String-loop completion of the minimal $\mathbb{Z}_3$ orbifold Pati–Salam quiver: anomaly-canceling terms, Fayet–Iliopoulos coefficients, and the symmetry-breaking scales

Dent and Kephart (arXiv:0705.1995) constructed a minimal three-family $\mathcal{N}=1$ supersymmetric Pati–Salam model as the quiver gauge theory dual to type IIB string theory on AdS$_5\otimes S^5/\mathbb{Z}_3$, with a spontaneous-symmetry-breaking chain down to the Standard Model, proton stability from unification near $10^{15}$ GeV, and a seesaw mechanism for neutrino masses. The construction assumes, but does not derive, string-loop-generated terms that cancel the anomalous $U(1)$s of the quiver, break them, and supply Fayet–Iliopoulos D-terms mediating supersymmetry breaking; it leaves the intermediate symmetry-breaking scales, the vectorlike-exotic spectrum, the proton lifetime, the seesaw scale, collider thresholds, and the stabilization of the tachyonic/moduli sector undetermined. No work computing these quantities for this model is known, so the problem remains open.

Background

The Pati–Salam model is a grand-unified framework based on the gauge group $SU(4)\times SU(2)\times SU(2)$, which treats lepton number as a fourth color: each fermion family fills the representations $(4,2,1)$ and $(\bar 4,1,2)$, including a right-handed neutrino. It is a maximal subgroup model of the $SO(10)$ grand unified theory, much as 'trinification' based on $SU(3)\times SU(3)\times SU(3)$ is a maximal subgroup model of $E_6$.

AdS/CFT duality identifies type IIB string theory on the product of five-dimensional anti-de Sitter space with a five-sphere, AdS$_5\otimes S^5$, with four-dimensional $\mathcal{N}=4$ supersymmetric $SU(N)$ Yang–Mills theory. The $\mathcal{N}=4$ theory is non-chiral and conformal. Orbifolding - quotienting the sphere by a finite group $\Gamma$ embedded in the $SU(4)$ R-symmetry of $\mathcal{N}=4$ - produces on the gauge-theory side a quiver gauge theory: a product gauge group whose matter sits in bifundamental representations of pairs of gauge-group factors, conveniently drawn as a diagram (a 'quiver') whose nodes are gauge groups and whose arrows are chiral multiplets (Kachru and Silverstein, Phys. Rev. Lett. 80, 4855 (1998)). Embedding $\Gamma$ inside $SU(3)\subset SU(4)$ leaves $\mathcal{N}=1$ supersymmetry and can generate chiral fermion content suitable for model building; Kephart and Päs (Phys. Lett. B 522, 315 (2001)) used this route to construct three-family supersymmetric models from $\mathbb{Z}_n$ orbifolds, the precursors of the model below.

Dent and Kephart (arXiv:0705.1995) chose $\Gamma=\mathbb{Z}_3$ with the embedding $4=(1,\alpha,\alpha,\alpha)$ and $N=4$, obtaining an $\mathcal{N}=1$ quiver based on $SU(4)^3$ (more precisely $U(4)^3$) with chiral multiplets $3[(4,\bar 4,1)+(1,4,\bar 4)+(\bar 4,1,4)]$ plus adjoints. A chain of vacuum expectation values (VEVs) for bifundamental scalars breaks the gauge group through $SU(4)\times SU(3)\times SU(3)\times U(1)_A$ to the Pati–Salam group plus two Abelian factors and then to the Standard Model, leaving exactly three chiral families $3[(4,2,1)+(\bar 4,1,2)]$; the remaining fermions are vectorlike (paired with conjugate representations, so they can become heavy near the Pati–Salam scale); the right-handed neutrinos acquire large Majorana masses (mass terms pairing a fermion with its own charge conjugate) usable in the seesaw mechanism (the suppression of light-neutrino masses by a heavy Majorana mass); and unification near $10^{15}$ GeV keeps proton decay below observational bounds.

Three ingredients are, however, assumed rather than derived. First, the Abelian $U(1)$ factors of $U(4)^3$ are anomalous in the quiver field theory - the mixed triangle-anomaly coefficients of type $U(1)_pU(1)_q^2$ and $U(1)_pSU(4)_q^2$ are nonzero - and consistency of the underlying string theory requires them to be canceled, in analogy with the Green–Schwarz mechanism (in which an axion-like field shifts so as to cancel an anomalous triangle divergence; Dine, Seiberg and Witten, Nucl. Phys. B 289, 589 (1987); Atick, Dixon and Sen, Nucl. Phys. B 292, 109 (1987); and for type IIB D3-brane vacua, Ibáñez, Rabadán and Uranga, Nucl. Phys. B 542, 112 (1999), and Aldazabal, Ibáñez and Quevedo, JHEP 0001, 031 (2000)). Dent and Kephart state that they 'assume the $U(1)$s are broken via terms of the form $\sum_p(m_p^2\phi_p^2+\lambda_p\phi_p^4)$ generated at the string loop level'. Second, they posit Fayet–Iliopoulos (FI) D-terms $L_{FI}=\kappa_p D_p$ - a linear term in the auxiliary field $D_p$ of the $U(1)_p$ vector supermultiplet, which drives scalar VEVs and can transmit supersymmetry breaking - with undetermined coefficients $\kappa_p$. String-loop corrections on the AdS dual correspond to $1/N$ and $\alpha'$ effects (the rank expansion and the expansion in inverse string tension) at strong 't Hooft coupling (the combination $g^2N$ of gauge coupling and rank held large), and no computation of the signs and magnitudes of $m_p^2$, $\lambda_p$, or $\kappa_p$ exists. Third, the tachyonic instabilities of the $U(1)$-charged fields, which the authors invoke to develop VEVs at finite values of moduli (scalar fields parameterizing the compactification) and to break conformal invariance, have no controlled dynamical treatment; related analyses (Armoni, JHEP 0704, 046 (2007); Dymarsky, Klebanov and Roiban, JHEP 0511, 038 (2005) for non-supersymmetric daughters) show that such questions are subtle. Consequently, the intermediate symmetry-breaking scales, the vectorlike-exotic masses, the proton lifetime prediction, the seesaw scale, and any Large Hadron Collider (LHC) thresholds are all undetermined.

Problem Statement

Complete the dynamical derivation of the minimal $\mathbb{Z}_3$ orbifold Pati–Salam quiver of Dent and Kephart from its type IIB string origin. Concretely: (i) derive, within a controlled approximation to type IIB string theory on AdS$_5\otimes S^5/\mathbb{Z}_3$ (or an explicitly identified controlled limit of it), the loop-generated terms that cancel the anomalous $U(1)$ triangle anomalies of the $U(4)^3$ quiver, together with the $U(1)$-breaking scalar potential $\sum_p(m_p^2\phi_p^2+\lambda_p\phi_p^4)$ and the Fayet–Iliopoulos coefficients $\kappa_p$, including their magnitudes and signs; (ii) use these derived quantities to determine quantitatively the intermediate symmetry-breaking scales, the vectorlike-exotic masses, the proton lifetime, the seesaw scale, and any LHC-accessible thresholds; and (iii) give a controlled dynamical account of the tachyonic/moduli sector that stabilizes the vacuum and realizes supersymmetry and conformal symmetry breaking in this compactification. Parts (ii) and (iii) presuppose (i): (ii) uses the quantities obtained in (i), and (iii) presupposes the $U(1)$-breaking terms of (i). The question may likewise be settled by showing, within this specific quiver, that the assumed string-loop terms cannot be generated or cannot break the $U(1)$s as assumed.

References

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