An Introduction to Grand Unified Theories Dibyakrupa Sahoo LECTURE – 2 Department of Physics, and Institute of Physics and Applied Physics, Yonsei University, Seoul, South Korea 20 July 2017 1 Contents ¶ Introduction · Generic features of grand unification ¸ Examples of grand unified theories (GUTs) ¹ Phenomenological influence of GUT inspired ideas º Conclusion 2 Introduction Need to go beyond SM v Standard Model (SM) is not the final theory of particle physics. v SM has excellent experimental agreement, but many shortcomings also: A gravity not included, A no explanation for neutrino mass and neutrino oscillation, A absence of candidates for dark matter, A no logic for several generations of fermions and their mass hierarchy, A no physical understanding of the plethora of parameters in the Lagrangian, A absence of any mechanism to generate observed baryon asymmetry in our universe, A no explanation for the observed smallness of CP violating θ term QCD. v We shall study grand unification and grand unified theories (GUTs) with an eye on how they help alleviate some of the above ailments of SM. 3 Introduction What is grand unification? v Does the SM gauge group GSM SU(3)C SU(2)L U(1)Y provide an unified description of electroweak≡ phenomena?× × v SU(2)L U(1)Y is product of two disconnected sets of gauge × transformations: SU(2)L comes with coupling strength g and U(1)Y with g0 which are not related by theory, i.e. the ratio, g0=g = tan θW, has to fixed experimentally. v A truly unified theory of electroweak interaction would use only one independent coupling constant instead of the above two. v The strong interaction is also not unified with electroweak interaction in SM. v The primary aim of grand unification is to describe strong, weak and electromagnetic interactions by a unified coupling constant via embedding SM gauge group inside a larger symmetry group. v Grand Unification Unification of strong and electroweak interactions (without≡ gravity) 4 Introduction Some excellent resources to start with: v Paul Langacker, Grand Unified Theories and Proton Decay, Phys. Rep. 72, 185 (1981). v G. Ross, Grand Unified Theories (Benjamin, 1985). v C. Patrignani et al. (Particle Data Group), Chapter 16, Chin. Phys. C, 40, 100001 (2016). 5 Generic features of grand unification Bird’s eye view of grand unification v Basic idea: The SM gauge group GSM is embedded in a larger symmetry group G. So strong and electroweak interactions are merged into one interaction and are described by one coupling constant. v Basic goal: To break the unifying group G into GSM by spontaneous symmetry breaking(s). v Constraint: The unifying group G must be large enough to contain GSM as a subgroup. Some possible unifying groups are, A Infinite series of groups: SU(n), SO(n), Sp(2n), A Exceptional groups: F4, E6, E7, E8. v Byproducts: A Additional symmetries of G restrict some arbitrary features of SM, e.g. family structure and fermion mass hierarchy. A New gauge bosons mediating new symmetries emerge which predict some observable phenomena. 6 Generic features of grand unification Ratios of Coupling constants in the Unifying Limit v Assume: A Total number of fermions in our theory = Nf . © total number of left-handed and right-handed fermions = 2Nf . A G GSM = SU(3)C SU(2)L U(1)Y . 9 ⊃ × × G has a single coupling constant gG. So the other coupling constants gS, g, and g0 are related to gG. v Aim: To find out how the coupling constants are related to one another.1 v Step 1: a maximal unifying group Gmax = SU(2Nf ) G GSM. Algebraic9 relationships amongst coupling constants derived⊃ ⊃ in Gmax is automatically valid in G. v Step 2: Denote the 2Nf 2Nf dimensional diagonal generators of × Gmax, SU(3)C, SU(2)L and U(1)Y by TG, TC, TL and TY respectively. (α) (i) TG denotes the αth generator of G. Similarly, TA with A C, L, Y 2 f g denotes the ith generator of SU(3)C, SU(2)L and U(1)Y . 1For details see, Jatinder K. Bajaj and G. Rajasekaran, “The electroweak mixing angle in unified gauge theories”, Pramana, Vol.14, No.5, May 1980, pp.395-409. 7 Generic features of grand unification Ratios of Coupling constants in the Unifying Limit (α) (i) (β) (j) v Step 3: It is known that gGTG = gCTC , gGTG = gTL and (γ) (k) gGTG = g0TY . There is no reason why α, β, and γ (or i, j and k) should be same. (α) (i) (α) (α) v Step 4: Also TG and TA are normalized such that Tr TG TG is (i) (i) independent of α and Tr TA TA is independent of i, so that: (α) (β) αβ 2 (i) (j) ij 2 Tr TG TG = δ Tr (TG) , Tr TA TA = δ Tr (TA) . 2 2 2 2 2 2 2 2 = gG Tr (TG) = gS Tr (TC) = g Tr (TL) = g0 Tr TQ , ) g2 Tr T 2 α g2 G ( A) G , where α X for X G, C, L, Y . = 2 = 2 = X = 4 ) gA Tr (TG) αA π 2 f g Here for A = C, L, Y we have gA = gS, g, g0 and αA = αS, αL, αY respectively. 2 2 αY Tr (TC) αY Tr (TL) 2 © = 2 and = 2 = tan θW. αS Tr (TY ) αL Tr (TY ) 8 Generic features of grand unification Ratios of Coupling constants in the Unifying Limit v Step 5: Since U(1)em SU(2)L U(1)Y we can construct a 2Nf 2Nf ⊂ × × dimensional diagonal generator TQ of U(1)em out of electric charges of all the fermions. The fine structure constant for U(1)em is denoted by α and following the previous step we get, α Tr T 2 ( L) sin2 θ . α = 2 = W L Tr TQ v Predicting ratios of coupling constants and finding sin θW: For this we must proceed as follows, A construct the required generators, A square the generators and evaluate their trace, A predict the ratio of coupling constants/fine structure constants from the relations derived before. v Note: The ratios of coupling constants as predicted above are valid at the unification scale. To determine the ratios at other energy scales, renormalization effects must be considered. 9 Generic features of grand unification Ratios of Coupling constants in the Unifying Limit: Example with doublets of fermions v Assumption: Quarks and leptons form doublets under the unifying uα ν group, i.e. for quarks we have and for leptons we have e . dα e Our discussion is independent of number of generations. − v Generators: 16 16 diagonal matrices, × 0 uL dL uR dR 1 ν e ν e z }| { z }| { eL −L z }| { z }| { eR −R B 2 2 2 1 1 1 z}|{ z}|{ 2 2 2 1 1 1 z}|{ z}|{C TQ = diag B , , , , , , 0 , 1 , , , , , , , 0 , 1 C @ 3 3 3 − 3 − 3 − 3 − 3 3 3 − 3 − 3 − 3 − A | {z } | {z } L R ν e 0 uL dL eL −L 1 ν e z }| { z }| { z}|{ z}|{ uR dR eR −R B 1 1 1 1 1 1 1 1 z }| { z }| { z}|{ z}|{C TL = diag B , , , , , , , , 0, 0, 0, 0, 0, 0, 0 , 0 C @ 2 2 2 − 2 − 2 − 2 2 − 2 | {z }A | {z } R L 0 uL dL uR dR 1 ν e ν e z }| { z }| { eL −L z }| { z }| { eR −R B 1 1 1 1 z}|{ z}|{ 1 1 1 1 z}|{ z}|{C TC = diag B , , 0, , , 0, 0 , 0 , , , 0, , , 0, 0 , 0 C @ 2 − 2 2 − 2 2 − 2 2 − 2 A | {z } | {z } L R 10 Generic features of grand unification Ratios of Coupling constants in the Unifying Limit: Example with doublets of fermions v Traces: 2 16 Tr T , Tr T 2 2, Tr T 2 2. Q = 3 ( L) = ( C) = v Predictions: α Tr T 2 3 ( C) , α = 2 = 8 S Tr TQ α Tr T 2 3 ( L) sin2 θ 0.375. α = 2 = W = 8 = L Tr TQ v Note 1: The above results are independent of the actual unifying group as long as it is a subgroup of Gmax which in this case is SU(16) and as long as the doublet scheme is maintained. v Note 2: Large renormalization corrections are required to bring 2 down the value of sin θW from 0.375 to the experimentally measured value 0.23126. 11 Generic features of grand unification Ratios of Coupling constants in the Unifying Limit: Example with triplets of fermions v Assumption: The quarks and leptons form triplets under G, i.e. for 0 α 1 0 1 u νe quarks we have B dα C and for leptons we have B e C. @ A @ − A xα E+ v Generators: 24 24 diagonal matrices, × 0 1 uL dL xL ν e E+ uR dR xR ν e E+ eL −L L eR −R R Bz }| { z }| { z }| { z }| { z }| { z }| { C B 2 2 2 1 1 1 1 1 1 z}|{ z}|{ z}|{ 2 2 2 1 1 1 1 1 1 z}|{ z}|{ z}|{C TQ = diag B , , , , , , , , , 0 , 1 , +1 , , , , , , , , , , 0 , 1 , +1 C B 3 3 3 − 3 − 3 − 3 − 3 − 3 − 3 − 3 3 3 − 3 − 3 − 3 − 3 − 3 − 3 − C @| {z } | {z }A L R 0 1 u d νe e L L x L −L E+ u d x νe e E+ B L L R R R R −R R C Bz }| { z }| { z}|{ z}|{ C B 1 1 1 1 1 1 z }| { 1 1 z}|{ z }| { z }| { z }| { z}|{ z}|{ z}|{C TL = diag B , , , , , , 0, 0, 0, , , 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0 , 0 C B 2 2 2 2 2 2 2 2 C B − − − − | {z }C @| {z } R A L 0 1 uL dL xL ν e E+ uR dR xR ν e E+ eL −L L eR −R R Bz }| { z }| { z }| { z }| { z }| { z }| { C B 1 1 1 1 1 1 z}|{ z}|{ z}|{ 1 1 1 1 1 1 z}|{ z}|{ z}|{C TC = diag B , , 0, , , 0, , , 0, 0 , 0 , 0 , , , 0, , , 0, , , 0, 0 , 0 , 0 C B 2 − 2 2 − 2 2 − 2 2 − 2 2 − 2 2 − 2 C @| {z } | {z }A L R 12 Generic features of grand unification Ratios of Coupling constants in the Unifying Limit: Example with triplets of fermions v Traces: 2 2 2 Tr TQ = 8, Tr (TL) = 2, Tr (TC) = 3.