Beyond the Standard Model

Transcription

1 Beyond the Standard Model The Standard Model Problems New Physics Supersymmetry Extended Gauge Structures Grand Unification Based on The Standard Model and Beyond, CRC Press, 1/09

2 Extended Gauge Structures Embed SU(3) SU() U(1) in larger group Bottom-up motivations µ problem (extra U(1) ) Alternative electroweak symmetry breaking Restore C, P (left-right -SU() L SU() R U(1) ) Family symmetries (probably gauged if continuous) Top-down motivations Grand unification TeV scale remnants of string or GUT symmetry Will focus on TeV-scale U(1) and on grand unification

3 The Physics of New U(1) Gauge Bosons Tevatron LHC LHC LC LC 15fb fb -1 14TeV 1ab -1 14TeV 0.5TeV 1ab -1 P - =0.8 P + = TeV 1ab -1 P - =0.8 P + =0.6 χ ψ η LR SSM σ m Z' [TeV] Motivations Basics The Standard TeV Scale Case Nonstandard Cases Experimental constraints and prospects Implications LHC/ILC, hep-ph/ (Recent review: PL )

4 Motivations Strings/GUTS (large underlying groups; U(n) in Type IIa) Harder to break U(1) factors than non-abelian (remnants) Supersymmetry: SU() U(1) and U(1) breaking scales both set by SUSY breaking scale (unless flat direction) µ problem Alternative electroweak model/breaking (TeV scale): DSB, Little Higgs, extra dimensions (Kaluza-Klein excitations, M R 1 TeV (10 17 cm/r)), left-right symmetry Connection to hidden sector (weak coupling, SUSY breaking/mediation) Extensive physics implications, especially for TeV scale

5 Standard Model Neutral Current L SM NC = gj µ 3 W 3µ + g J µ Y B µ = ej µ em A µ + g 1 J µ 1 Z0 1µ A µ = sin θ W W 3µ + cos θ W B µ Z µ = cos θ W W 3µ sin θ W B µ θ W tan 1 (g /g) e = g sin θ W g 1 = g / cos θ W J µ 1 = i f i γ µ [ɛ 1 L (i)p L + ɛ 1 R (i)p R]f i P L,R (1 γ5 ) ɛ 1 L (i) = t 3i L sin θ W q i ɛ 1 R (i) = sin θ W q i M Z 0 = 1 4 g 1 ν = M W cos θ W ν 46 GeV

6 Standard Model with Additional U(1) L NC = ej µ em A µ + g 1 J µ 1 Z0 1µ + }{{} SM n+1 α= g α J µ α Z0 αµ J µ α = i f i γ µ [ɛ α L (i)p L + ɛ α R (i)p R]f i ɛ α L,R (i) are U(1) α charges of the left and right handed components of fermion f i (chiral for ɛ α L (i) ɛα R (i)) g α V,A (i) = ɛα L (i) ± ɛα R (i) May specify left chiral charges for fermion f and antifermion f c ɛ α L (f) = Q αf ɛ α R (f) = Q αf c Q 1u = 1 3 sin θ W and Q 1u c = + 3 sin θ W

7 Mass and Mixing Mass matrix for single Z M Z Z = ( M Z 0 MZ ) Eg., SU() singlet S; doublets φ u = ( φ 0 u φ u ), φ d = ( φ + d φ 0 d ) M Z 0 = 1 4 g 1 ( ν u + ν d ) = 1 g 1g (Q u ν u Q d ν d ) M Z =g (Q u ν u + Q d ν d + Q S s ) ν u,d φ 0 u,d, s = S, ν = ( ν u + ν d ) (46 GeV)

8 Eigenvalues M1,, mixing angle θ tan θ = M Z 0 M 1 M M Z 0 For M Z (M Z 0, ) M 1 M Z 0 4 M Z M M M Z θ M Z C g g 1 M 1 M [ Qu ν u Q d ν d ] with C = ν u + ν d

9 Kinetic Mixing General kinetic energy term allowed by gauge invariance L kin 1 4 F 0µν 1 F 0 1µν 1 4 F 0µν F 0 µν sin χ F 0µν 1 F 0 µν Negligible effect on masses for M Z 0 M Z, but L g 1 J µ 1 Z 1µ + (g J µ g 1χJ µ 1 )Z µ Usually absent initially but induced by loops, e.g., nondegenerate heavy particles, in running couplings if heavy particles decouple, or by string-level loops (usually small) f Z 1 Z f

10 Anomalies and Exotics Must cancel triangle and mixed gravitational anomalies V 3 f V 1 V No solution except Q = 0 for family universal SM fermions Must introduce new fermions: SM singlets like ν c L (usually non-chiral under SM) D L + D R, ( E 0 E ) L + ( E 0 E ) or exotic SU() R Typeset by FoilTEX Supersymmetry: include Higgsinos and singlinos (partners of S)

11 The µ Problem In MSSM, introduce Higgsino mass parameter µ: W µ = µh u H d µ is supersymmetric. Natural scales: 0 or M P lanck GeV Phenomenologically, need µ SUSY breaking scale In Z models, U(1) may forbid elementary µ (if Q H u + Q Hd 0) If W µ = λ S SH u H d is allowed, then µ eff contributes to M Z λ S S, where S Can also forbid µ by discrete symmetries (NMSSM, nmssm, ), but simplest forms have domain wall problems U(1) is stringy version of NMSSM

12 Models Enormous number of models, distinguished by gauge coupling g, mass scale, charges Q, exotics, kinetic mixing, couplings to hidden sector No simple general parametrization Canonical models: TeV scale M Z with electroweak strength couplings Sequential Z SM Models based on T 3R and B L E 6 models Minimal Gauge Unification Models

13 Sequential Z SM Same couplings to fermions as the SM Z boson Reference model Hard to obtain in gauge theory unless complicated exotic sector Kaluza-Klein excitations with TeV extra dimensions

14 Models based on T 3R and B L Motivated by minimal exotics (only νl c left-right SU() L SU() R U(1) BL needed), SO(10), and T BL 1 (B L), T 3R = Y T BL = 1 [u R, ν R ], 1 [d R, e R ] For non-abelian embedding and no kinetic mixing Q LR = 3 5 [ αt 3R 1 ] α T BL α = g R g BL = q (g R /g) cot θ W 1 g = r 5 3 g tan θ W 0.46 More general: Q Y BL = ay + bt BL b(zy + T BL )

15 T 3R T BL Y QLR BL QY b Q 0 (z + 1) 6 6α u c L 1 1 α α 3 z 1 6 d c L L L e + L 1 ν c L α + 1 6α α 1 α α 1 α Additional fields needed to break U(1) 3 z 1 6 (z + 1) α z + 1 1

16 The E 6 models Example of anomaly free charges and exotics, based on E 6 SO(10) U(1) ψ and SO(10) SU(5) U(1) χ 3 7: 3 S fields, 3 exotic (D + D c ) pairs, 3 Higgs (or exotic lepton) pairs Supersymmetric version forbids µ term except χ model (SO(10)) SO(10) SU(5) 10Q χ 6Q ψ 15Q η (u, d, u c, e + ) (d c, ν, e ) ν c (D, H u ) 4 5 (D c, H d ) S 0 4 5

17 General: Q = cos θ E6 Q χ + sin θ E6 Q ψ (can add kinetic term ɛy ) g = r 5 3 g tan θ W λ 1/ g, λ g = O(1) SO(10) SU(5) Q I 10Q N 15Q S (u, d, u c, e + ) 0 1 1/ 5 (d c, ν, e ) 1 4 ν c (D, H u ) (D c, H d ) 1 3 7/ 1 1 S 1 5 5/ GUT Yukawas violated (proton decay), e.g., by string rearrangement Gauge unification requires additional non-chiral H u + H u Recent detailed studies (King, Moretti, Nevzorov)

18 Minimal Gauge Unification Models Supersymmetric models with µ eff and MSSM-like gauge unification(erler; Morrissey, Wells; PL; PL, Paz, Yavin) 3 ordinary families (with ν c ), one Higgs pair H u,d and n 55 (D i + L i ) and (Di c + Lc i ) pairs Q 55 Q ψ Q 55 Q ψ Q y 1/4 H u x 1/ u c x y 1/4 H d 1 x 1/ d c 1 + x y 1/4 S D 3/n 55 3/ L 1 3y 1/4 D i z 3/4 e + x + 3y 1/4 Di c 3/n 55 z 3/4 ν c 1 x + 3y 1/4 S L /n n S 1 1 L 55 i + x + 3y + 3z/ 1/ 4n 55 L c i /n 55 Q Li 1/

19 Other Models TeV scale dynamics (Little Higgs, un-unified, strong t t coupling, ) Kaluza-Klein excitations (large dimensions or Randall-Sundrum) Decoupled or doubly coupled (leptophobic, fermiophobic, weak coupling, low scale/massless, hidden sector [kinetic or HDO portal ], mediator, ) Secluded or intermediate scale SUSY (flat directions, Dirac m ν ) Family nonuniversal couplings (FCNC) String derived (may be T 3R, T BL, E 6 or random ) Stückelberg (no Higgs) Anomalous U(1) (string theories with large dimensions)

20 Experimental constraints and prospects Tevatron (CDF, D0): resonance in pp e + e, µ + µ, AB Z α in narrow width: dσ dy = 4π x 1 x 3M 3 α (f A q i (x 1 )f B q i (x )+f Ā q i (x 1 )f B q i (x ))Γ(Z α q i q i ) i Γ α f i Γ(Z α f i fi ) = g α C f i M α 4π ( ɛ α L (i) + ɛ α R (i)) x 1, = (M α / s)e ±y C fi = color factor

21 BR(Z µµ ) (pb) 95% C.L. Limits on σ SE Median 68% of SE 95% of SE Data Z I Z sec Z N Z ψ Z χ Z η Z SM M Z (TeV) (CDF dimuons: PRL 10, ) M Z

22 Low energy weak neutral current: Z exchange and Z Z mixing (still very important) L eff = 4G F (ρ eff J 1 + wj 1J + yj ) ρ eff =ρ 1 cos θ + ρ sin θ w = g cos θ sin θ(ρ 1 ρ ) g 1 «g y = (ρ 1 sin θ + ρ cos θ) ρ α M W /(M α cos θ W ) g 1 Z-pole (LEP, SLC): Z Z mixing (vertices; shift in M 1 ) V i = cos θg 1 V (i) + g g 1 sin θg V (i) A i = cos θg 1 A (i) + g g 1 sin θg A (i) LEP: four-fermi operator interfering with γ, Z

23 M Z [TeV] 6 5 CDF D0 LEP x M Z [TeV] 6 CDF D0 LEP 5 Z χ Z ψ sin θ zz x sin θ zz (Erler, PL, Munir, Peña)

24 Z M Z [GeV] sin θ ZZ χ min EW (this work) CDF DØ LEP sin θ ZZ sin θ min ZZ sin θ max ZZ Z χ 1, Z ψ Z η Z I 1, Z S 1, Z N Z R Z LR Z SM 1,401 1, , Z string 1, SM

25 Future Prospects Tevatron and LHC: pp( pp) Z e + e, µ + µ, jj, bb, tt, eµ, τ + τ Rates (total width) dependent on whether sparticle and exotic channels open (Kang, PL: Γ Z /M Z for E 6 ) LHC discovery to 4 5 TeV ILC: 5σ intererence effects up to 5 TeV

26 Tevatron 15fb -1 LHC χ ψ η LR SSM σ 100fb -1 14TeV LHC LC 1ab -1 14TeV LC 0.5TeV 1ab -1 P - =0.8 P + = TeV 1ab -1 P - =0.8 P + = m Z' [TeV] LHC/ILC, hep-ph/

27 LHC diagnostics to -.5 TeV Diagnostics of Z Couplings Forward-backward asymmetries and rapidity distributions in l + l l FB A Forward backward asymmetry measurement 1. Z χ a) Z 1 LHC, s=14tev ψ M Z =1.5TeV y ll >0.8 Z η Z LR -1 Z η, L=100fb dn / dy Rapidity distribution -1 Z η, 100fb Z η : fit uu fit dd sum Z ψ 1.45 TeV<M ll <1.55 TeV b) [GeV] M ll (LHC/ILC, hep-ph/ ) Y ll

28 Other two body decays Lineshape: σ Z B l, Γ Z τ polarization Associated production Z Z, Z W, Z γ Rare (but enhanced) decays Z W f 1 f (radiated W ) Z W + W, Zh, or W ± H : longitudinal W, Z small mixing compensated by Γ(Z W + W ) = g 1 θ M Z 19π MZ M Z «4 = g C M Z 19π (del Aguila, PL, Cvetič; Hewett, Rizzo; recent: Barger, Han, Walker; Godfrey, Martin; Petriello, Quackenbush)

29 Implications of a TeV-scale U(1) Natural Solution to µ problem W hsh u H d ( stringy version of NMSSM) µ eff = h S Extended Higgs sector (Barger, PL, Lee, Shaughnessy) Relaxed mass limits, couplings, parameters (e.g., tan β 1) Higgs singlets needed to break U(1) Doublet-singlet mixing, extended neutralino sector non-standard collider signatures Extended neutralino sector (BLLS) Additional neutralinos, non-standard couplings, e.g., light singlino-dominated, extended cascades Enhanced cold dark matter, g µ possibilities (even small tan β)

30 Exotics (anomaly-cancellation) (Kang, PL, Nelson; King, Moretti, Nevzorov) Non-chiral wrt SM but chiral wrt U(1) May decay by mixing; by diquark or leptoquark coupling; or be quasi-stable Z decays into sparticles/exotics (SUSY factory) (Baumgart, Hartman, Kilic, Wang; Cohen, Pierce; H.-S. Lee; Ali, Demir, Frank, Turan) Flavor changing neutral currents (for non-universal U(1) charges) Tree-level effects in B decay competing with SM loops (or with enhanced loops in MSSM with large tan β) B s B s mixing, B d penguins (Barger, Everett, Jiang, PL, Liu, Wagner)

31 Constraints on neutrino mass generation Various versions allow or exclude Type I or II seesaws, extended seesaw, small Dirac by HDO (Kang, PL, Li; King, Moretti, Nevzorov); small Dirac by non-holomorphic soft terms (Demir, Everett, PL) ; Majorana seesaw or small Dirac by string instantons (Cvetič talk) (also: King, Mohapatra, Valle talks) Large A term and possible tree-level CP violation (no new EDM constraints) electroweak baryogenesis (Kang, PL, Li, Liu)

32 Z Conclusions New Z are extremely well motivated TeV scale likely, especially in supersymmetry and alternative EWSB LHC discovery to 4-5 TeV, diagnostics to -.5 TeV Implications profound for particle physics and cosmology Possible portal to hidden/dark sector (massless, GeV, TeV)

33 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) (Review: PL Phys. Rep. 7, 185 (1981))

34 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 1030 yr for M X GeV (10 38 yr in SUSY, but faster p νk + )

35 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

36 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

37 The SU(n) Group (Type A Lie 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) L = n 1 representation matrices of dimension m m satisfying same Lie Algebra (commutation rules) as the T U (m) ( β) i β L e form m m dimensional representation of group (will suppress (m))

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

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

40 Lie algebra: [ L a b, Lc d] = δ a d L c b δc b La d Same algebra for generators T a b and other representations Field transformations: Let ψ c, c = 1,..., n transform as fundamental n, i.e. implies [ T a b, ψc] = ( L 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 L d ψ d = ( ) e i β L c ψ

41 Antifundamental χ a transforms as n, [ T a b, χ ] ( c = L a b (n ) ) d χ c d where L a b (n ) = ( L a ) T b = L b a ( L a b (n ) ) d ( L 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 L ) c χ c ψ c is SU(n) invariant Adjoint field (n n ): φ a b (U( β)φu ( β) ) a b

42 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 L 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

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

44 Antisymmetric n n representation: Let ψ ab = ψ ba be n(n 1) fields with [ T a b, ψcd] = (L a b )c e ψed (L 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)

45 The Georgi-Glashow SU(5) Model 5 1 = 4 generators Tb a 1 5 δa b T c c, a, b, c = 1,..., 5 ( P 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))

46 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()

47 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

48 Fermions still in highly reducible representation: each family of L-fields in (antifundamental and antisymmetric) }{{} 5 (3, 1, 1 3) + (1, }{{}, }{{ ) 1 } (χ L ) a (χ L ) α }{{} 10 ψ L ab= ψba L (3, 1, 3) }{{} ψ αβ L (χ L ) r + (3,, 1 6) }{{} ψ αr L + (1, 1, 1) }{{} ψ 45 L (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

49 10 : ψ ab L = 1 5 : χ La = 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 L L (family indices and weak-eigenstate superscript 0 suppressed)

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

51 Fermion Gauge Interactions [ ] 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 ) ]

52 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 = q 3 5 g 5) 3 5 Y is properly normalized generator, Tr(Li L j ) = δij

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

54 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 ) 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

55 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

56 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)

57 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)

58 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

59 Yukawa Couplings No Φ couplings to χ La, ψ ab L allowed by SU(5), but can have L Yuk = γ mn χ T mla C ψab nl H b + Γ mn ɛ abcde ψ T ab ml C ψcd nl He + h.c, (γ, Γ are family matrices (Γ symmetric); 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 0 H a 0 = ν 0 δ 5 a

60 First term: d L M d d R ē + L M et e }{{ + R +h.c. } ē L 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

61 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 (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))

62 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

63 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

64 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 4F 3 (F = number of families) ], b 3 = 1 4π [ 11 4F 3 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 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

65 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

66 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 )

67 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 (!, sin " W )! (G F, M W )! 3 m e m µ m b M Z M unification

68 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

69 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

70 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

71 Conclusions Standard model: SU() U(1) (extended to include ν masses) + QCD + general relativity Mathematically consistent, renormalizable theory Correct to cm However, too much arbitrariness and fine-tuning: many free parameters; gauge, fermion, Higgs, strong CP, gravity problems Missing ingredients: Higgs, neutrino mass, dark matter/energy, baryogenesis, suppression of FCNC/EDMs LHC era should be exciting!