Coupling Selection Rules in Heterotic Orbifold Compactifications

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1 Coupling Selection Rules in Heterotic Orbifold Compactifications Susha L. Parameswaran Leibniz Universität Hannover JHEP 1205 (2012) 008 with Tatsuo Kobayashi, Saúl Ramos-Sánchez & Ivonne Zavala see also 12XX.XXXX with Nana Cabo, Tatsuo Kobayashi, Damián Mayorga, Matthias Schmidt & Ivonne Zavala Coupling selection rules in heterotic orbifolds p.1

2 Coupling Selection Rules in Heterotic Orbifolds Heterotic orbifolds represent simple, globally consistent constructions with clear geometrical interpretation Can obtain the MSSM spectrum with no chiral exotics Buchmüller et al 06 Lebedev et al To understand dynamics (decoupling of vector-like exotics, moduli stabilization, susy breaking, quark and lepton masses...) we require couplings in LEEFT String couplings can be computed via free CFT Which couplings are vanishing is determined by string selection rules Coupling selection rules in heterotic orbifolds p.2

3 Plan Orbifold CFT basics Coupling selection rules from L-point correlation functions Conclusions Coupling selection rules in heterotic orbifolds p.3

4 Heterotic Orbifold Compactifications Heterotic string degrees of freedom: right-movers: XR M (σ τ), ψr M (σ τ) M = 1,...,10 left-movers: XL M (σ + τ), XL(σ I + τ) I = 1,...,16 Torus R 6 /Λ (factorizable); orbifold T 6 /Z N Orbifold boundary conditions X j (σ + π, τ) = (θ k X) j (σ, τ) + λ j, λ j Λ j, j = 1, 2,3 SU(3)/Z 3 for state in kth twisted sector States in string Hilbert space fields in CFT Vertex operator for emission of twisted bosonic field: 3Y V B = e φ ( X j ) Nj L( X j ) N Le j iqm H m e ipi X I σ j (k,f) j=1 Infer W Φ L via ψψφ L 2 - tree-level couplings V F V F V B...V B Coupling selection rules in heterotic orbifolds p.4

5 L-point Correlation Functions Correlation function factors into several parts: Dixon et al 87 Hamidi & Vafa F 3 pt = e i P 3 l=1 p I sh l XI e i P 3 l=1 qsh m l Hm 3Y ( X j ) P 3 l=1 N j L l ( Xj ) P 3 l=1 Nj L l σ j j=1 Each part has its own selection rule: gauge invariance: P 3 l=1 pi sh l = 0 H-momentum conservation: P 3 l=1 qm sh l = 0 (k 1,f 1 ) σj (k 2,f 2 ) σj (k 3,f 3 ) space group selection rule: boundary conditions allow twisted strings to join z 1 g 1 z 1 z 2 g 1 z 1 = z 2 z = z g 3 z 3 = g 2 z 2 z 2 g 2 g 1 g 2 g 3 ~ 11 Includes point group selection rule: coupling between θ k 1 θ k 2 θ k 3 twisted sectors allowed if k 1 + k 2 + k 3 = 0 mod N for Z N orbifold. Coupling selection rules in heterotic orbifolds p.5

6 Classical and Quantum Splitting The non-trivial part of the correlation function is: F = 3Y ( X j ) Nj L( X j ) N Lσ j j (k 1,f 1 ) σj (k 2,f 2 ) σj (k 3,f 3 ) j=1 Fields X j split into X j (z, z) = X j cl (z, z) + Xj qu(z, z) with classical instanton solution X j cl = 0 Classical solutions determined by local and global monodromy: X j cl (z) = aj ν j (z z 1 ) kj 1 1 (z z 2 ) kj 2 1 (z z 3 ) kj 3 1 X j cl ( z) = bj ν j ( z z 1 ) kj 1( z z 2 ) kj 2( z z 3 ) kj 3 z 1 z 2 z 3 and vanish if k j 1,2,3 are such that classical action does not converge. Coupling selection rules in heterotic orbifolds p.6

7 New String Coupling Selection Rule The correlation function splits as (in the jth plane): F j 3 pt = ( X j cl + Xj qu) Nj L( X j cl + X j qu) N j Lσ j (k 1,f 1 ) σj (k 2,f 2 ) σj (k 3,f 3 ) OPEs ( Xqu) j r ( X qu) j s σ j (k 1,f 1 ) σj (k 2,f 2 ) σj (k 3,f 3 ) = 0 unless r = s Local and global constraints on classical solutions depending on twisted sectors: either anti-holomorphic instantons vanish X j cl = 0 or holomorphic instantons vanish X j cl = 0 or both vanish X j cl = 0 = Xj cl Rule 5: only holomorphic instantons N j L N j L only anti-holomorphic instantons N j L N j L no instantons N j L = N j L Coupling selection rules in heterotic orbifolds p.7

8 Forgotten String Coupling Selection Rule Hamidi & Vafa 87 Font, Ibañez, Nilles & Quevedo 88 Assume e.g. only holomorphic instantons are allowed: F j 3 pt = X X j cl e S cl ( X j cl )Nj L N j LZ j qu Classical solutions are proportional to (shifted) lattice vectors: X j cl fj 2 f j 1 + λ j, λ j Λ j Twist invariance: N j L N j L = 0 mod P j for Z P j twist Together with H-momentum conservation this leads to R-charge conservation Font et al 88 Rule 4: for fields at same fixed point we have X j cl λj N j L N j L = 0 mod Kj SU(3) for Z K j lattice symmetry Coupling selection rules in heterotic orbifolds p.8

9 Conclusions To build realistic models we must understand couplings in LEEFT Selection rules for superpotential couplings can be identified via L-point string tree-level correlation functions Space group selection rule, gauge invariance and R-charge invariance are sufficient for non-oscillator couplings Couplings between excited massless states and higher order couplings involve oscillators Then the structure of worldsheet instanton solutions leads to additional stringy rules Rule 4: when torus lattice has extra symmetries beyond the orbifold twist Rule 5: when local and global constraints imply instanton solutions are vanishing These rules must be applied when computing allowed W couplings The ultimate objective is the full LEEFT, K, W,f a, ξ FI... Coupling selection rules in heterotic orbifolds p.9

10 Couplings in Explicit MSSM Model order no rules 4 & 5 with rules 4 & (+152) (+282) (+4587) Coupling selection rules in heterotic orbifolds p.10