(Supersymmetric) Grand Unification

(Supersymmetric) Grand Unification Jürgen Reuter Albert-Ludwigs-Universität Freiburg Uppsala, 15. May 2008

Literature General SUSY: M. Drees, R. Godbole, P. Roy, Sparticles, World Scientific, 2004 S. Martin, SUSY Primer, arxiv:hep-ph/9709356 H. Georgi, Lie Algebras in Particle Physics, Harvard University Press, 1992 R. Slansky, Group Theory for Unified Model Building, Phys. Rep. 79 (1981), 1. R. Mohapatra, Unification and Supersymmetry, Springer, 1986 P. Langacker, Grand Unified Theories, Phys. Rep. 72 (1981), 185. P. Nath, P. Fileviez Perez, Proton Stability..., arxiv:hep-ph/0601023. U. Amaldi, W. de Boer, H. Fürstenau, Comparison of grand unified theories with electroweak and strong coupling constants measured at LEP, Phys. Lett. B260, (1991), 447.

Scalar self-ia: (H H) (H H) 2 J. Reuter SUSY GUTs Uppsala, 15.05.2008 The Standard Model (SM) Theorist s View Renormalizable Quantum Field Theory (only with Higgs!) based on SU(3) c SU(2) w U(1) Y non-simple gauge group L L = ( ) ν l L Q L = ( ) u d L u c R d c R l c R [ν c R] Interactions: Gauge IA (covariant derivatives in kinetic terms): [( )] h + h 0 µ D µ = µ + i X k g k V a µ T a Yukawa IA: Y u Q L H uu R + Y d Q L H d d R + Y e L LH d e R ˆ+Y n L LH uν R

The group-theoretical bottom line Things to remember: Representations of SU(N) fundamental reps. φi N, ψ i N, adjoint reps. A j i N 2 1 SU(N) invariants: contract all indices φ iψ i φ ia j i ψj ɛ ijφ iξ j ɛ ijk φ iξ jη k ˆ Symmetry properties: Young tableaux The (group-theoretical) essence of the SM: remember: Q el. = T 3 + Y 2 Q L u c R d c R L L e c R H d H d ν c R (2, 3) 1 3 (1, 3) 4 3 (1, 3) 2 3 (2, 1) 1 (1, 1) 2 (2, 1) 1 (2, 1) 1 (1, 1) 0

Loose Ends/Deficiencies of the Standard Model Incompleteness Electroweak Symmetry Breaking Higgs boson Origin of neutrino masses Dark Matter: m DM 100 GeV Theoretical Dissatisfaction 28 free parameters strange fractional U(1) quantum numbers Hierarchy problem M H [GeV] 750 500 Λ 2 m 2 h = m 2 0 + Λ 2 (loop factors) 250 10 3 10 6 10 9 10 12 10 15 10 18 Λ[GeV]

Supersymmetry 800 connects gauge and space-time symmetries multiplets with equal-mass fermions and bosons SUSY is broken in Nature stabilizes the hierarchy Q J 1 1 2 0 Q m [GeV] 700 600 500 400 H 0, A 0 H± 300 χ 0 χ 0 4 3 χ ± 2 g ũl, dr ũr, dl t2 b2 b1 t1 Minimal Supersymmetric Standard Model (MSSM) Charginos, Neutralinos, Gluino Sleptons, Squarks, Sneutrinos R parity: discrete symmetry 200 ll νl τ2 χ 0 2 χ ± 1 LSP: Dark matter 100 0 h 0 lr τ1 χ 0 1 Superpartners have identical gauge quantum numbers

Supersymmetry: Symmetry between fermions and bosons Q boson = fermion Q fermion = boson Effectively: SM particles have SUSY partners (e.g. f L,R fl,r ) SUSY: additional contributions from scalar fields: f L,R f L,R H H H H Σ f H N f λ 2 f f L,R ( ) d 4 1 1 k + + terms without quadratic div. k 2 m 2 fl k 2 m 2 fr for Λ : Σ f H N f λ2 f Λ 2

quadratic divergences cancel for N fl = N fr = N f λ 2 f = λ 2 f complete correction vanishes if furthermore m f = m f Soft SUSY breaking: m 2 f = m 2 f + 2, λ 2 f = λ 2 f Σ f+ f H N f λ 2 f 2 +... correction stays acceptably small if mass splitting is of weak scale realized if mass scale of SUSY partners M SUSY < 1TeV SUSY at TeV scale provides attractive solution of hierarchy problem

Anatomy of MSSM Yukawa couplings Superpotential: W = ˆΦ 1 ˆΦ2 ˆΦ3 Φ 1 Φ 2 Φ 3 = (φ 1 φ 2 ) 2,..., ( ψ 1 ψ 2 ) φ3 NB: part of scalar potential from gauge kinetic terms (light Higgs) MSSM superpotential W MSSM = Y u u c QH u + Y d d c QH d + Y e e c LH d + µh u H d Ignorance about SUSY breaking shows up as soft-breaking terms : Gaugino and sparticle masses, trilinear scalar potential terms µ problem: EWSB demands µ O(100 GeV 1 TeV) Additional SUSY degrees of freedom modify vacuum polarization Unification of gauge couplings possible

(Gauge) Unification and the running of couplings

(Gauge) Unification and the running of couplings Renormalization group (RG) running of gauge couplings: dg a d log µ = g3 a 16π 2 B a SM B a = ( 41 10, 19 6, 7) MSSM B a = ( 33 5, 1, 3) 60 50 U(1) α i -1 40 30 20 10 SU(2) SU(3) Standard Model 0 10 2 10 4 10 6 10 8 10 10 10 12 10 14 10 16 10 18 µ (GeV)

(Gauge) Unification and the running of couplings Renormalization group (RG) running of gauge couplings: dg a d log µ = g3 a 16π 2 B a SM B a = ( 41 10, 19 6, 7) MSSM B a = ( 33 5, 1, 3) 60 50 40 U(1) 60 50 40 U(1) MSSM α i -1 30 SU(2) α i -1 30 SU(2) 20 10 SU(3) Standard Model 20 10 SU(3) 0 10 2 10 4 10 6 10 8 10 10 10 12 10 14 10 16 10 18 µ (GeV) 0 10 2 10 4 10 6 10 8 10 10 10 12 10 14 10 16 10 18 µ (GeV)

The prime example: (SUSY) SU(5) SU(5) MX SU(3) c SU(2) w U(1) Y MZ SU(5) has 5 2 1 = 24 generators: 24 (8, 1) 0 (1, 3) 0 (1, 1) 0 }{{}}{{}}{{} W B G β α SU(3) c U(1) em (3, 2) 5 3 }{{} X,Y (3, 2) 5 3 }{{} X,Ȳ

The prime example: (SUSY) SU(5) SU(5) MX SU(3) c SU(2) w U(1) Y MZ SU(5) has 5 2 1 = 24 generators: SU(3) c U(1) em 24 (8, 1) 0 (1, 3) 0 (1, 1) 0 }{{}}{{}}{{} W B G β α 24 A = g A a λa 2 = g 2G a λa GM 2 a=1 2 X X X Y Y Y 2 g 2 15 B (3, 2) 5 3 }{{} X,Y X X X (3, 2) 5 3 }{{} X,Ȳ Ȳ Ȳ Ȳ 2W a σ 2 2 0 2 +3 0 +3

Quantum numbers q Hypercharge: λ 12 3 Y = Y = 1 diag( 2, 2, 3, 3, 3) 2 5 2 3 Quantized hypercharges are fixed by non-abelian generator Weak Isospin: T1,2,3 = λ 9,10,11/2 Electric Charge: Q = T 3 + Y/2 = diag( 1, 1, 1, 1, 0) 3 3 3 Prediction for the weak mixing angle (with RGE running): α 1 (M Z) = 128.91(2), α s(m Z) = 0.1176(20), s 2 w(m Z) = 0.2312(3) non-susy: s 2 w(m Z) = 23 + α(m Z ) 109 134 α s(m Z 0.207 ) 201

Exercise 10: SUSY GUTs and sin 2 θ W The renormalization group equations for the coupling constants g i, i = 1, 2, 3 are given by: In the SM, the coefficients b i for the gauge groups SU(i) are: dα i d ln µ = bi α 2 i 2π. (1) b 3 = 11 + 4 3 Ng, b 2 = 22 3 + 4 3 Ng + 1 6 N H, b 1 = 20 9 Ng + 1 6 N H, where N g is the number of generations and N H the number of Higgs doublets. a) Why has b 1 a different form as b 2 and b 3? Show that the solution of (1) is:! 1 α = 1 i(µ) α bi i(µ 0) 4π ln µ 2 µ 2 0 (2) b) In a Grand Unified Theory (GUT) we should have the following relation at the GUT M X : q 5/3g 1(M X ) = g 2(M X ) = g 3(M X ) = g GUT. (3) In a GUT like SU(5) MX SU(3) c h SU(2) L U(1) Y, d c, L are together in a multiplet ( 5), as well as u c, e c, Q (10). Demand that tr T a T bi = 1 2 δab to find the normalization of hypercharge.

Furthermore, we have: α 1 = α(µ) α 2 = α(µ) (4) c 2 W (µ) s 2 W (µ) c) For the scale µ 0 = M X take the the GUT scale. Express s 2 W (µ) as a function of α(µ) ans ln(m GUT /µ). Replace the logarithm by a corresponding relation containing α(µ) and α 3(µ) α s(µ). Keep the b i explicite. c) Wo do you get with α(m Z ) 1/128 and α s(m Z ) 0.12 for s 2 W (M Z), M X and α GUT? Experimentally, the weak mixing angle has the value s 2 W = 0.2312(3). d) Due to the additional particle content of the MSSM the coefficients of the renormalization group equations for the coupling constants are changed with respect to the SM. In the MSSM they are: b 3 = 9 + 2N g, b 2 = 6 + 2N g + 1 2 N H, b 1 = 10 3 Ng 1 2 N H, Repeat the above calculations for the MSSM. What changes?

Fermions (Matter Superfields) The only possible way to group together the matter: d c 0 u c u c u d d c 5 = : d c 1 u c 0 u c u d 10 = : u c u c 0 u d l 2 u u u 0 e c ν l d d d e c 0 Remarks 5 = (3, 1) 2 3 (1, 2) 1 10 = (3, 2) 1 3 (3, 1) 4 3 (1, 1) 2 2 = = 2, (5 5)a = 10, (3 3) a = 3, ( ) a = Quarks and leptons in the same multiplet Fractional charges from tracelessness condition (color!) 5 and 10 have equal and opposite anomalies ν c must be SU(5) singlet

Interactions e e ν Leptoquark couplings d X u Y d Y (and SUSY vertices) u u Diquark couplings u X d Y (and SUSY vertices) Vector bosons induce e.g. decay p e + π 0 u u Y d e + d

The doublet-triplet splitting problem SU(5) breaking: Higgs Σ in adjoint 24 rep. 0 Σ 0 = w diag(1, 1, 1, 3 2, 3 2 ) M X = M Y = 5 2 2 g w other breaking mechanisms possible (e.g. orbifold)

The doublet-triplet splitting problem SU(5) breaking: Higgs Σ in adjoint 24 rep. 0 Σ 0 = w diag(1, 1, 1, 3 2, 3 2 ) M X = M Y = 5 2 2 g w other breaking mechanisms possible (e.g. orbifold) (MS)SM Higgs(es) included in 5 5 5 = : 0 1 D D D B @ h + C A h 0 5 = : 0 D c 1 D c D c B @ h C A h 0 5 = (3, 1) 2 3 (1, 2) 1 5 = (3, 1) 2 3 (1, 2) 1 D, D c coloured triplet Higgses with charges ± 1 3 also induces proton decay mh 100 GeV, m D 10 16 GeV Doublet-triplet splitting problem

Naive estimate of proton lifetime u u e + u u d d d Effective 4-fermion operator (analogy to muon decay) d L F = 4G F 2 (µγ κ ν µ )(ν e γ κ e) G 2 F = g2 2 8MW 2 192π3 G 2 F m5 µ τ(µ eν µ ν µ ) L GUT = 4G GUT 2 G GUT 2 = τ(p e + π 0 ) (uγu)(eγd) g 2 8MGUT 2 192π3 G 2 GUT m5 p Proton lifetime for α(m GUT ) 1/24 and M GUT 2 10 16 GeV: τ(p e + π 0 M ) 4 GUT [α(m GUT )] 2 m 10 31±1 years 5 p Compare: τ SM p 10 150 years (gravity-induced)

Proton decay crucial for GUT search Tracking calorimeter (SOUDAN) or RICH Cherenkovs Super-Kamiokande: 50 kt water RICH measure change and time for reconstruction γ γ π 0 P + e

Proton decay crucial for GUT search Tracking calorimeter (SOUDAN) or RICH Cherenkovs Super-Kamiokande: 50 kt water RICH measure change and time for reconstruction γ γ 0 π P + e Channel τ p(10 30 years) p invisible 0.21 p e + π 0 1600 p µ + π 0 473 p νπ + 25 p νk + 670 p e + η 0 312 p µ + η 0 126 p e + ρ 0 75 p µ + ρ 0 110 p νρ + 162 p e + ω 0 1000 p µ + ω 0 117 p e + K 0 150 p µ + K 0 1300 p νk + 2300 p e + γ 670 p µ + γ 478 New experiments: HyperK (1 Mt), UNO (650 kt), European project Fréjus (1 Mt) Precision: 10 years running = 10 34 10 35 years

SuperK abuse: Neutrino masses Long time no see: no proton decay until now Looking for neutrino appearance and disappearance events Neutrino oscillations discovered (end of 1990s): P (ν i ν j ) products of mixing matrices sin 2 ( m 2 i m2 j 2E L ) See-saw mechanism points to a high scale: Majorana mass term M R for ν c not forbidden by SM symmetries!! 0 v ν L (ν L, ν R) v M R ν R (5) v Neutrino masses: m ν Y 2 ν M R with M R 10 14 10 15 GeV

Yukawa coupling unification and all that... MSSM: negative sign of m H comes for free provided that... assume GUT scale (as motivated by coupling constant unification) take universal input parameters at the GUT scale (see msugra below) run down to the electroweak scale with RGEs ~ l ~ g H q ~ W ~ B ~ H Exactly one parameter turns negative: the µ in the Higgs potential But this only works if m t = 150... 200 GeV and M SUSY 1 TeV

Sfermion Systematics Off-diagonal element prop. to mass of partner quark (tan β v u/v d ) mixing important in stop sector (sbottom sector for large tan β) Taken into account also for stau sector Mixing makes (often) τ 1 lightest slepton (NLSP), t 1 lightest squark gauge invariance relation between m t 1, m t 2, θ t, m b1, m b2, θ b No right-handed sneutrinos (GUT-scale?) Characteristics from renormalization group equations: m 2 d = m 2 1 0 + K 3 + K 2 + L 36 K1 + D d L m 2 ũ = m 2 1 L 0 + K 3 + K 2 + 36 K1 + Dũ L K 1 0.15 m 2 1/2 m 2 ũ = m 2 4 R 0 + K 3 + 9 K1 + Dũ R m 2 dr = m 2 1 0 + K 3 + 9 K1 + D d R K 2 0.5 m 2 1/2 m 2 ẽ = m 2 1 L 0 + K 2 + 4 K1 + Dẽ L m 2 ν = m 2 1 0 + K 2 + 4 K1 + D ν K3 (4.5 6.5) m2 1/2 m 2 ẽ L = m 2 0 + K 1 + DẽR

The gluino Color octet fermion cannot mix with any other MSSM state Pure QCD interactions GUT-inspired models imply at any RG scale (up to tiny 2-loop corrections): M 3 = α s α sin2 θ W M 2 = 3 α s 5 α cos2 θ W M 1 Translates to mass relations near TeV scale: Crucial for SUSY detection: Proof its fermion nature Proof its Majorana nature Proof its octet nature M 3 : M 2 : M 1 6 : 2 : 1

The Question of Anomalies Anomaly: violation of a symmetry of the classical Lagrangian by quantum effects Global symmetries: most welcome π γγ, m η Local symmetry: a catastratophe (breakdown of very definition of the theory!) SM/MSSM anomaly-free accidentally: lepton and quark contributions cancel (in each gen.) Ameliorated slightly in SU(5): 5 and 10 cancel

Why chiral exotics? Unification verification only with megatons? What about colliders? SPA: super precision accurately Look for chiral exotics Physics beyond MSSM provides handle to GUT scale µ problem NMSSM trick Singlet superfield with TeV-scale VEV Doublet-triplet splitting problem, Longevity of the proton Try to keep D, D c superfields at TeV scale Need mechanism to prevent rapid proton decay Need to rearrange running for unification Flavour problem Flavour might help protecting the proton

Sketch of a model Superpotential: W= W MSSM + W D + W N W MSSM = Y u u c QH u + Y d d c QH d + Y e e c LH d W D = Y D Du c e c + Y Dc D c QL W S = Y S H SH u H d + Y S D SDD c U(1) S symmetry gauged at high energies, radiative breaking of global left-over U(1) 0 S 0 generates µh and µ D See-saw mechanism ν c at M ν 10 14 10 15 GeV m ν v2 M ν 10 1 10 2 ev At M ν left-right symmetric model: SU(2) L SU(2) ( ) ( ) u c ν c Q R =, L R = d c l c H = ( Hu H d ) D D c

RGE running 60 Running of 1/α, MSSM with 1 [(H u, D) (H d, D c )] SU(3) c SU(2) L SU(2) R 40 1/α 1 1/α 2 20 1/α 3 1/α 1 0 5 10 15 20 log (µ [GeV])

RGE running 60 Running of 1/α, MSSM with 1 [(H u, D) (H d, D c )] SU(4) c SU(2) L SU(2) R 40 1/α 1 1/α 2 20 1/α 3 1/α 1 0 5 10 15 20 log (µ [GeV])

RGE running 60 Running of 1/α, MSSM with 3 [(H u, D) (H d, D c )] SU(4) c SU(2) L SU(2) R 40 1/α 1 20 1/α 2 1/α 3 0 5 10 15 20 log (µ [GeV])

Matter-Higgs unification: E 6 Pati-Salam group SU(4) c SU(2) L SU(2) R Q L = (Q, L) = (4, 2, 1) Q R = ((u c, d c ), (ν c, l c )) = (4, 1, 2) H = (H u, H d ) = (1, 2, 2) D = (D, D c ) = (6, 1, 1) S = (1, 1, 1)

Matter-Higgs unification: E 6 Pati-Salam group SU(4) c SU(2) L SU(2) R Q L = (Q, L) = (4, 2, 1) Q R = ((u c, d c ), (ν c, l c )) = (4, 1, 2) H = (H u, H d ) = (1, 2, 2) D = (D, D c ) = (6, 1, 1) S = (1, 1, 1) Be radical: Embed everything in fundamental rep. of E 6 fundamental 27 contains exactly the stuff above adjoint: 78 Matter and Higgs fields in one big multiplet 3 generations of S and D 3 generations of Higgs fields: Higgs (VEV) vs. unhiggs (no VEV) Problem of FCNCs: Introduce Z2 symmetry (H parity)

Flavour Symmetry and proton decay Assume SU(3) F or SO(3) F flavour symmetry Left-right symmetry: SU(2)L SU(2) R, SU(3) c, SU(3) F Diquark couplings vanish identically: DQ LQ L = ɛ abc ɛ αβγ ɛ jk D a α (Q L) b βj (QL)c γk Baryon number is symmetry of the superpotential SU(2)R and SU(3) F breaking spurions? symmetry breaking by condensates linear/bilinear in fundamental reps.: D, D c couple to other quarks only as singlets Integrating out heavy fields: baryon number emerges as low-energy symmetry, flavour symmetry not Leptoquark couplings possible

A little bit of Pheno Next step: Provide a viable low-energy spectrum Extended MSSM Higgs sector relaxed Higgs bounds (light pseudoscalars) possibly large invisible decay ratio lightest unhiggs: H parity protected dark matter dark matter mix: interesting relic abundance (relaxes all neutralino bounds!) Pair production of unhiggses/unhiggsinos, cascade decays (Down-type) Leptoquarks, Leptoquarkinos 3 generations at TeV scale produced in gluon fusion, single production final states: tτ, bντ, tτ,... if flavor symmetry leaves traces: gq Dl enhanced, decays tµ, te Extended neutralino sector like in NMSSM no Z

Summary Grand Unification: Old theoretical idea (35 yrs.) All interactions in Nature essentially one (gravity?) Experimental hint from LEP (LHC?) High-scale recently supported by neutrino masses (see-saw?) Prediction: proton decay (lifetime?) No experimental evidence: τ p > 10 33 yrs. Search will become longer better: 10 35 yrs. Decay of heavy particles: source of missing CP violation Stabilization of large scale difference: Strong interactions disfavored (?) Supersymmetry (Unification seems perfect) Quasi-natural breaking chain: E 8 E 6 SO(10) SU(5) G SM Lots of open questions: FLAVOR!!!!! µ probl., doublet-triplet splitting probl. Yukawa coupling pattern GUT breaking: Higgs vs. boundary conditions

Some Unification needs time