CALABI-YAU STRING THEORY-CURRENT STATUS PRAN NATH Department of Physics, Northeastern University Boston, MA 02115, USA and R. Arnowitt Center for Theoretical Physics, Department of Physics Texas A & M University, College Station, TX 77843 ABSTRACT Existence of 2 and probably more new light exotic states which are SU(3)cxSU(2)LxU(1) neutral is predicted within the framework of the three generation Calabi-Yau string models which flux break to [SU(3)J at the compactification scale, have intermediate scale breakdown to the Standard Model and possess matter parity invariance. It is shown that these exotic states mediate u eï and T -> uY decays with decay rates that may lie in the observable range. The final lepton is found to be right handed polarized. These phenomenon are predicted as experimental test of this class of string theories. Standard Model gauge group SU(3)cxSU(2) xU(1) I. INTRODUCTION L is accomplished by spontaneous breaking at an Calabi-Yau compactification of the intermediate scale Mi<Mc~MP£* Proton Heterotic String has been a subject of stability requires that the Calabi-Yau theory sonsiderable interest over the recent years below MQ be matter parity (M2) invariant [M =C U where C acts on the C-Y co-ordinates since it leads to physically interesting 2 z and obeys the condition C2=1 , while U is three gerneration models''*^. While it does z defined by diag U =(111)x(-1-11)x(-1-11). not seem likely that the three generation z Calabi-yau models as originally defined as Spontaneous breaking of [SU(3)]3 at Mj is corresponding to symmetric points in the accomplished by the VEV growth of an SO(10) c moduli space can be phenomenologically singlet field N and an SU(5) singlet field v viable, it appears that extensions of these belonging to the lepton nonet L(1,3,3). Two models-'' which include the allowed moduli lepton generations (L_q ,L^) and two mirror lepton generations (L^,L ) enter _ in the dependence of the corresponding manifolds can 2 satisfy stringent phenomenological spontaneous breaking. Of these L-j , L-j are O even and L , L are C-odd. Detailed analysis constraints. In recent investigations an 2 2 shows that spontaneous breaking at Mj analysis valid for a wide class of three preserves matter parity and protects proton generation Calabi-Yau compact ifications has stability against rapid proton decay i.e. one been carried out^' . One of the remarkable has <v^>=0=<N >, <v^>=0=<N > while N , , v results arising from the analysis of the 2 2 1 2 and v grow VEVs. In the following we shall three generation models is the prediction of 2 use the notation i=(n,r)=(1,m; 2,s) where two and probably more exotic particles at low n=(1,m) are C-even and r=(2,s) are C-odd energy. In addition to the normal particle indices. spectrum of the minimal N=1 Supergravity models. Additionally beyond the Standard Model phenomena such as p •) eï and T -> pY Detailed analysis shows that the accessible in current and future experiments couplings among the leptons must satisfy are predicted. certain conditions in order for the theory to satisfy the phenomenological constraints of II. THREE GENERATION CALABI-YAU MODELS AND light Higgs and lepton mass heirarchy and THEIR REDUCTION TO THE STANDARD MODEL small neutrino masses. The [SU(3)]^ invariant couplings among leptons have the In Calabi-Yau compact ification, the form residual symmetry after compactification from 10 dimensions to M^xK, where M^ is the four dimensional Minkowskian space and K is the compact six dimensional Calabi-Yau manifold is EgTxE^xN=1 supergravity. Physically interesting three generation models are those It is found that the condition needed to where the Ec grandunification group reduces guarantee a pair of light Higgs is Aj^=0, 3 s while as discussed in Sec III a small me/mT to [SU(3)] l SU(3)cxSU(3)LxSU(3) J to the R ratio requires 6 /e~10~^ where 6 is defined Conference Speaker so that 376 LA=^ni^a,(f>a^ wnfre ni==(n2»n2^^a=^n1'^2^ and <j)a are the E6 H (EndT) singlet fields which remain light. The matrix mAg is block where diagonal i.e. 1^=4m^, matJ where m^, ma^ are soft SUSY breaking masses (which we expect to be 0(1 CP GeV)) and mat) is the mass matrix in the Eç H (EndT) singlet sector. (As already pointed out in Sec. II these singlet fields will in general receive masses After spontaneous breaking at Mj, one both from SUSY breaking and also from finds that the spectrum below Mj consists of instanton corrections). Finally Ae d a and the spectrum of the minimal N=1 supergravity An x stand for coupling constant matrices in theory. In addition one finds that there are their respective sectors. We note that the and 2-1* SU(3)cxSU(2)LxU(1 ) singlets from L, 2 first three terms in are the Standard L-j 2 generations and a number of H (EndT) Model interactions while the remaining singlets that remain light. The light interaction terms involve the singlet fields SU(3)cxSU(2)LxU(1) singlet states are ^2*^2 and <j>a. The orders of the An and A. couplings are as follows: A?ia<5 » *i1=(S 1 2 2 ? while A- =6 e and A^a =6 £ for p=2,3 (p=1,2,3 label the light leptons Hp). The low energy interaction structure arising from gaugino couplings is given by where (c,s)=(cose, sine) and tan0=e. Of the above n^, vc always remain light while r^.n^ may also remain light depending on their interactions with E^ singlets: The E^ singlets in addition to gaining masses from SUSY breaking effects, may also gain masses via instanton corrections i.e. In Eq (3.2) ë is the left selectron-sneutrino doublet. The interactions of Eq (3.2) are new, not present in the Standard Model. The mass matrix in the leptonic sector can be conveniently described by the form =(o (e),a ,3 ) where m^)= =0(o2/e), and where Rp=Mp^"'' and R is the overall radius of 0 m a 0(1) J the Calabi-Yau manifold. p2 p and m^ =Bp-a(1). If we define r=3/a and a*3=a$ eos<j), then for small <J> and III. EFFECTIVE LOW ENERGY INTERACTION large r one obtains After spontaneous breaking at Mj one must carry out a light and heavy mode expansion of all of the fields that enter the theory below MQ. The interaction structure of the effective low energy theory below Mj Lepton mass heirarchy indicates a small value can be obtained by integrating over the of ô2/e i.e. 0(10~3), r*3 and sin^0.2 superheavy modes of the theory. This structure consists of two parts: A IV. NEUTRINO MASSES superpotential term and a term in the Lagrangian arising from gaugino interactions. The superpotential term has the form In a basis where the neutral states that ix are labelled by (va, $A) where va=^v1 »v2,v3^ and ^A3^'^' the mass matrix has the form +L m L +w( A AB B seesaw) (3.D In the above we have used the notation 377 when one neglects the very small W(seesaw) the current experimental upper limits allows masses. Here maBare proportional to various for a sizable value of e i.e. e*0(1). In powers of e and 6 and are scaled by the Higgs this section we will show that an e~1 leads mass while MAB are the order of the SUSY to y->eY and T->UY branching ratios that may breaking mass 0(10^GeV) but could be larger lie in the observable range. Contribution to in the Eg singlet sector due to instanton these decays arise from the exchange of corrections as discussed earlier. A detailed SU(3)cxSU(2)LxU(1) singlets and from E6 analysis shows that the neutrino mass singlets. The decay branching ratios are eigenstates are mvi=Ai<H> /M where M~0(MAg) controlled by E and 6. The full e and 6 and are given in Table 1 dependence of y -» eY and of T -» y Y amplitudes arising from the exchange of the singlets is Table 1. Orders of A,- exhibited in Table 2. Tabel 2: e,6 orders of y -> eY and T yY amplitudes The effective Lagrangian that governs y->eY is The current experimental limit on neutrino masses from the Big Bang Cosmology is 2 £Mvi<100h eV where 1/2^1 is the experimental uncertainty in the determination of the Hubble constant, while the direct The largest contributions to a^ ^ arise from c experimental limit on the from Los Alamos the exchange of v and n-j particles which is m ^9.^ ev (95% CL)> These experimental have been proven to be always light. One results act as strong constraints on e and 6. finds There are two possibilities that arise naturally. First if n2 or no or both ^2^2 are light with masses ~ 1CH GeV, then the experimental bounds require e to be small i.e. e£0.03-0.05 . The second possibility which is natural is that theoretically one expects e ~ 1 since it is a ratio of two VEVS which are proportional to coupling constants where L(x) is the one loop function in the non-renormalizable interactions that generate these VEVS.