String Perturbation Theory Around Dynamically Shifted Vacuum

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In some string theories, e.g. SO(32) heterotic string theory on Calabi-Yau manifolds, a massless field with a tree level potential can acquire a tachyonic mass at the one loop level, forcing us to quantize the theory around a new background that is
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    a  r   X   i  v  :   1   4   0   4 .   6   2   5   4  v   1   [   h  e  p  -   t   h   ]   2   4   A  p  r   2   0   1   4 DAMTP-2014-27HRI/ST1406 String Perturbation Theory Around DynamicallyShifted Vacuum Roji Pius a , Arnab Rudra b and Ashoke Sen aa Harish-Chandra Research Institute Chhatnag Road, Jhusi, Allahabad 211019, India  b Department of Applied Mathematics and Theoretical Physics Wilberforce Road, Cambridge CB3 0WA, UK  E-mail: rojipius@mri.ernet.in, A.Rudra@damtp.cam.ac.uk, sen@mri.ernet.in Abstract In some string theories, e.g. SO(32) heterotic string theory on Calabi-Yau manifolds, amassless field with a tree level potential can acquire a tachyonic mass at the one loop level,forcing us to quantize the theory around a new background that is not a solution to the classicalequations of motion and hence is not described by a conformally invariant world-sheet theory.We describe a systematic procedure for carrying out string perturbation theory around suchbackgrounds.1  Contents 1 Introduction 22 Systematic construction of the new vacuum 43 General amplitudes at the new vacuum 104 Extension to superstring and heterotic string theories 155 Spurious infrared divergences 15A Effect of shifting a massless field 19 1 Introduction In many  N   = 1 supersymmetric compactification of string theory down to 3+1 dimensions, wehave  U  (1) gauge fields with Fayet-Iliopoulos (FI) terms generated at one loop [1–4] (see [5,6] for a recent perspective on this). By choosing suitable linear combination of these gauge fieldswe can ensure that only one gauge field has FI term. Typically there are also massless scalars φ i  charged under this U(1) gauge field. If   q  i  is the charge carried by  φ i  then the presence of the FI term generates a term in the potential of the form1 g 2  i q  i φ ∗ i φ i − C g 2  2 (1.1)where  C   is a numerical constant that determines the coefficient of the FI term and  g  is thestring coupling.  C   could be positive or negative and  q  i ’s for different fields could have differentsigns. As a result when we expand the potential in powers of   φ i  around the perturbativevacuum  φ i  = 0, some of these scalars can become tachyonic. 1 It is clear from the form of the effective potential that the correct procedure to compute physical quantities is to shift thecorresponding fields so that we have a new vacuum where   i  q  i  φ ∗ i  φ i  =  C g 2 , and quantizestring theory around this new background. However since classically the  C g 2 term is absentfrom this potential (1.1), this new vacuum is not a solution to the  classical   equations of motion. 1 It was shown in [2] that for any compactification of SO(32) heterotic string theory preserving (2,2) world-sheet supersymmetry, there is always at least one such tachyonic scalar, leading to the existence of a stablesupersymmetric vacuum. 2  As a result on-shell methods [7–11], which require that we begin with a conformally invariant world-sheet theory, is not suitable for carrying out a systematic perturbation expansion aroundthis new background.Although the above example provides the motivation for our analysis, we shall address thisin a more general context. At the same time we shall simplify our analysis by assuming thatonly one scalar field is involved instead of multiple scalar fields. So we consider a generalsituation in string theory where at tree level we have a massless real scalar with a non-zerofour point coupling represented by a potential Aφ 4 + ···  (1.2)where ··· denote higher order terms. We suppose further that at one loop the scalar receives anegative contribution  − 2 Cg 2 to its mass 2 . Here  A  and  C   are  g -independent constants. Thenthe total potential will be Aφ 4 − C g 2 φ 2 + ···  .  (1.3)This has a minimum at φ 2 = 12 C A g 2 + ···  .  (1.4)Our goal will be to understand how to systematically develop string perturbation theory aroundthis new background and also to correct the expectation value of   φ  due to higher order correc-tions. If we had an underlyng string field theory that is fully consistent at the quantum level, e.g.  the one described in [12], then that would provide a natural framework for addressingthis issue. Our method does not require the existence of an underlying string field theory,although the requirement of gluing compatibility of the local coordinate system that we shalluse is borrowed from string field theory.The method we shall describe can be used to address other similar problems in string theorywhere loop correction induces small shift in the vev of a massless field. For example supposewe have a massless field  χ  with a tree level cubic potential and suppose further that one loopcorrection generates a tadpole for this field. Then from the effective field theory approach itis clear that there is a nearby perturbative vacuum where the field  χ  is non-tachyonic. Usualstring perturbation theory does not tell us how to deal with this situation, but the method wedescribe below can be used in this case as well.There are of course also problems involving tadpoles of massless fields without tree levelpotential,  e.g.  of the kind discussed in [13] and many follow up papers. As of now our methoddoes not offer any new insight into such problems.3  The rest of the paper is organised as follows. In § 2 we describe the procedure for construct-ing amplitudes in the presence of a small shift in the vacuum expectation value of a masslessscalar following the procedure of [14]. We also discuss systematic procedure for determiningthe shift by requiring absence of tadpoles. In § 3 we show that the amount of shift in the scalar,needed to cancel the tadpole, depends on the choice of local coordinate system that we useto construct the amplitudes. However general physical amplitudes in the presence of the shiftare independent of the choice of local coordinate system as long as we use a gluing compatiblesystem of local coordinates for defining the amplitude. This is our main result. For simplicitywe restrict our analysis in  § 2 and  § 3 to bosonic string theory, but in  § 4 we discuss generaliza-tion of our analysis to include NS sector states in heterotic and superstring theories. In  § 5 wedescribe the procedure for regulating the spurious infrared divergences in loops, arising fromthe fact that the shift in the vacuum renders some of the srcinally massless states massive. 2 Systematic construction of the new vacuum We shall carry out our analysis under several simplifying assumptions. These are made mainlyto keep the analysis simple, but we believe that none of these (except 4) is necessary.1. We shall assume that there is a symmetry under which  φ →− φ  so that amplitudes withodd number of external  φ  fields vanish.2. We shall assume that  φ  does not mix with any other physical or unphysical states of mass level zero even when quantum corrections are included.3. Shifting the  φ  field can sometimes induce tadpoles in other massless fields. If there isa tree level potential for this field then we can cancel the tadpole by giving a vacuumexpectation value (vev) to that field and determine the required vev by following thesame procedure that we used to determine the shift in  φ . We shall assume that such asituation does not arise and that  φ  is the only field that needs to be shifted. Howeverextension of our analysis to this more general case should be straightforward.4. If on the other hand shifting  φ  leads to the tadpole of a massless field which has vanishingtree level potential then it is not in general possible to find a nearby vacuum where alltadpoles vanish. In this case the vacuum is perturbatively unstable. We shall assumethat this is not the case here.4  5. When the theory has other massless fields besides  φ  but their tadpoles vanish, then thesituation can be dealt with in the manner discussed in § 7.2 of  [7] and will not be discussedhere any further. 2 6. In this section and in  § 3 we shall restrict our analysis to the bosonic string theory.However the result can be generalized to include the case where  φ  is Neveu-Schwarz (NS)sector field in the heterotic string theory or NS-NS sector field in type IIA or IIB stringtheory. This is discussed briefly in  § 4.As discussed in detail in [17,18], for computing renormalized masses and S-matrix of general string states we need to work with off-shell string theory. This requires choosing a set of gluingcompatible local coordinate system on the (super-)Riemann surfaces. The result for off-shellamplitude depends on the choice of local coordinates, but the renormalized masses and S-matrix elements computed from it are independent of this choice. Our analysis will be carriedout in this context.The off-shell amplitudes do not directly compute the off-shell Green’s functions. Insteadthey compute truncated off-shell Green’s functions. If we denote by  G ( n ) ( k 1 ,b 1 ; ··· k n ,b n ) the n -point off-shell Green’s function of fields carrying quantum numbers  { b i }  and momenta  { k i } ,then the truncated off-shell Green’s functions are defined asΓ ( n ) ( k 1 ,b 1 ; ··· k n ,b n ) =  G ( n ) ( k 1 ,b 1 ; ··· k n ,b n ) n  i =1 ( k 2 i  +  m 2 b i ) ,  (2.1)where  m b  is the  tree level   mass of the state carrying quantum number  b . The usual on-shellamplitudes of string theory compute Γ ( n ) at  k 2 i  = − m 2 b i . This differs from the S-matrix elementsby multiplicative wave-function renormalization factors for each external state and also due tothe fact that the S-matrix elements require replacing  m 2 b i ’s by physical mass 2 ’s in this formula.However from the knowledge of off-shell amplitude Γ ( n ) defined in (2.1) we can extract thephysical S-matrix elements following the procedure described in [17,18]. Our goal is to study what happens when we switch on a vev of   φ . For this we shall firstconsider a slightly different situation. Suppose that  φ  is an exactly marginal deformation instring theory and furthermore that it remains marginal even under string loop corrections.In this case there is no potential for  φ  and we can give any vacuum expectation value  λ  to 2 In the special case of the D-term potential in supersymmetric theories discussed in  § 1 the dilaton tadpoledoes not vanish in the perturbative vacuum [1,7,15,16], but is expected to vanish in the shifted vacuum since the latter has zero energy density. 5
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