2018-12-17 LeifLönnblad BeyondtheStandardModel

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Problems with the SM Grand Unification Supersymmetry ˇ

Beyond the Standard Model

Leif Lönnblad

Institutionen för Astronomi och teoretisk fysik Lunds Universitet

2018-12-17

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Unconstrained Arbitrary ˇIncomplete

The standard model and why we hate it!

I There are too many free parameters. Twelve fermion masses, eight mixing parameters, three couplings and the higgs field parameters (µ, λ) ⇔ (m_h,m_Z). In total 25 parameters (26 assuming there is CP-violation in QCD) Wouldn’t it be much nicer if we had a theory where these could be predicted?

I Unnaturally (?) large scale ratios:

me/mν ∼ 10⁷, m_W/me∼ 10⁵, m_Pl/m_W ∼ 10¹⁷

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Unconstrained Arbitrary ˇIncomplete

Arbitrary

I Why are there three generations.

I Why are the left-handed fermions in SU(2) doublets and the right-handed in singlets?

I Why do we just have SU(3) × SU(2) × U(1)? Nature could have picked any symmetry!

I Why is there charge quantization? In principle Y in U(1) could be anything.

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ˆ Arbitrary Incomplete ˇFine-tuned?

Incomplete

I Where is the anti-matter?

I Where is gravity?

I Where is dark matter?

I Where is dark energy?

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ˆ Incomplete Fine-tuned?

Fine-tuned?

There is a problem with the Higgs mass.

Just as couplings are renormalized to be scale dependent, so are masses:

m → m 1 + α 3π

Z ∞ m²₀

dp² p² + . . .

!

→ m 1 + α 3π ln Λ²

m₀² + . . .

!

This comes from self-energy diagrams

= + + +

For the Higgs we find thatR _dp²

p² →R dp²and we have a

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Assuming the Higgs mass at the scale Λ is m_h(Λ), looking only at the top-loop we have

m_h²(m_Z) ∼m²_h(Λ) − (Λ²− m_t²)

If there is no physics below m_Pl, that means m_h(m_Z) ≈125 GeV comes from the subtraction between two huge numbers.

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Consensus

There must be something beyond the Standard Model!

The big questions: What? and Where?

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Consensus

There must be something beyond the Standard Model!

The big questions: What? and Where?

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SU(5)_GUT Multiplets ˇNew bosons

Grand Unification

We know about spontaneous symmetry breaking U(1)_Y × SU(2)L→ U(1)EM

Imagine that at a high scale all three forces are united into one under a common larger symmety group G_GUT.

For some reason this group is then spontaneously broken G_GUT→ SU(3)QCD× U(1)Y × SU(2)L

Let’s try G = SU(5)

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Grand Unification

We know about spontaneous symmetry breaking U(1)_Y × SU(2)L→ U(1)EM

Imagine that at a high scale all three forces are united into one under a common larger symmety group G_GUT.

For some reason this group is then spontaneously broken G_GUT→ SU(3)QCD× U(1)Y × SU(2)L

Let’s try G = SU(5)

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SU(5)

_GUT

I Simplest possible group

I Invented by Georgi and Glashow 1974

I Excluded by data — but still instructive

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The basic multiplet is given by a colour triplet and a weak doublet in an (anti-) quintet.

Since the weak doublet is left-handed, the quarks need to be left-handed and weak singlets, so we use the ¯d .

U5¯=









 d¯r

d¯_b d¯g





e⁻ ν_e







L

Group generators are traceless N × N matrices, where the diagonal generator will give the charge and requiresP Q_i =0.

This gives us charge quantisation and 3Qd¯+Q_e =0

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Where are the other quarks and leptons?

The quintet of right-handed fields:

U₅= (d_r,d_g,d_b,e⁺, ¯ν_e)_R

The anti-symmetric decuplet with ten left-handed fields

U₁₀= 1

√ 2











0 u¯_b −¯u_g

−¯u_b 0 u¯r

¯u_g −¯u_r 0





−ur −dr

−ug −dg

−u_b −d_b u_r u_g u_b

dr dg d_b

0 −e⁺ e⁺ 0







L

and the corresponding one for the right-handed fields.

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U₅= (d_r,d_g,d_b,e⁺, ¯ν_e)_R

U₁₀= 1

√ 2











0 u¯_b −¯u_g

−¯u_b 0 u¯r

¯u_g −¯u_r 0





−ur −dr

−ug −dg

dr dg d_b

0 −e⁺ e⁺ 0







L

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U₅= (d_r,d_g,d_b,e⁺, ¯ν_e)_R

U₁₀= 1

√ 2











0 u¯_b −¯u_g

−¯u_b 0 u¯r

u¯_g −¯u_r 0





−ur −dr

−ug −dg

dr dg d_b

0 −e⁺ e⁺ 0







L

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ˆ Multiplets New bosons ˇThe GUT scale

The gauge bosons

SU(5) has 5²− 1 = 24 generators

A =









 g_ij −^√^2B

30δ_ij





X¯r Y¯r

X¯_g Y¯_g X¯_b Y¯_b Xr Xg X_b

Y_r Y_g Y_b

W√³ 2 +^√^3B

30 W⁺

W⁻ −^W^√³

2+ ^√^3B

30

!







Giving us 12 new gauge bosons!

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ˆ New bosons The GUT scale Interactions

The GUT scale

At some large scale SU(5) is an exact symmetry with a single coupling g5.

We expect all SM couplings to come together at some high scale M_GUT.

Remember:

1

αi(M²) = 1

αi(µ²)+ b_i 4π lnM²

µ²

with b₃=11 − 4n_F/3, b₂=22/3 − 4n_F/3, b⁰₁= −4n_F/3

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We should have eg.

1

α₃(µ²) + b₃

4πlnM_GUT²

µ² = 1

α₂(µ²) + b₂

4π lnM_GUT² µ² using 1/α₂(m_Z²) ≈30 and 1/α₃(m²_Z) ≈10 we get

M_GUT∼ 10¹⁸ GeV

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Even if we haven’t specified the way the GUT is broken we should be able to estimate e.g. the ratio between g₁and g₂at lower energies.

Let’s look at the covariant derivative of SU(5) D^µ= ∂^µ− ig5T_aU_a^µ

And Pick out the parts relevant to the electro-weak sector using

B^µ = A^µcos θ_W +Z^µsin θ_W W₃^µ = −A^µsin θ_W +Z^µcos θ_W

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D^µ = ∂^µ− ig5(T₃W₃^µ+T₁B^µ+ . . .)

= ∂^µ− ig5sin θ_W(T₃+cot θ_WT₁)A^µ+ . . .

= ∂^µ− ieQA^µ+ . . .

Identify e = g₅sin θ_W and Q = T₃− cot θWT₁≡ T3+cT₁. Now, for any representation, R, of a group we have

orthogonality and equal normalization of the generators T_a, so that

Tr_RT_aT_b=N_Rδ_ab

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TrQ²=Tr(T₃+cT₁)²= (1 + c²)TrT₃² since TrT₃²=TrT₁²and

sin²θW = 1

1 + c² = TrT₃²

TrQ² = 0 + 0 + 0 + ¹₂2

+ ¹₂2 1

3

2

+ ¹₃2

+1 + 0

= 3 8

Including the running of the couplings we can get close to sin²θ_W ≈ 0.23 at around m_Z.

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Interactions

How does the new gauge bosons interact?

looking at the terms when we sanwich the gauge boson matrix, A, between the fundamental representations

U¯5AU5, U10¯AU5and U10¯AU₁₀ we get eg.

X → uu, X → e⁺d ,¯ Y → ud , Y → ¯d ¯ν_e, Y → e⁺u¯ Giving the charges 4/3 and 1/3 for X and Y .

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Decaying protons!

p = u ud → u Y → u ¯ue⁺ → π⁰e⁺ Remembering the muon width we estimate

Γp∝ α²₅m_p⁵ m⁴_Y

and with m_Y ∼ mGUT∼ 10¹⁵GeV we get τ_p ∼ 10^31±2years.

(The current limit p → π⁰e⁺is τ_p >10³³ years.)

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Grand Unificationˆ Supersymmetry (Super) String Theory

New particles R-parity ˇFine-tuning solved?

Supersymmetry

Postulate there being a symmetry between fermions and bosons, with an operator Q changing one into the other

|bii = Q|fii and |fii = ¯Q|b_ii but leaving any other quantum number unchanged.

The transformation is actually defined in terms of an algebra where

{Q, ¯Q} = Q ¯Q + ¯QQ = 2σ^µP_µ where P_µ=i∂_µis a translation.

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normal partner spin

q_L q˜_L 0 squarks

q_R q˜_R 0 (can mix)

l_L ˜l_L 0 sleptons

l_R ˜l_R 0 (also mix) ν_L ν˜_L 0 sneutrinos

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g g˜ 1/2 gluino

γ (˜γ) 1/2 (photino zino higgsino) Z⁰ ( ˜Z ) 1/2 all mix together into h⁰ 1/2 neutralinos ˜χ⁰_i

H⁰ ( ˜H) 1/2 2 + 3 + 1 + 1 + 1 = 8 spin states for the bosons A⁰ 1/2 gives four neutralinos with two spin states each.

W^± W˜^± 1/2 (wino higgsino) mix together H^± H˜^± 1/2 charginos ˜χ^±_i

We need an extra higgs doublet (4 new higgs particles):

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In the simplest version of SUSY (MSSM) we can derive mass relations for the Higgs particles, and get m_h<m_Z <m_H. But there are many ways of constructing SUSY.

If SUSY was an exact theory we would have m_q =m_˜_qand it would be easy to find the new particles.

Since we have not found any sparticles, SUSY is broken. There are many ways of breaking SUSY.

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If SUSY was an exact theory we would have mq =m˜qand it would be easy to find the new particles.

Since we have not found any sparticles, SUSY is broken.

There are many ways of breaking SUSY.

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If SUSY was an exact theory we would have mq =m˜qand it would be easy to find the new particles.

Since we have not found any sparticles, SUSY is broken.

There are many ways of breaking SUSY.

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R-parity

We can define a “parity” relating to SYSU, called R-parity R = (−1)^L+3B+2S

where L is lepton number, B is baryon number and S is spin.

All ordinary particles have R = +1 and their super-partners have R = −1.

If R-parity is conserved, sparticles can only be produced in pairs.

This also means that the lightest super-symmetric particle (LSP) is stable.

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Even if the masses are not the same, the couplings do not care if we have particles or sparticles. Hence we have that vertices such as eg.

W⁺→ e⁺νe, W⁺ → ˜e⁺˜νe, W˜⁺→ ˜e⁺νe, W˜⁺→ e⁺ν˜e

all have the same coupling (g₂). So as soon as we come above the mass-threshold for producing sparticles, they will be

produced at the same rate as their ordinary particle equivalents.

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ˆ R-parity Fine-tuning solved?

The Higgs mass revisited

m_h²(m_Z) ∼m²_h(Λ) − (Λ²− m_t²)

With SUSY the Higgs would also have self-energy loops from stop squarks (˜t), but since they are bosons, the sign of the loop is reversed

m²_h(m_Z) ∼m_h²(Λ) − (Λ²− m_t²) + (Λ²− m_˜²_t) ∼m²_h(Λ) − (m_˜²_t − m²_t) So, as long as m_˜_t is not too large the fine-tuning goes away.

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A dark matter candidate

If there is a stable LSP, which only interacts weakly (eg. ˜χ⁰) it would be produced copiously at the big bang and would basically still be around.

Even if R-parity is not conserved and we could have decays like

˜

γ → νγ, it could still contribute to dark matter if the decay is slow enough.

If R-parity is not conserved we could get lepton and/or baryon number violation (and proton decay).

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Desperately seeking SUSY

With R-parity conserved we would get characteristic decay chains of sparticles according to their mass hierarchy., eg.

u → d + [ ˜˜ χ⁺₁ → ν_τ+ [˜τ⁺→ τ⁺χ˜⁰₁]]

Should be easy to see at the LHC

(not seen yet)

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Desperately seeking SUSY

With R-parity conserved we would get characteristic decay chains of sparticles according to their mass hierarchy., eg.

u → d + [ ˜˜ χ⁺₁ → ν_τ+ [˜τ⁺→ τ⁺χ˜⁰₁]]

Should be easy to see at the LHC (not seen yet)

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The trouble with SUSY

I (more than) double number of particles

I (more than) double number of masses

I (more than) double number of mixing angles In total more than 100 free parameters to measure

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Strings Extra dimensions ˇLarge extra dimensions

A quantum theory of gravity

How do we include gravity in the standard model?

The naive way is to take General Relativity and reinterpret it as a Lagrange density

This leads to a spin-2 graviton (possibly with a supersymmetric spin 3/2 graviton) and a theory that is not renormalisable.

This is related to the fact that Field theory assumes particles to be point-like.

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(Super) String theory

Elementary particles are not point-like but vibration modes of one-dimensional objects –Strings.

I a point like particles describes a world line, x^µ(τ )

I a string will describe a world sheet, x^µ(τ, σ)

A string can be open or closed (x^µ(τ, σ +2π) = x^µ(τ, σ)).

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Since nothing is point-like there is a natural cutoff to ensure renormalisability.

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To combine string theory with QFT is tricky

I To avoid tachyons (m²<0) we need 26 space-time dimensions.

I Including SUSY we can get down to 10 if we have SO(32) or E₈× E₈symmetry.

I Good news (1):

E₈⊃ E6⊃ SO(10) ⊃ SU(5) ⊃ SU(3) × SU(2) × U(1) but a new zoo of particles only interacting with gravity

I Good news (2): Ony five such theories: type-I, type-IIA, type-IIB, heterotic SO(32) and heterotic E₈× E₈

I Good news (3)? Only special cases of 11-dimensional M-theory, with ∼ 10⁵⁰⁰different string theories consistent with (SuSY) SM.

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I Good news (1):

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I Good news (1):

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I Good news (1):

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Where are all the extra dimensions?

The universe extends only L ∼ m⁻¹_Pl in all but four dimensions – compactification.

Have we checked that there are only four dimensions?

Current limit is L . 1 µm.

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ˆ Extra dimensions Large extra dimensions

How do we check the number of dimensions?

Look at the gravitational potential in 4 + n dimensions V (r ) ∼ m

m_Plⁿ⁺² 1 rⁿ⁺¹,

Now, if the n extra dimensions are of size R, for distances much larger than that the potential would look like

V (r ) ∼ m mⁿ⁺²_Pl

1 Rⁿ

1 r

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we have found m_Pl4≈ 10¹⁹ GeV. But if the extra dimensions are large this means that the true Planck scale is much smaller

m_Pl∼ R⁻ⁿ⁺²ⁿ m

2 n+2

Pl4

could even be close to the scales reachable at the LHC.

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BSM phenomenology

1. Make stuff up

2. Check that it is consistent with the SM 3. Make prediction (for the LHC)

4. Convince the experiments to look for it 5.

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BSM phenomenology

1. Make stuff up

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BSM phenomenology

1. Make stuff up

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BSM phenomenology

1. Make stuff up

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BSM phenomenology

1. Make stuff up

4. Convince the experiments to look for it 5. Go to Stockholm and collect prize

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BSM phenomenology

1. Make stuff up

4. Convince the experiments to look for it 5. When they find nothing, goto 1.