Elements of Grand Unification

Transcription

1 Elements of Grand Unification Problems with the Standard Model The Two Paths Grand Unification The Georgi-Glashow SU(5) Model Beyond SU(5) arger Groups Supersymmetry Extra Dimensions Additional Implications To GUT or not to GUT

2 Reviews P, Grand Unified Theories and Proton Decay, Phys. 7, 185 (1981) Rep. G. Ross, Grand Unified Theories (Benjamin, 1985)

3 Problems with the Standard Model Standard model: SU(3) SU() U(1) (extended to include ν masses) + general relativity Mathematically consistent, renormalizable theory Correct to cm However, too much arbitrariness and fine-tuning (O(0) parameters, not including ν masses/mixings, which add at least 7 more, and electric charges)

4 Gauge Problem complicated gauge group with 3 couplings charge quantization ( q e = q p ) unexplained Possible solutions: strings; grand unification; magnetic monopoles (partial); anomaly constraints (partial) Fermion problem Fermion masses, mixings, families unexplained Neutrino masses, nature? CP violation inadequate to explain baryon asymmetry Possible solutions: strings; brane worlds; family symmetries; compositeness; radiative hierarchies. New sources of CP violation.

5 Higgs/hierarchy problem Expect M H = O(M W ) higher order corrections: δm H /M W 1034 Possible solutions: supersymmetry; dynamical symmetry breaking; large extra dimensions; ittle Higgs Strong CP problem Can add θ g 3π s F F to QCD (breaks, P, T, CP) d N θ < 10 9 but δθ weak 10 3 Possible solutions: spontaneously broken global U(1) (Peccei- Quinn) axion; unbroken global U(1) (massless u quark); spontaneously broken CP + other symmetries

6 Graviton problem gravity not unified quantum gravity not renormalizable cosmological constant: Λ SSB = 8πG N V > Λ obs (10 14 for GUTs, strings) Possible solutions: supergravity and Kaluza Klein unify strings yield finite gravity. Λ?

7 The Bang The Two Paths: Unification or Compositeness unification of interactions grand desert to unification (GUT) or Planck scale elementary Higgs, supersymmetry (SUSY), GUTs, strings possibility of probing to M P and very early universe hint from coupling constant unification tests light (< GeV) Higgs (EP, TeV, HC) absence of deviations in precision tests (usually) supersymmetry (HC) possible: m b, proton decay, ν mass, rare decays SUSY-safe: Z ; seq/mirror/exotic fermions; singlets variant versions: large dimensions, low fundamental scale, brane worlds

8 The Whimper onion-like layers composite fermions, scalars (dynamical sym. breaking) not like to atom nucleus +e p + n quark at most one more layer accessible (HC) rare decays (e.g., K µe) severe problem no realistic models effects (typically, few %) expected at EP & other precision observables (4-f ops; Zb b; ρ 0 ; S, T, U) anomalous V V V, new particles, future W W W W recent variant: ittle Higgs

9 Grand Unification Pati-Salam, 73; Georgi-Glashow, 74 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) Couplings meet at M X GeV (w/o SUSY) (works much better with SUSY M X GeV)

10 q X = 4 3 q Y = 1 3 q, q, l, l unified (in same multiplets) Charge quantization (no U(1) factors) 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 1030 yr for M X GeV (10 38 yr in SUSY, but faster p νk + )

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

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

13 The SU(n) Group (Type A ie algebra; n 1 generators; rank (# diagonal generators) = n 1 ) Group element: U( β) = e i β T β = n 1 real numbers T = n 1 generators (operators) = n 1 representation matrices of dimension m m satisfying same ie Algebra (commutation rules) as the T U (m) i β ( β) e form m m dimensional representation of group (will suppress (m))

14 Fundamental representation (n): i λi are n n Hermitian matrices with Tr i = 0 e i β are n n unitary matrices with det = 1 Normalize to Tr ( i j ) = δ ij For n =, λ i τ i (Pauli matrices) For n = 3, λ i Gell-Mann matrices Arbitrary n: construct i using non-hermitian basis Define matrices a b, where (a, b) label the matrix (they are not indices) by ( ) a b ( a c cd b) = d δa d δc b 1 n δa b δc d ( a b) = b a non-hermitian for a b a b +b a and i a b b a are Hermitian

15 ( ) For a b, a b lowering operators ba = 1, others 0, e.g., SU() raising and 1 = τ 1 + iτ = ( ) 1 = τ 1 iτ = ( ) a a is diagonal with ( ) a a a = n 1 a n ( a a) b b = 1 n for b a The n matrices a a are not independent ( a a a alternate diagonal basis = 0) use k 1 k(k + 1) diag (1 1 1 }{{} k k 0 } {{ 0} ), k = 1,..., n 1 n k 1

16 ie algebra: [ a b, c d] = δ a d c b δc b a d Same algebra for generators T a b and other representations Field transformations: et ψ c, c = 1,..., n transform as fundamental n, i.e. implies [ T a b, ψc] = ( a ) c b d ψd ψ c e i β T ψ c e i β T = U( β) c d ψd = (e ) i c β d ψ d = ( ) e i β c ψ

17 Antifundamental χ a transforms as n, [ T a b, χ ] ( c = a b (n ) ) d χ c d where a b (n ) = ( a ) T b = b a ( a b (n ) ) d ( a c b (n ) ) cd (χ c ɛ cd1 d n 1 φ d 1 d n 1, totally antisymmetric products of n 1 fundamentals) χ c e i β T χ c e i β T = χ d U ( β) d c = ( χe i β ) c χ c ψ c is SU(n) invariant Adjoint field (n n ): φ a b (U( β)φu ( β) ) a b

18 SU(n) Gauge Theory n 1 Hermitian generators T i, i = 1,, n 1 and corresponding gauge fields A i Define n n gauge matrix A = n 1 i=1 i A i = λ i A i (A a b (A) ab = Ab a n = : A = are non-hermitian for a b) A 3 A 1 A 1 A3, A 1 = A1 ia n = 3: A = A 3 + A8 6 A 1 A 1 3 A 1 A 3 1 A 3 A3 + A8 6 A 3 A 8 6, A 1 = A1 ia A 1 3 = A4 ia 5 A 3 = A6 ia 7

19 Covariant Derivatives Fundamental: [D µ ψ] a = = [ ] µ δ a b + ig( A µ ) a b [ µ δ a b + i g ] (A µ ) a b ψ b ψ b Antifundamental: [D µ χ] a = = [ ] µ δ a b + ig( A µ (n )) b a [ µ δ a b i g ] (A µ ) b a χ b χ b

20 Antisymmetric n n representation: et ψ ab = ψ ba be n(n 1) fields with [ T a b, ψcd] = ( a b )c e ψed ( a b )d e ψce [D µ ψ] ab = µ ψ ab + i g (A µ ) a c ψcb + i g (A µ ) b d ψad (singlet for n = ; 3 for n = 3; 10 for n = 5)

21 The Georgi-Glashow SU(5) Model 5 1 = 4 generators Tb a 1 5 δa b T c c, a, b, c = 1,..., 5 ( 5 a=1 [T a a 1 5 δa a T c c ] = 0) SU(5)contains SU(3) SU() U(1) SU(3): T α β 1 3 δα β T γ γ, α, β, γ = 1,, 3 SU(): Ts r 1 δr s T t t, r, s, t = 4, 5 U(1): 1 3 T α α + 1 T r r = 1 3 (T T + T 3 3) + 1 (T T 5 5) Q = T 3 + Y = 1 (T 4 4 T 5 5 ) + Y (Q and Y are part of simple group cannot pick arbitrarily (charge quantization))

22 4 gauge bosons (adjoint representation) decompose as 4 (8, 1, 0) }{{} G α β + (1, 3, 0) }{{} + (1, 1, 0) }{{} W ±,W 0 B under SU(3) SU() U(1) + (3,, 5 6) }{{} A α r + (3,, + 5 6) }{{} A r α 1 new gauge bosons, A α r, Ar α, α = 1,, 3, r = 4, 5, carry flavor and color A 4 α X α [3, Q X = 4 3] A α 4 X α [3, Q X = 4 3] A 5 α Y α [3, Q Y = 1 3] A α 5 Ȳ α [3, Q Ȳ = 1 3] (X, Y ) ; ( X, Ȳ ) under SU()

23 A = = 4 i=1 A i λi G 1 1 B 30 G 1 G 1 3 G 1 G B 30 G 3 X 1 Ȳ 1 X Ȳ G 3 1 G 3 G 3 3 B 30 X 3 Ȳ 3 X 1 X X 3 W 3 + 3B 30 W + Y 1 Y Y 3 W W 3 + 3B 30 with W ± = W 1 iw

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

25 10 : ψ ab = 1 5 : χ a = d c 1 d c d c 3 e ν e 0 u c 3 u c u 1 d 1 u c 3 0 u c 1 u d u c u c 1 0 u 3 d 3 u 1 u u 3 0 e + d 1 d d 3 e + 0 (family indices and weak-eigenstate superscript 0 suppressed)

26 R-fields (CP conjugates): ψr c Cψ T C = (representation dependent) charge conjugation matrix, e.g., iγ γ 0 5 : χ ca R = Cχ a T = 10 : ψ c Rab = T 1 Cψab = d 1 d d 3 e + ν c e 0 u 3 u u c 1 d c 1 u 3 0 u 1 u c d c u u 1 0 u c 3 d c 3 u c 1 u c u c 3 0 e d c 1 d c d c 3 e 0 R R

27 Fermion Gauge Interactions [ ] 8 f = g 5 ū G iλi u + d G iλi d i=1 }{{} (QCD with g s = g 5 ) [ 3 + g 5 (ū d) W iτ i ( u d ) + ( ν e ē) W iτ i i=1 }{{} (Weak SU() with g = g 5 ) U(1)Y X,Y + additional families ( νe e ) ]

28 U(1)Y = [ 3 5 g 5 1 ( ν Bν + ē Be ) (ū Bu + d Bd ) + 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(i j ) = δij

29 X,Y = g 5 [ d Rα X α e + R + d α X α e + d Rα Ȳ α ν c R ū α Ȳ α e + ] } {{} + g [ 5 ɛ αβγ ū cγ (eptoquark vertices) X α u β + ɛ αβγū cγ } {{ } (Diquark vertices) ] Ȳ α d β + H.C. q X = 4 3 q Y = 1 3

30 Proton Decay Combination of leptoquark and diquark vertices leads to baryon (B) and lepton () number violation (B 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 ) where α 5 g 5 4π τ > p yr and α 5 α M > X,Y GeV (GUT scale) Also bound neutron decay Additional mechanisms/modes in SUSY GUT

31 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

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

33 To break SU() U(1), introduce (fundamental) Higgs 5 H a = Hα 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 to give φ 0 = ν 0 with ν 0 46 GeV (weak scale) ν Problem : Need M H > GeV > 10 1 M φ to avoid too fast proton decay mediated by H α (doublet-triplet splitting problem)

34 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 0 ν and M H M φ, but requires fine-tuned cancellations

35 Yukawa Couplings No Φ couplings to χ a, ψ ab allowed by SU(5), but can have Yuk = γ mn χ T ma C ψab n H b + Γ mn ɛ abcde ψ T ab m C ψcd n He + HC (γ, Γ are family matrices (Γ symmetric); m, n = family indices; ɛ = antisymmetric with ɛ 1345 = 1; C = charge conjugation matrix, with ψ T Cη = ψ c R η ) Fermion masses from 0 H a 0 = ν 0 δ 5 a

36 First term: d M d d R ē + M et e }{{ + R + HC } ē M e e R M d = M et = 1 ν 0γ d and e have same mass matrices up to transpose (M d = M et M e used in lopsided models, but harder to implement in SO(10)) m d = m e, m s = m µ, m b = m τ at M X Appears to be a disaster, but

37 These run, mainly from gauge loops. masses larger at low energies Gluonic loops make quark [ md (Q ] ) ln m e (Q ) [ md (MX = ln ) ] m e (M }{{ X ) } =0 [ n q /3 ln αs (Q ] ) α 5 (MX + 3 [ α1 (Q ] ) ln n q α 5 (MX 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 is failure of model (need more complicated Higgs sector) Γ mn term gives independent M u (M u = M ν Dirac in SO(10) plus other (model dependent) relations))

38 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( i j ) = δij 3 5 Y is SU(5) generator, with g Y = 5 3 g }{{} g Y

39 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

40 Running α i = g i /4π 1 α i (M Z ) = 1 α i (M X ) b i ln M Z MX b 1 = F 3π, b = 1 [ 4π 3 F ] 3 (F = number of families), b 3 = 1 4π sin θ W (M Z ) = α(m Z ) 9α s (M Z ) 0.0 M X GeV Reasonable zeroth order prediction However, proton decay requires M X > GeV [ 11 F ] 3 Precise sin θ W (M Z ) 0.3 and α s (M Z ) 0.1 does not quite work, even with two-loop corrections SUSY

41 Beyond SU(5) arger 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 5} + {{ 10} SO(10) SU(5) family + 1 }{{} ν R ν R ν c = right-handed (singlet, sterile) neutrino M u = MDirac ν in simplest Higgs scheme Seesaw for large Majorana mass (need Higgs 16) Right mass scales, but small mixings in simplest schemes eptogenesis

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

43 Supersymmetric extensions: 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.

44 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

45 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 conserved and asymmetry with B = 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

46 To GUT or not to GUT String GUT MSSM (+ extended?) or String MSSM (+ extended?) gauge unification quantum numbers for family (15-plet) seesaw ν mass scale/leptogenesis m b /m τ large lepton mixings other fermion mass relations (need large Higgs representatons) additional GUT scale; no adjoints in simple heterotic hierarchies, e.g. doublet-triplet proton decay