Grand Unification. Strong, weak, electromagnetic unified at Q M X M Z Simple group SU(3) SU(2) U(1) Gravity not included

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1 Pati-Salam, 73; Georgi-Glashow, 74 Grand Unification Strong, weak, electromagnetic unified at Q M X M Z Simple group G M X SU(3) SU() U(1) Gravity not included (perhaps not ambitious enough) α(q ) α 3 Couplings meet at M X GeV (w/o SUSY) (works much better with SUSY M X GeV) α α 1 LE MZ desert unification M X Q P59 Spring, 013 1

2 e + u c u e + u c u e + u c u X Y Y d u u u d u d u u q X = 4 3 e + d c d e + d c d q Y = 1 3 X Y u u d u u d q, q, l, l unified (in same multiplets) Charge quantization (no U(1) factors) Typeset by FoilTEX 1 Proton decay mediated by new gauge bosons, e.g. p e + π 0 (other modes in SUSY GUTS) τ p M 4 X α m 5 p 109 yr for M X GeV (10 38 yr in SUSY, but faster p νk + ) P59 Spring, 013

3 The Georgi-Glashow SU(5) Model SU(5) M X SU(3) SU() U(1) M Z SU(3) U(1) Q In GUTs, SUSY, convenient to work with left-chiral fields for particles and antiparticles Right-chiral related by CP (or CP T ) d c L CP d R, u c L CP u R, e + L CP e R P59 Spring, 013 3

4 Each family in of SU(5) SU() ( ν 0 L e 0 L X, Y d 0c L ) ( e +0 L X, Y u 0 L d 0 L u 0c L ) SU() 5 10 ψ 0 weak eigenstate (mixture of mass eigenstates) Color (SU(3)) indices suppressed P59 Spring, 013 4

5 SU(5) Rank 4; fundamental 5 5: L i = λ i /, i = 1 4 Upper left 3 3 block SU(3) (a, b = 1,, 3 α and β) Lower right block SU() (a, b = 4, 5 r and s) 3 Normalized hypercharge generator Y 1 5 Y : L Y1 = 1 60 diag( 3 3), Tr L Y 1 = 1 Twelve additional generators (gauge bosons) L i αr or Li rα P59 Spring, 013 5

6 4 gauge bosons (adjoint representation) (SU(3), SU(), y = q t 3 ) 4 (8, 1, 0) }{{} G β α + (1, 3, 0) }{{} + (1, 1, 0) }{{} W ±,W 0 B + (3,, 5 6 ) }{{} A r α + (3,, ) }{{} A α r 1 new gauge bosons, A α r and Ar α, carry flavor and color A α 4 Xα [3,, q X = 4 3], A 4 α X α [3,, q X = 4 3] A α 5 Y α [3,, q Y = 1 3], A 5 α Ȳ α [3,, q Ȳ = 1 3], (X, Y ) ; ( X, Ȳ ) under SU() P59 Spring, 013 6

7 A = = 4 i=1 A i λi G 1 1 B 30 G 1 G 3 1 X 1 Ȳ 1 G 1 G B 30 G 3 X Ȳ G 1 3 G 3 G 3 3 B 30 X 3 Ȳ 3 X 1 X X 3 W 0 + 3B 30 W + Y 1 Y Y 3 W W 0 + 3B 30 (W ± = W 1 iw ) P59 Spring, 013 7

8 Fermions still in highly reducible representation: each family of L-fields in (antifundamental and antisymmetric) }{{} 5 (3, 1, 1 χ a 3 ) + (1,, 1 }{{} ) }{{} χ α χ r }{{} 10 (3, 1, ψ ab = ψ ba 3 ) 1 + (3,, }{{} 6 ) + (1, 1, 1) }{{}}{{} ψ 45 ψ αβ ψ αr (a, b = 1,..., 5; α, β = 1,, 3; r = 4, 5) SU() ( ν 0 L e 0 L X, Y d 0c L ) ( e +0 L X, Y u 0 L d 0 L u 0c L ) SU() 5 10 P59 Spring, 013 8

9 χ a = ( d c1 d c d c3 e ) T ν e L ψ ab = 1 0 u c3 u c u 1 d 1 u c3 0 u c1 u d u c u c1 0 u 3 d 3 u 1 u u 3 0 e + d 1 d d 3 e + 0 L (family indices and weak-eigenstate superscript 0 suppressed) P59 Spring, 013 9

10 Fermion Gauge Interactions L f = χ a i ( Dχ) a + ψ ab i ( Dψ) ab = L SM f + L XY f (D µ χ) a = µ χ a i g 5 (A µ ) a b χb (D µ ψ) ab = µ ψ ab + i g 5 (A µ ) c a ψ cb + i g 5 (A µ ) d b ψ ad P59 Spring,

11 [ ] 8 L f = g 5 ū G iλi u + d G iλi d i=1 }{{} QCD with g s = g 5 [ 3 + g 5 (ū d) L W iτ i ( u d ) + ( ν e ē) L W iτ i i=1 L L }{{} weak SU() with g = g 5 L U(1)Y L X,Y + additional families ( νe e ) ] u α G i µ i g s λi αβ γµ W i µ i g τ i 1 γ5 γ µ B µ ig 1 γ5 yγ µ u β P59 Spring,

12 L U(1)Y = [ 3 5 g 5 1 ( ν L Bν L + ē L Be L ) (ū L Bu L + d L Bd L ) + Bu R 1 ] d R Bd R ē R Be R 3ūR 3 }{{} weak U(1) Y with g = 3 5 g 5 3 ( 5 Y is properly normalized generator, Tr(Li L j ) = δij ) P59 Spring, 013 1

13 L XY f = g 5 [ē+ R Xα d Rα + ē + L Xα d Lα ν c R Y α d Rα ē + L Y α ] u Lα } {{} leptoquark vertices g [ ] 5 ɛ αβγ ū c Lγ X α u Lβ + ɛ αβγ ū c Lγ Ȳ α d Lβ +h.c. } {{} diquark vertices e + u c u e + u c u e + u c u X Y Y d u u u d u d u u q X = 4 3 e + d c d e + d c d q Y = 1 3 X Y u u d u u d P59 Spring,

14 Proton Decay Combination of leptoquark and diquark vertices leads to baryon (B) and lepton (L) number violation (B L conserved) p e + ūu, e + dd p e + π 0, e + ρ 0, e + ω, e + η, e + π + π, p ν du p νπ +, νρ +, νπ + π 0, Expect τ p M 4 X,Y α 5 m5 p (cf τ µ m 4 W /g4 m 5 µ ) τ p yr and α 5 α M X,Y (GUT scale) where α 5 g 5 4π GeV e + u c u e + u c u e + u c u X Y Y d u u u d u d u u e + d c d e + d c d Also bound neutron decay Additional mechanisms/modes in SUSY GUT X u u d Y u u d P59 Spring,

15 Spontaneous Symmetry Breaking Introduce adjoint Higgs, Φ = 4 i=1 φi λ i V (Φ) = µ Tr(Φ ) + a 4 [ Tr(Φ ) ] + b Tr (Φ4 ) (have assumed Φ Φ symmetry) Can take 0 Φ 0 = diagonal by SU(5) transformation 0 Φ 0 = 0 for µ < 0 P59 Spring,

16 For b > 0 minimum is at Φ = ν Φ , ν Φ = µ 15a + 7b with SU(5) M X SU(3) SU() U(1) M X = M Y = 5 8 g 5 ν Φ (Need a > 7 15 b for vacuum stability. SU(5) M X SU(4) U(1) for b < 0) P59 Spring,

17 To break SU() U(1), introduce (fundamental) Higgs 5 H a = H α φ + φ 0 H α = (3, 1, 1 3 ), color triplet with q H = 1/3 ( φ + φ 0 ) = (1,, 1 ), SM Higgs doublet Problem 1: Need φ 0 = ν, ν 46 GeV (weak scale) ν Φ Problem : Need M H GeV 10 1 M φ to avoid too fast proton decay mediated by H α (doublet-triplet splitting problem) P59 Spring,

18 V (Φ, H) = µ Tr(Φ ) + a 4 [ Tr(Φ ) ] b + Tr (Φ4 ) + µ 5 H H + λ 4 ( H H ) + αh H Tr ( Φ ) + βh Φ H (µ 5, λ, α terms don t split doublet from triplet, but β does) Can satisfy ν ν Φ and M H M φ, but requires fine-tuned cancellations P59 Spring,

19 Yukawa Couplings No Φ couplings to χ a, ψ ab allowed by SU(5), but can have L Y uk =γ mn χ at m C ψ ab H b + Γ mn ɛ abcde ψ T mab C ψ ncd H e + κ mn ν ct ml Cχa n H a + h.c. (γ, Γ = Γ T are family matrices; m, n = family indices; ɛ = antisymmetric with ɛ 1345 = 1; C= charge conjugation matrix, with ψ T L Cη L = ψ c R η L) Fermion masses from H a = δ 5 ν a P59 Spring,

20 L de = d L M d d R ē + L M et e + R +h.c. = d L M d d R ē L M e e R +h.c. M d = M et = 1 νγ d and e have same mass matrices up to transpose m d = m e, m s = m µ, m b = m τ at M X Appears to be a disaster, but masses run (mainly from gauge loops) P59 Spring, 013 0

21 Gluonic loops make quark masses larger at low energies [ ] [ md (Q ) md (M X ln = ln ) ] [ ] 4 m e (Q ) m e (MX }{{ ) F ln αs (Q ) α 5 (MX } ) + 3 [ ] 4F ln α1 (Q ) α 5 (MX ) 0 m b /m τ 5/1.7; works reasonably well (ordinary and SUSY GUT) But m e m µ 1 00 m d m s 1 0 (need more complicated Higgs sector) is failure of model Γ mn term gives independent M u (M u = M ν Dirac in SO(10) plus other (model dependent) relations) P59 Spring, 013 1

22 Gauge Unification Gauge couplings unified at M X (simple group) g 3 g s = g 5 g = g = g 5 5 g 1 = 3 g = g 5 Generators must have same normalization Tr(L i L j ) = δij 3 5 Y is SU(5) generator, with g Y = 5 3 g }{{} g Y P59 Spring, 013

23 Weak mixing angle at M X sin θ W (M X ) = g g + g = 3/ /5 = 3 8 SU(5) broken below M X (and some particles decouple) SU(3) SU() U(1) couplings run at different rates α(q ) α 3 LE α 1 α desert unification M Z M X Q P59 Spring, 013 3

24 Running α i = g i /4π b 1 = 1 α i (M Z ) = 1 α i (M X ) 4πb i ln M Z MX F 1π, b = 1 [ 16π 3 4F 3 ], b 3 = 1 16π (F = number of families; small Higgs contribution ignored) α(m Z ) α s (M Z ) = 3 8 [ 1 11α(M Z ) π ln M X ] MZ [ 11 4F 3, ŝ Z (M Z ) = α(mz ) 9α s (MZ ) ] With corrections: M X GeV, ŝ Z (M Z ) 0.1 Reasonable zeroth order prediction in 1970 s Present: ŝ Z (M Z ) 0.3; M X GeV (proton decay) SUSY P59 Spring, 013 4

25 Beyond SU(5) Larger gauge groups: SU(5) SO(10) E 6 SO(10) SU(5) U(1) χ Extra Z from U(1) χ may survive to TeV scale Family in one IRREP: 16 }{{} SO(10) }{{} SU(5) family + 1 }{{} ν R ν R ν c L = right-handed (singlet, sterile) neutrino M u = M ν Dirac in simplest Higgs scheme Seesaw for large Majorana mass (need Higgs 16) Right mass scales, but small mixings in simplest schemes Leptogenesis P59 Spring, 013 5

26 E 6 SO(10) U(1) ψ (second possible Z ) Family: 10 = }{{} 7 }{{} 16 E 6 SO(10) ( ) ( ) E 0 E 0 E + E } L {{ R} both doublets }{{} exotics + D L + D R }{{} both singlets 1 = S L (no charge, similar to ν c L ) P59 Spring, 013 6

27 Supersymmetric extensions: Extended spectrum Two Higgs W Φ,H = µ Φ Tr (Φ ) + λ Φ Tr (Φ 3 ) + µ H H u H d + λ ΦH H u ΦH d Different b i factors better gauge unification agreement M X GeV τ (p e + π 0 ) yr However, new dimension-5 operators involving superpartners may yield too rapid p νk +, etc. -1! i ! i Standard Model! 1 "1! "1! 3 " log 10 µ (GeV) Supersymmetric Standard Model M SUSY = M Z! 1 "1! "1! 3 " log 10 µ (GeV) P59 Spring, 013 7

28 Extra dimensions: new GUT-breaking mechanisms from boundary conditions String compactifications: direct compactifications have some ingredients of GUTs String MSSM (+ extended?) in 4D without intermediate GUT phase avoids doublet-triplet problem, and need for large representations for GUT breaking and fermion/neutrino masses/mixings P59 Spring, 013 8

29 Additional Implications Fermion mass textures: often done in SO(10) context, but need larger Higgs representations and family symmetries Baryogenesis: Baryon excess could be generated by out of equilibrium decays of H α (insufficient in minimal model) However, B L conserved and asymmetry with B L = 0 wiped out by electroweak sphalerons ( leptogenesis, EW baryogenesis, Affleck-Dine, ) Magnetic monopoles: Topologically stable gauge/higgs configurations with M M M X /α Greatly overclose Universe unless subsequent inflation P59 Spring, 013 9