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The Fayet-Iliopoulos Term in Type-I String Theory and M-Theory

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The Fayet-Iliopoulos Term in Type-I String Theory and M-Theory IASSNS-HEP-97/128 CERN-TH/98-198 hep-ph/9806426 The Fayet-Iliop oulos term in Typ e-I string theory and M-theory 1 John March-Russell Scho ol of Natural Sciences Institute for Advanced Study Princeton, NJ 08540, USA and 2 Theoretical Physics Division CERN, CH-1211 Geneva 23, Switzerland Abstract The magnitude of the Fayet-Iliop oulos term is calculated for compacti ca- tions of Typ e-I string theory and Horava-Witten M-theory in which there exists a pseudo-anomalous U(1) . Contrary to various conjectures, it is found that X in leading order in the p erturbative expansion around the weakly-coupled M- theory or Typ e-I limits, a result identical to that of the weakly-coupled E E 8 8 heterotic string is obtained. The result is indep endent of the values chosen for the Typ e-I string scale or the size of the M-theory 11th dimension, only dep ending up on Newton's constant and the uni ed gauge coupling. 1 Research supp orted in part by U.S. Department of Energy contract #DE-FG02-90ER40542, and by the W.M. KeckFoundation. Alfred P. Sloan Foundation Fellow. 2 On leave of absence from the Institute for Advanced Study after June 2nd 1998. Email: [email protected] 1 Intro duction One of the most phenomenologically useful features of many string compacti cations is the existence of a U(1) symmetry with apparent eld theoretic anomalies [1]. These anomalies are cancelled by a four-dimensional version of the Green-Schwarz mecha- nism [2], whichinvolves shifts of the mo del indep endent axion of string theory under gauge and general co ordinate transformations. Such shifts are able to comp ensate the eld-theoretic anomalies of the pseudo-anomalous U(1) symmetry if and only if X 2 it has equal U(1) G anomalies for all gauge groups G as well as with gravity (here X and in the following we omit for simplicity the ane level factors k which can easily G be reinstated if required). As originally noticed by Dine, Seib erg, and Witten [1], when this mechanism is combined with sup ersymmetry,aFayet-Iliop oulos (FI) term is induced in the D-term for U(1) . For the weakly-coupled E E heterotic string the magnitude of the X 8 8 induced FI term dep ends on the eld-theoretic anomaly co ecient, the string cou- p pling, g , and the reduced Planck mass M = M = 8 . A detailed calculation str Planck gives [1, 3, 4] 2 g Tr(Q ) X str 2 2 M ; (1) = 2 192 2 where, following convention, this is expressed in terms of the total U(1) (gravity) X 1 anomaly prop ortional to Tr(Q ). X This FI term allows many phenomenologically interesting applications. These 2 include, an alternate non-grand-uni ed theory explanation of the successful sin = w 3=8 relation at the uni cation scale [5], the p ossible avor universal communication of sup ersymmetry breaking from a hidden sector to the minimal sup ersymmetric standard mo del (MSSM) [6], and an explanation of the texture of the fermion masses and mixings [7]. Many such applications require for their success the existence of a hierarchy between the contributions induced by the presence of the FI term and those contributions arising from other sup ergravity or string theory mo des. Since the strength of sup ergravity corrections is controlled by the reduced Planck mass, the appropriate expansion parameter is exp ected to be 2 2 g Tr(Q ) X str " = : (2) 2 2 M 192 For the value of g wemay take the MSSM uni ed gauge coupling, however, a remain- str ing uncertainty in " is the size of the gravitational anomaly co ecient. For realistic string mo dels, Tr(Q ) is certainly quite large; for instance, in one typical semi-realistic X p example, (based on the free-fermionic construction) Tr(Q ) = 72= 3 [8]. However X since the gravitational anomaly is sensitive to all chiral states in the mo del, b oth in the MSSM and hidden sectors, additional matter b eyond the MSSM or large hidden 1 An interesting asp ect of this result is that it evades the \pro of " that a non-zero FI term is inconsistent with lo cal sup ersymmetry. 1 sectors can increase Tr(Q ) dramatically. Thus in the context of the weakly coupled X E E string the expansion parameter is unlikely to b e any smaller than " 1=100. 8 8 The precise value of the expansion parameter " is quite imp ortant phenomenologi- cally. For example, in the case that the soft masses of the MSSM squarks and sleptons are communicated by U(1) , the MSSM gauginos, b eing necessarily uncharged under X U(1) , only receive soft masses via sup ergravity. Thus the gaugino masses are sup- X pressed by the factor " relative to the soft masses of the MSSM squarks and sleptons (by assumption U(1) charged { at least for the rst and second generations) [6]. This X leads to either unacceptably light MSSM gauginos, or so heavy squarks and sleptons that naturalness, or stability of the U(1) and SU(3) color preserving minimum is em threatened. Although this problem is ameliorated when a non-zero F-comp onent for the dilaton sup er eld of string theory is included [9], it is presently unclear if a fully consistent mo del is p ossible (esp ecially the question of whether F -terms for the other, non-universal, mo duli are induced). In any case it is certainly interesting to ask how the U(1) scenarios b ecome mo di ed outside of the weakly-coupled E E case. X 8 8 Another of the prop osed uses of the string-induced FI term is D-term in ation [10]. In these mo dels the in ationary vacuum energy density is due to a non-zero D-term rather than the more usual F-term contributions. This has the advantage, over the F-term dominated mo dels of sup ersymmetric in ation of naturally p ossessing su- ciently at in aton p otentials for successful slow-roll in ation, even after the e ects of sp ontaneous sup ersymmetry breaking due to V 6=0 are taken into account. Since vac the magnitude of the FI term sets the size of the Hubble constant during in ation, as well as its rate of change along the slow-roll direction, it sets in turn the magni- tude of the uctuations in the cosmic microwave background radiation (CMBR). An analysis [10] shows that the observed amplitude of uctuations in the CMBR requires 15 17 > =6:610 GeV . This is to b e compared with the larger value, 2 10 GeV , string obtained by substituting the deduced value of the MSSM uni ed coupling into the weakly-coupled heterotic string theory prediction Eq.(1). The most attractive pro- p osal for the solution of this mismatch is that the weak-coupling prediction for is mo di ed in the strong-coupling (Horava-Witten M-theory, or Typ e-I) limit. This seems not unreasonable since it is well known that in such a limit manyE E weak- 8 8 coupling predictions are amended, most notably the prediction for the scale of string uni cation [11]. Given these various motivations for considering the FI term in more general con- texts, we show in the following sections that a simple calculation of the induced FI term in Typ e-I string theory and M-theory is p ossible. The calculation is quite gen- eral, b eing indep endent of almost all details of the compacti cation down to four dimensions. However it is imp ortant to keep in mind two assumptions under which we work: First, we assume that the standard mo del gauge group arises from within the p erturbative excitations of the theory (rather than from D-brane dynamics), and second, that there are not large corrections to the normalisation of the kinetic terms of various elds arising from corrections to the minimal Kahler p otential. Although 2 such corrections to the Kahler p otential are certainly p ossible as wemoveaway from the weakly coupled M-theory or Typ e-I domain into the domain of intermediate cou- pling, it seems very likely that the FI term itself is protected from such corrections by the surviving N =1 spacetime sup ersymmetry. 2 The FI term The calculation of the magnitude of the FI term induced in a string theory with an anomalous U(1) symmetry is greatly simpli ed by a judicious use of sup ersymmetry X together with the anomaly constraints. Since we are concerned with the phenomeno- logical applications of the FI-term, we are interested in compacti cations of either Typ e-I string theory or M-theory down to four dimensions, in which the low-energy limit is an N=1 sup ersymmetric eld theory. In this situation the dilaton and the mo del-indep endent axion a combine to form the lowest complex scalar comp onentof the dilaton chiral sup er eld, S = exp (2)+ia + :::. Expanded out in terms of comp onents, the coupling of the dilaton sup er eld to a gauge eld kinetic term of a gauge group G reads Z 1 1 a 1 a 2 a 2 a a 2 a a ~ (F ) + (D ) F F +:::; (3) d S W W +h:c: = 2 2 4 4e 2e 4 1 a lpa ~ where F = " F .
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