Elements of Grand Unification

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

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

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 10 16 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)

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.

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

Graviton problem gravity not unified quantum gravity not renormalizable cosmological constant: Λ SSB = 8πG N V > 10 50 Λ obs (10 14 for GUTs, strings) Possible solutions: supergravity and Kaluza Klein unify strings yield finite gravity. Λ?

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 (< 130 150 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

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

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 10 14 GeV (w/o SUSY) (works much better with SUSY M X 10 16 GeV)

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 10 14 GeV (10 38 yr in SUSY, but faster p νk + )

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

Each family in 5 + 10 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

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))

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

( ) For a b, a b lowering operators ba = 1, others 0, e.g., SU() raising and 1 = τ 1 + iτ = ( 0 1 0 0 ) 1 = τ 1 iτ = ( 0 0 1 0 ) 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

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

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

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

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

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)

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 1 1 + T + T 3 3) + 1 (T 4 4 + 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))

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()

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

Fermions still in highly reducible representation: each family of -fields in 5 + 10 (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

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)

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

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 ) ]

U(1)Y = [ 3 5 g 5 1 ( ν Bν + ē Be ) + 1 6 (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

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

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 10 33 yr and α 5 α M > X,Y 10 15 GeV (GUT scale) Also bound neutron decay Additional mechanisms/modes in SUSY GUT

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

For b > 0 minimum is at Φ = ν 0 0 0 0 0 ν 0 0 0 0 0 ν 0 0 0 0 0 3ν 0 0 0 0 0 3ν 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)

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) 10 13 ν Problem : Need M H > 10 14 GeV > 10 1 M φ to avoid too fast proton decay mediated by H α (doublet-triplet splitting problem)

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

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

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

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 [ 4 + 11 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))

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 5 3 5 Y

Weak mixing angle at M X sin θ W (M X ) = g g + g = 3/5 1 + 3/5 = 3 8 SU(5) broken below M X (and some particles decouple) SU(3) SU() U(1) couplings run at different rates

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 ) = 1 6 + 5 α(m Z ) 9α s (M Z ) 0.0 M X 10 14 GeV Reasonable zeroth order prediction However, proton decay requires M X > 10 15 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

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

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

Supersymmetric extensions: different b i factors Better gauge unification agreement M X 10 16 GeV τ (p e + π 0 ) 10 38 yr However, new dimension-5 operators involving superpartners may yield too rapid p νk +, etc.

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

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

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