Relative Scales of the GUT and Twin Sectors in an F-theory model

In this letter we analyze the relative scales for the GUT and twin sectors in the F-theory model discussed in Ref. \cite{Clemens-3}. There are a number of volume moduli in the model. The volume of the GUT surface in the visible sector {[}sector(1){]} (with the Wilson line GUT breaking) defines the GUT scale $M_{GUT}\sim2\times10^{16}~GeV$ as the unification scale with precise gauge coupling unification of $SU(3)\times SU(2)\times U(1)_{Y}$. We choose $\alpha_{GUT}^{-1}\sim24$. We are then free to choose the ratio $\alpha_{G}(2)/\alpha_{G}(1)=m_{1}/m_{2}$ with $m(1)$ and $m(2)$ independent volume moduli associated with the directions perpendicular to the two GUT surfaces. We then analyze the effective field theory of the twin sector(2), which may lead to a SUSY breaking gaugino condensate. Of course, all these results are subject to the self-consistent stabilization of the moduli.


Relative Scales in F-theory GUT
The effective low energy field theory of an F -theory GUT is defined on a real 10-dimensional manifold M 10 = R 3,1 × B 3 where B 3 is a smooth complex projective complex 3-fold with ample anti-canonical bundle, that is, a Fano threefold. Gravity fills all of this real 10-dimensional space-time while the GUT theory resides on a 7-brane, S GUT ×R 3,1 , with S GUT given by a smooth two-dimensional anti-canonical complex surface S GUT ⊆ B 3 . The GUT surface S GUT is defined by the vanishing of z ∈ H 0 K −1 B3 . 1.1. Semi-stable degeneration of the F -theory model. More precisely, as in [2] we consider the product B 3 = P [u0,v0] × B 2 B 2 is a fixed del Pezzo surface with Z 4 -symmetry. For the affine coordinates and consider the closure P [δ] of the subset Date: December 12, 2021.
Here B 2 is a divisor in D 6 × P 1 ≡ D 6 × P [k,l] endowed with its canonical toric metric g (6) as in (1.3) of [3] with respect to which the Z 4 -action T 0 as in Section 6.1 of [2] is isometric with finite fixpoint set. 1 We recall that D 6 is the blow-up of P 2 = P [a,b,c] at the three points where two of its coordinates vanish. To obtain B 2 we pick one general point in D 6 and blow up the four points of its orbit under the Z 4 symmetry generated by T 0 .
To do this we proceed as follows. Any four points of P [a,b,c] , no three of which are collinear, can be written as the common zeros of two homogeneous quadratic forms, q 1 (a, b, c) and q 2 (a, b, c). Since the intersection set is invariant under the action of T 0 , the q j can also be chosen to be invariant. Then that defines the blow up as a divisor in D 6 × P [k,l] by smoothly inserting a copy of P [k,l] at each the four common zeros of q 1 (a, b, c) and q 2 (a, b, c) in D 6 . We can define the metric g (2) on B 2 as the one induced from the product metric on D 6 × P [k,l] by restriction. Thus we have one additional real degree of freedom in the choice of the scaling constant on the standard SU (2)-invariant metric on P 1 . Summarizing, the metric induced on and all its submanifolds is the metric induced by inclusion from the product of the four given metrics on D 6 × P [k,l] × P 1 a × P 1 b . The canonical metric on D 6 is the standard metric given by its toric structure [3] and the metric on each of the three P 1 's is a positive real multiple of the standard SU (2)-invariant metric. We let b . This allows two additional scaling constants, the first giving volume m 1 to the standard SU (2)-invariant metric on P a and the second giving volume m 2 to the standard SU (2)-invariant metric on P b .
The Einstein-Hilbert action is given by As a consequence, the four-dimensional Planck constant is given by . The semi-stable limit of the F -theory geometry as δ goes to zero is the union of two components or 'gauge sectors' We call B (1) 3 with induced metric g 1 the 'visible sector' and B (2) 3 with induced metric g 2 the 'hidden or twin sector.' Thus 1.2. Asymptotic position of S GUT . As δ varies, the GUT surface S GUT,δ is defined by the vanishing of is a section in the (−1)-eigenspace for the Z 4 -action on Here the gauge action is given by where F i denotes the (limiting) curvature tensor of the Yang-Mills connection on the i-th gauge sector B (i) 3 and δ 2 (z 0 ) is the standard distribution supported on (See Appendix A for more detail on the decomposition S 1 ∪ S 2 of S GUT,0 .)

Scaling the effective 4-D theory
Hence in the effective 4-dimensional theory we should have GUT coupling constant . Therefore the relative size of the GUT coupling constants and the GUT scales for the visible and twin sectors depends on the relative sizes of the perpendicular directions in B 3,0 to B 2, .
Let's define the sector labeled (1) as the visible sector with GUT coupling constant, α G (1) −1 = 24 at the GUT scale M G (1) = 3×10 16 GeV . Then the twin sector is sector (2). The ratio αG (2) αG ( However the twin QCD coupling will become strong at a scale much greater than the visible QCD scale. The effective twin QCD theory has N C = 3 and N f lavors = 6. Hence it is described by Seiberg duality [7]. In the magnetic phase, we have an effective superpotential given by where q (q) are left-handed color triplets (anti-triplets) with the family index, i = 1, 2, 3, and SU (2) isospin index, a = 1, 2. When q 0 = q 0 = 0, the theory has a flat direction for the fundamental meson field, T i,a j,b . Note, since the twin electroweak group is gauged, we should identify T i,a j,1 ≡ (T ua ) i j and T i,a j,2 ≡ (T da ) i j . The twin supersymmetric SU (2) × U (1) Y gauge interactions introduce a quartic potential for Then all twin quarks and leptons obtain mass at the scale T and, moreover, the twin electroweak gauge symmetry is broken down to twin U (1) EM . For T ∼ M G (2), we find a twin gluino condensate occurs at the scale Λ tQCD = T exp(− 2π 9αG(2) ) ∼ 9 × 10 13 GeV . We expect that the effective 4D QCD Lagrangian contains a term where φ, b is the dilaton, axion fields, m i is as above and K(i) ∼ M 6 * M G (i) −4 [8,9,10]. As a consequence, the twin QCD condensate will contribute SUSY breaking effects to both the twin and visible sectors of the theory. In this local SUSY theory, we find an effective low energy SUSY breaking scale given by (2.4) m 3/2 = Λ 3 tQCD /M 2 P l ∼ 130 T eV. Of course, whether supersymmetry is broken (or not) depends on stabilizing all the moduli.
The low energy supersymmetric theory contains 3 families of twin neutrino superfields (assuming that the three right-handed neutrinos obtain mass near the GUT scale), 19 chiral charged and neutral Higgs superfields (which include the massless components of H u , H d and T u , T d ), and the twin photon superfield. Renormalizing from M G (2) we find α tEM (m 3/2 ) ∼ 1/105. There do not appear to be any portals to the twin sector. Clearly, the twin sector introduces new candidates for dark matter, but a complete analysis of the cosmological implications of this sector for the theory is beyond the scope of the present paper.

Conclusion
In conclusion, we have analyzed the relative scales of the visible and twin sectors in the global F -theory GUT with Wilson line breaking given in [1]. We have found that there is sufficient freedom to have independent GUT scales and couplings in order to have interesting physics coming from the twin sector. In particular, if we assume that the GUT coupling for the twin sector is larger than that of the visible sector, then it is possible to spontaneously break the twin electroweak theory at the GUT scale with all twin quarks and charged leptons obtaining mass at that scale. In addition, a twin gluino condensate can then occur at a scale of order Λ tQCD ∼ 9 × 10 13 GeV which leads to an effective low energy SUSY breaking scale, m 3/2 = Λ 3 tQCD /M 2 P l ∼ 130 T eV which affects both the twin and visible sectors. There is clearly more analysis that needs to be done on the consequences of these results, including the stabilization of moduli, which we leave for the future. For example, the model also includes 11 D 3 branes and fluxes which must be considered [11].