CHIRAL STRING IN A CURVED SPACE: 1 GRAVITATIONAL SELF-ACTION 0 0 2 Yu.V.GRATS∗,A.A.ROSSIKHINandA.O.SBOICHAKOV n a Department of Theoretical Physics, M.V. Lomonosov Moscow State University J 119899, Moscow, Russia 2 1 We analyze the effective action describing the linearised gravitational self-action for a classical superconducting string in a curved spacetime. It is shown that the divergent 1 part of the effective action is equal to zero for the both Nambu-Goto string and chiral v superconductingstring. 9 4 Nowitiswellunderstoodthattopologicaldefectsinquantumfieldtheorymayplay 0 an essential role in cosmology, and the vertex line defects describable on a macro- 1 scopic scale as cosmic strings are most likely to be actually generated at phase 0 1 transitions in the early Universe.1 Strings are predicted by many of the commonly 0 consideredfieldtheoreticalmodels,andtheir gravitationaleffects becomethe ques- / c tionofincreasinginterestduringlasttwentyyears. Indeed,theevolutionofacosmic q stringnetworkmayplayanessentialroleintheformationofthelargescalestructure - r oftheUniverseobservedatthepresenttime. Butpossiblecosmologicalapplications g is not the only reason for this consideration. There are some nontrivial features in : v the interaction of strings with gravity. One of them is the absence of gravitational i X bremsstrachlung from colliding straight gravitating segments2, while test particle r freely moving near Nambu-Goto string emits gravitational waves.3 Another one is a the absence of classical linearised self-interaction of a Nambu-Goto string with a gravitational field in a four-dimensional Minkowski spacetime.4,5 The last result is nontrivial. We know that ina lotof applicationsthe radiusof curvature ofa string is much larger that its transverse diameter. So, the string can be considered to be infinitely thin, andthe dynamicsofthe vertexline canbe describedwith the use of the Nambu-Gotoactionor its generalization,if the interactionwith anelectromag- netic field or the fields of axion and dilaton is taken into account. In these cases underlyinggaugetheoryisusedonlyforthefixingtheparametersinastringaction. Interaction with matter fields give rise to outgoing flaxes of corresponding quanta. ∗E-mailaddress: [email protected] 1 Atthesametimezerothicknessofstringsmustleadtothedivergenceofaself-force. Thisproblemiswellknowninclassicalelectrodynamics. Inthecaseofpointcharge inMinkowskispace suchasituationwasconsideredbyDirac6 who showedthatthe divergence in the self-action can be removed by the renormalization of the particle mass with the use of corresponding classical counterterm. Analogous problem has beenstudiedinalotofpapersforthestringinteractingwithlinearisedgravitational field onMinkowskibackground,with electromagnetic field, with dilaton and axion, see [4, 5] and the Refs. therein. From our point of view, the absence of classical linearisedgravitationalself-interactionisthe mostinterestingresultofthe previous consideration. Inthe caseofstraightNambu-Gotostring this conclusionis more or less obvious because Newtonian potential of such a string is equal to zero. But for the curved string, for the string in an external gravitational field or for the string with some internaldegreesof freedom,say a currentcarryingone,the answeris not obvious. So,wecansaythatevenatclassicallevelinteractionofcosmicstringswith gravitational field, and in particular gravitational self-interaction, is not properly investigated. Here we consider gravitational self-action of a chiral currentcarrying string, i.e. a superconducting string with an isotropic current. Chiral strings are of increasing interestbecausethissimplemodelmaybeconsideredasafirststeptowardstheun- derstandingofafulltheoryincludingsuperconductivityinstrings. ¿Fromtheother hand, vertex defects of this type arise naturally in some kinds of supersymmetric theory,andtherearesomeinterestingcosmologicalconsequencesoftheir existence. In particular, current can stabilize cosmic string, and chiral vortons are expected to be more stable than non-chiral string loops. We use the metric with the signature (+ ) and the system of units c=1. −−− Inordertodescribethe macroscopiceffects ofthecurrentsarisingfromthe pro- cesses of superconducting kind, Witten7 proposed the use of simple generalization of the Nambu-Goto model which is characterizedby the action 1 S = d2ζ√ γ µ+ γab∂ φ∂ φ , (1) str a b − − 2 ! Z whereζa a=0,1aretheinternalcoordinatesontheworldsheetwhichisimbedded in the four-dimensional spacetime with coordinates xµ and metric g , γ is the µν determinant of the induced metric γ =g xµxν. ab µν ,a ,b In this model there is an additionalinternal scalarfield φ in terms of which one can express the worldsheet supported conserved current 1 a = eabφ a =0 (2) ,b ;a J √ γ J − and energy-momentum tensor 1 =µγ +φ φ γ φ φ,c . (3) ab ab ,a ,b ab ,c T − 2 2 In Eq. (3) eab/√ γ stands for the two-dimensional Levi-Civita tensor, with e01 = − e10 = 1. − − Thesurfacecurrent aandenergy-momentumtensor arethetwo-dimensional ab J T tensorial fields defined on the worldsheet of a string, while corresponding four- dimensional sources which determine electromagnetic and gravitationalinteraction of the string read Jµ(x)= d2ζ√ γ a∂ xµδ(x,x(ζ)) , a − J Z Tαβ(x)= d2ζ√ γ ab(ζ)∂ xα∂ xβδ4(x,x(ζ)) . (4) a b − T Z And, as in the Dirac’s case, the ultraviolet divergence in a self-force, if it is, is the consequence of a distributional nature of the currents (4). Below we will consider the case of chiral string with a so-called null current. Thisconditiondoesnotmeanthatthecurrentitselfmustbeequaltozero,butthat it is timelike γ a b =0=γabφ φ . (5) ab ,a ,b J J For the chiral string the last term in Eq. (3) vanishes, but the energy-momentum tensor still differs from the one describing string of the Nambu-Goto type. So, one can suppose the existence of some differences in the interaction of a Nambu-Goto and currentcarrying strings with an external gravitationalfield. To allow for the interaction with a linearised gravitational perturbations, one must perform the replacement g g+h which is equivalent to the augmentation 7→ of the string Lagrangianin Eg. (1) by the interaction term 1 [g] [g] ab∂ xµ∂ xνh (x(ζ)) , (6) str str a b µν L 7→L − 2T As for the gravitational action, we must expand it in powers of h up to the µν second-order terms. Corresponding equation for the linear metric perturbations has the wellknown form δS [g] √ g gr δ = − T (x) . (7) δgµν(x) − 2 µν Second-order variational derivative of the gravitational action can be found for example in Ref.[7]. Substituting the solution of the Eq. (7) back into the total action, we obtain S [g,x,φ]=S [g]+S [g,x,φ]+ tot gr str + G d2ζ√ γ d2ζ′ γ′ µν(ζ)G¯gr (ζ,ζ′) µ′ν′(ζ′) , (8) 4 − − T µνµ′ν′ T Z Z p where for simplisity we use the notation µν =∂ xµ∂ xν ab. a b T T ThesecondandthethirdtermsintheEq.(8)dependonthestringvariablesand describe the dynamics of an infinitely thin cosmic string which interacts with it’s 3 linearised gravitational field. When gravitational radiation from the string is not taken into account G¯gr is the real part of the Feynman propagator. µνµ′ν′ In the Lorentz gauge, when (h 1/2g h);µ =0, this equation becomes µν µν − 1 δµαδνβ2+Hgrαβµν Ggαrβµ′ν′(x,x′)=−16π gµµ′gνν′ − 2gµνgµ′ν′ δ(x,x′) , (9) (cid:18) (cid:19) (cid:0) (cid:1) and H αβ takes the form gr µν 1 H αβ = 2Rα β 2δαRβ g gαβR+g Rαβ +δαδβR . (10) gr µν − µν − (µ ν)− 2 µν µν µ ν Proceeding along the same line as in Ref. [9] one can obtain the expression 1 G¯gµrνµ′ν′(x,x′)=2∆1/2(x,x′) ǫµµ′ǫνν′ − 2gµνgµ′ν′ δ(σ)+ (cid:18) (cid:19) ′ +θ( σ)vµνµ′ν′(x,x) . (11) − Inthelastequation∆(x,x′)=−(−g(x))−1/2det ∂µ2ν′σ(x,x′) (−g(x′))−1/2,ǫµµ′ is thebivectorofparalleltransport,σ isthehalfofgeodesicdistancebetweenpointsx (cid:0) (cid:1) ′ ′ and x, and vµνµ′ν′(x,x) denotes some regular function, which must be calculated for any particular case. ¿From Eq. (11) we can see that the part of the action we would expect to be divergent is G S = d2ζ γ(ζ) d2ζ′ γ(ζ′)∆1/2(ζ,ζ′)δ(σ(ζ,ζ′)) div 2 − − × Z Z p p µν(ζ) ǫµµ′ǫνν′ 1gµνgµ′ν′ µ′ν′(ζ′) . (12) ×T − 2 T (cid:18) (cid:19) In four spacetime dimensions the integral in Eq.(12) diverges logarithmically as ′ ζ ζ , and it must be renormalized. This integral can be estimated with the use → of the normal Riemann coordinates on the worldsheet with the origin at the point ζ or using any other regularizationscheme, see for example Ref. [5], 1 S =2Glog△ d2ζ√ γ µν g g g g γδ+S , (13) eff µγ νδ µν γδ fin δ − T − 2 T Z (cid:18) (cid:19) where δ is a short-range cutoff length, which is identified with the string radius, while∆mustbeintroducedbecauseofthelogarithmicdependenceofδ inEq.(13). This infrared regularization length corresponds to the distance at which conical geometry of string is disturbed by the background geometry. Now we have to substitute the explicit expression for µν into Eq. (13). We T then obtain ∆ S =Glog d2ζ√ γ 4 . (14) div δ − J Z 4 We calculated the divergent part of the effective action for a currentcarrying string interacting with gravitational field. Eq. (14) shows that linearised gravi- tational self-interaction in the presence of an external gravitational field is equal to zero for the both Nambu-Goto and superconducting chiral string. This result generalizes the one obtained in Refs. [4, 5]. The finite part of the action determines the effect of the topological self-action which is equal to zero in the Minkowski space. There is no general prescription for its calculation. Sometimes the finite part of the self-force can be calculated explicitly10,11 or with the use of perturbative technique.12 If 2 =0thesituationbecomessomemoreinteresting. Inthiscasethelogarith- J 6 mic divergent term (14) does not renormalize the bare action (1) because there is no term 4 in Eq. (1). It seems that this seems that the full noncontradictional ∼J theory of superconducting string must be a nonlinear theory. Acknowledgments This work was supported by the Russian Foundation for Basic Research, grant 99-02-16132. References 1. The Formation and Evolution of Cosmic Strings, eds. G. Gibbons, S. Hawking, and T. Vachaspaty (Cambridge UniversityPress, Cambridge, 1990). 2. D.V.Gal’tsov, Yu.V.Grats, and P.S. Letelier, Ann. of Phys. (NY) 224, 90 (1993). 3. A.N.Aliev and D.V. Gal’tsov, Ann. of Phys. (NY) 193, 142 (1989). 4. B. Carter and R.A.Battye, Phys. Lett. B430, 49 (1998). 5. A.Buonanno and T. Damour, Phys. Lett. B432, 51 (1998). 6. P.A.M. Dirac, Proc. Roy. Soc. A 167, 148 (1938). 7. B.S.DeWitt,DynamicalTheoryofGroupsandFields(GardonandBreach,NewYork, 1965). 8. E. Witten, Nucl. Phys. B249, 485 (1985). 9. B.S. DeWitt, R.W. Brehme, Ann. Phys. (NY) 9, 220 (1960). 10. Yu.V.Grats and A.A. Rossikhin, Theor. and Math. Phys. 123, 539 (2000). 11. E.R. Bezerra deMello, V.B. 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