Grand Unification M.Dine, Supersymmetry And String Theory: Beyond the Standard Model 6 2009 2 24 by Standard Model Coupling constant θ-parameter 8 Charge quantization. hypercharge charge Gauge group. simple semi-simple group Georgi and Glashow underlying gauge group geuge Standard Model gauge group Standard Model gauge SU(3) SU(2) U() gauge SU(N) rank N gauge gauge SU(5) GUT embed gauge adjoint gauge δa µ = iω a [T a, A µ ] where T a 5 5 traceless Hermitian matrices, 24 gauge generator Standard Model break up T a (T a ) j i T a SU(5) 5 (T a ) i j 5 j 5 q q 2 5 = q 3 L L 2 generator T a T a = ( λ a ) 2 0 0 0 T i = ( ) 0 0 σ 0 i 2
q quark L left-handed lepton. λ Gell-Mann SU(3), a = 8 σ Pauli, i =, 2, 3 U() T a T i 2 Ỹ = 60 traceless tr Ỹ 2 = 2 notation. 2 off-diagonal matrices 2 2 3 3 (Xa i )b J = δb a δi j raising and lowering operator generator gauge boson quark lepton baryon e.g. p π 0 e + gauge boson Ỹ ordinary hypercharge 5 5 d d 5 i 2 = d 3 L d right-handed down-type quark. 5 gauge generator T at L 2 (T at ) i j5 j hypercharge Y Y = 60 3 Ỹ -Gell-Mann Q = I 3 + Y 2 Dine (6.7) d Y gauge SU(5) Standard Model gauge coupling Weinberg θ W SU(5) gauge coupling g SU(3) SU(2) gauge coupling g hypercharge coupling g gỹ = g 2 Y 2
SU(5) gauge group 5 fermion gauge SU(2) U() Dine 2 notation Weibnerg sin 2 θ W = g 2 g 2 + g 2 = 3 8 Weinberg SU(5) GUT Standard Model 5 fermion SU(3) triplet SU(2) doublet left-handed quark 6 Q i a SU(3) triplet SU(2) singlet right-handed quark 6 u a, d a SU(3) singlet SU(2) doublet left-handed lepton 2 L i SU(3) singlet SU(2) singlet right-handed lepton e 5 5 0 SU(5) 0 0 2 5 f i f j = 2 ( f i f j + f j f i ) + } {{ } 2 ( f i f j f j f i ) } {{ } 5 i j 0 i j 0 SU(3) index right-handed up-type quark 0 SU(3) SU(2) left-handed quark 0 SU(2) right-handed lepton Standard Model left-handed u-quark SU(3) 3 SU(2) 2 I = + 2, Q = +2 3 Y = 3 left-handed d-quark SU(3) 3 SU(2) 2 I = 2, Q = 3 Y = 3 right-handed u-quark SU(3) 3 SU(2) singlet I = 0, Q = + 2 3 Y = 4 3 right-handed d-quark SU(3) 3 SU(2) singlet I = 0, Q = 3 Y = 2 3 left-handed neutrino SU(3) singlet SU(2) 2 I = + 2, Q = 0 Y = 3
left-handed electron SU(3) singlet SU(2) 2 I = 2, Q = Y = right-handed electron SU(3) SU(2) singlet I = 0, Q = Y = 2 0 hypercharge Y 0 generator 0 i j (T a ) i k 0 k j + (T a ) j k 0 ik T a Y 0 Y 0 = 0 Y 0 2 + Y 2 0 2 Y 0 3 + Y 3 0 3 Y 0 + Y 4 0 4 Y 0 + Y 5 0 5 0 Y 2 0 23 + Y 3 0 23 Y 2 0 24 + Y 4 0 24 Y 2 0 25 + Y 5 0 25 0 Y 3 0 34 + Y 4 0 34 Y 3 0 35 + Y 5 0 35 0 4 3 0 2 4 3 0 3 0 4 3 0 23 = 3 0 4 0 3 0 5 3 0 24 3 0 25 3 0 34 3 0 35 0 2 0 45 0 0 Y 4 0 45 + Y 5 0 45 0 hypercharge SU(5) 5 0 SU(3) SU(2) 0 hypercharge consistent 6. Cancellation of anomalies gauge boson 3 anomaly d abc = tr T a { T b, T c} Peskin Dine (6.) {T a, T b } = d abc T c 5 0 anomaly 5 generator d abc 5 = tr(t c T b T a + T b T c T a ) 4
0 2 generator 2 5 generator (5 i 5 j ) ( T { a T b, T c}) kl (5 k 5 l ) i j ( { T a T b, T c}) kl (T a ) j i 0 kl i j 0 0 i j (( { T a T b, T c}) kl ( T { a T b, T c}) ) lk 0 kl 2 i j i j kl i j 2 d abc = 2 ( δ i k δ j l δi l δ ( k) { j T a T b, T c}) kl i j 5 6.2 Renormalization of couplings SU(5) GUT high energy M GUT high-energy coupling low-energy running coupling constant ( α i (µ) = α GUT (M GUT) + bi 0 4π log β- -loop b i 0 b 0 = 3 C A 4 3 c(m) f N (m) f µ M GUT 3 c(m) φ N(m) φ m color fermion N (m) f N (m) φ m scalar C A SU(N) N Casimir operator for adjoint c f, c φ 2 tr(t a T b ) = c f δ ab for fermion scalar SU(3) C A = 3 c (i) f N (i) f = 2 6 = 3 RGB 3 u, d 2 fermion b 3 0 = 7 8 ) SU(2) C A = 2 6 5
Ross, Grand Unified Theories (984) Mohapatra, Unification and Supersymmetry running coupling constant Dine U() coupling SU(2) coupling M GUT 0 5 GeV SU(3) coupling 7 standard deviations standard deviation M GUT low-energy SUSY 6.3 Breaking to SU(3) SU(2) U() SU(5) scalar Φ adjoint scalar gauge δφ = ω a [T a, Φ] Higgs Higgs Φ Φ = vỹ SU(3),SU(2),U() generator Φ X SU(5) invariant 4 δ tr Φ 2 = 2ω a tr[t a, Φ]Φ = 0 δ tr Φ 4 = 4ω a tr[t a, Φ]Φ 3 = 0 SU(5) Φ V = m 2 tr Φ 2 + λ tr Φ 4 + λ ( tr Φ 2) 2 Φ = diag.(a, a 2, a 3, a 4, a 5 ) Φ traceless Lagrange multiplier minimum vỹ V tr Φ 2 = 2 v2 tr Φ 4 = v 4 60 2 (24 + 2 4 + 2 4 + 3 4 + 3 4 ) = 7 20 v4 V = 2 m2 v 2 + ( ) 7 4 30 λ + λ v 4 6
v = m 7 30 λ + λ fluctuation Φ SU(5) SU(3) SU(2) U() fluctuation δφ = (, ) + (8, ) + (, 3) + (3, 2) + (3, 2) ( ) SU(3) 2 SU(2) (,) singlet Ỹ (8,) SU(3) adjoint SU(2) singlet T a (,3) SU(3) singlet SU(2) adjoint T i (3,2) (3,2) off-diagonal X SU(2) 2 2 real representation (3,2) (3,2) δv leading order δφ = diag.(x, x 2, x 3, x 4, x 5 ) δ tr Φ 2 δ tr Φ 4 x leading order x + x 2 + x 3 x 4 + x 5 (,) leading order minimal 2 fluctuation 6.4 SU(2) U() breaking Higgs SU(2) U() fundamental scalar SU(5) 5 ( ) Hc H = H d H c color triplet H d ordinary Higgs doublet H V(H) = µ 2 H 2 + λ 4 H4 V H SU(5) SU(5) Goldstone boson 7
electroweak symmetry breaking scale µ λ Higgs triplet doublet mass W boson Z 0 boson baryon lepton H Φ coupling V Φ H = ΓH ΦH + λ H H tr Φ 2 + λ H Φ 2 H V(H) + V Φ H H c H d λ, λ. µ 2 = + 3 60 Γv + ɛ, λ = λ = λ = 0 Dine Φ m 2 H c = 5 60 Γv ɛ, m 2 H d = ɛ m Hc GUT m Hd electroweak scale Γv ɛ 0 3 hierarchy problem M W M GUT fermion mass SU(5) Yukawa coupling L Y = y ɛ i jklm H i 0 jk 0 lm + y 2 H i 5 j 0 i j eq.(6.2) index H v.e.v. y coupling up-type quark mass y 2 coupling lepton down-type quark mass L Y Q, L, u, d generation tree-level m τ = m b 3 lepton quark e.g. m d 0 e Higgs higher dim. operator SUSY GUT M GUT 0 2 M Pl quark lepton 0 4 M weak 8
6.5 6.6 pass 6.7 Other groups Grand unification SU(5) O(0) GUT O(0) 0 generator 0 0 real 45 rank 5 GUT commuting generator 6 Standard Model O(0) SU(5) O(0) generate x A, (A = 0) O(0) x A x A O(0) x z = x + ix 2, z 2 = x 3 + ix 4, z 3 = x 5 + ix 6, z i z i O(0) SU(5) O(0) 0 SU(5) 5 + 5 O(0) adjoint A AB SU(5) A AB = A iī + A i j + Aī j A iī 24 of SU(5) SU(5) adjoint singlet O(0) U() component A i j 0 of SU(5) Aī j 0 of SU(5) 45 O(0) 6 9
O(0) 0 γ- {Γ I, Γ J } = 2δ IJ γ- 6 approach 6 SU(5) 0 γ- fermion creation annihilation operator a = 2 (Γ + iγ 2 ) a 2 = 2 (Γ3 + iγ 4 ) a 3 = 2 (Γ5 + iγ 6 ) a = 2 (Γ iγ 2 ) a 2 = 2 (Γ3 iγ 4 ) a 3 = 2 (Γ5 iγ 6 ) {a i, a j } = {aī, a j } = 0 {a i, a j } = δ i j 5 fermion state Lie a i 0 state ground state ground a i 0 = 0 for i 0 creation operator aī O(0) 32 32 4 chirality γ 5 Γ O(0) generator S IJ = i 4 [ΓI, Γ J ] Γ O(0) chirality Γ Γ 0 = + 0 0 aī state Γ = + 0 aī state Γ = aī 0 SU(5) 5 aīa j a k 0 SU(5) 0 a a 2 a 3 a 4 a 5 0 SU(5) singlet 3 SU(5) 3 2 0 ɛ i jklm ī j k lm 0
O(0) 6 SU(5) 5 + 0 + 5 + 0 Standard Model SU(3) SU(2) singlet 0 Standard Model Standard Model coupling mass M GUT O(0) gauge mass left-handed neutrino coupling Standard Model right-handed neutrino seesaw neutrino mass