Grand Unified Theories and Beyond

Size: px

Start display at page:

Download "Grand Unified Theories and Beyond"

Ruth Norris
5 years ago
Views:

1 Grand Unified Theories and Beyond G. Zoupanos Department of Physics, NTUA Dubna, February 2014 G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

2 Antonio Pich: Particle Physics: The Standard Model (lectures) W. Hollik: Theory of Electroweak Interactions, Corfu Summer Institute, School and Workshop on Standard Model and Beyond, 2013 Aitchison I J R & Hey A J G: Gauge Theories In Particle Physics Volume 1: From Relativistic Quantum Mechanics To QED Francis Halzen-Alan D.Martin:Quarks and Leptons Tai-Pei Cheng,Ling-Fong Li: Gauge Theory of Elementary Particle Physics Abdelhak Djouadi: The Anatomy of Electro-Weak Symmetry Breaking, Tome I: The Higgs boson in the Standard Model K. Vagionakis: Particle Physics: An Introduction to the Basic Structure of Matter G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

3 Standard Model very successful low energy accessible part of a (more) Fundamental Theory of Elemental Particles. BUT with ad hoc Higgs sector ad hoc Yukawa couplings free parameters (> 20) Renormalization free parameters Traditional way of reducing the number of parameters: SYMMETRY Celebrated example: GUTs -e.g.minimal SU(5): sin 2 θ w (testable) m τ /m b (successful) However more SYMMETRY (e.g. SO(10), E 6, E 7, E 8) does not necessarily lead to more predictions for the SM parameters. Extreme case: Superstring Theories G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

4 On the other hand: LEP data N = 1 SU(5) N = 1 SU(5) MSSM MSSM best candidate for physics beyond SM. But with > 100 free parameters mostly in its SSB sector. Cures problem of quadratic divergencies of the SM (hierarchy problem). Restricts the Higgs sector leading to approximate prediction of the Higgs mass. G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

5 SM with two Higgs doublets V = 1 2 m2 1(H 1 H1) 1 2 m2 2(H 2 H2) 1 2 m2 3(H 1 H2 + h.c.) λ1(h 1 H1) λ2(h 2 H2)2 + λ 3(H 1 H1)(H 2 H2) + λ4(h 1 H2)(H 2 [ H1) ] λ5(h1h2)2 + [λ 6(H 1 H1) + λ7(h 2 H2)](H 1 H2) + h.c. Supersymmetry provides tree level relations among couplings: λ 1 = λ 2 = 1 4 (g 2 + g 2 ) λ 3 = 1 4 (g 2 g 2 ), λ 4 = 1 4 g 2 λ 5 = λ 6 = λ 7 = 0 with υ 1 =< ReH 0 1 >, υ 2 =< ReH 0 2 > and υ υ 2 2 = (246 Gev) 2,υ 2/υ 1 tan β = h 0, H 0, H ±, A 0 G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

6 At tree level: M 2 h 0,H 0 = 1 2 { [ ] } 1/2 MA 2 + MZ 2 (MA 2 + MZ 2 ) 2 4MAM 2 Z 2 cos 2 2β M 2 H ± = M2 W + M 2 A M h 0 < M Z cos 2β = M H 0 > M Z M H ± > M W Radiative Corrections M 2 h 0 M2 Z cos 2 2β + 3g 2 mt 4 log m2 t 1 m t π 2 MW 2 mt 4 G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

7 Grand Unified Theories The SU(5) Model The SU(5) Model (Georgi-Glashow) From group theory (e.g. Slansky, Physics Repts): SU(5) SU(3) SU(2) U(1) In addition: 5 = (3, 1) 2/3 + (1, 2) 1 fundamental rep : ψ i 10 = (3, 2) 1/3 + ( 3, 1) 4/3 + (1, 1) 2 24 = (8, 1) 0 + (3, 2) 5/3 + ( 3, 2) 5/3 + (1, 3) 0 + (1, 1) 0 adjoint rep antisymmetric tensor ψ ij = ψ ji 5 5 = = = = G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

8 Grand Unified Theories The SU(5) Model Recall: Dirac equation in electromagnetic field: (iγ µ µ qγ µ A µ(x) m) ψ(x) = 0 { (iγ µ µ + eγ µ A µ(x) m) Ψ(x) = 0 (iγ µ µ eγ µ A µ(x) m) Ψ c (x) = 0 Ψ(x) : electron Ψ c (x) : positron (antiparticle) Ψ c (x) = iγ 2 Ψ (x) = (Ψ R ) c = (Ψ c ) L Ψ c L Then we can write e.g. the quarks of the first family using the SU(3) c SU(2) L U(1) Y quantum numbers and only left-handed 2-component fields: u L, d L : (3, 2) 1/3 u c L : ( 3, 1) 4/3 d c L : ( 3, 1) 2/3 G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

9 Grand Unified Theories The SU(5) Model A comparison shows that one family of fermions of the SM can be accommodated in two SU(5) irreps (or three if ν R exists): or 5 : (Ψ i ) L = (d c1 d c2 d c3 e ν e) L and 5 : (Ψ i ) R = (d 1 d 2 d 3 e + ν c e ) L 0 u c3 u c2 u 1 d 1 10 : (χ ij ) L = 1 0 u c1 u 2 d u 3 d 3 0 e + 0 The combination 5 and 10 is anomaly free. G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

10 SU(5) Generators Grand Unified Theories The SU(5) Model {λ a }, a = 1, 2,..., 24 is a set of 24 (5 2 1) unitary, traceless 5 5 (fundamental rep) matrices with normalization: Tr(λ a λ b ) = 2δ ab satisfying the commutation relations: [ ] λ a 2, λb = ic abc λ c 2 2, C abc structure constants i.e. generalized Gell-Mann matrices. Examples: 0 0 λ a 0 0 λ a = , a = 1, 2,..., 8 Generators of SU(3) - Gell-Mann matrices (the corresponding gauge bosons are the gluons) G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

11 Grand Unified Theories The SU(5) Model λ 9,10 = σ 1,2, λ11 = diag(0, 0, 0, 1, 1) Generators of SU(2) - Pauli Matrices (the corresponding gauge bosons are the W ±, W 3) λ 12 = 1 15 diag( 2, 2, 2, 3, 3) Generator of U(1) (the corresponding gauge boson is B) 1 0 O 0 0 λ 13 = O, λ14 = i 0 O , etc. i 0 0 O G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

12 Charge Quantization Grand Unified Theories The SU(5) Model Property of simple non-abelian groups that the eigenvalues of the generators are discrete (recall e.g. SO(3)). In SU(5) the Q is one (linear combination) of the generators and therefore quantized. Since electric charge is an additive quantum number, Q must be some linear combination of the 4 diagonal generators of SU(5) (rank 4). Since Q commutes with SU(3) c generators (2 diagonal; rank 2) Q = T 3 + Y 2 = 1 2 λ11 + c 2 λ12 c is determined by comparing the eigenvalues of λ 12 with the hypecharge Y values of particles in 5 Then the traceless condition: c = (5/3) 1/2 Tr Q = 0 3q d + q e + = 0! G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

13 (More) SU(5) Physics Grand Unified Theories The SU(5) Model 1st generation of fermions SU(5) SU(3) c SU(2) L U(1) Y Gauge bosons = ( 3, 1) 2/3 + (1, 2) 1 = d c L + (ν e, e ) L 10 = (3, 2) 1/3 + ( 3, 1) 4/3 + (1, 1) 2 = (u, d) L + u c L + e + L 24 = (8, 1) 0 + (3, 2) 5/3 + ( 3, 2) 5/3 + (1, 3) 0 + (1, 1) 0 }{{}}{{}}{{}}{{}}{{} gluons X 4/3, Y 1/3 X 4/3, Y 1/3 W ±, W 3 B X 1 Y 1 gluons X 2 Y 2 G X 3 Y 3 X 1 X 2 X 3 W 3/ 2 Y 1 Y 2 Y 3 W 3/ 2 + B G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

14 Grand Unified Theories The SU(5) Model From covariant derivatives: L int = g5 2 G a µ(ūγ µ λ a u + dγ µ λ a d) g5 2 W µ( i Q L γ µ τ i Q L + Lγ µ τ i L) ( ) 1/2 g5 3 B µ f γ µ Yf 2 5 all fermions + interactions of X, Ys Note that g strong = g SU(2) = g 5 However, g 5λ 12 A 12 µ = g }{{} of SM YB µ and Y = (5/3) 1/2 λ 12 g = (3/5) 1/2 g 5 sin 2 θ W = g 2 g 2 SU(2) + g 2 = 3 8! G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

15 Grand Unified Theories Spontaneous Symmetry Breaking The SU(5) Model The first symmetry breaking SU(5) SU(3) c SU(2) L U(1) Y introducing a 24-plet of scalars Φ(x), with potential is achieved by V (Φ) = m1(trφ 2 2 ) + λ 1(TrΦ 2 ) 2 + λ 2(TrΦ 4 ) for λ 1 > 7/30, λ 2 > < Φ >= V 1 3/2 L F.Li 3/2 where V 2 = m 2 1/[15λ 1 + (7/2)λ 2] The gauge bosons X, Y obtain mass: m 2 X = m 2 Y = 25 8 g 2 5 V 2 G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

g5 2 Grand Unified Theories The SU(5) Model Examining further the interactions of

αβγ ūlγγ c µ u Lβ + h.c. [ ) ] Yµ,α ( d α γ µ νr c + ūl α γ µ el c + ɛ αβγ ūlβγ c µ d Lγ + h.

limits T p 10 31 y m X,Y 10 15 GeV m4 X g 4 5 m5 p G.

16 g5 2 Grand Unified Theories The SU(5) Model Examining further the interactions of X, Y we find: [ L XYint = g5 X α µ,α ( d R γ µ er c + 2 d ) ] L α γ µ el c + ɛ αβγ ūlγγ c µ u Lβ + h.c. [ ) ] Yµ,α ( d α γ µ νr c + ūl α γ µ el c + ɛ αβγ ūlβγ c µ d Lγ + h.c. leading to proton decay via diagrams such as: with strength g5 2 /mx 2,Y T p limits T p y m X,Y GeV m4 X g 4 5 m5 p G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

17 Gauge Hierarchy Problems Grand Unified Theories The SU(5) Model The second breaking in SU(5), i.e. SU(3) c SU(2) L U(1) Y SU(3) C U(1) e/m is due to a 5-plet of scalars. Then the complete potential is: with V = V (Φ) + V (H) +V (Φ, H) }{{} 5-plet V (H) = m 2 2H H + λ 3(H H) 2 V (H, Φ) = α(h H)(TrΦ 2 ) + βh Φ 2 H In turn the vev of Φ changes a bit: 1 < Φ >= V ɛ ɛ 2 G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

18 Grand Unified Theories The SU(5) Model when 0 < H >= υ for an appropriate range of parameters. In turn, the gauge boson (X, Y ) masses are superheavy: and m 2 X m 2 Y = 25 8 g 2 5 V 2 W, Z : m 2 W g 2 υ 2 4 (1 + ɛ), m2 Z g 2 υ 2 4 cos 2 θ w To keep the two scales in the theory ɛ 10 28! The triplet in the H 5-plet can mediate proton decay. Therefore should be superheavy, while the doublet has electroweak scale mass. G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

19 Grand Unified Theories The SU(5) Model Use of Renormalization Group Equations Our picture is that at scales above V we have a gauge invariant SU(5) theory, which at V breaks down spontaneously to the SM SU(3) c SU(2) L U(1) Y. Then the evolution of the three gauge couplings g 3, g 2, g 1 is controlled by the corresponding β-functions: with t = lnµ. Specifically: dg i dt = β i(g i ), i = 3, 2, 1 β 3(g 3) = g ( π 2 3 N f N ) H, 6 where N f is the number of 4-component colour triplet fermions and N H the number of colour triplet Higgs bosons. β 2(g 2) = g ( π N f N ) H 6 β 1(g 1) = N f g1 3 + Higgs boson contr. 16π2 G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

20 Grand Unified Theories The SU(5) Model In addition we have the boundary condition: ( ) g 3(M X ) = g 2(M X ) = g 1(M X ) = g 5 = g 5 3 We find: And follows: M X = (1.5) ±1 ( Λ MS 0.16GeV ) sin 2 θ W (M W ) = ± ± 0.006ln ( 0.16GeV Λ MS ) Experiment: sin 2 θ W = ± G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

21 Fermion Masses in SU(5) Grand Unified Theories The SU(5) Model Introduce Yukawa couplings: f 110 f 10 f 5 H + f 210 f 5 f 5 H < 5 H >= ( υ/ 2 ) i.e. m e/µ/τ = m d/s/b respectively υ 2 f 1ūu + υ 2 f 2( dd + ēe) holding at mass scales that SU(5) is a good symmetry, subject to significant renormalization corrections m b /m t 3 (!) at µ = µ th 10GeV. but m µ/m e }{{} 200 = m s/m d }{{} 20 which can be improved to m µ/m e = 9m s/m d by introducing a 45-plet of scalars. G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

22 The Supersymmetric SU(5) Grand Unified Theories The Supersymmetric SU(5) Motivation: Try to solve the hierarchy problem. Recall that tree level parameters were tuned to an accuracy to generate M x/m w in the scalar potential of SU(5). This relation is destroyed by renormalization effects. It has to be enforced order by order in perturbation theory. Supersymmetry can solve the technical problem. If it is exact, the mass parameters of the potential (in fact the whole superpotential) do not get renormalized (non-renorm. theorem). When SUSY is broken, corrections are finite and calculable ( m 2 square mass splitting in supermultiplet). G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

23 Grand Unified Theories The Supersymmetric SU(5) RGEs estimation of the GUT scale and proton lifetime gauge bosons the same larger number of spinors and scalars smaller - in absolute value - β function and therefore slower variation of the asymptotically free couplings { Mx,susy GeV = sin 2 θ w (m w ) = (!) Proton decay is out of reach with usual gauge boson exchange. There are new diagrams (based on dimension 5 operators - dressed by gauginos -) that become dominant. The resulting proton lifetime is compatible with presently known bounds. An important outcome is that the dominant decay mode is different from the SU(5) model and is: p K + v µ (Due to the anti-symmetry in the colour index that appears, the resulting operators are flavour non-diagonal) G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

24 Grand Unified Theories The Supersymmetric SU(5) Attempts towards Gauge - Yukawa Unification GUTs can also relate Yukawa couplings among themselves and might lead to predictions e.g. in SU(5): successful m τ /m b In SO 10 all elementary particles of each family (both chiralities and v R ) are in a commmon 16-plet representation. G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

25 Grand Unified Theories The Supersymmetric SU(5) Natural gradual extension Attempt to relate the couplings of the two sectors Gauge - Yukawa Unification Searching for a symmetry is needed - one that relates fields with different spins Supersymmetry Fayet BUT N = 2 GYU - functional relationship derived by some principle. In: Superstrings Composite models In principle such relations exist. In practice both have more problems than the SM. G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

26 Grand Unified Theories The Supersymmetric SU(5) Attempts to relate gauge and Yukawa couplings: Requiring absence of quadratic divergencies (Decker + Pestieu, Veltman) m 2 e +m 2 µ + m 2 τ + 3(m 2 u + m 2 d + m 2 c + m 2 s + m 2 t + m 2 b) = 3 2 m2 w m2 z m2 H Spontaneous symmetry breaking of SUSY via F-terms ( 1) 2J (2J + 1)mJ 2 = 0 J G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

27 Grand Unified Theories The Supersymmetric SU(5) G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

28 Veltman Grand Unified Theories The Supersymmetric SU(5) Dimensional regularization: quadratic divergencies manifest themselves as pole singularities in d = 2 In order to make them vanish, one imposes the condition: 1 [ 4 f (d) me 2 + mµ 2 + mτ 2 + 3(mu 2 + md 2 + mc 2 + ms 2 + mt 2 + mb) 2 = d 1 2 where f (d) = Tr[1] mw 2 + d 1 mz m2 H, Veltman chose f (d) = 4 and using SUSY arguments he put d = 4 ] Osland + Wu using point splitting reg Jack + Jones + Roberts using DRED found the same relation G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

29 Grand Unified Theories The Supersymmetric SU(5) Veltman 81: Requiring absence of quadratic divergencies, he found: me 2 + mµ 2 + mτ 2 + 3(mu 2 + md 2 + mc 2 + ms 2 + mt 2 + mb) 2 = 3 2 m2 w m2 z m2 H For m 2 H << m 2 w m t = 69GeV For mh 2 = mw 2 m t = 77.5GeV. For m 2 H = (316GeV ) 2 m t = 174GeV Ferrara, Girardello and Palumbo considering the spontaneous breaking of a SUSY theory found: ( 1) 2J (2J + 1)mJ 2 = 0 J G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

30 Chaichian et al Grand Unified Theories The Supersymmetric SU(5) G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

31 Grand Unified Theories Cancellation of Quadratic Divergencies Cancellation of Quadratic Divergencies Inami - Nishino - Watamura Deshpande - Johnson - Ma SUPERSYMMETRY Unique solution? G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

32 Standard Model Grand Unified Theories Cancellation of Quadratic Divergencies Pendleton - Ross infrared fixed point: For strong α 3, i.e. α 1 = α 2 = 0 dα 3 dt dy t dt P R : = 7α 2 3 = Yt 4π d dt (8α 3 92 ) Yt, Y t = h2 t 4π ( Yt α 3 m P R t = ) = 0 Y t = 2 9 α3 8π α3 v 100GeV 9 In the reduction scheme same result is obtained by the requirement that the system is described by a single coupling theory with a renormalized power series expansion in α 3. G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

33 Grand Unified Theories Cancellation of Quadratic Divergencies Infrared Quasi - fixed point: Vanishing β function for Y t 9 2 Y Q f t = 8α 3 m Q f t = 8m P R t 280GeV Quasi - fixed point would also become an exact fixed point if β 3 = 0 G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

34 SUSY S.M. Pendleton - Ross: Grand Unified Theories Cancellation of Quadratic Divergencies Quasi - fixed point: dα 3 = 3α3 2 dt ( ) dy t 16 α 3 = Y t dt 3 4π 6Yt ( ) d Yt = 0 Y susyp R t dt α 3 m susyp R t = = 7 18 α3 7 4πα3v sin β GeV sin β tan β = vu v d α 3 Y susyq f t = π m susyq f t 200GeV sin β Quasi - fixed point is reached if h t becomes strong at scales µ = GeV G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

35 Summary Grand Unified Theories Cancellation of Quadratic Divergencies Pendleton - Ross infrared fixed point ( ) d Yt = 0 dt m t 100GeV α 3 (Divergent (!) in 2 - loops, Zimmermann) Infrared quasi - fixed point (Hill) SUSY Pendleton - Ross d dt (Yt) = 0 mt 280GeV, Yt h2 t 4π m t 140GeV sin β, tan β = vu v d SUSY Quasi - fixed point (Berger et al, Carena et al) m t 200GeV sin β (If tan β > 2 m I.R. t 188GeV, Kubo et al) G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

36 Grand Unified Theories Cancellation of Quadratic Divergencies We attempt to reduce further the parameters of GUTs searching for renormalization group invariant relations among GUT s couplings holding beyond the unification scale G. Zoupanos (Department of Physics, NTUA ) Grand Unified Theories and Beyond Dubna, February / 36

Updates in the Finite N=1 SU(5) Model

Updates in the Finite N=1 SU(5) Model Gregory Patellis Physics Department, NTUA Sven Heinemeyer, Myriam Mondragon, Nicholas Tracas, George Zoupanos arxiv:1802.04666 March 31, 2018 Updates in the Finite