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Minimalist Approach to Moduli Stabilization in Type IIB and M-theory

Minimalist Approach to Moduli Stabilization in Type IIB and M-theory. Konstantin Bobkov Ohio State University. KB, V. Braun, P. Kumar, S. Raby: hep-th 1003.1982 B. Acharya, KB: hep-th/0810.3285 B. Acharya, KB, G. Kane, P. Kumar, J. Shao: hep-th/0701034

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Minimalist Approach to Moduli Stabilization in Type IIB and M-theory

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  1. Minimalist Approach to Moduli Stabilization in Type IIB and M-theory Konstantin Bobkov Ohio State University KB, V. Braun, P. Kumar, S. Raby: hep-th 1003.1982 B. Acharya, KB: hep-th/0810.3285 B. Acharya, KB, G. Kane, P. Kumar, J. Shao: hep-th/0701034 B. Acharya, KB, G. Kane, P. Kumar, D. Vaman: hep-th/0606262 March 20 , Great Lakes Strings/SPOCK

  2. Outline • Introduction and motivation • Kahler moduli fixing in Type IIB on CY • Moduli fixing in M-theory on G2 • Soft SUSY breaking terms (if time allows)

  3. Introduction and motivation Why do we study 4-dim String/M-theory vacua? • Non-Abelian gauge symmetry • Chiral fermions • Hierarchical Yukawa couplings • Dynamical supersymmetry breaking • Gauge coupling unification Great! But can we make genuine predictions even in principle?

  4. By compactifying to 4D, we obtain a multitude of scalar fields – moduli, parameterizing internal metric deformations, brane positions, etc. • Their masses must be large enough to be compatible with observations (BBN => Mmod > O(10)TeV ) • In String/M-theory all masses and couplings are functions of the moduli vevs • Once the moduli are stabilized, all couplings are completely fixed and are computable in principle • Can we stabilize all moduli so that we can make predictions?

  5. Moduli fixing in Type IIB Calabi-Yau orientifolds • Comlex structure moduli and the axio-dilaton can be stabilized at tree-level by turning on fluxes Dasgupta, Rajesh, Sethi; Giddings, Kachru, Polchinski; Larsen, O’Connell, Robbins • Kahler moduli can be stabilized by including non-perturbative effects (instantons, gaugino condensation) Kachru, Kallosh, Linde, Trivedi • Can obtain de Sitter vacua once supersymmetry breaking effects are included: alpha prime corrections (Balasubramanian, Berglund, Conlon, Quevedo), matter F-terms plus D-terms (Lebedev, Nilles, Ratz), anti D3-branes (KKLT)

  6. Let us concentrate on the Kahler moduli Axions (periods of the 4-form, transform under a shift symmetry) Volumes of 4-cycles (fluctuations of the metric) Volume of the k-th divisor (supersymmetric 4-cycle) Integers, specifying the homology of the k-th divisor

  7. Common practice: assume as many divisors contributing to the superpotential as there are Kahlermoduli and go to a basis where • For a k-th divisor D to contribute the instanton must possess precisely two fermionic zero modes: • Necessary condition: c+(D,OD)-c-(D,OD) =1 • Sufficient: h02(D)=0, h01(D)=0

  8. Naively, one needs at least as many rigid divisors to contribute to the superpotential as there are Kahlermoduli. If h1,1+(X)=O(100) this is a rather daunting task • Furthermore, visible sector divisor Dvisible supports chiral matter => extra zero modes (recall Mirjam’s talk) • We already know that <Fi>=0 in the visible sector • Hence, despite the success of the KKLT approach the problem seems to comes back Visible sector charged chiral matter

  9. Is it possible to stabilize all Kahlermoduli with fewer than h1,1+(X)? • Consider an extreme case with a single divisor D • A divisor D corresponds to a zero locus of a section of a holomorphic line bundle O(D). If D is ample, its Poincare dual [w] is inside the Kahler cone.

  10. The Kahler cone is given by a set of linear inequalities in the two-cycle volumes ti. After a linear change of coordinates • An ample divisor is transverse (has a positive intersection with) to all holomorphic curves • Hence, if a single ample divisor contributes to the superpotential, the vevs of all moduli are automatically inside the Kahler cone

  11. Recall: • SUSY extremum (pick for definiteness Wflux< 0, A > 0) • where • All two- and four-cycle volumes as well as the volume of the Calabi-Yau are paremeterized by tD!

  12. Introduce anti D3-branes to break SUSY Suppressed by the warp factor

  13. A three-parameter example CICY: • Define the orientifold action: • Demand that the polynomials transform as • Kahler potential: • Ample divisor with c+- c- = 1, h01=0, h02=0:

  14. Recall the general expressions for the moduli vevs • For the specific example we get: • All moduli are parameterized by a SINGLE parameter! • This approach to Kahler moduli fixing is highly constraining and therefore potentially predictive!

  15. There may exist additional non-perturbative contributions. How robust are the vacua we found? • Clearly, when N >O(1)M , the extra contributions are exponentially suppressed. • This is especially true if the leading term is due to a gaugino condensate with N~O(10), while the remaining terms come from instantons, i.e. M=1.

  16. M-theory on G2 manifolds • Consider M-theory compactifications on singular 7-dim manifolds X with G2holonomy (required for N=1 SUSY) • There are “only” b3(X) complexifiedmoduli: • Non-Abelian gauge fields are localized on three-dimensional submanifolds along which there is an orbifold singularity. Acharya. hep-th/9812205, hep-th/0011089 fluctuations of the metric (periods of the associative 3-form) Axions (periods of the 3-form, transform under a shift symmetry)

  17. Example: locally, M-theory on is the 11-dim SUGRA coupled to a 7-dim SU(N) gauge theory on . • Chiral fermions are localized at point-like isolated singularities . Atiyah-Witten. hep-th/0107177, Acharya-Witten. hep-th/0109152 • A particle localized at p will be charged under the gauge group supported along the associative three-cycle Q if associative (supersymmetric) 3-cycle

  18. Bulk Kahler potential Beasley-Witten: hep-th/0203061; Acharya, Denef, Valandro: hep-th/0502060 • We know that the 7-dim volume V7 is a homogeneous function of the moduli siof degree 7/3 • These properties and everything that follows from them are all we use! In particular, no specific choice of the 7-dim volume needs to be made in order to compute the soft SUSY breaking terms!

  19. Kahler potential for matter fields • Because chiral matter is localized at points in the seven extra dimensions, we expect that the corresponding kinetic terms should be “largely independent of bulk moduli fields” • A single conical singularity gives only N of SU(N) which gives a trivial superpotential • When W=0 there is a one-to one correspondence between holomorphic gauge-invariant operators (HGIO) and D-flat directions Taylor, Luty, hep-th/9506098 • One cannot construct any HGIO from a single N => no D-flat directions => no local moduli

  20. Matter Kahler metric from dim reduction • The physics of a conical singularity in M-theory does not involve any new scale, aside from the 11d Planck scale so the natural frame is the 11d Einstein frame. • Lagrangian density in 11d frame • where • Integrating over X gives Peaked at the position of the matter multiplet Kinetic term is “largely independent of bulk moduli”

  21. Weil rescaling into the 4d Einstein frame gives • The kinetic term is non-trivial in the 4d Einstein frame – the standard frame in which we define the Kahler potential. • Read off the Kahler potential from the kinetic term

  22. Matter Kahler metric from the finiteness of the physical (normalized) Yukawa couplings • In N=1 D=4 supergravity • where (in M-theory on G2) the superpotential Yukawa is • There are well-defined local models where manifold X is non-compact , i.e. • is finite

  23. Kahler potential for the visible sector matter • For the visible sector matter fields, the Kahler potential may generally include a tree-level interaction term with the hidden sector • In the above we assumed that the flavor structure of is completely determined by the matrix . Here this form is motivated by the suppression of the FCNCs from the bottom up. Another way to suppress FCNCs is to assume that .

  24. Expand the fundamental associative three-form F in a basis of harmonic three-forms {fi} • Choose the basis of {fi} such that for all Poincare dual four-cycles {bi} the periods of *F are positive definite • Then, we can ensure that all four-cycle volumes are positive because

  25. Consider an associative three-cycle Q • Key point: we implicitly assume that we study any G2manifold X containing an associative three-cycle Q supporting a gaugino condensate, such that in the above basis of harmonic three-forms {fi} the integers specifying the homology of Q are all positive definite

  26. Moduli stabilization and SUSY breaking • Superpotential is generated by strong dynamics in the hidden sectors (PQ symmetry => W is non-pert.) • where the gauge kinetic function • Most economical way to fix moduli and get dS vacua ! • An effective meson field • For and hidden sector gauge groups and Nf =1 flavor: • , , , where Integers specifying the homology of the hidden sector 3-cycle

  27. Inverse Kahler metric (first consider ): “angular” (scale-invariant) coordinates on the moduli space These contraction properties are completely general and are crucial for our ability to minimize the scalar potential and perform explicit computations of the soft terms!

  28. N=1 D=4 SUGRA scalar potential

  29. Minimize the potential with respect to the moduli and the meson to obtain a system of N+1 coupled equations • Solve in the limit when the volume VQof the hidden sector associative cycle Q is large • Moduli vevs: where • Parameters satisfy a system of N equations Integers specifying the homology of the hidden sector 3-cycle

  30. Alternatively, introducing a basis of dual variables • We find upon minimizing the scalar potential • The four-form is dynamically fixed by the homology of Q • Recasting the volume V7 in terms of tiwe can express • In Type IIB even more explicit Integers specifying the homology of the hidden sector 3-cycle

  31. Key point: • To find the quantities relevant for particle physics we do not need to know the explicit values of and si ! The following contraction properties are all we need to know!

  32. Potential at the minimum in the leading order as a function of the meson vev • where such that

  33. For we obtain , , • Can we actually tune ? • Recall • When the 3-cycle is a lens space S3/Zq, the KK thresholds are: Ray-Singer torsions Discrete Wilson line data (winding numbers, etc)

  34. Coarse discrete tuning is indeed possible! • But can we REALLY fine tune the CC? • In general, the total superpotential • Each Ak depends on the discrete data of its 3-cycle • For an SU(N) supported along S3/Zq there are • inequivalent discrete Wilson lines T. Friedmann, hep-th/9506098 • Fluxless discretuum !

  35. When M-P=3 and the CC is tuned , the volume of the hidden sector cycle is • the contributions of the leading condensates are fixed! • Consider another rigid 3-cycle Y, supporting an SU(M`) gauge group. Its volume is also proportional to M ! • The extra contributions to the superpotential can be suppressed relative to the leading ones when M >>M`

  36. Gravitino mass is fixed once coarse tuning is done • Extract from • For typical values , • Obtain where Friedmann and Witten, hep-th/0211269

  37. Moduli masses: • one is heavy • N are light • moduli are probably heavy enough to decay before BBN • For generic toy examples we numerically obtain the light moduli cannot decay into the gravitinos. Large entropy production at late times, but before BBN, helps to avoid the gravitino problem Acharya, et.al, hep-ph/0804.0863

  38. Generalizing to the case when • The main difference is in the moduli vevs • where, and is the same and parameters can be found from

  39. Computation of soft SUSY breaking terms • Since we stabilized the moduli we can compute the terms in the soft-breaking lagrangian Nilles: Phys. Rept. 110 (1984) 1, Brignole et.al.: hep-th/9707209 • Tree-level gaugino masses • where the SM gauge kinetic function Integers specifying the homology of the visible sector 3-cycle

  40. Use the contraction rules to obtain • The tree-level gaugino mass is always suppressed for the entire class of dS vacua obtained in our model • Depending on we find typically • Vary • However, stays virtually inert!

  41. Anomaly mediated gaugino masses: • Use • In the limit Konishi anomaly contribution vanishes. In this case we obtain a particular example of mirage pattern for gaugino masses Gaillard et. al.: hep-th/09905122, Bagger et. al.: hep-th/9911029

  42. Assume a SUSY GUT broken to MSSM • Require zero CC at tree-level and to obtain tree-level plus anomaly gaugino masses: • From KK threshold corrections • Ray-Singer torsion • is fixed from gauge coupling unification • For the case when c=0, the Konishi anomaly is large such that the tree-level and anomaly partially cancel Friedmann and Witten, hep-th/0211269

  43. - problem • where originates from • -parameter can vanish if there is a discrete symmetry, used to solve the doublet-triplet splitting problem • Since is unknown • If then typically expect • Heavy higgsinos and in superpotential from Kahler potential. (Giudice-Masiero) physical Witten, hep-ph/0201018

  44. From Higgs-Higgsino loops, at the weak scale get an extra contribution to M1 and M2. Pierce, et.al, hep-ph/9606211; Gherghetta et.al, hep-ph/9904378; Arkani-Hamed, et.al hep-ph/0601041 used that and When m ~ O(1)m3/2 , such contributions can change the type of the LSP, depending on the sign of m !

  45. In N=1 D=4 sugra, the unnormalized scalar masses: • In our construction we obtain (using the homogeneity) • For generic values of c the scalars are very heavy • Tree-level mass vanishes in the limit c=1 – get tachyons in the scalar spectrum. • We restrict our analysis to the values of c when

  46. Trilinear couplings (again use the homogeneity): • Anomaly: Membrane instanton volume: Gaillard-Nelson. hep-th/0004170

  47. Total trilinear couplings for M-P=3 and zero CC: • We assumed that the third generation Yukawas arise from colliding singularities and thus dropped the volume terms • For generic values of 0<c<1: Friedmann and Witten, hep-th/0211269 Atiyah-Witten. hep-th/0107177

  48. Electroweak Symmetry Breaking • In most models REWSB is accommodated but not predicted, i.e. one picks and then finds , which give the experimental value of • Recall • Radiative EWSB is generic in our construction for a large range of values • However, generically get • Getting requires fine tuning up to 0.01% level

  49. Gaugino masses at the EW scale (generated by SOFTSUSY package) • Pure Wino LSP is rapidly excluded as c is increased • For typical values 0<c<1 get pure Bino LSP • In the limit c=1, for large can have light higgsinos mixing with gauginos

  50. Benchmark spectra (masses are in GeV) GUT scale input

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