preprint TPJU-28-93 Classical Open String Models in 4-Dim Minkowski Spacetime ∗ 4 9 9 1 n Pawe l We¸grzyn a † J 3 Department of Field Theory 1 Institute of Physics, Jagellonian University 4 30-059 Cracow 16, Poland v 4 1 0 1 0 4 Abstract 9 / h Classical bosonic open string models in four-dimensional Minkowski t - p spacetime are discussed. A special attention is paid to the choice of e edge conditions, which can follow consistently from the action principle. h : We consider lagrangians that can depend on second order derivatives v i of worldsheet coordinates. A revised interpretation of the variational X problem for such string theories is given. We derive a general form of a r a boundary term that can be added to the open string action to control edge conditions and modify conservation laws. An extended boundary problem for minimal surfaces is examined. Following the treatment of this model in the geometric approach, we obtain that classical open string states correspond to solutions of a complex Liouville equation. In contrast to the Nambu-Goto case, the Liouville potential is finite and constant at worldsheet boundaries. The phase part of the potential defines topological sectors of solutions. December 1993 ∗Supported in part by the KBN under grant 2 P302 049 05 † e-mail address: [email protected] 1 Introduction. There is a common conviction that in order to gain more insight into the dynamical structure of QCD we need most likely to use some string representation of this theory. It is suggested by topological nature of 1/N expansion [1], area confinement law found in the strong coupling lattice expansion [2], the success of dual models in description of Regge phenomenology, the existence of flux-line solutions in confining gauge theories [3, 4]. More arguments are presented in recent reviews [5, 6]. In spite of numerous works, there is still a state of confusion about the existence of anexact, or even approximate, stringyreformulation of4-dimQCD atalldistance scales. Even at any specific scale, it is not evident what is the adequate set of string variables and fields and how they correspond to QCD gauge fields. Referring only to long distance scale, we usually adopt the naive, but lucid, picture of flux tubes regime. Apairofquarksintheconfiningphaseisjoinedbyacolourfluxconcentrated in a thin tube. If these quarks are kept sufficiently far apart, the flux tube behaves like a vibrating string. Using string variables as collective coordinates, one should in principle find flux tube excitations by some quantization of the string action. The question what kind of the string action should be employed to represent the flux tube has yet to be answered. It is conceivable that to the lowest order the action is just given by the Nambu-Goto action, which decribes an infinitely thin relativistic bosonic string with constant energy per unit length. As is well known, we cannot be satisfied with this first approximation because of some unacceptable features of the quantized Nambu-Goto string. Apart from the problems with conformal anomaly outside the critical dimension or tachyons and undesirable massless states in quantum spectra (which are presumably less embarrassing at the long distance scale [7]), all standard quantizations give the incorrect number of degrees of freedom if we confront it with QCD predictions [8]. Basically, there are two ways to modify the 4-dim Nambu-Goto action. In the first approach, keeping the conformal invariance we can place additional fields on the worldsheet [9, 10]. The conformal anomaly can be saturated due to the contri- bution of new conformal fields. Since we can hardly justify the assumption that only massless degrees of freedom are important at the hadronic scale, so the respecting of conformalsymmetry ishereratheracompromisetomakeourtheorymathematically tractable. The second kind of modifications of the Nambu-Goto action, advocated in many papers (e.g. [11]), is to introduce new action terms representing interac- tions between transverse string modes. The fact that Regge trajectories derived directly from fundamental quark models [12, 13, 14] depart somewhat from straight lines is a strong argument that vibrating string modes cannot be considered as free. Next, some couplingsbetween these modes(short-distance interactions) wouldcause preferring smooth string worldsheets, leading to a well-defined quantum theory. Un- fortunately, all such string ”self-interaction” terms involve higher order derivatives in their lagrangians. Theories with higher order derivatives usually reveal embar- rassing pathological features, like lack of the energy bound, tachyons already on 1 the classical level, unitarity violation due to the presence of negative norm states. Presumably, it means that one must regard any particular effective theory of this type with a limited range of validity. From technical point of view, such theories of strings are non-linear and cannot be linearized by a suit choice of gauge. Subse- quently, a string cannot be described as an infinite set of oscillators and there is no analogue of Virasoro algebra. The conformal symmetry is usually spoiled. All that makes the evaluation of physical observables technically difficult. In this paper, we discuss possible modifications of the Nambu-Goto model (or any other specific bosonic string model) by the change of boundary conditions for open strings. This aspect is not well explored in literature, even though the choice of boundary conditions can be crucial for defining relevant open string models. Let us give some examples: - Taking the usual hadronic string picture, we assume that quarks live only at the opposite endpoints of the string and communicate through their couplings to the string between. Then, to some extent the choice of worldsheet boundary conditions determines quarks trajectories. For instance, in the classical Nambu- Goto model they are rather peculiar, being boosted periodic light-like (null) curves. Undoubtedly, the unsolved problem how the quark masses and quantum numbers (spin, color) couple to the string variables, partly lies in the proper specification of string edge conditions. - It is obvious that any internal symmetry of the worldsheet is necessarily broken when worldsheet boundaries are included. Conformal transformations or the full set of all reparametrizations are examples of that. Correspondingly, conformal field theories defined onsurfaces with boundaries are usually endowed with only one copy of Virasoro algebra (instead of two, as for closed surfaces). Recently [10], on the same basis the chiral symmetry breaking mechanism has been included in hadronic string models. This simple observation that the existence of boundaries restricts the group of local worldsheet symmetries, indicates that physical observables can essentially depend on fields or currents evaluated on string boundaries. - One of the straightforward calculations to test some open string model against QCD expectations is to evaluate the static interquark potential. Asympotically at the long distances scale, this potential is linear and its slope can be related to the string tension. The first quantum corrections give an universal Coulomb term [15] (Casimir effect), being the function of the number of worldsheet fields and of their boundary conditions. In an approximation of flux-tube action by some conformal 2 string theory, onecanrepresent boundaryconditions bytheset ofrelevant conformal operators inserted at boundaries [16]. The physical states are now constructed with the help of both bulk and boundary operators. The Coulomb term depends on the effective conformal anomaly [17], being the the total conformal anomaly diminished by the weight of the lowest state. This weight is sensitive to the choice of boundary operators [9]. - The influence of worldsheet boundaries on critical string field theories has been discussed in recent papers (see [18, 19] and references therein). In the framework of BRST formalism in the critical dimension, we can consider either Neumann-type (e.g. standard Nambu-Goto edge equations) or Dirichlet-type boundary conditions imposed on worldsheet coordinates. With Dirichlet conditions, we have no physi- cal open strings, but the closed-string theory is radically modified, particularly the massless spectrum. Instead of characteristic exponential fall-off of fixed-angle scat- tering amplitudes for string models at high energies, we obtain for Dirichlet strings power-like behaviour, like for parton models. We see here that the special type of worldsheet boundaries, where these boundaries are mapped to single spacetime points, implies that some point-like structure may appear at high energies. In this work, we restrict ourselves to discuss open bosonic string models defined by local lagrangians densities that can depend on second order derivatives of world- sheet coordinates. In Section 2, we present general formulas suitable to perform classical analysis of such string models. In comparison with earlier works on this subject, a different interpretation of the variational problem for string actions with second order derivatives is given. Moreover, all derived classical formulas are explic- itly covariant with respect to reparametrization transformations. In Section 3, we derive ageneralformofa boundarytermthat canbeaddedto theaction, allowed by requirements of Poincare and reparametrization invariances. Such a term can mod- ify edge conditions for open strings while bulk equations of motion are preserved. Canonical conserved quantities are modified by some edge contributions. Section 4 is devoted to the classical analysis of the string model defined by the Nambu-Goto action with some new boundary terms added. It is argued that such an open string model can be a suitable modification of the Nambu-Goto model as far as hadronic string interpretation is concerned. We carry out the classical analysis using the ge- ometric approach, which is particularly convenient for our purposes. The classical open string configurations that extremize the extended action correspond to solu- tions of a complex Liouville equation. The relevant edge conditions for a Liouville 3 field are derived. The edge values are constant and finite there. Some preliminary discussion about physical consequences is made. In Appendix, the notation used throughout the paper is introduced and some basic mathematical definitions and equations of surface theory are collected. 2 String Lagrangians with Second Order Deriva- tives. In thissection, we introduce some general formulas appertainedto the classical anal- ysisofstringmodelsdefinedbylagrangianswhichdependonsecondorderderivatives of worldsheet radius vector. In comparison with previous papers (e.g. [20, 21]), all formulas presented below are explicitly covariant with respect to the reparametriza- tion, and we care especially with the correct derivation of edge conditions for open strings. Let us consider the general form of the bosonic string action, τ2 π S = dτ dσ . (1) string Zτ1 Z0 L It is convenient to represent the lagrangian density as = (X ;X ) = √ g (gab;X ; X ) , (2) string string µ,a µ,ab µ,a a b µ L L − L ∇ ∇ where issomescalarfunctionmadeupofitsspecifiedarguments. Havingthestring L lagrangian with second order derivatives written down in the above form, we can much easier perform mathematical calculations and keep the explicit reparametriza- tion invariance in all following steps. Toderive theclassicalequations ofmotion, wearetoevaluatethevariationofthe string action under the infinitesimal change of the worldsheet. Usually, the following boundary conditions are assumed, ˙ δX (τ ,σ) = δX (τ ,σ) = 0 , i = 1,2 . (3) µ i µ i There is some subtle problem at this point. The above requirements suggest the different interpretation of the variational problem in comparison with the usual Nambu-Goto case. In (3), not only initial and final string positions are fixed, but 4 also the initial and final velocities of string points. Therefore, if we consider some string at the time τ , another string at the time τ and some string trajectory being 1 2 a solution of Euler-Lagrange eqs. which interpolates between them, the solution does not extremize the string action unless we restrict possible deviations of the worldsheet to those that do not change its tangent vectors at the initial and final positions. Inotherwords, thestringinstantstateisspecified notonlybyitsposition, but also by its velocities. In fact, this modified interpretation is not true as the boundary conditions (3) are not quite proper for the string variational problem with second order derivatives. This point will be clarified below. The classical equations of motion following from (1) can be presented in the explicitly covariant form √ g Πa = 0 , (4) − ∇a µ where Πa is given by the following formula µ ∂ ∂ ∂ Πa = aX L +2 L gab cX + L . (5) µ −L∇ µ − ∂X,µa ∂gbc ∇ µ ∇b"∂( a bXµ)# ∇ ∇ For open strings, the edge conditions at σ = 0,π must be satisfied, ∂ √ gΠ1 +∂ √ g L = 0 , (6) − µ 0" − ∂( 0 1Xµ)# ∇ ∇ ∂ √ g L = 0 . (7) − ∂( Xµ) 1 1 ∇ ∇ For the sake of more convenient notation, here and throughout the paper we define and calculate the variational derivatives of with the formal assumption that g01 L and g10, Xµ and Xµ are independent variables. Thus, all variational 0 1 1 0 ∇ ∇ ∇ ∇ derivatives on r.h.s. of (5) are tensor objects with respect to the reparametrization invariance. The covariance of edge conditions becomes easy to check if we remind that in the presence of the worldsheet boundary any reparametrization transforma- tion σa σ˜a(τ,σ) must satisfy → σ˜(τ,0) = 0 , σ˜(τ,π) = π . (8) It is necessary in order to preserve the condition that the string parameter σ belongs to the interval [0,π]. In other case, performing the variation of the string action we 5 are forced to implement the variations due to the change of σ-interval, and the fact that the set of allowed reparametrization transformations is restricted for open strings manifests in additional Euler-Lagrange eqs. The derivation of (4) from the standard Euler-Lagrange variational equations is straightforward, so let us only cite the following identities used in this derivation ∂ L Xµ = 0 . (9) ∂( Xµ) ,c a b ∇ ∇ To prove the above identities for lagrangians which include only scalar constant pa- rameters, it is enough to notice that the scalar (with respect to both reparametriza- tion and Poincare transformations) function can be composed of the following L ”building blocks” gab , ǫµνρσ( X )( X )X X , Xµ X , a b µ c d ν ρ,e σ,f a b c d µ ∇ ∇ ∇ ∇ ∇ ∇ ∇ ∇ and refer to the trivial identities ( Xµ)X = 0 . a b µ,c ∇ ∇ In general, the origin of identities (9) lies in the reparametrization invariance of thestringaction(1). ThefullsetofallNoetheridentities(see(27-29))followingfrom the reparametrization invariance of the string action with second order derivatives has been derived in [22]. Let us return to the problem of boundary conditions (3) imposed on the varia- tions of the worldsheet. If we assumed only that δX (τ ,σ) = 0 , i = 1,2 , (10) µ i then using eqs. of motion (4) together with edge conditions (6,7) we would obtain the following result for the variation of the string action δS = πdσ √ g ∂L δX˙µ τ=τ2 . (11) Z0 − ∂(∇0∇0Xµ) (cid:12)τ=τ1 (cid:12) (cid:12) If g = 0 or the surface is locally flat then the following term vanishes, else we can choose parametrization in such a way that the four vectors (X˙ ,X′,X¨ ,X˙′) are µ µ µ µ 6 linearly independent at the point of the worldsheet with τ = τ . Then, we can write i down the general form of δX˙ as the linear combination of these vectors, µ δX˙ = a X˙ +a X′ +a X¨ +a X˙′ . (12) µ 1 µ 2 µ 3 µ 4 µ ˙ On the other hand, the variation δX induced by the change of parametrization µ σa σa +δσa is given by → δX˙ = X¨ δσ0 X˙′δσ1 . (13) µ − µ − µ It means that the variations of X˙ in the directions of X¨ and X˙′ are not important, µ µ µ because they can be removed by the change of parametrization. In turn, if we restrict ourselves to the ”physical” variations of the worldsheet, then with the help of identities (9) we conclude that the term (11) vanishes. Therefore, there are two ways to define properly the variational problem for string action functionals which depend on second order derivatives. One way is to assume boundary conditions (10) together with the additional requirements that the variations δX˙ in the directions of X¨ and X˙′ vanish, what in light of (13) means µ µ µ that the choice of the parametrization of the worldsheet is locally fixed at boundary points τ = τ . Other way is to take only the boundary conditions (10), as in the i Nambu-Goto case, and together with relevant eqs. of motion and edge conditions we obtain additional equations ∂ √ g L = 0 for τ = τ ,τ , (14) − ∂( Xµ) 1 2 0 0 ∇ ∇ which have no dynamical content and impose only some boundary constraints on the choice of worldsheet parametrization. Recapitulating, the interpretation of the variational problem for string actions with second order derivatives is the same as in the usual Nambu-Goto case. To derive the classical dynamics of strings from the variational principle it is just enough to consider the boundary conditions (10), i.e. to assume that the initial and final string positions are fixed. The appearence of the term (11) in the action variation and resulted equations reflect only the fact that the geometrical definitions of the initial and final string positions are not invariant. One more comment on the derivation of edge conditions should be made. They are an integral part of equations of motion. They arise as in the variational problem for open string worldsheets the whole boundary of the worldsheet is not fixed (like 7 in an ordinary Plateau problem for two-dimensional surfaces), but only a part of it composed of the initial and final string positions. The other part of the worldsheet boundary, defined by trajectories of string endpoints, is not fixed (the ends of open strings are free). However, we can use another equivalent method for the derivation of edge conditions. In the variational problem we can dispense with considering the edge variations (assuming that the whole worldsheet boundary is fixed), and the edge conditions are produced when we demand that there is no flow of the canonical Noether invariants through the string ends. In distinction with the Nambu-Goto case, for strings with second order derivatives it is not enough to assure only that the canonical momentum is conserved. We must check the same independently for the angular momentum, because of its ”spin part” induced by higher order derivatives. The comment on the latter method of the edge conditions derivation is relevant to the recent work of Boisseau and Letelier [23]. They make use of the internalgeometricaldescriptionofworldsheetstostudymodelsofstringswithsecond order derivatives. In this approach, they gain some new insight into the content of dynamical equations. However, their formalism should be corrected for open strings. The set of edge conditions derived from the conservation of total energy-momentum should be supplemented by additional conditions associated with the total angular momentum conservation. In particular, it changes some results of the work [23]. For example, the prediction that the endpoints of the Polyakov rigid string can travel with a speed less than the velocity of light is not valid. Just taking into account the missing set of edge conditions, we check again that these velocities must be light-like, what agrees with the independent proof of this fact given in [22]. In the last part of this section, we write down formulas for Noether invariants. The total momentum reads π P = dσ p , (15) µ µ Z0 where ∂ ∂ ∂ p = Lstring +∂ Lstring = √ gΠ0 ∂ √ g L . µ − ∂X,µ0 0 ∂X,µ00 ! − µ − 1" − ∂(∇0∇1Xµ)# The total angular momentum can be calculated from the following formula, π M = dσ m , (16) µν µν Z0 8 where ∂ string m = x p + L X = µν [µ ν] [µ ν],a ∂X ,0a ∂ ∂ = √ gX Π0 √ gX L ∂ √ gX L . − [µ ν] − − [µ,a∂( a 0Xν]) − 1" − [µ∂( 0 1Xν])# ∇ ∇ ∇ ∇ 3 Boundary Terms for String Actions. We discuss the general string action functional with some boundary term added, S = d2σ bulk d2σ ∂ Va . (17) Lstring − a Z Z The stationarity of this action results in some equations for the interior of the string following from bulk , and the role of the second action term is to ensure Lstring a more general set of edge conditions for an open string case. Below, we will find the general form of this term allowed by requirements of the locality, Poincare and reparametrization invariance. We restrict ourselves to string lagrangians which de- pend on not higher than second order derivatives, what implies that ∂Va µ X = 0 . (18) ∂Xµ ,abc ,bc The above identities give immediately the following equations ∂V0 ∂V0 ∂V1 ∂V1 ∂V1 ∂V0 = +2 = = +2 = 0 , (19) ∂Xµ ∂Xµ ∂Xµ ∂Xµ ∂Xµ ∂Xµ ,00 ,11 ,01 11 ,00 ,01 and their general solution is of the form Va = ǫabA˜cXµ +B˜a , (20) µ ,bc where A˜c and B˜a are some arbitrary functions which depend on X and their first µ µ derivatives. The translational invariance of the action requires that ∂(∂ Va) ∂Va a 0 = = ∂ , (21) ∂Xµ a ∂Xµ! 9