Brane SUSY breaking and the gravitino mass

Journal of High Energy Physics, Apr 2018

Abstract Supergravity models with spontaneously broken supersymmetry have been widely investigated over the years, together with some notable non-linear limits. Although in these models the gravitino becomes naturally massive absorbing the degrees of freedom of a Nambu-Goldstone fermion, there are cases in which the naive counting of degrees of freedom does not apply, in particular because of the absence of explicit gravitino mass terms in unitary gauge. The corresponding models require non-trivial de Sitter-like backgrounds, and it becomes of interest to clarify the fate of their Nambu-Goldstone modes. We elaborate on the fact that these non-trivial backgrounds can accommodate, consistently, gravitino fields carrying a number of degrees of freedom that is intermediate between those of massless and massive fields in a flat spacetime. For instance, in a simple supergravity model of this type with de Sitter background, the overall degrees of freedom of gravitino are as many as for a massive spin-3/2 field in flat spacetime, while the gravitino remains massless in the sense that it undergoes null-cone propagation in the stereographic picture. On the other hand, in the ten-dimensional USp(32) Type I Sugimoto model with “brane SUSY breaking”, which requires a more complicated background, the degrees of freedom of gravitino are half as many of those of a massive one, and yet it somehow behaves again as a massless one.

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Brane SUSY breaking and the gravitino mass

HJE Brane SUSY breaking and the gravitino mass so that the component @ 0 1 0 Hachioji , Tokyo 192-0397 , Japan 1 Department of Physics, Tokyo Metropolitan University Supergravity models with spontaneously broken supersymmetry have been widely investigated over the years, together with some notable non-linear limits. Although in these models the gravitino becomes naturally massive absorbing the degrees of freedom of a Nambu-Goldstone fermion, there are cases in which the naive counting of degrees of freedom does not apply, in particular because of the absence of explicit gravitino mass terms in unitary gauge. The corresponding models require non-trivial de Sitter-like backgrounds, and it becomes of interest to clarify the fate of their Nambu-Goldstone modes. We elaborate on the fact that these non-trivial backgrounds can accommodate, consistently, gravitino elds carrying a number of degrees of freedom that is intermediate between those of massless and massive elds in a at spacetime. For instance, in a simple supergravity model of this type with de Sitter background, the overall degrees of freedom of gravitino are as many as for a massive spin{3=2 eld in at spacetime, while the gravitino remains massless in the sense that it undergoes null-cone propagation in the stereographic picture. On the other hand, in the ten-dimensional USp(32) Type I Sugimoto model with \brane SUSY breaking", which requires a more complicated background, the degrees of freedom of gravitino are half as many of those of a massive one, and yet it somehow behaves again as a massless one. Supersymmetry Breaking; D-branes; Supergravity Models 1 Introduction 2 3 4 5 1 Massless propagation in de Sitter backgrounds Massless gravitino in non-linear supersymmetry Conclusions Introduction Akulov model [22], whose rst application to supergravity models was discussed in [23{25]. This framework remains of great interest, and its coupling to Supergravity was recently reconsidered in [26{29], in the light of constrained super elds [30{33], and in the light of non-BPS D-branes [34, 35]. As was explained in [25], one must introduce mass terms for the gravitino and Nambu-Goldstone fermion and modify accordingly its supersymmetry transformation in order to eliminate a cosmological constant term to arrive at at-space models. The systematics of these constructions, examined in detail in [36{39], led eventually to the no-scale models of [40], which also found a string realization in the presence of internal uxes [41]. Typically, the gravitino becomes massive absorbing the degrees of freedom of a NambuGoldstone fermion, a phenomenon that becomes manifest in a unitary gauge, but in the ten-dimensional and six-dimensional orientifold models [42{51] with \brane supersymmetry breaking" [52{57] supersymmetry is non-linearly realized and no explicit gravitino mass term is allowed [58, 59], since the gravitino is a Majorana-Weyl fermion.1 On the other 1\Brane supersymmetry breaking" is a way to break supersymmetry by brane con gurations without tachyon instabilities. For example, the simultaneous presence of branes and anti-branes in order to break cosmological term [60, 61]. Since no gravitino mass term emerges in a unitary gauge, the role of the Nambu-Goldstone fermion may appear confusing. Our aim here is to elaborate on the fate of the degrees of freedom of Nambu-Goldstone fermions in this second class of models. There are some interesting aspects in this story, since the curved spacetime is crucial in a proper account of the related degrees of freedom. Much e ort was devoted, over the years, to providing suitable de nitions of masses and degrees of freedom in de Sitter spacetime, in particular in [62{66]. A highlight of these works is that null-cone propagation takes naturally the place of masslessness in at space time [62], and the correspondence is illuminated by the special choice of \symmetric density in a at slicing coordinate system for a de Sitter spacetime. In section 4 we demonstrate that, in a simple model without gravitino mass term that will be introduced in the next section, the gravitino remains massless in this sense, displaying its null-cone propagation and also applying our new criterion. We then show that the gravitino in the Sugimoto model remains massless, in the sense speci ed above, via our new criterion, which applies insofar as the background can be understood as a cosmological evolution. The overall number of degrees of freedom of the massless gravitino is the sum of those of a massless gravitino in at spacetime and a Nambu-Goldstone fermion, which is half as many of those of a massive spin{3=2 eld. The last section contains some concluding remarks. 2 Supergravity models with non-linear supersymmetry Let us begin by considering pure supergravity e 1 2 { 2 { 1 4, which allows Majorana fermions, at the cost of leaving aside some members of the supergravity multiplet. In this paper we follow the conventions of [68]. This Lagrangian is invariant, up to a total divergence and higher powers of gravitino, under the transformations whose parameter is the Majorana spinor . The covariant derivative of the spinor eld involves the spin connection ! and reads HJEP04(218) where local Lorentz indices are denoted by Latin letters. Let us now introduce a Nambu Goldstone fermion eld that provides a non-linear realization of supersymmetry LNL = e 2f 2 D + O ( )4 ; whose lowest-order coupling to the gravitino involves the supersymmetry current e m = = 1 D ; m ; D = Lcurrent = e S = f S : = f : (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) (2.9) 1 2 1 4 e 1 2 The dimensionful quantity f de nes the scale of supersymmetry breaking and the NambuGoldstone fermion eld transforms, to lowest order, according to The total Lagrangian Lpure + LNL + Lcurrent L = 1 that the gravitino is a simple massless spin{3=2 eld, which conforms to the fate of the degrees of freedom introduced by Nambu-Goldstone fermion. The purpose of this paper is to elaborate on what happens to the gravitino and the Nambu-Goldstone fermion in more complicated models of broken Supersymmetry. negative cosmological constant, so that Lpure(massive) = m ; D + D 1 2 m 1 2 quires the introduction of a mass term for the Nambu-Goldstone fermion [70, 71], Lmass = e 1 The portion of the total Lagrangian that we are focussing on, Lpure(masive) +LNL +Lcurrent + Lmass, is then invariant under the new gauge transformation, up to higher order terms or contributions involving the other members of the gravity or matter multiplets. In particular, a model without cosmological constant term would require tuning 2f 2 = C, and thus linking the mass parameters of the gravitino and the Nambu-Goldstone fermion, m, to the supersymmetry breaking scale f . In a model of this type admitting a at spacetime background the gravitino has a mass term in the unitary gauge = 0, and in fact it is a conventional massive spin{3=2 eld. Models that are similar to some extent to those in eq. (2.9) are realized in orientifolds with brane supersymmetry breaking, and the ten-dimensional Sugimoto model is the simplest example in this class. At tree level the supersymmetry present in the original type IIB closed string sector is halved by the orientifold projection, as in the type-I superstring, while in the open sector supersymmetry is completely broken, or non-linearly realized. The low-energy e ective Lagrangian of the Sugimoto model was discussed in detail in [58, 59]. In Einstein frame and in unitary gauge it combines open and closed contributions, where Lclosed = L = Lclosed + Lopen; 1 1 p D ] p 2 1 2 D i F { 4 { describes the contribution from the closed sector, while Lopen = 1 describes the contribution from open sector. Beginning from the closed sector, model thus reduces to L = ( 1 2 210 e R 1 2 Now the eld equation of the Nambu-Goldstone fermion yields D 1 2 = 3 p 2 D ; which identi es the dilatino and the gamma trace of the gravitino. We see that there is no mass term for gravitino, and therefore this model belongs to the same class as the model of eq. (2.9). There is an additional reason for the absence of a gravitino mass term: in this ten-dimensional setting this eld is a Majorana-Weyl fermion eld, for which the mass term simply does not exist. The goal of this paper is to elaborate on what happens to the gravitino in this more complicated model, in the presence of non-trivial background elds. 3 Massless propagation in de Sitter backgrounds It is well known that one can de ne a D-dimensional de Sitter spacetime starting from the quadratic constraint = curvature tensors are described by the metric tensor g , independently of the coordinate system, as R = H2 ( g g ) ; R = R = (D 1)H2g ; (3.2) with ; ; ; 1, and the scalar curvature is a constant, with R = D(D 1)H2. In this paper use two concrete coordinate systems, the symmetric coordinates and the at slicing coordinates. The symmetric coordinates [67], the result from a stereographic projection of the generalized hyperboloid of eq. (3.1) to D-dimensional at spacetime, de ned via In this fashion the metric tensor takes the form with = (1 + s) 1, while = x 1 + s ; D = 1 is a corresponding vielbein. The Christo el symbols and the spin connection are then ds2 = g dx dx = dx0 2 + e2Hx0 ij dxidxj ; and a corresponding vielbein is e m = diag 1; eHx0 ; eHx0 ; : { 6 { (3.3) (3.4) (3.5) (3.6) (3.7) (3.8) (3.9) 2 H r2eHx0 ; (3.10) (3.11) (3.12) 1 4 ! mn mn = = mem = H2 2 H2 4 1 x ; = x x ; x ; respectively, where Since this coordinate system yields a conformally at metric, it is a convenient choice to investigate null-cone propagation and conformal covariance of the eld equations. The more conventional at slicing coordinates of de Sitter spacetime are de ned as 1 H 0 = sinh Hx0 + 2 H r2eHx0 ; i = xieHx0 ; D = cosh Hx0 1 H where r2 = ij xixj with i; j = 1; 1. As is well known, this coordinate system covers half of the whole de Sitter spacetime, the portion where space expands in time, which a ords a natural cosmological interpretation. In a Minkowski spacetime the Casimir operator p^ p^ of Poincare group, where p^ is the generator of spacetime translation, is used to de ne the mass of the eld. Since there is no such Casimir operator in the de Sitter SO(D; 1) group, there is no natural way to de ne the mass of elds. Still, null-cone propagation remains a convincing criterion for masslessness. Let us now brie y recall the massless criterion via null-cone propagation in the symmetric coordinate system [62]. To this end, let us consider a simple spin{1=2 spinor eld in de Sitter background, for which The eld equation can be rewritten as where is the rescaled eld de ned by Lspin 1=2 = e D w1=2 ; { 7 { (3.13) (3.14) (3.15) (3.16) (3.17) (3.18) (3.20) (3.21) with the conformal weight of the spinor eld w1=2 = (1 D)=2. One is thus led to null-cone propagation, and in this respect the spinor eld can be considered a massless eld in de Sitter background. A similar conclusion can be reached starting from the second-order eld equation. D D = g D D D(D 4 1) H2 = 3+D 2 (3.19) which leads consistently to null-cone propagation and thus points again to the notion of a massless spinor eld. The action of this model is invariant under the de Sitter SO(D; 2) conformal transformation. Since local Weyl invariance is enough for de Sitter conformal invariance [66], we show local Weyl invariance of this model. The Weyl transformation is de ned as e m ! W e m ; em ! 1 W e m ; to be combined with T = T00 = 1 2 / e D2 1 Hx0 ; D D 2 : o where W is the local Weyl scaling factor. Using the de nition of the gamma matrices in a non-trivial background, = mem , and the transformation of spin connection ! mn ! ! mn + (em e n en e W ; it is straightforward to see the invariance of the Lagrangian of eq. (3.15) provided one chooses the conformal weight w1=2 = (1 D)=2. Since conformal invariance re ects the absence of a speci c scale, it is natural to regard the spinor eld as a massless eld. We now turn to propose a massless criterion for fermion elds in the at slicing coordinate system, referring again to the above model of a spinor eld. The explicit form of the eld equation in the at slicing coordinate is D 2 1 H = 0: Here, we have also assumed that the eld depends only on time, since we anticipate the use of some physical arguments related to the homogeneous expansion of the Universe. The important property of the solution of this equation is that it is not oscillatory, and consequently does not contribute to the energy density T00, where The explicit form of T00 in our present setting (3.22) (3.23) (3.24) (3.25) (3.26) (3.27) (3.28) (3.29) clearly indicates that it could be non-vanishing for an oscillatory solution \rest mass" m. Hence, the non-oscillatory behavior of the Fermi eld can regarded as an / e imx0 with indication of masslessness. We can now propose a similar argument for the second-order eld equation, anticipating some steps that will prove useful for the application to the gravitino in the next section. The explicit form of the second-order eld equation is + (D D 2 Let us now perform the rescaling of the eld as in eq. (3.18) with for this choice of is that the metric can be described, in conformal time , as = eHx0 . The reason ds2 = 2 d 2 + ij dxidxj with = 1=jH j = eHx0 , so that this setting compares naturally with eq. (3.5). The eld equation now takes the simple form If we assume that a solution should be nite in the limit of x0 which is consistent with eq. (3.24). In this respect, a constant solution for the rescaled eld in the at slicing coordinate system can be a criterion of masslessness. ! 1, should be a constant, Consider the model of eq. (2.9) in the background de Sitter spacetime required by the cosmological constant 2f 2. The eld equations of the gravitino and the Nambu-Goldstone fermion are In the unitary gauge = 0, these equations reduce to one eld equation with one constraint: the combination that was discussed as a model of null-cone propagation without de Sitter conformal invariance in [62]. Applying to the eld equation gives another constraint Eliminating and x using the explicit forms of the two constraints yields nally the eld equation D 1 2 D = and if D 6= 2 the eld equation becomes rescaled eld with w3=2 = (3 D)=2, is now In the symmetric coordinate system the explicit form of this eld equation for the D D + f f This is a well-known conformally covariant eld equation. It was described in [62], and was derived in [72] in D = 4. The di erence with respect to the massless Rarita-Schwinger equation lies in the special value of the coe cient of the second term, which would be one for the massless Rarita-Schwinger equation rather than 2=D. Note that one can not take to obtain it we had to perform a division by H2 in order to eliminate the x a naive at limit H ! 0 in this conformally covariant eld equation, since in the process term. As a result, the number of degrees of freedom of this spin{3=2 eld undergoes a discontinuity in moving between at and de Sitter spacetimes. { 9 { (4.1) (4.2) (4.3) (4.4) (4.5) (4.6) (4.7) (4.8) (4.9) carrying spin{1=2 degrees of freedom propagates on the null cone. As a result, the other components of the eld ~ , with @ ~ = 0 (and also ~ = 0) satisfy the eld equation which means again null-cone propagation. Therefore, the gravitino of this model can be regarded as massless in de Sitter spacetime, although its degrees of freedom combine those of a massless spin{3=2 eld and of a massless spin{1=2 eld in a at Minkowski background. Let us stress that this is a di erent state of a airs from the \partially massless eld" in [73]. In the at slicing coordinate system, under the assumption that the eld depends only on time, the second order eld equation before rescaling the eld D D = g D D 1 2 R 1 4 R (4.10) (4.11) (4.12) (4.13) (4.14) (4.15) (4.16) (4.17) (4.18) (4.19) takes the form The constraint D = 0 now reads (D 5)(D 4 1 2 (D 5)(D 4 1) H2 0 = 0; 1) H2 i + H2 0i 0 H2 i = 0 : and a simple solution of the above constraint and = 0 is 0 = 0; i i = 0: In this fashion the second-order eld equation becomes D 2 This equation is very similar to eq. (3.27) for the spinor eld. Rescaling the eld according to where x 0 i ! w3=2 i; = eHx0 and w3=2 = (3 D)=2 is the conformal weight of vector-spinor eld, the eld equation takes nally the very simple form If we add the reasonable assumption that the solution should be nite in the limit of ! 1, i can only be a constant. This means that gravitino is massless, exactly as was the case for the spinor eld in previous section. The energy-momentum tensor of the eld before rescaling can be recast in the form T = 1 4 g D D D D 1 4 and the explicit form of T00 in our present setting is T00 = 1 We see that this energy density function vanishes for the non-oscillatory solutions, as HJEP04(218) pertains to elds with a vanishing \rest mass". We are now ready to investigate the behavior of the gravitino in the more complicated Sugimoto model. The relevant starting point was already introduced at the end of section 2, in eqs. (2.18) and (2.19). A time-dependent background, which can be interpreted as a cosmological evolution, was rst obtained in [60, 61] starting from the ansatz ds2 = e2B dx0 2 + e2A ij dxidxj ; for the metric, where A and B are functions only of x0. The gauge condition B = which corresponds to a convenient choice for the time coordinate reduces the equations for the background to (4.20) (4.21) (4.22) 3 =4 (4.23) (4.24) (4.25) This result of [60, 61] is a special instance of a class of solutions for Einstein gravity minimally coupled to a scalar eld in the presence of an exponential potential proportional to exp( ), which were studied in detail in [74]. The exponent selected by String Theory for \brane supersymmetry breaking", = 3=2 in the Einstein frame, has the key property of separating two regions of solutions with widely di erent behavior. As pointed out in [75{ 77], this value marks the onset of the \climbing phenomenon", according to which the scalar eld has no other option, when emerging from the initial singularity, than climbing up the potential before reaching a turning point and starting its descent. This is important, since >>:>36 A _ 2 8  + 9A_ B _ _ = 8A 8A_ B_ + 36 A 3; + where dots indicate derivatives with respect to the dimensionless time solution then reads = p E x0. The Notice that = eA can be regarded as the scale factor of cosmological expansion, while de nes implicitly the cosmic time via this dynamics sets naturally an upper bound on the string coupling (although there are no indications of a similar bound on 0 corrections [78]), and suggests that the resulting descent could help one model the onset of in ation. A climbing phase could provide the impulse to start in ation [79{91], and in general an early fast-roll would introduce a lowfrequency cut in the primordial power spectrum of scalar perturbations. This option was explored over the years in di erent contexts [92{111], but it arguably obtains, in \brane supersymmetry breaking", an enticing input from String Theory. There are also some signs, away from the Galactic plane, of an encouraging comparison with the lack of power apparently present in the low{` CMB [112{115]. There is an important link with the preceding case, since close to the turning point for the climbing scalar eld, where _ = 0 and A = 0, the Universe in this case bears some similarities with de Sitter spacetime with temporally constant A_ in = eA. The Christo el symbols and the spin connection for this background are 0 ij = p E A_ e2A 2B ij ; 0 respectively. The curvature tensors are described as where in the application to the original model one should set D = 10. The eld equations of the gravitino and the dilatino that follow from the Lagrangian of eq. (2.18) are R = R = h E 0 and are to be combined with the constraint of eq. (2.19). equation of the gravitino becomes Let us now try to nd a solution with vanishing dilatino = 0. In this case, the eld (4.26) (4.27) (4.28) (4.29) (4.30) (4.31) (4.32) (4.33) 0 0 1) h D D + + 1 p with the constraints = 0, which follow from the eld equation of the dilatino. Applying to the eld equation gives the additional constraint = 0, while the covariant divergence of the eld equation gives further additional constraint R It is simple to see that all of these four constraints are solved by 0 = 0 and i i = 0, using the explicit forms of the Christo el symbols, the spin connection and the Ricci tensor. Therefore, the problem is reduces to solve the eld equation D i = 0; where the eld is subject to the constraint i i = 0. The explicit form of the second order equation D D i = 0 is HJEP04(218) i + 7A_ + but rescaling the eld according to it takes a very simple form: i ! Using the explicit expression of the background in eq. (4.24), the eld equation reduces to whose solution can be expressed in terms of the incomplete -function, according to dd2 2i + 3 4 2 3 3 2 d d i = 0; i = C( 1 ) + C( 2 ) i i 1 4 ; 16 where C( 1 ) and C( 2 ) are integration constants that satisfy the conditions iC( 1 ) = 0 and iC( 2 ) = 0. They key point is that these solutions are not oscillatory. Moreover, if we i require a nite i at ! 1, the only option is with iC( 1 ) = 0, which points again to a massless gravitino in view of our new criterion, namely no \rest mass" or no contribution to the energy density T00. In this respect, we can conclude that the gravitino is massless in this cosmological vacuum, which somehow replaces at spacetime in the Sugimoto model. The next issue concerns the actual number of degrees of freedom that are carried by the gravitino in the unitary gauge. To begin with, a Majorana vector-spinor eld has 2[D=2] real degrees of freedom, or 320 in D = 10. Since 0 = 0 and i i = 0, this number is readily cut to 256 in D = 10. Moreover, is also a Weyl eld, so that number i = C( 1 ) i is again reduced by a factor of two, or to 128 degrees of freedom in D = 10. As usual, the rst order Dirac equation that is left in unitary gauge D amounts, in our background, to a condition 0 = 0, and considering projection operators P = (1 i 0)=2, one is nally left with 64 degrees of freedom. This number results precisely from the 56 degrees of freedom carried by the original gravitino and the 8 degrees of freedom carried by the Nambu-Goldstone fermion. To reiterate, the gravitino in the Sugimoto model behaves as a massless eld in its cosmological background of [60, 61], although it combines the degrees of freedom that a massless gravitino would describe in a at spacetime with those of the absorbed NambuGoldstone fermion. The two elds recover separate lives, consistently with this interpretation and known facts, if one turns o the tadpole potential. Moreover, this number is the half of the degrees of freedom that a massive spin{3=2 eld would have in a tendimensional Minkowski spacetime, as demanded by the orientifold projection that underlies the Sugimoto model. 5 Conclusions We have investigated the behavior of the gravitino in a class of orientifold models [42{51] with \brane supersymmetry breaking" [52{57], where supersymmetry is non-linearly realized while a gravitino mass term is not allowed. These models require non-trivial spacetime backgrounds, so that the notion of mass entails some subtleties and is di erent from the more familiar case of a at spacetime. In a de Sitter spacetime null-cone propagation in the symmetric coordinate system provides an accepted criterion of masslessness, and along these lines we have proposed a new criterion for fermions that applies in the at slicing coordinate system. The new criterion rests on a cosmological interpretation, and applies to more general background spacetimes. These include the spatially at geometries where \brane supersymmetry breaking" brings along the climbing mechanism [75{77]. This could provide the initial impulse to start an in ationary phase [79{91], and with a short in ation this type of dynamics could have had some bearing on the low{` CMB anomalies [112{115]. We have investigated the gravitino mass in the Sugimoto model, with reference to the background eld con guration of [75{77], which is more complicated than de Sitter spacetime, arriving at conclusions that are reasonable with a massless gravitino. The overall lesson is consistent with massless gravitinos in non-trivial backgrounds that entail di erent numbers of degrees of freedom than in at spacetime. Let us also stress that the \cosmological" criterion of masslessness in de Sitter-like backgrounds also applies to a conformal scalar eld theory with non-minimal coupling to the scalar curvature, consistently with our view. Even in this case, the eld equation for a suitably rescaled scalar eld allows non-oscillatory solutions for which the energy density vanishes, although there is also a di erent option, which is proportional to a D, where a = exp(Hx0) is the scale factor. The latter contribution is typical of massless radiation. There is a host of evidence that our Universe has never been exactly a at Minkowski spacetime, and that now it is close to a de Sitter spacetime. Fields with unusual numbers of degrees of freedom can thus be of interest, in principle, for Cosmology. A recent investigation along these lines can be found in [116{119], for example. Acknowledgments The author would like to thank S. Ferrara for helpful discussions and A. Sagnotti for helpful discussions, suggestions and a careful reading of the manuscript. The author would also like to thank Scuola Normale Superiore for the kind hospitality and for partial support while this work was in progress. Open Access. 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Noriaki Kitazawa. Brane SUSY breaking and the gravitino mass, Journal of High Energy Physics, 2018, 81, DOI: 10.1007/JHEP04(2018)081