ASPECTS OF FREE FERMIONIC HETEROTIC-STRING MODELS Alon E. Faraggi AEF, 990 s (top quark mass, CKM, MSSM w Cleaver and Nanopoulos) AEF, C. Kounnas, J. Rizos, 003-009 AEF, PLB00, work in progress with Angelantonj, Tsulaia Galileo Galilei Institute, Firenze, 6 May 009
DATA STANDARD MODEL SU(3) x SU() x U() Y SU(5) SO(0) [ ν e + D L c [ + [ c U + L u d + E c L [ + N c L + 5 0 6 STANDARD MODEL -> UNIFICATION ADDITIONAL EVIDENCE: Logarithmic running, proton longevity, neutrino masses PRIMARY GUIDES: 3 generations SO(0) embedding
Realistic free fermionic models Phenomenology of the Standard Model and string unification Top quark mass 75 80GeV PLB 74 (99) 47 Generation mass hierarchy NPB 407 (993) 57 CKM mixing Stringy seesaw mechanism Gauge coupling unification NPB 46 (994) 63 (with Halyo) PLB 307 (993) 3 (with Halyo) NPB 457 (993) 409 (with Dienes) Proton stability NPB 48 (994) Squark degeneracy NPB 56 (998) (with Pati) Minimal Superstring Standard Model PLB 455 (999) 35 (with Cleaver & Nanopoulos) Moduli fixing NPB 78 (005) 83
Free Fermionic Construction Left-Movers: ψ µ,, χ i, y i, ω i (i =,, 6) Right-Movers ȳ i, ω i i =,, 6 φ A=,,44 = η i i =,, 3 ψ,,5 φ,,8 f e iπα(f) f V V Z = all spin c ( α ) ( α β ) β Z structures Models Basis vectors + one loop phases
Model building Construction of the physical states b j j =,,N Ξ = j n jb j For α = ( α L ; α R ) Ξ H α α(f) = ± ; α(f) f,f, ν f,f = α(f) M L = + α L α L 8 + N L = + α R α R 8 + N R = M R ( 0) GSO projections e iπ( b i F α ) s α = δ α c ( α bi ) s α Q(f) = α(f)+f (f) U() charges
Calculation of Mass Terms nonvanishing correlators V f V f V b 3 V b N gauge & string invariant anomalous U() A TrQ A 0 D A =0=A + Q A k φ k D j =0= W = W N η i =0 N =3 Supersymmetric vacuum F = D =0. Q j k φ k nonrenormalizable terms effective renormalizable operators V f V f V b 3 V b N V f V f V 3 b V b 4 V b N M N 3
The NAHE set: = {ψ µ,χ,...,6,y,...,6,ω,...,6 ȳ,...,6, ω,...,6, η,,3, ψ,...,5, φ,...,8 } S = {ψ µ,χ,...,6 }, N =4Vacua b = {χ 34,χ 56,y 34,y 56 ȳ 34, ȳ 56, η, ψ,...,5 }, N =4 N = b = {χ,χ 56,y,ω 56 ȳ, ω 56, η, ψ,...,5 }, N = N = b 3 = {χ,χ 34,ω,ω 34 ω, ω 34, η 3, ψ,...,5 }, N = N = Z Z orbifold compactification = Gauge group SO(0) SO(6),,3 E 8
beyond the NAHE set Add {α, β, γ} ψ µ χ χ 34 χ 56 y 3 y 6 y 4 ȳ 4 y 5 ȳ 5 ȳ 3 ȳ 6 y ω 5 y ȳ ω 6 ω 6 ȳ ω 5 ω ω 4 ω ω ω 3 ω 3 ω ω 4 ψ,...,5 η η η 3 φ,...,8 α 0 0 0 0 0 0 0 0 0 0 00 0 0 000 β 0 0 0 0 0 0 0 0 0 0 00 0 0 000 γ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 number of generations is reduced to three SO(0) SU(3) SU() U() T3R U() B L U() Y = (B L) + T 3 R SO(0)! SO(6),,3 U(),,3 U(),,3
The massless spectrum Three twisted generations b, b, b 3 h,0,0 h,0,0 Untwisted Higgs doublets h 0,,0 h0,,0 h 30,0, h30,0, Sector b + b + α + β h αβ,,0,0,0,0 hαβ,,0,0,0,0 SO(0) singlets Sectors b j +γ j =,, 3 hidden matter multiplets standard SO(0) representations NAHE + { α, β, γ} exotic vector like matter superheavy Quasi-realistic phenomenology
Fermion mass hierarchy Fermion mass terms ( φ cgf i f j h M ) N 3 c - calculable coefficients g - gauge coupling f i, f j b j j =,, 3 h light Higgs multiplets M 0 8 GeV φ generalized VEVs, several sources
Top quark mass prediction only λ t = t c Q t h = g 0 at N =3 W 4 b c ( Q b h αβ ) Φ + τ c L ( τ h αβ Φ ) = λ b = λ τ = c b φ M λ b = λ τ = 0.35g 3 8 λ t Evolve λ t, λ b to low energies c τ φ M m t = λ t v = λ t v 0 sin β m b = λ b v = λ b v 0 cos β where v 0 = m W g (M Z ) = 46GeV and (v + v )=v 0 m t = λ t (m t ) v 0 tan β ( + tan β) = m t 75GeV PLB74(99)47
Cabibbo mixing PLB 307 (993) 305 Find anomaly free solution V ɛ V3 Φ αβ M 3 0 V M d V 3 Φ αβ ξ Φ ξ M 4 M 0 Φ 0 0 v, ξ M V V3 Φ αβ 5 g 6 M 3 = 64 π 3 3 0 4. ɛ<0 8.Fix ξ, ξ, Φ to fit m b, m s 0.98 0. 0 V 0. 0.98 0 0 0 Three generation mixing NPB 46 (994) 63 J 0 6
Correspondence with Z Z orbifold PLB 36 (994) 6 NAHE (ξ = { ψ,,5, η, η, η 3 } =) {,S,ξ,ξ, b,b } Gauge group: SO(4) 3 E 6 U() E 8 and 4 generations. toroidal compactification (6 L +6 R ) g ij, b ij 0 0 0 0 0 0 0 0 0 0 g ij = b 0 0 ij = 0 0 0 0 0 0 0 0 g ij i j 0 i=j g ij i j R i the free fermionic point G.G. SO() E 8 E 8 mod out by a Z Z with standard embedding SO(4) 3 E 6 U() E 8 with 4 generations Exact correspondence
In the realistic free fermionic models replace X = { ψ,,5, η, η, η 3 } = with γ = { ψ,,5, η, η, η 3, φ,,4 } = Then {, S, ξ = + b + b + b 3, γ} N=4 SUSY and SO() SO(6) SO(6) apply b b Z Z N= SUSY and SO(4) 3 SO(0) U() 3 SO(6) b, b, b 3 (3 8) 6 of SO(0) O b +γ, b +γ, b 3 +γ Alternatively, c ( ξ ξ ) (3 8) 6 of SO(6) H = +
Z X Z orbifolds A torus One complex parameter Z = Z + n e + m e T x T x T Three complex coordinates z, z and z 3 Z orbifold : Z = Z + Σ i m i e i 4 fixed points Z = { 0, / e, / e, / ( e + e ) } T x T x T Z X Z α : ( β : ( αβ : ( z, z, z3 ) > ( z, z, +z3 ) > 6 z, z, z3 ) > ( + z, z, z3 ) > 6 z, z, z3 ) > ( z, +z, z3 ) > 6 48 γ : ( z, z, z3 ) > (z+/, z+/,z3+/) 4
Classification of fermionic Z Z orbifolds (FKNR, FKR) Basis vectors: consistent modular blocks 4,8 periodic fermions = {ψ µ,χ,...,6,y,...,6,ω,...,6 ȳ,...,6, ω,...,6, η,,3, ψ,...,5, φ,...,8 } S = {ψ µ,χ,...,6 }, z = { φ,...,4 }, z = { φ 5,...,8 }, e i = {y i,ω i ȳ i, ω i },i=,...,6, N =4Vacua b = {χ 34,χ 56,y 34,y 56 ȳ 34, ȳ 56, η, ψ,...,5 }, N =4 N = b = {χ,χ 56,y,y 56 ȳ, ȳ 56, η, ψ,...,5 }, N = N = Vector bosons: NS, z,, z + z, x =+s + e i + z + z impose: Gauge group SO(0) U() 3 hidden
Independent phases c [ vi v j ] = exp[iπ(vi v j )]: upper block S e e e 3 e 4 e 5 e 6 z z b b ± ± ± ± ± ± ± ± ± ± S e ± ± ± ± ± ± ± ± ± e ± ± ± ± ± ± ± ± e 3 ± ± ± ± ± ± ± e 4 ± ± ± ± ± ± e 5 ± ± ± ± ± e 6 ± ± ± ± z ± ± ± z ± ± b ± b Apriori 55 independent coefficients 55 distinct vacua Impose: Gauge group SO(0) U() 3 SO(8) 40 independent coefficients
RESULTS: FKR I: Random sampling of phases. SO(0) U() 3 hidden FKR II: Complete classification. SO(0) U() 3 SO(8) RESULTS ARE SIMILAR
7 0 9 models 5% with 3 gen FKRI
Spinor vector duality: Invariance under exchange of #(6 + 6) < > #(0) 4 0 8 6 4 0 8 6 4 0 0 4 6 8 0 4 6 8 0 4 0 4 6 8 0 4 6 8 0 4 4 0 8 6 4 0 8 6 4 0 Symmetric under exchange of rows and columns Self-dual: #(6 + 6) = #(0) without E 6 symmetry
Future: Understand in geometrical terms: Progress: Tristan Catelin-Jullien, AEF, C. Kounnas, J. Rizos, NPB009 Operational understanding in terms of partition function free phases
NAHE-based partition functions: Question: T 6 Z Z 48 fixed points SO() Z Z 4 fixed points Z shift :48 4 Is this the same vacuum? In general, no. shift that reproduces the SO() lattice at the free fermionic point?
Possible shifts: A : X L,R X L,R + πr, ) A : X L,R X L,R + (πr ± πα R A 3 : X L,R X L,R ± πα R. Using the level-one SO(n) characters ( O n = ϑ n ) 3 η n + ϑn 4 η n S n = ( ϑ n η n + i nϑn η n (, V n = ϑ n ) 3 η n ϑn 4 η n, ) (, C n = ϑ n η n i nϑn η n, ).
The partition function of the heterotic string on SO() lattice: [ Z + =(V 8 S 8 ) O + V + S + C ] (Ō6 + S 6 )(Ō6 + S ) 6, and Z =(V 8 S 8 ) [( O + V ) (Ō6 Ō 6 + C ) 6 C6 + ( S + C ) ( S6 S6 + V ) 6 V6 + ( O V + V Ō )( S6 V6 + V ) 6 S6 + ( S C + C S )(Ō6 C6 + C )] 6 Ō 6. where ± refers to c ( ξ ξ ) = ± connected by : Z =Z + /a b, a =( ) F int L +F ξ, b =( ) F int L +F ξ.
Starting from: Z + =(V 8 S 8 ) 6 Λ m,n (Ō6 + S 6 )(Ō6 + S ) 6, m,n where as usual, for each circle, and p i L,R = m i R i ± n ir i α, Λ m,n = qα 4 p L q α 4 p R η. Add shifts : (A,A,A ), (A 3,A 3,A 3 ) (48 4 yes) (SO()? no)
Uniquely: g : (A,A, 0), h : (0,A,A ), where each A acts on a complex coordinate (48 4 yes) (SO()? yes)
A STRINGY DOUBLET-TRIPLET SPLITTING MECHANIS ψ µ χ χ 34 χ 56 y 3 y 6 y 4 ȳ 4 y 5 ȳ 5 ȳ 3 ȳ 6 y ω 5 y ȳ ω 6 ω 6 ȳ ω 5 ω ω 4 ω ω ω 3 ω 3 ω ω 4 ψ,...,5 η η η 3 φ α 0 0 0 0 0 0 0 0 0 0 00 0 0 0 β 0 0 0 0 0 0 0 0 0 0 00 0 0 00 γ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 NAHE χ j ψ,,5 η j + c.c. (5 + 5) j =0 j of SO(0) α, β SO(0) SO(6) SO(4) j = α L (T j ) α R(T j ) j =0 D j, D j j = h j, h j h, h,d, D, h, h,d, D are projected out h 3, h 3 remain in the spectrum
Conclusions Phenomenological string models produce interesting lessons Spinor vector duality relevance of non standard geometries Free Fermionic Models Z Z orbifold near the self dual point Duality & Self Duality String Vacuum Selection