 3 = − + − /< /< where H 1 (µ/r) and ∆ 2 (D 2) 1 1 1 − 1 M 2π M 2 D 3 . In the near-horizon region the metric takes the κ = √ 2< √ ,S= 2< √ . D−2 Λ Λ κ2 Λ D−2 2 AdS2 × S form [10]: (4.12) − 2− 4(D 3) µˇ ∆ Thus, we have all the necessary formulae to study the ds2 ≈− dt2 d r nonminimal mechanics of any given dilatonic 0-brane: K.G. Zloshchastiev / Physics Letters B 519 (2001) 111–120 117 expressions (2.2)–(2.4) and (4.8) relate supergravity that black holes arising in the theories like those de- with the approach of Section 2, whereas Eqs. (4.9) and scribed by Eqs. (3.1), (3.2) or (4.8) seem not to be the (4.10) give the desired solution. Let us consider the true black holes because of the “semi-completeness” two important examples, γ/δ2 = 1andγ/δ2 = 0: of the singularity [13]. Namely, all light-like extremals approach a singularity at an infinite value of a canon- CGHS-reduced 0-brane γ/δ2 = 1. We have ical parameter while time- and space-like ones hit a D − 3 singularity at a finite value. Thus, solution (4.14) de- γ = 4,δ=−2,a= 2 , D − 2 scribes either naked singularity or (quasi-)black hole

D − 3 2 depending on what is the sign of Φ0 hence if we are in- Λ = , (4.13) terested in solutions with positive Φ we must suppose µˇ the black hole case Φ0 > 0. Finally, let us say some −2φ and applying the transformation Φ = e , g˜µν = words about the value of C. If the black hole is formed 2φ e g¯µν one can see that the action (4.8) has the form and one calculates the surface gravity from the metric of that of the CGHS model [12] above by virtue of the naive Wick rotation argument and compares the result with that obtained from (4.12)√ SCGHS = =− at < 1 then one can conclude that C 2/ Λ. 1 2 −2φ 2 = −˜ ˜ + + However, below we will keep C unassigned for gen- 2 d x g e ΦRg 4(∂φ) Λ . 2k2 erality. Let us perform now the field-to-particle transition Then the static solution of the action (4.8) at γ/δ2 = 1 for this solution to obtain the action of the type (2.15) is given by remembering that Xa ={¯u, Φ} and signature of our Λ | |+ ¯ metric is opposite to that assumed in Section 2. To u¯ = Cρ +¯u ,Φ= eC ρ u0 + Φ , (4.14) s 0 s C2 0 obtain the first parameter µ which is given by Eq. (2.5) √ | | − we find that for our solution L(Xs ) = 2ΛeC ρ and where ρ = γ (r − vt), γ 1 = 1 − v2 and C<0, l l hence u¯0, Φ0 = 2M, v are some constants. We assume the absolute value of ρ in Φ to guarantee that the s 2 Λ true dilaton (which is the logarithm of Φ up to a µ =− . (4.17) 2 C coefficient) remains real. k2 Before considering the field-to-particle transition let To find the nonminimal coefficients α let us consider us clarify the global properties of this solution. The bar i ¯ the corresponding Sturm–Liouville eigenvalue prob- metric eus η is flat but the true metric, µν lem. The system (2.19) takes the form (no summation eCρ over a) ds2 = −dt2 + dr2 Φs Cρ − − = e ga (z) baga(z) 0, (4.18) = −dτ2 + dρ2 , (4.15) Φs where z = Cρ, b1 = cˆ /C and b2 = cˆ /C−1. One can → =−C 1 2 is not, and after the transformation ρ R Λ see that this system yield free modes and, therefore, no ln Λ eCρ + Φ can be written in the Schwarzschild- bound states are formed on the (nonclosed and infinite) C2 0 like form axis of ρ’s. In fact, the absence of any effective 2 interaction means that the particle action does not 2 C Λ R 2 ds =− 1 − Φ e C dτ contain nonminimal curvature-dependent terms, and Λ 0 the CGHS-reduced 0-brane is trivial in this sense. Λ Λ R −1 2 2 + 1 − Φ e C dR , (4.16) Another case, γ/δ = 0, appears to be more nontrivial 2 0 C in this connection. which exhibits the existence at Rsing =−∞ of either naked singularity or singularity under the horizon JT-reduced case γ/δ2 = 0. In this case the bar =−C RH Λ ln Φ0 if Φ0 > 0. It is necessary to point out metric coincides with the physical one, and the action 118 K.G. Zloshchastiev / Physics Letters B 519 (2001) 111–120

(3.2) becomes that of Jackiw–Teitelboim gravity where µ is given by Eq. (4.22). On the quantum level these particles should be 1 √ S = d2x −gΦ(R + Λ) (4.19) treated as outlined in the second part of Section 2, e.g., 2 g 2k2 the first-order quantum corrections to their masses are whereas the desired static solution is given by given for the particle S1 by Eq. (2.23) where c ≡ 3C/2 | |≡ and for the particle S2 by Eq. (2.24) where c C/2 C − eus = cosh 2 C/2 ρ ,g= eus η , or by Eq. (2.25) provided µ is small. Λ µν µν (4.20) 4.2. Non-dilatonic brane Φs = Φ∞ tanh C/2 |ρ| , (4.21) √ −1 2 In this case φ ≡ 0 and the action (4.1) becomes where ρ = γl(r − vt), γ = 1 − v and C = l singular. The true effective action for the non-dilatonic 2Λ3/2M>0, Φ∞ =|Φ∞| sign(ρ) > 0, v are some supergravity is constants. We assume the absolute value of ρ in Φs because of the same reason as in the previous 1 √ 1 S = dDx −G R − F F MN , case. Also similarly to the√ previous case we√ find that 2 G · |!| MN − 2kD 2 2 L(Xs ) =−2CΦ∞ sinh( C/2 |ρ|) cosh 3( C/2 |ρ|) and hence the first coefficient for the action (2.15) is (4.25) √ where F is the field strength of the U(1) gauge field −2 µ = k 2CΦ∞. (4.22) 2 AM . Let us take the ansatz for the D-dimensional metric in the form To find the nonminimal coefficients αi let us consider the corresponding Sturm–Liouville eigenvalue prob- G dxM dxN lem. The system (2.19) takes the form MN − 4 ϕ = g dxµ dxν + e D−2 dΩ2 , (4.26) − − − 2 + = µν D−2 g1 (z) 2 2tanh (z) b1 g1(z) 0, (4.23) = ν − − − 2 + = where µ,ν 0, 1, gµν and ϕ are functions of x .If g2 (z) 2 6tanh (z) b2 g2(z) 0, (4.24) √ one assumes that the gauge field A is electric then ∇ MN = where z = C/2 ρ and ba = 2caˆ /C. Using the as- resolving the Maxwell equation M F 0 one ymptotical technique described in Refs. [9,14,15] we obtains can find that the only solutions of the Sturm–Liouville √ = 2ϕ − bound-state problem are Ftr Qe g, (4.27) −1 where g = det(gµν ) and Q is the constant related g1 = const. · cosh (z), cˆ =−C/2 , 1 to the electric charge. Then the initial action after g = const. · tanh(z), cˆ = 0 , D−2 1 1 dimensional reduction on S takes the following −2 g = const. · sinh(z) cosh (z), cˆ = 3C/2 , form 2 2 −2 √ − g2 = const. · cosh (z), cˆ = 0 . 1 2 −2ϕ D 3 2 2 S = d x −g e Rg − 4 (∂ϕ) 2k2 D − 2 The solutions with zero eigenvalues should be omitted 2 2 since they do not fulfill the requirements of the 4 ϕ Q − (D − 2)(D − 3)e D−2 + e4ϕ , field-to-particle transition approach. The rest ones are 2 good so the field solution {us,Φs } can be viewed as the doublet superposition of the two particles with (4.28) rigidity: i.e., it can be reduced to the form (2.2)–(2.4) hence (being gauged) to the form (2.1) that means that 2 1 2 its localized static solutions can be interpreted as S1 =−µ ds x˙ 1 − k , C nonminimal particles as well (however, some subtle 1 thing exists here so let us demonstrate the approach S =−µ ds x˙2 1 + k2 , 2 3C step by step). The corresponding equations of motion K.G. Zloshchastiev / Physics Letters B 519 (2001) 111–120 119

=− 2 =−D−3 are where δ 2, γ/δ D−2 ,and D − (∂Φ)2 ∂ν Φ 2 3 D−3 δ Q R + − ∇ ¯ − − δ g − 2 ν Λ(Φ) ≡−Φ D 2 (D − 2)(D − 3)Φ D 2 − Φ , D 2 Φ Φ 2 2 δ Q δ u¯ − (D − 3)(D − 4)Φ D−2 − Φ = 0, (4.29) therefore, in the Liouville gauge g¯µν = e s ηµν the 2 Weyl-transformed action obtains the form (2.1) where 2 D − 3 gµν (∂Φ) ∂µΦ∂ν Φ ∇ ∇ + − D µ ν Φ ¯ C/2 − − D − 2 2 Φ Φ X1 ≡ eus = Φ D 2 cosh 2 C/2 ρ , D − 3 s − 2 gµν D 4 Q 1 + (D − 2)(D − 3)Φ D−2 − = 0, (4.30) X2 ≡ Φ ,U≡ Λ¯ X2 X2eX . 2 2Φ s However, it turns out that the effective (sigma-model) where Φ = eδϕ, δ ≡−2and∇ means the covariant Lagrangian vanishes on this doublet solution, there- derivative with respect to the metric g. fore, the non-dilatonic brane in the near-horizon re- In this Letter we are mainly interesting in gion cannot be described by the two-component sigma 0-branes, moreover, in the near-horizon region. The model. In fact, it is caused by the fact that the elec- D-dimensional 0-brane solution is given by tric component of this solution happens to be constant 2 −2 2 2 2 2 2 in this region hence the whole solution cannot be re- ds =−H dt + H D−3 dr + r dΩ − , D 2 garded as two-component. =− −1 At H , (4.31) To construct the appropriate one-component effec- − = where H = 1 + (µ/r)ˇ D 3. In the near-horizon region tive action let us fix Φ Φs rather than g. Then in the D−2 space–time becomes AdS2 × S and we have Liouville gauge the remaining equation (4.29) can be written as the Liouville equation 2(D−3) −2 2 r 2 r 2 2 2 ds =− dt + dr +ˇµ dΩ − , ηµν∂ ∂ u − Λ u = , µˇ µˇ D 2 µ ν e 0 (4.36) D−3 hence the appropriate one-component effective theory At =−(r/µ)ˇ , (4.32) is the Liouville one. The field-to-particle transition whereas ϕ becomes constant. for the Liouville theory was previously considered Therefore, we will seek for those static solutions in Ref. [15] so let us use the results obtained there. of Eqs. (4.29), (4.30) whose space–time part has the Defining D−2 desired AdS2 × S form. One can check that the following pair, β ≡ 1/2,m= 2Λ, ζ ≡ mC = 2ΛC, → 2 C S 2k2S, us −2 us e = cosh C/2 ρ ,gµν = e ηµν , (4.33) Λ we straightforwardly obtain the desired nonminimal − Λ/2 1 D/2 particle action −2ϕs e ≡ Φs = , (4.34) (D − 3)2 2 2 k Seff =−µ ds x˙ 1 − , (4.37) − − 1 where Λ = 2(D−3)2 (D 2)(D 3) D−3 , C is a positive ΛC Q2/2 √ = −2 constant and ρ is the same as in previous sections, where µ 2k2 2ΛC. Finally, also we immediately appears to be the solution of Eqs. (4.29), (4.30), obtain that the first-order quantum correction to the besides the gravitational part has the desired AdS form bare mass µ is given by Eq. (2.24) (or by Eq. (2.25) = → | |≡ Rgs Λ. if µ 0) where c ΛC/2. Further, performing the Weyl rescaling {g,ϕ}→ −γ/δ2 {¯g,Φ} such that gµν = Φ g¯µν we can rewrite the action (4.28): References 1 2 ¯ [1] R. Rajaraman, Solitons and Instantons, North-Holland, Ams- S = d x −¯gΦ Rg¯ + Λ(Φ) , (4.35) 2 terdam, 1988. 2k2 120 K.G. Zloshchastiev / Physics Letters B 519 (2001) 111–120

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