Reduction of Couplings and Gauge Yukawa Unification: in Finite Thories and in the MSSM

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1 Reduction of Couplings and Gauge Yukawa Unification: in Finite Thories and in the MSSM Myriam Mondragón Sven Heinemeyer Nick Tracas George Zoupanos CORFU 2013

2 What happens as we approach the Planck scale? or just as we go up in energy... How do we go from a fundamental theory to field theory as we know it? How are the gauge, Yukawa and Higgs sectors related at a more fundamental level? How do particles get their very different masses? What is the nature of the Higgs? Is there one or many? How this affects all the above? Where is the new physics??

3 Search for understanding relations between parameters addition of symmetries. N = 1 SUSY GUTs. Complementary approach: look for RGI relations among couplings at GUT scale Planck scale reduction of couplings resulting theory: less free parameters more predictive

4 Gauge Yukawa Unification GYU Remarkable: reduction of couplings provides a way to relate two previously unrelated sectors gauge and Yukawa couplings Reduction of couplings in third generation provides predictions for quark masses (top and bottom) Including soft breaking terms gives Higgs masses and SUSY spectrum Kubo, M.M., Olechowski, Tracas, Zoupanos (1995,1996,1997); Oehme (1995); Kobayashi, Kubo, Raby, Zhang (2005); Gogoladze, Mimura, Nandi (2003,2004); Gogoladze, Li, Senoguz, Shafi, Khalid, Raza (2006,2011); M.M., Tracas, Zoupanos (2013)

5 Gauge Yukawa Unification in Finite Theories Dimensionless sector of all-loop finite SU(5) model M top 178 GeV large tan β, heavy SUSY spectrum Kapetanakis, M.M., Zoupanos, Z.f.Physik (1993) M exp top 176 ± 18 GeV found in 1995 M exp top ±.09 GeV 2013 Very promising, a more detailed analysis was clearly needed Higgs mass GeV M exp H 126 ± 1 GeV 2012 Heinemeyer M.M., Zoupanos, JHEP, 2007, Phys.Lett.B (2013)

6 Gauge Yukawa Unification in the MSSM Possible to have a reduced system in the third generation compatible with quark masses large tan β, heavy SUSY spectrum Higgs mass GeV M.M., Tracas, Zoupanos, arxiv:

7 Reduction of Couplings see George Tsamis talk A RGI relation among couplings Φ(g 1,..., g N ) = 0 satisfies µ dφ/dµ = N β i Φ/ g i = 0. i=1 g i = coupling, β i its β function Finding the (N 1) independent Φ s is equivalent to solve the reduction equations (RE) β g (dg i /dg) = β i, i = 1,, N Reduced theory: only one independent coupling and its β function complete reduction: power series solution of RE g a = n=0 ρ (n) a g 2n+1

8 uniqueness of the solution can be investigated at one-loop valid at all loops Zimmermann, Oehme, Sibold (1984,1985) The complete reduction might be too restrictive, one may use fewer Φ s as RGI constraints Reduction of couplings is essential for finiteness finiteness: absence of renormalizations β N = 0 SUSY no-renormalization theorems only study one and two-loops guarantee that is gauge and reparameterization invariant to all loops

9 Finiteness A chiral, anomaly free, N = 1 globally supersymmetric gauge theory based on a group G with gauge coupling constant g has a superpotential W = 1 2 mij Φ i Φ j Cijk Φ i Φ j Φ k, Requiring one-loop finiteness β (1) g following conditions: T (R i ) = 3C 2 (G), i = 0 = γ j(1) i gives the 1 2 C ipqc jpq = 2δ j i g2 C 2 (R i ). C 2 (G) quadratic Casimir invariant, T (R i ) Dynkin index of R i, C ijk Yukawa coup., g gauge coup. restricts the particle content of the models relates the gauge and Yukawa sectors

10 One-loop finiteness two-loop finiteness Jones, Mezincescu and Yao (1984,1985) One-loop finiteness restricts the choice of irreps R i, as well as the Yukawa couplings Cannot be applied to the susy Standard Model (SSM): C 2 [U(1)] = 0 The finiteness conditions allow only SSB terms It is possible to achieve all-loop finiteness β n = 0: 1. One-loop finiteness conditions must be satisfied Lucchesi, Piguet, Sibold 2. The Yukawa couplings must be a formal power series in g, which is solution (isolated and non-degenerate) to the reduction equations

11 SUSY breaking soft terms

12 RGI in the Soft Supersymmetry Breaking Sector Supersymmetry is essential. It has to be broken, though... L SB = 1 6 hijk φ i φ j φ k bij φ i φ j (m2 ) j i φ i φ j + 1 M λλ + H.c. 2 h trilinear couplings (A), b ij bilinear couplings, m 2 squared scalar masses, M unified gaugino mass The RGI method has been extended to the SSB of these theories. One- and two-loop finiteness conditions for SSB have been known for some time Jack, Jones, et al. It is also possible to have all-loop RGI relations in the finite and non-finite cases Kazakov; Jack, Jones, Pickering

13 SSB terms depend only on g and the unified gaugino mass M universality conditions h = MC, m 2 M 2, b Mµ Very appealing! But too restrictive it leads to phenomenological problems: Charge and colour breaking vacua Incompatible with radiative electroweak breaking The lightest susy particle (LSP) is charged Brignole, Ibáñez, Muñoz Yoshioka; Kobayashi et al Possible to relax the universality condition to a sum-rule for the soft scalar masses better phenomenology. Kobayashi, Kubo, M.M., Zoupanos

14 Soft scalar sum-rule for the finite case Finiteness implies C ijk = g n=0 ρ ijk (n) g2n h ijk = MC ijk + = Mρ ijk (0) g + O(g5 ) If lowest order coefficients ρ ijk (0) and (m2 ) i j satisfy diagonality relations ρ ipq(0) ρ jpq (0) δj i, (m 2 ) i j = m 2 j δi j for all p and q. We find the the following soft scalar-mass sum rule, also to all-loops for i, j, k with ρ ijk (0) 0, where (1) is the two-loop correction =0 for universal choice ( m 2 i + m 2 j + m 2 k )/MM = 1 + g2 16π 2 (2) + O(g 4 ) Kazakov et al; Jack, Jones et al; Yamada; Hisano, Shifman; Kobayashi, Kubo, Zoupanos Also satisfied in certain class of orbifold models, where massive states are organized into N = 4 supermultiples

15 Several aspects of Finite Models have been studied SU(5) Finite Models studied extensively Rabi et al; Kazakov et al; López-Mercader, Quirós et al; M.M, Kapetanakis, Zoupanos; etc One of the above coincides with a non-standard Calabi-Yau SU(5) E 8 Greene et al; Kapetanakis, M.M., Zoupanos Finite theory from compactified string model also exists (albeit not good phenomenology) Ibáñez Criteria for getting finite theories from branes N = 2 finiteness Models involving three generations Hanany, Strassler, Uranga Frere, Mezincescu and Yao Babu, Enkhbat, Gogoladze Some models with SU(N) k finite 3 generations, good phenomenology with SU(3) 3 Ma, M.M, Zoupanos Relation between commutative field theories and finiteness studied Jack and Jones Proof of conformal invariance in finite theories Kazakov

16 SU(5) Finite Models We study two models with SU(5) gauge group. The matter content is {5 + 5} + 24 The models are finite to all-loops in the dimensionful and dimensionless sector. In addition: The soft scalar masses obey a sum rule At the M GUT scale the gauge symmetry is broken and we are left with the MSSM At the same time finiteness is broken The two Higgs doublets of the MSSM should mostly be made out of a pair of Higgs {5 + 5} which couple to the third generation The difference between the two models is the way the Higgses couple to the 24 Kapetanakis, Mondragón, Zoupanos; Kazakov et al.

17 The superpotential which describes the two models takes the form W = 3 [ 1 2 gu i 10 i 10 i H i + gi d 10 i 5 i H i ] + g23 u H 4 i=1 +g d H 4 + g d H a=1 g f a H a 24 H a + gλ 3 (24)3 find isolated and non-degenerate solution to the finiteness conditions The unique solution implies discrete symmetries We will do a partial reduction, only third generation

18 The finiteness relations give at the M GUT scale Model A Model B g 2 t = 8 5 g2 g 2 t = 4 5 g2 g 2 b,τ = 6 5 g2 m 2 H u + 2m 2 10 = M2 m 2 H d + m m2 10 = M2 g 2 b,τ = 3 5 g2 m 2 H u + 2m 2 10 = M2 m 2 H d 2m 2 10 = M2 3 m m2 10 = 4M2 3 3 free parameters: M, m 2 5 and m free parameters: M, m 2 5

19 Phenomenology The gauge symmetry is broken below M GUT, and what remains are boundary conditions of the form C i = κ i g, h = MC and the sum rule at M GUT, below that is the MSSM. Fix the value of m τ tan β M top and m bot We assume a unique susy breaking scale The LSP is neutral The solutions should be compatible with radiative electroweak breaking No fast proton decay We also Allow 5% variation of the Yukawa couplings at GUT scale due to threshold corrections Include radiative corrections to bottom and tau, plus resummation (very important!) Estimate theoretical uncertainties

20 We look for the solutions that satisfy the following constraints: Right masses for top and bottom fact of life The decay b sγ fact of life The branching ratio B s µ + µ fact of life Cold dark matter density Ω CDM h 2 loose constraint FeynHiggs MicroOmegas MicroOmegas MicroOmegas The anomalous magnetic moment of the muon g 2 see what we get The lightest MSSM Higgs boson mass The SUSY spectrum FeynHiggs, Suspect, FUT

21 TOP AND BOTTOM MASS M top [GeV] FUTA µ > 0 µ < 0 FUTB µ < 0 µ > 0 m bot (M Z ) FUTA FUTA µ > 0 µ < 0 FUTB FUTB M [GeV] M [GeV] FUTA: M top GeV Theoretical uncertainties 4% FUTB: M top GeV b and τ included, resummation done FUTB µ < 0 favoured

22 New experimental data We use the experimental values of M H to compare with our previous results (M H = GeV, 2007) and put extra constraints M exp H = 126 ± 2 ± 1 2 GeV theoretical, 1 GeV experimental We also use the current experimental value of B µ + µ Upper limit October 2012 BR(B s µ + µ ) = Experimental value November 2012 BR(B s µ + µ ) = ( (stat) (syst)) 10 9 We can now restrict (partly) our boundary conditions on M

23 Higgs mass M Higgs [GeV] M[GeV] FUTB: M Higgs = GeV with B Physics constraints Uncertainties ±3 GeV (FeynHiggs) Heinemeyer, M.M., Zoupanos (2007); Heinemeyer, M.M., Zoupanos (2013)

24 S-SPECTRUM M h =126±1 GeV M h =126±3 GeV masses [GeV] 1000 masses [GeV] 1000 h H A H ± stop 1,2 sbot 1,2 gl stau 1,2 cha 1,2 neu 1,2,3,4 h H A H ± stop 1,2 sbot 1,2 gl stau 1,2 cha 1,2 neu 1,2,3, SUSY spectrum with B physics constraints Challenging for LHC

25 Results When confronted with low-energy precision data only FUTB µ < 0 survives M top 173 GeV 4% M exp top exp = (173.2 ± 0.9)GeV m exp m bot (M Z ) 2.8 GeV 8 % bot (M Z ) = (2.83 ± 0.10)GeV M Higgs GeV 3 GeV M exp Higgs = 126 ± 1 tan β s-spectrum > 500 GeV In progress 3 families with discrete symmetry neutrino masses via R consistent with the exp bounds

26 Reduction of couplings in the MSSM The superpotential W = Y t H 2 Qt c + Y b H 1 Qb c + Y τ H 1 Lτ c + µh 1 H 2 with soft breaking terms, L SSB = φ m 2 φ φ φ + [ m 2 3 H 1H i=1 ] 1 2 M iλ i λ i + h.c + [h t H 2 Qt c + h b H 1 Qb c + h τ H 1 Lτ c + h.c.], then, reduction of couplings implies β Yt,b,τ = β g3 dy t,b,τ dg 3

27 Boundary conditions at the unification scale are given by Y 2 t g 2 3 4π = c 1 4π + c 2 Y 2 b g 2 3 4π = p 1 4π + p 2 ( g 2 ) 2 3 4π (1) ( g 2 ) 2 3 4π (2) c 1 = Kτ = Kτ 35 p 1 = Kτ = Kτ 35 c 2 = K τ Kτ K τ 3 4π K τ Kτ 2 p 2 = K τ Kτ K τ 3 4π K τ Kτ 2 where K τ = Y 2 τ /g 2 3 Y τ not reduced, its reduction gives imaginary values

28 Soft breaking terms The reduction of couplings in the SSB sector gives the following boundary conditions at the unification scale Yt 2 = c 1 g3 2 + c 2g3 4 /(4π) and Y b 2 = p 1g3 2 + p 2g3 4 /(4π) h t,b = MY t,b, m 2 3 = Mµ, M is unified gaugino mass m 2 H 2 + m 2 Q + m2 t c = M2, m 2 H 1 + m 2 Q + m2 b c = M2,

29 Allowed values of K τ tan β K τ Radiative corrections coming from SUSY breaking to the bottom and tau mass can be large (especially to bottom) They depend on the values of the SUSY masses (M), and tan β, and modify the allowed values of K τ = Y 2 τ /g 2 3.

30 M top vs M bot M b (M Z )[GeV] M t [GeV] Requiring the top and bottom masses within experimental bounds further constrains K τ = Yτ 2 /g3 2 with µ < 0. No such region exists for µ > 0 The central value (green dashed lines), 1 and 2σ deviation (orange and magenta lines respectively)

31 Higgs mass vs K τ (aka the bear ) M Higgs K τ SUSY spectrum and the Higgs mass, given the GYU conditions constrained by the third generation of quark masses

32 SUSY spectrum 1e Mass [GeV] 1000 M A, M H, M H+- stop1,2 sbot1,2 stau1,2 charginos neutralinos 100 M Higgs Heavy spectrum, 1.3TeV < M < 10TeV Will be constrained further by B physics and CDM

33 Results GYU in MSSM Possible to have reduction of couplings in MSSM Up to know only attempted in SM or in GUTs Reduced system further constrained by phenomenology: compatible with quark masses with µ < 0 SUSY spectrum, large tan β Higgs mass GeV

34 Conclusions Reduction of couplings: powerful implies Gauge Yukawa Unification Finiteness, interesting and predictive principle reduces greatly the number of free parameters completely finite theories i.e. including the SSB terms, that satisfy the sum rule Confronting the SU(5) FUT models with low-energy precision data does distinguish among models FUTB favoured Possible to have reduction of couplings in MSSM only solutions for µ < 0 compatible with quark masses Heavy SUSY spectrum, large tan β s-spectrum starts above 500 GeV prediction for the Higgs M h GeV Detailed study of SUSY masses and Higgs decays in progress