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

Home , Type I string 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

John March-Russell

Scho ol of Natural Sciences

Institute for Advanced Study

Princeton, NJ 08540, USA

and

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

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.

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.

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

it has equal U(1) G anomalies for all gauge groups G as well as with gravity (here

and in the following we omit for simplicity the ane level factors k which can easily

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-

pling, g , and the reduced Planck mass M = M = 8 . A detailed calculation

str Planck

gives [1, 3, 4]

g Tr(Q )

str

2 2

M ; (1) =

192

where, following convention, this is expressed in terms of the total U(1) (gravity)

anomaly prop ortional to Tr(Q ).

This FI term allows many phenomenologically interesting applications. These

include, an alternate non-grand-uni ed theory explanation of the successful sin =

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 )

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

example, (based on the free-fermionic construction) Tr(Q ) = 72= 3 [8]. However

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

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

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

U(1) , only receive soft masses via sup ergravity. Thus the gaugino masses are sup-

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

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

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

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

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

a lpa

where F = " F . Here four-dimensional Lorentz indices are denoted m;n;:::,

mnl p

and a;b;::: are adjoint indices of G. The exp ectation value of the dilaton sets the

value of the four-dimensional gauge couplings at an appropriate scale, here always

2 2

taken to b e the string uni cation scale, so he i = g .

unif

It is easiest to calculate the FI term by starting with the eld-theoretic expres-

sion for the U(1) G anomaly. Such an anomaly implies that under a U(1) gauge

X X

transformation v ! v + @ ! the low-energy e ective action is not invariant, but

m m m

transforms as

L = F F; (4)

where A is the anomaly co ecient. In a sup ersymmetric theory ! gets promoted to a

chiral sup er eld , and the U(1) vector sup er eld V transforms as V ! V + i(

)=2.

The apparent anomaly in the low-energy e ective action Eq.(4) is cancelled by

assigning a shift in the axion under U(1) gauge transformations. In terms of sup er-

elds,

S = i ; (5)

and then the shift in the gauge kinetic term Eq.(3) comp ensates the triangle anomaly

Eq.(4). However, the non-trivial transformation of S implies that the conventional

Kahler p otential term for the dilaton sup er eld ln(S + S ) is no longer invariant. The 3

dilaton sup er eld kinetic energy term must b e mo di ed to

2 4

C d ln S + S + V ; (6)

then the transformation of the U(1) vector sup er eld cancels the variation. (In

Eq.(6) C is an all imp ortant normalization factor that will b e calculated b elow from

the underlying Typ e-I or M-theory.) The most imp ortant feature of Eq.(6) from our

p oint view is that when this mo di ed kinetic term is expanded out in comp onent

form, a term linear in the U(1) D-term o ccurs:

4 2

1 e Ae

2 2 2

C (@) + (@a) + D + ::: : (7)

2 4 16

This is nothing other than the FI term, ,( has mass dimension 1 in the conventions

b eing followed here):

2 2 2 2

C Ahe i C Ag

unif

= = : (8)

2 2

16 16

Thus we see that sup ersymmetry allows us to calculate the FI term given the co e-

cient of the anomaly, A, and the normalization, C , of the mo del-indep endent axion

kinetic energy, in the basis that the axion itself is normalized so that it couples to

the G gauge eld strengths as in Eq.(3).

To xC it is necessary to consider the origin of the axion a in string (or M-) theory.

Within the framework of an e ective 10-dimensional string-derived sup ergravity, the

mo del indep endent axion arises from the 3-form eld strength H , which satis es

the Green-Schwarz mo di ed Bianchi identity

dH = tr F ^ F +trR ^ R: (9)

Once we compactify down to four dimensions we may de ne the e ective four-

dimensional axion a in terms of H via

2 q

H = M " @ a; (10)

mnp mnpq

where M is a dimensionful normalization factor intro duced so that a has mass dimen-

sion zero in four dimensions, and which can b e xed by demanding that a couples to

F F correctly.

The mo di ed Bianchi identity, Eq.(9), implies that a satis es the equation of

motion

2 2 a a

M @ a = F F +(Rdep endent terms); (11)

which is equivalent to the statement that a has the required axion-like interaction

term aF F in the e ective action up to a normalization factor. Furthermore, after

compacti cation, the low-energy four-dimensional kinetic energy for H

mnp

L = ::: + NH H + :::; (12)

e ;4d mnp 4

induces the kinetic energy term for the axion. To pro ceed we must convert this to

the normalization used in Eqs.(3), (7). Fixing the coupling of the axion to F F gives

2 2 4

the relation M =1=32N , while we nd that C = e =32N , and thus using Eq.(8)

= : (13)

512 g N

unif

Thus the calculation of the FI term is reduced to a computation of the co ecient,

N , of the H kinetic term in the four-dimensional low-energy e ective action after

compacti cation.

3 The low-energy e ective action in M-theory

The Horava-Witten [12] 11-dimensional low-energy sup ergravity action is, in comp o-

nent form, given by (see also [13]):

1 1 1 2

11 IJ KL I :::I

1 11

L = g d x R G G " C G G

11d IJ KL I I I I :::I I :::I

1 2 3 4 7 8 11

2 48 3456

1 1

10 2 2

+ fermi terms: (14) + d x tr F + tr R

2 2=3

8 (4 ) 2

i=1;2

This describ es a system in which the sup ergravity mo des propagate in the 11-d bulk

1 2

of M , while the E E sup er Yang-Mills (SYM) degrees of freedom are lo calized

8 8

to the two 10-dimensional orbifold hyp erplanes @M (i = 1; 2) of M at x = 0;

resp ectively. The normalization of the gravitational part of the action is appropriate

for the 11th dimension b eing an S , of circumference 2, with the elds satisfying

a b ab

Z orbifold conditions (the E generators ob ey tr(T T ) = ). The 4-form eld

2 8

strength G satis es a mo di ed Bianchi identity

IJ KL

2=3

3 2 1

11 1 1

(dG) = (x )(tr F F R R )

11;I J K L

[IJ KL]

2 4 2

2=3

1 3 2

11 2 2

(x )(tr F F R R ); (15)

[IJ KL]

2 4 2

and will provide, after compacti cation, the four-dimensional mo del-indep endent ax-

ion a.

Before we calculate the normalization of the axion kinetic term arising from this

action, it is necessary to eliminate and in terms of the 4d gauge and gravitational

couplings. We can compactify the theory down to four dimensions, preserving N =1

1 4

sup ersymmetry, by taking M = S R K , where K is a (p ossibly deformed)

Calabi-Yau manifold, and where, as usual, the spin connection is emb edded in the

E gauge connection. This leaves an E SYM theory which can b e further broken by

Wilson lines, etc. (We assume in the following analysis that the visible MSSM gauge 5

group arises from the i =1 b oundary theory.) Denote the volume of the Calabi-Yau

measured at the i = 1 b oundary by V (0). As shown by Horava and Witten, in leading

2=3 11

non-trivial order in the expansion in , the volume of K decreases linearly with x .

The maximum allowed value of is such that V (x = ) approximately equals the

Planck volume. In any case, compactifying the bulk sup ergravity action on S K

leads to

Z Z

1 1 V (0)

11 4 (4)

(4)

d x d x gR ! g 2R ; (16)

2 2

2 2 2

where the factor of V (0)=2 arises from averaging over the Calabi-Yau volume (V (0) >>

V () in the scenarios where the hidden E is strongly coupled). Comparing this to

the standard Einstein-Hilb ert action in four dimensions gives the usual result

G = : (17)

8 V (0)

Similarly reducing the i = 1 SYM b oundary theory leads to an expression for the

uni ed gauge coupling of the four-dimensional theory

unif

2 2=3

(4 )

= (18)

unif

2V (0)

It is imp ortant to recall that b eyond leading order quantum corrections mo dify these

classical relations; however, the ab ove expressions for G and suce to deter-

N unif

mine the FI term.

As discussed by Witten [11], the additional dep endence of G on the size of the

11th dimension relativeto (as compared to the standard weak-coupling E E

unif 8 8

result) allows one to somewhat adjust the string prediction of the uni cation scale.

Roughly sp eaking the four-dimensional MSSM uni cation scale can b e de ned as the

scale at which either string excitations, or the additional six dimensions in which

the gauge degrees of freedom propagate, rst b ecome apparent. An arbitrarily low

uni cation scale is not achievable within the Horava-Witten picture (at least if our

world is situated at the weakly coupled b oundary), since the second E SYM theory

moves into the strongly-coupled domain as increases. Indeed a calculation [11]

indicates that it is just p ossible to take the length of the 11th dimension long enough

so as to lower the string uni cation scale to match that inferred in the MSSM, M '

unif

2 10 GeV . In the following we will keep the volume of the Calabi-Yau space and

the length of the 11th dimension as free parameters.

Having completed this preliminary discussion, the e ective 3-form eld strength

H is de ned as G = H . Here the normalization factor is necessary so

11;mnp mnp

that the gauge dep endent piece of the mo di ed Bianchi identity for H restricted to

1 1

the i = 1 b oundary theory reads dH = tr F ^ F , as in Eq.(9). It follows from the

M-theory version of the Bianchi identity Eq.(15), that

2=3

3 2

= : (19)

4 4 6

The axion kinetic term arises from the G term in Eq.(14), which expressed in terms

of the correctly normalized 3-form eld strength reads, after compacti cation to four

dimensions:

Z Z

1 1

p p

11 2 2 4 2

d x gG ! V (0)A d x gH : (20)

AB C ;11 mnp

2 2

12 12

M M

Comparing to the general expression for the H kinetic energy leads gives N which,

using the relations Eqs.(17) and (18) between G , and ; , and V (0), may be

N unif

written

3 V (0)

N =

3 4=3 2=3

32 (4 )

= (21)

unif

Finally, substituting this into the expression for the FI term, Eq.(13), gives

2 2

Ag M

unif

: (22) =

192

This is identical to the value obtained for the weakly-coupled E E heterotic string,

8 8

independent of the size of the 11th dimension, or equivalently the uni cation scale in

this strongly-coupled E E picture.

8 8

4 Typ e-I string theory

For Typ e-I string theory, the situation is quite similar. We start from the 10-

dimensional sup ergravity Lagrangian derived from its' low-energy limit [14]:

7 2

I I

p e 8 e

10 a a

L = d x g R + F F

Typ eI

0 0

4 3

( ) 16( )

I I

+ H + ::: : (23)

7 2

2048 ( )

Here H is the 3-form eld strength, the 10d dilaton of Typ e-I string theory, and,

as usual, has dimension of an inverse mass-squared. If we compactify this theory

on a Calabi-Yau of volume V , the classical expressions for Newton's constant and the

4d uni ed gauge coupling that follow from comparing to the standard 4d action are

2 0 4

e ( )

G = ;

128 V

0 3

e ( )

= : (24)

unif

V 7

As recently emphasised by Witten [11], it is now the di erent dilaton dep endencies

of the gauge and gravitational pieces of the action that allow one to adjust the string

prediction of the uni cation scale to match the inferred MSSM uni cation scale.

However unlike the case in either the weakly-coupled E E heterotic string or the

8 8

standard Horava-Witten M-theory with E strongly coupled, the Typ e-I uni cation

scale can b e made arbitrarily low. In the Typ e-I case of interest, the uni cation scale

1=6

corresp onds to leading order to M ' 2 (V ), the mass of the rst Kaluza-Klein

unif

excitations, and as this is lowered, the typ e-I string coupling is also reduced, moving

the theory farther into the weakly coupled domain. (However one must b e careful that

the string worldsheet coupling is not b ecoming strong, see Ref. [15] for an extensive

discussion of these issues.)

Indep endentofV, the expressions Eq.(24) satisfy the standard Typ e-I relation b e-

2 7 0

tween the reduced Planck mass, M , and the gauge coupling, M = 128 e = .

unif

(Note that this relation di ers by numerical factors from that stated in Ref. [11].)

Solving for from Eqs.(24) gives

2 2

M V

unif

0 2

( ) = : (25)

As b efore, the quantity of interest for the calculation of the FI term is the co-

ecient of the compacti ed 3-form eld strength kinetic term expressed in terms of

, M and M . Using Eq.(25) leads to

unif unif

3V 3

N = = ; (26)

0 2

7 2 2 2

2048 ( ) 128 M

unif

and thus from Eq.(13) we nd that the FI term in Typ e-I string theory is given by

2 2

AM g

unif

= ; (27)

192

again identical to the result in the weakly-coupled E E heterotic string. This is

8 8

indep endent of the Typ e-I string scale, or equivalently the e ective four-dimensional

uni cation scale, at least as long as one stays in the domain of applicability of the

ab ove calculation.

5 Comments

We have shown that a quite simple calculation of the induced Fayet-Iliop oulos term

is p ossible in the contexts of Typ e-I string theory and Horava-Witten M-theory. For

xed Newton's constant and four-dimensional uni ed gauge coupling the value of the

FI term is indep endent of all details of the underlying Typ e-I string theory or M-

theory. This includes the mo duli of the compacti cation down to four dimensions,

the underlying string coupling, and in the case of M-theory the length of the 11th

dimension. 8

One signi cant caveat to this work is that we have always assumed that the

gauge degrees of freedom of the MSSM app ear in the p erturbative sp ectrum of the

Typ e-I string theory or Horava-Witten M-theory. Clearly this is not a necessity,

for the MSSM degrees of freedom could in principle arise from D-brane excitations.

Interestingly, FI terms for theories containing anomalous U(1)'s do arise in the D-

brane construction of six- and four-dimensional gauge theories (see for example the

discussions of Refs. [16] and [17]). It would certainly b e worthwhile to consider such

FI terms in the case of realistic (or semi-realistic) D-brane mo dels of the uni ed

MSSM interactions.

Acknowledgements

I wish to thank K. Dienes, C. Kolda, and N. Seib erg for discussions, and M. Dine

for encouraging me to write up these results. I also wish to thank P. Binetruy and

P. Ramond for discussions of their closely related work prior to publication.

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