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Threshold corrections and modular invariance in free fermionic strings
Physics Letters B 269 ( 1991 ) 84-90 North-Holland
PHYSICS LETTERS B
Threshold corrections and modular invariance in free fermionic strings S. Kalara
a,b J.L.
L o p e z a,b,I a n d D . V . N a n o p o u l o s a,b,c
a Center for Theoretical Physics, Department of Physics, Texas A&M University, College Station, TX 77843-4242, USA b Astroparticle Physics Group, Houston Advanced Research Center (HARC), The Woodlands, TX 77381, USA c Theoretical Physics Division, CERN, CH- 1211 Geneva 23, Switzerland
Received 22 July 1991
We examine the modular invariant properties of the low-energy effective action of free fermionic models. This entails a detailed comparison between the so-called string and supergravity bases, since only in the latter are the modular invariant properties evident, whereas the S-matrix calculations are performed in the former. We then adjoin these results to the K~hler anomaly method for obtaining the untwisted moduli dependence of the string threshold corrections to the gauge couplings. Finally we apply our results to the flipped SU ( 5 ) string model and determine the common unification scale for all the gauge couplings in the model.
1. Introduction String derived models o f particle interactions are endowed with an extraordinarily rich s y m m e t r y structure [ 1 ]. C o m p a r e d to o r d i n a r y grand unified theories, the string derived effective low-energy action has far m o r e predictive power and leads to penetrating insight into the connection between seemingly unrelated sectors of the theory [2,3]. Although the task o f constructing the low-energy effective string action which manifests all the symmetries o f the original string theory is by no means straightforward, over the past few years great strides have been m a d e in this direction [ 4 ]. One o f the genuinely stringy symmetries which has attracted a great deal o f attention is the so-called " m o d u l a r s y m m e try", a generalization o f the basic duality s y m m e t r y o f string models [ 5,6]. In a large class o f models, such as orbifold [6,7] and C a l a b i - Y a u compactifications [ 8 ], this is an infinite-dimensional discrete symmetry. The existence o f such s y m m e t r y has been known for quite some time [5]. However, its p h e n o m e n o logical implications have not been explored to their fullest extent [9-1 1 ]. Supported in part by an ICSC-World Laboratory Scholarship. 84
One o f the coveted predictions o f string theory is the unification o f all gauge couplings at some string scale M, that is g a ( M ) = g / x / ~ , where ka is the level o f the K a c - M o o d y algebra representing the gauge group Ga [ 12 ]. This relation holds for all hidden a n d / or observable gauge groups. F r o m the low-energy point o f view, this implies that the fine structure constant o~e,the strong coupling constant c~3, and the weak mixing angle sin20w, are all calculable from the knowledge o f M and g. In theory this scenario is realized by extrapolating g ~ ( M ) down to the electroweak scale using the standard renormalization group analysis [ 13 ]. However, in practice any precise calculation o f the low-energy observables requires the inclusion o f the effect o f the infinite n u m b e r o f massive string m o d e s (the so-called *'threshold effects") in the renormalization group equations [ 14,15 ]. Recent attempts to calculate explicitly these effects have led to some very interesting general results. A m o n g these is the connection between the threshold corrections and m o d u l a r invariance [ 16-19 ]. In this p a p e r we first review the status o f the traditional calculation o f the threshold corrections. We then examine the m o d u l a r invariance properties o f the low-energy effective action obtained through the S-matrix approach, especially in models derived in the free
0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
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fermionic formulation of superstring theory [20]. This entails a detailed comparison between the socalled string and supergravity bases, since only in the latter are the modular invariant properties evident, whereas the S-matrix calculations are performed in the former. We then adjoin these results to the K~ihler anomaly method for obtaining the untwisted moduli dependence of the threshold corrections. We finally apply our results to the flipped SU (5) string model and determine the common unification scale for all the gauge couplings in the model.
parameters of the model, especially on the moduli fields T [ 16,18,22 ]. Indeed, general arguments based on the geometry of the compactified space lead to the following generic expression in orbifold models [ 16 ]:
Aa=L-~Ca-~ Y.
In the one-loop approximation, the renormalized gauge coupling at a scale p is given by [ 15 ] 16n"2 g2(/z )
, 16~ 2
M2
=K~-~5--+ b a l n ~ -
+Aa,
(1)
where ba are the fl-function coefficients, and Aa are the threshold corrections to the inverse gauge coupling which arise due to the presence of the infinite set of massive string models. The reference scale M is renormalization scheme dependent, although to our level of approximation it is basically the same in the various schemes of interest and is given by [ 21 ] M = 1.03 × g × 1018G e V .
(2)
In ref. [ 15 ] Aa was calculated explicitly and is given by
Aa= F d2T{/A,(r, f)-ba] , J F
(4)
It has been shown that nontrivial dependence of Aa on the untwisted moduli fields Tz arises only if the theory contains sectors which preserve N = 2 supersymmetry [16,23,22]. This dependence is partially encoded in Aa which is given by [ 16,23 ]
Aa = 2. Threshold corrections. N = 2 method
24 October 1991
Z !mln'[Iq(iTz)14Re2~a I
T1]
,
(5)
where T~ are the untwisted moduli fields for each of the N - - 2 sectors of the orbifold model, and b~ are the corresponding N = 2 fl-function coefficients. (This expression is a one-loop effect and remains uncorrected at higher orders in the loop expansion [22 ]. ) The universal (gauge-independent) piece Y can also depend on untwisted (and twisted) moduli but its effect on observable physics can be masked out by absorbing it into the also unknown string coupling g. The gauge-dependent piece ca does not depend on the untwisted moduli (although it may depend on twisted moduli) and generally receives contributions from N = 2 and N = 1 sectors of the orbifold model. For free fermionic models (which by construction are at a particular point in the moduli space) a stronger statement can be made regarding the constant piece c,: the contribution from N = 1 sectors vanishes identically and thus only the N = 2 sectors contribute [ 21 ].
(3)
T2
where the integral is over the modulus T= rl + iT2 of the worldsheet torus, and it is restricted to the fundamental domain F = {r2 > 0; [Vii < ½; IV[ > l }; .~, is related to the string partition function by insertion of helicity and gauge group generators. Unfortunately, evaluation of Aa using eq. (3) requires the knowledge of the complete string spectrum, an information which is very often lacking or can only be obtained with great difficulty. In fact, a few examples have been worked out only in two classes of models (orbifolds [ 15 ] and free fermionic constructions [21 ] ). A more fruitful approach has been to examine the dependence of Aa on different
3. Threshold corrections, anomaly method
In a large class of models the information which is at our disposal are the low-energy effective lagrangian and the symmetries of the theory. It would be of great value ifAa could be extracted from this limited information. It has been pointed out that using the modular symmetry and the low-energy effective action a substantial portion of Aa can be inferred. An effective (bosonic) low-energy effective action is given by
85
V o l u m e 269, n u m b e r 1,2
S=
f(R d4x
-- ~
PHYSICS LETTERS B
+ -
1
4g~(T)
a
ltv
S= f d 4 x ( f d40~"reV~'
F I'"Fa
iOa Fa f f ~ , v . 4 _ I l~ * --l~v--a -- ~GHO TIOuTj
+ f d40
+ 3292
\ + ½gkl( T)OuCI)kOu CI)7 + V( T, ~)) ) ,
(6)
where 1/g~ is the inverse gauge coupling, 0~ is the vacuum angle, 7"/are scalar (moduli) fields, q~k are the matter fields, G~j is the metric of the space of moduli T~ and gkt(T) is the matter field metric. In terms of a manifestly supersymmetric action for the matter and moduli fields, the effective action can be rewritten as follows:
S= ~ d 4 x ( f d4O Kj( q)t, eVq~, T, "f )
+ f
d40 K2 ( T,
(7)
String theory saddles the above action with the modular symmetry [ 5,6 ]
aT-ib T~ - icT+ d'
ad-bc= 1, a, b, c, deY_.
(8)
However, when one tries to reconstruct the geometry of the moduli space from the S-matrix approach, one is forced to choose a coordinate system in which [24] K = ~ @,q:', ' '~- + Z t,{~+ .... i
(9)
I
The above form o f K i s a reflection of the fact that the vertex operators used to obtain the S-matrix elements are chosen to generate properly normalized fields. This is the so-called "string basis" as opposed to the traditional "supergravity basis" (primed fields refer to the string basis, unprimed ones to the supergravity basis). Interpretation of the t~ as the moduli fields in the string basis allows us to infer their K~ihlet function to be K(t) = - ~'~ In ( 1 - t~~). In the string basis typically one gets [24,6,3 ]
86
ti-+ (higher orders)
-'1"- f d 2 0 Wijkq~,q~iq~ + h '. c . '
+ (higher o r d e r s ) ) .
(10)
Passage from eq. (10) to eq. (7) clearly involves (possibly) nonlinear transformations. Furthermore, if one wishes to avail oneself of the machinery developed for noncompact supergravity models, where K (correct to first order) is chosen to be [25,6] K= -ln[T+
T) + f d20 f ~"Wg IV~,~,
+ f d2OW(T,cl))+h.c.).
24 October 1991
T+
(T+T)PcJScJSt+...]
,
( I1 )
the connection between t and T fields is definitely nonlinear and perhaps even nonholomorphic. Since the modular invariant properties of the theory are evident only in the supergravity basis, it is necessary to determine the precise relation between matter and moduli fields in these bases. By carefully keeping track of the modular transformations of t, T, q)', and 4, this connection has been unraveled in (2, 2) orbifold models [6,26]. We now do this for free fermionic models. From the connection between the fermionic formulation and orbifolds [ 3 ], one infers the existence of three untwisted moduli for the class of models of interest [ 3,21 ], which will be denoted in the supergravity basis by T~, I = 1, 2, 3 (related to the string basis fields ti). Defining F~= In (ic~ T~ + d~), the connection between the two bases becomes 1- T j ~ t , t~- I+T~
exp(F~-F/)
(12)
q), = @~ r ] ( Tj _~ T~jj~"'/2~ exp (iw'/0D, .I
~q)i lq exp( - w)k~D ,
(13)
J
where the arrows indicate the result of a modular transformation of 7"/, and the phases 0~ transform as exp(iO¢)~exp(icb~) e x p [ ½ ( F i - F ~ ) ] . In this same notation, we also have ( T 1 + T ¢ ) ~ ( T z + T z ) × exp[ - (F~+F~) ]. Also, the ~ are inert under mod-
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ular transformations [6]. The charges w} are those of the vertex operator for the scalar component of qO~ under the three U( 1 )'s which make up the N = 2 worldsheet supersymmetry U j( 1 ) current [ 3 ]. These charges are permutations of{ 1, 0, 0} for untwisted fields, and permutations of { ½, ~, 0} for twisted ones
[3]. Using the above connection, the K~hler function in the string basis (eq. ( 9 ) ) can be rewritten in the supergravity basis as follows (modulo a K~ihler transformation )
K=
[ - I n ( T, + T,)
24 October 1991
with a~=-~v-
~ c~"/(2w~- 1)~,
(17)
i
where c~ is the quadratic Casimir of the gauge group G~ and the sum over i runs over all matter fields in the N = 2 sector I which belong to representations of the gauge group with Casimir c~J. Under the modular transformations ~.~ yields an anomaly which can be cancelled by adding a counterterm [ 18 ]
<~Pc'=l ~a f
d2Ofa(WaWa)a+h'c"
(18)
I
+ Z H (T, + 7~,)-""aS, e'aS,~+ .... t
(14)
f~--
2
16n2~ °L~lntlZ(iTl).
(19)
/
Using eqs. (12) and (13), this K~ihler function is easily seen to be covariant under the modular transformations ~t of 7) (see eq. ( 8 ) ) with (15)
I
At first glance, the modular transformation appears to be a subset of the K~ihler transformations, i.e., if the theory is of the K~ihler class, then it would appear to be invariant under modular transformations. However, the K~ihler transformation is one-loop anomalous [17-19]. A one-loop triangle diagram among one Tfield and two vector fields generates the following nonlocal term [ 17-19,28]:
, j-
d40(
1
×D--D21n(T'+7~/)+h'c" ,
(16)
~ The effect of modular transformations on the superpotential is rather nontrivial due to the presence of nonrenormalizable terms, but it can be shown to be such that G = K + I n I WI 2 is modular invariant [27]. For the cubic terms this follows straightforwardly from eq. ( 13 ), i.e., qO,~k~q0
qO]qOk l~I e x p [ - - F ~ ( w ~ + w ~ + w ~ ) ]
= q~,qsfl>kH exp(-F~) . I
Thus
In [ Wl2~In [W( 2- ~
~,
3 a = 3 a ( n l ) "~ 3 a ( c t )
K~,K+ Z (F,+ff,).
1 1
The two pieces ~ and 5°ctthen contribute to the oneloop threshold corrections as follows [ 18 ]:
(F;+b))
.
= ~ . + y,
= - ~/ °~{ln[tq(iTDI4l+ln(T,+~)}.
(20)
Here Aa(nl) includes a portion (Y') of the universal piece Y which may depend on T. The effect of massive string states is encoded in ~w(nu (and consequently in Aa(no ) since its structure and the weights w' are determined by integrating out the massive fields, whereas A~(ct) contains the effect of mixing between massive and massless states induced by modular transformations. Hence, both Aa(n~) and 3a(ct) in tandem incorporate the effects of the tower of massive states. Alternatively put, even though an effective action looses most information about the heavier modes, some of it can be recaptured by demanding modular invariance. Another place where the universal piece Y derives its origin from is the Chern-Simons term in the effective action [17]. Due to the nonholomorphicity of the gauge kinetic function)Cab, supersymmetrization of the theory necessitates the existence of a ChernSimons term, which implements an anomaly cancellation mechanism similar to the ten-dimensional Green-Schwarz mechanism [ 17,19 ] ~2. This procedure is gauge-independent (but it may be N = 2 sector and thus T dependent [22] ) and hence contrib~2 This anomaly should not be confused with the modular transformation anomaly discussed above.
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utes to the universal piece Y by an amount Yc~sThe total contribution to the threshold corrections from all sources is then given by A~ = A ~ l ) + A . m ~+ G + ( Y - Y') + YGs
=3a +c~ + Y+ ro~.
(21)
For ~3 orbifolds A, = 0 [ 16 ]. From the general arguments of ref. [22], the T dependence o f Yos and Y (if any) must cancel each other out, leaving A~ completely independent of any moduli. As we shall see below, in the flipped SU (5) case the presence of YGs ensures that the universal piece Y remains independent of the untwisted moduli. To summarize, after including the Green-Schwarz type of term in the effective action, the anomaly method gives the correct Tdependence of the threshold corrections. Its failure to give ca (which in all known cases is a rather small contribution) is compensated by the sheer simplicity of the method.
4. Application to flipped SU(5) In this section we compute the threshold corrections in the flipped SU ( 5 ) string model [ 2 ] using the formulas for A, obtained in the N = 2 method (eq. ( 5 ) ) and the anomaly method (eq. (20) ). The model is specified by a set of eight basis vectors of boundary conditions for the worldsheet fermions: B = {S, ~, bl, b2, b3, b4, bs, ol}. It is convenient to consider the subset B4={S, ~, l = b l + b 2 + b 3 + ~ , b'4=b.+b4, b'5= b2 + b> ee } since it defines an N = 4 space-time supersymmetric model [22]. The three N = 2 sectors ( o f the underlying 2-2 × 7/2 orbifold ) are obtained by adding to B4 the vectors b~, b2 or b, +b2 respectively [21 ]. All the massless twisted sectors in the model then fall into three groups as follows: l=l: hi,4, hi, 4 +0~, hi. 4 q- 20L ,
•=2: b2,5, b2,5 +20< b2,5 + h i + b 4 + o / ,
1=3:
24 October 1991
The untwisted states (NS sector) also fall into the above groups according to their w} charge, which is a permutation of { 1, 0, 0} for these states. The b J coefficients for the S U ( 5 ) × U ( 1 ) × SO ( 10 ) h × SU (4) h gauge groups of the model can be easily obtained from the massless spectrum of the model [2] using ~b~ 1 ,1 = -c~e + ~_ic] "I. The result is: ½b;'={0, 0, - 1 } , lb',l = {7.5, 7.5, 6.5}, I b ' , ~ = { - 6 , - 6 , - 7 } , and ½b~= {0, 0, - 1}. The a~ coefficients (eq. ( 1 7 ) ) that go into the anomaly determination o f A , (eq. ( 2 0 ) ) can be calculated in a similar way. The twisted states have charges (0, I, ½), (I, 0, I ) , (½, I, 0) with respect to the U I ( 1 ) ( I = I , 2, 3) respectively, hence they effectively split into three groups contributing to a~1=1.2.3. These groups correspond precisely to the N = 2 sectors above. The untwisted states, on the other hand, contribute to all at, and in this they differ from the N = 2 case. However, despite the difference in the role of the untwisted states and the different formulas for ~ba ~ ,1 and a~, one can verify that the result is exactly the same, that is ~b~ J ,1 for I = 1, 2, 3, as expected from consistency arguments [ 18 ]. To make numerical estimates transparent, we will consider the case where the three moduli take the same value T, as is the case in free fermionic models. In this case the sum over I involves only the b~ and we have Z, ~~b a'1 = b [ 2 3 , 1 0 ] , w h e r e b ~ i s t h e N = l f l function coefficient. The threshold corrections then reduce to ~3
zJ.
=
- b a In[ Iq(iT)14 Re T] + G + Y
- -b~A(T)+c,+Y,
(22)
with b5 = - 1, b, =s,Q b l 0 = - 19, and b4 = -- 1. The constant piece c, is not determined by these methods but in free fermionic models it only receives contributions form the N = 2 sectors [21 ]. Furthermore, in symmetric orbifold models the N = 2 contributions to c, are just constants multiplying the logarithm in eq. ( 5 ) [ 16 ], i.e., c, = ~1½b~ .c = cba. In what follows we will assume this relation holds, even though fermionic constructions actually correspond to asymmetric orbifolds. The T-dependent function in A, can
S+b4 +bs, b3, 1~3+2c~, S+b2 +b4 +ol, b2 +b3 +bs -t-og .
88
~3 We have implicitly assumed that the contribution of Y~s to Y renders it independent of T.
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be evaluated from the known series expansion of the q-function, as follows:
- 3 ( T ) = ~1n R e T - l n ( R e T) -2
~ ln[l+exp(-4nnReT) n--I
- 2 e x p ( - 2 n n Re T)cos(2nn Im T ) ] .
(23)
Since T takes values in the fundamental domain = {Re T > 0, I Im T[ < ½, I TI > 1 }, Re T > ½,,/3 and the series of logarithms gives a < 1% correction to the first two terms in A(T) for Re T = ½x/3 and dies out very quickly for increasing values o f Re T. In fact, - A ( T ) ~ -~n Re T - I n ( R e T) is a monotonically increasing function of Re T with m i n i m u m value - A ( T ) = ~3nat Re T = 1, which is where the free fermionic models are constructed ~4. At this point in the moduli space we obtain ~At 5 = d l - A s ~ n ( b l b5 ) + C ( b l -- b5 ) = 22.5 (~3n+c), which compared with the exact calculation fiA1_5=24.13 [21] gives c = 0 . 0 2 5 , a rather small number. Let us recall that the only other existing calculation of threshold corrections [ 15 ] is for a Z3 orbifold model with gauge group S U ( 3 ) × E 6 X E 8 . In this c a s e ~'~3 8 receives contributions only from the ca piece of Aa since the Z3 orbifold does not have any N = 2 sectors. The numerical result is ~ d 3 _ s / ( b 3 - b 8 ) = ( c 3 - c s ) / ( b 3 - b s ) = 0.068, another small number. These two examples lure us into conjecturing that I ( c a - c a , ) / ( b , - b ~ , ) l << 1 in these classes of models. At the free fermionic point we then get A~=ba'~3n (dropping Ywhich can be be absorbed into 1/g2, and neglecting c) and thus eq. (1) can be rewritten as follows: 16n 2 167/"2 MZu ga2(/t) =k~ g2 + b a l n ~ /z,
(24)
where Msu = M e x p ( ~ n ) = 1.69M = 1 . 7 4 × g × 1018 G e V ,
(25)
is the c o m m o n unification scale for all gauge cou~4
This is true for Re T > 1. For 7,£ 3 < R e T < 1, A ( T ) / A ( t = 1 ) can be as small as 0.98, a negligible effect.
24 October 1991
plings in the flipped SU (5) string model. This result should be compared with an exact calculation at the particular point in the moduli space where the fermionic construction is valid, which gives M s u = 1 . 7 6 × g × 1018 GeV [21 ]. Thus, if our above conjecture is indeed true in general, this would open the way to simplified evaluations of threshold corrections in this class of models. From the expression for / I a (eq. ( 2 2 ) ) and our above remarks regarding A(T), it is clear that any point in the moduli space away from the fermionic formulation (i.e., T ¢ 1 ), leads to a larger value o f Aa and correspondingly Msv ( T:~ 1 ) > Msv ( T = 1 ) ~ 1.7 × g × 10 ~8 GeV. This implies among other things that for a model away from the free fermionic point, this disparity between the S U ( 2 ) and U( 1 ) gauge couplings at the scale MGux (defined by the partial unification of S U ( 3 ) and S U ( 2 ) ), which is o f great relevance to the prediction of sin20w, will be larger than that for the corresponding model at the free fermionic point. The latter is already larger than the one obtained in the approximation of neglecting all threshold effects [ 21 ].
5. Conclusions
One of the diacritical properties of the flipped SU (5) string model is the presence of an anomalous UA ( 1 ) factor. At the one-loop level, this anomaly may lead to a nonzero vacuum energy and the destabilization of the vacuum. This anomaly is cancelled by "shifting" the vacuum by giving vacuum expectation values to some of the singlet scalar fields. One may be concerned about the effect of this shifting on the evaluation o f Aa. However, the correction induced by the vacuum shift arises only at a scale (qb) which is typically an order of magnitude smaller that Msu [2 ]. In a related note, it is interesting to observe that the shifting of the vacuum, necessary to cancel the UA( 1 ) anomaly and which breaks many of the U ( 1 ) symmetries [2 ], also breaks the modular symmetry. A complete analysis of modular symmetry breaking is beyond the scope of this paper and will be presented elsewhere [ 27 ]. However, it is clear that since shifted fields carry nonzero charges under the modular symmetry PSL(2, Z) 3, part or all of this symmetry is broken. The breaking of modular symmetry 89
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at a high scale m a y h a v e great i n f l u e n c e in o u r u n d e r s t a n d i n g o f the low-energy e f f e c t i v e a c t i o n [ 11 ]. In c o n c l u s i o n , the scale Msu o b t a i n e d in the prev i o u s section is w h e r e gauge couplings for all the gauge groups c o n v e r g e to g. It should be n o t e d that this scale is c o n s i d e r a b l y larger t h a n MGUT ( w h e r e the extrapo l a t e d low-energy v a l u e s o f the S U ( 3 ) a n d S U ( 2 ) gauge c o u p l i n g s m e e t ) . Thus, to o b t a i n a precise pred i c t i o n for sin20w f u r t h e r e x t r a p o l a t i o n to m u c h higher energies will be necessary. A n o t h e r place where a h i g h e r v a l u e o f the u n i f i c a t i o n scale has n o n t r i v i a l c o n s e q u e n c e s is in the d e t e r m i n a t i o n o f the scales A, o f n o n a b e l i a n h i d d e n sector c o n d e n s a t i o n (e.g., S O ( 1 0 ) a n d S U ( 4 ) in the f l i p p e d S U ( 5 ) string m o d e l ). A reliable d e t e r m i n a t i o n of Ah w o u l d i m p a c t on o u r u n d e r s t a n d i n g o f t h e c o s m o l o g i c a l role p l a y e d by c r y p t o n s ( h a d r o n s o f the n o n a b e l i a n h i d d e n sector g r o u p s [29] ), a n d m a y e x p l a i n the b r e a k i n g o f s u p e r s y m m e t r y [ 9 ]. H e n c e the study o f the i n t e r p l a y o f m o d u l a r s y m m e t r y c o n t i n u e s to be r e m u n e r a t i v e .
Acknowledgement We w o u l d like to t h a n k I. A n t o n i a d i s , L. D i x o n , J. Ellis a n d J. L o u i s for useful discussions. T h i s w o r k has b e e n s u p p o r t e d in part by D O E g r a n t D E - F G 0 5 91-ER-40633.
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[ 8 ] P. Candelas, X. De La Ossa, P.S. Green and L. Parks, Nucl. Phys. B 355 (1991) 455; Phys. Lett. B 258 (1991) 118. [ 9] I. Antoniadis, J. Ellis, A.B. Lahanas and D.V. Nanopoulos, Phys. Lett. B241 (1990) 24; L. Dixon, in: Proc. Rice Meeting. eds. B. Bonner and H. Miettinen (World Scientific, Singapore, 1990) p. 811; C.P. Burguess and F. Quevedo, Phys. Rev. Len. 64 (1990) 2611; S. Ferrara, N. Magnoli, T.R. Taylor and G. Veneziano, Phys. Lett. B 245 (1990) 409; H.P. Nilles and M. Olechowsky, Phys. Lett. B 248 (1990) 268; P. Binetruy and M.K. Gaillard, Phys. Len. B 253 (1991)
119; D. Liist and T.R. Taylor, Phys. Lett. B 253 ( 1991 ) 335; B. de Carlos, J.A. Casas and C. Mufioz, Phys. Lett. B 263 (1991) 248. [ 10 ] M. Cvetic et al., Nucl Phys. B 361 ( 1991 ) 194. [ 11 ] L. Ib~tfiez and D. Li.ist, CERN preprint CERN-TH.6120/ 91. [ 12] P. Ginsparg, Phys. Left. B 197 (1987) 139. [ 13 ] See e.g.J. Ellis, J. Hagelin, S. Kelley and D.V. Nanopoulos, Nucl. Phys. B 311 (1988/89) 1. [ 14] J. Minahan, Nucl. Phys. B 298 (1988) 36. [ 15 ] V. Kaplunovsky, Nucl. Phys. B 307 (1988) 145. [ 16] L. Dixon, V. Kaplunovsky and J. Louis, Nucl. Phys. B 355 (1991) 649. [17] J. Derendinger, S. Ferrara, C. Kounnas and F. Zwirner, CERN preprint CERN-TH.6004/91. [ 18] J. Louis, SLAC preprint SLAC-PUB-5527 ( 1991 ). [ 19 ] G. Lopes Cardoso and B. Ovrut, University of Pennsylvania preprint UPR-0464T ( 1991 ). [20] I. Antoniadis, C. Bachas and C. Kounnas, Nucl. Phys. B 289 (1987) 87; I. Antoniadis and C. Bachas, Nucl. Phys. B 298 (1988) 586; H. Kawai, D.C. Lewellen and S.H.-H. Tye, Phys. Rev. Lett. 57 (1986) 1832; Phys. Rev. D 34 (1986) 3794; Nucl. Phys. B288 (1987) 1; R. Bluhm, L. Dolan and P. Goddard, Nucl. Phys. B 309 (1988) 330; H. Dreiner, J.L. Lopez, D.V. Nanopoulos and D. Reiss, Nucl. Phys. B 320 (1989) 401. [ 21 ] I. Antoniadis, J. Ellis, R. Lacaze and D.V. Nanopoulos, Phys. Lett. B 268 (1991) 188. [22] I. Antoniadis, K.S. Narain and T. Taylor, Phys. Lett. B 267 (1991) 37. [23] I. Antoniadis, Phys. Len. B 246 (1990) 377. [24] L. Dixon, V. Kaphmovsky and J. Louis, Nucl. Phys. B 329 (1990) 27. [25] For a review see A. Lahanas and D.V. Nanopoulos, Phys. Rep. 145 (1987) 1. [26] L. Ib~ifiez, W. Lerche, D. Liist and S. Theisen, Nucl. Phys. B352 (1991)435. [27] S. Kalara, J. L. Lopez and D.V. Nanopoulos, in preparation. [28] S.J. Gates, M. Grisaru, M. Ro6ek and W. Siegel, Superspace (Benjamin-Cummings, New York, 1983 ). [29] J. Ellis, J.L. Lopez and D.V. Nanopoulos, Phys. Lett. B 247 (1990) 257.
PHYSICS LETTERS B
Threshold corrections and modular invariance in free fermionic strings S. Kalara
a,b J.L.
L o p e z a,b,I a n d D . V . N a n o p o u l o s a,b,c
a Center for Theoretical Physics, Department of Physics, Texas A&M University, College Station, TX 77843-4242, USA b Astroparticle Physics Group, Houston Advanced Research Center (HARC), The Woodlands, TX 77381, USA c Theoretical Physics Division, CERN, CH- 1211 Geneva 23, Switzerland
Received 22 July 1991
We examine the modular invariant properties of the low-energy effective action of free fermionic models. This entails a detailed comparison between the so-called string and supergravity bases, since only in the latter are the modular invariant properties evident, whereas the S-matrix calculations are performed in the former. We then adjoin these results to the K~hler anomaly method for obtaining the untwisted moduli dependence of the string threshold corrections to the gauge couplings. Finally we apply our results to the flipped SU ( 5 ) string model and determine the common unification scale for all the gauge couplings in the model.
1. Introduction String derived models o f particle interactions are endowed with an extraordinarily rich s y m m e t r y structure [ 1 ]. C o m p a r e d to o r d i n a r y grand unified theories, the string derived effective low-energy action has far m o r e predictive power and leads to penetrating insight into the connection between seemingly unrelated sectors of the theory [2,3]. Although the task o f constructing the low-energy effective string action which manifests all the symmetries o f the original string theory is by no means straightforward, over the past few years great strides have been m a d e in this direction [ 4 ]. One o f the genuinely stringy symmetries which has attracted a great deal o f attention is the so-called " m o d u l a r s y m m e try", a generalization o f the basic duality s y m m e t r y o f string models [ 5,6]. In a large class o f models, such as orbifold [6,7] and C a l a b i - Y a u compactifications [ 8 ], this is an infinite-dimensional discrete symmetry. The existence o f such s y m m e t r y has been known for quite some time [5]. However, its p h e n o m e n o logical implications have not been explored to their fullest extent [9-1 1 ]. Supported in part by an ICSC-World Laboratory Scholarship. 84
One o f the coveted predictions o f string theory is the unification o f all gauge couplings at some string scale M, that is g a ( M ) = g / x / ~ , where ka is the level o f the K a c - M o o d y algebra representing the gauge group Ga [ 12 ]. This relation holds for all hidden a n d / or observable gauge groups. F r o m the low-energy point o f view, this implies that the fine structure constant o~e,the strong coupling constant c~3, and the weak mixing angle sin20w, are all calculable from the knowledge o f M and g. In theory this scenario is realized by extrapolating g ~ ( M ) down to the electroweak scale using the standard renormalization group analysis [ 13 ]. However, in practice any precise calculation o f the low-energy observables requires the inclusion o f the effect o f the infinite n u m b e r o f massive string m o d e s (the so-called *'threshold effects") in the renormalization group equations [ 14,15 ]. Recent attempts to calculate explicitly these effects have led to some very interesting general results. A m o n g these is the connection between the threshold corrections and m o d u l a r invariance [ 16-19 ]. In this p a p e r we first review the status o f the traditional calculation o f the threshold corrections. We then examine the m o d u l a r invariance properties o f the low-energy effective action obtained through the S-matrix approach, especially in models derived in the free
0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
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fermionic formulation of superstring theory [20]. This entails a detailed comparison between the socalled string and supergravity bases, since only in the latter are the modular invariant properties evident, whereas the S-matrix calculations are performed in the former. We then adjoin these results to the K~ihler anomaly method for obtaining the untwisted moduli dependence of the threshold corrections. We finally apply our results to the flipped SU (5) string model and determine the common unification scale for all the gauge couplings in the model.
parameters of the model, especially on the moduli fields T [ 16,18,22 ]. Indeed, general arguments based on the geometry of the compactified space lead to the following generic expression in orbifold models [ 16 ]:
Aa=L-~Ca-~ Y.
In the one-loop approximation, the renormalized gauge coupling at a scale p is given by [ 15 ] 16n"2 g2(/z )
, 16~ 2
M2
=K~-~5--+ b a l n ~ -
+Aa,
(1)
where ba are the fl-function coefficients, and Aa are the threshold corrections to the inverse gauge coupling which arise due to the presence of the infinite set of massive string models. The reference scale M is renormalization scheme dependent, although to our level of approximation it is basically the same in the various schemes of interest and is given by [ 21 ] M = 1.03 × g × 1018G e V .
(2)
In ref. [ 15 ] Aa was calculated explicitly and is given by
Aa= F d2T{/A,(r, f)-ba] , J F
(4)
It has been shown that nontrivial dependence of Aa on the untwisted moduli fields Tz arises only if the theory contains sectors which preserve N = 2 supersymmetry [16,23,22]. This dependence is partially encoded in Aa which is given by [ 16,23 ]
Aa = 2. Threshold corrections. N = 2 method
24 October 1991
Z !mln'[Iq(iTz)14Re2~a I
T1]
,
(5)
where T~ are the untwisted moduli fields for each of the N - - 2 sectors of the orbifold model, and b~ are the corresponding N = 2 fl-function coefficients. (This expression is a one-loop effect and remains uncorrected at higher orders in the loop expansion [22 ]. ) The universal (gauge-independent) piece Y can also depend on untwisted (and twisted) moduli but its effect on observable physics can be masked out by absorbing it into the also unknown string coupling g. The gauge-dependent piece ca does not depend on the untwisted moduli (although it may depend on twisted moduli) and generally receives contributions from N = 2 and N = 1 sectors of the orbifold model. For free fermionic models (which by construction are at a particular point in the moduli space) a stronger statement can be made regarding the constant piece c,: the contribution from N = 1 sectors vanishes identically and thus only the N = 2 sectors contribute [ 21 ].
(3)
T2
where the integral is over the modulus T= rl + iT2 of the worldsheet torus, and it is restricted to the fundamental domain F = {r2 > 0; [Vii < ½; IV[ > l }; .~, is related to the string partition function by insertion of helicity and gauge group generators. Unfortunately, evaluation of Aa using eq. (3) requires the knowledge of the complete string spectrum, an information which is very often lacking or can only be obtained with great difficulty. In fact, a few examples have been worked out only in two classes of models (orbifolds [ 15 ] and free fermionic constructions [21 ] ). A more fruitful approach has been to examine the dependence of Aa on different
3. Threshold corrections, anomaly method
In a large class of models the information which is at our disposal are the low-energy effective lagrangian and the symmetries of the theory. It would be of great value ifAa could be extracted from this limited information. It has been pointed out that using the modular symmetry and the low-energy effective action a substantial portion of Aa can be inferred. An effective (bosonic) low-energy effective action is given by
85
V o l u m e 269, n u m b e r 1,2
S=
f(R d4x
-- ~
PHYSICS LETTERS B
+ -
1
4g~(T)
a
ltv
S= f d 4 x ( f d40~"reV~'
F I'"Fa
iOa Fa f f ~ , v . 4 _ I l~ * --l~v--a -- ~GHO TIOuTj
+ f d40
+ 3292
\ + ½gkl( T)OuCI)kOu CI)7 + V( T, ~)) ) ,
(6)
where 1/g~ is the inverse gauge coupling, 0~ is the vacuum angle, 7"/are scalar (moduli) fields, q~k are the matter fields, G~j is the metric of the space of moduli T~ and gkt(T) is the matter field metric. In terms of a manifestly supersymmetric action for the matter and moduli fields, the effective action can be rewritten as follows:
S= ~ d 4 x ( f d4O Kj( q)t, eVq~, T, "f )
+ f
d40 K2 ( T,
(7)
String theory saddles the above action with the modular symmetry [ 5,6 ]
aT-ib T~ - icT+ d'
ad-bc= 1, a, b, c, deY_.
(8)
However, when one tries to reconstruct the geometry of the moduli space from the S-matrix approach, one is forced to choose a coordinate system in which [24] K = ~ @,q:', ' '~- + Z t,{~+ .... i
(9)
I
The above form o f K i s a reflection of the fact that the vertex operators used to obtain the S-matrix elements are chosen to generate properly normalized fields. This is the so-called "string basis" as opposed to the traditional "supergravity basis" (primed fields refer to the string basis, unprimed ones to the supergravity basis). Interpretation of the t~ as the moduli fields in the string basis allows us to infer their K~ihlet function to be K(t) = - ~'~ In ( 1 - t~~). In the string basis typically one gets [24,6,3 ]
86
ti-+ (higher orders)
-'1"- f d 2 0 Wijkq~,q~iq~ + h '. c . '
+ (higher o r d e r s ) ) .
(10)
Passage from eq. (10) to eq. (7) clearly involves (possibly) nonlinear transformations. Furthermore, if one wishes to avail oneself of the machinery developed for noncompact supergravity models, where K (correct to first order) is chosen to be [25,6] K= -ln[T+
T) + f d20 f ~"Wg IV~,~,
+ f d2OW(T,cl))+h.c.).
24 October 1991
T+
(T+T)PcJScJSt+...]
,
( I1 )
the connection between t and T fields is definitely nonlinear and perhaps even nonholomorphic. Since the modular invariant properties of the theory are evident only in the supergravity basis, it is necessary to determine the precise relation between matter and moduli fields in these bases. By carefully keeping track of the modular transformations of t, T, q)', and 4, this connection has been unraveled in (2, 2) orbifold models [6,26]. We now do this for free fermionic models. From the connection between the fermionic formulation and orbifolds [ 3 ], one infers the existence of three untwisted moduli for the class of models of interest [ 3,21 ], which will be denoted in the supergravity basis by T~, I = 1, 2, 3 (related to the string basis fields ti). Defining F~= In (ic~ T~ + d~), the connection between the two bases becomes 1- T j ~ t , t~- I+T~
exp(F~-F/)
(12)
q), = @~ r ] ( Tj _~ T~jj~"'/2~ exp (iw'/0D, .I
~q)i lq exp( - w)k~D ,
(13)
J
where the arrows indicate the result of a modular transformation of 7"/, and the phases 0~ transform as exp(iO¢)~exp(icb~) e x p [ ½ ( F i - F ~ ) ] . In this same notation, we also have ( T 1 + T ¢ ) ~ ( T z + T z ) × exp[ - (F~+F~) ]. Also, the ~ are inert under mod-
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PHYSICS LETTERS B
ular transformations [6]. The charges w} are those of the vertex operator for the scalar component of qO~ under the three U( 1 )'s which make up the N = 2 worldsheet supersymmetry U j( 1 ) current [ 3 ]. These charges are permutations of{ 1, 0, 0} for untwisted fields, and permutations of { ½, ~, 0} for twisted ones
[3]. Using the above connection, the K~hler function in the string basis (eq. ( 9 ) ) can be rewritten in the supergravity basis as follows (modulo a K~ihler transformation )
K=
[ - I n ( T, + T,)
24 October 1991
with a~=-~v-
~ c~"/(2w~- 1)~,
(17)
i
where c~ is the quadratic Casimir of the gauge group G~ and the sum over i runs over all matter fields in the N = 2 sector I which belong to representations of the gauge group with Casimir c~J. Under the modular transformations ~.~ yields an anomaly which can be cancelled by adding a counterterm [ 18 ]
<~Pc'=l ~a f
d2Ofa(WaWa)a+h'c"
(18)
I
+ Z H (T, + 7~,)-""aS, e'aS,~+ .... t
(14)
f~--
2
16n2~ °L~lntlZ(iTl).
(19)
/
Using eqs. (12) and (13), this K~ihler function is easily seen to be covariant under the modular transformations ~t of 7) (see eq. ( 8 ) ) with (15)
I
At first glance, the modular transformation appears to be a subset of the K~ihler transformations, i.e., if the theory is of the K~ihler class, then it would appear to be invariant under modular transformations. However, the K~ihler transformation is one-loop anomalous [17-19]. A one-loop triangle diagram among one Tfield and two vector fields generates the following nonlocal term [ 17-19,28]:
, j-
d40(
1
×D--D21n(T'+7~/)+h'c" ,
(16)
~ The effect of modular transformations on the superpotential is rather nontrivial due to the presence of nonrenormalizable terms, but it can be shown to be such that G = K + I n I WI 2 is modular invariant [27]. For the cubic terms this follows straightforwardly from eq. ( 13 ), i.e., qO,~k~q0
qO]qOk l~I e x p [ - - F ~ ( w ~ + w ~ + w ~ ) ]
= q~,qsfl>kH exp(-F~) . I
Thus
In [ Wl2~In [W( 2- ~
~,
3 a = 3 a ( n l ) "~ 3 a ( c t )
K~,K+ Z (F,+ff,).
1 1
The two pieces ~ and 5°ctthen contribute to the oneloop threshold corrections as follows [ 18 ]:
(F;+b))
.
= ~ . + y,
= - ~/ °~{ln[tq(iTDI4l+ln(T,+~)}.
(20)
Here Aa(nl) includes a portion (Y') of the universal piece Y which may depend on T. The effect of massive string states is encoded in ~w(nu (and consequently in Aa(no ) since its structure and the weights w' are determined by integrating out the massive fields, whereas A~(ct) contains the effect of mixing between massive and massless states induced by modular transformations. Hence, both Aa(n~) and 3a(ct) in tandem incorporate the effects of the tower of massive states. Alternatively put, even though an effective action looses most information about the heavier modes, some of it can be recaptured by demanding modular invariance. Another place where the universal piece Y derives its origin from is the Chern-Simons term in the effective action [17]. Due to the nonholomorphicity of the gauge kinetic function)Cab, supersymmetrization of the theory necessitates the existence of a ChernSimons term, which implements an anomaly cancellation mechanism similar to the ten-dimensional Green-Schwarz mechanism [ 17,19 ] ~2. This procedure is gauge-independent (but it may be N = 2 sector and thus T dependent [22] ) and hence contrib~2 This anomaly should not be confused with the modular transformation anomaly discussed above.
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utes to the universal piece Y by an amount Yc~sThe total contribution to the threshold corrections from all sources is then given by A~ = A ~ l ) + A . m ~+ G + ( Y - Y') + YGs
=3a +c~ + Y+ ro~.
(21)
For ~3 orbifolds A, = 0 [ 16 ]. From the general arguments of ref. [22], the T dependence o f Yos and Y (if any) must cancel each other out, leaving A~ completely independent of any moduli. As we shall see below, in the flipped SU (5) case the presence of YGs ensures that the universal piece Y remains independent of the untwisted moduli. To summarize, after including the Green-Schwarz type of term in the effective action, the anomaly method gives the correct Tdependence of the threshold corrections. Its failure to give ca (which in all known cases is a rather small contribution) is compensated by the sheer simplicity of the method.
4. Application to flipped SU(5) In this section we compute the threshold corrections in the flipped SU ( 5 ) string model [ 2 ] using the formulas for A, obtained in the N = 2 method (eq. ( 5 ) ) and the anomaly method (eq. (20) ). The model is specified by a set of eight basis vectors of boundary conditions for the worldsheet fermions: B = {S, ~, bl, b2, b3, b4, bs, ol}. It is convenient to consider the subset B4={S, ~, l = b l + b 2 + b 3 + ~ , b'4=b.+b4, b'5= b2 + b> ee } since it defines an N = 4 space-time supersymmetric model [22]. The three N = 2 sectors ( o f the underlying 2-2 × 7/2 orbifold ) are obtained by adding to B4 the vectors b~, b2 or b, +b2 respectively [21 ]. All the massless twisted sectors in the model then fall into three groups as follows: l=l: hi,4, hi, 4 +0~, hi. 4 q- 20L ,
•=2: b2,5, b2,5 +20< b2,5 + h i + b 4 + o / ,
1=3:
24 October 1991
The untwisted states (NS sector) also fall into the above groups according to their w} charge, which is a permutation of { 1, 0, 0} for these states. The b J coefficients for the S U ( 5 ) × U ( 1 ) × SO ( 10 ) h × SU (4) h gauge groups of the model can be easily obtained from the massless spectrum of the model [2] using ~b~ 1 ,1 = -c~e + ~_ic] "I. The result is: ½b;'={0, 0, - 1 } , lb',l = {7.5, 7.5, 6.5}, I b ' , ~ = { - 6 , - 6 , - 7 } , and ½b~= {0, 0, - 1}. The a~ coefficients (eq. ( 1 7 ) ) that go into the anomaly determination o f A , (eq. ( 2 0 ) ) can be calculated in a similar way. The twisted states have charges (0, I, ½), (I, 0, I ) , (½, I, 0) with respect to the U I ( 1 ) ( I = I , 2, 3) respectively, hence they effectively split into three groups contributing to a~1=1.2.3. These groups correspond precisely to the N = 2 sectors above. The untwisted states, on the other hand, contribute to all at, and in this they differ from the N = 2 case. However, despite the difference in the role of the untwisted states and the different formulas for ~ba ~ ,1 and a~, one can verify that the result is exactly the same, that is ~b~ J ,1 for I = 1, 2, 3, as expected from consistency arguments [ 18 ]. To make numerical estimates transparent, we will consider the case where the three moduli take the same value T, as is the case in free fermionic models. In this case the sum over I involves only the b~ and we have Z, ~~b a'1 = b [ 2 3 , 1 0 ] , w h e r e b ~ i s t h e N = l f l function coefficient. The threshold corrections then reduce to ~3
zJ.
=
- b a In[ Iq(iT)14 Re T] + G + Y
- -b~A(T)+c,+Y,
(22)
with b5 = - 1, b, =s,Q b l 0 = - 19, and b4 = -- 1. The constant piece c, is not determined by these methods but in free fermionic models it only receives contributions form the N = 2 sectors [21 ]. Furthermore, in symmetric orbifold models the N = 2 contributions to c, are just constants multiplying the logarithm in eq. ( 5 ) [ 16 ], i.e., c, = ~1½b~ .c = cba. In what follows we will assume this relation holds, even though fermionic constructions actually correspond to asymmetric orbifolds. The T-dependent function in A, can
S+b4 +bs, b3, 1~3+2c~, S+b2 +b4 +ol, b2 +b3 +bs -t-og .
88
~3 We have implicitly assumed that the contribution of Y~s to Y renders it independent of T.
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be evaluated from the known series expansion of the q-function, as follows:
- 3 ( T ) = ~1n R e T - l n ( R e T) -2
~ ln[l+exp(-4nnReT) n--I
- 2 e x p ( - 2 n n Re T)cos(2nn Im T ) ] .
(23)
Since T takes values in the fundamental domain = {Re T > 0, I Im T[ < ½, I TI > 1 }, Re T > ½,,/3 and the series of logarithms gives a < 1% correction to the first two terms in A(T) for Re T = ½x/3 and dies out very quickly for increasing values o f Re T. In fact, - A ( T ) ~ -~n Re T - I n ( R e T) is a monotonically increasing function of Re T with m i n i m u m value - A ( T ) = ~3nat Re T = 1, which is where the free fermionic models are constructed ~4. At this point in the moduli space we obtain ~At 5 = d l - A s ~ n ( b l b5 ) + C ( b l -- b5 ) = 22.5 (~3n+c), which compared with the exact calculation fiA1_5=24.13 [21] gives c = 0 . 0 2 5 , a rather small number. Let us recall that the only other existing calculation of threshold corrections [ 15 ] is for a Z3 orbifold model with gauge group S U ( 3 ) × E 6 X E 8 . In this c a s e ~'~3 8 receives contributions only from the ca piece of Aa since the Z3 orbifold does not have any N = 2 sectors. The numerical result is ~ d 3 _ s / ( b 3 - b 8 ) = ( c 3 - c s ) / ( b 3 - b s ) = 0.068, another small number. These two examples lure us into conjecturing that I ( c a - c a , ) / ( b , - b ~ , ) l << 1 in these classes of models. At the free fermionic point we then get A~=ba'~3n (dropping Ywhich can be be absorbed into 1/g2, and neglecting c) and thus eq. (1) can be rewritten as follows: 16n 2 167/"2 MZu ga2(/t) =k~ g2 + b a l n ~ /z,
(24)
where Msu = M e x p ( ~ n ) = 1.69M = 1 . 7 4 × g × 1018 G e V ,
(25)
is the c o m m o n unification scale for all gauge cou~4
This is true for Re T > 1. For 7,£ 3 < R e T < 1, A ( T ) / A ( t = 1 ) can be as small as 0.98, a negligible effect.
24 October 1991
plings in the flipped SU (5) string model. This result should be compared with an exact calculation at the particular point in the moduli space where the fermionic construction is valid, which gives M s u = 1 . 7 6 × g × 1018 GeV [21 ]. Thus, if our above conjecture is indeed true in general, this would open the way to simplified evaluations of threshold corrections in this class of models. From the expression for / I a (eq. ( 2 2 ) ) and our above remarks regarding A(T), it is clear that any point in the moduli space away from the fermionic formulation (i.e., T ¢ 1 ), leads to a larger value o f Aa and correspondingly Msv ( T:~ 1 ) > Msv ( T = 1 ) ~ 1.7 × g × 10 ~8 GeV. This implies among other things that for a model away from the free fermionic point, this disparity between the S U ( 2 ) and U( 1 ) gauge couplings at the scale MGux (defined by the partial unification of S U ( 3 ) and S U ( 2 ) ), which is o f great relevance to the prediction of sin20w, will be larger than that for the corresponding model at the free fermionic point. The latter is already larger than the one obtained in the approximation of neglecting all threshold effects [ 21 ].
5. Conclusions
One of the diacritical properties of the flipped SU (5) string model is the presence of an anomalous UA ( 1 ) factor. At the one-loop level, this anomaly may lead to a nonzero vacuum energy and the destabilization of the vacuum. This anomaly is cancelled by "shifting" the vacuum by giving vacuum expectation values to some of the singlet scalar fields. One may be concerned about the effect of this shifting on the evaluation o f Aa. However, the correction induced by the vacuum shift arises only at a scale (qb) which is typically an order of magnitude smaller that Msu [2 ]. In a related note, it is interesting to observe that the shifting of the vacuum, necessary to cancel the UA( 1 ) anomaly and which breaks many of the U ( 1 ) symmetries [2 ], also breaks the modular symmetry. A complete analysis of modular symmetry breaking is beyond the scope of this paper and will be presented elsewhere [ 27 ]. However, it is clear that since shifted fields carry nonzero charges under the modular symmetry PSL(2, Z) 3, part or all of this symmetry is broken. The breaking of modular symmetry 89
Volume 269, number 1,2
PHYSICS LETTERS B
at a high scale m a y h a v e great i n f l u e n c e in o u r u n d e r s t a n d i n g o f the low-energy e f f e c t i v e a c t i o n [ 11 ]. In c o n c l u s i o n , the scale Msu o b t a i n e d in the prev i o u s section is w h e r e gauge couplings for all the gauge groups c o n v e r g e to g. It should be n o t e d that this scale is c o n s i d e r a b l y larger t h a n MGUT ( w h e r e the extrapo l a t e d low-energy v a l u e s o f the S U ( 3 ) a n d S U ( 2 ) gauge c o u p l i n g s m e e t ) . Thus, to o b t a i n a precise pred i c t i o n for sin20w f u r t h e r e x t r a p o l a t i o n to m u c h higher energies will be necessary. A n o t h e r place where a h i g h e r v a l u e o f the u n i f i c a t i o n scale has n o n t r i v i a l c o n s e q u e n c e s is in the d e t e r m i n a t i o n o f the scales A, o f n o n a b e l i a n h i d d e n sector c o n d e n s a t i o n (e.g., S O ( 1 0 ) a n d S U ( 4 ) in the f l i p p e d S U ( 5 ) string m o d e l ). A reliable d e t e r m i n a t i o n of Ah w o u l d i m p a c t on o u r u n d e r s t a n d i n g o f t h e c o s m o l o g i c a l role p l a y e d by c r y p t o n s ( h a d r o n s o f the n o n a b e l i a n h i d d e n sector g r o u p s [29] ), a n d m a y e x p l a i n the b r e a k i n g o f s u p e r s y m m e t r y [ 9 ]. H e n c e the study o f the i n t e r p l a y o f m o d u l a r s y m m e t r y c o n t i n u e s to be r e m u n e r a t i v e .
Acknowledgement We w o u l d like to t h a n k I. A n t o n i a d i s , L. D i x o n , J. Ellis a n d J. L o u i s for useful discussions. T h i s w o r k has b e e n s u p p o r t e d in part by D O E g r a n t D E - F G 0 5 91-ER-40633.
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