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February 1,2008HUTP-95/A043hep-ph/9511307The Derivation of M GUT =1016GeV from A String Model Rulin Xiu ∗Lyman Laboratory of Physics Harvard University Cambridge,MA 02138Abstract We propose that gaugino condensation in the hidden sector may dynamically induce intermediate gauge symmetry breaking and spec-ify the compactification scale.We also show that this scheme makes possible affine level one grand unified string models with gauge break-ing scale M GUT ∼1016GeV.T -duality plays a crucial role in making this scheme possible,even though it is spontaneously broken.We also
discuss the generation of the large mass hierarchy between the string scale and the electroweak scale and the solution of the dilaton runaway problem in this scheme.
字符串常量的格式Precision electroweak measurements indicate that supersymmetric grand
unified models lead to a good agreement with a single unification scale of
M GUT=1016±0.3GeV[1]and a bestfit for supersymmetry breaking scale
M SUSY≃1TeV[1,2,3].In this paper,we explore the possibility of deriving these features from string models.
In string theory,all gauge interactions and gravity are unified[4,5].This
unification happens at around the string scale,which is related to the gauge
coupling constant at the string unification scale through the relation:M2S= 18πG
=1018GeV is the reduced Planck
mass,where G N is Newton’s constant.This indicates that the string gauge coupling constants unify at1017GeV–1018GeV for affine level one string models without string threshold corrections.It is not po
ssible to obtain a scale of1016GeV without invoking string threshold correction andfine-tuning the gauge coupling constant and the string threshold correction[8].
Given this situation,it is natural to consider string models with grand unified groups which break below the string scale.However,it is not known how to dynamically induce intermediate scale gauge symmetry breaking in string models.In fact,it has been shown that adjoint Higgs representations of low-energy gauge groups cannot exist in affine level one string models,the simplest and most well-studied class of models.This means that the usual breaking mechanisms from the grand unified group SU(5)or SO(10)to the standard model gauge group does not work in these models.Adjoint Higgs representations can exist in more complicated string models,for example higher affine level string models.However,it is much more difficult to build “realistic”models of this type.In the last few years,much effort has been devoted to this problem,with no convincing sucess so far.
In this letter,we propose that intermediate gauge symmetry breaking at 1016GeV may be induced by gaugino condensation in the hidden sector.In particular,we propose that some charged backgroundfield VEVs in the ob-servable E8sector can be turned on when gaugino condensation happens in the hidden sector,and that this can lead to the intermediate gauge symmetry
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breaking.Wefind that this stringy gauge symmetry breaking scheme makes affine level one grand unified string models possible.The resulting super-symmetry breaking scheme has some advantages over traditional approaches [9]and may even lead to determining the dilaton at a realistic value from gaugino condensation dynamics.
In the following,we will review some important features of gaugino con-densation in the heterotic string theory before describing our scheme.In the end,we will discuss about generating the large mass hierarchy between M GUT=1016GeV and M SUSY=1T eV in this kind of schemes.
Heterotic string theory has a hidden sector with gauge group E8.This E8(or its unbroken subgroup)is asymptotically free and will become strong at a“confining”scaleΛ.One expected result of this confinement is that the gaugino bilinear will get a VEV.From the10-dimensional effective lagrangian of the heterotic string,one can see that the gaugino bilinear couples linearly to the antisymmetric tensorfield strength,defined as[10]
H=dB−ω3Y+ω3L,(1)
where B is the2-index antisymmetric tensorfield,andω3Y andω3L are the Yang-Mills and Lorentz Chern-Simons symbols:
ω3Y=1
3
AAA),(2)
ω3L=1
3
ωωω).(3)
It was observed in[9]that the terms in the lagrangian that depend on H ijk and the gauge-invariant gaugino bilinear¯λΓijkλcombine into a perfect square:
δL∼(H ijk−√
Unfortunately,in this case c is integer quantized in planck units[11],so gaugino condensation is forced to take place near the Planck scale.It is proposed in refs.[12,13]that the necessary constant term comes from matter field VEVs.In this case,the H is not quantized.
In this letter,we propose a modified version of the mechanism proposed in refs.[9,12,13].In particular,we assume that the induced H VEV arises from the Chern–Simon term rather than from the antisymmetric tensor strength or matterfields.Furthermore,to satisfy the equation of motion,we require that the induced Chern–Simon term satisfies the condition:dω=F ij F ij=0, i.e.F ij=0.It appears that in all known string models
1
,(5)
m
where n and m are integers.However,we will discuss a possible mechanism to generating a small value for H below.
We now explain how the above proposal may make affine level one grand unified models possible.Since the induced charged background VEV’s do not correspond to any four-dimensional physical modes,they can be in the adjoint representation of the observable gauge interaction E6.They can break the grand unified string models to low-energy standard-like models through the string Higgs
effect[14].Specifically,the VEV’s of the charged background fields give masses to the matterfieldsΦvia a cubic superpotential of the form W∼AΦΦand break the gauge symmetry to the gauge subgroup that commutes with A .In this case,the masses of the matterfields as well as the gauge symmetry breaking scale will be of order A .Since it is determined by gaugino consensation dynamics,the VEV of A will be
A ∼M S e−S/2b0.(6) Wefix the dilaton VEV phenomenologically at S ≃2,the value consistent with the observed weak scale couplings.For affine level one string models with unbroken E6hidden sector gauge symmetry,we have b0=36/(16π2),
3
which gives A ∼1016GeV.(For E8hidden sector,we have b0=90/(16π2),
which gives A ∼1017GeV.)This scale can be identified with the grand
unification scale M GUT from the low-energy point of view.The heavy massive
fields generated in this mechanism will not substantially alter the weak-scale
predictions,because these masses are all on the order of M GUT.One can also
avoid the doublet-triplet problem by using the the pseudo-Goldstone schemes
in[15]since the induced charge backgroundfield VEVs play the exact same
role as the usual Higgsfields.
In the above,we show that to generate M GUT=1016GeV,we need the
induced charged background to be A =10−2M S.But it is usually not
natural to generate the small shift in the antisymmetricfield strength H.
In fact,in the case that these induced charged background VEVs satisfy
F ij =0,they are naturally Wilson lines.Since one hasπ1(K)=Z N(where K is the compact spacetime manifold),we recover eq.5.
In the following,we will propose a mechanism to make possible small
values of H .We assume that the compactification scale is at the gaugino
condensation scale rather than near the string scale,as is usually assumed.
In this case,
1
n
.(8) For orbifolds with the moduli backgroundfields
G ij=R2δij,G ij=R−2δij,(9)
one has A i dx i= A i G ij dx i=2πA
n ,A=
R
So in this scheme to have the small value for A,the compactification radius is forced to be large compared to the string scale.Through the charged background,the compactification radius is related to the gaugino conden-sation scale.One can think of this as gaugino condensation inducing the compactification of some dimensions and the charged background VEV’s at the gaugino condensation scale.
The fact that the compactification radius is on the same order as the gaugino condensation scale makes the dynamics of inducing the charged background VEV by the gaugino condensation appear natural,but it may ap-pear to invalidate our four-dimensional description of gaugino condensation. However,it has been proved perturbatively to all orders in string perturba-tion theory[16]that string models are invariant under T-duality[17],i.e. T→1/T or R→1/R.In our case,T-duality implies our string models with large radius can equally well be described by the theory with the small compactification radius.In the small-radius description,the gaugino conden-sation in the hidden sector can be well approximated by the four-dimensional field theory in which we know the gaugino condensation happens.With the application of T-duality,we can see that the same dynamics happens in our model.This is a nice demonstration that T-duality is indeed a powerful tool!
We have argued above that there is a real hope for the existence of affine level one grand unified string models with intermediate scale gauge symmetry breaking.But there are also some potential problems with this type of model which we now discuss.
First of all,if one appeals to large moduli,then the string threshold correction[6]to the gauge coupling constant could be large and spoil the possible predictions of both M GUT and the weak mixing angle in this scheme. To avoid this problem,we can restrict attention to string models without string threshold co
rrections.This kind of string models do exist.It has been show that for string models with no sectors that preserve N=2spacetime supersymmetry(for example,Z3and Z7orbifold models)the gauge coupling constant does not receive moduli-dependent string threshold correction[18,
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