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SUSY Breaking and Moduli Stabilization

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Supersymmetric Grand Unified Theories

Part of the book series: Lecture Notes in Physics ((LNP,volume 939))

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Abstract

In string models all couplings, i.e. both gauge and Yukawa couplings, depend on the values of moduli. In addition, the size of the extra 6 dimensions also depend on the size of moduli. Finally, the value of the GUT symmetry breaking scale and proton decay rates depend on the size of the extra dimensions. In addition, in order to stabilize all the moduli one must necessarily spontaneously break supersymmetry.

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Notes

  1. 1.

    Racetrack models for SUSY breaking are discussed in [473–475].

  2. 2.

    Note, the variation of the dilaton provides the Green-Schwarz cancelation of the U(1) A anomaly.

  3. 3.

    For an excellent review with many references, see [495].

  4. 4.

    These terms arise as a consequence of world-sheet instantons in a string calculation. In fact, world sheet instantons typically result in more general modular functions [488–494].

  5. 5.

    Note, the constants \(\gamma _{T_{i}},\ \gamma _{U}\) can quite generally have either sign, depending upon the modular weights of the fields at the particular vertex.

  6. 6.

    The Dynkin index ℓ a (rep I ) ≡ T R defined earlier in Eq. (5.38). Thus if T a I are the generators of the group G a in the representation I, then we have Tr(T a I T b I) = ℓ a (rep I )δ ab .

  7. 7.

    Note, for clarity, this is just a toy model which is not derived directly from any particular string model.

  8. 8.

    In fact, one of the SU(4) quark- anti-quark pairs remained massless in the two “benchmark” models.

  9. 9.

    There is a check on the consistency of this approach: at the end of the day, after calculating the VEVs of the scalars, we can verify that the mass terms for the quarks are indeed of the correct magnitude.

  10. 10.

    The meson field, \(Q_{a}\tilde{Q}_{a}\) is assumed to be diagonal and proportional to the identity in flavor space. Thus not breaking the \(SU(N_{f_{a}})\) flavor symmetry.

  11. 11.

    The coefficient A [Eq. (24.39)] is an implicit function of all other non-vanishing chiral singlet VEVs which would be necessary to satisfy the modular invariance constraints, i.e. A = A(〈ϕ I 〉). If one re-scales the U(1) A charges, \(q_{\phi _{i}},q_{\chi } \rightarrow q_{\phi _{i}}/r,q_{\chi }/r\), then the U(1) A constraint is satisfied with r = 15p (assuming no additional singlets in A). Otherwise we may let r and p be independent. This re-scaling does not affect our analysis, since the vacuum value of the ϕ i , χ term in the superpotential vanishes.

  12. 12.

    The fields entering w 0 have string scale mass.

  13. 13.

    Note, we have chosen to keep the form of the Kähler potential for this single T modulus with the factor of 3, so as to maintain the approximate no-scale behavior.

  14. 14.

    Note, the constants b,  b 2 can have either sign. For the case with b,  b 2 > 0 the superpotential for T is racetrack-like. However for b,  b 2 < 0 the scalar potential for T diverges as T goes to zero or infinity and compactification is guaranteed [509, 513].

  15. 15.

    Holomorphic gauge invariant monomials span the moduli space of supersymmetric vacua. One such monomial is necessary to cancel the Fayet-Illiopoulos D-term (see Sect. 24.5).

  16. 16.

    We have also found solutions for the case with N = 4,  N f  = 7 which is closer to the “mini-landscape” benchmark models. Note, when N f  > N we may still use the same formalism, since we assume that all the \(Q,\tilde{Q}\) s get mass much above the effective QCD scale.

  17. 17.

    Note the parameter relation r = 15p in Table 24.2 is derived using U(1) A invariance and the assumption that no other fields with non-vanishing U(1) A charge enter into the effective mass matrix for hidden sector quarks. We have also allowed for two cases where this relation is not satisfied.

  18. 18.

    The fields χ and ϕ 1 cannot be expressed in polar coordinates as they receive zero VEV, and cannot be canonically normalized in this basis.

  19. 19.

    Note, with just dilaton and moduli SUSY breaking we can define \(\frac{F^{S}} {(S+\bar{S})} = -\sqrt{3}m_{3/2}\sin (\theta )e^{-i\phi _{S}},\; \frac{F^{T_{i}}} {(T_{i}+\bar{T}_{i})} = -\sqrt{3}m_{3/2}\cos (\theta )e^{-i\phi _{i}}\varTheta _{i}\) with ∑ i = 1 3 Θ i 2 = 1. Then A IJK (0) is independent of the moduli VEVs, only depending on the mixing angles, θ,  ϕ S ,  Θ i ,  ϕ i .

  20. 20.

    In estimating this result, we have assumed that the mass terms of the Pauli-Villars fields do not depend on the SUSY breaking singlet field ϕ 2, and that the modular weights of the Pauli-Villars fields obey specific properties.

  21. 21.

    This is due to the assumed modular weight of the field ϕ 2.

  22. 22.

    In racetrack models F S is suppressed by more than an order of magnitude. In these cases \(F_{\phi _{2}}\) is dominant [466].

  23. 23.

    I am assuming that only Kähler moduli, T i ,  i = 1, 2, 3, contribute to SUSY breaking, i.e. the complex structure moduli, U i ,  i = 1, 2, 3, have vanishing F terms.

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  1. Department of Physics, The Ohio State University, Columbus, Ohio, USA

    Stuart Raby

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Raby, S. (2017). SUSY Breaking and Moduli Stabilization. In: Supersymmetric Grand Unified Theories. Lecture Notes in Physics, vol 939. Springer, Cham. https://doi.org/10.1007/978-3-319-55255-2_24

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