PHYSICAL REVIEW D 84, 085016 (2011) Role of space-time foam in breaking supersymmetry via the Barbero-Immirzi parameter
John Ellis and Nick E. Mavromatos Theoretical Particle Physics and Cosmology Group, Department of Physics, King’s College London, Strand, London WC2R 2LS, UK and Theory Division, Physics Department, CERN, CH-1211 Geneva 23, Switzerland (Received 31 August 2011; published 19 October 2011) We discuss how: (i) a dilaton/axion superfield can play the role of a Barbero-Immirzi field in four- dimensional conformal quantum supergravity theories, (ii) a fermionic component of such a dilaton/axion superfield may play the role of a Goldstino in the low-energy effective action obtained from a superstring theory with F-type global supersymmetry breaking, (iii) this global supersymmetry breaking is commu- nicated to the gravitational sector via the supergravity coupling of the Goldstino, and (iv) such a scenario may be realized explicitly in a D-foam model with D-particle defects fluctuating stochastically.
DOI: 10.1103/PhysRevD.84.085016 PACS numbers: 04.65.+e, 04.60.Pp, 12.60.Jv
I. INTRODUCTION low, this includes the supersymmetric extension of the Holst term proportional to the dual of the Riemann tensor, as well One of the most important issues in string theory and as the conventional Einstein action. The coefficient of this supersymmetry phenomenology is the mechanism of su- term, called the Barbero-Immirzi parameter , has some persymmetry breaking. It is generally thought to be non- superficial affinity with a term in a non-Abelian gauge perturbative, but developing the intuition and calculational theory. The supersymmetric completion of the Host term tools needed to understand the relevant aspects of non- involves a dilaton/axion/dilatino supermultiplet, and we perturbative string theory is an unmet challenge, as yet. explore whether this could play a role in local supersym- Given these shortcomings in our understanding, it is natu- metry breaking, with the dilatino becoming the Goldstino. ral to reason by analogy with better-understood nonpertur- For this to happen, there should presumably be some bative phenomena in field theory, notably in gauge theories stringy analogue of the instanton and related nonperturba- and specifically in (supersymmetric) QCD. Much intuition tive non-Abelian gauge theory configurations that play a has been obtained from the well-studied phenomenology key role in quark condensation and chiral symmetry break- of nonperturbative effects in QCD, which has been bol- ing in QCD, and gluino condensation in N 2 supersym- stered by explicit exact calculations in N ¼ 1 and N ¼ 2 metric gauge theories. As discussed elsewhere, D-particles supersymmetric gauge theories. Building on this experi- moving in the higher-dimensional bulk appear in a three- ence, promising scenarios have been proposed for super- dimensional brane world, such as the one we inhabit, to symmetry breaking via gaugino condensation induced by be localized at space-time events x , just like instantons. nonperturbative gauge dynamics. We have no objection to In order to develop the analogy further, one should identify such scenarios, but cannot resist trying to be more ambi- the dynamical interaction of matter with D-particles that tious, and develop a scenario for supersymmetry breaking might give rise to condensation. We have argued that rooted in intrinsically nonperturbative string dynamics. fermions without internal quantum numbers such as the From this point of view, it is natural to consider intrinsi- dilatino may indeed have nontrivial interactions with cally stringy analogues of nonperturbative aspects of gauge D-particles, and in this paper we sketch how such inter- theories. The latter have topologically nontrivial sectors actions might play a role in condensation and supersym- populated by gauge configurations such as instantons and metry breaking. We regard this is an illustrative example monopoles, whose contributions to the gauge functional how short-distance Planckian dynamics might play a role integral are weighted by arbitrary vacuum angle parame- in supersymmetry breaking. ters such as QCD. In the case of string theory, there are various classes of nonperturbative D-brane configurations II. QUANTUM GRAVITY, THE and analogues of gauge vacuum angle parameters. The BARBERO-IMMIRZI PARAMETER, questions then arise how they might lead to some conden- AND SUPERGRAVITY sation phenomenon, analogous to quark or gluino conden- sation, that might break supersymmetry. A. Introducing the Barbero-Immirzi parameter In this paper, our approach to these questions takes as its The Ashtekar formalism of general relativity (GR) [1] starting-point the effective Lagrangian that characterizes is a version of canonical quantization, which introduces the low-energy, large-distance limit of string theory. This self-dual SL(2,C) connections as the fundamental under- should be taken to have the most general form compatible lying variables, enabling the GR constraints to be reduced with basic symmetry principles, which we take to be N ¼ 1 to a polynomial form. However, the complex nature of local supersymmetry, i.e., supergravity. As we discuss be- the self-dual connections necessitates the introduction of
1550-7998=2011=84(8)=085016(20) 085016-1 Ó 2011 American Physical Society JOHN ELLIS AND NICK E. MAVROMATOS PHYSICAL REVIEW D 84, 085016 (2011) reality conditions, which complicate the quantization pro- Viewing gravity as a gauge theory, the second term (the cedure. For the moment, this hurdle prevents the emergence Holst modification to the Einstein action) has an arbitrary of a complete theory of quantum gravity from such a coefficient 1= that is somewhat analogous to the pa- procedure. In order to avoid this problem, Ashtekar [2] rameter in a non-Abelian gauge theory. The presence of the and Barbero [3] independently introduced real-valued second term, if it is nontrivial, induces an antisymmetric SU(2) or SO(3) connections (called Ashtekar-Barbero con- term in the connection, i.e., nontrivial torsion, nections) in a partially ‘‘gauge fixed’’vierbein formalism of ab ¼ ab þ ab GR. This led Ashtekar, Rovelli, and Smolin subsequently to ! !~ C ; (2) the development of the loop quantum gravity (LQG) ap- where !~ab denotes the torsion-free connection determined proach [4] to the quantization of GR, according to which ab by the tetrads and C is the torsion. In the limit ! 0, the one can construct a nonperturbative and space-time torsion term vanishes in a path integral over the Euclidean background-independent formalism for QG, which in gravity action, and this corresponds to the standard torsion- some explicit examples is free from the short-distance free limit of general relativity. singular behavior of GR. In pure gravity, the Holst term does not affect the gravi- The Ashtekar-Barbero connections contain a free pa- ton equation of motion, which takes the form rameter, , which arises when one expresses the Lorentz m ¼ connection of the noncompact group SO(3,1) in terms of D½ e 0; (3) a complex connection in the compact groups of SU(2) or SO(3). The existence of such a free parameter in the where connection of the Ashtekar formalism was implicit in the i D ¼ @ !ab ; (4) formalism of Barbero [3], but was put on a firm footing by 4 ab Immirzi [5], and is now termed the Barbero-Immirzi pa- ab i ½ a b rameter. The significance of this parameter became ob- with 2 ~ ; ~ , denotes the gravitational covariant ¼ vious after the observation that the area operator of LQG derivative. In view of the Bianchi identity, R½ 0,(3) depends on it, leading to a black-hole entropy in this implies that the torsion (Barbero-Immirzi) term in (1)is formalism of the form (in four-dimensional Planck units identically zero in pure general relativity in the absence of ¼ ¼ 0 A ab MP 1) S 4 , where 0 is a numerical factor depend- matter. In such a case, only the !~ term survives in the ing on the gauge group. The standard Bekenstein-Hawking connection (2). ¼ c entropy is recovered in the case 0 . This is no longer the case when fermions are present [7], in which case it can be shown that the torsion term is B. Fermionic torsion and the nonzero. This is because (2) is no longer satisfied, but one Barbero-Immirzi parameter has instead a nonvanishing right-hand side, as a result of nonzero fermionic currents. In this case, the gravitational Fermions induce a nontrivial torsion term into the gravi- action is augmented by a fermion contribution, tational action that involves, in the first-order formalism, i Z the dual of the Riemann curvature form. This introduces a ¼ þ 4 c ð Þc ð Þc c SGF Sgrav d x D ! D ! : new parameter in the action, namely, the Barbero-Immirzi 2 parameter [5], , introduced above. Specifically, using the (5) Palatini formalism of general relativity to express the four- Variation of the action (5) with respect to the connection dimensional Einstein-Hilbert action in terms of the vier- ab mn 1 ! (2), which incorporates torsion, yields beins em and the spin connection, ! , as is necessary in the presence of fermions, one can always add to the action ð Þ¼ 1cd D ee½aeb 8 pab Jcd; a term involving the dual of the curvature tensor, ?Rmn 1 mn Rpq , obtaining [6] ab 2 1 2 pq p 1 ¼ a b þ ab ; cd 2 þ 1 ½c d 2 cd (6) S ðe;!Þ¼S þ S ; 1 grav Einst Holst ¼ a f Z Jcd ea cdfjð Þ; 1 4 A ¼ 4 mn where SEinst d xee me nR ; and 16 GN f c f c Z where jð Þ 5 ~ is the fermionic axial current. The 1 1 A ¼ 4 mn pq SHolst d xee me n pqR : torsion can be found in this case as a consistent solution of 2 16 GN (6), and takes the form [7] (1) ab ½a b c ab d C ¼ 2 G ðe J e J Þ; N 2 þ 1 ðAÞ cd ðAÞ 1 (7) We remind the reader that Latin indices and quantities with a Ja ¼ c ~a ~ c : tilde refer to the flat Minkowski tangent space-time plane. ðAÞ 5
085016-2 ROLE OF SPACE-TIME FOAM IN BREAKING ... PHYSICAL REVIEW D 84, 085016 (2011) From a path-integral point of view, the use of the equations generation for the fermions, and thus chiral symmetry of motion is equivalent to integrating out the (nonpropa- breaking in the case of multiflavor interactions [9].2 gating) torsion field. However, the appearance of the Barbero-Immirzi pa- It is straightforward to show in this approach [7] that rameter in the effective action in the above approach [7], the effective Dirac action contains an axial current-current as a four-fermion interaction coupling, would invalidate interaction, with a coefficient that depends on the Barbero- the analogy of the Holst action with the instanton action Immirzi parameter. To this end, consider the Dirac action of QCD, in which the angle is purely topological.3 coupled to a connection field ! with torsion. Using the Moreover, the loop-quantum-gravity limit, in which explicit form (4) of the covariant derivative, and the fol- ! i, would lead to divergent four-fermion couplings lowing property of the product of three flat-space matri- (10), incompatible with the well-defined Ashtekar- ces in four space-time dimensions: Romano-Tate theory [11], a version of the canonical for- mulation of quantum gravity in which only the self-dual a b c ¼ ab c þ bc a ac b abcd parts of the curvature tensor contribute to the Lagrangian ~ ~ ~ ~ ~ ~ i ~5 ~d; (8) density. The above approach has been criticized [12] on the we may write the fermionic action as follows [8]: grounds of mathematical inconsistency, namely that, for an arbitrary value of the Immirzi parameter, the decom- a a 5 L ¼ ec ði ~ @a m þ ~ ~ BaÞc ; position of the torsion (7) into its irreducible parts, a trace (9) vector, a pseudoscalar axial vector, and a tensor part, fails Ba ¼ abcdðe @ e þ C Þ; b a c bcd for the following reason. Consider the contorsion tensor ab ¼ a b where Cbcd denotes the torsion part of the connection (2), C C e½ e ; (11) which cannot be expressed in terms of the vierbeins (tet- rads). We see that in this formalism the field B plays the and decompose it into its irreducible parts, namely, the a ¼ ¼ role of an axial ‘‘external field’’: its spatial components B~ trace vector C C , the pseudotrace axial vector S ¼ act as a ‘‘magnetic’’ field, while its temporal component C , and the tensor q (with q 0, 0 ¼ behaves as an axial ‘‘scalar potential’’ B . Even in flat q 0). The solution (7) would imply [12] space-times, the field B is nontrivial in the presence of 3 3 2 fermionic torsion. T ¼ Jð Þ;S¼ Jð Þ;q¼ 0: Substituting the expression (7) for the torsion into (9), 4 2 þ 1 A 2 þ 1 A (12) bi1i2i3 ¼ and using that ai1i2i3 3!, we straightforwardly arrive at an effective four-fermion Thirring-type interac- The first of these relations is inconsistent, as it equates a tion that is quadratic in the axial fermion current JðAÞ, of the form [7] Lorentz vector (T ) with a pseudovector (JðAÞ), which have different transformation properties under the Lorentz Z 3 2 group. As such, in this formulation, the Barbero-Immirzi S ¼ d4xe G J2 : Ja ¼ c ~a ~ c ; int 2 N 2 þ 1 ðAÞ ðAÞ 5 parameter cannot be arbitrary: the only consistent limiting values are (10) pffiffiffiffiffiffiffiffiffiffiffiffiffi either ! 0; or !1: (13) where e ¼ detðgÞ is the vierbein determinant. We note that the only remnant of the metric in this four-Fermi In the first limit there is no torsion at all, and in the second interaction is this vierbein determinant factor, as the rest limit the torsion is given by the fermionic axial vector, of the terms can be expressed in terms of flat space-time which is a result characteristic of the Einstein-Cartan (EC) quantities alone, as a result of the properties of the theory, vierbein. 1 We note that, when taking the complex conjugate of the ¼ c c d c C ab e abcd 5 ~ : (14) action, in other words when the covariant derivatives in the 4 Dirac equation are taken to act on both the fermionic fields In both the limits (13), the trace C of the contorsion tensor c and their conjugates c , the contributions of only one vanishes, and thus the theory is consistent. part of the torsion (7) survive in the Hermitian effective action (10), namely, that proportional to 2=ð 2 þ 1Þ and ab d 2 the dual of the axial current c jð Þ. This allows the Cosmological aspects of such an approach, in which the d A dynamical fermion condensate may be identified with the dark Barbero-Immirzi parameter to assume purely imaginary energy at late eras of the Universe, have been discussed in [10]. values, and thus yield attractive four-fermion interactions. 3We note, however, that instantons are argued to lead to Such interactions might then lead to dynamical mass effective chiral-symmetry-breaking multifermion interactions.
085016-3 JOHN ELLIS AND NICK E. MAVROMATOS PHYSICAL REVIEW D 84, 085016 (2011) Z In fact, once the torsion assumes the Einstein-Cartan S ¼ i d4x½I þ @ J ; form (14), it is straightforward to show, using (9), that the Holst NY ðAÞ 2 effective Dirac action contains an axial current-current 1 ¼ a ab interaction, with a fixed coefficient, independent of the INY C C a R ab 2 (19) Barbero-Immirzi parameter, ¼ @ C ; Z 3 EC ¼ 4 a ab 1 a b Sint d xe GJð ÞJðAÞa: (15) ¼ e e C 2 A where 2 ½ and is the contorsion tensor defined previously. Notice that in the last equality for the This is the limiting case of (10) when !þ1, but with Nieh-Yan topological invariant we took into account a real parameter. However, we stress once more, attractive the fact that, for the torsion (14), the term quadratic in a ¼ four-fermion interactions arise in the approach of [7] only the contorsion in INY vanishes: C C a 0. in the case of a purely imaginary Barbero-Immirzi parame- This implies that, in this limit, variations of the full ter [7]. Moreover, as we see below, this limit respects local gravitational action with respect to the connection will supersymmetry transformations. not affect the equations of motion. However, there is an To avoid the above-mentioned constraint on the Immirzi axial current-current term in the effective action, which is parameter , and thus incorporate in a consistent way the independent of the Barbero-Immirzi parameter [12–14]. limit ¼ i, nonminimal couplings of the Holst action to This term is a repulsive interaction of the form fermions were considered in [12–14], that allow for arbi- Z trary values of the Barbero-Immirzi parameter. In this way 4 3 a S ¼ d xe G Jð Þ J ; (20) the inconsistency in (12) is removed and the analogy of this int 2 N A a ðAÞ parameter with the angle of QCD is more complete. which coincides with the corresponding term in the Specifically, one may consider a nonminimal fermion cou- ¼ pling in the Holst action Einstein-Cartan theory. Thus, the case i is incorpo- rated trivially, as the effective action turns out to be inde- i Z pendent of the parameter in this limit. S ¼ d4xe½c D ð!Þc Holst 2 5 The topological nature of the Barbero-Immirzi parameter (16) at the classical level has been clarified in [16] via a canoni- ð Þc c D ! 5 ; cal Hamiltonian analysis. As pointed out in [17], however, subtleties arise at the quantum level. Specifically, the and combine it with the gravitational action (5) in the standard Dirac quantization procedure for solving the presence of fermions. One then observes [12] that varia- second-class constraints that characterize the relevant mod- tions of the action with respect to the irreducible compo- els before quantization proves insufficient to preserve the nents of the contorsion tensor, T , S , and q yield topological nature of the Barbero-Immirzi parameter. The Nieh-Yan invariant density (19) vanishes ‘‘strongly’’ in this 2 3 = case after implementation of the constraints. Nevertheless, T ¼ Jð Þ; 4 2 þ 1 A alternative procedures for quantization have been suggested = þ 1 (17) [17], in which the elimination of the second-class con- S ¼ 3 J ; 2 þ 1 ðAÞ straints before quantization is avoided, and thus the Nieh- Yan density is nonvanishing. This leads [17] to a consistent ¼ q 0: topological interpretation of the Barbero-Immirzi parame- ter, but also indicates the subtle differences between this The inconsistency with the Lorentz properties of the first parameter and the QCD (instanton) angle. equation, involving the trace vector T , is thereby avoided in the limit C. Promotion of the Barbero-Immirzi 1 parameter to an axion field ! ; (18) The promotion of the Barbero-Immirzi parameter to a space-time dynamical field was proposed in [18], and a in which case the torsion assumes the form of the Einstein- canonical formalism for its quantization in nonsupersym- Cartan theory (14). However, the important point is that the metric theories was developed in [19]. In this approach, the limit (18) may be taken for any value of the Immirzi total divergence of the Nieh-Yan topological invariant (19) parameter . In this limiting case, as noted in [12], the acquires dynamical meaning, resulting in a pseudoscalar modified Holst action is nothing but a total derivative, and field replacing the constant Barbero-Immirzi parameter. can be expressed solely in terms of topological invariants, The field is pseudoscalar because the Barbero-Immirzi namely, the so-called Nieh-Yan (NY) invariant density parameter couples to the dual of the curvature tensor. A [15], and a total divergence of the fermion axial current, canonical kinetic term for the induced field is obtained
085016-4 ROLE OF SPACE-TIME FOAM IN BREAKING ... PHYSICAL REVIEW D 84, 085016 (2011) if we define it in terms of the Barbero-Immirzi field ðx;~ tÞ the addition of appropriate fermion bilinears that are total by [18] derivatives, expressible in terms of the corresponding pffiffiffi Nieh-Yan invariant densities, as in the nonsupersymmetric ¼ 1ð Þ 3sinh 1= ; (21) case (19) discussed above [12]. In this way, Barbero- which implies that the on-shell gravitational equations in Immirzi terms do not affect the equations of motion that this case become those obtained from the Einstein-Hilbert satisfy the local N ¼ 1 supergravity transformations [20]. action in the presence of a scalar field, As discussed in [21], one may construct an N ¼ 1 supergravity version of the Holst action as follows. One 2 1 first adds the Holst action to the purely gravitational G ¼ ð@ Þð@ Þ g ð@ Þð@ Þ ; (22) 2 (Einstein-Hilbert) sector of the theory, 2 ¼ ¼ 2 where 8 GN 8 =MP is the gravitational constant, 1 1 L ¼ ab ab cd with M the four-dimensional Planck mass, and G is the G eea e R R ; P 16 G b cd ab standard Einstein tensor. The equation of motion of the N (23) ð Þ field is the standard Klein-Gordon equation, h ¼ 0, e det ea ; which includes the case of a constant Barbero-Immirzi ab ab ac cb where R ¼ @½ ! þ ! ! is the curvature tensor parameter as a trivial solution in which the standard gen- ½ ab eral relativity equations are recovered. obtained from the connection ! , which includes torsion It should be noted that consistency of the canonical when the system couples to fermions, and is the Barbero- formulation of this field extension of the Barbero-Immirzi Immirzi parameter, which is in general complex. Next, parameter [19] requires that in the Ashtekar-Barbero con- one elevates this to the N ¼ 1 supergravity Holst action nection only a constant Barbero-Immirzi parameter enters, [21] by coupling the gravitational action to the ordinary which may be identified with a vacuum expectation value of Lagrangian for a Majorana Rarita-Schwinger (RS) c the Barbero-Immirzi pseudoscalar field. The pseudoscalar spin-3=2 fermion field , plus a total derivative of the nature of the Barbero-Immirzi field makes it resemble an axial gravitino current density, proportional to a (complex) axion. The feature that this field appears in the effective parameter . In flat space-time this recipe would give [21] action only through its derivatives was argued in [18]tobe i L ¼ L ðordinaryÞþ @ ð c c Þ essential for realizing the Peccei-Quinn U(1) symmetry RS RS 4 characteristic of general axion fields. 1 i ¼ c 5 @ c : (24) 5 2 D. Local supersymmetry (supergravity) and Barbero-Immirzi terms The coupling to gravity is achieved by the usual minimal prescription of replacing ordinary derivatives in flat space The above Holst framework may be extended to super- by gravitationally covariant derivatives containing the gravity actions, with the corresponding supersymmetries (torsionful) connection, being preserved in the limit ! 0, as in the case of global supersymmetry. In the context of N ¼ 1 supergravity [20], i ab ab i a b @ ! D @ þ !ab ; ½ ~ ; ~ ; a generalization to include a Holst term with a Barbero- 2 4 Immirzi parameter would break the underlying local (25) ! 1 supersymmetry, except in the limits 0 or . In this so that the gravitational RS Lagrangian reads case, following the standard procedure of varying the Holst 1 i ! L ¼ c 5 c action with respect to the spin connection would yield a GRS 5 D : (26) torsion contribution to the total !, involving the gravitino 2 axial current. However, the supersymmetry of such an The N ¼ 1 supergravity Lagrangian is then obtained by action can be preserved by modifying the Holst action by adding the Holst gravitational action (23)to(26),
Z 1 L ¼ 4 ½ abð Þ cc ð Þc N¼1SG d x e abR ! 5 D ! 16 GN Z i 1 þ 4 ~ abð Þ c c ð Þc d x e abR ! D ! ; 16 GN 1 where ab ea eb ; (27) 2 ½ 1 R~ab abcdR : 2 cd
085016-5 JOHN ELLIS AND NICK E. MAVROMATOS PHYSICAL REVIEW D 84, 085016 (2011) ab Following [21], one may vary the action with respect to the In the case of N ¼ 1 supergravity, the torsion C ,given following quantity constructed out of the torsionful con- as a solution of (29) for ¼ 1, leads to the following ab nection ! : contorsion tensor [20]: 1 1 1 B ! cd! ; (28) C ðc Þ¼ 8 G ðc c þ c c ab 2 ab 2 ab cd 4 N c c Þ; (33) obtaining with the torsion given by T ¼ 1 C , such that ð1 þ Þ ð1 Þ 2 ½ D ðee e Þ¼ X þ cdX ½a b 2 þ ab 2 þ ab cd 1 1 1 a a a D½ ð!Þe ¼ 2T ¼ c c : (34) 1 ð1 Þ 2 ¼ 1 X þ cdX ; 2 þ 1 ab 2 þ 1 ab cd With such a torsion, an appropriate Fierz rearrangement a ¼ 1 L L implies T T a 0 and the Nieh-Yan invariant where X RS þ cd RS ab 1 þ 2 !ab 2 ab !cd can be expressed as a total derivative, exactly as in the Dirac spin-1=2 fermion case (19). As such, the N ¼ 1 i ¼ c c supergravity equations of motion are not affected, and 5 ab : (29) 4 the on-shell local supersymmetry transformations remain The reader will notice that (29) has formal analogies with intact for arbitrary values of the parameter . Explicitly, the action of Barbero-Immirzi-modified N ¼ 1 supergrav- the Dirac fermion case (6), in that similar structures appear ¼ containing the corresponding fermion current terms. In this ity in the limit 1= , with arbitrary, is invariant case they are related by Fierz identities to terms containing under the following transformations generated by a ¼ 1 c c Majorana local parameter ðxÞ [21]: the gravitino axial current Jgravitino 2 . As in (6), we observe that, for the purely imaginary 1 c ¼ D ; Barbero-Immirzi parameters and of interest to us here, the coefficient of the second term in the right-hand- i ea ¼ a c ; side of (29) is purely imaginary, and so does not contribute 2 (35) to the Hermitian effective Rarita-Schwinger action, leav- 1 ¼ ðC Cc Þ ing only contributions from the first term. In fact, apart Bab ab e ½a cb ; with from the overall coefficient, the structure of these 2 i four-fermion terms is the same as the torsion-induced 1 5 C e 1 D c ; four-fermion terms in standard N ¼ 1 supergravity [20]. 5 2 The usual four-fermion terms of N ¼ 1 supergravity in the where 2 ¼ 8 G is the gravitational coupling. The limit second-order formalism are obtained in the limit N ! i (purely imaginary Immirzi parameter) leads to chiral N ¼ 1 supergravity. Substitution of the torsion 1 ! 0; (30) (33) into the first-order action yields the standard N ¼ 1 supergravity four-fermion interaction terms [20]. The in which case the Lagrangian (27) differs from the standard existence of torsion-induced four-gravitino terms in the N ¼ 1 supergravity Lagrangian simply by the total deriva- effective Lagrangian in the ¼ 1 supergravity limit, tive of the axial gravitino current, for arbitrary values of the Immirzi parameter , is analo- gous to the axial current-current interactions in the L ¼ L ¼ ðsecond order formalismÞ N 1 Einstein-Cartan theory (15).4,5 1 We next remark that the promotion of the Barbero- þ @ ð c c Þ: (31) 4 Immirzi parameter to an axion field, as discussed in Sec. II C above, can also be applied to N ¼ 1 four- ¼ In the limit 1, the Holst modification of the gravity dimensional supergravity, with the axion field being pro- action, with the fermionic corrections, is nothing but the moted to a complex chiral axion-dilaton superfield, whose topological invariant Nieh-Yan density plus the total de- lowest component comprises a scalar and a pseudoscalar rivative of the gravitino current, in analogy with the Dirac fermion case (19) discussed previously [22], 4Extensions of this formalism to N ¼ 2 and N ¼ 4 supergrav- Z 1 1 ity models are also known [22], but we do not discuss them in the ¼ 4 þ ð c c Þ SN¼1 Holst d INY @ ; current article. 2 2 5The interpretation of the Barbero-Immirzi parameter as a (32) topological parameter in general supergravity theories has been confirmed by a detailed canonical analysis of the with INY given in terms of the torsion tensor as in (19). spin-3=2 fermionic action in [23].
085016-6 ROLE OF SPACE-TIME FOAM IN BREAKING ... PHYSICAL REVIEW D 84, 085016 (2011) field, which play the roles of the dilaton and axion, respec- Following the argument of [24], we note next that, under tively [24]. The incorporation of a Barbero-Immirzi field in superfield Weyl transformations (which comprise ordinary such a formalism can be done neatly by first complexifying Weyl transformations of component fields, chiral rotation, the gravitational coupling constant 2 of the standard and a superconformal symmetry transformation), the fol- N ¼ 1 supergravity theory, lowing transformation laws are obeyed by the superfields entering (39): 1 1 ! ð1 þ i Þ; (36) 2 2 1 E ! e3 E; R ! e 2 R r 2 e ; (40) 2 where is a real dimensionless parameter identified with the inverse of the Barbero-Immirzi parameter, ¼ 1= ,as where is an arbitrary covariantly chiral superfield discussed above. r ¼ 0, with r denoting a curved superspace cova- As is standard in supersymmetry, such complex cou- riant derivative. The above freedom under Weyl transforms plings are consistent, given that in the pertinent actions one allows the imposition of a holomorphic ‘‘gauge fixing,’’ always includes the complex conjugate, so the final action is real. The gauge-invariant action of N ¼ 1 supergravity Z ¼ ; (41) in superfield formalism reads [25] as the simplest condition. Other arbitrary functions of Z 3 are allowed in more complicated gauge fixings. Such am- S ¼ d4xd2 d2 E 1 SG 2 biguities may be fixed dynamically when one considers the Z 3 embedding of the effective action (39) in a more micro- ¼ d4xd2 ER þ H:c:; (37) scopic framework such as string theory. 2 2 The use of (41) results in the following form of the where in the second equality we used the chiral-superspace supergravity action, involving a chiral complex superfield formalism of N ¼ 1 supergravity, which is convenient for coupled to supergravity [we revert here to the full curved our discussion below [24]. In the above formulas, E 1 ¼ superspace notation ðx; ; Þ]: M SDetEA is a supervierbein density in full curved super- Z ¼ 4 2 1 þ ð þ Þ space ðx; theta; Þ, with SDet denoting a superdeterminant S d xd d E e E R [25], while and denote supersymmetric generaliza- Z tions of the volume element and the Lagrangian density in 3 d4xd d2 e K=3; (42) chiral superspace. Details of the formalism and the equiva- ¼ lence of the action (37) with the standard N 1 super- with a Ka¨hler potential gravity action [20] in component formalism are provided in [24,25] and will not be repeated here. 1 Kð ; Þ¼ 3ln e þ ½ þ : (43) For our purposes we note that the complexification (36), 3 when substituted into the chiral superspace action (37), yields the supersymmetric Holst term of [21](32), which In general such a simplified form may be modified by more for constant is a total derivative, thereby not affecting the complicated potentials of the dilaton-axion superfields, N ¼ 1 supergravity equations of motion. Promotion of the when such models are viewed as low-energy approxima- parameter to a field is achieved by replacing in (36)by tions to some string theory. In particular, when higher- a complex chiral superfield Zðx; Þ with scalar component order string loop corrections are taken into account, Zðx; Þj ¼ ’ðxÞþibðxÞ, where ’ðxÞ is the dilaton scalar nontrivial corrections to the dilaton/axion potentials may and bðxÞ is a four-dimensional axion pseudoscalar field be generated, that stabilize the fields to constant values. [24], E. Global supersymmetry breaking and the 3 ð1 þ i Þ!Zðx; Þ; (38) Immirzi dilaton/axion superfield 2 2 We now discuss how local supersymmetry may be bro- so that the N ¼ 1 supergravity action becomes ken by the superfield acquiring an appropriate vacuum Z expectation value. We first describe some generic consid- d4xd2 EZR þ H:c: (39) erations on supersymmetry breaking due to chiral matter superfields. In general there are two generic types, F and Thus, in this formalism the field Z plays the role of a Fayet-Iliopoulos D-term breaking, the former arising when ‘‘Lagrange multiplier’’ superfield, whose variations yield the F-term of a chiral superfield acquires a nontrivial the dilaton-axion field equations of motion. These include vacuum expectation value hFi¼f 0, and we start the constant Barbero-Immirzi case ( ¼ const) as a trivial with this case. Our initial considerations below refer to solution, in analogy with the nonsupersymmetric example generic chiral superfields, and we turn later to the specific discussed above. case of the dilaton/axion superfield.
085016-7 JOHN ELLIS AND NICK E. MAVROMATOS PHYSICAL REVIEW D 84, 085016 (2011) The F-term breaking of supersymmetry implies in gen- that the superfield is constrained. Comparing our eral [26] that the Ferrara-Zumino (FZ) current superfield normalization with that of [26], we see that the J _ J _ ¼ 1 multiplet is well defined in such a sce- supersymmetry-breaking parameterpfffiffiffiffiffi 2 in this case, in nario (we use standard superfield notation in what follows). units of the gravitational scale ¼ 8 ¼ 1. In the case of MP This (Lorentz vector) multiplet consists of the supersym- the simplest Ka¨hler potential (43), it therefore seems that ¼ metry current S , the energy-momentum tensor T the supersymmetry parameter f is fixed at the gravitational T , and an R-symmetry current, which is not necessarily scale (up to a numerical factor 1=2), which is to be expected, conserved. The current obeys a conservation equation given that the Barbero-Immirzi field is a gravitational ef- D _ J ¼ D ; (44) fect. However, this might be problematic from the point of _ view of low-energy (infrared) phenomenology, where one where is a complex chiral superfield, which will in our might like the supersymmetry-breaking scale to be far case be the dilaton-axion multiplet.6 In the absence of below the Planck mass. However, it goes without saying D-term supersymmetry breaking, the FZ current multiplet that the embedding of the Barbero-Immirzi model in string is gauge invariant. theory, which would entail a more complicated potential The existence of such a well-defined FZ current multi- and higher-order curvature terms, could change this condi- plet implies, by means of generic supersymmetry consid- tion, as we discuss later on. erations [26], that at low energies the solution of the system In our particular Barbero-Immirzi model, it is the fer- of supersymmetry transformations of the infrared limit of mionic partner of the dilaton-axion (the ‘‘dilatino/axino’’) ¼ þ þ the chiral superfield NL NL NL FNL that can be identified with the Goldstino field at low leads to the following expression (where NL stands for energies, where the constraint (46) is satisfied, Following nonlinear): the generic supersymmetry analysis of [26], i.e., substitut- ing the solution (45) with f ¼ 1=2 into (47), the effective G2 pffiffiffi ¼ þ þ 2 NL 2 G F; (45) Lagrangian for the Goldstino field G has the form of the 2F Volkov-Akulov effective Lagrangian for the nonlinear real- which leads to the constraint ization of supersymmetry [27], 2 ¼ 0: (46) 1 NL L ¼ þ þ 2 2 2 IR i@ G G G @ G A general discussion in [26] established that the fermionic 4 component of the chiral superfield plays the role of the 4G2G 2@2G2@2G 2: (48) Goldstino, and that at low energies its action is the Volkov- Akulov action of nonlinear supersymmetry [27]. Thus, in In the four-component formalism, this is equivalent to the F-type supersymmetry breaking the Goldstino always original Volkov-Akulov Lagrangian.7 resides in a chiral superfield. The above low-energy considerations can be extended In the Immirzi extension of N ¼ 1 supergravity (42) with [28] to the case of D-term supersymmetry breaking in a the (simplest) Ka¨hler potential (43), one may assume that rather nontrivial way. In such a case, the FZ multiplet is not the F-term of the dilaton/axion Immirzi superfield breaks well-defined, in the sense that it is not gauge invariant. This global supersymmetry, in which case the low-energy limit is not a pathology of the theory; it just means that the above will lead to the nonlinear constraint (46). Taking this con- considerations based on the definition of a FZ current straint into account implies that the action (42), in the flat cannot apply. Nevertheless, in the low-energy limit one space-time limit we start with, assumes the form can still define constrained dilaton superfields NL, so that Z the connection of the dilatino-axino with the Goldstino can ¼ 4 L S d x flat ; still be maintained. In fact, in the string literature [29] there Z Z (47) are dynamical supersymmetry models with anomalous 1 1 L ¼ 2 d2 d2 þ d2 þd2 : U(1) symmetry and anomaly cancellation by the Green- flat 2 2 Schwarz mechanism, where the D-terms may be signifi- cant, depending on how the dilaton is stabilized. In such This has the same form as the (lowest-order) nonderivative infrared effective action of [26], which resembles the trivial (free superfield) case of supersymmetry breaking, except 7In the original works of [27], the Lagrangian is written in terms of a four-component Majorana spin-1=2 Goldstino field L ¼ ð 2Þ ð þ 1 r Þ (called ), f det i 2f2 @ that, upon ex- 6We note that the dilaton is a real field, whereas we use here pansion of the determinant and fermionic truncation, yields (48) complex supermultiplets, which have complex scalar fields with when one passes into the two-component spinor formalism. The 2 degrees of freedom in the lowest component of the superfield. constant f expresses the strength of global supersymmetry As we discuss later in the article, we identify one of these with breaking and, as mentioned above, in our case this occurs at the usual dilaton while the other, which couples in the super- f ¼ 1=2 in Planck units. The Lagrangian is characterized by a gravity sector with the dual of the curvature tensor, plays the role nonlinear realization of global supersymmetry with infinitesimal ¼ þ 1 of a pseudoscalar axion. parameter , f i f @ .
085016-8 ROLE OF SPACE-TIME FOAM IN BREAKING ... PHYSICAL REVIEW D 84, 085016 (2011) scenarios, the dilatino also emerges as (mostly) the can promote the global supersymmetry to a local one, by Goldstino, as in the pure F-term case discussed above. allowing the parameter ðxÞ to depend on space-time One can discuss generalizations of the FZ current coordinates, and coupling the action (49) to that of N ¼ 1 multiplet to another supermultiplet, also containing the supergravity in such a way that the combined action is stress-energy tensor and the supersymmetry current in its invariant under the following supergravity transformations: components, whose conservation equation involves neces- ¼ 1 ðxÞþ ; sarily massless chiral scalar superfields. In string theories a ¼ ð Þ a c the role of such a massless superfield is played by the e i x ; (50) dilaton and other moduli fields [28]. From our point of 1 c ¼ 2 @ ðxÞþ : view, therefore, the incorporation of dilaton-axion super- fields in a Jordan-frame-like modification (42) of the stan- The action that changes by a divergence under these trans- ¼ dard supergravity action fits into the above picture. In the formations is the standard N 1 supergravity action plus i if low-energy regime of more general supergravity theories, 2 pffiffiffi L ¼ f e @ c þ ; (51) where a FZ multiplet cannot be defined, condensates of the 2 2 Goldstino field (which is the dilatino) still form in the which contains the coupling of the Goldstino to the grav- infrared, and correspond to the infrared dilaton (and axion) itino. The Goldstino can be gauged away [30] by a suitable fields. It is not clear, however, that such theories are con- redefinition of the gravitino field and the tetrad. One may sistent theories of quantum gravity, given that they are impose the gauge condition characterized by additional global R symmetries. Thus, c ¼ for our purposes below we concentrate on theories with 0; (52) well-defined FZ multiplets, and therefore primarily F-type but this leaves behind a negative cosmological constant supersymmetry-breaking models. term, so the total Lagrangian after these redefinitions reads L ¼ 2 þð ¼ Þ eff f e N 1 supergravity : (53) F. Coupling to supergravity The presence of four-gravitino interactions in the standard The coupling of the Goldstino to supergravity generates N ¼ 1 supergravity Lagrangian in the second-order for- a mass for the gravitino through the absorption of the malism, due to the fermionic contributions to the torsion in Goldstino, via the super-Higgs effect envisaged in [30]. the spin connection (33), implies also an induced gravitino ¼ According to this model, the N 1 supergravity theory is mass term that is generated dynamically. coupled to a Volkov-Akulov Majorana fermion that may To see this, one may simply linearize the appropriate arise from some spontaneous or dynamical breaking of four-gravitino mass terms of the N ¼ 1 supergravity ¼ supersymmetry. In our extension of N 1 supergravity Lagrangian in the second-order formalism [20], by means to include a Barbero-Immirzi field, the coupling is pro- of an auxiliary scalar field ðxÞ [31], vided by the Lagrangian (42) which, in view of the above 1 1 discussion on the identification of the dilatino as the L ¼ ð Þþ c c þ 2ð Þ eff 2 R e 5 D x Goldstino field in the case of broken supersymmetry, has pffiffiffiffiffiffi 2 at low energies the component form suggested in the super- 11 ðxÞðc c Þþ ; (54) Higgs effect scenario of [30], namely, a nonlinear Volkov- with 1 ½ ; , where the indicate terms we are Akulov Lagrangian coupled to N ¼ 1 supergravity. It is 4 instructive, and illuminating for what follows, to review not interested in, including other four-gravitino interac- ¼ explicitly this coupled system in component form.8 tions with 5 insertions, as well as other standard N 1 Thus, we consider a spontaneously broken supersym- interactions and auxiliary supergravity fields. On account metric theory with a Majorana Goldstino , whose action of the gauge fixing condition (52), we have takes the nonlinear form considered by Volkov and Akulov 1 c c ¼ c c ; (55) [27,30], 2 2 1 using the anticommutation properties of the Dirac matrices L ¼ f det þ i @ 9 2f2 . The formation of a condensate, 1 h ðxÞi hc c i; (56) ¼ f2 i @ þ : (49) 2 Here we keep the discussion general by allowing for an 9Sometimes [30] the gravitino mass term is defined in the 1 c c arbitrary value of the parameter f, as is possible in general presence of 5 as m 2 . It is easy to adapt to this models: see the discussion above. As discussed in [30], one case by linearizing the appropriate gravitino four-fermion terms that contain the square of such terms, which we did not indicate explicitly in the effective action (54). The analysis is exactly the 8We repeat that in superfield language this is just (42), upon same, whichever form of gravitino mass we seek to create 2 ¼ using the condition NL 0 [(46)]. dynamically.
085016-9 JOHN ELLIS AND NICK E. MAVROMATOS PHYSICAL REVIEW D 84, 085016 (2011) which should be independent of x because of the trans- 2 2 2 m3=2 44 ; (59) lation invariance of the vacuum, is possible by minimizing the effective action (54) along the lines in [31]. with 2 ¼ f2; cf. (58). Since the graviton remains massless We observe [30,31] that the formation of the condensate in this approach, supersymmetry is broken locally. may cancel the negative cosmological constant term. The The size of the condensate can be determined in condensate contributes to the vacuum energy a term of the principle by solving the appropriate gap equations in the form Minkowski background, following from the expression for Z the gravitino propagator and the imposition of the vanish- d4xe 2 > 0; (57) ing of the tadpole of the fluctuation of the linearizing auxiliary field 0ðxÞ. The analysis of [31] leads to the ¼ and at tree level one can fine-tune this term so as to cancel following gap equation for the gravitino field of N 1 the negative cosmological constant of the Volkov-Akulov supergravity, by passing to the appropriate Fourier k-space Lagrangian (51) and (53), which depends on the supersym- in the Minkowski space-time background: 2 metry breaking scale f , by setting Z 3 1 k2 1 ¼ d4k 4 þ ; (60) 2 ¼ f2: (58) 2 2 2 þ 2 44 m3=2 k m3=2 In Ref. [32], a one-loop effective potential analysis has demonstrated that such a cancellation occurs for a suitable with m3=2 given by (59). This equation has to be regular- value of the parameter f. Whether the situation persists ized in the UV by a cutoff , which is consistent with to higher orders, so that the cancellation of the effective broken supersymmetry. The spherical symmetry of the cosmological constant can be achieved exactly, is not integrand in (60) yields the following analytic expression known. Assuming this to be the case, or restricting our- for the integral [31], assuming a Euclidean formulation and selves to one-loop order, we may therefore consider the performing the usual analytic continuation back to quantization of the gravitino field in a Minkowski space- Minkowski space-time only at the very end of the compu- time background and discuss its dynamical mass genera- tations: tion in the same spirit as in the flat space-time prototype 3 1 4 2 case of the chiral-symmetry breaking four-fermion ¼ 2 þ 3 2 3m2 ln þ 1 : 10,11 2 2 3=2 2 Nambu-Jona-Lasinio model [33]. 44 2m3=2 m3=2 Enforcing this cancellation of the vacuum energy con- (61) tributions, one may write the effective action for the con- 0 0 densate fluctuations ðxÞ, ðxÞ¼ þ ðxÞ in a standard The cutoff cannot be determined in the low-energy fashion, by direct substitution in (54) and expansion effective field theory framework. In our case, one may around a Minkowski space-time, using the vierbein expan- use as the UV cutoff the supersymmetry breaking scale a ¼ a þ a sion e h and taking into account the gauge f. However, in our Barbero-Immirzi case as discussed so condition (52). From the resulting effective action at zeroth far, f is of the order of the Planck mass squared, f ¼ 2 ¼ 2 order in the gravitational field h, then, one can read off =2 MP=16 ; cf. the discussion following (47) and directly the dynamically generated gravitino mass as [31] reinstating units of MP, implying a gravitino mass of the order of the Planck mass, as was the case in [31]. Similar 10It should be remarked here, especially in connection with the results are obtained if one considers the one-loop effective D-foam model [34] discussed in the next section, that such a potential analysis of [32], which shows that up to that order cancellation of the cosmological constant may not characterize the effective potential for the field is always positive and the microscopic model, which may thus exhibit an (anti–) vanishes at a nontrivial minimum, for a value of the cutoff de Sitter background, depending on the net sign of the vacuum ¼ 1 and values of f and the gravitino mass of similar energy contributions. It is well-known [35] that the dynamical formation of chiral condensates via four-fermion interactions, as (Planckian) order to that indicated by the tree-level analy- of interest to us here, exhibits better UV behavior than in the sis described above. corresponding flat space-time case [33], in the sense that any However, these results are not consistent with our infra- potential UV infinities, which are regularized by means of an UV red analysis, where we expected a gravitino that is light cutoff , may be absorbed into the cosmological constant and compared to the Planck mass. Moreover, there is another Planck scale of the (anti–)de Sitter space-time. For chiral sym- metry breaking in the early Universe, for instance, this has been problem with such high-scale supersymmetry breaking. demonstrated explicitly in [9]. In our case, the regularization of For a dilaton vacuum expectation value (VEV) of the order UV infinities by a cutoff is consistent with the breaking of of the Planck scale, one cannot ignore the quantum supersymmetry, and one may attempt a similar analysis as in fluctuations of the gravitational field around the classical the spin-1=2 four-Fermi models of [9]. However, we do not do anti–de Sitter metric background, g0 , in which the this in the present work. 11We also note that in (anti–)de Sitter space-times even repul- Volkov-Akulov Lagrangian is formulated. It is for this sive interactions lead to condensates. reason that the above considerations have been disputed
085016-10 ROLE OF SPACE-TIME FOAM IN BREAKING ... PHYSICAL REVIEW D 84, 085016 (2011) Bulk in [36]. Indeed, in a linearized gravity approximation, (a) 0 g ¼ g þ h , integrating out the metric fluctuations h in the way suggested in [37] leads [36] to gauge- Open strings D−particles dependent imaginary parts in the effective action, indicat- ing an instability of the gravitino condensate.12 Hence it is desirable to be able to extend the Barbero- Immirzi formalism to incorporate light gravitinos, corre- sponding to supersymmetry breaking scales that are low compared to Planck mass, in which the gravitational fluc- tuations can be ignored. This could in principle be achieved by embedding the Barbero-Immirzi effective action (42) Brane world into a full string theory framework, with an appropriate V_perp dilaton potential to be generated by string loops. One such (b) concrete framework, that of D-particle space-time foam, is V_parall Open string discussed in the next section, but other examples may well be possible. In such a case, a low scale of supersymmetry OR D3-brane breaking f may be introduced, and the problem of fluctua- D-particle V_perp tions in the gravitational field could be evaded. The sim- V_parall plest way of achieving this is to rescale the dilaton/axion superfield field , ! f in (42) and (47), and embed the theory into higher dimensions, by assuming for in- FIG. 1. (a) Schematic representation of a generic D-particle stance a brane Universe propagating into a bulk space. In space-time foam model, in which matter particles are treated as such a case, the action (42) is nothing but part of an open strings propagating on a D3-brane, and the higher- effective action on the brane world in the Jordan frame. dimensional bulk space-time is punctured by D-particle defects. The only ingredient of the Barbero-Immirzi field that we Consistent supersymmetric D-particle foam models can be con- structed. No recoil and no brane motion yields zero vacuum maintain is that it is associated with the dilaton/axion energy and unbroken supersymmetry. Recoil contributions to excitation of this string theory. There is a potential for vacuum energy yield broken supersymmetry. (b) Details of the the dilaton field as already mentioned, which determines process whereby an open string state propagating on the the global supersymmetry properties of the dilaton F-term, D3-brane is captured by a D-particle defect, which then recoils. implying a low f. This process involves an intermediate composite state that 0 In such a scenario, the low-energy cutoff p,ffiffiffi taken to be of persists for a period t E, where E is the energy of the order of the supersymmetry-breaking scale f, is assumed incident string state. This distorts the surrounding space-time to be much lower than the four-dimensional Planck scale during the scattering process, leading to an effective refractive index, but not birefringence. Components of the recoil velocity MP, and also much higher than the mass of the gravitino m f M . We then obtain from (61) perpendicular to the D3-brane world lead to vacuum energy 3=2 P contributions and thus target-space supersymmetry breaking. 22 4 m2 2 : (62) 3=2 2 III. QUANTUM GRAVITY FOAM, DILATONS, AND 3 MP SUPERSYMMETRY BREAKING Consistency between (59) and (62) is achieved for A. D-particle foam as a quantum-gravity medium 2 4=3 f2. We now proceed to discuss a concrete example, that of a The model illustrated in the upper panel of Fig. 1 [34] D-brane model for stringy space-time foam, where such a will serve as our concrete D-brane approach to the phe- low supersymmetry-breaking scale may be realized. nomenology of space-time foam. In it, after appropriate compactification, our Universe is represented as a Dirichlet three-brane (D3-brane), on which conventional particles 12However, in our opinion, the gauge-invariant Batalin- propagate as open strings. This three-brane propagates Vilkovisky formalism for the effective action of N 1 super- in a 10-dimensional bulk space-time containing orientifold gravity theories, as used in [36,37], is not completely understood 13 at present. Hence, the gauge dependence of the imaginary parts, planes, that is punctured by D-particle defects. and the fact that in some gauges the imaginary parts are zero D-particles cross the D3-brane world as it moves through [38], might simply indicate our inadequate understanding of the bulk. To an observer on the D3-brane, these crossings low-energy supergravity theories, rather than proving that dy- namical mass generation of a gravitino mass of the order of the Planck mass via four-gravitino interactions is not possible in the above-described fashion. Nevertheless, for our purposes here we 13Since an isolated D-particle cannot exist [39] because of are primarily interested in relatively light gravitinos, as already string gauge flux conservation, the presence of a D-brane is mentioned. essential.
085016-11 JOHN ELLIS AND NICK E. MAVROMATOS PHYSICAL REVIEW D 84, 085016 (2011) constitute a realization of ‘‘space-time foam’’ with defects use a general parametrization of the foam fluctuations as at space-time events due to the D-particles traversing the follows: D3-brane; we term this structure ‘‘D-foam.’’ When the rk open strings encounter D-particles in the foam, their inter- v ¼ g p ; A s M A actions involve energy-momentum exchange that cause the s D-particles to recoil. A ¼ 1; 2; 3 )hrki¼0; If there is no relative motion of branes in the bulk, target- hr2i¼ 2 0 space supersymmetry, implies the vanishing of the ground- k k short short state energy of the configuration. However, if there is v? v ; relative bulk motion, supersymmetry is broken and there ¼ 4; ...9)hvshorti¼0; are nontrivial forces among the D-particles, as well as (65) h short shorti¼ 02 between the D-particles and the brane world and orienti- v v short 0; folds [34]. The resulting nonzero contribution to the energy ; ¼ 4...9; is proportional to v2 for transverse relative motions of 4 long long branes with different dimensionalities, and to v for branes v? v ; 14 of the same dimensionality. There is also a dependence ¼ )h longi¼ on the relative distances of the various branes. In particular, 4; ...9 v 0; the interaction of a single D-particle that lies far away from h long longi¼ 02 v v long 0; a D8-brane world, and moves adiabatically with a small ¼ velocity v? in a direction transverse to the brane, results in ; 4...9; the following potential [34]: h short;long short;longi¼ v vA 0: rðvlongÞ2 pffiffiffiffiffi V long ¼þ ? þ 0 In general, stochastic foam implies vanishing correlators of D0–D8 0 ;r ; (63) 8 odd powers of the recoil velocity vi, but nontrivial correla- tors of even powers. Above, h...i indicates averaging over where the indicate velocity-independent parts that are both quantum fluctuations and the ensemble of D-particles cancelled by the orientifold planes in the model of [34], in the foam, and indices A ¼ 1; 2; 3 denote the longitudinal and do not play any role in our discussion below. On the dimensions of the D3-brane world. The v , A ¼ 1; 2; 3, other hand, a D-particle close to the D8-brane, at a distance A pffiffiffiffiffi represent the recoil velocity components of the D-particle r0 0 and moving adiabatically in the perpendicular short during scattering. In the absence of any interaction with direction with a velocity v? , induces the following matter strings there is also a velocity vi expressing the potential: quantum fluctuations of individual D-particles. 0ðvshortÞ2 pffiffiffiffiffi In view of the opposite signs of the contributions (63) V short ¼ ? þ 0 D0–D8 03 ;r< ; (64) and (64), in a stochastic foam situation it is possible for the 12r effective potential on the brane world to vanish on average. where again the denote velocity-independent terms that However, the spontaneous breaking of global supersym- are cancelled in the model of [34], as mentioned previ- metry is possible even in such a case, as we discuss below, ously. This D-foam model involves different configurations and also the breaking of local supersymmetry through the of D-particles [34], and one must average over appropriate coupling to gravity, through the special role played by the populations and quantum fluctuations [40] of D-particles dilaton field explained above. with bulk (nine-dimensional) recoil velocities vi. To explore this possibility, we consider in detail the interaction of an open-string particle with a heavy, non- B. D-foam and the effective target space-time metric relativistic D-particle described as in [41] by the process of capture and release by the D-particle of a open string Recoil fluctuations along the D3-brane correspond to a representing a neutral flavored particle such as a neutrino; stochastic fraction of the incident momentum of the open- cf. the lower panel in Fig. 1, which may induce multi- string particle [34,40]. Because of the D-particle fluctua- fermion interactions of the form discussed in the previous tions in the direction transverse to the brane worlds, there section. As shown in [41], the recoil excitation of a would also be contributions to the potential energy of the D-particle during its nontrivial scattering (capture) with brane [cf. (63) and (64), where now the quantities involv- an open-string state propagating on the brane world (as in short;long ing v? should be averaged over populations of Fig. 1) results in a distortion of the neighboring space-time D-particles and quantum fluctuations]. We can plausibly which, for relatively long times after the scattering, is described by the following flat metric that depends on the 14There is no contribution to the energy of a D-brane world momentum transfer: from motions of other branes in directions parallel to its longi- 2 ¼ 2 i j i tudinal directions. ds dt ijdx dx 2uidx dt; (66)
085016-12 ROLE OF SPACE-TIME FOAM IN BREAKING ... PHYSICAL REVIEW D 84, 085016 (2011)
2 where huiujiD ¼ ðtÞ ij;i;j¼ 1; 2; 3; (70)
pi ui ¼ gs (67) we obtain for the average quantum foam induced metric at Ms each space-time point is the D-particle recoil velocity and pi the open-string- 2 X3 state momentum transfer during the scattering/capture pro- ðtÞ hhds2ii ¼ ð1 2ðtÞÞdt2 1 ðdxiÞ2: (71) cess. This metric is Finsler-like, in that the recoil velocity 3 i¼1 ui depends in general on the open-string particle velocity. The D-particles are assumed heavy, with their mass M g p , so that u 1 is nonrelativistic. This metric After rescaling the time coordinate appropriately, we may = s i i map the homogeneous and isotropic metric into a diagonal is nothing other than the induced metric, from the point of metric of cosmological type in a conformal time , view of a passive observer, under a Galilean transformation ! i ! i þ i t t, x x jujt. It is worth noticing that, from a X world-sheet -model point of view, which provides a first- hhds2ii ¼ Cð Þ d 2 ðdxiÞ2 ; quantized description of the dynamics of the composite i (72) string/recoiling-D-particle state following the scattering 2ð Þ event shown in the lower panel of Fig. 1, such a metric Cð Þ 1 ; 3 arises in the restoration of the world-sheet conformal sym- metry that was disturbed by the recoil operators. This ð Þ¼ ð ð ÞÞ restoration was achieved by means of a Liouville dressing where t denotes the time-dependent fluc- of the model, combined with the identification of target tuation function of the homogeneous and isotropic foam. time with (the world-sheet zero mode of) the Liouville The time dependence reflects the possibility of a density of field, in a way consistent with the logarithmic conformal defects crossing our brane world in Fig. 1 that is not algebra of recoil [41]. constant in cosmic time. The metric (66) can be diagonalized by means of appro- From a -model point of view, the metric (72) may be priate transformations. Equivalently, in a T-dual formal- viewed as the -model-frame metric. In this sense, the ism, the propagation of strings in a recoiling D-particle curvature kinetic term in the low-energy effective action, background, when the string excitations interact with the representing the dynamics of the problem, does not have D0-brane defect in the way described above, can be the canonical Einstein form, but rather the Jordan-frame mapped to the problem of a string in a constant ‘‘electric form field’’ background of the form [42] Z ¼ 2 4 2’ ð Þþ Sgrav MP d xee R g ; (73) ¼ F0i ui: (68) where ’ is the four-dimensional dilaton field and the The reader should notice that, in view of (67), the electric denote the rest of the terms in the low-energy effective field background is actually in the phase space of the string action, including kinetic terms of the field ’. Appropriate state, as it depends on the relevant momentum transport. compactification or other bulk effects are ignored for brev- Assuming for simplicity that the electric field is along the ity, and MP denotes the four-dimensional Planck mass. x1 direction, (68) implies that the string state propagates in This is in general different from the D-particle mass the diagonal metric Ms=gs, where Ms is the string mass scale, depending on X the details of the compactification. 2 ¼ð 2Þ 2 ð 2Þð 1Þ2 ð iÞ2 The physical metric [43] is related to the -model ds 1 u1 dt 1 u1 dx dx : (69) i 1 Jordan-frame metric by the following transformation (we restrict our discussion to four space-time dimensions in As a result of quantum collisions in a stochastic space-time what follows, as we consider a D3 brane world): foam, the recoil velocity fluctuates along different direc- phys ¼ 2’ ¼ 2’þlnCð Þ tions. This prompts us to consider the superposition of g e g e : (74) metrics (69) at each space-time point, averaged over pop- Thus, we see that one may absorb the time-dependent ulations of D-particle defects. Such a stochastic superpo- ð Þ 1 effects of the D-foam induced scale factor C in a sition is denoted by hh...ii ¼ h...iD, where d is the d redefinition of the dilaton field. number of spatial directions along which the recoil veloc- ity vector fluctuates; in our case we take d ¼ 3 and h...iD denotes averages over statistical ensembles of D-particle C. Cosmological-type backgrounds defects in the foam. Assuming homogeneous and isotropic Let us first discuss a case in which the dilaton ’ is foam backgrounds with stochastic fluctuations of the form stabilized to a constant by means of some unspecified
085016-13 JOHN ELLIS AND NICK E. MAVROMATOS PHYSICAL REVIEW D 84, 085016 (2011) potential—which is not known at present in string theory, where denote higher derivative terms in the low-energy given that it depends on string-loop effects over which string action, and Vð’Þ is the dilaton potential, which there is no control in general. In such a case, the confor- receives in general contributions also from higher string mally flat, homogeneous and isotropic physical space-time loops. The matter terms in the effective action couple to the metric (74) with (time) dependence only, assumes a dilaton via exponential factors, in general, form familiar in cosmology, and the physical effect of X Z ðiÞ ð Þ ð Þ3 4 i’L i ¼ the fluctuating D-particle foam on the brane world is to V ’ d xe matter; const; induce a cosmological background. The constancy of the i¼matter species dilaton ’ implies that in this simplified scenario the gravi- (77) tational sector of the low-energy theory consists only of cosmological backgrounds, over which propagate the mat- where the factors i are in general different for different ter fields, represented in the microscopic picture by open matter species, and are determined by scale symmetry. strings on the D3 brane world. Such terms also contribute to the dilaton potential in Whether the background corresponds to an expanding (76), which must be known precisely in order to see universe depends on the bulk density of the D-particles in whether the expression (75) is a consistent solution of the the model illustrated in Fig. 1. It is natural to expect that equations of motion. Depending on the exact form of the the stochastic fluctuations ð Þ2 of the foam-averaged dilaton potential, it is possible that there are constant, time- D-particle recoil velocities are proportional to the density independent solutions with 2 ¼ constant that correspond of the foam, in the sense that the higher the density the to a minimum of the dilaton potential. stronger the average of the fluctuations. In such a case, an Such a solution could have important implications for expanding effective metric (74) on the brane Universe is the spontaneous breaking of target-space supersymmetry obtained if the density of D-particles on the brane reduces since, depending on the form of the dilaton potential, the as time elapses. F-term of the dilaton chiral superfield may acquire a In the presence of string matter, then, the Einstein cos- vacuum expectation value that breaks global target-space mological equations for the background (74) with constant supersymmetry. The dilaton is one of the two fields of the dilaton ’ impose a consistency condition on the fluctua- chiral Barbero-Immirzi superfield, the other being a pseu- tions 2ð Þ, i.e., the scale factor Cð Þ, depending whether doscalar axion. The dilaton suffices for supersymmetry the era is dominated by matter or dark energy. Lack of breaking, and one may assume that the axion has a zero knowledge of the exact dilaton potential prevents a more expectation value in the D-foam background. However, explicit discussion of the corresponding low-energy action. this assumption is not binding, and more general D-foam backgrounds may be found with a nontrivial axion. In our D. D-foam-driven supersymmetry breaking case, in which the constant dilaton configuration arises in and the dilatino Goldstino field the presence of an appropriate stochastic fluctuation of the D-foam recoil velocities, it is the space-time foam that is However, more interesting for us is an alternative pos- responsible for the breaking of target supersymmetry, in- sibility that we now concentrate upon, namely, the possi- ducing dynamical breaking of local supersymmetry (su- bility of an equilibrium, nonexpanding Universe in the pergravity). The way this is achieved in principle has been presence of D-foam effects. Such a scenario is realized if described in the previous sections, and will now be dis- the dilaton field is time-dependent, but its entire time cussed in the specific D-foam context. dependence is of the form Since this D-foam scenario differs from standard ways 1 of breaking supersymmetry spontaneously in that the ’ ¼þ lnCð Þ: (75) 2 above-described background (72) is induced by stringy matter excitations interacting with the D-foam, one can We then see from (74) that the physical metric is just flat talk about a supersymmetry obstruction [44] rather than Minkowski, and all the low-energy effects of the foam are breaking, in the sense that the true vacuum of the theory absorbed by the dilaton. The gravitational sector of this (characterized by the absence of excitations) may still be theory consists of just a cosmological dilaton background, supersymmetric, but the effects of supersymmetry break- and the corresponding field-theory equations of motion are ing can be seen in the excitation spectrum via, e.g., mass given by just a flat-space equation of motion for the scalar splittings between fermions and bosons. field. As already mentioned above, the potential of Our discussion of a dynamically generated gravitino the dilaton in string theory is not known, in general, so mass (59) in an effectively Minkowski flat space-time we consider the following generic equation of motion for background applies intact in the case of D-foam, since the equilibrium dilaton in the presence of D-foam: the background of the physical metric (74) implied by the specific dilaton background (75) is indeed Vð’Þ h’ þ ¼ V0ð’Þ; (76) Minkowski. We recall that, from a microscopic brane ’ Universe point of view (see Fig. 1), in general one may
085016-14 ROLE OF SPACE-TIME FOAM IN BREAKING ... PHYSICAL REVIEW D 84, 085016 (2011) obtain (anti–)de Sitter-type constant vacuum energy con- following gravitational, dilaton, and four-gravitino terms tributions, as a result of the forces exerted on the three- in the notation and conventions of [45]: brane Universe by bulk D-particles, (63) and (64), upon 1 1 averaging over D-particle populations using (65). In such a e 1L ¼ ~ ½RðeÞ c R ð@ ~ Þðc c Þ case, the transverse distance r may be replaced by a 6 6 constant representing an averagep value,ffiffiffiffiffi usually assumed 1 0 V þ ~ ½c c ðc c þ 2c c Þ to be of order of the string length . 96 For simplicity, we assume that the foam recoil-velocity þ ; fluctuations of the short- and long-distance defects are of 1 the same order, R @ þ !abðeÞ þ auxiliary fields c ; 4 ab (80) short; long short; long 02 hv v i¼ ; (78) short long where the notation ... denotes appropriately antisym- metrized products of matrices, e denotes the vierbein, ! ðeÞ is the torsion-free spin connection, and the four- in which case the contributions from the foam to the gravitino terms arise from the torsion of the supergravity vacuum energy density on the brane can be negative, i.e., model.15 The denote other structures, including kinetic of anti–de Sitter type and compatible with supersymmetry terms for the scalar ~ and four-fermion terms coupling the 02 in the three-brane world, and of order , provided the gauginos of the model with the gravitinos, as well as effects of the bulk defect populations near the brane over- appropriate auxiliary fields, which are not of interest to come those of the distant D-particles, which is a possible us here. Complete expressions can be found in [45], to outcome of the bulk distribution of the D0-particles in the which we refer the interested reader. In that work, there are models. Note that in this picture, the quantum fluctuations contributions to the scalar superfield also from the super- h 2i¼ 2 of the D-particles along the three-brane, vk may in symmetrization of matter in the standard model. In our 0 general be different from 2. case, we concentrate only on the gravitational (D-foam) In such a case, the significance of the dilaton back- contributions to the (dilaton) scalar field ~ , which is one of ground (75), which enforces a Minkowski space-time the scalar components of the Barbero-Immirzi superfield. form for the background physical metric, is analogous to Comparing (80) with the string effective action in the so- the imposition of the cancellation of the cosmological called -model (or Jordan) frame, (73), we obtain the constant contributions (58)in[31]. In such a case, the following relation between the Barbero-Immirzi superfield supersymmetry-breaking scale associated with the VEV and the dilaton field ’ in our normalization (up to numeri- of the dilaton has the value cal factors that we do not write explicitly):
~ ¼ 2’ e ; (81) 2 1 h’i¼þ ln 1 ; (79) ~ 2 3 with constraints to ensure the positivity of . One can pass to the Einstein frame (whose metric is denoted by a superscript E), in which the curvature terms in (80) are canonically normalized with coefficient 1 , by redefining which can be expressed in terms of the bulk fluctuations 2 0 the metric (in the normalization of [45]), 2 (78), in order to enforce the cancellation of the effec- tive cosmological constant on the brane. This cancellation 1 E ¼ 2 2 ¼ ~ involves the string-loop-induced contribution to the dilaton g g ; > 0; (82) 3 potential, and cannot be derived from first principles, since the dilaton potential in string theory is not known. In this and then normalizing the kinetic terms of the gravitino respect, our situation is no different from any other ap- terms, by absorbing appropriate powers of ~ into the field proach to string phenomenology. c . The Einstein-frame version of the action (80) then The presence of a nontrivial dilaton potential in the reads [using (81) and reinstating units of the gravitational effective low-energy supergravity theory on the brane constant 2 ¼ 8 G for completeness] world prompts us to consider conformal supergravity mod- N els with more general nontrivial superpotentials than that in 15 ¼ 2 [24] in order to explore the possible effects of the Barbero- The action (80) reduces, for 6= (in our conventions and normalization), to the standard N ¼ 1 supergravity action— Immirzi (dilaton-axion) superfield. One such model is with the terms c R reducing to the standard kinetic term for the conformal supergravity model in the Jordan frame of the gravitino (24), using standard properties of the product of [45], which in component form contains among others the three matrices.
085016-15 JOHN ELLIS AND NICK E. MAVROMATOS PHYSICAL REVIEW D 84, 085016 (2011) 1 1 11 2 LEðeEÞ 1 ¼ REðeEÞþ c 0 DE c 0 e2’VE þ e 2’½ðc 0 c 0 Þ2 ðc 0 c 0 Þ2 2 2 2 5 16 5
33 þ 2 2’ðc 0 c 0 Þ2 þ e 5 64 pffiffiffiffiffiffi 1 1 11 ¼ Eð EÞþ c 0 E c 0 2’ E þ 2ð Þþ ð Þ ’ðc 0 c 0 Þþ 2ð Þ 2 R e 5 D e V x x e x 2p ffiffiffiffiffiffi 2 pffiffiffiffiffiffi 2 11 33 þ e ’ i ðxÞðc 0 c 0 Þþ e ’i ðc 0 c 0 Þþ ; (83) 2 5 2 5
0 where c denotes the canonically normalized gravitino Such backgrounds also arise in our nonequilibrium with standard kinetic term as in N ¼ 1 supergravity, and Liouville string approach to quantum gravity [34,46,47], the denote structures, including auxiliary fields, that in which the world-sheet zero mode of the Liouville dress- are not of interest here. In writing (83) we have expanded ing field plays the role of time. When such backgrounds the four-gravitino terms into detailed structures to exhibit are considered in conformal supergravity (83), which is a explicitly the terms that generate masses, and we linearize natural effective low-energy framework for string the four-gravitino terms. The condensate of interest to us is theory, the above considerations imply the presence of the VEV of the linearizing field ðxÞ. strong coupling four-gravitino interactions that could The reader should notice that the coefficients of the lead to stable condensates, unaffected by weak gravita- gravitino- interaction terms in (83) are now modified tional fluctuations. compared to (54) by dilaton-dependent factors e ’, This D-foam model provides a specific realization of the being proportional not to but to scenario for global and hence local supersymmetry break- ing via the VEVof a dilaton F-term that we proposed in the ~ e ’: (84) previous section, which is based on D-particle stochastic Using (79), we observe that the coefficients of the four- recoil velocity fluctuations 02 perpendicular to the gravitino terms in (83) are of order brane Universe; cf. Fig. 1 and (65). As we have seen (64) the contributions of forces exerted on our brane world by 11 2 2 1 1 : (85) neighboring bulk D-particles are negative and of anti– 16 3 de Sitter type, provided the density of such defects is We observe that the foam fluctuations tend to increase the more or less constant for late eras. The latter can be interaction as compared with the Minkowski background arranged by a suitable bulk density of D-particles in case [31]. In the weak foam backgrounds we have consid- the foam [46], that is consistent with the observed cosmol- ered in our work so far, the form of the induced metric ogy at late eras. In fact, using (64) and averaging deformation (72) implies that 2 1, otherwise the sig- over populations of nearby bulk D-particles we obtain a nature of the space and time components of the induced vacuum energy on the brane world of the form of a cos- 02 ¼hh 2 ii metric are reversed. This situation implies that the foam mological constant proportional to v? , where fluctuations are sufficiently weak that the gravitational v? denotes the velocity of the D-particles transverse fluctuations of the background cannot be ignored, which, to the brane world. The reader should recall that any according to the arguments of [36] and our previous dis- D-particle movement parallel to the brane world does not cussion, destabilizes the putative gravitino condensate. yield any contributions to the brane vacuum energy in our model. However, one may consider the embedding of the 0 D-foam model [34] in a cosmological framework [46], in The D-foam recoil velocity bulk fluctuations 2, 2 are which case strong negative dilaton backgrounds in the free parameters of the model at the present stage of our conformal supergravity framework are also allowed, which understanding. The quantum fluctuations of the D-particles include not only the foam contributions (79) but also much parallel to the brane Universe are associated with the stronger cosmological time dependences. In particular, for momentum transfer and hence influenced by the momen- late times in the history of the Universe, far away from the tum of the matter excitation, whereas those perpendicular inflationary phase, one may have ’ 1, in which case to it are related to properties of the foam per se. It is natural e 2’ 1. One such background is the cosmological run- then to assume that in our low-energy effective supersym- metric field theory framework the induced supersymmetry away linear-dilaton case [43], in which at asymptotically 0 late cosmic Robertson-Walker time t in the history of the breaking scale f is of order of the foam fluctuations 2, Universe, ! 1 2 ¼ ð 02Þ 4 ’ lnt : (86) f O MP; (87)
085016-16 ROLE OF SPACE-TIME FOAM IN BREAKING ... PHYSICAL REVIEW D 84, 085016 (2011) 2 ¼ c 2 c Z 2 2 where M V Ms , with Ms the string scale and V the 4 ~ P e 1V ¼ d4p ln 1 þ 11 þ f2 2 volume of thep compactffiffiffiffiffi extra dimensions in units of the eff ð2 Þ4 p2 0 ¼ string length 1=Ms. In this way, for sufficiently 1 121 1 02 ¼ 4 4 ð 2 2 2Þ small D-particle recoil fluctuations 1 one may ob- 3 ~ ln 11~ = tain supersymmetry-breaking scales that are low compared 4 2 2 to the four-dimensional Planck mass, in which case the þ 11~ 2 2 2 þ f2 2; (88) infrared limit used in our analysis in this paper is applicable. where is a cutoff used to regularize the UV divergences In this picture, the supersymmetry of the vacuum is of the nonrenormalizable effective low-energy N ¼ 1 su- broken spontaneously, in the sense that the recoil velocity pergravity theory. Following [32], one observes that for the fluctuations that affect the VEVof the F-term of the dilaton values field are vacuum properties, and supersymmetry transfor- mations are formally invariances of the effective 1 ¼ ~ 1; ¼ 1:796~ 2 (89) Lagrangian. This implies that the supersymmetric partner f of the dilaton is the Goldstino field, for the reasons dis- cussed above. The role of the initially negative (anti– the effective potential is non-negative and has a minimum de Sitter) cosmological constant of the Volkov-Akulov at zero, therefore leading to a vanishing cosmological con- field is then played in this framework by the vacuum stant on our brane world, consistent with a Minkowski energy (64) induced by nearby bulk D-particles [34]. background, allowing us to interpret terms of the form 0 0 As explained above, the coupling of the dilatino with the h ic c as corresponding to gravitino mass terms. rest of the low-energy conformal supergravity is described in The numerical analysis of [32] shows that the effective the infrared by the Volkov-Akulov Lagrangian (48), (49), and potential for the field acquires a minimum at (53), which realizes global supersymmetry nonlinearly, cou- ¼h i¼ pling to the Lagrangian (83) via the Goldstino/gravitino min 0:726 (90) coupling of (51). The Goldstino field is then eaten by the at which it vanishes; cf. Figure 2. The gravitino mass term gravitino, which becomes massive via the formation of in (83) then takes the following form; cf. (55): the condensate, as discussed in the previous section. In the specific D-foam model discussed in this section, the conden- 0 0 1 0 0 m c c ¼ m c c ; (91) sate contributes a positive term ( 2) to the brane vacuum 3=s 2 3=2 energy able to cancel the anti–de Sitter-type contributions 2 with due to f (87)in(49). In our low-energy string-induced pffiffiffiffiffiffi conformal supergravity framework, the full effective poten- m ¼ 11 ~ 1 ¼ 2:408~ 1 ¼ 3:227Oð 02ÞM M tial is complicated, since the precise form of the dilaton 3=2 min P P potential is not known in string theory. Hence, the cancella- (92) tion of the effective cosmological constant on the brane is an for a small foam contribution, where we used (87) and (89). issue that is far from being resolved here. Nevertheless, we now offer plausible arguments how such a cancellation may be achieved in our context. The V key point to realize is that, once the effective conformal supergravity action is considered in the runaway dilaton limit (86), the effective coupling of the four-gravitino terms is no longer given by the gravitational constant, but by the much stronger effective coupling (84). In practice we may consider the runaway dilaton limit as describing a long but finite time after, say, inflation, at which ’ 1 but finite; the Universe may enter a phase in which the 2 dilaton is frozen at a large but finite value. f
E. Analysis of one-loop effective potential (0,0) ρ In this limit the effective action (83) can lead to an FIG. 2. The one-loop effective potential for the linearizing effective potential that, at one-loop order, is obtained by field ðxÞ in the effective action (83), which is suitable for following exactly the analysis of [32], but replacing the dynamical breaking of local supersymmetry and the generation gravitational coupling in that work by the new coupling ~ of a gravitino mass. For certain values of the relevant parameters (84). The resulting effective potential for the field ðxÞ at the potential has a minimum at zero, indicating an effective one-loop order is then found to be [32] vanishing cosmological constant.
085016-17 JOHN ELLIS AND NICK E. MAVROMATOS PHYSICAL REVIEW D 84, 085016 (2011) We recall that cosmological and astrophysical con- connection in such space-time backgrounds that have a straints are compatible either with scenarios where the ðX0Þ structure. Such singularities are integrable, though, gravitino is the lightest supersymmetric particle and makes and hence they do not yield any nontrivial contributions to a contribution to dark matter, or with scenarios where the the Holst term (32). Nevertheless, the effective field theory gravitino weighs a few TeV or more, in which case it is close to the moment of impact may be quite different unstable and decays sufficiently early not to upset the from the effective N ¼ 1 supergravities we have discussed agreement between standard big-bang nucleosynthesis in this work, which are relevant for situations far away and astrophysical observations of light-element abundan- from that moment. In fact, it was argued in [49] that a ces. Indeed, the decays of a gravitino in this mass range metric of the form (93) arises as a consistent solution of the might even improve agreement with the measured abun- gravitational part of an effective (d þ 1)-dimensional ac- dance of 7Li [48]. tion on a d-dimensional brane world, involving a Gauss- The important physical difference of our case from that Bonnet curvature-squaredR combination coupled to a dþ1 2’ 2 2 2 of [32] lies in the fact that, as a result of the dilaton factors, dilaton field, d xe ðR 4R þ R Þþ . the effective coupling ~ appearing in (89) is much larger Supersymmetrizing such terms would lead to supergavities than the gravitational constant ¼ 1=MP. Because of this with higher-order curvature terms, of the form considered enhancement of the four-gravitino interactions in (83)by in low-energy actions of superstrings and in a field-theory dilaton factors e 2’ (84), one can argue that metric fluc- framework in [50]. These actions are not the subject of our tuations around the space-time foam background cannot present discussion, but are viewed as describing the short- destabilize the condensate, and thus the scenario presented distance properties of the foam, while the considerations in here implements the original arguments of [31], avoiding our work, based on conformal N ¼ 1 supergravities with the objections of [36]. Indeed, for sufficiently low conventional Einstein curvature terms, describe the long- supersymmetry-breaking scales (87), thepffiffiffi relevant excita- distance (infrared) properties of the D-foam. The issue tions are those with momenta below f MP, where whether other singular backgrounds could also produce quantum-gravity fluctuations are irrelevant. such contributions and break supersymmetry is an open Before closing this section we remark on another issue one that we do not discuss further here. associated with a potentially nontrivial role of the Immirzi- Nieh-Yan topological density (32) in the supergravity case, VI. CONCLUSIONS which involves the total derivative of the gravitino current. An on-shell solution of the gravitino field equations ob- In this paper we have highlighted the role in global and tained from the appropriate Lagrangian in a recoiling local supersymmetry breaking that might be played by the space-time foam background may contain singularities, Barbero-Immirzi parameter in an effective low-energy su- which could make such total derivative contributions non- pergravity theory valid in the infrared limit. We have trivial, in which case the Barbero-Immirzi parameter ap- stressed the potential analogies with the vacuum parame- pearing in the respective Holst actions could play a ter in a non-Abelian gauge theory. In that case nontrivial physical role. If such were the case, there would be a direct topological configurations are thought to play an important analogy of the Barbero-Immirzi parameter with the role in the formation of a chiral symmetry-breaking con- parameter of QCD in the presence of color magnetic densate. We have argued that nonperturbative events might monopoles. play an analogous role in a suitably compactified string In our recoiling D-particle background, the space-time theory. Specifically, we have pointed out that D-particles metric deformation induced by the recoil of a single moving through the bulk appear as space-time events when D-particle during its interaction with an open string exci- crossing the D-brane Universe, and have shown that inter- tation includes off-diagonal terms of the following form actions with open-string particle excitations that are asso- (before averaging over D-particle populations in the foam) ciated with such events could give rise to four-fermion [41]: interactions. Combined with a suitable string contribution to the effective potential of the dilaton component of the Barbero-Immirzi superfield multiplet, this mechanism may g ¼ u X0 ðX0Þ; (93) 0i i lead to condensation with vanishing net vacuum energy in the brane Universe that breaks supersymmetry both glob- where X0 is the time. For long times after the impact with ally and locally. In this scenario, the dilatino component of the defect, the induced metric takes the form (66), which is the Barbero-Immirzi supermultiplet is represented by a then averaged over D-particle populations in the foam to nonlinear effective Lagrangian of the Volkov-Akulov produce the situation we have analyzed in this work. type, and is eaten by the gravitino, giving it a mass. However, close to the moment of impact of the string state In closing, we emphasize that the general features of this on the D-particle defect, and its splitting as in Fig. 1, the Barbero-Immirzi mechanism for breaking supersymmetry space-time structure is that of a Rindler wedge [49]. may have wider validity than the specific D-particle recoil There are singularities of the induced metric Christoffel implementation that we have discussed here. Specifically,
085016-18 ROLE OF SPACE-TIME FOAM IN BREAKING ... PHYSICAL REVIEW D 84, 085016 (2011) in the framework of conformal supergravity [45] in the -model frame in the string picture of [43]), nonminimal Einstein frame (83), all we need for our arguments on couplings with the curvature that might lead to acceptable the dynamical breaking of local supersymmetry through slow-roll conditions for inflation in the early Universe the formation of condensates is some dynamical mecha- [45,51,52]. Connecting such nonminimally coupled scalar nism yielding a VEVof the dilaton field of appropriate size, multiplets to gravitino condensates, as discussed here, so that the induced effective coupling of the four-gravitino might provide a link to minimal inflation as suggested in interactions becomes stronger than the gravitational cou- [53]. However, this issue is a delicate one, depending on pling, overcoming its destabilizing effects. the details of the scalar field potential, which in our string In our D-foam framework we have used the supersym- dilaton case is not known. metric dilaton/axion multiplet identified with the complex Barbero-Immirzi superfield to provide, in the infrared, ACKNOWLEDGMENTS the Goldstino superfield associated with the breaking of supersymmetry. However, one may consider in addition This work was supported in part by the London Centre more general scalar fields, e.g., standard model scalar for Terauniverse Studies (LCTS), using funding from the multiplets such as Higgs fields, as in the analysis of [45]. European Research Council via the Advanced Investigator Such fields have, in the corresponding Jordan frame (or Grant No. 267352.
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